lst:=[
# ::Package:: 

# ::Title:: 
#Integration problems of the form 
#Cos[c+d x]^m (A+B Cos[c+d x]+C Cos[c+d x]^2) (a+b Cos[c+d x])^n


# ::Section:: 
#Cos[c+d x]^m (a+b Cos[c+d x])^n


# ::Subsection::Closed:: 
#Integrands of the form Cos[c+d x]^m (a+b Cos[c+d x])^n
#


# ::Subsubsection::Closed:: 
#n>0


[cos(c + d*x)^4*(a + b*cos(c + d*x))^2, x, 6, (1/16)*(6*a^2 + 5*b^2)*x + (16*a*b*sin(c + d*x))/(15*d) + ((6*a^2 + 5*b^2)*cos(c + d*x)*sin(c + d*x))/(16*d) + (8*a*b*cos(c + d*x)^2*sin(c + d*x))/(15*d) + ((6*a^2 + 5*b^2)*cos(c + d*x)^3*sin(c + d*x))/(24*d) + (2*a*b*cos(c + d*x)^4*sin(c + d*x))/(5*d) + (b^2*cos(c + d*x)^5*sin(c + d*x))/(6*d)],
[cos(c + d*x)^3*(a + b*cos(c + d*x))^2, x, 5, (3*a*b*x)/4 + (2*(5*a^2 + 4*b^2)*sin(c + d*x))/(15*d) + (3*a*b*cos(c + d*x)*sin(c + d*x))/(4*d) + ((5*a^2 + 4*b^2)*cos(c + d*x)^2*sin(c + d*x))/(15*d) + (a*b*cos(c + d*x)^3*sin(c + d*x))/(2*d) + (b^2*cos(c + d*x)^4*sin(c + d*x))/(5*d)],
[cos(c + d*x)^2*(a + b*cos(c + d*x))^2, x, 4, (1/8)*(4*a^2 + 3*b^2)*x + (4*a*b*sin(c + d*x))/(3*d) + ((4*a^2 + 3*b^2)*cos(c + d*x)*sin(c + d*x))/(8*d) + (2*a*b*cos(c + d*x)^2*sin(c + d*x))/(3*d) + (b^2*cos(c + d*x)^3*sin(c + d*x))/(4*d)],
[cos(c + d*x)^1*(a + b*cos(c + d*x))^2, x, 3, a*b*x + ((3*a^2 + 2*b^2)*sin(c + d*x))/(3*d) + (a*b*cos(c + d*x)*sin(c + d*x))/d + (b^2*cos(c + d*x)^2*sin(c + d*x))/(3*d)],
[cos(c + d*x)^0*(a + b*cos(c + d*x))^2, x, 2, (1/2)*(2*a^2 + b^2)*x + (2*a*b*sin(c + d*x))/d + (b^2*cos(c + d*x)*sin(c + d*x))/(2*d)],
[sec(c + d*x)^1*(a + b*cos(c + d*x))^2, x, 3, 2*a*b*x + (a^2*arctanh(sin(c + d*x)))/d + (b^2*sin(c + d*x))/d],
[sec(c + d*x)^2*(a + b*cos(c + d*x))^2, x, 3, b^2*x + (2*a*b*arctanh(sin(c + d*x)))/d + (a^2*tan(c + d*x))/d],
[sec(c + d*x)^3*(a + b*cos(c + d*x))^2, x, 4, ((a^2 + 2*b^2)*arctanh(sin(c + d*x)))/(2*d) + (2*a*b*tan(c + d*x))/d + (a^2*sec(c + d*x)*tan(c + d*x))/(2*d)],
[sec(c + d*x)^4*(a + b*cos(c + d*x))^2, x, 5, (a*b*arctanh(sin(c + d*x)))/d + ((2*a^2 + 3*b^2)*tan(c + d*x))/(3*d) + (a*b*sec(c + d*x)*tan(c + d*x))/d + (a^2*sec(c + d*x)^2*tan(c + d*x))/(3*d)],
[sec(c + d*x)^5*(a + b*cos(c + d*x))^2, x, 6, ((3*a^2 + 4*b^2)*arctanh(sin(c + d*x)))/(8*d) + (4*a*b*tan(c + d*x))/(3*d) + ((3*a^2 + 4*b^2)*sec(c + d*x)*tan(c + d*x))/(8*d) + (2*a*b*sec(c + d*x)^2*tan(c + d*x))/(3*d) + (a^2*sec(c + d*x)^3*tan(c + d*x))/(4*d)],
[sec(c + d*x)^6*(a + b*cos(c + d*x))^2, x, 7, (3*a*b*arctanh(sin(c + d*x)))/(4*d) + (2*(4*a^2 + 5*b^2)*tan(c + d*x))/(15*d) + (3*a*b*sec(c + d*x)*tan(c + d*x))/(4*d) + ((4*a^2 + 5*b^2)*sec(c + d*x)^2*tan(c + d*x))/(15*d) + (a*b*sec(c + d*x)^3*tan(c + d*x))/(2*d) + (a^2*sec(c + d*x)^4*tan(c + d*x))/(5*d)],


[cos(c + d*x)^3*(a + b*cos(c + d*x))^3, x, 6, (1/16)*b*(18*a^2 + 5*b^2)*x + (2*a*(5*a^2 + 12*b^2)*sin(c + d*x))/(15*d) + (b*(18*a^2 + 5*b^2)*cos(c + d*x)*sin(c + d*x))/(16*d) + (a*(5*a^2 + 12*b^2)*cos(c + d*x)^2*sin(c + d*x))/(15*d) + (b*(18*a^2 + 5*b^2)*cos(c + d*x)^3*sin(c + d*x))/(24*d) + (13*a*b^2*cos(c + d*x)^4*sin(c + d*x))/(30*d) + (b^2*cos(c + d*x)^4*(a + b*cos(c + d*x))*sin(c + d*x))/(6*d)],
[cos(c + d*x)^2*(a + b*cos(c + d*x))^3, x, 5, (1/8)*a*(4*a^2 + 9*b^2)*x - ((3*a^4 - 52*a^2*b^2 - 16*b^4)*sin(c + d*x))/(30*b*d) - (a*(6*a^2 - 71*b^2)*cos(c + d*x)*sin(c + d*x))/(120*d) - ((3*a^2 - 16*b^2)*(a + b*cos(c + d*x))^2*sin(c + d*x))/(60*b*d) - (a*(a + b*cos(c + d*x))^3*sin(c + d*x))/(20*b*d) + ((a + b*cos(c + d*x))^4*sin(c + d*x))/(5*b*d)],
[cos(c + d*x)^1*(a + b*cos(c + d*x))^3, x, 4, (3/8)*b*(4*a^2 + b^2)*x + (a*(a^2 + 4*b^2)*sin(c + d*x))/(2*d) + (b*(2*a^2 + 3*b^2)*cos(c + d*x)*sin(c + d*x))/(8*d) + (a*(a + b*cos(c + d*x))^2*sin(c + d*x))/(4*d) + ((a + b*cos(c + d*x))^3*sin(c + d*x))/(4*d)],
[cos(c + d*x)^0*(a + b*cos(c + d*x))^3, x, 3, (1/2)*a*(2*a^2 + 3*b^2)*x + (2*b*(4*a^2 + b^2)*sin(c + d*x))/(3*d) + (5*a*b^2*cos(c + d*x)*sin(c + d*x))/(6*d) + (b*(a + b*cos(c + d*x))^2*sin(c + d*x))/(3*d)],
[sec(c + d*x)^1*(a + b*cos(c + d*x))^3, x, 4, (1/2)*b*(6*a^2 + b^2)*x + (a^3*arctanh(sin(c + d*x)))/d + (5*a*b^2*sin(c + d*x))/(2*d) + (b^2*(a + b*cos(c + d*x))*sin(c + d*x))/(2*d)],
[sec(c + d*x)^2*(a + b*cos(c + d*x))^3, x, 4, 3*a*b^2*x + (3*a^2*b*arctanh(sin(c + d*x)))/d - (b*(a^2 - b^2)*sin(c + d*x))/d + (a^2*(a + b*cos(c + d*x))*tan(c + d*x))/d],
[sec(c + d*x)^3*(a + b*cos(c + d*x))^3, x, 4, b^3*x + (a*(a^2 + 6*b^2)*arctanh(sin(c + d*x)))/(2*d) + (5*a^2*b*tan(c + d*x))/(2*d) + (a^2*(a + b*cos(c + d*x))*sec(c + d*x)*tan(c + d*x))/(2*d)],
[sec(c + d*x)^4*(a + b*cos(c + d*x))^3, x, 5, (b*(3*a^2 + 2*b^2)*arctanh(sin(c + d*x)))/(2*d) + (a*(2*a^2 + 9*b^2)*tan(c + d*x))/(3*d) + (7*a^2*b*sec(c + d*x)*tan(c + d*x))/(6*d) + (a^2*(a + b*cos(c + d*x))*sec(c + d*x)^2*tan(c + d*x))/(3*d)],
[sec(c + d*x)^5*(a + b*cos(c + d*x))^3, x, 6, (3*a*(a^2 + 4*b^2)*arctanh(sin(c + d*x)))/(8*d) + (b*(2*a^2 + b^2)*tan(c + d*x))/d + (3*a*(a^2 + 4*b^2)*sec(c + d*x)*tan(c + d*x))/(8*d) + (3*a^2*b*sec(c + d*x)^2*tan(c + d*x))/(4*d) + (a^2*(a + b*cos(c + d*x))*sec(c + d*x)^3*tan(c + d*x))/(4*d)],
[sec(c + d*x)^6*(a + b*cos(c + d*x))^3, x, 7, (b*(9*a^2 + 4*b^2)*arctanh(sin(c + d*x)))/(8*d) + (2*a*(4*a^2 + 15*b^2)*tan(c + d*x))/(15*d) + (b*(9*a^2 + 4*b^2)*sec(c + d*x)*tan(c + d*x))/(8*d) + (a*(4*a^2 + 15*b^2)*sec(c + d*x)^2*tan(c + d*x))/(15*d) + (11*a^2*b*sec(c + d*x)^3*tan(c + d*x))/(20*d) + (a^2*(a + b*cos(c + d*x))*sec(c + d*x)^4*tan(c + d*x))/(5*d)],


[cos(c + d*x)^3*(a + b*cos(c + d*x))^4, x, 7, (1/4)*a*b*(6*a^2 + 5*b^2)*x + (2*(35*a^4 + 168*a^2*b^2 + 24*b^4)*sin(c + d*x))/(105*d) + (a*b*(6*a^2 + 5*b^2)*cos(c + d*x)*sin(c + d*x))/(4*d) + ((35*a^4 + 168*a^2*b^2 + 24*b^4)*cos(c + d*x)^2*sin(c + d*x))/(105*d) + (a*b*(6*a^2 + 5*b^2)*cos(c + d*x)^3*sin(c + d*x))/(6*d) + (b^2*(71*a^2 + 18*b^2)*cos(c + d*x)^4*sin(c + d*x))/(105*d) + (8*a*b^2*cos(c + d*x)^4*(a + b*cos(c + d*x))*sin(c + d*x))/(21*d) + (b^2*cos(c + d*x)^4*(a + b*cos(c + d*x))^2*sin(c + d*x))/(7*d)],
[cos(c + d*x)^2*(a + b*cos(c + d*x))^4, x, 6, (1/16)*(8*a^4 + 36*a^2*b^2 + 5*b^4)*x - (a*(4*a^4 - 121*a^2*b^2 - 128*b^4)*sin(c + d*x))/(60*b*d) - ((8*a^4 - 178*a^2*b^2 - 75*b^4)*cos(c + d*x)*sin(c + d*x))/(240*d) - (a*(4*a^2 - 53*b^2)*(a + b*cos(c + d*x))^2*sin(c + d*x))/(120*b*d) - ((4*a^2 - 25*b^2)*(a + b*cos(c + d*x))^3*sin(c + d*x))/(120*b*d) - (a*(a + b*cos(c + d*x))^4*sin(c + d*x))/(30*b*d) + ((a + b*cos(c + d*x))^5*sin(c + d*x))/(6*b*d)],
[cos(c + d*x)^1*(a + b*cos(c + d*x))^4, x, 5, (1/2)*a*b*(4*a^2 + 3*b^2)*x + (2*(3*a^4 + 28*a^2*b^2 + 4*b^4)*sin(c + d*x))/(15*d) + (a*b*(6*a^2 + 29*b^2)*cos(c + d*x)*sin(c + d*x))/(30*d) + ((3*a^2 + 4*b^2)*(a + b*cos(c + d*x))^2*sin(c + d*x))/(15*d) + (a*(a + b*cos(c + d*x))^3*sin(c + d*x))/(5*d) + ((a + b*cos(c + d*x))^4*sin(c + d*x))/(5*d)],
[cos(c + d*x)^0*(a + b*cos(c + d*x))^4, x, 4, (1/8)*(8*a^4 + 24*a^2*b^2 + 3*b^4)*x + (a*b*(19*a^2 + 16*b^2)*sin(c + d*x))/(6*d) + (b^2*(26*a^2 + 9*b^2)*cos(c + d*x)*sin(c + d*x))/(24*d) + (7*a*b*(a + b*cos(c + d*x))^2*sin(c + d*x))/(12*d) + (b*(a + b*cos(c + d*x))^3*sin(c + d*x))/(4*d)],
[sec(c + d*x)^1*(a + b*cos(c + d*x))^4, x, 5, 2*a*b*(2*a^2 + b^2)*x + (a^4*arctanh(sin(c + d*x)))/d + (b^2*(13*a^2 + 2*b^2)*sin(c + d*x))/(3*d) + (4*a*b^2*(a + b*cos(c + d*x))*sin(c + d*x))/(3*d) + (b^2*(a + b*cos(c + d*x))^2*sin(c + d*x))/(3*d)],
[sec(c + d*x)^2*(a + b*cos(c + d*x))^4, x, 5, (1/2)*b^2*(12*a^2 + b^2)*x + (4*a^3*b*arctanh(sin(c + d*x)))/d - (a*b*(2*a^2 - 7*b^2)*sin(c + d*x))/(2*d) - (b*(2*a^2 - b^2)*(a + b*cos(c + d*x))*sin(c + d*x))/(2*d) + (a^2*(a + b*cos(c + d*x))^2*tan(c + d*x))/d],
[sec(c + d*x)^3*(a + b*cos(c + d*x))^4, x, 5, 4*a*b^3*x + (a^2*(a^2 + 12*b^2)*arctanh(sin(c + d*x)))/(2*d) - (b^2*(7*a^2 - 2*b^2)*sin(c + d*x))/(2*d) + (3*a^2*b*(a + b*cos(c + d*x))*tan(c + d*x))/d + (a^2*(a + b*cos(c + d*x))^2*sec(c + d*x)*tan(c + d*x))/(2*d)],
[sec(c + d*x)^4*(a + b*cos(c + d*x))^4, x, 5, b^4*x + (2*a*b*(a^2 + 2*b^2)*arctanh(sin(c + d*x)))/d + (a^2*(2*a^2 + 13*b^2)*tan(c + d*x))/(3*d) + (4*a^2*b*(b + a*sec(c + d*x))*tan(c + d*x))/(3*d) + (a^2*(b + a*sec(c + d*x))^2*tan(c + d*x))/(3*d)],
[sec(c + d*x)^5*(a + b*cos(c + d*x))^4, x, 6, ((3*a^4 + 24*a^2*b^2 + 8*b^4)*arctanh(sin(c + d*x)))/(8*d) + (4*a*b*(2*a^2 + 3*b^2)*tan(c + d*x))/(3*d) + (a^2*(9*a^2 + 46*b^2)*sec(c + d*x)*tan(c + d*x))/(24*d) + (5*a^2*b*(a + b*cos(c + d*x))*sec(c + d*x)^2*tan(c + d*x))/(6*d) + (a^2*(a + b*cos(c + d*x))^2*sec(c + d*x)^3*tan(c + d*x))/(4*d)],
[sec(c + d*x)^6*(a + b*cos(c + d*x))^4, x, 7, (a*b*(3*a^2 + 4*b^2)*arctanh(sin(c + d*x)))/(2*d) + ((8*a^4 + 60*a^2*b^2 + 15*b^4)*tan(c + d*x))/(15*d) + (a*b*(3*a^2 + 4*b^2)*sec(c + d*x)*tan(c + d*x))/(2*d) + (2*a^2*(2*a^2 + 9*b^2)*sec(c + d*x)^2*tan(c + d*x))/(15*d) + (3*a^2*b*(a + b*cos(c + d*x))*sec(c + d*x)^3*tan(c + d*x))/(5*d) + (a^2*(a + b*cos(c + d*x))^2*sec(c + d*x)^4*tan(c + d*x))/(5*d)],
[sec(c + d*x)^7*(a + b*cos(c + d*x))^4, x, 8, ((5*a^4 + 36*a^2*b^2 + 8*b^4)*arctanh(sin(c + d*x)))/(16*d) + (8*a*b*(4*a^2 + 5*b^2)*tan(c + d*x))/(15*d) + ((5*a^4 + 36*a^2*b^2 + 8*b^4)*sec(c + d*x)*tan(c + d*x))/(16*d) + (4*a*b*(4*a^2 + 5*b^2)*sec(c + d*x)^2*tan(c + d*x))/(15*d) + (a^2*(25*a^2 + 104*b^2)*sec(c + d*x)^3*tan(c + d*x))/(120*d) + (7*a^2*b*(a + b*cos(c + d*x))*sec(c + d*x)^4*tan(c + d*x))/(15*d) + (a^2*(a + b*cos(c + d*x))^2*sec(c + d*x)^5*tan(c + d*x))/(6*d)],


# ::Subsubsection::Closed:: 
#n<0


[cos(c + d*x)^5/(a + b*cos(c + d*x)), x, 6, ((8*a^4 + 4*a^2*b^2 + 3*b^4)*x)/(8*b^5) - (2*a^5*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^5*sqrt(a^2 - b^2)*d) - (a*(3*a^2 + 2*b^2)*sin(c + d*x))/(3*b^4*d) + ((4*a^2 + 3*b^2)*cos(c + d*x)*sin(c + d*x))/(8*b^3*d) - (a*cos(c + d*x)^2*sin(c + d*x))/(3*b^2*d) + (cos(c + d*x)^3*sin(c + d*x))/(4*b*d)],
[cos(c + d*x)^4/(a + b*cos(c + d*x)), x, 5, -((a*(2*a^2 + b^2)*x)/(2*b^4)) + (2*a^4*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^4*sqrt(a^2 - b^2)*d) + ((3*a^2 + 2*b^2)*sin(c + d*x))/(3*b^3*d) - (a*cos(c + d*x)*sin(c + d*x))/(2*b^2*d) + (cos(c + d*x)^2*sin(c + d*x))/(3*b*d)],
[cos(c + d*x)^3/(a + b*cos(c + d*x)), x, 4, ((2*a^2 + b^2)*x)/(2*b^3) - (2*a^3*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^3*sqrt(a^2 - b^2)*d) - (a*sin(c + d*x))/(b^2*d) + (cos(c + d*x)*sin(c + d*x))/(2*b*d)],
[cos(c + d*x)^2/(a + b*cos(c + d*x)), x, 4, -((a*x)/b^2) + (2*a^2*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^2*sqrt(a^2 - b^2)*d) + sin(c + d*x)/(b*d)],
[cos(c + d*x)^1/(a + b*cos(c + d*x)), x, 2, x/b - (2*a*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b*sqrt(a^2 - b^2)*d)],
[cos(c + d*x)^0/(a + b*cos(c + d*x)), x, 1, (2*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(sqrt(a^2 - b^2)*d)],
[sec(c + d*x)^1/(a + b*cos(c + d*x)), x, 3, -((2*b*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a*sqrt(a^2 - b^2)*d)) + arctanh(sin(c + d*x))/(a*d)],
[sec(c + d*x)^2/(a + b*cos(c + d*x)), x, 5, (2*b^2*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^2*sqrt(a^2 - b^2)*d) - (b*arctanh(sin(c + d*x)))/(a^2*d) + tan(c + d*x)/(a*d)],
[sec(c + d*x)^3/(a + b*cos(c + d*x)), x, 5, -((2*b^3*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^3*sqrt(a^2 - b^2)*d)) + ((a^2 + 2*b^2)*arctanh(sin(c + d*x)))/(2*a^3*d) - (b*tan(c + d*x))/(a^2*d) + (sec(c + d*x)*tan(c + d*x))/(2*a*d)],
[sec(c + d*x)^4/(a + b*cos(c + d*x)), x, 6, (2*b^4*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^4*sqrt(a^2 - b^2)*d) - (b*(a^2 + 2*b^2)*arctanh(sin(c + d*x)))/(2*a^4*d) + ((2*a^2 + 3*b^2)*tan(c + d*x))/(3*a^3*d) - (b*sec(c + d*x)*tan(c + d*x))/(2*a^2*d) + (sec(c + d*x)^2*tan(c + d*x))/(3*a*d)],


[cos(c + d*x)^5/(a + b*cos(c + d*x))^2, x, 6, -((a*(4*a^2 + b^2)*x)/b^5) + (2*a^4*(4*a^2 - 5*b^2)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^5*(a^2 - b^2)^(3/2)*d) + ((12*a^4 - 7*a^2*b^2 - 2*b^4)*sin(c + d*x))/(3*b^4*(a^2 - b^2)*d) - (a*(2*a^2 - b^2)*cos(c + d*x)*sin(c + d*x))/(b^3*(a^2 - b^2)*d) + ((4*a^2 - b^2)*cos(c + d*x)^2*sin(c + d*x))/(3*b^2*(a^2 - b^2)*d) - (a^2*cos(c + d*x)^3*sin(c + d*x))/(b*(a^2 - b^2)*d*(a + b*cos(c + d*x)))],
[cos(c + d*x)^4/(a + b*cos(c + d*x))^2, x, 5, ((6*a^2 + b^2)*x)/(2*b^4) - (2*a^3*(3*a^2 - 4*b^2)*arctanh(((a - b)*tan((1/2)*(c + d*x)))/sqrt(-a^2 + b^2)))/(b^4*(-a^2 + b^2)^(3/2)*d) - (2*a*sin(c + d*x))/(b^3*d) + (cos(c + d*x)*sin(c + d*x))/(2*b^2*d) - (a^4*sin(c + d*x))/(b^3*(a^2 - b^2)*d*(a + b*cos(c + d*x))), ((6*a^2 + b^2)*x)/(2*b^4) - (2*a^3*(3*a^2 - 4*b^2)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^4*(a^2 - b^2)^(3/2)*d) - (a*(3*a^2 - 2*b^2)*sin(c + d*x))/(b^3*(a^2 - b^2)*d) + ((3*a^2 - b^2)*cos(c + d*x)*sin(c + d*x))/(2*b^2*(a^2 - b^2)*d) - (a^2*cos(c + d*x)^2*sin(c + d*x))/(b*(a^2 - b^2)*d*(a + b*cos(c + d*x)))],
[cos(c + d*x)^3/(a + b*cos(c + d*x))^2, x, 4, -((2*a*x)/b^3) + (2*a^2*(2*a^2 - 3*b^2)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^3*(a^2 - b^2)^(3/2)*d) + ((2*a^2 - b^2)*sin(c + d*x))/(b^2*(a^2 - b^2)*d) - (a^2*cos(c + d*x)*sin(c + d*x))/(b*(a^2 - b^2)*d*(a + b*cos(c + d*x)))],
[cos(c + d*x)^2/(a + b*cos(c + d*x))^2, x, 3, x/b^2 - (2*a*(a^2 - 2*b^2)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^2*(a^2 - b^2)^(3/2)*d) - (a^2*sin(c + d*x))/(b*(a^2 - b^2)*d*(a + b*cos(c + d*x)))],
[cos(c + d*x)^1/(a + b*cos(c + d*x))^2, x, 3, -((2*b*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/((a^2 - b^2)^(3/2)*d)) + (a*sin(c + d*x))/((a^2 - b^2)*d*(a + b*cos(c + d*x)))],
[cos(c + d*x)^0/(a + b*cos(c + d*x))^2, x, 3, (2*a*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/((a^2 - b^2)^(3/2)*d) - (b*sin(c + d*x))/((a^2 - b^2)*d*(a + b*cos(c + d*x)))],
[sec(c + d*x)^1/(a + b*cos(c + d*x))^2, x, 4, -((2*b*(2*a^2 - b^2)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^2*(a^2 - b^2)^(3/2)*d)) + arctanh(sin(c + d*x))/(a^2*d) + (b^2*sin(c + d*x))/(a*(a^2 - b^2)*d*(a + b*cos(c + d*x)))],
[sec(c + d*x)^2/(a + b*cos(c + d*x))^2, x, 5, (2*b^2*(3*a^2 - 2*b^2)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^3*(a^2 - b^2)^(3/2)*d) - (2*b*arctanh(sin(c + d*x)))/(a^3*d) + ((a^2 - 2*b^2)*tan(c + d*x))/(a^2*(a^2 - b^2)*d) + (b^2*tan(c + d*x))/(a*(a^2 - b^2)*d*(a + b*cos(c + d*x)))],
[sec(c + d*x)^3/(a + b*cos(c + d*x))^2, x, 6, -((2*b^3*(4*a^2 - 3*b^2)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^4*(a^2 - b^2)^(3/2)*d)) + ((a^2 + 6*b^2)*arctanh(sin(c + d*x)))/(2*a^4*d) - (b*(2*a^2 - 3*b^2)*tan(c + d*x))/(a^3*(a^2 - b^2)*d) + ((a^2 - 3*b^2)*sec(c + d*x)*tan(c + d*x))/(2*a^2*(a^2 - b^2)*d) + (b^2*sec(c + d*x)*tan(c + d*x))/(a*(a^2 - b^2)*d*(a + b*cos(c + d*x)))],
[sec(c + d*x)^4/(a + b*cos(c + d*x))^2, x, 7, (2*b^4*(5*a^2 - 4*b^2)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^5*(a^2 - b^2)^(3/2)*d) - (b*(a^2 + 4*b^2)*arctanh(sin(c + d*x)))/(a^5*d) + ((2*a^4 + 7*a^2*b^2 - 12*b^4)*tan(c + d*x))/(3*a^4*(a^2 - b^2)*d) - (b*(a^2 - 2*b^2)*sec(c + d*x)*tan(c + d*x))/(a^3*(a^2 - b^2)*d) + ((a^2 - 4*b^2)*sec(c + d*x)^2*tan(c + d*x))/(3*a^2*(a^2 - b^2)*d) + (b^2*sec(c + d*x)^2*tan(c + d*x))/(a*(a^2 - b^2)*d*(a + b*cos(c + d*x)))],


[cos(c + d*x)^5/(a + b*cos(c + d*x))^3, x, 6, ((12*a^2 + b^2)*x)/(2*b^5) - (a^3*(12*a^4 - 29*a^2*b^2 + 20*b^4)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^5*(a^2 - b^2)^(5/2)*d) - (3*a*(4*a^4 - 7*a^2*b^2 + 2*b^4)*sin(c + d*x))/(2*b^4*(a^2 - b^2)^2*d) + ((6*a^4 - 10*a^2*b^2 + b^4)*cos(c + d*x)*sin(c + d*x))/(2*b^3*(a^2 - b^2)^2*d) - (a^2*cos(c + d*x)^3*sin(c + d*x))/(2*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) - (a^2*(4*a^2 - 7*b^2)*cos(c + d*x)^2*sin(c + d*x))/(2*b^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))],
[cos(c + d*x)^4/(a + b*cos(c + d*x))^3, x, 5, -((3*a*x)/b^4) + (3*a^2*(2*a^4 - 5*a^2*b^2 + 4*b^4)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^4*(a^2 - b^2)^(5/2)*d) + ((6*a^4 - 11*a^2*b^2 + 2*b^4)*sin(c + d*x))/(2*b^3*(a^2 - b^2)^2*d) - (a^2*cos(c + d*x)^2*sin(c + d*x))/(2*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) - (3*a^2*(a^2 - 2*b^2)*cos(c + d*x)*sin(c + d*x))/(2*b^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))],
[cos(c + d*x)^3/(a + b*cos(c + d*x))^3, x, 4, x/b^3 - (a*(2*a^4 - 5*a^2*b^2 + 6*b^4)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^3*(a^2 - b^2)^(5/2)*d) - (a^2*cos(c + d*x)*sin(c + d*x))/(2*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) - (a^2*(2*a^2 - 5*b^2)*sin(c + d*x))/(2*b^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))],
[cos(c + d*x)^2/(a + b*cos(c + d*x))^3, x, 4, ((a^2 + 2*b^2)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/((a^2 - b^2)^(5/2)*d) - (a^2*sin(c + d*x))/(2*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) + (a*(a^2 - 4*b^2)*sin(c + d*x))/(2*b*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))],
[cos(c + d*x)^1/(a + b*cos(c + d*x))^3, x, 4, -((3*a*b*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/((a^2 - b^2)^(5/2)*d)) + (a*sin(c + d*x))/(2*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) + ((a^2 + 2*b^2)*sin(c + d*x))/(2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))],
[cos(c + d*x)^0/(a + b*cos(c + d*x))^3, x, 4, ((2*a^2 + b^2)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/((a^2 - b^2)^(5/2)*d) - (b*sin(c + d*x))/(2*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) - (3*a*b*sin(c + d*x))/(2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))],
[sec(c + d*x)^1/(a + b*cos(c + d*x))^3, x, 5, -((b*(6*a^4 - 5*a^2*b^2 + 2*b^4)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^3*(a^2 - b^2)^(5/2)*d)) + arctanh(sin(c + d*x))/(a^3*d) + (b^2*sin(c + d*x))/(2*a*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) + (b^2*(5*a^2 - 2*b^2)*sin(c + d*x))/(2*a^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))],
[sec(c + d*x)^2/(a + b*cos(c + d*x))^3, x, 6, (3*b^2*(4*a^4 - 5*a^2*b^2 + 2*b^4)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^4*(a^2 - b^2)^(5/2)*d) - (3*b*arctanh(sin(c + d*x)))/(a^4*d) + ((2*a^4 - 11*a^2*b^2 + 6*b^4)*tan(c + d*x))/(2*a^3*(a^2 - b^2)^2*d) + (b^2*tan(c + d*x))/(2*a*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) + (3*b^2*(2*a^2 - b^2)*tan(c + d*x))/(2*a^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))],
[sec(c + d*x)^3/(a + b*cos(c + d*x))^3, x, 7, -((b^3*(20*a^4 - 29*a^2*b^2 + 12*b^4)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^5*(a^2 - b^2)^(5/2)*d)) + ((a^2 + 12*b^2)*arctanh(sin(c + d*x)))/(2*a^5*d) - (3*b*(2*a^4 - 7*a^2*b^2 + 4*b^4)*tan(c + d*x))/(2*a^4*(a^2 - b^2)^2*d) + ((a^4 - 10*a^2*b^2 + 6*b^4)*sec(c + d*x)*tan(c + d*x))/(2*a^3*(a^2 - b^2)^2*d) + (b^2*sec(c + d*x)*tan(c + d*x))/(2*a*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) + (b^2*(7*a^2 - 4*b^2)*sec(c + d*x)*tan(c + d*x))/(2*a^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))],


[cos(c + d*x)^5/(a + b*cos(c + d*x))^4, x, 6, -((4*a*x)/b^5) + (a^2*(8*a^6 - 28*a^4*b^2 + 35*a^2*b^4 - 20*b^6)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^5*(a^2 - b^2)^(7/2)*d) + ((24*a^6 - 68*a^4*b^2 + 65*a^2*b^4 - 6*b^6)*sin(c + d*x))/(6*b^4*(a^2 - b^2)^3*d) - (a^2*cos(c + d*x)^3*sin(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^3) - (a^2*(4*a^2 - 9*b^2)*cos(c + d*x)^2*sin(c + d*x))/(6*b^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x))^2) - (a^2*(4*a^4 - 11*a^2*b^2 + 12*b^4)*cos(c + d*x)*sin(c + d*x))/(2*b^3*(a^2 - b^2)^3*d*(a + b*cos(c + d*x)))],
[cos(c + d*x)^4/(a + b*cos(c + d*x))^4, x, 5, x/b^4 - (a*(2*a^6 - 7*a^4*b^2 + 8*a^2*b^4 - 8*b^6)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^4*(a^2 - b^2)^(7/2)*d) - (a^2*cos(c + d*x)^2*sin(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^3) - (a^2*(3*a^2 - 8*b^2)*cos(c + d*x)*sin(c + d*x))/(6*b^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x))^2) - (a^2*(6*a^4 - 17*a^2*b^2 + 26*b^4)*sin(c + d*x))/(6*b^3*(a^2 - b^2)^3*d*(a + b*cos(c + d*x)))],
[cos(c + d*x)^3/(a + b*cos(c + d*x))^4, x, 5, -((b*(3*a^2 + 2*b^2)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/((a^2 - b^2)^(7/2)*d)) - (a^2*cos(c + d*x)*sin(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^3) - (a^2*(2*a^2 - 7*b^2)*sin(c + d*x))/(6*b^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x))^2) + (a*(2*a^4 - 5*a^2*b^2 + 18*b^4)*sin(c + d*x))/(6*b^2*(a^2 - b^2)^3*d*(a + b*cos(c + d*x)))],
[cos(c + d*x)^2/(a + b*cos(c + d*x))^4, x, 5, (a*(a^2 + 4*b^2)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/((a^2 - b^2)^(7/2)*d) - (a^2*sin(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^3) + (a*(a^2 - 6*b^2)*sin(c + d*x))/(6*b*(a^2 - b^2)^2*d*(a + b*cos(c + d*x))^2) + ((a^4 - 10*a^2*b^2 - 6*b^4)*sin(c + d*x))/(6*b*(a^2 - b^2)^3*d*(a + b*cos(c + d*x)))],
[cos(c + d*x)^1/(a + b*cos(c + d*x))^4, x, 5, -((b*(4*a^2 + b^2)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/((a^2 - b^2)^(7/2)*d)) + (a*sin(c + d*x))/(3*(a^2 - b^2)*d*(a + b*cos(c + d*x))^3) + ((2*a^2 + 3*b^2)*sin(c + d*x))/(6*(a^2 - b^2)^2*d*(a + b*cos(c + d*x))^2) + (a*(2*a^2 + 13*b^2)*sin(c + d*x))/(6*(a^2 - b^2)^3*d*(a + b*cos(c + d*x)))],
[cos(c + d*x)^0/(a + b*cos(c + d*x))^4, x, 5, (a*(2*a^2 + 3*b^2)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/((a^2 - b^2)^(7/2)*d) - (b*sin(c + d*x))/(3*(a^2 - b^2)*d*(a + b*cos(c + d*x))^3) - (5*a*b*sin(c + d*x))/(6*(a^2 - b^2)^2*d*(a + b*cos(c + d*x))^2) - (b*(11*a^2 + 4*b^2)*sin(c + d*x))/(6*(a^2 - b^2)^3*d*(a + b*cos(c + d*x)))],
[sec(c + d*x)^1/(a + b*cos(c + d*x))^4, x, 6, -((b*(8*a^6 - 8*a^4*b^2 + 7*a^2*b^4 - 2*b^6)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^4*(a^2 - b^2)^(7/2)*d)) + arctanh(sin(c + d*x))/(a^4*d) + (b^2*sin(c + d*x))/(3*a*(a^2 - b^2)*d*(a + b*cos(c + d*x))^3) + (b^2*(8*a^2 - 3*b^2)*sin(c + d*x))/(6*a^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x))^2) + (b^2*(26*a^4 - 17*a^2*b^2 + 6*b^4)*sin(c + d*x))/(6*a^3*(a^2 - b^2)^3*d*(a + b*cos(c + d*x)))],
[sec(c + d*x)^2/(a + b*cos(c + d*x))^4, x, 7, (b^2*(20*a^6 - 35*a^4*b^2 + 28*a^2*b^4 - 8*b^6)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^5*(a^2 - b^2)^(7/2)*d) - (4*b*arctanh(sin(c + d*x)))/(a^5*d) + ((6*a^6 - 65*a^4*b^2 + 68*a^2*b^4 - 24*b^6)*tan(c + d*x))/(6*a^4*(a^2 - b^2)^3*d) + (b^2*tan(c + d*x))/(3*a*(a^2 - b^2)*d*(a + b*cos(c + d*x))^3) + (b^2*(9*a^2 - 4*b^2)*tan(c + d*x))/(6*a^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x))^2) + (b^2*(12*a^4 - 11*a^2*b^2 + 4*b^4)*tan(c + d*x))/(2*a^3*(a^2 - b^2)^3*d*(a + b*cos(c + d*x)))],


[1/(3 + 5*cos(c + d*x)), x, 1, arctanh((1/2)*tan((1/2)*(c + d*x)))/(2*d)],
[1/(3 + 5*cos(c + d*x))^2, x, 3, -((3*arctanh((1/2)*tan((1/2)*(c + d*x))))/(32*d)) + (5*sin(c + d*x))/(16*d*(3 + 5*cos(c + d*x)))],
[1/(3 + 5*cos(c + d*x))^3, x, 4, (43*arctanh((1/2)*tan((1/2)*(c + d*x))))/(1024*d) + (5*sin(c + d*x))/(32*d*(3 + 5*cos(c + d*x))^2) - (45*sin(c + d*x))/(512*d*(3 + 5*cos(c + d*x)))],
[1/(3 + 5*cos(c + d*x))^4, x, 5, -((279*arctanh((1/2)*tan((1/2)*(c + d*x))))/(16384*d)) + (5*sin(c + d*x))/(48*d*(3 + 5*cos(c + d*x))^3) - (25*sin(c + d*x))/(512*d*(3 + 5*cos(c + d*x))^2) + (995*sin(c + d*x))/(24576*d*(3 + 5*cos(c + d*x)))],

[1/(5 + 3*cos(c + d*x)), x, 1, x/4 - arctan(sin(c + d*x)/(3 + cos(c + d*x)))/(2*d)],
[1/(5 + 3*cos(c + d*x))^2, x, 3, (5*x)/64 - (5*arctan(sin(c + d*x)/(3 + cos(c + d*x))))/(32*d) - (3*sin(c + d*x))/(16*d*(5 + 3*cos(c + d*x)))],
[1/(5 + 3*cos(c + d*x))^3, x, 4, (59*x)/2048 - (59*arctan(sin(c + d*x)/(3 + cos(c + d*x))))/(1024*d) - (3*sin(c + d*x))/(32*d*(5 + 3*cos(c + d*x))^2) - (45*sin(c + d*x))/(512*d*(5 + 3*cos(c + d*x)))],
[1/(5 + 3*cos(c + d*x))^4, x, 5, (385*x)/32768 - (385*arctan(sin(c + d*x)/(3 + cos(c + d*x))))/(16384*d) - sin(c + d*x)/(16*d*(5 + 3*cos(c + d*x))^3) - (25*sin(c + d*x))/(512*d*(5 + 3*cos(c + d*x))^2) - (311*sin(c + d*x))/(8192*d*(5 + 3*cos(c + d*x)))],


# ::Subsection::Closed:: 
#Integrands of the form Cos[c+d x]^m (a+b Cos[c+d x])^(n/2)


# ::Subsubsection::Closed:: 
#n>0


[cos(c + d*x)^3*sqrt(a + b*cos(c + d*x)), x, 8, (2*a*(a + b)*(8*a^2 + 19*b^2)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(105*b^3*d*sqrt(a + b*cos(c + d*x))) - (2*(8*a^4 + 17*a^2*b^2 - 25*b^4)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(105*b^3*d*sqrt(a + b*cos(c + d*x))) - (2*(4*a^2 - 25*b^2)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(105*b^2*d) + (2*a*cos(c + d*x)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(35*b*d) + (2*cos(c + d*x)^2*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(7*d)],
[cos(c + d*x)^2*sqrt(a + b*cos(c + d*x)), x, 7, -((2*(a + b)*(2*a^2 - 9*b^2)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(15*b^2*d*sqrt(a + b*cos(c + d*x)))) + (4*a*(a^2 - b^2)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(15*b^2*d*sqrt(a + b*cos(c + d*x))) - (4*a*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(15*b*d) + (2*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(5*b*d)],
[cos(c + d*x)^1*sqrt(a + b*cos(c + d*x)), x, 6, (2*a*(a + b)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(3*b*d*sqrt(a + b*cos(c + d*x))) - (2*(a^2 - b^2)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(3*b*d*sqrt(a + b*cos(c + d*x))) + (2*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*d)],
[cos(c + d*x)^0*sqrt(a + b*cos(c + d*x)), x, 2, (2*(a + b)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x)))],
[sec(c + d*x)^1*sqrt(a + b*cos(c + d*x)), x, 5, (2*b*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x))) + (2*a*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x)))],
[sec(c + d*x)^2*sqrt(a + b*cos(c + d*x)), x, 9, -(((a + b)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x)))) + (a*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x))) + (b*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x))) + (sqrt(a + b*cos(c + d*x))*tan(c + d*x))/d],
[sec(c + d*x)^3*sqrt(a + b*cos(c + d*x)), x, 10, -((b*(a + b)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(4*a*d*sqrt(a + b*cos(c + d*x)))) + (3*b*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(4*d*sqrt(a + b*cos(c + d*x))) + ((4*a^2 - b^2)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*b)/(a + b)))/(4*a*d*sqrt(a + b*cos(c + d*x))) + (b*sqrt(a + b*cos(c + d*x))*tan(c + d*x))/(4*a*d) + (sqrt(a + b*cos(c + d*x))*sec(c + d*x)*tan(c + d*x))/(2*d)],


[cos(c + d*x)^3*(a + b*cos(c + d*x))^(3/2), x, 9, (2*(a + b)*(8*a^4 + 33*a^2*b^2 + 147*b^4)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(315*b^3*d*sqrt(a + b*cos(c + d*x))) - (2*a*(8*a^4 + 31*a^2*b^2 - 39*b^4)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(315*b^3*d*sqrt(a + b*cos(c + d*x))) - (8*a*(a^2 - 22*b^2)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(315*b^2*d) + (2*(3*a^2 + 49*b^2)*cos(c + d*x)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(315*b*d) + (20*a*cos(c + d*x)^2*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(63*d) + (2*b*cos(c + d*x)^3*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(9*d)],
[cos(c + d*x)^2*(a + b*cos(c + d*x))^(3/2), x, 8, -((4*a*(a + b)*(3*a^2 - 41*b^2)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(105*b^2*d*sqrt(a + b*cos(c + d*x)))) + (2*(6*a^4 - 31*a^2*b^2 + 25*b^4)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(105*b^2*d*sqrt(a + b*cos(c + d*x))) - (2*(6*a^2 - 25*b^2)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(105*b*d) - (4*a*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(35*b*d) + (2*(a + b*cos(c + d*x))^(5/2)*sin(c + d*x))/(7*b*d)],
[cos(c + d*x)^1*(a + b*cos(c + d*x))^(3/2), x, 7, (2*(a + b)*(a^2 + 3*b^2)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(5*b*d*sqrt(a + b*cos(c + d*x))) - (2*a*(a^2 - b^2)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(5*b*d*sqrt(a + b*cos(c + d*x))) + (2*a*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(5*d) + (2*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(5*d)],
[cos(c + d*x)^0*(a + b*cos(c + d*x))^(3/2), x, 6, (8*a*(a + b)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(3*d*sqrt(a + b*cos(c + d*x))) - (2*(a^2 - b^2)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(3*d*sqrt(a + b*cos(c + d*x))) + (2*b*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*d)],
[sec(c + d*x)^1*(a + b*cos(c + d*x))^(3/2), x, 8, (2*b*(a + b)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x))) + (2*a*b*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x))) + (2*a^2*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x)))],
[sec(c + d*x)^2*(a + b*cos(c + d*x))^(3/2), x, 9, -((a*(a + b)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x)))) + ((a^2 + 2*b^2)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x))) + (3*a*b*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x))) + (a*sqrt(a + b*cos(c + d*x))*tan(c + d*x))/d],
[sec(c + d*x)^3*(a + b*cos(c + d*x))^(3/2), x, 10, -((5*b*(a + b)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(4*d*sqrt(a + b*cos(c + d*x)))) + (7*a*b*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(4*d*sqrt(a + b*cos(c + d*x))) + ((4*a^2 + 3*b^2)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*b)/(a + b)))/(4*d*sqrt(a + b*cos(c + d*x))) + (5*b*sqrt(a + b*cos(c + d*x))*tan(c + d*x))/(4*d) + (a*sqrt(a + b*cos(c + d*x))*sec(c + d*x)*tan(c + d*x))/(2*d)],


[cos(c + d*x)^3*(a + b*cos(c + d*x))^(5/2), x, 10, (2*a*(a + b)*(8*a^4 + 51*a^2*b^2 + 741*b^4)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(693*b^3*d*sqrt(a + b*cos(c + d*x))) - (2*(8*a^6 + 49*a^4*b^2 + 78*a^2*b^4 - 135*b^6)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(693*b^3*d*sqrt(a + b*cos(c + d*x))) - (2*(4*a^4 - 205*a^2*b^2 - 135*b^4)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(693*b^2*d) + (2*a*(3*a^2 + 229*b^2)*cos(c + d*x)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(693*b*d) + (2*(113*a^2 + 81*b^2)*cos(c + d*x)^2*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(693*d) + (46*a*b*cos(c + d*x)^3*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(99*d) + (2*b^2*cos(c + d*x)^4*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(11*d)],
[cos(c + d*x)^2*(a + b*cos(c + d*x))^(5/2), x, 9, -((2*(a + b)*(10*a^4 - 279*a^2*b^2 - 147*b^4)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(315*b^2*d*sqrt(a + b*cos(c + d*x)))) + (4*a*(5*a^4 - 62*a^2*b^2 + 57*b^4)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(315*b^2*d*sqrt(a + b*cos(c + d*x))) - (4*a*(5*a^2 - 57*b^2)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(315*b*d) - (2*(10*a^2 - 49*b^2)*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(315*b*d) - (4*a*(a + b*cos(c + d*x))^(5/2)*sin(c + d*x))/(63*b*d) + (2*(a + b*cos(c + d*x))^(7/2)*sin(c + d*x))/(9*b*d)],
[cos(c + d*x)^1*(a + b*cos(c + d*x))^(5/2), x, 8, (2*a*(a + b)*(3*a^2 + 29*b^2)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(21*b*d*sqrt(a + b*cos(c + d*x))) - (2*(3*a^4 + 2*a^2*b^2 - 5*b^4)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(21*b*d*sqrt(a + b*cos(c + d*x))) + (2*(3*a^2 + 5*b^2)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(21*d) + (2*a*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(7*d) + (2*(a + b*cos(c + d*x))^(5/2)*sin(c + d*x))/(7*d)],
[cos(c + d*x)^0*(a + b*cos(c + d*x))^(5/2), x, 7, (2*(a + b)*(23*a^2 + 9*b^2)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(15*d*sqrt(a + b*cos(c + d*x))) - (16*a*(a^2 - b^2)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(15*d*sqrt(a + b*cos(c + d*x))) + (16*a*b*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(15*d) + (2*b*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(5*d)],
[sec(c + d*x)^1*(a + b*cos(c + d*x))^(5/2), x, 9, (14*a*b*(a + b)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(3*d*sqrt(a + b*cos(c + d*x))) + (2*b*(2*a^2 + b^2)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(3*d*sqrt(a + b*cos(c + d*x))) + (2*a^3*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x))) + (2*b^2*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*d)],
[sec(c + d*x)^2*(a + b*cos(c + d*x))^(5/2), x, 9, -(((a + b)*(a^2 - 2*b^2)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x)))) + (a*(a^2 + 4*b^2)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x))) + (5*a^2*b*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x))) + (a^2*sqrt(a + b*cos(c + d*x))*tan(c + d*x))/d],
[sec(c + d*x)^3*(a + b*cos(c + d*x))^(5/2), x, 10, -((9*a*b*(a + b)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(4*d*sqrt(a + b*cos(c + d*x)))) + (b*(11*a^2 + 8*b^2)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(4*d*sqrt(a + b*cos(c + d*x))) + (a*(4*a^2 + 15*b^2)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*b)/(a + b)))/(4*d*sqrt(a + b*cos(c + d*x))) + (9*a*b*sqrt(a + b*cos(c + d*x))*tan(c + d*x))/(4*d) + (a^2*sqrt(a + b*cos(c + d*x))*sec(c + d*x)*tan(c + d*x))/(2*d)],
[sec(c + d*x)^4*(a + b*cos(c + d*x))^(5/2), x, 11, -(((a + b)*(16*a^2 + 33*b^2)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(24*d*sqrt(a + b*cos(c + d*x)))) + (a*(16*a^2 + 59*b^2)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(24*d*sqrt(a + b*cos(c + d*x))) + (5*b*(4*a^2 + b^2)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*b)/(a + b)))/(8*d*sqrt(a + b*cos(c + d*x))) + ((16*a^2 + 33*b^2)*sqrt(a + b*cos(c + d*x))*tan(c + d*x))/(24*d) + (13*a*b*sqrt(a + b*cos(c + d*x))*sec(c + d*x)*tan(c + d*x))/(12*d) + (a^2*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^2*tan(c + d*x))/(3*d)],


[(a + b*cos(c + d*x))^(7/2), x, 8, (32*a*(a + b)*(11*a^2 + 13*b^2)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(105*d*sqrt(a + b*cos(c + d*x))) - (2*(71*a^4 - 46*a^2*b^2 - 25*b^4)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(105*d*sqrt(a + b*cos(c + d*x))) + (2*b*(71*a^2 + 25*b^2)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(105*d) + (24*a*b*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(35*d) + (2*b*(a + b*cos(c + d*x))^(5/2)*sin(c + d*x))/(7*d)],


# ::Subsubsection::Closed:: 
#n<0


[cos(c + d*x)^3/sqrt(a + b*cos(c + d*x)), x, 7, (2*(a + b)*(8*a^2 + 9*b^2)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(15*b^3*d*sqrt(a + b*cos(c + d*x))) - (2*a*(8*a^2 + 7*b^2)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(15*b^3*d*sqrt(a + b*cos(c + d*x))) - (8*a*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(15*b^2*d) + (2*cos(c + d*x)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(5*b*d)],
[cos(c + d*x)^2/sqrt(a + b*cos(c + d*x)), x, 6, -((4*a*(a + b)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(3*b^2*d*sqrt(a + b*cos(c + d*x)))) + (2*(2*a^2 + b^2)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(3*b^2*d*sqrt(a + b*cos(c + d*x))) + (2*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*b*d)],
[cos(c + d*x)^1/sqrt(a + b*cos(c + d*x)), x, 5, (2*(a + b)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(b*d*sqrt(a + b*cos(c + d*x))) - (2*a*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(b*d*sqrt(a + b*cos(c + d*x)))],
[cos(c + d*x)^0/sqrt(a + b*cos(c + d*x)), x, 2, (2*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x)))],
[sec(c + d*x)^1/sqrt(a + b*cos(c + d*x)), x, 2, (2*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x)))],
[sec(c + d*x)^2/sqrt(a + b*cos(c + d*x)), x, 9, -(((a + b)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(a*d*sqrt(a + b*cos(c + d*x)))) + (sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x))) - (b*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*b)/(a + b)))/(a*d*sqrt(a + b*cos(c + d*x))) + (sqrt(a + b*cos(c + d*x))*tan(c + d*x))/(a*d)],
[sec(c + d*x)^3/sqrt(a + b*cos(c + d*x)), x, 10, (3*b*(a + b)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(4*a^2*d*sqrt(a + b*cos(c + d*x))) - (b*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(4*a*d*sqrt(a + b*cos(c + d*x))) + ((4*a^2 + 3*b^2)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*b)/(a + b)))/(4*a^2*d*sqrt(a + b*cos(c + d*x))) - (3*b*sqrt(a + b*cos(c + d*x))*tan(c + d*x))/(4*a^2*d) + (sqrt(a + b*cos(c + d*x))*sec(c + d*x)*tan(c + d*x))/(2*a*d)],


[cos(c + d*x)^4/(a + b*cos(c + d*x))^(3/2), x, 8, (2*(16*a^4 - 8*a^2*b^2 - 3*b^4)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(5*(a - b)*b^4*d*sqrt(a + b*cos(c + d*x))) - (8*a*(4*a^2 + b^2)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(5*b^4*d*sqrt(a + b*cos(c + d*x))) - (2*a^2*cos(c + d*x)^2*sin(c + d*x))/(b*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) - (2*a*(8*a^2 - 3*b^2)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(5*b^3*(a^2 - b^2)*d) + (2*(6*a^2 - b^2)*cos(c + d*x)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(5*b^2*(a^2 - b^2)*d)],
[cos(c + d*x)^3/(a + b*cos(c + d*x))^(3/2), x, 7, -((2*a*(8*a^2 - 5*b^2)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(3*(a - b)*b^3*d*sqrt(a + b*cos(c + d*x)))) + (2*(8*a^2 + b^2)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(3*b^3*d*sqrt(a + b*cos(c + d*x))) - (2*a^2*cos(c + d*x)*sin(c + d*x))/(b*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) + (2*(4*a^2 - b^2)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*b^2*(a^2 - b^2)*d)],
[cos(c + d*x)^2/(a + b*cos(c + d*x))^(3/2), x, 6, (2*(2*a^2 - b^2)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/((a - b)*b^2*d*sqrt(a + b*cos(c + d*x))) - (4*a*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(b^2*d*sqrt(a + b*cos(c + d*x))) - (2*a^2*sin(c + d*x))/(b*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x)))],
[cos(c + d*x)^1/(a + b*cos(c + d*x))^(3/2), x, 6, -((2*a*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/((a - b)*b*d*sqrt(a + b*cos(c + d*x)))) + (2*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(b*d*sqrt(a + b*cos(c + d*x))) + (2*a*sin(c + d*x))/((a^2 - b^2)*d*sqrt(a + b*cos(c + d*x)))],
[cos(c + d*x)^0/(a + b*cos(c + d*x))^(3/2), x, 4, (2*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/((a - b)*d*sqrt(a + b*cos(c + d*x))) - (2*b*sin(c + d*x))/((a^2 - b^2)*d*sqrt(a + b*cos(c + d*x)))],
[sec(c + d*x)^1/(a + b*cos(c + d*x))^(3/2), x, 7, -((2*b*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(a*(a - b)*d*sqrt(a + b*cos(c + d*x)))) + (2*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*b)/(a + b)))/(a*d*sqrt(a + b*cos(c + d*x))) + (2*b^2*sin(c + d*x))/(a*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x)))],
[sec(c + d*x)^2/(a + b*cos(c + d*x))^(3/2), x, 10, -(((a^2 - 3*b^2)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(a^2*(a - b)*d*sqrt(a + b*cos(c + d*x)))) + (sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(a*d*sqrt(a + b*cos(c + d*x))) - (3*b*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*b)/(a + b)))/(a^2*d*sqrt(a + b*cos(c + d*x))) + (2*b^2*tan(c + d*x))/(a*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) + ((a^2 - 3*b^2)*sqrt(a + b*cos(c + d*x))*tan(c + d*x))/(a^2*(a^2 - b^2)*d)],
[sec(c + d*x)^3/(a + b*cos(c + d*x))^(3/2), x, 11, (b*(7*a^2 - 15*b^2)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(4*a^3*(a - b)*d*sqrt(a + b*cos(c + d*x))) - (5*b*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(4*a^2*d*sqrt(a + b*cos(c + d*x))) + ((4*a^2 + 15*b^2)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*b)/(a + b)))/(4*a^3*d*sqrt(a + b*cos(c + d*x))) - (b*(7*a^2 - 15*b^2)*sqrt(a + b*cos(c + d*x))*tan(c + d*x))/(4*a^3*(a^2 - b^2)*d) + (2*b^2*sec(c + d*x)*tan(c + d*x))/(a*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) + ((a^2 - 5*b^2)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)*tan(c + d*x))/(2*a^2*(a^2 - b^2)*d)],


[cos(c + d*x)^5/(a + b*cos(c + d*x))^(5/2), x, 9, (2*(128*a^6 - 212*a^4*b^2 + 55*a^2*b^4 + 9*b^6)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(15*(a - b)^2*b^5*(a + b)*d*sqrt(a + b*cos(c + d*x))) - (2*a*(128*a^4 - 116*a^2*b^2 - 17*b^4)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(15*b^5*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) - (2*a^2*cos(c + d*x)^3*sin(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)) - (8*a^2*(2*a^2 - 3*b^2)*cos(c + d*x)^2*sin(c + d*x))/(3*b^2*(a^2 - b^2)^2*d*sqrt(a + b*cos(c + d*x))) - (4*a*(32*a^4 - 49*a^2*b^2 + 7*b^4)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(15*b^4*(a^2 - b^2)^2*d) + (2*(48*a^4 - 71*a^2*b^2 + 3*b^4)*cos(c + d*x)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(15*b^3*(a^2 - b^2)^2*d)],
[cos(c + d*x)^4/(a + b*cos(c + d*x))^(5/2), x, 8, -((8*a*(4*a^4 - 7*a^2*b^2 + 2*b^4)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(3*(a - b)^2*b^4*(a + b)*d*sqrt(a + b*cos(c + d*x)))) + (2*(16*a^4 - 16*a^2*b^2 - b^4)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(3*b^4*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) - (2*a^2*cos(c + d*x)^2*sin(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)) - (4*a^2*(3*a^2 - 5*b^2)*cos(c + d*x)*sin(c + d*x))/(3*b^2*(a^2 - b^2)^2*d*sqrt(a + b*cos(c + d*x))) + (2*(8*a^4 - 13*a^2*b^2 + b^4)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*b^3*(a^2 - b^2)^2*d)],
[cos(c + d*x)^3/(a + b*cos(c + d*x))^(5/2), x, 7, (2*(8*a^4 - 15*a^2*b^2 + 3*b^4)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(3*(a - b)^2*b^3*(a + b)*d*sqrt(a + b*cos(c + d*x))) - (2*a*(8*a^2 - 9*b^2)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(3*b^3*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) - (2*a^2*cos(c + d*x)*sin(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)) - (8*a^2*(a^2 - 2*b^2)*sin(c + d*x))/(3*b^2*(a^2 - b^2)^2*d*sqrt(a + b*cos(c + d*x)))],
[cos(c + d*x)^2/(a + b*cos(c + d*x))^(5/2), x, 7, -((4*a*(a^2 - 3*b^2)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(3*(a - b)^2*b^2*(a + b)*d*sqrt(a + b*cos(c + d*x)))) + (2*(2*a^2 - 3*b^2)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(3*b^2*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) - (2*a^2*sin(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)) + (4*a*(a^2 - 3*b^2)*sin(c + d*x))/(3*b*(a^2 - b^2)^2*d*sqrt(a + b*cos(c + d*x)))],
[cos(c + d*x)^1/(a + b*cos(c + d*x))^(5/2), x, 7, -((2*(a^2 + 3*b^2)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(3*(a - b)^2*b*(a + b)*d*sqrt(a + b*cos(c + d*x)))) + (2*a*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(3*b*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) + (2*a*sin(c + d*x))/(3*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)) + (2*(a^2 + 3*b^2)*sin(c + d*x))/(3*(a^2 - b^2)^2*d*sqrt(a + b*cos(c + d*x)))],
[cos(c + d*x)^0/(a + b*cos(c + d*x))^(5/2), x, 7, (8*a*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(3*(a - b)^2*(a + b)*d*sqrt(a + b*cos(c + d*x))) - (2*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(3*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) - (2*b*sin(c + d*x))/(3*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)) - (8*a*b*sin(c + d*x))/(3*(a^2 - b^2)^2*d*sqrt(a + b*cos(c + d*x)))],
[sec(c + d*x)^1/(a + b*cos(c + d*x))^(5/2), x, 10, -((2*b*(7*a^2 - 3*b^2)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(3*a^2*(a - b)^2*(a + b)*d*sqrt(a + b*cos(c + d*x)))) + (2*b*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(3*a*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) + (2*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*b)/(a + b)))/(a^2*d*sqrt(a + b*cos(c + d*x))) + (2*b^2*sin(c + d*x))/(3*a*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)) + (2*b^2*(7*a^2 - 3*b^2)*sin(c + d*x))/(3*a^2*(a^2 - b^2)^2*d*sqrt(a + b*cos(c + d*x)))],
[sec(c + d*x)^2/(a + b*cos(c + d*x))^(5/2), x, 11, -(((3*a^4 - 26*a^2*b^2 + 15*b^4)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(3*a^3*(a - b)^2*(a + b)*d*sqrt(a + b*cos(c + d*x)))) + ((3*a^2 - 5*b^2)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(3*a^2*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) - (5*b*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*b)/(a + b)))/(a^3*d*sqrt(a + b*cos(c + d*x))) + (2*b^2*tan(c + d*x))/(3*a*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)) + (2*b^2*(9*a^2 - 5*b^2)*tan(c + d*x))/(3*a^2*(a^2 - b^2)^2*d*sqrt(a + b*cos(c + d*x))) + ((3*a^4 - 26*a^2*b^2 + 15*b^4)*sqrt(a + b*cos(c + d*x))*tan(c + d*x))/(3*a^3*(a^2 - b^2)^2*d)],


[1/(a + b*cos(c + d*x))^(7/2), x, 8, (2*(23*a^2 + 9*b^2)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(15*(a - b)^3*(a + b)^2*d*sqrt(a + b*cos(c + d*x))) - (16*a*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(15*(a^2 - b^2)^2*d*sqrt(a + b*cos(c + d*x))) - (2*b*sin(c + d*x))/(5*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(5/2)) - (16*a*b*sin(c + d*x))/(15*(a^2 - b^2)^2*d*(a + b*cos(c + d*x))^(3/2)) - (2*b*(23*a^2 + 9*b^2)*sin(c + d*x))/(15*(a^2 - b^2)^3*d*sqrt(a + b*cos(c + d*x)))],


# ::Subsection::Closed:: 
#Integrands of the form Cos[c+d x]^(m/2) (a+b Cos[c+d x])^n


# ::Subsubsection::Closed:: 
#n>0


[cos(c + d*x)^(5/2)*(a + b*cos(c + d*x))^2, x, 7, (2*(9*a^2 + 7*b^2)*EllipticE((1/2)*(c + d*x), 2))/(15*d) + (20*a*b*EllipticF((1/2)*(c + d*x), 2))/(21*d) + (20*a*b*sqrt(cos(c + d*x))*sin(c + d*x))/(21*d) + (2*(9*a^2 + 7*b^2)*cos(c + d*x)^(3/2)*sin(c + d*x))/(45*d) + (4*a*b*cos(c + d*x)^(5/2)*sin(c + d*x))/(7*d) + (2*b^2*cos(c + d*x)^(7/2)*sin(c + d*x))/(9*d)],
[cos(c + d*x)^(3/2)*(a + b*cos(c + d*x))^2, x, 6, (12*a*b*EllipticE((1/2)*(c + d*x), 2))/(5*d) + (2*(7*a^2 + 5*b^2)*EllipticF((1/2)*(c + d*x), 2))/(21*d) + (2*(7*a^2 + 5*b^2)*sqrt(cos(c + d*x))*sin(c + d*x))/(21*d) + (4*a*b*cos(c + d*x)^(3/2)*sin(c + d*x))/(5*d) + (2*b^2*cos(c + d*x)^(5/2)*sin(c + d*x))/(7*d)],
[cos(c + d*x)^(1/2)*(a + b*cos(c + d*x))^2, x, 5, (2*(5*a^2 + 3*b^2)*EllipticE((1/2)*(c + d*x), 2))/(5*d) + (4*a*b*EllipticF((1/2)*(c + d*x), 2))/(3*d) + (4*a*b*sqrt(cos(c + d*x))*sin(c + d*x))/(3*d) + (2*b^2*cos(c + d*x)^(3/2)*sin(c + d*x))/(5*d)],
[cos(c + d*x)^(-1/2)*(a + b*cos(c + d*x))^2, x, 4, (4*a*b*EllipticE((1/2)*(c + d*x), 2))/d + (2*(3*a^2 + b^2)*EllipticF((1/2)*(c + d*x), 2))/(3*d) + (2*b^2*sqrt(cos(c + d*x))*sin(c + d*x))/(3*d)],
[cos(c + d*x)^(-3/2)*(a + b*cos(c + d*x))^2, x, 4, -((2*(a^2 - b^2)*EllipticE((1/2)*(c + d*x), 2))/d) + (4*a*b*EllipticF((1/2)*(c + d*x), 2))/d + (2*a^2*sin(c + d*x))/(d*sqrt(cos(c + d*x)))],
[cos(c + d*x)^(-5/2)*(a + b*cos(c + d*x))^2, x, 5, -((4*a*b*EllipticE((1/2)*(c + d*x), 2))/d) + (2*(a^2 + 3*b^2)*EllipticF((1/2)*(c + d*x), 2))/(3*d) + (2*a^2*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2)) + (4*a*b*sin(c + d*x))/(d*sqrt(cos(c + d*x)))],
[cos(c + d*x)^(-7/2)*(a + b*cos(c + d*x))^2, x, 6, -((2*(3*a^2 + 5*b^2)*EllipticE((1/2)*(c + d*x), 2))/(5*d)) + (4*a*b*EllipticF((1/2)*(c + d*x), 2))/(3*d) + (2*a^2*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2)) + (4*a*b*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2)) + (2*(3*a^2 + 5*b^2)*sin(c + d*x))/(5*d*sqrt(cos(c + d*x)))],


[cos(c + d*x)^(3/2)*(a + b*cos(c + d*x))^3, x, 7, (2*b*(27*a^2 + 7*b^2)*EllipticE((1/2)*(c + d*x), 2))/(15*d) + (2*a*(7*a^2 + 15*b^2)*EllipticF((1/2)*(c + d*x), 2))/(21*d) + (2*a*(7*a^2 + 15*b^2)*sqrt(cos(c + d*x))*sin(c + d*x))/(21*d) + (2*b*(27*a^2 + 7*b^2)*cos(c + d*x)^(3/2)*sin(c + d*x))/(45*d) + (40*a*b^2*cos(c + d*x)^(5/2)*sin(c + d*x))/(63*d) + (2*b^2*cos(c + d*x)^(5/2)*(a + b*cos(c + d*x))*sin(c + d*x))/(9*d)],
[cos(c + d*x)^(1/2)*(a + b*cos(c + d*x))^3, x, 6, (2*a*(5*a^2 + 9*b^2)*EllipticE((1/2)*(c + d*x), 2))/(5*d) + (2*b*(21*a^2 + 5*b^2)*EllipticF((1/2)*(c + d*x), 2))/(21*d) + (2*b*(21*a^2 + 5*b^2)*sqrt(cos(c + d*x))*sin(c + d*x))/(21*d) + (32*a*b^2*cos(c + d*x)^(3/2)*sin(c + d*x))/(35*d) + (2*b^2*cos(c + d*x)^(3/2)*(a + b*cos(c + d*x))*sin(c + d*x))/(7*d)],
[cos(c + d*x)^(-1/2)*(a + b*cos(c + d*x))^3, x, 5, (6*b*(5*a^2 + b^2)*EllipticE((1/2)*(c + d*x), 2))/(5*d) + (2*a*(a^2 + b^2)*EllipticF((1/2)*(c + d*x), 2))/d + (8*a*b^2*sqrt(cos(c + d*x))*sin(c + d*x))/(5*d) + (2*b^2*sqrt(cos(c + d*x))*(a + b*cos(c + d*x))*sin(c + d*x))/(5*d)],
[cos(c + d*x)^(-3/2)*(a + b*cos(c + d*x))^3, x, 5, -((2*a*(a^2 - 3*b^2)*EllipticE((1/2)*(c + d*x), 2))/d) + (2*b*(9*a^2 + b^2)*EllipticF((1/2)*(c + d*x), 2))/(3*d) - (2*b*(3*a^2 - b^2)*sqrt(cos(c + d*x))*sin(c + d*x))/(3*d) + (2*a^2*(a + b*cos(c + d*x))*sin(c + d*x))/(d*sqrt(cos(c + d*x)))],
[cos(c + d*x)^(-5/2)*(a + b*cos(c + d*x))^3, x, 5, -((2*b*(3*a^2 - b^2)*EllipticE((1/2)*(c + d*x), 2))/d) + (2*a*(a^2 + 9*b^2)*EllipticF((1/2)*(c + d*x), 2))/(3*d) + (16*a^2*b*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))) + (2*a^2*(a + b*cos(c + d*x))*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2))],
[cos(c + d*x)^(-7/2)*(a + b*cos(c + d*x))^3, x, 6, -((6*a*(a^2 + 5*b^2)*EllipticE((1/2)*(c + d*x), 2))/(5*d)) + (2*b*(a^2 + b^2)*EllipticF((1/2)*(c + d*x), 2))/d + (8*a^2*b*sin(c + d*x))/(5*d*cos(c + d*x)^(3/2)) + (6*a*(a^2 + 5*b^2)*sin(c + d*x))/(5*d*sqrt(cos(c + d*x))) + (2*a^2*(a + b*cos(c + d*x))*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2))],
[cos(c + d*x)^(-9/2)*(a + b*cos(c + d*x))^3, x, 7, -((2*b*(9*a^2 + 5*b^2)*EllipticE((1/2)*(c + d*x), 2))/(5*d)) + (2*a*(5*a^2 + 21*b^2)*EllipticF((1/2)*(c + d*x), 2))/(21*d) + (32*a^2*b*sin(c + d*x))/(35*d*cos(c + d*x)^(5/2)) + (2*a*(5*a^2 + 21*b^2)*sin(c + d*x))/(21*d*cos(c + d*x)^(3/2)) + (2*b*(9*a^2 + 5*b^2)*sin(c + d*x))/(5*d*sqrt(cos(c + d*x))) + (2*a^2*(a + b*cos(c + d*x))*sin(c + d*x))/(7*d*cos(c + d*x)^(7/2))],


# ::Subsubsection::Closed:: 
#n<0


[cos(c + d*x)^(5/2)/(a + b*cos(c + d*x)), x, 6, -((2*a*EllipticE((1/2)*(c + d*x), 2))/(b^2*d)) + (2*(3*a^2 + b^2)*EllipticF((1/2)*(c + d*x), 2))/(3*b^3*d) - (2*a^3*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2))/(b^3*(a + b)*d) + (2*sqrt(cos(c + d*x))*sin(c + d*x))/(3*b*d)],
[cos(c + d*x)^(3/2)/(a + b*cos(c + d*x)), x, 5, (2*EllipticE((1/2)*(c + d*x), 2))/(b*d) - (2*a*EllipticF((1/2)*(c + d*x), 2))/(b^2*d) + (2*a^2*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2))/(b^2*(a + b)*d)],
[cos(c + d*x)^(1/2)/(a + b*cos(c + d*x)), x, 3, (2*EllipticF((1/2)*(c + d*x), 2))/(b*d) - (2*a*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2))/(b*(a + b)*d)],
[cos(c + d*x)^(-1/2)/(a + b*cos(c + d*x)), x, 1, (2*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2))/((a + b)*d)],
[cos(c + d*x)^(-3/2)/(a + b*cos(c + d*x)), x, 5, -((2*EllipticE((1/2)*(c + d*x), 2))/(a*d)) - (2*b*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2))/(a*(a + b)*d) + (2*sin(c + d*x))/(a*d*sqrt(cos(c + d*x)))],
[cos(c + d*x)^(-5/2)/(a + b*cos(c + d*x)), x, 7, (2*b*EllipticE((1/2)*(c + d*x), 2))/(a^2*d) + (2*EllipticF((1/2)*(c + d*x), 2))/(3*a*d) + (2*b^2*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2))/(a^2*(a + b)*d) + (2*sin(c + d*x))/(3*a*d*cos(c + d*x)^(3/2)) - (2*b*sin(c + d*x))/(a^2*d*sqrt(cos(c + d*x)))],


[cos(c + d*x)^(7/2)/(a + b*cos(c + d*x))^2, x, 7, -((a*(5*a^2 - 4*b^2)*EllipticE((1/2)*(c + d*x), 2))/(b^3*(a^2 - b^2)*d)) + ((15*a^4 - 16*a^2*b^2 - 2*b^4)*EllipticF((1/2)*(c + d*x), 2))/(3*b^4*(a^2 - b^2)*d) - (a^3*(5*a^2 - 7*b^2)*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2))/((a - b)*b^4*(a + b)^2*d) + ((5*a^2 - 2*b^2)*sqrt(cos(c + d*x))*sin(c + d*x))/(3*b^2*(a^2 - b^2)*d) - (a^2*cos(c + d*x)^(3/2)*sin(c + d*x))/(b*(a^2 - b^2)*d*(a + b*cos(c + d*x)))],
[cos(c + d*x)^(5/2)/(a + b*cos(c + d*x))^2, x, 6, ((3*a^2 - 2*b^2)*EllipticE((1/2)*(c + d*x), 2))/(b^2*(a^2 - b^2)*d) - (a*(3*a^2 - 4*b^2)*EllipticF((1/2)*(c + d*x), 2))/(b^3*(a^2 - b^2)*d) + (a^2*(3*a^2 - 5*b^2)*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2))/((a - b)*b^3*(a + b)^2*d) - (a^2*sqrt(cos(c + d*x))*sin(c + d*x))/(b*(a^2 - b^2)*d*(a + b*cos(c + d*x)))],
[cos(c + d*x)^(3/2)/(a + b*cos(c + d*x))^2, x, 6, -((a*EllipticE((1/2)*(c + d*x), 2))/(b*(a^2 - b^2)*d)) + ((a^2 - 2*b^2)*EllipticF((1/2)*(c + d*x), 2))/(b^2*(a^2 - b^2)*d) - (a*(a^2 - 3*b^2)*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2))/((a - b)*b^2*(a + b)^2*d) + (a*sqrt(cos(c + d*x))*sin(c + d*x))/((a^2 - b^2)*d*(a + b*cos(c + d*x)))],
[cos(c + d*x)^(1/2)/(a + b*cos(c + d*x))^2, x, 6, EllipticE((1/2)*(c + d*x), 2)/((a^2 - b^2)*d) + (a*EllipticF((1/2)*(c + d*x), 2))/(b*(a^2 - b^2)*d) - ((a^2 + b^2)*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2))/((a - b)*b*(a + b)^2*d) - (b*sqrt(cos(c + d*x))*sin(c + d*x))/((a^2 - b^2)*d*(a + b*cos(c + d*x)))],
[cos(c + d*x)^(-1/2)/(a + b*cos(c + d*x))^2, x, 6, -((b*EllipticE((1/2)*(c + d*x), 2))/(a*(a^2 - b^2)*d)) - EllipticF((1/2)*(c + d*x), 2)/((a^2 - b^2)*d) + ((3*a^2 - b^2)*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2))/(a*(a - b)*(a + b)^2*d) + (b^2*sqrt(cos(c + d*x))*sin(c + d*x))/(a*(a^2 - b^2)*d*(a + b*cos(c + d*x)))],
[cos(c + d*x)^(-3/2)/(a + b*cos(c + d*x))^2, x, 7, -(((2*a^2 - 3*b^2)*EllipticE((1/2)*(c + d*x), 2))/(a^2*(a^2 - b^2)*d)) + (b*EllipticF((1/2)*(c + d*x), 2))/(a*(a^2 - b^2)*d) - (b*(5*a^2 - 3*b^2)*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2))/(a^2*(a - b)*(a + b)^2*d) + ((2*a^2 - 3*b^2)*sin(c + d*x))/(a^2*(a^2 - b^2)*d*sqrt(cos(c + d*x))) + (b^2*sin(c + d*x))/(a*(a^2 - b^2)*d*sqrt(cos(c + d*x))*(a + b*cos(c + d*x)))],
[cos(c + d*x)^(-5/2)/(a + b*cos(c + d*x))^2, x, 8, (b*(4*a^2 - 5*b^2)*EllipticE((1/2)*(c + d*x), 2))/(a^3*(a^2 - b^2)*d) + ((2*a^2 - 5*b^2)*EllipticF((1/2)*(c + d*x), 2))/(3*a^2*(a^2 - b^2)*d) + (b^2*(7*a^2 - 5*b^2)*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2))/(a^3*(a - b)*(a + b)^2*d) + ((2*a^2 - 5*b^2)*sin(c + d*x))/(3*a^2*(a^2 - b^2)*d*cos(c + d*x)^(3/2)) - (b*(4*a^2 - 5*b^2)*sin(c + d*x))/(a^3*(a^2 - b^2)*d*sqrt(cos(c + d*x))) + (b^2*sin(c + d*x))/(a*(a^2 - b^2)*d*cos(c + d*x)^(3/2)*(a + b*cos(c + d*x)))],


[cos(c + d*x)^(9/2)/(a + b*cos(c + d*x))^3, x, 8, -((a*(35*a^4 - 65*a^2*b^2 + 24*b^4)*EllipticE((1/2)*(c + d*x), 2))/(4*b^4*(a^2 - b^2)^2*d)) + ((105*a^6 - 223*a^4*b^2 + 128*a^2*b^4 + 8*b^6)*EllipticF((1/2)*(c + d*x), 2))/(12*b^5*(a^2 - b^2)^2*d) - (a^3*(35*a^4 - 86*a^2*b^2 + 63*b^4)*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2))/(4*(a - b)^2*b^5*(a + b)^3*d) + ((35*a^4 - 61*a^2*b^2 + 8*b^4)*sqrt(cos(c + d*x))*sin(c + d*x))/(12*b^3*(a^2 - b^2)^2*d) - (a^2*cos(c + d*x)^(5/2)*sin(c + d*x))/(2*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) - (a^2*(7*a^2 - 13*b^2)*cos(c + d*x)^(3/2)*sin(c + d*x))/(4*b^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))],
[cos(c + d*x)^(7/2)/(a + b*cos(c + d*x))^3, x, 7, ((15*a^4 - 29*a^2*b^2 + 8*b^4)*EllipticE((1/2)*(c + d*x), 2))/(4*b^3*(a^2 - b^2)^2*d) - (3*a*(5*a^4 - 11*a^2*b^2 + 8*b^4)*EllipticF((1/2)*(c + d*x), 2))/(4*b^4*(a^2 - b^2)^2*d) + (a^2*(15*a^4 - 38*a^2*b^2 + 35*b^4)*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2))/(4*(a - b)^2*b^4*(a + b)^3*d) - (a^2*cos(c + d*x)^(3/2)*sin(c + d*x))/(2*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) - (a^2*(5*a^2 - 11*b^2)*sqrt(cos(c + d*x))*sin(c + d*x))/(4*b^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))],
[cos(c + d*x)^(5/2)/(a + b*cos(c + d*x))^3, x, 7, -((3*a*(a^2 - 3*b^2)*EllipticE((1/2)*(c + d*x), 2))/(4*b^2*(a^2 - b^2)^2*d)) + ((3*a^4 - 5*a^2*b^2 + 8*b^4)*EllipticF((1/2)*(c + d*x), 2))/(4*b^3*(a^2 - b^2)^2*d) - (3*a*(a^4 - 2*a^2*b^2 + 5*b^4)*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2))/(4*(a - b)^2*b^3*(a + b)^3*d) - (a^2*sqrt(cos(c + d*x))*sin(c + d*x))/(2*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) + (3*a*(a^2 - 3*b^2)*sqrt(cos(c + d*x))*sin(c + d*x))/(4*b*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))],
[cos(c + d*x)^(3/2)/(a + b*cos(c + d*x))^3, x, 7, -(((a^2 + 5*b^2)*EllipticE((1/2)*(c + d*x), 2))/(4*b*(a^2 - b^2)^2*d)) + (a*(a^2 - 7*b^2)*EllipticF((1/2)*(c + d*x), 2))/(4*b^2*(a^2 - b^2)^2*d) - ((a^4 - 10*a^2*b^2 - 3*b^4)*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2))/(4*(a - b)^2*b^2*(a + b)^3*d) + (a*sqrt(cos(c + d*x))*sin(c + d*x))/(2*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) + ((a^2 + 5*b^2)*sqrt(cos(c + d*x))*sin(c + d*x))/(4*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))],
[cos(c + d*x)^(1/2)/(a + b*cos(c + d*x))^3, x, 7, ((5*a^2 + b^2)*EllipticE((1/2)*(c + d*x), 2))/(4*a*(a^2 - b^2)^2*d) + (3*(a^2 + b^2)*EllipticF((1/2)*(c + d*x), 2))/(4*b*(a^2 - b^2)^2*d) - ((3*a^4 + 10*a^2*b^2 - b^4)*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2))/(4*a*(a - b)^2*b*(a + b)^3*d) - (b*sqrt(cos(c + d*x))*sin(c + d*x))/(2*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) - (b*(5*a^2 + b^2)*sqrt(cos(c + d*x))*sin(c + d*x))/(4*a*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))],
[cos(c + d*x)^(-1/2)/(a + b*cos(c + d*x))^3, x, 7, -((3*b*(3*a^2 - b^2)*EllipticE((1/2)*(c + d*x), 2))/(4*a^2*(a^2 - b^2)^2*d)) - ((7*a^2 - b^2)*EllipticF((1/2)*(c + d*x), 2))/(4*a*(a^2 - b^2)^2*d) + (3*(5*a^4 - 2*a^2*b^2 + b^4)*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2))/(4*a^2*(a - b)^2*(a + b)^3*d) + (b^2*sqrt(cos(c + d*x))*sin(c + d*x))/(2*a*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) + (3*b^2*(3*a^2 - b^2)*sqrt(cos(c + d*x))*sin(c + d*x))/(4*a^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))],
[cos(c + d*x)^(-3/2)/(a + b*cos(c + d*x))^3, x, 8, -(((8*a^4 - 29*a^2*b^2 + 15*b^4)*EllipticE((1/2)*(c + d*x), 2))/(4*a^3*(a^2 - b^2)^2*d)) + (b*(11*a^2 - 5*b^2)*EllipticF((1/2)*(c + d*x), 2))/(4*a^2*(a^2 - b^2)^2*d) - (b*(35*a^4 - 38*a^2*b^2 + 15*b^4)*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2))/(4*a^3*(a - b)^2*(a + b)^3*d) + ((8*a^4 - 29*a^2*b^2 + 15*b^4)*sin(c + d*x))/(4*a^3*(a^2 - b^2)^2*d*sqrt(cos(c + d*x))) + (b^2*sin(c + d*x))/(2*a*(a^2 - b^2)*d*sqrt(cos(c + d*x))*(a + b*cos(c + d*x))^2) + (b^2*(11*a^2 - 5*b^2)*sin(c + d*x))/(4*a^2*(a^2 - b^2)^2*d*sqrt(cos(c + d*x))*(a + b*cos(c + d*x)))],
[cos(c + d*x)^(-5/2)/(a + b*cos(c + d*x))^3, x, 9, (b*(24*a^4 - 65*a^2*b^2 + 35*b^4)*EllipticE((1/2)*(c + d*x), 2))/(4*a^4*(a^2 - b^2)^2*d) + ((8*a^4 - 61*a^2*b^2 + 35*b^4)*EllipticF((1/2)*(c + d*x), 2))/(12*a^3*(a^2 - b^2)^2*d) + (b^2*(63*a^4 - 86*a^2*b^2 + 35*b^4)*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2))/(4*a^4*(a - b)^2*(a + b)^3*d) + ((8*a^4 - 61*a^2*b^2 + 35*b^4)*sin(c + d*x))/(12*a^3*(a^2 - b^2)^2*d*cos(c + d*x)^(3/2)) - (b*(24*a^4 - 65*a^2*b^2 + 35*b^4)*sin(c + d*x))/(4*a^4*(a^2 - b^2)^2*d*sqrt(cos(c + d*x))) + (b^2*sin(c + d*x))/(2*a*(a^2 - b^2)*d*cos(c + d*x)^(3/2)*(a + b*cos(c + d*x))^2) + (b^2*(13*a^2 - 7*b^2)*sin(c + d*x))/(4*a^2*(a^2 - b^2)^2*d*cos(c + d*x)^(3/2)*(a + b*cos(c + d*x)))],


# ::Subsection::Closed:: 
#Integrands of the form Cos[c+d x]^(m/2) (a+b Cos[c+d x])^(n/2)
#


# ::Subsubsection::Closed:: 
#n>0


[cos(c + d*x)^(3/2)*(a + b*cos(c + d*x))^(1/2), x, 7, (a*(a + b)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(4*b*d*sqrt(a + b*cos(c + d*x))) + ((a - 2*b)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(2*d*sqrt(a + b*cos(c + d*x))) - ((a^2 - 4*b^2)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(2*b*d*sqrt(a + b*cos(c + d*x))) + (sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(2*d) + (a*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(4*b*d)],
[cos(c + d*x)^(1/2)*(a + b*cos(c + d*x))^(1/2), x, 6, ((a + b)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(d*sqrt(a + b*cos(c + d*x))) - (2*a*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(d*sqrt(a + b*cos(c + d*x))) + (2*a*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(d*sqrt(a + b*cos(c + d*x))) + (sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/d],
[cos(c + d*x)^(-1/2)*(a + b*cos(c + d*x))^(1/2), x, 3, (2*(a - b)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(d*sqrt(a + b*cos(c + d*x))) + (4*b*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(d*sqrt(a + b*cos(c + d*x)))],
[cos(c + d*x)^(-3/2)*(a + b*cos(c + d*x))^(1/2), x, 4, -((2*(a + b)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(d*sqrt(a + b*cos(c + d*x)))) + (2*(a + b)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(d*sqrt(a + b*cos(c + d*x))) + (2*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(d*sqrt(cos(c + d*x)))],
[cos(c + d*x)^(-5/2)*(a + b*cos(c + d*x))^(1/2), x, 5, -((2*b*(a + b)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(3*a*d*sqrt(a + b*cos(c + d*x)))) + (2*(a + b)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(3*d*sqrt(a + b*cos(c + d*x))) + (2*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2)) + (2*b*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(3*a*d*sqrt(cos(c + d*x)))],
[cos(c + d*x)^(-7/2)*(a + b*cos(c + d*x))^(1/2), x, 6, -((2*(a + b)*(9*a^2 - 2*b^2)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(15*a^2*d*sqrt(a + b*cos(c + d*x)))) + (2*(9*a^2 + 7*a*b - 2*b^2)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(15*a*d*sqrt(a + b*cos(c + d*x))) + (2*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2)) + (2*b*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(15*a*d*cos(c + d*x)^(3/2)) + (2*(9*a^2 - 2*b^2)*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(15*a^2*d*sqrt(cos(c + d*x)))],


[cos(c + d*x)^(3/2)*(a + b*cos(c + d*x))^(3/2), x, 8, ((a + b)*(3*a^2 + 16*b^2)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(24*b*d*sqrt(a + b*cos(c + d*x))) + (a*(7*a - 26*b)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(12*d*sqrt(a + b*cos(c + d*x))) - (a*(a^2 - 12*b^2)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(4*b*d*sqrt(a + b*cos(c + d*x))) + (7*a*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(12*d) + (b*cos(c + d*x)^(3/2)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*d) + ((3*a^2 + 16*b^2)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(24*b*d)],
[cos(c + d*x)^(1/2)*(a + b*cos(c + d*x))^(3/2), x, 7, (5*a*(a + b)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(4*d*sqrt(a + b*cos(c + d*x))) - ((4*a^2 - a*b + 2*b^2)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(2*d*sqrt(a + b*cos(c + d*x))) + ((3*a^2 + 4*b^2)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(2*d*sqrt(a + b*cos(c + d*x))) + (b*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(2*d) + (5*a*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(4*d)],
[cos(c + d*x)^(-1/2)*(a + b*cos(c + d*x))^(3/2), x, 7, (b*(a + b)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(d*sqrt(a + b*cos(c + d*x))) + (2*a*(a - 2*b)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(d*sqrt(a + b*cos(c + d*x))) + (6*a*b*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(d*sqrt(a + b*cos(c + d*x))) + (b*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/d],
[cos(c + d*x)^(-3/2)*(a + b*cos(c + d*x))^(3/2), x, 6, -((2*a*(a + b)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(d*sqrt(a + b*cos(c + d*x)))) + (2*(a^2 + 2*a*b - b^2)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(d*sqrt(a + b*cos(c + d*x))) + (4*b^2*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(d*sqrt(a + b*cos(c + d*x))) + (2*a*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(d*sqrt(cos(c + d*x)))],
[cos(c + d*x)^(-5/2)*(a + b*cos(c + d*x))^(3/2), x, 5, -((8*b*(a + b)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(3*d*sqrt(a + b*cos(c + d*x)))) + (2*(a^2 + 4*a*b + 3*b^2)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(3*d*sqrt(a + b*cos(c + d*x))) + (2*a*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2)) + (8*b*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(3*d*sqrt(cos(c + d*x)))],
[cos(c + d*x)^(-7/2)*(a + b*cos(c + d*x))^(3/2), x, 6, -((2*(a + b)*(3*a^2 + b^2)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(5*a*d*sqrt(a + b*cos(c + d*x)))) + (2*(3*a^2 + 4*a*b + b^2)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(5*d*sqrt(a + b*cos(c + d*x))) + (2*a*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2)) + (4*b*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(5*d*cos(c + d*x)^(3/2)) + (2*(3*a^2 + b^2)*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(5*a*d*sqrt(cos(c + d*x)))],
[cos(c + d*x)^(-9/2)*(a + b*cos(c + d*x))^(3/2), x, 7, -((4*b*(a + b)*(41*a^2 - 3*b^2)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(105*a^2*d*sqrt(a + b*cos(c + d*x)))) + (2*(25*a^3 + 82*a^2*b + 51*a*b^2 - 6*b^3)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(105*a*d*sqrt(a + b*cos(c + d*x))) + (2*a*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(7*d*cos(c + d*x)^(7/2)) + (16*b*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(35*d*cos(c + d*x)^(5/2)) + (2*(25*a^2 + 3*b^2)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(105*a*d*cos(c + d*x)^(3/2)) + (4*b*(41*a^2 - 3*b^2)*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(105*a^2*d*sqrt(cos(c + d*x)))],


[cos(c + d*x)^(3/2)*(a + b*cos(c + d*x))^(5/2), x, 9, (a*(a + b)*(15*a^2 + 284*b^2)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(192*b*d*sqrt(a + b*cos(c + d*x))) + ((59*a^3 - 322*a^2*b + 36*a*b^2 - 72*b^3)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(96*d*sqrt(a + b*cos(c + d*x))) - ((5*a^4 - 120*a^2*b^2 - 48*b^4)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(32*b*d*sqrt(a + b*cos(c + d*x))) + ((59*a^2 + 36*b^2)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(96*d) + (17*a*b*cos(c + d*x)^(3/2)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(24*d) + (b^2*cos(c + d*x)^(5/2)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(4*d) + (a*(15*a^2 + 284*b^2)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(192*b*d)],
[cos(c + d*x)^(1/2)*(a + b*cos(c + d*x))^(5/2), x, 8, ((a + b)*(33*a^2 + 16*b^2)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(24*d*sqrt(a + b*cos(c + d*x))) - (a*(24*a^2 - 13*a*b + 38*b^2)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(12*d*sqrt(a + b*cos(c + d*x))) + (5*a*(a^2 + 4*b^2)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(4*d*sqrt(a + b*cos(c + d*x))) + (13*a*b*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(12*d) + (b^2*cos(c + d*x)^(3/2)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*d) + ((33*a^2 + 16*b^2)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(24*d)],
[cos(c + d*x)^(-1/2)*(a + b*cos(c + d*x))^(5/2), x, 7, (9*a*b*(a + b)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(4*d*sqrt(a + b*cos(c + d*x))) + ((4*a^3 - 12*a^2*b + a*b^2 - 2*b^3)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(2*d*sqrt(a + b*cos(c + d*x))) + (b*(15*a^2 + 4*b^2)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(2*d*sqrt(a + b*cos(c + d*x))) + (b^2*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(2*d) + (9*a*b*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(4*d)],
[cos(c + d*x)^(-3/2)*(a + b*cos(c + d*x))^(5/2), x, 7, -(((a + b)*(2*a^2 - b^2)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(d*sqrt(a + b*cos(c + d*x)))) + (2*a*(a^2 + 3*a*b - 3*b^2)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(d*sqrt(a + b*cos(c + d*x))) + (10*a*b^2*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(d*sqrt(a + b*cos(c + d*x))) + (2*a^2*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(d*sqrt(cos(c + d*x))) - ((2*a^2 - b^2)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/d],
[cos(c + d*x)^(-5/2)*(a + b*cos(c + d*x))^(5/2), x, 7, -((14*a*b*(a + b)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(3*d*sqrt(a + b*cos(c + d*x)))) + (2*(a^3 + 7*a^2*b + 9*a*b^2 - 3*b^3)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(3*d*sqrt(a + b*cos(c + d*x))) + (4*b^3*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(d*sqrt(a + b*cos(c + d*x))) + (2*a^2*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2)) + (14*a*b*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(3*d*sqrt(cos(c + d*x)))],
[cos(c + d*x)^(-7/2)*(a + b*cos(c + d*x))^(5/2), x, 6, -((2*(a + b)*(9*a^2 + 23*b^2)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(15*d*sqrt(a + b*cos(c + d*x)))) + (2*(9*a^3 + 17*a^2*b + 23*a*b^2 + 15*b^3)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(15*d*sqrt(a + b*cos(c + d*x))) + (2*a^2*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2)) + (22*a*b*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(15*d*cos(c + d*x)^(3/2)) + (2*(9*a^2 + 23*b^2)*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(15*d*sqrt(cos(c + d*x)))],
[cos(c + d*x)^(-9/2)*(a + b*cos(c + d*x))^(5/2), x, 7, -((2*b*(a + b)*(29*a^2 + 3*b^2)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(21*a*d*sqrt(a + b*cos(c + d*x)))) + (2*(5*a^3 + 29*a^2*b + 27*a*b^2 + 3*b^3)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(21*d*sqrt(a + b*cos(c + d*x))) + (2*a^2*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(7*d*cos(c + d*x)^(7/2)) + (6*a*b*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(7*d*cos(c + d*x)^(5/2)) + (2*(5*a^2 + 9*b^2)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(21*d*cos(c + d*x)^(3/2)) + (2*b*(29*a^2 + 3*b^2)*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(21*a*d*sqrt(cos(c + d*x)))],
[cos(c + d*x)^(-11/2)*(a + b*cos(c + d*x))^(5/2), x, 8, -((2*(a + b)*(147*a^4 + 279*a^2*b^2 - 10*b^4)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(315*a^2*d*sqrt(a + b*cos(c + d*x)))) + (2*(147*a^4 + 261*a^3*b + 279*a^2*b^2 + 155*a*b^3 - 10*b^4)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(315*a*d*sqrt(a + b*cos(c + d*x))) + (2*a^2*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(9*d*cos(c + d*x)^(9/2)) + (38*a*b*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(63*d*cos(c + d*x)^(7/2)) + (2*(49*a^2 + 75*b^2)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(315*d*cos(c + d*x)^(5/2)) + (2*b*(163*a^2 + 5*b^2)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(315*a*d*cos(c + d*x)^(3/2)) + (2*(147*a^4 + 279*a^2*b^2 - 10*b^4)*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(315*a^2*d*sqrt(cos(c + d*x)))],


# ::Subsubsection::Closed:: 
#n<0


[1/(sqrt(cos(c + d*x))*sqrt(2 + 3*cos(c + d*x))), x, 1, (2*EllipticF(arcsin(tan((c+d*x)/2)), 1/5))/(d*sqrt(5))],
[1/(sqrt(cos(c + d*x))*sqrt(-2 + 3*cos(c + d*x))), x, 1, (2*EllipticF(arcsin(tan((c+d*x)/2)), 5))/d],

[1/(sqrt(cos(c + d*x))*sqrt(2 - 3*cos(c + d*x))), x, 1, (2*sqrt((2 - 3*cos(c + d*x))/(-1 - cos(c + d*x)))*sqrt(1 + cos(c + d*x))*EllipticF(arcsin(tan((1/2)*(c + d*x))), 5))/(d*sqrt(2 - 3*cos(c + d*x)))],
[1/(sqrt(cos(c + d*x))*sqrt(-2 - 3*cos(c + d*x))), x, 1, (2*sqrt(-((-2 - 3*cos(c + d*x))/(1 + cos(c + d*x))))*sqrt(1 + cos(c + d*x))*EllipticF(arcsin(tan((1/2)*(c + d*x))), 1/5))/(sqrt(5)*d*sqrt(-2 - 3*cos(c + d*x)))],

[1/(sqrt(cos(c + d*x))*sqrt(3 + 2*cos(c + d*x))), x, 1, (2*sqrt(1 + cos(c + d*x))*sqrt((3 + 2*cos(c + d*x))/(1 + cos(c + d*x)))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -(1/5)))/(sqrt(5)*d*sqrt(3 + 2*cos(c + d*x)))],
[1/(sqrt(cos(c + d*x))*sqrt(3 - 2*cos(c + d*x))), x, 1, (2*sqrt(1 + cos(c + d*x))*sqrt((3 - 2*cos(c + d*x))/(1 + cos(c + d*x)))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -5))/(d*sqrt(3 - 2*cos(c + d*x)))],

[1/(sqrt(cos(c + d*x))*sqrt(-3 + 2*cos(c + d*x))), x, 1, (2*sqrt(1 + cos(c + d*x))*sqrt((-3 + 2*cos(c + d*x))/(-1 - cos(c + d*x)))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -5))/(d*sqrt(-3 + 2*cos(c + d*x)))],
[1/(sqrt(cos(c + d*x))*sqrt(-3 - 2*cos(c + d*x))), x, 1, (2*sqrt(-((-3 - 2*cos(c + d*x))/(1 + cos(c + d*x))))*sqrt(1 + cos(c + d*x))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -(1/5)))/(sqrt(5)*d*sqrt(-3 - 2*cos(c + d*x)))],


[sqrt(cos(c + d*x))/sqrt(2 + 3*cos(c + d*x)), x, 3, -((2*EllipticF(arcsin(tan((1/2)*(c + d*x))), 1/5))/(sqrt(5)*d)) + (4*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), 1/5))/(sqrt(5)*d)],
[sqrt(cos(c + d*x))/sqrt(-2 + 3*cos(c + d*x)), x, 3, -((2*EllipticF(arcsin(tan((1/2)*(c + d*x))), 5))/d) + (4*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), 5))/d],

[sqrt(cos(c + d*x))/sqrt(2 - 3*cos(c + d*x)), x, 3, -((2*sqrt((2 - 3*cos(c + d*x))/(-1 - cos(c + d*x)))*sqrt(1 + cos(c + d*x))*EllipticF(arcsin(tan((1/2)*(c + d*x))), 5))/(d*sqrt(2 - 3*cos(c + d*x)))) + (4*sqrt((2 - 3*cos(c + d*x))/(-1 - cos(c + d*x)))*sqrt(1 + cos(c + d*x))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), 5))/(d*sqrt(2 - 3*cos(c + d*x)))],
[sqrt(cos(c + d*x))/sqrt(-2 - 3*cos(c + d*x)), x, 3, -((2*sqrt(-((-2 - 3*cos(c + d*x))/(1 + cos(c + d*x))))*sqrt(1 + cos(c + d*x))*EllipticF(arcsin(tan((1/2)*(c + d*x))), 1/5))/(sqrt(5)*d*sqrt(-2 - 3*cos(c + d*x)))) + (4*sqrt(-((-2 - 3*cos(c + d*x))/(1 + cos(c + d*x))))*sqrt(1 + cos(c + d*x))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), 1/5))/(sqrt(5)*d*sqrt(-2 - 3*cos(c + d*x)))],

[sqrt(cos(c + d*x))/sqrt(3 + 2*cos(c + d*x)), x, 3, -((2*sqrt(1 + cos(c + d*x))*sqrt((3 + 2*cos(c + d*x))/(1 + cos(c + d*x)))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -(1/5)))/(sqrt(5)*d*sqrt(3 + 2*cos(c + d*x)))) + (4*sqrt(1 + cos(c + d*x))*sqrt((3 + 2*cos(c + d*x))/(1 + cos(c + d*x)))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -(1/5)))/(sqrt(5)*d*sqrt(3 + 2*cos(c + d*x)))],
[sqrt(cos(c + d*x))/sqrt(3 - 2*cos(c + d*x)), x, 3, -((2*sqrt((3 - 2*cos(c + d*x))/(1 + cos(c + d*x)))*sqrt(1 + cos(c + d*x))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -5))/(d*sqrt(3 - 2*cos(c + d*x)))) + (4*sqrt((3 - 2*cos(c + d*x))/(1 + cos(c + d*x)))*sqrt(1 + cos(c + d*x))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -5))/(d*sqrt(3 - 2*cos(c + d*x)))],

[sqrt(cos(c + d*x))/sqrt(-3 + 2*cos(c + d*x)), x, 3, -((2*sqrt(1 + cos(c + d*x))*sqrt((-3 + 2*cos(c + d*x))/(-1 - cos(c + d*x)))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -5))/(d*sqrt(-3 + 2*cos(c + d*x)))) + (4*sqrt(1 + cos(c + d*x))*sqrt((-3 + 2*cos(c + d*x))/(-1 - cos(c + d*x)))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -5))/(d*sqrt(-3 + 2*cos(c + d*x)))],
[sqrt(cos(c + d*x))/sqrt(-3 - 2*cos(c + d*x)), x, 3, -((2*sqrt(-((-3 - 2*cos(c + d*x))/(1 + cos(c + d*x))))*sqrt(1 + cos(c + d*x))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -(1/5)))/(sqrt(5)*d*sqrt(-3 - 2*cos(c + d*x)))) + (4*sqrt(-((-3 - 2*cos(c + d*x))/(1 + cos(c + d*x))))*sqrt(1 + cos(c + d*x))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -(1/5)))/(sqrt(5)*d*sqrt(-3 - 2*cos(c + d*x)))],


[cos(c + d*x)^(3/2)/(a + b*cos(c + d*x))^(1/2), x, 4, ((a + b)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(b*d*sqrt(a + b*cos(c + d*x))) - (2*a*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(b*d*sqrt(a + b*cos(c + d*x))) + (sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(b*d)],
[cos(c + d*x)^(1/2)/(a + b*cos(c + d*x))^(1/2), x, 3, -((2*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(d*sqrt(a + b*cos(c + d*x)))) + (4*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(d*sqrt(a + b*cos(c + d*x)))],
[cos(c + d*x)^(-1/2)/(a + b*cos(c + d*x))^(1/2), x, 1, (2*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(d*sqrt(a + b*cos(c + d*x)))],
[cos(c + d*x)^(-3/2)/(a + b*cos(c + d*x))^(1/2), x, 4, -((2*(a + b)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(a*d*sqrt(a + b*cos(c + d*x)))) + (2*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(d*sqrt(a + b*cos(c + d*x))) + (2*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(a*d*sqrt(cos(c + d*x)))],
[cos(c + d*x)^(-5/2)/(a + b*cos(c + d*x))^(1/2), x, 5, (4*b*(a + b)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(3*a^2*d*sqrt(a + b*cos(c + d*x))) + (2*(a - 2*b)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(3*a*d*sqrt(a + b*cos(c + d*x))) + (2*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*a*d*cos(c + d*x)^(3/2)) - (4*b*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(3*a^2*d*sqrt(cos(c + d*x)))],


[cos(c + d*x)^(5/2)/(a + b*cos(c + d*x))^(3/2), x, 7, ((3*a^2 - b^2)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/((a - b)*b^2*d*sqrt(a + b*cos(c + d*x))) - (2*a*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/((a - b)*b*d*sqrt(a + b*cos(c + d*x))) - (6*a*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(b^2*d*sqrt(a + b*cos(c + d*x))) - (2*a^2*sqrt(cos(c + d*x))*sin(c + d*x))/(b*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) + ((3*a^2 - b^2)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(b^2*(a^2 - b^2)*d)],
[cos(c + d*x)^(3/2)/(a + b*cos(c + d*x))^(3/2), x, 6, -((2*a*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/((a - b)*b*d*sqrt(a + b*cos(c + d*x)))) + (2*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/((a - b)*d*sqrt(a + b*cos(c + d*x))) + (4*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(b*d*sqrt(a + b*cos(c + d*x))) - (2*a*sqrt(cos(c + d*x))*tan((1/2)*(c + d*x)))/(b*(a + b)*d*sqrt(a + b*cos(c + d*x)))],
[cos(c + d*x)^(1/2)/(a + b*cos(c + d*x))^(3/2), x, 4, (2*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/((a - b)*d*sqrt(a + b*cos(c + d*x))) - (2*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/((a - b)*d*sqrt(a + b*cos(c + d*x))) + (2*sqrt(cos(c + d*x))*tan((1/2)*(c + d*x)))/((a + b)*d*sqrt(a + b*cos(c + d*x)))],
[cos(c + d*x)^(-1/2)/(a + b*cos(c + d*x))^(3/2), x, 4, -((2*b*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(a*(a - b)*d*sqrt(a + b*cos(c + d*x)))) + (2*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/((a - b)*d*sqrt(a + b*cos(c + d*x))) - (2*b*sqrt(cos(c + d*x))*tan((1/2)*(c + d*x)))/(a*(a + b)*d*sqrt(a + b*cos(c + d*x)))],
[cos(c + d*x)^(-3/2)/(a + b*cos(c + d*x))^(3/2), x, 5, -((2*(a^2 - 2*b^2)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(a^2*(a - b)*d*sqrt(a + b*cos(c + d*x)))) + (2*(a - 2*b)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(a*(a - b)*d*sqrt(a + b*cos(c + d*x))) + (2*b^2*sin(c + d*x))/(a*(a^2 - b^2)*d*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))) + (2*(a^2 - 2*b^2)*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(a^2*(a^2 - b^2)*d*sqrt(cos(c + d*x)))],
[cos(c + d*x)^(-5/2)/(a + b*cos(c + d*x))^(3/2), x, 6, (2*b*(5*a^2 - 8*b^2)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(3*a^3*(a - b)*d*sqrt(a + b*cos(c + d*x))) + (2*(a^2 - 6*a*b + 8*b^2)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(3*a^2*(a - b)*d*sqrt(a + b*cos(c + d*x))) + (2*b^2*sin(c + d*x))/(a*(a^2 - b^2)*d*cos(c + d*x)^(3/2)*sqrt(a + b*cos(c + d*x))) + (2*(a^2 - 4*b^2)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*a^2*(a^2 - b^2)*d*cos(c + d*x)^(3/2)) - (2*b*(5*a^2 - 8*b^2)*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(3*a^3*(a^2 - b^2)*d*sqrt(cos(c + d*x)))],


[cos(c + d*x)^(5/2)/(a + b*cos(c + d*x))^(5/2), x, 7, -((2*a*(3*a^2 - 7*b^2)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(3*(a - b)^2*b^2*(a + b)*d*sqrt(a + b*cos(c + d*x)))) + (2*(2*a^2 - 3*a*b - 3*b^2)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(3*(a - b)^2*b*(a + b)*d*sqrt(a + b*cos(c + d*x))) + (4*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(b^2*d*sqrt(a + b*cos(c + d*x))) - (2*a^2*sqrt(cos(c + d*x))*sin(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)) - (2*a*(3*a^2 - 7*b^2)*sqrt(cos(c + d*x))*tan((1/2)*(c + d*x)))/(3*(a - b)*b^2*(a + b)^2*d*sqrt(a + b*cos(c + d*x)))],
[cos(c + d*x)^(3/2)/(a + b*cos(c + d*x))^(5/2), x, 5, -((8*b*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(3*(a - b)^2*(a + b)*d*sqrt(a + b*cos(c + d*x)))) + (2*(a + 3*b)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(3*(a - b)^2*(a + b)*d*sqrt(a + b*cos(c + d*x))) + (2*a*sqrt(cos(c + d*x))*sin(c + d*x))/(3*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)) - (8*b*sqrt(cos(c + d*x))*tan((1/2)*(c + d*x)))/(3*(a - b)*(a + b)^2*d*sqrt(a + b*cos(c + d*x)))],
[cos(c + d*x)^(1/2)/(a + b*cos(c + d*x))^(5/2), x, 5, (2*(3*a^2 + b^2)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(3*a*(a - b)^2*(a + b)*d*sqrt(a + b*cos(c + d*x))) - (2*(3*a + b)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(3*(a - b)^2*(a + b)*d*sqrt(a + b*cos(c + d*x))) - (2*b*sqrt(cos(c + d*x))*sin(c + d*x))/(3*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)) + (2*(3*a^2 + b^2)*sqrt(cos(c + d*x))*tan((1/2)*(c + d*x)))/(3*a*(a - b)*(a + b)^2*d*sqrt(a + b*cos(c + d*x)))],
[cos(c + d*x)^(-1/2)/(a + b*cos(c + d*x))^(5/2), x, 5, -((4*b*(3*a^2 - b^2)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(3*a^2*(a - b)^2*(a + b)*d*sqrt(a + b*cos(c + d*x)))) + (2*(3*a^2 + 3*a*b - 2*b^2)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(3*a*(a - b)^2*(a + b)*d*sqrt(a + b*cos(c + d*x))) + (2*b^2*sqrt(cos(c + d*x))*sin(c + d*x))/(3*a*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)) - (4*b*(3*a^2 - b^2)*sqrt(cos(c + d*x))*tan((1/2)*(c + d*x)))/(3*a^2*(a - b)*(a + b)^2*d*sqrt(a + b*cos(c + d*x)))],
[cos(c + d*x)^(-3/2)/(a + b*cos(c + d*x))^(5/2), x, 6, -((2*(3*a^4 - 15*a^2*b^2 + 8*b^4)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(3*a^3*(a - b)^2*(a + b)*d*sqrt(a + b*cos(c + d*x)))) + (2*(3*a^3 - 9*a^2*b - 6*a*b^2 + 8*b^3)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(3*a^2*(a - b)^2*(a + b)*d*sqrt(a + b*cos(c + d*x))) + (2*b^2*sin(c + d*x))/(3*a*(a^2 - b^2)*d*sqrt(cos(c + d*x))*(a + b*cos(c + d*x))^(3/2)) + (8*b^2*(2*a^2 - b^2)*sin(c + d*x))/(3*a^2*(a^2 - b^2)^2*d*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))) + (2*(3*a^4 - 15*a^2*b^2 + 8*b^4)*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(3*a^3*(a^2 - b^2)^2*d*sqrt(cos(c + d*x)))],
[cos(c + d*x)^(-5/2)/(a + b*cos(c + d*x))^(5/2), x, 7, (8*b*(2*a^4 - 7*a^2*b^2 + 4*b^4)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(3*a^4*(a - b)^2*(a + b)*d*sqrt(a + b*cos(c + d*x))) + (2*(a^4 - 9*a^3*b + 16*a^2*b^2 + 12*a*b^3 - 16*b^4)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(3*a^3*(a - b)^2*(a + b)*d*sqrt(a + b*cos(c + d*x))) + (2*b^2*sin(c + d*x))/(3*a*(a^2 - b^2)*d*cos(c + d*x)^(3/2)*(a + b*cos(c + d*x))^(3/2)) + (4*b^2*(5*a^2 - 3*b^2)*sin(c + d*x))/(3*a^2*(a^2 - b^2)^2*d*cos(c + d*x)^(3/2)*sqrt(a + b*cos(c + d*x))) + (2*(a^4 - 13*a^2*b^2 + 8*b^4)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*a^3*(a^2 - b^2)^2*d*cos(c + d*x)^(3/2)) - (8*b*(2*a^4 - 7*a^2*b^2 + 4*b^4)*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(3*a^4*(a^2 - b^2)^2*d*sqrt(cos(c + d*x)))],


# ::Subsection::Closed:: 
#Integrands of the form Cos[c+d x]^(m/3) (a+b Cos[c+d x])^n


# ::Subsubsection:: 
#n>0


# ::Subsubsection::Closed:: 
#n<0


[cos(c + d*x)^(7/3)/(a + b*cos(c + d*x)), x, 1, (3*Int((a/3 + (1/3)*b*cos(c + d*x) - (1/3)*(4*a)*cos(c + d*x)^2)/(cos(c + d*x)^(2/3)*(a + b*cos(c + d*x))), x))/(4*b) + (3*cos(c + d*x)^(1/3)*sin(c + d*x))/(4*b*d)],
[cos(c + d*x)^(5/3)/(a + b*cos(c + d*x)), x, 0, Int(cos(c + d*x)^(5/3)/(a + b*cos(c + d*x)), x)],
[cos(c + d*x)^(4/3)/(a + b*cos(c + d*x)), x, 0, Int(cos(c + d*x)^(4/3)/(a + b*cos(c + d*x)), x)],
[cos(c + d*x)^(2/3)/(a + b*cos(c + d*x)), x, 0, Int(cos(c + d*x)^(2/3)/(a + b*cos(c + d*x)), x)],
[cos(c + d*x)^(1/3)/(a + b*cos(c + d*x)), x, 0, Int(cos(c + d*x)^(1/3)/(a + b*cos(c + d*x)), x)],
[cos(c + d*x)^(-1/3)/(a + b*cos(c + d*x)), x, 0, Int(1/(cos(c + d*x)^(1/3)*(a + b*cos(c + d*x))), x)],
[cos(c + d*x)^(-2/3)/(a + b*cos(c + d*x)), x, 0, Int(1/(cos(c + d*x)^(2/3)*(a + b*cos(c + d*x))), x)],
[cos(c + d*x)^(-4/3)/(a + b*cos(c + d*x)), x, 1, -((3*Int((b/3 + (1/3)*(2*a)*cos(c + d*x) + (1/3)*(2*b)*cos(c + d*x)^2)/(cos(c + d*x)^(1/3)*(a + b*cos(c + d*x))), x))/a) + (3*sin(c + d*x))/(a*d*cos(c + d*x)^(1/3))],
[cos(c + d*x)^(-5/3)/(a + b*cos(c + d*x)), x, 1, -((3*Int(((2*b)/3 + (1/3)*a*cos(c + d*x) + (1/3)*b*cos(c + d*x)^2)/(cos(c + d*x)^(2/3)*(a + b*cos(c + d*x))), x))/(2*a)) + (3*sin(c + d*x))/(2*a*d*cos(c + d*x)^(2/3))],
[cos(c + d*x)^(-7/3)/(a + b*cos(c + d*x)), x, 2, (9*Int(((1/9)*(a^2 + 4*b^2) + (a*b)*cos(c + d*x) + (1/9)*(8*b^2)*cos(c + d*x)^2)/(cos(c + d*x)^(1/3)*(a + b*cos(c + d*x))), x))/(4*a^2) + (3*sin(c + d*x))/(4*a*d*cos(c + d*x)^(4/3)) - (3*b*sin(c + d*x))/(a^2*d*cos(c + d*x)^(1/3))],


# ::Subsection::Closed:: 
#Integrands of the form Cos[c+d x]^(m/3) (a+b Cos[c+d x])^(n/2)


# ::Subsubsection:: 
#n>0


# ::Subsubsection::Closed:: 
#n<0


[cos(c + d*x)^(7/3)/sqrt(a + b*cos(c + d*x)), x, 1, (6*Int((a/3 + (1/6)*(5*b)*cos(c + d*x) - (1/3)*(4*a)*cos(c + d*x)^2)/(cos(c + d*x)^(2/3)*sqrt(a + b*cos(c + d*x))), x))/(11*b) + (6*cos(c + d*x)^(1/3)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(11*b*d)],
[cos(c + d*x)^(5/3)/sqrt(a + b*cos(c + d*x)), x, 0, Int(cos(c + d*x)^(5/3)/sqrt(a + b*cos(c + d*x)), x)],
[cos(c + d*x)^(4/3)/sqrt(a + b*cos(c + d*x)), x, 0, Int(cos(c + d*x)^(4/3)/sqrt(a + b*cos(c + d*x)), x)],
[cos(c + d*x)^(2/3)/sqrt(a + b*cos(c + d*x)), x, 0, Int(cos(c + d*x)^(2/3)/sqrt(a + b*cos(c + d*x)), x)],
[cos(c + d*x)^(1/3)/sqrt(a + b*cos(c + d*x)), x, 0, Int(cos(c + d*x)^(1/3)/sqrt(a + b*cos(c + d*x)), x)],
[cos(c + d*x)^(-1/3)/sqrt(a + b*cos(c + d*x)), x, 0, Int(1/(cos(c + d*x)^(1/3)*sqrt(a + b*cos(c + d*x))), x)],
[cos(c + d*x)^(-2/3)/sqrt(a + b*cos(c + d*x)), x, 0, Int(1/(cos(c + d*x)^(2/3)*sqrt(a + b*cos(c + d*x))), x)],
[cos(c + d*x)^(-4/3)/sqrt(a + b*cos(c + d*x)), x, 1, -((3*Int((-(b/6) + (1/3)*(2*a)*cos(c + d*x) + (1/6)*(7*b)*cos(c + d*x)^2)/(cos(c + d*x)^(1/3)*sqrt(a + b*cos(c + d*x))), x))/a) + (3*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(a*d*cos(c + d*x)^(1/3))],
[cos(c + d*x)^(-5/3)/sqrt(a + b*cos(c + d*x)), x, 1, -((3*Int((b/6 + (1/3)*a*cos(c + d*x) + (1/6)*(5*b)*cos(c + d*x)^2)/(cos(c + d*x)^(2/3)*sqrt(a + b*cos(c + d*x))), x))/(2*a)) + (3*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(2*a*d*cos(c + d*x)^(2/3))],
[cos(c + d*x)^(-7/3)/sqrt(a + b*cos(c + d*x)), x, 2, (9*Int(((1/36)*(4*a^2 - 5*b^2) + (1/2)*(a*b)*cos(c + d*x) + (1/36)*(35*b^2)*cos(c + d*x)^2)/(cos(c + d*x)^(1/3)*sqrt(a + b*cos(c + d*x))), x))/(4*a^2) + (3*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(4*a*d*cos(c + d*x)^(4/3)) - (15*b*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(8*a^2*d*cos(c + d*x)^(1/3))],


# ::Subsection::Closed:: 
#Integrands of the form Cos[c+d x]^m (a+b Cos[c+d x])^(n/3)
#


# ::Subsubsection::Closed:: 
#n>0


[(a + b*cos(c + d*x))^(5/3), x, 1, (3/5)*Int(((1/3)*(5*a^2 + 2*b^2) + (1/3)*(7*a*b)*cos(c + d*x))/(a + b*cos(c + d*x))^(1/3), x) + (3*b*(a + b*cos(c + d*x))^(2/3)*sin(c + d*x))/(5*d)],
[(a + b*cos(c + d*x))^(4/3), x, 1, (3/4)*Int(((1/3)*(4*a^2 + b^2) + (5/3)*a*b*cos(c + d*x))/(a + b*cos(c + d*x))^(2/3), x) + (3*b*(a + b*cos(c + d*x))^(1/3)*sin(c + d*x))/(4*d)],
[(a + b*cos(c + d*x))^(2/3), x, 0, Int((a + b*cos(c + d*x))^(2/3), x)],
[(a + b*cos(c + d*x))^(1/3), x, 0, Int((a + b*cos(c + d*x))^(1/3), x)],


[(a + b*cos(c + d*x))^(4/3) - (3*((1/3)*(4*a^2 + b^2) + (5/3)*a*b*cos(c + d*x)))/(4*(a + b*cos(c + d*x))^(2/3)), x, 2, (3*b*(a + b*cos(c + d*x))^(1/3)*sin(c + d*x))/(4*d)],


# ::Subsubsection::Closed:: 
#n<0


[1/(a + b*cos(c + d*x))^(1/3), x, 0, Int(1/(a + b*cos(c + d*x))^(1/3), x)],
[1/(a + b*cos(c + d*x))^(2/3), x, 0, Int(1/(a + b*cos(c + d*x))^(2/3), x)],
[1/(a + b*cos(c + d*x))^(4/3), x, 1, -((3*Int((-(a/3) - (1/3)*(2*b)*cos(c + d*x))/(a + b*cos(c + d*x))^(1/3), x))/(a^2 - b^2)) - (3*b*sin(c + d*x))/((a^2 - b^2)*d*(a + b*cos(c + d*x))^(1/3))],
[1/(a + b*cos(c + d*x))^(5/3), x, 1, Int(1/(a + b*cos(c + d*x))^(5/3), x)],


# ::Subsection::Closed:: 
#Integrands of the form Sec[c+d x]^(m/2) (a+b Cos[c+d x])^n


# ::Subsubsection::Closed:: 
#n>0


[sec(c + d*x)^(7/2)*(a + b*cos(c + d*x))^2, x, 7, -((2*(3*a^2 + 5*b^2)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d)) + (4*a*b*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) + (2*(3*a^2 + 5*b^2)*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d) + (4*a*b*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d) + (2*a^2*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d)],
[sec(c + d*x)^(5/2)*(a + b*cos(c + d*x))^2, x, 6, -((4*a*b*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/d) + (2*(a^2 + 3*b^2)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) + (4*a*b*sqrt(sec(c + d*x))*sin(c + d*x))/d + (2*a^2*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d)],
[sec(c + d*x)^(3/2)*(a + b*cos(c + d*x))^2, x, 5, -((2*(a^2 - b^2)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/d) + (4*a*b*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/d + (2*a^2*sqrt(sec(c + d*x))*sin(c + d*x))/d],
[sec(c + d*x)^(1/2)*(a + b*cos(c + d*x))^2, x, 5, (4*a*b*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/d + (2*(3*a^2 + b^2)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) + (2*b^2*sin(c + d*x))/(3*d*sqrt(sec(c + d*x)))],
[sec(c + d*x)^(-1/2)*(a + b*cos(c + d*x))^2, x, 6, (2*(5*a^2 + 3*b^2)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d) + (4*a*b*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) + (2*b^2*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2)) + (4*a*b*sin(c + d*x))/(3*d*sqrt(sec(c + d*x)))],
[sec(c + d*x)^(-3/2)*(a + b*cos(c + d*x))^2, x, 7, (12*a*b*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d) + (2*(7*a^2 + 5*b^2)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) + (2*b^2*sin(c + d*x))/(7*d*sec(c + d*x)^(5/2)) + (4*a*b*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2)) + (2*(7*a^2 + 5*b^2)*sin(c + d*x))/(21*d*sqrt(sec(c + d*x)))],
[sec(c + d*x)^(-5/2)*(a + b*cos(c + d*x))^2, x, 8, (2*(9*a^2 + 7*b^2)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(15*d) + (20*a*b*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) + (2*b^2*sin(c + d*x))/(9*d*sec(c + d*x)^(7/2)) + (4*a*b*sin(c + d*x))/(7*d*sec(c + d*x)^(5/2)) + (2*(9*a^2 + 7*b^2)*sin(c + d*x))/(45*d*sec(c + d*x)^(3/2)) + (20*a*b*sin(c + d*x))/(21*d*sqrt(sec(c + d*x)))],
[sec(c + d*x)^(-7/2)*(a + b*cos(c + d*x))^2, x, 9, (28*a*b*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(15*d) + (10*(11*a^2 + 9*b^2)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(231*d) + (2*b^2*sin(c + d*x))/(11*d*sec(c + d*x)^(9/2)) + (4*a*b*sin(c + d*x))/(9*d*sec(c + d*x)^(7/2)) + (2*(11*a^2 + 9*b^2)*sin(c + d*x))/(77*d*sec(c + d*x)^(5/2)) + (28*a*b*sin(c + d*x))/(45*d*sec(c + d*x)^(3/2)) + (10*(11*a^2 + 9*b^2)*sin(c + d*x))/(231*d*sqrt(sec(c + d*x)))],


[sec(c + d*x)^(9/2)*(a + b*cos(c + d*x))^3, x, 8, -((2*b*(9*a^2 + 5*b^2)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d)) + (2*a*(5*a^2 + 21*b^2)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) + (2*b*(9*a^2 + 5*b^2)*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d) + (2*a*(5*a^2 + 21*b^2)*sec(c + d*x)^(3/2)*sin(c + d*x))/(21*d) + (32*a^2*b*sec(c + d*x)^(5/2)*sin(c + d*x))/(35*d) + (2*a^2*(a + b*cos(c + d*x))*sec(c + d*x)^(7/2)*sin(c + d*x))/(7*d)],
[sec(c + d*x)^(7/2)*(a + b*cos(c + d*x))^3, x, 7, -((6*a*(a^2 + 5*b^2)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d)) + (2*b*(a^2 + b^2)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/d + (6*a*(a^2 + 5*b^2)*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d) + (8*a^2*b*sec(c + d*x)^(3/2)*sin(c + d*x))/(5*d) + (2*a^2*(a + b*cos(c + d*x))*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d)],
[sec(c + d*x)^(5/2)*(a + b*cos(c + d*x))^3, x, 6, -((2*b*(3*a^2 - b^2)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/d) + (2*a*(a^2 + 9*b^2)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) + (16*a^2*b*sqrt(sec(c + d*x))*sin(c + d*x))/(3*d) + (2*a^2*(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d)],
[sec(c + d*x)^(3/2)*(a + b*cos(c + d*x))^3, x, 6, -((2*a*(a^2 - 3*b^2)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/d) + (2*b*(9*a^2 + b^2)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) - (2*b*(3*a^2 - b^2)*sin(c + d*x))/(3*d*sqrt(sec(c + d*x))) + (2*a^2*(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*sin(c + d*x))/d],
[sec(c + d*x)^(1/2)*(a + b*cos(c + d*x))^3, x, 6, (6*b*(5*a^2 + b^2)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d) + (2*a*(a^2 + b^2)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/d + (8*a*b^2*sin(c + d*x))/(5*d*sqrt(sec(c + d*x))) + (2*b^2*(a + b*cos(c + d*x))*sin(c + d*x))/(5*d*sqrt(sec(c + d*x)))],
[sec(c + d*x)^(-1/2)*(a + b*cos(c + d*x))^3, x, 7, (2*a*(5*a^2 + 9*b^2)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d) + (2*b*(21*a^2 + 5*b^2)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) + (32*a*b^2*sin(c + d*x))/(35*d*sec(c + d*x)^(3/2)) + (2*b^2*(a + b*cos(c + d*x))*sin(c + d*x))/(7*d*sec(c + d*x)^(3/2)) + (2*b*(21*a^2 + 5*b^2)*sin(c + d*x))/(21*d*sqrt(sec(c + d*x)))],
[sec(c + d*x)^(-3/2)*(a + b*cos(c + d*x))^3, x, 8, (2*b*(27*a^2 + 7*b^2)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(15*d) + (2*a*(7*a^2 + 15*b^2)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) + (40*a*b^2*sin(c + d*x))/(63*d*sec(c + d*x)^(5/2)) + (2*b^2*(a + b*cos(c + d*x))*sin(c + d*x))/(9*d*sec(c + d*x)^(5/2)) + (2*b*(27*a^2 + 7*b^2)*sin(c + d*x))/(45*d*sec(c + d*x)^(3/2)) + (2*a*(7*a^2 + 15*b^2)*sin(c + d*x))/(21*d*sqrt(sec(c + d*x)))],


# ::Subsubsection::Closed:: 
#n<0


[sec(c + d*x)^(5/2)/(a + b*cos(c + d*x)), x, 8, (2*b*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a^2*d) + (2*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*a*d) + (2*b^2*sqrt(cos(c + d*x))*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a^2*(a + b)*d) - (2*b*sqrt(sec(c + d*x))*sin(c + d*x))/(a^2*d) + (2*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*a*d)],
[sec(c + d*x)^(3/2)/(a + b*cos(c + d*x)), x, 6, -((2*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a*d)) - (2*b*sqrt(cos(c + d*x))*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a*(a + b)*d) + (2*sqrt(sec(c + d*x))*sin(c + d*x))/(a*d)],
[sec(c + d*x)^(1/2)/(a + b*cos(c + d*x)), x, 2, (2*sqrt(cos(c + d*x))*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/((a + b)*d)],
[sec(c + d*x)^(-1/2)/(a + b*cos(c + d*x)), x, 4, (2*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(b*d) - (2*a*sqrt(cos(c + d*x))*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(b*(a + b)*d)],
[sec(c + d*x)^(-3/2)/(a + b*cos(c + d*x)), x, 6, (2*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(b*d) - (2*a*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(b^2*d) + (2*a^2*sqrt(cos(c + d*x))*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(b^2*(a + b)*d)],
[sec(c + d*x)^(-5/2)/(a + b*cos(c + d*x)), x, 7, -((2*a*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(b^2*d)) + (2*(3*a^2 + b^2)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*b^3*d) - (2*a^3*sqrt(cos(c + d*x))*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(b^3*(a + b)*d) + (2*sin(c + d*x))/(3*b*d*sqrt(sec(c + d*x)))],


[sec(c + d*x)^(5/2)/(a + b*cos(c + d*x))^2, x, 9, (b*(4*a^2 - 5*b^2)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a^3*(a^2 - b^2)*d) + ((2*a^2 - 5*b^2)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*a^2*(a^2 - b^2)*d) + (b^2*(7*a^2 - 5*b^2)*sqrt(cos(c + d*x))*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a^3*(a - b)*(a + b)^2*d) - (b*(4*a^2 - 5*b^2)*sqrt(sec(c + d*x))*sin(c + d*x))/(a^3*(a^2 - b^2)*d) + ((2*a^2 - 5*b^2)*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*a^2*(a^2 - b^2)*d) + (b^2*sec(c + d*x)^(3/2)*sin(c + d*x))/(a*(a^2 - b^2)*d*(a + b*cos(c + d*x)))],
[sec(c + d*x)^(3/2)/(a + b*cos(c + d*x))^2, x, 8, -(((2*a^2 - 3*b^2)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a^2*(a^2 - b^2)*d)) + (b*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a*(a^2 - b^2)*d) - (b*(5*a^2 - 3*b^2)*sqrt(cos(c + d*x))*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a^2*(a - b)*(a + b)^2*d) + ((2*a^2 - 3*b^2)*sqrt(sec(c + d*x))*sin(c + d*x))/(a^2*(a^2 - b^2)*d) + (b^2*sqrt(sec(c + d*x))*sin(c + d*x))/(a*(a^2 - b^2)*d*(a + b*cos(c + d*x)))],
[sec(c + d*x)^(1/2)/(a + b*cos(c + d*x))^2, x, 7, -((b*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a*(a^2 - b^2)*d)) - (sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/((a^2 - b^2)*d) + ((3*a^2 - b^2)*sqrt(cos(c + d*x))*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a*(a - b)*(a + b)^2*d) + (b^2*sin(c + d*x))/(a*(a^2 - b^2)*d*(a + b*cos(c + d*x))*sqrt(sec(c + d*x)))],
[sec(c + d*x)^(-1/2)/(a + b*cos(c + d*x))^2, x, 7, (sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/((a^2 - b^2)*d) + (a*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(b*(a^2 - b^2)*d) - ((a^2 + b^2)*sqrt(cos(c + d*x))*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/((a - b)*b*(a + b)^2*d) - (b*sin(c + d*x))/((a^2 - b^2)*d*(a + b*cos(c + d*x))*sqrt(sec(c + d*x)))],
[sec(c + d*x)^(-3/2)/(a + b*cos(c + d*x))^2, x, 7, -((a*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(b*(a^2 - b^2)*d)) + ((a^2 - 2*b^2)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(b^2*(a^2 - b^2)*d) - (a*(a^2 - 3*b^2)*sqrt(cos(c + d*x))*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/((a - b)*b^2*(a + b)^2*d) + (a*sin(c + d*x))/((a^2 - b^2)*d*(a + b*cos(c + d*x))*sqrt(sec(c + d*x)))],
[sec(c + d*x)^(-5/2)/(a + b*cos(c + d*x))^2, x, 7, ((3*a^2 - 2*b^2)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(b^2*(a^2 - b^2)*d) - (a*(3*a^2 - 4*b^2)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(b^3*(a^2 - b^2)*d) + (a^2*(3*a^2 - 5*b^2)*sqrt(cos(c + d*x))*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/((a - b)*b^3*(a + b)^2*d) - (a^2*sin(c + d*x))/(b*(a^2 - b^2)*d*(a + b*cos(c + d*x))*sqrt(sec(c + d*x)))],


[sec(c + d*x)^(5/2)/(a + b*cos(c + d*x))^3, x, 10, (b*(24*a^4 - 65*a^2*b^2 + 35*b^4)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(4*a^4*(a^2 - b^2)^2*d) + ((8*a^4 - 61*a^2*b^2 + 35*b^4)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(12*a^3*(a^2 - b^2)^2*d) + (b^2*(63*a^4 - 86*a^2*b^2 + 35*b^4)*sqrt(cos(c + d*x))*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(4*a^4*(a - b)^2*(a + b)^3*d) - (b*(24*a^4 - 65*a^2*b^2 + 35*b^4)*sqrt(sec(c + d*x))*sin(c + d*x))/(4*a^4*(a^2 - b^2)^2*d) + ((8*a^4 - 61*a^2*b^2 + 35*b^4)*sec(c + d*x)^(3/2)*sin(c + d*x))/(12*a^3*(a^2 - b^2)^2*d) + (b^2*sec(c + d*x)^(3/2)*sin(c + d*x))/(2*a*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) + (b^2*(13*a^2 - 7*b^2)*sec(c + d*x)^(3/2)*sin(c + d*x))/(4*a^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))],
[sec(c + d*x)^(3/2)/(a + b*cos(c + d*x))^3, x, 9, -(((8*a^4 - 29*a^2*b^2 + 15*b^4)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(4*a^3*(a^2 - b^2)^2*d)) + (b*(11*a^2 - 5*b^2)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(4*a^2*(a^2 - b^2)^2*d) - (b*(35*a^4 - 38*a^2*b^2 + 15*b^4)*sqrt(cos(c + d*x))*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(4*a^3*(a - b)^2*(a + b)^3*d) + ((8*a^4 - 29*a^2*b^2 + 15*b^4)*sqrt(sec(c + d*x))*sin(c + d*x))/(4*a^3*(a^2 - b^2)^2*d) + (b^2*sqrt(sec(c + d*x))*sin(c + d*x))/(2*a*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) + (b^2*(11*a^2 - 5*b^2)*sqrt(sec(c + d*x))*sin(c + d*x))/(4*a^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))],
[sec(c + d*x)^(1/2)/(a + b*cos(c + d*x))^3, x, 8, -((3*b*(3*a^2 - b^2)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(4*a^2*(a^2 - b^2)^2*d)) - ((7*a^2 - b^2)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(4*a*(a^2 - b^2)^2*d) + (3*(5*a^4 - 2*a^2*b^2 + b^4)*sqrt(cos(c + d*x))*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(4*a^2*(a - b)^2*(a + b)^3*d) + (b^2*sin(c + d*x))/(2*a*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2*sqrt(sec(c + d*x))) + (3*b^2*(3*a^2 - b^2)*sin(c + d*x))/(4*a^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x))*sqrt(sec(c + d*x)))],
[sec(c + d*x)^(-1/2)/(a + b*cos(c + d*x))^3, x, 8, ((5*a^2 + b^2)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(4*a*(a^2 - b^2)^2*d) + (3*(a^2 + b^2)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(4*b*(a^2 - b^2)^2*d) - ((3*a^4 + 10*a^2*b^2 - b^4)*sqrt(cos(c + d*x))*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(4*a*(a - b)^2*b*(a + b)^3*d) - (b*sin(c + d*x))/(2*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2*sqrt(sec(c + d*x))) - (b*(5*a^2 + b^2)*sin(c + d*x))/(4*a*(a^2 - b^2)^2*d*(a + b*cos(c + d*x))*sqrt(sec(c + d*x)))],
[sec(c + d*x)^(-3/2)/(a + b*cos(c + d*x))^3, x, 8, -(((a^2 + 5*b^2)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(4*b*(a^2 - b^2)^2*d)) + (a*(a^2 - 7*b^2)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(4*b^2*(a^2 - b^2)^2*d) - ((a^4 - 10*a^2*b^2 - 3*b^4)*sqrt(cos(c + d*x))*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(4*(a - b)^2*b^2*(a + b)^3*d) + (a*sin(c + d*x))/(2*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2*sqrt(sec(c + d*x))) + ((a^2 + 5*b^2)*sin(c + d*x))/(4*(a^2 - b^2)^2*d*(a + b*cos(c + d*x))*sqrt(sec(c + d*x)))],
[sec(c + d*x)^(-5/2)/(a + b*cos(c + d*x))^3, x, 8, -((3*a*(a^2 - 3*b^2)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(4*b^2*(a^2 - b^2)^2*d)) + ((3*a^4 - 5*a^2*b^2 + 8*b^4)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(4*b^3*(a^2 - b^2)^2*d) - (3*a*(a^4 - 2*a^2*b^2 + 5*b^4)*sqrt(cos(c + d*x))*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(4*(a - b)^2*b^3*(a + b)^3*d) - (a^2*sin(c + d*x))/(2*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2*sqrt(sec(c + d*x))) + (3*a*(a^2 - 3*b^2)*sin(c + d*x))/(4*b*(a^2 - b^2)^2*d*(a + b*cos(c + d*x))*sqrt(sec(c + d*x)))],


# ::Subsection::Closed:: 
#Integrands of the form Sec[c+d x]^(m/2) (a+b Cos[c+d x])^(n/2)
#


# ::Subsubsection::Closed:: 
#n>0


[sec(c + d*x)^(7/2)*(a + b*cos(c + d*x))^(1/2), x, 7, -((2*(a + b)*(9*a^2 - 2*b^2)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(15*a^2*d*sqrt(a + b*cos(c + d*x)))) + (2*(9*a^2 + 7*a*b - 2*b^2)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(15*a*d*sqrt(a + b*cos(c + d*x))) + (2*b*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(15*a*d) + (2*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d) + (2*(9*a^2 - 2*b^2)*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*tan((1/2)*(c + d*x)))/(15*a^2*d)],
[sec(c + d*x)^(5/2)*(a + b*cos(c + d*x))^(1/2), x, 6, -((2*b*(a + b)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(3*a*d*sqrt(a + b*cos(c + d*x)))) + (2*(a + b)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(3*d*sqrt(a + b*cos(c + d*x))) + (2*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d) + (2*b*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*tan((1/2)*(c + d*x)))/(3*a*d)],
[sec(c + d*x)^(3/2)*(a + b*cos(c + d*x))^(1/2), x, 5, -((2*(a + b)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*cos(c + d*x)))) + (2*(a + b)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*cos(c + d*x))) + (2*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*tan((1/2)*(c + d*x)))/d],
[sec(c + d*x)^(1/2)*(a + b*cos(c + d*x))^(1/2), x, 4, (2*(a - b)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*cos(c + d*x))) + (4*b*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*cos(c + d*x)))],
[sec(c + d*x)^(-1/2)*(a + b*cos(c + d*x))^(1/2), x, 7, ((a + b)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*cos(c + d*x))) - (2*a*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*cos(c + d*x))) + (2*a*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*cos(c + d*x))) + (sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(d*sqrt(sec(c + d*x)))],
[sec(c + d*x)^(-3/2)*(a + b*cos(c + d*x))^(1/2), x, 8, (a*(a + b)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(4*b*d*sqrt(a + b*cos(c + d*x))) + ((a - 2*b)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(2*d*sqrt(a + b*cos(c + d*x))) - ((a^2 - 4*b^2)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(2*b*d*sqrt(a + b*cos(c + d*x))) + (sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(2*d*sqrt(sec(c + d*x))) + (a*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(4*b*d*sqrt(sec(c + d*x)))],


[sec(c + d*x)^(9/2)*(a + b*cos(c + d*x))^(3/2), x, 8, -((4*b*(a + b)*(41*a^2 - 3*b^2)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(105*a^2*d*sqrt(a + b*cos(c + d*x)))) + (2*(25*a^3 + 82*a^2*b + 51*a*b^2 - 6*b^3)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(105*a*d*sqrt(a + b*cos(c + d*x))) + (2*(25*a^2 + 3*b^2)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(105*a*d) + (16*b*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(5/2)*sin(c + d*x))/(35*d) + (2*a*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(7/2)*sin(c + d*x))/(7*d) + (4*b*(41*a^2 - 3*b^2)*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*tan((1/2)*(c + d*x)))/(105*a^2*d)],
[sec(c + d*x)^(7/2)*(a + b*cos(c + d*x))^(3/2), x, 7, -((2*(a + b)*(3*a^2 + b^2)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(5*a*d*sqrt(a + b*cos(c + d*x)))) + (2*(3*a^2 + 4*a*b + b^2)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(5*d*sqrt(a + b*cos(c + d*x))) + (4*b*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(5*d) + (2*a*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d) + (2*(3*a^2 + b^2)*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*tan((1/2)*(c + d*x)))/(5*a*d)],
[sec(c + d*x)^(5/2)*(a + b*cos(c + d*x))^(3/2), x, 6, -((8*b*(a + b)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(3*d*sqrt(a + b*cos(c + d*x)))) + (2*(a^2 + 4*a*b + 3*b^2)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(3*d*sqrt(a + b*cos(c + d*x))) + (2*a*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d) + (8*b*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*tan((1/2)*(c + d*x)))/(3*d)],
[sec(c + d*x)^(3/2)*(a + b*cos(c + d*x))^(3/2), x, 7, -((2*a*(a + b)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*cos(c + d*x)))) + (2*(a^2 + 2*a*b - b^2)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*cos(c + d*x))) + (4*b^2*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*cos(c + d*x))) + (2*a*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*tan((1/2)*(c + d*x)))/d],
[sec(c + d*x)^(1/2)*(a + b*cos(c + d*x))^(3/2), x, 8, (b*(a + b)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*cos(c + d*x))) + (2*a*(a - 2*b)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*cos(c + d*x))) + (6*a*b*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*cos(c + d*x))) + (b*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(d*sqrt(sec(c + d*x)))],
[sec(c + d*x)^(-1/2)*(a + b*cos(c + d*x))^(3/2), x, 8, (5*a*(a + b)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(4*d*sqrt(a + b*cos(c + d*x))) - ((4*a^2 - a*b + 2*b^2)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(2*d*sqrt(a + b*cos(c + d*x))) + ((3*a^2 + 4*b^2)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(2*d*sqrt(a + b*cos(c + d*x))) + (b*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(2*d*sqrt(sec(c + d*x))) + (5*a*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(4*d*sqrt(sec(c + d*x)))],
[sec(c + d*x)^(-3/2)*(a + b*cos(c + d*x))^(3/2), x, 9, ((a + b)*(3*a^2 + 16*b^2)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(24*b*d*sqrt(a + b*cos(c + d*x))) + (a*(7*a - 26*b)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(12*d*sqrt(a + b*cos(c + d*x))) - (a*(a^2 - 12*b^2)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(4*b*d*sqrt(a + b*cos(c + d*x))) + (b*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*d*sec(c + d*x)^(3/2)) + (7*a*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(12*d*sqrt(sec(c + d*x))) + ((3*a^2 + 16*b^2)*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(24*b*d*sqrt(sec(c + d*x)))],


[sec(c + d*x)^(11/2)*(a + b*cos(c + d*x))^(5/2), x, 9, -((2*(a + b)*(147*a^4 + 279*a^2*b^2 - 10*b^4)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(315*a^2*d*sqrt(a + b*cos(c + d*x)))) + (2*(147*a^4 + 261*a^3*b + 279*a^2*b^2 + 155*a*b^3 - 10*b^4)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(315*a*d*sqrt(a + b*cos(c + d*x))) + (2*b*(163*a^2 + 5*b^2)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(315*a*d) + (2*(49*a^2 + 75*b^2)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(5/2)*sin(c + d*x))/(315*d) + (38*a*b*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(7/2)*sin(c + d*x))/(63*d) + (2*a^2*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(9/2)*sin(c + d*x))/(9*d) + (2*(147*a^4 + 279*a^2*b^2 - 10*b^4)*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*tan((1/2)*(c + d*x)))/(315*a^2*d)],
[sec(c + d*x)^(9/2)*(a + b*cos(c + d*x))^(5/2), x, 8, -((2*b*(a + b)*(29*a^2 + 3*b^2)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(21*a*d*sqrt(a + b*cos(c + d*x)))) + (2*(5*a^3 + 29*a^2*b + 27*a*b^2 + 3*b^3)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(21*d*sqrt(a + b*cos(c + d*x))) + (2*(5*a^2 + 9*b^2)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(21*d) + (6*a*b*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(5/2)*sin(c + d*x))/(7*d) + (2*a^2*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(7/2)*sin(c + d*x))/(7*d) + (2*b*(29*a^2 + 3*b^2)*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*tan((1/2)*(c + d*x)))/(21*a*d)],
[sec(c + d*x)^(7/2)*(a + b*cos(c + d*x))^(5/2), x, 7, -((2*(a + b)*(9*a^2 + 23*b^2)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(15*d*sqrt(a + b*cos(c + d*x)))) + (2*(9*a^3 + 17*a^2*b + 23*a*b^2 + 15*b^3)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(15*d*sqrt(a + b*cos(c + d*x))) + (22*a*b*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(15*d) + (2*a^2*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d) + (2*(9*a^2 + 23*b^2)*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*tan((1/2)*(c + d*x)))/(15*d)],
[sec(c + d*x)^(5/2)*(a + b*cos(c + d*x))^(5/2), x, 8, -((14*a*b*(a + b)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(3*d*sqrt(a + b*cos(c + d*x)))) + (2*(a^3 + 7*a^2*b + 9*a*b^2 - 3*b^3)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(3*d*sqrt(a + b*cos(c + d*x))) + (4*b^3*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*cos(c + d*x))) + (2*a^2*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d) + (14*a*b*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*tan((1/2)*(c + d*x)))/(3*d)],
[sec(c + d*x)^(3/2)*(a + b*cos(c + d*x))^(5/2), x, 8, -(((a + b)*(2*a^2 - b^2)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*cos(c + d*x)))) + (2*a*(a^2 + 3*a*b - 3*b^2)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*cos(c + d*x))) + (10*a*b^2*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*cos(c + d*x))) + (2*a^2*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*sin(c + d*x))/d - ((2*a^2 - b^2)*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(d*sqrt(sec(c + d*x)))],
[sec(c + d*x)^(1/2)*(a + b*cos(c + d*x))^(5/2), x, 8, (9*a*b*(a + b)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(4*d*sqrt(a + b*cos(c + d*x))) + ((4*a^3 - 12*a^2*b + a*b^2 - 2*b^3)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(2*d*sqrt(a + b*cos(c + d*x))) + (b*(15*a^2 + 4*b^2)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(2*d*sqrt(a + b*cos(c + d*x))) + (b^2*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(2*d*sqrt(sec(c + d*x))) + (9*a*b*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(4*d*sqrt(sec(c + d*x)))],
[sec(c + d*x)^(-1/2)*(a + b*cos(c + d*x))^(5/2), x, 9, ((a + b)*(33*a^2 + 16*b^2)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(24*d*sqrt(a + b*cos(c + d*x))) - (a*(24*a^2 - 13*a*b + 38*b^2)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(12*d*sqrt(a + b*cos(c + d*x))) + (5*a*(a^2 + 4*b^2)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(4*d*sqrt(a + b*cos(c + d*x))) + (b^2*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*d*sec(c + d*x)^(3/2)) + (13*a*b*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(12*d*sqrt(sec(c + d*x))) + ((33*a^2 + 16*b^2)*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(24*d*sqrt(sec(c + d*x)))],
[sec(c + d*x)^(-3/2)*(a + b*cos(c + d*x))^(5/2), x, 10, (a*(a + b)*(15*a^2 + 284*b^2)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(192*b*d*sqrt(a + b*cos(c + d*x))) + ((59*a^3 - 322*a^2*b + 36*a*b^2 - 72*b^3)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(96*d*sqrt(a + b*cos(c + d*x))) - ((5*a^4 - 120*a^2*b^2 - 48*b^4)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(32*b*d*sqrt(a + b*cos(c + d*x))) + (b^2*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(4*d*sec(c + d*x)^(5/2)) + (17*a*b*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(24*d*sec(c + d*x)^(3/2)) + ((59*a^2 + 36*b^2)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(96*d*sqrt(sec(c + d*x))) + (a*(15*a^2 + 284*b^2)*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(192*b*d*sqrt(sec(c + d*x)))],


# ::Subsubsection::Closed:: 
#n<0


[sec(c + d*x)^(5/2)/(a + b*cos(c + d*x))^(1/2), x, 6, (4*b*(a + b)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(3*a^2*d*sqrt(a + b*cos(c + d*x))) + (2*(a - 2*b)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(3*a*d*sqrt(a + b*cos(c + d*x))) + (2*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*a*d) - (4*b*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*tan((1/2)*(c + d*x)))/(3*a^2*d)],
[sec(c + d*x)^(3/2)/(a + b*cos(c + d*x))^(1/2), x, 5, -((2*(a + b)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(a*d*sqrt(a + b*cos(c + d*x)))) + (2*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*cos(c + d*x))) + (2*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*tan((1/2)*(c + d*x)))/(a*d)],
[sec(c + d*x)^(1/2)/(a + b*cos(c + d*x))^(1/2), x, 2, (2*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*cos(c + d*x)))],
[sec(c + d*x)^(-1/2)/(a + b*cos(c + d*x))^(1/2), x, 4, -((2*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*cos(c + d*x)))) + (4*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*cos(c + d*x)))],
[sec(c + d*x)^(-3/2)/(a + b*cos(c + d*x))^(1/2), x, 5, ((a + b)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(b*d*sqrt(a + b*cos(c + d*x))) - (2*a*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(b*d*sqrt(a + b*cos(c + d*x))) + (sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(b*d*sqrt(sec(c + d*x)))],
[sec(c + d*x)^(-5/2)/(a + b*cos(c + d*x))^(1/2), x, 8, -((3*a*(a + b)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(4*b^2*d*sqrt(a + b*cos(c + d*x)))) + ((a - 2*b)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(2*b*d*sqrt(a + b*cos(c + d*x))) + ((3*a^2 + 4*b^2)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(2*b^2*d*sqrt(a + b*cos(c + d*x))) + (sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(2*b*d*sqrt(sec(c + d*x))) - (3*a*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(4*b^2*d*sqrt(sec(c + d*x)))],


[sec(c + d*x)^(5/2)/(a + b*cos(c + d*x))^(3/2), x, 7, (2*b*(5*a^2 - 8*b^2)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(3*a^3*(a - b)*d*sqrt(a + b*cos(c + d*x))) + (2*(a^2 - 6*a*b + 8*b^2)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(3*a^2*(a - b)*d*sqrt(a + b*cos(c + d*x))) + (2*b^2*sec(c + d*x)^(3/2)*sin(c + d*x))/(a*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) + (2*(a^2 - 4*b^2)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*a^2*(a^2 - b^2)*d) - (2*b*(5*a^2 - 8*b^2)*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*tan((1/2)*(c + d*x)))/(3*a^3*(a^2 - b^2)*d)],
[sec(c + d*x)^(3/2)/(a + b*cos(c + d*x))^(3/2), x, 6, -((2*(a^2 - 2*b^2)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(a^2*(a - b)*d*sqrt(a + b*cos(c + d*x)))) + (2*(a - 2*b)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(a*(a - b)*d*sqrt(a + b*cos(c + d*x))) + (2*b^2*sqrt(sec(c + d*x))*sin(c + d*x))/(a*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) + (2*(a^2 - 2*b^2)*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*tan((1/2)*(c + d*x)))/(a^2*(a^2 - b^2)*d)],
[sec(c + d*x)^(1/2)/(a + b*cos(c + d*x))^(3/2), x, 5, -((2*b*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(a*(a - b)*d*sqrt(a + b*cos(c + d*x)))) + (2*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/((a - b)*d*sqrt(a + b*cos(c + d*x))) - (2*b*tan((1/2)*(c + d*x)))/(a*(a + b)*d*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x)))],
[sec(c + d*x)^(-1/2)/(a + b*cos(c + d*x))^(3/2), x, 5, (2*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/((a - b)*d*sqrt(a + b*cos(c + d*x))) - (2*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/((a - b)*d*sqrt(a + b*cos(c + d*x))) + (2*tan((1/2)*(c + d*x)))/((a + b)*d*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x)))],
[sec(c + d*x)^(-3/2)/(a + b*cos(c + d*x))^(3/2), x, 7, -((2*a*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/((a - b)*b*d*sqrt(a + b*cos(c + d*x)))) + (2*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/((a - b)*d*sqrt(a + b*cos(c + d*x))) + (4*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(b*d*sqrt(a + b*cos(c + d*x))) - (2*a*tan((1/2)*(c + d*x)))/(b*(a + b)*d*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x)))],
[sec(c + d*x)^(-5/2)/(a + b*cos(c + d*x))^(3/2), x, 8, ((3*a^2 - b^2)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/((a - b)*b^2*d*sqrt(a + b*cos(c + d*x))) - (2*a*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/((a - b)*b*d*sqrt(a + b*cos(c + d*x))) - (6*a*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(b^2*d*sqrt(a + b*cos(c + d*x))) - (2*a^2*sin(c + d*x))/(b*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))) + ((3*a^2 - b^2)*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(b^2*(a^2 - b^2)*d*sqrt(sec(c + d*x)))],


[sec(c + d*x)^(5/2)/(a + b*cos(c + d*x))^(5/2), x, 8, (8*b*(2*a^4 - 7*a^2*b^2 + 4*b^4)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(3*a^4*(a - b)^2*(a + b)*d*sqrt(a + b*cos(c + d*x))) + (2*(a^4 - 9*a^3*b + 16*a^2*b^2 + 12*a*b^3 - 16*b^4)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(3*a^3*(a - b)^2*(a + b)*d*sqrt(a + b*cos(c + d*x))) + (2*b^2*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*a*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)) + (4*b^2*(5*a^2 - 3*b^2)*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*a^2*(a^2 - b^2)^2*d*sqrt(a + b*cos(c + d*x))) + (2*(a^4 - 13*a^2*b^2 + 8*b^4)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*a^3*(a^2 - b^2)^2*d) - (8*b*(2*a^4 - 7*a^2*b^2 + 4*b^4)*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*tan((1/2)*(c + d*x)))/(3*a^4*(a^2 - b^2)^2*d)],
[sec(c + d*x)^(3/2)/(a + b*cos(c + d*x))^(5/2), x, 7, -((2*(3*a^4 - 15*a^2*b^2 + 8*b^4)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(3*a^3*(a - b)^2*(a + b)*d*sqrt(a + b*cos(c + d*x)))) + (2*(3*a^3 - 9*a^2*b - 6*a*b^2 + 8*b^3)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(3*a^2*(a - b)^2*(a + b)*d*sqrt(a + b*cos(c + d*x))) + (2*b^2*sqrt(sec(c + d*x))*sin(c + d*x))/(3*a*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)) + (8*b^2*(2*a^2 - b^2)*sqrt(sec(c + d*x))*sin(c + d*x))/(3*a^2*(a^2 - b^2)^2*d*sqrt(a + b*cos(c + d*x))) + (2*(3*a^4 - 15*a^2*b^2 + 8*b^4)*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*tan((1/2)*(c + d*x)))/(3*a^3*(a^2 - b^2)^2*d)],
[sec(c + d*x)^(1/2)/(a + b*cos(c + d*x))^(5/2), x, 6, -((4*b*(3*a^2 - b^2)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(3*a^2*(a - b)^2*(a + b)*d*sqrt(a + b*cos(c + d*x)))) + (2*(3*a^2 + 3*a*b - 2*b^2)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(3*a*(a - b)^2*(a + b)*d*sqrt(a + b*cos(c + d*x))) + (2*b^2*sin(c + d*x))/(3*a*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)*sqrt(sec(c + d*x))) - (4*b*(3*a^2 - b^2)*tan((1/2)*(c + d*x)))/(3*a^2*(a - b)*(a + b)^2*d*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x)))],
[sec(c + d*x)^(-1/2)/(a + b*cos(c + d*x))^(5/2), x, 6, (2*(3*a^2 + b^2)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(3*a*(a - b)^2*(a + b)*d*sqrt(a + b*cos(c + d*x))) - (2*(3*a + b)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(3*(a - b)^2*(a + b)*d*sqrt(a + b*cos(c + d*x))) - (2*b*sin(c + d*x))/(3*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)*sqrt(sec(c + d*x))) + (2*(3*a^2 + b^2)*tan((1/2)*(c + d*x)))/(3*a*(a - b)*(a + b)^2*d*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x)))],
[sec(c + d*x)^(-3/2)/(a + b*cos(c + d*x))^(5/2), x, 6, -((8*b*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(3*(a - b)^2*(a + b)*d*sqrt(a + b*cos(c + d*x)))) + (2*(a + 3*b)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(3*(a - b)^2*(a + b)*d*sqrt(a + b*cos(c + d*x))) + (2*a*sin(c + d*x))/(3*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)*sqrt(sec(c + d*x))) - (8*b*tan((1/2)*(c + d*x)))/(3*(a - b)*(a + b)^2*d*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x)))],
[sec(c + d*x)^(-5/2)/(a + b*cos(c + d*x))^(5/2), x, 8, -((2*a*(3*a^2 - 7*b^2)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(3*(a - b)^2*b^2*(a + b)*d*sqrt(a + b*cos(c + d*x)))) + (2*(2*a^2 - 3*a*b - 3*b^2)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(3*(a - b)^2*b*(a + b)*d*sqrt(a + b*cos(c + d*x))) + (4*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(b^2*d*sqrt(a + b*cos(c + d*x))) - (2*a^2*sin(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)*sqrt(sec(c + d*x))) - (2*a*(3*a^2 - 7*b^2)*tan((1/2)*(c + d*x)))/(3*(a - b)*b^2*(a + b)^2*d*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x)))],


# ::Section:: 
#Cos[c+d x]^m (A+B Cos[c+d x]) (a+b Cos[c+d x])^n


# ::Subsection::Closed:: 
#Integrands of the form Cos[c+d x]^m (A+B Cos[c+d x]) (a+b Cos[c+d x])^n
#


# ::Subsubsection::Closed:: 
#n>0


[cos(c + d*x)^2*(A + B*cos(c + d*x))*(a + b*cos(c + d*x)), x, 4, (1/8)*(4*a*A + 3*b*B)*x + (2*(A*b + a*B)*sin(c + d*x))/(3*d) + ((4*a*A + 3*b*B)*cos(c + d*x)*sin(c + d*x))/(8*d) + ((A*b + a*B)*cos(c + d*x)^2*sin(c + d*x))/(3*d) + (b*B*cos(c + d*x)^3*sin(c + d*x))/(4*d)],
[cos(c + d*x)^1*(A + B*cos(c + d*x))*(a + b*cos(c + d*x)), x, 3, (1/2)*(A*b + a*B)*x + ((3*a*A + 2*b*B)*sin(c + d*x))/(3*d) + ((A*b + a*B)*cos(c + d*x)*sin(c + d*x))/(2*d) + (b*B*cos(c + d*x)^2*sin(c + d*x))/(3*d)],
[cos(c + d*x)^0*(A + B*cos(c + d*x))*(a + b*cos(c + d*x)), x, 2, (1/2)*(2*a*A + b*B)*x + ((A*b + a*B)*sin(c + d*x))/d + (b*B*cos(c + d*x)*sin(c + d*x))/(2*d)],
[sec(c + d*x)^1*(A + B*cos(c + d*x))*(a + b*cos(c + d*x)), x, 3, (A*b + a*B)*x + (a*A*arctanh(sin(c + d*x)))/d + (b*B*sin(c + d*x))/d],
[sec(c + d*x)^2*(A + B*cos(c + d*x))*(a + b*cos(c + d*x)), x, 3, b*B*x + ((A*b + a*B)*arctanh(sin(c + d*x)))/d + (a*A*tan(c + d*x))/d],
[sec(c + d*x)^3*(A + B*cos(c + d*x))*(a + b*cos(c + d*x)), x, 4, ((a*A + 2*b*B)*arctanh(sin(c + d*x)))/(2*d) + ((A*b + a*B)*tan(c + d*x))/d + (a*A*sec(c + d*x)*tan(c + d*x))/(2*d)],
[sec(c + d*x)^4*(A + B*cos(c + d*x))*(a + b*cos(c + d*x)), x, 5, ((A*b + a*B)*arctanh(sin(c + d*x)))/(2*d) + ((2*a*A + 3*b*B)*tan(c + d*x))/(3*d) + ((A*b + a*B)*sec(c + d*x)*tan(c + d*x))/(2*d) + (a*A*sec(c + d*x)^2*tan(c + d*x))/(3*d)],
[sec(c + d*x)^5*(A + B*cos(c + d*x))*(a + b*cos(c + d*x)), x, 6, ((3*a*A + 4*b*B)*arctanh(sin(c + d*x)))/(8*d) + (2*(A*b + a*B)*tan(c + d*x))/(3*d) + ((3*a*A + 4*b*B)*sec(c + d*x)*tan(c + d*x))/(8*d) + ((A*b + a*B)*sec(c + d*x)^2*tan(c + d*x))/(3*d) + (a*A*sec(c + d*x)^3*tan(c + d*x))/(4*d)],


[cos(c + d*x)^2*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^2, x, 5, (1/8)*(4*a^2*A + 3*A*b^2 + 6*a*b*B)*x + (2*(10*a*A*b + 5*a^2*B + 4*b^2*B)*sin(c + d*x))/(15*d) + ((4*a^2*A + 3*A*b^2 + 6*a*b*B)*cos(c + d*x)*sin(c + d*x))/(8*d) + ((10*a*A*b + 5*a^2*B + 4*b^2*B)*cos(c + d*x)^2*sin(c + d*x))/(15*d) + (b*(5*A*b + 6*a*B)*cos(c + d*x)^3*sin(c + d*x))/(20*d) + (b*B*cos(c + d*x)^3*(a + b*cos(c + d*x))*sin(c + d*x))/(5*d)],
[cos(c + d*x)^1*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^2, x, 4, (1/8)*(8*a*A*b + 4*a^2*B + 3*b^2*B)*x + ((4*a^2*A*b + 4*A*b^3 - a^3*B + 8*a*b^2*B)*sin(c + d*x))/(6*b*d) + ((8*a*A*b - 2*a^2*B + 9*b^2*B)*cos(c + d*x)*sin(c + d*x))/(24*d) + ((4*A*b - a*B)*(a + b*cos(c + d*x))^2*sin(c + d*x))/(12*b*d) + (B*(a + b*cos(c + d*x))^3*sin(c + d*x))/(4*b*d)],
[cos(c + d*x)^0*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^2, x, 3, (1/2)*(2*a^2*A + A*b^2 + 2*a*b*B)*x + (2*(3*a*A*b + a^2*B + b^2*B)*sin(c + d*x))/(3*d) + (b*(3*A*b + 2*a*B)*cos(c + d*x)*sin(c + d*x))/(6*d) + (B*(a + b*cos(c + d*x))^2*sin(c + d*x))/(3*d)],
[sec(c + d*x)^1*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^2, x, 4, ((4*a*A*b + 2*a^2*B + b^2*B)*x)/2 + (a^2*A*arctanh(sin(c + d*x)))/d + (b*(2*A*b + 3*a*B)*sin(c + d*x))/(2*d) + (b*B*(a + b*cos(c + d*x))*sin(c + d*x))/(2*d)],
[sec(c + d*x)^2*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^2, x, 4, (A*b^2 + 2*a*b*B)*x + (a*(2*A*b + a*B)*arctanh(sin(c + d*x)))/d - (b*(a*A - b*B)*sin(c + d*x))/d + (a*A*(a + b*cos(c + d*x))*tan(c + d*x))/d],
[sec(c + d*x)^3*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^2, x, 4, b^2*B*x + ((a^2*A + 2*A*b^2 + 4*a*b*B)*arctanh(sin(c + d*x)))/(2*d) + (a*(3*A*b + 2*a*B)*tan(c + d*x))/(2*d) + (a*A*(b + a*sec(c + d*x))*tan(c + d*x))/(2*d)],
[sec(c + d*x)^4*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^2, x, 5, ((2*a*A*b + a^2*B + 2*b^2*B)*arctanh(sin(c + d*x)))/(2*d) + ((2*a^2*A + 3*A*b^2 + 6*a*b*B)*tan(c + d*x))/(3*d) + (a*(4*A*b + 3*a*B)*sec(c + d*x)*tan(c + d*x))/(6*d) + (a*A*(a + b*cos(c + d*x))*sec(c + d*x)^2*tan(c + d*x))/(3*d)],
[sec(c + d*x)^5*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^2, x, 6, ((3*a^2*A + 4*A*b^2 + 8*a*b*B)*arctanh(sin(c + d*x)))/(8*d) + ((4*a*A*b + 2*a^2*B + 3*b^2*B)*tan(c + d*x))/(3*d) + ((3*a^2*A + 4*A*b^2 + 8*a*b*B)*sec(c + d*x)*tan(c + d*x))/(8*d) + (a*(5*A*b + 4*a*B)*sec(c + d*x)^2*tan(c + d*x))/(12*d) + (a*A*(a + b*cos(c + d*x))*sec(c + d*x)^3*tan(c + d*x))/(4*d)],


[cos(c + d*x)^2*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^3, x, 6, (1/16)*(8*a^3*A + 18*a*A*b^2 + 18*a^2*b*B + 5*b^3*B)*x + (2*(15*a^2*A*b + 4*A*b^3 + 5*a^3*B + 12*a*b^2*B)*sin(c + d*x))/(15*d) + ((8*a^3*A + 18*a*A*b^2 + 18*a^2*b*B + 5*b^3*B)*cos(c + d*x)*sin(c + d*x))/(16*d) + ((15*a^2*A*b + 4*A*b^3 + 5*a^3*B + 12*a*b^2*B)*cos(c + d*x)^2*sin(c + d*x))/(15*d) + (b*(66*a*A*b + 38*a^2*B + 25*b^2*B)*cos(c + d*x)^3*sin(c + d*x))/(120*d) + (b*(3*A*b + 4*a*B)*cos(c + d*x)^3*(a + b*cos(c + d*x))*sin(c + d*x))/(15*d) + (b*B*cos(c + d*x)^3*(a + b*cos(c + d*x))^2*sin(c + d*x))/(6*d)],
[cos(c + d*x)^1*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^3, x, 5, (1/8)*(12*a^2*A*b + 3*A*b^3 + 4*a^3*B + 9*a*b^2*B)*x + ((15*a^3*A*b + 60*a*A*b^3 - 3*a^4*B + 52*a^2*b^2*B + 16*b^4*B)*sin(c + d*x))/(30*b*d) + ((30*a^2*A*b + 45*A*b^3 - 6*a^3*B + 71*a*b^2*B)*cos(c + d*x)*sin(c + d*x))/(120*d) + ((15*a*A*b - 3*a^2*B + 16*b^2*B)*(a + b*cos(c + d*x))^2*sin(c + d*x))/(60*b*d) + ((5*A*b - a*B)*(a + b*cos(c + d*x))^3*sin(c + d*x))/(20*b*d) + (B*(a + b*cos(c + d*x))^4*sin(c + d*x))/(5*b*d)],
[cos(c + d*x)^0*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^3, x, 4, (1/8)*(8*a^3*A + 12*a*A*b^2 + 12*a^2*b*B + 3*b^3*B)*x + ((16*a^2*A*b + 4*A*b^3 + 3*a^3*B + 12*a*b^2*B)*sin(c + d*x))/(6*d) + (b*(20*a*A*b + 6*a^2*B + 9*b^2*B)*cos(c + d*x)*sin(c + d*x))/(24*d) + ((4*A*b + 3*a*B)*(a + b*cos(c + d*x))^2*sin(c + d*x))/(12*d) + (B*(a + b*cos(c + d*x))^3*sin(c + d*x))/(4*d)],
[sec(c + d*x)^1*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^3, x, 5, ((6*a^2*A*b + A*b^3 + 2*a^3*B + 3*a*b^2*B)*x)/2 + (a^3*A*arctanh(sin(c + d*x)))/d + (b*(15*a*A*b + 11*a^2*B + 4*b^2*B)*sin(c + d*x))/(6*d) + (b*(3*A*b + 5*a*B)*(a + b*cos(c + d*x))*sin(c + d*x))/(6*d) + (b*B*(a + b*cos(c + d*x))^2*sin(c + d*x))/(3*d)],
[sec(c + d*x)^2*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^3, x, 5, (b*(6*a*A*b + 6*a^2*B + b^2*B)*x)/2 + (a^2*(3*A*b + a*B)*arctanh(sin(c + d*x)))/d - (b*(2*a^2*A - 2*A*b^2 - 5*a*b*B)*sin(c + d*x))/(2*d) - (b*(2*a*A - b*B)*(a + b*cos(c + d*x))*sin(c + d*x))/(2*d) + (a*A*(a + b*cos(c + d*x))^2*tan(c + d*x))/d],
[sec(c + d*x)^3*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^3, x, 5, (A*b^3 + 3*a*b^2*B)*x + (a*(a^2*A + 6*A*b^2 + 6*a*b*B)*arctanh(sin(c + d*x)))/(2*d) - (b*(5*a*A*b + 2*a^2*B - 2*b^2*B)*sin(c + d*x))/(2*d) + (a*(2*A*b + a*B)*(a + b*cos(c + d*x))*tan(c + d*x))/d + (a*A*(a + b*cos(c + d*x))^2*sec(c + d*x)*tan(c + d*x))/(2*d)],
[sec(c + d*x)^4*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^3, x, 5, b^3*B*x + ((3*a^2*A*b + 2*A*b^3 + a^3*B + 6*a*b^2*B)*arctanh(sin(c + d*x)))/(2*d) + (a*(4*a^2*A + 11*A*b^2 + 15*a*b*B)*tan(c + d*x))/(6*d) + (a*(5*A*b + 3*a*B)*(b + a*sec(c + d*x))*tan(c + d*x))/(6*d) + (a*A*(b + a*sec(c + d*x))^2*tan(c + d*x))/(3*d)],
[sec(c + d*x)^5*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^3, x, 6, ((3*a^3*A + 12*a*A*b^2 + 12*a^2*b*B + 8*b^3*B)*arctanh(sin(c + d*x)))/(8*d) + ((6*a^2*A*b + 3*A*b^3 + 2*a^3*B + 9*a*b^2*B)*tan(c + d*x))/(3*d) + (a*(9*a^2*A + 18*A*b^2 + 28*a*b*B)*sec(c + d*x)*tan(c + d*x))/(24*d) + (a*(3*A*b + 2*a*B)*(a + b*cos(c + d*x))*sec(c + d*x)^2*tan(c + d*x))/(6*d) + (a*A*(a + b*cos(c + d*x))^2*sec(c + d*x)^3*tan(c + d*x))/(4*d)],
[sec(c + d*x)^6*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^3, x, 7, ((9*a^2*A*b + 4*A*b^3 + 3*a^3*B + 12*a*b^2*B)*arctanh(sin(c + d*x)))/(8*d) + ((8*a^3*A + 30*a*A*b^2 + 30*a^2*b*B + 15*b^3*B)*tan(c + d*x))/(15*d) + ((9*a^2*A*b + 4*A*b^3 + 3*a^3*B + 12*a*b^2*B)*sec(c + d*x)*tan(c + d*x))/(8*d) + (a*(16*a^2*A + 27*A*b^2 + 45*a*b*B)*sec(c + d*x)^2*tan(c + d*x))/(60*d) + (a*(7*A*b + 5*a*B)*(a + b*cos(c + d*x))*sec(c + d*x)^3*tan(c + d*x))/(20*d) + (a*A*(a + b*cos(c + d*x))^2*sec(c + d*x)^4*tan(c + d*x))/(5*d)],


[cos(c + d*x)^2*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^4, x, 7, (1/16)*(8*a^4*A + 36*a^2*A*b^2 + 5*A*b^4 + 24*a^3*b*B + 20*a*b^3*B)*x + (2*(140*a^3*A*b + 112*a*A*b^3 + 35*a^4*B + 168*a^2*b^2*B + 24*b^4*B)*sin(c + d*x))/(105*d) + ((8*a^4*A + 36*a^2*A*b^2 + 5*A*b^4 + 24*a^3*b*B + 20*a*b^3*B)*cos(c + d*x)*sin(c + d*x))/(16*d) + ((140*a^3*A*b + 112*a*A*b^3 + 35*a^4*B + 168*a^2*b^2*B + 24*b^4*B)*cos(c + d*x)^2*sin(c + d*x))/(105*d) + (b*(728*a^2*A*b + 175*A*b^3 + 272*a^3*B + 556*a*b^2*B)*cos(c + d*x)^3*sin(c + d*x))/(840*d) + (b*(49*a*A*b + 31*a^2*B + 18*b^2*B)*cos(c + d*x)^3*(a + b*cos(c + d*x))*sin(c + d*x))/(105*d) + (b*(7*A*b + 10*a*B)*cos(c + d*x)^3*(a + b*cos(c + d*x))^2*sin(c + d*x))/(42*d) + (b*B*cos(c + d*x)^3*(a + b*cos(c + d*x))^3*sin(c + d*x))/(7*d)],
[cos(c + d*x)^1*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^4, x, 6, (1/16)*(32*a^3*A*b + 24*a*A*b^3 + 8*a^4*B + 36*a^2*b^2*B + 5*b^4*B)*x + ((24*a^4*A*b + 224*a^2*A*b^3 + 32*A*b^5 - 4*a^5*B + 121*a^3*b^2*B + 128*a*b^4*B)*sin(c + d*x))/(60*b*d) + ((48*a^3*A*b + 232*a*A*b^3 - 8*a^4*B + 178*a^2*b^2*B + 75*b^4*B)*cos(c + d*x)*sin(c + d*x))/(240*d) + ((24*a^2*A*b + 32*A*b^3 - 4*a^3*B + 53*a*b^2*B)*(a + b*cos(c + d*x))^2*sin(c + d*x))/(120*b*d) + ((24*a*A*b - 4*a^2*B + 25*b^2*B)*(a + b*cos(c + d*x))^3*sin(c + d*x))/(120*b*d) + ((6*A*b - a*B)*(a + b*cos(c + d*x))^4*sin(c + d*x))/(30*b*d) + (B*(a + b*cos(c + d*x))^5*sin(c + d*x))/(6*b*d)],
[cos(c + d*x)^0*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^4, x, 5, (1/8)*(8*a^4*A + 24*a^2*A*b^2 + 3*A*b^4 + 16*a^3*b*B + 12*a*b^3*B)*x + ((95*a^3*A*b + 80*a*A*b^3 + 12*a^4*B + 112*a^2*b^2*B + 16*b^4*B)*sin(c + d*x))/(30*d) + (b*(130*a^2*A*b + 45*A*b^3 + 24*a^3*B + 116*a*b^2*B)*cos(c + d*x)*sin(c + d*x))/(120*d) + ((35*a*A*b + 12*a^2*B + 16*b^2*B)*(a + b*cos(c + d*x))^2*sin(c + d*x))/(60*d) + ((5*A*b + 4*a*B)*(a + b*cos(c + d*x))^3*sin(c + d*x))/(20*d) + (B*(a + b*cos(c + d*x))^4*sin(c + d*x))/(5*d)],
[sec(c + d*x)^1*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^4, x, 6, (1/8)*(32*a^3*A*b + 16*a*A*b^3 + 8*a^4*B + 24*a^2*b^2*B + 3*b^4*B)*x + (a^4*A*arctanh(sin(c + d*x)))/d + (b*(104*a^2*A*b + 16*A*b^3 + 50*a^3*B + 55*a*b^2*B)*sin(c + d*x))/(24*d) + (b*(32*a*A*b + 26*a^2*B + 9*b^2*B)*(a + b*cos(c + d*x))*sin(c + d*x))/(24*d) + (b*(4*A*b + 7*a*B)*(a + b*cos(c + d*x))^2*sin(c + d*x))/(12*d) + (b*B*(a + b*cos(c + d*x))^3*sin(c + d*x))/(4*d)],
[sec(c + d*x)^2*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^4, x, 6, (1/2)*b*(12*a^2*A*b + A*b^3 + 8*a^3*B + 4*a*b^2*B)*x + (a^3*(4*A*b + a*B)*arctanh(sin(c + d*x)))/d - (b*(6*a^3*A - 21*a*A*b^2 - 26*a^2*b*B - 4*b^3*B)*sin(c + d*x))/(6*d) - (b*(6*a^2*A - 3*A*b^2 - 8*a*b*B)*(a + b*cos(c + d*x))*sin(c + d*x))/(6*d) - (b*(3*a*A - b*B)*(a + b*cos(c + d*x))^2*sin(c + d*x))/(3*d) + (a*A*(a + b*cos(c + d*x))^3*tan(c + d*x))/d],
[sec(c + d*x)^3*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^4, x, 6, (1/2)*b^2*(8*a*A*b + 12*a^2*B + b^2*B)*x + (a^2*(a^2*A + 12*A*b^2 + 8*a*b*B)*arctanh(sin(c + d*x)))/(2*d) - (b*(7*a^2*A*b - 2*A*b^3 + 2*a^3*B - 7*a*b^2*B)*sin(c + d*x))/(2*d) - (b*(6*a*A*b + 2*a^2*B - b^2*B)*(a + b*cos(c + d*x))*sin(c + d*x))/(2*d) + (a*(5*A*b + 2*a*B)*(a + b*cos(c + d*x))^2*tan(c + d*x))/(2*d) + (a*A*(a + b*cos(c + d*x))^3*sec(c + d*x)*tan(c + d*x))/(2*d)],
[sec(c + d*x)^4*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^4, x, 6, (A*b^4 + 4*a*b^3*B)*x + (a*(4*a^2*A*b + 8*A*b^3 + a^3*B + 12*a*b^2*B)*arctanh(sin(c + d*x)))/(2*d) - (b*(4*a^3*A + 26*a*A*b^2 + 21*a^2*b*B - 6*b^3*B)*sin(c + d*x))/(6*d) + (a*(2*a^2*A + 9*A*b^2 + 9*a*b*B)*(a + b*cos(c + d*x))*tan(c + d*x))/(3*d) + (a*(2*A*b + a*B)*(a + b*cos(c + d*x))^2*sec(c + d*x)*tan(c + d*x))/(2*d) + (a*A*(a + b*cos(c + d*x))^3*sec(c + d*x)^2*tan(c + d*x))/(3*d)],
[sec(c + d*x)^5*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^4, x, 6, b^4*B*x + ((3*a^4*A + 24*a^2*A*b^2 + 8*A*b^4 + 16*a^3*b*B + 32*a*b^3*B)*arctanh(sin(c + d*x)))/(8*d) + (a*(55*a^2*A*b + 50*A*b^3 + 16*a^3*B + 104*a*b^2*B)*tan(c + d*x))/(24*d) + (a*(9*a^2*A + 26*A*b^2 + 32*a*b*B)*(b + a*sec(c + d*x))*tan(c + d*x))/(24*d) + (a*(7*A*b + 4*a*B)*(b + a*sec(c + d*x))^2*tan(c + d*x))/(12*d) + (a*A*(b + a*sec(c + d*x))^3*tan(c + d*x))/(4*d)],
[sec(c + d*x)^6*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^4, x, 7, ((12*a^3*A*b + 16*a*A*b^3 + 3*a^4*B + 24*a^2*b^2*B + 8*b^4*B)*arctanh(sin(c + d*x)))/(8*d) + ((8*a^4*A + 60*a^2*A*b^2 + 15*A*b^4 + 40*a^3*b*B + 60*a*b^3*B)*tan(c + d*x))/(15*d) + (a*(148*a^2*A*b + 96*A*b^3 + 45*a^3*B + 230*a*b^2*B)*sec(c + d*x)*tan(c + d*x))/(120*d) + (a*(8*a^2*A + 18*A*b^2 + 25*a*b*B)*(a + b*cos(c + d*x))*sec(c + d*x)^2*tan(c + d*x))/(30*d) + (a*(8*A*b + 5*a*B)*(a + b*cos(c + d*x))^2*sec(c + d*x)^3*tan(c + d*x))/(20*d) + (a*A*(a + b*cos(c + d*x))^3*sec(c + d*x)^4*tan(c + d*x))/(5*d)],
[sec(c + d*x)^7*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^4, x, 8, ((5*a^4*A + 36*a^2*A*b^2 + 8*A*b^4 + 24*a^3*b*B + 32*a*b^3*B)*arctanh(sin(c + d*x)))/(16*d) + ((32*a^3*A*b + 40*a*A*b^3 + 8*a^4*B + 60*a^2*b^2*B + 15*b^4*B)*tan(c + d*x))/(15*d) + ((5*a^4*A + 36*a^2*A*b^2 + 8*A*b^4 + 24*a^3*b*B + 32*a*b^3*B)*sec(c + d*x)*tan(c + d*x))/(16*d) + (a*(103*a^2*A*b + 56*A*b^3 + 32*a^3*B + 144*a*b^2*B)*sec(c + d*x)^2*tan(c + d*x))/(120*d) + (a*(25*a^2*A + 48*A*b^2 + 72*a*b*B)*(a + b*cos(c + d*x))*sec(c + d*x)^3*tan(c + d*x))/(120*d) + (a*(3*A*b + 2*a*B)*(a + b*cos(c + d*x))^2*sec(c + d*x)^4*tan(c + d*x))/(10*d) + (a*A*(a + b*cos(c + d*x))^3*sec(c + d*x)^5*tan(c + d*x))/(6*d)],


# ::Subsubsection::Closed:: 
#n<0


[cos(c + d*x)^3*(A + B*cos(c + d*x))/(a + b*cos(c + d*x)), x, 5, ((2*a^2*A*b + A*b^3 - 2*a^3*B - a*b^2*B)*x)/(2*b^4) - (2*a^3*(A*b - a*B)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^4*sqrt(a^2 - b^2)*d) - ((3*a*A*b - 3*a^2*B - 2*b^2*B)*sin(c + d*x))/(3*b^3*d) + ((A*b - a*B)*cos(c + d*x)*sin(c + d*x))/(2*b^2*d) + (B*cos(c + d*x)^2*sin(c + d*x))/(3*b*d)],
[cos(c + d*x)^2*(A + B*cos(c + d*x))/(a + b*cos(c + d*x)), x, 4, -(((2*a*A*b - 2*a^2*B - b^2*B)*x)/(2*b^3)) + (2*a^2*(A*b - a*B)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^3*sqrt(a^2 - b^2)*d) + ((A*b - a*B)*sin(c + d*x))/(b^2*d) + (B*cos(c + d*x)*sin(c + d*x))/(2*b*d)],
[cos(c + d*x)^1*(A + B*cos(c + d*x))/(a + b*cos(c + d*x)), x, 3, ((A*b - a*B)*x)/b^2 - (2*a*(A*b - a*B)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^2*sqrt(a^2 - b^2)*d) + (B*sin(c + d*x))/(b*d)],
[cos(c + d*x)^0*(A + B*cos(c + d*x))/(a + b*cos(c + d*x)), x, 2, (B*x)/b + (2*(A*b - a*B)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b*sqrt(a^2 - b^2)*d)],
[sec(c + d*x)^1*(A + B*cos(c + d*x))/(a + b*cos(c + d*x)), x, 3, -((2*(A*b - a*B)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a*sqrt(a^2 - b^2)*d)) + (A*arctanh(sin(c + d*x)))/(a*d)],
[sec(c + d*x)^2*(A + B*cos(c + d*x))/(a + b*cos(c + d*x)), x, 5, (2*b*(A*b - a*B)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^2*sqrt(a^2 - b^2)*d) - ((A*b - a*B)*arctanh(sin(c + d*x)))/(a^2*d) + (A*tan(c + d*x))/(a*d)],
[sec(c + d*x)^3*(A + B*cos(c + d*x))/(a + b*cos(c + d*x)), x, 5, -((2*b^2*(A*b - a*B)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^3*sqrt(a^2 - b^2)*d)) + ((a^2*A + 2*A*b^2 - 2*a*b*B)*arctanh(sin(c + d*x)))/(2*a^3*d) - ((A*b - a*B)*tan(c + d*x))/(a^2*d) + (A*sec(c + d*x)*tan(c + d*x))/(2*a*d)],
[sec(c + d*x)^4*(A + B*cos(c + d*x))/(a + b*cos(c + d*x)), x, 6, (2*b^3*(A*b - a*B)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^4*sqrt(a^2 - b^2)*d) - ((a^2*A*b + 2*A*b^3 - a^3*B - 2*a*b^2*B)*arctanh(sin(c + d*x)))/(2*a^4*d) + ((2*a^2*A + 3*A*b^2 - 3*a*b*B)*tan(c + d*x))/(3*a^3*d) - ((A*b - a*B)*sec(c + d*x)*tan(c + d*x))/(2*a^2*d) + (A*sec(c + d*x)^2*tan(c + d*x))/(3*a*d)],


[cos(c + d*x)^3*(A + B*cos(c + d*x))/(a + b*cos(c + d*x))^2, x, 5, -(((4*a*A*b - 6*a^2*B - b^2*B)*x)/(2*b^4)) + (2*a^2*(2*a^2*A*b - 3*A*b^3 - 3*a^3*B + 4*a*b^2*B)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^4*(a^2 - b^2)^(3/2)*d) + ((2*a^2*A*b - A*b^3 - 3*a^3*B + 2*a*b^2*B)*sin(c + d*x))/(b^3*(a^2 - b^2)*d) - ((2*a*A*b - 3*a^2*B + b^2*B)*cos(c + d*x)*sin(c + d*x))/(2*b^2*(a^2 - b^2)*d) + (a*(A*b - a*B)*cos(c + d*x)^2*sin(c + d*x))/(b*(a^2 - b^2)*d*(a + b*cos(c + d*x)))],
[cos(c + d*x)^2*(A + B*cos(c + d*x))/(a + b*cos(c + d*x))^2, x, 4, ((A*b - 2*a*B)*x)/b^3 - (2*a*(a^2*A*b - 2*A*b^3 - 2*a^3*B + 3*a*b^2*B)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^3*(a^2 - b^2)^(3/2)*d) - ((a*A*b - 2*a^2*B + b^2*B)*sin(c + d*x))/(b^2*(a^2 - b^2)*d) + (a*(A*b - a*B)*cos(c + d*x)*sin(c + d*x))/(b*(a^2 - b^2)*d*(a + b*cos(c + d*x)))],
[cos(c + d*x)^1*(A + B*cos(c + d*x))/(a + b*cos(c + d*x))^2, x, 3, (B*x)/b^2 - (2*(A*b^3 + a^3*B - 2*a*b^2*B)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^2*(a^2 - b^2)^(3/2)*d) + (a*(A*b - a*B)*sin(c + d*x))/(b*(a^2 - b^2)*d*(a + b*cos(c + d*x)))],
[cos(c + d*x)^0*(A + B*cos(c + d*x))/(a + b*cos(c + d*x))^2, x, 3, (2*(a*A - b*B)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/((a^2 - b^2)^(3/2)*d) - ((A*b - a*B)*sin(c + d*x))/((a^2 - b^2)*d*(a + b*cos(c + d*x)))],
[sec(c + d*x)^1*(A + B*cos(c + d*x))/(a + b*cos(c + d*x))^2, x, 4, -((2*(2*a^2*A*b - A*b^3 - a^3*B)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^2*(a^2 - b^2)^(3/2)*d)) + (A*arctanh(sin(c + d*x)))/(a^2*d) + (b*(A*b - a*B)*sin(c + d*x))/(a*(a^2 - b^2)*d*(a + b*cos(c + d*x)))],
[sec(c + d*x)^2*(A + B*cos(c + d*x))/(a + b*cos(c + d*x))^2, x, 5, (2*b*(3*a^2*A*b - 2*A*b^3 - 2*a^3*B + a*b^2*B)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^3*(a^2 - b^2)^(3/2)*d) - ((2*A*b - a*B)*arctanh(sin(c + d*x)))/(a^3*d) + ((a^2*A - 2*A*b^2 + a*b*B)*tan(c + d*x))/(a^2*(a^2 - b^2)*d) + (b*(A*b - a*B)*tan(c + d*x))/(a*(a^2 - b^2)*d*(a + b*cos(c + d*x)))],
[sec(c + d*x)^3*(A + B*cos(c + d*x))/(a + b*cos(c + d*x))^2, x, 6, -((2*b^2*(4*a^2*A*b - 3*A*b^3 - 3*a^3*B + 2*a*b^2*B)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^4*(a^2 - b^2)^(3/2)*d)) + ((a^2*A + 6*A*b^2 - 4*a*b*B)*arctanh(sin(c + d*x)))/(2*a^4*d) - ((2*a^2*A*b - 3*A*b^3 - a^3*B + 2*a*b^2*B)*tan(c + d*x))/(a^3*(a^2 - b^2)*d) + ((a^2*A - 3*A*b^2 + 2*a*b*B)*sec(c + d*x)*tan(c + d*x))/(2*a^2*(a^2 - b^2)*d) + (b*(A*b - a*B)*sec(c + d*x)*tan(c + d*x))/(a*(a^2 - b^2)*d*(a + b*cos(c + d*x)))],


[cos(c + d*x)^4*(A + B*cos(c + d*x))/(a + b*cos(c + d*x))^3, x, 6, -(((6*a*A*b - 12*a^2*B - b^2*B)*x)/(2*b^5)) + (a^2*(6*a^4*A*b - 15*a^2*A*b^3 + 12*A*b^5 - 12*a^5*B + 29*a^3*b^2*B - 20*a*b^4*B)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^5*(a^2 - b^2)^(5/2)*d) + ((6*a^4*A*b - 11*a^2*A*b^3 + 2*A*b^5 - 12*a^5*B + 21*a^3*b^2*B - 6*a*b^4*B)*sin(c + d*x))/(2*b^4*(a^2 - b^2)^2*d) - ((3*a^3*A*b - 6*a*A*b^3 - 6*a^4*B + 10*a^2*b^2*B - b^4*B)*cos(c + d*x)*sin(c + d*x))/(2*b^3*(a^2 - b^2)^2*d) + (a*(A*b - a*B)*cos(c + d*x)^3*sin(c + d*x))/(2*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) + (a*(2*a^2*A*b - 5*A*b^3 - 4*a^3*B + 7*a*b^2*B)*cos(c + d*x)^2*sin(c + d*x))/(2*b^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))],
[cos(c + d*x)^3*(A + B*cos(c + d*x))/(a + b*cos(c + d*x))^3, x, 5, ((A*b - 3*a*B)*x)/b^4 - (a*(2*a^4*A*b - 5*a^2*A*b^3 + 6*A*b^5 - 6*a^5*B + 15*a^3*b^2*B - 12*a*b^4*B)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^4*(a^2 - b^2)^(5/2)*d) - ((2*a^3*A*b - 5*a*A*b^3 - 6*a^4*B + 11*a^2*b^2*B - 2*b^4*B)*sin(c + d*x))/(2*b^3*(a^2 - b^2)^2*d) + (a*(A*b - a*B)*cos(c + d*x)^2*sin(c + d*x))/(2*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) + (a*(a^2*A*b - 4*A*b^3 - 3*a^3*B + 6*a*b^2*B)*cos(c + d*x)*sin(c + d*x))/(2*b^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))],
[cos(c + d*x)^2*(A + B*cos(c + d*x))/(a + b*cos(c + d*x))^3, x, 4, (B*x)/b^3 + ((a^2*A*b^3 + 2*A*b^5 - 2*a^5*B + 5*a^3*b^2*B - 6*a*b^4*B)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^3*(a^2 - b^2)^(5/2)*d) + (a*(A*b - a*B)*cos(c + d*x)*sin(c + d*x))/(2*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) - (a*(3*A*b^3 + 2*a^3*B - 5*a*b^2*B)*sin(c + d*x))/(2*b^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))],
[cos(c + d*x)^1*(A + B*cos(c + d*x))/(a + b*cos(c + d*x))^3, x, 4, -(((3*a*A*b - a^2*B - 2*b^2*B)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/((a^2 - b^2)^(5/2)*d)) + (a*(A*b - a*B)*sin(c + d*x))/(2*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) + ((a^2*A*b + 2*A*b^3 + a^3*B - 4*a*b^2*B)*sin(c + d*x))/(2*b*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))],
[cos(c + d*x)^0*(A + B*cos(c + d*x))/(a + b*cos(c + d*x))^3, x, 4, ((2*a^2*A + A*b^2 - 3*a*b*B)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/((a^2 - b^2)^(5/2)*d) - ((A*b - a*B)*sin(c + d*x))/(2*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) - ((3*a*A*b - a^2*B - 2*b^2*B)*sin(c + d*x))/(2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))],
[sec(c + d*x)^1*(A + B*cos(c + d*x))/(a + b*cos(c + d*x))^3, x, 5, -(((6*a^4*A*b - 5*a^2*A*b^3 + 2*A*b^5 - 2*a^5*B - a^3*b^2*B)*arctan(((a - b)*tan((c + d*x)/2))/sqrt(a^2 - b^2)))/(a^3*(a^2 - b^2)^(5/2)*d)) + (A*arctanh(sin(c + d*x)))/(a^3*d) + (b*(A*b - a*B)*sin(c + d*x))/(2*a*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) + (b*(5*a^2*A*b - 2*A*b^3 - 3*a^3*B)*sin(c + d*x))/(2*a^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))],
[sec(c + d*x)^2*(A + B*cos(c + d*x))/(a + b*cos(c + d*x))^3, x, 6, (b*(12*a^4*A*b - 15*a^2*A*b^3 + 6*A*b^5 - 6*a^5*B + 5*a^3*b^2*B - 2*a*b^4*B)*arctan(((a - b)*tan((c + d*x)/2))/sqrt(a^2 - b^2)))/(a^4*(a^2 - b^2)^(5/2)*d) - ((3*A*b - a*B)*arctanh(sin(c + d*x)))/(a^4*d) + ((2*a^4*A - 11*a^2*A*b^2 + 6*A*b^4 + 5*a^3*b*B - 2*a*b^3*B)*tan(c + d*x))/(2*a^3*(a^2 - b^2)^2*d) + (b*(A*b - a*B)*tan(c + d*x))/(2*a*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) + (b*(6*a^2*A*b - 3*A*b^3 - 4*a^3*B + a*b^2*B)*tan(c + d*x))/(2*a^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))],
[sec(c + d*x)^3*(A + B*cos(c + d*x))/(a + b*cos(c + d*x))^3, x, 7, -((b^2*(20*a^4*A*b - 29*a^2*A*b^3 + 12*A*b^5 - 12*a^5*B + 15*a^3*b^2*B - 6*a*b^4*B)*arctan(((a - b)*tan((c + d*x)/2))/sqrt(a^2 - b^2)))/(a^5*(a^2 - b^2)^(5/2)*d)) + ((a^2*A + 12*A*b^2 - 6*a*b*B)*arctanh(sin(c + d*x)))/(2*a^5*d) - ((6*a^4*A*b - 21*a^2*A*b^3 + 12*A*b^5 - 2*a^5*B + 11*a^3*b^2*B - 6*a*b^4*B)*tan(c + d*x))/(2*a^4*(a^2 - b^2)^2*d) + ((a^4*A - 10*a^2*A*b^2 + 6*A*b^4 + 6*a^3*b*B - 3*a*b^3*B)*sec(c + d*x)*tan(c + d*x))/(2*a^3*(a^2 - b^2)^2*d) + (b*(A*b - a*B)*sec(c + d*x)*tan(c + d*x))/(2*a*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) + (b*(7*a^2*A*b - 4*A*b^3 - 5*a^3*B + 2*a*b^2*B)*sec(c + d*x)*tan(c + d*x))/(2*a^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))],


[cos(c + d*x)^4*(A + B*cos(c + d*x))/(a + b*cos(c + d*x))^4, x, 6, ((A*b - 4*a*B)*x)/b^5 - (a*(2*a^6*A*b - 7*a^4*A*b^3 + 8*a^2*A*b^5 - 8*A*b^7 - 8*a^7*B + 28*a^5*b^2*B - 35*a^3*b^4*B + 20*a*b^6*B)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^5*(a^2 - b^2)^(7/2)*d) - ((6*a^5*A*b - 17*a^3*A*b^3 + 26*a*A*b^5 - 24*a^6*B + 68*a^4*b^2*B - 65*a^2*b^4*B + 6*b^6*B)*sin(c + d*x))/(6*b^4*(a^2 - b^2)^3*d) + (a*(A*b - a*B)*cos(c + d*x)^3*sin(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^3) + (a*(a^2*A*b - 6*A*b^3 - 4*a^3*B + 9*a*b^2*B)*cos(c + d*x)^2*sin(c + d*x))/(6*b^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x))^2) + (a*(a^4*A*b - 2*a^2*A*b^3 + 6*A*b^5 - 4*a^5*B + 11*a^3*b^2*B - 12*a*b^4*B)*cos(c + d*x)*sin(c + d*x))/(2*b^3*(a^2 - b^2)^3*d*(a + b*cos(c + d*x)))],
[cos(c + d*x)^3*(A + B*cos(c + d*x))/(a + b*cos(c + d*x))^4, x, 5, (B*x)/b^4 - ((3*a^2*A*b^5 + 2*A*b^7 + 2*a^7*B - 7*a^5*b^2*B + 8*a^3*b^4*B - 8*a*b^6*B)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^4*(a^2 - b^2)^(7/2)*d) + (a*(A*b - a*B)*cos(c + d*x)^2*sin(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^3) - (a*(5*A*b^3 + 3*a^3*B - 8*a*b^2*B)*cos(c + d*x)*sin(c + d*x))/(6*b^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x))^2) + (a*(4*a^2*A*b^3 + 11*A*b^5 - 6*a^5*B + 17*a^3*b^2*B - 26*a*b^4*B)*sin(c + d*x))/(6*b^3*(a^2 - b^2)^3*d*(a + b*cos(c + d*x)))],
[cos(c + d*x)^2*(A + B*cos(c + d*x))/(a + b*cos(c + d*x))^4, x, 5, ((a^3*A + 4*a*A*b^2 - 3*a^2*b*B - 2*b^3*B)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/((a^2 - b^2)^(7/2)*d) + (a*(A*b - a*B)*cos(c + d*x)*sin(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^3) - (a*(a^2*A*b + 4*A*b^3 + 2*a^3*B - 7*a*b^2*B)*sin(c + d*x))/(6*b^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x))^2) + ((a^4*A*b - 10*a^2*A*b^3 - 6*A*b^5 + 2*a^5*B - 5*a^3*b^2*B + 18*a*b^4*B)*sin(c + d*x))/(6*b^2*(a^2 - b^2)^3*d*(a + b*cos(c + d*x)))],
[cos(c + d*x)^1*(A + B*cos(c + d*x))/(a + b*cos(c + d*x))^4, x, 5, -(((4*a^2*A*b + A*b^3 - a^3*B - 4*a*b^2*B)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/((a^2 - b^2)^(7/2)*d)) + (a*(A*b - a*B)*sin(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^3) + ((2*a^2*A*b + 3*A*b^3 + a^3*B - 6*a*b^2*B)*sin(c + d*x))/(6*b*(a^2 - b^2)^2*d*(a + b*cos(c + d*x))^2) + ((2*a^3*A*b + 13*a*A*b^3 + a^4*B - 10*a^2*b^2*B - 6*b^4*B)*sin(c + d*x))/(6*b*(a^2 - b^2)^3*d*(a + b*cos(c + d*x)))],
[cos(c + d*x)^0*(A + B*cos(c + d*x))/(a + b*cos(c + d*x))^4, x, 5, ((2*a^3*A + 3*a*A*b^2 - 4*a^2*b*B - b^3*B)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/((a^2 - b^2)^(7/2)*d) - ((A*b - a*B)*sin(c + d*x))/(3*(a^2 - b^2)*d*(a + b*cos(c + d*x))^3) - ((5*a*A*b - 2*a^2*B - 3*b^2*B)*sin(c + d*x))/(6*(a^2 - b^2)^2*d*(a + b*cos(c + d*x))^2) - ((11*a^2*A*b + 4*A*b^3 - 2*a^3*B - 13*a*b^2*B)*sin(c + d*x))/(6*(a^2 - b^2)^3*d*(a + b*cos(c + d*x)))],
[sec(c + d*x)^1*(A + B*cos(c + d*x))/(a + b*cos(c + d*x))^4, x, 6, -(((8*a^6*A*b - 8*a^4*A*b^3 + 7*a^2*A*b^5 - 2*A*b^7 - 2*a^7*B - 3*a^5*b^2*B)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^4*(a^2 - b^2)^(7/2)*d)) + (A*arctanh(sin(c + d*x)))/(a^4*d) + (b*(A*b - a*B)*sin(c + d*x))/(3*a*(a^2 - b^2)*d*(a + b*cos(c + d*x))^3) + (b*(8*a^2*A*b - 3*A*b^3 - 5*a^3*B)*sin(c + d*x))/(6*a^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x))^2) + (b*(26*a^4*A*b - 17*a^2*A*b^3 + 6*A*b^5 - 11*a^5*B - 4*a^3*b^2*B)*sin(c + d*x))/(6*a^3*(a^2 - b^2)^3*d*(a + b*cos(c + d*x)))],
[sec(c + d*x)^2*(A + B*cos(c + d*x))/(a + b*cos(c + d*x))^4, x, 7, (b*(20*a^6*A*b - 35*a^4*A*b^3 + 28*a^2*A*b^5 - 8*A*b^7 - 8*a^7*B + 8*a^5*b^2*B - 7*a^3*b^4*B + 2*a*b^6*B)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^5*(a^2 - b^2)^(7/2)*d) - ((4*A*b - a*B)*arctanh(sin(c + d*x)))/(a^5*d) + ((6*a^6*A - 65*a^4*A*b^2 + 68*a^2*A*b^4 - 24*A*b^6 + 26*a^5*b*B - 17*a^3*b^3*B + 6*a*b^5*B)*tan(c + d*x))/(6*a^4*(a^2 - b^2)^3*d) + (b*(A*b - a*B)*tan(c + d*x))/(3*a*(a^2 - b^2)*d*(a + b*cos(c + d*x))^3) + (b*(9*a^2*A*b - 4*A*b^3 - 6*a^3*B + a*b^2*B)*tan(c + d*x))/(6*a^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x))^2) + (b*(12*a^4*A*b - 11*a^2*A*b^3 + 4*A*b^5 - 6*a^5*B + 2*a^3*b^2*B - a*b^4*B)*tan(c + d*x))/(2*a^3*(a^2 - b^2)^3*d*(a + b*cos(c + d*x)))],
[sec(c + d*x)^3*(A + B*cos(c + d*x))/(a + b*cos(c + d*x))^4, x, 8, -((b^2*(40*a^6*A*b - 84*a^4*A*b^3 + 69*a^2*A*b^5 - 20*A*b^7 - 20*a^7*B + 35*a^5*b^2*B - 28*a^3*b^4*B + 8*a*b^6*B)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^6*(a^2 - b^2)^(7/2)*d)) + ((a^2*A + 20*A*b^2 - 8*a*b*B)*arctanh(sin(c + d*x)))/(2*a^6*d) - ((24*a^6*A*b - 146*a^4*A*b^3 + 167*a^2*A*b^5 - 60*A*b^7 - 6*a^7*B + 65*a^5*b^2*B - 68*a^3*b^4*B + 24*a*b^6*B)*tan(c + d*x))/(6*a^5*(a^2 - b^2)^3*d) + ((a^6*A - 23*a^4*A*b^2 + 27*a^2*A*b^4 - 10*A*b^6 + 12*a^5*b*B - 11*a^3*b^3*B + 4*a*b^5*B)*sec(c + d*x)*tan(c + d*x))/(2*a^4*(a^2 - b^2)^3*d) + (b*(A*b - a*B)*sec(c + d*x)*tan(c + d*x))/(3*a*(a^2 - b^2)*d*(a + b*cos(c + d*x))^3) + (b*(10*a^2*A*b - 5*A*b^3 - 7*a^3*B + 2*a*b^2*B)*sec(c + d*x)*tan(c + d*x))/(6*a^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x))^2) + (b*(48*a^4*A*b - 53*a^2*A*b^3 + 20*A*b^5 - 27*a^5*B + 20*a^3*b^2*B - 8*a*b^4*B)*sec(c + d*x)*tan(c + d*x))/(6*a^3*(a^2 - b^2)^3*d*(a + b*cos(c + d*x)))],


[cos(c + d*x)^3*(a*B + b*B*cos(c + d*x))/(a + b*cos(c + d*x)), x, 3, (B*sin(c + d*x))/d - (B*sin(c + d*x)^3)/(3*d)],
[cos(c + d*x)^2*(a*B + b*B*cos(c + d*x))/(a + b*cos(c + d*x)), x, 2, (B*x)/2 + (B*cos(c + d*x)*sin(c + d*x))/(2*d)],
[cos(c + d*x)^1*(a*B + b*B*cos(c + d*x))/(a + b*cos(c + d*x)), x, 2, (B*sin(c + d*x))/d],
[cos(c + d*x)^0*(a*B + b*B*cos(c + d*x))/(a + b*cos(c + d*x)), x, 1, B*x],
[sec(c + d*x)^1*(a*B + b*B*cos(c + d*x))/(a + b*cos(c + d*x)), x, 2, (B*arctanh(sin(c + d*x)))/d],
[sec(c + d*x)^2*(a*B + b*B*cos(c + d*x))/(a + b*cos(c + d*x)), x, 2, (B*tan(c + d*x))/d],
[sec(c + d*x)^3*(a*B + b*B*cos(c + d*x))/(a + b*cos(c + d*x)), x, 3, (B*arctanh(sin(c + d*x)))/(2*d) + (B*sec(c + d*x)*tan(c + d*x))/(2*d)],
[sec(c + d*x)^4*(a*B + b*B*cos(c + d*x))/(a + b*cos(c + d*x)), x, 3, (B*tan(c + d*x))/d + (B*tan(c + d*x)^3)/(3*d)],


[cos(c + d*x)^3*(a*B + b*B*cos(c + d*x))/(a + b*cos(c + d*x))^2, x, 5, ((2*a^2 + b^2)*B*x)/(2*b^3) - (2*a^3*B*arctan(((a - b)*tan((c + d*x)/2))/sqrt(a^2 - b^2)))/(b^3*sqrt(a^2 - b^2)*d) - (a*B*sin(c + d*x))/(b^2*d) + (B*cos(c + d*x)*sin(c + d*x))/(2*b*d)],
[cos(c + d*x)^2*(a*B + b*B*cos(c + d*x))/(a + b*cos(c + d*x))^2, x, 5, -((a*B*x)/b^2) + (2*a^2*B*arctan(((a - b)*tan((c + d*x)/2))/sqrt(a^2 - b^2)))/(b^2*sqrt(a^2 - b^2)*d) + (B*sin(c + d*x))/(b*d)],
[cos(c + d*x)^1*(a*B + b*B*cos(c + d*x))/(a + b*cos(c + d*x))^2, x, 3, (B*x)/b - (2*a*B*arctan(((a - b)*tan((c + d*x)/2))/sqrt(a^2 - b^2)))/(b*sqrt(a^2 - b^2)*d)],
[cos(c + d*x)^0*(a*B + b*B*cos(c + d*x))/(a + b*cos(c + d*x))^2, x, 2, (2*B*arctan(((a - b)*tan((c + d*x)/2))/sqrt(a^2 - b^2)))/(sqrt(a^2 - b^2)*d)],
[sec(c + d*x)^1*(a*B + b*B*cos(c + d*x))/(a + b*cos(c + d*x))^2, x, 4, (-2*b*B*arctan(((a - b)*tan((c + d*x)/2))/sqrt(a^2 - b^2)))/(a*sqrt(a^2 - b^2)*d) + (B*arctanh(sin(c + d*x)))/(a*d)],
[sec(c + d*x)^2*(a*B + b*B*cos(c + d*x))/(a + b*cos(c + d*x))^2, x, 6, (2*b^2*B*arctan(((a - b)*tan((c + d*x)/2))/sqrt(a^2 - b^2)))/(a^2*sqrt(a^2 - b^2)*d) - (b*B*arctanh(sin(c + d*x)))/(a^2*d) + (B*tan(c + d*x))/(a*d)],
[sec(c + d*x)^3*(a*B + b*B*cos(c + d*x))/(a + b*cos(c + d*x))^2, x, 6, (-2*b^3*B*arctan(((a - b)*tan((c + d*x)/2))/sqrt(a^2 - b^2)))/(a^3*sqrt(a^2 - b^2)*d) + ((a^2 + 2*b^2)*B*arctanh(sin(c + d*x)))/(2*a^3*d) - (b*B*tan(c + d*x))/(a^2*d) + (B*sec(c + d*x)*tan(c + d*x))/(2*a*d)],


[(b*B/a + B*cos(c + d*x))/(a + b*cos(c + d*x)), x, 2, (B*x)/b - (2*sqrt(a^2 - b^2)*B*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a*b*d)],
[(a + b*cos(c + d*x))/(b + a*cos(c + d*x))^2, x, 2, sin(c + d*x)/(d*(b + a*cos(c + d*x)))],
[(3 + cos(c + d*x))/(2 - cos(c + d*x)), x, 2, -x + (5*x)/sqrt(3) + (10*arctan(sin(c + d*x)/(2 + sqrt(3) - cos(c + d*x))))/(sqrt(3)*d)],


# ::Subsection::Closed:: 
#Integrands of the form Cos[c+d x]^m (A+B Cos[c+d x]) (a+b Cos[c+d x])^(n/2)


# ::Subsubsection::Closed:: 
#n>0


[cos(c + d*x)^3*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^(1/2), x, 9, (2*(a + b)*(24*a^3*A*b + 57*a*A*b^3 - 16*a^4*B - 24*a^2*b^2*B + 147*b^4*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(315*b^4*d*sqrt(a + b*cos(c + d*x))) - (2*(24*a^4*A*b + 51*a^2*A*b^3 - 75*A*b^5 - 16*a^5*B - 20*a^3*b^2*B + 36*a*b^4*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(315*b^4*d*sqrt(a + b*cos(c + d*x))) - (2*(12*a^2*A*b - 75*A*b^3 - 8*a^3*B - 13*a*b^2*B)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(315*b^3*d) + (2*(9*a*A*b - 6*a^2*B + 49*b^2*B)*cos(c + d*x)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(315*b^2*d) + (2*(9*A*b + a*B)*cos(c + d*x)^2*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(63*b*d) + (2*B*cos(c + d*x)^3*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(9*d)],
[cos(c + d*x)^2*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^(1/2), x, 8, -((2*(a + b)*(14*a^2*A*b - 63*A*b^3 - 8*a^3*B - 19*a*b^2*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(105*b^3*d*sqrt(a + b*cos(c + d*x)))) + (2*(14*a^3*A*b - 14*a*A*b^3 - 8*a^4*B - 17*a^2*b^2*B + 25*b^4*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(105*b^3*d*sqrt(a + b*cos(c + d*x))) + (2*(7*a*A*b - 4*a^2*B + 25*b^2*B)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(105*b^2*d) + (2*(7*A*b + a*B)*cos(c + d*x)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(35*b*d) + (2*B*cos(c + d*x)^2*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(7*d)],
[cos(c + d*x)^1*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^(1/2), x, 7, (2*(a + b)*(5*a*A*b - 2*a^2*B + 9*b^2*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(15*b^2*d*sqrt(a + b*cos(c + d*x))) - (2*(5*a^2*A*b - 5*A*b^3 - 2*a^3*B + 2*a*b^2*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(15*b^2*d*sqrt(a + b*cos(c + d*x))) + (2*(5*A*b - 2*a*B)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(15*b*d) + (2*B*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(5*b*d)],
[cos(c + d*x)^0*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^(1/2), x, 6, (2*(a + b)*(3*A*b + a*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(3*b*d*sqrt(a + b*cos(c + d*x))) - (2*(a^2 - b^2)*B*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(3*b*d*sqrt(a + b*cos(c + d*x))) + (2*B*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*d)],
[sec(c + d*x)^1*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^(1/2), x, 8, (2*(a + b)*B*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x))) + (2*A*b*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x))) + (2*a*A*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (c + d*x)/2, (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x)))],
[sec(c + d*x)^2*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^(1/2), x, 9, -((A*(a + b)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x)))) + ((a*A + 2*b*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x))) + ((A*b + 2*a*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (c + d*x)/2, (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x))) + (A*sqrt(a + b*cos(c + d*x))*tan(c + d*x))/d],
[sec(c + d*x)^3*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^(1/2), x, 10, -((a + b)*(A*b + 4*a*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*b)/(a + b)))/(4*a*d*sqrt(a + b*cos(c + d*x))) + ((3*A*b + 4*a*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*b)/(a + b)))/(4*d*sqrt(a + b*cos(c + d*x))) + ((4*a^2*A - A*b^2 + 4*a*b*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (c + d*x)/2, (2*b)/(a + b)))/(4*a*d*sqrt(a + b*cos(c + d*x))) + ((A*b + 4*a*B)*sqrt(a + b*cos(c + d*x))*tan(c + d*x))/(4*a*d) + (A*sqrt(a + b*cos(c + d*x))*sec(c + d*x)*tan(c + d*x))/(2*d)],
[sec(c + d*x)^4*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^(1/2), x, 11, -((a + b)*(16*a^2*A - 3*A*b^2 + 6*a*b*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*b)/(a + b)))/(24*a^2*d*sqrt(a + b*cos(c + d*x))) + ((16*a^2*A - A*b^2 + 18*a*b*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*b)/(a + b)))/(24*a*d*sqrt(a + b*cos(c + d*x))) + ((4*a^2*A*b + A*b^3 + 8*a^3*B - 2*a*b^2*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (c + d*x)/2, (2*b)/(a + b)))/(8*a^2*d*sqrt(a + b*cos(c + d*x))) + ((16*a^2*A - 3*A*b^2 + 6*a*b*B)*sqrt(a + b*cos(c + d*x))*tan(c + d*x))/(24*a^2*d) + ((A*b + 6*a*B)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)*tan(c + d*x))/(12*a*d) + (A*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^2*tan(c + d*x))/(3*d)],


[cos(c + d*x)^2*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^(3/2), x, 9, -((2*(a + b)*(18*a^3*A*b - 246*a*A*b^3 - 8*a^4*B - 33*a^2*b^2*B - 147*b^4*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(315*b^3*d*sqrt(a + b*cos(c + d*x)))) + (2*(18*a^4*A*b - 93*a^2*A*b^3 + 75*A*b^5 - 8*a^5*B - 31*a^3*b^2*B + 39*a*b^4*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(315*b^3*d*sqrt(a + b*cos(c + d*x))) + (2*(9*a^2*A*b + 75*A*b^3 - 4*a^3*B + 88*a*b^2*B)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(315*b^2*d) + (2*(72*a*A*b + 3*a^2*B + 49*b^2*B)*cos(c + d*x)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(315*b*d) + (2*(9*A*b + 10*a*B)*cos(c + d*x)^2*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(63*d) + (2*b*B*cos(c + d*x)^3*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(9*d)],
[cos(c + d*x)^1*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^(3/2), x, 8, (2*(a + b)*(21*a^2*A*b + 63*A*b^3 - 6*a^3*B + 82*a*b^2*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(105*b^2*d*sqrt(a + b*cos(c + d*x))) - (2*(21*a^3*A*b - 21*a*A*b^3 - 6*a^4*B + 31*a^2*b^2*B - 25*b^4*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(105*b^2*d*sqrt(a + b*cos(c + d*x))) + (2*(21*a*A*b - 6*a^2*B + 25*b^2*B)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(105*b*d) + (2*(7*A*b - 2*a*B)*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(35*b*d) + (2*B*(a + b*cos(c + d*x))^(5/2)*sin(c + d*x))/(7*b*d)],
[cos(c + d*x)^0*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^(3/2), x, 7, (2*(a + b)*(20*a*A*b + 3*a^2*B + 9*b^2*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(15*b*d*sqrt(a + b*cos(c + d*x))) - (2*(5*a^2*A*b - 5*A*b^3 + 3*a^3*B - 3*a*b^2*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(15*b*d*sqrt(a + b*cos(c + d*x))) + (2*(5*A*b + 3*a*B)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(15*d) + (2*B*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(5*d)],
[sec(c + d*x)^1*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^(3/2), x, 9, (2*(a + b)*(3*A*b + 4*a*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*b)/(a + b)))/(3*d*sqrt(a + b*cos(c + d*x))) + (2*(3*a*A*b - a^2*B + b^2*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*b)/(a + b)))/(3*d*sqrt(a + b*cos(c + d*x))) + (2*a^2*A*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (c + d*x)/2, (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x))) + (2*b*B*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*d)],
[sec(c + d*x)^2*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^(3/2), x, 9, -(((a + b)*(a*A - 2*b*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x)))) + ((a^2*A + 2*A*b^2 + 2*a*b*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x))) + (a*(3*A*b + 2*a*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (c + d*x)/2, (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x))) + (a*A*sqrt(a + b*cos(c + d*x))*tan(c + d*x))/d],
[sec(c + d*x)^3*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^(3/2), x, 10, -((a + b)*(5*A*b + 4*a*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*b)/(a + b)))/(4*d*sqrt(a + b*cos(c + d*x))) + ((7*a*A*b + 4*a^2*B + 8*b^2*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*b)/(a + b)))/(4*d*sqrt(a + b*cos(c + d*x))) + ((4*a^2*A + 3*A*b^2 + 12*a*b*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (c + d*x)/2, (2*b)/(a + b)))/(4*d*sqrt(a + b*cos(c + d*x))) + ((5*A*b + 4*a*B)*sqrt(a + b*cos(c + d*x))*tan(c + d*x))/(4*d) + (a*A*sqrt(a + b*cos(c + d*x))*sec(c + d*x)*tan(c + d*x))/(2*d)],
[sec(c + d*x)^4*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^(3/2), x, 11, -((a + b)*(16*a^2*A + 3*A*b^2 + 30*a*b*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*b)/(a + b)))/(24*a*d*sqrt(a + b*cos(c + d*x))) + ((16*a^2*A + 17*A*b^2 + 42*a*b*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*b)/(a + b)))/(24*d*sqrt(a + b*cos(c + d*x))) + ((12*a^2*A*b - A*b^3 + 8*a^3*B + 6*a*b^2*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (c + d*x)/2, (2*b)/(a + b)))/(8*a*d*sqrt(a + b*cos(c + d*x))) + ((16*a^2*A + 3*A*b^2 + 30*a*b*B)*sqrt(a + b*cos(c + d*x))*tan(c + d*x))/(24*a*d) + ((7*A*b + 6*a*B)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)*tan(c + d*x))/(12*d) + (a*A*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^2*tan(c + d*x))/(3*d)],


[cos(c + d*x)^2*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^(5/2), x, 10, -((2*(a + b)*(110*a^4*A*b - 3069*a^2*A*b^3 - 1617*A*b^5 - 40*a^5*B - 255*a^3*b^2*B - 3705*a*b^4*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(3465*b^3*d*sqrt(a + b*cos(c + d*x)))) + (2*(110*a^5*A*b - 1364*a^3*A*b^3 + 1254*a*A*b^5 - 40*a^6*B - 245*a^4*b^2*B - 390*a^2*b^4*B + 675*b^6*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(3465*b^3*d*sqrt(a + b*cos(c + d*x))) + (2*(55*a^3*A*b + 1793*a*A*b^3 - 20*a^4*B + 1025*a^2*b^2*B + 675*b^4*B)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3465*b^2*d) + (2*(825*a^2*A*b + 539*A*b^3 + 15*a^3*B + 1145*a*b^2*B)*cos(c + d*x)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3465*b*d) + (2*(209*a*A*b + 113*a^2*B + 81*b^2*B)*cos(c + d*x)^2*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(693*d) + (2*b*(11*A*b + 14*a*B)*cos(c + d*x)^3*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(99*d) + (2*b*B*cos(c + d*x)^3*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(11*d)],
[cos(c + d*x)^1*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^(5/2), x, 9, (2*(a + b)*(45*a^3*A*b + 435*a*A*b^3 - 10*a^4*B + 279*a^2*b^2*B + 147*b^4*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(315*b^2*d*sqrt(a + b*cos(c + d*x))) - (2*(45*a^4*A*b + 30*a^2*A*b^3 - 75*A*b^5 - 10*a^5*B + 124*a^3*b^2*B - 114*a*b^4*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(315*b^2*d*sqrt(a + b*cos(c + d*x))) + (2*(45*a^2*A*b + 75*A*b^3 - 10*a^3*B + 114*a*b^2*B)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(315*b*d) + (2*(45*a*A*b - 10*a^2*B + 49*b^2*B)*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(315*b*d) + (2*(9*A*b - 2*a*B)*(a + b*cos(c + d*x))^(5/2)*sin(c + d*x))/(63*b*d) + (2*B*(a + b*cos(c + d*x))^(7/2)*sin(c + d*x))/(9*b*d)],
[cos(c + d*x)^0*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^(5/2), x, 8, (2*(a + b)*(161*a^2*A*b + 63*A*b^3 + 15*a^3*B + 145*a*b^2*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(105*b*d*sqrt(a + b*cos(c + d*x))) - (2*(56*a^3*A*b - 56*a*A*b^3 + 15*a^4*B + 10*a^2*b^2*B - 25*b^4*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(105*b*d*sqrt(a + b*cos(c + d*x))) + (2*(56*a*A*b + 15*a^2*B + 25*b^2*B)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(105*d) + (2*(7*A*b + 5*a*B)*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(35*d) + (2*B*(a + b*cos(c + d*x))^(5/2)*sin(c + d*x))/(7*d)],
[sec(c + d*x)^1*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^(5/2), x, 10, (2*(a + b)*(35*a*A*b + 23*a^2*B + 9*b^2*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*b)/(a + b)))/(15*d*sqrt(a + b*cos(c + d*x))) + (2*(10*a^2*A*b + 5*A*b^3 - 8*a^3*B + 8*a*b^2*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*b)/(a + b)))/(15*d*sqrt(a + b*cos(c + d*x))) + (2*a^3*A*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (c + d*x)/2, (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x))) + (2*b*(5*A*b + 8*a*B)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(15*d) + (2*b*B*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(5*d)],
[sec(c + d*x)^2*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^(5/2), x, 10, -((a + b)*(3*a^2*A - 6*A*b^2 - 14*a*b*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*b)/(a + b)))/(3*d*sqrt(a + b*cos(c + d*x))) + ((3*a^3*A + 12*a*A*b^2 + 4*a^2*b*B + 2*b^3*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*b)/(a + b)))/(3*d*sqrt(a + b*cos(c + d*x))) + (a^2*(5*A*b + 2*a*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (c + d*x)/2, (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x))) - (b*(3*a*A - 2*b*B)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*d) + (a*A*(a + b*cos(c + d*x))^(3/2)*tan(c + d*x))/d],
[sec(c + d*x)^3*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^(5/2), x, 10, -((a + b)*(9*a*A*b + 4*a^2*B - 8*b^2*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*b)/(a + b)))/(4*d*sqrt(a + b*cos(c + d*x))) + ((11*a^2*A*b + 8*A*b^3 + 4*a^3*B + 16*a*b^2*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*b)/(a + b)))/(4*d*sqrt(a + b*cos(c + d*x))) + (a*(4*a^2*A + 15*A*b^2 + 20*a*b*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (c + d*x)/2, (2*b)/(a + b)))/(4*d*sqrt(a + b*cos(c + d*x))) + (a*(7*A*b + 4*a*B)*sqrt(a + b*cos(c + d*x))*tan(c + d*x))/(4*d) + (a*A*(a + b*cos(c + d*x))^(3/2)*sec(c + d*x)*tan(c + d*x))/(2*d)],
[sec(c + d*x)^4*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^(5/2), x, 11, -((a + b)*(16*a^2*A + 33*A*b^2 + 54*a*b*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*b)/(a + b)))/(24*d*sqrt(a + b*cos(c + d*x))) + ((16*a^3*A + 59*a*A*b^2 + 66*a^2*b*B + 48*b^3*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*b)/(a + b)))/(24*d*sqrt(a + b*cos(c + d*x))) + ((20*a^2*A*b + 5*A*b^3 + 8*a^3*B + 30*a*b^2*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (c + d*x)/2, (2*b)/(a + b)))/(8*d*sqrt(a + b*cos(c + d*x))) + ((16*a^2*A + 33*A*b^2 + 54*a*b*B)*sqrt(a + b*cos(c + d*x))*tan(c + d*x))/(24*d) + (a*(3*A*b + 2*a*B)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)*tan(c + d*x))/(4*d) + (a*A*(a + b*cos(c + d*x))^(3/2)*sec(c + d*x)^2*tan(c + d*x))/(3*d)],
[sec(c + d*x)^5*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^(5/2), x, 12, -(((a + b)*(284*a^2*A*b + 15*A*b^3 + 128*a^3*B + 264*a*b^2*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(192*a*d*sqrt(a + b*cos(c + d*x)))) + ((356*a^2*A*b + 133*A*b^3 + 128*a^3*B + 472*a*b^2*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(192*d*sqrt(a + b*cos(c + d*x))) + ((48*a^4*A + 120*a^2*A*b^2 - 5*A*b^4 + 160*a^3*b*B + 40*a*b^3*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*b)/(a + b)))/(64*a*d*sqrt(a + b*cos(c + d*x))) + ((284*a^2*A*b + 15*A*b^3 + 128*a^3*B + 264*a*b^2*B)*sqrt(a + b*cos(c + d*x))*tan(c + d*x))/(192*a*d) + ((36*a^2*A + 59*A*b^2 + 104*a*b*B)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)*tan(c + d*x))/(96*d) + (a*(11*A*b + 8*a*B)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^2*tan(c + d*x))/(24*d) + (a*A*(a + b*cos(c + d*x))^(3/2)*sec(c + d*x)^3*tan(c + d*x))/(4*d)],


# ::Subsubsection::Closed:: 
#n<0


[cos(c + d*x)^3*(A + B*cos(c + d*x))/(a + b*cos(c + d*x))^(1/2), x, 8, (2*(a + b)*(56*a^2*A*b + 63*A*b^3 - 48*a^3*B - 44*a*b^2*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(105*b^4*d*sqrt(a + b*cos(c + d*x))) - (2*(56*a^3*A*b + 49*a*A*b^3 - 48*a^4*B - 32*a^2*b^2*B - 25*b^4*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(105*b^4*d*sqrt(a + b*cos(c + d*x))) - (2*(28*a*A*b - 24*a^2*B - 25*b^2*B)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(105*b^3*d) + (2*(7*A*b - 6*a*B)*cos(c + d*x)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(35*b^2*d) + (2*B*cos(c + d*x)^2*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(7*b*d)],
[cos(c + d*x)^2*(A + B*cos(c + d*x))/(a + b*cos(c + d*x))^(1/2), x, 7, -((2*(a + b)*(10*a*A*b - 8*a^2*B - 9*b^2*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(15*b^3*d*sqrt(a + b*cos(c + d*x)))) + (2*(10*a^2*A*b + 5*A*b^3 - 8*a^3*B - 7*a*b^2*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(15*b^3*d*sqrt(a + b*cos(c + d*x))) + (2*(5*A*b - 4*a*B)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(15*b^2*d) + (2*B*cos(c + d*x)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(5*b*d)],
[cos(c + d*x)^1*(A + B*cos(c + d*x))/(a + b*cos(c + d*x))^(1/2), x, 6, (2*(a + b)*(3*A*b - 2*a*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(3*b^2*d*sqrt(a + b*cos(c + d*x))) - (2*(3*a*A*b - 2*a^2*B - b^2*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(3*b^2*d*sqrt(a + b*cos(c + d*x))) + (2*B*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*b*d)],
[cos(c + d*x)^0*(A + B*cos(c + d*x))/(a + b*cos(c + d*x))^(1/2), x, 5, (2*(a + b)*B*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(b*d*sqrt(a + b*cos(c + d*x))) + (2*(A*b - a*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(b*d*sqrt(a + b*cos(c + d*x)))],
[sec(c + d*x)^1*(A + B*cos(c + d*x))/(a + b*cos(c + d*x))^(1/2), x, 5, (2*B*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x))) + (2*A*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (c + d*x)/2, (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x)))],
[sec(c + d*x)^2*(A + B*cos(c + d*x))/(a + b*cos(c + d*x))^(1/2), x, 9, -((A*(a + b)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*b)/(a + b)))/(a*d*sqrt(a + b*cos(c + d*x)))) + (A*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x))) - ((A*b - 2*a*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (c + d*x)/2, (2*b)/(a + b)))/(a*d*sqrt(a + b*cos(c + d*x))) + (A*sqrt(a + b*cos(c + d*x))*tan(c + d*x))/(a*d)],
[sec(c + d*x)^3*(A + B*cos(c + d*x))/(a + b*cos(c + d*x))^(1/2), x, 10, ((a + b)*(3*A*b - 4*a*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*b)/(a + b)))/(4*a^2*d*sqrt(a + b*cos(c + d*x))) - ((A*b - 4*a*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*b)/(a + b)))/(4*a*d*sqrt(a + b*cos(c + d*x))) + ((4*a^2*A + 3*A*b^2 - 4*a*b*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (c + d*x)/2, (2*b)/(a + b)))/(4*a^2*d*sqrt(a + b*cos(c + d*x))) - ((3*A*b - 4*a*B)*sqrt(a + b*cos(c + d*x))*tan(c + d*x))/(4*a^2*d) + (A*sqrt(a + b*cos(c + d*x))*sec(c + d*x)*tan(c + d*x))/(2*a*d)],


[cos(c + d*x)^3*(A + B*cos(c + d*x))/(a + b*cos(c + d*x))^(3/2), x, 8, -((2*(40*a^3*A*b - 25*a*A*b^3 - 48*a^4*B + 24*a^2*b^2*B + 9*b^4*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(15*(a - b)*b^4*d*sqrt(a + b*cos(c + d*x)))) + (2*(40*a^2*A*b + 5*A*b^3 - 48*a^3*B - 12*a*b^2*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(15*b^4*d*sqrt(a + b*cos(c + d*x))) + (2*a*(A*b - a*B)*cos(c + d*x)^2*sin(c + d*x))/(b*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) + (2*(20*a^2*A*b - 5*A*b^3 - 24*a^3*B + 9*a*b^2*B)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(15*b^3*(a^2 - b^2)*d) - (2*(5*a*A*b - 6*a^2*B + b^2*B)*cos(c + d*x)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(5*b^2*(a^2 - b^2)*d)],
[cos(c + d*x)^2*(A + B*cos(c + d*x))/(a + b*cos(c + d*x))^(3/2), x, 7, (2*(6*a^2*A*b - 3*A*b^3 - 8*a^3*B + 5*a*b^2*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(3*(a - b)*b^3*d*sqrt(a + b*cos(c + d*x))) - (2*(6*a*A*b - 8*a^2*B - b^2*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(3*b^3*d*sqrt(a + b*cos(c + d*x))) + (2*a*(A*b - a*B)*cos(c + d*x)*sin(c + d*x))/(b*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) - (2*(3*a*A*b - 4*a^2*B + b^2*B)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*b^2*(a^2 - b^2)*d)],
[cos(c + d*x)^1*(A + B*cos(c + d*x))/(a + b*cos(c + d*x))^(3/2), x, 6, -((2*(a*A*b - 2*a^2*B + b^2*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/((a - b)*b^2*d*sqrt(a + b*cos(c + d*x)))) + (2*(A*b - 2*a*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(b^2*d*sqrt(a + b*cos(c + d*x))) + (2*a*(A*b - a*B)*sin(c + d*x))/(b*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x)))],
[cos(c + d*x)^0*(A + B*cos(c + d*x))/(a + b*cos(c + d*x))^(3/2), x, 6, (2*(A*b - a*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/((a - b)*b*d*sqrt(a + b*cos(c + d*x))) + (2*B*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(b*d*sqrt(a + b*cos(c + d*x))) - (2*(A*b - a*B)*sin(c + d*x))/((a^2 - b^2)*d*sqrt(a + b*cos(c + d*x)))],
[sec(c + d*x)^1*(A + B*cos(c + d*x))/(a + b*cos(c + d*x))^(3/2), x, 7, (-2*(A*b - a*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*b)/(a + b)))/(a*(a - b)*d*sqrt(a + b*cos(c + d*x))) + (2*A*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (c + d*x)/2, (2*b)/(a + b)))/(a*d*sqrt(a + b*cos(c + d*x))) + (2*b*(A*b - a*B)*sin(c + d*x))/(a*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x)))],
[sec(c + d*x)^2*(A + B*cos(c + d*x))/(a + b*cos(c + d*x))^(3/2), x, 10, -(((a^2*A - 3*A*b^2 + 2*a*b*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*b)/(a + b)))/(a^2*(a - b)*d*sqrt(a + b*cos(c + d*x)))) + (A*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*b)/(a + b)))/(a*d*sqrt(a + b*cos(c + d*x))) - ((3*A*b - 2*a*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (c + d*x)/2, (2*b)/(a + b)))/(a^2*d*sqrt(a + b*cos(c + d*x))) + (2*b*(A*b - a*B)*tan(c + d*x))/(a*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) + ((a^2*A - 3*A*b^2 + 2*a*b*B)*sqrt(a + b*cos(c + d*x))*tan(c + d*x))/(a^2*(a^2 - b^2)*d)],
[sec(c + d*x)^3*(A + B*cos(c + d*x))/(a + b*cos(c + d*x))^(3/2), x, 11, ((7*a^2*A*b - 15*A*b^3 - 4*a^3*B + 12*a*b^2*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*b)/(a + b)))/(4*a^3*(a - b)*d*sqrt(a + b*cos(c + d*x))) - ((5*A*b - 4*a*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*b)/(a + b)))/(4*a^2*d*sqrt(a + b*cos(c + d*x))) + ((4*a^2*A + 15*A*b^2 - 12*a*b*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (c + d*x)/2, (2*b)/(a + b)))/(4*a^3*d*sqrt(a + b*cos(c + d*x))) - ((7*a^2*A*b - 15*A*b^3 - 4*a^3*B + 12*a*b^2*B)*sqrt(a + b*cos(c + d*x))*tan(c + d*x))/(4*a^3*(a^2 - b^2)*d) + (2*b*(A*b - a*B)*sec(c + d*x)*tan(c + d*x))/(a*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) + ((a^2*A - 5*A*b^2 + 4*a*b*B)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)*tan(c + d*x))/(2*a^2*(a^2 - b^2)*d)],


[cos(c + d*x)^4*(A + B*cos(c + d*x))/(a + b*cos(c + d*x))^(5/2), x, 9, -((2*(80*a^5*A*b - 140*a^3*A*b^3 + 40*a*A*b^5 - 128*a^6*B + 212*a^4*b^2*B - 55*a^2*b^4*B - 9*b^6*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(15*(a - b)^2*b^5*(a + b)*d*sqrt(a + b*cos(c + d*x)))) + (2*(80*a^4*A*b - 80*a^2*A*b^3 - 5*A*b^5 - 128*a^5*B + 116*a^3*b^2*B + 17*a*b^4*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(15*b^5*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) + (2*a*(A*b - a*B)*cos(c + d*x)^3*sin(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)) + (2*a*(5*a^2*A*b - 9*A*b^3 - 8*a^3*B + 12*a*b^2*B)*cos(c + d*x)^2*sin(c + d*x))/(3*b^2*(a^2 - b^2)^2*d*sqrt(a + b*cos(c + d*x))) + (2*(40*a^4*A*b - 65*a^2*A*b^3 + 5*A*b^5 - 64*a^5*B + 98*a^3*b^2*B - 14*a*b^4*B)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(15*b^4*(a^2 - b^2)^2*d) - (2*(30*a^3*A*b - 50*a*A*b^3 - 48*a^4*B + 71*a^2*b^2*B - 3*b^4*B)*cos(c + d*x)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(15*b^3*(a^2 - b^2)^2*d)],
[cos(c + d*x)^3*(A + B*cos(c + d*x))/(a + b*cos(c + d*x))^(5/2), x, 8, (2*(8*a^4*A*b - 15*a^2*A*b^3 + 3*A*b^5 - 16*a^5*B + 28*a^3*b^2*B - 8*a*b^4*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(3*(a - b)^2*b^4*(a + b)*d*sqrt(a + b*cos(c + d*x))) - (2*(8*a^3*A*b - 9*a*A*b^3 - 16*a^4*B + 16*a^2*b^2*B + b^4*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(3*b^4*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) + (2*a*(A*b - a*B)*cos(c + d*x)^2*sin(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)) + (2*a*(3*a^2*A*b - 7*A*b^3 - 6*a^3*B + 10*a*b^2*B)*cos(c + d*x)*sin(c + d*x))/(3*b^2*(a^2 - b^2)^2*d*sqrt(a + b*cos(c + d*x))) - (2*(4*a^3*A*b - 8*a*A*b^3 - 8*a^4*B + 13*a^2*b^2*B - b^4*B)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*b^3*(a^2 - b^2)^2*d)],
[cos(c + d*x)^2*(A + B*cos(c + d*x))/(a + b*cos(c + d*x))^(5/2), x, 7, -((2*(2*a^3*A*b - 6*a*A*b^3 - 8*a^4*B + 15*a^2*b^2*B - 3*b^4*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(3*(a - b)^2*b^3*(a + b)*d*sqrt(a + b*cos(c + d*x)))) + (2*(2*a^2*A*b - 3*A*b^3 - 8*a^3*B + 9*a*b^2*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(3*b^3*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) + (2*a*(A*b - a*B)*cos(c + d*x)*sin(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)) + (2*a*(a^2*A*b - 5*A*b^3 - 4*a^3*B + 8*a*b^2*B)*sin(c + d*x))/(3*b^2*(a^2 - b^2)^2*d*sqrt(a + b*cos(c + d*x)))],
[cos(c + d*x)^1*(A + B*cos(c + d*x))/(a + b*cos(c + d*x))^(5/2), x, 7, -((2*(a^2*A*b + 3*A*b^3 + 2*a^3*B - 6*a*b^2*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(3*(a - b)^2*b^2*(a + b)*d*sqrt(a + b*cos(c + d*x)))) + (2*(a*A*b + 2*a^2*B - 3*b^2*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(3*b^2*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) + (2*a*(A*b - a*B)*sin(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)) + (2*(a^2*A*b + 3*A*b^3 + 2*a^3*B - 6*a*b^2*B)*sin(c + d*x))/(3*b*(a^2 - b^2)^2*d*sqrt(a + b*cos(c + d*x)))],
[cos(c + d*x)^0*(A + B*cos(c + d*x))/(a + b*cos(c + d*x))^(5/2), x, 7, (2*(4*a*A*b - a^2*B - 3*b^2*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(3*(a - b)^2*b*(a + b)*d*sqrt(a + b*cos(c + d*x))) - (2*(A*b - a*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(3*b*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) - (2*(A*b - a*B)*sin(c + d*x))/(3*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)) - (2*(4*a*A*b - a^2*B - 3*b^2*B)*sin(c + d*x))/(3*(a^2 - b^2)^2*d*sqrt(a + b*cos(c + d*x)))],
[sec(c + d*x)^1*(A + B*cos(c + d*x))/(a + b*cos(c + d*x))^(5/2), x, 10, (-2*(7*a^2*A*b - 3*A*b^3 - 4*a^3*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*b)/(a + b)))/(3*a^2*(a - b)^2*(a + b)*d*sqrt(a + b*cos(c + d*x))) + (2*(A*b - a*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*b)/(a + b)))/(3*a*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) + (2*A*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (c + d*x)/2, (2*b)/(a + b)))/(a^2*d*sqrt(a + b*cos(c + d*x))) + (2*b*(A*b - a*B)*sin(c + d*x))/(3*a*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)) + (2*b*(7*a^2*A*b - 3*A*b^3 - 4*a^3*B)*sin(c + d*x))/(3*a^2*(a^2 - b^2)^2*d*sqrt(a + b*cos(c + d*x)))],
[sec(c + d*x)^2*(A + B*cos(c + d*x))/(a + b*cos(c + d*x))^(5/2), x, 11, -((3*a^4*A - 26*a^2*A*b^2 + 15*A*b^4 + 14*a^3*b*B - 6*a*b^3*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*b)/(a + b)))/(3*a^3*(a - b)^2*(a + b)*d*sqrt(a + b*cos(c + d*x))) + ((3*a^2*A - 5*A*b^2 + 2*a*b*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*b)/(a + b)))/(3*a^2*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) - ((5*A*b - 2*a*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (c + d*x)/2, (2*b)/(a + b)))/(a^3*d*sqrt(a + b*cos(c + d*x))) + (2*b*(A*b - a*B)*tan(c + d*x))/(3*a*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)) + (2*b*(9*a^2*A*b - 5*A*b^3 - 6*a^3*B + 2*a*b^2*B)*tan(c + d*x))/(3*a^2*(a^2 - b^2)^2*d*sqrt(a + b*cos(c + d*x))) + ((3*a^4*A - 26*a^2*A*b^2 + 15*A*b^4 + 14*a^3*b*B - 6*a*b^3*B)*sqrt(a + b*cos(c + d*x))*tan(c + d*x))/(3*a^3*(a^2 - b^2)^2*d)],
# {Sec[c + d*x]^3*(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(5/2), x, 12, ((33*a^4*A*b - 170*a^2*A*b^3 + 105*A*b^5 - 12*a^5*B + 104*a^3*b^2*B - 60*a*b^4*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticE[(1/2)*(c + d*x), (2*b)/(a + b)])/(12*a^4*(a - b)^2*(a + b)*d*Sqrt[a + b*Cos[c + d*x]]) - ((27*a^2*A*b - 35*A*b^3 - 12*a^3*B + 20*a*b^2*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(1/2)*(c + d*x), (2*b)/(a + b)])/(12*a^3*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + ((4*a^2*A + 35*A*b^2 - 20*a*b*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (1/2)*(c + d*x), (2*b)/(a + b)])/(4*a^4*d*Sqrt[a + b*Cos[c + d*x]]) - ((33*a^4*A*b - 170*a^2*A*b^3 + 105*A*b^5 - 12*a^5*B + 104*a^3*b^2*B - 60*a*b^4*B)*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(12*a^4*(a^2 - b^2)^2*d) + (2*b*(A*b - a*B)*Sec[c + d*x]*Tan[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (2*b*(11*a^2*A*b - 7*A*b^3 - 8*a^3*B + 4*a*b^2*B)*Sec[c + d*x]*Tan[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]]) + ((3*a^4*A - 54*a^2*A*b^2 + 35*A*b^4 + 36*a^3*b*B - 20*a*b^3*B)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(6*a^3*(a^2 - b^2)^2*d)} 


[cos(c + d*x)^0*(a*B + b*B*cos(c + d*x))/(a + b*cos(c + d*x))^(1/2), x, 3, (2*(a + b)*B*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x)))],


[cos(c + d*x)^0*(a*B + b*B*cos(c + d*x))/(a + b*cos(c + d*x))^(3/2), x, 3, (2*B*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x)))],
[sec(c + d*x)^1*(a*B + b*B*cos(c + d*x))/(a + b*cos(c + d*x))^(3/2), x, 3, (2*B*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x)))],


[cos(c + d*x)^0*(a*B + b*B*cos(c + d*x))/(a + b*cos(c + d*x))^(5/2), x, 5, (2*B*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/((a - b)*d*sqrt(a + b*cos(c + d*x))) - (2*b*B*sin(c + d*x))/((a^2 - b^2)*d*sqrt(a + b*cos(c + d*x)))],
[sec(c + d*x)^1*(a*B + b*B*cos(c + d*x))/(a + b*cos(c + d*x))^(5/2), x, 8, -((2*b*B*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(a*(a - b)*d*sqrt(a + b*cos(c + d*x)))) + (2*B*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*b)/(a + b)))/(a*d*sqrt(a + b*cos(c + d*x))) + (2*b^2*B*sin(c + d*x))/(a*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x)))],


# ::Subsection::Closed:: 
#Integrands of the form Cos[c+d x]^(m/2) (A+B Cos[c+d x]) (a+b Cos[c+d x])^n


# ::Subsubsection::Closed:: 
#n>0


[cos(c + d*x)^(5/2)*(A + B*cos(c + d*x))*(a + b*cos(c + d*x)), x, 7, (2*(9*a*A + 7*b*B)*EllipticE((1/2)*(c + d*x), 2))/(15*d) + (10*(A*b + a*B)*EllipticF((1/2)*(c + d*x), 2))/(21*d) + (10*(A*b + a*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(21*d) + (2*(9*a*A + 7*b*B)*cos(c + d*x)^(3/2)*sin(c + d*x))/(45*d) + (2*(A*b + a*B)*cos(c + d*x)^(5/2)*sin(c + d*x))/(7*d) + (2*b*B*cos(c + d*x)^(7/2)*sin(c + d*x))/(9*d)],
[cos(c + d*x)^(3/2)*(A + B*cos(c + d*x))*(a + b*cos(c + d*x)), x, 6, (6*(A*b + a*B)*EllipticE((1/2)*(c + d*x), 2))/(5*d) + (2*(7*a*A + 5*b*B)*EllipticF((1/2)*(c + d*x), 2))/(21*d) + (2*(7*a*A + 5*b*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(21*d) + (2*(A*b + a*B)*cos(c + d*x)^(3/2)*sin(c + d*x))/(5*d) + (2*b*B*cos(c + d*x)^(5/2)*sin(c + d*x))/(7*d)],
[cos(c + d*x)^(1/2)*(A + B*cos(c + d*x))*(a + b*cos(c + d*x)), x, 5, (2*(5*a*A + 3*b*B)*EllipticE((1/2)*(c + d*x), 2))/(5*d) + (2*(A*b + a*B)*EllipticF((1/2)*(c + d*x), 2))/(3*d) + (2*(A*b + a*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(3*d) + (2*b*B*cos(c + d*x)^(3/2)*sin(c + d*x))/(5*d)],
[cos(c + d*x)^(-1/2)*(A + B*cos(c + d*x))*(a + b*cos(c + d*x)), x, 4, (2*(A*b + a*B)*EllipticE((1/2)*(c + d*x), 2))/d + (2*(3*a*A + b*B)*EllipticF((1/2)*(c + d*x), 2))/(3*d) + (2*b*B*sqrt(cos(c + d*x))*sin(c + d*x))/(3*d)],
[cos(c + d*x)^(-3/2)*(A + B*cos(c + d*x))*(a + b*cos(c + d*x)), x, 4, -((2*(a*A - b*B)*EllipticE((1/2)*(c + d*x), 2))/d) + (2*(A*b + a*B)*EllipticF((1/2)*(c + d*x), 2))/d + (2*a*A*sin(c + d*x))/(d*sqrt(cos(c + d*x)))],
[cos(c + d*x)^(-5/2)*(A + B*cos(c + d*x))*(a + b*cos(c + d*x)), x, 5, -((2*(A*b + a*B)*EllipticE((1/2)*(c + d*x), 2))/d) + (2*(a*A + 3*b*B)*EllipticF((1/2)*(c + d*x), 2))/(3*d) + (2*a*A*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2)) + (2*(A*b + a*B)*sin(c + d*x))/(d*sqrt(cos(c + d*x)))],
[cos(c + d*x)^(-7/2)*(A + B*cos(c + d*x))*(a + b*cos(c + d*x)), x, 6, -((2*(3*a*A + 5*b*B)*EllipticE((1/2)*(c + d*x), 2))/(5*d)) + (2*(A*b + a*B)*EllipticF((1/2)*(c + d*x), 2))/(3*d) + (2*a*A*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2)) + (2*(A*b + a*B)*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2)) + (2*(3*a*A + 5*b*B)*sin(c + d*x))/(5*d*sqrt(cos(c + d*x)))],


[cos(c + d*x)^(5/2)*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^2, x, 8, (2*(9*a^2*A + 7*A*b^2 + 14*a*b*B)*EllipticE((1/2)*(c + d*x), 2))/(15*d) + (10*(22*a*A*b + 11*a^2*B + 9*b^2*B)*EllipticF((1/2)*(c + d*x), 2))/(231*d) + (10*(22*a*A*b + 11*a^2*B + 9*b^2*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(231*d) + (2*(9*a^2*A + 7*A*b^2 + 14*a*b*B)*cos(c + d*x)^(3/2)*sin(c + d*x))/(45*d) + (2*(22*a*A*b + 11*a^2*B + 9*b^2*B)*cos(c + d*x)^(5/2)*sin(c + d*x))/(77*d) + (2*b*(11*A*b + 13*a*B)*cos(c + d*x)^(7/2)*sin(c + d*x))/(99*d) + (2*b*B*cos(c + d*x)^(7/2)*(a + b*cos(c + d*x))*sin(c + d*x))/(11*d)],
[cos(c + d*x)^(3/2)*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^2, x, 7, (2*(18*a*A*b + 9*a^2*B + 7*b^2*B)*EllipticE((1/2)*(c + d*x), 2))/(15*d) + (2*(7*a^2*A + 5*A*b^2 + 10*a*b*B)*EllipticF((1/2)*(c + d*x), 2))/(21*d) + (2*(7*a^2*A + 5*A*b^2 + 10*a*b*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(21*d) + (2*(18*a*A*b + 9*a^2*B + 7*b^2*B)*cos(c + d*x)^(3/2)*sin(c + d*x))/(45*d) + (2*b*(9*A*b + 11*a*B)*cos(c + d*x)^(5/2)*sin(c + d*x))/(63*d) + (2*b*B*cos(c + d*x)^(5/2)*(a + b*cos(c + d*x))*sin(c + d*x))/(9*d)],
[cos(c + d*x)^(1/2)*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^2, x, 6, (2*(5*a^2*A + 3*A*b^2 + 6*a*b*B)*EllipticE((1/2)*(c + d*x), 2))/(5*d) + (2*(14*a*A*b + 7*a^2*B + 5*b^2*B)*EllipticF((1/2)*(c + d*x), 2))/(21*d) + (2*(14*a*A*b + 7*a^2*B + 5*b^2*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(21*d) + (2*b*(7*A*b + 9*a*B)*cos(c + d*x)^(3/2)*sin(c + d*x))/(35*d) + (2*b*B*cos(c + d*x)^(3/2)*(a + b*cos(c + d*x))*sin(c + d*x))/(7*d)],
[cos(c + d*x)^(-1/2)*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^2, x, 5, (2*(10*a*A*b + 5*a^2*B + 3*b^2*B)*EllipticE((1/2)*(c + d*x), 2))/(5*d) + (2*(3*a^2*A + A*b^2 + 2*a*b*B)*EllipticF((1/2)*(c + d*x), 2))/(3*d) + (2*b*(5*A*b + 7*a*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(15*d) + (2*b*B*sqrt(cos(c + d*x))*(a + b*cos(c + d*x))*sin(c + d*x))/(5*d)],
[cos(c + d*x)^(-3/2)*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^2, x, 5, -((2*(a^2*A - A*b^2 - 2*a*b*B)*EllipticE((1/2)*(c + d*x), 2))/d) + (2*(6*a*A*b + 3*a^2*B + b^2*B)*EllipticF((1/2)*(c + d*x), 2))/(3*d) - (2*b*(3*a*A - b*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(3*d) + (2*a*A*(a + b*cos(c + d*x))*sin(c + d*x))/(d*sqrt(cos(c + d*x)))],
[cos(c + d*x)^(-5/2)*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^2, x, 5, -((2*(2*a*A*b + a^2*B - b^2*B)*EllipticE((1/2)*(c + d*x), 2))/d) + (2*(a^2*A + 3*A*b^2 + 6*a*b*B)*EllipticF((1/2)*(c + d*x), 2))/(3*d) + (2*a*(5*A*b + 3*a*B)*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))) + (2*a*A*(a + b*cos(c + d*x))*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2))],
[cos(c + d*x)^(-7/2)*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^2, x, 6, -((2*(3*a^2*A + 5*A*b^2 + 10*a*b*B)*EllipticE((1/2)*(c + d*x), 2))/(5*d)) + (2*(2*a*A*b + a^2*B + 3*b^2*B)*EllipticF((1/2)*(c + d*x), 2))/(3*d) + (2*a*(7*A*b + 5*a*B)*sin(c + d*x))/(15*d*cos(c + d*x)^(3/2)) + (2*(3*a^2*A + 5*A*b^2 + 10*a*b*B)*sin(c + d*x))/(5*d*sqrt(cos(c + d*x))) + (2*a*A*(a + b*cos(c + d*x))*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2))],


[cos(c + d*x)^(3/2)*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^3, x, 8, (2*(27*a^2*A*b + 7*A*b^3 + 9*a^3*B + 21*a*b^2*B)*EllipticE((1/2)*(c + d*x), 2))/(15*d) + (2*(77*a^3*A + 165*a*A*b^2 + 165*a^2*b*B + 45*b^3*B)*EllipticF((1/2)*(c + d*x), 2))/(231*d) + (2*(77*a^3*A + 165*a*A*b^2 + 165*a^2*b*B + 45*b^3*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(231*d) + (2*(27*a^2*A*b + 7*A*b^3 + 9*a^3*B + 21*a*b^2*B)*cos(c + d*x)^(3/2)*sin(c + d*x))/(45*d) + (2*b*(220*a*A*b + 129*a^2*B + 81*b^2*B)*cos(c + d*x)^(5/2)*sin(c + d*x))/(693*d) + (2*b*(11*A*b + 15*a*B)*cos(c + d*x)^(5/2)*(a + b*cos(c + d*x))*sin(c + d*x))/(99*d) + (2*b*B*cos(c + d*x)^(5/2)*(a + b*cos(c + d*x))^2*sin(c + d*x))/(11*d)],
[cos(c + d*x)^(1/2)*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^3, x, 7, (2*(15*a^3*A + 27*a*A*b^2 + 27*a^2*b*B + 7*b^3*B)*EllipticE((1/2)*(c + d*x), 2))/(15*d) + (2*(21*a^2*A*b + 5*A*b^3 + 7*a^3*B + 15*a*b^2*B)*EllipticF((1/2)*(c + d*x), 2))/(21*d) + (2*(21*a^2*A*b + 5*A*b^3 + 7*a^3*B + 15*a*b^2*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(21*d) + (2*b*(144*a*A*b + 89*a^2*B + 49*b^2*B)*cos(c + d*x)^(3/2)*sin(c + d*x))/(315*d) + (2*b*(9*A*b + 13*a*B)*cos(c + d*x)^(3/2)*(a + b*cos(c + d*x))*sin(c + d*x))/(63*d) + (2*b*B*cos(c + d*x)^(3/2)*(a + b*cos(c + d*x))^2*sin(c + d*x))/(9*d)],
[cos(c + d*x)^(-1/2)*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^3, x, 6, (2*(15*a^2*A*b + 3*A*b^3 + 5*a^3*B + 9*a*b^2*B)*EllipticE((1/2)*(c + d*x), 2))/(5*d) + (2*(21*a^3*A + 21*a*A*b^2 + 21*a^2*b*B + 5*b^3*B)*EllipticF((1/2)*(c + d*x), 2))/(21*d) + (2*b*(84*a*A*b + 57*a^2*B + 25*b^2*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(105*d) + (2*b*(7*A*b + 11*a*B)*sqrt(cos(c + d*x))*(a + b*cos(c + d*x))*sin(c + d*x))/(35*d) + (2*b*B*sqrt(cos(c + d*x))*(a + b*cos(c + d*x))^2*sin(c + d*x))/(7*d)],
[cos(c + d*x)^(-3/2)*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^3, x, 6, -((2*(5*a^3*A - 15*a*A*b^2 - 15*a^2*b*B - 3*b^3*B)*EllipticE((1/2)*(c + d*x), 2))/(5*d)) + (2*(9*a^2*A*b + A*b^3 + 3*a^3*B + 3*a*b^2*B)*EllipticF((1/2)*(c + d*x), 2))/(3*d) - (2*b*(15*a^2*A - 5*A*b^2 - 12*a*b*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(15*d) - (2*b*(5*a*A - b*B)*sqrt(cos(c + d*x))*(a + b*cos(c + d*x))*sin(c + d*x))/(5*d) + (2*a*A*(a + b*cos(c + d*x))^2*sin(c + d*x))/(d*sqrt(cos(c + d*x)))],
[cos(c + d*x)^(-5/2)*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^3, x, 6, -((2*(3*a^2*A*b - A*b^3 + a^3*B - 3*a*b^2*B)*EllipticE((1/2)*(c + d*x), 2))/d) + (2*(a^3*A + 9*a*A*b^2 + 9*a^2*b*B + b^3*B)*EllipticF((1/2)*(c + d*x), 2))/(3*d) - (2*b*(8*a*A*b + 3*a^2*B - b^2*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(3*d) + (2*a*(7*A*b + 3*a*B)*(a + b*cos(c + d*x))*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))) + (2*a*A*(a + b*cos(c + d*x))^2*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2))],
[cos(c + d*x)^(-7/2)*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^3, x, 6, -((2*(3*a^3*A + 15*a*A*b^2 + 15*a^2*b*B - 5*b^3*B)*EllipticE((1/2)*(c + d*x), 2))/(5*d)) + (2*(3*a^2*A*b + 3*A*b^3 + a^3*B + 9*a*b^2*B)*EllipticF((1/2)*(c + d*x), 2))/(3*d) + (2*a*(9*a^2*A + 33*A*b^2 + 40*a*b*B)*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))) + (2*a*(9*A*b + 5*a*B)*(a + b*cos(c + d*x))*sin(c + d*x))/(15*d*cos(c + d*x)^(3/2)) + (2*a*A*(a + b*cos(c + d*x))^2*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2))],


# ::Subsubsection::Closed:: 
#n<0


[cos(c + d*x)^(5/2)*(A + B*cos(c + d*x))/(a + b*cos(c + d*x)), x, 7, -((2*(5*a*A*b - 5*a^2*B - 3*b^2*B)*EllipticE((1/2)*(c + d*x), 2))/(5*b^3*d)) + (2*(3*a^2*A*b + A*b^3 - 3*a^3*B - a*b^2*B)*EllipticF((1/2)*(c + d*x), 2))/(3*b^4*d) - (2*a^3*(A*b - a*B)*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2))/(b^4*(a + b)*d) + (2*(A*b - a*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(3*b^2*d) + (2*B*cos(c + d*x)^(3/2)*sin(c + d*x))/(5*b*d)],
[cos(c + d*x)^(3/2)*(A + B*cos(c + d*x))/(a + b*cos(c + d*x)), x, 6, (2*(A*b - a*B)*EllipticE((1/2)*(c + d*x), 2))/(b^2*d) - (2*(3*a*A*b - 3*a^2*B - b^2*B)*EllipticF((1/2)*(c + d*x), 2))/(3*b^3*d) + (2*a^2*(A*b - a*B)*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2))/(b^3*(a + b)*d) + (2*B*sqrt(cos(c + d*x))*sin(c + d*x))/(3*b*d)],
[cos(c + d*x)^(1/2)*(A + B*cos(c + d*x))/(a + b*cos(c + d*x)), x, 5, (2*B*EllipticE((1/2)*(c + d*x), 2))/(b*d) + (2*(A*b - a*B)*EllipticF((1/2)*(c + d*x), 2))/(b^2*d) - (2*a*(A*b - a*B)*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2))/(b^2*(a + b)*d)],
[cos(c + d*x)^(-1/2)*(A + B*cos(c + d*x))/(a + b*cos(c + d*x)), x, 3, (2*B*EllipticF((1/2)*(c + d*x), 2))/(b*d) + (2*(A*b - a*B)*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2))/(b*(a + b)*d)],
[cos(c + d*x)^(-3/2)*(A + B*cos(c + d*x))/(a + b*cos(c + d*x)), x, 5, -((2*A*EllipticE((1/2)*(c + d*x), 2))/(a*d)) - (2*(A*b - a*B)*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2))/(a*(a + b)*d) + (2*A*sin(c + d*x))/(a*d*sqrt(cos(c + d*x)))],
[cos(c + d*x)^(-5/2)*(A + B*cos(c + d*x))/(a + b*cos(c + d*x)), x, 7, (2*(A*b - a*B)*EllipticE((1/2)*(c + d*x), 2))/(a^2*d) + (2*A*EllipticF((1/2)*(c + d*x), 2))/(3*a*d) + (2*b*(A*b - a*B)*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2))/(a^2*(a + b)*d) + (2*A*sin(c + d*x))/(3*a*d*cos(c + d*x)^(3/2)) - (2*(A*b - a*B)*sin(c + d*x))/(a^2*d*sqrt(cos(c + d*x)))],


[cos(c + d*x)^(5/2)*(A + B*cos(c + d*x))/(a + b*cos(c + d*x))^2, x, 7, ((3*a^2*A*b - 2*A*b^3 - 5*a^3*B + 4*a*b^2*B)*EllipticE((1/2)*(c + d*x), 2))/(b^3*(a^2 - b^2)*d) - ((9*a^3*A*b - 12*a*A*b^3 - 15*a^4*B + 16*a^2*b^2*B + 2*b^4*B)*EllipticF((1/2)*(c + d*x), 2))/(3*b^4*(a^2 - b^2)*d) + (a^2*(3*a^2*A*b - 5*A*b^3 - 5*a^3*B + 7*a*b^2*B)*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2))/((a - b)*b^4*(a + b)^2*d) - ((3*a*A*b - 5*a^2*B + 2*b^2*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(3*b^2*(a^2 - b^2)*d) + (a*(A*b - a*B)*cos(c + d*x)^(3/2)*sin(c + d*x))/(b*(a^2 - b^2)*d*(a + b*cos(c + d*x)))],
[cos(c + d*x)^(3/2)*(A + B*cos(c + d*x))/(a + b*cos(c + d*x))^2, x, 6, -(((a*A*b - 3*a^2*B + 2*b^2*B)*EllipticE((1/2)*(c + d*x), 2))/(b^2*(a^2 - b^2)*d)) + ((a^2*A*b - 2*A*b^3 - 3*a^3*B + 4*a*b^2*B)*EllipticF((1/2)*(c + d*x), 2))/(b^3*(a^2 - b^2)*d) - (a*(a^2*A*b - 3*A*b^3 - 3*a^3*B + 5*a*b^2*B)*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2))/((a - b)*b^3*(a + b)^2*d) + (a*(A*b - a*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(b*(a^2 - b^2)*d*(a + b*cos(c + d*x)))],
[cos(c + d*x)^(1/2)*(A + B*cos(c + d*x))/(a + b*cos(c + d*x))^2, x, 6, ((A*b - a*B)*EllipticE((1/2)*(c + d*x), 2))/(b*(a^2 - b^2)*d) + ((a*A*b + a^2*B - 2*b^2*B)*EllipticF((1/2)*(c + d*x), 2))/(b^2*(a^2 - b^2)*d) - ((a^2*A*b + A*b^3 + a^3*B - 3*a*b^2*B)*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2))/((a - b)*b^2*(a + b)^2*d) - ((A*b - a*B)*sqrt(cos(c + d*x))*sin(c + d*x))/((a^2 - b^2)*d*(a + b*cos(c + d*x)))],
[cos(c + d*x)^(-1/2)*(A + B*cos(c + d*x))/(a + b*cos(c + d*x))^2, x, 6, -(((A*b - a*B)*EllipticE((1/2)*(c + d*x), 2))/(a*(a^2 - b^2)*d)) - ((A*b - a*B)*EllipticF((1/2)*(c + d*x), 2))/(b*(a^2 - b^2)*d) + ((3*a^2*A*b - A*b^3 - a^3*B - a*b^2*B)*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2))/(a*(a - b)*b*(a + b)^2*d) + (b*(A*b - a*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(a*(a^2 - b^2)*d*(a + b*cos(c + d*x)))],
[cos(c + d*x)^(-3/2)*(A + B*cos(c + d*x))/(a + b*cos(c + d*x))^2, x, 7, -(((2*a^2*A - 3*A*b^2 + a*b*B)*EllipticE((1/2)*(c + d*x), 2))/(a^2*(a^2 - b^2)*d)) + ((A*b - a*B)*EllipticF((1/2)*(c + d*x), 2))/(a*(a^2 - b^2)*d) - ((5*a^2*A*b - 3*A*b^3 - 3*a^3*B + a*b^2*B)*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2))/(a^2*(a - b)*(a + b)^2*d) + ((2*a^2*A - 3*A*b^2 + a*b*B)*sin(c + d*x))/(a^2*(a^2 - b^2)*d*sqrt(cos(c + d*x))) + (b*(A*b - a*B)*sin(c + d*x))/(a*(a^2 - b^2)*d*sqrt(cos(c + d*x))*(a + b*cos(c + d*x)))],
[cos(c + d*x)^(-5/2)*(A + B*cos(c + d*x))/(a + b*cos(c + d*x))^2, x, 8, ((4*a^2*A*b - 5*A*b^3 - 2*a^3*B + 3*a*b^2*B)*EllipticE((1/2)*(c + d*x), 2))/(a^3*(a^2 - b^2)*d) + ((2*a^2*A - 5*A*b^2 + 3*a*b*B)*EllipticF((1/2)*(c + d*x), 2))/(3*a^2*(a^2 - b^2)*d) + (b*(7*a^2*A*b - 5*A*b^3 - 5*a^3*B + 3*a*b^2*B)*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2))/(a^3*(a - b)*(a + b)^2*d) + ((2*a^2*A - 5*A*b^2 + 3*a*b*B)*sin(c + d*x))/(3*a^2*(a^2 - b^2)*d*cos(c + d*x)^(3/2)) - ((4*a^2*A*b - 5*A*b^3 - 2*a^3*B + 3*a*b^2*B)*sin(c + d*x))/(a^3*(a^2 - b^2)*d*sqrt(cos(c + d*x))) + (b*(A*b - a*B)*sin(c + d*x))/(a*(a^2 - b^2)*d*cos(c + d*x)^(3/2)*(a + b*cos(c + d*x)))],


[cos(c + d*x)^(5/2)*(A + B*cos(c + d*x))/(a + b*cos(c + d*x))^3, x, 7, -(((3*a^3*A*b - 9*a*A*b^3 - 15*a^4*B + 29*a^2*b^2*B - 8*b^4*B)*EllipticE((1/2)*(c + d*x), 2))/(4*b^3*(a^2 - b^2)^2*d)) + ((3*a^4*A*b - 5*a^2*A*b^3 + 8*A*b^5 - 15*a^5*B + 33*a^3*b^2*B - 24*a*b^4*B)*EllipticF((1/2)*(c + d*x), 2))/(4*b^4*(a^2 - b^2)^2*d) - (a*(3*a^4*A*b - 6*a^2*A*b^3 + 15*A*b^5 - 15*a^5*B + 38*a^3*b^2*B - 35*a*b^4*B)*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2))/(4*(a - b)^2*b^4*(a + b)^3*d) + (a*(A*b - a*B)*cos(c + d*x)^(3/2)*sin(c + d*x))/(2*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) + (a*(a^2*A*b - 7*A*b^3 - 5*a^3*B + 11*a*b^2*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(4*b^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))],
[cos(c + d*x)^(3/2)*(A + B*cos(c + d*x))/(a + b*cos(c + d*x))^3, x, 7, -(((a^2*A*b + 5*A*b^3 + 3*a^3*B - 9*a*b^2*B)*EllipticE((1/2)*(c + d*x), 2))/(4*b^2*(a^2 - b^2)^2*d)) + ((a^3*A*b - 7*a*A*b^3 + 3*a^4*B - 5*a^2*b^2*B + 8*b^4*B)*EllipticF((1/2)*(c + d*x), 2))/(4*b^3*(a^2 - b^2)^2*d) - ((a^4*A*b - 10*a^2*A*b^3 - 3*A*b^5 + 3*a^5*B - 6*a^3*b^2*B + 15*a*b^4*B)*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2))/(4*(a - b)^2*b^3*(a + b)^3*d) + (a*(A*b - a*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(2*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) + ((a^2*A*b + 5*A*b^3 + 3*a^3*B - 9*a*b^2*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(4*b*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))],
[cos(c + d*x)^(1/2)*(A + B*cos(c + d*x))/(a + b*cos(c + d*x))^3, x, 7, ((5*a^2*A*b + A*b^3 - a^3*B - 5*a*b^2*B)*EllipticE((1/2)*(c + d*x), 2))/(4*a*b*(a^2 - b^2)^2*d) + ((3*a^2*A*b + 3*A*b^3 + a^3*B - 7*a*b^2*B)*EllipticF((1/2)*(c + d*x), 2))/(4*b^2*(a^2 - b^2)^2*d) - ((3*a^4*A*b + 10*a^2*A*b^3 - A*b^5 + a^5*B - 10*a^3*b^2*B - 3*a*b^4*B)*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2))/(4*a*(a - b)^2*b^2*(a + b)^3*d) - ((A*b - a*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(2*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) - ((5*a^2*A*b + A*b^3 - a^3*B - 5*a*b^2*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(4*a*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))],
[cos(c + d*x)^(-1/2)*(A + B*cos(c + d*x))/(a + b*cos(c + d*x))^3, x, 7, -(((9*a^2*A*b - 3*A*b^3 - 5*a^3*B - a*b^2*B)*EllipticE((1/2)*(c + d*x), 2))/(4*a^2*(a^2 - b^2)^2*d)) - ((7*a^2*A*b - A*b^3 - 3*a^3*B - 3*a*b^2*B)*EllipticF((1/2)*(c + d*x), 2))/(4*a*b*(a^2 - b^2)^2*d) + ((15*a^4*A*b - 6*a^2*A*b^3 + 3*A*b^5 - 3*a^5*B - 10*a^3*b^2*B + a*b^4*B)*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2))/(4*a^2*(a - b)^2*b*(a + b)^3*d) + (b*(A*b - a*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(2*a*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) + (b*(9*a^2*A*b - 3*A*b^3 - 5*a^3*B - a*b^2*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(4*a^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))],
[cos(c + d*x)^(-3/2)*(A + B*cos(c + d*x))/(a + b*cos(c + d*x))^3, x, 8, -(((8*a^4*A - 29*a^2*A*b^2 + 15*A*b^4 + 9*a^3*b*B - 3*a*b^3*B)*EllipticE((1/2)*(c + d*x), 2))/(4*a^3*(a^2 - b^2)^2*d)) + ((11*a^2*A*b - 5*A*b^3 - 7*a^3*B + a*b^2*B)*EllipticF((1/2)*(c + d*x), 2))/(4*a^2*(a^2 - b^2)^2*d) - ((35*a^4*A*b - 38*a^2*A*b^3 + 15*A*b^5 - 15*a^5*B + 6*a^3*b^2*B - 3*a*b^4*B)*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2))/(4*a^3*(a - b)^2*(a + b)^3*d) + ((8*a^4*A - 29*a^2*A*b^2 + 15*A*b^4 + 9*a^3*b*B - 3*a*b^3*B)*sin(c + d*x))/(4*a^3*(a^2 - b^2)^2*d*sqrt(cos(c + d*x))) + (b*(A*b - a*B)*sin(c + d*x))/(2*a*(a^2 - b^2)*d*sqrt(cos(c + d*x))*(a + b*cos(c + d*x))^2) + (b*(11*a^2*A*b - 5*A*b^3 - 7*a^3*B + a*b^2*B)*sin(c + d*x))/(4*a^2*(a^2 - b^2)^2*d*sqrt(cos(c + d*x))*(a + b*cos(c + d*x)))],
[cos(c + d*x)^(-5/2)*(A + B*cos(c + d*x))/(a + b*cos(c + d*x))^3, x, 9, ((24*a^4*A*b - 65*a^2*A*b^3 + 35*A*b^5 - 8*a^5*B + 29*a^3*b^2*B - 15*a*b^4*B)*EllipticE((1/2)*(c + d*x), 2))/(4*a^4*(a^2 - b^2)^2*d) + ((8*a^4*A - 61*a^2*A*b^2 + 35*A*b^4 + 33*a^3*b*B - 15*a*b^3*B)*EllipticF((1/2)*(c + d*x), 2))/(12*a^3*(a^2 - b^2)^2*d) + (b*(63*a^4*A*b - 86*a^2*A*b^3 + 35*A*b^5 - 35*a^5*B + 38*a^3*b^2*B - 15*a*b^4*B)*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2))/(4*a^4*(a - b)^2*(a + b)^3*d) + ((8*a^4*A - 61*a^2*A*b^2 + 35*A*b^4 + 33*a^3*b*B - 15*a*b^3*B)*sin(c + d*x))/(12*a^3*(a^2 - b^2)^2*d*cos(c + d*x)^(3/2)) - ((24*a^4*A*b - 65*a^2*A*b^3 + 35*A*b^5 - 8*a^5*B + 29*a^3*b^2*B - 15*a*b^4*B)*sin(c + d*x))/(4*a^4*(a^2 - b^2)^2*d*sqrt(cos(c + d*x))) + (b*(A*b - a*B)*sin(c + d*x))/(2*a*(a^2 - b^2)*d*cos(c + d*x)^(3/2)*(a + b*cos(c + d*x))^2) + (b*(13*a^2*A*b - 7*A*b^3 - 9*a^3*B + 3*a*b^2*B)*sin(c + d*x))/(4*a^2*(a^2 - b^2)^2*d*cos(c + d*x)^(3/2)*(a + b*cos(c + d*x)))],


[cos(c + d*x)^(5/2)*(a*B + b*B*cos(c + d*x))/(a + b*cos(c + d*x)), x, 3, (6*B*EllipticE((1/2)*(c + d*x), 2))/(5*d) + (2*B*cos(c + d*x)^(3/2)*sin(c + d*x))/(5*d)],
[cos(c + d*x)^(3/2)*(a*B + b*B*cos(c + d*x))/(a + b*cos(c + d*x)), x, 3, (2*B*EllipticF((1/2)*(c + d*x), 2))/(3*d) + (2*B*sqrt(cos(c + d*x))*sin(c + d*x))/(3*d)],
[cos(c + d*x)^(1/2)*(a*B + b*B*cos(c + d*x))/(a + b*cos(c + d*x)), x, 2, (2*B*EllipticE((1/2)*(c + d*x), 2))/d],
[cos(c + d*x)^(-1/2)*(a*B + b*B*cos(c + d*x))/(a + b*cos(c + d*x)), x, 2, (2*B*EllipticF((1/2)*(c + d*x), 2))/d],
[cos(c + d*x)^(-3/2)*(a*B + b*B*cos(c + d*x))/(a + b*cos(c + d*x)), x, 3, -((2*B*EllipticE((1/2)*(c + d*x), 2))/d) + (2*B*sin(c + d*x))/(d*sqrt(cos(c + d*x)))],
[cos(c + d*x)^(-5/2)*(a*B + b*B*cos(c + d*x))/(a + b*cos(c + d*x)), x, 3, (2*B*EllipticF((1/2)*(c + d*x), 2))/(3*d) + (2*B*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2))],


[cos(c + d*x)^(5/2)*(a*B + b*B*cos(c + d*x))/(a + b*cos(c + d*x))^2, x, 7, -((2*a*B*EllipticE((1/2)*(c + d*x), 2))/(b^2*d)) + (2*(3*a^2 + b^2)*B*EllipticF((1/2)*(c + d*x), 2))/(3*b^3*d) - (2*a^3*B*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2))/(b^3*(a + b)*d) + (2*B*sqrt(cos(c + d*x))*sin(c + d*x))/(3*b*d)],
[cos(c + d*x)^(3/2)*(a*B + b*B*cos(c + d*x))/(a + b*cos(c + d*x))^2, x, 6, (2*B*EllipticE((1/2)*(c + d*x), 2))/(b*d) - (2*a*B*EllipticF((1/2)*(c + d*x), 2))/(b^2*d) + (2*a^2*B*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2))/(b^2*(a + b)*d)],
[cos(c + d*x)^(1/2)*(a*B + b*B*cos(c + d*x))/(a + b*cos(c + d*x))^2, x, 4, (2*B*EllipticF((1/2)*(c + d*x), 2))/(b*d) - (2*a*B*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2))/(b*(a + b)*d)],
[cos(c + d*x)^(-1/2)*(a*B + b*B*cos(c + d*x))/(a + b*cos(c + d*x))^2, x, 2, (2*B*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2))/((a + b)*d)],
[cos(c + d*x)^(-3/2)*(a*B + b*B*cos(c + d*x))/(a + b*cos(c + d*x))^2, x, 6, -((2*B*EllipticE((1/2)*(c + d*x), 2))/(a*d)) - (2*b*B*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2))/(a*(a + b)*d) + (2*B*sin(c + d*x))/(a*d*sqrt(cos(c + d*x)))],
[cos(c + d*x)^(-5/2)*(a*B + b*B*cos(c + d*x))/(a + b*cos(c + d*x))^2, x, 8, (2*b*B*EllipticE((1/2)*(c + d*x), 2))/(a^2*d) + (2*B*EllipticF((1/2)*(c + d*x), 2))/(3*a*d) + (2*b^2*B*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2))/(a^2*(a + b)*d) + (2*B*sin(c + d*x))/(3*a*d*cos(c + d*x)^(3/2)) - (2*b*B*sin(c + d*x))/(a^2*d*sqrt(cos(c + d*x)))],


# ::Subsection::Closed:: 
#Integrands of the form Cos[c+d x]^(m/2) (A+B Cos[c+d x]) (a+b Cos[c+d x])^(n/2)
#


# ::Subsubsection::Closed:: 
#n>0


[cos(c + d*x)^(3/2)*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^(1/2), x, 8, ((a + b)*(6*a*A*b - 3*a^2*B + 16*b^2*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(24*b^2*d*sqrt(a + b*cos(c + d*x))) + ((6*a*A*b - 12*A*b^2 + a^2*B - 14*a*b*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(12*b*d*sqrt(a + b*cos(c + d*x))) - ((2*a^2*A*b - 8*A*b^3 - a^3*B - 4*a*b^2*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(4*b^2*d*sqrt(a + b*cos(c + d*x))) + ((6*A*b + a*B)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(12*b*d) + (B*cos(c + d*x)^(3/2)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*d) + ((6*a*A*b - 3*a^2*B + 16*b^2*B)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(24*b^2*d)],
[cos(c + d*x)^(1/2)*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^(1/2), x, 7, ((a + b)*(4*A*b + a*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(4*b*d*sqrt(a + b*cos(c + d*x))) - ((4*a*A - a*B + 2*b*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(2*d*sqrt(a + b*cos(c + d*x))) + ((4*a*A*b - a^2*B + 4*b^2*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(2*b*d*sqrt(a + b*cos(c + d*x))) + (B*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(2*d) + ((4*A*b + a*B)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(4*b*d)],
[cos(c + d*x)^(-1/2)*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^(1/2), x, 7, ((a + b)*B*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(d*sqrt(a + b*cos(c + d*x))) + (2*(a*A - A*b - a*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(d*sqrt(a + b*cos(c + d*x))) + (2*(2*A*b + a*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(d*sqrt(a + b*cos(c + d*x))) + (B*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/d],
[cos(c + d*x)^(-3/2)*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^(1/2), x, 6, -((2*A*(a + b)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(d*sqrt(a + b*cos(c + d*x)))) + (2*(a*A + A*b + a*B - b*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(d*sqrt(a + b*cos(c + d*x))) + (4*b*B*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(d*sqrt(a + b*cos(c + d*x))) + (2*A*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(d*sqrt(cos(c + d*x)))],
[cos(c + d*x)^(-5/2)*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^(1/2), x, 5, -((2*(a + b)*(A*b + 3*a*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(3*a*d*sqrt(a + b*cos(c + d*x)))) + (2*(a*A + A*b + 3*a*B + 3*b*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(3*d*sqrt(a + b*cos(c + d*x))) + (2*A*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2)) + (2*(A*b + 3*a*B)*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(3*a*d*sqrt(cos(c + d*x)))],
[cos(c + d*x)^(-7/2)*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^(1/2), x, 6, -((2*(a + b)*(9*a^2*A - 2*A*b^2 + 5*a*b*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(15*a^2*d*sqrt(a + b*cos(c + d*x)))) + (2*(9*a^2*A + 7*a*A*b - 2*A*b^2 + 5*a^2*B + 5*a*b*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(15*a*d*sqrt(a + b*cos(c + d*x))) + (2*A*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2)) + (2*(A*b + 5*a*B)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(15*a*d*cos(c + d*x)^(3/2)) + (2*(9*a^2*A - 2*A*b^2 + 5*a*b*B)*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(15*a^2*d*sqrt(cos(c + d*x)))],
[cos(c + d*x)^(-9/2)*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^(1/2), x, 7, -((2*(a + b)*(19*a^2*A*b + 8*A*b^3 + 63*a^3*B - 14*a*b^2*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(105*a^3*d*sqrt(a + b*cos(c + d*x)))) + (2*(25*a^3*A + 19*a^2*A*b + 2*a*A*b^2 + 8*A*b^3 + 63*a^3*B + 49*a^2*b*B - 14*a*b^2*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(105*a^2*d*sqrt(a + b*cos(c + d*x))) + (2*A*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(7*d*cos(c + d*x)^(7/2)) + (2*(A*b + 7*a*B)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(35*a*d*cos(c + d*x)^(5/2)) + (2*(25*a^2*A - 4*A*b^2 + 7*a*b*B)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(105*a^2*d*cos(c + d*x)^(3/2)) + (2*(19*a^2*A*b + 8*A*b^3 + 63*a^3*B - 14*a*b^2*B)*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(105*a^3*d*sqrt(cos(c + d*x)))],


[cos(c + d*x)^(3/2)*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^(3/2), x, 9, ((a + b)*(24*a^2*A*b + 128*A*b^3 - 9*a^3*B + 156*a*b^2*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(192*b^2*d*sqrt(a + b*cos(c + d*x))) + ((56*a^2*A*b - 208*a*A*b^2 + 3*a^3*B - 114*a^2*b*B + 36*a*b^2*B - 72*b^3*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(96*b*d*sqrt(a + b*cos(c + d*x))) - ((8*a^3*A*b - 96*a*A*b^3 - 3*a^4*B - 24*a^2*b^2*B - 48*b^4*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(32*b^2*d*sqrt(a + b*cos(c + d*x))) + ((56*a*A*b + 3*a^2*B + 36*b^2*B)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(96*b*d) + ((8*A*b + 9*a*B)*cos(c + d*x)^(3/2)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(24*d) + (b*B*cos(c + d*x)^(5/2)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(4*d) + ((24*a^2*A*b + 128*A*b^3 - 9*a^3*B + 156*a*b^2*B)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(192*b^2*d)],
[cos(c + d*x)^(1/2)*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^(3/2), x, 8, ((a + b)*(30*a*A*b + 3*a^2*B + 16*b^2*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(24*b*d*sqrt(a + b*cos(c + d*x))) - ((24*a^2*A - 6*a*A*b + 12*A*b^2 - 7*a^2*B + 26*a*b*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(12*d*sqrt(a + b*cos(c + d*x))) + ((6*a^2*A*b + 8*A*b^3 - a^3*B + 12*a*b^2*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(4*b*d*sqrt(a + b*cos(c + d*x))) + ((6*A*b + 7*a*B)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(12*d) + (b*B*cos(c + d*x)^(3/2)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*d) + ((30*a*A*b + 3*a^2*B + 16*b^2*B)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(24*b*d)],
[cos(c + d*x)^(-1/2)*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^(3/2), x, 7, ((a + b)*(4*A*b + 5*a*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(4*d*sqrt(a + b*cos(c + d*x))) + ((4*a^2*A - 8*a*A*b - 4*a^2*B + a*b*B - 2*b^2*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(2*d*sqrt(a + b*cos(c + d*x))) + ((12*a*A*b + 3*a^2*B + 4*b^2*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(2*d*sqrt(a + b*cos(c + d*x))) + (b*B*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(2*d) + ((4*A*b + 5*a*B)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(4*d)],
[cos(c + d*x)^(-3/2)*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^(3/2), x, 7, -(((a + b)*(2*a*A - b*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(d*sqrt(a + b*cos(c + d*x)))) + (2*(a^2*A + 2*a*A*b - A*b^2 + a^2*B - 2*a*b*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(d*sqrt(a + b*cos(c + d*x))) + (2*b*(2*A*b + 3*a*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(d*sqrt(a + b*cos(c + d*x))) + (2*a*A*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(d*sqrt(cos(c + d*x))) - ((2*a*A - b*B)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/d],
[cos(c + d*x)^(-5/2)*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^(3/2), x, 7, -((2*(a + b)*(4*A*b + 3*a*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(3*d*sqrt(a + b*cos(c + d*x)))) + (2*(a^2*A + 4*a*A*b + 3*A*b^2 + 3*a^2*B + 6*a*b*B - 3*b^2*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(3*d*sqrt(a + b*cos(c + d*x))) + (4*b^2*B*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(d*sqrt(a + b*cos(c + d*x))) + (2*a*A*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2)) + (2*(4*A*b + 3*a*B)*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(3*d*sqrt(cos(c + d*x)))],
[cos(c + d*x)^(-7/2)*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^(3/2), x, 6, -((2*(a + b)*(9*a^2*A + 3*A*b^2 + 20*a*b*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(15*a*d*sqrt(a + b*cos(c + d*x)))) + (2*(9*a^2*A + 12*a*A*b + 3*A*b^2 + 5*a^2*B + 20*a*b*B + 15*b^2*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(15*d*sqrt(a + b*cos(c + d*x))) + (2*a*A*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2)) + (2*(6*A*b + 5*a*B)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(15*d*cos(c + d*x)^(3/2)) + (2*(9*a^2*A + 3*A*b^2 + 20*a*b*B)*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(15*a*d*sqrt(cos(c + d*x)))],
[cos(c + d*x)^(-9/2)*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^(3/2), x, 7, -((2*(a + b)*(82*a^2*A*b - 6*A*b^3 + 63*a^3*B + 21*a*b^2*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(105*a^2*d*sqrt(a + b*cos(c + d*x)))) + (2*(25*a^3*A + 82*a^2*A*b + 51*a*A*b^2 - 6*A*b^3 + 63*a^3*B + 84*a^2*b*B + 21*a*b^2*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(105*a*d*sqrt(a + b*cos(c + d*x))) + (2*a*A*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(7*d*cos(c + d*x)^(7/2)) + (2*(8*A*b + 7*a*B)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(35*d*cos(c + d*x)^(5/2)) + (2*(25*a^2*A + 3*A*b^2 + 42*a*b*B)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(105*a*d*cos(c + d*x)^(3/2)) + (2*(82*a^2*A*b - 6*A*b^3 + 63*a^3*B + 21*a*b^2*B)*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(105*a^2*d*sqrt(cos(c + d*x)))],
[cos(c + d*x)^(-11/2)*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^(3/2), x, 8, -((2*(a + b)*(147*a^4*A + 33*a^2*A*b^2 + 8*A*b^4 + 246*a^3*b*B - 18*a*b^3*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(315*a^3*d*sqrt(a + b*cos(c + d*x)))) + (2*(147*a^4*A + 186*a^3*A*b + 33*a^2*A*b^2 + 2*a*A*b^3 + 8*A*b^4 + 75*a^4*B + 246*a^3*b*B + 153*a^2*b^2*B - 18*a*b^3*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(315*a^2*d*sqrt(a + b*cos(c + d*x))) + (2*a*A*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(9*d*cos(c + d*x)^(9/2)) + (2*(10*A*b + 9*a*B)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(63*d*cos(c + d*x)^(7/2)) + (2*(49*a^2*A + 3*A*b^2 + 72*a*b*B)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(315*a*d*cos(c + d*x)^(5/2)) + (2*(88*a^2*A*b - 4*A*b^3 + 75*a^3*B + 9*a*b^2*B)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(315*a^2*d*cos(c + d*x)^(3/2)) + (2*(147*a^4*A + 33*a^2*A*b^2 + 8*A*b^4 + 246*a^3*b*B - 18*a*b^3*B)*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(315*a^3*d*sqrt(cos(c + d*x)))],


[cos(c + d*x)^(3/2)*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^(5/2), x, 10, ((a + b)*(150*a^3*A*b + 2840*a*A*b^3 - 45*a^4*B + 1692*a^2*b^2*B + 1024*b^4*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(1920*b^2*d*sqrt(a + b*cos(c + d*x))) + ((590*a^3*A*b - 3220*a^2*A*b^2 + 360*a*A*b^3 - 720*A*b^4 + 15*a^4*B - 1146*a^3*b*B + 772*a^2*b^2*B - 2312*a*b^3*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(960*b*d*sqrt(a + b*cos(c + d*x))) - ((10*a^4*A*b - 240*a^2*A*b^3 - 96*A*b^5 - 3*a^5*B - 40*a^3*b^2*B - 240*a*b^4*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(64*b^2*d*sqrt(a + b*cos(c + d*x))) + ((590*a^2*A*b + 360*A*b^3 + 15*a^3*B + 772*a*b^2*B)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(960*b*d) + ((170*a*A*b + 93*a^2*B + 64*b^2*B)*cos(c + d*x)^(3/2)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(240*d) + (b*(10*A*b + 13*a*B)*cos(c + d*x)^(5/2)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(40*d) + (b*B*cos(c + d*x)^(5/2)*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(5*d) + ((150*a^3*A*b + 2840*a*A*b^3 - 45*a^4*B + 1692*a^2*b^2*B + 1024*b^4*B)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(1920*b^2*d)],
[cos(c + d*x)^(1/2)*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^(5/2), x, 9, ((a + b)*(264*a^2*A*b + 128*A*b^3 + 15*a^3*B + 284*a*b^2*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(192*b*d*sqrt(a + b*cos(c + d*x))) - ((192*a^3*A - 104*a^2*A*b + 304*a*A*b^2 - 59*a^3*B + 322*a^2*b*B - 36*a*b^2*B + 72*b^3*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(96*d*sqrt(a + b*cos(c + d*x))) + ((40*a^3*A*b + 160*a*A*b^3 - 5*a^4*B + 120*a^2*b^2*B + 48*b^4*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(32*b*d*sqrt(a + b*cos(c + d*x))) + ((104*a*A*b + 59*a^2*B + 36*b^2*B)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(96*d) + (b*(8*A*b + 11*a*B)*cos(c + d*x)^(3/2)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(24*d) + (b*B*cos(c + d*x)^(3/2)*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(4*d) + ((264*a^2*A*b + 128*A*b^3 + 15*a^3*B + 284*a*b^2*B)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(192*b*d)],
[cos(c + d*x)^(-1/2)*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^(5/2), x, 8, ((a + b)*(54*a*A*b + 33*a^2*B + 16*b^2*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(24*d*sqrt(a + b*cos(c + d*x))) + ((24*a^3*A - 72*a^2*A*b + 6*a*A*b^2 - 12*A*b^3 - 24*a^3*B + 13*a^2*b*B - 38*a*b^2*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(12*d*sqrt(a + b*cos(c + d*x))) + ((30*a^2*A*b + 8*A*b^3 + 5*a^3*B + 20*a*b^2*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(4*d*sqrt(a + b*cos(c + d*x))) + (b*(2*A*b + 3*a*B)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(4*d) + (b*B*sqrt(cos(c + d*x))*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(3*d) + ((54*a*A*b + 33*a^2*B + 16*b^2*B)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(24*d)],
[cos(c + d*x)^(-3/2)*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^(5/2), x, 8, -(((a + b)*(8*a^2*A - 4*A*b^2 - 9*a*b*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(4*d*sqrt(a + b*cos(c + d*x)))) + ((4*a^3*A + 12*a^2*A*b - 12*a*A*b^2 + 4*a^3*B - 12*a^2*b*B + a*b^2*B - 2*b^3*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(2*d*sqrt(a + b*cos(c + d*x))) + (b*(20*a*A*b + 15*a^2*B + 4*b^2*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(2*d*sqrt(a + b*cos(c + d*x))) - (b*(4*a*A - b*B)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(2*d) + (2*a*A*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(d*sqrt(cos(c + d*x))) - ((8*a^2*A - 4*A*b^2 - 9*a*b*B)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(4*d)],
[cos(c + d*x)^(-5/2)*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^(5/2), x, 8, -(((a + b)*(14*a*A*b + 6*a^2*B - 3*b^2*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(3*d*sqrt(a + b*cos(c + d*x)))) + (2*(a^3*A + 7*a^2*A*b + 9*a*A*b^2 - 3*A*b^3 + 3*a^3*B + 9*a^2*b*B - 9*a*b^2*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(3*d*sqrt(a + b*cos(c + d*x))) + (2*b^2*(2*A*b + 5*a*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(d*sqrt(a + b*cos(c + d*x))) + (2*a*(2*A*b + a*B)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(d*sqrt(cos(c + d*x))) + (2*a*A*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2)) - ((14*a*A*b + 6*a^2*B - 3*b^2*B)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(3*d)],
[cos(c + d*x)^(-7/2)*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^(5/2), x, 8, -((2*(a + b)*(9*a^2*A + 23*A*b^2 + 35*a*b*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(15*d*sqrt(a + b*cos(c + d*x)))) + (2*(9*a^3*A + 17*a^2*A*b + 23*a*A*b^2 + 15*A*b^3 + 5*a^3*B + 35*a^2*b*B + 45*a*b^2*B - 15*b^3*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(15*d*sqrt(a + b*cos(c + d*x))) + (4*b^3*B*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(d*sqrt(a + b*cos(c + d*x))) + (2*a*(8*A*b + 5*a*B)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(15*d*cos(c + d*x)^(3/2)) + (2*a*A*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2)) + (2*(9*a^2*A + 23*A*b^2 + 35*a*b*B)*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(15*d*sqrt(cos(c + d*x)))],
[cos(c + d*x)^(-9/2)*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^(5/2), x, 7, -((2*(a + b)*(145*a^2*A*b + 15*A*b^3 + 63*a^3*B + 161*a*b^2*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(105*a*d*sqrt(a + b*cos(c + d*x)))) + (2*(25*a^3*A + 145*a^2*A*b + 135*a*A*b^2 + 15*A*b^3 + 63*a^3*B + 119*a^2*b*B + 161*a*b^2*B + 105*b^3*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(105*d*sqrt(a + b*cos(c + d*x))) + (2*a*(10*A*b + 7*a*B)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(35*d*cos(c + d*x)^(5/2)) + (2*(25*a^2*A + 45*A*b^2 + 77*a*b*B)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(105*d*cos(c + d*x)^(3/2)) + (2*a*A*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(7*d*cos(c + d*x)^(7/2)) + (2*(145*a^2*A*b + 15*A*b^3 + 63*a^3*B + 161*a*b^2*B)*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(105*a*d*sqrt(cos(c + d*x)))],
[cos(c + d*x)^(-11/2)*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^(5/2), x, 8, -((2*(a + b)*(147*a^4*A + 279*a^2*A*b^2 - 10*A*b^4 + 435*a^3*b*B + 45*a*b^3*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(315*a^2*d*sqrt(a + b*cos(c + d*x)))) + (2*(147*a^4*A + 261*a^3*A*b + 279*a^2*A*b^2 + 155*a*A*b^3 - 10*A*b^4 + 75*a^4*B + 435*a^3*b*B + 405*a^2*b^2*B + 45*a*b^3*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(315*a*d*sqrt(a + b*cos(c + d*x))) + (2*a*(4*A*b + 3*a*B)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(21*d*cos(c + d*x)^(7/2)) + (2*(49*a^2*A + 75*A*b^2 + 135*a*b*B)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(315*d*cos(c + d*x)^(5/2)) + (2*(163*a^2*A*b + 5*A*b^3 + 75*a^3*B + 135*a*b^2*B)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(315*a*d*cos(c + d*x)^(3/2)) + (2*a*A*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(9*d*cos(c + d*x)^(9/2)) + (2*(147*a^4*A + 279*a^2*A*b^2 - 10*A*b^4 + 435*a^3*b*B + 45*a*b^3*B)*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(315*a^2*d*sqrt(cos(c + d*x)))],
[cos(c + d*x)^(-13/2)*(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^(5/2), x, 9, -((2*(a + b)*(3705*a^4*A*b + 255*a^2*A*b^3 + 40*A*b^5 + 1617*a^5*B + 3069*a^3*b^2*B - 110*a*b^4*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(3465*a^3*d*sqrt(a + b*cos(c + d*x)))) + (1/(3465*a^2*d*sqrt(a + b*cos(c + d*x))))*(2*(675*a^5*A + 3705*a^4*A*b + 3315*a^3*A*b^2 + 255*a^2*A*b^3 + 10*a*A*b^4 + 40*A*b^5 + 1617*a^5*B + 2871*a^4*b*B + 3069*a^3*b^2*B + 1705*a^2*b^3*B - 110*a*b^4*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))) + (2*a*(14*A*b + 11*a*B)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(99*d*cos(c + d*x)^(9/2)) + (2*(81*a^2*A + 113*A*b^2 + 209*a*b*B)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(693*d*cos(c + d*x)^(7/2)) + (2*(1145*a^2*A*b + 15*A*b^3 + 539*a^3*B + 825*a*b^2*B)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3465*a*d*cos(c + d*x)^(5/2)) + (2*(675*a^4*A + 1025*a^2*A*b^2 - 20*A*b^4 + 1793*a^3*b*B + 55*a*b^3*B)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3465*a^2*d*cos(c + d*x)^(3/2)) + (2*a*A*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(11*d*cos(c + d*x)^(11/2)) + (2*(3705*a^4*A*b + 255*a^2*A*b^3 + 40*A*b^5 + 1617*a^5*B + 3069*a^3*b^2*B - 110*a*b^4*B)*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(3465*a^3*d*sqrt(cos(c + d*x)))],


[cos(c + d*x)^(-5/2)*(3*b*(B/(2*a)) + B*cos(c + d*x))*(a + b*cos(c + d*x))^(5/2), x, 7, -((2*(a + b)*(a^2 + 3*b^2)*B*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(d*sqrt(a + b*cos(c + d*x)))) + ((2*a^4 + 7*a^3*b + a^2*b^2 + 9*a*b^3 - 3*b^4)*B*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(a*d*sqrt(a + b*cos(c + d*x))) + (2*b^2*(5*a^2 + 3*b^2)*B*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(a*d*sqrt(a + b*cos(c + d*x))) + (b*B*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(d*cos(c + d*x)^(3/2)) + (2*(a^2 + 3*b^2)*B*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(d*sqrt(cos(c + d*x)))],


# ::Subsubsection::Closed:: 
#n<0


[cos(c + d*x)^(3/2)*(A + B*cos(c + d*x))/(a + b*cos(c + d*x))^(1/2), x, 7, ((a + b)*(4*A*b - 3*a*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(4*b^2*d*sqrt(a + b*cos(c + d*x))) + ((a - 2*b)*B*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(2*b*d*sqrt(a + b*cos(c + d*x))) - ((4*a*A*b - 3*a^2*B - 4*b^2*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(2*b^2*d*sqrt(a + b*cos(c + d*x))) + (B*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(2*b*d) + ((4*A*b - 3*a*B)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(4*b^2*d)],
[cos(c + d*x)^(1/2)*(A + B*cos(c + d*x))/(a + b*cos(c + d*x))^(1/2), x, 7, ((a + b)*B*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(b*d*sqrt(a + b*cos(c + d*x))) - (2*A*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(d*sqrt(a + b*cos(c + d*x))) + (2*(2*A*b - a*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(b*d*sqrt(a + b*cos(c + d*x))) + (B*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(b*d)],
[cos(c + d*x)^(-1/2)*(A + B*cos(c + d*x))/(a + b*cos(c + d*x))^(1/2), x, 3, (2*(A - B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(d*sqrt(a + b*cos(c + d*x))) + (4*B*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(d*sqrt(a + b*cos(c + d*x)))],
[cos(c + d*x)^(-3/2)*(A + B*cos(c + d*x))/(a + b*cos(c + d*x))^(1/2), x, 4, -((2*A*(a + b)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(a*d*sqrt(a + b*cos(c + d*x)))) + (2*(A + B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(d*sqrt(a + b*cos(c + d*x))) + (2*A*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(a*d*sqrt(cos(c + d*x)))],
[cos(c + d*x)^(-5/2)*(A + B*cos(c + d*x))/(a + b*cos(c + d*x))^(1/2), x, 5, (2*(a + b)*(2*A*b - 3*a*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(3*a^2*d*sqrt(a + b*cos(c + d*x))) + (2*(a*A - 2*A*b + 3*a*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(3*a*d*sqrt(a + b*cos(c + d*x))) + (2*A*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*a*d*cos(c + d*x)^(3/2)) - (2*(2*A*b - 3*a*B)*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(3*a^2*d*sqrt(cos(c + d*x)))],
[cos(c + d*x)^(-7/2)*(A + B*cos(c + d*x))/(a + b*cos(c + d*x))^(1/2), x, 6, -((2*(a + b)*(9*a^2*A + 8*A*b^2 - 10*a*b*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(15*a^3*d*sqrt(a + b*cos(c + d*x)))) + (2*(9*a^2*A + 2*a*A*b + 8*A*b^2 + 5*a^2*B - 10*a*b*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(15*a^2*d*sqrt(a + b*cos(c + d*x))) + (2*A*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(5*a*d*cos(c + d*x)^(5/2)) - (2*(4*A*b - 5*a*B)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(15*a^2*d*cos(c + d*x)^(3/2)) + (2*(9*a^2*A + 8*A*b^2 - 10*a*b*B)*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(15*a^3*d*sqrt(cos(c + d*x)))],


[cos(c + d*x)^(3/2)*(A + B*cos(c + d*x))/(a + b*cos(c + d*x))^(3/2), x, 7, -(((2*a*A*b - 3*a^2*B + b^2*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/((a - b)*b^2*d*sqrt(a + b*cos(c + d*x)))) + (2*(A*b - a*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/((a - b)*b*d*sqrt(a + b*cos(c + d*x))) + (2*(2*A*b - 3*a*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(b^2*d*sqrt(a + b*cos(c + d*x))) + (2*a*(A*b - a*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(b*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) - ((2*a*A*b - 3*a^2*B + b^2*B)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(b^2*(a^2 - b^2)*d)],
[cos(c + d*x)^(1/2)*(A + B*cos(c + d*x))/(a + b*cos(c + d*x))^(3/2), x, 6, (2*(A*b - a*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/((a - b)*b*d*sqrt(a + b*cos(c + d*x))) - (2*(A - B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/((a - b)*d*sqrt(a + b*cos(c + d*x))) + (4*B*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(b*d*sqrt(a + b*cos(c + d*x))) + (2*(A*b - a*B)*sqrt(cos(c + d*x))*tan((1/2)*(c + d*x)))/(b*(a + b)*d*sqrt(a + b*cos(c + d*x)))],
[cos(c + d*x)^(-1/2)*(A + B*cos(c + d*x))/(a + b*cos(c + d*x))^(3/2), x, 4, -((2*(A*b - a*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(a*(a - b)*d*sqrt(a + b*cos(c + d*x)))) + (2*(A - B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/((a - b)*d*sqrt(a + b*cos(c + d*x))) - (2*(A*b - a*B)*sqrt(cos(c + d*x))*tan((1/2)*(c + d*x)))/(a*(a + b)*d*sqrt(a + b*cos(c + d*x)))],
[cos(c + d*x)^(-3/2)*(A + B*cos(c + d*x))/(a + b*cos(c + d*x))^(3/2), x, 5, -((2*(a^2*A - 2*A*b^2 + a*b*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(a^2*(a - b)*d*sqrt(a + b*cos(c + d*x)))) + (2*(a*A - 2*A*b + a*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(a*(a - b)*d*sqrt(a + b*cos(c + d*x))) + (2*b*(A*b - a*B)*sin(c + d*x))/(a*(a^2 - b^2)*d*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))) + (2*(a^2*A - 2*A*b^2 + a*b*B)*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(a^2*(a^2 - b^2)*d*sqrt(cos(c + d*x)))],
[cos(c + d*x)^(-5/2)*(A + B*cos(c + d*x))/(a + b*cos(c + d*x))^(3/2), x, 6, (2*(5*a^2*A*b - 8*A*b^3 - 3*a^3*B + 6*a*b^2*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(3*a^3*(a - b)*d*sqrt(a + b*cos(c + d*x))) + (2*(a^2*A - 6*a*A*b + 8*A*b^2 + 3*a^2*B - 6*a*b*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(3*a^2*(a - b)*d*sqrt(a + b*cos(c + d*x))) + (2*b*(A*b - a*B)*sin(c + d*x))/(a*(a^2 - b^2)*d*cos(c + d*x)^(3/2)*sqrt(a + b*cos(c + d*x))) + (2*(a^2*A - 4*A*b^2 + 3*a*b*B)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*a^2*(a^2 - b^2)*d*cos(c + d*x)^(3/2)) - (2*(5*a^2*A*b - 8*A*b^3 - 3*a^3*B + 6*a*b^2*B)*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(3*a^3*(a^2 - b^2)*d*sqrt(cos(c + d*x)))],


[cos(c + d*x)^(5/2)*(A + B*cos(c + d*x))/(a + b*cos(c + d*x))^(5/2), x, 8, -(((6*a^3*A*b - 14*a*A*b^3 - 15*a^4*B + 26*a^2*b^2*B - 3*b^4*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(3*(a - b)^2*b^3*(a + b)*d*sqrt(a + b*cos(c + d*x)))) + (2*(2*a^2*A*b - 3*a*A*b^2 - 3*A*b^3 - 5*a^3*B + 3*a^2*b*B + 6*a*b^2*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(3*(a - b)^2*b^2*(a + b)*d*sqrt(a + b*cos(c + d*x))) + (2*(2*A*b - 5*a*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(b^3*d*sqrt(a + b*cos(c + d*x))) + (2*a*(A*b - a*B)*cos(c + d*x)^(3/2)*sin(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)) + (2*a*(2*a^2*A*b - 6*A*b^3 - 5*a^3*B + 9*a*b^2*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(3*b^2*(a^2 - b^2)^2*d*sqrt(a + b*cos(c + d*x))) - ((6*a^3*A*b - 14*a*A*b^3 - 15*a^4*B + 26*a^2*b^2*B - 3*b^4*B)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(3*b^3*(a^2 - b^2)^2*d)],
[cos(c + d*x)^(3/2)*(A + B*cos(c + d*x))/(a + b*cos(c + d*x))^(5/2), x, 7, -((2*(4*A*b^3 + 3*a^3*B - 7*a*b^2*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(3*(a - b)^2*b^2*(a + b)*d*sqrt(a + b*cos(c + d*x)))) + (2*(a*A*b + 3*A*b^2 + 2*a^2*B - 3*a*b*B - 3*b^2*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(3*(a - b)^2*b*(a + b)*d*sqrt(a + b*cos(c + d*x))) + (4*B*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(b^2*d*sqrt(a + b*cos(c + d*x))) + (2*a*(A*b - a*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)) - (2*(4*A*b^3 + 3*a^3*B - 7*a*b^2*B)*sqrt(cos(c + d*x))*tan((1/2)*(c + d*x)))/(3*(a - b)*b^2*(a + b)^2*d*sqrt(a + b*cos(c + d*x)))],
[cos(c + d*x)^(1/2)*(A + B*cos(c + d*x))/(a + b*cos(c + d*x))^(5/2), x, 5, (2*(3*a^2*A + A*b^2 - 4*a*b*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(3*a*(a - b)^2*(a + b)*d*sqrt(a + b*cos(c + d*x))) - (2*(3*a*A + A*b - a*B - 3*b*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(3*(a - b)^2*(a + b)*d*sqrt(a + b*cos(c + d*x))) - (2*(A*b - a*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(3*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)) + (2*(3*a^2*A + A*b^2 - 4*a*b*B)*sqrt(cos(c + d*x))*tan((1/2)*(c + d*x)))/(3*a*(a - b)*(a + b)^2*d*sqrt(a + b*cos(c + d*x)))],
[cos(c + d*x)^(-1/2)*(A + B*cos(c + d*x))/(a + b*cos(c + d*x))^(5/2), x, 5, -((2*(6*a^2*A*b - 2*A*b^3 - 3*a^3*B - a*b^2*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(3*a^2*(a - b)^2*(a + b)*d*sqrt(a + b*cos(c + d*x)))) + (2*(3*a^2*A + 3*a*A*b - 2*A*b^2 - 3*a^2*B - a*b*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(3*a*(a - b)^2*(a + b)*d*sqrt(a + b*cos(c + d*x))) + (2*b*(A*b - a*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(3*a*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)) - (2*(6*a^2*A*b - 2*A*b^3 - 3*a^3*B - a*b^2*B)*sqrt(cos(c + d*x))*tan((1/2)*(c + d*x)))/(3*a^2*(a - b)*(a + b)^2*d*sqrt(a + b*cos(c + d*x)))],
[cos(c + d*x)^(-3/2)*(A + B*cos(c + d*x))/(a + b*cos(c + d*x))^(5/2), x, 6, -((2*(3*a^4*A - 15*a^2*A*b^2 + 8*A*b^4 + 6*a^3*b*B - 2*a*b^3*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(3*a^3*(a - b)^2*(a + b)*d*sqrt(a + b*cos(c + d*x)))) + (2*(3*a^3*A - 9*a^2*A*b - 6*a*A*b^2 + 8*A*b^3 + 3*a^3*B + 3*a^2*b*B - 2*a*b^2*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(3*a^2*(a - b)^2*(a + b)*d*sqrt(a + b*cos(c + d*x))) + (2*b*(A*b - a*B)*sin(c + d*x))/(3*a*(a^2 - b^2)*d*sqrt(cos(c + d*x))*(a + b*cos(c + d*x))^(3/2)) + (2*b*(8*a^2*A*b - 4*A*b^3 - 5*a^3*B + a*b^2*B)*sin(c + d*x))/(3*a^2*(a^2 - b^2)^2*d*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))) + (2*(3*a^4*A - 15*a^2*A*b^2 + 8*A*b^4 + 6*a^3*b*B - 2*a*b^3*B)*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(3*a^3*(a^2 - b^2)^2*d*sqrt(cos(c + d*x)))],
[cos(c + d*x)^(-5/2)*(A + B*cos(c + d*x))/(a + b*cos(c + d*x))^(5/2), x, 7, (2*(8*a^4*A*b - 28*a^2*A*b^3 + 16*A*b^5 - 3*a^5*B + 15*a^3*b^2*B - 8*a*b^4*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(3*a^4*(a - b)^2*(a + b)*d*sqrt(a + b*cos(c + d*x))) + (2*(a^4*A - 9*a^3*A*b + 16*a^2*A*b^2 + 12*a*A*b^3 - 16*A*b^4 + 3*a^4*B - 9*a^3*b*B - 6*a^2*b^2*B + 8*a*b^3*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(3*a^3*(a - b)^2*(a + b)*d*sqrt(a + b*cos(c + d*x))) + (2*b*(A*b - a*B)*sin(c + d*x))/(3*a*(a^2 - b^2)*d*cos(c + d*x)^(3/2)*(a + b*cos(c + d*x))^(3/2)) + (2*b*(10*a^2*A*b - 6*A*b^3 - 7*a^3*B + 3*a*b^2*B)*sin(c + d*x))/(3*a^2*(a^2 - b^2)^2*d*cos(c + d*x)^(3/2)*sqrt(a + b*cos(c + d*x))) + (2*(a^4*A - 13*a^2*A*b^2 + 8*A*b^4 + 8*a^3*b*B - 4*a*b^3*B)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*a^3*(a^2 - b^2)^2*d*cos(c + d*x)^(3/2)) - (2*(8*a^4*A*b - 28*a^2*A*b^3 + 16*A*b^5 - 3*a^5*B + 15*a^3*b^2*B - 8*a*b^4*B)*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(3*a^4*(a^2 - b^2)^2*d*sqrt(cos(c + d*x)))],


[cos(c + d*x)^(3/2)*(a*B + b*B*cos(c + d*x))/(a + b*cos(c + d*x))^(3/2), x, 5, ((a + b)*B*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(b*d*sqrt(a + b*cos(c + d*x))) - (2*a*B*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(b*d*sqrt(a + b*cos(c + d*x))) + (B*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(b*d)],
[cos(c + d*x)^(1/2)*(a*B + b*B*cos(c + d*x))/(a + b*cos(c + d*x))^(3/2), x, 4, -((2*B*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(d*sqrt(a + b*cos(c + d*x)))) + (4*B*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(d*sqrt(a + b*cos(c + d*x)))],
[cos(c + d*x)^(-1/2)*(a*B + b*B*cos(c + d*x))/(a + b*cos(c + d*x))^(3/2), x, 2, (2*B*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(d*sqrt(a + b*cos(c + d*x)))],
[cos(c + d*x)^(-3/2)*(a*B + b*B*cos(c + d*x))/(a + b*cos(c + d*x))^(3/2), x, 5, -((2*(a + b)*B*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(a*d*sqrt(a + b*cos(c + d*x)))) + (2*B*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(d*sqrt(a + b*cos(c + d*x))) + (2*B*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(a*d*sqrt(cos(c + d*x)))],


# ::Subsection::Closed:: 
#Integrands of the form Cos[c+d x]^m (A+B Cos[c+d x]) (a+b Cos[c+d x])^(n/3)
#


# ::Subsubsection::Closed:: 
#n>0


[(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^(2/3), x, 1, (3/5)*Int(((1/3)*(5*a*A + 2*b*B) + (1/3)*(5*A*b + 2*a*B)*cos(c + d*x))/(a + b*cos(c + d*x))^(1/3), x) + (3*B*(a + b*cos(c + d*x))^(2/3)*sin(c + d*x))/(5*d)],
[(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^(1/3), x, 1, Int((a + b*cos(c + d*x))^(1/3)*(A + B*cos(c + d*x)), x)],


# ::Subsubsection::Closed:: 
#n<0


[(A + B*cos(c + d*x))/(a + b*cos(c + d*x))^(1/3), x, 0, Int((A + B*cos(c + d*x))/(a + b*cos(c + d*x))^(1/3), x)],
[(A + B*cos(c + d*x))/(a + b*cos(c + d*x))^(2/3), x, 0, Int((A + B*cos(c + d*x))/(a + b*cos(c + d*x))^(2/3), x)],


# ::Subsection::Closed:: 
#Integrands of the form Sec[c+d x]^(m/2) (A+B Cos[c+d x]) (a+b Cos[c+d x])^n


# ::Subsubsection::Closed:: 
#n>0


[(a + b*cos(c + d*x))*(A + B*cos(c + d*x))*sec(c + d*x)^(7/2), x, 7, -((2*(3*a*A + 5*b*B)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d)) + (2*(A*b + a*B)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) + (2*(3*a*A + 5*b*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d) + (2*(A*b + a*B)*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d) + (2*a*A*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d)],
[(a + b*cos(c + d*x))*(A + B*cos(c + d*x))*sec(c + d*x)^(5/2), x, 6, (-2*(A*b + a*B)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/d + (2*(a*A + 3*b*B)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*d) + (2*(A*b + a*B)*sqrt(sec(c + d*x))*sin(c + d*x))/d + (2*a*A*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d)],
[(a + b*cos(c + d*x))*(A + B*cos(c + d*x))*sec(c + d*x)^(3/2), x, 5, (-2*(a*A - b*B)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/d + (2*(A*b + a*B)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/d + (2*a*A*sqrt(sec(c + d*x))*sin(c + d*x))/d],
[(a + b*cos(c + d*x))*(A + B*cos(c + d*x))*sqrt(sec(c + d*x)), x, 5, (2*(A*b + a*B)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/d + (2*(3*a*A + b*B)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*d) + (2*b*B*sin(c + d*x))/(3*d*sqrt(sec(c + d*x)))],
[((a + b*cos(c + d*x))*(A + B*cos(c + d*x)))/sqrt(sec(c + d*x)), x, 6, (2*(5*a*A + 3*b*B)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*d) + (2*(A*b + a*B)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*d) + (2*b*B*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2)) + (2*(A*b + a*B)*sin(c + d*x))/(3*d*sqrt(sec(c + d*x)))],
[((a + b*cos(c + d*x))*(A + B*cos(c + d*x)))/sec(c + d*x)^(3/2), x, 7, (6*(A*b + a*B)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*d) + (2*(7*a*A + 5*b*B)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(21*d) + (2*b*B*sin(c + d*x))/(7*d*sec(c + d*x)^(5/2)) + (2*(A*b + a*B)*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2)) + (2*(7*a*A + 5*b*B)*sin(c + d*x))/(21*d*sqrt(sec(c + d*x)))],


[(a + b*cos(c + d*x))^2*(A + B*cos(c + d*x))*sec(c + d*x)^(9/2), x, 8, -((2*(6*a*A*b + 3*a^2*B + 5*b^2*B)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d)) + (2*(5*a^2*A + 7*A*b^2 + 14*a*b*B)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) + (2*(6*a*A*b + 3*a^2*B + 5*b^2*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d) + (2*(5*a^2*A + 7*A*b^2 + 14*a*b*B)*sec(c + d*x)^(3/2)*sin(c + d*x))/(21*d) + (2*a*(9*A*b + 7*a*B)*sec(c + d*x)^(5/2)*sin(c + d*x))/(35*d) + (2*a*A*(a + b*cos(c + d*x))*sec(c + d*x)^(7/2)*sin(c + d*x))/(7*d)],
[(a + b*cos(c + d*x))^2*(A + B*cos(c + d*x))*sec(c + d*x)^(7/2), x, 7, -((2*(3*a^2*A + 5*A*b^2 + 10*a*b*B)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d)) + (2*(2*a*A*b + a^2*B + 3*b^2*B)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) + (2*(3*a^2*A + 5*A*b^2 + 10*a*b*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d) + (2*a*(7*A*b + 5*a*B)*sec(c + d*x)^(3/2)*sin(c + d*x))/(15*d) + (2*a*A*(a + b*cos(c + d*x))*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d)],
[(a + b*cos(c + d*x))^2*(A + B*cos(c + d*x))*sec(c + d*x)^(5/2), x, 6, (-2*(2*a*A*b + a^2*B - b^2*B)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/d + (2*(a^2*A + 3*A*b^2 + 6*a*b*B)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*d) + (2*a*(5*A*b + 3*a*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(3*d) + (2*a*A*(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d)],
[(a + b*cos(c + d*x))^2*(A + B*cos(c + d*x))*sec(c + d*x)^(3/2), x, 6, (-2*(a^2*A - A*b^2 - 2*a*b*B)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/d + (2*(6*a*A*b + 3*a^2*B + b^2*B)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*d) - (2*b*(3*a*A - b*B)*sin(c + d*x))/(3*d*sqrt(sec(c + d*x))) + (2*a*A*(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*sin(c + d*x))/d],
[(a + b*cos(c + d*x))^2*(A + B*cos(c + d*x))*sqrt(sec(c + d*x)), x, 6, (2*(10*a*A*b + 5*a^2*B + 3*b^2*B)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*d) + (2*(3*a^2*A + A*b^2 + 2*a*b*B)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*d) + (2*b*(5*A*b + 7*a*B)*sin(c + d*x))/(15*d*sqrt(sec(c + d*x))) + (2*b*B*(a + b*cos(c + d*x))*sin(c + d*x))/(5*d*sqrt(sec(c + d*x)))],
[((a + b*cos(c + d*x))^2*(A + B*cos(c + d*x)))/sqrt(sec(c + d*x)), x, 7, (2*(5*a^2*A + 3*A*b^2 + 6*a*b*B)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*d) + (2*(14*a*A*b + 7*a^2*B + 5*b^2*B)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(21*d) + (2*b*(7*A*b + 9*a*B)*sin(c + d*x))/(35*d*sec(c + d*x)^(3/2)) + (2*b*B*(a + b*cos(c + d*x))*sin(c + d*x))/(7*d*sec(c + d*x)^(3/2)) + (2*(14*a*A*b + 7*a^2*B + 5*b^2*B)*sin(c + d*x))/(21*d*sqrt(sec(c + d*x)))],
[((a + b*cos(c + d*x))^2*(A + B*cos(c + d*x)))/sec(c + d*x)^(3/2), x, 8, (2*(18*a*A*b + 9*a^2*B + 7*b^2*B)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(15*d) + (2*(7*a^2*A + 5*A*b^2 + 10*a*b*B)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(21*d) + (2*b*(9*A*b + 11*a*B)*sin(c + d*x))/(63*d*sec(c + d*x)^(5/2)) + (2*b*B*(a + b*cos(c + d*x))*sin(c + d*x))/(9*d*sec(c + d*x)^(5/2)) + (2*(18*a*A*b + 9*a^2*B + 7*b^2*B)*sin(c + d*x))/(45*d*sec(c + d*x)^(3/2)) + (2*(7*a^2*A + 5*A*b^2 + 10*a*b*B)*sin(c + d*x))/(21*d*sqrt(sec(c + d*x)))],


[(a + b*cos(c + d*x))^3*(A + B*cos(c + d*x))*sec(c + d*x)^(11/2), x, 9, -((2*(7*a^3*A + 27*a*A*b^2 + 27*a^2*b*B + 15*b^3*B)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(15*d)) + (2*(15*a^2*A*b + 7*A*b^3 + 5*a^3*B + 21*a*b^2*B)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) + (2*(7*a^3*A + 27*a*A*b^2 + 27*a^2*b*B + 15*b^3*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(15*d) + (2*(15*a^2*A*b + 7*A*b^3 + 5*a^3*B + 21*a*b^2*B)*sec(c + d*x)^(3/2)*sin(c + d*x))/(21*d) + (2*a*(49*a^2*A + 89*A*b^2 + 144*a*b*B)*sec(c + d*x)^(5/2)*sin(c + d*x))/(315*d) + (2*a*(13*A*b + 9*a*B)*(a + b*cos(c + d*x))*sec(c + d*x)^(7/2)*sin(c + d*x))/(63*d) + (2*a*A*(a + b*cos(c + d*x))^2*sec(c + d*x)^(9/2)*sin(c + d*x))/(9*d)],
[(a + b*cos(c + d*x))^3*(A + B*cos(c + d*x))*sec(c + d*x)^(9/2), x, 8, -((2*(9*a^2*A*b + 5*A*b^3 + 3*a^3*B + 15*a*b^2*B)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d)) + (2*(5*a^3*A + 21*a*A*b^2 + 21*a^2*b*B + 21*b^3*B)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) + (2*(9*a^2*A*b + 5*A*b^3 + 3*a^3*B + 15*a*b^2*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d) + (2*a*(25*a^2*A + 57*A*b^2 + 84*a*b*B)*sec(c + d*x)^(3/2)*sin(c + d*x))/(105*d) + (2*a*(11*A*b + 7*a*B)*(a + b*cos(c + d*x))*sec(c + d*x)^(5/2)*sin(c + d*x))/(35*d) + (2*a*A*(a + b*cos(c + d*x))^2*sec(c + d*x)^(7/2)*sin(c + d*x))/(7*d)],
[(a + b*cos(c + d*x))^3*(A + B*cos(c + d*x))*sec(c + d*x)^(7/2), x, 7, -((2*(3*a^3*A + 15*a*A*b^2 + 15*a^2*b*B - 5*b^3*B)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d)) + (2*(3*a^2*A*b + 3*A*b^3 + a^3*B + 9*a*b^2*B)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) + (2*a*(9*a^2*A + 33*A*b^2 + 40*a*b*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(15*d) + (2*a*(9*A*b + 5*a*B)*(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(15*d) + (2*a*A*(a + b*cos(c + d*x))^2*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d)],
[(a + b*cos(c + d*x))^3*(A + B*cos(c + d*x))*sec(c + d*x)^(5/2), x, 7, (-2*(3*a^2*A*b - A*b^3 + a^3*B - 3*a*b^2*B)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/d + (2*(a^3*A + 9*a*A*b^2 + 9*a^2*b*B + b^3*B)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*d) - (2*b*(8*a*A*b + 3*a^2*B - b^2*B)*sin(c + d*x))/(3*d*sqrt(sec(c + d*x))) + (2*a*(7*A*b + 3*a*B)*(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*sin(c + d*x))/(3*d) + (2*a*A*(a + b*cos(c + d*x))^2*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d)],
[(a + b*cos(c + d*x))^3*(A + B*cos(c + d*x))*sec(c + d*x)^(3/2), x, 7, (-2*(5*a^3*A - 15*a*A*b^2 - 15*a^2*b*B - 3*b^3*B)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*d) + (2*(9*a^2*A*b + A*b^3 + 3*a^3*B + 3*a*b^2*B)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*d) - (2*b*(15*a^2*A - 5*A*b^2 - 12*a*b*B)*sin(c + d*x))/(15*d*sqrt(sec(c + d*x))) - (2*b*(5*a*A - b*B)*(a + b*cos(c + d*x))*sin(c + d*x))/(5*d*sqrt(sec(c + d*x))) + (2*a*A*(a + b*cos(c + d*x))^2*sqrt(sec(c + d*x))*sin(c + d*x))/d],
[(a + b*cos(c + d*x))^3*(A + B*cos(c + d*x))*sqrt(sec(c + d*x)), x, 7, (2*(15*a^2*A*b + 3*A*b^3 + 5*a^3*B + 9*a*b^2*B)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*d) + (2*(21*a^3*A + 21*a*A*b^2 + 21*a^2*b*B + 5*b^3*B)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(21*d) + (2*b*(84*a*A*b + 57*a^2*B + 25*b^2*B)*sin(c + d*x))/(105*d*sqrt(sec(c + d*x))) + (2*b*(7*A*b + 11*a*B)*(a + b*cos(c + d*x))*sin(c + d*x))/(35*d*sqrt(sec(c + d*x))) + (2*b*B*(a + b*cos(c + d*x))^2*sin(c + d*x))/(7*d*sqrt(sec(c + d*x)))],
[((a + b*cos(c + d*x))^3*(A + B*cos(c + d*x)))/sqrt(sec(c + d*x)), x, 8, (2*(15*a^3*A + 27*a*A*b^2 + 27*a^2*b*B + 7*b^3*B)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(15*d) + (2*(21*a^2*A*b + 5*A*b^3 + 7*a^3*B + 15*a*b^2*B)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(21*d) + (2*b*(144*a*A*b + 89*a^2*B + 49*b^2*B)*sin(c + d*x))/(315*d*sec(c + d*x)^(3/2)) + (2*b*(9*A*b + 13*a*B)*(a + b*cos(c + d*x))*sin(c + d*x))/(63*d*sec(c + d*x)^(3/2)) + (2*b*B*(a + b*cos(c + d*x))^2*sin(c + d*x))/(9*d*sec(c + d*x)^(3/2)) + (2*(21*a^2*A*b + 5*A*b^3 + 7*a^3*B + 15*a*b^2*B)*sin(c + d*x))/(21*d*sqrt(sec(c + d*x)))],
[((a + b*cos(c + d*x))^3*(A + B*cos(c + d*x)))/sec(c + d*x)^(3/2), x, 9, (2*(27*a^2*A*b + 7*A*b^3 + 9*a^3*B + 21*a*b^2*B)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(15*d) + (2*(77*a^3*A + 165*a*A*b^2 + 165*a^2*b*B + 45*b^3*B)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(231*d) + (2*b*(220*a*A*b + 129*a^2*B + 81*b^2*B)*sin(c + d*x))/(693*d*sec(c + d*x)^(5/2)) + (2*b*(11*A*b + 15*a*B)*(a + b*cos(c + d*x))*sin(c + d*x))/(99*d*sec(c + d*x)^(5/2)) + (2*b*B*(a + b*cos(c + d*x))^2*sin(c + d*x))/(11*d*sec(c + d*x)^(5/2)) + (2*(27*a^2*A*b + 7*A*b^3 + 9*a^3*B + 21*a*b^2*B)*sin(c + d*x))/(45*d*sec(c + d*x)^(3/2)) + (2*(77*a^3*A + 165*a*A*b^2 + 165*a^2*b*B + 45*b^3*B)*sin(c + d*x))/(231*d*sqrt(sec(c + d*x)))],


# ::Subsubsection::Closed:: 
#n<0


[((A + B*cos(c + d*x))*sec(c + d*x)^(5/2))/(a + b*cos(c + d*x)), x, 8, (2*(A*b - a*B)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(a^2*d) + (2*A*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*a*d) + (2*b*(A*b - a*B)*sqrt(cos(c + d*x))*EllipticPi((2*b)/(a + b), (c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(a^2*(a + b)*d) - (2*(A*b - a*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(a^2*d) + (2*A*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*a*d)],
[((A + B*cos(c + d*x))*sec(c + d*x)^(3/2))/(a + b*cos(c + d*x)), x, 6, (-2*A*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(a*d) - (2*(A*b - a*B)*sqrt(cos(c + d*x))*EllipticPi((2*b)/(a + b), (c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(a*(a + b)*d) + (2*A*sqrt(sec(c + d*x))*sin(c + d*x))/(a*d)],
[((A + B*cos(c + d*x))*sqrt(sec(c + d*x)))/(a + b*cos(c + d*x)), x, 4, (2*B*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(b*d) + (2*(A*b - a*B)*sqrt(cos(c + d*x))*EllipticPi((2*b)/(a + b), (c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(b*(a + b)*d)],
[(A + B*cos(c + d*x))/((a + b*cos(c + d*x))*sqrt(sec(c + d*x))), x, 6, (2*B*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(b*d) + (2*(A*b - a*B)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(b^2*d) - (2*a*(A*b - a*B)*sqrt(cos(c + d*x))*EllipticPi((2*b)/(a + b), (c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(b^2*(a + b)*d)],
[(A + B*cos(c + d*x))/((a + b*cos(c + d*x))*sec(c + d*x)^(3/2)), x, 7, (2*(A*b - a*B)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(b^2*d) - (2*(3*a*A*b - 3*a^2*B - b^2*B)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*b^3*d) + (2*a^2*(A*b - a*B)*sqrt(cos(c + d*x))*EllipticPi((2*b)/(a + b), (c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(b^3*(a + b)*d) + (2*B*sin(c + d*x))/(3*b*d*sqrt(sec(c + d*x)))],
[(A + B*cos(c + d*x))/((a + b*cos(c + d*x))*sec(c + d*x)^(5/2)), x, 8, (-2*(5*a*A*b - 5*a^2*B - 3*b^2*B)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*b^3*d) + (2*(3*a^2*A*b + A*b^3 - 3*a^3*B - a*b^2*B)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*b^4*d) - (2*a^3*(A*b - a*B)*sqrt(cos(c + d*x))*EllipticPi((2*b)/(a + b), (c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(b^4*(a + b)*d) + (2*B*sin(c + d*x))/(5*b*d*sec(c + d*x)^(3/2)) + (2*(A*b - a*B)*sin(c + d*x))/(3*b^2*d*sqrt(sec(c + d*x)))],


[((A + B*cos(c + d*x))*sec(c + d*x)^(5/2))/(a + b*cos(c + d*x))^2, x, 9, ((4*a^2*A*b - 5*A*b^3 - 2*a^3*B + 3*a*b^2*B)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(a^3*(a^2 - b^2)*d) + ((2*a^2*A - 5*A*b^2 + 3*a*b*B)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*a^2*(a^2 - b^2)*d) + (b*(7*a^2*A*b - 5*A*b^3 - 5*a^3*B + 3*a*b^2*B)*sqrt(cos(c + d*x))*EllipticPi((2*b)/(a + b), (c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(a^3*(a - b)*(a + b)^2*d) - ((4*a^2*A*b - 5*A*b^3 - 2*a^3*B + 3*a*b^2*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(a^3*(a^2 - b^2)*d) + ((2*a^2*A - 5*A*b^2 + 3*a*b*B)*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*a^2*(a^2 - b^2)*d) + (b*(A*b - a*B)*sec(c + d*x)^(3/2)*sin(c + d*x))/(a*(a^2 - b^2)*d*(a + b*cos(c + d*x)))],
[((A + B*cos(c + d*x))*sec(c + d*x)^(3/2))/(a + b*cos(c + d*x))^2, x, 8, -(((2*a^2*A - 3*A*b^2 + a*b*B)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(a^2*(a^2 - b^2)*d)) + ((A*b - a*B)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(a*(a^2 - b^2)*d) - ((5*a^2*A*b - 3*A*b^3 - 3*a^3*B + a*b^2*B)*sqrt(cos(c + d*x))*EllipticPi((2*b)/(a + b), (c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(a^2*(a - b)*(a + b)^2*d) + ((2*a^2*A - 3*A*b^2 + a*b*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(a^2*(a^2 - b^2)*d) + (b*(A*b - a*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(a*(a^2 - b^2)*d*(a + b*cos(c + d*x)))],
[((A + B*cos(c + d*x))*sqrt(sec(c + d*x)))/(a + b*cos(c + d*x))^2, x, 7, -(((A*b - a*B)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(a*(a^2 - b^2)*d)) - ((A*b - a*B)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(b*(a^2 - b^2)*d) + ((3*a^2*A*b - A*b^3 - a^3*B - a*b^2*B)*sqrt(cos(c + d*x))*EllipticPi((2*b)/(a + b), (c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(a*(a - b)*b*(a + b)^2*d) + (b*(A*b - a*B)*sin(c + d*x))/(a*(a^2 - b^2)*d*(a + b*cos(c + d*x))*sqrt(sec(c + d*x)))],
[(A + B*cos(c + d*x))/((a + b*cos(c + d*x))^2*sqrt(sec(c + d*x))), x, 7, ((A*b - a*B)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(b*(a^2 - b^2)*d) + ((a*A*b + a^2*B - 2*b^2*B)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(b^2*(a^2 - b^2)*d) - ((a^2*A*b + A*b^3 + a^3*B - 3*a*b^2*B)*sqrt(cos(c + d*x))*EllipticPi((2*b)/(a + b), (c + d*x)/2, 2)*sqrt(sec(c + d*x)))/((a - b)*b^2*(a + b)^2*d) - ((A*b - a*B)*sin(c + d*x))/((a^2 - b^2)*d*(a + b*cos(c + d*x))*sqrt(sec(c + d*x)))],
[(A + B*cos(c + d*x))/((a + b*cos(c + d*x))^2*sec(c + d*x)^(3/2)), x, 7, -(((a*A*b - 3*a^2*B + 2*b^2*B)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(b^2*(a^2 - b^2)*d)) + ((a^2*A*b - 2*A*b^3 - 3*a^3*B + 4*a*b^2*B)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(b^3*(a^2 - b^2)*d) - (a*(a^2*A*b - 3*A*b^3 - 3*a^3*B + 5*a*b^2*B)*sqrt(cos(c + d*x))*EllipticPi((2*b)/(a + b), (c + d*x)/2, 2)*sqrt(sec(c + d*x)))/((a - b)*b^3*(a + b)^2*d) + (a*(A*b - a*B)*sin(c + d*x))/(b*(a^2 - b^2)*d*(a + b*cos(c + d*x))*sqrt(sec(c + d*x)))],
[(A + B*cos(c + d*x))/((a + b*cos(c + d*x))^2*sec(c + d*x)^(5/2)), x, 8, ((3*a^2*A*b - 2*A*b^3 - 5*a^3*B + 4*a*b^2*B)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(b^3*(a^2 - b^2)*d) - ((9*a^3*A*b - 12*a*A*b^3 - 15*a^4*B + 16*a^2*b^2*B + 2*b^4*B)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*b^4*(a^2 - b^2)*d) + (a^2*(3*a^2*A*b - 5*A*b^3 - 5*a^3*B + 7*a*b^2*B)*sqrt(cos(c + d*x))*EllipticPi((2*b)/(a + b), (c + d*x)/2, 2)*sqrt(sec(c + d*x)))/((a - b)*b^4*(a + b)^2*d) + (a*(A*b - a*B)*sin(c + d*x))/(b*(a^2 - b^2)*d*(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)) - ((3*a*A*b - 5*a^2*B + 2*b^2*B)*sin(c + d*x))/(3*b^2*(a^2 - b^2)*d*sqrt(sec(c + d*x)))],


[((A + B*cos(c + d*x))*sec(c + d*x)^(5/2))/(a + b*cos(c + d*x))^3, x, 10, ((24*a^4*A*b - 65*a^2*A*b^3 + 35*A*b^5 - 8*a^5*B + 29*a^3*b^2*B - 15*a*b^4*B)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(4*a^4*(a^2 - b^2)^2*d) + ((8*a^4*A - 61*a^2*A*b^2 + 35*A*b^4 + 33*a^3*b*B - 15*a*b^3*B)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(12*a^3*(a^2 - b^2)^2*d) + (b*(63*a^4*A*b - 86*a^2*A*b^3 + 35*A*b^5 - 35*a^5*B + 38*a^3*b^2*B - 15*a*b^4*B)*sqrt(cos(c + d*x))*EllipticPi((2*b)/(a + b), (c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(4*a^4*(a - b)^2*(a + b)^3*d) - ((24*a^4*A*b - 65*a^2*A*b^3 + 35*A*b^5 - 8*a^5*B + 29*a^3*b^2*B - 15*a*b^4*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(4*a^4*(a^2 - b^2)^2*d) + ((8*a^4*A - 61*a^2*A*b^2 + 35*A*b^4 + 33*a^3*b*B - 15*a*b^3*B)*sec(c + d*x)^(3/2)*sin(c + d*x))/(12*a^3*(a^2 - b^2)^2*d) + (b*(A*b - a*B)*sec(c + d*x)^(3/2)*sin(c + d*x))/(2*a*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) + (b*(13*a^2*A*b - 7*A*b^3 - 9*a^3*B + 3*a*b^2*B)*sec(c + d*x)^(3/2)*sin(c + d*x))/(4*a^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))],
[((A + B*cos(c + d*x))*sec(c + d*x)^(3/2))/(a + b*cos(c + d*x))^3, x, 9, -((8*a^4*A - 29*a^2*A*b^2 + 15*A*b^4 + 9*a^3*b*B - 3*a*b^3*B)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(4*a^3*(a^2 - b^2)^2*d) + ((11*a^2*A*b - 5*A*b^3 - 7*a^3*B + a*b^2*B)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(4*a^2*(a^2 - b^2)^2*d) - ((35*a^4*A*b - 38*a^2*A*b^3 + 15*A*b^5 - 15*a^5*B + 6*a^3*b^2*B - 3*a*b^4*B)*sqrt(cos(c + d*x))*EllipticPi((2*b)/(a + b), (c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(4*a^3*(a - b)^2*(a + b)^3*d) + ((8*a^4*A - 29*a^2*A*b^2 + 15*A*b^4 + 9*a^3*b*B - 3*a*b^3*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(4*a^3*(a^2 - b^2)^2*d) + (b*(A*b - a*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(2*a*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) + (b*(11*a^2*A*b - 5*A*b^3 - 7*a^3*B + a*b^2*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(4*a^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))],
[((A + B*cos(c + d*x))*sqrt(sec(c + d*x)))/(a + b*cos(c + d*x))^3, x, 8, -((9*a^2*A*b - 3*A*b^3 - 5*a^3*B - a*b^2*B)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(4*a^2*(a^2 - b^2)^2*d) - ((7*a^2*A*b - A*b^3 - 3*a^3*B - 3*a*b^2*B)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(4*a*b*(a^2 - b^2)^2*d) + ((15*a^4*A*b - 6*a^2*A*b^3 + 3*A*b^5 - 3*a^5*B - 10*a^3*b^2*B + a*b^4*B)*sqrt(cos(c + d*x))*EllipticPi((2*b)/(a + b), (c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(4*a^2*(a - b)^2*b*(a + b)^3*d) + (b*(A*b - a*B)*sin(c + d*x))/(2*a*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2*sqrt(sec(c + d*x))) + (b*(9*a^2*A*b - 3*A*b^3 - 5*a^3*B - a*b^2*B)*sin(c + d*x))/(4*a^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x))*sqrt(sec(c + d*x)))],
[(A + B*cos(c + d*x))/((a + b*cos(c + d*x))^3*sqrt(sec(c + d*x))), x, 8, ((5*a^2*A*b + A*b^3 - a^3*B - 5*a*b^2*B)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(4*a*b*(a^2 - b^2)^2*d) + ((3*a^2*A*b + 3*A*b^3 + a^3*B - 7*a*b^2*B)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(4*b^2*(a^2 - b^2)^2*d) - ((3*a^4*A*b + 10*a^2*A*b^3 - A*b^5 + a^5*B - 10*a^3*b^2*B - 3*a*b^4*B)*sqrt(cos(c + d*x))*EllipticPi((2*b)/(a + b), (c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(4*a*(a - b)^2*b^2*(a + b)^3*d) - ((A*b - a*B)*sin(c + d*x))/(2*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2*sqrt(sec(c + d*x))) - ((5*a^2*A*b + A*b^3 - a^3*B - 5*a*b^2*B)*sin(c + d*x))/(4*a*(a^2 - b^2)^2*d*(a + b*cos(c + d*x))*sqrt(sec(c + d*x)))],
[(A + B*cos(c + d*x))/((a + b*cos(c + d*x))^3*sec(c + d*x)^(3/2)), x, 8, -((a^2*A*b + 5*A*b^3 + 3*a^3*B - 9*a*b^2*B)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(4*b^2*(a^2 - b^2)^2*d) + ((a^3*A*b - 7*a*A*b^3 + 3*a^4*B - 5*a^2*b^2*B + 8*b^4*B)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(4*b^3*(a^2 - b^2)^2*d) - ((a^4*A*b - 10*a^2*A*b^3 - 3*A*b^5 + 3*a^5*B - 6*a^3*b^2*B + 15*a*b^4*B)*sqrt(cos(c + d*x))*EllipticPi((2*b)/(a + b), (c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(4*(a - b)^2*b^3*(a + b)^3*d) + (a*(A*b - a*B)*sin(c + d*x))/(2*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2*sqrt(sec(c + d*x))) + ((a^2*A*b + 5*A*b^3 + 3*a^3*B - 9*a*b^2*B)*sin(c + d*x))/(4*b*(a^2 - b^2)^2*d*(a + b*cos(c + d*x))*sqrt(sec(c + d*x)))],
[(A + B*cos(c + d*x))/((a + b*cos(c + d*x))^3*sec(c + d*x)^(5/2)), x, 8, -((3*a^3*A*b - 9*a*A*b^3 - 15*a^4*B + 29*a^2*b^2*B - 8*b^4*B)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(4*b^3*(a^2 - b^2)^2*d) + ((3*a^4*A*b - 5*a^2*A*b^3 + 8*A*b^5 - 15*a^5*B + 33*a^3*b^2*B - 24*a*b^4*B)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(4*b^4*(a^2 - b^2)^2*d) - (a*(3*a^4*A*b - 6*a^2*A*b^3 + 15*A*b^5 - 15*a^5*B + 38*a^3*b^2*B - 35*a*b^4*B)*sqrt(cos(c + d*x))*EllipticPi((2*b)/(a + b), (c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(4*(a - b)^2*b^4*(a + b)^3*d) + (a*(A*b - a*B)*sin(c + d*x))/(2*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2*sec(c + d*x)^(3/2)) + (a*(a^2*A*b - 7*A*b^3 - 5*a^3*B + 11*a*b^2*B)*sin(c + d*x))/(4*b^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x))*sqrt(sec(c + d*x)))],
[(A + B*cos(c + d*x))/((a + b*cos(c + d*x))^3*sec(c + d*x)^(7/2)), x, 9, ((15*a^4*A*b - 29*a^2*A*b^3 + 8*A*b^5 - 35*a^5*B + 65*a^3*b^2*B - 24*a*b^4*B)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(4*b^4*(a^2 - b^2)^2*d) - ((45*a^5*A*b - 99*a^3*A*b^3 + 72*a*A*b^5 - 105*a^6*B + 223*a^4*b^2*B - 128*a^2*b^4*B - 8*b^6*B)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(12*b^5*(a^2 - b^2)^2*d) + (a^2*(15*a^4*A*b - 38*a^2*A*b^3 + 35*A*b^5 - 35*a^5*B + 86*a^3*b^2*B - 63*a*b^4*B)*sqrt(cos(c + d*x))*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(4*(a - b)^2*b^5*(a + b)^3*d) + (a*(A*b - a*B)*sin(c + d*x))/(2*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2*sec(c + d*x)^(5/2)) + (a*(3*a^2*A*b - 9*A*b^3 - 7*a^3*B + 13*a*b^2*B)*sin(c + d*x))/(4*b^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)) - ((15*a^3*A*b - 33*a*A*b^3 - 35*a^4*B + 61*a^2*b^2*B - 8*b^4*B)*sin(c + d*x))/(12*b^3*(a^2 - b^2)^2*d*sqrt(sec(c + d*x)))],


[((a*B + b*B*cos(c + d*x))*sec(c + d*x)^(5/2))/(a + b*cos(c + d*x)), x, 4, (2*B*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) + (2*B*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d)],
[((a*B + b*B*cos(c + d*x))*sec(c + d*x)^(3/2))/(a + b*cos(c + d*x)), x, 4, -((2*B*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/d) + (2*B*sqrt(sec(c + d*x))*sin(c + d*x))/d],
[((a*B + b*B*cos(c + d*x))*sqrt(sec(c + d*x)))/(a + b*cos(c + d*x)), x, 3, (2*B*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/d],
[(a*B + b*B*cos(c + d*x))/((a + b*cos(c + d*x))*sqrt(sec(c + d*x))), x, 3, (2*B*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/d],
[(a*B + b*B*cos(c + d*x))/((a + b*cos(c + d*x))*sec(c + d*x)^(3/2)), x, 4, (2*B*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) + (2*B*sin(c + d*x))/(3*d*sqrt(sec(c + d*x)))],
[(a*B + b*B*cos(c + d*x))/((a + b*cos(c + d*x))*sec(c + d*x)^(5/2)), x, 4, (6*B*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d) + (2*B*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2))],


# ::Subsection::Closed:: 
#Integrands of the form Sec[c+d x]^(m/2) (A+B Cos[c+d x]) (a+b Cos[c+d x])^(n/2)
#


# ::Subsubsection::Closed:: 
#n>0


[sqrt(a + b*cos(c + d*x))*(A + B*cos(c + d*x))*sec(c + d*x)^(9/2), x, 8, -((2*(a + b)*(19*a^2*A*b + 8*A*b^3 + 63*a^3*B - 14*a*b^2*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(105*a^3*d*sqrt(a + b*cos(c + d*x)))) + (2*(25*a^3*A + 19*a^2*A*b + 2*a*A*b^2 + 8*A*b^3 + 63*a^3*B + 49*a^2*b*B - 14*a*b^2*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(105*a^2*d*sqrt(a + b*cos(c + d*x))) + (2*(25*a^2*A - 4*A*b^2 + 7*a*b*B)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(105*a^2*d) + (2*(A*b + 7*a*B)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(5/2)*sin(c + d*x))/(35*a*d) + (2*A*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(7/2)*sin(c + d*x))/(7*d) + (2*(19*a^2*A*b + 8*A*b^3 + 63*a^3*B - 14*a*b^2*B)*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*tan((1/2)*(c + d*x)))/(105*a^3*d)],
[sqrt(a + b*cos(c + d*x))*(A + B*cos(c + d*x))*sec(c + d*x)^(7/2), x, 7, -((2*(a + b)*(9*a^2*A - 2*A*b^2 + 5*a*b*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(15*a^2*d*sqrt(a + b*cos(c + d*x)))) + (2*(9*a^2*A + 7*a*A*b - 2*A*b^2 + 5*a^2*B + 5*a*b*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(15*a*d*sqrt(a + b*cos(c + d*x))) + (2*(A*b + 5*a*B)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(15*a*d) + (2*A*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d) + (2*(9*a^2*A - 2*A*b^2 + 5*a*b*B)*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*tan((1/2)*(c + d*x)))/(15*a^2*d)],
[sqrt(a + b*cos(c + d*x))*(A + B*cos(c + d*x))*sec(c + d*x)^(5/2), x, 6, (-2*(a + b)*(A*b + 3*a*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(3*a*d*sqrt(a + b*cos(c + d*x))) + (2*(a*A + A*b + 3*a*B + 3*b*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(3*d*sqrt(a + b*cos(c + d*x))) + (2*A*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d) + (2*(A*b + 3*a*B)*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*tan((c + d*x)/2))/(3*a*d)],
[sqrt(a + b*cos(c + d*x))*(A + B*cos(c + d*x))*sec(c + d*x)^(3/2), x, 7, -((2*A*(a + b)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*cos(c + d*x)))) + (2*(a*A + A*b + a*B - b*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*cos(c + d*x))) + (4*b*B*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*cos(c + d*x))) + (2*A*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*tan((1/2)*(c + d*x)))/d],
[sqrt(a + b*cos(c + d*x))*(A + B*cos(c + d*x))*sec(c + d*x)^(1/2), x, 8, ((a + b)*B*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*cos(c + d*x))) + (2*(a*A - A*b - a*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*cos(c + d*x))) + (2*(2*A*b + a*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*cos(c + d*x))) + (B*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(d*sqrt(sec(c + d*x)))],
[sqrt(a + b*cos(c + d*x))*(A + B*cos(c + d*x))/sec(c + d*x)^(1/2), x, 8, ((a + b)*(4*A*b + a*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(4*b*d*sqrt(a + b*cos(c + d*x))) - ((4*a*A - a*B + 2*b*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(2*d*sqrt(a + b*cos(c + d*x))) + ((4*a*A*b - a^2*B + 4*b^2*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(2*b*d*sqrt(a + b*cos(c + d*x))) + (B*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(2*d*sqrt(sec(c + d*x))) + ((4*A*b + a*B)*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(4*b*d*sqrt(sec(c + d*x)))],
[sqrt(a + b*cos(c + d*x))*(A + B*cos(c + d*x))/sec(c + d*x)^(3/2), x, 9, ((a + b)*(6*a*A*b - 3*a^2*B + 16*b^2*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(24*b^2*d*sqrt(a + b*cos(c + d*x))) + ((6*a*A*b - 12*A*b^2 + a^2*B - 14*a*b*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(12*b*d*sqrt(a + b*cos(c + d*x))) - ((2*a^2*A*b - 8*A*b^3 - a^3*B - 4*a*b^2*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(4*b^2*d*sqrt(a + b*cos(c + d*x))) + (B*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*d*sec(c + d*x)^(3/2)) + ((6*A*b + a*B)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(12*b*d*sqrt(sec(c + d*x))) + ((6*a*A*b - 3*a^2*B + 16*b^2*B)*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(24*b^2*d*sqrt(sec(c + d*x)))],


[(a + b*cos(c + d*x))^(3/2)*(A + B*cos(c + d*x))*sec(c + d*x)^(11/2), x, 9, -((2*(a + b)*(147*a^4*A + 33*a^2*A*b^2 + 8*A*b^4 + 246*a^3*b*B - 18*a*b^3*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(315*a^3*d*sqrt(a + b*cos(c + d*x)))) + (1/(315*a^2*d*sqrt(a + b*cos(c + d*x))))*(2*(147*a^4*A + 186*a^3*A*b + 33*a^2*A*b^2 + 2*a*A*b^3 + 8*A*b^4 + 75*a^4*B + 246*a^3*b*B + 153*a^2*b^2*B - 18*a*b^3*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x))) + (2*(88*a^2*A*b - 4*A*b^3 + 75*a^3*B + 9*a*b^2*B)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(315*a^2*d) + (2*(49*a^2*A + 3*A*b^2 + 72*a*b*B)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(5/2)*sin(c + d*x))/(315*a*d) + (2*(10*A*b + 9*a*B)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(7/2)*sin(c + d*x))/(63*d) + (2*a*A*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(9/2)*sin(c + d*x))/(9*d) + (2*(147*a^4*A + 33*a^2*A*b^2 + 8*A*b^4 + 246*a^3*b*B - 18*a*b^3*B)*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*tan((1/2)*(c + d*x)))/(315*a^3*d)],
[(a + b*cos(c + d*x))^(3/2)*(A + B*cos(c + d*x))*sec(c + d*x)^(9/2), x, 8, -((2*(a + b)*(82*a^2*A*b - 6*A*b^3 + 63*a^3*B + 21*a*b^2*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(105*a^2*d*sqrt(a + b*cos(c + d*x)))) + (2*(25*a^3*A + 82*a^2*A*b + 51*a*A*b^2 - 6*A*b^3 + 63*a^3*B + 84*a^2*b*B + 21*a*b^2*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(105*a*d*sqrt(a + b*cos(c + d*x))) + (2*(25*a^2*A + 3*A*b^2 + 42*a*b*B)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(105*a*d) + (2*(8*A*b + 7*a*B)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(5/2)*sin(c + d*x))/(35*d) + (2*a*A*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(7/2)*sin(c + d*x))/(7*d) + (2*(82*a^2*A*b - 6*A*b^3 + 63*a^3*B + 21*a*b^2*B)*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*tan((1/2)*(c + d*x)))/(105*a^2*d)],
[(a + b*cos(c + d*x))^(3/2)*(A + B*cos(c + d*x))*sec(c + d*x)^(7/2), x, 7, -((2*(a + b)*(9*a^2*A + 3*A*b^2 + 20*a*b*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(15*a*d*sqrt(a + b*cos(c + d*x)))) + (2*(9*a^2*A + 12*a*A*b + 3*A*b^2 + 5*a^2*B + 20*a*b*B + 15*b^2*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(15*d*sqrt(a + b*cos(c + d*x))) + (2*(6*A*b + 5*a*B)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(15*d) + (2*a*A*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d) + (2*(9*a^2*A + 3*A*b^2 + 20*a*b*B)*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*tan((1/2)*(c + d*x)))/(15*a*d)],
[(a + b*cos(c + d*x))^(3/2)*(A + B*cos(c + d*x))*sec(c + d*x)^(5/2), x, 8, (-2*(a + b)*(4*A*b + 3*a*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(3*d*sqrt(a + b*cos(c + d*x))) + (2*(a^2*A + 4*a*A*b + 3*A*b^2 + 3*a^2*B + 6*a*b*B - 3*b^2*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(3*d*sqrt(a + b*cos(c + d*x))) + (4*b^2*B*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*cos(c + d*x))) + (2*a*A*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d) + (2*(4*A*b + 3*a*B)*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*tan((c + d*x)/2))/(3*d)],
[(a + b*cos(c + d*x))^(3/2)*(A + B*cos(c + d*x))*sec(c + d*x)^(3/2), x, 8, -(((a + b)*(2*a*A - b*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*cos(c + d*x)))) + (2*(a^2*A + 2*a*A*b - A*b^2 + a^2*B - 2*a*b*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*cos(c + d*x))) + (2*b*(2*A*b + 3*a*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*cos(c + d*x))) + (2*a*A*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*sin(c + d*x))/d - ((2*a*A - b*B)*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(d*sqrt(sec(c + d*x)))],
[(a + b*cos(c + d*x))^(3/2)*(A + B*cos(c + d*x))*sec(c + d*x)^(1/2), x, 8, ((a + b)*(4*A*b + 5*a*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(4*d*sqrt(a + b*cos(c + d*x))) + ((4*a^2*A - 8*a*A*b - 4*a^2*B + a*b*B - 2*b^2*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(2*d*sqrt(a + b*cos(c + d*x))) + ((12*a*A*b + 3*a^2*B + 4*b^2*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(2*d*sqrt(a + b*cos(c + d*x))) + (b*B*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(2*d*sqrt(sec(c + d*x))) + ((4*A*b + 5*a*B)*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(4*d*sqrt(sec(c + d*x)))],
[(a + b*cos(c + d*x))^(3/2)*(A + B*cos(c + d*x))/sec(c + d*x)^(1/2), x, 9, ((a + b)*(30*a*A*b + 3*a^2*B + 16*b^2*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(24*b*d*sqrt(a + b*cos(c + d*x))) - ((24*a^2*A - 6*a*A*b + 12*A*b^2 - 7*a^2*B + 26*a*b*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(12*d*sqrt(a + b*cos(c + d*x))) + ((6*a^2*A*b + 8*A*b^3 - a^3*B + 12*a*b^2*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(4*b*d*sqrt(a + b*cos(c + d*x))) + (b*B*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*d*sec(c + d*x)^(3/2)) + ((6*A*b + 7*a*B)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(12*d*sqrt(sec(c + d*x))) + ((30*a*A*b + 3*a^2*B + 16*b^2*B)*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(24*b*d*sqrt(sec(c + d*x)))],
[(a + b*cos(c + d*x))^(3/2)*(A + B*cos(c + d*x))/sec(c + d*x)^(3/2), x, 10, ((a + b)*(24*a^2*A*b + 128*A*b^3 - 9*a^3*B + 156*a*b^2*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(192*b^2*d*sqrt(a + b*cos(c + d*x))) + ((56*a^2*A*b - 208*a*A*b^2 + 3*a^3*B - 114*a^2*b*B + 36*a*b^2*B - 72*b^3*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(96*b*d*sqrt(a + b*cos(c + d*x))) - ((8*a^3*A*b - 96*a*A*b^3 - 3*a^4*B - 24*a^2*b^2*B - 48*b^4*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(32*b^2*d*sqrt(a + b*cos(c + d*x))) + (b*B*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(4*d*sec(c + d*x)^(5/2)) + ((8*A*b + 9*a*B)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(24*d*sec(c + d*x)^(3/2)) + ((56*a*A*b + 3*a^2*B + 36*b^2*B)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(96*b*d*sqrt(sec(c + d*x))) + ((24*a^2*A*b + 128*A*b^3 - 9*a^3*B + 156*a*b^2*B)*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(192*b^2*d*sqrt(sec(c + d*x)))],


[(a + b*cos(c + d*x))^(5/2)*(A + B*cos(c + d*x))*sec(c + d*x)^(13/2), x, 10, -((2*(a + b)*(3705*a^4*A*b + 255*a^2*A*b^3 + 40*A*b^5 + 1617*a^5*B + 3069*a^3*b^2*B - 110*a*b^4*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(3465*a^3*d*sqrt(a + b*cos(c + d*x)))) + (1/(3465*a^2*d*sqrt(a + b*cos(c + d*x))))*(2*(675*a^5*A + 3705*a^4*A*b + 3315*a^3*A*b^2 + 255*a^2*A*b^3 + 10*a*A*b^4 + 40*A*b^5 + 1617*a^5*B + 2871*a^4*b*B + 3069*a^3*b^2*B + 1705*a^2*b^3*B - 110*a*b^4*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x))) + (2*(675*a^4*A + 1025*a^2*A*b^2 - 20*A*b^4 + 1793*a^3*b*B + 55*a*b^3*B)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(3465*a^2*d) + (2*(1145*a^2*A*b + 15*A*b^3 + 539*a^3*B + 825*a*b^2*B)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(5/2)*sin(c + d*x))/(3465*a*d) + (2*(81*a^2*A + 113*A*b^2 + 209*a*b*B)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(7/2)*sin(c + d*x))/(693*d) + (2*a*(14*A*b + 11*a*B)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(9/2)*sin(c + d*x))/(99*d) + (2*a*A*(a + b*cos(c + d*x))^(3/2)*sec(c + d*x)^(11/2)*sin(c + d*x))/(11*d) + (2*(3705*a^4*A*b + 255*a^2*A*b^3 + 40*A*b^5 + 1617*a^5*B + 3069*a^3*b^2*B - 110*a*b^4*B)*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*tan((1/2)*(c + d*x)))/(3465*a^3*d)],
[(a + b*cos(c + d*x))^(5/2)*(A + B*cos(c + d*x))*sec(c + d*x)^(11/2), x, 9, -((2*(a + b)*(147*a^4*A + 279*a^2*A*b^2 - 10*A*b^4 + 435*a^3*b*B + 45*a*b^3*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(315*a^2*d*sqrt(a + b*cos(c + d*x)))) + (1/(315*a*d*sqrt(a + b*cos(c + d*x))))*(2*(147*a^4*A + 261*a^3*A*b + 279*a^2*A*b^2 + 155*a*A*b^3 - 10*A*b^4 + 75*a^4*B + 435*a^3*b*B + 405*a^2*b^2*B + 45*a*b^3*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x))) + (2*(163*a^2*A*b + 5*A*b^3 + 75*a^3*B + 135*a*b^2*B)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(315*a*d) + (2*(49*a^2*A + 75*A*b^2 + 135*a*b*B)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(5/2)*sin(c + d*x))/(315*d) + (2*a*(4*A*b + 3*a*B)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(7/2)*sin(c + d*x))/(21*d) + (2*a*A*(a + b*cos(c + d*x))^(3/2)*sec(c + d*x)^(9/2)*sin(c + d*x))/(9*d) + (2*(147*a^4*A + 279*a^2*A*b^2 - 10*A*b^4 + 435*a^3*b*B + 45*a*b^3*B)*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*tan((1/2)*(c + d*x)))/(315*a^2*d)],
[(a + b*cos(c + d*x))^(5/2)*(A + B*cos(c + d*x))*sec(c + d*x)^(9/2), x, 8, -((2*(a + b)*(145*a^2*A*b + 15*A*b^3 + 63*a^3*B + 161*a*b^2*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(105*a*d*sqrt(a + b*cos(c + d*x)))) + (2*(25*a^3*A + 145*a^2*A*b + 135*a*A*b^2 + 15*A*b^3 + 63*a^3*B + 119*a^2*b*B + 161*a*b^2*B + 105*b^3*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(105*d*sqrt(a + b*cos(c + d*x))) + (2*(25*a^2*A + 45*A*b^2 + 77*a*b*B)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(105*d) + (2*a*(10*A*b + 7*a*B)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(5/2)*sin(c + d*x))/(35*d) + (2*a*A*(a + b*cos(c + d*x))^(3/2)*sec(c + d*x)^(7/2)*sin(c + d*x))/(7*d) + (2*(145*a^2*A*b + 15*A*b^3 + 63*a^3*B + 161*a*b^2*B)*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*tan((1/2)*(c + d*x)))/(105*a*d)],
[(a + b*cos(c + d*x))^(5/2)*(A + B*cos(c + d*x))*sec(c + d*x)^(7/2), x, 9, -((2*(a + b)*(9*a^2*A + 23*A*b^2 + 35*a*b*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(15*d*sqrt(a + b*cos(c + d*x)))) + (2*(9*a^3*A + 17*a^2*A*b + 23*a*A*b^2 + 15*A*b^3 + 5*a^3*B + 35*a^2*b*B + 45*a*b^2*B - 15*b^3*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(15*d*sqrt(a + b*cos(c + d*x))) + (4*b^3*B*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*cos(c + d*x))) + (2*a*(8*A*b + 5*a*B)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(15*d) + (2*a*A*(a + b*cos(c + d*x))^(3/2)*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d) + (2*(9*a^2*A + 23*A*b^2 + 35*a*b*B)*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*tan((1/2)*(c + d*x)))/(15*d)],
[(a + b*cos(c + d*x))^(5/2)*(A + B*cos(c + d*x))*sec(c + d*x)^(5/2), x, 9, -((a + b)*(14*a*A*b + 6*a^2*B - 3*b^2*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(3*d*sqrt(a + b*cos(c + d*x))) + (2*(a^3*A + 7*a^2*A*b + 9*a*A*b^2 - 3*A*b^3 + 3*a^3*B + 9*a^2*b*B - 9*a*b^2*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(3*d*sqrt(a + b*cos(c + d*x))) + (2*b^2*(2*A*b + 5*a*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*cos(c + d*x))) + (2*a*(2*A*b + a*B)*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*sin(c + d*x))/d + (2*a*A*(a + b*cos(c + d*x))^(3/2)*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d) - ((14*a*A*b + 6*a^2*B - 3*b^2*B)*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(3*d*sqrt(sec(c + d*x)))],
[(a + b*cos(c + d*x))^(5/2)*(A + B*cos(c + d*x))*sec(c + d*x)^(3/2), x, 9, -((a + b)*(8*a^2*A - 4*A*b^2 - 9*a*b*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(4*d*sqrt(a + b*cos(c + d*x))) + ((4*a^3*A + 12*a^2*A*b - 12*a*A*b^2 + 4*a^3*B - 12*a^2*b*B + a*b^2*B - 2*b^3*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(2*d*sqrt(a + b*cos(c + d*x))) + (b*(20*a*A*b + 15*a^2*B + 4*b^2*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(2*d*sqrt(a + b*cos(c + d*x))) - (b*(4*a*A - b*B)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(2*d*sqrt(sec(c + d*x))) + (2*a*A*(a + b*cos(c + d*x))^(3/2)*sqrt(sec(c + d*x))*sin(c + d*x))/d - ((8*a^2*A - 4*A*b^2 - 9*a*b*B)*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(4*d*sqrt(sec(c + d*x)))],
[(a + b*cos(c + d*x))^(5/2)*(A + B*cos(c + d*x))*sec(c + d*x)^(1/2), x, 9, ((a + b)*(54*a*A*b + 33*a^2*B + 16*b^2*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(24*d*sqrt(a + b*cos(c + d*x))) + ((24*a^3*A - 72*a^2*A*b + 6*a*A*b^2 - 12*A*b^3 - 24*a^3*B + 13*a^2*b*B - 38*a*b^2*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(12*d*sqrt(a + b*cos(c + d*x))) + ((30*a^2*A*b + 8*A*b^3 + 5*a^3*B + 20*a*b^2*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(4*d*sqrt(a + b*cos(c + d*x))) + (b*(2*A*b + 3*a*B)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(4*d*sqrt(sec(c + d*x))) + (b*B*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(3*d*sqrt(sec(c + d*x))) + ((54*a*A*b + 33*a^2*B + 16*b^2*B)*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(24*d*sqrt(sec(c + d*x)))],
[(a + b*cos(c + d*x))^(5/2)*(A + B*cos(c + d*x))/sec(c + d*x)^(1/2), x, 10, ((a + b)*(264*a^2*A*b + 128*A*b^3 + 15*a^3*B + 284*a*b^2*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(192*b*d*sqrt(a + b*cos(c + d*x))) - ((192*a^3*A - 104*a^2*A*b + 304*a*A*b^2 - 59*a^3*B + 322*a^2*b*B - 36*a*b^2*B + 72*b^3*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(96*d*sqrt(a + b*cos(c + d*x))) + ((40*a^3*A*b + 160*a*A*b^3 - 5*a^4*B + 120*a^2*b^2*B + 48*b^4*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(32*b*d*sqrt(a + b*cos(c + d*x))) + (b*(8*A*b + 11*a*B)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(24*d*sec(c + d*x)^(3/2)) + (b*B*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(4*d*sec(c + d*x)^(3/2)) + ((104*a*A*b + 59*a^2*B + 36*b^2*B)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(96*d*sqrt(sec(c + d*x))) + ((264*a^2*A*b + 128*A*b^3 + 15*a^3*B + 284*a*b^2*B)*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(192*b*d*sqrt(sec(c + d*x)))],
[(a + b*cos(c + d*x))^(5/2)*(A + B*cos(c + d*x))/sec(c + d*x)^(3/2), x, 11, ((a + b)*(150*a^3*A*b + 2840*a*A*b^3 - 45*a^4*B + 1692*a^2*b^2*B + 1024*b^4*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(1920*b^2*d*sqrt(a + b*cos(c + d*x))) + ((590*a^3*A*b - 3220*a^2*A*b^2 + 360*a*A*b^3 - 720*A*b^4 + 15*a^4*B - 1146*a^3*b*B + 772*a^2*b^2*B - 2312*a*b^3*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(960*b*d*sqrt(a + b*cos(c + d*x))) - ((10*a^4*A*b - 240*a^2*A*b^3 - 96*A*b^5 - 3*a^5*B - 40*a^3*b^2*B - 240*a*b^4*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(64*b^2*d*sqrt(a + b*cos(c + d*x))) + (b*(10*A*b + 13*a*B)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(40*d*sec(c + d*x)^(5/2)) + (b*B*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(5*d*sec(c + d*x)^(5/2)) + ((170*a*A*b + 93*a^2*B + 64*b^2*B)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(240*d*sec(c + d*x)^(3/2)) + ((590*a^2*A*b + 360*A*b^3 + 15*a^3*B + 772*a*b^2*B)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(960*b*d*sqrt(sec(c + d*x))) + ((150*a^3*A*b + 2840*a*A*b^3 - 45*a^4*B + 1692*a^2*b^2*B + 1024*b^4*B)*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(1920*b^2*d*sqrt(sec(c + d*x)))],


# ::Subsubsection::Closed:: 
#n<0


[sec(c + d*x)^(7/2)*(A + B*cos(c + d*x))/sqrt(a + b*cos(c + d*x)), x, 7, -((2*(a + b)*(9*a^2*A + 8*A*b^2 - 10*a*b*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(15*a^3*d*sqrt(a + b*cos(c + d*x)))) + (2*(9*a^2*A + 2*a*A*b + 8*A*b^2 + 5*a^2*B - 10*a*b*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(15*a^2*d*sqrt(a + b*cos(c + d*x))) - (2*(4*A*b - 5*a*B)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(15*a^2*d) + (2*A*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*a*d) + (2*(9*a^2*A + 8*A*b^2 - 10*a*b*B)*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*tan((1/2)*(c + d*x)))/(15*a^3*d)],
[sec(c + d*x)^(5/2)*(A + B*cos(c + d*x))/sqrt(a + b*cos(c + d*x)), x, 6, (2*(a + b)*(2*A*b - 3*a*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(3*a^2*d*sqrt(a + b*cos(c + d*x))) + (2*(a*A - 2*A*b + 3*a*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(3*a*d*sqrt(a + b*cos(c + d*x))) + (2*A*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*a*d) - (2*(2*A*b - 3*a*B)*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*tan((1/2)*(c + d*x)))/(3*a^2*d)],
[sec(c + d*x)^(3/2)*(A + B*cos(c + d*x))/sqrt(a + b*cos(c + d*x)), x, 5, -((2*A*(a + b)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(a*d*sqrt(a + b*cos(c + d*x)))) + (2*(A + B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*cos(c + d*x))) + (2*A*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*tan((1/2)*(c + d*x)))/(a*d)],
[sec(c + d*x)^(1/2)*(A + B*cos(c + d*x))/sqrt(a + b*cos(c + d*x)), x, 4, (2*(A - B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*cos(c + d*x))) + (4*B*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*cos(c + d*x)))],
[sec(c + d*x)^(-1/2)*(A + B*cos(c + d*x))/sqrt(a + b*cos(c + d*x)), x, 8, ((a + b)*B*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(b*d*sqrt(a + b*cos(c + d*x))) - (2*A*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*cos(c + d*x))) + (2*(2*A*b - a*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(b*d*sqrt(a + b*cos(c + d*x))) + (B*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(b*d*sqrt(sec(c + d*x)))],
[sec(c + d*x)^(-3/2)*(A + B*cos(c + d*x))/sqrt(a + b*cos(c + d*x)), x, 8, ((a + b)*(4*A*b - 3*a*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(4*b^2*d*sqrt(a + b*cos(c + d*x))) + ((a - 2*b)*B*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(2*b*d*sqrt(a + b*cos(c + d*x))) - ((4*a*A*b - 3*a^2*B - 4*b^2*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(2*b^2*d*sqrt(a + b*cos(c + d*x))) + (B*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(2*b*d*sqrt(sec(c + d*x))) + ((4*A*b - 3*a*B)*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(4*b^2*d*sqrt(sec(c + d*x)))],


[(A + B*cos(c + d*x))*sec(c + d*x)^(5/2)/(a + b*cos(c + d*x))^(3/2), x, 7, (2*(5*a^2*A*b - 8*A*b^3 - 3*a^3*B + 6*a*b^2*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(3*a^3*(a - b)*d*sqrt(a + b*cos(c + d*x))) + (2*(a^2*A - 6*a*A*b + 8*A*b^2 + 3*a^2*B - 6*a*b*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(3*a^2*(a - b)*d*sqrt(a + b*cos(c + d*x))) + (2*b*(A*b - a*B)*sec(c + d*x)^(3/2)*sin(c + d*x))/(a*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) + (2*(a^2*A - 4*A*b^2 + 3*a*b*B)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*a^2*(a^2 - b^2)*d) - (2*(5*a^2*A*b - 8*A*b^3 - 3*a^3*B + 6*a*b^2*B)*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*tan((c + d*x)/2))/(3*a^3*(a^2 - b^2)*d)],
[(A + B*cos(c + d*x))*sec(c + d*x)^(3/2)/(a + b*cos(c + d*x))^(3/2), x, 6, (-2*(a^2*A - 2*A*b^2 + a*b*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(a^2*(a - b)*d*sqrt(a + b*cos(c + d*x))) + (2*(a*A - 2*A*b + a*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(a*(a - b)*d*sqrt(a + b*cos(c + d*x))) + (2*b*(A*b - a*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(a*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) + (2*(a^2*A - 2*A*b^2 + a*b*B)*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*tan((c + d*x)/2))/(a^2*(a^2 - b^2)*d)],
[(A + B*cos(c + d*x))*sec(c + d*x)^(1/2)/(a + b*cos(c + d*x))^(3/2), x, 5, (-2*(A*b - a*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(a*(a - b)*d*sqrt(a + b*cos(c + d*x))) + (2*(A - B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/((a - b)*d*sqrt(a + b*cos(c + d*x))) - (2*(A*b - a*B)*tan((c + d*x)/2))/(a*(a + b)*d*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x)))],
[(A + B*cos(c + d*x))/(sec(c + d*x)^(1/2)*(a + b*cos(c + d*x))^(3/2)), x, 7, (2*(A*b - a*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/((a - b)*b*d*sqrt(a + b*cos(c + d*x))) - (2*(A - B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/((a - b)*d*sqrt(a + b*cos(c + d*x))) + (4*B*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(b*d*sqrt(a + b*cos(c + d*x))) + (2*(A*b - a*B)*tan((1/2)*(c + d*x)))/(b*(a + b)*d*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x)))],
[(A + B*cos(c + d*x))/(sec(c + d*x)^(3/2)*(a + b*cos(c + d*x))^(3/2)), x, 8, -(((2*a*A*b - 3*a^2*B + b^2*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/((a - b)*b^2*d*sqrt(a + b*cos(c + d*x)))) + (2*(A*b - a*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/((a - b)*b*d*sqrt(a + b*cos(c + d*x))) + (2*(2*A*b - 3*a*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(b^2*d*sqrt(a + b*cos(c + d*x))) + (2*a*(A*b - a*B)*sin(c + d*x))/(b*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))) - ((2*a*A*b - 3*a^2*B + b^2*B)*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(b^2*(a^2 - b^2)*d*sqrt(sec(c + d*x)))],


[(A + B*cos(c + d*x))*sec(c + d*x)^(5/2)/(a + b*cos(c + d*x))^(5/2), x, 8, (2*(8*a^4*A*b - 28*a^2*A*b^3 + 16*A*b^5 - 3*a^5*B + 15*a^3*b^2*B - 8*a*b^4*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(3*a^4*(a - b)^2*(a + b)*d*sqrt(a + b*cos(c + d*x))) + (2*(a^4*A - 9*a^3*A*b + 16*a^2*A*b^2 + 12*a*A*b^3 - 16*A*b^4 + 3*a^4*B - 9*a^3*b*B - 6*a^2*b^2*B + 8*a*b^3*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(3*a^3*(a - b)^2*(a + b)*d*sqrt(a + b*cos(c + d*x))) + (2*b*(A*b - a*B)*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*a*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)) + (2*b*(10*a^2*A*b - 6*A*b^3 - 7*a^3*B + 3*a*b^2*B)*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*a^2*(a^2 - b^2)^2*d*sqrt(a + b*cos(c + d*x))) + (2*(a^4*A - 13*a^2*A*b^2 + 8*A*b^4 + 8*a^3*b*B - 4*a*b^3*B)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*a^3*(a^2 - b^2)^2*d) - (2*(8*a^4*A*b - 28*a^2*A*b^3 + 16*A*b^5 - 3*a^5*B + 15*a^3*b^2*B - 8*a*b^4*B)*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*tan((c + d*x)/2))/(3*a^4*(a^2 - b^2)^2*d)],
[(A + B*cos(c + d*x))*sec(c + d*x)^(3/2)/(a + b*cos(c + d*x))^(5/2), x, 7, (-2*(3*a^4*A - 15*a^2*A*b^2 + 8*A*b^4 + 6*a^3*b*B - 2*a*b^3*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(3*a^3*(a - b)^2*(a + b)*d*sqrt(a + b*cos(c + d*x))) + (2*(3*a^3*A - 9*a^2*A*b - 6*a*A*b^2 + 8*A*b^3 + 3*a^3*B + 3*a^2*b*B - 2*a*b^2*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(3*a^2*(a - b)^2*(a + b)*d*sqrt(a + b*cos(c + d*x))) + (2*b*(A*b - a*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(3*a*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)) + (2*b*(8*a^2*A*b - 4*A*b^3 - 5*a^3*B + a*b^2*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(3*a^2*(a^2 - b^2)^2*d*sqrt(a + b*cos(c + d*x))) + (2*(3*a^4*A - 15*a^2*A*b^2 + 8*A*b^4 + 6*a^3*b*B - 2*a*b^3*B)*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*tan((c + d*x)/2))/(3*a^3*(a^2 - b^2)^2*d)],
[(A + B*cos(c + d*x))*sec(c + d*x)^(1/2)/(a + b*cos(c + d*x))^(5/2), x, 6, (-2*(6*a^2*A*b - 2*A*b^3 - 3*a^3*B - a*b^2*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(3*a^2*(a - b)^2*(a + b)*d*sqrt(a + b*cos(c + d*x))) + (2*(3*a^2*A + 3*a*A*b - 2*A*b^2 - 3*a^2*B - a*b*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(3*a*(a - b)^2*(a + b)*d*sqrt(a + b*cos(c + d*x))) + (2*b*(A*b - a*B)*sin(c + d*x))/(3*a*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)*sqrt(sec(c + d*x))) - (2*(6*a^2*A*b - 2*A*b^3 - 3*a^3*B - a*b^2*B)*tan((c + d*x)/2))/(3*a^2*(a - b)*(a + b)^2*d*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x)))],
[(A + B*cos(c + d*x))/((a + b*cos(c + d*x))^(5/2)*sec(c + d*x)^(1/2)), x, 6, (2*(3*a^2*A + A*b^2 - 4*a*b*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(3*a*(a - b)^2*(a + b)*d*sqrt(a + b*cos(c + d*x))) - (2*(3*a*A + A*b - a*B - 3*b*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(3*(a - b)^2*(a + b)*d*sqrt(a + b*cos(c + d*x))) - (2*(A*b - a*B)*sin(c + d*x))/(3*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)*sqrt(sec(c + d*x))) + (2*(3*a^2*A + A*b^2 - 4*a*b*B)*tan((c + d*x)/2))/(3*a*(a - b)*(a + b)^2*d*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x)))],
[(A + B*cos(c + d*x))/((a + b*cos(c + d*x))^(5/2)*sec(c + d*x)^(3/2)), x, 8, (-2*(4*A*b^3 + 3*a^3*B - 7*a*b^2*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(3*(a - b)^2*b^2*(a + b)*d*sqrt(a + b*cos(c + d*x))) + (2*(a*A*b + 3*A*b^2 + 2*a^2*B - 3*a*b*B - 3*b^2*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(3*(a - b)^2*b*(a + b)*d*sqrt(a + b*cos(c + d*x))) + (4*B*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(b^2*d*sqrt(a + b*cos(c + d*x))) + (2*a*(A*b - a*B)*sin(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)*sqrt(sec(c + d*x))) - (2*(4*A*b^3 + 3*a^3*B - 7*a*b^2*B)*tan((c + d*x)/2))/(3*(a - b)*b^2*(a + b)^2*d*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x)))],
[(A + B*cos(c + d*x))/((a + b*cos(c + d*x))^(5/2)*sec(c + d*x)^(5/2)), x, 9, -((6*a^3*A*b - 14*a*A*b^3 - 15*a^4*B + 26*a^2*b^2*B - 3*b^4*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(3*(a - b)^2*b^3*(a + b)*d*sqrt(a + b*cos(c + d*x))) + (2*(2*a^2*A*b - 3*a*A*b^2 - 3*A*b^3 - 5*a^3*B + 3*a^2*b*B + 6*a*b^2*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(3*(a - b)^2*b^2*(a + b)*d*sqrt(a + b*cos(c + d*x))) + (2*(2*A*b - 5*a*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(b^3*d*sqrt(a + b*cos(c + d*x))) + (2*a*(A*b - a*B)*sin(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)*sec(c + d*x)^(3/2)) + (2*a*(2*a^2*A*b - 6*A*b^3 - 5*a^3*B + 9*a*b^2*B)*sin(c + d*x))/(3*b^2*(a^2 - b^2)^2*d*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))) - ((6*a^3*A*b - 14*a*A*b^3 - 15*a^4*B + 26*a^2*b^2*B - 3*b^4*B)*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(3*b^3*(a^2 - b^2)^2*d*sqrt(sec(c + d*x)))],


[(a*B + b*B*cos(c + d*x))*sec(c + d*x)^(1/2)/(a + b*cos(c + d*x))^(3/2), x, 3, (2*B*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*cos(c + d*x)))],
[(a*B + b*B*cos(c + d*x))/(sec(c + d*x)^(1/2)*(a + b*cos(c + d*x))^(3/2)), x, 5, -((2*B*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*cos(c + d*x)))) + (4*B*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*cos(c + d*x)))],
[(a*B + b*B*cos(c + d*x))/(sec(c + d*x)^(3/2)*(a + b*cos(c + d*x))^(3/2)), x, 6, ((a + b)*B*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(b*d*sqrt(a + b*cos(c + d*x))) - (2*a*B*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(b*d*sqrt(a + b*cos(c + d*x))) + (B*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(b*d*sqrt(sec(c + d*x)))],


# ::Section:: 
#Cos[c+d x]^m (A+C Cos[c+d x]^2) (a+b Cos[c+d x])^n


# ::Subsection::Closed:: 
#Integrands of the form Cos[c+d x]^m (A+C Cos[c+d x]^2) (a+b Cos[c+d x])^n
#


# ::Subsubsection::Closed:: 
#n>0


[cos(c + d*x)^2*(a + b*cos(c + d*x))*(A + C*cos(c + d*x)^2), x, 5, (a*(4*A + 3*C)*x)/8 + (2*b*(5*A + 4*C)*sin(c + d*x))/(15*d) + (a*(4*A + 3*C)*cos(c + d*x)*sin(c + d*x))/(8*d) + (b*(5*A + 4*C)*cos(c + d*x)^2*sin(c + d*x))/(15*d) + (a*C*cos(c + d*x)^3*sin(c + d*x))/(20*d) + (C*cos(c + d*x)^3*(a + b*cos(c + d*x))*sin(c + d*x))/(5*d)],
[cos(c + d*x)^1*(a + b*cos(c + d*x))*(A + C*cos(c + d*x)^2), x, 4, (b*(4*A + 3*C)*x)/8 + (a*(3*A + 2*C)*sin(c + d*x))/(3*d) + (b*(4*A + 3*C)*cos(c + d*x)*sin(c + d*x))/(8*d) + (a*C*cos(c + d*x)^2*sin(c + d*x))/(12*d) + (C*cos(c + d*x)^2*(a + b*cos(c + d*x))*sin(c + d*x))/(4*d)],
[cos(c + d*x)^0*(a + b*cos(c + d*x))*(A + C*cos(c + d*x)^2), x, 3, (1/2)*a*(2*A + C)*x + ((3*A*b^2 - a^2*C + 2*b^2*C)*sin(c + d*x))/(3*b*d) - (a*C*cos(c + d*x)*sin(c + d*x))/(6*d) + (C*(a + b*cos(c + d*x))^2*sin(c + d*x))/(3*b*d)],
[sec(c + d*x)^1*(a + b*cos(c + d*x))*(A + C*cos(c + d*x)^2), x, 4, (1/2)*b*(2*A + C)*x + (a*A*arctanh(sin(c + d*x)))/d + (a*C*sin(c + d*x))/(2*d) + (C*(a + b*cos(c + d*x))*sin(c + d*x))/(2*d)],
[sec(c + d*x)^2*(a + b*cos(c + d*x))*(A + C*cos(c + d*x)^2), x, 4, a*C*x + (A*b*arctanh(sin(c + d*x)))/d - (b*(A - C)*sin(c + d*x))/d + (A*(a + b*cos(c + d*x))*tan(c + d*x))/d],
[sec(c + d*x)^3*(a + b*cos(c + d*x))*(A + C*cos(c + d*x)^2), x, 4, b*C*x + (a*(A + 2*C)*arctanh(sin(c + d*x)))/(2*d) + (A*b*tan(c + d*x))/(2*d) + (A*(b + a*sec(c + d*x))*tan(c + d*x))/(2*d)],
[sec(c + d*x)^4*(a + b*cos(c + d*x))*(A + C*cos(c + d*x)^2), x, 5, (b*(A + 2*C)*arctanh(sin(c + d*x)))/(2*d) + (a*(2*A + 3*C)*tan(c + d*x))/(3*d) + (A*b*sec(c + d*x)*tan(c + d*x))/(6*d) + (A*(a + b*cos(c + d*x))*sec(c + d*x)^2*tan(c + d*x))/(3*d)],
[sec(c + d*x)^5*(a + b*cos(c + d*x))*(A + C*cos(c + d*x)^2), x, 6, (a*(3*A + 4*C)*arctanh(sin(c + d*x)))/(8*d) + (b*(2*A + 3*C)*tan(c + d*x))/(3*d) + (a*(3*A + 4*C)*sec(c + d*x)*tan(c + d*x))/(8*d) + (A*b*sec(c + d*x)^2*tan(c + d*x))/(12*d) + (A*(a + b*cos(c + d*x))*sec(c + d*x)^3*tan(c + d*x))/(4*d)],
[sec(c + d*x)^6*(a + b*cos(c + d*x))*(A + C*cos(c + d*x)^2), x, 7, (b*(3*A + 4*C)*arctanh(sin(c + d*x)))/(8*d) + (2*a*(4*A + 5*C)*tan(c + d*x))/(15*d) + (b*(3*A + 4*C)*sec(c + d*x)*tan(c + d*x))/(8*d) + (a*(4*A + 5*C)*sec(c + d*x)^2*tan(c + d*x))/(15*d) + (A*b*sec(c + d*x)^3*tan(c + d*x))/(20*d) + (A*(a + b*cos(c + d*x))*sec(c + d*x)^4*tan(c + d*x))/(5*d)],


[cos(c + d*x)^2*(a + b*cos(c + d*x))^2*(A + C*cos(c + d*x)^2), x, 6, ((8*a^2*A + 6*A*b^2 + 6*a^2*C + 5*b^2*C)*x)/16 + (4*a*b*(5*A + 4*C)*sin(c + d*x))/(15*d) + ((8*a^2*A + 6*A*b^2 + 6*a^2*C + 5*b^2*C)*cos(c + d*x)*sin(c + d*x))/(16*d) + (2*a*b*(5*A + 4*C)*cos(c + d*x)^2*sin(c + d*x))/(15*d) + ((30*A*b^2 + 2*a^2*C + 25*b^2*C)*cos(c + d*x)^3*sin(c + d*x))/(120*d) + (a*C*cos(c + d*x)^3*(a + b*cos(c + d*x))*sin(c + d*x))/(15*d) + (C*cos(c + d*x)^3*(a + b*cos(c + d*x))^2*sin(c + d*x))/(6*d)],
[cos(c + d*x)^1*(a + b*cos(c + d*x))^2*(A + C*cos(c + d*x)^2), x, 5, (a*b*(4*A + 3*C)*x)/4 + ((15*a^2*A + 10*A*b^2 + 10*a^2*C + 8*b^2*C)*sin(c + d*x))/(15*d) + (a*b*(4*A + 3*C)*cos(c + d*x)*sin(c + d*x))/(4*d) + ((10*A*b^2 + a^2*C + 8*b^2*C)*cos(c + d*x)^2*sin(c + d*x))/(30*d) + (a*C*cos(c + d*x)^2*(a + b*cos(c + d*x))*sin(c + d*x))/(10*d) + (C*cos(c + d*x)^2*(a + b*cos(c + d*x))^2*sin(c + d*x))/(5*d)],
[cos(c + d*x)^0*(a + b*cos(c + d*x))^2*(A + C*cos(c + d*x)^2), x, 4, (1/8)*(8*a^2*A + 4*A*b^2 + 4*a^2*C + 3*b^2*C)*x + (a*(12*A*b^2 - a^2*C + 8*b^2*C)*sin(c + d*x))/(6*b*d) + ((12*A*b^2 - 2*a^2*C + 9*b^2*C)*cos(c + d*x)*sin(c + d*x))/(24*d) - (a*C*(a + b*cos(c + d*x))^2*sin(c + d*x))/(12*b*d) + (C*(a + b*cos(c + d*x))^3*sin(c + d*x))/(4*b*d)],
[sec(c + d*x)^1*(a + b*cos(c + d*x))^2*(A + C*cos(c + d*x)^2), x, 5, (2*a*A*b + a*b*C)*x + (a^2*A*arctanh(sin(c + d*x)))/d + ((3*A*b^2 + a^2*C + 2*b^2*C)*sin(c + d*x))/(3*d) + (a*C*(a + b*cos(c + d*x))*sin(c + d*x))/(3*d) + (C*(a + b*cos(c + d*x))^2*sin(c + d*x))/(3*d)],
[sec(c + d*x)^2*(a + b*cos(c + d*x))^2*(A + C*cos(c + d*x)^2), x, 5, (1/2)*(2*A*b^2 + 2*a^2*C + b^2*C)*x + (2*a*A*b*arctanh(sin(c + d*x)))/d - (a*b*(2*A - 3*C)*sin(c + d*x))/(2*d) - (b*(2*A - C)*(a + b*cos(c + d*x))*sin(c + d*x))/(2*d) + (A*(a + b*cos(c + d*x))^2*tan(c + d*x))/d],
[sec(c + d*x)^3*(a + b*cos(c + d*x))^2*(A + C*cos(c + d*x)^2), x, 5, 2*a*b*C*x + ((a^2*A + 2*A*b^2 + 2*a^2*C)*arctanh(sin(c + d*x)))/(2*d) - (b^2*(3*A - 2*C)*sin(c + d*x))/(2*d) + (A*b*(a + b*cos(c + d*x))*tan(c + d*x))/d + (A*(a + b*cos(c + d*x))^2*sec(c + d*x)*tan(c + d*x))/(2*d)],
[sec(c + d*x)^4*(a + b*cos(c + d*x))^2*(A + C*cos(c + d*x)^2), x, 5, b^2*C*x + (a*b*(A + 2*C)*arctanh(sin(c + d*x)))/d + ((2*a^2*A + A*b^2 + 3*a^2*C)*tan(c + d*x))/(3*d) + (A*b*(b + a*sec(c + d*x))*tan(c + d*x))/(3*d) + (A*(b + a*sec(c + d*x))^2*tan(c + d*x))/(3*d)],
[sec(c + d*x)^5*(a + b*cos(c + d*x))^2*(A + C*cos(c + d*x)^2), x, 6, ((3*a^2*A + 4*A*b^2 + 4*a^2*C + 8*b^2*C)*arctanh(sin(c + d*x)))/(8*d) + (2*a*b*(2*A + 3*C)*tan(c + d*x))/(3*d) + ((9*a^2*A + 2*A*b^2 + 12*a^2*C)*sec(c + d*x)*tan(c + d*x))/(24*d) + (A*b*(a + b*cos(c + d*x))*sec(c + d*x)^2*tan(c + d*x))/(6*d) + (A*(a + b*cos(c + d*x))^2*sec(c + d*x)^3*tan(c + d*x))/(4*d)],
[sec(c + d*x)^6*(a + b*cos(c + d*x))^2*(A + C*cos(c + d*x)^2), x, 7, (a*b*(3*A + 4*C)*arctanh(sin(c + d*x)))/(4*d) + ((8*a^2*A + 10*A*b^2 + 10*a^2*C + 15*b^2*C)*tan(c + d*x))/(15*d) + (a*b*(3*A + 4*C)*sec(c + d*x)*tan(c + d*x))/(4*d) + ((8*a^2*A + A*b^2 + 10*a^2*C)*sec(c + d*x)^2*tan(c + d*x))/(30*d) + (A*b*(a + b*cos(c + d*x))*sec(c + d*x)^3*tan(c + d*x))/(10*d) + (A*(a + b*cos(c + d*x))^2*sec(c + d*x)^4*tan(c + d*x))/(5*d)],


[cos(c + d*x)^1*(a + b*cos(c + d*x))^3*(A + C*cos(c + d*x)^2), x, 6, (b*(24*a^2*A + 6*A*b^2 + 18*a^2*C + 5*b^2*C)*x)/16 + (a*(15*a^2*A + 30*A*b^2 + 10*a^2*C + 24*b^2*C)*sin(c + d*x))/(15*d) + (b*(24*a^2*A + 6*A*b^2 + 18*a^2*C + 5*b^2*C)*cos(c + d*x)*sin(c + d*x))/(16*d) + (a*(90*A*b^2 + 2*a^2*C + 71*b^2*C)*cos(c + d*x)^2*sin(c + d*x))/(120*d) + ((30*A*b^2 + 6*a^2*C + 25*b^2*C)*cos(c + d*x)^2*(a + b*cos(c + d*x))*sin(c + d*x))/(120*d) + (a*C*cos(c + d*x)^2*(a + b*cos(c + d*x))^2*sin(c + d*x))/(10*d) + (C*cos(c + d*x)^2*(a + b*cos(c + d*x))^3*sin(c + d*x))/(6*d)],
[cos(c + d*x)^0*(a + b*cos(c + d*x))^3*(A + C*cos(c + d*x)^2), x, 5, (1/8)*a*(8*a^2*A + 12*A*b^2 + 4*a^2*C + 9*b^2*C)*x + ((80*a^2*A*b^2 + 20*A*b^4 - 3*a^4*C + 52*a^2*b^2*C + 16*b^4*C)*sin(c + d*x))/(30*b*d) + (a*(100*A*b^2 - 6*a^2*C + 71*b^2*C)*cos(c + d*x)*sin(c + d*x))/(120*d) + ((20*A*b^2 - 3*a^2*C + 16*b^2*C)*(a + b*cos(c + d*x))^2*sin(c + d*x))/(60*b*d) - (a*C*(a + b*cos(c + d*x))^3*sin(c + d*x))/(20*b*d) + (C*(a + b*cos(c + d*x))^4*sin(c + d*x))/(5*b*d)],
[sec(c + d*x)^1*(a + b*cos(c + d*x))^3*(A + C*cos(c + d*x)^2), x, 6, (1/8)*b*(24*a^2*A + 4*A*b^2 + 12*a^2*C + 3*b^2*C)*x + (a^3*A*arctanh(sin(c + d*x)))/d + (a*(20*A*b^2 + 2*a^2*C + 13*b^2*C)*sin(c + d*x))/(8*d) + ((4*A*b^2 + 2*a^2*C + 3*b^2*C)*(a + b*cos(c + d*x))*sin(c + d*x))/(8*d) + (a*C*(a + b*cos(c + d*x))^2*sin(c + d*x))/(4*d) + (C*(a + b*cos(c + d*x))^3*sin(c + d*x))/(4*d)],
[sec(c + d*x)^2*(a + b*cos(c + d*x))^3*(A + C*cos(c + d*x)^2), x, 6, (1/2)*a*(6*A*b^2 + 2*a^2*C + 3*b^2*C)*x + (3*a^2*A*b*arctanh(sin(c + d*x)))/d - (b*(6*a^2*A - 6*A*b^2 - 11*a^2*C - 4*b^2*C)*sin(c + d*x))/(6*d) - (a*b*(6*A - 5*C)*(a + b*cos(c + d*x))*sin(c + d*x))/(6*d) - (b*(3*A - C)*(a + b*cos(c + d*x))^2*sin(c + d*x))/(3*d) + (A*(a + b*cos(c + d*x))^3*tan(c + d*x))/d],
[sec(c + d*x)^3*(a + b*cos(c + d*x))^3*(A + C*cos(c + d*x)^2), x, 6, (1/2)*b*(2*A*b^2 + 6*a^2*C + b^2*C)*x + (a*(a^2*A + 6*A*b^2 + 2*a^2*C)*arctanh(sin(c + d*x)))/(2*d) - (5*a*b^2*(A - C)*sin(c + d*x))/(2*d) - (b^2*(4*A - C)*(a + b*cos(c + d*x))*sin(c + d*x))/(2*d) + (3*A*b*(a + b*cos(c + d*x))^2*tan(c + d*x))/(2*d) + (A*(a + b*cos(c + d*x))^3*sec(c + d*x)*tan(c + d*x))/(2*d)],
[sec(c + d*x)^4*(a + b*cos(c + d*x))^3*(A + C*cos(c + d*x)^2), x, 6, 3*a*b^2*C*x + (b*(3*a^2*A + 2*A*b^2 + 6*a^2*C)*arctanh(sin(c + d*x)))/(2*d) - (b*(4*a^2*A + 11*A*b^2 + 6*a^2*C - 6*b^2*C)*sin(c + d*x))/(6*d) + ((2*a^2*A + 3*A*b^2 + 3*a^2*C)*(a + b*cos(c + d*x))*tan(c + d*x))/(3*d) + (A*b*(a + b*cos(c + d*x))^2*sec(c + d*x)*tan(c + d*x))/(2*d) + (A*(a + b*cos(c + d*x))^3*sec(c + d*x)^2*tan(c + d*x))/(3*d)],
[sec(c + d*x)^5*(a + b*cos(c + d*x))^3*(A + C*cos(c + d*x)^2), x, 6, b^3*C*x + (a*(3*a^2*A + 12*A*b^2 + 4*a^2*C + 24*b^2*C)*arctanh(sin(c + d*x)))/(8*d) + (b*(13*a^2*A + 2*A*b^2 + 20*a^2*C)*tan(c + d*x))/(8*d) + ((3*a^2*A + 2*A*b^2 + 4*a^2*C)*(b + a*sec(c + d*x))*tan(c + d*x))/(8*d) + (A*b*(b + a*sec(c + d*x))^2*tan(c + d*x))/(4*d) + (A*(b + a*sec(c + d*x))^3*tan(c + d*x))/(4*d)],
[sec(c + d*x)^6*(a + b*cos(c + d*x))^3*(A + C*cos(c + d*x)^2), x, 7, (b*(9*a^2*A + 4*A*b^2 + 12*a^2*C + 8*b^2*C)*arctanh(sin(c + d*x)))/(8*d) + (a*(8*a^2*A + 30*A*b^2 + 10*a^2*C + 45*b^2*C)*tan(c + d*x))/(15*d) + (b*(103*a^2*A + 6*A*b^2 + 140*a^2*C)*sec(c + d*x)*tan(c + d*x))/(120*d) + ((8*a^2*A + 3*A*b^2 + 10*a^2*C)*(a + b*cos(c + d*x))*sec(c + d*x)^2*tan(c + d*x))/(30*d) + (3*A*b*(a + b*cos(c + d*x))^2*sec(c + d*x)^3*tan(c + d*x))/(20*d) + (A*(a + b*cos(c + d*x))^3*sec(c + d*x)^4*tan(c + d*x))/(5*d)],
[sec(c + d*x)^7*(a + b*cos(c + d*x))^3*(A + C*cos(c + d*x)^2), x, 8, (a*(5*a^2*A + 18*A*b^2 + 6*a^2*C + 24*b^2*C)*arctanh(sin(c + d*x)))/(16*d) + (b*(24*a^2*A + 10*A*b^2 + 30*a^2*C + 15*b^2*C)*tan(c + d*x))/(15*d) + (a*(5*a^2*A + 18*A*b^2 + 6*a^2*C + 24*b^2*C)*sec(c + d*x)*tan(c + d*x))/(16*d) + (b*(71*a^2*A + 2*A*b^2 + 90*a^2*C)*sec(c + d*x)^2*tan(c + d*x))/(120*d) + ((25*a^2*A + 6*A*b^2 + 30*a^2*C)*(a + b*cos(c + d*x))*sec(c + d*x)^3*tan(c + d*x))/(120*d) + (A*b*(a + b*cos(c + d*x))^2*sec(c + d*x)^4*tan(c + d*x))/(10*d) + (A*(a + b*cos(c + d*x))^3*sec(c + d*x)^5*tan(c + d*x))/(6*d)],


[cos(c + d*x)^1*(a + b*cos(c + d*x))^4*(A + C*cos(c + d*x)^2), x, 7, (a*b*(8*a^2*A + 6*A*b^2 + 6*a^2*C + 5*b^2*C)*x)/4 + ((105*a^4*A + 420*a^2*A*b^2 + 56*A*b^4 + 70*a^4*C + 336*a^2*b^2*C + 48*b^4*C)*sin(c + d*x))/(105*d) + (a*b*(8*a^2*A + 6*A*b^2 + 6*a^2*C + 5*b^2*C)*cos(c + d*x)*sin(c + d*x))/(4*d) + ((252*a^2*A*b^2 + 56*A*b^4 + 2*a^4*C + 197*a^2*b^2*C + 48*b^4*C)*cos(c + d*x)^2*sin(c + d*x))/(210*d) + (a*(126*A*b^2 + 6*a^2*C + 103*b^2*C)*cos(c + d*x)^2*(a + b*cos(c + d*x))*sin(c + d*x))/(210*d) + ((7*A*b^2 + 2*a^2*C + 6*b^2*C)*cos(c + d*x)^2*(a + b*cos(c + d*x))^2*sin(c + d*x))/(35*d) + (2*a*C*cos(c + d*x)^2*(a + b*cos(c + d*x))^3*sin(c + d*x))/(21*d) + (C*cos(c + d*x)^2*(a + b*cos(c + d*x))^4*sin(c + d*x))/(7*d)],
[cos(c + d*x)^0*(a + b*cos(c + d*x))^4*(A + C*cos(c + d*x)^2), x, 6, (1/16)*(16*a^4*A + 48*a^2*A*b^2 + 6*A*b^4 + 8*a^4*C + 36*a^2*b^2*C + 5*b^4*C)*x + (a*(190*a^2*A*b^2 + 160*A*b^4 - 4*a^4*C + 121*a^2*b^2*C + 128*b^4*C)*sin(c + d*x))/(60*b*d) + ((260*a^2*A*b^2 + 90*A*b^4 - 8*a^4*C + 178*a^2*b^2*C + 75*b^4*C)*cos(c + d*x)*sin(c + d*x))/(240*d) + (a*(70*A*b^2 - 4*a^2*C + 53*b^2*C)*(a + b*cos(c + d*x))^2*sin(c + d*x))/(120*b*d) + ((30*A*b^2 - 4*a^2*C + 25*b^2*C)*(a + b*cos(c + d*x))^3*sin(c + d*x))/(120*b*d) - (a*C*(a + b*cos(c + d*x))^4*sin(c + d*x))/(30*b*d) + (C*(a + b*cos(c + d*x))^5*sin(c + d*x))/(6*b*d)],
[sec(c + d*x)^1*(a + b*cos(c + d*x))^4*(A + C*cos(c + d*x)^2), x, 7, (1/2)*a*b*(8*a^2*A + 4*A*b^2 + 4*a^2*C + 3*b^2*C)*x + (a^4*A*arctanh(sin(c + d*x)))/d + ((130*a^2*A*b^2 + 20*A*b^4 + 6*a^4*C + 83*a^2*b^2*C + 16*b^4*C)*sin(c + d*x))/(30*d) + (a*(40*A*b^2 + 6*a^2*C + 29*b^2*C)*(a + b*cos(c + d*x))*sin(c + d*x))/(30*d) + ((5*A*b^2 + 3*a^2*C + 4*b^2*C)*(a + b*cos(c + d*x))^2*sin(c + d*x))/(15*d) + (a*C*(a + b*cos(c + d*x))^3*sin(c + d*x))/(5*d) + (C*(a + b*cos(c + d*x))^4*sin(c + d*x))/(5*d)],
[sec(c + d*x)^2*(a + b*cos(c + d*x))^4*(A + C*cos(c + d*x)^2), x, 7, (1/8)*(48*a^2*A*b^2 + 4*A*b^4 + 8*a^4*C + 24*a^2*b^2*C + 3*b^4*C)*x + (4*a^3*A*b*arctanh(sin(c + d*x)))/d - (a*b*(24*a^2*A - 84*A*b^2 - 50*a^2*C - 55*b^2*C)*sin(c + d*x))/(24*d) - (b*(24*a^2*A - 12*A*b^2 - 26*a^2*C - 9*b^2*C)*(a + b*cos(c + d*x))*sin(c + d*x))/(24*d) - (a*b*(12*A - 7*C)*(a + b*cos(c + d*x))^2*sin(c + d*x))/(12*d) - (b*(4*A - C)*(a + b*cos(c + d*x))^3*sin(c + d*x))/(4*d) + (A*(a + b*cos(c + d*x))^4*tan(c + d*x))/d],
[sec(c + d*x)^3*(a + b*cos(c + d*x))^4*(A + C*cos(c + d*x)^2), x, 7, 2*a*b*(2*A*b^2 + 2*a^2*C + b^2*C)*x + (a^2*(a^2*A + 12*A*b^2 + 2*a^2*C)*arctanh(sin(c + d*x)))/(2*d) - (b^2*(21*a^2*A - 6*A*b^2 - 26*a^2*C - 4*b^2*C)*sin(c + d*x))/(6*d) - (a*b^2*(9*A - 4*C)*(a + b*cos(c + d*x))*sin(c + d*x))/(3*d) - (b^2*(15*A - 2*C)*(a + b*cos(c + d*x))^2*sin(c + d*x))/(6*d) + (2*A*b*(a + b*cos(c + d*x))^3*tan(c + d*x))/d + (A*(a + b*cos(c + d*x))^4*sec(c + d*x)*tan(c + d*x))/(2*d)],
[sec(c + d*x)^4*(a + b*cos(c + d*x))^4*(A + C*cos(c + d*x)^2), x, 7, (1/2)*b^2*(2*A*b^2 + 12*a^2*C + b^2*C)*x + (2*a*b*(a^2*A + 2*A*b^2 + 2*a^2*C)*arctanh(sin(c + d*x)))/d - (a*b*(4*a^2*A + 26*A*b^2 + 6*a^2*C - 21*b^2*C)*sin(c + d*x))/(6*d) - (b*(4*a^2*A + 18*A*b^2 + 6*a^2*C - 3*b^2*C)*(a + b*cos(c + d*x))*sin(c + d*x))/(6*d) + ((2*a^2*A + 6*A*b^2 + 3*a^2*C)*(a + b*cos(c + d*x))^2*tan(c + d*x))/(3*d) + (2*A*b*(a + b*cos(c + d*x))^3*sec(c + d*x)*tan(c + d*x))/(3*d) + (A*(a + b*cos(c + d*x))^4*sec(c + d*x)^2*tan(c + d*x))/(3*d)],
[sec(c + d*x)^5*(a + b*cos(c + d*x))^4*(A + C*cos(c + d*x)^2), x, 7, 4*a*b^3*C*x + ((3*a^4*A + 24*a^2*A*b^2 + 8*A*b^4 + 4*a^4*C + 48*a^2*b^2*C)*arctanh(sin(c + d*x)))/(8*d) - (b^2*(55*a^2*A + 50*A*b^2 + 84*a^2*C - 24*b^2*C)*sin(c + d*x))/(24*d) + (b*(23*a^2*A + 12*A*b^2 + 36*a^2*C)*(a + b*cos(c + d*x))*tan(c + d*x))/(12*d) + ((3*a^2*A + 4*A*b^2 + 4*a^2*C)*(a + b*cos(c + d*x))^2*sec(c + d*x)*tan(c + d*x))/(8*d) + (A*b*(a + b*cos(c + d*x))^3*sec(c + d*x)^2*tan(c + d*x))/(3*d) + (A*(a + b*cos(c + d*x))^4*sec(c + d*x)^3*tan(c + d*x))/(4*d)],
[sec(c + d*x)^6*(a + b*cos(c + d*x))^4*(A + C*cos(c + d*x)^2), x, 7, b^4*C*x + (a*b*(3*a^2*A + 4*A*b^2 + 4*a^2*C + 8*b^2*C)*arctanh(sin(c + d*x)))/(2*d) + ((16*a^4*A + 83*a^2*A*b^2 + 6*A*b^4 + 20*a^4*C + 130*a^2*b^2*C)*tan(c + d*x))/(30*d) + (b*(29*a^2*A + 6*A*b^2 + 40*a^2*C)*(b + a*sec(c + d*x))*tan(c + d*x))/(30*d) + ((4*a^2*A + 3*A*b^2 + 5*a^2*C)*(b + a*sec(c + d*x))^2*tan(c + d*x))/(15*d) + (A*b*(b + a*sec(c + d*x))^3*tan(c + d*x))/(5*d) + (A*(b + a*sec(c + d*x))^4*tan(c + d*x))/(5*d)],
[sec(c + d*x)^7*(a + b*cos(c + d*x))^4*(A + C*cos(c + d*x)^2), x, 8, ((5*a^4*A + 36*a^2*A*b^2 + 8*A*b^4 + 6*a^4*C + 48*a^2*b^2*C + 16*b^4*C)*arctanh(sin(c + d*x)))/(16*d) + (4*a*b*(8*a^2*A + 10*A*b^2 + 10*a^2*C + 15*b^2*C)*tan(c + d*x))/(15*d) + ((75*a^4*A + 334*a^2*A*b^2 + 8*A*b^4 + 90*a^4*C + 460*a^2*b^2*C)*sec(c + d*x)*tan(c + d*x))/(240*d) + (b*(39*a^2*A + 4*A*b^2 + 50*a^2*C)*(a + b*cos(c + d*x))*sec(c + d*x)^2*tan(c + d*x))/(60*d) + ((25*a^2*A + 12*A*b^2 + 30*a^2*C)*(a + b*cos(c + d*x))^2*sec(c + d*x)^3*tan(c + d*x))/(120*d) + (2*A*b*(a + b*cos(c + d*x))^3*sec(c + d*x)^4*tan(c + d*x))/(15*d) + (A*(a + b*cos(c + d*x))^4*sec(c + d*x)^5*tan(c + d*x))/(6*d)],
[sec(c + d*x)^8*(a + b*cos(c + d*x))^4*(A + C*cos(c + d*x)^2), x, 9, (a*b*(5*a^2*A + 6*A*b^2 + 6*a^2*C + 8*b^2*C)*arctanh(sin(c + d*x)))/(4*d) + ((48*a^4*A + 336*a^2*A*b^2 + 70*A*b^4 + 56*a^4*C + 420*a^2*b^2*C + 105*b^4*C)*tan(c + d*x))/(105*d) + (a*b*(5*a^2*A + 6*A*b^2 + 6*a^2*C + 8*b^2*C)*sec(c + d*x)*tan(c + d*x))/(4*d) + ((48*a^4*A + 197*a^2*A*b^2 + 2*A*b^4 + 56*a^4*C + 252*a^2*b^2*C)*sec(c + d*x)^2*tan(c + d*x))/(210*d) + (b*(103*a^2*A + 6*A*b^2 + 126*a^2*C)*(a + b*cos(c + d*x))*sec(c + d*x)^3*tan(c + d*x))/(210*d) + ((6*a^2*A + 2*A*b^2 + 7*a^2*C)*(a + b*cos(c + d*x))^2*sec(c + d*x)^4*tan(c + d*x))/(35*d) + (2*A*b*(a + b*cos(c + d*x))^3*sec(c + d*x)^5*tan(c + d*x))/(21*d) + (A*(a + b*cos(c + d*x))^4*sec(c + d*x)^6*tan(c + d*x))/(7*d)],


[(a^2 - b^2*cos(c + d*x)^2)*(a + b*cos(c + d*x))^3, x, 5, (1/8)*a*(8*a^4 + 8*a^2*b^2 - 9*b^4)*x + (b*(83*a^4 - 32*a^2*b^2 - 16*b^4)*sin(c + d*x))/(30*d) + (a*b^2*(106*a^2 - 71*b^2)*cos(c + d*x)*sin(c + d*x))/(120*d) + (b*(23*a^2 - 16*b^2)*(a + b*cos(c + d*x))^2*sin(c + d*x))/(60*d) + (a*b*(a + b*cos(c + d*x))^3*sin(c + d*x))/(20*d) - (b*(a + b*cos(c + d*x))^4*sin(c + d*x))/(5*d)],
[(a^2 - b^2*cos(c + d*x)^2)*(a + b*cos(c + d*x))^2, x, 4, (1/8)*(8*a^4 - 3*b^4)*x + (a*b*(13*a^2 - 8*b^2)*sin(c + d*x))/(6*d) + (b^2*(14*a^2 - 9*b^2)*cos(c + d*x)*sin(c + d*x))/(24*d) + (a*b*(a + b*cos(c + d*x))^2*sin(c + d*x))/(12*d) - (b*(a + b*cos(c + d*x))^3*sin(c + d*x))/(4*d)],
[(a^2 - b^2*cos(c + d*x)^2)*(a + b*cos(c + d*x)), x, 3, (1/2)*a*(2*a^2 - b^2)*x + (2*b*(2*a^2 - b^2)*sin(c + d*x))/(3*d) + (a*b^2*cos(c + d*x)*sin(c + d*x))/(6*d) - (b*(a + b*cos(c + d*x))^2*sin(c + d*x))/(3*d)],


# ::Subsubsection::Closed:: 
#n<0


[cos(c + d*x)^3*(A + C*cos(c + d*x)^2)/(a + b*cos(c + d*x)), x, 6, ((8*a^2*A*b^2 + 4*A*b^4 + 8*a^4*C + 4*a^2*b^2*C + 3*b^4*C)*x)/(8*b^5) - (2*a^3*(A*b^2 + a^2*C)*arctan(((a - b)*tan((c + d*x)/2))/sqrt(a^2 - b^2)))/(b^5*sqrt(a^2 - b^2)*d) - (a*(3*A*b^2 + 3*a^2*C + 2*b^2*C)*sin(c + d*x))/(3*b^4*d) + ((4*A*b^2 + 4*a^2*C + 3*b^2*C)*cos(c + d*x)*sin(c + d*x))/(8*b^3*d) - (a*C*cos(c + d*x)^2*sin(c + d*x))/(3*b^2*d) + (C*cos(c + d*x)^3*sin(c + d*x))/(4*b*d)],
[cos(c + d*x)^2*(A + C*cos(c + d*x)^2)/(a + b*cos(c + d*x)), x, 5, -(a*(2*A*b^2 + 2*a^2*C + b^2*C)*x)/(2*b^4) + (2*a^2*(A*b^2 + a^2*C)*arctan(((a - b)*tan((c + d*x)/2))/sqrt(a^2 - b^2)))/(b^4*sqrt(a^2 - b^2)*d) + ((3*A*b^2 + 3*a^2*C + 2*b^2*C)*sin(c + d*x))/(3*b^3*d) - (a*C*cos(c + d*x)*sin(c + d*x))/(2*b^2*d) + (C*cos(c + d*x)^2*sin(c + d*x))/(3*b*d)],
[cos(c + d*x)^1*(A + C*cos(c + d*x)^2)/(a + b*cos(c + d*x)), x, 4, ((2*A*b^2 + 2*a^2*C + b^2*C)*x)/(2*b^3) - (2*a*(A*b^2 + a^2*C)*arctan(((a - b)*tan((c + d*x)/2))/sqrt(a^2 - b^2)))/(b^3*sqrt(a^2 - b^2)*d) - (a*C*sin(c + d*x))/(b^2*d) + (C*cos(c + d*x)*sin(c + d*x))/(2*b*d)],
[cos(c + d*x)^0*(A + C*cos(c + d*x)^2)/(a + b*cos(c + d*x)), x, 3, -((a*C*x)/b^2) + (2*(A*b^2 + a^2*C)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^2*sqrt(a^2 - b^2)*d) + (C*sin(c + d*x))/(b*d)],
[sec(c + d*x)^1*(A + C*cos(c + d*x)^2)/(a + b*cos(c + d*x)), x, 4, (C*x)/b - (2*(A*b^2 + a^2*C)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a*b*sqrt(a^2 - b^2)*d) + (A*arctanh(sin(c + d*x)))/(a*d)],
[sec(c + d*x)^2*(A + C*cos(c + d*x)^2)/(a + b*cos(c + d*x)), x, 4, (2*(A*b^2 + a^2*C)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^2*sqrt(a^2 - b^2)*d) - (A*b*arctanh(sin(c + d*x)))/(a^2*d) + (A*tan(c + d*x))/(a*d)],
[sec(c + d*x)^3*(A + C*cos(c + d*x)^2)/(a + b*cos(c + d*x)), x, 5, -((2*b*(A*b^2 + a^2*C)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^3*sqrt(a^2 - b^2)*d)) + ((a^2*A + 2*A*b^2 + 2*a^2*C)*arctanh(sin(c + d*x)))/(2*a^3*d) - (A*b*tan(c + d*x))/(a^2*d) + (A*sec(c + d*x)*tan(c + d*x))/(2*a*d)],
[sec(c + d*x)^4*(A + C*cos(c + d*x)^2)/(a + b*cos(c + d*x)), x, 6, (2*b^2*(A*b^2 + a^2*C)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^4*sqrt(a^2 - b^2)*d) - (b*(a^2*A + 2*A*b^2 + 2*a^2*C)*arctanh(sin(c + d*x)))/(2*a^4*d) + ((2*a^2*A + 3*A*b^2 + 3*a^2*C)*tan(c + d*x))/(3*a^3*d) - (A*b*sec(c + d*x)*tan(c + d*x))/(2*a^2*d) + (A*sec(c + d*x)^2*tan(c + d*x))/(3*a*d)],


[cos(c + d*x)^3*(A + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^2, x, 6, -((a*(2*A*b^2 + 4*a^2*C + b^2*C)*x)/b^5) + (2*a^2*(2*a^2*A*b^2 - 3*A*b^4 + 4*a^4*C - 5*a^2*b^2*C)*arctan(((a - b)*tan((c + d*x)/2))/sqrt(a^2 - b^2)))/(b^5*(a^2 - b^2)^(3/2)*d) + ((6*a^2*A*b^2 - 3*A*b^4 + 12*a^4*C - 7*a^2*b^2*C - 2*b^4*C)*sin(c + d*x))/(3*b^4*(a^2 - b^2)*d) - (a*(A*b^2 + 2*a^2*C - b^2*C)*cos(c + d*x)*sin(c + d*x))/(b^3*(a^2 - b^2)*d) + ((3*A*b^2 + 4*a^2*C - b^2*C)*cos(c + d*x)^2*sin(c + d*x))/(3*b^2*(a^2 - b^2)*d) - ((A*b^2 + a^2*C)*cos(c + d*x)^3*sin(c + d*x))/(b*(a^2 - b^2)*d*(a + b*cos(c + d*x)))],
[cos(c + d*x)^2*(A + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^2, x, 5, ((2*A*b^2 + 6*a^2*C + b^2*C)*x)/(2*b^4) - (2*a*(a^2*A*b^2 - 2*A*b^4 + 3*a^4*C - 4*a^2*b^2*C)*arctan(((a - b)*tan((c + d*x)/2))/sqrt(a^2 - b^2)))/(b^4*(a^2 - b^2)^(3/2)*d) - (a*(A*b^2 + 3*a^2*C - 2*b^2*C)*sin(c + d*x))/(b^3*(a^2 - b^2)*d) + ((2*A*b^2 + 3*a^2*C - b^2*C)*cos(c + d*x)*sin(c + d*x))/(2*b^2*(a^2 - b^2)*d) - ((A*b^2 + a^2*C)*cos(c + d*x)^2*sin(c + d*x))/(b*(a^2 - b^2)*d*(a + b*cos(c + d*x)))],
[cos(c + d*x)^1*(A + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^2, x, 4, (-2*a*C*x)/b^3 - (2*(A*b^4 - 2*a^4*C + 3*a^2*b^2*C)*arctan(((a - b)*tan((c + d*x)/2))/sqrt(a^2 - b^2)))/(b^3*(a^2 - b^2)^(3/2)*d) + ((A*b^2 + 2*a^2*C - b^2*C)*sin(c + d*x))/(b^2*(a^2 - b^2)*d) - ((A*b^2 + a^2*C)*cos(c + d*x)*sin(c + d*x))/(b*(a^2 - b^2)*d*(a + b*cos(c + d*x)))],
[cos(c + d*x)^0*(A + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^2, x, 3, (C*x)/b^2 + (2*a*(A*b^2 - a^2*C + 2*b^2*C)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^2*(a^2 - b^2)^(3/2)*d) - ((A*b^2 + a^2*C)*sin(c + d*x))/(b*(a^2 - b^2)*d*(a + b*cos(c + d*x)))],
[sec(c + d*x)^1*(A + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^2, x, 4, -((2*b*(2*a^2*A - A*b^2 + a^2*C)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^2*(a^2 - b^2)^(3/2)*d)) + (A*arctanh(sin(c + d*x)))/(a^2*d) + ((A*b^2 + a^2*C)*sin(c + d*x))/(a*(a^2 - b^2)*d*(a + b*cos(c + d*x)))],
[sec(c + d*x)^2*(A + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^2, x, 5, (2*(3*a^2*A*b^2 - 2*A*b^4 + a^4*C)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^3*(a^2 - b^2)^(3/2)*d) - (2*A*b*arctanh(sin(c + d*x)))/(a^3*d) + ((a^2*A - 2*A*b^2 - a^2*C)*tan(c + d*x))/(a^2*(a^2 - b^2)*d) + ((A*b^2 + a^2*C)*tan(c + d*x))/(a*(a^2 - b^2)*d*(a + b*cos(c + d*x)))],
[sec(c + d*x)^3*(A + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^2, x, 6, -((2*b*(4*a^2*A*b^2 - 3*A*b^4 + 2*a^4*C - a^2*b^2*C)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^4*(a^2 - b^2)^(3/2)*d)) + ((a^2*A + 6*A*b^2 + 2*a^2*C)*arctanh(sin(c + d*x)))/(2*a^4*d) - (b*(2*a^2*A - 3*A*b^2 - a^2*C)*tan(c + d*x))/(a^3*(a^2 - b^2)*d) + ((a^2*A - 3*A*b^2 - 2*a^2*C)*sec(c + d*x)*tan(c + d*x))/(2*a^2*(a^2 - b^2)*d) + ((A*b^2 + a^2*C)*sec(c + d*x)*tan(c + d*x))/(a*(a^2 - b^2)*d*(a + b*cos(c + d*x)))],
[sec(c + d*x)^4*(A + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^2, x, 7, (2*b^2*(5*a^2*A*b^2 - 4*A*b^4 + 3*a^4*C - 2*a^2*b^2*C)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^5*(a^2 - b^2)^(3/2)*d) - (b*(a^2*A + 4*A*b^2 + 2*a^2*C)*arctanh(sin(c + d*x)))/(a^5*d) + ((2*a^4*A + 7*a^2*A*b^2 - 12*A*b^4 + 3*a^4*C - 6*a^2*b^2*C)*tan(c + d*x))/(3*a^4*(a^2 - b^2)*d) - (b*(a^2*A - 2*A*b^2 - a^2*C)*sec(c + d*x)*tan(c + d*x))/(a^3*(a^2 - b^2)*d) + ((a^2*A - 4*A*b^2 - 3*a^2*C)*sec(c + d*x)^2*tan(c + d*x))/(3*a^2*(a^2 - b^2)*d) + ((A*b^2 + a^2*C)*sec(c + d*x)^2*tan(c + d*x))/(a*(a^2 - b^2)*d*(a + b*cos(c + d*x)))],


[cos(c + d*x)^3*(A + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^3, x, 6, ((2*A*b^2 + 12*a^2*C + b^2*C)*x)/(2*b^5) - (a*(2*a^4*A*b^2 - 5*a^2*A*b^4 + 6*A*b^6 + 12*a^6*C - 29*a^4*b^2*C + 20*a^2*b^4*C)*arctan(((a - b)*tan((c + d*x)/2))/sqrt(a^2 - b^2)))/(b^5*(a^2 - b^2)^(5/2)*d) - (a*(2*a^2*A*b^2 - 5*A*b^4 + 12*a^4*C - 21*a^2*b^2*C + 6*b^4*C)*sin(c + d*x))/(2*b^4*(a^2 - b^2)^2*d) + ((a^2*A*b^2 - 4*A*b^4 + 6*a^4*C - 10*a^2*b^2*C + b^4*C)*cos(c + d*x)*sin(c + d*x))/(2*b^3*(a^2 - b^2)^2*d) - ((A*b^2 + a^2*C)*cos(c + d*x)^3*sin(c + d*x))/(2*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) + ((3*A*b^4 - 4*a^4*C + 7*a^2*b^2*C)*cos(c + d*x)^2*sin(c + d*x))/(2*b^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))],
[cos(c + d*x)^2*(A + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^3, x, 5, (-3*a*C*x)/b^4 + ((a^2*A*b^4 + 2*A*b^6 + 6*a^6*C - 15*a^4*b^2*C + 12*a^2*b^4*C)*arctan(((a - b)*tan((c + d*x)/2))/sqrt(a^2 - b^2)))/(b^4*(a^2 - b^2)^(5/2)*d) - ((3*A*b^4 - 6*a^4*C + 11*a^2*b^2*C - 2*b^4*C)*sin(c + d*x))/(2*b^3*(a^2 - b^2)^2*d) - ((A*b^2 + a^2*C)*cos(c + d*x)^2*sin(c + d*x))/(2*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) + ((a^2*A*b^2 + 2*A*b^4 - 3*a^4*C + 6*a^2*b^2*C)*cos(c + d*x)*sin(c + d*x))/(2*b^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))],
[cos(c + d*x)^1*(A + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^3, x, 4, (C*x)/b^3 - (a*(3*A*b^4 + 2*a^4*C - 5*a^2*b^2*C + 6*b^4*C)*arctan(((a - b)*tan((c + d*x)/2))/sqrt(a^2 - b^2)))/(b^3*(a^2 - b^2)^(5/2)*d) - ((A*b^2 + a^2*C)*cos(c + d*x)*sin(c + d*x))/(2*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) + ((2*a^2*A*b^2 + A*b^4 - 2*a^4*C + 5*a^2*b^2*C)*sin(c + d*x))/(2*b^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))],
[cos(c + d*x)^0*(A + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^3, x, 4, ((2*a^2*A + A*b^2 + a^2*C + 2*b^2*C)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/((a^2 - b^2)^(5/2)*d) - ((A*b^2 + a^2*C)*sin(c + d*x))/(2*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) - (a*(3*A*b^2 - a^2*C + 4*b^2*C)*sin(c + d*x))/(2*b*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))],
[sec(c + d*x)^1*(A + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^3, x, 5, -((b*(6*a^4*A - 5*a^2*A*b^2 + 2*A*b^4 + 3*a^4*C)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^3*(a^2 - b^2)^(5/2)*d)) + (A*arctanh(sin(c + d*x)))/(a^3*d) + ((A*b^2 + a^2*C)*sin(c + d*x))/(2*a*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) + ((5*a^2*A*b^2 - 2*A*b^4 + a^4*C + 2*a^2*b^2*C)*sin(c + d*x))/(2*a^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))],
[sec(c + d*x)^2*(A + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^3, x, 6, ((12*a^4*A*b^2 - 15*a^2*A*b^4 + 6*A*b^6 + 2*a^6*C + a^4*b^2*C)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^4*(a^2 - b^2)^(5/2)*d) - (3*A*b*arctanh(sin(c + d*x)))/(a^4*d) + ((2*a^4*A - 11*a^2*A*b^2 + 6*A*b^4 - 3*a^4*C)*tan(c + d*x))/(2*a^3*(a^2 - b^2)^2*d) + ((A*b^2 + a^2*C)*tan(c + d*x))/(2*a*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) + ((6*a^2*A*b^2 - 3*A*b^4 + 2*a^4*C + a^2*b^2*C)*tan(c + d*x))/(2*a^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))],
[sec(c + d*x)^3*(A + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^3, x, 7, -((b*(20*a^4*A*b^2 - 29*a^2*A*b^4 + 12*A*b^6 + 6*a^6*C - 5*a^4*b^2*C + 2*a^2*b^4*C)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^5*(a^2 - b^2)^(5/2)*d)) + ((a^2*A + 12*A*b^2 + 2*a^2*C)*arctanh(sin(c + d*x)))/(2*a^5*d) - (b*(6*a^4*A - 21*a^2*A*b^2 + 12*A*b^4 - 5*a^4*C + 2*a^2*b^2*C)*tan(c + d*x))/(2*a^4*(a^2 - b^2)^2*d) + ((a^4*A - 10*a^2*A*b^2 + 6*A*b^4 - 4*a^4*C + a^2*b^2*C)*sec(c + d*x)*tan(c + d*x))/(2*a^3*(a^2 - b^2)^2*d) + ((A*b^2 + a^2*C)*sec(c + d*x)*tan(c + d*x))/(2*a*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) + ((7*a^2*A*b^2 - 4*A*b^4 + 3*a^4*C)*sec(c + d*x)*tan(c + d*x))/(2*a^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))],


[cos(c + d*x)^4*(A + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^4, x, 7, ((2*A*b^2 + 20*a^2*C + b^2*C)*x)/(2*b^6) - (a*(2*a^6*A*b^2 - 7*a^4*A*b^4 + 8*a^2*A*b^6 - 8*A*b^8 + 20*a^8*C - 69*a^6*b^2*C + 84*a^4*b^4*C - 40*a^2*b^6*C)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^6*(a^2 - b^2)^(7/2)*d) - (a*(6*a^4*A*b^2 - 17*a^2*A*b^4 + 26*A*b^6 + 60*a^6*C - 167*a^4*b^2*C + 146*a^2*b^4*C - 24*b^6*C)*sin(c + d*x))/(6*b^5*(a^2 - b^2)^3*d) + ((a^4*A*b^2 - 2*a^2*A*b^4 + 6*A*b^6 + 10*a^6*C - 27*a^4*b^2*C + 23*a^2*b^4*C - b^6*C)*cos(c + d*x)*sin(c + d*x))/(2*b^4*(a^2 - b^2)^3*d) - ((A*b^2 + a^2*C)*cos(c + d*x)^4*sin(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^3) + ((a^2*A*b^2 + 4*A*b^4 - 5*a^4*C + 10*a^2*b^2*C)*cos(c + d*x)^3*sin(c + d*x))/(6*b^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x))^2) - ((2*a^4*A*b^2 + a^2*A*b^4 + 12*A*b^6 + 20*a^6*C - 53*a^4*b^2*C + 48*a^2*b^4*C)*cos(c + d*x)^2*sin(c + d*x))/(6*b^3*(a^2 - b^2)^3*d*(a + b*cos(c + d*x)))],
[cos(c + d*x)^3*(A + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^4, x, 6, (-4*a*C*x)/b^5 - ((3*a^2*A*b^6 + 2*A*b^8 - 8*a^8*C + 28*a^6*b^2*C - 35*a^4*b^4*C + 20*a^2*b^6*C)*arctan(((a - b)*tan((c + d*x)/2))/sqrt(a^2 - b^2)))/(b^5*(a^2 - b^2)^(7/2)*d) + ((4*a^2*A*b^4 + 11*A*b^6 + 24*a^6*C - 68*a^4*b^2*C + 65*a^2*b^4*C - 6*b^6*C)*sin(c + d*x))/(6*b^4*(a^2 - b^2)^3*d) - ((A*b^2 + a^2*C)*cos(c + d*x)^3*sin(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^3) + ((2*a^2*A*b^2 + 3*A*b^4 - 4*a^4*C + 9*a^2*b^2*C)*cos(c + d*x)^2*sin(c + d*x))/(6*b^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x))^2) - ((3*a^2*A*b^4 + 2*A*b^6 + 4*a^6*C - 11*a^4*b^2*C + 12*a^2*b^4*C)*cos(c + d*x)*sin(c + d*x))/(2*b^3*(a^2 - b^2)^3*d*(a + b*cos(c + d*x)))],
[cos(c + d*x)^2*(A + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^4, x, 5, (C*x)/b^4 + (a*(a^2*A*b^4 + 4*A*b^6 - 2*a^6*C + 7*a^4*b^2*C - 8*a^2*b^4*C + 8*b^6*C)*arctan(((a - b)*tan((c + d*x)/2))/sqrt(a^2 - b^2)))/(b^4*(a^2 - b^2)^(7/2)*d) - ((A*b^2 + a^2*C)*cos(c + d*x)^2*sin(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^3) + ((3*a^2*A*b^2 + 2*A*b^4 - 3*a^4*C + 8*a^2*b^2*C)*cos(c + d*x)*sin(c + d*x))/(6*b^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x))^2) - ((13*a^2*A*b^4 + 2*A*b^6 + 6*a^6*C - 17*a^4*b^2*C + 26*a^2*b^4*C)*sin(c + d*x))/(6*b^3*(a^2 - b^2)^3*d*(a + b*cos(c + d*x)))],
[cos(c + d*x)^1*(A + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^4, x, 5, -((b*(4*a^2*A + A*b^2 + 3*a^2*C + 2*b^2*C)*arctan(((a - b)*tan((c + d*x)/2))/sqrt(a^2 - b^2)))/((a^2 - b^2)^(7/2)*d)) - ((A*b^2 + a^2*C)*cos(c + d*x)*sin(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^3) + ((4*a^2*A*b^2 + A*b^4 - 2*a^4*C + 7*a^2*b^2*C)*sin(c + d*x))/(6*b^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x))^2) + (a*(2*a^2*A*b^2 + 13*A*b^4 + 2*a^4*C - 5*a^2*b^2*C + 18*b^4*C)*sin(c + d*x))/(6*b^2*(a^2 - b^2)^3*d*(a + b*cos(c + d*x)))],
[cos(c + d*x)^0*(A + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^4, x, 5, (a*(2*a^2*A + 3*A*b^2 + a^2*C + 4*b^2*C)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/((a^2 - b^2)^(7/2)*d) - ((A*b^2 + a^2*C)*sin(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^3) - (a*(5*A*b^2 - a^2*C + 6*b^2*C)*sin(c + d*x))/(6*b*(a^2 - b^2)^2*d*(a + b*cos(c + d*x))^2) - ((11*a^2*A*b^2 + 4*A*b^4 - a^4*C + 10*a^2*b^2*C + 6*b^4*C)*sin(c + d*x))/(6*b*(a^2 - b^2)^3*d*(a + b*cos(c + d*x)))],
[sec(c + d*x)^1*(A + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^4, x, 6, -((b*(8*a^6*A - 8*a^4*A*b^2 + 7*a^2*A*b^4 - 2*A*b^6 + 4*a^6*C + a^4*b^2*C)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^4*(a^2 - b^2)^(7/2)*d)) + (A*arctanh(sin(c + d*x)))/(a^4*d) + ((A*b^2 + a^2*C)*sin(c + d*x))/(3*a*(a^2 - b^2)*d*(a + b*cos(c + d*x))^3) + ((8*a^2*A*b^2 - 3*A*b^4 + 2*a^4*C + 3*a^2*b^2*C)*sin(c + d*x))/(6*a^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x))^2) + ((26*a^4*A*b^2 - 17*a^2*A*b^4 + 6*A*b^6 + 2*a^6*C + 13*a^4*b^2*C)*sin(c + d*x))/(6*a^3*(a^2 - b^2)^3*d*(a + b*cos(c + d*x)))],
[sec(c + d*x)^2*(A + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^4, x, 7, ((20*a^6*A*b^2 - 35*a^4*A*b^4 + 28*a^2*A*b^6 - 8*A*b^8 + 2*a^8*C + 3*a^6*b^2*C)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^5*(a^2 - b^2)^(7/2)*d) - (4*A*b*arctanh(sin(c + d*x)))/(a^5*d) + ((6*a^6*A - 65*a^4*A*b^2 + 68*a^2*A*b^4 - 24*A*b^6 - 11*a^6*C - 4*a^4*b^2*C)*tan(c + d*x))/(6*a^4*(a^2 - b^2)^3*d) + ((A*b^2 + a^2*C)*tan(c + d*x))/(3*a*(a^2 - b^2)*d*(a + b*cos(c + d*x))^3) + ((9*a^2*A*b^2 - 4*A*b^4 + 3*a^4*C + 2*a^2*b^2*C)*tan(c + d*x))/(6*a^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x))^2) + ((12*a^4*A*b^2 - 11*a^2*A*b^4 + 4*A*b^6 + 2*a^6*C + 3*a^4*b^2*C)*tan(c + d*x))/(2*a^3*(a^2 - b^2)^3*d*(a + b*cos(c + d*x)))],
[sec(c + d*x)^3*(A + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^4, x, 8, -((b*(40*a^6*A*b^2 - 84*a^4*A*b^4 + 69*a^2*A*b^6 - 20*A*b^8 + 8*a^8*C - 8*a^6*b^2*C + 7*a^4*b^4*C - 2*a^2*b^6*C)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^6*(a^2 - b^2)^(7/2)*d)) + ((a^2*A + 20*A*b^2 + 2*a^2*C)*arctanh(sin(c + d*x)))/(2*a^6*d) - (b*(24*a^6*A - 146*a^4*A*b^2 + 167*a^2*A*b^4 - 60*A*b^6 - 26*a^6*C + 17*a^4*b^2*C - 6*a^2*b^4*C)*tan(c + d*x))/(6*a^5*(a^2 - b^2)^3*d) + ((a^6*A - 23*a^4*A*b^2 + 27*a^2*A*b^4 - 10*A*b^6 - 6*a^6*C + 2*a^4*b^2*C - a^2*b^4*C)*sec(c + d*x)*tan(c + d*x))/(2*a^4*(a^2 - b^2)^3*d) + ((A*b^2 + a^2*C)*sec(c + d*x)*tan(c + d*x))/(3*a*(a^2 - b^2)*d*(a + b*cos(c + d*x))^3) + ((10*a^2*A*b^2 - 5*A*b^4 + 4*a^4*C + a^2*b^2*C)*sec(c + d*x)*tan(c + d*x))/(6*a^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x))^2) + ((48*a^4*A*b^2 - 53*a^2*A*b^4 + 20*A*b^6 + 12*a^6*C + a^4*b^2*C + 2*a^2*b^4*C)*sec(c + d*x)*tan(c + d*x))/(6*a^3*(a^2 - b^2)^3*d*(a + b*cos(c + d*x)))],


[(a^2 - b^2*cos(c + d*x)^2)/(a + b*cos(c + d*x)), x, 2, a*x - (b*sin(c + d*x))/d],
[(a^2 - b^2*cos(c + d*x)^2)/(a + b*cos(c + d*x))^2, x, 3, -x + (4*a*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(sqrt(a^2 - b^2)*d)],
[(a^2 - b^2*cos(c + d*x)^2)/(a + b*cos(c + d*x))^3, x, 4, (2*(a^2 + b^2)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/((a^2 - b^2)^(3/2)*d) - (2*a*b*sin(c + d*x))/((a^2 - b^2)*d*(a + b*cos(c + d*x)))],
[(a^2 - b^2*cos(c + d*x)^2)/(a + b*cos(c + d*x))^4, x, 5, (2*a*(a^2 + 2*b^2)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/((a^2 - b^2)^(5/2)*d) - (a*b*sin(c + d*x))/((a^2 - b^2)*d*(a + b*cos(c + d*x))^2) - (b*(2*a^2 + b^2)*sin(c + d*x))/((a^2 - b^2)^2*d*(a + b*cos(c + d*x)))],


# ::Subsection::Closed:: 
#Integrands of the form Cos[c+d x]^m (A+C Cos[c+d x]^2) (a+b Cos[c+d x])^(n/2)


# ::Subsubsection::Closed:: 
#n>0


[cos(c + d*x)^2*sqrt(a + b*cos(c + d*x))*(A + C*cos(c + d*x)^2), x, 9, (-2*(a + b)*(42*a^2*A*b^2 - 189*A*b^4 + 16*a^4*C + 24*a^2*b^2*C - 147*b^4*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*b)/(a + b)))/(315*b^4*d*sqrt(a + b*cos(c + d*x))) + (4*a*(21*a^2*A*b^2 - 21*A*b^4 + 8*a^4*C + 10*a^2*b^2*C - 18*b^4*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*b)/(a + b)))/(315*b^4*d*sqrt(a + b*cos(c + d*x))) + (2*a*(21*A*b^2 + 8*a^2*C + 13*b^2*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(315*b^3*d) + (2*(63*A*b^2 - 6*a^2*C + 49*b^2*C)*cos(c + d*x)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(315*b^2*d) + (2*a*C*cos(c + d*x)^2*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(63*b*d) + (2*C*cos(c + d*x)^3*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(9*d)],
[cos(c + d*x)^1*sqrt(a + b*cos(c + d*x))*(A + C*cos(c + d*x)^2), x, 8, (2*a*(a + b)*(35*A*b^2 + 8*a^2*C + 19*b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*b)/(a + b)))/(105*b^3*d*sqrt(a + b*cos(c + d*x))) - (2*(35*a^2*A*b^2 - 35*A*b^4 + 8*a^4*C + 17*a^2*b^2*C - 25*b^4*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*b)/(a + b)))/(105*b^3*d*sqrt(a + b*cos(c + d*x))) + (2*(35*A*b^2 - 4*a^2*C + 25*b^2*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(105*b^2*d) + (2*a*C*cos(c + d*x)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(35*b*d) + (2*C*cos(c + d*x)^2*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(7*d)],
[cos(c + d*x)^0*sqrt(a + b*cos(c + d*x))*(A + C*cos(c + d*x)^2), x, 7, (2*(a + b)*(15*A*b^2 - 2*a^2*C + 9*b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(15*b^2*d*sqrt(a + b*cos(c + d*x))) + (4*a*(a^2 - b^2)*C*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(15*b^2*d*sqrt(a + b*cos(c + d*x))) - (4*a*C*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(15*b*d) + (2*C*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(5*b*d)],
[sec(c + d*x)^1*sqrt(a + b*cos(c + d*x))*(A + C*cos(c + d*x)^2), x, 9, (2*a*(a + b)*C*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(3*b*d*sqrt(a + b*cos(c + d*x))) + (2*(3*A*b^2 - a^2*C + b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(3*b*d*sqrt(a + b*cos(c + d*x))) + (2*a*A*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x))) + (2*C*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*d)],
[sec(c + d*x)^2*sqrt(a + b*cos(c + d*x))*(A + C*cos(c + d*x)^2), x, 9, -(((a + b)*(A - 2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x)))) + (a*A*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x))) + (A*b*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x))) + (A*sqrt(a + b*cos(c + d*x))*tan(c + d*x))/d],
[sec(c + d*x)^3*sqrt(a + b*cos(c + d*x))*(A + C*cos(c + d*x)^2), x, 10, -((A*b*(a + b)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(4*a*d*sqrt(a + b*cos(c + d*x)))) + (b*(3*A + 8*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(4*d*sqrt(a + b*cos(c + d*x))) + ((4*a^2*A - A*b^2 + 8*a^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*b)/(a + b)))/(4*a*d*sqrt(a + b*cos(c + d*x))) + (A*b*sqrt(a + b*cos(c + d*x))*tan(c + d*x))/(4*a*d) + (A*sqrt(a + b*cos(c + d*x))*sec(c + d*x)*tan(c + d*x))/(2*d)],
[sec(c + d*x)^4*sqrt(a + b*cos(c + d*x))*(A + C*cos(c + d*x)^2), x, 11, -(((a + b)*(16*a^2*A - 3*A*b^2 + 24*a^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(24*a^2*d*sqrt(a + b*cos(c + d*x)))) + ((16*a^2*A - A*b^2 + 24*a^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(24*a*d*sqrt(a + b*cos(c + d*x))) + (b*(4*a^2*A + A*b^2 + 8*a^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*b)/(a + b)))/(8*a^2*d*sqrt(a + b*cos(c + d*x))) + ((16*a^2*A - 3*A*b^2 + 24*a^2*C)*sqrt(a + b*cos(c + d*x))*tan(c + d*x))/(24*a^2*d) + (A*b*sqrt(a + b*cos(c + d*x))*sec(c + d*x)*tan(c + d*x))/(12*a*d) + (A*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^2*tan(c + d*x))/(3*d)],


[cos(c + d*x)^2*(a + b*cos(c + d*x))^(3/2)*(A + C*cos(c + d*x)^2), x, 10, (-4*a*(a + b)*(33*a^2*A*b^2 - 451*A*b^4 + 8*a^4*C + 18*a^2*b^2*C - 348*b^4*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*b)/(a + b)))/(1155*b^4*d*sqrt(a + b*cos(c + d*x))) + (2*(66*a^4*A*b^2 - 341*a^2*A*b^4 + 275*A*b^6 + 16*a^6*C + 32*a^4*b^2*C - 273*a^2*b^4*C + 225*b^6*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*b)/(a + b)))/(1155*b^4*d*sqrt(a + b*cos(c + d*x))) + (2*(33*a^2*A*b^2 + 275*A*b^4 + 8*a^4*C + 19*a^2*b^2*C + 225*b^4*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(1155*b^3*d) + (4*a*(132*A*b^2 - 3*a^2*C + 101*b^2*C)*cos(c + d*x)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(1155*b^2*d) + (2*(33*A*b^2 + a^2*C + 27*b^2*C)*cos(c + d*x)^2*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(231*b*d) + (2*a*C*cos(c + d*x)^3*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(33*d) + (2*C*cos(c + d*x)^3*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(11*d)],
[cos(c + d*x)^1*(a + b*cos(c + d*x))^(3/2)*(A + C*cos(c + d*x)^2), x, 9, (2*(a + b)*(63*a^2*A*b^2 + 189*A*b^4 + 8*a^4*C + 33*a^2*b^2*C + 147*b^4*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*b)/(a + b)))/(315*b^3*d*sqrt(a + b*cos(c + d*x))) - (2*a*(63*a^2*A*b^2 - 63*A*b^4 + 8*a^4*C + 31*a^2*b^2*C - 39*b^4*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*b)/(a + b)))/(315*b^3*d*sqrt(a + b*cos(c + d*x))) + (4*a*(63*A*b^2 - 2*a^2*C + 44*b^2*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(315*b^2*d) + (2*(63*A*b^2 + 3*a^2*C + 49*b^2*C)*cos(c + d*x)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(315*b*d) + (2*a*C*cos(c + d*x)^2*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(21*d) + (2*C*cos(c + d*x)^2*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(9*d)],
[cos(c + d*x)^0*(a + b*cos(c + d*x))^(3/2)*(A + C*cos(c + d*x)^2), x, 8, (4*a*(a + b)*(70*A*b^2 - 3*a^2*C + 41*b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(105*b^2*d*sqrt(a + b*cos(c + d*x))) - (2*(35*a^2*A*b^2 - 35*A*b^4 - 6*a^4*C + 31*a^2*b^2*C - 25*b^4*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(105*b^2*d*sqrt(a + b*cos(c + d*x))) + (2*(35*A*b^2 - 6*a^2*C + 25*b^2*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(105*b*d) - (4*a*C*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(35*b*d) + (2*C*(a + b*cos(c + d*x))^(5/2)*sin(c + d*x))/(7*b*d)],
[sec(c + d*x)^1*(a + b*cos(c + d*x))^(3/2)*(A + C*cos(c + d*x)^2), x, 10, (2*(a + b)*(5*A*b^2 + a^2*C + 3*b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(5*b*d*sqrt(a + b*cos(c + d*x))) + (2*a*(5*A*b^2 - a^2*C + b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(5*b*d*sqrt(a + b*cos(c + d*x))) + (2*a^2*A*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x))) + (2*a*C*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(5*d) + (2*C*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(5*d)],
[sec(c + d*x)^2*(a + b*cos(c + d*x))^(3/2)*(A + C*cos(c + d*x)^2), x, 10, -((a*(a + b)*(3*A - 8*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(3*d*sqrt(a + b*cos(c + d*x)))) + ((3*a^2*A + 6*A*b^2 - 2*a^2*C + 2*b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(3*d*sqrt(a + b*cos(c + d*x))) + (3*a*A*b*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x))) - (b*(3*A - 2*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*d) + (A*(a + b*cos(c + d*x))^(3/2)*tan(c + d*x))/d],
[sec(c + d*x)^3*(a + b*cos(c + d*x))^(3/2)*(A + C*cos(c + d*x)^2), x, 10, -((b*(a + b)*(5*A - 8*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(4*d*sqrt(a + b*cos(c + d*x)))) + (a*b*(7*A + 8*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(4*d*sqrt(a + b*cos(c + d*x))) + ((4*a^2*A + 3*A*b^2 + 8*a^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*b)/(a + b)))/(4*d*sqrt(a + b*cos(c + d*x))) + (3*A*b*sqrt(a + b*cos(c + d*x))*tan(c + d*x))/(4*d) + (A*(a + b*cos(c + d*x))^(3/2)*sec(c + d*x)*tan(c + d*x))/(2*d)],
[sec(c + d*x)^4*(a + b*cos(c + d*x))^(3/2)*(A + C*cos(c + d*x)^2), x, 11, -(((a + b)*(16*a^2*A + 3*A*b^2 + 24*a^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(24*a*d*sqrt(a + b*cos(c + d*x)))) + ((16*a^2*A + 17*A*b^2 + 24*a^2*C + 48*b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(24*d*sqrt(a + b*cos(c + d*x))) + (b*(12*a^2*A - A*b^2 + 24*a^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*b)/(a + b)))/(8*a*d*sqrt(a + b*cos(c + d*x))) + ((16*a^2*A + 3*A*b^2 + 24*a^2*C)*sqrt(a + b*cos(c + d*x))*tan(c + d*x))/(24*a*d) + (A*b*sqrt(a + b*cos(c + d*x))*sec(c + d*x)*tan(c + d*x))/(4*d) + (A*(a + b*cos(c + d*x))^(3/2)*sec(c + d*x)^2*tan(c + d*x))/(3*d)],
[sec(c + d*x)^5*(a + b*cos(c + d*x))^(3/2)*(A + C*cos(c + d*x)^2), x, 12, -((b*(a + b)*(52*a^2*A - 3*A*b^2 + 80*a^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(64*a^2*d*sqrt(a + b*cos(c + d*x)))) + (b*(76*a^2*A - A*b^2 + 112*a^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(64*a*d*sqrt(a + b*cos(c + d*x))) + ((48*a^4*A + 24*a^2*A*b^2 + 3*A*b^4 + 64*a^4*C + 48*a^2*b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*b)/(a + b)))/(64*a^2*d*sqrt(a + b*cos(c + d*x))) + (b*(52*a^2*A - 3*A*b^2 + 80*a^2*C)*sqrt(a + b*cos(c + d*x))*tan(c + d*x))/(64*a^2*d) + ((12*a^2*A + A*b^2 + 16*a^2*C)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)*tan(c + d*x))/(32*a*d) + (A*b*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^2*tan(c + d*x))/(8*d) + (A*(a + b*cos(c + d*x))^(3/2)*sec(c + d*x)^3*tan(c + d*x))/(4*d)],


[cos(c + d*x)^2*(a + b*cos(c + d*x))^(5/2)*(A + C*cos(c + d*x)^2), x, 11, (-2*(a + b)*(1430*a^4*A*b^2 - 39897*a^2*A*b^4 - 21021*A*b^6 + 240*a^6*C + 760*a^4*b^2*C - 30669*a^2*b^4*C - 17787*b^6*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*b)/(a + b)))/(45045*b^4*d*sqrt(a + b*cos(c + d*x))) + (4*a*(715*a^4*A*b^2 - 8866*a^2*A*b^4 + 8151*A*b^6 + 120*a^6*C + 350*a^4*b^2*C - 6992*a^2*b^4*C + 6522*b^6*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*b)/(a + b)))/(45045*b^4*d*sqrt(a + b*cos(c + d*x))) + (2*a*(715*a^2*A*b^2 + 23309*A*b^4 + 120*a^4*C + 395*a^2*b^2*C + 18973*b^4*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(45045*b^3*d) + (2*(10725*a^2*A*b^2 + 7007*A*b^4 - 90*a^4*C + 8145*a^2*b^2*C + 5929*b^4*C)*cos(c + d*x)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(45045*b^2*d) + (2*a*(2717*A*b^2 + 15*a^2*C + 2209*b^2*C)*cos(c + d*x)^2*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(9009*b*d) + (2*(143*A*b^2 + 15*a^2*C + 121*b^2*C)*cos(c + d*x)^3*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(1287*d) + (10*a*C*cos(c + d*x)^3*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(143*d) + (2*C*cos(c + d*x)^3*(a + b*cos(c + d*x))^(5/2)*sin(c + d*x))/(13*d)],
[cos(c + d*x)^1*(a + b*cos(c + d*x))^(5/2)*(A + C*cos(c + d*x)^2), x, 10, (2*a*(a + b)*(99*a^2*A*b^2 + 957*A*b^4 + 8*a^4*C + 51*a^2*b^2*C + 741*b^4*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*b)/(a + b)))/(693*b^3*d*sqrt(a + b*cos(c + d*x))) - (2*(99*a^4*A*b^2 + 66*a^2*A*b^4 - 165*A*b^6 + 8*a^6*C + 49*a^4*b^2*C + 78*a^2*b^4*C - 135*b^6*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*b)/(a + b)))/(693*b^3*d*sqrt(a + b*cos(c + d*x))) + (2*(297*a^2*A*b^2 + 165*A*b^4 - 4*a^4*C + 205*a^2*b^2*C + 135*b^4*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(693*b^2*d) + (2*a*(297*A*b^2 + 3*a^2*C + 229*b^2*C)*cos(c + d*x)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(693*b*d) + (2*(33*A*b^2 + 5*a^2*C + 27*b^2*C)*cos(c + d*x)^2*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(231*d) + (10*a*C*cos(c + d*x)^2*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(99*d) + (2*C*cos(c + d*x)^2*(a + b*cos(c + d*x))^(5/2)*sin(c + d*x))/(11*d)],
[cos(c + d*x)^0*(a + b*cos(c + d*x))^(5/2)*(A + C*cos(c + d*x)^2), x, 9, (2*(a + b)*(483*a^2*A*b^2 + 189*A*b^4 - 10*a^4*C + 279*a^2*b^2*C + 147*b^4*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(315*b^2*d*sqrt(a + b*cos(c + d*x))) - (4*a*(84*a^2*A*b^2 - 84*A*b^4 - 5*a^4*C + 62*a^2*b^2*C - 57*b^4*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(315*b^2*d*sqrt(a + b*cos(c + d*x))) + (4*a*(84*A*b^2 - 5*a^2*C + 57*b^2*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(315*b*d) + (2*(63*A*b^2 - 10*a^2*C + 49*b^2*C)*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(315*b*d) - (4*a*C*(a + b*cos(c + d*x))^(5/2)*sin(c + d*x))/(63*b*d) + (2*C*(a + b*cos(c + d*x))^(7/2)*sin(c + d*x))/(9*b*d)],
[sec(c + d*x)^1*(a + b*cos(c + d*x))^(5/2)*(A + C*cos(c + d*x)^2), x, 11, (2*a*(a + b)*(49*A*b^2 + 3*a^2*C + 29*b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(21*b*d*sqrt(a + b*cos(c + d*x))) + (2*(14*a^2*A*b^2 + 7*A*b^4 - 3*a^4*C - 2*a^2*b^2*C + 5*b^4*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(21*b*d*sqrt(a + b*cos(c + d*x))) + (2*a^3*A*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x))) + (2*(7*A*b^2 + 3*a^2*C + 5*b^2*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(21*d) + (2*a*C*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(7*d) + (2*C*(a + b*cos(c + d*x))^(5/2)*sin(c + d*x))/(7*d)],
[sec(c + d*x)^2*(a + b*cos(c + d*x))^(5/2)*(A + C*cos(c + d*x)^2), x, 11, -(((a + b)*(15*a^2*A - 30*A*b^2 - 46*a^2*C - 18*b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(15*d*sqrt(a + b*cos(c + d*x)))) + (a*(15*a^2*A + 60*A*b^2 - 16*a^2*C + 16*b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(15*d*sqrt(a + b*cos(c + d*x))) + (5*a^2*A*b*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x))) - (a*b*(15*A - 16*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(15*d) - (b*(5*A - 2*C)*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(5*d) + (A*(a + b*cos(c + d*x))^(5/2)*tan(c + d*x))/d],
[sec(c + d*x)^3*(a + b*cos(c + d*x))^(5/2)*(A + C*cos(c + d*x)^2), x, 11, -((a*b*(a + b)*(27*A - 56*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(12*d*sqrt(a + b*cos(c + d*x)))) + (b*(33*a^2*A + 24*A*b^2 + 16*a^2*C + 8*b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(12*d*sqrt(a + b*cos(c + d*x))) + (a*(4*a^2*A + 15*A*b^2 + 8*a^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*b)/(a + b)))/(4*d*sqrt(a + b*cos(c + d*x))) - (b^2*(21*A - 8*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(12*d) + (5*A*b*(a + b*cos(c + d*x))^(3/2)*tan(c + d*x))/(4*d) + (A*(a + b*cos(c + d*x))^(5/2)*sec(c + d*x)*tan(c + d*x))/(2*d)],
[sec(c + d*x)^4*(a + b*cos(c + d*x))^(5/2)*(A + C*cos(c + d*x)^2), x, 11, -(((a + b)*(16*a^2*A + 33*A*b^2 + 24*a^2*C - 48*b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(24*d*sqrt(a + b*cos(c + d*x)))) + (a*(16*a^2*A + 59*A*b^2 + 24*a^2*C + 96*b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(24*d*sqrt(a + b*cos(c + d*x))) + (5*b*(4*a^2*A + A*b^2 + 8*a^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*b)/(a + b)))/(8*d*sqrt(a + b*cos(c + d*x))) + ((16*a^2*A + 15*A*b^2 + 24*a^2*C)*sqrt(a + b*cos(c + d*x))*tan(c + d*x))/(24*d) + (5*A*b*(a + b*cos(c + d*x))^(3/2)*sec(c + d*x)*tan(c + d*x))/(12*d) + (A*(a + b*cos(c + d*x))^(5/2)*sec(c + d*x)^2*tan(c + d*x))/(3*d)],
[sec(c + d*x)^5*(a + b*cos(c + d*x))^(5/2)*(A + C*cos(c + d*x)^2), x, 12, -((b*(a + b)*(284*a^2*A + 15*A*b^2 + 432*a^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(192*a*d*sqrt(a + b*cos(c + d*x)))) + (b*(356*a^2*A + 133*A*b^2 + 528*a^2*C + 384*b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(192*d*sqrt(a + b*cos(c + d*x))) + ((48*a^4*A + 120*a^2*A*b^2 - 5*A*b^4 + 64*a^4*C + 240*a^2*b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*b)/(a + b)))/(64*a*d*sqrt(a + b*cos(c + d*x))) + (b*(284*a^2*A + 15*A*b^2 + 432*a^2*C)*sqrt(a + b*cos(c + d*x))*tan(c + d*x))/(192*a*d) + ((12*a^2*A + 5*A*b^2 + 16*a^2*C)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)*tan(c + d*x))/(32*d) + (5*A*b*(a + b*cos(c + d*x))^(3/2)*sec(c + d*x)^2*tan(c + d*x))/(24*d) + (A*(a + b*cos(c + d*x))^(5/2)*sec(c + d*x)^3*tan(c + d*x))/(4*d)],


[(a^2 - b^2*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(3/2), x, 8, (4*a*(a + b)*(73*a^2 - 41*b^2)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(105*d*sqrt(a + b*cos(c + d*x))) - (2*(41*a^4 - 66*a^2*b^2 + 25*b^4)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(105*d*sqrt(a + b*cos(c + d*x))) + (2*b*(41*a^2 - 25*b^2)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(105*d) + (4*a*b*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(35*d) - (2*b*(a + b*cos(c + d*x))^(5/2)*sin(c + d*x))/(7*d)],
[(a^2 - b^2*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(1/2), x, 7, (2*(a + b)*(17*a^2 - 9*b^2)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(15*d*sqrt(a + b*cos(c + d*x))) - (4*a*(a^2 - b^2)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(15*d*sqrt(a + b*cos(c + d*x))) + (4*a*b*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(15*d) - (2*b*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(5*d)],


# ::Subsubsection::Closed:: 
#n<0


[cos(c + d*x)^3*(A + C*cos(c + d*x)^2)/sqrt(a + b*cos(c + d*x)), x, 9, (2*(a + b)*(168*a^2*A*b^2 + 189*A*b^4 + 128*a^4*C + 108*a^2*b^2*C + 147*b^4*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*b)/(a + b)))/(315*b^5*d*sqrt(a + b*cos(c + d*x))) - (2*a*(168*a^2*A*b^2 + 147*A*b^4 + 128*a^4*C + 76*a^2*b^2*C + 111*b^4*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*b)/(a + b)))/(315*b^5*d*sqrt(a + b*cos(c + d*x))) - (4*a*(42*A*b^2 + 32*a^2*C + 31*b^2*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(315*b^4*d) + (2*(63*A*b^2 + 48*a^2*C + 49*b^2*C)*cos(c + d*x)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(315*b^3*d) - (16*a*C*cos(c + d*x)^2*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(63*b^2*d) + (2*C*cos(c + d*x)^3*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(9*b*d)],
[cos(c + d*x)^2*(A + C*cos(c + d*x)^2)/sqrt(a + b*cos(c + d*x)), x, 8, (-4*a*(a + b)*(35*A*b^2 + 24*a^2*C + 22*b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*b)/(a + b)))/(105*b^4*d*sqrt(a + b*cos(c + d*x))) + (2*(70*a^2*A*b^2 + 35*A*b^4 + 48*a^4*C + 32*a^2*b^2*C + 25*b^4*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*b)/(a + b)))/(105*b^4*d*sqrt(a + b*cos(c + d*x))) + (2*(35*A*b^2 + 24*a^2*C + 25*b^2*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(105*b^3*d) - (12*a*C*cos(c + d*x)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(35*b^2*d) + (2*C*cos(c + d*x)^2*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(7*b*d)],
[cos(c + d*x)^1*(A + C*cos(c + d*x)^2)/sqrt(a + b*cos(c + d*x)), x, 7, (2*(a + b)*(15*A*b^2 + 8*a^2*C + 9*b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*b)/(a + b)))/(15*b^3*d*sqrt(a + b*cos(c + d*x))) - (2*a*(15*A*b^2 + 8*a^2*C + 7*b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*b)/(a + b)))/(15*b^3*d*sqrt(a + b*cos(c + d*x))) - (8*a*C*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(15*b^2*d) + (2*C*cos(c + d*x)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(5*b*d)],
[cos(c + d*x)^0*(A + C*cos(c + d*x)^2)/sqrt(a + b*cos(c + d*x)), x, 6, -((4*a*(a + b)*C*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(3*b^2*d*sqrt(a + b*cos(c + d*x)))) + (2*(3*A*b^2 + 2*a^2*C + b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(3*b^2*d*sqrt(a + b*cos(c + d*x))) + (2*C*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*b*d)],
[sec(c + d*x)^1*(A + C*cos(c + d*x)^2)/sqrt(a + b*cos(c + d*x)), x, 8, (2*(a + b)*C*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(b*d*sqrt(a + b*cos(c + d*x))) - (2*a*C*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(b*d*sqrt(a + b*cos(c + d*x))) + (2*A*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x)))],
[sec(c + d*x)^2*(A + C*cos(c + d*x)^2)/sqrt(a + b*cos(c + d*x)), x, 9, -((A*(a + b)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(a*d*sqrt(a + b*cos(c + d*x)))) + ((A + 2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x))) - (A*b*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*b)/(a + b)))/(a*d*sqrt(a + b*cos(c + d*x))) + (A*sqrt(a + b*cos(c + d*x))*tan(c + d*x))/(a*d)],
[sec(c + d*x)^3*(A + C*cos(c + d*x)^2)/sqrt(a + b*cos(c + d*x)), x, 10, (3*A*b*(a + b)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(4*a^2*d*sqrt(a + b*cos(c + d*x))) - (A*b*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(4*a*d*sqrt(a + b*cos(c + d*x))) + ((4*a^2*A + 3*A*b^2 + 8*a^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*b)/(a + b)))/(4*a^2*d*sqrt(a + b*cos(c + d*x))) - (3*A*b*sqrt(a + b*cos(c + d*x))*tan(c + d*x))/(4*a^2*d) + (A*sqrt(a + b*cos(c + d*x))*sec(c + d*x)*tan(c + d*x))/(2*a*d)],
[sec(c + d*x)^4*(A + C*cos(c + d*x)^2)/sqrt(a + b*cos(c + d*x)), x, 11, -(((a + b)*(16*a^2*A + 15*A*b^2 + 24*a^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(24*a^3*d*sqrt(a + b*cos(c + d*x)))) + ((16*a^2*A + 5*A*b^2 + 24*a^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(24*a^2*d*sqrt(a + b*cos(c + d*x))) - (b*(4*a^2*A + 5*A*b^2 + 8*a^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*b)/(a + b)))/(8*a^3*d*sqrt(a + b*cos(c + d*x))) + ((16*a^2*A + 15*A*b^2 + 24*a^2*C)*sqrt(a + b*cos(c + d*x))*tan(c + d*x))/(24*a^3*d) - (5*A*b*sqrt(a + b*cos(c + d*x))*sec(c + d*x)*tan(c + d*x))/(12*a^2*d) + (A*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^2*tan(c + d*x))/(3*a*d)],


[cos(c + d*x)^3*(A + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(3/2), x, 9, (-2*a*(280*a^2*A*b^2 - 175*A*b^4 + 384*a^4*C - 172*a^2*b^2*C - 107*b^4*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*b)/(a + b)))/(105*(a - b)*b^5*d*sqrt(a + b*cos(c + d*x))) + (2*(280*a^2*A*b^2 + 35*A*b^4 + 384*a^4*C + 116*a^2*b^2*C + 25*b^4*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*b)/(a + b)))/(105*b^5*d*sqrt(a + b*cos(c + d*x))) - (2*(A*b^2 + a^2*C)*cos(c + d*x)^3*sin(c + d*x))/(b*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) + (2*(140*a^2*A*b^2 - 35*A*b^4 + 192*a^4*C - 62*a^2*b^2*C - 25*b^4*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(105*b^4*(a^2 - b^2)*d) - (2*a*(35*A*b^2 + 48*a^2*C - 13*b^2*C)*cos(c + d*x)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(35*b^3*(a^2 - b^2)*d) + (2*(7*A*b^2 + 8*a^2*C - b^2*C)*cos(c + d*x)^2*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(7*b^2*(a^2 - b^2)*d)],
[cos(c + d*x)^2*(A + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(3/2), x, 8, (2*(10*a^2*A*b^2 - 5*A*b^4 + 16*a^4*C - 8*a^2*b^2*C - 3*b^4*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*b)/(a + b)))/(5*(a - b)*b^4*d*sqrt(a + b*cos(c + d*x))) - (4*a*(5*A*b^2 + 8*a^2*C + 2*b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*b)/(a + b)))/(5*b^4*d*sqrt(a + b*cos(c + d*x))) - (2*(A*b^2 + a^2*C)*cos(c + d*x)^2*sin(c + d*x))/(b*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) - (2*a*(5*A*b^2 + 8*a^2*C - 3*b^2*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(5*b^3*(a^2 - b^2)*d) + (2*(5*A*b^2 + 6*a^2*C - b^2*C)*cos(c + d*x)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(5*b^2*(a^2 - b^2)*d)],
[cos(c + d*x)^1*(A + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(3/2), x, 7, (-2*a*(3*A*b^2 + 8*a^2*C - 5*b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*b)/(a + b)))/(3*(a - b)*b^3*d*sqrt(a + b*cos(c + d*x))) + (2*(3*A*b^2 + 8*a^2*C + b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*b)/(a + b)))/(3*b^3*d*sqrt(a + b*cos(c + d*x))) - (2*(A*b^2 + a^2*C)*cos(c + d*x)*sin(c + d*x))/(b*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) + (2*(3*A*b^2 + 4*a^2*C - b^2*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*b^2*(a^2 - b^2)*d)],
[cos(c + d*x)^0*(A + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(3/2), x, 6, (2*(A*b^2 + 2*a^2*C - b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/((a - b)*b^2*d*sqrt(a + b*cos(c + d*x))) - (4*a*C*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(b^2*d*sqrt(a + b*cos(c + d*x))) - (2*(A*b^2 + a^2*C)*sin(c + d*x))/(b*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x)))],
[sec(c + d*x)^1*(A + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(3/2), x, 9, -((2*(A*b^2 + a^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(a*(a - b)*b*d*sqrt(a + b*cos(c + d*x)))) + (2*C*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(b*d*sqrt(a + b*cos(c + d*x))) + (2*A*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*b)/(a + b)))/(a*d*sqrt(a + b*cos(c + d*x))) + (2*(A*b^2 + a^2*C)*sin(c + d*x))/(a*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x)))],
[sec(c + d*x)^2*(A + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(3/2), x, 10, -(((a^2*A - 3*A*b^2 - 2*a^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(a^2*(a - b)*d*sqrt(a + b*cos(c + d*x)))) + (A*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(a*d*sqrt(a + b*cos(c + d*x))) - (3*A*b*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*b)/(a + b)))/(a^2*d*sqrt(a + b*cos(c + d*x))) + (2*(A*b^2 + a^2*C)*tan(c + d*x))/(a*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) + ((a^2*A - 3*A*b^2 - 2*a^2*C)*sqrt(a + b*cos(c + d*x))*tan(c + d*x))/(a^2*(a^2 - b^2)*d)],
[sec(c + d*x)^3*(A + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(3/2), x, 11, (b*(7*a^2*A - 15*A*b^2 - 8*a^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(4*a^3*(a - b)*d*sqrt(a + b*cos(c + d*x))) - (5*A*b*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(4*a^2*d*sqrt(a + b*cos(c + d*x))) + ((4*a^2*A + 15*A*b^2 + 8*a^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*b)/(a + b)))/(4*a^3*d*sqrt(a + b*cos(c + d*x))) - (b*(7*a^2*A - 15*A*b^2 - 8*a^2*C)*sqrt(a + b*cos(c + d*x))*tan(c + d*x))/(4*a^3*(a^2 - b^2)*d) + (2*(A*b^2 + a^2*C)*sec(c + d*x)*tan(c + d*x))/(a*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) + ((a^2*A - 5*A*b^2 - 4*a^2*C)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)*tan(c + d*x))/(2*a^2*(a^2 - b^2)*d)],


[cos(c + d*x)^3*(A + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(5/2), x, 9, (2*(40*a^4*A*b^2 - 75*a^2*A*b^4 + 15*A*b^6 + 128*a^6*C - 212*a^4*b^2*C + 55*a^2*b^4*C + 9*b^6*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*b)/(a + b)))/(15*(a - b)^2*b^5*(a + b)*d*sqrt(a + b*cos(c + d*x))) - (2*a*(40*a^2*A*b^2 - 45*A*b^4 + 128*a^4*C - 116*a^2*b^2*C - 17*b^4*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*b)/(a + b)))/(15*b^5*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) - (2*(A*b^2 + a^2*C)*cos(c + d*x)^3*sin(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)) - (4*(a^2*A*b^2 - 3*A*b^4 + 4*a^4*C - 6*a^2*b^2*C)*cos(c + d*x)^2*sin(c + d*x))/(3*b^2*(a^2 - b^2)^2*d*sqrt(a + b*cos(c + d*x))) - (4*a*(10*a^2*A*b^2 - 20*A*b^4 + 32*a^4*C - 49*a^2*b^2*C + 7*b^4*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(15*b^4*(a^2 - b^2)^2*d) + (2*(15*a^2*A*b^2 - 35*A*b^4 + 48*a^4*C - 71*a^2*b^2*C + 3*b^4*C)*cos(c + d*x)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(15*b^3*(a^2 - b^2)^2*d)],
[cos(c + d*x)^2*(A + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(5/2), x, 8, (-4*a*(a^2*A*b^2 - 3*A*b^4 + 8*a^4*C - 14*a^2*b^2*C + 4*b^4*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*b)/(a + b)))/(3*(a - b)^2*b^4*(a + b)*d*sqrt(a + b*cos(c + d*x))) + (2*(2*a^2*A*b^2 - 3*A*b^4 + 16*a^4*C - 16*a^2*b^2*C - b^4*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*b)/(a + b)))/(3*b^4*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) - (2*(A*b^2 + a^2*C)*cos(c + d*x)^2*sin(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)) + (4*(2*A*b^4 - 3*a^4*C + 5*a^2*b^2*C)*cos(c + d*x)*sin(c + d*x))/(3*b^2*(a^2 - b^2)^2*d*sqrt(a + b*cos(c + d*x))) + (2*(a^2*A*b^2 - 5*A*b^4 + 8*a^4*C - 13*a^2*b^2*C + b^4*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*b^3*(a^2 - b^2)^2*d)],
[cos(c + d*x)^1*(A + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(5/2), x, 7, (-2*(a^2*A*b^2 + 3*A*b^4 - 8*a^4*C + 15*a^2*b^2*C - 3*b^4*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*b)/(a + b)))/(3*(a - b)^2*b^3*(a + b)*d*sqrt(a + b*cos(c + d*x))) + (2*a*(A*b^2 - 8*a^2*C + 9*b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*b)/(a + b)))/(3*b^3*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) - (2*(A*b^2 + a^2*C)*cos(c + d*x)*sin(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)) + (4*(a^2*A*b^2 + A*b^4 - 2*a^4*C + 4*a^2*b^2*C)*sin(c + d*x))/(3*b^2*(a^2 - b^2)^2*d*sqrt(a + b*cos(c + d*x)))],
[cos(c + d*x)^0*(A + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(5/2), x, 7, (4*a*(2*A*b^2 - a^2*C + 3*b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(3*(a - b)^2*b^2*(a + b)*d*sqrt(a + b*cos(c + d*x))) - (2*(A*b^2 - 2*a^2*C + 3*b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(3*b^2*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) - (2*(A*b^2 + a^2*C)*sin(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)) - (4*a*(2*A*b^2 - a^2*C + 3*b^2*C)*sin(c + d*x))/(3*b*(a^2 - b^2)^2*d*sqrt(a + b*cos(c + d*x)))],
[sec(c + d*x)^1*(A + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(5/2), x, 10, -((2*(7*a^2*A*b^2 - 3*A*b^4 + a^4*C + 3*a^2*b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(3*a^2*(a - b)^2*b*(a + b)*d*sqrt(a + b*cos(c + d*x)))) + (2*(A*b^2 + a^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(3*a*b*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) + (2*A*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*b)/(a + b)))/(a^2*d*sqrt(a + b*cos(c + d*x))) + (2*(A*b^2 + a^2*C)*sin(c + d*x))/(3*a*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)) + (2*(7*a^2*A*b^2 - 3*A*b^4 + a^4*C + 3*a^2*b^2*C)*sin(c + d*x))/(3*a^2*(a^2 - b^2)^2*d*sqrt(a + b*cos(c + d*x)))],
[sec(c + d*x)^2*(A + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(5/2), x, 11, -(((3*a^4*A - 26*a^2*A*b^2 + 15*A*b^4 - 8*a^4*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(3*a^3*(a - b)^2*(a + b)*d*sqrt(a + b*cos(c + d*x)))) + ((3*a^2*A - 5*A*b^2 - 2*a^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(3*a^2*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) - (5*A*b*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*b)/(a + b)))/(a^3*d*sqrt(a + b*cos(c + d*x))) + (2*(A*b^2 + a^2*C)*tan(c + d*x))/(3*a*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)) + (2*(9*a^2*A*b^2 - 5*A*b^4 + 3*a^4*C + a^2*b^2*C)*tan(c + d*x))/(3*a^2*(a^2 - b^2)^2*d*sqrt(a + b*cos(c + d*x))) + ((3*a^4*A - 26*a^2*A*b^2 + 15*A*b^4 - 8*a^4*C)*sqrt(a + b*cos(c + d*x))*tan(c + d*x))/(3*a^3*(a^2 - b^2)^2*d)],


[(A + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(7/2), x, 8, (2*(23*a^2*A*b^2 + 9*A*b^4 - 2*a^4*C + 19*a^2*b^2*C + 15*b^4*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(15*(a - b)^3*b^2*(a + b)^2*d*sqrt(a + b*cos(c + d*x))) - (4*a*(4*A*b^2 - a^2*C + 5*b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(15*b^2*(a^2 - b^2)^2*d*sqrt(a + b*cos(c + d*x))) - (2*(A*b^2 + a^2*C)*sin(c + d*x))/(5*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(5/2)) - (4*a*(4*A*b^2 - a^2*C + 5*b^2*C)*sin(c + d*x))/(15*b*(a^2 - b^2)^2*d*(a + b*cos(c + d*x))^(3/2)) - (2*(23*a^2*A*b^2 + 9*A*b^4 - 2*a^4*C + 19*a^2*b^2*C + 15*b^4*C)*sin(c + d*x))/(15*b*(a^2 - b^2)^3*d*sqrt(a + b*cos(c + d*x)))],


[(a^2 - b^2*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(1/2), x, 6, (4*a*(a + b)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(3*d*sqrt(a + b*cos(c + d*x))) + (2*(a^2 - b^2)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(3*d*sqrt(a + b*cos(c + d*x))) - (2*b*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*d)],
[(a^2 - b^2*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(3/2), x, 6, -((2*(a + b)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x)))) + (4*a*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x)))],
[(a^2 - b^2*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(5/2), x, 7, (4*a*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/((a - b)*d*sqrt(a + b*cos(c + d*x))) - (2*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x))) - (4*a*b*sin(c + d*x))/((a^2 - b^2)*d*sqrt(a + b*cos(c + d*x)))],
[(a^2 - b^2*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(7/2), x, 8, (2*(5*a^2 + 3*b^2)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(3*(a - b)^2*(a + b)*d*sqrt(a + b*cos(c + d*x))) - (4*a*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(3*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) - (4*a*b*sin(c + d*x))/(3*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)) - (2*b*(5*a^2 + 3*b^2)*sin(c + d*x))/(3*(a^2 - b^2)^2*d*sqrt(a + b*cos(c + d*x)))],


# ::Subsection::Closed:: 
#Integrands of the form Cos[c+d x]^(m/2) (A+C Cos[c+d x]^2) (a+b Cos[c+d x])^n


# ::Subsubsection::Closed:: 
#n>0


[cos(c + d*x)^(5/2)*(a + b*cos(c + d*x))*(A + C*cos(c + d*x)^2), x, 8, (2*a*(9*A + 7*C)*EllipticE((c + d*x)/2, 2))/(15*d) + (10*b*(11*A + 9*C)*EllipticF((c + d*x)/2, 2))/(231*d) + (10*b*(11*A + 9*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(231*d) + (2*a*(9*A + 7*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(45*d) + (2*b*(11*A + 9*C)*cos(c + d*x)^(5/2)*sin(c + d*x))/(77*d) + (4*a*C*cos(c + d*x)^(7/2)*sin(c + d*x))/(99*d) + (2*C*cos(c + d*x)^(7/2)*(a + b*cos(c + d*x))*sin(c + d*x))/(11*d)],
[cos(c + d*x)^(3/2)*(a + b*cos(c + d*x))*(A + C*cos(c + d*x)^2), x, 7, (2*b*(9*A + 7*C)*EllipticE((c + d*x)/2, 2))/(15*d) + (2*a*(7*A + 5*C)*EllipticF((c + d*x)/2, 2))/(21*d) + (2*a*(7*A + 5*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(21*d) + (2*b*(9*A + 7*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(45*d) + (4*a*C*cos(c + d*x)^(5/2)*sin(c + d*x))/(63*d) + (2*C*cos(c + d*x)^(5/2)*(a + b*cos(c + d*x))*sin(c + d*x))/(9*d)],
[sqrt(cos(c + d*x))*(a + b*cos(c + d*x))*(A + C*cos(c + d*x)^2), x, 6, (2*a*(5*A + 3*C)*EllipticE((c + d*x)/2, 2))/(5*d) + (2*b*(7*A + 5*C)*EllipticF((c + d*x)/2, 2))/(21*d) + (2*b*(7*A + 5*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(21*d) + (4*a*C*cos(c + d*x)^(3/2)*sin(c + d*x))/(35*d) + (2*C*cos(c + d*x)^(3/2)*(a + b*cos(c + d*x))*sin(c + d*x))/(7*d)],
[((a + b*cos(c + d*x))*(A + C*cos(c + d*x)^2))/sqrt(cos(c + d*x)), x, 5, (2*b*(5*A + 3*C)*EllipticE((c + d*x)/2, 2))/(5*d) + (2*a*(3*A + C)*EllipticF((c + d*x)/2, 2))/(3*d) + (4*a*C*sqrt(cos(c + d*x))*sin(c + d*x))/(15*d) + (2*C*sqrt(cos(c + d*x))*(a + b*cos(c + d*x))*sin(c + d*x))/(5*d)],
[((a + b*cos(c + d*x))*(A + C*cos(c + d*x)^2))/cos(c + d*x)^(3/2), x, 5, (-2*a*(A - C)*EllipticE((c + d*x)/2, 2))/d + (2*b*(3*A + C)*EllipticF((c + d*x)/2, 2))/(3*d) - (2*b*(3*A - C)*sqrt(cos(c + d*x))*sin(c + d*x))/(3*d) + (2*A*(a + b*cos(c + d*x))*sin(c + d*x))/(d*sqrt(cos(c + d*x)))],
[((a + b*cos(c + d*x))*(A + C*cos(c + d*x)^2))/cos(c + d*x)^(5/2), x, 5, (-2*b*(A - C)*EllipticE((c + d*x)/2, 2))/d + (2*a*(A + 3*C)*EllipticF((c + d*x)/2, 2))/(3*d) + (4*A*b*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))) + (2*A*(a + b*cos(c + d*x))*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2))],
[((a + b*cos(c + d*x))*(A + C*cos(c + d*x)^2))/cos(c + d*x)^(7/2), x, 6, (-2*a*(3*A + 5*C)*EllipticE((c + d*x)/2, 2))/(5*d) + (2*b*(A + 3*C)*EllipticF((c + d*x)/2, 2))/(3*d) + (4*A*b*sin(c + d*x))/(15*d*cos(c + d*x)^(3/2)) + (2*a*(3*A + 5*C)*sin(c + d*x))/(5*d*sqrt(cos(c + d*x))) + (2*A*(a + b*cos(c + d*x))*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2))],
[((a + b*cos(c + d*x))*(A + C*cos(c + d*x)^2))/cos(c + d*x)^(9/2), x, 7, (-2*b*(3*A + 5*C)*EllipticE((c + d*x)/2, 2))/(5*d) + (2*a*(5*A + 7*C)*EllipticF((c + d*x)/2, 2))/(21*d) + (4*A*b*sin(c + d*x))/(35*d*cos(c + d*x)^(5/2)) + (2*a*(5*A + 7*C)*sin(c + d*x))/(21*d*cos(c + d*x)^(3/2)) + (2*b*(3*A + 5*C)*sin(c + d*x))/(5*d*sqrt(cos(c + d*x))) + (2*A*(a + b*cos(c + d*x))*sin(c + d*x))/(7*d*cos(c + d*x)^(7/2))],


[cos(c + d*x)^(3/2)*(a + b*cos(c + d*x))^2*(A + C*cos(c + d*x)^2), x, 8, (4*a*b*(9*A + 7*C)*EllipticE((c + d*x)/2, 2))/(15*d) + (2*(77*a^2*A + 55*A*b^2 + 55*a^2*C + 45*b^2*C)*EllipticF((c + d*x)/2, 2))/(231*d) + (2*(77*a^2*A + 55*A*b^2 + 55*a^2*C + 45*b^2*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(231*d) + (4*a*b*(9*A + 7*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(45*d) + (2*(99*A*b^2 + 8*a^2*C + 81*b^2*C)*cos(c + d*x)^(5/2)*sin(c + d*x))/(693*d) + (8*a*C*cos(c + d*x)^(5/2)*(a + b*cos(c + d*x))*sin(c + d*x))/(99*d) + (2*C*cos(c + d*x)^(5/2)*(a + b*cos(c + d*x))^2*sin(c + d*x))/(11*d)],
[sqrt(cos(c + d*x))*(a + b*cos(c + d*x))^2*(A + C*cos(c + d*x)^2), x, 7, (2*(15*a^2*A + 9*A*b^2 + 9*a^2*C + 7*b^2*C)*EllipticE((c + d*x)/2, 2))/(15*d) + (4*a*b*(7*A + 5*C)*EllipticF((c + d*x)/2, 2))/(21*d) + (4*a*b*(7*A + 5*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(21*d) + (2*(63*A*b^2 + 8*a^2*C + 49*b^2*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(315*d) + (8*a*C*cos(c + d*x)^(3/2)*(a + b*cos(c + d*x))*sin(c + d*x))/(63*d) + (2*C*cos(c + d*x)^(3/2)*(a + b*cos(c + d*x))^2*sin(c + d*x))/(9*d)],
[((a + b*cos(c + d*x))^2*(A + C*cos(c + d*x)^2))/sqrt(cos(c + d*x)), x, 6, (4*a*b*(5*A + 3*C)*EllipticE((c + d*x)/2, 2))/(5*d) + (2*(21*a^2*A + 7*A*b^2 + 7*a^2*C + 5*b^2*C)*EllipticF((c + d*x)/2, 2))/(21*d) + (2*(35*A*b^2 + 8*a^2*C + 25*b^2*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(105*d) + (8*a*C*sqrt(cos(c + d*x))*(a + b*cos(c + d*x))*sin(c + d*x))/(35*d) + (2*C*sqrt(cos(c + d*x))*(a + b*cos(c + d*x))^2*sin(c + d*x))/(7*d)],
[((a + b*cos(c + d*x))^2*(A + C*cos(c + d*x)^2))/cos(c + d*x)^(3/2), x, 6, (-2*(5*a^2*A - 5*A*b^2 - 5*a^2*C - 3*b^2*C)*EllipticE((c + d*x)/2, 2))/(5*d) + (4*a*b*(3*A + C)*EllipticF((c + d*x)/2, 2))/(3*d) - (2*a*b*(15*A - 7*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(15*d) - (2*b*(5*A - C)*sqrt(cos(c + d*x))*(a + b*cos(c + d*x))*sin(c + d*x))/(5*d) + (2*A*(a + b*cos(c + d*x))^2*sin(c + d*x))/(d*sqrt(cos(c + d*x)))],
[((a + b*cos(c + d*x))^2*(A + C*cos(c + d*x)^2))/cos(c + d*x)^(5/2), x, 6, (-4*a*b*(A - C)*EllipticE((c + d*x)/2, 2))/d + (2*(a^2*A + 3*A*b^2 + 3*a^2*C + b^2*C)*EllipticF((c + d*x)/2, 2))/(3*d) - (2*b^2*(5*A - C)*sqrt(cos(c + d*x))*sin(c + d*x))/(3*d) + (8*A*b*(a + b*cos(c + d*x))*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))) + (2*A*(a + b*cos(c + d*x))^2*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2))],
[((a + b*cos(c + d*x))^2*(A + C*cos(c + d*x)^2))/cos(c + d*x)^(7/2), x, 6, (-2*(3*a^2*A + 5*A*b^2 + 5*a^2*C - 5*b^2*C)*EllipticE((c + d*x)/2, 2))/(5*d) + (4*a*b*(A + 3*C)*EllipticF((c + d*x)/2, 2))/(3*d) + (2*(9*a^2*A + 8*A*b^2 + 15*a^2*C)*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))) + (8*A*b*(a + b*cos(c + d*x))*sin(c + d*x))/(15*d*cos(c + d*x)^(3/2)) + (2*A*(a + b*cos(c + d*x))^2*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2))],
[((a + b*cos(c + d*x))^2*(A + C*cos(c + d*x)^2))/cos(c + d*x)^(9/2), x, 7, (-4*a*b*(3*A + 5*C)*EllipticE((c + d*x)/2, 2))/(5*d) + (2*(5*a^2*A + 7*A*b^2 + 7*a^2*C + 21*b^2*C)*EllipticF((c + d*x)/2, 2))/(21*d) + (2*(25*a^2*A + 8*A*b^2 + 35*a^2*C)*sin(c + d*x))/(105*d*cos(c + d*x)^(3/2)) + (4*a*b*(3*A + 5*C)*sin(c + d*x))/(5*d*sqrt(cos(c + d*x))) + (8*A*b*(a + b*cos(c + d*x))*sin(c + d*x))/(35*d*cos(c + d*x)^(5/2)) + (2*A*(a + b*cos(c + d*x))^2*sin(c + d*x))/(7*d*cos(c + d*x)^(7/2))],


[sqrt(cos(c + d*x))*(a + b*cos(c + d*x))^3*(A + C*cos(c + d*x)^2), x, 8, (2*a*(5*a^2*A + 9*A*b^2 + 3*a^2*C + 7*b^2*C)*EllipticE((c + d*x)/2, 2))/(5*d) + (2*b*(231*a^2*A + 55*A*b^2 + 165*a^2*C + 45*b^2*C)*EllipticF((c + d*x)/2, 2))/(231*d) + (2*b*(231*a^2*A + 55*A*b^2 + 165*a^2*C + 45*b^2*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(231*d) + (8*a*(132*A*b^2 + 4*a^2*C + 101*b^2*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(1155*d) + (2*(33*A*b^2 + 8*a^2*C + 27*b^2*C)*cos(c + d*x)^(3/2)*(a + b*cos(c + d*x))*sin(c + d*x))/(231*d) + (4*a*C*cos(c + d*x)^(3/2)*(a + b*cos(c + d*x))^2*sin(c + d*x))/(33*d) + (2*C*cos(c + d*x)^(3/2)*(a + b*cos(c + d*x))^3*sin(c + d*x))/(11*d)],
[((a + b*cos(c + d*x))^3*(A + C*cos(c + d*x)^2))/sqrt(cos(c + d*x)), x, 7, (2*b*(45*a^2*A + 9*A*b^2 + 27*a^2*C + 7*b^2*C)*EllipticE((c + d*x)/2, 2))/(15*d) + (2*a*(21*a^2*A + 21*A*b^2 + 7*a^2*C + 15*b^2*C)*EllipticF((c + d*x)/2, 2))/(21*d) + (8*a*(63*A*b^2 + 4*a^2*C + 44*b^2*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(315*d) + (2*(63*A*b^2 + 24*a^2*C + 49*b^2*C)*sqrt(cos(c + d*x))*(a + b*cos(c + d*x))*sin(c + d*x))/(315*d) + (4*a*C*sqrt(cos(c + d*x))*(a + b*cos(c + d*x))^2*sin(c + d*x))/(21*d) + (2*C*sqrt(cos(c + d*x))*(a + b*cos(c + d*x))^3*sin(c + d*x))/(9*d)],
[((a + b*cos(c + d*x))^3*(A + C*cos(c + d*x)^2))/cos(c + d*x)^(3/2), x, 7, (-2*a*(5*a^2*A - 15*A*b^2 - 5*a^2*C - 9*b^2*C)*EllipticE((c + d*x)/2, 2))/(5*d) + (2*b*(63*a^2*A + 7*A*b^2 + 21*a^2*C + 5*b^2*C)*EllipticF((c + d*x)/2, 2))/(21*d) - (2*b*(105*a^2*A - 35*A*b^2 - 57*a^2*C - 25*b^2*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(105*d) - (2*a*b*(35*A - 11*C)*sqrt(cos(c + d*x))*(a + b*cos(c + d*x))*sin(c + d*x))/(35*d) - (2*b*(7*A - C)*sqrt(cos(c + d*x))*(a + b*cos(c + d*x))^2*sin(c + d*x))/(7*d) + (2*A*(a + b*cos(c + d*x))^3*sin(c + d*x))/(d*sqrt(cos(c + d*x)))],
[((a + b*cos(c + d*x))^3*(A + C*cos(c + d*x)^2))/cos(c + d*x)^(5/2), x, 7, (-2*b*(15*a^2*A - 5*A*b^2 - 15*a^2*C - 3*b^2*C)*EllipticE((c + d*x)/2, 2))/(5*d) + (2*a*(a^2*A + 9*A*b^2 + 3*a^2*C + 3*b^2*C)*EllipticF((c + d*x)/2, 2))/(3*d) - (8*a*b^2*(10*A - 3*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(15*d) - (2*b^2*(35*A - 3*C)*sqrt(cos(c + d*x))*(a + b*cos(c + d*x))*sin(c + d*x))/(15*d) + (4*A*b*(a + b*cos(c + d*x))^2*sin(c + d*x))/(d*sqrt(cos(c + d*x))) + (2*A*(a + b*cos(c + d*x))^3*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2))],
[((a + b*cos(c + d*x))^3*(A + C*cos(c + d*x)^2))/cos(c + d*x)^(7/2), x, 7, (-2*a*(3*a^2*A + 15*A*b^2 + 5*a^2*C - 15*b^2*C)*EllipticE((c + d*x)/2, 2))/(5*d) + (2*b*(3*a^2*A + 3*A*b^2 + 9*a^2*C + b^2*C)*EllipticF((c + d*x)/2, 2))/(3*d) - (2*b*(9*a^2*A + 33*A*b^2 + 15*a^2*C - 5*b^2*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(15*d) + (2*(3*a^2*A + 8*A*b^2 + 5*a^2*C)*(a + b*cos(c + d*x))*sin(c + d*x))/(5*d*sqrt(cos(c + d*x))) + (4*A*b*(a + b*cos(c + d*x))^2*sin(c + d*x))/(5*d*cos(c + d*x)^(3/2)) + (2*A*(a + b*cos(c + d*x))^3*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2))],
[((a + b*cos(c + d*x))^3*(A + C*cos(c + d*x)^2))/cos(c + d*x)^(9/2), x, 7, (-2*b*(9*a^2*A + 5*A*b^2 + 15*a^2*C - 5*b^2*C)*EllipticE((c + d*x)/2, 2))/(5*d) + (2*a*(5*a^2*A + 21*A*b^2 + 7*a^2*C + 63*b^2*C)*EllipticF((c + d*x)/2, 2))/(21*d) + (8*b*(41*a^2*A + 12*A*b^2 + 70*a^2*C)*sin(c + d*x))/(105*d*sqrt(cos(c + d*x))) + (2*(25*a^2*A + 24*A*b^2 + 35*a^2*C)*(a + b*cos(c + d*x))*sin(c + d*x))/(105*d*cos(c + d*x)^(3/2)) + (12*A*b*(a + b*cos(c + d*x))^2*sin(c + d*x))/(35*d*cos(c + d*x)^(5/2)) + (2*A*(a + b*cos(c + d*x))^3*sin(c + d*x))/(7*d*cos(c + d*x)^(7/2))],
[((a + b*cos(c + d*x))^3*(A + C*cos(c + d*x)^2))/cos(c + d*x)^(11/2), x, 8, (-2*a*(7*a^2*A + 27*A*b^2 + 9*a^2*C + 45*b^2*C)*EllipticE((c + d*x)/2, 2))/(15*d) + (2*b*(15*a^2*A + 7*A*b^2 + 21*a^2*C + 21*b^2*C)*EllipticF((c + d*x)/2, 2))/(21*d) + (8*b*(44*a^2*A + 4*A*b^2 + 63*a^2*C)*sin(c + d*x))/(315*d*cos(c + d*x)^(3/2)) + (2*a*(7*a^2*A + 27*A*b^2 + 9*a^2*C + 45*b^2*C)*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))) + (2*(49*a^2*A + 24*A*b^2 + 63*a^2*C)*(a + b*cos(c + d*x))*sin(c + d*x))/(315*d*cos(c + d*x)^(5/2)) + (4*A*b*(a + b*cos(c + d*x))^2*sin(c + d*x))/(21*d*cos(c + d*x)^(7/2)) + (2*A*(a + b*cos(c + d*x))^3*sin(c + d*x))/(9*d*cos(c + d*x)^(9/2))],


[sqrt(cos(c + d*x))*(a + b*cos(c + d*x))^4*(A + C*cos(c + d*x)^2), x, 9, (2*(195*a^4*A + 702*a^2*A*b^2 + 91*A*b^4 + 117*a^4*C + 546*a^2*b^2*C + 77*b^4*C)*EllipticE((c + d*x)/2, 2))/(195*d) + (8*a*b*(77*a^2*A + 55*A*b^2 + 55*a^2*C + 45*b^2*C)*EllipticF((c + d*x)/2, 2))/(231*d) + (8*a*b*(77*a^2*A + 55*A*b^2 + 55*a^2*C + 45*b^2*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(231*d) + (2*(33319*a^2*A*b^2 + 7007*A*b^4 + 384*a^4*C + 25217*a^2*b^2*C + 5929*b^4*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(45045*d) + (4*a*(1573*A*b^2 + 96*a^2*C + 1259*b^2*C)*cos(c + d*x)^(3/2)*(a + b*cos(c + d*x))*sin(c + d*x))/(9009*d) + (2*(143*A*b^2 + 48*a^2*C + 121*b^2*C)*cos(c + d*x)^(3/2)*(a + b*cos(c + d*x))^2*sin(c + d*x))/(1287*d) + (16*a*C*cos(c + d*x)^(3/2)*(a + b*cos(c + d*x))^3*sin(c + d*x))/(143*d) + (2*C*cos(c + d*x)^(3/2)*(a + b*cos(c + d*x))^4*sin(c + d*x))/(13*d)],
[((a + b*cos(c + d*x))^4*(A + C*cos(c + d*x)^2))/sqrt(cos(c + d*x)), x, 8, (8*a*b*(15*a^2*A + 9*A*b^2 + 9*a^2*C + 7*b^2*C)*EllipticE((c + d*x)/2, 2))/(15*d) + (2*(231*a^4*A + 462*a^2*A*b^2 + 55*A*b^4 + 77*a^4*C + 330*a^2*b^2*C + 45*b^4*C)*EllipticF((c + d*x)/2, 2))/(231*d) + (2*(4653*a^2*A*b^2 + 825*A*b^4 + 128*a^4*C + 3199*a^2*b^2*C + 675*b^4*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(3465*d) + (4*a*(891*A*b^2 + 96*a^2*C + 673*b^2*C)*sqrt(cos(c + d*x))*(a + b*cos(c + d*x))*sin(c + d*x))/(3465*d) + (2*(33*A*b^2 + 16*a^2*C + 27*b^2*C)*sqrt(cos(c + d*x))*(a + b*cos(c + d*x))^2*sin(c + d*x))/(231*d) + (16*a*C*sqrt(cos(c + d*x))*(a + b*cos(c + d*x))^3*sin(c + d*x))/(99*d) + (2*C*sqrt(cos(c + d*x))*(a + b*cos(c + d*x))^4*sin(c + d*x))/(11*d)],
[((a + b*cos(c + d*x))^4*(A + C*cos(c + d*x)^2))/cos(c + d*x)^(3/2), x, 8, (-2*(15*a^4*A - 90*a^2*A*b^2 - 9*A*b^4 - 15*a^4*C - 54*a^2*b^2*C - 7*b^4*C)*EllipticE((c + d*x)/2, 2))/(15*d) + (8*a*b*(21*a^2*A + 7*A*b^2 + 7*a^2*C + 5*b^2*C)*EllipticF((c + d*x)/2, 2))/(21*d) - (2*a*b*(315*a^2*A - 357*A*b^2 - 187*a^2*C - 251*b^2*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(315*d) - (2*b*(315*a^2*A - 63*A*b^2 - 123*a^2*C - 49*b^2*C)*sqrt(cos(c + d*x))*(a + b*cos(c + d*x))*sin(c + d*x))/(315*d) - (2*a*b*(21*A - 5*C)*sqrt(cos(c + d*x))*(a + b*cos(c + d*x))^2*sin(c + d*x))/(21*d) - (2*b*(9*A - C)*sqrt(cos(c + d*x))*(a + b*cos(c + d*x))^3*sin(c + d*x))/(9*d) + (2*A*(a + b*cos(c + d*x))^4*sin(c + d*x))/(d*sqrt(cos(c + d*x)))],
[((a + b*cos(c + d*x))^4*(A + C*cos(c + d*x)^2))/cos(c + d*x)^(5/2), x, 8, (-8*a*b*(5*a^2*A - 5*A*b^2 - 5*a^2*C - 3*b^2*C)*EllipticE((c + d*x)/2, 2))/(5*d) + (2*(7*a^4*A + 126*a^2*A*b^2 + 7*A*b^4 + 21*a^4*C + 42*a^2*b^2*C + 5*b^4*C)*EllipticF((c + d*x)/2, 2))/(21*d) - (2*b^2*(385*a^2*A - 35*A*b^2 - 141*a^2*C - 25*b^2*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(105*d) - (4*a*b^2*(175*A - 27*C)*sqrt(cos(c + d*x))*(a + b*cos(c + d*x))*sin(c + d*x))/(105*d) - (2*b^2*(21*A - C)*sqrt(cos(c + d*x))*(a + b*cos(c + d*x))^2*sin(c + d*x))/(7*d) + (16*A*b*(a + b*cos(c + d*x))^3*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))) + (2*A*(a + b*cos(c + d*x))^4*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2))],
[((a + b*cos(c + d*x))^4*(A + C*cos(c + d*x)^2))/cos(c + d*x)^(7/2), x, 8, (-2*(3*a^4*A + 30*a^2*A*b^2 - 5*A*b^4 + 5*a^4*C - 30*a^2*b^2*C - 3*b^4*C)*EllipticE((c + d*x)/2, 2))/(5*d) + (8*a*b*(a^2*A + 3*A*b^2 + 3*a^2*C + b^2*C)*EllipticF((c + d*x)/2, 2))/(3*d) - (2*a*b*(9*a^2*A + 73*A*b^2 + 15*a^2*C - 17*b^2*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(15*d) - (2*b*(9*a^2*A + 59*A*b^2 + 15*a^2*C - 3*b^2*C)*sqrt(cos(c + d*x))*(a + b*cos(c + d*x))*sin(c + d*x))/(15*d) + (2*(3*a^2*A + 16*A*b^2 + 5*a^2*C)*(a + b*cos(c + d*x))^2*sin(c + d*x))/(5*d*sqrt(cos(c + d*x))) + (16*A*b*(a + b*cos(c + d*x))^3*sin(c + d*x))/(15*d*cos(c + d*x)^(3/2)) + (2*A*(a + b*cos(c + d*x))^4*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2))],
[((a + b*cos(c + d*x))^4*(A + C*cos(c + d*x)^2))/cos(c + d*x)^(9/2), x, 8, (-8*a*b*(3*a^2*A + 5*A*b^2 + 5*a^2*C - 5*b^2*C)*EllipticE((c + d*x)/2, 2))/(5*d) + (2*(5*a^4*A + 42*a^2*A*b^2 + 21*A*b^4 + 7*a^4*C + 126*a^2*b^2*C + 7*b^4*C)*EllipticF((c + d*x)/2, 2))/(21*d) - (2*b^2*(227*a^2*A + 279*A*b^2 + 385*a^2*C - 35*b^2*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(105*d) + (4*b*(101*a^2*A + 96*A*b^2 + 175*a^2*C)*(a + b*cos(c + d*x))*sin(c + d*x))/(105*d*sqrt(cos(c + d*x))) + (2*(25*a^2*A + 48*A*b^2 + 35*a^2*C)*(a + b*cos(c + d*x))^2*sin(c + d*x))/(105*d*cos(c + d*x)^(3/2)) + (16*A*b*(a + b*cos(c + d*x))^3*sin(c + d*x))/(35*d*cos(c + d*x)^(5/2)) + (2*A*(a + b*cos(c + d*x))^4*sin(c + d*x))/(7*d*cos(c + d*x)^(7/2))],
[((a + b*cos(c + d*x))^4*(A + C*cos(c + d*x)^2))/cos(c + d*x)^(11/2), x, 8, (-2*(7*a^4*A + 54*a^2*A*b^2 + 15*A*b^4 + 9*a^4*C + 90*a^2*b^2*C - 15*b^4*C)*EllipticE((c + d*x)/2, 2))/(15*d) + (8*a*b*(5*a^2*A + 7*A*b^2 + 7*a^2*C + 21*b^2*C)*EllipticF((c + d*x)/2, 2))/(21*d) + (2*(147*a^4*A + 883*a^2*A*b^2 + 128*A*b^4 + 189*a^4*C + 1533*a^2*b^2*C)*sin(c + d*x))/(315*d*sqrt(cos(c + d*x))) + (4*b*(101*a^2*A + 32*A*b^2 + 147*a^2*C)*(a + b*cos(c + d*x))*sin(c + d*x))/(315*d*cos(c + d*x)^(3/2)) + (2*(49*a^2*A + 48*A*b^2 + 63*a^2*C)*(a + b*cos(c + d*x))^2*sin(c + d*x))/(315*d*cos(c + d*x)^(5/2)) + (16*A*b*(a + b*cos(c + d*x))^3*sin(c + d*x))/(63*d*cos(c + d*x)^(7/2)) + (2*A*(a + b*cos(c + d*x))^4*sin(c + d*x))/(9*d*cos(c + d*x)^(9/2))],
[((a + b*cos(c + d*x))^4*(A + C*cos(c + d*x)^2))/cos(c + d*x)^(13/2), x, 9, (-8*a*b*(7*a^2*A + 9*A*b^2 + 9*a^2*C + 15*b^2*C)*EllipticE((c + d*x)/2, 2))/(15*d) + (2*(45*a^4*A + 330*a^2*A*b^2 + 77*A*b^4 + 55*a^4*C + 462*a^2*b^2*C + 231*b^4*C)*EllipticF((c + d*x)/2, 2))/(231*d) + (2*(675*a^4*A + 3199*a^2*A*b^2 + 128*A*b^4 + 825*a^4*C + 4653*a^2*b^2*C)*sin(c + d*x))/(3465*d*cos(c + d*x)^(3/2)) + (8*a*b*(7*a^2*A + 9*A*b^2 + 9*a^2*C + 15*b^2*C)*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))) + (4*b*(673*a^2*A + 96*A*b^2 + 891*a^2*C)*(a + b*cos(c + d*x))*sin(c + d*x))/(3465*d*cos(c + d*x)^(5/2)) + (2*(27*a^2*A + 16*A*b^2 + 33*a^2*C)*(a + b*cos(c + d*x))^2*sin(c + d*x))/(231*d*cos(c + d*x)^(7/2)) + (16*A*b*(a + b*cos(c + d*x))^3*sin(c + d*x))/(99*d*cos(c + d*x)^(9/2)) + (2*A*(a + b*cos(c + d*x))^4*sin(c + d*x))/(11*d*cos(c + d*x)^(11/2))],


# ::Subsubsection::Closed:: 
#n<0


[(cos(c + d*x)^(7/2)*(A + C*cos(c + d*x)^2))/(a + b*cos(c + d*x)), x, 9, (2*(15*a^2*A*b^2 + 9*A*b^4 + 15*a^4*C + 9*a^2*b^2*C + 7*b^4*C)*EllipticE((c + d*x)/2, 2))/(15*b^5*d) - (2*a*(21*a^2*A*b^2 + 7*A*b^4 + 21*a^4*C + 7*a^2*b^2*C + 5*b^4*C)*EllipticF((c + d*x)/2, 2))/(21*b^6*d) + (2*a^4*(A*b^2 + a^2*C)*EllipticPi((2*b)/(a + b), (c + d*x)/2, 2))/(b^6*(a + b)*d) - (2*a*(7*A*b^2 + 7*a^2*C + 5*b^2*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(21*b^4*d) + (2*(9*A*b^2 + 9*a^2*C + 7*b^2*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(45*b^3*d) - (2*a*C*cos(c + d*x)^(5/2)*sin(c + d*x))/(7*b^2*d) + (2*C*cos(c + d*x)^(7/2)*sin(c + d*x))/(9*b*d)],
[(cos(c + d*x)^(5/2)*(A + C*cos(c + d*x)^2))/(a + b*cos(c + d*x)), x, 8, (-2*a*(5*A*b^2 + 5*a^2*C + 3*b^2*C)*EllipticE((c + d*x)/2, 2))/(5*b^4*d) + (2*(21*a^2*A*b^2 + 7*A*b^4 + 21*a^4*C + 7*a^2*b^2*C + 5*b^4*C)*EllipticF((c + d*x)/2, 2))/(21*b^5*d) - (2*a^3*(A*b^2 + a^2*C)*EllipticPi((2*b)/(a + b), (c + d*x)/2, 2))/(b^5*(a + b)*d) + (2*(7*A*b^2 + 7*a^2*C + 5*b^2*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(21*b^3*d) - (2*a*C*cos(c + d*x)^(3/2)*sin(c + d*x))/(5*b^2*d) + (2*C*cos(c + d*x)^(5/2)*sin(c + d*x))/(7*b*d)],
[(cos(c + d*x)^(3/2)*(A + C*cos(c + d*x)^2))/(a + b*cos(c + d*x)), x, 7, (2*(5*A*b^2 + 5*a^2*C + 3*b^2*C)*EllipticE((c + d*x)/2, 2))/(5*b^3*d) - (2*a*(3*A*b^2 + 3*a^2*C + b^2*C)*EllipticF((c + d*x)/2, 2))/(3*b^4*d) + (2*a^2*(A*b^2 + a^2*C)*EllipticPi((2*b)/(a + b), (c + d*x)/2, 2))/(b^4*(a + b)*d) - (2*a*C*sqrt(cos(c + d*x))*sin(c + d*x))/(3*b^2*d) + (2*C*cos(c + d*x)^(3/2)*sin(c + d*x))/(5*b*d)],
[(sqrt(cos(c + d*x))*(A + C*cos(c + d*x)^2))/(a + b*cos(c + d*x)), x, 6, (-2*a*C*EllipticE((c + d*x)/2, 2))/(b^2*d) + (2*(3*A*b^2 + 3*a^2*C + b^2*C)*EllipticF((c + d*x)/2, 2))/(3*b^3*d) - (2*a*(A*b^2 + a^2*C)*EllipticPi((2*b)/(a + b), (c + d*x)/2, 2))/(b^3*(a + b)*d) + (2*C*sqrt(cos(c + d*x))*sin(c + d*x))/(3*b*d)],
[(A + C*cos(c + d*x)^2)/(sqrt(cos(c + d*x))*(a + b*cos(c + d*x))), x, 5, (2*C*EllipticE((c + d*x)/2, 2))/(b*d) - (2*a*C*EllipticF((c + d*x)/2, 2))/(b^2*d) + (2*(A*b^2 + a^2*C)*EllipticPi((2*b)/(a + b), (c + d*x)/2, 2))/(b^2*(a + b)*d)],
[(A + C*cos(c + d*x)^2)/(cos(c + d*x)^(3/2)*(a + b*cos(c + d*x))), x, 6, (-2*A*EllipticE((c + d*x)/2, 2))/(a*d) + (2*C*EllipticF((c + d*x)/2, 2))/(b*d) - (2*(A*b^2 + a^2*C)*EllipticPi((2*b)/(a + b), (c + d*x)/2, 2))/(a*b*(a + b)*d) + (2*A*sin(c + d*x))/(a*d*sqrt(cos(c + d*x)))],
[(A + C*cos(c + d*x)^2)/(cos(c + d*x)^(5/2)*(a + b*cos(c + d*x))), x, 7, (2*A*b*EllipticE((c + d*x)/2, 2))/(a^2*d) + (2*A*EllipticF((c + d*x)/2, 2))/(3*a*d) + (2*(A*b^2 + a^2*C)*EllipticPi((2*b)/(a + b), (c + d*x)/2, 2))/(a^2*(a + b)*d) + (2*A*sin(c + d*x))/(3*a*d*cos(c + d*x)^(3/2)) - (2*A*b*sin(c + d*x))/(a^2*d*sqrt(cos(c + d*x)))],
[(A + C*cos(c + d*x)^2)/(cos(c + d*x)^(7/2)*(a + b*cos(c + d*x))), x, 8, (-2*(3*a^2*A + 5*A*b^2 + 5*a^2*C)*EllipticE((c + d*x)/2, 2))/(5*a^3*d) - (2*A*b*EllipticF((c + d*x)/2, 2))/(3*a^2*d) - (2*b*(A*b^2 + a^2*C)*EllipticPi((2*b)/(a + b), (c + d*x)/2, 2))/(a^3*(a + b)*d) + (2*A*sin(c + d*x))/(5*a*d*cos(c + d*x)^(5/2)) - (2*A*b*sin(c + d*x))/(3*a^2*d*cos(c + d*x)^(3/2)) + (2*(3*a^2*A + 5*A*b^2 + 5*a^2*C)*sin(c + d*x))/(5*a^3*d*sqrt(cos(c + d*x)))],
[(A + C*cos(c + d*x)^2)/(cos(c + d*x)^(9/2)*(a + b*cos(c + d*x))), x, 9, (2*b*(3*a^2*A + 5*A*b^2 + 5*a^2*C)*EllipticE((c + d*x)/2, 2))/(5*a^4*d) + (2*(5*a^2*A + 7*A*b^2 + 7*a^2*C)*EllipticF((c + d*x)/2, 2))/(21*a^3*d) + (2*b^2*(A*b^2 + a^2*C)*EllipticPi((2*b)/(a + b), (c + d*x)/2, 2))/(a^4*(a + b)*d) + (2*A*sin(c + d*x))/(7*a*d*cos(c + d*x)^(7/2)) - (2*A*b*sin(c + d*x))/(5*a^2*d*cos(c + d*x)^(5/2)) + (2*(5*a^2*A + 7*A*b^2 + 7*a^2*C)*sin(c + d*x))/(21*a^3*d*cos(c + d*x)^(3/2)) - (2*b*(3*a^2*A + 5*A*b^2 + 5*a^2*C)*sin(c + d*x))/(5*a^4*d*sqrt(cos(c + d*x)))],
[(A + C*cos(c + d*x)^2)/(cos(c + d*x)^(11/2)*(a + b*cos(c + d*x))), x, 10, (-2*(7*a^4*A + 9*a^2*A*b^2 + 15*A*b^4 + 9*a^4*C + 15*a^2*b^2*C)*EllipticE((c + d*x)/2, 2))/(15*a^5*d) - (2*b*(5*a^2*A + 7*A*b^2 + 7*a^2*C)*EllipticF((c + d*x)/2, 2))/(21*a^4*d) - (2*b^3*(A*b^2 + a^2*C)*EllipticPi((2*b)/(a + b), (c + d*x)/2, 2))/(a^5*(a + b)*d) + (2*A*sin(c + d*x))/(9*a*d*cos(c + d*x)^(9/2)) - (2*A*b*sin(c + d*x))/(7*a^2*d*cos(c + d*x)^(7/2)) + (2*(7*a^2*A + 9*A*b^2 + 9*a^2*C)*sin(c + d*x))/(45*a^3*d*cos(c + d*x)^(5/2)) - (2*b*(5*a^2*A + 7*A*b^2 + 7*a^2*C)*sin(c + d*x))/(21*a^4*d*cos(c + d*x)^(3/2)) + (2*(7*a^4*A + 9*a^2*A*b^2 + 15*A*b^4 + 9*a^4*C + 15*a^2*b^2*C)*sin(c + d*x))/(15*a^5*d*sqrt(cos(c + d*x)))],


[(cos(c + d*x)^(5/2)*(A + C*cos(c + d*x)^2))/(a + b*cos(c + d*x))^2, x, 8, ((15*a^2*A*b^2 - 10*A*b^4 + 35*a^4*C - 24*a^2*b^2*C - 6*b^4*C)*EllipticE((c + d*x)/2, 2))/(5*b^4*(a^2 - b^2)*d) - (a*(9*a^2*A*b^2 - 12*A*b^4 + 21*a^4*C - 20*a^2*b^2*C - 4*b^4*C)*EllipticF((c + d*x)/2, 2))/(3*b^5*(a^2 - b^2)*d) + (a^2*(3*a^2*A*b^2 - 5*A*b^4 + 7*a^4*C - 9*a^2*b^2*C)*EllipticPi((2*b)/(a + b), (c + d*x)/2, 2))/((a - b)*b^5*(a + b)^2*d) - (a*(3*A*b^2 + 7*a^2*C - 4*b^2*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(3*b^3*(a^2 - b^2)*d) + ((5*A*b^2 + 7*a^2*C - 2*b^2*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(5*b^2*(a^2 - b^2)*d) - ((A*b^2 + a^2*C)*cos(c + d*x)^(5/2)*sin(c + d*x))/(b*(a^2 - b^2)*d*(a + b*cos(c + d*x)))],
[(cos(c + d*x)^(3/2)*(A + C*cos(c + d*x)^2))/(a + b*cos(c + d*x))^2, x, 7, -((a*(A*b^2 + 5*a^2*C - 4*b^2*C)*EllipticE((c + d*x)/2, 2))/(b^3*(a^2 - b^2)*d)) + ((3*a^2*A*b^2 - 6*A*b^4 + 15*a^4*C - 16*a^2*b^2*C - 2*b^4*C)*EllipticF((c + d*x)/2, 2))/(3*b^4*(a^2 - b^2)*d) - (a*(a^2*A*b^2 - 3*A*b^4 + 5*a^4*C - 7*a^2*b^2*C)*EllipticPi((2*b)/(a + b), (c + d*x)/2, 2))/((a - b)*b^4*(a + b)^2*d) + ((3*A*b^2 + 5*a^2*C - 2*b^2*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(3*b^2*(a^2 - b^2)*d) - ((A*b^2 + a^2*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(b*(a^2 - b^2)*d*(a + b*cos(c + d*x)))],
[(sqrt(cos(c + d*x))*(A + C*cos(c + d*x)^2))/(a + b*cos(c + d*x))^2, x, 6, ((A*b^2 + 3*a^2*C - 2*b^2*C)*EllipticE((c + d*x)/2, 2))/(b^2*(a^2 - b^2)*d) + (a*(A*b^2 - 3*a^2*C + 4*b^2*C)*EllipticF((c + d*x)/2, 2))/(b^3*(a^2 - b^2)*d) - ((a^2*A*b^2 + A*b^4 - 3*a^4*C + 5*a^2*b^2*C)*EllipticPi((2*b)/(a + b), (c + d*x)/2, 2))/((a - b)*b^3*(a + b)^2*d) - ((A*b^2 + a^2*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(b*(a^2 - b^2)*d*(a + b*cos(c + d*x)))],
[(A + C*cos(c + d*x)^2)/(sqrt(cos(c + d*x))*(a + b*cos(c + d*x))^2), x, 6, -(((A*b^2 + a^2*C)*EllipticE((c + d*x)/2, 2))/(a*b*(a^2 - b^2)*d)) - ((A*b^2 - a^2*C + 2*b^2*C)*EllipticF((c + d*x)/2, 2))/(b^2*(a^2 - b^2)*d) + ((3*a^2*A*b^2 - A*b^4 - a^4*C + 3*a^2*b^2*C)*EllipticPi((2*b)/(a + b), (c + d*x)/2, 2))/(a*(a - b)*b^2*(a + b)^2*d) + ((A*b^2 + a^2*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(a*(a^2 - b^2)*d*(a + b*cos(c + d*x)))],
[(A + C*cos(c + d*x)^2)/(cos(c + d*x)^(3/2)*(a + b*cos(c + d*x))^2), x, 7, -(((2*a^2*A - 3*A*b^2 - a^2*C)*EllipticE((c + d*x)/2, 2))/(a^2*(a^2 - b^2)*d)) + ((A*b^2 + a^2*C)*EllipticF((c + d*x)/2, 2))/(a*b*(a^2 - b^2)*d) - ((5*a^2*A*b^2 - 3*A*b^4 + a^4*C + a^2*b^2*C)*EllipticPi((2*b)/(a + b), (c + d*x)/2, 2))/(a^2*(a - b)*b*(a + b)^2*d) + ((2*a^2*A - 3*A*b^2 - a^2*C)*sin(c + d*x))/(a^2*(a^2 - b^2)*d*sqrt(cos(c + d*x))) + ((A*b^2 + a^2*C)*sin(c + d*x))/(a*(a^2 - b^2)*d*sqrt(cos(c + d*x))*(a + b*cos(c + d*x)))],
[(A + C*cos(c + d*x)^2)/(cos(c + d*x)^(5/2)*(a + b*cos(c + d*x))^2), x, 8, (b*(4*a^2*A - 5*A*b^2 - a^2*C)*EllipticE((c + d*x)/2, 2))/(a^3*(a^2 - b^2)*d) + ((2*a^2*A - 5*A*b^2 - 3*a^2*C)*EllipticF((c + d*x)/2, 2))/(3*a^2*(a^2 - b^2)*d) + ((7*a^2*A*b^2 - 5*A*b^4 + 3*a^4*C - a^2*b^2*C)*EllipticPi((2*b)/(a + b), (c + d*x)/2, 2))/(a^3*(a - b)*(a + b)^2*d) + ((2*a^2*A - 5*A*b^2 - 3*a^2*C)*sin(c + d*x))/(3*a^2*(a^2 - b^2)*d*cos(c + d*x)^(3/2)) - (b*(4*a^2*A - 5*A*b^2 - a^2*C)*sin(c + d*x))/(a^3*(a^2 - b^2)*d*sqrt(cos(c + d*x))) + ((A*b^2 + a^2*C)*sin(c + d*x))/(a*(a^2 - b^2)*d*cos(c + d*x)^(3/2)*(a + b*cos(c + d*x)))],
[(A + C*cos(c + d*x)^2)/(cos(c + d*x)^(7/2)*(a + b*cos(c + d*x))^2), x, 9, -((6*a^4*A + 24*a^2*A*b^2 - 35*A*b^4 + 10*a^4*C - 15*a^2*b^2*C)*EllipticE((c + d*x)/2, 2))/(5*a^4*(a^2 - b^2)*d) - (b*(4*a^2*A - 7*A*b^2 - 3*a^2*C)*EllipticF((c + d*x)/2, 2))/(3*a^3*(a^2 - b^2)*d) - (b*(9*a^2*A*b^2 - 7*A*b^4 + 5*a^4*C - 3*a^2*b^2*C)*EllipticPi((2*b)/(a + b), (c + d*x)/2, 2))/(a^4*(a - b)*(a + b)^2*d) + ((2*a^2*A - 7*A*b^2 - 5*a^2*C)*sin(c + d*x))/(5*a^2*(a^2 - b^2)*d*cos(c + d*x)^(5/2)) - (b*(4*a^2*A - 7*A*b^2 - 3*a^2*C)*sin(c + d*x))/(3*a^3*(a^2 - b^2)*d*cos(c + d*x)^(3/2)) + ((6*a^4*A + 24*a^2*A*b^2 - 35*A*b^4 + 10*a^4*C - 15*a^2*b^2*C)*sin(c + d*x))/(5*a^4*(a^2 - b^2)*d*sqrt(cos(c + d*x))) + ((A*b^2 + a^2*C)*sin(c + d*x))/(a*(a^2 - b^2)*d*cos(c + d*x)^(5/2)*(a + b*cos(c + d*x)))],


[(cos(c + d*x)^(5/2)*(A + C*cos(c + d*x)^2))/(a + b*cos(c + d*x))^3, x, 8, -(a*(3*a^2*A*b^2 - 9*A*b^4 + 35*a^4*C - 65*a^2*b^2*C + 24*b^4*C)*EllipticE((c + d*x)/2, 2))/(4*b^4*(a^2 - b^2)^2*d) + ((9*a^4*A*b^2 - 15*a^2*A*b^4 + 24*A*b^6 + 105*a^6*C - 223*a^4*b^2*C + 128*a^2*b^4*C + 8*b^6*C)*EllipticF((c + d*x)/2, 2))/(12*b^5*(a^2 - b^2)^2*d) - (a*(3*a^4*A*b^2 - 6*a^2*A*b^4 + 15*A*b^6 + 35*a^6*C - 86*a^4*b^2*C + 63*a^2*b^4*C)*EllipticPi((2*b)/(a + b), (c + d*x)/2, 2))/(4*(a - b)^2*b^5*(a + b)^3*d) + ((3*a^2*A*b^2 - 21*A*b^4 + 35*a^4*C - 61*a^2*b^2*C + 8*b^4*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(12*b^3*(a^2 - b^2)^2*d) - ((A*b^2 + a^2*C)*cos(c + d*x)^(5/2)*sin(c + d*x))/(2*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) + ((a^2*A*b^2 + 5*A*b^4 - 7*a^4*C + 13*a^2*b^2*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(4*b^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))],
[(cos(c + d*x)^(3/2)*(A + C*cos(c + d*x)^2))/(a + b*cos(c + d*x))^3, x, 7, -((a^2*A*b^2 + 5*A*b^4 - 15*a^4*C + 29*a^2*b^2*C - 8*b^4*C)*EllipticE((c + d*x)/2, 2))/(4*b^3*(a^2 - b^2)^2*d) + (a*(a^2*A*b^2 - 7*A*b^4 - 15*a^4*C + 33*a^2*b^2*C - 24*b^4*C)*EllipticF((c + d*x)/2, 2))/(4*b^4*(a^2 - b^2)^2*d) - ((a^4*A*b^2 - 10*a^2*A*b^4 - 3*A*b^6 - 15*a^6*C + 38*a^4*b^2*C - 35*a^2*b^4*C)*EllipticPi((2*b)/(a + b), (c + d*x)/2, 2))/(4*(a - b)^2*b^4*(a + b)^3*d) - ((A*b^2 + a^2*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(2*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) + ((3*a^2*A*b^2 + 3*A*b^4 - 5*a^4*C + 11*a^2*b^2*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(4*b^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))],
[(sqrt(cos(c + d*x))*(A + C*cos(c + d*x)^2))/(a + b*cos(c + d*x))^3, x, 7, ((5*a^2*A*b^2 + A*b^4 - 3*a^4*C + 9*a^2*b^2*C)*EllipticE((c + d*x)/2, 2))/(4*a*b^2*(a^2 - b^2)^2*d) + ((3*a^2*A*b^2 + 3*A*b^4 + 3*a^4*C - 5*a^2*b^2*C + 8*b^4*C)*EllipticF((c + d*x)/2, 2))/(4*b^3*(a^2 - b^2)^2*d) - ((3*a^4*A*b^2 + 10*a^2*A*b^4 - A*b^6 + 3*a^6*C - 6*a^4*b^2*C + 15*a^2*b^4*C)*EllipticPi((2*b)/(a + b), (c + d*x)/2, 2))/(4*a*(a - b)^2*b^3*(a + b)^3*d) - ((A*b^2 + a^2*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(2*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) - ((5*a^2*A*b^2 + A*b^4 - 3*a^4*C + 9*a^2*b^2*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(4*a*b*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))],
[(A + C*cos(c + d*x)^2)/(sqrt(cos(c + d*x))*(a + b*cos(c + d*x))^3), x, 7, -((9*a^2*A*b^2 - 3*A*b^4 + a^4*C + 5*a^2*b^2*C)*EllipticE((c + d*x)/2, 2))/(4*a^2*b*(a^2 - b^2)^2*d) - ((7*a^2*A*b^2 - A*b^4 - a^4*C + 7*a^2*b^2*C)*EllipticF((c + d*x)/2, 2))/(4*a*b^2*(a^2 - b^2)^2*d) + ((15*a^4*A*b^2 - 6*a^2*A*b^4 + 3*A*b^6 - a^6*C + 10*a^4*b^2*C + 3*a^2*b^4*C)*EllipticPi((2*b)/(a + b), (c + d*x)/2, 2))/(4*a^2*(a - b)^2*b^2*(a + b)^3*d) + ((A*b^2 + a^2*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(2*a*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) + ((9*a^2*A*b^2 - 3*A*b^4 + a^4*C + 5*a^2*b^2*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(4*a^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))],
[(A + C*cos(c + d*x)^2)/(cos(c + d*x)^(3/2)*(a + b*cos(c + d*x))^3), x, 8, -((8*a^4*A - 29*a^2*A*b^2 + 15*A*b^4 - 5*a^4*C - a^2*b^2*C)*EllipticE((c + d*x)/2, 2))/(4*a^3*(a^2 - b^2)^2*d) + ((11*a^2*A*b^2 - 5*A*b^4 + 3*a^4*C + 3*a^2*b^2*C)*EllipticF((c + d*x)/2, 2))/(4*a^2*b*(a^2 - b^2)^2*d) - ((35*a^4*A*b^2 - 38*a^2*A*b^4 + 15*A*b^6 + 3*a^6*C + 10*a^4*b^2*C - a^2*b^4*C)*EllipticPi((2*b)/(a + b), (c + d*x)/2, 2))/(4*a^3*(a - b)^2*b*(a + b)^3*d) + ((8*a^4*A - 29*a^2*A*b^2 + 15*A*b^4 - 5*a^4*C - a^2*b^2*C)*sin(c + d*x))/(4*a^3*(a^2 - b^2)^2*d*sqrt(cos(c + d*x))) + ((A*b^2 + a^2*C)*sin(c + d*x))/(2*a*(a^2 - b^2)*d*sqrt(cos(c + d*x))*(a + b*cos(c + d*x))^2) + ((11*a^2*A*b^2 - 5*A*b^4 + 3*a^4*C + 3*a^2*b^2*C)*sin(c + d*x))/(4*a^2*(a^2 - b^2)^2*d*sqrt(cos(c + d*x))*(a + b*cos(c + d*x)))],
[(A + C*cos(c + d*x)^2)/(cos(c + d*x)^(5/2)*(a + b*cos(c + d*x))^3), x, 9, (b*(24*a^4*A - 65*a^2*A*b^2 + 35*A*b^4 - 9*a^4*C + 3*a^2*b^2*C)*EllipticE((c + d*x)/2, 2))/(4*a^4*(a^2 - b^2)^2*d) + ((8*a^4*A - 61*a^2*A*b^2 + 35*A*b^4 - 21*a^4*C + 3*a^2*b^2*C)*EllipticF((c + d*x)/2, 2))/(12*a^3*(a^2 - b^2)^2*d) + ((63*a^4*A*b^2 - 86*a^2*A*b^4 + 35*A*b^6 + 15*a^6*C - 6*a^4*b^2*C + 3*a^2*b^4*C)*EllipticPi((2*b)/(a + b), (c + d*x)/2, 2))/(4*a^4*(a - b)^2*(a + b)^3*d) + ((8*a^4*A - 61*a^2*A*b^2 + 35*A*b^4 - 21*a^4*C + 3*a^2*b^2*C)*sin(c + d*x))/(12*a^3*(a^2 - b^2)^2*d*cos(c + d*x)^(3/2)) - (b*(24*a^4*A - 65*a^2*A*b^2 + 35*A*b^4 - 9*a^4*C + 3*a^2*b^2*C)*sin(c + d*x))/(4*a^4*(a^2 - b^2)^2*d*sqrt(cos(c + d*x))) + ((A*b^2 + a^2*C)*sin(c + d*x))/(2*a*(a^2 - b^2)*d*cos(c + d*x)^(3/2)*(a + b*cos(c + d*x))^2) + ((13*a^2*A*b^2 - 7*A*b^4 + 5*a^4*C + a^2*b^2*C)*sin(c + d*x))/(4*a^2*(a^2 - b^2)^2*d*cos(c + d*x)^(3/2)*(a + b*cos(c + d*x)))],


# ::Subsection::Closed:: 
#Integrands of the form Cos[c+d x]^(m/2) (A+C Cos[c+d x]^2) (a+b Cos[c+d x])^(n/2)
#


# ::Subsubsection::Closed:: 
#n>0


[sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*(A + C*cos(c + d*x)^2), x, 8, ((a + b)*(24*A*b^2 - 3*a^2*C + 16*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(24*b^2*d*sqrt(a + b*cos(c + d*x))) - (a*(24*A*b - a*C + 14*b*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(12*b*d*sqrt(a + b*cos(c + d*x))) + (a*(8*A*b^2 + a^2*C + 4*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(4*b^2*d*sqrt(a + b*cos(c + d*x))) + (a*C*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(12*b*d) + (C*cos(c + d*x)^(3/2)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*d) + ((24*A*b^2 - 3*a^2*C + 16*b^2*C)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(24*b^2*d)],
[(sqrt(a + b*cos(c + d*x))*(A + C*cos(c + d*x)^2))/sqrt(cos(c + d*x)), x, 7, (a*(a + b)*C*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(4*b*d*sqrt(a + b*cos(c + d*x))) + ((4*a*A - 4*A*b + a*C - 2*b*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(2*d*sqrt(a + b*cos(c + d*x))) + ((8*A*b^2 - a^2*C + 4*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(2*b*d*sqrt(a + b*cos(c + d*x))) + (C*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(2*d) + (a*C*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(4*b*d)],
[(sqrt(a + b*cos(c + d*x))*(A + C*cos(c + d*x)^2))/cos(c + d*x)^(3/2), x, 7, -(((a + b)*(2*A - C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(d*sqrt(a + b*cos(c + d*x)))) + (2*(a*A + A*b - a*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(d*sqrt(a + b*cos(c + d*x))) + (2*a*C*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(d*sqrt(a + b*cos(c + d*x))) + (2*A*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(d*sqrt(cos(c + d*x))) - ((2*A - C)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/d],
[(sqrt(a + b*cos(c + d*x))*(A + C*cos(c + d*x)^2))/cos(c + d*x)^(5/2), x, 7, (-2*A*b*(a + b)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(3*a*d*sqrt(a + b*cos(c + d*x))) + (2*(a*A + A*b + 3*a*C - 3*b*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(3*d*sqrt(a + b*cos(c + d*x))) + (4*b*C*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(d*sqrt(a + b*cos(c + d*x))) + (2*A*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2)) + (2*A*b*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(3*a*d*sqrt(cos(c + d*x)))],
[(sqrt(a + b*cos(c + d*x))*(A + C*cos(c + d*x)^2))/cos(c + d*x)^(7/2), x, 6, (-2*(a + b)*(9*a^2*A - 2*A*b^2 + 15*a^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(15*a^2*d*sqrt(a + b*cos(c + d*x))) + (2*(9*a^2*A + 7*a*A*b - 2*A*b^2 + 15*a^2*C + 15*a*b*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(15*a*d*sqrt(a + b*cos(c + d*x))) + (2*A*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2)) + (2*A*b*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(15*a*d*cos(c + d*x)^(3/2)) + (2*(9*a^2*A - 2*A*b^2 + 15*a^2*C)*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(15*a^2*d*sqrt(cos(c + d*x)))],
[(sqrt(a + b*cos(c + d*x))*(A + C*cos(c + d*x)^2))/cos(c + d*x)^(9/2), x, 7, (-2*b*(a + b)*(19*a^2*A + 8*A*b^2 + 35*a^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(105*a^3*d*sqrt(a + b*cos(c + d*x))) + (2*(25*a^3*A + 19*a^2*A*b + 2*a*A*b^2 + 8*A*b^3 + 35*a^3*C + 35*a^2*b*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(105*a^2*d*sqrt(a + b*cos(c + d*x))) + (2*A*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(7*d*cos(c + d*x)^(7/2)) + (2*A*b*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(35*a*d*cos(c + d*x)^(5/2)) + (2*(25*a^2*A - 4*A*b^2 + 35*a^2*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(105*a^2*d*cos(c + d*x)^(3/2)) + (2*b*(19*a^2*A + 8*A*b^2 + 35*a^2*C)*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(105*a^3*d*sqrt(cos(c + d*x)))],


[sqrt(cos(c + d*x))*(a + b*cos(c + d*x))^(3/2)*(A + C*cos(c + d*x)^2), x, 9, (a*(a + b)*(80*A*b^2 - 3*a^2*C + 52*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(64*b^2*d*sqrt(a + b*cos(c + d*x))) - ((64*a^2*A*b - 16*a*A*b^2 + 32*A*b^3 - a^3*C + 38*a^2*b*C - 12*a*b^2*C + 24*b^3*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(32*b*d*sqrt(a + b*cos(c + d*x))) + ((48*a^2*A*b^2 + 64*A*b^4 + 3*a^4*C + 24*a^2*b^2*C + 48*b^4*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(32*b^2*d*sqrt(a + b*cos(c + d*x))) + ((16*A*b^2 + a^2*C + 12*b^2*C)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(32*b*d) + (a*C*cos(c + d*x)^(3/2)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(8*d) + (C*cos(c + d*x)^(3/2)*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(4*d) + (a*(80*A*b^2 - 3*a^2*C + 52*b^2*C)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(64*b^2*d)],
[((a + b*cos(c + d*x))^(3/2)*(A + C*cos(c + d*x)^2))/sqrt(cos(c + d*x)), x, 8, ((a + b)*(24*A*b^2 + 3*a^2*C + 16*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(24*b*d*sqrt(a + b*cos(c + d*x))) + (a*(24*a*A - 48*A*b + 7*a*C - 26*b*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(12*d*sqrt(a + b*cos(c + d*x))) + (a*(24*A*b^2 - a^2*C + 12*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(4*b*d*sqrt(a + b*cos(c + d*x))) + (a*C*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(4*d) + (C*sqrt(cos(c + d*x))*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(3*d) + ((24*A*b^2 + 3*a^2*C + 16*b^2*C)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(24*b*d)],
[((a + b*cos(c + d*x))^(3/2)*(A + C*cos(c + d*x)^2))/cos(c + d*x)^(3/2), x, 8, -(a*(a + b)*(8*A - 5*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(4*d*sqrt(a + b*cos(c + d*x))) + ((4*a^2*A + 8*a*A*b - 4*A*b^2 - 4*a^2*C + a*b*C - 2*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(2*d*sqrt(a + b*cos(c + d*x))) + ((8*A*b^2 + 3*a^2*C + 4*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(2*d*sqrt(a + b*cos(c + d*x))) - (b*(4*A - C)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(2*d) + (2*A*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(d*sqrt(cos(c + d*x))) - (a*(8*A - 5*C)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(4*d)],
[((a + b*cos(c + d*x))^(3/2)*(A + C*cos(c + d*x)^2))/cos(c + d*x)^(5/2), x, 8, -(b*(a + b)*(8*A - 3*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(3*d*sqrt(a + b*cos(c + d*x))) + (2*(a^2*A + 4*a*A*b + 3*A*b^2 + 3*a^2*C - 6*a*b*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(3*d*sqrt(a + b*cos(c + d*x))) + (6*a*b*C*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(d*sqrt(a + b*cos(c + d*x))) + (2*A*b*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(d*sqrt(cos(c + d*x))) + (2*A*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2)) - (b*(8*A - 3*C)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(3*d)],
[((a + b*cos(c + d*x))^(3/2)*(A + C*cos(c + d*x)^2))/cos(c + d*x)^(7/2), x, 8, (-2*(a + b)*(3*a^2*A + A*b^2 + 5*a^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(5*a*d*sqrt(a + b*cos(c + d*x))) + (2*(3*a^2*A + 4*a*A*b + A*b^2 + 5*a^2*C + 10*a*b*C - 5*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(5*d*sqrt(a + b*cos(c + d*x))) + (4*b^2*C*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(d*sqrt(a + b*cos(c + d*x))) + (2*A*b*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(5*d*cos(c + d*x)^(3/2)) + (2*A*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2)) + (2*(3*a^2*A + A*b^2 + 5*a^2*C)*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(5*a*d*sqrt(cos(c + d*x)))],
[((a + b*cos(c + d*x))^(3/2)*(A + C*cos(c + d*x)^2))/cos(c + d*x)^(9/2), x, 7, (-4*b*(a + b)*(41*a^2*A - 3*A*b^2 + 70*a^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(105*a^2*d*sqrt(a + b*cos(c + d*x))) + (2*(25*a^3*A + 82*a^2*A*b + 51*a*A*b^2 - 6*A*b^3 + 35*a^3*C + 140*a^2*b*C + 105*a*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(105*a*d*sqrt(a + b*cos(c + d*x))) + (6*A*b*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(35*d*cos(c + d*x)^(5/2)) + (2*(25*a^2*A + 3*A*b^2 + 35*a^2*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(105*a*d*cos(c + d*x)^(3/2)) + (2*A*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(7*d*cos(c + d*x)^(7/2)) + (4*b*(41*a^2*A - 3*A*b^2 + 70*a^2*C)*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(105*a^2*d*sqrt(cos(c + d*x)))],
[((a + b*cos(c + d*x))^(3/2)*(A + C*cos(c + d*x)^2))/cos(c + d*x)^(11/2), x, 8, (-2*(a + b)*(147*a^4*A + 33*a^2*A*b^2 + 8*A*b^4 + 189*a^4*C + 63*a^2*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(315*a^3*d*sqrt(a + b*cos(c + d*x))) + (2*(147*a^4*A + 186*a^3*A*b + 33*a^2*A*b^2 + 2*a*A*b^3 + 8*A*b^4 + 189*a^4*C + 252*a^3*b*C + 63*a^2*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(315*a^2*d*sqrt(a + b*cos(c + d*x))) + (2*A*b*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(21*d*cos(c + d*x)^(7/2)) + (2*(49*a^2*A + 3*A*b^2 + 63*a^2*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(315*a*d*cos(c + d*x)^(5/2)) + (4*b*(44*a^2*A - 2*A*b^2 + 63*a^2*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(315*a^2*d*cos(c + d*x)^(3/2)) + (2*A*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(9*d*cos(c + d*x)^(9/2)) + (2*(147*a^4*A + 33*a^2*A*b^2 + 8*A*b^4 + 189*a^4*C + 63*a^2*b^2*C)*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(315*a^3*d*sqrt(cos(c + d*x)))],


[sqrt(cos(c + d*x))*(a + b*cos(c + d*x))^(5/2)*(A + C*cos(c + d*x)^2), x, 10, ((a + b)*(2640*a^2*A*b^2 + 1280*A*b^4 - 45*a^4*C + 1692*a^2*b^2*C + 1024*b^4*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(1920*b^2*d*sqrt(a + b*cos(c + d*x))) - (a*(1920*a^2*A*b - 1040*a*A*b^2 + 3040*A*b^3 - 15*a^3*C + 1146*a^2*b*C - 772*a*b^2*C + 2312*b^3*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(960*b*d*sqrt(a + b*cos(c + d*x))) + (a*(80*a^2*A*b^2 + 320*A*b^4 + 3*a^4*C + 40*a^2*b^2*C + 240*b^4*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(64*b^2*d*sqrt(a + b*cos(c + d*x))) + (a*(1040*A*b^2 + 15*a^2*C + 772*b^2*C)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(960*b*d) + ((80*A*b^2 + 15*a^2*C + 64*b^2*C)*cos(c + d*x)^(3/2)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(240*d) + (a*C*cos(c + d*x)^(3/2)*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(8*d) + (C*cos(c + d*x)^(3/2)*(a + b*cos(c + d*x))^(5/2)*sin(c + d*x))/(5*d) + ((2640*a^2*A*b^2 + 1280*A*b^4 - 45*a^4*C + 1692*a^2*b^2*C + 1024*b^4*C)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(1920*b^2*d)],
[((a + b*cos(c + d*x))^(5/2)*(A + C*cos(c + d*x)^2))/sqrt(cos(c + d*x)), x, 9, (a*(a + b)*(432*A*b^2 + 15*a^2*C + 284*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(192*b*d*sqrt(a + b*cos(c + d*x))) + ((192*a^3*A - 576*a^2*A*b + 48*a*A*b^2 - 96*A*b^3 + 59*a^3*C - 322*a^2*b*C + 36*a*b^2*C - 72*b^3*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(96*d*sqrt(a + b*cos(c + d*x))) + ((240*a^2*A*b^2 + 64*A*b^4 - 5*a^4*C + 120*a^2*b^2*C + 48*b^4*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(32*b*d*sqrt(a + b*cos(c + d*x))) + ((16*A*b^2 + 5*a^2*C + 12*b^2*C)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(32*d) + (5*a*C*sqrt(cos(c + d*x))*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(24*d) + (C*sqrt(cos(c + d*x))*(a + b*cos(c + d*x))^(5/2)*sin(c + d*x))/(4*d) + (a*(432*A*b^2 + 15*a^2*C + 284*b^2*C)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(192*b*d)],
[((a + b*cos(c + d*x))^(5/2)*(A + C*cos(c + d*x)^2))/cos(c + d*x)^(3/2), x, 9, -((a + b)*(48*a^2*A - 24*A*b^2 - 33*a^2*C - 16*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(24*d*sqrt(a + b*cos(c + d*x))) + (a*(24*a^2*A + 72*a*A*b - 72*A*b^2 - 24*a^2*C + 13*a*b*C - 38*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(12*d*sqrt(a + b*cos(c + d*x))) + (5*a*(8*A*b^2 + a^2*C + 4*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(4*d*sqrt(a + b*cos(c + d*x))) - (a*b*(8*A - 3*C)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(4*d) - (b*(6*A - C)*sqrt(cos(c + d*x))*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(3*d) + (2*A*(a + b*cos(c + d*x))^(5/2)*sin(c + d*x))/(d*sqrt(cos(c + d*x))) - ((48*a^2*A - 24*A*b^2 - 33*a^2*C - 16*b^2*C)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(24*d)],
[((a + b*cos(c + d*x))^(5/2)*(A + C*cos(c + d*x)^2))/cos(c + d*x)^(5/2), x, 9, -(a*b*(a + b)*(56*A - 27*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(12*d*sqrt(a + b*cos(c + d*x))) + ((4*a^3*A + 28*a^2*A*b + 36*a*A*b^2 - 12*A*b^3 + 12*a^3*C - 36*a^2*b*C + 3*a*b^2*C - 6*b^3*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(6*d*sqrt(a + b*cos(c + d*x))) + (b*(8*A*b^2 + 15*a^2*C + 4*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(2*d*sqrt(a + b*cos(c + d*x))) - (b^2*(8*A - C)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(2*d) + (10*A*b*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))) + (2*A*(a + b*cos(c + d*x))^(5/2)*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2)) - (a*b*(56*A - 27*C)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(12*d)],
[((a + b*cos(c + d*x))^(5/2)*(A + C*cos(c + d*x)^2))/cos(c + d*x)^(7/2), x, 9, -((a + b)*(18*a^2*A + 46*A*b^2 + 30*a^2*C - 15*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(15*d*sqrt(a + b*cos(c + d*x))) + (2*(9*a^3*A + 17*a^2*A*b + 23*a*A*b^2 + 15*A*b^3 + 15*a^3*C + 45*a^2*b*C - 45*a*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(15*d*sqrt(a + b*cos(c + d*x))) + (10*a*b^2*C*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(d*sqrt(a + b*cos(c + d*x))) + (2*(3*a^2*A + 5*A*b^2 + 5*a^2*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(5*d*sqrt(cos(c + d*x))) + (2*A*b*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2)) + (2*A*(a + b*cos(c + d*x))^(5/2)*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2)) - ((18*a^2*A + 46*A*b^2 + 30*a^2*C - 15*b^2*C)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(15*d)],
[((a + b*cos(c + d*x))^(5/2)*(A + C*cos(c + d*x)^2))/cos(c + d*x)^(9/2), x, 9, (-2*b*(a + b)*(29*a^2*A + 3*A*b^2 + 49*a^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(21*a*d*sqrt(a + b*cos(c + d*x))) + (2*(5*a^3*A + 29*a^2*A*b + 27*a*A*b^2 + 3*A*b^3 + 7*a^3*C + 49*a^2*b*C + 63*a*b^2*C - 21*b^3*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(21*d*sqrt(a + b*cos(c + d*x))) + (4*b^3*C*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(d*sqrt(a + b*cos(c + d*x))) + (2*(5*a^2*A + 3*A*b^2 + 7*a^2*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(21*d*cos(c + d*x)^(3/2)) + (2*A*b*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(7*d*cos(c + d*x)^(5/2)) + (2*A*(a + b*cos(c + d*x))^(5/2)*sin(c + d*x))/(7*d*cos(c + d*x)^(7/2)) + (2*b*(29*a^2*A + 3*A*b^2 + 49*a^2*C)*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(21*a*d*sqrt(cos(c + d*x)))],
[((a + b*cos(c + d*x))^(5/2)*(A + C*cos(c + d*x)^2))/cos(c + d*x)^(11/2), x, 8, (-2*(a + b)*(147*a^4*A + 279*a^2*A*b^2 - 10*A*b^4 + 189*a^4*C + 483*a^2*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(315*a^2*d*sqrt(a + b*cos(c + d*x))) + (2*(147*a^4*A + 261*a^3*A*b + 279*a^2*A*b^2 + 155*a*A*b^3 - 10*A*b^4 + 189*a^4*C + 357*a^3*b*C + 483*a^2*b^2*C + 315*a*b^3*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(315*a*d*sqrt(a + b*cos(c + d*x))) + (2*(49*a^2*A + 15*A*b^2 + 63*a^2*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(315*d*cos(c + d*x)^(5/2)) + (2*b*(163*a^2*A + 5*A*b^2 + 231*a^2*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(315*a*d*cos(c + d*x)^(3/2)) + (10*A*b*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(63*d*cos(c + d*x)^(7/2)) + (2*A*(a + b*cos(c + d*x))^(5/2)*sin(c + d*x))/(9*d*cos(c + d*x)^(9/2)) + (2*(147*a^4*A + 279*a^2*A*b^2 - 10*A*b^4 + 189*a^4*C + 483*a^2*b^2*C)*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(315*a^2*d*sqrt(cos(c + d*x)))],
[((a + b*cos(c + d*x))^(5/2)*(A + C*cos(c + d*x)^2))/cos(c + d*x)^(13/2), x, 9, -((2*b*(a + b)*(741*a^4*A + 51*a^2*A*b^2 + 8*A*b^4 + 957*a^4*C + 99*a^2*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(693*a^3*d*sqrt(a + b*cos(c + d*x)))) + (2*(135*a^5*A + 741*a^4*A*b + 663*a^3*A*b^2 + 51*a^2*A*b^3 + 2*a*A*b^4 + 8*A*b^5 + 165*a^5*C + 957*a^4*b*C + 891*a^3*b^2*C + 99*a^2*b^3*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(693*a^2*d*sqrt(a + b*cos(c + d*x))) + (2*(27*a^2*A + 5*A*b^2 + 33*a^2*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(231*d*cos(c + d*x)^(7/2)) + (2*b*(229*a^2*A + 3*A*b^2 + 297*a^2*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(693*a*d*cos(c + d*x)^(5/2)) + (2*(135*a^4*A + 205*a^2*A*b^2 - 4*A*b^4 + 165*a^4*C + 297*a^2*b^2*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(693*a^2*d*cos(c + d*x)^(3/2)) + (10*A*b*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(99*d*cos(c + d*x)^(9/2)) + (2*A*(a + b*cos(c + d*x))^(5/2)*sin(c + d*x))/(11*d*cos(c + d*x)^(11/2)) + (2*b*(741*a^4*A + 51*a^2*A*b^2 + 8*A*b^4 + 957*a^4*C + 99*a^2*b^2*C)*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(693*a^3*d*sqrt(cos(c + d*x)))],
# {((a + b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(15/2), x, 10, -((2*(a + b)*(17787*a^6*A + 30669*a^4*A*b^2 - 760*a^2*A*b^4 - 240*A*b^6 + 21021*a^6*C + 39897*a^4*b^2*C - 1430*a^2*b^4*C)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))])/(45045*a^4*d*Sqrt[a + b*Cos[c + d*x]])) + (1/(45045*a^3*d*Sqrt[a + b*Cos[c + d*x]]))*(2*(17787*a^6*A + 30831*a^5*A*b + 30669*a^4*A*b^2 + 16685*a^3*A*b^3 - 760*a^2*A*b^4 - 60*a*A*b^5 - 240*A*b^6 + 21021*a^6*C + 37323*a^5*b*C + 39897*a^4*b^2*C + 22165*a^3*b^3*C - 1430*a^2*b^4*C)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(1/2)*(c + d*x)]], -((a - b)/(a + b))]) + (2*(121*a^2*A + 15*A*b^2 + 143*a^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(1287*d*Cos[c + d*x]^(9/2)) + (2*b*(2209*a^2*A + 15*A*b^2 + 2717*a^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(9009*a*d*Cos[c + d*x]^(7/2)) + (2*(5929*a^4*A + 8145*a^2*A*b^2 - 90*A*b^4 + 7007*a^4*C + 10725*a^2*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(45045*a^2*d*Cos[c + d*x]^(5/2)) + (2*b*(18973*a^4*A + 395*a^2*A*b^2 + 120*A*b^4 + 23309*a^4*C + 715*a^2*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(45045*a^3*d*Cos[c + d*x]^(3/2)) + (10*A*b*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(143*d*Cos[c + d*x]^(11/2)) + (2*A*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(13*d*Cos[c + d*x]^(13/2)) + (2*(17787*a^6*A + 30669*a^4*A*b^2 - 760*a^2*A*b^4 - 240*A*b^6 + 21021*a^6*C + 39897*a^4*b^2*C - 1430*a^2*b^4*C)*Sqrt[a + b*Cos[c + d*x]]*Tan[(1/2)*(c + d*x)])/(45045*a^4*d*Sqrt[Cos[c + d*x]])} 


# ::Subsubsection::Closed:: 
#n<0


[(cos(c + d*x)^(3/2)*(A + C*cos(c + d*x)^2))/sqrt(a + b*cos(c + d*x)), x, 8, ((a + b)*(24*A*b^2 + 15*a^2*C + 16*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(24*b^3*d*sqrt(a + b*cos(c + d*x))) - (a*(5*a + 2*b)*C*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(12*b^2*d*sqrt(a + b*cos(c + d*x))) - (a*(8*A*b^2 + 5*a^2*C + 4*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(4*b^3*d*sqrt(a + b*cos(c + d*x))) - (5*a*C*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(12*b^2*d) + (C*cos(c + d*x)^(3/2)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*b*d) + ((24*A*b^2 + 15*a^2*C + 16*b^2*C)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(24*b^3*d)],
[(sqrt(cos(c + d*x))*(A + C*cos(c + d*x)^2))/sqrt(a + b*cos(c + d*x)), x, 7, (-3*a*(a + b)*C*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(4*b^2*d*sqrt(a + b*cos(c + d*x))) - ((4*A*b - a*C + 2*b*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(2*b*d*sqrt(a + b*cos(c + d*x))) + ((8*A*b^2 + 3*a^2*C + 4*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(2*b^2*d*sqrt(a + b*cos(c + d*x))) + (C*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(2*b*d) - (3*a*C*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(4*b^2*d)],
[(A + C*cos(c + d*x)^2)/(sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))), x, 6, ((a + b)*C*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(b*d*sqrt(a + b*cos(c + d*x))) + (2*A*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(d*sqrt(a + b*cos(c + d*x))) - (2*a*C*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(b*d*sqrt(a + b*cos(c + d*x))) + (C*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(b*d)],
[(A + C*cos(c + d*x)^2)/(cos(c + d*x)^(3/2)*sqrt(a + b*cos(c + d*x))), x, 6, (-2*A*(a + b)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(a*d*sqrt(a + b*cos(c + d*x))) + (2*(A - C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(d*sqrt(a + b*cos(c + d*x))) + (4*C*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(d*sqrt(a + b*cos(c + d*x))) + (2*A*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(a*d*sqrt(cos(c + d*x)))],
[(A + C*cos(c + d*x)^2)/(cos(c + d*x)^(5/2)*sqrt(a + b*cos(c + d*x))), x, 5, (4*A*b*(a + b)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(3*a^2*d*sqrt(a + b*cos(c + d*x))) + (2*(a*A - 2*A*b + 3*a*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(3*a*d*sqrt(a + b*cos(c + d*x))) + (2*A*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*a*d*cos(c + d*x)^(3/2)) - (4*A*b*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(3*a^2*d*sqrt(cos(c + d*x)))],
[(A + C*cos(c + d*x)^2)/(cos(c + d*x)^(7/2)*sqrt(a + b*cos(c + d*x))), x, 6, (-2*(a + b)*(9*a^2*A + 8*A*b^2 + 15*a^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(15*a^3*d*sqrt(a + b*cos(c + d*x))) + (2*(9*a^2*A + 2*a*A*b + 8*A*b^2 + 15*a^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(15*a^2*d*sqrt(a + b*cos(c + d*x))) + (2*A*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(5*a*d*cos(c + d*x)^(5/2)) - (8*A*b*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(15*a^2*d*cos(c + d*x)^(3/2)) + (2*(9*a^2*A + 8*A*b^2 + 15*a^2*C)*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(15*a^3*d*sqrt(cos(c + d*x)))],
[(A + C*cos(c + d*x)^2)/(cos(c + d*x)^(9/2)*sqrt(a + b*cos(c + d*x))), x, 7, (4*b*(a + b)*(22*a^2*A + 24*A*b^2 + 35*a^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(105*a^4*d*sqrt(a + b*cos(c + d*x))) + (2*(25*a^3*A - 44*a^2*A*b - 12*a*A*b^2 - 48*A*b^3 + 35*a^3*C - 70*a^2*b*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(105*a^3*d*sqrt(a + b*cos(c + d*x))) + (2*A*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(7*a*d*cos(c + d*x)^(7/2)) - (12*A*b*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(35*a^2*d*cos(c + d*x)^(5/2)) + (2*(25*a^2*A + 24*A*b^2 + 35*a^2*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(105*a^3*d*cos(c + d*x)^(3/2)) - (4*b*(22*a^2*A + 24*A*b^2 + 35*a^2*C)*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(105*a^4*d*sqrt(cos(c + d*x)))],

[(sqrt(cos(c + d*x))*(a*A + (A*b + a*B)*cos(c + d*x) + b*B*cos(c + d*x)^2))/sqrt(a + b*cos(c + d*x)), x, 7, ((a + b)*(4*A*b + a*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(4*b*d*sqrt(a + b*cos(c + d*x))) - ((4*a*A - a*B + 2*b*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(2*d*sqrt(a + b*cos(c + d*x))) + ((4*a*A*b - a^2*B + 4*b^2*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(2*b*d*sqrt(a + b*cos(c + d*x))) + (B*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(2*d) + ((4*A*b + a*B)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(4*b*d)],
[(a + a*cos(c + d*x) + 2*b*cos(c + d*x)^2)/(sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))), x, 2, (2*(a + b)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(d*sqrt(a + b*cos(c + d*x))) + (2*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/d],


[(cos(c + d*x)^(3/2)*(A + C*cos(c + d*x)^2))/(a + b*cos(c + d*x))^(3/2), x, 8, -(a*(8*A*b^2 + 15*a^2*C - 7*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(4*(a - b)*b^3*d*sqrt(a + b*cos(c + d*x))) + ((4*A*b^2 + 5*a^2*C - 3*a*b*C + 2*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(2*(a - b)*b^2*d*sqrt(a + b*cos(c + d*x))) + ((8*A*b^2 + 15*a^2*C + 4*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(2*b^3*d*sqrt(a + b*cos(c + d*x))) - (2*(A*b^2 + a^2*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(b*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) + ((4*A*b^2 + 5*a^2*C - b^2*C)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(2*b^2*(a^2 - b^2)*d) - (a*(8*A*b^2 + 15*a^2*C - 7*b^2*C)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(4*b^3*(a^2 - b^2)*d)],
[(sqrt(cos(c + d*x))*(A + C*cos(c + d*x)^2))/(a + b*cos(c + d*x))^(3/2), x, 7, ((2*A*b^2 + 3*a^2*C - b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/((a - b)*b^2*d*sqrt(a + b*cos(c + d*x))) - (2*(A*b + a*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/((a - b)*b*d*sqrt(a + b*cos(c + d*x))) - (6*a*C*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(b^2*d*sqrt(a + b*cos(c + d*x))) - (2*(A*b^2 + a^2*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(b*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) + ((2*A*b^2 + 3*a^2*C - b^2*C)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(b^2*(a^2 - b^2)*d)],
[(A + C*cos(c + d*x)^2)/(sqrt(cos(c + d*x))*(a + b*cos(c + d*x))^(3/2)), x, 6, (-2*(A*b^2 + a^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(a*(a - b)*b*d*sqrt(a + b*cos(c + d*x))) + (2*(A + C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/((a - b)*d*sqrt(a + b*cos(c + d*x))) + (4*C*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(b*d*sqrt(a + b*cos(c + d*x))) - (2*(A*b^2 + a^2*C)*sqrt(cos(c + d*x))*tan((c + d*x)/2))/(a*b*(a + b)*d*sqrt(a + b*cos(c + d*x)))],
[(A + C*cos(c + d*x)^2)/(cos(c + d*x)^(3/2)*(a + b*cos(c + d*x))^(3/2)), x, 5, (-2*(a^2*A - 2*A*b^2 - a^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(a^2*(a - b)*d*sqrt(a + b*cos(c + d*x))) + (2*(a*A - 2*A*b - a*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(a*(a - b)*d*sqrt(a + b*cos(c + d*x))) + (2*(A*b^2 + a^2*C)*sin(c + d*x))/(a*(a^2 - b^2)*d*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))) + (2*(a^2*A - 2*A*b^2 - a^2*C)*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(a^2*(a^2 - b^2)*d*sqrt(cos(c + d*x)))],
[(A + C*cos(c + d*x)^2)/(cos(c + d*x)^(5/2)*(a + b*cos(c + d*x))^(3/2)), x, 6, (2*b*(5*a^2*A - 8*A*b^2 - 3*a^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(3*a^3*(a - b)*d*sqrt(a + b*cos(c + d*x))) + (2*(a^2*A - 6*a*A*b + 8*A*b^2 + 3*a^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(3*a^2*(a - b)*d*sqrt(a + b*cos(c + d*x))) + (2*(A*b^2 + a^2*C)*sin(c + d*x))/(a*(a^2 - b^2)*d*cos(c + d*x)^(3/2)*sqrt(a + b*cos(c + d*x))) + (2*(a^2*A - 4*A*b^2 - 3*a^2*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*a^2*(a^2 - b^2)*d*cos(c + d*x)^(3/2)) - (2*b*(5*a^2*A - 8*A*b^2 - 3*a^2*C)*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(3*a^3*(a^2 - b^2)*d*sqrt(cos(c + d*x)))],
[(A + C*cos(c + d*x)^2)/(cos(c + d*x)^(7/2)*(a + b*cos(c + d*x))^(3/2)), x, 7, (-2*(3*a^4*A + 8*a^2*A*b^2 - 16*A*b^4 + 5*a^4*C - 10*a^2*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(5*a^4*(a - b)*d*sqrt(a + b*cos(c + d*x))) + (2*(3*a^3*A - 4*a^2*A*b + 12*a*A*b^2 - 16*A*b^3 + 5*a^3*C - 10*a^2*b*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(5*a^3*(a - b)*d*sqrt(a + b*cos(c + d*x))) + (2*(A*b^2 + a^2*C)*sin(c + d*x))/(a*(a^2 - b^2)*d*cos(c + d*x)^(5/2)*sqrt(a + b*cos(c + d*x))) + (2*(a^2*A - 6*A*b^2 - 5*a^2*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(5*a^2*(a^2 - b^2)*d*cos(c + d*x)^(5/2)) - (2*b*(3*a^2*A - 8*A*b^2 - 5*a^2*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(5*a^3*(a^2 - b^2)*d*cos(c + d*x)^(3/2)) + (2*(3*a^4*A + 8*a^2*A*b^2 - 16*A*b^4 + 5*a^4*C - 10*a^2*b^2*C)*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(5*a^4*(a^2 - b^2)*d*sqrt(cos(c + d*x)))],


[(cos(c + d*x)^(3/2)*(A + C*cos(c + d*x)^2))/(a + b*cos(c + d*x))^(5/2), x, 8, -((8*A*b^4 - 15*a^4*C + 26*a^2*b^2*C - 3*b^4*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(3*(a - b)^2*b^3*(a + b)*d*sqrt(a + b*cos(c + d*x))) + (2*(a*A*b^2 + 3*A*b^3 - 5*a^3*C + 3*a^2*b*C + 6*a*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(3*(a - b)^2*b^2*(a + b)*d*sqrt(a + b*cos(c + d*x))) - (10*a*C*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(b^3*d*sqrt(a + b*cos(c + d*x))) - (2*(A*b^2 + a^2*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)) + (2*(a^2*A*b^2 + 3*A*b^4 - 5*a^4*C + 9*a^2*b^2*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(3*b^2*(a^2 - b^2)^2*d*sqrt(a + b*cos(c + d*x))) - ((8*A*b^4 - 15*a^4*C + 26*a^2*b^2*C - 3*b^4*C)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(3*b^3*(a^2 - b^2)^2*d)],
[cos(c + d*x)^(1/2)*(A + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(5/2), x, 7, (2*(3*a^2*A*b^2 + A*b^4 - 3*a^4*C + 7*a^2*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(3*a*(a - b)^2*b^2*(a + b)*d*sqrt(a + b*cos(c + d*x))) - (2*(3*a*A*b + A*b^2 - 2*a^2*C + 3*a*b*C + 3*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(3*(a - b)^2*b*(a + b)*d*sqrt(a + b*cos(c + d*x))) + (4*C*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(b^2*d*sqrt(a + b*cos(c + d*x))) - (2*(A*b^2 + a^2*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)) + (2*(3*a^2*A*b^2 + A*b^4 - 3*a^4*C + 7*a^2*b^2*C)*sqrt(cos(c + d*x))*tan((1/2)*(c + d*x)))/(3*a*(a - b)*b^2*(a + b)^2*d*sqrt(a + b*cos(c + d*x)))],
[(A + C*cos(c + d*x)^2)/(sqrt(cos(c + d*x))*(a + b*cos(c + d*x))^(5/2)), x, 5, (-4*b*(3*a^2*A - A*b^2 + 2*a^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(3*a^2*(a - b)^2*(a + b)*d*sqrt(a + b*cos(c + d*x))) + (2*(3*a^2*A + 3*a*A*b - 2*A*b^2 + a^2*C + 3*a*b*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(3*a*(a - b)^2*(a + b)*d*sqrt(a + b*cos(c + d*x))) + (2*(A*b^2 + a^2*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(3*a*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)) - (4*b*(3*a^2*A - A*b^2 + 2*a^2*C)*sqrt(cos(c + d*x))*tan((c + d*x)/2))/(3*a^2*(a - b)*(a + b)^2*d*sqrt(a + b*cos(c + d*x)))],
[(A + C*cos(c + d*x)^2)/(cos(c + d*x)^(3/2)*(a + b*cos(c + d*x))^(5/2)), x, 6, (-2*(3*a^4*A - 15*a^2*A*b^2 + 8*A*b^4 - 3*a^4*C - a^2*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(3*a^3*(a - b)^2*(a + b)*d*sqrt(a + b*cos(c + d*x))) + (2*(3*a^3*A - 9*a^2*A*b - 6*a*A*b^2 + 8*A*b^3 - 3*a^3*C - a^2*b*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(3*a^2*(a - b)^2*(a + b)*d*sqrt(a + b*cos(c + d*x))) + (2*(A*b^2 + a^2*C)*sin(c + d*x))/(3*a*(a^2 - b^2)*d*sqrt(cos(c + d*x))*(a + b*cos(c + d*x))^(3/2)) + (4*(4*a^2*A*b^2 - 2*A*b^4 + a^4*C + a^2*b^2*C)*sin(c + d*x))/(3*a^2*(a^2 - b^2)^2*d*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))) + (2*(3*a^4*A - 15*a^2*A*b^2 + 8*A*b^4 - 3*a^4*C - a^2*b^2*C)*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(3*a^3*(a^2 - b^2)^2*d*sqrt(cos(c + d*x)))],
[(A + C*cos(c + d*x)^2)/(cos(c + d*x)^(5/2)*(a + b*cos(c + d*x))^(5/2)), x, 7, (4*b*(4*a^4*A - 14*a^2*A*b^2 + 8*A*b^4 - 3*a^4*C + a^2*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(3*a^4*(a - b)^2*(a + b)*d*sqrt(a + b*cos(c + d*x))) + (2*(a^4*A - 9*a^3*A*b + 16*a^2*A*b^2 + 12*a*A*b^3 - 16*A*b^4 + 3*a^4*C + 3*a^3*b*C - 2*a^2*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(3*a^3*(a - b)^2*(a + b)*d*sqrt(a + b*cos(c + d*x))) + (2*(A*b^2 + a^2*C)*sin(c + d*x))/(3*a*(a^2 - b^2)*d*cos(c + d*x)^(3/2)*(a + b*cos(c + d*x))^(3/2)) + (4*(5*a^2*A*b^2 - 3*A*b^4 + 2*a^4*C)*sin(c + d*x))/(3*a^2*(a^2 - b^2)^2*d*cos(c + d*x)^(3/2)*sqrt(a + b*cos(c + d*x))) + (2*(a^4*A - 13*a^2*A*b^2 + 8*A*b^4 - 5*a^4*C + a^2*b^2*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*a^3*(a^2 - b^2)^2*d*cos(c + d*x)^(3/2)) - (4*b*(4*a^4*A - 14*a^2*A*b^2 + 8*A*b^4 - 3*a^4*C + a^2*b^2*C)*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(3*a^4*(a^2 - b^2)^2*d*sqrt(cos(c + d*x)))],


# ::Subsection::Closed:: 
#Integrands of the form Cos[c+d x]^m (A+C Cos[c+d x]^2) (a+b Cos[c+d x])^(n/3)
#


# ::Subsubsection::Closed:: 
#n>0


[(A + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(2/3), x, 2, Int((A + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(2/3), x)],
[(A + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(1/3), x, 2, Int((A + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(1/3), x)],


# ::Subsubsection::Closed:: 
#n<0


[(A + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(1/3), x, 1, Int((A + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(1/3), x)],
[(A + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(2/3), x, 1, Int((A + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(2/3), x)],


# ::Subsection::Closed:: 
#Integrands of the form Sec[c+d x]^(m/2) (A+C Cos[c+d x]^2) (a+b Cos[c+d x])^n


# ::Subsubsection::Closed:: 
#n>0


[(a + b*cos(c + d*x))*(A + C*cos(c + d*x)^2)*sec(c + d*x)^(9/2), x, 8, -((2*b*(3*A + 5*C)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d)) + (2*a*(5*A + 7*C)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) + (2*b*(3*A + 5*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d) + (2*a*(5*A + 7*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(21*d) + (4*A*b*sec(c + d*x)^(5/2)*sin(c + d*x))/(35*d) + (2*A*(a + b*cos(c + d*x))*sec(c + d*x)^(7/2)*sin(c + d*x))/(7*d)],
[(a + b*cos(c + d*x))*(A + C*cos(c + d*x)^2)*sec(c + d*x)^(7/2), x, 7, -((2*a*(3*A + 5*C)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d)) + (2*b*(A + 3*C)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) + (2*a*(3*A + 5*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d) + (4*A*b*sec(c + d*x)^(3/2)*sin(c + d*x))/(15*d) + (2*A*(a + b*cos(c + d*x))*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d)],
[(a + b*cos(c + d*x))*(A + C*cos(c + d*x)^2)*sec(c + d*x)^(5/2), x, 6, -((2*b*(A - C)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/d) + (2*a*(A + 3*C)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) + (4*A*b*sqrt(sec(c + d*x))*sin(c + d*x))/(3*d) + (2*A*(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d)],
[(a + b*cos(c + d*x))*(A + C*cos(c + d*x)^2)*sec(c + d*x)^(3/2), x, 6, -((2*a*(A - C)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/d) + (2*b*(3*A + C)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) - (2*b*(3*A - C)*sin(c + d*x))/(3*d*sqrt(sec(c + d*x))) + (2*A*(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*sin(c + d*x))/d],
[(a + b*cos(c + d*x))*(A + C*cos(c + d*x)^2)*sec(c + d*x)^(1/2), x, 6, (2*b*(5*A + 3*C)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d) + (2*a*(3*A + C)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) + (4*a*C*sin(c + d*x))/(15*d*sqrt(sec(c + d*x))) + (2*C*(a + b*cos(c + d*x))*sin(c + d*x))/(5*d*sqrt(sec(c + d*x)))],
[((a + b*cos(c + d*x))*(A + C*cos(c + d*x)^2))/sec(c + d*x)^(1/2), x, 7, (2*a*(5*A + 3*C)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d) + (2*b*(7*A + 5*C)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) + (4*a*C*sin(c + d*x))/(35*d*sec(c + d*x)^(3/2)) + (2*C*(a + b*cos(c + d*x))*sin(c + d*x))/(7*d*sec(c + d*x)^(3/2)) + (2*b*(7*A + 5*C)*sin(c + d*x))/(21*d*sqrt(sec(c + d*x)))],
[((a + b*cos(c + d*x))*(A + C*cos(c + d*x)^2))/sec(c + d*x)^(3/2), x, 8, (2*b*(9*A + 7*C)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(15*d) + (2*a*(7*A + 5*C)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) + (4*a*C*sin(c + d*x))/(63*d*sec(c + d*x)^(5/2)) + (2*C*(a + b*cos(c + d*x))*sin(c + d*x))/(9*d*sec(c + d*x)^(5/2)) + (2*b*(9*A + 7*C)*sin(c + d*x))/(45*d*sec(c + d*x)^(3/2)) + (2*a*(7*A + 5*C)*sin(c + d*x))/(21*d*sqrt(sec(c + d*x)))],


[(a + b*cos(c + d*x))^2*(A + C*cos(c + d*x)^2)*sec(c + d*x)^(11/2), x, 9, -((2*(7*a^2*A + 9*A*b^2 + 9*a^2*C + 15*b^2*C)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(15*d)) + (4*a*b*(5*A + 7*C)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) + (2*(7*a^2*A + 9*A*b^2 + 9*a^2*C + 15*b^2*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(15*d) + (4*a*b*(5*A + 7*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(21*d) + (2*(49*a^2*A + 8*A*b^2 + 63*a^2*C)*sec(c + d*x)^(5/2)*sin(c + d*x))/(315*d) + (8*A*b*(a + b*cos(c + d*x))*sec(c + d*x)^(7/2)*sin(c + d*x))/(63*d) + (2*A*(a + b*cos(c + d*x))^2*sec(c + d*x)^(9/2)*sin(c + d*x))/(9*d)],
[(a + b*cos(c + d*x))^2*(A + C*cos(c + d*x)^2)*sec(c + d*x)^(9/2), x, 8, -((4*a*b*(3*A + 5*C)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d)) + (2*(5*a^2*A + 7*A*b^2 + 7*a^2*C + 21*b^2*C)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) + (4*a*b*(3*A + 5*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d) + (2*(25*a^2*A + 8*A*b^2 + 35*a^2*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(105*d) + (8*A*b*(a + b*cos(c + d*x))*sec(c + d*x)^(5/2)*sin(c + d*x))/(35*d) + (2*A*(a + b*cos(c + d*x))^2*sec(c + d*x)^(7/2)*sin(c + d*x))/(7*d)],
[(a + b*cos(c + d*x))^2*(A + C*cos(c + d*x)^2)*sec(c + d*x)^(7/2), x, 7, -((2*(3*a^2*A + 5*A*b^2 + 5*a^2*C - 5*b^2*C)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d)) + (4*a*b*(A + 3*C)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) + (2*(9*a^2*A + 8*A*b^2 + 15*a^2*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(15*d) + (8*A*b*(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(15*d) + (2*A*(a + b*cos(c + d*x))^2*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d)],
[(a + b*cos(c + d*x))^2*(A + C*cos(c + d*x)^2)*sec(c + d*x)^(5/2), x, 7, -((4*a*b*(A - C)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/d) + (2*(a^2*A + 3*A*b^2 + 3*a^2*C + b^2*C)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) - (2*b^2*(5*A - C)*sin(c + d*x))/(3*d*sqrt(sec(c + d*x))) + (8*A*b*(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*sin(c + d*x))/(3*d) + (2*A*(a + b*cos(c + d*x))^2*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d)],
[(a + b*cos(c + d*x))^2*(A + C*cos(c + d*x)^2)*sec(c + d*x)^(3/2), x, 7, -((2*(5*a^2*A - 5*A*b^2 - 5*a^2*C - 3*b^2*C)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d)) + (4*a*b*(3*A + C)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) - (2*a*b*(15*A - 7*C)*sin(c + d*x))/(15*d*sqrt(sec(c + d*x))) - (2*b*(5*A - C)*(a + b*cos(c + d*x))*sin(c + d*x))/(5*d*sqrt(sec(c + d*x))) + (2*A*(a + b*cos(c + d*x))^2*sqrt(sec(c + d*x))*sin(c + d*x))/d],
[(a + b*cos(c + d*x))^2*(A + C*cos(c + d*x)^2)*sec(c + d*x)^(1/2), x, 7, (4*a*b*(5*A + 3*C)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d) + (2*(21*a^2*A + 7*A*b^2 + 7*a^2*C + 5*b^2*C)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) + (2*(35*A*b^2 + 8*a^2*C + 25*b^2*C)*sin(c + d*x))/(105*d*sqrt(sec(c + d*x))) + (8*a*C*(a + b*cos(c + d*x))*sin(c + d*x))/(35*d*sqrt(sec(c + d*x))) + (2*C*(a + b*cos(c + d*x))^2*sin(c + d*x))/(7*d*sqrt(sec(c + d*x)))],
[((a + b*cos(c + d*x))^2*(A + C*cos(c + d*x)^2))/sec(c + d*x)^(1/2), x, 8, (2*(15*a^2*A + 9*A*b^2 + 9*a^2*C + 7*b^2*C)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(15*d) + (4*a*b*(7*A + 5*C)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) + (2*(63*A*b^2 + 8*a^2*C + 49*b^2*C)*sin(c + d*x))/(315*d*sec(c + d*x)^(3/2)) + (8*a*C*(a + b*cos(c + d*x))*sin(c + d*x))/(63*d*sec(c + d*x)^(3/2)) + (2*C*(a + b*cos(c + d*x))^2*sin(c + d*x))/(9*d*sec(c + d*x)^(3/2)) + (4*a*b*(7*A + 5*C)*sin(c + d*x))/(21*d*sqrt(sec(c + d*x)))],
[((a + b*cos(c + d*x))^2*(A + C*cos(c + d*x)^2))/sec(c + d*x)^(3/2), x, 9, (4*a*b*(9*A + 7*C)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(15*d) + (2*(77*a^2*A + 55*A*b^2 + 55*a^2*C + 45*b^2*C)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(231*d) + (2*(99*A*b^2 + 8*a^2*C + 81*b^2*C)*sin(c + d*x))/(693*d*sec(c + d*x)^(5/2)) + (8*a*C*(a + b*cos(c + d*x))*sin(c + d*x))/(99*d*sec(c + d*x)^(5/2)) + (2*C*(a + b*cos(c + d*x))^2*sin(c + d*x))/(11*d*sec(c + d*x)^(5/2)) + (4*a*b*(9*A + 7*C)*sin(c + d*x))/(45*d*sec(c + d*x)^(3/2)) + (2*(77*a^2*A + 55*A*b^2 + 55*a^2*C + 45*b^2*C)*sin(c + d*x))/(231*d*sqrt(sec(c + d*x)))],


[(a + b*cos(c + d*x))^3*(A + C*cos(c + d*x)^2)*sec(c + d*x)^(11/2), x, 9, -((2*a*(7*a^2*A + 27*A*b^2 + 9*a^2*C + 45*b^2*C)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(15*d)) + (2*b*(15*a^2*A + 7*A*b^2 + 21*a^2*C + 21*b^2*C)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) + (2*a*(7*a^2*A + 27*A*b^2 + 9*a^2*C + 45*b^2*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(15*d) + (8*b*(44*a^2*A + 4*A*b^2 + 63*a^2*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(315*d) + (2*(49*a^2*A + 24*A*b^2 + 63*a^2*C)*(a + b*cos(c + d*x))*sec(c + d*x)^(5/2)*sin(c + d*x))/(315*d) + (4*A*b*(a + b*cos(c + d*x))^2*sec(c + d*x)^(7/2)*sin(c + d*x))/(21*d) + (2*A*(a + b*cos(c + d*x))^3*sec(c + d*x)^(9/2)*sin(c + d*x))/(9*d)],
[(a + b*cos(c + d*x))^3*(A + C*cos(c + d*x)^2)*sec(c + d*x)^(9/2), x, 8, -((2*b*(9*a^2*A + 5*A*b^2 + 15*a^2*C - 5*b^2*C)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d)) + (2*a*(5*a^2*A + 21*A*b^2 + 7*a^2*C + 63*b^2*C)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) + (8*b*(41*a^2*A + 12*A*b^2 + 70*a^2*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(105*d) + (2*(25*a^2*A + 24*A*b^2 + 35*a^2*C)*(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(105*d) + (12*A*b*(a + b*cos(c + d*x))^2*sec(c + d*x)^(5/2)*sin(c + d*x))/(35*d) + (2*A*(a + b*cos(c + d*x))^3*sec(c + d*x)^(7/2)*sin(c + d*x))/(7*d)],
[(a + b*cos(c + d*x))^3*(A + C*cos(c + d*x)^2)*sec(c + d*x)^(7/2), x, 8, -((2*a*(3*a^2*A + 15*A*b^2 + 5*a^2*C - 15*b^2*C)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d)) + (2*b*(3*a^2*A + 3*A*b^2 + 9*a^2*C + b^2*C)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) - (2*b*(9*a^2*A + 33*A*b^2 + 15*a^2*C - 5*b^2*C)*sin(c + d*x))/(15*d*sqrt(sec(c + d*x))) + (2*(3*a^2*A + 8*A*b^2 + 5*a^2*C)*(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d) + (4*A*b*(a + b*cos(c + d*x))^2*sec(c + d*x)^(3/2)*sin(c + d*x))/(5*d) + (2*A*(a + b*cos(c + d*x))^3*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d)],
[(a + b*cos(c + d*x))^3*(A + C*cos(c + d*x)^2)*sec(c + d*x)^(5/2), x, 8, -((2*b*(15*a^2*A - 5*A*b^2 - 15*a^2*C - 3*b^2*C)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d)) + (2*a*(a^2*A + 9*A*b^2 + 3*a^2*C + 3*b^2*C)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) - (8*a*b^2*(10*A - 3*C)*sin(c + d*x))/(15*d*sqrt(sec(c + d*x))) - (2*b^2*(35*A - 3*C)*(a + b*cos(c + d*x))*sin(c + d*x))/(15*d*sqrt(sec(c + d*x))) + (4*A*b*(a + b*cos(c + d*x))^2*sqrt(sec(c + d*x))*sin(c + d*x))/d + (2*A*(a + b*cos(c + d*x))^3*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d)],
[(a + b*cos(c + d*x))^3*(A + C*cos(c + d*x)^2)*sec(c + d*x)^(3/2), x, 8, -((2*a*(5*a^2*A - 15*A*b^2 - 5*a^2*C - 9*b^2*C)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d)) + (2*b*(63*a^2*A + 7*A*b^2 + 21*a^2*C + 5*b^2*C)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) - (2*b*(105*a^2*A - 35*A*b^2 - 57*a^2*C - 25*b^2*C)*sin(c + d*x))/(105*d*sqrt(sec(c + d*x))) - (2*a*b*(35*A - 11*C)*(a + b*cos(c + d*x))*sin(c + d*x))/(35*d*sqrt(sec(c + d*x))) - (2*b*(7*A - C)*(a + b*cos(c + d*x))^2*sin(c + d*x))/(7*d*sqrt(sec(c + d*x))) + (2*A*(a + b*cos(c + d*x))^3*sqrt(sec(c + d*x))*sin(c + d*x))/d],
[(a + b*cos(c + d*x))^3*(A + C*cos(c + d*x)^2)*sec(c + d*x)^(1/2), x, 8, (2*b*(45*a^2*A + 9*A*b^2 + 27*a^2*C + 7*b^2*C)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(15*d) + (2*a*(21*a^2*A + 21*A*b^2 + 7*a^2*C + 15*b^2*C)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) + (8*a*(63*A*b^2 + 4*a^2*C + 44*b^2*C)*sin(c + d*x))/(315*d*sqrt(sec(c + d*x))) + (2*(63*A*b^2 + 24*a^2*C + 49*b^2*C)*(a + b*cos(c + d*x))*sin(c + d*x))/(315*d*sqrt(sec(c + d*x))) + (4*a*C*(a + b*cos(c + d*x))^2*sin(c + d*x))/(21*d*sqrt(sec(c + d*x))) + (2*C*(a + b*cos(c + d*x))^3*sin(c + d*x))/(9*d*sqrt(sec(c + d*x)))],
[((a + b*cos(c + d*x))^3*(A + C*cos(c + d*x)^2))/sec(c + d*x)^(1/2), x, 9, (2*a*(5*a^2*A + 9*A*b^2 + 3*a^2*C + 7*b^2*C)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d) + (2*b*(231*a^2*A + 55*A*b^2 + 165*a^2*C + 45*b^2*C)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(231*d) + (8*a*(132*A*b^2 + 4*a^2*C + 101*b^2*C)*sin(c + d*x))/(1155*d*sec(c + d*x)^(3/2)) + (2*(33*A*b^2 + 8*a^2*C + 27*b^2*C)*(a + b*cos(c + d*x))*sin(c + d*x))/(231*d*sec(c + d*x)^(3/2)) + (4*a*C*(a + b*cos(c + d*x))^2*sin(c + d*x))/(33*d*sec(c + d*x)^(3/2)) + (2*C*(a + b*cos(c + d*x))^3*sin(c + d*x))/(11*d*sec(c + d*x)^(3/2)) + (2*b*(231*a^2*A + 55*A*b^2 + 165*a^2*C + 45*b^2*C)*sin(c + d*x))/(231*d*sqrt(sec(c + d*x)))],
[((a + b*cos(c + d*x))^3*(A + C*cos(c + d*x)^2))/sec(c + d*x)^(3/2), x, 10, (2*b*(351*a^2*A + 91*A*b^2 + 273*a^2*C + 77*b^2*C)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(195*d) + (2*a*(77*a^2*A + 165*A*b^2 + 55*a^2*C + 135*b^2*C)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(231*d) + (8*a*(715*A*b^2 + 12*a^2*C + 578*b^2*C)*sin(c + d*x))/(9009*d*sec(c + d*x)^(5/2)) + (2*(143*A*b^2 + 24*a^2*C + 121*b^2*C)*(a + b*cos(c + d*x))*sin(c + d*x))/(1287*d*sec(c + d*x)^(5/2)) + (12*a*C*(a + b*cos(c + d*x))^2*sin(c + d*x))/(143*d*sec(c + d*x)^(5/2)) + (2*C*(a + b*cos(c + d*x))^3*sin(c + d*x))/(13*d*sec(c + d*x)^(5/2)) + (2*b*(351*a^2*A + 91*A*b^2 + 273*a^2*C + 77*b^2*C)*sin(c + d*x))/(585*d*sec(c + d*x)^(3/2)) + (2*a*(77*a^2*A + 165*A*b^2 + 55*a^2*C + 135*b^2*C)*sin(c + d*x))/(231*d*sqrt(sec(c + d*x)))],


[(a + b*cos(c + d*x))^4*(A + C*cos(c + d*x)^2)*sec(c + d*x)^(13/2), x, 10, -((8*a*b*(7*a^2*A + 9*A*b^2 + 9*a^2*C + 15*b^2*C)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(15*d)) + (2*(45*a^4*A + 330*a^2*A*b^2 + 77*A*b^4 + 55*a^4*C + 462*a^2*b^2*C + 231*b^4*C)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(231*d) + (8*a*b*(7*a^2*A + 9*A*b^2 + 9*a^2*C + 15*b^2*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(15*d) + (2*(675*a^4*A + 3199*a^2*A*b^2 + 128*A*b^4 + 825*a^4*C + 4653*a^2*b^2*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(3465*d) + (4*b*(673*a^2*A + 96*A*b^2 + 891*a^2*C)*(a + b*cos(c + d*x))*sec(c + d*x)^(5/2)*sin(c + d*x))/(3465*d) + (2*(27*a^2*A + 16*A*b^2 + 33*a^2*C)*(a + b*cos(c + d*x))^2*sec(c + d*x)^(7/2)*sin(c + d*x))/(231*d) + (16*A*b*(a + b*cos(c + d*x))^3*sec(c + d*x)^(9/2)*sin(c + d*x))/(99*d) + (2*A*(a + b*cos(c + d*x))^4*sec(c + d*x)^(11/2)*sin(c + d*x))/(11*d)],
[(a + b*cos(c + d*x))^4*(A + C*cos(c + d*x)^2)*sec(c + d*x)^(11/2), x, 9, -((2*(7*a^4*A + 54*a^2*A*b^2 + 15*A*b^4 + 9*a^4*C + 90*a^2*b^2*C - 15*b^4*C)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(15*d)) + (8*a*b*(5*a^2*A + 7*A*b^2 + 7*a^2*C + 21*b^2*C)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) + (2*(147*a^4*A + 883*a^2*A*b^2 + 128*A*b^4 + 189*a^4*C + 1533*a^2*b^2*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(315*d) + (4*b*(101*a^2*A + 32*A*b^2 + 147*a^2*C)*(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(315*d) + (2*(49*a^2*A + 48*A*b^2 + 63*a^2*C)*(a + b*cos(c + d*x))^2*sec(c + d*x)^(5/2)*sin(c + d*x))/(315*d) + (16*A*b*(a + b*cos(c + d*x))^3*sec(c + d*x)^(7/2)*sin(c + d*x))/(63*d) + (2*A*(a + b*cos(c + d*x))^4*sec(c + d*x)^(9/2)*sin(c + d*x))/(9*d)],
[(a + b*cos(c + d*x))^4*(A + C*cos(c + d*x)^2)*sec(c + d*x)^(9/2), x, 9, -((8*a*b*(3*a^2*A + 5*A*b^2 + 5*a^2*C - 5*b^2*C)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d)) + (2*(5*a^4*A + 42*a^2*A*b^2 + 21*A*b^4 + 7*a^4*C + 126*a^2*b^2*C + 7*b^4*C)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) - (2*b^2*(227*a^2*A + 279*A*b^2 + 385*a^2*C - 35*b^2*C)*sin(c + d*x))/(105*d*sqrt(sec(c + d*x))) + (4*b*(101*a^2*A + 96*A*b^2 + 175*a^2*C)*(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*sin(c + d*x))/(105*d) + (2*(25*a^2*A + 48*A*b^2 + 35*a^2*C)*(a + b*cos(c + d*x))^2*sec(c + d*x)^(3/2)*sin(c + d*x))/(105*d) + (16*A*b*(a + b*cos(c + d*x))^3*sec(c + d*x)^(5/2)*sin(c + d*x))/(35*d) + (2*A*(a + b*cos(c + d*x))^4*sec(c + d*x)^(7/2)*sin(c + d*x))/(7*d)],
[(a + b*cos(c + d*x))^4*(A + C*cos(c + d*x)^2)*sec(c + d*x)^(7/2), x, 9, -((2*(3*a^4*A + 30*a^2*A*b^2 - 5*A*b^4 + 5*a^4*C - 30*a^2*b^2*C - 3*b^4*C)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d)) + (8*a*b*(a^2*A + 3*A*b^2 + 3*a^2*C + b^2*C)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) - (2*a*b*(9*a^2*A + 73*A*b^2 + 15*a^2*C - 17*b^2*C)*sin(c + d*x))/(15*d*sqrt(sec(c + d*x))) - (2*b*(9*a^2*A + 59*A*b^2 + 15*a^2*C - 3*b^2*C)*(a + b*cos(c + d*x))*sin(c + d*x))/(15*d*sqrt(sec(c + d*x))) + (2*(3*a^2*A + 16*A*b^2 + 5*a^2*C)*(a + b*cos(c + d*x))^2*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d) + (16*A*b*(a + b*cos(c + d*x))^3*sec(c + d*x)^(3/2)*sin(c + d*x))/(15*d) + (2*A*(a + b*cos(c + d*x))^4*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d)],
[(a + b*cos(c + d*x))^4*(A + C*cos(c + d*x)^2)*sec(c + d*x)^(5/2), x, 9, -((8*a*b*(5*a^2*A - 5*A*b^2 - 5*a^2*C - 3*b^2*C)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d)) + (2*(7*a^4*A + 126*a^2*A*b^2 + 7*A*b^4 + 21*a^4*C + 42*a^2*b^2*C + 5*b^4*C)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) - (2*b^2*(385*a^2*A - 35*A*b^2 - 141*a^2*C - 25*b^2*C)*sin(c + d*x))/(105*d*sqrt(sec(c + d*x))) - (4*a*b^2*(175*A - 27*C)*(a + b*cos(c + d*x))*sin(c + d*x))/(105*d*sqrt(sec(c + d*x))) - (2*b^2*(21*A - C)*(a + b*cos(c + d*x))^2*sin(c + d*x))/(7*d*sqrt(sec(c + d*x))) + (16*A*b*(a + b*cos(c + d*x))^3*sqrt(sec(c + d*x))*sin(c + d*x))/(3*d) + (2*A*(a + b*cos(c + d*x))^4*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d)],
[(a + b*cos(c + d*x))^4*(A + C*cos(c + d*x)^2)*sec(c + d*x)^(3/2), x, 9, -((2*(15*a^4*A - 90*a^2*A*b^2 - 9*A*b^4 - 15*a^4*C - 54*a^2*b^2*C - 7*b^4*C)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(15*d)) + (8*a*b*(21*a^2*A + 7*A*b^2 + 7*a^2*C + 5*b^2*C)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) - (2*a*b*(315*a^2*A - 357*A*b^2 - 187*a^2*C - 251*b^2*C)*sin(c + d*x))/(315*d*sqrt(sec(c + d*x))) - (2*b*(315*a^2*A - 63*A*b^2 - 123*a^2*C - 49*b^2*C)*(a + b*cos(c + d*x))*sin(c + d*x))/(315*d*sqrt(sec(c + d*x))) - (2*a*b*(21*A - 5*C)*(a + b*cos(c + d*x))^2*sin(c + d*x))/(21*d*sqrt(sec(c + d*x))) - (2*b*(9*A - C)*(a + b*cos(c + d*x))^3*sin(c + d*x))/(9*d*sqrt(sec(c + d*x))) + (2*A*(a + b*cos(c + d*x))^4*sqrt(sec(c + d*x))*sin(c + d*x))/d],
[(a + b*cos(c + d*x))^4*(A + C*cos(c + d*x)^2)*sec(c + d*x)^(1/2), x, 9, (8*a*b*(15*a^2*A + 9*A*b^2 + 9*a^2*C + 7*b^2*C)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(15*d) + (2*(231*a^4*A + 462*a^2*A*b^2 + 55*A*b^4 + 77*a^4*C + 330*a^2*b^2*C + 45*b^4*C)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(231*d) + (2*(4653*a^2*A*b^2 + 825*A*b^4 + 128*a^4*C + 3199*a^2*b^2*C + 675*b^4*C)*sin(c + d*x))/(3465*d*sqrt(sec(c + d*x))) + (4*a*(891*A*b^2 + 96*a^2*C + 673*b^2*C)*(a + b*cos(c + d*x))*sin(c + d*x))/(3465*d*sqrt(sec(c + d*x))) + (2*(33*A*b^2 + 16*a^2*C + 27*b^2*C)*(a + b*cos(c + d*x))^2*sin(c + d*x))/(231*d*sqrt(sec(c + d*x))) + (16*a*C*(a + b*cos(c + d*x))^3*sin(c + d*x))/(99*d*sqrt(sec(c + d*x))) + (2*C*(a + b*cos(c + d*x))^4*sin(c + d*x))/(11*d*sqrt(sec(c + d*x)))],
[((a + b*cos(c + d*x))^4*(A + C*cos(c + d*x)^2))/sec(c + d*x)^(1/2), x, 10, (2*(195*a^4*A + 702*a^2*A*b^2 + 91*A*b^4 + 117*a^4*C + 546*a^2*b^2*C + 77*b^4*C)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(195*d) + (8*a*b*(77*a^2*A + 55*A*b^2 + 55*a^2*C + 45*b^2*C)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(231*d) + (2*(33319*a^2*A*b^2 + 7007*A*b^4 + 384*a^4*C + 25217*a^2*b^2*C + 5929*b^4*C)*sin(c + d*x))/(45045*d*sec(c + d*x)^(3/2)) + (4*a*(1573*A*b^2 + 96*a^2*C + 1259*b^2*C)*(a + b*cos(c + d*x))*sin(c + d*x))/(9009*d*sec(c + d*x)^(3/2)) + (2*(143*A*b^2 + 48*a^2*C + 121*b^2*C)*(a + b*cos(c + d*x))^2*sin(c + d*x))/(1287*d*sec(c + d*x)^(3/2)) + (16*a*C*(a + b*cos(c + d*x))^3*sin(c + d*x))/(143*d*sec(c + d*x)^(3/2)) + (2*C*(a + b*cos(c + d*x))^4*sin(c + d*x))/(13*d*sec(c + d*x)^(3/2)) + (8*a*b*(77*a^2*A + 55*A*b^2 + 55*a^2*C + 45*b^2*C)*sin(c + d*x))/(231*d*sqrt(sec(c + d*x)))],


# ::Subsubsection::Closed:: 
#n<0


[((A + C*cos(c + d*x)^2)*sec(c + d*x)^(7/2))/(a + b*cos(c + d*x)), x, 9, -((2*(3*a^2*A + 5*A*b^2 + 5*a^2*C)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*a^3*d)) - (2*A*b*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*a^2*d) - (2*b*(A*b^2 + a^2*C)*sqrt(cos(c + d*x))*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a^3*(a + b)*d) + (2*(3*a^2*A + 5*A*b^2 + 5*a^2*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(5*a^3*d) - (2*A*b*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*a^2*d) + (2*A*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*a*d)],
[((A + C*cos(c + d*x)^2)*sec(c + d*x)^(5/2))/(a + b*cos(c + d*x)), x, 8, (2*A*b*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a^2*d) + (2*A*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*a*d) + (2*(A*b^2 + a^2*C)*sqrt(cos(c + d*x))*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a^2*(a + b)*d) - (2*A*b*sqrt(sec(c + d*x))*sin(c + d*x))/(a^2*d) + (2*A*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*a*d)],
[((A + C*cos(c + d*x)^2)*sec(c + d*x)^(3/2))/(a + b*cos(c + d*x)), x, 7, -((2*A*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a*d)) + (2*C*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(b*d) - (2*(A*b^2 + a^2*C)*sqrt(cos(c + d*x))*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a*b*(a + b)*d) + (2*A*sqrt(sec(c + d*x))*sin(c + d*x))/(a*d)],
[((A + C*cos(c + d*x)^2)*sqrt(sec(c + d*x)))/(a + b*cos(c + d*x)), x, 6, (2*C*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(b*d) - (2*a*C*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(b^2*d) + (2*(A*b^2 + a^2*C)*sqrt(cos(c + d*x))*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(b^2*(a + b)*d)],
[(A + C*cos(c + d*x)^2)/((a + b*cos(c + d*x))*sqrt(sec(c + d*x))), x, 7, -((2*a*C*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(b^2*d)) + (2*(3*A*b^2 + 3*a^2*C + b^2*C)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*b^3*d) - (2*a*(A*b^2 + a^2*C)*sqrt(cos(c + d*x))*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(b^3*(a + b)*d) + (2*C*sin(c + d*x))/(3*b*d*sqrt(sec(c + d*x)))],
[(A + C*cos(c + d*x)^2)/((a + b*cos(c + d*x))*sec(c + d*x)^(3/2)), x, 8, (2*(5*A*b^2 + 5*a^2*C + 3*b^2*C)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*b^3*d) - (2*a*(3*A*b^2 + 3*a^2*C + b^2*C)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*b^4*d) + (2*a^2*(A*b^2 + a^2*C)*sqrt(cos(c + d*x))*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(b^4*(a + b)*d) + (2*C*sin(c + d*x))/(5*b*d*sec(c + d*x)^(3/2)) - (2*a*C*sin(c + d*x))/(3*b^2*d*sqrt(sec(c + d*x)))],
[(A + C*cos(c + d*x)^2)/((a + b*cos(c + d*x))*sec(c + d*x)^(5/2)), x, 9, -((2*a*(5*A*b^2 + 5*a^2*C + 3*b^2*C)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*b^4*d)) + (2*(21*a^2*A*b^2 + 7*A*b^4 + 21*a^4*C + 7*a^2*b^2*C + 5*b^4*C)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*b^5*d) - (2*a^3*(A*b^2 + a^2*C)*sqrt(cos(c + d*x))*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(b^5*(a + b)*d) + (2*C*sin(c + d*x))/(7*b*d*sec(c + d*x)^(5/2)) - (2*a*C*sin(c + d*x))/(5*b^2*d*sec(c + d*x)^(3/2)) + (2*(7*A*b^2 + 7*a^2*C + 5*b^2*C)*sin(c + d*x))/(21*b^3*d*sqrt(sec(c + d*x)))],


[((A + C*cos(c + d*x)^2)*sec(c + d*x)^(5/2))/(a + b*cos(c + d*x))^2, x, 9, (b*(4*a^2*A - 5*A*b^2 - a^2*C)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a^3*(a^2 - b^2)*d) + ((2*a^2*A - 5*A*b^2 - 3*a^2*C)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*a^2*(a^2 - b^2)*d) + ((7*a^2*A*b^2 - 5*A*b^4 + 3*a^4*C - a^2*b^2*C)*sqrt(cos(c + d*x))*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a^3*(a - b)*(a + b)^2*d) - (b*(4*a^2*A - 5*A*b^2 - a^2*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(a^3*(a^2 - b^2)*d) + ((2*a^2*A - 5*A*b^2 - 3*a^2*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*a^2*(a^2 - b^2)*d) + ((A*b^2 + a^2*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(a*(a^2 - b^2)*d*(a + b*cos(c + d*x)))],
[((A + C*cos(c + d*x)^2)*sec(c + d*x)^(3/2))/(a + b*cos(c + d*x))^2, x, 8, -(((2*a^2*A - 3*A*b^2 - a^2*C)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a^2*(a^2 - b^2)*d)) + ((A*b^2 + a^2*C)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a*b*(a^2 - b^2)*d) - ((5*a^2*A*b^2 - 3*A*b^4 + a^4*C + a^2*b^2*C)*sqrt(cos(c + d*x))*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a^2*(a - b)*b*(a + b)^2*d) + ((2*a^2*A - 3*A*b^2 - a^2*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(a^2*(a^2 - b^2)*d) + ((A*b^2 + a^2*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(a*(a^2 - b^2)*d*(a + b*cos(c + d*x)))],
[((A + C*cos(c + d*x)^2)*sqrt(sec(c + d*x)))/(a + b*cos(c + d*x))^2, x, 7, -(((A*b^2 + a^2*C)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a*b*(a^2 - b^2)*d)) - ((A*b^2 - a^2*C + 2*b^2*C)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(b^2*(a^2 - b^2)*d) + ((3*a^2*A*b^2 - A*b^4 - a^4*C + 3*a^2*b^2*C)*sqrt(cos(c + d*x))*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a*(a - b)*b^2*(a + b)^2*d) + ((A*b^2 + a^2*C)*sin(c + d*x))/(a*(a^2 - b^2)*d*(a + b*cos(c + d*x))*sqrt(sec(c + d*x)))],
[(A + C*cos(c + d*x)^2)/((a + b*cos(c + d*x))^2*sqrt(sec(c + d*x))), x, 7, ((A*b^2 + 3*a^2*C - 2*b^2*C)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(b^2*(a^2 - b^2)*d) + (a*(A*b^2 - 3*a^2*C + 4*b^2*C)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(b^3*(a^2 - b^2)*d) - ((a^2*A*b^2 + A*b^4 - 3*a^4*C + 5*a^2*b^2*C)*sqrt(cos(c + d*x))*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/((a - b)*b^3*(a + b)^2*d) - ((A*b^2 + a^2*C)*sin(c + d*x))/(b*(a^2 - b^2)*d*(a + b*cos(c + d*x))*sqrt(sec(c + d*x)))],
[(A + C*cos(c + d*x)^2)/((a + b*cos(c + d*x))^2*sec(c + d*x)^(3/2)), x, 8, -((a*(A*b^2 + 5*a^2*C - 4*b^2*C)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(b^3*(a^2 - b^2)*d)) + ((3*a^2*A*b^2 - 6*A*b^4 + 15*a^4*C - 16*a^2*b^2*C - 2*b^4*C)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*b^4*(a^2 - b^2)*d) - (a*(a^2*A*b^2 - 3*A*b^4 + 5*a^4*C - 7*a^2*b^2*C)*sqrt(cos(c + d*x))*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/((a - b)*b^4*(a + b)^2*d) - ((A*b^2 + a^2*C)*sin(c + d*x))/(b*(a^2 - b^2)*d*(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)) + ((3*A*b^2 + 5*a^2*C - 2*b^2*C)*sin(c + d*x))/(3*b^2*(a^2 - b^2)*d*sqrt(sec(c + d*x)))],
[(A + C*cos(c + d*x)^2)/((a + b*cos(c + d*x))^2*sec(c + d*x)^(5/2)), x, 9, ((15*a^2*A*b^2 - 10*A*b^4 + 35*a^4*C - 24*a^2*b^2*C - 6*b^4*C)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*b^4*(a^2 - b^2)*d) - (a*(9*a^2*A*b^2 - 12*A*b^4 + 21*a^4*C - 20*a^2*b^2*C - 4*b^4*C)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*b^5*(a^2 - b^2)*d) + (a^2*(3*a^2*A*b^2 - 5*A*b^4 + 7*a^4*C - 9*a^2*b^2*C)*sqrt(cos(c + d*x))*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/((a - b)*b^5*(a + b)^2*d) - ((A*b^2 + a^2*C)*sin(c + d*x))/(b*(a^2 - b^2)*d*(a + b*cos(c + d*x))*sec(c + d*x)^(5/2)) + ((5*A*b^2 + 7*a^2*C - 2*b^2*C)*sin(c + d*x))/(5*b^2*(a^2 - b^2)*d*sec(c + d*x)^(3/2)) - (a*(3*A*b^2 + 7*a^2*C - 4*b^2*C)*sin(c + d*x))/(3*b^3*(a^2 - b^2)*d*sqrt(sec(c + d*x)))],


[((A + C*cos(c + d*x)^2)*sec(c + d*x)^(5/2))/(a + b*cos(c + d*x))^3, x, 10, (b*(24*a^4*A - 65*a^2*A*b^2 + 35*A*b^4 - 9*a^4*C + 3*a^2*b^2*C)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(4*a^4*(a^2 - b^2)^2*d) + ((8*a^4*A - 61*a^2*A*b^2 + 35*A*b^4 - 21*a^4*C + 3*a^2*b^2*C)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(12*a^3*(a^2 - b^2)^2*d) + ((63*a^4*A*b^2 - 86*a^2*A*b^4 + 35*A*b^6 + 15*a^6*C - 6*a^4*b^2*C + 3*a^2*b^4*C)*sqrt(cos(c + d*x))*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(4*a^4*(a - b)^2*(a + b)^3*d) - (b*(24*a^4*A - 65*a^2*A*b^2 + 35*A*b^4 - 9*a^4*C + 3*a^2*b^2*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(4*a^4*(a^2 - b^2)^2*d) + ((8*a^4*A - 61*a^2*A*b^2 + 35*A*b^4 - 21*a^4*C + 3*a^2*b^2*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(12*a^3*(a^2 - b^2)^2*d) + ((A*b^2 + a^2*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(2*a*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) + ((13*a^2*A*b^2 - 7*A*b^4 + 5*a^4*C + a^2*b^2*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(4*a^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))],
[((A + C*cos(c + d*x)^2)*sec(c + d*x)^(3/2))/(a + b*cos(c + d*x))^3, x, 9, -(((8*a^4*A - 29*a^2*A*b^2 + 15*A*b^4 - 5*a^4*C - a^2*b^2*C)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(4*a^3*(a^2 - b^2)^2*d)) + ((11*a^2*A*b^2 - 5*A*b^4 + 3*a^4*C + 3*a^2*b^2*C)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(4*a^2*b*(a^2 - b^2)^2*d) - ((35*a^4*A*b^2 - 38*a^2*A*b^4 + 15*A*b^6 + 3*a^6*C + 10*a^4*b^2*C - a^2*b^4*C)*sqrt(cos(c + d*x))*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(4*a^3*(a - b)^2*b*(a + b)^3*d) + ((8*a^4*A - 29*a^2*A*b^2 + 15*A*b^4 - 5*a^4*C - a^2*b^2*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(4*a^3*(a^2 - b^2)^2*d) + ((A*b^2 + a^2*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(2*a*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) + ((11*a^2*A*b^2 - 5*A*b^4 + 3*a^4*C + 3*a^2*b^2*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(4*a^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))],
[((A + C*cos(c + d*x)^2)*sqrt(sec(c + d*x)))/(a + b*cos(c + d*x))^3, x, 8, -(((9*a^2*A*b^2 - 3*A*b^4 + a^4*C + 5*a^2*b^2*C)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(4*a^2*b*(a^2 - b^2)^2*d)) - ((7*a^2*A*b^2 - A*b^4 - a^4*C + 7*a^2*b^2*C)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(4*a*b^2*(a^2 - b^2)^2*d) + ((15*a^4*A*b^2 - 6*a^2*A*b^4 + 3*A*b^6 - a^6*C + 10*a^4*b^2*C + 3*a^2*b^4*C)*sqrt(cos(c + d*x))*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(4*a^2*(a - b)^2*b^2*(a + b)^3*d) + ((A*b^2 + a^2*C)*sin(c + d*x))/(2*a*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2*sqrt(sec(c + d*x))) + ((9*a^2*A*b^2 - 3*A*b^4 + a^4*C + 5*a^2*b^2*C)*sin(c + d*x))/(4*a^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x))*sqrt(sec(c + d*x)))],
[(A + C*cos(c + d*x)^2)/((a + b*cos(c + d*x))^3*sqrt(sec(c + d*x))), x, 8, ((5*a^2*A*b^2 + A*b^4 - 3*a^4*C + 9*a^2*b^2*C)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(4*a*b^2*(a^2 - b^2)^2*d) + ((3*a^2*A*b^2 + 3*A*b^4 + 3*a^4*C - 5*a^2*b^2*C + 8*b^4*C)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(4*b^3*(a^2 - b^2)^2*d) - ((3*a^4*A*b^2 + 10*a^2*A*b^4 - A*b^6 + 3*a^6*C - 6*a^4*b^2*C + 15*a^2*b^4*C)*sqrt(cos(c + d*x))*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(4*a*(a - b)^2*b^3*(a + b)^3*d) - ((A*b^2 + a^2*C)*sin(c + d*x))/(2*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2*sqrt(sec(c + d*x))) - ((5*a^2*A*b^2 + A*b^4 - 3*a^4*C + 9*a^2*b^2*C)*sin(c + d*x))/(4*a*b*(a^2 - b^2)^2*d*(a + b*cos(c + d*x))*sqrt(sec(c + d*x)))],
[(A + C*cos(c + d*x)^2)/((a + b*cos(c + d*x))^3*sec(c + d*x)^(3/2)), x, 8, -(((a^2*A*b^2 + 5*A*b^4 - 15*a^4*C + 29*a^2*b^2*C - 8*b^4*C)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(4*b^3*(a^2 - b^2)^2*d)) + (a*(a^2*A*b^2 - 7*A*b^4 - 15*a^4*C + 33*a^2*b^2*C - 24*b^4*C)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(4*b^4*(a^2 - b^2)^2*d) - ((a^4*A*b^2 - 10*a^2*A*b^4 - 3*A*b^6 - 15*a^6*C + 38*a^4*b^2*C - 35*a^2*b^4*C)*sqrt(cos(c + d*x))*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(4*(a - b)^2*b^4*(a + b)^3*d) - ((A*b^2 + a^2*C)*sin(c + d*x))/(2*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2*sec(c + d*x)^(3/2)) + ((3*a^2*A*b^2 + 3*A*b^4 - 5*a^4*C + 11*a^2*b^2*C)*sin(c + d*x))/(4*b^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x))*sqrt(sec(c + d*x)))],
[(A + C*cos(c + d*x)^2)/((a + b*cos(c + d*x))^3*sec(c + d*x)^(5/2)), x, 9, -((a*(3*a^2*A*b^2 - 9*A*b^4 + 35*a^4*C - 65*a^2*b^2*C + 24*b^4*C)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(4*b^4*(a^2 - b^2)^2*d)) + ((9*a^4*A*b^2 - 15*a^2*A*b^4 + 24*A*b^6 + 105*a^6*C - 223*a^4*b^2*C + 128*a^2*b^4*C + 8*b^6*C)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(12*b^5*(a^2 - b^2)^2*d) - (a*(3*a^4*A*b^2 - 6*a^2*A*b^4 + 15*A*b^6 + 35*a^6*C - 86*a^4*b^2*C + 63*a^2*b^4*C)*sqrt(cos(c + d*x))*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(4*(a - b)^2*b^5*(a + b)^3*d) - ((A*b^2 + a^2*C)*sin(c + d*x))/(2*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2*sec(c + d*x)^(5/2)) + ((a^2*A*b^2 + 5*A*b^4 - 7*a^4*C + 13*a^2*b^2*C)*sin(c + d*x))/(4*b^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)) + ((3*a^2*A*b^2 - 21*A*b^4 + 35*a^4*C - 61*a^2*b^2*C + 8*b^4*C)*sin(c + d*x))/(12*b^3*(a^2 - b^2)^2*d*sqrt(sec(c + d*x)))],
[(A + C*cos(c + d*x)^2)/((a + b*cos(c + d*x))^3*sec(c + d*x)^(7/2)), x, 10, ((75*a^4*A*b^2 - 145*a^2*A*b^4 + 40*A*b^6 + 315*a^6*C - 561*a^4*b^2*C + 192*a^2*b^4*C + 24*b^6*C)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(20*b^5*(a^2 - b^2)^2*d) - (a*(15*a^4*A*b^2 - 33*a^2*A*b^4 + 24*A*b^6 + 63*a^6*C - 129*a^4*b^2*C + 64*a^2*b^4*C + 8*b^6*C)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(4*b^6*(a^2 - b^2)^2*d) + (a^2*(15*a^4*A*b^2 - 38*a^2*A*b^4 + 35*A*b^6 + 63*a^6*C - 150*a^4*b^2*C + 99*a^2*b^4*C)*sqrt(cos(c + d*x))*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(4*(a - b)^2*b^6*(a + b)^3*d) - ((A*b^2 + a^2*C)*sin(c + d*x))/(2*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2*sec(c + d*x)^(7/2)) - ((a^2*A*b^2 - 7*A*b^4 + 9*a^4*C - 15*a^2*b^2*C)*sin(c + d*x))/(4*b^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x))*sec(c + d*x)^(5/2)) + ((15*a^2*A*b^2 - 45*A*b^4 + 63*a^4*C - 101*a^2*b^2*C + 8*b^4*C)*sin(c + d*x))/(20*b^3*(a^2 - b^2)^2*d*sec(c + d*x)^(3/2)) - (a*(5*a^2*A*b^2 - 11*A*b^4 + 21*a^4*C - 35*a^2*b^2*C + 8*b^4*C)*sin(c + d*x))/(4*b^4*(a^2 - b^2)^2*d*sqrt(sec(c + d*x)))],


# ::Subsection::Closed:: 
#Integrands of the form Sec[c+d x]^(m/2) (A+C Cos[c+d x]^2) (a+b Cos[c+d x])^(n/2)
#


# ::Subsubsection::Closed:: 
#n>0


[sqrt(a + b*cos(c + d*x))*(A + C*cos(c + d*x)^2)*sec(c + d*x)^(11/2), x, 9, -((2*(a + b)*(147*a^4*A - 24*a^2*A*b^2 - 16*A*b^4 + 189*a^4*C - 42*a^2*b^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(315*a^4*d*sqrt(a + b*cos(c + d*x)))) + (2*(147*a^4*A + 111*a^3*A*b - 24*a^2*A*b^2 - 4*a*A*b^3 - 16*A*b^4 + 189*a^4*C + 147*a^3*b*C - 42*a^2*b^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(315*a^3*d*sqrt(a + b*cos(c + d*x))) + (2*b*(13*a^2*A + 8*A*b^2 + 21*a^2*C)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(315*a^3*d) + (2*(49*a^2*A - 6*A*b^2 + 63*a^2*C)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(5/2)*sin(c + d*x))/(315*a^2*d) + (2*A*b*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(7/2)*sin(c + d*x))/(63*a*d) + (2*A*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(9/2)*sin(c + d*x))/(9*d) + (2*(147*a^4*A - 24*a^2*A*b^2 - 16*A*b^4 + 189*a^4*C - 42*a^2*b^2*C)*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*tan((1/2)*(c + d*x)))/(315*a^4*d)],
[sqrt(a + b*cos(c + d*x))*(A + C*cos(c + d*x)^2)*sec(c + d*x)^(9/2), x, 8, -((2*b*(a + b)*(19*a^2*A + 8*A*b^2 + 35*a^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(105*a^3*d*sqrt(a + b*cos(c + d*x)))) + (2*(25*a^3*A + 19*a^2*A*b + 2*a*A*b^2 + 8*A*b^3 + 35*a^3*C + 35*a^2*b*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(105*a^2*d*sqrt(a + b*cos(c + d*x))) + (2*(25*a^2*A - 4*A*b^2 + 35*a^2*C)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(105*a^2*d) + (2*A*b*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(5/2)*sin(c + d*x))/(35*a*d) + (2*A*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(7/2)*sin(c + d*x))/(7*d) + (2*b*(19*a^2*A + 8*A*b^2 + 35*a^2*C)*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*tan((1/2)*(c + d*x)))/(105*a^3*d)],
[sqrt(a + b*cos(c + d*x))*(A + C*cos(c + d*x)^2)*sec(c + d*x)^(7/2), x, 7, -((2*(a + b)*(9*a^2*A - 2*A*b^2 + 15*a^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(15*a^2*d*sqrt(a + b*cos(c + d*x)))) + (2*(9*a^2*A + 7*a*A*b - 2*A*b^2 + 15*a^2*C + 15*a*b*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(15*a*d*sqrt(a + b*cos(c + d*x))) + (2*A*b*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(15*a*d) + (2*A*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d) + (2*(9*a^2*A - 2*A*b^2 + 15*a^2*C)*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*tan((1/2)*(c + d*x)))/(15*a^2*d)],
[sqrt(a + b*cos(c + d*x))*(A + C*cos(c + d*x)^2)*sec(c + d*x)^(5/2), x, 8, -((2*A*b*(a + b)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(3*a*d*sqrt(a + b*cos(c + d*x)))) + (2*(a*A + A*b + 3*a*C - 3*b*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(3*d*sqrt(a + b*cos(c + d*x))) + (4*b*C*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*cos(c + d*x))) + (2*A*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d) + (2*A*b*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*tan((1/2)*(c + d*x)))/(3*a*d)],
[sqrt(a + b*cos(c + d*x))*(A + C*cos(c + d*x)^2)*sec(c + d*x)^(3/2), x, 8, -(((a + b)*(2*A - C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*cos(c + d*x)))) + (2*(a*A + A*b - a*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*cos(c + d*x))) + (2*a*C*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*cos(c + d*x))) + (2*A*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*sin(c + d*x))/d - ((2*A - C)*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(d*sqrt(sec(c + d*x)))],
[sqrt(a + b*cos(c + d*x))*(A + C*cos(c + d*x)^2)*sec(c + d*x)^(1/2), x, 8, (a*(a + b)*C*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(4*b*d*sqrt(a + b*cos(c + d*x))) + ((4*a*A - 4*A*b + a*C - 2*b*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(2*d*sqrt(a + b*cos(c + d*x))) + ((8*A*b^2 - a^2*C + 4*b^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(2*b*d*sqrt(a + b*cos(c + d*x))) + (C*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(2*d*sqrt(sec(c + d*x))) + (a*C*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(4*b*d*sqrt(sec(c + d*x)))],
[(sqrt(a + b*cos(c + d*x))*(A + C*cos(c + d*x)^2))/sec(c + d*x)^(1/2), x, 9, ((a + b)*(24*A*b^2 - 3*a^2*C + 16*b^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(24*b^2*d*sqrt(a + b*cos(c + d*x))) - (a*(24*A*b - a*C + 14*b*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(12*b*d*sqrt(a + b*cos(c + d*x))) + (a*(8*A*b^2 + a^2*C + 4*b^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(4*b^2*d*sqrt(a + b*cos(c + d*x))) + (C*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*d*sec(c + d*x)^(3/2)) + (a*C*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(12*b*d*sqrt(sec(c + d*x))) + ((24*A*b^2 - 3*a^2*C + 16*b^2*C)*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(24*b^2*d*sqrt(sec(c + d*x)))],
[(sqrt(a + b*cos(c + d*x))*(A + C*cos(c + d*x)^2))/sec(c + d*x)^(3/2), x, 10, (a*(a + b)*(48*A*b^2 + 15*a^2*C + 28*b^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(192*b^3*d*sqrt(a + b*cos(c + d*x))) + ((48*a*A*b^2 - 96*A*b^3 - 5*a^3*C - 2*a^2*b*C + 36*a*b^2*C - 72*b^3*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(96*b^2*d*sqrt(a + b*cos(c + d*x))) - ((16*a^2*A*b^2 - 64*A*b^4 + 5*a^4*C + 8*a^2*b^2*C - 48*b^4*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(32*b^3*d*sqrt(a + b*cos(c + d*x))) + (C*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(4*d*sec(c + d*x)^(5/2)) + (a*C*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(24*b*d*sec(c + d*x)^(3/2)) + ((48*A*b^2 - 5*a^2*C + 36*b^2*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(96*b^2*d*sqrt(sec(c + d*x))) + (a*(48*A*b^2 + 15*a^2*C + 28*b^2*C)*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(192*b^3*d*sqrt(sec(c + d*x)))],


[(a + b*cos(c + d*x))^(3/2)*(A + C*cos(c + d*x)^2)*sec(c + d*x)^(11/2), x, 9, -((2*(a + b)*(147*a^4*A + 33*a^2*A*b^2 + 8*A*b^4 + 189*a^4*C + 63*a^2*b^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(315*a^3*d*sqrt(a + b*cos(c + d*x)))) + (2*(147*a^4*A + 186*a^3*A*b + 33*a^2*A*b^2 + 2*a*A*b^3 + 8*A*b^4 + 189*a^4*C + 252*a^3*b*C + 63*a^2*b^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(315*a^2*d*sqrt(a + b*cos(c + d*x))) + (4*b*(44*a^2*A - 2*A*b^2 + 63*a^2*C)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(315*a^2*d) + (2*(49*a^2*A + 3*A*b^2 + 63*a^2*C)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(5/2)*sin(c + d*x))/(315*a*d) + (2*A*b*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(7/2)*sin(c + d*x))/(21*d) + (2*A*(a + b*cos(c + d*x))^(3/2)*sec(c + d*x)^(9/2)*sin(c + d*x))/(9*d) + (2*(147*a^4*A + 33*a^2*A*b^2 + 8*A*b^4 + 189*a^4*C + 63*a^2*b^2*C)*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*tan((1/2)*(c + d*x)))/(315*a^3*d)],
[(a + b*cos(c + d*x))^(3/2)*(A + C*cos(c + d*x)^2)*sec(c + d*x)^(9/2), x, 8, -((4*b*(a + b)*(41*a^2*A - 3*A*b^2 + 70*a^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(105*a^2*d*sqrt(a + b*cos(c + d*x)))) + (2*(25*a^3*A + 82*a^2*A*b + 51*a*A*b^2 - 6*A*b^3 + 35*a^3*C + 140*a^2*b*C + 105*a*b^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(105*a*d*sqrt(a + b*cos(c + d*x))) + (2*(25*a^2*A + 3*A*b^2 + 35*a^2*C)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(105*a*d) + (6*A*b*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(5/2)*sin(c + d*x))/(35*d) + (2*A*(a + b*cos(c + d*x))^(3/2)*sec(c + d*x)^(7/2)*sin(c + d*x))/(7*d) + (4*b*(41*a^2*A - 3*A*b^2 + 70*a^2*C)*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*tan((1/2)*(c + d*x)))/(105*a^2*d)],
[(a + b*cos(c + d*x))^(3/2)*(A + C*cos(c + d*x)^2)*sec(c + d*x)^(7/2), x, 9, -((2*(a + b)*(3*a^2*A + A*b^2 + 5*a^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(5*a*d*sqrt(a + b*cos(c + d*x)))) + (2*(3*a^2*A + 4*a*A*b + A*b^2 + 5*a^2*C + 10*a*b*C - 5*b^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(5*d*sqrt(a + b*cos(c + d*x))) + (4*b^2*C*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*cos(c + d*x))) + (2*A*b*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(5*d) + (2*A*(a + b*cos(c + d*x))^(3/2)*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d) + (2*(3*a^2*A + A*b^2 + 5*a^2*C)*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*tan((1/2)*(c + d*x)))/(5*a*d)],
[(a + b*cos(c + d*x))^(3/2)*(A + C*cos(c + d*x)^2)*sec(c + d*x)^(5/2), x, 9, -((b*(a + b)*(8*A - 3*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(3*d*sqrt(a + b*cos(c + d*x)))) + (2*(a^2*A + 4*a*A*b + 3*A*b^2 + 3*a^2*C - 6*a*b*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(3*d*sqrt(a + b*cos(c + d*x))) + (6*a*b*C*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*cos(c + d*x))) + (2*A*b*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*sin(c + d*x))/d + (2*A*(a + b*cos(c + d*x))^(3/2)*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d) - (b*(8*A - 3*C)*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(3*d*sqrt(sec(c + d*x)))],
[(a + b*cos(c + d*x))^(3/2)*(A + C*cos(c + d*x)^2)*sec(c + d*x)^(3/2), x, 9, -((a*(a + b)*(8*A - 5*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(4*d*sqrt(a + b*cos(c + d*x)))) + ((4*a^2*A + 8*a*A*b - 4*A*b^2 - 4*a^2*C + a*b*C - 2*b^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(2*d*sqrt(a + b*cos(c + d*x))) + ((8*A*b^2 + 3*a^2*C + 4*b^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(2*d*sqrt(a + b*cos(c + d*x))) - (b*(4*A - C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(2*d*sqrt(sec(c + d*x))) + (2*A*(a + b*cos(c + d*x))^(3/2)*sqrt(sec(c + d*x))*sin(c + d*x))/d - (a*(8*A - 5*C)*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(4*d*sqrt(sec(c + d*x)))],
[(a + b*cos(c + d*x))^(3/2)*(A + C*cos(c + d*x)^2)*sec(c + d*x)^(1/2), x, 9, ((a + b)*(24*A*b^2 + 3*a^2*C + 16*b^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(24*b*d*sqrt(a + b*cos(c + d*x))) + (a*(24*a*A - 48*A*b + 7*a*C - 26*b*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(12*d*sqrt(a + b*cos(c + d*x))) + (a*(24*A*b^2 - a^2*C + 12*b^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(4*b*d*sqrt(a + b*cos(c + d*x))) + (a*C*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(4*d*sqrt(sec(c + d*x))) + (C*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(3*d*sqrt(sec(c + d*x))) + ((24*A*b^2 + 3*a^2*C + 16*b^2*C)*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(24*b*d*sqrt(sec(c + d*x)))],
[((a + b*cos(c + d*x))^(3/2)*(A + C*cos(c + d*x)^2))/sec(c + d*x)^(1/2), x, 10, (a*(a + b)*(80*A*b^2 - 3*a^2*C + 52*b^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(64*b^2*d*sqrt(a + b*cos(c + d*x))) - ((64*a^2*A*b - 16*a*A*b^2 + 32*A*b^3 - a^3*C + 38*a^2*b*C - 12*a*b^2*C + 24*b^3*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(32*b*d*sqrt(a + b*cos(c + d*x))) + ((48*a^2*A*b^2 + 64*A*b^4 + 3*a^4*C + 24*a^2*b^2*C + 48*b^4*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(32*b^2*d*sqrt(a + b*cos(c + d*x))) + (a*C*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(8*d*sec(c + d*x)^(3/2)) + (C*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(4*d*sec(c + d*x)^(3/2)) + ((16*A*b^2 + a^2*C + 12*b^2*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(32*b*d*sqrt(sec(c + d*x))) + (a*(80*A*b^2 - 3*a^2*C + 52*b^2*C)*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(64*b^2*d*sqrt(sec(c + d*x)))],


[(a + b*cos(c + d*x))^(5/2)*(A + C*cos(c + d*x)^2)*sec(c + d*x)^(13/2), x, 10, -((2*b*(a + b)*(741*a^4*A + 51*a^2*A*b^2 + 8*A*b^4 + 957*a^4*C + 99*a^2*b^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(693*a^3*d*sqrt(a + b*cos(c + d*x)))) + (1/(693*a^2*d*sqrt(a + b*cos(c + d*x))))*(2*(135*a^5*A + 741*a^4*A*b + 663*a^3*A*b^2 + 51*a^2*A*b^3 + 2*a*A*b^4 + 8*A*b^5 + 165*a^5*C + 957*a^4*b*C + 891*a^3*b^2*C + 99*a^2*b^3*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x))) + (2*(135*a^4*A + 205*a^2*A*b^2 - 4*A*b^4 + 165*a^4*C + 297*a^2*b^2*C)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(693*a^2*d) + (2*b*(229*a^2*A + 3*A*b^2 + 297*a^2*C)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(5/2)*sin(c + d*x))/(693*a*d) + (2*(27*a^2*A + 5*A*b^2 + 33*a^2*C)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(7/2)*sin(c + d*x))/(231*d) + (10*A*b*(a + b*cos(c + d*x))^(3/2)*sec(c + d*x)^(9/2)*sin(c + d*x))/(99*d) + (2*A*(a + b*cos(c + d*x))^(5/2)*sec(c + d*x)^(11/2)*sin(c + d*x))/(11*d) + (2*b*(741*a^4*A + 51*a^2*A*b^2 + 8*A*b^4 + 957*a^4*C + 99*a^2*b^2*C)*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*tan((1/2)*(c + d*x)))/(693*a^3*d)],
[(a + b*cos(c + d*x))^(5/2)*(A + C*cos(c + d*x)^2)*sec(c + d*x)^(11/2), x, 9, -((2*(a + b)*(147*a^4*A + 279*a^2*A*b^2 - 10*A*b^4 + 189*a^4*C + 483*a^2*b^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(315*a^2*d*sqrt(a + b*cos(c + d*x)))) + (1/(315*a*d*sqrt(a + b*cos(c + d*x))))*(2*(147*a^4*A + 261*a^3*A*b + 279*a^2*A*b^2 + 155*a*A*b^3 - 10*A*b^4 + 189*a^4*C + 357*a^3*b*C + 483*a^2*b^2*C + 315*a*b^3*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x))) + (2*b*(163*a^2*A + 5*A*b^2 + 231*a^2*C)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(315*a*d) + (2*(49*a^2*A + 15*A*b^2 + 63*a^2*C)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(5/2)*sin(c + d*x))/(315*d) + (10*A*b*(a + b*cos(c + d*x))^(3/2)*sec(c + d*x)^(7/2)*sin(c + d*x))/(63*d) + (2*A*(a + b*cos(c + d*x))^(5/2)*sec(c + d*x)^(9/2)*sin(c + d*x))/(9*d) + (2*(147*a^4*A + 279*a^2*A*b^2 - 10*A*b^4 + 189*a^4*C + 483*a^2*b^2*C)*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*tan((1/2)*(c + d*x)))/(315*a^2*d)],
[(a + b*cos(c + d*x))^(5/2)*(A + C*cos(c + d*x)^2)*sec(c + d*x)^(9/2), x, 10, -((2*b*(a + b)*(29*a^2*A + 3*A*b^2 + 49*a^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(21*a*d*sqrt(a + b*cos(c + d*x)))) + (2*(5*a^3*A + 29*a^2*A*b + 27*a*A*b^2 + 3*A*b^3 + 7*a^3*C + 49*a^2*b*C + 63*a*b^2*C - 21*b^3*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(21*d*sqrt(a + b*cos(c + d*x))) + (4*b^3*C*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*cos(c + d*x))) + (2*(5*a^2*A + 3*A*b^2 + 7*a^2*C)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(21*d) + (2*A*b*(a + b*cos(c + d*x))^(3/2)*sec(c + d*x)^(5/2)*sin(c + d*x))/(7*d) + (2*A*(a + b*cos(c + d*x))^(5/2)*sec(c + d*x)^(7/2)*sin(c + d*x))/(7*d) + (2*b*(29*a^2*A + 3*A*b^2 + 49*a^2*C)*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*tan((1/2)*(c + d*x)))/(21*a*d)],
[(a + b*cos(c + d*x))^(5/2)*(A + C*cos(c + d*x)^2)*sec(c + d*x)^(7/2), x, 10, -(((a + b)*(18*a^2*A + 46*A*b^2 + 30*a^2*C - 15*b^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(15*d*sqrt(a + b*cos(c + d*x)))) + (2*(9*a^3*A + 17*a^2*A*b + 23*a*A*b^2 + 15*A*b^3 + 15*a^3*C + 45*a^2*b*C - 45*a*b^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(15*d*sqrt(a + b*cos(c + d*x))) + (10*a*b^2*C*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*cos(c + d*x))) + (2*(3*a^2*A + 5*A*b^2 + 5*a^2*C)*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d) + (2*A*b*(a + b*cos(c + d*x))^(3/2)*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d) + (2*A*(a + b*cos(c + d*x))^(5/2)*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d) - ((18*a^2*A + 46*A*b^2 + 30*a^2*C - 15*b^2*C)*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(15*d*sqrt(sec(c + d*x)))],
[(a + b*cos(c + d*x))^(5/2)*(A + C*cos(c + d*x)^2)*sec(c + d*x)^(5/2), x, 10, -((a*b*(a + b)*(56*A - 27*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(12*d*sqrt(a + b*cos(c + d*x)))) + ((4*a^3*A + 28*a^2*A*b + 36*a*A*b^2 - 12*A*b^3 + 12*a^3*C - 36*a^2*b*C + 3*a*b^2*C - 6*b^3*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(6*d*sqrt(a + b*cos(c + d*x))) + (b*(8*A*b^2 + 15*a^2*C + 4*b^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(2*d*sqrt(a + b*cos(c + d*x))) - (b^2*(8*A - C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(2*d*sqrt(sec(c + d*x))) + (10*A*b*(a + b*cos(c + d*x))^(3/2)*sqrt(sec(c + d*x))*sin(c + d*x))/(3*d) + (2*A*(a + b*cos(c + d*x))^(5/2)*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d) - (a*b*(56*A - 27*C)*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(12*d*sqrt(sec(c + d*x)))],
[(a + b*cos(c + d*x))^(5/2)*(A + C*cos(c + d*x)^2)*sec(c + d*x)^(3/2), x, 10, -(((a + b)*(48*a^2*A - 24*A*b^2 - 33*a^2*C - 16*b^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(24*d*sqrt(a + b*cos(c + d*x)))) + (a*(24*a^2*A + 72*a*A*b - 72*A*b^2 - 24*a^2*C + 13*a*b*C - 38*b^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(12*d*sqrt(a + b*cos(c + d*x))) + (5*a*(8*A*b^2 + a^2*C + 4*b^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(4*d*sqrt(a + b*cos(c + d*x))) - (a*b*(8*A - 3*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(4*d*sqrt(sec(c + d*x))) - (b*(6*A - C)*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(3*d*sqrt(sec(c + d*x))) + (2*A*(a + b*cos(c + d*x))^(5/2)*sqrt(sec(c + d*x))*sin(c + d*x))/d - ((48*a^2*A - 24*A*b^2 - 33*a^2*C - 16*b^2*C)*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(24*d*sqrt(sec(c + d*x)))],
[(a + b*cos(c + d*x))^(5/2)*(A + C*cos(c + d*x)^2)*sec(c + d*x)^(1/2), x, 10, (a*(a + b)*(432*A*b^2 + 15*a^2*C + 284*b^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(192*b*d*sqrt(a + b*cos(c + d*x))) + ((192*a^3*A - 576*a^2*A*b + 48*a*A*b^2 - 96*A*b^3 + 59*a^3*C - 322*a^2*b*C + 36*a*b^2*C - 72*b^3*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(96*d*sqrt(a + b*cos(c + d*x))) + ((240*a^2*A*b^2 + 64*A*b^4 - 5*a^4*C + 120*a^2*b^2*C + 48*b^4*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(32*b*d*sqrt(a + b*cos(c + d*x))) + ((16*A*b^2 + 5*a^2*C + 12*b^2*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(32*d*sqrt(sec(c + d*x))) + (5*a*C*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(24*d*sqrt(sec(c + d*x))) + (C*(a + b*cos(c + d*x))^(5/2)*sin(c + d*x))/(4*d*sqrt(sec(c + d*x))) + (a*(432*A*b^2 + 15*a^2*C + 284*b^2*C)*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(192*b*d*sqrt(sec(c + d*x)))],
[((a + b*cos(c + d*x))^(5/2)*(A + C*cos(c + d*x)^2))/sec(c + d*x)^(1/2), x, 11, ((a + b)*(2640*a^2*A*b^2 + 1280*A*b^4 - 45*a^4*C + 1692*a^2*b^2*C + 1024*b^4*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(1920*b^2*d*sqrt(a + b*cos(c + d*x))) - (a*(1920*a^2*A*b - 1040*a*A*b^2 + 3040*A*b^3 - 15*a^3*C + 1146*a^2*b*C - 772*a*b^2*C + 2312*b^3*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(960*b*d*sqrt(a + b*cos(c + d*x))) + (a*(80*a^2*A*b^2 + 320*A*b^4 + 3*a^4*C + 40*a^2*b^2*C + 240*b^4*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(64*b^2*d*sqrt(a + b*cos(c + d*x))) + ((80*A*b^2 + 15*a^2*C + 64*b^2*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(240*d*sec(c + d*x)^(3/2)) + (a*C*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(8*d*sec(c + d*x)^(3/2)) + (C*(a + b*cos(c + d*x))^(5/2)*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2)) + (a*(1040*A*b^2 + 15*a^2*C + 772*b^2*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(960*b*d*sqrt(sec(c + d*x))) + ((2640*a^2*A*b^2 + 1280*A*b^4 - 45*a^4*C + 1692*a^2*b^2*C + 1024*b^4*C)*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(1920*b^2*d*sqrt(sec(c + d*x)))],


# ::Subsubsection::Closed:: 
#n<0


[((A + C*cos(c + d*x)^2)*sec(c + d*x)^(9/2))/sqrt(a + b*cos(c + d*x)), x, 8, (4*b*(a + b)*(22*a^2*A + 24*A*b^2 + 35*a^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(105*a^4*d*sqrt(a + b*cos(c + d*x))) + (2*(25*a^3*A - 44*a^2*A*b - 12*a*A*b^2 - 48*A*b^3 + 35*a^3*C - 70*a^2*b*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(105*a^3*d*sqrt(a + b*cos(c + d*x))) + (2*(25*a^2*A + 24*A*b^2 + 35*a^2*C)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(105*a^3*d) - (12*A*b*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(5/2)*sin(c + d*x))/(35*a^2*d) + (2*A*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(7/2)*sin(c + d*x))/(7*a*d) - (4*b*(22*a^2*A + 24*A*b^2 + 35*a^2*C)*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*tan((1/2)*(c + d*x)))/(105*a^4*d)],
[((A + C*cos(c + d*x)^2)*sec(c + d*x)^(7/2))/sqrt(a + b*cos(c + d*x)), x, 7, -((2*(a + b)*(9*a^2*A + 8*A*b^2 + 15*a^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(15*a^3*d*sqrt(a + b*cos(c + d*x)))) + (2*(9*a^2*A + 2*a*A*b + 8*A*b^2 + 15*a^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(15*a^2*d*sqrt(a + b*cos(c + d*x))) - (8*A*b*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(15*a^2*d) + (2*A*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*a*d) + (2*(9*a^2*A + 8*A*b^2 + 15*a^2*C)*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*tan((1/2)*(c + d*x)))/(15*a^3*d)],
[((A + C*cos(c + d*x)^2)*sec(c + d*x)^(5/2))/sqrt(a + b*cos(c + d*x)), x, 6, (4*A*b*(a + b)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(3*a^2*d*sqrt(a + b*cos(c + d*x))) + (2*(a*A - 2*A*b + 3*a*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(3*a*d*sqrt(a + b*cos(c + d*x))) + (2*A*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*a*d) - (4*A*b*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*tan((1/2)*(c + d*x)))/(3*a^2*d)],
[((A + C*cos(c + d*x)^2)*sec(c + d*x)^(3/2))/sqrt(a + b*cos(c + d*x)), x, 7, -((2*A*(a + b)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(a*d*sqrt(a + b*cos(c + d*x)))) + (2*(A - C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*cos(c + d*x))) + (4*C*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*cos(c + d*x))) + (2*A*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*tan((1/2)*(c + d*x)))/(a*d)],
[((A + C*cos(c + d*x)^2)*sec(c + d*x)^(1/2))/sqrt(a + b*cos(c + d*x)), x, 7, ((a + b)*C*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(b*d*sqrt(a + b*cos(c + d*x))) + (2*A*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*cos(c + d*x))) - (2*a*C*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(b*d*sqrt(a + b*cos(c + d*x))) + (C*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(b*d*sqrt(sec(c + d*x)))],
[(A + C*cos(c + d*x)^2)/(sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(1/2)), x, 8, -((3*a*(a + b)*C*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(4*b^2*d*sqrt(a + b*cos(c + d*x)))) - ((4*A*b - a*C + 2*b*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(2*b*d*sqrt(a + b*cos(c + d*x))) + ((8*A*b^2 + 3*a^2*C + 4*b^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(2*b^2*d*sqrt(a + b*cos(c + d*x))) + (C*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(2*b*d*sqrt(sec(c + d*x))) - (3*a*C*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(4*b^2*d*sqrt(sec(c + d*x)))],
[(A + C*cos(c + d*x)^2)/(sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)), x, 9, ((a + b)*(24*A*b^2 + 15*a^2*C + 16*b^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(24*b^3*d*sqrt(a + b*cos(c + d*x))) - (a*(5*a + 2*b)*C*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(12*b^2*d*sqrt(a + b*cos(c + d*x))) - (a*(8*A*b^2 + 5*a^2*C + 4*b^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(4*b^3*d*sqrt(a + b*cos(c + d*x))) + (C*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*b*d*sec(c + d*x)^(3/2)) - (5*a*C*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(12*b^2*d*sqrt(sec(c + d*x))) + ((24*A*b^2 + 15*a^2*C + 16*b^2*C)*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(24*b^3*d*sqrt(sec(c + d*x)))],


[((A + C*cos(c + d*x)^2)*sec(c + d*x)^(7/2))/(a + b*cos(c + d*x))^(3/2), x, 8, -((2*(3*a^4*A + 8*a^2*A*b^2 - 16*A*b^4 + 5*a^4*C - 10*a^2*b^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(5*a^4*(a - b)*d*sqrt(a + b*cos(c + d*x)))) + (2*(3*a^3*A - 4*a^2*A*b + 12*a*A*b^2 - 16*A*b^3 + 5*a^3*C - 10*a^2*b*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(5*a^3*(a - b)*d*sqrt(a + b*cos(c + d*x))) - (2*b*(3*a^2*A - 8*A*b^2 - 5*a^2*C)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(5*a^3*(a^2 - b^2)*d) + (2*(A*b^2 + a^2*C)*sec(c + d*x)^(5/2)*sin(c + d*x))/(a*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) + (2*(a^2*A - 6*A*b^2 - 5*a^2*C)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*a^2*(a^2 - b^2)*d) + (2*(3*a^4*A + 8*a^2*A*b^2 - 16*A*b^4 + 5*a^4*C - 10*a^2*b^2*C)*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*tan((1/2)*(c + d*x)))/(5*a^4*(a^2 - b^2)*d)],
[((A + C*cos(c + d*x)^2)*sec(c + d*x)^(5/2))/(a + b*cos(c + d*x))^(3/2), x, 7, (2*b*(5*a^2*A - 8*A*b^2 - 3*a^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(3*a^3*(a - b)*d*sqrt(a + b*cos(c + d*x))) + (2*(a^2*A - 6*a*A*b + 8*A*b^2 + 3*a^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(3*a^2*(a - b)*d*sqrt(a + b*cos(c + d*x))) + (2*(A*b^2 + a^2*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(a*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) + (2*(a^2*A - 4*A*b^2 - 3*a^2*C)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*a^2*(a^2 - b^2)*d) - (2*b*(5*a^2*A - 8*A*b^2 - 3*a^2*C)*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*tan((1/2)*(c + d*x)))/(3*a^3*(a^2 - b^2)*d)],
[((A + C*cos(c + d*x)^2)*sec(c + d*x)^(3/2))/(a + b*cos(c + d*x))^(3/2), x, 6, -((2*(a^2*A - 2*A*b^2 - a^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(a^2*(a - b)*d*sqrt(a + b*cos(c + d*x)))) + (2*(a*A - 2*A*b - a*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(a*(a - b)*d*sqrt(a + b*cos(c + d*x))) + (2*(A*b^2 + a^2*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(a*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) + (2*(a^2*A - 2*A*b^2 - a^2*C)*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*tan((1/2)*(c + d*x)))/(a^2*(a^2 - b^2)*d)],
[((A + C*cos(c + d*x)^2)*sec(c + d*x)^(1/2))/(a + b*cos(c + d*x))^(3/2), x, 7, -((2*(A*b^2 + a^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(a*(a - b)*b*d*sqrt(a + b*cos(c + d*x)))) + (2*(A + C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/((a - b)*d*sqrt(a + b*cos(c + d*x))) + (4*C*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(b*d*sqrt(a + b*cos(c + d*x))) - (2*(A*b^2 + a^2*C)*tan((1/2)*(c + d*x)))/(a*b*(a + b)*d*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x)))],
[(A + C*cos(c + d*x)^2)/((a + b*cos(c + d*x))^(3/2)*sec(c + d*x)^(1/2)), x, 8, ((2*A*b^2 + 3*a^2*C - b^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/((a - b)*b^2*d*sqrt(a + b*cos(c + d*x))) - (2*(A*b + a*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/((a - b)*b*d*sqrt(a + b*cos(c + d*x))) - (6*a*C*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(b^2*d*sqrt(a + b*cos(c + d*x))) - (2*(A*b^2 + a^2*C)*sin(c + d*x))/(b*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))) + ((2*A*b^2 + 3*a^2*C - b^2*C)*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(b^2*(a^2 - b^2)*d*sqrt(sec(c + d*x)))],
[(A + C*cos(c + d*x)^2)/((a + b*cos(c + d*x))^(3/2)*sec(c + d*x)^(3/2)), x, 9, -((a*(8*A*b^2 + 15*a^2*C - 7*b^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(4*(a - b)*b^3*d*sqrt(a + b*cos(c + d*x)))) + ((4*A*b^2 + 5*a^2*C - 3*a*b*C + 2*b^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(2*(a - b)*b^2*d*sqrt(a + b*cos(c + d*x))) + ((8*A*b^2 + 15*a^2*C + 4*b^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(2*b^3*d*sqrt(a + b*cos(c + d*x))) - (2*(A*b^2 + a^2*C)*sin(c + d*x))/(b*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)) + ((4*A*b^2 + 5*a^2*C - b^2*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(2*b^2*(a^2 - b^2)*d*sqrt(sec(c + d*x))) - (a*(8*A*b^2 + 15*a^2*C - 7*b^2*C)*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(4*b^3*(a^2 - b^2)*d*sqrt(sec(c + d*x)))],


[((A + C*cos(c + d*x)^2)*sec(c + d*x)^(5/2))/(a + b*cos(c + d*x))^(5/2), x, 8, (4*b*(4*a^4*A - 14*a^2*A*b^2 + 8*A*b^4 - 3*a^4*C + a^2*b^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(3*a^4*(a - b)^2*(a + b)*d*sqrt(a + b*cos(c + d*x))) + (2*(a^4*A - 9*a^3*A*b + 16*a^2*A*b^2 + 12*a*A*b^3 - 16*A*b^4 + 3*a^4*C + 3*a^3*b*C - 2*a^2*b^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(3*a^3*(a - b)^2*(a + b)*d*sqrt(a + b*cos(c + d*x))) + (2*(A*b^2 + a^2*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*a*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)) + (4*(5*a^2*A*b^2 - 3*A*b^4 + 2*a^4*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*a^2*(a^2 - b^2)^2*d*sqrt(a + b*cos(c + d*x))) + (2*(a^4*A - 13*a^2*A*b^2 + 8*A*b^4 - 5*a^4*C + a^2*b^2*C)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*a^3*(a^2 - b^2)^2*d) - (4*b*(4*a^4*A - 14*a^2*A*b^2 + 8*A*b^4 - 3*a^4*C + a^2*b^2*C)*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*tan((1/2)*(c + d*x)))/(3*a^4*(a^2 - b^2)^2*d)],
[((A + C*cos(c + d*x)^2)*sec(c + d*x)^(3/2))/(a + b*cos(c + d*x))^(5/2), x, 7, -((2*(3*a^4*A - 15*a^2*A*b^2 + 8*A*b^4 - 3*a^4*C - a^2*b^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(3*a^3*(a - b)^2*(a + b)*d*sqrt(a + b*cos(c + d*x)))) + (2*(3*a^3*A - 9*a^2*A*b - 6*a*A*b^2 + 8*A*b^3 - 3*a^3*C - a^2*b*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(3*a^2*(a - b)^2*(a + b)*d*sqrt(a + b*cos(c + d*x))) + (2*(A*b^2 + a^2*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(3*a*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)) + (4*(4*a^2*A*b^2 - 2*A*b^4 + a^4*C + a^2*b^2*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(3*a^2*(a^2 - b^2)^2*d*sqrt(a + b*cos(c + d*x))) + (2*(3*a^4*A - 15*a^2*A*b^2 + 8*A*b^4 - 3*a^4*C - a^2*b^2*C)*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*tan((1/2)*(c + d*x)))/(3*a^3*(a^2 - b^2)^2*d)],
[((A + C*cos(c + d*x)^2)*sec(c + d*x)^(1/2))/(a + b*cos(c + d*x))^(5/2), x, 6, -((4*b*(3*a^2*A - A*b^2 + 2*a^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(3*a^2*(a - b)^2*(a + b)*d*sqrt(a + b*cos(c + d*x)))) + (2*(3*a^2*A + 3*a*A*b - 2*A*b^2 + a^2*C + 3*a*b*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(3*a*(a - b)^2*(a + b)*d*sqrt(a + b*cos(c + d*x))) + (2*(A*b^2 + a^2*C)*sin(c + d*x))/(3*a*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)*sqrt(sec(c + d*x))) - (4*b*(3*a^2*A - A*b^2 + 2*a^2*C)*tan((1/2)*(c + d*x)))/(3*a^2*(a - b)*(a + b)^2*d*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x)))],
[(A + C*cos(c + d*x)^2)/((a + b*cos(c + d*x))^(5/2)*sec(c + d*x)^(1/2)), x, 8, (2*(3*a^2*A*b^2 + A*b^4 - 3*a^4*C + 7*a^2*b^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(3*a*(a - b)^2*b^2*(a + b)*d*sqrt(a + b*cos(c + d*x))) - (2*(3*a*A*b + A*b^2 - 2*a^2*C + 3*a*b*C + 3*b^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(3*(a - b)^2*b*(a + b)*d*sqrt(a + b*cos(c + d*x))) + (4*C*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(b^2*d*sqrt(a + b*cos(c + d*x))) - (2*(A*b^2 + a^2*C)*sin(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)*sqrt(sec(c + d*x))) + (2*(3*a^2*A*b^2 + A*b^4 - 3*a^4*C + 7*a^2*b^2*C)*tan((1/2)*(c + d*x)))/(3*a*(a - b)*b^2*(a + b)^2*d*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x)))],
[(A + C*cos(c + d*x)^2)/((a + b*cos(c + d*x))^(5/2)*sec(c + d*x)^(3/2)), x, 9, -(((8*A*b^4 - 15*a^4*C + 26*a^2*b^2*C - 3*b^4*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(3*(a - b)^2*b^3*(a + b)*d*sqrt(a + b*cos(c + d*x)))) + (2*(a*A*b^2 + 3*A*b^3 - 5*a^3*C + 3*a^2*b*C + 6*a*b^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(3*(a - b)^2*b^2*(a + b)*d*sqrt(a + b*cos(c + d*x))) - (10*a*C*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(b^3*d*sqrt(a + b*cos(c + d*x))) - (2*(A*b^2 + a^2*C)*sin(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)*sec(c + d*x)^(3/2)) + (2*(a^2*A*b^2 + 3*A*b^4 - 5*a^4*C + 9*a^2*b^2*C)*sin(c + d*x))/(3*b^2*(a^2 - b^2)^2*d*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))) - ((8*A*b^4 - 15*a^4*C + 26*a^2*b^2*C - 3*b^4*C)*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(3*b^3*(a^2 - b^2)^2*d*sqrt(sec(c + d*x)))],


# ::Section:: 
#Cos[c+d x]^m (B Cos[c+d x]+C Cos[c+d x]^2) (a+b Cos[c+d x])^n


# ::Subsection::Closed:: 
#Integrands of the form Cos[c+d x]^m (B Cos[c+d x]+C Cos[c+d x]^2) (a+b Cos[c+d x])^n
#


# ::Subsubsection::Closed:: 
#n>0


[cos(c + d*x)^1*(B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x)), x, 4, (1/8)*(4*a*B + 3*b*C)*x + (2*(b*B + a*C)*sin(c + d*x))/(3*d) + ((4*a*B + 3*b*C)*cos(c + d*x)*sin(c + d*x))/(8*d) + ((4*b*B + a*C)*cos(c + d*x)^2*sin(c + d*x))/(12*d) + (C*cos(c + d*x)^2*(a + b*cos(c + d*x))*sin(c + d*x))/(4*d)],
[cos(c + d*x)^0*(B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x)), x, 3, (1/2)*(b*B + a*C)*x + ((3*a*b*B - a^2*C + 2*b^2*C)*sin(c + d*x))/(3*b*d) + ((3*b*B - a*C)*cos(c + d*x)*sin(c + d*x))/(6*d) + (C*(a + b*cos(c + d*x))^2*sin(c + d*x))/(3*b*d)],
[sec(c + d*x)^1*(B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x)), x, 3, (1/2)*(2*a*B + b*C)*x + ((2*b*B + a*C)*sin(c + d*x))/(2*d) + (C*(a + b*cos(c + d*x))*sin(c + d*x))/(2*d)],
[sec(c + d*x)^2*(B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x)), x, 4, (b*B + a*C)*x + (a*B*arctanh(sin(c + d*x)))/d + (b*C*sin(c + d*x))/d],
[sec(c + d*x)^3*(B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x)), x, 4, b*C*x + ((b*B + a*C)*arctanh(sin(c + d*x)))/d + (a*B*tan(c + d*x))/d],
[sec(c + d*x)^4*(B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x)), x, 5, ((a*B + 2*b*C)*arctanh(sin(c + d*x)))/(2*d) + ((b*B + a*C)*tan(c + d*x))/d + (a*B*sec(c + d*x)*tan(c + d*x))/(2*d)],
[sec(c + d*x)^5*(B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x)), x, 6, ((b*B + a*C)*arctanh(sin(c + d*x)))/(2*d) + ((2*a*B + 3*b*C)*tan(c + d*x))/(3*d) + ((b*B + a*C)*sec(c + d*x)*tan(c + d*x))/(2*d) + (a*B*sec(c + d*x)^2*tan(c + d*x))/(3*d)],
[sec(c + d*x)^6*(B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x)), x, 7, ((3*a*B + 4*b*C)*arctanh(sin(c + d*x)))/(8*d) + (2*(b*B + a*C)*tan(c + d*x))/(3*d) + ((3*a*B + 4*b*C)*sec(c + d*x)*tan(c + d*x))/(8*d) + ((b*B + a*C)*sec(c + d*x)^2*tan(c + d*x))/(3*d) + (a*B*sec(c + d*x)^3*tan(c + d*x))/(4*d)],


[cos(c + d*x)^1*(B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^2, x, 5, (1/8)*(4*a^2*B + 3*b^2*B + 6*a*b*C)*x + (2*(10*a*b*B + 5*a^2*C + 4*b^2*C)*sin(c + d*x))/(15*d) + ((4*a^2*B + 3*b^2*B + 6*a*b*C)*cos(c + d*x)*sin(c + d*x))/(8*d) + ((25*a*b*B + 2*a^2*C + 16*b^2*C)*cos(c + d*x)^2*sin(c + d*x))/(60*d) + ((5*b*B + 2*a*C)*cos(c + d*x)^2*(a + b*cos(c + d*x))*sin(c + d*x))/(20*d) + (C*cos(c + d*x)^2*(a + b*cos(c + d*x))^2*sin(c + d*x))/(5*d)],
[cos(c + d*x)^0*(B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^2, x, 4, (1/8)*(8*a*b*B + 4*a^2*C + 3*b^2*C)*x + ((4*a^2*b*B + 4*b^3*B - a^3*C + 8*a*b^2*C)*sin(c + d*x))/(6*b*d) + ((8*a*b*B - 2*a^2*C + 9*b^2*C)*cos(c + d*x)*sin(c + d*x))/(24*d) + ((4*b*B - a*C)*(a + b*cos(c + d*x))^2*sin(c + d*x))/(12*b*d) + (C*(a + b*cos(c + d*x))^3*sin(c + d*x))/(4*b*d)],
[sec(c + d*x)^1*(B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^2, x, 4, (1/2)*(2*a^2*B + b^2*B + 2*a*b*C)*x + ((9*a*b*B + 2*a^2*C + 4*b^2*C)*sin(c + d*x))/(6*d) + ((3*b*B + 2*a*C)*(a + b*cos(c + d*x))*sin(c + d*x))/(6*d) + (C*(a + b*cos(c + d*x))^2*sin(c + d*x))/(3*d)],
[sec(c + d*x)^2*(B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^2, x, 5, (1/2)*(4*a*b*B + 2*a^2*C + b^2*C)*x + (a^2*B*arctanh(sin(c + d*x)))/d + (b*(2*b*B + 3*a*C)*sin(c + d*x))/(2*d) + (b*C*(a + b*cos(c + d*x))*sin(c + d*x))/(2*d)],
[sec(c + d*x)^3*(B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^2, x, 5, (b^2*B + 2*a*b*C)*x + (a*(2*b*B + a*C)*arctanh(sin(c + d*x)))/d - (b*(a*B - b*C)*sin(c + d*x))/d + (a*B*(a + b*cos(c + d*x))*tan(c + d*x))/d],
[sec(c + d*x)^4*(B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^2, x, 5, b^2*C*x + ((a^2*B + 2*b^2*B + 4*a*b*C)*arctanh(sin(c + d*x)))/(2*d) + (a*(3*b*B + 2*a*C)*tan(c + d*x))/(2*d) + (a*B*(b + a*sec(c + d*x))*tan(c + d*x))/(2*d)],
[sec(c + d*x)^5*(B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^2, x, 6, ((2*a*b*B + a^2*C + 2*b^2*C)*arctanh(sin(c + d*x)))/(2*d) + ((2*a^2*B + 3*b^2*B + 6*a*b*C)*tan(c + d*x))/(3*d) + (a*(4*b*B + 3*a*C)*sec(c + d*x)*tan(c + d*x))/(6*d) + (a*B*(a + b*cos(c + d*x))*sec(c + d*x)^2*tan(c + d*x))/(3*d)],
[sec(c + d*x)^6*(B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^2, x, 7, ((3*a^2*B + 4*b^2*B + 8*a*b*C)*arctanh(sin(c + d*x)))/(8*d) + ((4*a*b*B + 2*a^2*C + 3*b^2*C)*tan(c + d*x))/(3*d) + ((3*a^2*B + 4*b^2*B + 8*a*b*C)*sec(c + d*x)*tan(c + d*x))/(8*d) + (a*(5*b*B + 4*a*C)*sec(c + d*x)^2*tan(c + d*x))/(12*d) + (a*B*(a + b*cos(c + d*x))*sec(c + d*x)^3*tan(c + d*x))/(4*d)],


[cos(c + d*x)^0*(B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^3, x, 5, (1/8)*(12*a^2*b*B + 3*b^3*B + 4*a^3*C + 9*a*b^2*C)*x + ((15*a^3*b*B + 60*a*b^3*B - 3*a^4*C + 52*a^2*b^2*C + 16*b^4*C)*sin(c + d*x))/(30*b*d) + ((30*a^2*b*B + 45*b^3*B - 6*a^3*C + 71*a*b^2*C)*cos(c + d*x)*sin(c + d*x))/(120*d) + ((15*a*b*B - 3*a^2*C + 16*b^2*C)*(a + b*cos(c + d*x))^2*sin(c + d*x))/(60*b*d) + ((5*b*B - a*C)*(a + b*cos(c + d*x))^3*sin(c + d*x))/(20*b*d) + (C*(a + b*cos(c + d*x))^4*sin(c + d*x))/(5*b*d)],
[sec(c + d*x)^1*(B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^3, x, 5, (1/8)*(8*a^3*B + 12*a*b^2*B + 12*a^2*b*C + 3*b^3*C)*x + ((44*a^2*b*B + 16*b^3*B + 6*a^3*C + 39*a*b^2*C)*sin(c + d*x))/(24*d) + ((20*a*b*B + 6*a^2*C + 9*b^2*C)*(a + b*cos(c + d*x))*sin(c + d*x))/(24*d) + ((4*b*B + 3*a*C)*(a + b*cos(c + d*x))^2*sin(c + d*x))/(12*d) + (C*(a + b*cos(c + d*x))^3*sin(c + d*x))/(4*d)],
[sec(c + d*x)^2*(B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^3, x, 6, (1/2)*(6*a^2*b*B + b^3*B + 2*a^3*C + 3*a*b^2*C)*x + (a^3*B*arctanh(sin(c + d*x)))/d + (b*(15*a*b*B + 11*a^2*C + 4*b^2*C)*sin(c + d*x))/(6*d) + (b*(3*b*B + 5*a*C)*(a + b*cos(c + d*x))*sin(c + d*x))/(6*d) + (b*C*(a + b*cos(c + d*x))^2*sin(c + d*x))/(3*d)],
[sec(c + d*x)^3*(B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^3, x, 6, (1/2)*b*(6*a*b*B + 6*a^2*C + b^2*C)*x + (a^2*(3*b*B + a*C)*arctanh(sin(c + d*x)))/d - (b*(2*a^2*B - 2*b^2*B - 5*a*b*C)*sin(c + d*x))/(2*d) - (b*(2*a*B - b*C)*(a + b*cos(c + d*x))*sin(c + d*x))/(2*d) + (a*B*(a + b*cos(c + d*x))^2*tan(c + d*x))/d],
[sec(c + d*x)^4*(B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^3, x, 6, (b^3*B + 3*a*b^2*C)*x + (a*(a^2*B + 6*b^2*B + 6*a*b*C)*arctanh(sin(c + d*x)))/(2*d) - (b*(5*a*b*B + 2*a^2*C - 2*b^2*C)*sin(c + d*x))/(2*d) + (a*(2*b*B + a*C)*(a + b*cos(c + d*x))*tan(c + d*x))/d + (a*B*(a + b*cos(c + d*x))^2*sec(c + d*x)*tan(c + d*x))/(2*d)],
[sec(c + d*x)^5*(B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^3, x, 6, b^3*C*x + ((3*a^2*b*B + 2*b^3*B + a^3*C + 6*a*b^2*C)*arctanh(sin(c + d*x)))/(2*d) + (a*(4*a^2*B + 11*b^2*B + 15*a*b*C)*tan(c + d*x))/(6*d) + (a*(5*b*B + 3*a*C)*(b + a*sec(c + d*x))*tan(c + d*x))/(6*d) + (a*B*(b + a*sec(c + d*x))^2*tan(c + d*x))/(3*d)],
[sec(c + d*x)^6*(B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^3, x, 7, ((3*a^3*B + 12*a*b^2*B + 12*a^2*b*C + 8*b^3*C)*arctanh(sin(c + d*x)))/(8*d) + ((6*a^2*b*B + 3*b^3*B + 2*a^3*C + 9*a*b^2*C)*tan(c + d*x))/(3*d) + (a*(9*a^2*B + 18*b^2*B + 28*a*b*C)*sec(c + d*x)*tan(c + d*x))/(24*d) + (a*(3*b*B + 2*a*C)*(a + b*cos(c + d*x))*sec(c + d*x)^2*tan(c + d*x))/(6*d) + (a*B*(a + b*cos(c + d*x))^2*sec(c + d*x)^3*tan(c + d*x))/(4*d)],
[sec(c + d*x)^7*(B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^3, x, 8, ((9*a^2*b*B + 4*b^3*B + 3*a^3*C + 12*a*b^2*C)*arctanh(sin(c + d*x)))/(8*d) + ((8*a^3*B + 30*a*b^2*B + 30*a^2*b*C + 15*b^3*C)*tan(c + d*x))/(15*d) + ((9*a^2*b*B + 4*b^3*B + 3*a^3*C + 12*a*b^2*C)*sec(c + d*x)*tan(c + d*x))/(8*d) + (a*(16*a^2*B + 27*b^2*B + 45*a*b*C)*sec(c + d*x)^2*tan(c + d*x))/(60*d) + (a*(7*b*B + 5*a*C)*(a + b*cos(c + d*x))*sec(c + d*x)^3*tan(c + d*x))/(20*d) + (a*B*(a + b*cos(c + d*x))^2*sec(c + d*x)^4*tan(c + d*x))/(5*d)],


# ::Subsubsection::Closed:: 
#n<0


[cos(c + d*x)^2*(B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x)), x, 5, ((2*a^2*b*B + b^3*B - 2*a^3*C - a*b^2*C)*x)/(2*b^4) - (2*a^3*(b*B - a*C)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^4*sqrt(a^2 - b^2)*d) - ((3*a*b*B - 3*a^2*C - 2*b^2*C)*sin(c + d*x))/(3*b^3*d) + ((b*B - a*C)*cos(c + d*x)*sin(c + d*x))/(2*b^2*d) + (C*cos(c + d*x)^2*sin(c + d*x))/(3*b*d)],
[cos(c + d*x)^1*(B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x)), x, 4, -(((2*a*b*B - 2*a^2*C - b^2*C)*x)/(2*b^3)) + (2*a^2*(b*B - a*C)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^3*sqrt(a^2 - b^2)*d) + ((b*B - a*C)*sin(c + d*x))/(b^2*d) + (C*cos(c + d*x)*sin(c + d*x))/(2*b*d)],
[cos(c + d*x)^0*(B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x)), x, 4, ((b*B - a*C)*x)/b^2 - (2*a*(b*B - a*C)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^2*sqrt(a^2 - b^2)*d) + (C*sin(c + d*x))/(b*d)],
[sec(c + d*x)^1*(B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x)), x, 4, (C*x)/b + (2*(b*B - a*C)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b*sqrt(a^2 - b^2)*d)],
[sec(c + d*x)^2*(B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x)), x, 4, -((2*(b*B - a*C)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a*sqrt(a^2 - b^2)*d)) + (B*arctanh(sin(c + d*x)))/(a*d)],
[sec(c + d*x)^3*(B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x)), x, 6, (2*b*(b*B - a*C)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^2*sqrt(a^2 - b^2)*d) - ((b*B - a*C)*arctanh(sin(c + d*x)))/(a^2*d) + (B*tan(c + d*x))/(a*d)],
[sec(c + d*x)^4*(B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x)), x, 6, -((2*b^2*(b*B - a*C)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^3*sqrt(a^2 - b^2)*d)) + ((a^2*B + 2*b^2*B - 2*a*b*C)*arctanh(sin(c + d*x)))/(2*a^3*d) - ((b*B - a*C)*tan(c + d*x))/(a^2*d) + (B*sec(c + d*x)*tan(c + d*x))/(2*a*d)],


[cos(c + d*x)^2*(B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^2, x, 5, -(((4*a*b*B - 6*a^2*C - b^2*C)*x)/(2*b^4)) + (2*a^2*(2*a^2*b*B - 3*b^3*B - 3*a^3*C + 4*a*b^2*C)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^4*(a^2 - b^2)^(3/2)*d) + ((2*a^2*b*B - b^3*B - 3*a^3*C + 2*a*b^2*C)*sin(c + d*x))/(b^3*(a^2 - b^2)*d) - ((2*a*b*B - 3*a^2*C + b^2*C)*cos(c + d*x)*sin(c + d*x))/(2*b^2*(a^2 - b^2)*d) + (a*(b*B - a*C)*cos(c + d*x)^2*sin(c + d*x))/(b*(a^2 - b^2)*d*(a + b*cos(c + d*x)))],
[cos(c + d*x)^1*(B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^2, x, 4, ((b*B - 2*a*C)*x)/b^3 - (2*a*(a^2*b*B - 2*b^3*B - 2*a^3*C + 3*a*b^2*C)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^3*(a^2 - b^2)^(3/2)*d) - ((a*b*B - 2*a^2*C + b^2*C)*sin(c + d*x))/(b^2*(a^2 - b^2)*d) + (a*(b*B - a*C)*cos(c + d*x)*sin(c + d*x))/(b*(a^2 - b^2)*d*(a + b*cos(c + d*x)))],
[cos(c + d*x)^0*(B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^2, x, 3, (C*x)/b^2 - (2*(b^3*B + a^3*C - 2*a*b^2*C)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^2*(a^2 - b^2)^(3/2)*d) + (a*(b*B - a*C)*sin(c + d*x))/(b*(a^2 - b^2)*d*(a + b*cos(c + d*x)))],
[sec(c + d*x)^1*(B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^2, x, 3, (2*(a*B - b*C)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/((a^2 - b^2)^(3/2)*d) - ((b*B - a*C)*sin(c + d*x))/((a^2 - b^2)*d*(a + b*cos(c + d*x)))],
[sec(c + d*x)^2*(B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^2, x, 5, -((2*(2*a^2*b*B - b^3*B - a^3*C)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^2*(a^2 - b^2)^(3/2)*d)) + (B*arctanh(sin(c + d*x)))/(a^2*d) + ((b*B - a*C)*tan(c + d*x))/(a*(a^2 - b^2)*d) - ((b*B - a*C)*tan(c + d*x))/((a^2 - b^2)*d*(a + b*cos(c + d*x)))],
[sec(c + d*x)^3*(B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^2, x, 6, (2*b*(3*a^2*b*B - 2*b^3*B - 2*a^3*C + a*b^2*C)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^3*(a^2 - b^2)^(3/2)*d) - ((2*b*B - a*C)*arctanh(sin(c + d*x)))/(a^3*d) + ((a^2*B - 2*b^2*B + a*b*C)*tan(c + d*x))/(a^2*(a^2 - b^2)*d) + ((b*B - a*C)*sec(c + d*x)*tan(c + d*x))/(a*(a^2 - b^2)*d) - ((b*B - a*C)*sec(c + d*x)*tan(c + d*x))/((a^2 - b^2)*d*(a + b*cos(c + d*x)))],


[cos(c + d*x)^3*(B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^3, x, 6, -(((6*a*b*B - 12*a^2*C - b^2*C)*x)/(2*b^5)) + (a^2*(6*a^4*b*B - 15*a^2*b^3*B + 12*b^5*B - 12*a^5*C + 29*a^3*b^2*C - 20*a*b^4*C)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^5*(a^2 - b^2)^(5/2)*d) + ((6*a^4*b*B - 11*a^2*b^3*B + 2*b^5*B - 12*a^5*C + 21*a^3*b^2*C - 6*a*b^4*C)*sin(c + d*x))/(2*b^4*(a^2 - b^2)^2*d) - ((3*a^3*b*B - 6*a*b^3*B - 6*a^4*C + 10*a^2*b^2*C - b^4*C)*cos(c + d*x)*sin(c + d*x))/(2*b^3*(a^2 - b^2)^2*d) + (a*(b*B - a*C)*cos(c + d*x)^3*sin(c + d*x))/(2*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) + (a*(2*a^2*b*B - 5*b^3*B - 4*a^3*C + 7*a*b^2*C)*cos(c + d*x)^2*sin(c + d*x))/(2*b^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))],
[cos(c + d*x)^2*(B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^3, x, 5, ((b*B - 3*a*C)*x)/b^4 - (a*(2*a^4*b*B - 5*a^2*b^3*B + 6*b^5*B - 6*a^5*C + 15*a^3*b^2*C - 12*a*b^4*C)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^4*(a^2 - b^2)^(5/2)*d) - ((2*a^3*b*B - 5*a*b^3*B - 6*a^4*C + 11*a^2*b^2*C - 2*b^4*C)*sin(c + d*x))/(2*b^3*(a^2 - b^2)^2*d) + (a*(b*B - a*C)*cos(c + d*x)^2*sin(c + d*x))/(2*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) + (a*(a^2*b*B - 4*b^3*B - 3*a^3*C + 6*a*b^2*C)*cos(c + d*x)*sin(c + d*x))/(2*b^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))],
[cos(c + d*x)^1*(B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^3, x, 4, (C*x)/b^3 + ((a^2*b^3*B + 2*b^5*B - 2*a^5*C + 5*a^3*b^2*C - 6*a*b^4*C)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^3*(a^2 - b^2)^(5/2)*d) + (a*(b*B - a*C)*cos(c + d*x)*sin(c + d*x))/(2*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) - (a*(3*b^3*B + 2*a^3*C - 5*a*b^2*C)*sin(c + d*x))/(2*b^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))],
[cos(c + d*x)^0*(B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^3, x, 4, -(((3*a*b*B - a^2*C - 2*b^2*C)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/((a^2 - b^2)^(5/2)*d)) + (a*(b*B - a*C)*sin(c + d*x))/(2*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) + ((a^2*b*B + 2*b^3*B + a^3*C - 4*a*b^2*C)*sin(c + d*x))/(2*b*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))],
[sec(c + d*x)^1*(B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^3, x, 4, ((2*a^2*B + b^2*B - 3*a*b*C)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/((a^2 - b^2)^(5/2)*d) - ((b*B - a*C)*sin(c + d*x))/(2*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) - ((3*a*b*B - a^2*C - 2*b^2*C)*sin(c + d*x))/(2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))],
[sec(c + d*x)^2*(B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^3, x, 6, -(((6*a^4*b*B - 5*a^2*b^3*B + 2*b^5*B - 2*a^5*C - a^3*b^2*C)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^3*(a^2 - b^2)^(5/2)*d)) + (B*arctanh(sin(c + d*x)))/(a^3*d) + ((5*a^2*b*B - 2*b^3*B - 3*a^3*C)*tan(c + d*x))/(2*a^2*(a^2 - b^2)^2*d) - ((b*B - a*C)*tan(c + d*x))/(2*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) - ((4*a^2*b*B - b^3*B - 2*a^3*C - a*b^2*C)*tan(c + d*x))/(2*a*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))],
[sec(c + d*x)^3*(B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^3, x, 7, (b*(12*a^4*b*B - 15*a^2*b^3*B + 6*b^5*B - 6*a^5*C + 5*a^3*b^2*C - 2*a*b^4*C)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^4*(a^2 - b^2)^(5/2)*d) - ((3*b*B - a*C)*arctanh(sin(c + d*x)))/(a^4*d) + ((2*a^4*B - 11*a^2*b^2*B + 6*b^4*B + 5*a^3*b*C - 2*a*b^3*C)*tan(c + d*x))/(2*a^3*(a^2 - b^2)^2*d) + ((6*a^2*b*B - 3*b^3*B - 4*a^3*C + a*b^2*C)*sec(c + d*x)*tan(c + d*x))/(2*a^2*(a^2 - b^2)^2*d) - ((b*B - a*C)*sec(c + d*x)*tan(c + d*x))/(2*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) - ((5*a^2*b*B - 2*b^3*B - 3*a^3*C)*sec(c + d*x)*tan(c + d*x))/(2*a*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))],


# ::Subsection::Closed:: 
#Integrands of the form Cos[c+d x]^m (B Cos[c+d x]+C Cos[c+d x]^2) (a+b Cos[c+d x])^(n/2)


# ::Subsubsection::Closed:: 
#n>0


[cos(c + d*x)^1*(B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(1/2), x, 8, (-2*(a + b)*(14*a^2*b*B - 63*b^3*B - 8*a^3*C - 19*a*b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*b)/(a + b)))/(105*b^3*d*sqrt(a + b*cos(c + d*x))) + (2*(14*a^3*b*B - 14*a*b^3*B - 8*a^4*C - 17*a^2*b^2*C + 25*b^4*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*b)/(a + b)))/(105*b^3*d*sqrt(a + b*cos(c + d*x))) + (2*(7*a*b*B - 4*a^2*C + 25*b^2*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(105*b^2*d) + (2*(7*b*B + a*C)*cos(c + d*x)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(35*b*d) + (2*C*cos(c + d*x)^2*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(7*d)],
[cos(c + d*x)^0*(B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(1/2), x, 7, (2*(a + b)*(5*a*b*B - 2*a^2*C + 9*b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(15*b^2*d*sqrt(a + b*cos(c + d*x))) - (2*(5*a^2*b*B - 5*b^3*B - 2*a^3*C + 2*a*b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(15*b^2*d*sqrt(a + b*cos(c + d*x))) + (2*(5*b*B - 2*a*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(15*b*d) + (2*C*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(5*b*d)],
[sec(c + d*x)^1*(B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(1/2), x, 7, (2*(a + b)*(3*b*B + a*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(3*b*d*sqrt(a + b*cos(c + d*x))) - (2*(a^2 - b^2)*C*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(3*b*d*sqrt(a + b*cos(c + d*x))) + (2*C*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*d)],
[sec(c + d*x)^2*(B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(1/2), x, 9, (2*(a + b)*C*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x))) + (2*b*B*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x))) + (2*a*B*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x)))],
[sec(c + d*x)^3*(B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(1/2), x, 10, -(((a + b)*B*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x)))) + ((a*B + 2*b*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x))) + ((b*B + 2*a*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x))) + (B*sqrt(a + b*cos(c + d*x))*tan(c + d*x))/d],
[sec(c + d*x)^4*(B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(1/2), x, 11, -(((a + b)*(b*B + 4*a*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(4*a*d*sqrt(a + b*cos(c + d*x)))) + ((3*b*B + 4*a*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(4*d*sqrt(a + b*cos(c + d*x))) + ((4*a^2*B - b^2*B + 4*a*b*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*b)/(a + b)))/(4*a*d*sqrt(a + b*cos(c + d*x))) + ((b*B + 4*a*C)*sqrt(a + b*cos(c + d*x))*tan(c + d*x))/(4*a*d) + (B*sqrt(a + b*cos(c + d*x))*sec(c + d*x)*tan(c + d*x))/(2*d)],


[cos(c + d*x)^1*(B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(3/2), x, 9, (-2*(a + b)*(18*a^3*b*B - 246*a*b^3*B - 8*a^4*C - 33*a^2*b^2*C - 147*b^4*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*b)/(a + b)))/(315*b^3*d*sqrt(a + b*cos(c + d*x))) + (2*(18*a^4*b*B - 93*a^2*b^3*B + 75*b^5*B - 8*a^5*C - 31*a^3*b^2*C + 39*a*b^4*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*b)/(a + b)))/(315*b^3*d*sqrt(a + b*cos(c + d*x))) + (2*(9*a^2*b*B + 75*b^3*B - 4*a^3*C + 88*a*b^2*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(315*b^2*d) + (2*(72*a*b*B + 3*a^2*C + 49*b^2*C)*cos(c + d*x)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(315*b*d) + (2*(3*b*B + a*C)*cos(c + d*x)^2*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(21*d) + (2*C*cos(c + d*x)^2*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(9*d)],
[cos(c + d*x)^0*(B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(3/2), x, 8, (2*(a + b)*(21*a^2*b*B + 63*b^3*B - 6*a^3*C + 82*a*b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(105*b^2*d*sqrt(a + b*cos(c + d*x))) - (2*(21*a^3*b*B - 21*a*b^3*B - 6*a^4*C + 31*a^2*b^2*C - 25*b^4*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(105*b^2*d*sqrt(a + b*cos(c + d*x))) + (2*(21*a*b*B - 6*a^2*C + 25*b^2*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(105*b*d) + (2*(7*b*B - 2*a*C)*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(35*b*d) + (2*C*(a + b*cos(c + d*x))^(5/2)*sin(c + d*x))/(7*b*d)],
[sec(c + d*x)^1*(B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(3/2), x, 8, (2*(a + b)*(20*a*b*B + 3*a^2*C + 9*b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(15*b*d*sqrt(a + b*cos(c + d*x))) - (2*(5*a^2*b*B - 5*b^3*B + 3*a^3*C - 3*a*b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(15*b*d*sqrt(a + b*cos(c + d*x))) + (2*(5*b*B + 3*a*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(15*d) + (2*C*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(5*d)],
[sec(c + d*x)^2*(B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(3/2), x, 10, (2*(a + b)*(3*b*B + 4*a*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(3*d*sqrt(a + b*cos(c + d*x))) + (2*(3*a*b*B - a^2*C + b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(3*d*sqrt(a + b*cos(c + d*x))) + (2*a^2*B*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x))) + (2*b*C*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*d)],
[sec(c + d*x)^3*(B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(3/2), x, 10, -(((a + b)*(a*B - 2*b*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x)))) + ((a^2*B + 2*b^2*B + 2*a*b*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x))) + (a*(3*b*B + 2*a*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x))) + (a*B*sqrt(a + b*cos(c + d*x))*tan(c + d*x))/d],
[sec(c + d*x)^4*(B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(3/2), x, 11, -(((a + b)*(5*b*B + 4*a*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(4*d*sqrt(a + b*cos(c + d*x)))) + ((7*a*b*B + 4*a^2*C + 8*b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(4*d*sqrt(a + b*cos(c + d*x))) + ((4*a^2*B + 3*b^2*B + 12*a*b*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*b)/(a + b)))/(4*d*sqrt(a + b*cos(c + d*x))) + ((5*b*B + 4*a*C)*sqrt(a + b*cos(c + d*x))*tan(c + d*x))/(4*d) + (a*B*sqrt(a + b*cos(c + d*x))*sec(c + d*x)*tan(c + d*x))/(2*d)],
[sec(c + d*x)^5*(B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(3/2), x, 12, -(((a + b)*(16*a^2*B + 3*b^2*B + 30*a*b*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(24*a*d*sqrt(a + b*cos(c + d*x)))) + ((16*a^2*B + 17*b^2*B + 42*a*b*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(24*d*sqrt(a + b*cos(c + d*x))) + ((12*a^2*b*B - b^3*B + 8*a^3*C + 6*a*b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*b)/(a + b)))/(8*a*d*sqrt(a + b*cos(c + d*x))) + ((16*a^2*B + 3*b^2*B + 30*a*b*C)*sqrt(a + b*cos(c + d*x))*tan(c + d*x))/(24*a*d) + ((7*b*B + 6*a*C)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)*tan(c + d*x))/(12*d) + (a*B*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^2*tan(c + d*x))/(3*d)],


[cos(c + d*x)^1*(B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(5/2), x, 10, (-2*(a + b)*(110*a^4*b*B - 3069*a^2*b^3*B - 1617*b^5*B - 40*a^5*C - 255*a^3*b^2*C - 3705*a*b^4*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*b)/(a + b)))/(3465*b^3*d*sqrt(a + b*cos(c + d*x))) + (2*(110*a^5*b*B - 1364*a^3*b^3*B + 1254*a*b^5*B - 40*a^6*C - 245*a^4*b^2*C - 390*a^2*b^4*C + 675*b^6*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*b)/(a + b)))/(3465*b^3*d*sqrt(a + b*cos(c + d*x))) + (2*(55*a^3*b*B + 1793*a*b^3*B - 20*a^4*C + 1025*a^2*b^2*C + 675*b^4*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3465*b^2*d) + (2*(825*a^2*b*B + 539*b^3*B + 15*a^3*C + 1145*a*b^2*C)*cos(c + d*x)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3465*b*d) + (2*(44*a*b*B + 5*a^2*C + 27*b^2*C)*cos(c + d*x)^2*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(231*d) + (2*(11*b*B + 5*a*C)*cos(c + d*x)^2*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(99*d) + (2*C*cos(c + d*x)^2*(a + b*cos(c + d*x))^(5/2)*sin(c + d*x))/(11*d)],
[cos(c + d*x)^0*(B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(5/2), x, 9, (2*(a + b)*(45*a^3*b*B + 435*a*b^3*B - 10*a^4*C + 279*a^2*b^2*C + 147*b^4*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(315*b^2*d*sqrt(a + b*cos(c + d*x))) - (2*(45*a^4*b*B + 30*a^2*b^3*B - 75*b^5*B - 10*a^5*C + 124*a^3*b^2*C - 114*a*b^4*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(315*b^2*d*sqrt(a + b*cos(c + d*x))) + (2*(45*a^2*b*B + 75*b^3*B - 10*a^3*C + 114*a*b^2*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(315*b*d) + (2*(45*a*b*B - 10*a^2*C + 49*b^2*C)*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(315*b*d) + (2*(9*b*B - 2*a*C)*(a + b*cos(c + d*x))^(5/2)*sin(c + d*x))/(63*b*d) + (2*C*(a + b*cos(c + d*x))^(7/2)*sin(c + d*x))/(9*b*d)],
[sec(c + d*x)^1*(B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(5/2), x, 9, (2*(a + b)*(161*a^2*b*B + 63*b^3*B + 15*a^3*C + 145*a*b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(105*b*d*sqrt(a + b*cos(c + d*x))) - (2*(56*a^3*b*B - 56*a*b^3*B + 15*a^4*C + 10*a^2*b^2*C - 25*b^4*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(105*b*d*sqrt(a + b*cos(c + d*x))) + (2*(56*a*b*B + 15*a^2*C + 25*b^2*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(105*d) + (2*(7*b*B + 5*a*C)*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(35*d) + (2*C*(a + b*cos(c + d*x))^(5/2)*sin(c + d*x))/(7*d)],
[sec(c + d*x)^2*(B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(5/2), x, 11, (2*(a + b)*(35*a*b*B + 23*a^2*C + 9*b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(15*d*sqrt(a + b*cos(c + d*x))) + (2*(10*a^2*b*B + 5*b^3*B - 8*a^3*C + 8*a*b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(15*d*sqrt(a + b*cos(c + d*x))) + (2*a^3*B*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x))) + (2*b*(5*b*B + 8*a*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(15*d) + (2*b*C*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(5*d)],
[sec(c + d*x)^3*(B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(5/2), x, 11, -(((a + b)*(3*a^2*B - 6*b^2*B - 14*a*b*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(3*d*sqrt(a + b*cos(c + d*x)))) + ((3*a^3*B + 12*a*b^2*B + 4*a^2*b*C + 2*b^3*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(3*d*sqrt(a + b*cos(c + d*x))) + (a^2*(5*b*B + 2*a*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x))) - (b*(3*a*B - 2*b*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*d) + (a*B*(a + b*cos(c + d*x))^(3/2)*tan(c + d*x))/d],
[sec(c + d*x)^4*(B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(5/2), x, 11, -(((a + b)*(9*a*b*B + 4*a^2*C - 8*b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(4*d*sqrt(a + b*cos(c + d*x)))) + ((11*a^2*b*B + 8*b^3*B + 4*a^3*C + 16*a*b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(4*d*sqrt(a + b*cos(c + d*x))) + (a*(4*a^2*B + 15*b^2*B + 20*a*b*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*b)/(a + b)))/(4*d*sqrt(a + b*cos(c + d*x))) + (a*(7*b*B + 4*a*C)*sqrt(a + b*cos(c + d*x))*tan(c + d*x))/(4*d) + (a*B*(a + b*cos(c + d*x))^(3/2)*sec(c + d*x)*tan(c + d*x))/(2*d)],
[sec(c + d*x)^5*(B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(5/2), x, 12, -(((a + b)*(16*a^2*B + 33*b^2*B + 54*a*b*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(24*d*sqrt(a + b*cos(c + d*x)))) + ((16*a^3*B + 59*a*b^2*B + 66*a^2*b*C + 48*b^3*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(24*d*sqrt(a + b*cos(c + d*x))) + ((20*a^2*b*B + 5*b^3*B + 8*a^3*C + 30*a*b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*b)/(a + b)))/(8*d*sqrt(a + b*cos(c + d*x))) + ((16*a^2*B + 33*b^2*B + 54*a*b*C)*sqrt(a + b*cos(c + d*x))*tan(c + d*x))/(24*d) + (a*(3*b*B + 2*a*C)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)*tan(c + d*x))/(4*d) + (a*B*(a + b*cos(c + d*x))^(3/2)*sec(c + d*x)^2*tan(c + d*x))/(3*d)],
[sec(c + d*x)^6*(B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(5/2), x, 13, -(((a + b)*(284*a^2*b*B + 15*b^3*B + 128*a^3*C + 264*a*b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(192*a*d*sqrt(a + b*cos(c + d*x)))) + ((356*a^2*b*B + 133*b^3*B + 128*a^3*C + 472*a*b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(192*d*sqrt(a + b*cos(c + d*x))) + ((48*a^4*B + 120*a^2*b^2*B - 5*b^4*B + 160*a^3*b*C + 40*a*b^3*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*b)/(a + b)))/(64*a*d*sqrt(a + b*cos(c + d*x))) + ((284*a^2*b*B + 15*b^3*B + 128*a^3*C + 264*a*b^2*C)*sqrt(a + b*cos(c + d*x))*tan(c + d*x))/(192*a*d) + ((36*a^2*B + 59*b^2*B + 104*a*b*C)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)*tan(c + d*x))/(96*d) + (a*(11*b*B + 8*a*C)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^2*tan(c + d*x))/(24*d) + (a*B*(a + b*cos(c + d*x))^(3/2)*sec(c + d*x)^3*tan(c + d*x))/(4*d)],


# ::Subsubsection::Closed:: 
#n<0


[cos(c + d*x)^1*(B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(1/2), x, 7, (-2*(a + b)*(10*a*b*B - 8*a^2*C - 9*b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*b)/(a + b)))/(15*b^3*d*sqrt(a + b*cos(c + d*x))) + (2*(10*a^2*b*B + 5*b^3*B - 8*a^3*C - 7*a*b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*b)/(a + b)))/(15*b^3*d*sqrt(a + b*cos(c + d*x))) + (2*(5*b*B - 4*a*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(15*b^2*d) + (2*C*cos(c + d*x)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(5*b*d)],
[cos(c + d*x)^0*(B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(1/2), x, 6, (2*(a + b)*(3*b*B - 2*a*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(3*b^2*d*sqrt(a + b*cos(c + d*x))) - (2*(3*a*b*B - 2*a^2*C - b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(3*b^2*d*sqrt(a + b*cos(c + d*x))) + (2*C*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*b*d)],
[sec(c + d*x)^1*(B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(1/2), x, 6, (2*(a + b)*C*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(b*d*sqrt(a + b*cos(c + d*x))) + (2*(b*B - a*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(b*d*sqrt(a + b*cos(c + d*x)))],
[sec(c + d*x)^2*(B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(1/2), x, 6, (2*C*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x))) + (2*B*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x)))],
[sec(c + d*x)^3*(B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(1/2), x, 10, -(((a + b)*B*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(a*d*sqrt(a + b*cos(c + d*x)))) + (B*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x))) - ((b*B - 2*a*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*b)/(a + b)))/(a*d*sqrt(a + b*cos(c + d*x))) + (B*sqrt(a + b*cos(c + d*x))*tan(c + d*x))/(a*d)],
[sec(c + d*x)^4*(B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(1/2), x, 11, ((a + b)*(3*b*B - 4*a*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(4*a^2*d*sqrt(a + b*cos(c + d*x))) - ((b*B - 4*a*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(4*a*d*sqrt(a + b*cos(c + d*x))) + ((4*a^2*B + 3*b^2*B - 4*a*b*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*b)/(a + b)))/(4*a^2*d*sqrt(a + b*cos(c + d*x))) - ((3*b*B - 4*a*C)*sqrt(a + b*cos(c + d*x))*tan(c + d*x))/(4*a^2*d) + (B*sqrt(a + b*cos(c + d*x))*sec(c + d*x)*tan(c + d*x))/(2*a*d)],


[cos(c + d*x)^2*(B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(3/2), x, 8, (-2*(40*a^3*b*B - 25*a*b^3*B - 48*a^4*C + 24*a^2*b^2*C + 9*b^4*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*b)/(a + b)))/(15*(a - b)*b^4*d*sqrt(a + b*cos(c + d*x))) + (2*(40*a^2*b*B + 5*b^3*B - 48*a^3*C - 12*a*b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*b)/(a + b)))/(15*b^4*d*sqrt(a + b*cos(c + d*x))) + (2*a*(b*B - a*C)*cos(c + d*x)^2*sin(c + d*x))/(b*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) + (2*(20*a^2*b*B - 5*b^3*B - 24*a^3*C + 9*a*b^2*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(15*b^3*(a^2 - b^2)*d) - (2*(5*a*b*B - 6*a^2*C + b^2*C)*cos(c + d*x)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(5*b^2*(a^2 - b^2)*d)],
[cos(c + d*x)^1*(B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(3/2), x, 7, (2*(6*a^2*b*B - 3*b^3*B - 8*a^3*C + 5*a*b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*b)/(a + b)))/(3*(a - b)*b^3*d*sqrt(a + b*cos(c + d*x))) - (2*(6*a*b*B - 8*a^2*C - b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*b)/(a + b)))/(3*b^3*d*sqrt(a + b*cos(c + d*x))) + (2*a*(b*B - a*C)*cos(c + d*x)*sin(c + d*x))/(b*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) - (2*(3*a*b*B - 4*a^2*C + b^2*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*b^2*(a^2 - b^2)*d)],
[cos(c + d*x)^0*(B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(3/2), x, 6, -((2*(a*b*B - 2*a^2*C + b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/((a - b)*b^2*d*sqrt(a + b*cos(c + d*x)))) + (2*(b*B - 2*a*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(b^2*d*sqrt(a + b*cos(c + d*x))) + (2*a*(b*B - a*C)*sin(c + d*x))/(b*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x)))],
[sec(c + d*x)^1*(B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(3/2), x, 7, (2*(b*B - a*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/((a - b)*b*d*sqrt(a + b*cos(c + d*x))) + (2*C*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(b*d*sqrt(a + b*cos(c + d*x))) - (2*(b*B - a*C)*sin(c + d*x))/((a^2 - b^2)*d*sqrt(a + b*cos(c + d*x)))],
[sec(c + d*x)^2*(B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(3/2), x, 8, -((2*(b*B - a*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(a*(a - b)*d*sqrt(a + b*cos(c + d*x)))) + (2*B*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*b)/(a + b)))/(a*d*sqrt(a + b*cos(c + d*x))) - (2*(b*B - a*C)*tan(c + d*x))/((a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) + (2*(b*B - a*C)*sqrt(a + b*cos(c + d*x))*tan(c + d*x))/(a*(a^2 - b^2)*d)],
[sec(c + d*x)^3*(B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(3/2), x, 11, -(((a^2*B - 3*b^2*B + 2*a*b*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(a^2*(a - b)*d*sqrt(a + b*cos(c + d*x)))) + (B*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(a*d*sqrt(a + b*cos(c + d*x))) - ((3*b*B - 2*a*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*b)/(a + b)))/(a^2*d*sqrt(a + b*cos(c + d*x))) + ((a^2*B - 3*b^2*B + 2*a*b*C)*sqrt(a + b*cos(c + d*x))*tan(c + d*x))/(a^2*(a^2 - b^2)*d) - (2*(b*B - a*C)*sec(c + d*x)*tan(c + d*x))/((a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) + (2*(b*B - a*C)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)*tan(c + d*x))/(a*(a^2 - b^2)*d)],


[cos(c + d*x)^2*(B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(5/2), x, 8, (2*(8*a^4*b*B - 15*a^2*b^3*B + 3*b^5*B - 16*a^5*C + 28*a^3*b^2*C - 8*a*b^4*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*b)/(a + b)))/(3*(a - b)^2*b^4*(a + b)*d*sqrt(a + b*cos(c + d*x))) - (2*(8*a^3*b*B - 9*a*b^3*B - 16*a^4*C + 16*a^2*b^2*C + b^4*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*b)/(a + b)))/(3*b^4*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) + (2*a*(b*B - a*C)*cos(c + d*x)^2*sin(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)) + (2*a*(3*a^2*b*B - 7*b^3*B - 6*a^3*C + 10*a*b^2*C)*cos(c + d*x)*sin(c + d*x))/(3*b^2*(a^2 - b^2)^2*d*sqrt(a + b*cos(c + d*x))) - (2*(4*a^3*b*B - 8*a*b^3*B - 8*a^4*C + 13*a^2*b^2*C - b^4*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*b^3*(a^2 - b^2)^2*d)],
[cos(c + d*x)^1*(B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(5/2), x, 7, (-2*(2*a^3*b*B - 6*a*b^3*B - 8*a^4*C + 15*a^2*b^2*C - 3*b^4*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*b)/(a + b)))/(3*(a - b)^2*b^3*(a + b)*d*sqrt(a + b*cos(c + d*x))) + (2*(2*a^2*b*B - 3*b^3*B - 8*a^3*C + 9*a*b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*b)/(a + b)))/(3*b^3*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) + (2*a*(b*B - a*C)*cos(c + d*x)*sin(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)) + (2*a*(a^2*b*B - 5*b^3*B - 4*a^3*C + 8*a*b^2*C)*sin(c + d*x))/(3*b^2*(a^2 - b^2)^2*d*sqrt(a + b*cos(c + d*x)))],
[cos(c + d*x)^0*(B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(5/2), x, 7, -((2*(a^2*b*B + 3*b^3*B + 2*a^3*C - 6*a*b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(3*(a - b)^2*b^2*(a + b)*d*sqrt(a + b*cos(c + d*x)))) + (2*(a*b*B + 2*a^2*C - 3*b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(3*b^2*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) + (2*a*(b*B - a*C)*sin(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)) + (2*(a^2*b*B + 3*b^3*B + 2*a^3*C - 6*a*b^2*C)*sin(c + d*x))/(3*b*(a^2 - b^2)^2*d*sqrt(a + b*cos(c + d*x)))],
[sec(c + d*x)^1*(B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(5/2), x, 8, (2*(4*a*b*B - a^2*C - 3*b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(3*(a - b)^2*b*(a + b)*d*sqrt(a + b*cos(c + d*x))) - (2*(b*B - a*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(3*b*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) - (2*(b*B - a*C)*sin(c + d*x))/(3*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)) - (2*(4*a*b*B - a^2*C - 3*b^2*C)*sin(c + d*x))/(3*(a^2 - b^2)^2*d*sqrt(a + b*cos(c + d*x)))],
[sec(c + d*x)^2*(B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(5/2), x, 11, -((2*(7*a^2*b*B - 3*b^3*B - 4*a^3*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(3*a^2*(a - b)^2*(a + b)*d*sqrt(a + b*cos(c + d*x)))) + (2*(b*B - a*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(3*a*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) + (2*B*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*b)/(a + b)))/(a^2*d*sqrt(a + b*cos(c + d*x))) - (2*(b*B - a*C)*tan(c + d*x))/(3*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)) - (2*(6*a^2*b*B - 2*b^3*B - 3*a^3*C - a*b^2*C)*tan(c + d*x))/(3*a*(a^2 - b^2)^2*d*sqrt(a + b*cos(c + d*x))) + (2*(7*a^2*b*B - 3*b^3*B - 4*a^3*C)*sqrt(a + b*cos(c + d*x))*tan(c + d*x))/(3*a^2*(a^2 - b^2)^2*d)],
[sec(c + d*x)^3*(B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(5/2), x, 12, -(((3*a^4*B - 26*a^2*b^2*B + 15*b^4*B + 14*a^3*b*C - 6*a*b^3*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(3*a^3*(a - b)^2*(a + b)*d*sqrt(a + b*cos(c + d*x)))) + ((3*a^2*B - 5*b^2*B + 2*a*b*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(3*a^2*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) - ((5*b*B - 2*a*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*b)/(a + b)))/(a^3*d*sqrt(a + b*cos(c + d*x))) + ((3*a^4*B - 26*a^2*b^2*B + 15*b^4*B + 14*a^3*b*C - 6*a*b^3*C)*sqrt(a + b*cos(c + d*x))*tan(c + d*x))/(3*a^3*(a^2 - b^2)^2*d) - (2*(b*B - a*C)*sec(c + d*x)*tan(c + d*x))/(3*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)) - (2*(8*a^2*b*B - 4*b^3*B - 5*a^3*C + a*b^2*C)*sec(c + d*x)*tan(c + d*x))/(3*a*(a^2 - b^2)^2*d*sqrt(a + b*cos(c + d*x))) + (2*(9*a^2*b*B - 5*b^3*B - 6*a^3*C + 2*a*b^2*C)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)*tan(c + d*x))/(3*a^2*(a^2 - b^2)^2*d)],


# ::Subsection::Closed:: 
#Integrands of the form Cos[c+d x]^(m/2) (B Cos[c+d x]+C Cos[c+d x]^2) (a+b Cos[c+d x])^n


# ::Subsubsection::Closed:: 
#n>0


[cos(c + d*x)^(3/2)*(a + b*cos(c + d*x))*(B*cos(c + d*x) + C*cos(c + d*x)^2), x, 7, (2*(9*a*B + 7*b*C)*EllipticE((c + d*x)/2, 2))/(15*d) + (10*(b*B + a*C)*EllipticF((c + d*x)/2, 2))/(21*d) + (10*(b*B + a*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(21*d) + (2*(9*a*B + 7*b*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(45*d) + (2*(9*b*B + 2*a*C)*cos(c + d*x)^(5/2)*sin(c + d*x))/(63*d) + (2*C*cos(c + d*x)^(5/2)*(a + b*cos(c + d*x))*sin(c + d*x))/(9*d)],
[sqrt(cos(c + d*x))*(a + b*cos(c + d*x))*(B*cos(c + d*x) + C*cos(c + d*x)^2), x, 6, (6*(b*B + a*C)*EllipticE((c + d*x)/2, 2))/(5*d) + (2*(7*a*B + 5*b*C)*EllipticF((c + d*x)/2, 2))/(21*d) + (2*(7*a*B + 5*b*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(21*d) + (2*(7*b*B + 2*a*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(35*d) + (2*C*cos(c + d*x)^(3/2)*(a + b*cos(c + d*x))*sin(c + d*x))/(7*d)],
[((a + b*cos(c + d*x))*(B*cos(c + d*x) + C*cos(c + d*x)^2))/sqrt(cos(c + d*x)), x, 5, (2*(5*a*B + 3*b*C)*EllipticE((c + d*x)/2, 2))/(5*d) + (2*(b*B + a*C)*EllipticF((c + d*x)/2, 2))/(3*d) + (2*(5*b*B + 2*a*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(15*d) + (2*C*sqrt(cos(c + d*x))*(a + b*cos(c + d*x))*sin(c + d*x))/(5*d)],
[((a + b*cos(c + d*x))*(B*cos(c + d*x) + C*cos(c + d*x)^2))/cos(c + d*x)^(3/2), x, 5, (2*(b*B + a*C)*EllipticE((c + d*x)/2, 2))/d + (2*(3*a*B + b*C)*EllipticF((c + d*x)/2, 2))/(3*d) + (2*b*C*sqrt(cos(c + d*x))*sin(c + d*x))/(3*d)],
[((a + b*cos(c + d*x))*(B*cos(c + d*x) + C*cos(c + d*x)^2))/cos(c + d*x)^(5/2), x, 5, (-2*(a*B - b*C)*EllipticE((c + d*x)/2, 2))/d + (2*(b*B + a*C)*EllipticF((c + d*x)/2, 2))/d + (2*a*B*sin(c + d*x))/(d*sqrt(cos(c + d*x)))],
[((a + b*cos(c + d*x))*(B*cos(c + d*x) + C*cos(c + d*x)^2))/cos(c + d*x)^(7/2), x, 6, (-2*(b*B + a*C)*EllipticE((c + d*x)/2, 2))/d + (2*(a*B + 3*b*C)*EllipticF((c + d*x)/2, 2))/(3*d) + (2*a*B*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2)) + (2*(b*B + a*C)*sin(c + d*x))/(d*sqrt(cos(c + d*x)))],
[((a + b*cos(c + d*x))*(B*cos(c + d*x) + C*cos(c + d*x)^2))/cos(c + d*x)^(9/2), x, 7, (-2*(3*a*B + 5*b*C)*EllipticE((c + d*x)/2, 2))/(5*d) + (2*(b*B + a*C)*EllipticF((c + d*x)/2, 2))/(3*d) + (2*a*B*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2)) + (2*(b*B + a*C)*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2)) + (2*(3*a*B + 5*b*C)*sin(c + d*x))/(5*d*sqrt(cos(c + d*x)))],


[cos(c + d*x)^(3/2)*(a + b*cos(c + d*x))^2*(B*cos(c + d*x) + C*cos(c + d*x)^2), x, 8, (2*(9*a^2*B + 7*b^2*B + 14*a*b*C)*EllipticE((c + d*x)/2, 2))/(15*d) + (10*(22*a*b*B + 11*a^2*C + 9*b^2*C)*EllipticF((c + d*x)/2, 2))/(231*d) + (10*(22*a*b*B + 11*a^2*C + 9*b^2*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(231*d) + (2*(9*a^2*B + 7*b^2*B + 14*a*b*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(45*d) + (2*(121*a*b*B + 8*a^2*C + 81*b^2*C)*cos(c + d*x)^(5/2)*sin(c + d*x))/(693*d) + (2*(11*b*B + 4*a*C)*cos(c + d*x)^(5/2)*(a + b*cos(c + d*x))*sin(c + d*x))/(99*d) + (2*C*cos(c + d*x)^(5/2)*(a + b*cos(c + d*x))^2*sin(c + d*x))/(11*d)],
[sqrt(cos(c + d*x))*(a + b*cos(c + d*x))^2*(B*cos(c + d*x) + C*cos(c + d*x)^2), x, 7, (2*(18*a*b*B + 9*a^2*C + 7*b^2*C)*EllipticE((c + d*x)/2, 2))/(15*d) + (2*(7*a^2*B + 5*b^2*B + 10*a*b*C)*EllipticF((c + d*x)/2, 2))/(21*d) + (2*(7*a^2*B + 5*b^2*B + 10*a*b*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(21*d) + (2*(81*a*b*B + 8*a^2*C + 49*b^2*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(315*d) + (2*(9*b*B + 4*a*C)*cos(c + d*x)^(3/2)*(a + b*cos(c + d*x))*sin(c + d*x))/(63*d) + (2*C*cos(c + d*x)^(3/2)*(a + b*cos(c + d*x))^2*sin(c + d*x))/(9*d)],
[((a + b*cos(c + d*x))^2*(B*cos(c + d*x) + C*cos(c + d*x)^2))/sqrt(cos(c + d*x)), x, 6, (2*(5*a^2*B + 3*b^2*B + 6*a*b*C)*EllipticE((c + d*x)/2, 2))/(5*d) + (2*(14*a*b*B + 7*a^2*C + 5*b^2*C)*EllipticF((c + d*x)/2, 2))/(21*d) + (2*(49*a*b*B + 8*a^2*C + 25*b^2*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(105*d) + (2*(7*b*B + 4*a*C)*sqrt(cos(c + d*x))*(a + b*cos(c + d*x))*sin(c + d*x))/(35*d) + (2*C*sqrt(cos(c + d*x))*(a + b*cos(c + d*x))^2*sin(c + d*x))/(7*d)],
[((a + b*cos(c + d*x))^2*(B*cos(c + d*x) + C*cos(c + d*x)^2))/cos(c + d*x)^(3/2), x, 6, (2*(10*a*b*B + 5*a^2*C + 3*b^2*C)*EllipticE((c + d*x)/2, 2))/(5*d) + (2*(3*a^2*B + b^2*B + 2*a*b*C)*EllipticF((c + d*x)/2, 2))/(3*d) + (2*b*(5*b*B + 7*a*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(15*d) + (2*b*C*sqrt(cos(c + d*x))*(a + b*cos(c + d*x))*sin(c + d*x))/(5*d)],
[((a + b*cos(c + d*x))^2*(B*cos(c + d*x) + C*cos(c + d*x)^2))/cos(c + d*x)^(5/2), x, 6, (-2*(a^2*B - b^2*B - 2*a*b*C)*EllipticE((c + d*x)/2, 2))/d + (2*(6*a*b*B + 3*a^2*C + b^2*C)*EllipticF((c + d*x)/2, 2))/(3*d) - (2*b*(3*a*B - b*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(3*d) + (2*a*B*(a + b*cos(c + d*x))*sin(c + d*x))/(d*sqrt(cos(c + d*x)))],
[((a + b*cos(c + d*x))^2*(B*cos(c + d*x) + C*cos(c + d*x)^2))/cos(c + d*x)^(7/2), x, 6, (-2*(2*a*b*B + a^2*C - b^2*C)*EllipticE((c + d*x)/2, 2))/d + (2*(a^2*B + 3*b^2*B + 6*a*b*C)*EllipticF((c + d*x)/2, 2))/(3*d) + (2*a*(5*b*B + 3*a*C)*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))) + (2*a*B*(a + b*cos(c + d*x))*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2))],
[((a + b*cos(c + d*x))^2*(B*cos(c + d*x) + C*cos(c + d*x)^2))/cos(c + d*x)^(9/2), x, 7, (-2*(3*a^2*B + 5*b^2*B + 10*a*b*C)*EllipticE((c + d*x)/2, 2))/(5*d) + (2*(2*a*b*B + a^2*C + 3*b^2*C)*EllipticF((c + d*x)/2, 2))/(3*d) + (2*a*(7*b*B + 5*a*C)*sin(c + d*x))/(15*d*cos(c + d*x)^(3/2)) + (2*(3*a^2*B + 5*b^2*B + 10*a*b*C)*sin(c + d*x))/(5*d*sqrt(cos(c + d*x))) + (2*a*B*(a + b*cos(c + d*x))*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2))],
[((a + b*cos(c + d*x))^2*(B*cos(c + d*x) + C*cos(c + d*x)^2))/cos(c + d*x)^(11/2), x, 8, -((2*(6*a*b*B + 3*a^2*C + 5*b^2*C)*EllipticE((1/2)*(c + d*x), 2))/(5*d)) + (2*(5*a^2*B + 7*b^2*B + 14*a*b*C)*EllipticF((1/2)*(c + d*x), 2))/(21*d) + (2*a*(9*b*B + 7*a*C)*sin(c + d*x))/(35*d*cos(c + d*x)^(5/2)) + (2*(5*a^2*B + 7*b^2*B + 14*a*b*C)*sin(c + d*x))/(21*d*cos(c + d*x)^(3/2)) + (2*(6*a*b*B + 3*a^2*C + 5*b^2*C)*sin(c + d*x))/(5*d*sqrt(cos(c + d*x))) + (2*a*B*(a + b*cos(c + d*x))*sin(c + d*x))/(7*d*cos(c + d*x)^(7/2))],


[sqrt(cos(c + d*x))*(a + b*cos(c + d*x))^3*(B*cos(c + d*x) + C*cos(c + d*x)^2), x, 8, (2*(27*a^2*b*B + 7*b^3*B + 9*a^3*C + 21*a*b^2*C)*EllipticE((c + d*x)/2, 2))/(15*d) + (2*(77*a^3*B + 165*a*b^2*B + 165*a^2*b*C + 45*b^3*C)*EllipticF((c + d*x)/2, 2))/(231*d) + (2*(77*a^3*B + 165*a*b^2*B + 165*a^2*b*C + 45*b^3*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(231*d) + (2*(979*a^2*b*B + 539*b^3*B + 48*a^3*C + 1212*a*b^2*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(3465*d) + (2*(143*a*b*B + 24*a^2*C + 81*b^2*C)*cos(c + d*x)^(3/2)*(a + b*cos(c + d*x))*sin(c + d*x))/(693*d) + (2*(11*b*B + 6*a*C)*cos(c + d*x)^(3/2)*(a + b*cos(c + d*x))^2*sin(c + d*x))/(99*d) + (2*C*cos(c + d*x)^(3/2)*(a + b*cos(c + d*x))^3*sin(c + d*x))/(11*d)],
[((a + b*cos(c + d*x))^3*(B*cos(c + d*x) + C*cos(c + d*x)^2))/sqrt(cos(c + d*x)), x, 7, (2*(15*a^3*B + 27*a*b^2*B + 27*a^2*b*C + 7*b^3*C)*EllipticE((c + d*x)/2, 2))/(15*d) + (2*(21*a^2*b*B + 5*b^3*B + 7*a^3*C + 15*a*b^2*C)*EllipticF((c + d*x)/2, 2))/(21*d) + (2*(171*a^2*b*B + 75*b^3*B + 16*a^3*C + 176*a*b^2*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(315*d) + (2*(99*a*b*B + 24*a^2*C + 49*b^2*C)*sqrt(cos(c + d*x))*(a + b*cos(c + d*x))*sin(c + d*x))/(315*d) + (2*(3*b*B + 2*a*C)*sqrt(cos(c + d*x))*(a + b*cos(c + d*x))^2*sin(c + d*x))/(21*d) + (2*C*sqrt(cos(c + d*x))*(a + b*cos(c + d*x))^3*sin(c + d*x))/(9*d)],
[((a + b*cos(c + d*x))^3*(B*cos(c + d*x) + C*cos(c + d*x)^2))/cos(c + d*x)^(3/2), x, 7, (2*(15*a^2*b*B + 3*b^3*B + 5*a^3*C + 9*a*b^2*C)*EllipticE((c + d*x)/2, 2))/(5*d) + (2*(21*a^3*B + 21*a*b^2*B + 21*a^2*b*C + 5*b^3*C)*EllipticF((c + d*x)/2, 2))/(21*d) + (2*b*(84*a*b*B + 57*a^2*C + 25*b^2*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(105*d) + (2*b*(7*b*B + 11*a*C)*sqrt(cos(c + d*x))*(a + b*cos(c + d*x))*sin(c + d*x))/(35*d) + (2*b*C*sqrt(cos(c + d*x))*(a + b*cos(c + d*x))^2*sin(c + d*x))/(7*d)],
[((a + b*cos(c + d*x))^3*(B*cos(c + d*x) + C*cos(c + d*x)^2))/cos(c + d*x)^(5/2), x, 7, (-2*(5*a^3*B - 15*a*b^2*B - 15*a^2*b*C - 3*b^3*C)*EllipticE((c + d*x)/2, 2))/(5*d) + (2*(9*a^2*b*B + b^3*B + 3*a^3*C + 3*a*b^2*C)*EllipticF((c + d*x)/2, 2))/(3*d) - (2*b*(15*a^2*B - 5*b^2*B - 12*a*b*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(15*d) - (2*b*(5*a*B - b*C)*sqrt(cos(c + d*x))*(a + b*cos(c + d*x))*sin(c + d*x))/(5*d) + (2*a*B*(a + b*cos(c + d*x))^2*sin(c + d*x))/(d*sqrt(cos(c + d*x)))],
[((a + b*cos(c + d*x))^3*(B*cos(c + d*x) + C*cos(c + d*x)^2))/cos(c + d*x)^(7/2), x, 7, (-2*(3*a^2*b*B - b^3*B + a^3*C - 3*a*b^2*C)*EllipticE((c + d*x)/2, 2))/d + (2*(a^3*B + 9*a*b^2*B + 9*a^2*b*C + b^3*C)*EllipticF((c + d*x)/2, 2))/(3*d) - (2*b*(8*a*b*B + 3*a^2*C - b^2*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(3*d) + (2*a*(7*b*B + 3*a*C)*(a + b*cos(c + d*x))*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))) + (2*a*B*(a + b*cos(c + d*x))^2*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2))],
[((a + b*cos(c + d*x))^3*(B*cos(c + d*x) + C*cos(c + d*x)^2))/cos(c + d*x)^(9/2), x, 7, (-2*(3*a^3*B + 15*a*b^2*B + 15*a^2*b*C - 5*b^3*C)*EllipticE((c + d*x)/2, 2))/(5*d) + (2*(3*a^2*b*B + 3*b^3*B + a^3*C + 9*a*b^2*C)*EllipticF((c + d*x)/2, 2))/(3*d) + (2*a*(9*a^2*B + 33*b^2*B + 40*a*b*C)*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))) + (2*a*(9*b*B + 5*a*C)*(a + b*cos(c + d*x))*sin(c + d*x))/(15*d*cos(c + d*x)^(3/2)) + (2*a*B*(a + b*cos(c + d*x))^2*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2))],
[((a + b*cos(c + d*x))^3*(B*cos(c + d*x) + C*cos(c + d*x)^2))/cos(c + d*x)^(11/2), x, 8, (-2*(9*a^2*b*B + 5*b^3*B + 3*a^3*C + 15*a*b^2*C)*EllipticE((c + d*x)/2, 2))/(5*d) + (2*(5*a^3*B + 21*a*b^2*B + 21*a^2*b*C + 21*b^3*C)*EllipticF((c + d*x)/2, 2))/(21*d) + (2*a*(25*a^2*B + 57*b^2*B + 84*a*b*C)*sin(c + d*x))/(105*d*cos(c + d*x)^(3/2)) + (2*(9*a^2*b*B + 5*b^3*B + 3*a^3*C + 15*a*b^2*C)*sin(c + d*x))/(5*d*sqrt(cos(c + d*x))) + (2*a*(11*b*B + 7*a*C)*(a + b*cos(c + d*x))*sin(c + d*x))/(35*d*cos(c + d*x)^(5/2)) + (2*a*B*(a + b*cos(c + d*x))^2*sin(c + d*x))/(7*d*cos(c + d*x)^(7/2))],
[((a + b*cos(c + d*x))^3*(B*cos(c + d*x) + C*cos(c + d*x)^2))/cos(c + d*x)^(13/2), x, 9, -((2*(7*a^3*B + 27*a*b^2*B + 27*a^2*b*C + 15*b^3*C)*EllipticE((1/2)*(c + d*x), 2))/(15*d)) + (2*(15*a^2*b*B + 7*b^3*B + 5*a^3*C + 21*a*b^2*C)*EllipticF((1/2)*(c + d*x), 2))/(21*d) + (2*a*(49*a^2*B + 89*b^2*B + 144*a*b*C)*sin(c + d*x))/(315*d*cos(c + d*x)^(5/2)) + (2*(15*a^2*b*B + 7*b^3*B + 5*a^3*C + 21*a*b^2*C)*sin(c + d*x))/(21*d*cos(c + d*x)^(3/2)) + (2*(7*a^3*B + 27*a*b^2*B + 27*a^2*b*C + 15*b^3*C)*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))) + (2*a*(13*b*B + 9*a*C)*(a + b*cos(c + d*x))*sin(c + d*x))/(63*d*cos(c + d*x)^(7/2)) + (2*a*B*(a + b*cos(c + d*x))^2*sin(c + d*x))/(9*d*cos(c + d*x)^(9/2))],


# ::Subsubsection::Closed:: 
#n<0


[cos(c + d*x)^(5/2)*(B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x)), x, 8, (2*(5*a^2*b*B + 3*b^3*B - 5*a^3*C - 3*a*b^2*C)*EllipticE((1/2)*(c + d*x), 2))/(5*b^4*d) - (2*(21*a^3*b*B + 7*a*b^3*B - 21*a^4*C - 7*a^2*b^2*C - 5*b^4*C)*EllipticF((1/2)*(c + d*x), 2))/(21*b^5*d) + (2*a^4*(b*B - a*C)*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2))/(b^5*(a + b)*d) - (2*(7*a*b*B - 7*a^2*C - 5*b^2*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(21*b^3*d) + (2*(b*B - a*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(5*b^2*d) + (2*C*cos(c + d*x)^(5/2)*sin(c + d*x))/(7*b*d)],
[cos(c + d*x)^(3/2)*(B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x)), x, 7, -((2*(5*a*b*B - 5*a^2*C - 3*b^2*C)*EllipticE((1/2)*(c + d*x), 2))/(5*b^3*d)) + (2*(3*a^2*b*B + b^3*B - 3*a^3*C - a*b^2*C)*EllipticF((1/2)*(c + d*x), 2))/(3*b^4*d) - (2*a^3*(b*B - a*C)*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2))/(b^4*(a + b)*d) + (2*(b*B - a*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(3*b^2*d) + (2*C*cos(c + d*x)^(3/2)*sin(c + d*x))/(5*b*d)],
[cos(c + d*x)^(1/2)*(B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x)), x, 6, (2*(b*B - a*C)*EllipticE((1/2)*(c + d*x), 2))/(b^2*d) - (2*(3*a*b*B - 3*a^2*C - b^2*C)*EllipticF((1/2)*(c + d*x), 2))/(3*b^3*d) + (2*a^2*(b*B - a*C)*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2))/(b^3*(a + b)*d) + (2*C*sqrt(cos(c + d*x))*sin(c + d*x))/(3*b*d)],
[cos(c + d*x)^(-1/2)*(B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x)), x, 6, (2*C*EllipticE((1/2)*(c + d*x), 2))/(b*d) + (2*(b*B - a*C)*EllipticF((1/2)*(c + d*x), 2))/(b^2*d) - (2*a*(b*B - a*C)*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2))/(b^2*(a + b)*d)],
[cos(c + d*x)^(-3/2)*(B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x)), x, 4, (2*C*EllipticF((1/2)*(c + d*x), 2))/(b*d) + (2*(b*B - a*C)*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2))/(b*(a + b)*d)],
[cos(c + d*x)^(-5/2)*(B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x)), x, 6, -((2*B*EllipticE((1/2)*(c + d*x), 2))/(a*d)) - (2*(b*B - a*C)*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2))/(a*(a + b)*d) + (2*B*sin(c + d*x))/(a*d*sqrt(cos(c + d*x)))],
[cos(c + d*x)^(-7/2)*(B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x)), x, 8, (2*(b*B - a*C)*EllipticE((1/2)*(c + d*x), 2))/(a^2*d) + (2*B*EllipticF((1/2)*(c + d*x), 2))/(3*a*d) + (2*b*(b*B - a*C)*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2))/(a^2*(a + b)*d) + (2*B*sin(c + d*x))/(3*a*d*cos(c + d*x)^(3/2)) - (2*(b*B - a*C)*sin(c + d*x))/(a^2*d*sqrt(cos(c + d*x)))],
[cos(c + d*x)^(-9/2)*(B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x)), x, 9, -((2*(3*a^2*B + 5*b^2*B - 5*a*b*C)*EllipticE((1/2)*(c + d*x), 2))/(5*a^3*d)) - (2*(b*B - a*C)*EllipticF((1/2)*(c + d*x), 2))/(3*a^2*d) - (2*b^2*(b*B - a*C)*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2))/(a^3*(a + b)*d) + (2*B*sin(c + d*x))/(5*a*d*cos(c + d*x)^(5/2)) - (2*(b*B - a*C)*sin(c + d*x))/(3*a^2*d*cos(c + d*x)^(3/2)) + (2*(3*a^2*B + 5*b^2*B - 5*a*b*C)*sin(c + d*x))/(5*a^3*d*sqrt(cos(c + d*x)))],


[(cos(c + d*x)^(5/2)*(B*cos(c + d*x) + C*cos(c + d*x)^2))/(a + b*cos(c + d*x))^2, x, 8, -((25*a^3*b*B - 20*a*b^3*B - 35*a^4*C + 24*a^2*b^2*C + 6*b^4*C)*EllipticE((c + d*x)/2, 2))/(5*b^4*(a^2 - b^2)*d) + ((15*a^4*b*B - 16*a^2*b^3*B - 2*b^5*B - 21*a^5*C + 20*a^3*b^2*C + 4*a*b^4*C)*EllipticF((c + d*x)/2, 2))/(3*b^5*(a^2 - b^2)*d) - (a^3*(5*a^2*b*B - 7*b^3*B - 7*a^3*C + 9*a*b^2*C)*EllipticPi((2*b)/(a + b), (c + d*x)/2, 2))/((a - b)*b^5*(a + b)^2*d) + ((5*a^2*b*B - 2*b^3*B - 7*a^3*C + 4*a*b^2*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(3*b^3*(a^2 - b^2)*d) - ((5*a*b*B - 7*a^2*C + 2*b^2*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(5*b^2*(a^2 - b^2)*d) + (a*(b*B - a*C)*cos(c + d*x)^(5/2)*sin(c + d*x))/(b*(a^2 - b^2)*d*(a + b*cos(c + d*x)))],
[(cos(c + d*x)^(3/2)*(B*cos(c + d*x) + C*cos(c + d*x)^2))/(a + b*cos(c + d*x))^2, x, 7, ((3*a^2*b*B - 2*b^3*B - 5*a^3*C + 4*a*b^2*C)*EllipticE((c + d*x)/2, 2))/(b^3*(a^2 - b^2)*d) - ((9*a^3*b*B - 12*a*b^3*B - 15*a^4*C + 16*a^2*b^2*C + 2*b^4*C)*EllipticF((c + d*x)/2, 2))/(3*b^4*(a^2 - b^2)*d) + (a^2*(3*a^2*b*B - 5*b^3*B - 5*a^3*C + 7*a*b^2*C)*EllipticPi((2*b)/(a + b), (c + d*x)/2, 2))/((a - b)*b^4*(a + b)^2*d) - ((3*a*b*B - 5*a^2*C + 2*b^2*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(3*b^2*(a^2 - b^2)*d) + (a*(b*B - a*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(b*(a^2 - b^2)*d*(a + b*cos(c + d*x)))],
[(sqrt(cos(c + d*x))*(B*cos(c + d*x) + C*cos(c + d*x)^2))/(a + b*cos(c + d*x))^2, x, 6, -(((a*b*B - 3*a^2*C + 2*b^2*C)*EllipticE((c + d*x)/2, 2))/(b^2*(a^2 - b^2)*d)) + ((a^2*b*B - 2*b^3*B - 3*a^3*C + 4*a*b^2*C)*EllipticF((c + d*x)/2, 2))/(b^3*(a^2 - b^2)*d) - (a*(a^2*b*B - 3*b^3*B - 3*a^3*C + 5*a*b^2*C)*EllipticPi((2*b)/(a + b), (c + d*x)/2, 2))/((a - b)*b^3*(a + b)^2*d) + (a*(b*B - a*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(b*(a^2 - b^2)*d*(a + b*cos(c + d*x)))],
[(B*cos(c + d*x) + C*cos(c + d*x)^2)/(sqrt(cos(c + d*x))*(a + b*cos(c + d*x))^2), x, 6, ((b*B - a*C)*EllipticE((c + d*x)/2, 2))/(b*(a^2 - b^2)*d) + ((a*b*B + a^2*C - 2*b^2*C)*EllipticF((c + d*x)/2, 2))/(b^2*(a^2 - b^2)*d) - ((a^2*b*B + b^3*B + a^3*C - 3*a*b^2*C)*EllipticPi((2*b)/(a + b), (c + d*x)/2, 2))/((a - b)*b^2*(a + b)^2*d) - ((b*B - a*C)*sqrt(cos(c + d*x))*sin(c + d*x))/((a^2 - b^2)*d*(a + b*cos(c + d*x)))],
[(B*cos(c + d*x) + C*cos(c + d*x)^2)/(cos(c + d*x)^(3/2)*(a + b*cos(c + d*x))^2), x, 7, -(((b*B - a*C)*EllipticE((c + d*x)/2, 2))/(a*(a^2 - b^2)*d)) - ((b*B - a*C)*EllipticF((c + d*x)/2, 2))/(b*(a^2 - b^2)*d) + ((3*a^2*b*B - b^3*B - a^3*C - a*b^2*C)*EllipticPi((2*b)/(a + b), (c + d*x)/2, 2))/(a*(a - b)*b*(a + b)^2*d) + ((b*B - a*C)*sin(c + d*x))/(a*(a^2 - b^2)*d*sqrt(cos(c + d*x))) - ((b*B - a*C)*sin(c + d*x))/((a^2 - b^2)*d*sqrt(cos(c + d*x))*(a + b*cos(c + d*x)))],
[(B*cos(c + d*x) + C*cos(c + d*x)^2)/(cos(c + d*x)^(5/2)*(a + b*cos(c + d*x))^2), x, 8, -(((2*a^2*B - 3*b^2*B + a*b*C)*EllipticE((c + d*x)/2, 2))/(a^2*(a^2 - b^2)*d)) + ((b*B - a*C)*EllipticF((c + d*x)/2, 2))/(a*(a^2 - b^2)*d) - ((5*a^2*b*B - 3*b^3*B - 3*a^3*C + a*b^2*C)*EllipticPi((2*b)/(a + b), (c + d*x)/2, 2))/(a^2*(a - b)*(a + b)^2*d) + ((b*B - a*C)*sin(c + d*x))/(a*(a^2 - b^2)*d*cos(c + d*x)^(3/2)) + ((2*a^2*B - 3*b^2*B + a*b*C)*sin(c + d*x))/(a^2*(a^2 - b^2)*d*sqrt(cos(c + d*x))) - ((b*B - a*C)*sin(c + d*x))/((a^2 - b^2)*d*cos(c + d*x)^(3/2)*(a + b*cos(c + d*x)))],
[(B*cos(c + d*x) + C*cos(c + d*x)^2)/(cos(c + d*x)^(7/2)*(a + b*cos(c + d*x))^2), x, 9, ((4*a^2*b*B - 5*b^3*B - 2*a^3*C + 3*a*b^2*C)*EllipticE((c + d*x)/2, 2))/(a^3*(a^2 - b^2)*d) + ((2*a^2*B - 5*b^2*B + 3*a*b*C)*EllipticF((c + d*x)/2, 2))/(3*a^2*(a^2 - b^2)*d) + (b*(7*a^2*b*B - 5*b^3*B - 5*a^3*C + 3*a*b^2*C)*EllipticPi((2*b)/(a + b), (c + d*x)/2, 2))/(a^3*(a - b)*(a + b)^2*d) + ((b*B - a*C)*sin(c + d*x))/(a*(a^2 - b^2)*d*cos(c + d*x)^(5/2)) + ((2*a^2*B - 5*b^2*B + 3*a*b*C)*sin(c + d*x))/(3*a^2*(a^2 - b^2)*d*cos(c + d*x)^(3/2)) - ((4*a^2*b*B - 5*b^3*B - 2*a^3*C + 3*a*b^2*C)*sin(c + d*x))/(a^3*(a^2 - b^2)*d*sqrt(cos(c + d*x))) - ((b*B - a*C)*sin(c + d*x))/((a^2 - b^2)*d*cos(c + d*x)^(5/2)*(a + b*cos(c + d*x)))],


[(cos(c + d*x)^(5/2)*(B*cos(c + d*x) + C*cos(c + d*x)^2))/(a + b*cos(c + d*x))^3, x, 8, ((15*a^4*b*B - 29*a^2*b^3*B + 8*b^5*B - 35*a^5*C + 65*a^3*b^2*C - 24*a*b^4*C)*EllipticE((c + d*x)/2, 2))/(4*b^4*(a^2 - b^2)^2*d) - ((45*a^5*b*B - 99*a^3*b^3*B + 72*a*b^5*B - 105*a^6*C + 223*a^4*b^2*C - 128*a^2*b^4*C - 8*b^6*C)*EllipticF((c + d*x)/2, 2))/(12*b^5*(a^2 - b^2)^2*d) + (a^2*(15*a^4*b*B - 38*a^2*b^3*B + 35*b^5*B - 35*a^5*C + 86*a^3*b^2*C - 63*a*b^4*C)*EllipticPi((2*b)/(a + b), (c + d*x)/2, 2))/(4*(a - b)^2*b^5*(a + b)^3*d) - ((15*a^3*b*B - 33*a*b^3*B - 35*a^4*C + 61*a^2*b^2*C - 8*b^4*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(12*b^3*(a^2 - b^2)^2*d) + (a*(b*B - a*C)*cos(c + d*x)^(5/2)*sin(c + d*x))/(2*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) + (a*(3*a^2*b*B - 9*b^3*B - 7*a^3*C + 13*a*b^2*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(4*b^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))],
[(cos(c + d*x)^(3/2)*(B*cos(c + d*x) + C*cos(c + d*x)^2))/(a + b*cos(c + d*x))^3, x, 7, -((3*a^3*b*B - 9*a*b^3*B - 15*a^4*C + 29*a^2*b^2*C - 8*b^4*C)*EllipticE((c + d*x)/2, 2))/(4*b^3*(a^2 - b^2)^2*d) + ((3*a^4*b*B - 5*a^2*b^3*B + 8*b^5*B - 15*a^5*C + 33*a^3*b^2*C - 24*a*b^4*C)*EllipticF((c + d*x)/2, 2))/(4*b^4*(a^2 - b^2)^2*d) - (a*(3*a^4*b*B - 6*a^2*b^3*B + 15*b^5*B - 15*a^5*C + 38*a^3*b^2*C - 35*a*b^4*C)*EllipticPi((2*b)/(a + b), (c + d*x)/2, 2))/(4*(a - b)^2*b^4*(a + b)^3*d) + (a*(b*B - a*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(2*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) + (a*(a^2*b*B - 7*b^3*B - 5*a^3*C + 11*a*b^2*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(4*b^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))],
[(sqrt(cos(c + d*x))*(B*cos(c + d*x) + C*cos(c + d*x)^2))/(a + b*cos(c + d*x))^3, x, 7, -((a^2*b*B + 5*b^3*B + 3*a^3*C - 9*a*b^2*C)*EllipticE((c + d*x)/2, 2))/(4*b^2*(a^2 - b^2)^2*d) + ((a^3*b*B - 7*a*b^3*B + 3*a^4*C - 5*a^2*b^2*C + 8*b^4*C)*EllipticF((c + d*x)/2, 2))/(4*b^3*(a^2 - b^2)^2*d) - ((a^4*b*B - 10*a^2*b^3*B - 3*b^5*B + 3*a^5*C - 6*a^3*b^2*C + 15*a*b^4*C)*EllipticPi((2*b)/(a + b), (c + d*x)/2, 2))/(4*(a - b)^2*b^3*(a + b)^3*d) + (a*(b*B - a*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(2*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) + ((a^2*b*B + 5*b^3*B + 3*a^3*C - 9*a*b^2*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(4*b*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))],
[(B*cos(c + d*x) + C*cos(c + d*x)^2)/(sqrt(cos(c + d*x))*(a + b*cos(c + d*x))^3), x, 7, ((5*a^2*b*B + b^3*B - a^3*C - 5*a*b^2*C)*EllipticE((c + d*x)/2, 2))/(4*a*b*(a^2 - b^2)^2*d) + ((3*a^2*b*B + 3*b^3*B + a^3*C - 7*a*b^2*C)*EllipticF((c + d*x)/2, 2))/(4*b^2*(a^2 - b^2)^2*d) - ((3*a^4*b*B + 10*a^2*b^3*B - b^5*B + a^5*C - 10*a^3*b^2*C - 3*a*b^4*C)*EllipticPi((2*b)/(a + b), (c + d*x)/2, 2))/(4*a*(a - b)^2*b^2*(a + b)^3*d) - ((b*B - a*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(2*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) - ((5*a^2*b*B + b^3*B - a^3*C - 5*a*b^2*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(4*a*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))],
[(B*cos(c + d*x) + C*cos(c + d*x)^2)/(cos(c + d*x)^(3/2)*(a + b*cos(c + d*x))^3), x, 8, -((9*a^2*b*B - 3*b^3*B - 5*a^3*C - a*b^2*C)*EllipticE((c + d*x)/2, 2))/(4*a^2*(a^2 - b^2)^2*d) - ((7*a^2*b*B - b^3*B - 3*a^3*C - 3*a*b^2*C)*EllipticF((c + d*x)/2, 2))/(4*a*b*(a^2 - b^2)^2*d) + ((15*a^4*b*B - 6*a^2*b^3*B + 3*b^5*B - 3*a^5*C - 10*a^3*b^2*C + a*b^4*C)*EllipticPi((2*b)/(a + b), (c + d*x)/2, 2))/(4*a^2*(a - b)^2*b*(a + b)^3*d) + ((9*a^2*b*B - 3*b^3*B - 5*a^3*C - a*b^2*C)*sin(c + d*x))/(4*a^2*(a^2 - b^2)^2*d*sqrt(cos(c + d*x))) - ((b*B - a*C)*sin(c + d*x))/(2*(a^2 - b^2)*d*sqrt(cos(c + d*x))*(a + b*cos(c + d*x))^2) - ((7*a^2*b*B - b^3*B - 3*a^3*C - 3*a*b^2*C)*sin(c + d*x))/(4*a*(a^2 - b^2)^2*d*sqrt(cos(c + d*x))*(a + b*cos(c + d*x)))],
[(B*cos(c + d*x) + C*cos(c + d*x)^2)/(cos(c + d*x)^(5/2)*(a + b*cos(c + d*x))^3), x, 9, -((8*a^4*B - 29*a^2*b^2*B + 15*b^4*B + 9*a^3*b*C - 3*a*b^3*C)*EllipticE((c + d*x)/2, 2))/(4*a^3*(a^2 - b^2)^2*d) + ((11*a^2*b*B - 5*b^3*B - 7*a^3*C + a*b^2*C)*EllipticF((c + d*x)/2, 2))/(4*a^2*(a^2 - b^2)^2*d) - ((35*a^4*b*B - 38*a^2*b^3*B + 15*b^5*B - 15*a^5*C + 6*a^3*b^2*C - 3*a*b^4*C)*EllipticPi((2*b)/(a + b), (c + d*x)/2, 2))/(4*a^3*(a - b)^2*(a + b)^3*d) + ((11*a^2*b*B - 5*b^3*B - 7*a^3*C + a*b^2*C)*sin(c + d*x))/(4*a^2*(a^2 - b^2)^2*d*cos(c + d*x)^(3/2)) + ((8*a^4*B - 29*a^2*b^2*B + 15*b^4*B + 9*a^3*b*C - 3*a*b^3*C)*sin(c + d*x))/(4*a^3*(a^2 - b^2)^2*d*sqrt(cos(c + d*x))) - ((b*B - a*C)*sin(c + d*x))/(2*(a^2 - b^2)*d*cos(c + d*x)^(3/2)*(a + b*cos(c + d*x))^2) - ((9*a^2*b*B - 3*b^3*B - 5*a^3*C - a*b^2*C)*sin(c + d*x))/(4*a*(a^2 - b^2)^2*d*cos(c + d*x)^(3/2)*(a + b*cos(c + d*x)))],


# ::Subsection::Closed:: 
#Integrands of the form Cos[c+d x]^(m/2) (B Cos[c+d x]+C Cos[c+d x]^2) (a+b Cos[c+d x])^(n/2)
#


# ::Subsubsection::Closed:: 
#n>0


[sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*(B*cos(c + d*x) + C*cos(c + d*x)^2), x, 8, ((a + b)*(6*a*b*B - 3*a^2*C + 16*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(24*b^2*d*sqrt(a + b*cos(c + d*x))) + ((6*a*b*B - 12*b^2*B + a^2*C - 14*a*b*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(12*b*d*sqrt(a + b*cos(c + d*x))) - ((2*a^2*b*B - 8*b^3*B - a^3*C - 4*a*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(4*b^2*d*sqrt(a + b*cos(c + d*x))) + ((6*b*B + a*C)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(12*b*d) + (C*cos(c + d*x)^(3/2)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*d) + ((6*a*b*B - 3*a^2*C + 16*b^2*C)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(24*b^2*d)],
[(sqrt(a + b*cos(c + d*x))*(B*cos(c + d*x) + C*cos(c + d*x)^2))/sqrt(cos(c + d*x)), x, 7, ((a + b)*(4*b*B + a*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(4*b*d*sqrt(a + b*cos(c + d*x))) - ((4*a*B - a*C + 2*b*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(2*d*sqrt(a + b*cos(c + d*x))) + ((4*a*b*B - a^2*C + 4*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(2*b*d*sqrt(a + b*cos(c + d*x))) + (C*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(2*d) + ((4*b*B + a*C)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(4*b*d)],
[(sqrt(a + b*cos(c + d*x))*(B*cos(c + d*x) + C*cos(c + d*x)^2))/cos(c + d*x)^(3/2), x, 7, ((a + b)*C*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(d*sqrt(a + b*cos(c + d*x))) + (2*(a*B - b*B - a*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(d*sqrt(a + b*cos(c + d*x))) + (2*(2*b*B + a*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(d*sqrt(a + b*cos(c + d*x))) + (C*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/d],
[(sqrt(a + b*cos(c + d*x))*(B*cos(c + d*x) + C*cos(c + d*x)^2))/cos(c + d*x)^(5/2), x, 7, (-2*(a + b)*B*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(d*sqrt(a + b*cos(c + d*x))) + (2*(a*B + b*B + a*C - b*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(d*sqrt(a + b*cos(c + d*x))) + (4*b*C*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(d*sqrt(a + b*cos(c + d*x))) + (2*B*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(d*sqrt(cos(c + d*x)))],
[(sqrt(a + b*cos(c + d*x))*(B*cos(c + d*x) + C*cos(c + d*x)^2))/cos(c + d*x)^(7/2), x, 6, (-2*(a + b)*(b*B + 3*a*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(3*a*d*sqrt(a + b*cos(c + d*x))) + (2*(a*B + b*B + 3*a*C + 3*b*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(3*d*sqrt(a + b*cos(c + d*x))) + (2*B*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2)) + (2*(b*B + 3*a*C)*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(3*a*d*sqrt(cos(c + d*x)))],
[(sqrt(a + b*cos(c + d*x))*(B*cos(c + d*x) + C*cos(c + d*x)^2))/cos(c + d*x)^(9/2), x, 7, (-2*(a + b)*(9*a^2*B - 2*b^2*B + 5*a*b*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(15*a^2*d*sqrt(a + b*cos(c + d*x))) + (2*(9*a^2*B + 7*a*b*B - 2*b^2*B + 5*a^2*C + 5*a*b*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(15*a*d*sqrt(a + b*cos(c + d*x))) + (2*B*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2)) + (2*(b*B + 5*a*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(15*a*d*cos(c + d*x)^(3/2)) + (2*(9*a^2*B - 2*b^2*B + 5*a*b*C)*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(15*a^2*d*sqrt(cos(c + d*x)))],
[(sqrt(a + b*cos(c + d*x))*(B*cos(c + d*x) + C*cos(c + d*x)^2))/cos(c + d*x)^(11/2), x, 8, -((2*(a + b)*(19*a^2*b*B + 8*b^3*B + 63*a^3*C - 14*a*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(105*a^3*d*sqrt(a + b*cos(c + d*x)))) + (2*(25*a^3*B + 19*a^2*b*B + 2*a*b^2*B + 8*b^3*B + 63*a^3*C + 49*a^2*b*C - 14*a*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(105*a^2*d*sqrt(a + b*cos(c + d*x))) + (2*B*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(7*d*cos(c + d*x)^(7/2)) + (2*(b*B + 7*a*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(35*a*d*cos(c + d*x)^(5/2)) + (2*(25*a^2*B - 4*b^2*B + 7*a*b*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(105*a^2*d*cos(c + d*x)^(3/2)) + (2*(19*a^2*b*B + 8*b^3*B + 63*a^3*C - 14*a*b^2*C)*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(105*a^3*d*sqrt(cos(c + d*x)))],


[sqrt(cos(c + d*x))*(a + b*cos(c + d*x))^(3/2)*(B*cos(c + d*x) + C*cos(c + d*x)^2), x, 9, ((a + b)*(24*a^2*b*B + 128*b^3*B - 9*a^3*C + 156*a*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(192*b^2*d*sqrt(a + b*cos(c + d*x))) + ((56*a^2*b*B - 208*a*b^2*B + 3*a^3*C - 114*a^2*b*C + 36*a*b^2*C - 72*b^3*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(96*b*d*sqrt(a + b*cos(c + d*x))) - ((8*a^3*b*B - 96*a*b^3*B - 3*a^4*C - 24*a^2*b^2*C - 48*b^4*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(32*b^2*d*sqrt(a + b*cos(c + d*x))) + ((56*a*b*B + 3*a^2*C + 36*b^2*C)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(96*b*d) + ((8*b*B + 3*a*C)*cos(c + d*x)^(3/2)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(24*d) + (C*cos(c + d*x)^(3/2)*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(4*d) + ((24*a^2*b*B + 128*b^3*B - 9*a^3*C + 156*a*b^2*C)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(192*b^2*d)],
[((a + b*cos(c + d*x))^(3/2)*(B*cos(c + d*x) + C*cos(c + d*x)^2))/sqrt(cos(c + d*x)), x, 8, ((a + b)*(30*a*b*B + 3*a^2*C + 16*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(24*b*d*sqrt(a + b*cos(c + d*x))) - ((24*a^2*B - 6*a*b*B + 12*b^2*B - 7*a^2*C + 26*a*b*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(12*d*sqrt(a + b*cos(c + d*x))) + ((6*a^2*b*B + 8*b^3*B - a^3*C + 12*a*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(4*b*d*sqrt(a + b*cos(c + d*x))) + ((2*b*B + a*C)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(4*d) + (C*sqrt(cos(c + d*x))*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(3*d) + ((30*a*b*B + 3*a^2*C + 16*b^2*C)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(24*b*d)],
[((a + b*cos(c + d*x))^(3/2)*(B*cos(c + d*x) + C*cos(c + d*x)^2))/cos(c + d*x)^(3/2), x, 8, ((a + b)*(4*b*B + 5*a*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(4*d*sqrt(a + b*cos(c + d*x))) + ((4*a^2*B - 8*a*b*B - 4*a^2*C + a*b*C - 2*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(2*d*sqrt(a + b*cos(c + d*x))) + ((12*a*b*B + 3*a^2*C + 4*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(2*d*sqrt(a + b*cos(c + d*x))) + (b*C*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(2*d) + ((4*b*B + 5*a*C)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(4*d)],
[((a + b*cos(c + d*x))^(3/2)*(B*cos(c + d*x) + C*cos(c + d*x)^2))/cos(c + d*x)^(5/2), x, 8, -(((a + b)*(2*a*B - b*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(d*sqrt(a + b*cos(c + d*x)))) + (2*(a^2*B + 2*a*b*B - b^2*B + a^2*C - 2*a*b*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(d*sqrt(a + b*cos(c + d*x))) + (2*b*(2*b*B + 3*a*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(d*sqrt(a + b*cos(c + d*x))) + (2*a*B*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(d*sqrt(cos(c + d*x))) - ((2*a*B - b*C)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/d],
[((a + b*cos(c + d*x))^(3/2)*(B*cos(c + d*x) + C*cos(c + d*x)^2))/cos(c + d*x)^(7/2), x, 8, (-2*(a + b)*(4*b*B + 3*a*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(3*d*sqrt(a + b*cos(c + d*x))) + (2*(a^2*B + 4*a*b*B + 3*b^2*B + 3*a^2*C + 6*a*b*C - 3*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(3*d*sqrt(a + b*cos(c + d*x))) + (4*b^2*C*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(d*sqrt(a + b*cos(c + d*x))) + (2*a*B*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2)) + (2*(4*b*B + 3*a*C)*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(3*d*sqrt(cos(c + d*x)))],
[((a + b*cos(c + d*x))^(3/2)*(B*cos(c + d*x) + C*cos(c + d*x)^2))/cos(c + d*x)^(9/2), x, 7, (-2*(a + b)*(9*a^2*B + 3*b^2*B + 20*a*b*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(15*a*d*sqrt(a + b*cos(c + d*x))) + (2*(9*a^2*B + 12*a*b*B + 3*b^2*B + 5*a^2*C + 20*a*b*C + 15*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(15*d*sqrt(a + b*cos(c + d*x))) + (2*a*B*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2)) + (2*(6*b*B + 5*a*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(15*d*cos(c + d*x)^(3/2)) + (2*(9*a^2*B + 3*b^2*B + 20*a*b*C)*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(15*a*d*sqrt(cos(c + d*x)))],
[((a + b*cos(c + d*x))^(3/2)*(B*cos(c + d*x) + C*cos(c + d*x)^2))/cos(c + d*x)^(11/2), x, 8, (-2*(a + b)*(82*a^2*b*B - 6*b^3*B + 63*a^3*C + 21*a*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(105*a^2*d*sqrt(a + b*cos(c + d*x))) + (2*(25*a^3*B + 82*a^2*b*B + 51*a*b^2*B - 6*b^3*B + 63*a^3*C + 84*a^2*b*C + 21*a*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(105*a*d*sqrt(a + b*cos(c + d*x))) + (2*a*B*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(7*d*cos(c + d*x)^(7/2)) + (2*(8*b*B + 7*a*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(35*d*cos(c + d*x)^(5/2)) + (2*(25*a^2*B + 3*b^2*B + 42*a*b*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(105*a*d*cos(c + d*x)^(3/2)) + (2*(82*a^2*b*B - 6*b^3*B + 63*a^3*C + 21*a*b^2*C)*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(105*a^2*d*sqrt(cos(c + d*x)))],


[sqrt(cos(c + d*x))*(a + b*cos(c + d*x))^(5/2)*(B*cos(c + d*x) + C*cos(c + d*x)^2), x, 10, ((a + b)*(150*a^3*b*B + 2840*a*b^3*B - 45*a^4*C + 1692*a^2*b^2*C + 1024*b^4*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(1920*b^2*d*sqrt(a + b*cos(c + d*x))) + ((590*a^3*b*B - 3220*a^2*b^2*B + 360*a*b^3*B - 720*b^4*B + 15*a^4*C - 1146*a^3*b*C + 772*a^2*b^2*C - 2312*a*b^3*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(960*b*d*sqrt(a + b*cos(c + d*x))) - ((10*a^4*b*B - 240*a^2*b^3*B - 96*b^5*B - 3*a^5*C - 40*a^3*b^2*C - 240*a*b^4*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(64*b^2*d*sqrt(a + b*cos(c + d*x))) + ((590*a^2*b*B + 360*b^3*B + 15*a^3*C + 772*a*b^2*C)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(960*b*d) + ((110*a*b*B + 15*a^2*C + 64*b^2*C)*cos(c + d*x)^(3/2)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(240*d) + ((2*b*B + a*C)*cos(c + d*x)^(3/2)*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(8*d) + (C*cos(c + d*x)^(3/2)*(a + b*cos(c + d*x))^(5/2)*sin(c + d*x))/(5*d) + ((150*a^3*b*B + 2840*a*b^3*B - 45*a^4*C + 1692*a^2*b^2*C + 1024*b^4*C)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(1920*b^2*d)],
[((a + b*cos(c + d*x))^(5/2)*(B*cos(c + d*x) + C*cos(c + d*x)^2))/sqrt(cos(c + d*x)), x, 9, ((a + b)*(264*a^2*b*B + 128*b^3*B + 15*a^3*C + 284*a*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(192*b*d*sqrt(a + b*cos(c + d*x))) - ((192*a^3*B - 104*a^2*b*B + 304*a*b^2*B - 59*a^3*C + 322*a^2*b*C - 36*a*b^2*C + 72*b^3*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(96*d*sqrt(a + b*cos(c + d*x))) + ((40*a^3*b*B + 160*a*b^3*B - 5*a^4*C + 120*a^2*b^2*C + 48*b^4*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(32*b*d*sqrt(a + b*cos(c + d*x))) + ((24*a*b*B + 5*a^2*C + 12*b^2*C)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(32*d) + ((8*b*B + 5*a*C)*sqrt(cos(c + d*x))*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(24*d) + (C*sqrt(cos(c + d*x))*(a + b*cos(c + d*x))^(5/2)*sin(c + d*x))/(4*d) + ((264*a^2*b*B + 128*b^3*B + 15*a^3*C + 284*a*b^2*C)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(192*b*d)],
[((a + b*cos(c + d*x))^(5/2)*(B*cos(c + d*x) + C*cos(c + d*x)^2))/cos(c + d*x)^(3/2), x, 9, ((a + b)*(54*a*b*B + 33*a^2*C + 16*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(24*d*sqrt(a + b*cos(c + d*x))) + ((24*a^3*B - 72*a^2*b*B + 6*a*b^2*B - 12*b^3*B - 24*a^3*C + 13*a^2*b*C - 38*a*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(12*d*sqrt(a + b*cos(c + d*x))) + ((30*a^2*b*B + 8*b^3*B + 5*a^3*C + 20*a*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(4*d*sqrt(a + b*cos(c + d*x))) + (b*(2*b*B + 3*a*C)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(4*d) + (b*C*sqrt(cos(c + d*x))*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(3*d) + ((54*a*b*B + 33*a^2*C + 16*b^2*C)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(24*d)],
[((a + b*cos(c + d*x))^(5/2)*(B*cos(c + d*x) + C*cos(c + d*x)^2))/cos(c + d*x)^(5/2), x, 9, -((a + b)*(8*a^2*B - 4*b^2*B - 9*a*b*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(4*d*sqrt(a + b*cos(c + d*x))) + ((4*a^3*B + 12*a^2*b*B - 12*a*b^2*B + 4*a^3*C - 12*a^2*b*C + a*b^2*C - 2*b^3*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(2*d*sqrt(a + b*cos(c + d*x))) + (b*(20*a*b*B + 15*a^2*C + 4*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(2*d*sqrt(a + b*cos(c + d*x))) - (b*(4*a*B - b*C)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(2*d) + (2*a*B*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(d*sqrt(cos(c + d*x))) - ((8*a^2*B - 4*b^2*B - 9*a*b*C)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(4*d)],
[((a + b*cos(c + d*x))^(5/2)*(B*cos(c + d*x) + C*cos(c + d*x)^2))/cos(c + d*x)^(7/2), x, 9, -((a + b)*(14*a*b*B + 6*a^2*C - 3*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(3*d*sqrt(a + b*cos(c + d*x))) + (2*(a^3*B + 7*a^2*b*B + 9*a*b^2*B - 3*b^3*B + 3*a^3*C + 9*a^2*b*C - 9*a*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(3*d*sqrt(a + b*cos(c + d*x))) + (2*b^2*(2*b*B + 5*a*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(d*sqrt(a + b*cos(c + d*x))) + (2*a*(2*b*B + a*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(d*sqrt(cos(c + d*x))) + (2*a*B*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2)) - ((14*a*b*B + 6*a^2*C - 3*b^2*C)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(3*d)],
[((a + b*cos(c + d*x))^(5/2)*(B*cos(c + d*x) + C*cos(c + d*x)^2))/cos(c + d*x)^(9/2), x, 9, (-2*(a + b)*(9*a^2*B + 23*b^2*B + 35*a*b*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(15*d*sqrt(a + b*cos(c + d*x))) + (2*(9*a^3*B + 17*a^2*b*B + 23*a*b^2*B + 15*b^3*B + 5*a^3*C + 35*a^2*b*C + 45*a*b^2*C - 15*b^3*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(15*d*sqrt(a + b*cos(c + d*x))) + (4*b^3*C*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(d*sqrt(a + b*cos(c + d*x))) + (2*a*(8*b*B + 5*a*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(15*d*cos(c + d*x)^(3/2)) + (2*a*B*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2)) + (2*(9*a^2*B + 23*b^2*B + 35*a*b*C)*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(15*d*sqrt(cos(c + d*x)))],
[((a + b*cos(c + d*x))^(5/2)*(B*cos(c + d*x) + C*cos(c + d*x)^2))/cos(c + d*x)^(11/2), x, 8, (-2*(a + b)*(145*a^2*b*B + 15*b^3*B + 63*a^3*C + 161*a*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(105*a*d*sqrt(a + b*cos(c + d*x))) + (2*(25*a^3*B + 145*a^2*b*B + 135*a*b^2*B + 15*b^3*B + 63*a^3*C + 119*a^2*b*C + 161*a*b^2*C + 105*b^3*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(105*d*sqrt(a + b*cos(c + d*x))) + (2*a*(10*b*B + 7*a*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(35*d*cos(c + d*x)^(5/2)) + (2*(25*a^2*B + 45*b^2*B + 77*a*b*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(105*d*cos(c + d*x)^(3/2)) + (2*a*B*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(7*d*cos(c + d*x)^(7/2)) + (2*(145*a^2*b*B + 15*b^3*B + 63*a^3*C + 161*a*b^2*C)*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(105*a*d*sqrt(cos(c + d*x)))],
[((a + b*cos(c + d*x))^(5/2)*(B*cos(c + d*x) + C*cos(c + d*x)^2))/cos(c + d*x)^(13/2), x, 9, -((2*(a + b)*(147*a^4*B + 279*a^2*b^2*B - 10*b^4*B + 435*a^3*b*C + 45*a*b^3*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(315*a^2*d*sqrt(a + b*cos(c + d*x)))) + (2*(147*a^4*B + 261*a^3*b*B + 279*a^2*b^2*B + 155*a*b^3*B - 10*b^4*B + 75*a^4*C + 435*a^3*b*C + 405*a^2*b^2*C + 45*a*b^3*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(315*a*d*sqrt(a + b*cos(c + d*x))) + (2*a*(4*b*B + 3*a*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(21*d*cos(c + d*x)^(7/2)) + (2*(49*a^2*B + 75*b^2*B + 135*a*b*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(315*d*cos(c + d*x)^(5/2)) + (2*(163*a^2*b*B + 5*b^3*B + 75*a^3*C + 135*a*b^2*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(315*a*d*cos(c + d*x)^(3/2)) + (2*a*B*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(9*d*cos(c + d*x)^(9/2)) + (2*(147*a^4*B + 279*a^2*b^2*B - 10*b^4*B + 435*a^3*b*C + 45*a*b^3*C)*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(315*a^2*d*sqrt(cos(c + d*x)))],
[((a + b*cos(c + d*x))^(5/2)*(B*cos(c + d*x) + C*cos(c + d*x)^2))/cos(c + d*x)^(15/2), x, 10, -((2*(a + b)*(3705*a^4*b*B + 255*a^2*b^3*B + 40*b^5*B + 1617*a^5*C + 3069*a^3*b^2*C - 110*a*b^4*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(3465*a^3*d*sqrt(a + b*cos(c + d*x)))) + (2*(675*a^5*B + 3705*a^4*b*B + 3315*a^3*b^2*B + 255*a^2*b^3*B + 10*a*b^4*B + 40*b^5*B + 1617*a^5*C + 2871*a^4*b*C + 3069*a^3*b^2*C + 1705*a^2*b^3*C - 110*a*b^4*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(3465*a^2*d*sqrt(a + b*cos(c + d*x))) + (2*a*(14*b*B + 11*a*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(99*d*cos(c + d*x)^(9/2)) + (2*(81*a^2*B + 113*b^2*B + 209*a*b*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(693*d*cos(c + d*x)^(7/2)) + (2*(1145*a^2*b*B + 15*b^3*B + 539*a^3*C + 825*a*b^2*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3465*a*d*cos(c + d*x)^(5/2)) + (2*(675*a^4*B + 1025*a^2*b^2*B - 20*b^4*B + 1793*a^3*b*C + 55*a*b^3*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3465*a^2*d*cos(c + d*x)^(3/2)) + (2*a*B*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(11*d*cos(c + d*x)^(11/2)) + (2*(3705*a^4*b*B + 255*a^2*b^3*B + 40*b^5*B + 1617*a^5*C + 3069*a^3*b^2*C - 110*a*b^4*C)*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(3465*a^3*d*sqrt(cos(c + d*x)))],


# ::Subsubsection::Closed:: 
#n<0


[(cos(c + d*x)^(3/2)*(B*cos(c + d*x) + C*cos(c + d*x)^2))/sqrt(a + b*cos(c + d*x)), x, 8, -((a + b)*(18*a*b*B - 15*a^2*C - 16*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(24*b^3*d*sqrt(a + b*cos(c + d*x))) + ((6*a*b*B - 12*b^2*B - 5*a^2*C - 2*a*b*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(12*b^2*d*sqrt(a + b*cos(c + d*x))) + ((6*a^2*b*B + 8*b^3*B - 5*a^3*C - 4*a*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(4*b^3*d*sqrt(a + b*cos(c + d*x))) + ((6*b*B - 5*a*C)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(12*b^2*d) + (C*cos(c + d*x)^(3/2)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*b*d) - ((18*a*b*B - 15*a^2*C - 16*b^2*C)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(24*b^3*d)],
[(sqrt(cos(c + d*x))*(B*cos(c + d*x) + C*cos(c + d*x)^2))/sqrt(a + b*cos(c + d*x)), x, 7, ((a + b)*(4*b*B - 3*a*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(4*b^2*d*sqrt(a + b*cos(c + d*x))) + ((a - 2*b)*C*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(2*b*d*sqrt(a + b*cos(c + d*x))) - ((4*a*b*B - 3*a^2*C - 4*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(2*b^2*d*sqrt(a + b*cos(c + d*x))) + (C*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(2*b*d) + ((4*b*B - 3*a*C)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(4*b^2*d)],
[(B*cos(c + d*x) + C*cos(c + d*x)^2)/(sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))), x, 6, ((a + b)*C*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(b*d*sqrt(a + b*cos(c + d*x))) - (2*B*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(d*sqrt(a + b*cos(c + d*x))) + (2*(2*b*B - a*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(b*d*sqrt(a + b*cos(c + d*x))) + (C*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(b*d)],
[(B*cos(c + d*x) + C*cos(c + d*x)^2)/(cos(c + d*x)^(3/2)*sqrt(a + b*cos(c + d*x))), x, 4, (2*(B - C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(d*sqrt(a + b*cos(c + d*x))) + (4*C*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(d*sqrt(a + b*cos(c + d*x)))],
[(B*cos(c + d*x) + C*cos(c + d*x)^2)/(cos(c + d*x)^(5/2)*sqrt(a + b*cos(c + d*x))), x, 5, (-2*(a + b)*B*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(a*d*sqrt(a + b*cos(c + d*x))) + (2*(B + C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(d*sqrt(a + b*cos(c + d*x))) + (2*B*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(a*d*sqrt(cos(c + d*x)))],
[(B*cos(c + d*x) + C*cos(c + d*x)^2)/(cos(c + d*x)^(7/2)*sqrt(a + b*cos(c + d*x))), x, 6, (2*(a + b)*(2*b*B - 3*a*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(3*a^2*d*sqrt(a + b*cos(c + d*x))) + (2*(a*B - 2*b*B + 3*a*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(3*a*d*sqrt(a + b*cos(c + d*x))) + (2*B*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*a*d*cos(c + d*x)^(3/2)) - (2*(2*b*B - 3*a*C)*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(3*a^2*d*sqrt(cos(c + d*x)))],
[(B*cos(c + d*x) + C*cos(c + d*x)^2)/(cos(c + d*x)^(9/2)*sqrt(a + b*cos(c + d*x))), x, 7, (-2*(a + b)*(9*a^2*B + 8*b^2*B - 10*a*b*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(15*a^3*d*sqrt(a + b*cos(c + d*x))) + (2*(9*a^2*B + 2*a*b*B + 8*b^2*B + 5*a^2*C - 10*a*b*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(15*a^2*d*sqrt(a + b*cos(c + d*x))) + (2*B*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(5*a*d*cos(c + d*x)^(5/2)) - (2*(4*b*B - 5*a*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(15*a^2*d*cos(c + d*x)^(3/2)) + (2*(9*a^2*B + 8*b^2*B - 10*a*b*C)*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(15*a^3*d*sqrt(cos(c + d*x)))],


[(cos(c + d*x)^(3/2)*(B*cos(c + d*x) + C*cos(c + d*x)^2))/(a + b*cos(c + d*x))^(3/2), x, 8, ((12*a^2*b*B - 4*b^3*B - 15*a^3*C + 7*a*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(4*(a - b)*b^3*d*sqrt(a + b*cos(c + d*x))) - ((4*a*b*B - 5*a^2*C + 3*a*b*C - 2*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(2*(a - b)*b^2*d*sqrt(a + b*cos(c + d*x))) - ((12*a*b*B - 15*a^2*C - 4*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(2*b^3*d*sqrt(a + b*cos(c + d*x))) + (2*a*(b*B - a*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(b*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) - ((4*a*b*B - 5*a^2*C + b^2*C)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(2*b^2*(a^2 - b^2)*d) + ((12*a^2*b*B - 4*b^3*B - 15*a^3*C + 7*a*b^2*C)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(4*b^3*(a^2 - b^2)*d)],
[(sqrt(cos(c + d*x))*(B*cos(c + d*x) + C*cos(c + d*x)^2))/(a + b*cos(c + d*x))^(3/2), x, 7, -(((2*a*b*B - 3*a^2*C + b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/((a - b)*b^2*d*sqrt(a + b*cos(c + d*x)))) + (2*(b*B - a*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/((a - b)*b*d*sqrt(a + b*cos(c + d*x))) + (2*(2*b*B - 3*a*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(b^2*d*sqrt(a + b*cos(c + d*x))) + (2*a*(b*B - a*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(b*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) - ((2*a*b*B - 3*a^2*C + b^2*C)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(b^2*(a^2 - b^2)*d)],
[(B*cos(c + d*x) + C*cos(c + d*x)^2)/(sqrt(cos(c + d*x))*(a + b*cos(c + d*x))^(3/2)), x, 6, (2*(b*B - a*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/((a - b)*b*d*sqrt(a + b*cos(c + d*x))) - (2*(B - C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/((a - b)*d*sqrt(a + b*cos(c + d*x))) + (4*C*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(b*d*sqrt(a + b*cos(c + d*x))) + (2*(b*B - a*C)*sqrt(cos(c + d*x))*tan((c + d*x)/2))/(b*(a + b)*d*sqrt(a + b*cos(c + d*x)))],
[(B*cos(c + d*x) + C*cos(c + d*x)^2)/(cos(c + d*x)^(3/2)*(a + b*cos(c + d*x))^(3/2)), x, 5, (-2*(b*B - a*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(a*(a - b)*d*sqrt(a + b*cos(c + d*x))) + (2*(B - C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/((a - b)*d*sqrt(a + b*cos(c + d*x))) - (2*(b*B - a*C)*sin(c + d*x))/((a^2 - b^2)*d*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))) + (2*(b*B - a*C)*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(a*(a^2 - b^2)*d*sqrt(cos(c + d*x)))],
[(B*cos(c + d*x) + C*cos(c + d*x)^2)/(cos(c + d*x)^(5/2)*(a + b*cos(c + d*x))^(3/2)), x, 6, (-2*(a^2*B - 2*b^2*B + a*b*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(a^2*(a - b)*d*sqrt(a + b*cos(c + d*x))) + (2*(a*B - 2*b*B + a*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(a*(a - b)*d*sqrt(a + b*cos(c + d*x))) - (2*(b*B - a*C)*sin(c + d*x))/((a^2 - b^2)*d*cos(c + d*x)^(3/2)*sqrt(a + b*cos(c + d*x))) + (2*(b*B - a*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(a*(a^2 - b^2)*d*cos(c + d*x)^(3/2)) + (2*(a^2*B - 2*b^2*B + a*b*C)*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(a^2*(a^2 - b^2)*d*sqrt(cos(c + d*x)))],
[(B*cos(c + d*x) + C*cos(c + d*x)^2)/(cos(c + d*x)^(7/2)*(a + b*cos(c + d*x))^(3/2)), x, 7, (2*(5*a^2*b*B - 8*b^3*B - 3*a^3*C + 6*a*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(3*a^3*(a - b)*d*sqrt(a + b*cos(c + d*x))) + (2*(a^2*B - 6*a*b*B + 8*b^2*B + 3*a^2*C - 6*a*b*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(3*a^2*(a - b)*d*sqrt(a + b*cos(c + d*x))) - (2*(b*B - a*C)*sin(c + d*x))/((a^2 - b^2)*d*cos(c + d*x)^(5/2)*sqrt(a + b*cos(c + d*x))) + (2*(b*B - a*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(a*(a^2 - b^2)*d*cos(c + d*x)^(5/2)) + (2*(a^2*B - 4*b^2*B + 3*a*b*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*a^2*(a^2 - b^2)*d*cos(c + d*x)^(3/2)) - (2*(5*a^2*b*B - 8*b^3*B - 3*a^3*C + 6*a*b^2*C)*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(3*a^3*(a^2 - b^2)*d*sqrt(cos(c + d*x)))],


[(cos(c + d*x)^(3/2)*(B*cos(c + d*x) + C*cos(c + d*x)^2))/(a + b*cos(c + d*x))^(5/2), x, 8, -((6*a^3*b*B - 14*a*b^3*B - 15*a^4*C + 26*a^2*b^2*C - 3*b^4*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(3*(a - b)^2*b^3*(a + b)*d*sqrt(a + b*cos(c + d*x))) + (2*(2*a^2*b*B - 3*a*b^2*B - 3*b^3*B - 5*a^3*C + 3*a^2*b*C + 6*a*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(3*(a - b)^2*b^2*(a + b)*d*sqrt(a + b*cos(c + d*x))) + (2*(2*b*B - 5*a*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(b^3*d*sqrt(a + b*cos(c + d*x))) + (2*a*(b*B - a*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)) + (2*a*(2*a^2*b*B - 6*b^3*B - 5*a^3*C + 9*a*b^2*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(3*b^2*(a^2 - b^2)^2*d*sqrt(a + b*cos(c + d*x))) - ((6*a^3*b*B - 14*a*b^3*B - 15*a^4*C + 26*a^2*b^2*C - 3*b^4*C)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(3*b^3*(a^2 - b^2)^2*d)],
[(sqrt(cos(c + d*x))*(B*cos(c + d*x) + C*cos(c + d*x)^2))/(a + b*cos(c + d*x))^(5/2), x, 7, (-2*(4*b^3*B + 3*a^3*C - 7*a*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(3*(a - b)^2*b^2*(a + b)*d*sqrt(a + b*cos(c + d*x))) + (2*(a*b*B + 3*b^2*B + 2*a^2*C - 3*a*b*C - 3*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(3*(a - b)^2*b*(a + b)*d*sqrt(a + b*cos(c + d*x))) + (4*C*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(b^2*d*sqrt(a + b*cos(c + d*x))) + (2*a*(b*B - a*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)) - (2*(4*b^3*B + 3*a^3*C - 7*a*b^2*C)*sqrt(cos(c + d*x))*tan((c + d*x)/2))/(3*(a - b)*b^2*(a + b)^2*d*sqrt(a + b*cos(c + d*x)))],
[(B*cos(c + d*x) + C*cos(c + d*x)^2)/(sqrt(cos(c + d*x))*(a + b*cos(c + d*x))^(5/2)), x, 5, (2*(3*a^2*B + b^2*B - 4*a*b*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(3*a*(a - b)^2*(a + b)*d*sqrt(a + b*cos(c + d*x))) - (2*(3*a*B + b*B - a*C - 3*b*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(3*(a - b)^2*(a + b)*d*sqrt(a + b*cos(c + d*x))) - (2*(b*B - a*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(3*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)) + (2*(3*a^2*B + b^2*B - 4*a*b*C)*sqrt(cos(c + d*x))*tan((c + d*x)/2))/(3*a*(a - b)*(a + b)^2*d*sqrt(a + b*cos(c + d*x)))],
[(B*cos(c + d*x) + C*cos(c + d*x)^2)/(cos(c + d*x)^(3/2)*(a + b*cos(c + d*x))^(5/2)), x, 6, (-2*(6*a^2*b*B - 2*b^3*B - 3*a^3*C - a*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(3*a^2*(a - b)^2*(a + b)*d*sqrt(a + b*cos(c + d*x))) + (2*(3*a^2*B + 3*a*b*B - 2*b^2*B - 3*a^2*C - a*b*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(3*a*(a - b)^2*(a + b)*d*sqrt(a + b*cos(c + d*x))) - (2*(b*B - a*C)*sin(c + d*x))/(3*(a^2 - b^2)*d*sqrt(cos(c + d*x))*(a + b*cos(c + d*x))^(3/2)) - (2*(5*a^2*b*B - b^3*B - 2*a^3*C - 2*a*b^2*C)*sin(c + d*x))/(3*a*(a^2 - b^2)^2*d*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))) + (2*(6*a^2*b*B - 2*b^3*B - 3*a^3*C - a*b^2*C)*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(3*a^2*(a^2 - b^2)^2*d*sqrt(cos(c + d*x)))],
[(B*cos(c + d*x) + C*cos(c + d*x)^2)/(cos(c + d*x)^(5/2)*(a + b*cos(c + d*x))^(5/2)), x, 7, (-2*(3*a^4*B - 15*a^2*b^2*B + 8*b^4*B + 6*a^3*b*C - 2*a*b^3*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(3*a^3*(a - b)^2*(a + b)*d*sqrt(a + b*cos(c + d*x))) + (2*(3*a^3*B - 9*a^2*b*B - 6*a*b^2*B + 8*b^3*B + 3*a^3*C + 3*a^2*b*C - 2*a*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b))))/(3*a^2*(a - b)^2*(a + b)*d*sqrt(a + b*cos(c + d*x))) - (2*(b*B - a*C)*sin(c + d*x))/(3*(a^2 - b^2)*d*cos(c + d*x)^(3/2)*(a + b*cos(c + d*x))^(3/2)) - (2*(7*a^2*b*B - 3*b^3*B - 4*a^3*C)*sin(c + d*x))/(3*a*(a^2 - b^2)^2*d*cos(c + d*x)^(3/2)*sqrt(a + b*cos(c + d*x))) + (2*(8*a^2*b*B - 4*b^3*B - 5*a^3*C + a*b^2*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*a^2*(a^2 - b^2)^2*d*cos(c + d*x)^(3/2)) + (2*(3*a^4*B - 15*a^2*b^2*B + 8*b^4*B + 6*a^3*b*C - 2*a*b^3*C)*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(3*a^3*(a^2 - b^2)^2*d*sqrt(cos(c + d*x)))],


# ::Subsection::Closed:: 
#Integrands of the form Cos[c+d x]^(m/2) (B Cos[c+d x]+C Cos[c+d x]^2) (a+b Cos[c+d x])^(n/3)
#


[(B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(2/3), x, 2, Int((B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(2/3), x)],
[(B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(1/3), x, 2, Int((B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(1/3), x)],
[(B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(1/3), x, 1, Int((B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(1/3), x)],
[(B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(2/3), x, 1, Int((B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(2/3), x)],


# ::Subsection:: 
#Integrands of the form Sec[c+d x]^(m/2) (B Cos[c+d x]+C Cos[c+d x]^2) (a+b Cos[c+d x])^n


# ::Subsection:: 
#Integrands of the form Sec[c+d x]^(m/2) (B Cos[c+d x]+C Cos[c+d x]^2) (a+b Cos[c+d x])^(n/2)
#


# ::Section:: 
#Cos[c+d x]^m (A+B Cos[c+d x]+C Cos[c+d x]^2) (a+b Cos[c+d x])^n


# ::Subsection::Closed:: 
#Integrands of the form Cos[c+d x]^m (A+B Cos[c+d x]+C Cos[c+d x]^2) (a+b Cos[c+d x])^n
#


# ::Subsubsection::Closed:: 
#n>0


[cos(c + d*x)^2*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x)), x, 5, (1/8)*(4*a*A + 3*b*B + 3*a*C)*x + (2*(5*A*b + 5*a*B + 4*b*C)*sin(c + d*x))/(15*d) + ((4*a*A + 3*b*B + 3*a*C)*cos(c + d*x)*sin(c + d*x))/(8*d) + ((5*A*b + 5*a*B + 4*b*C)*cos(c + d*x)^2*sin(c + d*x))/(15*d) + ((5*b*B + a*C)*cos(c + d*x)^3*sin(c + d*x))/(20*d) + (C*cos(c + d*x)^3*(a + b*cos(c + d*x))*sin(c + d*x))/(5*d)],
[cos(c + d*x)^1*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x)), x, 4, (1/8)*(4*A*b + 4*a*B + 3*b*C)*x + ((3*a*A + 2*b*B + 2*a*C)*sin(c + d*x))/(3*d) + ((4*A*b + 4*a*B + 3*b*C)*cos(c + d*x)*sin(c + d*x))/(8*d) + ((4*b*B + a*C)*cos(c + d*x)^2*sin(c + d*x))/(12*d) + (C*cos(c + d*x)^2*(a + b*cos(c + d*x))*sin(c + d*x))/(4*d)],
[cos(c + d*x)^0*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x)), x, 3, (1/2)*(2*a*A + b*B + a*C)*x + ((3*A*b^2 + 3*a*b*B - a^2*C + 2*b^2*C)*sin(c + d*x))/(3*b*d) + ((3*b*B - a*C)*cos(c + d*x)*sin(c + d*x))/(6*d) + (C*(a + b*cos(c + d*x))^2*sin(c + d*x))/(3*b*d)],
[sec(c + d*x)^1*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x)), x, 4, ((2*A*b + 2*a*B + b*C)*x)/2 + (a*A*arctanh(sin(c + d*x)))/d + ((2*b*B + a*C)*sin(c + d*x))/(2*d) + (C*(a + b*cos(c + d*x))*sin(c + d*x))/(2*d)],
[sec(c + d*x)^2*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x)), x, 4, (b*B + a*C)*x + ((A*b + a*B)*arctanh(sin(c + d*x)))/d - (b*(A - C)*sin(c + d*x))/d + (A*(a + b*cos(c + d*x))*tan(c + d*x))/d],
[sec(c + d*x)^3*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x)), x, 4, b*C*x + ((a*A + 2*b*B + 2*a*C)*arctanh(sin(c + d*x)))/(2*d) + ((A*b + 2*a*B)*tan(c + d*x))/(2*d) + (A*(b + a*sec(c + d*x))*tan(c + d*x))/(2*d)],
[sec(c + d*x)^4*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x)), x, 5, ((A*b + a*B + 2*b*C)*arctanh(sin(c + d*x)))/(2*d) + ((2*a*A + 3*b*B + 3*a*C)*tan(c + d*x))/(3*d) + ((A*b + 3*a*B)*sec(c + d*x)*tan(c + d*x))/(6*d) + (A*(a + b*cos(c + d*x))*sec(c + d*x)^2*tan(c + d*x))/(3*d)],
[sec(c + d*x)^5*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x)), x, 6, ((3*a*A + 4*b*B + 4*a*C)*arctanh(sin(c + d*x)))/(8*d) + ((2*A*b + 2*a*B + 3*b*C)*tan(c + d*x))/(3*d) + ((3*a*A + 4*b*B + 4*a*C)*sec(c + d*x)*tan(c + d*x))/(8*d) + ((A*b + 4*a*B)*sec(c + d*x)^2*tan(c + d*x))/(12*d) + (A*(a + b*cos(c + d*x))*sec(c + d*x)^3*tan(c + d*x))/(4*d)],
[sec(c + d*x)^6*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x)), x, 7, ((3*A*b + 3*a*B + 4*b*C)*arctanh(sin(c + d*x)))/(8*d) + (2*(4*a*A + 5*b*B + 5*a*C)*tan(c + d*x))/(15*d) + ((3*A*b + 3*a*B + 4*b*C)*sec(c + d*x)*tan(c + d*x))/(8*d) + ((4*a*A + 5*b*B + 5*a*C)*sec(c + d*x)^2*tan(c + d*x))/(15*d) + ((A*b + 5*a*B)*sec(c + d*x)^3*tan(c + d*x))/(20*d) + (A*(a + b*cos(c + d*x))*sec(c + d*x)^4*tan(c + d*x))/(5*d)],


[cos(c + d*x)^1*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^2, x, 5, (1/8)*(8*a*A*b + 4*a^2*B + 3*b^2*B + 6*a*b*C)*x + ((15*a^2*A + 10*A*b^2 + 20*a*b*B + 10*a^2*C + 8*b^2*C)*sin(c + d*x))/(15*d) + ((8*a*A*b + 4*a^2*B + 3*b^2*B + 6*a*b*C)*cos(c + d*x)*sin(c + d*x))/(8*d) + ((20*A*b^2 + 25*a*b*B + 2*a^2*C + 16*b^2*C)*cos(c + d*x)^2*sin(c + d*x))/(60*d) + ((5*b*B + 2*a*C)*cos(c + d*x)^2*(a + b*cos(c + d*x))*sin(c + d*x))/(20*d) + (C*cos(c + d*x)^2*(a + b*cos(c + d*x))^2*sin(c + d*x))/(5*d)],
[cos(c + d*x)^0*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^2, x, 4, (1/8)*(8*a^2*A + 4*A*b^2 + 8*a*b*B + 4*a^2*C + 3*b^2*C)*x + ((12*a*A*b^2 + 4*a^2*b*B + 4*b^3*B - a^3*C + 8*a*b^2*C)*sin(c + d*x))/(6*b*d) + ((12*A*b^2 + 8*a*b*B - 2*a^2*C + 9*b^2*C)*cos(c + d*x)*sin(c + d*x))/(24*d) + ((4*b*B - a*C)*(a + b*cos(c + d*x))^2*sin(c + d*x))/(12*b*d) + (C*(a + b*cos(c + d*x))^3*sin(c + d*x))/(4*b*d)],
[sec(c + d*x)^1*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^2, x, 5, ((4*a*A*b + 2*a^2*B + b^2*B + 2*a*b*C)*x)/2 + (a^2*A*arctanh(sin(c + d*x)))/d + ((6*A*b^2 + 9*a*b*B + 2*a^2*C + 4*b^2*C)*sin(c + d*x))/(6*d) + ((3*b*B + 2*a*C)*(a + b*cos(c + d*x))*sin(c + d*x))/(6*d) + (C*(a + b*cos(c + d*x))^2*sin(c + d*x))/(3*d)],
[sec(c + d*x)^2*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^2, x, 5, ((2*A*b^2 + 4*a*b*B + 2*a^2*C + b^2*C)*x)/2 + (a*(2*A*b + a*B)*arctanh(sin(c + d*x)))/d - (b*(2*a*A - 2*b*B - 3*a*C)*sin(c + d*x))/(2*d) - (b*(2*A - C)*(a + b*cos(c + d*x))*sin(c + d*x))/(2*d) + (A*(a + b*cos(c + d*x))^2*tan(c + d*x))/d],
[sec(c + d*x)^3*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^2, x, 5, (b^2*B + 2*a*b*C)*x + ((a^2*A + 2*A*b^2 + 4*a*b*B + 2*a^2*C)*arctanh(sin(c + d*x)))/(2*d) - (b*(3*A*b + 2*a*B - 2*b*C)*sin(c + d*x))/(2*d) + ((A*b + a*B)*(a + b*cos(c + d*x))*tan(c + d*x))/d + (A*(a + b*cos(c + d*x))^2*sec(c + d*x)*tan(c + d*x))/(2*d)],
[sec(c + d*x)^4*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^2, x, 5, b^2*C*x + ((2*a*A*b + a^2*B + 2*b^2*B + 4*a*b*C)*arctanh(sin(c + d*x)))/(2*d) + ((4*a^2*A + 2*A*b^2 + 9*a*b*B + 6*a^2*C)*tan(c + d*x))/(6*d) + ((2*A*b + 3*a*B)*(b + a*sec(c + d*x))*tan(c + d*x))/(6*d) + (A*(b + a*sec(c + d*x))^2*tan(c + d*x))/(3*d)],
[sec(c + d*x)^5*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^2, x, 6, ((3*a^2*A + 4*A*b^2 + 8*a*b*B + 4*a^2*C + 8*b^2*C)*arctanh(sin(c + d*x)))/(8*d) + ((4*a*A*b + 2*a^2*B + 3*b^2*B + 6*a*b*C)*tan(c + d*x))/(3*d) + ((9*a^2*A + 2*A*b^2 + 16*a*b*B + 12*a^2*C)*sec(c + d*x)*tan(c + d*x))/(24*d) + ((A*b + 2*a*B)*(a + b*cos(c + d*x))*sec(c + d*x)^2*tan(c + d*x))/(6*d) + (A*(a + b*cos(c + d*x))^2*sec(c + d*x)^3*tan(c + d*x))/(4*d)],
[sec(c + d*x)^6*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^2, x, 7, ((6*a*A*b + 3*a^2*B + 4*b^2*B + 8*a*b*C)*arctanh(sin(c + d*x)))/(8*d) + ((8*a^2*A + 10*A*b^2 + 20*a*b*B + 10*a^2*C + 15*b^2*C)*tan(c + d*x))/(15*d) + ((6*a*A*b + 3*a^2*B + 4*b^2*B + 8*a*b*C)*sec(c + d*x)*tan(c + d*x))/(8*d) + ((16*a^2*A + 2*A*b^2 + 25*a*b*B + 20*a^2*C)*sec(c + d*x)^2*tan(c + d*x))/(60*d) + ((2*A*b + 5*a*B)*(a + b*cos(c + d*x))*sec(c + d*x)^3*tan(c + d*x))/(20*d) + (A*(a + b*cos(c + d*x))^2*sec(c + d*x)^4*tan(c + d*x))/(5*d)],


[cos(c + d*x)^1*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^3, x, 6, (1/16)*(24*a^2*A*b + 6*A*b^3 + 8*a^3*B + 18*a*b^2*B + 18*a^2*b*C + 5*b^3*C)*x + ((15*a^3*A + 30*a*A*b^2 + 30*a^2*b*B + 8*b^3*B + 10*a^3*C + 24*a*b^2*C)*sin(c + d*x))/(15*d) + ((24*a^2*A*b + 6*A*b^3 + 8*a^3*B + 18*a*b^2*B + 18*a^2*b*C + 5*b^3*C)*cos(c + d*x)*sin(c + d*x))/(16*d) + ((90*a*A*b^2 + 54*a^2*b*B + 32*b^3*B + 2*a^3*C + 71*a*b^2*C)*cos(c + d*x)^2*sin(c + d*x))/(120*d) + ((30*A*b^2 + 42*a*b*B + 6*a^2*C + 25*b^2*C)*cos(c + d*x)^2*(a + b*cos(c + d*x))*sin(c + d*x))/(120*d) + ((2*b*B + a*C)*cos(c + d*x)^2*(a + b*cos(c + d*x))^2*sin(c + d*x))/(10*d) + (C*cos(c + d*x)^2*(a + b*cos(c + d*x))^3*sin(c + d*x))/(6*d)],
[cos(c + d*x)^0*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^3, x, 5, (1/8)*(8*a^3*A + 12*a*A*b^2 + 12*a^2*b*B + 3*b^3*B + 4*a^3*C + 9*a*b^2*C)*x + ((80*a^2*A*b^2 + 20*A*b^4 + 15*a^3*b*B + 60*a*b^3*B - 3*a^4*C + 52*a^2*b^2*C + 16*b^4*C)*sin(c + d*x))/(30*b*d) + ((100*a*A*b^2 + 30*a^2*b*B + 45*b^3*B - 6*a^3*C + 71*a*b^2*C)*cos(c + d*x)*sin(c + d*x))/(120*d) + ((20*A*b^2 + 15*a*b*B - 3*a^2*C + 16*b^2*C)*(a + b*cos(c + d*x))^2*sin(c + d*x))/(60*b*d) + ((5*b*B - a*C)*(a + b*cos(c + d*x))^3*sin(c + d*x))/(20*b*d) + (C*(a + b*cos(c + d*x))^4*sin(c + d*x))/(5*b*d)],
[sec(c + d*x)^1*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^3, x, 6, ((24*a^2*A*b + 4*A*b^3 + 8*a^3*B + 12*a*b^2*B + 12*a^2*b*C + 3*b^3*C)*x)/8 + (a^3*A*arctanh(sin(c + d*x)))/d + ((60*a*A*b^2 + 44*a^2*b*B + 16*b^3*B + 6*a^3*C + 39*a*b^2*C)*sin(c + d*x))/(24*d) + ((12*A*b^2 + 20*a*b*B + 6*a^2*C + 9*b^2*C)*(a + b*cos(c + d*x))*sin(c + d*x))/(24*d) + ((4*b*B + 3*a*C)*(a + b*cos(c + d*x))^2*sin(c + d*x))/(12*d) + (C*(a + b*cos(c + d*x))^3*sin(c + d*x))/(4*d)],
[sec(c + d*x)^2*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^3, x, 6, ((6*a*A*b^2 + 6*a^2*b*B + b^3*B + 2*a^3*C + 3*a*b^2*C)*x)/2 + (a^2*(3*A*b + a*B)*arctanh(sin(c + d*x)))/d - (b*(6*a^2*A - 6*A*b^2 - 15*a*b*B - 11*a^2*C - 4*b^2*C)*sin(c + d*x))/(6*d) - (b*(6*a*A - 3*b*B - 5*a*C)*(a + b*cos(c + d*x))*sin(c + d*x))/(6*d) - (b*(3*A - C)*(a + b*cos(c + d*x))^2*sin(c + d*x))/(3*d) + (A*(a + b*cos(c + d*x))^3*tan(c + d*x))/d],
[sec(c + d*x)^3*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^3, x, 6, (b*(2*A*b^2 + 6*a*b*B + 6*a^2*C + b^2*C)*x)/2 + (a*(a^2*A + 6*A*b^2 + 6*a*b*B + 2*a^2*C)*arctanh(sin(c + d*x)))/(2*d) - (b*(5*a*A*b + 2*a^2*B - 2*b^2*B - 5*a*b*C)*sin(c + d*x))/(2*d) - (b*(4*A*b + 2*a*B - b*C)*(a + b*cos(c + d*x))*sin(c + d*x))/(2*d) + ((3*A*b + 2*a*B)*(a + b*cos(c + d*x))^2*tan(c + d*x))/(2*d) + (A*(a + b*cos(c + d*x))^3*sec(c + d*x)*tan(c + d*x))/(2*d)],
[sec(c + d*x)^4*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^3, x, 6, (b^3*B + 3*a*b^2*C)*x + ((3*a^2*A*b + 2*A*b^3 + a^3*B + 6*a*b^2*B + 6*a^2*b*C)*arctanh(sin(c + d*x)))/(2*d) - (b*(4*a^2*A + 11*A*b^2 + 15*a*b*B + 6*a^2*C - 6*b^2*C)*sin(c + d*x))/(6*d) + ((2*a^2*A + 3*A*b^2 + 6*a*b*B + 3*a^2*C)*(a + b*cos(c + d*x))*tan(c + d*x))/(3*d) + ((A*b + a*B)*(a + b*cos(c + d*x))^2*sec(c + d*x)*tan(c + d*x))/(2*d) + (A*(a + b*cos(c + d*x))^3*sec(c + d*x)^2*tan(c + d*x))/(3*d)],
[sec(c + d*x)^5*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^3, x, 6, b^3*C*x + ((3*a^3*A + 12*a*A*b^2 + 12*a^2*b*B + 8*b^3*B + 4*a^3*C + 24*a*b^2*C)*arctanh(sin(c + d*x)))/(8*d) + ((39*a^2*A*b + 6*A*b^3 + 16*a^3*B + 44*a*b^2*B + 60*a^2*b*C)*tan(c + d*x))/(24*d) + ((9*a^2*A + 6*A*b^2 + 20*a*b*B + 12*a^2*C)*(b + a*sec(c + d*x))*tan(c + d*x))/(24*d) + ((3*A*b + 4*a*B)*(b + a*sec(c + d*x))^2*tan(c + d*x))/(12*d) + (A*(b + a*sec(c + d*x))^3*tan(c + d*x))/(4*d)],
[sec(c + d*x)^6*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^3, x, 7, ((9*a^2*A*b + 4*A*b^3 + 3*a^3*B + 12*a*b^2*B + 12*a^2*b*C + 8*b^3*C)*arctanh(sin(c + d*x)))/(8*d) + ((8*a^3*A + 30*a*A*b^2 + 30*a^2*b*B + 15*b^3*B + 10*a^3*C + 45*a*b^2*C)*tan(c + d*x))/(15*d) + ((103*a^2*A*b + 6*A*b^3 + 45*a^3*B + 90*a*b^2*B + 140*a^2*b*C)*sec(c + d*x)*tan(c + d*x))/(120*d) + ((8*a^2*A + 3*A*b^2 + 15*a*b*B + 10*a^2*C)*(a + b*cos(c + d*x))*sec(c + d*x)^2*tan(c + d*x))/(30*d) + ((3*A*b + 5*a*B)*(a + b*cos(c + d*x))^2*sec(c + d*x)^3*tan(c + d*x))/(20*d) + (A*(a + b*cos(c + d*x))^3*sec(c + d*x)^4*tan(c + d*x))/(5*d)],
[sec(c + d*x)^7*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^3, x, 8, ((5*a^3*A + 18*a*A*b^2 + 18*a^2*b*B + 8*b^3*B + 6*a^3*C + 24*a*b^2*C)*arctanh(sin(c + d*x)))/(16*d) + ((24*a^2*A*b + 10*A*b^3 + 8*a^3*B + 30*a*b^2*B + 30*a^2*b*C + 15*b^3*C)*tan(c + d*x))/(15*d) + ((5*a^3*A + 18*a*A*b^2 + 18*a^2*b*B + 8*b^3*B + 6*a^3*C + 24*a*b^2*C)*sec(c + d*x)*tan(c + d*x))/(16*d) + ((71*a^2*A*b + 2*A*b^3 + 32*a^3*B + 54*a*b^2*B + 90*a^2*b*C)*sec(c + d*x)^2*tan(c + d*x))/(120*d) + ((25*a^2*A + 6*A*b^2 + 42*a*b*B + 30*a^2*C)*(a + b*cos(c + d*x))*sec(c + d*x)^3*tan(c + d*x))/(120*d) + ((A*b + 2*a*B)*(a + b*cos(c + d*x))^2*sec(c + d*x)^4*tan(c + d*x))/(10*d) + (A*(a + b*cos(c + d*x))^3*sec(c + d*x)^5*tan(c + d*x))/(6*d)],


[cos(c + d*x)^1*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^4, x, 7, (1/16)*(32*a^3*A*b + 24*a*A*b^3 + 8*a^4*B + 36*a^2*b^2*B + 5*b^4*B + 24*a^3*b*C + 20*a*b^3*C)*x + ((105*a^4*A + 420*a^2*A*b^2 + 56*A*b^4 + 280*a^3*b*B + 224*a*b^3*B + 70*a^4*C + 336*a^2*b^2*C + 48*b^4*C)*sin(c + d*x))/(105*d) + ((32*a^3*A*b + 24*a*A*b^3 + 8*a^4*B + 36*a^2*b^2*B + 5*b^4*B + 24*a^3*b*C + 20*a*b^3*C)*cos(c + d*x)*sin(c + d*x))/(16*d) + ((1008*a^2*A*b^2 + 224*A*b^4 + 392*a^3*b*B + 721*a*b^3*B + 8*a^4*C + 788*a^2*b^2*C + 192*b^4*C)*cos(c + d*x)^2*sin(c + d*x))/(840*d) + ((504*a*A*b^2 + 336*a^2*b*B + 175*b^3*B + 24*a^3*C + 412*a*b^2*C)*cos(c + d*x)^2*(a + b*cos(c + d*x))*sin(c + d*x))/(840*d) + ((14*A*b^2 + 21*a*b*B + 4*a^2*C + 12*b^2*C)*cos(c + d*x)^2*(a + b*cos(c + d*x))^2*sin(c + d*x))/(70*d) + ((7*b*B + 4*a*C)*cos(c + d*x)^2*(a + b*cos(c + d*x))^3*sin(c + d*x))/(42*d) + (C*cos(c + d*x)^2*(a + b*cos(c + d*x))^4*sin(c + d*x))/(7*d)],
[cos(c + d*x)^0*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^4, x, 6, (1/16)*(16*a^4*A + 48*a^2*A*b^2 + 6*A*b^4 + 32*a^3*b*B + 24*a*b^3*B + 8*a^4*C + 36*a^2*b^2*C + 5*b^4*C)*x + ((190*a^3*A*b^2 + 160*a*A*b^4 + 24*a^4*b*B + 224*a^2*b^3*B + 32*b^5*B - 4*a^5*C + 121*a^3*b^2*C + 128*a*b^4*C)*sin(c + d*x))/(60*b*d) + ((260*a^2*A*b^2 + 90*A*b^4 + 48*a^3*b*B + 232*a*b^3*B - 8*a^4*C + 178*a^2*b^2*C + 75*b^4*C)*cos(c + d*x)*sin(c + d*x))/(240*d) + ((70*a*A*b^2 + 24*a^2*b*B + 32*b^3*B - 4*a^3*C + 53*a*b^2*C)*(a + b*cos(c + d*x))^2*sin(c + d*x))/(120*b*d) + ((30*A*b^2 + 24*a*b*B - 4*a^2*C + 25*b^2*C)*(a + b*cos(c + d*x))^3*sin(c + d*x))/(120*b*d) + ((6*b*B - a*C)*(a + b*cos(c + d*x))^4*sin(c + d*x))/(30*b*d) + (C*(a + b*cos(c + d*x))^5*sin(c + d*x))/(6*b*d)],
[sec(c + d*x)^1*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^4, x, 7, ((32*a^3*A*b + 16*a*A*b^3 + 8*a^4*B + 24*a^2*b^2*B + 3*b^4*B + 16*a^3*b*C + 12*a*b^3*C)*x)/8 + (a^4*A*arctanh(sin(c + d*x)))/d + ((520*a^2*A*b^2 + 80*A*b^4 + 250*a^3*b*B + 275*a*b^3*B + 24*a^4*C + 332*a^2*b^2*C + 64*b^4*C)*sin(c + d*x))/(120*d) + ((160*a*A*b^2 + 130*a^2*b*B + 45*b^3*B + 24*a^3*C + 116*a*b^2*C)*(a + b*cos(c + d*x))*sin(c + d*x))/(120*d) + ((20*A*b^2 + 35*a*b*B + 12*a^2*C + 16*b^2*C)*(a + b*cos(c + d*x))^2*sin(c + d*x))/(60*d) + ((5*b*B + 4*a*C)*(a + b*cos(c + d*x))^3*sin(c + d*x))/(20*d) + (C*(a + b*cos(c + d*x))^4*sin(c + d*x))/(5*d)],
[sec(c + d*x)^2*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^4, x, 7, ((48*a^2*A*b^2 + 4*A*b^4 + 32*a^3*b*B + 16*a*b^3*B + 8*a^4*C + 24*a^2*b^2*C + 3*b^4*C)*x)/8 + (a^3*(4*A*b + a*B)*arctanh(sin(c + d*x)))/d - (b*(24*a^3*A - 84*a*A*b^2 - 104*a^2*b*B - 16*b^3*B - 50*a^3*C - 55*a*b^2*C)*sin(c + d*x))/(24*d) - (b*(24*a^2*A - 12*A*b^2 - 32*a*b*B - 26*a^2*C - 9*b^2*C)*(a + b*cos(c + d*x))*sin(c + d*x))/(24*d) - (b*(12*a*A - 4*b*B - 7*a*C)*(a + b*cos(c + d*x))^2*sin(c + d*x))/(12*d) - (b*(4*A - C)*(a + b*cos(c + d*x))^3*sin(c + d*x))/(4*d) + (A*(a + b*cos(c + d*x))^4*tan(c + d*x))/d],
[sec(c + d*x)^3*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^4, x, 7, (b*(8*a*A*b^2 + 12*a^2*b*B + b^3*B + 8*a^3*C + 4*a*b^2*C)*x)/2 + (a^2*(a^2*A + 12*A*b^2 + 8*a*b*B + 2*a^2*C)*arctanh(sin(c + d*x)))/(2*d) - (b*(21*a^2*A*b - 6*A*b^3 + 6*a^3*B - 21*a*b^2*B - 26*a^2*b*C - 4*b^3*C)*sin(c + d*x))/(6*d) - (b*(18*a*A*b + 6*a^2*B - 3*b^2*B - 8*a*b*C)*(a + b*cos(c + d*x))*sin(c + d*x))/(6*d) - (b*(15*A*b + 6*a*B - 2*b*C)*(a + b*cos(c + d*x))^2*sin(c + d*x))/(6*d) + ((2*A*b + a*B)*(a + b*cos(c + d*x))^3*tan(c + d*x))/d + (A*(a + b*cos(c + d*x))^4*sec(c + d*x)*tan(c + d*x))/(2*d)],
[sec(c + d*x)^4*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^4, x, 7, (b^2*(2*A*b^2 + 8*a*b*B + 12*a^2*C + b^2*C)*x)/2 + (a*(4*a^2*A*b + 8*A*b^3 + a^3*B + 12*a*b^2*B + 8*a^2*b*C)*arctanh(sin(c + d*x)))/(2*d) - (b*(4*a^3*A + 26*a*A*b^2 + 21*a^2*b*B - 6*b^3*B + 6*a^3*C - 21*a*b^2*C)*sin(c + d*x))/(6*d) - (b*(4*a^2*A + 18*A*b^2 + 18*a*b*B + 6*a^2*C - 3*b^2*C)*(a + b*cos(c + d*x))*sin(c + d*x))/(6*d) + ((4*a^2*A + 12*A*b^2 + 15*a*b*B + 6*a^2*C)*(a + b*cos(c + d*x))^2*tan(c + d*x))/(6*d) + ((4*A*b + 3*a*B)*(a + b*cos(c + d*x))^3*sec(c + d*x)*tan(c + d*x))/(6*d) + (A*(a + b*cos(c + d*x))^4*sec(c + d*x)^2*tan(c + d*x))/(3*d)],
[sec(c + d*x)^5*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^4, x, 7, (b^4*B + 4*a*b^3*C)*x + ((3*a^4*A + 24*a^2*A*b^2 + 8*A*b^4 + 16*a^3*b*B + 32*a*b^3*B + 4*a^4*C + 48*a^2*b^2*C)*arctanh(sin(c + d*x)))/(8*d) - (b*(55*a^2*A*b + 50*A*b^3 + 16*a^3*B + 104*a*b^2*B + 84*a^2*b*C - 24*b^3*C)*sin(c + d*x))/(24*d) + ((23*a^2*A*b + 12*A*b^3 + 8*a^3*B + 36*a*b^2*B + 36*a^2*b*C)*(a + b*cos(c + d*x))*tan(c + d*x))/(12*d) + ((3*a^2*A + 4*A*b^2 + 8*a*b*B + 4*a^2*C)*(a + b*cos(c + d*x))^2*sec(c + d*x)*tan(c + d*x))/(8*d) + ((A*b + a*B)*(a + b*cos(c + d*x))^3*sec(c + d*x)^2*tan(c + d*x))/(3*d) + (A*(a + b*cos(c + d*x))^4*sec(c + d*x)^3*tan(c + d*x))/(4*d)],
[sec(c + d*x)^6*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^4, x, 7, b^4*C*x + ((12*a^3*A*b + 16*a*A*b^3 + 3*a^4*B + 24*a^2*b^2*B + 8*b^4*B + 16*a^3*b*C + 32*a*b^3*C)*arctanh(sin(c + d*x)))/(8*d) + ((64*a^4*A + 332*a^2*A*b^2 + 24*A*b^4 + 275*a^3*b*B + 250*a*b^3*B + 80*a^4*C + 520*a^2*b^2*C)*tan(c + d*x))/(120*d) + ((116*a^2*A*b + 24*A*b^3 + 45*a^3*B + 130*a*b^2*B + 160*a^2*b*C)*(b + a*sec(c + d*x))*tan(c + d*x))/(120*d) + ((16*a^2*A + 12*A*b^2 + 35*a*b*B + 20*a^2*C)*(b + a*sec(c + d*x))^2*tan(c + d*x))/(60*d) + ((4*A*b + 5*a*B)*(b + a*sec(c + d*x))^3*tan(c + d*x))/(20*d) + (A*(b + a*sec(c + d*x))^4*tan(c + d*x))/(5*d)],
[sec(c + d*x)^7*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^4, x, 8, ((5*a^4*A + 36*a^2*A*b^2 + 8*A*b^4 + 24*a^3*b*B + 32*a*b^3*B + 6*a^4*C + 48*a^2*b^2*C + 16*b^4*C)*arctanh(sin(c + d*x)))/(16*d) + ((32*a^3*A*b + 40*a*A*b^3 + 8*a^4*B + 60*a^2*b^2*B + 15*b^4*B + 40*a^3*b*C + 60*a*b^3*C)*tan(c + d*x))/(15*d) + ((75*a^4*A + 334*a^2*A*b^2 + 8*A*b^4 + 296*a^3*b*B + 192*a*b^3*B + 90*a^4*C + 460*a^2*b^2*C)*sec(c + d*x)*tan(c + d*x))/(240*d) + ((39*a^2*A*b + 4*A*b^3 + 16*a^3*B + 36*a*b^2*B + 50*a^2*b*C)*(a + b*cos(c + d*x))*sec(c + d*x)^2*tan(c + d*x))/(60*d) + ((25*a^2*A + 12*A*b^2 + 48*a*b*B + 30*a^2*C)*(a + b*cos(c + d*x))^2*sec(c + d*x)^3*tan(c + d*x))/(120*d) + ((2*A*b + 3*a*B)*(a + b*cos(c + d*x))^3*sec(c + d*x)^4*tan(c + d*x))/(15*d) + (A*(a + b*cos(c + d*x))^4*sec(c + d*x)^5*tan(c + d*x))/(6*d)],
[sec(c + d*x)^8*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^4, x, 9, ((20*a^3*A*b + 24*a*A*b^3 + 5*a^4*B + 36*a^2*b^2*B + 8*b^4*B + 24*a^3*b*C + 32*a*b^3*C)*arctanh(sin(c + d*x)))/(16*d) + ((48*a^4*A + 336*a^2*A*b^2 + 70*A*b^4 + 224*a^3*b*B + 280*a*b^3*B + 56*a^4*C + 420*a^2*b^2*C + 105*b^4*C)*tan(c + d*x))/(105*d) + ((20*a^3*A*b + 24*a*A*b^3 + 5*a^4*B + 36*a^2*b^2*B + 8*b^4*B + 24*a^3*b*C + 32*a*b^3*C)*sec(c + d*x)*tan(c + d*x))/(16*d) + ((192*a^4*A + 788*a^2*A*b^2 + 8*A*b^4 + 721*a^3*b*B + 392*a*b^3*B + 224*a^4*C + 1008*a^2*b^2*C)*sec(c + d*x)^2*tan(c + d*x))/(840*d) + ((412*a^2*A*b + 24*A*b^3 + 175*a^3*B + 336*a*b^2*B + 504*a^2*b*C)*(a + b*cos(c + d*x))*sec(c + d*x)^3*tan(c + d*x))/(840*d) + ((12*a^2*A + 4*A*b^2 + 21*a*b*B + 14*a^2*C)*(a + b*cos(c + d*x))^2*sec(c + d*x)^4*tan(c + d*x))/(70*d) + ((4*A*b + 7*a*B)*(a + b*cos(c + d*x))^3*sec(c + d*x)^5*tan(c + d*x))/(42*d) + (A*(a + b*cos(c + d*x))^4*sec(c + d*x)^6*tan(c + d*x))/(7*d)],


[(a*b*B - a^2*C + b^2*B*cos(c + d*x) + b^2*C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^3, x, 5, (1/8)*(8*a^4*b*B + 24*a^2*b^3*B + 3*b^5*B - 8*a^5*C - 8*a^3*b^2*C + 9*a*b^4*C)*x + (b*(95*a^3*b*B + 80*a*b^3*B - 83*a^4*C + 32*a^2*b^2*C + 16*b^4*C)*sin(c + d*x))/(30*d) + (b^2*(130*a^2*b*B + 45*b^3*B - 106*a^3*C + 71*a*b^2*C)*cos(c + d*x)*sin(c + d*x))/(120*d) + (b*(35*a*b*B - 23*a^2*C + 16*b^2*C)*(a + b*cos(c + d*x))^2*sin(c + d*x))/(60*d) + (b*(5*b*B - a*C)*(a + b*cos(c + d*x))^3*sin(c + d*x))/(20*d) + (b*C*(a + b*cos(c + d*x))^4*sin(c + d*x))/(5*d)],
[(a*b*B - a^2*C + b^2*B*cos(c + d*x) + b^2*C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^2, x, 4, (1/8)*(8*a^3*b*B + 12*a*b^3*B - 8*a^4*C + 3*b^4*C)*x + (b*(16*a^2*b*B + 4*b^3*B - 13*a^3*C + 8*a*b^2*C)*sin(c + d*x))/(6*d) + (b^2*(20*a*b*B - 14*a^2*C + 9*b^2*C)*cos(c + d*x)*sin(c + d*x))/(24*d) + (b*(4*b*B - a*C)*(a + b*cos(c + d*x))^2*sin(c + d*x))/(12*d) + (b*C*(a + b*cos(c + d*x))^3*sin(c + d*x))/(4*d)],
[(a*b*B - a^2*C + b^2*B*cos(c + d*x) + b^2*C*cos(c + d*x)^2)*(a + b*cos(c + d*x)), x, 3, (1/2)*(2*a^2*b*B + b^3*B - 2*a^3*C + a*b^2*C)*x + (2*b*(3*a*b*B - 2*a^2*C + b^2*C)*sin(c + d*x))/(3*d) + (b^2*(3*b*B - a*C)*cos(c + d*x)*sin(c + d*x))/(6*d) + (b*C*(a + b*cos(c + d*x))^2*sin(c + d*x))/(3*d)],


# ::Subsubsection::Closed:: 
#n<0


[cos(c + d*x)^3*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x)), x, 6, ((8*a^2*A*b^2 + 4*A*b^4 - 8*a^3*b*B - 4*a*b^3*B + 8*a^4*C + 4*a^2*b^2*C + 3*b^4*C)*x)/(8*b^5) - (2*a^3*(A*b^2 - a*b*B + a^2*C)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^5*sqrt(a^2 - b^2)*d) - ((3*a*A*b^2 - 3*a^2*b*B - 2*b^3*B + 3*a^3*C + 2*a*b^2*C)*sin(c + d*x))/(3*b^4*d) + ((4*A*b^2 - 4*a*b*B + 4*a^2*C + 3*b^2*C)*cos(c + d*x)*sin(c + d*x))/(8*b^3*d) + ((b*B - a*C)*cos(c + d*x)^2*sin(c + d*x))/(3*b^2*d) + (C*cos(c + d*x)^3*sin(c + d*x))/(4*b*d)],
[cos(c + d*x)^2*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x)), x, 5, -(((2*a*A*b^2 - 2*a^2*b*B - b^3*B + 2*a^3*C + a*b^2*C)*x)/(2*b^4)) + (2*a^2*(A*b^2 - a*b*B + a^2*C)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^4*sqrt(a^2 - b^2)*d) + ((3*A*b^2 - 3*a*b*B + 3*a^2*C + 2*b^2*C)*sin(c + d*x))/(3*b^3*d) + ((b*B - a*C)*cos(c + d*x)*sin(c + d*x))/(2*b^2*d) + (C*cos(c + d*x)^2*sin(c + d*x))/(3*b*d)],
[cos(c + d*x)^1*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x)), x, 4, ((2*A*b^2 - 2*a*b*B + 2*a^2*C + b^2*C)*x)/(2*b^3) - (2*a*(A*b^2 - a*b*B + a^2*C)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^3*sqrt(a^2 - b^2)*d) + ((b*B - a*C)*sin(c + d*x))/(b^2*d) + (C*cos(c + d*x)*sin(c + d*x))/(2*b*d)],
[cos(c + d*x)^0*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x)), x, 3, ((b*B - a*C)*x)/b^2 + (2*(A*b^2 - a*b*B + a^2*C)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^2*sqrt(a^2 - b^2)*d) + (C*sin(c + d*x))/(b*d)],
[sec(c + d*x)^1*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x)), x, 4, (C*x)/b - (2*(A*b^2 - a*b*B + a^2*C)*arctan(((a - b)*tan((c + d*x)/2))/sqrt(a^2 - b^2)))/(a*b*sqrt(a^2 - b^2)*d) + (A*arctanh(sin(c + d*x)))/(a*d)],
[sec(c + d*x)^2*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x)), x, 4, (2*(A*b^2 - a*b*B + a^2*C)*arctan(((a - b)*tan((c + d*x)/2))/sqrt(a^2 - b^2)))/(a^2*sqrt(a^2 - b^2)*d) - ((A*b - a*B)*arctanh(sin(c + d*x)))/(a^2*d) + (A*tan(c + d*x))/(a*d)],
[sec(c + d*x)^3*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x)), x, 5, (-2*b*(A*b^2 - a*b*B + a^2*C)*arctan(((a - b)*tan((c + d*x)/2))/sqrt(a^2 - b^2)))/(a^3*sqrt(a^2 - b^2)*d) + ((a^2*A + 2*A*b^2 - 2*a*b*B + 2*a^2*C)*arctanh(sin(c + d*x)))/(2*a^3*d) - ((A*b - a*B)*tan(c + d*x))/(a^2*d) + (A*sec(c + d*x)*tan(c + d*x))/(2*a*d)],
[sec(c + d*x)^4*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x)), x, 6, (2*b^2*(A*b^2 - a*b*B + a^2*C)*arctan(((a - b)*tan((c + d*x)/2))/sqrt(a^2 - b^2)))/(a^4*sqrt(a^2 - b^2)*d) - ((a^2*A*b + 2*A*b^3 - a^3*B - 2*a*b^2*B + 2*a^2*b*C)*arctanh(sin(c + d*x)))/(2*a^4*d) + ((2*a^2*A + 3*A*b^2 - 3*a*b*B + 3*a^2*C)*tan(c + d*x))/(3*a^3*d) - ((A*b - a*B)*sec(c + d*x)*tan(c + d*x))/(2*a^2*d) + (A*sec(c + d*x)^2*tan(c + d*x))/(3*a*d)],
[sec(c + d*x)^5*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x)), x, 7, -((2*b^3*(A*b^2 - a*b*B + a^2*C)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^5*sqrt(a^2 - b^2)*d)) + ((3*a^4*A + 4*a^2*A*b^2 + 8*A*b^4 - 4*a^3*b*B - 8*a*b^3*B + 4*a^4*C + 8*a^2*b^2*C)*arctanh(sin(c + d*x)))/(8*a^5*d) - ((2*a^2*A*b + 3*A*b^3 - 2*a^3*B - 3*a*b^2*B + 3*a^2*b*C)*tan(c + d*x))/(3*a^4*d) + ((3*a^2*A + 4*A*b^2 - 4*a*b*B + 4*a^2*C)*sec(c + d*x)*tan(c + d*x))/(8*a^3*d) - ((A*b - a*B)*sec(c + d*x)^2*tan(c + d*x))/(3*a^2*d) + (A*sec(c + d*x)^3*tan(c + d*x))/(4*a*d)],


[cos(c + d*x)^3*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^2, x, 6, -(((4*a*A*b^2 - 6*a^2*b*B - b^3*B + 8*a^3*C + 2*a*b^2*C)*x)/(2*b^5)) + (2*a^2*(2*a^2*A*b^2 - 3*A*b^4 - 3*a^3*b*B + 4*a*b^3*B + 4*a^4*C - 5*a^2*b^2*C)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^5*(a^2 - b^2)^(3/2)*d) + ((6*a^2*A*b^2 - 3*A*b^4 - 9*a^3*b*B + 6*a*b^3*B + 12*a^4*C - 7*a^2*b^2*C - 2*b^4*C)*sin(c + d*x))/(3*b^4*(a^2 - b^2)*d) - ((2*a*A*b^2 - 3*a^2*b*B + b^3*B + 4*a^3*C - 2*a*b^2*C)*cos(c + d*x)*sin(c + d*x))/(2*b^3*(a^2 - b^2)*d) + ((3*A*b^2 - 3*a*b*B + 4*a^2*C - b^2*C)*cos(c + d*x)^2*sin(c + d*x))/(3*b^2*(a^2 - b^2)*d) - ((A*b^2 - a*b*B + a^2*C)*cos(c + d*x)^3*sin(c + d*x))/(b*(a^2 - b^2)*d*(a + b*cos(c + d*x)))],
[cos(c + d*x)^2*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^2, x, 5, ((2*A*b^2 - 4*a*b*B + 6*a^2*C + b^2*C)*x)/(2*b^4) - (2*a*(a^2*A*b^2 - 2*A*b^4 - 2*a^3*b*B + 3*a*b^3*B + 3*a^4*C - 4*a^2*b^2*C)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^4*(a^2 - b^2)^(3/2)*d) - ((a*A*b^2 - 2*a^2*b*B + b^3*B + 3*a^3*C - 2*a*b^2*C)*sin(c + d*x))/(b^3*(a^2 - b^2)*d) + ((2*A*b^2 - 2*a*b*B + 3*a^2*C - b^2*C)*cos(c + d*x)*sin(c + d*x))/(2*b^2*(a^2 - b^2)*d) - ((A*b^2 - a*b*B + a^2*C)*cos(c + d*x)^2*sin(c + d*x))/(b*(a^2 - b^2)*d*(a + b*cos(c + d*x)))],
[cos(c + d*x)^1*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^2, x, 4, ((b*B - 2*a*C)*x)/b^3 - (2*(A*b^4 + a^3*b*B - 2*a*b^3*B - 2*a^4*C + 3*a^2*b^2*C)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^3*(a^2 - b^2)^(3/2)*d) + ((A*b^2 - a*b*B + 2*a^2*C - b^2*C)*sin(c + d*x))/(b^2*(a^2 - b^2)*d) - ((A*b^2 - a*b*B + a^2*C)*cos(c + d*x)*sin(c + d*x))/(b*(a^2 - b^2)*d*(a + b*cos(c + d*x)))],
[cos(c + d*x)^0*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^2, x, 3, (C*x)/b^2 + (2*(a*A*b^2 - b^3*B - a^3*C + 2*a*b^2*C)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^2*(a^2 - b^2)^(3/2)*d) - ((A*b^2 - a*b*B + a^2*C)*sin(c + d*x))/(b*(a^2 - b^2)*d*(a + b*cos(c + d*x)))],
[sec(c + d*x)^1*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^2, x, 4, (-2*(2*a^2*A*b - A*b^3 - a^3*B + a^2*b*C)*arctan(((a - b)*tan((c + d*x)/2))/sqrt(a^2 - b^2)))/(a^2*(a^2 - b^2)^(3/2)*d) + (A*arctanh(sin(c + d*x)))/(a^2*d) + ((A*b^2 - a*b*B + a^2*C)*sin(c + d*x))/(a*(a^2 - b^2)*d*(a + b*cos(c + d*x)))],
[sec(c + d*x)^2*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^2, x, 5, (2*(3*a^2*A*b^2 - 2*A*b^4 - 2*a^3*b*B + a*b^3*B + a^4*C)*arctan(((a - b)*tan((c + d*x)/2))/sqrt(a^2 - b^2)))/(a^3*(a^2 - b^2)^(3/2)*d) - ((2*A*b - a*B)*arctanh(sin(c + d*x)))/(a^3*d) + ((a^2*A - 2*A*b^2 + a*b*B - a^2*C)*tan(c + d*x))/(a^2*(a^2 - b^2)*d) + ((A*b^2 - a*b*B + a^2*C)*tan(c + d*x))/(a*(a^2 - b^2)*d*(a + b*cos(c + d*x)))],
[sec(c + d*x)^3*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^2, x, 6, (-2*b*(4*a^2*A*b^2 - 3*A*b^4 - 3*a^3*b*B + 2*a*b^3*B + 2*a^4*C - a^2*b^2*C)*arctan(((a - b)*tan((c + d*x)/2))/sqrt(a^2 - b^2)))/(a^4*(a^2 - b^2)^(3/2)*d) + ((a^2*A + 6*A*b^2 - 4*a*b*B + 2*a^2*C)*arctanh(sin(c + d*x)))/(2*a^4*d) - ((2*a^2*A*b - 3*A*b^3 - a^3*B + 2*a*b^2*B - a^2*b*C)*tan(c + d*x))/(a^3*(a^2 - b^2)*d) + ((a^2*A - 3*A*b^2 + 2*a*b*B - 2*a^2*C)*sec(c + d*x)*tan(c + d*x))/(2*a^2*(a^2 - b^2)*d) + ((A*b^2 - a*b*B + a^2*C)*sec(c + d*x)*tan(c + d*x))/(a*(a^2 - b^2)*d*(a + b*cos(c + d*x)))],
[sec(c + d*x)^4*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^2, x, 7, (2*b^2*(5*a^2*A*b^2 - 4*A*b^4 - 4*a^3*b*B + 3*a*b^3*B + 3*a^4*C - 2*a^2*b^2*C)*arctan(((a - b)*tan((c + d*x)/2))/sqrt(a^2 - b^2)))/(a^5*(a^2 - b^2)^(3/2)*d) - ((2*a^2*A*b + 8*A*b^3 - a^3*B - 6*a*b^2*B + 4*a^2*b*C)*arctanh(sin(c + d*x)))/(2*a^5*d) + ((2*a^4*A + 7*a^2*A*b^2 - 12*A*b^4 - 6*a^3*b*B + 9*a*b^3*B + 3*a^4*C - 6*a^2*b^2*C)*tan(c + d*x))/(3*a^4*(a^2 - b^2)*d) - ((2*a^2*A*b - 4*A*b^3 - a^3*B + 3*a*b^2*B - 2*a^2*b*C)*sec(c + d*x)*tan(c + d*x))/(2*a^3*(a^2 - b^2)*d) + ((a^2*A - 4*A*b^2 + 3*a*b*B - 3*a^2*C)*sec(c + d*x)^2*tan(c + d*x))/(3*a^2*(a^2 - b^2)*d) + ((A*b^2 - a*b*B + a^2*C)*sec(c + d*x)^2*tan(c + d*x))/(a*(a^2 - b^2)*d*(a + b*cos(c + d*x)))],


[cos(c + d*x)^3*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^3, x, 6, ((2*A*b^2 - 6*a*b*B + 12*a^2*C + b^2*C)*x)/(2*b^5) - (a*(2*a^4*A*b^2 - 5*a^2*A*b^4 + 6*A*b^6 - 6*a^5*b*B + 15*a^3*b^3*B - 12*a*b^5*B + 12*a^6*C - 29*a^4*b^2*C + 20*a^2*b^4*C)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^5*(a^2 - b^2)^(5/2)*d) - ((2*a^3*A*b^2 - 5*a*A*b^4 - 6*a^4*b*B + 11*a^2*b^3*B - 2*b^5*B + 12*a^5*C - 21*a^3*b^2*C + 6*a*b^4*C)*sin(c + d*x))/(2*b^4*(a^2 - b^2)^2*d) + ((a^2*A*b^2 - 4*A*b^4 - 3*a^3*b*B + 6*a*b^3*B + 6*a^4*C - 10*a^2*b^2*C + b^4*C)*cos(c + d*x)*sin(c + d*x))/(2*b^3*(a^2 - b^2)^2*d) - ((A*b^2 - a*b*B + a^2*C)*cos(c + d*x)^3*sin(c + d*x))/(2*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) + ((3*A*b^4 + 2*a^3*b*B - 5*a*b^3*B - 4*a^4*C + 7*a^2*b^2*C)*cos(c + d*x)^2*sin(c + d*x))/(2*b^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))],
[cos(c + d*x)^2*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^3, x, 5, ((b*B - 3*a*C)*x)/b^4 + ((a^2*A*b^4 + 2*A*b^6 - 2*a^5*b*B + 5*a^3*b^3*B - 6*a*b^5*B + 6*a^6*C - 15*a^4*b^2*C + 12*a^2*b^4*C)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^4*(a^2 - b^2)^(5/2)*d) - ((3*A*b^4 + 2*a^3*b*B - 5*a*b^3*B - 6*a^4*C + 11*a^2*b^2*C - 2*b^4*C)*sin(c + d*x))/(2*b^3*(a^2 - b^2)^2*d) - ((A*b^2 - a*b*B + a^2*C)*cos(c + d*x)^2*sin(c + d*x))/(2*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) + ((a^2*A*b^2 + 2*A*b^4 + a^3*b*B - 4*a*b^3*B - 3*a^4*C + 6*a^2*b^2*C)*cos(c + d*x)*sin(c + d*x))/(2*b^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))],
[cos(c + d*x)^1*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^3, x, 4, (C*x)/b^3 - ((3*a*A*b^4 - a^2*b^3*B - 2*b^5*B + 2*a^5*C - 5*a^3*b^2*C + 6*a*b^4*C)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^3*(a^2 - b^2)^(5/2)*d) - ((A*b^2 - a*b*B + a^2*C)*cos(c + d*x)*sin(c + d*x))/(2*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) + ((2*a^2*A*b^2 + A*b^4 - 3*a*b^3*B - 2*a^4*C + 5*a^2*b^2*C)*sin(c + d*x))/(2*b^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))],
[cos(c + d*x)^0*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^3, x, 4, ((2*a^2*A + A*b^2 - 3*a*b*B + a^2*C + 2*b^2*C)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/((a^2 - b^2)^(5/2)*d) - ((A*b^2 - a*b*B + a^2*C)*sin(c + d*x))/(2*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) - ((3*a*A*b^2 - a^2*b*B - 2*b^3*B - a^3*C + 4*a*b^2*C)*sin(c + d*x))/(2*b*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))],
[sec(c + d*x)^1*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^3, x, 5, -(((6*a^4*A*b - 5*a^2*A*b^3 + 2*A*b^5 - 2*a^5*B - a^3*b^2*B + 3*a^4*b*C)*arctan(((a - b)*tan((c + d*x)/2))/sqrt(a^2 - b^2)))/(a^3*(a^2 - b^2)^(5/2)*d)) + (A*arctanh(sin(c + d*x)))/(a^3*d) + ((A*b^2 - a*b*B + a^2*C)*sin(c + d*x))/(2*a*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) + ((5*a^2*A*b^2 - 2*A*b^4 - 3*a^3*b*B + a^4*C + 2*a^2*b^2*C)*sin(c + d*x))/(2*a^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))],
[sec(c + d*x)^2*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^3, x, 6, ((12*a^4*A*b^2 - 15*a^2*A*b^4 + 6*A*b^6 - 6*a^5*b*B + 5*a^3*b^3*B - 2*a*b^5*B + 2*a^6*C + a^4*b^2*C)*arctan(((a - b)*tan((c + d*x)/2))/sqrt(a^2 - b^2)))/(a^4*(a^2 - b^2)^(5/2)*d) - ((3*A*b - a*B)*arctanh(sin(c + d*x)))/(a^4*d) + ((2*a^4*A - 11*a^2*A*b^2 + 6*A*b^4 + 5*a^3*b*B - 2*a*b^3*B - 3*a^4*C)*tan(c + d*x))/(2*a^3*(a^2 - b^2)^2*d) + ((A*b^2 - a*b*B + a^2*C)*tan(c + d*x))/(2*a*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) + ((6*a^2*A*b^2 - 3*A*b^4 - 4*a^3*b*B + a*b^3*B + 2*a^4*C + a^2*b^2*C)*tan(c + d*x))/(2*a^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))],
[sec(c + d*x)^3*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^3, x, 7, -((b*(20*a^4*A*b^2 - 29*a^2*A*b^4 + 12*A*b^6 - 12*a^5*b*B + 15*a^3*b^3*B - 6*a*b^5*B + 6*a^6*C - 5*a^4*b^2*C + 2*a^2*b^4*C)*arctan(((a - b)*tan((c + d*x)/2))/sqrt(a^2 - b^2)))/(a^5*(a^2 - b^2)^(5/2)*d)) + ((a^2*A + 12*A*b^2 - 6*a*b*B + 2*a^2*C)*arctanh(sin(c + d*x)))/(2*a^5*d) - ((6*a^4*A*b - 21*a^2*A*b^3 + 12*A*b^5 - 2*a^5*B + 11*a^3*b^2*B - 6*a*b^4*B - 5*a^4*b*C + 2*a^2*b^3*C)*tan(c + d*x))/(2*a^4*(a^2 - b^2)^2*d) + ((a^4*A - 10*a^2*A*b^2 + 6*A*b^4 + 6*a^3*b*B - 3*a*b^3*B - 4*a^4*C + a^2*b^2*C)*sec(c + d*x)*tan(c + d*x))/(2*a^3*(a^2 - b^2)^2*d) + ((A*b^2 - a*b*B + a^2*C)*sec(c + d*x)*tan(c + d*x))/(2*a*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) + ((7*a^2*A*b^2 - 4*A*b^4 - 5*a^3*b*B + 2*a*b^3*B + 3*a^4*C)*sec(c + d*x)*tan(c + d*x))/(2*a^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))],


# {Cos[c + d*x]^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^4, x, 7, ((2*A*b^2 - 8*a*b*B + 20*a^2*C + b^2*C)*x)/(2*b^6) - (a*(2*a^6*A*b^2 - 7*a^4*A*b^4 + 8*a^2*A*b^6 - 8*A*b^8 - 8*a^7*b*B + 28*a^5*b^3*B - 35*a^3*b^5*B + 20*a*b^7*B + 20*a^8*C - 69*a^6*b^2*C + 84*a^4*b^4*C - 40*a^2*b^6*C)*ArcTan[((a - b)*Tan[(1/2)*(c + d*x)])/Sqrt[a^2 - b^2]])/(b^6*(a^2 - b^2)^(7/2)*d) - ((6*a^5*A*b^2 - 17*a^3*A*b^4 + 26*a*A*b^6 - 24*a^6*b*B + 68*a^4*b^3*B - 65*a^2*b^5*B + 6*b^7*B + 60*a^7*C - 167*a^5*b^2*C + 146*a^3*b^4*C - 24*a*b^6*C)*Sin[c + d*x])/(6*b^5*(a^2 - b^2)^3*d) + ((a^4*A*b^2 - 2*a^2*A*b^4 + 6*A*b^6 - 4*a^5*b*B + 11*a^3*b^3*B - 12*a*b^5*B + 10*a^6*C - 27*a^4*b^2*C + 23*a^2*b^4*C - b^6*C)*Cos[c + d*x]*Sin[c + d*x])/(2*b^4*(a^2 - b^2)^3*d) - ((A*b^2 - a*b*B + a^2*C)*Cos[c + d*x]^4*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) + ((a^2*A*b^2 + 4*A*b^4 + 2*a^3*b*B - 7*a*b^3*B - 5*a^4*C + 10*a^2*b^2*C)*Cos[c + d*x]^3*Sin[c + d*x])/(6*b^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) - ((2*a^4*A*b^2 + a^2*A*b^4 + 12*A*b^6 - 8*a^5*b*B + 20*a^3*b^3*B - 27*a*b^5*B + 20*a^6*C - 53*a^4*b^2*C + 48*a^2*b^4*C)*Cos[c + d*x]^2*Sin[c + d*x])/(6*b^3*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))} 
[cos(c + d*x)^3*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^4, x, 6, ((b*B - 4*a*C)*x)/b^5 - ((3*a^2*A*b^6 + 2*A*b^8 + 2*a^7*b*B - 7*a^5*b^3*B + 8*a^3*b^5*B - 8*a*b^7*B - 8*a^8*C + 28*a^6*b^2*C - 35*a^4*b^4*C + 20*a^2*b^6*C)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^5*(a^2 - b^2)^(7/2)*d) + ((4*a^2*A*b^4 + 11*A*b^6 - 6*a^5*b*B + 17*a^3*b^3*B - 26*a*b^5*B + 24*a^6*C - 68*a^4*b^2*C + 65*a^2*b^4*C - 6*b^6*C)*sin(c + d*x))/(6*b^4*(a^2 - b^2)^3*d) - ((A*b^2 - a*b*B + a^2*C)*cos(c + d*x)^3*sin(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^3) + ((2*a^2*A*b^2 + 3*A*b^4 + a^3*b*B - 6*a*b^3*B - 4*a^4*C + 9*a^2*b^2*C)*cos(c + d*x)^2*sin(c + d*x))/(6*b^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x))^2) - ((3*a^2*A*b^4 + 2*A*b^6 - a^5*b*B + 2*a^3*b^3*B - 6*a*b^5*B + 4*a^6*C - 11*a^4*b^2*C + 12*a^2*b^4*C)*cos(c + d*x)*sin(c + d*x))/(2*b^3*(a^2 - b^2)^3*d*(a + b*cos(c + d*x)))],
[cos(c + d*x)^2*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^4, x, 5, (C*x)/b^4 + ((a^3*A*b^4 + 4*a*A*b^6 - 3*a^2*b^5*B - 2*b^7*B - 2*a^7*C + 7*a^5*b^2*C - 8*a^3*b^4*C + 8*a*b^6*C)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b^4*(a^2 - b^2)^(7/2)*d) - ((A*b^2 - a*b*B + a^2*C)*cos(c + d*x)^2*sin(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^3) + ((3*a^2*A*b^2 + 2*A*b^4 - 5*a*b^3*B - 3*a^4*C + 8*a^2*b^2*C)*cos(c + d*x)*sin(c + d*x))/(6*b^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x))^2) - ((13*a^2*A*b^4 + 2*A*b^6 - 4*a^3*b^3*B - 11*a*b^5*B + 6*a^6*C - 17*a^4*b^2*C + 26*a^2*b^4*C)*sin(c + d*x))/(6*b^3*(a^2 - b^2)^3*d*(a + b*cos(c + d*x)))],
[cos(c + d*x)^1*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^4, x, 5, -(((4*a^2*A*b + A*b^3 - a^3*B - 4*a*b^2*B + 3*a^2*b*C + 2*b^3*C)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/((a^2 - b^2)^(7/2)*d)) - ((A*b^2 - a*b*B + a^2*C)*cos(c + d*x)*sin(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^3) + ((4*a^2*A*b^2 + A*b^4 - a^3*b*B - 4*a*b^3*B - 2*a^4*C + 7*a^2*b^2*C)*sin(c + d*x))/(6*b^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x))^2) + ((2*a^3*A*b^2 + 13*a*A*b^4 + a^4*b*B - 10*a^2*b^3*B - 6*b^5*B + 2*a^5*C - 5*a^3*b^2*C + 18*a*b^4*C)*sin(c + d*x))/(6*b^2*(a^2 - b^2)^3*d*(a + b*cos(c + d*x)))],
[cos(c + d*x)^0*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^4, x, 5, ((2*a^3*A + 3*a*A*b^2 - 4*a^2*b*B - b^3*B + a^3*C + 4*a*b^2*C)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/((a^2 - b^2)^(7/2)*d) - ((A*b^2 - a*b*B + a^2*C)*sin(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^3) - ((5*a*A*b^2 - 2*a^2*b*B - 3*b^3*B - a^3*C + 6*a*b^2*C)*sin(c + d*x))/(6*b*(a^2 - b^2)^2*d*(a + b*cos(c + d*x))^2) - ((11*a^2*A*b^2 + 4*A*b^4 - 2*a^3*b*B - 13*a*b^3*B - a^4*C + 10*a^2*b^2*C + 6*b^4*C)*sin(c + d*x))/(6*b*(a^2 - b^2)^3*d*(a + b*cos(c + d*x)))],
[sec(c + d*x)^1*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^4, x, 6, -(((8*a^6*A*b - 8*a^4*A*b^3 + 7*a^2*A*b^5 - 2*A*b^7 - 2*a^7*B - 3*a^5*b^2*B + 4*a^6*b*C + a^4*b^3*C)*arctan(((a - b)*tan((c + d*x)/2))/sqrt(a^2 - b^2)))/(a^4*(a^2 - b^2)^(7/2)*d)) + (A*arctanh(sin(c + d*x)))/(a^4*d) + ((A*b^2 - a*b*B + a^2*C)*sin(c + d*x))/(3*a*(a^2 - b^2)*d*(a + b*cos(c + d*x))^3) + ((8*a^2*A*b^2 - 3*A*b^4 - 5*a^3*b*B + 2*a^4*C + 3*a^2*b^2*C)*sin(c + d*x))/(6*a^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x))^2) + ((26*a^4*A*b^2 - 17*a^2*A*b^4 + 6*A*b^6 - 11*a^5*b*B - 4*a^3*b^3*B + 2*a^6*C + 13*a^4*b^2*C)*sin(c + d*x))/(6*a^3*(a^2 - b^2)^3*d*(a + b*cos(c + d*x)))],
[sec(c + d*x)^2*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^4, x, 7, ((20*a^6*A*b^2 - 35*a^4*A*b^4 + 28*a^2*A*b^6 - 8*A*b^8 - 8*a^7*b*B + 8*a^5*b^3*B - 7*a^3*b^5*B + 2*a*b^7*B + 2*a^8*C + 3*a^6*b^2*C)*arctan(((a - b)*tan((c + d*x)/2))/sqrt(a^2 - b^2)))/(a^5*(a^2 - b^2)^(7/2)*d) - ((4*A*b - a*B)*arctanh(sin(c + d*x)))/(a^5*d) + ((6*a^6*A - 65*a^4*A*b^2 + 68*a^2*A*b^4 - 24*A*b^6 + 26*a^5*b*B - 17*a^3*b^3*B + 6*a*b^5*B - 11*a^6*C - 4*a^4*b^2*C)*tan(c + d*x))/(6*a^4*(a^2 - b^2)^3*d) + ((A*b^2 - a*b*B + a^2*C)*tan(c + d*x))/(3*a*(a^2 - b^2)*d*(a + b*cos(c + d*x))^3) + ((9*a^2*A*b^2 - 4*A*b^4 - 6*a^3*b*B + a*b^3*B + 3*a^4*C + 2*a^2*b^2*C)*tan(c + d*x))/(6*a^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x))^2) + ((12*a^4*A*b^2 - 11*a^2*A*b^4 + 4*A*b^6 - 6*a^5*b*B + 2*a^3*b^3*B - a*b^5*B + 2*a^6*C + 3*a^4*b^2*C)*tan(c + d*x))/(2*a^3*(a^2 - b^2)^3*d*(a + b*cos(c + d*x)))],
[sec(c + d*x)^3*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^4, x, 8, ((6*a^2*b*(a^6*A - 23*a^4*A*b^2 + 27*a^2*A*b^4 - 10*A*b^6 + 12*a^5*b*B - 11*a^3*b^3*B + 4*a*b^5*B - 6*a^6*C + 2*a^4*b^2*C - a^2*b^4*C) - 6*b*(a^8*A + 17*a^6*A*b^2 - 57*a^4*A*b^4 + 59*a^2*A*b^6 - 20*A*b^8 - 8*a^7*b*B + 24*a^5*b^3*B - 24*a^3*b^5*B + 8*a*b^7*B + 2*a^8*C - 6*a^6*b^2*C + 6*a^4*b^4*C - 2*a^2*b^6*C))*arctan(((a - b)*tan((c + d*x)/2))/sqrt(a^2 - b^2)))/(6*a^6*(a^2 - b^2)^(7/2)*d) + ((a^8*A + 17*a^6*A*b^2 - 57*a^4*A*b^4 + 59*a^2*A*b^6 - 20*A*b^8 - 8*a^7*b*B + 24*a^5*b^3*B - 24*a^3*b^5*B + 8*a*b^7*B + 2*a^8*C - 6*a^6*b^2*C + 6*a^4*b^4*C - 2*a^2*b^6*C)*arctanh(sin(c + d*x)))/(2*a^6*(a^2 - b^2)^3*d) + ((-24*a^6*A*b + 146*a^4*A*b^3 - 167*a^2*A*b^5 + 60*A*b^7 + 6*a^7*B - 65*a^5*b^2*B + 68*a^3*b^4*B - 24*a*b^6*B + 26*a^6*b*C - 17*a^4*b^3*C + 6*a^2*b^5*C)*tan(c + d*x))/(6*a^5*(a^2 - b^2)^3*d) + ((a^6*A - 23*a^4*A*b^2 + 27*a^2*A*b^4 - 10*A*b^6 + 12*a^5*b*B - 11*a^3*b^3*B + 4*a*b^5*B - 6*a^6*C + 2*a^4*b^2*C - a^2*b^4*C)*sec(c + d*x)*tan(c + d*x))/(2*a^4*(a^2 - b^2)^3*d) + ((A*b^2 - a*b*B + a^2*C)*sec(c + d*x)*tan(c + d*x))/(3*a*(a^2 - b^2)*d*(a + b*cos(c + d*x))^3) - ((-4*a^2*(A*b^2 - a*b*B + a^2*C) + b^2*(-3*a^2*A + 5*A*b^2 - 2*a*b*B + 2*a^2*C) + 3*a^2*b*(-(A*b) + a*B - b*C))*sec(c + d*x)*tan(c + d*x))/(6*a^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x))^2) + ((-2*a^2*b*(-6*a^2*A*b + A*b^3 + 3*a^3*B + 2*a*b^2*B - 5*a^2*b*C) + 2*b^2*(3*a^4*A - 18*a^2*A*b^2 + 10*A*b^4 + 9*a^3*b*B - 4*a*b^3*B - 6*a^4*C + a^2*b^2*C) + 3*a^2*(10*a^2*A*b^2 - 5*A*b^4 - 7*a^3*b*B + 2*a*b^3*B + 4*a^4*C + a^2*b^2*C))*sec(c + d*x)*tan(c + d*x))/(6*a^3*(a^2 - b^2)^3*d*(a + b*cos(c + d*x)))],


[(a*b*B - a^2*C + b^2*B*cos(c + d*x) + b^2*C*cos(c + d*x)^2)/(a + b*cos(c + d*x)), x, 2, (b*B - a*C)*x + (b*C*sin(c + d*x))/d],
[(a*b*B - a^2*C + b^2*B*cos(c + d*x) + b^2*C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^2, x, 3, C*x + (2*(b*B - 2*a*C)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(sqrt(a^2 - b^2)*d)],
[(a*b*B - a^2*C + b^2*B*cos(c + d*x) + b^2*C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^3, x, 4, (2*(a*b*B - a^2*C - b^2*C)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/((a^2 - b^2)^(3/2)*d) - (b*(b*B - 2*a*C)*sin(c + d*x))/((a^2 - b^2)*d*(a + b*cos(c + d*x)))],
[(a*b*B - a^2*C + b^2*B*cos(c + d*x) + b^2*C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^4, x, 5, ((2*a^2*b*B + b^3*B - 2*a^3*C - 4*a*b^2*C)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/((a^2 - b^2)^(5/2)*d) - (b*(b*B - 2*a*C)*sin(c + d*x))/(2*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) - (b*(3*a*b*B - 4*a^2*C - 2*b^2*C)*sin(c + d*x))/(2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))],
[(a*b*B - a^2*C + b^2*B*cos(c + d*x) + b^2*C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^5, x, 6, ((2*a^3*b*B + 3*a*b^3*B - 2*a^4*C - 7*a^2*b^2*C - b^4*C)*arctan(((a - b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/((a^2 - b^2)^(7/2)*d) - (b*(b*B - 2*a*C)*sin(c + d*x))/(3*(a^2 - b^2)*d*(a + b*cos(c + d*x))^3) - (b*(5*a*b*B - 7*a^2*C - 3*b^2*C)*sin(c + d*x))/(6*(a^2 - b^2)^2*d*(a + b*cos(c + d*x))^2) - (b*(11*a^2*b*B + 4*b^3*B - 13*a^3*C - 17*a*b^2*C)*sin(c + d*x))/(6*(a^2 - b^2)^3*d*(a + b*cos(c + d*x)))],


# ::Subsection::Closed:: 
#Integrands of the form Cos[c+d x]^m (A+B Cos[c+d x]+C Cos[c+d x]^2) (a+b Cos[c+d x])^(n/2)


# ::Subsubsection::Closed:: 
#n>0


[cos(c + d*x)^2*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(1/2), x, 9, -((2*(a + b)*(42*a^2*A*b^2 - 189*A*b^4 - 24*a^3*b*B - 57*a*b^3*B + 16*a^4*C + 24*a^2*b^2*C - 147*b^4*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(315*b^4*d*sqrt(a + b*cos(c + d*x)))) + (2*(42*a^3*A*b^2 - 42*a*A*b^4 - 24*a^4*b*B - 51*a^2*b^3*B + 75*b^5*B + 16*a^5*C + 20*a^3*b^2*C - 36*a*b^4*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(315*b^4*d*sqrt(a + b*cos(c + d*x))) + (2*(21*a*A*b^2 - 12*a^2*b*B + 75*b^3*B + 8*a^3*C + 13*a*b^2*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(315*b^3*d) + (2*(63*A*b^2 + 9*a*b*B - 6*a^2*C + 49*b^2*C)*cos(c + d*x)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(315*b^2*d) + (2*(9*b*B + a*C)*cos(c + d*x)^2*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(63*b*d) + (2*C*cos(c + d*x)^3*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(9*d)],
[cos(c + d*x)^1*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(1/2), x, 8, (2*(a + b)*(35*a*A*b^2 - 14*a^2*b*B + 63*b^3*B + 8*a^3*C + 19*a*b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(105*b^3*d*sqrt(a + b*cos(c + d*x))) - (2*(35*a^2*A*b^2 - 35*A*b^4 - 14*a^3*b*B + 14*a*b^3*B + 8*a^4*C + 17*a^2*b^2*C - 25*b^4*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(105*b^3*d*sqrt(a + b*cos(c + d*x))) + (2*(35*A*b^2 + 7*a*b*B - 4*a^2*C + 25*b^2*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(105*b^2*d) + (2*(7*b*B + a*C)*cos(c + d*x)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(35*b*d) + (2*C*cos(c + d*x)^2*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(7*d)],
[cos(c + d*x)^0*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(1/2), x, 7, (2*(a + b)*(15*A*b^2 + 5*a*b*B - 2*a^2*C + 9*b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(15*b^2*d*sqrt(a + b*cos(c + d*x))) - (2*(5*a^2*b*B - 5*b^3*B - 2*a^3*C + 2*a*b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(15*b^2*d*sqrt(a + b*cos(c + d*x))) + (2*(5*b*B - 2*a*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(15*b*d) + (2*C*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(5*b*d)],
[sec(c + d*x)^1*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(1/2), x, 9, (2*(a + b)*(3*b*B + a*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*b)/(a + b)))/(3*b*d*sqrt(a + b*cos(c + d*x))) + (2*(3*A*b^2 - a^2*C + b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*b)/(a + b)))/(3*b*d*sqrt(a + b*cos(c + d*x))) + (2*a*A*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (c + d*x)/2, (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x))) + (2*C*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*d)],
[sec(c + d*x)^2*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(1/2), x, 9, -(((a + b)*(A - 2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x)))) + ((a*A + 2*b*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x))) + ((A*b + 2*a*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (c + d*x)/2, (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x))) + (A*sqrt(a + b*cos(c + d*x))*tan(c + d*x))/d],
[sec(c + d*x)^3*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(1/2), x, 10, -((a + b)*(A*b + 4*a*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*b)/(a + b)))/(4*a*d*sqrt(a + b*cos(c + d*x))) + ((3*A*b + 4*a*B + 8*b*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*b)/(a + b)))/(4*d*sqrt(a + b*cos(c + d*x))) + ((4*a^2*A - A*b^2 + 4*a*b*B + 8*a^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (c + d*x)/2, (2*b)/(a + b)))/(4*a*d*sqrt(a + b*cos(c + d*x))) + ((A*b + 4*a*B)*sqrt(a + b*cos(c + d*x))*tan(c + d*x))/(4*a*d) + (A*sqrt(a + b*cos(c + d*x))*sec(c + d*x)*tan(c + d*x))/(2*d)],
[sec(c + d*x)^4*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(1/2), x, 11, -((a + b)*(16*a^2*A - 3*A*b^2 + 6*a*b*B + 24*a^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*b)/(a + b)))/(24*a^2*d*sqrt(a + b*cos(c + d*x))) + ((16*a^2*A - A*b^2 + 18*a*b*B + 24*a^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*b)/(a + b)))/(24*a*d*sqrt(a + b*cos(c + d*x))) + ((4*a^2*A*b + A*b^3 + 8*a^3*B - 2*a*b^2*B + 8*a^2*b*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (c + d*x)/2, (2*b)/(a + b)))/(8*a^2*d*sqrt(a + b*cos(c + d*x))) + ((16*a^2*A - 3*A*b^2 + 6*a*b*B + 24*a^2*C)*sqrt(a + b*cos(c + d*x))*tan(c + d*x))/(24*a^2*d) + ((A*b + 6*a*B)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)*tan(c + d*x))/(12*a*d) + (A*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^2*tan(c + d*x))/(3*d)],


[cos(c + d*x)^2*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(3/2), x, 10, -((2*(a + b)*(198*a^3*A*b^2 - 2706*a*A*b^4 - 88*a^4*b*B - 363*a^2*b^3*B - 1617*b^5*B + 48*a^5*C + 108*a^3*b^2*C - 2088*a*b^4*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(3465*b^4*d*sqrt(a + b*cos(c + d*x)))) + (2*(198*a^4*A*b^2 - 1023*a^2*A*b^4 + 825*A*b^6 - 88*a^5*b*B - 341*a^3*b^3*B + 429*a*b^5*B + 48*a^6*C + 96*a^4*b^2*C - 819*a^2*b^4*C + 675*b^6*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(3465*b^4*d*sqrt(a + b*cos(c + d*x))) + (2*(99*a^2*A*b^2 + 825*A*b^4 - 44*a^3*b*B + 968*a*b^3*B + 24*a^4*C + 57*a^2*b^2*C + 675*b^4*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3465*b^3*d) + (2*(792*a*A*b^2 + 33*a^2*b*B + 539*b^3*B - 18*a^3*C + 606*a*b^2*C)*cos(c + d*x)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3465*b^2*d) + (2*(99*A*b^2 + 110*a*b*B + 3*a^2*C + 81*b^2*C)*cos(c + d*x)^2*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(693*b*d) + (2*(11*b*B + 3*a*C)*cos(c + d*x)^3*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(99*d) + (2*C*cos(c + d*x)^3*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(11*d)],
[cos(c + d*x)^1*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(3/2), x, 9, (2*(a + b)*(63*a^2*A*b^2 + 189*A*b^4 - 18*a^3*b*B + 246*a*b^3*B + 8*a^4*C + 33*a^2*b^2*C + 147*b^4*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(315*b^3*d*sqrt(a + b*cos(c + d*x))) - (2*(63*a^3*A*b^2 - 63*a*A*b^4 - 18*a^4*b*B + 93*a^2*b^3*B - 75*b^5*B + 8*a^5*C + 31*a^3*b^2*C - 39*a*b^4*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(315*b^3*d*sqrt(a + b*cos(c + d*x))) + (2*(126*a*A*b^2 + 9*a^2*b*B + 75*b^3*B - 4*a^3*C + 88*a*b^2*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(315*b^2*d) + (2*(63*A*b^2 + 72*a*b*B + 3*a^2*C + 49*b^2*C)*cos(c + d*x)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(315*b*d) + (2*(3*b*B + a*C)*cos(c + d*x)^2*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(21*d) + (2*C*cos(c + d*x)^2*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(9*d)],
[cos(c + d*x)^0*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(3/2), x, 8, (2*(a + b)*(140*a*A*b^2 + 21*a^2*b*B + 63*b^3*B - 6*a^3*C + 82*a*b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(105*b^2*d*sqrt(a + b*cos(c + d*x))) - (2*(35*a^2*A*b^2 - 35*A*b^4 + 21*a^3*b*B - 21*a*b^3*B - 6*a^4*C + 31*a^2*b^2*C - 25*b^4*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(105*b^2*d*sqrt(a + b*cos(c + d*x))) + (2*(35*A*b^2 + 21*a*b*B - 6*a^2*C + 25*b^2*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(105*b*d) + (2*(7*b*B - 2*a*C)*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(35*b*d) + (2*C*(a + b*cos(c + d*x))^(5/2)*sin(c + d*x))/(7*b*d)],
[sec(c + d*x)^1*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(3/2), x, 10, (2*(a + b)*(15*A*b^2 + 20*a*b*B + 3*a^2*C + 9*b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*b)/(a + b)))/(15*b*d*sqrt(a + b*cos(c + d*x))) + (2*(15*a*A*b^2 - 5*a^2*b*B + 5*b^3*B - 3*a^3*C + 3*a*b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*b)/(a + b)))/(15*b*d*sqrt(a + b*cos(c + d*x))) + (2*a^2*A*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (c + d*x)/2, (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x))) + (2*(5*b*B + 3*a*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(15*d) + (2*C*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(5*d)],
[sec(c + d*x)^2*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(3/2), x, 10, -((a + b)*(3*a*A - 6*b*B - 8*a*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*b)/(a + b)))/(3*d*sqrt(a + b*cos(c + d*x))) + ((3*a^2*A + 6*A*b^2 + 6*a*b*B - 2*a^2*C + 2*b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*b)/(a + b)))/(3*d*sqrt(a + b*cos(c + d*x))) + (a*(3*A*b + 2*a*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (c + d*x)/2, (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x))) - (b*(3*A - 2*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*d) + (A*(a + b*cos(c + d*x))^(3/2)*tan(c + d*x))/d],
[sec(c + d*x)^3*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(3/2), x, 10, -((a + b)*(5*A*b + 4*a*B - 8*b*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*b)/(a + b)))/(4*d*sqrt(a + b*cos(c + d*x))) + ((7*a*A*b + 4*a^2*B + 8*b^2*B + 8*a*b*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*b)/(a + b)))/(4*d*sqrt(a + b*cos(c + d*x))) + ((4*a^2*A + 3*A*b^2 + 12*a*b*B + 8*a^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (c + d*x)/2, (2*b)/(a + b)))/(4*d*sqrt(a + b*cos(c + d*x))) + ((3*A*b + 4*a*B)*sqrt(a + b*cos(c + d*x))*tan(c + d*x))/(4*d) + (A*(a + b*cos(c + d*x))^(3/2)*sec(c + d*x)*tan(c + d*x))/(2*d)],
[sec(c + d*x)^4*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(3/2), x, 11, -((a + b)*(16*a^2*A + 3*A*b^2 + 30*a*b*B + 24*a^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*b)/(a + b)))/(24*a*d*sqrt(a + b*cos(c + d*x))) + ((16*a^2*A + 17*A*b^2 + 42*a*b*B + 24*a^2*C + 48*b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*b)/(a + b)))/(24*d*sqrt(a + b*cos(c + d*x))) + ((12*a^2*A*b - A*b^3 + 8*a^3*B + 6*a*b^2*B + 24*a^2*b*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (c + d*x)/2, (2*b)/(a + b)))/(8*a*d*sqrt(a + b*cos(c + d*x))) + ((16*a^2*A + 3*A*b^2 + 30*a*b*B + 24*a^2*C)*sqrt(a + b*cos(c + d*x))*tan(c + d*x))/(24*a*d) + ((A*b + 2*a*B)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)*tan(c + d*x))/(4*d) + (A*(a + b*cos(c + d*x))^(3/2)*sec(c + d*x)^2*tan(c + d*x))/(3*d)],
[sec(c + d*x)^5*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(3/2), x, 12, -(((a + b)*(156*a^2*A*b - 9*A*b^3 + 128*a^3*B + 24*a*b^2*B + 240*a^2*b*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(192*a^2*d*sqrt(a + b*cos(c + d*x)))) + ((228*a^2*A*b - 3*A*b^3 + 128*a^3*B + 136*a*b^2*B + 336*a^2*b*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(192*a*d*sqrt(a + b*cos(c + d*x))) + ((48*a^4*A + 24*a^2*A*b^2 + 3*A*b^4 + 96*a^3*b*B - 8*a*b^3*B + 64*a^4*C + 48*a^2*b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*b)/(a + b)))/(64*a^2*d*sqrt(a + b*cos(c + d*x))) + ((156*a^2*A*b - 9*A*b^3 + 128*a^3*B + 24*a*b^2*B + 240*a^2*b*C)*sqrt(a + b*cos(c + d*x))*tan(c + d*x))/(192*a^2*d) + ((36*a^2*A + 3*A*b^2 + 56*a*b*B + 48*a^2*C)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)*tan(c + d*x))/(96*a*d) + ((3*A*b + 8*a*B)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^2*tan(c + d*x))/(24*d) + (A*(a + b*cos(c + d*x))^(3/2)*sec(c + d*x)^3*tan(c + d*x))/(4*d)],


[cos(c + d*x)^2*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(5/2), x, 11, -((2*(a + b)*(1430*a^4*A*b^2 - 39897*a^2*A*b^4 - 21021*A*b^6 - 520*a^5*b*B - 3315*a^3*b^3*B - 48165*a*b^5*B + 240*a^6*C + 760*a^4*b^2*C - 30669*a^2*b^4*C - 17787*b^6*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(45045*b^4*d*sqrt(a + b*cos(c + d*x)))) + (2*(1430*a^5*A*b^2 - 17732*a^3*A*b^4 + 16302*a*A*b^6 - 520*a^6*b*B - 3185*a^4*b^3*B - 5070*a^2*b^5*B + 8775*b^7*B + 240*a^7*C + 700*a^5*b^2*C - 13984*a^3*b^4*C + 13044*a*b^6*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(45045*b^4*d*sqrt(a + b*cos(c + d*x))) + (2*(715*a^3*A*b^2 + 23309*a*A*b^4 - 260*a^4*b*B + 13325*a^2*b^3*B + 8775*b^5*B + 120*a^5*C + 395*a^3*b^2*C + 18973*a*b^4*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(45045*b^3*d) + (2*(10725*a^2*A*b^2 + 7007*A*b^4 + 195*a^3*b*B + 14885*a*b^3*B - 90*a^4*C + 8145*a^2*b^2*C + 5929*b^4*C)*cos(c + d*x)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(45045*b^2*d) + (2*(2717*a*A*b^2 + 1469*a^2*b*B + 1053*b^3*B + 15*a^3*C + 2209*a*b^2*C)*cos(c + d*x)^2*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(9009*b*d) + (2*(143*A*b^2 + 182*a*b*B + 15*a^2*C + 121*b^2*C)*cos(c + d*x)^3*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(1287*d) + (2*(13*b*B + 5*a*C)*cos(c + d*x)^3*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(143*d) + (2*C*cos(c + d*x)^3*(a + b*cos(c + d*x))^(5/2)*sin(c + d*x))/(13*d)],
[cos(c + d*x)^1*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(5/2), x, 10, (2*(a + b)*(495*a^3*A*b^2 + 4785*a*A*b^4 - 110*a^4*b*B + 3069*a^2*b^3*B + 1617*b^5*B + 40*a^5*C + 255*a^3*b^2*C + 3705*a*b^4*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(3465*b^3*d*sqrt(a + b*cos(c + d*x))) - (2*(495*a^4*A*b^2 + 330*a^2*A*b^4 - 825*A*b^6 - 110*a^5*b*B + 1364*a^3*b^3*B - 1254*a*b^5*B + 40*a^6*C + 245*a^4*b^2*C + 390*a^2*b^4*C - 675*b^6*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(3465*b^3*d*sqrt(a + b*cos(c + d*x))) + (2*(1485*a^2*A*b^2 + 825*A*b^4 + 55*a^3*b*B + 1793*a*b^3*B - 20*a^4*C + 1025*a^2*b^2*C + 675*b^4*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3465*b^2*d) + (2*(1485*a*A*b^2 + 825*a^2*b*B + 539*b^3*B + 15*a^3*C + 1145*a*b^2*C)*cos(c + d*x)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3465*b*d) + (2*(33*A*b^2 + 44*a*b*B + 5*a^2*C + 27*b^2*C)*cos(c + d*x)^2*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(231*d) + (2*(11*b*B + 5*a*C)*cos(c + d*x)^2*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(99*d) + (2*C*cos(c + d*x)^2*(a + b*cos(c + d*x))^(5/2)*sin(c + d*x))/(11*d)],
[cos(c + d*x)^0*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(5/2), x, 9, (2*(a + b)*(483*a^2*A*b^2 + 189*A*b^4 + 45*a^3*b*B + 435*a*b^3*B - 10*a^4*C + 279*a^2*b^2*C + 147*b^4*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(315*b^2*d*sqrt(a + b*cos(c + d*x))) - (2*(168*a^3*A*b^2 - 168*a*A*b^4 + 45*a^4*b*B + 30*a^2*b^3*B - 75*b^5*B - 10*a^5*C + 124*a^3*b^2*C - 114*a*b^4*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(315*b^2*d*sqrt(a + b*cos(c + d*x))) + (2*(168*a*A*b^2 + 45*a^2*b*B + 75*b^3*B - 10*a^3*C + 114*a*b^2*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(315*b*d) + (2*(63*A*b^2 + 45*a*b*B - 10*a^2*C + 49*b^2*C)*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(315*b*d) + (2*(9*b*B - 2*a*C)*(a + b*cos(c + d*x))^(5/2)*sin(c + d*x))/(63*b*d) + (2*C*(a + b*cos(c + d*x))^(7/2)*sin(c + d*x))/(9*b*d)],
[sec(c + d*x)^1*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(5/2), x, 11, (2*(a + b)*(245*a*A*b^2 + 161*a^2*b*B + 63*b^3*B + 15*a^3*C + 145*a*b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*b)/(a + b)))/(105*b*d*sqrt(a + b*cos(c + d*x))) + (2*(70*a^2*A*b^2 + 35*A*b^4 - 56*a^3*b*B + 56*a*b^3*B - 15*a^4*C - 10*a^2*b^2*C + 25*b^4*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*b)/(a + b)))/(105*b*d*sqrt(a + b*cos(c + d*x))) + (2*a^3*A*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (c + d*x)/2, (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x))) + (2*(35*A*b^2 + 56*a*b*B + 15*a^2*C + 25*b^2*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(105*d) + (2*(7*b*B + 5*a*C)*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(35*d) + (2*C*(a + b*cos(c + d*x))^(5/2)*sin(c + d*x))/(7*d)],
[sec(c + d*x)^2*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(5/2), x, 11, -((a + b)*(15*a^2*A - 30*A*b^2 - 70*a*b*B - 46*a^2*C - 18*b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*b)/(a + b)))/(15*d*sqrt(a + b*cos(c + d*x))) + ((15*a^3*A + 60*a*A*b^2 + 20*a^2*b*B + 10*b^3*B - 16*a^3*C + 16*a*b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*b)/(a + b)))/(15*d*sqrt(a + b*cos(c + d*x))) + (a^2*(5*A*b + 2*a*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (c + d*x)/2, (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x))) - (b*(15*a*A - 10*b*B - 16*a*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(15*d) - (b*(5*A - 2*C)*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(5*d) + (A*(a + b*cos(c + d*x))^(5/2)*tan(c + d*x))/d],
[sec(c + d*x)^3*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(5/2), x, 11, -((a + b)*(27*a*A*b + 12*a^2*B - 24*b^2*B - 56*a*b*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*b)/(a + b)))/(12*d*sqrt(a + b*cos(c + d*x))) + ((33*a^2*A*b + 24*A*b^3 + 12*a^3*B + 48*a*b^2*B + 16*a^2*b*C + 8*b^3*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*b)/(a + b)))/(12*d*sqrt(a + b*cos(c + d*x))) + (a*(4*a^2*A + 15*A*b^2 + 20*a*b*B + 8*a^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (c + d*x)/2, (2*b)/(a + b)))/(4*d*sqrt(a + b*cos(c + d*x))) - (b*(21*A*b + 12*a*B - 8*b*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(12*d) + ((5*A*b + 4*a*B)*(a + b*cos(c + d*x))^(3/2)*tan(c + d*x))/(4*d) + (A*(a + b*cos(c + d*x))^(5/2)*sec(c + d*x)*tan(c + d*x))/(2*d)],
[sec(c + d*x)^4*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(5/2), x, 11, -((a + b)*(16*a^2*A + 33*A*b^2 + 54*a*b*B + 24*a^2*C - 48*b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*b)/(a + b)))/(24*d*sqrt(a + b*cos(c + d*x))) + ((16*a^3*A + 59*a*A*b^2 + 66*a^2*b*B + 48*b^3*B + 24*a^3*C + 96*a*b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*b)/(a + b)))/(24*d*sqrt(a + b*cos(c + d*x))) + ((20*a^2*A*b + 5*A*b^3 + 8*a^3*B + 30*a*b^2*B + 40*a^2*b*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (c + d*x)/2, (2*b)/(a + b)))/(8*d*sqrt(a + b*cos(c + d*x))) + ((16*a^2*A + 15*A*b^2 + 42*a*b*B + 24*a^2*C)*sqrt(a + b*cos(c + d*x))*tan(c + d*x))/(24*d) + ((5*A*b + 6*a*B)*(a + b*cos(c + d*x))^(3/2)*sec(c + d*x)*tan(c + d*x))/(12*d) + (A*(a + b*cos(c + d*x))^(5/2)*sec(c + d*x)^2*tan(c + d*x))/(3*d)],
[sec(c + d*x)^5*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(5/2), x, 12, -(((a + b)*(284*a^2*A*b + 15*A*b^3 + 128*a^3*B + 264*a*b^2*B + 432*a^2*b*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(192*a*d*sqrt(a + b*cos(c + d*x)))) + ((356*a^2*A*b + 133*A*b^3 + 128*a^3*B + 472*a*b^2*B + 528*a^2*b*C + 384*b^3*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(192*d*sqrt(a + b*cos(c + d*x))) + ((48*a^4*A + 120*a^2*A*b^2 - 5*A*b^4 + 160*a^3*b*B + 40*a*b^3*B + 64*a^4*C + 240*a^2*b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*b)/(a + b)))/(64*a*d*sqrt(a + b*cos(c + d*x))) + ((284*a^2*A*b + 15*A*b^3 + 128*a^3*B + 264*a*b^2*B + 432*a^2*b*C)*sqrt(a + b*cos(c + d*x))*tan(c + d*x))/(192*a*d) + ((12*a^2*A + 5*A*b^2 + 24*a*b*B + 16*a^2*C)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)*tan(c + d*x))/(32*d) + ((5*A*b + 8*a*B)*(a + b*cos(c + d*x))^(3/2)*sec(c + d*x)^2*tan(c + d*x))/(24*d) + (A*(a + b*cos(c + d*x))^(5/2)*sec(c + d*x)^3*tan(c + d*x))/(4*d)],
[sec(c + d*x)^6*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(5/2), x, 13, -(((a + b)*(1024*a^4*A + 1692*a^2*A*b^2 - 45*A*b^4 + 2840*a^3*b*B + 150*a*b^3*B + 1280*a^4*C + 2640*a^2*b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(1920*a^2*d*sqrt(a + b*cos(c + d*x)))) + ((1024*a^4*A + 3236*a^2*A*b^2 - 15*A*b^4 + 3560*a^3*b*B + 1330*a*b^3*B + 1280*a^4*C + 4720*a^2*b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(1920*a*d*sqrt(a + b*cos(c + d*x))) + ((240*a^4*A*b + 40*a^2*A*b^3 + 3*A*b^5 + 96*a^5*B + 240*a^3*b^2*B - 10*a*b^4*B + 320*a^4*b*C + 80*a^2*b^3*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*b)/(a + b)))/(128*a^2*d*sqrt(a + b*cos(c + d*x))) + ((1024*a^4*A + 1692*a^2*A*b^2 - 45*A*b^4 + 2840*a^3*b*B + 150*a*b^3*B + 1280*a^4*C + 2640*a^2*b^2*C)*sqrt(a + b*cos(c + d*x))*tan(c + d*x))/(1920*a^2*d) + ((772*a^2*A*b + 15*A*b^3 + 360*a^3*B + 590*a*b^2*B + 1040*a^2*b*C)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)*tan(c + d*x))/(960*a*d) + ((64*a^2*A + 15*A*b^2 + 110*a*b*B + 80*a^2*C)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^2*tan(c + d*x))/(240*d) + ((A*b + 2*a*B)*(a + b*cos(c + d*x))^(3/2)*sec(c + d*x)^3*tan(c + d*x))/(8*d) + (A*(a + b*cos(c + d*x))^(5/2)*sec(c + d*x)^4*tan(c + d*x))/(5*d)],


[(a*b*B - a^2*C + b^2*B*cos(c + d*x) + b^2*C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(3/2), x, 8, (2*(a + b)*(161*a^2*b*B + 63*b^3*B - 146*a^3*C + 82*a*b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(105*d*sqrt(a + b*cos(c + d*x))) - (2*(56*a^3*b*B - 56*a*b^3*B - 41*a^4*C + 66*a^2*b^2*C - 25*b^4*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(105*d*sqrt(a + b*cos(c + d*x))) + (2*b*(56*a*b*B - 41*a^2*C + 25*b^2*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(105*d) + (2*b*(7*b*B - 2*a*C)*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(35*d) + (2*b*C*(a + b*cos(c + d*x))^(5/2)*sin(c + d*x))/(7*d)],
[(a*b*B - a^2*C + b^2*B*cos(c + d*x) + b^2*C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(1/2), x, 7, (2*(a + b)*(20*a*b*B - 17*a^2*C + 9*b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(15*d*sqrt(a + b*cos(c + d*x))) - (2*(5*a^2*b*B - 5*b^3*B - 2*a^3*C + 2*a*b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(15*d*sqrt(a + b*cos(c + d*x))) + (2*b*(5*b*B - 2*a*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(15*d) + (2*b*C*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(5*d)],


# ::Subsubsection::Closed:: 
#n<0


[cos(c + d*x)^2*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(1/2), x, 8, -((2*(a + b)*(70*a*A*b^2 - 56*a^2*b*B - 63*b^3*B + 48*a^3*C + 44*a*b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(105*b^4*d*sqrt(a + b*cos(c + d*x)))) + (2*(70*a^2*A*b^2 + 35*A*b^4 - 56*a^3*b*B - 49*a*b^3*B + 48*a^4*C + 32*a^2*b^2*C + 25*b^4*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(105*b^4*d*sqrt(a + b*cos(c + d*x))) + (2*(35*A*b^2 - 28*a*b*B + 24*a^2*C + 25*b^2*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(105*b^3*d) + (2*(7*b*B - 6*a*C)*cos(c + d*x)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(35*b^2*d) + (2*C*cos(c + d*x)^2*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(7*b*d)],
[cos(c + d*x)^1*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(1/2), x, 7, (2*(a + b)*(15*A*b^2 - 10*a*b*B + 8*a^2*C + 9*b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(15*b^3*d*sqrt(a + b*cos(c + d*x))) - (2*(15*a*A*b^2 - 10*a^2*b*B - 5*b^3*B + 8*a^3*C + 7*a*b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(15*b^3*d*sqrt(a + b*cos(c + d*x))) + (2*(5*b*B - 4*a*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(15*b^2*d) + (2*C*cos(c + d*x)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(5*b*d)],
[cos(c + d*x)^0*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(1/2), x, 6, (2*(a + b)*(3*b*B - 2*a*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(3*b^2*d*sqrt(a + b*cos(c + d*x))) + (2*(3*A*b^2 - 3*a*b*B + 2*a^2*C + b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(3*b^2*d*sqrt(a + b*cos(c + d*x))) + (2*C*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*b*d)],
[sec(c + d*x)^1*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(1/2), x, 8, (2*(a + b)*C*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*b)/(a + b)))/(b*d*sqrt(a + b*cos(c + d*x))) + (2*(b*B - a*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*b)/(a + b)))/(b*d*sqrt(a + b*cos(c + d*x))) + (2*A*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (c + d*x)/2, (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x)))],
[sec(c + d*x)^2*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(1/2), x, 9, -((A*(a + b)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*b)/(a + b)))/(a*d*sqrt(a + b*cos(c + d*x)))) + ((A + 2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x))) - ((A*b - 2*a*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (c + d*x)/2, (2*b)/(a + b)))/(a*d*sqrt(a + b*cos(c + d*x))) + (A*sqrt(a + b*cos(c + d*x))*tan(c + d*x))/(a*d)],
[sec(c + d*x)^3*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(1/2), x, 10, ((a + b)*(3*A*b - 4*a*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*b)/(a + b)))/(4*a^2*d*sqrt(a + b*cos(c + d*x))) - ((A*b - 4*a*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*b)/(a + b)))/(4*a*d*sqrt(a + b*cos(c + d*x))) + ((4*a^2*A + 3*A*b^2 - 4*a*b*B + 8*a^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (c + d*x)/2, (2*b)/(a + b)))/(4*a^2*d*sqrt(a + b*cos(c + d*x))) - ((3*A*b - 4*a*B)*sqrt(a + b*cos(c + d*x))*tan(c + d*x))/(4*a^2*d) + (A*sqrt(a + b*cos(c + d*x))*sec(c + d*x)*tan(c + d*x))/(2*a*d)],
[sec(c + d*x)^4*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(1/2), x, 11, -((a + b)*(16*a^2*A + 15*A*b^2 - 18*a*b*B + 24*a^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*b)/(a + b)))/(24*a^3*d*sqrt(a + b*cos(c + d*x))) + ((16*a^2*A + 5*A*b^2 - 6*a*b*B + 24*a^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*b)/(a + b)))/(24*a^2*d*sqrt(a + b*cos(c + d*x))) - ((4*a^2*A*b + 5*A*b^3 - 8*a^3*B - 6*a*b^2*B + 8*a^2*b*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (c + d*x)/2, (2*b)/(a + b)))/(8*a^3*d*sqrt(a + b*cos(c + d*x))) + ((16*a^2*A + 15*A*b^2 - 18*a*b*B + 24*a^2*C)*sqrt(a + b*cos(c + d*x))*tan(c + d*x))/(24*a^3*d) - ((5*A*b - 6*a*B)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)*tan(c + d*x))/(12*a^2*d) + (A*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^2*tan(c + d*x))/(3*a*d)],


[cos(c + d*x)^2*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(3/2), x, 8, (2*(30*a^2*A*b^2 - 15*A*b^4 - 40*a^3*b*B + 25*a*b^3*B + 48*a^4*C - 24*a^2*b^2*C - 9*b^4*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(15*(a - b)*b^4*d*sqrt(a + b*cos(c + d*x))) - (2*(30*a*A*b^2 - 40*a^2*b*B - 5*b^3*B + 48*a^3*C + 12*a*b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(15*b^4*d*sqrt(a + b*cos(c + d*x))) - (2*(A*b^2 - a*b*B + a^2*C)*cos(c + d*x)^2*sin(c + d*x))/(b*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) - (2*(15*a*A*b^2 - 20*a^2*b*B + 5*b^3*B + 24*a^3*C - 9*a*b^2*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(15*b^3*(a^2 - b^2)*d) + (2*(5*A*b^2 - 5*a*b*B + 6*a^2*C - b^2*C)*cos(c + d*x)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(5*b^2*(a^2 - b^2)*d)],
[cos(c + d*x)^1*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(3/2), x, 7, -((2*(3*a*A*b^2 - 6*a^2*b*B + 3*b^3*B + 8*a^3*C - 5*a*b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(3*(a - b)*b^3*d*sqrt(a + b*cos(c + d*x)))) + (2*(3*A*b^2 - 6*a*b*B + 8*a^2*C + b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(3*b^3*d*sqrt(a + b*cos(c + d*x))) - (2*(A*b^2 - a*b*B + a^2*C)*cos(c + d*x)*sin(c + d*x))/(b*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) + (2*(3*A*b^2 - 3*a*b*B + 4*a^2*C - b^2*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*b^2*(a^2 - b^2)*d)],
[cos(c + d*x)^0*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(3/2), x, 6, (2*(A*b^2 - a*b*B + 2*a^2*C - b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/((a - b)*b^2*d*sqrt(a + b*cos(c + d*x))) + (2*(b*B - 2*a*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(b^2*d*sqrt(a + b*cos(c + d*x))) - (2*(A*b^2 - a*b*B + a^2*C)*sin(c + d*x))/(b*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x)))],
[sec(c + d*x)^1*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(3/2), x, 9, (-2*(A*b^2 - a*b*B + a^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*b)/(a + b)))/(a*(a - b)*b*d*sqrt(a + b*cos(c + d*x))) + (2*C*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*b)/(a + b)))/(b*d*sqrt(a + b*cos(c + d*x))) + (2*A*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (c + d*x)/2, (2*b)/(a + b)))/(a*d*sqrt(a + b*cos(c + d*x))) + (2*(A*b^2 - a*b*B + a^2*C)*sin(c + d*x))/(a*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x)))],
[sec(c + d*x)^2*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(3/2), x, 10, -(((a^2*A - 3*A*b^2 + 2*a*b*B - 2*a^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*b)/(a + b)))/(a^2*(a - b)*d*sqrt(a + b*cos(c + d*x)))) + (A*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*b)/(a + b)))/(a*d*sqrt(a + b*cos(c + d*x))) - ((3*A*b - 2*a*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (c + d*x)/2, (2*b)/(a + b)))/(a^2*d*sqrt(a + b*cos(c + d*x))) + (2*(A*b^2 - a*b*B + a^2*C)*tan(c + d*x))/(a*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) + ((a^2*A - 3*A*b^2 + 2*a*b*B - 2*a^2*C)*sqrt(a + b*cos(c + d*x))*tan(c + d*x))/(a^2*(a^2 - b^2)*d)],
[sec(c + d*x)^3*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(3/2), x, 11, ((7*a^2*A*b - 15*A*b^3 - 4*a^3*B + 12*a*b^2*B - 8*a^2*b*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*b)/(a + b)))/(4*a^3*(a - b)*d*sqrt(a + b*cos(c + d*x))) - ((5*A*b - 4*a*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*b)/(a + b)))/(4*a^2*d*sqrt(a + b*cos(c + d*x))) + ((4*a^2*A + 15*A*b^2 - 12*a*b*B + 8*a^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (c + d*x)/2, (2*b)/(a + b)))/(4*a^3*d*sqrt(a + b*cos(c + d*x))) - ((7*a^2*A*b - 15*A*b^3 - 4*a^3*B + 12*a*b^2*B - 8*a^2*b*C)*sqrt(a + b*cos(c + d*x))*tan(c + d*x))/(4*a^3*(a^2 - b^2)*d) + (2*(A*b^2 - a*b*B + a^2*C)*sec(c + d*x)*tan(c + d*x))/(a*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) + ((a^2*A - 5*A*b^2 + 4*a*b*B - 4*a^2*C)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)*tan(c + d*x))/(2*a^2*(a^2 - b^2)*d)],


[cos(c + d*x)^3*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(5/2), x, 9, (2*(40*a^4*A*b^2 - 75*a^2*A*b^4 + 15*A*b^6 - 80*a^5*b*B + 140*a^3*b^3*B - 40*a*b^5*B + 128*a^6*C - 212*a^4*b^2*C + 55*a^2*b^4*C + 9*b^6*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(15*(a - b)^2*b^5*(a + b)*d*sqrt(a + b*cos(c + d*x))) - (2*(40*a^3*A*b^2 - 45*a*A*b^4 - 80*a^4*b*B + 80*a^2*b^3*B + 5*b^5*B + 128*a^5*C - 116*a^3*b^2*C - 17*a*b^4*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(15*b^5*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) - (2*(A*b^2 - a*b*B + a^2*C)*cos(c + d*x)^3*sin(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)) - (2*(2*a^2*A*b^2 - 6*A*b^4 - 5*a^3*b*B + 9*a*b^3*B + 8*a^4*C - 12*a^2*b^2*C)*cos(c + d*x)^2*sin(c + d*x))/(3*b^2*(a^2 - b^2)^2*d*sqrt(a + b*cos(c + d*x))) - (2*(20*a^3*A*b^2 - 40*a*A*b^4 - 40*a^4*b*B + 65*a^2*b^3*B - 5*b^5*B + 64*a^5*C - 98*a^3*b^2*C + 14*a*b^4*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(15*b^4*(a^2 - b^2)^2*d) + (2*(15*a^2*A*b^2 - 35*A*b^4 - 30*a^3*b*B + 50*a*b^3*B + 48*a^4*C - 71*a^2*b^2*C + 3*b^4*C)*cos(c + d*x)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(15*b^3*(a^2 - b^2)^2*d)],
[cos(c + d*x)^2*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(5/2), x, 8, -((2*(2*a^3*A*b^2 - 6*a*A*b^4 - 8*a^4*b*B + 15*a^2*b^3*B - 3*b^5*B + 16*a^5*C - 28*a^3*b^2*C + 8*a*b^4*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(3*(a - b)^2*b^4*(a + b)*d*sqrt(a + b*cos(c + d*x)))) + (2*(2*a^2*A*b^2 - 3*A*b^4 - 8*a^3*b*B + 9*a*b^3*B + 16*a^4*C - 16*a^2*b^2*C - b^4*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(3*b^4*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) - (2*(A*b^2 - a*b*B + a^2*C)*cos(c + d*x)^2*sin(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)) + (2*(4*A*b^4 + 3*a^3*b*B - 7*a*b^3*B - 6*a^4*C + 10*a^2*b^2*C)*cos(c + d*x)*sin(c + d*x))/(3*b^2*(a^2 - b^2)^2*d*sqrt(a + b*cos(c + d*x))) + (2*(a^2*A*b^2 - 5*A*b^4 - 4*a^3*b*B + 8*a*b^3*B + 8*a^4*C - 13*a^2*b^2*C + b^4*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*b^3*(a^2 - b^2)^2*d)],
[cos(c + d*x)^1*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(5/2), x, 7, -((2*(a^2*A*b^2 + 3*A*b^4 + 2*a^3*b*B - 6*a*b^3*B - 8*a^4*C + 15*a^2*b^2*C - 3*b^4*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(3*(a - b)^2*b^3*(a + b)*d*sqrt(a + b*cos(c + d*x)))) + (2*(a*A*b^2 + 2*a^2*b*B - 3*b^3*B - 8*a^3*C + 9*a*b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(3*b^3*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) - (2*(A*b^2 - a*b*B + a^2*C)*cos(c + d*x)*sin(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)) + (2*(2*a^2*A*b^2 + 2*A*b^4 + a^3*b*B - 5*a*b^3*B - 4*a^4*C + 8*a^2*b^2*C)*sin(c + d*x))/(3*b^2*(a^2 - b^2)^2*d*sqrt(a + b*cos(c + d*x)))],
[cos(c + d*x)^0*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(5/2), x, 7, (2*(4*a*A*b^2 - a^2*b*B - 3*b^3*B - 2*a^3*C + 6*a*b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(3*(a - b)^2*b^2*(a + b)*d*sqrt(a + b*cos(c + d*x))) - (2*(A*b^2 - a*b*B - 2*a^2*C + 3*b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(3*b^2*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) - (2*(A*b^2 - a*b*B + a^2*C)*sin(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)) - (2*(4*a*A*b^2 - a^2*b*B - 3*b^3*B - 2*a^3*C + 6*a*b^2*C)*sin(c + d*x))/(3*b*(a^2 - b^2)^2*d*sqrt(a + b*cos(c + d*x)))],
[sec(c + d*x)^1*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(5/2), x, 10, (-2*(7*a^2*A*b^2 - 3*A*b^4 - 4*a^3*b*B + a^4*C + 3*a^2*b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*b)/(a + b)))/(3*a^2*(a - b)^2*b*(a + b)*d*sqrt(a + b*cos(c + d*x))) + (2*(A*b^2 - a*b*B + a^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*b)/(a + b)))/(3*a*b*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) + (2*A*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (c + d*x)/2, (2*b)/(a + b)))/(a^2*d*sqrt(a + b*cos(c + d*x))) + (2*(A*b^2 - a*b*B + a^2*C)*sin(c + d*x))/(3*a*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)) + (2*(7*a^2*A*b^2 - 3*A*b^4 - 4*a^3*b*B + a^4*C + 3*a^2*b^2*C)*sin(c + d*x))/(3*a^2*(a^2 - b^2)^2*d*sqrt(a + b*cos(c + d*x)))],
[sec(c + d*x)^2*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(5/2), x, 11, -((3*a^4*A - 26*a^2*A*b^2 + 15*A*b^4 + 14*a^3*b*B - 6*a*b^3*B - 8*a^4*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*b)/(a + b)))/(3*a^3*(a - b)^2*(a + b)*d*sqrt(a + b*cos(c + d*x))) + ((3*a^2*A - 5*A*b^2 + 2*a*b*B - 2*a^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*b)/(a + b)))/(3*a^2*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) - ((5*A*b - 2*a*B)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (c + d*x)/2, (2*b)/(a + b)))/(a^3*d*sqrt(a + b*cos(c + d*x))) + (2*(A*b^2 - a*b*B + a^2*C)*tan(c + d*x))/(3*a*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)) + (2*(9*a^2*A*b^2 - 5*A*b^4 - 6*a^3*b*B + 2*a*b^3*B + 3*a^4*C + a^2*b^2*C)*tan(c + d*x))/(3*a^2*(a^2 - b^2)^2*d*sqrt(a + b*cos(c + d*x))) + ((3*a^4*A - 26*a^2*A*b^2 + 15*A*b^4 + 14*a^3*b*B - 6*a*b^3*B - 8*a^4*C)*sqrt(a + b*cos(c + d*x))*tan(c + d*x))/(3*a^3*(a^2 - b^2)^2*d)],
[sec(c + d*x)^3*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(5/2), x, 12, ((33*a^4*A*b - 170*a^2*A*b^3 + 105*A*b^5 - 12*a^5*B + 104*a^3*b^2*B - 60*a*b^4*B - 56*a^4*b*C + 24*a^2*b^3*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*b)/(a + b)))/(12*a^4*(a - b)^2*(a + b)*d*sqrt(a + b*cos(c + d*x))) - ((27*a^2*A*b - 35*A*b^3 - 12*a^3*B + 20*a*b^2*B - 8*a^2*b*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*b)/(a + b)))/(12*a^3*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) + ((4*a^2*A + 35*A*b^2 - 20*a*b*B + 8*a^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticPi(2, (c + d*x)/2, (2*b)/(a + b)))/(4*a^4*d*sqrt(a + b*cos(c + d*x))) - ((33*a^4*A*b - 170*a^2*A*b^3 + 105*A*b^5 - 12*a^5*B + 104*a^3*b^2*B - 60*a*b^4*B - 56*a^4*b*C + 24*a^2*b^3*C)*sqrt(a + b*cos(c + d*x))*tan(c + d*x))/(12*a^4*(a^2 - b^2)^2*d) + (2*(A*b^2 - a*b*B + a^2*C)*sec(c + d*x)*tan(c + d*x))/(3*a*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)) + (2*(11*a^2*A*b^2 - 7*A*b^4 - 8*a^3*b*B + 4*a*b^3*B + 5*a^4*C - a^2*b^2*C)*sec(c + d*x)*tan(c + d*x))/(3*a^2*(a^2 - b^2)^2*d*sqrt(a + b*cos(c + d*x))) + ((3*a^4*A - 54*a^2*A*b^2 + 35*A*b^4 + 36*a^3*b*B - 20*a*b^3*B - 24*a^4*C + 8*a^2*b^2*C)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)*tan(c + d*x))/(6*a^3*(a^2 - b^2)^2*d)],


[(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(7/2), x, 8, (2*(23*a^2*A*b^2 + 9*A*b^4 - 3*a^3*b*B - 29*a*b^3*B - 2*a^4*C + 19*a^2*b^2*C + 15*b^4*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(15*(a - b)^3*b^2*(a + b)^2*d*sqrt(a + b*cos(c + d*x))) - (2*(8*a*A*b^2 - 3*a^2*b*B - 5*b^3*B - 2*a^3*C + 10*a*b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(15*b^2*(a^2 - b^2)^2*d*sqrt(a + b*cos(c + d*x))) - (2*(A*b^2 - a*b*B + a^2*C)*sin(c + d*x))/(5*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(5/2)) - (2*(8*a*A*b^2 - 3*a^2*b*B - 5*b^3*B - 2*a^3*C + 10*a*b^2*C)*sin(c + d*x))/(15*b*(a^2 - b^2)^2*d*(a + b*cos(c + d*x))^(3/2)) - (2*(23*a^2*A*b^2 + 9*A*b^4 - 3*a^3*b*B - 29*a*b^3*B - 2*a^4*C + 19*a^2*b^2*C + 15*b^4*C)*sin(c + d*x))/(15*b*(a^2 - b^2)^3*d*sqrt(a + b*cos(c + d*x)))],


[(a*b*B - a^2*C + b^2*B*cos(c + d*x) + b^2*C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(1/2), x, 6, (2*(a + b)*(3*b*B - 2*a*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(3*d*sqrt(a + b*cos(c + d*x))) - (2*(a^2 - b^2)*C*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(3*d*sqrt(a + b*cos(c + d*x))) + (2*b*C*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*d)],
[(a*b*B - a^2*C + b^2*B*cos(c + d*x) + b^2*C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(3/2), x, 6, (2*(a + b)*C*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x))) + (2*(b*B - 2*a*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x)))],
[(a*b*B - a^2*C + b^2*B*cos(c + d*x) + b^2*C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(5/2), x, 7, (2*(b*B - 2*a*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/((a - b)*d*sqrt(a + b*cos(c + d*x))) + (2*C*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x))) - (2*b*(b*B - 2*a*C)*sin(c + d*x))/((a^2 - b^2)*d*sqrt(a + b*cos(c + d*x)))],
[(a*b*B - a^2*C + b^2*B*cos(c + d*x) + b^2*C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(7/2), x, 8, (2*(4*a*b*B - 5*a^2*C - 3*b^2*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*b)/(a + b)))/(3*(a - b)^2*(a + b)*d*sqrt(a + b*cos(c + d*x))) - (2*(b*B - 2*a*C)*sqrt((a + b*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*b)/(a + b)))/(3*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) - (2*b*(b*B - 2*a*C)*sin(c + d*x))/(3*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)) - (2*b*(4*a*b*B - 5*a^2*C - 3*b^2*C)*sin(c + d*x))/(3*(a^2 - b^2)^2*d*sqrt(a + b*cos(c + d*x)))],


# ::Subsection::Closed:: 
#Integrands of the form Cos[c+d x]^(m/2) (A+B Cos[c+d x]+C Cos[c+d x]^2) (a+b Cos[c+d x])^n


# ::Subsubsection::Closed:: 
#n>0


[cos(c + d*x)^(3/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x)), x, 7, (2*(9*A*b + 9*a*B + 7*b*C)*EllipticE((1/2)*(c + d*x), 2))/(15*d) + (2*(7*a*A + 5*b*B + 5*a*C)*EllipticF((1/2)*(c + d*x), 2))/(21*d) + (2*(7*a*A + 5*b*B + 5*a*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(21*d) + (2*(9*A*b + 9*a*B + 7*b*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(45*d) + (2*(9*b*B + 2*a*C)*cos(c + d*x)^(5/2)*sin(c + d*x))/(63*d) + (2*C*cos(c + d*x)^(5/2)*(a + b*cos(c + d*x))*sin(c + d*x))/(9*d)],
[cos(c + d*x)^(1/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x)), x, 6, (2*(5*a*A + 3*b*B + 3*a*C)*EllipticE((1/2)*(c + d*x), 2))/(5*d) + (2*(7*A*b + 7*a*B + 5*b*C)*EllipticF((1/2)*(c + d*x), 2))/(21*d) + (2*(7*A*b + 7*a*B + 5*b*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(21*d) + (2*(7*b*B + 2*a*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(35*d) + (2*C*cos(c + d*x)^(3/2)*(a + b*cos(c + d*x))*sin(c + d*x))/(7*d)],
[cos(c + d*x)^(-1/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x)), x, 5, (2*(5*A*b + 5*a*B + 3*b*C)*EllipticE((1/2)*(c + d*x), 2))/(5*d) + (2*(3*a*A + b*B + a*C)*EllipticF((1/2)*(c + d*x), 2))/(3*d) + (2*(5*b*B + 2*a*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(15*d) + (2*C*sqrt(cos(c + d*x))*(a + b*cos(c + d*x))*sin(c + d*x))/(5*d)],
[cos(c + d*x)^(-3/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x)), x, 5, -((2*(a*A - b*B - a*C)*EllipticE((1/2)*(c + d*x), 2))/d) + (2*(3*A*b + 3*a*B + b*C)*EllipticF((1/2)*(c + d*x), 2))/(3*d) - (2*b*(3*A - C)*sqrt(cos(c + d*x))*sin(c + d*x))/(3*d) + (2*A*(a + b*cos(c + d*x))*sin(c + d*x))/(d*sqrt(cos(c + d*x)))],
[cos(c + d*x)^(-5/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x)), x, 5, -((2*(A*b + a*B - b*C)*EllipticE((1/2)*(c + d*x), 2))/d) + (2*(a*A + 3*b*B + 3*a*C)*EllipticF((1/2)*(c + d*x), 2))/(3*d) + (2*(2*A*b + 3*a*B)*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))) + (2*A*(a + b*cos(c + d*x))*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2))],
[cos(c + d*x)^(-7/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x)), x, 6, -((2*(3*a*A + 5*b*B + 5*a*C)*EllipticE((1/2)*(c + d*x), 2))/(5*d)) + (2*(A*b + a*B + 3*b*C)*EllipticF((1/2)*(c + d*x), 2))/(3*d) + (2*(2*A*b + 5*a*B)*sin(c + d*x))/(15*d*cos(c + d*x)^(3/2)) + (2*(3*a*A + 5*b*B + 5*a*C)*sin(c + d*x))/(5*d*sqrt(cos(c + d*x))) + (2*A*(a + b*cos(c + d*x))*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2))],
[cos(c + d*x)^(-9/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x)), x, 7, -((2*(3*A*b + 3*a*B + 5*b*C)*EllipticE((1/2)*(c + d*x), 2))/(5*d)) + (2*(5*a*A + 7*b*B + 7*a*C)*EllipticF((1/2)*(c + d*x), 2))/(21*d) + (2*(2*A*b + 7*a*B)*sin(c + d*x))/(35*d*cos(c + d*x)^(5/2)) + (2*(5*a*A + 7*b*B + 7*a*C)*sin(c + d*x))/(21*d*cos(c + d*x)^(3/2)) + (2*(3*A*b + 3*a*B + 5*b*C)*sin(c + d*x))/(5*d*sqrt(cos(c + d*x))) + (2*A*(a + b*cos(c + d*x))*sin(c + d*x))/(7*d*cos(c + d*x)^(7/2))],


[cos(c + d*x)^(3/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^2, x, 8, (2*(18*a*A*b + 9*a^2*B + 7*b^2*B + 14*a*b*C)*EllipticE((1/2)*(c + d*x), 2))/(15*d) + (2*(77*a^2*A + 55*A*b^2 + 110*a*b*B + 55*a^2*C + 45*b^2*C)*EllipticF((1/2)*(c + d*x), 2))/(231*d) + (2*(77*a^2*A + 55*A*b^2 + 110*a*b*B + 55*a^2*C + 45*b^2*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(231*d) + (2*(18*a*A*b + 9*a^2*B + 7*b^2*B + 14*a*b*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(45*d) + (2*(99*A*b^2 + 121*a*b*B + 8*a^2*C + 81*b^2*C)*cos(c + d*x)^(5/2)*sin(c + d*x))/(693*d) + (2*(11*b*B + 4*a*C)*cos(c + d*x)^(5/2)*(a + b*cos(c + d*x))*sin(c + d*x))/(99*d) + (2*C*cos(c + d*x)^(5/2)*(a + b*cos(c + d*x))^2*sin(c + d*x))/(11*d)],
[cos(c + d*x)^(1/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^2, x, 7, (2*(15*a^2*A + 9*A*b^2 + 18*a*b*B + 9*a^2*C + 7*b^2*C)*EllipticE((1/2)*(c + d*x), 2))/(15*d) + (2*(14*a*A*b + 7*a^2*B + 5*b^2*B + 10*a*b*C)*EllipticF((1/2)*(c + d*x), 2))/(21*d) + (2*(14*a*A*b + 7*a^2*B + 5*b^2*B + 10*a*b*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(21*d) + (2*(63*A*b^2 + 81*a*b*B + 8*a^2*C + 49*b^2*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(315*d) + (2*(9*b*B + 4*a*C)*cos(c + d*x)^(3/2)*(a + b*cos(c + d*x))*sin(c + d*x))/(63*d) + (2*C*cos(c + d*x)^(3/2)*(a + b*cos(c + d*x))^2*sin(c + d*x))/(9*d)],
[cos(c + d*x)^(-1/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^2, x, 6, (2*(10*a*A*b + 5*a^2*B + 3*b^2*B + 6*a*b*C)*EllipticE((1/2)*(c + d*x), 2))/(5*d) + (2*(21*a^2*A + 7*A*b^2 + 14*a*b*B + 7*a^2*C + 5*b^2*C)*EllipticF((1/2)*(c + d*x), 2))/(21*d) + (2*(35*A*b^2 + 49*a*b*B + 8*a^2*C + 25*b^2*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(105*d) + (2*(7*b*B + 4*a*C)*sqrt(cos(c + d*x))*(a + b*cos(c + d*x))*sin(c + d*x))/(35*d) + (2*C*sqrt(cos(c + d*x))*(a + b*cos(c + d*x))^2*sin(c + d*x))/(7*d)],
[cos(c + d*x)^(-3/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^2, x, 6, -((2*(5*a^2*A - 5*A*b^2 - 10*a*b*B - 5*a^2*C - 3*b^2*C)*EllipticE((1/2)*(c + d*x), 2))/(5*d)) + (2*(6*a*A*b + 3*a^2*B + b^2*B + 2*a*b*C)*EllipticF((1/2)*(c + d*x), 2))/(3*d) - (2*b*(15*a*A - 5*b*B - 7*a*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(15*d) - (2*b*(5*A - C)*sqrt(cos(c + d*x))*(a + b*cos(c + d*x))*sin(c + d*x))/(5*d) + (2*A*(a + b*cos(c + d*x))^2*sin(c + d*x))/(d*sqrt(cos(c + d*x)))],
[cos(c + d*x)^(-5/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^2, x, 6, -((2*(2*a*A*b + a^2*B - b^2*B - 2*a*b*C)*EllipticE((1/2)*(c + d*x), 2))/d) + (2*(a^2*A + 3*A*b^2 + 6*a*b*B + 3*a^2*C + b^2*C)*EllipticF((1/2)*(c + d*x), 2))/(3*d) - (2*b*(5*A*b + 3*a*B - b*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(3*d) + (2*(4*A*b + 3*a*B)*(a + b*cos(c + d*x))*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))) + (2*A*(a + b*cos(c + d*x))^2*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2))],
[cos(c + d*x)^(-7/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^2, x, 6, -((2*(3*a^2*A + 5*A*b^2 + 10*a*b*B + 5*a^2*C - 5*b^2*C)*EllipticE((1/2)*(c + d*x), 2))/(5*d)) + (2*(2*a*A*b + a^2*B + 3*b^2*B + 6*a*b*C)*EllipticF((1/2)*(c + d*x), 2))/(3*d) + (2*(9*a^2*A + 8*A*b^2 + 25*a*b*B + 15*a^2*C)*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))) + (2*(4*A*b + 5*a*B)*(a + b*cos(c + d*x))*sin(c + d*x))/(15*d*cos(c + d*x)^(3/2)) + (2*A*(a + b*cos(c + d*x))^2*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2))],
[cos(c + d*x)^(-9/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^2, x, 7, -((2*(6*a*A*b + 3*a^2*B + 5*b^2*B + 10*a*b*C)*EllipticE((1/2)*(c + d*x), 2))/(5*d)) + (2*(5*a^2*A + 7*A*b^2 + 14*a*b*B + 7*a^2*C + 21*b^2*C)*EllipticF((1/2)*(c + d*x), 2))/(21*d) + (2*(25*a^2*A + 8*A*b^2 + 49*a*b*B + 35*a^2*C)*sin(c + d*x))/(105*d*cos(c + d*x)^(3/2)) + (2*(6*a*A*b + 3*a^2*B + 5*b^2*B + 10*a*b*C)*sin(c + d*x))/(5*d*sqrt(cos(c + d*x))) + (2*(4*A*b + 7*a*B)*(a + b*cos(c + d*x))*sin(c + d*x))/(35*d*cos(c + d*x)^(5/2)) + (2*A*(a + b*cos(c + d*x))^2*sin(c + d*x))/(7*d*cos(c + d*x)^(7/2))],
[cos(c + d*x)^(-11/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^2, x, 8, -((2*(7*a^2*A + 9*A*b^2 + 18*a*b*B + 9*a^2*C + 15*b^2*C)*EllipticE((1/2)*(c + d*x), 2))/(15*d)) + (2*(10*a*A*b + 5*a^2*B + 7*b^2*B + 14*a*b*C)*EllipticF((1/2)*(c + d*x), 2))/(21*d) + (2*(49*a^2*A + 8*A*b^2 + 81*a*b*B + 63*a^2*C)*sin(c + d*x))/(315*d*cos(c + d*x)^(5/2)) + (2*(10*a*A*b + 5*a^2*B + 7*b^2*B + 14*a*b*C)*sin(c + d*x))/(21*d*cos(c + d*x)^(3/2)) + (2*(7*a^2*A + 9*A*b^2 + 18*a*b*B + 9*a^2*C + 15*b^2*C)*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))) + (2*(4*A*b + 9*a*B)*(a + b*cos(c + d*x))*sin(c + d*x))/(63*d*cos(c + d*x)^(7/2)) + (2*A*(a + b*cos(c + d*x))^2*sin(c + d*x))/(9*d*cos(c + d*x)^(9/2))],


[cos(c + d*x)^(1/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^3, x, 8, (2*(15*a^3*A + 27*a*A*b^2 + 27*a^2*b*B + 7*b^3*B + 9*a^3*C + 21*a*b^2*C)*EllipticE((1/2)*(c + d*x), 2))/(15*d) + (2*(231*a^2*A*b + 55*A*b^3 + 77*a^3*B + 165*a*b^2*B + 165*a^2*b*C + 45*b^3*C)*EllipticF((1/2)*(c + d*x), 2))/(231*d) + (2*(231*a^2*A*b + 55*A*b^3 + 77*a^3*B + 165*a*b^2*B + 165*a^2*b*C + 45*b^3*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(231*d) + (2*(1584*a*A*b^2 + 979*a^2*b*B + 539*b^3*B + 48*a^3*C + 1212*a*b^2*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(3465*d) + (2*(99*A*b^2 + 143*a*b*B + 24*a^2*C + 81*b^2*C)*cos(c + d*x)^(3/2)*(a + b*cos(c + d*x))*sin(c + d*x))/(693*d) + (2*(11*b*B + 6*a*C)*cos(c + d*x)^(3/2)*(a + b*cos(c + d*x))^2*sin(c + d*x))/(99*d) + (2*C*cos(c + d*x)^(3/2)*(a + b*cos(c + d*x))^3*sin(c + d*x))/(11*d)],
[cos(c + d*x)^(-1/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^3, x, 7, (2*(45*a^2*A*b + 9*A*b^3 + 15*a^3*B + 27*a*b^2*B + 27*a^2*b*C + 7*b^3*C)*EllipticE((1/2)*(c + d*x), 2))/(15*d) + (2*(21*a^3*A + 21*a*A*b^2 + 21*a^2*b*B + 5*b^3*B + 7*a^3*C + 15*a*b^2*C)*EllipticF((1/2)*(c + d*x), 2))/(21*d) + (2*(252*a*A*b^2 + 171*a^2*b*B + 75*b^3*B + 16*a^3*C + 176*a*b^2*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(315*d) + (2*(63*A*b^2 + 99*a*b*B + 24*a^2*C + 49*b^2*C)*sqrt(cos(c + d*x))*(a + b*cos(c + d*x))*sin(c + d*x))/(315*d) + (2*(3*b*B + 2*a*C)*sqrt(cos(c + d*x))*(a + b*cos(c + d*x))^2*sin(c + d*x))/(21*d) + (2*C*sqrt(cos(c + d*x))*(a + b*cos(c + d*x))^3*sin(c + d*x))/(9*d)],
[cos(c + d*x)^(-3/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^3, x, 7, -((1/(5*d))*(2*(5*a^3*A - 15*a*A*b^2 - 15*a^2*b*B - 3*b^3*B - 5*a^3*C - 9*a*b^2*C)*EllipticE((1/2)*(c + d*x), 2))) + (1/(21*d))*(2*(63*a^2*A*b + 7*A*b^3 + 21*a^3*B + 21*a*b^2*B + 21*a^2*b*C + 5*b^3*C)*EllipticF((1/2)*(c + d*x), 2)) - (2*b*(105*a^2*A - 35*A*b^2 - 84*a*b*B - 57*a^2*C - 25*b^2*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(105*d) - (2*b*(35*a*A - 7*b*B - 11*a*C)*sqrt(cos(c + d*x))*(a + b*cos(c + d*x))*sin(c + d*x))/(35*d) - (2*b*(7*A - C)*sqrt(cos(c + d*x))*(a + b*cos(c + d*x))^2*sin(c + d*x))/(7*d) + (2*A*(a + b*cos(c + d*x))^3*sin(c + d*x))/(d*sqrt(cos(c + d*x)))],
[cos(c + d*x)^(-5/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^3, x, 7, -((1/(5*d))*(2*(15*a^2*A*b - 5*A*b^3 + 5*a^3*B - 15*a*b^2*B - 15*a^2*b*C - 3*b^3*C)*EllipticE((1/2)*(c + d*x), 2))) + (2*(a^3*A + 9*a*A*b^2 + 9*a^2*b*B + b^3*B + 3*a^3*C + 3*a*b^2*C)*EllipticF((1/2)*(c + d*x), 2))/(3*d) - (2*b*(40*a*A*b + 15*a^2*B - 5*b^2*B - 12*a*b*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(15*d) - (2*b*(35*A*b + 15*a*B - 3*b*C)*sqrt(cos(c + d*x))*(a + b*cos(c + d*x))*sin(c + d*x))/(15*d) + (2*(2*A*b + a*B)*(a + b*cos(c + d*x))^2*sin(c + d*x))/(d*sqrt(cos(c + d*x))) + (2*A*(a + b*cos(c + d*x))^3*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2))],
[cos(c + d*x)^(-7/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^3, x, 7, -((2*(3*a^3*A + 15*a*A*b^2 + 15*a^2*b*B - 5*b^3*B + 5*a^3*C - 15*a*b^2*C)*EllipticE((1/2)*(c + d*x), 2))/(5*d)) + (2*(3*a^2*A*b + 3*A*b^3 + a^3*B + 9*a*b^2*B + 9*a^2*b*C + b^3*C)*EllipticF((1/2)*(c + d*x), 2))/(3*d) - (2*b*(9*a^2*A + 33*A*b^2 + 40*a*b*B + 15*a^2*C - 5*b^2*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(15*d) + (2*(9*a^2*A + 24*A*b^2 + 35*a*b*B + 15*a^2*C)*(a + b*cos(c + d*x))*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))) + (2*(6*A*b + 5*a*B)*(a + b*cos(c + d*x))^2*sin(c + d*x))/(15*d*cos(c + d*x)^(3/2)) + (2*A*(a + b*cos(c + d*x))^3*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2))],
[cos(c + d*x)^(-9/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^3, x, 7, -((2*(9*a^2*A*b + 5*A*b^3 + 3*a^3*B + 15*a*b^2*B + 15*a^2*b*C - 5*b^3*C)*EllipticE((1/2)*(c + d*x), 2))/(5*d)) + (2*(5*a^3*A + 21*a*A*b^2 + 21*a^2*b*B + 21*b^3*B + 7*a^3*C + 63*a*b^2*C)*EllipticF((1/2)*(c + d*x), 2))/(21*d) + (2*(164*a^2*A*b + 48*A*b^3 + 63*a^3*B + 231*a*b^2*B + 280*a^2*b*C)*sin(c + d*x))/(105*d*sqrt(cos(c + d*x))) + (2*(25*a^2*A + 24*A*b^2 + 63*a*b*B + 35*a^2*C)*(a + b*cos(c + d*x))*sin(c + d*x))/(105*d*cos(c + d*x)^(3/2)) + (2*(6*A*b + 7*a*B)*(a + b*cos(c + d*x))^2*sin(c + d*x))/(35*d*cos(c + d*x)^(5/2)) + (2*A*(a + b*cos(c + d*x))^3*sin(c + d*x))/(7*d*cos(c + d*x)^(7/2))],
[cos(c + d*x)^(-11/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^3, x, 8, -((2*(7*a^3*A + 27*a*A*b^2 + 27*a^2*b*B + 15*b^3*B + 9*a^3*C + 45*a*b^2*C)*EllipticE((1/2)*(c + d*x), 2))/(15*d)) + (2*(15*a^2*A*b + 7*A*b^3 + 5*a^3*B + 21*a*b^2*B + 21*a^2*b*C + 21*b^3*C)*EllipticF((1/2)*(c + d*x), 2))/(21*d) + (2*(176*a^2*A*b + 16*A*b^3 + 75*a^3*B + 171*a*b^2*B + 252*a^2*b*C)*sin(c + d*x))/(315*d*cos(c + d*x)^(3/2)) + (2*(7*a^3*A + 27*a*A*b^2 + 27*a^2*b*B + 15*b^3*B + 9*a^3*C + 45*a*b^2*C)*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))) + (2*(49*a^2*A + 24*A*b^2 + 99*a*b*B + 63*a^2*C)*(a + b*cos(c + d*x))*sin(c + d*x))/(315*d*cos(c + d*x)^(5/2)) + (2*(2*A*b + 3*a*B)*(a + b*cos(c + d*x))^2*sin(c + d*x))/(21*d*cos(c + d*x)^(7/2)) + (2*A*(a + b*cos(c + d*x))^3*sin(c + d*x))/(9*d*cos(c + d*x)^(9/2))],


[cos(c + d*x)^(1/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^4, x, 9, (2*(195*a^4*A + 702*a^2*A*b^2 + 91*A*b^4 + 468*a^3*b*B + 364*a*b^3*B + 117*a^4*C + 546*a^2*b^2*C + 77*b^4*C)*EllipticE((1/2)*(c + d*x), 2))/(195*d) + (2*(308*a^3*A*b + 220*a*A*b^3 + 77*a^4*B + 330*a^2*b^2*B + 45*b^4*B + 220*a^3*b*C + 180*a*b^3*C)*EllipticF((1/2)*(c + d*x), 2))/(231*d) + (2*(308*a^3*A*b + 220*a*A*b^3 + 77*a^4*B + 330*a^2*b^2*B + 45*b^4*B + 220*a^3*b*C + 180*a*b^3*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(231*d) + (2*(33319*a^2*A*b^2 + 7007*A*b^4 + 13351*a^3*b*B + 22763*a*b^3*B + 384*a^4*C + 25217*a^2*b^2*C + 5929*b^4*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(45045*d) + (2*(3146*a*A*b^2 + 2171*a^2*b*B + 1053*b^3*B + 192*a^3*C + 2518*a*b^2*C)*cos(c + d*x)^(3/2)*(a + b*cos(c + d*x))*sin(c + d*x))/(9009*d) + (2*(143*A*b^2 + 221*a*b*B + 48*a^2*C + 121*b^2*C)*cos(c + d*x)^(3/2)*(a + b*cos(c + d*x))^2*sin(c + d*x))/(1287*d) + (2*(13*b*B + 8*a*C)*cos(c + d*x)^(3/2)*(a + b*cos(c + d*x))^3*sin(c + d*x))/(143*d) + (2*C*cos(c + d*x)^(3/2)*(a + b*cos(c + d*x))^4*sin(c + d*x))/(13*d)],
[cos(c + d*x)^(-1/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^4, x, 8, (2*(60*a^3*A*b + 36*a*A*b^3 + 15*a^4*B + 54*a^2*b^2*B + 7*b^4*B + 36*a^3*b*C + 28*a*b^3*C)*EllipticE((1/2)*(c + d*x), 2))/(15*d) + (2*(231*a^4*A + 462*a^2*A*b^2 + 55*A*b^4 + 308*a^3*b*B + 220*a*b^3*B + 77*a^4*C + 330*a^2*b^2*C + 45*b^4*C)*EllipticF((1/2)*(c + d*x), 2))/(231*d) + (2*(4653*a^2*A*b^2 + 825*A*b^4 + 2057*a^3*b*B + 2761*a*b^3*B + 128*a^4*C + 3199*a^2*b^2*C + 675*b^4*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(3465*d) + (2*(1782*a*A*b^2 + 1353*a^2*b*B + 539*b^3*B + 192*a^3*C + 1346*a*b^2*C)*sqrt(cos(c + d*x))*(a + b*cos(c + d*x))*sin(c + d*x))/(3465*d) + (2*(33*A*b^2 + 55*a*b*B + 16*a^2*C + 27*b^2*C)*sqrt(cos(c + d*x))*(a + b*cos(c + d*x))^2*sin(c + d*x))/(231*d) + (2*(11*b*B + 8*a*C)*sqrt(cos(c + d*x))*(a + b*cos(c + d*x))^3*sin(c + d*x))/(99*d) + (2*C*sqrt(cos(c + d*x))*(a + b*cos(c + d*x))^4*sin(c + d*x))/(11*d)],
[cos(c + d*x)^(-3/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^4, x, 8, -((2*(15*a^4*A - 90*a^2*A*b^2 - 9*A*b^4 - 60*a^3*b*B - 36*a*b^3*B - 15*a^4*C - 54*a^2*b^2*C - 7*b^4*C)*EllipticE((1/2)*(c + d*x), 2))/(15*d)) + (2*(84*a^3*A*b + 28*a*A*b^3 + 21*a^4*B + 42*a^2*b^2*B + 5*b^4*B + 28*a^3*b*C + 20*a*b^3*C)*EllipticF((1/2)*(c + d*x), 2))/(21*d) - (2*b*(315*a^3*A - 357*a*A*b^2 - 423*a^2*b*B - 75*b^3*B - 187*a^3*C - 251*a*b^2*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(315*d) - (2*b*(315*a^2*A - 63*A*b^2 - 162*a*b*B - 123*a^2*C - 49*b^2*C)*sqrt(cos(c + d*x))*(a + b*cos(c + d*x))*sin(c + d*x))/(315*d) - (2*b*(21*a*A - 3*b*B - 5*a*C)*sqrt(cos(c + d*x))*(a + b*cos(c + d*x))^2*sin(c + d*x))/(21*d) - (2*b*(9*A - C)*sqrt(cos(c + d*x))*(a + b*cos(c + d*x))^3*sin(c + d*x))/(9*d) + (2*A*(a + b*cos(c + d*x))^4*sin(c + d*x))/(d*sqrt(cos(c + d*x)))],
[cos(c + d*x)^(-5/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^4, x, 8, -((2*(20*a^3*A*b - 20*a*A*b^3 + 5*a^4*B - 30*a^2*b^2*B - 3*b^4*B - 20*a^3*b*C - 12*a*b^3*C)*EllipticE((1/2)*(c + d*x), 2))/(5*d)) + (2*(7*a^4*A + 126*a^2*A*b^2 + 7*A*b^4 + 84*a^3*b*B + 28*a*b^3*B + 21*a^4*C + 42*a^2*b^2*C + 5*b^4*C)*EllipticF((1/2)*(c + d*x), 2))/(21*d) - (2*b*(385*a^2*A*b - 35*A*b^3 + 105*a^3*B - 119*a*b^2*B - 141*a^2*b*C - 25*b^3*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(105*d) - (2*b*(350*a*A*b + 105*a^2*B - 21*b^2*B - 54*a*b*C)*sqrt(cos(c + d*x))*(a + b*cos(c + d*x))*sin(c + d*x))/(105*d) - (2*b*(21*A*b + 7*a*B - b*C)*sqrt(cos(c + d*x))*(a + b*cos(c + d*x))^2*sin(c + d*x))/(7*d) + (2*(8*A*b + 3*a*B)*(a + b*cos(c + d*x))^3*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))) + (2*A*(a + b*cos(c + d*x))^4*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2))],
[cos(c + d*x)^(-7/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^4, x, 8, -((2*(3*a^4*A + 30*a^2*A*b^2 - 5*A*b^4 + 20*a^3*b*B - 20*a*b^3*B + 5*a^4*C - 30*a^2*b^2*C - 3*b^4*C)*EllipticE((1/2)*(c + d*x), 2))/(5*d)) + (2*(4*a^3*A*b + 12*a*A*b^3 + a^4*B + 18*a^2*b^2*B + b^4*B + 12*a^3*b*C + 4*a*b^3*C)*EllipticF((1/2)*(c + d*x), 2))/(3*d) - (2*b*(9*a^3*A + 73*a*A*b^2 + 55*a^2*b*B - 5*b^3*B + 15*a^3*C - 17*a*b^2*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(15*d) - (2*b*(9*a^2*A + 59*A*b^2 + 50*a*b*B + 15*a^2*C - 3*b^2*C)*sqrt(cos(c + d*x))*(a + b*cos(c + d*x))*sin(c + d*x))/(15*d) + (2*(3*a^2*A + 16*A*b^2 + 15*a*b*B + 5*a^2*C)*(a + b*cos(c + d*x))^2*sin(c + d*x))/(5*d*sqrt(cos(c + d*x))) + (2*(8*A*b + 5*a*B)*(a + b*cos(c + d*x))^3*sin(c + d*x))/(15*d*cos(c + d*x)^(3/2)) + (2*A*(a + b*cos(c + d*x))^4*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2))],
[cos(c + d*x)^(-9/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^4, x, 8, -((2*(12*a^3*A*b + 20*a*A*b^3 + 3*a^4*B + 30*a^2*b^2*B - 5*b^4*B + 20*a^3*b*C - 20*a*b^3*C)*EllipticE((1/2)*(c + d*x), 2))/(5*d)) + (2*(5*a^4*A + 42*a^2*A*b^2 + 21*A*b^4 + 28*a^3*b*B + 84*a*b^3*B + 7*a^4*C + 126*a^2*b^2*C + 7*b^4*C)*EllipticF((1/2)*(c + d*x), 2))/(21*d) - (2*b*(227*a^2*A*b + 279*A*b^3 + 63*a^3*B + 511*a*b^2*B + 385*a^2*b*C - 35*b^3*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(105*d) + (2*(202*a^2*A*b + 192*A*b^3 + 63*a^3*B + 413*a*b^2*B + 350*a^2*b*C)*(a + b*cos(c + d*x))*sin(c + d*x))/(105*d*sqrt(cos(c + d*x))) + (2*(25*a^2*A + 48*A*b^2 + 77*a*b*B + 35*a^2*C)*(a + b*cos(c + d*x))^2*sin(c + d*x))/(105*d*cos(c + d*x)^(3/2)) + (2*(8*A*b + 7*a*B)*(a + b*cos(c + d*x))^3*sin(c + d*x))/(35*d*cos(c + d*x)^(5/2)) + (2*A*(a + b*cos(c + d*x))^4*sin(c + d*x))/(7*d*cos(c + d*x)^(7/2))],
[cos(c + d*x)^(-11/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^4, x, 8, -((2*(7*a^4*A + 54*a^2*A*b^2 + 15*A*b^4 + 36*a^3*b*B + 60*a*b^3*B + 9*a^4*C + 90*a^2*b^2*C - 15*b^4*C)*EllipticE((1/2)*(c + d*x), 2))/(15*d)) + (2*(20*a^3*A*b + 28*a*A*b^3 + 5*a^4*B + 42*a^2*b^2*B + 21*b^4*B + 28*a^3*b*C + 84*a*b^3*C)*EllipticF((1/2)*(c + d*x), 2))/(21*d) + (2*(147*a^4*A + 883*a^2*A*b^2 + 128*A*b^4 + 681*a^3*b*B + 837*a*b^3*B + 189*a^4*C + 1533*a^2*b^2*C)*sin(c + d*x))/(315*d*sqrt(cos(c + d*x))) + (2*(202*a^2*A*b + 64*A*b^3 + 75*a^3*B + 261*a*b^2*B + 294*a^2*b*C)*(a + b*cos(c + d*x))*sin(c + d*x))/(315*d*cos(c + d*x)^(3/2)) + (2*(49*a^2*A + 48*A*b^2 + 117*a*b*B + 63*a^2*C)*(a + b*cos(c + d*x))^2*sin(c + d*x))/(315*d*cos(c + d*x)^(5/2)) + (2*(8*A*b + 9*a*B)*(a + b*cos(c + d*x))^3*sin(c + d*x))/(63*d*cos(c + d*x)^(7/2)) + (2*A*(a + b*cos(c + d*x))^4*sin(c + d*x))/(9*d*cos(c + d*x)^(9/2))],
[cos(c + d*x)^(-13/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^4, x, 9, -((2*(28*a^3*A*b + 36*a*A*b^3 + 7*a^4*B + 54*a^2*b^2*B + 15*b^4*B + 36*a^3*b*C + 60*a*b^3*C)*EllipticE((1/2)*(c + d*x), 2))/(15*d)) + (2*(45*a^4*A + 330*a^2*A*b^2 + 77*A*b^4 + 220*a^3*b*B + 308*a*b^3*B + 55*a^4*C + 462*a^2*b^2*C + 231*b^4*C)*EllipticF((1/2)*(c + d*x), 2))/(231*d) + (2*(675*a^4*A + 3199*a^2*A*b^2 + 128*A*b^4 + 2761*a^3*b*B + 2057*a*b^3*B + 825*a^4*C + 4653*a^2*b^2*C)*sin(c + d*x))/(3465*d*cos(c + d*x)^(3/2)) + (2*(28*a^3*A*b + 36*a*A*b^3 + 7*a^4*B + 54*a^2*b^2*B + 15*b^4*B + 36*a^3*b*C + 60*a*b^3*C)*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))) + (2*(1346*a^2*A*b + 192*A*b^3 + 539*a^3*B + 1353*a*b^2*B + 1782*a^2*b*C)*(a + b*cos(c + d*x))*sin(c + d*x))/(3465*d*cos(c + d*x)^(5/2)) + (2*(27*a^2*A + 16*A*b^2 + 55*a*b*B + 33*a^2*C)*(a + b*cos(c + d*x))^2*sin(c + d*x))/(231*d*cos(c + d*x)^(7/2)) + (2*(8*A*b + 11*a*B)*(a + b*cos(c + d*x))^3*sin(c + d*x))/(99*d*cos(c + d*x)^(9/2)) + (2*A*(a + b*cos(c + d*x))^4*sin(c + d*x))/(11*d*cos(c + d*x)^(11/2))],


# ::Subsubsection::Closed:: 
#n<0


[cos(c + d*x)^(5/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x)), x, 8, -((2*(5*a*A*b^2 - 5*a^2*b*B - 3*b^3*B + 5*a^3*C + 3*a*b^2*C)*EllipticE((1/2)*(c + d*x), 2))/(5*b^4*d)) + (2*(21*a^2*A*b^2 + 7*A*b^4 - 21*a^3*b*B - 7*a*b^3*B + 21*a^4*C + 7*a^2*b^2*C + 5*b^4*C)*EllipticF((1/2)*(c + d*x), 2))/(21*b^5*d) - (2*a^3*(A*b^2 - a*b*B + a^2*C)*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2))/(b^5*(a + b)*d) + (2*(7*A*b^2 - 7*a*b*B + 7*a^2*C + 5*b^2*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(21*b^3*d) + (2*(b*B - a*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(5*b^2*d) + (2*C*cos(c + d*x)^(5/2)*sin(c + d*x))/(7*b*d)],
[cos(c + d*x)^(3/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x)), x, 7, (2*(5*A*b^2 - 5*a*b*B + 5*a^2*C + 3*b^2*C)*EllipticE((1/2)*(c + d*x), 2))/(5*b^3*d) - (2*(3*a*A*b^2 - 3*a^2*b*B - b^3*B + 3*a^3*C + a*b^2*C)*EllipticF((1/2)*(c + d*x), 2))/(3*b^4*d) + (2*a^2*(A*b^2 - a*b*B + a^2*C)*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2))/(b^4*(a + b)*d) + (2*(b*B - a*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(3*b^2*d) + (2*C*cos(c + d*x)^(3/2)*sin(c + d*x))/(5*b*d)],
[cos(c + d*x)^(1/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x)), x, 6, (2*(b*B - a*C)*EllipticE((1/2)*(c + d*x), 2))/(b^2*d) + (2*(3*A*b^2 - 3*a*b*B + 3*a^2*C + b^2*C)*EllipticF((1/2)*(c + d*x), 2))/(3*b^3*d) - (2*a*(A*b^2 - a*b*B + a^2*C)*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2))/(b^3*(a + b)*d) + (2*C*sqrt(cos(c + d*x))*sin(c + d*x))/(3*b*d)],
[cos(c + d*x)^(-1/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x)), x, 5, (2*C*EllipticE((1/2)*(c + d*x), 2))/(b*d) + (2*(b*B - a*C)*EllipticF((1/2)*(c + d*x), 2))/(b^2*d) + (2*(A*b^2 - a*b*B + a^2*C)*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2))/(b^2*(a + b)*d)],
[cos(c + d*x)^(-3/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x)), x, 6, -((2*A*EllipticE((1/2)*(c + d*x), 2))/(a*d)) + (2*C*EllipticF((1/2)*(c + d*x), 2))/(b*d) - (2*(A*b^2 - a*b*B + a^2*C)*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2))/(a*b*(a + b)*d) + (2*A*sin(c + d*x))/(a*d*sqrt(cos(c + d*x)))],
[cos(c + d*x)^(-5/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x)), x, 7, (2*(A*b - a*B)*EllipticE((1/2)*(c + d*x), 2))/(a^2*d) + (2*A*EllipticF((1/2)*(c + d*x), 2))/(3*a*d) + (2*(A*b^2 - a*b*B + a^2*C)*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2))/(a^2*(a + b)*d) + (2*A*sin(c + d*x))/(3*a*d*cos(c + d*x)^(3/2)) - (2*(A*b - a*B)*sin(c + d*x))/(a^2*d*sqrt(cos(c + d*x)))],
[cos(c + d*x)^(-7/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x)), x, 8, -((2*(3*a^2*A + 5*A*b^2 - 5*a*b*B + 5*a^2*C)*EllipticE((1/2)*(c + d*x), 2))/(5*a^3*d)) - (2*(A*b - a*B)*EllipticF((1/2)*(c + d*x), 2))/(3*a^2*d) - (2*b*(A*b^2 - a*b*B + a^2*C)*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2))/(a^3*(a + b)*d) + (2*A*sin(c + d*x))/(5*a*d*cos(c + d*x)^(5/2)) - (2*(A*b - a*B)*sin(c + d*x))/(3*a^2*d*cos(c + d*x)^(3/2)) + (2*(3*a^2*A + 5*A*b^2 - 5*a*b*B + 5*a^2*C)*sin(c + d*x))/(5*a^3*d*sqrt(cos(c + d*x)))],
[cos(c + d*x)^(-9/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x)), x, 9, (2*(3*a^2*A*b + 5*A*b^3 - 3*a^3*B - 5*a*b^2*B + 5*a^2*b*C)*EllipticE((1/2)*(c + d*x), 2))/(5*a^4*d) + (2*(5*a^2*A + 7*A*b^2 - 7*a*b*B + 7*a^2*C)*EllipticF((1/2)*(c + d*x), 2))/(21*a^3*d) + (2*b^2*(A*b^2 - a*b*B + a^2*C)*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2))/(a^4*(a + b)*d) + (2*A*sin(c + d*x))/(7*a*d*cos(c + d*x)^(7/2)) - (2*(A*b - a*B)*sin(c + d*x))/(5*a^2*d*cos(c + d*x)^(5/2)) + (2*(5*a^2*A + 7*A*b^2 - 7*a*b*B + 7*a^2*C)*sin(c + d*x))/(21*a^3*d*cos(c + d*x)^(3/2)) - (2*(3*a^2*A*b + 5*A*b^3 - 3*a^3*B - 5*a*b^2*B + 5*a^2*b*C)*sin(c + d*x))/(5*a^4*d*sqrt(cos(c + d*x)))],


[cos(c + d*x)^(5/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^2, x, 8, ((15*a^2*A*b^2 - 10*A*b^4 - 25*a^3*b*B + 20*a*b^3*B + 35*a^4*C - 24*a^2*b^2*C - 6*b^4*C)*EllipticE((1/2)*(c + d*x), 2))/(5*b^4*(a^2 - b^2)*d) - ((9*a^3*A*b^2 - 12*a*A*b^4 - 15*a^4*b*B + 16*a^2*b^3*B + 2*b^5*B + 21*a^5*C - 20*a^3*b^2*C - 4*a*b^4*C)*EllipticF((1/2)*(c + d*x), 2))/(3*b^5*(a^2 - b^2)*d) + (a^2*(3*a^2*A*b^2 - 5*A*b^4 - 5*a^3*b*B + 7*a*b^3*B + 7*a^4*C - 9*a^2*b^2*C)*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2))/((a - b)*b^5*(a + b)^2*d) - ((3*a*A*b^2 - 5*a^2*b*B + 2*b^3*B + 7*a^3*C - 4*a*b^2*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(3*b^3*(a^2 - b^2)*d) + ((5*A*b^2 - 5*a*b*B + 7*a^2*C - 2*b^2*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(5*b^2*(a^2 - b^2)*d) - ((A*b^2 - a*b*B + a^2*C)*cos(c + d*x)^(5/2)*sin(c + d*x))/(b*(a^2 - b^2)*d*(a + b*cos(c + d*x)))],
[cos(c + d*x)^(3/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^2, x, 7, -(((a*A*b^2 - 3*a^2*b*B + 2*b^3*B + 5*a^3*C - 4*a*b^2*C)*EllipticE((1/2)*(c + d*x), 2))/(b^3*(a^2 - b^2)*d)) + ((3*a^2*A*b^2 - 6*A*b^4 - 9*a^3*b*B + 12*a*b^3*B + 15*a^4*C - 16*a^2*b^2*C - 2*b^4*C)*EllipticF((1/2)*(c + d*x), 2))/(3*b^4*(a^2 - b^2)*d) - (a*(a^2*A*b^2 - 3*A*b^4 - 3*a^3*b*B + 5*a*b^3*B + 5*a^4*C - 7*a^2*b^2*C)*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2))/((a - b)*b^4*(a + b)^2*d) + ((3*A*b^2 - 3*a*b*B + 5*a^2*C - 2*b^2*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(3*b^2*(a^2 - b^2)*d) - ((A*b^2 - a*b*B + a^2*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(b*(a^2 - b^2)*d*(a + b*cos(c + d*x)))],
[cos(c + d*x)^(1/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^2, x, 6, ((A*b^2 - a*b*B + 3*a^2*C - 2*b^2*C)*EllipticE((1/2)*(c + d*x), 2))/(b^2*(a^2 - b^2)*d) + ((a*A*b^2 + a^2*b*B - 2*b^3*B - 3*a^3*C + 4*a*b^2*C)*EllipticF((1/2)*(c + d*x), 2))/(b^3*(a^2 - b^2)*d) - ((a^2*A*b^2 + A*b^4 + a^3*b*B - 3*a*b^3*B - 3*a^4*C + 5*a^2*b^2*C)*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2))/((a - b)*b^3*(a + b)^2*d) - ((A*b^2 - a*b*B + a^2*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(b*(a^2 - b^2)*d*(a + b*cos(c + d*x)))],
[cos(c + d*x)^(-1/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^2, x, 6, -(((A*b^2 - a*b*B + a^2*C)*EllipticE((1/2)*(c + d*x), 2))/(a*b*(a^2 - b^2)*d)) - ((A*b^2 - a*b*B - a^2*C + 2*b^2*C)*EllipticF((1/2)*(c + d*x), 2))/(b^2*(a^2 - b^2)*d) + ((3*a^2*A*b^2 - A*b^4 - a^3*b*B - a*b^3*B - a^4*C + 3*a^2*b^2*C)*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2))/(a*(a - b)*b^2*(a + b)^2*d) + ((A*b^2 - a*b*B + a^2*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(a*(a^2 - b^2)*d*(a + b*cos(c + d*x)))],
[cos(c + d*x)^(-3/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^2, x, 7, -(((2*a^2*A - 3*A*b^2 + a*b*B - a^2*C)*EllipticE((1/2)*(c + d*x), 2))/(a^2*(a^2 - b^2)*d)) + ((A*b^2 - a*b*B + a^2*C)*EllipticF((1/2)*(c + d*x), 2))/(a*b*(a^2 - b^2)*d) - ((5*a^2*A*b^2 - 3*A*b^4 - 3*a^3*b*B + a*b^3*B + a^4*C + a^2*b^2*C)*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2))/(a^2*(a - b)*b*(a + b)^2*d) + ((2*a^2*A - 3*A*b^2 + a*b*B - a^2*C)*sin(c + d*x))/(a^2*(a^2 - b^2)*d*sqrt(cos(c + d*x))) + ((A*b^2 - a*b*B + a^2*C)*sin(c + d*x))/(a*(a^2 - b^2)*d*sqrt(cos(c + d*x))*(a + b*cos(c + d*x)))],
[cos(c + d*x)^(-5/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^2, x, 8, ((4*a^2*A*b - 5*A*b^3 - 2*a^3*B + 3*a*b^2*B - a^2*b*C)*EllipticE((1/2)*(c + d*x), 2))/(a^3*(a^2 - b^2)*d) + ((2*a^2*A - 5*A*b^2 + 3*a*b*B - 3*a^2*C)*EllipticF((1/2)*(c + d*x), 2))/(3*a^2*(a^2 - b^2)*d) + ((7*a^2*A*b^2 - 5*A*b^4 - 5*a^3*b*B + 3*a*b^3*B + 3*a^4*C - a^2*b^2*C)*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2))/(a^3*(a - b)*(a + b)^2*d) + ((2*a^2*A - 5*A*b^2 + 3*a*b*B - 3*a^2*C)*sin(c + d*x))/(3*a^2*(a^2 - b^2)*d*cos(c + d*x)^(3/2)) - ((4*a^2*A*b - 5*A*b^3 - 2*a^3*B + 3*a*b^2*B - a^2*b*C)*sin(c + d*x))/(a^3*(a^2 - b^2)*d*sqrt(cos(c + d*x))) + ((A*b^2 - a*b*B + a^2*C)*sin(c + d*x))/(a*(a^2 - b^2)*d*cos(c + d*x)^(3/2)*(a + b*cos(c + d*x)))],


[cos(c + d*x)^(7/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^3, x, 9, ((75*a^4*A*b^2 - 145*a^2*A*b^4 + 40*A*b^6 - 175*a^5*b*B + 325*a^3*b^3*B - 120*a*b^5*B + 315*a^6*C - 561*a^4*b^2*C + 192*a^2*b^4*C + 24*b^6*C)*EllipticE((1/2)*(c + d*x), 2))/(20*b^5*(a^2 - b^2)^2*d) - ((45*a^5*A*b^2 - 99*a^3*A*b^4 + 72*a*A*b^6 - 105*a^6*b*B + 223*a^4*b^3*B - 128*a^2*b^5*B - 8*b^7*B + 189*a^7*C - 387*a^5*b^2*C + 192*a^3*b^4*C + 24*a*b^6*C)*EllipticF((1/2)*(c + d*x), 2))/(12*b^6*(a^2 - b^2)^2*d) + (a^2*(15*a^4*A*b^2 - 38*a^2*A*b^4 + 35*A*b^6 - 35*a^5*b*B + 86*a^3*b^3*B - 63*a*b^5*B + 63*a^6*C - 150*a^4*b^2*C + 99*a^2*b^4*C)*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2))/(4*(a - b)^2*b^6*(a + b)^3*d) - ((15*a^3*A*b^2 - 33*a*A*b^4 - 35*a^4*b*B + 61*a^2*b^3*B - 8*b^5*B + 63*a^5*C - 105*a^3*b^2*C + 24*a*b^4*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(12*b^4*(a^2 - b^2)^2*d) + ((15*a^2*A*b^2 - 45*A*b^4 - 35*a^3*b*B + 65*a*b^3*B + 63*a^4*C - 101*a^2*b^2*C + 8*b^4*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(20*b^3*(a^2 - b^2)^2*d) - ((A*b^2 - a*b*B + a^2*C)*cos(c + d*x)^(7/2)*sin(c + d*x))/(2*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) - ((a^2*A*b^2 - 7*A*b^4 - 5*a^3*b*B + 11*a*b^3*B + 9*a^4*C - 15*a^2*b^2*C)*cos(c + d*x)^(5/2)*sin(c + d*x))/(4*b^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))],
[cos(c + d*x)^(5/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^3, x, 8, -(((3*a^3*A*b^2 - 9*a*A*b^4 - 15*a^4*b*B + 29*a^2*b^3*B - 8*b^5*B + 35*a^5*C - 65*a^3*b^2*C + 24*a*b^4*C)*EllipticE((1/2)*(c + d*x), 2))/(4*b^4*(a^2 - b^2)^2*d)) + ((9*a^4*A*b^2 - 15*a^2*A*b^4 + 24*A*b^6 - 45*a^5*b*B + 99*a^3*b^3*B - 72*a*b^5*B + 105*a^6*C - 223*a^4*b^2*C + 128*a^2*b^4*C + 8*b^6*C)*EllipticF((1/2)*(c + d*x), 2))/(12*b^5*(a^2 - b^2)^2*d) - (a*(3*a^4*A*b^2 - 6*a^2*A*b^4 + 15*A*b^6 - 15*a^5*b*B + 38*a^3*b^3*B - 35*a*b^5*B + 35*a^6*C - 86*a^4*b^2*C + 63*a^2*b^4*C)*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2))/(4*(a - b)^2*b^5*(a + b)^3*d) + ((3*a^2*A*b^2 - 21*A*b^4 - 15*a^3*b*B + 33*a*b^3*B + 35*a^4*C - 61*a^2*b^2*C + 8*b^4*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(12*b^3*(a^2 - b^2)^2*d) - ((A*b^2 - a*b*B + a^2*C)*cos(c + d*x)^(5/2)*sin(c + d*x))/(2*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) + ((a^2*A*b^2 + 5*A*b^4 + 3*a^3*b*B - 9*a*b^3*B - 7*a^4*C + 13*a^2*b^2*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(4*b^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))],
[cos(c + d*x)^(3/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^3, x, 7, -(((a^2*A*b^2 + 5*A*b^4 + 3*a^3*b*B - 9*a*b^3*B - 15*a^4*C + 29*a^2*b^2*C - 8*b^4*C)*EllipticE((1/2)*(c + d*x), 2))/(4*b^3*(a^2 - b^2)^2*d)) + ((a^3*A*b^2 - 7*a*A*b^4 + 3*a^4*b*B - 5*a^2*b^3*B + 8*b^5*B - 15*a^5*C + 33*a^3*b^2*C - 24*a*b^4*C)*EllipticF((1/2)*(c + d*x), 2))/(4*b^4*(a^2 - b^2)^2*d) - ((a^4*A*b^2 - 10*a^2*A*b^4 - 3*A*b^6 + 3*a^5*b*B - 6*a^3*b^3*B + 15*a*b^5*B - 15*a^6*C + 38*a^4*b^2*C - 35*a^2*b^4*C)*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2))/(4*(a - b)^2*b^4*(a + b)^3*d) - ((A*b^2 - a*b*B + a^2*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(2*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) + ((3*a^2*A*b^2 + 3*A*b^4 + a^3*b*B - 7*a*b^3*B - 5*a^4*C + 11*a^2*b^2*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(4*b^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))],
[cos(c + d*x)^(1/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^3, x, 7, ((5*a^2*A*b^2 + A*b^4 - a^3*b*B - 5*a*b^3*B - 3*a^4*C + 9*a^2*b^2*C)*EllipticE((1/2)*(c + d*x), 2))/(4*a*b^2*(a^2 - b^2)^2*d) + ((3*a^2*A*b^2 + 3*A*b^4 + a^3*b*B - 7*a*b^3*B + 3*a^4*C - 5*a^2*b^2*C + 8*b^4*C)*EllipticF((1/2)*(c + d*x), 2))/(4*b^3*(a^2 - b^2)^2*d) - ((3*a^4*A*b^2 + 10*a^2*A*b^4 - A*b^6 + a^5*b*B - 10*a^3*b^3*B - 3*a*b^5*B + 3*a^6*C - 6*a^4*b^2*C + 15*a^2*b^4*C)*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2))/(4*a*(a - b)^2*b^3*(a + b)^3*d) - ((A*b^2 - a*b*B + a^2*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(2*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) - ((5*a^2*A*b^2 + A*b^4 - a^3*b*B - 5*a*b^3*B - 3*a^4*C + 9*a^2*b^2*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(4*a*b*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))],
[cos(c + d*x)^(-1/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^3, x, 7, -(((9*a^2*A*b^2 - 3*A*b^4 - 5*a^3*b*B - a*b^3*B + a^4*C + 5*a^2*b^2*C)*EllipticE((1/2)*(c + d*x), 2))/(4*a^2*b*(a^2 - b^2)^2*d)) - ((7*a^2*A*b^2 - A*b^4 - 3*a^3*b*B - 3*a*b^3*B - a^4*C + 7*a^2*b^2*C)*EllipticF((1/2)*(c + d*x), 2))/(4*a*b^2*(a^2 - b^2)^2*d) + ((15*a^4*A*b^2 - 6*a^2*A*b^4 + 3*A*b^6 - 3*a^5*b*B - 10*a^3*b^3*B + a*b^5*B - a^6*C + 10*a^4*b^2*C + 3*a^2*b^4*C)*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2))/(4*a^2*(a - b)^2*b^2*(a + b)^3*d) + ((A*b^2 - a*b*B + a^2*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(2*a*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) + ((9*a^2*A*b^2 - 3*A*b^4 - 5*a^3*b*B - a*b^3*B + a^4*C + 5*a^2*b^2*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(4*a^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))],
[cos(c + d*x)^(-3/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^3, x, 8, -(((8*a^4*A - 29*a^2*A*b^2 + 15*A*b^4 + 9*a^3*b*B - 3*a*b^3*B - 5*a^4*C - a^2*b^2*C)*EllipticE((1/2)*(c + d*x), 2))/(4*a^3*(a^2 - b^2)^2*d)) + ((11*a^2*A*b^2 - 5*A*b^4 - 7*a^3*b*B + a*b^3*B + 3*a^4*C + 3*a^2*b^2*C)*EllipticF((1/2)*(c + d*x), 2))/(4*a^2*b*(a^2 - b^2)^2*d) - ((35*a^4*A*b^2 - 38*a^2*A*b^4 + 15*A*b^6 - 15*a^5*b*B + 6*a^3*b^3*B - 3*a*b^5*B + 3*a^6*C + 10*a^4*b^2*C - a^2*b^4*C)*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2))/(4*a^3*(a - b)^2*b*(a + b)^3*d) + ((8*a^4*A - 29*a^2*A*b^2 + 15*A*b^4 + 9*a^3*b*B - 3*a*b^3*B - 5*a^4*C - a^2*b^2*C)*sin(c + d*x))/(4*a^3*(a^2 - b^2)^2*d*sqrt(cos(c + d*x))) + ((A*b^2 - a*b*B + a^2*C)*sin(c + d*x))/(2*a*(a^2 - b^2)*d*sqrt(cos(c + d*x))*(a + b*cos(c + d*x))^2) + ((11*a^2*A*b^2 - 5*A*b^4 - 7*a^3*b*B + a*b^3*B + 3*a^4*C + 3*a^2*b^2*C)*sin(c + d*x))/(4*a^2*(a^2 - b^2)^2*d*sqrt(cos(c + d*x))*(a + b*cos(c + d*x)))],
[cos(c + d*x)^(-5/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^3, x, 9, ((24*a^4*A*b - 65*a^2*A*b^3 + 35*A*b^5 - 8*a^5*B + 29*a^3*b^2*B - 15*a*b^4*B - 9*a^4*b*C + 3*a^2*b^3*C)*EllipticE((1/2)*(c + d*x), 2))/(4*a^4*(a^2 - b^2)^2*d) + ((8*a^4*A - 61*a^2*A*b^2 + 35*A*b^4 + 33*a^3*b*B - 15*a*b^3*B - 21*a^4*C + 3*a^2*b^2*C)*EllipticF((1/2)*(c + d*x), 2))/(12*a^3*(a^2 - b^2)^2*d) + ((63*a^4*A*b^2 - 86*a^2*A*b^4 + 35*A*b^6 - 35*a^5*b*B + 38*a^3*b^3*B - 15*a*b^5*B + 15*a^6*C - 6*a^4*b^2*C + 3*a^2*b^4*C)*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2))/(4*a^4*(a - b)^2*(a + b)^3*d) + ((8*a^4*A - 61*a^2*A*b^2 + 35*A*b^4 + 33*a^3*b*B - 15*a*b^3*B - 21*a^4*C + 3*a^2*b^2*C)*sin(c + d*x))/(12*a^3*(a^2 - b^2)^2*d*cos(c + d*x)^(3/2)) - ((24*a^4*A*b - 65*a^2*A*b^3 + 35*A*b^5 - 8*a^5*B + 29*a^3*b^2*B - 15*a*b^4*B - 9*a^4*b*C + 3*a^2*b^3*C)*sin(c + d*x))/(4*a^4*(a^2 - b^2)^2*d*sqrt(cos(c + d*x))) + ((A*b^2 - a*b*B + a^2*C)*sin(c + d*x))/(2*a*(a^2 - b^2)*d*cos(c + d*x)^(3/2)*(a + b*cos(c + d*x))^2) + ((13*a^2*A*b^2 - 7*A*b^4 - 9*a^3*b*B + 3*a*b^3*B + 5*a^4*C + a^2*b^2*C)*sin(c + d*x))/(4*a^2*(a^2 - b^2)^2*d*cos(c + d*x)^(3/2)*(a + b*cos(c + d*x)))],


# ::Subsection::Closed:: 
#Integrands of the form Cos[c+d x]^(m/2) (A+B Cos[c+d x]+C Cos[c+d x]^2) (a+b Cos[c+d x])^(n/2)
#


# ::Subsubsection::Closed:: 
#n>0


[cos(c + d*x)^(1/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(1/2), x, 8, ((a + b)*(24*A*b^2 + 6*a*b*B - 3*a^2*C + 16*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(24*b^2*d*sqrt(a + b*cos(c + d*x))) - ((24*a*A*b - 6*a*b*B + 12*b^2*B - a^2*C + 14*a*b*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(12*b*d*sqrt(a + b*cos(c + d*x))) + ((8*a*A*b^2 - 2*a^2*b*B + 8*b^3*B + a^3*C + 4*a*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(4*b^2*d*sqrt(a + b*cos(c + d*x))) + ((6*b*B + a*C)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(12*b*d) + (C*cos(c + d*x)^(3/2)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*d) + ((24*A*b^2 + 6*a*b*B - 3*a^2*C + 16*b^2*C)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(24*b^2*d)],
[cos(c + d*x)^(-1/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(1/2), x, 7, ((a + b)*(4*b*B + a*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(4*b*d*sqrt(a + b*cos(c + d*x))) + ((4*a*A - 4*A*b - 4*a*B + a*C - 2*b*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(2*d*sqrt(a + b*cos(c + d*x))) + ((8*A*b^2 + 4*a*b*B - a^2*C + 4*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(2*b*d*sqrt(a + b*cos(c + d*x))) + (C*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(2*d) + ((4*b*B + a*C)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(4*b*d)],
[cos(c + d*x)^(-3/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(1/2), x, 7, -(((a + b)*(2*A - C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(d*sqrt(a + b*cos(c + d*x)))) + (2*(a*A + A*b + a*B - b*B - a*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(d*sqrt(a + b*cos(c + d*x))) + (2*(2*b*B + a*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(d*sqrt(a + b*cos(c + d*x))) + (2*A*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(d*sqrt(cos(c + d*x))) - ((2*A - C)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/d],
[cos(c + d*x)^(-5/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(1/2), x, 7, -((2*(a + b)*(A*b + 3*a*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(3*a*d*sqrt(a + b*cos(c + d*x)))) + (2*(a*A + A*b + 3*a*B + 3*b*B + 3*a*C - 3*b*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(3*d*sqrt(a + b*cos(c + d*x))) + (4*b*C*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(d*sqrt(a + b*cos(c + d*x))) + (2*A*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2)) + (2*(A*b + 3*a*B)*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(3*a*d*sqrt(cos(c + d*x)))],
[cos(c + d*x)^(-7/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(1/2), x, 6, -((2*(a + b)*(9*a^2*A - 2*A*b^2 + 5*a*b*B + 15*a^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(15*a^2*d*sqrt(a + b*cos(c + d*x)))) + (2*(9*a^2*A + 7*a*A*b - 2*A*b^2 + 5*a^2*B + 5*a*b*B + 15*a^2*C + 15*a*b*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(15*a*d*sqrt(a + b*cos(c + d*x))) + (2*A*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2)) + (2*(A*b + 5*a*B)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(15*a*d*cos(c + d*x)^(3/2)) + (2*(9*a^2*A - 2*A*b^2 + 5*a*b*B + 15*a^2*C)*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(15*a^2*d*sqrt(cos(c + d*x)))],
[cos(c + d*x)^(-9/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(1/2), x, 7, -((2*(a + b)*(19*a^2*A*b + 8*A*b^3 + 63*a^3*B - 14*a*b^2*B + 35*a^2*b*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(105*a^3*d*sqrt(a + b*cos(c + d*x)))) + (2*(25*a^3*A + 19*a^2*A*b + 2*a*A*b^2 + 8*A*b^3 + 63*a^3*B + 49*a^2*b*B - 14*a*b^2*B + 35*a^3*C + 35*a^2*b*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(105*a^2*d*sqrt(a + b*cos(c + d*x))) + (2*A*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(7*d*cos(c + d*x)^(7/2)) + (2*(A*b + 7*a*B)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(35*a*d*cos(c + d*x)^(5/2)) + (2*(25*a^2*A - 4*A*b^2 + 7*a*b*B + 35*a^2*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(105*a^2*d*cos(c + d*x)^(3/2)) + (2*(19*a^2*A*b + 8*A*b^3 + 63*a^3*B - 14*a*b^2*B + 35*a^2*b*C)*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(105*a^3*d*sqrt(cos(c + d*x)))],


[cos(c + d*x)^(1/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(3/2), x, 9, ((a + b)*(240*a*A*b^2 + 24*a^2*b*B + 128*b^3*B - 9*a^3*C + 156*a*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(192*b^2*d*sqrt(a + b*cos(c + d*x))) - ((192*a^2*A*b - 48*a*A*b^2 + 96*A*b^3 - 56*a^2*b*B + 208*a*b^2*B - 3*a^3*C + 114*a^2*b*C - 36*a*b^2*C + 72*b^3*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(96*b*d*sqrt(a + b*cos(c + d*x))) + ((48*a^2*A*b^2 + 64*A*b^4 - 8*a^3*b*B + 96*a*b^3*B + 3*a^4*C + 24*a^2*b^2*C + 48*b^4*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(32*b^2*d*sqrt(a + b*cos(c + d*x))) + ((48*A*b^2 + 56*a*b*B + 3*a^2*C + 36*b^2*C)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(96*b*d) + ((8*b*B + 3*a*C)*cos(c + d*x)^(3/2)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(24*d) + (C*cos(c + d*x)^(3/2)*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(4*d) + ((240*a*A*b^2 + 24*a^2*b*B + 128*b^3*B - 9*a^3*C + 156*a*b^2*C)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(192*b^2*d)],
[cos(c + d*x)^(-1/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(3/2), x, 8, ((a + b)*(24*A*b^2 + 30*a*b*B + 3*a^2*C + 16*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(24*b*d*sqrt(a + b*cos(c + d*x))) + ((24*a^2*A - 48*a*A*b - 24*a^2*B + 6*a*b*B - 12*b^2*B + 7*a^2*C - 26*a*b*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(12*d*sqrt(a + b*cos(c + d*x))) + ((24*a*A*b^2 + 6*a^2*b*B + 8*b^3*B - a^3*C + 12*a*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(4*b*d*sqrt(a + b*cos(c + d*x))) + ((2*b*B + a*C)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(4*d) + (C*sqrt(cos(c + d*x))*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(3*d) + ((24*A*b^2 + 30*a*b*B + 3*a^2*C + 16*b^2*C)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(24*b*d)],
[cos(c + d*x)^(-3/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(3/2), x, 8, -(((a + b)*(8*a*A - 4*b*B - 5*a*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(4*d*sqrt(a + b*cos(c + d*x)))) + ((4*a^2*A + 8*a*A*b - 4*A*b^2 + 4*a^2*B - 8*a*b*B - 4*a^2*C + a*b*C - 2*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(2*d*sqrt(a + b*cos(c + d*x))) + ((8*A*b^2 + 12*a*b*B + 3*a^2*C + 4*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(2*d*sqrt(a + b*cos(c + d*x))) - (b*(4*A - C)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(2*d) + (2*A*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(d*sqrt(cos(c + d*x))) - ((8*a*A - 4*b*B - 5*a*C)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(4*d)],
[cos(c + d*x)^(-5/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(3/2), x, 8, -(((a + b)*(8*A*b + 6*a*B - 3*b*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(3*d*sqrt(a + b*cos(c + d*x)))) + (2*(a^2*A + 4*a*A*b + 3*A*b^2 + 3*a^2*B + 6*a*b*B - 3*b^2*B + 3*a^2*C - 6*a*b*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(3*d*sqrt(a + b*cos(c + d*x))) + (2*b*(2*b*B + 3*a*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(d*sqrt(a + b*cos(c + d*x))) + (2*(A*b + a*B)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(d*sqrt(cos(c + d*x))) + (2*A*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2)) - ((8*A*b + 6*a*B - 3*b*C)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(3*d)],
[cos(c + d*x)^(-7/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(3/2), x, 8, -((2*(a + b)*(9*a^2*A + 3*A*b^2 + 20*a*b*B + 15*a^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(15*a*d*sqrt(a + b*cos(c + d*x)))) + (2*(9*a^2*A + 12*a*A*b + 3*A*b^2 + 5*a^2*B + 20*a*b*B + 15*b^2*B + 15*a^2*C + 30*a*b*C - 15*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(15*d*sqrt(a + b*cos(c + d*x))) + (4*b^2*C*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(d*sqrt(a + b*cos(c + d*x))) + (2*(3*A*b + 5*a*B)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(15*d*cos(c + d*x)^(3/2)) + (2*A*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2)) + (2*(9*a^2*A + 3*A*b^2 + 20*a*b*B + 15*a^2*C)*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(15*a*d*sqrt(cos(c + d*x)))],
[cos(c + d*x)^(-9/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(3/2), x, 7, -((2*(a + b)*(82*a^2*A*b - 6*A*b^3 + 63*a^3*B + 21*a*b^2*B + 140*a^2*b*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(105*a^2*d*sqrt(a + b*cos(c + d*x)))) + (2*(25*a^3*A + 82*a^2*A*b + 51*a*A*b^2 - 6*A*b^3 + 63*a^3*B + 84*a^2*b*B + 21*a*b^2*B + 35*a^3*C + 140*a^2*b*C + 105*a*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(105*a*d*sqrt(a + b*cos(c + d*x))) + (2*(3*A*b + 7*a*B)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(35*d*cos(c + d*x)^(5/2)) + (2*(25*a^2*A + 3*A*b^2 + 42*a*b*B + 35*a^2*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(105*a*d*cos(c + d*x)^(3/2)) + (2*A*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(7*d*cos(c + d*x)^(7/2)) + (2*(82*a^2*A*b - 6*A*b^3 + 63*a^3*B + 21*a*b^2*B + 140*a^2*b*C)*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(105*a^2*d*sqrt(cos(c + d*x)))],
[cos(c + d*x)^(-11/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(3/2), x, 8, -((2*(a + b)*(147*a^4*A + 33*a^2*A*b^2 + 8*A*b^4 + 246*a^3*b*B - 18*a*b^3*B + 189*a^4*C + 63*a^2*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(315*a^3*d*sqrt(a + b*cos(c + d*x)))) + (1/(315*a^2*d*sqrt(a + b*cos(c + d*x))))*(2*(147*a^4*A + 186*a^3*A*b + 33*a^2*A*b^2 + 2*a*A*b^3 + 8*A*b^4 + 75*a^4*B + 246*a^3*b*B + 153*a^2*b^2*B - 18*a*b^3*B + 189*a^4*C + 252*a^3*b*C + 63*a^2*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))) + (2*(A*b + 3*a*B)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(21*d*cos(c + d*x)^(7/2)) + (2*(49*a^2*A + 3*A*b^2 + 72*a*b*B + 63*a^2*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(315*a*d*cos(c + d*x)^(5/2)) + (2*(88*a^2*A*b - 4*A*b^3 + 75*a^3*B + 9*a*b^2*B + 126*a^2*b*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(315*a^2*d*cos(c + d*x)^(3/2)) + (2*A*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(9*d*cos(c + d*x)^(9/2)) + (2*(147*a^4*A + 33*a^2*A*b^2 + 8*A*b^4 + 246*a^3*b*B - 18*a*b^3*B + 189*a^4*C + 63*a^2*b^2*C)*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(315*a^3*d*sqrt(cos(c + d*x)))],


[cos(c + d*x)^(1/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(5/2), x, 10, ((a + b)*(2640*a^2*A*b^2 + 1280*A*b^4 + 150*a^3*b*B + 2840*a*b^3*B - 45*a^4*C + 1692*a^2*b^2*C + 1024*b^4*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(1920*b^2*d*sqrt(a + b*cos(c + d*x))) - (1/(960*b*d*sqrt(a + b*cos(c + d*x))))*((1920*a^3*A*b - 1040*a^2*A*b^2 + 3040*a*A*b^3 - 590*a^3*b*B + 3220*a^2*b^2*B - 360*a*b^3*B + 720*b^4*B - 15*a^4*C + 1146*a^3*b*C - 772*a^2*b^2*C + 2312*a*b^3*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))) + ((80*a^3*A*b^2 + 320*a*A*b^4 - 10*a^4*b*B + 240*a^2*b^3*B + 96*b^5*B + 3*a^5*C + 40*a^3*b^2*C + 240*a*b^4*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(64*b^2*d*sqrt(a + b*cos(c + d*x))) + ((1040*a*A*b^2 + 590*a^2*b*B + 360*b^3*B + 15*a^3*C + 772*a*b^2*C)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(960*b*d) + ((80*A*b^2 + 110*a*b*B + 15*a^2*C + 64*b^2*C)*cos(c + d*x)^(3/2)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(240*d) + ((2*b*B + a*C)*cos(c + d*x)^(3/2)*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(8*d) + (C*cos(c + d*x)^(3/2)*(a + b*cos(c + d*x))^(5/2)*sin(c + d*x))/(5*d) + ((2640*a^2*A*b^2 + 1280*A*b^4 + 150*a^3*b*B + 2840*a*b^3*B - 45*a^4*C + 1692*a^2*b^2*C + 1024*b^4*C)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(1920*b^2*d)],
[cos(c + d*x)^(-1/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(5/2), x, 9, ((a + b)*(432*a*A*b^2 + 264*a^2*b*B + 128*b^3*B + 15*a^3*C + 284*a*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(192*b*d*sqrt(a + b*cos(c + d*x))) + ((192*a^3*A - 576*a^2*A*b + 48*a*A*b^2 - 96*A*b^3 - 192*a^3*B + 104*a^2*b*B - 304*a*b^2*B + 59*a^3*C - 322*a^2*b*C + 36*a*b^2*C - 72*b^3*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(96*d*sqrt(a + b*cos(c + d*x))) + ((240*a^2*A*b^2 + 64*A*b^4 + 40*a^3*b*B + 160*a*b^3*B - 5*a^4*C + 120*a^2*b^2*C + 48*b^4*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(32*b*d*sqrt(a + b*cos(c + d*x))) + ((16*A*b^2 + 24*a*b*B + 5*a^2*C + 12*b^2*C)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(32*d) + ((8*b*B + 5*a*C)*sqrt(cos(c + d*x))*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(24*d) + (C*sqrt(cos(c + d*x))*(a + b*cos(c + d*x))^(5/2)*sin(c + d*x))/(4*d) + ((432*a*A*b^2 + 264*a^2*b*B + 128*b^3*B + 15*a^3*C + 284*a*b^2*C)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(192*b*d)],
[cos(c + d*x)^(-3/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(5/2), x, 9, -(((a + b)*(48*a^2*A - 24*A*b^2 - 54*a*b*B - 33*a^2*C - 16*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(24*d*sqrt(a + b*cos(c + d*x)))) + ((24*a^3*A + 72*a^2*A*b - 72*a*A*b^2 + 24*a^3*B - 72*a^2*b*B + 6*a*b^2*B - 12*b^3*B - 24*a^3*C + 13*a^2*b*C - 38*a*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(12*d*sqrt(a + b*cos(c + d*x))) + ((40*a*A*b^2 + 30*a^2*b*B + 8*b^3*B + 5*a^3*C + 20*a*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(4*d*sqrt(a + b*cos(c + d*x))) - (b*(8*a*A - 2*b*B - 3*a*C)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(4*d) - (b*(6*A - C)*sqrt(cos(c + d*x))*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(3*d) + (2*A*(a + b*cos(c + d*x))^(5/2)*sin(c + d*x))/(d*sqrt(cos(c + d*x))) - ((48*a^2*A - 24*A*b^2 - 54*a*b*B - 33*a^2*C - 16*b^2*C)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(24*d)],
[cos(c + d*x)^(-5/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(5/2), x, 9, -(((a + b)*(56*a*A*b + 24*a^2*B - 12*b^2*B - 27*a*b*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(12*d*sqrt(a + b*cos(c + d*x)))) + ((4*a^3*A + 28*a^2*A*b + 36*a*A*b^2 - 12*A*b^3 + 12*a^3*B + 36*a^2*b*B - 36*a*b^2*B + 12*a^3*C - 36*a^2*b*C + 3*a*b^2*C - 6*b^3*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(6*d*sqrt(a + b*cos(c + d*x))) + (b*(8*A*b^2 + 20*a*b*B + 15*a^2*C + 4*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(2*d*sqrt(a + b*cos(c + d*x))) - (b*(8*A*b + 4*a*B - b*C)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(2*d) + (2*(5*A*b + 3*a*B)*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))) + (2*A*(a + b*cos(c + d*x))^(5/2)*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2)) - ((56*a*A*b + 24*a^2*B - 12*b^2*B - 27*a*b*C)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(12*d)],
[cos(c + d*x)^(-7/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(5/2), x, 9, -(((a + b)*(18*a^2*A + 46*A*b^2 + 70*a*b*B + 30*a^2*C - 15*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(15*d*sqrt(a + b*cos(c + d*x)))) + (2*(9*a^3*A + 17*a^2*A*b + 23*a*A*b^2 + 15*A*b^3 + 5*a^3*B + 35*a^2*b*B + 45*a*b^2*B - 15*b^3*B + 15*a^3*C + 45*a^2*b*C - 45*a*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(15*d*sqrt(a + b*cos(c + d*x))) + (2*b^2*(2*b*B + 5*a*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(d*sqrt(a + b*cos(c + d*x))) + (2*(3*a^2*A + 5*A*b^2 + 10*a*b*B + 5*a^2*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(5*d*sqrt(cos(c + d*x))) + (2*(A*b + a*B)*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2)) + (2*A*(a + b*cos(c + d*x))^(5/2)*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2)) - ((18*a^2*A + 46*A*b^2 + 70*a*b*B + 30*a^2*C - 15*b^2*C)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(15*d)],
[cos(c + d*x)^(-9/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(5/2), x, 9, -((2*(a + b)*(145*a^2*A*b + 15*A*b^3 + 63*a^3*B + 161*a*b^2*B + 245*a^2*b*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(105*a*d*sqrt(a + b*cos(c + d*x)))) + (1/(105*d*sqrt(a + b*cos(c + d*x))))*(2*(25*a^3*A + 145*a^2*A*b + 135*a*A*b^2 + 15*A*b^3 + 63*a^3*B + 119*a^2*b*B + 161*a*b^2*B + 105*b^3*B + 35*a^3*C + 245*a^2*b*C + 315*a*b^2*C - 105*b^3*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))) + (4*b^3*C*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(d*sqrt(a + b*cos(c + d*x))) + (2*(25*a^2*A + 15*A*b^2 + 56*a*b*B + 35*a^2*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(105*d*cos(c + d*x)^(3/2)) + (2*(5*A*b + 7*a*B)*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(35*d*cos(c + d*x)^(5/2)) + (2*A*(a + b*cos(c + d*x))^(5/2)*sin(c + d*x))/(7*d*cos(c + d*x)^(7/2)) + (2*(145*a^2*A*b + 15*A*b^3 + 63*a^3*B + 161*a*b^2*B + 245*a^2*b*C)*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(105*a*d*sqrt(cos(c + d*x)))],
[cos(c + d*x)^(-11/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(5/2), x, 8, -((2*(a + b)*(147*a^4*A + 279*a^2*A*b^2 - 10*A*b^4 + 435*a^3*b*B + 45*a*b^3*B + 189*a^4*C + 483*a^2*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(315*a^2*d*sqrt(a + b*cos(c + d*x)))) + (1/(315*a*d*sqrt(a + b*cos(c + d*x))))*(2*(147*a^4*A + 261*a^3*A*b + 279*a^2*A*b^2 + 155*a*A*b^3 - 10*A*b^4 + 75*a^4*B + 435*a^3*b*B + 405*a^2*b^2*B + 45*a*b^3*B + 189*a^4*C + 357*a^3*b*C + 483*a^2*b^2*C + 315*a*b^3*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))) + (2*(49*a^2*A + 15*A*b^2 + 90*a*b*B + 63*a^2*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(315*d*cos(c + d*x)^(5/2)) + (2*(163*a^2*A*b + 5*A*b^3 + 75*a^3*B + 135*a*b^2*B + 231*a^2*b*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(315*a*d*cos(c + d*x)^(3/2)) + (2*(5*A*b + 9*a*B)*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(63*d*cos(c + d*x)^(7/2)) + (2*A*(a + b*cos(c + d*x))^(5/2)*sin(c + d*x))/(9*d*cos(c + d*x)^(9/2)) + (2*(147*a^4*A + 279*a^2*A*b^2 - 10*A*b^4 + 435*a^3*b*B + 45*a*b^3*B + 189*a^4*C + 483*a^2*b^2*C)*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(315*a^2*d*sqrt(cos(c + d*x)))],


# ::Subsubsection::Closed:: 
#n<0


[cos(c + d*x)^(3/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(1/2), x, 8, ((a + b)*(24*A*b^2 - 18*a*b*B + 15*a^2*C + 16*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(24*b^3*d*sqrt(a + b*cos(c + d*x))) + ((6*a*b*B - 12*b^2*B - 5*a^2*C - 2*a*b*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(12*b^2*d*sqrt(a + b*cos(c + d*x))) - ((8*a*A*b^2 - 6*a^2*b*B - 8*b^3*B + 5*a^3*C + 4*a*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(4*b^3*d*sqrt(a + b*cos(c + d*x))) + ((6*b*B - 5*a*C)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(12*b^2*d) + (C*cos(c + d*x)^(3/2)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*b*d) + ((24*A*b^2 - 18*a*b*B + 15*a^2*C + 16*b^2*C)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(24*b^3*d)],
[cos(c + d*x)^(1/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(1/2), x, 7, ((a + b)*(4*b*B - 3*a*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(4*b^2*d*sqrt(a + b*cos(c + d*x))) - ((4*A*b - a*C + 2*b*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(2*b*d*sqrt(a + b*cos(c + d*x))) + ((8*A*b^2 - 4*a*b*B + 3*a^2*C + 4*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(2*b^2*d*sqrt(a + b*cos(c + d*x))) + (C*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(2*b*d) + ((4*b*B - 3*a*C)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(4*b^2*d)],
[cos(c + d*x)^(-1/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(1/2), x, 6, ((a + b)*C*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(b*d*sqrt(a + b*cos(c + d*x))) + (2*(A - B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(d*sqrt(a + b*cos(c + d*x))) + (2*(2*b*B - a*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(b*d*sqrt(a + b*cos(c + d*x))) + (C*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(b*d)],
[cos(c + d*x)^(-3/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(1/2), x, 6, -((2*A*(a + b)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(a*d*sqrt(a + b*cos(c + d*x)))) + (2*(A + B - C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(d*sqrt(a + b*cos(c + d*x))) + (4*C*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(d*sqrt(a + b*cos(c + d*x))) + (2*A*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(a*d*sqrt(cos(c + d*x)))],
[cos(c + d*x)^(-5/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(1/2), x, 5, (2*(a + b)*(2*A*b - 3*a*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(3*a^2*d*sqrt(a + b*cos(c + d*x))) + (2*(a*A - 2*A*b + 3*a*B + 3*a*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(3*a*d*sqrt(a + b*cos(c + d*x))) + (2*A*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*a*d*cos(c + d*x)^(3/2)) - (2*(2*A*b - 3*a*B)*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(3*a^2*d*sqrt(cos(c + d*x)))],
[cos(c + d*x)^(-7/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(1/2), x, 6, -((2*(a + b)*(9*a^2*A + 8*A*b^2 - 10*a*b*B + 15*a^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(15*a^3*d*sqrt(a + b*cos(c + d*x)))) + (2*(9*a^2*A + 2*a*A*b + 8*A*b^2 + 5*a^2*B - 10*a*b*B + 15*a^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(15*a^2*d*sqrt(a + b*cos(c + d*x))) + (2*A*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(5*a*d*cos(c + d*x)^(5/2)) - (2*(4*A*b - 5*a*B)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(15*a^2*d*cos(c + d*x)^(3/2)) + (2*(9*a^2*A + 8*A*b^2 - 10*a*b*B + 15*a^2*C)*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(15*a^3*d*sqrt(cos(c + d*x)))],
[cos(c + d*x)^(-9/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(1/2), x, 7, (2*(a + b)*(44*a^2*A*b + 48*A*b^3 - 63*a^3*B - 56*a*b^2*B + 70*a^2*b*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(105*a^4*d*sqrt(a + b*cos(c + d*x))) + (2*(25*a^3*A - 44*a^2*A*b - 12*a*A*b^2 - 48*A*b^3 + 63*a^3*B + 14*a^2*b*B + 56*a*b^2*B + 35*a^3*C - 70*a^2*b*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(105*a^3*d*sqrt(a + b*cos(c + d*x))) + (2*A*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(7*a*d*cos(c + d*x)^(7/2)) - (2*(6*A*b - 7*a*B)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(35*a^2*d*cos(c + d*x)^(5/2)) + (2*(25*a^2*A + 24*A*b^2 - 28*a*b*B + 35*a^2*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(105*a^3*d*cos(c + d*x)^(3/2)) - (2*(44*a^2*A*b + 48*A*b^3 - 63*a^3*B - 56*a*b^2*B + 70*a^2*b*C)*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(105*a^4*d*sqrt(cos(c + d*x)))],

[cos(c + d*x)^(1/2)*(a*A + (A*b + a*B)*cos(c + d*x) + b*B*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(1/2), x, 7, ((a + b)*(4*A*b + a*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(4*b*d*sqrt(a + b*cos(c + d*x))) - ((4*a*A - a*B + 2*b*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(2*d*sqrt(a + b*cos(c + d*x))) + ((4*a*A*b - a^2*B + 4*b^2*B)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(2*b*d*sqrt(a + b*cos(c + d*x))) + (B*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(2*d) + ((4*A*b + a*B)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(4*b*d)],
[cos(c + d*x)^(-1/2)*(a + a*cos(c + d*x) + 2*b*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(1/2), x, 2, (2*(a + b)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(d*sqrt(a + b*cos(c + d*x))) + (2*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/d],


[cos(c + d*x)^(3/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(3/2), x, 8, -(((8*a*A*b^2 - 12*a^2*b*B + 4*b^3*B + 15*a^3*C - 7*a*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(4*(a - b)*b^3*d*sqrt(a + b*cos(c + d*x)))) + ((4*A*b^2 - 4*a*b*B + 5*a^2*C - 3*a*b*C + 2*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(2*(a - b)*b^2*d*sqrt(a + b*cos(c + d*x))) + ((8*A*b^2 - 12*a*b*B + 15*a^2*C + 4*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(2*b^3*d*sqrt(a + b*cos(c + d*x))) - (2*(A*b^2 - a*b*B + a^2*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(b*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) + ((4*A*b^2 - 4*a*b*B + 5*a^2*C - b^2*C)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(2*b^2*(a^2 - b^2)*d) - ((8*a*A*b^2 - 12*a^2*b*B + 4*b^3*B + 15*a^3*C - 7*a*b^2*C)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(4*b^3*(a^2 - b^2)*d)],
[cos(c + d*x)^(1/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(3/2), x, 7, ((2*A*b^2 - 2*a*b*B + 3*a^2*C - b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/((a - b)*b^2*d*sqrt(a + b*cos(c + d*x))) - (2*(A*b - b*B + a*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/((a - b)*b*d*sqrt(a + b*cos(c + d*x))) + (2*(2*b*B - 3*a*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(b^2*d*sqrt(a + b*cos(c + d*x))) - (2*(A*b^2 - a*b*B + a^2*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(b*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) + ((2*A*b^2 - 2*a*b*B + 3*a^2*C - b^2*C)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(b^2*(a^2 - b^2)*d)],
[cos(c + d*x)^(-1/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(3/2), x, 6, -((2*(A*b^2 - a*b*B + a^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(a*(a - b)*b*d*sqrt(a + b*cos(c + d*x)))) + (2*(A - B + C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/((a - b)*d*sqrt(a + b*cos(c + d*x))) + (4*C*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(b*d*sqrt(a + b*cos(c + d*x))) - (2*(A*b^2 - a*b*B + a^2*C)*sqrt(cos(c + d*x))*tan((1/2)*(c + d*x)))/(a*b*(a + b)*d*sqrt(a + b*cos(c + d*x)))],
[cos(c + d*x)^(-3/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(3/2), x, 5, -((2*(a^2*A - 2*A*b^2 + a*b*B - a^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(a^2*(a - b)*d*sqrt(a + b*cos(c + d*x)))) + (2*(a*A - 2*A*b + a*B - a*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(a*(a - b)*d*sqrt(a + b*cos(c + d*x))) + (2*(A*b^2 - a*b*B + a^2*C)*sin(c + d*x))/(a*(a^2 - b^2)*d*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))) + (2*(a^2*A - 2*A*b^2 + a*b*B - a^2*C)*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(a^2*(a^2 - b^2)*d*sqrt(cos(c + d*x)))],
[cos(c + d*x)^(-5/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(3/2), x, 6, (2*(5*a^2*A*b - 8*A*b^3 - 3*a^3*B + 6*a*b^2*B - 3*a^2*b*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(3*a^3*(a - b)*d*sqrt(a + b*cos(c + d*x))) + (2*(a^2*A - 6*a*A*b + 8*A*b^2 + 3*a^2*B - 6*a*b*B + 3*a^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(3*a^2*(a - b)*d*sqrt(a + b*cos(c + d*x))) + (2*(A*b^2 - a*b*B + a^2*C)*sin(c + d*x))/(a*(a^2 - b^2)*d*cos(c + d*x)^(3/2)*sqrt(a + b*cos(c + d*x))) + (2*(a^2*A - 4*A*b^2 + 3*a*b*B - 3*a^2*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*a^2*(a^2 - b^2)*d*cos(c + d*x)^(3/2)) - (2*(5*a^2*A*b - 8*A*b^3 - 3*a^3*B + 6*a*b^2*B - 3*a^2*b*C)*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(3*a^3*(a^2 - b^2)*d*sqrt(cos(c + d*x)))],
[cos(c + d*x)^(-7/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(3/2), x, 7, -((2*(9*a^4*A + 24*a^2*A*b^2 - 48*A*b^4 - 25*a^3*b*B + 40*a*b^3*B + 15*a^4*C - 30*a^2*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(15*a^4*(a - b)*d*sqrt(a + b*cos(c + d*x)))) + (2*(9*a^3*A - 12*a^2*A*b + 36*a*A*b^2 - 48*A*b^3 + 5*a^3*B - 30*a^2*b*B + 40*a*b^2*B + 15*a^3*C - 30*a^2*b*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(15*a^3*(a - b)*d*sqrt(a + b*cos(c + d*x))) + (2*(A*b^2 - a*b*B + a^2*C)*sin(c + d*x))/(a*(a^2 - b^2)*d*cos(c + d*x)^(5/2)*sqrt(a + b*cos(c + d*x))) + (2*(a^2*A - 6*A*b^2 + 5*a*b*B - 5*a^2*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(5*a^2*(a^2 - b^2)*d*cos(c + d*x)^(5/2)) - (2*(9*a^2*A*b - 24*A*b^3 - 5*a^3*B + 20*a*b^2*B - 15*a^2*b*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(15*a^3*(a^2 - b^2)*d*cos(c + d*x)^(3/2)) + (2*(9*a^4*A + 24*a^2*A*b^2 - 48*A*b^4 - 25*a^3*b*B + 40*a*b^3*B + 15*a^4*C - 30*a^2*b^2*C)*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(15*a^4*(a^2 - b^2)*d*sqrt(cos(c + d*x)))],


[cos(c + d*x)^(3/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(5/2), x, 8, -(((8*A*b^4 + 6*a^3*b*B - 14*a*b^3*B - 15*a^4*C + 26*a^2*b^2*C - 3*b^4*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(3*(a - b)^2*b^3*(a + b)*d*sqrt(a + b*cos(c + d*x)))) + (2*(a*A*b^2 + 3*A*b^3 + 2*a^2*b*B - 3*a*b^2*B - 3*b^3*B - 5*a^3*C + 3*a^2*b*C + 6*a*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(3*(a - b)^2*b^2*(a + b)*d*sqrt(a + b*cos(c + d*x))) + (2*(2*b*B - 5*a*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(b^3*d*sqrt(a + b*cos(c + d*x))) - (2*(A*b^2 - a*b*B + a^2*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)) + (2*(a^2*A*b^2 + 3*A*b^4 + 2*a^3*b*B - 6*a*b^3*B - 5*a^4*C + 9*a^2*b^2*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(3*b^2*(a^2 - b^2)^2*d*sqrt(a + b*cos(c + d*x))) - ((8*A*b^4 + 6*a^3*b*B - 14*a*b^3*B - 15*a^4*C + 26*a^2*b^2*C - 3*b^4*C)*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(3*b^3*(a^2 - b^2)^2*d)],
[cos(c + d*x)^(1/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(5/2), x, 7, (2*(3*a^2*A*b^2 + A*b^4 - 4*a*b^3*B - 3*a^4*C + 7*a^2*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(3*a*(a - b)^2*b^2*(a + b)*d*sqrt(a + b*cos(c + d*x))) - (2*(3*a*A*b + A*b^2 - a*b*B - 3*b^2*B - 2*a^2*C + 3*a*b*C + 3*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(3*(a - b)^2*b*(a + b)*d*sqrt(a + b*cos(c + d*x))) + (4*C*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(b^2*d*sqrt(a + b*cos(c + d*x))) - (2*(A*b^2 - a*b*B + a^2*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)) + (2*(3*a^2*A*b^2 + A*b^4 - 4*a*b^3*B - 3*a^4*C + 7*a^2*b^2*C)*sqrt(cos(c + d*x))*tan((1/2)*(c + d*x)))/(3*a*(a - b)*b^2*(a + b)^2*d*sqrt(a + b*cos(c + d*x)))],
[cos(c + d*x)^(-1/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(5/2), x, 5, -((2*(6*a^2*A*b - 2*A*b^3 - 3*a^3*B - a*b^2*B + 4*a^2*b*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(3*a^2*(a - b)^2*(a + b)*d*sqrt(a + b*cos(c + d*x)))) + (2*(3*a^2*A + 3*a*A*b - 2*A*b^2 - 3*a^2*B - a*b*B + a^2*C + 3*a*b*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(3*a*(a - b)^2*(a + b)*d*sqrt(a + b*cos(c + d*x))) + (2*(A*b^2 - a*b*B + a^2*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(3*a*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)) - (2*(6*a^2*A*b - 2*A*b^3 - 3*a^3*B - a*b^2*B + 4*a^2*b*C)*sqrt(cos(c + d*x))*tan((1/2)*(c + d*x)))/(3*a^2*(a - b)*(a + b)^2*d*sqrt(a + b*cos(c + d*x)))],
[cos(c + d*x)^(-3/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(5/2), x, 6, -((2*(3*a^4*A - 15*a^2*A*b^2 + 8*A*b^4 + 6*a^3*b*B - 2*a*b^3*B - 3*a^4*C - a^2*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(3*a^3*(a - b)^2*(a + b)*d*sqrt(a + b*cos(c + d*x)))) + (2*(3*a^3*A - 9*a^2*A*b - 6*a*A*b^2 + 8*A*b^3 + 3*a^3*B + 3*a^2*b*B - 2*a*b^2*B - 3*a^3*C - a^2*b*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(3*a^2*(a - b)^2*(a + b)*d*sqrt(a + b*cos(c + d*x))) + (2*(A*b^2 - a*b*B + a^2*C)*sin(c + d*x))/(3*a*(a^2 - b^2)*d*sqrt(cos(c + d*x))*(a + b*cos(c + d*x))^(3/2)) + (2*(8*a^2*A*b^2 - 4*A*b^4 - 5*a^3*b*B + a*b^3*B + 2*a^4*C + 2*a^2*b^2*C)*sin(c + d*x))/(3*a^2*(a^2 - b^2)^2*d*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))) + (2*(3*a^4*A - 15*a^2*A*b^2 + 8*A*b^4 + 6*a^3*b*B - 2*a*b^3*B - 3*a^4*C - a^2*b^2*C)*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(3*a^3*(a^2 - b^2)^2*d*sqrt(cos(c + d*x)))],
[cos(c + d*x)^(-5/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(5/2), x, 7, (2*(a + b)*(8*a^4*A*b - 28*a^2*A*b^3 + 16*A*b^5 - 3*a^5*B + 15*a^3*b^2*B - 8*a*b^4*B - 6*a^4*b*C + 2*a^2*b^3*C)*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b))))/(3*a^4*(a^2 - b^2)^2*d*sqrt(a + b*cos(c + d*x))) - (1/(9*a^3*(a^2 - b^2)^2*d*sqrt(a + b*cos(c + d*x))))*(16*((-(3/8))*a*(a^4*A + 7*a^2*A*b^2 - 4*A*b^4 - 6*a^3*b*B + 2*a*b^3*B + 3*a^4*C + a^2*b^2*C) + (3/8)*(8*a^4*A*b - 28*a^2*A*b^3 + 16*A*b^5 - 3*a^5*B + 15*a^3*b^2*B - 8*a*b^4*B - 6*a^4*b*C + 2*a^2*b^3*C))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))) + (2*(A*b^2 - a*b*B + a^2*C)*sin(c + d*x))/(3*a*(a^2 - b^2)*d*cos(c + d*x)^(3/2)*(a + b*cos(c + d*x))^(3/2)) - (4*((-(3/2))*b^2*(a^2*A - 2*A*b^2 + a*b*B - a^2*C) - 2*a^2*(A*b^2 - a*b*B + a^2*C) + (3/2)*a^2*b*((-A)*b + a*B - b*C))*sin(c + d*x))/(3*a^2*(a^2 - b^2)^2*d*cos(c + d*x)^(3/2)*sqrt(a + b*cos(c + d*x))) + (2*(a^4*A - 13*a^2*A*b^2 + 8*A*b^4 + 8*a^3*b*B - 4*a*b^3*B - 5*a^4*C + a^2*b^2*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*a^3*(a^2 - b^2)^2*d*cos(c + d*x)^(3/2)) - (2*(8*a^4*A*b - 28*a^2*A*b^3 + 16*A*b^5 - 3*a^5*B + 15*a^3*b^2*B - 8*a*b^4*B - 6*a^4*b*C + 2*a^2*b^3*C)*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(3*a^4*(a^2 - b^2)^2*d*sqrt(cos(c + d*x)))],


# ::Subsection::Closed:: 
#Integrands of the form Cos[c+d x]^(m/2) (A+B Cos[c+d x]+C Cos[c+d x]^2) (a+b Cos[c+d x])^(n/3)
#


[(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(2/3), x, 2, Int((A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(2/3), x)],
[(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(1/3), x, 2, Int((A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(1/3), x)],
[(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(1/3), x, 1, Int((A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(1/3), x)],
[(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(2/3), x, 1, Int((A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + b*cos(c + d*x))^(2/3), x)],


# ::Subsection::Closed:: 
#Integrands of the form Sec[c+d x]^(m/2) (A+B Cos[c+d x]+C Cos[c+d x]^2) (a+b Cos[c+d x])^n


# ::Subsubsection::Closed:: 
#n>0


[(a + b*cos(c + d*x))*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(9/2), x, 8, -((2*(3*A*b + 3*a*B + 5*b*C)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d)) + (2*(5*a*A + 7*b*B + 7*a*C)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) + (2*(3*A*b + 3*a*B + 5*b*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d) + (2*(5*a*A + 7*b*B + 7*a*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(21*d) + (2*(2*A*b + 7*a*B)*sec(c + d*x)^(5/2)*sin(c + d*x))/(35*d) + (2*A*(a + b*cos(c + d*x))*sec(c + d*x)^(7/2)*sin(c + d*x))/(7*d)],
[(a + b*cos(c + d*x))*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(7/2), x, 7, -((2*(3*a*A + 5*b*B + 5*a*C)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d)) + (2*(A*b + a*B + 3*b*C)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) + (2*(3*a*A + 5*b*B + 5*a*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d) + (2*(2*A*b + 5*a*B)*sec(c + d*x)^(3/2)*sin(c + d*x))/(15*d) + (2*A*(a + b*cos(c + d*x))*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d)],
[(a + b*cos(c + d*x))*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(5/2), x, 6, (-2*(A*b + a*B - b*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/d + (2*(a*A + 3*b*B + 3*a*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*d) + (2*(2*A*b + 3*a*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(3*d) + (2*A*(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d)],
[(a + b*cos(c + d*x))*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(3/2), x, 6, (-2*(a*A - b*B - a*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/d + (2*(3*A*b + 3*a*B + b*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*d) - (2*b*(3*A - C)*sin(c + d*x))/(3*d*sqrt(sec(c + d*x))) + (2*A*(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*sin(c + d*x))/d],
[(a + b*cos(c + d*x))*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(1/2), x, 6, (2*(5*A*b + 5*a*B + 3*b*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*d) + (2*(3*a*A + b*B + a*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*d) + (2*(5*b*B + 2*a*C)*sin(c + d*x))/(15*d*sqrt(sec(c + d*x))) + (2*C*(a + b*cos(c + d*x))*sin(c + d*x))/(5*d*sqrt(sec(c + d*x)))],
[((a + b*cos(c + d*x))*(A + B*cos(c + d*x) + C*cos(c + d*x)^2))/sec(c + d*x)^(1/2), x, 7, (2*(5*a*A + 3*b*B + 3*a*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*d) + (2*(7*A*b + 7*a*B + 5*b*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(21*d) + (2*(7*b*B + 2*a*C)*sin(c + d*x))/(35*d*sec(c + d*x)^(3/2)) + (2*C*(a + b*cos(c + d*x))*sin(c + d*x))/(7*d*sec(c + d*x)^(3/2)) + (2*(7*A*b + 7*a*B + 5*b*C)*sin(c + d*x))/(21*d*sqrt(sec(c + d*x)))],
[((a + b*cos(c + d*x))*(A + B*cos(c + d*x) + C*cos(c + d*x)^2))/sec(c + d*x)^(3/2), x, 8, (2*(9*A*b + 9*a*B + 7*b*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(15*d) + (2*(7*a*A + 5*b*B + 5*a*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(21*d) + (2*(9*b*B + 2*a*C)*sin(c + d*x))/(63*d*sec(c + d*x)^(5/2)) + (2*C*(a + b*cos(c + d*x))*sin(c + d*x))/(9*d*sec(c + d*x)^(5/2)) + (2*(9*A*b + 9*a*B + 7*b*C)*sin(c + d*x))/(45*d*sec(c + d*x)^(3/2)) + (2*(7*a*A + 5*b*B + 5*a*C)*sin(c + d*x))/(21*d*sqrt(sec(c + d*x)))],


[(a + b*cos(c + d*x))^2*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(11/2), x, 9, -((2*(7*a^2*A + 9*A*b^2 + 18*a*b*B + 9*a^2*C + 15*b^2*C)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(15*d)) + (2*(10*a*A*b + 5*a^2*B + 7*b^2*B + 14*a*b*C)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) + (2*(7*a^2*A + 9*A*b^2 + 18*a*b*B + 9*a^2*C + 15*b^2*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(15*d) + (2*(10*a*A*b + 5*a^2*B + 7*b^2*B + 14*a*b*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(21*d) + (2*(49*a^2*A + 8*A*b^2 + 81*a*b*B + 63*a^2*C)*sec(c + d*x)^(5/2)*sin(c + d*x))/(315*d) + (2*(4*A*b + 9*a*B)*(a + b*cos(c + d*x))*sec(c + d*x)^(7/2)*sin(c + d*x))/(63*d) + (2*A*(a + b*cos(c + d*x))^2*sec(c + d*x)^(9/2)*sin(c + d*x))/(9*d)],
[(a + b*cos(c + d*x))^2*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(9/2), x, 8, -((2*(6*a*A*b + 3*a^2*B + 5*b^2*B + 10*a*b*C)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d)) + (2*(5*a^2*A + 7*A*b^2 + 14*a*b*B + 7*a^2*C + 21*b^2*C)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) + (2*(6*a*A*b + 3*a^2*B + 5*b^2*B + 10*a*b*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d) + (2*(25*a^2*A + 8*A*b^2 + 49*a*b*B + 35*a^2*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(105*d) + (2*(4*A*b + 7*a*B)*(a + b*cos(c + d*x))*sec(c + d*x)^(5/2)*sin(c + d*x))/(35*d) + (2*A*(a + b*cos(c + d*x))^2*sec(c + d*x)^(7/2)*sin(c + d*x))/(7*d)],
[(a + b*cos(c + d*x))^2*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(7/2), x, 7, -((2*(3*a^2*A + 5*A*b^2 + 10*a*b*B + 5*a^2*C - 5*b^2*C)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d)) + (2*(2*a*A*b + a^2*B + 3*b^2*B + 6*a*b*C)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) + (2*(9*a^2*A + 8*A*b^2 + 25*a*b*B + 15*a^2*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(15*d) + (2*(4*A*b + 5*a*B)*(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(15*d) + (2*A*(a + b*cos(c + d*x))^2*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d)],
[(a + b*cos(c + d*x))^2*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(5/2), x, 7, -((2*(2*a*A*b + a^2*B - b^2*B - 2*a*b*C)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/d) + (2*(a^2*A + 3*A*b^2 + 6*a*b*B + 3*a^2*C + b^2*C)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) - (2*b*(5*A*b + 3*a*B - b*C)*sin(c + d*x))/(3*d*sqrt(sec(c + d*x))) + (2*(4*A*b + 3*a*B)*(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*sin(c + d*x))/(3*d) + (2*A*(a + b*cos(c + d*x))^2*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d)],
[(a + b*cos(c + d*x))^2*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(3/2), x, 7, (-2*(5*a^2*A - 5*A*b^2 - 10*a*b*B - 5*a^2*C - 3*b^2*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*d) + (2*(6*a*A*b + 3*a^2*B + b^2*B + 2*a*b*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*d) - (2*b*(15*a*A - 5*b*B - 7*a*C)*sin(c + d*x))/(15*d*sqrt(sec(c + d*x))) - (2*b*(5*A - C)*(a + b*cos(c + d*x))*sin(c + d*x))/(5*d*sqrt(sec(c + d*x))) + (2*A*(a + b*cos(c + d*x))^2*sqrt(sec(c + d*x))*sin(c + d*x))/d],
[(a + b*cos(c + d*x))^2*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(1/2), x, 7, (2*(10*a*A*b + 5*a^2*B + 3*b^2*B + 6*a*b*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*d) + (2*(21*a^2*A + 7*A*b^2 + 14*a*b*B + 7*a^2*C + 5*b^2*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(21*d) + (2*(35*A*b^2 + 49*a*b*B + 8*a^2*C + 25*b^2*C)*sin(c + d*x))/(105*d*sqrt(sec(c + d*x))) + (2*(7*b*B + 4*a*C)*(a + b*cos(c + d*x))*sin(c + d*x))/(35*d*sqrt(sec(c + d*x))) + (2*C*(a + b*cos(c + d*x))^2*sin(c + d*x))/(7*d*sqrt(sec(c + d*x)))],
[((a + b*cos(c + d*x))^2*(A + B*cos(c + d*x) + C*cos(c + d*x)^2))/sec(c + d*x)^(1/2), x, 8, (2*(15*a^2*A + 9*A*b^2 + 18*a*b*B + 9*a^2*C + 7*b^2*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(15*d) + (2*(14*a*A*b + 7*a^2*B + 5*b^2*B + 10*a*b*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(21*d) + (2*(63*A*b^2 + 81*a*b*B + 8*a^2*C + 49*b^2*C)*sin(c + d*x))/(315*d*sec(c + d*x)^(3/2)) + (2*(9*b*B + 4*a*C)*(a + b*cos(c + d*x))*sin(c + d*x))/(63*d*sec(c + d*x)^(3/2)) + (2*C*(a + b*cos(c + d*x))^2*sin(c + d*x))/(9*d*sec(c + d*x)^(3/2)) + (2*(14*a*A*b + 7*a^2*B + 5*b^2*B + 10*a*b*C)*sin(c + d*x))/(21*d*sqrt(sec(c + d*x)))],
[((a + b*cos(c + d*x))^2*(A + B*cos(c + d*x) + C*cos(c + d*x)^2))/sec(c + d*x)^(3/2), x, 9, (2*(18*a*A*b + 9*a^2*B + 7*b^2*B + 14*a*b*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(15*d) + (2*(77*a^2*A + 55*A*b^2 + 110*a*b*B + 55*a^2*C + 45*b^2*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(231*d) + (2*(99*A*b^2 + 121*a*b*B + 8*a^2*C + 81*b^2*C)*sin(c + d*x))/(693*d*sec(c + d*x)^(5/2)) + (2*(11*b*B + 4*a*C)*(a + b*cos(c + d*x))*sin(c + d*x))/(99*d*sec(c + d*x)^(5/2)) + (2*C*(a + b*cos(c + d*x))^2*sin(c + d*x))/(11*d*sec(c + d*x)^(5/2)) + (2*(18*a*A*b + 9*a^2*B + 7*b^2*B + 14*a*b*C)*sin(c + d*x))/(45*d*sec(c + d*x)^(3/2)) + (2*(77*a^2*A + 55*A*b^2 + 110*a*b*B + 55*a^2*C + 45*b^2*C)*sin(c + d*x))/(231*d*sqrt(sec(c + d*x)))],


[(a + b*cos(c + d*x))^3*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(11/2), x, 9, -((2*(7*a^3*A + 27*a*A*b^2 + 27*a^2*b*B + 15*b^3*B + 9*a^3*C + 45*a*b^2*C)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(15*d)) + (2*(15*a^2*A*b + 7*A*b^3 + 5*a^3*B + 21*a*b^2*B + 21*a^2*b*C + 21*b^3*C)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) + (2*(7*a^3*A + 27*a*A*b^2 + 27*a^2*b*B + 15*b^3*B + 9*a^3*C + 45*a*b^2*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(15*d) + (2*(176*a^2*A*b + 16*A*b^3 + 75*a^3*B + 171*a*b^2*B + 252*a^2*b*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(315*d) + (2*(49*a^2*A + 24*A*b^2 + 99*a*b*B + 63*a^2*C)*(a + b*cos(c + d*x))*sec(c + d*x)^(5/2)*sin(c + d*x))/(315*d) + (2*(2*A*b + 3*a*B)*(a + b*cos(c + d*x))^2*sec(c + d*x)^(7/2)*sin(c + d*x))/(21*d) + (2*A*(a + b*cos(c + d*x))^3*sec(c + d*x)^(9/2)*sin(c + d*x))/(9*d)],
[(a + b*cos(c + d*x))^3*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(9/2), x, 8, -((2*(9*a^2*A*b + 5*A*b^3 + 3*a^3*B + 15*a*b^2*B + 15*a^2*b*C - 5*b^3*C)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d)) + (2*(5*a^3*A + 21*a*A*b^2 + 21*a^2*b*B + 21*b^3*B + 7*a^3*C + 63*a*b^2*C)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) + (2*(164*a^2*A*b + 48*A*b^3 + 63*a^3*B + 231*a*b^2*B + 280*a^2*b*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(105*d) + (2*(25*a^2*A + 24*A*b^2 + 63*a*b*B + 35*a^2*C)*(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(105*d) + (2*(6*A*b + 7*a*B)*(a + b*cos(c + d*x))^2*sec(c + d*x)^(5/2)*sin(c + d*x))/(35*d) + (2*A*(a + b*cos(c + d*x))^3*sec(c + d*x)^(7/2)*sin(c + d*x))/(7*d)],
[(a + b*cos(c + d*x))^3*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(7/2), x, 8, -((2*(3*a^3*A + 15*a*A*b^2 + 15*a^2*b*B - 5*b^3*B + 5*a^3*C - 15*a*b^2*C)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d)) + (2*(3*a^2*A*b + 3*A*b^3 + a^3*B + 9*a*b^2*B + 9*a^2*b*C + b^3*C)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) - (2*b*(9*a^2*A + 33*A*b^2 + 40*a*b*B + 15*a^2*C - 5*b^2*C)*sin(c + d*x))/(15*d*sqrt(sec(c + d*x))) + (2*(9*a^2*A + 24*A*b^2 + 35*a*b*B + 15*a^2*C)*(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*sin(c + d*x))/(15*d) + (2*(6*A*b + 5*a*B)*(a + b*cos(c + d*x))^2*sec(c + d*x)^(3/2)*sin(c + d*x))/(15*d) + (2*A*(a + b*cos(c + d*x))^3*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d)],
[(a + b*cos(c + d*x))^3*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(5/2), x, 8, -((2*(15*a^2*A*b - 5*A*b^3 + 5*a^3*B - 15*a*b^2*B - 15*a^2*b*C - 3*b^3*C)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d)) + (2*(a^3*A + 9*a*A*b^2 + 9*a^2*b*B + b^3*B + 3*a^3*C + 3*a*b^2*C)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) - (2*b*(40*a*A*b + 15*a^2*B - 5*b^2*B - 12*a*b*C)*sin(c + d*x))/(15*d*sqrt(sec(c + d*x))) - (2*b*(35*A*b + 15*a*B - 3*b*C)*(a + b*cos(c + d*x))*sin(c + d*x))/(15*d*sqrt(sec(c + d*x))) + (2*(2*A*b + a*B)*(a + b*cos(c + d*x))^2*sqrt(sec(c + d*x))*sin(c + d*x))/d + (2*A*(a + b*cos(c + d*x))^3*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d)],
[(a + b*cos(c + d*x))^3*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(3/2), x, 8, -((2*(5*a^3*A - 15*a*A*b^2 - 15*a^2*b*B - 3*b^3*B - 5*a^3*C - 9*a*b^2*C)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d)) + (2*(63*a^2*A*b + 7*A*b^3 + 21*a^3*B + 21*a*b^2*B + 21*a^2*b*C + 5*b^3*C)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) - (2*b*(105*a^2*A - 35*A*b^2 - 84*a*b*B - 57*a^2*C - 25*b^2*C)*sin(c + d*x))/(105*d*sqrt(sec(c + d*x))) - (2*b*(35*a*A - 7*b*B - 11*a*C)*(a + b*cos(c + d*x))*sin(c + d*x))/(35*d*sqrt(sec(c + d*x))) - (2*b*(7*A - C)*(a + b*cos(c + d*x))^2*sin(c + d*x))/(7*d*sqrt(sec(c + d*x))) + (2*A*(a + b*cos(c + d*x))^3*sqrt(sec(c + d*x))*sin(c + d*x))/d],
[(a + b*cos(c + d*x))^3*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(1/2), x, 8, (2*(45*a^2*A*b + 9*A*b^3 + 15*a^3*B + 27*a*b^2*B + 27*a^2*b*C + 7*b^3*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(15*d) + (2*(21*a^3*A + 21*a*A*b^2 + 21*a^2*b*B + 5*b^3*B + 7*a^3*C + 15*a*b^2*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(21*d) + (2*(252*a*A*b^2 + 171*a^2*b*B + 75*b^3*B + 16*a^3*C + 176*a*b^2*C)*sin(c + d*x))/(315*d*sqrt(sec(c + d*x))) + (2*(63*A*b^2 + 99*a*b*B + 24*a^2*C + 49*b^2*C)*(a + b*cos(c + d*x))*sin(c + d*x))/(315*d*sqrt(sec(c + d*x))) + (2*(3*b*B + 2*a*C)*(a + b*cos(c + d*x))^2*sin(c + d*x))/(21*d*sqrt(sec(c + d*x))) + (2*C*(a + b*cos(c + d*x))^3*sin(c + d*x))/(9*d*sqrt(sec(c + d*x)))],
[((a + b*cos(c + d*x))^3*(A + B*cos(c + d*x) + C*cos(c + d*x)^2))/sec(c + d*x)^(1/2), x, 9, (2*(15*a^3*A + 27*a*A*b^2 + 27*a^2*b*B + 7*b^3*B + 9*a^3*C + 21*a*b^2*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(15*d) + (2*(231*a^2*A*b + 55*A*b^3 + 77*a^3*B + 165*a*b^2*B + 165*a^2*b*C + 45*b^3*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(231*d) + (2*(1584*a*A*b^2 + 979*a^2*b*B + 539*b^3*B + 48*a^3*C + 1212*a*b^2*C)*sin(c + d*x))/(3465*d*sec(c + d*x)^(3/2)) + (2*(99*A*b^2 + 143*a*b*B + 24*a^2*C + 81*b^2*C)*(a + b*cos(c + d*x))*sin(c + d*x))/(693*d*sec(c + d*x)^(3/2)) + (2*(11*b*B + 6*a*C)*(a + b*cos(c + d*x))^2*sin(c + d*x))/(99*d*sec(c + d*x)^(3/2)) + (2*C*(a + b*cos(c + d*x))^3*sin(c + d*x))/(11*d*sec(c + d*x)^(3/2)) + (2*(231*a^2*A*b + 55*A*b^3 + 77*a^3*B + 165*a*b^2*B + 165*a^2*b*C + 45*b^3*C)*sin(c + d*x))/(231*d*sqrt(sec(c + d*x)))],
[((a + b*cos(c + d*x))^3*(A + B*cos(c + d*x) + C*cos(c + d*x)^2))/sec(c + d*x)^(3/2), x, 10, (2*(351*a^2*A*b + 91*A*b^3 + 117*a^3*B + 273*a*b^2*B + 273*a^2*b*C + 77*b^3*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(195*d) + (2*(77*a^3*A + 165*a*A*b^2 + 165*a^2*b*B + 45*b^3*B + 55*a^3*C + 135*a*b^2*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(231*d) + (2*(2860*a*A*b^2 + 1677*a^2*b*B + 1053*b^3*B + 48*a^3*C + 2312*a*b^2*C)*sin(c + d*x))/(9009*d*sec(c + d*x)^(5/2)) + (2*(143*A*b^2 + 195*a*b*B + 24*a^2*C + 121*b^2*C)*(a + b*cos(c + d*x))*sin(c + d*x))/(1287*d*sec(c + d*x)^(5/2)) + (2*(13*b*B + 6*a*C)*(a + b*cos(c + d*x))^2*sin(c + d*x))/(143*d*sec(c + d*x)^(5/2)) + (2*C*(a + b*cos(c + d*x))^3*sin(c + d*x))/(13*d*sec(c + d*x)^(5/2)) + (2*(351*a^2*A*b + 91*A*b^3 + 117*a^3*B + 273*a*b^2*B + 273*a^2*b*C + 77*b^3*C)*sin(c + d*x))/(585*d*sec(c + d*x)^(3/2)) + (2*(77*a^3*A + 165*a*A*b^2 + 165*a^2*b*B + 45*b^3*B + 55*a^3*C + 135*a*b^2*C)*sin(c + d*x))/(231*d*sqrt(sec(c + d*x)))],


[(a + b*cos(c + d*x))^4*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(13/2), x, 10, -((2*(28*a^3*A*b + 36*a*A*b^3 + 7*a^4*B + 54*a^2*b^2*B + 15*b^4*B + 36*a^3*b*C + 60*a*b^3*C)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(15*d)) + (2*(45*a^4*A + 330*a^2*A*b^2 + 77*A*b^4 + 220*a^3*b*B + 308*a*b^3*B + 55*a^4*C + 462*a^2*b^2*C + 231*b^4*C)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(231*d) + (2*(28*a^3*A*b + 36*a*A*b^3 + 7*a^4*B + 54*a^2*b^2*B + 15*b^4*B + 36*a^3*b*C + 60*a*b^3*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(15*d) + (2*(675*a^4*A + 3199*a^2*A*b^2 + 128*A*b^4 + 2761*a^3*b*B + 2057*a*b^3*B + 825*a^4*C + 4653*a^2*b^2*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(3465*d) + (2*(1346*a^2*A*b + 192*A*b^3 + 539*a^3*B + 1353*a*b^2*B + 1782*a^2*b*C)*(a + b*cos(c + d*x))*sec(c + d*x)^(5/2)*sin(c + d*x))/(3465*d) + (2*(27*a^2*A + 16*A*b^2 + 55*a*b*B + 33*a^2*C)*(a + b*cos(c + d*x))^2*sec(c + d*x)^(7/2)*sin(c + d*x))/(231*d) + (2*(8*A*b + 11*a*B)*(a + b*cos(c + d*x))^3*sec(c + d*x)^(9/2)*sin(c + d*x))/(99*d) + (2*A*(a + b*cos(c + d*x))^4*sec(c + d*x)^(11/2)*sin(c + d*x))/(11*d)],
[(a + b*cos(c + d*x))^4*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(11/2), x, 9, -((2*(7*a^4*A + 54*a^2*A*b^2 + 15*A*b^4 + 36*a^3*b*B + 60*a*b^3*B + 9*a^4*C + 90*a^2*b^2*C - 15*b^4*C)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(15*d)) + (2*(20*a^3*A*b + 28*a*A*b^3 + 5*a^4*B + 42*a^2*b^2*B + 21*b^4*B + 28*a^3*b*C + 84*a*b^3*C)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) + (2*(147*a^4*A + 883*a^2*A*b^2 + 128*A*b^4 + 681*a^3*b*B + 837*a*b^3*B + 189*a^4*C + 1533*a^2*b^2*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(315*d) + (2*(202*a^2*A*b + 64*A*b^3 + 75*a^3*B + 261*a*b^2*B + 294*a^2*b*C)*(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(315*d) + (2*(49*a^2*A + 48*A*b^2 + 117*a*b*B + 63*a^2*C)*(a + b*cos(c + d*x))^2*sec(c + d*x)^(5/2)*sin(c + d*x))/(315*d) + (2*(8*A*b + 9*a*B)*(a + b*cos(c + d*x))^3*sec(c + d*x)^(7/2)*sin(c + d*x))/(63*d) + (2*A*(a + b*cos(c + d*x))^4*sec(c + d*x)^(9/2)*sin(c + d*x))/(9*d)],
[(a + b*cos(c + d*x))^4*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(9/2), x, 9, -((2*(12*a^3*A*b + 20*a*A*b^3 + 3*a^4*B + 30*a^2*b^2*B - 5*b^4*B + 20*a^3*b*C - 20*a*b^3*C)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d)) + (2*(5*a^4*A + 42*a^2*A*b^2 + 21*A*b^4 + 28*a^3*b*B + 84*a*b^3*B + 7*a^4*C + 126*a^2*b^2*C + 7*b^4*C)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) - (2*b*(227*a^2*A*b + 279*A*b^3 + 63*a^3*B + 511*a*b^2*B + 385*a^2*b*C - 35*b^3*C)*sin(c + d*x))/(105*d*sqrt(sec(c + d*x))) + (2*(202*a^2*A*b + 192*A*b^3 + 63*a^3*B + 413*a*b^2*B + 350*a^2*b*C)*(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*sin(c + d*x))/(105*d) + (2*(25*a^2*A + 48*A*b^2 + 77*a*b*B + 35*a^2*C)*(a + b*cos(c + d*x))^2*sec(c + d*x)^(3/2)*sin(c + d*x))/(105*d) + (2*(8*A*b + 7*a*B)*(a + b*cos(c + d*x))^3*sec(c + d*x)^(5/2)*sin(c + d*x))/(35*d) + (2*A*(a + b*cos(c + d*x))^4*sec(c + d*x)^(7/2)*sin(c + d*x))/(7*d)],
[(a + b*cos(c + d*x))^4*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(7/2), x, 9, -((2*(3*a^4*A + 30*a^2*A*b^2 - 5*A*b^4 + 20*a^3*b*B - 20*a*b^3*B + 5*a^4*C - 30*a^2*b^2*C - 3*b^4*C)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d)) + (2*(4*a^3*A*b + 12*a*A*b^3 + a^4*B + 18*a^2*b^2*B + b^4*B + 12*a^3*b*C + 4*a*b^3*C)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) - (2*b*(9*a^3*A + 73*a*A*b^2 + 55*a^2*b*B - 5*b^3*B + 15*a^3*C - 17*a*b^2*C)*sin(c + d*x))/(15*d*sqrt(sec(c + d*x))) - (2*b*(9*a^2*A + 59*A*b^2 + 50*a*b*B + 15*a^2*C - 3*b^2*C)*(a + b*cos(c + d*x))*sin(c + d*x))/(15*d*sqrt(sec(c + d*x))) + (2*(3*a^2*A + 16*A*b^2 + 15*a*b*B + 5*a^2*C)*(a + b*cos(c + d*x))^2*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d) + (2*(8*A*b + 5*a*B)*(a + b*cos(c + d*x))^3*sec(c + d*x)^(3/2)*sin(c + d*x))/(15*d) + (2*A*(a + b*cos(c + d*x))^4*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d)],
[(a + b*cos(c + d*x))^4*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(5/2), x, 9, -((2*(20*a^3*A*b - 20*a*A*b^3 + 5*a^4*B - 30*a^2*b^2*B - 3*b^4*B - 20*a^3*b*C - 12*a*b^3*C)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d)) + (2*(7*a^4*A + 126*a^2*A*b^2 + 7*A*b^4 + 84*a^3*b*B + 28*a*b^3*B + 21*a^4*C + 42*a^2*b^2*C + 5*b^4*C)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) - (2*b*(385*a^2*A*b - 35*A*b^3 + 105*a^3*B - 119*a*b^2*B - 141*a^2*b*C - 25*b^3*C)*sin(c + d*x))/(105*d*sqrt(sec(c + d*x))) - (2*b*(350*a*A*b + 105*a^2*B - 21*b^2*B - 54*a*b*C)*(a + b*cos(c + d*x))*sin(c + d*x))/(105*d*sqrt(sec(c + d*x))) - (2*b*(21*A*b + 7*a*B - b*C)*(a + b*cos(c + d*x))^2*sin(c + d*x))/(7*d*sqrt(sec(c + d*x))) + (2*(8*A*b + 3*a*B)*(a + b*cos(c + d*x))^3*sqrt(sec(c + d*x))*sin(c + d*x))/(3*d) + (2*A*(a + b*cos(c + d*x))^4*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d)],
[(a + b*cos(c + d*x))^4*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(3/2), x, 9, -((2*(15*a^4*A - 90*a^2*A*b^2 - 9*A*b^4 - 60*a^3*b*B - 36*a*b^3*B - 15*a^4*C - 54*a^2*b^2*C - 7*b^4*C)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(15*d)) + (2*(84*a^3*A*b + 28*a*A*b^3 + 21*a^4*B + 42*a^2*b^2*B + 5*b^4*B + 28*a^3*b*C + 20*a*b^3*C)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) - (2*b*(315*a^3*A - 357*a*A*b^2 - 423*a^2*b*B - 75*b^3*B - 187*a^3*C - 251*a*b^2*C)*sin(c + d*x))/(315*d*sqrt(sec(c + d*x))) - (2*b*(315*a^2*A - 63*A*b^2 - 162*a*b*B - 123*a^2*C - 49*b^2*C)*(a + b*cos(c + d*x))*sin(c + d*x))/(315*d*sqrt(sec(c + d*x))) - (2*b*(21*a*A - 3*b*B - 5*a*C)*(a + b*cos(c + d*x))^2*sin(c + d*x))/(21*d*sqrt(sec(c + d*x))) - (2*b*(9*A - C)*(a + b*cos(c + d*x))^3*sin(c + d*x))/(9*d*sqrt(sec(c + d*x))) + (2*A*(a + b*cos(c + d*x))^4*sqrt(sec(c + d*x))*sin(c + d*x))/d],
[(a + b*cos(c + d*x))^4*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(1/2), x, 9, (2*(60*a^3*A*b + 36*a*A*b^3 + 15*a^4*B + 54*a^2*b^2*B + 7*b^4*B + 36*a^3*b*C + 28*a*b^3*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(15*d) + (2*(231*a^4*A + 462*a^2*A*b^2 + 55*A*b^4 + 308*a^3*b*B + 220*a*b^3*B + 77*a^4*C + 330*a^2*b^2*C + 45*b^4*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(231*d) + (2*(4653*a^2*A*b^2 + 825*A*b^4 + 2057*a^3*b*B + 2761*a*b^3*B + 128*a^4*C + 3199*a^2*b^2*C + 675*b^4*C)*sin(c + d*x))/(3465*d*sqrt(sec(c + d*x))) + (2*(1782*a*A*b^2 + 1353*a^2*b*B + 539*b^3*B + 192*a^3*C + 1346*a*b^2*C)*(a + b*cos(c + d*x))*sin(c + d*x))/(3465*d*sqrt(sec(c + d*x))) + (2*(33*A*b^2 + 55*a*b*B + 16*a^2*C + 27*b^2*C)*(a + b*cos(c + d*x))^2*sin(c + d*x))/(231*d*sqrt(sec(c + d*x))) + (2*(11*b*B + 8*a*C)*(a + b*cos(c + d*x))^3*sin(c + d*x))/(99*d*sqrt(sec(c + d*x))) + (2*C*(a + b*cos(c + d*x))^4*sin(c + d*x))/(11*d*sqrt(sec(c + d*x)))],
[((a + b*cos(c + d*x))^4*(A + B*cos(c + d*x) + C*cos(c + d*x)^2))/sec(c + d*x)^(1/2), x, 10, (2*(195*a^4*A + 702*a^2*A*b^2 + 91*A*b^4 + 468*a^3*b*B + 364*a*b^3*B + 117*a^4*C + 546*a^2*b^2*C + 77*b^4*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(195*d) + (2*(308*a^3*A*b + 220*a*A*b^3 + 77*a^4*B + 330*a^2*b^2*B + 45*b^4*B + 220*a^3*b*C + 180*a*b^3*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(231*d) + (2*(33319*a^2*A*b^2 + 7007*A*b^4 + 13351*a^3*b*B + 22763*a*b^3*B + 384*a^4*C + 25217*a^2*b^2*C + 5929*b^4*C)*sin(c + d*x))/(45045*d*sec(c + d*x)^(3/2)) + (2*(3146*a*A*b^2 + 2171*a^2*b*B + 1053*b^3*B + 192*a^3*C + 2518*a*b^2*C)*(a + b*cos(c + d*x))*sin(c + d*x))/(9009*d*sec(c + d*x)^(3/2)) + (2*(143*A*b^2 + 221*a*b*B + 48*a^2*C + 121*b^2*C)*(a + b*cos(c + d*x))^2*sin(c + d*x))/(1287*d*sec(c + d*x)^(3/2)) + (2*(13*b*B + 8*a*C)*(a + b*cos(c + d*x))^3*sin(c + d*x))/(143*d*sec(c + d*x)^(3/2)) + (2*C*(a + b*cos(c + d*x))^4*sin(c + d*x))/(13*d*sec(c + d*x)^(3/2)) + (2*(308*a^3*A*b + 220*a*A*b^3 + 77*a^4*B + 330*a^2*b^2*B + 45*b^4*B + 220*a^3*b*C + 180*a*b^3*C)*sin(c + d*x))/(231*d*sqrt(sec(c + d*x)))],


# ::Subsubsection::Closed:: 
#n<0


[((A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(7/2))/(a + b*cos(c + d*x)), x, 9, (-2*(3*a^2*A + 5*A*b^2 - 5*a*b*B + 5*a^2*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*a^3*d) - (2*(A*b - a*B)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*a^2*d) - (2*b*(A*b^2 - a*b*B + a^2*C)*sqrt(cos(c + d*x))*EllipticPi((2*b)/(a + b), (c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(a^3*(a + b)*d) + (2*(3*a^2*A + 5*A*b^2 - 5*a*b*B + 5*a^2*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(5*a^3*d) - (2*(A*b - a*B)*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*a^2*d) + (2*A*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*a*d)],
[((A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(5/2))/(a + b*cos(c + d*x)), x, 8, (2*(A*b - a*B)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(a^2*d) + (2*A*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*a*d) + (2*(A*b^2 - a*b*B + a^2*C)*sqrt(cos(c + d*x))*EllipticPi((2*b)/(a + b), (c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(a^2*(a + b)*d) - (2*(A*b - a*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(a^2*d) + (2*A*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*a*d)],
[((A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(3/2))/(a + b*cos(c + d*x)), x, 7, (-2*A*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(a*d) + (2*C*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(b*d) - (2*(A*b^2 - a*b*B + a^2*C)*sqrt(cos(c + d*x))*EllipticPi((2*b)/(a + b), (c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(a*b*(a + b)*d) + (2*A*sqrt(sec(c + d*x))*sin(c + d*x))/(a*d)],
[((A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sqrt(sec(c + d*x)))/(a + b*cos(c + d*x)), x, 6, (2*C*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(b*d) + (2*(b*B - a*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(b^2*d) + (2*(A*b^2 - a*b*B + a^2*C)*sqrt(cos(c + d*x))*EllipticPi((2*b)/(a + b), (c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(b^2*(a + b)*d)],
[(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/((a + b*cos(c + d*x))*sqrt(sec(c + d*x))), x, 7, (2*(b*B - a*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(b^2*d) + (2*(3*A*b^2 - 3*a*b*B + 3*a^2*C + b^2*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*b^3*d) - (2*a*(A*b^2 - a*b*B + a^2*C)*sqrt(cos(c + d*x))*EllipticPi((2*b)/(a + b), (c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(b^3*(a + b)*d) + (2*C*sin(c + d*x))/(3*b*d*sqrt(sec(c + d*x)))],
[(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/((a + b*cos(c + d*x))*sec(c + d*x)^(3/2)), x, 8, (2*(5*A*b^2 - 5*a*b*B + 5*a^2*C + 3*b^2*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*b^3*d) - (2*(3*a*A*b^2 - 3*a^2*b*B - b^3*B + 3*a^3*C + a*b^2*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*b^4*d) + (2*a^2*(A*b^2 - a*b*B + a^2*C)*sqrt(cos(c + d*x))*EllipticPi((2*b)/(a + b), (c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(b^4*(a + b)*d) + (2*C*sin(c + d*x))/(5*b*d*sec(c + d*x)^(3/2)) + (2*(b*B - a*C)*sin(c + d*x))/(3*b^2*d*sqrt(sec(c + d*x)))],
[(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/((a + b*cos(c + d*x))*sec(c + d*x)^(5/2)), x, 9, (-2*(5*a*A*b^2 - 5*a^2*b*B - 3*b^3*B + 5*a^3*C + 3*a*b^2*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*b^4*d) + (2*(21*a^2*A*b^2 + 7*A*b^4 - 21*a^3*b*B - 7*a*b^3*B + 21*a^4*C + 7*a^2*b^2*C + 5*b^4*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(21*b^5*d) - (2*a^3*(A*b^2 - a*b*B + a^2*C)*sqrt(cos(c + d*x))*EllipticPi((2*b)/(a + b), (c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(b^5*(a + b)*d) + (2*C*sin(c + d*x))/(7*b*d*sec(c + d*x)^(5/2)) + (2*(b*B - a*C)*sin(c + d*x))/(5*b^2*d*sec(c + d*x)^(3/2)) + (2*(7*A*b^2 - 7*a*b*B + 7*a^2*C + 5*b^2*C)*sin(c + d*x))/(21*b^3*d*sqrt(sec(c + d*x)))],


[((A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(5/2))/(a + b*cos(c + d*x))^2, x, 9, ((4*a^2*A*b - 5*A*b^3 - 2*a^3*B + 3*a*b^2*B - a^2*b*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(a^3*(a^2 - b^2)*d) + ((2*a^2*A - 5*A*b^2 + 3*a*b*B - 3*a^2*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*a^2*(a^2 - b^2)*d) + ((7*a^2*A*b^2 - 5*A*b^4 - 5*a^3*b*B + 3*a*b^3*B + 3*a^4*C - a^2*b^2*C)*sqrt(cos(c + d*x))*EllipticPi((2*b)/(a + b), (c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(a^3*(a - b)*(a + b)^2*d) - ((4*a^2*A*b - 5*A*b^3 - 2*a^3*B + 3*a*b^2*B - a^2*b*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(a^3*(a^2 - b^2)*d) + ((2*a^2*A - 5*A*b^2 + 3*a*b*B - 3*a^2*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*a^2*(a^2 - b^2)*d) + ((A*b^2 - a*b*B + a^2*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(a*(a^2 - b^2)*d*(a + b*cos(c + d*x)))],
[((A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(3/2))/(a + b*cos(c + d*x))^2, x, 8, -(((2*a^2*A - 3*A*b^2 + a*b*B - a^2*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(a^2*(a^2 - b^2)*d)) + ((A*b^2 - a*b*B + a^2*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(a*b*(a^2 - b^2)*d) - ((5*a^2*A*b^2 - 3*A*b^4 - 3*a^3*b*B + a*b^3*B + a^4*C + a^2*b^2*C)*sqrt(cos(c + d*x))*EllipticPi((2*b)/(a + b), (c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(a^2*(a - b)*b*(a + b)^2*d) + ((2*a^2*A - 3*A*b^2 + a*b*B - a^2*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(a^2*(a^2 - b^2)*d) + ((A*b^2 - a*b*B + a^2*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(a*(a^2 - b^2)*d*(a + b*cos(c + d*x)))],
[((A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sqrt(sec(c + d*x)))/(a + b*cos(c + d*x))^2, x, 7, -(((A*b^2 - a*b*B + a^2*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(a*b*(a^2 - b^2)*d)) - ((A*b^2 - a*b*B - a^2*C + 2*b^2*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(b^2*(a^2 - b^2)*d) + ((3*a^2*A*b^2 - A*b^4 - a^3*b*B - a*b^3*B - a^4*C + 3*a^2*b^2*C)*sqrt(cos(c + d*x))*EllipticPi((2*b)/(a + b), (c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(a*(a - b)*b^2*(a + b)^2*d) + ((A*b^2 - a*b*B + a^2*C)*sin(c + d*x))/(a*(a^2 - b^2)*d*(a + b*cos(c + d*x))*sqrt(sec(c + d*x)))],
[(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/((a + b*cos(c + d*x))^2*sqrt(sec(c + d*x))), x, 7, ((A*b^2 - a*b*B + 3*a^2*C - 2*b^2*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(b^2*(a^2 - b^2)*d) + ((a*A*b^2 + a^2*b*B - 2*b^3*B - 3*a^3*C + 4*a*b^2*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(b^3*(a^2 - b^2)*d) - ((a^2*A*b^2 + A*b^4 + a^3*b*B - 3*a*b^3*B - 3*a^4*C + 5*a^2*b^2*C)*sqrt(cos(c + d*x))*EllipticPi((2*b)/(a + b), (c + d*x)/2, 2)*sqrt(sec(c + d*x)))/((a - b)*b^3*(a + b)^2*d) - ((A*b^2 - a*b*B + a^2*C)*sin(c + d*x))/(b*(a^2 - b^2)*d*(a + b*cos(c + d*x))*sqrt(sec(c + d*x)))],
[(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/((a + b*cos(c + d*x))^2*sec(c + d*x)^(3/2)), x, 8, -(((a*A*b^2 - 3*a^2*b*B + 2*b^3*B + 5*a^3*C - 4*a*b^2*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(b^3*(a^2 - b^2)*d)) + ((3*a^2*A*b^2 - 6*A*b^4 - 9*a^3*b*B + 12*a*b^3*B + 15*a^4*C - 16*a^2*b^2*C - 2*b^4*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*b^4*(a^2 - b^2)*d) - (a*(a^2*A*b^2 - 3*A*b^4 - 3*a^3*b*B + 5*a*b^3*B + 5*a^4*C - 7*a^2*b^2*C)*sqrt(cos(c + d*x))*EllipticPi((2*b)/(a + b), (c + d*x)/2, 2)*sqrt(sec(c + d*x)))/((a - b)*b^4*(a + b)^2*d) - ((A*b^2 - a*b*B + a^2*C)*sin(c + d*x))/(b*(a^2 - b^2)*d*(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)) + ((3*A*b^2 - 3*a*b*B + 5*a^2*C - 2*b^2*C)*sin(c + d*x))/(3*b^2*(a^2 - b^2)*d*sqrt(sec(c + d*x)))],
[(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/((a + b*cos(c + d*x))^2*sec(c + d*x)^(5/2)), x, 9, ((15*a^2*A*b^2 - 10*A*b^4 - 25*a^3*b*B + 20*a*b^3*B + 35*a^4*C - 24*a^2*b^2*C - 6*b^4*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*b^4*(a^2 - b^2)*d) - ((9*a^3*A*b^2 - 12*a*A*b^4 - 15*a^4*b*B + 16*a^2*b^3*B + 2*b^5*B + 21*a^5*C - 20*a^3*b^2*C - 4*a*b^4*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*b^5*(a^2 - b^2)*d) + (a^2*(3*a^2*A*b^2 - 5*A*b^4 - 5*a^3*b*B + 7*a*b^3*B + 7*a^4*C - 9*a^2*b^2*C)*sqrt(cos(c + d*x))*EllipticPi((2*b)/(a + b), (c + d*x)/2, 2)*sqrt(sec(c + d*x)))/((a - b)*b^5*(a + b)^2*d) - ((A*b^2 - a*b*B + a^2*C)*sin(c + d*x))/(b*(a^2 - b^2)*d*(a + b*cos(c + d*x))*sec(c + d*x)^(5/2)) + ((5*A*b^2 - 5*a*b*B + 7*a^2*C - 2*b^2*C)*sin(c + d*x))/(5*b^2*(a^2 - b^2)*d*sec(c + d*x)^(3/2)) - ((3*a*A*b^2 - 5*a^2*b*B + 2*b^3*B + 7*a^3*C - 4*a*b^2*C)*sin(c + d*x))/(3*b^3*(a^2 - b^2)*d*sqrt(sec(c + d*x)))],


[((A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(5/2))/(a + b*cos(c + d*x))^3, x, 10, ((24*a^4*A*b - 65*a^2*A*b^3 + 35*A*b^5 - 8*a^5*B + 29*a^3*b^2*B - 15*a*b^4*B - 9*a^4*b*C + 3*a^2*b^3*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(4*a^4*(a^2 - b^2)^2*d) + ((8*a^4*A - 61*a^2*A*b^2 + 35*A*b^4 + 33*a^3*b*B - 15*a*b^3*B - 21*a^4*C + 3*a^2*b^2*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(12*a^3*(a^2 - b^2)^2*d) + ((63*a^4*A*b^2 - 86*a^2*A*b^4 + 35*A*b^6 - 35*a^5*b*B + 38*a^3*b^3*B - 15*a*b^5*B + 15*a^6*C - 6*a^4*b^2*C + 3*a^2*b^4*C)*sqrt(cos(c + d*x))*EllipticPi((2*b)/(a + b), (c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(4*a^4*(a - b)^2*(a + b)^3*d) - ((24*a^4*A*b - 65*a^2*A*b^3 + 35*A*b^5 - 8*a^5*B + 29*a^3*b^2*B - 15*a*b^4*B - 9*a^4*b*C + 3*a^2*b^3*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(4*a^4*(a^2 - b^2)^2*d) + ((8*a^4*A - 61*a^2*A*b^2 + 35*A*b^4 + 33*a^3*b*B - 15*a*b^3*B - 21*a^4*C + 3*a^2*b^2*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(12*a^3*(a^2 - b^2)^2*d) + ((A*b^2 - a*b*B + a^2*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(2*a*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) + ((13*a^2*A*b^2 - 7*A*b^4 - 9*a^3*b*B + 3*a*b^3*B + 5*a^4*C + a^2*b^2*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(4*a^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))],
[((A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(3/2))/(a + b*cos(c + d*x))^3, x, 9, -((8*a^4*A - 29*a^2*A*b^2 + 15*A*b^4 + 9*a^3*b*B - 3*a*b^3*B - 5*a^4*C - a^2*b^2*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(4*a^3*(a^2 - b^2)^2*d) + ((11*a^2*A*b^2 - 5*A*b^4 - 7*a^3*b*B + a*b^3*B + 3*a^4*C + 3*a^2*b^2*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(4*a^2*b*(a^2 - b^2)^2*d) - ((35*a^4*A*b^2 - 38*a^2*A*b^4 + 15*A*b^6 - 15*a^5*b*B + 6*a^3*b^3*B - 3*a*b^5*B + 3*a^6*C + 10*a^4*b^2*C - a^2*b^4*C)*sqrt(cos(c + d*x))*EllipticPi((2*b)/(a + b), (c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(4*a^3*(a - b)^2*b*(a + b)^3*d) + ((8*a^4*A - 29*a^2*A*b^2 + 15*A*b^4 + 9*a^3*b*B - 3*a*b^3*B - 5*a^4*C - a^2*b^2*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(4*a^3*(a^2 - b^2)^2*d) + ((A*b^2 - a*b*B + a^2*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(2*a*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) + ((11*a^2*A*b^2 - 5*A*b^4 - 7*a^3*b*B + a*b^3*B + 3*a^4*C + 3*a^2*b^2*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(4*a^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))],
[((A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sqrt(sec(c + d*x)))/(a + b*cos(c + d*x))^3, x, 8, -((9*a^2*A*b^2 - 3*A*b^4 - 5*a^3*b*B - a*b^3*B + a^4*C + 5*a^2*b^2*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(4*a^2*b*(a^2 - b^2)^2*d) - ((7*a^2*A*b^2 - A*b^4 - 3*a^3*b*B - 3*a*b^3*B - a^4*C + 7*a^2*b^2*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(4*a*b^2*(a^2 - b^2)^2*d) + ((15*a^4*A*b^2 - 6*a^2*A*b^4 + 3*A*b^6 - 3*a^5*b*B - 10*a^3*b^3*B + a*b^5*B - a^6*C + 10*a^4*b^2*C + 3*a^2*b^4*C)*sqrt(cos(c + d*x))*EllipticPi((2*b)/(a + b), (c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(4*a^2*(a - b)^2*b^2*(a + b)^3*d) + ((A*b^2 - a*b*B + a^2*C)*sin(c + d*x))/(2*a*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2*sqrt(sec(c + d*x))) + ((9*a^2*A*b^2 - 3*A*b^4 - 5*a^3*b*B - a*b^3*B + a^4*C + 5*a^2*b^2*C)*sin(c + d*x))/(4*a^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x))*sqrt(sec(c + d*x)))],
[(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/((a + b*cos(c + d*x))^3*sqrt(sec(c + d*x))), x, 8, ((5*a^2*A*b^2 + A*b^4 - a^3*b*B - 5*a*b^3*B - 3*a^4*C + 9*a^2*b^2*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(4*a*b^2*(a^2 - b^2)^2*d) + ((3*a^2*A*b^2 + 3*A*b^4 + a^3*b*B - 7*a*b^3*B + 3*a^4*C - 5*a^2*b^2*C + 8*b^4*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(4*b^3*(a^2 - b^2)^2*d) - ((3*a^4*A*b^2 + 10*a^2*A*b^4 - A*b^6 + a^5*b*B - 10*a^3*b^3*B - 3*a*b^5*B + 3*a^6*C - 6*a^4*b^2*C + 15*a^2*b^4*C)*sqrt(cos(c + d*x))*EllipticPi((2*b)/(a + b), (c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(4*a*(a - b)^2*b^3*(a + b)^3*d) - ((A*b^2 - a*b*B + a^2*C)*sin(c + d*x))/(2*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2*sqrt(sec(c + d*x))) - ((5*a^2*A*b^2 + A*b^4 - a^3*b*B - 5*a*b^3*B - 3*a^4*C + 9*a^2*b^2*C)*sin(c + d*x))/(4*a*b*(a^2 - b^2)^2*d*(a + b*cos(c + d*x))*sqrt(sec(c + d*x)))],
[(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/((a + b*cos(c + d*x))^3*sec(c + d*x)^(3/2)), x, 8, -((a^2*A*b^2 + 5*A*b^4 + 3*a^3*b*B - 9*a*b^3*B - 15*a^4*C + 29*a^2*b^2*C - 8*b^4*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(4*b^3*(a^2 - b^2)^2*d) + ((a^3*A*b^2 - 7*a*A*b^4 + 3*a^4*b*B - 5*a^2*b^3*B + 8*b^5*B - 15*a^5*C + 33*a^3*b^2*C - 24*a*b^4*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(4*b^4*(a^2 - b^2)^2*d) - ((a^4*A*b^2 - 10*a^2*A*b^4 - 3*A*b^6 + 3*a^5*b*B - 6*a^3*b^3*B + 15*a*b^5*B - 15*a^6*C + 38*a^4*b^2*C - 35*a^2*b^4*C)*sqrt(cos(c + d*x))*EllipticPi((2*b)/(a + b), (c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(4*(a - b)^2*b^4*(a + b)^3*d) - ((A*b^2 - a*b*B + a^2*C)*sin(c + d*x))/(2*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2*sec(c + d*x)^(3/2)) + ((3*a^2*A*b^2 + 3*A*b^4 + a^3*b*B - 7*a*b^3*B - 5*a^4*C + 11*a^2*b^2*C)*sin(c + d*x))/(4*b^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x))*sqrt(sec(c + d*x)))],
[(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/((a + b*cos(c + d*x))^3*sec(c + d*x)^(5/2)), x, 9, -((3*a^3*A*b^2 - 9*a*A*b^4 - 15*a^4*b*B + 29*a^2*b^3*B - 8*b^5*B + 35*a^5*C - 65*a^3*b^2*C + 24*a*b^4*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(4*b^4*(a^2 - b^2)^2*d) + ((9*a^4*A*b^2 - 15*a^2*A*b^4 + 24*A*b^6 - 45*a^5*b*B + 99*a^3*b^3*B - 72*a*b^5*B + 105*a^6*C - 223*a^4*b^2*C + 128*a^2*b^4*C + 8*b^6*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(12*b^5*(a^2 - b^2)^2*d) - (a*(3*a^4*A*b^2 - 6*a^2*A*b^4 + 15*A*b^6 - 15*a^5*b*B + 38*a^3*b^3*B - 35*a*b^5*B + 35*a^6*C - 86*a^4*b^2*C + 63*a^2*b^4*C)*sqrt(cos(c + d*x))*EllipticPi((2*b)/(a + b), (c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(4*(a - b)^2*b^5*(a + b)^3*d) - ((A*b^2 - a*b*B + a^2*C)*sin(c + d*x))/(2*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2*sec(c + d*x)^(5/2)) + ((a^2*A*b^2 + 5*A*b^4 + 3*a^3*b*B - 9*a*b^3*B - 7*a^4*C + 13*a^2*b^2*C)*sin(c + d*x))/(4*b^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)) + ((3*a^2*A*b^2 - 21*A*b^4 - 15*a^3*b*B + 33*a*b^3*B + 35*a^4*C - 61*a^2*b^2*C + 8*b^4*C)*sin(c + d*x))/(12*b^3*(a^2 - b^2)^2*d*sqrt(sec(c + d*x)))],
[(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/((a + b*cos(c + d*x))^3*sec(c + d*x)^(7/2)), x, 10, ((75*a^4*A*b^2 - 145*a^2*A*b^4 + 40*A*b^6 - 175*a^5*b*B + 325*a^3*b^3*B - 120*a*b^5*B + 315*a^6*C - 561*a^4*b^2*C + 192*a^2*b^4*C + 24*b^6*C)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(20*b^5*(a^2 - b^2)^2*d) - ((45*a^5*A*b^2 - 99*a^3*A*b^4 + 72*a*A*b^6 - 105*a^6*b*B + 223*a^4*b^3*B - 128*a^2*b^5*B - 8*b^7*B + 189*a^7*C - 387*a^5*b^2*C + 192*a^3*b^4*C + 24*a*b^6*C)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(12*b^6*(a^2 - b^2)^2*d) + (a^2*(15*a^4*A*b^2 - 38*a^2*A*b^4 + 35*A*b^6 - 35*a^5*b*B + 86*a^3*b^3*B - 63*a*b^5*B + 63*a^6*C - 150*a^4*b^2*C + 99*a^2*b^4*C)*sqrt(cos(c + d*x))*EllipticPi((2*b)/(a + b), (1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(4*(a - b)^2*b^6*(a + b)^3*d) - ((A*b^2 - a*b*B + a^2*C)*sin(c + d*x))/(2*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2*sec(c + d*x)^(7/2)) - ((a^2*A*b^2 - 7*A*b^4 - 5*a^3*b*B + 11*a*b^3*B + 9*a^4*C - 15*a^2*b^2*C)*sin(c + d*x))/(4*b^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x))*sec(c + d*x)^(5/2)) + ((15*a^2*A*b^2 - 45*A*b^4 - 35*a^3*b*B + 65*a*b^3*B + 63*a^4*C - 101*a^2*b^2*C + 8*b^4*C)*sin(c + d*x))/(20*b^3*(a^2 - b^2)^2*d*sec(c + d*x)^(3/2)) - ((15*a^3*A*b^2 - 33*a*A*b^4 - 35*a^4*b*B + 61*a^2*b^3*B - 8*b^5*B + 63*a^5*C - 105*a^3*b^2*C + 24*a*b^4*C)*sin(c + d*x))/(12*b^4*(a^2 - b^2)^2*d*sqrt(sec(c + d*x)))],


# ::Subsection::Closed:: 
#Integrands of the form Sec[c+d x]^(m/2) (A+B Cos[c+d x]+C Cos[c+d x]^2) (a+b Cos[c+d x])^(n/2)
#


# ::Subsubsection::Closed:: 
#n>0


[sqrt(a + b*cos(c + d*x))*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(11/2), x, 9, -((2*(a + b)*(147*a^4*A - 24*a^2*A*b^2 - 16*A*b^4 + 57*a^3*b*B + 24*a*b^3*B + 189*a^4*C - 42*a^2*b^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(315*a^4*d*sqrt(a + b*cos(c + d*x)))) + (1/(315*a^3*d*sqrt(a + b*cos(c + d*x))))*(2*(147*a^4*A + 111*a^3*A*b - 24*a^2*A*b^2 - 4*a*A*b^3 - 16*A*b^4 + 75*a^4*B + 57*a^3*b*B + 6*a^2*b^2*B + 24*a*b^3*B + 189*a^4*C + 147*a^3*b*C - 42*a^2*b^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x))) + (2*(13*a^2*A*b + 8*A*b^3 + 75*a^3*B - 12*a*b^2*B + 21*a^2*b*C)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(315*a^3*d) + (2*(49*a^2*A - 6*A*b^2 + 9*a*b*B + 63*a^2*C)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(5/2)*sin(c + d*x))/(315*a^2*d) + (2*(A*b + 9*a*B)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(7/2)*sin(c + d*x))/(63*a*d) + (2*A*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(9/2)*sin(c + d*x))/(9*d) + (2*(147*a^4*A - 24*a^2*A*b^2 - 16*A*b^4 + 57*a^3*b*B + 24*a*b^3*B + 189*a^4*C - 42*a^2*b^2*C)*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*tan((1/2)*(c + d*x)))/(315*a^4*d)],
[sqrt(a + b*cos(c + d*x))*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(9/2), x, 8, -((2*(a + b)*(19*a^2*A*b + 8*A*b^3 + 63*a^3*B - 14*a*b^2*B + 35*a^2*b*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(105*a^3*d*sqrt(a + b*cos(c + d*x)))) + (2*(25*a^3*A + 19*a^2*A*b + 2*a*A*b^2 + 8*A*b^3 + 63*a^3*B + 49*a^2*b*B - 14*a*b^2*B + 35*a^3*C + 35*a^2*b*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(105*a^2*d*sqrt(a + b*cos(c + d*x))) + (2*(25*a^2*A - 4*A*b^2 + 7*a*b*B + 35*a^2*C)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(105*a^2*d) + (2*(A*b + 7*a*B)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(5/2)*sin(c + d*x))/(35*a*d) + (2*A*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(7/2)*sin(c + d*x))/(7*d) + (2*(19*a^2*A*b + 8*A*b^3 + 63*a^3*B - 14*a*b^2*B + 35*a^2*b*C)*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*tan((1/2)*(c + d*x)))/(105*a^3*d)],
[sqrt(a + b*cos(c + d*x))*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(7/2), x, 7, -((2*(a + b)*(9*a^2*A - 2*A*b^2 + 5*a*b*B + 15*a^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(15*a^2*d*sqrt(a + b*cos(c + d*x)))) + (2*(9*a^2*A + 7*a*A*b - 2*A*b^2 + 5*a^2*B + 5*a*b*B + 15*a^2*C + 15*a*b*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(15*a*d*sqrt(a + b*cos(c + d*x))) + (2*(A*b + 5*a*B)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(15*a*d) + (2*A*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d) + (2*(9*a^2*A - 2*A*b^2 + 5*a*b*B + 15*a^2*C)*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*tan((1/2)*(c + d*x)))/(15*a^2*d)],
[sqrt(a + b*cos(c + d*x))*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(5/2), x, 8, -((2*(a + b)*(A*b + 3*a*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(3*a*d*sqrt(a + b*cos(c + d*x)))) + (2*(a*A + A*b + 3*a*B + 3*b*B + 3*a*C - 3*b*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(3*d*sqrt(a + b*cos(c + d*x))) + (4*b*C*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*cos(c + d*x))) + (2*A*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d) + (2*(A*b + 3*a*B)*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*tan((1/2)*(c + d*x)))/(3*a*d)],
[sqrt(a + b*cos(c + d*x))*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(3/2), x, 8, -(((a + b)*(2*A - C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*cos(c + d*x)))) + (2*(a*A + A*b + a*B - b*B - a*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*cos(c + d*x))) + (2*(2*b*B + a*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*cos(c + d*x))) + (2*A*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*sin(c + d*x))/d - ((2*A - C)*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(d*sqrt(sec(c + d*x)))],
[sqrt(a + b*cos(c + d*x))*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(1/2), x, 8, ((a + b)*(4*b*B + a*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(4*b*d*sqrt(a + b*cos(c + d*x))) + ((4*a*A - 4*A*b - 4*a*B + a*C - 2*b*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(2*d*sqrt(a + b*cos(c + d*x))) + ((8*A*b^2 + 4*a*b*B - a^2*C + 4*b^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(2*b*d*sqrt(a + b*cos(c + d*x))) + (C*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(2*d*sqrt(sec(c + d*x))) + ((4*b*B + a*C)*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(4*b*d*sqrt(sec(c + d*x)))],
[(sqrt(a + b*cos(c + d*x))*(A + B*cos(c + d*x) + C*cos(c + d*x)^2))/sec(c + d*x)^(1/2), x, 9, ((a + b)*(24*A*b^2 + 6*a*b*B - 3*a^2*C + 16*b^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(24*b^2*d*sqrt(a + b*cos(c + d*x))) - ((24*a*A*b - 6*a*b*B + 12*b^2*B - a^2*C + 14*a*b*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(12*b*d*sqrt(a + b*cos(c + d*x))) + ((8*a*A*b^2 - 2*a^2*b*B + 8*b^3*B + a^3*C + 4*a*b^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(4*b^2*d*sqrt(a + b*cos(c + d*x))) + (C*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*d*sec(c + d*x)^(3/2)) + ((6*b*B + a*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(12*b*d*sqrt(sec(c + d*x))) + ((24*A*b^2 + 6*a*b*B - 3*a^2*C + 16*b^2*C)*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(24*b^2*d*sqrt(sec(c + d*x)))],
[(sqrt(a + b*cos(c + d*x))*(A + B*cos(c + d*x) + C*cos(c + d*x)^2))/sec(c + d*x)^(3/2), x, 10, ((a + b)*(48*a*A*b^2 - 24*a^2*b*B + 128*b^3*B + 15*a^3*C + 28*a*b^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(192*b^3*d*sqrt(a + b*cos(c + d*x))) + ((48*a*A*b^2 - 96*A*b^3 + 8*a^2*b*B - 112*a*b^2*B - 5*a^3*C - 2*a^2*b*C + 36*a*b^2*C - 72*b^3*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(96*b^2*d*sqrt(a + b*cos(c + d*x))) - ((16*a^2*A*b^2 - 64*A*b^4 - 8*a^3*b*B - 32*a*b^3*B + 5*a^4*C + 8*a^2*b^2*C - 48*b^4*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(32*b^3*d*sqrt(a + b*cos(c + d*x))) + (C*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(4*d*sec(c + d*x)^(5/2)) + ((8*b*B + a*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(24*b*d*sec(c + d*x)^(3/2)) + ((48*A*b^2 + 8*a*b*B - 5*a^2*C + 36*b^2*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(96*b^2*d*sqrt(sec(c + d*x))) + ((48*a*A*b^2 - 24*a^2*b*B + 128*b^3*B + 15*a^3*C + 28*a*b^2*C)*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(192*b^3*d*sqrt(sec(c + d*x)))],


[(a + b*cos(c + d*x))^(3/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(11/2), x, 9, -((2*(a + b)*(147*a^4*A + 33*a^2*A*b^2 + 8*A*b^4 + 246*a^3*b*B - 18*a*b^3*B + 189*a^4*C + 63*a^2*b^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(315*a^3*d*sqrt(a + b*cos(c + d*x)))) + (1/(315*a^2*d*sqrt(a + b*cos(c + d*x))))*(2*(147*a^4*A + 186*a^3*A*b + 33*a^2*A*b^2 + 2*a*A*b^3 + 8*A*b^4 + 75*a^4*B + 246*a^3*b*B + 153*a^2*b^2*B - 18*a*b^3*B + 189*a^4*C + 252*a^3*b*C + 63*a^2*b^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x))) + (2*(88*a^2*A*b - 4*A*b^3 + 75*a^3*B + 9*a*b^2*B + 126*a^2*b*C)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(315*a^2*d) + (2*(49*a^2*A + 3*A*b^2 + 72*a*b*B + 63*a^2*C)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(5/2)*sin(c + d*x))/(315*a*d) + (2*(A*b + 3*a*B)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(7/2)*sin(c + d*x))/(21*d) + (2*A*(a + b*cos(c + d*x))^(3/2)*sec(c + d*x)^(9/2)*sin(c + d*x))/(9*d) + (2*(147*a^4*A + 33*a^2*A*b^2 + 8*A*b^4 + 246*a^3*b*B - 18*a*b^3*B + 189*a^4*C + 63*a^2*b^2*C)*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*tan((1/2)*(c + d*x)))/(315*a^3*d)],
[(a + b*cos(c + d*x))^(3/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(9/2), x, 8, -((2*(a + b)*(82*a^2*A*b - 6*A*b^3 + 63*a^3*B + 21*a*b^2*B + 140*a^2*b*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(105*a^2*d*sqrt(a + b*cos(c + d*x)))) + (1/(105*a*d*sqrt(a + b*cos(c + d*x))))*(2*(25*a^3*A + 82*a^2*A*b + 51*a*A*b^2 - 6*A*b^3 + 63*a^3*B + 84*a^2*b*B + 21*a*b^2*B + 35*a^3*C + 140*a^2*b*C + 105*a*b^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x))) + (2*(25*a^2*A + 3*A*b^2 + 42*a*b*B + 35*a^2*C)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(105*a*d) + (2*(3*A*b + 7*a*B)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(5/2)*sin(c + d*x))/(35*d) + (2*A*(a + b*cos(c + d*x))^(3/2)*sec(c + d*x)^(7/2)*sin(c + d*x))/(7*d) + (2*(82*a^2*A*b - 6*A*b^3 + 63*a^3*B + 21*a*b^2*B + 140*a^2*b*C)*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*tan((1/2)*(c + d*x)))/(105*a^2*d)],
[(a + b*cos(c + d*x))^(3/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(7/2), x, 9, -((2*(a + b)*(9*a^2*A + 3*A*b^2 + 20*a*b*B + 15*a^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(15*a*d*sqrt(a + b*cos(c + d*x)))) + (2*(9*a^2*A + 12*a*A*b + 3*A*b^2 + 5*a^2*B + 20*a*b*B + 15*b^2*B + 15*a^2*C + 30*a*b*C - 15*b^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(15*d*sqrt(a + b*cos(c + d*x))) + (4*b^2*C*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*cos(c + d*x))) + (2*(3*A*b + 5*a*B)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(15*d) + (2*A*(a + b*cos(c + d*x))^(3/2)*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d) + (2*(9*a^2*A + 3*A*b^2 + 20*a*b*B + 15*a^2*C)*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*tan((1/2)*(c + d*x)))/(15*a*d)],
[(a + b*cos(c + d*x))^(3/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(5/2), x, 9, -(((a + b)*(8*A*b + 6*a*B - 3*b*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(3*d*sqrt(a + b*cos(c + d*x)))) + (2*(a^2*A + 4*a*A*b + 3*A*b^2 + 3*a^2*B + 6*a*b*B - 3*b^2*B + 3*a^2*C - 6*a*b*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(3*d*sqrt(a + b*cos(c + d*x))) + (2*b*(2*b*B + 3*a*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*cos(c + d*x))) + (2*(A*b + a*B)*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*sin(c + d*x))/d + (2*A*(a + b*cos(c + d*x))^(3/2)*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d) - ((8*A*b + 6*a*B - 3*b*C)*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(3*d*sqrt(sec(c + d*x)))],
[(a + b*cos(c + d*x))^(3/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(3/2), x, 9, -(((a + b)*(8*a*A - 4*b*B - 5*a*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(4*d*sqrt(a + b*cos(c + d*x)))) + ((4*a^2*A + 8*a*A*b - 4*A*b^2 + 4*a^2*B - 8*a*b*B - 4*a^2*C + a*b*C - 2*b^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(2*d*sqrt(a + b*cos(c + d*x))) + ((8*A*b^2 + 12*a*b*B + 3*a^2*C + 4*b^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(2*d*sqrt(a + b*cos(c + d*x))) - (b*(4*A - C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(2*d*sqrt(sec(c + d*x))) + (2*A*(a + b*cos(c + d*x))^(3/2)*sqrt(sec(c + d*x))*sin(c + d*x))/d - ((8*a*A - 4*b*B - 5*a*C)*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(4*d*sqrt(sec(c + d*x)))],
[(a + b*cos(c + d*x))^(3/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(1/2), x, 9, ((a + b)*(24*A*b^2 + 30*a*b*B + 3*a^2*C + 16*b^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(24*b*d*sqrt(a + b*cos(c + d*x))) + ((24*a^2*A - 48*a*A*b - 24*a^2*B + 6*a*b*B - 12*b^2*B + 7*a^2*C - 26*a*b*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(12*d*sqrt(a + b*cos(c + d*x))) + ((24*a*A*b^2 + 6*a^2*b*B + 8*b^3*B - a^3*C + 12*a*b^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(4*b*d*sqrt(a + b*cos(c + d*x))) + ((2*b*B + a*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(4*d*sqrt(sec(c + d*x))) + (C*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(3*d*sqrt(sec(c + d*x))) + ((24*A*b^2 + 30*a*b*B + 3*a^2*C + 16*b^2*C)*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(24*b*d*sqrt(sec(c + d*x)))],
[((a + b*cos(c + d*x))^(3/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2))/sec(c + d*x)^(1/2), x, 10, ((a + b)*(240*a*A*b^2 + 24*a^2*b*B + 128*b^3*B - 9*a^3*C + 156*a*b^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(192*b^2*d*sqrt(a + b*cos(c + d*x))) - ((192*a^2*A*b - 48*a*A*b^2 + 96*A*b^3 - 56*a^2*b*B + 208*a*b^2*B - 3*a^3*C + 114*a^2*b*C - 36*a*b^2*C + 72*b^3*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(96*b*d*sqrt(a + b*cos(c + d*x))) + ((48*a^2*A*b^2 + 64*A*b^4 - 8*a^3*b*B + 96*a*b^3*B + 3*a^4*C + 24*a^2*b^2*C + 48*b^4*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(32*b^2*d*sqrt(a + b*cos(c + d*x))) + ((8*b*B + 3*a*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(24*d*sec(c + d*x)^(3/2)) + (C*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(4*d*sec(c + d*x)^(3/2)) + ((48*A*b^2 + 56*a*b*B + 3*a^2*C + 36*b^2*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(96*b*d*sqrt(sec(c + d*x))) + ((240*a*A*b^2 + 24*a^2*b*B + 128*b^3*B - 9*a^3*C + 156*a*b^2*C)*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(192*b^2*d*sqrt(sec(c + d*x)))],


[(a + b*cos(c + d*x))^(5/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(13/2), x, 10, -((1/(3465*a^3*d*sqrt(a + b*cos(c + d*x))))*(2*(a + b)*(3705*a^4*A*b + 255*a^2*A*b^3 + 40*A*b^5 + 1617*a^5*B + 3069*a^3*b^2*B - 110*a*b^4*B + 4785*a^4*b*C + 495*a^2*b^3*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))) + (1/(3465*a^2*d*sqrt(a + b*cos(c + d*x))))*(2*(675*a^5*A + 3705*a^4*A*b + 3315*a^3*A*b^2 + 255*a^2*A*b^3 + 10*a*A*b^4 + 40*A*b^5 + 1617*a^5*B + 2871*a^4*b*B + 3069*a^3*b^2*B + 1705*a^2*b^3*B - 110*a*b^4*B + 825*a^5*C + 4785*a^4*b*C + 4455*a^3*b^2*C + 495*a^2*b^3*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x))) + (2*(675*a^4*A + 1025*a^2*A*b^2 - 20*A*b^4 + 1793*a^3*b*B + 55*a*b^3*B + 825*a^4*C + 1485*a^2*b^2*C)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(3465*a^2*d) + (2*(1145*a^2*A*b + 15*A*b^3 + 539*a^3*B + 825*a*b^2*B + 1485*a^2*b*C)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(5/2)*sin(c + d*x))/(3465*a*d) + (2*(27*a^2*A + 5*A*b^2 + 44*a*b*B + 33*a^2*C)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(7/2)*sin(c + d*x))/(231*d) + (2*(5*A*b + 11*a*B)*(a + b*cos(c + d*x))^(3/2)*sec(c + d*x)^(9/2)*sin(c + d*x))/(99*d) + (2*A*(a + b*cos(c + d*x))^(5/2)*sec(c + d*x)^(11/2)*sin(c + d*x))/(11*d) + (1/(3465*a^3*d))*(2*(3705*a^4*A*b + 255*a^2*A*b^3 + 40*A*b^5 + 1617*a^5*B + 3069*a^3*b^2*B - 110*a*b^4*B + 4785*a^4*b*C + 495*a^2*b^3*C)*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*tan((1/2)*(c + d*x)))],
[(a + b*cos(c + d*x))^(5/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(11/2), x, 9, -((2*(a + b)*(147*a^4*A + 279*a^2*A*b^2 - 10*A*b^4 + 435*a^3*b*B + 45*a*b^3*B + 189*a^4*C + 483*a^2*b^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(315*a^2*d*sqrt(a + b*cos(c + d*x)))) + (1/(315*a*d*sqrt(a + b*cos(c + d*x))))*(2*(147*a^4*A + 261*a^3*A*b + 279*a^2*A*b^2 + 155*a*A*b^3 - 10*A*b^4 + 75*a^4*B + 435*a^3*b*B + 405*a^2*b^2*B + 45*a*b^3*B + 189*a^4*C + 357*a^3*b*C + 483*a^2*b^2*C + 315*a*b^3*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x))) + (2*(163*a^2*A*b + 5*A*b^3 + 75*a^3*B + 135*a*b^2*B + 231*a^2*b*C)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(315*a*d) + (2*(49*a^2*A + 15*A*b^2 + 90*a*b*B + 63*a^2*C)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(5/2)*sin(c + d*x))/(315*d) + (2*(5*A*b + 9*a*B)*(a + b*cos(c + d*x))^(3/2)*sec(c + d*x)^(7/2)*sin(c + d*x))/(63*d) + (2*A*(a + b*cos(c + d*x))^(5/2)*sec(c + d*x)^(9/2)*sin(c + d*x))/(9*d) + (2*(147*a^4*A + 279*a^2*A*b^2 - 10*A*b^4 + 435*a^3*b*B + 45*a*b^3*B + 189*a^4*C + 483*a^2*b^2*C)*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*tan((1/2)*(c + d*x)))/(315*a^2*d)],
[(a + b*cos(c + d*x))^(5/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(9/2), x, 10, -((2*(a + b)*(145*a^2*A*b + 15*A*b^3 + 63*a^3*B + 161*a*b^2*B + 245*a^2*b*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(105*a*d*sqrt(a + b*cos(c + d*x)))) + (1/(105*d*sqrt(a + b*cos(c + d*x))))*(2*(25*a^3*A + 145*a^2*A*b + 135*a*A*b^2 + 15*A*b^3 + 63*a^3*B + 119*a^2*b*B + 161*a*b^2*B + 105*b^3*B + 35*a^3*C + 245*a^2*b*C + 315*a*b^2*C - 105*b^3*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x))) + (4*b^3*C*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*cos(c + d*x))) + (2*(25*a^2*A + 15*A*b^2 + 56*a*b*B + 35*a^2*C)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(105*d) + (2*(5*A*b + 7*a*B)*(a + b*cos(c + d*x))^(3/2)*sec(c + d*x)^(5/2)*sin(c + d*x))/(35*d) + (2*A*(a + b*cos(c + d*x))^(5/2)*sec(c + d*x)^(7/2)*sin(c + d*x))/(7*d) + (2*(145*a^2*A*b + 15*A*b^3 + 63*a^3*B + 161*a*b^2*B + 245*a^2*b*C)*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*tan((1/2)*(c + d*x)))/(105*a*d)],
[(a + b*cos(c + d*x))^(5/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(7/2), x, 10, -(((a + b)*(18*a^2*A + 46*A*b^2 + 70*a*b*B + 30*a^2*C - 15*b^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(15*d*sqrt(a + b*cos(c + d*x)))) + (2*(9*a^3*A + 17*a^2*A*b + 23*a*A*b^2 + 15*A*b^3 + 5*a^3*B + 35*a^2*b*B + 45*a*b^2*B - 15*b^3*B + 15*a^3*C + 45*a^2*b*C - 45*a*b^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(15*d*sqrt(a + b*cos(c + d*x))) + (2*b^2*(2*b*B + 5*a*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*cos(c + d*x))) + (2*(3*a^2*A + 5*A*b^2 + 10*a*b*B + 5*a^2*C)*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d) + (2*(A*b + a*B)*(a + b*cos(c + d*x))^(3/2)*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d) + (2*A*(a + b*cos(c + d*x))^(5/2)*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d) - ((18*a^2*A + 46*A*b^2 + 70*a*b*B + 30*a^2*C - 15*b^2*C)*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(15*d*sqrt(sec(c + d*x)))],
[(a + b*cos(c + d*x))^(5/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(5/2), x, 10, -(((a + b)*(56*a*A*b + 24*a^2*B - 12*b^2*B - 27*a*b*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(12*d*sqrt(a + b*cos(c + d*x)))) + ((4*a^3*A + 28*a^2*A*b + 36*a*A*b^2 - 12*A*b^3 + 12*a^3*B + 36*a^2*b*B - 36*a*b^2*B + 12*a^3*C - 36*a^2*b*C + 3*a*b^2*C - 6*b^3*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(6*d*sqrt(a + b*cos(c + d*x))) + (b*(8*A*b^2 + 20*a*b*B + 15*a^2*C + 4*b^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(2*d*sqrt(a + b*cos(c + d*x))) - (b*(8*A*b + 4*a*B - b*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(2*d*sqrt(sec(c + d*x))) + (2*(5*A*b + 3*a*B)*(a + b*cos(c + d*x))^(3/2)*sqrt(sec(c + d*x))*sin(c + d*x))/(3*d) + (2*A*(a + b*cos(c + d*x))^(5/2)*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d) - ((56*a*A*b + 24*a^2*B - 12*b^2*B - 27*a*b*C)*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(12*d*sqrt(sec(c + d*x)))],
[(a + b*cos(c + d*x))^(5/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(3/2), x, 10, -(((a + b)*(48*a^2*A - 24*A*b^2 - 54*a*b*B - 33*a^2*C - 16*b^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(24*d*sqrt(a + b*cos(c + d*x)))) + ((24*a^3*A + 72*a^2*A*b - 72*a*A*b^2 + 24*a^3*B - 72*a^2*b*B + 6*a*b^2*B - 12*b^3*B - 24*a^3*C + 13*a^2*b*C - 38*a*b^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(12*d*sqrt(a + b*cos(c + d*x))) + ((40*a*A*b^2 + 30*a^2*b*B + 8*b^3*B + 5*a^3*C + 20*a*b^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(4*d*sqrt(a + b*cos(c + d*x))) - (b*(8*a*A - 2*b*B - 3*a*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(4*d*sqrt(sec(c + d*x))) - (b*(6*A - C)*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(3*d*sqrt(sec(c + d*x))) + (2*A*(a + b*cos(c + d*x))^(5/2)*sqrt(sec(c + d*x))*sin(c + d*x))/d - ((48*a^2*A - 24*A*b^2 - 54*a*b*B - 33*a^2*C - 16*b^2*C)*sqrt(a + b*cos(c + d*x))*tan((1/2)*(c + d*x)))/(24*d*sqrt(sec(c + d*x)))],
[(a + b*cos(c + d*x))^(5/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(1/2), x, 10, ((a + b)*(432*a*A*b^2 + 264*a^2*b*B + 128*b^3*B + 15*a^3*C + 284*a*b^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(192*b*d*sqrt(a + b*cos(c + d*x))) + ((192*a^3*A - 576*a^2*A*b + 48*a*A*b^2 - 96*A*b^3 - 192*a^3*B + 104*a^2*b*B - 304*a*b^2*B + 59*a^3*C - 322*a^2*b*C + 36*a*b^2*C - 72*b^3*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(96*d*sqrt(a + b*cos(c + d*x))) + ((240*a^2*A*b^2 + 64*A*b^4 + 40*a^3*b*B + 160*a*b^3*B - 5*a^4*C + 120*a^2*b^2*C + 48*b^4*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(32*b*d*sqrt(a + b*cos(c + d*x))) + ((16*A*b^2 + 24*a*b*B + 5*a^2*C + 12*b^2*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(32*d*sqrt(sec(c + d*x))) + ((8*b*B + 5*a*C)*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(24*d*sqrt(sec(c + d*x))) + (C*(a + b*cos(c + d*x))^(5/2)*sin(c + d*x))/(4*d*sqrt(sec(c + d*x))) + ((432*a*A*b^2 + 264*a^2*b*B + 128*b^3*B + 15*a^3*C + 284*a*b^2*C)*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(192*b*d*sqrt(sec(c + d*x)))],
[((a + b*cos(c + d*x))^(5/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2))/sec(c + d*x)^(1/2), x, 11, ((a + b)*(2640*a^2*A*b^2 + 1280*A*b^4 + 150*a^3*b*B + 2840*a*b^3*B - 45*a^4*C + 1692*a^2*b^2*C + 1024*b^4*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(1920*b^2*d*sqrt(a + b*cos(c + d*x))) - ((1920*a^3*A*b - 1040*a^2*A*b^2 + 3040*a*A*b^3 - 590*a^3*b*B + 3220*a^2*b^2*B - 360*a*b^3*B + 720*b^4*B - 15*a^4*C + 1146*a^3*b*C - 772*a^2*b^2*C + 2312*a*b^3*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(960*b*d*sqrt(a + b*cos(c + d*x))) + ((80*a^3*A*b^2 + 320*a*A*b^4 - 10*a^4*b*B + 240*a^2*b^3*B + 96*b^5*B + 3*a^5*C + 40*a^3*b^2*C + 240*a*b^4*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(64*b^2*d*sqrt(a + b*cos(c + d*x))) + ((80*A*b^2 + 110*a*b*B + 15*a^2*C + 64*b^2*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(240*d*sec(c + d*x)^(3/2)) + ((2*b*B + a*C)*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(8*d*sec(c + d*x)^(3/2)) + (C*(a + b*cos(c + d*x))^(5/2)*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2)) + ((1040*a*A*b^2 + 590*a^2*b*B + 360*b^3*B + 15*a^3*C + 772*a*b^2*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(960*b*d*sqrt(sec(c + d*x))) + ((2640*a^2*A*b^2 + 1280*A*b^4 + 150*a^3*b*B + 2840*a*b^3*B - 45*a^4*C + 1692*a^2*b^2*C + 1024*b^4*C)*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(1920*b^2*d*sqrt(sec(c + d*x)))],


# ::Subsubsection::Closed:: 
#n<0


[((A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(9/2))/sqrt(a + b*cos(c + d*x)), x, 8, (2*(a + b)*(44*a^2*A*b + 48*A*b^3 - 63*a^3*B - 56*a*b^2*B + 70*a^2*b*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(105*a^4*d*sqrt(a + b*cos(c + d*x))) + (2*(25*a^3*A - 44*a^2*A*b - 12*a*A*b^2 - 48*A*b^3 + 63*a^3*B + 14*a^2*b*B + 56*a*b^2*B + 35*a^3*C - 70*a^2*b*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(105*a^3*d*sqrt(a + b*cos(c + d*x))) + (2*(25*a^2*A + 24*A*b^2 - 28*a*b*B + 35*a^2*C)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(105*a^3*d) - (2*(6*A*b - 7*a*B)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(5/2)*sin(c + d*x))/(35*a^2*d) + (2*A*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(7/2)*sin(c + d*x))/(7*a*d) - (2*(44*a^2*A*b + 48*A*b^3 - 63*a^3*B - 56*a*b^2*B + 70*a^2*b*C)*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*tan((1/2)*(c + d*x)))/(105*a^4*d)],
[((A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(7/2))/sqrt(a + b*cos(c + d*x)), x, 7, -((2*(a + b)*(9*a^2*A + 8*A*b^2 - 10*a*b*B + 15*a^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(15*a^3*d*sqrt(a + b*cos(c + d*x)))) + (2*(9*a^2*A + 2*a*A*b + 8*A*b^2 + 5*a^2*B - 10*a*b*B + 15*a^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(15*a^2*d*sqrt(a + b*cos(c + d*x))) - (2*(4*A*b - 5*a*B)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(15*a^2*d) + (2*A*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*a*d) + (2*(9*a^2*A + 8*A*b^2 - 10*a*b*B + 15*a^2*C)*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*tan((1/2)*(c + d*x)))/(15*a^3*d)],
[((A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(5/2))/sqrt(a + b*cos(c + d*x)), x, 6, (2*(a + b)*(2*A*b - 3*a*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(3*a^2*d*sqrt(a + b*cos(c + d*x))) + (2*(a*A - 2*A*b + 3*a*B + 3*a*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(3*a*d*sqrt(a + b*cos(c + d*x))) + (2*A*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*a*d) - (2*(2*A*b - 3*a*B)*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*tan((1/2)*(c + d*x)))/(3*a^2*d)],
[((A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(3/2))/sqrt(a + b*cos(c + d*x)), x, 7, (-2*A*(a + b)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(a*d*sqrt(a + b*cos(c + d*x))) + (2*(A + B - C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*cos(c + d*x))) + (4*C*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*cos(c + d*x))) + (2*A*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*tan((c + d*x)/2))/(a*d)],
[((A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(1/2))/sqrt(a + b*cos(c + d*x)), x, 7, ((a + b)*C*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(b*d*sqrt(a + b*cos(c + d*x))) + (2*(A - B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*cos(c + d*x))) + (2*(2*b*B - a*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(b*d*sqrt(a + b*cos(c + d*x))) + (C*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(b*d*sqrt(sec(c + d*x)))],
[(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(1/2)), x, 8, ((a + b)*(4*b*B - 3*a*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(4*b^2*d*sqrt(a + b*cos(c + d*x))) - ((4*A*b - a*C + 2*b*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(2*b*d*sqrt(a + b*cos(c + d*x))) + ((8*A*b^2 - 4*a*b*B + 3*a^2*C + 4*b^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(2*b^2*d*sqrt(a + b*cos(c + d*x))) + (C*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(2*b*d*sqrt(sec(c + d*x))) + ((4*b*B - 3*a*C)*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(4*b^2*d*sqrt(sec(c + d*x)))],
[(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)), x, 9, ((a + b)*(24*A*b^2 - 18*a*b*B + 15*a^2*C + 16*b^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(24*b^3*d*sqrt(a + b*cos(c + d*x))) + ((6*a*b*B - 12*b^2*B - 5*a^2*C - 2*a*b*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(12*b^2*d*sqrt(a + b*cos(c + d*x))) - ((8*a*A*b^2 - 6*a^2*b*B - 8*b^3*B + 5*a^3*C + 4*a*b^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(4*b^3*d*sqrt(a + b*cos(c + d*x))) + (C*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*b*d*sec(c + d*x)^(3/2)) + ((6*b*B - 5*a*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(12*b^2*d*sqrt(sec(c + d*x))) + ((24*A*b^2 - 18*a*b*B + 15*a^2*C + 16*b^2*C)*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(24*b^3*d*sqrt(sec(c + d*x)))],

[((a*A + (A*b + a*B)*cos(c + d*x) + b*B*cos(c + d*x)^2)*sqrt(sec(c + d*x)))/sqrt(a + b*cos(c + d*x)), x, 7, ((a + b)*B*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*cos(c + d*x))) + (2*(a*A - A*b - a*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*cos(c + d*x))) + (2*(2*A*b + a*B)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*cos(c + d*x))) + (B*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(d*sqrt(sec(c + d*x)))],


[((A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(7/2))/(a + b*cos(c + d*x))^(3/2), x, 8, -((2*(9*a^4*A + 24*a^2*A*b^2 - 48*A*b^4 - 25*a^3*b*B + 40*a*b^3*B + 15*a^4*C - 30*a^2*b^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(15*a^4*(a - b)*d*sqrt(a + b*cos(c + d*x)))) + (2*(9*a^3*A - 12*a^2*A*b + 36*a*A*b^2 - 48*A*b^3 + 5*a^3*B - 30*a^2*b*B + 40*a*b^2*B + 15*a^3*C - 30*a^2*b*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(15*a^3*(a - b)*d*sqrt(a + b*cos(c + d*x))) - (2*(9*a^2*A*b - 24*A*b^3 - 5*a^3*B + 20*a*b^2*B - 15*a^2*b*C)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(15*a^3*(a^2 - b^2)*d) + (2*(A*b^2 - a*b*B + a^2*C)*sec(c + d*x)^(5/2)*sin(c + d*x))/(a*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) + (2*(a^2*A - 6*A*b^2 + 5*a*b*B - 5*a^2*C)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*a^2*(a^2 - b^2)*d) + (2*(9*a^4*A + 24*a^2*A*b^2 - 48*A*b^4 - 25*a^3*b*B + 40*a*b^3*B + 15*a^4*C - 30*a^2*b^2*C)*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*tan((1/2)*(c + d*x)))/(15*a^4*(a^2 - b^2)*d)],
[((A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(5/2))/(a + b*cos(c + d*x))^(3/2), x, 7, (2*(5*a^2*A*b - 8*A*b^3 - 3*a^3*B + 6*a*b^2*B - 3*a^2*b*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(3*a^3*(a - b)*d*sqrt(a + b*cos(c + d*x))) + (2*(a^2*A - 6*a*A*b + 8*A*b^2 + 3*a^2*B - 6*a*b*B + 3*a^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(3*a^2*(a - b)*d*sqrt(a + b*cos(c + d*x))) + (2*(A*b^2 - a*b*B + a^2*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(a*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) + (2*(a^2*A - 4*A*b^2 + 3*a*b*B - 3*a^2*C)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*a^2*(a^2 - b^2)*d) - (2*(5*a^2*A*b - 8*A*b^3 - 3*a^3*B + 6*a*b^2*B - 3*a^2*b*C)*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*tan((1/2)*(c + d*x)))/(3*a^3*(a^2 - b^2)*d)],
[((A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(3/2))/(a + b*cos(c + d*x))^(3/2), x, 6, (-2*(a^2*A - 2*A*b^2 + a*b*B - a^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(a^2*(a - b)*d*sqrt(a + b*cos(c + d*x))) + (2*(a*A - 2*A*b + a*B - a*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(a*(a - b)*d*sqrt(a + b*cos(c + d*x))) + (2*(A*b^2 - a*b*B + a^2*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(a*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) + (2*(a^2*A - 2*A*b^2 + a*b*B - a^2*C)*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*tan((c + d*x)/2))/(a^2*(a^2 - b^2)*d)],
[((A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(1/2))/(a + b*cos(c + d*x))^(3/2), x, 7, (-2*(A*b^2 - a*b*B + a^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(a*(a - b)*b*d*sqrt(a + b*cos(c + d*x))) + (2*(A - B + C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/((a - b)*d*sqrt(a + b*cos(c + d*x))) + (4*C*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(b*d*sqrt(a + b*cos(c + d*x))) - (2*(A*b^2 - a*b*B + a^2*C)*tan((c + d*x)/2))/(a*b*(a + b)*d*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x)))],
[(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/((a + b*cos(c + d*x))^(3/2)*sec(c + d*x)^(1/2)), x, 8, ((2*A*b^2 - 2*a*b*B + 3*a^2*C - b^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/((a - b)*b^2*d*sqrt(a + b*cos(c + d*x))) - (2*(A*b - b*B + a*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/((a - b)*b*d*sqrt(a + b*cos(c + d*x))) + (2*(2*b*B - 3*a*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(b^2*d*sqrt(a + b*cos(c + d*x))) - (2*(A*b^2 - a*b*B + a^2*C)*sin(c + d*x))/(b*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))) + ((2*A*b^2 - 2*a*b*B + 3*a^2*C - b^2*C)*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(b^2*(a^2 - b^2)*d*sqrt(sec(c + d*x)))],
[(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/((a + b*cos(c + d*x))^(3/2)*sec(c + d*x)^(3/2)), x, 9, -((8*a*A*b^2 - 12*a^2*b*B + 4*b^3*B + 15*a^3*C - 7*a*b^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(4*(a - b)*b^3*d*sqrt(a + b*cos(c + d*x))) + ((4*A*b^2 - 4*a*b*B + 5*a^2*C - 3*a*b*C + 2*b^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(2*(a - b)*b^2*d*sqrt(a + b*cos(c + d*x))) + ((8*A*b^2 - 12*a*b*B + 15*a^2*C + 4*b^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(2*b^3*d*sqrt(a + b*cos(c + d*x))) - (2*(A*b^2 - a*b*B + a^2*C)*sin(c + d*x))/(b*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)) + ((4*A*b^2 - 4*a*b*B + 5*a^2*C - b^2*C)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(2*b^2*(a^2 - b^2)*d*sqrt(sec(c + d*x))) - ((8*a*A*b^2 - 12*a^2*b*B + 4*b^3*B + 15*a^3*C - 7*a*b^2*C)*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(4*b^3*(a^2 - b^2)*d*sqrt(sec(c + d*x)))],


[((A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(5/2))/(a + b*cos(c + d*x))^(5/2), x, 8, (2*(a + b)*(8*a^4*A*b - 28*a^2*A*b^3 + 16*A*b^5 - 3*a^5*B + 15*a^3*b^2*B - 8*a*b^4*B - 6*a^4*b*C + 2*a^2*b^3*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(3*a^4*(a^2 - b^2)^2*d*sqrt(a + b*cos(c + d*x))) - (1/(9*a^3*(a^2 - b^2)^2*d*sqrt(a + b*cos(c + d*x))))*(16*((-(3/8))*a*(a^4*A + 7*a^2*A*b^2 - 4*A*b^4 - 6*a^3*b*B + 2*a*b^3*B + 3*a^4*C + a^2*b^2*C) + (3/8)*(8*a^4*A*b - 28*a^2*A*b^3 + 16*A*b^5 - 3*a^5*B + 15*a^3*b^2*B - 8*a*b^4*B - 6*a^4*b*C + 2*a^2*b^3*C))*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x))) + (2*(A*b^2 - a*b*B + a^2*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*a*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)) - (4*((-(3/2))*b^2*(a^2*A - 2*A*b^2 + a*b*B - a^2*C) - 2*a^2*(A*b^2 - a*b*B + a^2*C) + (3/2)*a^2*b*((-A)*b + a*B - b*C))*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*a^2*(a^2 - b^2)^2*d*sqrt(a + b*cos(c + d*x))) + (1/(3*a^3*(a^2 - b^2)^2*d))*(2*(a^4*A - 13*a^2*A*b^2 + 8*A*b^4 + 8*a^3*b*B - 4*a*b^3*B - 5*a^4*C + a^2*b^2*C)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x)) - (1/(3*a^4*(a^2 - b^2)^2*d))*(2*(8*a^4*A*b - 28*a^2*A*b^3 + 16*A*b^5 - 3*a^5*B + 15*a^3*b^2*B - 8*a*b^4*B - 6*a^4*b*C + 2*a^2*b^3*C)*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*tan((1/2)*(c + d*x)))],
[((A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(3/2))/(a + b*cos(c + d*x))^(5/2), x, 7, (-2*(3*a^4*A - 15*a^2*A*b^2 + 8*A*b^4 + 6*a^3*b*B - 2*a*b^3*B - 3*a^4*C - a^2*b^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(3*a^3*(a - b)^2*(a + b)*d*sqrt(a + b*cos(c + d*x))) + (2*(3*a^3*A - 9*a^2*A*b - 6*a*A*b^2 + 8*A*b^3 + 3*a^3*B + 3*a^2*b*B - 2*a*b^2*B - 3*a^3*C - a^2*b*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(3*a^2*(a - b)^2*(a + b)*d*sqrt(a + b*cos(c + d*x))) + (2*(A*b^2 - a*b*B + a^2*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(3*a*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)) + (2*(8*a^2*A*b^2 - 4*A*b^4 - 5*a^3*b*B + a*b^3*B + 2*a^4*C + 2*a^2*b^2*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(3*a^2*(a^2 - b^2)^2*d*sqrt(a + b*cos(c + d*x))) + (2*(3*a^4*A - 15*a^2*A*b^2 + 8*A*b^4 + 6*a^3*b*B - 2*a*b^3*B - 3*a^4*C - a^2*b^2*C)*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*tan((c + d*x)/2))/(3*a^3*(a^2 - b^2)^2*d)],
[((A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(1/2))/(a + b*cos(c + d*x))^(5/2), x, 6, (-2*(6*a^2*A*b - 2*A*b^3 - 3*a^3*B - a*b^2*B + 4*a^2*b*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(3*a^2*(a - b)^2*(a + b)*d*sqrt(a + b*cos(c + d*x))) + (2*(3*a^2*A + 3*a*A*b - 2*A*b^2 - 3*a^2*B - a*b*B + a^2*C + 3*a*b*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(3*a*(a - b)^2*(a + b)*d*sqrt(a + b*cos(c + d*x))) + (2*(A*b^2 - a*b*B + a^2*C)*sin(c + d*x))/(3*a*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)*sqrt(sec(c + d*x))) - (2*(6*a^2*A*b - 2*A*b^3 - 3*a^3*B - a*b^2*B + 4*a^2*b*C)*tan((c + d*x)/2))/(3*a^2*(a - b)*(a + b)^2*d*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x)))],
[(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/((a + b*cos(c + d*x))^(5/2)*sec(c + d*x)^(1/2)), x, 8, (2*(3*a^2*A*b^2 + A*b^4 - 4*a*b^3*B - 3*a^4*C + 7*a^2*b^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(3*a*(a - b)^2*b^2*(a + b)*d*sqrt(a + b*cos(c + d*x))) - (2*(3*a*A*b + A*b^2 - a*b*B - 3*b^2*B - 2*a^2*C + 3*a*b*C + 3*b^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(3*(a - b)^2*b*(a + b)*d*sqrt(a + b*cos(c + d*x))) + (4*C*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(b^2*d*sqrt(a + b*cos(c + d*x))) - (2*(A*b^2 - a*b*B + a^2*C)*sin(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)*sqrt(sec(c + d*x))) + (2*(3*a^2*A*b^2 + A*b^4 - 4*a*b^3*B - 3*a^4*C + 7*a^2*b^2*C)*tan((c + d*x)/2))/(3*a*(a - b)*b^2*(a + b)^2*d*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x)))],
[(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/((a + b*cos(c + d*x))^(5/2)*sec(c + d*x)^(3/2)), x, 9, -((8*A*b^4 + 6*a^3*b*B - 14*a*b^3*B - 15*a^4*C + 26*a^2*b^2*C - 3*b^4*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(3*(a - b)^2*b^3*(a + b)*d*sqrt(a + b*cos(c + d*x))) + (2*(a*A*b^2 + 3*A*b^3 + 2*a^2*b*B - 3*a*b^2*B - 3*b^3*B - 5*a^3*C + 3*a^2*b*C + 6*a*b^2*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(3*(a - b)^2*b^2*(a + b)*d*sqrt(a + b*cos(c + d*x))) + (2*(2*b*B - 5*a*C)*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))*sqrt((a + b*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((c + d*x)/2)), -((a - b)/(a + b)))*sqrt(sec(c + d*x)))/(b^3*d*sqrt(a + b*cos(c + d*x))) - (2*(A*b^2 - a*b*B + a^2*C)*sin(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)*sec(c + d*x)^(3/2)) + (2*(a^2*A*b^2 + 3*A*b^4 + 2*a^3*b*B - 6*a*b^3*B - 5*a^4*C + 9*a^2*b^2*C)*sin(c + d*x))/(3*b^2*(a^2 - b^2)^2*d*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))) - ((8*A*b^4 + 6*a^3*b*B - 14*a*b^3*B - 15*a^4*C + 26*a^2*b^2*C - 3*b^4*C)*sqrt(a + b*cos(c + d*x))*tan((c + d*x)/2))/(3*b^3*(a^2 - b^2)^2*d*sqrt(sec(c + d*x)))]
]:
