lst:=[
# ::Package:: 

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


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


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


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


[cos(a + b*x)^8, x, 4, (35*x)/128 + (35*cos(a + b*x)*sin(a + b*x))/(128*b) + (35*cos(a + b*x)^3*sin(a + b*x))/(192*b) + (7*cos(a + b*x)^5*sin(a + b*x))/(48*b) + (cos(a + b*x)^7*sin(a + b*x))/(8*b)],
[cos(a + b*x)^7, x, 2, sin(a + b*x)/b - sin(a + b*x)^3/b + (3*sin(a + b*x)^5)/(5*b) - sin(a + b*x)^7/(7*b)],
[cos(a + b*x)^6, x, 3, (5*x)/16 + (5*cos(a + b*x)*sin(a + b*x))/(16*b) + (5*cos(a + b*x)^3*sin(a + b*x))/(24*b) + (cos(a + b*x)^5*sin(a + b*x))/(6*b)],
[cos(a + b*x)^5, x, 2, sin(a + b*x)/b - (2*sin(a + b*x)^3)/(3*b) + sin(a + b*x)^5/(5*b)],
[cos(a + b*x)^4, x, 2, (3*x)/8 + (3*cos(a + b*x)*sin(a + b*x))/(8*b) + (cos(a + b*x)^3*sin(a + b*x))/(4*b)],
[cos(a + b*x)^3, x, 2, sin(a + b*x)/b - sin(a + b*x)^3/(3*b)],
[cos(a + b*x)^2, x, 1, x/2 + (cos(a + b*x)*sin(a + b*x))/(2*b)],
[cos(a + b*x)^1, x, 1, sin(a + b*x)/b],
[sec(a + b*x)^1, x, 1, arctanh(sin(a + b*x))/b],
[sec(a + b*x)^2, x, 1, tan(a + b*x)/b],
[sec(a + b*x)^3, x, 2, arctanh(sin(a + b*x))/(2*b) + (sec(a + b*x)*tan(a + b*x))/(2*b)],
[sec(a + b*x)^4, x, 2, tan(a + b*x)/b + tan(a + b*x)^3/(3*b)],
[sec(a + b*x)^5, x, 3, (3*arctanh(sin(a + b*x)))/(8*b) + (3*sec(a + b*x)*tan(a + b*x))/(8*b) + (sec(a + b*x)^3*tan(a + b*x))/(4*b)],
[sec(a + b*x)^6, x, 3, tan(a + b*x)/b + (2*tan(a + b*x)^3)/(3*b) + tan(a + b*x)^5/(5*b)],
[sec(a + b*x)^7, x, 4, (5*arctanh(sin(a + b*x)))/(16*b) + (5*sec(a + b*x)*tan(a + b*x))/(16*b) + (5*sec(a + b*x)^3*tan(a + b*x))/(24*b) + (sec(a + b*x)^5*tan(a + b*x))/(6*b)],
[sec(a + b*x)^8, x, 3, tan(a + b*x)/b + tan(a + b*x)^3/b + (3*tan(a + b*x)^5)/(5*b) + tan(a + b*x)^7/(7*b)],
[sec(a + b*x)^9, x, 5, (35*arctanh(sin(a + b*x)))/(128*b) + (35*sec(a + b*x)*tan(a + b*x))/(128*b) + (35*sec(a + b*x)^3*tan(a + b*x))/(192*b) + (7*sec(a + b*x)^5*tan(a + b*x))/(48*b) + (sec(a + b*x)^7*tan(a + b*x))/(8*b)],


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


[cos(c + d*x)^4*(a + a*cos(c + d*x))^2, x, 6, (11*a^2*x)/16 + (16*a^2*sin(c + d*x))/(15*d) + (11*a^2*cos(c + d*x)*sin(c + d*x))/(16*d) + (8*a^2*cos(c + d*x)^2*sin(c + d*x))/(15*d) + (11*a^2*cos(c + d*x)^3*sin(c + d*x))/(24*d) + (2*a^2*cos(c + d*x)^4*sin(c + d*x))/(5*d) + (a^2*cos(c + d*x)^5*sin(c + d*x))/(6*d)],
[cos(c + d*x)^3*(a + a*cos(c + d*x))^2, x, 5, (3*a^2*x)/4 + (6*a^2*sin(c + d*x))/(5*d) + (3*a^2*cos(c + d*x)*sin(c + d*x))/(4*d) + (3*a^2*cos(c + d*x)^2*sin(c + d*x))/(5*d) + (a^2*cos(c + d*x)^3*sin(c + d*x))/(2*d) + (a^2*cos(c + d*x)^4*sin(c + d*x))/(5*d)],
[cos(c + d*x)^2*(a + a*cos(c + d*x))^2, x, 4, (7*a^2*x)/8 + (4*a^2*sin(c + d*x))/(3*d) + (7*a^2*cos(c + d*x)*sin(c + d*x))/(8*d) + (2*a^2*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)^1*(a + a*cos(c + d*x))^2, x, 3, a^2*x + (5*a^2*sin(c + d*x))/(3*d) + (a^2*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)^0*(a + a*cos(c + d*x))^2, x, 2, (3*a^2*x)/2 + (2*a^2*sin(c + d*x))/d + (a^2*cos(c + d*x)*sin(c + d*x))/(2*d)],
[sec(c + d*x)^1*(a + a*cos(c + d*x))^2, x, 3, 2*a^2*x + (a^2*arctanh(sin(c + d*x)))/d + (a^2*sin(c + d*x))/d],
[sec(c + d*x)^2*(a + a*cos(c + d*x))^2, x, 3, a^2*x + (2*a^2*arctanh(sin(c + d*x)))/d + (a^2*tan(c + d*x))/d],
[sec(c + d*x)^3*(a + a*cos(c + d*x))^2, x, 4, (3*a^2*arctanh(sin(c + d*x)))/(2*d) + (2*a^2*tan(c + d*x))/d + (a^2*sec(c + d*x)*tan(c + d*x))/(2*d)],
[sec(c + d*x)^4*(a + a*cos(c + d*x))^2, x, 5, (a^2*arctanh(sin(c + d*x)))/d + (5*a^2*tan(c + d*x))/(3*d) + (a^2*sec(c + d*x)*tan(c + d*x))/d + (a^2*sec(c + d*x)^2*tan(c + d*x))/(3*d)],
[sec(c + d*x)^5*(a + a*cos(c + d*x))^2, x, 6, (7*a^2*arctanh(sin(c + d*x)))/(8*d) + (4*a^2*tan(c + d*x))/(3*d) + (7*a^2*sec(c + d*x)*tan(c + d*x))/(8*d) + (2*a^2*sec(c + d*x)^2*tan(c + d*x))/(3*d) + (a^2*sec(c + d*x)^3*tan(c + d*x))/(4*d)],


[cos(c + d*x)^3*(a + a*cos(c + d*x))^3, x, 6, (23*a^3*x)/16 + (34*a^3*sin(c + d*x))/(15*d) + (23*a^3*cos(c + d*x)*sin(c + d*x))/(16*d) + (17*a^3*cos(c + d*x)^2*sin(c + d*x))/(15*d) + (23*a^3*cos(c + d*x)^3*sin(c + d*x))/(24*d) + (13*a^3*cos(c + d*x)^4*sin(c + d*x))/(30*d) + (a^2*cos(c + d*x)^4*(a + a*cos(c + d*x))*sin(c + d*x))/(6*d)],
[cos(c + d*x)^2*(a + a*cos(c + d*x))^3, x, 5, (13*a^3*x)/8 + (13*a^3*sin(c + d*x))/(6*d) + (13*a^3*cos(c + d*x)*sin(c + d*x))/(24*d) + (13*a*(a + a*cos(c + d*x))^2*sin(c + d*x))/(60*d) - ((a + a*cos(c + d*x))^3*sin(c + d*x))/(20*d) + ((a + a*cos(c + d*x))^4*sin(c + d*x))/(5*a*d)],
[cos(c + d*x)^1*(a + a*cos(c + d*x))^3, x, 4, (15*a^3*x)/8 + (5*a^3*sin(c + d*x))/(2*d) + (5*a^3*cos(c + d*x)*sin(c + d*x))/(8*d) + (a*(a + a*cos(c + d*x))^2*sin(c + d*x))/(4*d) + ((a + a*cos(c + d*x))^3*sin(c + d*x))/(4*d)],
[cos(c + d*x)^0*(a + a*cos(c + d*x))^3, x, 3, (5*a^3*x)/2 + (10*a^3*sin(c + d*x))/(3*d) + (5*a^3*cos(c + d*x)*sin(c + d*x))/(6*d) + (a*(a + a*cos(c + d*x))^2*sin(c + d*x))/(3*d)],
[sec(c + d*x)^1*(a + a*cos(c + d*x))^3, x, 4, (7*a^3*x)/2 + (a^3*arctanh(sin(c + d*x)))/d + (5*a^3*sin(c + d*x))/(2*d) + (a^2*(a + a*cos(c + d*x))*sin(c + d*x))/(2*d)],
[sec(c + d*x)^2*(a + a*cos(c + d*x))^3, x, 4, 3*a^3*x + (3*a^3*arctanh(sin(c + d*x)))/d + (a^2*(a + a*cos(c + d*x))*tan(c + d*x))/d],
[sec(c + d*x)^3*(a + a*cos(c + d*x))^3, x, 4, a^3*x + (7*a^3*arctanh(sin(c + d*x)))/(2*d) + (5*a^3*tan(c + d*x))/(2*d) + (a^2*(a + a*sec(c + d*x))*tan(c + d*x))/(2*d)],
[sec(c + d*x)^4*(a + a*cos(c + d*x))^3, x, 5, (5*a^3*arctanh(sin(c + d*x)))/(2*d) + (10*a^3*tan(c + d*x))/(3*d) + (5*a^3*sec(c + d*x)*tan(c + d*x))/(6*d) + (a*(a + a*sec(c + d*x))^2*tan(c + d*x))/(3*d)],
[sec(c + d*x)^5*(a + a*cos(c + d*x))^3, x, 6, (15*a^3*arctanh(sin(c + d*x)))/(8*d) + (5*a^3*tan(c + d*x))/(2*d) + (5*a^3*sec(c + d*x)*tan(c + d*x))/(8*d) + (a*(a + a*sec(c + d*x))^2*tan(c + d*x))/(4*d) + ((a + a*sec(c + d*x))^3*tan(c + d*x))/(4*d)],
[sec(c + d*x)^6*(a + a*cos(c + d*x))^3, x, 7, (13*a^3*arctanh(sin(c + d*x)))/(8*d) + (38*a^3*tan(c + d*x))/(15*d) + (13*a^3*sec(c + d*x)*tan(c + d*x))/(8*d) + (19*a^3*sec(c + d*x)^2*tan(c + d*x))/(15*d) + (11*a^3*sec(c + d*x)^3*tan(c + d*x))/(20*d) + (a^2*(a + a*cos(c + d*x))*sec(c + d*x)^4*tan(c + d*x))/(5*d)],


[cos(c + d*x)^2*(a + a*cos(c + d*x))^4, x, 6, (49*a^4*x)/16 + (49*a^4*sin(c + d*x))/(12*d) + (49*a^4*cos(c + d*x)*sin(c + d*x))/(48*d) + (49*a^2*(a + a*cos(c + d*x))^2*sin(c + d*x))/(120*d) + (7*a*(a + a*cos(c + d*x))^3*sin(c + d*x))/(40*d) - ((a + a*cos(c + d*x))^4*sin(c + d*x))/(30*d) + ((a + a*cos(c + d*x))^5*sin(c + d*x))/(6*a*d)],
[cos(c + d*x)^1*(a + a*cos(c + d*x))^4, x, 5, (7*a^4*x)/2 + (14*a^4*sin(c + d*x))/(3*d) + (7*a^4*cos(c + d*x)*sin(c + d*x))/(6*d) + (7*a^2*(a + a*cos(c + d*x))^2*sin(c + d*x))/(15*d) + (a*(a + a*cos(c + d*x))^3*sin(c + d*x))/(5*d) + ((a + a*cos(c + d*x))^4*sin(c + d*x))/(5*d)],
[cos(c + d*x)^0*(a + a*cos(c + d*x))^4, x, 4, (35*a^4*x)/8 + (35*a^4*sin(c + d*x))/(6*d) + (35*a^4*cos(c + d*x)*sin(c + d*x))/(24*d) + (7*a^2*(a + a*cos(c + d*x))^2*sin(c + d*x))/(12*d) + (a*(a + a*cos(c + d*x))^3*sin(c + d*x))/(4*d)],
[sec(c + d*x)^1*(a + a*cos(c + d*x))^4, x, 5, 6*a^4*x + (a^4*arctanh(sin(c + d*x)))/d + (5*a^4*sin(c + d*x))/d + (4*a^3*(a + a*cos(c + d*x))*sin(c + d*x))/(3*d) + (a^2*(a + a*cos(c + d*x))^2*sin(c + d*x))/(3*d)],
[sec(c + d*x)^2*(a + a*cos(c + d*x))^4, x, 5, (13*a^4*x)/2 + (4*a^4*arctanh(sin(c + d*x)))/d + (5*a^4*sin(c + d*x))/(2*d) - (a^3*(a + a*cos(c + d*x))*sin(c + d*x))/(2*d) + (a^2*(a + a*cos(c + d*x))^2*tan(c + d*x))/d],
[sec(c + d*x)^3*(a + a*cos(c + d*x))^4, x, 5, 4*a^4*x + (13*a^4*arctanh(sin(c + d*x)))/(2*d) - (5*a^4*sin(c + d*x))/(2*d) + (3*a^3*(a + a*cos(c + d*x))*tan(c + d*x))/d + (a^2*(a + a*cos(c + d*x))^2*sec(c + d*x)*tan(c + d*x))/(2*d)],
[sec(c + d*x)^4*(a + a*cos(c + d*x))^4, x, 5, a^4*x + (6*a^4*arctanh(sin(c + d*x)))/d + (5*a^4*tan(c + d*x))/d + (4*a^3*(a + a*sec(c + d*x))*tan(c + d*x))/(3*d) + (a^2*(a + a*sec(c + d*x))^2*tan(c + d*x))/(3*d)],
[sec(c + d*x)^5*(a + a*cos(c + d*x))^4, x, 6, (35*a^4*arctanh(sin(c + d*x)))/(8*d) + (35*a^4*tan(c + d*x))/(6*d) + (35*a^4*sec(c + d*x)*tan(c + d*x))/(24*d) + (7*a^2*(a + a*sec(c + d*x))^2*tan(c + d*x))/(12*d) + (a*(a + a*sec(c + d*x))^3*tan(c + d*x))/(4*d)],
[sec(c + d*x)^6*(a + a*cos(c + d*x))^4, x, 7, (7*a^4*arctanh(sin(c + d*x)))/(2*d) + (14*a^4*tan(c + d*x))/(3*d) + (7*a^4*sec(c + d*x)*tan(c + d*x))/(6*d) + (7*a^2*(a + a*sec(c + d*x))^2*tan(c + d*x))/(15*d) + (a*(a + a*sec(c + d*x))^3*tan(c + d*x))/(5*d) + ((a + a*sec(c + d*x))^4*tan(c + d*x))/(5*d)],
[sec(c + d*x)^7*(a + a*cos(c + d*x))^4, x, 8, (49*a^4*arctanh(sin(c + d*x)))/(16*d) + (24*a^4*tan(c + d*x))/(5*d) + (49*a^4*sec(c + d*x)*tan(c + d*x))/(16*d) + (12*a^4*sec(c + d*x)^2*tan(c + d*x))/(5*d) + (43*a^4*sec(c + d*x)^3*tan(c + d*x))/(40*d) + (7*a^3*(a + a*cos(c + d*x))*sec(c + d*x)^4*tan(c + d*x))/(15*d) + (a^2*(a + a*cos(c + d*x))^2*sec(c + d*x)^5*tan(c + d*x))/(6*d)],


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


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


[cos(c + d*x)^5/(a + a*cos(c + d*x))^2, x, 5, -((5*x)/a^2) + (8*sin(c + d*x))/(a^2*d) - (5*cos(c + d*x)*sin(c + d*x))/(a^2*d) + (4*cos(c + d*x)^2*sin(c + d*x))/(a^2*d) - (cos(c + d*x)^4*sin(c + d*x))/(3*d*(a + a*cos(c + d*x))^2) - (10*cos(c + d*x)^3*sin(c + d*x))/(3*a*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^4/(a + a*cos(c + d*x))^2, x, 4, (7*x)/(2*a^2) - (16*sin(c + d*x))/(3*a^2*d) + (7*cos(c + d*x)*sin(c + d*x))/(2*a^2*d) - (cos(c + d*x)^3*sin(c + d*x))/(3*d*(a + a*cos(c + d*x))^2) - (8*cos(c + d*x)^2*sin(c + d*x))/(3*a*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^3/(a + a*cos(c + d*x))^2, x, 4, -((2*x)/a^2) + (4*sin(c + d*x))/(3*a^2*d) - (cos(c + d*x)^2*sin(c + d*x))/(3*d*(a + a*cos(c + d*x))^2) + (2*sin(c + d*x))/(a*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^2/(a + a*cos(c + d*x))^2, x, 3, x/a^2 + sin(c + d*x)/(3*d*(a + a*cos(c + d*x))^2) - (5*sin(c + d*x))/(3*a*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^1/(a + a*cos(c + d*x))^2, x, 2, -(sin(c + d*x)/(3*d*(a + a*cos(c + d*x))^2)) + (2*sin(c + d*x))/(3*a*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^0/(a + a*cos(c + d*x))^2, x, 2, sin(c + d*x)/(3*d*(a + a*cos(c + d*x))^2) + sin(c + d*x)/(3*a*d*(a + a*cos(c + d*x)))],
[sec(c + d*x)^1/(a + a*cos(c + d*x))^2, x, 3, arctanh(sin(c + d*x))/(a^2*d) - sin(c + d*x)/(3*d*(a + a*cos(c + d*x))^2) - (4*sin(c + d*x))/(3*a*d*(a + a*cos(c + d*x)))],
[sec(c + d*x)^2/(a + a*cos(c + d*x))^2, x, 5, -((2*arctanh(sin(c + d*x)))/(a^2*d)) + (10*tan(c + d*x))/(3*a^2*d) - tan(c + d*x)/(3*d*(a + a*cos(c + d*x))^2) - (2*tan(c + d*x))/(a*d*(a + a*cos(c + d*x)))],
[sec(c + d*x)^3/(a + a*cos(c + d*x))^2, x, 6, (7*arctanh(sin(c + d*x)))/(2*a^2*d) - (16*tan(c + d*x))/(3*a^2*d) + (7*sec(c + d*x)*tan(c + d*x))/(2*a^2*d) - (sec(c + d*x)*tan(c + d*x))/(3*d*(a + a*cos(c + d*x))^2) - (8*sec(c + d*x)*tan(c + d*x))/(3*a*d*(a + a*cos(c + d*x)))],
[sec(c + d*x)^4/(a + a*cos(c + d*x))^2, x, 7, -((5*arctanh(sin(c + d*x)))/(a^2*d)) + (8*tan(c + d*x))/(a^2*d) - (5*sec(c + d*x)*tan(c + d*x))/(a^2*d) + (4*sec(c + d*x)^2*tan(c + d*x))/(a^2*d) - (sec(c + d*x)^2*tan(c + d*x))/(3*d*(a + a*cos(c + d*x))^2) - (10*sec(c + d*x)^2*tan(c + d*x))/(3*a*d*(a + a*cos(c + d*x)))],


[cos(c + d*x)^5/(a + a*cos(c + d*x))^3, x, 5, (13*x)/(2*a^3) - (152*sin(c + d*x))/(15*a^3*d) + (13*cos(c + d*x)*sin(c + d*x))/(2*a^3*d) - (cos(c + d*x)^4*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3) - (11*cos(c + d*x)^3*sin(c + d*x))/(15*a*d*(a + a*cos(c + d*x))^2) - (76*cos(c + d*x)^2*sin(c + d*x))/(15*a^2*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^4/(a + a*cos(c + d*x))^3, x, 5, -((3*x)/a^3) + (9*sin(c + d*x))/(5*a^3*d) - (cos(c + d*x)^3*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3) - (3*cos(c + d*x)^2*sin(c + d*x))/(5*a*d*(a + a*cos(c + d*x))^2) + (3*sin(c + d*x))/(a^2*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^3/(a + a*cos(c + d*x))^3, x, 4, x/a^3 - (cos(c + d*x)^2*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3) - (7*cos(c + d*x)*sin(c + d*x))/(15*a*d*(a + a*cos(c + d*x))^2) - (22*sin(c + d*x))/(15*a^2*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^2/(a + a*cos(c + d*x))^3, x, 3, sin(c + d*x)/(5*d*(a + a*cos(c + d*x))^3) - (8*sin(c + d*x))/(15*a*d*(a + a*cos(c + d*x))^2) + (7*sin(c + d*x))/(15*a^2*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^1/(a + a*cos(c + d*x))^3, x, 3, -(sin(c + d*x)/(5*d*(a + a*cos(c + d*x))^3)) + sin(c + d*x)/(5*a*d*(a + a*cos(c + d*x))^2) + sin(c + d*x)/(5*a^2*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^0/(a + a*cos(c + d*x))^3, x, 3, sin(c + d*x)/(5*d*(a + a*cos(c + d*x))^3) + (2*sin(c + d*x))/(15*a*d*(a + a*cos(c + d*x))^2) + (2*sin(c + d*x))/(15*a^2*d*(a + a*cos(c + d*x)))],
[sec(c + d*x)^1/(a + a*cos(c + d*x))^3, x, 4, arctanh(sin(c + d*x))/(a^3*d) - sin(c + d*x)/(5*d*(a + a*cos(c + d*x))^3) - (7*sin(c + d*x))/(15*a*d*(a + a*cos(c + d*x))^2) - (22*sin(c + d*x))/(15*a^2*d*(a + a*cos(c + d*x)))],
[sec(c + d*x)^2/(a + a*cos(c + d*x))^3, x, 6, -((3*arctanh(sin(c + d*x)))/(a^3*d)) + (24*tan(c + d*x))/(5*a^3*d) - tan(c + d*x)/(5*d*(a + a*cos(c + d*x))^3) - (3*tan(c + d*x))/(5*a*d*(a + a*cos(c + d*x))^2) - (3*tan(c + d*x))/(a^2*d*(a + a*cos(c + d*x)))],
[sec(c + d*x)^3/(a + a*cos(c + d*x))^3, x, 7, (13*arctanh(sin(c + d*x)))/(2*a^3*d) - (152*tan(c + d*x))/(15*a^3*d) + (13*sec(c + d*x)*tan(c + d*x))/(2*a^3*d) - (sec(c + d*x)*tan(c + d*x))/(5*d*(a + a*cos(c + d*x))^3) - (11*sec(c + d*x)*tan(c + d*x))/(15*a*d*(a + a*cos(c + d*x))^2) - (76*sec(c + d*x)*tan(c + d*x))/(15*a^2*d*(a + a*cos(c + d*x)))],


[cos(c + d*x)^6/(a + a*cos(c + d*x))^4, x, 6, (21*x)/(2*a^4) - (576*sin(c + d*x))/(35*a^4*d) + (21*cos(c + d*x)*sin(c + d*x))/(2*a^4*d) - (cos(c + d*x)^5*sin(c + d*x))/(7*d*(a + a*cos(c + d*x))^4) - (2*cos(c + d*x)^4*sin(c + d*x))/(5*a*d*(a + a*cos(c + d*x))^3) - (43*cos(c + d*x)^3*sin(c + d*x))/(35*a^2*d*(a + a*cos(c + d*x))^2) - (288*cos(c + d*x)^2*sin(c + d*x))/(35*a^3*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^5/(a + a*cos(c + d*x))^4, x, 6, -((4*x)/a^4) + (244*sin(c + d*x))/(105*a^4*d) - (cos(c + d*x)^4*sin(c + d*x))/(7*d*(a + a*cos(c + d*x))^4) - (12*cos(c + d*x)^3*sin(c + d*x))/(35*a*d*(a + a*cos(c + d*x))^3) - (88*cos(c + d*x)^2*sin(c + d*x))/(105*a^2*d*(a + a*cos(c + d*x))^2) + (4*sin(c + d*x))/(a^3*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^4/(a + a*cos(c + d*x))^4, x, 5, x/a^4 - (cos(c + d*x)^3*sin(c + d*x))/(7*d*(a + a*cos(c + d*x))^4) - (2*cos(c + d*x)^2*sin(c + d*x))/(7*a*d*(a + a*cos(c + d*x))^3) - (11*cos(c + d*x)*sin(c + d*x))/(21*a^2*d*(a + a*cos(c + d*x))^2) - (32*sin(c + d*x))/(21*a^3*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^3/(a + a*cos(c + d*x))^4, x, 4, (cos(c + d*x)^3*sin(c + d*x))/(7*d*(a + a*cos(c + d*x))^4) + (3*sin(c + d*x))/(35*a*d*(a + a*cos(c + d*x))^3) - (8*sin(c + d*x))/(35*a^2*d*(a + a*cos(c + d*x))^2) + sin(c + d*x)/(5*a^3*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^2/(a + a*cos(c + d*x))^4, x, 4, sin(c + d*x)/(7*d*(a + a*cos(c + d*x))^4) - (11*sin(c + d*x))/(35*a*d*(a + a*cos(c + d*x))^3) + (13*sin(c + d*x))/(105*a^2*d*(a + a*cos(c + d*x))^2) + (13*sin(c + d*x))/(105*a^3*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^1/(a + a*cos(c + d*x))^4, x, 4, -(sin(c + d*x)/(7*d*(a + a*cos(c + d*x))^4)) + (4*sin(c + d*x))/(35*a*d*(a + a*cos(c + d*x))^3) + (8*sin(c + d*x))/(105*a^2*d*(a + a*cos(c + d*x))^2) + (8*sin(c + d*x))/(105*a^3*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^0/(a + a*cos(c + d*x))^4, x, 4, sin(c + d*x)/(7*d*(a + a*cos(c + d*x))^4) + (3*sin(c + d*x))/(35*a*d*(a + a*cos(c + d*x))^3) + (2*sin(c + d*x))/(35*a^2*d*(a + a*cos(c + d*x))^2) + (2*sin(c + d*x))/(35*a^3*d*(a + a*cos(c + d*x)))],
[sec(c + d*x)^1/(a + a*cos(c + d*x))^4, x, 5, arctanh(sin(c + d*x))/(a^4*d) - sin(c + d*x)/(7*d*(a + a*cos(c + d*x))^4) - (2*sin(c + d*x))/(7*a*d*(a + a*cos(c + d*x))^3) - (11*sin(c + d*x))/(21*a^2*d*(a + a*cos(c + d*x))^2) - (32*sin(c + d*x))/(21*a^3*d*(a + a*cos(c + d*x)))],
[sec(c + d*x)^2/(a + a*cos(c + d*x))^4, x, 7, -((4*arctanh(sin(c + d*x)))/(a^4*d)) + (664*tan(c + d*x))/(105*a^4*d) - tan(c + d*x)/(7*d*(a + a*cos(c + d*x))^4) - (12*tan(c + d*x))/(35*a*d*(a + a*cos(c + d*x))^3) - (88*tan(c + d*x))/(105*a^2*d*(a + a*cos(c + d*x))^2) - (4*tan(c + d*x))/(a^3*d*(a + a*cos(c + d*x)))],
[sec(c + d*x)^3/(a + a*cos(c + d*x))^4, x, 8, (21*arctanh(sin(c + d*x)))/(2*a^4*d) - (576*tan(c + d*x))/(35*a^4*d) + (21*sec(c + d*x)*tan(c + d*x))/(2*a^4*d) - (sec(c + d*x)*tan(c + d*x))/(7*d*(a + a*cos(c + d*x))^4) - (2*sec(c + d*x)*tan(c + d*x))/(5*a*d*(a + a*cos(c + d*x))^3) - (43*sec(c + d*x)*tan(c + d*x))/(35*a^2*d*(a + a*cos(c + d*x))^2) - (288*sec(c + d*x)*tan(c + d*x))/(35*a^3*d*(a + a*cos(c + d*x)))],


[cos(c + d*x)^7/(a + a*cos(c + d*x))^5, x, 7, (31*x)/(2*a^5) - (7664*sin(c + d*x))/(315*a^5*d) + (31*cos(c + d*x)*sin(c + d*x))/(2*a^5*d) - (cos(c + d*x)^6*sin(c + d*x))/(9*d*(a + a*cos(c + d*x))^5) - (17*cos(c + d*x)^5*sin(c + d*x))/(63*a*d*(a + a*cos(c + d*x))^4) - (28*cos(c + d*x)^4*sin(c + d*x))/(45*a^2*d*(a + a*cos(c + d*x))^3) - (577*cos(c + d*x)^3*sin(c + d*x))/(315*a^3*d*(a + a*cos(c + d*x))^2) - (3832*cos(c + d*x)^2*sin(c + d*x))/(315*a^4*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^6/(a + a*cos(c + d*x))^5, x, 7, -((5*x)/a^5) + (181*sin(c + d*x))/(63*a^5*d) - (cos(c + d*x)^5*sin(c + d*x))/(9*d*(a + a*cos(c + d*x))^5) - (5*cos(c + d*x)^4*sin(c + d*x))/(21*a*d*(a + a*cos(c + d*x))^4) - (29*cos(c + d*x)^3*sin(c + d*x))/(63*a^2*d*(a + a*cos(c + d*x))^3) - (67*cos(c + d*x)^2*sin(c + d*x))/(63*a^3*d*(a + a*cos(c + d*x))^2) + (5*sin(c + d*x))/(a^4*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^5/(a + a*cos(c + d*x))^5, x, 6, x/a^5 - (cos(c + d*x)^4*sin(c + d*x))/(9*d*(a + a*cos(c + d*x))^5) - (13*cos(c + d*x)^3*sin(c + d*x))/(63*a*d*(a + a*cos(c + d*x))^4) - (34*cos(c + d*x)^2*sin(c + d*x))/(105*a^2*d*(a + a*cos(c + d*x))^3) - (173*cos(c + d*x)*sin(c + d*x))/(315*a^3*d*(a + a*cos(c + d*x))^2) - (488*sin(c + d*x))/(315*a^4*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^4/(a + a*cos(c + d*x))^5, x, 5, (cos(c + d*x)^4*sin(c + d*x))/(9*d*(a + a*cos(c + d*x))^5) + (4*cos(c + d*x)^3*sin(c + d*x))/(63*a*d*(a + a*cos(c + d*x))^4) + (4*sin(c + d*x))/(105*a^2*d*(a + a*cos(c + d*x))^3) - (32*sin(c + d*x))/(315*a^3*d*(a + a*cos(c + d*x))^2) + (4*sin(c + d*x))/(45*a^4*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^3/(a + a*cos(c + d*x))^5, x, 5, -((cos(c + d*x)^4*sin(c + d*x))/(9*d*(a + a*cos(c + d*x))^5)) + (5*cos(c + d*x)^3*sin(c + d*x))/(63*a*d*(a + a*cos(c + d*x))^4) + sin(c + d*x)/(21*a^2*d*(a + a*cos(c + d*x))^3) - (8*sin(c + d*x))/(63*a^3*d*(a + a*cos(c + d*x))^2) + sin(c + d*x)/(9*a^4*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^2/(a + a*cos(c + d*x))^5, x, 5, sin(c + d*x)/(9*d*(a + a*cos(c + d*x))^5) - (2*sin(c + d*x))/(9*a*d*(a + a*cos(c + d*x))^4) + sin(c + d*x)/(15*a^2*d*(a + a*cos(c + d*x))^3) + (2*sin(c + d*x))/(45*a^3*d*(a + a*cos(c + d*x))^2) + (2*sin(c + d*x))/(45*a^4*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^1/(a + a*cos(c + d*x))^5, x, 5, -(sin(c + d*x)/(9*d*(a + a*cos(c + d*x))^5)) + (5*sin(c + d*x))/(63*a*d*(a + a*cos(c + d*x))^4) + sin(c + d*x)/(21*a^2*d*(a + a*cos(c + d*x))^3) + (2*sin(c + d*x))/(63*a^3*d*(a + a*cos(c + d*x))^2) + (2*sin(c + d*x))/(63*a^4*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^0/(a + a*cos(c + d*x))^5, x, 5, sin(c + d*x)/(9*d*(a + a*cos(c + d*x))^5) + (4*sin(c + d*x))/(63*a*d*(a + a*cos(c + d*x))^4) + (4*sin(c + d*x))/(105*a^2*d*(a + a*cos(c + d*x))^3) + (8*sin(c + d*x))/(315*a^3*d*(a + a*cos(c + d*x))^2) + (8*sin(c + d*x))/(315*a^4*d*(a + a*cos(c + d*x)))],
[sec(c + d*x)^1/(a + a*cos(c + d*x))^5, x, 6, arctanh(sin(c + d*x))/(a^5*d) - sin(c + d*x)/(9*d*(a + a*cos(c + d*x))^5) - (13*sin(c + d*x))/(63*a*d*(a + a*cos(c + d*x))^4) - (34*sin(c + d*x))/(105*a^2*d*(a + a*cos(c + d*x))^3) - (173*sin(c + d*x))/(315*a^3*d*(a + a*cos(c + d*x))^2) - (488*sin(c + d*x))/(315*a^4*d*(a + a*cos(c + d*x)))],
[sec(c + d*x)^2/(a + a*cos(c + d*x))^5, x, 8, -((5*arctanh(sin(c + d*x)))/(a^5*d)) + (496*tan(c + d*x))/(63*a^5*d) - tan(c + d*x)/(9*d*(a + a*cos(c + d*x))^5) - (5*tan(c + d*x))/(21*a*d*(a + a*cos(c + d*x))^4) - (29*tan(c + d*x))/(63*a^2*d*(a + a*cos(c + d*x))^3) - (67*tan(c + d*x))/(63*a^3*d*(a + a*cos(c + d*x))^2) - (5*tan(c + d*x))/(a^4*d*(a + a*cos(c + d*x)))],
[sec(c + d*x)^3/(a + a*cos(c + d*x))^5, x, 9, (31*arctanh(sin(c + d*x)))/(2*a^5*d) - (7664*tan(c + d*x))/(315*a^5*d) + (31*sec(c + d*x)*tan(c + d*x))/(2*a^5*d) - (sec(c + d*x)*tan(c + d*x))/(9*d*(a + a*cos(c + d*x))^5) - (17*sec(c + d*x)*tan(c + d*x))/(63*a*d*(a + a*cos(c + d*x))^4) - (28*sec(c + d*x)*tan(c + d*x))/(45*a^2*d*(a + a*cos(c + d*x))^3) - (577*sec(c + d*x)*tan(c + d*x))/(315*a^3*d*(a + a*cos(c + d*x))^2) - (3832*sec(c + d*x)*tan(c + d*x))/(315*a^4*d*(a + a*cos(c + d*x)))],


[cos(c + d*x)^5/(a + a*cos(c + d*x))^6, x, 6, (cos(c + d*x)^5*sin(c + d*x))/(11*d*(a + a*cos(c + d*x))^6) + (5*cos(c + d*x)^4*sin(c + d*x))/(99*a*d*(a + a*cos(c + d*x))^5) + (20*cos(c + d*x)^3*sin(c + d*x))/(693*a^2*d*(a + a*cos(c + d*x))^4) + (4*sin(c + d*x))/(231*a^3*d*(a + a*cos(c + d*x))^3) - (32*sin(c + d*x))/(693*a^4*d*(a + a*cos(c + d*x))^2) + (4*sin(c + d*x))/(99*a^5*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^4/(a + a*cos(c + d*x))^6, x, 6, -((cos(c + d*x)^5*sin(c + d*x))/(11*d*(a + a*cos(c + d*x))^6)) + (2*cos(c + d*x)^4*sin(c + d*x))/(33*a*d*(a + a*cos(c + d*x))^5) + (8*cos(c + d*x)^3*sin(c + d*x))/(231*a^2*d*(a + a*cos(c + d*x))^4) + (8*sin(c + d*x))/(385*a^3*d*(a + a*cos(c + d*x))^3) - (64*sin(c + d*x))/(1155*a^4*d*(a + a*cos(c + d*x))^2) + (8*sin(c + d*x))/(165*a^5*d*(a + a*cos(c + d*x)))],


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


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


[cos(c + d*x)^4*(a + a*cos(c + d*x))^(1/2), x, 5, (32*a*sin(c + d*x))/(45*d*sqrt(a + a*cos(c + d*x))) + (16*a*cos(c + d*x)^3*sin(c + d*x))/(63*d*sqrt(a + a*cos(c + d*x))) + (2*a*cos(c + d*x)^4*sin(c + d*x))/(9*d*sqrt(a + a*cos(c + d*x))) - (64*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(315*d) + (32*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(105*a*d)],
[cos(c + d*x)^3*(a + a*cos(c + d*x))^(1/2), x, 4, (4*a*sin(c + d*x))/(5*d*sqrt(a + a*cos(c + d*x))) + (2*a*cos(c + d*x)^3*sin(c + d*x))/(7*d*sqrt(a + a*cos(c + d*x))) - (8*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(35*d) + (12*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(35*a*d)],
[cos(c + d*x)^2*(a + a*cos(c + d*x))^(1/2), x, 3, (14*a*sin(c + d*x))/(15*d*sqrt(a + a*cos(c + d*x))) - (4*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(15*d) + (2*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(5*a*d)],
[cos(c + d*x)^1*(a + a*cos(c + d*x))^(1/2), x, 2, (2*a*sin(c + d*x))/(3*d*sqrt(a + a*cos(c + d*x))) + (2*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(3*d)],
[cos(c + d*x)^0*(a + a*cos(c + d*x))^(1/2), x, 1, (2*a*sin(c + d*x))/(d*sqrt(a + a*cos(c + d*x)))],
[sec(c + d*x)^1*(a + a*cos(c + d*x))^(1/2), x, 1, (2*sqrt(2)*a*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(a + a*cos(c + d*x)))],
[sec(c + d*x)^2*(a + a*cos(c + d*x))^(1/2), x, 2, (sqrt(2)*a*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(a + a*cos(c + d*x))) + (a*tan(c + d*x))/(d*sqrt(a + a*cos(c + d*x)))],
[sec(c + d*x)^3*(a + a*cos(c + d*x))^(1/2), x, 3, (3*a*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(2*sqrt(2)*d*sqrt(a + a*cos(c + d*x))) + (3*a*tan(c + d*x))/(4*d*sqrt(a + a*cos(c + d*x))) + (a*sec(c + d*x)*tan(c + d*x))/(2*d*sqrt(a + a*cos(c + d*x)))],
[sec(c + d*x)^4*(a + a*cos(c + d*x))^(1/2), x, 4, (5*a*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(4*sqrt(2)*d*sqrt(a + a*cos(c + d*x))) + (5*a*tan(c + d*x))/(8*d*sqrt(a + a*cos(c + d*x))) + (5*a*sec(c + d*x)*tan(c + d*x))/(12*d*sqrt(a + a*cos(c + d*x))) + (a*sec(c + d*x)^2*tan(c + d*x))/(3*d*sqrt(a + a*cos(c + d*x)))],


[cos(c + d*x)^3*(a + a*cos(c + d*x))^(3/2), x, 6, (68*a^2*sin(c + d*x))/(45*d*sqrt(a + a*cos(c + d*x))) + (34*a^2*cos(c + d*x)^3*sin(c + d*x))/(63*d*sqrt(a + a*cos(c + d*x))) + (2*a^2*cos(c + d*x)^4*sin(c + d*x))/(9*d*sqrt(a + a*cos(c + d*x))) - (136*a*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(315*d) + (68*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(105*d)],
[cos(c + d*x)^2*(a + a*cos(c + d*x))^(3/2), x, 4, (152*a^2*sin(c + d*x))/(105*d*sqrt(a + a*cos(c + d*x))) + (38*a*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(105*d) - (4*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(35*d) + (2*(a + a*cos(c + d*x))^(5/2)*sin(c + d*x))/(7*a*d)],
[cos(c + d*x)^1*(a + a*cos(c + d*x))^(3/2), x, 3, (8*a^2*sin(c + d*x))/(5*d*sqrt(a + a*cos(c + d*x))) + (2*a*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(5*d) + (2*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(5*d)],
[cos(c + d*x)^0*(a + a*cos(c + d*x))^(3/2), x, 2, (8*a^2*sin(c + d*x))/(3*d*sqrt(a + a*cos(c + d*x))) + (2*a*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(3*d)],
[sec(c + d*x)^1*(a + a*cos(c + d*x))^(3/2), x, 3, (2*sqrt(2)*a^2*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(a + a*cos(c + d*x))) + (2*a^2*sin(c + d*x))/(d*sqrt(a + a*cos(c + d*x)))],
[sec(c + d*x)^2*(a + a*cos(c + d*x))^(3/2), x, 3, (3*sqrt(2)*a^2*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(a + a*cos(c + d*x))) + (a^2*tan(c + d*x))/(d*sqrt(a + a*cos(c + d*x)))],
[sec(c + d*x)^3*(a + a*cos(c + d*x))^(3/2), x, 4, (7*a^2*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(2*sqrt(2)*d*sqrt(a + a*cos(c + d*x))) + (7*a^2*tan(c + d*x))/(4*d*sqrt(a + a*cos(c + d*x))) + (a^2*sec(c + d*x)*tan(c + d*x))/(2*d*sqrt(a + a*cos(c + d*x)))],
[sec(c + d*x)^4*(a + a*cos(c + d*x))^(3/2), x, 5, (11*a^2*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(4*sqrt(2)*d*sqrt(a + a*cos(c + d*x))) + (11*a^2*tan(c + d*x))/(8*d*sqrt(a + a*cos(c + d*x))) + (11*a^2*sec(c + d*x)*tan(c + d*x))/(12*d*sqrt(a + a*cos(c + d*x))) + (a^2*sec(c + d*x)^2*tan(c + d*x))/(3*d*sqrt(a + a*cos(c + d*x)))],


[cos(c + d*x)^3*(a + a*cos(c + d*x))^(5/2), x, 6, (284*a^3*sin(c + d*x))/(99*d*sqrt(a + a*cos(c + d*x))) + (710*a^3*cos(c + d*x)^3*sin(c + d*x))/(693*d*sqrt(a + a*cos(c + d*x))) + (46*a^3*cos(c + d*x)^4*sin(c + d*x))/(99*d*sqrt(a + a*cos(c + d*x))) - (568*a^2*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(693*d) + (2*a^2*cos(c + d*x)^4*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(11*d) + (284*a*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(231*d)],
[cos(c + d*x)^2*(a + a*cos(c + d*x))^(5/2), x, 5, (832*a^3*sin(c + d*x))/(315*d*sqrt(a + a*cos(c + d*x))) + (208*a^2*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(315*d) + (26*a*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(105*d) - (4*(a + a*cos(c + d*x))^(5/2)*sin(c + d*x))/(63*d) + (2*(a + a*cos(c + d*x))^(7/2)*sin(c + d*x))/(9*a*d)],
[cos(c + d*x)^1*(a + a*cos(c + d*x))^(5/2), x, 4, (64*a^3*sin(c + d*x))/(21*d*sqrt(a + a*cos(c + d*x))) + (16*a^2*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(21*d) + (2*a*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(7*d) + (2*(a + a*cos(c + d*x))^(5/2)*sin(c + d*x))/(7*d)],
[cos(c + d*x)^0*(a + a*cos(c + d*x))^(5/2), x, 3, (64*a^3*sin(c + d*x))/(15*d*sqrt(a + a*cos(c + d*x))) + (16*a^2*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(15*d) + (2*a*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(5*d)],
[sec(c + d*x)^1*(a + a*cos(c + d*x))^(5/2), x, 3, (2*sqrt(2)*a^3*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(a + a*cos(c + d*x))) + (14*a^3*sin(c + d*x))/(3*d*sqrt(a + a*cos(c + d*x))) + (2*a^2*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(3*d)],
[sec(c + d*x)^2*(a + a*cos(c + d*x))^(5/2), x, 3, (5*sqrt(2)*a^3*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(a + a*cos(c + d*x))) + (a^3*sin(c + d*x))/(d*sqrt(a + a*cos(c + d*x))) + (a^2*sqrt(a + a*cos(c + d*x))*tan(c + d*x))/d],
[sec(c + d*x)^3*(a + a*cos(c + d*x))^(5/2), x, 3, (19*a^3*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(2*sqrt(2)*d*sqrt(a + a*cos(c + d*x))) + (9*a^3*tan(c + d*x))/(4*d*sqrt(a + a*cos(c + d*x))) + (a^2*sqrt(a + a*cos(c + d*x))*sec(c + d*x)*tan(c + d*x))/(2*d)],
[sec(c + d*x)^4*(a + a*cos(c + d*x))^(5/2), x, 4, (25*a^3*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(4*sqrt(2)*d*sqrt(a + a*cos(c + d*x))) + (25*a^3*tan(c + d*x))/(8*d*sqrt(a + a*cos(c + d*x))) + (13*a^3*sec(c + d*x)*tan(c + d*x))/(12*d*sqrt(a + a*cos(c + d*x))) + (a^2*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^2*tan(c + d*x))/(3*d)],
[sec(c + d*x)^5*(a + a*cos(c + d*x))^(5/2), x, 5, (163*a^3*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(32*sqrt(2)*d*sqrt(a + a*cos(c + d*x))) + (163*a^3*tan(c + d*x))/(64*d*sqrt(a + a*cos(c + d*x))) + (163*a^3*sec(c + d*x)*tan(c + d*x))/(96*d*sqrt(a + a*cos(c + d*x))) + (17*a^3*sec(c + d*x)^2*tan(c + d*x))/(24*d*sqrt(a + a*cos(c + d*x))) + (a^2*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^3*tan(c + d*x))/(4*d)],


[(a + a*cos(c + d*x))^(7/2), x, 4, (256*a^4*sin(c + d*x))/(35*d*sqrt(a + a*cos(c + d*x))) + (64*a^3*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(35*d) + (24*a^2*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(35*d) + (2*a*(a + a*cos(c + d*x))^(5/2)*sin(c + d*x))/(7*d)],


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


[cos(c + d*x)^4/(a + a*cos(c + d*x))^(1/2), x, 5, (2*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(d*sqrt(a + a*cos(c + d*x))) - (148*sin(c + d*x))/(105*d*sqrt(a + a*cos(c + d*x))) - (2*cos(c + d*x)^2*sin(c + d*x))/(35*d*sqrt(a + a*cos(c + d*x))) + (2*cos(c + d*x)^3*sin(c + d*x))/(7*d*sqrt(a + a*cos(c + d*x))) + (62*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(105*a*d)],
[cos(c + d*x)^3/(a + a*cos(c + d*x))^(1/2), x, 4, -((2*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(d*sqrt(a + a*cos(c + d*x)))) + (28*sin(c + d*x))/(15*d*sqrt(a + a*cos(c + d*x))) + (2*cos(c + d*x)^2*sin(c + d*x))/(5*d*sqrt(a + a*cos(c + d*x))) - (2*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(15*a*d)],
[cos(c + d*x)^2/(a + a*cos(c + d*x))^(1/2), x, 3, (2*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(d*sqrt(a + a*cos(c + d*x))) - (4*sin(c + d*x))/(3*d*sqrt(a + a*cos(c + d*x))) + (2*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(3*a*d)],
[cos(c + d*x)^1/(a + a*cos(c + d*x))^(1/2), x, 2, -((2*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(d*sqrt(a + a*cos(c + d*x)))) + (2*sin(c + d*x))/(d*sqrt(a + a*cos(c + d*x)))],
[cos(c + d*x)^0/(a + a*cos(c + d*x))^(1/2), x, 1, (2*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(d*sqrt(a + a*cos(c + d*x)))],
[sec(c + d*x)^1/sqrt(a + a*cos(c + d*x)), x, 3, (-2*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(a + a*cos(c + d*x))) + (2*sqrt(2)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(a + a*cos(c + d*x)))],
[sec(c + d*x)^2/sqrt(a + a*cos(c + d*x)), x, 4, (2*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(a + a*cos(c + d*x))) - (sqrt(2)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(a + a*cos(c + d*x))) + tan(c + d*x)/(d*sqrt(a + a*cos(c + d*x)))],
[sec(c + d*x)^3/sqrt(a + a*cos(c + d*x)), x, 5, (-2*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(a + a*cos(c + d*x))) + (7*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(2*sqrt(2)*d*sqrt(a + a*cos(c + d*x))) - tan(c + d*x)/(4*d*sqrt(a + a*cos(c + d*x))) + (sec(c + d*x)*tan(c + d*x))/(2*d*sqrt(a + a*cos(c + d*x)))],
[sec(c + d*x)^4/sqrt(a + a*cos(c + d*x)), x, 6, (2*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(a + a*cos(c + d*x))) - (9*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(4*sqrt(2)*d*sqrt(a + a*cos(c + d*x))) + (7*tan(c + d*x))/(8*d*sqrt(a + a*cos(c + d*x))) - (sec(c + d*x)*tan(c + d*x))/(12*d*sqrt(a + a*cos(c + d*x))) + (sec(c + d*x)^2*tan(c + d*x))/(3*d*sqrt(a + a*cos(c + d*x)))],


[cos(x)/sqrt(a + a*cos(x)), x, 2, -((2*arctanh(sin(x/2))*cos(x/2))/sqrt(a + a*cos(x))) + (2*sin(x))/sqrt(a + a*cos(x))],
[sec(x)/sqrt(a + a*cos(x)), x, 3, (-2*arctanh(sin(x/2))*cos(x/2))/sqrt(a + a*cos(x)) + (2*sqrt(2)*arctanh(sqrt(2)*sin(x/2))*cos(x/2))/sqrt(a + a*cos(x))],


[cos(c + d*x)^4/(a + a*cos(c + d*x))^(3/2), x, 5, -((15*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(2*a*d*sqrt(a + a*cos(c + d*x)))) - (cos(c + d*x)^3*sin(c + d*x))/(2*d*(a + a*cos(c + d*x))^(3/2)) + (31*sin(c + d*x))/(5*a*d*sqrt(a + a*cos(c + d*x))) + (9*cos(c + d*x)^2*sin(c + d*x))/(10*a*d*sqrt(a + a*cos(c + d*x))) - (13*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(10*a^2*d)],
[cos(c + d*x)^3/(a + a*cos(c + d*x))^(3/2), x, 4, (11*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(2*a*d*sqrt(a + a*cos(c + d*x))) - (cos(c + d*x)^2*sin(c + d*x))/(2*d*(a + a*cos(c + d*x))^(3/2)) - (13*sin(c + d*x))/(3*a*d*sqrt(a + a*cos(c + d*x))) + (7*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(6*a^2*d)],
[cos(c + d*x)^2/(a + a*cos(c + d*x))^(3/2), x, 3, -((7*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(2*a*d*sqrt(a + a*cos(c + d*x)))) + sin(c + d*x)/(2*d*(a + a*cos(c + d*x))^(3/2)) + (2*sin(c + d*x))/(a*d*sqrt(a + a*cos(c + d*x)))],
[cos(c + d*x)^1/(a + a*cos(c + d*x))^(3/2), x, 2, (3*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(2*a*d*sqrt(a + a*cos(c + d*x))) - sin(c + d*x)/(2*d*(a + a*cos(c + d*x))^(3/2))],
[cos(c + d*x)^0/(a + a*cos(c + d*x))^(3/2), x, 2, (arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(2*a*d*sqrt(a + a*cos(c + d*x))) + sin(c + d*x)/(2*d*(a + a*cos(c + d*x))^(3/2))],
[sec(c + d*x)^1/(a + a*cos(c + d*x))^(3/2), x, 4, (-5*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(2*a*d*sqrt(a + a*cos(c + d*x))) + (2*sqrt(2)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(a*d*sqrt(a + a*cos(c + d*x))) - sin(c + d*x)/(2*d*(a + a*cos(c + d*x))^(3/2))],
[sec(c + d*x)^2/(a + a*cos(c + d*x))^(3/2), x, 5, (9*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(2*a*d*sqrt(a + a*cos(c + d*x))) - (3*sqrt(2)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(a*d*sqrt(a + a*cos(c + d*x))) - tan(c + d*x)/(2*d*(a + a*cos(c + d*x))^(3/2)) + (3*tan(c + d*x))/(2*a*d*sqrt(a + a*cos(c + d*x)))],
[sec(c + d*x)^3/(a + a*cos(c + d*x))^(3/2), x, 6, (-13*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(2*a*d*sqrt(a + a*cos(c + d*x))) + (19*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(2*sqrt(2)*a*d*sqrt(a + a*cos(c + d*x))) - (7*tan(c + d*x))/(4*a*d*sqrt(a + a*cos(c + d*x))) - (sec(c + d*x)*tan(c + d*x))/(2*d*(a + a*cos(c + d*x))^(3/2)) + (sec(c + d*x)*tan(c + d*x))/(a*d*sqrt(a + a*cos(c + d*x)))],


[cos(c + d*x)^4/(a + a*cos(c + d*x))^(5/2), x, 5, (163*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(16*a^2*d*sqrt(a + a*cos(c + d*x))) - (cos(c + d*x)^3*sin(c + d*x))/(4*d*(a + a*cos(c + d*x))^(5/2)) - (17*cos(c + d*x)^2*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2)) - (197*sin(c + d*x))/(24*a^2*d*sqrt(a + a*cos(c + d*x))) + (95*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(48*a^3*d)],
[cos(c + d*x)^3/(a + a*cos(c + d*x))^(5/2), x, 4, -((75*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(16*a^2*d*sqrt(a + a*cos(c + d*x)))) - (cos(c + d*x)^2*sin(c + d*x))/(4*d*(a + a*cos(c + d*x))^(5/2)) - (13*cos(c + d*x)*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2)) + (49*sin(c + d*x))/(16*a^2*d*sqrt(a + a*cos(c + d*x)))],
[cos(c + d*x)^2/(a + a*cos(c + d*x))^(5/2), x, 3, (19*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(16*a^2*d*sqrt(a + a*cos(c + d*x))) + sin(c + d*x)/(4*d*(a + a*cos(c + d*x))^(5/2)) - (13*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2))],
[cos(c + d*x)^1/(a + a*cos(c + d*x))^(5/2), x, 3, (5*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(16*a^2*d*sqrt(a + a*cos(c + d*x))) - sin(c + d*x)/(4*d*(a + a*cos(c + d*x))^(5/2)) + (5*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2))],
[cos(c + d*x)^0/(a + a*cos(c + d*x))^(5/2), x, 3, (3*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(16*a^2*d*sqrt(a + a*cos(c + d*x))) + sin(c + d*x)/(4*d*(a + a*cos(c + d*x))^(5/2)) + (3*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2))],
[sec(c + d*x)^1/(a + a*cos(c + d*x))^(5/2), x, 5, (-43*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(16*a^2*d*sqrt(a + a*cos(c + d*x))) + (2*sqrt(2)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(a^2*d*sqrt(a + a*cos(c + d*x))) - sin(c + d*x)/(4*d*(a + a*cos(c + d*x))^(5/2)) - (11*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2))],
[sec(c + d*x)^2/(a + a*cos(c + d*x))^(5/2), x, 6, (115*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(16*a^2*d*sqrt(a + a*cos(c + d*x))) - (5*sqrt(2)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(a^2*d*sqrt(a + a*cos(c + d*x))) - tan(c + d*x)/(4*d*(a + a*cos(c + d*x))^(5/2)) - (15*tan(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2)) + (35*tan(c + d*x))/(16*a^2*d*sqrt(a + a*cos(c + d*x)))],


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


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


[cos(a + b*x)^(11/2), x, 4, (30*EllipticF((1/2)*(a + b*x), 2))/(77*b) + (30*sqrt(cos(a + b*x))*sin(a + b*x))/(77*b) + (18*cos(a + b*x)^(5/2)*sin(a + b*x))/(77*b) + (2*cos(a + b*x)^(9/2)*sin(a + b*x))/(11*b)],
[cos(a + b*x)^(9/2), x, 3, (14*EllipticE((1/2)*(a + b*x), 2))/(15*b) + (14*cos(a + b*x)^(3/2)*sin(a + b*x))/(45*b) + (2*cos(a + b*x)^(7/2)*sin(a + b*x))/(9*b)],
[cos(a + b*x)^(7/2), x, 3, (10*EllipticF((1/2)*(a + b*x), 2))/(21*b) + (10*sqrt(cos(a + b*x))*sin(a + b*x))/(21*b) + (2*cos(a + b*x)^(5/2)*sin(a + b*x))/(7*b)],
[cos(a + b*x)^(5/2), x, 2, (6*EllipticE((1/2)*(a + b*x), 2))/(5*b) + (2*cos(a + b*x)^(3/2)*sin(a + b*x))/(5*b)],
[cos(a + b*x)^(3/2), x, 2, (2*EllipticF((1/2)*(a + b*x), 2))/(3*b) + (2*sqrt(cos(a + b*x))*sin(a + b*x))/(3*b)],
[cos(a + b*x)^(1/2), x, 1, (2*EllipticE((1/2)*(a + b*x), 2))/b],
[1/cos(a + b*x)^(1/2), x, 1, (2*EllipticF((1/2)*(a + b*x), 2))/b],
[1/cos(a + b*x)^(3/2), x, 2, -((2*EllipticE((1/2)*(a + b*x), 2))/b) + (2*sin(a + b*x))/(b*sqrt(cos(a + b*x)))],
[1/cos(a + b*x)^(5/2), x, 2, (2*EllipticF((1/2)*(a + b*x), 2))/(3*b) + (2*sin(a + b*x))/(3*b*cos(a + b*x)^(3/2))],
[1/cos(a + b*x)^(7/2), x, 3, -((6*EllipticE((1/2)*(a + b*x), 2))/(5*b)) + (2*sin(a + b*x))/(5*b*cos(a + b*x)^(5/2)) + (6*sin(a + b*x))/(5*b*sqrt(cos(a + b*x)))],
[1/cos(a + b*x)^(9/2), x, 3, (10*EllipticF((1/2)*(a + b*x), 2))/(21*b) + (2*sin(a + b*x))/(7*b*cos(a + b*x)^(7/2)) + (10*sin(a + b*x))/(21*b*cos(a + b*x)^(3/2))],
[1/cos(a + b*x)^(11/2), x, 4, -((14*EllipticE((1/2)*(a + b*x), 2))/(15*b)) + (2*sin(a + b*x))/(9*b*cos(a + b*x)^(9/2)) + (14*sin(a + b*x))/(45*b*cos(a + b*x)^(5/2)) + (14*sin(a + b*x))/(15*b*sqrt(cos(a + b*x)))],


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


[cos(c + d*x)^(5/2)*(a + a*cos(c + d*x))^2, x, 7, (32*a^2*EllipticE((1/2)*(c + d*x), 2))/(15*d) + (20*a^2*EllipticF((1/2)*(c + d*x), 2))/(21*d) + (20*a^2*sqrt(cos(c + d*x))*sin(c + d*x))/(21*d) + (32*a^2*cos(c + d*x)^(3/2)*sin(c + d*x))/(45*d) + (4*a^2*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)^(3/2)*(a + a*cos(c + d*x))^2, x, 6, (12*a^2*EllipticE((1/2)*(c + d*x), 2))/(5*d) + (8*a^2*EllipticF((1/2)*(c + d*x), 2))/(7*d) + (8*a^2*sqrt(cos(c + d*x))*sin(c + d*x))/(7*d) + (4*a^2*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)^(1/2)*(a + a*cos(c + d*x))^2, x, 5, (16*a^2*EllipticE((1/2)*(c + d*x), 2))/(5*d) + (4*a^2*EllipticF((1/2)*(c + d*x), 2))/(3*d) + (4*a^2*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)^(-1/2)*(a + a*cos(c + d*x))^2, x, 4, (4*a^2*EllipticE((1/2)*(c + d*x), 2))/d + (8*a^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)^(-3/2)*(a + a*cos(c + d*x))^2, x, 3, (4*a^2*EllipticF((1/2)*(c + d*x), 2))/d + (2*a^2*sin(c + d*x))/(d*sqrt(cos(c + d*x)))],
[cos(c + d*x)^(-5/2)*(a + a*cos(c + d*x))^2, x, 5, -((4*a^2*EllipticE((1/2)*(c + d*x), 2))/d) + (8*a^2*EllipticF((1/2)*(c + d*x), 2))/(3*d) + (2*a^2*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2)) + (4*a^2*sin(c + d*x))/(d*sqrt(cos(c + d*x)))],
[cos(c + d*x)^(-7/2)*(a + a*cos(c + d*x))^2, x, 6, -((16*a^2*EllipticE((1/2)*(c + d*x), 2))/(5*d)) + (4*a^2*EllipticF((1/2)*(c + d*x), 2))/(3*d) + (2*a^2*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2)) + (4*a^2*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2)) + (16*a^2*sin(c + d*x))/(5*d*sqrt(cos(c + d*x)))],


[cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))^3, x, 7, (68*a^3*EllipticE((1/2)*(c + d*x), 2))/(15*d) + (44*a^3*EllipticF((1/2)*(c + d*x), 2))/(21*d) + (44*a^3*sqrt(cos(c + d*x))*sin(c + d*x))/(21*d) + (68*a^3*cos(c + d*x)^(3/2)*sin(c + d*x))/(45*d) + (40*a^3*cos(c + d*x)^(5/2)*sin(c + d*x))/(63*d) + (2*a^2*cos(c + d*x)^(5/2)*(a + a*cos(c + d*x))*sin(c + d*x))/(9*d)],
[cos(c + d*x)^(1/2)*(a + a*cos(c + d*x))^3, x, 6, (28*a^3*EllipticE((1/2)*(c + d*x), 2))/(5*d) + (52*a^3*EllipticF((1/2)*(c + d*x), 2))/(21*d) + (52*a^3*sqrt(cos(c + d*x))*sin(c + d*x))/(21*d) + (32*a^3*cos(c + d*x)^(3/2)*sin(c + d*x))/(35*d) + (2*a^2*cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))*sin(c + d*x))/(7*d)],
[cos(c + d*x)^(-1/2)*(a + a*cos(c + d*x))^3, x, 5, (36*a^3*EllipticE((1/2)*(c + d*x), 2))/(5*d) + (4*a^3*EllipticF((1/2)*(c + d*x), 2))/d + (8*a^3*sqrt(cos(c + d*x))*sin(c + d*x))/(5*d) + (2*a^2*sqrt(cos(c + d*x))*(a + a*cos(c + d*x))*sin(c + d*x))/(5*d)],
[cos(c + d*x)^(-3/2)*(a + a*cos(c + d*x))^3, x, 5, (4*a^3*EllipticE((1/2)*(c + d*x), 2))/d + (20*a^3*EllipticF((1/2)*(c + d*x), 2))/(3*d) - (4*a^3*sqrt(cos(c + d*x))*sin(c + d*x))/(3*d) + (2*a^2*(a + a*cos(c + d*x))*sin(c + d*x))/(d*sqrt(cos(c + d*x)))],
[cos(c + d*x)^(-5/2)*(a + a*cos(c + d*x))^3, x, 5, -((4*a^3*EllipticE((1/2)*(c + d*x), 2))/d) + (20*a^3*EllipticF((1/2)*(c + d*x), 2))/(3*d) + (16*a^3*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))) + (2*a^2*(a + a*cos(c + d*x))*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2))],
[cos(c + d*x)^(-7/2)*(a + a*cos(c + d*x))^3, x, 6, -((36*a^3*EllipticE((1/2)*(c + d*x), 2))/(5*d)) + (4*a^3*EllipticF((1/2)*(c + d*x), 2))/d + (8*a^3*sin(c + d*x))/(5*d*cos(c + d*x)^(3/2)) + (36*a^3*sin(c + d*x))/(5*d*sqrt(cos(c + d*x))) + (2*a^2*(a + a*cos(c + d*x))*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2))],
[cos(c + d*x)^(-9/2)*(a + a*cos(c + d*x))^3, x, 7, -((28*a^3*EllipticE((1/2)*(c + d*x), 2))/(5*d)) + (52*a^3*EllipticF((1/2)*(c + d*x), 2))/(21*d) + (32*a^3*sin(c + d*x))/(35*d*cos(c + d*x)^(5/2)) + (52*a^3*sin(c + d*x))/(21*d*cos(c + d*x)^(3/2)) + (28*a^3*sin(c + d*x))/(5*d*sqrt(cos(c + d*x))) + (2*a^2*(a + a*cos(c + d*x))*sin(c + d*x))/(7*d*cos(c + d*x)^(7/2))],


[cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))^4, x, 8, (128*a^4*EllipticE((1/2)*(c + d*x), 2))/(15*d) + (904*a^4*EllipticF((1/2)*(c + d*x), 2))/(231*d) + (904*a^4*sqrt(cos(c + d*x))*sin(c + d*x))/(231*d) + (128*a^4*cos(c + d*x)^(3/2)*sin(c + d*x))/(45*d) + (860*a^4*cos(c + d*x)^(5/2)*sin(c + d*x))/(693*d) + (52*a^3*cos(c + d*x)^(5/2)*(a + a*cos(c + d*x))*sin(c + d*x))/(99*d) + (2*a^2*cos(c + d*x)^(5/2)*(a + a*cos(c + d*x))^2*sin(c + d*x))/(11*d)],
[cos(c + d*x)^(1/2)*(a + a*cos(c + d*x))^4, x, 7, (152*a^4*EllipticE((1/2)*(c + d*x), 2))/(15*d) + (32*a^4*EllipticF((1/2)*(c + d*x), 2))/(7*d) + (32*a^4*sqrt(cos(c + d*x))*sin(c + d*x))/(7*d) + (188*a^4*cos(c + d*x)^(3/2)*sin(c + d*x))/(105*d) + (44*a^3*cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))*sin(c + d*x))/(63*d) + (2*a^2*cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))^2*sin(c + d*x))/(9*d)],
[cos(c + d*x)^(-1/2)*(a + a*cos(c + d*x))^4, x, 6, (64*a^4*EllipticE((1/2)*(c + d*x), 2))/(5*d) + (136*a^4*EllipticF((1/2)*(c + d*x), 2))/(21*d) + (332*a^4*sqrt(cos(c + d*x))*sin(c + d*x))/(105*d) + (36*a^3*sqrt(cos(c + d*x))*(a + a*cos(c + d*x))*sin(c + d*x))/(35*d) + (2*a^2*sqrt(cos(c + d*x))*(a + a*cos(c + d*x))^2*sin(c + d*x))/(7*d)],
[cos(c + d*x)^(-3/2)*(a + a*cos(c + d*x))^4, x, 6, (56*a^4*EllipticE((1/2)*(c + d*x), 2))/(5*d) + (32*a^4*EllipticF((1/2)*(c + d*x), 2))/(3*d) + (4*a^4*sqrt(cos(c + d*x))*sin(c + d*x))/(15*d) - (8*a^3*sqrt(cos(c + d*x))*(a + a*cos(c + d*x))*sin(c + d*x))/(5*d) + (2*a^2*(a + a*cos(c + d*x))^2*sin(c + d*x))/(d*sqrt(cos(c + d*x)))],
[cos(c + d*x)^(-5/2)*(a + a*cos(c + d*x))^4, x, 5, (40*a^4*EllipticF((1/2)*(c + d*x), 2))/(3*d) + (20*a^4*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))) + (2*a^2*(a + a*cos(c + d*x))^2*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2))],
[cos(c + d*x)^(-7/2)*(a + a*cos(c + d*x))^4, x, 6, -((56*a^4*EllipticE((1/2)*(c + d*x), 2))/(5*d)) + (32*a^4*EllipticF((1/2)*(c + d*x), 2))/(3*d) + (164*a^4*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))) + (28*a^3*(a + a*cos(c + d*x))*sin(c + d*x))/(15*d*cos(c + d*x)^(3/2)) + (2*a^2*(a + a*cos(c + d*x))^2*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2))],
[cos(c + d*x)^(-9/2)*(a + a*cos(c + d*x))^4, x, 7, -((64*a^4*EllipticE((1/2)*(c + d*x), 2))/(5*d)) + (136*a^4*EllipticF((1/2)*(c + d*x), 2))/(21*d) + (332*a^4*sin(c + d*x))/(105*d*cos(c + d*x)^(3/2)) + (64*a^4*sin(c + d*x))/(5*d*sqrt(cos(c + d*x))) + (36*a^3*(a + a*cos(c + d*x))*sin(c + d*x))/(35*d*cos(c + d*x)^(5/2)) + (2*a^2*(a + a*cos(c + d*x))^2*sin(c + d*x))/(7*d*cos(c + d*x)^(7/2))],


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


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


[cos(c + d*x)^(9/2)/(a + a*cos(c + d*x))^2, x, 7, (56*EllipticE((1/2)*(c + d*x), 2))/(5*a^2*d) - (5*EllipticF((1/2)*(c + d*x), 2))/(a^2*d) - (5*sqrt(cos(c + d*x))*sin(c + d*x))/(a^2*d) + (56*cos(c + d*x)^(3/2)*sin(c + d*x))/(15*a^2*d) - (cos(c + d*x)^(7/2)*sin(c + d*x))/(3*d*(a + a*cos(c + d*x))^2) - (3*cos(c + d*x)^(5/2)*sin(c + d*x))/(a*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^(7/2)/(a + a*cos(c + d*x))^2, x, 6, -((7*EllipticE((1/2)*(c + d*x), 2))/(a^2*d)) + (10*EllipticF((1/2)*(c + d*x), 2))/(3*a^2*d) + (10*sqrt(cos(c + d*x))*sin(c + d*x))/(3*a^2*d) - (cos(c + d*x)^(5/2)*sin(c + d*x))/(3*d*(a + a*cos(c + d*x))^2) - (7*cos(c + d*x)^(3/2)*sin(c + d*x))/(3*a*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^(5/2)/(a + a*cos(c + d*x))^2, x, 5, (4*EllipticE((1/2)*(c + d*x), 2))/(a^2*d) - (5*EllipticF((1/2)*(c + d*x), 2))/(3*a^2*d) - (cos(c + d*x)^(3/2)*sin(c + d*x))/(3*d*(a + a*cos(c + d*x))^2) - (5*sqrt(cos(c + d*x))*sin(c + d*x))/(3*a*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^(3/2)/(a + a*cos(c + d*x))^2, x, 5, -(EllipticE((1/2)*(c + d*x), 2)/(a^2*d)) + (2*EllipticF((1/2)*(c + d*x), 2))/(3*a^2*d) - (sqrt(cos(c + d*x))*sin(c + d*x))/(3*d*(a + a*cos(c + d*x))^2) + (sqrt(cos(c + d*x))*sin(c + d*x))/(a*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^(1/2)/(a + a*cos(c + d*x))^2, x, 2, EllipticF((1/2)*(c + d*x), 2)/(3*a^2*d) + (sqrt(cos(c + d*x))*sin(c + d*x))/(3*d*(a + a*cos(c + d*x))^2)],
[cos(c + d*x)^(-1/2)/(a + a*cos(c + d*x))^2, x, 5, EllipticE((1/2)*(c + d*x), 2)/(a^2*d) + (2*EllipticF((1/2)*(c + d*x), 2))/(3*a^2*d) - (sqrt(cos(c + d*x))*sin(c + d*x))/(3*d*(a + a*cos(c + d*x))^2) - (sqrt(cos(c + d*x))*sin(c + d*x))/(a*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^(-3/2)/(a + a*cos(c + d*x))^2, x, 6, -((4*EllipticE((1/2)*(c + d*x), 2))/(a^2*d)) - (5*EllipticF((1/2)*(c + d*x), 2))/(3*a^2*d) + (4*sin(c + d*x))/(a^2*d*sqrt(cos(c + d*x))) - sin(c + d*x)/(3*d*sqrt(cos(c + d*x))*(a + a*cos(c + d*x))^2) - (5*sin(c + d*x))/(3*a*d*sqrt(cos(c + d*x))*(a + a*cos(c + d*x)))],
[cos(c + d*x)^(-5/2)/(a + a*cos(c + d*x))^2, x, 7, (7*EllipticE((1/2)*(c + d*x), 2))/(a^2*d) + (10*EllipticF((1/2)*(c + d*x), 2))/(3*a^2*d) + (10*sin(c + d*x))/(3*a^2*d*cos(c + d*x)^(3/2)) - (7*sin(c + d*x))/(a^2*d*sqrt(cos(c + d*x))) - sin(c + d*x)/(3*d*cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))^2) - (7*sin(c + d*x))/(3*a*d*cos(c + d*x)^(3/2)*(a + a*cos(c + d*x)))],


[cos(c + d*x)^(11/2)/(a + a*cos(c + d*x))^3, x, 8, (231*EllipticE((1/2)*(c + d*x), 2))/(10*a^3*d) - (21*EllipticF((1/2)*(c + d*x), 2))/(2*a^3*d) - (21*sqrt(cos(c + d*x))*sin(c + d*x))/(2*a^3*d) + (77*cos(c + d*x)^(3/2)*sin(c + d*x))/(10*a^3*d) - (cos(c + d*x)^(9/2)*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3) - (4*cos(c + d*x)^(7/2)*sin(c + d*x))/(5*a*d*(a + a*cos(c + d*x))^2) - (63*cos(c + d*x)^(5/2)*sin(c + d*x))/(10*a^2*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^(9/2)/(a + a*cos(c + d*x))^3, x, 7, -((119*EllipticE((1/2)*(c + d*x), 2))/(10*a^3*d)) + (11*EllipticF((1/2)*(c + d*x), 2))/(2*a^3*d) + (11*sqrt(cos(c + d*x))*sin(c + d*x))/(2*a^3*d) - (cos(c + d*x)^(7/2)*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3) - (2*cos(c + d*x)^(5/2)*sin(c + d*x))/(3*a*d*(a + a*cos(c + d*x))^2) - (119*cos(c + d*x)^(3/2)*sin(c + d*x))/(30*a^2*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^(7/2)/(a + a*cos(c + d*x))^3, x, 6, (49*EllipticE((1/2)*(c + d*x), 2))/(10*a^3*d) - (13*EllipticF((1/2)*(c + d*x), 2))/(6*a^3*d) - (cos(c + d*x)^(5/2)*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3) - (8*cos(c + d*x)^(3/2)*sin(c + d*x))/(15*a*d*(a + a*cos(c + d*x))^2) - (13*sqrt(cos(c + d*x))*sin(c + d*x))/(6*a^2*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^(5/2)/(a + a*cos(c + d*x))^3, x, 6, -((9*EllipticE((1/2)*(c + d*x), 2))/(10*a^3*d)) + EllipticF((1/2)*(c + d*x), 2)/(2*a^3*d) - (cos(c + d*x)^(3/2)*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3) - (2*sqrt(cos(c + d*x))*sin(c + d*x))/(5*a*d*(a + a*cos(c + d*x))^2) + (9*sqrt(cos(c + d*x))*sin(c + d*x))/(10*a^2*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^(3/2)/(a + a*cos(c + d*x))^3, x, 6, -(EllipticE((1/2)*(c + d*x), 2)/(10*a^3*d)) + EllipticF((1/2)*(c + d*x), 2)/(6*a^3*d) - (sqrt(cos(c + d*x))*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3) + (4*sqrt(cos(c + d*x))*sin(c + d*x))/(15*a*d*(a + a*cos(c + d*x))^2) + (sqrt(cos(c + d*x))*sin(c + d*x))/(10*a^2*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^(1/2)/(a + a*cos(c + d*x))^3, x, 6, EllipticE((1/2)*(c + d*x), 2)/(10*a^3*d) + EllipticF((1/2)*(c + d*x), 2)/(6*a^3*d) + (sqrt(cos(c + d*x))*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3) + (sqrt(cos(c + d*x))*sin(c + d*x))/(15*a*d*(a + a*cos(c + d*x))^2) - (sqrt(cos(c + d*x))*sin(c + d*x))/(10*a^2*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^(-1/2)/(a + a*cos(c + d*x))^3, x, 6, (9*EllipticE((1/2)*(c + d*x), 2))/(10*a^3*d) + EllipticF((1/2)*(c + d*x), 2)/(2*a^3*d) - (sqrt(cos(c + d*x))*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3) - (2*sqrt(cos(c + d*x))*sin(c + d*x))/(5*a*d*(a + a*cos(c + d*x))^2) - (9*sqrt(cos(c + d*x))*sin(c + d*x))/(10*a^2*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^(-3/2)/(a + a*cos(c + d*x))^3, x, 7, -((49*EllipticE((1/2)*(c + d*x), 2))/(10*a^3*d)) - (13*EllipticF((1/2)*(c + d*x), 2))/(6*a^3*d) + (49*sin(c + d*x))/(10*a^3*d*sqrt(cos(c + d*x))) - sin(c + d*x)/(5*d*sqrt(cos(c + d*x))*(a + a*cos(c + d*x))^3) - (8*sin(c + d*x))/(15*a*d*sqrt(cos(c + d*x))*(a + a*cos(c + d*x))^2) - (13*sin(c + d*x))/(6*a^2*d*sqrt(cos(c + d*x))*(a + a*cos(c + d*x)))],
[cos(c + d*x)^(-5/2)/(a + a*cos(c + d*x))^3, x, 8, (119*EllipticE((1/2)*(c + d*x), 2))/(10*a^3*d) + (11*EllipticF((1/2)*(c + d*x), 2))/(2*a^3*d) + (11*sin(c + d*x))/(2*a^3*d*cos(c + d*x)^(3/2)) - (119*sin(c + d*x))/(10*a^3*d*sqrt(cos(c + d*x))) - sin(c + d*x)/(5*d*cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))^3) - (2*sin(c + d*x))/(3*a*d*cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))^2) - (119*sin(c + d*x))/(30*a^2*d*cos(c + d*x)^(3/2)*(a + a*cos(c + d*x)))],


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


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


[cos(c + d*x)^(5/2)*(a + a*cos(c + d*x))^(1/2), x, 4, (5*sqrt(a)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(8*d) + (5*a*sqrt(cos(c + d*x))*sin(c + d*x))/(8*d*sqrt(a + a*cos(c + d*x))) + (5*a*cos(c + d*x)^(3/2)*sin(c + d*x))/(12*d*sqrt(a + a*cos(c + d*x))) + (a*cos(c + d*x)^(5/2)*sin(c + d*x))/(3*d*sqrt(a + a*cos(c + d*x)))],
[cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))^(1/2), x, 3, (3*sqrt(a)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(4*d) + (3*a*sqrt(cos(c + d*x))*sin(c + d*x))/(4*d*sqrt(a + a*cos(c + d*x))) + (a*cos(c + d*x)^(3/2)*sin(c + d*x))/(2*d*sqrt(a + a*cos(c + d*x)))],
[cos(c + d*x)^(1/2)*(a + a*cos(c + d*x))^(1/2), x, 2, (sqrt(a)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/d + (a*sqrt(cos(c + d*x))*sin(c + d*x))/(d*sqrt(a + a*cos(c + d*x)))],
[cos(c + d*x)^(-1/2)*(a + a*cos(c + d*x))^(1/2), x, 1, (2*sqrt(a)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/d],
[cos(c + d*x)^(-3/2)*(a + a*cos(c + d*x))^(1/2), x, 1, (2*a*sin(c + d*x))/(d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))],
[cos(c + d*x)^(-5/2)*(a + a*cos(c + d*x))^(1/2), x, 2, (2*a*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2)*sqrt(a + a*cos(c + d*x))) + (4*a*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))],
[cos(c + d*x)^(-7/2)*(a + a*cos(c + d*x))^(1/2), x, 3, (2*a*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2)*sqrt(a + a*cos(c + d*x))) + (8*a*sin(c + d*x))/(15*d*cos(c + d*x)^(3/2)*sqrt(a + a*cos(c + d*x))) + (16*a*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))],
[cos(c + d*x)^(-9/2)*(a + a*cos(c + d*x))^(1/2), x, 4, (2*a*sin(c + d*x))/(7*d*cos(c + d*x)^(7/2)*sqrt(a + a*cos(c + d*x))) + (12*a*sin(c + d*x))/(35*d*cos(c + d*x)^(5/2)*sqrt(a + a*cos(c + d*x))) + (16*a*sin(c + d*x))/(35*d*cos(c + d*x)^(3/2)*sqrt(a + a*cos(c + d*x))) + (32*a*sin(c + d*x))/(35*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))],


[cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))^(3/2), x, 5, (11*a^(3/2)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(8*d) + (11*a^2*sqrt(cos(c + d*x))*sin(c + d*x))/(8*d*sqrt(a + a*cos(c + d*x))) + (11*a^2*cos(c + d*x)^(3/2)*sin(c + d*x))/(12*d*sqrt(a + a*cos(c + d*x))) + (a^2*cos(c + d*x)^(5/2)*sin(c + d*x))/(3*d*sqrt(a + a*cos(c + d*x)))],
[cos(c + d*x)^(1/2)*(a + a*cos(c + d*x))^(3/2), x, 4, (7*a^(3/2)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(4*d) + (7*a^2*sqrt(cos(c + d*x))*sin(c + d*x))/(4*d*sqrt(a + a*cos(c + d*x))) + (a^2*cos(c + d*x)^(3/2)*sin(c + d*x))/(2*d*sqrt(a + a*cos(c + d*x)))],
[cos(c + d*x)^(-1/2)*(a + a*cos(c + d*x))^(3/2), x, 3, (3*a^(3/2)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/d + (a^2*sqrt(cos(c + d*x))*sin(c + d*x))/(d*sqrt(a + a*cos(c + d*x)))],
[cos(c + d*x)^(-3/2)*(a + a*cos(c + d*x))^(3/2), x, 3, (2*a^(3/2)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/d + (2*a^2*sin(c + d*x))/(d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))],
[cos(c + d*x)^(-5/2)*(a + a*cos(c + d*x))^(3/2), x, 2, (8*a^2*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))) + (2*a*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2))],
[cos(c + d*x)^(-7/2)*(a + a*cos(c + d*x))^(3/2), x, 3, (8*a^2*sin(c + d*x))/(5*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))) + (2*a*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(5*d*cos(c + d*x)^(3/2)) + (2*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2))],
[cos(c + d*x)^(-9/2)*(a + a*cos(c + d*x))^(3/2), x, 5, (2*a^2*sin(c + d*x))/(7*d*cos(c + d*x)^(7/2)*sqrt(a + a*cos(c + d*x))) + (26*a^2*sin(c + d*x))/(35*d*cos(c + d*x)^(5/2)*sqrt(a + a*cos(c + d*x))) + (104*a^2*sin(c + d*x))/(105*d*cos(c + d*x)^(3/2)*sqrt(a + a*cos(c + d*x))) + (208*a^2*sin(c + d*x))/(105*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))],


[cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))^(5/2), x, 5, (163*a^(5/2)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(64*d) + (163*a^3*sqrt(cos(c + d*x))*sin(c + d*x))/(64*d*sqrt(a + a*cos(c + d*x))) + (163*a^3*cos(c + d*x)^(3/2)*sin(c + d*x))/(96*d*sqrt(a + a*cos(c + d*x))) + (17*a^3*cos(c + d*x)^(5/2)*sin(c + d*x))/(24*d*sqrt(a + a*cos(c + d*x))) + (a^2*cos(c + d*x)^(5/2)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(4*d)],
[cos(c + d*x)^(1/2)*(a + a*cos(c + d*x))^(5/2), x, 4, (25*a^(5/2)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(8*d) + (25*a^3*sqrt(cos(c + d*x))*sin(c + d*x))/(8*d*sqrt(a + a*cos(c + d*x))) + (13*a^3*cos(c + d*x)^(3/2)*sin(c + d*x))/(12*d*sqrt(a + a*cos(c + d*x))) + (a^2*cos(c + d*x)^(3/2)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(3*d)],
[cos(c + d*x)^(-1/2)*(a + a*cos(c + d*x))^(5/2), x, 3, (19*a^(5/2)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(4*d) + (9*a^3*sqrt(cos(c + d*x))*sin(c + d*x))/(4*d*sqrt(a + a*cos(c + d*x))) + (a^2*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(2*d)],
[cos(c + d*x)^(-3/2)*(a + a*cos(c + d*x))^(5/2), x, 3, (5*a^(5/2)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/d - (a^3*sqrt(cos(c + d*x))*sin(c + d*x))/(d*sqrt(a + a*cos(c + d*x))) + (2*a^2*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(d*sqrt(cos(c + d*x)))],
[cos(c + d*x)^(-5/2)*(a + a*cos(c + d*x))^(5/2), x, 3, (2*a^(5/2)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/d + (14*a^3*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))) + (2*a^2*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2))],
[cos(c + d*x)^(-7/2)*(a + a*cos(c + d*x))^(5/2), x, 3, (64*a^3*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))) + (16*a^2*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(15*d*cos(c + d*x)^(3/2)) + (2*a*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2))],
[cos(c + d*x)^(-9/2)*(a + a*cos(c + d*x))^(5/2), x, 4, (64*a^3*sin(c + d*x))/(21*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))) + (16*a^2*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(21*d*cos(c + d*x)^(3/2)) + (2*a*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(7*d*cos(c + d*x)^(5/2)) + (2*(a + a*cos(c + d*x))^(5/2)*sin(c + d*x))/(7*d*cos(c + d*x)^(7/2))],
[cos(c + d*x)^(-11/2)*(a + a*cos(c + d*x))^(5/2), x, 5, (38*a^3*sin(c + d*x))/(63*d*cos(c + d*x)^(7/2)*sqrt(a + a*cos(c + d*x))) + (146*a^3*sin(c + d*x))/(105*d*cos(c + d*x)^(5/2)*sqrt(a + a*cos(c + d*x))) + (584*a^3*sin(c + d*x))/(315*d*cos(c + d*x)^(3/2)*sqrt(a + a*cos(c + d*x))) + (1168*a^3*sin(c + d*x))/(315*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))) + (2*a^2*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(9*d*cos(c + d*x)^(9/2))],


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


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


[cos(c + d*x)^(5/2)/(1 + cos(c + d*x))^(1/2), x, 5, -((sqrt(2)*arcsin(tan((1/2)*(c + d*x))))/d) + (7*arctan(sin(c + d*x)/(sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x)))))/(4*d) - (sqrt(cos(c + d*x))*sin(c + d*x))/(4*d*sqrt(1 + cos(c + d*x))) + (cos(c + d*x)^(3/2)*sin(c + d*x))/(2*d*sqrt(1 + cos(c + d*x)))],
[cos(c + d*x)^(3/2)/(1 + cos(c + d*x))^(1/2), x, 4, (sqrt(2)*arcsin(tan((1/2)*(c + d*x))))/d - arctan(sin(c + d*x)/(sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))))/d + (sqrt(cos(c + d*x))*sin(c + d*x))/(d*sqrt(1 + cos(c + d*x)))],
[cos(c + d*x)^(1/2)/(1 + cos(c + d*x))^(1/2), x, 3, -((sqrt(2)*arcsin(tan((1/2)*(c + d*x))))/d) + (2*arctan(sin(c + d*x)/(sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x)))))/d],
[cos(c + d*x)^(-1/2)/(1 + cos(c + d*x))^(1/2), x, 1, (sqrt(2)*arcsin(tan((1/2)*(c + d*x))))/d],
[cos(c + d*x)^(-3/2)/(1 + cos(c + d*x))^(1/2), x, 2, -((sqrt(2)*arcsin(tan((1/2)*(c + d*x))))/d) + (2*sin(c + d*x))/(d*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x)))],
[cos(c + d*x)^(-5/2)/(1 + cos(c + d*x))^(1/2), x, 3, (sqrt(2)*arcsin(tan((1/2)*(c + d*x))))/d + (2*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2)*sqrt(1 + cos(c + d*x))) - (2*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x)))],
[cos(c + d*x)^(-7/2)/(1 + cos(c + d*x))^(1/2), x, 4, -((sqrt(2)*arcsin(tan((1/2)*(c + d*x))))/d) + (2*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2)*sqrt(1 + cos(c + d*x))) - (2*sin(c + d*x))/(15*d*cos(c + d*x)^(3/2)*sqrt(1 + cos(c + d*x))) + (26*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x)))],


[1/(sqrt(1 + cos(x))*sqrt(cos(x))), x, 1, sqrt(2)*arcsin(tan(x/2))],
[1/(sqrt(a + a*cos(x))*sqrt(cos(x))), x, 1, (sqrt(2)*arctan((sqrt(a)*sin(x))/(sqrt(2)*sqrt(cos(x))*sqrt(a + a*cos(x)))))/sqrt(a)],


[cos(c + d*x)^(5/2)/(a + a*cos(c + d*x))^(1/2), x, 5, (7*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(4*sqrt(a)*d) - (sqrt(2)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(sqrt(a)*d) - (sqrt(cos(c + d*x))*sin(c + d*x))/(4*d*sqrt(a + a*cos(c + d*x))) + (cos(c + d*x)^(3/2)*sin(c + d*x))/(2*d*sqrt(a + a*cos(c + d*x)))],
[cos(c + d*x)^(3/2)/(a + a*cos(c + d*x))^(1/2), x, 4, -(arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))/(sqrt(a)*d)) + (sqrt(2)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(sqrt(a)*d) + (sqrt(cos(c + d*x))*sin(c + d*x))/(d*sqrt(a + a*cos(c + d*x)))],
[cos(c + d*x)^(1/2)/(a + a*cos(c + d*x))^(1/2), x, 3, (2*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(sqrt(a)*d) - (sqrt(2)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(sqrt(a)*d)],
[cos(c + d*x)^(-1/2)/(a + a*cos(c + d*x))^(1/2), x, 1, (sqrt(2)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(sqrt(a)*d)],
[cos(c + d*x)^(-3/2)/(a + a*cos(c + d*x))^(1/2), x, 2, -((sqrt(2)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(sqrt(a)*d)) + (2*sin(c + d*x))/(d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))],
[cos(c + d*x)^(-5/2)/(a + a*cos(c + d*x))^(1/2), x, 3, (sqrt(2)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(sqrt(a)*d) + (2*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2)*sqrt(a + a*cos(c + d*x))) - (2*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))],
[cos(c + d*x)^(-7/2)/(a + a*cos(c + d*x))^(1/2), x, 4, -((sqrt(2)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(sqrt(a)*d)) + (2*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2)*sqrt(a + a*cos(c + d*x))) - (2*sin(c + d*x))/(15*d*cos(c + d*x)^(3/2)*sqrt(a + a*cos(c + d*x))) + (26*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))],


[cos(c + d*x)^(5/2)/(a + a*cos(c + d*x))^(3/2), x, 5, -((3*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(a^(3/2)*d)) + (9*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d) - (cos(c + d*x)^(3/2)*sin(c + d*x))/(2*d*(a + a*cos(c + d*x))^(3/2)) + (3*sqrt(cos(c + d*x))*sin(c + d*x))/(2*a*d*sqrt(a + a*cos(c + d*x)))],
[cos(c + d*x)^(3/2)/(a + a*cos(c + d*x))^(3/2), x, 4, (2*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(a^(3/2)*d) - (5*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d) - (sqrt(cos(c + d*x))*sin(c + d*x))/(2*d*(a + a*cos(c + d*x))^(3/2))],
[cos(c + d*x)^(1/2)/(a + a*cos(c + d*x))^(3/2), x, 2, arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))/(2*sqrt(2)*a^(3/2)*d) + (sqrt(cos(c + d*x))*sin(c + d*x))/(2*d*(a + a*cos(c + d*x))^(3/2))],
[cos(c + d*x)^(-1/2)/(a + a*cos(c + d*x))^(3/2), x, 2, (3*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d) - (sqrt(cos(c + d*x))*sin(c + d*x))/(2*d*(a + a*cos(c + d*x))^(3/2))],
[cos(c + d*x)^(-3/2)/(a + a*cos(c + d*x))^(3/2), x, 3, -((7*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d)) - sin(c + d*x)/(2*d*sqrt(cos(c + d*x))*(a + a*cos(c + d*x))^(3/2)) + (5*sin(c + d*x))/(2*a*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))],
[cos(c + d*x)^(-5/2)/(a + a*cos(c + d*x))^(3/2), x, 4, (11*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d) - sin(c + d*x)/(2*d*cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))^(3/2)) + (7*sin(c + d*x))/(6*a*d*cos(c + d*x)^(3/2)*sqrt(a + a*cos(c + d*x))) - (19*sin(c + d*x))/(6*a*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))],


[cos(c + d*x)^(7/2)/(a + a*cos(c + d*x))^(5/2), x, 6, -((5*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(a^(5/2)*d)) + (115*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d) - (cos(c + d*x)^(5/2)*sin(c + d*x))/(4*d*(a + a*cos(c + d*x))^(5/2)) - (15*cos(c + d*x)^(3/2)*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2)) + (35*sqrt(cos(c + d*x))*sin(c + d*x))/(16*a^2*d*sqrt(a + a*cos(c + d*x)))],
[cos(c + d*x)^(5/2)/(a + a*cos(c + d*x))^(5/2), x, 5, (2*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(a^(5/2)*d) - (43*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d) - (cos(c + d*x)^(3/2)*sin(c + d*x))/(4*d*(a + a*cos(c + d*x))^(5/2)) - (11*sqrt(cos(c + d*x))*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2))],
[cos(c + d*x)^(3/2)/(a + a*cos(c + d*x))^(5/2), x, 3, (3*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d) + (cos(c + d*x)^(3/2)*sin(c + d*x))/(4*d*(a + a*cos(c + d*x))^(5/2)) + (3*sqrt(cos(c + d*x))*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2))],
[cos(c + d*x)^(1/2)/(a + a*cos(c + d*x))^(5/2), x, 3, (5*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d) - (cos(c + d*x)^(3/2)*sin(c + d*x))/(4*d*(a + a*cos(c + d*x))^(5/2)) + (5*sqrt(cos(c + d*x))*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2))],
[cos(c + d*x)^(-1/2)/(a + a*cos(c + d*x))^(5/2), x, 3, (19*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d) - (sqrt(cos(c + d*x))*sin(c + d*x))/(4*d*(a + a*cos(c + d*x))^(5/2)) - (9*sqrt(cos(c + d*x))*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2))],
[cos(c + d*x)^(-3/2)/(a + a*cos(c + d*x))^(5/2), x, 4, -((75*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d)) - sin(c + d*x)/(4*d*sqrt(cos(c + d*x))*(a + a*cos(c + d*x))^(5/2)) - (13*sin(c + d*x))/(16*a*d*sqrt(cos(c + d*x))*(a + a*cos(c + d*x))^(3/2)) + (49*sin(c + d*x))/(16*a^2*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))],
[cos(c + d*x)^(-5/2)/(a + a*cos(c + d*x))^(5/2), x, 5, (163*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d) - sin(c + d*x)/(4*d*cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))^(5/2)) - (17*sin(c + d*x))/(16*a*d*cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))^(3/2)) + (95*sin(c + d*x))/(48*a^2*d*cos(c + d*x)^(3/2)*sqrt(a + a*cos(c + d*x))) - (299*sin(c + d*x))/(48*a^2*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))],


[cos(c + d*x)^(9/2)/(a + a*cos(c + d*x))^(7/2), x, 7, -((7*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(a^(7/2)*d)) + (637*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(64*sqrt(2)*a^(7/2)*d) - (cos(c + d*x)^(7/2)*sin(c + d*x))/(6*d*(a + a*cos(c + d*x))^(7/2)) - (7*cos(c + d*x)^(5/2)*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(5/2)) - (259*cos(c + d*x)^(3/2)*sin(c + d*x))/(192*a^2*d*(a + a*cos(c + d*x))^(3/2)) + (189*sqrt(cos(c + d*x))*sin(c + d*x))/(64*a^3*d*sqrt(a + a*cos(c + d*x)))],
[cos(c + d*x)^(7/2)/(a + a*cos(c + d*x))^(7/2), x, 6, (2*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(a^(7/2)*d) - (177*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(64*sqrt(2)*a^(7/2)*d) - (cos(c + d*x)^(5/2)*sin(c + d*x))/(6*d*(a + a*cos(c + d*x))^(7/2)) - (17*cos(c + d*x)^(3/2)*sin(c + d*x))/(48*a*d*(a + a*cos(c + d*x))^(5/2)) - (49*sqrt(cos(c + d*x))*sin(c + d*x))/(64*a^2*d*(a + a*cos(c + d*x))^(3/2))],
[cos(c + d*x)^(5/2)/(a + a*cos(c + d*x))^(7/2), x, 4, (5*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(64*sqrt(2)*a^(7/2)*d) + (cos(c + d*x)^(5/2)*sin(c + d*x))/(6*d*(a + a*cos(c + d*x))^(7/2)) + (5*cos(c + d*x)^(3/2)*sin(c + d*x))/(48*a*d*(a + a*cos(c + d*x))^(5/2)) + (5*sqrt(cos(c + d*x))*sin(c + d*x))/(64*a^2*d*(a + a*cos(c + d*x))^(3/2))],
[cos(c + d*x)^(3/2)/(a + a*cos(c + d*x))^(7/2), x, 4, (7*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(64*sqrt(2)*a^(7/2)*d) - (cos(c + d*x)^(5/2)*sin(c + d*x))/(6*d*(a + a*cos(c + d*x))^(7/2)) + (7*cos(c + d*x)^(3/2)*sin(c + d*x))/(48*a*d*(a + a*cos(c + d*x))^(5/2)) + (7*sqrt(cos(c + d*x))*sin(c + d*x))/(64*a^2*d*(a + a*cos(c + d*x))^(3/2))],
[cos(c + d*x)^(1/2)/(a + a*cos(c + d*x))^(7/2), x, 4, (13*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(64*sqrt(2)*a^(7/2)*d) + (sqrt(cos(c + d*x))*sin(c + d*x))/(6*d*(a + a*cos(c + d*x))^(7/2)) + (sqrt(cos(c + d*x))*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(5/2)) - (5*sqrt(cos(c + d*x))*sin(c + d*x))/(192*a^2*d*(a + a*cos(c + d*x))^(3/2))],
[cos(c + d*x)^(-1/2)/(a + a*cos(c + d*x))^(7/2), x, 4, (63*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(64*sqrt(2)*a^(7/2)*d) - (sqrt(cos(c + d*x))*sin(c + d*x))/(6*d*(a + a*cos(c + d*x))^(7/2)) - (5*sqrt(cos(c + d*x))*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(5/2)) - (103*sqrt(cos(c + d*x))*sin(c + d*x))/(192*a^2*d*(a + a*cos(c + d*x))^(3/2))],
[cos(c + d*x)^(-3/2)/(a + a*cos(c + d*x))^(7/2), x, 5, -((363*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(64*sqrt(2)*a^(7/2)*d)) - sin(c + d*x)/(6*d*sqrt(cos(c + d*x))*(a + a*cos(c + d*x))^(7/2)) - (19*sin(c + d*x))/(48*a*d*sqrt(cos(c + d*x))*(a + a*cos(c + d*x))^(5/2)) - (199*sin(c + d*x))/(192*a^2*d*sqrt(cos(c + d*x))*(a + a*cos(c + d*x))^(3/2)) + (691*sin(c + d*x))/(192*a^3*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))],
[cos(c + d*x)^(-5/2)/(a + a*cos(c + d*x))^(7/2), x, 6, (1015*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(64*sqrt(2)*a^(7/2)*d) - sin(c + d*x)/(6*d*cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))^(7/2)) - (23*sin(c + d*x))/(48*a*d*cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))^(5/2)) - (109*sin(c + d*x))/(64*a^2*d*cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))^(3/2)) + (193*sin(c + d*x))/(64*a^3*d*cos(c + d*x)^(3/2)*sqrt(a + a*cos(c + d*x))) - (629*sin(c + d*x))/(64*a^3*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))],


[cos(c + d*x)^(7/2)/(a + a*cos(c + d*x))^(9/2), x, 5, (35*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(1024*sqrt(2)*a^(9/2)*d) + (cos(c + d*x)^(7/2)*sin(c + d*x))/(8*d*(a + a*cos(c + d*x))^(9/2)) + (7*cos(c + d*x)^(5/2)*sin(c + d*x))/(96*a*d*(a + a*cos(c + d*x))^(7/2)) + (35*cos(c + d*x)^(3/2)*sin(c + d*x))/(768*a^2*d*(a + a*cos(c + d*x))^(5/2)) + (35*sqrt(cos(c + d*x))*sin(c + d*x))/(1024*a^3*d*(a + a*cos(c + d*x))^(3/2))],
[cos(c + d*x)^(5/2)/(a + a*cos(c + d*x))^(9/2), x, 5, (45*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(1024*sqrt(2)*a^(9/2)*d) - (cos(c + d*x)^(7/2)*sin(c + d*x))/(8*d*(a + a*cos(c + d*x))^(9/2)) + (3*cos(c + d*x)^(5/2)*sin(c + d*x))/(32*a*d*(a + a*cos(c + d*x))^(7/2)) + (15*cos(c + d*x)^(3/2)*sin(c + d*x))/(256*a^2*d*(a + a*cos(c + d*x))^(5/2)) + (45*sqrt(cos(c + d*x))*sin(c + d*x))/(1024*a^3*d*(a + a*cos(c + d*x))^(3/2))],


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


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


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


[cos(c + d*x)^(3/2)*(1 - cos(c + d*x))^(1/2), x, 3, -((3*arctanh(sin(c + d*x)/(sqrt(1 - cos(c + d*x))*sqrt(cos(c + d*x)))))/(4*d)) + (3*sqrt(cos(c + d*x))*sin(c + d*x))/(4*d*sqrt(1 - cos(c + d*x))) - (cos(c + d*x)^(3/2)*sin(c + d*x))/(2*d*sqrt(1 - cos(c + d*x)))],
[cos(c + d*x)^(1/2)*(1 - cos(c + d*x))^(1/2), x, 2, arctanh(sin(c + d*x)/(sqrt(1 - cos(c + d*x))*sqrt(cos(c + d*x))))/d - (sqrt(cos(c + d*x))*sin(c + d*x))/(d*sqrt(1 - cos(c + d*x)))],
[cos(c + d*x)^(-1/2)*(1 - cos(c + d*x))^(1/2), x, 1, -((2*arctanh(sin(c + d*x)/(sqrt(1 - cos(c + d*x))*sqrt(cos(c + d*x)))))/d)],
[cos(c + d*x)^(-3/2)*(1 - cos(c + d*x))^(1/2), x, 1, (2*sin(c + d*x))/(d*sqrt(1 - cos(c + d*x))*sqrt(cos(c + d*x)))],
[cos(c + d*x)^(-5/2)*(1 - cos(c + d*x))^(1/2), x, 2, (2*sin(c + d*x))/(3*d*sqrt(1 - cos(c + d*x))*cos(c + d*x)^(3/2)) - (4*sin(c + d*x))/(3*d*sqrt(1 - cos(c + d*x))*sqrt(cos(c + d*x)))],
[cos(c + d*x)^(-7/2)*(1 - cos(c + d*x))^(1/2), x, 3, (2*sin(c + d*x))/(5*d*sqrt(1 - cos(c + d*x))*cos(c + d*x)^(5/2)) - (8*sin(c + d*x))/(15*d*sqrt(1 - cos(c + d*x))*cos(c + d*x)^(3/2)) + (16*sin(c + d*x))/(15*d*sqrt(1 - cos(c + d*x))*sqrt(cos(c + d*x)))],


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


[cos(c + d*x)^(5/2)/(1 - cos(c + d*x))^(1/2), x, 5, (7*arctanh(sin(c + d*x)/(sqrt(1 - cos(c + d*x))*sqrt(cos(c + d*x)))))/(4*d) - (sqrt(2)*arctanh(sin(c + d*x)/(sqrt(2)*sqrt(1 - cos(c + d*x))*sqrt(cos(c + d*x)))))/d + (sqrt(cos(c + d*x))*sin(c + d*x))/(4*d*sqrt(1 - cos(c + d*x))) + (cos(c + d*x)^(3/2)*sin(c + d*x))/(2*d*sqrt(1 - cos(c + d*x)))],
[cos(c + d*x)^(3/2)/(1 - cos(c + d*x))^(1/2), x, 4, arctanh(sin(c + d*x)/(sqrt(1 - cos(c + d*x))*sqrt(cos(c + d*x))))/d - (sqrt(2)*arctanh(sin(c + d*x)/(sqrt(2)*sqrt(1 - cos(c + d*x))*sqrt(cos(c + d*x)))))/d + (sqrt(cos(c + d*x))*sin(c + d*x))/(d*sqrt(1 - cos(c + d*x)))],
[cos(c + d*x)^(1/2)/(1 - cos(c + d*x))^(1/2), x, 3, (2*arctanh(sin(c + d*x)/(sqrt(1 - cos(c + d*x))*sqrt(cos(c + d*x)))))/d - (sqrt(2)*arctanh(sin(c + d*x)/(sqrt(2)*sqrt(1 - cos(c + d*x))*sqrt(cos(c + d*x)))))/d],
[cos(c + d*x)^(-1/2)/(1 - cos(c + d*x))^(1/2), x, 1, -((sqrt(2)*arctanh(sin(c + d*x)/(sqrt(2)*sqrt(1 - cos(c + d*x))*sqrt(cos(c + d*x)))))/d)],
[cos(c + d*x)^(-3/2)/(1 - cos(c + d*x))^(1/2), x, 2, -((sqrt(2)*arctanh(sin(c + d*x)/(sqrt(2)*sqrt(1 - cos(c + d*x))*sqrt(cos(c + d*x)))))/d) + (2*sin(c + d*x))/(d*sqrt(1 - cos(c + d*x))*sqrt(cos(c + d*x)))],
[cos(c + d*x)^(-5/2)/(1 - cos(c + d*x))^(1/2), x, 3, -((sqrt(2)*arctanh(sin(c + d*x)/(sqrt(2)*sqrt(1 - cos(c + d*x))*sqrt(cos(c + d*x)))))/d) + (2*sin(c + d*x))/(3*d*sqrt(1 - cos(c + d*x))*cos(c + d*x)^(3/2)) + (2*sin(c + d*x))/(3*d*sqrt(1 - cos(c + d*x))*sqrt(cos(c + d*x)))],


[1/(sqrt(1 - cos(x))*sqrt(cos(x))), x, 1, (-sqrt(2))*arctanh(sin(x)/(sqrt(2)*sqrt(1 - cos(x))*sqrt(cos(x))))],
[1/(sqrt(a - a*cos(x))*sqrt(cos(x))), x, 1, -((sqrt(2)*arctan((sqrt(-a)*sin(x))/(sqrt(2)*sqrt(cos(x))*sqrt(a - a*cos(x)))))/sqrt(-a))],


[cos(c + d*x)^(5/2)/(a - a*cos(c + d*x))^(1/2), x, 5, (7*arctan((sqrt(-a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a - a*cos(c + d*x)))))/(4*sqrt(-a)*d) - (sqrt(2)*arctan((sqrt(-a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a - a*cos(c + d*x)))))/(sqrt(-a)*d) + (sqrt(cos(c + d*x))*sin(c + d*x))/(4*d*sqrt(a - a*cos(c + d*x))) + (cos(c + d*x)^(3/2)*sin(c + d*x))/(2*d*sqrt(a - a*cos(c + d*x)))],
[cos(c + d*x)^(3/2)/(a - a*cos(c + d*x))^(1/2), x, 4, arctan((sqrt(-a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a - a*cos(c + d*x))))/(sqrt(-a)*d) - (sqrt(2)*arctan((sqrt(-a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a - a*cos(c + d*x)))))/(sqrt(-a)*d) + (sqrt(cos(c + d*x))*sin(c + d*x))/(d*sqrt(a - a*cos(c + d*x)))],
[cos(c + d*x)^(1/2)/(a - a*cos(c + d*x))^(1/2), x, 3, (2*arctan((sqrt(-a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a - a*cos(c + d*x)))))/(sqrt(-a)*d) - (sqrt(2)*arctan((sqrt(-a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a - a*cos(c + d*x)))))/(sqrt(-a)*d)],
[cos(c + d*x)^(-1/2)/(a - a*cos(c + d*x))^(1/2), x, 1, -((sqrt(2)*arctan((sqrt(-a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a - a*cos(c + d*x)))))/(sqrt(-a)*d))],
[cos(c + d*x)^(-3/2)/(a - a*cos(c + d*x))^(1/2), x, 2, -((sqrt(2)*arctan((sqrt(-a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a - a*cos(c + d*x)))))/(sqrt(-a)*d)) + (2*sin(c + d*x))/(d*sqrt(cos(c + d*x))*sqrt(a - a*cos(c + d*x)))],
[cos(c + d*x)^(-5/2)/(a - a*cos(c + d*x))^(1/2), x, 3, -((sqrt(2)*arctan((sqrt(-a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a - a*cos(c + d*x)))))/(sqrt(-a)*d)) + (2*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2)*sqrt(a - a*cos(c + d*x))) + (2*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))*sqrt(a - a*cos(c + d*x)))],
[cos(c + d*x)^(-7/2)/(a - a*cos(c + d*x))^(1/2), x, 4, -((sqrt(2)*arctan((sqrt(-a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a - a*cos(c + d*x)))))/(sqrt(-a)*d)) + (2*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2)*sqrt(a - a*cos(c + d*x))) + (2*sin(c + d*x))/(15*d*cos(c + d*x)^(3/2)*sqrt(a - a*cos(c + d*x))) + (26*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))*sqrt(a - a*cos(c + d*x)))],


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


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


[cos(c + d*x)^(4/3)*(a + a*cos(c + d*x))^(1/3), x, 2, Int(cos(c + d*x)^(4/3)*(a + a*cos(c + d*x))^(1/3), x)],
[cos(c + d*x)^(4/3)*(a + a*cos(c + d*x))^(2/3), x, 2, Int(((5*a^2)/9 + (7/9)*a^2*cos(c + d*x))/(cos(c + d*x)^(2/3)*(a + a*cos(c + d*x))^(1/3)), x)/(2*a) + (5*a*cos(c + d*x)^(1/3)*sin(c + d*x))/(6*d*(a + a*cos(c + d*x))^(1/3)) + (a*cos(c + d*x)^(4/3)*sin(c + d*x))/(2*d*(a + a*cos(c + d*x))^(1/3))],
[cos(c + d*x)^(5/3)*(a + a*cos(c + d*x))^(2/3), x, 2, (9*Int(((4*a^2)/3 + (14/9)*a^2*cos(c + d*x))/(cos(c + d*x)^(1/3)*(a + a*cos(c + d*x))^(1/3)), x))/(28*a) + (9*a*cos(c + d*x)^(2/3)*sin(c + d*x))/(14*d*(a + a*cos(c + d*x))^(1/3)) + (3*a*cos(c + d*x)^(5/3)*sin(c + d*x))/(7*d*(a + a*cos(c + d*x))^(1/3))],


# ::Subsubsection:: 
#n<0


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


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


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


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


[(a + a*cos(c + d*x))^3*sec(c + d*x)^(9/2), x, 8, -((28*a^3*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d)) + (52*a^3*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) + (28*a^3*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d) + (52*a^3*sec(c + d*x)^(3/2)*sin(c + d*x))/(21*d) + (32*a^3*sec(c + d*x)^(5/2)*sin(c + d*x))/(35*d) + (2*a^2*(a + a*cos(c + d*x))*sec(c + d*x)^(7/2)*sin(c + d*x))/(7*d)],
[(a + a*cos(c + d*x))^3*sec(c + d*x)^(7/2), x, 7, -((36*a^3*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d)) + (4*a^3*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/d + (36*a^3*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d) + (8*a^3*sec(c + d*x)^(3/2)*sin(c + d*x))/(5*d) + (2*a^2*(a + a*cos(c + d*x))*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d)],
[(a + a*cos(c + d*x))^3*sec(c + d*x)^(5/2), x, 6, -((4*a^3*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/d) + (20*a^3*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) + (16*a^3*sqrt(sec(c + d*x))*sin(c + d*x))/(3*d) + (2*a^2*(a + a*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d)],
[(a + a*cos(c + d*x))^3*sec(c + d*x)^(3/2), x, 6, (4*a^3*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/d + (20*a^3*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*d) - (4*a^3*sin(c + d*x))/(3*d*sqrt(sec(c + d*x))) + (2*a^2*(a + a*cos(c + d*x))*sqrt(sec(c + d*x))*sin(c + d*x))/d],
[(a + a*cos(c + d*x))^3*sec(c + d*x)^(1/2), x, 6, (36*a^3*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*d) + (4*a^3*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/d + (8*a^3*sin(c + d*x))/(5*d*sqrt(sec(c + d*x))) + (2*a^2*(a + a*cos(c + d*x))*sin(c + d*x))/(5*d*sqrt(sec(c + d*x)))],
[(a + a*cos(c + d*x))^3/sec(c + d*x)^(1/2), x, 7, (28*a^3*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*d) + (52*a^3*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(21*d) + (32*a^3*sin(c + d*x))/(35*d*sec(c + d*x)^(3/2)) + (2*a^2*(a + a*cos(c + d*x))*sin(c + d*x))/(7*d*sec(c + d*x)^(3/2)) + (52*a^3*sin(c + d*x))/(21*d*sqrt(sec(c + d*x)))],
[(a + a*cos(c + d*x))^3/sec(c + d*x)^(3/2), x, 8, (68*a^3*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(15*d) + (44*a^3*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(21*d) + (40*a^3*sin(c + d*x))/(63*d*sec(c + d*x)^(5/2)) + (2*a^2*(a + a*cos(c + d*x))*sin(c + d*x))/(9*d*sec(c + d*x)^(5/2)) + (68*a^3*sin(c + d*x))/(45*d*sec(c + d*x)^(3/2)) + (44*a^3*sin(c + d*x))/(21*d*sqrt(sec(c + d*x)))],


[(a + a*cos(c + d*x))^4*sec(c + d*x)^(9/2), x, 8, -((64*a^4*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d)) + (136*a^4*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) + (64*a^4*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d) + (332*a^4*sec(c + d*x)^(3/2)*sin(c + d*x))/(105*d) + (36*a^3*(a + a*cos(c + d*x))*sec(c + d*x)^(5/2)*sin(c + d*x))/(35*d) + (2*a^2*(a + a*cos(c + d*x))^2*sec(c + d*x)^(7/2)*sin(c + d*x))/(7*d)],
[(a + a*cos(c + d*x))^4*sec(c + d*x)^(7/2), x, 7, -((56*a^4*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d)) + (32*a^4*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) + (164*a^4*sqrt(sec(c + d*x))*sin(c + d*x))/(15*d) + (28*a^3*(a + a*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(15*d) + (2*a^2*(a + a*cos(c + d*x))^2*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d)],
[(a + a*cos(c + d*x))^4*sec(c + d*x)^(5/2), x, 6, (40*a^4*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) + (20*a^4*sqrt(sec(c + d*x))*sin(c + d*x))/(3*d) + (2*a^2*(a + a*cos(c + d*x))^2*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d)],
[(a + a*cos(c + d*x))^4*sec(c + d*x)^(3/2), x, 7, (56*a^4*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d) + (32*a^4*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) + (4*a^4*sin(c + d*x))/(15*d*sqrt(sec(c + d*x))) - (8*a^3*(a + a*cos(c + d*x))*sin(c + d*x))/(5*d*sqrt(sec(c + d*x))) + (2*a^2*(a + a*cos(c + d*x))^2*sqrt(sec(c + d*x))*sin(c + d*x))/d],
[(a + a*cos(c + d*x))^4*sec(c + d*x)^(1/2), x, 7, (64*a^4*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d) + (136*a^4*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) + (332*a^4*sin(c + d*x))/(105*d*sqrt(sec(c + d*x))) + (36*a^3*(a + a*cos(c + d*x))*sin(c + d*x))/(35*d*sqrt(sec(c + d*x))) + (2*a^2*(a + a*cos(c + d*x))^2*sin(c + d*x))/(7*d*sqrt(sec(c + d*x)))],
[(a + a*cos(c + d*x))^4/sec(c + d*x)^(1/2), x, 8, (152*a^4*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(15*d) + (32*a^4*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(7*d) + (188*a^4*sin(c + d*x))/(105*d*sec(c + d*x)^(3/2)) + (44*a^3*(a + a*cos(c + d*x))*sin(c + d*x))/(63*d*sec(c + d*x)^(3/2)) + (2*a^2*(a + a*cos(c + d*x))^2*sin(c + d*x))/(9*d*sec(c + d*x)^(3/2)) + (32*a^4*sin(c + d*x))/(7*d*sqrt(sec(c + d*x)))],


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


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


[sec(c + d*x)^(5/2)/(a + a*cos(c + d*x))^2, x, 8, (7*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(a^2*d) + (10*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*a^2*d) - (7*sqrt(sec(c + d*x))*sin(c + d*x))/(a^2*d) + (10*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*a^2*d) - (sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d*(a + a*cos(c + d*x))^2) - (7*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*a*d*(a + a*cos(c + d*x)))],
[sec(c + d*x)^(3/2)/(a + a*cos(c + d*x))^2, x, 7, (-4*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(a^2*d) - (5*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*a^2*d) + (4*sqrt(sec(c + d*x))*sin(c + d*x))/(a^2*d) - (sqrt(sec(c + d*x))*sin(c + d*x))/(3*d*(a + a*cos(c + d*x))^2) - (5*sqrt(sec(c + d*x))*sin(c + d*x))/(3*a*d*(a + a*cos(c + d*x)))],
[sec(c + d*x)^(1/2)/(a + a*cos(c + d*x))^2, x, 6, (sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(a^2*d) + (2*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*a^2*d) - sin(c + d*x)/(3*d*(a + a*cos(c + d*x))^2*sqrt(sec(c + d*x))) - sin(c + d*x)/(a*d*(a + a*cos(c + d*x))*sqrt(sec(c + d*x)))],
[1/((a + a*cos(c + d*x))^2*sec(c + d*x)^(1/2)), x, 3, (sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*a^2*d) + sin(c + d*x)/(3*d*(a + a*cos(c + d*x))^2*sqrt(sec(c + d*x)))],
[1/((a + a*cos(c + d*x))^2*sec(c + d*x)^(3/2)), x, 6, -((sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(a^2*d)) + (2*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*a^2*d) - sin(c + d*x)/(3*d*(a + a*cos(c + d*x))^2*sqrt(sec(c + d*x))) + sin(c + d*x)/(a*d*(a + a*cos(c + d*x))*sqrt(sec(c + d*x)))],
[1/((a + a*cos(c + d*x))^2*sec(c + d*x)^(5/2)), x, 6, (4*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(a^2*d) - (5*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*a^2*d) - sin(c + d*x)/(3*d*(a + a*cos(c + d*x))^2*sec(c + d*x)^(3/2)) - (5*sin(c + d*x))/(3*a*d*(a + a*cos(c + d*x))*sqrt(sec(c + d*x)))],
[1/((a + a*cos(c + d*x))^2*sec(c + d*x)^(7/2)), x, 7, -((7*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a^2*d)) + (10*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*a^2*d) - sin(c + d*x)/(3*d*(a + a*cos(c + d*x))^2*sec(c + d*x)^(5/2)) - (7*sin(c + d*x))/(3*a*d*(a + a*cos(c + d*x))*sec(c + d*x)^(3/2)) + (10*sin(c + d*x))/(3*a^2*d*sqrt(sec(c + d*x)))],
[1/((a + a*cos(c + d*x))^2*sec(c + d*x)^(9/2)), x, 8, (56*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*a^2*d) - (5*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a^2*d) - sin(c + d*x)/(3*d*(a + a*cos(c + d*x))^2*sec(c + d*x)^(7/2)) - (3*sin(c + d*x))/(a*d*(a + a*cos(c + d*x))*sec(c + d*x)^(5/2)) + (56*sin(c + d*x))/(15*a^2*d*sec(c + d*x)^(3/2)) - (5*sin(c + d*x))/(a^2*d*sqrt(sec(c + d*x)))],


[sec(c + d*x)^(5/2)/(a + a*cos(c + d*x))^3, x, 9, (119*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(10*a^3*d) + (11*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(2*a^3*d) - (119*sqrt(sec(c + d*x))*sin(c + d*x))/(10*a^3*d) + (11*sec(c + d*x)^(3/2)*sin(c + d*x))/(2*a^3*d) - (sec(c + d*x)^(3/2)*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3) - (2*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*a*d*(a + a*cos(c + d*x))^2) - (119*sec(c + d*x)^(3/2)*sin(c + d*x))/(30*a^2*d*(a + a*cos(c + d*x)))],
[sec(c + d*x)^(3/2)/(a + a*cos(c + d*x))^3, x, 8, (-49*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(10*a^3*d) - (13*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(6*a^3*d) + (49*sqrt(sec(c + d*x))*sin(c + d*x))/(10*a^3*d) - (sqrt(sec(c + d*x))*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3) - (8*sqrt(sec(c + d*x))*sin(c + d*x))/(15*a*d*(a + a*cos(c + d*x))^2) - (13*sqrt(sec(c + d*x))*sin(c + d*x))/(6*a^2*d*(a + a*cos(c + d*x)))],
[sec(c + d*x)^(1/2)/(a + a*cos(c + d*x))^3, x, 7, (9*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(10*a^3*d) + (sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(2*a^3*d) - sin(c + d*x)/(5*d*(a + a*cos(c + d*x))^3*sqrt(sec(c + d*x))) - (2*sin(c + d*x))/(5*a*d*(a + a*cos(c + d*x))^2*sqrt(sec(c + d*x))) - (9*sin(c + d*x))/(10*a^2*d*(a + a*cos(c + d*x))*sqrt(sec(c + d*x)))],
[1/((a + a*cos(c + d*x))^3*sec(c + d*x)^(1/2)), x, 7, (sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(10*a^3*d) + (sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(6*a^3*d) + sin(c + d*x)/(5*d*(a + a*cos(c + d*x))^3*sqrt(sec(c + d*x))) + sin(c + d*x)/(15*a*d*(a + a*cos(c + d*x))^2*sqrt(sec(c + d*x))) - sin(c + d*x)/(10*a^2*d*(a + a*cos(c + d*x))*sqrt(sec(c + d*x)))],
[1/((a + a*cos(c + d*x))^3*sec(c + d*x)^(3/2)), x, 7, -(sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(10*a^3*d) + (sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(6*a^3*d) - sin(c + d*x)/(5*d*(a + a*cos(c + d*x))^3*sqrt(sec(c + d*x))) + (4*sin(c + d*x))/(15*a*d*(a + a*cos(c + d*x))^2*sqrt(sec(c + d*x))) + sin(c + d*x)/(10*a^2*d*(a + a*cos(c + d*x))*sqrt(sec(c + d*x)))],
[1/((a + a*cos(c + d*x))^3*sec(c + d*x)^(5/2)), x, 7, (-9*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(10*a^3*d) + (sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(2*a^3*d) - sin(c + d*x)/(5*d*(a + a*cos(c + d*x))^3*sec(c + d*x)^(3/2)) - (2*sin(c + d*x))/(5*a*d*(a + a*cos(c + d*x))^2*sqrt(sec(c + d*x))) + (9*sin(c + d*x))/(10*a^2*d*(a + a*cos(c + d*x))*sqrt(sec(c + d*x)))],
[1/((a + a*cos(c + d*x))^3*sec(c + d*x)^(7/2)), x, 7, (49*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(10*a^3*d) - (13*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(6*a^3*d) - sin(c + d*x)/(5*d*(a + a*cos(c + d*x))^3*sec(c + d*x)^(5/2)) - (8*sin(c + d*x))/(15*a*d*(a + a*cos(c + d*x))^2*sec(c + d*x)^(3/2)) - (13*sin(c + d*x))/(6*a^2*d*(a + a*cos(c + d*x))*sqrt(sec(c + d*x)))],
[1/((a + a*cos(c + d*x))^3*sec(c + d*x)^(9/2)), x, 8, -((119*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(10*a^3*d)) + (11*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(2*a^3*d) - sin(c + d*x)/(5*d*(a + a*cos(c + d*x))^3*sec(c + d*x)^(7/2)) - (2*sin(c + d*x))/(3*a*d*(a + a*cos(c + d*x))^2*sec(c + d*x)^(5/2)) - (119*sin(c + d*x))/(30*a^2*d*(a + a*cos(c + d*x))*sec(c + d*x)^(3/2)) + (11*sin(c + d*x))/(2*a^3*d*sqrt(sec(c + d*x)))],
[1/((a + a*cos(c + d*x))^3*sec(c + d*x)^(11/2)), x, 9, (231*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(10*a^3*d) - (21*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(2*a^3*d) - sin(c + d*x)/(5*d*(a + a*cos(c + d*x))^3*sec(c + d*x)^(9/2)) - (4*sin(c + d*x))/(5*a*d*(a + a*cos(c + d*x))^2*sec(c + d*x)^(7/2)) - (63*sin(c + d*x))/(10*a^2*d*(a + a*cos(c + d*x))*sec(c + d*x)^(5/2)) + (77*sin(c + d*x))/(10*a^3*d*sec(c + d*x)^(3/2)) - (21*sin(c + d*x))/(2*a^3*d*sqrt(sec(c + d*x)))],


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


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


[sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(9/2), x, 4, (32*a*sqrt(sec(c + d*x))*sin(c + d*x))/(35*d*sqrt(a + a*cos(c + d*x))) + (16*a*sec(c + d*x)^(3/2)*sin(c + d*x))/(35*d*sqrt(a + a*cos(c + d*x))) + (12*a*sec(c + d*x)^(5/2)*sin(c + d*x))/(35*d*sqrt(a + a*cos(c + d*x))) + (2*a*sec(c + d*x)^(7/2)*sin(c + d*x))/(7*d*sqrt(a + a*cos(c + d*x)))],
[sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(7/2), x, 3, (16*a*sqrt(sec(c + d*x))*sin(c + d*x))/(15*d*sqrt(a + a*cos(c + d*x))) + (8*a*sec(c + d*x)^(3/2)*sin(c + d*x))/(15*d*sqrt(a + a*cos(c + d*x))) + (2*a*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d*sqrt(a + a*cos(c + d*x)))],
[sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(5/2), x, 2, (4*a*sqrt(sec(c + d*x))*sin(c + d*x))/(3*d*sqrt(a + a*cos(c + d*x))) + (2*a*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d*sqrt(a + a*cos(c + d*x)))],
[sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(3/2), x, 1, (2*a*sqrt(sec(c + d*x))*sin(c + d*x))/(d*sqrt(a + a*cos(c + d*x)))],
[sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(1/2), x, 2, (2*sqrt(a)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/d],
[sqrt(a + a*cos(c + d*x))/sec(c + d*x)^(1/2), x, 3, (sqrt(a)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/d + (a*sin(c + d*x))/(d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x)))],
[sqrt(a + a*cos(c + d*x))/sec(c + d*x)^(3/2), x, 4, (3*sqrt(a)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(4*d) + (a*sin(c + d*x))/(2*d*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(3/2)) + (3*a*sin(c + d*x))/(4*d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x)))],


[(a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^(9/2), x, 5, (208*a^2*sqrt(sec(c + d*x))*sin(c + d*x))/(105*d*sqrt(a + a*cos(c + d*x))) + (104*a^2*sec(c + d*x)^(3/2)*sin(c + d*x))/(105*d*sqrt(a + a*cos(c + d*x))) + (26*a^2*sec(c + d*x)^(5/2)*sin(c + d*x))/(35*d*sqrt(a + a*cos(c + d*x))) + (2*a^2*sec(c + d*x)^(7/2)*sin(c + d*x))/(7*d*sqrt(a + a*cos(c + d*x)))],
[(a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^(7/2), x, 3, (8*a^2*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d*sqrt(a + a*cos(c + d*x))) + (2*a*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(5*d) + (2*(a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d)],
[(a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^(5/2), x, 2, (8*a^2*sqrt(sec(c + d*x))*sin(c + d*x))/(3*d*sqrt(a + a*cos(c + d*x))) + (2*a*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d)],
[(a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^(3/2), x, 4, (2*a^(3/2)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/d + (2*a^2*sqrt(sec(c + d*x))*sin(c + d*x))/(d*sqrt(a + a*cos(c + d*x)))],
[(a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^(1/2), x, 4, (3*a^(3/2)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/d + (a^2*sin(c + d*x))/(d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x)))],
[(a + a*cos(c + d*x))^(3/2)/sec(c + d*x)^(1/2), x, 5, (7*a^(3/2)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(4*d) + (a^2*sin(c + d*x))/(2*d*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(3/2)) + (7*a^2*sin(c + d*x))/(4*d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x)))],
[(a + a*cos(c + d*x))^(3/2)/sec(c + d*x)^(3/2), x, 6, (11*a^(3/2)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(8*d) + (a^2*sin(c + d*x))/(3*d*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(5/2)) + (11*a^2*sin(c + d*x))/(12*d*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(3/2)) + (11*a^2*sin(c + d*x))/(8*d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x)))],


[(a + a*cos(c + d*x))^(5/2)*sec(c + d*x)^(11/2), x, 5, (1168*a^3*sqrt(sec(c + d*x))*sin(c + d*x))/(315*d*sqrt(a + a*cos(c + d*x))) + (584*a^3*sec(c + d*x)^(3/2)*sin(c + d*x))/(315*d*sqrt(a + a*cos(c + d*x))) + (146*a^3*sec(c + d*x)^(5/2)*sin(c + d*x))/(105*d*sqrt(a + a*cos(c + d*x))) + (38*a^3*sec(c + d*x)^(7/2)*sin(c + d*x))/(63*d*sqrt(a + a*cos(c + d*x))) + (2*a^2*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(9/2)*sin(c + d*x))/(9*d)],
[(a + a*cos(c + d*x))^(5/2)*sec(c + d*x)^(9/2), x, 4, (64*a^3*sqrt(sec(c + d*x))*sin(c + d*x))/(21*d*sqrt(a + a*cos(c + d*x))) + (16*a^2*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(21*d) + (2*a*(a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^(5/2)*sin(c + d*x))/(7*d) + (2*(a + a*cos(c + d*x))^(5/2)*sec(c + d*x)^(7/2)*sin(c + d*x))/(7*d)],
[(a + a*cos(c + d*x))^(5/2)*sec(c + d*x)^(7/2), x, 3, (64*a^3*sqrt(sec(c + d*x))*sin(c + d*x))/(15*d*sqrt(a + a*cos(c + d*x))) + (16*a^2*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(15*d) + (2*a*(a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d)],
[(a + a*cos(c + d*x))^(5/2)*sec(c + d*x)^(5/2), x, 4, (2*a^(5/2)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/d + (14*a^3*sqrt(sec(c + d*x))*sin(c + d*x))/(3*d*sqrt(a + a*cos(c + d*x))) + (2*a^2*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d)],
[(a + a*cos(c + d*x))^(5/2)*sec(c + d*x)^(3/2), x, 4, (5*a^(5/2)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/d - (a^3*sin(c + d*x))/(d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x))) + (2*a^2*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x))*sin(c + d*x))/d],
[(a + a*cos(c + d*x))^(5/2)*sec(c + d*x)^(1/2), x, 4, (19*a^(5/2)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(4*d) + (9*a^3*sin(c + d*x))/(4*d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x))) + (a^2*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(2*d*sqrt(sec(c + d*x)))],
[(a + a*cos(c + d*x))^(5/2)/sec(c + d*x)^(1/2), x, 5, (25*a^(5/2)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(8*d) + (13*a^3*sin(c + d*x))/(12*d*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(3/2)) + (a^2*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(3*d*sec(c + d*x)^(3/2)) + (25*a^3*sin(c + d*x))/(8*d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x)))],
[(a + a*cos(c + d*x))^(5/2)/sec(c + d*x)^(3/2), x, 6, (163*a^(5/2)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(64*d) + (17*a^3*sin(c + d*x))/(24*d*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(5/2)) + (a^2*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(4*d*sec(c + d*x)^(5/2)) + (163*a^3*sin(c + d*x))/(96*d*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(3/2)) + (163*a^3*sin(c + d*x))/(64*d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x)))],


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


[sec(c + d*x)^(7/2)/sqrt(1 + cos(c + d*x)), x, 5, -((sqrt(2)*arcsin(tan((1/2)*(c + d*x)))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/d) + (26*sqrt(sec(c + d*x))*sin(c + d*x))/(15*d*sqrt(1 + cos(c + d*x))) - (2*sec(c + d*x)^(3/2)*sin(c + d*x))/(15*d*sqrt(1 + cos(c + d*x))) + (2*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d*sqrt(1 + cos(c + d*x)))],
[sec(c + d*x)^(5/2)/sqrt(1 + cos(c + d*x)), x, 4, (sqrt(2)*arcsin(tan((c + d*x)/2))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/d - (2*sqrt(sec(c + d*x))*sin(c + d*x))/(3*d*sqrt(1 + cos(c + d*x))) + (2*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d*sqrt(1 + cos(c + d*x)))],
[sec(c + d*x)^(3/2)/sqrt(1 + cos(c + d*x)), x, 3, -((sqrt(2)*arcsin(tan((c + d*x)/2))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/d) + (2*sqrt(sec(c + d*x))*sin(c + d*x))/(d*sqrt(1 + cos(c + d*x)))],
[sec(c + d*x)^(1/2)/sqrt(1 + cos(c + d*x)), x, 2, (sqrt(2)*arcsin(tan((c + d*x)/2))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/d],
[1/(sqrt(1 + cos(c + d*x))*sec(c + d*x)^(1/2)), x, 5, -((sqrt(2)*arcsin(tan((1/2)*(c + d*x)))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/d) + (2*arctan(sin(c + d*x)/(sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/d],
[1/(sqrt(1 + cos(c + d*x))*sec(c + d*x)^(3/2)), x, 5, (sqrt(2)*arcsin(tan((1/2)*(c + d*x)))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/d - (arctan(sin(c + d*x)/(sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/d + sin(c + d*x)/(d*sqrt(1 + cos(c + d*x))*sqrt(sec(c + d*x)))],


[sec(c + d*x)^(7/2)/sqrt(a + a*cos(c + d*x)), x, 5, -((sqrt(2)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(sqrt(a)*d)) + (26*sqrt(sec(c + d*x))*sin(c + d*x))/(15*d*sqrt(a + a*cos(c + d*x))) - (2*sec(c + d*x)^(3/2)*sin(c + d*x))/(15*d*sqrt(a + a*cos(c + d*x))) + (2*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d*sqrt(a + a*cos(c + d*x)))],
[sec(c + d*x)^(5/2)/sqrt(a + a*cos(c + d*x)), x, 4, (sqrt(2)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(sqrt(a)*d) - (2*sqrt(sec(c + d*x))*sin(c + d*x))/(3*d*sqrt(a + a*cos(c + d*x))) + (2*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d*sqrt(a + a*cos(c + d*x)))],
[sec(c + d*x)^(3/2)/sqrt(a + a*cos(c + d*x)), x, 3, -((sqrt(2)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(sqrt(a)*d)) + (2*sqrt(sec(c + d*x))*sin(c + d*x))/(d*sqrt(a + a*cos(c + d*x)))],
[sec(c + d*x)^(1/2)/sqrt(a + a*cos(c + d*x)), x, 2, (sqrt(2)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(sqrt(a)*d)],
[1/(sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(1/2)), x, 5, (2*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(sqrt(a)*d) - (sqrt(2)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(sqrt(a)*d)],
[1/(sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(3/2)), x, 5, -((arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(sqrt(a)*d)) + (sqrt(2)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(sqrt(a)*d) + sin(c + d*x)/(d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x)))],


[sec(c + d*x)^(5/2)/(a + a*cos(c + d*x))^(3/2), x, 5, (11*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(2*sqrt(2)*a^(3/2)*d) - (19*sqrt(sec(c + d*x))*sin(c + d*x))/(6*a*d*sqrt(a + a*cos(c + d*x))) - (sec(c + d*x)^(3/2)*sin(c + d*x))/(2*d*(a + a*cos(c + d*x))^(3/2)) + (7*sec(c + d*x)^(3/2)*sin(c + d*x))/(6*a*d*sqrt(a + a*cos(c + d*x)))],
[sec(c + d*x)^(3/2)/(a + a*cos(c + d*x))^(3/2), x, 4, -((7*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(2*sqrt(2)*a^(3/2)*d)) - (sqrt(sec(c + d*x))*sin(c + d*x))/(2*d*(a + a*cos(c + d*x))^(3/2)) + (5*sqrt(sec(c + d*x))*sin(c + d*x))/(2*a*d*sqrt(a + a*cos(c + d*x)))],
[sec(c + d*x)^(1/2)/(a + a*cos(c + d*x))^(3/2), x, 3, (3*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(2*sqrt(2)*a^(3/2)*d) - sin(c + d*x)/(2*d*(a + a*cos(c + d*x))^(3/2)*sqrt(sec(c + d*x)))],
[1/((a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^(1/2)), x, 3, (arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(2*sqrt(2)*a^(3/2)*d) + sin(c + d*x)/(2*d*(a + a*cos(c + d*x))^(3/2)*sqrt(sec(c + d*x)))],
[1/((a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^(3/2)), x, 5, (2*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(a^(3/2)*d) - (5*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(2*sqrt(2)*a^(3/2)*d) - sin(c + d*x)/(2*d*(a + a*cos(c + d*x))^(3/2)*sqrt(sec(c + d*x)))],
[1/((a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^(5/2)), x, 6, -((3*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(a^(3/2)*d)) + (9*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(2*sqrt(2)*a^(3/2)*d) - sin(c + d*x)/(2*d*(a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^(3/2)) + (3*sin(c + d*x))/(2*a*d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x)))],


[sec(c + d*x)^(5/2)/(a + a*cos(c + d*x))^(5/2), x, 6, (163*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(16*sqrt(2)*a^(5/2)*d) - (299*sqrt(sec(c + d*x))*sin(c + d*x))/(48*a^2*d*sqrt(a + a*cos(c + d*x))) - (sec(c + d*x)^(3/2)*sin(c + d*x))/(4*d*(a + a*cos(c + d*x))^(5/2)) - (17*sec(c + d*x)^(3/2)*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2)) + (95*sec(c + d*x)^(3/2)*sin(c + d*x))/(48*a^2*d*sqrt(a + a*cos(c + d*x)))],
[sec(c + d*x)^(3/2)/(a + a*cos(c + d*x))^(5/2), x, 5, -((75*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(16*sqrt(2)*a^(5/2)*d)) - (sqrt(sec(c + d*x))*sin(c + d*x))/(4*d*(a + a*cos(c + d*x))^(5/2)) - (13*sqrt(sec(c + d*x))*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2)) + (49*sqrt(sec(c + d*x))*sin(c + d*x))/(16*a^2*d*sqrt(a + a*cos(c + d*x)))],
[sec(c + d*x)^(1/2)/(a + a*cos(c + d*x))^(5/2), x, 4, (19*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(16*sqrt(2)*a^(5/2)*d) - sin(c + d*x)/(4*d*(a + a*cos(c + d*x))^(5/2)*sqrt(sec(c + d*x))) - (9*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2)*sqrt(sec(c + d*x)))],
[1/((a + a*cos(c + d*x))^(5/2)*sec(c + d*x)^(1/2)), x, 4, (5*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(16*sqrt(2)*a^(5/2)*d) - sin(c + d*x)/(4*d*(a + a*cos(c + d*x))^(5/2)*sec(c + d*x)^(3/2)) + (5*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2)*sqrt(sec(c + d*x)))],
[1/((a + a*cos(c + d*x))^(5/2)*sec(c + d*x)^(3/2)), x, 4, (3*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(16*sqrt(2)*a^(5/2)*d) + sin(c + d*x)/(4*d*(a + a*cos(c + d*x))^(5/2)*sec(c + d*x)^(3/2)) + (3*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2)*sqrt(sec(c + d*x)))],
[1/((a + a*cos(c + d*x))^(5/2)*sec(c + d*x)^(5/2)), x, 6, (2*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(a^(5/2)*d) - (43*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(16*sqrt(2)*a^(5/2)*d) - sin(c + d*x)/(4*d*(a + a*cos(c + d*x))^(5/2)*sec(c + d*x)^(3/2)) - (11*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2)*sqrt(sec(c + d*x)))],
[1/((a + a*cos(c + d*x))^(5/2)*sec(c + d*x)^(7/2)), x, 7, -((5*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(a^(5/2)*d)) + (115*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(16*sqrt(2)*a^(5/2)*d) - sin(c + d*x)/(4*d*(a + a*cos(c + d*x))^(5/2)*sec(c + d*x)^(5/2)) - (15*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^(3/2)) + (35*sin(c + d*x))/(16*a^2*d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x)))],


[sec(c + d*x)^(5/2)/(a + a*cos(c + d*x))^(7/2), x, 7, (1015*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(64*sqrt(2)*a^(7/2)*d) - (629*sqrt(sec(c + d*x))*sin(c + d*x))/(64*a^3*d*sqrt(a + a*cos(c + d*x))) - (sec(c + d*x)^(3/2)*sin(c + d*x))/(6*d*(a + a*cos(c + d*x))^(7/2)) - (23*sec(c + d*x)^(3/2)*sin(c + d*x))/(48*a*d*(a + a*cos(c + d*x))^(5/2)) - (109*sec(c + d*x)^(3/2)*sin(c + d*x))/(64*a^2*d*(a + a*cos(c + d*x))^(3/2)) + (193*sec(c + d*x)^(3/2)*sin(c + d*x))/(64*a^3*d*sqrt(a + a*cos(c + d*x)))],
[sec(c + d*x)^(3/2)/(a + a*cos(c + d*x))^(7/2), x, 6, -((363*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(64*sqrt(2)*a^(7/2)*d)) - (sqrt(sec(c + d*x))*sin(c + d*x))/(6*d*(a + a*cos(c + d*x))^(7/2)) - (19*sqrt(sec(c + d*x))*sin(c + d*x))/(48*a*d*(a + a*cos(c + d*x))^(5/2)) - (199*sqrt(sec(c + d*x))*sin(c + d*x))/(192*a^2*d*(a + a*cos(c + d*x))^(3/2)) + (691*sqrt(sec(c + d*x))*sin(c + d*x))/(192*a^3*d*sqrt(a + a*cos(c + d*x)))],
[sec(c + d*x)^(1/2)/(a + a*cos(c + d*x))^(7/2), x, 5, (63*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(64*sqrt(2)*a^(7/2)*d) - sin(c + d*x)/(6*d*(a + a*cos(c + d*x))^(7/2)*sqrt(sec(c + d*x))) - (5*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(5/2)*sqrt(sec(c + d*x))) - (103*sin(c + d*x))/(192*a^2*d*(a + a*cos(c + d*x))^(3/2)*sqrt(sec(c + d*x)))],
[1/((a + a*cos(c + d*x))^(7/2)*sec(c + d*x)^(1/2)), x, 5, (13*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(64*sqrt(2)*a^(7/2)*d) + sin(c + d*x)/(6*d*(a + a*cos(c + d*x))^(7/2)*sqrt(sec(c + d*x))) + sin(c + d*x)/(16*a*d*(a + a*cos(c + d*x))^(5/2)*sqrt(sec(c + d*x))) - (5*sin(c + d*x))/(192*a^2*d*(a + a*cos(c + d*x))^(3/2)*sqrt(sec(c + d*x)))],
[1/((a + a*cos(c + d*x))^(7/2)*sec(c + d*x)^(3/2)), x, 5, (7*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(64*sqrt(2)*a^(7/2)*d) - sin(c + d*x)/(6*d*(a + a*cos(c + d*x))^(7/2)*sec(c + d*x)^(5/2)) + (7*sin(c + d*x))/(48*a*d*(a + a*cos(c + d*x))^(5/2)*sec(c + d*x)^(3/2)) + (7*sin(c + d*x))/(64*a^2*d*(a + a*cos(c + d*x))^(3/2)*sqrt(sec(c + d*x)))],
[1/((a + a*cos(c + d*x))^(7/2)*sec(c + d*x)^(5/2)), x, 5, (5*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(64*sqrt(2)*a^(7/2)*d) + sin(c + d*x)/(6*d*(a + a*cos(c + d*x))^(7/2)*sec(c + d*x)^(5/2)) + (5*sin(c + d*x))/(48*a*d*(a + a*cos(c + d*x))^(5/2)*sec(c + d*x)^(3/2)) + (5*sin(c + d*x))/(64*a^2*d*(a + a*cos(c + d*x))^(3/2)*sqrt(sec(c + d*x)))],
[1/((a + a*cos(c + d*x))^(7/2)*sec(c + d*x)^(7/2)), x, 7, (2*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(a^(7/2)*d) - (177*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(64*sqrt(2)*a^(7/2)*d) - sin(c + d*x)/(6*d*(a + a*cos(c + d*x))^(7/2)*sec(c + d*x)^(5/2)) - (17*sin(c + d*x))/(48*a*d*(a + a*cos(c + d*x))^(5/2)*sec(c + d*x)^(3/2)) - (49*sin(c + d*x))/(64*a^2*d*(a + a*cos(c + d*x))^(3/2)*sqrt(sec(c + d*x)))],
[1/((a + a*cos(c + d*x))^(7/2)*sec(c + d*x)^(9/2)), x, 8, -((7*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(a^(7/2)*d)) + (637*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(64*sqrt(2)*a^(7/2)*d) - sin(c + d*x)/(6*d*(a + a*cos(c + d*x))^(7/2)*sec(c + d*x)^(7/2)) - (7*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(5/2)*sec(c + d*x)^(5/2)) - (259*sin(c + d*x))/(192*a^2*d*(a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^(3/2)) + (189*sin(c + d*x))/(64*a^3*d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x)))],


[1/((a + a*cos(c + d*x))^(9/2)*sec(c + d*x)^(5/2)), x, 6, (45*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(1024*sqrt(2)*a^(9/2)*d) - sin(c + d*x)/(8*d*(a + a*cos(c + d*x))^(9/2)*sec(c + d*x)^(7/2)) + (3*sin(c + d*x))/(32*a*d*(a + a*cos(c + d*x))^(7/2)*sec(c + d*x)^(5/2)) + (15*sin(c + d*x))/(256*a^2*d*(a + a*cos(c + d*x))^(5/2)*sec(c + d*x)^(3/2)) + (45*sin(c + d*x))/(1024*a^3*d*(a + a*cos(c + d*x))^(3/2)*sqrt(sec(c + d*x)))],
[1/((a + a*cos(c + d*x))^(9/2)*sec(c + d*x)^(7/2)), x, 6, (35*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(1024*sqrt(2)*a^(9/2)*d) + sin(c + d*x)/(8*d*(a + a*cos(c + d*x))^(9/2)*sec(c + d*x)^(7/2)) + (7*sin(c + d*x))/(96*a*d*(a + a*cos(c + d*x))^(7/2)*sec(c + d*x)^(5/2)) + (35*sin(c + d*x))/(768*a^2*d*(a + a*cos(c + d*x))^(5/2)*sec(c + d*x)^(3/2)) + (35*sin(c + d*x))/(1024*a^3*d*(a + a*cos(c + d*x))^(3/2)*sqrt(sec(c + d*x)))],


# ::Subsection::Closed:: 
#Integrands of the form Sec[c+d x]^(m/4) / (a+a Cos[c+d x])^(n/2)


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


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


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


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


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


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


[cos(c + d*x)^3*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^2, x, 6, (1/16)*a^2*(12*A + 11*B)*x + (2*a^2*(9*A + 8*B)*sin(c + d*x))/(15*d) + (a^2*(12*A + 11*B)*cos(c + d*x)*sin(c + d*x))/(16*d) + (a^2*(9*A + 8*B)*cos(c + d*x)^2*sin(c + d*x))/(15*d) + (a^2*(12*A + 11*B)*cos(c + d*x)^3*sin(c + d*x))/(24*d) + (a^2*(6*A + 7*B)*cos(c + d*x)^4*sin(c + d*x))/(30*d) + (a*B*cos(c + d*x)^4*(a + a*cos(c + d*x))*sin(c + d*x))/(6*d)],
[cos(c + d*x)^2*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^2, x, 5, (1/8)*a^2*(7*A + 6*B)*x + (2*a^2*(10*A + 9*B)*sin(c + d*x))/(15*d) + (a^2*(7*A + 6*B)*cos(c + d*x)*sin(c + d*x))/(8*d) + (a^2*(10*A + 9*B)*cos(c + d*x)^2*sin(c + d*x))/(15*d) + (a^2*(5*A + 6*B)*cos(c + d*x)^3*sin(c + d*x))/(20*d) + (a*B*cos(c + d*x)^3*(a + a*cos(c + d*x))*sin(c + d*x))/(5*d)],
[cos(c + d*x)^1*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^2, x, 4, (1/8)*a^2*(8*A + 7*B)*x + (a^2*(8*A + 7*B)*sin(c + d*x))/(6*d) + (a^2*(8*A + 7*B)*cos(c + d*x)*sin(c + d*x))/(24*d) + ((4*A - B)*(a + a*cos(c + d*x))^2*sin(c + d*x))/(12*d) + (B*(a + a*cos(c + d*x))^3*sin(c + d*x))/(4*a*d)],
[cos(c + d*x)^0*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^2, x, 3, (1/2)*a^2*(3*A + 2*B)*x + (2*a^2*(3*A + 2*B)*sin(c + d*x))/(3*d) + (a^2*(3*A + 2*B)*cos(c + d*x)*sin(c + d*x))/(6*d) + (B*(a + a*cos(c + d*x))^2*sin(c + d*x))/(3*d)],
[sec(c + d*x)^1*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^2, x, 4, (1/2)*a^2*(4*A + 3*B)*x + (a^2*A*arctanh(sin(c + d*x)))/d + (a^2*(2*A + 3*B)*sin(c + d*x))/(2*d) + (a*B*(a + a*cos(c + d*x))*sin(c + d*x))/(2*d)],
[sec(c + d*x)^2*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^2, x, 4, (a^2*A + 2*a^2*B)*x + (a^2*(2*A + B)*arctanh(sin(c + d*x)))/d - (a^2*(A - B)*sin(c + d*x))/d + (a*A*(a + a*cos(c + d*x))*tan(c + d*x))/d],
[sec(c + d*x)^3*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^2, x, 4, a^2*B*x + (a^2*(3*A + 4*B)*arctanh(sin(c + d*x)))/(2*d) + (a^2*(3*A + 2*B)*tan(c + d*x))/(2*d) + (a*A*(a + a*sec(c + d*x))*tan(c + d*x))/(2*d)],
[sec(c + d*x)^4*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^2, x, 5, (a^2*(2*A + 3*B)*arctanh(sin(c + d*x)))/(2*d) + (2*a^2*(2*A + 3*B)*tan(c + d*x))/(3*d) + (a^2*(2*A + 3*B)*sec(c + d*x)*tan(c + d*x))/(6*d) + (A*(a + a*sec(c + d*x))^2*tan(c + d*x))/(3*d)],
[sec(c + d*x)^5*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^2, x, 6, (a^2*(7*A + 8*B)*arctanh(sin(c + d*x)))/(8*d) + (a^2*(4*A + 5*B)*tan(c + d*x))/(3*d) + (a^2*(7*A + 8*B)*sec(c + d*x)*tan(c + d*x))/(8*d) + (a^2*(5*A + 4*B)*sec(c + d*x)^2*tan(c + d*x))/(12*d) + (a*A*(a + a*cos(c + d*x))*sec(c + d*x)^3*tan(c + d*x))/(4*d)],


[cos(c + d*x)^2*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^3, x, 6, (1/16)*a^3*(26*A + 23*B)*x + (2*a^3*(19*A + 17*B)*sin(c + d*x))/(15*d) + (a^3*(26*A + 23*B)*cos(c + d*x)*sin(c + d*x))/(16*d) + (a^3*(19*A + 17*B)*cos(c + d*x)^2*sin(c + d*x))/(15*d) + (a^3*(22*A + 21*B)*cos(c + d*x)^3*sin(c + d*x))/(40*d) + (a^2*(3*A + 4*B)*cos(c + d*x)^3*(a + a*cos(c + d*x))*sin(c + d*x))/(15*d) + (a*B*cos(c + d*x)^3*(a + a*cos(c + d*x))^2*sin(c + d*x))/(6*d)],
[cos(c + d*x)^1*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^3, x, 5, (1/8)*a^3*(15*A + 13*B)*x + (a^3*(15*A + 13*B)*sin(c + d*x))/(6*d) + (a^3*(15*A + 13*B)*cos(c + d*x)*sin(c + d*x))/(24*d) + (a*(15*A + 13*B)*(a + a*cos(c + d*x))^2*sin(c + d*x))/(60*d) + ((5*A - B)*(a + a*cos(c + d*x))^3*sin(c + d*x))/(20*d) + (B*(a + a*cos(c + d*x))^4*sin(c + d*x))/(5*a*d)],
[cos(c + d*x)^0*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^3, x, 4, (5/8)*a^3*(4*A + 3*B)*x + (5*a^3*(4*A + 3*B)*sin(c + d*x))/(6*d) + (5*a^3*(4*A + 3*B)*cos(c + d*x)*sin(c + d*x))/(24*d) + (a*(4*A + 3*B)*(a + a*cos(c + d*x))^2*sin(c + d*x))/(12*d) + (B*(a + a*cos(c + d*x))^3*sin(c + d*x))/(4*d)],
[sec(c + d*x)^1*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^3, x, 5, (1/2)*a^3*(7*A + 5*B)*x + (a^3*A*arctanh(sin(c + d*x)))/d + (5*a^3*(A + B)*sin(c + d*x))/(2*d) + (a^2*(3*A + 5*B)*(a + a*cos(c + d*x))*sin(c + d*x))/(6*d) + (a*B*(a + a*cos(c + d*x))^2*sin(c + d*x))/(3*d)],
[sec(c + d*x)^2*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^3, x, 5, (1/2)*a^3*(6*A + 7*B)*x + (a^3*(3*A + B)*arctanh(sin(c + d*x)))/d + (5*a^3*B*sin(c + d*x))/(2*d) - (a^2*(2*A - B)*(a + a*cos(c + d*x))*sin(c + d*x))/(2*d) + (a*A*(a + a*cos(c + d*x))^2*tan(c + d*x))/d],
[sec(c + d*x)^3*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^3, x, 5, (a^3*A + 3*a^3*B)*x + (a^3*(7*A + 6*B)*arctanh(sin(c + d*x)))/(2*d) - (5*a^3*A*sin(c + d*x))/(2*d) + (a^2*(2*A + B)*(a + a*cos(c + d*x))*tan(c + d*x))/d + (a*A*(a + a*cos(c + d*x))^2*sec(c + d*x)*tan(c + d*x))/(2*d)],
[sec(c + d*x)^4*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^3, x, 5, a^3*B*x + (a^3*(5*A + 7*B)*arctanh(sin(c + d*x)))/(2*d) + (5*a^3*(A + B)*tan(c + d*x))/(2*d) + (a^2*(5*A + 3*B)*(a + a*sec(c + d*x))*tan(c + d*x))/(6*d) + (a*A*(a + a*sec(c + d*x))^2*tan(c + d*x))/(3*d)],
[sec(c + d*x)^5*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^3, x, 6, (5*a^3*(3*A + 4*B)*arctanh(sin(c + d*x)))/(8*d) + (5*a^3*(3*A + 4*B)*tan(c + d*x))/(6*d) + (5*a^3*(3*A + 4*B)*sec(c + d*x)*tan(c + d*x))/(24*d) + (a*(3*A + 4*B)*(a + a*sec(c + d*x))^2*tan(c + d*x))/(12*d) + (A*(a + a*sec(c + d*x))^3*tan(c + d*x))/(4*d)],
[sec(c + d*x)^6*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^3, x, 7, (a^3*(13*A + 15*B)*arctanh(sin(c + d*x)))/(8*d) + (a^3*(38*A + 45*B)*tan(c + d*x))/(15*d) + (a^3*(13*A + 15*B)*sec(c + d*x)*tan(c + d*x))/(8*d) + (a^3*(43*A + 45*B)*sec(c + d*x)^2*tan(c + d*x))/(60*d) + (a^2*(7*A + 5*B)*(a + a*cos(c + d*x))*sec(c + d*x)^3*tan(c + d*x))/(20*d) + (a*A*(a + a*cos(c + d*x))^2*sec(c + d*x)^4*tan(c + d*x))/(5*d)],


[cos(c + d*x)^2*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^4, x, 7, (1/16)*a^4*(49*A + 44*B)*x + (2*a^4*(252*A + 227*B)*sin(c + d*x))/(105*d) + (a^4*(49*A + 44*B)*cos(c + d*x)*sin(c + d*x))/(16*d) + (a^4*(252*A + 227*B)*cos(c + d*x)^2*sin(c + d*x))/(105*d) + (a^4*(301*A + 276*B)*cos(c + d*x)^3*sin(c + d*x))/(280*d) + (7*a^3*(A + B)*cos(c + d*x)^3*(a + a*cos(c + d*x))*sin(c + d*x))/(15*d) + (a^2*(7*A + 10*B)*cos(c + d*x)^3*(a + a*cos(c + d*x))^2*sin(c + d*x))/(42*d) + (a*B*cos(c + d*x)^3*(a + a*cos(c + d*x))^3*sin(c + d*x))/(7*d)],
[cos(c + d*x)^1*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^4, x, 6, (7/16)*a^4*(8*A + 7*B)*x + (7*a^4*(8*A + 7*B)*sin(c + d*x))/(12*d) + (7*a^4*(8*A + 7*B)*cos(c + d*x)*sin(c + d*x))/(48*d) + (7*a^2*(8*A + 7*B)*(a + a*cos(c + d*x))^2*sin(c + d*x))/(120*d) + (a*(8*A + 7*B)*(a + a*cos(c + d*x))^3*sin(c + d*x))/(40*d) + ((6*A - B)*(a + a*cos(c + d*x))^4*sin(c + d*x))/(30*d) + (B*(a + a*cos(c + d*x))^5*sin(c + d*x))/(6*a*d)],
[cos(c + d*x)^0*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^4, x, 5, (7/8)*a^4*(5*A + 4*B)*x + (7*a^4*(5*A + 4*B)*sin(c + d*x))/(6*d) + (7*a^4*(5*A + 4*B)*cos(c + d*x)*sin(c + d*x))/(24*d) + (7*a^2*(5*A + 4*B)*(a + a*cos(c + d*x))^2*sin(c + d*x))/(60*d) + (a*(5*A + 4*B)*(a + a*cos(c + d*x))^3*sin(c + d*x))/(20*d) + (B*(a + a*cos(c + d*x))^4*sin(c + d*x))/(5*d)],
[sec(c + d*x)^1*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^4, x, 6, (1/8)*a^4*(48*A + 35*B)*x + (a^4*A*arctanh(sin(c + d*x)))/d + (5*a^4*(8*A + 7*B)*sin(c + d*x))/(8*d) + (a^3*(32*A + 35*B)*(a + a*cos(c + d*x))*sin(c + d*x))/(24*d) + (a^2*(4*A + 7*B)*(a + a*cos(c + d*x))^2*sin(c + d*x))/(12*d) + (a*B*(a + a*cos(c + d*x))^3*sin(c + d*x))/(4*d)],
[sec(c + d*x)^2*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^4, x, 6, (1/2)*a^4*(13*A + 12*B)*x + (a^4*(4*A + B)*arctanh(sin(c + d*x)))/d + (5*a^4*(A + 2*B)*sin(c + d*x))/(2*d) - (a^3*(3*A - 8*B)*(a + a*cos(c + d*x))*sin(c + d*x))/(6*d) - (a^2*(3*A - B)*(a + a*cos(c + d*x))^2*sin(c + d*x))/(3*d) + (a*A*(a + a*cos(c + d*x))^3*tan(c + d*x))/d],
[sec(c + d*x)^3*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^4, x, 6, (1/2)*a^4*(8*A + 13*B)*x + (a^4*(13*A + 8*B)*arctanh(sin(c + d*x)))/(2*d) - (5*a^4*(A - B)*sin(c + d*x))/(2*d) - (a^3*(6*A + B)*(a + a*cos(c + d*x))*sin(c + d*x))/(2*d) + (a^2*(5*A + 2*B)*(a + a*cos(c + d*x))^2*tan(c + d*x))/(2*d) + (a*A*(a + a*cos(c + d*x))^3*sec(c + d*x)*tan(c + d*x))/(2*d)],
[sec(c + d*x)^4*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^4, x, 6, (a^4*A + 4*a^4*B)*x + (a^4*(12*A + 13*B)*arctanh(sin(c + d*x)))/(2*d) - (5*a^4*(2*A + B)*sin(c + d*x))/(2*d) + (a^3*(11*A + 9*B)*(a + a*cos(c + d*x))*tan(c + d*x))/(3*d) + (a^2*(2*A + B)*(a + a*cos(c + d*x))^2*sec(c + d*x)*tan(c + d*x))/(2*d) + (a*A*(a + a*cos(c + d*x))^3*sec(c + d*x)^2*tan(c + d*x))/(3*d)],
[sec(c + d*x)^5*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^4, x, 6, a^4*B*x + (a^4*(35*A + 48*B)*arctanh(sin(c + d*x)))/(8*d) + (5*a^4*(7*A + 8*B)*tan(c + d*x))/(8*d) + (a^3*(35*A + 32*B)*(a + a*sec(c + d*x))*tan(c + d*x))/(24*d) + (a^2*(7*A + 4*B)*(a + a*sec(c + d*x))^2*tan(c + d*x))/(12*d) + (a*A*(a + a*sec(c + d*x))^3*tan(c + d*x))/(4*d)],
[sec(c + d*x)^6*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^4, x, 7, (7*a^4*(4*A + 5*B)*arctanh(sin(c + d*x)))/(8*d) + (7*a^4*(4*A + 5*B)*tan(c + d*x))/(6*d) + (7*a^4*(4*A + 5*B)*sec(c + d*x)*tan(c + d*x))/(24*d) + (7*a^2*(4*A + 5*B)*(a + a*sec(c + d*x))^2*tan(c + d*x))/(60*d) + (a*(4*A + 5*B)*(a + a*sec(c + d*x))^3*tan(c + d*x))/(20*d) + (A*(a + a*sec(c + d*x))^4*tan(c + d*x))/(5*d)],
[sec(c + d*x)^7*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^4, x, 8, (7*a^4*(7*A + 8*B)*arctanh(sin(c + d*x)))/(16*d) + (a^4*(72*A + 83*B)*tan(c + d*x))/(15*d) + (7*a^4*(7*A + 8*B)*sec(c + d*x)*tan(c + d*x))/(16*d) + (a^4*(159*A + 176*B)*sec(c + d*x)^2*tan(c + d*x))/(120*d) + (a^3*(73*A + 72*B)*(a + a*cos(c + d*x))*sec(c + d*x)^3*tan(c + d*x))/(120*d) + (a^2*(3*A + 2*B)*(a + a*cos(c + d*x))^2*sec(c + d*x)^4*tan(c + d*x))/(10*d) + (a*A*(a + a*cos(c + d*x))^3*sec(c + d*x)^5*tan(c + d*x))/(6*d)],


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


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


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


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


[cos(c + d*x)^5*(A + B*cos(c + d*x))/(a + a*cos(c + d*x))^3, x, 6, ((13*A - 23*B)*x)/(2*a^3) - (8*(19*A - 34*B)*sin(c + d*x))/(15*a^3*d) + ((13*A - 23*B)*cos(c + d*x)*sin(c + d*x))/(2*a^3*d) - (4*(19*A - 34*B)*cos(c + d*x)^2*sin(c + d*x))/(15*a^3*d) + ((A - B)*cos(c + d*x)^5*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3) + ((8*A - 13*B)*cos(c + d*x)^4*sin(c + d*x))/(15*a*d*(a + a*cos(c + d*x))^2) + ((13*A - 23*B)*cos(c + d*x)^3*sin(c + d*x))/(3*a^2*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^4*(A + B*cos(c + d*x))/(a + a*cos(c + d*x))^3, x, 5, -(((6*A - 13*B)*x)/(2*a^3)) + (8*(9*A - 19*B)*sin(c + d*x))/(15*a^3*d) - ((6*A - 13*B)*cos(c + d*x)*sin(c + d*x))/(2*a^3*d) + ((A - B)*cos(c + d*x)^4*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3) + ((6*A - 11*B)*cos(c + d*x)^3*sin(c + d*x))/(15*a*d*(a + a*cos(c + d*x))^2) + (4*(9*A - 19*B)*cos(c + d*x)^2*sin(c + d*x))/(15*a^2*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^3*(A + B*cos(c + d*x))/(a + a*cos(c + d*x))^3, x, 5, ((A - 3*B)*x)/a^3 - ((7*A - 27*B)*sin(c + d*x))/(15*a^3*d) + ((A - B)*cos(c + d*x)^3*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3) + ((4*A - 9*B)*cos(c + d*x)^2*sin(c + d*x))/(15*a*d*(a + a*cos(c + d*x))^2) - ((A - 3*B)*sin(c + d*x))/(a^2*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^2*(A + B*cos(c + d*x))/(a + a*cos(c + d*x))^3, x, 4, (B*x)/a^3 + ((A - B)*cos(c + d*x)^2*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3) + ((2*A - 7*B)*cos(c + d*x)*sin(c + d*x))/(15*a*d*(a + a*cos(c + d*x))^2) + (2*(A - 11*B)*sin(c + d*x))/(15*a^2*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^1*(A + B*cos(c + d*x))/(a + a*cos(c + d*x))^3, x, 3, ((A - B)*cos(c + d*x)*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3) - (B*sin(c + d*x))/(3*a*d*(a + a*cos(c + d*x))^2) + ((3*A + 7*B)*sin(c + d*x))/(15*a^2*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^0*(A + B*cos(c + d*x))/(a + a*cos(c + d*x))^3, x, 3, ((A - B)*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3) + ((2*A + 3*B)*sin(c + d*x))/(15*a*d*(a + a*cos(c + d*x))^2) + ((2*A + 3*B)*sin(c + d*x))/(15*a^2*d*(a + a*cos(c + d*x)))],
[sec(c + d*x)^1*(A + B*cos(c + d*x))/(a + a*cos(c + d*x))^3, x, 4, (A*arctanh(sin(c + d*x)))/(a^3*d) - ((A - B)*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3) - ((7*A - 2*B)*sin(c + d*x))/(15*a*d*(a + a*cos(c + d*x))^2) - (2*(11*A - B)*sin(c + d*x))/(15*a^2*d*(a + a*cos(c + d*x)))],
[sec(c + d*x)^2*(A + B*cos(c + d*x))/(a + a*cos(c + d*x))^3, x, 6, -(((3*A - B)*arctanh(sin(c + d*x)))/(a^3*d)) + (2*(36*A - 11*B)*tan(c + d*x))/(15*a^3*d) - ((A - B)*tan(c + d*x))/(5*d*(a + a*cos(c + d*x))^3) - ((9*A - 4*B)*tan(c + d*x))/(15*a*d*(a + a*cos(c + d*x))^2) - ((3*A - B)*tan(c + d*x))/(a^2*d*(a + a*cos(c + d*x)))],
[sec(c + d*x)^3*(A + B*cos(c + d*x))/(a + a*cos(c + d*x))^3, x, 7, ((13*A - 6*B)*arctanh(sin(c + d*x)))/(2*a^3*d) - (8*(19*A - 9*B)*tan(c + d*x))/(15*a^3*d) + ((13*A - 6*B)*sec(c + d*x)*tan(c + d*x))/(2*a^3*d) - ((A - B)*sec(c + d*x)*tan(c + d*x))/(5*d*(a + a*cos(c + d*x))^3) - ((11*A - 6*B)*sec(c + d*x)*tan(c + d*x))/(15*a*d*(a + a*cos(c + d*x))^2) - (4*(19*A - 9*B)*sec(c + d*x)*tan(c + d*x))/(15*a^2*d*(a + a*cos(c + d*x)))],


[cos(c + d*x)^5*(A + B*cos(c + d*x))/(a + a*cos(c + d*x))^4, x, 6, -(((8*A - 21*B)*x)/(2*a^4)) + (8*(83*A - 216*B)*sin(c + d*x))/(105*a^4*d) - ((8*A - 21*B)*cos(c + d*x)*sin(c + d*x))/(2*a^4*d) + ((A - B)*cos(c + d*x)^5*sin(c + d*x))/(7*d*(a + a*cos(c + d*x))^4) + ((A - 2*B)*cos(c + d*x)^4*sin(c + d*x))/(5*a*d*(a + a*cos(c + d*x))^3) + ((52*A - 129*B)*cos(c + d*x)^3*sin(c + d*x))/(105*a^2*d*(a + a*cos(c + d*x))^2) + (4*(83*A - 216*B)*cos(c + d*x)^2*sin(c + d*x))/(105*a^3*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^4*(A + B*cos(c + d*x))/(a + a*cos(c + d*x))^4, x, 6, ((A - 4*B)*x)/a^4 - ((55*A - 244*B)*sin(c + d*x))/(105*a^4*d) + ((A - B)*cos(c + d*x)^4*sin(c + d*x))/(7*d*(a + a*cos(c + d*x))^4) + ((5*A - 12*B)*cos(c + d*x)^3*sin(c + d*x))/(35*a*d*(a + a*cos(c + d*x))^3) + ((25*A - 88*B)*cos(c + d*x)^2*sin(c + d*x))/(105*a^2*d*(a + a*cos(c + d*x))^2) - ((A - 4*B)*sin(c + d*x))/(a^3*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^3*(A + B*cos(c + d*x))/(a + a*cos(c + d*x))^4, x, 5, (B*x)/a^4 + ((A - B)*cos(c + d*x)^3*sin(c + d*x))/(7*d*(a + a*cos(c + d*x))^4) + ((3*A - 10*B)*cos(c + d*x)^2*sin(c + d*x))/(35*a*d*(a + a*cos(c + d*x))^3) + ((6*A - 55*B)*cos(c + d*x)*sin(c + d*x))/(105*a^2*d*(a + a*cos(c + d*x))^2) + (2*(3*A - 80*B)*sin(c + d*x))/(105*a^3*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^2*(A + B*cos(c + d*x))/(a + a*cos(c + d*x))^4, x, 4, -(((A - B)*cos(c + d*x)^3*sin(c + d*x))/(7*d*(a + a*cos(c + d*x))^4)) + ((4*A + 3*B)*sin(c + d*x))/(35*a*d*(a + a*cos(c + d*x))^3) - (8*(4*A + 3*B)*sin(c + d*x))/(105*a^2*d*(a + a*cos(c + d*x))^2) + ((4*A + 3*B)*sin(c + d*x))/(15*a^3*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^1*(A + B*cos(c + d*x))/(a + a*cos(c + d*x))^4, x, 4, ((A - B)*cos(c + d*x)*sin(c + d*x))/(7*d*(a + a*cos(c + d*x))^4) - ((A + 6*B)*sin(c + d*x))/(35*a*d*(a + a*cos(c + d*x))^3) + ((8*A + 13*B)*sin(c + d*x))/(105*a^2*d*(a + a*cos(c + d*x))^2) + ((8*A + 13*B)*sin(c + d*x))/(105*a^3*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^0*(A + B*cos(c + d*x))/(a + a*cos(c + d*x))^4, x, 4, ((A - B)*sin(c + d*x))/(7*d*(a + a*cos(c + d*x))^4) + ((3*A + 4*B)*sin(c + d*x))/(35*a*d*(a + a*cos(c + d*x))^3) + (2*(3*A + 4*B)*sin(c + d*x))/(105*a^2*d*(a + a*cos(c + d*x))^2) + (2*(3*A + 4*B)*sin(c + d*x))/(105*a^3*d*(a + a*cos(c + d*x)))],
[sec(c + d*x)^1*(A + B*cos(c + d*x))/(a + a*cos(c + d*x))^4, x, 5, (A*arctanh(sin(c + d*x)))/(a^4*d) - ((A - B)*sin(c + d*x))/(7*d*(a + a*cos(c + d*x))^4) - ((10*A - 3*B)*sin(c + d*x))/(35*a*d*(a + a*cos(c + d*x))^3) - ((55*A - 6*B)*sin(c + d*x))/(105*a^2*d*(a + a*cos(c + d*x))^2) - (2*(80*A - 3*B)*sin(c + d*x))/(105*a^3*d*(a + a*cos(c + d*x)))],
[sec(c + d*x)^2*(A + B*cos(c + d*x))/(a + a*cos(c + d*x))^4, x, 7, -(((4*A - B)*arctanh(sin(c + d*x)))/(a^4*d)) + (8*(83*A - 20*B)*tan(c + d*x))/(105*a^4*d) - ((A - B)*tan(c + d*x))/(7*d*(a + a*cos(c + d*x))^4) - ((12*A - 5*B)*tan(c + d*x))/(35*a*d*(a + a*cos(c + d*x))^3) - ((88*A - 25*B)*tan(c + d*x))/(105*a^2*d*(a + a*cos(c + d*x))^2) - ((4*A - B)*tan(c + d*x))/(a^3*d*(a + a*cos(c + d*x)))],
[sec(c + d*x)^3*(A + B*cos(c + d*x))/(a + a*cos(c + d*x))^4, x, 8, ((21*A - 8*B)*arctanh(sin(c + d*x)))/(2*a^4*d) - (8*(216*A - 83*B)*tan(c + d*x))/(105*a^4*d) + ((21*A - 8*B)*sec(c + d*x)*tan(c + d*x))/(2*a^4*d) - ((A - B)*sec(c + d*x)*tan(c + d*x))/(7*d*(a + a*cos(c + d*x))^4) - ((2*A - B)*sec(c + d*x)*tan(c + d*x))/(5*a*d*(a + a*cos(c + d*x))^3) - ((129*A - 52*B)*sec(c + d*x)*tan(c + d*x))/(105*a^2*d*(a + a*cos(c + d*x))^2) - (4*(216*A - 83*B)*sec(c + d*x)*tan(c + d*x))/(105*a^3*d*(a + a*cos(c + d*x)))],


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


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


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


[cos(c + d*x)^3*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^(1/2), x, 5, (4*a*(9*A + 8*B)*sin(c + d*x))/(45*d*sqrt(a + a*cos(c + d*x))) + (2*a*(9*A + 8*B)*cos(c + d*x)^3*sin(c + d*x))/(63*d*sqrt(a + a*cos(c + d*x))) + (2*a*B*cos(c + d*x)^4*sin(c + d*x))/(9*d*sqrt(a + a*cos(c + d*x))) - (8*(9*A + 8*B)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(315*d) + (4*(9*A + 8*B)*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(105*a*d)],
[cos(c + d*x)^2*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^(1/2), x, 4, (2*a*(7*A + 6*B)*sin(c + d*x))/(15*d*sqrt(a + a*cos(c + d*x))) + (2*a*B*cos(c + d*x)^3*sin(c + d*x))/(7*d*sqrt(a + a*cos(c + d*x))) - (4*(7*A + 6*B)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(105*d) + (2*(7*A + 6*B)*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(35*a*d)],
[cos(c + d*x)^1*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^(1/2), x, 3, (2*a*(5*A + 7*B)*sin(c + d*x))/(15*d*sqrt(a + a*cos(c + d*x))) + (2*(5*A - 2*B)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(15*d) + (2*B*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(5*a*d)],
[cos(c + d*x)^0*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^(1/2), x, 2, (2*a*(3*A + B)*sin(c + d*x))/(3*d*sqrt(a + a*cos(c + d*x))) + (2*B*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(3*d)],
[sec(c + d*x)^1*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^(1/2), x, 2, (2*sqrt(2)*a*A*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(d*sqrt(a + a*cos(c + d*x))) + (2*a*B*sin(c + d*x))/(d*sqrt(a + a*cos(c + d*x)))],
[sec(c + d*x)^2*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^(1/2), x, 2, (sqrt(2)*a*(A + 2*B)*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(d*sqrt(a + a*cos(c + d*x))) + (a*A*tan(c + d*x))/(d*sqrt(a + a*cos(c + d*x)))],
[sec(c + d*x)^3*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^(1/2), x, 3, (a*(3*A + 4*B)*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(2*sqrt(2)*d*sqrt(a + a*cos(c + d*x))) + (a*(3*A + 4*B)*tan(c + d*x))/(4*d*sqrt(a + a*cos(c + d*x))) + (a*A*sec(c + d*x)*tan(c + d*x))/(2*d*sqrt(a + a*cos(c + d*x)))],
[sec(c + d*x)^4*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^(1/2), x, 4, (a*(5*A + 6*B)*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(4*sqrt(2)*d*sqrt(a + a*cos(c + d*x))) + (a*(5*A + 6*B)*tan(c + d*x))/(8*d*sqrt(a + a*cos(c + d*x))) + (a*(5*A + 6*B)*sec(c + d*x)*tan(c + d*x))/(12*d*sqrt(a + a*cos(c + d*x))) + (a*A*sec(c + d*x)^2*tan(c + d*x))/(3*d*sqrt(a + a*cos(c + d*x)))],


[cos(c + d*x)^3*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^(3/2), x, 6, (4*a^2*(187*A + 168*B)*sin(c + d*x))/(495*d*sqrt(a + a*cos(c + d*x))) + (2*a^2*(187*A + 168*B)*cos(c + d*x)^3*sin(c + d*x))/(693*d*sqrt(a + a*cos(c + d*x))) + (2*a^2*(11*A + 12*B)*cos(c + d*x)^4*sin(c + d*x))/(99*d*sqrt(a + a*cos(c + d*x))) - (8*a*(187*A + 168*B)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(3465*d) + (2*a*B*cos(c + d*x)^4*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(11*d) + (4*(187*A + 168*B)*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(1155*d)],
[cos(c + d*x)^2*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^(3/2), x, 5, (2*a^2*(39*A + 34*B)*sin(c + d*x))/(45*d*sqrt(a + a*cos(c + d*x))) + (2*a^2*(9*A + 10*B)*cos(c + d*x)^3*sin(c + d*x))/(63*d*sqrt(a + a*cos(c + d*x))) - (4*a*(39*A + 34*B)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(315*d) + (2*a*B*cos(c + d*x)^3*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(9*d) + (2*(39*A + 34*B)*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(105*d)],
[cos(c + d*x)^1*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^(3/2), x, 4, (8*a^2*(21*A + 19*B)*sin(c + d*x))/(105*d*sqrt(a + a*cos(c + d*x))) + (2*a*(21*A + 19*B)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(105*d) + (2*(7*A - 2*B)*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(35*d) + (2*B*(a + a*cos(c + d*x))^(5/2)*sin(c + d*x))/(7*a*d)],
[cos(c + d*x)^0*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^(3/2), x, 3, (8*a^2*(5*A + 3*B)*sin(c + d*x))/(15*d*sqrt(a + a*cos(c + d*x))) + (2*a*(5*A + 3*B)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(15*d) + (2*B*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(5*d)],
[sec(c + d*x)^1*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^(3/2), x, 3, (2*sqrt(2)*a^2*A*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(d*sqrt(a + a*cos(c + d*x))) + (2*a^2*(3*A + 4*B)*sin(c + d*x))/(3*d*sqrt(a + a*cos(c + d*x))) + (2*a*B*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(3*d)],
[sec(c + d*x)^2*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^(3/2), x, 3, (sqrt(2)*a^2*(3*A + 2*B)*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(d*sqrt(a + a*cos(c + d*x))) - (a^2*(A - 2*B)*sin(c + d*x))/(d*sqrt(a + a*cos(c + d*x))) + (a*A*sqrt(a + a*cos(c + d*x))*tan(c + d*x))/d],
[sec(c + d*x)^3*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^(3/2), x, 3, (a^2*(7*A + 12*B)*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(2*sqrt(2)*d*sqrt(a + a*cos(c + d*x))) + (a^2*(5*A + 4*B)*tan(c + d*x))/(4*d*sqrt(a + a*cos(c + d*x))) + (a*A*sqrt(a + a*cos(c + d*x))*sec(c + d*x)*tan(c + d*x))/(2*d)],
[sec(c + d*x)^4*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^(3/2), x, 4, (a^2*(11*A + 14*B)*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(4*sqrt(2)*d*sqrt(a + a*cos(c + d*x))) + (a^2*(11*A + 14*B)*tan(c + d*x))/(8*d*sqrt(a + a*cos(c + d*x))) + (a^2*(7*A + 6*B)*sec(c + d*x)*tan(c + d*x))/(12*d*sqrt(a + a*cos(c + d*x))) + (a*A*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^2*tan(c + d*x))/(3*d)],
[sec(c + d*x)^5*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^(3/2), x, 5, (a^2*(75*A + 88*B)*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(32*sqrt(2)*d*sqrt(a + a*cos(c + d*x))) + (a^2*(75*A + 88*B)*tan(c + d*x))/(64*d*sqrt(a + a*cos(c + d*x))) + (a^2*(75*A + 88*B)*sec(c + d*x)*tan(c + d*x))/(96*d*sqrt(a + a*cos(c + d*x))) + (a^2*(9*A + 8*B)*sec(c + d*x)^2*tan(c + d*x))/(24*d*sqrt(a + a*cos(c + d*x))) + (a*A*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^3*tan(c + d*x))/(4*d)],


[cos(c + d*x)^2*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^(5/2), x, 6, (2*a^3*(803*A + 710*B)*sin(c + d*x))/(495*d*sqrt(a + a*cos(c + d*x))) + (2*a^3*(209*A + 194*B)*cos(c + d*x)^3*sin(c + d*x))/(693*d*sqrt(a + a*cos(c + d*x))) - (4*a^2*(803*A + 710*B)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(3465*d) + (2*a^2*(11*A + 14*B)*cos(c + d*x)^3*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(99*d) + (2*a*(803*A + 710*B)*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(1155*d) + (2*a*B*cos(c + d*x)^3*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(11*d)],
[cos(c + d*x)^1*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^(5/2), x, 5, (64*a^3*(15*A + 13*B)*sin(c + d*x))/(315*d*sqrt(a + a*cos(c + d*x))) + (16*a^2*(15*A + 13*B)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(315*d) + (2*a*(15*A + 13*B)*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(105*d) + (2*(9*A - 2*B)*(a + a*cos(c + d*x))^(5/2)*sin(c + d*x))/(63*d) + (2*B*(a + a*cos(c + d*x))^(7/2)*sin(c + d*x))/(9*a*d)],
[cos(c + d*x)^0*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^(5/2), x, 4, (64*a^3*(7*A + 5*B)*sin(c + d*x))/(105*d*sqrt(a + a*cos(c + d*x))) + (16*a^2*(7*A + 5*B)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(105*d) + (2*a*(7*A + 5*B)*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(35*d) + (2*B*(a + a*cos(c + d*x))^(5/2)*sin(c + d*x))/(7*d)],
[sec(c + d*x)^1*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^(5/2), x, 4, (2*sqrt(2)*a^3*A*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(d*sqrt(a + a*cos(c + d*x))) + (2*a^3*(35*A + 32*B)*sin(c + d*x))/(15*d*sqrt(a + a*cos(c + d*x))) + (2*a^2*(5*A + 8*B)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(15*d) + (2*a*B*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(5*d)],
[sec(c + d*x)^2*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^(5/2), x, 4, (sqrt(2)*a^3*(5*A + 2*B)*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(d*sqrt(a + a*cos(c + d*x))) + (a^3*(3*A + 14*B)*sin(c + d*x))/(3*d*sqrt(a + a*cos(c + d*x))) - (a^2*(3*A - 2*B)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(3*d) + (a*A*(a + a*cos(c + d*x))^(3/2)*tan(c + d*x))/d],
[sec(c + d*x)^3*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^(5/2), x, 4, (a^3*(19*A + 20*B)*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(2*sqrt(2)*d*sqrt(a + a*cos(c + d*x))) - (a^3*(9*A - 4*B)*sin(c + d*x))/(4*d*sqrt(a + a*cos(c + d*x))) + (a^2*(7*A + 4*B)*sqrt(a + a*cos(c + d*x))*tan(c + d*x))/(4*d) + (a*A*(a + a*cos(c + d*x))^(3/2)*sec(c + d*x)*tan(c + d*x))/(2*d)],
[sec(c + d*x)^4*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^(5/2), x, 4, (a^3*(25*A + 38*B)*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(4*sqrt(2)*d*sqrt(a + a*cos(c + d*x))) + (a^3*(49*A + 54*B)*tan(c + d*x))/(24*d*sqrt(a + a*cos(c + d*x))) + (a^2*(3*A + 2*B)*sqrt(a + a*cos(c + d*x))*sec(c + d*x)*tan(c + d*x))/(4*d) + (a*A*(a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^2*tan(c + d*x))/(3*d)],
[sec(c + d*x)^5*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^(5/2), x, 5, (a^3*(163*A + 200*B)*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(32*sqrt(2)*d*sqrt(a + a*cos(c + d*x))) + (a^3*(163*A + 200*B)*tan(c + d*x))/(64*d*sqrt(a + a*cos(c + d*x))) + (a^3*(95*A + 104*B)*sec(c + d*x)*tan(c + d*x))/(96*d*sqrt(a + a*cos(c + d*x))) + (a^2*(11*A + 8*B)*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^2*tan(c + d*x))/(24*d) + (a*A*(a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^3*tan(c + d*x))/(4*d)],
[sec(c + d*x)^6*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^(5/2), x, 6, (a^3*(283*A + 326*B)*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(64*sqrt(2)*d*sqrt(a + a*cos(c + d*x))) + (a^3*(283*A + 326*B)*tan(c + d*x))/(128*d*sqrt(a + a*cos(c + d*x))) + (a^3*(283*A + 326*B)*sec(c + d*x)*tan(c + d*x))/(192*d*sqrt(a + a*cos(c + d*x))) + (a^3*(157*A + 170*B)*sec(c + d*x)^2*tan(c + d*x))/(240*d*sqrt(a + a*cos(c + d*x))) + (a^2*(13*A + 10*B)*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^3*tan(c + d*x))/(40*d) + (a*A*(a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^4*tan(c + d*x))/(5*d)],


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


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


[cos(c + d*x)^3*(A + B*cos(c + d*x))/(a + a*cos(c + d*x))^(1/2), x, 5, -((2*(A - B)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(d*sqrt(a + a*cos(c + d*x)))) + (4*(49*A - 37*B)*sin(c + d*x))/(105*d*sqrt(a + a*cos(c + d*x))) + (2*(7*A - B)*cos(c + d*x)^2*sin(c + d*x))/(35*d*sqrt(a + a*cos(c + d*x))) + (2*B*cos(c + d*x)^3*sin(c + d*x))/(7*d*sqrt(a + a*cos(c + d*x))) - (2*(7*A - 31*B)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(105*a*d)],
[cos(c + d*x)^2*(A + B*cos(c + d*x))/(a + a*cos(c + d*x))^(1/2), x, 4, (2*(A - B)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(d*sqrt(a + a*cos(c + d*x))) - (4*(5*A - 7*B)*sin(c + d*x))/(15*d*sqrt(a + a*cos(c + d*x))) + (2*B*cos(c + d*x)^2*sin(c + d*x))/(5*d*sqrt(a + a*cos(c + d*x))) + (2*(5*A - B)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(15*a*d)],
[cos(c + d*x)^1*(A + B*cos(c + d*x))/(a + a*cos(c + d*x))^(1/2), x, 3, -((2*(A - B)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(d*sqrt(a + a*cos(c + d*x)))) + (2*(3*A - 2*B)*sin(c + d*x))/(3*d*sqrt(a + a*cos(c + d*x))) + (2*B*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(3*a*d)],
[cos(c + d*x)^0*(A + B*cos(c + d*x))/(a + a*cos(c + d*x))^(1/2), x, 2, (2*(A - B)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(d*sqrt(a + a*cos(c + d*x))) + (2*B*sin(c + d*x))/(d*sqrt(a + a*cos(c + d*x)))],
[sec(c + d*x)^1*(A + B*cos(c + d*x))/(a + a*cos(c + d*x))^(1/2), x, 3, -((2*(A - B)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(d*sqrt(a + a*cos(c + d*x)))) + (2*sqrt(2)*A*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(d*sqrt(a + a*cos(c + d*x)))],
[sec(c + d*x)^2*(A + B*cos(c + d*x))/(a + a*cos(c + d*x))^(1/2), x, 4, (2*(A - B)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(d*sqrt(a + a*cos(c + d*x))) - (sqrt(2)*(A - 2*B)*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(d*sqrt(a + a*cos(c + d*x))) + (A*tan(c + d*x))/(d*sqrt(a + a*cos(c + d*x)))],
[sec(c + d*x)^3*(A + B*cos(c + d*x))/(a + a*cos(c + d*x))^(1/2), x, 5, -((2*(A - B)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(d*sqrt(a + a*cos(c + d*x)))) + ((7*A - 4*B)*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(2*sqrt(2)*d*sqrt(a + a*cos(c + d*x))) - ((A - 4*B)*tan(c + d*x))/(4*d*sqrt(a + a*cos(c + d*x))) + (A*sec(c + d*x)*tan(c + d*x))/(2*d*sqrt(a + a*cos(c + d*x)))],


[cos(c + d*x)^4*(A + B*cos(c + d*x))/(a + a*cos(c + d*x))^(3/2), x, 6, -(((15*A - 19*B)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(2*a*d*sqrt(a + a*cos(c + d*x)))) + ((A - B)*cos(c + d*x)^4*sin(c + d*x))/(2*d*(a + a*cos(c + d*x))^(3/2)) + ((651*A - 799*B)*sin(c + d*x))/(105*a*d*sqrt(a + a*cos(c + d*x))) + ((63*A - 67*B)*cos(c + d*x)^2*sin(c + d*x))/(70*a*d*sqrt(a + a*cos(c + d*x))) - ((7*A - 11*B)*cos(c + d*x)^3*sin(c + d*x))/(14*a*d*sqrt(a + a*cos(c + d*x))) - ((273*A - 397*B)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(210*a^2*d)],
[cos(c + d*x)^3*(A + B*cos(c + d*x))/(a + a*cos(c + d*x))^(3/2), x, 5, ((11*A - 15*B)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(2*a*d*sqrt(a + a*cos(c + d*x))) + ((A - B)*cos(c + d*x)^3*sin(c + d*x))/(2*d*(a + a*cos(c + d*x))^(3/2)) - ((65*A - 93*B)*sin(c + d*x))/(15*a*d*sqrt(a + a*cos(c + d*x))) - ((5*A - 9*B)*cos(c + d*x)^2*sin(c + d*x))/(10*a*d*sqrt(a + a*cos(c + d*x))) + ((35*A - 39*B)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(30*a^2*d)],
[cos(c + d*x)^2*(A + B*cos(c + d*x))/(a + a*cos(c + d*x))^(3/2), x, 4, -(((7*A - 11*B)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(2*a*d*sqrt(a + a*cos(c + d*x)))) + ((A - B)*cos(c + d*x)^2*sin(c + d*x))/(2*d*(a + a*cos(c + d*x))^(3/2)) + ((9*A - 13*B)*sin(c + d*x))/(3*a*d*sqrt(a + a*cos(c + d*x))) - ((3*A - 7*B)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(6*a^2*d)],
[cos(c + d*x)^1*(A + B*cos(c + d*x))/(a + a*cos(c + d*x))^(3/2), x, 3, ((3*A - 7*B)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(2*a*d*sqrt(a + a*cos(c + d*x))) + ((A - B)*cos(c + d*x)*sin(c + d*x))/(2*d*(a + a*cos(c + d*x))^(3/2)) - ((A - 5*B)*sin(c + d*x))/(2*a*d*sqrt(a + a*cos(c + d*x)))],
[cos(c + d*x)^0*(A + B*cos(c + d*x))/(a + a*cos(c + d*x))^(3/2), x, 2, ((A + 3*B)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(2*a*d*sqrt(a + a*cos(c + d*x))) + ((A - B)*sin(c + d*x))/(2*d*(a + a*cos(c + d*x))^(3/2))],
[sec(c + d*x)^1*(A + B*cos(c + d*x))/(a + a*cos(c + d*x))^(3/2), x, 4, -(((5*A - B)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(2*a*d*sqrt(a + a*cos(c + d*x)))) + (2*sqrt(2)*A*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(a*d*sqrt(a + a*cos(c + d*x))) - ((A - B)*sin(c + d*x))/(2*d*(a + a*cos(c + d*x))^(3/2))],
[sec(c + d*x)^2*(A + B*cos(c + d*x))/(a + a*cos(c + d*x))^(3/2), x, 5, ((9*A - 5*B)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(2*a*d*sqrt(a + a*cos(c + d*x))) - (sqrt(2)*(3*A - 2*B)*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(a*d*sqrt(a + a*cos(c + d*x))) - ((A - B)*tan(c + d*x))/(2*d*(a + a*cos(c + d*x))^(3/2)) + ((3*A - B)*tan(c + d*x))/(2*a*d*sqrt(a + a*cos(c + d*x)))],
[sec(c + d*x)^3*(A + B*cos(c + d*x))/(a + a*cos(c + d*x))^(3/2), x, 6, -(((13*A - 9*B)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(2*a*d*sqrt(a + a*cos(c + d*x)))) + ((19*A - 12*B)*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(2*sqrt(2)*a*d*sqrt(a + a*cos(c + d*x))) - ((7*A - 6*B)*tan(c + d*x))/(4*a*d*sqrt(a + a*cos(c + d*x))) - ((A - B)*sec(c + d*x)*tan(c + d*x))/(2*d*(a + a*cos(c + d*x))^(3/2)) + ((2*A - B)*sec(c + d*x)*tan(c + d*x))/(2*a*d*sqrt(a + a*cos(c + d*x)))],


[cos(c + d*x)^4*(A + B*cos(c + d*x))/(a + a*cos(c + d*x))^(5/2), x, 6, ((163*A - 283*B)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(16*a^2*d*sqrt(a + a*cos(c + d*x))) + ((A - B)*cos(c + d*x)^4*sin(c + d*x))/(4*d*(a + a*cos(c + d*x))^(5/2)) + ((13*A - 21*B)*cos(c + d*x)^3*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2)) - ((985*A - 1729*B)*sin(c + d*x))/(120*a^2*d*sqrt(a + a*cos(c + d*x))) - ((85*A - 157*B)*cos(c + d*x)^2*sin(c + d*x))/(80*a^2*d*sqrt(a + a*cos(c + d*x))) + ((475*A - 787*B)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(240*a^3*d)],
[cos(c + d*x)^3*(A + B*cos(c + d*x))/(a + a*cos(c + d*x))^(5/2), x, 5, -(((75*A - 163*B)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(16*a^2*d*sqrt(a + a*cos(c + d*x)))) + ((A - B)*cos(c + d*x)^3*sin(c + d*x))/(4*d*(a + a*cos(c + d*x))^(5/2)) + ((9*A - 17*B)*cos(c + d*x)^2*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2)) + ((93*A - 197*B)*sin(c + d*x))/(24*a^2*d*sqrt(a + a*cos(c + d*x))) - ((39*A - 95*B)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(48*a^3*d)],
[cos(c + d*x)^2*(A + B*cos(c + d*x))/(a + a*cos(c + d*x))^(5/2), x, 4, ((19*A - 75*B)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(16*a^2*d*sqrt(a + a*cos(c + d*x))) + ((A - B)*cos(c + d*x)^2*sin(c + d*x))/(4*d*(a + a*cos(c + d*x))^(5/2)) + ((5*A - 13*B)*cos(c + d*x)*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2)) - ((9*A - 49*B)*sin(c + d*x))/(16*a^2*d*sqrt(a + a*cos(c + d*x)))],
[cos(c + d*x)^1*(A + B*cos(c + d*x))/(a + a*cos(c + d*x))^(5/2), x, 3, ((5*A + 19*B)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(16*a^2*d*sqrt(a + a*cos(c + d*x))) + ((A - B)*cos(c + d*x)*sin(c + d*x))/(4*d*(a + a*cos(c + d*x))^(5/2)) + ((A - 9*B)*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2))],
[cos(c + d*x)^0*(A + B*cos(c + d*x))/(a + a*cos(c + d*x))^(5/2), x, 3, ((3*A + 5*B)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(16*a^2*d*sqrt(a + a*cos(c + d*x))) + ((A - B)*sin(c + d*x))/(4*d*(a + a*cos(c + d*x))^(5/2)) + ((3*A + 5*B)*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2))],
[sec(c + d*x)^1*(A + B*cos(c + d*x))/(a + a*cos(c + d*x))^(5/2), x, 5, -(((43*A - 3*B)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(16*a^2*d*sqrt(a + a*cos(c + d*x)))) + (2*sqrt(2)*A*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(a^2*d*sqrt(a + a*cos(c + d*x))) - ((A - B)*sin(c + d*x))/(4*d*(a + a*cos(c + d*x))^(5/2)) - ((11*A - 3*B)*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2))],
[sec(c + d*x)^2*(A + B*cos(c + d*x))/(a + a*cos(c + d*x))^(5/2), x, 6, ((115*A - 43*B)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(16*a^2*d*sqrt(a + a*cos(c + d*x))) - (sqrt(2)*(5*A - 2*B)*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(a^2*d*sqrt(a + a*cos(c + d*x))) - ((A - B)*tan(c + d*x))/(4*d*(a + a*cos(c + d*x))^(5/2)) - ((15*A - 7*B)*tan(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2)) + ((35*A - 11*B)*tan(c + d*x))/(16*a^2*d*sqrt(a + a*cos(c + d*x)))],
[sec(c + d*x)^3*(A + B*cos(c + d*x))/(a + a*cos(c + d*x))^(5/2), x, 7, -(((219*A - 115*B)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(16*a^2*d*sqrt(a + a*cos(c + d*x)))) + ((39*A - 20*B)*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(2*sqrt(2)*a^2*d*sqrt(a + a*cos(c + d*x))) - (7*(9*A - 5*B)*tan(c + d*x))/(16*a^2*d*sqrt(a + a*cos(c + d*x))) - ((A - B)*sec(c + d*x)*tan(c + d*x))/(4*d*(a + a*cos(c + d*x))^(5/2)) - ((19*A - 11*B)*sec(c + d*x)*tan(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2)) + ((31*A - 15*B)*sec(c + d*x)*tan(c + d*x))/(16*a^2*d*sqrt(a + a*cos(c + d*x)))],


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


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


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


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


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


[cos(c + d*x)^(3/2)*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^2, x, 7, (4*a^2*(9*A + 8*B)*EllipticE((1/2)*(c + d*x), 2))/(15*d) + (4*a^2*(6*A + 5*B)*EllipticF((1/2)*(c + d*x), 2))/(21*d) + (4*a^2*(6*A + 5*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(21*d) + (4*a^2*(9*A + 8*B)*cos(c + d*x)^(3/2)*sin(c + d*x))/(45*d) + (2*a^2*(9*A + 11*B)*cos(c + d*x)^(5/2)*sin(c + d*x))/(63*d) + (2*a*B*cos(c + d*x)^(5/2)*(a + a*cos(c + d*x))*sin(c + d*x))/(9*d)],
[cos(c + d*x)^(1/2)*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^2, x, 6, (4*a^2*(4*A + 3*B)*EllipticE((1/2)*(c + d*x), 2))/(5*d) + (4*a^2*(7*A + 6*B)*EllipticF((1/2)*(c + d*x), 2))/(21*d) + (4*a^2*(7*A + 6*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(21*d) + (2*a^2*(7*A + 9*B)*cos(c + d*x)^(3/2)*sin(c + d*x))/(35*d) + (2*a*B*cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))*sin(c + d*x))/(7*d)],
[cos(c + d*x)^(-1/2)*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^2, x, 5, (4*a^2*(5*A + 4*B)*EllipticE((1/2)*(c + d*x), 2))/(5*d) + (4*a^2*(2*A + B)*EllipticF((1/2)*(c + d*x), 2))/(3*d) + (2*a^2*(5*A + 7*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(15*d) + (2*a*B*sqrt(cos(c + d*x))*(a + a*cos(c + d*x))*sin(c + d*x))/(5*d)],
[cos(c + d*x)^(-3/2)*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^2, x, 5, (4*a^2*B*EllipticE((1/2)*(c + d*x), 2))/d + (4*a^2*(3*A + 2*B)*EllipticF((1/2)*(c + d*x), 2))/(3*d) - (2*a^2*(3*A - B)*sqrt(cos(c + d*x))*sin(c + d*x))/(3*d) + (2*a*A*(a + a*cos(c + d*x))*sin(c + d*x))/(d*sqrt(cos(c + d*x)))],
[cos(c + d*x)^(-5/2)*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^2, x, 5, -((4*a^2*A*EllipticE((1/2)*(c + d*x), 2))/d) + (4*a^2*(2*A + 3*B)*EllipticF((1/2)*(c + d*x), 2))/(3*d) + (2*a^2*(5*A + 3*B)*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))) + (2*a*A*(a + a*cos(c + d*x))*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2))],
[cos(c + d*x)^(-7/2)*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^2, x, 6, -((4*a^2*(4*A + 5*B)*EllipticE((1/2)*(c + d*x), 2))/(5*d)) + (4*a^2*(A + 2*B)*EllipticF((1/2)*(c + d*x), 2))/(3*d) + (2*a^2*(7*A + 5*B)*sin(c + d*x))/(15*d*cos(c + d*x)^(3/2)) + (4*a^2*(4*A + 5*B)*sin(c + d*x))/(5*d*sqrt(cos(c + d*x))) + (2*a*A*(a + a*cos(c + d*x))*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2))],
[cos(c + d*x)^(-9/2)*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^2, x, 7, -((4*a^2*(3*A + 4*B)*EllipticE((1/2)*(c + d*x), 2))/(5*d)) + (4*a^2*(6*A + 7*B)*EllipticF((1/2)*(c + d*x), 2))/(21*d) + (2*a^2*(9*A + 7*B)*sin(c + d*x))/(35*d*cos(c + d*x)^(5/2)) + (4*a^2*(6*A + 7*B)*sin(c + d*x))/(21*d*cos(c + d*x)^(3/2)) + (4*a^2*(3*A + 4*B)*sin(c + d*x))/(5*d*sqrt(cos(c + d*x))) + (2*a*A*(a + a*cos(c + d*x))*sin(c + d*x))/(7*d*cos(c + d*x)^(7/2))],


[cos(c + d*x)^(3/2)*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^3, x, 8, (4*a^3*(17*A + 15*B)*EllipticE((1/2)*(c + d*x), 2))/(15*d) + (4*a^3*(121*A + 105*B)*EllipticF((1/2)*(c + d*x), 2))/(231*d) + (4*a^3*(121*A + 105*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(231*d) + (4*a^3*(17*A + 15*B)*cos(c + d*x)^(3/2)*sin(c + d*x))/(45*d) + (20*a^3*(22*A + 21*B)*cos(c + d*x)^(5/2)*sin(c + d*x))/(693*d) + (2*a^2*(11*A + 15*B)*cos(c + d*x)^(5/2)*(a + a*cos(c + d*x))*sin(c + d*x))/(99*d) + (2*a*B*cos(c + d*x)^(5/2)*(a + a*cos(c + d*x))^2*sin(c + d*x))/(11*d)],
[cos(c + d*x)^(1/2)*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^3, x, 7, (4*a^3*(21*A + 17*B)*EllipticE((1/2)*(c + d*x), 2))/(15*d) + (4*a^3*(13*A + 11*B)*EllipticF((1/2)*(c + d*x), 2))/(21*d) + (4*a^3*(13*A + 11*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(21*d) + (4*a^3*(24*A + 23*B)*cos(c + d*x)^(3/2)*sin(c + d*x))/(105*d) + (2*a^2*(9*A + 13*B)*cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))*sin(c + d*x))/(63*d) + (2*a*B*cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))^2*sin(c + d*x))/(9*d)],
[cos(c + d*x)^(-1/2)*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^3, x, 6, (4*a^3*(9*A + 7*B)*EllipticE((1/2)*(c + d*x), 2))/(5*d) + (4*a^3*(21*A + 13*B)*EllipticF((1/2)*(c + d*x), 2))/(21*d) + (4*a^3*(42*A + 41*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(105*d) + (2*a^2*(7*A + 11*B)*sqrt(cos(c + d*x))*(a + a*cos(c + d*x))*sin(c + d*x))/(35*d) + (2*a*B*sqrt(cos(c + d*x))*(a + a*cos(c + d*x))^2*sin(c + d*x))/(7*d)],
[cos(c + d*x)^(-3/2)*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^3, x, 6, (4*a^3*(5*A + 9*B)*EllipticE((1/2)*(c + d*x), 2))/(5*d) + (4*a^3*(5*A + 3*B)*EllipticF((1/2)*(c + d*x), 2))/(3*d) - (4*a^3*(5*A - 6*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(15*d) - (2*a^2*(5*A - B)*sqrt(cos(c + d*x))*(a + a*cos(c + d*x))*sin(c + d*x))/(5*d) + (2*a*A*(a + a*cos(c + d*x))^2*sin(c + d*x))/(d*sqrt(cos(c + d*x)))],
[cos(c + d*x)^(-5/2)*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^3, x, 6, -((4*a^3*(A - B)*EllipticE((1/2)*(c + d*x), 2))/d) + (20*a^3*(A + B)*EllipticF((1/2)*(c + d*x), 2))/(3*d) - (4*a^3*(4*A + B)*sqrt(cos(c + d*x))*sin(c + d*x))/(3*d) + (2*a^2*(7*A + 3*B)*(a + a*cos(c + d*x))*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))) + (2*a*A*(a + a*cos(c + d*x))^2*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2))],
[cos(c + d*x)^(-7/2)*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^3, x, 6, -((4*a^3*(9*A + 5*B)*EllipticE((1/2)*(c + d*x), 2))/(5*d)) + (4*a^3*(3*A + 5*B)*EllipticF((1/2)*(c + d*x), 2))/(3*d) + (4*a^3*(21*A + 20*B)*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))) + (2*a^2*(9*A + 5*B)*(a + a*cos(c + d*x))*sin(c + d*x))/(15*d*cos(c + d*x)^(3/2)) + (2*a*A*(a + a*cos(c + d*x))^2*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2))],
[cos(c + d*x)^(-9/2)*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^3, x, 7, -((4*a^3*(7*A + 9*B)*EllipticE((1/2)*(c + d*x), 2))/(5*d)) + (4*a^3*(13*A + 21*B)*EllipticF((1/2)*(c + d*x), 2))/(21*d) + (4*a^3*(41*A + 42*B)*sin(c + d*x))/(105*d*cos(c + d*x)^(3/2)) + (4*a^3*(7*A + 9*B)*sin(c + d*x))/(5*d*sqrt(cos(c + d*x))) + (2*a^2*(11*A + 7*B)*(a + a*cos(c + d*x))*sin(c + d*x))/(35*d*cos(c + d*x)^(5/2)) + (2*a*A*(a + a*cos(c + d*x))^2*sin(c + d*x))/(7*d*cos(c + d*x)^(7/2))],
[cos(c + d*x)^(-11/2)*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^3, x, 8, -((4*a^3*(17*A + 21*B)*EllipticE((1/2)*(c + d*x), 2))/(15*d)) + (4*a^3*(11*A + 13*B)*EllipticF((1/2)*(c + d*x), 2))/(21*d) + (4*a^3*(23*A + 24*B)*sin(c + d*x))/(105*d*cos(c + d*x)^(5/2)) + (4*a^3*(11*A + 13*B)*sin(c + d*x))/(21*d*cos(c + d*x)^(3/2)) + (4*a^3*(17*A + 21*B)*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))) + (2*a^2*(13*A + 9*B)*(a + a*cos(c + d*x))*sin(c + d*x))/(63*d*cos(c + d*x)^(7/2)) + (2*a*A*(a + a*cos(c + d*x))^2*sin(c + d*x))/(9*d*cos(c + d*x)^(9/2))],


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


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


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


[cos(c + d*x)^(9/2)*(A + B*cos(c + d*x))/(a + a*cos(c + d*x))^3, x, 8, -((7*(17*A - 33*B)*EllipticE((1/2)*(c + d*x), 2))/(10*a^3*d)) + ((11*A - 21*B)*EllipticF((1/2)*(c + d*x), 2))/(2*a^3*d) + ((11*A - 21*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(2*a^3*d) - (7*(17*A - 33*B)*cos(c + d*x)^(3/2)*sin(c + d*x))/(30*a^3*d) + ((A - B)*cos(c + d*x)^(9/2)*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3) + ((7*A - 12*B)*cos(c + d*x)^(7/2)*sin(c + d*x))/(15*a*d*(a + a*cos(c + d*x))^2) + (3*(11*A - 21*B)*cos(c + d*x)^(5/2)*sin(c + d*x))/(10*a^2*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^(7/2)*(A + B*cos(c + d*x))/(a + a*cos(c + d*x))^3, x, 7, (7*(7*A - 17*B)*EllipticE((1/2)*(c + d*x), 2))/(10*a^3*d) - ((13*A - 33*B)*EllipticF((1/2)*(c + d*x), 2))/(6*a^3*d) - ((13*A - 33*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(6*a^3*d) + ((A - B)*cos(c + d*x)^(7/2)*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3) + ((A - 2*B)*cos(c + d*x)^(5/2)*sin(c + d*x))/(3*a*d*(a + a*cos(c + d*x))^2) + (7*(7*A - 17*B)*cos(c + d*x)^(3/2)*sin(c + d*x))/(30*a^2*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^(5/2)*(A + B*cos(c + d*x))/(a + a*cos(c + d*x))^3, x, 6, -(((9*A - 49*B)*EllipticE((1/2)*(c + d*x), 2))/(10*a^3*d)) + ((3*A - 13*B)*EllipticF((1/2)*(c + d*x), 2))/(6*a^3*d) + ((A - B)*cos(c + d*x)^(5/2)*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3) + ((3*A - 8*B)*cos(c + d*x)^(3/2)*sin(c + d*x))/(15*a*d*(a + a*cos(c + d*x))^2) + ((3*A - 13*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(6*a^2*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^(3/2)*(A + B*cos(c + d*x))/(a + a*cos(c + d*x))^3, x, 6, -(((A + 9*B)*EllipticE((1/2)*(c + d*x), 2))/(10*a^3*d)) + ((A + 3*B)*EllipticF((1/2)*(c + d*x), 2))/(6*a^3*d) + ((A - B)*cos(c + d*x)^(3/2)*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3) + ((A - 6*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(15*a*d*(a + a*cos(c + d*x))^2) + ((A + 9*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(10*a^2*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^(1/2)*(A + B*cos(c + d*x))/(a + a*cos(c + d*x))^3, x, 6, ((A - B)*EllipticE((1/2)*(c + d*x), 2))/(10*a^3*d) + ((A + B)*EllipticF((1/2)*(c + d*x), 2))/(6*a^3*d) + ((A - B)*sqrt(cos(c + d*x))*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3) + ((A + 4*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(15*a*d*(a + a*cos(c + d*x))^2) - ((A - B)*sqrt(cos(c + d*x))*sin(c + d*x))/(10*a^2*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^(-1/2)*(A + B*cos(c + d*x))/(a + a*cos(c + d*x))^3, x, 6, ((9*A + B)*EllipticE((1/2)*(c + d*x), 2))/(10*a^3*d) + ((3*A + B)*EllipticF((1/2)*(c + d*x), 2))/(6*a^3*d) - ((A - B)*sqrt(cos(c + d*x))*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3) - ((6*A - B)*sqrt(cos(c + d*x))*sin(c + d*x))/(15*a*d*(a + a*cos(c + d*x))^2) - ((9*A + B)*sqrt(cos(c + d*x))*sin(c + d*x))/(10*a^2*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^(-3/2)*(A + B*cos(c + d*x))/(a + a*cos(c + d*x))^3, x, 7, -(((49*A - 9*B)*EllipticE((1/2)*(c + d*x), 2))/(10*a^3*d)) - ((13*A - 3*B)*EllipticF((1/2)*(c + d*x), 2))/(6*a^3*d) + ((49*A - 9*B)*sin(c + d*x))/(10*a^3*d*sqrt(cos(c + d*x))) - ((A - B)*sin(c + d*x))/(5*d*sqrt(cos(c + d*x))*(a + a*cos(c + d*x))^3) - ((8*A - 3*B)*sin(c + d*x))/(15*a*d*sqrt(cos(c + d*x))*(a + a*cos(c + d*x))^2) - ((13*A - 3*B)*sin(c + d*x))/(6*a^2*d*sqrt(cos(c + d*x))*(a + a*cos(c + d*x)))],
[cos(c + d*x)^(-5/2)*(A + B*cos(c + d*x))/(a + a*cos(c + d*x))^3, x, 8, (7*(17*A - 7*B)*EllipticE((1/2)*(c + d*x), 2))/(10*a^3*d) + ((33*A - 13*B)*EllipticF((1/2)*(c + d*x), 2))/(6*a^3*d) + ((33*A - 13*B)*sin(c + d*x))/(6*a^3*d*cos(c + d*x)^(3/2)) - (7*(17*A - 7*B)*sin(c + d*x))/(10*a^3*d*sqrt(cos(c + d*x))) - ((A - B)*sin(c + d*x))/(5*d*cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))^3) - ((2*A - B)*sin(c + d*x))/(3*a*d*cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))^2) - (7*(17*A - 7*B)*sin(c + d*x))/(30*a^2*d*cos(c + d*x)^(3/2)*(a + a*cos(c + d*x)))],


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


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


[cos(c + d*x)^(5/2)*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^(1/2), x, 5, (5*sqrt(a)*(8*A + 7*B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(64*d) + (5*a*(8*A + 7*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(64*d*sqrt(a + a*cos(c + d*x))) + (5*a*(8*A + 7*B)*cos(c + d*x)^(3/2)*sin(c + d*x))/(96*d*sqrt(a + a*cos(c + d*x))) + (a*(8*A + 7*B)*cos(c + d*x)^(5/2)*sin(c + d*x))/(24*d*sqrt(a + a*cos(c + d*x))) + (a*B*cos(c + d*x)^(7/2)*sin(c + d*x))/(4*d*sqrt(a + a*cos(c + d*x)))],
[cos(c + d*x)^(3/2)*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^(1/2), x, 4, (sqrt(a)*(6*A + 5*B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(8*d) + (a*(6*A + 5*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(8*d*sqrt(a + a*cos(c + d*x))) + (a*(6*A + 5*B)*cos(c + d*x)^(3/2)*sin(c + d*x))/(12*d*sqrt(a + a*cos(c + d*x))) + (a*B*cos(c + d*x)^(5/2)*sin(c + d*x))/(3*d*sqrt(a + a*cos(c + d*x)))],
[cos(c + d*x)^(1/2)*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^(1/2), x, 3, (sqrt(a)*(4*A + 3*B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(4*d) + (a*(4*A + 3*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(4*d*sqrt(a + a*cos(c + d*x))) + (a*B*cos(c + d*x)^(3/2)*sin(c + d*x))/(2*d*sqrt(a + a*cos(c + d*x)))],
[cos(c + d*x)^(-1/2)*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^(1/2), x, 2, (sqrt(a)*(2*A + B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/d + (a*B*sqrt(cos(c + d*x))*sin(c + d*x))/(d*sqrt(a + a*cos(c + d*x)))],
[cos(c + d*x)^(-3/2)*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^(1/2), x, 2, (2*sqrt(a)*B*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/d + (2*a*A*sin(c + d*x))/(d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))],
[cos(c + d*x)^(-5/2)*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^(1/2), x, 2, (2*a*A*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2)*sqrt(a + a*cos(c + d*x))) + (2*a*(2*A + 3*B)*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))],
[cos(c + d*x)^(-7/2)*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^(1/2), x, 3, (2*a*A*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2)*sqrt(a + a*cos(c + d*x))) + (2*a*(4*A + 5*B)*sin(c + d*x))/(15*d*cos(c + d*x)^(3/2)*sqrt(a + a*cos(c + d*x))) + (4*a*(4*A + 5*B)*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))],
[cos(c + d*x)^(-9/2)*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^(1/2), x, 4, (2*a*A*sin(c + d*x))/(7*d*cos(c + d*x)^(7/2)*sqrt(a + a*cos(c + d*x))) + (2*a*(6*A + 7*B)*sin(c + d*x))/(35*d*cos(c + d*x)^(5/2)*sqrt(a + a*cos(c + d*x))) + (8*a*(6*A + 7*B)*sin(c + d*x))/(105*d*cos(c + d*x)^(3/2)*sqrt(a + a*cos(c + d*x))) + (16*a*(6*A + 7*B)*sin(c + d*x))/(105*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))],


[cos(c + d*x)^(3/2)*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^(3/2), x, 5, (a^(3/2)*(88*A + 75*B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(64*d) + (a^2*(88*A + 75*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(64*d*sqrt(a + a*cos(c + d*x))) + (a^2*(88*A + 75*B)*cos(c + d*x)^(3/2)*sin(c + d*x))/(96*d*sqrt(a + a*cos(c + d*x))) + (a^2*(8*A + 9*B)*cos(c + d*x)^(5/2)*sin(c + d*x))/(24*d*sqrt(a + a*cos(c + d*x))) + (a*B*cos(c + d*x)^(5/2)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(4*d)],
[cos(c + d*x)^(1/2)*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^(3/2), x, 4, (a^(3/2)*(14*A + 11*B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(8*d) + (a^2*(14*A + 11*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(8*d*sqrt(a + a*cos(c + d*x))) + (a^2*(6*A + 7*B)*cos(c + d*x)^(3/2)*sin(c + d*x))/(12*d*sqrt(a + a*cos(c + d*x))) + (a*B*cos(c + d*x)^(3/2)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(3*d)],
[cos(c + d*x)^(-1/2)*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^(3/2), x, 3, (a^(3/2)*(12*A + 7*B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(4*d) + (a^2*(4*A + 5*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(4*d*sqrt(a + a*cos(c + d*x))) + (a*B*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(2*d)],
[cos(c + d*x)^(-3/2)*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^(3/2), x, 3, (a^(3/2)*(2*A + 3*B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/d - (a^2*(2*A - B)*sqrt(cos(c + d*x))*sin(c + d*x))/(d*sqrt(a + a*cos(c + d*x))) + (2*a*A*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(d*sqrt(cos(c + d*x)))],
[cos(c + d*x)^(-5/2)*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^(3/2), x, 3, (2*a^(3/2)*B*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/d + (2*a^2*(4*A + 3*B)*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))) + (2*a*A*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2))],
[cos(c + d*x)^(-7/2)*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^(3/2), x, 3, (8*a^2*(3*A + 5*B)*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))) + (2*a*(3*A + 5*B)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(15*d*cos(c + d*x)^(3/2)) + (2*A*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2))],
[cos(c + d*x)^(-9/2)*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^(3/2), x, 4, (2*a^2*(8*A + 7*B)*sin(c + d*x))/(35*d*cos(c + d*x)^(5/2)*sqrt(a + a*cos(c + d*x))) + (2*a^2*(52*A + 63*B)*sin(c + d*x))/(105*d*cos(c + d*x)^(3/2)*sqrt(a + a*cos(c + d*x))) + (4*a^2*(52*A + 63*B)*sin(c + d*x))/(105*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))) + (2*a*A*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(7*d*cos(c + d*x)^(7/2))],
[cos(c + d*x)^(-11/2)*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^(3/2), x, 5, (2*a^2*(10*A + 9*B)*sin(c + d*x))/(63*d*cos(c + d*x)^(7/2)*sqrt(a + a*cos(c + d*x))) + (2*a^2*(34*A + 39*B)*sin(c + d*x))/(105*d*cos(c + d*x)^(5/2)*sqrt(a + a*cos(c + d*x))) + (8*a^2*(34*A + 39*B)*sin(c + d*x))/(315*d*cos(c + d*x)^(3/2)*sqrt(a + a*cos(c + d*x))) + (16*a^2*(34*A + 39*B)*sin(c + d*x))/(315*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))) + (2*a*A*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(9*d*cos(c + d*x)^(9/2))],


[cos(c + d*x)^(3/2)*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^(5/2), x, 6, (a^(5/2)*(326*A + 283*B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(128*d) + (a^3*(326*A + 283*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(128*d*sqrt(a + a*cos(c + d*x))) + (a^3*(326*A + 283*B)*cos(c + d*x)^(3/2)*sin(c + d*x))/(192*d*sqrt(a + a*cos(c + d*x))) + (a^3*(170*A + 157*B)*cos(c + d*x)^(5/2)*sin(c + d*x))/(240*d*sqrt(a + a*cos(c + d*x))) + (a^2*(10*A + 13*B)*cos(c + d*x)^(5/2)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(40*d) + (a*B*cos(c + d*x)^(5/2)*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(5*d)],
[cos(c + d*x)^(1/2)*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^(5/2), x, 5, (a^(5/2)*(200*A + 163*B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(64*d) + (a^3*(200*A + 163*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(64*d*sqrt(a + a*cos(c + d*x))) + (a^3*(104*A + 95*B)*cos(c + d*x)^(3/2)*sin(c + d*x))/(96*d*sqrt(a + a*cos(c + d*x))) + (a^2*(8*A + 11*B)*cos(c + d*x)^(3/2)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(24*d) + (a*B*cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(4*d)],
[cos(c + d*x)^(-1/2)*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^(5/2), x, 4, (a^(5/2)*(38*A + 25*B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(8*d) + (a^3*(54*A + 49*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(24*d*sqrt(a + a*cos(c + d*x))) + (a^2*(2*A + 3*B)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(4*d) + (a*B*sqrt(cos(c + d*x))*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(3*d)],
[cos(c + d*x)^(-3/2)*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^(5/2), x, 4, (a^(5/2)*(20*A + 19*B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(4*d) - (a^3*(4*A - 9*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(4*d*sqrt(a + a*cos(c + d*x))) - (a^2*(4*A - B)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(2*d) + (2*a*A*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(d*sqrt(cos(c + d*x)))],
[cos(c + d*x)^(-5/2)*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^(5/2), x, 4, (a^(5/2)*(2*A + 5*B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/d - (a^3*(14*A + 3*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(3*d*sqrt(a + a*cos(c + d*x))) + (2*a^2*(2*A + B)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(d*sqrt(cos(c + d*x))) + (2*a*A*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2))],
[cos(c + d*x)^(-7/2)*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^(5/2), x, 4, (2*a^(5/2)*B*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/d + (2*a^3*(32*A + 35*B)*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))) + (2*a^2*(8*A + 5*B)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(15*d*cos(c + d*x)^(3/2)) + (2*a*A*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2))],
[cos(c + d*x)^(-9/2)*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^(5/2), x, 4, (64*a^3*(5*A + 7*B)*sin(c + d*x))/(105*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))) + (16*a^2*(5*A + 7*B)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(105*d*cos(c + d*x)^(3/2)) + (2*a*(5*A + 7*B)*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(35*d*cos(c + d*x)^(5/2)) + (2*A*(a + a*cos(c + d*x))^(5/2)*sin(c + d*x))/(7*d*cos(c + d*x)^(7/2))],
[cos(c + d*x)^(-11/2)*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^(5/2), x, 5, (2*a^3*(124*A + 135*B)*sin(c + d*x))/(315*d*cos(c + d*x)^(5/2)*sqrt(a + a*cos(c + d*x))) + (2*a^3*(292*A + 345*B)*sin(c + d*x))/(315*d*cos(c + d*x)^(3/2)*sqrt(a + a*cos(c + d*x))) + (4*a^3*(292*A + 345*B)*sin(c + d*x))/(315*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))) + (2*a^2*(4*A + 3*B)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(21*d*cos(c + d*x)^(7/2)) + (2*a*A*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(9*d*cos(c + d*x)^(9/2))],
[cos(c + d*x)^(-13/2)*(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^(5/2), x, 6, (2*a^3*(194*A + 209*B)*sin(c + d*x))/(693*d*cos(c + d*x)^(7/2)*sqrt(a + a*cos(c + d*x))) + (2*a^3*(710*A + 803*B)*sin(c + d*x))/(1155*d*cos(c + d*x)^(5/2)*sqrt(a + a*cos(c + d*x))) + (8*a^3*(710*A + 803*B)*sin(c + d*x))/(3465*d*cos(c + d*x)^(3/2)*sqrt(a + a*cos(c + d*x))) + (16*a^3*(710*A + 803*B)*sin(c + d*x))/(3465*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))) + (2*a^2*(14*A + 11*B)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(99*d*cos(c + d*x)^(9/2)) + (2*a*A*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(11*d*cos(c + d*x)^(11/2))],


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


[cos(c + d*x)^(3/2)*(A + B*cos(c + d*x))/(a + a*cos(c + d*x))^(1/2), x, 5, -(((4*A - 7*B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(4*sqrt(a)*d)) + (sqrt(2)*(A - B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(sqrt(a)*d) + ((4*A - B)*sqrt(cos(c + d*x))*sin(c + d*x))/(4*d*sqrt(a + a*cos(c + d*x))) + (B*cos(c + d*x)^(3/2)*sin(c + d*x))/(2*d*sqrt(a + a*cos(c + d*x)))],
[cos(c + d*x)^(1/2)*(A + B*cos(c + d*x))/(a + a*cos(c + d*x))^(1/2), x, 4, ((2*A - B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(sqrt(a)*d) - (sqrt(2)*(A - B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(sqrt(a)*d) + (B*sqrt(cos(c + d*x))*sin(c + d*x))/(d*sqrt(a + a*cos(c + d*x)))],
[cos(c + d*x)^(-1/2)*(A + B*cos(c + d*x))/(a + a*cos(c + d*x))^(1/2), x, 3, (2*B*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(sqrt(a)*d) + (sqrt(2)*(A - B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(sqrt(a)*d)],
[cos(c + d*x)^(-3/2)*(A + B*cos(c + d*x))/(a + a*cos(c + d*x))^(1/2), x, 2, -((sqrt(2)*(A - B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(sqrt(a)*d)) + (2*A*sin(c + d*x))/(d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))],
[cos(c + d*x)^(-5/2)*(A + B*cos(c + d*x))/(a + a*cos(c + d*x))^(1/2), x, 3, (sqrt(2)*(A - B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(sqrt(a)*d) + (2*A*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2)*sqrt(a + a*cos(c + d*x))) - (2*(A - 3*B)*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))],
[cos(c + d*x)^(-7/2)*(A + B*cos(c + d*x))/(a + a*cos(c + d*x))^(1/2), x, 4, -((sqrt(2)*(A - B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(sqrt(a)*d)) + (2*A*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2)*sqrt(a + a*cos(c + d*x))) - (2*(A - 5*B)*sin(c + d*x))/(15*d*cos(c + d*x)^(3/2)*sqrt(a + a*cos(c + d*x))) + (2*(13*A - 5*B)*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))],


[cos(c + d*x)^(3/2)*(A + B*cos(c + d*x))/(a + a*cos(c + d*x))^(3/2), x, 5, ((2*A - 3*B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(a^(3/2)*d) - ((5*A - 9*B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d) + ((A - B)*cos(c + d*x)^(3/2)*sin(c + d*x))/(2*d*(a + a*cos(c + d*x))^(3/2)) - ((A - 3*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(2*a*d*sqrt(a + a*cos(c + d*x)))],
[cos(c + d*x)^(1/2)*(A + B*cos(c + d*x))/(a + a*cos(c + d*x))^(3/2), x, 4, (2*B*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(a^(3/2)*d) + ((A - 5*B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d) + ((A - B)*sqrt(cos(c + d*x))*sin(c + d*x))/(2*d*(a + a*cos(c + d*x))^(3/2))],
[cos(c + d*x)^(-1/2)*(A + B*cos(c + d*x))/(a + a*cos(c + d*x))^(3/2), x, 2, ((3*A + B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d) - ((A - B)*sqrt(cos(c + d*x))*sin(c + d*x))/(2*d*(a + a*cos(c + d*x))^(3/2))],
[cos(c + d*x)^(-3/2)*(A + B*cos(c + d*x))/(a + a*cos(c + d*x))^(3/2), x, 3, -(((7*A - 3*B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d)) - ((A - B)*sin(c + d*x))/(2*d*sqrt(cos(c + d*x))*(a + a*cos(c + d*x))^(3/2)) + ((5*A - B)*sin(c + d*x))/(2*a*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))],
[cos(c + d*x)^(-5/2)*(A + B*cos(c + d*x))/(a + a*cos(c + d*x))^(3/2), x, 4, ((11*A - 7*B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d) - ((A - B)*sin(c + d*x))/(2*d*cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))^(3/2)) + ((7*A - 3*B)*sin(c + d*x))/(6*a*d*cos(c + d*x)^(3/2)*sqrt(a + a*cos(c + d*x))) - ((19*A - 15*B)*sin(c + d*x))/(6*a*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))],


[cos(c + d*x)^(5/2)*(A + B*cos(c + d*x))/(a + a*cos(c + d*x))^(5/2), x, 6, ((2*A - 5*B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(a^(5/2)*d) - ((43*A - 115*B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d) + ((A - B)*cos(c + d*x)^(5/2)*sin(c + d*x))/(4*d*(a + a*cos(c + d*x))^(5/2)) + ((7*A - 15*B)*cos(c + d*x)^(3/2)*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2)) - ((11*A - 35*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(16*a^2*d*sqrt(a + a*cos(c + d*x)))],
[cos(c + d*x)^(3/2)*(A + B*cos(c + d*x))/(a + a*cos(c + d*x))^(5/2), x, 5, (2*B*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(a^(5/2)*d) + ((3*A - 43*B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d) + ((A - B)*cos(c + d*x)^(3/2)*sin(c + d*x))/(4*d*(a + a*cos(c + d*x))^(5/2)) + ((3*A - 11*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2))],
[cos(c + d*x)^(1/2)*(A + B*cos(c + d*x))/(a + a*cos(c + d*x))^(5/2), x, 3, ((5*A + 3*B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d) - ((A - B)*cos(c + d*x)^(3/2)*sin(c + d*x))/(4*d*(a + a*cos(c + d*x))^(5/2)) + ((5*A + 3*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2))],
[cos(c + d*x)^(-1/2)*(A + B*cos(c + d*x))/(a + a*cos(c + d*x))^(5/2), x, 3, ((19*A + 5*B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d) - ((A - B)*sqrt(cos(c + d*x))*sin(c + d*x))/(4*d*(a + a*cos(c + d*x))^(5/2)) - ((9*A - B)*sqrt(cos(c + d*x))*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2))],
[cos(c + d*x)^(-3/2)*(A + B*cos(c + d*x))/(a + a*cos(c + d*x))^(5/2), x, 4, -(((75*A - 19*B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d)) - ((A - B)*sin(c + d*x))/(4*d*sqrt(cos(c + d*x))*(a + a*cos(c + d*x))^(5/2)) - ((13*A - 5*B)*sin(c + d*x))/(16*a*d*sqrt(cos(c + d*x))*(a + a*cos(c + d*x))^(3/2)) + ((49*A - 9*B)*sin(c + d*x))/(16*a^2*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))],
[cos(c + d*x)^(-5/2)*(A + B*cos(c + d*x))/(a + a*cos(c + d*x))^(5/2), x, 5, ((163*A - 75*B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d) - ((A - B)*sin(c + d*x))/(4*d*cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))^(5/2)) - ((17*A - 9*B)*sin(c + d*x))/(16*a*d*cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))^(3/2)) + ((95*A - 39*B)*sin(c + d*x))/(48*a^2*d*cos(c + d*x)^(3/2)*sqrt(a + a*cos(c + d*x))) - ((299*A - 147*B)*sin(c + d*x))/(48*a^2*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))],


[cos(c + d*x)^(7/2)*(A + B*cos(c + d*x))/(a + a*cos(c + d*x))^(7/2), x, 7, ((2*A - 7*B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(a^(7/2)*d) - ((177*A - 637*B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(64*sqrt(2)*a^(7/2)*d) + ((A - B)*cos(c + d*x)^(7/2)*sin(c + d*x))/(6*d*(a + a*cos(c + d*x))^(7/2)) + ((3*A - 7*B)*cos(c + d*x)^(5/2)*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(5/2)) + ((79*A - 259*B)*cos(c + d*x)^(3/2)*sin(c + d*x))/(192*a^2*d*(a + a*cos(c + d*x))^(3/2)) - (7*(7*A - 27*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(64*a^3*d*sqrt(a + a*cos(c + d*x)))],
[cos(c + d*x)^(5/2)*(A + B*cos(c + d*x))/(a + a*cos(c + d*x))^(7/2), x, 6, (2*B*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(a^(7/2)*d) + ((5*A - 177*B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(64*sqrt(2)*a^(7/2)*d) + ((A - B)*cos(c + d*x)^(5/2)*sin(c + d*x))/(6*d*(a + a*cos(c + d*x))^(7/2)) + ((5*A - 17*B)*cos(c + d*x)^(3/2)*sin(c + d*x))/(48*a*d*(a + a*cos(c + d*x))^(5/2)) + ((5*A - 49*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(64*a^2*d*(a + a*cos(c + d*x))^(3/2))],
[cos(c + d*x)^(3/2)*(A + B*cos(c + d*x))/(a + a*cos(c + d*x))^(7/2), x, 4, ((7*A + 5*B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(64*sqrt(2)*a^(7/2)*d) - ((A - B)*cos(c + d*x)^(5/2)*sin(c + d*x))/(6*d*(a + a*cos(c + d*x))^(7/2)) + ((7*A + 5*B)*cos(c + d*x)^(3/2)*sin(c + d*x))/(48*a*d*(a + a*cos(c + d*x))^(5/2)) + ((7*A + 5*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(64*a^2*d*(a + a*cos(c + d*x))^(3/2))],
[cos(c + d*x)^(1/2)*(A + B*cos(c + d*x))/(a + a*cos(c + d*x))^(7/2), x, 4, ((13*A + 7*B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(64*sqrt(2)*a^(7/2)*d) + ((A - B)*sqrt(cos(c + d*x))*sin(c + d*x))/(6*d*(a + a*cos(c + d*x))^(7/2)) + ((A + 3*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(5/2)) - ((5*A - 17*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(192*a^2*d*(a + a*cos(c + d*x))^(3/2))],
[cos(c + d*x)^(-1/2)*(A + B*cos(c + d*x))/(a + a*cos(c + d*x))^(7/2), x, 4, ((63*A + 13*B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(64*sqrt(2)*a^(7/2)*d) - ((A - B)*sqrt(cos(c + d*x))*sin(c + d*x))/(6*d*(a + a*cos(c + d*x))^(7/2)) - ((5*A - B)*sqrt(cos(c + d*x))*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(5/2)) - ((103*A + 5*B)*sqrt(cos(c + d*x))*sin(c + d*x))/(192*a^2*d*(a + a*cos(c + d*x))^(3/2))],
[cos(c + d*x)^(-3/2)*(A + B*cos(c + d*x))/(a + a*cos(c + d*x))^(7/2), x, 5, -((3*(121*A - 21*B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(64*sqrt(2)*a^(7/2)*d)) - ((A - B)*sin(c + d*x))/(6*d*sqrt(cos(c + d*x))*(a + a*cos(c + d*x))^(7/2)) - ((19*A - 7*B)*sin(c + d*x))/(48*a*d*sqrt(cos(c + d*x))*(a + a*cos(c + d*x))^(5/2)) - ((199*A - 43*B)*sin(c + d*x))/(192*a^2*d*sqrt(cos(c + d*x))*(a + a*cos(c + d*x))^(3/2)) + ((691*A - 103*B)*sin(c + d*x))/(192*a^3*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))],
[cos(c + d*x)^(-5/2)*(A + B*cos(c + d*x))/(a + a*cos(c + d*x))^(7/2), x, 6, ((1015*A - 363*B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(64*sqrt(2)*a^(7/2)*d) - ((A - B)*sin(c + d*x))/(6*d*cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))^(7/2)) - ((23*A - 11*B)*sin(c + d*x))/(48*a*d*cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))^(5/2)) - ((109*A - 41*B)*sin(c + d*x))/(64*a^2*d*cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))^(3/2)) + ((579*A - 199*B)*sin(c + d*x))/(192*a^3*d*cos(c + d*x)^(3/2)*sqrt(a + a*cos(c + d*x))) - ((1887*A - 691*B)*sin(c + d*x))/(192*a^3*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))],


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


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


[(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^(2/3), x, 2, (1/10)*a*(5*A + 2*B)*Int(1/(a + a*cos(c + d*x))^(1/3), x) + (3*a*(5*A + 2*B)*sin(c + d*x))/(10*d*(a + a*cos(c + d*x))^(1/3)) + (3*B*(a + a*cos(c + d*x))^(2/3)*sin(c + d*x))/(5*d)],
[(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^(1/3), x, 2, (-(1/4))*a*(4*A + B)*Int(1/(a + a*sin(c + Pi/2 + d*x))^(2/3), x) + (3*a*(4*A + B)*sin(c + d*x))/(4*d*(a + a*cos(c + d*x))^(2/3)) + (3*B*(a + a*cos(c + d*x))^(1/3)*sin(c + d*x))/(4*d)],


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


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


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


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


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


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


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


[(a + a*cos(c + d*x))^3*(A + B*cos(c + d*x))*sec(c + d*x)^(13/2), x, 10, (-4*a^3*(15*A + 17*B)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(15*d) + (4*a^3*(105*A + 121*B)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(231*d) + (4*a^3*(15*A + 17*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(15*d) + (4*a^3*(105*A + 121*B)*sec(c + d*x)^(3/2)*sin(c + d*x))/(231*d) + (4*a^3*(15*A + 17*B)*sec(c + d*x)^(5/2)*sin(c + d*x))/(45*d) + (20*a^3*(21*A + 22*B)*sec(c + d*x)^(7/2)*sin(c + d*x))/(693*d) + (2*a^2*(15*A + 11*B)*(a + a*cos(c + d*x))*sec(c + d*x)^(9/2)*sin(c + d*x))/(99*d) + (2*a*A*(a + a*cos(c + d*x))^2*sec(c + d*x)^(11/2)*sin(c + d*x))/(11*d)],
[(a + a*cos(c + d*x))^3*(A + B*cos(c + d*x))*sec(c + d*x)^(11/2), x, 9, (-4*a^3*(17*A + 21*B)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(15*d) + (4*a^3*(11*A + 13*B)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(21*d) + (4*a^3*(17*A + 21*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(15*d) + (4*a^3*(11*A + 13*B)*sec(c + d*x)^(3/2)*sin(c + d*x))/(21*d) + (4*a^3*(23*A + 24*B)*sec(c + d*x)^(5/2)*sin(c + d*x))/(105*d) + (2*a^2*(13*A + 9*B)*(a + a*cos(c + d*x))*sec(c + d*x)^(7/2)*sin(c + d*x))/(63*d) + (2*a*A*(a + a*cos(c + d*x))^2*sec(c + d*x)^(9/2)*sin(c + d*x))/(9*d)],
[(a + a*cos(c + d*x))^3*(A + B*cos(c + d*x))*sec(c + d*x)^(9/2), x, 8, (-4*a^3*(7*A + 9*B)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*d) + (4*a^3*(13*A + 21*B)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(21*d) + (4*a^3*(7*A + 9*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d) + (4*a^3*(41*A + 42*B)*sec(c + d*x)^(3/2)*sin(c + d*x))/(105*d) + (2*a^2*(11*A + 7*B)*(a + a*cos(c + d*x))*sec(c + d*x)^(5/2)*sin(c + d*x))/(35*d) + (2*a*A*(a + a*cos(c + d*x))^2*sec(c + d*x)^(7/2)*sin(c + d*x))/(7*d)],
[(a + a*cos(c + d*x))^3*(A + B*cos(c + d*x))*sec(c + d*x)^(7/2), x, 7, (-4*a^3*(9*A + 5*B)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*d) + (4*a^3*(3*A + 5*B)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*d) + (4*a^3*(21*A + 20*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(15*d) + (2*a^2*(9*A + 5*B)*(a + a*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(15*d) + (2*a*A*(a + a*cos(c + d*x))^2*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d)],
[(a + a*cos(c + d*x))^3*(A + B*cos(c + d*x))*sec(c + d*x)^(5/2), x, 7, (-4*a^3*(A - B)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/d + (20*a^3*(A + B)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*d) - (4*a^3*(4*A + B)*sin(c + d*x))/(3*d*sqrt(sec(c + d*x))) + (2*a^2*(7*A + 3*B)*(a + a*cos(c + d*x))*sqrt(sec(c + d*x))*sin(c + d*x))/(3*d) + (2*a*A*(a + a*cos(c + d*x))^2*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d)],
[(a + a*cos(c + d*x))^3*(A + B*cos(c + d*x))*sec(c + d*x)^(3/2), x, 7, (4*a^3*(5*A + 9*B)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*d) + (4*a^3*(5*A + 3*B)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*d) - (4*a^3*(5*A - 6*B)*sin(c + d*x))/(15*d*sqrt(sec(c + d*x))) - (2*a^2*(5*A - B)*(a + a*cos(c + d*x))*sin(c + d*x))/(5*d*sqrt(sec(c + d*x))) + (2*a*A*(a + a*cos(c + d*x))^2*sqrt(sec(c + d*x))*sin(c + d*x))/d],
[(a + a*cos(c + d*x))^3*(A + B*cos(c + d*x))*sqrt(sec(c + d*x)), x, 7, (4*a^3*(9*A + 7*B)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*d) + (4*a^3*(21*A + 13*B)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(21*d) + (4*a^3*(42*A + 41*B)*sin(c + d*x))/(105*d*sqrt(sec(c + d*x))) + (2*a^2*(7*A + 11*B)*(a + a*cos(c + d*x))*sin(c + d*x))/(35*d*sqrt(sec(c + d*x))) + (2*a*B*(a + a*cos(c + d*x))^2*sin(c + d*x))/(7*d*sqrt(sec(c + d*x)))],
[((a + a*cos(c + d*x))^3*(A + B*cos(c + d*x)))/sqrt(sec(c + d*x)), x, 8, (4*a^3*(21*A + 17*B)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(15*d) + (4*a^3*(13*A + 11*B)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(21*d) + (4*a^3*(24*A + 23*B)*sin(c + d*x))/(105*d*sec(c + d*x)^(3/2)) + (2*a^2*(9*A + 13*B)*(a + a*cos(c + d*x))*sin(c + d*x))/(63*d*sec(c + d*x)^(3/2)) + (2*a*B*(a + a*cos(c + d*x))^2*sin(c + d*x))/(9*d*sec(c + d*x)^(3/2)) + (4*a^3*(13*A + 11*B)*sin(c + d*x))/(21*d*sqrt(sec(c + d*x)))],
[((a + a*cos(c + d*x))^3*(A + B*cos(c + d*x)))/sec(c + d*x)^(3/2), x, 9, (4*a^3*(17*A + 15*B)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(15*d) + (4*a^3*(121*A + 105*B)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(231*d) + (20*a^3*(22*A + 21*B)*sin(c + d*x))/(693*d*sec(c + d*x)^(5/2)) + (2*a^2*(11*A + 15*B)*(a + a*cos(c + d*x))*sin(c + d*x))/(99*d*sec(c + d*x)^(5/2)) + (2*a*B*(a + a*cos(c + d*x))^2*sin(c + d*x))/(11*d*sec(c + d*x)^(5/2)) + (4*a^3*(17*A + 15*B)*sin(c + d*x))/(45*d*sec(c + d*x)^(3/2)) + (4*a^3*(121*A + 105*B)*sin(c + d*x))/(231*d*sqrt(sec(c + d*x)))],


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


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


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


[((A + B*cos(c + d*x))*sec(c + d*x)^(5/2))/(a + a*cos(c + d*x))^3, x, 9, (7*(17*A - 7*B)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(10*a^3*d) + ((33*A - 13*B)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(6*a^3*d) - (7*(17*A - 7*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(10*a^3*d) + ((33*A - 13*B)*sec(c + d*x)^(3/2)*sin(c + d*x))/(6*a^3*d) - ((A - B)*sec(c + d*x)^(3/2)*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3) - ((2*A - B)*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*a*d*(a + a*cos(c + d*x))^2) - (7*(17*A - 7*B)*sec(c + d*x)^(3/2)*sin(c + d*x))/(30*a^2*d*(a + a*cos(c + d*x)))],
[((A + B*cos(c + d*x))*sec(c + d*x)^(3/2))/(a + a*cos(c + d*x))^3, x, 8, -((49*A - 9*B)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(10*a^3*d) - ((13*A - 3*B)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(6*a^3*d) + ((49*A - 9*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(10*a^3*d) - ((A - B)*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3) - ((8*A - 3*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(15*a*d*(a + a*cos(c + d*x))^2) - ((13*A - 3*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(6*a^2*d*(a + a*cos(c + d*x)))],
[((A + B*cos(c + d*x))*sqrt(sec(c + d*x)))/(a + a*cos(c + d*x))^3, x, 7, ((9*A + B)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(10*a^3*d) + ((3*A + B)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(6*a^3*d) - ((A - B)*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3*sqrt(sec(c + d*x))) - ((6*A - B)*sin(c + d*x))/(15*a*d*(a + a*cos(c + d*x))^2*sqrt(sec(c + d*x))) - ((9*A + B)*sin(c + d*x))/(10*a^2*d*(a + a*cos(c + d*x))*sqrt(sec(c + d*x)))],
[(A + B*cos(c + d*x))/((a + a*cos(c + d*x))^3*sqrt(sec(c + d*x))), x, 7, ((A - B)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(10*a^3*d) + ((A + B)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(6*a^3*d) + ((A - B)*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3*sqrt(sec(c + d*x))) + ((A + 4*B)*sin(c + d*x))/(15*a*d*(a + a*cos(c + d*x))^2*sqrt(sec(c + d*x))) - ((A - B)*sin(c + d*x))/(10*a^2*d*(a + a*cos(c + d*x))*sqrt(sec(c + d*x)))],
[(A + B*cos(c + d*x))/((a + a*cos(c + d*x))^3*sec(c + d*x)^(3/2)), x, 7, -((A + 9*B)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(10*a^3*d) + ((A + 3*B)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(6*a^3*d) + ((A - B)*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3*sec(c + d*x)^(3/2)) + ((A - 6*B)*sin(c + d*x))/(15*a*d*(a + a*cos(c + d*x))^2*sqrt(sec(c + d*x))) + ((A + 9*B)*sin(c + d*x))/(10*a^2*d*(a + a*cos(c + d*x))*sqrt(sec(c + d*x)))],
[(A + B*cos(c + d*x))/((a + a*cos(c + d*x))^3*sec(c + d*x)^(5/2)), x, 7, -((9*A - 49*B)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(10*a^3*d) + ((3*A - 13*B)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(6*a^3*d) + ((A - B)*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3*sec(c + d*x)^(5/2)) + ((3*A - 8*B)*sin(c + d*x))/(15*a*d*(a + a*cos(c + d*x))^2*sec(c + d*x)^(3/2)) + ((3*A - 13*B)*sin(c + d*x))/(6*a^2*d*(a + a*cos(c + d*x))*sqrt(sec(c + d*x)))],
[(A + B*cos(c + d*x))/((a + a*cos(c + d*x))^3*sec(c + d*x)^(7/2)), x, 8, (7*(7*A - 17*B)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(10*a^3*d) - ((13*A - 33*B)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(6*a^3*d) + ((A - B)*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3*sec(c + d*x)^(7/2)) + ((A - 2*B)*sin(c + d*x))/(3*a*d*(a + a*cos(c + d*x))^2*sec(c + d*x)^(5/2)) + (7*(7*A - 17*B)*sin(c + d*x))/(30*a^2*d*(a + a*cos(c + d*x))*sec(c + d*x)^(3/2)) - ((13*A - 33*B)*sin(c + d*x))/(6*a^3*d*sqrt(sec(c + d*x)))],
[(A + B*cos(c + d*x))/((a + a*cos(c + d*x))^3*sec(c + d*x)^(9/2)), x, 9, -((7*(17*A - 33*B)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(10*a^3*d)) + ((11*A - 21*B)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(2*a^3*d) + ((A - B)*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3*sec(c + d*x)^(9/2)) + ((7*A - 12*B)*sin(c + d*x))/(15*a*d*(a + a*cos(c + d*x))^2*sec(c + d*x)^(7/2)) + (3*(11*A - 21*B)*sin(c + d*x))/(10*a^2*d*(a + a*cos(c + d*x))*sec(c + d*x)^(5/2)) - (7*(17*A - 33*B)*sin(c + d*x))/(30*a^3*d*sec(c + d*x)^(3/2)) + ((11*A - 21*B)*sin(c + d*x))/(2*a^3*d*sqrt(sec(c + d*x)))],


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


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


[sqrt(a + a*cos(c + d*x))*(A + B*cos(c + d*x))*sec(c + d*x)^(11/2), x, 5, (32*a*(8*A + 9*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(315*d*sqrt(a + a*cos(c + d*x))) + (16*a*(8*A + 9*B)*sec(c + d*x)^(3/2)*sin(c + d*x))/(315*d*sqrt(a + a*cos(c + d*x))) + (4*a*(8*A + 9*B)*sec(c + d*x)^(5/2)*sin(c + d*x))/(105*d*sqrt(a + a*cos(c + d*x))) + (2*a*(8*A + 9*B)*sec(c + d*x)^(7/2)*sin(c + d*x))/(63*d*sqrt(a + a*cos(c + d*x))) + (2*a*A*sec(c + d*x)^(9/2)*sin(c + d*x))/(9*d*sqrt(a + a*cos(c + d*x)))],
[sqrt(a + a*cos(c + d*x))*(A + B*cos(c + d*x))*sec(c + d*x)^(9/2), x, 4, (16*a*(6*A + 7*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(105*d*sqrt(a + a*cos(c + d*x))) + (8*a*(6*A + 7*B)*sec(c + d*x)^(3/2)*sin(c + d*x))/(105*d*sqrt(a + a*cos(c + d*x))) + (2*a*(6*A + 7*B)*sec(c + d*x)^(5/2)*sin(c + d*x))/(35*d*sqrt(a + a*cos(c + d*x))) + (2*a*A*sec(c + d*x)^(7/2)*sin(c + d*x))/(7*d*sqrt(a + a*cos(c + d*x)))],
[sqrt(a + a*cos(c + d*x))*(A + B*cos(c + d*x))*sec(c + d*x)^(7/2), x, 3, (4*a*(4*A + 5*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(15*d*sqrt(a + a*cos(c + d*x))) + (2*a*(4*A + 5*B)*sec(c + d*x)^(3/2)*sin(c + d*x))/(15*d*sqrt(a + a*cos(c + d*x))) + (2*a*A*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d*sqrt(a + a*cos(c + d*x)))],
[sqrt(a + a*cos(c + d*x))*(A + B*cos(c + d*x))*sec(c + d*x)^(5/2), x, 2, (2*a*(2*A + 3*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(3*d*sqrt(a + a*cos(c + d*x))) + (2*a*A*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d*sqrt(a + a*cos(c + d*x)))],
[sqrt(a + a*cos(c + d*x))*(A + B*cos(c + d*x))*sec(c + d*x)^(3/2), x, 3, (2*sqrt(a)*B*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/d + (2*a*A*sqrt(sec(c + d*x))*sin(c + d*x))/(d*sqrt(a + a*cos(c + d*x)))],
[sqrt(a + a*cos(c + d*x))*(A + B*cos(c + d*x))*sqrt(sec(c + d*x)), x, 3, (sqrt(a)*(2*A + B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/d + (a*B*sin(c + d*x))/(d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x)))],
[(sqrt(a + a*cos(c + d*x))*(A + B*cos(c + d*x)))/sqrt(sec(c + d*x)), x, 4, (sqrt(a)*(4*A + 3*B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(4*d) + (a*B*sin(c + d*x))/(2*d*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(3/2)) + (a*(4*A + 3*B)*sin(c + d*x))/(4*d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x)))],
[(sqrt(a + a*cos(c + d*x))*(A + B*cos(c + d*x)))/sec(c + d*x)^(3/2), x, 5, (sqrt(a)*(6*A + 5*B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(8*d) + (a*B*sin(c + d*x))/(3*d*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(5/2)) + (a*(6*A + 5*B)*sin(c + d*x))/(12*d*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(3/2)) + (a*(6*A + 5*B)*sin(c + d*x))/(8*d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x)))],


[(a + a*cos(c + d*x))^(3/2)*(A + B*cos(c + d*x))*sec(c + d*x)^(13/2), x, 6, (32*a^2*(168*A + 187*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(3465*d*sqrt(a + a*cos(c + d*x))) + (16*a^2*(168*A + 187*B)*sec(c + d*x)^(3/2)*sin(c + d*x))/(3465*d*sqrt(a + a*cos(c + d*x))) + (4*a^2*(168*A + 187*B)*sec(c + d*x)^(5/2)*sin(c + d*x))/(1155*d*sqrt(a + a*cos(c + d*x))) + (2*a^2*(168*A + 187*B)*sec(c + d*x)^(7/2)*sin(c + d*x))/(693*d*sqrt(a + a*cos(c + d*x))) + (2*a^2*(12*A + 11*B)*sec(c + d*x)^(9/2)*sin(c + d*x))/(99*d*sqrt(a + a*cos(c + d*x))) + (2*a*A*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(11/2)*sin(c + d*x))/(11*d)],
[(a + a*cos(c + d*x))^(3/2)*(A + B*cos(c + d*x))*sec(c + d*x)^(11/2), x, 5, (16*a^2*(34*A + 39*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(315*d*sqrt(a + a*cos(c + d*x))) + (8*a^2*(34*A + 39*B)*sec(c + d*x)^(3/2)*sin(c + d*x))/(315*d*sqrt(a + a*cos(c + d*x))) + (2*a^2*(34*A + 39*B)*sec(c + d*x)^(5/2)*sin(c + d*x))/(105*d*sqrt(a + a*cos(c + d*x))) + (2*a^2*(10*A + 9*B)*sec(c + d*x)^(7/2)*sin(c + d*x))/(63*d*sqrt(a + a*cos(c + d*x))) + (2*a*A*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(9/2)*sin(c + d*x))/(9*d)],
[(a + a*cos(c + d*x))^(3/2)*(A + B*cos(c + d*x))*sec(c + d*x)^(9/2), x, 4, (4*a^2*(52*A + 63*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(105*d*sqrt(a + a*cos(c + d*x))) + (2*a^2*(52*A + 63*B)*sec(c + d*x)^(3/2)*sin(c + d*x))/(105*d*sqrt(a + a*cos(c + d*x))) + (2*a^2*(8*A + 7*B)*sec(c + d*x)^(5/2)*sin(c + d*x))/(35*d*sqrt(a + a*cos(c + d*x))) + (2*a*A*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(7/2)*sin(c + d*x))/(7*d)],
[(a + a*cos(c + d*x))^(3/2)*(A + B*cos(c + d*x))*sec(c + d*x)^(7/2), x, 3, (8*a^2*(3*A + 5*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(15*d*sqrt(a + a*cos(c + d*x))) + (2*a*(3*A + 5*B)*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(15*d) + (2*A*(a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d)],
[(a + a*cos(c + d*x))^(3/2)*(A + B*cos(c + d*x))*sec(c + d*x)^(5/2), x, 4, (2*a^(3/2)*B*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/d + (2*a^2*(4*A + 3*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(3*d*sqrt(a + a*cos(c + d*x))) + (2*a*A*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d)],
[(a + a*cos(c + d*x))^(3/2)*(A + B*cos(c + d*x))*sec(c + d*x)^(3/2), x, 4, (a^(3/2)*(2*A + 3*B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/d - (a^2*(2*A - B)*sin(c + d*x))/(d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x))) + (2*a*A*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x))*sin(c + d*x))/d],
[(a + a*cos(c + d*x))^(3/2)*(A + B*cos(c + d*x))*sqrt(sec(c + d*x)), x, 4, (a^(3/2)*(12*A + 7*B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(4*d) + (a^2*(4*A + 5*B)*sin(c + d*x))/(4*d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x))) + (a*B*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(2*d*sqrt(sec(c + d*x)))],
[((a + a*cos(c + d*x))^(3/2)*(A + B*cos(c + d*x)))/sqrt(sec(c + d*x)), x, 5, (a^(3/2)*(14*A + 11*B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(8*d) + (a^2*(6*A + 7*B)*sin(c + d*x))/(12*d*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(3/2)) + (a*B*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(3*d*sec(c + d*x)^(3/2)) + (a^2*(14*A + 11*B)*sin(c + d*x))/(8*d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x)))],
[((a + a*cos(c + d*x))^(3/2)*(A + B*cos(c + d*x)))/sec(c + d*x)^(3/2), x, 6, (a^(3/2)*(88*A + 75*B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(64*d) + (a^2*(8*A + 9*B)*sin(c + d*x))/(24*d*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(5/2)) + (a*B*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(4*d*sec(c + d*x)^(5/2)) + (a^2*(88*A + 75*B)*sin(c + d*x))/(96*d*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(3/2)) + (a^2*(88*A + 75*B)*sin(c + d*x))/(64*d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x)))],


[(a + a*cos(c + d*x))^(5/2)*(A + B*cos(c + d*x))*sec(c + d*x)^(15/2), x, 7, (32*a^3*(4184*A + 4615*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(45045*d*sqrt(a + a*cos(c + d*x))) + (16*a^3*(4184*A + 4615*B)*sec(c + d*x)^(3/2)*sin(c + d*x))/(45045*d*sqrt(a + a*cos(c + d*x))) + (4*a^3*(4184*A + 4615*B)*sec(c + d*x)^(5/2)*sin(c + d*x))/(15015*d*sqrt(a + a*cos(c + d*x))) + (2*a^3*(4184*A + 4615*B)*sec(c + d*x)^(7/2)*sin(c + d*x))/(9009*d*sqrt(a + a*cos(c + d*x))) + (2*a^3*(280*A + 299*B)*sec(c + d*x)^(9/2)*sin(c + d*x))/(1287*d*sqrt(a + a*cos(c + d*x))) + (2*a^2*(16*A + 13*B)*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(11/2)*sin(c + d*x))/(143*d) + (2*a*A*(a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^(13/2)*sin(c + d*x))/(13*d)],
[(a + a*cos(c + d*x))^(5/2)*(A + B*cos(c + d*x))*sec(c + d*x)^(13/2), x, 6, (16*a^3*(710*A + 803*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(3465*d*sqrt(a + a*cos(c + d*x))) + (8*a^3*(710*A + 803*B)*sec(c + d*x)^(3/2)*sin(c + d*x))/(3465*d*sqrt(a + a*cos(c + d*x))) + (2*a^3*(710*A + 803*B)*sec(c + d*x)^(5/2)*sin(c + d*x))/(1155*d*sqrt(a + a*cos(c + d*x))) + (2*a^3*(194*A + 209*B)*sec(c + d*x)^(7/2)*sin(c + d*x))/(693*d*sqrt(a + a*cos(c + d*x))) + (2*a^2*(14*A + 11*B)*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(9/2)*sin(c + d*x))/(99*d) + (2*a*A*(a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^(11/2)*sin(c + d*x))/(11*d)],
[(a + a*cos(c + d*x))^(5/2)*(A + B*cos(c + d*x))*sec(c + d*x)^(11/2), x, 5, (4*a^3*(292*A + 345*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(315*d*sqrt(a + a*cos(c + d*x))) + (2*a^3*(292*A + 345*B)*sec(c + d*x)^(3/2)*sin(c + d*x))/(315*d*sqrt(a + a*cos(c + d*x))) + (2*a^3*(124*A + 135*B)*sec(c + d*x)^(5/2)*sin(c + d*x))/(315*d*sqrt(a + a*cos(c + d*x))) + (2*a^2*(4*A + 3*B)*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(7/2)*sin(c + d*x))/(21*d) + (2*a*A*(a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^(9/2)*sin(c + d*x))/(9*d)],
[(a + a*cos(c + d*x))^(5/2)*(A + B*cos(c + d*x))*sec(c + d*x)^(9/2), x, 4, (64*a^3*(5*A + 7*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(105*d*sqrt(a + a*cos(c + d*x))) + (16*a^2*(5*A + 7*B)*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(105*d) + (2*a*(5*A + 7*B)*(a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^(5/2)*sin(c + d*x))/(35*d) + (2*A*(a + a*cos(c + d*x))^(5/2)*sec(c + d*x)^(7/2)*sin(c + d*x))/(7*d)],
[(a + a*cos(c + d*x))^(5/2)*(A + B*cos(c + d*x))*sec(c + d*x)^(7/2), x, 5, (2*a^(5/2)*B*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/d + (2*a^3*(32*A + 35*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(15*d*sqrt(a + a*cos(c + d*x))) + (2*a^2*(8*A + 5*B)*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(15*d) + (2*a*A*(a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d)],
[(a + a*cos(c + d*x))^(5/2)*(A + B*cos(c + d*x))*sec(c + d*x)^(5/2), x, 5, (a^(5/2)*(2*A + 5*B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/d - (a^3*(14*A + 3*B)*sin(c + d*x))/(3*d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x))) + (2*a^2*(2*A + B)*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x))*sin(c + d*x))/d + (2*a*A*(a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d)],
[(a + a*cos(c + d*x))^(5/2)*(A + B*cos(c + d*x))*sec(c + d*x)^(3/2), x, 5, (a^(5/2)*(20*A + 19*B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(4*d) - (a^3*(4*A - 9*B)*sin(c + d*x))/(4*d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x))) - (a^2*(4*A - B)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(2*d*sqrt(sec(c + d*x))) + (2*a*A*(a + a*cos(c + d*x))^(3/2)*sqrt(sec(c + d*x))*sin(c + d*x))/d],
[(a + a*cos(c + d*x))^(5/2)*(A + B*cos(c + d*x))*sqrt(sec(c + d*x)), x, 5, (a^(5/2)*(38*A + 25*B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(8*d) + (a^3*(54*A + 49*B)*sin(c + d*x))/(24*d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x))) + (a^2*(2*A + 3*B)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(4*d*sqrt(sec(c + d*x))) + (a*B*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(3*d*sqrt(sec(c + d*x)))],
[((a + a*cos(c + d*x))^(5/2)*(A + B*cos(c + d*x)))/sqrt(sec(c + d*x)), x, 6, (a^(5/2)*(200*A + 163*B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(64*d) + (a^3*(104*A + 95*B)*sin(c + d*x))/(96*d*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(3/2)) + (a^2*(8*A + 11*B)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(24*d*sec(c + d*x)^(3/2)) + (a*B*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(4*d*sec(c + d*x)^(3/2)) + (a^3*(200*A + 163*B)*sin(c + d*x))/(64*d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x)))],
[((a + a*cos(c + d*x))^(5/2)*(A + B*cos(c + d*x)))/sec(c + d*x)^(3/2), x, 7, (a^(5/2)*(326*A + 283*B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(128*d) + (a^3*(170*A + 157*B)*sin(c + d*x))/(240*d*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(5/2)) + (a^2*(10*A + 13*B)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(40*d*sec(c + d*x)^(5/2)) + (a*B*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(5*d*sec(c + d*x)^(5/2)) + (a^3*(326*A + 283*B)*sin(c + d*x))/(192*d*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(3/2)) + (a^3*(326*A + 283*B)*sin(c + d*x))/(128*d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x)))],


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


[((A + B*cos(c + d*x))*sec(c + d*x)^(11/2))/sqrt(a + a*cos(c + d*x)), x, 7, -((sqrt(2)*(A - B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(sqrt(a)*d)) + (2*(257*A - 129*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(315*d*sqrt(a + a*cos(c + d*x))) - (2*(29*A - 93*B)*sec(c + d*x)^(3/2)*sin(c + d*x))/(315*d*sqrt(a + a*cos(c + d*x))) + (2*(19*A - 3*B)*sec(c + d*x)^(5/2)*sin(c + d*x))/(105*d*sqrt(a + a*cos(c + d*x))) - (2*(A - 9*B)*sec(c + d*x)^(7/2)*sin(c + d*x))/(63*d*sqrt(a + a*cos(c + d*x))) + (2*A*sec(c + d*x)^(9/2)*sin(c + d*x))/(9*d*sqrt(a + a*cos(c + d*x)))],
[((A + B*cos(c + d*x))*sec(c + d*x)^(9/2))/sqrt(a + a*cos(c + d*x)), x, 6, (sqrt(2)*(A - B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(sqrt(a)*d) - (2*(43*A - 91*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(105*d*sqrt(a + a*cos(c + d*x))) + (2*(31*A - 7*B)*sec(c + d*x)^(3/2)*sin(c + d*x))/(105*d*sqrt(a + a*cos(c + d*x))) - (2*(A - 7*B)*sec(c + d*x)^(5/2)*sin(c + d*x))/(35*d*sqrt(a + a*cos(c + d*x))) + (2*A*sec(c + d*x)^(7/2)*sin(c + d*x))/(7*d*sqrt(a + a*cos(c + d*x)))],
[((A + B*cos(c + d*x))*sec(c + d*x)^(7/2))/sqrt(a + a*cos(c + d*x)), x, 5, -((sqrt(2)*(A - B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(sqrt(a)*d)) + (2*(13*A - 5*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(15*d*sqrt(a + a*cos(c + d*x))) - (2*(A - 5*B)*sec(c + d*x)^(3/2)*sin(c + d*x))/(15*d*sqrt(a + a*cos(c + d*x))) + (2*A*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d*sqrt(a + a*cos(c + d*x)))],
[((A + B*cos(c + d*x))*sec(c + d*x)^(5/2))/sqrt(a + a*cos(c + d*x)), x, 4, (sqrt(2)*(A - B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(sqrt(a)*d) - (2*(A - 3*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(3*d*sqrt(a + a*cos(c + d*x))) + (2*A*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d*sqrt(a + a*cos(c + d*x)))],
[((A + B*cos(c + d*x))*sec(c + d*x)^(3/2))/sqrt(a + a*cos(c + d*x)), x, 3, -((sqrt(2)*(A - B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(sqrt(a)*d)) + (2*A*sqrt(sec(c + d*x))*sin(c + d*x))/(d*sqrt(a + a*cos(c + d*x)))],
[((A + B*cos(c + d*x))*sec(c + d*x)^(1/2))/sqrt(a + a*cos(c + d*x)), x, 4, (2*B*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(sqrt(a)*d) + (sqrt(2)*(A - B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(sqrt(a)*d)],
[(A + B*cos(c + d*x))/(sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(1/2)), x, 5, ((2*A - B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(sqrt(a)*d) - (sqrt(2)*(A - B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(sqrt(a)*d) + (B*sin(c + d*x))/(d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x)))],
[(A + B*cos(c + d*x))/(sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(3/2)), x, 6, -((4*A - 7*B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(4*sqrt(a)*d) + (sqrt(2)*(A - B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(sqrt(a)*d) + (B*sin(c + d*x))/(2*d*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(3/2)) + ((4*A - B)*sin(c + d*x))/(4*d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x)))],

[((a*A + (A*b + a*B)*cos(c + d*x) + b*B*cos(c + d*x)^2)*sqrt(sec(c + d*x)))/sqrt(a + a*cos(c + d*x)), x, 5, ((2*A*b + 2*a*B - b*B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(sqrt(a)*d) + (sqrt(2)*(a*A - A*b - a*B + b*B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(sqrt(a)*d) + (b*B*sin(c + d*x))/(d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x)))],


[((A + B*cos(c + d*x))*sec(c + d*x)^(9/2))/(a + a*cos(c + d*x))^(3/2), x, 7, ((19*A - 15*B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(2*sqrt(2)*a^(3/2)*d) - ((1201*A - 1029*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(210*a*d*sqrt(a + a*cos(c + d*x))) + ((397*A - 273*B)*sec(c + d*x)^(3/2)*sin(c + d*x))/(210*a*d*sqrt(a + a*cos(c + d*x))) - ((67*A - 63*B)*sec(c + d*x)^(5/2)*sin(c + d*x))/(70*a*d*sqrt(a + a*cos(c + d*x))) - ((A - B)*sec(c + d*x)^(7/2)*sin(c + d*x))/(2*d*(a + a*cos(c + d*x))^(3/2)) + ((11*A - 7*B)*sec(c + d*x)^(7/2)*sin(c + d*x))/(14*a*d*sqrt(a + a*cos(c + d*x)))],
[((A + B*cos(c + d*x))*sec(c + d*x)^(7/2))/(a + a*cos(c + d*x))^(3/2), x, 6, -((15*A - 11*B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(2*sqrt(2)*a^(3/2)*d) + ((147*A - 95*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(30*a*d*sqrt(a + a*cos(c + d*x))) - ((39*A - 35*B)*sec(c + d*x)^(3/2)*sin(c + d*x))/(30*a*d*sqrt(a + a*cos(c + d*x))) - ((A - B)*sec(c + d*x)^(5/2)*sin(c + d*x))/(2*d*(a + a*cos(c + d*x))^(3/2)) + ((9*A - 5*B)*sec(c + d*x)^(5/2)*sin(c + d*x))/(10*a*d*sqrt(a + a*cos(c + d*x)))],
[((A + B*cos(c + d*x))*sec(c + d*x)^(5/2))/(a + a*cos(c + d*x))^(3/2), x, 5, ((11*A - 7*B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(2*sqrt(2)*a^(3/2)*d) - ((19*A - 15*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(6*a*d*sqrt(a + a*cos(c + d*x))) - ((A - B)*sec(c + d*x)^(3/2)*sin(c + d*x))/(2*d*(a + a*cos(c + d*x))^(3/2)) + ((7*A - 3*B)*sec(c + d*x)^(3/2)*sin(c + d*x))/(6*a*d*sqrt(a + a*cos(c + d*x)))],
[((A + B*cos(c + d*x))*sec(c + d*x)^(3/2))/(a + a*cos(c + d*x))^(3/2), x, 4, -((7*A - 3*B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(2*sqrt(2)*a^(3/2)*d) - ((A - B)*sqrt(sec(c + d*x))*sin(c + d*x))/(2*d*(a + a*cos(c + d*x))^(3/2)) + ((5*A - B)*sqrt(sec(c + d*x))*sin(c + d*x))/(2*a*d*sqrt(a + a*cos(c + d*x)))],
[((A + B*cos(c + d*x))*sec(c + d*x)^(1/2))/(a + a*cos(c + d*x))^(3/2), x, 3, ((3*A + B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(2*sqrt(2)*a^(3/2)*d) - ((A - B)*sin(c + d*x))/(2*d*(a + a*cos(c + d*x))^(3/2)*sqrt(sec(c + d*x)))],
[(A + B*cos(c + d*x))/((a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^(1/2)), x, 5, (2*B*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(a^(3/2)*d) + ((A - 5*B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(2*sqrt(2)*a^(3/2)*d) + ((A - B)*sin(c + d*x))/(2*d*(a + a*cos(c + d*x))^(3/2)*sqrt(sec(c + d*x)))],
[(A + B*cos(c + d*x))/((a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^(3/2)), x, 6, ((2*A - 3*B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(a^(3/2)*d) - ((5*A - 9*B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(2*sqrt(2)*a^(3/2)*d) + ((A - B)*sin(c + d*x))/(2*d*(a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^(3/2)) - ((A - 3*B)*sin(c + d*x))/(2*a*d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x)))],


[((A + B*cos(c + d*x))*sec(c + d*x)^(7/2))/(a + a*cos(c + d*x))^(5/2), x, 7, -((283*A - 163*B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(16*sqrt(2)*a^(5/2)*d) + ((2671*A - 1495*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(240*a^2*d*sqrt(a + a*cos(c + d*x))) - ((787*A - 475*B)*sec(c + d*x)^(3/2)*sin(c + d*x))/(240*a^2*d*sqrt(a + a*cos(c + d*x))) - ((A - B)*sec(c + d*x)^(5/2)*sin(c + d*x))/(4*d*(a + a*cos(c + d*x))^(5/2)) - ((21*A - 13*B)*sec(c + d*x)^(5/2)*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2)) + ((157*A - 85*B)*sec(c + d*x)^(5/2)*sin(c + d*x))/(80*a^2*d*sqrt(a + a*cos(c + d*x)))],
[((A + B*cos(c + d*x))*sec(c + d*x)^(5/2))/(a + a*cos(c + d*x))^(5/2), x, 6, ((163*A - 75*B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(16*sqrt(2)*a^(5/2)*d) - ((299*A - 147*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(48*a^2*d*sqrt(a + a*cos(c + d*x))) - ((A - B)*sec(c + d*x)^(3/2)*sin(c + d*x))/(4*d*(a + a*cos(c + d*x))^(5/2)) - ((17*A - 9*B)*sec(c + d*x)^(3/2)*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2)) + ((95*A - 39*B)*sec(c + d*x)^(3/2)*sin(c + d*x))/(48*a^2*d*sqrt(a + a*cos(c + d*x)))],
[((A + B*cos(c + d*x))*sec(c + d*x)^(3/2))/(a + a*cos(c + d*x))^(5/2), x, 5, -((75*A - 19*B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(16*sqrt(2)*a^(5/2)*d) - ((A - B)*sqrt(sec(c + d*x))*sin(c + d*x))/(4*d*(a + a*cos(c + d*x))^(5/2)) - ((13*A - 5*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2)) + ((49*A - 9*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(16*a^2*d*sqrt(a + a*cos(c + d*x)))],
[((A + B*cos(c + d*x))*sec(c + d*x)^(1/2))/(a + a*cos(c + d*x))^(5/2), x, 4, ((19*A + 5*B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(16*sqrt(2)*a^(5/2)*d) - ((A - B)*sin(c + d*x))/(4*d*(a + a*cos(c + d*x))^(5/2)*sqrt(sec(c + d*x))) - ((9*A - B)*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2)*sqrt(sec(c + d*x)))],
[(A + B*cos(c + d*x))/((a + a*cos(c + d*x))^(5/2)*sec(c + d*x)^(1/2)), x, 4, ((5*A + 3*B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(16*sqrt(2)*a^(5/2)*d) - ((A - B)*sin(c + d*x))/(4*d*(a + a*cos(c + d*x))^(5/2)*sec(c + d*x)^(3/2)) + ((5*A + 3*B)*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2)*sqrt(sec(c + d*x)))],
[(A + B*cos(c + d*x))/((a + a*cos(c + d*x))^(5/2)*sec(c + d*x)^(3/2)), x, 6, (2*B*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(a^(5/2)*d) + ((3*A - 43*B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(16*sqrt(2)*a^(5/2)*d) + ((A - B)*sin(c + d*x))/(4*d*(a + a*cos(c + d*x))^(5/2)*sec(c + d*x)^(3/2)) + ((3*A - 11*B)*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2)*sqrt(sec(c + d*x)))],
[(A + B*cos(c + d*x))/((a + a*cos(c + d*x))^(5/2)*sec(c + d*x)^(5/2)), x, 7, ((2*A - 5*B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(a^(5/2)*d) - ((43*A - 115*B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(16*sqrt(2)*a^(5/2)*d) + ((A - B)*sin(c + d*x))/(4*d*(a + a*cos(c + d*x))^(5/2)*sec(c + d*x)^(5/2)) + ((7*A - 15*B)*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^(3/2)) - ((11*A - 35*B)*sin(c + d*x))/(16*a^2*d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x)))],


[((A + B*cos(c + d*x))*sec(c + d*x)^(5/2))/(a + a*cos(c + d*x))^(7/2), x, 7, ((1015*A - 363*B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(64*sqrt(2)*a^(7/2)*d) - ((1887*A - 691*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(192*a^3*d*sqrt(a + a*cos(c + d*x))) - ((A - B)*sec(c + d*x)^(3/2)*sin(c + d*x))/(6*d*(a + a*cos(c + d*x))^(7/2)) - ((23*A - 11*B)*sec(c + d*x)^(3/2)*sin(c + d*x))/(48*a*d*(a + a*cos(c + d*x))^(5/2)) - ((109*A - 41*B)*sec(c + d*x)^(3/2)*sin(c + d*x))/(64*a^2*d*(a + a*cos(c + d*x))^(3/2)) + ((579*A - 199*B)*sec(c + d*x)^(3/2)*sin(c + d*x))/(192*a^3*d*sqrt(a + a*cos(c + d*x)))],
[((A + B*cos(c + d*x))*sec(c + d*x)^(3/2))/(a + a*cos(c + d*x))^(7/2), x, 6, -((3*(121*A - 21*B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(64*sqrt(2)*a^(7/2)*d)) - ((A - B)*sqrt(sec(c + d*x))*sin(c + d*x))/(6*d*(a + a*cos(c + d*x))^(7/2)) - ((19*A - 7*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(48*a*d*(a + a*cos(c + d*x))^(5/2)) - ((199*A - 43*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(192*a^2*d*(a + a*cos(c + d*x))^(3/2)) + ((691*A - 103*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(192*a^3*d*sqrt(a + a*cos(c + d*x)))],
[((A + B*cos(c + d*x))*sec(c + d*x)^(1/2))/(a + a*cos(c + d*x))^(7/2), x, 5, ((63*A + 13*B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(64*sqrt(2)*a^(7/2)*d) - ((A - B)*sin(c + d*x))/(6*d*(a + a*cos(c + d*x))^(7/2)*sqrt(sec(c + d*x))) - ((5*A - B)*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(5/2)*sqrt(sec(c + d*x))) - ((103*A + 5*B)*sin(c + d*x))/(192*a^2*d*(a + a*cos(c + d*x))^(3/2)*sqrt(sec(c + d*x)))],
[(A + B*cos(c + d*x))/((a + a*cos(c + d*x))^(7/2)*sec(c + d*x)^(1/2)), x, 5, ((13*A + 7*B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(64*sqrt(2)*a^(7/2)*d) + ((A - B)*sin(c + d*x))/(6*d*(a + a*cos(c + d*x))^(7/2)*sqrt(sec(c + d*x))) + ((A + 3*B)*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(5/2)*sqrt(sec(c + d*x))) - ((5*A - 17*B)*sin(c + d*x))/(192*a^2*d*(a + a*cos(c + d*x))^(3/2)*sqrt(sec(c + d*x)))],
[(A + B*cos(c + d*x))/((a + a*cos(c + d*x))^(7/2)*sec(c + d*x)^(3/2)), x, 5, ((7*A + 5*B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(64*sqrt(2)*a^(7/2)*d) - ((A - B)*sin(c + d*x))/(6*d*(a + a*cos(c + d*x))^(7/2)*sec(c + d*x)^(5/2)) + ((7*A + 5*B)*sin(c + d*x))/(48*a*d*(a + a*cos(c + d*x))^(5/2)*sec(c + d*x)^(3/2)) + ((7*A + 5*B)*sin(c + d*x))/(64*a^2*d*(a + a*cos(c + d*x))^(3/2)*sqrt(sec(c + d*x)))],
[(A + B*cos(c + d*x))/((a + a*cos(c + d*x))^(7/2)*sec(c + d*x)^(5/2)), x, 7, (2*B*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(a^(7/2)*d) + ((5*A - 177*B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(64*sqrt(2)*a^(7/2)*d) + ((A - B)*sin(c + d*x))/(6*d*(a + a*cos(c + d*x))^(7/2)*sec(c + d*x)^(5/2)) + ((5*A - 17*B)*sin(c + d*x))/(48*a*d*(a + a*cos(c + d*x))^(5/2)*sec(c + d*x)^(3/2)) + ((5*A - 49*B)*sin(c + d*x))/(64*a^2*d*(a + a*cos(c + d*x))^(3/2)*sqrt(sec(c + d*x)))],
[(A + B*cos(c + d*x))/((a + a*cos(c + d*x))^(7/2)*sec(c + d*x)^(7/2)), x, 8, ((2*A - 7*B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(a^(7/2)*d) - ((177*A - 637*B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(64*sqrt(2)*a^(7/2)*d) + ((A - B)*sin(c + d*x))/(6*d*(a + a*cos(c + d*x))^(7/2)*sec(c + d*x)^(7/2)) + ((3*A - 7*B)*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(5/2)*sec(c + d*x)^(5/2)) + ((79*A - 259*B)*sin(c + d*x))/(192*a^2*d*(a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^(3/2)) - (7*(7*A - 27*B)*sin(c + d*x))/(64*a^3*d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x)))],


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


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


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


[cos(c + d*x)^6*(A + C*cos(c + d*x)^2), x, 4, (5/128)*(8*A + 7*C)*x + (5*(8*A + 7*C)*cos(c + d*x)*sin(c + d*x))/(128*d) + (5*(8*A + 7*C)*cos(c + d*x)^3*sin(c + d*x))/(192*d) + ((8*A + 7*C)*cos(c + d*x)^5*sin(c + d*x))/(48*d) + (C*cos(c + d*x)^7*sin(c + d*x))/(8*d)],
[cos(c + d*x)^5*(A + C*cos(c + d*x)^2), x, 3, ((7*A + 6*C)*sin(c + d*x))/(7*d) + (C*cos(c + d*x)^6*sin(c + d*x))/(7*d) - (2*(7*A + 6*C)*sin(c + d*x)^3)/(21*d) + ((7*A + 6*C)*sin(c + d*x)^5)/(35*d)],
[cos(c + d*x)^4*(A + C*cos(c + d*x)^2), x, 3, (1/16)*(6*A + 5*C)*x + ((6*A + 5*C)*cos(c + d*x)*sin(c + d*x))/(16*d) + ((6*A + 5*C)*cos(c + d*x)^3*sin(c + d*x))/(24*d) + (C*cos(c + d*x)^5*sin(c + d*x))/(6*d)],
[cos(c + d*x)^3*(A + C*cos(c + d*x)^2), x, 3, ((5*A + 4*C)*sin(c + d*x))/(5*d) + (C*cos(c + d*x)^4*sin(c + d*x))/(5*d) - ((5*A + 4*C)*sin(c + d*x)^3)/(15*d)],
[cos(c + d*x)^2*(A + C*cos(c + d*x)^2), x, 2, (1/8)*(4*A + 3*C)*x + ((4*A + 3*C)*cos(c + d*x)*sin(c + d*x))/(8*d) + (C*cos(c + d*x)^3*sin(c + d*x))/(4*d)],
[cos(c + d*x)^1*(A + C*cos(c + d*x)^2), x, 2, ((3*A + 2*C)*sin(c + d*x))/(3*d) + (C*cos(c + d*x)^2*sin(c + d*x))/(3*d)],
[cos(c + d*x)^0*(A + C*cos(c + d*x)^2), x, 2, A*x + (C*x)/2 + (C*cos(c + d*x)*sin(c + d*x))/(2*d)],
[sec(c + d*x)^1*(A + C*cos(c + d*x)^2), x, 2, (A*arctanh(sin(c + d*x)))/d + (C*sin(c + d*x))/d],
[sec(c + d*x)^2*(A + C*cos(c + d*x)^2), x, 2, C*x + (A*tan(c + d*x))/d],
[sec(c + d*x)^3*(A + C*cos(c + d*x)^2), x, 2, ((A + 2*C)*arctanh(sin(c + d*x)))/(2*d) + (A*sec(c + d*x)*tan(c + d*x))/(2*d)],
[sec(c + d*x)^4*(A + C*cos(c + d*x)^2), x, 2, ((2*A + 3*C)*tan(c + d*x))/(3*d) + (A*sec(c + d*x)^2*tan(c + d*x))/(3*d)],
[sec(c + d*x)^5*(A + C*cos(c + d*x)^2), x, 3, ((3*A + 4*C)*arctanh(sin(c + d*x)))/(8*d) + ((3*A + 4*C)*sec(c + d*x)*tan(c + d*x))/(8*d) + (A*sec(c + d*x)^3*tan(c + d*x))/(4*d)],
[sec(c + d*x)^6*(A + C*cos(c + d*x)^2), x, 3, ((4*A + 5*C)*tan(c + d*x))/(5*d) + (A*sec(c + d*x)^4*tan(c + d*x))/(5*d) + ((4*A + 5*C)*tan(c + d*x)^3)/(15*d)],
[sec(c + d*x)^7*(A + C*cos(c + d*x)^2), x, 4, ((5*A + 6*C)*arctanh(sin(c + d*x)))/(16*d) + ((5*A + 6*C)*sec(c + d*x)*tan(c + d*x))/(16*d) + ((5*A + 6*C)*sec(c + d*x)^3*tan(c + d*x))/(24*d) + (A*sec(c + d*x)^5*tan(c + d*x))/(6*d)],
[sec(c + d*x)^8*(A + C*cos(c + d*x)^2), x, 3, ((6*A + 7*C)*tan(c + d*x))/(7*d) + (A*sec(c + d*x)^6*tan(c + d*x))/(7*d) + (2*(6*A + 7*C)*tan(c + d*x)^3)/(21*d) + ((6*A + 7*C)*tan(c + d*x)^5)/(35*d)],

[cos(c + d*x)^m*((-C)*((m + 1)/(m + 2)) + C*cos(c + d*x)^2), x, 1, (C*cos(c + d*x)^(1 + m)*sin(c + d*x))/(d*(2 + m))],
[cos(c + d*x)^m*(A + (-A)*((m + 2)/(m + 1))*cos(c + d*x)^2), x, 1, -((A*cos(c + d*x)^(1 + m)*sin(c + d*x))/(d*(1 + m)))],


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


[cos(c + d*x)^2*(A + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^2, x, 6, (1/16)*a^2*(14*A + 11*C)*x + (4*a^2*(5*A + 4*C)*sin(c + d*x))/(15*d) + (a^2*(14*A + 11*C)*cos(c + d*x)*sin(c + d*x))/(16*d) + (2*a^2*(5*A + 4*C)*cos(c + d*x)^2*sin(c + d*x))/(15*d) + (a^2*(10*A + 9*C)*cos(c + d*x)^3*sin(c + d*x))/(40*d) + (a*C*cos(c + d*x)^3*(a + a*cos(c + d*x))*sin(c + d*x))/(15*d) + (C*cos(c + d*x)^3*(a + a*cos(c + d*x))^2*sin(c + d*x))/(6*d)],
[cos(c + d*x)^1*(A + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^2, x, 5, (1/4)*a^2*(4*A + 3*C)*x + (a^2*(4*A + 3*C)*sin(c + d*x))/(3*d) + (a^2*(4*A + 3*C)*cos(c + d*x)*sin(c + d*x))/(12*d) + ((10*A + 3*C)*(a + a*cos(c + d*x))^2*sin(c + d*x))/(30*d) + (C*cos(c + d*x)^2*(a + a*cos(c + d*x))^2*sin(c + d*x))/(5*d) + (C*(a + a*cos(c + d*x))^3*sin(c + d*x))/(10*a*d)],
[cos(c + d*x)^0*(A + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^2, x, 4, (1/8)*a^2*(12*A + 7*C)*x + (a^2*(12*A + 7*C)*sin(c + d*x))/(6*d) + (a^2*(12*A + 7*C)*cos(c + d*x)*sin(c + d*x))/(24*d) - (C*(a + a*cos(c + d*x))^2*sin(c + d*x))/(12*d) + (C*(a + a*cos(c + d*x))^3*sin(c + d*x))/(4*a*d)],
[sec(c + d*x)^1*(A + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^2, x, 5, (2*a^2*A + a^2*C)*x + (a^2*A*arctanh(sin(c + d*x)))/d + (a^2*(A + C)*sin(c + d*x))/d + (a*C*(a + a*cos(c + d*x))*sin(c + d*x))/(3*d) + (C*(a + a*cos(c + d*x))^2*sin(c + d*x))/(3*d)],
[sec(c + d*x)^2*(A + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^2, x, 5, (1/2)*a^2*(2*A + 3*C)*x + (2*a^2*A*arctanh(sin(c + d*x)))/d - (a^2*(2*A - 3*C)*sin(c + d*x))/(2*d) - (a*(2*A - C)*(a + a*cos(c + d*x))*sin(c + d*x))/(2*d) + (A*(a + a*cos(c + d*x))^2*tan(c + d*x))/d],
[sec(c + d*x)^3*(A + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^2, x, 5, 2*a^2*C*x + (a^2*(3*A + 2*C)*arctanh(sin(c + d*x)))/(2*d) - (a^2*(3*A - 2*C)*sin(c + d*x))/(2*d) + (a*A*(a + a*cos(c + d*x))*tan(c + d*x))/d + (A*(a + a*cos(c + d*x))^2*sec(c + d*x)*tan(c + d*x))/(2*d)],
[sec(c + d*x)^4*(A + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^2, x, 5, a^2*C*x + (a^2*(A + 2*C)*arctanh(sin(c + d*x)))/d + (a^2*(A + C)*tan(c + d*x))/d + (a*A*(a + a*sec(c + d*x))*tan(c + d*x))/(3*d) + (A*(a + a*sec(c + d*x))^2*tan(c + d*x))/(3*d)],
[sec(c + d*x)^5*(A + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^2, x, 6, (a^2*(7*A + 12*C)*arctanh(sin(c + d*x)))/(8*d) + (a^2*(7*A + 12*C)*tan(c + d*x))/(6*d) + (a^2*(7*A + 12*C)*sec(c + d*x)*tan(c + d*x))/(24*d) + (A*(a + a*cos(c + d*x))^2*sec(c + d*x)^3*tan(c + d*x))/(4*d) + (A*(a + a*sec(c + d*x))^2*tan(c + d*x))/(6*d)],
[sec(c + d*x)^6*(A + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^2, x, 7, (a^2*(3*A + 4*C)*arctanh(sin(c + d*x)))/(4*d) + (a^2*(18*A + 25*C)*tan(c + d*x))/(15*d) + (a^2*(3*A + 4*C)*sec(c + d*x)*tan(c + d*x))/(4*d) + (a^2*(9*A + 10*C)*sec(c + d*x)^2*tan(c + d*x))/(30*d) + (a*A*(a + a*cos(c + d*x))*sec(c + d*x)^3*tan(c + d*x))/(10*d) + (A*(a + a*cos(c + d*x))^2*sec(c + d*x)^4*tan(c + d*x))/(5*d)],


[cos(c + d*x)^2*(A + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^3, x, 7, (1/16)*a^3*(26*A + 21*C)*x + (2*a^3*(133*A + 108*C)*sin(c + d*x))/(105*d) + (a^3*(26*A + 21*C)*cos(c + d*x)*sin(c + d*x))/(16*d) + (a^3*(133*A + 108*C)*cos(c + d*x)^2*sin(c + d*x))/(105*d) + (a^3*(154*A + 129*C)*cos(c + d*x)^3*sin(c + d*x))/(280*d) + (a^2*(A + C)*cos(c + d*x)^3*(a + a*cos(c + d*x))*sin(c + d*x))/(5*d) + (a*C*cos(c + d*x)^3*(a + a*cos(c + d*x))^2*sin(c + d*x))/(14*d) + (C*cos(c + d*x)^3*(a + a*cos(c + d*x))^3*sin(c + d*x))/(7*d)],
[cos(c + d*x)^1*(A + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^3, x, 6, (1/16)*a^3*(30*A + 23*C)*x + (a^3*(30*A + 23*C)*sin(c + d*x))/(12*d) + (a^3*(30*A + 23*C)*cos(c + d*x)*sin(c + d*x))/(48*d) + (a*(30*A + 23*C)*(a + a*cos(c + d*x))^2*sin(c + d*x))/(120*d) + ((30*A + 7*C)*(a + a*cos(c + d*x))^3*sin(c + d*x))/(120*d) + (C*cos(c + d*x)^2*(a + a*cos(c + d*x))^3*sin(c + d*x))/(6*d) + (C*(a + a*cos(c + d*x))^4*sin(c + d*x))/(10*a*d)],
[cos(c + d*x)^0*(A + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^3, x, 5, (1/8)*a^3*(20*A + 13*C)*x + (a^3*(20*A + 13*C)*sin(c + d*x))/(6*d) + (a^3*(20*A + 13*C)*cos(c + d*x)*sin(c + d*x))/(24*d) + (a*(20*A + 13*C)*(a + a*cos(c + d*x))^2*sin(c + d*x))/(60*d) - (C*(a + a*cos(c + d*x))^3*sin(c + d*x))/(20*d) + (C*(a + a*cos(c + d*x))^4*sin(c + d*x))/(5*a*d)],
[sec(c + d*x)^1*(A + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^3, x, 6, (1/8)*a^3*(28*A + 15*C)*x + (a^3*A*arctanh(sin(c + d*x)))/d + (5*a^3*(4*A + 3*C)*sin(c + d*x))/(8*d) + (a^2*(4*A + 5*C)*(a + a*cos(c + d*x))*sin(c + d*x))/(8*d) + (a*C*(a + a*cos(c + d*x))^2*sin(c + d*x))/(4*d) + (C*(a + a*cos(c + d*x))^3*sin(c + d*x))/(4*d)],
[sec(c + d*x)^2*(A + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^3, x, 6, (1/2)*a^3*(6*A + 5*C)*x + (3*a^3*A*arctanh(sin(c + d*x)))/d + (5*a^3*C*sin(c + d*x))/(2*d) - (a^2*(6*A - 5*C)*(a + a*cos(c + d*x))*sin(c + d*x))/(6*d) - (a*(3*A - C)*(a + a*cos(c + d*x))^2*sin(c + d*x))/(3*d) + (A*(a + a*cos(c + d*x))^3*tan(c + d*x))/d],
[sec(c + d*x)^3*(A + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^3, x, 6, (1/2)*a^3*(2*A + 7*C)*x + (a^3*(7*A + 2*C)*arctanh(sin(c + d*x)))/(2*d) - (5*a^3*(A - C)*sin(c + d*x))/(2*d) - (a^2*(4*A - C)*(a + a*cos(c + d*x))*sin(c + d*x))/(2*d) + (3*a*A*(a + a*cos(c + d*x))^2*tan(c + d*x))/(2*d) + (A*(a + a*cos(c + d*x))^3*sec(c + d*x)*tan(c + d*x))/(2*d)],
[sec(c + d*x)^4*(A + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^3, x, 6, 3*a^3*C*x + (a^3*(5*A + 6*C)*arctanh(sin(c + d*x)))/(2*d) - (5*a^3*A*sin(c + d*x))/(2*d) + (a^2*(5*A + 3*C)*(a + a*cos(c + d*x))*tan(c + d*x))/(3*d) + (a*A*(a + a*cos(c + d*x))^2*sec(c + d*x)*tan(c + d*x))/(2*d) + (A*(a + a*cos(c + d*x))^3*sec(c + d*x)^2*tan(c + d*x))/(3*d)],
[sec(c + d*x)^5*(A + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^3, x, 6, a^3*C*x + (a^3*(15*A + 28*C)*arctanh(sin(c + d*x)))/(8*d) + (5*a^3*(3*A + 4*C)*tan(c + d*x))/(8*d) + (a^2*(5*A + 4*C)*(a + a*sec(c + d*x))*tan(c + d*x))/(8*d) + (a*A*(a + a*sec(c + d*x))^2*tan(c + d*x))/(4*d) + (A*(a + a*sec(c + d*x))^3*tan(c + d*x))/(4*d)],
[sec(c + d*x)^6*(A + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^3, x, 7, (a^3*(13*A + 20*C)*arctanh(sin(c + d*x)))/(8*d) + (a^3*(13*A + 20*C)*tan(c + d*x))/(6*d) + (a^3*(13*A + 20*C)*sec(c + d*x)*tan(c + d*x))/(24*d) + (A*(a + a*cos(c + d*x))^3*sec(c + d*x)^4*tan(c + d*x))/(5*d) + (a*(13*A + 20*C)*(a + a*sec(c + d*x))^2*tan(c + d*x))/(60*d) + (3*A*(a + a*sec(c + d*x))^3*tan(c + d*x))/(20*d)],
[sec(c + d*x)^7*(A + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^3, x, 8, (a^3*(23*A + 30*C)*arctanh(sin(c + d*x)))/(16*d) + (a^3*(34*A + 45*C)*tan(c + d*x))/(15*d) + (a^3*(23*A + 30*C)*sec(c + d*x)*tan(c + d*x))/(16*d) + (a^3*(73*A + 90*C)*sec(c + d*x)^2*tan(c + d*x))/(120*d) + (a^2*(31*A + 30*C)*(a + a*cos(c + d*x))*sec(c + d*x)^3*tan(c + d*x))/(120*d) + (a*A*(a + a*cos(c + d*x))^2*sec(c + d*x)^4*tan(c + d*x))/(10*d) + (A*(a + a*cos(c + d*x))^3*sec(c + d*x)^5*tan(c + d*x))/(6*d)],


[cos(c + d*x)^2*(A + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^4, x, 8, (1/128)*a^4*(392*A + 323*C)*x + (8*a^4*(63*A + 52*C)*sin(c + d*x))/(105*d) + (a^4*(392*A + 323*C)*cos(c + d*x)*sin(c + d*x))/(128*d) + (4*a^4*(63*A + 52*C)*cos(c + d*x)^2*sin(c + d*x))/(105*d) + (a^4*(2408*A + 2007*C)*cos(c + d*x)^3*sin(c + d*x))/(2240*d) + (7*a^3*(8*A + 7*C)*cos(c + d*x)^3*(a + a*cos(c + d*x))*sin(c + d*x))/(120*d) + (a^2*(56*A + 61*C)*cos(c + d*x)^3*(a + a*cos(c + d*x))^2*sin(c + d*x))/(336*d) + (a*C*cos(c + d*x)^3*(a + a*cos(c + d*x))^3*sin(c + d*x))/(14*d) + (C*cos(c + d*x)^3*(a + a*cos(c + d*x))^4*sin(c + d*x))/(8*d)],
[cos(c + d*x)^1*(A + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^4, x, 7, (1/4)*a^4*(14*A + 11*C)*x + (a^4*(14*A + 11*C)*sin(c + d*x))/(3*d) + (a^4*(14*A + 11*C)*cos(c + d*x)*sin(c + d*x))/(12*d) + (a^2*(14*A + 11*C)*(a + a*cos(c + d*x))^2*sin(c + d*x))/(30*d) + (a*(14*A + 11*C)*(a + a*cos(c + d*x))^3*sin(c + d*x))/(70*d) + ((21*A + 4*C)*(a + a*cos(c + d*x))^4*sin(c + d*x))/(105*d) + (C*cos(c + d*x)^2*(a + a*cos(c + d*x))^4*sin(c + d*x))/(7*d) + (2*C*(a + a*cos(c + d*x))^5*sin(c + d*x))/(21*a*d)],
[cos(c + d*x)^0*(A + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^4, x, 6, (7/16)*a^4*(10*A + 7*C)*x + (7*a^4*(10*A + 7*C)*sin(c + d*x))/(12*d) + (7*a^4*(10*A + 7*C)*cos(c + d*x)*sin(c + d*x))/(48*d) + (7*a^2*(10*A + 7*C)*(a + a*cos(c + d*x))^2*sin(c + d*x))/(120*d) + (a*(10*A + 7*C)*(a + a*cos(c + d*x))^3*sin(c + d*x))/(40*d) - (C*(a + a*cos(c + d*x))^4*sin(c + d*x))/(30*d) + (C*(a + a*cos(c + d*x))^5*sin(c + d*x))/(6*a*d)],
[sec(c + d*x)^1*(A + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^4, x, 7, (1/2)*a^4*(12*A + 7*C)*x + (a^4*A*arctanh(sin(c + d*x)))/d + (a^4*(10*A + 7*C)*sin(c + d*x))/(2*d) + (a^3*(8*A + 7*C)*(a + a*cos(c + d*x))*sin(c + d*x))/(6*d) + (a^2*(5*A + 7*C)*(a + a*cos(c + d*x))^2*sin(c + d*x))/(15*d) + (a*C*(a + a*cos(c + d*x))^3*sin(c + d*x))/(5*d) + (C*(a + a*cos(c + d*x))^4*sin(c + d*x))/(5*d)],
[sec(c + d*x)^2*(A + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^4, x, 7, (1/8)*a^4*(52*A + 35*C)*x + (4*a^4*A*arctanh(sin(c + d*x)))/d + (5*a^4*(4*A + 7*C)*sin(c + d*x))/(8*d) - (a^3*(12*A - 35*C)*(a + a*cos(c + d*x))*sin(c + d*x))/(24*d) - (a^2*(12*A - 7*C)*(a + a*cos(c + d*x))^2*sin(c + d*x))/(12*d) - (a*(4*A - C)*(a + a*cos(c + d*x))^3*sin(c + d*x))/(4*d) + (A*(a + a*cos(c + d*x))^4*tan(c + d*x))/d],
[sec(c + d*x)^3*(A + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^4, x, 7, 2*a^4*(2*A + 3*C)*x + (a^4*(13*A + 2*C)*arctanh(sin(c + d*x)))/(2*d) - (5*a^4*(A - 2*C)*sin(c + d*x))/(2*d) - (a^3*(9*A - 4*C)*(a + a*cos(c + d*x))*sin(c + d*x))/(3*d) - (a^2*(15*A - 2*C)*(a + a*cos(c + d*x))^2*sin(c + d*x))/(6*d) + (2*a*A*(a + a*cos(c + d*x))^3*tan(c + d*x))/d + (A*(a + a*cos(c + d*x))^4*sec(c + d*x)*tan(c + d*x))/(2*d)],
[sec(c + d*x)^4*(A + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^4, x, 7, (1/2)*a^4*(2*A + 13*C)*x + (2*a^4*(3*A + 2*C)*arctanh(sin(c + d*x)))/d - (5*a^4*(2*A - C)*sin(c + d*x))/(2*d) - (a^3*(22*A + 3*C)*(a + a*cos(c + d*x))*sin(c + d*x))/(6*d) + (a^2*(8*A + 3*C)*(a + a*cos(c + d*x))^2*tan(c + d*x))/(3*d) + (2*a*A*(a + a*cos(c + d*x))^3*sec(c + d*x)*tan(c + d*x))/(3*d) + (A*(a + a*cos(c + d*x))^4*sec(c + d*x)^2*tan(c + d*x))/(3*d)],
[sec(c + d*x)^5*(A + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^4, x, 7, 4*a^4*C*x + (a^4*(35*A + 52*C)*arctanh(sin(c + d*x)))/(8*d) - (5*a^4*(7*A + 4*C)*sin(c + d*x))/(8*d) + (a^3*(35*A + 36*C)*(a + a*cos(c + d*x))*tan(c + d*x))/(12*d) + (a^2*(7*A + 4*C)*(a + a*cos(c + d*x))^2*sec(c + d*x)*tan(c + d*x))/(8*d) + (a*A*(a + a*cos(c + d*x))^3*sec(c + d*x)^2*tan(c + d*x))/(3*d) + (A*(a + a*cos(c + d*x))^4*sec(c + d*x)^3*tan(c + d*x))/(4*d)],
[sec(c + d*x)^6*(A + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^4, x, 7, a^4*C*x + (a^4*(7*A + 12*C)*arctanh(sin(c + d*x)))/(2*d) + (a^4*(7*A + 10*C)*tan(c + d*x))/(2*d) + (a^3*(7*A + 8*C)*(a + a*sec(c + d*x))*tan(c + d*x))/(6*d) + (a^2*(7*A + 5*C)*(a + a*sec(c + d*x))^2*tan(c + d*x))/(15*d) + (a*A*(a + a*sec(c + d*x))^3*tan(c + d*x))/(5*d) + (A*(a + a*sec(c + d*x))^4*tan(c + d*x))/(5*d)],
[sec(c + d*x)^7*(A + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^4, x, 8, (7*a^4*(7*A + 10*C)*arctanh(sin(c + d*x)))/(16*d) + (7*a^4*(7*A + 10*C)*tan(c + d*x))/(12*d) + (7*a^4*(7*A + 10*C)*sec(c + d*x)*tan(c + d*x))/(48*d) + (A*(a + a*cos(c + d*x))^4*sec(c + d*x)^5*tan(c + d*x))/(6*d) + (7*a^2*(7*A + 10*C)*(a + a*sec(c + d*x))^2*tan(c + d*x))/(120*d) + (a*(7*A + 10*C)*(a + a*sec(c + d*x))^3*tan(c + d*x))/(40*d) + (2*A*(a + a*sec(c + d*x))^4*tan(c + d*x))/(15*d)],
[sec(c + d*x)^8*(A + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^4, x, 9, (a^4*(11*A + 14*C)*arctanh(sin(c + d*x)))/(4*d) + (a^4*(454*A + 581*C)*tan(c + d*x))/(105*d) + (a^4*(11*A + 14*C)*sec(c + d*x)*tan(c + d*x))/(4*d) + (a^4*(247*A + 308*C)*sec(c + d*x)^2*tan(c + d*x))/(210*d) + (a^3*(109*A + 126*C)*(a + a*cos(c + d*x))*sec(c + d*x)^3*tan(c + d*x))/(210*d) + (a^2*(8*A + 7*C)*(a + a*cos(c + d*x))^2*sec(c + d*x)^4*tan(c + d*x))/(35*d) + (2*a*A*(a + a*cos(c + d*x))^3*sec(c + d*x)^5*tan(c + d*x))/(21*d) + (A*(a + a*cos(c + d*x))^4*sec(c + d*x)^6*tan(c + d*x))/(7*d)],


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


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


[cos(c + d*x)^4*(A + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^2, x, 6, ((28*A + 55*C)*x)/(8*a^2) - (16*(A + 2*C)*sin(c + d*x))/(3*a^2*d) + ((28*A + 55*C)*cos(c + d*x)*sin(c + d*x))/(8*a^2*d) - (8*(A + 2*C)*cos(c + d*x)^2*sin(c + d*x))/(3*a^2*d) + ((28*A + 55*C)*cos(c + d*x)^3*sin(c + d*x))/(12*a^2*d) - ((A + C)*cos(c + d*x)^5*sin(c + d*x))/(3*d*(a + a*cos(c + d*x))^2) - (2*(A + 2*C)*cos(c + d*x)^4*sin(c + d*x))/(a*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^3*(A + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^2, x, 5, -(((2*A + 5*C)*x)/a^2) + (2*(5*A + 12*C)*sin(c + d*x))/(3*a^2*d) - ((2*A + 5*C)*cos(c + d*x)*sin(c + d*x))/(a^2*d) + ((5*A + 12*C)*cos(c + d*x)^2*sin(c + d*x))/(3*a^2*d) - ((A + C)*cos(c + d*x)^4*sin(c + d*x))/(3*d*(a + a*cos(c + d*x))^2) - (2*(2*A + 5*C)*cos(c + d*x)^3*sin(c + d*x))/(3*a*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^2*(A + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^2, x, 4, ((2*A + 7*C)*x)/(2*a^2) - (4*(A + 4*C)*sin(c + d*x))/(3*a^2*d) + ((2*A + 7*C)*cos(c + d*x)*sin(c + d*x))/(2*a^2*d) - ((A + C)*cos(c + d*x)^3*sin(c + d*x))/(3*d*(a + a*cos(c + d*x))^2) - (2*(A + 4*C)*cos(c + d*x)^2*sin(c + d*x))/(3*a*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^1*(A + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^2, x, 4, -((2*C*x)/a^2) + ((A + 4*C)*sin(c + d*x))/(3*a^2*d) - ((A + C)*cos(c + d*x)^2*sin(c + d*x))/(3*d*(a + a*cos(c + d*x))^2) + (2*C*sin(c + d*x))/(a*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^0*(A + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^2, x, 3, (C*x)/a^2 + ((A + C)*sin(c + d*x))/(3*d*(a + a*cos(c + d*x))^2) + ((A - 5*C)*sin(c + d*x))/(3*a*d*(a + a*cos(c + d*x)))],
[sec(c + d*x)^1*(A + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^2, x, 3, (A*arctanh(sin(c + d*x)))/(a^2*d) - ((A + C)*sin(c + d*x))/(3*d*(a + a*cos(c + d*x))^2) - (2*(2*A - C)*sin(c + d*x))/(3*a*d*(a + a*cos(c + d*x)))],
[sec(c + d*x)^2*(A + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^2, x, 5, -((2*A*arctanh(sin(c + d*x)))/(a^2*d)) + ((10*A + C)*tan(c + d*x))/(3*a^2*d) - ((A + C)*tan(c + d*x))/(3*d*(a + a*cos(c + d*x))^2) - (2*A*tan(c + d*x))/(a*d*(a + a*cos(c + d*x)))],
[sec(c + d*x)^3*(A + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^2, x, 6, ((7*A + 2*C)*arctanh(sin(c + d*x)))/(2*a^2*d) - (4*(4*A + C)*tan(c + d*x))/(3*a^2*d) + ((7*A + 2*C)*sec(c + d*x)*tan(c + d*x))/(2*a^2*d) - ((A + C)*sec(c + d*x)*tan(c + d*x))/(3*d*(a + a*cos(c + d*x))^2) - (2*(4*A + C)*sec(c + d*x)*tan(c + d*x))/(3*a*d*(a + a*cos(c + d*x)))],
[sec(c + d*x)^4*(A + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^2, x, 7, -(((5*A + 2*C)*arctanh(sin(c + d*x)))/(a^2*d)) + (2*(12*A + 5*C)*tan(c + d*x))/(3*a^2*d) - ((5*A + 2*C)*sec(c + d*x)*tan(c + d*x))/(a^2*d) + ((12*A + 5*C)*sec(c + d*x)^2*tan(c + d*x))/(3*a^2*d) - ((A + C)*sec(c + d*x)^2*tan(c + d*x))/(3*d*(a + a*cos(c + d*x))^2) - (2*(5*A + 2*C)*sec(c + d*x)^2*tan(c + d*x))/(3*a*d*(a + a*cos(c + d*x)))],


[cos(c + d*x)^4*(A + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^3, x, 6, -(((6*A + 23*C)*x)/(2*a^3)) + (8*(9*A + 34*C)*sin(c + d*x))/(15*a^3*d) - ((6*A + 23*C)*cos(c + d*x)*sin(c + d*x))/(2*a^3*d) + (4*(9*A + 34*C)*cos(c + d*x)^2*sin(c + d*x))/(15*a^3*d) - ((A + C)*cos(c + d*x)^5*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3) - ((3*A + 13*C)*cos(c + d*x)^4*sin(c + d*x))/(15*a*d*(a + a*cos(c + d*x))^2) - ((6*A + 23*C)*cos(c + d*x)^3*sin(c + d*x))/(3*a^2*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^3*(A + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^3, x, 5, ((2*A + 13*C)*x)/(2*a^3) - (2*(11*A + 76*C)*sin(c + d*x))/(15*a^3*d) + ((2*A + 13*C)*cos(c + d*x)*sin(c + d*x))/(2*a^3*d) - ((A + C)*cos(c + d*x)^4*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3) - ((A + 11*C)*cos(c + d*x)^3*sin(c + d*x))/(15*a*d*(a + a*cos(c + d*x))^2) - ((11*A + 76*C)*cos(c + d*x)^2*sin(c + d*x))/(15*a^2*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^2*(A + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^3, x, 5, -((3*C*x)/a^3) + ((2*A + 27*C)*sin(c + d*x))/(15*a^3*d) - ((A + C)*cos(c + d*x)^3*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3) + ((A - 9*C)*cos(c + d*x)^2*sin(c + d*x))/(15*a*d*(a + a*cos(c + d*x))^2) + (3*C*sin(c + d*x))/(a^2*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^1*(A + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^3, x, 4, (C*x)/a^3 - ((A + C)*cos(c + d*x)^2*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3) + ((3*A - 7*C)*cos(c + d*x)*sin(c + d*x))/(15*a*d*(a + a*cos(c + d*x))^2) + ((3*A - 22*C)*sin(c + d*x))/(15*a^2*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^0*(A + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^3, x, 3, ((A + C)*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3) + (2*(A - 4*C)*sin(c + d*x))/(15*a*d*(a + a*cos(c + d*x))^2) + ((2*A + 7*C)*sin(c + d*x))/(15*a^2*d*(a + a*cos(c + d*x)))],
[sec(c + d*x)^1*(A + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^3, x, 4, (A*arctanh(sin(c + d*x)))/(a^3*d) - ((A + C)*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3) - ((7*A - 3*C)*sin(c + d*x))/(15*a*d*(a + a*cos(c + d*x))^2) - ((22*A - 3*C)*sin(c + d*x))/(15*a^2*d*(a + a*cos(c + d*x)))],
[sec(c + d*x)^2*(A + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^3, x, 6, -((3*A*arctanh(sin(c + d*x)))/(a^3*d)) + (2*(36*A + C)*tan(c + d*x))/(15*a^3*d) - ((A + C)*tan(c + d*x))/(5*d*(a + a*cos(c + d*x))^3) - ((9*A - C)*tan(c + d*x))/(15*a*d*(a + a*cos(c + d*x))^2) - (3*A*tan(c + d*x))/(a^2*d*(a + a*cos(c + d*x)))],
[sec(c + d*x)^3*(A + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^3, x, 7, ((13*A + 2*C)*arctanh(sin(c + d*x)))/(2*a^3*d) - (2*(76*A + 11*C)*tan(c + d*x))/(15*a^3*d) + ((13*A + 2*C)*sec(c + d*x)*tan(c + d*x))/(2*a^3*d) - ((A + C)*sec(c + d*x)*tan(c + d*x))/(5*d*(a + a*cos(c + d*x))^3) - ((11*A + C)*sec(c + d*x)*tan(c + d*x))/(15*a*d*(a + a*cos(c + d*x))^2) - ((76*A + 11*C)*sec(c + d*x)*tan(c + d*x))/(15*a^2*d*(a + a*cos(c + d*x)))],
[sec(c + d*x)^4*(A + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^3, x, 8, -(((23*A + 6*C)*arctanh(sin(c + d*x)))/(2*a^3*d)) + (8*(34*A + 9*C)*tan(c + d*x))/(15*a^3*d) - ((23*A + 6*C)*sec(c + d*x)*tan(c + d*x))/(2*a^3*d) + (4*(34*A + 9*C)*sec(c + d*x)^2*tan(c + d*x))/(15*a^3*d) - ((A + C)*sec(c + d*x)^2*tan(c + d*x))/(5*d*(a + a*cos(c + d*x))^3) - ((13*A + 3*C)*sec(c + d*x)^2*tan(c + d*x))/(15*a*d*(a + a*cos(c + d*x))^2) - ((23*A + 6*C)*sec(c + d*x)^2*tan(c + d*x))/(3*a^2*d*(a + a*cos(c + d*x)))],


[cos(c + d*x)^4*(A + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^4, x, 6, ((2*A + 21*C)*x)/(2*a^4) - (32*(5*A + 54*C)*sin(c + d*x))/(105*a^4*d) + ((2*A + 21*C)*cos(c + d*x)*sin(c + d*x))/(2*a^4*d) - ((A + C)*cos(c + d*x)^5*sin(c + d*x))/(7*d*(a + a*cos(c + d*x))^4) - (2*C*cos(c + d*x)^4*sin(c + d*x))/(5*a*d*(a + a*cos(c + d*x))^3) - ((10*A + 129*C)*cos(c + d*x)^3*sin(c + d*x))/(105*a^2*d*(a + a*cos(c + d*x))^2) - (16*(5*A + 54*C)*cos(c + d*x)^2*sin(c + d*x))/(105*a^3*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^3*(A + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^4, x, 6, -((4*C*x)/a^4) + (2*(3*A + 122*C)*sin(c + d*x))/(105*a^4*d) - ((A + C)*cos(c + d*x)^4*sin(c + d*x))/(7*d*(a + a*cos(c + d*x))^4) + (2*(A - 6*C)*cos(c + d*x)^3*sin(c + d*x))/(35*a*d*(a + a*cos(c + d*x))^3) + ((3*A - 88*C)*cos(c + d*x)^2*sin(c + d*x))/(105*a^2*d*(a + a*cos(c + d*x))^2) + (4*C*sin(c + d*x))/(a^3*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^2*(A + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^4, x, 5, (C*x)/a^4 - ((A + C)*cos(c + d*x)^3*sin(c + d*x))/(7*d*(a + a*cos(c + d*x))^4) + (2*(2*A - 5*C)*cos(c + d*x)^2*sin(c + d*x))/(35*a*d*(a + a*cos(c + d*x))^3) + ((8*A - 55*C)*cos(c + d*x)*sin(c + d*x))/(105*a^2*d*(a + a*cos(c + d*x))^2) + (8*(A - 20*C)*sin(c + d*x))/(105*a^3*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^1*(A + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^4, x, 4, -(((A + C)*cos(c + d*x)^2*sin(c + d*x))/(7*d*(a + a*cos(c + d*x))^4)) + (2*(3*A - 4*C)*cos(c + d*x)*sin(c + d*x))/(35*a*d*(a + a*cos(c + d*x))^3) + ((A - 6*C)*sin(c + d*x))/(21*a^2*d*(a + a*cos(c + d*x))^2) + (4*(2*A + 9*C)*sin(c + d*x))/(105*a^3*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^0*(A + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^4, x, 4, ((A + C)*sin(c + d*x))/(7*d*(a + a*cos(c + d*x))^4) + ((3*A - 11*C)*sin(c + d*x))/(35*a*d*(a + a*cos(c + d*x))^3) + ((6*A + 13*C)*sin(c + d*x))/(105*a^2*d*(a + a*cos(c + d*x))^2) + ((6*A + 13*C)*sin(c + d*x))/(105*a^3*d*(a + a*cos(c + d*x)))],
[sec(c + d*x)^1*(A + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^4, x, 5, (A*arctanh(sin(c + d*x)))/(a^4*d) - ((A + C)*sin(c + d*x))/(7*d*(a + a*cos(c + d*x))^4) - (2*(5*A - 2*C)*sin(c + d*x))/(35*a*d*(a + a*cos(c + d*x))^3) - ((55*A - 8*C)*sin(c + d*x))/(105*a^2*d*(a + a*cos(c + d*x))^2) - (8*(20*A - C)*sin(c + d*x))/(105*a^3*d*(a + a*cos(c + d*x)))],
[sec(c + d*x)^2*(A + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^4, x, 7, -((4*A*arctanh(sin(c + d*x)))/(a^4*d)) + (2*(332*A + 3*C)*tan(c + d*x))/(105*a^4*d) - ((A + C)*tan(c + d*x))/(7*d*(a + a*cos(c + d*x))^4) - (2*(6*A - C)*tan(c + d*x))/(35*a*d*(a + a*cos(c + d*x))^3) - ((88*A - 3*C)*tan(c + d*x))/(105*a^2*d*(a + a*cos(c + d*x))^2) - (4*A*tan(c + d*x))/(a^3*d*(a + a*cos(c + d*x)))],
[sec(c + d*x)^3*(A + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^4, x, 8, ((21*A + 2*C)*arctanh(sin(c + d*x)))/(2*a^4*d) - (32*(54*A + 5*C)*tan(c + d*x))/(105*a^4*d) + ((21*A + 2*C)*sec(c + d*x)*tan(c + d*x))/(2*a^4*d) - ((A + C)*sec(c + d*x)*tan(c + d*x))/(7*d*(a + a*cos(c + d*x))^4) - (2*A*sec(c + d*x)*tan(c + d*x))/(5*a*d*(a + a*cos(c + d*x))^3) - ((129*A + 10*C)*sec(c + d*x)*tan(c + d*x))/(105*a^2*d*(a + a*cos(c + d*x))^2) - (16*(54*A + 5*C)*sec(c + d*x)*tan(c + d*x))/(105*a^3*d*(a + a*cos(c + d*x)))],
[sec(c + d*x)^4*(A + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^4, x, 9, -((2*(11*A + 2*C)*arctanh(sin(c + d*x)))/(a^4*d)) + (8*(454*A + 83*C)*tan(c + d*x))/(105*a^4*d) - (2*(11*A + 2*C)*sec(c + d*x)*tan(c + d*x))/(a^4*d) + (4*(454*A + 83*C)*sec(c + d*x)^2*tan(c + d*x))/(105*a^4*d) - ((A + C)*sec(c + d*x)^2*tan(c + d*x))/(7*d*(a + a*cos(c + d*x))^4) - (2*(8*A + C)*sec(c + d*x)^2*tan(c + d*x))/(35*a*d*(a + a*cos(c + d*x))^3) - ((178*A + 31*C)*sec(c + d*x)^2*tan(c + d*x))/(105*a^2*d*(a + a*cos(c + d*x))^2) - (4*(11*A + 2*C)*sec(c + d*x)^2*tan(c + d*x))/(3*a^3*d*(a + a*cos(c + d*x)))],


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


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


[cos(c + d*x)^3*sqrt(a + a*cos(c + d*x))*(A + C*cos(c + d*x)^2), x, 6, (4*a*(99*A + 80*C)*sin(c + d*x))/(495*d*sqrt(a + a*cos(c + d*x))) + (2*a*(99*A + 80*C)*cos(c + d*x)^3*sin(c + d*x))/(693*d*sqrt(a + a*cos(c + d*x))) + (2*a*C*cos(c + d*x)^4*sin(c + d*x))/(99*d*sqrt(a + a*cos(c + d*x))) - (8*(99*A + 80*C)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(3465*d) + (2*C*cos(c + d*x)^4*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(11*d) + (4*(99*A + 80*C)*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(1155*a*d)],
[cos(c + d*x)^2*sqrt(a + a*cos(c + d*x))*(A + C*cos(c + d*x)^2), x, 5, (2*a*(21*A + 16*C)*sin(c + d*x))/(45*d*sqrt(a + a*cos(c + d*x))) + (2*a*C*cos(c + d*x)^3*sin(c + d*x))/(63*d*sqrt(a + a*cos(c + d*x))) - (4*(21*A + 16*C)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(315*d) + (2*C*cos(c + d*x)^3*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(9*d) + (2*(21*A + 16*C)*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(105*a*d)],
[cos(c + d*x)^1*sqrt(a + a*cos(c + d*x))*(A + C*cos(c + d*x)^2), x, 4, (2*a*(35*A + 27*C)*sin(c + d*x))/(105*d*sqrt(a + a*cos(c + d*x))) + (2*(35*A + 18*C)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(105*d) + (2*C*cos(c + d*x)^2*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(7*d) + (2*C*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(35*a*d)],
[cos(c + d*x)^0*sqrt(a + a*cos(c + d*x))*(A + C*cos(c + d*x)^2), x, 3, (2*a*(15*A + 7*C)*sin(c + d*x))/(15*d*sqrt(a + a*cos(c + d*x))) - (4*C*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(15*d) + (2*C*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(5*a*d)],
[sqrt(a + a*cos(c + d*x))*(A + C*cos(c + d*x)^2)*sec(c + d*x), x, 3, (2*sqrt(2)*a*A*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(a + a*cos(c + d*x))) + (2*a*C*sin(c + d*x))/(3*d*sqrt(a + a*cos(c + d*x))) + (2*C*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(3*d)],
[sqrt(a + a*cos(c + d*x))*(A + C*cos(c + d*x)^2)*sec(c + d*x)^2, x, 3, (sqrt(2)*a*A*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(a + a*cos(c + d*x))) - (a*(A - 2*C)*sin(c + d*x))/(d*sqrt(a + a*cos(c + d*x))) + (A*sqrt(a + a*cos(c + d*x))*tan(c + d*x))/d],
[sqrt(a + a*cos(c + d*x))*(A + C*cos(c + d*x)^2)*sec(c + d*x)^3, x, 3, (a*(3*A + 8*C)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(2*sqrt(2)*d*sqrt(a + a*cos(c + d*x))) + (a*A*tan(c + d*x))/(4*d*sqrt(a + a*cos(c + d*x))) + (A*sqrt(a + a*cos(c + d*x))*sec(c + d*x)*tan(c + d*x))/(2*d)],
[sqrt(a + a*cos(c + d*x))*(A + C*cos(c + d*x)^2)*sec(c + d*x)^4, x, 4, (a*(5*A + 8*C)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(4*sqrt(2)*d*sqrt(a + a*cos(c + d*x))) + (a*(5*A + 8*C)*tan(c + d*x))/(8*d*sqrt(a + a*cos(c + d*x))) + (a*A*sec(c + d*x)*tan(c + d*x))/(12*d*sqrt(a + a*cos(c + d*x))) + (A*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^2*tan(c + d*x))/(3*d)],
[sqrt(a + a*cos(c + d*x))*(A + C*cos(c + d*x)^2)*sec(c + d*x)^5, x, 5, (a*(35*A + 48*C)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(32*sqrt(2)*d*sqrt(a + a*cos(c + d*x))) + (a*(35*A + 48*C)*tan(c + d*x))/(64*d*sqrt(a + a*cos(c + d*x))) + (a*(35*A + 48*C)*sec(c + d*x)*tan(c + d*x))/(96*d*sqrt(a + a*cos(c + d*x))) + (a*A*sec(c + d*x)^2*tan(c + d*x))/(24*d*sqrt(a + a*cos(c + d*x))) + (A*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^3*tan(c + d*x))/(4*d)],


[cos(c + d*x)^2*(a + a*cos(c + d*x))^(3/2)*(A + C*cos(c + d*x)^2), x, 6, (2*a^2*(143*A + 112*C)*sin(c + d*x))/(165*d*sqrt(a + a*cos(c + d*x))) + (2*a^2*(33*A + 28*C)*cos(c + d*x)^3*sin(c + d*x))/(231*d*sqrt(a + a*cos(c + d*x))) - (4*a*(143*A + 112*C)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(1155*d) + (2*a*C*cos(c + d*x)^3*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(33*d) + (2*(143*A + 112*C)*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(385*d) + (2*C*cos(c + d*x)^3*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(11*d)],
[cos(c + d*x)^1*(a + a*cos(c + d*x))^(3/2)*(A + C*cos(c + d*x)^2), x, 5, (8*a^2*(63*A + 47*C)*sin(c + d*x))/(315*d*sqrt(a + a*cos(c + d*x))) + (2*a*(63*A + 47*C)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(315*d) + (2*(63*A + 22*C)*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(315*d) + (2*C*cos(c + d*x)^2*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(9*d) + (2*C*(a + a*cos(c + d*x))^(5/2)*sin(c + d*x))/(21*a*d)],
[cos(c + d*x)^0*(a + a*cos(c + d*x))^(3/2)*(A + C*cos(c + d*x)^2), x, 4, (8*a^2*(35*A + 19*C)*sin(c + d*x))/(105*d*sqrt(a + a*cos(c + d*x))) + (2*a*(35*A + 19*C)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(105*d) - (4*C*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(35*d) + (2*C*(a + a*cos(c + d*x))^(5/2)*sin(c + d*x))/(7*a*d)],
[(a + a*cos(c + d*x))^(3/2)*(A + C*cos(c + d*x)^2)*sec(c + d*x), x, 4, (2*sqrt(2)*a^2*A*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(a + a*cos(c + d*x))) + (2*a^2*(5*A + 4*C)*sin(c + d*x))/(5*d*sqrt(a + a*cos(c + d*x))) + (2*a*C*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(5*d) + (2*C*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(5*d)],
[(a + a*cos(c + d*x))^(3/2)*(A + C*cos(c + d*x)^2)*sec(c + d*x)^2, x, 4, (3*sqrt(2)*a^2*A*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(a + a*cos(c + d*x))) - (a^2*(3*A - 8*C)*sin(c + d*x))/(3*d*sqrt(a + a*cos(c + d*x))) - (a*(3*A - 2*C)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(3*d) + (A*(a + a*cos(c + d*x))^(3/2)*tan(c + d*x))/d],
[(a + a*cos(c + d*x))^(3/2)*(A + C*cos(c + d*x)^2)*sec(c + d*x)^3, x, 4, (a^2*(7*A + 8*C)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(2*sqrt(2)*d*sqrt(a + a*cos(c + d*x))) - (a^2*(5*A - 8*C)*sin(c + d*x))/(4*d*sqrt(a + a*cos(c + d*x))) + (3*a*A*sqrt(a + a*cos(c + d*x))*tan(c + d*x))/(4*d) + (A*(a + a*cos(c + d*x))^(3/2)*sec(c + d*x)*tan(c + d*x))/(2*d)],
[(a + a*cos(c + d*x))^(3/2)*(A + C*cos(c + d*x)^2)*sec(c + d*x)^4, x, 4, (a^2*(11*A + 24*C)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(4*sqrt(2)*d*sqrt(a + a*cos(c + d*x))) + (a^2*(19*A + 24*C)*tan(c + d*x))/(24*d*sqrt(a + a*cos(c + d*x))) + (a*A*sqrt(a + a*cos(c + d*x))*sec(c + d*x)*tan(c + d*x))/(4*d) + (A*(a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^2*tan(c + d*x))/(3*d)],
[(a + a*cos(c + d*x))^(3/2)*(A + C*cos(c + d*x)^2)*sec(c + d*x)^5, x, 5, (a^2*(75*A + 112*C)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(32*sqrt(2)*d*sqrt(a + a*cos(c + d*x))) + (a^2*(75*A + 112*C)*tan(c + d*x))/(64*d*sqrt(a + a*cos(c + d*x))) + (a^2*(13*A + 16*C)*sec(c + d*x)*tan(c + d*x))/(32*d*sqrt(a + a*cos(c + d*x))) + (a*A*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^2*tan(c + d*x))/(8*d) + (A*(a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^3*tan(c + d*x))/(4*d)],
[(a + a*cos(c + d*x))^(3/2)*(A + C*cos(c + d*x)^2)*sec(c + d*x)^6, x, 6, (a^2*(133*A + 176*C)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(64*sqrt(2)*d*sqrt(a + a*cos(c + d*x))) + (a^2*(133*A + 176*C)*tan(c + d*x))/(128*d*sqrt(a + a*cos(c + d*x))) + (a^2*(133*A + 176*C)*sec(c + d*x)*tan(c + d*x))/(192*d*sqrt(a + a*cos(c + d*x))) + (a^2*(67*A + 80*C)*sec(c + d*x)^2*tan(c + d*x))/(240*d*sqrt(a + a*cos(c + d*x))) + (3*a*A*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^3*tan(c + d*x))/(40*d) + (A*(a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^4*tan(c + d*x))/(5*d)],


[cos(c + d*x)^2*(a + a*cos(c + d*x))^(5/2)*(A + C*cos(c + d*x)^2), x, 7, (2*a^3*(10439*A + 8368*C)*sin(c + d*x))/(6435*d*sqrt(a + a*cos(c + d*x))) + (2*a^3*(2717*A + 2224*C)*cos(c + d*x)^3*sin(c + d*x))/(9009*d*sqrt(a + a*cos(c + d*x))) - (4*a^2*(10439*A + 8368*C)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(45045*d) + (2*a^2*(143*A + 136*C)*cos(c + d*x)^3*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(1287*d) + (2*a*(10439*A + 8368*C)*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(15015*d) + (10*a*C*cos(c + d*x)^3*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(143*d) + (2*C*cos(c + d*x)^3*(a + a*cos(c + d*x))^(5/2)*sin(c + d*x))/(13*d)],
[cos(c + d*x)^1*(a + a*cos(c + d*x))^(5/2)*(A + C*cos(c + d*x)^2), x, 6, (64*a^3*(33*A + 25*C)*sin(c + d*x))/(693*d*sqrt(a + a*cos(c + d*x))) + (16*a^2*(33*A + 25*C)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(693*d) + (2*a*(33*A + 25*C)*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(231*d) + (2*(99*A + 26*C)*(a + a*cos(c + d*x))^(5/2)*sin(c + d*x))/(693*d) + (2*C*cos(c + d*x)^2*(a + a*cos(c + d*x))^(5/2)*sin(c + d*x))/(11*d) + (10*C*(a + a*cos(c + d*x))^(7/2)*sin(c + d*x))/(99*a*d)],
[cos(c + d*x)^0*(a + a*cos(c + d*x))^(5/2)*(A + C*cos(c + d*x)^2), x, 5, (64*a^3*(21*A + 13*C)*sin(c + d*x))/(315*d*sqrt(a + a*cos(c + d*x))) + (16*a^2*(21*A + 13*C)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(315*d) + (2*a*(21*A + 13*C)*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(105*d) - (4*C*(a + a*cos(c + d*x))^(5/2)*sin(c + d*x))/(63*d) + (2*C*(a + a*cos(c + d*x))^(7/2)*sin(c + d*x))/(9*a*d)],
[(a + a*cos(c + d*x))^(5/2)*(A + C*cos(c + d*x)^2)*sec(c + d*x), x, 5, (2*sqrt(2)*a^3*A*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(a + a*cos(c + d*x))) + (2*a^3*(49*A + 32*C)*sin(c + d*x))/(21*d*sqrt(a + a*cos(c + d*x))) + (2*a^2*(7*A + 8*C)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(21*d) + (2*a*C*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(7*d) + (2*C*(a + a*cos(c + d*x))^(5/2)*sin(c + d*x))/(7*d)],
[(a + a*cos(c + d*x))^(5/2)*(A + C*cos(c + d*x)^2)*sec(c + d*x)^2, x, 5, (5*sqrt(2)*a^3*A*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(a + a*cos(c + d*x))) + (a^3*(15*A + 64*C)*sin(c + d*x))/(15*d*sqrt(a + a*cos(c + d*x))) - (a^2*(15*A - 16*C)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(15*d) - (a*(5*A - 2*C)*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(5*d) + (A*(a + a*cos(c + d*x))^(5/2)*tan(c + d*x))/d],
[(a + a*cos(c + d*x))^(5/2)*(A + C*cos(c + d*x)^2)*sec(c + d*x)^3, x, 5, (a^3*(19*A + 8*C)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(2*sqrt(2)*d*sqrt(a + a*cos(c + d*x))) - (a^3*(27*A - 56*C)*sin(c + d*x))/(12*d*sqrt(a + a*cos(c + d*x))) - (a^2*(21*A - 8*C)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(12*d) + (5*a*A*(a + a*cos(c + d*x))^(3/2)*tan(c + d*x))/(4*d) + (A*(a + a*cos(c + d*x))^(5/2)*sec(c + d*x)*tan(c + d*x))/(2*d)],
[(a + a*cos(c + d*x))^(5/2)*(A + C*cos(c + d*x)^2)*sec(c + d*x)^4, x, 5, (5*a^3*(5*A + 8*C)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(4*sqrt(2)*d*sqrt(a + a*cos(c + d*x))) - (a^3*(49*A - 24*C)*sin(c + d*x))/(24*d*sqrt(a + a*cos(c + d*x))) + (a^2*(31*A + 24*C)*sqrt(a + a*cos(c + d*x))*tan(c + d*x))/(24*d) + (5*a*A*(a + a*cos(c + d*x))^(3/2)*sec(c + d*x)*tan(c + d*x))/(12*d) + (A*(a + a*cos(c + d*x))^(5/2)*sec(c + d*x)^2*tan(c + d*x))/(3*d)],
[(a + a*cos(c + d*x))^(5/2)*(A + C*cos(c + d*x)^2)*sec(c + d*x)^5, x, 5, (a^3*(163*A + 304*C)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(32*sqrt(2)*d*sqrt(a + a*cos(c + d*x))) + (a^3*(299*A + 432*C)*tan(c + d*x))/(192*d*sqrt(a + a*cos(c + d*x))) + (a^2*(17*A + 16*C)*sqrt(a + a*cos(c + d*x))*sec(c + d*x)*tan(c + d*x))/(32*d) + (5*a*A*(a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^2*tan(c + d*x))/(24*d) + (A*(a + a*cos(c + d*x))^(5/2)*sec(c + d*x)^3*tan(c + d*x))/(4*d)],
[(a + a*cos(c + d*x))^(5/2)*(A + C*cos(c + d*x)^2)*sec(c + d*x)^6, x, 6, (a^3*(283*A + 400*C)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(64*sqrt(2)*d*sqrt(a + a*cos(c + d*x))) + (a^3*(283*A + 400*C)*tan(c + d*x))/(128*d*sqrt(a + a*cos(c + d*x))) + (a^3*(787*A + 1040*C)*sec(c + d*x)*tan(c + d*x))/(960*d*sqrt(a + a*cos(c + d*x))) + (a^2*(79*A + 80*C)*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^2*tan(c + d*x))/(240*d) + (a*A*(a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^3*tan(c + d*x))/(8*d) + (A*(a + a*cos(c + d*x))^(5/2)*sec(c + d*x)^4*tan(c + d*x))/(5*d)],
[(a + a*cos(c + d*x))^(5/2)*(A + C*cos(c + d*x)^2)*sec(c + d*x)^7, x, 7, (a^3*(1015*A + 1304*C)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(256*sqrt(2)*d*sqrt(a + a*cos(c + d*x))) + (a^3*(1015*A + 1304*C)*tan(c + d*x))/(512*d*sqrt(a + a*cos(c + d*x))) + (a^3*(1015*A + 1304*C)*sec(c + d*x)*tan(c + d*x))/(768*d*sqrt(a + a*cos(c + d*x))) + (a^3*(109*A + 136*C)*sec(c + d*x)^2*tan(c + d*x))/(192*d*sqrt(a + a*cos(c + d*x))) + (a^2*(23*A + 24*C)*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^3*tan(c + d*x))/(96*d) + (a*A*(a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^4*tan(c + d*x))/(12*d) + (A*(a + a*cos(c + d*x))^(5/2)*sec(c + d*x)^5*tan(c + d*x))/(6*d)],


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


[cos(c + d*x)^3*(A + C*cos(c + d*x)^2)/sqrt(a + a*cos(c + d*x)), x, 6, -((2*(A + C)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(d*sqrt(a + a*cos(c + d*x)))) + (4*(147*A + 143*C)*sin(c + d*x))/(315*d*sqrt(a + a*cos(c + d*x))) + (2*(21*A + 19*C)*cos(c + d*x)^2*sin(c + d*x))/(105*d*sqrt(a + a*cos(c + d*x))) - (2*C*cos(c + d*x)^3*sin(c + d*x))/(63*d*sqrt(a + a*cos(c + d*x))) + (2*C*cos(c + d*x)^4*sin(c + d*x))/(9*d*sqrt(a + a*cos(c + d*x))) - (2*(21*A + 29*C)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(315*a*d)],
[cos(c + d*x)^2*(A + C*cos(c + d*x)^2)/sqrt(a + a*cos(c + d*x)), x, 5, (2*(A + C)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(d*sqrt(a + a*cos(c + d*x))) - (4*(35*A + 37*C)*sin(c + d*x))/(105*d*sqrt(a + a*cos(c + d*x))) - (2*C*cos(c + d*x)^2*sin(c + d*x))/(35*d*sqrt(a + a*cos(c + d*x))) + (2*C*cos(c + d*x)^3*sin(c + d*x))/(7*d*sqrt(a + a*cos(c + d*x))) + (2*(35*A + 31*C)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(105*a*d)],
[cos(c + d*x)^1*(A + C*cos(c + d*x)^2)/sqrt(a + a*cos(c + d*x)), x, 4, -((2*(A + C)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(d*sqrt(a + a*cos(c + d*x)))) + (2*(15*A + 14*C)*sin(c + d*x))/(15*d*sqrt(a + a*cos(c + d*x))) + (2*C*cos(c + d*x)^2*sin(c + d*x))/(5*d*sqrt(a + a*cos(c + d*x))) - (2*C*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(15*a*d)],
[cos(c + d*x)^0*(A + C*cos(c + d*x)^2)/sqrt(a + a*cos(c + d*x)), x, 3, (2*(A + C)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(a + a*cos(c + d*x))) - (4*C*sin(c + d*x))/(3*d*sqrt(a + a*cos(c + d*x))) + (2*C*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(3*a*d)],
[((A + C*cos(c + d*x)^2)*sec(c + d*x))/sqrt(a + a*cos(c + d*x)), x, 4, (-2*(A + C)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(a + a*cos(c + d*x))) + (2*sqrt(2)*A*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(a + a*cos(c + d*x))) + (2*C*sin(c + d*x))/(d*sqrt(a + a*cos(c + d*x)))],
[((A + C*cos(c + d*x)^2)*sec(c + d*x)^2)/sqrt(a + a*cos(c + d*x)), x, 4, (2*(A + C)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(a + a*cos(c + d*x))) - (sqrt(2)*A*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(a + a*cos(c + d*x))) + (A*tan(c + d*x))/(d*sqrt(a + a*cos(c + d*x)))],
[((A + C*cos(c + d*x)^2)*sec(c + d*x)^3)/sqrt(a + a*cos(c + d*x)), x, 5, (-2*(A + C)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(a + a*cos(c + d*x))) + ((7*A + 8*C)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(2*sqrt(2)*d*sqrt(a + a*cos(c + d*x))) - (A*tan(c + d*x))/(4*d*sqrt(a + a*cos(c + d*x))) + (A*sec(c + d*x)*tan(c + d*x))/(2*d*sqrt(a + a*cos(c + d*x)))],
[((A + C*cos(c + d*x)^2)*sec(c + d*x)^4)/sqrt(a + a*cos(c + d*x)), x, 6, (2*(A + C)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(a + a*cos(c + d*x))) - ((9*A + 8*C)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(4*sqrt(2)*d*sqrt(a + a*cos(c + d*x))) + ((7*A + 8*C)*tan(c + d*x))/(8*d*sqrt(a + a*cos(c + d*x))) - (A*sec(c + d*x)*tan(c + d*x))/(12*d*sqrt(a + a*cos(c + d*x))) + (A*sec(c + d*x)^2*tan(c + d*x))/(3*d*sqrt(a + a*cos(c + d*x)))],
[((A + C*cos(c + d*x)^2)*sec(c + d*x)^5)/sqrt(a + a*cos(c + d*x)), x, 7, (-2*(A + C)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(a + a*cos(c + d*x))) + ((107*A + 112*C)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(32*sqrt(2)*d*sqrt(a + a*cos(c + d*x))) - ((21*A + 16*C)*tan(c + d*x))/(64*d*sqrt(a + a*cos(c + d*x))) + ((43*A + 48*C)*sec(c + d*x)*tan(c + d*x))/(96*d*sqrt(a + a*cos(c + d*x))) - (A*sec(c + d*x)^2*tan(c + d*x))/(24*d*sqrt(a + a*cos(c + d*x))) + (A*sec(c + d*x)^3*tan(c + d*x))/(4*d*sqrt(a + a*cos(c + d*x)))],


[cos(c + d*x)^3*(A + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^(3/2), x, 6, ((11*A + 19*C)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(2*a*d*sqrt(a + a*cos(c + d*x))) - ((A + C)*cos(c + d*x)^4*sin(c + d*x))/(2*d*(a + a*cos(c + d*x))^(3/2)) - ((455*A + 799*C)*sin(c + d*x))/(105*a*d*sqrt(a + a*cos(c + d*x))) - ((35*A + 67*C)*cos(c + d*x)^2*sin(c + d*x))/(70*a*d*sqrt(a + a*cos(c + d*x))) + ((7*A + 11*C)*cos(c + d*x)^3*sin(c + d*x))/(14*a*d*sqrt(a + a*cos(c + d*x))) + ((245*A + 397*C)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(210*a^2*d)],
[cos(c + d*x)^2*(A + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^(3/2), x, 5, -(((7*A + 15*C)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(2*a*d*sqrt(a + a*cos(c + d*x)))) - ((A + C)*cos(c + d*x)^3*sin(c + d*x))/(2*d*(a + a*cos(c + d*x))^(3/2)) + ((15*A + 31*C)*sin(c + d*x))/(5*a*d*sqrt(a + a*cos(c + d*x))) + ((5*A + 9*C)*cos(c + d*x)^2*sin(c + d*x))/(10*a*d*sqrt(a + a*cos(c + d*x))) - ((5*A + 13*C)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(10*a^2*d)],
[cos(c + d*x)^1*(A + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^(3/2), x, 4, ((3*A + 11*C)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(2*a*d*sqrt(a + a*cos(c + d*x))) - ((A + C)*cos(c + d*x)^2*sin(c + d*x))/(2*d*(a + a*cos(c + d*x))^(3/2)) - ((3*A + 13*C)*sin(c + d*x))/(3*a*d*sqrt(a + a*cos(c + d*x))) + ((3*A + 7*C)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(6*a^2*d)],
[cos(c + d*x)^0*(A + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^(3/2), x, 3, ((A - 7*C)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(2*a*d*sqrt(a + a*cos(c + d*x))) + ((A + C)*sin(c + d*x))/(2*d*(a + a*cos(c + d*x))^(3/2)) + (2*C*sin(c + d*x))/(a*d*sqrt(a + a*cos(c + d*x)))],
[((A + C*cos(c + d*x)^2)*sec(c + d*x))/(a + a*cos(c + d*x))^(3/2), x, 4, -((5*A - 3*C)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(2*a*d*sqrt(a + a*cos(c + d*x))) + (2*sqrt(2)*A*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(a*d*sqrt(a + a*cos(c + d*x))) - ((A + C)*sin(c + d*x))/(2*d*(a + a*cos(c + d*x))^(3/2))],
[((A + C*cos(c + d*x)^2)*sec(c + d*x)^2)/(a + a*cos(c + d*x))^(3/2), x, 5, ((9*A + C)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(2*a*d*sqrt(a + a*cos(c + d*x))) - (3*sqrt(2)*A*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(a*d*sqrt(a + a*cos(c + d*x))) - ((A + C)*tan(c + d*x))/(2*d*(a + a*cos(c + d*x))^(3/2)) + ((3*A + C)*tan(c + d*x))/(2*a*d*sqrt(a + a*cos(c + d*x)))],
[((A + C*cos(c + d*x)^2)*sec(c + d*x)^3)/(a + a*cos(c + d*x))^(3/2), x, 6, -((13*A + 5*C)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(2*a*d*sqrt(a + a*cos(c + d*x))) + ((19*A + 8*C)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(2*sqrt(2)*a*d*sqrt(a + a*cos(c + d*x))) - ((7*A + 2*C)*tan(c + d*x))/(4*a*d*sqrt(a + a*cos(c + d*x))) - ((A + C)*sec(c + d*x)*tan(c + d*x))/(2*d*(a + a*cos(c + d*x))^(3/2)) + ((2*A + C)*sec(c + d*x)*tan(c + d*x))/(2*a*d*sqrt(a + a*cos(c + d*x)))],
[((A + C*cos(c + d*x)^2)*sec(c + d*x)^4)/(a + a*cos(c + d*x))^(3/2), x, 7, ((17*A + 9*C)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(2*a*d*sqrt(a + a*cos(c + d*x))) - ((47*A + 24*C)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(4*sqrt(2)*a*d*sqrt(a + a*cos(c + d*x))) + (3*(7*A + 4*C)*tan(c + d*x))/(8*a*d*sqrt(a + a*cos(c + d*x))) - ((13*A + 6*C)*sec(c + d*x)*tan(c + d*x))/(12*a*d*sqrt(a + a*cos(c + d*x))) - ((A + C)*sec(c + d*x)^2*tan(c + d*x))/(2*d*(a + a*cos(c + d*x))^(3/2)) + ((5*A + 3*C)*sec(c + d*x)^2*tan(c + d*x))/(6*a*d*sqrt(a + a*cos(c + d*x)))],


[cos(c + d*x)^3*(A + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^(5/2), x, 6, -(((75*A + 283*C)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(16*a^2*d*sqrt(a + a*cos(c + d*x)))) - ((A + C)*cos(c + d*x)^4*sin(c + d*x))/(4*d*(a + a*cos(c + d*x))^(5/2)) - ((5*A + 21*C)*cos(c + d*x)^3*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2)) + ((465*A + 1729*C)*sin(c + d*x))/(120*a^2*d*sqrt(a + a*cos(c + d*x))) + ((45*A + 157*C)*cos(c + d*x)^2*sin(c + d*x))/(80*a^2*d*sqrt(a + a*cos(c + d*x))) - ((195*A + 787*C)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(240*a^3*d)],
[cos(c + d*x)^2*(A + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^(5/2), x, 5, ((19*A + 163*C)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(16*a^2*d*sqrt(a + a*cos(c + d*x))) - ((A + C)*cos(c + d*x)^3*sin(c + d*x))/(4*d*(a + a*cos(c + d*x))^(5/2)) - ((A + 17*C)*cos(c + d*x)^2*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2)) - ((21*A + 197*C)*sin(c + d*x))/(24*a^2*d*sqrt(a + a*cos(c + d*x))) + (5*(3*A + 19*C)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(48*a^3*d)],
[cos(c + d*x)^1*(A + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^(5/2), x, 4, (5*(A - 15*C)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(16*a^2*d*sqrt(a + a*cos(c + d*x))) - ((A + C)*cos(c + d*x)^2*sin(c + d*x))/(4*d*(a + a*cos(c + d*x))^(5/2)) + ((3*A - 13*C)*cos(c + d*x)*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2)) + ((A + 49*C)*sin(c + d*x))/(16*a^2*d*sqrt(a + a*cos(c + d*x)))],
[cos(c + d*x)^0*(A + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^(5/2), x, 3, ((3*A + 19*C)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(16*a^2*d*sqrt(a + a*cos(c + d*x))) + ((A + C)*sin(c + d*x))/(4*d*(a + a*cos(c + d*x))^(5/2)) + ((3*A - 13*C)*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2))],
[((A + C*cos(c + d*x)^2)*sec(c + d*x))/(a + a*cos(c + d*x))^(5/2), x, 5, -((43*A - 5*C)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(16*a^2*d*sqrt(a + a*cos(c + d*x))) + (2*sqrt(2)*A*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(a^2*d*sqrt(a + a*cos(c + d*x))) - ((A + C)*sin(c + d*x))/(4*d*(a + a*cos(c + d*x))^(5/2)) - ((11*A - 5*C)*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2))],
[((A + C*cos(c + d*x)^2)*sec(c + d*x)^2)/(a + a*cos(c + d*x))^(5/2), x, 6, ((115*A + 3*C)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(16*a^2*d*sqrt(a + a*cos(c + d*x))) - (5*sqrt(2)*A*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(a^2*d*sqrt(a + a*cos(c + d*x))) - ((A + C)*tan(c + d*x))/(4*d*(a + a*cos(c + d*x))^(5/2)) - ((15*A - C)*tan(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2)) + ((35*A + 3*C)*tan(c + d*x))/(16*a^2*d*sqrt(a + a*cos(c + d*x)))],
[((A + C*cos(c + d*x)^2)*sec(c + d*x)^3)/(a + a*cos(c + d*x))^(5/2), x, 7, -((219*A + 43*C)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(16*a^2*d*sqrt(a + a*cos(c + d*x))) + ((39*A + 8*C)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(2*sqrt(2)*a^2*d*sqrt(a + a*cos(c + d*x))) - ((63*A + 11*C)*tan(c + d*x))/(16*a^2*d*sqrt(a + a*cos(c + d*x))) - ((A + C)*sec(c + d*x)*tan(c + d*x))/(4*d*(a + a*cos(c + d*x))^(5/2)) - ((19*A + 3*C)*sec(c + d*x)*tan(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2)) + ((31*A + 7*C)*sec(c + d*x)*tan(c + d*x))/(16*a^2*d*sqrt(a + a*cos(c + d*x)))],


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


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


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

[(-1 + 3*cos(c + d*x)^2)/cos(c + d*x)^(1/2), x, 1, (2*sqrt(cos(c + d*x))*sin(c + d*x))/d],


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


[cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))^2*(A + C*cos(c + d*x)^2), x, 8, (4*a^2*(9*A + 7*C)*EllipticE((c + d*x)/2, 2))/(15*d) + (8*a^2*(33*A + 25*C)*EllipticF((c + d*x)/2, 2))/(231*d) + (8*a^2*(33*A + 25*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(231*d) + (4*a^2*(9*A + 7*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(45*d) + (2*a^2*(99*A + 89*C)*cos(c + d*x)^(5/2)*sin(c + d*x))/(693*d) + (8*a*C*cos(c + d*x)^(5/2)*(a + a*cos(c + d*x))*sin(c + d*x))/(99*d) + (2*C*cos(c + d*x)^(5/2)*(a + a*cos(c + d*x))^2*sin(c + d*x))/(11*d)],
[sqrt(cos(c + d*x))*(a + a*cos(c + d*x))^2*(A + C*cos(c + d*x)^2), x, 7, (16*a^2*(3*A + 2*C)*EllipticE((c + d*x)/2, 2))/(15*d) + (4*a^2*(7*A + 5*C)*EllipticF((c + d*x)/2, 2))/(21*d) + (4*a^2*(7*A + 5*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(21*d) + (2*a^2*(21*A + 19*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(105*d) + (8*a*C*cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))*sin(c + d*x))/(63*d) + (2*C*cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))^2*sin(c + d*x))/(9*d)],
[((a + a*cos(c + d*x))^2*(A + C*cos(c + d*x)^2))/sqrt(cos(c + d*x)), x, 6, (4*a^2*(5*A + 3*C)*EllipticE((c + d*x)/2, 2))/(5*d) + (8*a^2*(7*A + 3*C)*EllipticF((c + d*x)/2, 2))/(21*d) + (2*a^2*(35*A + 33*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(105*d) + (8*a*C*sqrt(cos(c + d*x))*(a + a*cos(c + d*x))*sin(c + d*x))/(35*d) + (2*C*sqrt(cos(c + d*x))*(a + a*cos(c + d*x))^2*sin(c + d*x))/(7*d)],
[((a + a*cos(c + d*x))^2*(A + C*cos(c + d*x)^2))/cos(c + d*x)^(3/2), x, 6, (16*a^2*C*EllipticE((c + d*x)/2, 2))/(5*d) + (4*a^2*(3*A + C)*EllipticF((c + d*x)/2, 2))/(3*d) - (2*a^2*(15*A - 7*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(15*d) - (2*a*(5*A - C)*sqrt(cos(c + d*x))*(a + a*cos(c + d*x))*sin(c + d*x))/(5*d) + (2*A*(a + a*cos(c + d*x))^2*sin(c + d*x))/(d*sqrt(cos(c + d*x)))],
[((a + a*cos(c + d*x))^2*(A + C*cos(c + d*x)^2))/cos(c + d*x)^(5/2), x, 6, (-4*a^2*(A - C)*EllipticE((c + d*x)/2, 2))/d + (8*a^2*(A + C)*EllipticF((c + d*x)/2, 2))/(3*d) - (2*a^2*(5*A - C)*sqrt(cos(c + d*x))*sin(c + d*x))/(3*d) + (8*a*A*(a + a*cos(c + d*x))*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))) + (2*A*(a + a*cos(c + d*x))^2*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2))],
[((a + a*cos(c + d*x))^2*(A + C*cos(c + d*x)^2))/cos(c + d*x)^(7/2), x, 6, (-16*a^2*A*EllipticE((c + d*x)/2, 2))/(5*d) + (4*a^2*(A + 3*C)*EllipticF((c + d*x)/2, 2))/(3*d) + (2*a^2*(17*A + 15*C)*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))) + (8*a*A*(a + a*cos(c + d*x))*sin(c + d*x))/(15*d*cos(c + d*x)^(3/2)) + (2*A*(a + a*cos(c + d*x))^2*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2))],
[((a + a*cos(c + d*x))^2*(A + C*cos(c + d*x)^2))/cos(c + d*x)^(9/2), x, 7, (-4*a^2*(3*A + 5*C)*EllipticE((c + d*x)/2, 2))/(5*d) + (8*a^2*(3*A + 7*C)*EllipticF((c + d*x)/2, 2))/(21*d) + (2*a^2*(33*A + 35*C)*sin(c + d*x))/(105*d*cos(c + d*x)^(3/2)) + (4*a^2*(3*A + 5*C)*sin(c + d*x))/(5*d*sqrt(cos(c + d*x))) + (8*a*A*(a + a*cos(c + d*x))*sin(c + d*x))/(35*d*cos(c + d*x)^(5/2)) + (2*A*(a + a*cos(c + d*x))^2*sin(c + d*x))/(7*d*cos(c + d*x)^(7/2))],
[((a + a*cos(c + d*x))^2*(A + C*cos(c + d*x)^2))/cos(c + d*x)^(11/2), x, 8, (-16*a^2*(2*A + 3*C)*EllipticE((c + d*x)/2, 2))/(15*d) + (4*a^2*(5*A + 7*C)*EllipticF((c + d*x)/2, 2))/(21*d) + (2*a^2*(19*A + 21*C)*sin(c + d*x))/(105*d*cos(c + d*x)^(5/2)) + (4*a^2*(5*A + 7*C)*sin(c + d*x))/(21*d*cos(c + d*x)^(3/2)) + (16*a^2*(2*A + 3*C)*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))) + (8*a*A*(a + a*cos(c + d*x))*sin(c + d*x))/(63*d*cos(c + d*x)^(7/2)) + (2*A*(a + a*cos(c + d*x))^2*sin(c + d*x))/(9*d*cos(c + d*x)^(9/2))],


[cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))^3*(A + C*cos(c + d*x)^2), x, 9, (4*a^3*(221*A + 175*C)*EllipticE((c + d*x)/2, 2))/(195*d) + (4*a^3*(121*A + 95*C)*EllipticF((c + d*x)/2, 2))/(231*d) + (4*a^3*(121*A + 95*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(231*d) + (4*a^3*(221*A + 175*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(585*d) + (40*a^3*(143*A + 118*C)*cos(c + d*x)^(5/2)*sin(c + d*x))/(9009*d) + (2*a^2*(143*A + 145*C)*cos(c + d*x)^(5/2)*(a + a*cos(c + d*x))*sin(c + d*x))/(1287*d) + (12*a*C*cos(c + d*x)^(5/2)*(a + a*cos(c + d*x))^2*sin(c + d*x))/(143*d) + (2*C*cos(c + d*x)^(5/2)*(a + a*cos(c + d*x))^3*sin(c + d*x))/(13*d)],
[sqrt(cos(c + d*x))*(a + a*cos(c + d*x))^3*(A + C*cos(c + d*x)^2), x, 8, (4*a^3*(7*A + 5*C)*EllipticE((c + d*x)/2, 2))/(5*d) + (4*a^3*(143*A + 105*C)*EllipticF((c + d*x)/2, 2))/(231*d) + (4*a^3*(143*A + 105*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(231*d) + (8*a^3*(44*A + 35*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(385*d) + (2*a^2*(33*A + 35*C)*cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))*sin(c + d*x))/(231*d) + (4*a*C*cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))^2*sin(c + d*x))/(33*d) + (2*C*cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))^3*sin(c + d*x))/(11*d)],
[((a + a*cos(c + d*x))^3*(A + C*cos(c + d*x)^2))/sqrt(cos(c + d*x)), x, 7, (4*a^3*(27*A + 17*C)*EllipticE((c + d*x)/2, 2))/(15*d) + (4*a^3*(21*A + 11*C)*EllipticF((c + d*x)/2, 2))/(21*d) + (8*a^3*(21*A + 16*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(105*d) + (2*a^2*(63*A + 73*C)*sqrt(cos(c + d*x))*(a + a*cos(c + d*x))*sin(c + d*x))/(315*d) + (4*a*C*sqrt(cos(c + d*x))*(a + a*cos(c + d*x))^2*sin(c + d*x))/(21*d) + (2*C*sqrt(cos(c + d*x))*(a + a*cos(c + d*x))^3*sin(c + d*x))/(9*d)],
[((a + a*cos(c + d*x))^3*(A + C*cos(c + d*x)^2))/cos(c + d*x)^(3/2), x, 7, (4*a^3*(5*A + 7*C)*EllipticE((c + d*x)/2, 2))/(5*d) + (4*a^3*(35*A + 13*C)*EllipticF((c + d*x)/2, 2))/(21*d) - (4*a^3*(35*A - 41*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(105*d) - (2*a^2*(35*A - 11*C)*sqrt(cos(c + d*x))*(a + a*cos(c + d*x))*sin(c + d*x))/(35*d) - (2*a*(7*A - C)*sqrt(cos(c + d*x))*(a + a*cos(c + d*x))^2*sin(c + d*x))/(7*d) + (2*A*(a + a*cos(c + d*x))^3*sin(c + d*x))/(d*sqrt(cos(c + d*x)))],
[((a + a*cos(c + d*x))^3*(A + C*cos(c + d*x)^2))/cos(c + d*x)^(5/2), x, 7, (-4*a^3*(5*A - 9*C)*EllipticE((c + d*x)/2, 2))/(5*d) + (4*a^3*(5*A + 3*C)*EllipticF((c + d*x)/2, 2))/(3*d) - (8*a^3*(10*A - 3*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(15*d) - (2*a^2*(35*A - 3*C)*sqrt(cos(c + d*x))*(a + a*cos(c + d*x))*sin(c + d*x))/(15*d) + (4*a*A*(a + a*cos(c + d*x))^2*sin(c + d*x))/(d*sqrt(cos(c + d*x))) + (2*A*(a + a*cos(c + d*x))^3*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2))],
[((a + a*cos(c + d*x))^3*(A + C*cos(c + d*x)^2))/cos(c + d*x)^(7/2), x, 7, (-4*a^3*(9*A - 5*C)*EllipticE((c + d*x)/2, 2))/(5*d) + (4*a^3*(3*A + 5*C)*EllipticF((c + d*x)/2, 2))/(3*d) - (4*a^3*(21*A + 5*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(15*d) + (2*a^2*(11*A + 5*C)*(a + a*cos(c + d*x))*sin(c + d*x))/(5*d*sqrt(cos(c + d*x))) + (4*a*A*(a + a*cos(c + d*x))^2*sin(c + d*x))/(5*d*cos(c + d*x)^(3/2)) + (2*A*(a + a*cos(c + d*x))^3*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2))],
[((a + a*cos(c + d*x))^3*(A + C*cos(c + d*x)^2))/cos(c + d*x)^(9/2), x, 7, (-4*a^3*(7*A + 5*C)*EllipticE((c + d*x)/2, 2))/(5*d) + (4*a^3*(13*A + 35*C)*EllipticF((c + d*x)/2, 2))/(21*d) + (8*a^3*(53*A + 70*C)*sin(c + d*x))/(105*d*sqrt(cos(c + d*x))) + (2*a^2*(7*A + 5*C)*(a + a*cos(c + d*x))*sin(c + d*x))/(15*d*cos(c + d*x)^(3/2)) + (12*a*A*(a + a*cos(c + d*x))^2*sin(c + d*x))/(35*d*cos(c + d*x)^(5/2)) + (2*A*(a + a*cos(c + d*x))^3*sin(c + d*x))/(7*d*cos(c + d*x)^(7/2))],
[((a + a*cos(c + d*x))^3*(A + C*cos(c + d*x)^2))/cos(c + d*x)^(11/2), x, 8, (-4*a^3*(17*A + 27*C)*EllipticE((c + d*x)/2, 2))/(15*d) + (4*a^3*(11*A + 21*C)*EllipticF((c + d*x)/2, 2))/(21*d) + (8*a^3*(16*A + 21*C)*sin(c + d*x))/(105*d*cos(c + d*x)^(3/2)) + (4*a^3*(17*A + 27*C)*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))) + (2*a^2*(73*A + 63*C)*(a + a*cos(c + d*x))*sin(c + d*x))/(315*d*cos(c + d*x)^(5/2)) + (4*a*A*(a + a*cos(c + d*x))^2*sin(c + d*x))/(21*d*cos(c + d*x)^(7/2)) + (2*A*(a + a*cos(c + d*x))^3*sin(c + d*x))/(9*d*cos(c + d*x)^(9/2))],
[((a + a*cos(c + d*x))^3*(A + C*cos(c + d*x)^2))/cos(c + d*x)^(13/2), x, 9, (-4*a^3*(5*A + 7*C)*EllipticE((c + d*x)/2, 2))/(5*d) + (4*a^3*(105*A + 143*C)*EllipticF((c + d*x)/2, 2))/(231*d) + (8*a^3*(35*A + 44*C)*sin(c + d*x))/(385*d*cos(c + d*x)^(5/2)) + (4*a^3*(105*A + 143*C)*sin(c + d*x))/(231*d*cos(c + d*x)^(3/2)) + (4*a^3*(5*A + 7*C)*sin(c + d*x))/(5*d*sqrt(cos(c + d*x))) + (2*a^2*(35*A + 33*C)*(a + a*cos(c + d*x))*sin(c + d*x))/(231*d*cos(c + d*x)^(7/2)) + (4*a*A*(a + a*cos(c + d*x))^2*sin(c + d*x))/(33*d*cos(c + d*x)^(9/2)) + (2*A*(a + a*cos(c + d*x))^3*sin(c + d*x))/(11*d*cos(c + d*x)^(11/2))],


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


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


[(cos(c + d*x)^(5/2)*(A + C*cos(c + d*x)^2))/(a + a*cos(c + d*x))^2, x, 7, (4*(5*A + 14*C)*EllipticE((c + d*x)/2, 2))/(5*a^2*d) - (5*(A + 3*C)*EllipticF((c + d*x)/2, 2))/(3*a^2*d) - (5*(A + 3*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(3*a^2*d) + (4*(5*A + 14*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(15*a^2*d) - ((A + C)*cos(c + d*x)^(7/2)*sin(c + d*x))/(3*d*(a + a*cos(c + d*x))^2) - ((A + 3*C)*cos(c + d*x)^(5/2)*sin(c + d*x))/(a*d*(a + a*cos(c + d*x)))],
[(cos(c + d*x)^(3/2)*(A + C*cos(c + d*x)^2))/(a + a*cos(c + d*x))^2, x, 6, -(((A + 7*C)*EllipticE((c + d*x)/2, 2))/(a^2*d)) + (2*(A + 5*C)*EllipticF((c + d*x)/2, 2))/(3*a^2*d) + (2*(A + 5*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(3*a^2*d) - ((A + C)*cos(c + d*x)^(5/2)*sin(c + d*x))/(3*d*(a + a*cos(c + d*x))^2) - ((A + 7*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(3*a*d*(a + a*cos(c + d*x)))],
[(sqrt(cos(c + d*x))*(A + C*cos(c + d*x)^2))/(a + a*cos(c + d*x))^2, x, 5, (4*C*EllipticE((c + d*x)/2, 2))/(a^2*d) + ((A - 5*C)*EllipticF((c + d*x)/2, 2))/(3*a^2*d) - ((A + C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(3*d*(a + a*cos(c + d*x))^2) + ((A - 5*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(3*a*d*(a + a*cos(c + d*x)))],
[(A + C*cos(c + d*x)^2)/(sqrt(cos(c + d*x))*(a + a*cos(c + d*x))^2), x, 5, ((A - C)*EllipticE((c + d*x)/2, 2))/(a^2*d) + (2*(A + C)*EllipticF((c + d*x)/2, 2))/(3*a^2*d) - ((A + C)*sqrt(cos(c + d*x))*sin(c + d*x))/(3*d*(a + a*cos(c + d*x))^2) - ((A - C)*sqrt(cos(c + d*x))*sin(c + d*x))/(a*d*(a + a*cos(c + d*x)))],
[(A + C*cos(c + d*x)^2)/(cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))^2), x, 6, (-4*A*EllipticE((c + d*x)/2, 2))/(a^2*d) - ((5*A - C)*EllipticF((c + d*x)/2, 2))/(3*a^2*d) + (4*A*sin(c + d*x))/(a^2*d*sqrt(cos(c + d*x))) - ((A + C)*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))*(a + a*cos(c + d*x))^2) - ((5*A - C)*sin(c + d*x))/(3*a*d*sqrt(cos(c + d*x))*(a + a*cos(c + d*x)))],
[(A + C*cos(c + d*x)^2)/(cos(c + d*x)^(5/2)*(a + a*cos(c + d*x))^2), x, 7, ((7*A + C)*EllipticE((c + d*x)/2, 2))/(a^2*d) + (2*(5*A + C)*EllipticF((c + d*x)/2, 2))/(3*a^2*d) + (2*(5*A + C)*sin(c + d*x))/(3*a^2*d*cos(c + d*x)^(3/2)) - ((7*A + C)*sin(c + d*x))/(a^2*d*sqrt(cos(c + d*x))) - ((A + C)*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))^2) - ((7*A + C)*sin(c + d*x))/(3*a*d*cos(c + d*x)^(3/2)*(a + a*cos(c + d*x)))],


[(cos(c + d*x)^(7/2)*(A + C*cos(c + d*x)^2))/(a + a*cos(c + d*x))^3, x, 8, (7*(7*A + 33*C)*EllipticE((c + d*x)/2, 2))/(10*a^3*d) - ((13*A + 63*C)*EllipticF((c + d*x)/2, 2))/(6*a^3*d) - ((13*A + 63*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(6*a^3*d) + (7*(7*A + 33*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(30*a^3*d) - ((A + C)*cos(c + d*x)^(9/2)*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3) - (2*(A + 6*C)*cos(c + d*x)^(7/2)*sin(c + d*x))/(15*a*d*(a + a*cos(c + d*x))^2) - ((13*A + 63*C)*cos(c + d*x)^(5/2)*sin(c + d*x))/(10*a^2*d*(a + a*cos(c + d*x)))],
[(cos(c + d*x)^(5/2)*(A + C*cos(c + d*x)^2))/(a + a*cos(c + d*x))^3, x, 7, -((9*A + 119*C)*EllipticE((c + d*x)/2, 2))/(10*a^3*d) + ((A + 11*C)*EllipticF((c + d*x)/2, 2))/(2*a^3*d) + ((A + 11*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(2*a^3*d) - ((A + C)*cos(c + d*x)^(7/2)*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3) - (2*C*cos(c + d*x)^(5/2)*sin(c + d*x))/(3*a*d*(a + a*cos(c + d*x))^2) - ((9*A + 119*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(30*a^2*d*(a + a*cos(c + d*x)))],
[(cos(c + d*x)^(3/2)*(A + C*cos(c + d*x)^2))/(a + a*cos(c + d*x))^3, x, 6, -((A - 49*C)*EllipticE((c + d*x)/2, 2))/(10*a^3*d) + ((A - 13*C)*EllipticF((c + d*x)/2, 2))/(6*a^3*d) - ((A + C)*cos(c + d*x)^(5/2)*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3) + (2*(A - 4*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(15*a*d*(a + a*cos(c + d*x))^2) + ((A - 13*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(6*a^2*d*(a + a*cos(c + d*x)))],
[(sqrt(cos(c + d*x))*(A + C*cos(c + d*x)^2))/(a + a*cos(c + d*x))^3, x, 6, ((A - 9*C)*EllipticE((c + d*x)/2, 2))/(10*a^3*d) + ((A + 3*C)*EllipticF((c + d*x)/2, 2))/(6*a^3*d) - ((A + C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3) + (2*(2*A - 3*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(15*a*d*(a + a*cos(c + d*x))^2) - ((A - 9*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(10*a^2*d*(a + a*cos(c + d*x)))],
[(A + C*cos(c + d*x)^2)/(sqrt(cos(c + d*x))*(a + a*cos(c + d*x))^3), x, 6, ((9*A - C)*EllipticE((c + d*x)/2, 2))/(10*a^3*d) + ((3*A + C)*EllipticF((c + d*x)/2, 2))/(6*a^3*d) - ((A + C)*sqrt(cos(c + d*x))*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3) - (2*(3*A - 2*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(15*a*d*(a + a*cos(c + d*x))^2) - ((9*A - C)*sqrt(cos(c + d*x))*sin(c + d*x))/(10*a^2*d*(a + a*cos(c + d*x)))],
[(A + C*cos(c + d*x)^2)/(cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))^3), x, 7, -((49*A - C)*EllipticE((c + d*x)/2, 2))/(10*a^3*d) - ((13*A - C)*EllipticF((c + d*x)/2, 2))/(6*a^3*d) + ((49*A - C)*sin(c + d*x))/(10*a^3*d*sqrt(cos(c + d*x))) - ((A + C)*sin(c + d*x))/(5*d*sqrt(cos(c + d*x))*(a + a*cos(c + d*x))^3) - (2*(4*A - C)*sin(c + d*x))/(15*a*d*sqrt(cos(c + d*x))*(a + a*cos(c + d*x))^2) - ((13*A - C)*sin(c + d*x))/(6*a^2*d*sqrt(cos(c + d*x))*(a + a*cos(c + d*x)))],
[(A + C*cos(c + d*x)^2)/(cos(c + d*x)^(5/2)*(a + a*cos(c + d*x))^3), x, 8, ((119*A + 9*C)*EllipticE((c + d*x)/2, 2))/(10*a^3*d) + ((11*A + C)*EllipticF((c + d*x)/2, 2))/(2*a^3*d) + ((11*A + C)*sin(c + d*x))/(2*a^3*d*cos(c + d*x)^(3/2)) - ((119*A + 9*C)*sin(c + d*x))/(10*a^3*d*sqrt(cos(c + d*x))) - ((A + C)*sin(c + d*x))/(5*d*cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))^3) - (2*A*sin(c + d*x))/(3*a*d*cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))^2) - ((119*A + 9*C)*sin(c + d*x))/(30*a^2*d*cos(c + d*x)^(3/2)*(a + a*cos(c + d*x)))],


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


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


[cos(c + d*x)^(3/2)*sqrt(a + a*cos(c + d*x))*(A + C*cos(c + d*x)^2), x, 5, (sqrt(a)*(48*A + 35*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(64*d) + (a*(48*A + 35*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(64*d*sqrt(a + a*cos(c + d*x))) + (a*(48*A + 35*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(96*d*sqrt(a + a*cos(c + d*x))) + (a*C*cos(c + d*x)^(5/2)*sin(c + d*x))/(24*d*sqrt(a + a*cos(c + d*x))) + (C*cos(c + d*x)^(5/2)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(4*d)],
[sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))*(A + C*cos(c + d*x)^2), x, 4, (sqrt(a)*(8*A + 5*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(8*d) + (a*(8*A + 5*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(8*d*sqrt(a + a*cos(c + d*x))) + (a*C*cos(c + d*x)^(3/2)*sin(c + d*x))/(12*d*sqrt(a + a*cos(c + d*x))) + (C*cos(c + d*x)^(3/2)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(3*d)],
[(sqrt(a + a*cos(c + d*x))*(A + C*cos(c + d*x)^2))/sqrt(cos(c + d*x)), x, 3, (sqrt(a)*(8*A + 3*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(4*d) + (a*C*sqrt(cos(c + d*x))*sin(c + d*x))/(4*d*sqrt(a + a*cos(c + d*x))) + (C*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(2*d)],
[(sqrt(a + a*cos(c + d*x))*(A + C*cos(c + d*x)^2))/cos(c + d*x)^(3/2), x, 3, (sqrt(a)*C*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/d - (a*(2*A - C)*sqrt(cos(c + d*x))*sin(c + d*x))/(d*sqrt(a + a*cos(c + d*x))) + (2*A*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(d*sqrt(cos(c + d*x)))],
[(sqrt(a + a*cos(c + d*x))*(A + C*cos(c + d*x)^2))/cos(c + d*x)^(5/2), x, 3, (2*sqrt(a)*C*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/d + (2*a*A*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))) + (2*A*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2))],
[(sqrt(a + a*cos(c + d*x))*(A + C*cos(c + d*x)^2))/cos(c + d*x)^(7/2), x, 3, (2*a*A*sin(c + d*x))/(15*d*cos(c + d*x)^(3/2)*sqrt(a + a*cos(c + d*x))) + (2*a*(8*A + 15*C)*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))) + (2*A*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2))],
[(sqrt(a + a*cos(c + d*x))*(A + C*cos(c + d*x)^2))/cos(c + d*x)^(9/2), x, 4, (2*a*A*sin(c + d*x))/(35*d*cos(c + d*x)^(5/2)*sqrt(a + a*cos(c + d*x))) + (2*a*(24*A + 35*C)*sin(c + d*x))/(105*d*cos(c + d*x)^(3/2)*sqrt(a + a*cos(c + d*x))) + (4*a*(24*A + 35*C)*sin(c + d*x))/(105*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))) + (2*A*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(7*d*cos(c + d*x)^(7/2))],
[(sqrt(a + a*cos(c + d*x))*(A + C*cos(c + d*x)^2))/cos(c + d*x)^(11/2), x, 5, (2*a*A*sin(c + d*x))/(63*d*cos(c + d*x)^(7/2)*sqrt(a + a*cos(c + d*x))) + (2*a*(16*A + 21*C)*sin(c + d*x))/(105*d*cos(c + d*x)^(5/2)*sqrt(a + a*cos(c + d*x))) + (8*a*(16*A + 21*C)*sin(c + d*x))/(315*d*cos(c + d*x)^(3/2)*sqrt(a + a*cos(c + d*x))) + (16*a*(16*A + 21*C)*sin(c + d*x))/(315*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))) + (2*A*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(9*d*cos(c + d*x)^(9/2))],


[cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))^(3/2)*(A + C*cos(c + d*x)^2), x, 6, (a^(3/2)*(176*A + 133*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(128*d) + (a^2*(176*A + 133*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(128*d*sqrt(a + a*cos(c + d*x))) + (a^2*(176*A + 133*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(192*d*sqrt(a + a*cos(c + d*x))) + (a^2*(80*A + 67*C)*cos(c + d*x)^(5/2)*sin(c + d*x))/(240*d*sqrt(a + a*cos(c + d*x))) + (3*a*C*cos(c + d*x)^(5/2)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(40*d) + (C*cos(c + d*x)^(5/2)*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(5*d)],
[sqrt(cos(c + d*x))*(a + a*cos(c + d*x))^(3/2)*(A + C*cos(c + d*x)^2), x, 5, (a^(3/2)*(112*A + 75*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(64*d) + (a^2*(112*A + 75*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(64*d*sqrt(a + a*cos(c + d*x))) + (a^2*(16*A + 13*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(32*d*sqrt(a + a*cos(c + d*x))) + (a*C*cos(c + d*x)^(3/2)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(8*d) + (C*cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(4*d)],
[((a + a*cos(c + d*x))^(3/2)*(A + C*cos(c + d*x)^2))/sqrt(cos(c + d*x)), x, 4, (a^(3/2)*(24*A + 11*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(8*d) + (a^2*(24*A + 19*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(24*d*sqrt(a + a*cos(c + d*x))) + (a*C*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(4*d) + (C*sqrt(cos(c + d*x))*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(3*d)],
[((a + a*cos(c + d*x))^(3/2)*(A + C*cos(c + d*x)^2))/cos(c + d*x)^(3/2), x, 4, (a^(3/2)*(8*A + 7*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(4*d) - (a^2*(8*A - 5*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(4*d*sqrt(a + a*cos(c + d*x))) - (a*(4*A - C)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(2*d) + (2*A*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(d*sqrt(cos(c + d*x)))],
[((a + a*cos(c + d*x))^(3/2)*(A + C*cos(c + d*x)^2))/cos(c + d*x)^(5/2), x, 4, (3*a^(3/2)*C*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/d - (a^2*(8*A - 3*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(3*d*sqrt(a + a*cos(c + d*x))) + (2*a*A*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(d*sqrt(cos(c + d*x))) + (2*A*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2))],
[((a + a*cos(c + d*x))^(3/2)*(A + C*cos(c + d*x)^2))/cos(c + d*x)^(7/2), x, 4, (2*a^(3/2)*C*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/d + (2*a^2*(4*A + 5*C)*sin(c + d*x))/(5*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))) + (2*a*A*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(5*d*cos(c + d*x)^(3/2)) + (2*A*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2))],
[((a + a*cos(c + d*x))^(3/2)*(A + C*cos(c + d*x)^2))/cos(c + d*x)^(9/2), x, 4, (8*a^2*(19*A + 35*C)*sin(c + d*x))/(105*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))) + (2*a*(19*A + 35*C)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(105*d*cos(c + d*x)^(3/2)) + (2*A*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(7*d*cos(c + d*x)^(7/2)) + (6*A*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(35*d*cos(c + d*x)^(5/2))],
[((a + a*cos(c + d*x))^(3/2)*(A + C*cos(c + d*x)^2))/cos(c + d*x)^(11/2), x, 5, (2*a^2*(52*A + 63*C)*sin(c + d*x))/(315*d*cos(c + d*x)^(5/2)*sqrt(a + a*cos(c + d*x))) + (2*a^2*(136*A + 189*C)*sin(c + d*x))/(315*d*cos(c + d*x)^(3/2)*sqrt(a + a*cos(c + d*x))) + (4*a^2*(136*A + 189*C)*sin(c + d*x))/(315*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))) + (2*a*A*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(21*d*cos(c + d*x)^(7/2)) + (2*A*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(9*d*cos(c + d*x)^(9/2))],
[((a + a*cos(c + d*x))^(3/2)*(A + C*cos(c + d*x)^2))/cos(c + d*x)^(13/2), x, 6, (2*a^2*(28*A + 33*C)*sin(c + d*x))/(231*d*cos(c + d*x)^(7/2)*sqrt(a + a*cos(c + d*x))) + (2*a^2*(112*A + 143*C)*sin(c + d*x))/(385*d*cos(c + d*x)^(5/2)*sqrt(a + a*cos(c + d*x))) + (8*a^2*(112*A + 143*C)*sin(c + d*x))/(1155*d*cos(c + d*x)^(3/2)*sqrt(a + a*cos(c + d*x))) + (16*a^2*(112*A + 143*C)*sin(c + d*x))/(1155*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))) + (2*a*A*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(33*d*cos(c + d*x)^(9/2)) + (2*A*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(11*d*cos(c + d*x)^(11/2))],


[cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))^(5/2)*(A + C*cos(c + d*x)^2), x, 7, (a^(5/2)*(1304*A + 1015*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(512*d) + (a^3*(1304*A + 1015*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(512*d*sqrt(a + a*cos(c + d*x))) + (a^3*(1304*A + 1015*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(768*d*sqrt(a + a*cos(c + d*x))) + (a^3*(136*A + 109*C)*cos(c + d*x)^(5/2)*sin(c + d*x))/(192*d*sqrt(a + a*cos(c + d*x))) + (a^2*(24*A + 23*C)*cos(c + d*x)^(5/2)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(96*d) + (a*C*cos(c + d*x)^(5/2)*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(12*d) + (C*cos(c + d*x)^(5/2)*(a + a*cos(c + d*x))^(5/2)*sin(c + d*x))/(6*d)],
[sqrt(cos(c + d*x))*(a + a*cos(c + d*x))^(5/2)*(A + C*cos(c + d*x)^2), x, 6, (a^(5/2)*(400*A + 283*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(128*d) + (a^3*(400*A + 283*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(128*d*sqrt(a + a*cos(c + d*x))) + (a^3*(1040*A + 787*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(960*d*sqrt(a + a*cos(c + d*x))) + (a^2*(80*A + 79*C)*cos(c + d*x)^(3/2)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(240*d) + (a*C*cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(8*d) + (C*cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))^(5/2)*sin(c + d*x))/(5*d)],
[((a + a*cos(c + d*x))^(5/2)*(A + C*cos(c + d*x)^2))/sqrt(cos(c + d*x)), x, 5, (a^(5/2)*(304*A + 163*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(64*d) + (a^3*(432*A + 299*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(192*d*sqrt(a + a*cos(c + d*x))) + (a^2*(16*A + 17*C)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(32*d) + (5*a*C*sqrt(cos(c + d*x))*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(24*d) + (C*sqrt(cos(c + d*x))*(a + a*cos(c + d*x))^(5/2)*sin(c + d*x))/(4*d)],
[((a + a*cos(c + d*x))^(5/2)*(A + C*cos(c + d*x)^2))/cos(c + d*x)^(3/2), x, 5, (5*a^(5/2)*(8*A + 5*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(8*d) - (a^3*(24*A - 49*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(24*d*sqrt(a + a*cos(c + d*x))) - (a^2*(8*A - 3*C)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(4*d) - (a*(6*A - C)*sqrt(cos(c + d*x))*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(3*d) + (2*A*(a + a*cos(c + d*x))^(5/2)*sin(c + d*x))/(d*sqrt(cos(c + d*x)))],
[((a + a*cos(c + d*x))^(5/2)*(A + C*cos(c + d*x)^2))/cos(c + d*x)^(5/2), x, 5, (a^(5/2)*(8*A + 19*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(4*d) - (a^3*(56*A - 27*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(12*d*sqrt(a + a*cos(c + d*x))) - (a^2*(8*A - C)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(2*d) + (10*a*A*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))) + (2*A*(a + a*cos(c + d*x))^(5/2)*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2))],
[((a + a*cos(c + d*x))^(5/2)*(A + C*cos(c + d*x)^2))/cos(c + d*x)^(7/2), x, 5, (5*a^(5/2)*C*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/d - (a^3*(64*A + 15*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(15*d*sqrt(a + a*cos(c + d*x))) + (2*a^2*(8*A + 5*C)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(5*d*sqrt(cos(c + d*x))) + (2*a*A*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2)) + (2*A*(a + a*cos(c + d*x))^(5/2)*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2))],
[((a + a*cos(c + d*x))^(5/2)*(A + C*cos(c + d*x)^2))/cos(c + d*x)^(9/2), x, 5, (2*a^(5/2)*C*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/d + (2*a^3*(32*A + 49*C)*sin(c + d*x))/(21*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))) + (2*a^2*(8*A + 7*C)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(21*d*cos(c + d*x)^(3/2)) + (2*a*A*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(7*d*cos(c + d*x)^(5/2)) + (2*A*(a + a*cos(c + d*x))^(5/2)*sin(c + d*x))/(7*d*cos(c + d*x)^(7/2))],
[((a + a*cos(c + d*x))^(5/2)*(A + C*cos(c + d*x)^2))/cos(c + d*x)^(11/2), x, 5, (64*a^3*(13*A + 21*C)*sin(c + d*x))/(315*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))) + (16*a^2*(13*A + 21*C)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(315*d*cos(c + d*x)^(3/2)) + (2*a*(13*A + 21*C)*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(105*d*cos(c + d*x)^(5/2)) + (2*A*(a + a*cos(c + d*x))^(5/2)*sin(c + d*x))/(9*d*cos(c + d*x)^(9/2)) + (10*A*(a + a*cos(c + d*x))^(5/2)*sin(c + d*x))/(63*d*cos(c + d*x)^(7/2))],
[((a + a*cos(c + d*x))^(5/2)*(A + C*cos(c + d*x)^2))/cos(c + d*x)^(13/2), x, 6, (2*a^3*(232*A + 297*C)*sin(c + d*x))/(693*d*cos(c + d*x)^(5/2)*sqrt(a + a*cos(c + d*x))) + (2*a^3*(568*A + 759*C)*sin(c + d*x))/(693*d*cos(c + d*x)^(3/2)*sqrt(a + a*cos(c + d*x))) + (4*a^3*(568*A + 759*C)*sin(c + d*x))/(693*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))) + (2*a^2*(32*A + 33*C)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(231*d*cos(c + d*x)^(7/2)) + (10*a*A*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(99*d*cos(c + d*x)^(9/2)) + (2*A*(a + a*cos(c + d*x))^(5/2)*sin(c + d*x))/(11*d*cos(c + d*x)^(11/2))],
[((a + a*cos(c + d*x))^(5/2)*(A + C*cos(c + d*x)^2))/cos(c + d*x)^(15/2), x, 7, (2*a^3*(2224*A + 2717*C)*sin(c + d*x))/(9009*d*cos(c + d*x)^(7/2)*sqrt(a + a*cos(c + d*x))) + (2*a^3*(8368*A + 10439*C)*sin(c + d*x))/(15015*d*cos(c + d*x)^(5/2)*sqrt(a + a*cos(c + d*x))) + (8*a^3*(8368*A + 10439*C)*sin(c + d*x))/(45045*d*cos(c + d*x)^(3/2)*sqrt(a + a*cos(c + d*x))) + (16*a^3*(8368*A + 10439*C)*sin(c + d*x))/(45045*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))) + (2*a^2*(136*A + 143*C)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(1287*d*cos(c + d*x)^(9/2)) + (10*a*A*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(143*d*cos(c + d*x)^(11/2)) + (2*A*(a + a*cos(c + d*x))^(5/2)*sin(c + d*x))/(13*d*cos(c + d*x)^(13/2))],


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


[(cos(c + d*x)^(3/2)*(A + C*cos(c + d*x)^2))/sqrt(a + a*cos(c + d*x)), x, 6, -((8*A + 9*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(8*sqrt(a)*d) + (sqrt(2)*(A + C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(sqrt(a)*d) + ((8*A + 7*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(8*d*sqrt(a + a*cos(c + d*x))) - (C*cos(c + d*x)^(3/2)*sin(c + d*x))/(12*d*sqrt(a + a*cos(c + d*x))) + (C*cos(c + d*x)^(5/2)*sin(c + d*x))/(3*d*sqrt(a + a*cos(c + d*x)))],
[(sqrt(cos(c + d*x))*(A + C*cos(c + d*x)^2))/sqrt(a + a*cos(c + d*x)), x, 5, ((8*A + 7*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(4*sqrt(a)*d) - (sqrt(2)*(A + C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(sqrt(a)*d) - (C*sqrt(cos(c + d*x))*sin(c + d*x))/(4*d*sqrt(a + a*cos(c + d*x))) + (C*cos(c + d*x)^(3/2)*sin(c + d*x))/(2*d*sqrt(a + a*cos(c + d*x)))],
[(A + C*cos(c + d*x)^2)/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))), x, 4, -((C*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(sqrt(a)*d)) + (sqrt(2)*(A + C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(sqrt(a)*d) + (C*sqrt(cos(c + d*x))*sin(c + d*x))/(d*sqrt(a + a*cos(c + d*x)))],
[(A + C*cos(c + d*x)^2)/(cos(c + d*x)^(3/2)*sqrt(a + a*cos(c + d*x))), x, 4, (2*C*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(sqrt(a)*d) - (sqrt(2)*(A + C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(sqrt(a)*d) + (2*A*sin(c + d*x))/(d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))],
[(A + C*cos(c + d*x)^2)/(cos(c + d*x)^(5/2)*sqrt(a + a*cos(c + d*x))), x, 3, (sqrt(2)*(A + C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(sqrt(a)*d) + (2*A*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2)*sqrt(a + a*cos(c + d*x))) - (2*A*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))],
[(A + C*cos(c + d*x)^2)/(cos(c + d*x)^(7/2)*sqrt(a + a*cos(c + d*x))), x, 4, -((sqrt(2)*(A + C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(sqrt(a)*d)) + (2*A*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2)*sqrt(a + a*cos(c + d*x))) - (2*A*sin(c + d*x))/(15*d*cos(c + d*x)^(3/2)*sqrt(a + a*cos(c + d*x))) + (2*(13*A + 15*C)*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))],
[(A + C*cos(c + d*x)^2)/(cos(c + d*x)^(9/2)*sqrt(a + a*cos(c + d*x))), x, 5, (sqrt(2)*(A + C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(sqrt(a)*d) + (2*A*sin(c + d*x))/(7*d*cos(c + d*x)^(7/2)*sqrt(a + a*cos(c + d*x))) - (2*A*sin(c + d*x))/(35*d*cos(c + d*x)^(5/2)*sqrt(a + a*cos(c + d*x))) + (2*(31*A + 35*C)*sin(c + d*x))/(105*d*cos(c + d*x)^(3/2)*sqrt(a + a*cos(c + d*x))) - (2*(43*A + 35*C)*sin(c + d*x))/(105*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))],


[(cos(c + d*x)^(3/2)*(A + C*cos(c + d*x)^2))/(a + a*cos(c + d*x))^(3/2), x, 6, ((8*A + 19*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(4*a^(3/2)*d) - ((5*A + 13*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d) - ((A + C)*cos(c + d*x)^(5/2)*sin(c + d*x))/(2*d*(a + a*cos(c + d*x))^(3/2)) - ((2*A + 7*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(4*a*d*sqrt(a + a*cos(c + d*x))) + ((A + 2*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(2*a*d*sqrt(a + a*cos(c + d*x)))],
[(sqrt(cos(c + d*x))*(A + C*cos(c + d*x)^2))/(a + a*cos(c + d*x))^(3/2), x, 5, (-3*C*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(a^(3/2)*d) + ((A + 9*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d) - ((A + C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(2*d*(a + a*cos(c + d*x))^(3/2)) + ((A + 3*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(2*a*d*sqrt(a + a*cos(c + d*x)))],
[(A + C*cos(c + d*x)^2)/(sqrt(cos(c + d*x))*(a + a*cos(c + d*x))^(3/2)), x, 4, (2*C*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(a^(3/2)*d) + ((3*A - 5*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d) - ((A + C)*sqrt(cos(c + d*x))*sin(c + d*x))/(2*d*(a + a*cos(c + d*x))^(3/2))],
[(A + C*cos(c + d*x)^2)/(cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))^(3/2)), x, 3, -((7*A - C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d) - ((A + C)*sin(c + d*x))/(2*d*sqrt(cos(c + d*x))*(a + a*cos(c + d*x))^(3/2)) + ((5*A + C)*sin(c + d*x))/(2*a*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))],
[(A + C*cos(c + d*x)^2)/(cos(c + d*x)^(5/2)*(a + a*cos(c + d*x))^(3/2)), x, 4, ((11*A + 3*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d) - ((A + C)*sin(c + d*x))/(2*d*cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))^(3/2)) + ((7*A + 3*C)*sin(c + d*x))/(6*a*d*cos(c + d*x)^(3/2)*sqrt(a + a*cos(c + d*x))) - ((19*A + 3*C)*sin(c + d*x))/(6*a*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))],
[(A + C*cos(c + d*x)^2)/(cos(c + d*x)^(7/2)*(a + a*cos(c + d*x))^(3/2)), x, 5, -((15*A + 7*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d) - ((A + C)*sin(c + d*x))/(2*d*cos(c + d*x)^(5/2)*(a + a*cos(c + d*x))^(3/2)) + ((9*A + 5*C)*sin(c + d*x))/(10*a*d*cos(c + d*x)^(5/2)*sqrt(a + a*cos(c + d*x))) - ((13*A + 5*C)*sin(c + d*x))/(10*a*d*cos(c + d*x)^(3/2)*sqrt(a + a*cos(c + d*x))) + ((49*A + 25*C)*sin(c + d*x))/(10*a*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))],


[(cos(c + d*x)^(3/2)*(A + C*cos(c + d*x)^2))/(a + a*cos(c + d*x))^(5/2), x, 6, (-5*C*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(a^(5/2)*d) + ((3*A + 115*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d) - ((A + C)*cos(c + d*x)^(5/2)*sin(c + d*x))/(4*d*(a + a*cos(c + d*x))^(5/2)) + ((A - 15*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2)) + ((3*A + 35*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(16*a^2*d*sqrt(a + a*cos(c + d*x)))],
[(sqrt(cos(c + d*x))*(A + C*cos(c + d*x)^2))/(a + a*cos(c + d*x))^(5/2), x, 5, (2*C*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(a^(5/2)*d) + ((5*A - 43*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d) - ((A + C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(4*d*(a + a*cos(c + d*x))^(5/2)) + ((5*A - 11*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2))],
[(A + C*cos(c + d*x)^2)/(sqrt(cos(c + d*x))*(a + a*cos(c + d*x))^(5/2)), x, 3, ((19*A + 3*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d) - ((A + C)*sqrt(cos(c + d*x))*sin(c + d*x))/(4*d*(a + a*cos(c + d*x))^(5/2)) - ((9*A - 7*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2))],
[(A + C*cos(c + d*x)^2)/(cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))^(5/2)), x, 4, (-5*(15*A - C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d) - ((A + C)*sin(c + d*x))/(4*d*sqrt(cos(c + d*x))*(a + a*cos(c + d*x))^(5/2)) - ((13*A - 3*C)*sin(c + d*x))/(16*a*d*sqrt(cos(c + d*x))*(a + a*cos(c + d*x))^(3/2)) + ((49*A + C)*sin(c + d*x))/(16*a^2*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))],
[(A + C*cos(c + d*x)^2)/(cos(c + d*x)^(5/2)*(a + a*cos(c + d*x))^(5/2)), x, 5, ((163*A + 19*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d) - ((A + C)*sin(c + d*x))/(4*d*cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))^(5/2)) - ((17*A + C)*sin(c + d*x))/(16*a*d*cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))^(3/2)) + (5*(19*A + 3*C)*sin(c + d*x))/(48*a^2*d*cos(c + d*x)^(3/2)*sqrt(a + a*cos(c + d*x))) - ((299*A + 27*C)*sin(c + d*x))/(48*a^2*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))],


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


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


[(A + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^(2/3), x, 3, (1/80)*a*(40*A + 19*C)*Int(1/(a + a*cos(c + d*x))^(1/3), x) + (3*a*(40*A + 19*C)*sin(c + d*x))/(80*d*(a + a*cos(c + d*x))^(1/3)) - (9*C*(a + a*cos(c + d*x))^(2/3)*sin(c + d*x))/(40*d) + (3*C*(a + a*cos(c + d*x))^(5/3)*sin(c + d*x))/(8*a*d)],
[(A + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^(1/3), x, 3, (-(1/28))*a*(28*A + 13*C)*Int(1/(a + a*sin(c + Pi/2 + d*x))^(2/3), x) + (3*a*(28*A + 13*C)*sin(c + d*x))/(28*d*(a + a*cos(c + d*x))^(2/3)) - (9*C*(a + a*cos(c + d*x))^(1/3)*sin(c + d*x))/(28*d) + (3*C*(a + a*cos(c + d*x))^(4/3)*sin(c + d*x))/(7*a*d)],


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


[(A + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^(1/3), x, 2, (1/10)*(10*A + 7*C)*Int(1/(a + a*cos(c + d*x))^(1/3), x) - (9*C*sin(c + d*x))/(10*d*(a + a*cos(c + d*x))^(1/3)) + (3*C*(a + a*cos(c + d*x))^(2/3)*sin(c + d*x))/(5*a*d)],
[(A + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^(2/3), x, 2, (1/4)*(4*A + 7*C)*Int(1/(a + a*sin(c + Pi/2 + d*x))^(2/3), x) - (9*C*sin(c + d*x))/(4*d*(a + a*cos(c + d*x))^(2/3)) + (3*C*(a + a*cos(c + d*x))^(1/3)*sin(c + d*x))/(4*a*d)],


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


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


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


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


[(a + a*cos(c + d*x))^2*(A + C*cos(c + d*x)^2)*sec(c + d*x)^(11/2), x, 9, (-16*a^2*(2*A + 3*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(15*d) + (4*a^2*(5*A + 7*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(21*d) + (16*a^2*(2*A + 3*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(15*d) + (4*a^2*(5*A + 7*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(21*d) + (2*a^2*(19*A + 21*C)*sec(c + d*x)^(5/2)*sin(c + d*x))/(105*d) + (8*a*A*(a + a*cos(c + d*x))*sec(c + d*x)^(7/2)*sin(c + d*x))/(63*d) + (2*A*(a + a*cos(c + d*x))^2*sec(c + d*x)^(9/2)*sin(c + d*x))/(9*d)],
[(a + a*cos(c + d*x))^2*(A + C*cos(c + d*x)^2)*sec(c + d*x)^(9/2), x, 8, (-4*a^2*(3*A + 5*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*d) + (8*a^2*(3*A + 7*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(21*d) + (4*a^2*(3*A + 5*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d) + (2*a^2*(33*A + 35*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(105*d) + (8*a*A*(a + a*cos(c + d*x))*sec(c + d*x)^(5/2)*sin(c + d*x))/(35*d) + (2*A*(a + a*cos(c + d*x))^2*sec(c + d*x)^(7/2)*sin(c + d*x))/(7*d)],
[(a + a*cos(c + d*x))^2*(A + C*cos(c + d*x)^2)*sec(c + d*x)^(7/2), x, 7, (-16*a^2*A*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*d) + (4*a^2*(A + 3*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*d) + (2*a^2*(17*A + 15*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(15*d) + (8*a*A*(a + a*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(15*d) + (2*A*(a + a*cos(c + d*x))^2*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d)],
[(a + a*cos(c + d*x))^2*(A + C*cos(c + d*x)^2)*sec(c + d*x)^(5/2), x, 7, (-4*a^2*(A - C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/d + (8*a^2*(A + C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*d) - (2*a^2*(5*A - C)*sin(c + d*x))/(3*d*sqrt(sec(c + d*x))) + (8*a*A*(a + a*cos(c + d*x))*sqrt(sec(c + d*x))*sin(c + d*x))/(3*d) + (2*A*(a + a*cos(c + d*x))^2*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d)],
[(a + a*cos(c + d*x))^2*(A + C*cos(c + d*x)^2)*sec(c + d*x)^(3/2), x, 7, (16*a^2*C*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*d) + (4*a^2*(3*A + C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*d) - (2*a^2*(15*A - 7*C)*sin(c + d*x))/(15*d*sqrt(sec(c + d*x))) - (2*a*(5*A - C)*(a + a*cos(c + d*x))*sin(c + d*x))/(5*d*sqrt(sec(c + d*x))) + (2*A*(a + a*cos(c + d*x))^2*sqrt(sec(c + d*x))*sin(c + d*x))/d],
[(a + a*cos(c + d*x))^2*(A + C*cos(c + d*x)^2)*sqrt(sec(c + d*x)), x, 7, (4*a^2*(5*A + 3*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*d) + (8*a^2*(7*A + 3*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(21*d) + (2*a^2*(35*A + 33*C)*sin(c + d*x))/(105*d*sqrt(sec(c + d*x))) + (8*a*C*(a + a*cos(c + d*x))*sin(c + d*x))/(35*d*sqrt(sec(c + d*x))) + (2*C*(a + a*cos(c + d*x))^2*sin(c + d*x))/(7*d*sqrt(sec(c + d*x)))],
[((a + a*cos(c + d*x))^2*(A + C*cos(c + d*x)^2))/sqrt(sec(c + d*x)), x, 8, (16*a^2*(3*A + 2*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(15*d) + (4*a^2*(7*A + 5*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(21*d) + (2*a^2*(21*A + 19*C)*sin(c + d*x))/(105*d*sec(c + d*x)^(3/2)) + (8*a*C*(a + a*cos(c + d*x))*sin(c + d*x))/(63*d*sec(c + d*x)^(3/2)) + (2*C*(a + a*cos(c + d*x))^2*sin(c + d*x))/(9*d*sec(c + d*x)^(3/2)) + (4*a^2*(7*A + 5*C)*sin(c + d*x))/(21*d*sqrt(sec(c + d*x)))],
[((a + a*cos(c + d*x))^2*(A + C*cos(c + d*x)^2))/sec(c + d*x)^(3/2), x, 9, (4*a^2*(9*A + 7*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(15*d) + (8*a^2*(33*A + 25*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(231*d) + (2*a^2*(99*A + 89*C)*sin(c + d*x))/(693*d*sec(c + d*x)^(5/2)) + (8*a*C*(a + a*cos(c + d*x))*sin(c + d*x))/(99*d*sec(c + d*x)^(5/2)) + (2*C*(a + a*cos(c + d*x))^2*sin(c + d*x))/(11*d*sec(c + d*x)^(5/2)) + (4*a^2*(9*A + 7*C)*sin(c + d*x))/(45*d*sec(c + d*x)^(3/2)) + (8*a^2*(33*A + 25*C)*sin(c + d*x))/(231*d*sqrt(sec(c + d*x)))],


[(a + a*cos(c + d*x))^3*(A + C*cos(c + d*x)^2)*sec(c + d*x)^(13/2), x, 10, (-4*a^3*(5*A + 7*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*d) + (4*a^3*(105*A + 143*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(231*d) + (4*a^3*(5*A + 7*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d) + (4*a^3*(105*A + 143*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(231*d) + (8*a^3*(35*A + 44*C)*sec(c + d*x)^(5/2)*sin(c + d*x))/(385*d) + (2*a^2*(35*A + 33*C)*(a + a*cos(c + d*x))*sec(c + d*x)^(7/2)*sin(c + d*x))/(231*d) + (4*a*A*(a + a*cos(c + d*x))^2*sec(c + d*x)^(9/2)*sin(c + d*x))/(33*d) + (2*A*(a + a*cos(c + d*x))^3*sec(c + d*x)^(11/2)*sin(c + d*x))/(11*d)],
[(a + a*cos(c + d*x))^3*(A + C*cos(c + d*x)^2)*sec(c + d*x)^(11/2), x, 9, (-4*a^3*(17*A + 27*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(15*d) + (4*a^3*(11*A + 21*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(21*d) + (4*a^3*(17*A + 27*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(15*d) + (8*a^3*(16*A + 21*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(105*d) + (2*a^2*(73*A + 63*C)*(a + a*cos(c + d*x))*sec(c + d*x)^(5/2)*sin(c + d*x))/(315*d) + (4*a*A*(a + a*cos(c + d*x))^2*sec(c + d*x)^(7/2)*sin(c + d*x))/(21*d) + (2*A*(a + a*cos(c + d*x))^3*sec(c + d*x)^(9/2)*sin(c + d*x))/(9*d)],
[(a + a*cos(c + d*x))^3*(A + C*cos(c + d*x)^2)*sec(c + d*x)^(9/2), x, 8, (-4*a^3*(7*A + 5*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*d) + (4*a^3*(13*A + 35*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(21*d) + (8*a^3*(53*A + 70*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(105*d) + (2*a^2*(7*A + 5*C)*(a + a*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(15*d) + (12*a*A*(a + a*cos(c + d*x))^2*sec(c + d*x)^(5/2)*sin(c + d*x))/(35*d) + (2*A*(a + a*cos(c + d*x))^3*sec(c + d*x)^(7/2)*sin(c + d*x))/(7*d)],
[(a + a*cos(c + d*x))^3*(A + C*cos(c + d*x)^2)*sec(c + d*x)^(7/2), x, 8, (-4*a^3*(9*A - 5*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*d) + (4*a^3*(3*A + 5*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*d) - (4*a^3*(21*A + 5*C)*sin(c + d*x))/(15*d*sqrt(sec(c + d*x))) + (2*a^2*(11*A + 5*C)*(a + a*cos(c + d*x))*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d) + (4*a*A*(a + a*cos(c + d*x))^2*sec(c + d*x)^(3/2)*sin(c + d*x))/(5*d) + (2*A*(a + a*cos(c + d*x))^3*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d)],
[(a + a*cos(c + d*x))^3*(A + C*cos(c + d*x)^2)*sec(c + d*x)^(5/2), x, 8, (-4*a^3*(5*A - 9*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*d) + (4*a^3*(5*A + 3*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*d) - (8*a^3*(10*A - 3*C)*sin(c + d*x))/(15*d*sqrt(sec(c + d*x))) - (2*a^2*(35*A - 3*C)*(a + a*cos(c + d*x))*sin(c + d*x))/(15*d*sqrt(sec(c + d*x))) + (4*a*A*(a + a*cos(c + d*x))^2*sqrt(sec(c + d*x))*sin(c + d*x))/d + (2*A*(a + a*cos(c + d*x))^3*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d)],
[(a + a*cos(c + d*x))^3*(A + C*cos(c + d*x)^2)*sec(c + d*x)^(3/2), x, 8, (4*a^3*(5*A + 7*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*d) + (4*a^3*(35*A + 13*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(21*d) - (4*a^3*(35*A - 41*C)*sin(c + d*x))/(105*d*sqrt(sec(c + d*x))) - (2*a^2*(35*A - 11*C)*(a + a*cos(c + d*x))*sin(c + d*x))/(35*d*sqrt(sec(c + d*x))) - (2*a*(7*A - C)*(a + a*cos(c + d*x))^2*sin(c + d*x))/(7*d*sqrt(sec(c + d*x))) + (2*A*(a + a*cos(c + d*x))^3*sqrt(sec(c + d*x))*sin(c + d*x))/d],
[(a + a*cos(c + d*x))^3*(A + C*cos(c + d*x)^2)*sqrt(sec(c + d*x)), x, 8, (4*a^3*(27*A + 17*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(15*d) + (4*a^3*(21*A + 11*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(21*d) + (8*a^3*(21*A + 16*C)*sin(c + d*x))/(105*d*sqrt(sec(c + d*x))) + (2*a^2*(63*A + 73*C)*(a + a*cos(c + d*x))*sin(c + d*x))/(315*d*sqrt(sec(c + d*x))) + (4*a*C*(a + a*cos(c + d*x))^2*sin(c + d*x))/(21*d*sqrt(sec(c + d*x))) + (2*C*(a + a*cos(c + d*x))^3*sin(c + d*x))/(9*d*sqrt(sec(c + d*x)))],
[((a + a*cos(c + d*x))^3*(A + C*cos(c + d*x)^2))/sqrt(sec(c + d*x)), x, 9, (4*a^3*(7*A + 5*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*d) + (4*a^3*(143*A + 105*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(231*d) + (8*a^3*(44*A + 35*C)*sin(c + d*x))/(385*d*sec(c + d*x)^(3/2)) + (2*a^2*(33*A + 35*C)*(a + a*cos(c + d*x))*sin(c + d*x))/(231*d*sec(c + d*x)^(3/2)) + (4*a*C*(a + a*cos(c + d*x))^2*sin(c + d*x))/(33*d*sec(c + d*x)^(3/2)) + (2*C*(a + a*cos(c + d*x))^3*sin(c + d*x))/(11*d*sec(c + d*x)^(3/2)) + (4*a^3*(143*A + 105*C)*sin(c + d*x))/(231*d*sqrt(sec(c + d*x)))],
[((a + a*cos(c + d*x))^3*(A + C*cos(c + d*x)^2))/sec(c + d*x)^(3/2), x, 10, (4*a^3*(221*A + 175*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(195*d) + (4*a^3*(121*A + 95*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(231*d) + (40*a^3*(143*A + 118*C)*sin(c + d*x))/(9009*d*sec(c + d*x)^(5/2)) + (2*a^2*(143*A + 145*C)*(a + a*cos(c + d*x))*sin(c + d*x))/(1287*d*sec(c + d*x)^(5/2)) + (12*a*C*(a + a*cos(c + d*x))^2*sin(c + d*x))/(143*d*sec(c + d*x)^(5/2)) + (2*C*(a + a*cos(c + d*x))^3*sin(c + d*x))/(13*d*sec(c + d*x)^(5/2)) + (4*a^3*(221*A + 175*C)*sin(c + d*x))/(585*d*sec(c + d*x)^(3/2)) + (4*a^3*(121*A + 95*C)*sin(c + d*x))/(231*d*sqrt(sec(c + d*x)))],


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


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


[((A + C*cos(c + d*x)^2)*sec(c + d*x)^(5/2))/(a + a*cos(c + d*x))^2, x, 8, ((7*A + C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(a^2*d) + (2*(5*A + C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*a^2*d) - ((7*A + C)*sqrt(sec(c + d*x))*sin(c + d*x))/(a^2*d) + (2*(5*A + C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*a^2*d) - ((A + C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d*(a + a*cos(c + d*x))^2) - ((7*A + C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*a*d*(a + a*cos(c + d*x)))],
[((A + C*cos(c + d*x)^2)*sec(c + d*x)^(3/2))/(a + a*cos(c + d*x))^2, x, 7, (-4*A*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(a^2*d) - ((5*A - C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*a^2*d) + (4*A*sqrt(sec(c + d*x))*sin(c + d*x))/(a^2*d) - ((A + C)*sqrt(sec(c + d*x))*sin(c + d*x))/(3*d*(a + a*cos(c + d*x))^2) - ((5*A - C)*sqrt(sec(c + d*x))*sin(c + d*x))/(3*a*d*(a + a*cos(c + d*x)))],
[((A + C*cos(c + d*x)^2)*sqrt(sec(c + d*x)))/(a + a*cos(c + d*x))^2, x, 6, ((A - C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(a^2*d) + (2*(A + C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*a^2*d) - ((A + C)*sin(c + d*x))/(3*d*(a + a*cos(c + d*x))^2*sqrt(sec(c + d*x))) - ((A - C)*sin(c + d*x))/(a*d*(a + a*cos(c + d*x))*sqrt(sec(c + d*x)))],
[(A + C*cos(c + d*x)^2)/((a + a*cos(c + d*x))^2*sqrt(sec(c + d*x))), x, 6, (4*C*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(a^2*d) + ((A - 5*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*a^2*d) - ((A + C)*sin(c + d*x))/(3*d*(a + a*cos(c + d*x))^2*sec(c + d*x)^(3/2)) + ((A - 5*C)*sin(c + d*x))/(3*a*d*(a + a*cos(c + d*x))*sqrt(sec(c + d*x)))],
[(A + C*cos(c + d*x)^2)/((a + a*cos(c + d*x))^2*sec(c + d*x)^(3/2)), x, 7, -(((A + 7*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(a^2*d)) + (2*(A + 5*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*a^2*d) - ((A + C)*sin(c + d*x))/(3*d*(a + a*cos(c + d*x))^2*sec(c + d*x)^(5/2)) - ((A + 7*C)*sin(c + d*x))/(3*a*d*(a + a*cos(c + d*x))*sec(c + d*x)^(3/2)) + (2*(A + 5*C)*sin(c + d*x))/(3*a^2*d*sqrt(sec(c + d*x)))],
[(A + C*cos(c + d*x)^2)/((a + a*cos(c + d*x))^2*sec(c + d*x)^(5/2)), x, 8, (4*(5*A + 14*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*a^2*d) - (5*(A + 3*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*a^2*d) - ((A + C)*sin(c + d*x))/(3*d*(a + a*cos(c + d*x))^2*sec(c + d*x)^(7/2)) - ((A + 3*C)*sin(c + d*x))/(a*d*(a + a*cos(c + d*x))*sec(c + d*x)^(5/2)) + (4*(5*A + 14*C)*sin(c + d*x))/(15*a^2*d*sec(c + d*x)^(3/2)) - (5*(A + 3*C)*sin(c + d*x))/(3*a^2*d*sqrt(sec(c + d*x)))],


[((A + C*cos(c + d*x)^2)*sec(c + d*x)^(5/2))/(a + a*cos(c + d*x))^3, x, 9, ((119*A + 9*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(10*a^3*d) + ((11*A + C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(2*a^3*d) - ((119*A + 9*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(10*a^3*d) + ((11*A + C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(2*a^3*d) - ((A + C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3) - (2*A*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*a*d*(a + a*cos(c + d*x))^2) - ((119*A + 9*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(30*a^2*d*(a + a*cos(c + d*x)))],
[((A + C*cos(c + d*x)^2)*sec(c + d*x)^(3/2))/(a + a*cos(c + d*x))^3, x, 8, -((49*A - C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(10*a^3*d) - ((13*A - C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(6*a^3*d) + ((49*A - C)*sqrt(sec(c + d*x))*sin(c + d*x))/(10*a^3*d) - ((A + C)*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3) - (2*(4*A - C)*sqrt(sec(c + d*x))*sin(c + d*x))/(15*a*d*(a + a*cos(c + d*x))^2) - ((13*A - C)*sqrt(sec(c + d*x))*sin(c + d*x))/(6*a^2*d*(a + a*cos(c + d*x)))],
[((A + C*cos(c + d*x)^2)*sqrt(sec(c + d*x)))/(a + a*cos(c + d*x))^3, x, 7, ((9*A - C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(10*a^3*d) + ((3*A + C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(6*a^3*d) - ((A + C)*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3*sqrt(sec(c + d*x))) - (2*(3*A - 2*C)*sin(c + d*x))/(15*a*d*(a + a*cos(c + d*x))^2*sqrt(sec(c + d*x))) - ((9*A - C)*sin(c + d*x))/(10*a^2*d*(a + a*cos(c + d*x))*sqrt(sec(c + d*x)))],
[(A + C*cos(c + d*x)^2)/((a + a*cos(c + d*x))^3*sqrt(sec(c + d*x))), x, 7, ((A - 9*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(10*a^3*d) + ((A + 3*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(6*a^3*d) - ((A + C)*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3*sec(c + d*x)^(3/2)) + (2*(2*A - 3*C)*sin(c + d*x))/(15*a*d*(a + a*cos(c + d*x))^2*sqrt(sec(c + d*x))) - ((A - 9*C)*sin(c + d*x))/(10*a^2*d*(a + a*cos(c + d*x))*sqrt(sec(c + d*x)))],
[(A + C*cos(c + d*x)^2)/((a + a*cos(c + d*x))^3*sec(c + d*x)^(3/2)), x, 7, -((A - 49*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(10*a^3*d) + ((A - 13*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(6*a^3*d) - ((A + C)*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3*sec(c + d*x)^(5/2)) + (2*(A - 4*C)*sin(c + d*x))/(15*a*d*(a + a*cos(c + d*x))^2*sec(c + d*x)^(3/2)) + ((A - 13*C)*sin(c + d*x))/(6*a^2*d*(a + a*cos(c + d*x))*sqrt(sec(c + d*x)))],
[(A + C*cos(c + d*x)^2)/((a + a*cos(c + d*x))^3*sec(c + d*x)^(5/2)), x, 8, -((9*A + 119*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(10*a^3*d) + ((A + 11*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(2*a^3*d) - ((A + C)*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3*sec(c + d*x)^(7/2)) - (2*C*sin(c + d*x))/(3*a*d*(a + a*cos(c + d*x))^2*sec(c + d*x)^(5/2)) - ((9*A + 119*C)*sin(c + d*x))/(30*a^2*d*(a + a*cos(c + d*x))*sec(c + d*x)^(3/2)) + ((A + 11*C)*sin(c + d*x))/(2*a^3*d*sqrt(sec(c + d*x)))],
[(A + C*cos(c + d*x)^2)/((a + a*cos(c + d*x))^3*sec(c + d*x)^(7/2)), x, 9, (7*(7*A + 33*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(10*a^3*d) - ((13*A + 63*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(6*a^3*d) - ((A + C)*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3*sec(c + d*x)^(9/2)) - (2*(A + 6*C)*sin(c + d*x))/(15*a*d*(a + a*cos(c + d*x))^2*sec(c + d*x)^(7/2)) - ((13*A + 63*C)*sin(c + d*x))/(10*a^2*d*(a + a*cos(c + d*x))*sec(c + d*x)^(5/2)) + (7*(7*A + 33*C)*sin(c + d*x))/(30*a^3*d*sec(c + d*x)^(3/2)) - ((13*A + 63*C)*sin(c + d*x))/(6*a^3*d*sqrt(sec(c + d*x)))],


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


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


[sqrt(a + a*cos(c + d*x))*(A + C*cos(c + d*x)^2)*sec(c + d*x)^(11/2), x, 5, (16*a*(16*A + 21*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(315*d*sqrt(a + a*cos(c + d*x))) + (8*a*(16*A + 21*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(315*d*sqrt(a + a*cos(c + d*x))) + (2*a*(16*A + 21*C)*sec(c + d*x)^(5/2)*sin(c + d*x))/(105*d*sqrt(a + a*cos(c + d*x))) + (2*a*A*sec(c + d*x)^(7/2)*sin(c + d*x))/(63*d*sqrt(a + a*cos(c + d*x))) + (2*A*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(9/2)*sin(c + d*x))/(9*d)],
[sqrt(a + a*cos(c + d*x))*(A + C*cos(c + d*x)^2)*sec(c + d*x)^(9/2), x, 4, (4*a*(24*A + 35*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(105*d*sqrt(a + a*cos(c + d*x))) + (2*a*(24*A + 35*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(105*d*sqrt(a + a*cos(c + d*x))) + (2*a*A*sec(c + d*x)^(5/2)*sin(c + d*x))/(35*d*sqrt(a + a*cos(c + d*x))) + (2*A*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(7/2)*sin(c + d*x))/(7*d)],
[sqrt(a + a*cos(c + d*x))*(A + C*cos(c + d*x)^2)*sec(c + d*x)^(7/2), x, 3, (2*a*(8*A + 15*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(15*d*sqrt(a + a*cos(c + d*x))) + (2*a*A*sec(c + d*x)^(3/2)*sin(c + d*x))/(15*d*sqrt(a + a*cos(c + d*x))) + (2*A*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d)],
[sqrt(a + a*cos(c + d*x))*(A + C*cos(c + d*x)^2)*sec(c + d*x)^(5/2), x, 4, (2*sqrt(a)*C*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/d + (2*a*A*sqrt(sec(c + d*x))*sin(c + d*x))/(3*d*sqrt(a + a*cos(c + d*x))) + (2*A*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d)],
[sqrt(a + a*cos(c + d*x))*(A + C*cos(c + d*x)^2)*sec(c + d*x)^(3/2), x, 4, (sqrt(a)*C*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/d - (a*(2*A - C)*sin(c + d*x))/(d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x))) + (2*A*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x))*sin(c + d*x))/d],
[sqrt(a + a*cos(c + d*x))*(A + C*cos(c + d*x)^2)*sqrt(sec(c + d*x)), x, 4, (sqrt(a)*(8*A + 3*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(4*d) + (a*C*sin(c + d*x))/(4*d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x))) + (C*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(2*d*sqrt(sec(c + d*x)))],
[(sqrt(a + a*cos(c + d*x))*(A + C*cos(c + d*x)^2))/sqrt(sec(c + d*x)), x, 5, (sqrt(a)*(8*A + 5*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(8*d) + (a*C*sin(c + d*x))/(12*d*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(3/2)) + (C*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(3*d*sec(c + d*x)^(3/2)) + (a*(8*A + 5*C)*sin(c + d*x))/(8*d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x)))],
[(sqrt(a + a*cos(c + d*x))*(A + C*cos(c + d*x)^2))/sec(c + d*x)^(3/2), x, 6, (sqrt(a)*(48*A + 35*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(64*d) + (a*C*sin(c + d*x))/(24*d*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(5/2)) + (C*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(4*d*sec(c + d*x)^(5/2)) + (a*(48*A + 35*C)*sin(c + d*x))/(96*d*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(3/2)) + (a*(48*A + 35*C)*sin(c + d*x))/(64*d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x)))],


[(a + a*cos(c + d*x))^(3/2)*(A + C*cos(c + d*x)^2)*sec(c + d*x)^(13/2), x, 6, (16*a^2*(112*A + 143*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(1155*d*sqrt(a + a*cos(c + d*x))) + (8*a^2*(112*A + 143*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(1155*d*sqrt(a + a*cos(c + d*x))) + (2*a^2*(112*A + 143*C)*sec(c + d*x)^(5/2)*sin(c + d*x))/(385*d*sqrt(a + a*cos(c + d*x))) + (2*a^2*(28*A + 33*C)*sec(c + d*x)^(7/2)*sin(c + d*x))/(231*d*sqrt(a + a*cos(c + d*x))) + (2*a*A*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(9/2)*sin(c + d*x))/(33*d) + (2*A*(a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^(11/2)*sin(c + d*x))/(11*d)],
[(a + a*cos(c + d*x))^(3/2)*(A + C*cos(c + d*x)^2)*sec(c + d*x)^(11/2), x, 5, (4*a^2*(136*A + 189*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(315*d*sqrt(a + a*cos(c + d*x))) + (2*a^2*(136*A + 189*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(315*d*sqrt(a + a*cos(c + d*x))) + (2*a^2*(52*A + 63*C)*sec(c + d*x)^(5/2)*sin(c + d*x))/(315*d*sqrt(a + a*cos(c + d*x))) + (2*a*A*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(7/2)*sin(c + d*x))/(21*d) + (2*A*(a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^(9/2)*sin(c + d*x))/(9*d)],
[(a + a*cos(c + d*x))^(3/2)*(A + C*cos(c + d*x)^2)*sec(c + d*x)^(9/2), x, 4, (8*a^2*(19*A + 35*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(105*d*sqrt(a + a*cos(c + d*x))) + (2*a*(19*A + 35*C)*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(105*d) + (6*A*(a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^(5/2)*sin(c + d*x))/(35*d) + (2*A*(a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^(7/2)*sin(c + d*x))/(7*d)],
[(a + a*cos(c + d*x))^(3/2)*(A + C*cos(c + d*x)^2)*sec(c + d*x)^(7/2), x, 5, (2*a^(3/2)*C*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/d + (2*a^2*(4*A + 5*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d*sqrt(a + a*cos(c + d*x))) + (2*a*A*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(5*d) + (2*A*(a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d)],
[(a + a*cos(c + d*x))^(3/2)*(A + C*cos(c + d*x)^2)*sec(c + d*x)^(5/2), x, 5, (3*a^(3/2)*C*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/d - (a^2*(8*A - 3*C)*sin(c + d*x))/(3*d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x))) + (2*a*A*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x))*sin(c + d*x))/d + (2*A*(a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d)],
[(a + a*cos(c + d*x))^(3/2)*(A + C*cos(c + d*x)^2)*sec(c + d*x)^(3/2), x, 5, (a^(3/2)*(8*A + 7*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(4*d) - (a^2*(8*A - 5*C)*sin(c + d*x))/(4*d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x))) - (a*(4*A - C)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(2*d*sqrt(sec(c + d*x))) + (2*A*(a + a*cos(c + d*x))^(3/2)*sqrt(sec(c + d*x))*sin(c + d*x))/d],
[(a + a*cos(c + d*x))^(3/2)*(A + C*cos(c + d*x)^2)*sqrt(sec(c + d*x)), x, 5, (a^(3/2)*(24*A + 11*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(8*d) + (a^2*(24*A + 19*C)*sin(c + d*x))/(24*d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x))) + (a*C*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(4*d*sqrt(sec(c + d*x))) + (C*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(3*d*sqrt(sec(c + d*x)))],
[((a + a*cos(c + d*x))^(3/2)*(A + C*cos(c + d*x)^2))/sqrt(sec(c + d*x)), x, 6, (a^(3/2)*(112*A + 75*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(64*d) + (a^2*(16*A + 13*C)*sin(c + d*x))/(32*d*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(3/2)) + (a*C*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(8*d*sec(c + d*x)^(3/2)) + (C*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(4*d*sec(c + d*x)^(3/2)) + (a^2*(112*A + 75*C)*sin(c + d*x))/(64*d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x)))],
[((a + a*cos(c + d*x))^(3/2)*(A + C*cos(c + d*x)^2))/sec(c + d*x)^(3/2), x, 7, (a^(3/2)*(176*A + 133*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(128*d) + (a^2*(80*A + 67*C)*sin(c + d*x))/(240*d*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(5/2)) + (3*a*C*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(40*d*sec(c + d*x)^(5/2)) + (C*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(5*d*sec(c + d*x)^(5/2)) + (a^2*(176*A + 133*C)*sin(c + d*x))/(192*d*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(3/2)) + (a^2*(176*A + 133*C)*sin(c + d*x))/(128*d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x)))],


[(a + a*cos(c + d*x))^(5/2)*(A + C*cos(c + d*x)^2)*sec(c + d*x)^(15/2), x, 7, (16*a^3*(8368*A + 10439*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(45045*d*sqrt(a + a*cos(c + d*x))) + (8*a^3*(8368*A + 10439*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(45045*d*sqrt(a + a*cos(c + d*x))) + (2*a^3*(8368*A + 10439*C)*sec(c + d*x)^(5/2)*sin(c + d*x))/(15015*d*sqrt(a + a*cos(c + d*x))) + (2*a^3*(2224*A + 2717*C)*sec(c + d*x)^(7/2)*sin(c + d*x))/(9009*d*sqrt(a + a*cos(c + d*x))) + (2*a^2*(136*A + 143*C)*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(9/2)*sin(c + d*x))/(1287*d) + (10*a*A*(a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^(11/2)*sin(c + d*x))/(143*d) + (2*A*(a + a*cos(c + d*x))^(5/2)*sec(c + d*x)^(13/2)*sin(c + d*x))/(13*d)],
[(a + a*cos(c + d*x))^(5/2)*(A + C*cos(c + d*x)^2)*sec(c + d*x)^(13/2), x, 6, (4*a^3*(568*A + 759*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(693*d*sqrt(a + a*cos(c + d*x))) + (2*a^3*(568*A + 759*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(693*d*sqrt(a + a*cos(c + d*x))) + (2*a^3*(232*A + 297*C)*sec(c + d*x)^(5/2)*sin(c + d*x))/(693*d*sqrt(a + a*cos(c + d*x))) + (2*a^2*(32*A + 33*C)*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(7/2)*sin(c + d*x))/(231*d) + (10*a*A*(a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^(9/2)*sin(c + d*x))/(99*d) + (2*A*(a + a*cos(c + d*x))^(5/2)*sec(c + d*x)^(11/2)*sin(c + d*x))/(11*d)],
[(a + a*cos(c + d*x))^(5/2)*(A + C*cos(c + d*x)^2)*sec(c + d*x)^(11/2), x, 5, (64*a^3*(13*A + 21*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(315*d*sqrt(a + a*cos(c + d*x))) + (16*a^2*(13*A + 21*C)*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(315*d) + (2*a*(13*A + 21*C)*(a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^(5/2)*sin(c + d*x))/(105*d) + (10*A*(a + a*cos(c + d*x))^(5/2)*sec(c + d*x)^(7/2)*sin(c + d*x))/(63*d) + (2*A*(a + a*cos(c + d*x))^(5/2)*sec(c + d*x)^(9/2)*sin(c + d*x))/(9*d)],
[(a + a*cos(c + d*x))^(5/2)*(A + C*cos(c + d*x)^2)*sec(c + d*x)^(9/2), x, 6, (2*a^(5/2)*C*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/d + (2*a^3*(32*A + 49*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(21*d*sqrt(a + a*cos(c + d*x))) + (2*a^2*(8*A + 7*C)*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(21*d) + (2*a*A*(a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^(5/2)*sin(c + d*x))/(7*d) + (2*A*(a + a*cos(c + d*x))^(5/2)*sec(c + d*x)^(7/2)*sin(c + d*x))/(7*d)],
[(a + a*cos(c + d*x))^(5/2)*(A + C*cos(c + d*x)^2)*sec(c + d*x)^(7/2), x, 6, (5*a^(5/2)*C*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/d - (a^3*(64*A + 15*C)*sin(c + d*x))/(15*d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x))) + (2*a^2*(8*A + 5*C)*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d) + (2*a*A*(a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d) + (2*A*(a + a*cos(c + d*x))^(5/2)*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d)],
[(a + a*cos(c + d*x))^(5/2)*(A + C*cos(c + d*x)^2)*sec(c + d*x)^(5/2), x, 6, (a^(5/2)*(8*A + 19*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(4*d) - (a^3*(56*A - 27*C)*sin(c + d*x))/(12*d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x))) - (a^2*(8*A - C)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(2*d*sqrt(sec(c + d*x))) + (10*a*A*(a + a*cos(c + d*x))^(3/2)*sqrt(sec(c + d*x))*sin(c + d*x))/(3*d) + (2*A*(a + a*cos(c + d*x))^(5/2)*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d)],
[(a + a*cos(c + d*x))^(5/2)*(A + C*cos(c + d*x)^2)*sec(c + d*x)^(3/2), x, 6, (5*a^(5/2)*(8*A + 5*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(8*d) - (a^3*(24*A - 49*C)*sin(c + d*x))/(24*d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x))) - (a^2*(8*A - 3*C)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(4*d*sqrt(sec(c + d*x))) - (a*(6*A - C)*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(3*d*sqrt(sec(c + d*x))) + (2*A*(a + a*cos(c + d*x))^(5/2)*sqrt(sec(c + d*x))*sin(c + d*x))/d],
[(a + a*cos(c + d*x))^(5/2)*(A + C*cos(c + d*x)^2)*sqrt(sec(c + d*x)), x, 6, (a^(5/2)*(304*A + 163*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(64*d) + (a^3*(432*A + 299*C)*sin(c + d*x))/(192*d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x))) + (a^2*(16*A + 17*C)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(32*d*sqrt(sec(c + d*x))) + (5*a*C*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(24*d*sqrt(sec(c + d*x))) + (C*(a + a*cos(c + d*x))^(5/2)*sin(c + d*x))/(4*d*sqrt(sec(c + d*x)))],
[((a + a*cos(c + d*x))^(5/2)*(A + C*cos(c + d*x)^2))/sqrt(sec(c + d*x)), x, 7, (a^(5/2)*(400*A + 283*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(128*d) + (a^3*(1040*A + 787*C)*sin(c + d*x))/(960*d*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(3/2)) + (a^2*(80*A + 79*C)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(240*d*sec(c + d*x)^(3/2)) + (a*C*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(8*d*sec(c + d*x)^(3/2)) + (C*(a + a*cos(c + d*x))^(5/2)*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2)) + (a^3*(400*A + 283*C)*sin(c + d*x))/(128*d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x)))],
[((a + a*cos(c + d*x))^(5/2)*(A + C*cos(c + d*x)^2))/sec(c + d*x)^(3/2), x, 8, (a^(5/2)*(1304*A + 1015*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(512*d) + (a^3*(136*A + 109*C)*sin(c + d*x))/(192*d*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(5/2)) + (a^2*(24*A + 23*C)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(96*d*sec(c + d*x)^(5/2)) + (a*C*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(12*d*sec(c + d*x)^(5/2)) + (C*(a + a*cos(c + d*x))^(5/2)*sin(c + d*x))/(6*d*sec(c + d*x)^(5/2)) + (a^3*(1304*A + 1015*C)*sin(c + d*x))/(768*d*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(3/2)) + (a^3*(1304*A + 1015*C)*sin(c + d*x))/(512*d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x)))],


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


[((A + C*cos(c + d*x)^2)*sec(c + d*x)^(11/2))/sqrt(a + a*cos(c + d*x)), x, 7, -((sqrt(2)*(A + C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(sqrt(a)*d)) + (2*(257*A + 273*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(315*d*sqrt(a + a*cos(c + d*x))) - (2*(29*A + 21*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(315*d*sqrt(a + a*cos(c + d*x))) + (2*(19*A + 21*C)*sec(c + d*x)^(5/2)*sin(c + d*x))/(105*d*sqrt(a + a*cos(c + d*x))) - (2*A*sec(c + d*x)^(7/2)*sin(c + d*x))/(63*d*sqrt(a + a*cos(c + d*x))) + (2*A*sec(c + d*x)^(9/2)*sin(c + d*x))/(9*d*sqrt(a + a*cos(c + d*x)))],
[((A + C*cos(c + d*x)^2)*sec(c + d*x)^(9/2))/sqrt(a + a*cos(c + d*x)), x, 6, (sqrt(2)*(A + C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(sqrt(a)*d) - (2*(43*A + 35*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(105*d*sqrt(a + a*cos(c + d*x))) + (2*(31*A + 35*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(105*d*sqrt(a + a*cos(c + d*x))) - (2*A*sec(c + d*x)^(5/2)*sin(c + d*x))/(35*d*sqrt(a + a*cos(c + d*x))) + (2*A*sec(c + d*x)^(7/2)*sin(c + d*x))/(7*d*sqrt(a + a*cos(c + d*x)))],
[((A + C*cos(c + d*x)^2)*sec(c + d*x)^(7/2))/sqrt(a + a*cos(c + d*x)), x, 5, -((sqrt(2)*(A + C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(sqrt(a)*d)) + (2*(13*A + 15*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(15*d*sqrt(a + a*cos(c + d*x))) - (2*A*sec(c + d*x)^(3/2)*sin(c + d*x))/(15*d*sqrt(a + a*cos(c + d*x))) + (2*A*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d*sqrt(a + a*cos(c + d*x)))],
[((A + C*cos(c + d*x)^2)*sec(c + d*x)^(5/2))/sqrt(a + a*cos(c + d*x)), x, 4, (sqrt(2)*(A + C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(sqrt(a)*d) - (2*A*sqrt(sec(c + d*x))*sin(c + d*x))/(3*d*sqrt(a + a*cos(c + d*x))) + (2*A*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d*sqrt(a + a*cos(c + d*x)))],
[((A + C*cos(c + d*x)^2)*sec(c + d*x)^(3/2))/sqrt(a + a*cos(c + d*x)), x, 5, (2*C*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(sqrt(a)*d) - (sqrt(2)*(A + C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(sqrt(a)*d) + (2*A*sqrt(sec(c + d*x))*sin(c + d*x))/(d*sqrt(a + a*cos(c + d*x)))],
[((A + C*cos(c + d*x)^2)*sqrt(sec(c + d*x)))/sqrt(a + a*cos(c + d*x)), x, 5, -((C*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(sqrt(a)*d)) + (sqrt(2)*(A + C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(sqrt(a)*d) + (C*sin(c + d*x))/(d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x)))],
[(A + C*cos(c + d*x)^2)/(sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x))), x, 6, ((8*A + 7*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(4*sqrt(a)*d) - (sqrt(2)*(A + C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(sqrt(a)*d) + (C*sin(c + d*x))/(2*d*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(3/2)) - (C*sin(c + d*x))/(4*d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x)))],
[(A + C*cos(c + d*x)^2)/(sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(3/2)), x, 7, -((8*A + 9*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(8*sqrt(a)*d) + (sqrt(2)*(A + C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(sqrt(a)*d) + (C*sin(c + d*x))/(3*d*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(5/2)) - (C*sin(c + d*x))/(12*d*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(3/2)) + ((8*A + 7*C)*sin(c + d*x))/(8*d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x)))],


[((A + C*cos(c + d*x)^2)*sec(c + d*x)^(9/2))/(a + a*cos(c + d*x))^(3/2), x, 7, ((19*A + 11*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(2*sqrt(2)*a^(3/2)*d) - ((1201*A + 665*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(210*a*d*sqrt(a + a*cos(c + d*x))) + ((397*A + 245*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(210*a*d*sqrt(a + a*cos(c + d*x))) - ((67*A + 35*C)*sec(c + d*x)^(5/2)*sin(c + d*x))/(70*a*d*sqrt(a + a*cos(c + d*x))) - ((A + C)*sec(c + d*x)^(7/2)*sin(c + d*x))/(2*d*(a + a*cos(c + d*x))^(3/2)) + ((11*A + 7*C)*sec(c + d*x)^(7/2)*sin(c + d*x))/(14*a*d*sqrt(a + a*cos(c + d*x)))],
[((A + C*cos(c + d*x)^2)*sec(c + d*x)^(7/2))/(a + a*cos(c + d*x))^(3/2), x, 6, -((15*A + 7*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(2*sqrt(2)*a^(3/2)*d) + ((49*A + 25*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(10*a*d*sqrt(a + a*cos(c + d*x))) - ((13*A + 5*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(10*a*d*sqrt(a + a*cos(c + d*x))) - ((A + C)*sec(c + d*x)^(5/2)*sin(c + d*x))/(2*d*(a + a*cos(c + d*x))^(3/2)) + ((9*A + 5*C)*sec(c + d*x)^(5/2)*sin(c + d*x))/(10*a*d*sqrt(a + a*cos(c + d*x)))],
[((A + C*cos(c + d*x)^2)*sec(c + d*x)^(5/2))/(a + a*cos(c + d*x))^(3/2), x, 5, ((11*A + 3*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(2*sqrt(2)*a^(3/2)*d) - ((19*A + 3*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(6*a*d*sqrt(a + a*cos(c + d*x))) - ((A + C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(2*d*(a + a*cos(c + d*x))^(3/2)) + ((7*A + 3*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(6*a*d*sqrt(a + a*cos(c + d*x)))],
[((A + C*cos(c + d*x)^2)*sec(c + d*x)^(3/2))/(a + a*cos(c + d*x))^(3/2), x, 4, -((7*A - C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(2*sqrt(2)*a^(3/2)*d) - ((A + C)*sqrt(sec(c + d*x))*sin(c + d*x))/(2*d*(a + a*cos(c + d*x))^(3/2)) + ((5*A + C)*sqrt(sec(c + d*x))*sin(c + d*x))/(2*a*d*sqrt(a + a*cos(c + d*x)))],
[((A + C*cos(c + d*x)^2)*sqrt(sec(c + d*x)))/(a + a*cos(c + d*x))^(3/2), x, 5, (2*C*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(a^(3/2)*d) + ((3*A - 5*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(2*sqrt(2)*a^(3/2)*d) - ((A + C)*sin(c + d*x))/(2*d*(a + a*cos(c + d*x))^(3/2)*sqrt(sec(c + d*x)))],
[(A + C*cos(c + d*x)^2)/((a + a*cos(c + d*x))^(3/2)*sqrt(sec(c + d*x))), x, 6, (-3*C*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(a^(3/2)*d) + ((A + 9*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(2*sqrt(2)*a^(3/2)*d) - ((A + C)*sin(c + d*x))/(2*d*(a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^(3/2)) + ((A + 3*C)*sin(c + d*x))/(2*a*d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x)))],
[(A + C*cos(c + d*x)^2)/((a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^(3/2)), x, 7, ((8*A + 19*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(4*a^(3/2)*d) - ((5*A + 13*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(2*sqrt(2)*a^(3/2)*d) - ((A + C)*sin(c + d*x))/(2*d*(a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^(5/2)) + ((A + 2*C)*sin(c + d*x))/(2*a*d*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(3/2)) - ((2*A + 7*C)*sin(c + d*x))/(4*a*d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x)))],


[((A + C*cos(c + d*x)^2)*sec(c + d*x)^(7/2))/(a + a*cos(c + d*x))^(5/2), x, 7, -((283*A + 75*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(16*sqrt(2)*a^(5/2)*d) + ((2671*A + 735*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(240*a^2*d*sqrt(a + a*cos(c + d*x))) - ((787*A + 195*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(240*a^2*d*sqrt(a + a*cos(c + d*x))) - ((A + C)*sec(c + d*x)^(5/2)*sin(c + d*x))/(4*d*(a + a*cos(c + d*x))^(5/2)) - ((21*A + 5*C)*sec(c + d*x)^(5/2)*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2)) + ((157*A + 45*C)*sec(c + d*x)^(5/2)*sin(c + d*x))/(80*a^2*d*sqrt(a + a*cos(c + d*x)))],
[((A + C*cos(c + d*x)^2)*sec(c + d*x)^(5/2))/(a + a*cos(c + d*x))^(5/2), x, 6, ((163*A + 19*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(16*sqrt(2)*a^(5/2)*d) - ((299*A + 27*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(48*a^2*d*sqrt(a + a*cos(c + d*x))) - ((A + C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(4*d*(a + a*cos(c + d*x))^(5/2)) - ((17*A + C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2)) + (5*(19*A + 3*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(48*a^2*d*sqrt(a + a*cos(c + d*x)))],
[((A + C*cos(c + d*x)^2)*sec(c + d*x)^(3/2))/(a + a*cos(c + d*x))^(5/2), x, 5, (-5*(15*A - C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(16*sqrt(2)*a^(5/2)*d) - ((A + C)*sqrt(sec(c + d*x))*sin(c + d*x))/(4*d*(a + a*cos(c + d*x))^(5/2)) - ((13*A - 3*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2)) + ((49*A + C)*sqrt(sec(c + d*x))*sin(c + d*x))/(16*a^2*d*sqrt(a + a*cos(c + d*x)))],
[((A + C*cos(c + d*x)^2)*sqrt(sec(c + d*x)))/(a + a*cos(c + d*x))^(5/2), x, 4, ((19*A + 3*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(16*sqrt(2)*a^(5/2)*d) - ((A + C)*sin(c + d*x))/(4*d*(a + a*cos(c + d*x))^(5/2)*sqrt(sec(c + d*x))) - ((9*A - 7*C)*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2)*sqrt(sec(c + d*x)))],
[(A + C*cos(c + d*x)^2)/((a + a*cos(c + d*x))^(5/2)*sqrt(sec(c + d*x))), x, 6, (2*C*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(a^(5/2)*d) + ((5*A - 43*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(16*sqrt(2)*a^(5/2)*d) - ((A + C)*sin(c + d*x))/(4*d*(a + a*cos(c + d*x))^(5/2)*sec(c + d*x)^(3/2)) + ((5*A - 11*C)*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2)*sqrt(sec(c + d*x)))],
[(A + C*cos(c + d*x)^2)/((a + a*cos(c + d*x))^(5/2)*sec(c + d*x)^(3/2)), x, 7, (-5*C*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(a^(5/2)*d) + ((3*A + 115*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(16*sqrt(2)*a^(5/2)*d) - ((A + C)*sin(c + d*x))/(4*d*(a + a*cos(c + d*x))^(5/2)*sec(c + d*x)^(5/2)) + ((A - 15*C)*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^(3/2)) + ((3*A + 35*C)*sin(c + d*x))/(16*a^2*d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x)))],
[(A + C*cos(c + d*x)^2)/((a + a*cos(c + d*x))^(5/2)*sec(c + d*x)^(5/2)), x, 8, ((8*A + 39*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(4*a^(5/2)*d) - ((43*A + 219*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(16*sqrt(2)*a^(5/2)*d) - ((A + C)*sin(c + d*x))/(4*d*(a + a*cos(c + d*x))^(5/2)*sec(c + d*x)^(7/2)) - ((3*A + 19*C)*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^(5/2)) + ((7*A + 31*C)*sin(c + d*x))/(16*a^2*d*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(3/2)) - ((11*A + 63*C)*sin(c + d*x))/(16*a^2*d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x)))],


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


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


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


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


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


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


[cos(c + d*x)^1*(B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^3, x, 7, (1/16)*a^3*(26*B + 23*C)*x + (2*a^3*(19*B + 17*C)*sin(c + d*x))/(15*d) + (a^3*(26*B + 23*C)*cos(c + d*x)*sin(c + d*x))/(16*d) + (a^3*(19*B + 17*C)*cos(c + d*x)^2*sin(c + d*x))/(15*d) + (a^3*(22*B + 21*C)*cos(c + d*x)^3*sin(c + d*x))/(40*d) + (a^2*(3*B + 4*C)*cos(c + d*x)^3*(a + a*cos(c + d*x))*sin(c + d*x))/(15*d) + (a*C*cos(c + d*x)^3*(a + a*cos(c + d*x))^2*sin(c + d*x))/(6*d)],
[cos(c + d*x)^0*(B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^3, x, 5, (1/8)*a^3*(15*B + 13*C)*x + (a^3*(15*B + 13*C)*sin(c + d*x))/(6*d) + (a^3*(15*B + 13*C)*cos(c + d*x)*sin(c + d*x))/(24*d) + (a*(15*B + 13*C)*(a + a*cos(c + d*x))^2*sin(c + d*x))/(60*d) + ((5*B - C)*(a + a*cos(c + d*x))^3*sin(c + d*x))/(20*d) + (C*(a + a*cos(c + d*x))^4*sin(c + d*x))/(5*a*d)],
[sec(c + d*x)^1*(B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^3, x, 5, (5/8)*a^3*(4*B + 3*C)*x + (5*a^3*(4*B + 3*C)*sin(c + d*x))/(6*d) + (5*a^3*(4*B + 3*C)*cos(c + d*x)*sin(c + d*x))/(24*d) + (a*(4*B + 3*C)*(a + a*cos(c + d*x))^2*sin(c + d*x))/(12*d) + (C*(a + a*cos(c + d*x))^3*sin(c + d*x))/(4*d)],
[sec(c + d*x)^2*(B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^3, x, 6, (1/2)*a^3*(7*B + 5*C)*x + (a^3*B*arctanh(sin(c + d*x)))/d + (5*a^3*(B + C)*sin(c + d*x))/(2*d) + (a^2*(3*B + 5*C)*(a + a*cos(c + d*x))*sin(c + d*x))/(6*d) + (a*C*(a + a*cos(c + d*x))^2*sin(c + d*x))/(3*d)],
[sec(c + d*x)^3*(B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^3, x, 6, (1/2)*a^3*(6*B + 7*C)*x + (a^3*(3*B + C)*arctanh(sin(c + d*x)))/d + (5*a^3*C*sin(c + d*x))/(2*d) - (a^2*(2*B - C)*(a + a*cos(c + d*x))*sin(c + d*x))/(2*d) + (a*B*(a + a*cos(c + d*x))^2*tan(c + d*x))/d],
[sec(c + d*x)^4*(B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^3, x, 6, (a^3*B + 3*a^3*C)*x + (a^3*(7*B + 6*C)*arctanh(sin(c + d*x)))/(2*d) - (5*a^3*B*sin(c + d*x))/(2*d) + (a^2*(2*B + C)*(a + a*cos(c + d*x))*tan(c + d*x))/d + (a*B*(a + a*cos(c + d*x))^2*sec(c + d*x)*tan(c + d*x))/(2*d)],
[sec(c + d*x)^5*(B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^3, x, 6, a^3*C*x + (a^3*(5*B + 7*C)*arctanh(sin(c + d*x)))/(2*d) + (5*a^3*(B + C)*tan(c + d*x))/(2*d) + (a^2*(5*B + 3*C)*(a + a*sec(c + d*x))*tan(c + d*x))/(6*d) + (a*B*(a + a*sec(c + d*x))^2*tan(c + d*x))/(3*d)],
[sec(c + d*x)^6*(B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^3, x, 7, (5*a^3*(3*B + 4*C)*arctanh(sin(c + d*x)))/(8*d) + (5*a^3*(3*B + 4*C)*tan(c + d*x))/(6*d) + (5*a^3*(3*B + 4*C)*sec(c + d*x)*tan(c + d*x))/(24*d) + (a*(3*B + 4*C)*(a + a*sec(c + d*x))^2*tan(c + d*x))/(12*d) + (B*(a + a*sec(c + d*x))^3*tan(c + d*x))/(4*d)],
[sec(c + d*x)^7*(B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^3, x, 8, (a^3*(13*B + 15*C)*arctanh(sin(c + d*x)))/(8*d) + (a^3*(38*B + 45*C)*tan(c + d*x))/(15*d) + (a^3*(13*B + 15*C)*sec(c + d*x)*tan(c + d*x))/(8*d) + (a^3*(43*B + 45*C)*sec(c + d*x)^2*tan(c + d*x))/(60*d) + (a^2*(7*B + 5*C)*(a + a*cos(c + d*x))*sec(c + d*x)^3*tan(c + d*x))/(20*d) + (a*B*(a + a*cos(c + d*x))^2*sec(c + d*x)^4*tan(c + d*x))/(5*d)],


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


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


[cos(c + d*x)^3*(B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^2, x, 6, ((7*B - 10*C)*x)/(2*a^2) - (8*(2*B - 3*C)*sin(c + d*x))/(3*a^2*d) + ((7*B - 10*C)*cos(c + d*x)*sin(c + d*x))/(2*a^2*d) - (4*(2*B - 3*C)*cos(c + d*x)^2*sin(c + d*x))/(3*a^2*d) + ((B - C)*cos(c + d*x)^4*sin(c + d*x))/(3*d*(a + a*cos(c + d*x))^2) + ((7*B - 10*C)*cos(c + d*x)^3*sin(c + d*x))/(3*a*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^2*(B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^2, x, 5, -(((4*B - 7*C)*x)/(2*a^2)) + (2*(5*B - 8*C)*sin(c + d*x))/(3*a^2*d) - ((4*B - 7*C)*cos(c + d*x)*sin(c + d*x))/(2*a^2*d) + ((B - C)*cos(c + d*x)^3*sin(c + d*x))/(3*d*(a + a*cos(c + d*x))^2) + ((5*B - 8*C)*cos(c + d*x)^2*sin(c + d*x))/(3*a*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^1*(B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^2, x, 5, ((B - 2*C)*x)/a^2 - ((B - 4*C)*sin(c + d*x))/(3*a^2*d) + ((B - C)*cos(c + d*x)^2*sin(c + d*x))/(3*d*(a + a*cos(c + d*x))^2) - ((B - 2*C)*sin(c + d*x))/(a*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^0*(B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^2, x, 3, (C*x)/a^2 - ((B - C)*sin(c + d*x))/(3*d*(a + a*cos(c + d*x))^2) + ((2*B - 5*C)*sin(c + d*x))/(3*a*d*(a + a*cos(c + d*x)))],
[sec(c + d*x)^1*(B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^2, x, 3, ((B - C)*sin(c + d*x))/(3*d*(a + a*cos(c + d*x))^2) + ((B + 2*C)*sin(c + d*x))/(3*a*d*(a + a*cos(c + d*x)))],
[sec(c + d*x)^2*(B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^2, x, 4, (B*arctanh(sin(c + d*x)))/(a^2*d) - ((B - C)*sin(c + d*x))/(3*d*(a + a*cos(c + d*x))^2) - ((4*B - C)*sin(c + d*x))/(3*a*d*(a + a*cos(c + d*x)))],
[sec(c + d*x)^3*(B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^2, x, 6, -(((2*B - C)*arctanh(sin(c + d*x)))/(a^2*d)) + (2*(5*B - 2*C)*tan(c + d*x))/(3*a^2*d) - ((B - C)*tan(c + d*x))/(3*d*(a + a*cos(c + d*x))^2) - ((2*B - C)*tan(c + d*x))/(a*d*(a + a*cos(c + d*x)))],
[sec(c + d*x)^4*(B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^2, x, 7, ((7*B - 4*C)*arctanh(sin(c + d*x)))/(2*a^2*d) - (2*(8*B - 5*C)*tan(c + d*x))/(3*a^2*d) + ((7*B - 4*C)*sec(c + d*x)*tan(c + d*x))/(2*a^2*d) - ((B - C)*sec(c + d*x)*tan(c + d*x))/(3*d*(a + a*cos(c + d*x))^2) - ((8*B - 5*C)*sec(c + d*x)*tan(c + d*x))/(3*a*d*(a + a*cos(c + d*x)))],


[cos(c + d*x)^3*(B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^3, x, 6, -(((6*B - 13*C)*x)/(2*a^3)) + (8*(9*B - 19*C)*sin(c + d*x))/(15*a^3*d) - ((6*B - 13*C)*cos(c + d*x)*sin(c + d*x))/(2*a^3*d) + ((B - C)*cos(c + d*x)^4*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3) + ((6*B - 11*C)*cos(c + d*x)^3*sin(c + d*x))/(15*a*d*(a + a*cos(c + d*x))^2) + (4*(9*B - 19*C)*cos(c + d*x)^2*sin(c + d*x))/(15*a^2*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^2*(B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^3, x, 6, ((B - 3*C)*x)/a^3 - ((7*B - 27*C)*sin(c + d*x))/(15*a^3*d) + ((B - C)*cos(c + d*x)^3*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3) + ((4*B - 9*C)*cos(c + d*x)^2*sin(c + d*x))/(15*a*d*(a + a*cos(c + d*x))^2) - ((B - 3*C)*sin(c + d*x))/(a^2*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^1*(B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^3, x, 5, (C*x)/a^3 + ((B - C)*cos(c + d*x)^2*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3) + ((2*B - 7*C)*cos(c + d*x)*sin(c + d*x))/(15*a*d*(a + a*cos(c + d*x))^2) + (2*(B - 11*C)*sin(c + d*x))/(15*a^2*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^0*(B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^3, x, 3, -(((B - C)*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3)) + ((3*B - 8*C)*sin(c + d*x))/(15*a*d*(a + a*cos(c + d*x))^2) + ((3*B + 7*C)*sin(c + d*x))/(15*a^2*d*(a + a*cos(c + d*x)))],
[sec(c + d*x)^1*(B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^3, x, 4, ((B - C)*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3) + ((2*B + 3*C)*sin(c + d*x))/(15*a*d*(a + a*cos(c + d*x))^2) + ((2*B + 3*C)*sin(c + d*x))/(15*a^2*d*(a + a*cos(c + d*x)))],
[sec(c + d*x)^2*(B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^3, x, 5, (B*arctanh(sin(c + d*x)))/(a^3*d) - ((B - C)*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3) - ((7*B - 2*C)*sin(c + d*x))/(15*a*d*(a + a*cos(c + d*x))^2) - (2*(11*B - C)*sin(c + d*x))/(15*a^2*d*(a + a*cos(c + d*x)))],
[sec(c + d*x)^3*(B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^3, x, 7, -(((3*B - C)*arctanh(sin(c + d*x)))/(a^3*d)) + (2*(36*B - 11*C)*tan(c + d*x))/(15*a^3*d) - ((B - C)*tan(c + d*x))/(5*d*(a + a*cos(c + d*x))^3) - ((9*B - 4*C)*tan(c + d*x))/(15*a*d*(a + a*cos(c + d*x))^2) - ((3*B - C)*tan(c + d*x))/(a^2*d*(a + a*cos(c + d*x)))],
[sec(c + d*x)^4*(B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^3, x, 8, ((13*B - 6*C)*arctanh(sin(c + d*x)))/(2*a^3*d) - (8*(19*B - 9*C)*tan(c + d*x))/(15*a^3*d) + ((13*B - 6*C)*sec(c + d*x)*tan(c + d*x))/(2*a^3*d) - ((B - C)*sec(c + d*x)*tan(c + d*x))/(5*d*(a + a*cos(c + d*x))^3) - ((11*B - 6*C)*sec(c + d*x)*tan(c + d*x))/(15*a*d*(a + a*cos(c + d*x))^2) - (4*(19*B - 9*C)*sec(c + d*x)*tan(c + d*x))/(15*a^2*d*(a + a*cos(c + d*x)))],


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


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


[(B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^(1/2), x, 3, (2*a*(5*B + 7*C)*sin(c + d*x))/(15*d*sqrt(a + a*cos(c + d*x))) + (2*(5*B - 2*C)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(15*d) + (2*C*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(5*a*d)],


[(B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^(3/2), x, 4, (8*a^2*(21*B + 19*C)*sin(c + d*x))/(105*d*sqrt(a + a*cos(c + d*x))) + (2*a*(21*B + 19*C)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(105*d) + (2*(7*B - 2*C)*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(35*d) + (2*C*(a + a*cos(c + d*x))^(5/2)*sin(c + d*x))/(7*a*d)],


[(B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^(5/2), x, 5, (64*a^3*(15*B + 13*C)*sin(c + d*x))/(315*d*sqrt(a + a*cos(c + d*x))) + (16*a^2*(15*B + 13*C)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(315*d) + (2*a*(15*B + 13*C)*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(105*d) + (2*(9*B - 2*C)*(a + a*cos(c + d*x))^(5/2)*sin(c + d*x))/(63*d) + (2*C*(a + a*cos(c + d*x))^(7/2)*sin(c + d*x))/(9*a*d)],


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


[(B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^(1/2), x, 3, -((2*(B - C)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(d*sqrt(a + a*cos(c + d*x)))) + (2*(3*B - 2*C)*sin(c + d*x))/(3*d*sqrt(a + a*cos(c + d*x))) + (2*C*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(3*a*d)],


[(B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^(3/2), x, 3, ((3*B - 7*C)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(2*a*d*sqrt(a + a*cos(c + d*x))) - ((B - C)*sin(c + d*x))/(2*d*(a + a*cos(c + d*x))^(3/2)) + (2*C*sin(c + d*x))/(a*d*sqrt(a + a*cos(c + d*x)))],


[(B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^(5/2), x, 3, ((5*B + 19*C)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(16*a^2*d*sqrt(a + a*cos(c + d*x))) - ((B - C)*sin(c + d*x))/(4*d*(a + a*cos(c + d*x))^(5/2)) + ((5*B - 13*C)*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2))],


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


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


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


# ::Subsubsection:: 
#n<0


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


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


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


[(B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(9/2), x, 8, -((6*B*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d)) + (2*C*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) + (6*B*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d) + (2*C*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d) + (2*B*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d)],
[(B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(7/2), x, 7, -((2*C*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/d) + (2*B*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) + (2*C*sqrt(sec(c + d*x))*sin(c + d*x))/d + (2*B*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d)],
[(B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(5/2), x, 6, -((2*B*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/d) + (2*C*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/d + (2*B*sqrt(sec(c + d*x))*sin(c + d*x))/d],
[(B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(3/2), x, 5, (2*C*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/d + (2*B*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/d],
[(B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(1/2), x, 5, (2*B*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/d + (2*C*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) + (2*C*sin(c + d*x))/(3*d*sqrt(sec(c + d*x)))],
[(B*cos(c + d*x) + C*cos(c + d*x)^2)/sec(c + d*x)^(1/2), x, 6, (6*C*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d) + (2*B*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) + (2*C*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2)) + (2*B*sin(c + d*x))/(3*d*sqrt(sec(c + d*x)))],
[(B*cos(c + d*x) + C*cos(c + d*x)^2)/sec(c + d*x)^(3/2), x, 7, (6*B*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d) + (10*C*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) + (2*C*sin(c + d*x))/(7*d*sec(c + d*x)^(5/2)) + (2*B*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2)) + (10*C*sin(c + d*x))/(21*d*sqrt(sec(c + d*x)))],


# ::Subsubsection:: 
#n<0


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


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


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


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


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


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


[cos(c + d*x)^2*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^2, x, 6, (1/16)*a^2*(14*A + 12*B + 11*C)*x + (2*a^2*(10*A + 9*B + 8*C)*sin(c + d*x))/(15*d) + (a^2*(14*A + 12*B + 11*C)*cos(c + d*x)*sin(c + d*x))/(16*d) + (a^2*(10*A + 9*B + 8*C)*cos(c + d*x)^2*sin(c + d*x))/(15*d) + (a^2*(10*A + 12*B + 9*C)*cos(c + d*x)^3*sin(c + d*x))/(40*d) + (a*(3*B + C)*cos(c + d*x)^3*(a + a*cos(c + d*x))*sin(c + d*x))/(15*d) + (C*cos(c + d*x)^3*(a + a*cos(c + d*x))^2*sin(c + d*x))/(6*d)],
[cos(c + d*x)^1*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^2, x, 5, (1/8)*a^2*(8*A + 7*B + 6*C)*x + (a^2*(8*A + 7*B + 6*C)*sin(c + d*x))/(6*d) + (a^2*(8*A + 7*B + 6*C)*cos(c + d*x)*sin(c + d*x))/(24*d) + ((20*A - 5*B + 6*C)*(a + a*cos(c + d*x))^2*sin(c + d*x))/(60*d) + (C*cos(c + d*x)^2*(a + a*cos(c + d*x))^2*sin(c + d*x))/(5*d) + ((5*B + 2*C)*(a + a*cos(c + d*x))^3*sin(c + d*x))/(20*a*d)],
[cos(c + d*x)^0*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^2, x, 4, (1/8)*a^2*(12*A + 8*B + 7*C)*x + (a^2*(12*A + 8*B + 7*C)*sin(c + d*x))/(6*d) + (a^2*(12*A + 8*B + 7*C)*cos(c + d*x)*sin(c + d*x))/(24*d) + ((4*B - C)*(a + a*cos(c + d*x))^2*sin(c + d*x))/(12*d) + (C*(a + a*cos(c + d*x))^3*sin(c + d*x))/(4*a*d)],
[sec(c + d*x)^1*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^2, x, 5, (1/2)*a^2*(4*A + 3*B + 2*C)*x + (a^2*A*arctanh(sin(c + d*x)))/d + (a^2*(2*A + 3*B + 2*C)*sin(c + d*x))/(2*d) + (a*(3*B + 2*C)*(a + a*cos(c + d*x))*sin(c + d*x))/(6*d) + (C*(a + a*cos(c + d*x))^2*sin(c + d*x))/(3*d)],
[sec(c + d*x)^2*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^2, x, 5, (1/2)*a^2*(2*A + 4*B + 3*C)*x + (a^2*(2*A + B)*arctanh(sin(c + d*x)))/d - (a^2*(2*A - 2*B - 3*C)*sin(c + d*x))/(2*d) - (a*(2*A - C)*(a + a*cos(c + d*x))*sin(c + d*x))/(2*d) + (A*(a + a*cos(c + d*x))^2*tan(c + d*x))/d],
[sec(c + d*x)^3*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^2, x, 5, (a^2*B + 2*a^2*C)*x + (a^2*(3*A + 4*B + 2*C)*arctanh(sin(c + d*x)))/(2*d) - (a^2*(3*A + 2*B - 2*C)*sin(c + d*x))/(2*d) + (a*(A + B)*(a + a*cos(c + d*x))*tan(c + d*x))/d + (A*(a + a*cos(c + d*x))^2*sec(c + d*x)*tan(c + d*x))/(2*d)],
[sec(c + d*x)^4*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^2, x, 5, a^2*C*x + (a^2*(2*A + 3*B + 4*C)*arctanh(sin(c + d*x)))/(2*d) + (a^2*(2*A + 3*B + 2*C)*tan(c + d*x))/(2*d) + (a*(2*A + 3*B)*(a + a*sec(c + d*x))*tan(c + d*x))/(6*d) + (A*(a + a*sec(c + d*x))^2*tan(c + d*x))/(3*d)],
[sec(c + d*x)^5*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^2, x, 6, (a^2*(7*A + 8*B + 12*C)*arctanh(sin(c + d*x)))/(8*d) + (a^2*(7*A + 8*B + 12*C)*tan(c + d*x))/(6*d) + (a^2*(7*A + 8*B + 12*C)*sec(c + d*x)*tan(c + d*x))/(24*d) + (A*(a + a*cos(c + d*x))^2*sec(c + d*x)^3*tan(c + d*x))/(4*d) + ((A + 2*B)*(a + a*sec(c + d*x))^2*tan(c + d*x))/(6*d)],
[sec(c + d*x)^6*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^2, x, 7, (a^2*(6*A + 7*B + 8*C)*arctanh(sin(c + d*x)))/(8*d) + (a^2*(18*A + 20*B + 25*C)*tan(c + d*x))/(15*d) + (a^2*(6*A + 7*B + 8*C)*sec(c + d*x)*tan(c + d*x))/(8*d) + (a^2*(18*A + 25*B + 20*C)*sec(c + d*x)^2*tan(c + d*x))/(60*d) + (a*(2*A + 5*B)*(a + a*cos(c + d*x))*sec(c + d*x)^3*tan(c + d*x))/(20*d) + (A*(a + a*cos(c + d*x))^2*sec(c + d*x)^4*tan(c + d*x))/(5*d)],


[cos(c + d*x)^2*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^3, x, 7, (1/16)*a^3*(26*A + 23*B + 21*C)*x + (2*a^3*(133*A + 119*B + 108*C)*sin(c + d*x))/(105*d) + (a^3*(26*A + 23*B + 21*C)*cos(c + d*x)*sin(c + d*x))/(16*d) + (a^3*(133*A + 119*B + 108*C)*cos(c + d*x)^2*sin(c + d*x))/(105*d) + (a^3*(154*A + 147*B + 129*C)*cos(c + d*x)^3*sin(c + d*x))/(280*d) + (a^2*(3*A + 4*B + 3*C)*cos(c + d*x)^3*(a + a*cos(c + d*x))*sin(c + d*x))/(15*d) + (a*(7*B + 3*C)*cos(c + d*x)^3*(a + a*cos(c + d*x))^2*sin(c + d*x))/(42*d) + (C*cos(c + d*x)^3*(a + a*cos(c + d*x))^3*sin(c + d*x))/(7*d)],
[cos(c + d*x)^1*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^3, x, 6, (1/16)*a^3*(30*A + 26*B + 23*C)*x + (a^3*(30*A + 26*B + 23*C)*sin(c + d*x))/(12*d) + (a^3*(30*A + 26*B + 23*C)*cos(c + d*x)*sin(c + d*x))/(48*d) + (a*(30*A + 26*B + 23*C)*(a + a*cos(c + d*x))^2*sin(c + d*x))/(120*d) + ((30*A - 6*B + 7*C)*(a + a*cos(c + d*x))^3*sin(c + d*x))/(120*d) + (C*cos(c + d*x)^2*(a + a*cos(c + d*x))^3*sin(c + d*x))/(6*d) + ((2*B + C)*(a + a*cos(c + d*x))^4*sin(c + d*x))/(10*a*d)],
[cos(c + d*x)^0*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^3, x, 5, (1/8)*a^3*(20*A + 15*B + 13*C)*x + (a^3*(20*A + 15*B + 13*C)*sin(c + d*x))/(6*d) + (a^3*(20*A + 15*B + 13*C)*cos(c + d*x)*sin(c + d*x))/(24*d) + (a*(20*A + 15*B + 13*C)*(a + a*cos(c + d*x))^2*sin(c + d*x))/(60*d) + ((5*B - C)*(a + a*cos(c + d*x))^3*sin(c + d*x))/(20*d) + (C*(a + a*cos(c + d*x))^4*sin(c + d*x))/(5*a*d)],
[sec(c + d*x)^1*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^3, x, 6, (1/8)*a^3*(28*A + 20*B + 15*C)*x + (a^3*A*arctanh(sin(c + d*x)))/d + (5*a^3*(4*A + 4*B + 3*C)*sin(c + d*x))/(8*d) + (a^2*(12*A + 20*B + 15*C)*(a + a*cos(c + d*x))*sin(c + d*x))/(24*d) + (a*(4*B + 3*C)*(a + a*cos(c + d*x))^2*sin(c + d*x))/(12*d) + (C*(a + a*cos(c + d*x))^3*sin(c + d*x))/(4*d)],
[sec(c + d*x)^2*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^3, x, 6, (1/2)*a^3*(6*A + 7*B + 5*C)*x + (a^3*(3*A + B)*arctanh(sin(c + d*x)))/d + (5*a^3*(B + C)*sin(c + d*x))/(2*d) - (a^2*(6*A - 3*B - 5*C)*(a + a*cos(c + d*x))*sin(c + d*x))/(6*d) - (a*(3*A - C)*(a + a*cos(c + d*x))^2*sin(c + d*x))/(3*d) + (A*(a + a*cos(c + d*x))^3*tan(c + d*x))/d],
[sec(c + d*x)^3*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^3, x, 6, (1/2)*a^3*(2*A + 6*B + 7*C)*x + (a^3*(7*A + 6*B + 2*C)*arctanh(sin(c + d*x)))/(2*d) - (5*a^3*(A - C)*sin(c + d*x))/(2*d) - (a^2*(4*A + 2*B - C)*(a + a*cos(c + d*x))*sin(c + d*x))/(2*d) + (a*(3*A + 2*B)*(a + a*cos(c + d*x))^2*tan(c + d*x))/(2*d) + (A*(a + a*cos(c + d*x))^3*sec(c + d*x)*tan(c + d*x))/(2*d)],
[sec(c + d*x)^4*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^3, x, 6, (a^3*B + 3*a^3*C)*x + (a^3*(5*A + 7*B + 6*C)*arctanh(sin(c + d*x)))/(2*d) - (5*a^3*(A + B)*sin(c + d*x))/(2*d) + (a^2*(5*A + 6*B + 3*C)*(a + a*cos(c + d*x))*tan(c + d*x))/(3*d) + (a*(A + B)*(a + a*cos(c + d*x))^2*sec(c + d*x)*tan(c + d*x))/(2*d) + (A*(a + a*cos(c + d*x))^3*sec(c + d*x)^2*tan(c + d*x))/(3*d)],
[sec(c + d*x)^5*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^3, x, 6, a^3*C*x + (a^3*(15*A + 20*B + 28*C)*arctanh(sin(c + d*x)))/(8*d) + (5*a^3*(3*A + 4*B + 4*C)*tan(c + d*x))/(8*d) + (a^2*(15*A + 20*B + 12*C)*(a + a*sec(c + d*x))*tan(c + d*x))/(24*d) + (a*(3*A + 4*B)*(a + a*sec(c + d*x))^2*tan(c + d*x))/(12*d) + (A*(a + a*sec(c + d*x))^3*tan(c + d*x))/(4*d)],
[sec(c + d*x)^6*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^3, x, 7, (a^3*(13*A + 15*B + 20*C)*arctanh(sin(c + d*x)))/(8*d) + (a^3*(13*A + 15*B + 20*C)*tan(c + d*x))/(6*d) + (a^3*(13*A + 15*B + 20*C)*sec(c + d*x)*tan(c + d*x))/(24*d) + (A*(a + a*cos(c + d*x))^3*sec(c + d*x)^4*tan(c + d*x))/(5*d) + (a*(13*A + 15*B + 20*C)*(a + a*sec(c + d*x))^2*tan(c + d*x))/(60*d) + ((3*A + 5*B)*(a + a*sec(c + d*x))^3*tan(c + d*x))/(20*d)],
[sec(c + d*x)^7*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^3, x, 8, (a^3*(23*A + 26*B + 30*C)*arctanh(sin(c + d*x)))/(16*d) + (a^3*(34*A + 38*B + 45*C)*tan(c + d*x))/(15*d) + (a^3*(23*A + 26*B + 30*C)*sec(c + d*x)*tan(c + d*x))/(16*d) + (a^3*(73*A + 86*B + 90*C)*sec(c + d*x)^2*tan(c + d*x))/(120*d) + (a^2*(31*A + 42*B + 30*C)*(a + a*cos(c + d*x))*sec(c + d*x)^3*tan(c + d*x))/(120*d) + (a*(A + 2*B)*(a + a*cos(c + d*x))^2*sec(c + d*x)^4*tan(c + d*x))/(10*d) + (A*(a + a*cos(c + d*x))^3*sec(c + d*x)^5*tan(c + d*x))/(6*d)],


[cos(c + d*x)^2*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^4, x, 8, (1/128)*a^4*(392*A + 352*B + 323*C)*x + (2*a^4*(252*A + 227*B + 208*C)*sin(c + d*x))/(105*d) + (a^4*(392*A + 352*B + 323*C)*cos(c + d*x)*sin(c + d*x))/(128*d) + (a^4*(252*A + 227*B + 208*C)*cos(c + d*x)^2*sin(c + d*x))/(105*d) + (a^4*(2408*A + 2208*B + 2007*C)*cos(c + d*x)^3*sin(c + d*x))/(2240*d) + (7*a^3*(8*A + 8*B + 7*C)*cos(c + d*x)^3*(a + a*cos(c + d*x))*sin(c + d*x))/(120*d) + (a^2*(56*A + 80*B + 61*C)*cos(c + d*x)^3*(a + a*cos(c + d*x))^2*sin(c + d*x))/(336*d) + (a*(2*B + C)*cos(c + d*x)^3*(a + a*cos(c + d*x))^3*sin(c + d*x))/(14*d) + (C*cos(c + d*x)^3*(a + a*cos(c + d*x))^4*sin(c + d*x))/(8*d)],
[cos(c + d*x)^1*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^4, x, 7, (1/16)*a^4*(56*A + 49*B + 44*C)*x + (a^4*(56*A + 49*B + 44*C)*sin(c + d*x))/(12*d) + (a^4*(56*A + 49*B + 44*C)*cos(c + d*x)*sin(c + d*x))/(48*d) + (a^2*(56*A + 49*B + 44*C)*(a + a*cos(c + d*x))^2*sin(c + d*x))/(120*d) + (a*(56*A + 49*B + 44*C)*(a + a*cos(c + d*x))^3*sin(c + d*x))/(280*d) + ((42*A - 7*B + 8*C)*(a + a*cos(c + d*x))^4*sin(c + d*x))/(210*d) + (C*cos(c + d*x)^2*(a + a*cos(c + d*x))^4*sin(c + d*x))/(7*d) + ((7*B + 4*C)*(a + a*cos(c + d*x))^5*sin(c + d*x))/(42*a*d)],
[cos(c + d*x)^0*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^4, x, 6, (7/16)*a^4*(10*A + 8*B + 7*C)*x + (7*a^4*(10*A + 8*B + 7*C)*sin(c + d*x))/(12*d) + (7*a^4*(10*A + 8*B + 7*C)*cos(c + d*x)*sin(c + d*x))/(48*d) + (7*a^2*(10*A + 8*B + 7*C)*(a + a*cos(c + d*x))^2*sin(c + d*x))/(120*d) + (a*(10*A + 8*B + 7*C)*(a + a*cos(c + d*x))^3*sin(c + d*x))/(40*d) + ((6*B - C)*(a + a*cos(c + d*x))^4*sin(c + d*x))/(30*d) + (C*(a + a*cos(c + d*x))^5*sin(c + d*x))/(6*a*d)],
[sec(c + d*x)^1*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^4, x, 7, (1/8)*a^4*(48*A + 35*B + 28*C)*x + (a^4*A*arctanh(sin(c + d*x)))/d + (a^4*(40*A + 35*B + 28*C)*sin(c + d*x))/(8*d) + (a^3*(32*A + 35*B + 28*C)*(a + a*cos(c + d*x))*sin(c + d*x))/(24*d) + (a^2*(20*A + 35*B + 28*C)*(a + a*cos(c + d*x))^2*sin(c + d*x))/(60*d) + (a*(5*B + 4*C)*(a + a*cos(c + d*x))^3*sin(c + d*x))/(20*d) + (C*(a + a*cos(c + d*x))^4*sin(c + d*x))/(5*d)],
[sec(c + d*x)^2*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^4, x, 7, (1/8)*a^4*(52*A + 48*B + 35*C)*x + (a^4*(4*A + B)*arctanh(sin(c + d*x)))/d + (5*a^4*(4*A + 8*B + 7*C)*sin(c + d*x))/(8*d) - (a^3*(12*A - 32*B - 35*C)*(a + a*cos(c + d*x))*sin(c + d*x))/(24*d) - (a^2*(12*A - 4*B - 7*C)*(a + a*cos(c + d*x))^2*sin(c + d*x))/(12*d) - (a*(4*A - C)*(a + a*cos(c + d*x))^3*sin(c + d*x))/(4*d) + (A*(a + a*cos(c + d*x))^4*tan(c + d*x))/d],
[sec(c + d*x)^3*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^4, x, 7, (1/2)*a^4*(8*A + 13*B + 12*C)*x + (a^4*(13*A + 8*B + 2*C)*arctanh(sin(c + d*x)))/(2*d) - (5*a^4*(A - B - 2*C)*sin(c + d*x))/(2*d) - (a^3*(18*A + 3*B - 8*C)*(a + a*cos(c + d*x))*sin(c + d*x))/(6*d) - (a^2*(15*A + 6*B - 2*C)*(a + a*cos(c + d*x))^2*sin(c + d*x))/(6*d) + (a*(2*A + B)*(a + a*cos(c + d*x))^3*tan(c + d*x))/d + (A*(a + a*cos(c + d*x))^4*sec(c + d*x)*tan(c + d*x))/(2*d)],
[sec(c + d*x)^4*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^4, x, 7, (1/2)*a^4*(2*A + 8*B + 13*C)*x + (a^4*(12*A + 13*B + 8*C)*arctanh(sin(c + d*x)))/(2*d) - (5*a^4*(2*A + B - C)*sin(c + d*x))/(2*d) - (a^3*(22*A + 18*B + 3*C)*(a + a*cos(c + d*x))*sin(c + d*x))/(6*d) + (a^2*(16*A + 15*B + 6*C)*(a + a*cos(c + d*x))^2*tan(c + d*x))/(6*d) + (a*(4*A + 3*B)*(a + a*cos(c + d*x))^3*sec(c + d*x)*tan(c + d*x))/(6*d) + (A*(a + a*cos(c + d*x))^4*sec(c + d*x)^2*tan(c + d*x))/(3*d)],
[sec(c + d*x)^5*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^4, x, 7, (a^4*B + 4*a^4*C)*x + (a^4*(35*A + 48*B + 52*C)*arctanh(sin(c + d*x)))/(8*d) - (5*a^4*(7*A + 8*B + 4*C)*sin(c + d*x))/(8*d) + (a^3*(35*A + 44*B + 36*C)*(a + a*cos(c + d*x))*tan(c + d*x))/(12*d) + (a^2*(7*A + 8*B + 4*C)*(a + a*cos(c + d*x))^2*sec(c + d*x)*tan(c + d*x))/(8*d) + (a*(A + B)*(a + a*cos(c + d*x))^3*sec(c + d*x)^2*tan(c + d*x))/(3*d) + (A*(a + a*cos(c + d*x))^4*sec(c + d*x)^3*tan(c + d*x))/(4*d)],
[sec(c + d*x)^6*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^4, x, 7, a^4*C*x + (a^4*(28*A + 35*B + 48*C)*arctanh(sin(c + d*x)))/(8*d) + (a^4*(28*A + 35*B + 40*C)*tan(c + d*x))/(8*d) + (a^3*(28*A + 35*B + 32*C)*(a + a*sec(c + d*x))*tan(c + d*x))/(24*d) + (a^2*(28*A + 35*B + 20*C)*(a + a*sec(c + d*x))^2*tan(c + d*x))/(60*d) + (a*(4*A + 5*B)*(a + a*sec(c + d*x))^3*tan(c + d*x))/(20*d) + (A*(a + a*sec(c + d*x))^4*tan(c + d*x))/(5*d)],
[sec(c + d*x)^7*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^4, x, 8, (7*a^4*(7*A + 8*B + 10*C)*arctanh(sin(c + d*x)))/(16*d) + (7*a^4*(7*A + 8*B + 10*C)*tan(c + d*x))/(12*d) + (7*a^4*(7*A + 8*B + 10*C)*sec(c + d*x)*tan(c + d*x))/(48*d) + (A*(a + a*cos(c + d*x))^4*sec(c + d*x)^5*tan(c + d*x))/(6*d) + (7*a^2*(7*A + 8*B + 10*C)*(a + a*sec(c + d*x))^2*tan(c + d*x))/(120*d) + (a*(7*A + 8*B + 10*C)*(a + a*sec(c + d*x))^3*tan(c + d*x))/(40*d) + ((2*A + 3*B)*(a + a*sec(c + d*x))^4*tan(c + d*x))/(15*d)],
[sec(c + d*x)^8*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^4, x, 9, (a^4*(44*A + 49*B + 56*C)*arctanh(sin(c + d*x)))/(16*d) + (a^4*(454*A + 504*B + 581*C)*tan(c + d*x))/(105*d) + (a^4*(44*A + 49*B + 56*C)*sec(c + d*x)*tan(c + d*x))/(16*d) + (a^4*(988*A + 1113*B + 1232*C)*sec(c + d*x)^2*tan(c + d*x))/(840*d) + (a^3*(436*A + 511*B + 504*C)*(a + a*cos(c + d*x))*sec(c + d*x)^3*tan(c + d*x))/(840*d) + (a^2*(16*A + 21*B + 14*C)*(a + a*cos(c + d*x))^2*sec(c + d*x)^4*tan(c + d*x))/(70*d) + (a*(4*A + 7*B)*(a + a*cos(c + d*x))^3*sec(c + d*x)^5*tan(c + d*x))/(42*d) + (A*(a + a*cos(c + d*x))^4*sec(c + d*x)^6*tan(c + d*x))/(7*d)],


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


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


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


[cos(c + d*x)^4*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^3, x, 6, -(((6*A - 13*B + 23*C)*x)/(2*a^3)) + (8*(9*A - 19*B + 34*C)*sin(c + d*x))/(15*a^3*d) - ((6*A - 13*B + 23*C)*cos(c + d*x)*sin(c + d*x))/(2*a^3*d) + (4*(9*A - 19*B + 34*C)*cos(c + d*x)^2*sin(c + d*x))/(15*a^3*d) - ((A - B + C)*cos(c + d*x)^5*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3) - ((3*A - 8*B + 13*C)*cos(c + d*x)^4*sin(c + d*x))/(15*a*d*(a + a*cos(c + d*x))^2) - ((6*A - 13*B + 23*C)*cos(c + d*x)^3*sin(c + d*x))/(3*a^2*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^3*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^3, x, 5, ((2*A - 6*B + 13*C)*x)/(2*a^3) - (2*(11*A - 36*B + 76*C)*sin(c + d*x))/(15*a^3*d) + ((2*A - 6*B + 13*C)*cos(c + d*x)*sin(c + d*x))/(2*a^3*d) - ((A - B + C)*cos(c + d*x)^4*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3) - ((A - 6*B + 11*C)*cos(c + d*x)^3*sin(c + d*x))/(15*a*d*(a + a*cos(c + d*x))^2) - ((11*A - 36*B + 76*C)*cos(c + d*x)^2*sin(c + d*x))/(15*a^2*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^2*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^3, x, 5, ((B - 3*C)*x)/a^3 + ((2*A - 7*B + 27*C)*sin(c + d*x))/(15*a^3*d) - ((A - B + C)*cos(c + d*x)^3*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3) + ((A + 4*B - 9*C)*cos(c + d*x)^2*sin(c + d*x))/(15*a*d*(a + a*cos(c + d*x))^2) - ((B - 3*C)*sin(c + d*x))/(a^2*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^1*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^3, x, 4, (C*x)/a^3 - ((A - B + C)*cos(c + d*x)^2*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3) + ((3*A + 2*B - 7*C)*cos(c + d*x)*sin(c + d*x))/(15*a*d*(a + a*cos(c + d*x))^2) + ((3*A + 2*B - 22*C)*sin(c + d*x))/(15*a^2*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^0*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^3, x, 3, ((A - B + C)*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3) + ((2*A + 3*B - 8*C)*sin(c + d*x))/(15*a*d*(a + a*cos(c + d*x))^2) + ((2*A + 3*B + 7*C)*sin(c + d*x))/(15*a^2*d*(a + a*cos(c + d*x)))],
[sec(c + d*x)^1*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^3, x, 4, (A*arctanh(sin(c + d*x)))/(a^3*d) - ((A - B + C)*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3) - ((7*A - 2*B - 3*C)*sin(c + d*x))/(15*a*d*(a + a*cos(c + d*x))^2) - ((22*A - 2*B - 3*C)*sin(c + d*x))/(15*a^2*d*(a + a*cos(c + d*x)))],
[sec(c + d*x)^2*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^3, x, 6, -(((3*A - B)*arctanh(sin(c + d*x)))/(a^3*d)) + (2*(36*A - 11*B + C)*tan(c + d*x))/(15*a^3*d) - ((A - B + C)*tan(c + d*x))/(5*d*(a + a*cos(c + d*x))^3) - ((9*A - 4*B - C)*tan(c + d*x))/(15*a*d*(a + a*cos(c + d*x))^2) - ((3*A - B)*tan(c + d*x))/(a^2*d*(a + a*cos(c + d*x)))],
[sec(c + d*x)^3*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^3, x, 7, ((13*A - 6*B + 2*C)*arctanh(sin(c + d*x)))/(2*a^3*d) - (2*(76*A - 36*B + 11*C)*tan(c + d*x))/(15*a^3*d) + ((13*A - 6*B + 2*C)*sec(c + d*x)*tan(c + d*x))/(2*a^3*d) - ((A - B + C)*sec(c + d*x)*tan(c + d*x))/(5*d*(a + a*cos(c + d*x))^3) - ((11*A - 6*B + C)*sec(c + d*x)*tan(c + d*x))/(15*a*d*(a + a*cos(c + d*x))^2) - ((76*A - 36*B + 11*C)*sec(c + d*x)*tan(c + d*x))/(15*a^2*d*(a + a*cos(c + d*x)))],
[sec(c + d*x)^4*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^3, x, 8, -(((23*A - 13*B + 6*C)*arctanh(sin(c + d*x)))/(2*a^3*d)) + (8*(34*A - 19*B + 9*C)*tan(c + d*x))/(15*a^3*d) - ((23*A - 13*B + 6*C)*sec(c + d*x)*tan(c + d*x))/(2*a^3*d) + (4*(34*A - 19*B + 9*C)*sec(c + d*x)^2*tan(c + d*x))/(15*a^3*d) - ((A - B + C)*sec(c + d*x)^2*tan(c + d*x))/(5*d*(a + a*cos(c + d*x))^3) - ((13*A - 8*B + 3*C)*sec(c + d*x)^2*tan(c + d*x))/(15*a*d*(a + a*cos(c + d*x))^2) - ((23*A - 13*B + 6*C)*sec(c + d*x)^2*tan(c + d*x))/(3*a^2*d*(a + a*cos(c + d*x)))],


[cos(c + d*x)^4*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^4, x, 6, ((2*A - 8*B + 21*C)*x)/(2*a^4) - (8*(20*A - 83*B + 216*C)*sin(c + d*x))/(105*a^4*d) + ((2*A - 8*B + 21*C)*cos(c + d*x)*sin(c + d*x))/(2*a^4*d) - ((A - B + C)*cos(c + d*x)^5*sin(c + d*x))/(7*d*(a + a*cos(c + d*x))^4) + ((B - 2*C)*cos(c + d*x)^4*sin(c + d*x))/(5*a*d*(a + a*cos(c + d*x))^3) - ((10*A - 52*B + 129*C)*cos(c + d*x)^3*sin(c + d*x))/(105*a^2*d*(a + a*cos(c + d*x))^2) - (4*(20*A - 83*B + 216*C)*cos(c + d*x)^2*sin(c + d*x))/(105*a^3*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^3*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^4, x, 6, ((B - 4*C)*x)/a^4 + ((6*A - 55*B + 244*C)*sin(c + d*x))/(105*a^4*d) - ((A - B + C)*cos(c + d*x)^4*sin(c + d*x))/(7*d*(a + a*cos(c + d*x))^4) + ((2*A + 5*B - 12*C)*cos(c + d*x)^3*sin(c + d*x))/(35*a*d*(a + a*cos(c + d*x))^3) + ((3*A + 25*B - 88*C)*cos(c + d*x)^2*sin(c + d*x))/(105*a^2*d*(a + a*cos(c + d*x))^2) - ((B - 4*C)*sin(c + d*x))/(a^3*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^2*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^4, x, 5, (C*x)/a^4 - ((A - B + C)*cos(c + d*x)^3*sin(c + d*x))/(7*d*(a + a*cos(c + d*x))^4) + ((4*A + 3*B - 10*C)*cos(c + d*x)^2*sin(c + d*x))/(35*a*d*(a + a*cos(c + d*x))^3) + ((8*A + 6*B - 55*C)*cos(c + d*x)*sin(c + d*x))/(105*a^2*d*(a + a*cos(c + d*x))^2) + (2*(4*A + 3*B - 80*C)*sin(c + d*x))/(105*a^3*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^1*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^4, x, 4, -(((A - B + C)*cos(c + d*x)^2*sin(c + d*x))/(7*d*(a + a*cos(c + d*x))^4)) + ((6*A + B - 8*C)*cos(c + d*x)*sin(c + d*x))/(35*a*d*(a + a*cos(c + d*x))^3) + ((A - B - 6*C)*sin(c + d*x))/(21*a^2*d*(a + a*cos(c + d*x))^2) + ((8*A + 13*B + 36*C)*sin(c + d*x))/(105*a^3*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^0*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^4, x, 4, ((A - B + C)*sin(c + d*x))/(7*d*(a + a*cos(c + d*x))^4) + ((3*A + 4*B - 11*C)*sin(c + d*x))/(35*a*d*(a + a*cos(c + d*x))^3) + ((6*A + 8*B + 13*C)*sin(c + d*x))/(105*a^2*d*(a + a*cos(c + d*x))^2) + ((6*A + 8*B + 13*C)*sin(c + d*x))/(105*a^3*d*(a + a*cos(c + d*x)))],
[sec(c + d*x)^1*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^4, x, 5, (A*arctanh(sin(c + d*x)))/(a^4*d) - ((A - B + C)*sin(c + d*x))/(7*d*(a + a*cos(c + d*x))^4) - ((10*A - 3*B - 4*C)*sin(c + d*x))/(35*a*d*(a + a*cos(c + d*x))^3) - ((55*A - 6*B - 8*C)*sin(c + d*x))/(105*a^2*d*(a + a*cos(c + d*x))^2) - (2*(80*A - 3*B - 4*C)*sin(c + d*x))/(105*a^3*d*(a + a*cos(c + d*x)))],
[sec(c + d*x)^2*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^4, x, 7, -(((4*A - B)*arctanh(sin(c + d*x)))/(a^4*d)) + (2*(332*A - 80*B + 3*C)*tan(c + d*x))/(105*a^4*d) - ((A - B + C)*tan(c + d*x))/(7*d*(a + a*cos(c + d*x))^4) - ((12*A - 5*B - 2*C)*tan(c + d*x))/(35*a*d*(a + a*cos(c + d*x))^3) - ((88*A - 25*B - 3*C)*tan(c + d*x))/(105*a^2*d*(a + a*cos(c + d*x))^2) - ((4*A - B)*tan(c + d*x))/(a^3*d*(a + a*cos(c + d*x)))],
[sec(c + d*x)^3*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^4, x, 8, ((21*A - 8*B + 2*C)*arctanh(sin(c + d*x)))/(2*a^4*d) - (8*(216*A - 83*B + 20*C)*tan(c + d*x))/(105*a^4*d) + ((21*A - 8*B + 2*C)*sec(c + d*x)*tan(c + d*x))/(2*a^4*d) - ((A - B + C)*sec(c + d*x)*tan(c + d*x))/(7*d*(a + a*cos(c + d*x))^4) - ((2*A - B)*sec(c + d*x)*tan(c + d*x))/(5*a*d*(a + a*cos(c + d*x))^3) - ((129*A - 52*B + 10*C)*sec(c + d*x)*tan(c + d*x))/(105*a^2*d*(a + a*cos(c + d*x))^2) - (4*(216*A - 83*B + 20*C)*sec(c + d*x)*tan(c + d*x))/(105*a^3*d*(a + a*cos(c + d*x)))],
[sec(c + d*x)^4*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^4, x, 9, -(((44*A - 21*B + 8*C)*arctanh(sin(c + d*x)))/(2*a^4*d)) + (8*(454*A - 216*B + 83*C)*tan(c + d*x))/(105*a^4*d) - ((44*A - 21*B + 8*C)*sec(c + d*x)*tan(c + d*x))/(2*a^4*d) + (4*(454*A - 216*B + 83*C)*sec(c + d*x)^2*tan(c + d*x))/(105*a^4*d) - ((A - B + C)*sec(c + d*x)^2*tan(c + d*x))/(7*d*(a + a*cos(c + d*x))^4) - ((16*A - 9*B + 2*C)*sec(c + d*x)^2*tan(c + d*x))/(35*a*d*(a + a*cos(c + d*x))^3) - ((178*A - 87*B + 31*C)*sec(c + d*x)^2*tan(c + d*x))/(105*a^2*d*(a + a*cos(c + d*x))^2) - ((44*A - 21*B + 8*C)*sec(c + d*x)^2*tan(c + d*x))/(3*a^3*d*(a + a*cos(c + d*x)))],


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


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


[cos(c + d*x)^3*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^(1/2), x, 6, (4*a*(99*A + 88*B + 80*C)*sin(c + d*x))/(495*d*sqrt(a + a*cos(c + d*x))) + (2*a*(99*A + 88*B + 80*C)*cos(c + d*x)^3*sin(c + d*x))/(693*d*sqrt(a + a*cos(c + d*x))) + (2*a*(11*B + C)*cos(c + d*x)^4*sin(c + d*x))/(99*d*sqrt(a + a*cos(c + d*x))) - (8*(99*A + 88*B + 80*C)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(3465*d) + (2*C*cos(c + d*x)^4*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(11*d) + (4*(99*A + 88*B + 80*C)*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(1155*a*d)],
[cos(c + d*x)^2*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^(1/2), x, 5, (2*a*(21*A + 18*B + 16*C)*sin(c + d*x))/(45*d*sqrt(a + a*cos(c + d*x))) + (2*a*(9*B + C)*cos(c + d*x)^3*sin(c + d*x))/(63*d*sqrt(a + a*cos(c + d*x))) - (4*(21*A + 18*B + 16*C)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(315*d) + (2*C*cos(c + d*x)^3*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(9*d) + (2*(21*A + 18*B + 16*C)*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(105*a*d)],
[cos(c + d*x)^1*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^(1/2), x, 4, (2*a*(35*A + 49*B + 27*C)*sin(c + d*x))/(105*d*sqrt(a + a*cos(c + d*x))) + (2*(35*A - 14*B + 18*C)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(105*d) + (2*C*cos(c + d*x)^2*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(7*d) + (2*(7*B + C)*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(35*a*d)],
[cos(c + d*x)^0*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^(1/2), x, 3, (2*a*(15*A + 5*B + 7*C)*sin(c + d*x))/(15*d*sqrt(a + a*cos(c + d*x))) + (2*(5*B - 2*C)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(15*d) + (2*C*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(5*a*d)],
[sec(c + d*x)^1*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^(1/2), x, 3, (2*sqrt(2)*a*A*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(a + a*cos(c + d*x))) + (2*a*(3*B + C)*sin(c + d*x))/(3*d*sqrt(a + a*cos(c + d*x))) + (2*C*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(3*d)],
[sec(c + d*x)^2*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^(1/2), x, 3, (sqrt(2)*a*(A + 2*B)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(a + a*cos(c + d*x))) - (a*(A - 2*C)*sin(c + d*x))/(d*sqrt(a + a*cos(c + d*x))) + (A*sqrt(a + a*cos(c + d*x))*tan(c + d*x))/d],
[sec(c + d*x)^3*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^(1/2), x, 3, (a*(3*A + 4*B + 8*C)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(2*sqrt(2)*d*sqrt(a + a*cos(c + d*x))) + (a*(A + 4*B)*tan(c + d*x))/(4*d*sqrt(a + a*cos(c + d*x))) + (A*sqrt(a + a*cos(c + d*x))*sec(c + d*x)*tan(c + d*x))/(2*d)],
[sec(c + d*x)^4*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^(1/2), x, 4, (a*(5*A + 6*B + 8*C)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(4*sqrt(2)*d*sqrt(a + a*cos(c + d*x))) + (a*(5*A + 6*B + 8*C)*tan(c + d*x))/(8*d*sqrt(a + a*cos(c + d*x))) + (a*(A + 6*B)*sec(c + d*x)*tan(c + d*x))/(12*d*sqrt(a + a*cos(c + d*x))) + (A*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^2*tan(c + d*x))/(3*d)],
[sec(c + d*x)^5*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^(1/2), x, 5, (a*(35*A + 40*B + 48*C)*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(32*sqrt(2)*d*sqrt(a + a*cos(c + d*x))) + (a*(35*A + 40*B + 48*C)*tan(c + d*x))/(64*d*sqrt(a + a*cos(c + d*x))) + (a*(35*A + 40*B + 48*C)*sec(c + d*x)*tan(c + d*x))/(96*d*sqrt(a + a*cos(c + d*x))) + (a*(A + 8*B)*sec(c + d*x)^2*tan(c + d*x))/(24*d*sqrt(a + a*cos(c + d*x))) + (A*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^3*tan(c + d*x))/(4*d)],


[cos(c + d*x)^2*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^(3/2), x, 6, (2*a^2*(429*A + 374*B + 336*C)*sin(c + d*x))/(495*d*sqrt(a + a*cos(c + d*x))) + (2*a^2*(99*A + 110*B + 84*C)*cos(c + d*x)^3*sin(c + d*x))/(693*d*sqrt(a + a*cos(c + d*x))) - (4*a*(429*A + 374*B + 336*C)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(3465*d) + (2*a*(11*B + 3*C)*cos(c + d*x)^3*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(99*d) + (2*(429*A + 374*B + 336*C)*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(1155*d) + (2*C*cos(c + d*x)^3*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(11*d)],
[cos(c + d*x)^1*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^(3/2), x, 5, (8*a^2*(63*A + 57*B + 47*C)*sin(c + d*x))/(315*d*sqrt(a + a*cos(c + d*x))) + (2*a*(63*A + 57*B + 47*C)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(315*d) + (2*(63*A - 18*B + 22*C)*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(315*d) + (2*C*cos(c + d*x)^2*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(9*d) + (2*(3*B + C)*(a + a*cos(c + d*x))^(5/2)*sin(c + d*x))/(21*a*d)],
[cos(c + d*x)^0*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^(3/2), x, 4, (8*a^2*(35*A + 21*B + 19*C)*sin(c + d*x))/(105*d*sqrt(a + a*cos(c + d*x))) + (2*a*(35*A + 21*B + 19*C)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(105*d) + (2*(7*B - 2*C)*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(35*d) + (2*C*(a + a*cos(c + d*x))^(5/2)*sin(c + d*x))/(7*a*d)],
[sec(c + d*x)^1*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^(3/2), x, 4, (2*sqrt(2)*a^2*A*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(a + a*cos(c + d*x))) + (2*a^2*(15*A + 20*B + 12*C)*sin(c + d*x))/(15*d*sqrt(a + a*cos(c + d*x))) + (2*a*(5*B + 3*C)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(15*d) + (2*C*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(5*d)],
[sec(c + d*x)^2*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^(3/2), x, 4, (sqrt(2)*a^2*(3*A + 2*B)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(a + a*cos(c + d*x))) - (a^2*(3*A - 6*B - 8*C)*sin(c + d*x))/(3*d*sqrt(a + a*cos(c + d*x))) - (a*(3*A - 2*C)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(3*d) + (A*(a + a*cos(c + d*x))^(3/2)*tan(c + d*x))/d],
[sec(c + d*x)^3*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^(3/2), x, 4, (a^2*(7*A + 12*B + 8*C)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(2*sqrt(2)*d*sqrt(a + a*cos(c + d*x))) - (a^2*(5*A + 4*B - 8*C)*sin(c + d*x))/(4*d*sqrt(a + a*cos(c + d*x))) + (a*(3*A + 4*B)*sqrt(a + a*cos(c + d*x))*tan(c + d*x))/(4*d) + (A*(a + a*cos(c + d*x))^(3/2)*sec(c + d*x)*tan(c + d*x))/(2*d)],
[sec(c + d*x)^4*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^(3/2), x, 4, (a^2*(11*A + 14*B + 24*C)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(4*sqrt(2)*d*sqrt(a + a*cos(c + d*x))) + (a^2*(19*A + 30*B + 24*C)*tan(c + d*x))/(24*d*sqrt(a + a*cos(c + d*x))) + (a*(A + 2*B)*sqrt(a + a*cos(c + d*x))*sec(c + d*x)*tan(c + d*x))/(4*d) + (A*(a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^2*tan(c + d*x))/(3*d)],
[sec(c + d*x)^5*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^(3/2), x, 5, (a^2*(75*A + 88*B + 112*C)*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(32*sqrt(2)*d*sqrt(a + a*cos(c + d*x))) + (a^2*(75*A + 88*B + 112*C)*tan(c + d*x))/(64*d*sqrt(a + a*cos(c + d*x))) + (a^2*(39*A + 56*B + 48*C)*sec(c + d*x)*tan(c + d*x))/(96*d*sqrt(a + a*cos(c + d*x))) + (a*(3*A + 8*B)*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^2*tan(c + d*x))/(24*d) + (A*(a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^3*tan(c + d*x))/(4*d)],
[sec(c + d*x)^6*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^(3/2), x, 6, (a^2*(133*A + 150*B + 176*C)*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(64*sqrt(2)*d*sqrt(a + a*cos(c + d*x))) + (a^2*(133*A + 150*B + 176*C)*tan(c + d*x))/(128*d*sqrt(a + a*cos(c + d*x))) + (a^2*(133*A + 150*B + 176*C)*sec(c + d*x)*tan(c + d*x))/(192*d*sqrt(a + a*cos(c + d*x))) + (a^2*(67*A + 90*B + 80*C)*sec(c + d*x)^2*tan(c + d*x))/(240*d*sqrt(a + a*cos(c + d*x))) + (a*(3*A + 10*B)*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^3*tan(c + d*x))/(40*d) + (A*(a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^4*tan(c + d*x))/(5*d)],


[cos(c + d*x)^2*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^(5/2), x, 7, (2*a^3*(10439*A + 9230*B + 8368*C)*sin(c + d*x))/(6435*d*sqrt(a + a*cos(c + d*x))) + (2*a^3*(2717*A + 2522*B + 2224*C)*cos(c + d*x)^3*sin(c + d*x))/(9009*d*sqrt(a + a*cos(c + d*x))) - (4*a^2*(10439*A + 9230*B + 8368*C)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(45045*d) + (2*a^2*(143*A + 182*B + 136*C)*cos(c + d*x)^3*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(1287*d) + (2*a*(10439*A + 9230*B + 8368*C)*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(15015*d) + (2*a*(13*B + 5*C)*cos(c + d*x)^3*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(143*d) + (2*C*cos(c + d*x)^3*(a + a*cos(c + d*x))^(5/2)*sin(c + d*x))/(13*d)],
[cos(c + d*x)^1*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^(5/2), x, 6, (64*a^3*(165*A + 143*B + 125*C)*sin(c + d*x))/(3465*d*sqrt(a + a*cos(c + d*x))) + (16*a^2*(165*A + 143*B + 125*C)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(3465*d) + (2*a*(165*A + 143*B + 125*C)*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(1155*d) + (2*(99*A - 22*B + 26*C)*(a + a*cos(c + d*x))^(5/2)*sin(c + d*x))/(693*d) + (2*C*cos(c + d*x)^2*(a + a*cos(c + d*x))^(5/2)*sin(c + d*x))/(11*d) + (2*(11*B + 5*C)*(a + a*cos(c + d*x))^(7/2)*sin(c + d*x))/(99*a*d)],
[cos(c + d*x)^0*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^(5/2), x, 5, (64*a^3*(21*A + 15*B + 13*C)*sin(c + d*x))/(315*d*sqrt(a + a*cos(c + d*x))) + (16*a^2*(21*A + 15*B + 13*C)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(315*d) + (2*a*(21*A + 15*B + 13*C)*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(105*d) + (2*(9*B - 2*C)*(a + a*cos(c + d*x))^(5/2)*sin(c + d*x))/(63*d) + (2*C*(a + a*cos(c + d*x))^(7/2)*sin(c + d*x))/(9*a*d)],
[sec(c + d*x)^1*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^(5/2), x, 5, (2*sqrt(2)*a^3*A*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(a + a*cos(c + d*x))) + (2*a^3*(245*A + 224*B + 160*C)*sin(c + d*x))/(105*d*sqrt(a + a*cos(c + d*x))) + (2*a^2*(35*A + 56*B + 40*C)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(105*d) + (2*a*(7*B + 5*C)*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(35*d) + (2*C*(a + a*cos(c + d*x))^(5/2)*sin(c + d*x))/(7*d)],
[sec(c + d*x)^2*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^(5/2), x, 5, (sqrt(2)*a^3*(5*A + 2*B)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(d*sqrt(a + a*cos(c + d*x))) + (a^3*(15*A + 70*B + 64*C)*sin(c + d*x))/(15*d*sqrt(a + a*cos(c + d*x))) - (a^2*(15*A - 10*B - 16*C)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(15*d) - (a*(5*A - 2*C)*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(5*d) + (A*(a + a*cos(c + d*x))^(5/2)*tan(c + d*x))/d],
[sec(c + d*x)^3*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^(5/2), x, 5, (a^3*(19*A + 20*B + 8*C)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(2*sqrt(2)*d*sqrt(a + a*cos(c + d*x))) - (a^3*(27*A - 12*B - 56*C)*sin(c + d*x))/(12*d*sqrt(a + a*cos(c + d*x))) - (a^2*(21*A + 12*B - 8*C)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(12*d) + (a*(5*A + 4*B)*(a + a*cos(c + d*x))^(3/2)*tan(c + d*x))/(4*d) + (A*(a + a*cos(c + d*x))^(5/2)*sec(c + d*x)*tan(c + d*x))/(2*d)],
[sec(c + d*x)^4*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^(5/2), x, 5, (a^3*(25*A + 38*B + 40*C)*arctanh(sqrt(2)*sin((c + d*x)/2))*cos((c + d*x)/2))/(4*sqrt(2)*d*sqrt(a + a*cos(c + d*x))) - (a^3*(49*A + 54*B - 24*C)*sin(c + d*x))/(24*d*sqrt(a + a*cos(c + d*x))) + (a^2*(31*A + 42*B + 24*C)*sqrt(a + a*cos(c + d*x))*tan(c + d*x))/(24*d) + (a*(5*A + 6*B)*(a + a*cos(c + d*x))^(3/2)*sec(c + d*x)*tan(c + d*x))/(12*d) + (A*(a + a*cos(c + d*x))^(5/2)*sec(c + d*x)^2*tan(c + d*x))/(3*d)],
[sec(c + d*x)^5*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^(5/2), x, 5, (a^3*(163*A + 200*B + 304*C)*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(32*sqrt(2)*d*sqrt(a + a*cos(c + d*x))) + (a^3*(299*A + 392*B + 432*C)*tan(c + d*x))/(192*d*sqrt(a + a*cos(c + d*x))) + (a^2*(17*A + 24*B + 16*C)*sqrt(a + a*cos(c + d*x))*sec(c + d*x)*tan(c + d*x))/(32*d) + (a*(5*A + 8*B)*(a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^2*tan(c + d*x))/(24*d) + (A*(a + a*cos(c + d*x))^(5/2)*sec(c + d*x)^3*tan(c + d*x))/(4*d)],
[sec(c + d*x)^6*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^(5/2), x, 6, (a^3*(283*A + 326*B + 400*C)*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(64*sqrt(2)*d*sqrt(a + a*cos(c + d*x))) + (a^3*(283*A + 326*B + 400*C)*tan(c + d*x))/(128*d*sqrt(a + a*cos(c + d*x))) + (a^3*(787*A + 950*B + 1040*C)*sec(c + d*x)*tan(c + d*x))/(960*d*sqrt(a + a*cos(c + d*x))) + (a^2*(79*A + 110*B + 80*C)*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^2*tan(c + d*x))/(240*d) + (a*(A + 2*B)*(a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^3*tan(c + d*x))/(8*d) + (A*(a + a*cos(c + d*x))^(5/2)*sec(c + d*x)^4*tan(c + d*x))/(5*d)],
[sec(c + d*x)^7*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^(5/2), x, 7, (a^3*(1015*A + 1132*B + 1304*C)*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(256*sqrt(2)*d*sqrt(a + a*cos(c + d*x))) + (a^3*(1015*A + 1132*B + 1304*C)*tan(c + d*x))/(512*d*sqrt(a + a*cos(c + d*x))) + (a^3*(1015*A + 1132*B + 1304*C)*sec(c + d*x)*tan(c + d*x))/(768*d*sqrt(a + a*cos(c + d*x))) + (a^3*(545*A + 628*B + 680*C)*sec(c + d*x)^2*tan(c + d*x))/(960*d*sqrt(a + a*cos(c + d*x))) + (a^2*(115*A + 156*B + 120*C)*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^3*tan(c + d*x))/(480*d) + (a*(5*A + 12*B)*(a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^4*tan(c + d*x))/(60*d) + (A*(a + a*cos(c + d*x))^(5/2)*sec(c + d*x)^5*tan(c + d*x))/(6*d)],


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


[cos(c + d*x)^3*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^(1/2), x, 6, -((2*(A - B + C)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(d*sqrt(a + a*cos(c + d*x)))) + (4*(147*A - 111*B + 143*C)*sin(c + d*x))/(315*d*sqrt(a + a*cos(c + d*x))) + (2*(21*A - 3*B + 19*C)*cos(c + d*x)^2*sin(c + d*x))/(105*d*sqrt(a + a*cos(c + d*x))) + (2*(9*B - C)*cos(c + d*x)^3*sin(c + d*x))/(63*d*sqrt(a + a*cos(c + d*x))) + (2*C*cos(c + d*x)^4*sin(c + d*x))/(9*d*sqrt(a + a*cos(c + d*x))) - (2*(21*A - 93*B + 29*C)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(315*a*d)],
[cos(c + d*x)^2*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^(1/2), x, 5, (2*(A - B + C)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(d*sqrt(a + a*cos(c + d*x))) - (4*(35*A - 49*B + 37*C)*sin(c + d*x))/(105*d*sqrt(a + a*cos(c + d*x))) + (2*(7*B - C)*cos(c + d*x)^2*sin(c + d*x))/(35*d*sqrt(a + a*cos(c + d*x))) + (2*C*cos(c + d*x)^3*sin(c + d*x))/(7*d*sqrt(a + a*cos(c + d*x))) + (2*(35*A - 7*B + 31*C)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(105*a*d)],
[cos(c + d*x)^1*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^(1/2), x, 4, -((2*(A - B + C)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(d*sqrt(a + a*cos(c + d*x)))) + (2*(15*A - 10*B + 14*C)*sin(c + d*x))/(15*d*sqrt(a + a*cos(c + d*x))) + (2*C*cos(c + d*x)^2*sin(c + d*x))/(5*d*sqrt(a + a*cos(c + d*x))) + (2*(5*B - C)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(15*a*d)],
[cos(c + d*x)^0*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^(1/2), x, 3, (2*(A - B + C)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(d*sqrt(a + a*cos(c + d*x))) + (2*(3*B - 2*C)*sin(c + d*x))/(3*d*sqrt(a + a*cos(c + d*x))) + (2*C*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(3*a*d)],
[sec(c + d*x)^1*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^(1/2), x, 4, -((2*(A - B + C)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(d*sqrt(a + a*cos(c + d*x)))) + (2*sqrt(2)*A*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(d*sqrt(a + a*cos(c + d*x))) + (2*C*sin(c + d*x))/(d*sqrt(a + a*cos(c + d*x)))],
[sec(c + d*x)^2*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^(1/2), x, 4, (2*(A - B + C)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(d*sqrt(a + a*cos(c + d*x))) - (sqrt(2)*(A - 2*B)*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(d*sqrt(a + a*cos(c + d*x))) + (A*tan(c + d*x))/(d*sqrt(a + a*cos(c + d*x)))],
[sec(c + d*x)^3*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^(1/2), x, 5, -((2*(A - B + C)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(d*sqrt(a + a*cos(c + d*x)))) + ((7*A - 4*B + 8*C)*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(2*sqrt(2)*d*sqrt(a + a*cos(c + d*x))) - ((A - 4*B)*tan(c + d*x))/(4*d*sqrt(a + a*cos(c + d*x))) + (A*sec(c + d*x)*tan(c + d*x))/(2*d*sqrt(a + a*cos(c + d*x)))],
[sec(c + d*x)^4*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^(1/2), x, 6, (2*(A - B + C)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(d*sqrt(a + a*cos(c + d*x))) - ((9*A - 14*B + 8*C)*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(4*sqrt(2)*d*sqrt(a + a*cos(c + d*x))) + ((7*A - 2*B + 8*C)*tan(c + d*x))/(8*d*sqrt(a + a*cos(c + d*x))) - ((A - 6*B)*sec(c + d*x)*tan(c + d*x))/(12*d*sqrt(a + a*cos(c + d*x))) + (A*sec(c + d*x)^2*tan(c + d*x))/(3*d*sqrt(a + a*cos(c + d*x)))],
[sec(c + d*x)^5*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^(1/2), x, 7, -((2*(A - B + C)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(d*sqrt(a + a*cos(c + d*x)))) + ((107*A - 72*B + 112*C)*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(32*sqrt(2)*d*sqrt(a + a*cos(c + d*x))) - ((21*A - 56*B + 16*C)*tan(c + d*x))/(64*d*sqrt(a + a*cos(c + d*x))) + ((43*A - 8*B + 48*C)*sec(c + d*x)*tan(c + d*x))/(96*d*sqrt(a + a*cos(c + d*x))) - ((A - 8*B)*sec(c + d*x)^2*tan(c + d*x))/(24*d*sqrt(a + a*cos(c + d*x))) + (A*sec(c + d*x)^3*tan(c + d*x))/(4*d*sqrt(a + a*cos(c + d*x)))],


[cos(c + d*x)^3*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^(3/2), x, 6, ((11*A - 15*B + 19*C)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(2*a*d*sqrt(a + a*cos(c + d*x))) - ((A - B + C)*cos(c + d*x)^4*sin(c + d*x))/(2*d*(a + a*cos(c + d*x))^(3/2)) - ((455*A - 651*B + 799*C)*sin(c + d*x))/(105*a*d*sqrt(a + a*cos(c + d*x))) - ((35*A - 63*B + 67*C)*cos(c + d*x)^2*sin(c + d*x))/(70*a*d*sqrt(a + a*cos(c + d*x))) + ((7*A - 7*B + 11*C)*cos(c + d*x)^3*sin(c + d*x))/(14*a*d*sqrt(a + a*cos(c + d*x))) + ((245*A - 273*B + 397*C)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(210*a^2*d)],
[cos(c + d*x)^2*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^(3/2), x, 5, -(((7*A - 11*B + 15*C)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(2*a*d*sqrt(a + a*cos(c + d*x)))) - ((A - B + C)*cos(c + d*x)^3*sin(c + d*x))/(2*d*(a + a*cos(c + d*x))^(3/2)) + ((45*A - 65*B + 93*C)*sin(c + d*x))/(15*a*d*sqrt(a + a*cos(c + d*x))) + ((5*A - 5*B + 9*C)*cos(c + d*x)^2*sin(c + d*x))/(10*a*d*sqrt(a + a*cos(c + d*x))) - ((15*A - 35*B + 39*C)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(30*a^2*d)],
[cos(c + d*x)^1*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^(3/2), x, 4, ((3*A - 7*B + 11*C)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(2*a*d*sqrt(a + a*cos(c + d*x))) - ((A - B + C)*cos(c + d*x)^2*sin(c + d*x))/(2*d*(a + a*cos(c + d*x))^(3/2)) - ((3*A - 9*B + 13*C)*sin(c + d*x))/(3*a*d*sqrt(a + a*cos(c + d*x))) + ((3*A - 3*B + 7*C)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(6*a^2*d)],
[cos(c + d*x)^0*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^(3/2), x, 3, ((A + 3*B - 7*C)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(2*a*d*sqrt(a + a*cos(c + d*x))) + ((A - B + C)*sin(c + d*x))/(2*d*(a + a*cos(c + d*x))^(3/2)) + (2*C*sin(c + d*x))/(a*d*sqrt(a + a*cos(c + d*x)))],
[sec(c + d*x)^1*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^(3/2), x, 4, -(((5*A - B - 3*C)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(2*a*d*sqrt(a + a*cos(c + d*x)))) + (2*sqrt(2)*A*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(a*d*sqrt(a + a*cos(c + d*x))) - ((A - B + C)*sin(c + d*x))/(2*d*(a + a*cos(c + d*x))^(3/2))],
[sec(c + d*x)^2*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^(3/2), x, 5, ((9*A - 5*B + C)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(2*a*d*sqrt(a + a*cos(c + d*x))) - (sqrt(2)*(3*A - 2*B)*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(a*d*sqrt(a + a*cos(c + d*x))) - ((A - B + C)*tan(c + d*x))/(2*d*(a + a*cos(c + d*x))^(3/2)) + ((3*A - B + C)*tan(c + d*x))/(2*a*d*sqrt(a + a*cos(c + d*x)))],
[sec(c + d*x)^3*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^(3/2), x, 6, -(((13*A - 9*B + 5*C)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(2*a*d*sqrt(a + a*cos(c + d*x)))) + ((19*A - 12*B + 8*C)*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(2*sqrt(2)*a*d*sqrt(a + a*cos(c + d*x))) - ((7*A - 6*B + 2*C)*tan(c + d*x))/(4*a*d*sqrt(a + a*cos(c + d*x))) - ((A - B + C)*sec(c + d*x)*tan(c + d*x))/(2*d*(a + a*cos(c + d*x))^(3/2)) + ((2*A - B + C)*sec(c + d*x)*tan(c + d*x))/(2*a*d*sqrt(a + a*cos(c + d*x)))],
[sec(c + d*x)^4*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^(3/2), x, 7, ((17*A - 13*B + 9*C)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(2*a*d*sqrt(a + a*cos(c + d*x))) - ((47*A - 38*B + 24*C)*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(4*sqrt(2)*a*d*sqrt(a + a*cos(c + d*x))) + ((21*A - 14*B + 12*C)*tan(c + d*x))/(8*a*d*sqrt(a + a*cos(c + d*x))) - ((13*A - 12*B + 6*C)*sec(c + d*x)*tan(c + d*x))/(12*a*d*sqrt(a + a*cos(c + d*x))) - ((A - B + C)*sec(c + d*x)^2*tan(c + d*x))/(2*d*(a + a*cos(c + d*x))^(3/2)) + ((5*A - 3*B + 3*C)*sec(c + d*x)^2*tan(c + d*x))/(6*a*d*sqrt(a + a*cos(c + d*x)))],


[cos(c + d*x)^3*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^(5/2), x, 6, -(((75*A - 163*B + 283*C)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(16*a^2*d*sqrt(a + a*cos(c + d*x)))) - ((A - B + C)*cos(c + d*x)^4*sin(c + d*x))/(4*d*(a + a*cos(c + d*x))^(5/2)) - ((5*A - 13*B + 21*C)*cos(c + d*x)^3*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2)) + ((465*A - 985*B + 1729*C)*sin(c + d*x))/(120*a^2*d*sqrt(a + a*cos(c + d*x))) + ((45*A - 85*B + 157*C)*cos(c + d*x)^2*sin(c + d*x))/(80*a^2*d*sqrt(a + a*cos(c + d*x))) - ((195*A - 475*B + 787*C)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(240*a^3*d)],
[cos(c + d*x)^2*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^(5/2), x, 5, ((19*A - 75*B + 163*C)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(16*a^2*d*sqrt(a + a*cos(c + d*x))) - ((A - B + C)*cos(c + d*x)^3*sin(c + d*x))/(4*d*(a + a*cos(c + d*x))^(5/2)) - ((A - 9*B + 17*C)*cos(c + d*x)^2*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2)) - ((21*A - 93*B + 197*C)*sin(c + d*x))/(24*a^2*d*sqrt(a + a*cos(c + d*x))) + ((15*A - 39*B + 95*C)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(48*a^3*d)],
[cos(c + d*x)^1*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^(5/2), x, 4, ((5*A + 19*B - 75*C)*arctanh(sin((c + d*x)/2))*cos((c + d*x)/2))/(16*a^2*d*sqrt(a + a*cos(c + d*x))) - ((A - B + C)*cos(c + d*x)^2*sin(c + d*x))/(4*d*(a + a*cos(c + d*x))^(5/2)) + ((3*A + 5*B - 13*C)*cos(c + d*x)*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2)) + ((A - 9*B + 49*C)*sin(c + d*x))/(16*a^2*d*sqrt(a + a*cos(c + d*x)))],
[cos(c + d*x)^0*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^(5/2), x, 3, ((3*A + 5*B + 19*C)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(16*a^2*d*sqrt(a + a*cos(c + d*x))) + ((A - B + C)*sin(c + d*x))/(4*d*(a + a*cos(c + d*x))^(5/2)) + ((3*A + 5*B - 13*C)*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2))],
[sec(c + d*x)^1*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^(5/2), x, 5, -(((43*A - 3*B - 5*C)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(16*a^2*d*sqrt(a + a*cos(c + d*x)))) + (2*sqrt(2)*A*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(a^2*d*sqrt(a + a*cos(c + d*x))) - ((A - B + C)*sin(c + d*x))/(4*d*(a + a*cos(c + d*x))^(5/2)) - ((11*A - 3*B - 5*C)*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2))],
[sec(c + d*x)^2*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^(5/2), x, 6, ((115*A - 43*B + 3*C)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(16*a^2*d*sqrt(a + a*cos(c + d*x))) - (sqrt(2)*(5*A - 2*B)*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(a^2*d*sqrt(a + a*cos(c + d*x))) - ((A - B + C)*tan(c + d*x))/(4*d*(a + a*cos(c + d*x))^(5/2)) - ((15*A - 7*B - C)*tan(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2)) + ((35*A - 11*B + 3*C)*tan(c + d*x))/(16*a^2*d*sqrt(a + a*cos(c + d*x)))],
[sec(c + d*x)^3*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^(5/2), x, 7, -(((219*A - 115*B + 43*C)*arctanh(sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(16*a^2*d*sqrt(a + a*cos(c + d*x)))) + ((39*A - 20*B + 8*C)*arctanh(sqrt(2)*sin((1/2)*(c + d*x)))*cos((1/2)*(c + d*x)))/(2*sqrt(2)*a^2*d*sqrt(a + a*cos(c + d*x))) - ((63*A - 35*B + 11*C)*tan(c + d*x))/(16*a^2*d*sqrt(a + a*cos(c + d*x))) - ((A - B + C)*sec(c + d*x)*tan(c + d*x))/(4*d*(a + a*cos(c + d*x))^(5/2)) - ((19*A - 11*B + 3*C)*sec(c + d*x)*tan(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2)) + ((31*A - 15*B + 7*C)*sec(c + d*x)*tan(c + d*x))/(16*a^2*d*sqrt(a + a*cos(c + d*x)))],


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


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


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


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


[cos(c + d*x)^(3/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^2, x, 8, (4*a^2*(9*A + 8*B + 7*C)*EllipticE((1/2)*(c + d*x), 2))/(15*d) + (4*a^2*(66*A + 55*B + 50*C)*EllipticF((1/2)*(c + d*x), 2))/(231*d) + (4*a^2*(66*A + 55*B + 50*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(231*d) + (4*a^2*(9*A + 8*B + 7*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(45*d) + (2*a^2*(99*A + 121*B + 89*C)*cos(c + d*x)^(5/2)*sin(c + d*x))/(693*d) + (2*a*(11*B + 4*C)*cos(c + d*x)^(5/2)*(a + a*cos(c + d*x))*sin(c + d*x))/(99*d) + (2*C*cos(c + d*x)^(5/2)*(a + a*cos(c + d*x))^2*sin(c + d*x))/(11*d)],
[cos(c + d*x)^(1/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^2, x, 7, (4*a^2*(12*A + 9*B + 8*C)*EllipticE((1/2)*(c + d*x), 2))/(15*d) + (4*a^2*(7*A + 6*B + 5*C)*EllipticF((1/2)*(c + d*x), 2))/(21*d) + (4*a^2*(7*A + 6*B + 5*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(21*d) + (2*a^2*(21*A + 27*B + 19*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(105*d) + (2*a*(9*B + 4*C)*cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))*sin(c + d*x))/(63*d) + (2*C*cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))^2*sin(c + d*x))/(9*d)],
[cos(c + d*x)^(-1/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^2, x, 6, (4*a^2*(5*A + 4*B + 3*C)*EllipticE((1/2)*(c + d*x), 2))/(5*d) + (4*a^2*(14*A + 7*B + 6*C)*EllipticF((1/2)*(c + d*x), 2))/(21*d) + (2*a^2*(35*A + 49*B + 33*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(105*d) + (2*a*(7*B + 4*C)*sqrt(cos(c + d*x))*(a + a*cos(c + d*x))*sin(c + d*x))/(35*d) + (2*C*sqrt(cos(c + d*x))*(a + a*cos(c + d*x))^2*sin(c + d*x))/(7*d)],
[cos(c + d*x)^(-3/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^2, x, 6, (4*a^2*(5*B + 4*C)*EllipticE((1/2)*(c + d*x), 2))/(5*d) + (4*a^2*(3*A + 2*B + C)*EllipticF((1/2)*(c + d*x), 2))/(3*d) - (2*a^2*(15*A - 5*B - 7*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(15*d) - (2*a*(5*A - C)*sqrt(cos(c + d*x))*(a + a*cos(c + d*x))*sin(c + d*x))/(5*d) + (2*A*(a + a*cos(c + d*x))^2*sin(c + d*x))/(d*sqrt(cos(c + d*x)))],
[cos(c + d*x)^(-5/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^2, x, 6, -((4*a^2*(A - C)*EllipticE((1/2)*(c + d*x), 2))/d) + (4*a^2*(2*A + 3*B + 2*C)*EllipticF((1/2)*(c + d*x), 2))/(3*d) - (2*a^2*(5*A + 3*B - C)*sqrt(cos(c + d*x))*sin(c + d*x))/(3*d) + (2*a*(4*A + 3*B)*(a + a*cos(c + d*x))*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))) + (2*A*(a + a*cos(c + d*x))^2*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2))],
[cos(c + d*x)^(-7/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^2, x, 6, -((4*a^2*(4*A + 5*B)*EllipticE((1/2)*(c + d*x), 2))/(5*d)) + (4*a^2*(A + 2*B + 3*C)*EllipticF((1/2)*(c + d*x), 2))/(3*d) + (2*a^2*(17*A + 25*B + 15*C)*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))) + (2*a*(4*A + 5*B)*(a + a*cos(c + d*x))*sin(c + d*x))/(15*d*cos(c + d*x)^(3/2)) + (2*A*(a + a*cos(c + d*x))^2*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2))],
[cos(c + d*x)^(-9/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^2, x, 7, -((4*a^2*(3*A + 4*B + 5*C)*EllipticE((1/2)*(c + d*x), 2))/(5*d)) + (4*a^2*(6*A + 7*B + 14*C)*EllipticF((1/2)*(c + d*x), 2))/(21*d) + (2*a^2*(33*A + 49*B + 35*C)*sin(c + d*x))/(105*d*cos(c + d*x)^(3/2)) + (4*a^2*(3*A + 4*B + 5*C)*sin(c + d*x))/(5*d*sqrt(cos(c + d*x))) + (2*a*(4*A + 7*B)*(a + a*cos(c + d*x))*sin(c + d*x))/(35*d*cos(c + d*x)^(5/2)) + (2*A*(a + a*cos(c + d*x))^2*sin(c + d*x))/(7*d*cos(c + d*x)^(7/2))],
[cos(c + d*x)^(-11/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^2, x, 8, -((4*a^2*(8*A + 9*B + 12*C)*EllipticE((1/2)*(c + d*x), 2))/(15*d)) + (4*a^2*(5*A + 6*B + 7*C)*EllipticF((1/2)*(c + d*x), 2))/(21*d) + (2*a^2*(19*A + 27*B + 21*C)*sin(c + d*x))/(105*d*cos(c + d*x)^(5/2)) + (4*a^2*(5*A + 6*B + 7*C)*sin(c + d*x))/(21*d*cos(c + d*x)^(3/2)) + (4*a^2*(8*A + 9*B + 12*C)*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))) + (2*a*(4*A + 9*B)*(a + a*cos(c + d*x))*sin(c + d*x))/(63*d*cos(c + d*x)^(7/2)) + (2*A*(a + a*cos(c + d*x))^2*sin(c + d*x))/(9*d*cos(c + d*x)^(9/2))],


[cos(c + d*x)^(3/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^3, x, 9, (4*a^3*(221*A + 195*B + 175*C)*EllipticE((1/2)*(c + d*x), 2))/(195*d) + (4*a^3*(121*A + 105*B + 95*C)*EllipticF((1/2)*(c + d*x), 2))/(231*d) + (4*a^3*(121*A + 105*B + 95*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(231*d) + (4*a^3*(221*A + 195*B + 175*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(585*d) + (20*a^3*(286*A + 273*B + 236*C)*cos(c + d*x)^(5/2)*sin(c + d*x))/(9009*d) + (2*a^2*(143*A + 195*B + 145*C)*cos(c + d*x)^(5/2)*(a + a*cos(c + d*x))*sin(c + d*x))/(1287*d) + (2*a*(13*B + 6*C)*cos(c + d*x)^(5/2)*(a + a*cos(c + d*x))^2*sin(c + d*x))/(143*d) + (2*C*cos(c + d*x)^(5/2)*(a + a*cos(c + d*x))^3*sin(c + d*x))/(13*d)],
[cos(c + d*x)^(1/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^3, x, 8, (4*a^3*(21*A + 17*B + 15*C)*EllipticE((1/2)*(c + d*x), 2))/(15*d) + (4*a^3*(143*A + 121*B + 105*C)*EllipticF((1/2)*(c + d*x), 2))/(231*d) + (4*a^3*(143*A + 121*B + 105*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(231*d) + (4*a^3*(264*A + 253*B + 210*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(1155*d) + (2*a^2*(99*A + 143*B + 105*C)*cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))*sin(c + d*x))/(693*d) + (2*a*(11*B + 6*C)*cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))^2*sin(c + d*x))/(99*d) + (2*C*cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))^3*sin(c + d*x))/(11*d)],
[cos(c + d*x)^(-1/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^3, x, 7, (4*a^3*(27*A + 21*B + 17*C)*EllipticE((1/2)*(c + d*x), 2))/(15*d) + (4*a^3*(21*A + 13*B + 11*C)*EllipticF((1/2)*(c + d*x), 2))/(21*d) + (4*a^3*(42*A + 41*B + 32*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(105*d) + (2*a^2*(63*A + 99*B + 73*C)*sqrt(cos(c + d*x))*(a + a*cos(c + d*x))*sin(c + d*x))/(315*d) + (2*a*(3*B + 2*C)*sqrt(cos(c + d*x))*(a + a*cos(c + d*x))^2*sin(c + d*x))/(21*d) + (2*C*sqrt(cos(c + d*x))*(a + a*cos(c + d*x))^3*sin(c + d*x))/(9*d)],
[cos(c + d*x)^(-3/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^3, x, 7, (4*a^3*(5*A + 9*B + 7*C)*EllipticE((1/2)*(c + d*x), 2))/(5*d) + (4*a^3*(35*A + 21*B + 13*C)*EllipticF((1/2)*(c + d*x), 2))/(21*d) - (4*a^3*(35*A - 42*B - 41*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(105*d) - (2*a^2*(35*A - 7*B - 11*C)*sqrt(cos(c + d*x))*(a + a*cos(c + d*x))*sin(c + d*x))/(35*d) - (2*a*(7*A - C)*sqrt(cos(c + d*x))*(a + a*cos(c + d*x))^2*sin(c + d*x))/(7*d) + (2*A*(a + a*cos(c + d*x))^3*sin(c + d*x))/(d*sqrt(cos(c + d*x)))],
[cos(c + d*x)^(-5/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^3, x, 7, -((4*a^3*(5*A - 5*B - 9*C)*EllipticE((1/2)*(c + d*x), 2))/(5*d)) + (4*a^3*(5*A + 5*B + 3*C)*EllipticF((1/2)*(c + d*x), 2))/(3*d) - (4*a^3*(20*A + 5*B - 6*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(15*d) - (2*a^2*(35*A + 15*B - 3*C)*sqrt(cos(c + d*x))*(a + a*cos(c + d*x))*sin(c + d*x))/(15*d) + (2*a*(2*A + B)*(a + a*cos(c + d*x))^2*sin(c + d*x))/(d*sqrt(cos(c + d*x))) + (2*A*(a + a*cos(c + d*x))^3*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2))],
[cos(c + d*x)^(-7/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^3, x, 7, -((4*a^3*(9*A + 5*B - 5*C)*EllipticE((1/2)*(c + d*x), 2))/(5*d)) + (4*a^3*(3*A + 5*B + 5*C)*EllipticF((1/2)*(c + d*x), 2))/(3*d) - (4*a^3*(21*A + 20*B + 5*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(15*d) + (2*a^2*(33*A + 35*B + 15*C)*(a + a*cos(c + d*x))*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))) + (2*a*(6*A + 5*B)*(a + a*cos(c + d*x))^2*sin(c + d*x))/(15*d*cos(c + d*x)^(3/2)) + (2*A*(a + a*cos(c + d*x))^3*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2))],
[cos(c + d*x)^(-9/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^3, x, 7, -((4*a^3*(7*A + 9*B + 5*C)*EllipticE((1/2)*(c + d*x), 2))/(5*d)) + (4*a^3*(13*A + 21*B + 35*C)*EllipticF((1/2)*(c + d*x), 2))/(21*d) + (4*a^3*(106*A + 147*B + 140*C)*sin(c + d*x))/(105*d*sqrt(cos(c + d*x))) + (2*a^2*(7*A + 9*B + 5*C)*(a + a*cos(c + d*x))*sin(c + d*x))/(15*d*cos(c + d*x)^(3/2)) + (2*a*(6*A + 7*B)*(a + a*cos(c + d*x))^2*sin(c + d*x))/(35*d*cos(c + d*x)^(5/2)) + (2*A*(a + a*cos(c + d*x))^3*sin(c + d*x))/(7*d*cos(c + d*x)^(7/2))],
[cos(c + d*x)^(-11/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^3, x, 8, -((4*a^3*(17*A + 21*B + 27*C)*EllipticE((1/2)*(c + d*x), 2))/(15*d)) + (4*a^3*(11*A + 13*B + 21*C)*EllipticF((1/2)*(c + d*x), 2))/(21*d) + (4*a^3*(32*A + 41*B + 42*C)*sin(c + d*x))/(105*d*cos(c + d*x)^(3/2)) + (4*a^3*(17*A + 21*B + 27*C)*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))) + (2*a^2*(73*A + 99*B + 63*C)*(a + a*cos(c + d*x))*sin(c + d*x))/(315*d*cos(c + d*x)^(5/2)) + (2*a*(2*A + 3*B)*(a + a*cos(c + d*x))^2*sin(c + d*x))/(21*d*cos(c + d*x)^(7/2)) + (2*A*(a + a*cos(c + d*x))^3*sin(c + d*x))/(9*d*cos(c + d*x)^(9/2))],
[cos(c + d*x)^(-13/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^3, x, 9, -((4*a^3*(15*A + 17*B + 21*C)*EllipticE((1/2)*(c + d*x), 2))/(15*d)) + (4*a^3*(105*A + 121*B + 143*C)*EllipticF((1/2)*(c + d*x), 2))/(231*d) + (4*a^3*(210*A + 253*B + 264*C)*sin(c + d*x))/(1155*d*cos(c + d*x)^(5/2)) + (4*a^3*(105*A + 121*B + 143*C)*sin(c + d*x))/(231*d*cos(c + d*x)^(3/2)) + (4*a^3*(15*A + 17*B + 21*C)*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))) + (2*a^2*(105*A + 143*B + 99*C)*(a + a*cos(c + d*x))*sin(c + d*x))/(693*d*cos(c + d*x)^(7/2)) + (2*a*(6*A + 11*B)*(a + a*cos(c + d*x))^2*sin(c + d*x))/(99*d*cos(c + d*x)^(9/2)) + (2*A*(a + a*cos(c + d*x))^3*sin(c + d*x))/(11*d*cos(c + d*x)^(11/2))],


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


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


[cos(c + d*x)^(5/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^2, x, 7, ((20*A - 35*B + 56*C)*EllipticE((1/2)*(c + d*x), 2))/(5*a^2*d) - (5*(A - 2*B + 3*C)*EllipticF((1/2)*(c + d*x), 2))/(3*a^2*d) - (5*(A - 2*B + 3*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(3*a^2*d) + ((20*A - 35*B + 56*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(15*a^2*d) - ((A - B + C)*cos(c + d*x)^(7/2)*sin(c + d*x))/(3*d*(a + a*cos(c + d*x))^2) - ((A - 2*B + 3*C)*cos(c + d*x)^(5/2)*sin(c + d*x))/(a*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^(3/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^2, x, 6, -(((A - 4*B + 7*C)*EllipticE((1/2)*(c + d*x), 2))/(a^2*d)) + ((2*A - 5*B + 10*C)*EllipticF((1/2)*(c + d*x), 2))/(3*a^2*d) + ((2*A - 5*B + 10*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(3*a^2*d) - ((A - B + C)*cos(c + d*x)^(5/2)*sin(c + d*x))/(3*d*(a + a*cos(c + d*x))^2) - ((A - 4*B + 7*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(3*a*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^(1/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^2, x, 5, -(((B - 4*C)*EllipticE((1/2)*(c + d*x), 2))/(a^2*d)) + ((A + 2*B - 5*C)*EllipticF((1/2)*(c + d*x), 2))/(3*a^2*d) - ((A - B + C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(3*d*(a + a*cos(c + d*x))^2) + ((A + 2*B - 5*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(3*a*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^(-1/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^2, x, 5, ((A - C)*EllipticE((1/2)*(c + d*x), 2))/(a^2*d) + ((2*A + B + 2*C)*EllipticF((1/2)*(c + d*x), 2))/(3*a^2*d) - ((A - B + C)*sqrt(cos(c + d*x))*sin(c + d*x))/(3*d*(a + a*cos(c + d*x))^2) - ((A - C)*sqrt(cos(c + d*x))*sin(c + d*x))/(a*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^(-3/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^2, x, 6, -(((4*A - B)*EllipticE((1/2)*(c + d*x), 2))/(a^2*d)) - ((5*A - 2*B - C)*EllipticF((1/2)*(c + d*x), 2))/(3*a^2*d) + ((4*A - B)*sin(c + d*x))/(a^2*d*sqrt(cos(c + d*x))) - ((A - B + C)*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))*(a + a*cos(c + d*x))^2) - ((5*A - 2*B - C)*sin(c + d*x))/(3*a*d*sqrt(cos(c + d*x))*(a + a*cos(c + d*x)))],
[cos(c + d*x)^(-5/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^2, x, 7, ((7*A - 4*B + C)*EllipticE((1/2)*(c + d*x), 2))/(a^2*d) + ((10*A - 5*B + 2*C)*EllipticF((1/2)*(c + d*x), 2))/(3*a^2*d) + ((10*A - 5*B + 2*C)*sin(c + d*x))/(3*a^2*d*cos(c + d*x)^(3/2)) - ((7*A - 4*B + C)*sin(c + d*x))/(a^2*d*sqrt(cos(c + d*x))) - ((A - B + C)*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))^2) - ((7*A - 4*B + C)*sin(c + d*x))/(3*a*d*cos(c + d*x)^(3/2)*(a + a*cos(c + d*x)))],


[cos(c + d*x)^(7/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^3, x, 8, (7*(7*A - 17*B + 33*C)*EllipticE((1/2)*(c + d*x), 2))/(10*a^3*d) - ((13*A - 33*B + 63*C)*EllipticF((1/2)*(c + d*x), 2))/(6*a^3*d) - ((13*A - 33*B + 63*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(6*a^3*d) + (7*(7*A - 17*B + 33*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(30*a^3*d) - ((A - B + C)*cos(c + d*x)^(9/2)*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3) - ((2*A - 7*B + 12*C)*cos(c + d*x)^(7/2)*sin(c + d*x))/(15*a*d*(a + a*cos(c + d*x))^2) - ((13*A - 33*B + 63*C)*cos(c + d*x)^(5/2)*sin(c + d*x))/(10*a^2*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^(5/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^3, x, 7, -(((9*A - 49*B + 119*C)*EllipticE((1/2)*(c + d*x), 2))/(10*a^3*d)) + ((3*A - 13*B + 33*C)*EllipticF((1/2)*(c + d*x), 2))/(6*a^3*d) + ((3*A - 13*B + 33*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(6*a^3*d) - ((A - B + C)*cos(c + d*x)^(7/2)*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3) + ((B - 2*C)*cos(c + d*x)^(5/2)*sin(c + d*x))/(3*a*d*(a + a*cos(c + d*x))^2) - ((9*A - 49*B + 119*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(30*a^2*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^(3/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^3, x, 6, -(((A + 9*B - 49*C)*EllipticE((1/2)*(c + d*x), 2))/(10*a^3*d)) + ((A + 3*B - 13*C)*EllipticF((1/2)*(c + d*x), 2))/(6*a^3*d) - ((A - B + C)*cos(c + d*x)^(5/2)*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3) + ((2*A + 3*B - 8*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(15*a*d*(a + a*cos(c + d*x))^2) + ((A + 3*B - 13*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(6*a^2*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^(1/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^3, x, 6, ((A - B - 9*C)*EllipticE((1/2)*(c + d*x), 2))/(10*a^3*d) + ((A + B + 3*C)*EllipticF((1/2)*(c + d*x), 2))/(6*a^3*d) - ((A - B + C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3) + ((4*A + B - 6*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(15*a*d*(a + a*cos(c + d*x))^2) - ((A - B - 9*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(10*a^2*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^(-1/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^3, x, 6, ((9*A + B - C)*EllipticE((1/2)*(c + d*x), 2))/(10*a^3*d) + ((3*A + B + C)*EllipticF((1/2)*(c + d*x), 2))/(6*a^3*d) - ((A - B + C)*sqrt(cos(c + d*x))*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3) - ((6*A - B - 4*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(15*a*d*(a + a*cos(c + d*x))^2) - ((9*A + B - C)*sqrt(cos(c + d*x))*sin(c + d*x))/(10*a^2*d*(a + a*cos(c + d*x)))],
[cos(c + d*x)^(-3/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^3, x, 7, -(((49*A - 9*B - C)*EllipticE((1/2)*(c + d*x), 2))/(10*a^3*d)) - ((13*A - 3*B - C)*EllipticF((1/2)*(c + d*x), 2))/(6*a^3*d) + ((49*A - 9*B - C)*sin(c + d*x))/(10*a^3*d*sqrt(cos(c + d*x))) - ((A - B + C)*sin(c + d*x))/(5*d*sqrt(cos(c + d*x))*(a + a*cos(c + d*x))^3) - ((8*A - 3*B - 2*C)*sin(c + d*x))/(15*a*d*sqrt(cos(c + d*x))*(a + a*cos(c + d*x))^2) - ((13*A - 3*B - C)*sin(c + d*x))/(6*a^2*d*sqrt(cos(c + d*x))*(a + a*cos(c + d*x)))],
[cos(c + d*x)^(-5/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^3, x, 8, ((119*A - 49*B + 9*C)*EllipticE((1/2)*(c + d*x), 2))/(10*a^3*d) + ((33*A - 13*B + 3*C)*EllipticF((1/2)*(c + d*x), 2))/(6*a^3*d) + ((33*A - 13*B + 3*C)*sin(c + d*x))/(6*a^3*d*cos(c + d*x)^(3/2)) - ((119*A - 49*B + 9*C)*sin(c + d*x))/(10*a^3*d*sqrt(cos(c + d*x))) - ((A - B + C)*sin(c + d*x))/(5*d*cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))^3) - ((2*A - B)*sin(c + d*x))/(3*a*d*cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))^2) - ((119*A - 49*B + 9*C)*sin(c + d*x))/(30*a^2*d*cos(c + d*x)^(3/2)*(a + a*cos(c + d*x)))],


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


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


[sqrt(a + a*cos(c + d*x))*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*cos(c + d*x)^(3/2), x, 5, (sqrt(a)*(48*A + 40*B + 35*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(64*d) + (a*(48*A + 40*B + 35*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(64*d*sqrt(a + a*cos(c + d*x))) + (a*(48*A + 40*B + 35*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(96*d*sqrt(a + a*cos(c + d*x))) + (a*(8*B + C)*cos(c + d*x)^(5/2)*sin(c + d*x))/(24*d*sqrt(a + a*cos(c + d*x))) + (C*cos(c + d*x)^(5/2)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(4*d)],
[sqrt(a + a*cos(c + d*x))*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*cos(c + d*x)^(1/2), x, 4, (sqrt(a)*(8*A + 6*B + 5*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(8*d) + (a*(8*A + 6*B + 5*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(8*d*sqrt(a + a*cos(c + d*x))) + (a*(6*B + C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(12*d*sqrt(a + a*cos(c + d*x))) + (C*cos(c + d*x)^(3/2)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(3*d)],
[sqrt(a + a*cos(c + d*x))*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/cos(c + d*x)^(1/2), x, 3, (sqrt(a)*(8*A + 4*B + 3*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(4*d) + (a*(4*B + C)*sqrt(cos(c + d*x))*sin(c + d*x))/(4*d*sqrt(a + a*cos(c + d*x))) + (C*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(2*d)],
[sqrt(a + a*cos(c + d*x))*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/cos(c + d*x)^(3/2), x, 3, (sqrt(a)*(2*B + C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/d - (a*(2*A - C)*sqrt(cos(c + d*x))*sin(c + d*x))/(d*sqrt(a + a*cos(c + d*x))) + (2*A*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(d*sqrt(cos(c + d*x)))],
[sqrt(a + a*cos(c + d*x))*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/cos(c + d*x)^(5/2), x, 3, (2*sqrt(a)*C*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/d + (2*a*(A + 3*B)*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))) + (2*A*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2))],
[sqrt(a + a*cos(c + d*x))*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/cos(c + d*x)^(7/2), x, 3, (2*a*(A + 5*B)*sin(c + d*x))/(15*d*cos(c + d*x)^(3/2)*sqrt(a + a*cos(c + d*x))) + (2*a*(8*A + 10*B + 15*C)*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))) + (2*A*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2))],
[sqrt(a + a*cos(c + d*x))*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/cos(c + d*x)^(9/2), x, 4, (2*a*(A + 7*B)*sin(c + d*x))/(35*d*cos(c + d*x)^(5/2)*sqrt(a + a*cos(c + d*x))) + (2*a*(24*A + 28*B + 35*C)*sin(c + d*x))/(105*d*cos(c + d*x)^(3/2)*sqrt(a + a*cos(c + d*x))) + (4*a*(24*A + 28*B + 35*C)*sin(c + d*x))/(105*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))) + (2*A*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(7*d*cos(c + d*x)^(7/2))],
[sqrt(a + a*cos(c + d*x))*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/cos(c + d*x)^(11/2), x, 5, (2*a*(A + 9*B)*sin(c + d*x))/(63*d*cos(c + d*x)^(7/2)*sqrt(a + a*cos(c + d*x))) + (2*a*(16*A + 18*B + 21*C)*sin(c + d*x))/(105*d*cos(c + d*x)^(5/2)*sqrt(a + a*cos(c + d*x))) + (8*a*(16*A + 18*B + 21*C)*sin(c + d*x))/(315*d*cos(c + d*x)^(3/2)*sqrt(a + a*cos(c + d*x))) + (16*a*(16*A + 18*B + 21*C)*sin(c + d*x))/(315*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))) + (2*A*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(9*d*cos(c + d*x)^(9/2))],


[(a + a*cos(c + d*x))^(3/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*cos(c + d*x)^(3/2), x, 6, (a^(3/2)*(176*A + 150*B + 133*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(128*d) + (a^2*(176*A + 150*B + 133*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(128*d*sqrt(a + a*cos(c + d*x))) + (a^2*(176*A + 150*B + 133*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(192*d*sqrt(a + a*cos(c + d*x))) + (a^2*(80*A + 90*B + 67*C)*cos(c + d*x)^(5/2)*sin(c + d*x))/(240*d*sqrt(a + a*cos(c + d*x))) + (a*(10*B + 3*C)*cos(c + d*x)^(5/2)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(40*d) + (C*cos(c + d*x)^(5/2)*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(5*d)],
[(a + a*cos(c + d*x))^(3/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*cos(c + d*x)^(1/2), x, 5, (a^(3/2)*(112*A + 88*B + 75*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(64*d) + (a^2*(112*A + 88*B + 75*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(64*d*sqrt(a + a*cos(c + d*x))) + (a^2*(48*A + 56*B + 39*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(96*d*sqrt(a + a*cos(c + d*x))) + (a*(8*B + 3*C)*cos(c + d*x)^(3/2)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(24*d) + (C*cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(4*d)],
[(a + a*cos(c + d*x))^(3/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/cos(c + d*x)^(1/2), x, 4, (a^(3/2)*(24*A + 14*B + 11*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(8*d) + (a^2*(24*A + 30*B + 19*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(24*d*sqrt(a + a*cos(c + d*x))) + (a*(2*B + C)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(4*d) + (C*sqrt(cos(c + d*x))*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(3*d)],
[(a + a*cos(c + d*x))^(3/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/cos(c + d*x)^(3/2), x, 4, (a^(3/2)*(8*A + 12*B + 7*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(4*d) - (a^2*(8*A - 4*B - 5*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(4*d*sqrt(a + a*cos(c + d*x))) - (a*(4*A - C)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(2*d) + (2*A*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(d*sqrt(cos(c + d*x)))],
[(a + a*cos(c + d*x))^(3/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/cos(c + d*x)^(5/2), x, 4, (a^(3/2)*(2*B + 3*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/d - (a^2*(8*A + 6*B - 3*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(3*d*sqrt(a + a*cos(c + d*x))) + (2*a*(A + B)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(d*sqrt(cos(c + d*x))) + (2*A*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2))],
[(a + a*cos(c + d*x))^(3/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/cos(c + d*x)^(7/2), x, 4, (2*a^(3/2)*C*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/d + (2*a^2*(12*A + 20*B + 15*C)*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))) + (2*a*(3*A + 5*B)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(15*d*cos(c + d*x)^(3/2)) + (2*A*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2))],
[(a + a*cos(c + d*x))^(3/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/cos(c + d*x)^(9/2), x, 4, (8*a^2*(19*A + 21*B + 35*C)*sin(c + d*x))/(105*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))) + (2*a*(19*A + 21*B + 35*C)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(105*d*cos(c + d*x)^(3/2)) + (2*A*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(7*d*cos(c + d*x)^(7/2)) + (2*(3*A + 7*B)*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(35*d*cos(c + d*x)^(5/2))],
[(a + a*cos(c + d*x))^(3/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/cos(c + d*x)^(11/2), x, 5, (2*a^2*(52*A + 72*B + 63*C)*sin(c + d*x))/(315*d*cos(c + d*x)^(5/2)*sqrt(a + a*cos(c + d*x))) + (2*a^2*(136*A + 156*B + 189*C)*sin(c + d*x))/(315*d*cos(c + d*x)^(3/2)*sqrt(a + a*cos(c + d*x))) + (4*a^2*(136*A + 156*B + 189*C)*sin(c + d*x))/(315*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))) + (2*a*(A + 3*B)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(21*d*cos(c + d*x)^(7/2)) + (2*A*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(9*d*cos(c + d*x)^(9/2))],
[(a + a*cos(c + d*x))^(3/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/cos(c + d*x)^(13/2), x, 6, (2*a^2*(84*A + 110*B + 99*C)*sin(c + d*x))/(693*d*cos(c + d*x)^(7/2)*sqrt(a + a*cos(c + d*x))) + (2*a^2*(336*A + 374*B + 429*C)*sin(c + d*x))/(1155*d*cos(c + d*x)^(5/2)*sqrt(a + a*cos(c + d*x))) + (8*a^2*(336*A + 374*B + 429*C)*sin(c + d*x))/(3465*d*cos(c + d*x)^(3/2)*sqrt(a + a*cos(c + d*x))) + (16*a^2*(336*A + 374*B + 429*C)*sin(c + d*x))/(3465*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))) + (2*a*(3*A + 11*B)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(99*d*cos(c + d*x)^(9/2)) + (2*A*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(11*d*cos(c + d*x)^(11/2))],


[(a + a*cos(c + d*x))^(5/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*cos(c + d*x)^(3/2), x, 7, (a^(5/2)*(1304*A + 1132*B + 1015*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(512*d) + (a^3*(1304*A + 1132*B + 1015*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(512*d*sqrt(a + a*cos(c + d*x))) + (a^3*(1304*A + 1132*B + 1015*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(768*d*sqrt(a + a*cos(c + d*x))) + (a^3*(680*A + 628*B + 545*C)*cos(c + d*x)^(5/2)*sin(c + d*x))/(960*d*sqrt(a + a*cos(c + d*x))) + (a^2*(120*A + 156*B + 115*C)*cos(c + d*x)^(5/2)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(480*d) + (a*(12*B + 5*C)*cos(c + d*x)^(5/2)*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(60*d) + (C*cos(c + d*x)^(5/2)*(a + a*cos(c + d*x))^(5/2)*sin(c + d*x))/(6*d)],
[(a + a*cos(c + d*x))^(5/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*cos(c + d*x)^(1/2), x, 6, (a^(5/2)*(400*A + 326*B + 283*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(128*d) + (a^3*(400*A + 326*B + 283*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(128*d*sqrt(a + a*cos(c + d*x))) + (a^3*(1040*A + 950*B + 787*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(960*d*sqrt(a + a*cos(c + d*x))) + (a^2*(80*A + 110*B + 79*C)*cos(c + d*x)^(3/2)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(240*d) + (a*(2*B + C)*cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(8*d) + (C*cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))^(5/2)*sin(c + d*x))/(5*d)],
[(a + a*cos(c + d*x))^(5/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/cos(c + d*x)^(1/2), x, 5, (a^(5/2)*(304*A + 200*B + 163*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(64*d) + (a^3*(432*A + 392*B + 299*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(192*d*sqrt(a + a*cos(c + d*x))) + (a^2*(16*A + 24*B + 17*C)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(32*d) + (a*(8*B + 5*C)*sqrt(cos(c + d*x))*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(24*d) + (C*sqrt(cos(c + d*x))*(a + a*cos(c + d*x))^(5/2)*sin(c + d*x))/(4*d)],
[(a + a*cos(c + d*x))^(5/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/cos(c + d*x)^(3/2), x, 5, (a^(5/2)*(40*A + 38*B + 25*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(8*d) - (a^3*(24*A - 54*B - 49*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(24*d*sqrt(a + a*cos(c + d*x))) - (a^2*(8*A - 2*B - 3*C)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(4*d) - (a*(6*A - C)*sqrt(cos(c + d*x))*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(3*d) + (2*A*(a + a*cos(c + d*x))^(5/2)*sin(c + d*x))/(d*sqrt(cos(c + d*x)))],
[(a + a*cos(c + d*x))^(5/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/cos(c + d*x)^(5/2), x, 5, (a^(5/2)*(8*A + 20*B + 19*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(4*d) - (a^3*(56*A + 12*B - 27*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(12*d*sqrt(a + a*cos(c + d*x))) - (a^2*(8*A + 4*B - C)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(2*d) + (2*a*(5*A + 3*B)*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))) + (2*A*(a + a*cos(c + d*x))^(5/2)*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2))],
[(a + a*cos(c + d*x))^(5/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/cos(c + d*x)^(7/2), x, 5, (a^(5/2)*(2*B + 5*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/d - (a^3*(64*A + 70*B + 15*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(15*d*sqrt(a + a*cos(c + d*x))) + (2*a^2*(8*A + 10*B + 5*C)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(5*d*sqrt(cos(c + d*x))) + (2*a*(A + B)*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2)) + (2*A*(a + a*cos(c + d*x))^(5/2)*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2))],
[(a + a*cos(c + d*x))^(5/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/cos(c + d*x)^(9/2), x, 5, (2*a^(5/2)*C*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/d + (2*a^3*(160*A + 224*B + 245*C)*sin(c + d*x))/(105*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))) + (2*a^2*(40*A + 56*B + 35*C)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(105*d*cos(c + d*x)^(3/2)) + (2*a*(5*A + 7*B)*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(35*d*cos(c + d*x)^(5/2)) + (2*A*(a + a*cos(c + d*x))^(5/2)*sin(c + d*x))/(7*d*cos(c + d*x)^(7/2))],
[(a + a*cos(c + d*x))^(5/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/cos(c + d*x)^(11/2), x, 5, (64*a^3*(13*A + 15*B + 21*C)*sin(c + d*x))/(315*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))) + (16*a^2*(13*A + 15*B + 21*C)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(315*d*cos(c + d*x)^(3/2)) + (2*a*(13*A + 15*B + 21*C)*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(105*d*cos(c + d*x)^(5/2)) + (2*A*(a + a*cos(c + d*x))^(5/2)*sin(c + d*x))/(9*d*cos(c + d*x)^(9/2)) + (2*(5*A + 9*B)*(a + a*cos(c + d*x))^(5/2)*sin(c + d*x))/(63*d*cos(c + d*x)^(7/2))],
[(a + a*cos(c + d*x))^(5/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/cos(c + d*x)^(13/2), x, 6, (2*a^3*(1160*A + 1364*B + 1485*C)*sin(c + d*x))/(3465*d*cos(c + d*x)^(5/2)*sqrt(a + a*cos(c + d*x))) + (2*a^3*(2840*A + 3212*B + 3795*C)*sin(c + d*x))/(3465*d*cos(c + d*x)^(3/2)*sqrt(a + a*cos(c + d*x))) + (4*a^3*(2840*A + 3212*B + 3795*C)*sin(c + d*x))/(3465*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))) + (2*a^2*(32*A + 44*B + 33*C)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(231*d*cos(c + d*x)^(7/2)) + (2*a*(5*A + 11*B)*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(99*d*cos(c + d*x)^(9/2)) + (2*A*(a + a*cos(c + d*x))^(5/2)*sin(c + d*x))/(11*d*cos(c + d*x)^(11/2))],
[(a + a*cos(c + d*x))^(5/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/cos(c + d*x)^(15/2), x, 7, (2*a^3*(2224*A + 2522*B + 2717*C)*sin(c + d*x))/(9009*d*cos(c + d*x)^(7/2)*sqrt(a + a*cos(c + d*x))) + (2*a^3*(8368*A + 9230*B + 10439*C)*sin(c + d*x))/(15015*d*cos(c + d*x)^(5/2)*sqrt(a + a*cos(c + d*x))) + (8*a^3*(8368*A + 9230*B + 10439*C)*sin(c + d*x))/(45045*d*cos(c + d*x)^(3/2)*sqrt(a + a*cos(c + d*x))) + (16*a^3*(8368*A + 9230*B + 10439*C)*sin(c + d*x))/(45045*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))) + (2*a^2*(136*A + 182*B + 143*C)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(1287*d*cos(c + d*x)^(9/2)) + (2*a*(5*A + 13*B)*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(143*d*cos(c + d*x)^(11/2)) + (2*A*(a + a*cos(c + d*x))^(5/2)*sin(c + d*x))/(13*d*cos(c + d*x)^(13/2))],


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


[(cos(c + d*x)^(3/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2))/sqrt(a + a*cos(c + d*x)), x, 6, -(((8*A - 14*B + 9*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(8*sqrt(a)*d)) + (sqrt(2)*(A - B + C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(sqrt(a)*d) + ((8*A - 2*B + 7*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(8*d*sqrt(a + a*cos(c + d*x))) + ((6*B - C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(12*d*sqrt(a + a*cos(c + d*x))) + (C*cos(c + d*x)^(5/2)*sin(c + d*x))/(3*d*sqrt(a + a*cos(c + d*x)))],
[(cos(c + d*x)^(1/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2))/sqrt(a + a*cos(c + d*x)), x, 5, ((8*A - 4*B + 7*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(4*sqrt(a)*d) - (sqrt(2)*(A - B + C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(sqrt(a)*d) + ((4*B - C)*sqrt(cos(c + d*x))*sin(c + d*x))/(4*d*sqrt(a + a*cos(c + d*x))) + (C*cos(c + d*x)^(3/2)*sin(c + d*x))/(2*d*sqrt(a + a*cos(c + d*x)))],
[(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(cos(c + d*x)^(1/2)*sqrt(a + a*cos(c + d*x))), x, 4, ((2*B - C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(sqrt(a)*d) + (sqrt(2)*(A - B + C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(sqrt(a)*d) + (C*sqrt(cos(c + d*x))*sin(c + d*x))/(d*sqrt(a + a*cos(c + d*x)))],
[(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(cos(c + d*x)^(3/2)*sqrt(a + a*cos(c + d*x))), x, 4, (2*C*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(sqrt(a)*d) - (sqrt(2)*(A - B + C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(sqrt(a)*d) + (2*A*sin(c + d*x))/(d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))],
[(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(cos(c + d*x)^(5/2)*sqrt(a + a*cos(c + d*x))), x, 3, (sqrt(2)*(A - B + C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(sqrt(a)*d) + (2*A*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2)*sqrt(a + a*cos(c + d*x))) - (2*(A - 3*B)*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))],
[(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(cos(c + d*x)^(7/2)*sqrt(a + a*cos(c + d*x))), x, 4, -((sqrt(2)*(A - B + C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(sqrt(a)*d)) + (2*A*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2)*sqrt(a + a*cos(c + d*x))) - (2*(A - 5*B)*sin(c + d*x))/(15*d*cos(c + d*x)^(3/2)*sqrt(a + a*cos(c + d*x))) + (2*(13*A - 5*B + 15*C)*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))],
[(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(cos(c + d*x)^(9/2)*sqrt(a + a*cos(c + d*x))), x, 5, (sqrt(2)*(A - B + C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(sqrt(a)*d) + (2*A*sin(c + d*x))/(7*d*cos(c + d*x)^(7/2)*sqrt(a + a*cos(c + d*x))) - (2*(A - 7*B)*sin(c + d*x))/(35*d*cos(c + d*x)^(5/2)*sqrt(a + a*cos(c + d*x))) + (2*(31*A - 7*B + 35*C)*sin(c + d*x))/(105*d*cos(c + d*x)^(3/2)*sqrt(a + a*cos(c + d*x))) - (2*(43*A - 91*B + 35*C)*sin(c + d*x))/(105*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))],

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


[(cos(c + d*x)^(3/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2))/(a + a*cos(c + d*x))^(3/2), x, 6, ((8*A - 12*B + 19*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(4*a^(3/2)*d) - ((5*A - 9*B + 13*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d) - ((A - B + C)*cos(c + d*x)^(5/2)*sin(c + d*x))/(2*d*(a + a*cos(c + d*x))^(3/2)) - ((2*A - 6*B + 7*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(4*a*d*sqrt(a + a*cos(c + d*x))) + ((A - B + 2*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(2*a*d*sqrt(a + a*cos(c + d*x)))],
[(cos(c + d*x)^(1/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2))/(a + a*cos(c + d*x))^(3/2), x, 5, ((2*B - 3*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(a^(3/2)*d) + ((A - 5*B + 9*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d) - ((A - B + C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(2*d*(a + a*cos(c + d*x))^(3/2)) + ((A - B + 3*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(2*a*d*sqrt(a + a*cos(c + d*x)))],
[(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(cos(c + d*x)^(1/2)*(a + a*cos(c + d*x))^(3/2)), x, 4, (2*C*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(a^(3/2)*d) + ((3*A + B - 5*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d) - ((A - B + C)*sqrt(cos(c + d*x))*sin(c + d*x))/(2*d*(a + a*cos(c + d*x))^(3/2))],
[(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))^(3/2)), x, 3, -(((7*A - 3*B - C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d)) - ((A - B + C)*sin(c + d*x))/(2*d*sqrt(cos(c + d*x))*(a + a*cos(c + d*x))^(3/2)) + ((5*A - B + C)*sin(c + d*x))/(2*a*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))],
[(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(cos(c + d*x)^(5/2)*(a + a*cos(c + d*x))^(3/2)), x, 4, ((11*A - 7*B + 3*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d) - ((A - B + C)*sin(c + d*x))/(2*d*cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))^(3/2)) + ((7*A - 3*B + 3*C)*sin(c + d*x))/(6*a*d*cos(c + d*x)^(3/2)*sqrt(a + a*cos(c + d*x))) - ((19*A - 15*B + 3*C)*sin(c + d*x))/(6*a*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))],
[(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(cos(c + d*x)^(7/2)*(a + a*cos(c + d*x))^(3/2)), x, 5, -(((15*A - 11*B + 7*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d)) - ((A - B + C)*sin(c + d*x))/(2*d*cos(c + d*x)^(5/2)*(a + a*cos(c + d*x))^(3/2)) + ((9*A - 5*B + 5*C)*sin(c + d*x))/(10*a*d*cos(c + d*x)^(5/2)*sqrt(a + a*cos(c + d*x))) - ((39*A - 35*B + 15*C)*sin(c + d*x))/(30*a*d*cos(c + d*x)^(3/2)*sqrt(a + a*cos(c + d*x))) + ((147*A - 95*B + 75*C)*sin(c + d*x))/(30*a*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))],


[(cos(c + d*x)^(3/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2))/(a + a*cos(c + d*x))^(5/2), x, 6, ((2*B - 5*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(a^(5/2)*d) + ((3*A - 43*B + 115*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d) - ((A - B + C)*cos(c + d*x)^(5/2)*sin(c + d*x))/(4*d*(a + a*cos(c + d*x))^(5/2)) + ((A + 7*B - 15*C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2)) + ((3*A - 11*B + 35*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(16*a^2*d*sqrt(a + a*cos(c + d*x)))],
[(cos(c + d*x)^(1/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2))/(a + a*cos(c + d*x))^(5/2), x, 5, (2*C*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(a^(5/2)*d) + ((5*A + 3*B - 43*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d) - ((A - B + C)*cos(c + d*x)^(3/2)*sin(c + d*x))/(4*d*(a + a*cos(c + d*x))^(5/2)) + ((5*A + 3*B - 11*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2))],
[(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(cos(c + d*x)^(1/2)*(a + a*cos(c + d*x))^(5/2)), x, 3, ((19*A + 5*B + 3*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d) - ((A - B + C)*sqrt(cos(c + d*x))*sin(c + d*x))/(4*d*(a + a*cos(c + d*x))^(5/2)) - ((9*A - B - 7*C)*sqrt(cos(c + d*x))*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2))],
[(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))^(5/2)), x, 4, -(((75*A - 19*B - 5*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d)) - ((A - B + C)*sin(c + d*x))/(4*d*sqrt(cos(c + d*x))*(a + a*cos(c + d*x))^(5/2)) - ((13*A - 5*B - 3*C)*sin(c + d*x))/(16*a*d*sqrt(cos(c + d*x))*(a + a*cos(c + d*x))^(3/2)) + ((49*A - 9*B + C)*sin(c + d*x))/(16*a^2*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))],
[(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(cos(c + d*x)^(5/2)*(a + a*cos(c + d*x))^(5/2)), x, 5, ((163*A - 75*B + 19*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d) - ((A - B + C)*sin(c + d*x))/(4*d*cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))^(5/2)) - ((17*A - 9*B + C)*sin(c + d*x))/(16*a*d*cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))^(3/2)) + ((95*A - 39*B + 15*C)*sin(c + d*x))/(48*a^2*d*cos(c + d*x)^(3/2)*sqrt(a + a*cos(c + d*x))) - ((299*A - 147*B + 27*C)*sin(c + d*x))/(48*a^2*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))],


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


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


[(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^(2/3), x, 3, (1/80)*a*(40*A + 16*B + 19*C)*Int(1/(a + a*cos(c + d*x))^(1/3), x) + (3*a*(40*A + 16*B + 19*C)*sin(c + d*x))/(80*d*(a + a*cos(c + d*x))^(1/3)) + (3*(8*B - 3*C)*(a + a*cos(c + d*x))^(2/3)*sin(c + d*x))/(40*d) + (3*C*(a + a*cos(c + d*x))^(5/3)*sin(c + d*x))/(8*a*d)],
[(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^(1/3), x, 3, (-(1/28))*a*(28*A + 7*B + 13*C)*Int(1/(a + a*sin(c + Pi/2 + d*x))^(2/3), x) + (3*a*(28*A + 7*B + 13*C)*sin(c + d*x))/(28*d*(a + a*cos(c + d*x))^(2/3)) + (3*(7*B - 3*C)*(a + a*cos(c + d*x))^(1/3)*sin(c + d*x))/(28*d) + (3*C*(a + a*cos(c + d*x))^(4/3)*sin(c + d*x))/(7*a*d)],


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


[(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^(1/3), x, 2, (1/10)*(10*A - 5*B + 7*C)*Int(1/(a + a*cos(c + d*x))^(1/3), x) + (3*(5*B - 3*C)*sin(c + d*x))/(10*d*(a + a*cos(c + d*x))^(1/3)) + (3*C*(a + a*cos(c + d*x))^(2/3)*sin(c + d*x))/(5*a*d)],
[(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(a + a*cos(c + d*x))^(2/3), x, 2, (1/4)*(4*A - 8*B + 7*C)*Int(1/(a + a*sin(c + Pi/2 + d*x))^(2/3), x) + (3*(4*B - 3*C)*sin(c + d*x))/(4*d*(a + a*cos(c + d*x))^(2/3)) + (3*C*(a + a*cos(c + d*x))^(1/3)*sin(c + d*x))/(4*a*d)],


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


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


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


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


[(a + a*cos(c + d*x))^2*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(11/2), x, 9, -((4*a^2*(8*A + 9*B + 12*C)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(15*d)) + (4*a^2*(5*A + 6*B + 7*C)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) + (4*a^2*(8*A + 9*B + 12*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(15*d) + (4*a^2*(5*A + 6*B + 7*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(21*d) + (2*a^2*(19*A + 27*B + 21*C)*sec(c + d*x)^(5/2)*sin(c + d*x))/(105*d) + (2*a*(4*A + 9*B)*(a + a*cos(c + d*x))*sec(c + d*x)^(7/2)*sin(c + d*x))/(63*d) + (2*A*(a + a*cos(c + d*x))^2*sec(c + d*x)^(9/2)*sin(c + d*x))/(9*d)],
[(a + a*cos(c + d*x))^2*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(9/2), x, 8, -((4*a^2*(3*A + 4*B + 5*C)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d)) + (4*a^2*(6*A + 7*B + 14*C)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) + (4*a^2*(3*A + 4*B + 5*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d) + (2*a^2*(33*A + 49*B + 35*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(105*d) + (2*a*(4*A + 7*B)*(a + a*cos(c + d*x))*sec(c + d*x)^(5/2)*sin(c + d*x))/(35*d) + (2*A*(a + a*cos(c + d*x))^2*sec(c + d*x)^(7/2)*sin(c + d*x))/(7*d)],
[(a + a*cos(c + d*x))^2*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(7/2), x, 7, -((4*a^2*(4*A + 5*B)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d)) + (4*a^2*(A + 2*B + 3*C)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) + (2*a^2*(17*A + 25*B + 15*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(15*d) + (2*a*(4*A + 5*B)*(a + a*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(15*d) + (2*A*(a + a*cos(c + d*x))^2*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d)],
[(a + a*cos(c + d*x))^2*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(5/2), x, 7, (-4*a^2*(A - C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/d + (4*a^2*(2*A + 3*B + 2*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*d) - (2*a^2*(5*A + 3*B - C)*sin(c + d*x))/(3*d*sqrt(sec(c + d*x))) + (2*a*(4*A + 3*B)*(a + a*cos(c + d*x))*sqrt(sec(c + d*x))*sin(c + d*x))/(3*d) + (2*A*(a + a*cos(c + d*x))^2*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d)],
[(a + a*cos(c + d*x))^2*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(3/2), x, 7, (4*a^2*(5*B + 4*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*d) + (4*a^2*(3*A + 2*B + C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*d) - (2*a^2*(15*A - 5*B - 7*C)*sin(c + d*x))/(15*d*sqrt(sec(c + d*x))) - (2*a*(5*A - C)*(a + a*cos(c + d*x))*sin(c + d*x))/(5*d*sqrt(sec(c + d*x))) + (2*A*(a + a*cos(c + d*x))^2*sqrt(sec(c + d*x))*sin(c + d*x))/d],
[(a + a*cos(c + d*x))^2*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sqrt(sec(c + d*x)), x, 7, (4*a^2*(5*A + 4*B + 3*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*d) + (4*a^2*(14*A + 7*B + 6*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(21*d) + (2*a^2*(35*A + 49*B + 33*C)*sin(c + d*x))/(105*d*sqrt(sec(c + d*x))) + (2*a*(7*B + 4*C)*(a + a*cos(c + d*x))*sin(c + d*x))/(35*d*sqrt(sec(c + d*x))) + (2*C*(a + a*cos(c + d*x))^2*sin(c + d*x))/(7*d*sqrt(sec(c + d*x)))],
[(a + a*cos(c + d*x))^2*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/sqrt(sec(c + d*x)), x, 8, (4*a^2*(12*A + 9*B + 8*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(15*d) + (4*a^2*(7*A + 6*B + 5*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(21*d) + (2*a^2*(21*A + 27*B + 19*C)*sin(c + d*x))/(105*d*sec(c + d*x)^(3/2)) + (2*a*(9*B + 4*C)*(a + a*cos(c + d*x))*sin(c + d*x))/(63*d*sec(c + d*x)^(3/2)) + (2*C*(a + a*cos(c + d*x))^2*sin(c + d*x))/(9*d*sec(c + d*x)^(3/2)) + (4*a^2*(7*A + 6*B + 5*C)*sin(c + d*x))/(21*d*sqrt(sec(c + d*x)))],
[(a + a*cos(c + d*x))^2*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/sec(c + d*x)^(3/2), x, 9, (4*a^2*(9*A + 8*B + 7*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(15*d) + (4*a^2*(66*A + 55*B + 50*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(231*d) + (2*a^2*(99*A + 121*B + 89*C)*sin(c + d*x))/(693*d*sec(c + d*x)^(5/2)) + (2*a*(11*B + 4*C)*(a + a*cos(c + d*x))*sin(c + d*x))/(99*d*sec(c + d*x)^(5/2)) + (2*C*(a + a*cos(c + d*x))^2*sin(c + d*x))/(11*d*sec(c + d*x)^(5/2)) + (4*a^2*(9*A + 8*B + 7*C)*sin(c + d*x))/(45*d*sec(c + d*x)^(3/2)) + (4*a^2*(66*A + 55*B + 50*C)*sin(c + d*x))/(231*d*sqrt(sec(c + d*x)))],


[(a + a*cos(c + d*x))^3*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(13/2), x, 10, -((4*a^3*(15*A + 17*B + 21*C)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(15*d)) + (4*a^3*(105*A + 121*B + 143*C)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(231*d) + (4*a^3*(15*A + 17*B + 21*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(15*d) + (4*a^3*(105*A + 121*B + 143*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(231*d) + (4*a^3*(210*A + 253*B + 264*C)*sec(c + d*x)^(5/2)*sin(c + d*x))/(1155*d) + (2*a^2*(105*A + 143*B + 99*C)*(a + a*cos(c + d*x))*sec(c + d*x)^(7/2)*sin(c + d*x))/(693*d) + (2*a*(6*A + 11*B)*(a + a*cos(c + d*x))^2*sec(c + d*x)^(9/2)*sin(c + d*x))/(99*d) + (2*A*(a + a*cos(c + d*x))^3*sec(c + d*x)^(11/2)*sin(c + d*x))/(11*d)],
[(a + a*cos(c + d*x))^3*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(11/2), x, 9, -((4*a^3*(17*A + 21*B + 27*C)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(15*d)) + (4*a^3*(11*A + 13*B + 21*C)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) + (4*a^3*(17*A + 21*B + 27*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(15*d) + (4*a^3*(32*A + 41*B + 42*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(105*d) + (2*a^2*(73*A + 99*B + 63*C)*(a + a*cos(c + d*x))*sec(c + d*x)^(5/2)*sin(c + d*x))/(315*d) + (2*a*(2*A + 3*B)*(a + a*cos(c + d*x))^2*sec(c + d*x)^(7/2)*sin(c + d*x))/(21*d) + (2*A*(a + a*cos(c + d*x))^3*sec(c + d*x)^(9/2)*sin(c + d*x))/(9*d)],
[(a + a*cos(c + d*x))^3*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(9/2), x, 8, -((4*a^3*(7*A + 9*B + 5*C)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d)) + (4*a^3*(13*A + 21*B + 35*C)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) + (4*a^3*(106*A + 147*B + 140*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(105*d) + (2*a^2*(7*A + 9*B + 5*C)*(a + a*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(15*d) + (2*a*(6*A + 7*B)*(a + a*cos(c + d*x))^2*sec(c + d*x)^(5/2)*sin(c + d*x))/(35*d) + (2*A*(a + a*cos(c + d*x))^3*sec(c + d*x)^(7/2)*sin(c + d*x))/(7*d)],
[(a + a*cos(c + d*x))^3*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(7/2), x, 8, -((4*a^3*(9*A + 5*B - 5*C)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d)) + (4*a^3*(3*A + 5*B + 5*C)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) - (4*a^3*(21*A + 20*B + 5*C)*sin(c + d*x))/(15*d*sqrt(sec(c + d*x))) + (2*a^2*(33*A + 35*B + 15*C)*(a + a*cos(c + d*x))*sqrt(sec(c + d*x))*sin(c + d*x))/(15*d) + (2*a*(6*A + 5*B)*(a + a*cos(c + d*x))^2*sec(c + d*x)^(3/2)*sin(c + d*x))/(15*d) + (2*A*(a + a*cos(c + d*x))^3*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d)],
[(a + a*cos(c + d*x))^3*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(5/2), x, 8, (-4*a^3*(5*A - 5*B - 9*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*d) + (4*a^3*(5*A + 5*B + 3*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*d) - (4*a^3*(20*A + 5*B - 6*C)*sin(c + d*x))/(15*d*sqrt(sec(c + d*x))) - (2*a^2*(35*A + 15*B - 3*C)*(a + a*cos(c + d*x))*sin(c + d*x))/(15*d*sqrt(sec(c + d*x))) + (2*a*(2*A + B)*(a + a*cos(c + d*x))^2*sqrt(sec(c + d*x))*sin(c + d*x))/d + (2*A*(a + a*cos(c + d*x))^3*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d)],
[(a + a*cos(c + d*x))^3*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(3/2), x, 8, (4*a^3*(5*A + 9*B + 7*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*d) + (4*a^3*(35*A + 21*B + 13*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(21*d) - (4*a^3*(35*A - 42*B - 41*C)*sin(c + d*x))/(105*d*sqrt(sec(c + d*x))) - (2*a^2*(35*A - 7*B - 11*C)*(a + a*cos(c + d*x))*sin(c + d*x))/(35*d*sqrt(sec(c + d*x))) - (2*a*(7*A - C)*(a + a*cos(c + d*x))^2*sin(c + d*x))/(7*d*sqrt(sec(c + d*x))) + (2*A*(a + a*cos(c + d*x))^3*sqrt(sec(c + d*x))*sin(c + d*x))/d],
[(a + a*cos(c + d*x))^3*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sqrt(sec(c + d*x)), x, 8, (4*a^3*(27*A + 21*B + 17*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(15*d) + (4*a^3*(21*A + 13*B + 11*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(21*d) + (4*a^3*(42*A + 41*B + 32*C)*sin(c + d*x))/(105*d*sqrt(sec(c + d*x))) + (2*a^2*(63*A + 99*B + 73*C)*(a + a*cos(c + d*x))*sin(c + d*x))/(315*d*sqrt(sec(c + d*x))) + (2*a*(3*B + 2*C)*(a + a*cos(c + d*x))^2*sin(c + d*x))/(21*d*sqrt(sec(c + d*x))) + (2*C*(a + a*cos(c + d*x))^3*sin(c + d*x))/(9*d*sqrt(sec(c + d*x)))],
[(a + a*cos(c + d*x))^3*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/sqrt(sec(c + d*x)), x, 9, (4*a^3*(21*A + 17*B + 15*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(15*d) + (4*a^3*(143*A + 121*B + 105*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(231*d) + (4*a^3*(264*A + 253*B + 210*C)*sin(c + d*x))/(1155*d*sec(c + d*x)^(3/2)) + (2*a^2*(99*A + 143*B + 105*C)*(a + a*cos(c + d*x))*sin(c + d*x))/(693*d*sec(c + d*x)^(3/2)) + (2*a*(11*B + 6*C)*(a + a*cos(c + d*x))^2*sin(c + d*x))/(99*d*sec(c + d*x)^(3/2)) + (2*C*(a + a*cos(c + d*x))^3*sin(c + d*x))/(11*d*sec(c + d*x)^(3/2)) + (4*a^3*(143*A + 121*B + 105*C)*sin(c + d*x))/(231*d*sqrt(sec(c + d*x)))],
[(a + a*cos(c + d*x))^3*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/sec(c + d*x)^(3/2), x, 10, (4*a^3*(221*A + 195*B + 175*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(195*d) + (4*a^3*(121*A + 105*B + 95*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(231*d) + (20*a^3*(286*A + 273*B + 236*C)*sin(c + d*x))/(9009*d*sec(c + d*x)^(5/2)) + (2*a^2*(143*A + 195*B + 145*C)*(a + a*cos(c + d*x))*sin(c + d*x))/(1287*d*sec(c + d*x)^(5/2)) + (2*a*(13*B + 6*C)*(a + a*cos(c + d*x))^2*sin(c + d*x))/(143*d*sec(c + d*x)^(5/2)) + (2*C*(a + a*cos(c + d*x))^3*sin(c + d*x))/(13*d*sec(c + d*x)^(5/2)) + (4*a^3*(221*A + 195*B + 175*C)*sin(c + d*x))/(585*d*sec(c + d*x)^(3/2)) + (4*a^3*(121*A + 105*B + 95*C)*sin(c + d*x))/(231*d*sqrt(sec(c + d*x)))],


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


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


[(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(5/2)/(a + a*cos(c + d*x))^2, x, 8, ((7*A - 4*B + C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(a^2*d) + ((10*A - 5*B + 2*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*a^2*d) - ((7*A - 4*B + C)*sqrt(sec(c + d*x))*sin(c + d*x))/(a^2*d) + ((10*A - 5*B + 2*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*a^2*d) - ((A - B + C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d*(a + a*cos(c + d*x))^2) - ((7*A - 4*B + C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*a*d*(a + a*cos(c + d*x)))],
[(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(3/2)/(a + a*cos(c + d*x))^2, x, 7, -(((4*A - B)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(a^2*d)) - ((5*A - 2*B - C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*a^2*d) + ((4*A - B)*sqrt(sec(c + d*x))*sin(c + d*x))/(a^2*d) - ((A - B + C)*sqrt(sec(c + d*x))*sin(c + d*x))/(3*d*(a + a*cos(c + d*x))^2) - ((5*A - 2*B - C)*sqrt(sec(c + d*x))*sin(c + d*x))/(3*a*d*(a + a*cos(c + d*x)))],
[(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sqrt(sec(c + d*x))/(a + a*cos(c + d*x))^2, x, 6, ((A - C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(a^2*d) + ((2*A + B + 2*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*a^2*d) - ((A - B + C)*sin(c + d*x))/(3*d*(a + a*cos(c + d*x))^2*sqrt(sec(c + d*x))) - ((A - C)*sin(c + d*x))/(a*d*(a + a*cos(c + d*x))*sqrt(sec(c + d*x)))],
[(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/((a + a*cos(c + d*x))^2*sqrt(sec(c + d*x))), x, 6, -(((B - 4*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(a^2*d)) + ((A + 2*B - 5*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*a^2*d) - ((A - B + C)*sin(c + d*x))/(3*d*(a + a*cos(c + d*x))^2*sec(c + d*x)^(3/2)) + ((A + 2*B - 5*C)*sin(c + d*x))/(3*a*d*(a + a*cos(c + d*x))*sqrt(sec(c + d*x)))],
[(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/((a + a*cos(c + d*x))^2*sec(c + d*x)^(3/2)), x, 7, -(((A - 4*B + 7*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(a^2*d)) + ((2*A - 5*B + 10*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*a^2*d) - ((A - B + C)*sin(c + d*x))/(3*d*(a + a*cos(c + d*x))^2*sec(c + d*x)^(5/2)) - ((A - 4*B + 7*C)*sin(c + d*x))/(3*a*d*(a + a*cos(c + d*x))*sec(c + d*x)^(3/2)) + ((2*A - 5*B + 10*C)*sin(c + d*x))/(3*a^2*d*sqrt(sec(c + d*x)))],
[(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/((a + a*cos(c + d*x))^2*sec(c + d*x)^(5/2)), x, 8, ((20*A - 35*B + 56*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*a^2*d) - (5*(A - 2*B + 3*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*a^2*d) - ((A - B + C)*sin(c + d*x))/(3*d*(a + a*cos(c + d*x))^2*sec(c + d*x)^(7/2)) - ((A - 2*B + 3*C)*sin(c + d*x))/(a*d*(a + a*cos(c + d*x))*sec(c + d*x)^(5/2)) + ((20*A - 35*B + 56*C)*sin(c + d*x))/(15*a^2*d*sec(c + d*x)^(3/2)) - (5*(A - 2*B + 3*C)*sin(c + d*x))/(3*a^2*d*sqrt(sec(c + d*x)))],


[(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(5/2)/(a + a*cos(c + d*x))^3, x, 9, ((119*A - 49*B + 9*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(10*a^3*d) + ((33*A - 13*B + 3*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(6*a^3*d) - ((119*A - 49*B + 9*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(10*a^3*d) + ((33*A - 13*B + 3*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(6*a^3*d) - ((A - B + C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3) - ((2*A - B)*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*a*d*(a + a*cos(c + d*x))^2) - ((119*A - 49*B + 9*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(30*a^2*d*(a + a*cos(c + d*x)))],
[(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(3/2)/(a + a*cos(c + d*x))^3, x, 8, -((49*A - 9*B - C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(10*a^3*d) - ((13*A - 3*B - C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(6*a^3*d) + ((49*A - 9*B - C)*sqrt(sec(c + d*x))*sin(c + d*x))/(10*a^3*d) - ((A - B + C)*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3) - ((8*A - 3*B - 2*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(15*a*d*(a + a*cos(c + d*x))^2) - ((13*A - 3*B - C)*sqrt(sec(c + d*x))*sin(c + d*x))/(6*a^2*d*(a + a*cos(c + d*x)))],
[(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sqrt(sec(c + d*x))/(a + a*cos(c + d*x))^3, x, 7, ((9*A + B - C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(10*a^3*d) + ((3*A + B + C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(6*a^3*d) - ((A - B + C)*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3*sqrt(sec(c + d*x))) - ((6*A - B - 4*C)*sin(c + d*x))/(15*a*d*(a + a*cos(c + d*x))^2*sqrt(sec(c + d*x))) - ((9*A + B - C)*sin(c + d*x))/(10*a^2*d*(a + a*cos(c + d*x))*sqrt(sec(c + d*x)))],
[(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/((a + a*cos(c + d*x))^3*sqrt(sec(c + d*x))), x, 7, ((A - B - 9*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(10*a^3*d) + ((A + B + 3*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(6*a^3*d) - ((A - B + C)*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3*sec(c + d*x)^(3/2)) + ((4*A + B - 6*C)*sin(c + d*x))/(15*a*d*(a + a*cos(c + d*x))^2*sqrt(sec(c + d*x))) - ((A - B - 9*C)*sin(c + d*x))/(10*a^2*d*(a + a*cos(c + d*x))*sqrt(sec(c + d*x)))],
[(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/((a + a*cos(c + d*x))^3*sec(c + d*x)^(3/2)), x, 7, -((A + 9*B - 49*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(10*a^3*d) + ((A + 3*B - 13*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(6*a^3*d) - ((A - B + C)*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3*sec(c + d*x)^(5/2)) + ((2*A + 3*B - 8*C)*sin(c + d*x))/(15*a*d*(a + a*cos(c + d*x))^2*sec(c + d*x)^(3/2)) + ((A + 3*B - 13*C)*sin(c + d*x))/(6*a^2*d*(a + a*cos(c + d*x))*sqrt(sec(c + d*x)))],
[(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/((a + a*cos(c + d*x))^3*sec(c + d*x)^(5/2)), x, 8, -((9*A - 49*B + 119*C)*sqrt(cos(c + d*x))*EllipticE((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(10*a^3*d) + ((3*A - 13*B + 33*C)*sqrt(cos(c + d*x))*EllipticF((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(6*a^3*d) - ((A - B + C)*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3*sec(c + d*x)^(7/2)) + ((B - 2*C)*sin(c + d*x))/(3*a*d*(a + a*cos(c + d*x))^2*sec(c + d*x)^(5/2)) - ((9*A - 49*B + 119*C)*sin(c + d*x))/(30*a^2*d*(a + a*cos(c + d*x))*sec(c + d*x)^(3/2)) + ((3*A - 13*B + 33*C)*sin(c + d*x))/(6*a^3*d*sqrt(sec(c + d*x)))],
[(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/((a + a*cos(c + d*x))^3*sec(c + d*x)^(7/2)), x, 9, (7*(7*A - 17*B + 33*C)*sqrt(cos(c + d*x))*EllipticE((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(10*a^3*d) - ((13*A - 33*B + 63*C)*sqrt(cos(c + d*x))*EllipticF((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(6*a^3*d) - ((A - B + C)*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3*sec(c + d*x)^(9/2)) - ((2*A - 7*B + 12*C)*sin(c + d*x))/(15*a*d*(a + a*cos(c + d*x))^2*sec(c + d*x)^(7/2)) - ((13*A - 33*B + 63*C)*sin(c + d*x))/(10*a^2*d*(a + a*cos(c + d*x))*sec(c + d*x)^(5/2)) + (7*(7*A - 17*B + 33*C)*sin(c + d*x))/(30*a^3*d*sec(c + d*x)^(3/2)) - ((13*A - 33*B + 63*C)*sin(c + d*x))/(6*a^3*d*sqrt(sec(c + d*x)))],


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


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


[sqrt(a + a*cos(c + d*x))*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(11/2), x, 5, (16*a*(16*A + 18*B + 21*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(315*d*sqrt(a + a*cos(c + d*x))) + (8*a*(16*A + 18*B + 21*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(315*d*sqrt(a + a*cos(c + d*x))) + (2*a*(16*A + 18*B + 21*C)*sec(c + d*x)^(5/2)*sin(c + d*x))/(105*d*sqrt(a + a*cos(c + d*x))) + (2*a*(A + 9*B)*sec(c + d*x)^(7/2)*sin(c + d*x))/(63*d*sqrt(a + a*cos(c + d*x))) + (2*A*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(9/2)*sin(c + d*x))/(9*d)],
[sqrt(a + a*cos(c + d*x))*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(9/2), x, 4, (4*a*(24*A + 28*B + 35*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(105*d*sqrt(a + a*cos(c + d*x))) + (2*a*(24*A + 28*B + 35*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(105*d*sqrt(a + a*cos(c + d*x))) + (2*a*(A + 7*B)*sec(c + d*x)^(5/2)*sin(c + d*x))/(35*d*sqrt(a + a*cos(c + d*x))) + (2*A*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(7/2)*sin(c + d*x))/(7*d)],
[sqrt(a + a*cos(c + d*x))*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(7/2), x, 3, (2*a*(8*A + 10*B + 15*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(15*d*sqrt(a + a*cos(c + d*x))) + (2*a*(A + 5*B)*sec(c + d*x)^(3/2)*sin(c + d*x))/(15*d*sqrt(a + a*cos(c + d*x))) + (2*A*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d)],
[sqrt(a + a*cos(c + d*x))*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(5/2), x, 4, (2*sqrt(a)*C*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/d + (2*a*(A + 3*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(3*d*sqrt(a + a*cos(c + d*x))) + (2*A*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d)],
[sqrt(a + a*cos(c + d*x))*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(3/2), x, 4, (sqrt(a)*(2*B + C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/d - (a*(2*A - C)*sin(c + d*x))/(d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x))) + (2*A*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x))*sin(c + d*x))/d],
[sqrt(a + a*cos(c + d*x))*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(1/2), x, 4, (sqrt(a)*(8*A + 4*B + 3*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(4*d) + (a*(4*B + C)*sin(c + d*x))/(4*d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x))) + (C*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(2*d*sqrt(sec(c + d*x)))],
[sqrt(a + a*cos(c + d*x))*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/sec(c + d*x)^(1/2), x, 5, (sqrt(a)*(8*A + 6*B + 5*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(8*d) + (a*(6*B + C)*sin(c + d*x))/(12*d*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(3/2)) + (C*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(3*d*sec(c + d*x)^(3/2)) + (a*(8*A + 6*B + 5*C)*sin(c + d*x))/(8*d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x)))],
[sqrt(a + a*cos(c + d*x))*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/sec(c + d*x)^(3/2), x, 6, (sqrt(a)*(48*A + 40*B + 35*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(64*d) + (a*(8*B + C)*sin(c + d*x))/(24*d*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(5/2)) + (C*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(4*d*sec(c + d*x)^(5/2)) + (a*(48*A + 40*B + 35*C)*sin(c + d*x))/(96*d*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(3/2)) + (a*(48*A + 40*B + 35*C)*sin(c + d*x))/(64*d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x)))],


[(a + a*cos(c + d*x))^(3/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(13/2), x, 6, (16*a^2*(336*A + 374*B + 429*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(3465*d*sqrt(a + a*cos(c + d*x))) + (8*a^2*(336*A + 374*B + 429*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(3465*d*sqrt(a + a*cos(c + d*x))) + (2*a^2*(336*A + 374*B + 429*C)*sec(c + d*x)^(5/2)*sin(c + d*x))/(1155*d*sqrt(a + a*cos(c + d*x))) + (2*a^2*(84*A + 110*B + 99*C)*sec(c + d*x)^(7/2)*sin(c + d*x))/(693*d*sqrt(a + a*cos(c + d*x))) + (2*a*(3*A + 11*B)*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(9/2)*sin(c + d*x))/(99*d) + (2*A*(a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^(11/2)*sin(c + d*x))/(11*d)],
[(a + a*cos(c + d*x))^(3/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(11/2), x, 5, (4*a^2*(136*A + 156*B + 189*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(315*d*sqrt(a + a*cos(c + d*x))) + (2*a^2*(136*A + 156*B + 189*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(315*d*sqrt(a + a*cos(c + d*x))) + (2*a^2*(52*A + 72*B + 63*C)*sec(c + d*x)^(5/2)*sin(c + d*x))/(315*d*sqrt(a + a*cos(c + d*x))) + (2*a*(A + 3*B)*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(7/2)*sin(c + d*x))/(21*d) + (2*A*(a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^(9/2)*sin(c + d*x))/(9*d)],
[(a + a*cos(c + d*x))^(3/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(9/2), x, 4, (8*a^2*(19*A + 21*B + 35*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(105*d*sqrt(a + a*cos(c + d*x))) + (2*a*(19*A + 21*B + 35*C)*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(105*d) + (2*(3*A + 7*B)*(a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^(5/2)*sin(c + d*x))/(35*d) + (2*A*(a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^(7/2)*sin(c + d*x))/(7*d)],
[(a + a*cos(c + d*x))^(3/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(7/2), x, 5, (2*a^(3/2)*C*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/d + (2*a^2*(12*A + 20*B + 15*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(15*d*sqrt(a + a*cos(c + d*x))) + (2*a*(3*A + 5*B)*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(15*d) + (2*A*(a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d)],
[(a + a*cos(c + d*x))^(3/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(5/2), x, 5, (a^(3/2)*(2*B + 3*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/d - (a^2*(8*A + 6*B - 3*C)*sin(c + d*x))/(3*d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x))) + (2*a*(A + B)*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x))*sin(c + d*x))/d + (2*A*(a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d)],
[(a + a*cos(c + d*x))^(3/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(3/2), x, 5, (a^(3/2)*(8*A + 12*B + 7*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(4*d) - (a^2*(8*A - 4*B - 5*C)*sin(c + d*x))/(4*d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x))) - (a*(4*A - C)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(2*d*sqrt(sec(c + d*x))) + (2*A*(a + a*cos(c + d*x))^(3/2)*sqrt(sec(c + d*x))*sin(c + d*x))/d],
[(a + a*cos(c + d*x))^(3/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(1/2), x, 5, (a^(3/2)*(24*A + 14*B + 11*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(8*d) + (a^2*(24*A + 30*B + 19*C)*sin(c + d*x))/(24*d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x))) + (a*(2*B + C)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(4*d*sqrt(sec(c + d*x))) + (C*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(3*d*sqrt(sec(c + d*x)))],
[(a + a*cos(c + d*x))^(3/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/sec(c + d*x)^(1/2), x, 6, (a^(3/2)*(112*A + 88*B + 75*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(64*d) + (a^2*(48*A + 56*B + 39*C)*sin(c + d*x))/(96*d*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(3/2)) + (a*(8*B + 3*C)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(24*d*sec(c + d*x)^(3/2)) + (C*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(4*d*sec(c + d*x)^(3/2)) + (a^2*(112*A + 88*B + 75*C)*sin(c + d*x))/(64*d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x)))],
[(a + a*cos(c + d*x))^(3/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/sec(c + d*x)^(3/2), x, 7, (a^(3/2)*(176*A + 150*B + 133*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(128*d) + (a^2*(80*A + 90*B + 67*C)*sin(c + d*x))/(240*d*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(5/2)) + (a*(10*B + 3*C)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(40*d*sec(c + d*x)^(5/2)) + (C*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(5*d*sec(c + d*x)^(5/2)) + (a^2*(176*A + 150*B + 133*C)*sin(c + d*x))/(192*d*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(3/2)) + (a^2*(176*A + 150*B + 133*C)*sin(c + d*x))/(128*d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x)))],


[(a + a*cos(c + d*x))^(5/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(15/2), x, 7, (16*a^3*(8368*A + 9230*B + 10439*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(45045*d*sqrt(a + a*cos(c + d*x))) + (8*a^3*(8368*A + 9230*B + 10439*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(45045*d*sqrt(a + a*cos(c + d*x))) + (2*a^3*(8368*A + 9230*B + 10439*C)*sec(c + d*x)^(5/2)*sin(c + d*x))/(15015*d*sqrt(a + a*cos(c + d*x))) + (2*a^3*(2224*A + 2522*B + 2717*C)*sec(c + d*x)^(7/2)*sin(c + d*x))/(9009*d*sqrt(a + a*cos(c + d*x))) + (2*a^2*(136*A + 182*B + 143*C)*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(9/2)*sin(c + d*x))/(1287*d) + (2*a*(5*A + 13*B)*(a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^(11/2)*sin(c + d*x))/(143*d) + (2*A*(a + a*cos(c + d*x))^(5/2)*sec(c + d*x)^(13/2)*sin(c + d*x))/(13*d)],
[(a + a*cos(c + d*x))^(5/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(13/2), x, 6, (4*a^3*(2840*A + 3212*B + 3795*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(3465*d*sqrt(a + a*cos(c + d*x))) + (2*a^3*(2840*A + 3212*B + 3795*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(3465*d*sqrt(a + a*cos(c + d*x))) + (2*a^3*(1160*A + 1364*B + 1485*C)*sec(c + d*x)^(5/2)*sin(c + d*x))/(3465*d*sqrt(a + a*cos(c + d*x))) + (2*a^2*(32*A + 44*B + 33*C)*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(7/2)*sin(c + d*x))/(231*d) + (2*a*(5*A + 11*B)*(a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^(9/2)*sin(c + d*x))/(99*d) + (2*A*(a + a*cos(c + d*x))^(5/2)*sec(c + d*x)^(11/2)*sin(c + d*x))/(11*d)],
[(a + a*cos(c + d*x))^(5/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(11/2), x, 5, (64*a^3*(13*A + 15*B + 21*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(315*d*sqrt(a + a*cos(c + d*x))) + (16*a^2*(13*A + 15*B + 21*C)*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(315*d) + (2*a*(13*A + 15*B + 21*C)*(a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^(5/2)*sin(c + d*x))/(105*d) + (2*(5*A + 9*B)*(a + a*cos(c + d*x))^(5/2)*sec(c + d*x)^(7/2)*sin(c + d*x))/(63*d) + (2*A*(a + a*cos(c + d*x))^(5/2)*sec(c + d*x)^(9/2)*sin(c + d*x))/(9*d)],
[(a + a*cos(c + d*x))^(5/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(9/2), x, 6, (2*a^(5/2)*C*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/d + (2*a^3*(160*A + 224*B + 245*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(105*d*sqrt(a + a*cos(c + d*x))) + (2*a^2*(40*A + 56*B + 35*C)*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(105*d) + (2*a*(5*A + 7*B)*(a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^(5/2)*sin(c + d*x))/(35*d) + (2*A*(a + a*cos(c + d*x))^(5/2)*sec(c + d*x)^(7/2)*sin(c + d*x))/(7*d)],
[(a + a*cos(c + d*x))^(5/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(7/2), x, 6, (a^(5/2)*(2*B + 5*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/d - (a^3*(64*A + 70*B + 15*C)*sin(c + d*x))/(15*d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x))) + (2*a^2*(8*A + 10*B + 5*C)*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d) + (2*a*(A + B)*(a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d) + (2*A*(a + a*cos(c + d*x))^(5/2)*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d)],
[(a + a*cos(c + d*x))^(5/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(5/2), x, 6, (a^(5/2)*(8*A + 20*B + 19*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(4*d) - (a^3*(56*A + 12*B - 27*C)*sin(c + d*x))/(12*d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x))) - (a^2*(8*A + 4*B - C)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(2*d*sqrt(sec(c + d*x))) + (2*a*(5*A + 3*B)*(a + a*cos(c + d*x))^(3/2)*sqrt(sec(c + d*x))*sin(c + d*x))/(3*d) + (2*A*(a + a*cos(c + d*x))^(5/2)*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d)],
[(a + a*cos(c + d*x))^(5/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(3/2), x, 6, (a^(5/2)*(40*A + 38*B + 25*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(8*d) - (a^3*(24*A - 54*B - 49*C)*sin(c + d*x))/(24*d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x))) - (a^2*(8*A - 2*B - 3*C)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(4*d*sqrt(sec(c + d*x))) - (a*(6*A - C)*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(3*d*sqrt(sec(c + d*x))) + (2*A*(a + a*cos(c + d*x))^(5/2)*sqrt(sec(c + d*x))*sin(c + d*x))/d],
[(a + a*cos(c + d*x))^(5/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(1/2), x, 6, (a^(5/2)*(304*A + 200*B + 163*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(64*d) + (a^3*(432*A + 392*B + 299*C)*sin(c + d*x))/(192*d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x))) + (a^2*(16*A + 24*B + 17*C)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(32*d*sqrt(sec(c + d*x))) + (a*(8*B + 5*C)*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(24*d*sqrt(sec(c + d*x))) + (C*(a + a*cos(c + d*x))^(5/2)*sin(c + d*x))/(4*d*sqrt(sec(c + d*x)))],
[(a + a*cos(c + d*x))^(5/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/sec(c + d*x)^(1/2), x, 7, (a^(5/2)*(400*A + 326*B + 283*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(128*d) + (a^3*(1040*A + 950*B + 787*C)*sin(c + d*x))/(960*d*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(3/2)) + (a^2*(80*A + 110*B + 79*C)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(240*d*sec(c + d*x)^(3/2)) + (a*(2*B + C)*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(8*d*sec(c + d*x)^(3/2)) + (C*(a + a*cos(c + d*x))^(5/2)*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2)) + (a^3*(400*A + 326*B + 283*C)*sin(c + d*x))/(128*d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x)))],
[(a + a*cos(c + d*x))^(5/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/sec(c + d*x)^(3/2), x, 8, (a^(5/2)*(1304*A + 1132*B + 1015*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(512*d) + (a^3*(680*A + 628*B + 545*C)*sin(c + d*x))/(960*d*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(5/2)) + (a^2*(120*A + 156*B + 115*C)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(480*d*sec(c + d*x)^(5/2)) + (a*(12*B + 5*C)*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(60*d*sec(c + d*x)^(5/2)) + (C*(a + a*cos(c + d*x))^(5/2)*sin(c + d*x))/(6*d*sec(c + d*x)^(5/2)) + (a^3*(1304*A + 1132*B + 1015*C)*sin(c + d*x))/(768*d*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(3/2)) + (a^3*(1304*A + 1132*B + 1015*C)*sin(c + d*x))/(512*d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x)))],


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


[(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(11/2)/sqrt(a + a*cos(c + d*x)), x, 7, -((sqrt(2)*(A - B + C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(sqrt(a)*d)) + (2*(257*A - 129*B + 273*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(315*d*sqrt(a + a*cos(c + d*x))) - (2*(29*A - 93*B + 21*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(315*d*sqrt(a + a*cos(c + d*x))) + (2*(19*A - 3*B + 21*C)*sec(c + d*x)^(5/2)*sin(c + d*x))/(105*d*sqrt(a + a*cos(c + d*x))) - (2*(A - 9*B)*sec(c + d*x)^(7/2)*sin(c + d*x))/(63*d*sqrt(a + a*cos(c + d*x))) + (2*A*sec(c + d*x)^(9/2)*sin(c + d*x))/(9*d*sqrt(a + a*cos(c + d*x)))],
[(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(9/2)/sqrt(a + a*cos(c + d*x)), x, 6, (sqrt(2)*(A - B + C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(sqrt(a)*d) - (2*(43*A - 91*B + 35*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(105*d*sqrt(a + a*cos(c + d*x))) + (2*(31*A - 7*B + 35*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(105*d*sqrt(a + a*cos(c + d*x))) - (2*(A - 7*B)*sec(c + d*x)^(5/2)*sin(c + d*x))/(35*d*sqrt(a + a*cos(c + d*x))) + (2*A*sec(c + d*x)^(7/2)*sin(c + d*x))/(7*d*sqrt(a + a*cos(c + d*x)))],
[(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(7/2)/sqrt(a + a*cos(c + d*x)), x, 5, -((sqrt(2)*(A - B + C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(sqrt(a)*d)) + (2*(13*A - 5*B + 15*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(15*d*sqrt(a + a*cos(c + d*x))) - (2*(A - 5*B)*sec(c + d*x)^(3/2)*sin(c + d*x))/(15*d*sqrt(a + a*cos(c + d*x))) + (2*A*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d*sqrt(a + a*cos(c + d*x)))],
[(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(5/2)/sqrt(a + a*cos(c + d*x)), x, 4, (sqrt(2)*(A - B + C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(sqrt(a)*d) - (2*(A - 3*B)*sqrt(sec(c + d*x))*sin(c + d*x))/(3*d*sqrt(a + a*cos(c + d*x))) + (2*A*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d*sqrt(a + a*cos(c + d*x)))],
[(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(3/2)/sqrt(a + a*cos(c + d*x)), x, 5, (2*C*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(sqrt(a)*d) - (sqrt(2)*(A - B + C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(sqrt(a)*d) + (2*A*sqrt(sec(c + d*x))*sin(c + d*x))/(d*sqrt(a + a*cos(c + d*x)))],
[(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(1/2)/sqrt(a + a*cos(c + d*x)), x, 5, ((2*B - C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(sqrt(a)*d) + (sqrt(2)*(A - B + C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(sqrt(a)*d) + (C*sin(c + d*x))/(d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x)))],
[(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(1/2)), x, 6, ((8*A - 4*B + 7*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(4*sqrt(a)*d) - (sqrt(2)*(A - B + C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(sqrt(a)*d) + (C*sin(c + d*x))/(2*d*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(3/2)) + ((4*B - C)*sin(c + d*x))/(4*d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x)))],
[(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(3/2)), x, 7, -(((8*A - 14*B + 9*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(8*sqrt(a)*d)) + (sqrt(2)*(A - B + C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(sqrt(a)*d) + (C*sin(c + d*x))/(3*d*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(5/2)) + ((6*B - C)*sin(c + d*x))/(12*d*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(3/2)) + ((8*A - 2*B + 7*C)*sin(c + d*x))/(8*d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x)))],

[((a*A + (A*b + a*B)*cos(c + d*x) + b*B*cos(c + d*x)^2)*sqrt(sec(c + d*x)))/sqrt(a + a*cos(c + d*x)), x, 5, ((2*A*b + 2*a*B - b*B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(sqrt(a)*d) + (sqrt(2)*(a*A - A*b - a*B + b*B)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(sqrt(a)*d) + (b*B*sin(c + d*x))/(d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x)))],


[(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(9/2)/(a + a*cos(c + d*x))^(3/2), x, 7, ((19*A - 15*B + 11*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(2*sqrt(2)*a^(3/2)*d) - ((1201*A - 1029*B + 665*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(210*a*d*sqrt(a + a*cos(c + d*x))) + ((397*A - 273*B + 245*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(210*a*d*sqrt(a + a*cos(c + d*x))) - ((67*A - 63*B + 35*C)*sec(c + d*x)^(5/2)*sin(c + d*x))/(70*a*d*sqrt(a + a*cos(c + d*x))) - ((A - B + C)*sec(c + d*x)^(7/2)*sin(c + d*x))/(2*d*(a + a*cos(c + d*x))^(3/2)) + ((11*A - 7*B + 7*C)*sec(c + d*x)^(7/2)*sin(c + d*x))/(14*a*d*sqrt(a + a*cos(c + d*x)))],
[(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(7/2)/(a + a*cos(c + d*x))^(3/2), x, 6, -(((15*A - 11*B + 7*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(2*sqrt(2)*a^(3/2)*d)) + ((147*A - 95*B + 75*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(30*a*d*sqrt(a + a*cos(c + d*x))) - ((39*A - 35*B + 15*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(30*a*d*sqrt(a + a*cos(c + d*x))) - ((A - B + C)*sec(c + d*x)^(5/2)*sin(c + d*x))/(2*d*(a + a*cos(c + d*x))^(3/2)) + ((9*A - 5*B + 5*C)*sec(c + d*x)^(5/2)*sin(c + d*x))/(10*a*d*sqrt(a + a*cos(c + d*x)))],
[(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(5/2)/(a + a*cos(c + d*x))^(3/2), x, 5, ((11*A - 7*B + 3*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(2*sqrt(2)*a^(3/2)*d) - ((19*A - 15*B + 3*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(6*a*d*sqrt(a + a*cos(c + d*x))) - ((A - B + C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(2*d*(a + a*cos(c + d*x))^(3/2)) + ((7*A - 3*B + 3*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(6*a*d*sqrt(a + a*cos(c + d*x)))],
[(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(3/2)/(a + a*cos(c + d*x))^(3/2), x, 4, -(((7*A - 3*B - C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(2*sqrt(2)*a^(3/2)*d)) - ((A - B + C)*sqrt(sec(c + d*x))*sin(c + d*x))/(2*d*(a + a*cos(c + d*x))^(3/2)) + ((5*A - B + C)*sqrt(sec(c + d*x))*sin(c + d*x))/(2*a*d*sqrt(a + a*cos(c + d*x)))],
[(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(1/2)/(a + a*cos(c + d*x))^(3/2), x, 5, (2*C*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(a^(3/2)*d) + ((3*A + B - 5*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(2*sqrt(2)*a^(3/2)*d) - ((A - B + C)*sin(c + d*x))/(2*d*(a + a*cos(c + d*x))^(3/2)*sqrt(sec(c + d*x)))],
[(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/((a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^(1/2)), x, 6, ((2*B - 3*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(a^(3/2)*d) + ((A - 5*B + 9*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(2*sqrt(2)*a^(3/2)*d) - ((A - B + C)*sin(c + d*x))/(2*d*(a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^(3/2)) + ((A - B + 3*C)*sin(c + d*x))/(2*a*d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x)))],
[(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/((a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^(3/2)), x, 7, ((8*A - 12*B + 19*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(4*a^(3/2)*d) - ((5*A - 9*B + 13*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(2*sqrt(2)*a^(3/2)*d) - ((A - B + C)*sin(c + d*x))/(2*d*(a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^(5/2)) + ((A - B + 2*C)*sin(c + d*x))/(2*a*d*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(3/2)) - ((2*A - 6*B + 7*C)*sin(c + d*x))/(4*a*d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x)))],


[(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(7/2)/(a + a*cos(c + d*x))^(5/2), x, 7, -(((283*A - 163*B + 75*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(16*sqrt(2)*a^(5/2)*d)) + ((2671*A - 1495*B + 735*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(240*a^2*d*sqrt(a + a*cos(c + d*x))) - ((787*A - 475*B + 195*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(240*a^2*d*sqrt(a + a*cos(c + d*x))) - ((A - B + C)*sec(c + d*x)^(5/2)*sin(c + d*x))/(4*d*(a + a*cos(c + d*x))^(5/2)) - ((21*A - 13*B + 5*C)*sec(c + d*x)^(5/2)*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2)) + ((157*A - 85*B + 45*C)*sec(c + d*x)^(5/2)*sin(c + d*x))/(80*a^2*d*sqrt(a + a*cos(c + d*x)))],
[(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(5/2)/(a + a*cos(c + d*x))^(5/2), x, 6, ((163*A - 75*B + 19*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(16*sqrt(2)*a^(5/2)*d) - ((299*A - 147*B + 27*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(48*a^2*d*sqrt(a + a*cos(c + d*x))) - ((A - B + C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(4*d*(a + a*cos(c + d*x))^(5/2)) - ((17*A - 9*B + C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2)) + ((95*A - 39*B + 15*C)*sec(c + d*x)^(3/2)*sin(c + d*x))/(48*a^2*d*sqrt(a + a*cos(c + d*x)))],
[(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(3/2)/(a + a*cos(c + d*x))^(5/2), x, 5, -(((75*A - 19*B - 5*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(16*sqrt(2)*a^(5/2)*d)) - ((A - B + C)*sqrt(sec(c + d*x))*sin(c + d*x))/(4*d*(a + a*cos(c + d*x))^(5/2)) - ((13*A - 5*B - 3*C)*sqrt(sec(c + d*x))*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2)) + ((49*A - 9*B + C)*sqrt(sec(c + d*x))*sin(c + d*x))/(16*a^2*d*sqrt(a + a*cos(c + d*x)))],
[(A + B*cos(c + d*x) + C*cos(c + d*x)^2)*sec(c + d*x)^(1/2)/(a + a*cos(c + d*x))^(5/2), x, 4, ((19*A + 5*B + 3*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(16*sqrt(2)*a^(5/2)*d) - ((A - B + C)*sin(c + d*x))/(4*d*(a + a*cos(c + d*x))^(5/2)*sqrt(sec(c + d*x))) - ((9*A - B - 7*C)*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2)*sqrt(sec(c + d*x)))],
[(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/((a + a*cos(c + d*x))^(5/2)*sec(c + d*x)^(1/2)), x, 6, (2*C*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(a^(5/2)*d) + ((5*A + 3*B - 43*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(16*sqrt(2)*a^(5/2)*d) - ((A - B + C)*sin(c + d*x))/(4*d*(a + a*cos(c + d*x))^(5/2)*sec(c + d*x)^(3/2)) + ((5*A + 3*B - 11*C)*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2)*sqrt(sec(c + d*x)))],
[(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/((a + a*cos(c + d*x))^(5/2)*sec(c + d*x)^(3/2)), x, 7, ((2*B - 5*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(a^(5/2)*d) + ((3*A - 43*B + 115*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(16*sqrt(2)*a^(5/2)*d) - ((A - B + C)*sin(c + d*x))/(4*d*(a + a*cos(c + d*x))^(5/2)*sec(c + d*x)^(5/2)) + ((A + 7*B - 15*C)*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^(3/2)) + ((3*A - 11*B + 35*C)*sin(c + d*x))/(16*a^2*d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x)))],
[(A + B*cos(c + d*x) + C*cos(c + d*x)^2)/((a + a*cos(c + d*x))^(5/2)*sec(c + d*x)^(5/2)), x, 8, ((8*A - 20*B + 39*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(4*a^(5/2)*d) - ((43*A - 115*B + 219*C)*arctan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(16*sqrt(2)*a^(5/2)*d) - ((A - B + C)*sin(c + d*x))/(4*d*(a + a*cos(c + d*x))^(5/2)*sec(c + d*x)^(7/2)) - ((3*A - 11*B + 19*C)*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^(5/2)) + ((7*A - 15*B + 31*C)*sin(c + d*x))/(16*a^2*d*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(3/2)) - ((11*A - 35*B + 63*C)*sin(c + d*x))/(16*a^2*d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x)))]
]:
