lst:=[
# ::Package:: 

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


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


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


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


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


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


[sec(c + d*x)^3*(a + b*sec(c + d*x))^4, x, 7, ((8*a^4 + 36*a^2*b^2 + 5*b^4)*arctanh(sin(c + d*x)))/(16*d) + (8*a*b*(5*a^2 + 4*b^2)*tan(c + d*x))/(15*d) + ((8*a^4 + 36*a^2*b^2 + 5*b^4)*sec(c + d*x)*tan(c + d*x))/(16*d) + (4*a*b*(5*a^2 + 4*b^2)*sec(c + d*x)^2*tan(c + d*x))/(15*d) + (b^2*(104*a^2 + 25*b^2)*sec(c + d*x)^3*tan(c + d*x))/(120*d) + (7*a*b^2*sec(c + d*x)^3*(a + b*sec(c + d*x))*tan(c + d*x))/(15*d) + (b^2*sec(c + d*x)^3*(a + b*sec(c + d*x))^2*tan(c + d*x))/(6*d)],
[sec(c + d*x)^2*(a + b*sec(c + d*x))^4, x, 7, (a*b*(4*a^2 + 3*b^2)*arctanh(sin(c + d*x)))/(2*d) + ((6*a^4 + 83*a^2*b^2 + 16*b^4)*tan(c + d*x))/(30*d) + (a*(6*a^2 + 29*b^2)*(a + b*sec(c + d*x))*tan(c + d*x))/(30*d) + ((3*a^2 + 4*b^2)*(a + b*sec(c + d*x))^2*tan(c + d*x))/(15*d) + (a*(a + b*sec(c + d*x))^3*tan(c + d*x))/(5*d) + ((a + b*sec(c + d*x))^4*tan(c + d*x))/(5*d)],
[sec(c + d*x)^1*(a + b*sec(c + d*x))^4, x, 6, ((8*a^4 + 24*a^2*b^2 + 3*b^4)*arctanh(sin(c + d*x)))/(8*d) + (5*a*b*(10*a^2 + 11*b^2)*tan(c + d*x))/(24*d) + (b*(26*a^2 + 9*b^2)*(a + b*sec(c + d*x))*tan(c + d*x))/(24*d) + (7*a*b*(a + b*sec(c + d*x))^2*tan(c + d*x))/(12*d) + (b*(a + b*sec(c + d*x))^3*tan(c + d*x))/(4*d)],
[sec(c + d*x)^0*(a + b*sec(c + d*x))^4, x, 5, a^4*x + (2*a*b*(2*a^2 + b^2)*arctanh(sin(c + d*x)))/d + (b^2*(13*a^2 + 2*b^2)*tan(c + d*x))/(3*d) + (4*a*b^2*(a + b*sec(c + d*x))*tan(c + d*x))/(3*d) + (b^2*(a + b*sec(c + d*x))^2*tan(c + d*x))/(3*d)],
[cos(c + d*x)^1*(a + b*sec(c + d*x))^4, x, 5, 4*a^3*b*x + (b^2*(12*a^2 + b^2)*arctanh(sin(c + d*x)))/(2*d) + (a*(2*a^2 - b^2)*(a + b*sec(c + d*x))*sin(c + d*x))/(2*d) + (b^2*(a + b*sec(c + d*x))^2*sin(c + d*x))/(2*d) - (a*b*(2*a^2 - 7*b^2)*tan(c + d*x))/(2*d)],
[cos(c + d*x)^2*(a + b*sec(c + d*x))^4, x, 5, (a^4*x)/2 + 6*a^2*b^2*x + (4*a*b^3*arctanh(sin(c + d*x)))/d + (3*a^2*b*(a + b*sec(c + d*x))*sin(c + d*x))/d + (a^2*cos(c + d*x)*(a + b*sec(c + d*x))^2*sin(c + d*x))/(2*d) - (b^2*(7*a^2 - 2*b^2)*tan(c + d*x))/(2*d)],
[cos(c + d*x)^3*(a + b*sec(c + d*x))^4, x, 5, 2*a*b*(a^2 + 2*b^2)*x + (b^4*arctanh(sin(c + d*x)))/d + (a^2*(2*a^2 + 13*b^2)*sin(c + d*x))/(3*d) + (4*a^2*b*(b + a*cos(c + d*x))*sin(c + d*x))/(3*d) + (a^2*(b + a*cos(c + d*x))^2*sin(c + d*x))/(3*d)],
[cos(c + d*x)^4*(a + b*sec(c + d*x))^4, x, 5, (1/8)*(3*a^4 + 24*a^2*b^2 + 8*b^4)*x + (4*a*b*(2*a^2 + 3*b^2)*sin(c + d*x))/(3*d) + (a^2*(9*a^2 + 46*b^2)*cos(c + d*x)*sin(c + d*x))/(24*d) + (5*a^2*b*cos(c + d*x)^2*(a + b*sec(c + d*x))*sin(c + d*x))/(6*d) + (a^2*cos(c + d*x)^3*(a + b*sec(c + d*x))^2*sin(c + d*x))/(4*d)],
[cos(c + d*x)^5*(a + b*sec(c + d*x))^4, x, 6, (1/2)*a*b*(3*a^2 + 4*b^2)*x + ((8*a^4 + 60*a^2*b^2 + 15*b^4)*sin(c + d*x))/(15*d) + (a*b*(3*a^2 + 4*b^2)*cos(c + d*x)*sin(c + d*x))/(2*d) + (2*a^2*(2*a^2 + 9*b^2)*cos(c + d*x)^2*sin(c + d*x))/(15*d) + (3*a^2*b*cos(c + d*x)^3*(a + b*sec(c + d*x))*sin(c + d*x))/(5*d) + (a^2*cos(c + d*x)^4*(a + b*sec(c + d*x))^2*sin(c + d*x))/(5*d)],
[cos(c + d*x)^6*(a + b*sec(c + d*x))^4, x, 7, (1/16)*(5*a^4 + 36*a^2*b^2 + 8*b^4)*x + (8*a*b*(4*a^2 + 5*b^2)*sin(c + d*x))/(15*d) + ((5*a^4 + 36*a^2*b^2 + 8*b^4)*cos(c + d*x)*sin(c + d*x))/(16*d) + (4*a*b*(4*a^2 + 5*b^2)*cos(c + d*x)^2*sin(c + d*x))/(15*d) + (a^2*(25*a^2 + 104*b^2)*cos(c + d*x)^3*sin(c + d*x))/(120*d) + (7*a^2*b*cos(c + d*x)^4*(a + b*sec(c + d*x))*sin(c + d*x))/(15*d) + (a^2*cos(c + d*x)^5*(a + b*sec(c + d*x))^2*sin(c + d*x))/(6*d)],


[(a + b*sec(c + d*x))^5, x, 6, a^5*x + (b*(40*a^4 + 40*a^2*b^2 + 3*b^4)*arctanh(sin(c + d*x)))/(8*d) + (a*b^2*(154*a^2 + 71*b^2)*tan(c + d*x))/(24*d) + (b^2*(58*a^2 + 9*b^2)*(a + b*sec(c + d*x))*tan(c + d*x))/(24*d) + (11*a*b^2*(a + b*sec(c + d*x))^2*tan(c + d*x))/(12*d) + (b^2*(a + b*sec(c + d*x))^3*tan(c + d*x))/(4*d)],


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


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


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


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


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


[1/(3 + 5*sec(c + d*x)), x, 2, -(x/12) + (5*arctan(sin(c + d*x)/(3 + cos(c + d*x))))/(6*d)],
[1/(3 + 5*sec(c + d*x))^2, x, 3, (29*x)/576 + (35*arctan(sin(c + d*x)/(3 + cos(c + d*x))))/(288*d) - (25*tan(c + d*x))/(48*d*(3 + 5*sec(c + d*x)))],
[1/(3 + 5*sec(c + d*x))^3, x, 4, -((1007*x)/55296) + (3055*arctan(sin(c + d*x)/(3 + cos(c + d*x))))/(27648*d) - (25*tan(c + d*x))/(96*d*(3 + 5*sec(c + d*x))^2) - (125*tan(c + d*x))/(4608*d*(3 + 5*sec(c + d*x)))],
[1/(3 + 5*sec(c + d*x))^4, x, 5, (21553*x)/2654208 + (11215*arctan(sin(c + d*x)/(3 + cos(c + d*x))))/(1327104*d) - (25*tan(c + d*x))/(144*d*(3 + 5*sec(c + d*x))^3) - (25*tan(c + d*x))/(4608*d*(3 + 5*sec(c + d*x))^2) - (16925*tan(c + d*x))/(221184*d*(3 + 5*sec(c + d*x)))],

[1/(5 + 3*sec(c + d*x)), x, 2, x/5 - (3*arctanh((1/2)*tan((1/2)*(c + d*x))))/(10*d)],
[1/(5 + 3*sec(c + d*x))^2, x, 3, x/25 - (123*arctanh((1/2)*tan((1/2)*(c + d*x))))/(800*d) + (9*tan(c + d*x))/(80*d*(5 + 3*sec(c + d*x)))],
[1/(5 + 3*sec(c + d*x))^3, x, 4, x/125 - (8361*arctanh((1/2)*tan((1/2)*(c + d*x))))/(128000*d) + (9*tan(c + d*x))/(160*d*(5 + 3*sec(c + d*x))^2) + (963*tan(c + d*x))/(12800*d*(5 + 3*sec(c + d*x)))],
[1/(5 + 3*sec(c + d*x))^4, x, 5, x/625 - (278151*arctanh((1/2)*tan((1/2)*(c + d*x))))/(10240000*d) + (3*tan(c + d*x))/(80*d*(5 + 3*sec(c + d*x))^3) + (519*tan(c + d*x))/(12800*d*(5 + 3*sec(c + d*x))^2) + (38733*tan(c + d*x))/(1024000*d*(5 + 3*sec(c + d*x)))],


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


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


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


[sec(c + d*x)^4*(a + b*sec(c + d*x))^(3/2), x, 9, -((2*(a + b)*(8*a^4 + 33*a^2*b^2 + 147*b^4)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(315*b^3*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))) + (2*(8*a^4 + 2*a^3*b + 33*a^2*b^2 + 186*a*b^3 + 147*b^4)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(315*b^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(8*a^4 + 33*a^2*b^2 + 147*b^4)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(315*b^3*d) - (8*a*(a^2 - 22*b^2)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(315*b^2*d) + (2*(3*a^2 + 49*b^2)*sec(c + d*x)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(315*b*d) + (20*a*sec(c + d*x)^2*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(63*d) + (2*b*sec(c + d*x)^3*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(9*d)],
[sec(c + d*x)^3*(a + b*sec(c + d*x))^(3/2), x, 8, (4*a*(a + b)*(3*a^2 - 41*b^2)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(105*b^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*(6*a^3 - 51*a^2*b - 82*a*b^2 - 25*b^3)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(105*b*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (4*a*(3*a^2 - 41*b^2)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(105*b^2*d) + (2*(3*a^2 + 25*b^2)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(105*b*d) + (16*a*sec(c + d*x)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(35*d) + (2*b*sec(c + d*x)^2*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(7*d)],
[sec(c + d*x)^2*(a + b*sec(c + d*x))^(3/2), x, 7, -((2*(a + b)*(a^2 + 3*b^2)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(5*b*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))) + (2*(a^2 + 4*a*b + 3*b^2)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(a^2 + 3*b^2)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(5*b*d) + (2*a*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(5*d) + (2*(a + b*sec(c + d*x))^(3/2)*tan(c + d*x))/(5*d)],
[sec(c + d*x)^1*(a + b*sec(c + d*x))^(3/2), x, 6, -((8*a*(a + b)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))) + (2*(3*a^2 + 4*a*b + b^2)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (8*a*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(3*d) + (2*b*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(3*d)],
[sec(c + d*x)^0*(a + b*sec(c + d*x))^(3/2), x, 7, -((2*b*(a + b)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))) - (2*(a^2 - 2*a*b - b^2)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (4*a^2*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*b*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/d],
[cos(c + d*x)^1*(a + b*sec(c + d*x))^(3/2), x, 8, (a*(a + b)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + ((-2*a^2*b + 2*a*b*(-3*a + 2*b))*sqrt(1 + cos(c + d*x))*sqrt((b + a*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*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (6*a*b*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (a*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/d],
[cos(c + d*x)^2*(a + b*sec(c + d*x))^(3/2), x, 8, (5*b*(a + b)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - ((2*a^2 - a*b + 4*b^2)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + ((4*a^2 + 3*b^2)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (a*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(2*d) + (5*b*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(4*d)],


[sec(c + d*x)^4*(a + b*sec(c + d*x))^(5/2), x, 10, -((2*a*(a + b)*(8*a^4 + 51*a^2*b^2 + 741*b^4)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(693*b^3*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))) + (2*(8*a^5 + 2*a^4*b + 51*a^3*b^2 + 663*a^2*b^3 + 741*a*b^4 + 135*b^5)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(693*b^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*a*(8*a^4 + 51*a^2*b^2 + 741*b^4)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(693*b^3*d) - (2*(4*a^4 - 205*a^2*b^2 - 135*b^4)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(693*b^2*d) + (2*a*(3*a^2 + 229*b^2)*sec(c + d*x)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(693*b*d) + (2*(113*a^2 + 81*b^2)*sec(c + d*x)^2*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(693*d) + (46*a*b*sec(c + d*x)^3*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(99*d) + (2*b^2*sec(c + d*x)^4*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(11*d)],
[sec(c + d*x)^3*(a + b*sec(c + d*x))^(5/2), x, 9, (2*(a + b)*(10*a^4 - 279*a^2*b^2 - 147*b^4)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(315*b^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*(10*a^4 - 155*a^3*b - 279*a^2*b^2 - 261*a*b^3 - 147*b^4)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(315*b*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*(10*a^4 - 279*a^2*b^2 - 147*b^4)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(315*b^2*d) + (2*a*(5*a^2 + 163*b^2)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(315*b*d) + (2*(75*a^2 + 49*b^2)*sec(c + d*x)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(315*d) + (38*a*b*sec(c + d*x)^2*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(63*d) + (2*b^2*sec(c + d*x)^3*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(9*d)],
[sec(c + d*x)^2*(a + b*sec(c + d*x))^(5/2), x, 8, -((2*a*(a + b)*(3*a^2 + 29*b^2)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(21*b*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))) + (2*(3*a^3 + 27*a^2*b + 29*a*b^2 + 5*b^3)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*a*(3*a^2 + 29*b^2)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(21*b*d) + (2*(3*a^2 + 5*b^2)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(21*d) + (2*a*(a + b*sec(c + d*x))^(3/2)*tan(c + d*x))/(7*d) + (2*(a + b*sec(c + d*x))^(5/2)*tan(c + d*x))/(7*d)],
[sec(c + d*x)^1*(a + b*sec(c + d*x))^(5/2), x, 7, -((2*(a + b)*(23*a^2 + 9*b^2)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))) + (2*(15*a^3 + 23*a^2*b + 17*a*b^2 + 9*b^3)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(23*a^2 + 9*b^2)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(15*d) + (16*a*b*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(15*d) + (2*b*(a + b*sec(c + d*x))^(3/2)*tan(c + d*x))/(5*d)],
[sec(c + d*x)^0*(a + b*sec(c + d*x))^(5/2), x, 8, -((14*a*b*(a + b)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))) - (2*(3*a^3 - 9*a^2*b - 7*a*b^2 - b^3)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (4*a^3*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (14*a*b*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(3*d) + (2*b^2*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(3*d)],
[cos(c + d*x)^1*(a + b*sec(c + d*x))^(5/2), x, 8, ((a + b)*(a^2 - 2*b^2)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*b*(3*a^2 - 3*a*b - b^2)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (10*a^2*b*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*b^2*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/d + ((a^2 - 2*b^2)*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/d],
[cos(c + d*x)^2*(a + b*sec(c + d*x))^(5/2), x, 8, (9*a*b*(a + b)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - ((2*a^3 - a^2*b + 12*a*b^2 - 4*b^3)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (a*(4*a^2 + 15*b^2)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (a^2*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(2*d) + (9*a*b*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(4*d)],
[cos(c + d*x)^3*(a + b*sec(c + d*x))^(5/2), x, 9, ((a + b)*(16*a^2 + 33*b^2)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (b*(38*a^2 - 13*a*b + 24*b^2)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (5*b*(4*a^2 + b^2)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (13*a*b*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(12*d) + (a^2*cos(c + d*x)^2*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(3*d) + ((16*a^2 + 33*b^2)*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(24*d)],
[cos(c + d*x)^4*(a + b*sec(c + d*x))^(5/2), x, 10, (b*(a + b)*(284*a^2 + 15*b^2)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(192*a*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - ((72*a^3 - 36*a^2*b + 322*a*b^2 - 59*b^3)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + ((48*a^4 + 120*a^2*b^2 - 5*b^4)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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*a*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + ((36*a^2 + 59*b^2)*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(96*d) + (17*a*b*cos(c + d*x)^2*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(24*d) + (a^2*cos(c + d*x)^3*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(4*d) + (b*(284*a^2 + 15*b^2)*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(192*a*d)],


[(a + b*sec(c + d*x))^(7/2), x, 9, -((2*b*(a + b)*(58*a^2 + 9*b^2)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))) - (2*(15*a^4 - 60*a^3*b - 58*a^2*b^2 - 22*a*b^3 - 9*b^4)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (4*a^4*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*b*(58*a^2 + 9*b^2)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(15*d) + (26*a*b^2*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(15*d) + (2*b^2*(a + b*sec(c + d*x))^(3/2)*tan(c + d*x))/(5*d)],


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


[sec(c + d*x)^5/sqrt(a + b*sec(c + d*x)), x, 8, (8*a*(a + b)*(12*a^2 + 11*b^2)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(105*b^4*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*(48*a^3 + 12*a^2*b + 44*a*b^2 - 25*b^3)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(105*b^3*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (8*a*(12*a^2 + 11*b^2)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(105*b^4*d) + (2*(24*a^2 + 25*b^2)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(105*b^3*d) - (12*a*sec(c + d*x)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(35*b^2*d) + (2*sec(c + d*x)^2*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(7*b*d)],
[sec(c + d*x)^4/sqrt(a + b*sec(c + d*x)), x, 7, -((2*(a + b)*(8*a^2 + 9*b^2)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(15*b^3*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))) + (2*(8*a^2 + 2*a*b + 9*b^2)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(15*b^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(8*a^2 + 9*b^2)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(15*b^3*d) - (8*a*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(15*b^2*d) + (2*sec(c + d*x)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(5*b*d)],
[sec(c + d*x)^3/sqrt(a + b*sec(c + d*x)), x, 6, (4*a*(a + b)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(3*b^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*(2*a - b)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(3*b*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (4*a*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(3*b^2*d) + (2*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(3*b*d)],
[sec(c + d*x)^2/sqrt(a + b*sec(c + d*x)), x, 5, -((2*(a + b)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))) + (2*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(b*d)],
[sec(c + d*x)^1/sqrt(a + b*sec(c + d*x)), x, 2, (2*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))],
[sec(c + d*x)^0/sqrt(a + b*sec(c + d*x)), x, 4, -((2*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))) + (4*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))],
[cos(c + d*x)^1/sqrt(a + b*sec(c + d*x)), x, 5, ((a + b)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*b*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (cos(c + d*x)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(a*d)],
[cos(c + d*x)^2/sqrt(a + b*sec(c + d*x)), x, 8, -((3*b*(a + b)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))) - ((2*a - b)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + ((4*a^2 + 3*b^2)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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*a^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (cos(c + d*x)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(2*a*d) - (3*b*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(4*a^2*d)],


[sec(c + d*x)^5/(a + b*sec(c + d*x))^(3/2), x, 8, -((2*(16*a^4 - 8*a^2*b^2 - 3*b^4)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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 - b)*b^4*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))) + (2*(16*a^3 - 12*a^2*b + 4*a*b^2 - 3*b^3)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(5*(a - b)*b^3*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(16*a^4 - 8*a^2*b^2 - 3*b^4)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(5*b^4*(a^2 - b^2)*d) - (2*a^2*sec(c + d*x)^2*tan(c + d*x))/(b*(a^2 - b^2)*d*sqrt(a + b*sec(c + d*x))) - (2*a*(8*a^2 - 3*b^2)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(5*b^3*(a^2 - b^2)*d) + (2*(6*a^2 - b^2)*sec(c + d*x)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(5*b^2*(a^2 - b^2)*d)],
[sec(c + d*x)^4/(a + b*sec(c + d*x))^(3/2), x, 7, (2*a*(8*a^2 - 5*b^2)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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)*b^3*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*(8*a^2 - 6*a*b + b^2)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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)*b^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*a*(8*a^2 - 5*b^2)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(3*b^3*(a^2 - b^2)*d) - (2*a^2*sec(c + d*x)*tan(c + d*x))/(b*(a^2 - b^2)*d*sqrt(a + b*sec(c + d*x))) + (2*(4*a^2 - b^2)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(3*b^2*(a^2 - b^2)*d)],
[sec(c + d*x)^3/(a + b*sec(c + d*x))^(3/2), x, 6, -((2*(2*a^2 - b^2)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))) + (2*(2*a - b)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(2*a^2 - b^2)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(b^2*(a^2 - b^2)*d) - (2*a^2*tan(c + d*x))/(b*(a^2 - b^2)*d*sqrt(a + b*sec(c + d*x)))],
[sec(c + d*x)^2/(a + b*sec(c + d*x))^(3/2), x, 5, (2*a*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*a*tan((1/2)*(c + d*x)))/(b*(a + b)*d*sqrt(a + b*sec(c + d*x)))],
[sec(c + d*x)^1/(a + b*sec(c + d*x))^(3/2), x, 5, -((2*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))) + (2*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*tan((1/2)*(c + d*x)))/((a + b)*d*sqrt(a + b*sec(c + d*x)))],
[sec(c + d*x)^0/(a + b*sec(c + d*x))^(3/2), x, 7, (2*b*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (4*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*b*tan((1/2)*(c + d*x)))/(a*(a + b)*d*sqrt(a + b*sec(c + d*x)))],
[cos(c + d*x)^1/(a + b*sec(c + d*x))^(3/2), x, 8, ((a^2 - 3*b^2)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*b*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (6*b*sqrt(1 + cos(c + d*x))*sqrt((b + a*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^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*b^2*sin(c + d*x))/(a*(a^2 - b^2)*d*sqrt(a + b*sec(c + d*x))) + ((a^2 - 3*b^2)*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(a^2*(a^2 - b^2)*d)],
[cos(c + d*x)^2/(a + b*sec(c + d*x))^(3/2), x, 9, -((b*(7*a^2 - 15*b^2)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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^3*(a - b)*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))) - ((2*a^2 - 3*a*b + 5*b^2)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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^2*(a - b)*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + ((4*a^2 + 15*b^2)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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*a^3*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*b^2*cos(c + d*x)*sin(c + d*x))/(a*(a^2 - b^2)*d*sqrt(a + b*sec(c + d*x))) + ((a^2 - 5*b^2)*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(2*a^2*(a^2 - b^2)*d) - (b*(7*a^2 - 15*b^2)*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(4*a^3*(a^2 - b^2)*d)],


[sec(c + d*x)^5/(a + b*sec(c + d*x))^(5/2), x, 8, (8*a*(4*a^4 - 7*a^2*b^2 + 2*b^4)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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^4*(a + b)*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*(16*a^4 - 12*a^3*b - 16*a^2*b^2 + 9*a*b^3 - b^4)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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^3*(a + b)*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (8*a*(4*a^4 - 7*a^2*b^2 + 2*b^4)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(3*b^4*(a^2 - b^2)^2*d) - (2*a^2*sec(c + d*x)^2*tan(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*sec(c + d*x))^(3/2)) - (4*a^2*(3*a^2 - 5*b^2)*sec(c + d*x)*tan(c + d*x))/(3*b^2*(a^2 - b^2)^2*d*sqrt(a + b*sec(c + d*x))) + (2*(8*a^4 - 13*a^2*b^2 + b^4)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(3*b^3*(a^2 - b^2)^2*d)],
[sec(c + d*x)^4/(a + b*sec(c + d*x))^(5/2), x, 7, -((2*(8*a^4 - 15*a^2*b^2 + 3*b^4)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))) + (2*(8*a^3 - 6*a^2*b - 9*a*b^2 + 3*b^3)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(8*a^4 - 15*a^2*b^2 + 3*b^4)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(3*b^3*(a^2 - b^2)^2*d) - (2*a^2*sec(c + d*x)*tan(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*sec(c + d*x))^(3/2)) - (8*a^2*(a^2 - 2*b^2)*tan(c + d*x))/(3*b^2*(a^2 - b^2)^2*d*sqrt(a + b*sec(c + d*x)))],
[sec(c + d*x)^3/(a + b*sec(c + d*x))^(5/2), x, 6, (4*a*(a^2 - 3*b^2)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*(2*a^2 - 3*a*b - 3*b^2)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (4*a*(a^2 - 3*b^2)*tan((1/2)*(c + d*x)))/(3*(a - b)*b^2*(a + b)^2*d*sqrt(a + b*sec(c + d*x))) - (2*a^2*tan(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*sec(c + d*x))^(3/2))],
[sec(c + d*x)^2/(a + b*sec(c + d*x))^(5/2), x, 6, (2*(a^2 + 3*b^2)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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*(a + b)*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*(a + 3*b)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*(a^2 + 3*b^2)*tan((1/2)*(c + d*x)))/(3*(a - b)*b*(a + b)^2*d*sqrt(a + b*sec(c + d*x))) + (2*a*tan(c + d*x))/(3*(a^2 - b^2)*d*(a + b*sec(c + d*x))^(3/2))],
[sec(c + d*x)^1/(a + b*sec(c + d*x))^(5/2), x, 6, -((8*a*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))) + (2*(3*a + b)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (8*a*tan((1/2)*(c + d*x)))/(3*(a - b)*(a + b)^2*d*sqrt(a + b*sec(c + d*x))) - (2*b*tan(c + d*x))/(3*(a^2 - b^2)*d*(a + b*sec(c + d*x))^(3/2))],
[sec(c + d*x)^0/(a + b*sec(c + d*x))^(5/2), x, 8, (2*b*(7*a^2 - 3*b^2)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*(3*a^2 + 3*a*b - 2*b^2)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (4*sqrt(1 + cos(c + d*x))*sqrt((b + a*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^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*b*(7*a^2 - 3*b^2)*tan((1/2)*(c + d*x)))/(3*a^2*(a - b)*(a + b)^2*d*sqrt(a + b*sec(c + d*x))) + (2*b^2*tan(c + d*x))/(3*a*(a^2 - b^2)*d*(a + b*sec(c + d*x))^(3/2))],
[cos(c + d*x)^1/(a + b*sec(c + d*x))^(5/2), x, 9, ((3*a^4 - 26*a^2*b^2 + 15*b^4)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*b*(6*a^2 + 3*a*b - 5*b^2)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (10*b*sqrt(1 + cos(c + d*x))*sqrt((b + a*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^3*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*b^2*sin(c + d*x))/(3*a*(a^2 - b^2)*d*(a + b*sec(c + d*x))^(3/2)) + (2*b^2*(9*a^2 - 5*b^2)*sin(c + d*x))/(3*a^2*(a^2 - b^2)^2*d*sqrt(a + b*sec(c + d*x))) + ((3*a^4 - 26*a^2*b^2 + 15*b^4)*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(3*a^3*(a^2 - b^2)^2*d)],
[cos(c + d*x)^2/(a + b*sec(c + d*x))^(5/2), x, 10, -((b*(33*a^4 - 170*a^2*b^2 + 105*b^4)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(12*a^4*(a - b)^2*(a + b)*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))) - ((6*a^4 - 9*a^3*b + 33*a^2*b^2 + 21*a*b^3 - 35*b^4)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(6*a^3*(a - b)^2*(a + b)*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + ((4*a^2 + 35*b^2)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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*a^4*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*b^2*cos(c + d*x)*sin(c + d*x))/(3*a*(a^2 - b^2)*d*(a + b*sec(c + d*x))^(3/2)) + (2*b^2*(11*a^2 - 7*b^2)*cos(c + d*x)*sin(c + d*x))/(3*a^2*(a^2 - b^2)^2*d*sqrt(a + b*sec(c + d*x))) + ((3*a^4 - 54*a^2*b^2 + 35*b^4)*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(6*a^3*(a^2 - b^2)^2*d) - (b*(33*a^4 - 170*a^2*b^2 + 105*b^4)*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(12*a^4*(a^2 - b^2)^2*d)],


[1/(a + b*sec(c + d*x))^(7/2), x, 9, (2*b*(58*a^4 - 41*a^2*b^2 + 15*b^4)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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*(a - b)^3*(a + b)^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*(15*a^4 + 30*a^3*b - 17*a^2*b^2 - 6*a*b^3 + 10*b^4)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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*(a - b)^3*(a + b)^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (4*sqrt(1 + cos(c + d*x))*sqrt((b + a*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^3*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*b*(58*a^4 - 41*a^2*b^2 + 15*b^4)*tan((1/2)*(c + d*x)))/(15*a^3*(a - b)^2*(a + b)^3*d*sqrt(a + b*sec(c + d*x))) + (2*b^2*tan(c + d*x))/(5*a*(a^2 - b^2)*d*(a + b*sec(c + d*x))^(5/2)) + (2*b^2*(13*a^2 - 5*b^2)*tan(c + d*x))/(15*a^2*(a^2 - b^2)^2*d*(a + b*sec(c + d*x))^(3/2))],


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


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


[sec(c + d*x)^(5/2)*(a + b*sec(c + d*x))^2, x, 8, -((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) + (12*a*b*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d) + (2*(7*a^2 + 5*b^2)*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))/(5*d) + (2*b^2*sec(c + d*x)^(7/2)*sin(c + d*x))/(7*d)],
[sec(c + d*x)^(3/2)*(a + b*sec(c + d*x))^2, x, 7, -((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*(5*a^2 + 3*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*b^2*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d)],
[sec(c + d*x)^(1/2)*(a + b*sec(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*(3*a^2 + 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*b^2*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d)],
[sec(c + d*x)^(-1/2)*(a + b*sec(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*b^2*sqrt(sec(c + d*x))*sin(c + d*x))/d],
[sec(c + d*x)^(-3/2)*(a + b*sec(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*(a^2 + 3*b^2)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) + (2*a^2*sin(c + d*x))/(3*d*sqrt(sec(c + d*x)))],
[sec(c + d*x)^(-5/2)*(a + b*sec(c + d*x))^2, x, 6, (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*a^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)^(-7/2)*(a + b*sec(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*(5*a^2 + 7*b^2)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) + (2*a^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*(5*a^2 + 7*b^2)*sin(c + d*x))/(21*d*sqrt(sec(c + d*x)))],


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


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


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


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


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


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


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


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


[sec(c + d*x)^(3/2)*(a + b*sec(c + d*x))^(3/2), x, 11, -((5*a*(a + b)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(4*d*sqrt(a + b*sec(c + d*x)))) + (7*a*b*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(4*d*sqrt(a + b*sec(c + d*x))) + ((3*a^2 + 4*b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(4*d*sqrt(a + b*sec(c + d*x))) + (5*a*sqrt(sec(c + d*x))*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(4*d) + (b*sec(c + d*x)^(3/2)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(2*d)],
[sec(c + d*x)^(1/2)*(a + b*sec(c + d*x))^(3/2), x, 10, -((b*(a + b)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*sec(c + d*x)))) + ((2*a^2 + b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*sec(c + d*x))) + (3*a*b*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*sec(c + d*x))) + (b*sqrt(sec(c + d*x))*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/d],
[sec(c + d*x)^(-1/2)*(a + b*sec(c + d*x))^(3/2), x, 9, (2*a*(a + b)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*sec(c + d*x))) + (2*a*b*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*sec(c + d*x))) + (2*b^2*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*sec(c + d*x)))],
[sec(c + d*x)^(-3/2)*(a + b*sec(c + d*x))^(3/2), x, 7, (8*b*(a + b)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(3*d*sqrt(a + b*sec(c + d*x))) + (2*(a^2 - b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(3*d*sqrt(a + b*sec(c + d*x))) + (2*a*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(3*d*sqrt(sec(c + d*x)))],
[sec(c + d*x)^(-5/2)*(a + b*sec(c + d*x))^(3/2), x, 8, (2*(a + b)*(3*a^2 + b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(5*a*d*sqrt(a + b*sec(c + d*x))) + (2*b*(a^2 - b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(5*a*d*sqrt(a + b*sec(c + d*x))) + (2*b*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(5*d*sqrt(sec(c + d*x))) + (2*(a + b*sec(c + d*x))^(3/2)*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2))],
[sec(c + d*x)^(-7/2)*(a + b*sec(c + d*x))^(3/2), x, 9, (4*b*(a + b)*(41*a^2 - 3*b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(105*a^2*d*sqrt(a + b*sec(c + d*x))) + (2*(25*a^4 - 31*a^2*b^2 + 6*b^4)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(105*a^2*d*sqrt(a + b*sec(c + d*x))) + (2*a*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(7*d*sec(c + d*x)^(5/2)) + (16*b*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(35*d*sec(c + d*x)^(3/2)) + (2*(25*a^2 + 3*b^2)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(105*a*d*sqrt(sec(c + d*x)))],


[sec(c + d*x)^(3/2)*(a + b*sec(c + d*x))^(5/2), x, 12, -(((a + b)*(33*a^2 + 16*b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(24*d*sqrt(a + b*sec(c + d*x)))) + (b*(59*a^2 + 16*b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(24*d*sqrt(a + b*sec(c + d*x))) + (5*a*(a^2 + 4*b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(8*d*sqrt(a + b*sec(c + d*x))) + ((33*a^2 + 16*b^2)*sqrt(sec(c + d*x))*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(24*d) + (13*a*b*sec(c + d*x)^(3/2)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(12*d) + (b^2*sec(c + d*x)^(5/2)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(3*d)],
[sec(c + d*x)^(1/2)*(a + b*sec(c + d*x))^(5/2), x, 11, -((9*a*b*(a + b)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(4*d*sqrt(a + b*sec(c + d*x)))) + (a*(8*a^2 + 11*b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(4*d*sqrt(a + b*sec(c + d*x))) + (b*(15*a^2 + 4*b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(4*d*sqrt(a + b*sec(c + d*x))) + (9*a*b*sqrt(sec(c + d*x))*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(4*d) + (b^2*sec(c + d*x)^(3/2)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(2*d)],
[sec(c + d*x)^(-1/2)*(a + b*sec(c + d*x))^(5/2), x, 10, ((a + b)*(2*a^2 - b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*sec(c + d*x))) + (b*(4*a^2 + b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*sec(c + d*x))) + (5*a*b^2*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*sec(c + d*x))) + (b^2*sqrt(sec(c + d*x))*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/d],
[sec(c + d*x)^(-3/2)*(a + b*sec(c + d*x))^(5/2), x, 10, (14*a*b*(a + b)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(3*d*sqrt(a + b*sec(c + d*x))) + (2*a*(a^2 + 2*b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(3*d*sqrt(a + b*sec(c + d*x))) + (2*b^3*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*sec(c + d*x))) + (2*a^2*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(3*d*sqrt(sec(c + d*x)))],
[sec(c + d*x)^(-5/2)*(a + b*sec(c + d*x))^(5/2), x, 8, (2*(a + b)*(9*a^2 + 23*b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(15*d*sqrt(a + b*sec(c + d*x))) + (16*b*(a^2 - b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(15*d*sqrt(a + b*sec(c + d*x))) + (2*a^2*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2)) + (22*a*b*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(15*d*sqrt(sec(c + d*x)))],
[sec(c + d*x)^(-7/2)*(a + b*sec(c + d*x))^(5/2), x, 9, (2*b*(a + b)*(29*a^2 + 3*b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(21*a*d*sqrt(a + b*sec(c + d*x))) + (2*(5*a^4 - 2*a^2*b^2 - 3*b^4)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(21*a*d*sqrt(a + b*sec(c + d*x))) + (2*a^2*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(7*d*sec(c + d*x)^(5/2)) + (6*a*b*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(7*d*sec(c + d*x)^(3/2)) + (2*(5*a^2 + 9*b^2)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(21*d*sqrt(sec(c + d*x)))],
[sec(c + d*x)^(-9/2)*(a + b*sec(c + d*x))^(5/2), x, 10, (2*(a + b)*(147*a^4 + 279*a^2*b^2 - 10*b^4)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(315*a^2*d*sqrt(a + b*sec(c + d*x))) + (4*b*(57*a^4 - 62*a^2*b^2 + 5*b^4)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(315*a^2*d*sqrt(a + b*sec(c + d*x))) + (2*a^2*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(9*d*sec(c + d*x)^(7/2)) + (38*a*b*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(63*d*sec(c + d*x)^(5/2)) + (2*(49*a^2 + 75*b^2)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(315*d*sec(c + d*x)^(3/2)) + (2*b*(163*a^2 + 5*b^2)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(315*a*d*sqrt(sec(c + d*x)))],


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


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


[sec(c + d*x)^(7/2)/(a + b*sec(c + d*x))^(3/2), x, 11, -(((3*a^2 - b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/((a - b)*b^2*d*sqrt(a + b*sec(c + d*x)))) + (sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(b*d*sqrt(a + b*sec(c + d*x))) - (3*a*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(b^2*d*sqrt(a + b*sec(c + d*x))) - (2*a^2*sec(c + d*x)^(3/2)*sin(c + d*x))/(b*(a^2 - b^2)*d*sqrt(a + b*sec(c + d*x))) + ((3*a^2 - b^2)*sqrt(sec(c + d*x))*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(b^2*(a^2 - b^2)*d)],
[sec(c + d*x)^(5/2)/(a + b*sec(c + d*x))^(3/2), x, 8, (2*a*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/((a - b)*b*d*sqrt(a + b*sec(c + d*x))) + (2*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(b*d*sqrt(a + b*sec(c + d*x))) - (2*a^2*sqrt(sec(c + d*x))*sin(c + d*x))/(b*(a^2 - b^2)*d*sqrt(a + b*sec(c + d*x)))],
[sec(c + d*x)^(3/2)/(a + b*sec(c + d*x))^(3/2), x, 5, -((2*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/((a - b)*d*sqrt(a + b*sec(c + d*x)))) + (2*a*sqrt(sec(c + d*x))*sin(c + d*x))/((a^2 - b^2)*d*sqrt(a + b*sec(c + d*x)))],
[sec(c + d*x)^(1/2)/(a + b*sec(c + d*x))^(3/2), x, 7, (2*b*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(a*(a - b)*d*sqrt(a + b*sec(c + d*x))) + (2*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(a*d*sqrt(a + b*sec(c + d*x))) - (2*b*sqrt(sec(c + d*x))*sin(c + d*x))/((a^2 - b^2)*d*sqrt(a + b*sec(c + d*x)))],
[sec(c + d*x)^(-1/2)/(a + b*sec(c + d*x))^(3/2), x, 7, (2*(a^2 - 2*b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(a^2*(a - b)*d*sqrt(a + b*sec(c + d*x))) - (4*b*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(a^2*d*sqrt(a + b*sec(c + d*x))) + (2*b^2*sqrt(sec(c + d*x))*sin(c + d*x))/(a*(a^2 - b^2)*d*sqrt(a + b*sec(c + d*x)))],
[sec(c + d*x)^(-3/2)/(a + b*sec(c + d*x))^(3/2), x, 8, -((2*b*(5*a^2 - 8*b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(3*a^3*(a - b)*d*sqrt(a + b*sec(c + d*x)))) + (2*(a^2 + 8*b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(3*a^3*d*sqrt(a + b*sec(c + d*x))) + (2*b^2*sin(c + d*x))/(a*(a^2 - b^2)*d*sqrt(sec(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(a^2 - 4*b^2)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(3*a^2*(a^2 - b^2)*d*sqrt(sec(c + d*x)))],
[sec(c + d*x)^(-5/2)/(a + b*sec(c + d*x))^(3/2), x, 9, (2*(3*a^4 + 8*a^2*b^2 - 16*b^4)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(5*a^4*(a - b)*d*sqrt(a + b*sec(c + d*x))) - (8*b*(a^2 + 4*b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(5*a^4*d*sqrt(a + b*sec(c + d*x))) + (2*b^2*sin(c + d*x))/(a*(a^2 - b^2)*d*sec(c + d*x)^(3/2)*sqrt(a + b*sec(c + d*x))) + (2*(a^2 - 6*b^2)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(5*a^2*(a^2 - b^2)*d*sec(c + d*x)^(3/2)) - (2*b*(3*a^2 - 8*b^2)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(5*a^3*(a^2 - b^2)*d*sqrt(sec(c + d*x)))],


[sec(c + d*x)^(9/2)/(a + b*sec(c + d*x))^(5/2), x, 12, -(((15*a^4 - 26*a^2*b^2 + 3*b^4)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(3*(a - b)^2*b^3*(a + b)*d*sqrt(a + b*sec(c + d*x)))) + ((5*a^2 - 3*b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(3*b^2*(a^2 - b^2)*d*sqrt(a + b*sec(c + d*x))) - (5*a*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(b^3*d*sqrt(a + b*sec(c + d*x))) - (2*a^2*sec(c + d*x)^(5/2)*sin(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*sec(c + d*x))^(3/2)) - (2*a^2*(5*a^2 - 9*b^2)*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*b^2*(a^2 - b^2)^2*d*sqrt(a + b*sec(c + d*x))) + ((15*a^4 - 26*a^2*b^2 + 3*b^4)*sqrt(sec(c + d*x))*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(3*b^3*(a^2 - b^2)^2*d)],
[sec(c + d*x)^(7/2)/(a + b*sec(c + d*x))^(5/2), x, 11, (2*a*(3*a^2 - 7*b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(3*(a - b)^2*b^2*(a + b)*d*sqrt(a + b*sec(c + d*x))) - (2*a*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(3*b*(a^2 - b^2)*d*sqrt(a + b*sec(c + d*x))) + (2*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(b^2*d*sqrt(a + b*sec(c + d*x))) - (2*a^2*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*sec(c + d*x))^(3/2)) - (2*a^2*(3*a^2 - 7*b^2)*sqrt(sec(c + d*x))*sin(c + d*x))/(3*b^2*(a^2 - b^2)^2*d*sqrt(a + b*sec(c + d*x)))],
[sec(c + d*x)^(5/2)/(a + b*sec(c + d*x))^(5/2), x, 8, (8*b*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(3*(a - b)^2*(a + b)*d*sqrt(a + b*sec(c + d*x))) + (2*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(3*(a^2 - b^2)*d*sqrt(a + b*sec(c + d*x))) - (2*a^2*sqrt(sec(c + d*x))*sin(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*sec(c + d*x))^(3/2)) + (2*a*(a^2 - 5*b^2)*sqrt(sec(c + d*x))*sin(c + d*x))/(3*b*(a^2 - b^2)^2*d*sqrt(a + b*sec(c + d*x)))],
[sec(c + d*x)^(3/2)/(a + b*sec(c + d*x))^(5/2), x, 8, -((2*(3*a^2 + b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(3*a*(a - b)^2*(a + b)*d*sqrt(a + b*sec(c + d*x)))) - (2*b*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(3*a*(a^2 - b^2)*d*sqrt(a + b*sec(c + d*x))) + (2*a*sqrt(sec(c + d*x))*sin(c + d*x))/(3*(a^2 - b^2)*d*(a + b*sec(c + d*x))^(3/2)) + (4*(a^2 + b^2)*sqrt(sec(c + d*x))*sin(c + d*x))/(3*(a^2 - b^2)^2*d*sqrt(a + b*sec(c + d*x)))],
[sec(c + d*x)^(1/2)/(a + b*sec(c + d*x))^(5/2), x, 8, (4*b*(3*a^2 - b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(3*a^2*(a - b)^2*(a + b)*d*sqrt(a + b*sec(c + d*x))) + (2*(3*a^2 - 2*b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(3*a^2*(a^2 - b^2)*d*sqrt(a + b*sec(c + d*x))) - (2*b*sqrt(sec(c + d*x))*sin(c + d*x))/(3*(a^2 - b^2)*d*(a + b*sec(c + d*x))^(3/2)) - (2*b*(5*a^2 - b^2)*sqrt(sec(c + d*x))*sin(c + d*x))/(3*a*(a^2 - b^2)^2*d*sqrt(a + b*sec(c + d*x)))],
[sec(c + d*x)^(-1/2)/(a + b*sec(c + d*x))^(5/2), x, 8, (2*(3*a^4 - 15*a^2*b^2 + 8*b^4)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(3*a^3*(a - b)^2*(a + b)*d*sqrt(a + b*sec(c + d*x))) - (2*b*(9*a^2 - 8*b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(3*a^3*(a^2 - b^2)*d*sqrt(a + b*sec(c + d*x))) + (2*b^2*sqrt(sec(c + d*x))*sin(c + d*x))/(3*a*(a^2 - b^2)*d*(a + b*sec(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*sec(c + d*x)))],
[sec(c + d*x)^(-3/2)/(a + b*sec(c + d*x))^(5/2), x, 9, -((8*b*(2*a^4 - 7*a^2*b^2 + 4*b^4)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(3*a^4*(a - b)^2*(a + b)*d*sqrt(a + b*sec(c + d*x)))) + (2*(a^4 + 16*a^2*b^2 - 16*b^4)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(3*a^4*(a^2 - b^2)*d*sqrt(a + b*sec(c + d*x))) + (2*b^2*sin(c + d*x))/(3*a*(a^2 - b^2)*d*sqrt(sec(c + d*x))*(a + b*sec(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*sqrt(sec(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(a^4 - 13*a^2*b^2 + 8*b^4)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(3*a^3*(a^2 - b^2)^2*d*sqrt(sec(c + d*x)))],
[sec(c + d*x)^(-5/2)/(a + b*sec(c + d*x))^(5/2), x, 10, (2*(9*a^6 + 55*a^4*b^2 - 212*a^2*b^4 + 128*b^6)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(15*a^5*(a - b)^2*(a + b)*d*sqrt(a + b*sec(c + d*x))) - (2*b*(17*a^4 + 116*a^2*b^2 - 128*b^4)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(15*a^5*(a^2 - b^2)*d*sqrt(a + b*sec(c + d*x))) + (2*b^2*sin(c + d*x))/(3*a*(a^2 - b^2)*d*sec(c + d*x)^(3/2)*(a + b*sec(c + d*x))^(3/2)) + (8*b^2*(3*a^2 - 2*b^2)*sin(c + d*x))/(3*a^2*(a^2 - b^2)^2*d*sec(c + d*x)^(3/2)*sqrt(a + b*sec(c + d*x))) + (2*(3*a^4 - 71*a^2*b^2 + 48*b^4)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(15*a^3*(a^2 - b^2)^2*d*sec(c + d*x)^(3/2)) - (4*b*(7*a^4 - 49*a^2*b^2 + 32*b^4)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(15*a^4*(a^2 - b^2)^2*d*sqrt(sec(c + d*x)))],


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

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

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

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


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

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

[sqrt(sec(c + d*x))/sqrt(3 + 2*sec(c + d*x)), x, 2, (2*sqrt(2 + 3*cos(c + d*x))*EllipticF((1/2)*(c + d*x), 6/5)*sqrt(sec(c + d*x)))/(sqrt(5)*d*sqrt(3 + 2*sec(c + d*x)))],
[sqrt(sec(c + d*x))/sqrt(3 - 2*sec(c + d*x)), x, 2, (2*sqrt(-2 + 3*cos(c + d*x))*EllipticF((1/2)*(c + d*x), 6)*sqrt(sec(c + d*x)))/(d*sqrt(3 - 2*sec(c + d*x)))],

[sqrt(sec(c + d*x))/sqrt(-3 + 2*sec(c + d*x)), x, 3, (2*sqrt(-2 + 3*cos(c + d*x))*EllipticF((1/2)*(c + d*x), 6)*sqrt(sec(c + d*x)))/(d*sqrt(-3 + 2*sec(c + d*x)))],
[sqrt(sec(c + d*x))/sqrt(-3 - 2*sec(c + d*x)), x, 3, (2*sqrt(2 + 3*cos(c + d*x))*EllipticF((1/2)*(c + d*x), 6/5)*sqrt(sec(c + d*x)))/(sqrt(5)*d*sqrt(-3 - 2*sec(c + d*x)))],


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


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


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


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


# ::Subsubsection:: 
#n>0


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


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


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


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


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


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


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


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


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


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


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


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


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


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


[cos(c + d*x)^(7/2)*(a + b*sec(c + d*x))^(3/2), x, 9, (4*b*(a + b)*(41*a^2 - 3*b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*a)/(a + b)))/(105*a^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(25*a^4 - 31*a^2*b^2 + 6*b^4)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*a)/(a + b)))/(105*a^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(25*a^2 + 3*b^2)*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(105*a*d) + (16*b*cos(c + d*x)^(3/2)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(35*d) + (2*a*cos(c + d*x)^(5/2)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(7*d)],
[cos(c + d*x)^(5/2)*(a + b*sec(c + d*x))^(3/2), x, 8, (2*(a + b)*(3*a^2 + b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*a)/(a + b)))/(5*a*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*b*(a^2 - b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*a)/(a + b)))/(5*a*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*b*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(5*d) + (2*cos(c + d*x)^(3/2)*(a + b*sec(c + d*x))^(3/2)*sin(c + d*x))/(5*d)],
[cos(c + d*x)^(3/2)*(a + b*sec(c + d*x))^(3/2), x, 7, (8*b*(a + b)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*a)/(a + b)))/(3*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(a^2 - b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*a)/(a + b)))/(3*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*a*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(3*d)],
[cos(c + d*x)^(1/2)*(a + b*sec(c + d*x))^(3/2), x, 9, (2*a*(a + b)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*a)/(a + b)))/(d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*a*b*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*a)/(a + b)))/(d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*b^2*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*a)/(a + b)))/(d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))],
[cos(c + d*x)^(-1/2)*(a + b*sec(c + d*x))^(3/2), x, 10, -((b*(a + b)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*a)/(a + b)))/(d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))) + ((2*a^2 + b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*a)/(a + b)))/(d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (3*a*b*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*a)/(a + b)))/(d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (b*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(d*sqrt(cos(c + d*x)))],
[cos(c + d*x)^(-3/2)*(a + b*sec(c + d*x))^(3/2), x, 11, -((5*a*(a + b)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*a)/(a + b)))/(4*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))) + (7*a*b*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*a)/(a + b)))/(4*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + ((3*a^2 + 4*b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*a)/(a + b)))/(4*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (b*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(2*d*cos(c + d*x)^(3/2)) + (5*a*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(4*d*sqrt(cos(c + d*x)))],


[cos(c + d*x)^(9/2)*(a + b*sec(c + d*x))^(5/2), x, 10, (2*(a + b)*(147*a^4 + 279*a^2*b^2 - 10*b^4)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*a)/(a + b)))/(315*a^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (4*b*(57*a^4 - 62*a^2*b^2 + 5*b^4)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*a)/(a + b)))/(315*a^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*b*(163*a^2 + 5*b^2)*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(315*a*d) + (2*(49*a^2 + 75*b^2)*cos(c + d*x)^(3/2)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(315*d) + (38*a*b*cos(c + d*x)^(5/2)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(63*d) + (2*a^2*cos(c + d*x)^(7/2)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(9*d)],
[cos(c + d*x)^(7/2)*(a + b*sec(c + d*x))^(5/2), x, 9, (2*b*(a + b)*(29*a^2 + 3*b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*a)/(a + b)))/(21*a*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(5*a^4 - 2*a^2*b^2 - 3*b^4)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*a)/(a + b)))/(21*a*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(5*a^2 + 9*b^2)*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(21*d) + (6*a*b*cos(c + d*x)^(3/2)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(7*d) + (2*a^2*cos(c + d*x)^(5/2)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(7*d)],
[cos(c + d*x)^(5/2)*(a + b*sec(c + d*x))^(5/2), x, 8, (2*(a + b)*(9*a^2 + 23*b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*a)/(a + b)))/(15*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (16*b*(a^2 - b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*a)/(a + b)))/(15*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (22*a*b*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(15*d) + (2*a^2*cos(c + d*x)^(3/2)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(5*d)],
[cos(c + d*x)^(3/2)*(a + b*sec(c + d*x))^(5/2), x, 10, (14*a*b*(a + b)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*a)/(a + b)))/(3*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*a*(a^2 + 2*b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*a)/(a + b)))/(3*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*b^3*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*a)/(a + b)))/(d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*a^2*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(3*d)],
[cos(c + d*x)^(1/2)*(a + b*sec(c + d*x))^(5/2), x, 10, ((a + b)*(2*a^2 - b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*a)/(a + b)))/(d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (b*(4*a^2 + b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*a)/(a + b)))/(d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (5*a*b^2*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*a)/(a + b)))/(d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (b^2*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(d*sqrt(cos(c + d*x)))],
[cos(c + d*x)^(-1/2)*(a + b*sec(c + d*x))^(5/2), x, 11, -((9*a*b*(a + b)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*a)/(a + b)))/(4*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))) + (a*(8*a^2 + 11*b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*a)/(a + b)))/(4*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (b*(15*a^2 + 4*b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*a)/(a + b)))/(4*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (b^2*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(2*d*cos(c + d*x)^(3/2)) + (9*a*b*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(4*d*sqrt(cos(c + d*x)))],
[cos(c + d*x)^(-3/2)*(a + b*sec(c + d*x))^(5/2), x, 12, -(((a + b)*(33*a^2 + 16*b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*a)/(a + b)))/(24*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))) + (b*(59*a^2 + 16*b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*a)/(a + b)))/(24*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (5*a*(a^2 + 4*b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*a)/(a + b)))/(8*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (b^2*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(3*d*cos(c + d*x)^(5/2)) + (13*a*b*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(12*d*cos(c + d*x)^(3/2)) + ((33*a^2 + 16*b^2)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(24*d*sqrt(cos(c + d*x)))],


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


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


[cos(c + d*x)^(5/2)/(a + b*sec(c + d*x))^(3/2), x, 9, (2*(3*a^4 + 8*a^2*b^2 - 16*b^4)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*a)/(a + b)))/(5*a^4*(a - b)*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (8*b*(a^2 + 4*b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*a)/(a + b)))/(5*a^4*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*b^2*cos(c + d*x)^(3/2)*sin(c + d*x))/(a*(a^2 - b^2)*d*sqrt(a + b*sec(c + d*x))) - (2*b*(3*a^2 - 8*b^2)*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(5*a^3*(a^2 - b^2)*d) + (2*(a^2 - 6*b^2)*cos(c + d*x)^(3/2)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(5*a^2*(a^2 - b^2)*d)],
[cos(c + d*x)^(3/2)/(a + b*sec(c + d*x))^(3/2), x, 8, -((2*b*(5*a^2 - 8*b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*a)/(a + b)))/(3*a^3*(a - b)*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))) + (2*(a^2 + 8*b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*a)/(a + b)))/(3*a^3*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*b^2*sqrt(cos(c + d*x))*sin(c + d*x))/(a*(a^2 - b^2)*d*sqrt(a + b*sec(c + d*x))) + (2*(a^2 - 4*b^2)*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(3*a^2*(a^2 - b^2)*d)],
[cos(c + d*x)^(1/2)/(a + b*sec(c + d*x))^(3/2), x, 7, (2*(a^2 - 2*b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*a)/(a + b)))/(a^2*(a - b)*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (4*b*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*a)/(a + b)))/(a^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*b^2*sin(c + d*x))/(a*(a^2 - b^2)*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))],
[cos(c + d*x)^(-1/2)/(a + b*sec(c + d*x))^(3/2), x, 7, (2*b*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*a)/(a + b)))/(a*(a - b)*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*a)/(a + b)))/(a*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*b*sin(c + d*x))/((a^2 - b^2)*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))],
[cos(c + d*x)^(-3/2)/(a + b*sec(c + d*x))^(3/2), x, 5, -((2*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*a)/(a + b)))/((a - b)*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))) + (2*a*sin(c + d*x))/((a^2 - b^2)*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))],
[cos(c + d*x)^(-5/2)/(a + b*sec(c + d*x))^(3/2), x, 8, (2*a*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*a)/(a + b)))/((a - b)*b*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*a)/(a + b)))/(b*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*a^2*sin(c + d*x))/(b*(a^2 - b^2)*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))],
[cos(c + d*x)^(-7/2)/(a + b*sec(c + d*x))^(3/2), x, 11, -(((3*a^2 - b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*a)/(a + b)))/((a - b)*b^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))) + (sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*a)/(a + b)))/(b*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (3*a*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*a)/(a + b)))/(b^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*a^2*sin(c + d*x))/(b*(a^2 - b^2)*d*cos(c + d*x)^(3/2)*sqrt(a + b*sec(c + d*x))) + ((3*a^2 - b^2)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(b^2*(a^2 - b^2)*d*sqrt(cos(c + d*x)))],


[cos(c + d*x)^(3/2)/(a + b*sec(c + d*x))^(5/2), x, 9, -((8*b*(2*a^4 - 7*a^2*b^2 + 4*b^4)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*a)/(a + b)))/(3*a^4*(a - b)^2*(a + b)*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))) + (2*(a^4 + 16*a^2*b^2 - 16*b^4)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*a)/(a + b)))/(3*a^4*(a^2 - b^2)*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*b^2*sqrt(cos(c + d*x))*sin(c + d*x))/(3*a*(a^2 - b^2)*d*(a + b*sec(c + d*x))^(3/2)) + (4*b^2*(5*a^2 - 3*b^2)*sqrt(cos(c + d*x))*sin(c + d*x))/(3*a^2*(a^2 - b^2)^2*d*sqrt(a + b*sec(c + d*x))) + (2*(a^4 - 13*a^2*b^2 + 8*b^4)*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(3*a^3*(a^2 - b^2)^2*d)],
[cos(c + d*x)^(1/2)/(a + b*sec(c + d*x))^(5/2), x, 8, (2*(3*a^4 - 15*a^2*b^2 + 8*b^4)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*a)/(a + b)))/(3*a^3*(a - b)^2*(a + b)*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*b*(9*a^2 - 8*b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*a)/(a + b)))/(3*a^3*(a^2 - b^2)*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*b^2*sin(c + d*x))/(3*a*(a^2 - b^2)*d*sqrt(cos(c + d*x))*(a + b*sec(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*sec(c + d*x)))],
[cos(c + d*x)^(-1/2)/(a + b*sec(c + d*x))^(5/2), x, 8, (4*b*(3*a^2 - b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*a)/(a + b)))/(3*a^2*(a - b)^2*(a + b)*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(3*a^2 - 2*b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*a)/(a + b)))/(3*a^2*(a^2 - b^2)*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*b*sin(c + d*x))/(3*(a^2 - b^2)*d*sqrt(cos(c + d*x))*(a + b*sec(c + d*x))^(3/2)) - (2*b*(5*a^2 - b^2)*sin(c + d*x))/(3*a*(a^2 - b^2)^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))],
[cos(c + d*x)^(-3/2)/(a + b*sec(c + d*x))^(5/2), x, 8, -((2*(3*a^2 + b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*a)/(a + b)))/(3*a*(a - b)^2*(a + b)*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))) - (2*b*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*a)/(a + b)))/(3*a*(a^2 - b^2)*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*a*sin(c + d*x))/(3*(a^2 - b^2)*d*sqrt(cos(c + d*x))*(a + b*sec(c + d*x))^(3/2)) + (4*(a^2 + b^2)*sin(c + d*x))/(3*(a^2 - b^2)^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))],
[cos(c + d*x)^(-5/2)/(a + b*sec(c + d*x))^(5/2), x, 8, (8*b*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*a)/(a + b)))/(3*(a - b)^2*(a + b)*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*a)/(a + b)))/(3*(a^2 - b^2)*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*a^2*sin(c + d*x))/(3*b*(a^2 - b^2)*d*sqrt(cos(c + d*x))*(a + b*sec(c + d*x))^(3/2)) + (2*a*(a^2 - 5*b^2)*sin(c + d*x))/(3*b*(a^2 - b^2)^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))],
[cos(c + d*x)^(-7/2)/(a + b*sec(c + d*x))^(5/2), x, 11, (2*a*(3*a^2 - 7*b^2)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*a)/(a + b)))/(3*(a - b)^2*b^2*(a + b)*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*a*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*a)/(a + b)))/(3*b*(a^2 - b^2)*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*a)/(a + b)))/(b^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*a^2*sin(c + d*x))/(3*b*(a^2 - b^2)*d*cos(c + d*x)^(3/2)*(a + b*sec(c + d*x))^(3/2)) - (2*a^2*(3*a^2 - 7*b^2)*sin(c + d*x))/(3*b^2*(a^2 - b^2)^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))],


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


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


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


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


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


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


[sec(c + d*x)^2*(A + B*sec(c + d*x))*(a + b*sec(c + d*x))^4, x, 7, ((32*a^3*A*b + 24*a*A*b^3 + 8*a^4*B + 36*a^2*b^2*B + 5*b^4*B)*arctanh(sin(c + d*x)))/(16*d) + ((15*a^4*A + 60*a^2*A*b^2 + 8*A*b^4 + 40*a^3*b*B + 32*a*b^3*B)*tan(c + d*x))/(15*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)*sec(c + d*x)*tan(c + d*x))/(16*d) + (b*(144*a^2*A*b + 32*A*b^3 + 56*a^3*B + 103*a*b^2*B)*sec(c + d*x)^2*tan(c + d*x))/(120*d) + (b*(72*a*A*b + 48*a^2*B + 25*b^2*B)*sec(c + d*x)^2*(a + b*sec(c + d*x))*tan(c + d*x))/(120*d) + (b*(2*A*b + 3*a*B)*sec(c + d*x)^2*(a + b*sec(c + d*x))^2*tan(c + d*x))/(10*d) + (b*B*sec(c + d*x)^2*(a + b*sec(c + d*x))^3*tan(c + d*x))/(6*d)],
[sec(c + d*x)^1*(A + B*sec(c + d*x))*(a + b*sec(c + d*x))^4, x, 6, ((8*a^4*A + 24*a^2*A*b^2 + 3*A*b^4 + 16*a^3*b*B + 12*a*b^3*B)*arctanh(sin(c + d*x)))/(8*d) + ((95*a^3*A*b + 80*a*A*b^3 + 12*a^4*B + 112*a^2*b^2*B + 16*b^4*B)*tan(c + d*x))/(30*d) + (b*(130*a^2*A*b + 45*A*b^3 + 24*a^3*B + 116*a*b^2*B)*sec(c + d*x)*tan(c + d*x))/(120*d) + ((35*a*A*b + 12*a^2*B + 16*b^2*B)*(a + b*sec(c + d*x))^2*tan(c + d*x))/(60*d) + ((5*A*b + 4*a*B)*(a + b*sec(c + d*x))^3*tan(c + d*x))/(20*d) + (B*(a + b*sec(c + d*x))^4*tan(c + d*x))/(5*d)],
[sec(c + d*x)^0*(A + B*sec(c + d*x))*(a + b*sec(c + d*x))^4, x, 6, a^4*A*x + ((32*a^3*A*b + 16*a*A*b^3 + 8*a^4*B + 24*a^2*b^2*B + 3*b^4*B)*arctanh(sin(c + d*x)))/(8*d) + (b*(104*a^2*A*b + 16*A*b^3 + 50*a^3*B + 55*a*b^2*B)*tan(c + d*x))/(24*d) + (b*(32*a*A*b + 26*a^2*B + 9*b^2*B)*(a + b*sec(c + d*x))*tan(c + d*x))/(24*d) + (b*(4*A*b + 7*a*B)*(a + b*sec(c + d*x))^2*tan(c + d*x))/(12*d) + (b*B*(a + b*sec(c + d*x))^3*tan(c + d*x))/(4*d)],
[cos(c + d*x)^1*(A + B*sec(c + d*x))*(a + b*sec(c + d*x))^4, x, 6, a^3*(4*A*b + a*B)*x + (b*(12*a^2*A*b + A*b^3 + 8*a^3*B + 4*a*b^2*B)*arctanh(sin(c + d*x)))/(2*d) + (a*(3*a*A - b*B)*(a + b*sec(c + d*x))^2*sin(c + d*x))/(3*d) + (b*B*(a + b*sec(c + d*x))^3*sin(c + d*x))/(3*d) - (b*(6*a^3*A - 21*a*A*b^2 - 26*a^2*b*B - 4*b^3*B)*tan(c + d*x))/(6*d) - (b*(6*a^2*A - 3*A*b^2 - 8*a*b*B)*(a + b*sec(c + d*x))*tan(c + d*x))/(6*d)],
[cos(c + d*x)^2*(A + B*sec(c + d*x))*(a + b*sec(c + d*x))^4, x, 6, (1/2)*a^2*(a^2*A + 12*A*b^2 + 8*a*b*B)*x + (b^2*(8*a*A*b + 12*a^2*B + b^2*B)*arctanh(sin(c + d*x)))/(2*d) + (a*(5*A*b + 2*a*B)*(a + b*sec(c + d*x))^2*sin(c + d*x))/(2*d) + (a*A*cos(c + d*x)*(a + b*sec(c + d*x))^3*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)*tan(c + d*x))/(2*d) - (b*(6*a*A*b + 2*a^2*B - b^2*B)*(a + b*sec(c + d*x))*tan(c + d*x))/(2*d)],
[cos(c + d*x)^3*(A + B*sec(c + d*x))*(a + b*sec(c + d*x))^4, x, 6, (1/2)*a*(4*a^2*A*b + 8*A*b^3 + a^3*B + 12*a*b^2*B)*x + (b^3*(A*b + 4*a*B)*arctanh(sin(c + d*x)))/d + (a*(2*a^2*A + 9*A*b^2 + 9*a*b*B)*(a + b*sec(c + d*x))*sin(c + d*x))/(3*d) + (a*(2*A*b + a*B)*cos(c + d*x)*(a + b*sec(c + d*x))^2*sin(c + d*x))/(2*d) + (a*A*cos(c + d*x)^2*(a + b*sec(c + d*x))^3*sin(c + d*x))/(3*d) - (b*(4*a^3*A + 26*a*A*b^2 + 21*a^2*b*B - 6*b^3*B)*tan(c + d*x))/(6*d)],
[cos(c + d*x)^4*(A + B*sec(c + d*x))*(a + b*sec(c + d*x))^4, x, 6, (1/8)*(3*a^4*A + 24*a^2*A*b^2 + 8*A*b^4 + 16*a^3*b*B + 32*a*b^3*B)*x + (b^4*B*arctanh(sin(c + d*x)))/d + (a*(55*a^2*A*b + 50*A*b^3 + 16*a^3*B + 104*a*b^2*B)*sin(c + d*x))/(24*d) + (a*(9*a^2*A + 26*A*b^2 + 32*a*b*B)*(b + a*cos(c + d*x))*sin(c + d*x))/(24*d) + (a*(7*A*b + 4*a*B)*(b + a*cos(c + d*x))^2*sin(c + d*x))/(12*d) + (a*A*(b + a*cos(c + d*x))^3*sin(c + d*x))/(4*d)],
[cos(c + d*x)^5*(A + B*sec(c + d*x))*(a + b*sec(c + d*x))^4, x, 6, (1/8)*(12*a^3*A*b + 16*a*A*b^3 + 3*a^4*B + 24*a^2*b^2*B + 8*b^4*B)*x + ((8*a^4*A + 60*a^2*A*b^2 + 15*A*b^4 + 40*a^3*b*B + 60*a*b^3*B)*sin(c + d*x))/(15*d) + (a*(148*a^2*A*b + 96*A*b^3 + 45*a^3*B + 230*a*b^2*B)*cos(c + d*x)*sin(c + d*x))/(120*d) + (a*(8*a^2*A + 18*A*b^2 + 25*a*b*B)*cos(c + d*x)^2*(a + b*sec(c + d*x))*sin(c + d*x))/(30*d) + (a*(8*A*b + 5*a*B)*cos(c + d*x)^3*(a + b*sec(c + d*x))^2*sin(c + d*x))/(20*d) + (a*A*cos(c + d*x)^4*(a + b*sec(c + d*x))^3*sin(c + d*x))/(5*d)],
[cos(c + d*x)^6*(A + B*sec(c + d*x))*(a + b*sec(c + d*x))^4, x, 7, (1/16)*(5*a^4*A + 36*a^2*A*b^2 + 8*A*b^4 + 24*a^3*b*B + 32*a*b^3*B)*x + ((32*a^3*A*b + 40*a*A*b^3 + 8*a^4*B + 60*a^2*b^2*B + 15*b^4*B)*sin(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)*cos(c + d*x)*sin(c + d*x))/(16*d) + (a*(103*a^2*A*b + 56*A*b^3 + 32*a^3*B + 144*a*b^2*B)*cos(c + d*x)^2*sin(c + d*x))/(120*d) + (a*(25*a^2*A + 48*A*b^2 + 72*a*b*B)*cos(c + d*x)^3*(a + b*sec(c + d*x))*sin(c + d*x))/(120*d) + (a*(3*A*b + 2*a*B)*cos(c + d*x)^4*(a + b*sec(c + d*x))^2*sin(c + d*x))/(10*d) + (a*A*cos(c + d*x)^5*(a + b*sec(c + d*x))^3*sin(c + d*x))/(6*d)],


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


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


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


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


[sec(c + d*x)^5*(A + B*sec(c + d*x))/(a + b*sec(c + d*x))^4, x, 8, ((A*b - 4*a*B)*arctanh(sin(c + d*x)))/(b^5*d) + (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)*arctanh(((-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)*tan(c + d*x))/(6*b^4*(a^2 - b^2)^3*d) + (a*(A*b - a*B)*sec(c + d*x)^3*tan(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*sec(c + d*x))^3) + (a*(a^2*A*b - 6*A*b^3 - 4*a^3*B + 9*a*b^2*B)*sec(c + d*x)^2*tan(c + d*x))/(6*b^2*(a^2 - b^2)^2*d*(a + b*sec(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)*sec(c + d*x)*tan(c + d*x))/(2*b^3*(a^2 - b^2)^3*d*(a + b*sec(c + d*x)))],
[sec(c + d*x)^4*(A + B*sec(c + d*x))/(a + b*sec(c + d*x))^4, x, 7, (B*arctanh(sin(c + d*x)))/(b^4*d) + ((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)*arctanh(((-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)*sec(c + d*x)^2*tan(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*sec(c + d*x))^3) - (a*(5*A*b^3 + 3*a^3*B - 8*a*b^2*B)*sec(c + d*x)*tan(c + d*x))/(6*b^2*(a^2 - b^2)^2*d*(a + b*sec(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)*tan(c + d*x))/(6*b^3*(a^2 - b^2)^3*d*(a + b*sec(c + d*x)))],
[sec(c + d*x)^3*(A + B*sec(c + d*x))/(a + b*sec(c + d*x))^4, x, 6, -(((a^3*A + 4*a*A*b^2 - 3*a^2*b*B - 2*b^3*B)*arctanh(((-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)*sec(c + d*x)*tan(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*sec(c + d*x))^3) - (a*(a^2*A*b + 4*A*b^3 + 2*a^3*B - 7*a*b^2*B)*tan(c + d*x))/(6*b^2*(a^2 - b^2)^2*d*(a + b*sec(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)*tan(c + d*x))/(6*b^2*(a^2 - b^2)^3*d*(a + b*sec(c + d*x)))],
[sec(c + d*x)^2*(A + B*sec(c + d*x))/(a + b*sec(c + d*x))^4, x, 6, ((4*a^2*A*b + A*b^3 - a^3*B - 4*a*b^2*B)*arctanh(((-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)*tan(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*sec(c + d*x))^3) + ((2*a^2*A*b + 3*A*b^3 + a^3*B - 6*a*b^2*B)*tan(c + d*x))/(6*b*(a^2 - b^2)^2*d*(a + b*sec(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)*tan(c + d*x))/(6*b*(a^2 - b^2)^3*d*(a + b*sec(c + d*x)))],
[sec(c + d*x)^1*(A + B*sec(c + d*x))/(a + b*sec(c + d*x))^4, x, 6, -(((2*a^3*A + 3*a*A*b^2 - 4*a^2*b*B - b^3*B)*arctanh(((-a + b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/((a^2 - b^2)^(7/2)*d)) - ((A*b - a*B)*tan(c + d*x))/(3*(a^2 - b^2)*d*(a + b*sec(c + d*x))^3) - ((5*a*A*b - 2*a^2*B - 3*b^2*B)*tan(c + d*x))/(6*(a^2 - b^2)^2*d*(a + b*sec(c + d*x))^2) - ((11*a^2*A*b + 4*A*b^3 - 2*a^3*B - 13*a*b^2*B)*tan(c + d*x))/(6*(a^2 - b^2)^3*d*(a + b*sec(c + d*x)))],
[sec(c + d*x)^0*(A + B*sec(c + d*x))/(a + b*sec(c + d*x))^4, x, 5, (A*x)/a^4 + ((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)*arctanh(((-a + b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^4*(a^2 - b^2)^(7/2)*d) + (b*(A*b - a*B)*tan(c + d*x))/(3*a*(a^2 - b^2)*d*(a + b*sec(c + d*x))^3) + (b*(8*a^2*A*b - 3*A*b^3 - 5*a^3*B)*tan(c + d*x))/(6*a^2*(a^2 - b^2)^2*d*(a + b*sec(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)*tan(c + d*x))/(6*a^3*(a^2 - b^2)^3*d*(a + b*sec(c + d*x)))],
[cos(c + d*x)^1*(A + B*sec(c + d*x))/(a + b*sec(c + d*x))^4, x, 7, -(((4*A*b - a*B)*x)/a^5) - (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)*arctanh(((-a + b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^5*(a^2 - b^2)^(7/2)*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)*sin(c + d*x))/(6*a^4*(a^2 - b^2)^3*d) + (b*(A*b - a*B)*sin(c + d*x))/(3*a*(a^2 - b^2)*d*(a + b*sec(c + d*x))^3) + (b*(9*a^2*A*b - 4*A*b^3 - 6*a^3*B + a*b^2*B)*sin(c + d*x))/(6*a^2*(a^2 - b^2)^2*d*(a + b*sec(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)*sin(c + d*x))/(2*a^3*(a^2 - b^2)^3*d*(a + b*sec(c + d*x)))],
[cos(c + d*x)^2*(A + B*sec(c + d*x))/(a + b*sec(c + d*x))^4, x, 8, ((a^2*A + 20*A*b^2 - 8*a*b*B)*x)/(2*a^6) + (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)*arctanh(((-a + b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^6*(a^2 - b^2)^(7/2)*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)*sin(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)*cos(c + d*x)*sin(c + d*x))/(2*a^4*(a^2 - b^2)^3*d) + (b*(A*b - a*B)*cos(c + d*x)*sin(c + d*x))/(3*a*(a^2 - b^2)*d*(a + b*sec(c + d*x))^3) + (b*(10*a^2*A*b - 5*A*b^3 - 7*a^3*B + 2*a*b^2*B)*cos(c + d*x)*sin(c + d*x))/(6*a^2*(a^2 - b^2)^2*d*(a + b*sec(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)*cos(c + d*x)*sin(c + d*x))/(6*a^3*(a^2 - b^2)^3*d*(a + b*sec(c + d*x)))],


[(b*B/a + B*sec(c + d*x))/(a + b*sec(c + d*x)), x, 2, (b*B*x)/a^2 - (2*sqrt(a^2 - b^2)*B*arctanh(((-a + b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^2*d)],
[(a*B/b + B*sec(c + d*x))/(a + b*sec(c + d*x)), x, 1, (B*x)/b],

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


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


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


[sec(c + d*x)^4*sqrt(a + b*sec(c + d*x))*(A + B*sec(c + d*x)), 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(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((-a + b)/(a + b))))/(315*b^4*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(24*a^3*A*b + 6*a^2*A*b^2 + 57*a*A*b^3 + 75*A*b^4 - 16*a^4*B - 4*a^3*b*B - 24*a^2*b^2*B + 111*a*b^3*B + 147*b^4*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((-a + b)/(a + b))))/(315*b^3*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(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*sec(c + d*x))*tan((c + d*x)/2))/(315*b^4*d) - (2*(12*a^2*A*b - 75*A*b^3 - 8*a^3*B - 13*a*b^2*B)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(315*b^3*d) + (2*(9*a*A*b - 6*a^2*B + 49*b^2*B)*sec(c + d*x)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(315*b^2*d) + (2*(9*A*b + a*B)*sec(c + d*x)^2*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(63*b*d) + (2*B*sec(c + d*x)^3*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(9*d)],
[sec(c + d*x)^3*sqrt(a + b*sec(c + d*x))*(A + B*sec(c + d*x)), x, 8, (2*(a + b)*(14*a^2*A*b - 63*A*b^3 - 8*a^3*B - 19*a*b^2*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((-a + b)/(a + b))))/(105*b^3*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*(14*a^2*A*b - 49*a*A*b^2 - 63*A*b^3 - 8*a^3*B - 2*a^2*b*B - 19*a*b^2*B - 25*b^3*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((-a + b)/(a + b))))/(105*b^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*(14*a^2*A*b - 63*A*b^3 - 8*a^3*B - 19*a*b^2*B)*sqrt(a + b*sec(c + d*x))*tan((c + d*x)/2))/(105*b^3*d) + (2*(7*a*A*b - 4*a^2*B + 25*b^2*B)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(105*b^2*d) + (2*(7*A*b + a*B)*sec(c + d*x)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(35*b*d) + (2*B*sec(c + d*x)^2*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(7*d)],
[sec(c + d*x)^2*sqrt(a + b*sec(c + d*x))*(A + B*sec(c + d*x)), x, 7, (-2*(a + b)*(5*a*A*b - 2*a^2*B + 9*b^2*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((-a + b)/(a + b))))/(15*b^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(5*a*A*b + 5*A*b^2 - 2*a^2*B + 7*a*b*B + 9*b^2*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((-a + b)/(a + b))))/(15*b*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(5*a*A*b - 2*a^2*B + 9*b^2*B)*sqrt(a + b*sec(c + d*x))*tan((c + d*x)/2))/(15*b^2*d) + (2*(5*A*b + a*B)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(15*b*d) + (2*B*sec(c + d*x)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(5*d)],
[sec(c + d*x)^1*sqrt(a + b*sec(c + d*x))*(A + B*sec(c + d*x)), x, 6, (-2*(a + b)*(3*A*b + a*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((-a + b)/(a + b))))/(3*b*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(3*a*A + 3*A*b + a*B + b*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(3*A*b + a*B)*sqrt(a + b*sec(c + d*x))*tan((c + d*x)/2))/(3*b*d) + (2*B*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(3*d)],
[sec(c + d*x)^0*sqrt(a + b*sec(c + d*x))*(A + B*sec(c + d*x)), x, 7, -((2*(a + b)*B*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))) - (2*(a*A - A*b - a*B - b*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (4*a*A*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*B*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/d],
[cos(c + d*x)^1*sqrt(a + b*sec(c + d*x))*(A + B*sec(c + d*x)), x, 8, (A*(a + b)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*(A*b + a*B - b*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(A*b + 2*a*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (A*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/d],
[cos(c + d*x)^2*sqrt(a + b*sec(c + d*x))*(A + B*sec(c + d*x)), x, 8, ((a + b)*(A*b + 4*a*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - ((2*a*A - A*b + 4*b*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + ((4*a^2*A - A*b^2 + 4*a*b*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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*a*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (A*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(2*d) + ((A*b + 4*a*B)*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(4*a*d)],
[cos(c + d*x)^3*sqrt(a + b*sec(c + d*x))*(A + B*sec(c + d*x)), x, 9, ((a + b)*(16*a^2*A - 3*A*b^2 + 6*a*b*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(24*a^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - ((14*a*A*b - A*b^2 + 12*a^2*B - 6*a*b*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(12*a*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + ((4*a^2*A*b + A*b^3 + 8*a^3*B - 2*a*b^2*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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*a^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + ((A*b + 6*a*B)*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(12*a*d) + (A*cos(c + d*x)^2*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(3*d) + ((16*a^2*A - 3*A*b^2 + 6*a*b*B)*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(24*a^2*d)],


[sec(c + d*x)^3*(a + b*sec(c + d*x))^(3/2)*(A + B*sec(c + d*x)), 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(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((-a + b)/(a + b))))/(315*b^3*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*(18*a^3*A*b - 153*a^2*A*b^2 - 246*a*A*b^3 - 75*A*b^4 - 8*a^4*B - 2*a^3*b*B - 33*a^2*b^2*B - 186*a*b^3*B - 147*b^4*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((-a + b)/(a + b))))/(315*b^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*(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*sec(c + d*x))*tan((c + d*x)/2))/(315*b^3*d) + (2*(9*a^2*A*b + 75*A*b^3 - 4*a^3*B + 88*a*b^2*B)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(315*b^2*d) + (2*(72*a*A*b + 3*a^2*B + 49*b^2*B)*sec(c + d*x)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(315*b*d) + (2*(9*A*b + 10*a*B)*sec(c + d*x)^2*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(63*d) + (2*b*B*sec(c + d*x)^3*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(9*d)],
[sec(c + d*x)^2*(a + b*sec(c + d*x))^(3/2)*(A + B*sec(c + d*x)), x, 8, (-2*(a + b)*(21*a^2*A*b + 63*A*b^3 - 6*a^3*B + 82*a*b^2*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((-a + b)/(a + b))))/(105*b^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(21*a^2*A*b + 84*a*A*b^2 + 63*A*b^3 - 6*a^3*B + 51*a^2*b*B + 82*a*b^2*B + 25*b^3*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((-a + b)/(a + b))))/(105*b*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(21*a^2*A*b + 63*A*b^3 - 6*a^3*B + 82*a*b^2*B)*sqrt(a + b*sec(c + d*x))*tan((c + d*x)/2))/(105*b^2*d) + (2*(42*a*A*b + 3*a^2*B + 25*b^2*B)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(105*b*d) + (2*(7*A*b + 8*a*B)*sec(c + d*x)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(35*d) + (2*b*B*sec(c + d*x)^2*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(7*d)],
[sec(c + d*x)^1*(a + b*sec(c + d*x))^(3/2)*(A + B*sec(c + d*x)), x, 7, -((2*(a + b)*(20*a*A*b + 3*a^2*B + 9*b^2*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(15*b*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))) + (2*(15*a^2*A + 20*a*A*b + 5*A*b^2 + 3*a^2*B + 12*a*b*B + 9*b^2*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(20*a*A*b + 3*a^2*B + 9*b^2*B)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(15*b*d) + (2*(5*A*b + 3*a*B)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(15*d) + (2*B*(a + b*sec(c + d*x))^(3/2)*tan(c + d*x))/(5*d)],
[sec(c + d*x)^0*(a + b*sec(c + d*x))^(3/2)*(A + B*sec(c + d*x)), x, 8, (-2*(a + b)*(3*A*b + 4*a*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*(3*a^2*A - 6*a*A*b - 3*A*b^2 - 3*a^2*B - 4*a*b*B - b^2*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (4*a^2*A*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(3*A*b + 4*a*B)*sqrt(a + b*sec(c + d*x))*tan((c + d*x)/2))/(3*d) + (2*b*B*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(3*d)],
[cos(c + d*x)^1*(a + b*sec(c + d*x))^(3/2)*(A + B*sec(c + d*x)), x, 8, ((a + b)*(a*A - 2*b*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*(2*a*A*b - A*b^2 + a^2*B - 2*a*b*B - b^2*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*a*(3*A*b + 2*a*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*b*B*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/d + ((a*A - 2*b*B)*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/d],
[cos(c + d*x)^2*(a + b*sec(c + d*x))^(3/2)*(A + B*sec(c + d*x)), x, 8, ((a + b)*(5*A*b + 4*a*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - ((2*a^2*A - a*A*b + 4*A*b^2 + 8*a*b*B - 4*b^2*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + ((4*a^2*A + 3*A*b^2 + 12*a*b*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (a*A*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(2*d) + ((5*A*b + 4*a*B)*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(4*d)],
[cos(c + d*x)^3*(a + b*sec(c + d*x))^(3/2)*(A + B*sec(c + d*x)), x, 9, ((a + b)*(16*a^2*A + 3*A*b^2 + 30*a*b*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(24*a*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - ((26*a*A*b - 7*A*b^2 + 12*a^2*B - 6*a*b*B + 24*b^2*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + ((12*a^2*A*b - A*b^3 + 8*a^3*B + 6*a*b^2*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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*a*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + ((7*A*b + 6*a*B)*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(12*d) + (a*A*cos(c + d*x)^2*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(3*d) + ((16*a^2*A + 3*A*b^2 + 30*a*b*B)*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(24*a*d)],


[sec(c + d*x)^3*(a + b*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x)), 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(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((-a + b)/(a + b))))/(3465*b^3*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*(110*a^4*A*b - 1705*a^3*A*b^2 - 3069*a^2*A*b^3 - 2871*a*A*b^4 - 1617*A*b^5 - 40*a^5*B - 10*a^4*b*B - 255*a^3*b^2*B - 3315*a^2*b^3*B - 3705*a*b^4*B - 675*b^5*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((-a + b)/(a + b))))/(3465*b^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*(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*sec(c + d*x))*tan((c + d*x)/2))/(3465*b^3*d) + (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*sec(c + d*x))*tan(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)*sec(c + d*x)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(3465*b*d) + (2*(209*a*A*b + 113*a^2*B + 81*b^2*B)*sec(c + d*x)^2*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(693*d) + (2*b*(11*A*b + 14*a*B)*sec(c + d*x)^3*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(99*d) + (2*b*B*sec(c + d*x)^3*(a + b*sec(c + d*x))^(3/2)*tan(c + d*x))/(11*d)],
[sec(c + d*x)^2*(a + b*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x)), 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(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((-a + b)/(a + b))))/(315*b^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(45*a^3*A*b + 405*a^2*A*b^2 + 435*a*A*b^3 + 75*A*b^4 - 10*a^4*B + 155*a^3*b*B + 279*a^2*b^2*B + 261*a*b^3*B + 147*b^4*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((-a + b)/(a + b))))/(315*b*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(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*sec(c + d*x))*tan((c + d*x)/2))/(315*b^2*d) + (2*(135*a^2*A*b + 75*A*b^3 + 5*a^3*B + 163*a*b^2*B)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(315*b*d) + (2*(135*a*A*b + 75*a^2*B + 49*b^2*B)*sec(c + d*x)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(315*d) + (2*b*(3*A*b + 4*a*B)*sec(c + d*x)^2*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(21*d) + (2*b*B*sec(c + d*x)^2*(a + b*sec(c + d*x))^(3/2)*tan(c + d*x))/(9*d)],
[sec(c + d*x)^1*(a + b*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x)), x, 8, -((2*(a + b)*(161*a^2*A*b + 63*A*b^3 + 15*a^3*B + 145*a*b^2*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(105*b*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))) + (2*(105*a^3*A + 161*a^2*A*b + 119*a*A*b^2 + 63*A*b^3 + 15*a^3*B + 135*a^2*b*B + 145*a*b^2*B + 25*b^3*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(161*a^2*A*b + 63*A*b^3 + 15*a^3*B + 145*a*b^2*B)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(105*b*d) + (2*(56*a*A*b + 15*a^2*B + 25*b^2*B)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(105*d) + (2*(7*A*b + 5*a*B)*(a + b*sec(c + d*x))^(3/2)*tan(c + d*x))/(35*d) + (2*B*(a + b*sec(c + d*x))^(5/2)*tan(c + d*x))/(7*d)],
[sec(c + d*x)^0*(a + b*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x)), x, 9, (-2*(a + b)*(35*a*A*b + 23*a^2*B + 9*b^2*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*(15*a^3*A - 45*a^2*A*b - 35*a*A*b^2 - 5*A*b^3 - 15*a^3*B - 23*a^2*b*B - 17*a*b^2*B - 9*b^3*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (4*a^3*A*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(35*a*A*b + 23*a^2*B + 9*b^2*B)*sqrt(a + b*sec(c + d*x))*tan((c + d*x)/2))/(15*d) + (2*b*(5*A*b + 8*a*B)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(15*d) + (2*b*B*(a + b*sec(c + d*x))^(3/2)*tan(c + d*x))/(5*d)],
[cos(c + d*x)^1*(a + b*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x)), x, 9, ((a + b)*(3*a^2*A - 6*A*b^2 - 14*a*b*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*(9*a^2*A*b - 9*a*A*b^2 - 3*A*b^3 + 3*a^3*B - 9*a^2*b*B - 7*a*b^2*B - b^3*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*a^2*(5*A*b + 2*a*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (a*(3*a*A - 2*b*B)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(3*d) + (2*b*B*(a + b*sec(c + d*x))^(3/2)*sin(c + d*x))/(3*d) - ((3*a^2*A - 6*A*b^2 - 14*a*b*B)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(3*d)],
[cos(c + d*x)^2*(a + b*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x)), x, 9, ((a + b)*(9*a*A*b + 4*a^2*B - 8*b^2*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - ((2*a^3*A - a^2*A*b + 12*a*A*b^2 - 4*A*b^3 + 12*a^2*b*B - 12*a*b^2*B - 4*b^3*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (a*(4*a^2*A + 15*A*b^2 + 20*a*b*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (a*(7*A*b + 4*a*B)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(4*d) + (a*A*cos(c + d*x)*(a + b*sec(c + d*x))^(3/2)*sin(c + d*x))/(2*d) - ((9*a*A*b + 4*a^2*B - 8*b^2*B)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(4*d)],
[cos(c + d*x)^3*(a + b*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x)), x, 9, ((a + b)*(16*a^2*A + 33*A*b^2 + 54*a*b*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - ((38*a^2*A*b - 13*a*A*b^2 + 24*A*b^3 + 12*a^3*B - 6*a^2*b*B + 72*a*b^2*B - 24*b^3*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + ((20*a^2*A*b + 5*A*b^3 + 8*a^3*B + 30*a*b^2*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (a*(3*A*b + 2*a*B)*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(4*d) + (a*A*cos(c + d*x)^2*(a + b*sec(c + d*x))^(3/2)*sin(c + d*x))/(3*d) + ((16*a^2*A + 33*A*b^2 + 54*a*b*B)*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(24*d)],
[cos(c + d*x)^4*(a + b*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x)), x, 10, ((a + b)*(284*a^2*A*b + 15*A*b^3 + 128*a^3*B + 264*a*b^2*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(192*a*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - ((72*a^3*A - 36*a^2*A*b + 322*a*A*b^2 - 59*A*b^3 + 304*a^2*b*B - 104*a*b^2*B + 192*b^3*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(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(1 + cos(c + d*x))*sqrt((b + a*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*a*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + ((36*a^2*A + 59*A*b^2 + 104*a*b*B)*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(96*d) + (a*(11*A*b + 8*a*B)*cos(c + d*x)^2*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(24*d) + (a*A*cos(c + d*x)^3*(a + b*sec(c + d*x))^(3/2)*sin(c + d*x))/(4*d) + ((284*a^2*A*b + 15*A*b^3 + 128*a^3*B + 264*a*b^2*B)*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(192*a*d)],


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


[sec(c + d*x)^3*(A + B*sec(c + d*x))/sqrt(a + b*sec(c + d*x)), x, 7, (2*(a + b)*(10*a*A*b - 8*a^2*B - 9*b^2*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((-a + b)/(a + b))))/(15*b^3*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*(10*a*A*b - 5*A*b^2 - 8*a^2*B - 2*a*b*B - 9*b^2*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((-a + b)/(a + b))))/(15*b^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*(10*a*A*b - 8*a^2*B - 9*b^2*B)*sqrt(a + b*sec(c + d*x))*tan((c + d*x)/2))/(15*b^3*d) + (2*(5*A*b - 4*a*B)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(15*b^2*d) + (2*B*sec(c + d*x)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(5*b*d)],
[sec(c + d*x)^2*(A + B*sec(c + d*x))/sqrt(a + b*sec(c + d*x)), x, 6, (-2*(a + b)*(3*A*b - 2*a*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((-a + b)/(a + b))))/(3*b^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(3*A*b - 2*a*B + b*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((-a + b)/(a + b))))/(3*b*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(3*A*b - 2*a*B)*sqrt(a + b*sec(c + d*x))*tan((c + d*x)/2))/(3*b^2*d) + (2*B*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(3*b*d)],
[sec(c + d*x)^1*(A + B*sec(c + d*x))/sqrt(a + b*sec(c + d*x)), x, 5, (-2*(a + b)*B*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(A + B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((-a + b)/(a + b))))/(d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*B*sqrt(a + b*sec(c + d*x))*tan((c + d*x)/2))/(b*d)],
[sec(c + d*x)^0*(A + B*sec(c + d*x))/sqrt(a + b*sec(c + d*x)), x, 4, (-2*(A - B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((-a + b)/(a + b))))/(d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (4*A*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))],
[cos(c + d*x)^1*(A + B*sec(c + d*x))/sqrt(a + b*sec(c + d*x)), x, 8, (A*(a + b)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*B*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*(A*b - 2*a*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (A*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(a*d)],
[cos(c + d*x)^2*(A + B*sec(c + d*x))/sqrt(a + b*sec(c + d*x)), x, 8, -(((a + b)*(3*A*b - 4*a*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))) - (A*(2*a - b)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + ((4*a^2*A + 3*A*b^2 - 4*a*b*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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*a^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (A*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(2*a*d) - ((3*A*b - 4*a*B)*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(4*a^2*d)],
[cos(c + d*x)^3*(A + B*sec(c + d*x))/sqrt(a + b*sec(c + d*x)), x, 9, ((a + b)*(16*a^2*A + 15*A*b^2 - 18*a*b*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(24*a^3*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - ((2*a*A*b + 5*A*b^2 + 12*a^2*B - 6*a*b*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(12*a^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - ((4*a^2*A*b + 5*A*b^3 - 8*a^3*B - 6*a*b^2*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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*a^3*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - ((5*A*b - 6*a*B)*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(12*a^2*d) + (A*cos(c + d*x)^2*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(3*a*d) + ((16*a^2*A + 15*A*b^2 - 18*a*b*B)*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(24*a^3*d)],


[sec(c + d*x)^3*(A + B*sec(c + d*x))/(a + b*sec(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(1 + cos(c + d*x))*sqrt((b + a*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)*b^3*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(6*a*A*b - 3*A*b^2 - 8*a^2*B + 6*a*b*B - b^2*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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)*b^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(6*a^2*A*b - 3*A*b^3 - 8*a^3*B + 5*a*b^2*B)*sqrt(a + b*sec(c + d*x))*tan((c + d*x)/2))/(3*b^3*(a^2 - b^2)*d) + (2*a*(A*b - a*B)*sec(c + d*x)*tan(c + d*x))/(b*(a^2 - b^2)*d*sqrt(a + b*sec(c + d*x))) - (2*(3*a*A*b - 4*a^2*B + b^2*B)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(3*b^2*(a^2 - b^2)*d)],
[sec(c + d*x)^2*(A + B*sec(c + d*x))/(a + b*sec(c + d*x))^(3/2), x, 6, (2*(a*A*b - 2*a^2*B + b^2*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*(A*b - 2*a*B + b*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*(a*A*b - 2*a^2*B + b^2*B)*sqrt(a + b*sec(c + d*x))*tan((c + d*x)/2))/(b^2*(a^2 - b^2)*d) + (2*a*(A*b - a*B)*tan(c + d*x))/(b*(a^2 - b^2)*d*sqrt(a + b*sec(c + d*x)))],
[sec(c + d*x)^1*(A + B*sec(c + d*x))/(a + b*sec(c + d*x))^(3/2), x, 5, (-2*(A*b - a*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(A - B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(A*b - a*B)*tan((c + d*x)/2))/(b*(a + b)*d*sqrt(a + b*sec(c + d*x)))],
[sec(c + d*x)^0*(A + B*sec(c + d*x))/(a + b*sec(c + d*x))^(3/2), x, 7, (2*(A*b - a*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*(A - B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (4*A*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*(A*b - a*B)*tan((1/2)*(c + d*x)))/(a*(a + b)*d*sqrt(a + b*sec(c + d*x)))],
[cos(c + d*x)^1*(A + B*sec(c + d*x))/(a + b*sec(c + d*x))^(3/2), x, 8, ((a^2*A - 3*A*b^2 + 2*a*b*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(A*b - a*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*(3*A*b - 2*a*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*b*(A*b - a*B)*sin(c + d*x))/(a*(a^2 - b^2)*d*sqrt(a + b*sec(c + d*x))) + ((a^2*A - 3*A*b^2 + 2*a*b*B)*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(a^2*(a^2 - b^2)*d)],
[cos(c + d*x)^2*(A + B*sec(c + d*x))/(a + b*sec(c + d*x))^(3/2), x, 9, -(((7*a^2*A*b - 15*A*b^3 - 4*a^3*B + 12*a*b^2*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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^3*(a - b)*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))) - ((2*a^2*A - 3*a*A*b + 5*A*b^2 - 4*a*b*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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^2*(a - b)*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + ((4*a^2*A + 15*A*b^2 - 12*a*b*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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*a^3*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*b*(A*b - a*B)*cos(c + d*x)*sin(c + d*x))/(a*(a^2 - b^2)*d*sqrt(a + b*sec(c + d*x))) + ((a^2*A - 5*A*b^2 + 4*a*b*B)*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(2*a^2*(a^2 - b^2)*d) - ((7*a^2*A*b - 15*A*b^3 - 4*a^3*B + 12*a*b^2*B)*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(4*a^3*(a^2 - b^2)*d)],
[cos(c + d*x)^3*(A + B*sec(c + d*x))/(a + b*sec(c + d*x))^(3/2), x, 10, ((16*a^4*A + 41*a^2*A*b^2 - 105*A*b^4 - 42*a^3*b*B + 90*a*b^3*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(24*a^4*(a - b)*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + ((10*a^2*A*b - 21*a*A*b^2 + 35*A*b^3 - 12*a^3*B + 18*a^2*b*B - 30*a*b^2*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(12*a^3*(a - b)*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - ((12*a^2*A*b + 35*A*b^3 - 8*a^3*B - 30*a*b^2*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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*a^4*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*b*(A*b - a*B)*cos(c + d*x)^2*sin(c + d*x))/(a*(a^2 - b^2)*d*sqrt(a + b*sec(c + d*x))) - ((11*a^2*A*b - 35*A*b^3 - 6*a^3*B + 30*a*b^2*B)*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(12*a^3*(a^2 - b^2)*d) + ((a^2*A - 7*A*b^2 + 6*a*b*B)*cos(c + d*x)^2*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(3*a^2*(a^2 - b^2)*d) + ((16*a^4*A + 41*a^2*A*b^2 - 105*A*b^4 - 42*a^3*b*B + 90*a*b^3*B)*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(24*a^4*(a^2 - b^2)*d)],


[sec(c + d*x)^4*(A + B*sec(c + d*x))/(a + b*sec(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(1 + cos(c + d*x))*sqrt((b + a*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^4*(a + b)*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(8*a^3*A*b - 6*a^2*A*b^2 - 9*a*A*b^3 + 3*A*b^4 - 16*a^4*B + 12*a^3*b*B + 16*a^2*b^2*B - 9*a*b^3*B + b^4*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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^3*(a + b)*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (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*sec(c + d*x))*tan((c + d*x)/2))/(3*b^4*(a^2 - b^2)^2*d) + (2*a*(A*b - a*B)*sec(c + d*x)^2*tan(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*sec(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)*sec(c + d*x)*tan(c + d*x))/(3*b^2*(a^2 - b^2)^2*d*sqrt(a + b*sec(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*sec(c + d*x))*tan(c + d*x))/(3*b^3*(a^2 - b^2)^2*d)],
[sec(c + d*x)^3*(A + B*sec(c + d*x))/(a + b*sec(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(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*(2*a^2*A*b - 3*a*A*b^2 - 3*A*b^3 - 8*a^3*B + 6*a^2*b*B + 9*a*b^2*B - 3*b^3*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (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*sec(c + d*x))*tan((c + d*x)/2))/(3*b^3*(a^2 - b^2)^2*d) + (2*a*(A*b - a*B)*sec(c + d*x)*tan(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*sec(c + d*x))^(3/2)) + (2*a*(a^2*A*b - 5*A*b^3 - 4*a^3*B + 8*a*b^2*B)*tan(c + d*x))/(3*b^2*(a^2 - b^2)^2*d*sqrt(a + b*sec(c + d*x)))],
[sec(c + d*x)^2*(A + B*sec(c + d*x))/(a + b*sec(c + d*x))^(5/2), x, 6, (2*(a^2*A*b + 3*A*b^3 + 2*a^3*B - 6*a*b^2*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(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((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*(a^2*A*b + 3*A*b^3 + 2*a^3*B - 6*a*b^2*B)*tan((c + d*x)/2))/(3*(a - b)*b^2*(a + b)^2*d*sqrt(a + b*sec(c + d*x))) + (2*a*(A*b - a*B)*tan(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*sec(c + d*x))^(3/2))],
[sec(c + d*x)^1*(A + B*sec(c + d*x))/(a + b*sec(c + d*x))^(5/2), x, 6, (-2*(4*a*A*b - a^2*B - 3*b^2*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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*(a + b)*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(3*a*A + A*b - a*B - 3*b*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(4*a*A*b - a^2*B - 3*b^2*B)*tan((c + d*x)/2))/(3*(a - b)*b*(a + b)^2*d*sqrt(a + b*sec(c + d*x))) - (2*(A*b - a*B)*tan(c + d*x))/(3*(a^2 - b^2)*d*(a + b*sec(c + d*x))^(3/2))],
[sec(c + d*x)^0*(A + B*sec(c + d*x))/(a + b*sec(c + d*x))^(5/2), x, 8, (2*(7*a^2*A*b - 3*A*b^3 - 4*a^3*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(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((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (4*A*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticPi(-1, arcsin(tan((c + d*x)/2)), -((-a + b)/(a + b))))/(a^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*(7*a^2*A*b - 3*A*b^3 - 4*a^3*B)*tan((c + d*x)/2))/(3*a^2*(a - b)*(a + b)^2*d*sqrt(a + b*sec(c + d*x))) + (2*b*(A*b - a*B)*tan(c + d*x))/(3*a*(a^2 - b^2)*d*(a + b*sec(c + d*x))^(3/2))],
[cos(c + d*x)^1*(A + B*sec(c + d*x))/(a + b*sec(c + d*x))^(5/2), x, 9, ((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(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(6*a^2*A*b + 3*a*A*b^2 - 5*A*b^3 - 3*a^3*B - 3*a^2*b*B + 2*a*b^2*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*(5*A*b - 2*a*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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^3*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*b*(A*b - a*B)*sin(c + d*x))/(3*a*(a^2 - b^2)*d*(a + b*sec(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)*sin(c + d*x))/(3*a^2*(a^2 - b^2)^2*d*sqrt(a + b*sec(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)*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(3*a^3*(a^2 - b^2)^2*d)],
[cos(c + d*x)^2*(A + B*sec(c + d*x))/(a + b*sec(c + d*x))^(5/2), x, 10, -(((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(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(12*a^4*(a - b)^2*(a + b)*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))) - ((6*a^4*A - 9*a^3*A*b + 33*a^2*A*b^2 + 21*a*A*b^3 - 35*A*b^4 - 24*a^3*b*B - 12*a^2*b^2*B + 20*a*b^3*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(6*a^3*(a - b)^2*(a + b)*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + ((4*a^2*A + 35*A*b^2 - 20*a*b*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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*a^4*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*b*(A*b - a*B)*cos(c + d*x)*sin(c + d*x))/(3*a*(a^2 - b^2)*d*(a + b*sec(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)*cos(c + d*x)*sin(c + d*x))/(3*a^2*(a^2 - b^2)^2*d*sqrt(a + b*sec(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)*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(6*a^3*(a^2 - b^2)^2*d) - ((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)*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(12*a^4*(a^2 - b^2)^2*d)],


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


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


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


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


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


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


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


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


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


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


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


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


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


[sec(c + d*x)^(3/2)*(a + b*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x)), x, 13, -((a + b)*(264*a^2*A*b + 128*A*b^3 + 15*a^3*B + 284*a*b^2*B)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*a)/(a + b))*sqrt(sec(c + d*x)))/(192*b*d*sqrt(a + b*sec(c + d*x))) + ((472*a^2*A*b + 128*A*b^3 + 133*a^3*B + 356*a*b^2*B)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*a)/(a + b))*sqrt(sec(c + d*x)))/(192*d*sqrt(a + b*sec(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((b + a*cos(c + d*x))/(a + b))*EllipticPi(2, (c + d*x)/2, (2*a)/(a + b))*sqrt(sec(c + d*x)))/(64*b*d*sqrt(a + b*sec(c + d*x))) + ((264*a^2*A*b + 128*A*b^3 + 15*a^3*B + 284*a*b^2*B)*sqrt(sec(c + d*x))*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(192*b*d) + ((104*a*A*b + 59*a^2*B + 36*b^2*B)*sec(c + d*x)^(3/2)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(96*d) + (b*(8*A*b + 11*a*B)*sec(c + d*x)^(5/2)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(24*d) + (b*B*sec(c + d*x)^(5/2)*(a + b*sec(c + d*x))^(3/2)*sin(c + d*x))/(4*d)],
[sqrt(sec(c + d*x))*(a + b*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x)), x, 12, -((a + b)*(54*a*A*b + 33*a^2*B + 16*b^2*B)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*a)/(a + b))*sqrt(sec(c + d*x)))/(24*d*sqrt(a + b*sec(c + d*x))) + ((48*a^3*A + 66*a*A*b^2 + 59*a^2*b*B + 16*b^3*B)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*a)/(a + b))*sqrt(sec(c + d*x)))/(24*d*sqrt(a + b*sec(c + d*x))) + ((30*a^2*A*b + 8*A*b^3 + 5*a^3*B + 20*a*b^2*B)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticPi(2, (c + d*x)/2, (2*a)/(a + b))*sqrt(sec(c + d*x)))/(8*d*sqrt(a + b*sec(c + d*x))) + ((54*a*A*b + 33*a^2*B + 16*b^2*B)*sqrt(sec(c + d*x))*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(24*d) + (b*(2*A*b + 3*a*B)*sec(c + d*x)^(3/2)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(4*d) + (b*B*sec(c + d*x)^(3/2)*(a + b*sec(c + d*x))^(3/2)*sin(c + d*x))/(3*d)],
[((a + b*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x)))/sqrt(sec(c + d*x)), x, 11, ((a + b)*(8*a^2*A - 4*A*b^2 - 9*a*b*B)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*a)/(a + b))*sqrt(sec(c + d*x)))/(4*d*sqrt(a + b*sec(c + d*x))) + ((16*a^2*A*b + 4*A*b^3 + 8*a^3*B + 11*a*b^2*B)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*a)/(a + b))*sqrt(sec(c + d*x)))/(4*d*sqrt(a + b*sec(c + d*x))) + (b*(20*a*A*b + 15*a^2*B + 4*b^2*B)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticPi(2, (c + d*x)/2, (2*a)/(a + b))*sqrt(sec(c + d*x)))/(4*d*sqrt(a + b*sec(c + d*x))) + (b*(4*A*b + 7*a*B)*sqrt(sec(c + d*x))*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(4*d) + (b*B*sqrt(sec(c + d*x))*(a + b*sec(c + d*x))^(3/2)*sin(c + d*x))/(2*d)],
[((a + b*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x)))/sec(c + d*x)^(3/2), x, 11, ((a + b)*(14*a*A*b + 6*a^2*B - 3*b^2*B)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(3*d*sqrt(a + b*sec(c + d*x))) + ((2*a^3*A + 4*a*A*b^2 + 12*a^2*b*B + 3*b^3*B)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(3*d*sqrt(a + b*sec(c + d*x))) + (b^2*(2*A*b + 5*a*B)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*sec(c + d*x))) - (b*(2*a*A - 3*b*B)*sqrt(sec(c + d*x))*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(3*d) + (2*a*A*(a + b*sec(c + d*x))^(3/2)*sin(c + d*x))/(3*d*sqrt(sec(c + d*x)))],
[((a + b*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x)))/sec(c + d*x)^(5/2), x, 11, (2*(a + b)*(9*a^2*A + 23*A*b^2 + 35*a*b*B)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*a)/(a + b))*sqrt(sec(c + d*x)))/(15*d*sqrt(a + b*sec(c + d*x))) + (2*(8*a^2*A*b - 8*A*b^3 + 5*a^3*B + 10*a*b^2*B)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*a)/(a + b))*sqrt(sec(c + d*x)))/(15*d*sqrt(a + b*sec(c + d*x))) + (2*b^3*B*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticPi(2, (c + d*x)/2, (2*a)/(a + b))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*sec(c + d*x))) + (2*a*(8*A*b + 5*a*B)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(15*d*sqrt(sec(c + d*x))) + (2*a*A*(a + b*sec(c + d*x))^(3/2)*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2))],
[((a + b*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x)))/sec(c + d*x)^(7/2), x, 9, (2*(a + b)*(145*a^2*A*b + 15*A*b^3 + 63*a^3*B + 161*a*b^2*B)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(105*a*d*sqrt(a + b*sec(c + d*x))) + (2*(25*a^4*A - 10*a^2*A*b^2 - 15*A*b^4 + 56*a^3*b*B - 56*a*b^3*B)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(105*a*d*sqrt(a + b*sec(c + d*x))) + (2*a*(10*A*b + 7*a*B)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(35*d*sec(c + d*x)^(3/2)) + (2*(25*a^2*A + 45*A*b^2 + 77*a*b*B)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(105*d*sqrt(sec(c + d*x))) + (2*a*A*(a + b*sec(c + d*x))^(3/2)*sin(c + d*x))/(7*d*sec(c + d*x)^(5/2))],
[((a + b*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x)))/sec(c + d*x)^(9/2), x, 10, (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((b + a*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(315*a^2*d*sqrt(a + b*sec(c + d*x))) + (2*(114*a^4*A*b - 124*a^2*A*b^3 + 10*A*b^5 + 75*a^5*B - 30*a^3*b^2*B - 45*a*b^4*B)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(315*a^2*d*sqrt(a + b*sec(c + d*x))) + (2*a*(4*A*b + 3*a*B)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(21*d*sec(c + d*x)^(5/2)) + (2*(49*a^2*A + 75*A*b^2 + 135*a*b*B)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(315*d*sec(c + d*x)^(3/2)) + (2*(163*a^2*A*b + 5*A*b^3 + 75*a^3*B + 135*a*b^2*B)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(315*a*d*sqrt(sec(c + d*x))) + (2*a*A*(a + b*sec(c + d*x))^(3/2)*sin(c + d*x))/(9*d*sec(c + d*x)^(7/2))],
[((a + b*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x)))/sec(c + d*x)^(11/2), x, 11, (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((b + a*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(3465*a^3*d*sqrt(a + b*sec(c + d*x))) + (2*(675*a^6*A - 390*a^4*A*b^2 - 245*a^2*A*b^4 - 40*A*b^6 + 1254*a^5*b*B - 1364*a^3*b^3*B + 110*a*b^5*B)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(3465*a^3*d*sqrt(a + b*sec(c + d*x))) + (2*a*(14*A*b + 11*a*B)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(99*d*sec(c + d*x)^(7/2)) + (2*(81*a^2*A + 113*A*b^2 + 209*a*b*B)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(693*d*sec(c + d*x)^(5/2)) + (2*(1145*a^2*A*b + 15*A*b^3 + 539*a^3*B + 825*a*b^2*B)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(3465*a*d*sec(c + d*x)^(3/2)) + (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*sec(c + d*x))*sin(c + d*x))/(3465*a^2*d*sqrt(sec(c + d*x))) + (2*a*A*(a + b*sec(c + d*x))^(3/2)*sin(c + d*x))/(11*d*sec(c + d*x)^(9/2))],


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


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


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


[(sec(c + d*x)^(5/2)*(A + B*sec(c + d*x)))/(a + b*sec(c + d*x))^(5/2), x, 11, (2*(4*A*b^3 + 3*a^3*B - 7*a*b^2*B)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*a)/(a + b))*sqrt(sec(c + d*x)))/(3*(a - b)^2*b^2*(a + b)*d*sqrt(a + b*sec(c + d*x))) + (2*(A*b - a*B)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*a)/(a + b))*sqrt(sec(c + d*x)))/(3*b*(a^2 - b^2)*d*sqrt(a + b*sec(c + d*x))) + (2*B*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticPi(2, (c + d*x)/2, (2*a)/(a + b))*sqrt(sec(c + d*x)))/(b^2*d*sqrt(a + b*sec(c + d*x))) + (2*a*(A*b - a*B)*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*sec(c + d*x))^(3/2)) - (2*a*(4*A*b^3 + 3*a^3*B - 7*a*b^2*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(3*b^2*(a^2 - b^2)^2*d*sqrt(a + b*sec(c + d*x)))],
[(sec(c + d*x)^(3/2)*(A + B*sec(c + d*x)))/(a + b*sec(c + d*x))^(5/2), x, 8, -((2*(3*a^2*A + A*b^2 - 4*a*b*B)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(3*a*(a - b)^2*(a + b)*d*sqrt(a + b*sec(c + d*x)))) - (2*(A*b - a*B)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(3*a*(a^2 - b^2)*d*sqrt(a + b*sec(c + d*x))) + (2*a*(A*b - a*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*sec(c + d*x))^(3/2)) + (2*(2*a^2*A*b + 2*A*b^3 + a^3*B - 5*a*b^2*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(3*b*(a^2 - b^2)^2*d*sqrt(a + b*sec(c + d*x)))],
[(sqrt(sec(c + d*x))*(A + B*sec(c + d*x)))/(a + b*sec(c + d*x))^(5/2), x, 8, (2*(6*a^2*A*b - 2*A*b^3 - 3*a^3*B - a*b^2*B)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(3*a^2*(a - b)^2*(a + b)*d*sqrt(a + b*sec(c + d*x))) + (2*(3*a^2*A - 2*A*b^2 - a*b*B)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(3*a^2*(a^2 - b^2)*d*sqrt(a + b*sec(c + d*x))) - (2*(A*b - a*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(3*(a^2 - b^2)*d*(a + b*sec(c + d*x))^(3/2)) - (2*(5*a^2*A*b - A*b^3 - 2*a^3*B - 2*a*b^2*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(3*a*(a^2 - b^2)^2*d*sqrt(a + b*sec(c + d*x)))],
[(A + B*sec(c + d*x))/(sqrt(sec(c + d*x))*(a + b*sec(c + d*x))^(5/2)), x, 8, (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((b + a*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*a)/(a + b))*sqrt(sec(c + d*x)))/(3*a^3*(a - b)^2*(a + b)*d*sqrt(a + b*sec(c + d*x))) - (2*(9*a^2*A*b - 8*A*b^3 - 3*a^3*B + 2*a*b^2*B)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*a)/(a + b))*sqrt(sec(c + d*x)))/(3*a^3*(a^2 - b^2)*d*sqrt(a + b*sec(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*sec(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*sec(c + d*x)))],
[(A + B*sec(c + d*x))/(sec(c + d*x)^(3/2)*(a + b*sec(c + d*x))^(5/2)), x, 9, (-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((b + a*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*a)/(a + b))*sqrt(sec(c + d*x)))/(3*a^4*(a - b)^2*(a + b)*d*sqrt(a + b*sec(c + d*x))) + (2*(a^4*A + 16*a^2*A*b^2 - 16*A*b^4 - 9*a^3*b*B + 8*a*b^3*B)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*a)/(a + b))*sqrt(sec(c + d*x)))/(3*a^4*(a^2 - b^2)*d*sqrt(a + b*sec(c + d*x))) + (2*b*(A*b - a*B)*sin(c + d*x))/(3*a*(a^2 - b^2)*d*sqrt(sec(c + d*x))*(a + b*sec(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*sqrt(sec(c + d*x))*sqrt(a + b*sec(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*sec(c + d*x))*sin(c + d*x))/(3*a^3*(a^2 - b^2)^2*d*sqrt(sec(c + d*x)))],
[(A + B*sec(c + d*x))/(sec(c + d*x)^(5/2)*(a + b*sec(c + d*x))^(5/2)), x, 10, (2*(9*a^6*A + 55*a^4*A*b^2 - 212*a^2*A*b^4 + 128*A*b^6 - 40*a^5*b*B + 140*a^3*b^3*B - 80*a*b^5*B)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*a)/(a + b))*sqrt(sec(c + d*x)))/(15*a^5*(a - b)^2*(a + b)*d*sqrt(a + b*sec(c + d*x))) - (2*(17*a^4*A*b + 116*a^2*A*b^3 - 128*A*b^5 - 5*a^5*B - 80*a^3*b^2*B + 80*a*b^4*B)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*a)/(a + b))*sqrt(sec(c + d*x)))/(15*a^5*(a^2 - b^2)*d*sqrt(a + b*sec(c + d*x))) + (2*b*(A*b - a*B)*sin(c + d*x))/(3*a*(a^2 - b^2)*d*sec(c + d*x)^(3/2)*(a + b*sec(c + d*x))^(3/2)) + (2*b*(12*a^2*A*b - 8*A*b^3 - 9*a^3*B + 5*a*b^2*B)*sin(c + d*x))/(3*a^2*(a^2 - b^2)^2*d*sec(c + d*x)^(3/2)*sqrt(a + b*sec(c + d*x))) + (2*(3*a^4*A - 71*a^2*A*b^2 + 48*A*b^4 + 50*a^3*b*B - 30*a*b^3*B)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(15*a^3*(a^2 - b^2)^2*d*sec(c + d*x)^(3/2)) - (2*(14*a^4*A*b - 98*a^2*A*b^3 + 64*A*b^5 - 5*a^5*B + 65*a^3*b^2*B - 40*a*b^4*B)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(15*a^4*(a^2 - b^2)^2*d*sqrt(sec(c + d*x)))],


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


[cos(c + d*x)^(11/2)*(a + b*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x)), x, 11, (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((b + a*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*a)/(a + b)))/(3465*a^3*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(675*a^6*A - 390*a^4*A*b^2 - 245*a^2*A*b^4 - 40*A*b^6 + 1254*a^5*b*B - 1364*a^3*b^3*B + 110*a*b^5*B)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*a)/(a + b)))/(3465*a^3*d*sqrt(cos(c + d*x))*sqrt(a + b*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))*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)*cos(c + d*x)^(3/2)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(3465*a*d) + (2*(81*a^2*A + 113*A*b^2 + 209*a*b*B)*cos(c + d*x)^(5/2)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(693*d) + (2*a*(14*A*b + 11*a*B)*cos(c + d*x)^(7/2)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(99*d) + (2*a*A*cos(c + d*x)^(9/2)*(a + b*sec(c + d*x))^(3/2)*sin(c + d*x))/(11*d)],
[cos(c + d*x)^(9/2)*(a + b*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x)), x, 10, (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((b + a*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*a)/(a + b)))/(315*a^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(114*a^4*A*b - 124*a^2*A*b^3 + 10*A*b^5 + 75*a^5*B - 30*a^3*b^2*B - 45*a*b^4*B)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*a)/(a + b)))/(315*a^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(163*a^2*A*b + 5*A*b^3 + 75*a^3*B + 135*a*b^2*B)*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(315*a*d) + (2*(49*a^2*A + 75*A*b^2 + 135*a*b*B)*cos(c + d*x)^(3/2)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(315*d) + (2*a*(4*A*b + 3*a*B)*cos(c + d*x)^(5/2)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(21*d) + (2*a*A*cos(c + d*x)^(7/2)*(a + b*sec(c + d*x))^(3/2)*sin(c + d*x))/(9*d)],
[cos(c + d*x)^(7/2)*(a + b*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x)), x, 9, (2*(a + b)*(145*a^2*A*b + 15*A*b^3 + 63*a^3*B + 161*a*b^2*B)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*a)/(a + b)))/(105*a*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(25*a^4*A - 10*a^2*A*b^2 - 15*A*b^4 + 56*a^3*b*B - 56*a*b^3*B)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*a)/(a + b)))/(105*a*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(25*a^2*A + 45*A*b^2 + 77*a*b*B)*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(105*d) + (2*a*(10*A*b + 7*a*B)*cos(c + d*x)^(3/2)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(35*d) + (2*a*A*cos(c + d*x)^(5/2)*(a + b*sec(c + d*x))^(3/2)*sin(c + d*x))/(7*d)],
[cos(c + d*x)^(5/2)*(a + b*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x)), x, 11, (2*(a + b)*(9*a^2*A + 23*A*b^2 + 35*a*b*B)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*a)/(a + b)))/(15*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(8*a^2*A*b - 8*A*b^3 + 5*a^3*B + 10*a*b^2*B)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*a)/(a + b)))/(15*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*b^3*B*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*a)/(a + b)))/(d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*a*(8*A*b + 5*a*B)*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(15*d) + (2*a*A*cos(c + d*x)^(3/2)*(a + b*sec(c + d*x))^(3/2)*sin(c + d*x))/(5*d)],
[cos(c + d*x)^(3/2)*(a + b*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x)), x, 11, ((a + b)*(14*a*A*b + 6*a^2*B - 3*b^2*B)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*a)/(a + b)))/(3*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + ((2*a^3*A + 4*a*A*b^2 + 12*a^2*b*B + 3*b^3*B)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*a)/(a + b)))/(3*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (b^2*(2*A*b + 5*a*B)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticPi(2, (c + d*x)/2, (2*a)/(a + b)))/(d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (b*(2*a*A - 3*b*B)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))) + (2*a*A*sqrt(cos(c + d*x))*(a + b*sec(c + d*x))^(3/2)*sin(c + d*x))/(3*d)],
[cos(c + d*x)^(1/2)*(a + b*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x)), x, 11, ((a + b)*(8*a^2*A - 4*A*b^2 - 9*a*b*B)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*a)/(a + b)))/(4*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + ((16*a^2*A*b + 4*A*b^3 + 8*a^3*B + 11*a*b^2*B)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*a)/(a + b)))/(4*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (b*(20*a*A*b + 15*a^2*B + 4*b^2*B)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticPi(2, (c + d*x)/2, (2*a)/(a + b)))/(4*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (b*(4*A*b + 7*a*B)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(4*d*sqrt(cos(c + d*x))) + (b*B*(a + b*sec(c + d*x))^(3/2)*sin(c + d*x))/(2*d*sqrt(cos(c + d*x)))],
[((a + b*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x)))/cos(c + d*x)^(1/2), x, 12, -((a + b)*(54*a*A*b + 33*a^2*B + 16*b^2*B)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*a)/(a + b)))/(24*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + ((48*a^3*A + 66*a*A*b^2 + 59*a^2*b*B + 16*b^3*B)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*a)/(a + b)))/(24*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + ((30*a^2*A*b + 8*A*b^3 + 5*a^3*B + 20*a*b^2*B)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticPi(2, (c + d*x)/2, (2*a)/(a + b)))/(8*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (b*(2*A*b + 3*a*B)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(4*d*cos(c + d*x)^(3/2)) + ((54*a*A*b + 33*a^2*B + 16*b^2*B)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(24*d*sqrt(cos(c + d*x))) + (b*B*(a + b*sec(c + d*x))^(3/2)*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2))],
[((a + b*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x)))/cos(c + d*x)^(3/2), x, 13, -((a + b)*(264*a^2*A*b + 128*A*b^3 + 15*a^3*B + 284*a*b^2*B)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*a)/(a + b)))/(192*b*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + ((472*a^2*A*b + 128*A*b^3 + 133*a^3*B + 356*a*b^2*B)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*a)/(a + b)))/(192*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(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((b + a*cos(c + d*x))/(a + b))*EllipticPi(2, (c + d*x)/2, (2*a)/(a + b)))/(64*b*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (b*(8*A*b + 11*a*B)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(24*d*cos(c + d*x)^(5/2)) + ((104*a*A*b + 59*a^2*B + 36*b^2*B)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(96*d*cos(c + d*x)^(3/2)) + ((264*a^2*A*b + 128*A*b^3 + 15*a^3*B + 284*a*b^2*B)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(192*b*d*sqrt(cos(c + d*x))) + (b*B*(a + b*sec(c + d*x))^(3/2)*sin(c + d*x))/(4*d*cos(c + d*x)^(5/2))],


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


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


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


[(cos(c + d*x)^(5/2)*(A + B*sec(c + d*x)))/(a + b*sec(c + d*x))^(5/2), x, 10, (2*(9*a^6*A + 55*a^4*A*b^2 - 212*a^2*A*b^4 + 128*A*b^6 - 40*a^5*b*B + 140*a^3*b^3*B - 80*a*b^5*B)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*a)/(a + b)))/(15*a^5*(a - b)^2*(a + b)*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*(17*a^4*A*b + 116*a^2*A*b^3 - 128*A*b^5 - 5*a^5*B - 80*a^3*b^2*B + 80*a*b^4*B)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*a)/(a + b)))/(15*a^5*(a^2 - b^2)*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*b*(A*b - a*B)*cos(c + d*x)^(3/2)*sin(c + d*x))/(3*a*(a^2 - b^2)*d*(a + b*sec(c + d*x))^(3/2)) + (2*b*(12*a^2*A*b - 8*A*b^3 - 9*a^3*B + 5*a*b^2*B)*cos(c + d*x)^(3/2)*sin(c + d*x))/(3*a^2*(a^2 - b^2)^2*d*sqrt(a + b*sec(c + d*x))) - (2*(14*a^4*A*b - 98*a^2*A*b^3 + 64*A*b^5 - 5*a^5*B + 65*a^3*b^2*B - 40*a*b^4*B)*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(15*a^4*(a^2 - b^2)^2*d) + (2*(3*a^4*A - 71*a^2*A*b^2 + 48*A*b^4 + 50*a^3*b*B - 30*a*b^3*B)*cos(c + d*x)^(3/2)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(15*a^3*(a^2 - b^2)^2*d)],
[(cos(c + d*x)^(3/2)*(A + B*sec(c + d*x)))/(a + b*sec(c + d*x))^(5/2), x, 9, (-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((b + a*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*a)/(a + b)))/(3*a^4*(a - b)^2*(a + b)*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(a^4*A + 16*a^2*A*b^2 - 16*A*b^4 - 9*a^3*b*B + 8*a*b^3*B)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*a)/(a + b)))/(3*a^4*(a^2 - b^2)*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(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*sec(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)*sqrt(cos(c + d*x))*sin(c + d*x))/(3*a^2*(a^2 - b^2)^2*d*sqrt(a + b*sec(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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(3*a^3*(a^2 - b^2)^2*d)],
[(cos(c + d*x)^(1/2)*(A + B*sec(c + d*x)))/(a + b*sec(c + d*x))^(5/2), x, 8, (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((b + a*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*a)/(a + b)))/(3*a^3*(a - b)^2*(a + b)*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*(9*a^2*A*b - 8*A*b^3 - 3*a^3*B + 2*a*b^2*B)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*a)/(a + b)))/(3*a^3*(a^2 - b^2)*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(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*sec(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*sec(c + d*x)))],
[(A + B*sec(c + d*x))/(cos(c + d*x)^(1/2)*(a + b*sec(c + d*x))^(5/2)), x, 8, (2*(6*a^2*A*b - 2*A*b^3 - 3*a^3*B - a*b^2*B)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*a)/(a + b)))/(3*a^2*(a - b)^2*(a + b)*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(3*a^2*A - 2*A*b^2 - a*b*B)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*a)/(a + b)))/(3*a^2*(a^2 - b^2)*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*(A*b - a*B)*sin(c + d*x))/(3*(a^2 - b^2)*d*sqrt(cos(c + d*x))*(a + b*sec(c + d*x))^(3/2)) - (2*(5*a^2*A*b - A*b^3 - 2*a^3*B - 2*a*b^2*B)*sin(c + d*x))/(3*a*(a^2 - b^2)^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))],
[(A + B*sec(c + d*x))/(cos(c + d*x)^(3/2)*(a + b*sec(c + d*x))^(5/2)), x, 8, (-2*(3*a^2*A + A*b^2 - 4*a*b*B)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*a)/(a + b)))/(3*a*(a - b)^2*(a + b)*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*(A*b - a*B)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*a)/(a + b)))/(3*a*(a^2 - b^2)*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*a*(A*b - a*B)*sin(c + d*x))/(3*b*(a^2 - b^2)*d*sqrt(cos(c + d*x))*(a + b*sec(c + d*x))^(3/2)) + (2*(2*a^2*A*b + 2*A*b^3 + a^3*B - 5*a*b^2*B)*sin(c + d*x))/(3*b*(a^2 - b^2)^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))],
[(A + B*sec(c + d*x))/(cos(c + d*x)^(5/2)*(a + b*sec(c + d*x))^(5/2)), x, 11, (2*(4*A*b^3 + 3*a^3*B - 7*a*b^2*B)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*a)/(a + b)))/(3*(a - b)^2*b^2*(a + b)*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(A*b - a*B)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*a)/(a + b)))/(3*b*(a^2 - b^2)*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*B*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticPi(2, (c + d*x)/2, (2*a)/(a + b)))/(b^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*a*(A*b - a*B)*sin(c + d*x))/(3*b*(a^2 - b^2)*d*cos(c + d*x)^(3/2)*(a + b*sec(c + d*x))^(3/2)) - (2*a*(4*A*b^3 + 3*a^3*B - 7*a*b^2*B)*sin(c + d*x))/(3*b^2*(a^2 - b^2)^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))],
[(A + B*sec(c + d*x))/(cos(c + d*x)^(7/2)*(a + b*sec(c + d*x))^(5/2)), x, 12, ((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((b + a*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*a)/(a + b)))/(3*(a - b)^2*b^3*(a + b)*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - ((2*a*A*b - 5*a^2*B + 3*b^2*B)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*a)/(a + b)))/(3*b^2*(a^2 - b^2)*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + ((2*A*b - 5*a*B)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*a)/(a + b)))/(b^3*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*a*(A*b - a*B)*sin(c + d*x))/(3*b*(a^2 - b^2)*d*cos(c + d*x)^(5/2)*(a + b*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*cos(c + d*x)^(3/2)*sqrt(a + b*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*sec(c + d*x))*sin(c + d*x))/(3*b^3*(a^2 - b^2)^2*d*sqrt(cos(c + d*x)))],


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


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


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


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


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


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


[sec(c + d*x)^2*(A + C*sec(c + d*x)^2)*(a + b*sec(c + d*x))^4, x, 8, (a*b*(8*a^2*A + 6*A*b^2 + 6*a^2*C + 5*b^2*C)*arctanh(sin(c + d*x)))/(4*d) + ((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)*tan(c + d*x))/(105*d) + (a*b*(8*a^2*A + 6*A*b^2 + 6*a^2*C + 5*b^2*C)*sec(c + d*x)*tan(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)*sec(c + d*x)^2*tan(c + d*x))/(210*d) + (a*(126*A*b^2 + 6*a^2*C + 103*b^2*C)*sec(c + d*x)^2*(a + b*sec(c + d*x))*tan(c + d*x))/(210*d) + ((7*A*b^2 + 2*a^2*C + 6*b^2*C)*sec(c + d*x)^2*(a + b*sec(c + d*x))^2*tan(c + d*x))/(35*d) + (2*a*C*sec(c + d*x)^2*(a + b*sec(c + d*x))^3*tan(c + d*x))/(21*d) + (C*sec(c + d*x)^2*(a + b*sec(c + d*x))^4*tan(c + d*x))/(7*d)],
[sec(c + d*x)^1*(A + C*sec(c + d*x)^2)*(a + b*sec(c + d*x))^4, x, 7, ((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)*arctanh(sin(c + d*x)))/(16*d) + (4*a*b*(15*a^2*A + 10*A*b^2 + 10*a^2*C + 8*b^2*C)*tan(c + d*x))/(15*d) + ((460*a^2*A*b^2 + 90*A*b^4 + 8*a^4*C + 334*a^2*b^2*C + 75*b^4*C)*sec(c + d*x)*tan(c + d*x))/(240*d) + (a*(50*A*b^2 + 4*a^2*C + 39*b^2*C)*sec(c + d*x)*(a + b*sec(c + d*x))*tan(c + d*x))/(60*d) + ((30*A*b^2 + 12*a^2*C + 25*b^2*C)*sec(c + d*x)*(a + b*sec(c + d*x))^2*tan(c + d*x))/(120*d) + (2*a*C*sec(c + d*x)*(a + b*sec(c + d*x))^3*tan(c + d*x))/(15*d) + (C*sec(c + d*x)*(a + b*sec(c + d*x))^4*tan(c + d*x))/(6*d)],
[sec(c + d*x)^0*(A + C*sec(c + d*x)^2)*(a + b*sec(c + d*x))^4, x, 7, a^4*A*x + (a*b*(8*a^2*A + 4*A*b^2 + 4*a^2*C + 3*b^2*C)*arctanh(sin(c + d*x)))/(2*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)*tan(c + d*x))/(30*d) + (a*(40*A*b^2 + 6*a^2*C + 29*b^2*C)*(a + b*sec(c + d*x))*tan(c + d*x))/(30*d) + ((5*A*b^2 + 3*a^2*C + 4*b^2*C)*(a + b*sec(c + d*x))^2*tan(c + d*x))/(15*d) + (a*C*(a + b*sec(c + d*x))^3*tan(c + d*x))/(5*d) + (C*(a + b*sec(c + d*x))^4*tan(c + d*x))/(5*d)],
[cos(c + d*x)^1*(A + C*sec(c + d*x)^2)*(a + b*sec(c + d*x))^4, x, 7, 4*a^3*A*b*x + ((48*a^2*A*b^2 + 4*A*b^4 + 8*a^4*C + 24*a^2*b^2*C + 3*b^4*C)*arctanh(sin(c + d*x)))/(8*d) + (A*(a + b*sec(c + d*x))^4*sin(c + d*x))/d - (a*b*(24*a^2*A - 84*A*b^2 - 50*a^2*C - 55*b^2*C)*tan(c + d*x))/(24*d) - (b*(24*a^2*A - 12*A*b^2 - 26*a^2*C - 9*b^2*C)*(a + b*sec(c + d*x))*tan(c + d*x))/(24*d) - (a*b*(12*A - 7*C)*(a + b*sec(c + d*x))^2*tan(c + d*x))/(12*d) - (b*(4*A - C)*(a + b*sec(c + d*x))^3*tan(c + d*x))/(4*d)],
[cos(c + d*x)^2*(A + C*sec(c + d*x)^2)*(a + b*sec(c + d*x))^4, x, 7, (1/2)*a^2*(a^2*A + 12*A*b^2 + 2*a^2*C)*x + (2*a*b*(2*A*b^2 + 2*a^2*C + b^2*C)*arctanh(sin(c + d*x)))/d + (2*A*b*(a + b*sec(c + d*x))^3*sin(c + d*x))/d + (A*cos(c + d*x)*(a + b*sec(c + d*x))^4*sin(c + d*x))/(2*d) - (b^2*(21*a^2*A - 6*A*b^2 - 26*a^2*C - 4*b^2*C)*tan(c + d*x))/(6*d) - (a*b^2*(9*A - 4*C)*(a + b*sec(c + d*x))*tan(c + d*x))/(3*d) - (b^2*(15*A - 2*C)*(a + b*sec(c + d*x))^2*tan(c + d*x))/(6*d)],
[cos(c + d*x)^3*(A + C*sec(c + d*x)^2)*(a + b*sec(c + d*x))^4, x, 7, 2*a*b*(a^2*A + 2*A*b^2 + 2*a^2*C)*x + (b^2*(2*A*b^2 + 12*a^2*C + b^2*C)*arctanh(sin(c + d*x)))/(2*d) + ((2*a^2*A + 6*A*b^2 + 3*a^2*C)*(a + b*sec(c + d*x))^2*sin(c + d*x))/(3*d) + (2*A*b*cos(c + d*x)*(a + b*sec(c + d*x))^3*sin(c + d*x))/(3*d) + (A*cos(c + d*x)^2*(a + b*sec(c + d*x))^4*sin(c + d*x))/(3*d) - (a*b*(4*a^2*A + 26*A*b^2 + 6*a^2*C - 21*b^2*C)*tan(c + d*x))/(6*d) - (b*(4*a^2*A + 18*A*b^2 + 6*a^2*C - 3*b^2*C)*(a + b*sec(c + d*x))*tan(c + d*x))/(6*d)],
[cos(c + d*x)^4*(A + C*sec(c + d*x)^2)*(a + b*sec(c + d*x))^4, x, 7, (1/8)*(3*a^4*A + 24*a^2*A*b^2 + 8*A*b^4 + 4*a^4*C + 48*a^2*b^2*C)*x + (4*a*b^3*C*arctanh(sin(c + d*x)))/d + (b*(23*a^2*A + 12*A*b^2 + 36*a^2*C)*(a + b*sec(c + d*x))*sin(c + d*x))/(12*d) + ((3*a^2*A + 4*A*b^2 + 4*a^2*C)*cos(c + d*x)*(a + b*sec(c + d*x))^2*sin(c + d*x))/(8*d) + (A*b*cos(c + d*x)^2*(a + b*sec(c + d*x))^3*sin(c + d*x))/(3*d) + (A*cos(c + d*x)^3*(a + b*sec(c + d*x))^4*sin(c + d*x))/(4*d) - (b^2*(55*a^2*A + 50*A*b^2 + 84*a^2*C - 24*b^2*C)*tan(c + d*x))/(24*d)],
[cos(c + d*x)^5*(A + C*sec(c + d*x)^2)*(a + b*sec(c + d*x))^4, x, 7, (1/2)*a*b*(3*a^2*A + 4*A*b^2 + 4*a^2*C + 8*b^2*C)*x + (b^4*C*arctanh(sin(c + d*x)))/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)*sin(c + d*x))/(30*d) + (b*(29*a^2*A + 6*A*b^2 + 40*a^2*C)*(b + a*cos(c + d*x))*sin(c + d*x))/(30*d) + ((4*a^2*A + 3*A*b^2 + 5*a^2*C)*(b + a*cos(c + d*x))^2*sin(c + d*x))/(15*d) + (A*b*(b + a*cos(c + d*x))^3*sin(c + d*x))/(5*d) + (A*(b + a*cos(c + d*x))^4*sin(c + d*x))/(5*d)],
[cos(c + d*x)^6*(A + C*sec(c + d*x)^2)*(a + b*sec(c + d*x))^4, x, 7, (1/16)*(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)*x + (4*a*b*(8*a^2*A + 10*A*b^2 + 10*a^2*C + 15*b^2*C)*sin(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)*cos(c + d*x)*sin(c + d*x))/(240*d) + (b*(39*a^2*A + 4*A*b^2 + 50*a^2*C)*cos(c + d*x)^2*(a + b*sec(c + d*x))*sin(c + d*x))/(60*d) + ((25*a^2*A + 12*A*b^2 + 30*a^2*C)*cos(c + d*x)^3*(a + b*sec(c + d*x))^2*sin(c + d*x))/(120*d) + (2*A*b*cos(c + d*x)^4*(a + b*sec(c + d*x))^3*sin(c + d*x))/(15*d) + (A*cos(c + d*x)^5*(a + b*sec(c + d*x))^4*sin(c + d*x))/(6*d)],
[cos(c + d*x)^7*(A + C*sec(c + d*x)^2)*(a + b*sec(c + d*x))^4, x, 8, (1/4)*a*b*(5*a^2*A + 6*A*b^2 + 6*a^2*C + 8*b^2*C)*x + ((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)*sin(c + d*x))/(105*d) + (a*b*(5*a^2*A + 6*A*b^2 + 6*a^2*C + 8*b^2*C)*cos(c + d*x)*sin(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)*cos(c + d*x)^2*sin(c + d*x))/(210*d) + (b*(103*a^2*A + 6*A*b^2 + 126*a^2*C)*cos(c + d*x)^3*(a + b*sec(c + d*x))*sin(c + d*x))/(210*d) + ((6*a^2*A + 2*A*b^2 + 7*a^2*C)*cos(c + d*x)^4*(a + b*sec(c + d*x))^2*sin(c + d*x))/(35*d) + (2*A*b*cos(c + d*x)^5*(a + b*sec(c + d*x))^3*sin(c + d*x))/(21*d) + (A*cos(c + d*x)^6*(a + b*sec(c + d*x))^4*sin(c + d*x))/(7*d)],


[(a^2 - b^2*sec(c + d*x)^2)*(a + b*sec(c + d*x))^3, x, 6, a^5*x + (b*(24*a^4 - 8*a^2*b^2 - 3*b^4)*arctanh(sin(c + d*x)))/(8*d) + (a*b^2*(18*a^2 - 13*b^2)*tan(c + d*x))/(8*d) + (b^2*(2*a^2 - 3*b^2)*(a + b*sec(c + d*x))*tan(c + d*x))/(8*d) - (a*b^2*(a + b*sec(c + d*x))^2*tan(c + d*x))/(4*d) - (b^2*(a + b*sec(c + d*x))^3*tan(c + d*x))/(4*d)],
[(a^2 - b^2*sec(c + d*x)^2)*(a + b*sec(c + d*x))^2, x, 5, a^4*x + (a*b*(2*a^2 - b^2)*arctanh(sin(c + d*x)))/d + (2*b^2*(a^2 - b^2)*tan(c + d*x))/(3*d) - (a*b^2*(a + b*sec(c + d*x))*tan(c + d*x))/(3*d) - (b^2*(a + b*sec(c + d*x))^2*tan(c + d*x))/(3*d)],
[(a^2 - b^2*sec(c + d*x)^2)*(a + b*sec(c + d*x)), x, 4, a^3*x + (b*(2*a^2 - b^2)*arctanh(sin(c + d*x)))/(2*d) - (a*b^2*tan(c + d*x))/(2*d) - (b^2*(a + b*sec(c + d*x))*tan(c + d*x))/(2*d)],


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


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


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


[sec(c + d*x)^4*(A + C*sec(c + d*x)^2)/(a + b*sec(c + d*x))^3, x, 8, ((2*A*b^2 + 12*a^2*C + b^2*C)*arctanh(sin(c + d*x)))/(2*b^5*d) + (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)*arctanh(((-a + b)*tan((1/2)*(c + d*x)))/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)*tan(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)*sec(c + d*x)*tan(c + d*x))/(2*b^3*(a^2 - b^2)^2*d) - ((A*b^2 + a^2*C)*sec(c + d*x)^3*tan(c + d*x))/(2*b*(a^2 - b^2)*d*(a + b*sec(c + d*x))^2) + ((3*A*b^4 - 4*a^4*C + 7*a^2*b^2*C)*sec(c + d*x)^2*tan(c + d*x))/(2*b^2*(a^2 - b^2)^2*d*(a + b*sec(c + d*x)))],
[sec(c + d*x)^3*(A + C*sec(c + d*x)^2)/(a + b*sec(c + d*x))^3, x, 7, -((3*a*C*arctanh(sin(c + d*x)))/(b^4*d)) - ((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)*arctanh(((-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 - 6*a^4*C + 11*a^2*b^2*C - 2*b^4*C)*tan(c + d*x))/(2*b^3*(a^2 - b^2)^2*d) - ((A*b^2 + a^2*C)*sec(c + d*x)^2*tan(c + d*x))/(2*b*(a^2 - b^2)*d*(a + b*sec(c + d*x))^2) + ((a^2*A*b^2 + 2*A*b^4 - 3*a^4*C + 6*a^2*b^2*C)*sec(c + d*x)*tan(c + d*x))/(2*b^2*(a^2 - b^2)^2*d*(a + b*sec(c + d*x)))],
[sec(c + d*x)^2*(A + C*sec(c + d*x)^2)/(a + b*sec(c + d*x))^3, x, 6, (C*arctanh(sin(c + d*x)))/(b^3*d) + (a*(3*A*b^4 + 2*a^4*C - 5*a^2*b^2*C + 6*b^4*C)*arctanh(((-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^2*C)*sec(c + d*x)*tan(c + d*x))/(2*b*(a^2 - b^2)*d*(a + b*sec(c + d*x))^2) + ((2*a^2*A*b^2 + A*b^4 - 2*a^4*C + 5*a^2*b^2*C)*tan(c + d*x))/(2*b^2*(a^2 - b^2)^2*d*(a + b*sec(c + d*x)))],
[sec(c + d*x)^1*(A + C*sec(c + d*x)^2)/(a + b*sec(c + d*x))^3, x, 5, -(((2*a^2*A + A*b^2 + a^2*C + 2*b^2*C)*arctanh(((-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)*tan(c + d*x))/(2*b*(a^2 - b^2)*d*(a + b*sec(c + d*x))^2) - (a*(3*A*b^2 - a^2*C + 4*b^2*C)*tan(c + d*x))/(2*b*(a^2 - b^2)^2*d*(a + b*sec(c + d*x)))],
[sec(c + d*x)^0*(A + C*sec(c + d*x)^2)/(a + b*sec(c + d*x))^3, x, 4, (A*x)/a^3 + (b*(6*a^4*A - 5*a^2*A*b^2 + 2*A*b^4 + 3*a^4*C)*arctanh(((-a + b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^3*(a^2 - b^2)^(5/2)*d) + ((A*b^2 + a^2*C)*tan(c + d*x))/(2*a*(a^2 - b^2)*d*(a + b*sec(c + d*x))^2) + ((5*a^2*A*b^2 - 2*A*b^4 + a^4*C + 2*a^2*b^2*C)*tan(c + d*x))/(2*a^2*(a^2 - b^2)^2*d*(a + b*sec(c + d*x)))],
[cos(c + d*x)^1*(A + C*sec(c + d*x)^2)/(a + b*sec(c + d*x))^3, x, 6, -((3*A*b*x)/a^4) - ((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)*arctanh(((-a + b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^4*(a^2 - b^2)^(5/2)*d) + ((2*a^4*A - 11*a^2*A*b^2 + 6*A*b^4 - 3*a^4*C)*sin(c + d*x))/(2*a^3*(a^2 - b^2)^2*d) + ((A*b^2 + a^2*C)*sin(c + d*x))/(2*a*(a^2 - b^2)*d*(a + b*sec(c + d*x))^2) + ((6*a^2*A*b^2 - 3*A*b^4 + 2*a^4*C + a^2*b^2*C)*sin(c + d*x))/(2*a^2*(a^2 - b^2)^2*d*(a + b*sec(c + d*x)))],
[cos(c + d*x)^2*(A + C*sec(c + d*x)^2)/(a + b*sec(c + d*x))^3, x, 7, ((a^2*A + 12*A*b^2 + 2*a^2*C)*x)/(2*a^5) + (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)*arctanh(((-a + b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^5*(a^2 - b^2)^(5/2)*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)*sin(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)*cos(c + d*x)*sin(c + d*x))/(2*a^3*(a^2 - b^2)^2*d) + ((A*b^2 + a^2*C)*cos(c + d*x)*sin(c + d*x))/(2*a*(a^2 - b^2)*d*(a + b*sec(c + d*x))^2) + ((7*a^2*A*b^2 - 4*A*b^4 + 3*a^4*C)*cos(c + d*x)*sin(c + d*x))/(2*a^2*(a^2 - b^2)^2*d*(a + b*sec(c + d*x)))],


[sec(c + d*x)^4*(A + C*sec(c + d*x)^2)/(a + b*sec(c + d*x))^4, x, 8, -((4*a*C*arctanh(sin(c + d*x)))/(b^5*d)) + ((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)*arctanh(((-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 + 24*a^6*C - 68*a^4*b^2*C + 65*a^2*b^4*C - 6*b^6*C)*tan(c + d*x))/(6*b^4*(a^2 - b^2)^3*d) - ((A*b^2 + a^2*C)*sec(c + d*x)^3*tan(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*sec(c + d*x))^3) + ((2*a^2*A*b^2 + 3*A*b^4 - 4*a^4*C + 9*a^2*b^2*C)*sec(c + d*x)^2*tan(c + d*x))/(6*b^2*(a^2 - b^2)^2*d*(a + b*sec(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)*sec(c + d*x)*tan(c + d*x))/(2*b^3*(a^2 - b^2)^3*d*(a + b*sec(c + d*x)))],
[sec(c + d*x)^3*(A + C*sec(c + d*x)^2)/(a + b*sec(c + d*x))^4, x, 7, (C*arctanh(sin(c + d*x)))/(b^4*d) - (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)*arctanh(((-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^2*C)*sec(c + d*x)^2*tan(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*sec(c + d*x))^3) + ((3*a^2*A*b^2 + 2*A*b^4 - 3*a^4*C + 8*a^2*b^2*C)*sec(c + d*x)*tan(c + d*x))/(6*b^2*(a^2 - b^2)^2*d*(a + b*sec(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)*tan(c + d*x))/(6*b^3*(a^2 - b^2)^3*d*(a + b*sec(c + d*x)))],
[sec(c + d*x)^2*(A + C*sec(c + d*x)^2)/(a + b*sec(c + d*x))^4, x, 6, (b*(4*a^2*A + A*b^2 + 3*a^2*C + 2*b^2*C)*arctanh(((-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)*sec(c + d*x)*tan(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*sec(c + d*x))^3) + ((4*a^2*A*b^2 + A*b^4 - 2*a^4*C + 7*a^2*b^2*C)*tan(c + d*x))/(6*b^2*(a^2 - b^2)^2*d*(a + b*sec(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)*tan(c + d*x))/(6*b^2*(a^2 - b^2)^3*d*(a + b*sec(c + d*x)))],
[sec(c + d*x)^1*(A + C*sec(c + d*x)^2)/(a + b*sec(c + d*x))^4, x, 6, -((a*(2*a^2*A + 3*A*b^2 + a^2*C + 4*b^2*C)*arctanh(((-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)*tan(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*sec(c + d*x))^3) - (a*(5*A*b^2 - a^2*C + 6*b^2*C)*tan(c + d*x))/(6*b*(a^2 - b^2)^2*d*(a + b*sec(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)*tan(c + d*x))/(6*b*(a^2 - b^2)^3*d*(a + b*sec(c + d*x)))],
[sec(c + d*x)^0*(A + C*sec(c + d*x)^2)/(a + b*sec(c + d*x))^4, x, 5, (A*x)/a^4 + (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)*arctanh(((-a + b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^4*(a^2 - b^2)^(7/2)*d) + ((A*b^2 + a^2*C)*tan(c + d*x))/(3*a*(a^2 - b^2)*d*(a + b*sec(c + d*x))^3) + ((8*a^2*A*b^2 - 3*A*b^4 + 2*a^4*C + 3*a^2*b^2*C)*tan(c + d*x))/(6*a^2*(a^2 - b^2)^2*d*(a + b*sec(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)*tan(c + d*x))/(6*a^3*(a^2 - b^2)^3*d*(a + b*sec(c + d*x)))],
[cos(c + d*x)^1*(A + C*sec(c + d*x)^2)/(a + b*sec(c + d*x))^4, x, 7, -((4*A*b*x)/a^5) - ((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)*arctanh(((-a + b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^5*(a^2 - b^2)^(7/2)*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)*sin(c + d*x))/(6*a^4*(a^2 - b^2)^3*d) + ((A*b^2 + a^2*C)*sin(c + d*x))/(3*a*(a^2 - b^2)*d*(a + b*sec(c + d*x))^3) + ((9*a^2*A*b^2 - 4*A*b^4 + 3*a^4*C + 2*a^2*b^2*C)*sin(c + d*x))/(6*a^2*(a^2 - b^2)^2*d*(a + b*sec(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)*sin(c + d*x))/(2*a^3*(a^2 - b^2)^3*d*(a + b*sec(c + d*x)))],
[cos(c + d*x)^2*(A + C*sec(c + d*x)^2)/(a + b*sec(c + d*x))^4, x, 8, ((a^2*A + 20*A*b^2 + 2*a^2*C)*x)/(2*a^6) + (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)*arctanh(((-a + b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^6*(a^2 - b^2)^(7/2)*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)*sin(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)*cos(c + d*x)*sin(c + d*x))/(2*a^4*(a^2 - b^2)^3*d) + ((A*b^2 + a^2*C)*cos(c + d*x)*sin(c + d*x))/(3*a*(a^2 - b^2)*d*(a + b*sec(c + d*x))^3) + ((10*a^2*A*b^2 - 5*A*b^4 + 4*a^4*C + a^2*b^2*C)*cos(c + d*x)*sin(c + d*x))/(6*a^2*(a^2 - b^2)^2*d*(a + b*sec(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)*cos(c + d*x)*sin(c + d*x))/(6*a^3*(a^2 - b^2)^3*d*(a + b*sec(c + d*x)))],


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


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


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


[sec(c + d*x)^3*(A + C*sec(c + d*x)^2)*sqrt(a + b*sec(c + d*x)), 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(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(315*b^4*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*(42*a^2*A*b^2 - 147*a*A*b^3 - 189*A*b^4 + 16*a^4*C + 4*a^3*b*C + 24*a^2*b^2*C - 111*a*b^3*C - 147*b^4*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(315*b^3*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*(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*sec(c + d*x))*tan((1/2)*(c + d*x)))/(315*b^4*d) + (2*a*(21*A*b^2 + 8*a^2*C + 13*b^2*C)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(315*b^3*d) + (2*(63*A*b^2 - 6*a^2*C + 49*b^2*C)*sec(c + d*x)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(315*b^2*d) + (2*a*C*sec(c + d*x)^2*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(63*b*d) + (2*C*sec(c + d*x)^3*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(9*d)],
[sec(c + d*x)^2*(A + C*sec(c + d*x)^2)*sqrt(a + b*sec(c + d*x)), x, 8, -((2*a*(a + b)*(35*A*b^2 + 8*a^2*C + 19*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(105*b^3*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))) + (2*(35*a*A*b^2 + 35*A*b^3 + 8*a^3*C + 2*a^2*b*C + 19*a*b^2*C + 25*b^3*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(105*b^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*a*(35*A*b^2 + 8*a^2*C + 19*b^2*C)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(105*b^3*d) + (2*(35*A*b^2 - 4*a^2*C + 25*b^2*C)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(105*b^2*d) + (2*a*C*sec(c + d*x)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(35*b*d) + (2*C*sec(c + d*x)^2*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(7*d)],
[sec(c + d*x)^1*(A + C*sec(c + d*x)^2)*sqrt(a + b*sec(c + d*x)), x, 7, -((2*(a + b)*(15*A*b^2 - 2*a^2*C + 9*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(15*b^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))) + (2*(15*a*A*b + 15*A*b^2 - 2*a^2*C + 7*a*b*C + 9*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(15*b*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(15*A*b^2 - 2*a^2*C + 9*b^2*C)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(15*b^2*d) + (2*a*C*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(15*b*d) + (2*C*sec(c + d*x)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(5*d)],
[sec(c + d*x)^0*(A + C*sec(c + d*x)^2)*sqrt(a + b*sec(c + d*x)), x, 8, -((2*a*(a + b)*C*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(3*b*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))) - (2*(3*a*A - 3*A*b - a*C - b*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (4*a*A*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*a*C*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(3*b*d) + (2*C*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(3*d)],
[cos(c + d*x)^1*(A + C*sec(c + d*x)^2)*sqrt(a + b*sec(c + d*x)), x, 8, ((a + b)*(A - 2*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*(A*b - a*C - b*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*A*b*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (A*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/d - ((A - 2*C)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/d],
[cos(c + d*x)^2*(A + C*sec(c + d*x)^2)*sqrt(a + b*sec(c + d*x)), x, 8, (A*b*(a + b)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - ((2*a*A - A*b + 4*a*C - 4*b*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + ((4*a^2*A - A*b^2 + 8*a^2*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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*a*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (A*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(2*d) + (A*b*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(4*a*d)],
[cos(c + d*x)^3*(A + C*sec(c + d*x)^2)*sqrt(a + b*sec(c + d*x)), x, 9, ((a + b)*(16*a^2*A - 3*A*b^2 + 24*a^2*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(24*a^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (b*(14*a*A - A*b + 24*a*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(12*a*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (b*(4*a^2*A + A*b^2 + 8*a^2*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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*a^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (A*b*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(12*a*d) + (A*cos(c + d*x)^2*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(3*d) + ((16*a^2*A - 3*A*b^2 + 24*a^2*C)*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(24*a^2*d)],
[cos(c + d*x)^4*(A + C*sec(c + d*x)^2)*sqrt(a + b*sec(c + d*x)), x, 10, (b*(a + b)*(28*a^2*A + 15*A*b^2 + 48*a^2*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(192*a^3*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - ((72*a^3*A - 36*a^2*A*b + 2*a*A*b^2 + 5*A*b^3 + 96*a^3*C - 48*a^2*b*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(96*a^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + ((48*a^4*A - 8*a^2*A*b^2 - 5*A*b^4 + 64*a^4*C - 16*a^2*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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*a^3*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + ((36*a^2*A - 5*A*b^2 + 48*a^2*C)*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(96*a^2*d) + (A*b*cos(c + d*x)^2*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(24*a*d) + (A*cos(c + d*x)^3*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(4*d) + (b*(28*a^2*A + 15*A*b^2 + 48*a^2*C)*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(192*a^3*d)],


[sec(c + d*x)^3*(A + C*sec(c + d*x)^2)*(a + b*sec(c + d*x))^(3/2), 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(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(1155*b^4*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*(66*a^3*A*b^2 - 561*a^2*A*b^3 - 902*a*A*b^4 - 275*A*b^5 + 16*a^5*C + 4*a^4*b*C + 36*a^3*b^2*C - 423*a^2*b^3*C - 696*a*b^4*C - 225*b^5*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(1155*b^3*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (4*a*(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*sec(c + d*x))*tan((1/2)*(c + d*x)))/(1155*b^4*d) + (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*sec(c + d*x))*tan(c + d*x))/(1155*b^3*d) + (4*a*(132*A*b^2 - 3*a^2*C + 101*b^2*C)*sec(c + d*x)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(1155*b^2*d) + (2*(33*A*b^2 + a^2*C + 27*b^2*C)*sec(c + d*x)^2*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(231*b*d) + (2*a*C*sec(c + d*x)^3*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(33*d) + (2*C*sec(c + d*x)^3*(a + b*sec(c + d*x))^(3/2)*tan(c + d*x))/(11*d)],
[sec(c + d*x)^2*(A + C*sec(c + d*x)^2)*(a + b*sec(c + d*x))^(3/2), 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(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(315*b^3*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))) + (2*(63*a^2*A*b^2 + 252*a*A*b^3 + 189*A*b^4 + 8*a^4*C + 2*a^3*b*C + 33*a^2*b^2*C + 186*a*b^3*C + 147*b^4*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(315*b^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(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*sec(c + d*x))*tan((1/2)*(c + d*x)))/(315*b^3*d) + (4*a*(63*A*b^2 - 2*a^2*C + 44*b^2*C)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(315*b^2*d) + (2*(63*A*b^2 + 3*a^2*C + 49*b^2*C)*sec(c + d*x)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(315*b*d) + (2*a*C*sec(c + d*x)^2*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(21*d) + (2*C*sec(c + d*x)^2*(a + b*sec(c + d*x))^(3/2)*tan(c + d*x))/(9*d)],
[sec(c + d*x)^1*(A + C*sec(c + d*x)^2)*(a + b*sec(c + d*x))^(3/2), x, 8, -((4*a*(a + b)*(70*A*b^2 - 3*a^2*C + 41*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(105*b^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))) + (2*(105*a^2*A*b + 140*a*A*b^2 + 35*A*b^3 - 6*a^3*C + 51*a^2*b*C + 82*a*b^2*C + 25*b^3*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(105*b*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (4*a*(70*A*b^2 - 3*a^2*C + 41*b^2*C)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(105*b^2*d) + (2*(35*A*b^2 + 3*a^2*C + 25*b^2*C)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(105*b*d) + (6*a*C*sec(c + d*x)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(35*d) + (2*C*sec(c + d*x)*(a + b*sec(c + d*x))^(3/2)*tan(c + d*x))/(7*d)],
[sec(c + d*x)^0*(A + C*sec(c + d*x)^2)*(a + b*sec(c + d*x))^(3/2), x, 9, -((2*(a + b)*(5*A*b^2 + a^2*C + 3*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(5*b*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))) - (2*(5*a^2*A - 10*a*A*b - 5*A*b^2 - a^2*C - 4*a*b*C - 3*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (4*a^2*A*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(5*A*b^2 + a^2*C + 3*b^2*C)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(5*b*d) + (2*a*C*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(5*d) + (2*C*(a + b*sec(c + d*x))^(3/2)*tan(c + d*x))/(5*d)],
[cos(c + d*x)^1*(A + C*sec(c + d*x)^2)*(a + b*sec(c + d*x))^(3/2), x, 9, (a*(a + b)*(3*A - 8*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*(6*a*A*b - 3*A*b^2 - 3*a^2*C - 4*a*b*C - b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (6*a*A*b*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (A*(a + b*sec(c + d*x))^(3/2)*sin(c + d*x))/d - (a*(3*A - 8*C)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(3*d) - (b*(3*A - 2*C)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(3*d)],
[cos(c + d*x)^2*(A + C*sec(c + d*x)^2)*(a + b*sec(c + d*x))^(3/2), x, 9, (b*(a + b)*(5*A - 8*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - ((2*a^2*A - a*A*b + 4*A*b^2 + 4*a^2*C - 8*a*b*C - 4*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + ((4*a^2*A + 3*A*b^2 + 8*a^2*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (3*A*b*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(4*d) + (A*cos(c + d*x)*(a + b*sec(c + d*x))^(3/2)*sin(c + d*x))/(2*d) - (b*(5*A - 8*C)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(4*d)],
[cos(c + d*x)^3*(A + C*sec(c + d*x)^2)*(a + b*sec(c + d*x))^(3/2), x, 9, ((a + b)*(16*a^2*A + 3*A*b^2 + 24*a^2*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(24*a*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (b*(26*a*A - 7*A*b + 48*a*C - 24*b*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (b*(12*a^2*A - A*b^2 + 24*a^2*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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*a*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (A*b*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(4*d) + (A*cos(c + d*x)^2*(a + b*sec(c + d*x))^(3/2)*sin(c + d*x))/(3*d) + ((16*a^2*A + 3*A*b^2 + 24*a^2*C)*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(24*a*d)],
[cos(c + d*x)^4*(A + C*sec(c + d*x)^2)*(a + b*sec(c + d*x))^(3/2), x, 10, (b*(a + b)*(52*a^2*A - 3*A*b^2 + 80*a^2*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(64*a^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - ((24*a^3*A - 12*a^2*A*b + 38*a*A*b^2 - A*b^3 + 32*a^3*C - 16*a^2*b*C + 64*a*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(32*a*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(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(1 + cos(c + d*x))*sqrt((b + a*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*a^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + ((12*a^2*A + A*b^2 + 16*a^2*C)*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(32*a*d) + (A*b*cos(c + d*x)^2*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(8*d) + (A*cos(c + d*x)^3*(a + b*sec(c + d*x))^(3/2)*sin(c + d*x))/(4*d) + (b*(52*a^2*A - 3*A*b^2 + 80*a^2*C)*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(64*a^2*d)],


[sec(c + d*x)^3*(A + C*sec(c + d*x)^2)*(a + b*sec(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 + 240*a^6*C + 760*a^4*b^2*C - 30669*a^2*b^4*C - 17787*b^6*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(45045*b^4*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (1/(45045*b^3*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))))*(2*(1430*a^4*A*b^2 - 22165*a^3*A*b^3 - 39897*a^2*A*b^4 - 37323*a*A*b^5 - 21021*A*b^6 + 240*a^6*C + 60*a^5*b*C + 760*a^4*b^2*C - 16685*a^3*b^3*C - 30669*a^2*b^4*C - 30831*a*b^5*C - 17787*b^6*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b)))) - (2*(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*sec(c + d*x))*tan((1/2)*(c + d*x)))/(45045*b^4*d) + (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*sec(c + d*x))*tan(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)*sec(c + d*x)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(45045*b^2*d) + (2*a*(2717*A*b^2 + 15*a^2*C + 2209*b^2*C)*sec(c + d*x)^2*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(9009*b*d) + (2*(143*A*b^2 + 15*a^2*C + 121*b^2*C)*sec(c + d*x)^3*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(1287*d) + (10*a*C*sec(c + d*x)^3*(a + b*sec(c + d*x))^(3/2)*tan(c + d*x))/(143*d) + (2*C*sec(c + d*x)^3*(a + b*sec(c + d*x))^(5/2)*tan(c + d*x))/(13*d)],
[sec(c + d*x)^2*(A + C*sec(c + d*x)^2)*(a + b*sec(c + d*x))^(5/2), 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(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(693*b^3*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))) + (2*(99*a^3*A*b^2 + 891*a^2*A*b^3 + 957*a*A*b^4 + 165*A*b^5 + 8*a^5*C + 2*a^4*b*C + 51*a^3*b^2*C + 663*a^2*b^3*C + 741*a*b^4*C + 135*b^5*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(693*b^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*a*(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*sec(c + d*x))*tan((1/2)*(c + d*x)))/(693*b^3*d) + (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*sec(c + d*x))*tan(c + d*x))/(693*b^2*d) + (2*a*(297*A*b^2 + 3*a^2*C + 229*b^2*C)*sec(c + d*x)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(693*b*d) + (2*(33*A*b^2 + 5*a^2*C + 27*b^2*C)*sec(c + d*x)^2*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(231*d) + (10*a*C*sec(c + d*x)^2*(a + b*sec(c + d*x))^(3/2)*tan(c + d*x))/(99*d) + (2*C*sec(c + d*x)^2*(a + b*sec(c + d*x))^(5/2)*tan(c + d*x))/(11*d)],
[sec(c + d*x)^1*(A + C*sec(c + d*x)^2)*(a + b*sec(c + d*x))^(5/2), 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(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(315*b^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))) + (2*(315*a^3*A*b + 483*a^2*A*b^2 + 357*a*A*b^3 + 189*A*b^4 - 10*a^4*C + 155*a^3*b*C + 279*a^2*b^2*C + 261*a*b^3*C + 147*b^4*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(315*b*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(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*sec(c + d*x))*tan((1/2)*(c + d*x)))/(315*b^2*d) + (2*a*(231*A*b^2 + 5*a^2*C + 163*b^2*C)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(315*b*d) + (2*(63*A*b^2 + 15*a^2*C + 49*b^2*C)*sec(c + d*x)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(315*d) + (10*a*C*sec(c + d*x)*(a + b*sec(c + d*x))^(3/2)*tan(c + d*x))/(63*d) + (2*C*sec(c + d*x)*(a + b*sec(c + d*x))^(5/2)*tan(c + d*x))/(9*d)],
[sec(c + d*x)^0*(A + C*sec(c + d*x)^2)*(a + b*sec(c + d*x))^(5/2), x, 10, -((2*a*(a + b)*(49*A*b^2 + 3*a^2*C + 29*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(21*b*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))) - (2*(21*a^3*A - 63*a^2*A*b - 49*a*A*b^2 - 7*A*b^3 - 3*a^3*C - 27*a^2*b*C - 29*a*b^2*C - 5*b^3*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (4*a^3*A*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*a*(49*A*b^2 + 3*a^2*C + 29*b^2*C)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(21*b*d) + (2*(7*A*b^2 + 3*a^2*C + 5*b^2*C)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(21*d) + (2*a*C*(a + b*sec(c + d*x))^(3/2)*tan(c + d*x))/(7*d) + (2*C*(a + b*sec(c + d*x))^(5/2)*tan(c + d*x))/(7*d)],
[cos(c + d*x)^1*(A + C*sec(c + d*x)^2)*(a + b*sec(c + d*x))^(5/2), x, 10, ((a + b)*(15*a^2*A - 30*A*b^2 - 46*a^2*C - 18*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*(45*a^2*A*b - 45*a*A*b^2 - 15*A*b^3 - 15*a^3*C - 23*a^2*b*C - 17*a*b^2*C - 9*b^3*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (10*a^2*A*b*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (A*(a + b*sec(c + d*x))^(5/2)*sin(c + d*x))/d - ((15*a^2*A - 30*A*b^2 - 46*a^2*C - 18*b^2*C)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(15*d) - (a*b*(15*A - 16*C)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(15*d) - (b*(5*A - 2*C)*(a + b*sec(c + d*x))^(3/2)*tan(c + d*x))/(5*d)],
[cos(c + d*x)^2*(A + C*sec(c + d*x)^2)*(a + b*sec(c + d*x))^(5/2), x, 10, (a*b*(a + b)*(27*A - 56*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - ((6*a^3*A - 3*a^2*A*b + 36*a*A*b^2 - 12*A*b^3 + 12*a^3*C - 36*a^2*b*C - 28*a*b^2*C - 4*b^3*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (a*(4*a^2*A + 15*A*b^2 + 8*a^2*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (5*A*b*(a + b*sec(c + d*x))^(3/2)*sin(c + d*x))/(4*d) + (A*cos(c + d*x)*(a + b*sec(c + d*x))^(5/2)*sin(c + d*x))/(2*d) - (a*b*(27*A - 56*C)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(12*d) - (b^2*(21*A - 8*C)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(12*d)],
[cos(c + d*x)^3*(A + C*sec(c + d*x)^2)*(a + b*sec(c + d*x))^(5/2), x, 10, ((a + b)*(16*a^2*A + 33*A*b^2 + 24*a^2*C - 48*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (b*(38*a^2*A - 13*a*A*b + 24*A*b^2 + 72*a^2*C - 72*a*b*C - 24*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (5*b*(4*a^2*A + A*b^2 + 8*a^2*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + ((16*a^2*A + 15*A*b^2 + 24*a^2*C)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(24*d) + (5*A*b*cos(c + d*x)*(a + b*sec(c + d*x))^(3/2)*sin(c + d*x))/(12*d) + (A*cos(c + d*x)^2*(a + b*sec(c + d*x))^(5/2)*sin(c + d*x))/(3*d) - ((16*a^2*A + 33*A*b^2 + 24*a^2*C - 48*b^2*C)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(24*d)],
[cos(c + d*x)^4*(A + C*sec(c + d*x)^2)*(a + b*sec(c + d*x))^(5/2), x, 10, (b*(a + b)*(284*a^2*A + 15*A*b^2 + 432*a^2*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(192*a*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - ((72*a^3*A - 36*a^2*A*b + 322*a*A*b^2 - 59*A*b^3 + 96*a^3*C - 48*a^2*b*C + 576*a*b^2*C - 192*b^3*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(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(1 + cos(c + d*x))*sqrt((b + a*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*a*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + ((12*a^2*A + 5*A*b^2 + 16*a^2*C)*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(32*d) + (5*A*b*cos(c + d*x)^2*(a + b*sec(c + d*x))^(3/2)*sin(c + d*x))/(24*d) + (A*cos(c + d*x)^3*(a + b*sec(c + d*x))^(5/2)*sin(c + d*x))/(4*d) + (b*(284*a^2*A + 15*A*b^2 + 432*a^2*C)*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(192*a*d)],


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


[sec(c + d*x)^3*(A + C*sec(c + d*x)^2)/sqrt(a + b*sec(c + d*x)), x, 8, (4*a*(a + b)*(35*A*b^2 + 24*a^2*C + 22*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(105*b^4*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*(70*a*A*b^2 - 35*A*b^3 + 48*a^3*C + 12*a^2*b*C + 44*a*b^2*C - 25*b^3*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(105*b^3*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (4*a*(35*A*b^2 + 24*a^2*C + 22*b^2*C)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(105*b^4*d) + (2*(35*A*b^2 + 24*a^2*C + 25*b^2*C)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(105*b^3*d) - (12*a*C*sec(c + d*x)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(35*b^2*d) + (2*C*sec(c + d*x)^2*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(7*b*d)],
[sec(c + d*x)^2*(A + C*sec(c + d*x)^2)/sqrt(a + b*sec(c + d*x)), x, 7, -((2*(a + b)*(15*A*b^2 + 8*a^2*C + 9*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(15*b^3*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))) + (2*(15*A*b^2 + 8*a^2*C + 2*a*b*C + 9*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(15*b^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(15*A*b^2 + 8*a^2*C + 9*b^2*C)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(15*b^3*d) - (8*a*C*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(15*b^2*d) + (2*C*sec(c + d*x)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(5*b*d)],
[sec(c + d*x)^1*(A + C*sec(c + d*x)^2)/sqrt(a + b*sec(c + d*x)), x, 6, (4*a*(a + b)*C*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(3*b^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(3*A*b - 2*a*C + b*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(3*b*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (4*a*C*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(3*b^2*d) + (2*C*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(3*b*d)],
[sec(c + d*x)^0*(A + C*sec(c + d*x)^2)/sqrt(a + b*sec(c + d*x)), x, 7, -((2*(a + b)*C*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))) - (2*(A - C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (4*A*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*C*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(b*d)],
[cos(c + d*x)^1*(A + C*sec(c + d*x)^2)/sqrt(a + b*sec(c + d*x)), x, 7, (A*(a + b)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*C*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*A*b*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (A*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(a*d)],
[cos(c + d*x)^2*(A + C*sec(c + d*x)^2)/sqrt(a + b*sec(c + d*x)), x, 8, -((3*A*b*(a + b)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))) - ((2*a*A - A*b + 4*a*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + ((4*a^2*A + 3*A*b^2 + 8*a^2*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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*a^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (A*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(2*a*d) - (3*A*b*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(4*a^2*d)],
[cos(c + d*x)^3*(A + C*sec(c + d*x)^2)/sqrt(a + b*sec(c + d*x)), x, 9, ((a + b)*(16*a^2*A + 15*A*b^2 + 24*a^2*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(24*a^3*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (A*b*(2*a + 5*b)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(12*a^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (b*(4*a^2*A + 5*A*b^2 + 8*a^2*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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*a^3*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (5*A*b*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(12*a^2*d) + (A*cos(c + d*x)^2*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(3*a*d) + ((16*a^2*A + 15*A*b^2 + 24*a^2*C)*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(24*a^3*d)],


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


[sec(c + d*x)^3*(A + C*sec(c + d*x)^2)/(a + b*sec(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(1 + cos(c + d*x))*sqrt((b + a*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^4*(a + b)*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*(2*a^2*A*b^2 - 3*a*A*b^3 - 3*A*b^4 + 16*a^4*C - 12*a^3*b*C - 16*a^2*b^2*C + 9*a*b^3*C - b^4*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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^3*(a + b)*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (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*sec(c + d*x))*tan((1/2)*(c + d*x)))/(3*b^4*(a^2 - b^2)^2*d) - (2*(A*b^2 + a^2*C)*sec(c + d*x)^2*tan(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*sec(c + d*x))^(3/2)) + (4*(2*A*b^4 - 3*a^4*C + 5*a^2*b^2*C)*sec(c + d*x)*tan(c + d*x))/(3*b^2*(a^2 - b^2)^2*d*sqrt(a + b*sec(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*sec(c + d*x))*tan(c + d*x))/(3*b^3*(a^2 - b^2)^2*d)],
[sec(c + d*x)^2*(A + C*sec(c + d*x)^2)/(a + b*sec(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(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*(a*A*b^2 + 3*A*b^3 - 8*a^3*C + 6*a^2*b*C + 9*a*b^2*C - 3*b^3*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (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*sec(c + d*x))*tan((1/2)*(c + d*x)))/(3*b^3*(a^2 - b^2)^2*d) - (2*(A*b^2 + a^2*C)*sec(c + d*x)*tan(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*sec(c + d*x))^(3/2)) + (4*(a^2*A*b^2 + A*b^4 - 2*a^4*C + 4*a^2*b^2*C)*tan(c + d*x))/(3*b^2*(a^2 - b^2)^2*d*sqrt(a + b*sec(c + d*x)))],
[sec(c + d*x)^1*(A + C*sec(c + d*x)^2)/(a + b*sec(c + d*x))^(5/2), x, 6, -((4*a*(2*A*b^2 - a^2*C + 3*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(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((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (4*a*(2*A*b^2 - a^2*C + 3*b^2*C)*tan((1/2)*(c + d*x)))/(3*(a - b)*b^2*(a + b)^2*d*sqrt(a + b*sec(c + d*x))) - (2*(A*b^2 + a^2*C)*tan(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*sec(c + d*x))^(3/2))],
[sec(c + d*x)^0*(A + C*sec(c + d*x)^2)/(a + b*sec(c + d*x))^(5/2), x, 8, (2*(7*a^2*A*b^2 - 3*A*b^4 + a^4*C + 3*a^2*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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*b*(a + b)*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(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((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (4*A*sqrt(1 + cos(c + d*x))*sqrt((b + a*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^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*(7*a^2*A*b^2 - 3*A*b^4 + a^4*C + 3*a^2*b^2*C)*tan((1/2)*(c + d*x)))/(3*a^2*(a - b)*b*(a + b)^2*d*sqrt(a + b*sec(c + d*x))) + (2*(A*b^2 + a^2*C)*tan(c + d*x))/(3*a*(a^2 - b^2)*d*(a + b*sec(c + d*x))^(3/2))],
[cos(c + d*x)^1*(A + C*sec(c + d*x)^2)/(a + b*sec(c + d*x))^(5/2), x, 9, ((3*a^4*A - 26*a^2*A*b^2 + 15*A*b^4 - 8*a^4*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(6*a^2*A*b + 3*a*A*b^2 - 5*A*b^3 + 3*a^3*C + a^2*b*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (10*A*b*sqrt(1 + cos(c + d*x))*sqrt((b + a*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^3*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(A*b^2 + a^2*C)*sin(c + d*x))/(3*a*(a^2 - b^2)*d*(a + b*sec(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)*sin(c + d*x))/(3*a^2*(a^2 - b^2)^2*d*sqrt(a + b*sec(c + d*x))) + ((3*a^4*A - 26*a^2*A*b^2 + 15*A*b^4 - 8*a^4*C)*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(3*a^3*(a^2 - b^2)^2*d)],
[cos(c + d*x)^2*(A + C*sec(c + d*x)^2)/(a + b*sec(c + d*x))^(5/2), x, 10, -((b*(33*a^4*A - 170*a^2*A*b^2 + 105*A*b^4 - 56*a^4*C + 24*a^2*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(12*a^4*(a - b)^2*(a + b)*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))) - ((6*a^4*A - 9*a^3*A*b + 33*a^2*A*b^2 + 21*a*A*b^3 - 35*A*b^4 + 12*a^4*C + 12*a^3*b*C - 8*a^2*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(6*a^3*(a - b)^2*(a + b)*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + ((4*a^2*A + 35*A*b^2 + 8*a^2*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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*a^4*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(A*b^2 + a^2*C)*cos(c + d*x)*sin(c + d*x))/(3*a*(a^2 - b^2)*d*(a + b*sec(c + d*x))^(3/2)) + (2*(11*a^2*A*b^2 - 7*A*b^4 + 5*a^4*C - a^2*b^2*C)*cos(c + d*x)*sin(c + d*x))/(3*a^2*(a^2 - b^2)^2*d*sqrt(a + b*sec(c + d*x))) + ((3*a^4*A - 54*a^2*A*b^2 + 35*A*b^4 - 24*a^4*C + 8*a^2*b^2*C)*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(6*a^3*(a^2 - b^2)^2*d) - (b*(33*a^4*A - 170*a^2*A*b^2 + 105*A*b^4 - 56*a^4*C + 24*a^2*b^2*C)*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(12*a^4*(a^2 - b^2)^2*d)],


[sec(c + d*x)^0*(A + C*sec(c + d*x)^2)/(a + b*sec(c + d*x))^(7/2), x, 9, (2*(58*a^4*A*b^2 - 41*a^2*A*b^4 + 15*A*b^6 + 3*a^6*C + 29*a^4*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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*(a - b)^3*b*(a + b)^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*(15*a^4*A + 30*a^3*A*b - 17*a^2*A*b^2 - 6*a*A*b^3 + 10*A*b^4 + 3*a^4*C + 24*a^3*b*C + 5*a^2*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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*(a - b)^3*(a + b)^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (4*A*sqrt(1 + cos(c + d*x))*sqrt((b + a*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^3*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*(58*a^4*A*b^2 - 41*a^2*A*b^4 + 15*A*b^6 + 3*a^6*C + 29*a^4*b^2*C)*tan((1/2)*(c + d*x)))/(15*a^3*(a - b)^2*b*(a + b)^3*d*sqrt(a + b*sec(c + d*x))) + (2*(A*b^2 + a^2*C)*tan(c + d*x))/(5*a*(a^2 - b^2)*d*(a + b*sec(c + d*x))^(5/2)) + (2*(13*a^2*A*b^2 - 5*A*b^4 + 3*a^4*C + 5*a^2*b^2*C)*tan(c + d*x))/(15*a^2*(a^2 - b^2)^2*d*(a + b*sec(c + d*x))^(3/2))],


[(a^2 - b^2*sec(c + d*x)^2)*(a + b*sec(c + d*x))^(3/2), x, 9, -((2*b*(a + b)*(4*a^2 - 3*b^2)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(5*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))) - (2*(5*a^4 - 10*a^3*b - 4*a^2*b^2 + 4*a*b^3 + 3*b^4)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (4*a^4*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*b*(4*a^2 - 3*b^2)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(5*d) - (2*a*b^2*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(5*d) - (2*b^2*(a + b*sec(c + d*x))^(3/2)*tan(c + d*x))/(5*d)],
[(a^2 - b^2*sec(c + d*x)^2)*(a + b*sec(c + d*x))^(1/2), x, 8, (2*a*b*(a + b)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*(3*a^3 - 3*a^2*b + a*b^2 + b^3)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (4*a^3*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*a*b*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(3*d) - (2*b^2*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(3*d)],
[(a^2 - b^2*sec(c + d*x)^2)/(a + b*sec(c + d*x))^(1/2), x, 7, (2*b*(a + b)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*(a^2 + b^2)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (4*a^2*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*b*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/d],
[(a^2 - b^2*sec(c + d*x)^2)/(a + b*sec(c + d*x))^(3/2), x, 5, -((2*(a + b)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))) + (4*a*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))],
[(a^2 - b^2*sec(c + d*x)^2)/(a + b*sec(c + d*x))^(5/2), x, 8, (4*b*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*(a + b)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (4*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (4*b*tan((1/2)*(c + d*x)))/((a + b)*d*sqrt(a + b*sec(c + d*x)))],
[(a^2 - b^2*sec(c + d*x)^2)/(a + b*sec(c + d*x))^(7/2), x, 9, (2*b*(11*a^2 - 3*b^2)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*(3*a^2 + 6*a*b - b^2)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (4*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*b*(11*a^2 - 3*b^2)*tan((1/2)*(c + d*x)))/(3*a*(a - b)*(a + b)^2*d*sqrt(a + b*sec(c + d*x))) + (4*b^2*tan(c + d*x))/(3*(a^2 - b^2)*d*(a + b*sec(c + d*x))^(3/2))],


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


# ::Subsubsection:: 
#n>0


# ::Subsubsection:: 
#n<0


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


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


# ::Subsubsection:: 
#n>0


# ::Subsubsection:: 
#n<0


[(A + C*sec(c + d*x)^2)/(sqrt(sec(c + d*x))*sqrt(a + b*sec(c + d*x))), x, 9, (2*A*(a + b)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(a*d*sqrt(a + b*sec(c + d*x))) - (2*A*b*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(a*d*sqrt(a + b*sec(c + d*x))) + (2*C*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*sec(c + d*x)))],


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


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


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


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


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


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


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


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


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


[sec(c + d*x)^2*(B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + b*sec(c + d*x))^3, x, 7, ((8*a^3*B + 18*a*b^2*B + 18*a^2*b*C + 5*b^3*C)*arctanh(sin(c + d*x)))/(16*d) + (2*(15*a^2*b*B + 4*b^3*B + 5*a^3*C + 12*a*b^2*C)*tan(c + d*x))/(15*d) + ((8*a^3*B + 18*a*b^2*B + 18*a^2*b*C + 5*b^3*C)*sec(c + d*x)*tan(c + d*x))/(16*d) + ((54*a^2*b*B + 32*b^3*B + 2*a^3*C + 71*a*b^2*C)*sec(c + d*x)^2*tan(c + d*x))/(120*d) + ((42*a*b*B + 6*a^2*C + 25*b^2*C)*sec(c + d*x)^2*(a + b*sec(c + d*x))*tan(c + d*x))/(120*d) + ((2*b*B + a*C)*sec(c + d*x)^2*(a + b*sec(c + d*x))^2*tan(c + d*x))/(10*d) + (C*sec(c + d*x)^2*(a + b*sec(c + d*x))^3*tan(c + d*x))/(6*d)],
[sec(c + d*x)^1*(B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + b*sec(c + d*x))^3, x, 6, ((12*a^2*b*B + 3*b^3*B + 4*a^3*C + 9*a*b^2*C)*arctanh(sin(c + d*x)))/(8*d) + ((15*a^3*B + 30*a*b^2*B + 30*a^2*b*C + 8*b^3*C)*tan(c + d*x))/(15*d) + ((90*a^2*b*B + 45*b^3*B + 6*a^3*C + 103*a*b^2*C)*sec(c + d*x)*tan(c + d*x))/(120*d) + ((15*a*b*B + 3*a^2*C + 8*b^2*C)*sec(c + d*x)*(a + b*sec(c + d*x))*tan(c + d*x))/(30*d) + ((5*b*B + 3*a*C)*sec(c + d*x)*(a + b*sec(c + d*x))^2*tan(c + d*x))/(20*d) + (C*sec(c + d*x)*(a + b*sec(c + d*x))^3*tan(c + d*x))/(5*d)],
[sec(c + d*x)^0*(B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + b*sec(c + d*x))^3, x, 6, ((8*a^3*B + 12*a*b^2*B + 12*a^2*b*C + 3*b^3*C)*arctanh(sin(c + d*x)))/(8*d) + ((44*a^2*b*B + 16*b^3*B + 6*a^3*C + 39*a*b^2*C)*tan(c + d*x))/(24*d) + ((20*a*b*B + 6*a^2*C + 9*b^2*C)*(a + b*sec(c + d*x))*tan(c + d*x))/(24*d) + ((4*b*B + 3*a*C)*(a + b*sec(c + d*x))^2*tan(c + d*x))/(12*d) + (C*(a + b*sec(c + d*x))^3*tan(c + d*x))/(4*d)],
[cos(c + d*x)^1*(B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + b*sec(c + d*x))^3, x, 6, a^3*B*x + ((6*a^2*b*B + b^3*B + 2*a^3*C + 3*a*b^2*C)*arctanh(sin(c + d*x)))/(2*d) + (b*(15*a*b*B + 11*a^2*C + 4*b^2*C)*tan(c + d*x))/(6*d) + (b*(3*b*B + 5*a*C)*(a + b*sec(c + d*x))*tan(c + d*x))/(6*d) + (b*C*(a + b*sec(c + d*x))^2*tan(c + d*x))/(3*d)],
[cos(c + d*x)^2*(B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + b*sec(c + d*x))^3, x, 6, a^2*(3*b*B + a*C)*x + (b*(6*a*b*B + 6*a^2*C + b^2*C)*arctanh(sin(c + d*x)))/(2*d) + (a*B*(a + b*sec(c + d*x))^2*sin(c + d*x))/d - (b*(2*a^2*B - 2*b^2*B - 5*a*b*C)*tan(c + d*x))/(2*d) - (b*(2*a*B - b*C)*(a + b*sec(c + d*x))*tan(c + d*x))/(2*d)],
[cos(c + d*x)^3*(B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + b*sec(c + d*x))^3, x, 6, (1/2)*a*(a^2*B + 6*b^2*B + 6*a*b*C)*x + (b^2*(b*B + 3*a*C)*arctanh(sin(c + d*x)))/d + (a*(2*b*B + a*C)*(a + b*sec(c + d*x))*sin(c + d*x))/d + (a*B*cos(c + d*x)*(a + b*sec(c + d*x))^2*sin(c + d*x))/(2*d) - (b*(5*a*b*B + 2*a^2*C - 2*b^2*C)*tan(c + d*x))/(2*d)],
[cos(c + d*x)^4*(B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + b*sec(c + d*x))^3, x, 6, (1/2)*(3*a^2*b*B + 2*b^3*B + a^3*C + 6*a*b^2*C)*x + (b^3*C*arctanh(sin(c + d*x)))/d + (a*(4*a^2*B + 11*b^2*B + 15*a*b*C)*sin(c + d*x))/(6*d) + (a*(5*b*B + 3*a*C)*(b + a*cos(c + d*x))*sin(c + d*x))/(6*d) + (a*B*(b + a*cos(c + d*x))^2*sin(c + d*x))/(3*d)],
[cos(c + d*x)^5*(B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + b*sec(c + d*x))^3, x, 6, (1/8)*(3*a^3*B + 12*a*b^2*B + 12*a^2*b*C + 8*b^3*C)*x + ((6*a^2*b*B + 3*b^3*B + 2*a^3*C + 9*a*b^2*C)*sin(c + d*x))/(3*d) + (a*(9*a^2*B + 18*b^2*B + 28*a*b*C)*cos(c + d*x)*sin(c + d*x))/(24*d) + (a*(3*b*B + 2*a*C)*cos(c + d*x)^2*(a + b*sec(c + d*x))*sin(c + d*x))/(6*d) + (a*B*cos(c + d*x)^3*(a + b*sec(c + d*x))^2*sin(c + d*x))/(4*d)],
[cos(c + d*x)^6*(B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + b*sec(c + d*x))^3, x, 7, (1/8)*(9*a^2*b*B + 4*b^3*B + 3*a^3*C + 12*a*b^2*C)*x + ((8*a^3*B + 30*a*b^2*B + 30*a^2*b*C + 15*b^3*C)*sin(c + d*x))/(15*d) + ((9*a^2*b*B + 4*b^3*B + 3*a^3*C + 12*a*b^2*C)*cos(c + d*x)*sin(c + d*x))/(8*d) + (a*(16*a^2*B + 27*b^2*B + 45*a*b*C)*cos(c + d*x)^2*sin(c + d*x))/(60*d) + (a*(7*b*B + 5*a*C)*cos(c + d*x)^3*(a + b*sec(c + d*x))*sin(c + d*x))/(20*d) + (a*B*cos(c + d*x)^4*(a + b*sec(c + d*x))^2*sin(c + d*x))/(5*d)],


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


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


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


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


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


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


[sec(c + d*x)^0*(B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + b*sec(c + d*x))^(1/2), x, 6, -((2*(a + b)*(3*b*B + a*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(3*b*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))) + (2*(3*a*B + 3*b*B + a*C + b*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(3*b*B + a*C)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(3*b*d) + (2*C*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(3*d)],


[sec(c + d*x)^0*(B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + b*sec(c + d*x))^(3/2), x, 7, -((2*(a + b)*(20*a*b*B + 3*a^2*C + 9*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(15*b*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))) + (2*(15*a^2*B + 20*a*b*B + 5*b^2*B + 3*a^2*C + 12*a*b*C + 9*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(20*a*b*B + 3*a^2*C + 9*b^2*C)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(15*b*d) + (2*(5*b*B + 3*a*C)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(15*d) + (2*C*(a + b*sec(c + d*x))^(3/2)*tan(c + d*x))/(5*d)],


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


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


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


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


[sec(c + d*x)^0*(B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + b*sec(c + d*x))^(7/2), x, 7, -((2*(23*a^2*b*B + 9*b^3*B - 3*a^3*C - 29*a*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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 - b)^3*b*(a + b)^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))) + (2*(15*a^2*B + 8*a*b*B + 9*b^2*B - 3*a^2*C - 24*a*b*C - 5*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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 - b)^3*(a + b)^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(23*a^2*b*B + 9*b^3*B - 3*a^3*C - 29*a*b^2*C)*tan((1/2)*(c + d*x)))/(15*(a - b)^2*b*(a + b)^3*d*sqrt(a + b*sec(c + d*x))) - (2*(b*B - a*C)*tan(c + d*x))/(5*(a^2 - b^2)*d*(a + b*sec(c + d*x))^(5/2)) - (2*(8*a*b*B - 3*a^2*C - 5*b^2*C)*tan(c + d*x))/(15*(a^2 - b^2)^2*d*(a + b*sec(c + d*x))^(3/2))],


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


# ::Subsubsection:: 
#n>0


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


[(B*sec(c + d*x) + C*sec(c + d*x)^2)/(sqrt(sec(c + d*x))*(a + b*sec(c + d*x))), x, 5, (2*B*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a*d) - (2*(b*B - a*C)*sqrt(cos(c + d*x))*EllipticPi((2*a)/(a + b), (1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a*(a + b)*d)],


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


# ::Subsubsection:: 
#n>0


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


[(B*sec(c + d*x) + C*sec(c + d*x)^2)/(sqrt(sec(c + d*x))*sqrt(a + b*sec(c + d*x))), x, 6, (2*B*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*sec(c + d*x))) + (2*C*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*sec(c + d*x)))],


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


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


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


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


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


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


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


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


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


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


[sec(c + d*x)^2*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + b*sec(c + d*x))^4, x, 8, ((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)*arctanh(sin(c + d*x)))/(16*d) + ((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)*tan(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)*sec(c + d*x)*tan(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)*sec(c + d*x)^2*tan(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)*sec(c + d*x)^2*(a + b*sec(c + d*x))*tan(c + d*x))/(840*d) + ((14*A*b^2 + 21*a*b*B + 4*a^2*C + 12*b^2*C)*sec(c + d*x)^2*(a + b*sec(c + d*x))^2*tan(c + d*x))/(70*d) + ((7*b*B + 4*a*C)*sec(c + d*x)^2*(a + b*sec(c + d*x))^3*tan(c + d*x))/(42*d) + (C*sec(c + d*x)^2*(a + b*sec(c + d*x))^4*tan(c + d*x))/(7*d)],
[sec(c + d*x)^1*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + b*sec(c + d*x))^4, x, 7, ((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)*arctanh(sin(c + d*x)))/(16*d) + ((60*a^3*A*b + 40*a*A*b^3 + 15*a^4*B + 60*a^2*b^2*B + 8*b^4*B + 40*a^3*b*C + 32*a*b^3*C)*tan(c + d*x))/(15*d) + ((460*a^2*A*b^2 + 90*A*b^4 + 192*a^3*b*B + 296*a*b^3*B + 8*a^4*C + 334*a^2*b^2*C + 75*b^4*C)*sec(c + d*x)*tan(c + d*x))/(240*d) + ((50*a*A*b^2 + 36*a^2*b*B + 16*b^3*B + 4*a^3*C + 39*a*b^2*C)*sec(c + d*x)*(a + b*sec(c + d*x))*tan(c + d*x))/(60*d) + ((30*A*b^2 + 48*a*b*B + 12*a^2*C + 25*b^2*C)*sec(c + d*x)*(a + b*sec(c + d*x))^2*tan(c + d*x))/(120*d) + ((3*b*B + 2*a*C)*sec(c + d*x)*(a + b*sec(c + d*x))^3*tan(c + d*x))/(15*d) + (C*sec(c + d*x)*(a + b*sec(c + d*x))^4*tan(c + d*x))/(6*d)],
[sec(c + d*x)^0*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + b*sec(c + d*x))^4, x, 7, a^4*A*x + ((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)*arctanh(sin(c + d*x)))/(8*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)*tan(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*sec(c + d*x))*tan(c + d*x))/(120*d) + ((20*A*b^2 + 35*a*b*B + 12*a^2*C + 16*b^2*C)*(a + b*sec(c + d*x))^2*tan(c + d*x))/(60*d) + ((5*b*B + 4*a*C)*(a + b*sec(c + d*x))^3*tan(c + d*x))/(20*d) + (C*(a + b*sec(c + d*x))^4*tan(c + d*x))/(5*d)],
[cos(c + d*x)^1*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + b*sec(c + d*x))^4, x, 7, a^3*(4*A*b + a*B)*x + ((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)*arctanh(sin(c + d*x)))/(8*d) + (A*(a + b*sec(c + d*x))^4*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)*tan(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*sec(c + d*x))*tan(c + d*x))/(24*d) - (b*(12*a*A - 4*b*B - 7*a*C)*(a + b*sec(c + d*x))^2*tan(c + d*x))/(12*d) - (b*(4*A - C)*(a + b*sec(c + d*x))^3*tan(c + d*x))/(4*d)],
[cos(c + d*x)^2*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + b*sec(c + d*x))^4, x, 7, (1/2)*a^2*(a^2*A + 12*A*b^2 + 8*a*b*B + 2*a^2*C)*x + (b*(8*a*A*b^2 + 12*a^2*b*B + b^3*B + 8*a^3*C + 4*a*b^2*C)*arctanh(sin(c + d*x)))/(2*d) + ((2*A*b + a*B)*(a + b*sec(c + d*x))^3*sin(c + d*x))/d + (A*cos(c + d*x)*(a + b*sec(c + d*x))^4*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)*tan(c + d*x))/(6*d) - (b*(18*a*A*b + 6*a^2*B - 3*b^2*B - 8*a*b*C)*(a + b*sec(c + d*x))*tan(c + d*x))/(6*d) - (b*(15*A*b + 6*a*B - 2*b*C)*(a + b*sec(c + d*x))^2*tan(c + d*x))/(6*d)],
[cos(c + d*x)^3*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + b*sec(c + d*x))^4, x, 7, (1/2)*a*(4*a^2*A*b + 8*A*b^3 + a^3*B + 12*a*b^2*B + 8*a^2*b*C)*x + (b^2*(2*A*b^2 + 8*a*b*B + 12*a^2*C + b^2*C)*arctanh(sin(c + d*x)))/(2*d) + ((4*a^2*A + 12*A*b^2 + 15*a*b*B + 6*a^2*C)*(a + b*sec(c + d*x))^2*sin(c + d*x))/(6*d) + ((4*A*b + 3*a*B)*cos(c + d*x)*(a + b*sec(c + d*x))^3*sin(c + d*x))/(6*d) + (A*cos(c + d*x)^2*(a + b*sec(c + d*x))^4*sin(c + d*x))/(3*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)*tan(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*sec(c + d*x))*tan(c + d*x))/(6*d)],
[cos(c + d*x)^4*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + b*sec(c + d*x))^4, x, 7, (1/8)*(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)*x + (b^3*(b*B + 4*a*C)*arctanh(sin(c + d*x)))/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*sec(c + d*x))*sin(c + d*x))/(12*d) + ((3*a^2*A + 4*A*b^2 + 8*a*b*B + 4*a^2*C)*cos(c + d*x)*(a + b*sec(c + d*x))^2*sin(c + d*x))/(8*d) + ((A*b + a*B)*cos(c + d*x)^2*(a + b*sec(c + d*x))^3*sin(c + d*x))/(3*d) + (A*cos(c + d*x)^3*(a + b*sec(c + d*x))^4*sin(c + d*x))/(4*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)*tan(c + d*x))/(24*d)],
[cos(c + d*x)^5*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + b*sec(c + d*x))^4, x, 7, (1/8)*(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)*x + (b^4*C*arctanh(sin(c + d*x)))/d + (1/(120*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)*sin(c + d*x)) + (1/(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*cos(c + d*x))*sin(c + d*x)) + ((16*a^2*A + 12*A*b^2 + 35*a*b*B + 20*a^2*C)*(b + a*cos(c + d*x))^2*sin(c + d*x))/(60*d) + ((4*A*b + 5*a*B)*(b + a*cos(c + d*x))^3*sin(c + d*x))/(20*d) + (A*(b + a*cos(c + d*x))^4*sin(c + d*x))/(5*d)],
[cos(c + d*x)^6*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + b*sec(c + d*x))^4, x, 7, (1/16)*(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)*x + (1/(15*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)*sin(c + d*x)) + (1/(240*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)*cos(c + d*x)*sin(c + d*x)) + (1/(60*d))*((39*a^2*A*b + 4*A*b^3 + 16*a^3*B + 36*a*b^2*B + 50*a^2*b*C)*cos(c + d*x)^2*(a + b*sec(c + d*x))*sin(c + d*x)) + (1/(120*d))*((25*a^2*A + 12*A*b^2 + 48*a*b*B + 30*a^2*C)*cos(c + d*x)^3*(a + b*sec(c + d*x))^2*sin(c + d*x)) + ((2*A*b + 3*a*B)*cos(c + d*x)^4*(a + b*sec(c + d*x))^3*sin(c + d*x))/(15*d) + (A*cos(c + d*x)^5*(a + b*sec(c + d*x))^4*sin(c + d*x))/(6*d)],
[cos(c + d*x)^7*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + b*sec(c + d*x))^4, x, 8, (1/16)*(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)*x + ((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)*sin(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)*cos(c + d*x)*sin(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)*cos(c + d*x)^2*sin(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)*cos(c + d*x)^3*(a + b*sec(c + d*x))*sin(c + d*x))/(840*d) + ((12*a^2*A + 4*A*b^2 + 21*a*b*B + 14*a^2*C)*cos(c + d*x)^4*(a + b*sec(c + d*x))^2*sin(c + d*x))/(70*d) + ((4*A*b + 7*a*B)*cos(c + d*x)^5*(a + b*sec(c + d*x))^3*sin(c + d*x))/(42*d) + (A*cos(c + d*x)^6*(a + b*sec(c + d*x))^4*sin(c + d*x))/(7*d)],


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


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


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


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


[sec(c + d*x)^4*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + b*sec(c + d*x))^4, x, 8, ((b*B - 4*a*C)*arctanh(sin(c + d*x)))/(b^5*d) + ((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)*arctanh(((-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)*tan(c + d*x))/(6*b^4*(a^2 - b^2)^3*d) - ((A*b^2 - a*b*B + a^2*C)*sec(c + d*x)^3*tan(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*sec(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)*sec(c + d*x)^2*tan(c + d*x))/(6*b^2*(a^2 - b^2)^2*d*(a + b*sec(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)*sec(c + d*x)*tan(c + d*x))/(2*b^3*(a^2 - b^2)^3*d*(a + b*sec(c + d*x)))],
[sec(c + d*x)^3*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + b*sec(c + d*x))^4, x, 7, (C*arctanh(sin(c + d*x)))/(b^4*d) - ((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)*arctanh(((-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)*sec(c + d*x)^2*tan(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*sec(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)*sec(c + d*x)*tan(c + d*x))/(6*b^2*(a^2 - b^2)^2*d*(a + b*sec(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)*tan(c + d*x))/(6*b^3*(a^2 - b^2)^3*d*(a + b*sec(c + d*x)))],
[sec(c + d*x)^2*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + b*sec(c + d*x))^4, x, 6, ((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)*arctanh(((-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)*sec(c + d*x)*tan(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*sec(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)*tan(c + d*x))/(6*b^2*(a^2 - b^2)^2*d*(a + b*sec(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)*tan(c + d*x))/(6*b^2*(a^2 - b^2)^3*d*(a + b*sec(c + d*x)))],
[sec(c + d*x)^1*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + b*sec(c + d*x))^4, x, 6, -(((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)*arctanh(((-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)*tan(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*sec(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)*tan(c + d*x))/(6*b*(a^2 - b^2)^2*d*(a + b*sec(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)*tan(c + d*x))/(6*b*(a^2 - b^2)^3*d*(a + b*sec(c + d*x)))],
[sec(c + d*x)^0*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + b*sec(c + d*x))^4, x, 5, (A*x)/a^4 + ((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)*arctanh(((-a + b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^4*(a^2 - b^2)^(7/2)*d) + ((A*b^2 - a*b*B + a^2*C)*tan(c + d*x))/(3*a*(a^2 - b^2)*d*(a + b*sec(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)*tan(c + d*x))/(6*a^2*(a^2 - b^2)^2*d*(a + b*sec(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)*tan(c + d*x))/(6*a^3*(a^2 - b^2)^3*d*(a + b*sec(c + d*x)))],
[cos(c + d*x)^1*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + b*sec(c + d*x))^4, x, 7, -(((4*A*b - a*B)*x)/a^5) - ((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)*arctanh(((-a + b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^5*(a^2 - b^2)^(7/2)*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)*sin(c + d*x))/(6*a^4*(a^2 - b^2)^3*d) + ((A*b^2 - a*b*B + a^2*C)*sin(c + d*x))/(3*a*(a^2 - b^2)*d*(a + b*sec(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)*sin(c + d*x))/(6*a^2*(a^2 - b^2)^2*d*(a + b*sec(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)*sin(c + d*x))/(2*a^3*(a^2 - b^2)^3*d*(a + b*sec(c + d*x)))],
[cos(c + d*x)^2*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + b*sec(c + d*x))^4, x, 8, ((a^2*A + 20*A*b^2 - 8*a*b*B + 2*a^2*C)*x)/(2*a^6) + (b*(40*a^6*A*b^2 - 84*a^4*A*b^4 + 69*a^2*A*b^6 - 20*A*b^8 - 20*a^7*b*B + 35*a^5*b^3*B - 28*a^3*b^5*B + 8*a*b^7*B + 8*a^8*C - 8*a^6*b^2*C + 7*a^4*b^4*C - 2*a^2*b^6*C)*arctanh(((-a + b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^6*(a^2 - b^2)^(7/2)*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)*sin(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)*cos(c + d*x)*sin(c + d*x))/(2*a^4*(a^2 - b^2)^3*d) + ((A*b^2 - a*b*B + a^2*C)*cos(c + d*x)*sin(c + d*x))/(3*a*(a^2 - b^2)*d*(a + b*sec(c + d*x))^3) + ((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)*cos(c + d*x)*sin(c + d*x))/(6*a^2*(a^2 - b^2)^2*d*(a + b*sec(c + d*x))^2) + ((48*a^4*A*b^2 - 53*a^2*A*b^4 + 20*A*b^6 - 27*a^5*b*B + 20*a^3*b^3*B - 8*a*b^5*B + 12*a^6*C + a^4*b^2*C + 2*a^2*b^4*C)*cos(c + d*x)*sin(c + d*x))/(6*a^3*(a^2 - b^2)^3*d*(a + b*sec(c + d*x)))],


[(a*b*B - a^2*C + b^2*B*sec(c + d*x) + b^2*C*sec(c + d*x)^2)*(a + b*sec(c + d*x))^3, x, 6, a^4*(b*B - a*C)*x + (b*(32*a^3*b*B + 16*a*b^3*B - 24*a^4*C + 8*a^2*b^2*C + 3*b^4*C)*arctanh(sin(c + d*x)))/(8*d) + (b^2*(104*a^2*b*B + 16*b^3*B - 54*a^3*C + 39*a*b^2*C)*tan(c + d*x))/(24*d) + (b^2*(32*a*b*B - 6*a^2*C + 9*b^2*C)*(a + b*sec(c + d*x))*tan(c + d*x))/(24*d) + (b^2*(4*b*B + 3*a*C)*(a + b*sec(c + d*x))^2*tan(c + d*x))/(12*d) + (b^2*C*(a + b*sec(c + d*x))^3*tan(c + d*x))/(4*d)],
[(a*b*B - a^2*C + b^2*B*sec(c + d*x) + b^2*C*sec(c + d*x)^2)*(a + b*sec(c + d*x))^2, x, 5, a^3*(b*B - a*C)*x + (b*(6*a^2*b*B + b^3*B - 4*a^3*C + 2*a*b^2*C)*arctanh(sin(c + d*x)))/(2*d) + (b^2*(15*a*b*B - 4*a^2*C + 4*b^2*C)*tan(c + d*x))/(6*d) + (b^2*(3*b*B + 2*a*C)*(a + b*sec(c + d*x))*tan(c + d*x))/(6*d) + (b^2*C*(a + b*sec(c + d*x))^2*tan(c + d*x))/(3*d)],
[(a*b*B - a^2*C + b^2*B*sec(c + d*x) + b^2*C*sec(c + d*x)^2)*(a + b*sec(c + d*x)), x, 4, a^2*(b*B - a*C)*x + (b*(4*a*b*B - 2*a^2*C + b^2*C)*arctanh(sin(c + d*x)))/(2*d) + (b^2*(2*b*B + a*C)*tan(c + d*x))/(2*d) + (b^2*C*(a + b*sec(c + d*x))*tan(c + d*x))/(2*d)],
[(a*b*B - a^2*C + b^2*B*sec(c + d*x) + b^2*C*sec(c + d*x)^2)/(a + b*sec(c + d*x)), x, 3, (b*B - a*C)*x + (b*C*arctanh(sin(c + d*x)))/d],
[(a*b*B - a^2*C + b^2*B*sec(c + d*x) + b^2*C*sec(c + d*x)^2)/(a + b*sec(c + d*x))^2, x, 3, ((b*B - a*C)*x)/a + (2*b*(b*B - 2*a*C)*arctanh(((-a + b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a*sqrt(a^2 - b^2)*d)],
[(a*b*B - a^2*C + b^2*B*sec(c + d*x) + b^2*C*sec(c + d*x)^2)/(a + b*sec(c + d*x))^3, x, 4, ((b*B - a*C)*x)/a^2 + (2*b*(2*a^2*b*B - b^3*B - 3*a^3*C + a*b^2*C)*arctanh(((-a + b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^2*(a^2 - b^2)^(3/2)*d) + (b^2*(b*B - 2*a*C)*tan(c + d*x))/(a*(a^2 - b^2)*d*(a + b*sec(c + d*x)))],
[(a*b*B - a^2*C + b^2*B*sec(c + d*x) + b^2*C*sec(c + d*x)^2)/(a + b*sec(c + d*x))^4, x, 5, ((b*B - a*C)*x)/a^3 + (b*(6*a^4*b*B - 5*a^2*b^3*B + 2*b^5*B - 8*a^5*C + 4*a^3*b^2*C - 2*a*b^4*C)*arctanh(((-a + b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^3*(a^2 - b^2)^(5/2)*d) + (b^2*(b*B - 2*a*C)*tan(c + d*x))/(2*a*(a^2 - b^2)*d*(a + b*sec(c + d*x))^2) + (b^2*(5*a^2*b*B - 2*b^3*B - 8*a^3*C + 2*a*b^2*C)*tan(c + d*x))/(2*a^2*(a^2 - b^2)^2*d*(a + b*sec(c + d*x)))],
[(a*b*B - a^2*C + b^2*B*sec(c + d*x) + b^2*C*sec(c + d*x)^2)/(a + b*sec(c + d*x))^5, x, 6, ((b*B - a*C)*x)/a^4 + (b*(8*a^6*b*B - 8*a^4*b^3*B + 7*a^2*b^5*B - 2*b^7*B - 10*a^7*C + 5*a^5*b^2*C - 7*a^3*b^4*C + 2*a*b^6*C)*arctanh(((-a + b)*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(a^4*(a^2 - b^2)^(7/2)*d) + (b^2*(b*B - 2*a*C)*tan(c + d*x))/(3*a*(a^2 - b^2)*d*(a + b*sec(c + d*x))^3) + (b^2*(8*a^2*b*B - 3*b^3*B - 13*a^3*C + 3*a*b^2*C)*tan(c + d*x))/(6*a^2*(a^2 - b^2)^2*d*(a + b*sec(c + d*x))^2) + (b^2*(26*a^4*b*B - 17*a^2*b^3*B + 6*b^5*B - 37*a^5*C + 13*a^3*b^2*C - 6*a*b^4*C)*tan(c + d*x))/(6*a^3*(a^2 - b^2)^3*d*(a + b*sec(c + d*x)))],


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


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


[sec(c + d*x)^3*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + b*sec(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(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((-a + b)/(a + b))))/(315*b^4*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*(42*a^2*A*b^2 - 147*a*A*b^3 - 189*A*b^4 - 24*a^3*b*B - 6*a^2*b^2*B - 57*a*b^3*B - 75*b^4*B + 16*a^4*C + 4*a^3*b*C + 24*a^2*b^2*C - 111*a*b^3*C - 147*b^4*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((-a + b)/(a + b))))/(315*b^3*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*(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*sec(c + d*x))*tan((c + d*x)/2))/(315*b^4*d) + (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*sec(c + d*x))*tan(c + d*x))/(315*b^3*d) + (2*(63*A*b^2 + 9*a*b*B - 6*a^2*C + 49*b^2*C)*sec(c + d*x)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(315*b^2*d) + (2*(9*b*B + a*C)*sec(c + d*x)^2*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(63*b*d) + (2*C*sec(c + d*x)^3*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(9*d)],
[sec(c + d*x)^2*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + b*sec(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(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((-a + b)/(a + b))))/(105*b^3*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(35*a*A*b^2 + 35*A*b^3 - 14*a^2*b*B + 49*a*b^2*B + 63*b^3*B + 8*a^3*C + 2*a^2*b*C + 19*a*b^2*C + 25*b^3*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((-a + b)/(a + b))))/(105*b^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(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*sec(c + d*x))*tan((c + d*x)/2))/(105*b^3*d) + (2*(35*A*b^2 + 7*a*b*B - 4*a^2*C + 25*b^2*C)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(105*b^2*d) + (2*(7*b*B + a*C)*sec(c + d*x)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(35*b*d) + (2*C*sec(c + d*x)^2*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(7*d)],
[sec(c + d*x)^1*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + b*sec(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(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((-a + b)/(a + b))))/(15*b^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(15*a*A*b + 15*A*b^2 + 5*a*b*B + 5*b^2*B - 2*a^2*C + 7*a*b*C + 9*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((-a + b)/(a + b))))/(15*b*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(15*A*b^2 + 5*a*b*B - 2*a^2*C + 9*b^2*C)*sqrt(a + b*sec(c + d*x))*tan((c + d*x)/2))/(15*b^2*d) + (2*(5*b*B + a*C)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(15*b*d) + (2*C*sec(c + d*x)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(5*d)],
[sec(c + d*x)^0*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + b*sec(c + d*x))^(1/2), x, 8, -((2*(a + b)*(3*b*B + a*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(3*b*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))) - (2*(3*a*A - 3*A*b - 3*a*B - 3*b*B - a*C - b*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (4*a*A*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(3*b*B + a*C)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(3*b*d) + (2*C*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(3*d)],
[cos(c + d*x)^1*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + b*sec(c + d*x))^(1/2), x, 8, ((a + b)*(A - 2*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*(A*b + a*B - b*B - a*C - b*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(A*b + 2*a*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (A*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/d - ((A - 2*C)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/d],
[cos(c + d*x)^2*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + b*sec(c + d*x))^(1/2), x, 8, ((a + b)*(A*b + 4*a*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - ((2*a*A - A*b + 4*b*B + 4*a*C - 4*b*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + ((4*a^2*A - A*b^2 + 4*a*b*B + 8*a^2*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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*a*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (A*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(2*d) + ((A*b + 4*a*B)*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(4*a*d)],
[cos(c + d*x)^3*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + b*sec(c + d*x))^(1/2), x, 9, ((a + b)*(16*a^2*A - 3*A*b^2 + 6*a*b*B + 24*a^2*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(24*a^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - ((14*a*A*b - A*b^2 + 12*a^2*B - 6*a*b*B + 24*a*b*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(12*a*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(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(1 + cos(c + d*x))*sqrt((b + a*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*a^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + ((A*b + 6*a*B)*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(12*a*d) + (A*cos(c + d*x)^2*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(3*d) + ((16*a^2*A - 3*A*b^2 + 6*a*b*B + 24*a^2*C)*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(24*a^2*d)],


[sec(c + d*x)^3*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + b*sec(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(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((-a + b)/(a + b))))/(3465*b^4*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*(198*a^3*A*b^2 - 1683*a^2*A*b^3 - 2706*a*A*b^4 - 825*A*b^5 - 88*a^4*b*B - 22*a^3*b^2*B - 363*a^2*b^3*B - 2046*a*b^4*B - 1617*b^5*B + 48*a^5*C + 12*a^4*b*C + 108*a^3*b^2*C - 1269*a^2*b^3*C - 2088*a*b^4*C - 675*b^5*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((-a + b)/(a + b))))/(3465*b^3*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*(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*sec(c + d*x))*tan((c + d*x)/2))/(3465*b^4*d) + (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*sec(c + d*x))*tan(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)*sec(c + d*x)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(3465*b^2*d) + (2*(99*A*b^2 + 110*a*b*B + 3*a^2*C + 81*b^2*C)*sec(c + d*x)^2*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(693*b*d) + (2*(11*b*B + 3*a*C)*sec(c + d*x)^3*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(99*d) + (2*C*sec(c + d*x)^3*(a + b*sec(c + d*x))^(3/2)*tan(c + d*x))/(11*d)],
[sec(c + d*x)^2*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + b*sec(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(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((-a + b)/(a + b))))/(315*b^3*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(63*a^2*A*b^2 + 252*a*A*b^3 + 189*A*b^4 - 18*a^3*b*B + 153*a^2*b^2*B + 246*a*b^3*B + 75*b^4*B + 8*a^4*C + 2*a^3*b*C + 33*a^2*b^2*C + 186*a*b^3*C + 147*b^4*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((-a + b)/(a + b))))/(315*b^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(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*sec(c + d*x))*tan((c + d*x)/2))/(315*b^3*d) + (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*sec(c + d*x))*tan(c + d*x))/(315*b^2*d) + (2*(63*A*b^2 + 72*a*b*B + 3*a^2*C + 49*b^2*C)*sec(c + d*x)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(315*b*d) + (2*(3*b*B + a*C)*sec(c + d*x)^2*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(21*d) + (2*C*sec(c + d*x)^2*(a + b*sec(c + d*x))^(3/2)*tan(c + d*x))/(9*d)],
[sec(c + d*x)^1*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + b*sec(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(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((-a + b)/(a + b))))/(105*b^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(105*a^2*A*b + 140*a*A*b^2 + 35*A*b^3 + 21*a^2*b*B + 84*a*b^2*B + 63*b^3*B - 6*a^3*C + 51*a^2*b*C + 82*a*b^2*C + 25*b^3*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((-a + b)/(a + b))))/(105*b*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(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*sec(c + d*x))*tan((c + d*x)/2))/(105*b^2*d) + (2*(35*A*b^2 + 42*a*b*B + 3*a^2*C + 25*b^2*C)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(105*b*d) + (2*(7*b*B + 3*a*C)*sec(c + d*x)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(35*d) + (2*C*sec(c + d*x)*(a + b*sec(c + d*x))^(3/2)*tan(c + d*x))/(7*d)],
[sec(c + d*x)^0*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + b*sec(c + d*x))^(3/2), x, 9, -((2*(a + b)*(15*A*b^2 + 20*a*b*B + 3*a^2*C + 9*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(15*b*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))) - (2*(15*a^2*A - 30*a*A*b - 15*A*b^2 - 15*a^2*B - 20*a*b*B - 5*b^2*B - 3*a^2*C - 12*a*b*C - 9*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (4*a^2*A*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(15*A*b^2 + 20*a*b*B + 3*a^2*C + 9*b^2*C)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(15*b*d) + (2*(5*b*B + 3*a*C)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(15*d) + (2*C*(a + b*sec(c + d*x))^(3/2)*tan(c + d*x))/(5*d)],
[cos(c + d*x)^1*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + b*sec(c + d*x))^(3/2), x, 9, ((a + b)*(3*a*A - 6*b*B - 8*a*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*(6*a*A*b - 3*A*b^2 + 3*a^2*B - 6*a*b*B - 3*b^2*B - 3*a^2*C - 4*a*b*C - b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*a*(3*A*b + 2*a*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (A*(a + b*sec(c + d*x))^(3/2)*sin(c + d*x))/d - ((3*a*A - 6*b*B - 8*a*C)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(3*d) - (b*(3*A - 2*C)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(3*d)],
[cos(c + d*x)^2*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + b*sec(c + d*x))^(3/2), x, 9, ((a + b)*(5*A*b + 4*a*B - 8*b*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - ((2*a^2*A - a*A*b + 4*A*b^2 + 8*a*b*B - 4*b^2*B + 4*a^2*C - 8*a*b*C - 4*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + ((4*a^2*A + 3*A*b^2 + 12*a*b*B + 8*a^2*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + ((3*A*b + 4*a*B)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(4*d) + (A*cos(c + d*x)*(a + b*sec(c + d*x))^(3/2)*sin(c + d*x))/(2*d) - ((5*A*b + 4*a*B - 8*b*C)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(4*d)],
[cos(c + d*x)^3*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + b*sec(c + d*x))^(3/2), x, 9, ((a + b)*(16*a^2*A + 3*A*b^2 + 30*a*b*B + 24*a^2*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(24*a*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - ((26*a*A*b - 7*A*b^2 + 12*a^2*B - 6*a*b*B + 24*b^2*B + 48*a*b*C - 24*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(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(1 + cos(c + d*x))*sqrt((b + a*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*a*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + ((A*b + 2*a*B)*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(4*d) + (A*cos(c + d*x)^2*(a + b*sec(c + d*x))^(3/2)*sin(c + d*x))/(3*d) + ((16*a^2*A + 3*A*b^2 + 30*a*b*B + 24*a^2*C)*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(24*a*d)],
[cos(c + d*x)^4*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + b*sec(c + d*x))^(3/2), x, 10, ((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(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(192*a^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - ((72*a^3*A - 36*a^2*A*b + 114*a*A*b^2 - 3*A*b^3 + 208*a^2*b*B - 56*a*b^2*B + 96*a^3*C - 48*a^2*b*C + 192*a*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(96*a*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(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(1 + cos(c + d*x))*sqrt((b + a*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*a^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + ((36*a^2*A + 3*A*b^2 + 56*a*b*B + 48*a^2*C)*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(96*a*d) + ((3*A*b + 8*a*B)*cos(c + d*x)^2*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(24*d) + (A*cos(c + d*x)^3*(a + b*sec(c + d*x))^(3/2)*sin(c + d*x))/(4*d) + ((156*a^2*A*b - 9*A*b^3 + 128*a^3*B + 24*a*b^2*B + 240*a^2*b*C)*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(192*a^2*d)],


[sec(c + d*x)^2*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + b*sec(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(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((-a + b)/(a + b))))/(3465*b^3*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(495*a^3*A*b^2 + 4455*a^2*A*b^3 + 4785*a*A*b^4 + 825*A*b^5 - 110*a^4*b*B + 1705*a^3*b^2*B + 3069*a^2*b^3*B + 2871*a*b^4*B + 1617*b^5*B + 40*a^5*C + 10*a^4*b*C + 255*a^3*b^2*C + 3315*a^2*b^3*C + 3705*a*b^4*C + 675*b^5*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((-a + b)/(a + b))))/(3465*b^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(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*sec(c + d*x))*tan((c + d*x)/2))/(3465*b^3*d) + (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*sec(c + d*x))*tan(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)*sec(c + d*x)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(3465*b*d) + (2*(33*A*b^2 + 44*a*b*B + 5*a^2*C + 27*b^2*C)*sec(c + d*x)^2*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(231*d) + (2*(11*b*B + 5*a*C)*sec(c + d*x)^2*(a + b*sec(c + d*x))^(3/2)*tan(c + d*x))/(99*d) + (2*C*sec(c + d*x)^2*(a + b*sec(c + d*x))^(5/2)*tan(c + d*x))/(11*d)],
[sec(c + d*x)^1*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + b*sec(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(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((-a + b)/(a + b))))/(315*b^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(315*a^3*A*b + 483*a^2*A*b^2 + 357*a*A*b^3 + 189*A*b^4 + 45*a^3*b*B + 405*a^2*b^2*B + 435*a*b^3*B + 75*b^4*B - 10*a^4*C + 155*a^3*b*C + 279*a^2*b^2*C + 261*a*b^3*C + 147*b^4*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((-a + b)/(a + b))))/(315*b*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(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*sec(c + d*x))*tan((c + d*x)/2))/(315*b^2*d) + (2*(231*a*A*b^2 + 135*a^2*b*B + 75*b^3*B + 5*a^3*C + 163*a*b^2*C)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(315*b*d) + (2*(63*A*b^2 + 90*a*b*B + 15*a^2*C + 49*b^2*C)*sec(c + d*x)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(315*d) + (2*(9*b*B + 5*a*C)*sec(c + d*x)*(a + b*sec(c + d*x))^(3/2)*tan(c + d*x))/(63*d) + (2*C*sec(c + d*x)*(a + b*sec(c + d*x))^(5/2)*tan(c + d*x))/(9*d)],
[sec(c + d*x)^0*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + b*sec(c + d*x))^(5/2), x, 10, -((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(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(105*b*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))) - (2*(105*a^3*A - 315*a^2*A*b - 245*a*A*b^2 - 35*A*b^3 - 105*a^3*B - 161*a^2*b*B - 119*a*b^2*B - 63*b^3*B - 15*a^3*C - 135*a^2*b*C - 145*a*b^2*C - 25*b^3*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (4*a^3*A*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(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*sec(c + d*x))*tan((1/2)*(c + d*x)))/(105*b*d) + (2*(35*A*b^2 + 56*a*b*B + 15*a^2*C + 25*b^2*C)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(105*d) + (2*(7*b*B + 5*a*C)*(a + b*sec(c + d*x))^(3/2)*tan(c + d*x))/(35*d) + (2*C*(a + b*sec(c + d*x))^(5/2)*tan(c + d*x))/(7*d)],
[cos(c + d*x)^1*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + b*sec(c + d*x))^(5/2), x, 10, ((a + b)*(15*a^2*A - 30*A*b^2 - 70*a*b*B - 46*a^2*C - 18*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*(45*a^2*A*b - 45*a*A*b^2 - 15*A*b^3 + 15*a^3*B - 45*a^2*b*B - 35*a*b^2*B - 5*b^3*B - 15*a^3*C - 23*a^2*b*C - 17*a*b^2*C - 9*b^3*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*a^2*(5*A*b + 2*a*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (A*(a + b*sec(c + d*x))^(5/2)*sin(c + d*x))/d - ((15*a^2*A - 30*A*b^2 - 70*a*b*B - 46*a^2*C - 18*b^2*C)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(15*d) - (b*(15*a*A - 10*b*B - 16*a*C)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(15*d) - (b*(5*A - 2*C)*(a + b*sec(c + d*x))^(3/2)*tan(c + d*x))/(5*d)],
[cos(c + d*x)^2*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + b*sec(c + d*x))^(5/2), x, 10, ((a + b)*(27*a*A*b + 12*a^2*B - 24*b^2*B - 56*a*b*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - ((6*a^3*A - 3*a^2*A*b + 36*a*A*b^2 - 12*A*b^3 + 36*a^2*b*B - 36*a*b^2*B - 12*b^3*B + 12*a^3*C - 36*a^2*b*C - 28*a*b^2*C - 4*b^3*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (a*(4*a^2*A + 15*A*b^2 + 20*a*b*B + 8*a^2*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + ((5*A*b + 4*a*B)*(a + b*sec(c + d*x))^(3/2)*sin(c + d*x))/(4*d) + (A*cos(c + d*x)*(a + b*sec(c + d*x))^(5/2)*sin(c + d*x))/(2*d) - ((27*a*A*b + 12*a^2*B - 24*b^2*B - 56*a*b*C)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(12*d) - (b*(21*A*b + 12*a*B - 8*b*C)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(12*d)],
[cos(c + d*x)^3*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + b*sec(c + d*x))^(5/2), x, 10, ((a + b)*(16*a^2*A + 33*A*b^2 + 54*a*b*B + 24*a^2*C - 48*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - ((38*a^2*A*b - 13*a*A*b^2 + 24*A*b^3 + 12*a^3*B - 6*a^2*b*B + 72*a*b^2*B - 24*b^3*B + 72*a^2*b*C - 72*a*b^2*C - 24*b^3*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(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(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + ((16*a^2*A + 15*A*b^2 + 42*a*b*B + 24*a^2*C)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(24*d) + ((5*A*b + 6*a*B)*cos(c + d*x)*(a + b*sec(c + d*x))^(3/2)*sin(c + d*x))/(12*d) + (A*cos(c + d*x)^2*(a + b*sec(c + d*x))^(5/2)*sin(c + d*x))/(3*d) - ((16*a^2*A + 33*A*b^2 + 54*a*b*B + 24*a^2*C - 48*b^2*C)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(24*d)],
[cos(c + d*x)^4*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + b*sec(c + d*x))^(5/2), x, 10, ((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(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(192*a*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - ((72*a^3*A - 36*a^2*A*b + 322*a*A*b^2 - 59*A*b^3 + 304*a^2*b*B - 104*a*b^2*B + 192*b^3*B + 96*a^3*C - 48*a^2*b*C + 576*a*b^2*C - 192*b^3*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(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(1 + cos(c + d*x))*sqrt((b + a*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*a*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + ((12*a^2*A + 5*A*b^2 + 24*a*b*B + 16*a^2*C)*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(32*d) + ((5*A*b + 8*a*B)*cos(c + d*x)^2*(a + b*sec(c + d*x))^(3/2)*sin(c + d*x))/(24*d) + (A*cos(c + d*x)^3*(a + b*sec(c + d*x))^(5/2)*sin(c + d*x))/(4*d) + ((284*a^2*A*b + 15*A*b^3 + 128*a^3*B + 264*a*b^2*B + 432*a^2*b*C)*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(192*a*d)],
[cos(c + d*x)^5*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + b*sec(c + d*x))^(5/2), x, 11, ((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(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(1920*a^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - ((2312*a^3*A*b - 772*a^2*A*b^2 + 1146*a*A*b^3 - 15*A*b^4 + 720*a^4*B - 360*a^3*b*B + 3220*a^2*b^2*B - 590*a*b^3*B + 3040*a^3*b*C - 1040*a^2*b^2*C + 1920*a*b^3*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(960*a*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(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(1 + cos(c + d*x))*sqrt((b + a*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*a^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + ((772*a^2*A*b + 15*A*b^3 + 360*a^3*B + 590*a*b^2*B + 1040*a^2*b*C)*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(960*a*d) + ((64*a^2*A + 15*A*b^2 + 110*a*b*B + 80*a^2*C)*cos(c + d*x)^2*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(240*d) + ((A*b + 2*a*B)*cos(c + d*x)^3*(a + b*sec(c + d*x))^(3/2)*sin(c + d*x))/(8*d) + (A*cos(c + d*x)^4*(a + b*sec(c + d*x))^(5/2)*sin(c + d*x))/(5*d) + ((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)*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(1920*a^2*d)],


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


[sec(c + d*x)^3*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/sqrt(a + b*sec(c + d*x)), 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(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((-a + b)/(a + b))))/(105*b^4*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*(70*a*A*b^2 - 35*A*b^3 - 56*a^2*b*B - 14*a*b^2*B - 63*b^3*B + 48*a^3*C + 12*a^2*b*C + 44*a*b^2*C - 25*b^3*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((-a + b)/(a + b))))/(105*b^3*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*(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*sec(c + d*x))*tan((c + d*x)/2))/(105*b^4*d) + (2*(35*A*b^2 - 28*a*b*B + 24*a^2*C + 25*b^2*C)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(105*b^3*d) + (2*(7*b*B - 6*a*C)*sec(c + d*x)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(35*b^2*d) + (2*C*sec(c + d*x)^2*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(7*b*d)],
[sec(c + d*x)^2*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/sqrt(a + b*sec(c + d*x)), x, 7, (-2*(a + b)*(15*A*b^2 - 10*a*b*B + 8*a^2*C + 9*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((-a + b)/(a + b))))/(15*b^3*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(15*A*b^2 - 10*a*b*B + 5*b^2*B + 8*a^2*C + 2*a*b*C + 9*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((-a + b)/(a + b))))/(15*b^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(15*A*b^2 - 10*a*b*B + 8*a^2*C + 9*b^2*C)*sqrt(a + b*sec(c + d*x))*tan((c + d*x)/2))/(15*b^3*d) + (2*(5*b*B - 4*a*C)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(15*b^2*d) + (2*C*sec(c + d*x)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(5*b*d)],
[sec(c + d*x)^1*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/sqrt(a + b*sec(c + d*x)), x, 6, (-2*(a + b)*(3*b*B - 2*a*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((-a + b)/(a + b))))/(3*b^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(3*A*b + 3*b*B - 2*a*C + b*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((-a + b)/(a + b))))/(3*b*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(3*b*B - 2*a*C)*sqrt(a + b*sec(c + d*x))*tan((c + d*x)/2))/(3*b^2*d) + (2*C*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(3*b*d)],
[sec(c + d*x)^0*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/sqrt(a + b*sec(c + d*x)), x, 7, -((2*(a + b)*C*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))) - (2*(A - B - C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (4*A*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*C*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(b*d)],
[cos(c + d*x)^1*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/sqrt(a + b*sec(c + d*x)), x, 7, (A*(a + b)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*(B - C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*(A*b - 2*a*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (A*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(a*d)],
[cos(c + d*x)^2*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/sqrt(a + b*sec(c + d*x)), x, 8, -(((a + b)*(3*A*b - 4*a*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))) - ((2*a*A - A*b + 4*a*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + ((4*a^2*A + 3*A*b^2 - 4*a*b*B + 8*a^2*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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*a^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (A*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(2*a*d) - ((3*A*b - 4*a*B)*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(4*a^2*d)],
[cos(c + d*x)^3*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/sqrt(a + b*sec(c + d*x)), x, 9, ((a + b)*(16*a^2*A + 15*A*b^2 - 18*a*b*B + 24*a^2*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(24*a^3*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - ((2*a*A*b + 5*A*b^2 + 12*a^2*B - 6*a*b*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(12*a^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(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(1 + cos(c + d*x))*sqrt((b + a*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*a^3*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - ((5*A*b - 6*a*B)*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(12*a^2*d) + (A*cos(c + d*x)^2*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(3*a*d) + ((16*a^2*A + 15*A*b^2 - 18*a*b*B + 24*a^2*C)*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(24*a^3*d)],


[sec(c + d*x)^3*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + b*sec(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(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((c + d*x)/2)), -((-a + b)/(a + b))))/(15*(a - b)*b^4*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(30*a*A*b^2 - 15*A*b^3 - 40*a^2*b*B + 30*a*b^2*B - 5*b^3*B + 48*a^3*C - 36*a^2*b*C + 12*a*b^2*C - 9*b^3*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((c + d*x)/2)), -((-a + b)/(a + b))))/(15*(a - b)*b^3*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (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*sec(c + d*x))*tan((c + d*x)/2))/(15*b^4*(a^2 - b^2)*d) - (2*(A*b^2 - a*b*B + a^2*C)*sec(c + d*x)^2*tan(c + d*x))/(b*(a^2 - b^2)*d*sqrt(a + b*sec(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*sec(c + d*x))*tan(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)*sec(c + d*x)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(5*b^2*(a^2 - b^2)*d)],
[sec(c + d*x)^2*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + b*sec(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(1 + cos(c + d*x))*sqrt((b + a*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)*b^3*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*(3*A*b^2 - 6*a*b*B + 3*b^2*B + 8*a^2*C - 6*a*b*C + b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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)*b^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (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*sec(c + d*x))*tan((c + d*x)/2))/(3*b^3*(a^2 - b^2)*d) - (2*(A*b^2 - a*b*B + a^2*C)*sec(c + d*x)*tan(c + d*x))/(b*(a^2 - b^2)*d*sqrt(a + b*sec(c + d*x))) + (2*(3*A*b^2 - 3*a*b*B + 4*a^2*C - b^2*C)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(3*b^2*(a^2 - b^2)*d)],
[sec(c + d*x)^1*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + b*sec(c + d*x))^(3/2), x, 6, (-2*(A*b^2 - a*b*B + 2*a^2*C - b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(A*b - b*B + 2*a*C - b*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(A*b^2 - a*b*B + 2*a^2*C - b^2*C)*sqrt(a + b*sec(c + d*x))*tan((c + d*x)/2))/(b^2*(a^2 - b^2)*d) - (2*(A*b^2 - a*b*B + a^2*C)*tan(c + d*x))/(b*(a^2 - b^2)*d*sqrt(a + b*sec(c + d*x)))],
[sec(c + d*x)^0*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + b*sec(c + d*x))^(3/2), x, 7, (2*(A*b^2 - a*b*B + a^2*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*(A - B + C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (4*A*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*(A*b^2 - a*b*B + a^2*C)*tan((1/2)*(c + d*x)))/(a*b*(a + b)*d*sqrt(a + b*sec(c + d*x)))],
[cos(c + d*x)^1*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + b*sec(c + d*x))^(3/2), x, 8, ((a^2*A - 3*A*b^2 + 2*a*b*B - 2*a^2*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(A*b - a*B + a*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*(3*A*b - 2*a*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(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*sec(c + d*x))) + ((a^2*A - 3*A*b^2 + 2*a*b*B - 2*a^2*C)*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(a^2*(a^2 - b^2)*d)],
[cos(c + d*x)^2*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + b*sec(c + d*x))^(3/2), x, 9, -(((7*a^2*A*b - 15*A*b^3 - 4*a^3*B + 12*a*b^2*B - 8*a^2*b*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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^3*(a - b)*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))) - ((2*a^2*A - 3*a*A*b + 5*A*b^2 - 4*a*b*B + 4*a^2*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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^2*(a - b)*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + ((4*a^2*A + 15*A*b^2 - 12*a*b*B + 8*a^2*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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*a^3*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(A*b^2 - a*b*B + a^2*C)*cos(c + d*x)*sin(c + d*x))/(a*(a^2 - b^2)*d*sqrt(a + b*sec(c + d*x))) + ((a^2*A - 5*A*b^2 + 4*a*b*B - 4*a^2*C)*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(2*a^2*(a^2 - b^2)*d) - ((7*a^2*A*b - 15*A*b^3 - 4*a^3*B + 12*a*b^2*B - 8*a^2*b*C)*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(4*a^3*(a^2 - b^2)*d)],


[sec(c + d*x)^3*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + b*sec(c + d*x))^(5/2), x, 8, -((2*(a + b)*(-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(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(3*b^4*(a^2 - b^2)^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))) + (1/(9*b^3*(a^2 - b^2)^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))))*(16*((3/8)*b*(a^2*A*b^2 + 3*A*b^4 + 2*a^3*b*B - 6*a*b^3*B - 4*a^4*C + 7*a^2*b^2*C + b^4*C) + (3/8)*(-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(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b)))) + (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*sec(c + d*x))*tan((1/2)*(c + d*x)))/(3*b^4*(a^2 - b^2)^2*d) - (2*(A*b^2 - a*b*B + a^2*C)*sec(c + d*x)^2*tan(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*sec(c + d*x))^(3/2)) + (4*((-(3/2))*a*b^2*((-a)*A + b*B - a*C) + 2*b^2*(A*b^2 - a*b*B + a^2*C) - (3/2)*a^2*(A*b^2 - a*b*B + 2*a^2*C - b^2*C))*sec(c + d*x)*tan(c + d*x))/(3*b^2*(a^2 - b^2)^2*d*sqrt(a + b*sec(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*sec(c + d*x))*tan(c + d*x))/(3*b^3*(a^2 - b^2)^2*d)],
[sec(c + d*x)^2*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + b*sec(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(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(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 - 8*a^3*C + 6*a^2*b*C + 9*a*b^2*C - 3*b^3*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (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*sec(c + d*x))*tan((c + d*x)/2))/(3*b^3*(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*b*(a^2 - b^2)*d*(a + b*sec(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)*tan(c + d*x))/(3*b^2*(a^2 - b^2)^2*d*sqrt(a + b*sec(c + d*x)))],
[sec(c + d*x)^1*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + b*sec(c + d*x))^(5/2), x, 6, (-2*(4*a*A*b^2 - a^2*b*B - 3*b^3*B - 2*a^3*C + 6*a*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(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((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(4*a*A*b^2 - a^2*b*B - 3*b^3*B - 2*a^3*C + 6*a*b^2*C)*tan((c + d*x)/2))/(3*(a - b)*b^2*(a + b)^2*d*sqrt(a + b*sec(c + d*x))) - (2*(A*b^2 - a*b*B + a^2*C)*tan(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*sec(c + d*x))^(3/2))],
[sec(c + d*x)^0*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + b*sec(c + d*x))^(5/2), x, 8, (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(1 + cos(c + d*x))*sqrt((b + a*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*b*(a + b)*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(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((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (4*A*sqrt(1 + cos(c + d*x))*sqrt((b + a*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^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (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)*tan((1/2)*(c + d*x)))/(3*a^2*(a - b)*b*(a + b)^2*d*sqrt(a + b*sec(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*sec(c + d*x))^(3/2))],
[cos(c + d*x)^1*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + b*sec(c + d*x))^(5/2), x, 9, ((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(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(6*a^2*A*b + 3*a*A*b^2 - 5*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((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*(5*A*b - 2*a*B)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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^3*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(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*sec(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)*sin(c + d*x))/(3*a^2*(a^2 - b^2)^2*d*sqrt(a + b*sec(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)*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(3*a^3*(a^2 - b^2)^2*d)],
[cos(c + d*x)^2*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + b*sec(c + d*x))^(5/2), x, 10, -(((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(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticE(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(12*a^4*(a - b)^2*(a + b)*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))) - ((6*a^4*A - 9*a^3*A*b + 33*a^2*A*b^2 + 21*a*A*b^3 - 35*A*b^4 - 24*a^3*b*B - 12*a^2*b^2*B + 20*a*b^3*B + 12*a^4*C + 12*a^3*b*C - 8*a^2*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*cos(c + d*x))/((a + b)*(1 + cos(c + d*x))))*EllipticF(arcsin(tan((1/2)*(c + d*x))), -((-a + b)/(a + b))))/(6*a^3*(a - b)^2*(a + b)*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + ((4*a^2*A + 35*A*b^2 - 20*a*b*B + 8*a^2*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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*a^4*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(A*b^2 - a*b*B + a^2*C)*cos(c + d*x)*sin(c + d*x))/(3*a*(a^2 - b^2)*d*(a + b*sec(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)*cos(c + d*x)*sin(c + d*x))/(3*a^2*(a^2 - b^2)^2*d*sqrt(a + b*sec(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)*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(6*a^3*(a^2 - b^2)^2*d) - ((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)*cos(c + d*x)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(12*a^4*(a^2 - b^2)^2*d)],


[(a*b*B - a^2*C + b^2*B*sec(c + d*x) + b^2*C*sec(c + d*x)^2)*(a + b*sec(c + d*x))^(3/2), x, 9, -((2*b*(a + b)*(35*a*b*B - 12*a^2*C + 9*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))) - (2*(15*a^3*b*B - 45*a^2*b^2*B - 35*a*b^3*B - 5*b^4*B - 15*a^4*C + 30*a^3*b*C + 12*a^2*b^2*C - 12*a*b^3*C - 9*b^4*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (4*a^3*(b*B - a*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*b*(35*a*b*B - 12*a^2*C + 9*b^2*C)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(15*d) + (2*b^2*(5*b*B + 3*a*C)*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(15*d) + (2*b^2*C*(a + b*sec(c + d*x))^(3/2)*tan(c + d*x))/(5*d)],
[(a*b*B - a^2*C + b^2*B*sec(c + d*x) + b^2*C*sec(c + d*x)^2)*(a + b*sec(c + d*x))^(1/2), x, 8, -((2*b*(a + b)*(3*b*B + a*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))) - (2*(3*a^2*b*B - 6*a*b^2*B - 3*b^3*B - 3*a^3*C + 3*a^2*b*C - a*b^2*C - b^3*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (4*a^2*(b*B - a*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*b*(3*b*B + a*C)*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/(3*d) + (2*b^2*C*sqrt(a + b*sec(c + d*x))*tan(c + d*x))/(3*d)],
[(a*b*B - a^2*C + b^2*B*sec(c + d*x) + b^2*C*sec(c + d*x)^2)/(a + b*sec(c + d*x))^(1/2), x, 7, -((2*b*(a + b)*C*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))) - (2*(a*b*B - b^2*B - a^2*C - b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (4*a*(b*B - a*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*b*C*sqrt(a + b*sec(c + d*x))*tan((1/2)*(c + d*x)))/d],
[(a*b*B - a^2*C + b^2*B*sec(c + d*x) + b^2*C*sec(c + d*x)^2)/(a + b*sec(c + d*x))^(3/2), x, 5, -((2*(b*B - a*C - b*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))) + (4*(b*B - a*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))],
[(a*b*B - a^2*C + b^2*B*sec(c + d*x) + b^2*C*sec(c + d*x)^2)/(a + b*sec(c + d*x))^(5/2), x, 8, (2*b*(b*B - 2*a*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*(b*B - a*C - b*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (4*(b*B - a*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*b*(b*B - 2*a*C)*tan((1/2)*(c + d*x)))/(a*(a + b)*d*sqrt(a + b*sec(c + d*x)))],
[(a*b*B - a^2*C + b^2*B*sec(c + d*x) + b^2*C*sec(c + d*x)^2)/(a + b*sec(c + d*x))^(7/2), x, 9, (2*b*(7*a^2*b*B - 3*b^3*B - 11*a^3*C + 3*a*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*(3*a^2*b*B + 3*a*b^2*B - 2*b^3*B - 3*a^3*C - 6*a^2*b*C + a*b^2*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (4*(b*B - a*C)*sqrt(1 + cos(c + d*x))*sqrt((b + a*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^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*b*(7*a^2*b*B - 3*b^3*B - 11*a^3*C + 3*a*b^2*C)*tan((1/2)*(c + d*x)))/(3*a^2*(a - b)*(a + b)^2*d*sqrt(a + b*sec(c + d*x))) + (2*b^2*(b*B - 2*a*C)*tan(c + d*x))/(3*a*(a^2 - b^2)*d*(a + b*sec(c + d*x))^(3/2))],


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


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


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


[sec(c + d*x)^(3/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + b*sec(c + d*x))^2, x, 9, -((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((1/2)*(c + d*x), 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((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) + (2*(15*a^2*A + 9*A*b^2 + 18*a*b*B + 9*a^2*C + 7*b^2*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(15*d) + (2*(14*a*A*b + 7*a^2*B + 5*b^2*B + 10*a*b*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(21*d) + (2*(63*A*b^2 + 81*a*b*B + 8*a^2*C + 49*b^2*C)*sec(c + d*x)^(5/2)*sin(c + d*x))/(315*d) + (2*(9*b*B + 4*a*C)*sec(c + d*x)^(5/2)*(a + b*sec(c + d*x))*sin(c + d*x))/(63*d) + (2*C*sec(c + d*x)^(5/2)*(a + b*sec(c + d*x))^2*sin(c + d*x))/(9*d)],
[sec(c + d*x)^(1/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + b*sec(c + d*x))^2, x, 8, -((2*(10*a*A*b + 5*a^2*B + 3*b^2*B + 6*a*b*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 + 14*a*b*B + 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*(10*a*A*b + 5*a^2*B + 3*b^2*B + 6*a*b*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d) + (2*(35*A*b^2 + 49*a*b*B + 8*a^2*C + 25*b^2*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(105*d) + (2*(7*b*B + 4*a*C)*sec(c + d*x)^(3/2)*(a + b*sec(c + d*x))*sin(c + d*x))/(35*d) + (2*C*sec(c + d*x)^(3/2)*(a + b*sec(c + d*x))^2*sin(c + d*x))/(7*d)],
[sec(c + d*x)^(-1/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + b*sec(c + d*x))^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((1/2)*(c + d*x), 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((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) + (2*(15*A*b^2 + 25*a*b*B + 8*a^2*C + 9*b^2*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(15*d) + (2*(5*b*B + 4*a*C)*sqrt(sec(c + d*x))*(a + b*sec(c + d*x))*sin(c + d*x))/(15*d) + (2*C*sqrt(sec(c + d*x))*(a + b*sec(c + d*x))^2*sin(c + d*x))/(5*d)],
[sec(c + d*x)^(-3/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + b*sec(c + d*x))^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*(a*A - 3*b*B - 5*a*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(3*d) - (2*b*(A - C)*sqrt(sec(c + d*x))*(a + b*sec(c + d*x))*sin(c + d*x))/(3*d) + (2*A*(a + b*sec(c + d*x))^2*sin(c + d*x))/(3*d*sqrt(sec(c + d*x)))],
[sec(c + d*x)^(-5/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + b*sec(c + d*x))^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*b*(7*A*b + 5*a*B - 15*b*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(15*d) + (2*(4*A*b + 5*a*B)*(a + b*sec(c + d*x))*sin(c + d*x))/(15*d*sqrt(sec(c + d*x))) + (2*A*(a + b*sec(c + d*x))^2*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2))],
[sec(c + d*x)^(-7/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + b*sec(c + d*x))^2, x, 7, (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*(25*a^2*A + 8*A*b^2 + 49*a*b*B + 35*a^2*C)*sin(c + d*x))/(105*d*sqrt(sec(c + d*x))) + (2*(4*A*b + 7*a*B)*(a + b*sec(c + d*x))*sin(c + d*x))/(35*d*sec(c + d*x)^(3/2)) + (2*A*(a + b*sec(c + d*x))^2*sin(c + d*x))/(7*d*sec(c + d*x)^(5/2))],
[sec(c + d*x)^(-9/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + b*sec(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)*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*(49*a^2*A + 8*A*b^2 + 81*a*b*B + 63*a^2*C)*sin(c + d*x))/(315*d*sec(c + d*x)^(3/2)) + (2*(10*a*A*b + 5*a^2*B + 7*b^2*B + 14*a*b*C)*sin(c + d*x))/(21*d*sqrt(sec(c + d*x))) + (2*(4*A*b + 9*a*B)*(a + b*sec(c + d*x))*sin(c + d*x))/(63*d*sec(c + d*x)^(5/2)) + (2*A*(a + b*sec(c + d*x))^2*sin(c + d*x))/(9*d*sec(c + d*x)^(7/2))],


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


[sec(c + d*x)^(1/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + b*sec(c + d*x))^4, x, 10, -((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((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 + 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((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(231*d) + (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(sec(c + d*x))*sin(c + d*x))/(15*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)*sec(c + d*x)^(3/2)*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)*sec(c + d*x)^(3/2)*(a + b*sec(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)*sec(c + d*x)^(3/2)*(a + b*sec(c + d*x))^2*sin(c + d*x))/(231*d) + (2*(11*b*B + 8*a*C)*sec(c + d*x)^(3/2)*(a + b*sec(c + d*x))^3*sin(c + d*x))/(99*d) + (2*C*sec(c + d*x)^(3/2)*(a + b*sec(c + d*x))^4*sin(c + d*x))/(11*d)],
[sec(c + d*x)^(-1/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + b*sec(c + d*x))^4, 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*(1533*a^2*A*b^2 + 189*A*b^4 + 837*a^3*b*B + 681*a*b^3*B + 128*a^4*C + 883*a^2*b^2*C + 147*b^4*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(315*d) + (2*(294*a*A*b^2 + 261*a^2*b*B + 75*b^3*B + 64*a^3*C + 202*a*b^2*C)*sqrt(sec(c + d*x))*(a + b*sec(c + d*x))*sin(c + d*x))/(315*d) + (2*(63*A*b^2 + 117*a*b*B + 48*a^2*C + 49*b^2*C)*sqrt(sec(c + d*x))*(a + b*sec(c + d*x))^2*sin(c + d*x))/(315*d) + (2*(9*b*B + 8*a*C)*sqrt(sec(c + d*x))*(a + b*sec(c + d*x))^3*sin(c + d*x))/(63*d) + (2*C*sqrt(sec(c + d*x))*(a + b*sec(c + d*x))^4*sin(c + d*x))/(9*d)],
[sec(c + d*x)^(-3/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + b*sec(c + d*x))^4, 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*(35*a^3*A - 385*a*A*b^2 - 511*a^2*b*B - 63*b^3*B - 279*a^3*C - 227*a*b^2*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(105*d) - (2*b*(35*a^2*A - 35*A*b^2 - 98*a*b*B - 87*a^2*C - 25*b^2*C)*sqrt(sec(c + d*x))*(a + b*sec(c + d*x))*sin(c + d*x))/(105*d) - (2*b*(35*a*A - 21*b*B - 39*a*C)*sqrt(sec(c + d*x))*(a + b*sec(c + d*x))^2*sin(c + d*x))/(105*d) - (2*b*(7*A - 3*C)*sqrt(sec(c + d*x))*(a + b*sec(c + d*x))^3*sin(c + d*x))/(21*d) + (2*A*(a + b*sec(c + d*x))^4*sin(c + d*x))/(3*d*sqrt(sec(c + d*x)))],
[sec(c + d*x)^(-5/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + b*sec(c + d*x))^4, 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*(17*a^2*A*b - 15*A*b^3 + 5*a^3*B - 55*a*b^2*B - 73*a^2*b*C - 9*b^3*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(15*d) - (2*b*(14*a*A*b + 5*a^2*B - 5*b^2*B - 14*a*b*C)*sqrt(sec(c + d*x))*(a + b*sec(c + d*x))*sin(c + d*x))/(15*d) - (2*b*(11*A*b + 5*a*B - 3*b*C)*sqrt(sec(c + d*x))*(a + b*sec(c + d*x))^2*sin(c + d*x))/(15*d) + (2*(8*A*b + 5*a*B)*(a + b*sec(c + d*x))^3*sin(c + d*x))/(15*d*sqrt(sec(c + d*x))) + (2*A*(a + b*sec(c + d*x))^4*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2))],
[sec(c + d*x)^(-7/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + b*sec(c + d*x))^4, 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*(25*a^3*A + 141*a*A*b^2 + 119*a^2*b*B - 105*b^3*B + 35*a^3*C - 385*a*b^2*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(105*d) - (2*b*(25*a^2*A + 87*A*b^2 + 98*a*b*B + 35*a^2*C - 35*b^2*C)*sqrt(sec(c + d*x))*(a + b*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*sec(c + d*x))^2*sin(c + d*x))/(105*d*sqrt(sec(c + d*x))) + (2*(8*A*b + 7*a*B)*(a + b*sec(c + d*x))^3*sin(c + d*x))/(35*d*sec(c + d*x)^(3/2)) + (2*A*(a + b*sec(c + d*x))^4*sin(c + d*x))/(7*d*sec(c + d*x)^(5/2))],
[sec(c + d*x)^(-9/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + b*sec(c + d*x))^4, 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*b*(251*a^2*A*b + 187*A*b^3 + 75*a^3*B + 423*a*b^2*B + 357*a^2*b*C - 315*b^3*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*sec(c + d*x))*sin(c + d*x))/(315*d*sqrt(sec(c + d*x))) + (2*(49*a^2*A + 48*A*b^2 + 117*a*b*B + 63*a^2*C)*(a + b*sec(c + d*x))^2*sin(c + d*x))/(315*d*sec(c + d*x)^(3/2)) + (2*(8*A*b + 9*a*B)*(a + b*sec(c + d*x))^3*sin(c + d*x))/(63*d*sec(c + d*x)^(5/2)) + (2*A*(a + b*sec(c + d*x))^4*sin(c + d*x))/(9*d*sec(c + d*x)^(7/2))],
[sec(c + d*x)^(-11/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + b*sec(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)*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*(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*sqrt(sec(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*sec(c + d*x))*sin(c + d*x))/(3465*d*sec(c + d*x)^(3/2)) + (2*(27*a^2*A + 16*A*b^2 + 55*a*b*B + 33*a^2*C)*(a + b*sec(c + d*x))^2*sin(c + d*x))/(231*d*sec(c + d*x)^(5/2)) + (2*(8*A*b + 11*a*B)*(a + b*sec(c + d*x))^3*sin(c + d*x))/(99*d*sec(c + d*x)^(7/2)) + (2*A*(a + b*sec(c + d*x))^4*sin(c + d*x))/(11*d*sec(c + d*x)^(9/2))],
[sec(c + d*x)^(-13/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)*(a + b*sec(c + d*x))^4, x, 10, (2*(77*a^4*A + 546*a^2*A*b^2 + 117*A*b^4 + 364*a^3*b*B + 468*a*b^3*B + 91*a^4*C + 702*a^2*b^2*C + 195*b^4*C)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(195*d) + (2*(180*a^3*A*b + 220*a*A*b^3 + 45*a^4*B + 330*a^2*b^2*B + 77*b^4*B + 220*a^3*b*C + 308*a*b^3*C)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(231*d) + (2*(5929*a^4*A + 25217*a^2*A*b^2 + 384*A*b^4 + 22763*a^3*b*B + 13351*a*b^3*B + 7007*a^4*C + 33319*a^2*b^2*C)*sin(c + d*x))/(45045*d*sec(c + d*x)^(3/2)) + (2*(180*a^3*A*b + 220*a*A*b^3 + 45*a^4*B + 330*a^2*b^2*B + 77*b^4*B + 220*a^3*b*C + 308*a*b^3*C)*sin(c + d*x))/(231*d*sqrt(sec(c + d*x))) + (2*(2518*a^2*A*b + 192*A*b^3 + 1053*a^3*B + 2171*a*b^2*B + 3146*a^2*b*C)*(a + b*sec(c + d*x))*sin(c + d*x))/(9009*d*sec(c + d*x)^(5/2)) + (2*(121*a^2*A + 48*A*b^2 + 221*a*b*B + 143*a^2*C)*(a + b*sec(c + d*x))^2*sin(c + d*x))/(1287*d*sec(c + d*x)^(7/2)) + (2*(8*A*b + 13*a*B)*(a + b*sec(c + d*x))^3*sin(c + d*x))/(143*d*sec(c + d*x)^(9/2)) + (2*A*(a + b*sec(c + d*x))^4*sin(c + d*x))/(13*d*sec(c + d*x)^(11/2))],


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


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


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


[sec(c + d*x)^(7/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + b*sec(c + d*x))^3, x, 11, ((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((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(4*b^4*(a^2 - b^2)^2*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))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(12*b^3*(a^2 - b^2)^2*d) + ((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*a)/(a + b), (1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(4*(a - b)^2*b^4*(a + b)^3*d) - ((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(sec(c + d*x))*sin(c + d*x))/(4*b^4*(a^2 - b^2)^2*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)*sec(c + d*x)^(3/2)*sin(c + d*x))/(12*b^3*(a^2 - b^2)^2*d) - ((A*b^2 - a*b*B + a^2*C)*sec(c + d*x)^(7/2)*sin(c + d*x))/(2*b*(a^2 - b^2)*d*(a + b*sec(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)*sec(c + d*x)^(5/2)*sin(c + d*x))/(4*b^2*(a^2 - b^2)^2*d*(a + b*sec(c + d*x)))],
[sec(c + d*x)^(5/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + b*sec(c + d*x))^3, x, 10, ((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((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(4*b^3*(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 - 5*a^4*C + 11*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) + ((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*a)/(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^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(sec(c + d*x))*sin(c + d*x))/(4*b^3*(a^2 - b^2)^2*d) - ((A*b^2 - a*b*B + a^2*C)*sec(c + d*x)^(5/2)*sin(c + d*x))/(2*b*(a^2 - b^2)*d*(a + b*sec(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)*sec(c + d*x)^(3/2)*sin(c + d*x))/(4*b^2*(a^2 - b^2)^2*d*(a + b*sec(c + d*x)))],
[sec(c + d*x)^(3/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + b*sec(c + d*x))^3, x, 9, -(((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((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(4*a*b^2*(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((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(4*a^2*b*(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*a)/(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*b*B + a^2*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(2*b*(a^2 - b^2)*d*(a + b*sec(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(sec(c + d*x))*sin(c + d*x))/(4*b^2*(a^2 - b^2)^2*d*(a + b*sec(c + d*x)))],
[sec(c + d*x)^(1/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + b*sec(c + d*x))^3, x, 9, ((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((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(4*a^2*b*(a^2 - b^2)^2*d) + ((8*a^4*A - 5*a^2*A*b^2 + 3*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((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(4*a^3*(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*a)/(a + b), (1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(4*a^3*(a - b)^2*b*(a + b)^3*d) - ((A*b^2 - a*b*B + a^2*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(2*b*(a^2 - b^2)*d*(a + b*sec(c + d*x))^2) - ((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(sec(c + d*x))*sin(c + d*x))/(4*a*b*(a^2 - b^2)^2*d*(a + b*sec(c + d*x)))],
[sec(c + d*x)^(-1/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + b*sec(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((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(4*a^3*(a^2 - b^2)^2*d) - ((24*a^4*A*b - 33*a^2*A*b^3 + 15*A*b^5 - 8*a^5*B + 5*a^3*b^2*B - 3*a*b^4*B + 7*a^4*b*C - a^2*b^3*C)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(4*a^4*(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*a)/(a + b), (1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(4*a^4*(a - b)^2*(a + b)^3*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*sec(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*sec(c + d*x)))],
[sec(c + d*x)^(-3/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + b*sec(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((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(4*a^4*(a^2 - b^2)^2*d)) + ((8*a^6*A + 128*a^4*A*b^2 - 223*a^2*A*b^4 + 105*A*b^6 - 72*a^5*b*B + 99*a^3*b^3*B - 45*a*b^5*B + 24*a^6*C - 15*a^4*b^2*C + 9*a^2*b^4*C)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(12*a^5*(a^2 - b^2)^2*d) - (b*(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*a)/(a + b), (1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(4*a^5*(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*sqrt(sec(c + d*x))) + ((A*b^2 - a*b*B + a^2*C)*sin(c + d*x))/(2*a*(a^2 - b^2)*d*sqrt(sec(c + d*x))*(a + b*sec(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*sqrt(sec(c + d*x))*(a + b*sec(c + d*x)))],


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


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


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


[sec(c + d*x)^(3/2)*(a + b*sec(c + d*x))^(3/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 13, -(((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((b + a*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(192*b^2*d*sqrt(a + b*sec(c + d*x)))) + ((336*a*A*b^2 + 136*a^2*b*B + 128*b^3*B - 3*a^3*C + 228*a*b^2*C)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(192*b*d*sqrt(a + b*sec(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((b + a*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(64*b^2*d*sqrt(a + b*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(sec(c + d*x))*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(192*b^2*d) + ((48*A*b^2 + 56*a*b*B + 3*a^2*C + 36*b^2*C)*sec(c + d*x)^(3/2)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(96*b*d) + ((8*b*B + 3*a*C)*sec(c + d*x)^(5/2)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(24*d) + (C*sec(c + d*x)^(5/2)*(a + b*sec(c + d*x))^(3/2)*sin(c + d*x))/(4*d)],
[sqrt(sec(c + d*x))*(a + b*sec(c + d*x))^(3/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 12, -((a + b)*(24*A*b^2 + 30*a*b*B + 3*a^2*C + 16*b^2*C)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*a)/(a + b))*sqrt(sec(c + d*x)))/(24*b*d*sqrt(a + b*sec(c + d*x))) + ((48*a^2*A + 24*A*b^2 + 42*a*b*B + 17*a^2*C + 16*b^2*C)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*a)/(a + b))*sqrt(sec(c + d*x)))/(24*d*sqrt(a + b*sec(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((b + a*cos(c + d*x))/(a + b))*EllipticPi(2, (c + d*x)/2, (2*a)/(a + b))*sqrt(sec(c + d*x)))/(8*b*d*sqrt(a + b*sec(c + d*x))) + ((24*A*b^2 + 30*a*b*B + 3*a^2*C + 16*b^2*C)*sqrt(sec(c + d*x))*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(24*b*d) + ((2*b*B + a*C)*sec(c + d*x)^(3/2)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(4*d) + (C*sec(c + d*x)^(3/2)*(a + b*sec(c + d*x))^(3/2)*sin(c + d*x))/(3*d)],
[((a + b*sec(c + d*x))^(3/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/sqrt(sec(c + d*x)), x, 11, ((a + b)*(8*a*A - 4*b*B - 5*a*C)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*a)/(a + b))*sqrt(sec(c + d*x)))/(4*d*sqrt(a + b*sec(c + d*x))) + ((8*a*A*b + 8*a^2*B + 4*b^2*B + 7*a*b*C)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*a)/(a + b))*sqrt(sec(c + d*x)))/(4*d*sqrt(a + b*sec(c + d*x))) + ((8*A*b^2 + 12*a*b*B + 3*a^2*C + 4*b^2*C)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticPi(2, (c + d*x)/2, (2*a)/(a + b))*sqrt(sec(c + d*x)))/(4*d*sqrt(a + b*sec(c + d*x))) + ((4*b*B + 3*a*C)*sqrt(sec(c + d*x))*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(4*d) + (C*sqrt(sec(c + d*x))*(a + b*sec(c + d*x))^(3/2)*sin(c + d*x))/(2*d)],
[((a + b*sec(c + d*x))^(3/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/sec(c + d*x)^(3/2), x, 11, ((a + b)*(8*A*b + 6*a*B - 3*b*C)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(3*d*sqrt(a + b*sec(c + d*x))) + ((2*a^2*A - 2*A*b^2 + 6*a*b*B + 6*a^2*C + 3*b^2*C)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(3*d*sqrt(a + b*sec(c + d*x))) + (b*(2*b*B + 3*a*C)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*sec(c + d*x))) - (b*(2*A - 3*C)*sqrt(sec(c + d*x))*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(3*d) + (2*A*(a + b*sec(c + d*x))^(3/2)*sin(c + d*x))/(3*d*sqrt(sec(c + d*x)))],
[((a + b*sec(c + d*x))^(3/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/sec(c + d*x)^(5/2), x, 11, (2*(a + b)*(9*a^2*A + 3*A*b^2 + 20*a*b*B + 15*a^2*C)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*a)/(a + b))*sqrt(sec(c + d*x)))/(15*a*d*sqrt(a + b*sec(c + d*x))) + (2*(3*a^2*A*b - 3*A*b^3 + 5*a^3*B - 5*a*b^2*B + 15*a^2*b*C)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*a)/(a + b))*sqrt(sec(c + d*x)))/(15*a*d*sqrt(a + b*sec(c + d*x))) + (2*b^2*C*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticPi(2, (c + d*x)/2, (2*a)/(a + b))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*sec(c + d*x))) + (2*(3*A*b + 5*a*B)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(15*d*sqrt(sec(c + d*x))) + (2*A*(a + b*sec(c + d*x))^(3/2)*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2))],
[((a + b*sec(c + d*x))^(3/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/sec(c + d*x)^(7/2), x, 9, (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((b + a*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(105*a^2*d*sqrt(a + b*sec(c + d*x))) + (2*(25*a^4*A - 31*a^2*A*b^2 + 6*A*b^4 + 21*a^3*b*B - 21*a*b^3*B + 35*a^4*C - 35*a^2*b^2*C)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(105*a^2*d*sqrt(a + b*sec(c + d*x))) + (2*(3*A*b + 7*a*B)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(35*d*sec(c + d*x)^(3/2)) + (2*(25*a^2*A + 3*A*b^2 + 42*a*b*B + 35*a^2*C)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(105*a*d*sqrt(sec(c + d*x))) + (2*A*(a + b*sec(c + d*x))^(3/2)*sin(c + d*x))/(7*d*sec(c + d*x)^(5/2))],
[((a + b*sec(c + d*x))^(3/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/sec(c + d*x)^(9/2), x, 10, (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((b + a*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*a)/(a + b))*sqrt(sec(c + d*x)))/(315*a^3*d*sqrt(a + b*sec(c + d*x))) + (2*(39*a^4*A*b - 31*a^2*A*b^3 - 8*A*b^5 + 75*a^5*B - 93*a^3*b^2*B + 18*a*b^4*B + 63*a^4*b*C - 63*a^2*b^3*C)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*a)/(a + b))*sqrt(sec(c + d*x)))/(315*a^3*d*sqrt(a + b*sec(c + d*x))) + (2*(A*b + 3*a*B)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(21*d*sec(c + d*x)^(5/2)) + (2*(49*a^2*A + 3*A*b^2 + 72*a*b*B + 63*a^2*C)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(315*a*d*sec(c + d*x)^(3/2)) + (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*sec(c + d*x))*sin(c + d*x))/(315*a^2*d*sqrt(sec(c + d*x))) + (2*A*(a + b*sec(c + d*x))^(3/2)*sin(c + d*x))/(9*d*sec(c + d*x)^(7/2))],


[sqrt(sec(c + d*x))*(a + b*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 13, -((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((b + a*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*a)/(a + b))*sqrt(sec(c + d*x)))/(192*b*d*sqrt(a + b*sec(c + d*x))) + ((384*a^3*A + 528*a*A*b^2 + 472*a^2*b*B + 128*b^3*B + 133*a^3*C + 356*a*b^2*C)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*a)/(a + b))*sqrt(sec(c + d*x)))/(192*d*sqrt(a + b*sec(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((b + a*cos(c + d*x))/(a + b))*EllipticPi(2, (c + d*x)/2, (2*a)/(a + b))*sqrt(sec(c + d*x)))/(64*b*d*sqrt(a + b*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(sec(c + d*x))*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(192*b*d) + ((16*A*b^2 + 24*a*b*B + 5*a^2*C + 12*b^2*C)*sec(c + d*x)^(3/2)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(32*d) + ((8*b*B + 5*a*C)*sec(c + d*x)^(3/2)*(a + b*sec(c + d*x))^(3/2)*sin(c + d*x))/(24*d) + (C*sec(c + d*x)^(3/2)*(a + b*sec(c + d*x))^(5/2)*sin(c + d*x))/(4*d)],
[((a + b*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/sqrt(sec(c + d*x)), x, 12, ((a + b)*(48*a^2*A - 24*A*b^2 - 54*a*b*B - 33*a^2*C - 16*b^2*C)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*a)/(a + b))*sqrt(sec(c + d*x)))/(24*d*sqrt(a + b*sec(c + d*x))) + ((96*a^2*A*b + 24*A*b^3 + 48*a^3*B + 66*a*b^2*B + 59*a^2*b*C + 16*b^3*C)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*a)/(a + b))*sqrt(sec(c + d*x)))/(24*d*sqrt(a + b*sec(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((b + a*cos(c + d*x))/(a + b))*EllipticPi(2, (c + d*x)/2, (2*a)/(a + b))*sqrt(sec(c + d*x)))/(8*d*sqrt(a + b*sec(c + d*x))) + ((24*A*b^2 + 42*a*b*B + 15*a^2*C + 16*b^2*C)*sqrt(sec(c + d*x))*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(24*d) + ((6*b*B + 5*a*C)*sqrt(sec(c + d*x))*(a + b*sec(c + d*x))^(3/2)*sin(c + d*x))/(12*d) + (C*sqrt(sec(c + d*x))*(a + b*sec(c + d*x))^(5/2)*sin(c + d*x))/(3*d)],
[((a + b*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/sec(c + d*x)^(3/2), x, 12, ((a + b)*(56*a*A*b + 24*a^2*B - 12*b^2*B - 27*a*b*C)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(12*d*sqrt(a + b*sec(c + d*x))) + ((8*a^3*A + 16*a*A*b^2 + 48*a^2*b*B + 12*b^3*B + 24*a^3*C + 33*a*b^2*C)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(12*d*sqrt(a + b*sec(c + d*x))) + (b*(8*A*b^2 + 20*a*b*B + 15*a^2*C + 4*b^2*C)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(4*d*sqrt(a + b*sec(c + d*x))) - (b*(8*a*A - 12*b*B - 21*a*C)*sqrt(sec(c + d*x))*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(12*d) - (b*(4*A - 3*C)*sqrt(sec(c + d*x))*(a + b*sec(c + d*x))^(3/2)*sin(c + d*x))/(6*d) + (2*A*(a + b*sec(c + d*x))^(5/2)*sin(c + d*x))/(3*d*sqrt(sec(c + d*x)))],
[((a + b*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/sec(c + d*x)^(5/2), x, 12, ((a + b)*(18*a^2*A + 46*A*b^2 + 70*a*b*B + 30*a^2*C - 15*b^2*C)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(15*d*sqrt(a + b*sec(c + d*x))) + ((16*a^2*A*b - 16*A*b^3 + 10*a^3*B + 20*a*b^2*B + 60*a^2*b*C + 15*b^3*C)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(15*d*sqrt(a + b*sec(c + d*x))) + (b^2*(2*b*B + 5*a*C)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*sec(c + d*x))) - (b*(16*A*b + 10*a*B - 15*b*C)*sqrt(sec(c + d*x))*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(15*d) + (2*(A*b + a*B)*(a + b*sec(c + d*x))^(3/2)*sin(c + d*x))/(3*d*sqrt(sec(c + d*x))) + (2*A*(a + b*sec(c + d*x))^(5/2)*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2))],
[((a + b*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/sec(c + d*x)^(7/2), x, 12, (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((b + a*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*a)/(a + b))*sqrt(sec(c + d*x)))/(105*a*d*sqrt(a + b*sec(c + d*x))) + (2*(25*a^4*A - 10*a^2*A*b^2 - 15*A*b^4 + 56*a^3*b*B - 56*a*b^3*B + 35*a^4*C + 70*a^2*b^2*C)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*a)/(a + b))*sqrt(sec(c + d*x)))/(105*a*d*sqrt(a + b*sec(c + d*x))) + (2*b^3*C*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticPi(2, (c + d*x)/2, (2*a)/(a + b))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*sec(c + d*x))) + (2*(25*a^2*A + 15*A*b^2 + 56*a*b*B + 35*a^2*C)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(105*d*sqrt(sec(c + d*x))) + (2*(5*A*b + 7*a*B)*(a + b*sec(c + d*x))^(3/2)*sin(c + d*x))/(35*d*sec(c + d*x)^(3/2)) + (2*A*(a + b*sec(c + d*x))^(5/2)*sin(c + d*x))/(7*d*sec(c + d*x)^(5/2))],
[((a + b*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/sec(c + d*x)^(9/2), x, 10, (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((b + a*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(315*a^2*d*sqrt(a + b*sec(c + d*x))) + (2*(114*a^4*A*b - 124*a^2*A*b^3 + 10*A*b^5 + 75*a^5*B - 30*a^3*b^2*B - 45*a*b^4*B + 168*a^4*b*C - 168*a^2*b^3*C)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(315*a^2*d*sqrt(a + b*sec(c + d*x))) + (2*(49*a^2*A + 15*A*b^2 + 90*a*b*B + 63*a^2*C)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(315*d*sec(c + d*x)^(3/2)) + (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*sec(c + d*x))*sin(c + d*x))/(315*a*d*sqrt(sec(c + d*x))) + (2*(5*A*b + 9*a*B)*(a + b*sec(c + d*x))^(3/2)*sin(c + d*x))/(63*d*sec(c + d*x)^(5/2)) + (2*A*(a + b*sec(c + d*x))^(5/2)*sin(c + d*x))/(9*d*sec(c + d*x)^(7/2))],
[((a + b*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/sec(c + d*x)^(11/2), x, 11, (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((b + a*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*a)/(a + b))*sqrt(sec(c + d*x)))/(3465*a^3*d*sqrt(a + b*sec(c + d*x))) + (2*(675*a^6*A - 390*a^4*A*b^2 - 245*a^2*A*b^4 - 40*A*b^6 + 1254*a^5*b*B - 1364*a^3*b^3*B + 110*a*b^5*B + 825*a^6*C - 330*a^4*b^2*C - 495*a^2*b^4*C)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*a)/(a + b))*sqrt(sec(c + d*x)))/(3465*a^3*d*sqrt(a + b*sec(c + d*x))) + (2*(27*a^2*A + 5*A*b^2 + 44*a*b*B + 33*a^2*C)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(231*d*sec(c + d*x)^(5/2)) + (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*sec(c + d*x))*sin(c + d*x))/(3465*a*d*sec(c + d*x)^(3/2)) + (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 + 825*a^4*C + 1485*a^2*b^2*C)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(3465*a^2*d*sqrt(sec(c + d*x))) + (2*(5*A*b + 11*a*B)*(a + b*sec(c + d*x))^(3/2)*sin(c + d*x))/(99*d*sec(c + d*x)^(7/2)) + (2*A*(a + b*sec(c + d*x))^(5/2)*sin(c + d*x))/(11*d*sec(c + d*x)^(9/2))],


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


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

[(sqrt(sec(c + d*x))*(a*A + (A*b + a*B)*sec(c + d*x) + b*B*sec(c + d*x)^2))/sqrt(a + b*sec(c + d*x)), x, 10, -(((a + b)*B*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*a)/(a + b))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*sec(c + d*x)))) + ((2*a*A + b*B)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*a)/(a + b))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*sec(c + d*x))) + ((2*A*b + a*B)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticPi(2, (c + d*x)/2, (2*a)/(a + b))*sqrt(sec(c + d*x)))/(d*sqrt(a + b*sec(c + d*x))) + (B*sqrt(sec(c + d*x))*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/d],


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


[(sec(c + d*x)^(5/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/(a + b*sec(c + d*x))^(5/2), x, 12, ((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((b + a*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*a)/(a + b))*sqrt(sec(c + d*x)))/(3*(a - b)^2*b^3*(a + b)*d*sqrt(a + b*sec(c + d*x))) + ((2*A*b^2 - 2*a*b*B + 5*a^2*C - 3*b^2*C)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*a)/(a + b))*sqrt(sec(c + d*x)))/(3*b^2*(a^2 - b^2)*d*sqrt(a + b*sec(c + d*x))) + ((2*b*B - 5*a*C)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticPi(2, (c + d*x)/2, (2*a)/(a + b))*sqrt(sec(c + d*x)))/(b^3*d*sqrt(a + b*sec(c + d*x))) - (2*(A*b^2 - a*b*B + a^2*C)*sec(c + d*x)^(5/2)*sin(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*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)*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*b^2*(a^2 - b^2)^2*d*sqrt(a + b*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(sec(c + d*x))*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(3*b^3*(a^2 - b^2)^2*d)],
[(sec(c + d*x)^(3/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/(a + b*sec(c + d*x))^(5/2), x, 11, (-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((b + a*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*a)/(a + b))*sqrt(sec(c + d*x)))/(3*a*(a - b)^2*b^2*(a + b)*d*sqrt(a + b*sec(c + d*x))) - (2*(A*b^2 - a*b*B + a^2*C)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*a)/(a + b))*sqrt(sec(c + d*x)))/(3*a*b*(a^2 - b^2)*d*sqrt(a + b*sec(c + d*x))) + (2*C*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticPi(2, (c + d*x)/2, (2*a)/(a + b))*sqrt(sec(c + d*x)))/(b^2*d*sqrt(a + b*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*b*(a^2 - b^2)*d*(a + b*sec(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(sec(c + d*x))*sin(c + d*x))/(3*b^2*(a^2 - b^2)^2*d*sqrt(a + b*sec(c + d*x)))],
[(sqrt(sec(c + d*x))*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/(a + b*sec(c + d*x))^(5/2), x, 8, (2*(6*a^2*A*b - 2*A*b^3 - 3*a^3*B - a*b^2*B + 4*a^2*b*C)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(3*a^2*(a - b)^2*(a + b)*d*sqrt(a + b*sec(c + d*x))) + (2*(3*a^2*A - 2*A*b^2 - a*b*B + a^2*C)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*a)/(a + b))*sqrt(sec(c + d*x)))/(3*a^2*(a^2 - b^2)*d*sqrt(a + b*sec(c + d*x))) - (2*(A*b^2 - a*b*B + a^2*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*sec(c + d*x))^(3/2)) - (2*(5*a^2*A*b^2 - A*b^4 - 2*a^3*b*B - 2*a*b^3*B - a^4*C + 5*a^2*b^2*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(3*a*b*(a^2 - b^2)^2*d*sqrt(a + b*sec(c + d*x)))],
[(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(sqrt(sec(c + d*x))*(a + b*sec(c + d*x))^(5/2)), x, 8, (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((b + a*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*a)/(a + b))*sqrt(sec(c + d*x)))/(3*a^3*(a - b)^2*(a + b)*d*sqrt(a + b*sec(c + d*x))) - (2*(9*a^2*A*b - 8*A*b^3 - 3*a^3*B + 2*a*b^2*B + a^2*b*C)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*a)/(a + b))*sqrt(sec(c + d*x)))/(3*a^3*(a^2 - b^2)*d*sqrt(a + b*sec(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*sec(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*sec(c + d*x)))],
[(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(sec(c + d*x)^(3/2)*(a + b*sec(c + d*x))^(5/2)), x, 9, (-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((b + a*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*a)/(a + b))*sqrt(sec(c + d*x)))/(3*a^4*(a - b)^2*(a + b)*d*sqrt(a + b*sec(c + d*x))) + (2*(a^4*A + 16*a^2*A*b^2 - 16*A*b^4 - 9*a^3*b*B + 8*a*b^3*B + 3*a^4*C - 2*a^2*b^2*C)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*a)/(a + b))*sqrt(sec(c + d*x)))/(3*a^4*(a^2 - b^2)*d*sqrt(a + b*sec(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(sec(c + d*x))*(a + b*sec(c + d*x))^(3/2)) + (2*(10*a^2*A*b^2 - 6*A*b^4 - 7*a^3*b*B + 3*a*b^3*B + 4*a^4*C)*sin(c + d*x))/(3*a^2*(a^2 - b^2)^2*d*sqrt(sec(c + d*x))*sqrt(a + b*sec(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*sec(c + d*x))*sin(c + d*x))/(3*a^3*(a^2 - b^2)^2*d*sqrt(sec(c + d*x)))],
[(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(sec(c + d*x)^(5/2)*(a + b*sec(c + d*x))^(5/2)), x, 10, (2*(9*a^6*A + 55*a^4*A*b^2 - 212*a^2*A*b^4 + 128*A*b^6 - 40*a^5*b*B + 140*a^3*b^3*B - 80*a*b^5*B + 15*a^6*C - 75*a^4*b^2*C + 40*a^2*b^4*C)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*a)/(a + b))*sqrt(sec(c + d*x)))/(15*a^5*(a - b)^2*(a + b)*d*sqrt(a + b*sec(c + d*x))) - (2*(17*a^4*A*b + 116*a^2*A*b^3 - 128*A*b^5 - 5*a^5*B - 80*a^3*b^2*B + 80*a*b^4*B + 45*a^4*b*C - 40*a^2*b^3*C)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*a)/(a + b))*sqrt(sec(c + d*x)))/(15*a^5*(a^2 - b^2)*d*sqrt(a + b*sec(c + d*x))) + (2*(A*b^2 - a*b*B + a^2*C)*sin(c + d*x))/(3*a*(a^2 - b^2)*d*sec(c + d*x)^(3/2)*(a + b*sec(c + d*x))^(3/2)) + (2*(12*a^2*A*b^2 - 8*A*b^4 - 9*a^3*b*B + 5*a*b^3*B + 6*a^4*C - 2*a^2*b^2*C)*sin(c + d*x))/(3*a^2*(a^2 - b^2)^2*d*sec(c + d*x)^(3/2)*sqrt(a + b*sec(c + d*x))) + (2*(3*a^4*A - 71*a^2*A*b^2 + 48*A*b^4 + 50*a^3*b*B - 30*a*b^3*B - 35*a^4*C + 15*a^2*b^2*C)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(15*a^3*(a^2 - b^2)^2*d*sec(c + d*x)^(3/2)) - (2*(14*a^4*A*b - 98*a^2*A*b^3 + 64*A*b^5 - 5*a^5*B + 65*a^3*b^2*B - 40*a*b^4*B - 40*a^4*b*C + 20*a^2*b^3*C)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(15*a^4*(a^2 - b^2)^2*d*sqrt(sec(c + d*x)))],


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


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


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


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


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


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


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


[cos(c + d*x)^(13/2)*(a + b*sec(c + d*x))^4*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 9, (2*(77*a^4*A + 546*a^2*A*b^2 + 117*A*b^4 + 364*a^3*b*B + 468*a*b^3*B + 91*a^4*C + 702*a^2*b^2*C + 195*b^4*C)*EllipticE((1/2)*(c + d*x), 2))/(195*d) + (2*(180*a^3*A*b + 220*a*A*b^3 + 45*a^4*B + 330*a^2*b^2*B + 77*b^4*B + 220*a^3*b*C + 308*a*b^3*C)*EllipticF((1/2)*(c + d*x), 2))/(231*d) + (2*(180*a^3*A*b + 220*a*A*b^3 + 45*a^4*B + 330*a^2*b^2*B + 77*b^4*B + 220*a^3*b*C + 308*a*b^3*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(231*d) + (2*(5929*a^4*A + 25217*a^2*A*b^2 + 384*A*b^4 + 22763*a^3*b*B + 13351*a*b^3*B + 7007*a^4*C + 33319*a^2*b^2*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(45045*d) + (2*(2518*a^2*A*b + 192*A*b^3 + 1053*a^3*B + 2171*a*b^2*B + 3146*a^2*b*C)*cos(c + d*x)^(5/2)*(a + b*sec(c + d*x))*sin(c + d*x))/(9009*d) + (2*(121*a^2*A + 48*A*b^2 + 221*a*b*B + 143*a^2*C)*cos(c + d*x)^(7/2)*(a + b*sec(c + d*x))^2*sin(c + d*x))/(1287*d) + (2*(8*A*b + 13*a*B)*cos(c + d*x)^(9/2)*(a + b*sec(c + d*x))^3*sin(c + d*x))/(143*d) + (2*A*cos(c + d*x)^(11/2)*(a + b*sec(c + d*x))^4*sin(c + d*x))/(13*d)],
[cos(c + d*x)^(11/2)*(a + b*sec(c + d*x))^4*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), x, 8, (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)*sqrt(cos(c + d*x))*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)*cos(c + d*x)^(3/2)*(a + b*sec(c + d*x))*sin(c + d*x))/(3465*d) + (2*(27*a^2*A + 16*A*b^2 + 55*a*b*B + 33*a^2*C)*cos(c + d*x)^(5/2)*(a + b*sec(c + d*x))^2*sin(c + d*x))/(231*d) + (2*(8*A*b + 11*a*B)*cos(c + d*x)^(7/2)*(a + b*sec(c + d*x))^3*sin(c + d*x))/(99*d) + (2*A*cos(c + d*x)^(9/2)*(a + b*sec(c + d*x))^4*sin(c + d*x))/(11*d)],
[cos(c + d*x)^(9/2)*(a + b*sec(c + d*x))^4*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), 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*b*(251*a^2*A*b + 187*A*b^3 + 75*a^3*B + 423*a*b^2*B + 357*a^2*b*C - 315*b^3*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)*sqrt(cos(c + d*x))*(a + b*sec(c + d*x))*sin(c + d*x))/(315*d) + (2*(49*a^2*A + 48*A*b^2 + 117*a*b*B + 63*a^2*C)*cos(c + d*x)^(3/2)*(a + b*sec(c + d*x))^2*sin(c + d*x))/(315*d) + (2*(8*A*b + 9*a*B)*cos(c + d*x)^(5/2)*(a + b*sec(c + d*x))^3*sin(c + d*x))/(63*d) + (2*A*cos(c + d*x)^(7/2)*(a + b*sec(c + d*x))^4*sin(c + d*x))/(9*d)],
[cos(c + d*x)^(7/2)*(a + b*sec(c + d*x))^4*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), 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*(25*a^3*A + 141*a*A*b^2 + 119*a^2*b*B - 105*b^3*B + 35*a^3*C - 385*a*b^2*C)*sin(c + d*x))/(105*d*sqrt(cos(c + d*x))) - (2*b*(25*a^2*A + 87*A*b^2 + 98*a*b*B + 35*a^2*C - 35*b^2*C)*(a + b*sec(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)*sqrt(cos(c + d*x))*(a + b*sec(c + d*x))^2*sin(c + d*x))/(105*d) + (2*(8*A*b + 7*a*B)*cos(c + d*x)^(3/2)*(a + b*sec(c + d*x))^3*sin(c + d*x))/(35*d) + (2*A*cos(c + d*x)^(5/2)*(a + b*sec(c + d*x))^4*sin(c + d*x))/(7*d)],
[cos(c + d*x)^(5/2)*(a + b*sec(c + d*x))^4*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), 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*(17*a^2*A*b - 15*A*b^3 + 5*a^3*B - 55*a*b^2*B - 73*a^2*b*C - 9*b^3*C)*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))) - (2*b*(14*a*A*b + 5*a^2*B - 5*b^2*B - 14*a*b*C)*(a + b*sec(c + d*x))*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))) - (2*b*(11*A*b + 5*a*B - 3*b*C)*(a + b*sec(c + d*x))^2*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))) + (2*(8*A*b + 5*a*B)*sqrt(cos(c + d*x))*(a + b*sec(c + d*x))^3*sin(c + d*x))/(15*d) + (2*A*cos(c + d*x)^(3/2)*(a + b*sec(c + d*x))^4*sin(c + d*x))/(5*d)],
[cos(c + d*x)^(3/2)*(a + b*sec(c + d*x))^4*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), 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*(35*a^3*A - 385*a*A*b^2 - 511*a^2*b*B - 63*b^3*B - 279*a^3*C - 227*a*b^2*C)*sin(c + d*x))/(105*d*sqrt(cos(c + d*x))) - (2*b*(35*a^2*A - 35*A*b^2 - 98*a*b*B - 87*a^2*C - 25*b^2*C)*(a + b*sec(c + d*x))*sin(c + d*x))/(105*d*sqrt(cos(c + d*x))) - (2*b*(35*a*A - 21*b*B - 39*a*C)*(a + b*sec(c + d*x))^2*sin(c + d*x))/(105*d*sqrt(cos(c + d*x))) - (2*b*(7*A - 3*C)*(a + b*sec(c + d*x))^3*sin(c + d*x))/(21*d*sqrt(cos(c + d*x))) + (2*A*sqrt(cos(c + d*x))*(a + b*sec(c + d*x))^4*sin(c + d*x))/(3*d)],
[cos(c + d*x)^(1/2)*(a + b*sec(c + d*x))^4*(A + B*sec(c + d*x) + C*sec(c + d*x)^2), 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*(1533*a^2*A*b^2 + 189*A*b^4 + 837*a^3*b*B + 681*a*b^3*B + 128*a^4*C + 883*a^2*b^2*C + 147*b^4*C)*sin(c + d*x))/(315*d*sqrt(cos(c + d*x))) + (2*(294*a*A*b^2 + 261*a^2*b*B + 75*b^3*B + 64*a^3*C + 202*a*b^2*C)*(a + b*sec(c + d*x))*sin(c + d*x))/(315*d*sqrt(cos(c + d*x))) + (2*(63*A*b^2 + 117*a*b*B + 48*a^2*C + 49*b^2*C)*(a + b*sec(c + d*x))^2*sin(c + d*x))/(315*d*sqrt(cos(c + d*x))) + (2*(9*b*B + 8*a*C)*(a + b*sec(c + d*x))^3*sin(c + d*x))/(63*d*sqrt(cos(c + d*x))) + (2*C*(a + b*sec(c + d*x))^4*sin(c + d*x))/(9*d*sqrt(cos(c + d*x)))],
[((a + b*sec(c + d*x))^4*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/cos(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)*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)*sin(c + d*x))/(3465*d*cos(c + d*x)^(3/2)) + (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)*sin(c + d*x))/(15*d*sqrt(cos(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*sec(c + d*x))*sin(c + d*x))/(3465*d*cos(c + d*x)^(3/2)) + (2*(33*A*b^2 + 55*a*b*B + 16*a^2*C + 27*b^2*C)*(a + b*sec(c + d*x))^2*sin(c + d*x))/(231*d*cos(c + d*x)^(3/2)) + (2*(11*b*B + 8*a*C)*(a + b*sec(c + d*x))^3*sin(c + d*x))/(99*d*cos(c + d*x)^(3/2)) + (2*C*(a + b*sec(c + d*x))^4*sin(c + d*x))/(11*d*cos(c + d*x)^(3/2))],


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


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


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


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


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


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


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


[(a + b*sec(c + d*x))^(3/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)*cos(c + d*x)^(9/2), x, 10, (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((b + a*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*a)/(a + b)))/(315*a^3*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(39*a^4*A*b - 31*a^2*A*b^3 - 8*A*b^5 + 75*a^5*B - 93*a^3*b^2*B + 18*a*b^4*B + 63*a^4*b*C - 63*a^2*b^3*C)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*a)/(a + b)))/(315*a^3*d*sqrt(cos(c + d*x))*sqrt(a + b*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))*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)*cos(c + d*x)^(3/2)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(315*a*d) + (2*(A*b + 3*a*B)*cos(c + d*x)^(5/2)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(21*d) + (2*A*cos(c + d*x)^(7/2)*(a + b*sec(c + d*x))^(3/2)*sin(c + d*x))/(9*d)],
[(a + b*sec(c + d*x))^(3/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)*cos(c + d*x)^(7/2), x, 9, (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((b + a*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*a)/(a + b)))/(105*a^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(25*a^4*A - 31*a^2*A*b^2 + 6*A*b^4 + 21*a^3*b*B - 21*a*b^3*B + 35*a^4*C - 35*a^2*b^2*C)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*a)/(a + b)))/(105*a^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(25*a^2*A + 3*A*b^2 + 42*a*b*B + 35*a^2*C)*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(105*a*d) + (2*(3*A*b + 7*a*B)*cos(c + d*x)^(3/2)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(35*d) + (2*A*cos(c + d*x)^(5/2)*(a + b*sec(c + d*x))^(3/2)*sin(c + d*x))/(7*d)],
[(a + b*sec(c + d*x))^(3/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)*cos(c + d*x)^(5/2), x, 11, (2*(a + b)*(9*a^2*A + 3*A*b^2 + 20*a*b*B + 15*a^2*C)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*a)/(a + b)))/(15*a*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(3*a^2*A*b - 3*A*b^3 + 5*a^3*B - 5*a*b^2*B + 15*a^2*b*C)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*a)/(a + b)))/(15*a*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*b^2*C*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*a)/(a + b)))/(d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(3*A*b + 5*a*B)*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(15*d) + (2*A*cos(c + d*x)^(3/2)*(a + b*sec(c + d*x))^(3/2)*sin(c + d*x))/(5*d)],
[(a + b*sec(c + d*x))^(3/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)*cos(c + d*x)^(3/2), x, 11, ((a + b)*(8*A*b + 6*a*B - 3*b*C)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*a)/(a + b)))/(3*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + ((2*a^2*A - 2*A*b^2 + 6*a*b*B + 6*a^2*C + 3*b^2*C)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*a)/(a + b)))/(3*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (b*(2*b*B + 3*a*C)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*a)/(a + b)))/(d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (b*(2*A - 3*C)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))) + (2*A*sqrt(cos(c + d*x))*(a + b*sec(c + d*x))^(3/2)*sin(c + d*x))/(3*d)],
[(a + b*sec(c + d*x))^(3/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)*cos(c + d*x)^(1/2), x, 11, ((a + b)*(8*a*A - 4*b*B - 5*a*C)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*a)/(a + b)))/(4*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + ((8*a*A*b + 8*a^2*B + 4*b^2*B + 7*a*b*C)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*a)/(a + b)))/(4*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + ((8*A*b^2 + 12*a*b*B + 3*a^2*C + 4*b^2*C)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticPi(2, (c + d*x)/2, (2*a)/(a + b)))/(4*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + ((4*b*B + 3*a*C)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(4*d*sqrt(cos(c + d*x))) + (C*(a + b*sec(c + d*x))^(3/2)*sin(c + d*x))/(2*d*sqrt(cos(c + d*x)))],
[(a + b*sec(c + d*x))^(3/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/cos(c + d*x)^(1/2), x, 12, -((a + b)*(24*A*b^2 + 30*a*b*B + 3*a^2*C + 16*b^2*C)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*a)/(a + b)))/(24*b*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + ((48*a^2*A + 24*A*b^2 + 42*a*b*B + 17*a^2*C + 16*b^2*C)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*a)/(a + b)))/(24*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(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((b + a*cos(c + d*x))/(a + b))*EllipticPi(2, (c + d*x)/2, (2*a)/(a + b)))/(8*b*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + ((2*b*B + a*C)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(4*d*cos(c + d*x)^(3/2)) + ((24*A*b^2 + 30*a*b*B + 3*a^2*C + 16*b^2*C)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(24*b*d*sqrt(cos(c + d*x))) + (C*(a + b*sec(c + d*x))^(3/2)*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2))],
[(a + b*sec(c + d*x))^(3/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/cos(c + d*x)^(3/2), x, 13, -((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((b + a*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*a)/(a + b)))/(192*b^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + ((336*a*A*b^2 + 136*a^2*b*B + 128*b^3*B - 3*a^3*C + 228*a*b^2*C)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*a)/(a + b)))/(192*b*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(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((b + a*cos(c + d*x))/(a + b))*EllipticPi(2, (c + d*x)/2, (2*a)/(a + b)))/(64*b^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + ((8*b*B + 3*a*C)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(24*d*cos(c + d*x)^(5/2)) + ((48*A*b^2 + 56*a*b*B + 3*a^2*C + 36*b^2*C)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(96*b*d*cos(c + d*x)^(3/2)) + ((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*sec(c + d*x))*sin(c + d*x))/(192*b^2*d*sqrt(cos(c + d*x))) + (C*(a + b*sec(c + d*x))^(3/2)*sin(c + d*x))/(4*d*cos(c + d*x)^(5/2))],


[(a + b*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)*cos(c + d*x)^(11/2), x, 11, (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((b + a*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*a)/(a + b)))/(3465*a^3*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(675*a^6*A - 390*a^4*A*b^2 - 245*a^2*A*b^4 - 40*A*b^6 + 1254*a^5*b*B - 1364*a^3*b^3*B + 110*a*b^5*B + 825*a^6*C - 330*a^4*b^2*C - 495*a^2*b^4*C)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*a)/(a + b)))/(3465*a^3*d*sqrt(cos(c + d*x))*sqrt(a + b*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))*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)*cos(c + d*x)^(3/2)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(3465*a*d) + (2*(27*a^2*A + 5*A*b^2 + 44*a*b*B + 33*a^2*C)*cos(c + d*x)^(5/2)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(231*d) + (2*(5*A*b + 11*a*B)*cos(c + d*x)^(7/2)*(a + b*sec(c + d*x))^(3/2)*sin(c + d*x))/(99*d) + (2*A*cos(c + d*x)^(9/2)*(a + b*sec(c + d*x))^(5/2)*sin(c + d*x))/(11*d)],
[(a + b*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)*cos(c + d*x)^(9/2), x, 10, (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((b + a*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*a)/(a + b)))/(315*a^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(114*a^4*A*b - 124*a^2*A*b^3 + 10*A*b^5 + 75*a^5*B - 30*a^3*b^2*B - 45*a*b^4*B + 168*a^4*b*C - 168*a^2*b^3*C)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*a)/(a + b)))/(315*a^2*d*sqrt(cos(c + d*x))*sqrt(a + b*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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(315*a*d) + (2*(49*a^2*A + 15*A*b^2 + 90*a*b*B + 63*a^2*C)*cos(c + d*x)^(3/2)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(315*d) + (2*(5*A*b + 9*a*B)*cos(c + d*x)^(5/2)*(a + b*sec(c + d*x))^(3/2)*sin(c + d*x))/(63*d) + (2*A*cos(c + d*x)^(7/2)*(a + b*sec(c + d*x))^(5/2)*sin(c + d*x))/(9*d)],
[(a + b*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)*cos(c + d*x)^(7/2), x, 12, (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((b + a*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*a)/(a + b)))/(105*a*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(25*a^4*A - 10*a^2*A*b^2 - 15*A*b^4 + 56*a^3*b*B - 56*a*b^3*B + 35*a^4*C + 70*a^2*b^2*C)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*a)/(a + b)))/(105*a*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*b^3*C*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*a)/(a + b)))/(d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(25*a^2*A + 15*A*b^2 + 56*a*b*B + 35*a^2*C)*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(105*d) + (2*(5*A*b + 7*a*B)*cos(c + d*x)^(3/2)*(a + b*sec(c + d*x))^(3/2)*sin(c + d*x))/(35*d) + (2*A*cos(c + d*x)^(5/2)*(a + b*sec(c + d*x))^(5/2)*sin(c + d*x))/(7*d)],
[(a + b*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)*cos(c + d*x)^(5/2), x, 12, ((a + b)*(18*a^2*A + 46*A*b^2 + 70*a*b*B + 30*a^2*C - 15*b^2*C)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*a)/(a + b)))/(15*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + ((16*a^2*A*b - 16*A*b^3 + 10*a^3*B + 20*a*b^2*B + 60*a^2*b*C + 15*b^3*C)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*a)/(a + b)))/(15*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (b^2*(2*b*B + 5*a*C)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*a)/(a + b)))/(d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (b*(16*A*b + 10*a*B - 15*b*C)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))) + (2*(A*b + a*B)*sqrt(cos(c + d*x))*(a + b*sec(c + d*x))^(3/2)*sin(c + d*x))/(3*d) + (2*A*cos(c + d*x)^(3/2)*(a + b*sec(c + d*x))^(5/2)*sin(c + d*x))/(5*d)],
[(a + b*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)*cos(c + d*x)^(3/2), x, 12, ((a + b)*(56*a*A*b + 24*a^2*B - 12*b^2*B - 27*a*b*C)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*a)/(a + b)))/(12*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + ((8*a^3*A + 16*a*A*b^2 + 48*a^2*b*B + 12*b^3*B + 24*a^3*C + 33*a*b^2*C)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*a)/(a + b)))/(12*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (b*(8*A*b^2 + 20*a*b*B + 15*a^2*C + 4*b^2*C)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticPi(2, (1/2)*(c + d*x), (2*a)/(a + b)))/(4*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (b*(8*a*A - 12*b*B - 21*a*C)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(12*d*sqrt(cos(c + d*x))) - (b*(4*A - 3*C)*(a + b*sec(c + d*x))^(3/2)*sin(c + d*x))/(6*d*sqrt(cos(c + d*x))) + (2*A*sqrt(cos(c + d*x))*(a + b*sec(c + d*x))^(5/2)*sin(c + d*x))/(3*d)],
[(a + b*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)*cos(c + d*x)^(1/2), x, 12, ((a + b)*(48*a^2*A - 24*A*b^2 - 54*a*b*B - 33*a^2*C - 16*b^2*C)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*a)/(a + b)))/(24*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + ((96*a^2*A*b + 24*A*b^3 + 48*a^3*B + 66*a*b^2*B + 59*a^2*b*C + 16*b^3*C)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*a)/(a + b)))/(24*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(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((b + a*cos(c + d*x))/(a + b))*EllipticPi(2, (c + d*x)/2, (2*a)/(a + b)))/(8*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + ((24*A*b^2 + 42*a*b*B + 15*a^2*C + 16*b^2*C)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(24*d*sqrt(cos(c + d*x))) + ((6*b*B + 5*a*C)*(a + b*sec(c + d*x))^(3/2)*sin(c + d*x))/(12*d*sqrt(cos(c + d*x))) + (C*(a + b*sec(c + d*x))^(5/2)*sin(c + d*x))/(3*d*sqrt(cos(c + d*x)))],
[(a + b*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/cos(c + d*x)^(1/2), x, 13, -((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((b + a*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*a)/(a + b)))/(192*b*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + ((384*a^3*A + 528*a*A*b^2 + 472*a^2*b*B + 128*b^3*B + 133*a^3*C + 356*a*b^2*C)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*a)/(a + b)))/(192*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(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((b + a*cos(c + d*x))/(a + b))*EllipticPi(2, (c + d*x)/2, (2*a)/(a + b)))/(64*b*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + ((16*A*b^2 + 24*a*b*B + 5*a^2*C + 12*b^2*C)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(32*d*cos(c + d*x)^(3/2)) + ((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*sec(c + d*x))*sin(c + d*x))/(192*b*d*sqrt(cos(c + d*x))) + ((8*b*B + 5*a*C)*(a + b*sec(c + d*x))^(3/2)*sin(c + d*x))/(24*d*cos(c + d*x)^(3/2)) + (C*(a + b*sec(c + d*x))^(5/2)*sin(c + d*x))/(4*d*cos(c + d*x)^(3/2))],
[(a + b*sec(c + d*x))^(5/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/cos(c + d*x)^(3/2), x, 14, -((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((b + a*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*a)/(a + b)))/(1920*b^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + ((4720*a^2*A*b^2 + 1280*A*b^4 + 1330*a^3*b*B + 3560*a*b^3*B - 15*a^4*C + 3236*a^2*b^2*C + 1024*b^4*C)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*a)/(a + b)))/(1920*b*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(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((b + a*cos(c + d*x))/(a + b))*EllipticPi(2, (c + d*x)/2, (2*a)/(a + b)))/(128*b^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + ((80*A*b^2 + 110*a*b*B + 15*a^2*C + 64*b^2*C)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(240*d*cos(c + d*x)^(5/2)) + ((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*sec(c + d*x))*sin(c + d*x))/(960*b*d*cos(c + d*x)^(3/2)) + ((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*sec(c + d*x))*sin(c + d*x))/(1920*b^2*d*sqrt(cos(c + d*x))) + ((2*b*B + a*C)*(a + b*sec(c + d*x))^(3/2)*sin(c + d*x))/(8*d*cos(c + d*x)^(5/2)) + (C*(a + b*sec(c + d*x))^(5/2)*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2))],


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


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

[(cos(c + d*x)^(1/2)*(a*A + (A*b + a*B)*sec(c + d*x) + b*B*sec(c + d*x)^2))/sqrt(a + b*sec(c + d*x)), x, 9, (2*A*(a + b)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*a)/(a + b)))/(d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*a*B*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*a)/(a + b)))/(d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*b*B*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticPi(2, (c + d*x)/2, (2*a)/(a + b)))/(d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))],


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


[(cos(c + d*x)^(5/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/(a + b*sec(c + d*x))^(5/2), x, 10, (2*(9*a^6*A + 55*a^4*A*b^2 - 212*a^2*A*b^4 + 128*A*b^6 - 40*a^5*b*B + 140*a^3*b^3*B - 80*a*b^5*B + 15*a^6*C - 75*a^4*b^2*C + 40*a^2*b^4*C)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*a)/(a + b)))/(15*a^5*(a - b)^2*(a + b)*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*(17*a^4*A*b + 116*a^2*A*b^3 - 128*A*b^5 - 5*a^5*B - 80*a^3*b^2*B + 80*a*b^4*B + 45*a^4*b*C - 40*a^2*b^3*C)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*a)/(a + b)))/(15*a^5*(a^2 - b^2)*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(A*b^2 - a*b*B + a^2*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(3*a*(a^2 - b^2)*d*(a + b*sec(c + d*x))^(3/2)) + (2*(12*a^2*A*b^2 - 8*A*b^4 - 9*a^3*b*B + 5*a*b^3*B + 6*a^4*C - 2*a^2*b^2*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(3*a^2*(a^2 - b^2)^2*d*sqrt(a + b*sec(c + d*x))) - (2*(14*a^4*A*b - 98*a^2*A*b^3 + 64*A*b^5 - 5*a^5*B + 65*a^3*b^2*B - 40*a*b^4*B - 40*a^4*b*C + 20*a^2*b^3*C)*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(15*a^4*(a^2 - b^2)^2*d) + (2*(3*a^4*A - 71*a^2*A*b^2 + 48*A*b^4 + 50*a^3*b*B - 30*a*b^3*B - 35*a^4*C + 15*a^2*b^2*C)*cos(c + d*x)^(3/2)*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(15*a^3*(a^2 - b^2)^2*d)],
[(cos(c + d*x)^(3/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/(a + b*sec(c + d*x))^(5/2), x, 9, (-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((b + a*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*a)/(a + b)))/(3*a^4*(a - b)^2*(a + b)*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(a^4*A + 16*a^2*A*b^2 - 16*A*b^4 - 9*a^3*b*B + 8*a*b^3*B + 3*a^4*C - 2*a^2*b^2*C)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*a)/(a + b)))/(3*a^4*(a^2 - b^2)*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(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*sec(c + d*x))^(3/2)) + (2*(10*a^2*A*b^2 - 6*A*b^4 - 7*a^3*b*B + 3*a*b^3*B + 4*a^4*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(3*a^2*(a^2 - b^2)^2*d*sqrt(a + b*sec(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(cos(c + d*x))*sqrt(a + b*sec(c + d*x))*sin(c + d*x))/(3*a^3*(a^2 - b^2)^2*d)],
[(cos(c + d*x)^(1/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)^2))/(a + b*sec(c + d*x))^(5/2), x, 8, (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((b + a*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*a)/(a + b)))/(3*a^3*(a - b)^2*(a + b)*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*(9*a^2*A*b - 8*A*b^3 - 3*a^3*B + 2*a*b^2*B + a^2*b*C)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*a)/(a + b)))/(3*a^3*(a^2 - b^2)*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(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*sec(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*sec(c + d*x)))],
[(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(cos(c + d*x)^(1/2)*(a + b*sec(c + d*x))^(5/2)), x, 8, (2*(6*a^2*A*b - 2*A*b^3 - 3*a^3*B - a*b^2*B + 4*a^2*b*C)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticE((1/2)*(c + d*x), (2*a)/(a + b)))/(3*a^2*(a - b)^2*(a + b)*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*(3*a^2*A - 2*A*b^2 - a*b*B + a^2*C)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((1/2)*(c + d*x), (2*a)/(a + b)))/(3*a^2*(a^2 - b^2)*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*(A*b^2 - a*b*B + a^2*C)*sin(c + d*x))/(3*b*(a^2 - b^2)*d*sqrt(cos(c + d*x))*(a + b*sec(c + d*x))^(3/2)) - (2*(5*a^2*A*b^2 - A*b^4 - 2*a^3*b*B - 2*a*b^3*B - a^4*C + 5*a^2*b^2*C)*sin(c + d*x))/(3*a*b*(a^2 - b^2)^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))],
[(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(cos(c + d*x)^(3/2)*(a + b*sec(c + d*x))^(5/2)), x, 11, (-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((b + a*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*a)/(a + b)))/(3*a*(a - b)^2*b^2*(a + b)*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*(A*b^2 - a*b*B + a^2*C)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*a)/(a + b)))/(3*a*b*(a^2 - b^2)*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + (2*C*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticPi(2, (c + d*x)/2, (2*a)/(a + b)))/(b^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*(A*b^2 - a*b*B + a^2*C)*sin(c + d*x))/(3*b*(a^2 - b^2)*d*cos(c + d*x)^(3/2)*(a + b*sec(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)*sin(c + d*x))/(3*b^2*(a^2 - b^2)^2*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x)))],
[(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(cos(c + d*x)^(5/2)*(a + b*sec(c + d*x))^(5/2)), x, 12, ((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((b + a*cos(c + d*x))/(a + b))*EllipticE((c + d*x)/2, (2*a)/(a + b)))/(3*(a - b)^2*b^3*(a + b)*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + ((2*A*b^2 - 2*a*b*B + 5*a^2*C - 3*b^2*C)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticF((c + d*x)/2, (2*a)/(a + b)))/(3*b^2*(a^2 - b^2)*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) + ((2*b*B - 5*a*C)*sqrt((b + a*cos(c + d*x))/(a + b))*EllipticPi(2, (c + d*x)/2, (2*a)/(a + b)))/(b^3*d*sqrt(cos(c + d*x))*sqrt(a + b*sec(c + d*x))) - (2*(A*b^2 - a*b*B + a^2*C)*sin(c + d*x))/(3*b*(a^2 - b^2)*d*cos(c + d*x)^(5/2)*(a + b*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*cos(c + d*x)^(3/2)*sqrt(a + b*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*sec(c + d*x))*sin(c + d*x))/(3*b^3*(a^2 - b^2)^2*d*sqrt(cos(c + d*x)))]
]:
