# Maple integration test file: "4 Trig functions\4.1 Sine\4.1.2.2 (g cos)^p (a+b sin)^m (c+d sin)^n.txt"

lst:=[

# Integrands of the form (g Cos[e+f x])^p (a+a Sin[e+f x])^m (c-c Sin[e+f x])^n

# Integrands of the form Cos[e+f x]^2 (a+a Sin[e+f x])^m (c-c Sin[e+f x])^n

# Integrands of the form Cos[e+f x]^2 (a+a Sin[e+f x])^m (c-c Sin[e+f x])^n

# Integrands of the form Cos[e+f x]^2 (a+a Sin[e+f x])^m (c-c Sin[e+f x])^(n/2)

# Integrands of the form Cos[e+f x]^2 (a+a Sin[e+f x])^(m/2) (c-c Sin[e+f x])^(n/2)

# m>0
[cos(e+f*x)^2*(c-c*sin(e+f*x))^(7/2)*sqrt(a+a*sin(e+f*x)),x,3,-1/15*a*cos(e+f*x)*(c-c*sin(e+f*x))^(9/2)/(c*f*sqrt(a+a*sin(e+f*x)))-1/6*cos(e+f*x)*(c-c*sin(e+f*x))^(9/2)*sqrt(a+a*sin(e+f*x))/(c*f)],
[cos(e+f*x)^2*(c-c*sin(e+f*x))^(5/2)*sqrt(a+a*sin(e+f*x)),x,3,-1/10*a*cos(e+f*x)*(c-c*sin(e+f*x))^(7/2)/(c*f*sqrt(a+a*sin(e+f*x)))-1/5*cos(e+f*x)*(c-c*sin(e+f*x))^(7/2)*sqrt(a+a*sin(e+f*x))/(c*f)],
[cos(e+f*x)^2*(c-c*sin(e+f*x))^(3/2)*sqrt(a+a*sin(e+f*x)),x,3,-1/6*a*cos(e+f*x)*(c-c*sin(e+f*x))^(5/2)/(c*f*sqrt(a+a*sin(e+f*x)))-1/4*cos(e+f*x)*(c-c*sin(e+f*x))^(5/2)*sqrt(a+a*sin(e+f*x))/(c*f)],
[cos(e+f*x)^2*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)),x,3,-1/3*a*cos(e+f*x)*(c-c*sin(e+f*x))^(3/2)/(c*f*sqrt(a+a*sin(e+f*x)))-1/3*cos(e+f*x)*(c-c*sin(e+f*x))^(3/2)*sqrt(a+a*sin(e+f*x))/(c*f)],
[cos(e+f*x)^2*sqrt(a+a*sin(e+f*x))/sqrt(c-c*sin(e+f*x)),x,2,1/2*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)/(a*f*sqrt(c-c*sin(e+f*x)))],
[cos(e+f*x)^2*sqrt(a+a*sin(e+f*x))/(c-c*sin(e+f*x))^(3/2),x,5,-2*a*cos(e+f*x)*log(1-sin(e+f*x))/(c*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))-cos(e+f*x)*sqrt(a+a*sin(e+f*x))/(c*f*sqrt(c-c*sin(e+f*x)))],
[cos(e+f*x)^2*sqrt(a+a*sin(e+f*x))/(c-c*sin(e+f*x))^(5/2),x,5,cos(e+f*x)*sqrt(a+a*sin(e+f*x))/(c*f*(c-c*sin(e+f*x))^(3/2))+a*cos(e+f*x)*log(1-sin(e+f*x))/(c^2*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))],
[cos(e+f*x)^2*sqrt(a+a*sin(e+f*x))/(c-c*sin(e+f*x))^(7/2),x,2,1/4*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)/(a*c*f*(c-c*sin(e+f*x))^(5/2))],
[cos(e+f*x)^2*(a+a*sin(e+f*x))^(3/2)*(c-c*sin(e+f*x))^(7/2),x,4,-1/7*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)*(c-c*sin(e+f*x))^(9/2)/(c*f)-4/105*a^2*cos(e+f*x)*(c-c*sin(e+f*x))^(9/2)/(c*f*sqrt(a+a*sin(e+f*x)))-2/21*a*cos(e+f*x)*(c-c*sin(e+f*x))^(9/2)*sqrt(a+a*sin(e+f*x))/(c*f)],
[cos(e+f*x)^2*(a+a*sin(e+f*x))^(3/2)*(c-c*sin(e+f*x))^(5/2),x,4,-1/6*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)*(c-c*sin(e+f*x))^(7/2)/(c*f)-1/15*a^2*cos(e+f*x)*(c-c*sin(e+f*x))^(7/2)/(c*f*sqrt(a+a*sin(e+f*x)))-2/15*a*cos(e+f*x)*(c-c*sin(e+f*x))^(7/2)*sqrt(a+a*sin(e+f*x))/(c*f)],
[cos(e+f*x)^2*(a+a*sin(e+f*x))^(3/2)*(c-c*sin(e+f*x))^(3/2),x,4,-1/5*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)*(c-c*sin(e+f*x))^(5/2)/(c*f)-2/15*a^2*cos(e+f*x)*(c-c*sin(e+f*x))^(5/2)/(c*f*sqrt(a+a*sin(e+f*x)))-1/5*a*cos(e+f*x)*(c-c*sin(e+f*x))^(5/2)*sqrt(a+a*sin(e+f*x))/(c*f)],
[cos(e+f*x)^2*(a+a*sin(e+f*x))^(3/2)*sqrt(c-c*sin(e+f*x)),x,3,1/6*c*cos(e+f*x)*(a+a*sin(e+f*x))^(5/2)/(a*f*sqrt(c-c*sin(e+f*x)))+1/4*cos(e+f*x)*(a+a*sin(e+f*x))^(5/2)*sqrt(c-c*sin(e+f*x))/(a*f)],
[cos(e+f*x)^2*(a+a*sin(e+f*x))^(3/2)/sqrt(c-c*sin(e+f*x)),x,2,1/3*cos(e+f*x)*(a+a*sin(e+f*x))^(5/2)/(a*f*sqrt(c-c*sin(e+f*x)))],
[cos(e+f*x)^2*(a+a*sin(e+f*x))^(3/2)/(c-c*sin(e+f*x))^(3/2),x,6,-1/2*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)/(c*f*sqrt(c-c*sin(e+f*x)))-4*a^2*cos(e+f*x)*log(1-sin(e+f*x))/(c*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))-2*a*cos(e+f*x)*sqrt(a+a*sin(e+f*x))/(c*f*sqrt(c-c*sin(e+f*x)))],
[cos(e+f*x)^2*(a+a*sin(e+f*x))^(3/2)/(c-c*sin(e+f*x))^(5/2),x,6,cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)/(c*f*(c-c*sin(e+f*x))^(3/2))+4*a^2*cos(e+f*x)*log(1-sin(e+f*x))/(c^2*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))+2*a*cos(e+f*x)*sqrt(a+a*sin(e+f*x))/(c^2*f*sqrt(c-c*sin(e+f*x)))],
[cos(e+f*x)^2*(a+a*sin(e+f*x))^(3/2)/(c-c*sin(e+f*x))^(7/2),x,6,1/2*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)/(c*f*(c-c*sin(e+f*x))^(5/2))-a*cos(e+f*x)*sqrt(a+a*sin(e+f*x))/(c^2*f*(c-c*sin(e+f*x))^(3/2))-a^2*cos(e+f*x)*log(1-sin(e+f*x))/(c^3*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))],
[cos(e+f*x)^2*(a+a*sin(e+f*x))^(3/2)/(c-c*sin(e+f*x))^(9/2),x,2,1/6*cos(e+f*x)*(a+a*sin(e+f*x))^(5/2)/(a*c*f*(c-c*sin(e+f*x))^(7/2))],
[cos(e+f*x)^2*(a+a*sin(e+f*x))^(3/2)/(c-c*sin(e+f*x))^(11/2),x,3,1/8*cos(e+f*x)*(a+a*sin(e+f*x))^(5/2)/(a*c*f*(c-c*sin(e+f*x))^(9/2))+1/48*cos(e+f*x)*(a+a*sin(e+f*x))^(5/2)/(a*c^2*f*(c-c*sin(e+f*x))^(7/2))],
[cos(e+f*x)^2*(a+a*sin(e+f*x))^(5/2)*(c-c*sin(e+f*x))^(7/2),x,5,-3/28*a*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)*(c-c*sin(e+f*x))^(9/2)/(c*f)-1/8*cos(e+f*x)*(a+a*sin(e+f*x))^(5/2)*(c-c*sin(e+f*x))^(9/2)/(c*f)-1/35*a^3*cos(e+f*x)*(c-c*sin(e+f*x))^(9/2)/(c*f*sqrt(a+a*sin(e+f*x)))-1/14*a^2*cos(e+f*x)*(c-c*sin(e+f*x))^(9/2)*sqrt(a+a*sin(e+f*x))/(c*f)],
[cos(e+f*x)^2*(a+a*sin(e+f*x))^(5/2)*(c-c*sin(e+f*x))^(5/2),x,5,-1/7*a*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)*(c-c*sin(e+f*x))^(7/2)/(c*f)-1/7*cos(e+f*x)*(a+a*sin(e+f*x))^(5/2)*(c-c*sin(e+f*x))^(7/2)/(c*f)-2/35*a^3*cos(e+f*x)*(c-c*sin(e+f*x))^(7/2)/(c*f*sqrt(a+a*sin(e+f*x)))-4/35*a^2*cos(e+f*x)*(c-c*sin(e+f*x))^(7/2)*sqrt(a+a*sin(e+f*x))/(c*f)],
[cos(e+f*x)^2*(a+a*sin(e+f*x))^(5/2)*(c-c*sin(e+f*x))^(3/2),x,4,1/6*cos(e+f*x)*(a+a*sin(e+f*x))^(7/2)*(c-c*sin(e+f*x))^(3/2)/(a*f)+1/15*c^2*cos(e+f*x)*(a+a*sin(e+f*x))^(7/2)/(a*f*sqrt(c-c*sin(e+f*x)))+2/15*c*cos(e+f*x)*(a+a*sin(e+f*x))^(7/2)*sqrt(c-c*sin(e+f*x))/(a*f)],
[cos(e+f*x)^2*(a+a*sin(e+f*x))^(5/2)*sqrt(c-c*sin(e+f*x)),x,3,1/10*c*cos(e+f*x)*(a+a*sin(e+f*x))^(7/2)/(a*f*sqrt(c-c*sin(e+f*x)))+1/5*cos(e+f*x)*(a+a*sin(e+f*x))^(7/2)*sqrt(c-c*sin(e+f*x))/(a*f)],
[cos(e+f*x)^2*(a+a*sin(e+f*x))^(5/2)/sqrt(c-c*sin(e+f*x)),x,2,1/4*cos(e+f*x)*(a+a*sin(e+f*x))^(7/2)/(a*f*sqrt(c-c*sin(e+f*x)))],
[cos(e+f*x)^2*(a+a*sin(e+f*x))^(5/2)/(c-c*sin(e+f*x))^(3/2),x,7,-a*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)/(c*f*sqrt(c-c*sin(e+f*x)))-1/3*cos(e+f*x)*(a+a*sin(e+f*x))^(5/2)/(c*f*sqrt(c-c*sin(e+f*x)))-8*a^3*cos(e+f*x)*log(1-sin(e+f*x))/(c*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))-4*a^2*cos(e+f*x)*sqrt(a+a*sin(e+f*x))/(c*f*sqrt(c-c*sin(e+f*x)))],
[cos(e+f*x)^2*(a+a*sin(e+f*x))^(5/2)/(c-c*sin(e+f*x))^(5/2),x,7,cos(e+f*x)*(a+a*sin(e+f*x))^(5/2)/(c*f*(c-c*sin(e+f*x))^(3/2))+3/2*a*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)/(c^2*f*sqrt(c-c*sin(e+f*x)))+12*a^3*cos(e+f*x)*log(1-sin(e+f*x))/(c^2*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))+6*a^2*cos(e+f*x)*sqrt(a+a*sin(e+f*x))/(c^2*f*sqrt(c-c*sin(e+f*x)))],
[cos(e+f*x)^2*(a+a*sin(e+f*x))^(5/2)/(c-c*sin(e+f*x))^(7/2),x,7,1/2*cos(e+f*x)*(a+a*sin(e+f*x))^(5/2)/(c*f*(c-c*sin(e+f*x))^(5/2))-3/2*a*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)/(c^2*f*(c-c*sin(e+f*x))^(3/2))-6*a^3*cos(e+f*x)*log(1-sin(e+f*x))/(c^3*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))-3*a^2*cos(e+f*x)*sqrt(a+a*sin(e+f*x))/(c^3*f*sqrt(c-c*sin(e+f*x)))],
[cos(e+f*x)^2*(a+a*sin(e+f*x))^(5/2)/(c-c*sin(e+f*x))^(9/2),x,7,1/3*cos(e+f*x)*(a+a*sin(e+f*x))^(5/2)/(c*f*(c-c*sin(e+f*x))^(7/2))-1/2*a*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)/(c^2*f*(c-c*sin(e+f*x))^(5/2))+a^2*cos(e+f*x)*sqrt(a+a*sin(e+f*x))/(c^3*f*(c-c*sin(e+f*x))^(3/2))+a^3*cos(e+f*x)*log(1-sin(e+f*x))/(c^4*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))],
[cos(e+f*x)^2*(a+a*sin(e+f*x))^(5/2)/(c-c*sin(e+f*x))^(11/2),x,2,1/8*cos(e+f*x)*(a+a*sin(e+f*x))^(7/2)/(a*c*f*(c-c*sin(e+f*x))^(9/2))],
[cos(e+f*x)^2*(a+a*sin(e+f*x))^(5/2)/(c-c*sin(e+f*x))^(13/2),x,3,1/10*cos(e+f*x)*(a+a*sin(e+f*x))^(7/2)/(a*c*f*(c-c*sin(e+f*x))^(11/2))+1/80*cos(e+f*x)*(a+a*sin(e+f*x))^(7/2)/(a*c^2*f*(c-c*sin(e+f*x))^(9/2))],
[cos(e+f*x)^2*(a+a*sin(e+f*x))^(7/2)*(c-c*sin(e+f*x))^(9/2),x,6,-1/15*a^2*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)*(c-c*sin(e+f*x))^(11/2)/(c*f)-4/45*a*cos(e+f*x)*(a+a*sin(e+f*x))^(5/2)*(c-c*sin(e+f*x))^(11/2)/(c*f)-1/10*cos(e+f*x)*(a+a*sin(e+f*x))^(7/2)*(c-c*sin(e+f*x))^(11/2)/(c*f)-4/315*a^4*cos(e+f*x)*(c-c*sin(e+f*x))^(11/2)/(c*f*sqrt(a+a*sin(e+f*x)))-4/105*a^3*cos(e+f*x)*(c-c*sin(e+f*x))^(11/2)*sqrt(a+a*sin(e+f*x))/(c*f)],
[cos(e+f*x)^2*(a+a*sin(e+f*x))^(7/2)*(c-c*sin(e+f*x))^(7/2),x,6,-2/21*a^2*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)*(c-c*sin(e+f*x))^(9/2)/(c*f)-1/9*a*cos(e+f*x)*(a+a*sin(e+f*x))^(5/2)*(c-c*sin(e+f*x))^(9/2)/(c*f)-1/9*cos(e+f*x)*(a+a*sin(e+f*x))^(7/2)*(c-c*sin(e+f*x))^(9/2)/(c*f)-8/315*a^4*cos(e+f*x)*(c-c*sin(e+f*x))^(9/2)/(c*f*sqrt(a+a*sin(e+f*x)))-4/63*a^3*cos(e+f*x)*(c-c*sin(e+f*x))^(9/2)*sqrt(a+a*sin(e+f*x))/(c*f)],
[cos(e+f*x)^2*(a+a*sin(e+f*x))^(7/2)*(c-c*sin(e+f*x))^(5/2),x,5,3/28*c*cos(e+f*x)*(a+a*sin(e+f*x))^(9/2)*(c-c*sin(e+f*x))^(3/2)/(a*f)+1/8*cos(e+f*x)*(a+a*sin(e+f*x))^(9/2)*(c-c*sin(e+f*x))^(5/2)/(a*f)+1/35*c^3*cos(e+f*x)*(a+a*sin(e+f*x))^(9/2)/(a*f*sqrt(c-c*sin(e+f*x)))+1/14*c^2*cos(e+f*x)*(a+a*sin(e+f*x))^(9/2)*sqrt(c-c*sin(e+f*x))/(a*f)],
[cos(e+f*x)^2*(a+a*sin(e+f*x))^(7/2)*(c-c*sin(e+f*x))^(3/2),x,4,1/7*cos(e+f*x)*(a+a*sin(e+f*x))^(9/2)*(c-c*sin(e+f*x))^(3/2)/(a*f)+4/105*c^2*cos(e+f*x)*(a+a*sin(e+f*x))^(9/2)/(a*f*sqrt(c-c*sin(e+f*x)))+2/21*c*cos(e+f*x)*(a+a*sin(e+f*x))^(9/2)*sqrt(c-c*sin(e+f*x))/(a*f)],
[cos(e+f*x)^2*(a+a*sin(e+f*x))^(7/2)*sqrt(c-c*sin(e+f*x)),x,3,1/15*c*cos(e+f*x)*(a+a*sin(e+f*x))^(9/2)/(a*f*sqrt(c-c*sin(e+f*x)))+1/6*cos(e+f*x)*(a+a*sin(e+f*x))^(9/2)*sqrt(c-c*sin(e+f*x))/(a*f)],
[cos(e+f*x)^2*(a+a*sin(e+f*x))^(7/2)/sqrt(c-c*sin(e+f*x)),x,2,1/5*cos(e+f*x)*(a+a*sin(e+f*x))^(9/2)/(a*f*sqrt(c-c*sin(e+f*x)))],
[cos(e+f*x)^2*(a+a*sin(e+f*x))^(7/2)/(c-c*sin(e+f*x))^(3/2),x,8,-2*a^2*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)/(c*f*sqrt(c-c*sin(e+f*x)))-2/3*a*cos(e+f*x)*(a+a*sin(e+f*x))^(5/2)/(c*f*sqrt(c-c*sin(e+f*x)))-1/4*cos(e+f*x)*(a+a*sin(e+f*x))^(7/2)/(c*f*sqrt(c-c*sin(e+f*x)))-16*a^4*cos(e+f*x)*log(1-sin(e+f*x))/(c*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))-8*a^3*cos(e+f*x)*sqrt(a+a*sin(e+f*x))/(c*f*sqrt(c-c*sin(e+f*x)))],
[cos(e+f*x)^2*(a+a*sin(e+f*x))^(7/2)/(c-c*sin(e+f*x))^(5/2),x,8,cos(e+f*x)*(a+a*sin(e+f*x))^(7/2)/(c*f*(c-c*sin(e+f*x))^(3/2))+4*a^2*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)/(c^2*f*sqrt(c-c*sin(e+f*x)))+4/3*a*cos(e+f*x)*(a+a*sin(e+f*x))^(5/2)/(c^2*f*sqrt(c-c*sin(e+f*x)))+32*a^4*cos(e+f*x)*log(1-sin(e+f*x))/(c^2*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))+16*a^3*cos(e+f*x)*sqrt(a+a*sin(e+f*x))/(c^2*f*sqrt(c-c*sin(e+f*x)))],
[cos(e+f*x)^2*(a+a*sin(e+f*x))^(7/2)/(c-c*sin(e+f*x))^(7/2),x,8,1/2*cos(e+f*x)*(a+a*sin(e+f*x))^(7/2)/(c*f*(c-c*sin(e+f*x))^(5/2))-2*a*cos(e+f*x)*(a+a*sin(e+f*x))^(5/2)/(c^2*f*(c-c*sin(e+f*x))^(3/2))-3*a^2*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)/(c^3*f*sqrt(c-c*sin(e+f*x)))-24*a^4*cos(e+f*x)*log(1-sin(e+f*x))/(c^3*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))-12*a^3*cos(e+f*x)*sqrt(a+a*sin(e+f*x))/(c^3*f*sqrt(c-c*sin(e+f*x)))],
[cos(e+f*x)^2*(a+a*sin(e+f*x))^(7/2)/(c-c*sin(e+f*x))^(9/2),x,8,1/3*cos(e+f*x)*(a+a*sin(e+f*x))^(7/2)/(c*f*(c-c*sin(e+f*x))^(7/2))-2/3*a*cos(e+f*x)*(a+a*sin(e+f*x))^(5/2)/(c^2*f*(c-c*sin(e+f*x))^(5/2))+2*a^2*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)/(c^3*f*(c-c*sin(e+f*x))^(3/2))+8*a^4*cos(e+f*x)*log(1-sin(e+f*x))/(c^4*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))+4*a^3*cos(e+f*x)*sqrt(a+a*sin(e+f*x))/(c^4*f*sqrt(c-c*sin(e+f*x)))],
[cos(e+f*x)^2*(a+a*sin(e+f*x))^(7/2)/(c-c*sin(e+f*x))^(11/2),x,8,1/4*cos(e+f*x)*(a+a*sin(e+f*x))^(7/2)/(c*f*(c-c*sin(e+f*x))^(9/2))-1/3*a*cos(e+f*x)*(a+a*sin(e+f*x))^(5/2)/(c^2*f*(c-c*sin(e+f*x))^(7/2))+1/2*a^2*cos(e+f*x)*(a+a*sin(e+f*x))^(3/2)/(c^3*f*(c-c*sin(e+f*x))^(5/2))-a^3*cos(e+f*x)*sqrt(a+a*sin(e+f*x))/(c^4*f*(c-c*sin(e+f*x))^(3/2))-a^4*cos(e+f*x)*log(1-sin(e+f*x))/(c^5*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))],
[cos(e+f*x)^2*(a+a*sin(e+f*x))^(7/2)/(c-c*sin(e+f*x))^(13/2),x,2,1/10*cos(e+f*x)*(a+a*sin(e+f*x))^(9/2)/(a*c*f*(c-c*sin(e+f*x))^(11/2))],
[cos(e+f*x)^2*(a+a*sin(e+f*x))^(7/2)/(c-c*sin(e+f*x))^(15/2),x,3,1/12*cos(e+f*x)*(a+a*sin(e+f*x))^(9/2)/(a*c*f*(c-c*sin(e+f*x))^(13/2))+1/120*cos(e+f*x)*(a+a*sin(e+f*x))^(9/2)/(a*c^2*f*(c-c*sin(e+f*x))^(11/2))],
[cos(e+f*x)^2*(a+a*sin(e+f*x))^(7/2)/(c-c*sin(e+f*x))^(17/2),x,4,1/14*cos(e+f*x)*(a+a*sin(e+f*x))^(9/2)/(a*c*f*(c-c*sin(e+f*x))^(15/2))+1/84*cos(e+f*x)*(a+a*sin(e+f*x))^(9/2)/(a*c^2*f*(c-c*sin(e+f*x))^(13/2))+1/840*cos(e+f*x)*(a+a*sin(e+f*x))^(9/2)/(a*c^3*f*(c-c*sin(e+f*x))^(11/2))],

# m<0
[cos(e+f*x)^2*(c-c*sin(e+f*x))^(5/2)/sqrt(a+a*sin(e+f*x)),x,2,-1/4*cos(e+f*x)*(c-c*sin(e+f*x))^(7/2)/(c*f*sqrt(a+a*sin(e+f*x)))],
[cos(e+f*x)^2*(c-c*sin(e+f*x))^(3/2)/sqrt(a+a*sin(e+f*x)),x,2,-1/3*cos(e+f*x)*(c-c*sin(e+f*x))^(5/2)/(c*f*sqrt(a+a*sin(e+f*x)))],
[cos(e+f*x)^2*sqrt(c-c*sin(e+f*x))/sqrt(a+a*sin(e+f*x)),x,2,-1/2*cos(e+f*x)*(c-c*sin(e+f*x))^(3/2)/(c*f*sqrt(a+a*sin(e+f*x)))],
[cos(e+f*x)^2/(sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x))),x,2,-cos(e+f*x)*sqrt(c-c*sin(e+f*x))/(c*f*sqrt(a+a*sin(e+f*x)))],
[cos(e+f*x)^2/((c-c*sin(e+f*x))^(3/2)*sqrt(a+a*sin(e+f*x))),x,4,-cos(e+f*x)*log(1-sin(e+f*x))/(c*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))],
[cos(e+f*x)^2/((c-c*sin(e+f*x))^(5/2)*sqrt(a+a*sin(e+f*x))),x,2,cos(e+f*x)/(c*f*(c-c*sin(e+f*x))^(3/2)*sqrt(a+a*sin(e+f*x)))],
[cos(e+f*x)^2*(c-c*sin(e+f*x))^(7/2)/(a+a*sin(e+f*x))^(3/2),x,8,2*c^2*cos(e+f*x)*(c-c*sin(e+f*x))^(3/2)/(a*f*sqrt(a+a*sin(e+f*x)))+2/3*c*cos(e+f*x)*(c-c*sin(e+f*x))^(5/2)/(a*f*sqrt(a+a*sin(e+f*x)))+1/4*cos(e+f*x)*(c-c*sin(e+f*x))^(7/2)/(a*f*sqrt(a+a*sin(e+f*x)))+16*c^4*cos(e+f*x)*log(1+sin(e+f*x))/(a*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))+8*c^3*cos(e+f*x)*sqrt(c-c*sin(e+f*x))/(a*f*sqrt(a+a*sin(e+f*x)))],
[cos(e+f*x)^2*(c-c*sin(e+f*x))^(5/2)/(a+a*sin(e+f*x))^(3/2),x,7,c*cos(e+f*x)*(c-c*sin(e+f*x))^(3/2)/(a*f*sqrt(a+a*sin(e+f*x)))+1/3*cos(e+f*x)*(c-c*sin(e+f*x))^(5/2)/(a*f*sqrt(a+a*sin(e+f*x)))+8*c^3*cos(e+f*x)*log(1+sin(e+f*x))/(a*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))+4*c^2*cos(e+f*x)*sqrt(c-c*sin(e+f*x))/(a*f*sqrt(a+a*sin(e+f*x)))],
[cos(e+f*x)^2*(c-c*sin(e+f*x))^(3/2)/(a+a*sin(e+f*x))^(3/2),x,6,1/2*cos(e+f*x)*(c-c*sin(e+f*x))^(3/2)/(a*f*sqrt(a+a*sin(e+f*x)))+4*c^2*cos(e+f*x)*log(1+sin(e+f*x))/(a*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))+2*c*cos(e+f*x)*sqrt(c-c*sin(e+f*x))/(a*f*sqrt(a+a*sin(e+f*x)))],
[cos(e+f*x)^2*sqrt(c-c*sin(e+f*x))/(a+a*sin(e+f*x))^(3/2),x,5,2*c*cos(e+f*x)*log(1+sin(e+f*x))/(a*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))+cos(e+f*x)*sqrt(c-c*sin(e+f*x))/(a*f*sqrt(a+a*sin(e+f*x)))],
[cos(e+f*x)^2/((a+a*sin(e+f*x))^(3/2)*sqrt(c-c*sin(e+f*x))),x,4,cos(e+f*x)*log(1+sin(e+f*x))/(a*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))],
[cos(e+f*x)^2/((a+a*sin(e+f*x))^(3/2)*(c-c*sin(e+f*x))^(3/2)),x,3,arctanh(sin(e+f*x))*cos(e+f*x)/(a*c*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))],
[cos(e+f*x)^2/((a+a*sin(e+f*x))^(3/2)*(c-c*sin(e+f*x))^(5/2)),x,4,1/2*cos(e+f*x)/(a*c*f*(c-c*sin(e+f*x))^(3/2)*sqrt(a+a*sin(e+f*x)))+1/2*arctanh(sin(e+f*x))*cos(e+f*x)/(a*c^2*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))],
[cos(e+f*x)^2*(c-c*sin(e+f*x))^(9/2)/(a+a*sin(e+f*x))^(5/2),x,9,-cos(e+f*x)*(c-c*sin(e+f*x))^(9/2)/(a*f*(a+a*sin(e+f*x))^(3/2))-10*c^3*cos(e+f*x)*(c-c*sin(e+f*x))^(3/2)/(a^2*f*sqrt(a+a*sin(e+f*x)))-10/3*c^2*cos(e+f*x)*(c-c*sin(e+f*x))^(5/2)/(a^2*f*sqrt(a+a*sin(e+f*x)))-5/4*c*cos(e+f*x)*(c-c*sin(e+f*x))^(7/2)/(a^2*f*sqrt(a+a*sin(e+f*x)))-80*c^5*cos(e+f*x)*log(1+sin(e+f*x))/(a^2*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))-40*c^4*cos(e+f*x)*sqrt(c-c*sin(e+f*x))/(a^2*f*sqrt(a+a*sin(e+f*x)))],
[cos(e+f*x)^2*(c-c*sin(e+f*x))^(7/2)/(a+a*sin(e+f*x))^(5/2),x,8,-cos(e+f*x)*(c-c*sin(e+f*x))^(7/2)/(a*f*(a+a*sin(e+f*x))^(3/2))-4*c^2*cos(e+f*x)*(c-c*sin(e+f*x))^(3/2)/(a^2*f*sqrt(a+a*sin(e+f*x)))-4/3*c*cos(e+f*x)*(c-c*sin(e+f*x))^(5/2)/(a^2*f*sqrt(a+a*sin(e+f*x)))-32*c^4*cos(e+f*x)*log(1+sin(e+f*x))/(a^2*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))-16*c^3*cos(e+f*x)*sqrt(c-c*sin(e+f*x))/(a^2*f*sqrt(a+a*sin(e+f*x)))],
[cos(e+f*x)^2*(c-c*sin(e+f*x))^(5/2)/(a+a*sin(e+f*x))^(5/2),x,7,-cos(e+f*x)*(c-c*sin(e+f*x))^(5/2)/(a*f*(a+a*sin(e+f*x))^(3/2))-3/2*c*cos(e+f*x)*(c-c*sin(e+f*x))^(3/2)/(a^2*f*sqrt(a+a*sin(e+f*x)))-12*c^3*cos(e+f*x)*log(1+sin(e+f*x))/(a^2*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))-6*c^2*cos(e+f*x)*sqrt(c-c*sin(e+f*x))/(a^2*f*sqrt(a+a*sin(e+f*x)))],
[cos(e+f*x)^2*(c-c*sin(e+f*x))^(3/2)/(a+a*sin(e+f*x))^(5/2),x,6,-cos(e+f*x)*(c-c*sin(e+f*x))^(3/2)/(a*f*(a+a*sin(e+f*x))^(3/2))-4*c^2*cos(e+f*x)*log(1+sin(e+f*x))/(a^2*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))-2*c*cos(e+f*x)*sqrt(c-c*sin(e+f*x))/(a^2*f*sqrt(a+a*sin(e+f*x)))],
[cos(e+f*x)^2*sqrt(c-c*sin(e+f*x))/(a+a*sin(e+f*x))^(5/2),x,5,-c*cos(e+f*x)*log(1+sin(e+f*x))/(a^2*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))-cos(e+f*x)*sqrt(c-c*sin(e+f*x))/(a*f*(a+a*sin(e+f*x))^(3/2))],
[cos(e+f*x)^2/((a+a*sin(e+f*x))^(5/2)*sqrt(c-c*sin(e+f*x))),x,2,-cos(e+f*x)/(a*f*(a+a*sin(e+f*x))^(3/2)*sqrt(c-c*sin(e+f*x)))],
[cos(e+f*x)^2/((a+a*sin(e+f*x))^(5/2)*(c-c*sin(e+f*x))^(3/2)),x,4,-1/2*cos(e+f*x)/(a*c*f*(a+a*sin(e+f*x))^(3/2)*sqrt(c-c*sin(e+f*x)))+1/2*arctanh(sin(e+f*x))*cos(e+f*x)/(a^2*c*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))],
[cos(e+f*x)^2/((a+a*sin(e+f*x))^(5/2)*(c-c*sin(e+f*x))^(5/2)),x,5,-1/2*cos(e+f*x)/(a*c*f*(a+a*sin(e+f*x))^(3/2)*(c-c*sin(e+f*x))^(3/2))+1/2*cos(e+f*x)/(a^2*c*f*(c-c*sin(e+f*x))^(3/2)*sqrt(a+a*sin(e+f*x)))+1/2*arctanh(sin(e+f*x))*cos(e+f*x)/(a^2*c^2*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))],

# Integrands of the form Cos[e+f x]^2 (a+a Sin[e+f x])^m (c-c Sin[e+f x])^n with m and/or n symbolic
[cos(e+f*x)^2*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^n,x,5,2^(3/2+n)*c^2*cos(e+f*x)^3*hypergeom([1/2*(3+2*m),1/2*(-1-2*n)],[1/2*(5+2*m)],1/2*(1+sin(e+f*x)))*(1-sin(e+f*x))^(1/2-n)*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^(-2+n)/(f*(3+2*m))],
[cos(e+f*x)^2*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^3,x,4,-1/9*2^(3/2+m)*a^4*c^3*cos(e+f*x)^9*hypergeom([9/2,-1/2-m],[11/2],1/2*(1-sin(e+f*x)))*(1+sin(e+f*x))^(-1/2-m)*(a+a*sin(e+f*x))^(-4+m)/f],
[cos(e+f*x)^2*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^2,x,4,-1/7*2^(3/2+m)*a^3*c^2*cos(e+f*x)^7*hypergeom([7/2,-1/2-m],[9/2],1/2*(1-sin(e+f*x)))*(1+sin(e+f*x))^(-1/2-m)*(a+a*sin(e+f*x))^(-3+m)/f],
[cos(e+f*x)^2*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x)),x,4,-1/5*2^(3/2+m)*a^2*c*cos(e+f*x)^5*hypergeom([5/2,-1/2-m],[7/2],1/2*(1-sin(e+f*x)))*(1+sin(e+f*x))^(-1/2-m)*(a+a*sin(e+f*x))^(-2+m)/f],
[cos(e+f*x)^2*(a+a*sin(e+f*x))^m,x,3,-1/3*2^(3/2+m)*a*cos(e+f*x)^3*hypergeom([3/2,-1/2-m],[5/2],1/2*(1-sin(e+f*x)))*(1+sin(e+f*x))^(-1/2-m)*(a+a*sin(e+f*x))^(-1+m)/f],
[cos(e+f*x)^2*(a+a*sin(e+f*x))^m/(c-c*sin(e+f*x)),x,3,-2^(3/2+m)*cos(e+f*x)*hypergeom([1/2,-1/2-m],[3/2],1/2*(1-sin(e+f*x)))*(1+sin(e+f*x))^(-1/2-m)*(a+a*sin(e+f*x))^m/(c*f)],
[cos(e+f*x)^2*(a+a*sin(e+f*x))^m/(c-c*sin(e+f*x))^2,x,4,2^(3/2+m)*hypergeom([-1/2,-1/2-m],[1/2],1/2*(1-sin(e+f*x)))*sec(e+f*x)*(1+sin(e+f*x))^(-1/2-m)*(a+a*sin(e+f*x))^(1+m)/(a*c^2*f)],
[cos(e+f*x)^2*(a+a*sin(e+f*x))^m/(c-c*sin(e+f*x))^3,x,4,1/3*2^(3/2+m)*hypergeom([-3/2,-1/2-m],[-1/2],1/2*(1-sin(e+f*x)))*sec(e+f*x)^3*(1+sin(e+f*x))^(-1/2-m)*(a+a*sin(e+f*x))^(2+m)/(a^2*c^3*f)],
[cos(e+f*x)^2*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^(5/2),x,5,24*c*cos(e+f*x)*(a+a*sin(e+f*x))^(1+m)*(c-c*sin(e+f*x))^(3/2)/(a*f*(63+32*m+4*m^2))+2*cos(e+f*x)*(a+a*sin(e+f*x))^(1+m)*(c-c*sin(e+f*x))^(5/2)/(a*f*(9+2*m))+768*c^3*cos(e+f*x)*(a+a*sin(e+f*x))^(1+m)/(a*f*(15+16*m+4*m^2)*(63+32*m+4*m^2)*sqrt(c-c*sin(e+f*x)))+192*c^2*cos(e+f*x)*(a+a*sin(e+f*x))^(1+m)*sqrt(c-c*sin(e+f*x))/(a*f*(315+286*m+84*m^2+8*m^3))],
[cos(e+f*x)^2*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^(3/2),x,4,2*cos(e+f*x)*(a+a*sin(e+f*x))^(1+m)*(c-c*sin(e+f*x))^(3/2)/(a*f*(7+2*m))+64*c^2*cos(e+f*x)*(a+a*sin(e+f*x))^(1+m)/(a*f*(105+142*m+60*m^2+8*m^3)*sqrt(c-c*sin(e+f*x)))+16*c*cos(e+f*x)*(a+a*sin(e+f*x))^(1+m)*sqrt(c-c*sin(e+f*x))/(a*f*(35+24*m+4*m^2))],
[cos(e+f*x)^2*(a+a*sin(e+f*x))^m*sqrt(c-c*sin(e+f*x)),x,3,8*c*cos(e+f*x)*(a+a*sin(e+f*x))^(1+m)/(a*f*(15+16*m+4*m^2)*sqrt(c-c*sin(e+f*x)))+2*cos(e+f*x)*(a+a*sin(e+f*x))^(1+m)*sqrt(c-c*sin(e+f*x))/(a*f*(5+2*m))],
[cos(e+f*x)^2*(a+a*sin(e+f*x))^m/sqrt(c-c*sin(e+f*x)),x,2,2*cos(e+f*x)*(a+a*sin(e+f*x))^(1+m)/(a*f*(3+2*m)*sqrt(c-c*sin(e+f*x)))],
[cos(e+f*x)^2*(a+a*sin(e+f*x))^m/(c-c*sin(e+f*x))^(3/2),x,4,cos(e+f*x)*hypergeom([1,3/2+m],[5/2+m],1/2*(1+sin(e+f*x)))*(a+a*sin(e+f*x))^(1+m)/(a*c*f*(3+2*m)*sqrt(c-c*sin(e+f*x)))],
[cos(e+f*x)^2*(a+a*sin(e+f*x))^m/(c-c*sin(e+f*x))^(5/2),x,4,1/2*cos(e+f*x)*hypergeom([2,3/2+m],[5/2+m],1/2*(1+sin(e+f*x)))*(a+a*sin(e+f*x))^(1+m)/(a*c^2*f*(3+2*m)*sqrt(c-c*sin(e+f*x)))],

#  The same rules should be used to integrate the following two problems: 
[cos(e+f*x)^2*(a+a*sin(e+f*x))^m/sqrt(c-c*sin(e+f*x)),x,2,2*cos(e+f*x)*(a+a*sin(e+f*x))^(1+m)/(a*f*(3+2*m)*sqrt(c-c*sin(e+f*x)))],
[cos(e+f*x)^2*(c+c*sin(e+f*x))^m/sqrt(a-a*sin(e+f*x)),x,2,2*cos(e+f*x)*(c+c*sin(e+f*x))^(1+m)/(c*f*(3+2*m)*sqrt(a-a*sin(e+f*x)))],
[cos(e+f*x)^2*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^(-5-m),x,4,cos(e+f*x)*(a+a*sin(e+f*x))^(1+m)*(c-c*sin(e+f*x))^(-4-m)/(a*c*f*(7+2*m))+2*cos(e+f*x)*(a+a*sin(e+f*x))^(1+m)*(c-c*sin(e+f*x))^(-3-m)/(a*c^2*f*(35+24*m+4*m^2))+2*cos(e+f*x)*(a+a*sin(e+f*x))^(1+m)*(c-c*sin(e+f*x))^(-2-m)/(a*c^3*f*(105+142*m+60*m^2+8*m^3))],
[cos(e+f*x)^2*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^(-4-m),x,3,cos(e+f*x)*(a+a*sin(e+f*x))^(1+m)*(c-c*sin(e+f*x))^(-3-m)/(a*c*f*(5+2*m))+cos(e+f*x)*(a+a*sin(e+f*x))^(1+m)*(c-c*sin(e+f*x))^(-2-m)/(a*c^2*f*(15+16*m+4*m^2))],
[cos(e+f*x)^2*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^(-3-m),x,2,cos(e+f*x)*(a+a*sin(e+f*x))^(1+m)*(c-c*sin(e+f*x))^(-2-m)/(a*c*f*(3+2*m))],
[cos(e+f*x)^2*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^(-2-m),x,5,2^(-1/2-m)*cos(e+f*x)^3*hypergeom([1/2*(3+2*m),1/2*(3+2*m)],[1/2*(5+2*m)],1/2*(1+sin(e+f*x)))*(1-sin(e+f*x))^(1/2+m)*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^(-2-m)/(f*(3+2*m))],
[cos(e+f*x)^2*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^(-1-m),x,5,2^(1/2-m)*c*cos(e+f*x)^3*hypergeom([1/2*(1+2*m),1/2*(3+2*m)],[1/2*(5+2*m)],1/2*(1+sin(e+f*x)))*(1-sin(e+f*x))^(1/2+m)*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^(-2-m)/(f*(3+2*m))],
[cos(e+f*x)^2*(a+a*sin(e+f*x))^m/(c-c*sin(e+f*x))^m,x,5,2^(3/2-m)*c^2*cos(e+f*x)^3*hypergeom([1/2*(-1+2*m),1/2*(3+2*m)],[1/2*(5+2*m)],1/2*(1+sin(e+f*x)))*(1-sin(e+f*x))^(1/2+m)*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^(-2-m)/(f*(3+2*m))],
[cos(e+f*x)^2*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^(1-m),x,5,2^(5/2-m)*c^3*cos(e+f*x)^3*hypergeom([1/2*(-3+2*m),1/2*(3+2*m)],[1/2*(5+2*m)],1/2*(1+sin(e+f*x)))*(1-sin(e+f*x))^(1/2+m)*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^(-2-m)/(f*(3+2*m))],

# Integrands of the form (g Cos[e+f x])^(3/2) (a+a Sin[e+f x])^m (c-c Sin[e+f x])^n

# Integrands of the form (g Cos[e+f x])^(3/2) (a+a Sin[e+f x])^m (c-c Sin[e+f x])^n

# Integrands of the form (g Cos[e+f x])^(3/2) (a+a Sin[e+f x])^m (c-c Sin[e+f x])^(n/2)

# Integrands of the form (g Cos[e+f x])^(3/2) (a+a Sin[e+f x])^(m/2) (c-c Sin[e+f x])^(n/2)

# m>0
[(g*cos(e+f*x))^(3/2)*(c-c*sin(e+f*x))^(7/2)*sqrt(a+a*sin(e+f*x)),x,8,10/77*a*c^2*(g*cos(e+f*x))^(5/2)*(c-c*sin(e+f*x))^(3/2)/(f*g*sqrt(a+a*sin(e+f*x)))+2/33*a*c*(g*cos(e+f*x))^(5/2)*(c-c*sin(e+f*x))^(5/2)/(f*g*sqrt(a+a*sin(e+f*x)))-2/11*a*(g*cos(e+f*x))^(5/2)*(c-c*sin(e+f*x))^(7/2)/(f*g*sqrt(a+a*sin(e+f*x)))+2/3*a*c^4*(g*cos(e+f*x))^(5/2)/(f*g*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))+2*a*c^4*g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(g*cos(e+f*x))/(f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))+2/7*a*c^3*(g*cos(e+f*x))^(5/2)*sqrt(c-c*sin(e+f*x))/(f*g*sqrt(a+a*sin(e+f*x)))],
[(g*cos(e+f*x))^(3/2)*(c-c*sin(e+f*x))^(5/2)*sqrt(a+a*sin(e+f*x)),x,7,2/21*a*c*(g*cos(e+f*x))^(5/2)*(c-c*sin(e+f*x))^(3/2)/(f*g*sqrt(a+a*sin(e+f*x)))-2/9*a*(g*cos(e+f*x))^(5/2)*(c-c*sin(e+f*x))^(5/2)/(f*g*sqrt(a+a*sin(e+f*x)))+22/45*a*c^3*(g*cos(e+f*x))^(5/2)/(f*g*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))+22/15*a*c^3*g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(g*cos(e+f*x))/(f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))+22/105*a*c^2*(g*cos(e+f*x))^(5/2)*sqrt(c-c*sin(e+f*x))/(f*g*sqrt(a+a*sin(e+f*x)))],
[(g*cos(e+f*x))^(3/2)*(c-c*sin(e+f*x))^(3/2)*sqrt(a+a*sin(e+f*x)),x,6,-2/7*a*(g*cos(e+f*x))^(5/2)*(c-c*sin(e+f*x))^(3/2)/(f*g*sqrt(a+a*sin(e+f*x)))+2/5*a*c^2*(g*cos(e+f*x))^(5/2)/(f*g*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))+6/5*a*c^2*g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(g*cos(e+f*x))/(f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))+6/35*a*c*(g*cos(e+f*x))^(5/2)*sqrt(c-c*sin(e+f*x))/(f*g*sqrt(a+a*sin(e+f*x)))],
[(g*cos(e+f*x))^(3/2)*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)),x,5,2/5*a*c*(g*cos(e+f*x))^(5/2)/(f*g*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))+6/5*a*c*g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(g*cos(e+f*x))/(f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))-2/5*a*(g*cos(e+f*x))^(5/2)*sqrt(c-c*sin(e+f*x))/(f*g*sqrt(a+a*sin(e+f*x)))],
[(g*cos(e+f*x))^(3/2)*sqrt(a+a*sin(e+f*x))/sqrt(c-c*sin(e+f*x)),x,4,-2/3*a*(g*cos(e+f*x))^(5/2)/(f*g*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))+2*a*g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(g*cos(e+f*x))/(f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))],
[(g*cos(e+f*x))^(3/2)*sqrt(a+a*sin(e+f*x))/(c-c*sin(e+f*x))^(3/2),x,4,4*a*(g*cos(e+f*x))^(5/2)/(f*g*(c-c*sin(e+f*x))^(3/2)*sqrt(a+a*sin(e+f*x)))-6*a*g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(g*cos(e+f*x))/(c*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))],
[(g*cos(e+f*x))^(3/2)*sqrt(a+a*sin(e+f*x))/(c-c*sin(e+f*x))^(5/2),x,5,4/5*a*(g*cos(e+f*x))^(5/2)/(f*g*(c-c*sin(e+f*x))^(5/2)*sqrt(a+a*sin(e+f*x)))-6/5*a*(g*cos(e+f*x))^(5/2)/(c*f*g*(c-c*sin(e+f*x))^(3/2)*sqrt(a+a*sin(e+f*x)))+6/5*a*g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(g*cos(e+f*x))/(c^2*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))],
[(g*cos(e+f*x))^(3/2)*sqrt(a+a*sin(e+f*x))/(c-c*sin(e+f*x))^(7/2),x,6,4/9*a*(g*cos(e+f*x))^(5/2)/(f*g*(c-c*sin(e+f*x))^(7/2)*sqrt(a+a*sin(e+f*x)))-2/15*a*(g*cos(e+f*x))^(5/2)/(c*f*g*(c-c*sin(e+f*x))^(5/2)*sqrt(a+a*sin(e+f*x)))-2/15*a*(g*cos(e+f*x))^(5/2)/(c^2*f*g*(c-c*sin(e+f*x))^(3/2)*sqrt(a+a*sin(e+f*x)))+2/15*a*g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(g*cos(e+f*x))/(c^3*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))],
[(g*cos(e+f*x))^(3/2)*sqrt(a+a*sin(e+f*x))/(c-c*sin(e+f*x))^(9/2),x,7,4/13*a*(g*cos(e+f*x))^(5/2)/(f*g*(c-c*sin(e+f*x))^(9/2)*sqrt(a+a*sin(e+f*x)))-2/39*a*(g*cos(e+f*x))^(5/2)/(c*f*g*(c-c*sin(e+f*x))^(7/2)*sqrt(a+a*sin(e+f*x)))-2/65*a*(g*cos(e+f*x))^(5/2)/(c^2*f*g*(c-c*sin(e+f*x))^(5/2)*sqrt(a+a*sin(e+f*x)))-2/65*a*(g*cos(e+f*x))^(5/2)/(c^3*f*g*(c-c*sin(e+f*x))^(3/2)*sqrt(a+a*sin(e+f*x)))+2/65*a*g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(g*cos(e+f*x))/(c^4*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))],
[(g*cos(e+f*x))^(3/2)*(a+a*sin(e+f*x))^(3/2)*(c-c*sin(e+f*x))^(5/2),x,8,2/33*a^2*c*(g*cos(e+f*x))^(5/2)*(c-c*sin(e+f*x))^(3/2)/(f*g*sqrt(a+a*sin(e+f*x)))-14/99*a^2*(g*cos(e+f*x))^(5/2)*(c-c*sin(e+f*x))^(5/2)/(f*g*sqrt(a+a*sin(e+f*x)))-2/11*a*(g*cos(e+f*x))^(5/2)*(c-c*sin(e+f*x))^(5/2)*sqrt(a+a*sin(e+f*x))/(f*g)+14/45*a^2*c^3*(g*cos(e+f*x))^(5/2)/(f*g*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))+14/15*a^2*c^3*g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(g*cos(e+f*x))/(f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))+2/15*a^2*c^2*(g*cos(e+f*x))^(5/2)*sqrt(c-c*sin(e+f*x))/(f*g*sqrt(a+a*sin(e+f*x)))],
[(g*cos(e+f*x))^(3/2)*(a+a*sin(e+f*x))^(3/2)*(c-c*sin(e+f*x))^(3/2),x,7,-2/9*a^2*(g*cos(e+f*x))^(5/2)*(c-c*sin(e+f*x))^(3/2)/(f*g*sqrt(a+a*sin(e+f*x)))-2/9*a*(g*cos(e+f*x))^(5/2)*(c-c*sin(e+f*x))^(3/2)*sqrt(a+a*sin(e+f*x))/(f*g)+14/45*a^2*c^2*(g*cos(e+f*x))^(5/2)/(f*g*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))+14/15*a^2*c^2*g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(g*cos(e+f*x))/(f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))+2/15*a^2*c*(g*cos(e+f*x))^(5/2)*sqrt(c-c*sin(e+f*x))/(f*g*sqrt(a+a*sin(e+f*x)))],
[(g*cos(e+f*x))^(3/2)*(a+a*sin(e+f*x))^(3/2)*sqrt(c-c*sin(e+f*x)),x,6,2/7*c*(g*cos(e+f*x))^(5/2)*(a+a*sin(e+f*x))^(3/2)/(f*g*sqrt(c-c*sin(e+f*x)))-2/5*a^2*c*(g*cos(e+f*x))^(5/2)/(f*g*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))+6/5*a^2*c*g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(g*cos(e+f*x))/(f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))-6/35*a*c*(g*cos(e+f*x))^(5/2)*sqrt(a+a*sin(e+f*x))/(f*g*sqrt(c-c*sin(e+f*x)))],
[(g*cos(e+f*x))^(3/2)*(a+a*sin(e+f*x))^(3/2)/sqrt(c-c*sin(e+f*x)),x,5,-14/15*a^2*(g*cos(e+f*x))^(5/2)/(f*g*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))+14/5*a^2*g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(g*cos(e+f*x))/(f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))-2/5*a*(g*cos(e+f*x))^(5/2)*sqrt(a+a*sin(e+f*x))/(f*g*sqrt(c-c*sin(e+f*x)))],
[(g*cos(e+f*x))^(3/2)*(a+a*sin(e+f*x))^(3/2)/(c-c*sin(e+f*x))^(3/2),x,5,4*a*(g*cos(e+f*x))^(5/2)*sqrt(a+a*sin(e+f*x))/(f*g*(c-c*sin(e+f*x))^(3/2))+14/3*a^2*(g*cos(e+f*x))^(5/2)/(c*f*g*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))-14*a^2*g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(g*cos(e+f*x))/(c*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))],
[(g*cos(e+f*x))^(3/2)*(a+a*sin(e+f*x))^(3/2)/(c-c*sin(e+f*x))^(5/2),x,5,-28/5*a^2*(g*cos(e+f*x))^(5/2)/(c*f*g*(c-c*sin(e+f*x))^(3/2)*sqrt(a+a*sin(e+f*x)))+4/5*a*(g*cos(e+f*x))^(5/2)*sqrt(a+a*sin(e+f*x))/(f*g*(c-c*sin(e+f*x))^(5/2))+42/5*a^2*g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(g*cos(e+f*x))/(c^2*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))],
[(g*cos(e+f*x))^(3/2)*(a+a*sin(e+f*x))^(3/2)/(c-c*sin(e+f*x))^(7/2),x,6,-28/45*a^2*(g*cos(e+f*x))^(5/2)/(c*f*g*(c-c*sin(e+f*x))^(5/2)*sqrt(a+a*sin(e+f*x)))+14/15*a^2*(g*cos(e+f*x))^(5/2)/(c^2*f*g*(c-c*sin(e+f*x))^(3/2)*sqrt(a+a*sin(e+f*x)))+4/9*a*(g*cos(e+f*x))^(5/2)*sqrt(a+a*sin(e+f*x))/(f*g*(c-c*sin(e+f*x))^(7/2))-14/15*a^2*g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(g*cos(e+f*x))/(c^3*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))],
[(g*cos(e+f*x))^(3/2)*(a+a*sin(e+f*x))^(3/2)/(c-c*sin(e+f*x))^(9/2),x,7,-28/117*a^2*(g*cos(e+f*x))^(5/2)/(c*f*g*(c-c*sin(e+f*x))^(7/2)*sqrt(a+a*sin(e+f*x)))+14/195*a^2*(g*cos(e+f*x))^(5/2)/(c^2*f*g*(c-c*sin(e+f*x))^(5/2)*sqrt(a+a*sin(e+f*x)))+14/195*a^2*(g*cos(e+f*x))^(5/2)/(c^3*f*g*(c-c*sin(e+f*x))^(3/2)*sqrt(a+a*sin(e+f*x)))+4/13*a*(g*cos(e+f*x))^(5/2)*sqrt(a+a*sin(e+f*x))/(f*g*(c-c*sin(e+f*x))^(9/2))-14/195*a^2*g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(g*cos(e+f*x))/(c^4*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))],
[(g*cos(e+f*x))^(3/2)*(a+a*sin(e+f*x))^(3/2)/(c-c*sin(e+f*x))^(11/2),x,8,-28/221*a^2*(g*cos(e+f*x))^(5/2)/(c*f*g*(c-c*sin(e+f*x))^(9/2)*sqrt(a+a*sin(e+f*x)))+14/663*a^2*(g*cos(e+f*x))^(5/2)/(c^2*f*g*(c-c*sin(e+f*x))^(7/2)*sqrt(a+a*sin(e+f*x)))+14/1105*a^2*(g*cos(e+f*x))^(5/2)/(c^3*f*g*(c-c*sin(e+f*x))^(5/2)*sqrt(a+a*sin(e+f*x)))+14/1105*a^2*(g*cos(e+f*x))^(5/2)/(c^4*f*g*(c-c*sin(e+f*x))^(3/2)*sqrt(a+a*sin(e+f*x)))+4/17*a*(g*cos(e+f*x))^(5/2)*sqrt(a+a*sin(e+f*x))/(f*g*(c-c*sin(e+f*x))^(11/2))-14/1105*a^2*g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(g*cos(e+f*x))/(c^5*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))],
[(g*cos(e+f*x))^(3/2)*(a+a*sin(e+f*x))^(5/2)*(c-c*sin(e+f*x))^(5/2),x,9,-2/13*a*(g*cos(e+f*x))^(5/2)*(a+a*sin(e+f*x))^(3/2)*(c-c*sin(e+f*x))^(5/2)/(f*g)+2/39*a^3*c*(g*cos(e+f*x))^(5/2)*(c-c*sin(e+f*x))^(3/2)/(f*g*sqrt(a+a*sin(e+f*x)))-14/117*a^3*(g*cos(e+f*x))^(5/2)*(c-c*sin(e+f*x))^(5/2)/(f*g*sqrt(a+a*sin(e+f*x)))-2/13*a^2*(g*cos(e+f*x))^(5/2)*(c-c*sin(e+f*x))^(5/2)*sqrt(a+a*sin(e+f*x))/(f*g)+154/585*a^3*c^3*(g*cos(e+f*x))^(5/2)/(f*g*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))+154/195*a^3*c^3*g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(g*cos(e+f*x))/(f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))+22/195*a^3*c^2*(g*cos(e+f*x))^(5/2)*sqrt(c-c*sin(e+f*x))/(f*g*sqrt(a+a*sin(e+f*x)))],
[(g*cos(e+f*x))^(3/2)*(a+a*sin(e+f*x))^(5/2)*(c-c*sin(e+f*x))^(3/2),x,8,-2/33*a*c^2*(g*cos(e+f*x))^(5/2)*(a+a*sin(e+f*x))^(3/2)/(f*g*sqrt(c-c*sin(e+f*x)))+14/99*c^2*(g*cos(e+f*x))^(5/2)*(a+a*sin(e+f*x))^(5/2)/(f*g*sqrt(c-c*sin(e+f*x)))-14/45*a^3*c^2*(g*cos(e+f*x))^(5/2)/(f*g*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))+14/15*a^3*c^2*g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(g*cos(e+f*x))/(f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))-2/15*a^2*c^2*(g*cos(e+f*x))^(5/2)*sqrt(a+a*sin(e+f*x))/(f*g*sqrt(c-c*sin(e+f*x)))+2/11*c*(g*cos(e+f*x))^(5/2)*(a+a*sin(e+f*x))^(5/2)*sqrt(c-c*sin(e+f*x))/(f*g)],
[(g*cos(e+f*x))^(3/2)*(a+a*sin(e+f*x))^(5/2)*sqrt(c-c*sin(e+f*x)),x,7,-2/21*a*c*(g*cos(e+f*x))^(5/2)*(a+a*sin(e+f*x))^(3/2)/(f*g*sqrt(c-c*sin(e+f*x)))+2/9*c*(g*cos(e+f*x))^(5/2)*(a+a*sin(e+f*x))^(5/2)/(f*g*sqrt(c-c*sin(e+f*x)))-22/45*a^3*c*(g*cos(e+f*x))^(5/2)/(f*g*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))+22/15*a^3*c*g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(g*cos(e+f*x))/(f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))-22/105*a^2*c*(g*cos(e+f*x))^(5/2)*sqrt(a+a*sin(e+f*x))/(f*g*sqrt(c-c*sin(e+f*x)))],
[(g*cos(e+f*x))^(3/2)*(a+a*sin(e+f*x))^(5/2)/sqrt(c-c*sin(e+f*x)),x,6,-2/7*a*(g*cos(e+f*x))^(5/2)*(a+a*sin(e+f*x))^(3/2)/(f*g*sqrt(c-c*sin(e+f*x)))-22/15*a^3*(g*cos(e+f*x))^(5/2)/(f*g*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))+22/5*a^3*g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(g*cos(e+f*x))/(f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))-22/35*a^2*(g*cos(e+f*x))^(5/2)*sqrt(a+a*sin(e+f*x))/(f*g*sqrt(c-c*sin(e+f*x)))],
[(g*cos(e+f*x))^(3/2)*(a+a*sin(e+f*x))^(5/2)/(c-c*sin(e+f*x))^(3/2),x,6,4*a*(g*cos(e+f*x))^(5/2)*(a+a*sin(e+f*x))^(3/2)/(f*g*(c-c*sin(e+f*x))^(3/2))+154/15*a^3*(g*cos(e+f*x))^(5/2)/(c*f*g*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))-154/5*a^3*g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(g*cos(e+f*x))/(c*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))+22/5*a^2*(g*cos(e+f*x))^(5/2)*sqrt(a+a*sin(e+f*x))/(c*f*g*sqrt(c-c*sin(e+f*x)))],
[(g*cos(e+f*x))^(3/2)*(a+a*sin(e+f*x))^(5/2)/(c-c*sin(e+f*x))^(5/2),x,6,4/5*a*(g*cos(e+f*x))^(5/2)*(a+a*sin(e+f*x))^(3/2)/(f*g*(c-c*sin(e+f*x))^(5/2))-44/5*a^2*(g*cos(e+f*x))^(5/2)*sqrt(a+a*sin(e+f*x))/(c*f*g*(c-c*sin(e+f*x))^(3/2))-154/15*a^3*(g*cos(e+f*x))^(5/2)/(c^2*f*g*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))+154/5*a^3*g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(g*cos(e+f*x))/(c^2*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))],
[(g*cos(e+f*x))^(3/2)*(a+a*sin(e+f*x))^(5/2)/(c-c*sin(e+f*x))^(7/2),x,6,4/9*a*(g*cos(e+f*x))^(5/2)*(a+a*sin(e+f*x))^(3/2)/(f*g*(c-c*sin(e+f*x))^(7/2))+308/45*a^3*(g*cos(e+f*x))^(5/2)/(c^2*f*g*(c-c*sin(e+f*x))^(3/2)*sqrt(a+a*sin(e+f*x)))-44/45*a^2*(g*cos(e+f*x))^(5/2)*sqrt(a+a*sin(e+f*x))/(c*f*g*(c-c*sin(e+f*x))^(5/2))-154/15*a^3*g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(g*cos(e+f*x))/(c^3*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))],
[(g*cos(e+f*x))^(3/2)*(a+a*sin(e+f*x))^(5/2)/(c-c*sin(e+f*x))^(9/2),x,7,4/13*a*(g*cos(e+f*x))^(5/2)*(a+a*sin(e+f*x))^(3/2)/(f*g*(c-c*sin(e+f*x))^(9/2))+308/585*a^3*(g*cos(e+f*x))^(5/2)/(c^2*f*g*(c-c*sin(e+f*x))^(5/2)*sqrt(a+a*sin(e+f*x)))-154/195*a^3*(g*cos(e+f*x))^(5/2)/(c^3*f*g*(c-c*sin(e+f*x))^(3/2)*sqrt(a+a*sin(e+f*x)))-44/117*a^2*(g*cos(e+f*x))^(5/2)*sqrt(a+a*sin(e+f*x))/(c*f*g*(c-c*sin(e+f*x))^(7/2))+154/195*a^3*g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(g*cos(e+f*x))/(c^4*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))],
[(g*cos(e+f*x))^(3/2)*(a+a*sin(e+f*x))^(5/2)/(c-c*sin(e+f*x))^(11/2),x,8,4/17*a*(g*cos(e+f*x))^(5/2)*(a+a*sin(e+f*x))^(3/2)/(f*g*(c-c*sin(e+f*x))^(11/2))+308/1989*a^3*(g*cos(e+f*x))^(5/2)/(c^2*f*g*(c-c*sin(e+f*x))^(7/2)*sqrt(a+a*sin(e+f*x)))-154/3315*a^3*(g*cos(e+f*x))^(5/2)/(c^3*f*g*(c-c*sin(e+f*x))^(5/2)*sqrt(a+a*sin(e+f*x)))-154/3315*a^3*(g*cos(e+f*x))^(5/2)/(c^4*f*g*(c-c*sin(e+f*x))^(3/2)*sqrt(a+a*sin(e+f*x)))-44/221*a^2*(g*cos(e+f*x))^(5/2)*sqrt(a+a*sin(e+f*x))/(c*f*g*(c-c*sin(e+f*x))^(9/2))+154/3315*a^3*g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(g*cos(e+f*x))/(c^5*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))],
[(g*cos(e+f*x))^(3/2)*(a+a*sin(e+f*x))^(5/2)/(c-c*sin(e+f*x))^(13/2),x,9,4/21*a*(g*cos(e+f*x))^(5/2)*(a+a*sin(e+f*x))^(3/2)/(f*g*(c-c*sin(e+f*x))^(13/2))+44/663*a^3*(g*cos(e+f*x))^(5/2)/(c^2*f*g*(c-c*sin(e+f*x))^(9/2)*sqrt(a+a*sin(e+f*x)))-22/1989*a^3*(g*cos(e+f*x))^(5/2)/(c^3*f*g*(c-c*sin(e+f*x))^(7/2)*sqrt(a+a*sin(e+f*x)))-22/3315*a^3*(g*cos(e+f*x))^(5/2)/(c^4*f*g*(c-c*sin(e+f*x))^(5/2)*sqrt(a+a*sin(e+f*x)))-22/3315*a^3*(g*cos(e+f*x))^(5/2)/(c^5*f*g*(c-c*sin(e+f*x))^(3/2)*sqrt(a+a*sin(e+f*x)))-44/357*a^2*(g*cos(e+f*x))^(5/2)*sqrt(a+a*sin(e+f*x))/(c*f*g*(c-c*sin(e+f*x))^(11/2))+22/3315*a^3*g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(g*cos(e+f*x))/(c^6*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))],
[(g*cos(e+f*x))^(3/2)*(a+a*sin(e+f*x))^(7/2)*(c-c*sin(e+f*x))^(5/2),x,10,2/15*c*(g*cos(e+f*x))^(5/2)*(a+a*sin(e+f*x))^(7/2)*(c-c*sin(e+f*x))^(3/2)/(f*g)-2/39*a^2*c^3*(g*cos(e+f*x))^(5/2)*(a+a*sin(e+f*x))^(3/2)/(f*g*sqrt(c-c*sin(e+f*x)))-14/585*a*c^3*(g*cos(e+f*x))^(5/2)*(a+a*sin(e+f*x))^(5/2)/(f*g*sqrt(c-c*sin(e+f*x)))+14/195*c^3*(g*cos(e+f*x))^(5/2)*(a+a*sin(e+f*x))^(7/2)/(f*g*sqrt(c-c*sin(e+f*x)))-154/585*a^4*c^3*(g*cos(e+f*x))^(5/2)/(f*g*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))+154/195*a^4*c^3*g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(g*cos(e+f*x))/(f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))-22/195*a^3*c^3*(g*cos(e+f*x))^(5/2)*sqrt(a+a*sin(e+f*x))/(f*g*sqrt(c-c*sin(e+f*x)))+22/195*c^2*(g*cos(e+f*x))^(5/2)*(a+a*sin(e+f*x))^(7/2)*sqrt(c-c*sin(e+f*x))/(f*g)],
[(g*cos(e+f*x))^(3/2)*(a+a*sin(e+f*x))^(7/2)*(c-c*sin(e+f*x))^(3/2),x,9,-10/143*a^2*c^2*(g*cos(e+f*x))^(5/2)*(a+a*sin(e+f*x))^(3/2)/(f*g*sqrt(c-c*sin(e+f*x)))-14/429*a*c^2*(g*cos(e+f*x))^(5/2)*(a+a*sin(e+f*x))^(5/2)/(f*g*sqrt(c-c*sin(e+f*x)))+14/143*c^2*(g*cos(e+f*x))^(5/2)*(a+a*sin(e+f*x))^(7/2)/(f*g*sqrt(c-c*sin(e+f*x)))-14/39*a^4*c^2*(g*cos(e+f*x))^(5/2)/(f*g*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))+14/13*a^4*c^2*g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(g*cos(e+f*x))/(f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))-2/13*a^3*c^2*(g*cos(e+f*x))^(5/2)*sqrt(a+a*sin(e+f*x))/(f*g*sqrt(c-c*sin(e+f*x)))+2/13*c*(g*cos(e+f*x))^(5/2)*(a+a*sin(e+f*x))^(7/2)*sqrt(c-c*sin(e+f*x))/(f*g)],
[(g*cos(e+f*x))^(3/2)*(a+a*sin(e+f*x))^(7/2)*sqrt(c-c*sin(e+f*x)),x,8,-10/77*a^2*c*(g*cos(e+f*x))^(5/2)*(a+a*sin(e+f*x))^(3/2)/(f*g*sqrt(c-c*sin(e+f*x)))-2/33*a*c*(g*cos(e+f*x))^(5/2)*(a+a*sin(e+f*x))^(5/2)/(f*g*sqrt(c-c*sin(e+f*x)))+2/11*c*(g*cos(e+f*x))^(5/2)*(a+a*sin(e+f*x))^(7/2)/(f*g*sqrt(c-c*sin(e+f*x)))-2/3*a^4*c*(g*cos(e+f*x))^(5/2)/(f*g*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))+2*a^4*c*g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(g*cos(e+f*x))/(f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))-2/7*a^3*c*(g*cos(e+f*x))^(5/2)*sqrt(a+a*sin(e+f*x))/(f*g*sqrt(c-c*sin(e+f*x)))],
[(g*cos(e+f*x))^(3/2)*(a+a*sin(e+f*x))^(7/2)/sqrt(c-c*sin(e+f*x)),x,7,-10/21*a^2*(g*cos(e+f*x))^(5/2)*(a+a*sin(e+f*x))^(3/2)/(f*g*sqrt(c-c*sin(e+f*x)))-2/9*a*(g*cos(e+f*x))^(5/2)*(a+a*sin(e+f*x))^(5/2)/(f*g*sqrt(c-c*sin(e+f*x)))-22/9*a^4*(g*cos(e+f*x))^(5/2)/(f*g*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))+22/3*a^4*g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(g*cos(e+f*x))/(f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))-22/21*a^3*(g*cos(e+f*x))^(5/2)*sqrt(a+a*sin(e+f*x))/(f*g*sqrt(c-c*sin(e+f*x)))],
[(g*cos(e+f*x))^(3/2)*(a+a*sin(e+f*x))^(7/2)/(c-c*sin(e+f*x))^(3/2),x,7,4*a*(g*cos(e+f*x))^(5/2)*(a+a*sin(e+f*x))^(5/2)/(f*g*(c-c*sin(e+f*x))^(3/2))+30/7*a^2*(g*cos(e+f*x))^(5/2)*(a+a*sin(e+f*x))^(3/2)/(c*f*g*sqrt(c-c*sin(e+f*x)))+22*a^4*(g*cos(e+f*x))^(5/2)/(c*f*g*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))-66*a^4*g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(g*cos(e+f*x))/(c*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))+66/7*a^3*(g*cos(e+f*x))^(5/2)*sqrt(a+a*sin(e+f*x))/(c*f*g*sqrt(c-c*sin(e+f*x)))],
[(g*cos(e+f*x))^(3/2)*(a+a*sin(e+f*x))^(7/2)/(c-c*sin(e+f*x))^(5/2),x,7,4/5*a*(g*cos(e+f*x))^(5/2)*(a+a*sin(e+f*x))^(5/2)/(f*g*(c-c*sin(e+f*x))^(5/2))-12*a^2*(g*cos(e+f*x))^(5/2)*(a+a*sin(e+f*x))^(3/2)/(c*f*g*(c-c*sin(e+f*x))^(3/2))-154/5*a^4*(g*cos(e+f*x))^(5/2)/(c^2*f*g*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))+462/5*a^4*g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(g*cos(e+f*x))/(c^2*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))-66/5*a^3*(g*cos(e+f*x))^(5/2)*sqrt(a+a*sin(e+f*x))/(c^2*f*g*sqrt(c-c*sin(e+f*x)))],
[(g*cos(e+f*x))^(3/2)*(a+a*sin(e+f*x))^(7/2)/(c-c*sin(e+f*x))^(7/2),x,7,4/9*a*(g*cos(e+f*x))^(5/2)*(a+a*sin(e+f*x))^(5/2)/(f*g*(c-c*sin(e+f*x))^(7/2))-4/3*a^2*(g*cos(e+f*x))^(5/2)*(a+a*sin(e+f*x))^(3/2)/(c*f*g*(c-c*sin(e+f*x))^(5/2))+44/3*a^3*(g*cos(e+f*x))^(5/2)*sqrt(a+a*sin(e+f*x))/(c^2*f*g*(c-c*sin(e+f*x))^(3/2))+154/9*a^4*(g*cos(e+f*x))^(5/2)/(c^3*f*g*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))-154/3*a^4*g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(g*cos(e+f*x))/(c^3*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))],
[(g*cos(e+f*x))^(3/2)*(a+a*sin(e+f*x))^(7/2)/(c-c*sin(e+f*x))^(9/2),x,7,4/13*a*(g*cos(e+f*x))^(5/2)*(a+a*sin(e+f*x))^(5/2)/(f*g*(c-c*sin(e+f*x))^(9/2))-20/39*a^2*(g*cos(e+f*x))^(5/2)*(a+a*sin(e+f*x))^(3/2)/(c*f*g*(c-c*sin(e+f*x))^(7/2))-308/39*a^4*(g*cos(e+f*x))^(5/2)/(c^3*f*g*(c-c*sin(e+f*x))^(3/2)*sqrt(a+a*sin(e+f*x)))+44/39*a^3*(g*cos(e+f*x))^(5/2)*sqrt(a+a*sin(e+f*x))/(c^2*f*g*(c-c*sin(e+f*x))^(5/2))+154/13*a^4*g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(g*cos(e+f*x))/(c^4*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))],
[(g*cos(e+f*x))^(3/2)*(a+a*sin(e+f*x))^(7/2)/(c-c*sin(e+f*x))^(11/2),x,8,4/17*a*(g*cos(e+f*x))^(5/2)*(a+a*sin(e+f*x))^(5/2)/(f*g*(c-c*sin(e+f*x))^(11/2))-60/221*a^2*(g*cos(e+f*x))^(5/2)*(a+a*sin(e+f*x))^(3/2)/(c*f*g*(c-c*sin(e+f*x))^(9/2))-308/663*a^4*(g*cos(e+f*x))^(5/2)/(c^3*f*g*(c-c*sin(e+f*x))^(5/2)*sqrt(a+a*sin(e+f*x)))+154/221*a^4*(g*cos(e+f*x))^(5/2)/(c^4*f*g*(c-c*sin(e+f*x))^(3/2)*sqrt(a+a*sin(e+f*x)))+220/663*a^3*(g*cos(e+f*x))^(5/2)*sqrt(a+a*sin(e+f*x))/(c^2*f*g*(c-c*sin(e+f*x))^(7/2))-154/221*a^4*g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(g*cos(e+f*x))/(c^5*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))],
[(g*cos(e+f*x))^(3/2)*(a+a*sin(e+f*x))^(7/2)/(c-c*sin(e+f*x))^(13/2),x,9,4/21*a*(g*cos(e+f*x))^(5/2)*(a+a*sin(e+f*x))^(5/2)/(f*g*(c-c*sin(e+f*x))^(13/2))-20/119*a^2*(g*cos(e+f*x))^(5/2)*(a+a*sin(e+f*x))^(3/2)/(c*f*g*(c-c*sin(e+f*x))^(11/2))-220/1989*a^4*(g*cos(e+f*x))^(5/2)/(c^3*f*g*(c-c*sin(e+f*x))^(7/2)*sqrt(a+a*sin(e+f*x)))+22/663*a^4*(g*cos(e+f*x))^(5/2)/(c^4*f*g*(c-c*sin(e+f*x))^(5/2)*sqrt(a+a*sin(e+f*x)))+22/663*a^4*(g*cos(e+f*x))^(5/2)/(c^5*f*g*(c-c*sin(e+f*x))^(3/2)*sqrt(a+a*sin(e+f*x)))+220/1547*a^3*(g*cos(e+f*x))^(5/2)*sqrt(a+a*sin(e+f*x))/(c^2*f*g*(c-c*sin(e+f*x))^(9/2))-22/663*a^4*g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(g*cos(e+f*x))/(c^6*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))],
[(g*cos(e+f*x))^(3/2)*(a+a*sin(e+f*x))^(7/2)/(c-c*sin(e+f*x))^(15/2),x,10,4/25*a*(g*cos(e+f*x))^(5/2)*(a+a*sin(e+f*x))^(5/2)/(f*g*(c-c*sin(e+f*x))^(15/2))-4/35*a^2*(g*cos(e+f*x))^(5/2)*(a+a*sin(e+f*x))^(3/2)/(c*f*g*(c-c*sin(e+f*x))^(13/2))-44/1105*a^4*(g*cos(e+f*x))^(5/2)/(c^3*f*g*(c-c*sin(e+f*x))^(9/2)*sqrt(a+a*sin(e+f*x)))+22/3315*a^4*(g*cos(e+f*x))^(5/2)/(c^4*f*g*(c-c*sin(e+f*x))^(7/2)*sqrt(a+a*sin(e+f*x)))+22/5525*a^4*(g*cos(e+f*x))^(5/2)/(c^5*f*g*(c-c*sin(e+f*x))^(5/2)*sqrt(a+a*sin(e+f*x)))+22/5525*a^4*(g*cos(e+f*x))^(5/2)/(c^6*f*g*(c-c*sin(e+f*x))^(3/2)*sqrt(a+a*sin(e+f*x)))+44/595*a^3*(g*cos(e+f*x))^(5/2)*sqrt(a+a*sin(e+f*x))/(c^2*f*g*(c-c*sin(e+f*x))^(11/2))-22/5525*a^4*g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(g*cos(e+f*x))/(c^7*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))],

# m<0
[(g*cos(e+f*x))^(3/2)*(c-c*sin(e+f*x))^(5/2)/sqrt(a+a*sin(e+f*x)),x,6,2/7*c*(g*cos(e+f*x))^(5/2)*(c-c*sin(e+f*x))^(3/2)/(f*g*sqrt(a+a*sin(e+f*x)))+22/15*c^3*(g*cos(e+f*x))^(5/2)/(f*g*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))+22/5*c^3*g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(g*cos(e+f*x))/(f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))+22/35*c^2*(g*cos(e+f*x))^(5/2)*sqrt(c-c*sin(e+f*x))/(f*g*sqrt(a+a*sin(e+f*x)))],
[(g*cos(e+f*x))^(3/2)*(c-c*sin(e+f*x))^(3/2)/sqrt(a+a*sin(e+f*x)),x,5,14/15*c^2*(g*cos(e+f*x))^(5/2)/(f*g*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))+14/5*c^2*g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(g*cos(e+f*x))/(f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))+2/5*c*(g*cos(e+f*x))^(5/2)*sqrt(c-c*sin(e+f*x))/(f*g*sqrt(a+a*sin(e+f*x)))],
[(g*cos(e+f*x))^(3/2)*sqrt(c-c*sin(e+f*x))/sqrt(a+a*sin(e+f*x)),x,4,2/3*c*(g*cos(e+f*x))^(5/2)/(f*g*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))+2*c*g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(g*cos(e+f*x))/(f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))],
[(g*cos(e+f*x))^(3/2)/(sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x))),x,3,2*g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(g*cos(e+f*x))/(f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))],
[(g*cos(e+f*x))^(3/2)/((c-c*sin(e+f*x))^(3/2)*sqrt(a+a*sin(e+f*x))),x,4,2*(g*cos(e+f*x))^(5/2)/(f*g*(c-c*sin(e+f*x))^(3/2)*sqrt(a+a*sin(e+f*x)))-2*g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(g*cos(e+f*x))/(c*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))],
[(g*cos(e+f*x))^(3/2)/((c-c*sin(e+f*x))^(5/2)*sqrt(a+a*sin(e+f*x))),x,5,2/5*(g*cos(e+f*x))^(5/2)/(f*g*(c-c*sin(e+f*x))^(5/2)*sqrt(a+a*sin(e+f*x)))+2/5*(g*cos(e+f*x))^(5/2)/(c*f*g*(c-c*sin(e+f*x))^(3/2)*sqrt(a+a*sin(e+f*x)))-2/5*g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(g*cos(e+f*x))/(c^2*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))],
[(g*cos(e+f*x))^(3/2)/((c-c*sin(e+f*x))^(7/2)*sqrt(a+a*sin(e+f*x))),x,6,2/9*(g*cos(e+f*x))^(5/2)/(f*g*(c-c*sin(e+f*x))^(7/2)*sqrt(a+a*sin(e+f*x)))+2/15*(g*cos(e+f*x))^(5/2)/(c*f*g*(c-c*sin(e+f*x))^(5/2)*sqrt(a+a*sin(e+f*x)))+2/15*(g*cos(e+f*x))^(5/2)/(c^2*f*g*(c-c*sin(e+f*x))^(3/2)*sqrt(a+a*sin(e+f*x)))-2/15*g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(g*cos(e+f*x))/(c^3*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))],
[(g*cos(e+f*x))^(3/2)*(c-c*sin(e+f*x))^(7/2)/(a+a*sin(e+f*x))^(3/2),x,7,-4*c*(g*cos(e+f*x))^(5/2)*(c-c*sin(e+f*x))^(5/2)/(f*g*(a+a*sin(e+f*x))^(3/2))-30/7*c^2*(g*cos(e+f*x))^(5/2)*(c-c*sin(e+f*x))^(3/2)/(a*f*g*sqrt(a+a*sin(e+f*x)))-22*c^4*(g*cos(e+f*x))^(5/2)/(a*f*g*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))-66*c^4*g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(g*cos(e+f*x))/(a*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))-66/7*c^3*(g*cos(e+f*x))^(5/2)*sqrt(c-c*sin(e+f*x))/(a*f*g*sqrt(a+a*sin(e+f*x)))],
[(g*cos(e+f*x))^(3/2)*(c-c*sin(e+f*x))^(5/2)/(a+a*sin(e+f*x))^(3/2),x,6,-4*c*(g*cos(e+f*x))^(5/2)*(c-c*sin(e+f*x))^(3/2)/(f*g*(a+a*sin(e+f*x))^(3/2))-154/15*c^3*(g*cos(e+f*x))^(5/2)/(a*f*g*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))-154/5*c^3*g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(g*cos(e+f*x))/(a*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))-22/5*c^2*(g*cos(e+f*x))^(5/2)*sqrt(c-c*sin(e+f*x))/(a*f*g*sqrt(a+a*sin(e+f*x)))],
[(g*cos(e+f*x))^(3/2)*(c-c*sin(e+f*x))^(3/2)/(a+a*sin(e+f*x))^(3/2),x,5,-14/3*c^2*(g*cos(e+f*x))^(5/2)/(a*f*g*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))-14*c^2*g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(g*cos(e+f*x))/(a*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))-4*c*(g*cos(e+f*x))^(5/2)*sqrt(c-c*sin(e+f*x))/(f*g*(a+a*sin(e+f*x))^(3/2))],
[(g*cos(e+f*x))^(3/2)*sqrt(c-c*sin(e+f*x))/(a+a*sin(e+f*x))^(3/2),x,4,-4*c*(g*cos(e+f*x))^(5/2)/(f*g*(a+a*sin(e+f*x))^(3/2)*sqrt(c-c*sin(e+f*x)))-6*c*g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(g*cos(e+f*x))/(a*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))],
[(g*cos(e+f*x))^(3/2)/((a+a*sin(e+f*x))^(3/2)*sqrt(c-c*sin(e+f*x))),x,4,-2*(g*cos(e+f*x))^(5/2)/(f*g*(a+a*sin(e+f*x))^(3/2)*sqrt(c-c*sin(e+f*x)))-2*g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(g*cos(e+f*x))/(a*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))],
[(g*cos(e+f*x))^(3/2)/((a+a*sin(e+f*x))^(3/2)*(c-c*sin(e+f*x))^(3/2)),x,5,-2*(g*cos(e+f*x))^(5/2)/(f*g*(a+a*sin(e+f*x))^(3/2)*(c-c*sin(e+f*x))^(3/2))+2*(g*cos(e+f*x))^(5/2)/(a*f*g*(c-c*sin(e+f*x))^(3/2)*sqrt(a+a*sin(e+f*x)))-2*g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(g*cos(e+f*x))/(a*c*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))],
[(g*cos(e+f*x))^(3/2)/((a+a*sin(e+f*x))^(3/2)*(c-c*sin(e+f*x))^(5/2)),x,6,-2*(g*cos(e+f*x))^(5/2)/(f*g*(a+a*sin(e+f*x))^(3/2)*(c-c*sin(e+f*x))^(5/2))+6/5*(g*cos(e+f*x))^(5/2)/(a*f*g*(c-c*sin(e+f*x))^(5/2)*sqrt(a+a*sin(e+f*x)))+6/5*(g*cos(e+f*x))^(5/2)/(a*c*f*g*(c-c*sin(e+f*x))^(3/2)*sqrt(a+a*sin(e+f*x)))-6/5*g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(g*cos(e+f*x))/(a*c^2*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))],
[(g*cos(e+f*x))^(3/2)/((a+a*sin(e+f*x))^(3/2)*(c-c*sin(e+f*x))^(7/2)),x,7,-2*(g*cos(e+f*x))^(5/2)/(f*g*(a+a*sin(e+f*x))^(3/2)*(c-c*sin(e+f*x))^(7/2))+10/9*(g*cos(e+f*x))^(5/2)/(a*f*g*(c-c*sin(e+f*x))^(7/2)*sqrt(a+a*sin(e+f*x)))+2/3*(g*cos(e+f*x))^(5/2)/(a*c*f*g*(c-c*sin(e+f*x))^(5/2)*sqrt(a+a*sin(e+f*x)))+2/3*(g*cos(e+f*x))^(5/2)/(a*c^2*f*g*(c-c*sin(e+f*x))^(3/2)*sqrt(a+a*sin(e+f*x)))-2/3*g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(g*cos(e+f*x))/(a*c^3*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))],
[(g*cos(e+f*x))^(3/2)*(c-c*sin(e+f*x))^(9/2)/(a+a*sin(e+f*x))^(5/2),x,8,76/5*c^2*(g*cos(e+f*x))^(5/2)*(c-c*sin(e+f*x))^(5/2)/(a*f*g*(a+a*sin(e+f*x))^(3/2))-4/5*c*(g*cos(e+f*x))^(5/2)*(c-c*sin(e+f*x))^(7/2)/(f*g*(a+a*sin(e+f*x))^(5/2))+114/7*c^3*(g*cos(e+f*x))^(5/2)*(c-c*sin(e+f*x))^(3/2)/(a^2*f*g*sqrt(a+a*sin(e+f*x)))+418/5*c^5*(g*cos(e+f*x))^(5/2)/(a^2*f*g*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))+1254/5*c^5*g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(g*cos(e+f*x))/(a^2*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))+1254/35*c^4*(g*cos(e+f*x))^(5/2)*sqrt(c-c*sin(e+f*x))/(a^2*f*g*sqrt(a+a*sin(e+f*x)))],
[(g*cos(e+f*x))^(3/2)*(c-c*sin(e+f*x))^(7/2)/(a+a*sin(e+f*x))^(5/2),x,7,12*c^2*(g*cos(e+f*x))^(5/2)*(c-c*sin(e+f*x))^(3/2)/(a*f*g*(a+a*sin(e+f*x))^(3/2))-4/5*c*(g*cos(e+f*x))^(5/2)*(c-c*sin(e+f*x))^(5/2)/(f*g*(a+a*sin(e+f*x))^(5/2))+154/5*c^4*(g*cos(e+f*x))^(5/2)/(a^2*f*g*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))+462/5*c^4*g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(g*cos(e+f*x))/(a^2*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))+66/5*c^3*(g*cos(e+f*x))^(5/2)*sqrt(c-c*sin(e+f*x))/(a^2*f*g*sqrt(a+a*sin(e+f*x)))],
[(g*cos(e+f*x))^(3/2)*(c-c*sin(e+f*x))^(5/2)/(a+a*sin(e+f*x))^(5/2),x,6,-4/5*c*(g*cos(e+f*x))^(5/2)*(c-c*sin(e+f*x))^(3/2)/(f*g*(a+a*sin(e+f*x))^(5/2))+154/15*c^3*(g*cos(e+f*x))^(5/2)/(a^2*f*g*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))+154/5*c^3*g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(g*cos(e+f*x))/(a^2*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))+44/5*c^2*(g*cos(e+f*x))^(5/2)*sqrt(c-c*sin(e+f*x))/(a*f*g*(a+a*sin(e+f*x))^(3/2))],
[(g*cos(e+f*x))^(3/2)*(c-c*sin(e+f*x))^(3/2)/(a+a*sin(e+f*x))^(5/2),x,5,28/5*c^2*(g*cos(e+f*x))^(5/2)/(a*f*g*(a+a*sin(e+f*x))^(3/2)*sqrt(c-c*sin(e+f*x)))+42/5*c^2*g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(g*cos(e+f*x))/(a^2*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))-4/5*c*(g*cos(e+f*x))^(5/2)*sqrt(c-c*sin(e+f*x))/(f*g*(a+a*sin(e+f*x))^(5/2))],
[(g*cos(e+f*x))^(3/2)*sqrt(c-c*sin(e+f*x))/(a+a*sin(e+f*x))^(5/2),x,5,-4/5*c*(g*cos(e+f*x))^(5/2)/(f*g*(a+a*sin(e+f*x))^(5/2)*sqrt(c-c*sin(e+f*x)))+6/5*c*(g*cos(e+f*x))^(5/2)/(a*f*g*(a+a*sin(e+f*x))^(3/2)*sqrt(c-c*sin(e+f*x)))+6/5*c*g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(g*cos(e+f*x))/(a^2*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))],
[(g*cos(e+f*x))^(3/2)/((a+a*sin(e+f*x))^(5/2)*sqrt(c-c*sin(e+f*x))),x,5,-2/5*(g*cos(e+f*x))^(5/2)/(f*g*(a+a*sin(e+f*x))^(5/2)*sqrt(c-c*sin(e+f*x)))-2/5*(g*cos(e+f*x))^(5/2)/(a*f*g*(a+a*sin(e+f*x))^(3/2)*sqrt(c-c*sin(e+f*x)))-2/5*g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(g*cos(e+f*x))/(a^2*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))],
[(g*cos(e+f*x))^(3/2)/((a+a*sin(e+f*x))^(5/2)*(c-c*sin(e+f*x))^(3/2)),x,6,2*(g*cos(e+f*x))^(5/2)/(f*g*(a+a*sin(e+f*x))^(5/2)*(c-c*sin(e+f*x))^(3/2))-6/5*(g*cos(e+f*x))^(5/2)/(c*f*g*(a+a*sin(e+f*x))^(5/2)*sqrt(c-c*sin(e+f*x)))-6/5*(g*cos(e+f*x))^(5/2)/(a*c*f*g*(a+a*sin(e+f*x))^(3/2)*sqrt(c-c*sin(e+f*x)))-6/5*g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(g*cos(e+f*x))/(a^2*c*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))],
[(g*cos(e+f*x))^(3/2)/((a+a*sin(e+f*x))^(5/2)*(c-c*sin(e+f*x))^(5/2)),x,7,-2/5*(g*cos(e+f*x))^(5/2)/(f*g*(a+a*sin(e+f*x))^(5/2)*(c-c*sin(e+f*x))^(5/2))-2*(g*cos(e+f*x))^(5/2)/(a*f*g*(a+a*sin(e+f*x))^(3/2)*(c-c*sin(e+f*x))^(5/2))+6/5*(g*cos(e+f*x))^(5/2)/(a^2*f*g*(c-c*sin(e+f*x))^(5/2)*sqrt(a+a*sin(e+f*x)))+6/5*(g*cos(e+f*x))^(5/2)/(a^2*c*f*g*(c-c*sin(e+f*x))^(3/2)*sqrt(a+a*sin(e+f*x)))-6/5*g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(g*cos(e+f*x))/(a^2*c^2*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))],
[(g*cos(e+f*x))^(3/2)/((a+a*sin(e+f*x))^(5/2)*(c-c*sin(e+f*x))^(7/2)),x,8,-2/5*(g*cos(e+f*x))^(5/2)/(f*g*(a+a*sin(e+f*x))^(5/2)*(c-c*sin(e+f*x))^(7/2))-14/5*(g*cos(e+f*x))^(5/2)/(a*f*g*(a+a*sin(e+f*x))^(3/2)*(c-c*sin(e+f*x))^(7/2))+14/9*(g*cos(e+f*x))^(5/2)/(a^2*f*g*(c-c*sin(e+f*x))^(7/2)*sqrt(a+a*sin(e+f*x)))+14/15*(g*cos(e+f*x))^(5/2)/(a^2*c*f*g*(c-c*sin(e+f*x))^(5/2)*sqrt(a+a*sin(e+f*x)))+14/15*(g*cos(e+f*x))^(5/2)/(a^2*c^2*f*g*(c-c*sin(e+f*x))^(3/2)*sqrt(a+a*sin(e+f*x)))-14/15*g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))*sqrt(g*cos(e+f*x))/(a^2*c^3*f*sqrt(a+a*sin(e+f*x))*sqrt(c-c*sin(e+f*x)))],

# Integrands of the form (g Cos[e+f x])^(3/2) (a+a Sin[e+f x])^m (c-c Sin[e+f x])^n with m and/or n symbolic
[(g*cos(e+f*x))^(3/2)*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^n,x,4,2^(9/4+n)*c*(g*cos(e+f*x))^(5/2)*hypergeom([1/4*(5+4*m),1/4*(-1-4*n)],[1/4*(9+4*m)],1/2*(1+sin(e+f*x)))*(1-sin(e+f*x))^(-1/4-n)*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^(-1+n)/(f*g*(5+4*m))],
[(g*cos(e+f*x))^(3/2)*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^3,x,4,-1/17*2^(9/4+m)*a^4*c^3*(g*cos(e+f*x))^(17/2)*hypergeom([17/4,-1/4-m],[21/4],1/2*(1-sin(e+f*x)))*(1+sin(e+f*x))^(-1/4-m)*(a+a*sin(e+f*x))^(-4+m)/(f*g^7)],
[(g*cos(e+f*x))^(3/2)*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^2,x,4,-1/13*2^(9/4+m)*a^3*c^2*(g*cos(e+f*x))^(13/2)*hypergeom([13/4,-1/4-m],[17/4],1/2*(1-sin(e+f*x)))*(1+sin(e+f*x))^(-1/4-m)*(a+a*sin(e+f*x))^(-3+m)/(f*g^5)],
[(g*cos(e+f*x))^(3/2)*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x)),x,4,-1/9*2^(9/4+m)*a^2*c*(g*cos(e+f*x))^(9/2)*hypergeom([9/4,-1/4-m],[13/4],1/2*(1-sin(e+f*x)))*(1+sin(e+f*x))^(-1/4-m)*(a+a*sin(e+f*x))^(-2+m)/(f*g^3)],
[(g*cos(e+f*x))^(3/2)*(a+a*sin(e+f*x))^m,x,3,-1/5*2^(9/4+m)*a*(g*cos(e+f*x))^(5/2)*hypergeom([5/4,-1/4-m],[9/4],1/2*(1-sin(e+f*x)))*(1+sin(e+f*x))^(-1/4-m)*(a+a*sin(e+f*x))^(-1+m)/(f*g)],
[(g*cos(e+f*x))^(3/2)*(a+a*sin(e+f*x))^m/(c-c*sin(e+f*x)),x,4,-2^(9/4+m)*g*hypergeom([1/4,-1/4-m],[5/4],1/2*(1-sin(e+f*x)))*(1+sin(e+f*x))^(-1/4-m)*(a+a*sin(e+f*x))^m*sqrt(g*cos(e+f*x))/(c*f)],
[(g*cos(e+f*x))^(3/2)*(a+a*sin(e+f*x))^m/(c-c*sin(e+f*x))^2,x,4,1/3*2^(9/4+m)*g^3*hypergeom([-3/4,-1/4-m],[1/4],1/2*(1-sin(e+f*x)))*(1+sin(e+f*x))^(-1/4-m)*(a+a*sin(e+f*x))^(1+m)/(a*c^2*f*(g*cos(e+f*x))^(3/2))],
[(g*cos(e+f*x))^(3/2)*(a+a*sin(e+f*x))^m/(c-c*sin(e+f*x))^3,x,4,1/7*2^(9/4+m)*g^5*hypergeom([-7/4,-1/4-m],[-3/4],1/2*(1-sin(e+f*x)))*(1+sin(e+f*x))^(-1/4-m)*(a+a*sin(e+f*x))^(2+m)/(a^2*c^3*f*(g*cos(e+f*x))^(7/2))],
[(g*cos(e+f*x))^(3/2)*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^(5/2),x,4,-1/15*2^(9/4+m)*a^3*c^2*(g*cos(e+f*x))^(15/2)*hypergeom([15/4,-1/4-m],[19/4],1/2*(1-sin(e+f*x)))*sec(e+f*x)*(1+sin(e+f*x))^(-1/4-m)*(a+a*sin(e+f*x))^(-3+m)*sqrt(c-c*sin(e+f*x))/(f*g^6)],
[(g*cos(e+f*x))^(3/2)*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^(3/2),x,4,-1/11*2^(9/4+m)*a^2*c*(g*cos(e+f*x))^(11/2)*hypergeom([11/4,-1/4-m],[15/4],1/2*(1-sin(e+f*x)))*sec(e+f*x)*(1+sin(e+f*x))^(-1/4-m)*(a+a*sin(e+f*x))^(-2+m)*sqrt(c-c*sin(e+f*x))/(f*g^4)],
[(g*cos(e+f*x))^(3/2)*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^(1/2),x,4,-1/7*2^(9/4+m)*a*(g*cos(e+f*x))^(7/2)*hypergeom([7/4,-1/4-m],[11/4],1/2*(1-sin(e+f*x)))*sec(e+f*x)*(1+sin(e+f*x))^(-1/4-m)*(a+a*sin(e+f*x))^(-1+m)*sqrt(c-c*sin(e+f*x))/(f*g^2)],
[(g*cos(e+f*x))^(3/2)*(a+a*sin(e+f*x))^m/(c-c*sin(e+f*x))^(1/2),x,4,-1/3*2^(9/4+m)*a*cos(e+f*x)*(g*cos(e+f*x))^(3/2)*hypergeom([3/4,-1/4-m],[7/4],1/2*(1-sin(e+f*x)))*(1+sin(e+f*x))^(-1/4-m)*(a+a*sin(e+f*x))^(-1+m)/(f*sqrt(c-c*sin(e+f*x)))],
[(g*cos(e+f*x))^(3/2)*(a+a*sin(e+f*x))^m/(c-c*sin(e+f*x))^(3/2),x,4,2^(9/4+m)*g^2*cos(e+f*x)*hypergeom([-1/4,-1/4-m],[3/4],1/2*(1-sin(e+f*x)))*(1+sin(e+f*x))^(-1/4-m)*(a+a*sin(e+f*x))^m/(c*f*sqrt(g*cos(e+f*x))*sqrt(c-c*sin(e+f*x)))],
[(g*cos(e+f*x))^(3/2)*(a+a*sin(e+f*x))^m/(c-c*sin(e+f*x))^(5/2),x,4,1/5*2^(9/4+m)*g^4*cos(e+f*x)*hypergeom([-5/4,-1/4-m],[-1/4],1/2*(1-sin(e+f*x)))*(1+sin(e+f*x))^(-1/4-m)*(a+a*sin(e+f*x))^(1+m)/(a*c^2*f*(g*cos(e+f*x))^(5/2)*sqrt(c-c*sin(e+f*x)))],

#  The same rules should be used to integrate the following two problems: 
[(g*cos(e+f*x))^(3/2)*(a+a*sin(e+f*x))^m/sqrt(c-c*sin(e+f*x)),x,4,-1/3*2^(9/4+m)*a*cos(e+f*x)*(g*cos(e+f*x))^(3/2)*hypergeom([3/4,-1/4-m],[7/4],1/2*(1-sin(e+f*x)))*(1+sin(e+f*x))^(-1/4-m)*(a+a*sin(e+f*x))^(-1+m)/(f*sqrt(c-c*sin(e+f*x)))],
[(g*cos(e+f*x))^(3/2)*(c+c*sin(e+f*x))^m/sqrt(a-a*sin(e+f*x)),x,4,-1/3*2^(9/4+m)*c*cos(e+f*x)*(g*cos(e+f*x))^(3/2)*hypergeom([3/4,-1/4-m],[7/4],1/2*(1-sin(e+f*x)))*(1+sin(e+f*x))^(-1/4-m)*(c+c*sin(e+f*x))^(-1+m)/(f*sqrt(a-a*sin(e+f*x)))],
[(g*cos(e+f*x))^(3/2)*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^(-3-m),x,4,2^(-3/4-m)*(g*cos(e+f*x))^(5/2)*hypergeom([1/4*(5+4*m),1/4*(11+4*m)],[1/4*(9+4*m)],1/2*(1+sin(e+f*x)))*(1-sin(e+f*x))^(-1/4+m)*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^(-1-m)/(c^2*f*g*(5+4*m))],
[(g*cos(e+f*x))^(3/2)*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^(-2-m),x,4,2^(1/4-m)*(g*cos(e+f*x))^(5/2)*hypergeom([1/4*(5+4*m),1/4*(7+4*m)],[1/4*(9+4*m)],1/2*(1+sin(e+f*x)))*(1-sin(e+f*x))^(-1/4+m)*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^(-1-m)/(c*f*g*(5+4*m))],
[(g*cos(e+f*x))^(3/2)*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^(-1-m),x,4,2^(5/4-m)*(g*cos(e+f*x))^(5/2)*hypergeom([1/4*(3+4*m),1/4*(5+4*m)],[1/4*(9+4*m)],1/2*(1+sin(e+f*x)))*(1-sin(e+f*x))^(-1/4+m)*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^(-1-m)/(f*g*(5+4*m))],
[(g*cos(e+f*x))^(3/2)*(a+a*sin(e+f*x))^m/(c-c*sin(e+f*x))^m,x,4,2^(9/4-m)*c*(g*cos(e+f*x))^(5/2)*hypergeom([1/4*(-1+4*m),1/4*(5+4*m)],[1/4*(9+4*m)],1/2*(1+sin(e+f*x)))*(1-sin(e+f*x))^(-1/4+m)*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^(-1-m)/(f*g*(5+4*m))],
[(g*cos(e+f*x))^(3/2)*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^(1-m),x,4,2^(13/4-m)*c^2*(g*cos(e+f*x))^(5/2)*hypergeom([1/4*(-5+4*m),1/4*(5+4*m)],[1/4*(9+4*m)],1/2*(1+sin(e+f*x)))*(1-sin(e+f*x))^(-1/4+m)*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^(-1-m)/(f*g*(5+4*m))],
[(g*cos(e+f*x))^(3/2)*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^(2-m),x,4,2^(17/4-m)*c^3*(g*cos(e+f*x))^(5/2)*hypergeom([1/4*(-9+4*m),1/4*(5+4*m)],[1/4*(9+4*m)],1/2*(1+sin(e+f*x)))*(1-sin(e+f*x))^(-1/4+m)*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^(-1-m)/(f*g*(5+4*m))],

# Integrands of the form (g Cos[e+f x])^p (a+a Sin[e+f x])^m (c-c Sin[e+f x])^n with m, n and p symbolic
[(g*cos(e+f*x))^p*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^n,x,4,2^(1/2+n+1/2*p)*c*(g*cos(e+f*x))^(1+p)*hypergeom([1/2*(1-2*n-p),1/2*(1+2*m+p)],[1/2*(3+2*m+p)],1/2*(1+sin(e+f*x)))*(1-sin(e+f*x))^(1/2*(1-2*n-p))*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^(-1+n)/(f*g*(1+2*m+p))],
[(g*cos(e+f*x))^(1-2*m)*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^(-1+m),x,4,-g*log(1-sin(e+f*x))*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^m/(c*f*(g*cos(e+f*x))^(2*m))],
[(g*cos(e+f*x))^(5-2*m)*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^n,x,3,-8*a^3*(g*cos(e+f*x))^(6-2*m)*(a+a*sin(e+f*x))^(-3+m)*(c-c*sin(e+f*x))^n/(f*g*(3-m+n)*(4-m+n)*(5-m+n))-4*a^2*(g*cos(e+f*x))^(6-2*m)*(a+a*sin(e+f*x))^(-2+m)*(c-c*sin(e+f*x))^n/(f*g*(4-m+n)*(5-m+n))-a*(g*cos(e+f*x))^(6-2*m)*(a+a*sin(e+f*x))^(-1+m)*(c-c*sin(e+f*x))^n/(f*g*(5-m+n))],
[(g*cos(e+f*x))^(3-2*m)*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^n,x,2,-2*a^2*(g*cos(e+f*x))^(4-2*m)*(a+a*sin(e+f*x))^(-2+m)*(c-c*sin(e+f*x))^n/(f*g*(2-m+n)*(3-m+n))-a*(g*cos(e+f*x))^(4-2*m)*(a+a*sin(e+f*x))^(-1+m)*(c-c*sin(e+f*x))^n/(f*g*(3-m+n))],
[(g*cos(e+f*x))^(1-2*m)*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^n,x,1,-a*(g*cos(e+f*x))^(2-2*m)*(a+a*sin(e+f*x))^(-1+m)*(c-c*sin(e+f*x))^n/(f*g*(1-m+n))],
[(g*cos(e+f*x))^(-1-2*m)*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^n,x,4,1/2*hypergeom([1,-m+n],[1-m+n],1/2*(1-sin(e+f*x)))*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^n/(f*g*(m-n)*(g*cos(e+f*x))^(2*m))],
[(g*cos(e+f*x))^(-3-2*m)*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^n,x,4,1/4*c*hypergeom([2,-1-m+n],[-m+n],1/2*(1-sin(e+f*x)))*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^(-1+n)/(f*g^3*(1+m-n)*(g*cos(e+f*x))^(2*m))],
[(g*cos(e+f*x))^(-5-2*m)*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^n,x,4,1/8*c^2*hypergeom([3,-2-m+n],[-1-m+n],1/2*(1-sin(e+f*x)))*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^(-2+n)/(f*g^5*(2+m-n)*(g*cos(e+f*x))^(2*m))],
[(g*cos(e+f*x))^(-1-2*m)*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^m,x,3,arctanh(sin(e+f*x))*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^m/(f*g*(g*cos(e+f*x))^(2*m))],
[(g*cos(e+f*x))^(-1-m-n)*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^(3+n),x,4,2^(3-1/2*m+1/2*n)*c^3*(g*cos(e+f*x))^(-m-n)*hypergeom([1/2*(-4+m-n),1/2*(m-n)],[1/2*(2+m-n)],1/2*(1+sin(e+f*x)))*(1-sin(e+f*x))^(1/2*(m-n))*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^n/(f*g*(m-n))],
[(g*cos(e+f*x))^(-1-m-n)*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^(2+n),x,4,2^(2-1/2*m+1/2*n)*c^2*(g*cos(e+f*x))^(-m-n)*hypergeom([1/2*(-2+m-n),1/2*(m-n)],[1/2*(2+m-n)],1/2*(1+sin(e+f*x)))*(1-sin(e+f*x))^(1/2*(m-n))*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^n/(f*g*(m-n))],
[(g*cos(e+f*x))^(-1-m-n)*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^(1+n),x,4,2^(1-1/2*m+1/2*n)*c*(g*cos(e+f*x))^(-m-n)*hypergeom([1/2*(m-n),1/2*(m-n)],[1/2*(2+m-n)],1/2*(1+sin(e+f*x)))*(1-sin(e+f*x))^(1/2*(m-n))*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^n/(f*g*(m-n))],
[(g*cos(e+f*x))^(-1-m-n)*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^n,x,1,(g*cos(e+f*x))^(-m-n)*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^n/(f*g*(m-n))],
[(g*cos(e+f*x))^(-1-m-n)*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^(-1+n),x,2,(g*cos(e+f*x))^(-m-n)*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^(-1+n)/(f*g*(2+m-n))+(g*cos(e+f*x))^(-m-n)*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^n/(c*f*g*(m-n)*(2+m-n))],
[(g*cos(e+f*x))^(-1-m-n)*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^(-2+n),x,3,(g*cos(e+f*x))^(-m-n)*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^(-2+n)/(f*g*(4+m-n))+2*(g*cos(e+f*x))^(-m-n)*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^(-1+n)/(c*f*g*(2+m-n)*(4+m-n))+2*(g*cos(e+f*x))^(-m-n)*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^n/(c^2*f*g*(m-n)*(2+m-n)*(4+m-n))],
[(g*cos(e+f*x))^(-1-m-n)*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^(-3+n),x,4,(g*cos(e+f*x))^(-m-n)*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^(-3+n)/(f*g*(6+m-n))+3*(g*cos(e+f*x))^(-m-n)*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^(-2+n)/(c*f*g*(4+m-n)*(6+m-n))+6*(g*cos(e+f*x))^(-m-n)*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^(-1+n)/(c^2*f*g*(2+m-n)*(4+m-n)*(6+m-n))+6*(g*cos(e+f*x))^(-m-n)*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^n/(c^3*f*g*(m-n)*(2+m-n)*(4+m-n)*(6+m-n))],

# Integrands of the form (g Sec[e+f x])^p (a+a Sin[e+f x])^m (c-c Sin[e+f x])^n

# Integrands of the form (g Sec[e+f x])^p (a+a Sin[e+f x])^m (c-c Sin[e+f x])^n with m, n and p symbolic
[(g*sec(e+f*x))^p*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^n,x,5,2^(1/2+n-1/2*p)*c*cos(e+f*x)*hypergeom([1/2*(1+2*m-p),1/2*(1-2*n+p)],[1/2*(3+2*m-p)],1/2*(1+sin(e+f*x)))*(g*sec(e+f*x))^p*(1-sin(e+f*x))^(1/2*(1-2*n+p))*(a+a*sin(e+f*x))^m*(c-c*sin(e+f*x))^(-1+n)/(f*(1+2*m-p))],

# Integrands of the form (g Cos[e+f x])^p (d Sin[e+f x])^n (a+a Sin[e+f x])^m

# Integrands of the form Cos[e+f x]^1 (d Sin[e+f x])^n (a+a Sin[e+f x])^m

# Integrands of the form Cos[e+f x] Sin[e+f x]^n (a+a Sin[e+f x])^m

# m>0
[cos(c+d*x)*sin(c+d*x)^2*(a+a*sin(c+d*x)),x,4,1/3*a*sin(c+d*x)^3/d+1/4*a*sin(c+d*x)^4/d],
[cos(c+d*x)*sin(c+d*x)*(a+a*sin(c+d*x)),x,4,1/2*a*sin(c+d*x)^2/d+1/3*a*sin(c+d*x)^3/d],
[cos(c+d*x)*csc(c+d*x)*(a+a*sin(c+d*x)),x,3,a*log(sin(c+d*x))/d+a*sin(c+d*x)/d],
[cos(c+d*x)*csc(c+d*x)^2*(a+a*sin(c+d*x)),x,4,-a*csc(c+d*x)/d+a*log(sin(c+d*x))/d],
[cos(c+d*x)*csc(c+d*x)^3*(a+a*sin(c+d*x)),x,3,-1/2*csc(c+d*x)^2*(a+a*sin(c+d*x))^2/(a*d)],
[cos(c+d*x)*csc(c+d*x)^4*(a+a*sin(c+d*x)),x,4,-1/2*a*csc(c+d*x)^2/d-1/3*a*csc(c+d*x)^3/d],
[cos(c+d*x)*csc(c+d*x)^5*(a+a*sin(c+d*x)),x,4,-1/3*a*csc(c+d*x)^3/d-1/4*a*csc(c+d*x)^4/d],
[cos(c+d*x)*sin(c+d*x)^2*(a+a*sin(c+d*x))^2,x,4,1/3*a^2*sin(c+d*x)^3/d+1/2*a^2*sin(c+d*x)^4/d+1/5*a^2*sin(c+d*x)^5/d],
[cos(c+d*x)*sin(c+d*x)*(a+a*sin(c+d*x))^2,x,4,1/2*a^2*sin(c+d*x)^2/d+2/3*a^2*sin(c+d*x)^3/d+1/4*a^2*sin(c+d*x)^4/d],
[cos(c+d*x)*csc(c+d*x)*(a+a*sin(c+d*x))^2,x,3,a^2*log(sin(c+d*x))/d+2*a^2*sin(c+d*x)/d+1/2*a^2*sin(c+d*x)^2/d],
[cos(c+d*x)*csc(c+d*x)^2*(a+a*sin(c+d*x))^2,x,4,-a^2*csc(c+d*x)/d+2*a^2*log(sin(c+d*x))/d+a^2*sin(c+d*x)/d],
[cos(c+d*x)*csc(c+d*x)^3*(a+a*sin(c+d*x))^2,x,4,-2*a^2*csc(c+d*x)/d-1/2*a^2*csc(c+d*x)^2/d+a^2*log(sin(c+d*x))/d],
[cos(c+d*x)*csc(c+d*x)^4*(a+a*sin(c+d*x))^2,x,3,-1/3*csc(c+d*x)^3*(a+a*sin(c+d*x))^3/(a*d)],
[cos(c+d*x)*csc(c+d*x)^5*(a+a*sin(c+d*x))^2,x,4,-1/2*a^2*csc(c+d*x)^2/d-2/3*a^2*csc(c+d*x)^3/d-1/4*a^2*csc(c+d*x)^4/d],
[cos(c+d*x)*csc(c+d*x)^6*(a+a*sin(c+d*x))^2,x,4,-1/3*a^2*csc(c+d*x)^3/d-1/2*a^2*csc(c+d*x)^4/d-1/5*a^2*csc(c+d*x)^5/d],
[cos(c+d*x)*csc(c+d*x)^7*(a+a*sin(c+d*x))^2,x,4,-1/4*a^2*csc(c+d*x)^4/d-2/5*a^2*csc(c+d*x)^5/d-1/6*a^2*csc(c+d*x)^6/d],
[cos(c+d*x)*sin(c+d*x)^3*(a+a*sin(c+d*x))^3,x,4,1/4*a^3*sin(c+d*x)^4/d+3/5*a^3*sin(c+d*x)^5/d+1/2*a^3*sin(c+d*x)^6/d+1/7*a^3*sin(c+d*x)^7/d],
[cos(c+d*x)*sin(c+d*x)^2*(a+a*sin(c+d*x))^3,x,4,1/3*a^3*sin(c+d*x)^3/d+3/4*a^3*sin(c+d*x)^4/d+3/5*a^3*sin(c+d*x)^5/d+1/6*a^3*sin(c+d*x)^6/d],
[cos(c+d*x)*sin(c+d*x)*(a+a*sin(c+d*x))^3,x,4,-1/4*(a+a*sin(c+d*x))^4/(a*d)+1/5*(a+a*sin(c+d*x))^5/(a^2*d)],
[cos(c+d*x)*csc(c+d*x)*(a+a*sin(c+d*x))^3,x,3,a^3*log(sin(c+d*x))/d+3*a^3*sin(c+d*x)/d+3/2*a^3*sin(c+d*x)^2/d+1/3*a^3*sin(c+d*x)^3/d],
[cos(c+d*x)*csc(c+d*x)^2*(a+a*sin(c+d*x))^3,x,4,-a^3*csc(c+d*x)/d+3*a^3*log(sin(c+d*x))/d+3*a^3*sin(c+d*x)/d+1/2*a^3*sin(c+d*x)^2/d],
[cos(c+d*x)*csc(c+d*x)^3*(a+a*sin(c+d*x))^3,x,4,-3*a^3*csc(c+d*x)/d-1/2*a^3*csc(c+d*x)^2/d+3*a^3*log(sin(c+d*x))/d+a^3*sin(c+d*x)/d],
[cos(c+d*x)*csc(c+d*x)^4*(a+a*sin(c+d*x))^3,x,4,-3*a^3*csc(c+d*x)/d-3/2*a^3*csc(c+d*x)^2/d-1/3*a^3*csc(c+d*x)^3/d+a^3*log(sin(c+d*x))/d],
[cos(c+d*x)*csc(c+d*x)^5*(a+a*sin(c+d*x))^3,x,3,-1/4*csc(c+d*x)^4*(a+a*sin(c+d*x))^4/(a*d)],
[cos(c+d*x)*csc(c+d*x)^6*(a+a*sin(c+d*x))^3,x,4,1/20*csc(c+d*x)^4*(a+a*sin(c+d*x))^4/(a*d)-1/5*csc(c+d*x)^5*(a+a*sin(c+d*x))^4/(a*d)],
[cos(c+d*x)*csc(c+d*x)^7*(a+a*sin(c+d*x))^3,x,4,-1/3*a^3*csc(c+d*x)^3/d-3/4*a^3*csc(c+d*x)^4/d-3/5*a^3*csc(c+d*x)^5/d-1/6*a^3*csc(c+d*x)^6/d],
[cos(c+d*x)*csc(c+d*x)^8*(a+a*sin(c+d*x))^3,x,4,-1/4*a^3*csc(c+d*x)^4/d-3/5*a^3*csc(c+d*x)^5/d-1/2*a^3*csc(c+d*x)^6/d-1/7*a^3*csc(c+d*x)^7/d],
[cos(c+d*x)*sin(c+d*x)^4*(a+a*sin(c+d*x))^4,x,4,1/5*a^4*sin(c+d*x)^5/d+2/3*a^4*sin(c+d*x)^6/d+6/7*a^4*sin(c+d*x)^7/d+1/2*a^4*sin(c+d*x)^8/d+1/9*a^4*sin(c+d*x)^9/d],
[cos(c+d*x)*sin(c+d*x)^3*(a+a*sin(c+d*x))^4,x,4,1/4*a^4*sin(c+d*x)^4/d+4/5*a^4*sin(c+d*x)^5/d+a^4*sin(c+d*x)^6/d+4/7*a^4*sin(c+d*x)^7/d+1/8*a^4*sin(c+d*x)^8/d],
[cos(c+d*x)*sin(c+d*x)^2*(a+a*sin(c+d*x))^4,x,4,1/5*(a+a*sin(c+d*x))^5/(a*d)-1/3*(a+a*sin(c+d*x))^6/(a^2*d)+1/7*(a+a*sin(c+d*x))^7/(a^3*d)],
[cos(c+d*x)*sin(c+d*x)*(a+a*sin(c+d*x))^4,x,4,-1/5*(a+a*sin(c+d*x))^5/(a*d)+1/6*(a+a*sin(c+d*x))^6/(a^2*d)],
[cos(c+d*x)*csc(c+d*x)*(a+a*sin(c+d*x))^4,x,3,a^4*log(sin(c+d*x))/d+4*a^4*sin(c+d*x)/d+3*a^4*sin(c+d*x)^2/d+4/3*a^4*sin(c+d*x)^3/d+1/4*a^4*sin(c+d*x)^4/d],
[cos(c+d*x)*csc(c+d*x)^2*(a+a*sin(c+d*x))^4,x,4,-a^4*csc(c+d*x)/d+4*a^4*log(sin(c+d*x))/d+6*a^4*sin(c+d*x)/d+2*a^4*sin(c+d*x)^2/d+1/3*a^4*sin(c+d*x)^3/d],
[cos(c+d*x)*csc(c+d*x)^3*(a+a*sin(c+d*x))^4,x,4,-4*a^4*csc(c+d*x)/d-1/2*a^4*csc(c+d*x)^2/d+6*a^4*log(sin(c+d*x))/d+4*a^4*sin(c+d*x)/d+1/2*a^4*sin(c+d*x)^2/d],

# m<0
[cos(c+d*x)*sin(c+d*x)^4/(a+a*sin(c+d*x)),x,4,log(1+sin(c+d*x))/(a*d)-sin(c+d*x)/(a*d)+1/2*sin(c+d*x)^2/(a*d)-1/3*sin(c+d*x)^3/(a*d)+1/4*sin(c+d*x)^4/(a*d)],
[cos(c+d*x)*sin(c+d*x)^3/(a+a*sin(c+d*x)),x,4,-log(1+sin(c+d*x))/(a*d)+sin(c+d*x)/(a*d)-1/2*sin(c+d*x)^2/(a*d)+1/3*sin(c+d*x)^3/(a*d)],
[cos(c+d*x)*sin(c+d*x)^2/(a+a*sin(c+d*x)),x,4,log(1+sin(c+d*x))/(a*d)-sin(c+d*x)/(a*d)+1/2*sin(c+d*x)^2/(a*d)],
[cos(c+d*x)*sin(c+d*x)/(a+a*sin(c+d*x)),x,4,-log(1+sin(c+d*x))/(a*d)+sin(c+d*x)/(a*d)],
[cos(c+d*x)*csc(c+d*x)/(a+a*sin(c+d*x)),x,4,log(sin(c+d*x))/(a*d)-log(1+sin(c+d*x))/(a*d)],
[cos(c+d*x)*csc(c+d*x)^2/(a+a*sin(c+d*x)),x,4,-csc(c+d*x)/(a*d)-log(sin(c+d*x))/(a*d)+log(1+sin(c+d*x))/(a*d)],
[cos(c+d*x)*csc(c+d*x)^3/(a+a*sin(c+d*x)),x,4,csc(c+d*x)/(a*d)-1/2*csc(c+d*x)^2/(a*d)+log(sin(c+d*x))/(a*d)-log(1+sin(c+d*x))/(a*d)],
[cos(c+d*x)*csc(c+d*x)^4/(a+a*sin(c+d*x)),x,4,-csc(c+d*x)/(a*d)+1/2*csc(c+d*x)^2/(a*d)-1/3*csc(c+d*x)^3/(a*d)-log(sin(c+d*x))/(a*d)+log(1+sin(c+d*x))/(a*d)],
[cos(c+d*x)*sin(c+d*x)^4/(a+a*sin(c+d*x))^2,x,4,-4*log(1+sin(c+d*x))/(a^2*d)+3*sin(c+d*x)/(a^2*d)-sin(c+d*x)^2/(a^2*d)+1/3*sin(c+d*x)^3/(a^2*d)+(-1)/(d*(a^2+a^2*sin(c+d*x)))],
[cos(c+d*x)*sin(c+d*x)^3/(a+a*sin(c+d*x))^2,x,4,3*log(1+sin(c+d*x))/(a^2*d)-2*sin(c+d*x)/(a^2*d)+1/2*sin(c+d*x)^2/(a^2*d)+1/(d*(a^2+a^2*sin(c+d*x)))],
[cos(c+d*x)*sin(c+d*x)^2/(a+a*sin(c+d*x))^2,x,4,-2*log(1+sin(c+d*x))/(a^2*d)+sin(c+d*x)/(a^2*d)+(-1)/(d*(a^2+a^2*sin(c+d*x)))],
[cos(c+d*x)*sin(c+d*x)/(a+a*sin(c+d*x))^2,x,4,log(1+sin(c+d*x))/(a^2*d)+1/(d*(a^2+a^2*sin(c+d*x)))],
[cos(c+d*x)*csc(c+d*x)/(a+a*sin(c+d*x))^2,x,3,log(sin(c+d*x))/(a^2*d)-log(1+sin(c+d*x))/(a^2*d)+1/(d*(a^2+a^2*sin(c+d*x)))],
[cos(c+d*x)*csc(c+d*x)^2/(a+a*sin(c+d*x))^2,x,4,-csc(c+d*x)/(a^2*d)-2*log(sin(c+d*x))/(a^2*d)+2*log(1+sin(c+d*x))/(a^2*d)+(-1)/(d*(a^2+a^2*sin(c+d*x)))],
[cos(c+d*x)*csc(c+d*x)^3/(a+a*sin(c+d*x))^2,x,4,2*csc(c+d*x)/(a^2*d)-1/2*csc(c+d*x)^2/(a^2*d)+3*log(sin(c+d*x))/(a^2*d)-3*log(1+sin(c+d*x))/(a^2*d)+1/(d*(a^2+a^2*sin(c+d*x)))],
[cos(c+d*x)*csc(c+d*x)^4/(a+a*sin(c+d*x))^2,x,4,-3*csc(c+d*x)/(a^2*d)+csc(c+d*x)^2/(a^2*d)-1/3*csc(c+d*x)^3/(a^2*d)-4*log(sin(c+d*x))/(a^2*d)+4*log(1+sin(c+d*x))/(a^2*d)+(-1)/(d*(a^2+a^2*sin(c+d*x)))],
[cos(c+d*x)*sin(c+d*x)^5/(a+a*sin(c+d*x))^3,x,4,-10*log(1+sin(c+d*x))/(a^3*d)+6*sin(c+d*x)/(a^3*d)-3/2*sin(c+d*x)^2/(a^3*d)+1/3*sin(c+d*x)^3/(a^3*d)+1/2/(a*d*(a+a*sin(c+d*x))^2)+(-5)/(d*(a^3+a^3*sin(c+d*x)))],
[cos(c+d*x)*sin(c+d*x)^4/(a+a*sin(c+d*x))^3,x,4,6*log(1+sin(c+d*x))/(a^3*d)-3*sin(c+d*x)/(a^3*d)+1/2*sin(c+d*x)^2/(a^3*d)+(-1/2)/(a*d*(a+a*sin(c+d*x))^2)+4/(d*(a^3+a^3*sin(c+d*x)))],
[cos(c+d*x)*sin(c+d*x)^3/(a+a*sin(c+d*x))^3,x,4,-3*log(1+sin(c+d*x))/(a^3*d)+sin(c+d*x)/(a^3*d)+1/2/(a*d*(a+a*sin(c+d*x))^2)+(-3)/(d*(a^3+a^3*sin(c+d*x)))],
[cos(c+d*x)*sin(c+d*x)^2/(a+a*sin(c+d*x))^3,x,4,log(1+sin(c+d*x))/(a^3*d)+(-1/2)/(a*d*(a+a*sin(c+d*x))^2)+2/(d*(a^3+a^3*sin(c+d*x)))],
[cos(c+d*x)*sin(c+d*x)/(a+a*sin(c+d*x))^3,x,3,1/2*sin(c+d*x)^2/(a*d*(a+a*sin(c+d*x))^2)],
[cos(c+d*x)*csc(c+d*x)/(a+a*sin(c+d*x))^3,x,3,log(sin(c+d*x))/(a^3*d)-log(1+sin(c+d*x))/(a^3*d)+1/2/(a*d*(a+a*sin(c+d*x))^2)+1/(d*(a^3+a^3*sin(c+d*x)))],
[cos(c+d*x)*csc(c+d*x)^2/(a+a*sin(c+d*x))^3,x,4,-csc(c+d*x)/(a^3*d)-3*log(sin(c+d*x))/(a^3*d)+3*log(1+sin(c+d*x))/(a^3*d)+(-1/2)/(a*d*(a+a*sin(c+d*x))^2)+(-2)/(d*(a^3+a^3*sin(c+d*x)))],
[cos(c+d*x)*csc(c+d*x)^3/(a+a*sin(c+d*x))^3,x,4,3*csc(c+d*x)/(a^3*d)-1/2*csc(c+d*x)^2/(a^3*d)+6*log(sin(c+d*x))/(a^3*d)-6*log(1+sin(c+d*x))/(a^3*d)+1/2/(a*d*(a+a*sin(c+d*x))^2)+3/(d*(a^3+a^3*sin(c+d*x)))],
[cos(c+d*x)*csc(c+d*x)^4/(a+a*sin(c+d*x))^3,x,4,-6*csc(c+d*x)/(a^3*d)+3/2*csc(c+d*x)^2/(a^3*d)-1/3*csc(c+d*x)^3/(a^3*d)-10*log(sin(c+d*x))/(a^3*d)+10*log(1+sin(c+d*x))/(a^3*d)+(-1/2)/(a*d*(a+a*sin(c+d*x))^2)+(-4)/(d*(a^3+a^3*sin(c+d*x)))],
[cos(c+d*x)*sin(c+d*x)^5/(a+a*sin(c+d*x))^4,x,4,10*log(1+sin(c+d*x))/(a^4*d)-4*sin(c+d*x)/(a^4*d)+1/2*sin(c+d*x)^2/(a^4*d)+1/3/(a*d*(a+a*sin(c+d*x))^3)+(-5/2)/(d*(a^2+a^2*sin(c+d*x))^2)+10/(d*(a^4+a^4*sin(c+d*x)))],
[cos(c+d*x)*sin(c+d*x)^4/(a+a*sin(c+d*x))^4,x,4,-4*log(1+sin(c+d*x))/(a^4*d)+sin(c+d*x)/(a^4*d)+(-1/3)/(a*d*(a+a*sin(c+d*x))^3)+2/(d*(a^2+a^2*sin(c+d*x))^2)+(-6)/(d*(a^4+a^4*sin(c+d*x)))],
[cos(c+d*x)*sin(c+d*x)^3/(a+a*sin(c+d*x))^4,x,4,log(1+sin(c+d*x))/(a^4*d)+1/3/(a*d*(a+a*sin(c+d*x))^3)+(-3/2)/(d*(a^2+a^2*sin(c+d*x))^2)+3/(d*(a^4+a^4*sin(c+d*x)))],
[cos(c+d*x)*sin(c+d*x)^2/(a+a*sin(c+d*x))^4,x,3,1/3*sin(c+d*x)^3/(a*d*(a+a*sin(c+d*x))^3)],
[cos(c+d*x)*sin(c+d*x)/(a+a*sin(c+d*x))^4,x,4,1/3/(a*d*(a+a*sin(c+d*x))^3)+(-1/2)/(d*(a^2+a^2*sin(c+d*x))^2)],
[cos(c+d*x)*csc(c+d*x)/(a+a*sin(c+d*x))^4,x,3,log(sin(c+d*x))/(a^4*d)-log(1+sin(c+d*x))/(a^4*d)+1/3/(a*d*(a+a*sin(c+d*x))^3)+1/2/(d*(a^2+a^2*sin(c+d*x))^2)+1/(d*(a^4+a^4*sin(c+d*x)))],
[cos(c+d*x)*csc(c+d*x)^2/(a+a*sin(c+d*x))^4,x,4,-csc(c+d*x)/(a^4*d)-4*log(sin(c+d*x))/(a^4*d)+4*log(1+sin(c+d*x))/(a^4*d)+(-1/3)/(a*d*(a+a*sin(c+d*x))^3)+(-1)/(d*(a^2+a^2*sin(c+d*x))^2)+(-3)/(d*(a^4+a^4*sin(c+d*x)))],
[cos(c+d*x)*csc(c+d*x)^3/(a+a*sin(c+d*x))^4,x,4,4*csc(c+d*x)/(a^4*d)-1/2*csc(c+d*x)^2/(a^4*d)+10*log(sin(c+d*x))/(a^4*d)-10*log(1+sin(c+d*x))/(a^4*d)+1/3/(a*d*(a+a*sin(c+d*x))^3)+3/2/(d*(a^2+a^2*sin(c+d*x))^2)+6/(d*(a^4+a^4*sin(c+d*x)))],

# Integrands of the form Cos[e+f x] Sin[e+f x]^n (a+a Sin[e+f x])^(m/2)

# m>0
[cos(c+d*x)*csc(c+d*x)*(a+a*sin(c+d*x))^(1/2),x,4,-2*arctanh(sqrt(a+a*sin(c+d*x))/sqrt(a))*sqrt(a)/d+2*sqrt(a+a*sin(c+d*x))/d],

# m<0

# Integrands of the form Cos[e+f x] Sin[e+f x]^n (a+a Sin[e+f x])^m with n symbolic
[cos(c+d*x)*sin(c+d*x)^n*(a+a*sin(c+d*x))^4,x,3,a^4*sin(c+d*x)^(1+n)/(d*(1+n))+4*a^4*sin(c+d*x)^(2+n)/(d*(2+n))+6*a^4*sin(c+d*x)^(3+n)/(d*(3+n))+4*a^4*sin(c+d*x)^(4+n)/(d*(4+n))+a^4*sin(c+d*x)^(5+n)/(d*(5+n))],
[cos(c+d*x)*sin(c+d*x)^n*(a+a*sin(c+d*x))^3,x,3,a^3*sin(c+d*x)^(1+n)/(d*(1+n))+3*a^3*sin(c+d*x)^(2+n)/(d*(2+n))+3*a^3*sin(c+d*x)^(3+n)/(d*(3+n))+a^3*sin(c+d*x)^(4+n)/(d*(4+n))],
[cos(c+d*x)*sin(c+d*x)^n*(a+a*sin(c+d*x))^2,x,3,a^2*sin(c+d*x)^(1+n)/(d*(1+n))+2*a^2*sin(c+d*x)^(2+n)/(d*(2+n))+a^2*sin(c+d*x)^(3+n)/(d*(3+n))],
[cos(c+d*x)*sin(c+d*x)^n*(a+a*sin(c+d*x)),x,3,a*sin(c+d*x)^(1+n)/(d*(1+n))+a*sin(c+d*x)^(2+n)/(d*(2+n))],
[cos(c+d*x)*sin(c+d*x)^n/(a+a*sin(c+d*x)),x,2,hypergeom([1,1+n],[2+n],-sin(c+d*x))*sin(c+d*x)^(1+n)/(a*d*(1+n))],
[cos(c+d*x)*sin(c+d*x)^n/(a+a*sin(c+d*x))^2,x,2,hypergeom([2,1+n],[2+n],-sin(c+d*x))*sin(c+d*x)^(1+n)/(a^2*d*(1+n))],
[cos(c+d*x)*sin(c+d*x)^n/(a+a*sin(c+d*x))^3,x,2,hypergeom([3,1+n],[2+n],-sin(c+d*x))*sin(c+d*x)^(1+n)/(a^3*d*(1+n))],
[cos(c+d*x)*sin(c+d*x)^n/(a+a*sin(c+d*x))^4,x,2,hypergeom([4,1+n],[2+n],-sin(c+d*x))*sin(c+d*x)^(1+n)/(a^4*d*(1+n))],

# Integrands of the form Cos[e+f x]^2 (d Sin[e+f x])^n (a+a Sin[e+f x])^m

# Integrands of the form Cos[e+f x]^2 Sin[e+f x]^n (a+a Sin[e+f x])^m

# m>0
[cos(c+d*x)^2*sin(c+d*x)^3*(a+a*sin(c+d*x)),x,8,1/16*a*x-1/3*a*cos(c+d*x)^3/d+1/5*a*cos(c+d*x)^5/d+1/16*a*cos(c+d*x)*sin(c+d*x)/d-1/8*a*cos(c+d*x)^3*sin(c+d*x)/d-1/6*a*cos(c+d*x)^3*sin(c+d*x)^3/d],
[cos(c+d*x)^2*sin(c+d*x)^2*(a+a*sin(c+d*x)),x,7,1/8*a*x-1/3*a*cos(c+d*x)^3/d+1/5*a*cos(c+d*x)^5/d+1/8*a*cos(c+d*x)*sin(c+d*x)/d-1/4*a*cos(c+d*x)^3*sin(c+d*x)/d],
[cos(c+d*x)^2*sin(c+d*x)*(a+a*sin(c+d*x)),x,6,1/8*a*x-1/3*a*cos(c+d*x)^3/d+1/8*a*cos(c+d*x)*sin(c+d*x)/d-1/4*a*cos(c+d*x)^3*sin(c+d*x)/d],
[cos(c+d*x)^2*csc(c+d*x)*(a+a*sin(c+d*x)),x,6,1/2*a*x-a*arctanh(cos(c+d*x))/d+a*cos(c+d*x)/d+1/2*a*cos(c+d*x)*sin(c+d*x)/d],
[cos(c+d*x)^2*csc(c+d*x)^2*(a+a*sin(c+d*x)),x,7,-a*x-a*arctanh(cos(c+d*x))/d+a*cos(c+d*x)/d-a*cot(c+d*x)/d],
[cos(c+d*x)^2*csc(c+d*x)^3*(a+a*sin(c+d*x)),x,5,-a*x+1/2*a*arctanh(cos(c+d*x))/d-a*cot(c+d*x)/d-1/2*a*cot(c+d*x)*csc(c+d*x)/d],
[cos(c+d*x)^2*csc(c+d*x)^4*(a+a*sin(c+d*x)),x,5,1/2*a*arctanh(cos(c+d*x))/d-1/3*a*cot(c+d*x)^3/d-1/2*a*cot(c+d*x)*csc(c+d*x)/d],
[cos(c+d*x)^2*csc(c+d*x)^5*(a+a*sin(c+d*x)),x,6,1/8*a*arctanh(cos(c+d*x))/d-1/3*a*cot(c+d*x)^3/d+1/8*a*cot(c+d*x)*csc(c+d*x)/d-1/4*a*cot(c+d*x)*csc(c+d*x)^3/d],
[cos(c+d*x)^2*csc(c+d*x)^6*(a+a*sin(c+d*x)),x,7,1/8*a*arctanh(cos(c+d*x))/d-1/3*a*cot(c+d*x)^3/d-1/5*a*cot(c+d*x)^5/d+1/8*a*cot(c+d*x)*csc(c+d*x)/d-1/4*a*cot(c+d*x)*csc(c+d*x)^3/d],
[cos(c+d*x)^2*sin(c+d*x)^3*(a+a*sin(c+d*x))^2,x,12,1/8*a^2*x-2/3*a^2*cos(c+d*x)^3/d+3/5*a^2*cos(c+d*x)^5/d-1/7*a^2*cos(c+d*x)^7/d+1/8*a^2*cos(c+d*x)*sin(c+d*x)/d-1/4*a^2*cos(c+d*x)^3*sin(c+d*x)/d-1/3*a^2*cos(c+d*x)^3*sin(c+d*x)^3/d],
[cos(c+d*x)^2*sin(c+d*x)^2*(a+a*sin(c+d*x))^2,x,5,3/16*a^2*x-1/10*a^2*cos(c+d*x)^5/d+3/16*a^2*cos(c+d*x)*sin(c+d*x)/d+1/8*a^2*cos(c+d*x)^3*sin(c+d*x)/d-1/6*cos(c+d*x)^3*(a+a*sin(c+d*x))^3/(a*d)],
[cos(c+d*x)^2*sin(c+d*x)*(a+a*sin(c+d*x))^2,x,5,1/4*a^2*x-2/3*a^2*cos(c+d*x)^3/d+1/5*a^2*cos(c+d*x)^5/d+1/4*a^2*cos(c+d*x)*sin(c+d*x)/d-1/2*a^2*cos(c+d*x)^3*sin(c+d*x)/d,1/4*a^2*x-1/6*a^2*cos(c+d*x)^3/d+1/4*a^2*cos(c+d*x)*sin(c+d*x)/d-1/5*cos(c+d*x)^3*(a+a*sin(c+d*x))^2/d-1/10*cos(c+d*x)^3*(a^2+a^2*sin(c+d*x))/d],
[cos(c+d*x)^2*csc(c+d*x)*(a+a*sin(c+d*x))^2,x,9,a^2*x-a^2*arctanh(cos(c+d*x))/d+a^2*cos(c+d*x)/d-1/3*a^2*cos(c+d*x)^3/d+a^2*cos(c+d*x)*sin(c+d*x)/d],
[cos(c+d*x)^2*csc(c+d*x)^2*(a+a*sin(c+d*x))^2,x,8,-1/2*a^2*x-2*a^2*arctanh(cos(c+d*x))/d+2*a^2*cos(c+d*x)/d-a^2*cot(c+d*x)/d+1/2*a^2*cos(c+d*x)*sin(c+d*x)/d],
[cos(c+d*x)^2*csc(c+d*x)^3*(a+a*sin(c+d*x))^2,x,7,-2*a^2*x-1/2*a^2*arctanh(cos(c+d*x))/d+a^2*cos(c+d*x)/d-2*a^2*cot(c+d*x)/d-1/2*a^2*cot(c+d*x)*csc(c+d*x)/d],
[cos(c+d*x)^2*csc(c+d*x)^4*(a+a*sin(c+d*x))^2,x,8,-a^2*x+a^2*arctanh(cos(c+d*x))/d-a^2*cot(c+d*x)/d-1/3*a^2*cot(c+d*x)^3/d-a^2*cot(c+d*x)*csc(c+d*x)/d],
[cos(c+d*x)^2*csc(c+d*x)^5*(a+a*sin(c+d*x))^2,x,9,5/8*a^2*arctanh(cos(c+d*x))/d-2/3*a^2*cot(c+d*x)^3/d-3/8*a^2*cot(c+d*x)*csc(c+d*x)/d-1/4*a^2*cot(c+d*x)*csc(c+d*x)^3/d],
[cos(c+d*x)^2*csc(c+d*x)^6*(a+a*sin(c+d*x))^2,x,10,1/4*a^2*arctanh(cos(c+d*x))/d-2/3*a^2*cot(c+d*x)^3/d-1/5*a^2*cot(c+d*x)^5/d+1/4*a^2*cot(c+d*x)*csc(c+d*x)/d-1/2*a^2*cot(c+d*x)*csc(c+d*x)^3/d],
[cos(c+d*x)^2*csc(c+d*x)^7*(a+a*sin(c+d*x))^2,x,12,3/16*a^2*arctanh(cos(c+d*x))/d-2/3*a^2*cot(c+d*x)^3/d-2/5*a^2*cot(c+d*x)^5/d+3/16*a^2*cot(c+d*x)*csc(c+d*x)/d-5/24*a^2*cot(c+d*x)*csc(c+d*x)^3/d-1/6*a^2*cot(c+d*x)*csc(c+d*x)^5/d],
[cos(c+d*x)^2*sin(c+d*x)^2*(a+a*sin(c+d*x))^3,x,15,5/16*a^3*x-4/3*a^3*cos(c+d*x)^3/d+a^3*cos(c+d*x)^5/d-1/7*a^3*cos(c+d*x)^7/d+5/16*a^3*cos(c+d*x)*sin(c+d*x)/d-5/8*a^3*cos(c+d*x)^3*sin(c+d*x)/d-1/2*a^3*cos(c+d*x)^3*sin(c+d*x)^3/d],
[cos(c+d*x)^2*sin(c+d*x)*(a+a*sin(c+d*x))^3,x,6,7/16*a^3*x-4/3*a^3*cos(c+d*x)^3/d+3/5*a^3*cos(c+d*x)^5/d+7/16*a^3*cos(c+d*x)*sin(c+d*x)/d-7/8*a^3*cos(c+d*x)^3*sin(c+d*x)/d-1/6*a^3*cos(c+d*x)^3*sin(c+d*x)^3/d,7/16*a^3*x-7/24*a^3*cos(c+d*x)^3/d+7/16*a^3*cos(c+d*x)*sin(c+d*x)/d-1/10*a*cos(c+d*x)^3*(a+a*sin(c+d*x))^2/d-1/6*cos(c+d*x)^3*(a+a*sin(c+d*x))^3/d-7/40*cos(c+d*x)^3*(a^3+a^3*sin(c+d*x))/d],
[cos(c+d*x)^2*csc(c+d*x)*(a+a*sin(c+d*x))^3,x,12,13/8*a^3*x-a^3*arctanh(cos(c+d*x))/d+a^3*cos(c+d*x)/d-a^3*cos(c+d*x)^3/d+13/8*a^3*cos(c+d*x)*sin(c+d*x)/d-1/4*a^3*cos(c+d*x)^3*sin(c+d*x)/d],
[cos(c+d*x)^2*csc(c+d*x)^2*(a+a*sin(c+d*x))^3,x,10,1/2*a^3*x-3*a^3*arctanh(cos(c+d*x))/d+3*a^3*cos(c+d*x)/d-1/3*a^3*cos(c+d*x)^3/d-a^3*cot(c+d*x)/d+3/2*a^3*cos(c+d*x)*sin(c+d*x)/d],
[cos(c+d*x)^2*csc(c+d*x)^3*(a+a*sin(c+d*x))^3,x,10,-5/2*a^3*x-5/2*a^3*arctanh(cos(c+d*x))/d+3*a^3*cos(c+d*x)/d-3*a^3*cot(c+d*x)/d-1/2*a^3*cot(c+d*x)*csc(c+d*x)/d+1/2*a^3*cos(c+d*x)*sin(c+d*x)/d],
[cos(c+d*x)^2*csc(c+d*x)^4*(a+a*sin(c+d*x))^3,x,10,-3*a^3*x+1/2*a^3*arctanh(cos(c+d*x))/d+a^3*cos(c+d*x)/d-3*a^3*cot(c+d*x)/d-1/3*a^3*cot(c+d*x)^3/d-3/2*a^3*cot(c+d*x)*csc(c+d*x)/d],
[cos(c+d*x)^2*csc(c+d*x)^5*(a+a*sin(c+d*x))^3,x,11,-a^3*x+13/8*a^3*arctanh(cos(c+d*x))/d-a^3*cot(c+d*x)/d-a^3*cot(c+d*x)^3/d-11/8*a^3*cot(c+d*x)*csc(c+d*x)/d-1/4*a^3*cot(c+d*x)*csc(c+d*x)^3/d],
[cos(c+d*x)^2*csc(c+d*x)^6*(a+a*sin(c+d*x))^3,x,12,7/8*a^3*arctanh(cos(c+d*x))/d-4/3*a^3*cot(c+d*x)^3/d-1/5*a^3*cot(c+d*x)^5/d-1/8*a^3*cot(c+d*x)*csc(c+d*x)/d-3/4*a^3*cot(c+d*x)*csc(c+d*x)^3/d],
[cos(c+d*x)^2*csc(c+d*x)^7*(a+a*sin(c+d*x))^3,x,14,7/16*a^3*arctanh(cos(c+d*x))/d-4/3*a^3*cot(c+d*x)^3/d-3/5*a^3*cot(c+d*x)^5/d+7/16*a^3*cot(c+d*x)*csc(c+d*x)/d-17/24*a^3*cot(c+d*x)*csc(c+d*x)^3/d-1/6*a^3*cot(c+d*x)*csc(c+d*x)^5/d],
[cos(c+d*x)^2*(a+a*sin(c+d*x))^4,x,6,21/16*a^4*x-7/8*a^4*cos(c+d*x)^3/d+21/16*a^4*cos(c+d*x)*sin(c+d*x)/d-1/6*a*cos(c+d*x)^3*(a+a*sin(c+d*x))^3/d-3/10*cos(c+d*x)^3*(a^2+a^2*sin(c+d*x))^2/d-21/40*cos(c+d*x)^3*(a^4+a^4*sin(c+d*x))/d],
[cos(c+d*x)^2*csc(c+d*x)*(a+a*sin(c+d*x))^4,x,15,5/2*a^4*x-a^4*arctanh(cos(c+d*x))/d+a^4*cos(c+d*x)/d-7/3*a^4*cos(c+d*x)^3/d+1/5*a^4*cos(c+d*x)^5/d+5/2*a^4*cos(c+d*x)*sin(c+d*x)/d-a^4*cos(c+d*x)^3*sin(c+d*x)/d],
[cos(c+d*x)^2*csc(c+d*x)^2*(a+a*sin(c+d*x))^4,x,12,17/8*a^4*x-4*a^4*arctanh(cos(c+d*x))/d+4*a^4*cos(c+d*x)/d-4/3*a^4*cos(c+d*x)^3/d-a^4*cot(c+d*x)/d+23/8*a^4*cos(c+d*x)*sin(c+d*x)/d+1/4*a^4*cos(c+d*x)*sin(c+d*x)^3/d],

# m<0
[cos(c+d*x)^2*sin(c+d*x)^4/(a+a*sin(c+d*x)),x,6,3/8*x/a+cos(c+d*x)/(a*d)-2/3*cos(c+d*x)^3/(a*d)+1/5*cos(c+d*x)^5/(a*d)-3/8*cos(c+d*x)*sin(c+d*x)/(a*d)-1/4*cos(c+d*x)*sin(c+d*x)^3/(a*d)],
[cos(c+d*x)^2*sin(c+d*x)^3/(a+a*sin(c+d*x)),x,6,-3/8*x/a-cos(c+d*x)/(a*d)+1/3*cos(c+d*x)^3/(a*d)+3/8*cos(c+d*x)*sin(c+d*x)/(a*d)+1/4*cos(c+d*x)*sin(c+d*x)^3/(a*d)],
[cos(c+d*x)^2*sin(c+d*x)^2/(a+a*sin(c+d*x)),x,5,1/2*x/a+cos(c+d*x)/(a*d)-1/3*cos(c+d*x)^3/(a*d)-1/2*cos(c+d*x)*sin(c+d*x)/(a*d)],
[cos(c+d*x)^2*sin(c+d*x)/(a+a*sin(c+d*x)),x,4,-1/2*x/a-cos(c+d*x)/(a*d)+1/2*cos(c+d*x)*sin(c+d*x)/(a*d)],
[cos(c+d*x)^2*csc(c+d*x)/(a+a*sin(c+d*x)),x,3,-x/a-arctanh(cos(c+d*x))/(a*d)],
[cos(c+d*x)^2*csc(c+d*x)^2/(a+a*sin(c+d*x)),x,4,arctanh(cos(c+d*x))/(a*d)-cot(c+d*x)/(a*d)],
[cos(c+d*x)^2*csc(c+d*x)^3/(a+a*sin(c+d*x)),x,5,-1/2*arctanh(cos(c+d*x))/(a*d)+cot(c+d*x)/(a*d)-1/2*cot(c+d*x)*csc(c+d*x)/(a*d)],
[cos(c+d*x)^2*csc(c+d*x)^4/(a+a*sin(c+d*x)),x,5,1/2*arctanh(cos(c+d*x))/(a*d)-cot(c+d*x)/(a*d)-1/3*cot(c+d*x)^3/(a*d)+1/2*cot(c+d*x)*csc(c+d*x)/(a*d)],
[cos(c+d*x)^2*csc(c+d*x)^5/(a+a*sin(c+d*x)),x,6,-3/8*arctanh(cos(c+d*x))/(a*d)+cot(c+d*x)/(a*d)+1/3*cot(c+d*x)^3/(a*d)-3/8*cot(c+d*x)*csc(c+d*x)/(a*d)-1/4*cot(c+d*x)*csc(c+d*x)^3/(a*d)],
[cos(c+d*x)^2*csc(c+d*x)^6/(a+a*sin(c+d*x)),x,6,3/8*arctanh(cos(c+d*x))/(a*d)-cot(c+d*x)/(a*d)-2/3*cot(c+d*x)^3/(a*d)-1/5*cot(c+d*x)^5/(a*d)+3/8*cot(c+d*x)*csc(c+d*x)/(a*d)+1/4*cot(c+d*x)*csc(c+d*x)^3/(a*d)],
[cos(c+d*x)^2*sin(c+d*x)^4/(a+a*sin(c+d*x))^2,x,12,-27/8*x/a^2-4*cos(c+d*x)/(a^2*d)+2/3*cos(c+d*x)^3/(a^2*d)+11/8*cos(c+d*x)*sin(c+d*x)/(a^2*d)+1/4*cos(c+d*x)*sin(c+d*x)^3/(a^2*d)-2*cos(c+d*x)/(a^2*d*(1+sin(c+d*x)))],
[cos(c+d*x)^2*sin(c+d*x)^3/(a+a*sin(c+d*x))^2,x,9,3*x/a^2+3*cos(c+d*x)/(a^2*d)-1/3*cos(c+d*x)^3/(a^2*d)-cos(c+d*x)*sin(c+d*x)/(a^2*d)+2*cos(c+d*x)/(a^2*d*(1+sin(c+d*x)))],
[cos(c+d*x)^2*sin(c+d*x)^2/(a+a*sin(c+d*x))^2,x,8,-5/2*x/a^2-2*cos(c+d*x)/(a^2*d)+1/2*cos(c+d*x)*sin(c+d*x)/(a^2*d)-2*cos(c+d*x)/(a^2*d*(1+sin(c+d*x)))],
[cos(c+d*x)^2*sin(c+d*x)/(a+a*sin(c+d*x))^2,x,3,2*x/a^2+cos(c+d*x)/(a^2*d)+2*cos(c+d*x)/(d*(a^2+a^2*sin(c+d*x)))],
[cos(c+d*x)^2*csc(c+d*x)/(a+a*sin(c+d*x))^2,x,5,-arctanh(cos(c+d*x))/(a^2*d)+2*cos(c+d*x)/(a^2*d*(1+sin(c+d*x)))],
[cos(c+d*x)^2*csc(c+d*x)^2/(a+a*sin(c+d*x))^2,x,9,2*arctanh(cos(c+d*x))/(a^2*d)-3*cot(c+d*x)/(a^2*d)+2*cot(c+d*x)/(a^2*d*(1+sin(c+d*x)))],
[cos(c+d*x)^2*csc(c+d*x)^3/(a+a*sin(c+d*x))^2,x,9,-5/2*arctanh(cos(c+d*x))/(a^2*d)+2*cot(c+d*x)/(a^2*d)-1/2*cot(c+d*x)*csc(c+d*x)/(a^2*d)+2*cos(c+d*x)/(a^2*d*(1+sin(c+d*x)))],
[cos(c+d*x)^2*csc(c+d*x)^4/(a+a*sin(c+d*x))^2,x,11,3*arctanh(cos(c+d*x))/(a^2*d)-3*cot(c+d*x)/(a^2*d)-1/3*cot(c+d*x)^3/(a^2*d)+cot(c+d*x)*csc(c+d*x)/(a^2*d)-2*cos(c+d*x)/(a^2*d*(1+sin(c+d*x)))],
[cos(c+d*x)^2*sin(c+d*x)^3/(a+a*sin(c+d*x))^3,x,9,-11/2*x/a^3-3*cos(c+d*x)/(a^3*d)+1/2*cos(c+d*x)*sin(c+d*x)/(a^3*d)+2/3*cos(c+d*x)/(a^3*d*(1+sin(c+d*x))^2)-19/3*cos(c+d*x)/(a^3*d*(1+sin(c+d*x)))],
[cos(c+d*x)^2*sin(c+d*x)^2/(a+a*sin(c+d*x))^3,x,4,3*x/a^3+3*cos(c+d*x)/(a^3*d)-1/3*cos(c+d*x)^3/(d*(a+a*sin(c+d*x))^3)+2*cos(c+d*x)^3/(a*d*(a+a*sin(c+d*x))^2)],
[cos(c+d*x)^2*sin(c+d*x)/(a+a*sin(c+d*x))^3,x,3,-x/a^3-7/3*cos(c+d*x)/(a^3*d*(1+sin(c+d*x)))+2/3*cos(c+d*x)/(a*d*(a+a*sin(c+d*x))^2)],
[cos(c+d*x)^2*csc(c+d*x)/(a+a*sin(c+d*x))^3,x,7,-arctanh(cos(c+d*x))/(a^3*d)+2/3*cos(c+d*x)/(a^3*d*(1+sin(c+d*x))^2)+5/3*cos(c+d*x)/(a^3*d*(1+sin(c+d*x)))],
[cos(c+d*x)^2*csc(c+d*x)^2/(a+a*sin(c+d*x))^3,x,13,3*arctanh(cos(c+d*x))/(a^3*d)-14/3*cot(c+d*x)/(a^3*d)+2/3*cot(c+d*x)/(a^3*d*(1+sin(c+d*x))^2)+3*cot(c+d*x)/(a^3*d*(1+sin(c+d*x)))],
[cos(c+d*x)^2*csc(c+d*x)^3/(a+a*sin(c+d*x))^3,x,11,-11/2*arctanh(cos(c+d*x))/(a^3*d)+3*cot(c+d*x)/(a^3*d)-1/2*cot(c+d*x)*csc(c+d*x)/(a^3*d)+2/3*cos(c+d*x)/(a^3*d*(1+sin(c+d*x))^2)+17/3*cos(c+d*x)/(a^3*d*(1+sin(c+d*x)))],
[cos(e+f*x)^2*sin(e+f*x)/(a+a*sin(e+f*x))^6,x,5,2/9*cos(e+f*x)/(a*f*(a+a*sin(e+f*x))^5)-19/63*cos(e+f*x)/(a^2*f*(a+a*sin(e+f*x))^4)+2/105*cos(e+f*x)/(f*(a^2+a^2*sin(e+f*x))^3)+4/315*cos(e+f*x)/(f*(a^3+a^3*sin(e+f*x))^2)+4/315*cos(e+f*x)/(f*(a^6+a^6*sin(e+f*x)))],

# Integrands of the form Cos[e+f x]^2 Sin[e+f x]^n (a+a Sin[e+f x])^(m/2)

# m>0
[cos(c+d*x)^2*sin(c+d*x)^3*sqrt(a+a*sin(c+d*x)),x,7,-76/1155*cos(c+d*x)*(a+a*sin(c+d*x))^(3/2)/(a*d)-76/495*a*cos(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))-38/693*a*cos(c+d*x)*sin(c+d*x)^3/(d*sqrt(a+a*sin(c+d*x)))+2/99*a*cos(c+d*x)*sin(c+d*x)^4/(d*sqrt(a+a*sin(c+d*x)))+152/3465*cos(c+d*x)*sqrt(a+a*sin(c+d*x))/d+2/11*cos(c+d*x)*sin(c+d*x)^4*sqrt(a+a*sin(c+d*x))/d],
[cos(c+d*x)^2*sin(c+d*x)^2*sqrt(a+a*sin(c+d*x)),x,4,-8/63*a^2*cos(c+d*x)^3/(d*(a+a*sin(c+d*x))^(3/2))-2/9*cos(c+d*x)^3*(a+a*sin(c+d*x))^(3/2)/(a*d)-2/21*a*cos(c+d*x)^3/(d*sqrt(a+a*sin(c+d*x)))+4/21*cos(c+d*x)^3*sqrt(a+a*sin(c+d*x))/d],
[cos(c+d*x)^2*sin(c+d*x)*sqrt(a+a*sin(c+d*x)),x,3,-8/105*a^2*cos(c+d*x)^3/(d*(a+a*sin(c+d*x))^(3/2))-2/35*a*cos(c+d*x)^3/(d*sqrt(a+a*sin(c+d*x)))-2/7*cos(c+d*x)^3*sqrt(a+a*sin(c+d*x))/d],
[cos(c+d*x)^2*csc(c+d*x)*sqrt(a+a*sin(c+d*x)),x,5,-2*arctanh(cos(c+d*x)*sqrt(a)/sqrt(a+a*sin(c+d*x)))*sqrt(a)/d+2/3*a*cos(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))+2/3*cos(c+d*x)*sqrt(a+a*sin(c+d*x))/d],
[cos(c+d*x)^2*csc(c+d*x)^2*sqrt(a+a*sin(c+d*x)),x,4,-arctanh(cos(c+d*x)*sqrt(a)/sqrt(a+a*sin(c+d*x)))*sqrt(a)/d+3*a*cos(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))-cot(c+d*x)*sqrt(a+a*sin(c+d*x))/d],
[cos(c+d*x)^2*csc(c+d*x)^3*sqrt(a+a*sin(c+d*x)),x,5,5/4*arctanh(cos(c+d*x)*sqrt(a)/sqrt(a+a*sin(c+d*x)))*sqrt(a)/d-1/4*a*cot(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))-1/2*cot(c+d*x)*csc(c+d*x)*sqrt(a+a*sin(c+d*x))/d],
[cos(c+d*x)^2*csc(c+d*x)^4*sqrt(a+a*sin(c+d*x)),x,6,3/8*arctanh(cos(c+d*x)*sqrt(a)/sqrt(a+a*sin(c+d*x)))*sqrt(a)/d+3/8*a*cot(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))-1/12*a*cot(c+d*x)*csc(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))-1/3*cot(c+d*x)*csc(c+d*x)^2*sqrt(a+a*sin(c+d*x))/d],
[cos(c+d*x)^2*sin(c+d*x)^3*(a+a*sin(c+d*x))^(3/2),x,8,-1724/15015*cos(c+d*x)*(a+a*sin(c+d*x))^(3/2)/d+2/13*cos(c+d*x)*sin(c+d*x)^4*(a+a*sin(c+d*x))^(3/2)/d-1724/6435*a^2*cos(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))-862/9009*a^2*cos(c+d*x)*sin(c+d*x)^3/(d*sqrt(a+a*sin(c+d*x)))-38/1287*a^2*cos(c+d*x)*sin(c+d*x)^4/(d*sqrt(a+a*sin(c+d*x)))+3448/45045*a*cos(c+d*x)*sqrt(a+a*sin(c+d*x))/d+6/143*a*cos(c+d*x)*sin(c+d*x)^4*sqrt(a+a*sin(c+d*x))/d],
[cos(c+d*x)^2*sin(c+d*x)^2*(a+a*sin(c+d*x))^(3/2),x,5,-64/385*a^3*cos(c+d*x)^3/(d*(a+a*sin(c+d*x))^(3/2))+4/33*cos(c+d*x)^3*(a+a*sin(c+d*x))^(3/2)/d-2/11*cos(c+d*x)^3*(a+a*sin(c+d*x))^(5/2)/(a*d)-48/385*a^2*cos(c+d*x)^3/(d*sqrt(a+a*sin(c+d*x)))-6/77*a*cos(c+d*x)^3*sqrt(a+a*sin(c+d*x))/d],
[cos(c+d*x)^2*sin(c+d*x)*(a+a*sin(c+d*x))^(3/2),x,4,-64/315*a^3*cos(c+d*x)^3/(d*(a+a*sin(c+d*x))^(3/2))-2/9*cos(c+d*x)^3*(a+a*sin(c+d*x))^(3/2)/d-16/105*a^2*cos(c+d*x)^3/(d*sqrt(a+a*sin(c+d*x)))-2/21*a*cos(c+d*x)^3*sqrt(a+a*sin(c+d*x))/d],
[cos(c+d*x)^2*csc(c+d*x)*(a+a*sin(c+d*x))^(3/2),x,6,-2*a^(3/2)*arctanh(cos(c+d*x)*sqrt(a)/sqrt(a+a*sin(c+d*x)))/d+2/5*cos(c+d*x)*(a+a*sin(c+d*x))^(3/2)/d-2/5*a^2*cos(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))+2/5*a*cos(c+d*x)*sqrt(a+a*sin(c+d*x))/d],
[cos(c+d*x)^2*csc(c+d*x)^2*(a+a*sin(c+d*x))^(3/2),x,5,-3*a^(3/2)*arctanh(cos(c+d*x)*sqrt(a)/sqrt(a+a*sin(c+d*x)))/d-cot(c+d*x)*(a+a*sin(c+d*x))^(3/2)/d+11/3*a^2*cos(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))+5/3*a*cos(c+d*x)*sqrt(a+a*sin(c+d*x))/d],
[cos(c+d*x)^2*csc(c+d*x)^3*(a+a*sin(c+d*x))^(3/2),x,6,1/4*a^(3/2)*arctanh(cos(c+d*x)*sqrt(a)/sqrt(a+a*sin(c+d*x)))/d-1/2*cot(c+d*x)*csc(c+d*x)*(a+a*sin(c+d*x))^(3/2)/d+13/4*a^2*cos(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))-3/4*a*cot(c+d*x)*sqrt(a+a*sin(c+d*x))/d],
[cos(c+d*x)^2*csc(c+d*x)^4*(a+a*sin(c+d*x))^(3/2),x,6,13/8*a^(3/2)*arctanh(cos(c+d*x)*sqrt(a)/sqrt(a+a*sin(c+d*x)))/d-1/3*cot(c+d*x)*csc(c+d*x)^2*(a+a*sin(c+d*x))^(3/2)/d+5/24*a^2*cot(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))-1/4*a*cot(c+d*x)*csc(c+d*x)*sqrt(a+a*sin(c+d*x))/d],

# m<0
[cos(c+d*x)^2*sin(c+d*x)^3/sqrt(a+a*sin(c+d*x)),x,6,-4/105*cos(c+d*x)*(a+a*sin(c+d*x))^(3/2)/(a^2*d)-4/45*cos(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))-2/63*cos(c+d*x)*sin(c+d*x)^3/(d*sqrt(a+a*sin(c+d*x)))+2/9*cos(c+d*x)*sin(c+d*x)^4/(d*sqrt(a+a*sin(c+d*x)))+8/315*cos(c+d*x)*sqrt(a+a*sin(c+d*x))/(a*d)],
[cos(c+d*x)^2*sin(c+d*x)^2/sqrt(a+a*sin(c+d*x)),x,4,-22/105*a*cos(c+d*x)^3/(d*(a+a*sin(c+d*x))^(3/2))+12/35*cos(c+d*x)^3/(d*sqrt(a+a*sin(c+d*x)))-2/7*cos(c+d*x)^3*sqrt(a+a*sin(c+d*x))/(a*d)],
[cos(c+d*x)^2*sin(c+d*x)/sqrt(a+a*sin(c+d*x)),x,2,2/15*a*cos(c+d*x)^3/(d*(a+a*sin(c+d*x))^(3/2))-2/5*cos(c+d*x)^3/(d*sqrt(a+a*sin(c+d*x)))],
[cos(c+d*x)^2*csc(c+d*x)/sqrt(a+a*sin(c+d*x)),x,4,-2*arctanh(cos(c+d*x)*sqrt(a)/sqrt(a+a*sin(c+d*x)))/(d*sqrt(a))+2*cos(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))],
[cos(c+d*x)^2*csc(c+d*x)^2/sqrt(a+a*sin(c+d*x)),x,4,arctanh(cos(c+d*x)*sqrt(a)/sqrt(a+a*sin(c+d*x)))/(d*sqrt(a))-cot(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))],
[cos(c+d*x)^2*csc(c+d*x)^3/sqrt(a+a*sin(c+d*x)),x,5,1/4*arctanh(cos(c+d*x)*sqrt(a)/sqrt(a+a*sin(c+d*x)))/(d*sqrt(a))+1/4*cot(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))-1/2*cot(c+d*x)*csc(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))],
[cos(c+d*x)^2*csc(c+d*x)^4/sqrt(a+a*sin(c+d*x)),x,6,1/8*arctanh(cos(c+d*x)*sqrt(a)/sqrt(a+a*sin(c+d*x)))/(d*sqrt(a))+1/8*cot(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))+1/12*cot(c+d*x)*csc(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))-1/3*cot(c+d*x)*csc(c+d*x)^2/(d*sqrt(a+a*sin(c+d*x)))],
[cos(c+d*x)^2*sin(c+d*x)^3/(a+a*sin(c+d*x))^(3/2),x,8,2*arctanh(cos(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(c+d*x))))*sqrt(2)/(a^(3/2)*d)-344/105*cos(c+d*x)/(a*d*sqrt(a+a*sin(c+d*x)))-16/35*cos(c+d*x)*sin(c+d*x)^2/(a*d*sqrt(a+a*sin(c+d*x)))+2/7*cos(c+d*x)*sin(c+d*x)^3/(a*d*sqrt(a+a*sin(c+d*x)))+76/105*cos(c+d*x)*sqrt(a+a*sin(c+d*x))/(a^2*d)],
[cos(c+d*x)^2*sin(c+d*x)^2/(a+a*sin(c+d*x))^(3/2),x,5,-2*arctanh(cos(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(c+d*x))))*sqrt(2)/(a^(3/2)*d)+18/5*cos(c+d*x)/(a*d*sqrt(a+a*sin(c+d*x)))-2/5*cos(c+d*x)^3/(a*d*sqrt(a+a*sin(c+d*x)))-4/5*cos(c+d*x)*sqrt(a+a*sin(c+d*x))/(a^2*d)],
[cos(c+d*x)^2*sin(c+d*x)/(a+a*sin(c+d*x))^(3/2),x,4,2*arctanh(cos(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(c+d*x))))*sqrt(2)/(a^(3/2)*d)-10/3*cos(c+d*x)/(a*d*sqrt(a+a*sin(c+d*x)))+2/3*cos(c+d*x)*sqrt(a+a*sin(c+d*x))/(a^2*d)],
[cos(c+d*x)^2*csc(c+d*x)/(a+a*sin(c+d*x))^(3/2),x,6,-2*arctanh(cos(c+d*x)*sqrt(a)/sqrt(a+a*sin(c+d*x)))/(a^(3/2)*d)+2*arctanh(cos(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(c+d*x))))*sqrt(2)/(a^(3/2)*d)],
[cos(c+d*x)^2*csc(c+d*x)^2/(a+a*sin(c+d*x))^(3/2),x,6,3*arctanh(cos(c+d*x)*sqrt(a)/sqrt(a+a*sin(c+d*x)))/(a^(3/2)*d)-2*arctanh(cos(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(c+d*x))))*sqrt(2)/(a^(3/2)*d)-cot(c+d*x)/(a*d*sqrt(a+a*sin(c+d*x)))],
[cos(c+d*x)^2*csc(c+d*x)^3/(a+a*sin(c+d*x))^(3/2),x,8,-11/4*arctanh(cos(c+d*x)*sqrt(a)/sqrt(a+a*sin(c+d*x)))/(a^(3/2)*d)+2*arctanh(cos(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(c+d*x))))*sqrt(2)/(a^(3/2)*d)+5/4*cot(c+d*x)/(a*d*sqrt(a+a*sin(c+d*x)))-1/2*cot(c+d*x)*csc(c+d*x)/(a*d*sqrt(a+a*sin(c+d*x)))],
[cos(c+d*x)^2*csc(c+d*x)^4/(a+a*sin(c+d*x))^(3/2),x,9,23/8*arctanh(cos(c+d*x)*sqrt(a)/sqrt(a+a*sin(c+d*x)))/(a^(3/2)*d)-2*arctanh(cos(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(c+d*x))))*sqrt(2)/(a^(3/2)*d)-9/8*cot(c+d*x)/(a*d*sqrt(a+a*sin(c+d*x)))+7/12*cot(c+d*x)*csc(c+d*x)/(a*d*sqrt(a+a*sin(c+d*x)))-1/3*cot(c+d*x)*csc(c+d*x)^2/(a*d*sqrt(a+a*sin(c+d*x)))],

# Integrands of the form Cos[e+f x]^3 (d Sin[e+f x])^n (a+a Sin[e+f x])^m

# Integrands of the form Cos[e+f x]^3 Sin[e+f x]^n (a+a Sin[e+f x])^m

# m>0
[cos(c+d*x)^3*sin(c+d*x)^3*(a+a*sin(c+d*x)),x,4,1/4*a*sin(c+d*x)^4/d+1/5*a*sin(c+d*x)^5/d-1/6*a*sin(c+d*x)^6/d-1/7*a*sin(c+d*x)^7/d],
[cos(c+d*x)^3*sin(c+d*x)^2*(a+a*sin(c+d*x)),x,4,1/3*a*sin(c+d*x)^3/d+1/4*a*sin(c+d*x)^4/d-1/5*a*sin(c+d*x)^5/d-1/6*a*sin(c+d*x)^6/d],
[cos(c+d*x)^3*sin(c+d*x)*(a+a*sin(c+d*x)),x,6,-1/4*a*cos(c+d*x)^4/d+1/3*a*sin(c+d*x)^3/d-1/5*a*sin(c+d*x)^5/d],
[cos(c+d*x)^3*(a+a*sin(c+d*x)),x,3,2/3*(a+a*sin(c+d*x))^3/(a^2*d)-1/4*(a+a*sin(c+d*x))^4/(a^3*d)],
[cos(c+d*x)^3*csc(c+d*x)*(a+a*sin(c+d*x)),x,4,a*log(sin(c+d*x))/d+a*sin(c+d*x)/d-1/2*a*sin(c+d*x)^2/d-1/3*a*sin(c+d*x)^3/d],
[cos(c+d*x)^3*csc(c+d*x)^2*(a+a*sin(c+d*x)),x,4,-a*csc(c+d*x)/d+a*log(sin(c+d*x))/d-a*sin(c+d*x)/d-1/2*a*sin(c+d*x)^2/d],
[cos(c+d*x)^3*csc(c+d*x)^3*(a+a*sin(c+d*x)),x,3,-a*csc(c+d*x)/d-1/2*a*csc(c+d*x)^2/d-a*log(sin(c+d*x))/d-a*sin(c+d*x)/d],

# m<0
[cos(c+d*x)^3*sin(c+d*x)^2/(a+a*sin(c+d*x)),x,4,1/3*sin(c+d*x)^3/(a*d)-1/4*sin(c+d*x)^4/(a*d)],
[cos(c+d*x)^3*sin(c+d*x)/(a+a*sin(c+d*x)),x,5,1/2*sin(c+d*x)^2/(a*d)-1/3*sin(c+d*x)^3/(a*d)],
[cos(c+d*x)^3/(a+a*sin(c+d*x)),x,2,sin(c+d*x)/(a*d)-1/2*sin(c+d*x)^2/(a*d)],
[cos(c+d*x)^3*csc(c+d*x)/(a+a*sin(c+d*x)),x,4,log(sin(c+d*x))/(a*d)-sin(c+d*x)/(a*d)],
[cos(c+d*x)^3*csc(c+d*x)^2/(a+a*sin(c+d*x)),x,4,-csc(c+d*x)/(a*d)-log(sin(c+d*x))/(a*d)],
[cos(c+d*x)^3*csc(c+d*x)^3/(a+a*sin(c+d*x)),x,5,csc(c+d*x)/(a*d)-1/2*csc(c+d*x)^2/(a*d)],
[cos(c+d*x)^3*csc(c+d*x)^4/(a+a*sin(c+d*x)),x,4,1/2*csc(c+d*x)^2/(a*d)-1/3*csc(c+d*x)^3/(a*d)],
[cos(c+d*x)^3*csc(c+d*x)^5/(a+a*sin(c+d*x)),x,4,1/3*csc(c+d*x)^3/(a*d)-1/4*csc(c+d*x)^4/(a*d)],

# Integrands of the form Cos[e+f x]^3 Sin[e+f x]^n (a+a Sin[e+f x])^(m/2)

# Integrands of the form Cos[e+f x]^4 (d Sin[e+f x])^n (a+a Sin[e+f x])^m

# Integrands of the form Cos[e+f x]^4 Sin[e+f x]^n (a+a Sin[e+f x])^m

# m>0
[cos(c+d*x)^4*sin(c+d*x)^4*(a+a*sin(c+d*x)),x,9,3/128*a*x-1/5*a*cos(c+d*x)^5/d+2/7*a*cos(c+d*x)^7/d-1/9*a*cos(c+d*x)^9/d+3/128*a*cos(c+d*x)*sin(c+d*x)/d+1/64*a*cos(c+d*x)^3*sin(c+d*x)/d-1/16*a*cos(c+d*x)^5*sin(c+d*x)/d-1/8*a*cos(c+d*x)^5*sin(c+d*x)^3/d],
[cos(c+d*x)^4*sin(c+d*x)^3*(a+a*sin(c+d*x)),x,9,3/128*a*x-1/5*a*cos(c+d*x)^5/d+1/7*a*cos(c+d*x)^7/d+3/128*a*cos(c+d*x)*sin(c+d*x)/d+1/64*a*cos(c+d*x)^3*sin(c+d*x)/d-1/16*a*cos(c+d*x)^5*sin(c+d*x)/d-1/8*a*cos(c+d*x)^5*sin(c+d*x)^3/d],
[cos(c+d*x)^4*sin(c+d*x)^2*(a+a*sin(c+d*x)),x,8,1/16*a*x-1/5*a*cos(c+d*x)^5/d+1/7*a*cos(c+d*x)^7/d+1/16*a*cos(c+d*x)*sin(c+d*x)/d+1/24*a*cos(c+d*x)^3*sin(c+d*x)/d-1/6*a*cos(c+d*x)^5*sin(c+d*x)/d],
[cos(c+d*x)^4*sin(c+d*x)*(a+a*sin(c+d*x)),x,7,1/16*a*x-1/5*a*cos(c+d*x)^5/d+1/16*a*cos(c+d*x)*sin(c+d*x)/d+1/24*a*cos(c+d*x)^3*sin(c+d*x)/d-1/6*a*cos(c+d*x)^5*sin(c+d*x)/d],
[cos(c+d*x)^4*csc(c+d*x)*(a+a*sin(c+d*x)),x,8,3/8*a*x-a*arctanh(cos(c+d*x))/d+a*cos(c+d*x)/d+1/3*a*cos(c+d*x)^3/d+3/8*a*cos(c+d*x)*sin(c+d*x)/d+1/4*a*cos(c+d*x)^3*sin(c+d*x)/d],
[cos(c+d*x)^4*csc(c+d*x)^2*(a+a*sin(c+d*x)),x,9,-3/2*a*x-a*arctanh(cos(c+d*x))/d+a*cos(c+d*x)/d+1/3*a*cos(c+d*x)^3/d-3/2*a*cot(c+d*x)/d+1/2*a*cos(c+d*x)^2*cot(c+d*x)/d],
[cos(c+d*x)^4*csc(c+d*x)^3*(a+a*sin(c+d*x)),x,9,-3/2*a*x+3/2*a*arctanh(cos(c+d*x))/d-3/2*a*cos(c+d*x)/d-3/2*a*cot(c+d*x)/d+1/2*a*cos(c+d*x)^2*cot(c+d*x)/d-1/2*a*cos(c+d*x)*cot(c+d*x)^2/d],
[cos(c+d*x)^4*csc(c+d*x)^4*(a+a*sin(c+d*x)),x,9,a*x+3/2*a*arctanh(cos(c+d*x))/d-3/2*a*cos(c+d*x)/d+a*cot(c+d*x)/d-1/2*a*cos(c+d*x)*cot(c+d*x)^2/d-1/3*a*cot(c+d*x)^3/d],
[cos(c+d*x)^4*csc(c+d*x)^5*(a+a*sin(c+d*x)),x,7,a*x-3/8*a*arctanh(cos(c+d*x))/d+a*cot(c+d*x)/d-1/3*a*cot(c+d*x)^3/d+3/8*a*cot(c+d*x)*csc(c+d*x)/d-1/4*a*cot(c+d*x)^3*csc(c+d*x)/d],
[cos(c+d*x)^4*csc(c+d*x)^6*(a+a*sin(c+d*x)),x,6,-3/8*a*arctanh(cos(c+d*x))/d-1/5*a*cot(c+d*x)^5/d+3/8*a*cot(c+d*x)*csc(c+d*x)/d-1/4*a*cot(c+d*x)^3*csc(c+d*x)/d],
[cos(c+d*x)^4*csc(c+d*x)^7*(a+a*sin(c+d*x)),x,7,-1/16*a*arctanh(cos(c+d*x))/d-1/5*a*cot(c+d*x)^5/d-1/16*a*cot(c+d*x)*csc(c+d*x)/d+1/8*a*cot(c+d*x)*csc(c+d*x)^3/d-1/6*a*cot(c+d*x)^3*csc(c+d*x)^3/d],
[cos(c+d*x)^4*csc(c+d*x)^8*(a+a*sin(c+d*x)),x,8,-1/16*a*arctanh(cos(c+d*x))/d-1/5*a*cot(c+d*x)^5/d-1/7*a*cot(c+d*x)^7/d-1/16*a*cot(c+d*x)*csc(c+d*x)/d+1/8*a*cot(c+d*x)*csc(c+d*x)^3/d-1/6*a*cot(c+d*x)^3*csc(c+d*x)^3/d],
[cos(c+d*x)^4*csc(c+d*x)^9*(a+a*sin(c+d*x)),x,9,-3/128*a*arctanh(cos(c+d*x))/d-1/5*a*cot(c+d*x)^5/d-1/7*a*cot(c+d*x)^7/d-3/128*a*cot(c+d*x)*csc(c+d*x)/d-1/64*a*cot(c+d*x)*csc(c+d*x)^3/d+1/16*a*cot(c+d*x)*csc(c+d*x)^5/d-1/8*a*cot(c+d*x)^3*csc(c+d*x)^5/d],
[cos(c+d*x)^4*sin(c+d*x)^4*(a+a*sin(c+d*x))^2,x,16,9/256*a^2*x-2/5*a^2*cos(c+d*x)^5/d+4/7*a^2*cos(c+d*x)^7/d-2/9*a^2*cos(c+d*x)^9/d+9/256*a^2*cos(c+d*x)*sin(c+d*x)/d+3/128*a^2*cos(c+d*x)^3*sin(c+d*x)/d-3/32*a^2*cos(c+d*x)^5*sin(c+d*x)/d-3/16*a^2*cos(c+d*x)^5*sin(c+d*x)^3/d-1/10*a^2*cos(c+d*x)^5*sin(c+d*x)^5/d],
[cos(c+d*x)^4*sin(c+d*x)^3*(a+a*sin(c+d*x))^2,x,13,3/64*a^2*x-2/5*a^2*cos(c+d*x)^5/d+3/7*a^2*cos(c+d*x)^7/d-1/9*a^2*cos(c+d*x)^9/d+3/64*a^2*cos(c+d*x)*sin(c+d*x)/d+1/32*a^2*cos(c+d*x)^3*sin(c+d*x)/d-1/8*a^2*cos(c+d*x)^5*sin(c+d*x)/d-1/4*a^2*cos(c+d*x)^5*sin(c+d*x)^3/d],
[cos(c+d*x)^4*sin(c+d*x)^2*(a+a*sin(c+d*x))^2,x,14,11/128*a^2*x-2/5*a^2*cos(c+d*x)^5/d+2/7*a^2*cos(c+d*x)^7/d+11/128*a^2*cos(c+d*x)*sin(c+d*x)/d+11/192*a^2*cos(c+d*x)^3*sin(c+d*x)/d-11/48*a^2*cos(c+d*x)^5*sin(c+d*x)/d-1/8*a^2*cos(c+d*x)^5*sin(c+d*x)^3/d],
[cos(c+d*x)^4*sin(c+d*x)*(a+a*sin(c+d*x))^2,x,6,1/8*a^2*x-1/15*a^2*cos(c+d*x)^5/d+1/8*a^2*cos(c+d*x)*sin(c+d*x)/d+1/12*a^2*cos(c+d*x)^3*sin(c+d*x)/d-1/7*cos(c+d*x)^5*(a+a*sin(c+d*x))^2/d-1/21*cos(c+d*x)^5*(a^2+a^2*sin(c+d*x))/d],
[cos(c+d*x)^4*csc(c+d*x)*(a+a*sin(c+d*x))^2,x,11,3/4*a^2*x-a^2*arctanh(cos(c+d*x))/d+a^2*cos(c+d*x)/d+1/3*a^2*cos(c+d*x)^3/d-1/5*a^2*cos(c+d*x)^5/d+3/4*a^2*cos(c+d*x)*sin(c+d*x)/d+1/2*a^2*cos(c+d*x)^3*sin(c+d*x)/d],
[cos(c+d*x)^4*csc(c+d*x)^2*(a+a*sin(c+d*x))^2,x,13,-9/8*a^2*x-2*a^2*arctanh(cos(c+d*x))/d+2*a^2*cos(c+d*x)/d+2/3*a^2*cos(c+d*x)^3/d-a^2*cot(c+d*x)/d+1/8*a^2*cos(c+d*x)*sin(c+d*x)/d-1/4*a^2*cos(c+d*x)*sin(c+d*x)^3/d],
[cos(c+d*x)^4*csc(c+d*x)^3*(a+a*sin(c+d*x))^2,x,12,-3*a^2*x+1/2*a^2*arctanh(cos(c+d*x))/d+1/3*a^2*cos(c+d*x)^3/d-2*a^2*cot(c+d*x)/d-1/2*a^2*cot(c+d*x)*csc(c+d*x)/d-a^2*cos(c+d*x)*sin(c+d*x)/d],
[cos(c+d*x)^4*csc(c+d*x)^4*(a+a*sin(c+d*x))^2,x,12,-1/2*a^2*x+3*a^2*arctanh(cos(c+d*x))/d-2*a^2*cos(c+d*x)/d-1/3*a^2*cot(c+d*x)^3/d-a^2*cot(c+d*x)*csc(c+d*x)/d-1/2*a^2*cos(c+d*x)*sin(c+d*x)/d],
[cos(c+d*x)^4*csc(c+d*x)^5*(a+a*sin(c+d*x))^2,x,13,2*a^2*x+9/8*a^2*arctanh(cos(c+d*x))/d-a^2*cos(c+d*x)/d+2*a^2*cot(c+d*x)/d-2/3*a^2*cot(c+d*x)^3/d+1/8*a^2*cot(c+d*x)*csc(c+d*x)/d-1/4*a^2*cot(c+d*x)*csc(c+d*x)^3/d],
[cos(c+d*x)^4*csc(c+d*x)^6*(a+a*sin(c+d*x))^2,x,10,a^2*x-3/4*a^2*arctanh(cos(c+d*x))/d+a^2*cot(c+d*x)/d-1/3*a^2*cot(c+d*x)^3/d-1/5*a^2*cot(c+d*x)^5/d+3/4*a^2*cot(c+d*x)*csc(c+d*x)/d-1/2*a^2*cot(c+d*x)^3*csc(c+d*x)/d],
[cos(c+d*x)^4*csc(c+d*x)^7*(a+a*sin(c+d*x))^2,x,11,-7/16*a^2*arctanh(cos(c+d*x))/d-2/5*a^2*cot(c+d*x)^5/d+5/16*a^2*cot(c+d*x)*csc(c+d*x)/d-1/4*a^2*cot(c+d*x)^3*csc(c+d*x)/d+1/8*a^2*cot(c+d*x)*csc(c+d*x)^3/d-1/6*a^2*cot(c+d*x)^3*csc(c+d*x)^3/d],
[cos(c+d*x)^4*csc(c+d*x)^9*(a+a*sin(c+d*x))^2,x,14,-11/128*a^2*arctanh(cos(c+d*x))/d-2/5*a^2*cot(c+d*x)^5/d-2/7*a^2*cot(c+d*x)^7/d-11/128*a^2*cot(c+d*x)*csc(c+d*x)/d+7/64*a^2*cot(c+d*x)*csc(c+d*x)^3/d-1/6*a^2*cot(c+d*x)^3*csc(c+d*x)^3/d+1/16*a^2*cot(c+d*x)*csc(c+d*x)^5/d-1/8*a^2*cot(c+d*x)^3*csc(c+d*x)^5/d],
[cos(c+d*x)^4*csc(c+d*x)^10*(a+a*sin(c+d*x))^2,x,13,-3/64*a^2*arctanh(cos(c+d*x))/d-2/5*a^2*cot(c+d*x)^5/d-3/7*a^2*cot(c+d*x)^7/d-1/9*a^2*cot(c+d*x)^9/d-3/64*a^2*cot(c+d*x)*csc(c+d*x)/d-1/32*a^2*cot(c+d*x)*csc(c+d*x)^3/d+1/8*a^2*cot(c+d*x)*csc(c+d*x)^5/d-1/4*a^2*cot(c+d*x)^3*csc(c+d*x)^5/d],
[cos(c+d*x)^4*csc(c+d*x)^11*(a+a*sin(c+d*x))^2,x,16,-9/256*a^2*arctanh(cos(c+d*x))/d-2/5*a^2*cot(c+d*x)^5/d-4/7*a^2*cot(c+d*x)^7/d-2/9*a^2*cot(c+d*x)^9/d-9/256*a^2*cot(c+d*x)*csc(c+d*x)/d-3/128*a^2*cot(c+d*x)*csc(c+d*x)^3/d+9/160*a^2*cot(c+d*x)*csc(c+d*x)^5/d-1/8*a^2*cot(c+d*x)^3*csc(c+d*x)^5/d+3/80*a^2*cot(c+d*x)*csc(c+d*x)^7/d-1/10*a^2*cot(c+d*x)^3*csc(c+d*x)^7/d],
[cos(c+d*x)^4*sin(c+d*x)^4*(a+a*sin(c+d*x))^3,x,19,15/256*a^3*x-4/5*a^3*cos(c+d*x)^5/d+9/7*a^3*cos(c+d*x)^7/d-2/3*a^3*cos(c+d*x)^9/d+1/11*a^3*cos(c+d*x)^11/d+15/256*a^3*cos(c+d*x)*sin(c+d*x)/d+5/128*a^3*cos(c+d*x)^3*sin(c+d*x)/d-5/32*a^3*cos(c+d*x)^5*sin(c+d*x)/d-5/16*a^3*cos(c+d*x)^5*sin(c+d*x)^3/d-3/10*a^3*cos(c+d*x)^5*sin(c+d*x)^5/d],
[cos(c+d*x)^4*sin(c+d*x)^3*(a+a*sin(c+d*x))^3,x,19,21/256*a^3*x-4/5*a^3*cos(c+d*x)^5/d+a^3*cos(c+d*x)^7/d-1/3*a^3*cos(c+d*x)^9/d+21/256*a^3*cos(c+d*x)*sin(c+d*x)/d+7/128*a^3*cos(c+d*x)^3*sin(c+d*x)/d-7/32*a^3*cos(c+d*x)^5*sin(c+d*x)/d-7/16*a^3*cos(c+d*x)^5*sin(c+d*x)^3/d-1/10*a^3*cos(c+d*x)^5*sin(c+d*x)^5/d],
[cos(c+d*x)^4*sin(c+d*x)^2*(a+a*sin(c+d*x))^3,x,17,17/128*a^3*x-4/5*a^3*cos(c+d*x)^5/d+5/7*a^3*cos(c+d*x)^7/d-1/9*a^3*cos(c+d*x)^9/d+17/128*a^3*cos(c+d*x)*sin(c+d*x)/d+17/192*a^3*cos(c+d*x)^3*sin(c+d*x)/d-17/48*a^3*cos(c+d*x)^5*sin(c+d*x)/d-3/8*a^3*cos(c+d*x)^5*sin(c+d*x)^3/d],
[cos(c+d*x)^4*sin(c+d*x)*(a+a*sin(c+d*x))^3,x,7,27/128*a^3*x-9/80*a^3*cos(c+d*x)^5/d+27/128*a^3*cos(c+d*x)*sin(c+d*x)/d+9/64*a^3*cos(c+d*x)^3*sin(c+d*x)/d-3/56*a*cos(c+d*x)^5*(a+a*sin(c+d*x))^2/d-1/8*cos(c+d*x)^5*(a+a*sin(c+d*x))^3/d-9/112*cos(c+d*x)^5*(a^3+a^3*sin(c+d*x))/d],
[cos(c+d*x)^4*csc(c+d*x)*(a+a*sin(c+d*x))^3,x,15,19/16*a^3*x-a^3*arctanh(cos(c+d*x))/d+a^3*cos(c+d*x)/d+1/3*a^3*cos(c+d*x)^3/d-3/5*a^3*cos(c+d*x)^5/d+19/16*a^3*cos(c+d*x)*sin(c+d*x)/d+19/24*a^3*cos(c+d*x)^3*sin(c+d*x)/d-1/6*a^3*cos(c+d*x)^5*sin(c+d*x)/d],
[cos(c+d*x)^4*csc(c+d*x)^2*(a+a*sin(c+d*x))^3,x,15,-3/8*a^3*x-3*a^3*arctanh(cos(c+d*x))/d+3*a^3*cos(c+d*x)/d+a^3*cos(c+d*x)^3/d-1/5*a^3*cos(c+d*x)^5/d-a^3*cot(c+d*x)/d+11/8*a^3*cos(c+d*x)*sin(c+d*x)/d-3/4*a^3*cos(c+d*x)*sin(c+d*x)^3/d],
[cos(c+d*x)^4*csc(c+d*x)^3*(a+a*sin(c+d*x))^3,x,15,-33/8*a^3*x-3/2*a^3*arctanh(cos(c+d*x))/d+2*a^3*cos(c+d*x)/d+a^3*cos(c+d*x)^3/d-3*a^3*cot(c+d*x)/d-1/2*a^3*cot(c+d*x)*csc(c+d*x)/d-7/8*a^3*cos(c+d*x)*sin(c+d*x)/d-1/4*a^3*cos(c+d*x)*sin(c+d*x)^3/d],
[cos(c+d*x)^4*csc(c+d*x)^4*(a+a*sin(c+d*x))^3,x,14,-7/2*a^3*x+7/2*a^3*arctanh(cos(c+d*x))/d-2*a^3*cos(c+d*x)/d+1/3*a^3*cos(c+d*x)^3/d-2*a^3*cot(c+d*x)/d-1/3*a^3*cot(c+d*x)^3/d-3/2*a^3*cot(c+d*x)*csc(c+d*x)/d-3/2*a^3*cos(c+d*x)*sin(c+d*x)/d],
[cos(c+d*x)^4*csc(c+d*x)^5*(a+a*sin(c+d*x))^3,x,15,3/2*a^3*x+33/8*a^3*arctanh(cos(c+d*x))/d-3*a^3*cos(c+d*x)/d+2*a^3*cot(c+d*x)/d-a^3*cot(c+d*x)^3/d-7/8*a^3*cot(c+d*x)*csc(c+d*x)/d-1/4*a^3*cot(c+d*x)*csc(c+d*x)^3/d-1/2*a^3*cos(c+d*x)*sin(c+d*x)/d],
[cos(c+d*x)^4*csc(c+d*x)^6*(a+a*sin(c+d*x))^3,x,15,3*a^3*x+3/8*a^3*arctanh(cos(c+d*x))/d-a^3*cos(c+d*x)/d+3*a^3*cot(c+d*x)/d-a^3*cot(c+d*x)^3/d-1/5*a^3*cot(c+d*x)^5/d+11/8*a^3*cot(c+d*x)*csc(c+d*x)/d-3/4*a^3*cot(c+d*x)*csc(c+d*x)^3/d],
[cos(c+d*x)^4*csc(c+d*x)^7*(a+a*sin(c+d*x))^3,x,14,a^3*x-19/16*a^3*arctanh(cos(c+d*x))/d+a^3*cot(c+d*x)/d-1/3*a^3*cot(c+d*x)^3/d-3/5*a^3*cot(c+d*x)^5/d+17/16*a^3*cot(c+d*x)*csc(c+d*x)/d-3/4*a^3*cot(c+d*x)^3*csc(c+d*x)/d+1/8*a^3*cot(c+d*x)*csc(c+d*x)^3/d-1/6*a^3*cot(c+d*x)^3*csc(c+d*x)^3/d],
[cos(c+d*x)^4*csc(c+d*x)^8*(a+a*sin(c+d*x))^3,x,14,-9/16*a^3*arctanh(cos(c+d*x))/d-4/5*a^3*cot(c+d*x)^5/d-1/7*a^3*cot(c+d*x)^7/d+3/16*a^3*cot(c+d*x)*csc(c+d*x)/d-1/4*a^3*cot(c+d*x)^3*csc(c+d*x)/d+3/8*a^3*cot(c+d*x)*csc(c+d*x)^3/d-1/2*a^3*cot(c+d*x)^3*csc(c+d*x)^3/d],
[cos(c+d*x)^4*csc(c+d*x)^9*(a+a*sin(c+d*x))^3,x,16,-27/128*a^3*arctanh(cos(c+d*x))/d-4/5*a^3*cot(c+d*x)^5/d-3/7*a^3*cot(c+d*x)^7/d-27/128*a^3*cot(c+d*x)*csc(c+d*x)/d+23/64*a^3*cot(c+d*x)*csc(c+d*x)^3/d-1/2*a^3*cot(c+d*x)^3*csc(c+d*x)^3/d+1/16*a^3*cot(c+d*x)*csc(c+d*x)^5/d-1/8*a^3*cot(c+d*x)^3*csc(c+d*x)^5/d],
[cos(c+d*x)^4*csc(c+d*x)^10*(a+a*sin(c+d*x))^3,x,17,-17/128*a^3*arctanh(cos(c+d*x))/d-4/5*a^3*cot(c+d*x)^5/d-5/7*a^3*cot(c+d*x)^7/d-1/9*a^3*cot(c+d*x)^9/d-17/128*a^3*cot(c+d*x)*csc(c+d*x)/d+5/64*a^3*cot(c+d*x)*csc(c+d*x)^3/d-1/6*a^3*cot(c+d*x)^3*csc(c+d*x)^3/d+3/16*a^3*cot(c+d*x)*csc(c+d*x)^5/d-3/8*a^3*cot(c+d*x)^3*csc(c+d*x)^5/d],
[cos(c+d*x)^4*csc(c+d*x)^11*(a+a*sin(c+d*x))^3,x,19,-21/256*a^3*arctanh(cos(c+d*x))/d-4/5*a^3*cot(c+d*x)^5/d-a^3*cot(c+d*x)^7/d-1/3*a^3*cot(c+d*x)^9/d-21/256*a^3*cot(c+d*x)*csc(c+d*x)/d-7/128*a^3*cot(c+d*x)*csc(c+d*x)^3/d+29/160*a^3*cot(c+d*x)*csc(c+d*x)^5/d-3/8*a^3*cot(c+d*x)^3*csc(c+d*x)^5/d+3/80*a^3*cot(c+d*x)*csc(c+d*x)^7/d-1/10*a^3*cot(c+d*x)^3*csc(c+d*x)^7/d],
[cos(c+d*x)^4*sin(c+d*x)^2*(a+a*sin(c+d*x))^4,x,8,55/256*a^4*x-11/112*a^4*cos(c+d*x)^7/d+55/256*a^4*cos(c+d*x)*sin(c+d*x)/d+55/384*a^4*cos(c+d*x)^3*sin(c+d*x)/d+11/96*a^4*cos(c+d*x)^5*sin(c+d*x)/d-1/10*cos(c+d*x)^5*(a+a*sin(c+d*x))^5/(a*d)-1/18*cos(c+d*x)^7*(a^2+a^2*sin(c+d*x))^2/d-11/144*cos(c+d*x)^7*(a^4+a^4*sin(c+d*x))/d],
[cos(c+d*x)^4*csc(c+d*x)^4*(a+a*sin(c+d*x))^4,x,17,-61/8*a^4*x+2*a^4*arctanh(cos(c+d*x))/d+4/3*a^4*cos(c+d*x)^3/d-5*a^4*cot(c+d*x)/d-1/3*a^4*cot(c+d*x)^3/d-2*a^4*cot(c+d*x)*csc(c+d*x)/d-19/8*a^4*cos(c+d*x)*sin(c+d*x)/d-1/4*a^4*cos(c+d*x)*sin(c+d*x)^3/d],

# m<0
[cos(c+d*x)^4*sin(c+d*x)^4/(a+a*sin(c+d*x)),x,8,1/16*x/a+1/3*cos(c+d*x)^3/(a*d)-2/5*cos(c+d*x)^5/(a*d)+1/7*cos(c+d*x)^7/(a*d)+1/16*cos(c+d*x)*sin(c+d*x)/(a*d)-1/8*cos(c+d*x)^3*sin(c+d*x)/(a*d)-1/6*cos(c+d*x)^3*sin(c+d*x)^3/(a*d)],
[cos(c+d*x)^4*sin(c+d*x)^3/(a+a*sin(c+d*x)),x,8,-1/16*x/a-1/3*cos(c+d*x)^3/(a*d)+1/5*cos(c+d*x)^5/(a*d)-1/16*cos(c+d*x)*sin(c+d*x)/(a*d)+1/8*cos(c+d*x)^3*sin(c+d*x)/(a*d)+1/6*cos(c+d*x)^3*sin(c+d*x)^3/(a*d)],
[cos(c+d*x)^4*sin(c+d*x)^2/(a+a*sin(c+d*x)),x,7,1/8*x/a+1/3*cos(c+d*x)^3/(a*d)-1/5*cos(c+d*x)^5/(a*d)+1/8*cos(c+d*x)*sin(c+d*x)/(a*d)-1/4*cos(c+d*x)^3*sin(c+d*x)/(a*d)],
[cos(c+d*x)^4*sin(c+d*x)/(a+a*sin(c+d*x)),x,6,-1/8*x/a-1/3*cos(c+d*x)^3/(a*d)-1/8*cos(c+d*x)*sin(c+d*x)/(a*d)+1/4*cos(c+d*x)^3*sin(c+d*x)/(a*d)],
[cos(c+d*x)^4*csc(c+d*x)/(a+a*sin(c+d*x)),x,6,-1/2*x/a-arctanh(cos(c+d*x))/(a*d)+cos(c+d*x)/(a*d)-1/2*cos(c+d*x)*sin(c+d*x)/(a*d)],
[cos(c+d*x)^4*csc(c+d*x)^2/(a+a*sin(c+d*x)),x,6,-x/a+arctanh(cos(c+d*x))/(a*d)-cos(c+d*x)/(a*d)-cot(c+d*x)/(a*d)],
[cos(c+d*x)^4*csc(c+d*x)^3/(a+a*sin(c+d*x)),x,5,x/a+1/2*arctanh(cos(c+d*x))/(a*d)+cot(c+d*x)/(a*d)-1/2*cot(c+d*x)*csc(c+d*x)/(a*d)],
[cos(c+d*x)^4*csc(c+d*x)^4/(a+a*sin(c+d*x)),x,5,-1/2*arctanh(cos(c+d*x))/(a*d)-1/3*cot(c+d*x)^3/(a*d)+1/2*cot(c+d*x)*csc(c+d*x)/(a*d)],
[cos(c+d*x)^4*csc(c+d*x)^5/(a+a*sin(c+d*x)),x,6,1/8*arctanh(cos(c+d*x))/(a*d)+1/3*cot(c+d*x)^3/(a*d)+1/8*cot(c+d*x)*csc(c+d*x)/(a*d)-1/4*cot(c+d*x)*csc(c+d*x)^3/(a*d)],
[cos(c+d*x)^4*csc(c+d*x)^6/(a+a*sin(c+d*x)),x,7,-1/8*arctanh(cos(c+d*x))/(a*d)-1/3*cot(c+d*x)^3/(a*d)-1/5*cot(c+d*x)^5/(a*d)-1/8*cot(c+d*x)*csc(c+d*x)/(a*d)+1/4*cot(c+d*x)*csc(c+d*x)^3/(a*d)],
[cos(c+d*x)^4*csc(c+d*x)^7/(a+a*sin(c+d*x)),x,8,1/16*arctanh(cos(c+d*x))/(a*d)+1/3*cot(c+d*x)^3/(a*d)+1/5*cot(c+d*x)^5/(a*d)+1/16*cot(c+d*x)*csc(c+d*x)/(a*d)+1/24*cot(c+d*x)*csc(c+d*x)^3/(a*d)-1/6*cot(c+d*x)*csc(c+d*x)^5/(a*d)],
[cos(c+d*x)^4*sin(c+d*x)^5/(a+a*sin(c+d*x))^2,x,11,-5/8*x/a^2-2*cos(c+d*x)/(a^2*d)+5/3*cos(c+d*x)^3/(a^2*d)-4/5*cos(c+d*x)^5/(a^2*d)+1/7*cos(c+d*x)^7/(a^2*d)+5/8*cos(c+d*x)*sin(c+d*x)/(a^2*d)+5/12*cos(c+d*x)*sin(c+d*x)^3/(a^2*d)+1/3*cos(c+d*x)*sin(c+d*x)^5/(a^2*d)],
[cos(c+d*x)^4*sin(c+d*x)^4/(a+a*sin(c+d*x))^2,x,12,11/16*x/a^2+2*cos(c+d*x)/(a^2*d)-4/3*cos(c+d*x)^3/(a^2*d)+2/5*cos(c+d*x)^5/(a^2*d)-11/16*cos(c+d*x)*sin(c+d*x)/(a^2*d)-11/24*cos(c+d*x)*sin(c+d*x)^3/(a^2*d)-1/6*cos(c+d*x)*sin(c+d*x)^5/(a^2*d)],
[cos(c+d*x)^4*sin(c+d*x)^3/(a+a*sin(c+d*x))^2,x,10,-3/4*x/a^2-2*cos(c+d*x)/(a^2*d)+cos(c+d*x)^3/(a^2*d)-1/5*cos(c+d*x)^5/(a^2*d)+3/4*cos(c+d*x)*sin(c+d*x)/(a^2*d)+1/2*cos(c+d*x)*sin(c+d*x)^3/(a^2*d)],
[cos(c+d*x)^4*sin(c+d*x)^2/(a+a*sin(c+d*x))^2,x,10,7/8*x/a^2+2*cos(c+d*x)/(a^2*d)-2/3*cos(c+d*x)^3/(a^2*d)-7/8*cos(c+d*x)*sin(c+d*x)/(a^2*d)-1/4*cos(c+d*x)*sin(c+d*x)^3/(a^2*d)],
[cos(c+d*x)^4*sin(c+d*x)/(a+a*sin(c+d*x))^2,x,4,-x/a^2-2/3*cos(c+d*x)^3/(a^2*d)-cos(c+d*x)*sin(c+d*x)/(a^2*d)-cos(c+d*x)^5/(d*(a+a*sin(c+d*x))^2)],
[cos(c+d*x)^4*csc(c+d*x)/(a+a*sin(c+d*x))^2,x,4,-2*x/a^2-arctanh(cos(c+d*x))/(a^2*d)-cos(c+d*x)/(a^2*d)],
[cos(c+d*x)^4*csc(c+d*x)^2/(a+a*sin(c+d*x))^2,x,6,x/a^2+2*arctanh(cos(c+d*x))/(a^2*d)-cot(c+d*x)/(a^2*d)],
[cos(c+d*x)^4*csc(c+d*x)^3/(a+a*sin(c+d*x))^2,x,8,-3/2*arctanh(cos(c+d*x))/(a^2*d)+2*cot(c+d*x)/(a^2*d)-1/2*cot(c+d*x)*csc(c+d*x)/(a^2*d)],
[cos(c+d*x)^4*csc(c+d*x)^4/(a+a*sin(c+d*x))^2,x,9,arctanh(cos(c+d*x))/(a^2*d)-2*cot(c+d*x)/(a^2*d)-1/3*cot(c+d*x)^3/(a^2*d)+cot(c+d*x)*csc(c+d*x)/(a^2*d)],
[cos(c+d*x)^4*csc(c+d*x)^5/(a+a*sin(c+d*x))^2,x,10,-7/8*arctanh(cos(c+d*x))/(a^2*d)+2*cot(c+d*x)/(a^2*d)+2/3*cot(c+d*x)^3/(a^2*d)-7/8*cot(c+d*x)*csc(c+d*x)/(a^2*d)-1/4*cot(c+d*x)*csc(c+d*x)^3/(a^2*d)],
[cos(c+d*x)^4*csc(c+d*x)^6/(a+a*sin(c+d*x))^2,x,10,3/4*arctanh(cos(c+d*x))/(a^2*d)-2*cot(c+d*x)/(a^2*d)-cot(c+d*x)^3/(a^2*d)-1/5*cot(c+d*x)^5/(a^2*d)+3/4*cot(c+d*x)*csc(c+d*x)/(a^2*d)+1/2*cot(c+d*x)*csc(c+d*x)^3/(a^2*d)],
[cos(c+d*x)^4*csc(c+d*x)^7/(a+a*sin(c+d*x))^2,x,12,-11/16*arctanh(cos(c+d*x))/(a^2*d)+2*cot(c+d*x)/(a^2*d)+4/3*cot(c+d*x)^3/(a^2*d)+2/5*cot(c+d*x)^5/(a^2*d)-11/16*cot(c+d*x)*csc(c+d*x)/(a^2*d)-11/24*cot(c+d*x)*csc(c+d*x)^3/(a^2*d)-1/6*cot(c+d*x)*csc(c+d*x)^5/(a^2*d)],
[cos(c+d*x)^4*sin(c+d*x)^3/(a+a*sin(c+d*x))^3,x,12,51/8*x/a^3+7*cos(c+d*x)/(a^3*d)-cos(c+d*x)^3/(a^3*d)-19/8*cos(c+d*x)*sin(c+d*x)/(a^3*d)-1/4*cos(c+d*x)*sin(c+d*x)^3/(a^3*d)+4*cos(c+d*x)/(a^3*d*(1+sin(c+d*x)))],
[cos(c+d*x)^4*sin(c+d*x)^2/(a+a*sin(c+d*x))^3,x,9,-11/2*x/a^3-5*cos(c+d*x)/(a^3*d)+1/3*cos(c+d*x)^3/(a^3*d)+3/2*cos(c+d*x)*sin(c+d*x)/(a^3*d)-4*cos(c+d*x)/(a^3*d*(1+sin(c+d*x)))],
[cos(c+d*x)^4*sin(c+d*x)/(a+a*sin(c+d*x))^3,x,4,9/2*x/a^3+9/2*cos(c+d*x)/(a^3*d)+cos(c+d*x)^5/(d*(a+a*sin(c+d*x))^3)+3/2*cos(c+d*x)^3/(d*(a^3+a^3*sin(c+d*x)))],
[cos(c+d*x)^4*csc(c+d*x)/(a+a*sin(c+d*x))^3,x,5,x/a^3-arctanh(cos(c+d*x))/(a^3*d)+4*cos(c+d*x)/(a^3*d*(1+sin(c+d*x)))],
[cos(c+d*x)^4*csc(c+d*x)^2/(a+a*sin(c+d*x))^3,x,7,3*arctanh(cos(c+d*x))/(a^3*d)-cot(c+d*x)/(a^3*d)-4*cos(c+d*x)/(a^3*d*(1+sin(c+d*x)))],
[cos(c+d*x)^4*csc(c+d*x)^3/(a+a*sin(c+d*x))^3,x,9,-9/2*arctanh(cos(c+d*x))/(a^3*d)+3*cot(c+d*x)/(a^3*d)-1/2*cot(c+d*x)*csc(c+d*x)/(a^3*d)+4*cos(c+d*x)/(a^3*d*(1+sin(c+d*x)))],
[cos(c+d*x)^4*csc(c+d*x)^4/(a+a*sin(c+d*x))^3,x,12,11/2*arctanh(cos(c+d*x))/(a^3*d)-13*cot(c+d*x)/(a^3*d)-13/3*cot(c+d*x)^3/(a^3*d)+11/2*cot(c+d*x)*csc(c+d*x)/(a^3*d)+4*cot(c+d*x)*csc(c+d*x)^2/(a^3*d*(1+sin(c+d*x)))],
[cos(c+d*x)^4*csc(c+d*x)^5/(a+a*sin(c+d*x))^3,x,14,-51/8*arctanh(cos(c+d*x))/(a^3*d)+7*cot(c+d*x)/(a^3*d)+cot(c+d*x)^3/(a^3*d)-19/8*cot(c+d*x)*csc(c+d*x)/(a^3*d)-1/4*cot(c+d*x)*csc(c+d*x)^3/(a^3*d)+4*cos(c+d*x)/(a^3*d*(1+sin(c+d*x)))],
[cos(e+f*x)^4*sin(e+f*x)/(a+a*sin(e+f*x))^6,x,2,1/7*cos(e+f*x)^5/(f*(a+a*sin(e+f*x))^6)-6/35*cos(e+f*x)^5/(a*f*(a+a*sin(e+f*x))^5)],
[cos(e+f*x)^4*sin(e+f*x)^2/(a+a*sin(e+f*x))^7,x,18,-1/18*a*cos(e+f*x)^7/(f*(a+a*sin(e+f*x))^8)+25/126*cos(e+f*x)^5/(a*f*(a+a*sin(e+f*x))^6)-47/315*cos(e+f*x)^5/(a^2*f*(a+a*sin(e+f*x))^5),-4/9*cos(e+f*x)/(a^7*f*(1+sin(e+f*x))^5)+92/63*cos(e+f*x)/(a^7*f*(1+sin(e+f*x))^4)-181/105*cos(e+f*x)/(a^7*f*(1+sin(e+f*x))^3)+268/315*cos(e+f*x)/(a^7*f*(1+sin(e+f*x))^2)-47/315*cos(e+f*x)/(a^7*f*(1+sin(e+f*x)))],
[cos(e+f*x)^4*sin(e+f*x)^3/(a+a*sin(e+f*x))^8,x,24,4/11*cos(e+f*x)/(a^8*f*(1+sin(e+f*x))^6)-52/33*cos(e+f*x)/(a^8*f*(1+sin(e+f*x))^5)+617/231*cos(e+f*x)/(a^8*f*(1+sin(e+f*x))^4)-846/385*cos(e+f*x)/(a^8*f*(1+sin(e+f*x))^3)+1003/1155*cos(e+f*x)/(a^8*f*(1+sin(e+f*x))^2)-152/1155*cos(e+f*x)/(a^8*f*(1+sin(e+f*x)))],

# Integrands of the form Cos[e+f x]^4 Sin[e+f x]^n (a+a Sin[e+f x])^(m/2)

# m>0
[cos(c+d*x)^4*sin(c+d*x)^2*sqrt(a+a*sin(c+d*x)),x,5,-1472/45045*a^3*cos(c+d*x)^5/(d*(a+a*sin(c+d*x))^(5/2))-368/9009*a^2*cos(c+d*x)^5/(d*(a+a*sin(c+d*x))^(3/2))-2/13*cos(c+d*x)^5*(a+a*sin(c+d*x))^(3/2)/(a*d)-46/1287*a*cos(c+d*x)^5/(d*sqrt(a+a*sin(c+d*x)))+20/143*cos(c+d*x)^5*sqrt(a+a*sin(c+d*x))/d],
[cos(c+d*x)^4*sin(c+d*x)*sqrt(a+a*sin(c+d*x)),x,4,-64/3465*a^3*cos(c+d*x)^5/(d*(a+a*sin(c+d*x))^(5/2))-16/693*a^2*cos(c+d*x)^5/(d*(a+a*sin(c+d*x))^(3/2))-2/99*a*cos(c+d*x)^5/(d*sqrt(a+a*sin(c+d*x)))-2/11*cos(c+d*x)^5*sqrt(a+a*sin(c+d*x))/d],
[cos(c+d*x)^4*csc(c+d*x)*sqrt(a+a*sin(c+d*x)),x,9,-12/35*cos(c+d*x)*(a+a*sin(c+d*x))^(3/2)/(a*d)-2*arctanh(cos(c+d*x)*sqrt(a)/sqrt(a+a*sin(c+d*x)))*sqrt(a)/d+8/15*a*cos(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))-2/7*a*cos(c+d*x)*sin(c+d*x)^3/(d*sqrt(a+a*sin(c+d*x)))+164/105*cos(c+d*x)*sqrt(a+a*sin(c+d*x))/d],
[cos(c+d*x)^4*csc(c+d*x)^2*sqrt(a+a*sin(c+d*x)),x,8,-2/5*cos(c+d*x)*(a+a*sin(c+d*x))^(3/2)/(a*d)-arctanh(cos(c+d*x)*sqrt(a)/sqrt(a+a*sin(c+d*x)))*sqrt(a)/d+61/15*a*cos(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))+4/15*cos(c+d*x)*sqrt(a+a*sin(c+d*x))/d-cot(c+d*x)*sqrt(a+a*sin(c+d*x))/d],
[cos(c+d*x)^4*csc(c+d*x)^3*sqrt(a+a*sin(c+d*x)),x,7,13/4*arctanh(cos(c+d*x)*sqrt(a)/sqrt(a+a*sin(c+d*x)))*sqrt(a)/d-2/3*a*cos(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))-1/4*a*cot(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))-2/3*cos(c+d*x)*sqrt(a+a*sin(c+d*x))/d-1/2*cot(c+d*x)*csc(c+d*x)*sqrt(a+a*sin(c+d*x))/d],
[cos(c+d*x)^4*csc(c+d*x)^4*sqrt(a+a*sin(c+d*x)),x,7,11/8*arctanh(cos(c+d*x)*sqrt(a)/sqrt(a+a*sin(c+d*x)))*sqrt(a)/d-2*a*cos(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))+11/8*a*cot(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))-1/12*a*cot(c+d*x)*csc(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))-1/3*cot(c+d*x)*csc(c+d*x)^2*sqrt(a+a*sin(c+d*x))/d],
[cos(c+d*x)^4*csc(c+d*x)^5*sqrt(a+a*sin(c+d*x)),x,9,-67/64*arctanh(cos(c+d*x)*sqrt(a)/sqrt(a+a*sin(c+d*x)))*sqrt(a)/d+61/64*a*cot(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))+61/96*a*cot(c+d*x)*csc(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))-1/24*a*cot(c+d*x)*csc(c+d*x)^2/(d*sqrt(a+a*sin(c+d*x)))-1/4*cot(c+d*x)*csc(c+d*x)^3*sqrt(a+a*sin(c+d*x))/d],
[cos(c+d*x)^4*csc(c+d*x)^6*sqrt(a+a*sin(c+d*x)),x,11,-31/128*arctanh(cos(c+d*x)*sqrt(a)/sqrt(a+a*sin(c+d*x)))*sqrt(a)/d-31/128*a*cot(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))+97/192*a*cot(c+d*x)*csc(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))+97/240*a*cot(c+d*x)*csc(c+d*x)^2/(d*sqrt(a+a*sin(c+d*x)))-1/40*a*cot(c+d*x)*csc(c+d*x)^3/(d*sqrt(a+a*sin(c+d*x)))-1/5*cot(c+d*x)*csc(c+d*x)^4*sqrt(a+a*sin(c+d*x))/d],
[cos(c+d*x)^4*csc(c+d*x)^7*sqrt(a+a*sin(c+d*x)),x,13,-55/512*arctanh(cos(c+d*x)*sqrt(a)/sqrt(a+a*sin(c+d*x)))*sqrt(a)/d-55/512*a*cot(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))-55/768*a*cot(c+d*x)*csc(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))+329/960*a*cot(c+d*x)*csc(c+d*x)^2/(d*sqrt(a+a*sin(c+d*x)))+47/160*a*cot(c+d*x)*csc(c+d*x)^3/(d*sqrt(a+a*sin(c+d*x)))-1/60*a*cot(c+d*x)*csc(c+d*x)^4/(d*sqrt(a+a*sin(c+d*x)))-1/6*cot(c+d*x)*csc(c+d*x)^5*sqrt(a+a*sin(c+d*x))/d],
[cos(c+d*x)^4*csc(c+d*x)^8*sqrt(a+a*sin(c+d*x)),x,15,-61/1024*arctanh(cos(c+d*x)*sqrt(a)/sqrt(a+a*sin(c+d*x)))*sqrt(a)/d-61/1024*a*cot(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))-61/1536*a*cot(c+d*x)*csc(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))-61/1920*a*cot(c+d*x)*csc(c+d*x)^2/(d*sqrt(a+a*sin(c+d*x)))+579/2240*a*cot(c+d*x)*csc(c+d*x)^3/(d*sqrt(a+a*sin(c+d*x)))+193/840*a*cot(c+d*x)*csc(c+d*x)^4/(d*sqrt(a+a*sin(c+d*x)))-1/84*a*cot(c+d*x)*csc(c+d*x)^5/(d*sqrt(a+a*sin(c+d*x)))-1/7*cot(c+d*x)*csc(c+d*x)^6*sqrt(a+a*sin(c+d*x))/d],
[cos(c+d*x)^4*sin(c+d*x)^2*(a+a*sin(c+d*x))^(3/2),x,6,-256/6435*a^4*cos(c+d*x)^5/(d*(a+a*sin(c+d*x))^(5/2))-64/1287*a^3*cos(c+d*x)^5/(d*(a+a*sin(c+d*x))^(3/2))+4/39*cos(c+d*x)^5*(a+a*sin(c+d*x))^(3/2)/d-2/15*cos(c+d*x)^5*(a+a*sin(c+d*x))^(5/2)/(a*d)-56/1287*a^2*cos(c+d*x)^5/(d*sqrt(a+a*sin(c+d*x)))-14/429*a*cos(c+d*x)^5*sqrt(a+a*sin(c+d*x))/d],
[cos(c+d*x)^4*sin(c+d*x)*(a+a*sin(c+d*x))^(3/2),x,5,-256/5005*a^4*cos(c+d*x)^5/(d*(a+a*sin(c+d*x))^(5/2))-64/1001*a^3*cos(c+d*x)^5/(d*(a+a*sin(c+d*x))^(3/2))-2/13*cos(c+d*x)^5*(a+a*sin(c+d*x))^(3/2)/d-8/143*a^2*cos(c+d*x)^5/(d*sqrt(a+a*sin(c+d*x)))-6/143*a*cos(c+d*x)^5*sqrt(a+a*sin(c+d*x))/d],
[cos(c+d*x)^4*csc(c+d*x)*(a+a*sin(c+d*x))^(3/2),x,12,-2*a^(3/2)*arctanh(cos(c+d*x)*sqrt(a)/sqrt(a+a*sin(c+d*x)))/d+16/105*cos(c+d*x)*(a+a*sin(c+d*x))^(3/2)/d-14/45*a^2*cos(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))-34/63*a^2*cos(c+d*x)*sin(c+d*x)^3/(d*sqrt(a+a*sin(c+d*x)))-2/9*a^2*cos(c+d*x)*sin(c+d*x)^4/(d*sqrt(a+a*sin(c+d*x)))+388/315*a*cos(c+d*x)*sqrt(a+a*sin(c+d*x))/d],
[cos(c+d*x)^4*csc(c+d*x)^2*(a+a*sin(c+d*x))^(3/2),x,10,-3*a^(3/2)*arctanh(cos(c+d*x)*sqrt(a)/sqrt(a+a*sin(c+d*x)))/d+4/35*cos(c+d*x)*(a+a*sin(c+d*x))^(3/2)/d-cot(c+d*x)*(a+a*sin(c+d*x))^(3/2)/d-2/7*cos(c+d*x)*(a+a*sin(c+d*x))^(5/2)/(a*d)+171/35*a^2*cos(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))+69/35*a*cos(c+d*x)*sqrt(a+a*sin(c+d*x))/d],
[cos(c+d*x)^4*csc(c+d*x)^3*(a+a*sin(c+d*x))^(3/2),x,9,9/4*a^(3/2)*arctanh(cos(c+d*x)*sqrt(a)/sqrt(a+a*sin(c+d*x)))/d-2/5*cos(c+d*x)*(a+a*sin(c+d*x))^(3/2)/d-1/2*cot(c+d*x)*csc(c+d*x)*(a+a*sin(c+d*x))^(3/2)/d+73/20*a^2*cos(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))-2/5*a*cos(c+d*x)*sqrt(a+a*sin(c+d*x))/d-3/4*a*cot(c+d*x)*sqrt(a+a*sin(c+d*x))/d],
[cos(c+d*x)^4*csc(c+d*x)^4*(a+a*sin(c+d*x))^(3/2),x,8,37/8*a^(3/2)*arctanh(cos(c+d*x)*sqrt(a)/sqrt(a+a*sin(c+d*x)))/d-1/3*cot(c+d*x)*csc(c+d*x)^2*(a+a*sin(c+d*x))^(3/2)/d-8/3*a^2*cos(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))+29/24*a^2*cot(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))-2/3*a*cos(c+d*x)*sqrt(a+a*sin(c+d*x))/d-1/4*a*cot(c+d*x)*csc(c+d*x)*sqrt(a+a*sin(c+d*x))/d],
[cos(c+d*x)^4*csc(c+d*x)^5*(a+a*sin(c+d*x))^(3/2),x,11,21/64*a^(3/2)*arctanh(cos(c+d*x)*sqrt(a)/sqrt(a+a*sin(c+d*x)))/d-1/4*cot(c+d*x)*csc(c+d*x)^3*(a+a*sin(c+d*x))^(3/2)/d-2*a^2*cos(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))+149/64*a^2*cot(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))+19/32*a^2*cot(c+d*x)*csc(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))-1/8*a*cot(c+d*x)*csc(c+d*x)^2*sqrt(a+a*sin(c+d*x))/d],
[cos(c+d*x)^4*csc(c+d*x)^6*(a+a*sin(c+d*x))^(3/2),x,12,-165/128*a^(3/2)*arctanh(cos(c+d*x)*sqrt(a)/sqrt(a+a*sin(c+d*x)))/d-1/5*cot(c+d*x)*csc(c+d*x)^4*(a+a*sin(c+d*x))^(3/2)/d+91/128*a^2*cot(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))+73/64*a^2*cot(c+d*x)*csc(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))+31/80*a^2*cot(c+d*x)*csc(c+d*x)^2/(d*sqrt(a+a*sin(c+d*x)))-3/40*a*cot(c+d*x)*csc(c+d*x)^3*sqrt(a+a*sin(c+d*x))/d],
[cos(c+d*x)^4*csc(c+d*x)^7*(a+a*sin(c+d*x))^(3/2),x,14,-179/512*a^(3/2)*arctanh(cos(c+d*x)*sqrt(a)/sqrt(a+a*sin(c+d*x)))/d-1/6*cot(c+d*x)*csc(c+d*x)^5*(a+a*sin(c+d*x))^(3/2)/d-179/512*a^2*cot(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))+111/256*a^2*cot(c+d*x)*csc(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))+239/320*a^2*cot(c+d*x)*csc(c+d*x)^2/(d*sqrt(a+a*sin(c+d*x)))+137/480*a^2*cot(c+d*x)*csc(c+d*x)^3/(d*sqrt(a+a*sin(c+d*x)))-1/20*a*cot(c+d*x)*csc(c+d*x)^4*sqrt(a+a*sin(c+d*x))/d],
[cos(c+d*x)^4*csc(c+d*x)^8*(a+a*sin(c+d*x))^(3/2),x,16,-171/1024*a^(3/2)*arctanh(cos(c+d*x)*sqrt(a)/sqrt(a+a*sin(c+d*x)))/d-1/7*cot(c+d*x)*csc(c+d*x)^6*(a+a*sin(c+d*x))^(3/2)/d-171/1024*a^2*cot(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))-57/512*a^2*cot(c+d*x)*csc(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))+199/640*a^2*cot(c+d*x)*csc(c+d*x)^2/(d*sqrt(a+a*sin(c+d*x)))+1237/2240*a^2*cot(c+d*x)*csc(c+d*x)^3/(d*sqrt(a+a*sin(c+d*x)))+9/40*a^2*cot(c+d*x)*csc(c+d*x)^4/(d*sqrt(a+a*sin(c+d*x)))-1/28*a*cot(c+d*x)*csc(c+d*x)^5*sqrt(a+a*sin(c+d*x))/d],
[cos(c+d*x)^4*csc(c+d*x)^9*(a+a*sin(c+d*x))^(3/2),x,18,-1587/16384*a^(3/2)*arctanh(cos(c+d*x)*sqrt(a)/sqrt(a+a*sin(c+d*x)))/d-1/8*cot(c+d*x)*csc(c+d*x)^7*(a+a*sin(c+d*x))^(3/2)/d-1587/16384*a^2*cot(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))-529/8192*a^2*cot(c+d*x)*csc(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))-529/10240*a^2*cot(c+d*x)*csc(c+d*x)^2/(d*sqrt(a+a*sin(c+d*x)))+8653/35840*a^2*cot(c+d*x)*csc(c+d*x)^3/(d*sqrt(a+a*sin(c+d*x)))+1957/4480*a^2*cot(c+d*x)*csc(c+d*x)^4/(d*sqrt(a+a*sin(c+d*x)))+83/448*a^2*cot(c+d*x)*csc(c+d*x)^5/(d*sqrt(a+a*sin(c+d*x)))-3/112*a*cot(c+d*x)*csc(c+d*x)^6*sqrt(a+a*sin(c+d*x))/d],

# m<0
[cos(c+d*x)^4*sin(c+d*x)^2/sqrt(a+a*sin(c+d*x)),x,5,-152/3465*a^2*cos(c+d*x)^5/(d*(a+a*sin(c+d*x))^(5/2))-38/693*a*cos(c+d*x)^5/(d*(a+a*sin(c+d*x))^(3/2))+20/99*cos(c+d*x)^5/(d*sqrt(a+a*sin(c+d*x)))-2/11*cos(c+d*x)^5*sqrt(a+a*sin(c+d*x))/(a*d)],
[cos(c+d*x)^4*sin(c+d*x)/sqrt(a+a*sin(c+d*x)),x,3,8/315*a^2*cos(c+d*x)^5/(d*(a+a*sin(c+d*x))^(5/2))+2/63*a*cos(c+d*x)^5/(d*(a+a*sin(c+d*x))^(3/2))-2/9*cos(c+d*x)^5/(d*sqrt(a+a*sin(c+d*x)))],
[cos(c+d*x)^4*csc(c+d*x)/sqrt(a+a*sin(c+d*x)),x,13,-2*arctanh(cos(c+d*x)*sqrt(a)/sqrt(a+a*sin(c+d*x)))/(d*sqrt(a))+32/15*cos(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))-2/5*cos(c+d*x)*sin(c+d*x)^2/(d*sqrt(a+a*sin(c+d*x)))+2/15*cos(c+d*x)*sqrt(a+a*sin(c+d*x))/(a*d)],
[cos(c+d*x)^4*csc(c+d*x)^2/sqrt(a+a*sin(c+d*x)),x,11,arctanh(cos(c+d*x)*sqrt(a)/sqrt(a+a*sin(c+d*x)))/(d*sqrt(a))+4/3*cos(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))-cot(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))-2/3*cos(c+d*x)*sqrt(a+a*sin(c+d*x))/(a*d)],
[cos(c+d*x)^4*csc(c+d*x)^3/sqrt(a+a*sin(c+d*x)),x,11,9/4*arctanh(cos(c+d*x)*sqrt(a)/sqrt(a+a*sin(c+d*x)))/(d*sqrt(a))-2*cos(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))+1/4*cot(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))-1/2*cot(c+d*x)*csc(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))],
[cos(c+d*x)^4*csc(c+d*x)^4/sqrt(a+a*sin(c+d*x)),x,11,-7/8*arctanh(cos(c+d*x)*sqrt(a)/sqrt(a+a*sin(c+d*x)))/(d*sqrt(a))+9/8*cot(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))+1/12*cot(c+d*x)*csc(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))-1/3*cot(c+d*x)*csc(c+d*x)^2/(d*sqrt(a+a*sin(c+d*x)))],
[cos(c+d*x)^4*csc(c+d*x)^5/sqrt(a+a*sin(c+d*x)),x,15,-11/64*arctanh(cos(c+d*x)*sqrt(a)/sqrt(a+a*sin(c+d*x)))/(d*sqrt(a))-11/64*cot(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))+53/96*cot(c+d*x)*csc(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))+1/24*cot(c+d*x)*csc(c+d*x)^2/(d*sqrt(a+a*sin(c+d*x)))-1/4*cot(c+d*x)*csc(c+d*x)^3/(d*sqrt(a+a*sin(c+d*x)))],
[cos(c+d*x)^4*csc(c+d*x)^6/sqrt(a+a*sin(c+d*x)),x,17,-9/128*arctanh(cos(c+d*x)*sqrt(a)/sqrt(a+a*sin(c+d*x)))/(d*sqrt(a))-9/128*cot(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))-3/64*cot(c+d*x)*csc(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))+29/80*cot(c+d*x)*csc(c+d*x)^2/(d*sqrt(a+a*sin(c+d*x)))+1/40*cot(c+d*x)*csc(c+d*x)^3/(d*sqrt(a+a*sin(c+d*x)))-1/5*cot(c+d*x)*csc(c+d*x)^4/(d*sqrt(a+a*sin(c+d*x)))],
[cos(c+d*x)^4*sin(c+d*x)^3/(a+a*sin(c+d*x))^(3/2),x,12,-4/385*cos(c+d*x)*(a+a*sin(c+d*x))^(3/2)/(a^3*d)-4/165*cos(c+d*x)/(a*d*sqrt(a+a*sin(c+d*x)))-2/231*cos(c+d*x)*sin(c+d*x)^3/(a*d*sqrt(a+a*sin(c+d*x)))+14/33*cos(c+d*x)*sin(c+d*x)^4/(a*d*sqrt(a+a*sin(c+d*x)))+8/1155*cos(c+d*x)*sqrt(a+a*sin(c+d*x))/(a^2*d)-2/11*cos(c+d*x)*sin(c+d*x)^4*sqrt(a+a*sin(c+d*x))/(a^2*d)],
[cos(c+d*x)^4*sin(c+d*x)^2/(a+a*sin(c+d*x))^(3/2),x,4,-46/315*a*cos(c+d*x)^5/(d*(a+a*sin(c+d*x))^(5/2))+20/63*cos(c+d*x)^5/(d*(a+a*sin(c+d*x))^(3/2))-2/9*cos(c+d*x)^5/(a*d*sqrt(a+a*sin(c+d*x)))],
[cos(c+d*x)^4*sin(c+d*x)/(a+a*sin(c+d*x))^(3/2),x,2,6/35*a*cos(c+d*x)^5/(d*(a+a*sin(c+d*x))^(5/2))-2/7*cos(c+d*x)^5/(d*(a+a*sin(c+d*x))^(3/2))],
[cos(c+d*x)^4*csc(c+d*x)/(a+a*sin(c+d*x))^(3/2),x,6,-2*arctanh(cos(c+d*x)*sqrt(a)/sqrt(a+a*sin(c+d*x)))/(a^(3/2)*d)+10/3*cos(c+d*x)/(a*d*sqrt(a+a*sin(c+d*x)))-2/3*cos(c+d*x)*sqrt(a+a*sin(c+d*x))/(a^2*d)],
[cos(c+d*x)^4*csc(c+d*x)^2/(a+a*sin(c+d*x))^(3/2),x,9,3*arctanh(cos(c+d*x)*sqrt(a)/sqrt(a+a*sin(c+d*x)))/(a^(3/2)*d)-cos(c+d*x)/(a*d*sqrt(a+a*sin(c+d*x)))-cot(c+d*x)*sqrt(a+a*sin(c+d*x))/(a^2*d)],
[cos(c+d*x)^4*csc(c+d*x)^3/(a+a*sin(c+d*x))^(3/2),x,8,-3/4*arctanh(cos(c+d*x)*sqrt(a)/sqrt(a+a*sin(c+d*x)))/(a^(3/2)*d)+7/4*cot(c+d*x)/(a*d*sqrt(a+a*sin(c+d*x)))-1/2*cot(c+d*x)*csc(c+d*x)*sqrt(a+a*sin(c+d*x))/(a^2*d)],
[cos(c+d*x)^4*csc(c+d*x)^4/(a+a*sin(c+d*x))^(3/2),x,10,-1/8*arctanh(cos(c+d*x)*sqrt(a)/sqrt(a+a*sin(c+d*x)))/(a^(3/2)*d)-1/8*cot(c+d*x)/(a*d*sqrt(a+a*sin(c+d*x)))+11/12*cot(c+d*x)*csc(c+d*x)/(a*d*sqrt(a+a*sin(c+d*x)))-1/3*cot(c+d*x)*csc(c+d*x)^2*sqrt(a+a*sin(c+d*x))/(a^2*d)],
[cos(c+d*x)^4*csc(c+d*x)^5/(a+a*sin(c+d*x))^(3/2),x,12,-3/64*arctanh(cos(c+d*x)*sqrt(a)/sqrt(a+a*sin(c+d*x)))/(a^(3/2)*d)-3/64*cot(c+d*x)/(a*d*sqrt(a+a*sin(c+d*x)))-1/32*cot(c+d*x)*csc(c+d*x)/(a*d*sqrt(a+a*sin(c+d*x)))+5/8*cot(c+d*x)*csc(c+d*x)^2/(a*d*sqrt(a+a*sin(c+d*x)))-1/4*cot(c+d*x)*csc(c+d*x)^3*sqrt(a+a*sin(c+d*x))/(a^2*d)],
[cos(c+d*x)^4*csc(c+d*x)^6/(a+a*sin(c+d*x))^(3/2),x,14,-3/128*arctanh(cos(c+d*x)*sqrt(a)/sqrt(a+a*sin(c+d*x)))/(a^(3/2)*d)-3/128*cot(c+d*x)/(a*d*sqrt(a+a*sin(c+d*x)))-1/64*cot(c+d*x)*csc(c+d*x)/(a*d*sqrt(a+a*sin(c+d*x)))-1/80*cot(c+d*x)*csc(c+d*x)^2/(a*d*sqrt(a+a*sin(c+d*x)))+19/40*cot(c+d*x)*csc(c+d*x)^3/(a*d*sqrt(a+a*sin(c+d*x)))-1/5*cot(c+d*x)*csc(c+d*x)^4*sqrt(a+a*sin(c+d*x))/(a^2*d)],
[cos(c+d*x)^4*sin(c+d*x)^4/(a+a*sin(c+d*x))^(5/2),x,18,-4*arctanh(cos(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(c+d*x))))*sqrt(2)/(a^(5/2)*d)+4496/693*cos(c+d*x)/(a^2*d*sqrt(a+a*sin(c+d*x)))+200/231*cos(c+d*x)*sin(c+d*x)^2/(a^2*d*sqrt(a+a*sin(c+d*x)))-424/693*cos(c+d*x)*sin(c+d*x)^3/(a^2*d*sqrt(a+a*sin(c+d*x)))+46/99*cos(c+d*x)*sin(c+d*x)^4/(a^2*d*sqrt(a+a*sin(c+d*x)))-2/11*cos(c+d*x)*sin(c+d*x)^5/(a^2*d*sqrt(a+a*sin(c+d*x)))-1048/693*cos(c+d*x)*sqrt(a+a*sin(c+d*x))/(a^3*d)],
[cos(c+d*x)^4*sin(c+d*x)^3/(a+a*sin(c+d*x))^(5/2),x,16,4*arctanh(cos(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(c+d*x))))*sqrt(2)/(a^(5/2)*d)-2048/315*cos(c+d*x)/(a^2*d*sqrt(a+a*sin(c+d*x)))-92/105*cos(c+d*x)*sin(c+d*x)^2/(a^2*d*sqrt(a+a*sin(c+d*x)))+38/63*cos(c+d*x)*sin(c+d*x)^3/(a^2*d*sqrt(a+a*sin(c+d*x)))-2/9*cos(c+d*x)*sin(c+d*x)^4/(a^2*d*sqrt(a+a*sin(c+d*x)))+472/315*cos(c+d*x)*sqrt(a+a*sin(c+d*x))/(a^3*d)],
[cos(c+d*x)^4*sin(c+d*x)^2/(a+a*sin(c+d*x))^(5/2),x,6,4/7*cos(c+d*x)^5/(d*(a+a*sin(c+d*x))^(5/2))+2/3*cos(c+d*x)^3/(a*d*(a+a*sin(c+d*x))^(3/2))-2/7*cos(c+d*x)^5/(a*d*(a+a*sin(c+d*x))^(3/2))-4*arctanh(cos(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(c+d*x))))*sqrt(2)/(a^(5/2)*d)+4*cos(c+d*x)/(a^2*d*sqrt(a+a*sin(c+d*x)))],
[cos(c+d*x)^4*sin(c+d*x)/(a+a*sin(c+d*x))^(5/2),x,5,-2/5*cos(c+d*x)^5/(d*(a+a*sin(c+d*x))^(5/2))-2/3*cos(c+d*x)^3/(a*d*(a+a*sin(c+d*x))^(3/2))+4*arctanh(cos(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(c+d*x))))*sqrt(2)/(a^(5/2)*d)-4*cos(c+d*x)/(a^2*d*sqrt(a+a*sin(c+d*x)))],
[cos(c+d*x)^4*csc(c+d*x)/(a+a*sin(c+d*x))^(5/2),x,9,-2*arctanh(cos(c+d*x)*sqrt(a)/sqrt(a+a*sin(c+d*x)))/(a^(5/2)*d)+4*arctanh(cos(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(c+d*x))))*sqrt(2)/(a^(5/2)*d)-2*cos(c+d*x)/(a^2*d*sqrt(a+a*sin(c+d*x)))],
[cos(c+d*x)^4*csc(c+d*x)^2/(a+a*sin(c+d*x))^(5/2),x,12,5*arctanh(cos(c+d*x)*sqrt(a)/sqrt(a+a*sin(c+d*x)))/(a^(5/2)*d)-4*arctanh(cos(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(c+d*x))))*sqrt(2)/(a^(5/2)*d)-cot(c+d*x)/(a^2*d*sqrt(a+a*sin(c+d*x)))],
[cos(c+d*x)^4*csc(c+d*x)^3/(a+a*sin(c+d*x))^(5/2),x,14,-23/4*arctanh(cos(c+d*x)*sqrt(a)/sqrt(a+a*sin(c+d*x)))/(a^(5/2)*d)+4*arctanh(cos(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(c+d*x))))*sqrt(2)/(a^(5/2)*d)+9/4*cot(c+d*x)/(a^2*d*sqrt(a+a*sin(c+d*x)))-1/2*cot(c+d*x)*csc(c+d*x)/(a^2*d*sqrt(a+a*sin(c+d*x)))],
[cos(c+d*x)^4*csc(c+d*x)^4/(a+a*sin(c+d*x))^(5/2),x,16,45/8*arctanh(cos(c+d*x)*sqrt(a)/sqrt(a+a*sin(c+d*x)))/(a^(5/2)*d)-4*arctanh(cos(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(c+d*x))))*sqrt(2)/(a^(5/2)*d)-19/8*cot(c+d*x)/(a^2*d*sqrt(a+a*sin(c+d*x)))+13/12*cot(c+d*x)*csc(c+d*x)/(a^2*d*sqrt(a+a*sin(c+d*x)))-1/3*cot(c+d*x)*csc(c+d*x)^2/(a^2*d*sqrt(a+a*sin(c+d*x)))],
[cos(c+d*x)^4*csc(c+d*x)^5/(a+a*sin(c+d*x))^(5/2),x,18,-363/64*arctanh(cos(c+d*x)*sqrt(a)/sqrt(a+a*sin(c+d*x)))/(a^(5/2)*d)+4*arctanh(cos(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(c+d*x))))*sqrt(2)/(a^(5/2)*d)+149/64*cot(c+d*x)/(a^2*d*sqrt(a+a*sin(c+d*x)))-107/96*cot(c+d*x)*csc(c+d*x)/(a^2*d*sqrt(a+a*sin(c+d*x)))+17/24*cot(c+d*x)*csc(c+d*x)^2/(a^2*d*sqrt(a+a*sin(c+d*x)))-1/4*cot(c+d*x)*csc(c+d*x)^3/(a^2*d*sqrt(a+a*sin(c+d*x)))],

# Integrands of the form Cos[e+f x]^4 Sin[e+f x]^n (a+a Sin[e+f x])^m with n symbolic
[cos(c+d*x)^4*sin(c+d*x)^n*(a+a*sin(c+d*x))^2,x,5,a^2*cos(c+d*x)*hypergeom([-3/2,1/2*(1+n)],[1/2*(3+n)],sin(c+d*x)^2)*sin(c+d*x)^(1+n)/(d*(1+n)*sqrt(cos(c+d*x)^2))+2*a^2*cos(c+d*x)*hypergeom([-3/2,1/2*(2+n)],[1/2*(4+n)],sin(c+d*x)^2)*sin(c+d*x)^(2+n)/(d*(2+n)*sqrt(cos(c+d*x)^2))+a^2*cos(c+d*x)*hypergeom([-3/2,1/2*(3+n)],[1/2*(5+n)],sin(c+d*x)^2)*sin(c+d*x)^(3+n)/(d*(3+n)*sqrt(cos(c+d*x)^2))],
[cos(c+d*x)^4*sin(c+d*x)^n*(a+a*sin(c+d*x)),x,3,a*cos(c+d*x)*hypergeom([-3/2,1/2*(1+n)],[1/2*(3+n)],sin(c+d*x)^2)*sin(c+d*x)^(1+n)/(d*(1+n)*sqrt(cos(c+d*x)^2))+a*cos(c+d*x)*hypergeom([-3/2,1/2*(2+n)],[1/2*(4+n)],sin(c+d*x)^2)*sin(c+d*x)^(2+n)/(d*(2+n)*sqrt(cos(c+d*x)^2))],
[cos(c+d*x)^4*sin(c+d*x)^n/(a+a*sin(c+d*x)),x,3,cos(c+d*x)*hypergeom([-1/2,1/2*(1+n)],[1/2*(3+n)],sin(c+d*x)^2)*sin(c+d*x)^(1+n)/(a*d*(1+n)*sqrt(cos(c+d*x)^2))-cos(c+d*x)*hypergeom([-1/2,1/2*(2+n)],[1/2*(4+n)],sin(c+d*x)^2)*sin(c+d*x)^(2+n)/(a*d*(2+n)*sqrt(cos(c+d*x)^2))],
[cos(c+d*x)^4*sin(c+d*x)^n/(a+a*sin(c+d*x))^2,x,5,-cos(c+d*x)*sin(c+d*x)^(1+n)/(a^2*d*(2+n))+(3+2*n)*cos(c+d*x)*hypergeom([1/2,1/2*(1+n)],[1/2*(3+n)],sin(c+d*x)^2)*sin(c+d*x)^(1+n)/(a^2*d*(1+n)*(2+n)*sqrt(cos(c+d*x)^2))-2*cos(c+d*x)*hypergeom([1/2,1/2*(2+n)],[1/2*(4+n)],sin(c+d*x)^2)*sin(c+d*x)^(2+n)/(a^2*d*(2+n)*sqrt(cos(c+d*x)^2))],

# Integrands of the form Cos[e+f x]^5 (d Sin[e+f x])^n (a+a Sin[e+f x])^m

# Integrands of the form Cos[e+f x]^5 Sin[e+f x]^n (a+a Sin[e+f x])^m

# m>0
[cos(c+d*x)^5*sin(c+d*x)^5*(a+a*sin(c+d*x)),x,4,1/6*a*sin(c+d*x)^6/d+1/7*a*sin(c+d*x)^7/d-1/4*a*sin(c+d*x)^8/d-2/9*a*sin(c+d*x)^9/d+1/10*a*sin(c+d*x)^10/d+1/11*a*sin(c+d*x)^11/d],
[cos(c+d*x)^5*sin(c+d*x)^4*(a+a*sin(c+d*x)),x,4,1/5*a*sin(c+d*x)^5/d+1/6*a*sin(c+d*x)^6/d-2/7*a*sin(c+d*x)^7/d-1/4*a*sin(c+d*x)^8/d+1/9*a*sin(c+d*x)^9/d+1/10*a*sin(c+d*x)^10/d],
[cos(c+d*x)^5*sin(c+d*x)^3*(a+a*sin(c+d*x)),x,7,-1/6*a*cos(c+d*x)^6/d+1/8*a*cos(c+d*x)^8/d+1/5*a*sin(c+d*x)^5/d-2/7*a*sin(c+d*x)^7/d+1/9*a*sin(c+d*x)^9/d],
[cos(c+d*x)^5*sin(c+d*x)^2*(a+a*sin(c+d*x)),x,7,-1/6*a*cos(c+d*x)^6/d+1/8*a*cos(c+d*x)^8/d+1/3*a*sin(c+d*x)^3/d-2/5*a*sin(c+d*x)^5/d+1/7*a*sin(c+d*x)^7/d],
[cos(c+d*x)^5*sin(c+d*x)*(a+a*sin(c+d*x)),x,6,-1/6*a*cos(c+d*x)^6/d+1/3*a*sin(c+d*x)^3/d-2/5*a*sin(c+d*x)^5/d+1/7*a*sin(c+d*x)^7/d],
[cos(c+d*x)^5*csc(c+d*x)*(a+a*sin(c+d*x)),x,4,a*log(sin(c+d*x))/d+a*sin(c+d*x)/d-a*sin(c+d*x)^2/d-2/3*a*sin(c+d*x)^3/d+1/4*a*sin(c+d*x)^4/d+1/5*a*sin(c+d*x)^5/d],
[cos(c+d*x)^5*csc(c+d*x)^2*(a+a*sin(c+d*x)),x,4,-a*csc(c+d*x)/d+a*log(sin(c+d*x))/d-2*a*sin(c+d*x)/d-a*sin(c+d*x)^2/d+1/3*a*sin(c+d*x)^3/d+1/4*a*sin(c+d*x)^4/d],
[cos(c+d*x)^5*csc(c+d*x)^3*(a+a*sin(c+d*x)),x,4,-a*csc(c+d*x)/d-1/2*a*csc(c+d*x)^2/d-2*a*log(sin(c+d*x))/d-2*a*sin(c+d*x)/d+1/2*a*sin(c+d*x)^2/d+1/3*a*sin(c+d*x)^3/d],
[cos(c+d*x)^5*csc(c+d*x)^4*(a+a*sin(c+d*x)),x,4,2*a*csc(c+d*x)/d-1/2*a*csc(c+d*x)^2/d-1/3*a*csc(c+d*x)^3/d-2*a*log(sin(c+d*x))/d+a*sin(c+d*x)/d+1/2*a*sin(c+d*x)^2/d],
[cos(c+d*x)^5*csc(c+d*x)^5*(a+a*sin(c+d*x)),x,3,2*a*csc(c+d*x)/d+a*csc(c+d*x)^2/d-1/3*a*csc(c+d*x)^3/d-1/4*a*csc(c+d*x)^4/d+a*log(sin(c+d*x))/d+a*sin(c+d*x)/d],
[cos(c+d*x)^5*csc(c+d*x)^6*(a+a*sin(c+d*x)),x,4,-a*csc(c+d*x)/d+a*csc(c+d*x)^2/d+2/3*a*csc(c+d*x)^3/d-1/4*a*csc(c+d*x)^4/d-1/5*a*csc(c+d*x)^5/d+a*log(sin(c+d*x))/d],
[cos(c+d*x)^5*csc(c+d*x)^7*(a+a*sin(c+d*x)),x,6,-1/6*a*cot(c+d*x)^6/d-a*csc(c+d*x)/d+2/3*a*csc(c+d*x)^3/d-1/5*a*csc(c+d*x)^5/d],
[cos(c+d*x)^5*csc(c+d*x)^8*(a+a*sin(c+d*x)),x,6,-1/6*a*cot(c+d*x)^6/d-1/3*a*csc(c+d*x)^3/d+2/5*a*csc(c+d*x)^5/d-1/7*a*csc(c+d*x)^7/d],
[cos(c+d*x)^5*csc(c+d*x)^9*(a+a*sin(c+d*x)),x,7,-1/6*a*cot(c+d*x)^6/d-1/8*a*cot(c+d*x)^8/d-1/3*a*csc(c+d*x)^3/d+2/5*a*csc(c+d*x)^5/d-1/7*a*csc(c+d*x)^7/d],
[cos(c+d*x)^5*csc(c+d*x)^10*(a+a*sin(c+d*x)),x,7,-1/6*a*cot(c+d*x)^6/d-1/8*a*cot(c+d*x)^8/d-1/5*a*csc(c+d*x)^5/d+2/7*a*csc(c+d*x)^7/d-1/9*a*csc(c+d*x)^9/d],
[cos(c+d*x)^5*csc(c+d*x)^11*(a+a*sin(c+d*x)),x,4,-1/5*a*csc(c+d*x)^5/d-1/6*a*csc(c+d*x)^6/d+2/7*a*csc(c+d*x)^7/d+1/4*a*csc(c+d*x)^8/d-1/9*a*csc(c+d*x)^9/d-1/10*a*csc(c+d*x)^10/d],
[cos(c+d*x)^5*csc(c+d*x)^12*(a+a*sin(c+d*x)),x,4,-1/6*a*csc(c+d*x)^6/d-1/7*a*csc(c+d*x)^7/d+1/4*a*csc(c+d*x)^8/d+2/9*a*csc(c+d*x)^9/d-1/10*a*csc(c+d*x)^10/d-1/11*a*csc(c+d*x)^11/d],
[cos(c+d*x)^5*sin(c+d*x)^3*(a+a*sin(c+d*x))^2,x,4,1/4*a^2*sin(c+d*x)^4/d+2/5*a^2*sin(c+d*x)^5/d-1/6*a^2*sin(c+d*x)^6/d-4/7*a^2*sin(c+d*x)^7/d-1/8*a^2*sin(c+d*x)^8/d+2/9*a^2*sin(c+d*x)^9/d+1/10*a^2*sin(c+d*x)^10/d],
[cos(c+d*x)^5*sin(c+d*x)^2*(a+a*sin(c+d*x))^2,x,4,4/5*(a+a*sin(c+d*x))^5/(a^3*d)-2*(a+a*sin(c+d*x))^6/(a^4*d)+13/7*(a+a*sin(c+d*x))^7/(a^5*d)-3/4*(a+a*sin(c+d*x))^8/(a^6*d)+1/9*(a+a*sin(c+d*x))^9/(a^7*d)],
[cos(c+d*x)^5*sin(c+d*x)*(a+a*sin(c+d*x))^2,x,4,-4/5*(a+a*sin(c+d*x))^5/(a^3*d)+4/3*(a+a*sin(c+d*x))^6/(a^4*d)-5/7*(a+a*sin(c+d*x))^7/(a^5*d)+1/8*(a+a*sin(c+d*x))^8/(a^6*d)],
[cos(c+d*x)^5*csc(c+d*x)*(a+a*sin(c+d*x))^2,x,4,a^2*log(sin(c+d*x))/d+2*a^2*sin(c+d*x)/d-1/2*a^2*sin(c+d*x)^2/d-4/3*a^2*sin(c+d*x)^3/d-1/4*a^2*sin(c+d*x)^4/d+2/5*a^2*sin(c+d*x)^5/d+1/6*a^2*sin(c+d*x)^6/d],
[cos(c+d*x)^5*csc(c+d*x)^2*(a+a*sin(c+d*x))^2,x,4,-a^2*csc(c+d*x)/d+2*a^2*log(sin(c+d*x))/d-a^2*sin(c+d*x)/d-2*a^2*sin(c+d*x)^2/d-1/3*a^2*sin(c+d*x)^3/d+1/2*a^2*sin(c+d*x)^4/d+1/5*a^2*sin(c+d*x)^5/d],
[cos(c+d*x)^5*csc(c+d*x)^3*(a+a*sin(c+d*x))^2,x,4,-2*a^2*csc(c+d*x)/d-1/2*a^2*csc(c+d*x)^2/d-a^2*log(sin(c+d*x))/d-4*a^2*sin(c+d*x)/d-1/2*a^2*sin(c+d*x)^2/d+2/3*a^2*sin(c+d*x)^3/d+1/4*a^2*sin(c+d*x)^4/d],
[cos(c+d*x)^5*csc(c+d*x)^4*(a+a*sin(c+d*x))^2,x,4,a^2*csc(c+d*x)/d-a^2*csc(c+d*x)^2/d-1/3*a^2*csc(c+d*x)^3/d-4*a^2*log(sin(c+d*x))/d-a^2*sin(c+d*x)/d+a^2*sin(c+d*x)^2/d+1/3*a^2*sin(c+d*x)^3/d],
[cos(c+d*x)^5*csc(c+d*x)^5*(a+a*sin(c+d*x))^2,x,3,4*a^2*csc(c+d*x)/d+1/2*a^2*csc(c+d*x)^2/d-2/3*a^2*csc(c+d*x)^3/d-1/4*a^2*csc(c+d*x)^4/d-a^2*log(sin(c+d*x))/d+2*a^2*sin(c+d*x)/d+1/2*a^2*sin(c+d*x)^2/d],
[cos(c+d*x)^5*csc(c+d*x)^6*(a+a*sin(c+d*x))^2,x,4,a^2*csc(c+d*x)/d+2*a^2*csc(c+d*x)^2/d+1/3*a^2*csc(c+d*x)^3/d-1/2*a^2*csc(c+d*x)^4/d-1/5*a^2*csc(c+d*x)^5/d+2*a^2*log(sin(c+d*x))/d+a^2*sin(c+d*x)/d],
[cos(c+d*x)^5*csc(c+d*x)^7*(a+a*sin(c+d*x))^2,x,4,-2*a^2*csc(c+d*x)/d+1/2*a^2*csc(c+d*x)^2/d+4/3*a^2*csc(c+d*x)^3/d+1/4*a^2*csc(c+d*x)^4/d-2/5*a^2*csc(c+d*x)^5/d-1/6*a^2*csc(c+d*x)^6/d+a^2*log(sin(c+d*x))/d],
[cos(c+d*x)^5*sin(c+d*x)^2*(a+a*sin(c+d*x))^3,x,4,2/3*(a+a*sin(c+d*x))^6/(a^3*d)-12/7*(a+a*sin(c+d*x))^7/(a^4*d)+13/8*(a+a*sin(c+d*x))^8/(a^5*d)-2/3*(a+a*sin(c+d*x))^9/(a^6*d)+1/10*(a+a*sin(c+d*x))^10/(a^7*d)],
[cos(c+d*x)^5*sin(c+d*x)*(a+a*sin(c+d*x))^3,x,4,-2/3*(a+a*sin(c+d*x))^6/(a^3*d)+8/7*(a+a*sin(c+d*x))^7/(a^4*d)-5/8*(a+a*sin(c+d*x))^8/(a^5*d)+1/9*(a+a*sin(c+d*x))^9/(a^6*d)],
[cos(c+d*x)^5*csc(c+d*x)*(a+a*sin(c+d*x))^3,x,4,a^3*log(sin(c+d*x))/d+3*a^3*sin(c+d*x)/d+1/2*a^3*sin(c+d*x)^2/d-5/3*a^3*sin(c+d*x)^3/d-5/4*a^3*sin(c+d*x)^4/d+1/5*a^3*sin(c+d*x)^5/d+1/2*a^3*sin(c+d*x)^6/d+1/7*a^3*sin(c+d*x)^7/d],
[cos(c+d*x)^5*csc(c+d*x)^2*(a+a*sin(c+d*x))^3,x,4,-a^3*csc(c+d*x)/d+3*a^3*log(sin(c+d*x))/d+a^3*sin(c+d*x)/d-5/2*a^3*sin(c+d*x)^2/d-5/3*a^3*sin(c+d*x)^3/d+1/4*a^3*sin(c+d*x)^4/d+3/5*a^3*sin(c+d*x)^5/d+1/6*a^3*sin(c+d*x)^6/d],
[cos(c+d*x)^5*csc(c+d*x)^3*(a+a*sin(c+d*x))^3,x,4,-3*a^3*csc(c+d*x)/d-1/2*a^3*csc(c+d*x)^2/d+a^3*log(sin(c+d*x))/d-5*a^3*sin(c+d*x)/d-5/2*a^3*sin(c+d*x)^2/d+1/3*a^3*sin(c+d*x)^3/d+3/4*a^3*sin(c+d*x)^4/d+1/5*a^3*sin(c+d*x)^5/d],
[cos(c+d*x)^5*csc(c+d*x)^4*(a+a*sin(c+d*x))^3,x,4,-a^3*csc(c+d*x)/d-3/2*a^3*csc(c+d*x)^2/d-1/3*a^3*csc(c+d*x)^3/d-5*a^3*log(sin(c+d*x))/d-5*a^3*sin(c+d*x)/d+1/2*a^3*sin(c+d*x)^2/d+a^3*sin(c+d*x)^3/d+1/4*a^3*sin(c+d*x)^4/d],
[cos(c+d*x)^5*csc(c+d*x)^5*(a+a*sin(c+d*x))^3,x,3,5*a^3*csc(c+d*x)/d-1/2*a^3*csc(c+d*x)^2/d-a^3*csc(c+d*x)^3/d-1/4*a^3*csc(c+d*x)^4/d-5*a^3*log(sin(c+d*x))/d+a^3*sin(c+d*x)/d+3/2*a^3*sin(c+d*x)^2/d+1/3*a^3*sin(c+d*x)^3/d],
[cos(c+d*x)^5*csc(c+d*x)^6*(a+a*sin(c+d*x))^3,x,4,5*a^3*csc(c+d*x)/d+5/2*a^3*csc(c+d*x)^2/d-1/3*a^3*csc(c+d*x)^3/d-3/4*a^3*csc(c+d*x)^4/d-1/5*a^3*csc(c+d*x)^5/d+a^3*log(sin(c+d*x))/d+3*a^3*sin(c+d*x)/d+1/2*a^3*sin(c+d*x)^2/d],
[cos(c+d*x)^5*csc(c+d*x)^7*(a+a*sin(c+d*x))^3,x,4,-a^3*csc(c+d*x)/d+5/2*a^3*csc(c+d*x)^2/d+5/3*a^3*csc(c+d*x)^3/d-1/4*a^3*csc(c+d*x)^4/d-3/5*a^3*csc(c+d*x)^5/d-1/6*a^3*csc(c+d*x)^6/d+3*a^3*log(sin(c+d*x))/d+a^3*sin(c+d*x)/d],
[cos(c+d*x)^5*csc(c+d*x)^4*(a+a*sin(c+d*x))^4,x,4,-4*a^4*csc(c+d*x)/d-2*a^4*csc(c+d*x)^2/d-1/3*a^4*csc(c+d*x)^3/d-4*a^4*log(sin(c+d*x))/d-10*a^4*sin(c+d*x)/d-2*a^4*sin(c+d*x)^2/d+4/3*a^4*sin(c+d*x)^3/d+a^4*sin(c+d*x)^4/d+1/5*a^4*sin(c+d*x)^5/d],
[cos(c+d*x)^5*csc(c+d*x)^5*(a+a*sin(c+d*x))^4,x,3,4*a^4*csc(c+d*x)/d-2*a^4*csc(c+d*x)^2/d-4/3*a^4*csc(c+d*x)^3/d-1/4*a^4*csc(c+d*x)^4/d-10*a^4*log(sin(c+d*x))/d-4*a^4*sin(c+d*x)/d+2*a^4*sin(c+d*x)^2/d+4/3*a^4*sin(c+d*x)^3/d+1/4*a^4*sin(c+d*x)^4/d],
[cos(c+d*x)^5*csc(c+d*x)^6*(a+a*sin(c+d*x))^4,x,4,10*a^4*csc(c+d*x)/d+2*a^4*csc(c+d*x)^2/d-4/3*a^4*csc(c+d*x)^3/d-a^4*csc(c+d*x)^4/d-1/5*a^4*csc(c+d*x)^5/d-4*a^4*log(sin(c+d*x))/d+4*a^4*sin(c+d*x)/d+2*a^4*sin(c+d*x)^2/d+1/3*a^4*sin(c+d*x)^3/d],

# m<0
[cos(c+d*x)^5*sin(c+d*x)^3/(a+a*sin(c+d*x)),x,4,1/4*sin(c+d*x)^4/(a*d)-1/5*sin(c+d*x)^5/(a*d)-1/6*sin(c+d*x)^6/(a*d)+1/7*sin(c+d*x)^7/(a*d)],
[cos(c+d*x)^5*sin(c+d*x)^2/(a+a*sin(c+d*x)),x,7,1/3*sin(c+d*x)^3/(a*d)-1/4*sin(c+d*x)^4/(a*d)-1/5*sin(c+d*x)^5/(a*d)+1/6*sin(c+d*x)^6/(a*d)],
[cos(c+d*x)^5*sin(c+d*x)/(a+a*sin(c+d*x)),x,6,-1/4*cos(c+d*x)^4/(a*d)-1/3*sin(c+d*x)^3/(a*d)+1/5*sin(c+d*x)^5/(a*d)],
[cos(c+d*x)^5*csc(c+d*x)/(a+a*sin(c+d*x)),x,4,log(sin(c+d*x))/(a*d)-sin(c+d*x)/(a*d)-1/2*sin(c+d*x)^2/(a*d)+1/3*sin(c+d*x)^3/(a*d)],
[cos(c+d*x)^5*csc(c+d*x)^2/(a+a*sin(c+d*x)),x,4,-csc(c+d*x)/(a*d)-log(sin(c+d*x))/(a*d)-sin(c+d*x)/(a*d)+1/2*sin(c+d*x)^2/(a*d)],
[cos(c+d*x)^5*csc(c+d*x)^3/(a+a*sin(c+d*x)),x,4,csc(c+d*x)/(a*d)-1/2*csc(c+d*x)^2/(a*d)-log(sin(c+d*x))/(a*d)+sin(c+d*x)/(a*d)],
[cos(c+d*x)^5*csc(c+d*x)^4/(a+a*sin(c+d*x)),x,4,csc(c+d*x)/(a*d)+1/2*csc(c+d*x)^2/(a*d)-1/3*csc(c+d*x)^3/(a*d)+log(sin(c+d*x))/(a*d)],
[cos(c+d*x)^5*csc(c+d*x)^5/(a+a*sin(c+d*x)),x,5,-1/4*cot(c+d*x)^4/(a*d)-csc(c+d*x)/(a*d)+1/3*csc(c+d*x)^3/(a*d)],
[cos(c+d*x)^5*csc(c+d*x)^6/(a+a*sin(c+d*x)),x,6,1/4*cot(c+d*x)^4/(a*d)+1/3*csc(c+d*x)^3/(a*d)-1/5*csc(c+d*x)^5/(a*d)],
[cos(c+d*x)^5*csc(c+d*x)^7/(a+a*sin(c+d*x)),x,4,-1/3*csc(c+d*x)^3/(a*d)+1/4*csc(c+d*x)^4/(a*d)+1/5*csc(c+d*x)^5/(a*d)-1/6*csc(c+d*x)^6/(a*d)],
[cos(c+d*x)^5*csc(c+d*x)^8/(a+a*sin(c+d*x)),x,4,-1/4*csc(c+d*x)^4/(a*d)+1/5*csc(c+d*x)^5/(a*d)+1/6*csc(c+d*x)^6/(a*d)-1/7*csc(c+d*x)^7/(a*d)],
[cos(c+d*x)^5*sin(c+d*x)^3/(a+a*sin(c+d*x))^2,x,4,1/4*sin(c+d*x)^4/(a^2*d)-2/5*sin(c+d*x)^5/(a^2*d)+1/6*sin(c+d*x)^6/(a^2*d)],
[cos(c+d*x)^5*sin(c+d*x)^2/(a+a*sin(c+d*x))^2,x,4,1/3*sin(c+d*x)^3/(a^2*d)-1/2*sin(c+d*x)^4/(a^2*d)+1/5*sin(c+d*x)^5/(a^2*d)],
[cos(c+d*x)^5*sin(c+d*x)/(a+a*sin(c+d*x))^2,x,4,1/2*sin(c+d*x)^2/(a^2*d)-2/3*sin(c+d*x)^3/(a^2*d)+1/4*sin(c+d*x)^4/(a^2*d)],
[cos(c+d*x)^5*csc(c+d*x)/(a+a*sin(c+d*x))^2,x,4,log(sin(c+d*x))/(a^2*d)-2*sin(c+d*x)/(a^2*d)+1/2*sin(c+d*x)^2/(a^2*d)],
[cos(c+d*x)^5*csc(c+d*x)^2/(a+a*sin(c+d*x))^2,x,4,-csc(c+d*x)/(a^2*d)-2*log(sin(c+d*x))/(a^2*d)+sin(c+d*x)/(a^2*d)],
[cos(c+d*x)^5*csc(c+d*x)^3/(a+a*sin(c+d*x))^2,x,4,2*csc(c+d*x)/(a^2*d)-1/2*csc(c+d*x)^2/(a^2*d)+log(sin(c+d*x))/(a^2*d)],
[cos(c+d*x)^5*csc(c+d*x)^4/(a+a*sin(c+d*x))^2,x,3,-1/3*csc(c+d*x)^3*(a-a*sin(c+d*x))^3/(a^5*d)],
[cos(c+d*x)^5*csc(c+d*x)^5/(a+a*sin(c+d*x))^2,x,3,-1/2*csc(c+d*x)^2/(a^2*d)+2/3*csc(c+d*x)^3/(a^2*d)-1/4*csc(c+d*x)^4/(a^2*d)],
[cos(c+d*x)^5*csc(c+d*x)^6/(a+a*sin(c+d*x))^2,x,4,-1/3*csc(c+d*x)^3/(a^2*d)+1/2*csc(c+d*x)^4/(a^2*d)-1/5*csc(c+d*x)^5/(a^2*d)],
[cos(c+d*x)^5*csc(c+d*x)^7/(a+a*sin(c+d*x))^2,x,4,-1/4*csc(c+d*x)^4/(a^2*d)+2/5*csc(c+d*x)^5/(a^2*d)-1/6*csc(c+d*x)^6/(a^2*d)],
[cos(c+d*x)^5*sin(c+d*x)^3/(a+a*sin(c+d*x))^3,x,4,-4*log(1+sin(c+d*x))/(a^3*d)+4*sin(c+d*x)/(a^3*d)-2*sin(c+d*x)^2/(a^3*d)+4/3*sin(c+d*x)^3/(a^3*d)-3/4*sin(c+d*x)^4/(a^3*d)+1/5*sin(c+d*x)^5/(a^3*d)],
[cos(c+d*x)^5*sin(c+d*x)^2/(a+a*sin(c+d*x))^3,x,4,4*log(1+sin(c+d*x))/(a^3*d)-4*sin(c+d*x)/(a^3*d)+2*sin(c+d*x)^2/(a^3*d)-sin(c+d*x)^3/(a^3*d)+1/4*sin(c+d*x)^4/(a^3*d)],
[cos(c+d*x)^5*sin(c+d*x)/(a+a*sin(c+d*x))^3,x,4,-4*log(1+sin(c+d*x))/(a^3*d)+4*sin(c+d*x)/(a^3*d)-3/2*sin(c+d*x)^2/(a^3*d)+1/3*sin(c+d*x)^3/(a^3*d)],
[cos(c+d*x)^5*csc(c+d*x)/(a+a*sin(c+d*x))^3,x,4,log(sin(c+d*x))/(a^3*d)-4*log(1+sin(c+d*x))/(a^3*d)+sin(c+d*x)/(a^3*d)],
[cos(c+d*x)^5*csc(c+d*x)^2/(a+a*sin(c+d*x))^3,x,4,-csc(c+d*x)/(a^3*d)-3*log(sin(c+d*x))/(a^3*d)+4*log(1+sin(c+d*x))/(a^3*d)],
[cos(c+d*x)^5*csc(c+d*x)^3/(a+a*sin(c+d*x))^3,x,4,3*csc(c+d*x)/(a^3*d)-1/2*csc(c+d*x)^2/(a^3*d)+4*log(sin(c+d*x))/(a^3*d)-4*log(1+sin(c+d*x))/(a^3*d)],
[cos(c+d*x)^5*csc(c+d*x)^4/(a+a*sin(c+d*x))^3,x,4,-4*csc(c+d*x)/(a^3*d)+3/2*csc(c+d*x)^2/(a^3*d)-1/3*csc(c+d*x)^3/(a^3*d)-4*log(sin(c+d*x))/(a^3*d)+4*log(1+sin(c+d*x))/(a^3*d)],
[cos(c+d*x)^5*csc(c+d*x)^5/(a+a*sin(c+d*x))^3,x,3,4*csc(c+d*x)/(a^3*d)-2*csc(c+d*x)^2/(a^3*d)+csc(c+d*x)^3/(a^3*d)-1/4*csc(c+d*x)^4/(a^3*d)+4*log(sin(c+d*x))/(a^3*d)-4*log(1+sin(c+d*x))/(a^3*d)],
[cos(c+d*x)^5*csc(c+d*x)^6/(a+a*sin(c+d*x))^3,x,4,-4*csc(c+d*x)/(a^3*d)+2*csc(c+d*x)^2/(a^3*d)-4/3*csc(c+d*x)^3/(a^3*d)+3/4*csc(c+d*x)^4/(a^3*d)-1/5*csc(c+d*x)^5/(a^3*d)-4*log(sin(c+d*x))/(a^3*d)+4*log(1+sin(c+d*x))/(a^3*d)],
[cos(c+d*x)^5*csc(c+d*x)^5/(a+a*sin(c+d*x))^4,x,3,12*csc(c+d*x)/(a^4*d)-4*csc(c+d*x)^2/(a^4*d)+4/3*csc(c+d*x)^3/(a^4*d)-1/4*csc(c+d*x)^4/(a^4*d)+16*log(sin(c+d*x))/(a^4*d)-16*log(1+sin(c+d*x))/(a^4*d)+4/(d*(a^4+a^4*sin(c+d*x)))],
[cos(c+d*x)^5*csc(c+d*x)^6/(a+a*sin(c+d*x))^4,x,4,-16*csc(c+d*x)/(a^4*d)+6*csc(c+d*x)^2/(a^4*d)-8/3*csc(c+d*x)^3/(a^4*d)+csc(c+d*x)^4/(a^4*d)-1/5*csc(c+d*x)^5/(a^4*d)-20*log(sin(c+d*x))/(a^4*d)+20*log(1+sin(c+d*x))/(a^4*d)+(-4)/(d*(a^4+a^4*sin(c+d*x)))],

# Integrands of the form Cos[e+f x]^5 Sin[e+f x]^n (a+a Sin[e+f x])^(m/2)

# Integrands of the form Cos[e+f x]^5 Sin[e+f x]^n (a+a Sin[e+f x])^m with n symbolic
[cos(c+d*x)^5*sin(c+d*x)^n*(a+a*sin(c+d*x))^3,x,3,a^3*sin(c+d*x)^(1+n)/(d*(1+n))+3*a^3*sin(c+d*x)^(2+n)/(d*(2+n))+a^3*sin(c+d*x)^(3+n)/(d*(3+n))-5*a^3*sin(c+d*x)^(4+n)/(d*(4+n))-5*a^3*sin(c+d*x)^(5+n)/(d*(5+n))+a^3*sin(c+d*x)^(6+n)/(d*(6+n))+3*a^3*sin(c+d*x)^(7+n)/(d*(7+n))+a^3*sin(c+d*x)^(8+n)/(d*(8+n))],
[cos(c+d*x)^5*sin(c+d*x)^n*(a+a*sin(c+d*x))^2,x,3,a^2*sin(c+d*x)^(1+n)/(d*(1+n))+2*a^2*sin(c+d*x)^(2+n)/(d*(2+n))-a^2*sin(c+d*x)^(3+n)/(d*(3+n))-4*a^2*sin(c+d*x)^(4+n)/(d*(4+n))-a^2*sin(c+d*x)^(5+n)/(d*(5+n))+2*a^2*sin(c+d*x)^(6+n)/(d*(6+n))+a^2*sin(c+d*x)^(7+n)/(d*(7+n))],
[cos(c+d*x)^5*sin(c+d*x)^n*(a+a*sin(c+d*x)),x,3,a*sin(c+d*x)^(1+n)/(d*(1+n))+a*sin(c+d*x)^(2+n)/(d*(2+n))-2*a*sin(c+d*x)^(3+n)/(d*(3+n))-2*a*sin(c+d*x)^(4+n)/(d*(4+n))+a*sin(c+d*x)^(5+n)/(d*(5+n))+a*sin(c+d*x)^(6+n)/(d*(6+n))],
[cos(c+d*x)^5*sin(c+d*x)^n/(a+a*sin(c+d*x)),x,3,sin(c+d*x)^(1+n)/(a*d*(1+n))-sin(c+d*x)^(2+n)/(a*d*(2+n))-sin(c+d*x)^(3+n)/(a*d*(3+n))+sin(c+d*x)^(4+n)/(a*d*(4+n))],
[cos(c+d*x)^5*sin(c+d*x)^n/(a+a*sin(c+d*x))^2,x,3,sin(c+d*x)^(1+n)/(a^2*d*(1+n))-2*sin(c+d*x)^(2+n)/(a^2*d*(2+n))+sin(c+d*x)^(3+n)/(a^2*d*(3+n))],
[cos(c+d*x)^5*sin(c+d*x)^n/(a+a*sin(c+d*x))^3,x,4,-3*sin(c+d*x)^(1+n)/(a^3*d*(1+n))+4*hypergeom([1,1+n],[2+n],-sin(c+d*x))*sin(c+d*x)^(1+n)/(a^3*d*(1+n))+sin(c+d*x)^(2+n)/(a^3*d*(2+n))],
[cos(c+d*x)^5*sin(c+d*x)^n/(a+a*sin(c+d*x))^4,x,4,sin(c+d*x)^(1+n)/(a^4*d*(1+n))-4*hypergeom([1,1+n],[2+n],-sin(c+d*x))*sin(c+d*x)^(1+n)/(a^4*d)+4*sin(c+d*x)^(1+n)/(d*(a^4+a^4*sin(c+d*x)))],

# Integrands of the form Cos[e+f x]^6 (d Sin[e+f x])^n (a+a Sin[e+f x])^m

# Integrands of the form Cos[e+f x]^6 Sin[e+f x]^n (a+a Sin[e+f x])^m

# m>0
[cos(c+d*x)^6*sin(c+d*x)^4*(a+a*sin(c+d*x)),x,10,3/256*a*x-1/7*a*cos(c+d*x)^7/d+2/9*a*cos(c+d*x)^9/d-1/11*a*cos(c+d*x)^11/d+3/256*a*cos(c+d*x)*sin(c+d*x)/d+1/128*a*cos(c+d*x)^3*sin(c+d*x)/d+1/160*a*cos(c+d*x)^5*sin(c+d*x)/d-3/80*a*cos(c+d*x)^7*sin(c+d*x)/d-1/10*a*cos(c+d*x)^7*sin(c+d*x)^3/d],
[cos(c+d*x)^6*sin(c+d*x)^3*(a+a*sin(c+d*x)),x,10,3/256*a*x-1/7*a*cos(c+d*x)^7/d+1/9*a*cos(c+d*x)^9/d+3/256*a*cos(c+d*x)*sin(c+d*x)/d+1/128*a*cos(c+d*x)^3*sin(c+d*x)/d+1/160*a*cos(c+d*x)^5*sin(c+d*x)/d-3/80*a*cos(c+d*x)^7*sin(c+d*x)/d-1/10*a*cos(c+d*x)^7*sin(c+d*x)^3/d],
[cos(c+d*x)^6*sin(c+d*x)^2*(a+a*sin(c+d*x)),x,9,5/128*a*x-1/7*a*cos(c+d*x)^7/d+1/9*a*cos(c+d*x)^9/d+5/128*a*cos(c+d*x)*sin(c+d*x)/d+5/192*a*cos(c+d*x)^3*sin(c+d*x)/d+1/48*a*cos(c+d*x)^5*sin(c+d*x)/d-1/8*a*cos(c+d*x)^7*sin(c+d*x)/d],
[cos(c+d*x)^6*sin(c+d*x)*(a+a*sin(c+d*x)),x,8,5/128*a*x-1/7*a*cos(c+d*x)^7/d+5/128*a*cos(c+d*x)*sin(c+d*x)/d+5/192*a*cos(c+d*x)^3*sin(c+d*x)/d+1/48*a*cos(c+d*x)^5*sin(c+d*x)/d-1/8*a*cos(c+d*x)^7*sin(c+d*x)/d],
[cos(c+d*x)^6*csc(c+d*x)*(a+a*sin(c+d*x)),x,9,5/16*a*x-a*arctanh(cos(c+d*x))/d+a*cos(c+d*x)/d+1/3*a*cos(c+d*x)^3/d+1/5*a*cos(c+d*x)^5/d+5/16*a*cos(c+d*x)*sin(c+d*x)/d+5/24*a*cos(c+d*x)^3*sin(c+d*x)/d+1/6*a*cos(c+d*x)^5*sin(c+d*x)/d],
[cos(c+d*x)^6*csc(c+d*x)^2*(a+a*sin(c+d*x)),x,10,-15/8*a*x-a*arctanh(cos(c+d*x))/d+a*cos(c+d*x)/d+1/3*a*cos(c+d*x)^3/d+1/5*a*cos(c+d*x)^5/d-15/8*a*cot(c+d*x)/d+5/8*a*cos(c+d*x)^2*cot(c+d*x)/d+1/4*a*cos(c+d*x)^4*cot(c+d*x)/d],
[cos(c+d*x)^6*csc(c+d*x)^3*(a+a*sin(c+d*x)),x,11,-15/8*a*x+5/2*a*arctanh(cos(c+d*x))/d-5/2*a*cos(c+d*x)/d-5/6*a*cos(c+d*x)^3/d-15/8*a*cot(c+d*x)/d+5/8*a*cos(c+d*x)^2*cot(c+d*x)/d+1/4*a*cos(c+d*x)^4*cot(c+d*x)/d-1/2*a*cos(c+d*x)^3*cot(c+d*x)^2/d],
[cos(c+d*x)^6*csc(c+d*x)^4*(a+a*sin(c+d*x)),x,11,5/2*a*x+5/2*a*arctanh(cos(c+d*x))/d-5/2*a*cos(c+d*x)/d-5/6*a*cos(c+d*x)^3/d+5/2*a*cot(c+d*x)/d-1/2*a*cos(c+d*x)^3*cot(c+d*x)^2/d-5/6*a*cot(c+d*x)^3/d+1/2*a*cos(c+d*x)^2*cot(c+d*x)^3/d],
[cos(c+d*x)^6*csc(c+d*x)^5*(a+a*sin(c+d*x)),x,11,5/2*a*x-15/8*a*arctanh(cos(c+d*x))/d+15/8*a*cos(c+d*x)/d+5/2*a*cot(c+d*x)/d+5/8*a*cos(c+d*x)*cot(c+d*x)^2/d-5/6*a*cot(c+d*x)^3/d+1/2*a*cos(c+d*x)^2*cot(c+d*x)^3/d-1/4*a*cos(c+d*x)*cot(c+d*x)^4/d],
[cos(c+d*x)^6*csc(c+d*x)^6*(a+a*sin(c+d*x)),x,11,-a*x-15/8*a*arctanh(cos(c+d*x))/d+15/8*a*cos(c+d*x)/d-a*cot(c+d*x)/d+5/8*a*cos(c+d*x)*cot(c+d*x)^2/d+1/3*a*cot(c+d*x)^3/d-1/4*a*cos(c+d*x)*cot(c+d*x)^4/d-1/5*a*cot(c+d*x)^5/d],
[cos(c+d*x)^6*csc(c+d*x)^7*(a+a*sin(c+d*x)),x,9,-a*x+5/16*a*arctanh(cos(c+d*x))/d-a*cot(c+d*x)/d+1/3*a*cot(c+d*x)^3/d-1/5*a*cot(c+d*x)^5/d-5/16*a*cot(c+d*x)*csc(c+d*x)/d+5/24*a*cot(c+d*x)^3*csc(c+d*x)/d-1/6*a*cot(c+d*x)^5*csc(c+d*x)/d],
[cos(c+d*x)^6*csc(c+d*x)^8*(a+a*sin(c+d*x)),x,7,5/16*a*arctanh(cos(c+d*x))/d-1/7*a*cot(c+d*x)^7/d-5/16*a*cot(c+d*x)*csc(c+d*x)/d+5/24*a*cot(c+d*x)^3*csc(c+d*x)/d-1/6*a*cot(c+d*x)^5*csc(c+d*x)/d],
[cos(c+d*x)^6*csc(c+d*x)^9*(a+a*sin(c+d*x)),x,8,5/128*a*arctanh(cos(c+d*x))/d-1/7*a*cot(c+d*x)^7/d+5/128*a*cot(c+d*x)*csc(c+d*x)/d-5/64*a*cot(c+d*x)*csc(c+d*x)^3/d+5/48*a*cot(c+d*x)^3*csc(c+d*x)^3/d-1/8*a*cot(c+d*x)^5*csc(c+d*x)^3/d],
[cos(c+d*x)^6*csc(c+d*x)^10*(a+a*sin(c+d*x)),x,9,5/128*a*arctanh(cos(c+d*x))/d-1/7*a*cot(c+d*x)^7/d-1/9*a*cot(c+d*x)^9/d+5/128*a*cot(c+d*x)*csc(c+d*x)/d-5/64*a*cot(c+d*x)*csc(c+d*x)^3/d+5/48*a*cot(c+d*x)^3*csc(c+d*x)^3/d-1/8*a*cot(c+d*x)^5*csc(c+d*x)^3/d],
[cos(c+d*x)^6*csc(c+d*x)^11*(a+a*sin(c+d*x)),x,10,3/256*a*arctanh(cos(c+d*x))/d-1/7*a*cot(c+d*x)^7/d-1/9*a*cot(c+d*x)^9/d+3/256*a*cot(c+d*x)*csc(c+d*x)/d+1/128*a*cot(c+d*x)*csc(c+d*x)^3/d-1/32*a*cot(c+d*x)*csc(c+d*x)^5/d+1/16*a*cot(c+d*x)^3*csc(c+d*x)^5/d-1/10*a*cot(c+d*x)^5*csc(c+d*x)^5/d],
[cos(c+d*x)^6*csc(c+d*x)^12*(a+a*sin(c+d*x)),x,10,3/256*a*arctanh(cos(c+d*x))/d-1/7*a*cot(c+d*x)^7/d-2/9*a*cot(c+d*x)^9/d-1/11*a*cot(c+d*x)^11/d+3/256*a*cot(c+d*x)*csc(c+d*x)/d+1/128*a*cot(c+d*x)*csc(c+d*x)^3/d-1/32*a*cot(c+d*x)*csc(c+d*x)^5/d+1/16*a*cot(c+d*x)^3*csc(c+d*x)^5/d-1/10*a*cot(c+d*x)^5*csc(c+d*x)^5/d],
[cos(c+d*x)^6*sin(c+d*x)^4*(a+a*sin(c+d*x))^2,x,18,17/1024*a^2*x-2/7*a^2*cos(c+d*x)^7/d+4/9*a^2*cos(c+d*x)^9/d-2/11*a^2*cos(c+d*x)^11/d+17/1024*a^2*cos(c+d*x)*sin(c+d*x)/d+17/1536*a^2*cos(c+d*x)^3*sin(c+d*x)/d+17/1920*a^2*cos(c+d*x)^5*sin(c+d*x)/d-17/320*a^2*cos(c+d*x)^7*sin(c+d*x)/d-17/120*a^2*cos(c+d*x)^7*sin(c+d*x)^3/d-1/12*a^2*cos(c+d*x)^7*sin(c+d*x)^5/d],
[cos(c+d*x)^6*sin(c+d*x)^3*(a+a*sin(c+d*x))^2,x,14,3/128*a^2*x-2/7*a^2*cos(c+d*x)^7/d+1/3*a^2*cos(c+d*x)^9/d-1/11*a^2*cos(c+d*x)^11/d+3/128*a^2*cos(c+d*x)*sin(c+d*x)/d+1/64*a^2*cos(c+d*x)^3*sin(c+d*x)/d+1/80*a^2*cos(c+d*x)^5*sin(c+d*x)/d-3/40*a^2*cos(c+d*x)^7*sin(c+d*x)/d-1/5*a^2*cos(c+d*x)^7*sin(c+d*x)^3/d],
[cos(c+d*x)^6*sin(c+d*x)^2*(a+a*sin(c+d*x))^2,x,16,13/256*a^2*x-2/7*a^2*cos(c+d*x)^7/d+2/9*a^2*cos(c+d*x)^9/d+13/256*a^2*cos(c+d*x)*sin(c+d*x)/d+13/384*a^2*cos(c+d*x)^3*sin(c+d*x)/d+13/480*a^2*cos(c+d*x)^5*sin(c+d*x)/d-13/80*a^2*cos(c+d*x)^7*sin(c+d*x)/d-1/10*a^2*cos(c+d*x)^7*sin(c+d*x)^3/d],
[cos(c+d*x)^6*sin(c+d*x)*(a+a*sin(c+d*x))^2,x,7,5/64*a^2*x-1/28*a^2*cos(c+d*x)^7/d+5/64*a^2*cos(c+d*x)*sin(c+d*x)/d+5/96*a^2*cos(c+d*x)^3*sin(c+d*x)/d+1/24*a^2*cos(c+d*x)^5*sin(c+d*x)/d-1/9*cos(c+d*x)^7*(a+a*sin(c+d*x))^2/d-1/36*cos(c+d*x)^7*(a^2+a^2*sin(c+d*x))/d],
[cos(c+d*x)^6*csc(c+d*x)*(a+a*sin(c+d*x))^2,x,12,5/8*a^2*x-a^2*arctanh(cos(c+d*x))/d+a^2*cos(c+d*x)/d+1/3*a^2*cos(c+d*x)^3/d+1/5*a^2*cos(c+d*x)^5/d-1/7*a^2*cos(c+d*x)^7/d+5/8*a^2*cos(c+d*x)*sin(c+d*x)/d+5/12*a^2*cos(c+d*x)^3*sin(c+d*x)/d+1/3*a^2*cos(c+d*x)^5*sin(c+d*x)/d],
[cos(c+d*x)^6*csc(c+d*x)^2*(a+a*sin(c+d*x))^2,x,17,-25/16*a^2*x-2*a^2*arctanh(cos(c+d*x))/d+2*a^2*cos(c+d*x)/d+2/3*a^2*cos(c+d*x)^3/d+2/5*a^2*cos(c+d*x)^5/d-a^2*cot(c+d*x)/d-7/16*a^2*cos(c+d*x)*sin(c+d*x)/d-7/24*a^2*cos(c+d*x)*sin(c+d*x)^3/d+1/6*a^2*cos(c+d*x)*sin(c+d*x)^5/d],
[cos(c+d*x)^6*csc(c+d*x)^3*(a+a*sin(c+d*x))^2,x,16,-15/4*a^2*x+3/2*a^2*arctanh(cos(c+d*x))/d-a^2*cos(c+d*x)/d+1/5*a^2*cos(c+d*x)^5/d-2*a^2*cot(c+d*x)/d-1/2*a^2*cot(c+d*x)*csc(c+d*x)/d-9/4*a^2*cos(c+d*x)*sin(c+d*x)/d+1/2*a^2*cos(c+d*x)*sin(c+d*x)^3/d],
[cos(c+d*x)^6*csc(c+d*x)^4*(a+a*sin(c+d*x))^2,x,17,5/8*a^2*x+5*a^2*arctanh(cos(c+d*x))/d-4*a^2*cos(c+d*x)/d-2/3*a^2*cos(c+d*x)^3/d+a^2*cot(c+d*x)/d-1/3*a^2*cot(c+d*x)^3/d-a^2*cot(c+d*x)*csc(c+d*x)/d-5/8*a^2*cos(c+d*x)*sin(c+d*x)/d+1/4*a^2*cos(c+d*x)*sin(c+d*x)^3/d],
[cos(c+d*x)^6*csc(c+d*x)^5*(a+a*sin(c+d*x))^2,x,16,5*a^2*x+5/8*a^2*arctanh(cos(c+d*x))/d-a^2*cos(c+d*x)/d-1/3*a^2*cos(c+d*x)^3/d+4*a^2*cot(c+d*x)/d-2/3*a^2*cot(c+d*x)^3/d+5/8*a^2*cot(c+d*x)*csc(c+d*x)/d-1/4*a^2*cot(c+d*x)*csc(c+d*x)^3/d+a^2*cos(c+d*x)*sin(c+d*x)/d],
[cos(c+d*x)^6*csc(c+d*x)^6*(a+a*sin(c+d*x))^2,x,15,3/2*a^2*x-15/4*a^2*arctanh(cos(c+d*x))/d+2*a^2*cos(c+d*x)/d+a^2*cot(c+d*x)/d-1/5*a^2*cot(c+d*x)^5/d+9/4*a^2*cot(c+d*x)*csc(c+d*x)/d-1/2*a^2*cot(c+d*x)*csc(c+d*x)^3/d+1/2*a^2*cos(c+d*x)*sin(c+d*x)/d],
[cos(c+d*x)^6*csc(c+d*x)^7*(a+a*sin(c+d*x))^2,x,17,-2*a^2*x-25/16*a^2*arctanh(cos(c+d*x))/d+a^2*cos(c+d*x)/d-2*a^2*cot(c+d*x)/d+2/3*a^2*cot(c+d*x)^3/d-2/5*a^2*cot(c+d*x)^5/d+7/16*a^2*cot(c+d*x)*csc(c+d*x)/d+7/24*a^2*cot(c+d*x)*csc(c+d*x)^3/d-1/6*a^2*cot(c+d*x)*csc(c+d*x)^5/d],
[cos(c+d*x)^6*csc(c+d*x)^8*(a+a*sin(c+d*x))^2,x,12,-a^2*x+5/8*a^2*arctanh(cos(c+d*x))/d-a^2*cot(c+d*x)/d+1/3*a^2*cot(c+d*x)^3/d-1/5*a^2*cot(c+d*x)^5/d-1/7*a^2*cot(c+d*x)^7/d-5/8*a^2*cot(c+d*x)*csc(c+d*x)/d+5/12*a^2*cot(c+d*x)^3*csc(c+d*x)/d-1/3*a^2*cot(c+d*x)^5*csc(c+d*x)/d],
[cos(c+d*x)^6*csc(c+d*x)^9*(a+a*sin(c+d*x))^2,x,13,45/128*a^2*arctanh(cos(c+d*x))/d-2/7*a^2*cot(c+d*x)^7/d-35/128*a^2*cot(c+d*x)*csc(c+d*x)/d+5/24*a^2*cot(c+d*x)^3*csc(c+d*x)/d-1/6*a^2*cot(c+d*x)^5*csc(c+d*x)/d-5/64*a^2*cot(c+d*x)*csc(c+d*x)^3/d+5/48*a^2*cot(c+d*x)^3*csc(c+d*x)^3/d-1/8*a^2*cot(c+d*x)^5*csc(c+d*x)^3/d],
[cos(c+d*x)^6*csc(c+d*x)^10*(a+a*sin(c+d*x))^2,x,12,5/64*a^2*arctanh(cos(c+d*x))/d-2/7*a^2*cot(c+d*x)^7/d-1/9*a^2*cot(c+d*x)^9/d+5/64*a^2*cot(c+d*x)*csc(c+d*x)/d-5/32*a^2*cot(c+d*x)*csc(c+d*x)^3/d+5/24*a^2*cot(c+d*x)^3*csc(c+d*x)^3/d-1/4*a^2*cot(c+d*x)^5*csc(c+d*x)^3/d],
[cos(c+d*x)^6*csc(c+d*x)^11*(a+a*sin(c+d*x))^2,x,16,13/256*a^2*arctanh(cos(c+d*x))/d-2/7*a^2*cot(c+d*x)^7/d-2/9*a^2*cot(c+d*x)^9/d+13/256*a^2*cot(c+d*x)*csc(c+d*x)/d-9/128*a^2*cot(c+d*x)*csc(c+d*x)^3/d+5/48*a^2*cot(c+d*x)^3*csc(c+d*x)^3/d-1/8*a^2*cot(c+d*x)^5*csc(c+d*x)^3/d-1/32*a^2*cot(c+d*x)*csc(c+d*x)^5/d+1/16*a^2*cot(c+d*x)^3*csc(c+d*x)^5/d-1/10*a^2*cot(c+d*x)^5*csc(c+d*x)^5/d],
[cos(c+d*x)^6*csc(c+d*x)^12*(a+a*sin(c+d*x))^2,x,14,3/128*a^2*arctanh(cos(c+d*x))/d-2/7*a^2*cot(c+d*x)^7/d-1/3*a^2*cot(c+d*x)^9/d-1/11*a^2*cot(c+d*x)^11/d+3/128*a^2*cot(c+d*x)*csc(c+d*x)/d+1/64*a^2*cot(c+d*x)*csc(c+d*x)^3/d-1/16*a^2*cot(c+d*x)*csc(c+d*x)^5/d+1/8*a^2*cot(c+d*x)^3*csc(c+d*x)^5/d-1/5*a^2*cot(c+d*x)^5*csc(c+d*x)^5/d],
[cos(c+d*x)^6*csc(c+d*x)^13*(a+a*sin(c+d*x))^2,x,18,17/1024*a^2*arctanh(cos(c+d*x))/d-2/7*a^2*cot(c+d*x)^7/d-4/9*a^2*cot(c+d*x)^9/d-2/11*a^2*cot(c+d*x)^11/d+17/1024*a^2*cot(c+d*x)*csc(c+d*x)/d+17/1536*a^2*cot(c+d*x)*csc(c+d*x)^3/d-11/384*a^2*cot(c+d*x)*csc(c+d*x)^5/d+1/16*a^2*cot(c+d*x)^3*csc(c+d*x)^5/d-1/10*a^2*cot(c+d*x)^5*csc(c+d*x)^5/d-1/64*a^2*cot(c+d*x)*csc(c+d*x)^7/d+1/24*a^2*cot(c+d*x)^3*csc(c+d*x)^7/d-1/12*a^2*cot(c+d*x)^5*csc(c+d*x)^7/d],
[cos(c+d*x)^6*sin(c+d*x)^4*(a+a*sin(c+d*x))^3,x,21,27/1024*a^3*x-4/7*a^3*cos(c+d*x)^7/d+a^3*cos(c+d*x)^9/d-6/11*a^3*cos(c+d*x)^11/d+1/13*a^3*cos(c+d*x)^13/d+27/1024*a^3*cos(c+d*x)*sin(c+d*x)/d+9/512*a^3*cos(c+d*x)^3*sin(c+d*x)/d+9/640*a^3*cos(c+d*x)^5*sin(c+d*x)/d-27/320*a^3*cos(c+d*x)^7*sin(c+d*x)/d-9/40*a^3*cos(c+d*x)^7*sin(c+d*x)^3/d-1/4*a^3*cos(c+d*x)^7*sin(c+d*x)^5/d],
[cos(c+d*x)^6*sin(c+d*x)^3*(a+a*sin(c+d*x))^3,x,21,41/1024*a^3*x-4/7*a^3*cos(c+d*x)^7/d+7/9*a^3*cos(c+d*x)^9/d-3/11*a^3*cos(c+d*x)^11/d+41/1024*a^3*cos(c+d*x)*sin(c+d*x)/d+41/1536*a^3*cos(c+d*x)^3*sin(c+d*x)/d+41/1920*a^3*cos(c+d*x)^5*sin(c+d*x)/d-41/320*a^3*cos(c+d*x)^7*sin(c+d*x)/d-41/120*a^3*cos(c+d*x)^7*sin(c+d*x)^3/d-1/12*a^3*cos(c+d*x)^7*sin(c+d*x)^5/d],
[cos(c+d*x)^6*sin(c+d*x)^2*(a+a*sin(c+d*x))^3,x,19,19/256*a^3*x-4/7*a^3*cos(c+d*x)^7/d+5/9*a^3*cos(c+d*x)^9/d-1/11*a^3*cos(c+d*x)^11/d+19/256*a^3*cos(c+d*x)*sin(c+d*x)/d+19/384*a^3*cos(c+d*x)^3*sin(c+d*x)/d+19/480*a^3*cos(c+d*x)^5*sin(c+d*x)/d-19/80*a^3*cos(c+d*x)^7*sin(c+d*x)/d-3/10*a^3*cos(c+d*x)^7*sin(c+d*x)^3/d],
[cos(c+d*x)^6*sin(c+d*x)*(a+a*sin(c+d*x))^3,x,8,33/256*a^3*x-33/560*a^3*cos(c+d*x)^7/d+33/256*a^3*cos(c+d*x)*sin(c+d*x)/d+11/128*a^3*cos(c+d*x)^3*sin(c+d*x)/d+11/160*a^3*cos(c+d*x)^5*sin(c+d*x)/d-1/30*a*cos(c+d*x)^7*(a+a*sin(c+d*x))^2/d-1/10*cos(c+d*x)^7*(a+a*sin(c+d*x))^3/d-11/240*cos(c+d*x)^7*(a^3+a^3*sin(c+d*x))/d],
[cos(c+d*x)^6*csc(c+d*x)*(a+a*sin(c+d*x))^3,x,17,125/128*a^3*x-a^3*arctanh(cos(c+d*x))/d+a^3*cos(c+d*x)/d+1/3*a^3*cos(c+d*x)^3/d+1/5*a^3*cos(c+d*x)^5/d-3/7*a^3*cos(c+d*x)^7/d+125/128*a^3*cos(c+d*x)*sin(c+d*x)/d+125/192*a^3*cos(c+d*x)^3*sin(c+d*x)/d+25/48*a^3*cos(c+d*x)^5*sin(c+d*x)/d-1/8*a^3*cos(c+d*x)^7*sin(c+d*x)/d],
[cos(c+d*x)^6*csc(c+d*x)^2*(a+a*sin(c+d*x))^3,x,19,-15/16*a^3*x-3*a^3*arctanh(cos(c+d*x))/d+3*a^3*cos(c+d*x)/d+a^3*cos(c+d*x)^3/d+3/5*a^3*cos(c+d*x)^5/d-1/7*a^3*cos(c+d*x)^7/d-a^3*cot(c+d*x)/d+15/16*a^3*cos(c+d*x)*sin(c+d*x)/d-11/8*a^3*cos(c+d*x)*sin(c+d*x)^3/d+1/2*a^3*cos(c+d*x)*sin(c+d*x)^5/d],
[cos(c+d*x)^6*csc(c+d*x)^3*(a+a*sin(c+d*x))^3,x,17,-85/16*a^3*x-1/2*a^3*arctanh(cos(c+d*x))/d+a^3*cos(c+d*x)/d+2/3*a^3*cos(c+d*x)^3/d+3/5*a^3*cos(c+d*x)^5/d-3*a^3*cot(c+d*x)/d-1/2*a^3*cot(c+d*x)*csc(c+d*x)/d-43/16*a^3*cos(c+d*x)*sin(c+d*x)/d+5/24*a^3*cos(c+d*x)*sin(c+d*x)^3/d+1/6*a^3*cos(c+d*x)*sin(c+d*x)^5/d],
[cos(c+d*x)^6*csc(c+d*x)^4*(a+a*sin(c+d*x))^3,x,15,-25/8*a^3*x+13/2*a^3*arctanh(cos(c+d*x))/d-5*a^3*cos(c+d*x)/d-2/3*a^3*cos(c+d*x)^3/d+1/5*a^3*cos(c+d*x)^5/d-a^3*cot(c+d*x)/d-1/3*a^3*cot(c+d*x)^3/d-3/2*a^3*cot(c+d*x)*csc(c+d*x)/d-23/8*a^3*cos(c+d*x)*sin(c+d*x)/d+3/4*a^3*cos(c+d*x)*sin(c+d*x)^3/d],
[cos(c+d*x)^6*csc(c+d*x)^5*(a+a*sin(c+d*x))^3,x,16,45/8*a^3*x+45/8*a^3*arctanh(cos(c+d*x))/d-5*a^3*cos(c+d*x)/d-a^3*cos(c+d*x)^3/d+5*a^3*cot(c+d*x)/d-a^3*cot(c+d*x)^3/d-3/8*a^3*cot(c+d*x)*csc(c+d*x)/d-1/4*a^3*cot(c+d*x)*csc(c+d*x)^3/d+3/8*a^3*cos(c+d*x)*sin(c+d*x)/d+1/4*a^3*cos(c+d*x)*sin(c+d*x)^3/d],
[cos(c+d*x)^6*csc(c+d*x)^6*(a+a*sin(c+d*x))^3,x,16,13/2*a^3*x-25/8*a^3*arctanh(cos(c+d*x))/d+a^3*cos(c+d*x)/d-1/3*a^3*cos(c+d*x)^3/d+5*a^3*cot(c+d*x)/d-2/3*a^3*cot(c+d*x)^3/d-1/5*a^3*cot(c+d*x)^5/d+23/8*a^3*cot(c+d*x)*csc(c+d*x)/d-3/4*a^3*cot(c+d*x)*csc(c+d*x)^3/d+3/2*a^3*cos(c+d*x)*sin(c+d*x)/d],
[cos(c+d*x)^6*csc(c+d*x)^7*(a+a*sin(c+d*x))^3,x,18,-1/2*a^3*x-85/16*a^3*arctanh(cos(c+d*x))/d+3*a^3*cos(c+d*x)/d-a^3*cot(c+d*x)/d+2/3*a^3*cot(c+d*x)^3/d-3/5*a^3*cot(c+d*x)^5/d+43/16*a^3*cot(c+d*x)*csc(c+d*x)/d-5/24*a^3*cot(c+d*x)*csc(c+d*x)^3/d-1/6*a^3*cot(c+d*x)*csc(c+d*x)^5/d+1/2*a^3*cos(c+d*x)*sin(c+d*x)/d],
[cos(c+d*x)^6*csc(c+d*x)^8*(a+a*sin(c+d*x))^3,x,18,-3*a^3*x-15/16*a^3*arctanh(cos(c+d*x))/d+a^3*cos(c+d*x)/d-3*a^3*cot(c+d*x)/d+a^3*cot(c+d*x)^3/d-3/5*a^3*cot(c+d*x)^5/d-1/7*a^3*cot(c+d*x)^7/d-15/16*a^3*cot(c+d*x)*csc(c+d*x)/d+11/8*a^3*cot(c+d*x)*csc(c+d*x)^3/d-1/2*a^3*cot(c+d*x)*csc(c+d*x)^5/d],
[cos(c+d*x)^6*csc(c+d*x)^9*(a+a*sin(c+d*x))^3,x,17,-a^3*x+125/128*a^3*arctanh(cos(c+d*x))/d-a^3*cot(c+d*x)/d+1/3*a^3*cot(c+d*x)^3/d-1/5*a^3*cot(c+d*x)^5/d-3/7*a^3*cot(c+d*x)^7/d-115/128*a^3*cot(c+d*x)*csc(c+d*x)/d+5/8*a^3*cot(c+d*x)^3*csc(c+d*x)/d-1/2*a^3*cot(c+d*x)^5*csc(c+d*x)/d-5/64*a^3*cot(c+d*x)*csc(c+d*x)^3/d+5/48*a^3*cot(c+d*x)^3*csc(c+d*x)^3/d-1/8*a^3*cot(c+d*x)^5*csc(c+d*x)^3/d],
[cos(c+d*x)^6*csc(c+d*x)^10*(a+a*sin(c+d*x))^3,x,16,55/128*a^3*arctanh(cos(c+d*x))/d-4/7*a^3*cot(c+d*x)^7/d-1/9*a^3*cot(c+d*x)^9/d-25/128*a^3*cot(c+d*x)*csc(c+d*x)/d+5/24*a^3*cot(c+d*x)^3*csc(c+d*x)/d-1/6*a^3*cot(c+d*x)^5*csc(c+d*x)/d-15/64*a^3*cot(c+d*x)*csc(c+d*x)^3/d+5/16*a^3*cot(c+d*x)^3*csc(c+d*x)^3/d-3/8*a^3*cot(c+d*x)^5*csc(c+d*x)^3/d],
[cos(c+d*x)^6*csc(c+d*x)^11*(a+a*sin(c+d*x))^3,x,18,33/256*a^3*arctanh(cos(c+d*x))/d-4/7*a^3*cot(c+d*x)^7/d-1/3*a^3*cot(c+d*x)^9/d+33/256*a^3*cot(c+d*x)*csc(c+d*x)/d-29/128*a^3*cot(c+d*x)*csc(c+d*x)^3/d+5/16*a^3*cot(c+d*x)^3*csc(c+d*x)^3/d-3/8*a^3*cot(c+d*x)^5*csc(c+d*x)^3/d-1/32*a^3*cot(c+d*x)*csc(c+d*x)^5/d+1/16*a^3*cot(c+d*x)^3*csc(c+d*x)^5/d-1/10*a^3*cot(c+d*x)^5*csc(c+d*x)^5/d],
[cos(c+d*x)^6*csc(c+d*x)^12*(a+a*sin(c+d*x))^3,x,19,19/256*a^3*arctanh(cos(c+d*x))/d-4/7*a^3*cot(c+d*x)^7/d-5/9*a^3*cot(c+d*x)^9/d-1/11*a^3*cot(c+d*x)^11/d+19/256*a^3*cot(c+d*x)*csc(c+d*x)/d-7/128*a^3*cot(c+d*x)*csc(c+d*x)^3/d+5/48*a^3*cot(c+d*x)^3*csc(c+d*x)^3/d-1/8*a^3*cot(c+d*x)^5*csc(c+d*x)^3/d-3/32*a^3*cot(c+d*x)*csc(c+d*x)^5/d+3/16*a^3*cot(c+d*x)^3*csc(c+d*x)^5/d-3/10*a^3*cot(c+d*x)^5*csc(c+d*x)^5/d],
[cos(c+d*x)^6*csc(c+d*x)^13*(a+a*sin(c+d*x))^3,x,21,41/1024*a^3*arctanh(cos(c+d*x))/d-4/7*a^3*cot(c+d*x)^7/d-7/9*a^3*cot(c+d*x)^9/d-3/11*a^3*cot(c+d*x)^11/d+41/1024*a^3*cot(c+d*x)*csc(c+d*x)/d+41/1536*a^3*cot(c+d*x)*csc(c+d*x)^3/d-35/384*a^3*cot(c+d*x)*csc(c+d*x)^5/d+3/16*a^3*cot(c+d*x)^3*csc(c+d*x)^5/d-3/10*a^3*cot(c+d*x)^5*csc(c+d*x)^5/d-1/64*a^3*cot(c+d*x)*csc(c+d*x)^7/d+1/24*a^3*cot(c+d*x)^3*csc(c+d*x)^7/d-1/12*a^3*cot(c+d*x)^5*csc(c+d*x)^7/d],
[cos(c+d*x)^6*csc(c+d*x)^14*(a+a*sin(c+d*x))^3,x,21,27/1024*a^3*arctanh(cos(c+d*x))/d-4/7*a^3*cot(c+d*x)^7/d-a^3*cot(c+d*x)^9/d-6/11*a^3*cot(c+d*x)^11/d-1/13*a^3*cot(c+d*x)^13/d+27/1024*a^3*cot(c+d*x)*csc(c+d*x)/d+9/512*a^3*cot(c+d*x)*csc(c+d*x)^3/d-3/128*a^3*cot(c+d*x)*csc(c+d*x)^5/d+1/16*a^3*cot(c+d*x)^3*csc(c+d*x)^5/d-1/10*a^3*cot(c+d*x)^5*csc(c+d*x)^5/d-3/64*a^3*cot(c+d*x)*csc(c+d*x)^7/d+1/8*a^3*cot(c+d*x)^3*csc(c+d*x)^7/d-1/4*a^3*cot(c+d*x)^5*csc(c+d*x)^7/d],
[cos(c+d*x)^6*csc(c+d*x)^4*(a+a*sin(c+d*x))^4,x,22,-135/16*a^4*x+6*a^4*arctanh(cos(c+d*x))/d-4*a^4*cos(c+d*x)/d+4/5*a^4*cos(c+d*x)^5/d-4*a^4*cot(c+d*x)/d-1/3*a^4*cot(c+d*x)^3/d-2*a^4*cot(c+d*x)*csc(c+d*x)/d-89/16*a^4*cos(c+d*x)*sin(c+d*x)/d+23/24*a^4*cos(c+d*x)*sin(c+d*x)^3/d+1/6*a^4*cos(c+d*x)*sin(c+d*x)^5/d],

# m<0
[cos(c+d*x)^6*sin(c+d*x)^4/(a+a*sin(c+d*x)),x,9,3/128*x/a+1/5*cos(c+d*x)^5/(a*d)-2/7*cos(c+d*x)^7/(a*d)+1/9*cos(c+d*x)^9/(a*d)+3/128*cos(c+d*x)*sin(c+d*x)/(a*d)+1/64*cos(c+d*x)^3*sin(c+d*x)/(a*d)-1/16*cos(c+d*x)^5*sin(c+d*x)/(a*d)-1/8*cos(c+d*x)^5*sin(c+d*x)^3/(a*d)],
[cos(c+d*x)^6*sin(c+d*x)^3/(a+a*sin(c+d*x)),x,9,-3/128*x/a-1/5*cos(c+d*x)^5/(a*d)+1/7*cos(c+d*x)^7/(a*d)-3/128*cos(c+d*x)*sin(c+d*x)/(a*d)-1/64*cos(c+d*x)^3*sin(c+d*x)/(a*d)+1/16*cos(c+d*x)^5*sin(c+d*x)/(a*d)+1/8*cos(c+d*x)^5*sin(c+d*x)^3/(a*d)],
[cos(c+d*x)^6*sin(c+d*x)^2/(a+a*sin(c+d*x)),x,8,1/16*x/a+1/5*cos(c+d*x)^5/(a*d)-1/7*cos(c+d*x)^7/(a*d)+1/16*cos(c+d*x)*sin(c+d*x)/(a*d)+1/24*cos(c+d*x)^3*sin(c+d*x)/(a*d)-1/6*cos(c+d*x)^5*sin(c+d*x)/(a*d)],
[cos(c+d*x)^6*sin(c+d*x)/(a+a*sin(c+d*x)),x,7,-1/16*x/a-1/5*cos(c+d*x)^5/(a*d)-1/16*cos(c+d*x)*sin(c+d*x)/(a*d)-1/24*cos(c+d*x)^3*sin(c+d*x)/(a*d)+1/6*cos(c+d*x)^5*sin(c+d*x)/(a*d)],
[cos(c+d*x)^6*csc(c+d*x)/(a+a*sin(c+d*x)),x,8,-3/8*x/a-arctanh(cos(c+d*x))/(a*d)+cos(c+d*x)/(a*d)+1/3*cos(c+d*x)^3/(a*d)-3/8*cos(c+d*x)*sin(c+d*x)/(a*d)-1/4*cos(c+d*x)^3*sin(c+d*x)/(a*d)],
[cos(c+d*x)^6*csc(c+d*x)^2/(a+a*sin(c+d*x)),x,9,-3/2*x/a+arctanh(cos(c+d*x))/(a*d)-cos(c+d*x)/(a*d)-1/3*cos(c+d*x)^3/(a*d)-3/2*cot(c+d*x)/(a*d)+1/2*cos(c+d*x)^2*cot(c+d*x)/(a*d)],
[cos(c+d*x)^6*csc(c+d*x)^3/(a+a*sin(c+d*x)),x,9,3/2*x/a+3/2*arctanh(cos(c+d*x))/(a*d)-3/2*cos(c+d*x)/(a*d)+3/2*cot(c+d*x)/(a*d)-1/2*cos(c+d*x)^2*cot(c+d*x)/(a*d)-1/2*cos(c+d*x)*cot(c+d*x)^2/(a*d)],
[cos(c+d*x)^6*csc(c+d*x)^4/(a+a*sin(c+d*x)),x,8,x/a-3/2*arctanh(cos(c+d*x))/(a*d)+3/2*cos(c+d*x)/(a*d)+cot(c+d*x)/(a*d)+1/2*cos(c+d*x)*cot(c+d*x)^2/(a*d)-1/3*cot(c+d*x)^3/(a*d)],
[cos(c+d*x)^6*csc(c+d*x)^5/(a+a*sin(c+d*x)),x,7,-x/a-3/8*arctanh(cos(c+d*x))/(a*d)-cot(c+d*x)/(a*d)+1/3*cot(c+d*x)^3/(a*d)+3/8*cot(c+d*x)*csc(c+d*x)/(a*d)-1/4*cot(c+d*x)^3*csc(c+d*x)/(a*d)],
[cos(c+d*x)^6*csc(c+d*x)^6/(a+a*sin(c+d*x)),x,6,3/8*arctanh(cos(c+d*x))/(a*d)-1/5*cot(c+d*x)^5/(a*d)-3/8*cot(c+d*x)*csc(c+d*x)/(a*d)+1/4*cot(c+d*x)^3*csc(c+d*x)/(a*d)],
[cos(c+d*x)^6*sin(c+d*x)^3/(a+a*sin(c+d*x))^2,x,13,-1/8*x/a^2-2/3*cos(c+d*x)^3/(a^2*d)+3/5*cos(c+d*x)^5/(a^2*d)-1/7*cos(c+d*x)^7/(a^2*d)-1/8*cos(c+d*x)*sin(c+d*x)/(a^2*d)+1/4*cos(c+d*x)^3*sin(c+d*x)/(a^2*d)+1/3*cos(c+d*x)^3*sin(c+d*x)^3/(a^2*d)],
[cos(c+d*x)^6*sin(c+d*x)^2/(a+a*sin(c+d*x))^2,x,6,3/16*x/a^2+1/10*cos(c+d*x)^5/(a^2*d)+3/16*cos(c+d*x)*sin(c+d*x)/(a^2*d)+1/8*cos(c+d*x)^3*sin(c+d*x)/(a^2*d)+1/6*cos(c+d*x)^3*(a-a*sin(c+d*x))^3/(a^5*d)],
[cos(c+d*x)^6*sin(c+d*x)/(a+a*sin(c+d*x))^2,x,5,-1/4*x/a^2-2/15*cos(c+d*x)^5/(a^2*d)-1/4*cos(c+d*x)*sin(c+d*x)/(a^2*d)-1/6*cos(c+d*x)^3*sin(c+d*x)/(a^2*d)-1/3*cos(c+d*x)^7/(d*(a+a*sin(c+d*x))^2)],
[cos(c+d*x)^6*csc(c+d*x)/(a+a*sin(c+d*x))^2,x,10,-x/a^2-arctanh(cos(c+d*x))/(a^2*d)+cos(c+d*x)/(a^2*d)-1/3*cos(c+d*x)^3/(a^2*d)-cos(c+d*x)*sin(c+d*x)/(a^2*d)],
[cos(c+d*x)^6*csc(c+d*x)^2/(a+a*sin(c+d*x))^2,x,9,-1/2*x/a^2+2*arctanh(cos(c+d*x))/(a^2*d)-2*cos(c+d*x)/(a^2*d)-cot(c+d*x)/(a^2*d)+1/2*cos(c+d*x)*sin(c+d*x)/(a^2*d)],
[cos(c+d*x)^6*csc(c+d*x)^3/(a+a*sin(c+d*x))^2,x,8,2*x/a^2-1/2*arctanh(cos(c+d*x))/(a^2*d)+cos(c+d*x)/(a^2*d)+2*cot(c+d*x)/(a^2*d)-1/2*cot(c+d*x)*csc(c+d*x)/(a^2*d)],
[cos(c+d*x)^6*csc(c+d*x)^4/(a+a*sin(c+d*x))^2,x,9,-x/a^2-arctanh(cos(c+d*x))/(a^2*d)-cot(c+d*x)/(a^2*d)-1/3*cot(c+d*x)^3/(a^2*d)+cot(c+d*x)*csc(c+d*x)/(a^2*d)],
[cos(c+d*x)^6*csc(c+d*x)^5/(a+a*sin(c+d*x))^2,x,10,5/8*arctanh(cos(c+d*x))/(a^2*d)+2/3*cot(c+d*x)^3/(a^2*d)-3/8*cot(c+d*x)*csc(c+d*x)/(a^2*d)-1/4*cot(c+d*x)*csc(c+d*x)^3/(a^2*d)],
[cos(c+d*x)^6*csc(c+d*x)^6/(a+a*sin(c+d*x))^2,x,11,-1/4*arctanh(cos(c+d*x))/(a^2*d)-2/3*cot(c+d*x)^3/(a^2*d)-1/5*cot(c+d*x)^5/(a^2*d)-1/4*cot(c+d*x)*csc(c+d*x)/(a^2*d)+1/2*cot(c+d*x)*csc(c+d*x)^3/(a^2*d)],
[cos(c+d*x)^6*csc(c+d*x)^7/(a+a*sin(c+d*x))^2,x,13,3/16*arctanh(cos(c+d*x))/(a^2*d)+2/3*cot(c+d*x)^3/(a^2*d)+2/5*cot(c+d*x)^5/(a^2*d)+3/16*cot(c+d*x)*csc(c+d*x)/(a^2*d)-5/24*cot(c+d*x)*csc(c+d*x)^3/(a^2*d)-1/6*cot(c+d*x)*csc(c+d*x)^5/(a^2*d)],
[cos(c+d*x)^6*sin(c+d*x)^3/(a+a*sin(c+d*x))^3,x,14,-23/16*x/a^3-4*cos(c+d*x)/(a^3*d)+7/3*cos(c+d*x)^3/(a^3*d)-3/5*cos(c+d*x)^5/(a^3*d)+23/16*cos(c+d*x)*sin(c+d*x)/(a^3*d)+23/24*cos(c+d*x)*sin(c+d*x)^3/(a^3*d)+1/6*cos(c+d*x)*sin(c+d*x)^5/(a^3*d)],
[cos(c+d*x)^6*sin(c+d*x)^2/(a+a*sin(c+d*x))^3,x,12,13/8*x/a^3+4*cos(c+d*x)/(a^3*d)-5/3*cos(c+d*x)^3/(a^3*d)+1/5*cos(c+d*x)^5/(a^3*d)-13/8*cos(c+d*x)*sin(c+d*x)/(a^3*d)-3/4*cos(c+d*x)*sin(c+d*x)^3/(a^3*d)],
[cos(c+d*x)^6*sin(c+d*x)/(a+a*sin(c+d*x))^3,x,5,-15/8*x/a^3-4*cos(c+d*x)/(a^3*d)+cos(c+d*x)^3/(a^3*d)+15/8*cos(c+d*x)*sin(c+d*x)/(a^3*d)+1/4*cos(c+d*x)*sin(c+d*x)^3/(a^3*d),-15/8*x/a^3-5/4*cos(c+d*x)^3/(a^3*d)-15/8*cos(c+d*x)*sin(c+d*x)/(a^3*d)-cos(c+d*x)^7/(d*(a+a*sin(c+d*x))^3)-3/4*cos(c+d*x)^5/(d*(a^3+a^3*sin(c+d*x)))],
[cos(c+d*x)^6*csc(c+d*x)/(a+a*sin(c+d*x))^3,x,7,-7/2*x/a^3-arctanh(cos(c+d*x))/(a^3*d)-3*cos(c+d*x)/(a^3*d)+1/2*cos(c+d*x)*sin(c+d*x)/(a^3*d)],
[cos(c+d*x)^6*csc(c+d*x)^2/(a+a*sin(c+d*x))^3,x,7,3*x/a^3+3*arctanh(cos(c+d*x))/(a^3*d)+cos(c+d*x)/(a^3*d)-cot(c+d*x)/(a^3*d)],
[cos(c+d*x)^6*csc(c+d*x)^3/(a+a*sin(c+d*x))^3,x,8,-x/a^3-7/2*arctanh(cos(c+d*x))/(a^3*d)+3*cot(c+d*x)/(a^3*d)-1/2*cot(c+d*x)*csc(c+d*x)/(a^3*d)],
[cos(c+d*x)^6*csc(c+d*x)^4/(a+a*sin(c+d*x))^3,x,10,5/2*arctanh(cos(c+d*x))/(a^3*d)-4*cot(c+d*x)/(a^3*d)-1/3*cot(c+d*x)^3/(a^3*d)+3/2*cot(c+d*x)*csc(c+d*x)/(a^3*d)],
[cos(c+d*x)^6*csc(c+d*x)^5/(a+a*sin(c+d*x))^3,x,12,-15/8*arctanh(cos(c+d*x))/(a^3*d)+4*cot(c+d*x)/(a^3*d)+cot(c+d*x)^3/(a^3*d)-15/8*cot(c+d*x)*csc(c+d*x)/(a^3*d)-1/4*cot(c+d*x)*csc(c+d*x)^3/(a^3*d)],
[cos(c+d*x)^6*csc(c+d*x)^6/(a+a*sin(c+d*x))^3,x,12,13/8*arctanh(cos(c+d*x))/(a^3*d)-4*cot(c+d*x)/(a^3*d)-5/3*cot(c+d*x)^3/(a^3*d)-1/5*cot(c+d*x)^5/(a^3*d)+13/8*cot(c+d*x)*csc(c+d*x)/(a^3*d)+3/4*cot(c+d*x)*csc(c+d*x)^3/(a^3*d)],

# Integrands of the form Cos[e+f x]^6 Sin[e+f x]^n (a+a Sin[e+f x])^(m/2) with n symbolic

# m>0

# m<0
[cos(c+d*x)^6*sin(c+d*x)^n*(a+a*sin(c+d*x))^3,x,6,a^3*cos(c+d*x)*hypergeom([-5/2,1/2*(1+n)],[1/2*(3+n)],sin(c+d*x)^2)*sin(c+d*x)^(1+n)/(d*(1+n)*sqrt(cos(c+d*x)^2))+3*a^3*cos(c+d*x)*hypergeom([-5/2,1/2*(2+n)],[1/2*(4+n)],sin(c+d*x)^2)*sin(c+d*x)^(2+n)/(d*(2+n)*sqrt(cos(c+d*x)^2))+3*a^3*cos(c+d*x)*hypergeom([-5/2,1/2*(3+n)],[1/2*(5+n)],sin(c+d*x)^2)*sin(c+d*x)^(3+n)/(d*(3+n)*sqrt(cos(c+d*x)^2))+a^3*cos(c+d*x)*hypergeom([-5/2,1/2*(4+n)],[1/2*(6+n)],sin(c+d*x)^2)*sin(c+d*x)^(4+n)/(d*(4+n)*sqrt(cos(c+d*x)^2))],
[cos(c+d*x)^6*sin(c+d*x)^n*(a+a*sin(c+d*x))^2,x,5,a^2*cos(c+d*x)*hypergeom([-5/2,1/2*(1+n)],[1/2*(3+n)],sin(c+d*x)^2)*sin(c+d*x)^(1+n)/(d*(1+n)*sqrt(cos(c+d*x)^2))+2*a^2*cos(c+d*x)*hypergeom([-5/2,1/2*(2+n)],[1/2*(4+n)],sin(c+d*x)^2)*sin(c+d*x)^(2+n)/(d*(2+n)*sqrt(cos(c+d*x)^2))+a^2*cos(c+d*x)*hypergeom([-5/2,1/2*(3+n)],[1/2*(5+n)],sin(c+d*x)^2)*sin(c+d*x)^(3+n)/(d*(3+n)*sqrt(cos(c+d*x)^2))],
[cos(c+d*x)^6*sin(c+d*x)^n*(a+a*sin(c+d*x)),x,3,a*cos(c+d*x)*hypergeom([-5/2,1/2*(1+n)],[1/2*(3+n)],sin(c+d*x)^2)*sin(c+d*x)^(1+n)/(d*(1+n)*sqrt(cos(c+d*x)^2))+a*cos(c+d*x)*hypergeom([-5/2,1/2*(2+n)],[1/2*(4+n)],sin(c+d*x)^2)*sin(c+d*x)^(2+n)/(d*(2+n)*sqrt(cos(c+d*x)^2))],

# Integrands of the form Cos[e+f x]^7 (d Sin[e+f x])^n (a+a Sin[e+f x])^m

# Integrands of the form Cos[e+f x]^7 Sin[e+f x]^n (a+a Sin[e+f x])^m

# m>0
[cos(c+d*x)^7*sin(c+d*x)^6*(a+a*sin(c+d*x)),x,4,1/7*a*sin(c+d*x)^7/d+1/8*a*sin(c+d*x)^8/d-1/3*a*sin(c+d*x)^9/d-3/10*a*sin(c+d*x)^10/d+3/11*a*sin(c+d*x)^11/d+1/4*a*sin(c+d*x)^12/d-1/13*a*sin(c+d*x)^13/d-1/14*a*sin(c+d*x)^14/d],
[cos(c+d*x)^7*sin(c+d*x)^5*(a+a*sin(c+d*x)),x,8,-1/8*a*cos(c+d*x)^8/d+1/5*a*cos(c+d*x)^10/d-1/12*a*cos(c+d*x)^12/d+1/7*a*sin(c+d*x)^7/d-1/3*a*sin(c+d*x)^9/d+3/11*a*sin(c+d*x)^11/d-1/13*a*sin(c+d*x)^13/d],
[cos(c+d*x)^7*sin(c+d*x)^4*(a+a*sin(c+d*x)),x,8,-1/8*a*cos(c+d*x)^8/d+1/5*a*cos(c+d*x)^10/d-1/12*a*cos(c+d*x)^12/d+1/5*a*sin(c+d*x)^5/d-3/7*a*sin(c+d*x)^7/d+1/3*a*sin(c+d*x)^9/d-1/11*a*sin(c+d*x)^11/d],
[cos(c+d*x)^7*sin(c+d*x)^3*(a+a*sin(c+d*x)),x,7,-1/8*a*cos(c+d*x)^8/d+1/10*a*cos(c+d*x)^10/d+1/5*a*sin(c+d*x)^5/d-3/7*a*sin(c+d*x)^7/d+1/3*a*sin(c+d*x)^9/d-1/11*a*sin(c+d*x)^11/d],
[cos(c+d*x)^7*sin(c+d*x)^2*(a+a*sin(c+d*x)),x,7,-1/8*a*cos(c+d*x)^8/d+1/10*a*cos(c+d*x)^10/d+1/3*a*sin(c+d*x)^3/d-3/5*a*sin(c+d*x)^5/d+3/7*a*sin(c+d*x)^7/d-1/9*a*sin(c+d*x)^9/d],
[cos(c+d*x)^7*sin(c+d*x)*(a+a*sin(c+d*x)),x,6,-1/8*a*cos(c+d*x)^8/d+1/3*a*sin(c+d*x)^3/d-3/5*a*sin(c+d*x)^5/d+3/7*a*sin(c+d*x)^7/d-1/9*a*sin(c+d*x)^9/d],
[cos(c+d*x)^7*csc(c+d*x)*(a+a*sin(c+d*x)),x,4,a*log(sin(c+d*x))/d+a*sin(c+d*x)/d-3/2*a*sin(c+d*x)^2/d-a*sin(c+d*x)^3/d+3/4*a*sin(c+d*x)^4/d+3/5*a*sin(c+d*x)^5/d-1/6*a*sin(c+d*x)^6/d-1/7*a*sin(c+d*x)^7/d],
[cos(c+d*x)^7*csc(c+d*x)^2*(a+a*sin(c+d*x)),x,4,-a*csc(c+d*x)/d+a*log(sin(c+d*x))/d-3*a*sin(c+d*x)/d-3/2*a*sin(c+d*x)^2/d+a*sin(c+d*x)^3/d+3/4*a*sin(c+d*x)^4/d-1/5*a*sin(c+d*x)^5/d-1/6*a*sin(c+d*x)^6/d],
[cos(c+d*x)^7*csc(c+d*x)^3*(a+a*sin(c+d*x)),x,4,-a*csc(c+d*x)/d-1/2*a*csc(c+d*x)^2/d-3*a*log(sin(c+d*x))/d-3*a*sin(c+d*x)/d+3/2*a*sin(c+d*x)^2/d+a*sin(c+d*x)^3/d-1/4*a*sin(c+d*x)^4/d-1/5*a*sin(c+d*x)^5/d],
[cos(c+d*x)^7*csc(c+d*x)^4*(a+a*sin(c+d*x)),x,4,3*a*csc(c+d*x)/d-1/2*a*csc(c+d*x)^2/d-1/3*a*csc(c+d*x)^3/d-3*a*log(sin(c+d*x))/d+3*a*sin(c+d*x)/d+3/2*a*sin(c+d*x)^2/d-1/3*a*sin(c+d*x)^3/d-1/4*a*sin(c+d*x)^4/d],
[cos(c+d*x)^7*csc(c+d*x)^5*(a+a*sin(c+d*x)),x,4,3*a*csc(c+d*x)/d+3/2*a*csc(c+d*x)^2/d-1/3*a*csc(c+d*x)^3/d-1/4*a*csc(c+d*x)^4/d+3*a*log(sin(c+d*x))/d+3*a*sin(c+d*x)/d-1/2*a*sin(c+d*x)^2/d-1/3*a*sin(c+d*x)^3/d],
[cos(c+d*x)^7*csc(c+d*x)^6*(a+a*sin(c+d*x)),x,4,-3*a*csc(c+d*x)/d+3/2*a*csc(c+d*x)^2/d+a*csc(c+d*x)^3/d-1/4*a*csc(c+d*x)^4/d-1/5*a*csc(c+d*x)^5/d+3*a*log(sin(c+d*x))/d-a*sin(c+d*x)/d-1/2*a*sin(c+d*x)^2/d],
[cos(c+d*x)^7*csc(c+d*x)^7*(a+a*sin(c+d*x)),x,3,-3*a*csc(c+d*x)/d-3/2*a*csc(c+d*x)^2/d+a*csc(c+d*x)^3/d+3/4*a*csc(c+d*x)^4/d-1/5*a*csc(c+d*x)^5/d-1/6*a*csc(c+d*x)^6/d-a*log(sin(c+d*x))/d-a*sin(c+d*x)/d],
[cos(c+d*x)^7*csc(c+d*x)^8*(a+a*sin(c+d*x)),x,4,a*csc(c+d*x)/d-3/2*a*csc(c+d*x)^2/d-a*csc(c+d*x)^3/d+3/4*a*csc(c+d*x)^4/d+3/5*a*csc(c+d*x)^5/d-1/6*a*csc(c+d*x)^6/d-1/7*a*csc(c+d*x)^7/d-a*log(sin(c+d*x))/d],
[cos(c+d*x)^7*csc(c+d*x)^9*(a+a*sin(c+d*x)),x,6,-1/8*a*cot(c+d*x)^8/d+a*csc(c+d*x)/d-a*csc(c+d*x)^3/d+3/5*a*csc(c+d*x)^5/d-1/7*a*csc(c+d*x)^7/d],
[cos(c+d*x)^7*csc(c+d*x)^10*(a+a*sin(c+d*x)),x,6,-1/8*a*cot(c+d*x)^8/d+1/3*a*csc(c+d*x)^3/d-3/5*a*csc(c+d*x)^5/d+3/7*a*csc(c+d*x)^7/d-1/9*a*csc(c+d*x)^9/d],
[cos(c+d*x)^7*csc(c+d*x)^11*(a+a*sin(c+d*x)),x,7,-1/8*a*cot(c+d*x)^8/d-1/10*a*cot(c+d*x)^10/d+1/3*a*csc(c+d*x)^3/d-3/5*a*csc(c+d*x)^5/d+3/7*a*csc(c+d*x)^7/d-1/9*a*csc(c+d*x)^9/d],
[cos(c+d*x)^7*csc(c+d*x)^12*(a+a*sin(c+d*x)),x,7,-1/8*a*cot(c+d*x)^8/d-1/10*a*cot(c+d*x)^10/d+1/5*a*csc(c+d*x)^5/d-3/7*a*csc(c+d*x)^7/d+1/3*a*csc(c+d*x)^9/d-1/11*a*csc(c+d*x)^11/d],
[cos(c+d*x)^7*csc(c+d*x)^13*(a+a*sin(c+d*x)),x,8,-1/8*a*cot(c+d*x)^8/d-1/5*a*cot(c+d*x)^10/d-1/12*a*cot(c+d*x)^12/d+1/5*a*csc(c+d*x)^5/d-3/7*a*csc(c+d*x)^7/d+1/3*a*csc(c+d*x)^9/d-1/11*a*csc(c+d*x)^11/d],
[cos(c+d*x)^7*csc(c+d*x)^14*(a+a*sin(c+d*x)),x,8,-1/8*a*cot(c+d*x)^8/d-1/5*a*cot(c+d*x)^10/d-1/12*a*cot(c+d*x)^12/d+1/7*a*csc(c+d*x)^7/d-1/3*a*csc(c+d*x)^9/d+3/11*a*csc(c+d*x)^11/d-1/13*a*csc(c+d*x)^13/d],
[cos(c+d*x)^7*csc(c+d*x)^15*(a+a*sin(c+d*x)),x,4,1/7*a*csc(c+d*x)^7/d+1/8*a*csc(c+d*x)^8/d-1/3*a*csc(c+d*x)^9/d-3/10*a*csc(c+d*x)^10/d+3/11*a*csc(c+d*x)^11/d+1/4*a*csc(c+d*x)^12/d-1/13*a*csc(c+d*x)^13/d-1/14*a*csc(c+d*x)^14/d],

# m<0
[cos(c+d*x)^7*sin(c+d*x)^6/(a+a*sin(c+d*x)),x,4,1/7*sin(c+d*x)^7/(a*d)-1/8*sin(c+d*x)^8/(a*d)-2/9*sin(c+d*x)^9/(a*d)+1/5*sin(c+d*x)^10/(a*d)+1/11*sin(c+d*x)^11/(a*d)-1/12*sin(c+d*x)^12/(a*d)],
[cos(c+d*x)^7*sin(c+d*x)^5/(a+a*sin(c+d*x)),x,4,1/6*sin(c+d*x)^6/(a*d)-1/7*sin(c+d*x)^7/(a*d)-1/4*sin(c+d*x)^8/(a*d)+2/9*sin(c+d*x)^9/(a*d)+1/10*sin(c+d*x)^10/(a*d)-1/11*sin(c+d*x)^11/(a*d)],
[cos(c+d*x)^7*sin(c+d*x)^4/(a+a*sin(c+d*x)),x,4,1/5*sin(c+d*x)^5/(a*d)-1/6*sin(c+d*x)^6/(a*d)-2/7*sin(c+d*x)^7/(a*d)+1/4*sin(c+d*x)^8/(a*d)+1/9*sin(c+d*x)^9/(a*d)-1/10*sin(c+d*x)^10/(a*d)],
[cos(c+d*x)^7*sin(c+d*x)^3/(a+a*sin(c+d*x)),x,7,-1/6*cos(c+d*x)^6/(a*d)+1/8*cos(c+d*x)^8/(a*d)-1/5*sin(c+d*x)^5/(a*d)+2/7*sin(c+d*x)^7/(a*d)-1/9*sin(c+d*x)^9/(a*d)],
[cos(c+d*x)^7*sin(c+d*x)^2/(a+a*sin(c+d*x)),x,7,1/6*cos(c+d*x)^6/(a*d)-1/8*cos(c+d*x)^8/(a*d)+1/3*sin(c+d*x)^3/(a*d)-2/5*sin(c+d*x)^5/(a*d)+1/7*sin(c+d*x)^7/(a*d)],
[cos(c+d*x)^7*sin(c+d*x)/(a+a*sin(c+d*x)),x,6,-1/6*cos(c+d*x)^6/(a*d)-1/3*sin(c+d*x)^3/(a*d)+2/5*sin(c+d*x)^5/(a*d)-1/7*sin(c+d*x)^7/(a*d)],
[cos(c+d*x)^7/(a+a*sin(c+d*x)),x,3,-(a-a*sin(c+d*x))^4/(a^5*d)+4/5*(a-a*sin(c+d*x))^5/(a^6*d)-1/6*(a-a*sin(c+d*x))^6/(a^7*d)],
[cos(c+d*x)^7*csc(c+d*x)/(a+a*sin(c+d*x)),x,4,log(sin(c+d*x))/(a*d)-sin(c+d*x)/(a*d)-sin(c+d*x)^2/(a*d)+2/3*sin(c+d*x)^3/(a*d)+1/4*sin(c+d*x)^4/(a*d)-1/5*sin(c+d*x)^5/(a*d)],
[cos(c+d*x)^7*csc(c+d*x)^2/(a+a*sin(c+d*x)),x,4,-csc(c+d*x)/(a*d)-log(sin(c+d*x))/(a*d)-2*sin(c+d*x)/(a*d)+sin(c+d*x)^2/(a*d)+1/3*sin(c+d*x)^3/(a*d)-1/4*sin(c+d*x)^4/(a*d)],
[cos(c+d*x)^7*csc(c+d*x)^3/(a+a*sin(c+d*x)),x,4,csc(c+d*x)/(a*d)-1/2*csc(c+d*x)^2/(a*d)-2*log(sin(c+d*x))/(a*d)+2*sin(c+d*x)/(a*d)+1/2*sin(c+d*x)^2/(a*d)-1/3*sin(c+d*x)^3/(a*d)],
[cos(c+d*x)^7*csc(c+d*x)^4/(a+a*sin(c+d*x)),x,4,2*csc(c+d*x)/(a*d)+1/2*csc(c+d*x)^2/(a*d)-1/3*csc(c+d*x)^3/(a*d)+2*log(sin(c+d*x))/(a*d)+sin(c+d*x)/(a*d)-1/2*sin(c+d*x)^2/(a*d)],
[cos(c+d*x)^7*csc(c+d*x)^5/(a+a*sin(c+d*x)),x,4,-2*csc(c+d*x)/(a*d)+csc(c+d*x)^2/(a*d)+1/3*csc(c+d*x)^3/(a*d)-1/4*csc(c+d*x)^4/(a*d)+log(sin(c+d*x))/(a*d)-sin(c+d*x)/(a*d)],
[cos(c+d*x)^7*csc(c+d*x)^6/(a+a*sin(c+d*x)),x,4,-csc(c+d*x)/(a*d)-csc(c+d*x)^2/(a*d)+2/3*csc(c+d*x)^3/(a*d)+1/4*csc(c+d*x)^4/(a*d)-1/5*csc(c+d*x)^5/(a*d)-log(sin(c+d*x))/(a*d)],
[cos(c+d*x)^7*csc(c+d*x)^7/(a+a*sin(c+d*x)),x,6,-1/6*cot(c+d*x)^6/(a*d)+csc(c+d*x)/(a*d)-2/3*csc(c+d*x)^3/(a*d)+1/5*csc(c+d*x)^5/(a*d)],
[cos(c+d*x)^7*csc(c+d*x)^8/(a+a*sin(c+d*x)),x,6,1/6*cot(c+d*x)^6/(a*d)-1/3*csc(c+d*x)^3/(a*d)+2/5*csc(c+d*x)^5/(a*d)-1/7*csc(c+d*x)^7/(a*d)],
[cos(c+d*x)^7*csc(c+d*x)^9/(a+a*sin(c+d*x)),x,7,-1/6*cot(c+d*x)^6/(a*d)-1/8*cot(c+d*x)^8/(a*d)+1/3*csc(c+d*x)^3/(a*d)-2/5*csc(c+d*x)^5/(a*d)+1/7*csc(c+d*x)^7/(a*d)],
[cos(c+d*x)^7*csc(c+d*x)^10/(a+a*sin(c+d*x)),x,7,1/6*cot(c+d*x)^6/(a*d)+1/8*cot(c+d*x)^8/(a*d)-1/5*csc(c+d*x)^5/(a*d)+2/7*csc(c+d*x)^7/(a*d)-1/9*csc(c+d*x)^9/(a*d)],
[cos(c+d*x)^7*csc(c+d*x)^11/(a+a*sin(c+d*x)),x,4,1/5*csc(c+d*x)^5/(a*d)-1/6*csc(c+d*x)^6/(a*d)-2/7*csc(c+d*x)^7/(a*d)+1/4*csc(c+d*x)^8/(a*d)+1/9*csc(c+d*x)^9/(a*d)-1/10*csc(c+d*x)^10/(a*d)],
[cos(c+d*x)^7*csc(c+d*x)^12/(a+a*sin(c+d*x)),x,4,1/6*csc(c+d*x)^6/(a*d)-1/7*csc(c+d*x)^7/(a*d)-1/4*csc(c+d*x)^8/(a*d)+2/9*csc(c+d*x)^9/(a*d)+1/10*csc(c+d*x)^10/(a*d)-1/11*csc(c+d*x)^11/(a*d)],
[cos(c+d*x)^7*csc(c+d*x)^13/(a+a*sin(c+d*x)),x,4,1/7*csc(c+d*x)^7/(a*d)-1/8*csc(c+d*x)^8/(a*d)-2/9*csc(c+d*x)^9/(a*d)+1/5*csc(c+d*x)^10/(a*d)+1/11*csc(c+d*x)^11/(a*d)-1/12*csc(c+d*x)^12/(a*d)],

# Integrands of the form Cos[e+f x]^7 Sin[e+f x]^n (a+a Sin[e+f x])^(m/2)

# Integrands of the form Cos[e+f x]^7 Sin[e+f x]^n (a+a Sin[e+f x])^m win n symbolic
[cos(c+d*x)^7*sin(c+d*x)^n*(a+a*sin(c+d*x))^3,x,3,a^3*sin(c+d*x)^(1+n)/(d*(1+n))+3*a^3*sin(c+d*x)^(2+n)/(d*(2+n))-8*a^3*sin(c+d*x)^(4+n)/(d*(4+n))-6*a^3*sin(c+d*x)^(5+n)/(d*(5+n))+6*a^3*sin(c+d*x)^(6+n)/(d*(6+n))+8*a^3*sin(c+d*x)^(7+n)/(d*(7+n))-3*a^3*sin(c+d*x)^(9+n)/(d*(9+n))-a^3*sin(c+d*x)^(10+n)/(d*(10+n))],
[cos(c+d*x)^7*sin(c+d*x)^n*(a+a*sin(c+d*x))^2,x,3,a^2*sin(c+d*x)^(1+n)/(d*(1+n))+2*a^2*sin(c+d*x)^(2+n)/(d*(2+n))-2*a^2*sin(c+d*x)^(3+n)/(d*(3+n))-6*a^2*sin(c+d*x)^(4+n)/(d*(4+n))+6*a^2*sin(c+d*x)^(6+n)/(d*(6+n))+2*a^2*sin(c+d*x)^(7+n)/(d*(7+n))-2*a^2*sin(c+d*x)^(8+n)/(d*(8+n))-a^2*sin(c+d*x)^(9+n)/(d*(9+n))],
[cos(c+d*x)^7*sin(c+d*x)^n*(a+a*sin(c+d*x)),x,3,a*sin(c+d*x)^(1+n)/(d*(1+n))+a*sin(c+d*x)^(2+n)/(d*(2+n))-3*a*sin(c+d*x)^(3+n)/(d*(3+n))-3*a*sin(c+d*x)^(4+n)/(d*(4+n))+3*a*sin(c+d*x)^(5+n)/(d*(5+n))+3*a*sin(c+d*x)^(6+n)/(d*(6+n))-a*sin(c+d*x)^(7+n)/(d*(7+n))-a*sin(c+d*x)^(8+n)/(d*(8+n))],
[cos(c+d*x)^7*sin(c+d*x)^n/(a+a*sin(c+d*x)),x,3,sin(c+d*x)^(1+n)/(a*d*(1+n))-sin(c+d*x)^(2+n)/(a*d*(2+n))-2*sin(c+d*x)^(3+n)/(a*d*(3+n))+2*sin(c+d*x)^(4+n)/(a*d*(4+n))+sin(c+d*x)^(5+n)/(a*d*(5+n))-sin(c+d*x)^(6+n)/(a*d*(6+n))],
[cos(c+d*x)^7*sin(c+d*x)^n/(a+a*sin(c+d*x))^2,x,3,sin(c+d*x)^(1+n)/(a^2*d*(1+n))-2*sin(c+d*x)^(2+n)/(a^2*d*(2+n))+2*sin(c+d*x)^(4+n)/(a^2*d*(4+n))-sin(c+d*x)^(5+n)/(a^2*d*(5+n))],
[cos(c+d*x)^7*sin(c+d*x)^n/(a+a*sin(c+d*x))^3,x,3,sin(c+d*x)^(1+n)/(a^3*d*(1+n))-3*sin(c+d*x)^(2+n)/(a^3*d*(2+n))+3*sin(c+d*x)^(3+n)/(a^3*d*(3+n))-sin(c+d*x)^(4+n)/(a^3*d*(4+n))],
[cos(c+d*x)^7*sin(c+d*x)^n/(a+a*sin(c+d*x))^4,x,8,-7*sin(c+d*x)^(1+n)/(a^4*d*(1+n))+8*hypergeom([1,1+n],[2+n],-sin(c+d*x))*sin(c+d*x)^(1+n)/(a^4*d*(1+n))+4*sin(c+d*x)^(2+n)/(a^4*d*(2+n))-sin(c+d*x)^(3+n)/(a^4*d*(3+n))],
[cos(c+d*x)^7*sin(c+d*x)^n/(a+a*sin(c+d*x))^5,x,4,-4*(3+2*n)*hypergeom([1,1+n],[2+n],-sin(c+d*x))*sin(c+d*x)^(1+n)/(a^5*d*(1+n))-sin(c+d*x)^(1+n)*(a-a*sin(c+d*x))^2/(d*(2+n)*(a^7+a^7*sin(c+d*x)))+sin(c+d*x)^(1+n)*(a*(27+30*n+8*n^2)+a*(7+2*n)*sin(c+d*x))/(d*(2+3*n+n^2)*(a^6+a^6*sin(c+d*x)))],

# Integrands of the form Cos[e+f x]^8 (d Sin[e+f x])^n (a+a Sin[e+f x])^m

# Integrands of the form Cos[e+f x]^8 Sin[e+f x]^n (a+a Sin[e+f x])^m

# m>0

# m<0
[cos(c+d*x)^8*sin(c+d*x)^5/(a+a*sin(c+d*x)),x,11,-5/1024*x/a-1/7*cos(c+d*x)^7/(a*d)+2/9*cos(c+d*x)^9/(a*d)-1/11*cos(c+d*x)^11/(a*d)-5/1024*cos(c+d*x)*sin(c+d*x)/(a*d)-5/1536*cos(c+d*x)^3*sin(c+d*x)/(a*d)-1/384*cos(c+d*x)^5*sin(c+d*x)/(a*d)+1/64*cos(c+d*x)^7*sin(c+d*x)/(a*d)+1/24*cos(c+d*x)^7*sin(c+d*x)^3/(a*d)+1/12*cos(c+d*x)^7*sin(c+d*x)^5/(a*d)],
[cos(c+d*x)^8*sin(c+d*x)^4/(a+a*sin(c+d*x)),x,10,3/256*x/a+1/7*cos(c+d*x)^7/(a*d)-2/9*cos(c+d*x)^9/(a*d)+1/11*cos(c+d*x)^11/(a*d)+3/256*cos(c+d*x)*sin(c+d*x)/(a*d)+1/128*cos(c+d*x)^3*sin(c+d*x)/(a*d)+1/160*cos(c+d*x)^5*sin(c+d*x)/(a*d)-3/80*cos(c+d*x)^7*sin(c+d*x)/(a*d)-1/10*cos(c+d*x)^7*sin(c+d*x)^3/(a*d)],
[cos(c+d*x)^8*sin(c+d*x)^3/(a+a*sin(c+d*x)),x,10,-3/256*x/a-1/7*cos(c+d*x)^7/(a*d)+1/9*cos(c+d*x)^9/(a*d)-3/256*cos(c+d*x)*sin(c+d*x)/(a*d)-1/128*cos(c+d*x)^3*sin(c+d*x)/(a*d)-1/160*cos(c+d*x)^5*sin(c+d*x)/(a*d)+3/80*cos(c+d*x)^7*sin(c+d*x)/(a*d)+1/10*cos(c+d*x)^7*sin(c+d*x)^3/(a*d)],
[cos(c+d*x)^8*sin(c+d*x)^2/(a+a*sin(c+d*x)),x,9,5/128*x/a+1/7*cos(c+d*x)^7/(a*d)-1/9*cos(c+d*x)^9/(a*d)+5/128*cos(c+d*x)*sin(c+d*x)/(a*d)+5/192*cos(c+d*x)^3*sin(c+d*x)/(a*d)+1/48*cos(c+d*x)^5*sin(c+d*x)/(a*d)-1/8*cos(c+d*x)^7*sin(c+d*x)/(a*d)],
[cos(c+d*x)^8*sin(c+d*x)/(a+a*sin(c+d*x)),x,8,-5/128*x/a-1/7*cos(c+d*x)^7/(a*d)-5/128*cos(c+d*x)*sin(c+d*x)/(a*d)-5/192*cos(c+d*x)^3*sin(c+d*x)/(a*d)-1/48*cos(c+d*x)^5*sin(c+d*x)/(a*d)+1/8*cos(c+d*x)^7*sin(c+d*x)/(a*d)],
[cos(c+d*x)^8*csc(c+d*x)/(a+a*sin(c+d*x)),x,9,-5/16*x/a-arctanh(cos(c+d*x))/(a*d)+cos(c+d*x)/(a*d)+1/3*cos(c+d*x)^3/(a*d)+1/5*cos(c+d*x)^5/(a*d)-5/16*cos(c+d*x)*sin(c+d*x)/(a*d)-5/24*cos(c+d*x)^3*sin(c+d*x)/(a*d)-1/6*cos(c+d*x)^5*sin(c+d*x)/(a*d)],
[cos(c+d*x)^8*csc(c+d*x)^2/(a+a*sin(c+d*x)),x,10,-15/8*x/a+arctanh(cos(c+d*x))/(a*d)-cos(c+d*x)/(a*d)-1/3*cos(c+d*x)^3/(a*d)-1/5*cos(c+d*x)^5/(a*d)-15/8*cot(c+d*x)/(a*d)+5/8*cos(c+d*x)^2*cot(c+d*x)/(a*d)+1/4*cos(c+d*x)^4*cot(c+d*x)/(a*d)],
[cos(c+d*x)^8*csc(c+d*x)^3/(a+a*sin(c+d*x)),x,11,15/8*x/a+5/2*arctanh(cos(c+d*x))/(a*d)-5/2*cos(c+d*x)/(a*d)-5/6*cos(c+d*x)^3/(a*d)+15/8*cot(c+d*x)/(a*d)-5/8*cos(c+d*x)^2*cot(c+d*x)/(a*d)-1/4*cos(c+d*x)^4*cot(c+d*x)/(a*d)-1/2*cos(c+d*x)^3*cot(c+d*x)^2/(a*d)],
[cos(c+d*x)^8*csc(c+d*x)^4/(a+a*sin(c+d*x)),x,11,5/2*x/a-5/2*arctanh(cos(c+d*x))/(a*d)+5/2*cos(c+d*x)/(a*d)+5/6*cos(c+d*x)^3/(a*d)+5/2*cot(c+d*x)/(a*d)+1/2*cos(c+d*x)^3*cot(c+d*x)^2/(a*d)-5/6*cot(c+d*x)^3/(a*d)+1/2*cos(c+d*x)^2*cot(c+d*x)^3/(a*d)],
[cos(c+d*x)^8*csc(c+d*x)^5/(a+a*sin(c+d*x)),x,11,-5/2*x/a-15/8*arctanh(cos(c+d*x))/(a*d)+15/8*cos(c+d*x)/(a*d)-5/2*cot(c+d*x)/(a*d)+5/8*cos(c+d*x)*cot(c+d*x)^2/(a*d)+5/6*cot(c+d*x)^3/(a*d)-1/2*cos(c+d*x)^2*cot(c+d*x)^3/(a*d)-1/4*cos(c+d*x)*cot(c+d*x)^4/(a*d)],
[cos(c+d*x)^8*csc(c+d*x)^6/(a+a*sin(c+d*x)),x,10,-x/a+15/8*arctanh(cos(c+d*x))/(a*d)-15/8*cos(c+d*x)/(a*d)-cot(c+d*x)/(a*d)-5/8*cos(c+d*x)*cot(c+d*x)^2/(a*d)+1/3*cot(c+d*x)^3/(a*d)+1/4*cos(c+d*x)*cot(c+d*x)^4/(a*d)-1/5*cot(c+d*x)^5/(a*d)],
[cos(c+d*x)^8*csc(c+d*x)^7/(a+a*sin(c+d*x)),x,9,x/a+5/16*arctanh(cos(c+d*x))/(a*d)+cot(c+d*x)/(a*d)-1/3*cot(c+d*x)^3/(a*d)+1/5*cot(c+d*x)^5/(a*d)-5/16*cot(c+d*x)*csc(c+d*x)/(a*d)+5/24*cot(c+d*x)^3*csc(c+d*x)/(a*d)-1/6*cot(c+d*x)^5*csc(c+d*x)/(a*d)],
[cos(c+d*x)^8*csc(c+d*x)^8/(a+a*sin(c+d*x)),x,7,-5/16*arctanh(cos(c+d*x))/(a*d)-1/7*cot(c+d*x)^7/(a*d)+5/16*cot(c+d*x)*csc(c+d*x)/(a*d)-5/24*cot(c+d*x)^3*csc(c+d*x)/(a*d)+1/6*cot(c+d*x)^5*csc(c+d*x)/(a*d)],
[cos(c+d*x)^8*csc(c+d*x)^9/(a+a*sin(c+d*x)),x,8,5/128*arctanh(cos(c+d*x))/(a*d)+1/7*cot(c+d*x)^7/(a*d)+5/128*cot(c+d*x)*csc(c+d*x)/(a*d)-5/64*cot(c+d*x)*csc(c+d*x)^3/(a*d)+5/48*cot(c+d*x)^3*csc(c+d*x)^3/(a*d)-1/8*cot(c+d*x)^5*csc(c+d*x)^3/(a*d)],
[cos(c+d*x)^8*csc(c+d*x)^10/(a+a*sin(c+d*x)),x,9,-5/128*arctanh(cos(c+d*x))/(a*d)-1/7*cot(c+d*x)^7/(a*d)-1/9*cot(c+d*x)^9/(a*d)-5/128*cot(c+d*x)*csc(c+d*x)/(a*d)+5/64*cot(c+d*x)*csc(c+d*x)^3/(a*d)-5/48*cot(c+d*x)^3*csc(c+d*x)^3/(a*d)+1/8*cot(c+d*x)^5*csc(c+d*x)^3/(a*d)],
[cos(c+d*x)^8*csc(c+d*x)^11/(a+a*sin(c+d*x)),x,10,3/256*arctanh(cos(c+d*x))/(a*d)+1/7*cot(c+d*x)^7/(a*d)+1/9*cot(c+d*x)^9/(a*d)+3/256*cot(c+d*x)*csc(c+d*x)/(a*d)+1/128*cot(c+d*x)*csc(c+d*x)^3/(a*d)-1/32*cot(c+d*x)*csc(c+d*x)^5/(a*d)+1/16*cot(c+d*x)^3*csc(c+d*x)^5/(a*d)-1/10*cot(c+d*x)^5*csc(c+d*x)^5/(a*d)],
[cos(c+d*x)^8*csc(c+d*x)^12/(a+a*sin(c+d*x)),x,10,-3/256*arctanh(cos(c+d*x))/(a*d)-1/7*cot(c+d*x)^7/(a*d)-2/9*cot(c+d*x)^9/(a*d)-1/11*cot(c+d*x)^11/(a*d)-3/256*cot(c+d*x)*csc(c+d*x)/(a*d)-1/128*cot(c+d*x)*csc(c+d*x)^3/(a*d)+1/32*cot(c+d*x)*csc(c+d*x)^5/(a*d)-1/16*cot(c+d*x)^3*csc(c+d*x)^5/(a*d)+1/10*cot(c+d*x)^5*csc(c+d*x)^5/(a*d)],
[cos(c+d*x)^8*sin(c+d*x)^5/(a+a*sin(c+d*x))^2,x,15,-3/128*x/a^2-2/5*cos(c+d*x)^5/(a^2*d)+5/7*cos(c+d*x)^7/(a^2*d)-4/9*cos(c+d*x)^9/(a^2*d)+1/11*cos(c+d*x)^11/(a^2*d)-3/128*cos(c+d*x)*sin(c+d*x)/(a^2*d)-1/64*cos(c+d*x)^3*sin(c+d*x)/(a^2*d)+1/16*cos(c+d*x)^5*sin(c+d*x)/(a^2*d)+1/8*cos(c+d*x)^5*sin(c+d*x)^3/(a^2*d)+1/5*cos(c+d*x)^5*sin(c+d*x)^5/(a^2*d)],
[cos(c+d*x)^8*sin(c+d*x)^4/(a+a*sin(c+d*x))^2,x,17,9/256*x/a^2+2/5*cos(c+d*x)^5/(a^2*d)-4/7*cos(c+d*x)^7/(a^2*d)+2/9*cos(c+d*x)^9/(a^2*d)+9/256*cos(c+d*x)*sin(c+d*x)/(a^2*d)+3/128*cos(c+d*x)^3*sin(c+d*x)/(a^2*d)-3/32*cos(c+d*x)^5*sin(c+d*x)/(a^2*d)-3/16*cos(c+d*x)^5*sin(c+d*x)^3/(a^2*d)-1/10*cos(c+d*x)^5*sin(c+d*x)^5/(a^2*d)],
[cos(c+d*x)^8*sin(c+d*x)^3/(a+a*sin(c+d*x))^2,x,14,-3/64*x/a^2-2/5*cos(c+d*x)^5/(a^2*d)+3/7*cos(c+d*x)^7/(a^2*d)-1/9*cos(c+d*x)^9/(a^2*d)-3/64*cos(c+d*x)*sin(c+d*x)/(a^2*d)-1/32*cos(c+d*x)^3*sin(c+d*x)/(a^2*d)+1/8*cos(c+d*x)^5*sin(c+d*x)/(a^2*d)+1/4*cos(c+d*x)^5*sin(c+d*x)^3/(a^2*d)],
[cos(c+d*x)^8*sin(c+d*x)^2/(a+a*sin(c+d*x))^2,x,15,11/128*x/a^2+2/5*cos(c+d*x)^5/(a^2*d)-2/7*cos(c+d*x)^7/(a^2*d)+11/128*cos(c+d*x)*sin(c+d*x)/(a^2*d)+11/192*cos(c+d*x)^3*sin(c+d*x)/(a^2*d)-11/48*cos(c+d*x)^5*sin(c+d*x)/(a^2*d)-1/8*cos(c+d*x)^5*sin(c+d*x)^3/(a^2*d)],
[cos(c+d*x)^8*sin(c+d*x)/(a+a*sin(c+d*x))^2,x,6,-1/8*x/a^2-2/35*cos(c+d*x)^7/(a^2*d)-1/8*cos(c+d*x)*sin(c+d*x)/(a^2*d)-1/12*cos(c+d*x)^3*sin(c+d*x)/(a^2*d)-1/15*cos(c+d*x)^5*sin(c+d*x)/(a^2*d)-1/5*cos(c+d*x)^9/(d*(a+a*sin(c+d*x))^2)],
[cos(c+d*x)^8*csc(c+d*x)/(a+a*sin(c+d*x))^2,x,12,-3/4*x/a^2-arctanh(cos(c+d*x))/(a^2*d)+cos(c+d*x)/(a^2*d)+1/3*cos(c+d*x)^3/(a^2*d)-1/5*cos(c+d*x)^5/(a^2*d)-3/4*cos(c+d*x)*sin(c+d*x)/(a^2*d)-1/2*cos(c+d*x)^3*sin(c+d*x)/(a^2*d)],
[cos(c+d*x)^8*csc(c+d*x)^2/(a+a*sin(c+d*x))^2,x,14,-9/8*x/a^2+2*arctanh(cos(c+d*x))/(a^2*d)-2*cos(c+d*x)/(a^2*d)-2/3*cos(c+d*x)^3/(a^2*d)-cot(c+d*x)/(a^2*d)+1/8*cos(c+d*x)*sin(c+d*x)/(a^2*d)-1/4*cos(c+d*x)*sin(c+d*x)^3/(a^2*d)],
[cos(c+d*x)^8*csc(c+d*x)^3/(a+a*sin(c+d*x))^2,x,13,3*x/a^2+1/2*arctanh(cos(c+d*x))/(a^2*d)+1/3*cos(c+d*x)^3/(a^2*d)+2*cot(c+d*x)/(a^2*d)-1/2*cot(c+d*x)*csc(c+d*x)/(a^2*d)+cos(c+d*x)*sin(c+d*x)/(a^2*d)],
[cos(c+d*x)^8*csc(c+d*x)^4/(a+a*sin(c+d*x))^2,x,13,-1/2*x/a^2-3*arctanh(cos(c+d*x))/(a^2*d)+2*cos(c+d*x)/(a^2*d)-1/3*cot(c+d*x)^3/(a^2*d)+cot(c+d*x)*csc(c+d*x)/(a^2*d)-1/2*cos(c+d*x)*sin(c+d*x)/(a^2*d)],
[cos(c+d*x)^8*csc(c+d*x)^5/(a+a*sin(c+d*x))^2,x,14,-2*x/a^2+9/8*arctanh(cos(c+d*x))/(a^2*d)-cos(c+d*x)/(a^2*d)-2*cot(c+d*x)/(a^2*d)+2/3*cot(c+d*x)^3/(a^2*d)+1/8*cot(c+d*x)*csc(c+d*x)/(a^2*d)-1/4*cot(c+d*x)*csc(c+d*x)^3/(a^2*d)],
[cos(c+d*x)^8*csc(c+d*x)^6/(a+a*sin(c+d*x))^2,x,11,x/a^2+3/4*arctanh(cos(c+d*x))/(a^2*d)+cot(c+d*x)/(a^2*d)-1/3*cot(c+d*x)^3/(a^2*d)-1/5*cot(c+d*x)^5/(a^2*d)-3/4*cot(c+d*x)*csc(c+d*x)/(a^2*d)+1/2*cot(c+d*x)^3*csc(c+d*x)/(a^2*d)],
[cos(c+d*x)^8*csc(c+d*x)^7/(a+a*sin(c+d*x))^2,x,12,-7/16*arctanh(cos(c+d*x))/(a^2*d)+2/5*cot(c+d*x)^5/(a^2*d)+5/16*cot(c+d*x)*csc(c+d*x)/(a^2*d)-1/4*cot(c+d*x)^3*csc(c+d*x)/(a^2*d)+1/8*cot(c+d*x)*csc(c+d*x)^3/(a^2*d)-1/6*cot(c+d*x)^3*csc(c+d*x)^3/(a^2*d)],
[cos(c+d*x)^8*csc(c+d*x)^8/(a+a*sin(c+d*x))^2,x,19,1/8*arctanh(cos(c+d*x))/(a^2*d)-2/5*cot(c+d*x)^5/(a^2*d)-1/7*cot(c+d*x)^7/(a^2*d)+1/8*cot(c+d*x)*csc(c+d*x)/(a^2*d)-7/12*cot(c+d*x)*csc(c+d*x)^3/(a^2*d)+1/3*cot(c+d*x)*csc(c+d*x)^5/(a^2*d)],
[cos(c+d*x)^8*csc(c+d*x)^9/(a+a*sin(c+d*x))^2,x,15,-11/128*arctanh(cos(c+d*x))/(a^2*d)+2/5*cot(c+d*x)^5/(a^2*d)+2/7*cot(c+d*x)^7/(a^2*d)-11/128*cot(c+d*x)*csc(c+d*x)/(a^2*d)+7/64*cot(c+d*x)*csc(c+d*x)^3/(a^2*d)-1/6*cot(c+d*x)^3*csc(c+d*x)^3/(a^2*d)+1/16*cot(c+d*x)*csc(c+d*x)^5/(a^2*d)-1/8*cot(c+d*x)^3*csc(c+d*x)^5/(a^2*d)],
[cos(c+d*x)^8*csc(c+d*x)^10/(a+a*sin(c+d*x))^2,x,14,3/64*arctanh(cos(c+d*x))/(a^2*d)-2/5*cot(c+d*x)^5/(a^2*d)-3/7*cot(c+d*x)^7/(a^2*d)-1/9*cot(c+d*x)^9/(a^2*d)+3/64*cot(c+d*x)*csc(c+d*x)/(a^2*d)+1/32*cot(c+d*x)*csc(c+d*x)^3/(a^2*d)-1/8*cot(c+d*x)*csc(c+d*x)^5/(a^2*d)+1/4*cot(c+d*x)^3*csc(c+d*x)^5/(a^2*d)],
[cos(c+d*x)^8*csc(c+d*x)^11/(a+a*sin(c+d*x))^2,x,17,-9/256*arctanh(cos(c+d*x))/(a^2*d)+2/5*cot(c+d*x)^5/(a^2*d)+4/7*cot(c+d*x)^7/(a^2*d)+2/9*cot(c+d*x)^9/(a^2*d)-9/256*cot(c+d*x)*csc(c+d*x)/(a^2*d)-3/128*cot(c+d*x)*csc(c+d*x)^3/(a^2*d)+9/160*cot(c+d*x)*csc(c+d*x)^5/(a^2*d)-1/8*cot(c+d*x)^3*csc(c+d*x)^5/(a^2*d)+3/80*cot(c+d*x)*csc(c+d*x)^7/(a^2*d)-1/10*cot(c+d*x)^3*csc(c+d*x)^7/(a^2*d)],
[cos(c+d*x)^8*csc(c+d*x)^12/(a+a*sin(c+d*x))^2,x,15,3/128*arctanh(cos(c+d*x))/(a^2*d)-2/5*cot(c+d*x)^5/(a^2*d)-5/7*cot(c+d*x)^7/(a^2*d)-4/9*cot(c+d*x)^9/(a^2*d)-1/11*cot(c+d*x)^11/(a^2*d)+3/128*cot(c+d*x)*csc(c+d*x)/(a^2*d)+1/64*cot(c+d*x)*csc(c+d*x)^3/(a^2*d)+1/80*cot(c+d*x)*csc(c+d*x)^5/(a^2*d)-3/40*cot(c+d*x)*csc(c+d*x)^7/(a^2*d)+1/5*cot(c+d*x)^3*csc(c+d*x)^7/(a^2*d)],
[cos(c+d*x)^8*sin(c+d*x)^3/(a+a*sin(c+d*x))^3,x,18,-29/128*x/a^3-4/3*cos(c+d*x)^3/(a^3*d)+7/5*cos(c+d*x)^5/(a^3*d)-3/7*cos(c+d*x)^7/(a^3*d)-29/128*cos(c+d*x)*sin(c+d*x)/(a^3*d)+29/64*cos(c+d*x)^3*sin(c+d*x)/(a^3*d)+29/48*cos(c+d*x)^3*sin(c+d*x)^3/(a^3*d)+1/8*cos(c+d*x)^3*sin(c+d*x)^5/(a^3*d)],
[cos(c+d*x)^8*sin(c+d*x)^2/(a+a*sin(c+d*x))^3,x,16,5/16*x/a^3+4/3*cos(c+d*x)^3/(a^3*d)-cos(c+d*x)^5/(a^3*d)+1/7*cos(c+d*x)^7/(a^3*d)+5/16*cos(c+d*x)*sin(c+d*x)/(a^3*d)-5/8*cos(c+d*x)^3*sin(c+d*x)/(a^3*d)-1/2*cos(c+d*x)^3*sin(c+d*x)^3/(a^3*d)],
[cos(c+d*x)^8*sin(c+d*x)/(a+a*sin(c+d*x))^3,x,6,-7/16*x/a^3-7/30*cos(c+d*x)^5/(a^3*d)-7/16*cos(c+d*x)*sin(c+d*x)/(a^3*d)-7/24*cos(c+d*x)^3*sin(c+d*x)/(a^3*d)-1/3*cos(c+d*x)^9/(d*(a+a*sin(c+d*x))^3)-1/6*cos(c+d*x)^7/(d*(a^3+a^3*sin(c+d*x)))],
[cos(c+d*x)^8*csc(c+d*x)/(a+a*sin(c+d*x))^3,x,13,-13/8*x/a^3-arctanh(cos(c+d*x))/(a^3*d)+cos(c+d*x)/(a^3*d)-cos(c+d*x)^3/(a^3*d)-13/8*cos(c+d*x)*sin(c+d*x)/(a^3*d)+1/4*cos(c+d*x)^3*sin(c+d*x)/(a^3*d)],
[cos(c+d*x)^8*csc(c+d*x)^2/(a+a*sin(c+d*x))^3,x,11,1/2*x/a^3+3*arctanh(cos(c+d*x))/(a^3*d)-3*cos(c+d*x)/(a^3*d)+1/3*cos(c+d*x)^3/(a^3*d)-cot(c+d*x)/(a^3*d)+3/2*cos(c+d*x)*sin(c+d*x)/(a^3*d)],
[cos(c+d*x)^8*csc(c+d*x)^3/(a+a*sin(c+d*x))^3,x,11,5/2*x/a^3-5/2*arctanh(cos(c+d*x))/(a^3*d)+3*cos(c+d*x)/(a^3*d)+3*cot(c+d*x)/(a^3*d)-1/2*cot(c+d*x)*csc(c+d*x)/(a^3*d)-1/2*cos(c+d*x)*sin(c+d*x)/(a^3*d)],
[cos(c+d*x)^8*csc(c+d*x)^4/(a+a*sin(c+d*x))^3,x,11,-3*x/a^3-1/2*arctanh(cos(c+d*x))/(a^3*d)-cos(c+d*x)/(a^3*d)-3*cot(c+d*x)/(a^3*d)-1/3*cot(c+d*x)^3/(a^3*d)+3/2*cot(c+d*x)*csc(c+d*x)/(a^3*d)],
[cos(c+d*x)^8*csc(c+d*x)^5/(a+a*sin(c+d*x))^3,x,12,x/a^3+13/8*arctanh(cos(c+d*x))/(a^3*d)+cot(c+d*x)/(a^3*d)+cot(c+d*x)^3/(a^3*d)-11/8*cot(c+d*x)*csc(c+d*x)/(a^3*d)-1/4*cot(c+d*x)*csc(c+d*x)^3/(a^3*d)],
[cos(c+d*x)^8*csc(c+d*x)^6/(a+a*sin(c+d*x))^3,x,13,-7/8*arctanh(cos(c+d*x))/(a^3*d)-4/3*cot(c+d*x)^3/(a^3*d)-1/5*cot(c+d*x)^5/(a^3*d)+1/8*cot(c+d*x)*csc(c+d*x)/(a^3*d)+3/4*cot(c+d*x)*csc(c+d*x)^3/(a^3*d)],
[cos(c+d*x)^8*csc(c+d*x)^7/(a+a*sin(c+d*x))^3,x,15,7/16*arctanh(cos(c+d*x))/(a^3*d)+4/3*cot(c+d*x)^3/(a^3*d)+3/5*cot(c+d*x)^5/(a^3*d)+7/16*cot(c+d*x)*csc(c+d*x)/(a^3*d)-17/24*cot(c+d*x)*csc(c+d*x)^3/(a^3*d)-1/6*cot(c+d*x)*csc(c+d*x)^5/(a^3*d)],
[cos(c+d*x)^8*csc(c+d*x)^8/(a+a*sin(c+d*x))^3,x,17,-5/16*arctanh(cos(c+d*x))/(a^3*d)-4/3*cot(c+d*x)^3/(a^3*d)-cot(c+d*x)^5/(a^3*d)-1/7*cot(c+d*x)^7/(a^3*d)-5/16*cot(c+d*x)*csc(c+d*x)/(a^3*d)+1/8*cot(c+d*x)*csc(c+d*x)^3/(a^3*d)+1/2*cot(c+d*x)*csc(c+d*x)^5/(a^3*d)],
[cos(c+d*x)^8*csc(c+d*x)^9/(a+a*sin(c+d*x))^3,x,18,29/128*arctanh(cos(c+d*x))/(a^3*d)+4/3*cot(c+d*x)^3/(a^3*d)+7/5*cot(c+d*x)^5/(a^3*d)+3/7*cot(c+d*x)^7/(a^3*d)+29/128*cot(c+d*x)*csc(c+d*x)/(a^3*d)+29/192*cot(c+d*x)*csc(c+d*x)^3/(a^3*d)-23/48*cot(c+d*x)*csc(c+d*x)^5/(a^3*d)-1/8*cot(c+d*x)*csc(c+d*x)^7/(a^3*d)],

# Integrands of the form (g Sec[e+f x])^p (d Sin[e+f x])^n (a+a Sin[e+f x])^m

# Integrands of the form Sec[e+f x]^1 (d Sin[e+f x])^n (a+a Sin[e+f x])^m

# Integrands of the form Sec[e+f x]^2 (d Sin[e+f x])^n (a+a Sin[e+f x])^m

# Integrands of the form Sec[e+f x]^2 Sin[e+f x]^n (a+a Sin[e+f x])^m

# m>0
[sec(c+d*x)^2*sin(c+d*x)^4*(a+a*sin(c+d*x)),x,8,-3/2*a*x+2*a*cos(c+d*x)/d-1/3*a*cos(c+d*x)^3/d+a*sec(c+d*x)/d+3/2*a*tan(c+d*x)/d-1/2*a*sin(c+d*x)^2*tan(c+d*x)/d],
[sec(c+d*x)^2*sin(c+d*x)^3*(a+a*sin(c+d*x)),x,8,-3/2*a*x+a*cos(c+d*x)/d+a*sec(c+d*x)/d+3/2*a*tan(c+d*x)/d-1/2*a*sin(c+d*x)^2*tan(c+d*x)/d],
[sec(c+d*x)^2*sin(c+d*x)^2*(a+a*sin(c+d*x)),x,5,-a*x+a*cos(c+d*x)/d+a*cos(c+d*x)/(d*(1-sin(c+d*x)))],
[sec(c+d*x)^2*sin(c+d*x)*(a+a*sin(c+d*x)),x,5,-a*x+a*sec(c+d*x)/d+a*tan(c+d*x)/d],
[csc(c+d*x)*sec(c+d*x)^2*(a+a*sin(c+d*x)),x,6,-a*arctanh(cos(c+d*x))/d+a*sec(c+d*x)/d+a*tan(c+d*x)/d],
[csc(c+d*x)^2*sec(c+d*x)^2*(a+a*sin(c+d*x)),x,7,-a*arctanh(cos(c+d*x))/d-a*cot(c+d*x)/d+a*sec(c+d*x)/d+a*tan(c+d*x)/d],
[csc(c+d*x)^3*sec(c+d*x)^2*(a+a*sin(c+d*x)),x,8,-3/2*a*arctanh(cos(c+d*x))/d-a*cot(c+d*x)/d+3/2*a*sec(c+d*x)/d-1/2*a*csc(c+d*x)^2*sec(c+d*x)/d+a*tan(c+d*x)/d],
[csc(c+d*x)^4*sec(c+d*x)^2*(a+a*sin(c+d*x)),x,8,-3/2*a*arctanh(cos(c+d*x))/d-2*a*cot(c+d*x)/d-1/3*a*cot(c+d*x)^3/d+3/2*a*sec(c+d*x)/d-1/2*a*csc(c+d*x)^2*sec(c+d*x)/d+a*tan(c+d*x)/d],
[sec(c+d*x)^2*sin(c+d*x)^3*(a+a*sin(c+d*x))^2,x,12,-3*a^2*x+3*a^2*cos(c+d*x)/d-1/3*a^2*cos(c+d*x)^3/d+2*a^2*sec(c+d*x)/d+3*a^2*tan(c+d*x)/d-a^2*sin(c+d*x)^2*tan(c+d*x)/d],
[sec(c+d*x)^2*sin(c+d*x)^2*(a+a*sin(c+d*x))^2,x,6,-5/2*a^2*x+2*a^2*cos(c+d*x)/d+2*a^2*cos(c+d*x)/(d*(1-sin(c+d*x)))+1/2*a^2*cos(c+d*x)*sin(c+d*x)/d],
[sec(c+d*x)^2*sin(c+d*x)*(a+a*sin(c+d*x))^2,x,3,-2*a^2*x+2*a^2*cos(c+d*x)/d+sec(c+d*x)*(a+a*sin(c+d*x))^2/d],
[csc(c+d*x)*sec(c+d*x)^2*(a+a*sin(c+d*x))^2,x,9,-a^2*arctanh(cos(c+d*x))/d+2*a^2*sec(c+d*x)/d+2*a^2*tan(c+d*x)/d],
[csc(c+d*x)^2*sec(c+d*x)^2*(a+a*sin(c+d*x))^2,x,10,-2*a^2*arctanh(cos(c+d*x))/d-a^2*cot(c+d*x)/d+2*a^2*sec(c+d*x)/d+2*a^2*tan(c+d*x)/d],
[csc(c+d*x)^3*sec(c+d*x)^2*(a+a*sin(c+d*x))^2,x,12,-5/2*a^2*arctanh(cos(c+d*x))/d-2*a^2*cot(c+d*x)/d+5/2*a^2*sec(c+d*x)/d-1/2*a^2*csc(c+d*x)^2*sec(c+d*x)/d+2*a^2*tan(c+d*x)/d],
[sec(c+d*x)^2*sin(c+d*x)^3*(a+a*sin(c+d*x))^3,x,11,-51/8*a^3*x+7*a^3*cos(c+d*x)/d-a^3*cos(c+d*x)^3/d+4*a^3*cos(c+d*x)/(d*(1-sin(c+d*x)))+19/8*a^3*cos(c+d*x)*sin(c+d*x)/d+1/4*a^3*cos(c+d*x)*sin(c+d*x)^3/d],
[sec(c+d*x)^2*sin(c+d*x)^2*(a+a*sin(c+d*x))^3,x,8,-11/2*a^3*x+5*a^3*cos(c+d*x)/d-1/3*a^3*cos(c+d*x)^3/d+4*a^3*cos(c+d*x)/(d*(1-sin(c+d*x)))+3/2*a^3*cos(c+d*x)*sin(c+d*x)/d],
[sec(c+d*x)^2*sin(c+d*x)*(a+a*sin(c+d*x))^3,x,2,-9/2*a^3*x+6*a^3*cos(c+d*x)/d+3/2*a^3*cos(c+d*x)*sin(c+d*x)/d+sec(c+d*x)*(a+a*sin(c+d*x))^3/d],
[csc(c+d*x)*sec(c+d*x)^2*(a+a*sin(c+d*x))^3,x,4,-a^3*x-a^3*arctanh(cos(c+d*x))/d+4*a^3*cos(c+d*x)/(d*(1-sin(c+d*x)))],
[csc(c+d*x)^2*sec(c+d*x)^2*(a+a*sin(c+d*x))^3,x,6,-3*a^3*arctanh(cos(c+d*x))/d-a^3*cot(c+d*x)/d+4*a^3*cos(c+d*x)/(d*(1-sin(c+d*x)))],
[csc(c+d*x)^3*sec(c+d*x)^2*(a+a*sin(c+d*x))^3,x,8,-9/2*a^3*arctanh(cos(c+d*x))/d-3*a^3*cot(c+d*x)/d-1/2*a^3*cot(c+d*x)*csc(c+d*x)/d+4*a^3*cos(c+d*x)/(d*(1-sin(c+d*x)))],
[csc(c+d*x)^4*sec(c+d*x)^2*(a+a*sin(c+d*x))^3,x,10,-11/2*a^3*arctanh(cos(c+d*x))/d-5*a^3*cot(c+d*x)/d-1/3*a^3*cot(c+d*x)^3/d-3/2*a^3*cot(c+d*x)*csc(c+d*x)/d+4*a^3*cos(c+d*x)/(d*(1-sin(c+d*x)))],

# m<0
[sec(c+d*x)^2*sin(c+d*x)^4/(a+a*sin(c+d*x)),x,7,x/a+cos(c+d*x)/(a*d)+2*sec(c+d*x)/(a*d)-1/3*sec(c+d*x)^3/(a*d)-tan(c+d*x)/(a*d)+1/3*tan(c+d*x)^3/(a*d)],
[sec(c+d*x)^2*sin(c+d*x)^3/(a+a*sin(c+d*x)),x,6,-x/a-sec(c+d*x)/(a*d)+1/3*sec(c+d*x)^3/(a*d)+tan(c+d*x)/(a*d)-1/3*tan(c+d*x)^3/(a*d)],
[sec(c+d*x)^2*sin(c+d*x)^2/(a+a*sin(c+d*x)),x,5,sec(c+d*x)/(a*d)-1/3*sec(c+d*x)^3/(a*d)+1/3*tan(c+d*x)^3/(a*d)],
[sec(c+d*x)^2*sin(c+d*x)/(a+a*sin(c+d*x)),x,5,1/3*sec(c+d*x)^3/(a*d)-1/3*tan(c+d*x)^3/(a*d)],
[csc(c+d*x)*sec(c+d*x)^2/(a+a*sin(c+d*x)),x,7,-arctanh(cos(c+d*x))/(a*d)+sec(c+d*x)/(a*d)+1/3*sec(c+d*x)^3/(a*d)-tan(c+d*x)/(a*d)-1/3*tan(c+d*x)^3/(a*d)],
[csc(c+d*x)^2*sec(c+d*x)^2/(a+a*sin(c+d*x)),x,8,arctanh(cos(c+d*x))/(a*d)-cot(c+d*x)/(a*d)-sec(c+d*x)/(a*d)-1/3*sec(c+d*x)^3/(a*d)+2*tan(c+d*x)/(a*d)+1/3*tan(c+d*x)^3/(a*d)],
[sec(c+d*x)^2*sin(c+d*x)^6/(a+a*sin(c+d*x))^2,x,15,-9/2*x/a^2-2*cos(c+d*x)/(a^2*d)-6*sec(c+d*x)/(a^2*d)+2*sec(c+d*x)^3/(a^2*d)-2/5*sec(c+d*x)^5/(a^2*d)+9/2*tan(c+d*x)/(a^2*d)-3/2*tan(c+d*x)^3/(a^2*d)+9/10*tan(c+d*x)^5/(a^2*d)-1/2*sin(c+d*x)^2*tan(c+d*x)^5/(a^2*d)],
[sec(c+d*x)^2*sin(c+d*x)^5/(a+a*sin(c+d*x))^2,x,13,2*x/a^2+cos(c+d*x)/(a^2*d)+4*sec(c+d*x)/(a^2*d)-5/3*sec(c+d*x)^3/(a^2*d)+2/5*sec(c+d*x)^5/(a^2*d)-2*tan(c+d*x)/(a^2*d)+2/3*tan(c+d*x)^3/(a^2*d)-2/5*tan(c+d*x)^5/(a^2*d)],
[sec(c+d*x)^2*sin(c+d*x)^4/(a+a*sin(c+d*x))^2,x,12,-x/a^2-2*sec(c+d*x)/(a^2*d)+4/3*sec(c+d*x)^3/(a^2*d)-2/5*sec(c+d*x)^5/(a^2*d)+tan(c+d*x)/(a^2*d)-1/3*tan(c+d*x)^3/(a^2*d)+2/5*tan(c+d*x)^5/(a^2*d)],
[sec(c+d*x)^2*sin(c+d*x)^3/(a+a*sin(c+d*x))^2,x,11,sec(c+d*x)/(a^2*d)-sec(c+d*x)^3/(a^2*d)+2/5*sec(c+d*x)^5/(a^2*d)-2/5*tan(c+d*x)^5/(a^2*d)],
[sec(c+d*x)^2*sin(c+d*x)^2/(a+a*sin(c+d*x))^2,x,10,2/3*sec(c+d*x)^3/(a^2*d)-2/5*sec(c+d*x)^5/(a^2*d)+1/3*tan(c+d*x)^3/(a^2*d)+2/5*tan(c+d*x)^5/(a^2*d)],
[sec(c+d*x)^2*sin(c+d*x)/(a+a*sin(c+d*x))^2,x,4,1/5*sec(c+d*x)/(d*(a+a*sin(c+d*x))^2)-2/15*sec(c+d*x)/(d*(a^2+a^2*sin(c+d*x)))+4/15*tan(c+d*x)/(a^2*d)],
[csc(c+d*x)*sec(c+d*x)^2/(a+a*sin(c+d*x))^2,x,11,-arctanh(cos(c+d*x))/(a^2*d)+sec(c+d*x)/(a^2*d)+1/3*sec(c+d*x)^3/(a^2*d)+2/5*sec(c+d*x)^5/(a^2*d)-2*tan(c+d*x)/(a^2*d)-4/3*tan(c+d*x)^3/(a^2*d)-2/5*tan(c+d*x)^5/(a^2*d)],
[csc(c+d*x)^2*sec(c+d*x)^2/(a+a*sin(c+d*x))^2,x,12,2*arctanh(cos(c+d*x))/(a^2*d)-cot(c+d*x)/(a^2*d)-2*sec(c+d*x)/(a^2*d)-2/3*sec(c+d*x)^3/(a^2*d)-2/5*sec(c+d*x)^5/(a^2*d)+4*tan(c+d*x)/(a^2*d)+5/3*tan(c+d*x)^3/(a^2*d)+2/5*tan(c+d*x)^5/(a^2*d)],
[csc(c+d*x)^3*sec(c+d*x)^2/(a+a*sin(c+d*x))^2,x,15,-9/2*arctanh(cos(c+d*x))/(a^2*d)+2*cot(c+d*x)/(a^2*d)+9/2*sec(c+d*x)/(a^2*d)+3/2*sec(c+d*x)^3/(a^2*d)+9/10*sec(c+d*x)^5/(a^2*d)-1/2*csc(c+d*x)^2*sec(c+d*x)^5/(a^2*d)-6*tan(c+d*x)/(a^2*d)-2*tan(c+d*x)^3/(a^2*d)-2/5*tan(c+d*x)^5/(a^2*d)],
[sec(c+d*x)^2*sin(c+d*x)^6/(a+a*sin(c+d*x))^3,x,16,3*x/a^3+cos(c+d*x)/(a^3*d)+7*sec(c+d*x)/(a^3*d)-5*sec(c+d*x)^3/(a^3*d)+13/5*sec(c+d*x)^5/(a^3*d)-4/7*sec(c+d*x)^7/(a^3*d)-3*tan(c+d*x)/(a^3*d)+tan(c+d*x)^3/(a^3*d)-3/5*tan(c+d*x)^5/(a^3*d)+4/7*tan(c+d*x)^7/(a^3*d)],
[sec(c+d*x)^2*sin(c+d*x)^5/(a+a*sin(c+d*x))^3,x,16,-x/a^3-3*sec(c+d*x)/(a^3*d)+10/3*sec(c+d*x)^3/(a^3*d)-11/5*sec(c+d*x)^5/(a^3*d)+4/7*sec(c+d*x)^7/(a^3*d)+tan(c+d*x)/(a^3*d)-1/3*tan(c+d*x)^3/(a^3*d)+1/5*tan(c+d*x)^5/(a^3*d)-4/7*tan(c+d*x)^7/(a^3*d)],
[sec(c+d*x)^2*sin(c+d*x)^4/(a+a*sin(c+d*x))^3,x,14,sec(c+d*x)/(a^3*d)-2*sec(c+d*x)^3/(a^3*d)+9/5*sec(c+d*x)^5/(a^3*d)-4/7*sec(c+d*x)^7/(a^3*d)+1/5*tan(c+d*x)^5/(a^3*d)+4/7*tan(c+d*x)^7/(a^3*d)],
[sec(c+d*x)^2*sin(c+d*x)^3/(a+a*sin(c+d*x))^3,x,14,sec(c+d*x)^3/(a^3*d)-7/5*sec(c+d*x)^5/(a^3*d)+4/7*sec(c+d*x)^7/(a^3*d)-3/5*tan(c+d*x)^5/(a^3*d)-4/7*tan(c+d*x)^7/(a^3*d)],
[sec(c+d*x)^2*sin(c+d*x)^2/(a+a*sin(c+d*x))^3,x,14,-1/3*sec(c+d*x)^3/(a^3*d)+sec(c+d*x)^5/(a^3*d)-4/7*sec(c+d*x)^7/(a^3*d)+1/3*tan(c+d*x)^3/(a^3*d)+tan(c+d*x)^5/(a^3*d)+4/7*tan(c+d*x)^7/(a^3*d)],
[sec(c+d*x)^2*sin(c+d*x)/(a+a*sin(c+d*x))^3,x,5,1/7*sec(c+d*x)/(d*(a+a*sin(c+d*x))^3)-3/35*sec(c+d*x)/(a*d*(a+a*sin(c+d*x))^2)-3/35*sec(c+d*x)/(d*(a^3+a^3*sin(c+d*x)))+6/35*tan(c+d*x)/(a^3*d)],
[csc(c+d*x)*sec(c+d*x)^2/(a+a*sin(c+d*x))^3,x,14,-arctanh(cos(c+d*x))/(a^3*d)+sec(c+d*x)/(a^3*d)+1/3*sec(c+d*x)^3/(a^3*d)+1/5*sec(c+d*x)^5/(a^3*d)+4/7*sec(c+d*x)^7/(a^3*d)-3*tan(c+d*x)/(a^3*d)-10/3*tan(c+d*x)^3/(a^3*d)-11/5*tan(c+d*x)^5/(a^3*d)-4/7*tan(c+d*x)^7/(a^3*d)],
[csc(c+d*x)^2*sec(c+d*x)^2/(a+a*sin(c+d*x))^3,x,14,3*arctanh(cos(c+d*x))/(a^3*d)-cot(c+d*x)/(a^3*d)-3*sec(c+d*x)/(a^3*d)-sec(c+d*x)^3/(a^3*d)-3/5*sec(c+d*x)^5/(a^3*d)-4/7*sec(c+d*x)^7/(a^3*d)+7*tan(c+d*x)/(a^3*d)+5*tan(c+d*x)^3/(a^3*d)+13/5*tan(c+d*x)^5/(a^3*d)+4/7*tan(c+d*x)^7/(a^3*d)],

# Integrands of the form Sec[e+f x]^2 Sin[e+f x]^n (a+a Sin[e+f x])^(m/2)

# m>0
#  {Sec[c + d*x]^2*Sin[c + d*x]^3*Sqrt[a + a*Sin[c + d*x]], x, 7, -((76*a*Cos[c + d*x])/(495*d*Sqrt[a + a*Sin[c + d*x]])) - (38*a*Cos[c + d*x]*Sin[c + d*x]^3)/(693*d*Sqrt[a + a*Sin[c + d*x]]) + (2*a*Cos[c + d*x]*Sin[c + d*x]^4)/(99*d*Sqrt[a + a*Sin[c + d*x]]) + (152*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(3465*d) + (2*Cos[c + d*x]*Sin[c + d*x]^4*Sqrt[a + a*Sin[c + d*x]])/(11*d) - (76*Cos[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/(1155*a*d)}
# {Sec[c + d*x]^2*Sin[c + d*x]^2*Sqrt[a + a*Sin[c + d*x]], x, 6, -((2*a*Cos[c + d*x])/(9*d*Sqrt[a + a*Sin[c + d*x]])) + (2*a*Cos[c + d*x]*Sin[c + d*x]^3)/(63*d*Sqrt[a + a*Sin[c + d*x]]) + (4*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(63*d) + (2*Cos[c + d*x]*Sin[c + d*x]^3*Sqrt[a + a*Sin[c + d*x]])/(9*d) - (2*Cos[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/(21*a*d)}
# {Sec[c + d*x]^2*Sin[c + d*x]^1*Sqrt[a + a*Sin[c + d*x]], x, 5, -((16*a*Cos[c + d*x])/(105*d*Sqrt[a + a*Sin[c + d*x]])) - (4*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(105*d) - (18*Cos[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/(35*a*d) + (2*Cos[c + d*x]*(a + a*Sin[c + d*x])^(5/2))/(7*a^2*d)}
# {Sec[c + d*x]^2*Csc[c + d*x]^1*Sqrt[a + a*Sin[c + d*x]], x, 4, -((2*Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/d) + (2*a*Cos[c + d*x])/(3*d*Sqrt[a + a*Sin[c + d*x]]) + (2*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(3*d)}
# {Sec[c + d*x]^2*Csc[c + d*x]^2*Sqrt[a + a*Sin[c + d*x]], x, 4, -((Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/d) + (3*a*Cos[c + d*x])/(d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/d}
# {Sec[c + d*x]^2*Csc[c + d*x]^3*Sqrt[a + a*Sin[c + d*x]], x, 4, (5*Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(4*d) - (a*Cot[c + d*x])/(4*d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(2*d)}

# {Sec[c + d*x]^2*Csc[c + d*x]^4*Sqrt[a + a*Sin[c + d*x]], x, 5, (3*Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(8*d) + (3*a*Cot[c + d*x])/(8*d*Sqrt[a + a*Sin[c + d*x]]) - (a*Cot[c + d*x]*Csc[c + d*x])/(12*d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x]^2*Sqrt[a + a*Sin[c + d*x]])/(3*d)} 
#  {Sec[c + d*x]^2*Sin[c + d*x]^3*(a + a*Sin[c + d*x])^(3/2), x, 8, -((1724*a^2*Cos[c + d*x])/(6435*d*Sqrt[a + a*Sin[c + d*x]])) - (862*a^2*Cos[c + d*x]*Sin[c + d*x]^3)/(9009*d*Sqrt[a + a*Sin[c + d*x]]) - (38*a^2*Cos[c + d*x]*Sin[c + d*x]^4)/(1287*d*Sqrt[a + a*Sin[c + d*x]]) + (3448*a*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(45045*d) + (6*a*Cos[c + d*x]*Sin[c + d*x]^4*Sqrt[a + a*Sin[c + d*x]])/(143*d) - (1724*Cos[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/(15015*d) + (2*Cos[c + d*x]*Sin[c + d*x]^4*(a + a*Sin[c + d*x])^(3/2))/(13*d)}
# {Sec[c + d*x]^2*Sin[c + d*x]^2*(a + a*Sin[c + d*x])^(3/2), x, 7, -((62*a^2*Cos[c + d*x])/(165*d*Sqrt[a + a*Sin[c + d*x]])) - (10*a^2*Cos[c + d*x]*Sin[c + d*x]^3)/(231*d*Sqrt[a + a*Sin[c + d*x]]) + (124*a*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(1155*d) + (2*a*Cos[c + d*x]*Sin[c + d*x]^3*Sqrt[a + a*Sin[c + d*x]])/(33*d) - (62*Cos[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/(385*d) + (2*Cos[c + d*x]*Sin[c + d*x]^3*(a + a*Sin[c + d*x])^(3/2))/(11*d)}
# {Sec[c + d*x]^2*Sin[c + d*x]^1*(a + a*Sin[c + d*x])^(3/2), x, 6, -((128*a^2*Cos[c + d*x])/(315*d*Sqrt[a + a*Sin[c + d*x]])) - (32*a*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(315*d) - (4*Cos[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/(105*d) - (22*Cos[c + d*x]*(a + a*Sin[c + d*x])^(5/2))/(63*a*d) + (2*Cos[c + d*x]*(a + a*Sin[c + d*x])^(7/2))/(9*a^2*d)}
# {Sec[c + d*x]^2*Csc[c + d*x]^1*(a + a*Sin[c + d*x])^(3/2), x, 5, -((2*a^(3/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/d) - (2*a^2*Cos[c + d*x])/(5*d*Sqrt[a + a*Sin[c + d*x]]) + (2*a*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(5*d) + (2*Cos[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/(5*d)}
# {Sec[c + d*x]^2*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^(3/2), x, 5, -((3*a^(3/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/d) + (11*a^2*Cos[c + d*x])/(3*d*Sqrt[a + a*Sin[c + d*x]]) + (5*a*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(3*d) - (Cot[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/d}
# {Sec[c + d*x]^2*Csc[c + d*x]^3*(a + a*Sin[c + d*x])^(3/2), x, 5, (a^(3/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(4*d) + (13*a^2*Cos[c + d*x])/(4*d*Sqrt[a + a*Sin[c + d*x]]) - (3*a*Cot[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(4*d) - (Cot[c + d*x]*Csc[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/(2*d)}

# {Sec[c + d*x]^2*Csc[c + d*x]^4*(a + a*Sin[c + d*x])^(3/2), x, 5, (13*a^(3/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(8*d) + (5*a^2*Cot[c + d*x])/(24*d*Sqrt[a + a*Sin[c + d*x]]) - (a*Cot[c + d*x]*Csc[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(4*d) - (Cot[c + d*x]*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^(3/2))/(3*d)} 

# m<0
#  {Sec[c + d*x]^2*Sin[c + d*x]^3/Sqrt[a + a*Sin[c + d*x]], x, 6, -((4*Cos[c + d*x])/(45*d*Sqrt[a + a*Sin[c + d*x]])) - (2*Cos[c + d*x]*Sin[c + d*x]^3)/(63*d*Sqrt[a + a*Sin[c + d*x]]) + (2*Cos[c + d*x]*Sin[c + d*x]^4)/(9*d*Sqrt[a + a*Sin[c + d*x]]) + (8*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(315*a*d) - (4*Cos[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/(105*a^2*d)}
# {Sec[c + d*x]^2*Sin[c + d*x]^2/Sqrt[a + a*Sin[c + d*x]], x, 5, -((2*Cos[c + d*x])/(15*d*Sqrt[a + a*Sin[c + d*x]])) + (2*Cos[c + d*x]*Sin[c + d*x]^3)/(7*d*Sqrt[a + a*Sin[c + d*x]]) + (4*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(105*a*d) - (2*Cos[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/(35*a^2*d)}
# {Sec[c + d*x]^2*Sin[c + d*x]^1/Sqrt[a + a*Sin[c + d*x]], x, 4, (4*Cos[c + d*x])/(15*d*Sqrt[a + a*Sin[c + d*x]]) - (14*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(15*a*d) + (2*Cos[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/(5*a^2*d)}
# {Sec[c + d*x]^2*Csc[c + d*x]^1/Sqrt[a + a*Sin[c + d*x]], x, 3, -((2*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(Sqrt[a]*d)) + (2*Cos[c + d*x])/(d*Sqrt[a + a*Sin[c + d*x]])}
# {Sec[c + d*x]^2*Csc[c + d*x]^2/Sqrt[a + a*Sin[c + d*x]], x, 3, ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]]/(Sqrt[a]*d) - Cot[c + d*x]/(d*Sqrt[a + a*Sin[c + d*x]])}
# {Sec[c + d*x]^2*Csc[c + d*x]^3/Sqrt[a + a*Sin[c + d*x]], x, 4, ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]]/(4*Sqrt[a]*d) + Cot[c + d*x]/(4*d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x])/(2*d*Sqrt[a + a*Sin[c + d*x]])}

# {Sec[c + d*x]^2*Csc[c + d*x]^4/Sqrt[a + a*Sin[c + d*x]], x, 5, ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]]/(8*Sqrt[a]*d) + Cot[c + d*x]/(8*d*Sqrt[a + a*Sin[c + d*x]]) + (Cot[c + d*x]*Csc[c + d*x])/(12*d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x]^2)/(3*d*Sqrt[a + a*Sin[c + d*x]])} 
#  {Sec[c + d*x]^2*Sin[c + d*x]^3/(a + a*Sin[c + d*x])^(3/2), x, 6, (2*Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(a^(3/2)*d) - (344*Cos[c + d*x])/(105*a*d*Sqrt[a + a*Sin[c + d*x]]) - (16*Cos[c + d*x]*Sin[c + d*x]^2)/(35*a*d*Sqrt[a + a*Sin[c + d*x]]) + (2*Cos[c + d*x]*Sin[c + d*x]^3)/(7*a*d*Sqrt[a + a*Sin[c + d*x]]) + (76*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(105*a^2*d)}
# {Sec[c + d*x]^2*Sin[c + d*x]^2/(a + a*Sin[c + d*x])^(3/2), x, 5, -((2*Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(a^(3/2)*d)) + (16*Cos[c + d*x])/(5*a*d*Sqrt[a + a*Sin[c + d*x]]) + (2*Cos[c + d*x]*Sin[c + d*x]^2)/(5*a*d*Sqrt[a + a*Sin[c + d*x]]) - (4*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(5*a^2*d)}
# {Sec[c + d*x]^2*Sin[c + d*x]^1/(a + a*Sin[c + d*x])^(3/2), x, 4, (2*Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(a^(3/2)*d) - (10*Cos[c + d*x])/(3*a*d*Sqrt[a + a*Sin[c + d*x]]) + (2*Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(3*a^2*d)}
# {Sec[c + d*x]^2*Csc[c + d*x]^1/(a + a*Sin[c + d*x])^(3/2), x, 4, -((2*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(a^(3/2)*d)) + (2*Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(a^(3/2)*d)}
# {Sec[c + d*x]^2*Csc[c + d*x]^2/(a + a*Sin[c + d*x])^(3/2), x, 5, (3*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(a^(3/2)*d) - (2*Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(a^(3/2)*d) - Cot[c + d*x]/(a*d*Sqrt[a + a*Sin[c + d*x]])}
# {Sec[c + d*x]^2*Csc[c + d*x]^3/(a + a*Sin[c + d*x])^(3/2), x, 6, -((11*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(4*a^(3/2)*d)) + (2*Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(a^(3/2)*d) + (5*Cot[c + d*x])/(4*a*d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x])/(2*a*d*Sqrt[a + a*Sin[c + d*x]])}

# {Sec[c + d*x]^2*Csc[c + d*x]^4/(a + a*Sin[c + d*x])^(3/2), x, 7, (23*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(8*a^(3/2)*d) - (2*Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(a^(3/2)*d) - (9*Cot[c + d*x])/(8*a*d*Sqrt[a + a*Sin[c + d*x]]) + (7*Cot[c + d*x]*Csc[c + d*x])/(12*a*d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x]^2)/(3*a*d*Sqrt[a + a*Sin[c + d*x]])} 

# Integrands of the form Sec[e+f x]^3 (d Sin[e+f x])^n (a+a Sin[e+f x])^m

# Integrands of the form Sec[e+f x]^4 (d Sin[e+f x])^n (a+a Sin[e+f x])^m

# Integrands of the form Sec[e+f x]^4 Sin[e+f x]^n (a+a Sin[e+f x])^m

# m>0
[sec(c+d*x)^4*sin(c+d*x)^6*(a+a*sin(c+d*x)),x,9,5/2*a*x-3*a*cos(c+d*x)/d+1/3*a*cos(c+d*x)^3/d-3*a*sec(c+d*x)/d+1/3*a*sec(c+d*x)^3/d-5/2*a*tan(c+d*x)/d+5/6*a*tan(c+d*x)^3/d-1/2*a*sin(c+d*x)^2*tan(c+d*x)^3/d],
[sec(c+d*x)^4*sin(c+d*x)^5*(a+a*sin(c+d*x)),x,9,5/2*a*x-a*cos(c+d*x)/d-2*a*sec(c+d*x)/d+1/3*a*sec(c+d*x)^3/d-5/2*a*tan(c+d*x)/d+5/6*a*tan(c+d*x)^3/d-1/2*a*sin(c+d*x)^2*tan(c+d*x)^3/d],
[sec(c+d*x)^4*sin(c+d*x)^4*(a+a*sin(c+d*x)),x,8,a*x-a*cos(c+d*x)/d-2*a*sec(c+d*x)/d+1/3*a*sec(c+d*x)^3/d-a*tan(c+d*x)/d+1/3*a*tan(c+d*x)^3/d],
[sec(c+d*x)^4*sin(c+d*x)^3*(a+a*sin(c+d*x)),x,6,a*x-a*sec(c+d*x)/d+1/3*a*sec(c+d*x)^3/d-a*tan(c+d*x)/d+1/3*a*tan(c+d*x)^3/d],
[sec(c+d*x)^4*sin(c+d*x)^2*(a+a*sin(c+d*x)),x,5,-a*sec(c+d*x)/d+1/3*a*sec(c+d*x)^3/d+1/3*a*tan(c+d*x)^3/d],
[sec(c+d*x)^4*sin(c+d*x)*(a+a*sin(c+d*x)),x,5,1/3*a*sec(c+d*x)^3/d+1/3*a*tan(c+d*x)^3/d],
[csc(c+d*x)*sec(c+d*x)^4*(a+a*sin(c+d*x)),x,7,-a*arctanh(cos(c+d*x))/d+a*sec(c+d*x)/d+1/3*a*sec(c+d*x)^3/d+a*tan(c+d*x)/d+1/3*a*tan(c+d*x)^3/d],
[csc(c+d*x)^2*sec(c+d*x)^4*(a+a*sin(c+d*x)),x,8,-a*arctanh(cos(c+d*x))/d-a*cot(c+d*x)/d+a*sec(c+d*x)/d+1/3*a*sec(c+d*x)^3/d+2*a*tan(c+d*x)/d+1/3*a*tan(c+d*x)^3/d],
[csc(c+d*x)^3*sec(c+d*x)^4*(a+a*sin(c+d*x)),x,9,-5/2*a*arctanh(cos(c+d*x))/d-a*cot(c+d*x)/d+5/2*a*sec(c+d*x)/d+5/6*a*sec(c+d*x)^3/d-1/2*a*csc(c+d*x)^2*sec(c+d*x)^3/d+2*a*tan(c+d*x)/d+1/3*a*tan(c+d*x)^3/d],
[sec(c+d*x)^4*sin(c+d*x)^4*(a+a*sin(c+d*x))^2,x,4,7/2*a^2*x-2*a^2*cos(c+d*x)/d+1/3*a^2*cos(c+d*x)/(d*(1-sin(c+d*x))^2)-11/3*a^2*cos(c+d*x)/(d*(1-sin(c+d*x)))-1/2*a^2*cos(c+d*x)*sin(c+d*x)/d,7/2*a^2*x-16/3*a^2*cos(c+d*x)/d-7/2*a^2*cos(c+d*x)*sin(c+d*x)/d-8/3*a^2*cos(c+d*x)*sin(c+d*x)^2/(d*(1-sin(c+d*x)))+1/3*a^4*cos(c+d*x)*sin(c+d*x)^3/(d*(a-a*sin(c+d*x))^2)],
[sec(c+d*x)^4*sin(c+d*x)^3*(a+a*sin(c+d*x))^2,x,7,2*a^2*x-4/3*a^2*cos(c+d*x)/d-2*a^2*cos(c+d*x)/(d*(1-sin(c+d*x)))+1/3*a^4*cos(c+d*x)*sin(c+d*x)^2/(d*(a-a*sin(c+d*x))^2)],
[sec(c+d*x)^4*sin(c+d*x)^2*(a+a*sin(c+d*x))^2,x,4,a^2*x-5/3*a^2*cos(c+d*x)/(d*(1-sin(c+d*x)))+1/3*a^4*cos(c+d*x)/(d*(a-a*sin(c+d*x))^2)],
[sec(c+d*x)^4*sin(c+d*x)*(a+a*sin(c+d*x))^2,x,4,-2/3*a^2*sec(c+d*x)/d+1/3*sec(c+d*x)^3*(a+a*sin(c+d*x))^2/d-2/3*a^2*tan(c+d*x)/d],
[csc(c+d*x)*sec(c+d*x)^4*(a+a*sin(c+d*x))^2,x,5,-a^2*arctanh(cos(c+d*x))/d+4/3*a^2*cos(c+d*x)/(d*(1-sin(c+d*x)))+1/3*a^4*cos(c+d*x)/(d*(a-a*sin(c+d*x))^2)],
[csc(c+d*x)^2*sec(c+d*x)^4*(a+a*sin(c+d*x))^2,x,7,-2*a^2*arctanh(cos(c+d*x))/d-10/3*a^2*cot(c+d*x)/d+2*a^2*cot(c+d*x)/(d*(1-sin(c+d*x)))+1/3*a^4*cot(c+d*x)/(d*(a-a*sin(c+d*x))^2)],
[csc(c+d*x)^3*sec(c+d*x)^4*(a+a*sin(c+d*x))^2,x,8,-7/2*a^2*arctanh(cos(c+d*x))/d-16/3*a^2*cot(c+d*x)/d-7/2*a^2*cot(c+d*x)*csc(c+d*x)/d+8/3*a^2*cot(c+d*x)*csc(c+d*x)/(d*(1-sin(c+d*x)))+1/3*a^4*cot(c+d*x)*csc(c+d*x)/(d*(a-a*sin(c+d*x))^2)],
[sec(c+d*x)^4*sin(c+d*x)^4*(a+a*sin(c+d*x))^3,x,10,17/2*a^3*x-6*a^3*cos(c+d*x)/d+1/3*a^3*cos(c+d*x)^3/d+2/3*a^3*cos(c+d*x)/(d*(1-sin(c+d*x))^2)-25/3*a^3*cos(c+d*x)/(d*(1-sin(c+d*x)))-3/2*a^3*cos(c+d*x)*sin(c+d*x)/d],
[sec(c+d*x)^4*sin(c+d*x)^3*(a+a*sin(c+d*x))^3,x,8,11/2*a^3*x-3*a^3*cos(c+d*x)/d+2/3*a^3*cos(c+d*x)/(d*(1-sin(c+d*x))^2)-19/3*a^3*cos(c+d*x)/(d*(1-sin(c+d*x)))-1/2*a^3*cos(c+d*x)*sin(c+d*x)/d],
[sec(c+d*x)^4*sin(c+d*x)^2*(a+a*sin(c+d*x))^3,x,5,3*a^3*x-3*a^3*cos(c+d*x)/d-2*a^5*cos(c+d*x)^3/(d*(a-a*sin(c+d*x))^2)+1/3*sec(c+d*x)^3*(a+a*sin(c+d*x))^3/d],
[sec(c+d*x)^4*sin(c+d*x)*(a+a*sin(c+d*x))^3,x,4,a^3*x+1/3*sec(c+d*x)^3*(a+a*sin(c+d*x))^3/d-2*a^5*cos(c+d*x)/(d*(a^2-a^2*sin(c+d*x)))],
[csc(c+d*x)*sec(c+d*x)^4*(a+a*sin(c+d*x))^3,x,6,-a^3*arctanh(cos(c+d*x))/d+2/3*a^3*cos(c+d*x)/(d*(1-sin(c+d*x))^2)+5/3*a^3*cos(c+d*x)/(d*(1-sin(c+d*x)))],
[csc(c+d*x)^2*sec(c+d*x)^4*(a+a*sin(c+d*x))^3,x,8,-3*a^3*arctanh(cos(c+d*x))/d-a^3*cot(c+d*x)/d+2/3*a^3*cos(c+d*x)/(d*(1-sin(c+d*x))^2)+11/3*a^3*cos(c+d*x)/(d*(1-sin(c+d*x)))],
[csc(c+d*x)^3*sec(c+d*x)^4*(a+a*sin(c+d*x))^3,x,10,-11/2*a^3*arctanh(cos(c+d*x))/d-3*a^3*cot(c+d*x)/d-1/2*a^3*cot(c+d*x)*csc(c+d*x)/d+2/3*a^3*cos(c+d*x)/(d*(1-sin(c+d*x))^2)+17/3*a^3*cos(c+d*x)/(d*(1-sin(c+d*x)))],
[csc(c+d*x)^4*sec(c+d*x)^4*(a+a*sin(c+d*x))^3,x,12,-17/2*a^3*arctanh(cos(c+d*x))/d-6*a^3*cot(c+d*x)/d-1/3*a^3*cot(c+d*x)^3/d-3/2*a^3*cot(c+d*x)*csc(c+d*x)/d+2/3*a^3*cos(c+d*x)/(d*(1-sin(c+d*x))^2)+23/3*a^3*cos(c+d*x)/(d*(1-sin(c+d*x)))],
[sec(c+d*x)^4*sin(c+d*x)^4*(a+a*sin(c+d*x))^4,x,13,163/8*a^4*x-16*a^4*cos(c+d*x)/d+4/3*a^4*cos(c+d*x)^3/d+4/3*a^4*cos(c+d*x)/(d*(1-sin(c+d*x))^2)-56/3*a^4*cos(c+d*x)/(d*(1-sin(c+d*x)))-35/8*a^4*cos(c+d*x)*sin(c+d*x)/d-1/4*a^4*cos(c+d*x)*sin(c+d*x)^3/d],
[sec(c+d*x)^4*sin(c+d*x)^2*(a+a*sin(c+d*x))^4,x,8,17/2*a^4*x-4*a^4*cos(c+d*x)/d+4/3*a^4*cos(c+d*x)/(d*(1-sin(c+d*x))^2)-32/3*a^4*cos(c+d*x)/(d*(1-sin(c+d*x)))-1/2*a^4*cos(c+d*x)*sin(c+d*x)/d],

# m<0
[sec(c+d*x)^4*sin(c+d*x)^6/(a+a*sin(c+d*x)),x,8,-x/a-cos(c+d*x)/(a*d)-3*sec(c+d*x)/(a*d)+sec(c+d*x)^3/(a*d)-1/5*sec(c+d*x)^5/(a*d)+tan(c+d*x)/(a*d)-1/3*tan(c+d*x)^3/(a*d)+1/5*tan(c+d*x)^5/(a*d)],
[sec(c+d*x)^4*sin(c+d*x)^5/(a+a*sin(c+d*x)),x,8,x/a+sec(c+d*x)/(a*d)-2/3*sec(c+d*x)^3/(a*d)+1/5*sec(c+d*x)^5/(a*d)-tan(c+d*x)/(a*d)+1/3*tan(c+d*x)^3/(a*d)-1/5*tan(c+d*x)^5/(a*d)],
[sec(c+d*x)^4*sin(c+d*x)^4/(a+a*sin(c+d*x)),x,6,-sec(c+d*x)/(a*d)+2/3*sec(c+d*x)^3/(a*d)-1/5*sec(c+d*x)^5/(a*d)+1/5*tan(c+d*x)^5/(a*d)],
[sec(c+d*x)^4*sin(c+d*x)^3/(a+a*sin(c+d*x)),x,6,-1/3*sec(c+d*x)^3/(a*d)+1/5*sec(c+d*x)^5/(a*d)-1/5*tan(c+d*x)^5/(a*d)],
[sec(c+d*x)^4*sin(c+d*x)^2/(a+a*sin(c+d*x)),x,7,1/3*sec(c+d*x)^3/(a*d)-1/5*sec(c+d*x)^5/(a*d)+1/3*tan(c+d*x)^3/(a*d)+1/5*tan(c+d*x)^5/(a*d)],
[sec(c+d*x)^4*sin(c+d*x)/(a+a*sin(c+d*x)),x,6,1/5*sec(c+d*x)^5/(a*d)-1/3*tan(c+d*x)^3/(a*d)-1/5*tan(c+d*x)^5/(a*d)],
[csc(c+d*x)*sec(c+d*x)^4/(a+a*sin(c+d*x)),x,7,-arctanh(cos(c+d*x))/(a*d)+sec(c+d*x)/(a*d)+1/3*sec(c+d*x)^3/(a*d)+1/5*sec(c+d*x)^5/(a*d)-tan(c+d*x)/(a*d)-2/3*tan(c+d*x)^3/(a*d)-1/5*tan(c+d*x)^5/(a*d)],
[csc(c+d*x)^2*sec(c+d*x)^4/(a+a*sin(c+d*x)),x,8,arctanh(cos(c+d*x))/(a*d)-cot(c+d*x)/(a*d)-sec(c+d*x)/(a*d)-1/3*sec(c+d*x)^3/(a*d)-1/5*sec(c+d*x)^5/(a*d)+3*tan(c+d*x)/(a*d)+tan(c+d*x)^3/(a*d)+1/5*tan(c+d*x)^5/(a*d)],
[sec(c+d*x)^4*sin(c+d*x)^7/(a+a*sin(c+d*x))^2,x,14,-2*x/a^2-cos(c+d*x)/(a^2*d)-5*sec(c+d*x)/(a^2*d)+3*sec(c+d*x)^3/(a^2*d)-7/5*sec(c+d*x)^5/(a^2*d)+2/7*sec(c+d*x)^7/(a^2*d)+2*tan(c+d*x)/(a^2*d)-2/3*tan(c+d*x)^3/(a^2*d)+2/5*tan(c+d*x)^5/(a^2*d)-2/7*tan(c+d*x)^7/(a^2*d)],
[sec(c+d*x)^4*sin(c+d*x)^6/(a+a*sin(c+d*x))^2,x,13,x/a^2+2*sec(c+d*x)/(a^2*d)-2*sec(c+d*x)^3/(a^2*d)+6/5*sec(c+d*x)^5/(a^2*d)-2/7*sec(c+d*x)^7/(a^2*d)-tan(c+d*x)/(a^2*d)+1/3*tan(c+d*x)^3/(a^2*d)-1/5*tan(c+d*x)^5/(a^2*d)+2/7*tan(c+d*x)^7/(a^2*d)],
[sec(c+d*x)^4*sin(c+d*x)^5/(a+a*sin(c+d*x))^2,x,11,-sec(c+d*x)/(a^2*d)+4/3*sec(c+d*x)^3/(a^2*d)-sec(c+d*x)^5/(a^2*d)+2/7*sec(c+d*x)^7/(a^2*d)-2/7*tan(c+d*x)^7/(a^2*d)],
[sec(c+d*x)^4*sin(c+d*x)^4/(a+a*sin(c+d*x))^2,x,10,-2/3*sec(c+d*x)^3/(a^2*d)+4/5*sec(c+d*x)^5/(a^2*d)-2/7*sec(c+d*x)^7/(a^2*d)+1/5*tan(c+d*x)^5/(a^2*d)+2/7*tan(c+d*x)^7/(a^2*d)],
[sec(c+d*x)^4*sin(c+d*x)^3/(a+a*sin(c+d*x))^2,x,12,1/3*sec(c+d*x)^3/(a^2*d)-3/5*sec(c+d*x)^5/(a^2*d)+2/7*sec(c+d*x)^7/(a^2*d)-2/5*tan(c+d*x)^5/(a^2*d)-2/7*tan(c+d*x)^7/(a^2*d)],
[sec(c+d*x)^4*sin(c+d*x)^2/(a+a*sin(c+d*x))^2,x,12,2/5*sec(c+d*x)^5/(a^2*d)-2/7*sec(c+d*x)^7/(a^2*d)+1/3*tan(c+d*x)^3/(a^2*d)+3/5*tan(c+d*x)^5/(a^2*d)+2/7*tan(c+d*x)^7/(a^2*d)],
[sec(c+d*x)^4*sin(c+d*x)/(a+a*sin(c+d*x))^2,x,4,1/7*sec(c+d*x)^3/(d*(a+a*sin(c+d*x))^2)-2/35*sec(c+d*x)^3/(d*(a^2+a^2*sin(c+d*x)))+8/35*tan(c+d*x)/(a^2*d)+8/105*tan(c+d*x)^3/(a^2*d)],
[csc(c+d*x)*sec(c+d*x)^4/(a+a*sin(c+d*x))^2,x,11,-arctanh(cos(c+d*x))/(a^2*d)+sec(c+d*x)/(a^2*d)+1/3*sec(c+d*x)^3/(a^2*d)+1/5*sec(c+d*x)^5/(a^2*d)+2/7*sec(c+d*x)^7/(a^2*d)-2*tan(c+d*x)/(a^2*d)-2*tan(c+d*x)^3/(a^2*d)-6/5*tan(c+d*x)^5/(a^2*d)-2/7*tan(c+d*x)^7/(a^2*d)],
[csc(c+d*x)^2*sec(c+d*x)^4/(a+a*sin(c+d*x))^2,x,12,2*arctanh(cos(c+d*x))/(a^2*d)-cot(c+d*x)/(a^2*d)-2*sec(c+d*x)/(a^2*d)-2/3*sec(c+d*x)^3/(a^2*d)-2/5*sec(c+d*x)^5/(a^2*d)-2/7*sec(c+d*x)^7/(a^2*d)+5*tan(c+d*x)/(a^2*d)+3*tan(c+d*x)^3/(a^2*d)+7/5*tan(c+d*x)^5/(a^2*d)+2/7*tan(c+d*x)^7/(a^2*d)],
[csc(c+d*x)^3*sec(c+d*x)^4/(a+a*sin(c+d*x))^2,x,15,-11/2*arctanh(cos(c+d*x))/(a^2*d)+2*cot(c+d*x)/(a^2*d)+11/2*sec(c+d*x)/(a^2*d)+11/6*sec(c+d*x)^3/(a^2*d)+11/10*sec(c+d*x)^5/(a^2*d)+11/14*sec(c+d*x)^7/(a^2*d)-1/2*csc(c+d*x)^2*sec(c+d*x)^7/(a^2*d)-8*tan(c+d*x)/(a^2*d)-4*tan(c+d*x)^3/(a^2*d)-8/5*tan(c+d*x)^5/(a^2*d)-2/7*tan(c+d*x)^7/(a^2*d)],
[sec(c+d*x)^4*sin(c+d*x)^7/(a+a*sin(c+d*x))^3,x,17,x/a^3+3*sec(c+d*x)/(a^3*d)-13/3*sec(c+d*x)^3/(a^3*d)+21/5*sec(c+d*x)^5/(a^3*d)-15/7*sec(c+d*x)^7/(a^3*d)+4/9*sec(c+d*x)^9/(a^3*d)-tan(c+d*x)/(a^3*d)+1/3*tan(c+d*x)^3/(a^3*d)-1/5*tan(c+d*x)^5/(a^3*d)+1/7*tan(c+d*x)^7/(a^3*d)-4/9*tan(c+d*x)^9/(a^3*d)],
[sec(c+d*x)^4*sin(c+d*x)^6/(a+a*sin(c+d*x))^3,x,14,-sec(c+d*x)/(a^3*d)+7/3*sec(c+d*x)^3/(a^3*d)-3*sec(c+d*x)^5/(a^3*d)+13/7*sec(c+d*x)^7/(a^3*d)-4/9*sec(c+d*x)^9/(a^3*d)+1/7*tan(c+d*x)^7/(a^3*d)+4/9*tan(c+d*x)^9/(a^3*d)],
[sec(c+d*x)^4*sin(c+d*x)^5/(a+a*sin(c+d*x))^3,x,14,-sec(c+d*x)^3/(a^3*d)+2*sec(c+d*x)^5/(a^3*d)-11/7*sec(c+d*x)^7/(a^3*d)+4/9*sec(c+d*x)^9/(a^3*d)-3/7*tan(c+d*x)^7/(a^3*d)-4/9*tan(c+d*x)^9/(a^3*d)],
[sec(c+d*x)^4*sin(c+d*x)^4/(a+a*sin(c+d*x))^3,x,14,1/3*sec(c+d*x)^3/(a^3*d)-6/5*sec(c+d*x)^5/(a^3*d)+9/7*sec(c+d*x)^7/(a^3*d)-4/9*sec(c+d*x)^9/(a^3*d)+1/5*tan(c+d*x)^5/(a^3*d)+5/7*tan(c+d*x)^7/(a^3*d)+4/9*tan(c+d*x)^9/(a^3*d)],
[sec(c+d*x)^4*sin(c+d*x)^3/(a+a*sin(c+d*x))^3,x,15,3/5*sec(c+d*x)^5/(a^3*d)-sec(c+d*x)^7/(a^3*d)+4/9*sec(c+d*x)^9/(a^3*d)-3/5*tan(c+d*x)^5/(a^3*d)-tan(c+d*x)^7/(a^3*d)-4/9*tan(c+d*x)^9/(a^3*d)],
[sec(c+d*x)^4*sin(c+d*x)^2/(a+a*sin(c+d*x))^3,x,15,-1/5*sec(c+d*x)^5/(a^3*d)+5/7*sec(c+d*x)^7/(a^3*d)-4/9*sec(c+d*x)^9/(a^3*d)+1/3*tan(c+d*x)^3/(a^3*d)+6/5*tan(c+d*x)^5/(a^3*d)+9/7*tan(c+d*x)^7/(a^3*d)+4/9*tan(c+d*x)^9/(a^3*d)],
[sec(c+d*x)^4*sin(c+d*x)/(a+a*sin(c+d*x))^3,x,5,1/9*sec(c+d*x)^3/(d*(a+a*sin(c+d*x))^3)-1/21*sec(c+d*x)^3/(a*d*(a+a*sin(c+d*x))^2)-1/21*sec(c+d*x)^3/(d*(a^3+a^3*sin(c+d*x)))+4/21*tan(c+d*x)/(a^3*d)+4/63*tan(c+d*x)^3/(a^3*d)],
[csc(c+d*x)*sec(c+d*x)^4/(a+a*sin(c+d*x))^3,x,14,-arctanh(cos(c+d*x))/(a^3*d)+sec(c+d*x)/(a^3*d)+1/3*sec(c+d*x)^3/(a^3*d)+1/5*sec(c+d*x)^5/(a^3*d)+1/7*sec(c+d*x)^7/(a^3*d)+4/9*sec(c+d*x)^9/(a^3*d)-3*tan(c+d*x)/(a^3*d)-13/3*tan(c+d*x)^3/(a^3*d)-21/5*tan(c+d*x)^5/(a^3*d)-15/7*tan(c+d*x)^7/(a^3*d)-4/9*tan(c+d*x)^9/(a^3*d)],
[csc(c+d*x)^2*sec(c+d*x)^4/(a+a*sin(c+d*x))^3,x,14,3*arctanh(cos(c+d*x))/(a^3*d)-cot(c+d*x)/(a^3*d)-3*sec(c+d*x)/(a^3*d)-sec(c+d*x)^3/(a^3*d)-3/5*sec(c+d*x)^5/(a^3*d)-3/7*sec(c+d*x)^7/(a^3*d)-4/9*sec(c+d*x)^9/(a^3*d)+8*tan(c+d*x)/(a^3*d)+22/3*tan(c+d*x)^3/(a^3*d)+28/5*tan(c+d*x)^5/(a^3*d)+17/7*tan(c+d*x)^7/(a^3*d)+4/9*tan(c+d*x)^9/(a^3*d)],
[sec(c+d*x)^4*sin(c+d*x)^4/(a+a*sin(c+d*x))^4,x,17,4/5*sec(c+d*x)^5/(a^4*d)-16/7*sec(c+d*x)^7/(a^4*d)+20/9*sec(c+d*x)^9/(a^4*d)-8/11*sec(c+d*x)^11/(a^4*d)+1/5*tan(c+d*x)^5/(a^4*d)+9/7*tan(c+d*x)^7/(a^4*d)+16/9*tan(c+d*x)^9/(a^4*d)+8/11*tan(c+d*x)^11/(a^4*d)],
[sec(c+d*x)^4*sin(c+d*x)^3/(a+a*sin(c+d*x))^4,x,18,-1/5*sec(c+d*x)^5/(a^4*d)+9/7*sec(c+d*x)^7/(a^4*d)-16/9*sec(c+d*x)^9/(a^4*d)+8/11*sec(c+d*x)^11/(a^4*d)-4/5*tan(c+d*x)^5/(a^4*d)-16/7*tan(c+d*x)^7/(a^4*d)-20/9*tan(c+d*x)^9/(a^4*d)-8/11*tan(c+d*x)^11/(a^4*d)],
[sec(c+d*x)^4*sin(c+d*x)^2/(a+a*sin(c+d*x))^4,x,8,-4/7*sec(c+d*x)^7/(a^4*d)+4/3*sec(c+d*x)^9/(a^4*d)-8/11*sec(c+d*x)^11/(a^4*d)+1/3*tan(c+d*x)^3/(a^4*d)+2*tan(c+d*x)^5/(a^4*d)+25/7*tan(c+d*x)^7/(a^4*d)+8/3*tan(c+d*x)^9/(a^4*d)+8/11*tan(c+d*x)^11/(a^4*d),-1/22*a*sec(c+d*x)/(d*(a+a*sin(c+d*x))^5)-1/33*sec(c+d*x)/(d*(a+a*sin(c+d*x))^4)-5/231*sec(c+d*x)/(a*d*(a+a*sin(c+d*x))^3)+1/6*sec(c+d*x)^3/(a*d*(a+a*sin(c+d*x))^3)-4/231*sec(c+d*x)/(d*(a^2+a^2*sin(c+d*x))^2)-4/231*sec(c+d*x)/(d*(a^4+a^4*sin(c+d*x)))+8/231*tan(c+d*x)/(a^4*d)],

# Integrands of the form Sec[e+f x]^4 Sin[e+f x]^n (a+a Sin[e+f x])^(m/2)

# Integrands of the form Sec[e+f x]^5 (d Sin[e+f x])^n (a+a Sin[e+f x])^m

# Integrands of the form Sec[e+f x]^5 Sin[e+f x]^n (a+a Sin[e+f x])^m

# m>0
[sec(c+d*x)^5*sin(c+d*x)^6*(a+a*sin(c+d*x)),x,4,-39/16*a*log(1-sin(c+d*x))/d-9/16*a*log(1+sin(c+d*x))/d-a*sin(c+d*x)/d-1/2*a*sin(c+d*x)^2/d+1/8*a^3/(d*(a-a*sin(c+d*x))^2)-5/4*a^2/(d*(a-a*sin(c+d*x)))-1/8*a^2/(d*(a+a*sin(c+d*x)))],
[sec(c+d*x)^5*sin(c+d*x)^5*(a+a*sin(c+d*x)),x,3,-23/16*a*log(1-sin(c+d*x))/d+7/16*a*log(1+sin(c+d*x))/d-a*sin(c+d*x)/d+1/8*a^3/(d*(a-a*sin(c+d*x))^2)-a^2/(d*(a-a*sin(c+d*x)))+1/8*a^2/(d*(a+a*sin(c+d*x)))],
[sec(c+d*x)^5*sin(c+d*x)^4*(a+a*sin(c+d*x)),x,4,-11/16*a*log(1-sin(c+d*x))/d-5/16*a*log(1+sin(c+d*x))/d+1/8*a^3/(d*(a-a*sin(c+d*x))^2)-3/4*a^2/(d*(a-a*sin(c+d*x)))-1/8*a^2/(d*(a+a*sin(c+d*x)))],
[sec(c+d*x)^5*sin(c+d*x)^3*(a+a*sin(c+d*x)),x,5,3/8*a*arctanh(sin(c+d*x))/d+1/8*a^3/(d*(a-a*sin(c+d*x))^2)-1/2*a^2/(d*(a-a*sin(c+d*x)))+1/8*a^2/(d*(a+a*sin(c+d*x)))],
[sec(c+d*x)^5*sin(c+d*x)^2*(a+a*sin(c+d*x)),x,5,-1/8*a*arctanh(sin(c+d*x))/d+1/8*a^3/(d*(a-a*sin(c+d*x))^2)-1/4*a^2/(d*(a-a*sin(c+d*x)))-1/8*a^2/(d*(a+a*sin(c+d*x)))],
[sec(c+d*x)^5*sin(c+d*x)*(a+a*sin(c+d*x)),x,5,-1/8*a*arctanh(sin(c+d*x))/d+1/8*a^3/(d*(a-a*sin(c+d*x))^2)+1/8*a^2/(d*(a+a*sin(c+d*x)))],
[csc(c+d*x)*sec(c+d*x)^5*(a+a*sin(c+d*x)),x,4,-11/16*a*log(1-sin(c+d*x))/d+a*log(sin(c+d*x))/d-5/16*a*log(1+sin(c+d*x))/d+1/8*a^3/(d*(a-a*sin(c+d*x))^2)+1/2*a^2/(d*(a-a*sin(c+d*x)))+1/8*a^2/(d*(a+a*sin(c+d*x)))],
[csc(c+d*x)^2*sec(c+d*x)^5*(a+a*sin(c+d*x)),x,4,-a*csc(c+d*x)/d-23/16*a*log(1-sin(c+d*x))/d+a*log(sin(c+d*x))/d+7/16*a*log(1+sin(c+d*x))/d+1/8*a^3/(d*(a-a*sin(c+d*x))^2)+3/4*a^2/(d*(a-a*sin(c+d*x)))-1/8*a^2/(d*(a+a*sin(c+d*x)))],
[csc(c+d*x)^3*sec(c+d*x)^5*(a+a*sin(c+d*x)),x,4,-a*csc(c+d*x)/d-1/2*a*csc(c+d*x)^2/d-39/16*a*log(1-sin(c+d*x))/d+3*a*log(sin(c+d*x))/d-9/16*a*log(1+sin(c+d*x))/d+1/8*a^3/(d*(a-a*sin(c+d*x))^2)+a^2/(d*(a-a*sin(c+d*x)))+1/8*a^2/(d*(a+a*sin(c+d*x)))],
[csc(c+d*x)^4*sec(c+d*x)^5*(a+a*sin(c+d*x)),x,4,-3*a*csc(c+d*x)/d-1/2*a*csc(c+d*x)^2/d-1/3*a*csc(c+d*x)^3/d-59/16*a*log(1-sin(c+d*x))/d+3*a*log(sin(c+d*x))/d+11/16*a*log(1+sin(c+d*x))/d+1/8*a^3/(d*(a-a*sin(c+d*x))^2)+5/4*a^2/(d*(a-a*sin(c+d*x)))-1/8*a^2/(d*(a+a*sin(c+d*x)))],
[sec(c+d*x)^5*sin(c+d*x)^5*(a+a*sin(c+d*x))^2,x,3,-31/8*a^2*log(1-sin(c+d*x))/d-1/8*a^2*log(1+sin(c+d*x))/d-2*a^2*sin(c+d*x)/d-1/2*a^2*sin(c+d*x)^2/d+1/4*a^4/(d*(a-a*sin(c+d*x))^2)-9/4*a^3/(d*(a-a*sin(c+d*x)))],
[sec(c+d*x)^5*sin(c+d*x)^4*(a+a*sin(c+d*x))^2,x,4,-17/8*a^2*log(1-sin(c+d*x))/d+1/8*a^2*log(1+sin(c+d*x))/d-a^2*sin(c+d*x)/d+1/4*a^4/(d*(a-a*sin(c+d*x))^2)-7/4*a^3/(d*(a-a*sin(c+d*x)))],
[sec(c+d*x)^5*sin(c+d*x)^3*(a+a*sin(c+d*x))^2,x,4,-7/8*a^2*log(1-sin(c+d*x))/d-1/8*a^2*log(1+sin(c+d*x))/d+1/4*a^4/(d*(a-a*sin(c+d*x))^2)-5/4*a^3/(d*(a-a*sin(c+d*x)))],
[sec(c+d*x)^5*sin(c+d*x)^2*(a+a*sin(c+d*x))^2,x,5,1/4*a^2*arctanh(sin(c+d*x))/d+1/4*a^4/(d*(a-a*sin(c+d*x))^2)-3/4*a^3/(d*(a-a*sin(c+d*x)))],
[sec(c+d*x)^5*sin(c+d*x)*(a+a*sin(c+d*x))^2,x,5,-1/4*a^2*arctanh(sin(c+d*x))/d+1/4*a^4/(d*(a-a*sin(c+d*x))^2)-1/4*a^3/(d*(a-a*sin(c+d*x)))],
[csc(c+d*x)*sec(c+d*x)^5*(a+a*sin(c+d*x))^2,x,4,-7/8*a^2*log(1-sin(c+d*x))/d+a^2*log(sin(c+d*x))/d-1/8*a^2*log(1+sin(c+d*x))/d+1/4*a^4/(d*(a-a*sin(c+d*x))^2)+3/4*a^3/(d*(a-a*sin(c+d*x)))],
[csc(c+d*x)^2*sec(c+d*x)^5*(a+a*sin(c+d*x))^2,x,4,-a^2*csc(c+d*x)/d-17/8*a^2*log(1-sin(c+d*x))/d+2*a^2*log(sin(c+d*x))/d+1/8*a^2*log(1+sin(c+d*x))/d+1/4*a^4/(d*(a-a*sin(c+d*x))^2)+5/4*a^3/(d*(a-a*sin(c+d*x)))],
[csc(c+d*x)^3*sec(c+d*x)^5*(a+a*sin(c+d*x))^2,x,4,-2*a^2*csc(c+d*x)/d-1/2*a^2*csc(c+d*x)^2/d-31/8*a^2*log(1-sin(c+d*x))/d+4*a^2*log(sin(c+d*x))/d-1/8*a^2*log(1+sin(c+d*x))/d+1/4*a^4/(d*(a-a*sin(c+d*x))^2)+7/4*a^3/(d*(a-a*sin(c+d*x)))],
[csc(c+d*x)^4*sec(c+d*x)^5*(a+a*sin(c+d*x))^2,x,4,-4*a^2*csc(c+d*x)/d-a^2*csc(c+d*x)^2/d-1/3*a^2*csc(c+d*x)^3/d-49/8*a^2*log(1-sin(c+d*x))/d+6*a^2*log(sin(c+d*x))/d+1/8*a^2*log(1+sin(c+d*x))/d+1/4*a^4/(d*(a-a*sin(c+d*x))^2)+9/4*a^3/(d*(a-a*sin(c+d*x)))],
[sec(c+d*x)^5*sin(c+d*x)^5*(a+a*sin(c+d*x))^3,x,3,-10*a^3*log(1-sin(c+d*x))/d-6*a^3*sin(c+d*x)/d-3/2*a^3*sin(c+d*x)^2/d-1/3*a^3*sin(c+d*x)^3/d+1/2*a^5/(d*(a-a*sin(c+d*x))^2)-5*a^4/(d*(a-a*sin(c+d*x)))],
[sec(c+d*x)^5*sin(c+d*x)^4*(a+a*sin(c+d*x))^3,x,4,-6*a^3*log(1-sin(c+d*x))/d-3*a^3*sin(c+d*x)/d-1/2*a^3*sin(c+d*x)^2/d+1/2*a^5/(d*(a-a*sin(c+d*x))^2)-4*a^4/(d*(a-a*sin(c+d*x)))],
[sec(c+d*x)^5*sin(c+d*x)^3*(a+a*sin(c+d*x))^3,x,4,-3*a^3*log(1-sin(c+d*x))/d-a^3*sin(c+d*x)/d+1/2*a^5/(d*(a-a*sin(c+d*x))^2)-3*a^4/(d*(a-a*sin(c+d*x)))],
[sec(c+d*x)^5*sin(c+d*x)^2*(a+a*sin(c+d*x))^3,x,4,-a^3*log(1-sin(c+d*x))/d+1/2*a^5/(d*(a-a*sin(c+d*x))^2)-2*a^4/(d*(a-a*sin(c+d*x)))],
[sec(c+d*x)^5*sin(c+d*x)*(a+a*sin(c+d*x))^3,x,3,1/2*a^5*sin(c+d*x)^2/(d*(a-a*sin(c+d*x))^2)],
[csc(c+d*x)*sec(c+d*x)^5*(a+a*sin(c+d*x))^3,x,4,-a^3*log(1-sin(c+d*x))/d+a^3*log(sin(c+d*x))/d+1/2*a^5/(d*(a-a*sin(c+d*x))^2)+a^4/(d*(a-a*sin(c+d*x)))],
[csc(c+d*x)^2*sec(c+d*x)^5*(a+a*sin(c+d*x))^3,x,4,-a^3*csc(c+d*x)/d-3*a^3*log(1-sin(c+d*x))/d+3*a^3*log(sin(c+d*x))/d+1/2*a^5/(d*(a-a*sin(c+d*x))^2)+2*a^4/(d*(a-a*sin(c+d*x)))],
[csc(c+d*x)^3*sec(c+d*x)^5*(a+a*sin(c+d*x))^3,x,4,-3*a^3*csc(c+d*x)/d-1/2*a^3*csc(c+d*x)^2/d-6*a^3*log(1-sin(c+d*x))/d+6*a^3*log(sin(c+d*x))/d+1/2*a^5/(d*(a-a*sin(c+d*x))^2)+3*a^4/(d*(a-a*sin(c+d*x)))],

# m<0

# Integrands of the form Sec[e+f x]^5 Sin[e+f x]^n (a+a Sin[e+f x])^(m/2)

# Integrands of the form Sec[e+f x]^6 (d Sin[e+f x])^n (a+a Sin[e+f x])^m

# Integrands of the form Sec[e+f x]^7 (d Sin[e+f x])^n (a+a Sin[e+f x])^m

# Integrands of the form Sec[e+f x]^7 Sin[e+f x]^n (a+a Sin[e+f x])^m

# m>0

# m<0
[sec(c+d*x)^7*sin(c+d*x)^11/(a+a*sin(c+d*x)),x,4,515/256*log(1-sin(c+d*x))/(a*d)-1795/256*log(1+sin(c+d*x))/(a*d)+5*sin(c+d*x)/(a*d)-1/2*sin(c+d*x)^2/(a*d)+1/3*sin(c+d*x)^3/(a*d)+1/96*a^2/(d*(a-a*sin(c+d*x))^3)-17/128*a/(d*(a-a*sin(c+d*x))^2)+125/128/(d*(a-a*sin(c+d*x)))+1/64*a^3/(d*(a+a*sin(c+d*x))^4)-3/16*a^2/(d*(a+a*sin(c+d*x))^3)+71/64*a/(d*(a+a*sin(c+d*x))^2)+(-5)/(d*(a+a*sin(c+d*x)))],
[sec(c+d*x)^7*sin(c+d*x)^10/(a+a*sin(c+d*x)),x,4,325/256*log(1-sin(c+d*x))/(a*d)+955/256*log(1+sin(c+d*x))/(a*d)-sin(c+d*x)/(a*d)+1/2*sin(c+d*x)^2/(a*d)+1/96*a^2/(d*(a-a*sin(c+d*x))^3)-15/128*a/(d*(a-a*sin(c+d*x))^2)+95/128/(d*(a-a*sin(c+d*x)))-1/64*a^3/(d*(a+a*sin(c+d*x))^4)+1/6*a^2/(d*(a+a*sin(c+d*x))^3)-55/64*a/(d*(a+a*sin(c+d*x))^2)+105/32/(d*(a+a*sin(c+d*x)))],
[sec(c+d*x)^7*sin(c+d*x)^9/(a+a*sin(c+d*x)),x,4,187/256*log(1-sin(c+d*x))/(a*d)-443/256*log(1+sin(c+d*x))/(a*d)+sin(c+d*x)/(a*d)+1/96*a^2/(d*(a-a*sin(c+d*x))^3)-13/128*a/(d*(a-a*sin(c+d*x))^2)+69/128/(d*(a-a*sin(c+d*x)))+1/64*a^3/(d*(a+a*sin(c+d*x))^4)-7/48*a^2/(d*(a+a*sin(c+d*x))^3)+41/64*a/(d*(a+a*sin(c+d*x))^2)+(-2)/(d*(a+a*sin(c+d*x)))],
[sec(c+d*x)^7*sin(c+d*x)^8/(a+a*sin(c+d*x)),x,4,93/256*log(1-sin(c+d*x))/(a*d)+163/256*log(1+sin(c+d*x))/(a*d)+1/96*a^2/(d*(a-a*sin(c+d*x))^3)-11/128*a/(d*(a-a*sin(c+d*x))^2)+47/128/(d*(a-a*sin(c+d*x)))-1/64*a^3/(d*(a+a*sin(c+d*x))^4)+1/8*a^2/(d*(a+a*sin(c+d*x))^3)-29/64*a/(d*(a+a*sin(c+d*x))^2)+35/32/(d*(a+a*sin(c+d*x)))],
[sec(c+d*x)^7*sin(c+d*x)^7/(a+a*sin(c+d*x)),x,8,-35/128*arctanh(sin(c+d*x))/(a*d)+35/128*sec(c+d*x)*tan(c+d*x)/(a*d)-35/192*sec(c+d*x)*tan(c+d*x)^3/(a*d)+7/48*sec(c+d*x)*tan(c+d*x)^5/(a*d)-1/8*sec(c+d*x)*tan(c+d*x)^7/(a*d)+1/8*tan(c+d*x)^8/(a*d)],
[sec(c+d*x)^7*sin(c+d*x)^6/(a+a*sin(c+d*x)),x,8,-5/128*arctanh(sin(c+d*x))/(a*d)-5/128*sec(c+d*x)*tan(c+d*x)/(a*d)+5/64*sec(c+d*x)^3*tan(c+d*x)/(a*d)-5/48*sec(c+d*x)^3*tan(c+d*x)^3/(a*d)+1/8*sec(c+d*x)^3*tan(c+d*x)^5/(a*d)-1/8*tan(c+d*x)^8/(a*d)],
[sec(c+d*x)^7*sin(c+d*x)^5/(a+a*sin(c+d*x)),x,9,5/128*arctanh(sin(c+d*x))/(a*d)+5/128*sec(c+d*x)*tan(c+d*x)/(a*d)-5/64*sec(c+d*x)^3*tan(c+d*x)/(a*d)+5/48*sec(c+d*x)^3*tan(c+d*x)^3/(a*d)-1/8*sec(c+d*x)^3*tan(c+d*x)^5/(a*d)+1/6*tan(c+d*x)^6/(a*d)+1/8*tan(c+d*x)^8/(a*d)],
[sec(c+d*x)^7*sin(c+d*x)^4/(a+a*sin(c+d*x)),x,9,3/128*arctanh(sin(c+d*x))/(a*d)+3/128*sec(c+d*x)*tan(c+d*x)/(a*d)+1/64*sec(c+d*x)^3*tan(c+d*x)/(a*d)-1/16*sec(c+d*x)^5*tan(c+d*x)/(a*d)+1/8*sec(c+d*x)^5*tan(c+d*x)^3/(a*d)-1/6*tan(c+d*x)^6/(a*d)-1/8*tan(c+d*x)^8/(a*d)],
[sec(c+d*x)^7*sin(c+d*x)^3/(a+a*sin(c+d*x)),x,9,-3/128*arctanh(sin(c+d*x))/(a*d)-1/6*sec(c+d*x)^6/(a*d)+1/8*sec(c+d*x)^8/(a*d)-3/128*sec(c+d*x)*tan(c+d*x)/(a*d)-1/64*sec(c+d*x)^3*tan(c+d*x)/(a*d)+1/16*sec(c+d*x)^5*tan(c+d*x)/(a*d)-1/8*sec(c+d*x)^5*tan(c+d*x)^3/(a*d)],
[sec(c+d*x)^7*sin(c+d*x)^2/(a+a*sin(c+d*x)),x,9,-5/128*arctanh(sin(c+d*x))/(a*d)+1/6*sec(c+d*x)^6/(a*d)-1/8*sec(c+d*x)^8/(a*d)-5/128*sec(c+d*x)*tan(c+d*x)/(a*d)-5/192*sec(c+d*x)^3*tan(c+d*x)/(a*d)-1/48*sec(c+d*x)^5*tan(c+d*x)/(a*d)+1/8*sec(c+d*x)^7*tan(c+d*x)/(a*d)],
[sec(c+d*x)^7*sin(c+d*x)/(a+a*sin(c+d*x)),x,8,5/128*arctanh(sin(c+d*x))/(a*d)+1/8*sec(c+d*x)^8/(a*d)+5/128*sec(c+d*x)*tan(c+d*x)/(a*d)+5/192*sec(c+d*x)^3*tan(c+d*x)/(a*d)+1/48*sec(c+d*x)^5*tan(c+d*x)/(a*d)-1/8*sec(c+d*x)^7*tan(c+d*x)/(a*d)],
[sec(c+d*x)^7/(a+a*sin(c+d*x)),x,4,35/128*arctanh(sin(c+d*x))/(a*d)+1/96*a^2/(d*(a-a*sin(c+d*x))^3)+5/128*a/(d*(a-a*sin(c+d*x))^2)+15/128/(d*(a-a*sin(c+d*x)))-1/64*a^3/(d*(a+a*sin(c+d*x))^4)-1/24*a^2/(d*(a+a*sin(c+d*x))^3)-5/64*a/(d*(a+a*sin(c+d*x))^2)+(-5/32)/(d*(a+a*sin(c+d*x)))],
[csc(c+d*x)*sec(c+d*x)^7/(a+a*sin(c+d*x)),x,4,-93/256*log(1-sin(c+d*x))/(a*d)+log(sin(c+d*x))/(a*d)-163/256*log(1+sin(c+d*x))/(a*d)+1/96*a^2/(d*(a-a*sin(c+d*x))^3)+7/128*a/(d*(a-a*sin(c+d*x))^2)+29/128/(d*(a-a*sin(c+d*x)))+1/64*a^3/(d*(a+a*sin(c+d*x))^4)+1/16*a^2/(d*(a+a*sin(c+d*x))^3)+11/64*a/(d*(a+a*sin(c+d*x))^2)+1/2/(d*(a+a*sin(c+d*x)))],
[csc(c+d*x)^2*sec(c+d*x)^7/(a+a*sin(c+d*x)),x,4,-csc(c+d*x)/(a*d)-187/256*log(1-sin(c+d*x))/(a*d)-log(sin(c+d*x))/(a*d)+443/256*log(1+sin(c+d*x))/(a*d)+1/96*a^2/(d*(a-a*sin(c+d*x))^3)+9/128*a/(d*(a-a*sin(c+d*x))^2)+47/128/(d*(a-a*sin(c+d*x)))-1/64*a^3/(d*(a+a*sin(c+d*x))^4)-1/12*a^2/(d*(a+a*sin(c+d*x))^3)-19/64*a/(d*(a+a*sin(c+d*x))^2)+(-35/32)/(d*(a+a*sin(c+d*x)))],
[csc(c+d*x)^3*sec(c+d*x)^7/(a+a*sin(c+d*x)),x,4,csc(c+d*x)/(a*d)-1/2*csc(c+d*x)^2/(a*d)-325/256*log(1-sin(c+d*x))/(a*d)+5*log(sin(c+d*x))/(a*d)-955/256*log(1+sin(c+d*x))/(a*d)+1/96*a^2/(d*(a-a*sin(c+d*x))^3)+11/128*a/(d*(a-a*sin(c+d*x))^2)+69/128/(d*(a-a*sin(c+d*x)))+1/64*a^3/(d*(a+a*sin(c+d*x))^4)+5/48*a^2/(d*(a+a*sin(c+d*x))^3)+29/64*a/(d*(a+a*sin(c+d*x))^2)+2/(d*(a+a*sin(c+d*x)))],
[csc(c+d*x)^4*sec(c+d*x)^7/(a+a*sin(c+d*x)),x,4,-5*csc(c+d*x)/(a*d)+1/2*csc(c+d*x)^2/(a*d)-1/3*csc(c+d*x)^3/(a*d)-515/256*log(1-sin(c+d*x))/(a*d)-5*log(sin(c+d*x))/(a*d)+1795/256*log(1+sin(c+d*x))/(a*d)+1/96*a^2/(d*(a-a*sin(c+d*x))^3)+13/128*a/(d*(a-a*sin(c+d*x))^2)+95/128/(d*(a-a*sin(c+d*x)))-1/64*a^3/(d*(a+a*sin(c+d*x))^4)-1/8*a^2/(d*(a+a*sin(c+d*x))^3)-41/64*a/(d*(a+a*sin(c+d*x))^2)+(-105/32)/(d*(a+a*sin(c+d*x)))],

# Integrands of the form Sec[e+f x]^3 Sin[e+f x]^n (a+a Sin[e+f x])^(m/2)

# Integrands of the form Sec[e+f x]^8 (d Sin[e+f x])^n (a+a Sin[e+f x])^m

# Integrands of the form Sec[e+f x]^8 Sin[e+f x]^n (a+a Sin[e+f x])^m

# m>0
[sec(c+d*x)^8*sin(c+d*x)^3*(a+a*sin(c+d*x))^2,x,11,1/3*a^2*sec(c+d*x)^3/d-3/5*a^2*sec(c+d*x)^5/d+2/7*a^2*sec(c+d*x)^7/d+2/5*a^2*tan(c+d*x)^5/d+2/7*a^2*tan(c+d*x)^7/d],

# m<0

# Integrands of the form Sec[e+f x]^9 (d Sin[e+f x])^n (a+a Sin[e+f x])^m

# Integrands of the form Sec[e+f x]^9 Sin[e+f x]^n (a+a Sin[e+f x])^m

# m>0

# m<0
[sec(c+d*x)^9*sin(c+d*x)^12/(a+a*sin(c+d*x)),x,4,-843/512*log(1-sin(c+d*x))/(a*d)-2229/512*log(1+sin(c+d*x))/(a*d)+sin(c+d*x)/(a*d)-1/2*sin(c+d*x)^2/(a*d)+1/256*a^3/(d*(a-a*sin(c+d*x))^4)-3/64*a^2/(d*(a-a*sin(c+d*x))^3)+141/512*a/(d*(a-a*sin(c+d*x))^2)+(-39/32)/(d*(a-a*sin(c+d*x)))-1/160*a^4/(d*(a+a*sin(c+d*x))^5)+19/256*a^3/(d*(a+a*sin(c+d*x))^4)-53/128*a^2/(d*(a+a*sin(c+d*x))^3)+765/512*a/(d*(a+a*sin(c+d*x))^2)+(-1155/256)/(d*(a+a*sin(c+d*x)))],
[sec(c+d*x)^9*sin(c+d*x)^11/(a+a*sin(c+d*x)),x,4,-437/512*log(1-sin(c+d*x))/(a*d)+949/512*log(1+sin(c+d*x))/(a*d)-sin(c+d*x)/(a*d)+1/256*a^3/(d*(a-a*sin(c+d*x))^4)-1/24*a^2/(d*(a-a*sin(c+d*x))^3)+109/512*a/(d*(a-a*sin(c+d*x))^2)+(-203/256)/(d*(a-a*sin(c+d*x)))+1/160*a^4/(d*(a+a*sin(c+d*x))^5)-17/256*a^3/(d*(a+a*sin(c+d*x))^4)+125/384*a^2/(d*(a+a*sin(c+d*x))^3)-515/512*a/(d*(a+a*sin(c+d*x))^2)+5/2/(d*(a+a*sin(c+d*x)))],
[sec(c+d*x)^9*sin(c+d*x)^10/(a+a*sin(c+d*x)),x,4,-193/512*log(1-sin(c+d*x))/(a*d)-319/512*log(1+sin(c+d*x))/(a*d)+1/256*a^3/(d*(a-a*sin(c+d*x))^4)-7/192*a^2/(d*(a-a*sin(c+d*x))^3)+81/512*a/(d*(a-a*sin(c+d*x))^2)+(-61/128)/(d*(a-a*sin(c+d*x)))-1/160*a^4/(d*(a+a*sin(c+d*x))^5)+15/256*a^3/(d*(a+a*sin(c+d*x))^4)-95/384*a^2/(d*(a+a*sin(c+d*x))^3)+325/512*a/(d*(a+a*sin(c+d*x))^2)+(-315/256)/(d*(a+a*sin(c+d*x)))],
[sec(c+d*x)^9*sin(c+d*x)^9/(a+a*sin(c+d*x)),x,9,63/256*arctanh(sin(c+d*x))/(a*d)-63/256*sec(c+d*x)*tan(c+d*x)/(a*d)+21/128*sec(c+d*x)*tan(c+d*x)^3/(a*d)-21/160*sec(c+d*x)*tan(c+d*x)^5/(a*d)+9/80*sec(c+d*x)*tan(c+d*x)^7/(a*d)-1/10*sec(c+d*x)*tan(c+d*x)^9/(a*d)+1/10*tan(c+d*x)^10/(a*d)],
[sec(c+d*x)^9*sin(c+d*x)^8/(a+a*sin(c+d*x)),x,9,7/256*arctanh(sin(c+d*x))/(a*d)+7/256*sec(c+d*x)*tan(c+d*x)/(a*d)-7/128*sec(c+d*x)^3*tan(c+d*x)/(a*d)+7/96*sec(c+d*x)^3*tan(c+d*x)^3/(a*d)-7/80*sec(c+d*x)^3*tan(c+d*x)^5/(a*d)+1/10*sec(c+d*x)^3*tan(c+d*x)^7/(a*d)-1/10*tan(c+d*x)^10/(a*d)],
[sec(c+d*x)^9*sin(c+d*x)^7/(a+a*sin(c+d*x)),x,10,-7/256*arctanh(sin(c+d*x))/(a*d)-7/256*sec(c+d*x)*tan(c+d*x)/(a*d)+7/128*sec(c+d*x)^3*tan(c+d*x)/(a*d)-7/96*sec(c+d*x)^3*tan(c+d*x)^3/(a*d)+7/80*sec(c+d*x)^3*tan(c+d*x)^5/(a*d)-1/10*sec(c+d*x)^3*tan(c+d*x)^7/(a*d)+1/8*tan(c+d*x)^8/(a*d)+1/10*tan(c+d*x)^10/(a*d)],
[sec(c+d*x)^9*sin(c+d*x)^6/(a+a*sin(c+d*x)),x,10,-3/256*arctanh(sin(c+d*x))/(a*d)-3/256*sec(c+d*x)*tan(c+d*x)/(a*d)-1/128*sec(c+d*x)^3*tan(c+d*x)/(a*d)+1/32*sec(c+d*x)^5*tan(c+d*x)/(a*d)-1/16*sec(c+d*x)^5*tan(c+d*x)^3/(a*d)+1/10*sec(c+d*x)^5*tan(c+d*x)^5/(a*d)-1/8*tan(c+d*x)^8/(a*d)-1/10*tan(c+d*x)^10/(a*d)],
[sec(c+d*x)^9*sin(c+d*x)^5/(a+a*sin(c+d*x)),x,11,3/256*arctanh(sin(c+d*x))/(a*d)+1/6*sec(c+d*x)^6/(a*d)-1/4*sec(c+d*x)^8/(a*d)+1/10*sec(c+d*x)^10/(a*d)+3/256*sec(c+d*x)*tan(c+d*x)/(a*d)+1/128*sec(c+d*x)^3*tan(c+d*x)/(a*d)-1/32*sec(c+d*x)^5*tan(c+d*x)/(a*d)+1/16*sec(c+d*x)^5*tan(c+d*x)^3/(a*d)-1/10*sec(c+d*x)^5*tan(c+d*x)^5/(a*d)],
[sec(c+d*x)^9*sin(c+d*x)^4/(a+a*sin(c+d*x)),x,11,3/256*arctanh(sin(c+d*x))/(a*d)-1/6*sec(c+d*x)^6/(a*d)+1/4*sec(c+d*x)^8/(a*d)-1/10*sec(c+d*x)^10/(a*d)+3/256*sec(c+d*x)*tan(c+d*x)/(a*d)+1/128*sec(c+d*x)^3*tan(c+d*x)/(a*d)+1/160*sec(c+d*x)^5*tan(c+d*x)/(a*d)-3/80*sec(c+d*x)^7*tan(c+d*x)/(a*d)+1/10*sec(c+d*x)^7*tan(c+d*x)^3/(a*d)],
[sec(c+d*x)^9*sin(c+d*x)^3/(a+a*sin(c+d*x)),x,10,-3/256*arctanh(sin(c+d*x))/(a*d)-1/8*sec(c+d*x)^8/(a*d)+1/10*sec(c+d*x)^10/(a*d)-3/256*sec(c+d*x)*tan(c+d*x)/(a*d)-1/128*sec(c+d*x)^3*tan(c+d*x)/(a*d)-1/160*sec(c+d*x)^5*tan(c+d*x)/(a*d)+3/80*sec(c+d*x)^7*tan(c+d*x)/(a*d)-1/10*sec(c+d*x)^7*tan(c+d*x)^3/(a*d)],
[sec(c+d*x)^9*sin(c+d*x)^2/(a+a*sin(c+d*x)),x,10,-7/256*arctanh(sin(c+d*x))/(a*d)+1/8*sec(c+d*x)^8/(a*d)-1/10*sec(c+d*x)^10/(a*d)-7/256*sec(c+d*x)*tan(c+d*x)/(a*d)-7/384*sec(c+d*x)^3*tan(c+d*x)/(a*d)-7/480*sec(c+d*x)^5*tan(c+d*x)/(a*d)-1/80*sec(c+d*x)^7*tan(c+d*x)/(a*d)+1/10*sec(c+d*x)^9*tan(c+d*x)/(a*d)],
[sec(c+d*x)^9*sin(c+d*x)/(a+a*sin(c+d*x)),x,9,7/256*arctanh(sin(c+d*x))/(a*d)+1/10*sec(c+d*x)^10/(a*d)+7/256*sec(c+d*x)*tan(c+d*x)/(a*d)+7/384*sec(c+d*x)^3*tan(c+d*x)/(a*d)+7/480*sec(c+d*x)^5*tan(c+d*x)/(a*d)+1/80*sec(c+d*x)^7*tan(c+d*x)/(a*d)-1/10*sec(c+d*x)^9*tan(c+d*x)/(a*d)],
[sec(c+d*x)^9/(a+a*sin(c+d*x)),x,4,63/256*arctanh(sin(c+d*x))/(a*d)+1/256*a^3/(d*(a-a*sin(c+d*x))^4)+1/64*a^2/(d*(a-a*sin(c+d*x))^3)+21/512*a/(d*(a-a*sin(c+d*x))^2)+7/64/(d*(a-a*sin(c+d*x)))-1/160*a^4/(d*(a+a*sin(c+d*x))^5)-5/256*a^3/(d*(a+a*sin(c+d*x))^4)-5/128*a^2/(d*(a+a*sin(c+d*x))^3)-35/512*a/(d*(a+a*sin(c+d*x))^2)+(-35/256)/(d*(a+a*sin(c+d*x)))],
[csc(c+d*x)*sec(c+d*x)^9/(a+a*sin(c+d*x)),x,4,-193/512*log(1-sin(c+d*x))/(a*d)+log(sin(c+d*x))/(a*d)-319/512*log(1+sin(c+d*x))/(a*d)+1/256*a^3/(d*(a-a*sin(c+d*x))^4)+1/48*a^2/(d*(a-a*sin(c+d*x))^3)+37/512*a/(d*(a-a*sin(c+d*x))^2)+65/256/(d*(a-a*sin(c+d*x)))+1/160*a^4/(d*(a+a*sin(c+d*x))^5)+7/256*a^3/(d*(a+a*sin(c+d*x))^4)+29/384*a^2/(d*(a+a*sin(c+d*x))^3)+93/512*a/(d*(a+a*sin(c+d*x))^2)+1/2/(d*(a+a*sin(c+d*x)))],
[csc(c+d*x)^2*sec(c+d*x)^9/(a+a*sin(c+d*x)),x,4,-csc(c+d*x)/(a*d)-437/512*log(1-sin(c+d*x))/(a*d)-log(sin(c+d*x))/(a*d)+949/512*log(1+sin(c+d*x))/(a*d)+1/256*a^3/(d*(a-a*sin(c+d*x))^4)+5/192*a^2/(d*(a-a*sin(c+d*x))^3)+57/512*a/(d*(a-a*sin(c+d*x))^2)+61/128/(d*(a-a*sin(c+d*x)))-1/160*a^4/(d*(a+a*sin(c+d*x))^5)-9/256*a^3/(d*(a+a*sin(c+d*x))^4)-47/384*a^2/(d*(a+a*sin(c+d*x))^3)-187/512*a/(d*(a+a*sin(c+d*x))^2)+(-315/256)/(d*(a+a*sin(c+d*x)))],
[csc(c+d*x)^3*sec(c+d*x)^9/(a+a*sin(c+d*x)),x,4,csc(c+d*x)/(a*d)-1/2*csc(c+d*x)^2/(a*d)-843/512*log(1-sin(c+d*x))/(a*d)+6*log(sin(c+d*x))/(a*d)-2229/512*log(1+sin(c+d*x))/(a*d)+1/256*a^3/(d*(a-a*sin(c+d*x))^4)+1/32*a^2/(d*(a-a*sin(c+d*x))^3)+81/512*a/(d*(a-a*sin(c+d*x))^2)+203/256/(d*(a-a*sin(c+d*x)))+1/160*a^4/(d*(a+a*sin(c+d*x))^5)+11/256*a^3/(d*(a+a*sin(c+d*x))^4)+23/128*a^2/(d*(a+a*sin(c+d*x))^3)+325/512*a/(d*(a+a*sin(c+d*x))^2)+5/2/(d*(a+a*sin(c+d*x)))],

# Integrands of the form (g Sec[e+f x])^p (d Sin[e+f x])^n (a+a Sin[e+f x])^m with m, n and p symbolic
[(g*sec(e+f*x))^p*(d*sin(e+f*x))^n*(a+a*sin(e+f*x))^m,x,5,AppellF1(1+n,1/2*(1+p),1/2*(1-2*m+p),2+n,sin(e+f*x),-sin(e+f*x))*sec(e+f*x)*(g*sec(e+f*x))^p*(1-sin(e+f*x))^(1/2*(1+p))*(d*sin(e+f*x))^(1+n)*(1+sin(e+f*x))^(1/2*(1-2*m+p))*(a+a*sin(e+f*x))^m/(d*f*(1+n))],

# Integrands of the form (g Cos[e+f x])^p (a+a Sin[e+f x])^m (c+d Sin[e+f x])^n

# Integrands of the form Cos[e+f x]^1 (a+a Sin[e+f x])^m (c+d Sin[e+f x])^n

# Integrands of the form Cos[e+f x] (a+a Sin[e+f x])^m (c+d Sin[e+f x])^n with m and/or n symbolic
[cos(e+f*x)*(a+a*sin(e+f*x))^m*(c+d*sin(e+f*x))^n,x,3,hypergeom([1+m,-n],[2+m],-d*(1+sin(e+f*x))/(c-d))*(a+a*sin(e+f*x))^(1+m)*(c+d*sin(e+f*x))^n/(a*f*(1+m)*((c+d*sin(e+f*x))/(c-d))^n)],
[cos(e+f*x)*(a+a*sin(e+f*x))^4*(c+d*sin(e+f*x))^n,x,3,a^4*(c-d)^4*(c+d*sin(e+f*x))^(1+n)/(d^5*f*(1+n))-4*a^4*(c-d)^3*(c+d*sin(e+f*x))^(2+n)/(d^5*f*(2+n))+6*a^4*(c-d)^2*(c+d*sin(e+f*x))^(3+n)/(d^5*f*(3+n))-4*a^4*(c-d)*(c+d*sin(e+f*x))^(4+n)/(d^5*f*(4+n))+a^4*(c+d*sin(e+f*x))^(5+n)/(d^5*f*(5+n))],
[cos(e+f*x)*(a+a*sin(e+f*x))^3*(c+d*sin(e+f*x))^n,x,3,-a^3*(c-d)^3*(c+d*sin(e+f*x))^(1+n)/(d^4*f*(1+n))+3*a^3*(c-d)^2*(c+d*sin(e+f*x))^(2+n)/(d^4*f*(2+n))-3*a^3*(c-d)*(c+d*sin(e+f*x))^(3+n)/(d^4*f*(3+n))+a^3*(c+d*sin(e+f*x))^(4+n)/(d^4*f*(4+n))],
[cos(e+f*x)*(a+a*sin(e+f*x))^2*(c+d*sin(e+f*x))^n,x,3,a^2*(c-d)^2*(c+d*sin(e+f*x))^(1+n)/(d^3*f*(1+n))-2*a^2*(c-d)*(c+d*sin(e+f*x))^(2+n)/(d^3*f*(2+n))+a^2*(c+d*sin(e+f*x))^(3+n)/(d^3*f*(3+n))],
[cos(e+f*x)*(a+a*sin(e+f*x))*(c+d*sin(e+f*x))^n,x,3,-a*(c-d)*(c+d*sin(e+f*x))^(1+n)/(d^2*f*(1+n))+a*(c+d*sin(e+f*x))^(2+n)/(d^2*f*(2+n))],
[cos(e+f*x)*(c+d*sin(e+f*x))^n/(a+a*sin(e+f*x)),x,2,-hypergeom([1,1+n],[2+n],(c+d*sin(e+f*x))/(c-d))*(c+d*sin(e+f*x))^(1+n)/(a*(c-d)*f*(1+n))],
[cos(e+f*x)*(c+d*sin(e+f*x))^n/(a+a*sin(e+f*x))^2,x,2,d*hypergeom([2,1+n],[2+n],(c+d*sin(e+f*x))/(c-d))*(c+d*sin(e+f*x))^(1+n)/(a^2*(c-d)^2*f*(1+n))],
[cos(e+f*x)*(c+d*sin(e+f*x))^n/(a+a*sin(e+f*x))^3,x,2,-d^2*hypergeom([3,1+n],[2+n],(c+d*sin(e+f*x))/(c-d))*(c+d*sin(e+f*x))^(1+n)/(a^3*(c-d)^3*f*(1+n))],
[cos(e+f*x)*(a+a*sin(e+f*x))^m*(c+d*sin(e+f*x))^4,x,3,(c-d)^4*(a+a*sin(e+f*x))^(1+m)/(a*f*(1+m))+4*(c-d)^3*d*(a+a*sin(e+f*x))^(2+m)/(a^2*f*(2+m))+6*(c-d)^2*d^2*(a+a*sin(e+f*x))^(3+m)/(a^3*f*(3+m))+4*(c-d)*d^3*(a+a*sin(e+f*x))^(4+m)/(a^4*f*(4+m))+d^4*(a+a*sin(e+f*x))^(5+m)/(a^5*f*(5+m))],
[cos(e+f*x)*(a+a*sin(e+f*x))^m*(c+d*sin(e+f*x))^3,x,3,(c-d)^3*(a+a*sin(e+f*x))^(1+m)/(a*f*(1+m))+3*(c-d)^2*d*(a+a*sin(e+f*x))^(2+m)/(a^2*f*(2+m))+3*(c-d)*d^2*(a+a*sin(e+f*x))^(3+m)/(a^3*f*(3+m))+d^3*(a+a*sin(e+f*x))^(4+m)/(a^4*f*(4+m))],
[cos(e+f*x)*(a+a*sin(e+f*x))^m*(c+d*sin(e+f*x))^2,x,3,(c-d)^2*(a+a*sin(e+f*x))^(1+m)/(a*f*(1+m))+2*(c-d)*d*(a+a*sin(e+f*x))^(2+m)/(a^2*f*(2+m))+d^2*(a+a*sin(e+f*x))^(3+m)/(a^3*f*(3+m))],
[cos(e+f*x)*(a+a*sin(e+f*x))^m*(c+d*sin(e+f*x)),x,3,(c-d)*(a+a*sin(e+f*x))^(1+m)/(a*f*(1+m))+d*(a+a*sin(e+f*x))^(2+m)/(a^2*f*(2+m))],
[cos(e+f*x)*(a+a*sin(e+f*x))^m/(c+d*sin(e+f*x)),x,2,hypergeom([1,1+m],[2+m],-d*(1+sin(e+f*x))/(c-d))*(a+a*sin(e+f*x))^(1+m)/(a*(c-d)*f*(1+m))],
[cos(e+f*x)*(a+a*sin(e+f*x))^m/(c+d*sin(e+f*x))^2,x,2,hypergeom([2,1+m],[2+m],-d*(1+sin(e+f*x))/(c-d))*(a+a*sin(e+f*x))^(1+m)/(a*(c-d)^2*f*(1+m))],
[cos(e+f*x)*(a+a*sin(e+f*x))^m/(c+d*sin(e+f*x))^3,x,2,hypergeom([3,1+m],[2+m],-d*(1+sin(e+f*x))/(c-d))*(a+a*sin(e+f*x))^(1+m)/(a*(c-d)^3*f*(1+m))],
[cos(c+d*x)*sin(c+d*x)^n*(a+a*sin(c+d*x))^m,x,3,-hypergeom([1,2+m+n],[2+m],1+sin(c+d*x))*sin(c+d*x)^(1+n)*(a+a*sin(c+d*x))^(1+m)/(a*d*(1+m)),hypergeom([1+n,-m],[2+n],-sin(c+d*x))*sin(c+d*x)^(1+n)*(a+a*sin(c+d*x))^m/(d*(1+n)*(1+sin(c+d*x))^m)],
[cos(c+d*x)*sin(c+d*x)^4*(a+a*sin(c+d*x))^m,x,4,(a+a*sin(c+d*x))^(1+m)/(a*d*(1+m))-4*(a+a*sin(c+d*x))^(2+m)/(a^2*d*(2+m))+6*(a+a*sin(c+d*x))^(3+m)/(a^3*d*(3+m))-4*(a+a*sin(c+d*x))^(4+m)/(a^4*d*(4+m))+(a+a*sin(c+d*x))^(5+m)/(a^5*d*(5+m))],
[cos(c+d*x)*sin(c+d*x)^3*(a+a*sin(c+d*x))^m,x,4,-(a+a*sin(c+d*x))^(1+m)/(a*d*(1+m))+3*(a+a*sin(c+d*x))^(2+m)/(a^2*d*(2+m))-3*(a+a*sin(c+d*x))^(3+m)/(a^3*d*(3+m))+(a+a*sin(c+d*x))^(4+m)/(a^4*d*(4+m))],
[cos(c+d*x)*sin(c+d*x)^2*(a+a*sin(c+d*x))^m,x,4,(a+a*sin(c+d*x))^(1+m)/(a*d*(1+m))-2*(a+a*sin(c+d*x))^(2+m)/(a^2*d*(2+m))+(a+a*sin(c+d*x))^(3+m)/(a^3*d*(3+m))],
[cos(c+d*x)*sin(c+d*x)*(a+a*sin(c+d*x))^m,x,4,-(a+a*sin(c+d*x))^(1+m)/(a*d*(1+m))+(a+a*sin(c+d*x))^(2+m)/(a^2*d*(2+m))],
[cos(c+d*x)*csc(c+d*x)*(a+a*sin(c+d*x))^m,x,2,-hypergeom([1,1+m],[2+m],1+sin(c+d*x))*(a+a*sin(c+d*x))^(1+m)/(a*d*(1+m))],
[cos(c+d*x)*csc(c+d*x)^2*(a+a*sin(c+d*x))^m,x,3,hypergeom([2,1+m],[2+m],1+sin(c+d*x))*(a+a*sin(c+d*x))^(1+m)/(a*d*(1+m))],
[cos(c+d*x)*csc(c+d*x)^3*(a+a*sin(c+d*x))^m,x,3,-hypergeom([3,1+m],[2+m],1+sin(c+d*x))*(a+a*sin(c+d*x))^(1+m)/(a*d*(1+m))],

# Integrands of the form Cos[e+f x]^2 (a+a Sin[e+f x])^m (c+d Sin[e+f x])^n

# Integrands of the form Cos[e+f x]^2 (a+a Sin[e+f x])^m (c+d Sin[e+f x])^n
[cos(e+f*x)^2*(a+a*sin(e+f*x))*(c+d*sin(e+f*x)),x,4,1/8*a*(4*c+d)*x-1/3*a*(c+d)*cos(e+f*x)^3/f+1/8*a*(4*c+d)*cos(e+f*x)*sin(e+f*x)/f-1/4*a*d*cos(e+f*x)^3*sin(e+f*x)/f,1/8*a*(4*c+d)*x-1/12*a*(4*c+d)*cos(e+f*x)^3/f+1/8*a*(4*c+d)*cos(e+f*x)*sin(e+f*x)/f-1/4*d*cos(e+f*x)^3*(a+a*sin(e+f*x))/f],

# Integrands of the form Cos[e+f x]^2 (a+a Sin[e+f x])^(m/2) (c+d Sin[e+f x])^n
[cos(e+f*x)^2/((a+a*sin(e+f*x))^(3/2)*(c+d*sin(e+f*x))),x,6,-2*arctanh(cos(e+f*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(e+f*x))))*sqrt(2)/(a^(3/2)*(c-d)*f)+2*arctanh(cos(e+f*x)*sqrt(a)*sqrt(d)/(sqrt(c+d)*sqrt(a+a*sin(e+f*x))))*sqrt(c+d)/(a^(3/2)*(c-d)*f*sqrt(d))],

# Integrands of the form Cos[e+f x]^2 (a+a Sin[e+f x])^(m/2) (c+d Sin[e+f x])^(n/2)
[cos(e+f*x)^2/((a+a*sin(e+f*x))^(3/2)*sqrt(c+d*sin(e+f*x))),x,6,-2*arctanh(cos(e+f*x)*sqrt(a)*sqrt(c-d)/(sqrt(2)*sqrt(a+a*sin(e+f*x))*sqrt(c+d*sin(e+f*x))))*sqrt(2)/(a^(3/2)*f*sqrt(c-d))+2*arctan(cos(e+f*x)*sqrt(a)*sqrt(d)/(sqrt(a+a*sin(e+f*x))*sqrt(c+d*sin(e+f*x))))/(a^(3/2)*f*sqrt(d))],

# Integrands of the form Cos[e+f x]^2 (a+a Sin[e+f x])^m (c+d Sin[e+f x])^n with m and/or n symbolic
[cos(e+f*x)^2*(a+a*sin(e+f*x))^m*(c+d*sin(e+f*x))^n,x,4,2*AppellF1(3/2+m,-1/2,-n,5/2+m,1/2*(1+sin(e+f*x)),-d*(1+sin(e+f*x))/(c-d))*cos(e+f*x)*(a+a*sin(e+f*x))^(1+m)*(c+d*sin(e+f*x))^n*sqrt(2)/(a*f*(3+2*m)*((c+d*sin(e+f*x))/(c-d))^n*sqrt(1-sin(e+f*x)))],
[cos(e+f*x)^2*(a+a*sin(e+f*x))^3*(c+d*sin(e+f*x))^n,x,3,-16/3*a^3*AppellF1(3/2,-7/2,-n,5/2,1/2*(1-sin(e+f*x)),d*(1-sin(e+f*x))/(c+d))*cos(e+f*x)*(1-sin(e+f*x))*(c+d*sin(e+f*x))^n*sqrt(2)/(f*((c+d*sin(e+f*x))/(c+d))^n*sqrt(1+sin(e+f*x)))],
[cos(e+f*x)^2*(a+a*sin(e+f*x))^2*(c+d*sin(e+f*x))^n,x,3,-8/3*a^2*AppellF1(3/2,-5/2,-n,5/2,1/2*(1-sin(e+f*x)),d*(1-sin(e+f*x))/(c+d))*cos(e+f*x)*(1-sin(e+f*x))*(c+d*sin(e+f*x))^n*sqrt(2)/(f*((c+d*sin(e+f*x))/(c+d))^n*sqrt(1+sin(e+f*x)))],
[cos(e+f*x)^2*(a+a*sin(e+f*x))*(c+d*sin(e+f*x))^n,x,3,-4/3*a*AppellF1(3/2,-3/2,-n,5/2,1/2*(1-sin(e+f*x)),d*(1-sin(e+f*x))/(c+d))*cos(e+f*x)*(1-sin(e+f*x))*(c+d*sin(e+f*x))^n*sqrt(2)/(f*((c+d*sin(e+f*x))/(c+d))^n*sqrt(1+sin(e+f*x)))],
[cos(e+f*x)^2*(c+d*sin(e+f*x))^n/(a+a*sin(e+f*x)),x,4,-1/3*AppellF1(3/2,1/2,-n,5/2,1/2*(1-sin(e+f*x)),d*(1-sin(e+f*x))/(c+d))*cos(e+f*x)*(1-sin(e+f*x))*(c+d*sin(e+f*x))^n*sqrt(2)/(a*f*((c+d*sin(e+f*x))/(c+d))^n*sqrt(1+sin(e+f*x)))],
[cos(e+f*x)^2*(c+d*sin(e+f*x))^n/(a+a*sin(e+f*x))^2,x,3,-1/3*AppellF1(3/2,3/2,-n,5/2,1/2*(1-sin(e+f*x)),d*(1-sin(e+f*x))/(c+d))*cos(e+f*x)*(1-sin(e+f*x))*(c+d*sin(e+f*x))^n/(a^2*f*((c+d*sin(e+f*x))/(c+d))^n*sqrt(2)*sqrt(1+sin(e+f*x)))],
[cos(e+f*x)^2*(c+d*sin(e+f*x))^n/(a+a*sin(e+f*x))^3,x,3,-1/6*AppellF1(3/2,5/2,-n,5/2,1/2*(1-sin(e+f*x)),d*(1-sin(e+f*x))/(c+d))*cos(e+f*x)*(1-sin(e+f*x))*(c+d*sin(e+f*x))^n/(a^3*f*((c+d*sin(e+f*x))/(c+d))^n*sqrt(2)*sqrt(1+sin(e+f*x)))],

# Integrands of the form Cos[e+f x]^4 (a+a Sin[e+f x])^m (c+d Sin[e+f x])^n

# Integrands of the form Cos[e+f x]^4 (a+a Sin[e+f x])^m (c+d Sin[e+f x])^n with m and/or n symbolic
[cos(e+f*x)^4*(a+a*sin(e+f*x))^m*(c+d*sin(e+f*x))^n,x,4,4*AppellF1(5/2+m,-3/2,-n,7/2+m,1/2*(1+sin(e+f*x)),-d*(1+sin(e+f*x))/(c-d))*cos(e+f*x)*(a+a*sin(e+f*x))^(2+m)*(c+d*sin(e+f*x))^n*sqrt(2)/(a^2*f*(5+2*m)*((c+d*sin(e+f*x))/(c-d))^n*sqrt(1-sin(e+f*x)))],
[cos(e+f*x)^4*(a+a*sin(e+f*x))^2*(c+d*sin(e+f*x))^n,x,3,-16/5*a^2*AppellF1(5/2,-7/2,-n,7/2,1/2*(1-sin(e+f*x)),d*(1-sin(e+f*x))/(c+d))*cos(e+f*x)*(1-sin(e+f*x))^2*(c+d*sin(e+f*x))^n*sqrt(2)/(f*((c+d*sin(e+f*x))/(c+d))^n*sqrt(1+sin(e+f*x)))],
[cos(e+f*x)^4*(a+a*sin(e+f*x))*(c+d*sin(e+f*x))^n,x,3,-8/5*a*AppellF1(5/2,-5/2,-n,7/2,1/2*(1-sin(e+f*x)),d*(1-sin(e+f*x))/(c+d))*cos(e+f*x)^3*(1-sin(e+f*x))*(c+d*sin(e+f*x))^n*sqrt(2)/(f*(1+sin(e+f*x))^(3/2)*((c+d*sin(e+f*x))/(c+d))^n)],
[cos(e+f*x)^4*(c+d*sin(e+f*x))^n/(a+a*sin(e+f*x)),x,3,-2/5*AppellF1(5/2,-1/2,-n,7/2,1/2*(1-sin(e+f*x)),d*(1-sin(e+f*x))/(c+d))*cos(e+f*x)*(1-sin(e+f*x))^2*(c+d*sin(e+f*x))^n*sqrt(2)/(a*f*((c+d*sin(e+f*x))/(c+d))^n*sqrt(1+sin(e+f*x)))],
[cos(e+f*x)^4*(c+d*sin(e+f*x))^n/(a+a*sin(e+f*x))^2,x,4,-1/5*AppellF1(5/2,1/2,-n,7/2,1/2*(1-sin(e+f*x)),d*(1-sin(e+f*x))/(c+d))*cos(e+f*x)*(1-sin(e+f*x))^2*(c+d*sin(e+f*x))^n*sqrt(2)/(a^2*f*((c+d*sin(e+f*x))/(c+d))^n*sqrt(1+sin(e+f*x)))],
[cos(e+f*x)^4*(c+d*sin(e+f*x))^n/(a+a*sin(e+f*x))^3,x,3,-1/5*AppellF1(5/2,3/2,-n,7/2,1/2*(1-sin(e+f*x)),d*(1-sin(e+f*x))/(c+d))*cos(e+f*x)*(1-sin(e+f*x))^2*(c+d*sin(e+f*x))^n/(a^3*f*((c+d*sin(e+f*x))/(c+d))^n*sqrt(2)*sqrt(1+sin(e+f*x)))],
[cos(e+f*x)^4*(c+d*sin(e+f*x))^n/(a+a*sin(e+f*x))^4,x,3,-1/10*AppellF1(5/2,5/2,-n,7/2,1/2*(1-sin(e+f*x)),d*(1-sin(e+f*x))/(c+d))*cos(e+f*x)*(1-sin(e+f*x))^2*(c+d*sin(e+f*x))^n/(a^4*f*((c+d*sin(e+f*x))/(c+d))^n*sqrt(2)*sqrt(1+sin(e+f*x)))],
[cos(e+f*x)^4*(c+d*sin(e+f*x))^n/(a+a*sin(e+f*x))^5,x,3,-1/20*AppellF1(5/2,7/2,-n,7/2,1/2*(1-sin(e+f*x)),d*(1-sin(e+f*x))/(c+d))*cos(e+f*x)*(1-sin(e+f*x))^2*(c+d*sin(e+f*x))^n/(a^5*f*((c+d*sin(e+f*x))/(c+d))^n*sqrt(2)*sqrt(1+sin(e+f*x)))],

# Integrands of the form (g Cos[e+f x])^p (a+a Sin[e+f x])^m (c+d Sin[e+f x])

# Integrands of the form (e Cos[e+f x])^m (a+a Sin[e+f x])^n (A+B Sin[e+f x])

# n>0
[cos(c+d*x)^7*(a+a*sin(c+d*x))*(A+B*sin(c+d*x)),x,3,8/5*(A-B)*(a+a*sin(c+d*x))^5/(a^4*d)-2/3*(3*A-5*B)*(a+a*sin(c+d*x))^6/(a^5*d)+6/7*(A-3*B)*(a+a*sin(c+d*x))^7/(a^6*d)-1/8*(A-7*B)*(a+a*sin(c+d*x))^8/(a^7*d)-1/9*B*(a+a*sin(c+d*x))^9/(a^8*d)],
[cos(c+d*x)^5*(a+a*sin(c+d*x))*(A+B*sin(c+d*x)),x,3,(A-B)*(a+a*sin(c+d*x))^4/(a^3*d)-4/5*(A-2*B)*(a+a*sin(c+d*x))^5/(a^4*d)+1/6*(A-5*B)*(a+a*sin(c+d*x))^6/(a^5*d)+1/7*B*(a+a*sin(c+d*x))^7/(a^6*d)],
[cos(c+d*x)^3*(a+a*sin(c+d*x))*(A+B*sin(c+d*x)),x,3,2/3*(A-B)*(a+a*sin(c+d*x))^3/(a^2*d)-1/4*(A-3*B)*(a+a*sin(c+d*x))^4/(a^3*d)-1/5*B*(a+a*sin(c+d*x))^5/(a^4*d)],
[cos(c+d*x)*(a+a*sin(c+d*x))*(A+B*sin(c+d*x)),x,3,a*A*sin(c+d*x)/d+1/2*a*(A+B)*sin(c+d*x)^2/d+1/3*a*B*sin(c+d*x)^3/d],
[sec(c+d*x)*(a+a*sin(c+d*x))*(A+B*sin(c+d*x)),x,3,-a*(A+B)*log(1-sin(c+d*x))/d-a*B*sin(c+d*x)/d],
[sec(c+d*x)^3*(a+a*sin(c+d*x))*(A+B*sin(c+d*x)),x,4,1/2*a*(A-B)*arctanh(sin(c+d*x))/d+1/2*a^2*(A+B)/(d*(a-a*sin(c+d*x)))],
[sec(c+d*x)^5*(a+a*sin(c+d*x))*(A+B*sin(c+d*x)),x,4,1/8*a*(3*A-B)*arctanh(sin(c+d*x))/d+1/8*a^3*(A+B)/(d*(a-a*sin(c+d*x))^2)+1/4*a^2*A/(d*(a-a*sin(c+d*x)))-1/8*a^2*(A-B)/(d*(a+a*sin(c+d*x)))],
[sec(c+d*x)^7*(a+a*sin(c+d*x))*(A+B*sin(c+d*x)),x,4,1/16*a*(5*A-B)*arctanh(sin(c+d*x))/d+1/24*a^4*(A+B)/(d*(a-a*sin(c+d*x))^3)+1/32*a^3*(3*A+B)/(d*(a-a*sin(c+d*x))^2)+3/16*a^2*A/(d*(a-a*sin(c+d*x)))-1/32*a^3*(A-B)/(d*(a+a*sin(c+d*x))^2)-1/16*a^2*(2*A-B)/(d*(a+a*sin(c+d*x)))],
[cos(c+d*x)^6*(a+a*sin(c+d*x))*(A+B*sin(c+d*x)),x,6,5/128*a*(8*A+B)*x-1/56*a*(8*A+B)*cos(c+d*x)^7/d+5/128*a*(8*A+B)*cos(c+d*x)*sin(c+d*x)/d+5/192*a*(8*A+B)*cos(c+d*x)^3*sin(c+d*x)/d+1/48*a*(8*A+B)*cos(c+d*x)^5*sin(c+d*x)/d-1/8*B*cos(c+d*x)^7*(a+a*sin(c+d*x))/d],
[cos(c+d*x)^4*(a+a*sin(c+d*x))*(A+B*sin(c+d*x)),x,5,1/16*a*(6*A+B)*x-1/30*a*(6*A+B)*cos(c+d*x)^5/d+1/16*a*(6*A+B)*cos(c+d*x)*sin(c+d*x)/d+1/24*a*(6*A+B)*cos(c+d*x)^3*sin(c+d*x)/d-1/6*B*cos(c+d*x)^5*(a+a*sin(c+d*x))/d],
[cos(c+d*x)^2*(a+a*sin(c+d*x))*(A+B*sin(c+d*x)),x,4,1/8*a*(4*A+B)*x-1/12*a*(4*A+B)*cos(c+d*x)^3/d+1/8*a*(4*A+B)*cos(c+d*x)*sin(c+d*x)/d-1/4*B*cos(c+d*x)^3*(a+a*sin(c+d*x))/d],
[sec(c+d*x)^2*(a+a*sin(c+d*x))*(A+B*sin(c+d*x)),x,2,-a*B*x+(A+B)*sec(c+d*x)*(a+a*sin(c+d*x))/d],
[sec(c+d*x)^4*(a+a*sin(c+d*x))*(A+B*sin(c+d*x)),x,3,1/3*(A+B)*sec(c+d*x)^3*(a+a*sin(c+d*x))/d+1/3*a*(2*A-B)*tan(c+d*x)/d],
[sec(c+d*x)^6*(a+a*sin(c+d*x))*(A+B*sin(c+d*x)),x,3,1/5*(A+B)*sec(c+d*x)^5*(a+a*sin(c+d*x))/d+1/5*a*(4*A-B)*tan(c+d*x)/d+1/15*a*(4*A-B)*tan(c+d*x)^3/d],
[sec(c+d*x)^8*(a+a*sin(c+d*x))*(A+B*sin(c+d*x)),x,3,1/7*(A+B)*sec(c+d*x)^7*(a+a*sin(c+d*x))/d+1/7*a*(6*A-B)*tan(c+d*x)/d+2/21*a*(6*A-B)*tan(c+d*x)^3/d+1/35*a*(6*A-B)*tan(c+d*x)^5/d],
[sec(c+d*x)^10*(a+a*sin(c+d*x))*(A+B*sin(c+d*x)),x,3,1/9*(A+B)*sec(c+d*x)^9*(a+a*sin(c+d*x))/d+1/9*a*(8*A-B)*tan(c+d*x)/d+1/9*a*(8*A-B)*tan(c+d*x)^3/d+1/15*a*(8*A-B)*tan(c+d*x)^5/d+1/63*a*(8*A-B)*tan(c+d*x)^7/d],
[cos(c+d*x)^7*(a+a*sin(c+d*x))^2*(A+B*sin(c+d*x)),x,3,4/3*(A-B)*(a+a*sin(c+d*x))^6/(a^4*d)-4/7*(3*A-5*B)*(a+a*sin(c+d*x))^7/(a^5*d)+3/4*(A-3*B)*(a+a*sin(c+d*x))^8/(a^6*d)-1/9*(A-7*B)*(a+a*sin(c+d*x))^9/(a^7*d)-1/10*B*(a+a*sin(c+d*x))^10/(a^8*d)],
[cos(c+d*x)^5*(a+a*sin(c+d*x))^2*(A+B*sin(c+d*x)),x,3,4/5*(A-B)*(a+a*sin(c+d*x))^5/(a^3*d)-2/3*(A-2*B)*(a+a*sin(c+d*x))^6/(a^4*d)+1/7*(A-5*B)*(a+a*sin(c+d*x))^7/(a^5*d)+1/8*B*(a+a*sin(c+d*x))^8/(a^6*d)],
[cos(c+d*x)^3*(a+a*sin(c+d*x))^2*(A+B*sin(c+d*x)),x,3,1/2*(A-B)*(a+a*sin(c+d*x))^4/(a^2*d)-1/5*(A-3*B)*(a+a*sin(c+d*x))^5/(a^3*d)-1/6*B*(a+a*sin(c+d*x))^6/(a^4*d)],
[cos(c+d*x)*(a+a*sin(c+d*x))^2*(A+B*sin(c+d*x)),x,3,1/3*(A-B)*(a+a*sin(c+d*x))^3/(a*d)+1/4*B*(a+a*sin(c+d*x))^4/(a^2*d)],
[sec(c+d*x)*(a+a*sin(c+d*x))^2*(A+B*sin(c+d*x)),x,3,-2*a^2*(A+B)*log(1-sin(c+d*x))/d-a^2*(A+B)*sin(c+d*x)/d-1/2*B*(a+a*sin(c+d*x))^2/d],
[sec(c+d*x)^3*(a+a*sin(c+d*x))^2*(A+B*sin(c+d*x)),x,3,a^2*B*log(1-sin(c+d*x))/d+a^3*(A+B)/(d*(a-a*sin(c+d*x)))],
[sec(c+d*x)^5*(a+a*sin(c+d*x))^2*(A+B*sin(c+d*x)),x,4,1/4*a^2*(A-B)*arctanh(sin(c+d*x))/d+1/4*a^4*(A+B)/(d*(a-a*sin(c+d*x))^2)+1/4*a^3*(A-B)/(d*(a-a*sin(c+d*x)))],
[sec(c+d*x)^7*(a+a*sin(c+d*x))^2*(A+B*sin(c+d*x)),x,4,1/8*a^2*(2*A-B)*arctanh(sin(c+d*x))/d+1/12*a^5*(A+B)/(d*(a-a*sin(c+d*x))^3)+1/8*a^4*A/(d*(a-a*sin(c+d*x))^2)+1/16*a^3*(3*A-B)/(d*(a-a*sin(c+d*x)))-1/16*a^3*(A-B)/(d*(a+a*sin(c+d*x)))],
[cos(c+d*x)^6*(a+a*sin(c+d*x))^2*(A+B*sin(c+d*x)),x,7,5/128*a^2*(9*A+2*B)*x-1/56*a^2*(9*A+2*B)*cos(c+d*x)^7/d+5/128*a^2*(9*A+2*B)*cos(c+d*x)*sin(c+d*x)/d+5/192*a^2*(9*A+2*B)*cos(c+d*x)^3*sin(c+d*x)/d+1/48*a^2*(9*A+2*B)*cos(c+d*x)^5*sin(c+d*x)/d-1/9*B*cos(c+d*x)^7*(a+a*sin(c+d*x))^2/d-1/72*(9*A+2*B)*cos(c+d*x)^7*(a^2+a^2*sin(c+d*x))/d],
[cos(c+d*x)^4*(a+a*sin(c+d*x))^2*(A+B*sin(c+d*x)),x,6,1/16*a^2*(7*A+2*B)*x-1/30*a^2*(7*A+2*B)*cos(c+d*x)^5/d+1/16*a^2*(7*A+2*B)*cos(c+d*x)*sin(c+d*x)/d+1/24*a^2*(7*A+2*B)*cos(c+d*x)^3*sin(c+d*x)/d-1/7*B*cos(c+d*x)^5*(a+a*sin(c+d*x))^2/d-1/42*(7*A+2*B)*cos(c+d*x)^5*(a^2+a^2*sin(c+d*x))/d],
[cos(c+d*x)^2*(a+a*sin(c+d*x))^2*(A+B*sin(c+d*x)),x,5,1/8*a^2*(5*A+2*B)*x-1/12*a^2*(5*A+2*B)*cos(c+d*x)^3/d+1/8*a^2*(5*A+2*B)*cos(c+d*x)*sin(c+d*x)/d-1/5*B*cos(c+d*x)^3*(a+a*sin(c+d*x))^2/d-1/20*(5*A+2*B)*cos(c+d*x)^3*(a^2+a^2*sin(c+d*x))/d],
[sec(c+d*x)^2*(a+a*sin(c+d*x))^2*(A+B*sin(c+d*x)),x,3,-a^2*(A+2*B)*x+a^2*(A+2*B)*cos(c+d*x)/d+(A+B)*sec(c+d*x)*(a+a*sin(c+d*x))^2/d],
[sec(c+d*x)^4*(a+a*sin(c+d*x))^2*(A+B*sin(c+d*x)),x,4,1/3*a^2*(A-2*B)*sec(c+d*x)/d+1/3*(A+B)*sec(c+d*x)^3*(a+a*sin(c+d*x))^2/d+1/3*a^2*(A-2*B)*tan(c+d*x)/d],
[sec(c+d*x)^6*(a+a*sin(c+d*x))^2*(A+B*sin(c+d*x)),x,4,1/15*a^2*(3*A-2*B)*sec(c+d*x)^3/d+1/5*(A+B)*sec(c+d*x)^5*(a+a*sin(c+d*x))^2/d+1/5*a^2*(3*A-2*B)*tan(c+d*x)/d+1/15*a^2*(3*A-2*B)*tan(c+d*x)^3/d],
[sec(c+d*x)^8*(a+a*sin(c+d*x))^2*(A+B*sin(c+d*x)),x,4,1/35*a^2*(5*A-2*B)*sec(c+d*x)^5/d+1/7*(A+B)*sec(c+d*x)^7*(a+a*sin(c+d*x))^2/d+1/7*a^2*(5*A-2*B)*tan(c+d*x)/d+2/21*a^2*(5*A-2*B)*tan(c+d*x)^3/d+1/35*a^2*(5*A-2*B)*tan(c+d*x)^5/d],
[sec(c+d*x)^10*(a+a*sin(c+d*x))^2*(A+B*sin(c+d*x)),x,4,1/63*a^2*(7*A-2*B)*sec(c+d*x)^7/d+1/9*(A+B)*sec(c+d*x)^9*(a+a*sin(c+d*x))^2/d+1/9*a^2*(7*A-2*B)*tan(c+d*x)/d+1/9*a^2*(7*A-2*B)*tan(c+d*x)^3/d+1/15*a^2*(7*A-2*B)*tan(c+d*x)^5/d+1/63*a^2*(7*A-2*B)*tan(c+d*x)^7/d],
[sec(c+d*x)^12*(a+a*sin(c+d*x))^2*(A+B*sin(c+d*x)),x,4,1/99*a^2*(9*A-2*B)*sec(c+d*x)^9/d+1/11*(A+B)*sec(c+d*x)^11*(a+a*sin(c+d*x))^2/d+1/11*a^2*(9*A-2*B)*tan(c+d*x)/d+4/33*a^2*(9*A-2*B)*tan(c+d*x)^3/d+6/55*a^2*(9*A-2*B)*tan(c+d*x)^5/d+4/77*a^2*(9*A-2*B)*tan(c+d*x)^7/d+1/99*a^2*(9*A-2*B)*tan(c+d*x)^9/d],
[cos(c+d*x)^7*(a+a*sin(c+d*x))^3*(A+B*sin(c+d*x)),x,3,8/7*(A-B)*(a+a*sin(c+d*x))^7/(a^4*d)-1/2*(3*A-5*B)*(a+a*sin(c+d*x))^8/(a^5*d)+2/3*(A-3*B)*(a+a*sin(c+d*x))^9/(a^6*d)-1/10*(A-7*B)*(a+a*sin(c+d*x))^10/(a^7*d)-1/11*B*(a+a*sin(c+d*x))^11/(a^8*d)],
[cos(c+d*x)^5*(a+a*sin(c+d*x))^3*(A+B*sin(c+d*x)),x,3,2/3*(A-B)*(a+a*sin(c+d*x))^6/(a^3*d)-4/7*(A-2*B)*(a+a*sin(c+d*x))^7/(a^4*d)+1/8*(A-5*B)*(a+a*sin(c+d*x))^8/(a^5*d)+1/9*B*(a+a*sin(c+d*x))^9/(a^6*d)],
[cos(c+d*x)^3*(a+a*sin(c+d*x))^3*(A+B*sin(c+d*x)),x,3,2/5*(A-B)*(a+a*sin(c+d*x))^5/(a^2*d)-1/6*(A-3*B)*(a+a*sin(c+d*x))^6/(a^3*d)-1/7*B*(a+a*sin(c+d*x))^7/(a^4*d)],
[cos(c+d*x)*(a+a*sin(c+d*x))^3*(A+B*sin(c+d*x)),x,3,1/4*(A-B)*(a+a*sin(c+d*x))^4/(a*d)+1/5*B*(a+a*sin(c+d*x))^5/(a^2*d)],
[sec(c+d*x)*(a+a*sin(c+d*x))^3*(A+B*sin(c+d*x)),x,3,-4*a^3*(A+B)*log(1-sin(c+d*x))/d-3*a^3*(A+B)*sin(c+d*x)/d-1/2*a^3*(A+B)*sin(c+d*x)^2/d-1/3*B*(a+a*sin(c+d*x))^3/d],
[sec(c+d*x)^3*(a+a*sin(c+d*x))^3*(A+B*sin(c+d*x)),x,3,a^3*(A+3*B)*log(1-sin(c+d*x))/d+a^3*B*sin(c+d*x)/d+2*a^4*(A+B)/(d*(a-a*sin(c+d*x)))],
[sec(c+d*x)^5*(a+a*sin(c+d*x))^3*(A+B*sin(c+d*x)),x,2,1/2*a^3*(a*A+a*B*sin(c+d*x))^2/((A+B)*d*(a-a*sin(c+d*x))^2)],
[sec(c+d*x)^7*(a+a*sin(c+d*x))^3*(A+B*sin(c+d*x)),x,4,1/8*a^3*(A-B)*arctanh(sin(c+d*x))/d+1/6*a^6*(A+B)/(d*(a-a*sin(c+d*x))^3)+1/8*a^5*(A-B)/(d*(a-a*sin(c+d*x))^2)+1/8*a^4*(A-B)/(d*(a-a*sin(c+d*x)))],
[sec(c+d*x)^9*(a+a*sin(c+d*x))^3*(A+B*sin(c+d*x)),x,4,1/32*a^3*(5*A-3*B)*arctanh(sin(c+d*x))/d+1/16*a^7*(A+B)/(d*(a-a*sin(c+d*x))^4)+1/12*a^6*A/(d*(a-a*sin(c+d*x))^3)+1/32*a^5*(3*A-B)/(d*(a-a*sin(c+d*x))^2)+1/16*a^4*(2*A-B)/(d*(a-a*sin(c+d*x)))-1/32*a^4*(A-B)/(d*(a+a*sin(c+d*x)))],
[cos(c+d*x)^6*(a+a*sin(c+d*x))^3*(A+B*sin(c+d*x)),x,8,11/256*a^3*(10*A+3*B)*x-11/560*a^3*(10*A+3*B)*cos(c+d*x)^7/d+11/256*a^3*(10*A+3*B)*cos(c+d*x)*sin(c+d*x)/d+11/384*a^3*(10*A+3*B)*cos(c+d*x)^3*sin(c+d*x)/d+11/480*a^3*(10*A+3*B)*cos(c+d*x)^5*sin(c+d*x)/d-1/90*a*(10*A+3*B)*cos(c+d*x)^7*(a+a*sin(c+d*x))^2/d-1/10*B*cos(c+d*x)^7*(a+a*sin(c+d*x))^3/d-11/720*(10*A+3*B)*cos(c+d*x)^7*(a^3+a^3*sin(c+d*x))/d],
[cos(c+d*x)^4*(a+a*sin(c+d*x))^3*(A+B*sin(c+d*x)),x,7,9/128*a^3*(8*A+3*B)*x-3/80*a^3*(8*A+3*B)*cos(c+d*x)^5/d+9/128*a^3*(8*A+3*B)*cos(c+d*x)*sin(c+d*x)/d+3/64*a^3*(8*A+3*B)*cos(c+d*x)^3*sin(c+d*x)/d-1/56*a*(8*A+3*B)*cos(c+d*x)^5*(a+a*sin(c+d*x))^2/d-1/8*B*cos(c+d*x)^5*(a+a*sin(c+d*x))^3/d-3/112*(8*A+3*B)*cos(c+d*x)^5*(a^3+a^3*sin(c+d*x))/d],
[cos(c+d*x)^2*(a+a*sin(c+d*x))^3*(A+B*sin(c+d*x)),x,6,7/16*a^3*(2*A+B)*x-7/24*a^3*(2*A+B)*cos(c+d*x)^3/d+7/16*a^3*(2*A+B)*cos(c+d*x)*sin(c+d*x)/d-1/10*a*(2*A+B)*cos(c+d*x)^3*(a+a*sin(c+d*x))^2/d-1/6*B*cos(c+d*x)^3*(a+a*sin(c+d*x))^3/d-7/40*(2*A+B)*cos(c+d*x)^3*(a^3+a^3*sin(c+d*x))/d],
[sec(c+d*x)^2*(a+a*sin(c+d*x))^3*(A+B*sin(c+d*x)),x,2,-3/2*a^3*(2*A+3*B)*x+2*a^3*(2*A+3*B)*cos(c+d*x)/d+1/2*a^3*(2*A+3*B)*cos(c+d*x)*sin(c+d*x)/d+(A+B)*sec(c+d*x)*(a+a*sin(c+d*x))^3/d],
[sec(c+d*x)^4*(a+a*sin(c+d*x))^3*(A+B*sin(c+d*x)),x,4,a^3*B*x+1/3*(A+B)*sec(c+d*x)^3*(a+a*sin(c+d*x))^3/d-2*a^5*B*cos(c+d*x)/(d*(a^2-a^2*sin(c+d*x)))],
[sec(c+d*x)^6*(a+a*sin(c+d*x))^3*(A+B*sin(c+d*x)),x,4,1/15*a^5*(2*A-3*B)*cos(c+d*x)/(d*(a-a*sin(c+d*x))^2)+1/5*(A+B)*sec(c+d*x)^5*(a+a*sin(c+d*x))^3/d+1/15*a^5*(2*A-3*B)*cos(c+d*x)/(d*(a^2-a^2*sin(c+d*x)))],
[sec(c+d*x)^8*(a+a*sin(c+d*x))^3*(A+B*sin(c+d*x)),x,4,1/7*(A+B)*sec(c+d*x)^7*(a+a*sin(c+d*x))^3/d+2/35*(4*A-3*B)*sec(c+d*x)^5*(a^3+a^3*sin(c+d*x))/d+3/35*a^3*(4*A-3*B)*tan(c+d*x)/d+1/35*a^3*(4*A-3*B)*tan(c+d*x)^3/d],
[sec(c+d*x)^10*(a+a*sin(c+d*x))^3*(A+B*sin(c+d*x)),x,4,1/9*(A+B)*sec(c+d*x)^9*(a+a*sin(c+d*x))^3/d+2/21*(2*A-B)*sec(c+d*x)^7*(a^3+a^3*sin(c+d*x))/d+5/21*a^3*(2*A-B)*tan(c+d*x)/d+10/63*a^3*(2*A-B)*tan(c+d*x)^3/d+1/21*a^3*(2*A-B)*tan(c+d*x)^5/d],

# n<0
[cos(c+d*x)^7*(A+B*sin(c+d*x))/(a+a*sin(c+d*x)),x,3,-(A+B)*(a-a*sin(c+d*x))^4/(a^5*d)+4/5*(A+2*B)*(a-a*sin(c+d*x))^5/(a^6*d)-1/6*(A+5*B)*(a-a*sin(c+d*x))^6/(a^7*d)+1/7*B*(a-a*sin(c+d*x))^7/(a^8*d)],
[cos(c+d*x)^5*(A+B*sin(c+d*x))/(a+a*sin(c+d*x)),x,3,-2/3*(A+B)*(a-a*sin(c+d*x))^3/(a^4*d)+1/4*(A+3*B)*(a-a*sin(c+d*x))^4/(a^5*d)-1/5*B*(a-a*sin(c+d*x))^5/(a^6*d)],
[cos(c+d*x)^3*(A+B*sin(c+d*x))/(a+a*sin(c+d*x)),x,3,A*sin(c+d*x)/(a*d)-1/2*(A-B)*sin(c+d*x)^2/(a*d)-1/3*B*sin(c+d*x)^3/(a*d)],
[cos(c+d*x)*(A+B*sin(c+d*x))/(a+a*sin(c+d*x)),x,3,(A-B)*log(1+sin(c+d*x))/(a*d)+B*sin(c+d*x)/(a*d)],
[sec(c+d*x)*(A+B*sin(c+d*x))/(a+a*sin(c+d*x)),x,4,1/2*(A+B)*arctanh(sin(c+d*x))/(a*d)+1/2*(-A+B)/(d*(a+a*sin(c+d*x)))],
[sec(c+d*x)^3*(A+B*sin(c+d*x))/(a+a*sin(c+d*x)),x,4,1/8*(3*A+B)*arctanh(sin(c+d*x))/(a*d)+1/8*(A+B)/(d*(a-a*sin(c+d*x)))-1/8*a*(A-B)/(d*(a+a*sin(c+d*x))^2)-1/4*A/(d*(a+a*sin(c+d*x)))],
[sec(c+d*x)^5*(A+B*sin(c+d*x))/(a+a*sin(c+d*x)),x,4,1/16*(5*A+B)*arctanh(sin(c+d*x))/(a*d)+1/32*a*(A+B)/(d*(a-a*sin(c+d*x))^2)+1/16*(2*A+B)/(d*(a-a*sin(c+d*x)))-1/24*a^2*(A-B)/(d*(a+a*sin(c+d*x))^3)-1/32*a*(3*A-B)/(d*(a+a*sin(c+d*x))^2)-3/16*A/(d*(a+a*sin(c+d*x)))],
[sec(c+d*x)^7*(A+B*sin(c+d*x))/(a+a*sin(c+d*x)),x,4,5/128*(7*A+B)*arctanh(sin(c+d*x))/(a*d)+1/96*a^2*(A+B)/(d*(a-a*sin(c+d*x))^3)+1/128*a*(5*A+3*B)/(d*(a-a*sin(c+d*x))^2)+5/128*(3*A+B)/(d*(a-a*sin(c+d*x)))-1/64*a^3*(A-B)/(d*(a+a*sin(c+d*x))^4)-1/48*a^2*(2*A-B)/(d*(a+a*sin(c+d*x))^3)-1/64*a*(5*A-B)/(d*(a+a*sin(c+d*x))^2)-5/32*A/(d*(a+a*sin(c+d*x)))],
[cos(c+d*x)^7*(A+B*sin(c+d*x))/(a+a*sin(c+d*x))^2,x,3,-1/2*(A+B)*(a-a*sin(c+d*x))^4/(a^6*d)+1/5*(A+3*B)*(a-a*sin(c+d*x))^5/(a^7*d)-1/6*B*(a-a*sin(c+d*x))^6/(a^8*d)],
[cos(c+d*x)^5*(A+B*sin(c+d*x))/(a+a*sin(c+d*x))^2,x,3,-1/3*(A+B)*(a-a*sin(c+d*x))^3/(a^5*d)+1/4*B*(a-a*sin(c+d*x))^4/(a^6*d)],
[cos(c+d*x)^3*(A+B*sin(c+d*x))/(a+a*sin(c+d*x))^2,x,3,2*(A-B)*log(1+sin(c+d*x))/(a^2*d)-(A-B)*sin(c+d*x)/(a^2*d)-1/2*B*(a-a*sin(c+d*x))^2/(a^4*d)],
[cos(c+d*x)*(A+B*sin(c+d*x))/(a+a*sin(c+d*x))^2,x,3,B*log(1+sin(c+d*x))/(a^2*d)+(-A+B)/(d*(a^2+a^2*sin(c+d*x)))],
[sec(c+d*x)*(A+B*sin(c+d*x))/(a+a*sin(c+d*x))^2,x,4,1/4*(A+B)*arctanh(sin(c+d*x))/(a^2*d)+1/4*(-A+B)/(d*(a+a*sin(c+d*x))^2)+1/4*(-A-B)/(d*(a^2+a^2*sin(c+d*x)))],
[sec(c+d*x)^3*(A+B*sin(c+d*x))/(a+a*sin(c+d*x))^2,x,4,1/8*(2*A+B)*arctanh(sin(c+d*x))/(a^2*d)-1/12*a*(A-B)/(d*(a+a*sin(c+d*x))^3)-1/8*A/(d*(a+a*sin(c+d*x))^2)+1/16*(A+B)/(d*(a^2-a^2*sin(c+d*x)))+1/16*(-3*A-B)/(d*(a^2+a^2*sin(c+d*x)))],
[sec(c+d*x)^5*(A+B*sin(c+d*x))/(a+a*sin(c+d*x))^2,x,4,5/64*(3*A+B)*arctanh(sin(c+d*x))/(a^2*d)+1/64*(A+B)/(d*(a-a*sin(c+d*x))^2)-1/32*a^2*(A-B)/(d*(a+a*sin(c+d*x))^4)-1/48*a*(3*A-B)/(d*(a+a*sin(c+d*x))^3)-3/32*A/(d*(a+a*sin(c+d*x))^2)+1/64*(5*A+3*B)/(d*(a^2-a^2*sin(c+d*x)))+1/32*(-5*A-B)/(d*(a^2+a^2*sin(c+d*x)))],
[sec(c+d*x)^7*(A+B*sin(c+d*x))/(a+a*sin(c+d*x))^2,x,4,7/128*(4*A+B)*arctanh(sin(c+d*x))/(a^2*d)+1/192*a*(A+B)/(d*(a-a*sin(c+d*x))^3)+1/128*(3*A+2*B)/(d*(a-a*sin(c+d*x))^2)-1/80*a^3*(A-B)/(d*(a+a*sin(c+d*x))^5)-1/64*a^2*(2*A-B)/(d*(a+a*sin(c+d*x))^4)-1/96*a*(5*A-B)/(d*(a+a*sin(c+d*x))^3)-5/64*A/(d*(a+a*sin(c+d*x))^2)+3/256*(7*A+3*B)/(d*(a^2-a^2*sin(c+d*x)))-5/256*(7*A+B)/(d*(a^2+a^2*sin(c+d*x)))],

# Integrands of the form (e Cos[e+f x])^m (a+a Sin[e+f x])^(n/2) (A+B Sin[e+f x])

# Integrands of the form (e Cos[e+f x])^(m/2) (a+a Sin[e+f x])^n (A+B Sin[e+f x])

# Integrands of the form (e Cos[e+f x])^(m/2) (a+a Sin[e+f x])^(n/2) (A+B Sin[e+f x])

# Integrands of the form (e Cos[e+f x])^m (a+a Sin[e+f x])^n (A+B Sin[e+f x]) with n symbolic
[(g*cos(e+f*x))^p*(a+a*sin(e+f*x))^m*(A+B*sin(e+f*x)),x,4,-2^(1/2*(1+2*m+p))*a*(B*m+A*(1+m+p))*(g*cos(e+f*x))^(1+p)*hypergeom([1/2*(1-2*m-p),1/2*(1+p)],[1/2*(3+p)],1/2*(1-sin(e+f*x)))*(1+sin(e+f*x))^(1/2*(1-2*m-p))*(a+a*sin(e+f*x))^(-1+m)/(f*g*(1+p)*(1+m+p))-B*(g*cos(e+f*x))^(1+p)*(a+a*sin(e+f*x))^m/(f*g*(1+m+p))],
[cos(e+f*x)^7*(a+a*sin(e+f*x))^m*(A+B*sin(e+f*x)),x,3,8*(A-B)*(a+a*sin(e+f*x))^(4+m)/(a^4*f*(4+m))-4*(3*A-5*B)*(a+a*sin(e+f*x))^(5+m)/(a^5*f*(5+m))+6*(A-3*B)*(a+a*sin(e+f*x))^(6+m)/(a^6*f*(6+m))-(A-7*B)*(a+a*sin(e+f*x))^(7+m)/(a^7*f*(7+m))-B*(a+a*sin(e+f*x))^(8+m)/(a^8*f*(8+m))],
[cos(e+f*x)^5*(a+a*sin(e+f*x))^m*(A+B*sin(e+f*x)),x,3,4*(A-B)*(a+a*sin(e+f*x))^(3+m)/(a^3*f*(3+m))-4*(A-2*B)*(a+a*sin(e+f*x))^(4+m)/(a^4*f*(4+m))+(A-5*B)*(a+a*sin(e+f*x))^(5+m)/(a^5*f*(5+m))+B*(a+a*sin(e+f*x))^(6+m)/(a^6*f*(6+m))],
[cos(e+f*x)^3*(a+a*sin(e+f*x))^m*(A+B*sin(e+f*x)),x,3,2*(A-B)*(a+a*sin(e+f*x))^(2+m)/(a^2*f*(2+m))-(A-3*B)*(a+a*sin(e+f*x))^(3+m)/(a^3*f*(3+m))-B*(a+a*sin(e+f*x))^(4+m)/(a^4*f*(4+m))],
[cos(e+f*x)*(a+a*sin(e+f*x))^m*(A+B*sin(e+f*x)),x,3,(A-B)*(a+a*sin(e+f*x))^(1+m)/(a*f*(1+m))+B*(a+a*sin(e+f*x))^(2+m)/(a^2*f*(2+m))],
[sec(e+f*x)*(a+a*sin(e+f*x))^m*(A+B*sin(e+f*x)),x,3,1/2*(A-B)*(a+a*sin(e+f*x))^m/(f*m)+1/4*(A+B)*hypergeom([1,1+m],[2+m],1/2*(1+sin(e+f*x)))*(a+a*sin(e+f*x))^(1+m)/(a*f*(1+m))],
[sec(e+f*x)^3*(a+a*sin(e+f*x))^m*(A+B*sin(e+f*x)),x,3,-1/4*a*(A*(2-m)-B*m)*hypergeom([1,-1+m],[m],1/2*(1+sin(e+f*x)))*(a+a*sin(e+f*x))^(-1+m)/(f*(1-m))+1/2*a^2*(A+B)*(a+a*sin(e+f*x))^(-1+m)/(f*(a-a*sin(e+f*x)))],
[sec(e+f*x)^5*(a+a*sin(e+f*x))^m*(A+B*sin(e+f*x)),x,3,-1/16*a^2*(A*(4-m)-B*m)*hypergeom([2,-2+m],[-1+m],1/2*(1+sin(e+f*x)))*(a+a*sin(e+f*x))^(-2+m)/(f*(2-m))+1/4*a^4*(A+B)*(a+a*sin(e+f*x))^(-2+m)/(f*(a-a*sin(e+f*x))^2)],
[cos(e+f*x)^6*(a+a*sin(e+f*x))^m*(A+B*sin(e+f*x)),x,4,-1/7*2^(7/2+m)*a^3*(B*m+A*(7+m))*cos(e+f*x)^7*hypergeom([7/2,-5/2-m],[9/2],1/2*(1-sin(e+f*x)))*(1+sin(e+f*x))^(-1/2-m)*(a+a*sin(e+f*x))^(-3+m)/(f*(7+m))-B*cos(e+f*x)^7*(a+a*sin(e+f*x))^m/(f*(7+m))],
[cos(e+f*x)^4*(a+a*sin(e+f*x))^m*(A+B*sin(e+f*x)),x,4,-1/5*2^(5/2+m)*a^2*(B*m+A*(5+m))*cos(e+f*x)^5*hypergeom([5/2,-3/2-m],[7/2],1/2*(1-sin(e+f*x)))*(1+sin(e+f*x))^(-1/2-m)*(a+a*sin(e+f*x))^(-2+m)/(f*(5+m))-B*cos(e+f*x)^5*(a+a*sin(e+f*x))^m/(f*(5+m))],
[cos(e+f*x)^2*(a+a*sin(e+f*x))^m*(A+B*sin(e+f*x)),x,4,-1/3*2^(3/2+m)*a*(B*m+A*(3+m))*cos(e+f*x)^3*hypergeom([3/2,-1/2-m],[5/2],1/2*(1-sin(e+f*x)))*(1+sin(e+f*x))^(-1/2-m)*(a+a*sin(e+f*x))^(-1+m)/(f*(3+m))-B*cos(e+f*x)^3*(a+a*sin(e+f*x))^m/(f*(3+m))],
[sec(e+f*x)^2*(a+a*sin(e+f*x))^m*(A+B*sin(e+f*x)),x,4,B*sec(e+f*x)*(a+a*sin(e+f*x))^m/(f*(1-m))+2^(-1/2+m)*(A*(1-m)-B*m)*hypergeom([-1/2,3/2-m],[1/2],1/2*(1-sin(e+f*x)))*sec(e+f*x)*(1+sin(e+f*x))^(1/2-m)*(a+a*sin(e+f*x))^m/(f*(1-m))],
[sec(e+f*x)^4*(a+a*sin(e+f*x))^m*(A+B*sin(e+f*x)),x,4,B*sec(e+f*x)^3*(a+a*sin(e+f*x))^m/(f*(3-m))+1/3*2^(-3/2+m)*(A*(3-m)-B*m)*hypergeom([-3/2,5/2-m],[-1/2],1/2*(1-sin(e+f*x)))*sec(e+f*x)^3*(1+sin(e+f*x))^(1/2-m)*(a+a*sin(e+f*x))^(1+m)/(a*f*(3-m))],
[sec(e+f*x)^6*(a+a*sin(e+f*x))^m*(A+B*sin(e+f*x)),x,4,B*sec(e+f*x)^5*(a+a*sin(e+f*x))^m/(f*(5-m))+1/5*2^(-5/2+m)*(A*(5-m)-B*m)*hypergeom([-5/2,7/2-m],[-3/2],1/2*(1-sin(e+f*x)))*sec(e+f*x)^5*(1+sin(e+f*x))^(1/2-m)*(a+a*sin(e+f*x))^(2+m)/(a^2*f*(5-m))],
[(g*cos(e+f*x))^p*(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^(-4-p),x,4,(A+B)*(g*cos(e+f*x))^(1+p)*(c-c*sin(e+f*x))^(-4-p)/(f*g*(7+p))+(3*A-B*(4+p))*(g*cos(e+f*x))^(1+p)*(c-c*sin(e+f*x))^(-3-p)/(c*f*g*(35+12*p+p^2))+2*(3*A-B*(4+p))*(g*cos(e+f*x))^(1+p)*(c-c*sin(e+f*x))^(-2-p)/(c^2*f*g*(5+p)*(21+10*p+p^2))+2*(3*A-B*(4+p))*(g*cos(e+f*x))^(1+p)*(c-c*sin(e+f*x))^(-1-p)/(c^3*f*g*(5+6*p+p^2)*(21+10*p+p^2))],
[(g*cos(e+f*x))^p*(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^(-3-p),x,3,(A+B)*(g*cos(e+f*x))^(1+p)*(c-c*sin(e+f*x))^(-3-p)/(f*g*(5+p))+(2*A-B*(3+p))*(g*cos(e+f*x))^(1+p)*(c-c*sin(e+f*x))^(-2-p)/(c*f*g*(15+8*p+p^2))+(2*A-B*(3+p))*(g*cos(e+f*x))^(1+p)*(c-c*sin(e+f*x))^(-1-p)/(c^2*f*g*(3+p)*(5+6*p+p^2))],
[(g*cos(e+f*x))^p*(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^(-2-p),x,2,(A+B)*(g*cos(e+f*x))^(1+p)*(c-c*sin(e+f*x))^(-2-p)/(f*g*(3+p))+(A-B*(2+p))*(g*cos(e+f*x))^(1+p)*(c-c*sin(e+f*x))^(-1-p)/(c*f*g*(3+4*p+p^2))],
[(g*cos(e+f*x))^p*(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^(-1-p),x,4,(A+B)*(g*cos(e+f*x))^(1+p)*(c-c*sin(e+f*x))^(-1-p)/(f*g*(1+p))-2^(1/2-1/2*p)*B*(g*cos(e+f*x))^(1+p)*hypergeom([1/2*(1+p),1/2*(1+p)],[1/2*(3+p)],1/2*(1+sin(e+f*x)))*(1-sin(e+f*x))^(1/2*(1+p))*(c-c*sin(e+f*x))^(-1-p)/(f*g*(1+p))],
[(g*cos(e+f*x))^p*(A+B*sin(e+f*x))/(c-c*sin(e+f*x))^p,x,4,2^(1/2-1/2*p)*c*(A+B*p)*(g*cos(e+f*x))^(1+p)*hypergeom([1/2*(1+p),1/2*(1+p)],[1/2*(3+p)],1/2*(1+sin(e+f*x)))*(1-sin(e+f*x))^(1/2*(1+p))*(c-c*sin(e+f*x))^(-1-p)/(f*g*(1+p))-B*(g*cos(e+f*x))^(1+p)/(f*g*(c-c*sin(e+f*x))^p)],
[(g*cos(e+f*x))^p*(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^(1-p),x,4,2^(1/2-1/2*p)*c^2*(2*A-B*(1-p))*(g*cos(e+f*x))^(1+p)*hypergeom([1/2*(-1+p),1/2*(1+p)],[1/2*(3+p)],1/2*(1+sin(e+f*x)))*(1-sin(e+f*x))^(1/2*(1+p))*(c-c*sin(e+f*x))^(-1-p)/(f*g*(1+p))-1/2*B*(g*cos(e+f*x))^(1+p)*(c-c*sin(e+f*x))^(1-p)/(f*g)],
[(g*cos(e+f*x))^p*(A+B*sin(e+f*x))*(c-c*sin(e+f*x))^(2-p),x,4,1/3*2^(5/2-1/2*p)*c^3*(3*A-B*(2-p))*(g*cos(e+f*x))^(1+p)*hypergeom([1/2*(-3+p),1/2*(1+p)],[1/2*(3+p)],1/2*(1+sin(e+f*x)))*(1-sin(e+f*x))^(1/2*(1+p))*(c-c*sin(e+f*x))^(-1-p)/(f*g*(1+p))-1/3*B*(g*cos(e+f*x))^(1+p)*(c-c*sin(e+f*x))^(2-p)/(f*g)],
[(g*cos(e+f*x))^p*(a+a*sin(e+f*x))^m*(A*m-A*(1+m+p)*sin(e+f*x)),x,1,A*(g*cos(e+f*x))^(1+p)*(a+a*sin(e+f*x))^m/(f*g)],
[(g*cos(e+f*x))^p*(a-a*sin(e+f*x))^m*(A*m+A*(1+m+p)*sin(e+f*x)),x,1,-A*(g*cos(e+f*x))^(1+p)*(a-a*sin(e+f*x))^m/(f*g)],

# Integrands of the form (g Cos[e+f x])^p (a+a Sin[e+f x])^m (c+d Sin[e+f x])^n with m, n and p symbolic
[(g*cos(e+f*x))^p*(a+a*sin(e+f*x))^m*(c+d*sin(e+f*x))^n,x,4,2^(1/2*(1+p))*g*AppellF1(1/2*(1+2*m+p),1/2*(1-p),-n,1/2*(3+2*m+p),1/2*(1+sin(e+f*x)),-d*(1+sin(e+f*x))/(c-d))*(g*cos(e+f*x))^(-1+p)*(1-sin(e+f*x))^(1/2*(1-p))*(a+a*sin(e+f*x))^(1+m)*(c+d*sin(e+f*x))^n/(a*f*(1+2*m+p)*((c+d*sin(e+f*x))/(c-d))^n)],
[(g*cos(e+f*x))^p*(a+a*sin(e+f*x))^2*(c+d*sin(e+f*x))^n,x,3,-2^(5/2+1/2*p)*a^2*AppellF1(1/2*(1+p),1/2*(-3-p),-n,1/2*(3+p),1/2*(1-sin(e+f*x)),d*(1-sin(e+f*x))/(c+d))*(g*cos(e+f*x))^(1+p)*(1+sin(e+f*x))^(2+1/2*(-5-p))*(c+d*sin(e+f*x))^n/(f*g*(1+p)*((c+d*sin(e+f*x))/(c+d))^n),-2^(1/2*(5+p))*a^2*g*AppellF1(1/2*(1+p),1/2*(-3-p),-n,1/2*(3+p),1/2*(1-sin(e+f*x)),d*(1-sin(e+f*x))/(c+d))*(g*cos(e+f*x))^(-1+p)*(1-sin(e+f*x))*(1+sin(e+f*x))^(1/2*(1-p))*(c+d*sin(e+f*x))^n/(f*(1+p)*((c+d*sin(e+f*x))/(c+d))^n)],
[(g*cos(e+f*x))^p*(a+a*sin(e+f*x))*(c+d*sin(e+f*x))^n,x,3,-2^(3/2+1/2*p)*a*AppellF1(1/2*(1+p),1/2*(-1-p),-n,1/2*(3+p),1/2*(1-sin(e+f*x)),d*(1-sin(e+f*x))/(c+d))*(g*cos(e+f*x))^(1+p)*(1+sin(e+f*x))^(1/2*(-1-p))*(c+d*sin(e+f*x))^n/(f*g*(1+p)*((c+d*sin(e+f*x))/(c+d))^n),-2^(1/2*(3+p))*a*g*AppellF1(1/2*(1+p),1/2*(-1-p),-n,1/2*(3+p),1/2*(1-sin(e+f*x)),d*(1-sin(e+f*x))/(c+d))*(g*cos(e+f*x))^(-1+p)*(1-sin(e+f*x))*(1+sin(e+f*x))^(1/2*(1-p))*(c+d*sin(e+f*x))^n/(f*(1+p)*((c+d*sin(e+f*x))/(c+d))^n)],
[(g*cos(e+f*x))^p*(c+d*sin(e+f*x))^n/(a+a*sin(e+f*x)),x,3,-2^(-1/2+1/2*p)*AppellF1(1/2*(1+p),1/2*(3-p),-n,1/2*(3+p),1/2*(1-sin(e+f*x)),d*(1-sin(e+f*x))/(c+d))*(g*cos(e+f*x))^(1+p)*(1+sin(e+f*x))^(-1+1/2*(1-p))*(c+d*sin(e+f*x))^n/(a*f*g*(1+p)*((c+d*sin(e+f*x))/(c+d))^n),-2^(-1/2+1/2*p)*g*AppellF1(1/2*(1+p),1/2*(3-p),-n,1/2*(3+p),1/2*(1-sin(e+f*x)),d*(1-sin(e+f*x))/(c+d))*(g*cos(e+f*x))^(-1+p)*(1-sin(e+f*x))*(1+sin(e+f*x))^(1/2*(1-p))*(c+d*sin(e+f*x))^n/(a*f*(1+p)*((c+d*sin(e+f*x))/(c+d))^n)],
[(g*cos(e+f*x))^p*(c+d*sin(e+f*x))^n/(a+a*sin(e+f*x))^2,x,3,-2^(-3/2+1/2*p)*AppellF1(1/2*(1+p),1/2*(5-p),-n,1/2*(3+p),1/2*(1-sin(e+f*x)),d*(1-sin(e+f*x))/(c+d))*(g*cos(e+f*x))^(1+p)*(1+sin(e+f*x))^(-2+1/2*(3-p))*(c+d*sin(e+f*x))^n/(a^2*f*g*(1+p)*((c+d*sin(e+f*x))/(c+d))^n),-2^(1/2*(-3+p))*g*AppellF1(1/2*(1+p),1/2*(5-p),-n,1/2*(3+p),1/2*(1-sin(e+f*x)),d*(1-sin(e+f*x))/(c+d))*(g*cos(e+f*x))^(-1+p)*(1-sin(e+f*x))*(1+sin(e+f*x))^(1/2*(1-p))*(c+d*sin(e+f*x))^n/(a^2*f*(1+p)*((c+d*sin(e+f*x))/(c+d))^n)],
[(g*cos(e+f*x))^p*(c+d*sin(e+f*x))^n/(a+a*sin(e+f*x))^3,x,3,-2^(-5/2+1/2*p)*AppellF1(1/2*(1+p),1/2*(7-p),-n,1/2*(3+p),1/2*(1-sin(e+f*x)),d*(1-sin(e+f*x))/(c+d))*(g*cos(e+f*x))^(1+p)*(1+sin(e+f*x))^(-3+1/2*(5-p))*(c+d*sin(e+f*x))^n/(a^3*f*g*(1+p)*((c+d*sin(e+f*x))/(c+d))^n),-2^(1/2*(-5+p))*g*AppellF1(1/2*(1+p),1/2*(7-p),-n,1/2*(3+p),1/2*(1-sin(e+f*x)),d*(1-sin(e+f*x))/(c+d))*(g*cos(e+f*x))^(-1+p)*(1-sin(e+f*x))*(1+sin(e+f*x))^(1/2*(1-p))*(c+d*sin(e+f*x))^n/(a^3*f*(1+p)*((c+d*sin(e+f*x))/(c+d))^n)],
[(g*cos(e+f*x))^p*(c+d*sin(e+f*x))^n/(a+a*sin(e+f*x))^4,x,3,-2^(-7/2+1/2*p)*AppellF1(1/2*(1+p),1/2*(9-p),-n,1/2*(3+p),1/2*(1-sin(e+f*x)),d*(1-sin(e+f*x))/(c+d))*(g*cos(e+f*x))^(1+p)*(1+sin(e+f*x))^(-4+1/2*(7-p))*(c+d*sin(e+f*x))^n/(a^4*f*g*(1+p)*((c+d*sin(e+f*x))/(c+d))^n),-2^(1/2*(-7+p))*g*AppellF1(1/2*(1+p),1/2*(9-p),-n,1/2*(3+p),1/2*(1-sin(e+f*x)),d*(1-sin(e+f*x))/(c+d))*(g*cos(e+f*x))^(-1+p)*(1-sin(e+f*x))*(1+sin(e+f*x))^(1/2*(1-p))*(c+d*sin(e+f*x))^n/(a^4*f*(1+p)*((c+d*sin(e+f*x))/(c+d))^n)],

# Integrands of the form (g Sec[e+f x])^p (a+a Sin[e+f x])^m (c+d Sin[e+f x])^n

# Integrands of the form (g Sec[e+f x])^p (a+a Sin[e+f x])^m (c+d Sin[e+f x])^n with m, n and p symbolic
[(g*sec(e+f*x))^p*(a+a*sin(e+f*x))^m*(c+d*sin(e+f*x))^n,x,5,2^(1/2-1/2*p)*AppellF1(1/2*(1+2*m-p),1/2*(1+p),-n,1/2*(3+2*m-p),1/2*(1+sin(e+f*x)),-d*(1+sin(e+f*x))/(c-d))*sec(e+f*x)*(g*sec(e+f*x))^p*(1-sin(e+f*x))^(1/2*(1+p))*(a+a*sin(e+f*x))^(1+m)*(c+d*sin(e+f*x))^n/(a*f*(1+2*m-p)*((c+d*sin(e+f*x))/(c-d))^n)],

# Integrands of the form (g Cos[e+f x])^p (d Sin[e+f x])^n (a+b Sin[e+f x])^m

# Integrands of the form Cos[e+f x]^1 (d Sin[e+f x])^n (a+b Sin[e+f x])^m

# Integrands of the form Cos[e+f x]^2 (d Sin[e+f x])^n (a+b Sin[e+f x])^m

# Integrands of the form Cos[e+f x]^2 Sin[e+f x]^n (a+b Sin[e+f x])^m

# m>0
[cos(c+d*x)^2*sin(c+d*x)^3*(a+b*sin(c+d*x)),x,8,1/16*b*x-1/3*a*cos(c+d*x)^3/d+1/5*a*cos(c+d*x)^5/d+1/16*b*cos(c+d*x)*sin(c+d*x)/d-1/8*b*cos(c+d*x)^3*sin(c+d*x)/d-1/6*b*cos(c+d*x)^3*sin(c+d*x)^3/d],
[cos(c+d*x)^2*sin(c+d*x)^2*(a+b*sin(c+d*x)),x,7,1/8*a*x-1/3*b*cos(c+d*x)^3/d+1/5*b*cos(c+d*x)^5/d+1/8*a*cos(c+d*x)*sin(c+d*x)/d-1/4*a*cos(c+d*x)^3*sin(c+d*x)/d],
[cos(c+d*x)^2*sin(c+d*x)*(a+b*sin(c+d*x)),x,6,1/8*b*x-1/3*a*cos(c+d*x)^3/d+1/8*b*cos(c+d*x)*sin(c+d*x)/d-1/4*b*cos(c+d*x)^3*sin(c+d*x)/d],
[cos(c+d*x)^2*csc(c+d*x)*(a+b*sin(c+d*x)),x,6,1/2*b*x-a*arctanh(cos(c+d*x))/d+a*cos(c+d*x)/d+1/2*b*cos(c+d*x)*sin(c+d*x)/d],
[cos(c+d*x)^2*csc(c+d*x)^2*(a+b*sin(c+d*x)),x,7,-a*x-b*arctanh(cos(c+d*x))/d+b*cos(c+d*x)/d-a*cot(c+d*x)/d],
[cos(c+d*x)^2*csc(c+d*x)^3*(a+b*sin(c+d*x)),x,5,-b*x+1/2*a*arctanh(cos(c+d*x))/d-b*cot(c+d*x)/d-1/2*a*cot(c+d*x)*csc(c+d*x)/d],
[cos(c+d*x)^2*csc(c+d*x)^4*(a+b*sin(c+d*x)),x,5,1/2*b*arctanh(cos(c+d*x))/d-1/3*a*cot(c+d*x)^3/d-1/2*b*cot(c+d*x)*csc(c+d*x)/d],
[cos(c+d*x)^2*csc(c+d*x)^5*(a+b*sin(c+d*x)),x,6,1/8*a*arctanh(cos(c+d*x))/d-1/3*b*cot(c+d*x)^3/d+1/8*a*cot(c+d*x)*csc(c+d*x)/d-1/4*a*cot(c+d*x)*csc(c+d*x)^3/d],
[cos(c+d*x)^2*csc(c+d*x)^6*(a+b*sin(c+d*x)),x,7,1/8*b*arctanh(cos(c+d*x))/d-1/3*a*cot(c+d*x)^3/d-1/5*a*cot(c+d*x)^5/d+1/8*b*cot(c+d*x)*csc(c+d*x)/d-1/4*b*cot(c+d*x)*csc(c+d*x)^3/d],
[cos(c+d*x)^2*sin(c+d*x)^3*(a+b*sin(c+d*x))^2,x,10,1/8*a*b*x-1/35*(7*a^2+4*b^2)*cos(c+d*x)/d+1/105*(7*a^2+4*b^2)*cos(c+d*x)^3/d-1/8*a*b*cos(c+d*x)*sin(c+d*x)/d-1/12*a*b*cos(c+d*x)*sin(c+d*x)^3/d+1/35*(2*a^2-b^2)*cos(c+d*x)*sin(c+d*x)^4/d+1/21*a*b*cos(c+d*x)*sin(c+d*x)^5/d+1/7*cos(c+d*x)*sin(c+d*x)^4*(a+b*sin(c+d*x))^2/d],
[cos(c+d*x)^2*sin(c+d*x)^2*(a+b*sin(c+d*x))^2,x,9,1/16*(2*a^2+b^2)*x-2/5*a*b*cos(c+d*x)/d+2/15*a*b*cos(c+d*x)^3/d-1/16*(2*a^2+b^2)*cos(c+d*x)*sin(c+d*x)/d+1/24*(2*a^2-b^2)*cos(c+d*x)*sin(c+d*x)^3/d+1/15*a*b*cos(c+d*x)*sin(c+d*x)^4/d+1/6*cos(c+d*x)*sin(c+d*x)^3*(a+b*sin(c+d*x))^2/d],
[cos(c+d*x)^2*sin(c+d*x)*(a+b*sin(c+d*x))^2,x,5,1/4*a*b*x-1/30*(a^2+4*b^2)*cos(c+d*x)^3/d+1/4*a*b*cos(c+d*x)*sin(c+d*x)/d-1/10*a*cos(c+d*x)^3*(a+b*sin(c+d*x))/d-1/5*cos(c+d*x)^3*(a+b*sin(c+d*x))^2/d],
[cos(c+d*x)^2*csc(c+d*x)*(a+b*sin(c+d*x))^2,x,6,a*b*x-a^2*arctanh(cos(c+d*x))/d+1/3*(2*a^2-b^2)*cos(c+d*x)/d+1/3*a*b*cos(c+d*x)*sin(c+d*x)/d+1/3*cos(c+d*x)*(a+b*sin(c+d*x))^2/d],
[cos(c+d*x)^2*csc(c+d*x)^2*(a+b*sin(c+d*x))^2,x,9,-a^2*x+1/2*b^2*x-2*a*b*arctanh(cos(c+d*x))/d+2*a*b*cos(c+d*x)/d-a^2*cot(c+d*x)/d+1/2*b^2*cos(c+d*x)*sin(c+d*x)/d],
[cos(c+d*x)^2*csc(c+d*x)^3*(a+b*sin(c+d*x))^2,x,6,-2*a*b*x+1/2*(a^2-2*b^2)*arctanh(cos(c+d*x))/d+3/2*b^2*cos(c+d*x)/d-a*b*cot(c+d*x)/d-1/2*cot(c+d*x)*csc(c+d*x)*(a+b*sin(c+d*x))^2/d],
[cos(c+d*x)^2*csc(c+d*x)^4*(a+b*sin(c+d*x))^2,x,6,-b^2*x+a*b*arctanh(cos(c+d*x))/d+1/3*(a^2-2*b^2)*cot(c+d*x)/d-1/3*a*b*cot(c+d*x)*csc(c+d*x)/d-1/3*cot(c+d*x)*csc(c+d*x)^2*(a+b*sin(c+d*x))^2/d],
[cos(c+d*x)^2*csc(c+d*x)^5*(a+b*sin(c+d*x))^2,x,8,1/8*(a^2+4*b^2)*arctanh(cos(c+d*x))/d+2/3*a*b*cot(c+d*x)/d+1/8*(a^2-2*b^2)*cot(c+d*x)*csc(c+d*x)/d-1/6*a*b*cot(c+d*x)*csc(c+d*x)^2/d-1/4*cot(c+d*x)*csc(c+d*x)^3*(a+b*sin(c+d*x))^2/d],
[cos(c+d*x)^2*csc(c+d*x)^6*(a+b*sin(c+d*x))^2,x,9,1/4*a*b*arctanh(cos(c+d*x))/d+1/15*(2*a^2+5*b^2)*cot(c+d*x)/d+1/4*a*b*cot(c+d*x)*csc(c+d*x)/d+1/15*(a^2-2*b^2)*cot(c+d*x)*csc(c+d*x)^2/d-1/10*a*b*cot(c+d*x)*csc(c+d*x)^3/d-1/5*cot(c+d*x)*csc(c+d*x)^4*(a+b*sin(c+d*x))^2/d],
[cos(c+d*x)^2*csc(c+d*x)^7*(a+b*sin(c+d*x))^2,x,9,1/16*(a^2+2*b^2)*arctanh(cos(c+d*x))/d+2/5*a*b*cot(c+d*x)/d+2/15*a*b*cot(c+d*x)^3/d+1/16*(a^2+2*b^2)*cot(c+d*x)*csc(c+d*x)/d+1/24*(a^2-2*b^2)*cot(c+d*x)*csc(c+d*x)^3/d-1/15*a*b*cot(c+d*x)*csc(c+d*x)^4/d-1/6*cot(c+d*x)*csc(c+d*x)^5*(a+b*sin(c+d*x))^2/d],
[cos(c+d*x)^2*sin(c+d*x)^2*(a+b*sin(c+d*x))^3,x,10,1/16*a*(2*a^2+3*b^2)*x-1/35*b*(21*a^2+4*b^2)*cos(c+d*x)/d+1/105*b*(21*a^2+4*b^2)*cos(c+d*x)^3/d-1/16*a*(2*a^2+3*b^2)*cos(c+d*x)*sin(c+d*x)/d+1/56*a*(2*a^2-7*b^2)*cos(c+d*x)*sin(c+d*x)^3/d+1/35*b*(a^2-b^2)*cos(c+d*x)*sin(c+d*x)^4/d+1/14*a*cos(c+d*x)*sin(c+d*x)^3*(a+b*sin(c+d*x))^2/d+1/7*cos(c+d*x)*sin(c+d*x)^3*(a+b*sin(c+d*x))^3/d],
[cos(c+d*x)^2*sin(c+d*x)*(a+b*sin(c+d*x))^3,x,6,1/16*b*(6*a^2+b^2)*x-1/120*a*(2*a^2+33*b^2)*cos(c+d*x)^3/d+1/16*b*(6*a^2+b^2)*cos(c+d*x)*sin(c+d*x)/d-1/40*(2*a^2+5*b^2)*cos(c+d*x)^3*(a+b*sin(c+d*x))/d-1/10*a*cos(c+d*x)^3*(a+b*sin(c+d*x))^2/d-1/6*cos(c+d*x)^3*(a+b*sin(c+d*x))^3/d],
[cos(c+d*x)^2*csc(c+d*x)*(a+b*sin(c+d*x))^3,x,7,1/8*b*(12*a^2+b^2)*x-a^3*arctanh(cos(c+d*x))/d+1/2*a*(a^2-2*b^2)*cos(c+d*x)/d+1/8*b*(2*a^2-b^2)*cos(c+d*x)*sin(c+d*x)/d+1/4*a*cos(c+d*x)*(a+b*sin(c+d*x))^2/d+1/4*cos(c+d*x)*(a+b*sin(c+d*x))^3/d],
[cos(c+d*x)^2*csc(c+d*x)^2*(a+b*sin(c+d*x))^3,x,11,-a^3*x+3/2*a*b^2*x-3*a^2*b*arctanh(cos(c+d*x))/d+3*a^2*b*cos(c+d*x)/d-1/3*b^3*cos(c+d*x)^3/d-a^3*cot(c+d*x)/d+3/2*a*b^2*cos(c+d*x)*sin(c+d*x)/d],
[cos(c+d*x)^2*csc(c+d*x)^3*(a+b*sin(c+d*x))^3,x,7,-1/2*b*(6*a^2-b^2)*x+1/2*a*(a^2-6*b^2)*arctanh(cos(c+d*x))/d+15/2*a*b^2*cos(c+d*x)/d+5/2*b^3*cos(c+d*x)*sin(c+d*x)/d-3/2*b*cot(c+d*x)*(a+b*sin(c+d*x))^2/d-1/2*cot(c+d*x)*csc(c+d*x)*(a+b*sin(c+d*x))^3/d],
[cos(c+d*x)^2*csc(c+d*x)^4*(a+b*sin(c+d*x))^3,x,7,-3*a*b^2*x+1/2*b*(3*a^2-2*b^2)*arctanh(cos(c+d*x))/d+11/6*b^3*cos(c+d*x)/d+1/3*a*(a^2-3*b^2)*cot(c+d*x)/d-1/2*b*cot(c+d*x)*csc(c+d*x)*(a+b*sin(c+d*x))^2/d-1/3*cot(c+d*x)*csc(c+d*x)^2*(a+b*sin(c+d*x))^3/d],
[cos(c+d*x)^2*csc(c+d*x)^5*(a+b*sin(c+d*x))^3,x,7,-b^3*x+1/8*a*(a^2+12*b^2)*arctanh(cos(c+d*x))/d+1/2*b*(2*a^2-b^2)*cot(c+d*x)/d+1/8*a*(a^2-2*b^2)*cot(c+d*x)*csc(c+d*x)/d-1/4*b*cot(c+d*x)*csc(c+d*x)^2*(a+b*sin(c+d*x))^2/d-1/4*cot(c+d*x)*csc(c+d*x)^3*(a+b*sin(c+d*x))^3/d],
[cos(c+d*x)^2*csc(c+d*x)^6*(a+b*sin(c+d*x))^3,x,9,1/8*b*(3*a^2+4*b^2)*arctanh(cos(c+d*x))/d+1/15*a*(2*a^2+15*b^2)*cot(c+d*x)/d+3/40*b*(5*a^2-2*b^2)*cot(c+d*x)*csc(c+d*x)/d+1/30*a*(2*a^2-3*b^2)*cot(c+d*x)*csc(c+d*x)^2/d-3/20*b*cot(c+d*x)*csc(c+d*x)^3*(a+b*sin(c+d*x))^2/d-1/5*cot(c+d*x)*csc(c+d*x)^4*(a+b*sin(c+d*x))^3/d],
[cos(c+d*x)^2*csc(c+d*x)^7*(a+b*sin(c+d*x))^3,x,10,1/16*a*(a^2+6*b^2)*arctanh(cos(c+d*x))/d+1/15*b*(6*a^2+5*b^2)*cot(c+d*x)/d+1/16*a*(a^2+6*b^2)*cot(c+d*x)*csc(c+d*x)/d+1/15*b*(3*a^2-b^2)*cot(c+d*x)*csc(c+d*x)^2/d+1/120*a*(5*a^2-6*b^2)*cot(c+d*x)*csc(c+d*x)^3/d-1/10*b*cot(c+d*x)*csc(c+d*x)^4*(a+b*sin(c+d*x))^2/d-1/6*cot(c+d*x)*csc(c+d*x)^5*(a+b*sin(c+d*x))^3/d],

# m<0
[cos(c+d*x)^2*sin(c+d*x)^3/(a+b*sin(c+d*x))^2,x,9,a*(4*a^2-b^2)*x/b^5+1/3*(12*a^2-b^2)*cos(c+d*x)/(b^4*d)-2*a*cos(c+d*x)*sin(c+d*x)/(b^3*d)+4/3*cos(c+d*x)*sin(c+d*x)^2/(b^2*d)-cos(c+d*x)*sin(c+d*x)^3/(b*d*(a+b*sin(c+d*x)))-2*a^2*(4*a^2-3*b^2)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(b^5*d*sqrt(a^2-b^2))],
[cos(c+d*x)^2*sin(c+d*x)^2/(a+b*sin(c+d*x))^2,x,8,-1/2*(6*a^2-b^2)*x/b^4-3*a*cos(c+d*x)/(b^3*d)+3/2*cos(c+d*x)*sin(c+d*x)/(b^2*d)-cos(c+d*x)*sin(c+d*x)^2/(b*d*(a+b*sin(c+d*x)))+2*a*(3*a^2-2*b^2)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(b^4*d*sqrt(a^2-b^2))],
[cos(c+d*x)^2*sin(c+d*x)/(a+b*sin(c+d*x))^2,x,5,2*a*x/b^3+cos(c+d*x)*(2*a+b*sin(c+d*x))/(b^2*d*(a+b*sin(c+d*x)))-2*(2*a^2-b^2)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(b^3*d*sqrt(a^2-b^2))],
[cos(c+d*x)^2*csc(c+d*x)/(a+b*sin(c+d*x))^2,x,8,-arctanh(cos(c+d*x))/(a^2*d)+cos(c+d*x)/(a*d*(a+b*sin(c+d*x)))-2*b*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(a^2*d*sqrt(a^2-b^2))],
[cos(c+d*x)^2*csc(c+d*x)^2/(a+b*sin(c+d*x))^2,x,8,2*b*arctanh(cos(c+d*x))/(a^3*d)-2*cot(c+d*x)/(a^2*d)+cot(c+d*x)/(a*d*(a+b*sin(c+d*x)))-2*(a^2-2*b^2)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(a^3*d*sqrt(a^2-b^2))],
[cos(c+d*x)^2*csc(c+d*x)^3/(a+b*sin(c+d*x))^2,x,9,1/2*(a^2-6*b^2)*arctanh(cos(c+d*x))/(a^4*d)+3*b*cot(c+d*x)/(a^3*d)-3/2*cot(c+d*x)*csc(c+d*x)/(a^2*d)+cot(c+d*x)*csc(c+d*x)/(a*d*(a+b*sin(c+d*x)))+2*b*(2*a^2-3*b^2)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(a^4*d*sqrt(a^2-b^2))],
[cos(c+d*x)^2*csc(c+d*x)^4/(a+b*sin(c+d*x))^2,x,10,-b*(a^2-4*b^2)*arctanh(cos(c+d*x))/(a^5*d)+1/3*(a^2-12*b^2)*cot(c+d*x)/(a^4*d)+2*b*cot(c+d*x)*csc(c+d*x)/(a^3*d)-4/3*cot(c+d*x)*csc(c+d*x)^2/(a^2*d)+cot(c+d*x)*csc(c+d*x)^2/(a*d*(a+b*sin(c+d*x)))-2*b^2*(3*a^2-4*b^2)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(a^5*d*sqrt(a^2-b^2))],
[cos(c+d*x)^2*sin(c+d*x)^3/(a+b*sin(c+d*x))^3,x,9,-1/2*(12*a^2-b^2)*x/b^5+a*(12*a^4-19*a^2*b^2+6*b^4)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(b^5*(a^2-b^2)^(3/2)*d)-1/2*a*(12*a^2-11*b^2)*cos(c+d*x)/(b^4*(a^2-b^2)*d)+1/2*(6*a^2-5*b^2)*cos(c+d*x)*sin(c+d*x)/(b^3*(a^2-b^2)*d)-1/2*cos(c+d*x)*sin(c+d*x)^3/(b*d*(a+b*sin(c+d*x))^2)-1/2*(4*a^2-3*b^2)*cos(c+d*x)*sin(c+d*x)^2/(b^2*(a^2-b^2)*d*(a+b*sin(c+d*x)))],
[cos(c+d*x)^2*sin(c+d*x)^2/(a+b*sin(c+d*x))^3,x,8,3*a*x/b^4-(6*a^4-9*a^2*b^2+2*b^4)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(b^4*(a^2-b^2)^(3/2)*d)+3/2*cos(c+d*x)/(b^3*d)-1/2*cos(c+d*x)*sin(c+d*x)^2/(b*d*(a+b*sin(c+d*x))^2)+1/2*a*(3*a^2-2*b^2)*cos(c+d*x)/(b^3*(a^2-b^2)*d*(a+b*sin(c+d*x)))],
[cos(c+d*x)^2*sin(c+d*x)/(a+b*sin(c+d*x))^3,x,6,-x/b^3+a*(2*a^2-3*b^2)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(b^3*(a^2-b^2)^(3/2)*d)-1/2*a*cos(c+d*x)^3/((a^2-b^2)*d*(a+b*sin(c+d*x))^2)-1/2*cos(c+d*x)*(2*(a^2-b^2)+a*b*sin(c+d*x))/(b^2*(a^2-b^2)*d*(a+b*sin(c+d*x)))],
[cos(c+d*x)^2*csc(c+d*x)/(a+b*sin(c+d*x))^3,x,8,-b*(3*a^2-2*b^2)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(a^3*(a^2-b^2)^(3/2)*d)-arctanh(cos(c+d*x))/(a^3*d)+1/2*cos(c+d*x)/(a*d*(a+b*sin(c+d*x))^2)+1/2*(a^2-2*b^2)*cos(c+d*x)/(a^2*(a^2-b^2)*d*(a+b*sin(c+d*x)))],
[cos(c+d*x)^2*csc(c+d*x)^2/(a+b*sin(c+d*x))^3,x,9,-(2*a^4-9*a^2*b^2+6*b^4)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(a^4*(a^2-b^2)^(3/2)*d)+3*b*arctanh(cos(c+d*x))/(a^4*d)-1/2*(5*a^2-6*b^2)*cot(c+d*x)/(a^3*(a^2-b^2)*d)+1/2*cot(c+d*x)/(a*d*(a+b*sin(c+d*x))^2)+1/2*(2*a^2-3*b^2)*cot(c+d*x)/(a^2*(a^2-b^2)*d*(a+b*sin(c+d*x)))],
[cos(c+d*x)^2*csc(c+d*x)^3/(a+b*sin(c+d*x))^3,x,10,b*(6*a^4-19*a^2*b^2+12*b^4)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(a^5*(a^2-b^2)^(3/2)*d)+1/2*(a^2-12*b^2)*arctanh(cos(c+d*x))/(a^5*d)+1/2*b*(11*a^2-12*b^2)*cot(c+d*x)/(a^4*(a^2-b^2)*d)-1/2*(5*a^2-6*b^2)*cot(c+d*x)*csc(c+d*x)/(a^3*(a^2-b^2)*d)+1/2*cot(c+d*x)*csc(c+d*x)/(a*d*(a+b*sin(c+d*x))^2)+1/2*(3*a^2-4*b^2)*cot(c+d*x)*csc(c+d*x)/(a^2*(a^2-b^2)*d*(a+b*sin(c+d*x)))],

# Integrands of the form Cos[e+f x]^2 Sin[e+f x]^n (a+b Sin[e+f x])^(m/2)

# Integrands of the form Cos[e+f x]^2 (d Sin[e+f x])^(n/2) (a+b Sin[e+f x])^(m/2)
[cos(e+f*x)^2/((a+b*sin(e+f*x))^(5/2)*sqrt(d*sin(e+f*x))),x,5,2/3*cos(e+f*x)*sqrt(d*sin(e+f*x))/(a*d*f*(a+b*sin(e+f*x))^(3/2))+4/3*b*cos(e+f*x)/(a*(a^2-b^2)*f*sqrt(d*sin(e+f*x))*sqrt(a+b*sin(e+f*x)))-4/3*b*EllipticE(sqrt(d)*sqrt(a+b*sin(e+f*x))/(sqrt(a+b)*sqrt(d*sin(e+f*x))),sqrt((-a-b)/(a-b)))*sqrt(a*(1-csc(e+f*x))/(a+b))*sqrt(a*(1+csc(e+f*x))/(a-b))*tan(e+f*x)/(a^3*f*sqrt(a+b)*sqrt(d))-4/3*EllipticF(sqrt(d)*sqrt(a+b*sin(e+f*x))/(sqrt(a+b)*sqrt(d*sin(e+f*x))),sqrt((-a-b)/(a-b)))*sqrt(a*(1-csc(e+f*x))/(a+b))*sqrt(a*(1+csc(e+f*x))/(a-b))*tan(e+f*x)/(a^2*f*sqrt(a+b)*sqrt(d))],

# Integrands of the form Cos[e+f x]^3 (d Sin[e+f x])^n (a+b Sin[e+f x])^m

# Integrands of the form Cos[e+f x]^4 (d Sin[e+f x])^n (a+b Sin[e+f x])^m

# Integrands of the form Cos[e+f x]^4 Sin[e+f x]^n (a+b Sin[e+f x])^m

# m>0
[cos(c+d*x)^4*sin(c+d*x)^4*(a+b*sin(c+d*x)),x,9,3/128*a*x-1/5*b*cos(c+d*x)^5/d+2/7*b*cos(c+d*x)^7/d-1/9*b*cos(c+d*x)^9/d+3/128*a*cos(c+d*x)*sin(c+d*x)/d+1/64*a*cos(c+d*x)^3*sin(c+d*x)/d-1/16*a*cos(c+d*x)^5*sin(c+d*x)/d-1/8*a*cos(c+d*x)^5*sin(c+d*x)^3/d],
[cos(c+d*x)^4*sin(c+d*x)^3*(a+b*sin(c+d*x)),x,9,3/128*b*x-1/5*a*cos(c+d*x)^5/d+1/7*a*cos(c+d*x)^7/d+3/128*b*cos(c+d*x)*sin(c+d*x)/d+1/64*b*cos(c+d*x)^3*sin(c+d*x)/d-1/16*b*cos(c+d*x)^5*sin(c+d*x)/d-1/8*b*cos(c+d*x)^5*sin(c+d*x)^3/d],
[cos(c+d*x)^4*sin(c+d*x)^2*(a+b*sin(c+d*x)),x,8,1/16*a*x-1/5*b*cos(c+d*x)^5/d+1/7*b*cos(c+d*x)^7/d+1/16*a*cos(c+d*x)*sin(c+d*x)/d+1/24*a*cos(c+d*x)^3*sin(c+d*x)/d-1/6*a*cos(c+d*x)^5*sin(c+d*x)/d],
[cos(c+d*x)^4*sin(c+d*x)*(a+b*sin(c+d*x)),x,7,1/16*b*x-1/5*a*cos(c+d*x)^5/d+1/16*b*cos(c+d*x)*sin(c+d*x)/d+1/24*b*cos(c+d*x)^3*sin(c+d*x)/d-1/6*b*cos(c+d*x)^5*sin(c+d*x)/d],
[cos(c+d*x)^4*csc(c+d*x)*(a+b*sin(c+d*x)),x,8,3/8*b*x-a*arctanh(cos(c+d*x))/d+a*cos(c+d*x)/d+1/3*a*cos(c+d*x)^3/d+3/8*b*cos(c+d*x)*sin(c+d*x)/d+1/4*b*cos(c+d*x)^3*sin(c+d*x)/d],
[cos(c+d*x)^4*csc(c+d*x)^2*(a+b*sin(c+d*x)),x,9,-3/2*a*x-b*arctanh(cos(c+d*x))/d+b*cos(c+d*x)/d+1/3*b*cos(c+d*x)^3/d-3/2*a*cot(c+d*x)/d+1/2*a*cos(c+d*x)^2*cot(c+d*x)/d],
[cos(c+d*x)^4*csc(c+d*x)^3*(a+b*sin(c+d*x)),x,9,-3/2*b*x+3/2*a*arctanh(cos(c+d*x))/d-3/2*a*cos(c+d*x)/d-3/2*b*cot(c+d*x)/d+1/2*b*cos(c+d*x)^2*cot(c+d*x)/d-1/2*a*cos(c+d*x)*cot(c+d*x)^2/d],
[cos(c+d*x)^4*csc(c+d*x)^4*(a+b*sin(c+d*x)),x,9,a*x+3/2*b*arctanh(cos(c+d*x))/d-3/2*b*cos(c+d*x)/d+a*cot(c+d*x)/d-1/2*b*cos(c+d*x)*cot(c+d*x)^2/d-1/3*a*cot(c+d*x)^3/d],
[cos(c+d*x)^4*csc(c+d*x)^5*(a+b*sin(c+d*x)),x,7,b*x-3/8*a*arctanh(cos(c+d*x))/d+b*cot(c+d*x)/d-1/3*b*cot(c+d*x)^3/d+3/8*a*cot(c+d*x)*csc(c+d*x)/d-1/4*a*cot(c+d*x)^3*csc(c+d*x)/d],
[cos(c+d*x)^4*csc(c+d*x)^6*(a+b*sin(c+d*x)),x,6,-3/8*b*arctanh(cos(c+d*x))/d-1/5*a*cot(c+d*x)^5/d+3/8*b*cot(c+d*x)*csc(c+d*x)/d-1/4*b*cot(c+d*x)^3*csc(c+d*x)/d],
[cos(c+d*x)^4*csc(c+d*x)^7*(a+b*sin(c+d*x)),x,7,-1/16*a*arctanh(cos(c+d*x))/d-1/5*b*cot(c+d*x)^5/d-1/16*a*cot(c+d*x)*csc(c+d*x)/d+1/8*a*cot(c+d*x)*csc(c+d*x)^3/d-1/6*a*cot(c+d*x)^3*csc(c+d*x)^3/d],
[cos(c+d*x)^4*csc(c+d*x)^8*(a+b*sin(c+d*x)),x,8,-1/16*b*arctanh(cos(c+d*x))/d-1/5*a*cot(c+d*x)^5/d-1/7*a*cot(c+d*x)^7/d-1/16*b*cot(c+d*x)*csc(c+d*x)/d+1/8*b*cot(c+d*x)*csc(c+d*x)^3/d-1/6*b*cot(c+d*x)^3*csc(c+d*x)^3/d],
[cos(c+d*x)^4*csc(c+d*x)^9*(a+b*sin(c+d*x)),x,9,-3/128*a*arctanh(cos(c+d*x))/d-1/5*b*cot(c+d*x)^5/d-1/7*b*cot(c+d*x)^7/d-3/128*a*cot(c+d*x)*csc(c+d*x)/d-1/64*a*cot(c+d*x)*csc(c+d*x)^3/d+1/16*a*cot(c+d*x)*csc(c+d*x)^5/d-1/8*a*cot(c+d*x)^3*csc(c+d*x)^5/d],
[cos(c+d*x)^4*sin(c+d*x)^3*(a+b*sin(c+d*x))^2,x,10,3/64*a*b*x-1/105*(9*a^2+4*b^2)*cos(c+d*x)/d+1/315*(9*a^2+4*b^2)*cos(c+d*x)^3/d-3/64*a*b*cos(c+d*x)*sin(c+d*x)/d-1/32*a*b*cos(c+d*x)*sin(c+d*x)^3/d-1/630*(15*a^4-44*a^2*b^2+6*b^4)*cos(c+d*x)*sin(c+d*x)^4/(b^2*d)-1/504*a*(10*a^2-29*b^2)*cos(c+d*x)*sin(c+d*x)^5/(b*d)-5/252*(3*a^2-8*b^2)*cos(c+d*x)*sin(c+d*x)^4*(a+b*sin(c+d*x))^2/(b^2*d)+1/12*a*cos(c+d*x)*sin(c+d*x)^4*(a+b*sin(c+d*x))^3/(b^2*d)-1/9*cos(c+d*x)*sin(c+d*x)^5*(a+b*sin(c+d*x))^3/(b*d)],
[cos(c+d*x)^4*sin(c+d*x)^2*(a+b*sin(c+d*x))^2,x,9,1/128*(8*a^2+3*b^2)*x-6/35*a*b*cos(c+d*x)/d+2/35*a*b*cos(c+d*x)^3/d-1/128*(8*a^2+3*b^2)*cos(c+d*x)*sin(c+d*x)/d-1/1344*(40*a^4-140*a^2*b^2+21*b^4)*cos(c+d*x)*sin(c+d*x)^3/(b^2*d)-1/840*a*(20*a^2-69*b^2)*cos(c+d*x)*sin(c+d*x)^4/(b*d)-1/336*(20*a^2-63*b^2)*cos(c+d*x)*sin(c+d*x)^3*(a+b*sin(c+d*x))^2/(b^2*d)+5/56*a*cos(c+d*x)*sin(c+d*x)^3*(a+b*sin(c+d*x))^3/(b^2*d)-1/8*cos(c+d*x)*sin(c+d*x)^4*(a+b*sin(c+d*x))^3/(b*d)],
[cos(c+d*x)^4*sin(c+d*x)*(a+b*sin(c+d*x))^2,x,6,1/8*a*b*x-1/105*(a^2+6*b^2)*cos(c+d*x)^5/d+1/8*a*b*cos(c+d*x)*sin(c+d*x)/d+1/12*a*b*cos(c+d*x)^3*sin(c+d*x)/d-1/21*a*cos(c+d*x)^5*(a+b*sin(c+d*x))/d-1/7*cos(c+d*x)^5*(a+b*sin(c+d*x))^2/d],
[cos(c+d*x)^4*csc(c+d*x)*(a+b*sin(c+d*x))^2,x,6,3/4*a*b*x-a^2*arctanh(cos(c+d*x))/d+a^2*cos(c+d*x)/d+1/3*a^2*cos(c+d*x)^3/d-1/5*b^2*cos(c+d*x)^5/d+3/4*a*b*cos(c+d*x)*sin(c+d*x)/d+1/2*a*b*cos(c+d*x)^3*sin(c+d*x)/d,3/4*a*b*x-a^2*arctanh(cos(c+d*x))/d-1/15*(a^4-14*a^2*b^2+3*b^4)*cos(c+d*x)/(b^2*d)-1/60*a*(2*a^2-27*b^2)*cos(c+d*x)*sin(c+d*x)/(b*d)-1/30*(a^2-12*b^2)*cos(c+d*x)*(a+b*sin(c+d*x))^2/(b^2*d)+1/10*a*cos(c+d*x)*(a+b*sin(c+d*x))^3/(b^2*d)-1/5*cos(c+d*x)*sin(c+d*x)*(a+b*sin(c+d*x))^3/(b*d)],
[cos(c+d*x)^4*csc(c+d*x)^2*(a+b*sin(c+d*x))^2,x,6,-3/8*(4*a^2-b^2)*x-2*a*b*arctanh(cos(c+d*x))/d+1/6*a*(a^2+28*b^2)*cos(c+d*x)/(b*d)+1/24*(2*a^2+39*b^2)*cos(c+d*x)*sin(c+d*x)/d+1/12*(a^2+12*b^2)*cos(c+d*x)*(a+b*sin(c+d*x))^2/(a*b*d)-1/4*cos(c+d*x)*(a+b*sin(c+d*x))^3/(b*d)-cot(c+d*x)*(a+b*sin(c+d*x))^3/(a*d)],
[cos(c+d*x)^4*csc(c+d*x)^3*(a+b*sin(c+d*x))^2,x,6,-3*a*b*x+1/2*(3*a^2-2*b^2)*arctanh(cos(c+d*x))/d-1/6*(4*a^2-23*b^2)*cos(c+d*x)/d-1/3*b*(a^2-3*b^2)*cos(c+d*x)*sin(c+d*x)/(a*d)-1/6*(2*a^2-3*b^2)*cos(c+d*x)*(a+b*sin(c+d*x))^2/(a^2*d)-1/2*b*cot(c+d*x)*(a+b*sin(c+d*x))^3/(a^2*d)-1/2*cot(c+d*x)*csc(c+d*x)*(a+b*sin(c+d*x))^3/(a*d)],
[cos(c+d*x)^4*csc(c+d*x)^4*(a+b*sin(c+d*x))^2,x,13,a^2*x-3/2*b^2*x+3*a*b*arctanh(cos(c+d*x))/d-3*a*b*cos(c+d*x)/d+a^2*cot(c+d*x)/d-3/2*b^2*cot(c+d*x)/d+1/2*b^2*cos(c+d*x)^2*cot(c+d*x)/d-a*b*cos(c+d*x)*cot(c+d*x)^2/d-1/3*a^2*cot(c+d*x)^3/d],
[cos(c+d*x)^4*csc(c+d*x)^5*(a+b*sin(c+d*x))^2,x,6,2*a*b*x-3/8*(a^2-4*b^2)*arctanh(cos(c+d*x))/d-1/24*b^2*(39*a^2+2*b^2)*cos(c+d*x)/(a^2*d)+17/12*a*b*cot(c+d*x)/d+5/8*cot(c+d*x)*csc(c+d*x)*(a+b*sin(c+d*x))^2/d+1/12*b*cot(c+d*x)*csc(c+d*x)^2*(a+b*sin(c+d*x))^3/(a^2*d)-1/4*cot(c+d*x)*csc(c+d*x)^3*(a+b*sin(c+d*x))^3/(a*d)],
[cos(c+d*x)^4*csc(c+d*x)^6*(a+b*sin(c+d*x))^2,x,6,b^2*x-3/4*a*b*arctanh(cos(c+d*x))/d-1/15*(3*a^4-14*a^2*b^2+b^4)*cot(c+d*x)/(a^2*d)+1/60*b*(27*a^2-2*b^2)*cot(c+d*x)*csc(c+d*x)/(a*d)+1/30*(12*a^2-b^2)*cot(c+d*x)*csc(c+d*x)^2*(a+b*sin(c+d*x))^2/(a^2*d)+1/10*b*cot(c+d*x)*csc(c+d*x)^3*(a+b*sin(c+d*x))^3/(a^2*d)-1/5*cot(c+d*x)*csc(c+d*x)^4*(a+b*sin(c+d*x))^3/(a*d)],
[cos(c+d*x)^4*csc(c+d*x)^7*(a+b*sin(c+d*x))^2,x,8,-1/16*(a^2+6*b^2)*arctanh(cos(c+d*x))/d-2/5*a*b*cot(c+d*x)/d-1/240*(15*a^4-80*a^2*b^2+12*b^4)*cot(c+d*x)*csc(c+d*x)/(a^2*d)+1/60*b*(13*a^2-2*b^2)*cot(c+d*x)*csc(c+d*x)^2/(a*d)+1/120*(35*a^2-6*b^2)*cot(c+d*x)*csc(c+d*x)^3*(a+b*sin(c+d*x))^2/(a^2*d)+1/10*b*cot(c+d*x)*csc(c+d*x)^4*(a+b*sin(c+d*x))^3/(a^2*d)-1/6*cot(c+d*x)*csc(c+d*x)^5*(a+b*sin(c+d*x))^3/(a*d)],
[cos(c+d*x)^4*csc(c+d*x)^8*(a+b*sin(c+d*x))^2,x,9,-1/8*a*b*arctanh(cos(c+d*x))/d-1/35*(2*a^2+7*b^2)*cot(c+d*x)/d-1/8*a*b*cot(c+d*x)*csc(c+d*x)/d-1/105*(3*a^4-18*a^2*b^2+4*b^4)*cot(c+d*x)*csc(c+d*x)^2/(a^2*d)+1/420*b*(53*a^2-12*b^2)*cot(c+d*x)*csc(c+d*x)^3/(a*d)+2/35*(4*a^2-b^2)*cot(c+d*x)*csc(c+d*x)^4*(a+b*sin(c+d*x))^2/(a^2*d)+2/21*b*cot(c+d*x)*csc(c+d*x)^5*(a+b*sin(c+d*x))^3/(a^2*d)-1/7*cot(c+d*x)*csc(c+d*x)^6*(a+b*sin(c+d*x))^3/(a*d)],
[cos(c+d*x)^4*sin(c+d*x)^2*(a+b*sin(c+d*x))^3,x,10,1/128*a*(8*a^2+9*b^2)*x-1/105*b*(27*a^2+4*b^2)*cos(c+d*x)/d+1/315*b*(27*a^2+4*b^2)*cos(c+d*x)^3/d-1/128*a*(8*a^2+9*b^2)*cos(c+d*x)*sin(c+d*x)/d-1/4032*a*(40*a^4-188*a^2*b^2+189*b^4)*cos(c+d*x)*sin(c+d*x)^3/(b^2*d)-1/2520*(20*a^4-93*a^2*b^2+24*b^4)*cos(c+d*x)*sin(c+d*x)^4/(b*d)-1/1008*a*(20*a^2-87*b^2)*cos(c+d*x)*sin(c+d*x)^3*(a+b*sin(c+d*x))^2/(b^2*d)-5/126*(a^2-4*b^2)*cos(c+d*x)*sin(c+d*x)^3*(a+b*sin(c+d*x))^3/(b^2*d)+5/72*a*cos(c+d*x)*sin(c+d*x)^3*(a+b*sin(c+d*x))^4/(b^2*d)-1/9*cos(c+d*x)*sin(c+d*x)^4*(a+b*sin(c+d*x))^4/(b*d)],
[cos(c+d*x)^4*sin(c+d*x)*(a+b*sin(c+d*x))^3,x,7,3/128*b*(8*a^2+b^2)*x-1/560*a*(2*a^2+61*b^2)*cos(c+d*x)^5/d+3/128*b*(8*a^2+b^2)*cos(c+d*x)*sin(c+d*x)/d+1/64*b*(8*a^2+b^2)*cos(c+d*x)^3*sin(c+d*x)/d-1/112*(2*a^2+7*b^2)*cos(c+d*x)^5*(a+b*sin(c+d*x))/d-3/56*a*cos(c+d*x)^5*(a+b*sin(c+d*x))^2/d-1/8*cos(c+d*x)^5*(a+b*sin(c+d*x))^3/d],
[cos(c+d*x)^4*csc(c+d*x)*(a+b*sin(c+d*x))^3,x,7,1/16*b*(18*a^2+b^2)*x-a^3*arctanh(cos(c+d*x))/d-1/60*a*(2*a^4-43*a^2*b^2+36*b^4)*cos(c+d*x)/(b^2*d)-1/240*(4*a^4-84*a^2*b^2+15*b^4)*cos(c+d*x)*sin(c+d*x)/(b*d)-1/120*a*(2*a^2-39*b^2)*cos(c+d*x)*(a+b*sin(c+d*x))^2/(b^2*d)-1/120*(2*a^2-35*b^2)*cos(c+d*x)*(a+b*sin(c+d*x))^3/(b^2*d)+1/15*a*cos(c+d*x)*(a+b*sin(c+d*x))^4/(b^2*d)-1/6*cos(c+d*x)*sin(c+d*x)*(a+b*sin(c+d*x))^4/(b*d)],
[cos(c+d*x)^4*csc(c+d*x)^2*(a+b*sin(c+d*x))^3,x,7,-3/8*a*(4*a^2-3*b^2)*x-3*a^2*b*arctanh(cos(c+d*x))/d+1/10*(a^4+56*a^2*b^2-2*b^4)*cos(c+d*x)/(b*d)+1/40*a*(2*a^2+83*b^2)*cos(c+d*x)*sin(c+d*x)/d+1/20*(a^2+28*b^2)*cos(c+d*x)*(a+b*sin(c+d*x))^2/(b*d)+1/20*(a^2+20*b^2)*cos(c+d*x)*(a+b*sin(c+d*x))^3/(a*b*d)-1/5*cos(c+d*x)*(a+b*sin(c+d*x))^4/(b*d)-cot(c+d*x)*(a+b*sin(c+d*x))^4/(a*d)],
[cos(c+d*x)^4*csc(c+d*x)^3*(a+b*sin(c+d*x))^3,x,7,-3/8*b*(12*a^2-b^2)*x+3/2*a*(a^2-2*b^2)*arctanh(cos(c+d*x))/d-1/2*a*(a^2-17*b^2)*cos(c+d*x)/d-1/8*b*(2*a^2-21*b^2)*cos(c+d*x)*sin(c+d*x)/d-1/4*(a^2-6*b^2)*cos(c+d*x)*(a+b*sin(c+d*x))^2/(a*d)-1/4*(a^2-4*b^2)*cos(c+d*x)*(a+b*sin(c+d*x))^3/(a^2*d)-b*cot(c+d*x)*(a+b*sin(c+d*x))^4/(a^2*d)-1/2*cot(c+d*x)*csc(c+d*x)*(a+b*sin(c+d*x))^4/(a*d)],
[cos(c+d*x)^4*csc(c+d*x)^4*(a+b*sin(c+d*x))^3,x,17,a^3*x-9/2*a*b^2*x+9/2*a^2*b*arctanh(cos(c+d*x))/d-b^3*arctanh(cos(c+d*x))/d-9/2*a^2*b*cos(c+d*x)/d+b^3*cos(c+d*x)/d+1/3*b^3*cos(c+d*x)^3/d+a^3*cot(c+d*x)/d-9/2*a*b^2*cot(c+d*x)/d+3/2*a*b^2*cos(c+d*x)^2*cot(c+d*x)/d-3/2*a^2*b*cos(c+d*x)*cot(c+d*x)^2/d-1/3*a^3*cot(c+d*x)^3/d],
[cos(c+d*x)^4*csc(c+d*x)^5*(a+b*sin(c+d*x))^3,x,7,3/2*b*(2*a^2-b^2)*x-3/8*a*(a^2-12*b^2)*arctanh(cos(c+d*x))/d-1/8*b^2*(73*a^2-2*b^2)*cos(c+d*x)/(a*d)-13/4*b^3*cos(c+d*x)*sin(c+d*x)/d+17/8*b*cot(c+d*x)*(a+b*sin(c+d*x))^2/d+5/8*cot(c+d*x)*csc(c+d*x)*(a+b*sin(c+d*x))^3/d-1/4*cot(c+d*x)*csc(c+d*x)^3*(a+b*sin(c+d*x))^4/(a*d)],
[cos(c+d*x)^4*csc(c+d*x)^6*(a+b*sin(c+d*x))^3,x,7,3*a*b^2*x-3/8*b*(3*a^2-4*b^2)*arctanh(cos(c+d*x))/d-1/40*b^3*(83*a^2+2*b^2)*cos(c+d*x)/(a^2*d)-1/20*a*(4*a^2-29*b^2)*cot(c+d*x)/d+27/40*b*cot(c+d*x)*csc(c+d*x)*(a+b*sin(c+d*x))^2/d+2/5*cot(c+d*x)*csc(c+d*x)^2*(a+b*sin(c+d*x))^3/d+1/20*b*cot(c+d*x)*csc(c+d*x)^3*(a+b*sin(c+d*x))^4/(a^2*d)-1/5*cot(c+d*x)*csc(c+d*x)^4*(a+b*sin(c+d*x))^4/(a*d)],
[cos(c+d*x)^4*csc(c+d*x)^7*(a+b*sin(c+d*x))^3,x,7,b^3*x-1/16*a*(a^2+18*b^2)*arctanh(cos(c+d*x))/d-1/60*b*(36*a^4-43*a^2*b^2+2*b^4)*cot(c+d*x)/(a^2*d)-1/240*(15*a^4-84*a^2*b^2+4*b^4)*cot(c+d*x)*csc(c+d*x)/(a*d)+1/120*b*(39*a^2-2*b^2)*cot(c+d*x)*csc(c+d*x)^2*(a+b*sin(c+d*x))^2/(a^2*d)+1/120*(35*a^2-2*b^2)*cot(c+d*x)*csc(c+d*x)^3*(a+b*sin(c+d*x))^3/(a^2*d)+1/15*b*cot(c+d*x)*csc(c+d*x)^4*(a+b*sin(c+d*x))^4/(a^2*d)-1/6*cot(c+d*x)*csc(c+d*x)^5*(a+b*sin(c+d*x))^4/(a*d)],
[cos(c+d*x)^4*csc(c+d*x)^8*(a+b*sin(c+d*x))^3,x,9,-3/16*b*(a^2+2*b^2)*arctanh(cos(c+d*x))/d-1/35*a*(2*a^2+21*b^2)*cot(c+d*x)/d-1/560*b*(105*a^4-116*a^2*b^2+12*b^4)*cot(c+d*x)*csc(c+d*x)/(a^2*d)-1/140*(4*a^4-19*a^2*b^2+2*b^4)*cot(c+d*x)*csc(c+d*x)^2/(a*d)+1/280*b*(53*a^2-6*b^2)*cot(c+d*x)*csc(c+d*x)^3*(a+b*sin(c+d*x))^2/(a^2*d)+1/35*(8*a^2-b^2)*cot(c+d*x)*csc(c+d*x)^4*(a+b*sin(c+d*x))^3/(a^2*d)+1/14*b*cot(c+d*x)*csc(c+d*x)^5*(a+b*sin(c+d*x))^4/(a^2*d)-1/7*cot(c+d*x)*csc(c+d*x)^6*(a+b*sin(c+d*x))^4/(a*d)],
[cos(c+d*x)^4*csc(c+d*x)^9*(a+b*sin(c+d*x))^3,x,10,-3/128*a*(a^2+8*b^2)*arctanh(cos(c+d*x))/d-1/35*b*(6*a^2+7*b^2)*cot(c+d*x)/d-3/128*a*(a^2+8*b^2)*cot(c+d*x)*csc(c+d*x)/d-1/280*b*(24*a^4-25*a^2*b^2+4*b^4)*cot(c+d*x)*csc(c+d*x)^2/(a^2*d)-1/2240*(35*a^4-148*a^2*b^2+24*b^4)*cot(c+d*x)*csc(c+d*x)^3/(a*d)+3/560*b*(23*a^2-4*b^2)*cot(c+d*x)*csc(c+d*x)^4*(a+b*sin(c+d*x))^2/(a^2*d)+1/112*(21*a^2-4*b^2)*cot(c+d*x)*csc(c+d*x)^5*(a+b*sin(c+d*x))^3/(a^2*d)+1/14*b*cot(c+d*x)*csc(c+d*x)^6*(a+b*sin(c+d*x))^4/(a^2*d)-1/8*cot(c+d*x)*csc(c+d*x)^7*(a+b*sin(c+d*x))^4/(a*d)],

# m<0
[cos(c+d*x)^4*sin(c+d*x)^3/(a+b*sin(c+d*x))^2,x,9,-3/4*a*(8*a^4-8*a^2*b^2+b^4)*x/b^7-1/5*(30*a^4-25*a^2*b^2+b^4)*cos(c+d*x)/(b^6*d)+3/4*a*(4*a^2-3*b^2)*cos(c+d*x)*sin(c+d*x)/(b^5*d)-1/5*(10*a^2-7*b^2)*cos(c+d*x)*sin(c+d*x)^2/(b^4*d)+1/2*(3*a^2-2*b^2)*cos(c+d*x)*sin(c+d*x)^3/(a*b^3*d)-1/5*cos(c+d*x)*sin(c+d*x)^4/(b^2*d)-(a^2-b^2)*cos(c+d*x)*sin(c+d*x)^4/(a*b^2*d*(a+b*sin(c+d*x)))+6*a^2*(2*a^4-3*a^2*b^2+b^4)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(b^7*d*sqrt(a^2-b^2))],
[cos(c+d*x)^4*sin(c+d*x)^2/(a+b*sin(c+d*x))^2,x,8,1/8*(40*a^4-36*a^2*b^2+3*b^4)*x/b^6+1/3*a*(15*a^2-11*b^2)*cos(c+d*x)/(b^5*d)-1/8*(20*a^2-13*b^2)*cos(c+d*x)*sin(c+d*x)/(b^4*d)+1/3*(5*a^2-3*b^2)*cos(c+d*x)*sin(c+d*x)^2/(a*b^3*d)-1/4*cos(c+d*x)*sin(c+d*x)^3/(b^2*d)-(a^2-b^2)*cos(c+d*x)*sin(c+d*x)^3/(a*b^2*d*(a+b*sin(c+d*x)))-2*a*(5*a^4-7*a^2*b^2+2*b^4)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(b^6*d*sqrt(a^2-b^2))],
[cos(c+d*x)^4*sin(c+d*x)/(a+b*sin(c+d*x))^2,x,6,-a*(4*a^2-3*b^2)*x/b^5+1/3*cos(c+d*x)^3*(4*a+b*sin(c+d*x))/(b^2*d*(a+b*sin(c+d*x)))-cos(c+d*x)*(4*a^2-b^2-2*a*b*sin(c+d*x))/(b^4*d)+2*(4*a^4-5*a^2*b^2+b^4)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(b^5*d*sqrt(a^2-b^2))],
[cos(c+d*x)^4*csc(c+d*x)/(a+b*sin(c+d*x))^2,x,6,-2*a*x/b^3-arctanh(cos(c+d*x))/(a^2*d)-cos(c+d*x)/(b^2*d)-(a^2-b^2)*cos(c+d*x)/(a*b^2*d*(a+b*sin(c+d*x)))+2*(2*a^2+b^2)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))*sqrt(a^2-b^2)/(a^2*b^3*d)],
[cos(c+d*x)^4*csc(c+d*x)^2/(a+b*sin(c+d*x))^2,x,6,x/b^2+2*b*arctanh(cos(c+d*x))/(a^3*d)+(a^2-2*b^2)*cos(c+d*x)/(a^2*b*d*(a+b*sin(c+d*x)))-cot(c+d*x)/(a*d*(a+b*sin(c+d*x)))-2*(a^4+a^2*b^2-2*b^4)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(a^3*b^2*d*sqrt(a^2-b^2))],
[cos(c+d*x)^4*csc(c+d*x)^3/(a+b*sin(c+d*x))^2,x,7,3/2*(a^2-2*b^2)*arctanh(cos(c+d*x))/(a^4*d)-1/2*cos(c+d*x)/(a^2*d*(1-cos(c+d*x)^2))+2*b*cot(c+d*x)/(a^3*d)-(a^2-b^2)*cos(c+d*x)/(a^3*d*(a+b*sin(c+d*x)))+6*b*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))*sqrt(a^2-b^2)/(a^4*d),3/2*(a^2-2*b^2)*arctanh(cos(c+d*x))/(a^4*d)-(a^2-3*b^2)*cot(c+d*x)/(a^3*b*d)+1/2*(2*a^2-3*b^2)*cot(c+d*x)/(a^2*b*d*(a+b*sin(c+d*x)))-1/2*cot(c+d*x)*csc(c+d*x)/(a*d*(a+b*sin(c+d*x)))+6*b*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))*sqrt(a^2-b^2)/(a^4*d)],
[cos(c+d*x)^4*csc(c+d*x)^4/(a+b*sin(c+d*x))^2,x,8,-b*(3*a^2-4*b^2)*arctanh(cos(c+d*x))/(a^5*d)+1/3*(7*a^2-12*b^2)*cot(c+d*x)/(a^4*d)-(a^2-2*b^2)*cot(c+d*x)*csc(c+d*x)/(a^3*b*d)+1/3*(3*a^2-4*b^2)*cot(c+d*x)*csc(c+d*x)/(a^2*b*d*(a+b*sin(c+d*x)))-1/3*cot(c+d*x)*csc(c+d*x)^2/(a*d*(a+b*sin(c+d*x)))+2*(a^4-5*a^2*b^2+4*b^4)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(a^5*d*sqrt(a^2-b^2))],
[cos(c+d*x)^4*csc(c+d*x)^5/(a+b*sin(c+d*x))^2,x,9,-1/8*(3*a^4-36*a^2*b^2+40*b^4)*arctanh(cos(c+d*x))/(a^6*d)-1/3*b*(11*a^2-15*b^2)*cot(c+d*x)/(a^5*d)+1/8*(13*a^2-20*b^2)*cot(c+d*x)*csc(c+d*x)/(a^4*d)-1/3*(3*a^2-5*b^2)*cot(c+d*x)*csc(c+d*x)^2/(a^3*b*d)+1/4*(4*a^2-5*b^2)*cot(c+d*x)*csc(c+d*x)^2/(a^2*b*d*(a+b*sin(c+d*x)))-1/4*cot(c+d*x)*csc(c+d*x)^3/(a*d*(a+b*sin(c+d*x)))-2*b*(2*a^4-7*a^2*b^2+5*b^4)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(a^6*d*sqrt(a^2-b^2))],
[cos(c+d*x)^4*sin(c+d*x)^3/(a+b*sin(c+d*x))^3,x,9,3/8*(40*a^4-24*a^2*b^2+b^4)*x/b^7+1/2*a*(30*a^2-13*b^2)*cos(c+d*x)/(b^6*d)-3/8*(20*a^2-7*b^2)*cos(c+d*x)*sin(c+d*x)/(b^5*d)+1/2*(10*a^2-3*b^2)*cos(c+d*x)*sin(c+d*x)^2/(a*b^4*d)-1/4*(15*a^2-4*b^2)*cos(c+d*x)*sin(c+d*x)^3/(a^2*b^3*d)-1/2*(a^2-b^2)*cos(c+d*x)*sin(c+d*x)^4/(a*b^2*d*(a+b*sin(c+d*x))^2)+1/2*(7*a^2-2*b^2)*cos(c+d*x)*sin(c+d*x)^4/(a^2*b^2*d*(a+b*sin(c+d*x)))-3*a*(10*a^4-11*a^2*b^2+2*b^4)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(b^7*d*sqrt(a^2-b^2))],
[cos(c+d*x)^4*sin(c+d*x)^2/(a+b*sin(c+d*x))^3,x,8,1/2*a*(9-20*a^2/b^2)*x/b^4-1/6*(60*a^2-17*b^2)*cos(c+d*x)/(b^5*d)+(5*a^2-b^2)*cos(c+d*x)*sin(c+d*x)/(a*b^4*d)-1/6*(20*a^2-3*b^2)*cos(c+d*x)*sin(c+d*x)^2/(a^2*b^3*d)-1/2*(a^2-b^2)*cos(c+d*x)*sin(c+d*x)^3/(a*b^2*d*(a+b*sin(c+d*x))^2)+1/2*(6*a^2-b^2)*cos(c+d*x)*sin(c+d*x)^3/(a^2*b^2*d*(a+b*sin(c+d*x)))+(20*a^4-19*a^2*b^2+2*b^4)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(b^6*d*sqrt(a^2-b^2))],
[cos(c+d*x)^4*sin(c+d*x)/(a+b*sin(c+d*x))^3,x,6,3/2*(4*a^2-b^2)*x/b^5+1/2*cos(c+d*x)^3*(2*a+b*sin(c+d*x))/(b^2*d*(a+b*sin(c+d*x))^2)+3/2*cos(c+d*x)*(4*a^2-b^2+2*a*b*sin(c+d*x))/(b^4*d*(a+b*sin(c+d*x)))-3*a*(4*a^2-3*b^2)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(b^5*d*sqrt(a^2-b^2))],
[cos(c+d*x)^4*csc(c+d*x)/(a+b*sin(c+d*x))^3,x,6,x/b^3-arctanh(cos(c+d*x))/(a^3*d)-1/2*(a^2-b^2)*cos(c+d*x)/(a*b^2*d*(a+b*sin(c+d*x))^2)+1/2*(3*a^2+2*b^2)*cos(c+d*x)/(a^2*b^2*d*(a+b*sin(c+d*x)))-(2*a^4-a^2*b^2+2*b^4)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(a^3*b^3*d*sqrt(a^2-b^2))],
[cos(c+d*x)^4*csc(c+d*x)^2/(a+b*sin(c+d*x))^3,x,7,3*b*arctanh(cos(c+d*x))/(a^4*d)+1/2*(a^2-3*b^2)*cos(c+d*x)/(a^2*b*d*(a+b*sin(c+d*x))^2)-cot(c+d*x)/(a*d*(a+b*sin(c+d*x))^2)-1/2*(a^2+6*b^2)*cos(c+d*x)/(a^3*b*d*(a+b*sin(c+d*x)))-3*(a^2-2*b^2)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(a^4*d*sqrt(a^2-b^2))],
[cos(c+d*x)^4*csc(c+d*x)^3/(a+b*sin(c+d*x))^3,x,8,3/2*(a^2-4*b^2)*arctanh(cos(c+d*x))/(a^5*d)-1/2*(a^2-12*b^2)*cot(c+d*x)/(a^4*b*d)+1/2*(a^2-2*b^2)*cot(c+d*x)/(a^2*b*d*(a+b*sin(c+d*x))^2)-1/2*cot(c+d*x)*csc(c+d*x)/(a*d*(a+b*sin(c+d*x))^2)-3*b*cot(c+d*x)/(a^3*d*(a+b*sin(c+d*x)))+3*b*(3*a^2-4*b^2)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(a^5*d*sqrt(a^2-b^2))],
[cos(c+d*x)^4*csc(c+d*x)^4/(a+b*sin(c+d*x))^3,x,9,-1/2*b*(9*a^2-20*b^2)*arctanh(cos(c+d*x))/(a^6*d)+1/6*(17*a^2-60*b^2)*cot(c+d*x)/(a^5*d)-(a^2-5*b^2)*cot(c+d*x)*csc(c+d*x)/(a^4*b*d)+1/6*(3*a^2-5*b^2)*cot(c+d*x)*csc(c+d*x)/(a^2*b*d*(a+b*sin(c+d*x))^2)-1/3*cot(c+d*x)*csc(c+d*x)^2/(a*d*(a+b*sin(c+d*x))^2)+1/6*(3*a^2-20*b^2)*cot(c+d*x)*csc(c+d*x)/(a^3*b*d*(a+b*sin(c+d*x)))+(2*a^4-19*a^2*b^2+20*b^4)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(a^6*d*sqrt(a^2-b^2))],
[cos(c+d*x)^4*csc(c+d*x)^5/(a+b*sin(c+d*x))^3,x,10,-3/8*(a^4-24*a^2*b^2+40*b^4)*arctanh(cos(c+d*x))/(a^7*d)-1/2*b*(13*a^2-30*b^2)*cot(c+d*x)/(a^6*d)+3/8*(7*a^2-20*b^2)*cot(c+d*x)*csc(c+d*x)/(a^5*d)-1/2*(3*a^2-10*b^2)*cot(c+d*x)*csc(c+d*x)^2/(a^4*b*d)+1/4*(2*a^2-3*b^2)*cot(c+d*x)*csc(c+d*x)^2/(a^2*b*d*(a+b*sin(c+d*x))^2)-1/4*cot(c+d*x)*csc(c+d*x)^3/(a*d*(a+b*sin(c+d*x))^2)+1/4*(4*a^2-15*b^2)*cot(c+d*x)*csc(c+d*x)^2/(a^3*b*d*(a+b*sin(c+d*x)))-3*b*(2*a^4-11*a^2*b^2+10*b^4)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(a^7*d*sqrt(a^2-b^2))],

# Integrands of the form Cos[e+f x]^4 Sin[e+f x]^n (a+b Sin[e+f x])^(m/2)

# m>0
[cos(c+d*x)^4*sin(c+d*x)^2*sqrt(a+b*sin(c+d*x)),x,10,-8/45045*(480*a^4-937*a^2*b^2+231*b^4)*cos(c+d*x)*(a+b*sin(c+d*x))^(3/2)/(b^5*d)+8/3003*a*(40*a^2-81*b^2)*cos(c+d*x)*sin(c+d*x)*(a+b*sin(c+d*x))^(3/2)/(b^4*d)-10/1287*(16*a^2-33*b^2)*cos(c+d*x)*sin(c+d*x)^2*(a+b*sin(c+d*x))^(3/2)/(b^3*d)+20/143*a*cos(c+d*x)*sin(c+d*x)^3*(a+b*sin(c+d*x))^(3/2)/(b^2*d)-2/13*cos(c+d*x)*sin(c+d*x)^4*(a+b*sin(c+d*x))^(3/2)/(b*d)+16/45045*a*(160*a^4-279*a^2*b^2+27*b^4)*cos(c+d*x)*sqrt(a+b*sin(c+d*x))/(b^5*d)-8/45045*(320*a^6-798*a^4*b^2+435*a^2*b^4-693*b^6)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*sin(c+d*x))/(b^6*d*sqrt((a+b*sin(c+d*x))/(a+b)))+16/45045*a*(160*a^6-439*a^4*b^2+306*a^2*b^4-27*b^6)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(b^6*d*sqrt(a+b*sin(c+d*x)))],
[cos(c+d*x)^4*sin(c+d*x)*sqrt(a+b*sin(c+d*x)),x,8,-2/11*cos(c+d*x)^5*sqrt(a+b*sin(c+d*x))/d-2/693*cos(c+d*x)^3*(8*a^2-9*b^2-7*a*b*sin(c+d*x))*sqrt(a+b*sin(c+d*x))/(b^2*d)+4/3465*cos(c+d*x)*(32*a^4-69*a^2*b^2+45*b^4-24*a*b*(a^2-2*b^2)*sin(c+d*x))*sqrt(a+b*sin(c+d*x))/(b^4*d)+8/3465*a*(32*a^4-93*a^2*b^2+93*b^4)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*sin(c+d*x))/(b^5*d*sqrt((a+b*sin(c+d*x))/(a+b)))-8/3465*(32*a^6-101*a^4*b^2+114*a^2*b^4-45*b^6)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(b^5*d*sqrt(a+b*sin(c+d*x)))],
[cos(c+d*x)^3*cot(c+d*x)*sqrt(a+b*sin(c+d*x)),x,10,8/35*a*cos(c+d*x)*(a+b*sin(c+d*x))^(3/2)/(b^2*d)-2/7*cos(c+d*x)*sin(c+d*x)*(a+b*sin(c+d*x))^(3/2)/(b*d)-2/105*(8*a^2-45*b^2)*cos(c+d*x)*sqrt(a+b*sin(c+d*x))/(b^2*d)+2/105*a*(8*a^2-51*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*sin(c+d*x))/(b^3*d*sqrt((a+b*sin(c+d*x))/(a+b)))-2/105*(8*a^4-53*a^2*b^2-60*b^4)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(b^3*d*sqrt(a+b*sin(c+d*x)))+2*a*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2,sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(d*sqrt(a+b*sin(c+d*x)))],
[cos(c+d*x)^2*cot(c+d*x)^2*sqrt(a+b*sin(c+d*x)),x,10,-2/5*cos(c+d*x)*(a+b*sin(c+d*x))^(3/2)/(b*d)-cot(c+d*x)*(a+b*sin(c+d*x))^(3/2)/(a*d)+1/15*(4*a^2+15*b^2)*cos(c+d*x)*sqrt(a+b*sin(c+d*x))/(a*b*d)-1/15*(4*a^2+57*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*sin(c+d*x))/(b^2*d*sqrt((a+b*sin(c+d*x))/(a+b)))+1/15*a*(4*a^2+11*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(b^2*d*sqrt(a+b*sin(c+d*x)))+b*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2,sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(d*sqrt(a+b*sin(c+d*x)))],
[cos(c+d*x)*cot(c+d*x)^3*sqrt(a+b*sin(c+d*x)),x,10,1/4*b*cot(c+d*x)*(a+b*sin(c+d*x))^(3/2)/(a^2*d)-1/2*cot(c+d*x)*csc(c+d*x)*(a+b*sin(c+d*x))^(3/2)/(a*d)-1/12*(8*a^2+3*b^2)*cos(c+d*x)*sqrt(a+b*sin(c+d*x))/(a^2*d)+1/12*(8*a^2-3*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*sin(c+d*x))/(a*b*d*sqrt((a+b*sin(c+d*x))/(a+b)))-1/12*(8*a^2+31*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(b*d*sqrt(a+b*sin(c+d*x)))-1/4*(12*a^2+b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2,sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(a*d*sqrt(a+b*sin(c+d*x)))],
[cot(c+d*x)^4*sqrt(a+b*sin(c+d*x)),x,10,1/4*b*cot(c+d*x)*csc(c+d*x)*(a+b*sin(c+d*x))^(3/2)/(a^2*d)-1/3*cot(c+d*x)*csc(c+d*x)^2*(a+b*sin(c+d*x))^(3/2)/(a*d)+1/24*(32*a^2-3*b^2)*cot(c+d*x)*sqrt(a+b*sin(c+d*x))/(a^2*d)+1/24*(80*a^2+3*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*sin(c+d*x))/(a^2*d*sqrt((a+b*sin(c+d*x))/(a+b)))-1/24*(32*a^2+b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(a*d*sqrt(a+b*sin(c+d*x)))-1/8*b*(12*a^2-b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2,sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(a^2*d*sqrt(a+b*sin(c+d*x)))],
[cot(c+d*x)^4*csc(c+d*x)*sqrt(a+b*sin(c+d*x)),x,11,5/24*b*cot(c+d*x)*csc(c+d*x)^2*(a+b*sin(c+d*x))^(3/2)/(a^2*d)-1/4*cot(c+d*x)*csc(c+d*x)^3*(a+b*sin(c+d*x))^(3/2)/(a*d)+1/192*b*(68*a^2-15*b^2)*cot(c+d*x)*sqrt(a+b*sin(c+d*x))/(a^3*d)+5/32*(4*a^2-b^2)*cot(c+d*x)*csc(c+d*x)*sqrt(a+b*sin(c+d*x))/(a^2*d)+1/192*b*(68*a^2-15*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*sin(c+d*x))/(a^3*d*sqrt((a+b*sin(c+d*x))/(a+b)))+1/192*b*(196*a^2+5*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(a^2*d*sqrt(a+b*sin(c+d*x)))+1/64*(48*a^4+24*a^2*b^2-5*b^4)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2,sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(a^3*d*sqrt(a+b*sin(c+d*x)))],
[cot(c+d*x)^4*csc(c+d*x)^2*sqrt(a+b*sin(c+d*x)),x,12,7/40*b*cot(c+d*x)*csc(c+d*x)^3*(a+b*sin(c+d*x))^(3/2)/(a^2*d)-1/5*cot(c+d*x)*csc(c+d*x)^4*(a+b*sin(c+d*x))^(3/2)/(a*d)-1/1920*(384*a^4+332*a^2*b^2-105*b^4)*cot(c+d*x)*sqrt(a+b*sin(c+d*x))/(a^4*d)+1/960*b*(108*a^2-35*b^2)*cot(c+d*x)*csc(c+d*x)*sqrt(a+b*sin(c+d*x))/(a^3*d)+1/240*(96*a^2-35*b^2)*cot(c+d*x)*csc(c+d*x)^2*sqrt(a+b*sin(c+d*x))/(a^2*d)-1/1920*(384*a^4+332*a^2*b^2-105*b^4)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*sin(c+d*x))/(a^4*d*sqrt((a+b*sin(c+d*x))/(a+b)))+1/1920*(384*a^4+116*a^2*b^2-35*b^4)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(a^3*d*sqrt(a+b*sin(c+d*x)))+1/128*b*(48*a^4-24*a^2*b^2+7*b^4)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2,sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(a^4*d*sqrt(a+b*sin(c+d*x)))],
[cos(c+d*x)^4*sin(c+d*x)^2*(a+b*sin(c+d*x))^(3/2),x,11,16/45045*a*(32*a^4-47*a^2*b^2-27*b^4)*cos(c+d*x)*(a+b*sin(c+d*x))^(3/2)/(b^5*d)-8/45045*(160*a^4-375*a^2*b^2+117*b^4)*cos(c+d*x)*(a+b*sin(c+d*x))^(5/2)/(b^5*d)+8/1287*a*(8*a^2-21*b^2)*cos(c+d*x)*sin(c+d*x)*(a+b*sin(c+d*x))^(5/2)/(b^4*d)-2/2145*(80*a^2-221*b^2)*cos(c+d*x)*sin(c+d*x)^2*(a+b*sin(c+d*x))^(5/2)/(b^3*d)+4/39*a*cos(c+d*x)*sin(c+d*x)^3*(a+b*sin(c+d*x))^(5/2)/(b^2*d)-2/15*cos(c+d*x)*sin(c+d*x)^4*(a+b*sin(c+d*x))^(5/2)/(b*d)+8/45045*(64*a^6-174*a^4*b^2+81*a^2*b^4-195*b^6)*cos(c+d*x)*sqrt(a+b*sin(c+d*x))/(b^5*d)-16/45045*a*(32*a^6-111*a^4*b^2+102*a^2*b^4-471*b^6)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*sin(c+d*x))/(b^6*d*sqrt((a+b*sin(c+d*x))/(a+b)))+8/45045*(64*a^8-238*a^6*b^2+255*a^4*b^4-276*a^2*b^6+195*b^8)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(b^6*d*sqrt(a+b*sin(c+d*x)))],
[cos(c+d*x)^4*sin(c+d*x)*(a+b*sin(c+d*x))^(3/2),x,9,-2/13*cos(c+d*x)^5*(a+b*sin(c+d*x))^(3/2)/d-6/143*a*cos(c+d*x)^5*sqrt(a+b*sin(c+d*x))/d-2/3003*cos(c+d*x)^3*(4*a*(2*a^2-5*b^2)-7*b*(a^2+11*b^2)*sin(c+d*x))*sqrt(a+b*sin(c+d*x))/(b^2*d)+4/15015*cos(c+d*x)*(a*(32*a^4-113*a^2*b^2+177*b^4)-3*b*(8*a^4-27*a^2*b^2-77*b^4)*sin(c+d*x))*sqrt(a+b*sin(c+d*x))/(b^4*d)+8/15015*(32*a^6-137*a^4*b^2+258*a^2*b^4+231*b^6)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*sin(c+d*x))/(b^5*d*sqrt((a+b*sin(c+d*x))/(a+b)))-8/15015*a*(32*a^6-145*a^4*b^2+290*a^2*b^4-177*b^6)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(b^5*d*sqrt(a+b*sin(c+d*x)))],
[cos(c+d*x)^3*cot(c+d*x)*(a+b*sin(c+d*x))^(3/2),x,11,-2/315*(8*a^2-77*b^2)*cos(c+d*x)*(a+b*sin(c+d*x))^(3/2)/(b^2*d)+8/63*a*cos(c+d*x)*(a+b*sin(c+d*x))^(5/2)/(b^2*d)-2/9*cos(c+d*x)*sin(c+d*x)*(a+b*sin(c+d*x))^(5/2)/(b*d)-2/315*a*(8*a^2-87*b^2)*cos(c+d*x)*sqrt(a+b*sin(c+d*x))/(b^2*d)+2/315*(8*a^4-93*a^2*b^2+84*b^4)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*sin(c+d*x))/(b^3*d*sqrt((a+b*sin(c+d*x))/(a+b)))-2/315*a*(8*a^4-95*a^2*b^2-228*b^4)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(b^3*d*sqrt(a+b*sin(c+d*x)))+2*a^2*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2,sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(d*sqrt(a+b*sin(c+d*x)))],
[cos(c+d*x)^2*cot(c+d*x)^2*(a+b*sin(c+d*x))^(3/2),x,11,1/35*(4*a^2+35*b^2)*cos(c+d*x)*(a+b*sin(c+d*x))^(3/2)/(a*b*d)-2/7*cos(c+d*x)*(a+b*sin(c+d*x))^(5/2)/(b*d)-cot(c+d*x)*(a+b*sin(c+d*x))^(5/2)/(a*d)+1/35*(4*a^2+65*b^2)*cos(c+d*x)*sqrt(a+b*sin(c+d*x))/(b*d)-1/35*a*(4*a^2+167*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*sin(c+d*x))/(b^2*d*sqrt((a+b*sin(c+d*x))/(a+b)))+1/35*(4*a^4+61*a^2*b^2+40*b^4)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(b^2*d*sqrt(a+b*sin(c+d*x)))+3*a*b*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2,sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(d*sqrt(a+b*sin(c+d*x)))],
[cos(c+d*x)*cot(c+d*x)^3*(a+b*sin(c+d*x))^(3/2),x,11,-1/20*(8*a^2-5*b^2)*cos(c+d*x)*(a+b*sin(c+d*x))^(3/2)/(a^2*d)-1/4*b*cot(c+d*x)*(a+b*sin(c+d*x))^(5/2)/(a^2*d)-1/2*cot(c+d*x)*csc(c+d*x)*(a+b*sin(c+d*x))^(5/2)/(a*d)-1/20*(8*a^2-15*b^2)*cos(c+d*x)*sqrt(a+b*sin(c+d*x))/(a*d)+1/20*(8*a^2-81*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*sin(c+d*x))/(b*d*sqrt((a+b*sin(c+d*x))/(a+b)))-1/20*a*(8*a^2+37*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(b*d*sqrt(a+b*sin(c+d*x)))-3/4*(4*a^2-b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2,sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(d*sqrt(a+b*sin(c+d*x)))],
[cot(c+d*x)^4*(a+b*sin(c+d*x))^(3/2),x,11,1/24*(32*a^2+b^2)*cot(c+d*x)*(a+b*sin(c+d*x))^(3/2)/(a^2*d)+1/12*b*cot(c+d*x)*csc(c+d*x)*(a+b*sin(c+d*x))^(5/2)/(a^2*d)-1/3*cot(c+d*x)*csc(c+d*x)^2*(a+b*sin(c+d*x))^(5/2)/(a*d)-1/8*b*(16*a^2+b^2)*cos(c+d*x)*sqrt(a+b*sin(c+d*x))/(a^2*d)+1/8*(32*a^2-b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*sin(c+d*x))/(a*d*sqrt((a+b*sin(c+d*x))/(a+b)))-1/8*(16*a^2+21*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(d*sqrt(a+b*sin(c+d*x)))-1/8*b*(36*a^2+b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2,sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(a*d*sqrt(a+b*sin(c+d*x)))],
[cot(c+d*x)^4*csc(c+d*x)*(a+b*sin(c+d*x))^(3/2),x,11,1/32*(20*a^2-b^2)*cot(c+d*x)*csc(c+d*x)*(a+b*sin(c+d*x))^(3/2)/(a^2*d)+1/8*b*cot(c+d*x)*csc(c+d*x)^2*(a+b*sin(c+d*x))^(5/2)/(a^2*d)-1/4*cot(c+d*x)*csc(c+d*x)^3*(a+b*sin(c+d*x))^(5/2)/(a*d)+1/64*b*(68*a^2-3*b^2)*cot(c+d*x)*sqrt(a+b*sin(c+d*x))/(a^2*d)+1/64*b*(236*a^2+3*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*sin(c+d*x))/(a^2*d*sqrt((a+b*sin(c+d*x))/(a+b)))-1/64*b*(20*a^2+b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(a*d*sqrt(a+b*sin(c+d*x)))+3/64*(16*a^4-24*a^2*b^2+b^4)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2,sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(a^2*d*sqrt(a+b*sin(c+d*x)))],
[cot(c+d*x)^4*csc(c+d*x)^2*(a+b*sin(c+d*x))^(3/2),x,12,1/80*(32*a^2-5*b^2)*cot(c+d*x)*csc(c+d*x)^2*(a+b*sin(c+d*x))^(3/2)/(a^2*d)+1/8*b*cot(c+d*x)*csc(c+d*x)^3*(a+b*sin(c+d*x))^(5/2)/(a^2*d)-1/5*cot(c+d*x)*csc(c+d*x)^4*(a+b*sin(c+d*x))^(5/2)/(a*d)-1/640*(128*a^4-116*a^2*b^2+15*b^4)*cot(c+d*x)*sqrt(a+b*sin(c+d*x))/(a^3*d)+3/320*b*(36*a^2-5*b^2)*cot(c+d*x)*csc(c+d*x)*sqrt(a+b*sin(c+d*x))/(a^2*d)-1/640*(128*a^4-116*a^2*b^2+15*b^4)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*sin(c+d*x))/(a^3*d*sqrt((a+b*sin(c+d*x))/(a+b)))+1/640*(128*a^4+692*a^2*b^2+5*b^4)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(a^2*d*sqrt(a+b*sin(c+d*x)))+3/128*b*(48*a^4+8*a^2*b^2-b^4)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2,sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(a^3*d*sqrt(a+b*sin(c+d*x)))],
[cot(c+d*x)^4*csc(c+d*x)^3*(a+b*sin(c+d*x))^(3/2),x,13,7/96*(4*a^2-b^2)*cot(c+d*x)*csc(c+d*x)^3*(a+b*sin(c+d*x))^(3/2)/(a^2*d)+7/60*b*cot(c+d*x)*csc(c+d*x)^4*(a+b*sin(c+d*x))^(5/2)/(a^2*d)-1/6*cot(c+d*x)*csc(c+d*x)^5*(a+b*sin(c+d*x))^(5/2)/(a*d)-1/7680*b*(2064*a^4+512*a^2*b^2-105*b^4)*cot(c+d*x)*sqrt(a+b*sin(c+d*x))/(a^4*d)-1/3840*(240*a^4-168*a^2*b^2+35*b^4)*cot(c+d*x)*csc(c+d*x)*sqrt(a+b*sin(c+d*x))/(a^3*d)+1/960*b*(156*a^2-35*b^2)*cot(c+d*x)*csc(c+d*x)^2*sqrt(a+b*sin(c+d*x))/(a^2*d)-1/7680*b*(2064*a^4+512*a^2*b^2-105*b^4)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*sin(c+d*x))/(a^4*d*sqrt((a+b*sin(c+d*x))/(a+b)))+1/7680*b*(2544*a^4+176*a^2*b^2-35*b^4)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(a^3*d*sqrt(a+b*sin(c+d*x)))+1/512*(64*a^6+144*a^4*b^2-36*a^2*b^4+7*b^6)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2,sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(a^4*d*sqrt(a+b*sin(c+d*x)))],
[cos(c+d*x)^4*sin(c+d*x)*(a+b*sin(c+d*x))^(5/2),x,10,-2/39*a*cos(c+d*x)^5*(a+b*sin(c+d*x))^(3/2)/d-2/15*cos(c+d*x)^5*(a+b*sin(c+d*x))^(5/2)/d-2/429*(3*a^2+13*b^2)*cos(c+d*x)^5*sqrt(a+b*sin(c+d*x))/d-2/9009*cos(c+d*x)^3*(8*a^4-33*a^2*b^2-39*b^4-7*a*b*(a^2+63*b^2)*sin(c+d*x))*sqrt(a+b*sin(c+d*x))/(b^2*d)+4/45045*cos(c+d*x)*(32*a^6-165*a^4*b^2+450*a^2*b^4+195*b^6-24*a*b*(a^4-5*a^2*b^2-60*b^4)*sin(c+d*x))*sqrt(a+b*sin(c+d*x))/(b^4*d)+8/45045*a*(32*a^6-189*a^4*b^2+570*a^2*b^4+1635*b^6)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*sin(c+d*x))/(b^5*d*sqrt((a+b*sin(c+d*x))/(a+b)))-8/45045*(32*a^8-197*a^6*b^2+615*a^4*b^4-255*a^2*b^6-195*b^8)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(b^5*d*sqrt(a+b*sin(c+d*x)))],
[cos(c+d*x)^3*cot(c+d*x)*(a+b*sin(c+d*x))^(5/2),x,12,-2/693*a*(8*a^2-131*b^2)*cos(c+d*x)*(a+b*sin(c+d*x))^(3/2)/(b^2*d)-2/693*(8*a^2-117*b^2)*cos(c+d*x)*(a+b*sin(c+d*x))^(5/2)/(b^2*d)+8/99*a*cos(c+d*x)*(a+b*sin(c+d*x))^(7/2)/(b^2*d)-2/11*cos(c+d*x)*sin(c+d*x)*(a+b*sin(c+d*x))^(7/2)/(b*d)-2/693*(8*a^4-141*a^2*b^2+36*b^4)*cos(c+d*x)*sqrt(a+b*sin(c+d*x))/(b^2*d)+2/693*a*(8*a^4-147*a^2*b^2+444*b^4)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*sin(c+d*x))/(b^3*d*sqrt((a+b*sin(c+d*x))/(a+b)))-2/693*(8*a^6-149*a^4*b^2-516*a^2*b^4-36*b^6)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(b^3*d*sqrt(a+b*sin(c+d*x)))+2*a^3*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2,sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(d*sqrt(a+b*sin(c+d*x)))],
[cos(c+d*x)^2*cot(c+d*x)^2*(a+b*sin(c+d*x))^(5/2),x,12,1/315*(20*a^2+469*b^2)*cos(c+d*x)*(a+b*sin(c+d*x))^(3/2)/(b*d)+1/63*(4*a^2+63*b^2)*cos(c+d*x)*(a+b*sin(c+d*x))^(5/2)/(a*b*d)-2/9*cos(c+d*x)*(a+b*sin(c+d*x))^(7/2)/(b*d)-cot(c+d*x)*(a+b*sin(c+d*x))^(7/2)/(a*d)+1/315*a*(20*a^2+759*b^2)*cos(c+d*x)*sqrt(a+b*sin(c+d*x))/(b*d)-1/315*(20*a^4+1689*a^2*b^2-168*b^4)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*sin(c+d*x))/(b^2*d*sqrt((a+b*sin(c+d*x))/(a+b)))+1/315*a*(20*a^4+739*a^2*b^2+816*b^4)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(b^2*d*sqrt(a+b*sin(c+d*x)))+5*a^2*b*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2,sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(d*sqrt(a+b*sin(c+d*x)))],
[cos(c+d*x)*cot(c+d*x)^3*(a+b*sin(c+d*x))^(5/2),x,12,-1/28*(8*a^2-35*b^2)*cos(c+d*x)*(a+b*sin(c+d*x))^(3/2)/(a*d)-1/28*(8*a^2-21*b^2)*cos(c+d*x)*(a+b*sin(c+d*x))^(5/2)/(a^2*d)-3/4*b*cot(c+d*x)*(a+b*sin(c+d*x))^(7/2)/(a^2*d)-1/2*cot(c+d*x)*csc(c+d*x)*(a+b*sin(c+d*x))^(7/2)/(a*d)-1/28*(8*a^2-73*b^2)*cos(c+d*x)*sqrt(a+b*sin(c+d*x))/d+1/28*a*(8*a^2-247*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*sin(c+d*x))/(b*d*sqrt((a+b*sin(c+d*x))/(a+b)))-1/28*(8*a^4+3*a^2*b^2-32*b^4)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(b*d*sqrt(a+b*sin(c+d*x)))-3/4*a*(4*a^2-5*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2,sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(d*sqrt(a+b*sin(c+d*x)))],
[cot(c+d*x)^4*(a+b*sin(c+d*x))^(5/2),x,12,-1/120*b*(208*a^2-25*b^2)*cos(c+d*x)*(a+b*sin(c+d*x))^(3/2)/(a^2*d)+1/24*(32*a^2-3*b^2)*cot(c+d*x)*(a+b*sin(c+d*x))^(5/2)/(a^2*d)-1/12*b*cot(c+d*x)*csc(c+d*x)*(a+b*sin(c+d*x))^(7/2)/(a^2*d)-1/3*cot(c+d*x)*csc(c+d*x)^2*(a+b*sin(c+d*x))^(7/2)/(a*d)-1/40*b*(96*a^2-25*b^2)*cos(c+d*x)*sqrt(a+b*sin(c+d*x))/(a*d)+1/40*(176*a^2-167*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*sin(c+d*x))/(d*sqrt((a+b*sin(c+d*x))/(a+b)))-1/40*a*(96*a^2+179*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(d*sqrt(a+b*sin(c+d*x)))-5/8*b*(12*a^2-b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2,sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(d*sqrt(a+b*sin(c+d*x)))],
[cot(c+d*x)^4*csc(c+d*x)*(a+b*sin(c+d*x))^(5/2),x,12,5/192*b*(68*a^2+b^2)*cot(c+d*x)*(a+b*sin(c+d*x))^(3/2)/(a^2*d)+1/96*(60*a^2+b^2)*cot(c+d*x)*csc(c+d*x)*(a+b*sin(c+d*x))^(5/2)/(a^2*d)+1/24*b*cot(c+d*x)*csc(c+d*x)^2*(a+b*sin(c+d*x))^(7/2)/(a^2*d)-1/4*cot(c+d*x)*csc(c+d*x)^3*(a+b*sin(c+d*x))^(7/2)/(a*d)-1/64*b^2*(196*a^2+5*b^2)*cos(c+d*x)*sqrt(a+b*sin(c+d*x))/(a^2*d)+1/64*b*(492*a^2-5*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*sin(c+d*x))/(a*d*sqrt((a+b*sin(c+d*x))/(a+b)))-1/64*b*(148*a^2+169*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(d*sqrt(a+b*sin(c+d*x)))+1/64*(48*a^4-360*a^2*b^2-5*b^4)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2,sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(a*d*sqrt(a+b*sin(c+d*x)))],
[cot(c+d*x)^4*csc(c+d*x)^2*(a+b*sin(c+d*x))^(5/2),x,12,1/64*b*(36*a^2-b^2)*cot(c+d*x)*csc(c+d*x)*(a+b*sin(c+d*x))^(3/2)/(a^2*d)+1/80*(32*a^2-b^2)*cot(c+d*x)*csc(c+d*x)^2*(a+b*sin(c+d*x))^(5/2)/(a^2*d)+3/40*b*cot(c+d*x)*csc(c+d*x)^3*(a+b*sin(c+d*x))^(7/2)/(a^2*d)-1/5*cot(c+d*x)*csc(c+d*x)^4*(a+b*sin(c+d*x))^(7/2)/(a*d)-1/640*(128*a^4-580*a^2*b^2+15*b^4)*cot(c+d*x)*sqrt(a+b*sin(c+d*x))/(a^2*d)-1/640*(128*a^4-2476*a^2*b^2-15*b^4)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*sin(c+d*x))/(a^2*d*sqrt((a+b*sin(c+d*x))/(a+b)))+1/640*(128*a^4+492*a^2*b^2-5*b^4)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(a*d*sqrt(a+b*sin(c+d*x)))+3/128*b*(80*a^4-40*a^2*b^2+b^4)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2,sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(a^2*d*sqrt(a+b*sin(c+d*x)))],
[cot(c+d*x)^4*csc(c+d*x)^3*(a+b*sin(c+d*x))^(5/2),x,13,1/192*b*(52*a^2-5*b^2)*cot(c+d*x)*csc(c+d*x)^2*(a+b*sin(c+d*x))^(3/2)/(a^2*d)+1/96*(28*a^2-3*b^2)*cot(c+d*x)*csc(c+d*x)^3*(a+b*sin(c+d*x))^(5/2)/(a^2*d)+1/12*b*cot(c+d*x)*csc(c+d*x)^4*(a+b*sin(c+d*x))^(7/2)/(a^2*d)-1/6*cot(c+d*x)*csc(c+d*x)^5*(a+b*sin(c+d*x))^(7/2)/(a*d)-1/1536*b*(720*a^4-176*a^2*b^2+15*b^4)*cot(c+d*x)*sqrt(a+b*sin(c+d*x))/(a^3*d)-1/256*(16*a^4-56*a^2*b^2+5*b^4)*cot(c+d*x)*csc(c+d*x)*sqrt(a+b*sin(c+d*x))/(a^2*d)-1/1536*b*(720*a^4-176*a^2*b^2+15*b^4)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*sin(c+d*x))/(a^3*d*sqrt((a+b*sin(c+d*x))/(a+b)))+1/1536*b*(816*a^4+1696*a^2*b^2+5*b^4)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(a^2*d*sqrt(a+b*sin(c+d*x)))+1/512*(64*a^6+720*a^4*b^2+60*a^2*b^4-5*b^6)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2,sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(a^3*d*sqrt(a+b*sin(c+d*x)))],

# m<0
[cos(c+d*x)^4*sin(c+d*x)^3/sqrt(a+b*sin(c+d*x)),x,10,64/15015*a*(80*a^4-118*a^2*b^2+17*b^4)*cos(c+d*x)*sqrt(a+b*sin(c+d*x))/(b^6*d)-8/15015*(480*a^4-683*a^2*b^2+77*b^4)*cos(c+d*x)*sin(c+d*x)*sqrt(a+b*sin(c+d*x))/(b^5*d)+4/3003*a*(160*a^2-223*b^2)*cos(c+d*x)*sin(c+d*x)^2*sqrt(a+b*sin(c+d*x))/(b^4*d)-10/429*(8*a^2-11*b^2)*cos(c+d*x)*sin(c+d*x)^3*sqrt(a+b*sin(c+d*x))/(b^3*d)+24/143*a*cos(c+d*x)*sin(c+d*x)^4*sqrt(a+b*sin(c+d*x))/(b^2*d)-2/13*cos(c+d*x)*sin(c+d*x)^5*sqrt(a+b*sin(c+d*x))/(b*d)+8/15015*(1280*a^6-2048*a^4*b^2+453*a^2*b^4+231*b^6)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*sin(c+d*x))/(b^7*d*sqrt((a+b*sin(c+d*x))/(a+b)))-8/15015*a*(1280*a^6-2368*a^4*b^2+875*a^2*b^4+213*b^6)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(b^7*d*sqrt(a+b*sin(c+d*x)))],
[cos(c+d*x)^4*sin(c+d*x)^2/sqrt(a+b*sin(c+d*x)),x,9,-8/3465*(160*a^4-247*a^2*b^2+45*b^4)*cos(c+d*x)*sqrt(a+b*sin(c+d*x))/(b^5*d)+8/3465*a*(120*a^2-179*b^2)*cos(c+d*x)*sin(c+d*x)*sqrt(a+b*sin(c+d*x))/(b^4*d)-2/693*(80*a^2-117*b^2)*cos(c+d*x)*sin(c+d*x)^2*sqrt(a+b*sin(c+d*x))/(b^3*d)+20/99*a*cos(c+d*x)*sin(c+d*x)^3*sqrt(a+b*sin(c+d*x))/(b^2*d)-2/11*cos(c+d*x)*sin(c+d*x)^4*sqrt(a+b*sin(c+d*x))/(b*d)-16/3465*a*(160*a^4-267*a^2*b^2+69*b^4)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*sin(c+d*x))/(b^6*d*sqrt((a+b*sin(c+d*x))/(a+b)))+8/3465*(320*a^6-614*a^4*b^2+249*a^2*b^4+45*b^6)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(b^6*d*sqrt(a+b*sin(c+d*x)))],
[cos(c+d*x)^4*sin(c+d*x)/sqrt(a+b*sin(c+d*x)),x,7,-2/63*cos(c+d*x)^3*(8*a-7*b*sin(c+d*x))*sqrt(a+b*sin(c+d*x))/(b^2*d)+4/315*cos(c+d*x)*(a*(32*a^2-33*b^2)-3*b*(8*a^2-7*b^2)*sin(c+d*x))*sqrt(a+b*sin(c+d*x))/(b^4*d)+8/315*(32*a^4-57*a^2*b^2+21*b^4)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*sin(c+d*x))/(b^5*d*sqrt((a+b*sin(c+d*x))/(a+b)))-8/315*a*(32*a^4-65*a^2*b^2+33*b^4)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(b^5*d*sqrt(a+b*sin(c+d*x)))],
[cos(c+d*x)^3*cot(c+d*x)/sqrt(a+b*sin(c+d*x)),x,9,8/15*a*cos(c+d*x)*sqrt(a+b*sin(c+d*x))/(b^2*d)-2/5*cos(c+d*x)*sin(c+d*x)*sqrt(a+b*sin(c+d*x))/(b*d)+2/15*(8*a^2-21*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*sin(c+d*x))/(b^3*d*sqrt((a+b*sin(c+d*x))/(a+b)))-2/15*a*(8*a^2-23*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(b^3*d*sqrt(a+b*sin(c+d*x)))+2*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2,sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(d*sqrt(a+b*sin(c+d*x)))],
[cos(c+d*x)^2*cot(c+d*x)^2/sqrt(a+b*sin(c+d*x)),x,9,-2/3*cos(c+d*x)*sqrt(a+b*sin(c+d*x))/(b*d)-cot(c+d*x)*sqrt(a+b*sin(c+d*x))/(a*d)-1/3*(4*a^2+3*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*sin(c+d*x))/(a*b^2*d*sqrt((a+b*sin(c+d*x))/(a+b)))+1/3*(4*a^2-7*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(b^2*d*sqrt(a+b*sin(c+d*x)))-b*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2,sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(a*d*sqrt(a+b*sin(c+d*x)))],
[cos(c+d*x)*cot(c+d*x)^3/sqrt(a+b*sin(c+d*x)),x,9,3/4*b*cot(c+d*x)*sqrt(a+b*sin(c+d*x))/(a^2*d)-1/2*cot(c+d*x)*csc(c+d*x)*sqrt(a+b*sin(c+d*x))/(a*d)+1/4*(8*a^2+3*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*sin(c+d*x))/(a^2*b*d*sqrt((a+b*sin(c+d*x))/(a+b)))-1/4*(8*a^2+b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(a*b*d*sqrt(a+b*sin(c+d*x)))-3/4*(4*a^2-b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2,sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(a^2*d*sqrt(a+b*sin(c+d*x)))],
[cot(c+d*x)^4/sqrt(a+b*sin(c+d*x)),x,10,1/24*(32*a^2-15*b^2)*cot(c+d*x)*sqrt(a+b*sin(c+d*x))/(a^3*d)+5/12*b*cot(c+d*x)*csc(c+d*x)*sqrt(a+b*sin(c+d*x))/(a^2*d)-1/3*cot(c+d*x)*csc(c+d*x)^2*sqrt(a+b*sin(c+d*x))/(a*d)+1/24*(32*a^2-15*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*sin(c+d*x))/(a^3*d*sqrt((a+b*sin(c+d*x))/(a+b)))+1/24*(16*a^2+5*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(a^2*d*sqrt(a+b*sin(c+d*x)))+1/8*b*(12*a^2-5*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2,sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(a^3*d*sqrt(a+b*sin(c+d*x)))],
[cot(c+d*x)^4*csc(c+d*x)/sqrt(a+b*sin(c+d*x)),x,11,-1/192*b*(188*a^2-105*b^2)*cot(c+d*x)*sqrt(a+b*sin(c+d*x))/(a^4*d)+5/96*(12*a^2-7*b^2)*cot(c+d*x)*csc(c+d*x)*sqrt(a+b*sin(c+d*x))/(a^3*d)+7/24*b*cot(c+d*x)*csc(c+d*x)^2*sqrt(a+b*sin(c+d*x))/(a^2*d)-1/4*cot(c+d*x)*csc(c+d*x)^3*sqrt(a+b*sin(c+d*x))/(a*d)-1/192*b*(188*a^2-105*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*sin(c+d*x))/(a^4*d*sqrt((a+b*sin(c+d*x))/(a+b)))+1/192*b*(68*a^2-35*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(a^3*d*sqrt(a+b*sin(c+d*x)))+1/64*(48*a^4-72*a^2*b^2+35*b^4)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2,sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(a^4*d*sqrt(a+b*sin(c+d*x)))],
[cos(c+d*x)^4*sin(c+d*x)^3/(a+b*sin(c+d*x))^(3/2),x,10,-2*(a^2-b^2)*cos(c+d*x)*sin(c+d*x)^4/(a*b^2*d*sqrt(a+b*sin(c+d*x)))-8/1155*(640*a^4-592*a^2*b^2+15*b^4)*cos(c+d*x)*sqrt(a+b*sin(c+d*x))/(b^6*d)+8/1155*a*(480*a^2-419*b^2)*cos(c+d*x)*sin(c+d*x)*sqrt(a+b*sin(c+d*x))/(b^5*d)-20/231*(32*a^2-27*b^2)*cos(c+d*x)*sin(c+d*x)^2*sqrt(a+b*sin(c+d*x))/(b^4*d)+2/33*(40*a^2-33*b^2)*cos(c+d*x)*sin(c+d*x)^3*sqrt(a+b*sin(c+d*x))/(a*b^3*d)-2/11*cos(c+d*x)*sin(c+d*x)^4*sqrt(a+b*sin(c+d*x))/(b^2*d)-8/1155*a*(1280*a^4-1344*a^2*b^2+123*b^4)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*sin(c+d*x))/(b^7*d*sqrt((a+b*sin(c+d*x))/(a+b)))+8/1155*(1280*a^6-1664*a^4*b^2+369*a^2*b^4+15*b^6)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(b^7*d*sqrt(a+b*sin(c+d*x)))],
[cos(c+d*x)^4*sin(c+d*x)^2/(a+b*sin(c+d*x))^(3/2),x,9,-2*(a^2-b^2)*cos(c+d*x)*sin(c+d*x)^3/(a*b^2*d*sqrt(a+b*sin(c+d*x)))+8/315*a*(160*a^2-139*b^2)*cos(c+d*x)*sqrt(a+b*sin(c+d*x))/(b^5*d)-16/315*(60*a^2-49*b^2)*cos(c+d*x)*sin(c+d*x)*sqrt(a+b*sin(c+d*x))/(b^4*d)+2/63*(80*a^2-63*b^2)*cos(c+d*x)*sin(c+d*x)^2*sqrt(a+b*sin(c+d*x))/(a*b^3*d)-2/9*cos(c+d*x)*sin(c+d*x)^3*sqrt(a+b*sin(c+d*x))/(b^2*d)+8/315*(320*a^4-318*a^2*b^2+21*b^4)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*sin(c+d*x))/(b^6*d*sqrt((a+b*sin(c+d*x))/(a+b)))-16/315*a*(160*a^4-199*a^2*b^2+39*b^4)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(b^6*d*sqrt(a+b*sin(c+d*x)))],
[cos(c+d*x)^4*sin(c+d*x)/(a+b*sin(c+d*x))^(3/2),x,7,2/7*cos(c+d*x)^3*(8*a+b*sin(c+d*x))/(b^2*d*sqrt(a+b*sin(c+d*x)))-4/35*cos(c+d*x)*(32*a^2-5*b^2-24*a*b*sin(c+d*x))*sqrt(a+b*sin(c+d*x))/(b^4*d)-8/35*a*(32*a^2-29*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*sin(c+d*x))/(b^5*d*sqrt((a+b*sin(c+d*x))/(a+b)))+8/35*(32*a^4-37*a^2*b^2+5*b^4)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(b^5*d*sqrt(a+b*sin(c+d*x)))],
[cos(c+d*x)^3*cot(c+d*x)/(a+b*sin(c+d*x))^(3/2),x,9,-2*(a^2-b^2)*cos(c+d*x)/(a*b^2*d*sqrt(a+b*sin(c+d*x)))-2/3*cos(c+d*x)*sqrt(a+b*sin(c+d*x))/(b^2*d)-2/3*(8*a^2-3*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*sin(c+d*x))/(a*b^3*d*sqrt((a+b*sin(c+d*x))/(a+b)))+2/3*(8*a^2-5*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(b^3*d*sqrt(a+b*sin(c+d*x)))+2*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2,sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(a*d*sqrt(a+b*sin(c+d*x)))],
[cos(c+d*x)^2*cot(c+d*x)^2/(a+b*sin(c+d*x))^(3/2),x,9,(2*a^2-3*b^2)*cos(c+d*x)/(a^2*b*d*sqrt(a+b*sin(c+d*x)))-cot(c+d*x)/(a*d*sqrt(a+b*sin(c+d*x)))+(4*a^2-3*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*sin(c+d*x))/(a^2*b^2*d*sqrt((a+b*sin(c+d*x))/(a+b)))-(4*a^2-b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(a*b^2*d*sqrt(a+b*sin(c+d*x)))-3*b*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2,sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(a^2*d*sqrt(a+b*sin(c+d*x)))],
[cos(c+d*x)*cot(c+d*x)^3/(a+b*sin(c+d*x))^(3/2),x,10,1/2*(4*a^2-5*b^2)*cot(c+d*x)/(a^2*b*d*sqrt(a+b*sin(c+d*x)))-1/2*cot(c+d*x)*csc(c+d*x)/(a*d*sqrt(a+b*sin(c+d*x)))-1/4*(8*a^2-15*b^2)*cot(c+d*x)*sqrt(a+b*sin(c+d*x))/(a^3*b*d)-1/4*(8*a^2-15*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*sin(c+d*x))/(a^3*b*d*sqrt((a+b*sin(c+d*x))/(a+b)))+1/4*(8*a^2-5*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(a^2*b*d*sqrt(a+b*sin(c+d*x)))-3/4*(4*a^2-5*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2,sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(a^3*d*sqrt(a+b*sin(c+d*x)))],
[cot(c+d*x)^4/(a+b*sin(c+d*x))^(3/2),x,11,1/3*(6*a^2-7*b^2)*cot(c+d*x)*csc(c+d*x)/(a^2*b*d*sqrt(a+b*sin(c+d*x)))-1/3*cot(c+d*x)*csc(c+d*x)^2/(a*d*sqrt(a+b*sin(c+d*x)))+5/24*(16*a^2-21*b^2)*cot(c+d*x)*sqrt(a+b*sin(c+d*x))/(a^4*d)-1/12*(24*a^2-35*b^2)*cot(c+d*x)*csc(c+d*x)*sqrt(a+b*sin(c+d*x))/(a^3*b*d)+5/24*(16*a^2-21*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*sin(c+d*x))/(a^4*d*sqrt((a+b*sin(c+d*x))/(a+b)))-1/24*(32*a^2-35*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(a^3*d*sqrt(a+b*sin(c+d*x)))+1/8*b*(36*a^2-35*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2,sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(a^4*d*sqrt(a+b*sin(c+d*x)))],
[cos(c+d*x)^4*sin(c+d*x)^3/(a+b*sin(c+d*x))^(5/2),x,10,-2/3*(a^2-b^2)*cos(c+d*x)*sin(c+d*x)^4/(a*b^2*d*(a+b*sin(c+d*x))^(3/2))+2/3*(13*a^2-5*b^2)*cos(c+d*x)*sin(c+d*x)^4/(a^2*b^2*d*sqrt(a+b*sin(c+d*x)))+128/315*a*(40*a^2-19*b^2)*cos(c+d*x)*sqrt(a+b*sin(c+d*x))/(b^6*d)-8/315*(480*a^2-203*b^2)*cos(c+d*x)*sin(c+d*x)*sqrt(a+b*sin(c+d*x))/(b^5*d)+4/63*(160*a^2-63*b^2)*cos(c+d*x)*sin(c+d*x)^2*sqrt(a+b*sin(c+d*x))/(a*b^4*d)-10/9*(8*a^2-3*b^2)*cos(c+d*x)*sin(c+d*x)^3*sqrt(a+b*sin(c+d*x))/(a^2*b^3*d)+8/315*(1280*a^4-768*a^2*b^2+21*b^4)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*sin(c+d*x))/(b^7*d*sqrt((a+b*sin(c+d*x))/(a+b)))-8/315*a*(1280*a^4-1088*a^2*b^2+123*b^4)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(b^7*d*sqrt(a+b*sin(c+d*x)))],
[cos(c+d*x)^4*sin(c+d*x)^2/(a+b*sin(c+d*x))^(5/2),x,9,-2/3*(a^2-b^2)*cos(c+d*x)*sin(c+d*x)^3/(a*b^2*d*(a+b*sin(c+d*x))^(3/2))+2/3*(11*a^2-3*b^2)*cos(c+d*x)*sin(c+d*x)^3/(a^2*b^2*d*sqrt(a+b*sin(c+d*x)))-8/21*(32*a^2-11*b^2)*cos(c+d*x)*sqrt(a+b*sin(c+d*x))/(b^5*d)+8/21*(24*a^2-7*b^2)*cos(c+d*x)*sin(c+d*x)*sqrt(a+b*sin(c+d*x))/(a*b^4*d)-2/21*(80*a^2-21*b^2)*cos(c+d*x)*sin(c+d*x)^2*sqrt(a+b*sin(c+d*x))/(a^2*b^3*d)-16/21*a*(32*a^2-15*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*sin(c+d*x))/(b^6*d*sqrt((a+b*sin(c+d*x))/(a+b)))+8/21*(64*a^4-46*a^2*b^2+3*b^4)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(b^6*d*sqrt(a+b*sin(c+d*x)))],
[cos(c+d*x)^4*sin(c+d*x)/(a+b*sin(c+d*x))^(5/2),x,7,2/15*cos(c+d*x)^3*(8*a+3*b*sin(c+d*x))/(b^2*d*(a+b*sin(c+d*x))^(3/2))+4/15*cos(c+d*x)*(32*a^2-9*b^2+8*a*b*sin(c+d*x))/(b^4*d*sqrt(a+b*sin(c+d*x)))+8/15*(32*a^2-9*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*sin(c+d*x))/(b^5*d*sqrt((a+b*sin(c+d*x))/(a+b)))-8/15*a*(32*a^2-17*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(b^5*d*sqrt(a+b*sin(c+d*x)))],
[cos(c+d*x)^3*cot(c+d*x)/(a+b*sin(c+d*x))^(5/2),x,9,-2/3*(a^2-b^2)*cos(c+d*x)/(a*b^2*d*(a+b*sin(c+d*x))^(3/2))+2/3*(5*a^2+3*b^2)*cos(c+d*x)/(a^2*b^2*d*sqrt(a+b*sin(c+d*x)))+2/3*(8*a^2+3*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*sin(c+d*x))/(a^2*b^3*d*sqrt((a+b*sin(c+d*x))/(a+b)))-2/3*(8*a^2+b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(a*b^3*d*sqrt(a+b*sin(c+d*x)))+2*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2,sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(a^2*d*sqrt(a+b*sin(c+d*x)))],
[cos(c+d*x)^2*cot(c+d*x)^2/(a+b*sin(c+d*x))^(5/2),x,10,1/3*(2*a^2-5*b^2)*cos(c+d*x)/(a^2*b*d*(a+b*sin(c+d*x))^(3/2))-cot(c+d*x)/(a*d*(a+b*sin(c+d*x))^(3/2))-1/3*(4*a^2+15*b^2)*cos(c+d*x)/(a^3*b*d*sqrt(a+b*sin(c+d*x)))-1/3*(4*a^2+15*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*sin(c+d*x))/(a^3*b^2*d*sqrt((a+b*sin(c+d*x))/(a+b)))+1/3*(4*a^2+5*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(a^2*b^2*d*sqrt(a+b*sin(c+d*x)))-5*b*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2,sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(a^3*d*sqrt(a+b*sin(c+d*x)))],
[cos(c+d*x)*cot(c+d*x)^3/(a+b*sin(c+d*x))^(5/2),x,11,1/6*(4*a^2-7*b^2)*cot(c+d*x)/(a^2*b*d*(a+b*sin(c+d*x))^(3/2))-1/2*cot(c+d*x)*csc(c+d*x)/(a*d*(a+b*sin(c+d*x))^(3/2))-1/12*(8*a^2-105*b^2)*cos(c+d*x)/(a^4*d*sqrt(a+b*sin(c+d*x)))-1/12*(8*a^2-35*b^2)*cot(c+d*x)/(a^3*b*d*sqrt(a+b*sin(c+d*x)))-1/12*(8*a^2-105*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*sin(c+d*x))/(a^4*b*d*sqrt((a+b*sin(c+d*x))/(a+b)))+1/12*(8*a^2-35*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(a^3*b*d*sqrt(a+b*sin(c+d*x)))-1/4*(12*a^2-35*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2,sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(a^4*d*sqrt(a+b*sin(c+d*x)))],
[cot(c+d*x)^4/(a+b*sin(c+d*x))^(5/2),x,12,1/3*(2*a^2-3*b^2)*cot(c+d*x)*csc(c+d*x)/(a^2*b*d*(a+b*sin(c+d*x))^(3/2))-1/3*cot(c+d*x)*csc(c+d*x)^2/(a*d*(a+b*sin(c+d*x))^(3/2))+1/8*b*(32*a^2-105*b^2)*cos(c+d*x)/(a^5*d*sqrt(a+b*sin(c+d*x)))+1/8*(16*a^2-35*b^2)*cot(c+d*x)/(a^4*d*sqrt(a+b*sin(c+d*x)))-1/12*(8*a^2-21*b^2)*cot(c+d*x)*csc(c+d*x)/(a^3*b*d*sqrt(a+b*sin(c+d*x)))+1/8*(32*a^2-105*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*sin(c+d*x))/(a^5*d*sqrt((a+b*sin(c+d*x))/(a+b)))-1/8*(16*a^2-35*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(a^4*d*sqrt(a+b*sin(c+d*x)))+15/8*b*(4*a^2-7*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticPi(sin(1/2*(c-1/2*Pi+d*x)),2,sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(a^5*d*sqrt(a+b*sin(c+d*x)))],

# Integrands of the form Cos[e+f x]^4 (d Sin[e+f x])^(n/2) (a+b Sin[e+f x])^(m/2)
[cos(e+f*x)^4/((a+b*sin(e+f*x))^(9/2)*sqrt(d*sin(e+f*x))),x,8,2/7*cos(e+f*x)^3*sqrt(d*sin(e+f*x))/(a*d*f*(a+b*sin(e+f*x))^(7/2))+12/35*cos(e+f*x)*sqrt(d*sin(e+f*x))/(a^2*d*f*(a+b*sin(e+f*x))^(5/2))+8/35*(a^2-2*b^2)*cos(e+f*x)*sqrt(d*sin(e+f*x))/(a^3*(a^2-b^2)*d*f*(a+b*sin(e+f*x))^(3/2))+32/35*b*(2*a^2-b^2)*cos(e+f*x)/(a^3*(a^2-b^2)^2*f*sqrt(d*sin(e+f*x))*sqrt(a+b*sin(e+f*x)))-32/35*b*(2*a^2-b^2)*EllipticE(sqrt(d)*sqrt(a+b*sin(e+f*x))/(sqrt(a+b)*sqrt(d*sin(e+f*x))),sqrt((-a-b)/(a-b)))*sqrt(a*(1-csc(e+f*x))/(a+b))*sqrt(a*(1+csc(e+f*x))/(a-b))*tan(e+f*x)/(a^5*(a-b)*(a+b)^(3/2)*f*sqrt(d))-8/35*(5*a^2-3*a*b-4*b^2)*EllipticF(sqrt(d)*sqrt(a+b*sin(e+f*x))/(sqrt(a+b)*sqrt(d*sin(e+f*x))),sqrt((-a-b)/(a-b)))*sqrt(a*(1-csc(e+f*x))/(a+b))*sqrt(a*(1+csc(e+f*x))/(a-b))*tan(e+f*x)/(a^4*(a-b)*(a+b)^(3/2)*f*sqrt(d))],

# Integrands of the form Cos[e+f x]^4 (d Sin[e+f x])^(n/3) (a+b Sin[e+f x])^(m/2)
[cos(c+d*x)^4*sin(c+d*x)^(1/3)/sqrt(a+b*sin(c+d*x)),x,0,Unintegrable(cos(c+d*x)^4*sin(c+d*x)^(1/3)/sqrt(a+b*sin(c+d*x)),x)],

# Integrands of the form Cos[e+f x]^4 Sin[e+f x]^n (a+b Sin[e+f x])^m with n and/or p symbolic
[cos(c+d*x)^4*sin(c+d*x)^n*(a+b*sin(c+d*x))^p,x,0,Unintegrable(cos(c+d*x)^4*sin(c+d*x)^n*(a+b*sin(c+d*x))^p,x)],
[cos(c+d*x)^4*sin(c+d*x)^(-3-p)*(a+b*sin(c+d*x))^p,x,0,Unintegrable(cos(c+d*x)^4*sin(c+d*x)^(-3-p)*(a+b*sin(c+d*x))^p,x)],
[cos(c+d*x)^4*sin(c+d*x)^(-4-p)*(a+b*sin(c+d*x))^p,x,0,Unintegrable(cos(c+d*x)^4*sin(c+d*x)^(-4-p)*(a+b*sin(c+d*x))^p,x)],
[cos(c+d*x)^4*sin(c+d*x)^n*(a+b*sin(c+d*x))^3,x,8,-3*a*(2*a^4*(6+5*n+n^2)+3*b^4*(35+12*n+n^2)-2*a^2*b^2*(58+16*n+n^2))*cos(c+d*x)*sin(c+d*x)^(1+n)/(b^2*d*(5+n)*(6+n)*(7+n)*(8+6*n+n^2))-3*(2*a^4*(6+5*n+n^2)+b^4*(24+10*n+n^2)-2*a^2*b^2*(57+16*n+n^2))*cos(c+d*x)*sin(c+d*x)^(2+n)/(b*d*(3+n)*(4+n)*(5+n)*(6+n)*(7+n))-3*a*(a^2*(6+5*n+n^2)-b^2*(53+15*n+n^2))*cos(c+d*x)*sin(c+d*x)^(1+n)*(a+b*sin(c+d*x))^2/(b^2*d*(4+n)*(5+n)*(6+n)*(7+n))-(a^2*(2+n)*(3+n)-b^2*(6+n)*(8+n))*cos(c+d*x)*sin(c+d*x)^(1+n)*(a+b*sin(c+d*x))^3/(b^2*d*(5+n)*(6+n)*(7+n))+a*(3+n)*cos(c+d*x)*sin(c+d*x)^(1+n)*(a+b*sin(c+d*x))^4/(b^2*d*(6+n)*(7+n))-cos(c+d*x)*sin(c+d*x)^(2+n)*(a+b*sin(c+d*x))^4/(b*d*(7+n))+3*a*(3*b^2*(1+n)+a^2*(6+n))*cos(c+d*x)*hypergeom([1/2,1/2*(1+n)],[1/2*(3+n)],sin(c+d*x)^2)*sin(c+d*x)^(1+n)/(d*(6+n)*(8+14*n+7*n^2+n^3)*sqrt(cos(c+d*x)^2))+3*b*(b^2*(2+n)+3*a^2*(7+n))*cos(c+d*x)*hypergeom([1/2,1/2*(2+n)],[1/2*(4+n)],sin(c+d*x)^2)*sin(c+d*x)^(2+n)/(d*(5+n)*(7+n)*(6+5*n+n^2)*sqrt(cos(c+d*x)^2))],
[cos(c+d*x)^4*sin(c+d*x)^n*(a+b*sin(c+d*x))^2,x,7,-(3*b^4*(5+n)+2*a^4*(6+5*n+n^2)-2*a^2*b^2*(40+13*n+n^2))*cos(c+d*x)*sin(c+d*x)^(1+n)/(b^2*d*(5+n)*(6+n)*(8+6*n+n^2))-2*a*(a^2*(6+5*n+n^2)-b^2*(39+13*n+n^2))*cos(c+d*x)*sin(c+d*x)^(2+n)/(b*d*(3+n)*(4+n)*(5+n)*(6+n))-(a^2*(2+n)*(3+n)-b^2*(5+n)*(7+n))*cos(c+d*x)*sin(c+d*x)^(1+n)*(a+b*sin(c+d*x))^2/(b^2*d*(4+n)*(5+n)*(6+n))+a*(3+n)*cos(c+d*x)*sin(c+d*x)^(1+n)*(a+b*sin(c+d*x))^3/(b^2*d*(5+n)*(6+n))-cos(c+d*x)*sin(c+d*x)^(2+n)*(a+b*sin(c+d*x))^3/(b*d*(6+n))+3*(b^2*(1+n)+a^2*(6+n))*cos(c+d*x)*hypergeom([1/2,1/2*(1+n)],[1/2*(3+n)],sin(c+d*x)^2)*sin(c+d*x)^(1+n)/(d*(4+n)*(6+n)*(2+3*n+n^2)*sqrt(cos(c+d*x)^2))+6*a*b*cos(c+d*x)*hypergeom([1/2,1/2*(2+n)],[1/2*(4+n)],sin(c+d*x)^2)*sin(c+d*x)^(2+n)/(d*(5+n)*(6+5*n+n^2)*sqrt(cos(c+d*x)^2))],
[cos(c+d*x)^4*sin(c+d*x)^n*(a+b*sin(c+d*x)),x,3,a*cos(c+d*x)*hypergeom([-3/2,1/2*(1+n)],[1/2*(3+n)],sin(c+d*x)^2)*sin(c+d*x)^(1+n)/(d*(1+n)*sqrt(cos(c+d*x)^2))+b*cos(c+d*x)*hypergeom([-3/2,1/2*(2+n)],[1/2*(4+n)],sin(c+d*x)^2)*sin(c+d*x)^(2+n)/(d*(2+n)*sqrt(cos(c+d*x)^2))],

# Integrands of the form Cos[e+f x]^5 (d Sin[e+f x])^n (a+b Sin[e+f x])^m

# Integrands of the form Cos[e+f x]^5 Sin[e+f x]^n (a+b Sin[e+f x])^m

# m>0
[cos(c+d*x)^5*sin(c+d*x)^5*(a+b*sin(c+d*x)),x,4,1/6*a*sin(c+d*x)^6/d+1/7*b*sin(c+d*x)^7/d-1/4*a*sin(c+d*x)^8/d-2/9*b*sin(c+d*x)^9/d+1/10*a*sin(c+d*x)^10/d+1/11*b*sin(c+d*x)^11/d],
[cos(c+d*x)^5*sin(c+d*x)^4*(a+b*sin(c+d*x)),x,4,1/5*a*sin(c+d*x)^5/d+1/6*b*sin(c+d*x)^6/d-2/7*a*sin(c+d*x)^7/d-1/4*b*sin(c+d*x)^8/d+1/9*a*sin(c+d*x)^9/d+1/10*b*sin(c+d*x)^10/d],
[cos(c+d*x)^5*sin(c+d*x)^3*(a+b*sin(c+d*x)),x,7,-1/6*a*cos(c+d*x)^6/d+1/8*a*cos(c+d*x)^8/d+1/5*b*sin(c+d*x)^5/d-2/7*b*sin(c+d*x)^7/d+1/9*b*sin(c+d*x)^9/d],
[cos(c+d*x)^5*sin(c+d*x)^2*(a+b*sin(c+d*x)),x,7,-1/6*b*cos(c+d*x)^6/d+1/8*b*cos(c+d*x)^8/d+1/3*a*sin(c+d*x)^3/d-2/5*a*sin(c+d*x)^5/d+1/7*a*sin(c+d*x)^7/d],
[cos(c+d*x)^5*sin(c+d*x)*(a+b*sin(c+d*x)),x,6,-1/6*a*cos(c+d*x)^6/d+1/3*b*sin(c+d*x)^3/d-2/5*b*sin(c+d*x)^5/d+1/7*b*sin(c+d*x)^7/d],
[cos(c+d*x)^5*csc(c+d*x)*(a+b*sin(c+d*x)),x,4,a*log(sin(c+d*x))/d+b*sin(c+d*x)/d-a*sin(c+d*x)^2/d-2/3*b*sin(c+d*x)^3/d+1/4*a*sin(c+d*x)^4/d+1/5*b*sin(c+d*x)^5/d],
[cos(c+d*x)^5*csc(c+d*x)^2*(a+b*sin(c+d*x)),x,4,-a*csc(c+d*x)/d+b*log(sin(c+d*x))/d-2*a*sin(c+d*x)/d-b*sin(c+d*x)^2/d+1/3*a*sin(c+d*x)^3/d+1/4*b*sin(c+d*x)^4/d],
[cos(c+d*x)^5*csc(c+d*x)^3*(a+b*sin(c+d*x)),x,4,-b*csc(c+d*x)/d-1/2*a*csc(c+d*x)^2/d-2*a*log(sin(c+d*x))/d-2*b*sin(c+d*x)/d+1/2*a*sin(c+d*x)^2/d+1/3*b*sin(c+d*x)^3/d],
[cos(c+d*x)^5*csc(c+d*x)^4*(a+b*sin(c+d*x)),x,4,2*a*csc(c+d*x)/d-1/2*b*csc(c+d*x)^2/d-1/3*a*csc(c+d*x)^3/d-2*b*log(sin(c+d*x))/d+a*sin(c+d*x)/d+1/2*b*sin(c+d*x)^2/d],
[cos(c+d*x)^5*csc(c+d*x)^5*(a+b*sin(c+d*x)),x,3,2*b*csc(c+d*x)/d+a*csc(c+d*x)^2/d-1/3*b*csc(c+d*x)^3/d-1/4*a*csc(c+d*x)^4/d+a*log(sin(c+d*x))/d+b*sin(c+d*x)/d],
[cos(c+d*x)^5*csc(c+d*x)^6*(a+b*sin(c+d*x)),x,4,-a*csc(c+d*x)/d+b*csc(c+d*x)^2/d+2/3*a*csc(c+d*x)^3/d-1/4*b*csc(c+d*x)^4/d-1/5*a*csc(c+d*x)^5/d+b*log(sin(c+d*x))/d],
[cos(c+d*x)^5*csc(c+d*x)^7*(a+b*sin(c+d*x)),x,6,-1/6*a*cot(c+d*x)^6/d-b*csc(c+d*x)/d+2/3*b*csc(c+d*x)^3/d-1/5*b*csc(c+d*x)^5/d],
[cos(c+d*x)^5*csc(c+d*x)^8*(a+b*sin(c+d*x)),x,6,-1/6*b*cot(c+d*x)^6/d-1/3*a*csc(c+d*x)^3/d+2/5*a*csc(c+d*x)^5/d-1/7*a*csc(c+d*x)^7/d],
[cos(c+d*x)^5*csc(c+d*x)^9*(a+b*sin(c+d*x)),x,7,-1/6*a*cot(c+d*x)^6/d-1/8*a*cot(c+d*x)^8/d-1/3*b*csc(c+d*x)^3/d+2/5*b*csc(c+d*x)^5/d-1/7*b*csc(c+d*x)^7/d],
[cos(c+d*x)^5*csc(c+d*x)^10*(a+b*sin(c+d*x)),x,7,-1/6*b*cot(c+d*x)^6/d-1/8*b*cot(c+d*x)^8/d-1/5*a*csc(c+d*x)^5/d+2/7*a*csc(c+d*x)^7/d-1/9*a*csc(c+d*x)^9/d],
[cos(c+d*x)^5*csc(c+d*x)^11*(a+b*sin(c+d*x)),x,4,-1/5*b*csc(c+d*x)^5/d-1/6*a*csc(c+d*x)^6/d+2/7*b*csc(c+d*x)^7/d+1/4*a*csc(c+d*x)^8/d-1/9*b*csc(c+d*x)^9/d-1/10*a*csc(c+d*x)^10/d],
[cos(c+d*x)^5*csc(c+d*x)^12*(a+b*sin(c+d*x)),x,4,-1/6*b*csc(c+d*x)^6/d-1/7*a*csc(c+d*x)^7/d+1/4*b*csc(c+d*x)^8/d+2/9*a*csc(c+d*x)^9/d-1/10*b*csc(c+d*x)^10/d-1/11*a*csc(c+d*x)^11/d],
[cos(c+d*x)^5*sin(c+d*x)^2*(a+b*sin(c+d*x))^2,x,4,1/3*a^2*sin(c+d*x)^3/d+1/2*a*b*sin(c+d*x)^4/d-1/5*(2*a^2-b^2)*sin(c+d*x)^5/d-2/3*a*b*sin(c+d*x)^6/d+1/7*(a^2-2*b^2)*sin(c+d*x)^7/d+1/4*a*b*sin(c+d*x)^8/d+1/9*b^2*sin(c+d*x)^9/d],
[cos(c+d*x)^5*sin(c+d*x)*(a+b*sin(c+d*x))^2,x,4,1/2*a^2*sin(c+d*x)^2/d+2/3*a*b*sin(c+d*x)^3/d-1/4*(2*a^2-b^2)*sin(c+d*x)^4/d-4/5*a*b*sin(c+d*x)^5/d+1/6*(a^2-2*b^2)*sin(c+d*x)^6/d+2/7*a*b*sin(c+d*x)^7/d+1/8*b^2*sin(c+d*x)^8/d],
[cos(c+d*x)^5*csc(c+d*x)*(a+b*sin(c+d*x))^2,x,4,a^2*log(sin(c+d*x))/d+2*a*b*sin(c+d*x)/d-1/2*(2*a^2-b^2)*sin(c+d*x)^2/d-4/3*a*b*sin(c+d*x)^3/d+1/4*(a^2-2*b^2)*sin(c+d*x)^4/d+2/5*a*b*sin(c+d*x)^5/d+1/6*b^2*sin(c+d*x)^6/d],
[cos(c+d*x)^5*csc(c+d*x)^2*(a+b*sin(c+d*x))^2,x,4,-a^2*csc(c+d*x)/d+2*a*b*log(sin(c+d*x))/d-(2*a^2-b^2)*sin(c+d*x)/d-2*a*b*sin(c+d*x)^2/d+1/3*(a^2-2*b^2)*sin(c+d*x)^3/d+1/2*a*b*sin(c+d*x)^4/d+1/5*b^2*sin(c+d*x)^5/d],
[cos(c+d*x)^5*csc(c+d*x)^3*(a+b*sin(c+d*x))^2,x,4,-2*a*b*csc(c+d*x)/d-1/2*a^2*csc(c+d*x)^2/d-(2*a^2-b^2)*log(sin(c+d*x))/d-4*a*b*sin(c+d*x)/d+1/2*(a^2-2*b^2)*sin(c+d*x)^2/d+2/3*a*b*sin(c+d*x)^3/d+1/4*b^2*sin(c+d*x)^4/d],
[cos(c+d*x)^5*csc(c+d*x)^4*(a+b*sin(c+d*x))^2,x,4,(2*a^2-b^2)*csc(c+d*x)/d-a*b*csc(c+d*x)^2/d-1/3*a^2*csc(c+d*x)^3/d-4*a*b*log(sin(c+d*x))/d+(a^2-2*b^2)*sin(c+d*x)/d+a*b*sin(c+d*x)^2/d+1/3*b^2*sin(c+d*x)^3/d],
[cos(c+d*x)^5*csc(c+d*x)^5*(a+b*sin(c+d*x))^2,x,3,4*a*b*csc(c+d*x)/d+1/2*(2*a^2-b^2)*csc(c+d*x)^2/d-2/3*a*b*csc(c+d*x)^3/d-1/4*a^2*csc(c+d*x)^4/d+(a^2-2*b^2)*log(sin(c+d*x))/d+2*a*b*sin(c+d*x)/d+1/2*b^2*sin(c+d*x)^2/d],
[cos(c+d*x)^5*csc(c+d*x)^6*(a+b*sin(c+d*x))^2,x,4,-(a^2-2*b^2)*csc(c+d*x)/d+2*a*b*csc(c+d*x)^2/d+1/3*(2*a^2-b^2)*csc(c+d*x)^3/d-1/2*a*b*csc(c+d*x)^4/d-1/5*a^2*csc(c+d*x)^5/d+2*a*b*log(sin(c+d*x))/d+b^2*sin(c+d*x)/d],
[cos(c+d*x)^5*csc(c+d*x)^7*(a+b*sin(c+d*x))^2,x,4,-2*a*b*csc(c+d*x)/d-1/2*(a^2-2*b^2)*csc(c+d*x)^2/d+4/3*a*b*csc(c+d*x)^3/d+1/4*(2*a^2-b^2)*csc(c+d*x)^4/d-2/5*a*b*csc(c+d*x)^5/d-1/6*a^2*csc(c+d*x)^6/d+b^2*log(sin(c+d*x))/d],
[cos(c+d*x)^5*csc(c+d*x)^8*(a+b*sin(c+d*x))^2,x,4,-b^2*csc(c+d*x)/d-a*b*csc(c+d*x)^2/d-1/3*(a^2-2*b^2)*csc(c+d*x)^3/d+a*b*csc(c+d*x)^4/d+1/5*(2*a^2-b^2)*csc(c+d*x)^5/d-1/3*a*b*csc(c+d*x)^6/d-1/7*a^2*csc(c+d*x)^7/d],
[cos(c+d*x)^5*csc(c+d*x)^9*(a+b*sin(c+d*x))^2,x,4,-1/2*b^2*csc(c+d*x)^2/d-2/3*a*b*csc(c+d*x)^3/d-1/4*(a^2-2*b^2)*csc(c+d*x)^4/d+4/5*a*b*csc(c+d*x)^5/d+1/6*(2*a^2-b^2)*csc(c+d*x)^6/d-2/7*a*b*csc(c+d*x)^7/d-1/8*a^2*csc(c+d*x)^8/d],

# m<0
[cos(c+d*x)^5*sin(c+d*x)^3/(a+b*sin(c+d*x))^2,x,4,a^2*(7*a^4-10*a^2*b^2+3*b^4)*log(a+b*sin(c+d*x))/(b^8*d)-2*a*(3*a^4-4*a^2*b^2+b^4)*sin(c+d*x)/(b^7*d)+1/2*(5*a^4-6*a^2*b^2+b^4)*sin(c+d*x)^2/(b^6*d)-4/3*a*(a^2-b^2)*sin(c+d*x)^3/(b^5*d)+1/4*(3*a^2-2*b^2)*sin(c+d*x)^4/(b^4*d)-2/5*a*sin(c+d*x)^5/(b^3*d)+1/6*sin(c+d*x)^6/(b^2*d)+a^3*(a^2-b^2)^2/(b^8*d*(a+b*sin(c+d*x)))],
[cos(c+d*x)^5*sin(c+d*x)^2/(a+b*sin(c+d*x))^2,x,4,-2*a*(3*a^4-4*a^2*b^2+b^4)*log(a+b*sin(c+d*x))/(b^7*d)+(5*a^4-6*a^2*b^2+b^4)*sin(c+d*x)/(b^6*d)-2*a*(a^2-b^2)*sin(c+d*x)^2/(b^5*d)-1/3*(2-3*a^2/b^2)*sin(c+d*x)^3/(b^2*d)-1/2*a*sin(c+d*x)^4/(b^3*d)+1/5*sin(c+d*x)^5/(b^2*d)-a^2*(a^2-b^2)^2/(b^7*d*(a+b*sin(c+d*x)))],
[cos(c+d*x)^5*sin(c+d*x)/(a+b*sin(c+d*x))^2,x,4,(5*a^4-6*a^2*b^2+b^4)*log(a+b*sin(c+d*x))/(b^6*d)-4*a*(a^2-b^2)*sin(c+d*x)/(b^5*d)+1/2*(3*a^2-2*b^2)*sin(c+d*x)^2/(b^4*d)-2/3*a*sin(c+d*x)^3/(b^3*d)+1/4*sin(c+d*x)^4/(b^2*d)+a*(a^2-b^2)^2/(b^6*d*(a+b*sin(c+d*x)))],
[cos(c+d*x)^5*csc(c+d*x)/(a+b*sin(c+d*x))^2,x,4,log(sin(c+d*x))/(a^2*d)+(a^2-b^2)*(3*a^2+b^2)*log(a+b*sin(c+d*x))/(a^2*b^4*d)-2*a*sin(c+d*x)/(b^3*d)+1/2*sin(c+d*x)^2/(b^2*d)+(a^2-b^2)^2/(a*b^4*d*(a+b*sin(c+d*x)))],
[cos(c+d*x)^5*csc(c+d*x)^2/(a+b*sin(c+d*x))^2,x,4,-csc(c+d*x)/(a^2*d)-2*b*log(sin(c+d*x))/(a^3*d)-2*(a^4-b^4)*log(a+b*sin(c+d*x))/(a^3*b^3*d)+sin(c+d*x)/(b^2*d)-(a^2-b^2)^2/(a^2*b^3*d*(a+b*sin(c+d*x)))],
[cos(c+d*x)^5*csc(c+d*x)^3/(a+b*sin(c+d*x))^2,x,4,2*b*csc(c+d*x)/(a^3*d)-1/2*csc(c+d*x)^2/(a^2*d)-(2*a^2-3*b^2)*log(sin(c+d*x))/(a^4*d)+(a^4+2*a^2*b^2-3*b^4)*log(a+b*sin(c+d*x))/(a^4*b^2*d)+(a^2-b^2)^2/(a^3*b^2*d*(a+b*sin(c+d*x)))],
[cos(c+d*x)^5*csc(c+d*x)^4/(a+b*sin(c+d*x))^2,x,4,(2*a^2-3*b^2)*csc(c+d*x)/(a^4*d)+b*csc(c+d*x)^2/(a^3*d)-1/3*csc(c+d*x)^3/(a^2*d)+4*b*(a^2-b^2)*log(sin(c+d*x))/(a^5*d)-4*b*(a^2-b^2)*log(a+b*sin(c+d*x))/(a^5*d)-(a^2-b^2)^2/(a^4*b*d*(a+b*sin(c+d*x)))],
[cos(c+d*x)^5*csc(c+d*x)^5/(a+b*sin(c+d*x))^2,x,3,-4*b*(a^2-b^2)*csc(c+d*x)/(a^5*d)+1/2*(2*a^2-3*b^2)*csc(c+d*x)^2/(a^4*d)+2/3*b*csc(c+d*x)^3/(a^3*d)-1/4*csc(c+d*x)^4/(a^2*d)+(a^4-6*a^2*b^2+5*b^4)*log(sin(c+d*x))/(a^6*d)-(a^4-6*a^2*b^2+5*b^4)*log(a+b*sin(c+d*x))/(a^6*d)+(a^2-b^2)^2/(a^5*d*(a+b*sin(c+d*x)))],
[cos(c+d*x)^5*csc(c+d*x)^6/(a+b*sin(c+d*x))^2,x,4,-(a^4-6*a^2*b^2+5*b^4)*csc(c+d*x)/(a^6*d)-2*b*(a^2-b^2)*csc(c+d*x)^2/(a^5*d)+1/3*(2*a^2-3*b^2)*csc(c+d*x)^3/(a^4*d)+1/2*b*csc(c+d*x)^4/(a^3*d)-1/5*csc(c+d*x)^5/(a^2*d)-2*b*(a^4-4*a^2*b^2+3*b^4)*log(sin(c+d*x))/(a^7*d)+2*b*(a^4-4*a^2*b^2+3*b^4)*log(a+b*sin(c+d*x))/(a^7*d)-b*(a^2-b^2)^2/(a^6*d*(a+b*sin(c+d*x)))],

# Integrands of the form Cos[e+f x]^5 Sin[e+f x]^n (a+b Sin[e+f x])^(m/2)

# Integrands of the form Cos[e+f x]^5 Sin[e+f x]^n (a+b Sin[e+f x])^m with n symbolic
[cos(c+d*x)^5*sin(c+d*x)^n*(a+b*sin(c+d*x))^2,x,3,a^2*sin(c+d*x)^(1+n)/(d*(1+n))+2*a*b*sin(c+d*x)^(2+n)/(d*(2+n))-(2*a^2-b^2)*sin(c+d*x)^(3+n)/(d*(3+n))-4*a*b*sin(c+d*x)^(4+n)/(d*(4+n))+(a^2-2*b^2)*sin(c+d*x)^(5+n)/(d*(5+n))+2*a*b*sin(c+d*x)^(6+n)/(d*(6+n))+b^2*sin(c+d*x)^(7+n)/(d*(7+n))],
[cos(c+d*x)^5*sin(c+d*x)^n*(a+b*sin(c+d*x)),x,3,a*sin(c+d*x)^(1+n)/(d*(1+n))+b*sin(c+d*x)^(2+n)/(d*(2+n))-2*a*sin(c+d*x)^(3+n)/(d*(3+n))-2*b*sin(c+d*x)^(4+n)/(d*(4+n))+a*sin(c+d*x)^(5+n)/(d*(5+n))+b*sin(c+d*x)^(6+n)/(d*(6+n))],
[cos(c+d*x)^5*sin(c+d*x)^n/(a+b*sin(c+d*x)),x,5,-a*(a^2-2*b^2)*sin(c+d*x)^(1+n)/(b^4*d*(1+n))+(a^2-b^2)^2*hypergeom([1,1+n],[2+n],-b*sin(c+d*x)/a)*sin(c+d*x)^(1+n)/(a*b^4*d*(1+n))+(a^2-2*b^2)*sin(c+d*x)^(2+n)/(b^3*d*(2+n))-a*sin(c+d*x)^(3+n)/(b^2*d*(3+n))+sin(c+d*x)^(4+n)/(b*d*(4+n))],
[cos(c+d*x)^5*sin(c+d*x)^n/(a+b*sin(c+d*x))^2,x,5,(3*a^2-2*b^2)*sin(c+d*x)^(1+n)/(b^4*d*(1+n))+(a^2-b^2)*(b^2*n-a^2*(4+n))*hypergeom([1,1+n],[2+n],-b*sin(c+d*x)/a)*sin(c+d*x)^(1+n)/(a^2*b^4*d*(1+n))-2*a*sin(c+d*x)^(2+n)/(b^3*d*(2+n))+sin(c+d*x)^(3+n)/(b^2*d*(3+n))+(a^2-b^2)^2*sin(c+d*x)^(1+n)/(a*b^4*d*(a+b*sin(c+d*x)))],

# Integrands of the form Cos[e+f x]^6 (d Sin[e+f x])^n (a+b Sin[e+f x])^m

# Integrands of the form Cos[e+f x]^6 Sin[e+f x]^n (a+b Sin[e+f x])^m

# m>0
[cos(c+d*x)^6*sin(c+d*x)^5*(a+b*sin(c+d*x))^2,x,12,5/512*a*b*x-1/7*(a^2+b^2)*cos(c+d*x)^7/d+1/9*(2*a^2+3*b^2)*cos(c+d*x)^9/d-1/11*(a^2+3*b^2)*cos(c+d*x)^11/d+1/13*b^2*cos(c+d*x)^13/d+5/512*a*b*cos(c+d*x)*sin(c+d*x)/d+5/768*a*b*cos(c+d*x)^3*sin(c+d*x)/d+1/192*a*b*cos(c+d*x)^5*sin(c+d*x)/d-1/32*a*b*cos(c+d*x)^7*sin(c+d*x)/d-1/12*a*b*cos(c+d*x)^7*sin(c+d*x)^3/d-1/6*a*b*cos(c+d*x)^7*sin(c+d*x)^5/d],
[cos(c+d*x)^6*sin(c+d*x)^4*(a+b*sin(c+d*x))^2,x,12,1/1024*(12*a^2+5*b^2)*x-2/7*a*b*cos(c+d*x)^7/d+4/9*a*b*cos(c+d*x)^9/d-2/11*a*b*cos(c+d*x)^11/d+1/1024*(12*a^2+5*b^2)*cos(c+d*x)*sin(c+d*x)/d+1/1536*(12*a^2+5*b^2)*cos(c+d*x)^3*sin(c+d*x)/d+1/1920*(12*a^2+5*b^2)*cos(c+d*x)^5*sin(c+d*x)/d-1/320*(44*a^2+45*b^2)*cos(c+d*x)^7*sin(c+d*x)/d+1/120*(12*a^2+25*b^2)*cos(c+d*x)^9*sin(c+d*x)/d-1/12*b^2*cos(c+d*x)^11*sin(c+d*x)/d],
[cos(c+d*x)^6*sin(c+d*x)^3*(a+b*sin(c+d*x))^2,x,11,3/128*a*b*x-1/7*(a^2+b^2)*cos(c+d*x)^7/d+1/9*(a^2+2*b^2)*cos(c+d*x)^9/d-1/11*b^2*cos(c+d*x)^11/d+3/128*a*b*cos(c+d*x)*sin(c+d*x)/d+1/64*a*b*cos(c+d*x)^3*sin(c+d*x)/d+1/80*a*b*cos(c+d*x)^5*sin(c+d*x)/d-3/40*a*b*cos(c+d*x)^7*sin(c+d*x)/d-1/5*a*b*cos(c+d*x)^7*sin(c+d*x)^3/d],
[cos(c+d*x)^6*sin(c+d*x)^2*(a+b*sin(c+d*x))^2,x,11,1/256*(10*a^2+3*b^2)*x-2/7*a*b*cos(c+d*x)^7/d+2/9*a*b*cos(c+d*x)^9/d+1/256*(10*a^2+3*b^2)*cos(c+d*x)*sin(c+d*x)/d+1/384*(10*a^2+3*b^2)*cos(c+d*x)^3*sin(c+d*x)/d+1/480*(10*a^2+3*b^2)*cos(c+d*x)^5*sin(c+d*x)/d-1/80*(10*a^2+11*b^2)*cos(c+d*x)^7*sin(c+d*x)/d+1/10*b^2*cos(c+d*x)^9*sin(c+d*x)/d],
[cos(c+d*x)^6*sin(c+d*x)*(a+b*sin(c+d*x))^2,x,7,5/64*a*b*x-1/252*(a^2+8*b^2)*cos(c+d*x)^7/d+5/64*a*b*cos(c+d*x)*sin(c+d*x)/d+5/96*a*b*cos(c+d*x)^3*sin(c+d*x)/d+1/24*a*b*cos(c+d*x)^5*sin(c+d*x)/d-1/36*a*cos(c+d*x)^7*(a+b*sin(c+d*x))/d-1/9*cos(c+d*x)^7*(a+b*sin(c+d*x))^2/d],
[cos(c+d*x)^6*csc(c+d*x)*(a+b*sin(c+d*x))^2,x,10,5/8*a*b*x-a^2*arctanh(cos(c+d*x))/d+a^2*cos(c+d*x)/d+1/3*a^2*cos(c+d*x)^3/d+1/5*a^2*cos(c+d*x)^5/d-1/7*b^2*cos(c+d*x)^7/d+5/8*a*b*cos(c+d*x)*sin(c+d*x)/d+5/12*a*b*cos(c+d*x)^3*sin(c+d*x)/d+1/3*a*b*cos(c+d*x)^5*sin(c+d*x)/d],
[cos(c+d*x)^6*csc(c+d*x)^2*(a+b*sin(c+d*x))^2,x,11,-5/16*(6*a^2-b^2)*x-2*a*b*arctanh(cos(c+d*x))/d+2*a*b*cos(c+d*x)/d+2/3*a*b*cos(c+d*x)^3/d+2/5*a*b*cos(c+d*x)^5/d-a^2*cot(c+d*x)/d-1/16*(14*a^2-5*b^2)*cos(c+d*x)*sin(c+d*x)/d-1/24*(6*a^2-5*b^2)*cos(c+d*x)^3*sin(c+d*x)/d+1/6*b^2*cos(c+d*x)^5*sin(c+d*x)/d],
[cos(c+d*x)^6*csc(c+d*x)^3*(a+b*sin(c+d*x))^2,x,11,-15/4*a*b*x+1/2*(5*a^2-2*b^2)*arctanh(cos(c+d*x))/d-(2*a^2-b^2)*cos(c+d*x)/d-1/3*(a^2-b^2)*cos(c+d*x)^3/d+1/5*b^2*cos(c+d*x)^5/d-15/4*a*b*cot(c+d*x)/d+5/4*a*b*cos(c+d*x)^2*cot(c+d*x)/d+1/2*a*b*cos(c+d*x)^4*cot(c+d*x)/d-1/2*a^2*cot(c+d*x)*csc(c+d*x)/d],
[cos(c+d*x)^6*csc(c+d*x)^4*(a+b*sin(c+d*x))^2,x,12,5/8*(4*a^2-3*b^2)*x+5*a*b*arctanh(cos(c+d*x))/d-5*a*b*cos(c+d*x)/d-5/3*a*b*cos(c+d*x)^3/d+(2*a^2-b^2)*cot(c+d*x)/d-a*b*cos(c+d*x)^3*cot(c+d*x)^2/d-1/3*a^2*cot(c+d*x)^3/d+1/8*(4*a^2-7*b^2)*cos(c+d*x)*sin(c+d*x)/d-1/4*b^2*cos(c+d*x)^3*sin(c+d*x)/d],
[cos(c+d*x)^6*csc(c+d*x)^5*(a+b*sin(c+d*x))^2,x,12,5*a*b*x-5/8*(3*a^2-4*b^2)*arctanh(cos(c+d*x))/d+(a^2-2*b^2)*cos(c+d*x)/d-1/3*b^2*cos(c+d*x)^3/d+5*a*b*cot(c+d*x)/d-5/3*a*b*cot(c+d*x)^3/d+a*b*cos(c+d*x)^2*cot(c+d*x)^3/d+1/8*(9*a^2-4*b^2)*cot(c+d*x)*csc(c+d*x)/d-1/4*a^2*cot(c+d*x)*csc(c+d*x)^3/d],
[cos(c+d*x)^6*csc(c+d*x)^6*(a+b*sin(c+d*x))^2,x,16,-a^2*x+5/2*b^2*x-15/4*a*b*arctanh(cos(c+d*x))/d+15/4*a*b*cos(c+d*x)/d-a^2*cot(c+d*x)/d+5/2*b^2*cot(c+d*x)/d+5/4*a*b*cos(c+d*x)*cot(c+d*x)^2/d+1/3*a^2*cot(c+d*x)^3/d-5/6*b^2*cot(c+d*x)^3/d+1/2*b^2*cos(c+d*x)^2*cot(c+d*x)^3/d-1/2*a*b*cos(c+d*x)*cot(c+d*x)^4/d-1/5*a^2*cot(c+d*x)^5/d],
[cos(c+d*x)^6*csc(c+d*x)^7*(a+b*sin(c+d*x))^2,x,11,-2*a*b*x+5/16*(a^2-6*b^2)*arctanh(cos(c+d*x))/d+b^2*cos(c+d*x)/d-2*a*b*cot(c+d*x)/d+2/3*a*b*cot(c+d*x)^3/d-2/5*a*b*cot(c+d*x)^5/d-1/16*(11*a^2-18*b^2)*cot(c+d*x)*csc(c+d*x)/d+1/24*(13*a^2-6*b^2)*cot(c+d*x)*csc(c+d*x)^3/d-1/6*a^2*cot(c+d*x)*csc(c+d*x)^5/d],
[cos(c+d*x)^6*csc(c+d*x)^8*(a+b*sin(c+d*x))^2,x,9,-b^2*x+5/8*a*b*arctanh(cos(c+d*x))/d-b^2*cot(c+d*x)/d+1/3*b^2*cot(c+d*x)^3/d-1/5*b^2*cot(c+d*x)^5/d-1/7*a^2*cot(c+d*x)^7/d-5/8*a*b*cot(c+d*x)*csc(c+d*x)/d+5/12*a*b*cot(c+d*x)^3*csc(c+d*x)/d-1/3*a*b*cot(c+d*x)^5*csc(c+d*x)/d],
[cos(c+d*x)^6*csc(c+d*x)^9*(a+b*sin(c+d*x))^2,x,9,5/128*(a^2+8*b^2)*arctanh(cos(c+d*x))/d-2/7*a*b*cot(c+d*x)^7/d+1/128*(5*a^2-88*b^2)*cot(c+d*x)*csc(c+d*x)/d-1/192*(59*a^2-104*b^2)*cot(c+d*x)*csc(c+d*x)^3/d+1/48*(17*a^2-8*b^2)*cot(c+d*x)*csc(c+d*x)^5/d-1/8*a^2*cot(c+d*x)*csc(c+d*x)^7/d],
[cos(c+d*x)^6*csc(c+d*x)^10*(a+b*sin(c+d*x))^2,x,9,5/64*a*b*arctanh(cos(c+d*x))/d-1/7*(a^2+b^2)*cot(c+d*x)^7/d-1/9*a^2*cot(c+d*x)^9/d+5/64*a*b*cot(c+d*x)*csc(c+d*x)/d-5/32*a*b*cot(c+d*x)*csc(c+d*x)^3/d+5/24*a*b*cot(c+d*x)^3*csc(c+d*x)^3/d-1/4*a*b*cot(c+d*x)^5*csc(c+d*x)^3/d],
[cos(c+d*x)^6*csc(c+d*x)^11*(a+b*sin(c+d*x))^2,x,11,1/256*(3*a^2+10*b^2)*arctanh(cos(c+d*x))/d-2/7*a*b*cot(c+d*x)^7/d-2/9*a*b*cot(c+d*x)^9/d+1/256*(3*a^2+10*b^2)*cot(c+d*x)*csc(c+d*x)/d+1/384*(3*a^2-118*b^2)*cot(c+d*x)*csc(c+d*x)^3/d-1/480*(93*a^2-170*b^2)*cot(c+d*x)*csc(c+d*x)^5/d+1/80*(21*a^2-10*b^2)*cot(c+d*x)*csc(c+d*x)^7/d-1/10*a^2*cot(c+d*x)*csc(c+d*x)^9/d],
[cos(c+d*x)^6*csc(c+d*x)^12*(a+b*sin(c+d*x))^2,x,10,3/128*a*b*arctanh(cos(c+d*x))/d-1/7*(a^2+b^2)*cot(c+d*x)^7/d-1/9*(2*a^2+b^2)*cot(c+d*x)^9/d-1/11*a^2*cot(c+d*x)^11/d+3/128*a*b*cot(c+d*x)*csc(c+d*x)/d+1/64*a*b*cot(c+d*x)*csc(c+d*x)^3/d-1/16*a*b*cot(c+d*x)*csc(c+d*x)^5/d+1/8*a*b*cot(c+d*x)^3*csc(c+d*x)^5/d-1/5*a*b*cot(c+d*x)^5*csc(c+d*x)^5/d],

# m<0
[cos(c+d*x)^6*sin(c+d*x)^3/(a+b*sin(c+d*x))^2,x,11,1/8*a*(64*a^6-120*a^4*b^2+60*a^2*b^4-5*b^6)*x/b^9-2*a^2*(8*a^2-3*b^2)*(a^2-b^2)^(3/2)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(b^9*d)+1/105*(840*a^6-1435*a^4*b^2+588*a^2*b^4-15*b^6)*cos(c+d*x)/(b^8*d)-1/8*a*(32*a^4-52*a^2*b^2+19*b^4)*cos(c+d*x)*sin(c+d*x)/(b^7*d)+1/105*(280*a^4-441*a^2*b^2+150*b^4)*cos(c+d*x)*sin(c+d*x)^2/(b^6*d)-1/12*(24*a^4-37*a^2*b^2+12*b^4)*cos(c+d*x)*sin(c+d*x)^3/(a*b^5*d)+1/140*(224*a^4-340*a^2*b^2+105*b^4)*cos(c+d*x)*sin(c+d*x)^4/(a^2*b^4*d)+1/4*cos(c+d*x)*sin(c+d*x)^4/(a*d*(a+b*sin(c+d*x)))-3/20*b*cos(c+d*x)*sin(c+d*x)^5/(a^2*d*(a+b*sin(c+d*x)))-1/15*(20*a^4-30*a^2*b^2+9*b^4)*cos(c+d*x)*sin(c+d*x)^5/(a^2*b^3*d*(a+b*sin(c+d*x)))-4/21*a*cos(c+d*x)*sin(c+d*x)^6/(b^2*d*(a+b*sin(c+d*x)))+1/7*cos(c+d*x)*sin(c+d*x)^7/(b*d*(a+b*sin(c+d*x)))],
[cos(c+d*x)^6*sin(c+d*x)^2/(a+b*sin(c+d*x))^2,x,10,-1/16*(112*a^6-200*a^4*b^2+90*a^2*b^4-5*b^6)*x/b^8+2*a*(7*a^2-2*b^2)*(a^2-b^2)^(3/2)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(b^8*d)-1/15*a*(105*a^4-170*a^2*b^2+61*b^4)*cos(c+d*x)/(b^7*d)+1/16*(56*a^4-86*a^2*b^2+27*b^4)*cos(c+d*x)*sin(c+d*x)/(b^6*d)-1/15*(35*a^4-52*a^2*b^2+15*b^4)*cos(c+d*x)*sin(c+d*x)^2/(a*b^5*d)+1/24*(42*a^4-61*a^2*b^2+16*b^4)*cos(c+d*x)*sin(c+d*x)^3/(a^2*b^4*d)+1/3*cos(c+d*x)*sin(c+d*x)^3/(a*d*(a+b*sin(c+d*x)))-1/6*b*cos(c+d*x)*sin(c+d*x)^4/(a^2*d*(a+b*sin(c+d*x)))-1/10*(14*a^4-20*a^2*b^2+5*b^4)*cos(c+d*x)*sin(c+d*x)^4/(a^2*b^3*d*(a+b*sin(c+d*x)))-7/30*a*cos(c+d*x)*sin(c+d*x)^5/(b^2*d*(a+b*sin(c+d*x)))+1/6*cos(c+d*x)*sin(c+d*x)^6/(b*d*(a+b*sin(c+d*x)))],
[cos(c+d*x)^6*sin(c+d*x)/(a+b*sin(c+d*x))^2,x,7,1/4*a*(24*a^4-40*a^2*b^2+15*b^4)*x/b^7-2*(a^2-b^2)^(3/2)*(6*a^2-b^2)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(b^7*d)+1/5*cos(c+d*x)^5*(6*a+b*sin(c+d*x))/(b^2*d*(a+b*sin(c+d*x)))-1/6*cos(c+d*x)^3*(2*(6*a^2-b^2)-9*a*b*sin(c+d*x))/(b^4*d)+1/4*cos(c+d*x)*(4*(6*a^4-7*a^2*b^2+b^4)-a*b*(12*a^2-11*b^2)*sin(c+d*x))/(b^6*d)],
[cos(c+d*x)^6*csc(c+d*x)/(a+b*sin(c+d*x))^2,x,16,a*x/b^3+2*a*(2*a^2-3*b^2)*x/b^5+2*(a^2-b^2)^(3/2)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(b^5*d)-2*(a^2-b^2)^(3/2)*(5*a^2+b^2)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(a^2*b^5*d)-arctanh(cos(c+d*x))/(a^2*d)+cos(c+d*x)/(b^2*d)+3*(a^2-b^2)*cos(c+d*x)/(b^4*d)-1/3*cos(c+d*x)^3/(b^2*d)-a*cos(c+d*x)*sin(c+d*x)/(b^3*d)+(a^2-b^2)^2*cos(c+d*x)/(a*b^4*d*(a+b*sin(c+d*x)))],
[cos(c+d*x)^6*csc(c+d*x)^2/(a+b*sin(c+d*x))^2,x,16,-1/2*x/b^2-3*(a^2-b^2)*x/b^4-2*(a^2-b^2)^(3/2)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(a*b^4*d)+2*b*arctanh(cos(c+d*x))/(a^3*d)-2*a*cos(c+d*x)/(b^3*d)-cot(c+d*x)/(a^2*d)+1/2*cos(c+d*x)*sin(c+d*x)/(b^2*d)-(a^2-b^2)^2*cos(c+d*x)/(a^2*b^3*d*(a+b*sin(c+d*x)))+4*(2*a^6-3*a^4*b^2+b^6)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(a^3*b^4*d*sqrt(a^2-b^2))],
[cos(c+d*x)^6*csc(c+d*x)^3/(a+b*sin(c+d*x))^2,x,16,2*a*x/b^3+2*(a^2-b^2)^(3/2)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(a^2*b^3*d)-6*(a^2-b^2)^(3/2)*(a^2+b^2)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(a^4*b^3*d)-1/2*arctanh(cos(c+d*x))/(a^2*d)+3*(a^2-b^2)*arctanh(cos(c+d*x))/(a^4*d)+cos(c+d*x)/(b^2*d)+2*b*cot(c+d*x)/(a^3*d)-1/2*cot(c+d*x)*csc(c+d*x)/(a^2*d)+(a^2-b^2)^2*cos(c+d*x)/(a^3*b^2*d*(a+b*sin(c+d*x)))],
[cos(c+d*x)^6*csc(c+d*x)^4/(a+b*sin(c+d*x))^2,x,17,-x/b^2-2*(a^2-b^2)^(3/2)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(a^3*b^2*d)+b*arctanh(cos(c+d*x))/(a^3*d)-2*b*(3*a^2-2*b^2)*arctanh(cos(c+d*x))/(a^5*d)-cot(c+d*x)/(a^2*d)+3*(a^2-b^2)*cot(c+d*x)/(a^4*d)-1/3*cot(c+d*x)^3/(a^2*d)+b*cot(c+d*x)*csc(c+d*x)/(a^3*d)-(a^2-b^2)^2*cos(c+d*x)/(a^4*b*d*(a+b*sin(c+d*x)))+4*(a^6-3*a^2*b^4+2*b^6)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(a^5*b^2*d*sqrt(a^2-b^2))],
[cos(c+d*x)^6*csc(c+d*x)^5/(a+b*sin(c+d*x))^2,x,9,-10*b*(a^2-b^2)^(3/2)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(a^6*d)-5/8*(3*a^4-12*a^2*b^2+8*b^4)*arctanh(cos(c+d*x))/(a^6*d)+1/3*(3*a^4-20*a^2*b^2+15*b^4)*cot(c+d*x)/(a^5*b*d)+5/8*(5*a^2-4*b^2)*cot(c+d*x)*csc(c+d*x)/(a^4*d)-cot(c+d*x)/(b*d*(a+b*sin(c+d*x)))-1/3*(6*a^2-5*b^2)*cot(c+d*x)*csc(c+d*x)/(a^3*d*(a+b*sin(c+d*x)))+5/12*b*cot(c+d*x)*csc(c+d*x)^2/(a^2*d*(a+b*sin(c+d*x)))-1/4*cot(c+d*x)*csc(c+d*x)^3/(a*d*(a+b*sin(c+d*x)))],
[cos(c+d*x)^6*csc(c+d*x)^6/(a+b*sin(c+d*x))^2,x,10,-2*(a^2-6*b^2)*(a^2-b^2)^(3/2)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(a^7*d)+1/4*b*(15*a^4-40*a^2*b^2+24*b^4)*arctanh(cos(c+d*x))/(a^7*d)-1/15*(38*a^4-135*a^2*b^2+90*b^4)*cot(c+d*x)/(a^6*d)+1/4*(4*a^4-17*a^2*b^2+12*b^4)*cot(c+d*x)*csc(c+d*x)/(a^5*b*d)-1/30*(15*a^4-82*a^2*b^2+60*b^4)*cot(c+d*x)*csc(c+d*x)^2/(a^4*b^2*d)-1/2*cot(c+d*x)*csc(c+d*x)/(b*d*(a+b*sin(c+d*x)))+1/6*a*cot(c+d*x)*csc(c+d*x)^2/(b^2*d*(a+b*sin(c+d*x)))+1/6*(2*a^4-12*a^2*b^2+9*b^4)*cot(c+d*x)*csc(c+d*x)^2/(a^3*b^2*d*(a+b*sin(c+d*x)))+3/10*b*cot(c+d*x)*csc(c+d*x)^3/(a^2*d*(a+b*sin(c+d*x)))-1/5*cot(c+d*x)*csc(c+d*x)^4/(a*d*(a+b*sin(c+d*x)))],
[cos(c+d*x)^6*csc(c+d*x)^7/(a+b*sin(c+d*x))^2,x,11,2*b*(2*a^2-7*b^2)*(a^2-b^2)^(3/2)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(a^8*d)+1/16*(5*a^6-90*a^4*b^2+200*a^2*b^4-112*b^6)*arctanh(cos(c+d*x))/(a^8*d)+1/15*b*(61*a^4-170*a^2*b^2+105*b^4)*cot(c+d*x)/(a^7*d)-1/16*(27*a^4-86*a^2*b^2+56*b^4)*cot(c+d*x)*csc(c+d*x)/(a^6*d)+1/15*(15*a^4-52*a^2*b^2+35*b^4)*cot(c+d*x)*csc(c+d*x)^2/(a^5*b*d)-1/24*(16*a^4-61*a^2*b^2+42*b^4)*cot(c+d*x)*csc(c+d*x)^3/(a^4*b^2*d)-1/3*cot(c+d*x)*csc(c+d*x)^2/(b*d*(a+b*sin(c+d*x)))+1/6*a*cot(c+d*x)*csc(c+d*x)^3/(b^2*d*(a+b*sin(c+d*x)))+1/10*(5*a^4-20*a^2*b^2+14*b^4)*cot(c+d*x)*csc(c+d*x)^3/(a^3*b^2*d*(a+b*sin(c+d*x)))+7/30*b*cot(c+d*x)*csc(c+d*x)^4/(a^2*d*(a+b*sin(c+d*x)))-1/6*cot(c+d*x)*csc(c+d*x)^5/(a*d*(a+b*sin(c+d*x)))],
[cos(c+d*x)^6*sin(c+d*x)^3/(a+b*sin(c+d*x))^3,x,11,-1/16*(448*a^6-600*a^4*b^2+180*a^2*b^4-5*b^6)*x/b^9-1/30*a*(840*a^4-985*a^2*b^2+213*b^4)*cos(c+d*x)/(b^8*d)+1/16*(224*a^4-244*a^2*b^2+43*b^4)*cos(c+d*x)*sin(c+d*x)/(b^7*d)-1/30*(280*a^4-291*a^2*b^2+45*b^4)*cos(c+d*x)*sin(c+d*x)^2/(a*b^6*d)+1/24*(168*a^4-169*a^2*b^2+24*b^4)*cos(c+d*x)*sin(c+d*x)^3/(a^2*b^5*d)+1/4*cos(c+d*x)*sin(c+d*x)^4/(a*d*(a+b*sin(c+d*x))^2)-1/10*b*cos(c+d*x)*sin(c+d*x)^5/(a^2*d*(a+b*sin(c+d*x))^2)-1/60*(56*a^4-60*a^2*b^2+9*b^4)*cos(c+d*x)*sin(c+d*x)^5/(a^2*b^3*d*(a+b*sin(c+d*x))^2)-4/15*a*cos(c+d*x)*sin(c+d*x)^6/(b^2*d*(a+b*sin(c+d*x))^2)+1/6*cos(c+d*x)*sin(c+d*x)^7/(b*d*(a+b*sin(c+d*x))^2)-1/20*(112*a^4-110*a^2*b^2+15*b^4)*cos(c+d*x)*sin(c+d*x)^4/(a^2*b^4*d*(a+b*sin(c+d*x)))+a*(56*a^4-47*a^2*b^2+6*b^4)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))*sqrt(a^2-b^2)/(b^9*d)],
[cos(c+d*x)^6*sin(c+d*x)^2/(a+b*sin(c+d*x))^3,x,10,1/8*a*(168*a^4-200*a^2*b^2+45*b^4)*x/b^8+1/30*(630*a^4-645*a^2*b^2+91*b^4)*cos(c+d*x)/(b^7*d)-1/8*(84*a^4-79*a^2*b^2+8*b^4)*cos(c+d*x)*sin(c+d*x)/(a*b^6*d)+1/30*(210*a^4-187*a^2*b^2+15*b^4)*cos(c+d*x)*sin(c+d*x)^2/(a^2*b^5*d)+1/3*cos(c+d*x)*sin(c+d*x)^3/(a*d*(a+b*sin(c+d*x))^2)-1/12*b*cos(c+d*x)*sin(c+d*x)^4/(a^2*d*(a+b*sin(c+d*x))^2)-1/60*(63*a^4-60*a^2*b^2+5*b^4)*cos(c+d*x)*sin(c+d*x)^4/(a^2*b^3*d*(a+b*sin(c+d*x))^2)-7/20*a*cos(c+d*x)*sin(c+d*x)^5/(b^2*d*(a+b*sin(c+d*x))^2)+1/5*cos(c+d*x)*sin(c+d*x)^6/(b*d*(a+b*sin(c+d*x))^2)-1/12*(63*a^4-54*a^2*b^2+4*b^4)*cos(c+d*x)*sin(c+d*x)^3/(a^2*b^4*d*(a+b*sin(c+d*x)))-(42*a^4-29*a^2*b^2+2*b^4)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))*sqrt(a^2-b^2)/(b^8*d)],
[cos(c+d*x)^6*sin(c+d*x)/(a+b*sin(c+d*x))^3,x,7,-15/8*(8*a^4-8*a^2*b^2+b^4)*x/b^7+1/4*cos(c+d*x)^5*(3*a+b*sin(c+d*x))/(b^2*d*(a+b*sin(c+d*x))^2)+5/4*cos(c+d*x)^3*(4*a^2-b^2+a*b*sin(c+d*x))/(b^4*d*(a+b*sin(c+d*x)))-15/8*cos(c+d*x)*(4*a*(2*a^2-b^2)-b*(4*a^2-b^2)*sin(c+d*x))/(b^6*d)+15*a*(2*a^4-3*a^2*b^2+b^4)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(b^7*d*sqrt(a^2-b^2))],
[cos(c+d*x)^6*csc(c+d*x)/(a+b*sin(c+d*x))^3,x,20,-1/2*x/b^3-3*(2*a^2-b^2)*x/b^5-arctanh(cos(c+d*x))/(a^3*d)-3*a*cos(c+d*x)/(b^4*d)+1/2*cos(c+d*x)*sin(c+d*x)/(b^3*d)+1/2*(a^2-b^2)^2*cos(c+d*x)/(a*b^4*d*(a+b*sin(c+d*x))^2)+3/2*(a^2-b^2)*cos(c+d*x)/(b^4*d*(a+b*sin(c+d*x)))-(a^2-b^2)*(5*a^2+b^2)*cos(c+d*x)/(a^2*b^4*d*(a+b*sin(c+d*x)))+2*(10*a^6-9*a^4*b^2-b^6)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(a^3*b^5*d*sqrt(a^2-b^2))+(2*a^2+b^2)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))*sqrt(a^2-b^2)/(a*b^5*d)-2*(5*a^2+b^2)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))*sqrt(a^2-b^2)/(a*b^5*d)],
[cos(c+d*x)^6*csc(c+d*x)^2/(a+b*sin(c+d*x))^3,x,20,3*a*x/b^4+3*b*arctanh(cos(c+d*x))/(a^4*d)+cos(c+d*x)/(b^3*d)-cot(c+d*x)/(a^3*d)-1/2*(a^2-b^2)^2*cos(c+d*x)/(a^2*b^3*d*(a+b*sin(c+d*x))^2)-3/2*(a^2-b^2)*cos(c+d*x)/(a*b^3*d*(a+b*sin(c+d*x)))+2*(a^2-b^2)*(2*a^2+b^2)*cos(c+d*x)/(a^3*b^3*d*(a+b*sin(c+d*x)))-6*(2*a^6-a^4*b^2-b^6)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(a^4*b^4*d*sqrt(a^2-b^2))+3*(2*a^2+b^2)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))*sqrt(a^2-b^2)/(a^2*b^4*d)],
[cos(c+d*x)^6*csc(c+d*x)^3/(a+b*sin(c+d*x))^3,x,21,-x/b^3-1/2*arctanh(cos(c+d*x))/(a^3*d)+3*(a^2-2*b^2)*arctanh(cos(c+d*x))/(a^5*d)+3*b*cot(c+d*x)/(a^4*d)-1/2*cot(c+d*x)*csc(c+d*x)/(a^3*d)+1/2*(a^2-b^2)^2*cos(c+d*x)/(a^3*b^2*d*(a+b*sin(c+d*x))^2)+3/2*(a^2-b^2)*cos(c+d*x)/(a^2*b^2*d*(a+b*sin(c+d*x)))-3*(a^4-b^4)*cos(c+d*x)/(a^4*b^2*d*(a+b*sin(c+d*x)))+6*(a^6+a^2*b^4-2*b^6)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(a^5*b^3*d*sqrt(a^2-b^2))-6*(a^2+b^2)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))*sqrt(a^2-b^2)/(a^3*b^3*d)+(2*a^2+b^2)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))*sqrt(a^2-b^2)/(a^3*b^3*d)],
[cos(c+d*x)^6*csc(c+d*x)^4/(a+b*sin(c+d*x))^3,x,9,-5/2*b*(3*a^2-4*b^2)*arctanh(cos(c+d*x))/(a^6*d)+1/6*(3*a^4+35*a^2*b^2-60*b^4)*cot(c+d*x)/(a^5*b^2*d)-cos(c+d*x)/(b*d*(a+b*sin(c+d*x))^2)-1/2*a*cot(c+d*x)/(b^2*d*(a+b*sin(c+d*x))^2)-1/3*(3*a^2-5*b^2)*cot(c+d*x)/(a^3*d*(a+b*sin(c+d*x))^2)+5/6*b*cot(c+d*x)*csc(c+d*x)/(a^2*d*(a+b*sin(c+d*x))^2)-1/3*cot(c+d*x)*csc(c+d*x)^2/(a*d*(a+b*sin(c+d*x))^2)-5/2*(a^2-2*b^2)*cot(c+d*x)/(a^4*d*(a+b*sin(c+d*x)))+5*(a^2-4*b^2)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))*sqrt(a^2-b^2)/(a^6*d)],
[cos(c+d*x)^6*csc(c+d*x)^5/(a+b*sin(c+d*x))^3,x,10,-15/8*(a^4-8*a^2*b^2+8*b^4)*arctanh(cos(c+d*x))/(a^7*d)+1/2*(a^4-25*a^2*b^2+30*b^4)*cot(c+d*x)/(a^6*b*d)+15/8*(3*a^2-4*b^2)*cot(c+d*x)*csc(c+d*x)/(a^5*d)-1/2*cot(c+d*x)/(b*d*(a+b*sin(c+d*x))^2)-1/4*(4*a^2-5*b^2)*cot(c+d*x)*csc(c+d*x)/(a^3*d*(a+b*sin(c+d*x))^2)+1/2*b*cot(c+d*x)*csc(c+d*x)^2/(a^2*d*(a+b*sin(c+d*x))^2)-1/4*cot(c+d*x)*csc(c+d*x)^3/(a*d*(a+b*sin(c+d*x))^2)-1/2*(7*a^2-10*b^2)*cot(c+d*x)*csc(c+d*x)/(a^4*d*(a+b*sin(c+d*x)))-15*b*(a^2-2*b^2)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))*sqrt(a^2-b^2)/(a^7*d)],
[cos(c+d*x)^6*csc(c+d*x)^6/(a+b*sin(c+d*x))^3,x,11,1/8*b*(45*a^4-200*a^2*b^2+168*b^4)*arctanh(cos(c+d*x))/(a^8*d)-1/30*(91*a^4-645*a^2*b^2+630*b^4)*cot(c+d*x)/(a^7*d)+1/8*(8*a^4-79*a^2*b^2+84*b^4)*cot(c+d*x)*csc(c+d*x)/(a^6*b*d)-1/30*(15*a^4-187*a^2*b^2+210*b^4)*cot(c+d*x)*csc(c+d*x)^2/(a^5*b^2*d)-1/3*cot(c+d*x)*csc(c+d*x)/(b*d*(a+b*sin(c+d*x))^2)+1/12*a*cot(c+d*x)*csc(c+d*x)^2/(b^2*d*(a+b*sin(c+d*x))^2)+1/60*(5*a^4-60*a^2*b^2+63*b^4)*cot(c+d*x)*csc(c+d*x)^2/(a^3*b^2*d*(a+b*sin(c+d*x))^2)+7/20*b*cot(c+d*x)*csc(c+d*x)^3/(a^2*d*(a+b*sin(c+d*x))^2)-1/5*cot(c+d*x)*csc(c+d*x)^4/(a*d*(a+b*sin(c+d*x))^2)+1/12*(4*a^4-54*a^2*b^2+63*b^4)*cot(c+d*x)*csc(c+d*x)^2/(a^4*b^2*d*(a+b*sin(c+d*x)))-(2*a^4-29*a^2*b^2+42*b^4)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))*sqrt(a^2-b^2)/(a^8*d)],
[cos(c+d*x)^6*csc(c+d*x)^8/(a+b*sin(c+d*x))^3,x,13,-3/16*b*(5*a^6-100*a^4*b^2+280*a^2*b^4-192*b^6)*arctanh(cos(c+d*x))/(a^10*d)+1/70*(10*a^6-889*a^4*b^2+3255*a^2*b^4-2520*b^6)*cot(c+d*x)/(a^9*d)+3/16*b*(27*a^4-116*a^2*b^2+96*b^4)*cot(c+d*x)*csc(c+d*x)/(a^8*d)-1/70*(205*a^4-973*a^2*b^2+840*b^4)*cot(c+d*x)*csc(c+d*x)^2/(a^7*d)+1/8*(16*a^4-81*a^2*b^2+72*b^4)*cot(c+d*x)*csc(c+d*x)^3/(a^6*b*d)-3/70*(35*a^4-185*a^2*b^2+168*b^4)*cot(c+d*x)*csc(c+d*x)^4/(a^5*b^2*d)-1/5*cot(c+d*x)*csc(c+d*x)^3/(b*d*(a+b*sin(c+d*x))^2)+1/10*a*cot(c+d*x)*csc(c+d*x)^4/(b^2*d*(a+b*sin(c+d*x))^2)+1/35*(7*a^4-35*a^2*b^2+30*b^4)*cot(c+d*x)*csc(c+d*x)^4/(a^3*b^2*d*(a+b*sin(c+d*x))^2)+3/14*b*cot(c+d*x)*csc(c+d*x)^5/(a^2*d*(a+b*sin(c+d*x))^2)-1/7*cot(c+d*x)*csc(c+d*x)^6/(a*d*(a+b*sin(c+d*x))^2)+1/10*(12*a^4-65*a^2*b^2+60*b^4)*cot(c+d*x)*csc(c+d*x)^4/(a^4*b^2*d*(a+b*sin(c+d*x)))-3*b^2*(4*a^4-23*a^2*b^2+24*b^4)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))*sqrt(a^2-b^2)/(a^10*d)],

# Integrands of the form Cos[e+f x]^6 Sin[e+f x]^n (a+b Sin[e+f x])^(m/2)

# Integrands of the form Cos[e+f x]^6 (d Sin[e+f x])^(n/2) (a+b Sin[e+f x])^(m/2)
[cos(e+f*x)^6/((a+b*sin(e+f*x))^(13/2)*sqrt(d*sin(e+f*x))),x,8,2/11*cos(e+f*x)^5*sqrt(d*sin(e+f*x))/(a*d*f*(a+b*sin(e+f*x))^(11/2))-20/99*(a^2-b^2)*cos(e+f*x)*sqrt(d*sin(e+f*x))/(a^2*b^2*d*f*(a+b*sin(e+f*x))^(9/2))+80/693*(3*a^2+2*b^2)*cos(e+f*x)*sqrt(d*sin(e+f*x))/(a^3*b^2*d*f*(a+b*sin(e+f*x))^(7/2))-4/231*(5*a^4-17*a^2*b^2+16*b^4)*cos(e+f*x)*sqrt(d*sin(e+f*x))/(a^4*b^2*(a^2-b^2)*d*f*(a+b*sin(e+f*x))^(5/2))-8/693*(5*a^6-22*a^4*b^2+65*a^2*b^4-32*b^6)*cos(e+f*x)*sqrt(d*sin(e+f*x))/(a^5*b^2*(a^2-b^2)^2*d*f*(a+b*sin(e+f*x))^(3/2))+16/693*b*(93*a^4-93*a^2*b^2+32*b^4)*cos(e+f*x)/(a^5*(a^2-b^2)^3*f*sqrt(d*sin(e+f*x))*sqrt(a+b*sin(e+f*x)))-16/693*b*(93*a^4-93*a^2*b^2+32*b^4)*EllipticE(sqrt(d)*sqrt(a+b*sin(e+f*x))/(sqrt(a+b)*sqrt(d*sin(e+f*x))),sqrt((-a-b)/(a-b)))*sqrt(a*(1-csc(e+f*x))/(a+b))*sqrt(a*(1+csc(e+f*x))/(a-b))*tan(e+f*x)/(a^7*(a-b)^2*(a+b)^(5/2)*f*sqrt(d))-16/693*(45*a^4-48*a^3*b-69*a^2*b^2+24*a*b^3+32*b^4)*EllipticF(sqrt(d)*sqrt(a+b*sin(e+f*x))/(sqrt(a+b)*sqrt(d*sin(e+f*x))),sqrt((-a-b)/(a-b)))*sqrt(a*(1-csc(e+f*x))/(a+b))*sqrt(a*(1+csc(e+f*x))/(a-b))*tan(e+f*x)/(a^6*(a-b)^2*(a+b)^(5/2)*f*sqrt(d))],

# Integrands of the form (g Cos[e+f x])^p (d Sin[e+f x])^n (a+b Sin[e+f x])^2

# p>0

# p<0
[(a+b*sin(e+f*x))^2/((g*cos(e+f*x))^(5/2)*sqrt(d*sin(e+f*x))),x,9,4/3*a*b*(d*sin(e+f*x))^(3/2)/(d^2*f*g*(g*cos(e+f*x))^(3/2))+2/3*(a^2+b^2)*sqrt(d*sin(e+f*x))/(d*f*g*(g*cos(e+f*x))^(3/2))+1/3*(2*a^2-b^2)*sqrt(cos(1/4*(4*e-Pi)+f*x)^2)/cos(1/4*(4*e-Pi)+f*x)*EllipticF(sin(1/4*(4*e-Pi)+f*x),sqrt(2))*sqrt(sin(2*e+2*f*x))/(f*g^2*sqrt(g*cos(e+f*x))*sqrt(d*sin(e+f*x))),4/3*a*b*(d*sin(e+f*x))^(3/2)/(d^2*f*g*(g*cos(e+f*x))^(3/2))-2/3*(2*a^2-b^2)*(1-csc(e+f*x)^2)^(3/4)*sqrt(cos(1/2*arccsc(sin(e+f*x)))^2)/cos(1/2*arccsc(sin(e+f*x)))*EllipticF(sin(1/2*arccsc(sin(e+f*x))),sqrt(2))*(d*sin(e+f*x))^(3/2)/(d^2*f*g*(g*cos(e+f*x))^(3/2))+2/3*(a^2+b^2)*sqrt(d*sin(e+f*x))/(d*f*g*(g*cos(e+f*x))^(3/2))],
[(a+b*sin(e+f*x))^2/((g*cos(e+f*x))^(7/2)*sqrt(d*sin(e+f*x))),x,6,8/5*a*b*(d*sin(e+f*x))^(3/2)/(d^2*f*g^3*sqrt(g*cos(e+f*x)))+2/5*(a+b*sin(e+f*x))^2*sqrt(d*sin(e+f*x))/(d*f*g*(g*cos(e+f*x))^(5/2))+8/5*a^2*sqrt(d*sin(e+f*x))/(d*f*g^3*sqrt(g*cos(e+f*x)))-8/5*a*b*sqrt(cos(e-1/4*Pi+f*x)^2)/cos(e-1/4*Pi+f*x)*EllipticE(sin(e-1/4*Pi+f*x),sqrt(2))*sqrt(g*cos(e+f*x))*sqrt(d*sin(e+f*x))/(d*f*g^4*sqrt(sin(2*e+2*f*x)))],

# Integrands of the form (g Cos[e+f x])^p (d Sin[e+f x])^n / (a+b Sin[e+f x])^1

# Integrands of the form Cos[e+f x]^p Sin[e+f x]^n / (a+b Sin[e+f x])

# p>0
[cos(c+d*x)*sin(c+d*x)^3/(a+b*sin(c+d*x)),x,4,-a^3*log(a+b*sin(c+d*x))/(b^4*d)+a^2*sin(c+d*x)/(b^3*d)-1/2*a*sin(c+d*x)^2/(b^2*d)+1/3*sin(c+d*x)^3/(b*d)],
[cos(c+d*x)*sin(c+d*x)^2/(a+b*sin(c+d*x)),x,4,a^2*log(a+b*sin(c+d*x))/(b^3*d)-a*sin(c+d*x)/(b^2*d)+1/2*sin(c+d*x)^2/(b*d)],
[cos(c+d*x)*sin(c+d*x)/(a+b*sin(c+d*x)),x,4,-a*log(a+b*sin(c+d*x))/(b^2*d)+sin(c+d*x)/(b*d)],
[cos(c+d*x)*csc(c+d*x)/(a+b*sin(c+d*x)),x,4,log(sin(c+d*x))/(a*d)-log(a+b*sin(c+d*x))/(a*d)],
[cos(c+d*x)*csc(c+d*x)^2/(a+b*sin(c+d*x)),x,4,-csc(c+d*x)/(a*d)-b*log(sin(c+d*x))/(a^2*d)+b*log(a+b*sin(c+d*x))/(a^2*d)],
[cos(c+d*x)*csc(c+d*x)^3/(a+b*sin(c+d*x)),x,4,b*csc(c+d*x)/(a^2*d)-1/2*csc(c+d*x)^2/(a*d)+b^2*log(sin(c+d*x))/(a^3*d)-b^2*log(a+b*sin(c+d*x))/(a^3*d)],
[cos(c+d*x)^2*sin(c+d*x)^4/(a+b*sin(c+d*x)),x,10,1/8*a*(8*a^4-4*a^2*b^2-b^4)*x/b^6+1/15*(15*a^4-5*a^2*b^2-2*b^4)*cos(c+d*x)/(b^5*d)-1/8*a*(4*a^2-b^2)*cos(c+d*x)*sin(c+d*x)/(b^4*d)+1/15*(5*a^2-b^2)*cos(c+d*x)*sin(c+d*x)^2/(b^3*d)-1/4*a*cos(c+d*x)*sin(c+d*x)^3/(b^2*d)+1/5*cos(c+d*x)*sin(c+d*x)^4/(b*d)-2*a^4*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))*sqrt(a^2-b^2)/(b^6*d)],
[cos(c+d*x)^2*sin(c+d*x)^3/(a+b*sin(c+d*x)),x,9,-1/8*(8*a^4-4*a^2*b^2-b^4)*x/b^5-1/3*a*(3*a^2-b^2)*cos(c+d*x)/(b^4*d)+1/8*(4*a^2-b^2)*cos(c+d*x)*sin(c+d*x)/(b^3*d)-1/3*a*cos(c+d*x)*sin(c+d*x)^2/(b^2*d)+1/4*cos(c+d*x)*sin(c+d*x)^3/(b*d)+2*a^3*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))*sqrt(a^2-b^2)/(b^5*d)],
[cos(c+d*x)^2*sin(c+d*x)^2/(a+b*sin(c+d*x)),x,8,1/2*a*(2*a^2-b^2)*x/b^4+1/3*(3*a^2-b^2)*cos(c+d*x)/(b^3*d)-1/2*a*cos(c+d*x)*sin(c+d*x)/(b^2*d)+1/3*cos(c+d*x)*sin(c+d*x)^2/(b*d)-2*a^2*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))*sqrt(a^2-b^2)/(b^4*d)],
[cos(c+d*x)^2*sin(c+d*x)/(a+b*sin(c+d*x)),x,5,-1/2*(2*a^2-b^2)*x/b^3-1/2*cos(c+d*x)*(2*a-b*sin(c+d*x))/(b^2*d)+2*a*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))*sqrt(a^2-b^2)/(b^3*d)],
[cos(c+d*x)^2*csc(c+d*x)/(a+b*sin(c+d*x)),x,6,-x/b-arctanh(cos(c+d*x))/(a*d)+2*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))*sqrt(a^2-b^2)/(a*b*d)],
[cos(c+d*x)^2*csc(c+d*x)^2/(a+b*sin(c+d*x)),x,7,b*arctanh(cos(c+d*x))/(a^2*d)-cot(c+d*x)/(a*d)-2*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))*sqrt(a^2-b^2)/(a^2*d)],
[cos(c+d*x)^2*csc(c+d*x)^3/(a+b*sin(c+d*x)),x,8,1/2*(a^2-2*b^2)*arctanh(cos(c+d*x))/(a^3*d)+b*cot(c+d*x)/(a^2*d)-1/2*cot(c+d*x)*csc(c+d*x)/(a*d)+2*b*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))*sqrt(a^2-b^2)/(a^3*d)],
[cos(c+d*x)^2*csc(c+d*x)^4/(a+b*sin(c+d*x)),x,9,-1/2*b*(a^2-2*b^2)*arctanh(cos(c+d*x))/(a^4*d)+1/3*(a^2-3*b^2)*cot(c+d*x)/(a^3*d)+1/2*b*cot(c+d*x)*csc(c+d*x)/(a^2*d)-1/3*cot(c+d*x)*csc(c+d*x)^2/(a*d)-2*b^2*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))*sqrt(a^2-b^2)/(a^4*d)],
[cos(c+d*x)^2*csc(c+d*x)^5/(a+b*sin(c+d*x)),x,10,1/8*(a^4+4*a^2*b^2-8*b^4)*arctanh(cos(c+d*x))/(a^5*d)-1/3*b*(a^2-3*b^2)*cot(c+d*x)/(a^4*d)+1/8*(a^2-4*b^2)*cot(c+d*x)*csc(c+d*x)/(a^3*d)+1/3*b*cot(c+d*x)*csc(c+d*x)^2/(a^2*d)-1/4*cot(c+d*x)*csc(c+d*x)^3/(a*d)+2*b^3*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))*sqrt(a^2-b^2)/(a^5*d)],
[cos(c+d*x)^2*csc(c+d*x)^6/(a+b*sin(c+d*x)),x,11,-1/8*b*(a^4+4*a^2*b^2-8*b^4)*arctanh(cos(c+d*x))/(a^6*d)+1/15*(2*a^4+5*a^2*b^2-15*b^4)*cot(c+d*x)/(a^5*d)-1/8*b*(a^2-4*b^2)*cot(c+d*x)*csc(c+d*x)/(a^4*d)+1/15*(a^2-5*b^2)*cot(c+d*x)*csc(c+d*x)^2/(a^3*d)+1/4*b*cot(c+d*x)*csc(c+d*x)^3/(a^2*d)-1/5*cot(c+d*x)*csc(c+d*x)^4/(a*d)-2*b^4*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))*sqrt(a^2-b^2)/(a^6*d)],
[cos(c+d*x)^3*sin(c+d*x)^3/(a+b*sin(c+d*x)),x,4,a^3*(a^2-b^2)*log(a+b*sin(c+d*x))/(b^6*d)-a^2*(a^2-b^2)*sin(c+d*x)/(b^5*d)+1/2*a*(a^2-b^2)*sin(c+d*x)^2/(b^4*d)-1/3*(a^2-b^2)*sin(c+d*x)^3/(b^3*d)+1/4*a*sin(c+d*x)^4/(b^2*d)-1/5*sin(c+d*x)^5/(b*d)],
[cos(c+d*x)^3*sin(c+d*x)^2/(a+b*sin(c+d*x)),x,4,-a^2*(a^2-b^2)*log(a+b*sin(c+d*x))/(b^5*d)+a*(a^2-b^2)*sin(c+d*x)/(b^4*d)-1/2*(a^2-b^2)*sin(c+d*x)^2/(b^3*d)+1/3*a*sin(c+d*x)^3/(b^2*d)-1/4*sin(c+d*x)^4/(b*d)],
[cos(c+d*x)^3*sin(c+d*x)/(a+b*sin(c+d*x)),x,4,a*(a^2-b^2)*log(a+b*sin(c+d*x))/(b^4*d)-(a^2-b^2)*sin(c+d*x)/(b^3*d)+1/2*a*sin(c+d*x)^2/(b^2*d)-1/3*sin(c+d*x)^3/(b*d)],
[cos(c+d*x)^3*csc(c+d*x)/(a+b*sin(c+d*x)),x,4,log(sin(c+d*x))/(a*d)+(a^2-b^2)*log(a+b*sin(c+d*x))/(a*b^2*d)-sin(c+d*x)/(b*d)],
[cos(c+d*x)^3*csc(c+d*x)^2/(a+b*sin(c+d*x)),x,4,-csc(c+d*x)/(a*d)-b*log(sin(c+d*x))/(a^2*d)-(1-b^2/a^2)*log(a+b*sin(c+d*x))/(b*d)],
[cos(c+d*x)^3*csc(c+d*x)^3/(a+b*sin(c+d*x)),x,3,b*csc(c+d*x)/(a^2*d)-1/2*csc(c+d*x)^2/(a*d)-(a^2-b^2)*log(sin(c+d*x))/(a^3*d)+(a^2-b^2)*log(a+b*sin(c+d*x))/(a^3*d)],
[cos(c+d*x)^4*sin(c+d*x)^3/(a+b*sin(c+d*x)),x,9,1/16*(16*a^6-24*a^4*b^2+6*a^2*b^4+b^6)*x/b^7-2*a^3*(a^2-b^2)^(3/2)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(b^7*d)+1/15*a*(15*a^4-20*a^2*b^2+3*b^4)*cos(c+d*x)/(b^6*d)-1/16*(8*a^4-10*a^2*b^2+b^4)*cos(c+d*x)*sin(c+d*x)/(b^5*d)+1/15*a*(5*a^2-6*b^2)*cos(c+d*x)*sin(c+d*x)^2/(b^4*d)-1/24*(6*a^2-7*b^2)*cos(c+d*x)*sin(c+d*x)^3/(b^3*d)+1/5*a*cos(c+d*x)*sin(c+d*x)^4/(b^2*d)-1/6*cos(c+d*x)*sin(c+d*x)^5/(b*d)],
[cos(c+d*x)^4*sin(c+d*x)^2/(a+b*sin(c+d*x)),x,8,-1/8*a*(8*a^4-12*a^2*b^2+3*b^4)*x/b^6+2*a^2*(a^2-b^2)^(3/2)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(b^6*d)-1/15*(15*a^4-20*a^2*b^2+3*b^4)*cos(c+d*x)/(b^5*d)+1/8*a*(4*a^2-5*b^2)*cos(c+d*x)*sin(c+d*x)/(b^4*d)-1/15*(5*a^2-6*b^2)*cos(c+d*x)*sin(c+d*x)^2/(b^3*d)+1/4*a*cos(c+d*x)*sin(c+d*x)^3/(b^2*d)-1/5*cos(c+d*x)*sin(c+d*x)^4/(b*d)],
[cos(c+d*x)^4*sin(c+d*x)/(a+b*sin(c+d*x)),x,6,1/8*(8*a^4-12*a^2*b^2+3*b^4)*x/b^5-2*a*(a^2-b^2)^(3/2)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(b^5*d)-1/12*cos(c+d*x)^3*(4*a-3*b*sin(c+d*x))/(b^2*d)+1/8*cos(c+d*x)*(8*a*(a^2-b^2)-b*(4*a^2-3*b^2)*sin(c+d*x))/(b^4*d)],
[cos(c+d*x)^4*csc(c+d*x)/(a+b*sin(c+d*x)),x,6,1/2*(2*a^2-3*b^2)*x/b^3-2*(a^2-b^2)^(3/2)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(a*b^3*d)-arctanh(cos(c+d*x))/(a*d)+a*cos(c+d*x)/(b^2*d)-1/2*cos(c+d*x)*sin(c+d*x)/(b*d)],
[cos(c+d*x)^4*csc(c+d*x)^2/(a+b*sin(c+d*x)),x,6,-a*x/b^2+2*(a^2-b^2)^(3/2)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(a^2*b^2*d)+b*arctanh(cos(c+d*x))/(a^2*d)-cos(c+d*x)/(b*d)-cot(c+d*x)/(a*d)],
[cos(c+d*x)^4*csc(c+d*x)^3/(a+b*sin(c+d*x)),x,6,x/b-2*(a^2-b^2)^(3/2)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(a^3*b*d)+1/2*(3*a^2-2*b^2)*arctanh(cos(c+d*x))/(a^3*d)+b*cot(c+d*x)/(a^2*d)-1/2*cot(c+d*x)*csc(c+d*x)/(a*d)],
[cos(c+d*x)^4*csc(c+d*x)^4/(a+b*sin(c+d*x)),x,7,2*(a^2-b^2)^(3/2)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(a^4*d)-1/2*b*(3*a^2-2*b^2)*arctanh(cos(c+d*x))/(a^4*d)+1/3*(4*a^2-3*b^2)*cot(c+d*x)/(a^3*d)+1/2*b*cot(c+d*x)*csc(c+d*x)/(a^2*d)-1/3*cot(c+d*x)*csc(c+d*x)^2/(a*d)],
[cos(c+d*x)^4*csc(c+d*x)^5/(a+b*sin(c+d*x)),x,8,-2*b*(a^2-b^2)^(3/2)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(a^5*d)-1/8*(3*a^4-12*a^2*b^2+8*b^4)*arctanh(cos(c+d*x))/(a^5*d)-1/3*b*(4*a^2-3*b^2)*cot(c+d*x)/(a^4*d)+1/8*(5*a^2-4*b^2)*cot(c+d*x)*csc(c+d*x)/(a^3*d)+1/3*b*cot(c+d*x)*csc(c+d*x)^2/(a^2*d)-1/4*cot(c+d*x)*csc(c+d*x)^3/(a*d)],
[cos(c+d*x)^4*csc(c+d*x)^6/(a+b*sin(c+d*x)),x,9,2*b^2*(a^2-b^2)^(3/2)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(a^6*d)+1/8*b*(3*a^4-12*a^2*b^2+8*b^4)*arctanh(cos(c+d*x))/(a^6*d)-1/15*(3*a^4-20*a^2*b^2+15*b^4)*cot(c+d*x)/(a^5*d)-1/8*b*(5*a^2-4*b^2)*cot(c+d*x)*csc(c+d*x)/(a^4*d)+1/15*(6*a^2-5*b^2)*cot(c+d*x)*csc(c+d*x)^2/(a^3*d)+1/4*b*cot(c+d*x)*csc(c+d*x)^3/(a^2*d)-1/5*cot(c+d*x)*csc(c+d*x)^4/(a*d)],
[cos(c+d*x)^5*sin(c+d*x)^3/(a+b*sin(c+d*x)),x,4,-a^3*(a^2-b^2)^2*log(a+b*sin(c+d*x))/(b^8*d)+a^2*(a^2-b^2)^2*sin(c+d*x)/(b^7*d)-1/2*a*(a^2-b^2)^2*sin(c+d*x)^2/(b^6*d)+1/3*(a^2-b^2)^2*sin(c+d*x)^3/(b^5*d)-1/4*a*(a^2-2*b^2)*sin(c+d*x)^4/(b^4*d)+1/5*(a^2-2*b^2)*sin(c+d*x)^5/(b^3*d)-1/6*a*sin(c+d*x)^6/(b^2*d)+1/7*sin(c+d*x)^7/(b*d)],
[cos(c+d*x)^5*sin(c+d*x)^2/(a+b*sin(c+d*x)),x,4,a^2*(a^2-b^2)^2*log(a+b*sin(c+d*x))/(b^7*d)-a*(a^2-b^2)^2*sin(c+d*x)/(b^6*d)+1/2*(a^2-b^2)^2*sin(c+d*x)^2/(b^5*d)-1/3*a*(a^2-2*b^2)*sin(c+d*x)^3/(b^4*d)+1/4*(a^2-2*b^2)*sin(c+d*x)^4/(b^3*d)-1/5*a*sin(c+d*x)^5/(b^2*d)+1/6*sin(c+d*x)^6/(b*d)],
[cos(c+d*x)^5*sin(c+d*x)/(a+b*sin(c+d*x)),x,4,-a*(a^2-b^2)^2*log(a+b*sin(c+d*x))/(b^6*d)+(a^2-b^2)^2*sin(c+d*x)/(b^5*d)-1/2*a*(a^2-2*b^2)*sin(c+d*x)^2/(b^4*d)+1/3*(a^2-2*b^2)*sin(c+d*x)^3/(b^3*d)-1/4*a*sin(c+d*x)^4/(b^2*d)+1/5*sin(c+d*x)^5/(b*d)],
[cos(c+d*x)^5*csc(c+d*x)/(a+b*sin(c+d*x)),x,4,log(sin(c+d*x))/(a*d)-(a^2-b^2)^2*log(a+b*sin(c+d*x))/(a*b^4*d)+(a^2-2*b^2)*sin(c+d*x)/(b^3*d)-1/2*a*sin(c+d*x)^2/(b^2*d)+1/3*sin(c+d*x)^3/(b*d)],
[cos(c+d*x)^5*csc(c+d*x)^2/(a+b*sin(c+d*x)),x,4,-csc(c+d*x)/(a*d)-b*log(sin(c+d*x))/(a^2*d)+(a^2-b^2)^2*log(a+b*sin(c+d*x))/(a^2*b^3*d)-a*sin(c+d*x)/(b^2*d)+1/2*sin(c+d*x)^2/(b*d)],
[cos(c+d*x)^5*csc(c+d*x)^3/(a+b*sin(c+d*x)),x,4,b*csc(c+d*x)/(a^2*d)-1/2*csc(c+d*x)^2/(a*d)-(2*a^2-b^2)*log(sin(c+d*x))/(a^3*d)-(a^2-b^2)^2*log(a+b*sin(c+d*x))/(a^3*b^2*d)+sin(c+d*x)/(b*d)],
[cos(c+d*x)^5*csc(c+d*x)^4/(a+b*sin(c+d*x)),x,4,(2*a^2-b^2)*csc(c+d*x)/(a^3*d)+1/2*b*csc(c+d*x)^2/(a^2*d)-1/3*csc(c+d*x)^3/(a*d)+b*(2*a^2-b^2)*log(sin(c+d*x))/(a^4*d)+(a^2-b^2)^2*log(a+b*sin(c+d*x))/(a^4*b*d)],
[cos(c+d*x)^5*csc(c+d*x)^5/(a+b*sin(c+d*x)),x,3,-b*(2*a^2-b^2)*csc(c+d*x)/(a^4*d)+1/2*(2*a^2-b^2)*csc(c+d*x)^2/(a^3*d)+1/3*b*csc(c+d*x)^3/(a^2*d)-1/4*csc(c+d*x)^4/(a*d)+(a^2-b^2)^2*log(sin(c+d*x))/(a^5*d)-(a^2-b^2)^2*log(a+b*sin(c+d*x))/(a^5*d)],
[cos(c+d*x)^5*csc(c+d*x)^6/(a+b*sin(c+d*x)),x,4,-(a^2-b^2)^2*csc(c+d*x)/(a^5*d)-1/2*b*(2*a^2-b^2)*csc(c+d*x)^2/(a^4*d)+1/3*(2*a^2-b^2)*csc(c+d*x)^3/(a^3*d)+1/4*b*csc(c+d*x)^4/(a^2*d)-1/5*csc(c+d*x)^5/(a*d)-b*(a^2-b^2)^2*log(sin(c+d*x))/(a^6*d)+b*(a^2-b^2)^2*log(a+b*sin(c+d*x))/(a^6*d)],
[cos(c+d*x)^5*csc(c+d*x)^7/(a+b*sin(c+d*x)),x,4,b*(a^2-b^2)^2*csc(c+d*x)/(a^6*d)-1/2*(a^2-b^2)^2*csc(c+d*x)^2/(a^5*d)-1/3*b*(2*a^2-b^2)*csc(c+d*x)^3/(a^4*d)+1/4*(2*a^2-b^2)*csc(c+d*x)^4/(a^3*d)+1/5*b*csc(c+d*x)^5/(a^2*d)-1/6*csc(c+d*x)^6/(a*d)+b^2*(a^2-b^2)^2*log(sin(c+d*x))/(a^7*d)-b^2*(a^2-b^2)^2*log(a+b*sin(c+d*x))/(a^7*d)],
[cos(c+d*x)^6*sin(c+d*x)^3/(a+b*sin(c+d*x)),x,11,-1/128*(128*a^8-320*a^6*b^2+240*a^4*b^4-40*a^2*b^6-5*b^8)*x/b^9+2*a^3*(a^2-b^2)^(5/2)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(b^9*d)-1/105*a*(105*a^6-245*a^4*b^2+161*a^2*b^4-15*b^6)*cos(c+d*x)/(b^8*d)+1/128*(64*a^6-144*a^4*b^2+88*a^2*b^4-5*b^6)*cos(c+d*x)*sin(c+d*x)/(b^7*d)-1/105*a*(35*a^4-77*a^2*b^2+45*b^4)*cos(c+d*x)*sin(c+d*x)^2/(b^6*d)+1/192*(48*a^4-104*a^2*b^2+59*b^4)*cos(c+d*x)*sin(c+d*x)^3/(b^5*d)+1/4*cos(c+d*x)*sin(c+d*x)^4/(a*d)-1/140*(28*a^4-60*a^2*b^2+35*b^4)*cos(c+d*x)*sin(c+d*x)^4/(a*b^4*d)-1/5*b*cos(c+d*x)*sin(c+d*x)^5/(a^2*d)+1/240*(40*a^4-85*a^2*b^2+48*b^4)*cos(c+d*x)*sin(c+d*x)^5/(a^2*b^3*d)-1/7*a*cos(c+d*x)*sin(c+d*x)^6/(b^2*d)+1/8*cos(c+d*x)*sin(c+d*x)^7/(b*d)],
[cos(c+d*x)^6*sin(c+d*x)^2/(a+b*sin(c+d*x)),x,10,1/16*a*(16*a^6-40*a^4*b^2+30*a^2*b^4-5*b^6)*x/b^8-2*a^2*(a^2-b^2)^(5/2)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(b^8*d)+1/105*(105*a^6-245*a^4*b^2+161*a^2*b^4-15*b^6)*cos(c+d*x)/(b^7*d)-1/16*a*(8*a^4-18*a^2*b^2+11*b^4)*cos(c+d*x)*sin(c+d*x)/(b^6*d)+1/105*(35*a^4-77*a^2*b^2+45*b^4)*cos(c+d*x)*sin(c+d*x)^2/(b^5*d)+1/3*cos(c+d*x)*sin(c+d*x)^3/(a*d)-1/24*(6*a^4-13*a^2*b^2+8*b^4)*cos(c+d*x)*sin(c+d*x)^3/(a*b^4*d)-1/4*b*cos(c+d*x)*sin(c+d*x)^4/(a^2*d)+1/140*(28*a^4-60*a^2*b^2+35*b^4)*cos(c+d*x)*sin(c+d*x)^4/(a^2*b^3*d)-1/6*a*cos(c+d*x)*sin(c+d*x)^5/(b^2*d)+1/7*cos(c+d*x)*sin(c+d*x)^6/(b*d)],
[cos(c+d*x)^6*sin(c+d*x)/(a+b*sin(c+d*x)),x,7,-1/16*(16*a^6-40*a^4*b^2+30*a^2*b^4-5*b^6)*x/b^7+2*a*(a^2-b^2)^(5/2)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(b^7*d)-1/30*cos(c+d*x)^5*(6*a-5*b*sin(c+d*x))/(b^2*d)+1/24*cos(c+d*x)^3*(8*a*(a^2-b^2)-b*(6*a^2-5*b^2)*sin(c+d*x))/(b^4*d)-1/16*cos(c+d*x)*(16*a*(a^2-b^2)^2-b*(8*a^4-14*a^2*b^2+5*b^4)*sin(c+d*x))/(b^6*d)],
[cos(c+d*x)^6*csc(c+d*x)/(a+b*sin(c+d*x)),x,14,-3/8*x/b-1/2*(a^2-3*b^2)*x/b^3-(a^4-3*a^2*b^2+3*b^4)*x/b^5+2*(a^2-b^2)^(5/2)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(a*b^5*d)-arctanh(cos(c+d*x))/(a*d)-a*cos(c+d*x)/(b^2*d)-a*(a^2-3*b^2)*cos(c+d*x)/(b^4*d)+1/3*a*cos(c+d*x)^3/(b^2*d)+3/8*cos(c+d*x)*sin(c+d*x)/(b*d)+1/2*(a^2-3*b^2)*cos(c+d*x)*sin(c+d*x)/(b^3*d)+1/4*cos(c+d*x)*sin(c+d*x)^3/(b*d)],
[cos(c+d*x)^6*csc(c+d*x)^2/(a+b*sin(c+d*x)),x,13,1/2*a*x/b^2+a*(a^2-3*b^2)*x/b^4-2*(a^2-b^2)^(5/2)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(a^2*b^4*d)+b*arctanh(cos(c+d*x))/(a^2*d)+cos(c+d*x)/(b*d)+(a^2-3*b^2)*cos(c+d*x)/(b^3*d)-1/3*cos(c+d*x)^3/(b*d)-cot(c+d*x)/(a*d)-1/2*a*cos(c+d*x)*sin(c+d*x)/(b^2*d)],
[cos(c+d*x)^6*csc(c+d*x)^3/(a+b*sin(c+d*x)),x,6,-1/2*(2*a^2-5*b^2)*x/b^3+2*(a^2-b^2)^(5/2)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(a^3*b^3*d)+1/2*(5*a^2-2*b^2)*arctanh(cos(c+d*x))/(a^3*d)-a*cos(c+d*x)/(b^2*d)+b*cot(c+d*x)/(a^2*d)-1/2*cot(c+d*x)*csc(c+d*x)/(a*d)+1/2*cos(c+d*x)*sin(c+d*x)/(b*d)],
[cos(c+d*x)^6*csc(c+d*x)^4/(a+b*sin(c+d*x)),x,13,a*x/b^2-2*(a^2-b^2)^(5/2)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(a^4*b^2*d)+1/2*b*arctanh(cos(c+d*x))/(a^2*d)-b*(3*a^2-b^2)*arctanh(cos(c+d*x))/(a^4*d)+cos(c+d*x)/(b*d)-cot(c+d*x)/(a*d)+(3*a^2-b^2)*cot(c+d*x)/(a^3*d)-1/3*cot(c+d*x)^3/(a*d)+1/2*b*cot(c+d*x)*csc(c+d*x)/(a^2*d)],
[cos(c+d*x)^6*csc(c+d*x)^5/(a+b*sin(c+d*x)),x,15,-x/b+2*(a^2-b^2)^(5/2)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(a^5*b*d)-1/8*(15*a^4-20*a^2*b^2+8*b^4)*arctanh(cos(c+d*x))/(a^5*d)+b*(-2*a^2+b^2)*cot(c+d*x)/(a^4*d)+1/3*b*cot(c+d*x)^3/(a^2*d)+1/8*(7*a^2-4*b^2)*cot(c+d*x)*csc(c+d*x)/(a^3*d)-1/4*cot(c+d*x)^3*csc(c+d*x)/(a*d),-x/b+2*(a^2-b^2)^(5/2)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(a^5*b*d)-3/8*arctanh(cos(c+d*x))/(a*d)+1/2*(3*a^2-b^2)*arctanh(cos(c+d*x))/(a^3*d)-(3*a^4-3*a^2*b^2+b^4)*arctanh(cos(c+d*x))/(a^5*d)+b*cot(c+d*x)/(a^2*d)-b*(3*a^2-b^2)*cot(c+d*x)/(a^4*d)+1/3*b*cot(c+d*x)^3/(a^2*d)-3/8*cot(c+d*x)*csc(c+d*x)/(a*d)+1/2*(3*a^2-b^2)*cot(c+d*x)*csc(c+d*x)/(a^3*d)-1/4*cot(c+d*x)*csc(c+d*x)^3/(a*d)],
[cos(c+d*x)^6*csc(c+d*x)^6/(a+b*sin(c+d*x)),x,9,-2*(a^2-b^2)^(5/2)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(a^6*d)+1/8*b*(15*a^4-20*a^2*b^2+8*b^4)*arctanh(cos(c+d*x))/(a^6*d)-1/15*(23*a^4-35*a^2*b^2+15*b^4)*cot(c+d*x)/(a^5*d)+1/8*b*(-9*a^2+4*b^2)*cot(c+d*x)*csc(c+d*x)/(a^4*d)+1/15*(11*a^2-5*b^2)*cot(c+d*x)*csc(c+d*x)^2/(a^3*d)+1/4*b*cot(c+d*x)*csc(c+d*x)^3/(a^2*d)-1/5*cot(c+d*x)*csc(c+d*x)^4/(a*d),-2*(a^2-b^2)^(5/2)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(a^6*d)+1/8*b*(15*a^4-20*a^2*b^2+8*b^4)*arctanh(cos(c+d*x))/(a^6*d)-1/15*(23*a^4-35*a^2*b^2+15*b^4)*cot(c+d*x)/(a^5*d)-cot(c+d*x)*csc(c+d*x)/(b*d)+1/8*(8*a^4-9*a^2*b^2+4*b^4)*cot(c+d*x)*csc(c+d*x)/(a^4*b*d)+1/2*a*cot(c+d*x)*csc(c+d*x)^2/(b^2*d)-1/30*(15*a^4-22*a^2*b^2+10*b^4)*cot(c+d*x)*csc(c+d*x)^2/(a^3*b^2*d)+1/4*b*cot(c+d*x)*csc(c+d*x)^3/(a^2*d)-1/5*cot(c+d*x)*csc(c+d*x)^4/(a*d)],
[cos(c+d*x)^6*csc(c+d*x)^7/(a+b*sin(c+d*x)),x,10,2*b*(a^2-b^2)^(5/2)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(a^7*d)+1/16*(5*a^6-30*a^4*b^2+40*a^2*b^4-16*b^6)*arctanh(cos(c+d*x))/(a^7*d)+1/15*b*(23*a^4-35*a^2*b^2+15*b^4)*cot(c+d*x)/(a^6*d)-1/16*(11*a^4-18*a^2*b^2+8*b^4)*cot(c+d*x)*csc(c+d*x)/(a^5*d)-1/2*cot(c+d*x)*csc(c+d*x)^2/(b*d)+1/30*(15*a^4-22*a^2*b^2+10*b^4)*cot(c+d*x)*csc(c+d*x)^2/(a^4*b*d)+1/3*a*cot(c+d*x)*csc(c+d*x)^3/(b^2*d)-1/24*(8*a^4-13*a^2*b^2+6*b^4)*cot(c+d*x)*csc(c+d*x)^3/(a^3*b^2*d)+1/5*b*cot(c+d*x)*csc(c+d*x)^4/(a^2*d)-1/6*cot(c+d*x)*csc(c+d*x)^5/(a*d)],
[cos(c+d*x)^6*csc(c+d*x)^8/(a+b*sin(c+d*x)),x,11,-2*b^2*(a^2-b^2)^(5/2)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(a^8*d)-1/16*b*(5*a^6-30*a^4*b^2+40*a^2*b^4-16*b^6)*arctanh(cos(c+d*x))/(a^8*d)+1/105*(15*a^6-161*a^4*b^2+245*a^2*b^4-105*b^6)*cot(c+d*x)/(a^7*d)+1/16*b*(11*a^4-18*a^2*b^2+8*b^4)*cot(c+d*x)*csc(c+d*x)/(a^6*d)-1/105*(45*a^4-77*a^2*b^2+35*b^4)*cot(c+d*x)*csc(c+d*x)^2/(a^5*d)-1/3*cot(c+d*x)*csc(c+d*x)^3/(b*d)+1/24*(8*a^4-13*a^2*b^2+6*b^4)*cot(c+d*x)*csc(c+d*x)^3/(a^4*b*d)+1/4*a*cot(c+d*x)*csc(c+d*x)^4/(b^2*d)-1/140*(35*a^4-60*a^2*b^2+28*b^4)*cot(c+d*x)*csc(c+d*x)^4/(a^3*b^2*d)+1/6*b*cot(c+d*x)*csc(c+d*x)^5/(a^2*d)-1/7*cot(c+d*x)*csc(c+d*x)^6/(a*d)],
[cos(c+d*x)^6*csc(c+d*x)^9/(a+b*sin(c+d*x)),x,12,2*b^3*(a^2-b^2)^(5/2)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(a^9*d)+1/128*(5*a^8+40*a^6*b^2-240*a^4*b^4+320*a^2*b^6-128*b^8)*arctanh(cos(c+d*x))/(a^9*d)-1/105*b*(15*a^6-161*a^4*b^2+245*a^2*b^4-105*b^6)*cot(c+d*x)/(a^8*d)+1/128*(5*a^6-88*a^4*b^2+144*a^2*b^4-64*b^6)*cot(c+d*x)*csc(c+d*x)/(a^7*d)+1/105*b*(45*a^4-77*a^2*b^2+35*b^4)*cot(c+d*x)*csc(c+d*x)^2/(a^6*d)-1/192*(59*a^4-104*a^2*b^2+48*b^4)*cot(c+d*x)*csc(c+d*x)^3/(a^5*d)-1/4*cot(c+d*x)*csc(c+d*x)^4/(b*d)+1/140*(35*a^4-60*a^2*b^2+28*b^4)*cot(c+d*x)*csc(c+d*x)^4/(a^4*b*d)+1/5*a*cot(c+d*x)*csc(c+d*x)^5/(b^2*d)-1/240*(48*a^4-85*a^2*b^2+40*b^4)*cot(c+d*x)*csc(c+d*x)^5/(a^3*b^2*d)+1/7*b*cot(c+d*x)*csc(c+d*x)^6/(a^2*d)-1/8*cot(c+d*x)*csc(c+d*x)^7/(a*d)],

# p<0
[sec(c+d*x)*sin(c+d*x)^3/(a+b*sin(c+d*x)),x,4,-1/2*log(1-sin(c+d*x))/((a+b)*d)-1/2*log(1+sin(c+d*x))/((a-b)*d)+a^3*log(a+b*sin(c+d*x))/(b^2*(a^2-b^2)*d)-sin(c+d*x)/(b*d)],
[sec(c+d*x)*sin(c+d*x)^2/(a+b*sin(c+d*x)),x,4,-1/2*log(1-sin(c+d*x))/((a+b)*d)+1/2*log(1+sin(c+d*x))/((a-b)*d)-a^2*log(a+b*sin(c+d*x))/(b*(a^2-b^2)*d)],
[sec(c+d*x)*sin(c+d*x)/(a+b*sin(c+d*x)),x,3,-1/2*log(1-sin(c+d*x))/((a+b)*d)-1/2*log(1+sin(c+d*x))/((a-b)*d)+a*log(a+b*sin(c+d*x))/((a^2-b^2)*d)],
[csc(c+d*x)*sec(c+d*x)/(a+b*sin(c+d*x)),x,4,-1/2*log(1-sin(c+d*x))/((a+b)*d)+log(sin(c+d*x))/(a*d)-1/2*log(1+sin(c+d*x))/((a-b)*d)+b^2*log(a+b*sin(c+d*x))/(a*(a^2-b^2)*d)],
[csc(c+d*x)^2*sec(c+d*x)/(a+b*sin(c+d*x)),x,4,-csc(c+d*x)/(a*d)-1/2*log(1-sin(c+d*x))/((a+b)*d)-b*log(sin(c+d*x))/(a^2*d)+1/2*log(1+sin(c+d*x))/((a-b)*d)-b^3*log(a+b*sin(c+d*x))/(a^2*(a^2-b^2)*d)],
[csc(c+d*x)^3*sec(c+d*x)/(a+b*sin(c+d*x)),x,4,b*csc(c+d*x)/(a^2*d)-1/2*csc(c+d*x)^2/(a*d)-1/2*log(1-sin(c+d*x))/((a+b)*d)+(a^2+b^2)*log(sin(c+d*x))/(a^3*d)-1/2*log(1+sin(c+d*x))/((a-b)*d)+b^4*log(a+b*sin(c+d*x))/(a^3*(a^2-b^2)*d)],
[sec(c+d*x)^2*sin(c+d*x)^5/(a+b*sin(c+d*x)),x,14,3/2*b*x/(a^2-b^2)-1/2*a^2*(2*a^2+b^2)*x/(b^3*(a^2-b^2))+2*a^5*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(b^3*(a^2-b^2)^(3/2)*d)+a*cos(c+d*x)/((a^2-b^2)*d)-a^3*cos(c+d*x)/(b^2*(a^2-b^2)*d)+a*sec(c+d*x)/((a^2-b^2)*d)+1/2*a^2*cos(c+d*x)*sin(c+d*x)/(b*(a^2-b^2)*d)-3/2*b*tan(c+d*x)/((a^2-b^2)*d)+1/2*b*sin(c+d*x)^2*tan(c+d*x)/((a^2-b^2)*d)],
[sec(c+d*x)^2*sin(c+d*x)^4/(a+b*sin(c+d*x)),x,12,-a*x/(a^2-b^2)+a^3*x/(b^2*(a^2-b^2))-2*a^4*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(b^2*(a^2-b^2)^(3/2)*d)+a^2*cos(c+d*x)/(b*(a^2-b^2)*d)-b*cos(c+d*x)/((a^2-b^2)*d)-b*sec(c+d*x)/((a^2-b^2)*d)+a*tan(c+d*x)/((a^2-b^2)*d)],
[sec(c+d*x)^2*sin(c+d*x)^3/(a+b*sin(c+d*x)),x,9,-a^2*x/(b*(a^2-b^2))+b*x/(a^2-b^2)+2*a^3*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(b*(a^2-b^2)^(3/2)*d)+a*sec(c+d*x)/((a^2-b^2)*d)-b*tan(c+d*x)/((a^2-b^2)*d)],
[sec(c+d*x)^2*sin(c+d*x)^2/(a+b*sin(c+d*x)),x,8,-2*a^2*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/((a^2-b^2)^(3/2)*d)-b*sec(c+d*x)/((a^2-b^2)*d)+a*tan(c+d*x)/((a^2-b^2)*d)],
[sec(c+d*x)^2*sin(c+d*x)/(a+b*sin(c+d*x)),x,5,2*a*b*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/((a^2-b^2)^(3/2)*d)+sec(c+d*x)*(a-b*sin(c+d*x))/((a^2-b^2)*d)],
[csc(c+d*x)*sec(c+d*x)^2/(a+b*sin(c+d*x)),x,10,2*b^3*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(a*(a^2-b^2)^(3/2)*d)-arctanh(cos(c+d*x))/(a*d)+sec(c+d*x)/(a*d)+b*sec(c+d*x)*(b-a*sin(c+d*x))/(a*(a^2-b^2)*d)],
[csc(c+d*x)^2*sec(c+d*x)^2/(a+b*sin(c+d*x)),x,13,-2*b^4*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(a^2*(a^2-b^2)^(3/2)*d)+b*arctanh(cos(c+d*x))/(a^2*d)-cot(c+d*x)/(a*d)-b*sec(c+d*x)/((a^2-b^2)*d)+a*tan(c+d*x)/((a^2-b^2)*d),-2*b^4*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(a^2*(a^2-b^2)^(3/2)*d)+b*arctanh(cos(c+d*x))/(a^2*d)-cot(c+d*x)/(a*d)-b*sec(c+d*x)/(a^2*d)-b^2*sec(c+d*x)*(b-a*sin(c+d*x))/(a^2*(a^2-b^2)*d)+tan(c+d*x)/(a*d)],
[csc(c+d*x)^3*sec(c+d*x)^2/(a+b*sin(c+d*x)),x,17,2*b^5*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(a^3*(a^2-b^2)^(3/2)*d)-1/2*(3*a^2+2*b^2)*arctanh(cos(c+d*x))/(a^3*d)+b*cot(c+d*x)/(a^2*d)+1/2*(3*a^2-b^2)*sec(c+d*x)/(a*(a^2-b^2)*d)-1/2*csc(c+d*x)^2*sec(c+d*x)/(a*d)-b*tan(c+d*x)/((a^2-b^2)*d),2*b^5*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(a^3*(a^2-b^2)^(3/2)*d)-3/2*arctanh(cos(c+d*x))/(a*d)-b^2*arctanh(cos(c+d*x))/(a^3*d)+b*cot(c+d*x)/(a^2*d)+3/2*sec(c+d*x)/(a*d)+b^2*sec(c+d*x)/(a^3*d)-1/2*csc(c+d*x)^2*sec(c+d*x)/(a*d)+b^3*sec(c+d*x)*(b-a*sin(c+d*x))/(a^3*(a^2-b^2)*d)-b*tan(c+d*x)/(a^2*d)],
[sec(c+d*x)^3*sin(c+d*x)^3/(a+b*sin(c+d*x)),x,4,1/4*(2*a+b)*log(1-sin(c+d*x))/((a+b)^2*d)+1/4*(2*a-b)*log(1+sin(c+d*x))/((a-b)^2*d)-a^3*log(a+b*sin(c+d*x))/((a^2-b^2)^2*d)+1/2*sec(c+d*x)^2*(a-b*sin(c+d*x))/((a^2-b^2)*d)],
[sec(c+d*x)^3*sin(c+d*x)^2/(a+b*sin(c+d*x)),x,5,1/4*a*log(1-sin(c+d*x))/((a+b)^2*d)-1/4*a*log(1+sin(c+d*x))/((a-b)^2*d)+a^2*b*log(a+b*sin(c+d*x))/((a^2-b^2)^2*d)-1/2*sec(c+d*x)^2*(b-a*sin(c+d*x))/((a^2-b^2)*d)],
[sec(c+d*x)^3*sin(c+d*x)/(a+b*sin(c+d*x)),x,5,-1/4*b*log(1-sin(c+d*x))/((a+b)^2*d)+1/4*b*log(1+sin(c+d*x))/((a-b)^2*d)-a*b^2*log(a+b*sin(c+d*x))/((a^2-b^2)^2*d)+1/2*sec(c+d*x)^2*(a-b*sin(c+d*x))/((a^2-b^2)*d)],
[csc(c+d*x)*sec(c+d*x)^3/(a+b*sin(c+d*x)),x,4,-1/4*(2*a+3*b)*log(1-sin(c+d*x))/((a+b)^2*d)+log(sin(c+d*x))/(a*d)-1/4*(2*a-3*b)*log(1+sin(c+d*x))/((a-b)^2*d)-b^4*log(a+b*sin(c+d*x))/(a*(a^2-b^2)^2*d)+1/4/((a+b)*d*(1-sin(c+d*x)))+1/4/((a-b)*d*(1+sin(c+d*x)))],
[csc(c+d*x)^2*sec(c+d*x)^3/(a+b*sin(c+d*x)),x,4,-csc(c+d*x)/(a*d)-1/4*(3*a+4*b)*log(1-sin(c+d*x))/((a+b)^2*d)-b*log(sin(c+d*x))/(a^2*d)+1/4*(3*a-4*b)*log(1+sin(c+d*x))/((a-b)^2*d)+b^5*log(a+b*sin(c+d*x))/(a^2*(a^2-b^2)^2*d)+1/4/((a+b)*d*(1-sin(c+d*x)))+(-1/4)/((a-b)*d*(1+sin(c+d*x)))],
[csc(c+d*x)^3*sec(c+d*x)^3/(a+b*sin(c+d*x)),x,4,b*csc(c+d*x)/(a^2*d)-1/2*csc(c+d*x)^2/(a*d)-1/4*(4*a+5*b)*log(1-sin(c+d*x))/((a+b)^2*d)+(2*a^2+b^2)*log(sin(c+d*x))/(a^3*d)-1/4*(4*a-5*b)*log(1+sin(c+d*x))/((a-b)^2*d)-b^6*log(a+b*sin(c+d*x))/(a^3*(a^2-b^2)^2*d)+1/4/((a+b)*d*(1-sin(c+d*x)))+1/4/((a-b)*d*(1+sin(c+d*x)))],
[sec(c+d*x)^4*sin(c+d*x)^4/(a+b*sin(c+d*x)),x,13,2*a^4*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/((a^2-b^2)^(5/2)*d)+a^2*b*sec(c+d*x)/((a^2-b^2)^2*d)+b*sec(c+d*x)/((a^2-b^2)*d)-1/3*b*sec(c+d*x)^3/((a^2-b^2)*d)-a^3*tan(c+d*x)/((a^2-b^2)^2*d)+1/3*a*tan(c+d*x)^3/((a^2-b^2)*d)],
[sec(c+d*x)^4*sin(c+d*x)^3/(a+b*sin(c+d*x)),x,10,-2*a^3*b*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/((a^2-b^2)^(5/2)*d)+1/3*a*sec(c+d*x)^3/((a^2-b^2)*d)-a^2*sec(c+d*x)*(a-b*sin(c+d*x))/((a^2-b^2)^2*d)-1/3*b*tan(c+d*x)^3/((a^2-b^2)*d)],
[sec(c+d*x)^4*sin(c+d*x)^2/(a+b*sin(c+d*x)),x,10,2*a^2*b^2*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/((a^2-b^2)^(5/2)*d)-1/3*b*sec(c+d*x)^3/((a^2-b^2)*d)+a^2*sec(c+d*x)*(b-a*sin(c+d*x))/((a^2-b^2)^2*d)+a*tan(c+d*x)/((a^2-b^2)*d)+1/3*a*tan(c+d*x)^3/((a^2-b^2)*d)],
[sec(c+d*x)^4*sin(c+d*x)/(a+b*sin(c+d*x)),x,6,-2*a*b^3*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/((a^2-b^2)^(5/2)*d)+1/3*sec(c+d*x)^3*(a-b*sin(c+d*x))/((a^2-b^2)*d)-1/3*sec(c+d*x)*(3*a*b^2-b*(a^2+2*b^2)*sin(c+d*x))/((a^2-b^2)^2*d)],
[csc(c+d*x)*sec(c+d*x)^4/(a+b*sin(c+d*x)),x,12,-2*b^5*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(a*(a^2-b^2)^(5/2)*d)-arctanh(cos(c+d*x))/(a*d)+sec(c+d*x)/(a*d)+1/3*sec(c+d*x)^3/(a*d)+1/3*b*sec(c+d*x)^3*(b-a*sin(c+d*x))/(a*(a^2-b^2)*d)-1/3*b*sec(c+d*x)*(3*b^3+a*(2*a^2-5*b^2)*sin(c+d*x))/(a*(a^2-b^2)^2*d)],
[csc(c+d*x)^2*sec(c+d*x)^4/(a+b*sin(c+d*x)),x,15,2*b^6*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(a^2*(a^2-b^2)^(5/2)*d)+b*arctanh(cos(c+d*x))/(a^2*d)-cot(c+d*x)/(a*d)+b*(-a^2+2*b^2)*sec(c+d*x)/((a^2-b^2)^2*d)+1/3*b*sec(c+d*x)^3*(-a+b*sin(c+d*x))/(a*(a^2-b^2)*d)+1/3*(6*a^4-10*a^2*b^2+b^4)*tan(c+d*x)/(a*(a^2-b^2)^2*d)+1/3*tan(c+d*x)^3/(a*d),2*b^6*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(a^2*(a^2-b^2)^(5/2)*d)+b*arctanh(cos(c+d*x))/(a^2*d)-cot(c+d*x)/(a*d)-b*sec(c+d*x)/(a^2*d)-1/3*b*sec(c+d*x)^3/(a^2*d)-1/3*b^2*sec(c+d*x)^3*(b-a*sin(c+d*x))/(a^2*(a^2-b^2)*d)+1/3*b^2*sec(c+d*x)*(3*b^3+a*(2*a^2-5*b^2)*sin(c+d*x))/(a^2*(a^2-b^2)^2*d)+2*tan(c+d*x)/(a*d)+1/3*tan(c+d*x)^3/(a*d)],
[csc(c+d*x)^3*sec(c+d*x)^4/(a+b*sin(c+d*x)),x,20,-2*b^7*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(a^3*(a^2-b^2)^(5/2)*d)-5/2*arctanh(cos(c+d*x))/(a*d)-b^2*arctanh(cos(c+d*x))/(a^3*d)+b*cot(c+d*x)/(a^2*d)+5/2*sec(c+d*x)/(a*d)+b^2*sec(c+d*x)/(a^3*d)+5/6*sec(c+d*x)^3/(a*d)+1/3*b^2*sec(c+d*x)^3/(a^3*d)-1/2*csc(c+d*x)^2*sec(c+d*x)^3/(a*d)+1/3*b^3*sec(c+d*x)^3*(b-a*sin(c+d*x))/(a^3*(a^2-b^2)*d)-1/3*b^3*sec(c+d*x)*(3*b^3+a*(2*a^2-5*b^2)*sin(c+d*x))/(a^3*(a^2-b^2)^2*d)-2*b*tan(c+d*x)/(a^2*d)-1/3*b*tan(c+d*x)^3/(a^2*d)],
[sec(c+d*x)^5*sin(c+d*x)^8/(a+b*sin(c+d*x)),x,6,-1/16*(35*a^2+57*a*b+24*b^2)*log(1-sin(c+d*x))/((a+b)^3*d)+1/16*(35*a^2-57*a*b+24*b^2)*log(1+sin(c+d*x))/((a-b)^3*d)-a^8*log(a+b*sin(c+d*x))/(b^3*(a^2-b^2)^3*d)+a*sin(c+d*x)/(b^2*d)-1/2*sin(c+d*x)^2/(b*d)-1/4*sec(c+d*x)^4*(b-a*sin(c+d*x))/((a^2-b^2)*d)+1/8*sec(c+d*x)^2*(4*b*(4*a^2-3*b^2)-a*(13*a^2-9*b^2)*sin(c+d*x))/((a^2-b^2)^2*d)],
[sec(c+d*x)^5*sin(c+d*x)^7/(a+b*sin(c+d*x)),x,6,-1/16*(24*a^2+37*a*b+15*b^2)*log(1-sin(c+d*x))/((a+b)^3*d)-1/16*(24*a^2-37*a*b+15*b^2)*log(1+sin(c+d*x))/((a-b)^3*d)+a^7*log(a+b*sin(c+d*x))/(b^2*(a^2-b^2)^3*d)-sin(c+d*x)/(b*d)+1/4*sec(c+d*x)^4*(a-b*sin(c+d*x))/((a^2-b^2)*d)-1/8*sec(c+d*x)^2*(4*a*(3*a^2-2*b^2)-b*(13*a^2-9*b^2)*sin(c+d*x))/((a^2-b^2)^2*d)],
[sec(c+d*x)^5*sin(c+d*x)^6/(a+b*sin(c+d*x)),x,6,-1/16*(15*a^2+21*a*b+8*b^2)*log(1-sin(c+d*x))/((a+b)^3*d)+1/16*(15*a^2-21*a*b+8*b^2)*log(1+sin(c+d*x))/((a-b)^3*d)-a^6*log(a+b*sin(c+d*x))/(b*(a^2-b^2)^3*d)-1/4*sec(c+d*x)^4*(b-a*sin(c+d*x))/((a^2-b^2)*d)+1/8*sec(c+d*x)^2*(4*b*(3*a^2-2*b^2)-a*(9*a^2-5*b^2)*sin(c+d*x))/((a^2-b^2)^2*d)],
[sec(c+d*x)^5*sin(c+d*x)^5/(a+b*sin(c+d*x)),x,5,-1/16*(8*a^2+9*a*b+3*b^2)*log(1-sin(c+d*x))/((a+b)^3*d)-1/16*(8*a^2-9*a*b+3*b^2)*log(1+sin(c+d*x))/((a-b)^3*d)+a^5*log(a+b*sin(c+d*x))/((a^2-b^2)^3*d)+1/4*sec(c+d*x)^4*(a-b*sin(c+d*x))/((a^2-b^2)*d)-1/8*sec(c+d*x)^2*(4*a*(2*a^2-b^2)-b*(9*a^2-5*b^2)*sin(c+d*x))/((a^2-b^2)^2*d)],
[sec(c+d*x)^5*sin(c+d*x)^4/(a+b*sin(c+d*x)),x,6,-1/16*a*(3*a+b)*log(1-sin(c+d*x))/((a+b)^3*d)+1/16*a*(3*a-b)*log(1+sin(c+d*x))/((a-b)^3*d)-a^4*b*log(a+b*sin(c+d*x))/((a^2-b^2)^3*d)-1/4*sec(c+d*x)^4*(b-a*sin(c+d*x))/((a^2-b^2)*d)+1/8*sec(c+d*x)^2*(4*b*(2*a^2-b^2)-a*(5*a^2-b^2)*sin(c+d*x))/((a^2-b^2)^2*d)],
[sec(c+d*x)^5*sin(c+d*x)^3/(a+b*sin(c+d*x)),x,6,1/16*b*(3*a+b)*log(1-sin(c+d*x))/((a+b)^3*d)-1/16*(3*a-b)*b*log(1+sin(c+d*x))/((a-b)^3*d)+a^3*b^2*log(a+b*sin(c+d*x))/((a^2-b^2)^3*d)+1/4*sec(c+d*x)^4*(a-b*sin(c+d*x))/((a^2-b^2)*d)-1/8*sec(c+d*x)^2*(4*a^3-b*(5*a^2-b^2)*sin(c+d*x))/((a^2-b^2)^2*d)],
[sec(c+d*x)^5*sin(c+d*x)^2/(a+b*sin(c+d*x)),x,6,1/16*a*(a+3*b)*log(1-sin(c+d*x))/((a+b)^3*d)-1/16*a*(a-3*b)*log(1+sin(c+d*x))/((a-b)^3*d)-a^2*b^3*log(a+b*sin(c+d*x))/((a^2-b^2)^3*d)-1/4*sec(c+d*x)^4*(b-a*sin(c+d*x))/((a^2-b^2)*d)+1/8*a*sec(c+d*x)^2*(4*a*b-(a^2+3*b^2)*sin(c+d*x))/((a^2-b^2)^2*d)],
[sec(c+d*x)^5*sin(c+d*x)/(a+b*sin(c+d*x)),x,6,-1/16*b*(a+3*b)*log(1-sin(c+d*x))/((a+b)^3*d)+1/16*(a-3*b)*b*log(1+sin(c+d*x))/((a-b)^3*d)+a*b^4*log(a+b*sin(c+d*x))/((a^2-b^2)^3*d)+1/4*sec(c+d*x)^4*(a-b*sin(c+d*x))/((a^2-b^2)*d)-1/8*sec(c+d*x)^2*(4*a*b^2-b*(a^2+3*b^2)*sin(c+d*x))/((a^2-b^2)^2*d)],
[csc(c+d*x)*sec(c+d*x)^5/(a+b*sin(c+d*x)),x,4,-1/16*(8*a^2+21*a*b+15*b^2)*log(1-sin(c+d*x))/((a+b)^3*d)+log(sin(c+d*x))/(a*d)-1/16*(8*a^2-21*a*b+15*b^2)*log(1+sin(c+d*x))/((a-b)^3*d)+b^6*log(a+b*sin(c+d*x))/(a*(a^2-b^2)^3*d)+1/16/((a+b)*d*(1-sin(c+d*x))^2)+1/16*(5*a+7*b)/((a+b)^2*d*(1-sin(c+d*x)))+1/16/((a-b)*d*(1+sin(c+d*x))^2)+1/16*(5*a-7*b)/((a-b)^2*d*(1+sin(c+d*x)))],
[csc(c+d*x)^2*sec(c+d*x)^5/(a+b*sin(c+d*x)),x,4,-csc(c+d*x)/(a*d)-1/16*(15*a^2+37*a*b+24*b^2)*log(1-sin(c+d*x))/((a+b)^3*d)-b*log(sin(c+d*x))/(a^2*d)+1/16*(15*a^2-37*a*b+24*b^2)*log(1+sin(c+d*x))/((a-b)^3*d)-b^7*log(a+b*sin(c+d*x))/(a^2*(a^2-b^2)^3*d)+1/16/((a+b)*d*(1-sin(c+d*x))^2)+1/16*(7*a+9*b)/((a+b)^2*d*(1-sin(c+d*x)))+(-1/16)/((a-b)*d*(1+sin(c+d*x))^2)+1/16*(-7*a+9*b)/((a-b)^2*d*(1+sin(c+d*x)))],
[csc(c+d*x)^3*sec(c+d*x)^5/(a+b*sin(c+d*x)),x,4,b*csc(c+d*x)/(a^2*d)-1/2*csc(c+d*x)^2/(a*d)-1/16*(24*a^2+57*a*b+35*b^2)*log(1-sin(c+d*x))/((a+b)^3*d)+(3*a^2+b^2)*log(sin(c+d*x))/(a^3*d)-1/16*(24*a^2-57*a*b+35*b^2)*log(1+sin(c+d*x))/((a-b)^3*d)+b^8*log(a+b*sin(c+d*x))/(a^3*(a^2-b^2)^3*d)+1/16/((a+b)*d*(1-sin(c+d*x))^2)+1/16*(9*a+11*b)/((a+b)^2*d*(1-sin(c+d*x)))+1/16/((a-b)*d*(1+sin(c+d*x))^2)+1/16*(9*a-11*b)/((a-b)^2*d*(1+sin(c+d*x)))],

# Integrands of the form Cos[e+f x]^p (d Sin[e+f x])^(n/2) / (a+b Sin[e+f x])

# Integrands of the form (g Cos[e+f x])^(p/2) Sin[e+f x]^n / (a+b Sin[e+f x])

# p>0
[sin(e+f*x)^4*sqrt(g*cos(e+f*x))/(a+b*sin(e+f*x)),x,21,-2/3*a^2*(g*cos(e+f*x))^(3/2)/(b^3*f*g)-2/3*(g*cos(e+f*x))^(3/2)/(b*f*g)+2/7*(g*cos(e+f*x))^(7/2)/(b*f*g^3)+2/5*a*(g*cos(e+f*x))^(3/2)*sin(e+f*x)/(b^2*f*g)+a^4*arctan(sqrt(b)*sqrt(g*cos(e+f*x))/((-a^2+b^2)^(1/4)*sqrt(g)))*sqrt(g)/(b^(9/2)*(-a^2+b^2)^(1/4)*f)-a^4*arctanh(sqrt(b)*sqrt(g*cos(e+f*x))/((-a^2+b^2)^(1/4)*sqrt(g)))*sqrt(g)/(b^(9/2)*(-a^2+b^2)^(1/4)*f)+a^5*g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(e+f*x))/(b^5*f*(b-sqrt(-a^2+b^2))*sqrt(g*cos(e+f*x)))+a^5*g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(e+f*x))/(b^5*f*(b+sqrt(-a^2+b^2))*sqrt(g*cos(e+f*x)))-2*a^3*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(g*cos(e+f*x))/(b^4*f*sqrt(cos(e+f*x)))-4/5*a*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(g*cos(e+f*x))/(b^2*f*sqrt(cos(e+f*x)))],
[sin(e+f*x)^3*sqrt(g*cos(e+f*x))/(a+b*sin(e+f*x)),x,18,2/3*a*(g*cos(e+f*x))^(3/2)/(b^2*f*g)-2/5*(g*cos(e+f*x))^(3/2)*sin(e+f*x)/(b*f*g)-a^3*arctan(sqrt(b)*sqrt(g*cos(e+f*x))/((-a^2+b^2)^(1/4)*sqrt(g)))*sqrt(g)/(b^(7/2)*(-a^2+b^2)^(1/4)*f)+a^3*arctanh(sqrt(b)*sqrt(g*cos(e+f*x))/((-a^2+b^2)^(1/4)*sqrt(g)))*sqrt(g)/(b^(7/2)*(-a^2+b^2)^(1/4)*f)-a^4*g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(e+f*x))/(b^4*f*(b-sqrt(-a^2+b^2))*sqrt(g*cos(e+f*x)))-a^4*g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(e+f*x))/(b^4*f*(b+sqrt(-a^2+b^2))*sqrt(g*cos(e+f*x)))+2*a^2*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(g*cos(e+f*x))/(b^3*f*sqrt(cos(e+f*x)))+4/5*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(g*cos(e+f*x))/(b*f*sqrt(cos(e+f*x)))],
[sin(e+f*x)^2*sqrt(g*cos(e+f*x))/(a+b*sin(e+f*x)),x,15,-2/3*(g*cos(e+f*x))^(3/2)/(b*f*g)+a^2*arctan(sqrt(b)*sqrt(g*cos(e+f*x))/((-a^2+b^2)^(1/4)*sqrt(g)))*sqrt(g)/(b^(5/2)*(-a^2+b^2)^(1/4)*f)-a^2*arctanh(sqrt(b)*sqrt(g*cos(e+f*x))/((-a^2+b^2)^(1/4)*sqrt(g)))*sqrt(g)/(b^(5/2)*(-a^2+b^2)^(1/4)*f)+a^3*g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(e+f*x))/(b^3*f*(b-sqrt(-a^2+b^2))*sqrt(g*cos(e+f*x)))+a^3*g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(e+f*x))/(b^3*f*(b+sqrt(-a^2+b^2))*sqrt(g*cos(e+f*x)))-2*a*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(g*cos(e+f*x))/(b^2*f*sqrt(cos(e+f*x)))],
[sin(e+f*x)*sqrt(g*cos(e+f*x))/(a+b*sin(e+f*x)),x,12,-a*arctan(sqrt(b)*sqrt(g*cos(e+f*x))/((-a^2+b^2)^(1/4)*sqrt(g)))*sqrt(g)/(b^(3/2)*(-a^2+b^2)^(1/4)*f)+a*arctanh(sqrt(b)*sqrt(g*cos(e+f*x))/((-a^2+b^2)^(1/4)*sqrt(g)))*sqrt(g)/(b^(3/2)*(-a^2+b^2)^(1/4)*f)-a^2*g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(e+f*x))/(b^2*f*(b-sqrt(-a^2+b^2))*sqrt(g*cos(e+f*x)))-a^2*g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(e+f*x))/(b^2*f*(b+sqrt(-a^2+b^2))*sqrt(g*cos(e+f*x)))+2*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(g*cos(e+f*x))/(b*f*sqrt(cos(e+f*x)))],
[csc(e+f*x)*sqrt(g*cos(e+f*x))/(a+b*sin(e+f*x)),x,16,arctan(sqrt(g*cos(e+f*x))/sqrt(g))*sqrt(g)/(a*f)-arctanh(sqrt(g*cos(e+f*x))/sqrt(g))*sqrt(g)/(a*f)-arctan(sqrt(b)*sqrt(g*cos(e+f*x))/((-a^2+b^2)^(1/4)*sqrt(g)))*sqrt(b)*sqrt(g)/(a*(-a^2+b^2)^(1/4)*f)+arctanh(sqrt(b)*sqrt(g*cos(e+f*x))/((-a^2+b^2)^(1/4)*sqrt(g)))*sqrt(b)*sqrt(g)/(a*(-a^2+b^2)^(1/4)*f)-g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(e+f*x))/(f*(b-sqrt(-a^2+b^2))*sqrt(g*cos(e+f*x)))-g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(e+f*x))/(f*(b+sqrt(-a^2+b^2))*sqrt(g*cos(e+f*x)))],
[csc(e+f*x)^2*sqrt(g*cos(e+f*x))/(a+b*sin(e+f*x)),x,19,-(g*cos(e+f*x))^(3/2)*csc(e+f*x)/(a*f*g)-b*arctan(sqrt(g*cos(e+f*x))/sqrt(g))*sqrt(g)/(a^2*f)+b^(3/2)*arctan(sqrt(b)*sqrt(g*cos(e+f*x))/((-a^2+b^2)^(1/4)*sqrt(g)))*sqrt(g)/(a^2*(-a^2+b^2)^(1/4)*f)+b*arctanh(sqrt(g*cos(e+f*x))/sqrt(g))*sqrt(g)/(a^2*f)-b^(3/2)*arctanh(sqrt(b)*sqrt(g*cos(e+f*x))/((-a^2+b^2)^(1/4)*sqrt(g)))*sqrt(g)/(a^2*(-a^2+b^2)^(1/4)*f)+b*g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(e+f*x))/(a*f*(b-sqrt(-a^2+b^2))*sqrt(g*cos(e+f*x)))+b*g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(e+f*x))/(a*f*(b+sqrt(-a^2+b^2))*sqrt(g*cos(e+f*x)))-sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(g*cos(e+f*x))/(a*f*sqrt(cos(e+f*x)))],
[csc(e+f*x)^3*sqrt(g*cos(e+f*x))/(a+b*sin(e+f*x)),x,25,b*(g*cos(e+f*x))^(3/2)*csc(e+f*x)/(a^2*f*g)-1/2*(g*cos(e+f*x))^(3/2)*csc(e+f*x)^2/(a*f*g)+1/4*arctan(sqrt(g*cos(e+f*x))/sqrt(g))*sqrt(g)/(a*f)+b^2*arctan(sqrt(g*cos(e+f*x))/sqrt(g))*sqrt(g)/(a^3*f)-b^(5/2)*arctan(sqrt(b)*sqrt(g*cos(e+f*x))/((-a^2+b^2)^(1/4)*sqrt(g)))*sqrt(g)/(a^3*(-a^2+b^2)^(1/4)*f)-1/4*arctanh(sqrt(g*cos(e+f*x))/sqrt(g))*sqrt(g)/(a*f)-b^2*arctanh(sqrt(g*cos(e+f*x))/sqrt(g))*sqrt(g)/(a^3*f)+b^(5/2)*arctanh(sqrt(b)*sqrt(g*cos(e+f*x))/((-a^2+b^2)^(1/4)*sqrt(g)))*sqrt(g)/(a^3*(-a^2+b^2)^(1/4)*f)-b^2*g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(e+f*x))/(a^2*f*(b-sqrt(-a^2+b^2))*sqrt(g*cos(e+f*x)))-b^2*g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(e+f*x))/(a^2*f*(b+sqrt(-a^2+b^2))*sqrt(g*cos(e+f*x)))+b*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(g*cos(e+f*x))/(a^2*f*sqrt(cos(e+f*x)))],
[(g*cos(e+f*x))^(3/2)*sin(e+f*x)^3/(a+b*sin(e+f*x)),x,24,a^3*(-a^2+b^2)^(1/4)*g^(3/2)*arctan(sqrt(b)*sqrt(g*cos(e+f*x))/((-a^2+b^2)^(1/4)*sqrt(g)))/(b^(9/2)*f)+a^3*(-a^2+b^2)^(1/4)*g^(3/2)*arctanh(sqrt(b)*sqrt(g*cos(e+f*x))/((-a^2+b^2)^(1/4)*sqrt(g)))/(b^(9/2)*f)+2/5*a*(g*cos(e+f*x))^(5/2)/(b^2*f*g)-2/7*(g*cos(e+f*x))^(5/2)*sin(e+f*x)/(b*f*g)-2*a^4*g^2*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticF(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))/(b^5*f*sqrt(g*cos(e+f*x)))+2/3*a^2*g^2*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticF(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))/(b^3*f*sqrt(g*cos(e+f*x)))+4/21*g^2*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticF(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))/(b*f*sqrt(g*cos(e+f*x)))+a^4*(a^2-b^2)*g^2*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(e+f*x))/(b^5*f*(a^2-b*(b-sqrt(-a^2+b^2)))*sqrt(g*cos(e+f*x)))+a^4*(a^2-b^2)*g^2*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(e+f*x))/(b^5*f*(a^2-b*(b+sqrt(-a^2+b^2)))*sqrt(g*cos(e+f*x)))-2*a^3*g*sqrt(g*cos(e+f*x))/(b^4*f)+2/3*a^2*g*sin(e+f*x)*sqrt(g*cos(e+f*x))/(b^3*f)+4/21*g*sin(e+f*x)*sqrt(g*cos(e+f*x))/(b*f)],
[(g*cos(e+f*x))^(3/2)*sin(e+f*x)^2/(a+b*sin(e+f*x)),x,20,-a^2*(-a^2+b^2)^(1/4)*g^(3/2)*arctan(sqrt(b)*sqrt(g*cos(e+f*x))/((-a^2+b^2)^(1/4)*sqrt(g)))/(b^(7/2)*f)-a^2*(-a^2+b^2)^(1/4)*g^(3/2)*arctanh(sqrt(b)*sqrt(g*cos(e+f*x))/((-a^2+b^2)^(1/4)*sqrt(g)))/(b^(7/2)*f)-2/5*(g*cos(e+f*x))^(5/2)/(b*f*g)+2*a^3*g^2*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticF(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))/(b^4*f*sqrt(g*cos(e+f*x)))-2/3*a*g^2*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticF(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))/(b^2*f*sqrt(g*cos(e+f*x)))-a^3*(a^2-b^2)*g^2*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(e+f*x))/(b^4*f*(a^2-b*(b-sqrt(-a^2+b^2)))*sqrt(g*cos(e+f*x)))-a^3*(a^2-b^2)*g^2*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(e+f*x))/(b^4*f*(a^2-b*(b+sqrt(-a^2+b^2)))*sqrt(g*cos(e+f*x)))+2*a^2*g*sqrt(g*cos(e+f*x))/(b^3*f)-2/3*a*g*sin(e+f*x)*sqrt(g*cos(e+f*x))/(b^2*f)],
[(g*cos(e+f*x))^(3/2)*sin(e+f*x)/(a+b*sin(e+f*x)),x,13,a*(-a^2+b^2)^(1/4)*g^(3/2)*arctan(sqrt(b)*sqrt(g*cos(e+f*x))/((-a^2+b^2)^(1/4)*sqrt(g)))/(b^(5/2)*f)+a*(-a^2+b^2)^(1/4)*g^(3/2)*arctanh(sqrt(b)*sqrt(g*cos(e+f*x))/((-a^2+b^2)^(1/4)*sqrt(g)))/(b^(5/2)*f)-2/3*(3*a^2-b^2)*g^2*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticF(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))/(b^3*f*sqrt(g*cos(e+f*x)))+a^2*(a^2-b^2)*g^2*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(e+f*x))/(b^3*f*(a^2-b*(b-sqrt(-a^2+b^2)))*sqrt(g*cos(e+f*x)))+a^2*(a^2-b^2)*g^2*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(e+f*x))/(b^3*f*(a^2-b*(b+sqrt(-a^2+b^2)))*sqrt(g*cos(e+f*x)))-2/3*g*(3*a-b*sin(e+f*x))*sqrt(g*cos(e+f*x))/(b^2*f)],
[(g*cos(e+f*x))^(3/2)*csc(e+f*x)/(a+b*sin(e+f*x)),x,21,-g^(3/2)*arctan(sqrt(g*cos(e+f*x))/sqrt(g))/(a*f)-g^(3/2)*arctanh(sqrt(g*cos(e+f*x))/sqrt(g))/(a*f)+(-a^2+b^2)^(1/4)*g^(3/2)*arctan(sqrt(b)*sqrt(g*cos(e+f*x))/((-a^2+b^2)^(1/4)*sqrt(g)))/(a*f*sqrt(b))+(-a^2+b^2)^(1/4)*g^(3/2)*arctanh(sqrt(b)*sqrt(g*cos(e+f*x))/((-a^2+b^2)^(1/4)*sqrt(g)))/(a*f*sqrt(b))-2*g^2*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticF(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))/(b*f*sqrt(g*cos(e+f*x)))+(a^2-b^2)*g^2*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(e+f*x))/(b*f*(a^2-b*(b-sqrt(-a^2+b^2)))*sqrt(g*cos(e+f*x)))+(a^2-b^2)*g^2*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(e+f*x))/(b*f*(a^2-b*(b+sqrt(-a^2+b^2)))*sqrt(g*cos(e+f*x)))],
[(g*cos(e+f*x))^(3/2)*csc(e+f*x)^2/(a+b*sin(e+f*x)),x,24,b*g^(3/2)*arctan(sqrt(g*cos(e+f*x))/sqrt(g))/(a^2*f)+b*g^(3/2)*arctanh(sqrt(g*cos(e+f*x))/sqrt(g))/(a^2*f)-(-a^2+b^2)^(1/4)*g^(3/2)*arctan(sqrt(b)*sqrt(g*cos(e+f*x))/((-a^2+b^2)^(1/4)*sqrt(g)))*sqrt(b)/(a^2*f)-(-a^2+b^2)^(1/4)*g^(3/2)*arctanh(sqrt(b)*sqrt(g*cos(e+f*x))/((-a^2+b^2)^(1/4)*sqrt(g)))*sqrt(b)/(a^2*f)+g^2*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticF(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))/(a*f*sqrt(g*cos(e+f*x)))-(a^2-b^2)*g^2*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(e+f*x))/(a*f*(a^2-b*(b-sqrt(-a^2+b^2)))*sqrt(g*cos(e+f*x)))-(a^2-b^2)*g^2*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(e+f*x))/(a*f*(a^2-b*(b+sqrt(-a^2+b^2)))*sqrt(g*cos(e+f*x)))-g*csc(e+f*x)*sqrt(g*cos(e+f*x))/(a*f)],
[(g*cos(e+f*x))^(3/2)*csc(e+f*x)^3/(a+b*sin(e+f*x)),x,30,1/4*g^(3/2)*arctan(sqrt(g*cos(e+f*x))/sqrt(g))/(a*f)-b^2*g^(3/2)*arctan(sqrt(g*cos(e+f*x))/sqrt(g))/(a^3*f)+b^(3/2)*(-a^2+b^2)^(1/4)*g^(3/2)*arctan(sqrt(b)*sqrt(g*cos(e+f*x))/((-a^2+b^2)^(1/4)*sqrt(g)))/(a^3*f)+1/4*g^(3/2)*arctanh(sqrt(g*cos(e+f*x))/sqrt(g))/(a*f)-b^2*g^(3/2)*arctanh(sqrt(g*cos(e+f*x))/sqrt(g))/(a^3*f)+b^(3/2)*(-a^2+b^2)^(1/4)*g^(3/2)*arctanh(sqrt(b)*sqrt(g*cos(e+f*x))/((-a^2+b^2)^(1/4)*sqrt(g)))/(a^3*f)-b*g^2*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticF(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))/(a^2*f*sqrt(g*cos(e+f*x)))+b*(a^2-b^2)*g^2*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(e+f*x))/(a^2*f*(a^2-b*(b-sqrt(-a^2+b^2)))*sqrt(g*cos(e+f*x)))+b*(a^2-b^2)*g^2*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(e+f*x))/(a^2*f*(a^2-b*(b+sqrt(-a^2+b^2)))*sqrt(g*cos(e+f*x)))+b*g*csc(e+f*x)*sqrt(g*cos(e+f*x))/(a^2*f)-1/2*g*csc(e+f*x)^2*sqrt(g*cos(e+f*x))/(a*f)],
[(g*cos(e+f*x))^(5/2)*sin(e+f*x)^3/(a+b*sin(e+f*x)),x,24,-a^3*(-a^2+b^2)^(3/4)*g^(5/2)*arctan(sqrt(b)*sqrt(g*cos(e+f*x))/((-a^2+b^2)^(1/4)*sqrt(g)))/(b^(11/2)*f)+a^3*(-a^2+b^2)^(3/4)*g^(5/2)*arctanh(sqrt(b)*sqrt(g*cos(e+f*x))/((-a^2+b^2)^(1/4)*sqrt(g)))/(b^(11/2)*f)-2/3*a^3*g*(g*cos(e+f*x))^(3/2)/(b^4*f)+2/7*a*(g*cos(e+f*x))^(7/2)/(b^2*f*g)+2/5*a^2*g*(g*cos(e+f*x))^(3/2)*sin(e+f*x)/(b^3*f)+4/45*g*(g*cos(e+f*x))^(3/2)*sin(e+f*x)/(b*f)-2/9*(g*cos(e+f*x))^(7/2)*sin(e+f*x)/(b*f*g)+a^4*(a^2-b^2)*g^3*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(e+f*x))/(b^6*f*(b-sqrt(-a^2+b^2))*sqrt(g*cos(e+f*x)))+a^4*(a^2-b^2)*g^3*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(e+f*x))/(b^6*f*(b+sqrt(-a^2+b^2))*sqrt(g*cos(e+f*x)))-2*a^4*g^2*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(g*cos(e+f*x))/(b^5*f*sqrt(cos(e+f*x)))+6/5*a^2*g^2*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(g*cos(e+f*x))/(b^3*f*sqrt(cos(e+f*x)))+4/15*g^2*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(g*cos(e+f*x))/(b*f*sqrt(cos(e+f*x)))],
[(g*cos(e+f*x))^(5/2)*sin(e+f*x)^2/(a+b*sin(e+f*x)),x,20,a^2*(-a^2+b^2)^(3/4)*g^(5/2)*arctan(sqrt(b)*sqrt(g*cos(e+f*x))/((-a^2+b^2)^(1/4)*sqrt(g)))/(b^(9/2)*f)-a^2*(-a^2+b^2)^(3/4)*g^(5/2)*arctanh(sqrt(b)*sqrt(g*cos(e+f*x))/((-a^2+b^2)^(1/4)*sqrt(g)))/(b^(9/2)*f)+2/3*a^2*g*(g*cos(e+f*x))^(3/2)/(b^3*f)-2/7*(g*cos(e+f*x))^(7/2)/(b*f*g)-2/5*a*g*(g*cos(e+f*x))^(3/2)*sin(e+f*x)/(b^2*f)-a^3*(a^2-b^2)*g^3*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(e+f*x))/(b^5*f*(b-sqrt(-a^2+b^2))*sqrt(g*cos(e+f*x)))-a^3*(a^2-b^2)*g^3*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(e+f*x))/(b^5*f*(b+sqrt(-a^2+b^2))*sqrt(g*cos(e+f*x)))+2*a^3*g^2*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(g*cos(e+f*x))/(b^4*f*sqrt(cos(e+f*x)))-6/5*a*g^2*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(g*cos(e+f*x))/(b^2*f*sqrt(cos(e+f*x)))],
[(g*cos(e+f*x))^(5/2)*sin(e+f*x)/(a+b*sin(e+f*x)),x,13,-a*(-a^2+b^2)^(3/4)*g^(5/2)*arctan(sqrt(b)*sqrt(g*cos(e+f*x))/((-a^2+b^2)^(1/4)*sqrt(g)))/(b^(7/2)*f)+a*(-a^2+b^2)^(3/4)*g^(5/2)*arctanh(sqrt(b)*sqrt(g*cos(e+f*x))/((-a^2+b^2)^(1/4)*sqrt(g)))/(b^(7/2)*f)-2/15*g*(g*cos(e+f*x))^(3/2)*(5*a-3*b*sin(e+f*x))/(b^2*f)+a^2*(a^2-b^2)*g^3*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(e+f*x))/(b^4*f*(b-sqrt(-a^2+b^2))*sqrt(g*cos(e+f*x)))+a^2*(a^2-b^2)*g^3*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(e+f*x))/(b^4*f*(b+sqrt(-a^2+b^2))*sqrt(g*cos(e+f*x)))-2/5*(5*a^2-3*b^2)*g^2*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(g*cos(e+f*x))/(b^3*f*sqrt(cos(e+f*x)))],
[(g*cos(e+f*x))^(5/2)*csc(e+f*x)/(a+b*sin(e+f*x)),x,21,g^(5/2)*arctan(sqrt(g*cos(e+f*x))/sqrt(g))/(a*f)-(-a^2+b^2)^(3/4)*g^(5/2)*arctan(sqrt(b)*sqrt(g*cos(e+f*x))/((-a^2+b^2)^(1/4)*sqrt(g)))/(a*b^(3/2)*f)-g^(5/2)*arctanh(sqrt(g*cos(e+f*x))/sqrt(g))/(a*f)+(-a^2+b^2)^(3/4)*g^(5/2)*arctanh(sqrt(b)*sqrt(g*cos(e+f*x))/((-a^2+b^2)^(1/4)*sqrt(g)))/(a*b^(3/2)*f)+(a^2-b^2)*g^3*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(e+f*x))/(b^2*f*(b-sqrt(-a^2+b^2))*sqrt(g*cos(e+f*x)))+(a^2-b^2)*g^3*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(e+f*x))/(b^2*f*(b+sqrt(-a^2+b^2))*sqrt(g*cos(e+f*x)))-2*g^2*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(g*cos(e+f*x))/(b*f*sqrt(cos(e+f*x)))],
[(g*cos(e+f*x))^(5/2)*csc(e+f*x)^2/(a+b*sin(e+f*x)),x,24,-b*g^(5/2)*arctan(sqrt(g*cos(e+f*x))/sqrt(g))/(a^2*f)+b*g^(5/2)*arctanh(sqrt(g*cos(e+f*x))/sqrt(g))/(a^2*f)-g*(g*cos(e+f*x))^(3/2)*csc(e+f*x)/(a*f)+(-a^2+b^2)^(3/4)*g^(5/2)*arctan(sqrt(b)*sqrt(g*cos(e+f*x))/((-a^2+b^2)^(1/4)*sqrt(g)))/(a^2*f*sqrt(b))-(-a^2+b^2)^(3/4)*g^(5/2)*arctanh(sqrt(b)*sqrt(g*cos(e+f*x))/((-a^2+b^2)^(1/4)*sqrt(g)))/(a^2*f*sqrt(b))-(a^2-b^2)*g^3*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(e+f*x))/(a*b*f*(b-sqrt(-a^2+b^2))*sqrt(g*cos(e+f*x)))-(a^2-b^2)*g^3*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(e+f*x))/(a*b*f*(b+sqrt(-a^2+b^2))*sqrt(g*cos(e+f*x)))-g^2*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(g*cos(e+f*x))/(a*f*sqrt(cos(e+f*x)))],
[(g*cos(e+f*x))^(5/2)*csc(e+f*x)^3/(a+b*sin(e+f*x)),x,30,-3/4*g^(5/2)*arctan(sqrt(g*cos(e+f*x))/sqrt(g))/(a*f)+b^2*g^(5/2)*arctan(sqrt(g*cos(e+f*x))/sqrt(g))/(a^3*f)+3/4*g^(5/2)*arctanh(sqrt(g*cos(e+f*x))/sqrt(g))/(a*f)-b^2*g^(5/2)*arctanh(sqrt(g*cos(e+f*x))/sqrt(g))/(a^3*f)+b*g*(g*cos(e+f*x))^(3/2)*csc(e+f*x)/(a^2*f)-1/2*g*(g*cos(e+f*x))^(3/2)*csc(e+f*x)^2/(a*f)-(-a^2+b^2)^(3/4)*g^(5/2)*arctan(sqrt(b)*sqrt(g*cos(e+f*x))/((-a^2+b^2)^(1/4)*sqrt(g)))*sqrt(b)/(a^3*f)+(-a^2+b^2)^(3/4)*g^(5/2)*arctanh(sqrt(b)*sqrt(g*cos(e+f*x))/((-a^2+b^2)^(1/4)*sqrt(g)))*sqrt(b)/(a^3*f)+(a^2-b^2)*g^3*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(e+f*x))/(a^2*f*(b-sqrt(-a^2+b^2))*sqrt(g*cos(e+f*x)))+(a^2-b^2)*g^3*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(e+f*x))/(a^2*f*(b+sqrt(-a^2+b^2))*sqrt(g*cos(e+f*x)))+b*g^2*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(g*cos(e+f*x))/(a^2*f*sqrt(cos(e+f*x)))],

# p<0
[sin(e+f*x)^4/((a+b*sin(e+f*x))*sqrt(g*cos(e+f*x))),x,23,2/5*(g*cos(e+f*x))^(5/2)/(b*f*g^3)-a^4*arctan(sqrt(b)*sqrt(g*cos(e+f*x))/((-a^2+b^2)^(1/4)*sqrt(g)))/(b^(7/2)*(-a^2+b^2)^(3/4)*f*sqrt(g))-a^4*arctanh(sqrt(b)*sqrt(g*cos(e+f*x))/((-a^2+b^2)^(1/4)*sqrt(g)))/(b^(7/2)*(-a^2+b^2)^(3/4)*f*sqrt(g))-2*a^3*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticF(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))/(b^4*f*sqrt(g*cos(e+f*x)))-4/3*a*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticF(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))/(b^2*f*sqrt(g*cos(e+f*x)))+a^5*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(e+f*x))/(b^4*f*(a^2-b*(b-sqrt(-a^2+b^2)))*sqrt(g*cos(e+f*x)))+a^5*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(e+f*x))/(b^4*f*(a^2-b*(b+sqrt(-a^2+b^2)))*sqrt(g*cos(e+f*x)))-2*a^2*sqrt(g*cos(e+f*x))/(b^3*f*g)-2*sqrt(g*cos(e+f*x))/(b*f*g)+2/3*a*sin(e+f*x)*sqrt(g*cos(e+f*x))/(b^2*f*g)],
[sin(e+f*x)^3/((a+b*sin(e+f*x))*sqrt(g*cos(e+f*x))),x,19,a^3*arctan(sqrt(b)*sqrt(g*cos(e+f*x))/((-a^2+b^2)^(1/4)*sqrt(g)))/(b^(5/2)*(-a^2+b^2)^(3/4)*f*sqrt(g))+a^3*arctanh(sqrt(b)*sqrt(g*cos(e+f*x))/((-a^2+b^2)^(1/4)*sqrt(g)))/(b^(5/2)*(-a^2+b^2)^(3/4)*f*sqrt(g))+2*a^2*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticF(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))/(b^3*f*sqrt(g*cos(e+f*x)))+4/3*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticF(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))/(b*f*sqrt(g*cos(e+f*x)))-a^4*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(e+f*x))/(b^3*f*(a^2-b*(b-sqrt(-a^2+b^2)))*sqrt(g*cos(e+f*x)))-a^4*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(e+f*x))/(b^3*f*(a^2-b*(b+sqrt(-a^2+b^2)))*sqrt(g*cos(e+f*x)))+2*a*sqrt(g*cos(e+f*x))/(b^2*f*g)-2/3*sin(e+f*x)*sqrt(g*cos(e+f*x))/(b*f*g)],
[sin(e+f*x)^2/((a+b*sin(e+f*x))*sqrt(g*cos(e+f*x))),x,15,-a^2*arctan(sqrt(b)*sqrt(g*cos(e+f*x))/((-a^2+b^2)^(1/4)*sqrt(g)))/(b^(3/2)*(-a^2+b^2)^(3/4)*f*sqrt(g))-a^2*arctanh(sqrt(b)*sqrt(g*cos(e+f*x))/((-a^2+b^2)^(1/4)*sqrt(g)))/(b^(3/2)*(-a^2+b^2)^(3/4)*f*sqrt(g))-2*a*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticF(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))/(b^2*f*sqrt(g*cos(e+f*x)))+a^3*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(e+f*x))/(b^2*f*(a^2-b*(b-sqrt(-a^2+b^2)))*sqrt(g*cos(e+f*x)))+a^3*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(e+f*x))/(b^2*f*(a^2-b*(b+sqrt(-a^2+b^2)))*sqrt(g*cos(e+f*x)))-2*sqrt(g*cos(e+f*x))/(b*f*g)],
[sin(e+f*x)/((a+b*sin(e+f*x))*sqrt(g*cos(e+f*x))),x,12,a*arctan(sqrt(b)*sqrt(g*cos(e+f*x))/((-a^2+b^2)^(1/4)*sqrt(g)))/((-a^2+b^2)^(3/4)*f*sqrt(b)*sqrt(g))+a*arctanh(sqrt(b)*sqrt(g*cos(e+f*x))/((-a^2+b^2)^(1/4)*sqrt(g)))/((-a^2+b^2)^(3/4)*f*sqrt(b)*sqrt(g))+2*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticF(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))/(b*f*sqrt(g*cos(e+f*x)))-a^2*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(e+f*x))/(b*f*(a^2-b^2+b*sqrt(-a^2+b^2))*sqrt(g*cos(e+f*x)))-a^2*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(e+f*x))/(b*f*(a^2-b*(b+sqrt(-a^2+b^2)))*sqrt(g*cos(e+f*x)))],
[csc(e+f*x)/((a+b*sin(e+f*x))*sqrt(g*cos(e+f*x))),x,16,-arctan(sqrt(g*cos(e+f*x))/sqrt(g))/(a*f*sqrt(g))+b^(3/2)*arctan(sqrt(b)*sqrt(g*cos(e+f*x))/((-a^2+b^2)^(1/4)*sqrt(g)))/(a*(-a^2+b^2)^(3/4)*f*sqrt(g))-arctanh(sqrt(g*cos(e+f*x))/sqrt(g))/(a*f*sqrt(g))+b^(3/2)*arctanh(sqrt(b)*sqrt(g*cos(e+f*x))/((-a^2+b^2)^(1/4)*sqrt(g)))/(a*(-a^2+b^2)^(3/4)*f*sqrt(g))-b*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(e+f*x))/(f*(a^2-b*(b-sqrt(-a^2+b^2)))*sqrt(g*cos(e+f*x)))-b*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(e+f*x))/(f*(a^2-b*(b+sqrt(-a^2+b^2)))*sqrt(g*cos(e+f*x)))],
[csc(e+f*x)^2/((a+b*sin(e+f*x))*sqrt(g*cos(e+f*x))),x,19,b*arctan(sqrt(g*cos(e+f*x))/sqrt(g))/(a^2*f*sqrt(g))-b^(5/2)*arctan(sqrt(b)*sqrt(g*cos(e+f*x))/((-a^2+b^2)^(1/4)*sqrt(g)))/(a^2*(-a^2+b^2)^(3/4)*f*sqrt(g))+b*arctanh(sqrt(g*cos(e+f*x))/sqrt(g))/(a^2*f*sqrt(g))-b^(5/2)*arctanh(sqrt(b)*sqrt(g*cos(e+f*x))/((-a^2+b^2)^(1/4)*sqrt(g)))/(a^2*(-a^2+b^2)^(3/4)*f*sqrt(g))+sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticF(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))/(a*f*sqrt(g*cos(e+f*x)))+b^2*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(e+f*x))/(a*f*(a^2-b^2+b*sqrt(-a^2+b^2))*sqrt(g*cos(e+f*x)))+b^2*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(e+f*x))/(a*f*(a^2-b*(b+sqrt(-a^2+b^2)))*sqrt(g*cos(e+f*x)))-csc(e+f*x)*sqrt(g*cos(e+f*x))/(a*f*g)],
[csc(e+f*x)^3/((a+b*sin(e+f*x))*sqrt(g*cos(e+f*x))),x,25,-3/4*arctan(sqrt(g*cos(e+f*x))/sqrt(g))/(a*f*sqrt(g))-b^2*arctan(sqrt(g*cos(e+f*x))/sqrt(g))/(a^3*f*sqrt(g))+b^(7/2)*arctan(sqrt(b)*sqrt(g*cos(e+f*x))/((-a^2+b^2)^(1/4)*sqrt(g)))/(a^3*(-a^2+b^2)^(3/4)*f*sqrt(g))-3/4*arctanh(sqrt(g*cos(e+f*x))/sqrt(g))/(a*f*sqrt(g))-b^2*arctanh(sqrt(g*cos(e+f*x))/sqrt(g))/(a^3*f*sqrt(g))+b^(7/2)*arctanh(sqrt(b)*sqrt(g*cos(e+f*x))/((-a^2+b^2)^(1/4)*sqrt(g)))/(a^3*(-a^2+b^2)^(3/4)*f*sqrt(g))-b*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticF(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))/(a^2*f*sqrt(g*cos(e+f*x)))-b^3*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(e+f*x))/(a^2*f*(a^2-b*(b-sqrt(-a^2+b^2)))*sqrt(g*cos(e+f*x)))-b^3*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(e+f*x))/(a^2*f*(a^2-b*(b+sqrt(-a^2+b^2)))*sqrt(g*cos(e+f*x)))+b*csc(e+f*x)*sqrt(g*cos(e+f*x))/(a^2*f*g)-1/2*csc(e+f*x)^2*sqrt(g*cos(e+f*x))/(a*f*g)],
[sin(e+f*x)^4/((g*cos(e+f*x))^(3/2)*(a+b*sin(e+f*x))),x,22,a^4*arctan(sqrt(b)*sqrt(g*cos(e+f*x))/((-a^2+b^2)^(1/4)*sqrt(g)))/(b^(5/2)*(-a^2+b^2)^(5/4)*f*g^(3/2))-a^4*arctanh(sqrt(b)*sqrt(g*cos(e+f*x))/((-a^2+b^2)^(1/4)*sqrt(g)))/(b^(5/2)*(-a^2+b^2)^(5/4)*f*g^(3/2))+2/3*a^2*(g*cos(e+f*x))^(3/2)/(b*(a^2-b^2)*f*g^3)-2/3*b*(g*cos(e+f*x))^(3/2)/((a^2-b^2)*f*g^3)-2*b/((a^2-b^2)*f*g*sqrt(g*cos(e+f*x)))+2*a*sin(e+f*x)/((a^2-b^2)*f*g*sqrt(g*cos(e+f*x)))-a^5*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(e+f*x))/(b^3*(a^2-b^2)*f*g*(b-sqrt(-a^2+b^2))*sqrt(g*cos(e+f*x)))-a^5*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(e+f*x))/(b^3*(a^2-b^2)*f*g*(b+sqrt(-a^2+b^2))*sqrt(g*cos(e+f*x)))-4*a*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(g*cos(e+f*x))/((a^2-b^2)*f*g^2*sqrt(cos(e+f*x)))+2*a^3*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(g*cos(e+f*x))/(b^2*(a^2-b^2)*f*g^2*sqrt(cos(e+f*x)))],
[sin(e+f*x)^3/((g*cos(e+f*x))^(3/2)*(a+b*sin(e+f*x))),x,18,-a^3*arctan(sqrt(b)*sqrt(g*cos(e+f*x))/((-a^2+b^2)^(1/4)*sqrt(g)))/(b^(3/2)*(-a^2+b^2)^(5/4)*f*g^(3/2))+a^3*arctanh(sqrt(b)*sqrt(g*cos(e+f*x))/((-a^2+b^2)^(1/4)*sqrt(g)))/(b^(3/2)*(-a^2+b^2)^(5/4)*f*g^(3/2))+2*a/((a^2-b^2)*f*g*sqrt(g*cos(e+f*x)))-2*b*sin(e+f*x)/((a^2-b^2)*f*g*sqrt(g*cos(e+f*x)))+a^4*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(e+f*x))/(b^2*(a^2-b^2)*f*g*(b-sqrt(-a^2+b^2))*sqrt(g*cos(e+f*x)))+a^4*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(e+f*x))/(b^2*(a^2-b^2)*f*g*(b+sqrt(-a^2+b^2))*sqrt(g*cos(e+f*x)))-2*a^2*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(g*cos(e+f*x))/(b*(a^2-b^2)*f*g^2*sqrt(cos(e+f*x)))+4*b*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(g*cos(e+f*x))/((a^2-b^2)*f*g^2*sqrt(cos(e+f*x)))],
[sin(e+f*x)^2/((g*cos(e+f*x))^(3/2)*(a+b*sin(e+f*x))),x,15,a^2*arctan(sqrt(b)*sqrt(g*cos(e+f*x))/((-a^2+b^2)^(1/4)*sqrt(g)))/((-a^2+b^2)^(5/4)*f*g^(3/2)*sqrt(b))-a^2*arctanh(sqrt(b)*sqrt(g*cos(e+f*x))/((-a^2+b^2)^(1/4)*sqrt(g)))/((-a^2+b^2)^(5/4)*f*g^(3/2)*sqrt(b))-2*b/((a^2-b^2)*f*g*sqrt(g*cos(e+f*x)))+2*a*sin(e+f*x)/((a^2-b^2)*f*g*sqrt(g*cos(e+f*x)))-a^3*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(e+f*x))/(b*(a^2-b^2)*f*g*(b-sqrt(-a^2+b^2))*sqrt(g*cos(e+f*x)))-a^3*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(e+f*x))/(b*(a^2-b^2)*f*g*(b+sqrt(-a^2+b^2))*sqrt(g*cos(e+f*x)))-2*a*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(g*cos(e+f*x))/((a^2-b^2)*f*g^2*sqrt(cos(e+f*x)))],
[sin(e+f*x)/((g*cos(e+f*x))^(3/2)*(a+b*sin(e+f*x))),x,13,-a*arctan(sqrt(b)*sqrt(g*cos(e+f*x))/((-a^2+b^2)^(1/4)*sqrt(g)))*sqrt(b)/((-a^2+b^2)^(5/4)*f*g^(3/2))+a*arctanh(sqrt(b)*sqrt(g*cos(e+f*x))/((-a^2+b^2)^(1/4)*sqrt(g)))*sqrt(b)/((-a^2+b^2)^(5/4)*f*g^(3/2))+2*(a-b*sin(e+f*x))/((a^2-b^2)*f*g*sqrt(g*cos(e+f*x)))+a^2*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(e+f*x))/((a^2-b^2)*f*g*(b-sqrt(-a^2+b^2))*sqrt(g*cos(e+f*x)))+a^2*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(e+f*x))/((a^2-b^2)*f*g*(b+sqrt(-a^2+b^2))*sqrt(g*cos(e+f*x)))+2*b*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(g*cos(e+f*x))/((a^2-b^2)*f*g^2*sqrt(cos(e+f*x)))],
[csc(e+f*x)/((g*cos(e+f*x))^(3/2)*(a+b*sin(e+f*x))),x,21,arctan(sqrt(g*cos(e+f*x))/sqrt(g))/(a*f*g^(3/2))-b^(5/2)*arctan(sqrt(b)*sqrt(g*cos(e+f*x))/((-a^2+b^2)^(1/4)*sqrt(g)))/(a*(-a^2+b^2)^(5/4)*f*g^(3/2))-arctanh(sqrt(g*cos(e+f*x))/sqrt(g))/(a*f*g^(3/2))+b^(5/2)*arctanh(sqrt(b)*sqrt(g*cos(e+f*x))/((-a^2+b^2)^(1/4)*sqrt(g)))/(a*(-a^2+b^2)^(5/4)*f*g^(3/2))+2/(a*f*g*sqrt(g*cos(e+f*x)))+2*b*(b-a*sin(e+f*x))/(a*(a^2-b^2)*f*g*sqrt(g*cos(e+f*x)))+b^2*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(e+f*x))/((a^2-b^2)*f*g*(b-sqrt(-a^2+b^2))*sqrt(g*cos(e+f*x)))+b^2*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(e+f*x))/((a^2-b^2)*f*g*(b+sqrt(-a^2+b^2))*sqrt(g*cos(e+f*x)))+2*b*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(g*cos(e+f*x))/((a^2-b^2)*f*g^2*sqrt(cos(e+f*x)))],
[csc(e+f*x)^2/((g*cos(e+f*x))^(3/2)*(a+b*sin(e+f*x))),x,25,-b*arctan(sqrt(g*cos(e+f*x))/sqrt(g))/(a^2*f*g^(3/2))+b^(7/2)*arctan(sqrt(b)*sqrt(g*cos(e+f*x))/((-a^2+b^2)^(1/4)*sqrt(g)))/(a^2*(-a^2+b^2)^(5/4)*f*g^(3/2))+b*arctanh(sqrt(g*cos(e+f*x))/sqrt(g))/(a^2*f*g^(3/2))-b^(7/2)*arctanh(sqrt(b)*sqrt(g*cos(e+f*x))/((-a^2+b^2)^(1/4)*sqrt(g)))/(a^2*(-a^2+b^2)^(5/4)*f*g^(3/2))-2*b/(a^2*f*g*sqrt(g*cos(e+f*x)))-csc(e+f*x)/(a*f*g*sqrt(g*cos(e+f*x)))+3*sin(e+f*x)/(a*f*g*sqrt(g*cos(e+f*x)))-2*b^2*(b-a*sin(e+f*x))/(a^2*(a^2-b^2)*f*g*sqrt(g*cos(e+f*x)))-b^3*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(e+f*x))/(a*(a^2-b^2)*f*g*(b-sqrt(-a^2+b^2))*sqrt(g*cos(e+f*x)))-b^3*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(e+f*x))/(a*(a^2-b^2)*f*g*(b+sqrt(-a^2+b^2))*sqrt(g*cos(e+f*x)))-3*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(g*cos(e+f*x))/(a*f*g^2*sqrt(cos(e+f*x)))-2*b^2*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2))*sqrt(g*cos(e+f*x))/(a*(a^2-b^2)*f*g^2*sqrt(cos(e+f*x)))],
[sin(e+f*x)^4/((g*cos(e+f*x))^(5/2)*(a+b*sin(e+f*x))),x,22,-a^4*arctan(sqrt(b)*sqrt(g*cos(e+f*x))/((-a^2+b^2)^(1/4)*sqrt(g)))/(b^(3/2)*(-a^2+b^2)^(7/4)*f*g^(5/2))-a^4*arctanh(sqrt(b)*sqrt(g*cos(e+f*x))/((-a^2+b^2)^(1/4)*sqrt(g)))/(b^(3/2)*(-a^2+b^2)^(7/4)*f*g^(5/2))-2/3*b/((a^2-b^2)*f*g*(g*cos(e+f*x))^(3/2))+2/3*a*sin(e+f*x)/((a^2-b^2)*f*g*(g*cos(e+f*x))^(3/2))-4/3*a*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticF(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))/((a^2-b^2)*f*g^2*sqrt(g*cos(e+f*x)))+2*a^3*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticF(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))/(b^2*(a^2-b^2)*f*g^2*sqrt(g*cos(e+f*x)))-a^5*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(e+f*x))/(b^2*(a^2-b^2)*f*g^2*(a^2-b*(b-sqrt(-a^2+b^2)))*sqrt(g*cos(e+f*x)))-a^5*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(e+f*x))/(b^2*(a^2-b^2)*f*g^2*(a^2-b*(b+sqrt(-a^2+b^2)))*sqrt(g*cos(e+f*x)))+2*a^2*sqrt(g*cos(e+f*x))/(b*(a^2-b^2)*f*g^3)-2*b*sqrt(g*cos(e+f*x))/((a^2-b^2)*f*g^3)],
[sin(e+f*x)^3/((g*cos(e+f*x))^(5/2)*(a+b*sin(e+f*x))),x,18,2/3*a/((a^2-b^2)*f*g*(g*cos(e+f*x))^(3/2))-2/3*b*sin(e+f*x)/((a^2-b^2)*f*g*(g*cos(e+f*x))^(3/2))+a^3*arctan(sqrt(b)*sqrt(g*cos(e+f*x))/((-a^2+b^2)^(1/4)*sqrt(g)))/((-a^2+b^2)^(7/4)*f*g^(5/2)*sqrt(b))+a^3*arctanh(sqrt(b)*sqrt(g*cos(e+f*x))/((-a^2+b^2)^(1/4)*sqrt(g)))/((-a^2+b^2)^(7/4)*f*g^(5/2)*sqrt(b))-2*a^2*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticF(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))/(b*(a^2-b^2)*f*g^2*sqrt(g*cos(e+f*x)))+4/3*b*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticF(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))/((a^2-b^2)*f*g^2*sqrt(g*cos(e+f*x)))+a^4*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(e+f*x))/(b*(a^2-b^2)*f*g^2*(a^2-b*(b-sqrt(-a^2+b^2)))*sqrt(g*cos(e+f*x)))+a^4*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(e+f*x))/(b*(a^2-b^2)*f*g^2*(a^2-b*(b+sqrt(-a^2+b^2)))*sqrt(g*cos(e+f*x)))],
[sin(e+f*x)^2/((g*cos(e+f*x))^(5/2)*(a+b*sin(e+f*x))),x,15,-2/3*b/((a^2-b^2)*f*g*(g*cos(e+f*x))^(3/2))+2/3*a*sin(e+f*x)/((a^2-b^2)*f*g*(g*cos(e+f*x))^(3/2))-a^2*arctan(sqrt(b)*sqrt(g*cos(e+f*x))/((-a^2+b^2)^(1/4)*sqrt(g)))*sqrt(b)/((-a^2+b^2)^(7/4)*f*g^(5/2))-a^2*arctanh(sqrt(b)*sqrt(g*cos(e+f*x))/((-a^2+b^2)^(1/4)*sqrt(g)))*sqrt(b)/((-a^2+b^2)^(7/4)*f*g^(5/2))+2/3*a*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticF(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))/((a^2-b^2)*f*g^2*sqrt(g*cos(e+f*x)))-a^3*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(e+f*x))/((a^2-b^2)*f*g^2*(a^2-b*(b-sqrt(-a^2+b^2)))*sqrt(g*cos(e+f*x)))-a^3*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(e+f*x))/((a^2-b^2)*f*g^2*(a^2-b*(b+sqrt(-a^2+b^2)))*sqrt(g*cos(e+f*x)))],
[sin(e+f*x)/((g*cos(e+f*x))^(5/2)*(a+b*sin(e+f*x))),x,13,a*b^(3/2)*arctan(sqrt(b)*sqrt(g*cos(e+f*x))/((-a^2+b^2)^(1/4)*sqrt(g)))/((-a^2+b^2)^(7/4)*f*g^(5/2))+a*b^(3/2)*arctanh(sqrt(b)*sqrt(g*cos(e+f*x))/((-a^2+b^2)^(1/4)*sqrt(g)))/((-a^2+b^2)^(7/4)*f*g^(5/2))+2/3*(a-b*sin(e+f*x))/((a^2-b^2)*f*g*(g*cos(e+f*x))^(3/2))-2/3*b*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticF(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))/((a^2-b^2)*f*g^2*sqrt(g*cos(e+f*x)))+a^2*b*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(e+f*x))/((a^2-b^2)*f*g^2*(a^2-b*(b-sqrt(-a^2+b^2)))*sqrt(g*cos(e+f*x)))+a^2*b*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(e+f*x))/((a^2-b^2)*f*g^2*(a^2-b*(b+sqrt(-a^2+b^2)))*sqrt(g*cos(e+f*x)))],
[csc(e+f*x)/((g*cos(e+f*x))^(5/2)*(a+b*sin(e+f*x))),x,21,-arctan(sqrt(g*cos(e+f*x))/sqrt(g))/(a*f*g^(5/2))+b^(7/2)*arctan(sqrt(b)*sqrt(g*cos(e+f*x))/((-a^2+b^2)^(1/4)*sqrt(g)))/(a*(-a^2+b^2)^(7/4)*f*g^(5/2))-arctanh(sqrt(g*cos(e+f*x))/sqrt(g))/(a*f*g^(5/2))+b^(7/2)*arctanh(sqrt(b)*sqrt(g*cos(e+f*x))/((-a^2+b^2)^(1/4)*sqrt(g)))/(a*(-a^2+b^2)^(7/4)*f*g^(5/2))+2/3/(a*f*g*(g*cos(e+f*x))^(3/2))+2/3*b*(b-a*sin(e+f*x))/(a*(a^2-b^2)*f*g*(g*cos(e+f*x))^(3/2))-2/3*b*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticF(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))/((a^2-b^2)*f*g^2*sqrt(g*cos(e+f*x)))+b^3*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(e+f*x))/((a^2-b^2)*f*g^2*(a^2-b*(b-sqrt(-a^2+b^2)))*sqrt(g*cos(e+f*x)))+b^3*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(e+f*x))/((a^2-b^2)*f*g^2*(a^2-b*(b+sqrt(-a^2+b^2)))*sqrt(g*cos(e+f*x)))],
[csc(e+f*x)^2/((g*cos(e+f*x))^(5/2)*(a+b*sin(e+f*x))),x,25,b*arctan(sqrt(g*cos(e+f*x))/sqrt(g))/(a^2*f*g^(5/2))-b^(9/2)*arctan(sqrt(b)*sqrt(g*cos(e+f*x))/((-a^2+b^2)^(1/4)*sqrt(g)))/(a^2*(-a^2+b^2)^(7/4)*f*g^(5/2))+b*arctanh(sqrt(g*cos(e+f*x))/sqrt(g))/(a^2*f*g^(5/2))-b^(9/2)*arctanh(sqrt(b)*sqrt(g*cos(e+f*x))/((-a^2+b^2)^(1/4)*sqrt(g)))/(a^2*(-a^2+b^2)^(7/4)*f*g^(5/2))-2/3*b/(a^2*f*g*(g*cos(e+f*x))^(3/2))-csc(e+f*x)/(a*f*g*(g*cos(e+f*x))^(3/2))+5/3*sin(e+f*x)/(a*f*g*(g*cos(e+f*x))^(3/2))-2/3*b^2*(b-a*sin(e+f*x))/(a^2*(a^2-b^2)*f*g*(g*cos(e+f*x))^(3/2))+5/3*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticF(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))/(a*f*g^2*sqrt(g*cos(e+f*x)))+2/3*b^2*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticF(sin(1/2*(e+f*x)),sqrt(2))*sqrt(cos(e+f*x))/(a*(a^2-b^2)*f*g^2*sqrt(g*cos(e+f*x)))-b^4*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(e+f*x))/(a*(a^2-b^2)*f*g^2*(a^2-b*(b-sqrt(-a^2+b^2)))*sqrt(g*cos(e+f*x)))-b^4*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(e+f*x))/(a*(a^2-b^2)*f*g^2*(a^2-b*(b+sqrt(-a^2+b^2)))*sqrt(g*cos(e+f*x)))],

# Integrands of the form (g Cos[e+f x])^(p/2) (d Sin[e+f x])^(n/2) / (a+b Sin[e+f x])

# p>0
[(d*sin(e+f*x))^(5/2)*sqrt(g*cos(e+f*x))/(a+b*sin(e+f*x)),x,31,a^2*d^(5/2)*arctan(1-sqrt(2)*sqrt(d)*sqrt(g*cos(e+f*x))/(sqrt(g)*sqrt(d*sin(e+f*x))))*sqrt(g)/(b^3*f*sqrt(2))+1/4*d^(5/2)*arctan(1-sqrt(2)*sqrt(d)*sqrt(g*cos(e+f*x))/(sqrt(g)*sqrt(d*sin(e+f*x))))*sqrt(g)/(b*f*sqrt(2))-a^2*d^(5/2)*arctan(1+sqrt(2)*sqrt(d)*sqrt(g*cos(e+f*x))/(sqrt(g)*sqrt(d*sin(e+f*x))))*sqrt(g)/(b^3*f*sqrt(2))-1/4*d^(5/2)*arctan(1+sqrt(2)*sqrt(d)*sqrt(g*cos(e+f*x))/(sqrt(g)*sqrt(d*sin(e+f*x))))*sqrt(g)/(b*f*sqrt(2))-1/2*a^2*d^(5/2)*log(sqrt(g)+cot(e+f*x)*sqrt(g)-sqrt(2)*sqrt(d)*sqrt(g*cos(e+f*x))/sqrt(d*sin(e+f*x)))*sqrt(g)/(b^3*f*sqrt(2))-1/8*d^(5/2)*log(sqrt(g)+cot(e+f*x)*sqrt(g)-sqrt(2)*sqrt(d)*sqrt(g*cos(e+f*x))/sqrt(d*sin(e+f*x)))*sqrt(g)/(b*f*sqrt(2))+1/2*a^2*d^(5/2)*log(sqrt(g)+cot(e+f*x)*sqrt(g)+sqrt(2)*sqrt(d)*sqrt(g*cos(e+f*x))/sqrt(d*sin(e+f*x)))*sqrt(g)/(b^3*f*sqrt(2))+1/8*d^(5/2)*log(sqrt(g)+cot(e+f*x)*sqrt(g)+sqrt(2)*sqrt(d)*sqrt(g*cos(e+f*x))/sqrt(d*sin(e+f*x)))*sqrt(g)/(b*f*sqrt(2))-2*a^3*d^3*EllipticPi(sqrt(g*cos(e+f*x))/(sqrt(g)*sqrt(1+sin(e+f*x))),-sqrt(-a+b)/sqrt(a+b),I)*sqrt(2)*sqrt(g)*sqrt(sin(e+f*x))/(b^3*f*sqrt(-a+b)*sqrt(a+b)*sqrt(d*sin(e+f*x)))+2*a^3*d^3*EllipticPi(sqrt(g*cos(e+f*x))/(sqrt(g)*sqrt(1+sin(e+f*x))),sqrt(-a+b)/sqrt(a+b),I)*sqrt(2)*sqrt(g)*sqrt(sin(e+f*x))/(b^3*f*sqrt(-a+b)*sqrt(a+b)*sqrt(d*sin(e+f*x)))-1/2*d^2*(g*cos(e+f*x))^(3/2)*sqrt(d*sin(e+f*x))/(b*f*g)-a*d^2*sqrt(cos(e-1/4*Pi+f*x)^2)/cos(e-1/4*Pi+f*x)*EllipticE(sin(e-1/4*Pi+f*x),sqrt(2))*sqrt(g*cos(e+f*x))*sqrt(d*sin(e+f*x))/(b^2*f*sqrt(sin(2*e+2*f*x)))],
[(d*sin(e+f*x))^(3/2)*sqrt(g*cos(e+f*x))/(a+b*sin(e+f*x)),x,19,-a*d^(3/2)*arctan(1-sqrt(2)*sqrt(d)*sqrt(g*cos(e+f*x))/(sqrt(g)*sqrt(d*sin(e+f*x))))*sqrt(g)/(b^2*f*sqrt(2))+a*d^(3/2)*arctan(1+sqrt(2)*sqrt(d)*sqrt(g*cos(e+f*x))/(sqrt(g)*sqrt(d*sin(e+f*x))))*sqrt(g)/(b^2*f*sqrt(2))+1/2*a*d^(3/2)*log(sqrt(g)+cot(e+f*x)*sqrt(g)-sqrt(2)*sqrt(d)*sqrt(g*cos(e+f*x))/sqrt(d*sin(e+f*x)))*sqrt(g)/(b^2*f*sqrt(2))-1/2*a*d^(3/2)*log(sqrt(g)+cot(e+f*x)*sqrt(g)+sqrt(2)*sqrt(d)*sqrt(g*cos(e+f*x))/sqrt(d*sin(e+f*x)))*sqrt(g)/(b^2*f*sqrt(2))+2*a^2*d^2*EllipticPi(sqrt(g*cos(e+f*x))/(sqrt(g)*sqrt(1+sin(e+f*x))),-sqrt(-a+b)/sqrt(a+b),I)*sqrt(2)*sqrt(g)*sqrt(sin(e+f*x))/(b^2*f*sqrt(-a+b)*sqrt(a+b)*sqrt(d*sin(e+f*x)))-2*a^2*d^2*EllipticPi(sqrt(g*cos(e+f*x))/(sqrt(g)*sqrt(1+sin(e+f*x))),sqrt(-a+b)/sqrt(a+b),I)*sqrt(2)*sqrt(g)*sqrt(sin(e+f*x))/(b^2*f*sqrt(-a+b)*sqrt(a+b)*sqrt(d*sin(e+f*x)))+d*sqrt(cos(e-1/4*Pi+f*x)^2)/cos(e-1/4*Pi+f*x)*EllipticE(sin(e-1/4*Pi+f*x),sqrt(2))*sqrt(g*cos(e+f*x))*sqrt(d*sin(e+f*x))/(b*f*sqrt(sin(2*e+2*f*x)))],
[(d*sin(e+f*x))^(1/2)*sqrt(g*cos(e+f*x))/(a+b*sin(e+f*x)),x,16,arctan(1-sqrt(2)*sqrt(d)*sqrt(g*cos(e+f*x))/(sqrt(g)*sqrt(d*sin(e+f*x))))*sqrt(d)*sqrt(g)/(b*f*sqrt(2))-arctan(1+sqrt(2)*sqrt(d)*sqrt(g*cos(e+f*x))/(sqrt(g)*sqrt(d*sin(e+f*x))))*sqrt(d)*sqrt(g)/(b*f*sqrt(2))-1/2*log(sqrt(g)+cot(e+f*x)*sqrt(g)-sqrt(2)*sqrt(d)*sqrt(g*cos(e+f*x))/sqrt(d*sin(e+f*x)))*sqrt(d)*sqrt(g)/(b*f*sqrt(2))+1/2*log(sqrt(g)+cot(e+f*x)*sqrt(g)+sqrt(2)*sqrt(d)*sqrt(g*cos(e+f*x))/sqrt(d*sin(e+f*x)))*sqrt(d)*sqrt(g)/(b*f*sqrt(2))-2*a*d*EllipticPi(sqrt(g*cos(e+f*x))/(sqrt(g)*sqrt(1+sin(e+f*x))),-sqrt(-a+b)/sqrt(a+b),I)*sqrt(2)*sqrt(g)*sqrt(sin(e+f*x))/(b*f*sqrt(-a+b)*sqrt(a+b)*sqrt(d*sin(e+f*x)))+2*a*d*EllipticPi(sqrt(g*cos(e+f*x))/(sqrt(g)*sqrt(1+sin(e+f*x))),sqrt(-a+b)/sqrt(a+b),I)*sqrt(2)*sqrt(g)*sqrt(sin(e+f*x))/(b*f*sqrt(-a+b)*sqrt(a+b)*sqrt(d*sin(e+f*x)))],
[sqrt(g*cos(e+f*x))/((d*sin(e+f*x))^(1/2)*(a+b*sin(e+f*x))),x,5,2*EllipticPi(sqrt(g*cos(e+f*x))/(sqrt(g)*sqrt(1+sin(e+f*x))),-sqrt(-a+b)/sqrt(a+b),I)*sqrt(2)*sqrt(g)*sqrt(sin(e+f*x))/(f*sqrt(-a+b)*sqrt(a+b)*sqrt(d*sin(e+f*x)))-2*EllipticPi(sqrt(g*cos(e+f*x))/(sqrt(g)*sqrt(1+sin(e+f*x))),sqrt(-a+b)/sqrt(a+b),I)*sqrt(2)*sqrt(g)*sqrt(sin(e+f*x))/(f*sqrt(-a+b)*sqrt(a+b)*sqrt(d*sin(e+f*x)))],
[sqrt(g*cos(e+f*x))/((d*sin(e+f*x))^(3/2)*(a+b*sin(e+f*x))),x,9,-2*(g*cos(e+f*x))^(3/2)/(a*d*f*g*sqrt(d*sin(e+f*x)))-2*b*EllipticPi(sqrt(g*cos(e+f*x))/(sqrt(g)*sqrt(1+sin(e+f*x))),-sqrt(-a+b)/sqrt(a+b),I)*sqrt(2)*sqrt(g)*sqrt(sin(e+f*x))/(a*d*f*sqrt(-a+b)*sqrt(a+b)*sqrt(d*sin(e+f*x)))+2*b*EllipticPi(sqrt(g*cos(e+f*x))/(sqrt(g)*sqrt(1+sin(e+f*x))),sqrt(-a+b)/sqrt(a+b),I)*sqrt(2)*sqrt(g)*sqrt(sin(e+f*x))/(a*d*f*sqrt(-a+b)*sqrt(a+b)*sqrt(d*sin(e+f*x)))-2*sqrt(cos(e-1/4*Pi+f*x)^2)/cos(e-1/4*Pi+f*x)*EllipticE(sin(e-1/4*Pi+f*x),sqrt(2))*sqrt(g*cos(e+f*x))*sqrt(d*sin(e+f*x))/(a*d^2*f*sqrt(sin(2*e+2*f*x)))],
[sqrt(g*cos(e+f*x))/((d*sin(e+f*x))^(5/2)*(a+b*sin(e+f*x))),x,11,-2/3*(g*cos(e+f*x))^(3/2)/(a*d*f*g*(d*sin(e+f*x))^(3/2))+2*b*(g*cos(e+f*x))^(3/2)/(a^2*d^2*f*g*sqrt(d*sin(e+f*x)))+2*b^2*EllipticPi(sqrt(g*cos(e+f*x))/(sqrt(g)*sqrt(1+sin(e+f*x))),-sqrt(-a+b)/sqrt(a+b),I)*sqrt(2)*sqrt(g)*sqrt(sin(e+f*x))/(a^2*d^2*f*sqrt(-a+b)*sqrt(a+b)*sqrt(d*sin(e+f*x)))-2*b^2*EllipticPi(sqrt(g*cos(e+f*x))/(sqrt(g)*sqrt(1+sin(e+f*x))),sqrt(-a+b)/sqrt(a+b),I)*sqrt(2)*sqrt(g)*sqrt(sin(e+f*x))/(a^2*d^2*f*sqrt(-a+b)*sqrt(a+b)*sqrt(d*sin(e+f*x)))+2*b*sqrt(cos(e-1/4*Pi+f*x)^2)/cos(e-1/4*Pi+f*x)*EllipticE(sin(e-1/4*Pi+f*x),sqrt(2))*sqrt(g*cos(e+f*x))*sqrt(d*sin(e+f*x))/(a^2*d^3*f*sqrt(sin(2*e+2*f*x)))],
[sqrt(g*cos(e+f*x))/((d*sin(e+f*x))^(7/2)*(a+b*sin(e+f*x))),x,16,-2/5*(g*cos(e+f*x))^(3/2)/(a*d*f*g*(d*sin(e+f*x))^(5/2))+2/3*b*(g*cos(e+f*x))^(3/2)/(a^2*d^2*f*g*(d*sin(e+f*x))^(3/2))-4/5*(g*cos(e+f*x))^(3/2)/(a*d^3*f*g*sqrt(d*sin(e+f*x)))-2*b^2*(g*cos(e+f*x))^(3/2)/(a^3*d^3*f*g*sqrt(d*sin(e+f*x)))-2*b^3*EllipticPi(sqrt(g*cos(e+f*x))/(sqrt(g)*sqrt(1+sin(e+f*x))),-sqrt(-a+b)/sqrt(a+b),I)*sqrt(2)*sqrt(g)*sqrt(sin(e+f*x))/(a^3*d^3*f*sqrt(-a+b)*sqrt(a+b)*sqrt(d*sin(e+f*x)))+2*b^3*EllipticPi(sqrt(g*cos(e+f*x))/(sqrt(g)*sqrt(1+sin(e+f*x))),sqrt(-a+b)/sqrt(a+b),I)*sqrt(2)*sqrt(g)*sqrt(sin(e+f*x))/(a^3*d^3*f*sqrt(-a+b)*sqrt(a+b)*sqrt(d*sin(e+f*x)))-4/5*sqrt(cos(e-1/4*Pi+f*x)^2)/cos(e-1/4*Pi+f*x)*EllipticE(sin(e-1/4*Pi+f*x),sqrt(2))*sqrt(g*cos(e+f*x))*sqrt(d*sin(e+f*x))/(a*d^4*f*sqrt(sin(2*e+2*f*x)))-2*b^2*sqrt(cos(e-1/4*Pi+f*x)^2)/cos(e-1/4*Pi+f*x)*EllipticE(sin(e-1/4*Pi+f*x),sqrt(2))*sqrt(g*cos(e+f*x))*sqrt(d*sin(e+f*x))/(a^3*d^4*f*sqrt(sin(2*e+2*f*x)))],
[sqrt(g*cos(e+f*x))/((d*sin(e+f*x))^(9/2)*(a+b*sin(e+f*x))),x,19,-2/7*(g*cos(e+f*x))^(3/2)/(a*d*f*g*(d*sin(e+f*x))^(7/2))+2/5*b*(g*cos(e+f*x))^(3/2)/(a^2*d^2*f*g*(d*sin(e+f*x))^(5/2))-8/21*(g*cos(e+f*x))^(3/2)/(a*d^3*f*g*(d*sin(e+f*x))^(3/2))-2/3*b^2*(g*cos(e+f*x))^(3/2)/(a^3*d^3*f*g*(d*sin(e+f*x))^(3/2))+4/5*b*(g*cos(e+f*x))^(3/2)/(a^2*d^4*f*g*sqrt(d*sin(e+f*x)))+2*b^3*(g*cos(e+f*x))^(3/2)/(a^4*d^4*f*g*sqrt(d*sin(e+f*x)))+2*b^4*EllipticPi(sqrt(g*cos(e+f*x))/(sqrt(g)*sqrt(1+sin(e+f*x))),-sqrt(-a+b)/sqrt(a+b),I)*sqrt(2)*sqrt(g)*sqrt(sin(e+f*x))/(a^4*d^4*f*sqrt(-a+b)*sqrt(a+b)*sqrt(d*sin(e+f*x)))-2*b^4*EllipticPi(sqrt(g*cos(e+f*x))/(sqrt(g)*sqrt(1+sin(e+f*x))),sqrt(-a+b)/sqrt(a+b),I)*sqrt(2)*sqrt(g)*sqrt(sin(e+f*x))/(a^4*d^4*f*sqrt(-a+b)*sqrt(a+b)*sqrt(d*sin(e+f*x)))+4/5*b*sqrt(cos(e-1/4*Pi+f*x)^2)/cos(e-1/4*Pi+f*x)*EllipticE(sin(e-1/4*Pi+f*x),sqrt(2))*sqrt(g*cos(e+f*x))*sqrt(d*sin(e+f*x))/(a^2*d^5*f*sqrt(sin(2*e+2*f*x)))+2*b^3*sqrt(cos(e-1/4*Pi+f*x)^2)/cos(e-1/4*Pi+f*x)*EllipticE(sin(e-1/4*Pi+f*x),sqrt(2))*sqrt(g*cos(e+f*x))*sqrt(d*sin(e+f*x))/(a^4*d^5*f*sqrt(sin(2*e+2*f*x)))],
[(g*cos(e+f*x))^(3/2)*(d*sin(e+f*x))^(3/2)/(a+b*sin(e+f*x)),x,31,3/4*d^(3/2)*g^(3/2)*arctan(1-sqrt(2)*sqrt(g)*sqrt(d*sin(e+f*x))/(sqrt(d)*sqrt(g*cos(e+f*x))))/(b*f*sqrt(2))+(a^2-b^2)*d^(3/2)*g^(3/2)*arctan(1-sqrt(2)*sqrt(g)*sqrt(d*sin(e+f*x))/(sqrt(d)*sqrt(g*cos(e+f*x))))/(b^3*f*sqrt(2))-3/4*d^(3/2)*g^(3/2)*arctan(1+sqrt(2)*sqrt(g)*sqrt(d*sin(e+f*x))/(sqrt(d)*sqrt(g*cos(e+f*x))))/(b*f*sqrt(2))-(a^2-b^2)*d^(3/2)*g^(3/2)*arctan(1+sqrt(2)*sqrt(g)*sqrt(d*sin(e+f*x))/(sqrt(d)*sqrt(g*cos(e+f*x))))/(b^3*f*sqrt(2))-3/8*d^(3/2)*g^(3/2)*log(sqrt(d)-sqrt(2)*sqrt(g)*sqrt(d*sin(e+f*x))/sqrt(g*cos(e+f*x))+sqrt(d)*tan(e+f*x))/(b*f*sqrt(2))-1/2*(a^2-b^2)*d^(3/2)*g^(3/2)*log(sqrt(d)-sqrt(2)*sqrt(g)*sqrt(d*sin(e+f*x))/sqrt(g*cos(e+f*x))+sqrt(d)*tan(e+f*x))/(b^3*f*sqrt(2))+3/8*d^(3/2)*g^(3/2)*log(sqrt(d)+sqrt(2)*sqrt(g)*sqrt(d*sin(e+f*x))/sqrt(g*cos(e+f*x))+sqrt(d)*tan(e+f*x))/(b*f*sqrt(2))+1/2*(a^2-b^2)*d^(3/2)*g^(3/2)*log(sqrt(d)+sqrt(2)*sqrt(g)*sqrt(d*sin(e+f*x))/sqrt(g*cos(e+f*x))+sqrt(d)*tan(e+f*x))/(b^3*f*sqrt(2))+2*a*d^(3/2)*g^2*EllipticPi(sqrt(d*sin(e+f*x))/(sqrt(d)*sqrt(1+cos(e+f*x))),-a/(b-sqrt(-a^2+b^2)),I)*sqrt(2)*sqrt(-a^2+b^2)*sqrt(cos(e+f*x))/(b^3*f*sqrt(g*cos(e+f*x)))-2*a*d^(3/2)*g^2*EllipticPi(sqrt(d*sin(e+f*x))/(sqrt(d)*sqrt(1+cos(e+f*x))),-a/(b+sqrt(-a^2+b^2)),I)*sqrt(2)*sqrt(-a^2+b^2)*sqrt(cos(e+f*x))/(b^3*f*sqrt(g*cos(e+f*x)))+1/2*g*(d*sin(e+f*x))^(3/2)*sqrt(g*cos(e+f*x))/(b*f)-a*d*g*sqrt(g*cos(e+f*x))*sqrt(d*sin(e+f*x))/(b^2*f)+1/2*a*d^2*g^2*sqrt(cos(e-1/4*Pi+f*x)^2)/cos(e-1/4*Pi+f*x)*EllipticF(sin(e-1/4*Pi+f*x),sqrt(2))*sqrt(sin(2*e+2*f*x))/(b^2*f*sqrt(g*cos(e+f*x))*sqrt(d*sin(e+f*x)))],
[(g*cos(e+f*x))^(3/2)*(d*sin(e+f*x))^(1/2)/(a+b*sin(e+f*x)),x,19,-a*g^(3/2)*arctan(1-sqrt(2)*sqrt(g)*sqrt(d*sin(e+f*x))/(sqrt(d)*sqrt(g*cos(e+f*x))))*sqrt(d)/(b^2*f*sqrt(2))+a*g^(3/2)*arctan(1+sqrt(2)*sqrt(g)*sqrt(d*sin(e+f*x))/(sqrt(d)*sqrt(g*cos(e+f*x))))*sqrt(d)/(b^2*f*sqrt(2))+1/2*a*g^(3/2)*log(sqrt(d)-sqrt(2)*sqrt(g)*sqrt(d*sin(e+f*x))/sqrt(g*cos(e+f*x))+sqrt(d)*tan(e+f*x))*sqrt(d)/(b^2*f*sqrt(2))-1/2*a*g^(3/2)*log(sqrt(d)+sqrt(2)*sqrt(g)*sqrt(d*sin(e+f*x))/sqrt(g*cos(e+f*x))+sqrt(d)*tan(e+f*x))*sqrt(d)/(b^2*f*sqrt(2))-2*g^2*EllipticPi(sqrt(d*sin(e+f*x))/(sqrt(d)*sqrt(1+cos(e+f*x))),-a/(b-sqrt(-a^2+b^2)),I)*sqrt(2)*sqrt(-a^2+b^2)*sqrt(d)*sqrt(cos(e+f*x))/(b^2*f*sqrt(g*cos(e+f*x)))+2*g^2*EllipticPi(sqrt(d*sin(e+f*x))/(sqrt(d)*sqrt(1+cos(e+f*x))),-a/(b+sqrt(-a^2+b^2)),I)*sqrt(2)*sqrt(-a^2+b^2)*sqrt(d)*sqrt(cos(e+f*x))/(b^2*f*sqrt(g*cos(e+f*x)))+g*sqrt(g*cos(e+f*x))*sqrt(d*sin(e+f*x))/(b*f)-1/2*d*g^2*sqrt(cos(e-1/4*Pi+f*x)^2)/cos(e-1/4*Pi+f*x)*EllipticF(sin(e-1/4*Pi+f*x),sqrt(2))*sqrt(sin(2*e+2*f*x))/(b*f*sqrt(g*cos(e+f*x))*sqrt(d*sin(e+f*x)))],
[(g*cos(e+f*x))^(3/2)/((d*sin(e+f*x))^(1/2)*(a+b*sin(e+f*x))),x,18,g^(3/2)*arctan(1-sqrt(2)*sqrt(g)*sqrt(d*sin(e+f*x))/(sqrt(d)*sqrt(g*cos(e+f*x))))/(b*f*sqrt(2)*sqrt(d))-g^(3/2)*arctan(1+sqrt(2)*sqrt(g)*sqrt(d*sin(e+f*x))/(sqrt(d)*sqrt(g*cos(e+f*x))))/(b*f*sqrt(2)*sqrt(d))-1/2*g^(3/2)*log(sqrt(d)-sqrt(2)*sqrt(g)*sqrt(d*sin(e+f*x))/sqrt(g*cos(e+f*x))+sqrt(d)*tan(e+f*x))/(b*f*sqrt(2)*sqrt(d))+1/2*g^(3/2)*log(sqrt(d)+sqrt(2)*sqrt(g)*sqrt(d*sin(e+f*x))/sqrt(g*cos(e+f*x))+sqrt(d)*tan(e+f*x))/(b*f*sqrt(2)*sqrt(d))+2*g^2*EllipticPi(sqrt(d*sin(e+f*x))/(sqrt(d)*sqrt(1+cos(e+f*x))),-a/(b-sqrt(-a^2+b^2)),I)*sqrt(2)*sqrt(-a^2+b^2)*sqrt(cos(e+f*x))/(a*b*f*sqrt(d)*sqrt(g*cos(e+f*x)))-2*g^2*EllipticPi(sqrt(d*sin(e+f*x))/(sqrt(d)*sqrt(1+cos(e+f*x))),-a/(b+sqrt(-a^2+b^2)),I)*sqrt(2)*sqrt(-a^2+b^2)*sqrt(cos(e+f*x))/(a*b*f*sqrt(d)*sqrt(g*cos(e+f*x)))+g^2*sqrt(cos(e-1/4*Pi+f*x)^2)/cos(e-1/4*Pi+f*x)*EllipticF(sin(e-1/4*Pi+f*x),sqrt(2))*sqrt(sin(2*e+2*f*x))/(a*f*sqrt(g*cos(e+f*x))*sqrt(d*sin(e+f*x)))],
[(g*cos(e+f*x))^(3/2)/((d*sin(e+f*x))^(3/2)*(a+b*sin(e+f*x))),x,8,-2*g^2*EllipticPi(sqrt(d*sin(e+f*x))/(sqrt(d)*sqrt(1+cos(e+f*x))),-a/(b-sqrt(-a^2+b^2)),I)*sqrt(2)*sqrt(-a^2+b^2)*sqrt(cos(e+f*x))/(a^2*d^(3/2)*f*sqrt(g*cos(e+f*x)))+2*g^2*EllipticPi(sqrt(d*sin(e+f*x))/(sqrt(d)*sqrt(1+cos(e+f*x))),-a/(b+sqrt(-a^2+b^2)),I)*sqrt(2)*sqrt(-a^2+b^2)*sqrt(cos(e+f*x))/(a^2*d^(3/2)*f*sqrt(g*cos(e+f*x)))-2*g*sqrt(g*cos(e+f*x))/(a*d*f*sqrt(d*sin(e+f*x)))-b*g^2*sqrt(cos(e-1/4*Pi+f*x)^2)/cos(e-1/4*Pi+f*x)*EllipticF(sin(e-1/4*Pi+f*x),sqrt(2))*sqrt(sin(2*e+2*f*x))/(a^2*d*f*sqrt(g*cos(e+f*x))*sqrt(d*sin(e+f*x)))],
[(g*cos(e+f*x))^(3/2)/((d*sin(e+f*x))^(5/2)*(a+b*sin(e+f*x))),x,12,2*b*g^2*EllipticPi(sqrt(d*sin(e+f*x))/(sqrt(d)*sqrt(1+cos(e+f*x))),-a/(b-sqrt(-a^2+b^2)),I)*sqrt(2)*sqrt(-a^2+b^2)*sqrt(cos(e+f*x))/(a^3*d^(5/2)*f*sqrt(g*cos(e+f*x)))-2*b*g^2*EllipticPi(sqrt(d*sin(e+f*x))/(sqrt(d)*sqrt(1+cos(e+f*x))),-a/(b+sqrt(-a^2+b^2)),I)*sqrt(2)*sqrt(-a^2+b^2)*sqrt(cos(e+f*x))/(a^3*d^(5/2)*f*sqrt(g*cos(e+f*x)))-2/3*g*sqrt(g*cos(e+f*x))/(a*d*f*(d*sin(e+f*x))^(3/2))+2*b*g*sqrt(g*cos(e+f*x))/(a^2*d^2*f*sqrt(d*sin(e+f*x)))+2/3*g^2*sqrt(cos(e-1/4*Pi+f*x)^2)/cos(e-1/4*Pi+f*x)*EllipticF(sin(e-1/4*Pi+f*x),sqrt(2))*sqrt(sin(2*e+2*f*x))/(a*d^2*f*sqrt(g*cos(e+f*x))*sqrt(d*sin(e+f*x)))-(a^2-b^2)*g^2*sqrt(cos(e-1/4*Pi+f*x)^2)/cos(e-1/4*Pi+f*x)*EllipticF(sin(e-1/4*Pi+f*x),sqrt(2))*sqrt(sin(2*e+2*f*x))/(a^3*d^2*f*sqrt(g*cos(e+f*x))*sqrt(d*sin(e+f*x)))],
[(g*cos(e+f*x))^(3/2)/((d*sin(e+f*x))^(7/2)*(a+b*sin(e+f*x))),x,15,-2*b^2*g^2*EllipticPi(sqrt(d*sin(e+f*x))/(sqrt(d)*sqrt(1+cos(e+f*x))),-a/(b-sqrt(-a^2+b^2)),I)*sqrt(2)*sqrt(-a^2+b^2)*sqrt(cos(e+f*x))/(a^4*d^(7/2)*f*sqrt(g*cos(e+f*x)))+2*b^2*g^2*EllipticPi(sqrt(d*sin(e+f*x))/(sqrt(d)*sqrt(1+cos(e+f*x))),-a/(b+sqrt(-a^2+b^2)),I)*sqrt(2)*sqrt(-a^2+b^2)*sqrt(cos(e+f*x))/(a^4*d^(7/2)*f*sqrt(g*cos(e+f*x)))-2/5*g*sqrt(g*cos(e+f*x))/(a*d*f*(d*sin(e+f*x))^(5/2))+2/3*b*g*sqrt(g*cos(e+f*x))/(a^2*d^2*f*(d*sin(e+f*x))^(3/2))-8/5*g*sqrt(g*cos(e+f*x))/(a*d^3*f*sqrt(d*sin(e+f*x)))+2*(a^2-b^2)*g*sqrt(g*cos(e+f*x))/(a^3*d^3*f*sqrt(d*sin(e+f*x)))-2/3*b*g^2*sqrt(cos(e-1/4*Pi+f*x)^2)/cos(e-1/4*Pi+f*x)*EllipticF(sin(e-1/4*Pi+f*x),sqrt(2))*sqrt(sin(2*e+2*f*x))/(a^2*d^3*f*sqrt(g*cos(e+f*x))*sqrt(d*sin(e+f*x)))+b*(a^2-b^2)*g^2*sqrt(cos(e-1/4*Pi+f*x)^2)/cos(e-1/4*Pi+f*x)*EllipticF(sin(e-1/4*Pi+f*x),sqrt(2))*sqrt(sin(2*e+2*f*x))/(a^4*d^3*f*sqrt(g*cos(e+f*x))*sqrt(d*sin(e+f*x)))],
[(g*cos(e+f*x))^(3/2)/((d*sin(e+f*x))^(9/2)*(a+b*sin(e+f*x))),x,20,2*b^3*g^2*EllipticPi(sqrt(d*sin(e+f*x))/(sqrt(d)*sqrt(1+cos(e+f*x))),-a/(b-sqrt(-a^2+b^2)),I)*sqrt(2)*sqrt(-a^2+b^2)*sqrt(cos(e+f*x))/(a^5*d^(9/2)*f*sqrt(g*cos(e+f*x)))-2*b^3*g^2*EllipticPi(sqrt(d*sin(e+f*x))/(sqrt(d)*sqrt(1+cos(e+f*x))),-a/(b+sqrt(-a^2+b^2)),I)*sqrt(2)*sqrt(-a^2+b^2)*sqrt(cos(e+f*x))/(a^5*d^(9/2)*f*sqrt(g*cos(e+f*x)))-2/7*g*sqrt(g*cos(e+f*x))/(a*d*f*(d*sin(e+f*x))^(7/2))+2/5*b*g*sqrt(g*cos(e+f*x))/(a^2*d^2*f*(d*sin(e+f*x))^(5/2))-4/7*g*sqrt(g*cos(e+f*x))/(a*d^3*f*(d*sin(e+f*x))^(3/2))+2/3*(a^2-b^2)*g*sqrt(g*cos(e+f*x))/(a^3*d^3*f*(d*sin(e+f*x))^(3/2))+8/5*b*g*sqrt(g*cos(e+f*x))/(a^2*d^4*f*sqrt(d*sin(e+f*x)))-2*b*(a^2-b^2)*g*sqrt(g*cos(e+f*x))/(a^4*d^4*f*sqrt(d*sin(e+f*x)))+4/7*g^2*sqrt(cos(e-1/4*Pi+f*x)^2)/cos(e-1/4*Pi+f*x)*EllipticF(sin(e-1/4*Pi+f*x),sqrt(2))*sqrt(sin(2*e+2*f*x))/(a*d^4*f*sqrt(g*cos(e+f*x))*sqrt(d*sin(e+f*x)))-2/3*(a^2-b^2)*g^2*sqrt(cos(e-1/4*Pi+f*x)^2)/cos(e-1/4*Pi+f*x)*EllipticF(sin(e-1/4*Pi+f*x),sqrt(2))*sqrt(sin(2*e+2*f*x))/(a^3*d^4*f*sqrt(g*cos(e+f*x))*sqrt(d*sin(e+f*x)))-b^2*(a^2-b^2)*g^2*sqrt(cos(e-1/4*Pi+f*x)^2)/cos(e-1/4*Pi+f*x)*EllipticF(sin(e-1/4*Pi+f*x),sqrt(2))*sqrt(sin(2*e+2*f*x))/(a^5*d^4*f*sqrt(g*cos(e+f*x))*sqrt(d*sin(e+f*x)))],
[(g*cos(e+f*x))^(5/2)*(d*sin(e+f*x))^(1/2)/(a+b*sin(e+f*x)),x,31,-1/4*g^(5/2)*arctan(1-sqrt(2)*sqrt(d)*sqrt(g*cos(e+f*x))/(sqrt(g)*sqrt(d*sin(e+f*x))))*sqrt(d)/(b*f*sqrt(2))-(a^2-b^2)*g^(5/2)*arctan(1-sqrt(2)*sqrt(d)*sqrt(g*cos(e+f*x))/(sqrt(g)*sqrt(d*sin(e+f*x))))*sqrt(d)/(b^3*f*sqrt(2))+1/4*g^(5/2)*arctan(1+sqrt(2)*sqrt(d)*sqrt(g*cos(e+f*x))/(sqrt(g)*sqrt(d*sin(e+f*x))))*sqrt(d)/(b*f*sqrt(2))+(a^2-b^2)*g^(5/2)*arctan(1+sqrt(2)*sqrt(d)*sqrt(g*cos(e+f*x))/(sqrt(g)*sqrt(d*sin(e+f*x))))*sqrt(d)/(b^3*f*sqrt(2))+1/8*g^(5/2)*log(sqrt(g)+cot(e+f*x)*sqrt(g)-sqrt(2)*sqrt(d)*sqrt(g*cos(e+f*x))/sqrt(d*sin(e+f*x)))*sqrt(d)/(b*f*sqrt(2))+1/2*(a^2-b^2)*g^(5/2)*log(sqrt(g)+cot(e+f*x)*sqrt(g)-sqrt(2)*sqrt(d)*sqrt(g*cos(e+f*x))/sqrt(d*sin(e+f*x)))*sqrt(d)/(b^3*f*sqrt(2))-1/8*g^(5/2)*log(sqrt(g)+cot(e+f*x)*sqrt(g)+sqrt(2)*sqrt(d)*sqrt(g*cos(e+f*x))/sqrt(d*sin(e+f*x)))*sqrt(d)/(b*f*sqrt(2))-1/2*(a^2-b^2)*g^(5/2)*log(sqrt(g)+cot(e+f*x)*sqrt(g)+sqrt(2)*sqrt(d)*sqrt(g*cos(e+f*x))/sqrt(d*sin(e+f*x)))*sqrt(d)/(b^3*f*sqrt(2))-2*a*d*g^(5/2)*EllipticPi(sqrt(g*cos(e+f*x))/(sqrt(g)*sqrt(1+sin(e+f*x))),-sqrt(-a+b)/sqrt(a+b),I)*sqrt(2)*sqrt(-a+b)*sqrt(a+b)*sqrt(sin(e+f*x))/(b^3*f*sqrt(d*sin(e+f*x)))+2*a*d*g^(5/2)*EllipticPi(sqrt(g*cos(e+f*x))/(sqrt(g)*sqrt(1+sin(e+f*x))),sqrt(-a+b)/sqrt(a+b),I)*sqrt(2)*sqrt(-a+b)*sqrt(a+b)*sqrt(sin(e+f*x))/(b^3*f*sqrt(d*sin(e+f*x)))+1/2*g*(g*cos(e+f*x))^(3/2)*sqrt(d*sin(e+f*x))/(b*f)+a*g^2*sqrt(cos(e-1/4*Pi+f*x)^2)/cos(e-1/4*Pi+f*x)*EllipticE(sin(e-1/4*Pi+f*x),sqrt(2))*sqrt(g*cos(e+f*x))*sqrt(d*sin(e+f*x))/(b^2*f*sqrt(sin(2*e+2*f*x)))],
[(g*cos(e+f*x))^(5/2)/((d*sin(e+f*x))^(1/2)*(a+b*sin(e+f*x))),x,19,a*g^(5/2)*arctan(1-sqrt(2)*sqrt(d)*sqrt(g*cos(e+f*x))/(sqrt(g)*sqrt(d*sin(e+f*x))))/(b^2*f*sqrt(2)*sqrt(d))-a*g^(5/2)*arctan(1+sqrt(2)*sqrt(d)*sqrt(g*cos(e+f*x))/(sqrt(g)*sqrt(d*sin(e+f*x))))/(b^2*f*sqrt(2)*sqrt(d))-1/2*a*g^(5/2)*log(sqrt(g)+cot(e+f*x)*sqrt(g)-sqrt(2)*sqrt(d)*sqrt(g*cos(e+f*x))/sqrt(d*sin(e+f*x)))/(b^2*f*sqrt(2)*sqrt(d))+1/2*a*g^(5/2)*log(sqrt(g)+cot(e+f*x)*sqrt(g)+sqrt(2)*sqrt(d)*sqrt(g*cos(e+f*x))/sqrt(d*sin(e+f*x)))/(b^2*f*sqrt(2)*sqrt(d))+2*g^(5/2)*EllipticPi(sqrt(g*cos(e+f*x))/(sqrt(g)*sqrt(1+sin(e+f*x))),-sqrt(-a+b)/sqrt(a+b),I)*sqrt(2)*sqrt(-a+b)*sqrt(a+b)*sqrt(sin(e+f*x))/(b^2*f*sqrt(d*sin(e+f*x)))-2*g^(5/2)*EllipticPi(sqrt(g*cos(e+f*x))/(sqrt(g)*sqrt(1+sin(e+f*x))),sqrt(-a+b)/sqrt(a+b),I)*sqrt(2)*sqrt(-a+b)*sqrt(a+b)*sqrt(sin(e+f*x))/(b^2*f*sqrt(d*sin(e+f*x)))-g^2*sqrt(cos(e-1/4*Pi+f*x)^2)/cos(e-1/4*Pi+f*x)*EllipticE(sin(e-1/4*Pi+f*x),sqrt(2))*sqrt(g*cos(e+f*x))*sqrt(d*sin(e+f*x))/(b*d*f*sqrt(sin(2*e+2*f*x)))],
[(g*cos(e+f*x))^(5/2)/((d*sin(e+f*x))^(3/2)*(a+b*sin(e+f*x))),x,20,-g^(5/2)*arctan(1-sqrt(2)*sqrt(d)*sqrt(g*cos(e+f*x))/(sqrt(g)*sqrt(d*sin(e+f*x))))/(b*d^(3/2)*f*sqrt(2))+g^(5/2)*arctan(1+sqrt(2)*sqrt(d)*sqrt(g*cos(e+f*x))/(sqrt(g)*sqrt(d*sin(e+f*x))))/(b*d^(3/2)*f*sqrt(2))+1/2*g^(5/2)*log(sqrt(g)+cot(e+f*x)*sqrt(g)-sqrt(2)*sqrt(d)*sqrt(g*cos(e+f*x))/sqrt(d*sin(e+f*x)))/(b*d^(3/2)*f*sqrt(2))-1/2*g^(5/2)*log(sqrt(g)+cot(e+f*x)*sqrt(g)+sqrt(2)*sqrt(d)*sqrt(g*cos(e+f*x))/sqrt(d*sin(e+f*x)))/(b*d^(3/2)*f*sqrt(2))-2*g*(g*cos(e+f*x))^(3/2)/(a*d*f*sqrt(d*sin(e+f*x)))-2*g^(5/2)*EllipticPi(sqrt(g*cos(e+f*x))/(sqrt(g)*sqrt(1+sin(e+f*x))),-sqrt(-a+b)/sqrt(a+b),I)*sqrt(2)*sqrt(-a+b)*sqrt(a+b)*sqrt(sin(e+f*x))/(a*b*d*f*sqrt(d*sin(e+f*x)))+2*g^(5/2)*EllipticPi(sqrt(g*cos(e+f*x))/(sqrt(g)*sqrt(1+sin(e+f*x))),sqrt(-a+b)/sqrt(a+b),I)*sqrt(2)*sqrt(-a+b)*sqrt(a+b)*sqrt(sin(e+f*x))/(a*b*d*f*sqrt(d*sin(e+f*x)))-2*g^2*sqrt(cos(e-1/4*Pi+f*x)^2)/cos(e-1/4*Pi+f*x)*EllipticE(sin(e-1/4*Pi+f*x),sqrt(2))*sqrt(g*cos(e+f*x))*sqrt(d*sin(e+f*x))/(a*d^2*f*sqrt(sin(2*e+2*f*x)))],
[(g*cos(e+f*x))^(5/2)/((d*sin(e+f*x))^(5/2)*(a+b*sin(e+f*x))),x,10,-2/3*g*(g*cos(e+f*x))^(3/2)/(a*d*f*(d*sin(e+f*x))^(3/2))+2*b*g*(g*cos(e+f*x))^(3/2)/(a^2*d^2*f*sqrt(d*sin(e+f*x)))+2*g^(5/2)*EllipticPi(sqrt(g*cos(e+f*x))/(sqrt(g)*sqrt(1+sin(e+f*x))),-sqrt(-a+b)/sqrt(a+b),I)*sqrt(2)*sqrt(-a+b)*sqrt(a+b)*sqrt(sin(e+f*x))/(a^2*d^2*f*sqrt(d*sin(e+f*x)))-2*g^(5/2)*EllipticPi(sqrt(g*cos(e+f*x))/(sqrt(g)*sqrt(1+sin(e+f*x))),sqrt(-a+b)/sqrt(a+b),I)*sqrt(2)*sqrt(-a+b)*sqrt(a+b)*sqrt(sin(e+f*x))/(a^2*d^2*f*sqrt(d*sin(e+f*x)))+2*b*g^2*sqrt(cos(e-1/4*Pi+f*x)^2)/cos(e-1/4*Pi+f*x)*EllipticE(sin(e-1/4*Pi+f*x),sqrt(2))*sqrt(g*cos(e+f*x))*sqrt(d*sin(e+f*x))/(a^2*d^3*f*sqrt(sin(2*e+2*f*x)))],
[(g*cos(e+f*x))^(5/2)/((d*sin(e+f*x))^(7/2)*(a+b*sin(e+f*x))),x,15,-2/5*g*(g*cos(e+f*x))^(3/2)/(a*d*f*(d*sin(e+f*x))^(5/2))+2/3*b*g*(g*cos(e+f*x))^(3/2)/(a^2*d^2*f*(d*sin(e+f*x))^(3/2))-4/5*g*(g*cos(e+f*x))^(3/2)/(a*d^3*f*sqrt(d*sin(e+f*x)))+2*(a^2-b^2)*g*(g*cos(e+f*x))^(3/2)/(a^3*d^3*f*sqrt(d*sin(e+f*x)))-2*b*g^(5/2)*EllipticPi(sqrt(g*cos(e+f*x))/(sqrt(g)*sqrt(1+sin(e+f*x))),-sqrt(-a+b)/sqrt(a+b),I)*sqrt(2)*sqrt(-a+b)*sqrt(a+b)*sqrt(sin(e+f*x))/(a^3*d^3*f*sqrt(d*sin(e+f*x)))+2*b*g^(5/2)*EllipticPi(sqrt(g*cos(e+f*x))/(sqrt(g)*sqrt(1+sin(e+f*x))),sqrt(-a+b)/sqrt(a+b),I)*sqrt(2)*sqrt(-a+b)*sqrt(a+b)*sqrt(sin(e+f*x))/(a^3*d^3*f*sqrt(d*sin(e+f*x)))-4/5*g^2*sqrt(cos(e-1/4*Pi+f*x)^2)/cos(e-1/4*Pi+f*x)*EllipticE(sin(e-1/4*Pi+f*x),sqrt(2))*sqrt(g*cos(e+f*x))*sqrt(d*sin(e+f*x))/(a*d^4*f*sqrt(sin(2*e+2*f*x)))+2*(a^2-b^2)*g^2*sqrt(cos(e-1/4*Pi+f*x)^2)/cos(e-1/4*Pi+f*x)*EllipticE(sin(e-1/4*Pi+f*x),sqrt(2))*sqrt(g*cos(e+f*x))*sqrt(d*sin(e+f*x))/(a^3*d^4*f*sqrt(sin(2*e+2*f*x)))],
[(g*cos(e+f*x))^(5/2)/((d*sin(e+f*x))^(9/2)*(a+b*sin(e+f*x))),x,18,-2/7*g*(g*cos(e+f*x))^(3/2)/(a*d*f*(d*sin(e+f*x))^(7/2))+2/5*b*g*(g*cos(e+f*x))^(3/2)/(a^2*d^2*f*(d*sin(e+f*x))^(5/2))-8/21*g*(g*cos(e+f*x))^(3/2)/(a*d^3*f*(d*sin(e+f*x))^(3/2))+2/3*(a^2-b^2)*g*(g*cos(e+f*x))^(3/2)/(a^3*d^3*f*(d*sin(e+f*x))^(3/2))+4/5*b*g*(g*cos(e+f*x))^(3/2)/(a^2*d^4*f*sqrt(d*sin(e+f*x)))-2*b*(a^2-b^2)*g*(g*cos(e+f*x))^(3/2)/(a^4*d^4*f*sqrt(d*sin(e+f*x)))+2*b^2*g^(5/2)*EllipticPi(sqrt(g*cos(e+f*x))/(sqrt(g)*sqrt(1+sin(e+f*x))),-sqrt(-a+b)/sqrt(a+b),I)*sqrt(2)*sqrt(-a+b)*sqrt(a+b)*sqrt(sin(e+f*x))/(a^4*d^4*f*sqrt(d*sin(e+f*x)))-2*b^2*g^(5/2)*EllipticPi(sqrt(g*cos(e+f*x))/(sqrt(g)*sqrt(1+sin(e+f*x))),sqrt(-a+b)/sqrt(a+b),I)*sqrt(2)*sqrt(-a+b)*sqrt(a+b)*sqrt(sin(e+f*x))/(a^4*d^4*f*sqrt(d*sin(e+f*x)))+4/5*b*g^2*sqrt(cos(e-1/4*Pi+f*x)^2)/cos(e-1/4*Pi+f*x)*EllipticE(sin(e-1/4*Pi+f*x),sqrt(2))*sqrt(g*cos(e+f*x))*sqrt(d*sin(e+f*x))/(a^2*d^5*f*sqrt(sin(2*e+2*f*x)))-2*b*(a^2-b^2)*g^2*sqrt(cos(e-1/4*Pi+f*x)^2)/cos(e-1/4*Pi+f*x)*EllipticE(sin(e-1/4*Pi+f*x),sqrt(2))*sqrt(g*cos(e+f*x))*sqrt(d*sin(e+f*x))/(a^4*d^5*f*sqrt(sin(2*e+2*f*x)))],
[(g*cos(e+f*x))^(5/2)/((d*sin(e+f*x))^(11/2)*(a+b*sin(e+f*x))),x,24,-2/9*g*(g*cos(e+f*x))^(3/2)/(a*d*f*(d*sin(e+f*x))^(9/2))+2/7*b*g*(g*cos(e+f*x))^(3/2)/(a^2*d^2*f*(d*sin(e+f*x))^(7/2))-4/15*g*(g*cos(e+f*x))^(3/2)/(a*d^3*f*(d*sin(e+f*x))^(5/2))+2/5*(a^2-b^2)*g*(g*cos(e+f*x))^(3/2)/(a^3*d^3*f*(d*sin(e+f*x))^(5/2))+8/21*b*g*(g*cos(e+f*x))^(3/2)/(a^2*d^4*f*(d*sin(e+f*x))^(3/2))-2/3*b*(a^2-b^2)*g*(g*cos(e+f*x))^(3/2)/(a^4*d^4*f*(d*sin(e+f*x))^(3/2))-8/15*g*(g*cos(e+f*x))^(3/2)/(a*d^5*f*sqrt(d*sin(e+f*x)))+4/5*(a^2-b^2)*g*(g*cos(e+f*x))^(3/2)/(a^3*d^5*f*sqrt(d*sin(e+f*x)))+2*b^2*(a^2-b^2)*g*(g*cos(e+f*x))^(3/2)/(a^5*d^5*f*sqrt(d*sin(e+f*x)))-2*b^3*g^(5/2)*EllipticPi(sqrt(g*cos(e+f*x))/(sqrt(g)*sqrt(1+sin(e+f*x))),-sqrt(-a+b)/sqrt(a+b),I)*sqrt(2)*sqrt(-a+b)*sqrt(a+b)*sqrt(sin(e+f*x))/(a^5*d^5*f*sqrt(d*sin(e+f*x)))+2*b^3*g^(5/2)*EllipticPi(sqrt(g*cos(e+f*x))/(sqrt(g)*sqrt(1+sin(e+f*x))),sqrt(-a+b)/sqrt(a+b),I)*sqrt(2)*sqrt(-a+b)*sqrt(a+b)*sqrt(sin(e+f*x))/(a^5*d^5*f*sqrt(d*sin(e+f*x)))-8/15*g^2*sqrt(cos(e-1/4*Pi+f*x)^2)/cos(e-1/4*Pi+f*x)*EllipticE(sin(e-1/4*Pi+f*x),sqrt(2))*sqrt(g*cos(e+f*x))*sqrt(d*sin(e+f*x))/(a*d^6*f*sqrt(sin(2*e+2*f*x)))+4/5*(a^2-b^2)*g^2*sqrt(cos(e-1/4*Pi+f*x)^2)/cos(e-1/4*Pi+f*x)*EllipticE(sin(e-1/4*Pi+f*x),sqrt(2))*sqrt(g*cos(e+f*x))*sqrt(d*sin(e+f*x))/(a^3*d^6*f*sqrt(sin(2*e+2*f*x)))+2*b^2*(a^2-b^2)*g^2*sqrt(cos(e-1/4*Pi+f*x)^2)/cos(e-1/4*Pi+f*x)*EllipticE(sin(e-1/4*Pi+f*x),sqrt(2))*sqrt(g*cos(e+f*x))*sqrt(d*sin(e+f*x))/(a^5*d^6*f*sqrt(sin(2*e+2*f*x)))],

# p<0
[(d*sin(e+f*x))^(5/2)/((a+b*sin(e+f*x))*sqrt(g*cos(e+f*x))),x,19,a*d^(5/2)*arctan(1-sqrt(2)*sqrt(g)*sqrt(d*sin(e+f*x))/(sqrt(d)*sqrt(g*cos(e+f*x))))/(b^2*f*sqrt(2)*sqrt(g))-a*d^(5/2)*arctan(1+sqrt(2)*sqrt(g)*sqrt(d*sin(e+f*x))/(sqrt(d)*sqrt(g*cos(e+f*x))))/(b^2*f*sqrt(2)*sqrt(g))-1/2*a*d^(5/2)*log(sqrt(d)-sqrt(2)*sqrt(g)*sqrt(d*sin(e+f*x))/sqrt(g*cos(e+f*x))+sqrt(d)*tan(e+f*x))/(b^2*f*sqrt(2)*sqrt(g))+1/2*a*d^(5/2)*log(sqrt(d)+sqrt(2)*sqrt(g)*sqrt(d*sin(e+f*x))/sqrt(g*cos(e+f*x))+sqrt(d)*tan(e+f*x))/(b^2*f*sqrt(2)*sqrt(g))-2*a^2*d^(5/2)*EllipticPi(sqrt(d*sin(e+f*x))/(sqrt(d)*sqrt(1+cos(e+f*x))),-a/(b-sqrt(-a^2+b^2)),I)*sqrt(2)*sqrt(cos(e+f*x))/(b^2*f*sqrt(-a^2+b^2)*sqrt(g*cos(e+f*x)))+2*a^2*d^(5/2)*EllipticPi(sqrt(d*sin(e+f*x))/(sqrt(d)*sqrt(1+cos(e+f*x))),-a/(b+sqrt(-a^2+b^2)),I)*sqrt(2)*sqrt(cos(e+f*x))/(b^2*f*sqrt(-a^2+b^2)*sqrt(g*cos(e+f*x)))-d^2*sqrt(g*cos(e+f*x))*sqrt(d*sin(e+f*x))/(b*f*g)+1/2*d^3*sqrt(cos(e-1/4*Pi+f*x)^2)/cos(e-1/4*Pi+f*x)*EllipticF(sin(e-1/4*Pi+f*x),sqrt(2))*sqrt(sin(2*e+2*f*x))/(b*f*sqrt(g*cos(e+f*x))*sqrt(d*sin(e+f*x)))],
[(d*sin(e+f*x))^(3/2)/((a+b*sin(e+f*x))*sqrt(g*cos(e+f*x))),x,15,-d^(3/2)*arctan(1-sqrt(2)*sqrt(g)*sqrt(d*sin(e+f*x))/(sqrt(d)*sqrt(g*cos(e+f*x))))/(b*f*sqrt(2)*sqrt(g))+d^(3/2)*arctan(1+sqrt(2)*sqrt(g)*sqrt(d*sin(e+f*x))/(sqrt(d)*sqrt(g*cos(e+f*x))))/(b*f*sqrt(2)*sqrt(g))+1/2*d^(3/2)*log(sqrt(d)-sqrt(2)*sqrt(g)*sqrt(d*sin(e+f*x))/sqrt(g*cos(e+f*x))+sqrt(d)*tan(e+f*x))/(b*f*sqrt(2)*sqrt(g))-1/2*d^(3/2)*log(sqrt(d)+sqrt(2)*sqrt(g)*sqrt(d*sin(e+f*x))/sqrt(g*cos(e+f*x))+sqrt(d)*tan(e+f*x))/(b*f*sqrt(2)*sqrt(g))+2*a*d^(3/2)*EllipticPi(sqrt(d*sin(e+f*x))/(sqrt(d)*sqrt(1+cos(e+f*x))),-a/(b-sqrt(-a^2+b^2)),I)*sqrt(2)*sqrt(cos(e+f*x))/(b*f*sqrt(-a^2+b^2)*sqrt(g*cos(e+f*x)))-2*a*d^(3/2)*EllipticPi(sqrt(d*sin(e+f*x))/(sqrt(d)*sqrt(1+cos(e+f*x))),-a/(b+sqrt(-a^2+b^2)),I)*sqrt(2)*sqrt(cos(e+f*x))/(b*f*sqrt(-a^2+b^2)*sqrt(g*cos(e+f*x)))],
[(d*sin(e+f*x))^(1/2)/((a+b*sin(e+f*x))*sqrt(g*cos(e+f*x))),x,4,-2*EllipticPi(sqrt(d*sin(e+f*x))/(sqrt(d)*sqrt(1+cos(e+f*x))),-a/(b-sqrt(-a^2+b^2)),I)*sqrt(2)*sqrt(d)*sqrt(cos(e+f*x))/(f*sqrt(-a^2+b^2)*sqrt(g*cos(e+f*x)))+2*EllipticPi(sqrt(d*sin(e+f*x))/(sqrt(d)*sqrt(1+cos(e+f*x))),-a/(b+sqrt(-a^2+b^2)),I)*sqrt(2)*sqrt(d)*sqrt(cos(e+f*x))/(f*sqrt(-a^2+b^2)*sqrt(g*cos(e+f*x)))],
[1/((d*sin(e+f*x))^(1/2)*(a+b*sin(e+f*x))*sqrt(g*cos(e+f*x))),x,7,2*b*EllipticPi(sqrt(d*sin(e+f*x))/(sqrt(d)*sqrt(1+cos(e+f*x))),-a/(b-sqrt(-a^2+b^2)),I)*sqrt(2)*sqrt(cos(e+f*x))/(a*f*sqrt(-a^2+b^2)*sqrt(d)*sqrt(g*cos(e+f*x)))-2*b*EllipticPi(sqrt(d*sin(e+f*x))/(sqrt(d)*sqrt(1+cos(e+f*x))),-a/(b+sqrt(-a^2+b^2)),I)*sqrt(2)*sqrt(cos(e+f*x))/(a*f*sqrt(-a^2+b^2)*sqrt(d)*sqrt(g*cos(e+f*x)))+sqrt(cos(e-1/4*Pi+f*x)^2)/cos(e-1/4*Pi+f*x)*EllipticF(sin(e-1/4*Pi+f*x),sqrt(2))*sqrt(sin(2*e+2*f*x))/(a*f*sqrt(g*cos(e+f*x))*sqrt(d*sin(e+f*x)))],
[1/((d*sin(e+f*x))^(3/2)*(a+b*sin(e+f*x))*sqrt(g*cos(e+f*x))),x,9,-2*b^2*EllipticPi(sqrt(d*sin(e+f*x))/(sqrt(d)*sqrt(1+cos(e+f*x))),-a/(b-sqrt(-a^2+b^2)),I)*sqrt(2)*sqrt(cos(e+f*x))/(a^2*d^(3/2)*f*sqrt(-a^2+b^2)*sqrt(g*cos(e+f*x)))+2*b^2*EllipticPi(sqrt(d*sin(e+f*x))/(sqrt(d)*sqrt(1+cos(e+f*x))),-a/(b+sqrt(-a^2+b^2)),I)*sqrt(2)*sqrt(cos(e+f*x))/(a^2*d^(3/2)*f*sqrt(-a^2+b^2)*sqrt(g*cos(e+f*x)))-2*sqrt(g*cos(e+f*x))/(a*d*f*g*sqrt(d*sin(e+f*x)))-b*sqrt(cos(e-1/4*Pi+f*x)^2)/cos(e-1/4*Pi+f*x)*EllipticF(sin(e-1/4*Pi+f*x),sqrt(2))*sqrt(sin(2*e+2*f*x))/(a^2*d*f*sqrt(g*cos(e+f*x))*sqrt(d*sin(e+f*x)))],
[1/((d*sin(e+f*x))^(5/2)*(a+b*sin(e+f*x))*sqrt(g*cos(e+f*x))),x,13,2*b^3*EllipticPi(sqrt(d*sin(e+f*x))/(sqrt(d)*sqrt(1+cos(e+f*x))),-a/(b-sqrt(-a^2+b^2)),I)*sqrt(2)*sqrt(cos(e+f*x))/(a^3*d^(5/2)*f*sqrt(-a^2+b^2)*sqrt(g*cos(e+f*x)))-2*b^3*EllipticPi(sqrt(d*sin(e+f*x))/(sqrt(d)*sqrt(1+cos(e+f*x))),-a/(b+sqrt(-a^2+b^2)),I)*sqrt(2)*sqrt(cos(e+f*x))/(a^3*d^(5/2)*f*sqrt(-a^2+b^2)*sqrt(g*cos(e+f*x)))-2/3*sqrt(g*cos(e+f*x))/(a*d*f*g*(d*sin(e+f*x))^(3/2))+2*b*sqrt(g*cos(e+f*x))/(a^2*d^2*f*g*sqrt(d*sin(e+f*x)))+2/3*sqrt(cos(e-1/4*Pi+f*x)^2)/cos(e-1/4*Pi+f*x)*EllipticF(sin(e-1/4*Pi+f*x),sqrt(2))*sqrt(sin(2*e+2*f*x))/(a*d^2*f*sqrt(g*cos(e+f*x))*sqrt(d*sin(e+f*x)))+b^2*sqrt(cos(e-1/4*Pi+f*x)^2)/cos(e-1/4*Pi+f*x)*EllipticF(sin(e-1/4*Pi+f*x),sqrt(2))*sqrt(sin(2*e+2*f*x))/(a^3*d^2*f*sqrt(g*cos(e+f*x))*sqrt(d*sin(e+f*x)))],
[(d*sin(e+f*x))^(5/2)/((g*cos(e+f*x))^(3/2)*(a+b*sin(e+f*x))),x,31,-a^2*d^(5/2)*arctan(1-sqrt(2)*sqrt(d)*sqrt(g*cos(e+f*x))/(sqrt(g)*sqrt(d*sin(e+f*x))))/(b*(a^2-b^2)*f*g^(3/2)*sqrt(2))+b*d^(5/2)*arctan(1-sqrt(2)*sqrt(d)*sqrt(g*cos(e+f*x))/(sqrt(g)*sqrt(d*sin(e+f*x))))/((a^2-b^2)*f*g^(3/2)*sqrt(2))+a^2*d^(5/2)*arctan(1+sqrt(2)*sqrt(d)*sqrt(g*cos(e+f*x))/(sqrt(g)*sqrt(d*sin(e+f*x))))/(b*(a^2-b^2)*f*g^(3/2)*sqrt(2))-b*d^(5/2)*arctan(1+sqrt(2)*sqrt(d)*sqrt(g*cos(e+f*x))/(sqrt(g)*sqrt(d*sin(e+f*x))))/((a^2-b^2)*f*g^(3/2)*sqrt(2))+1/2*a^2*d^(5/2)*log(sqrt(g)+cot(e+f*x)*sqrt(g)-sqrt(2)*sqrt(d)*sqrt(g*cos(e+f*x))/sqrt(d*sin(e+f*x)))/(b*(a^2-b^2)*f*g^(3/2)*sqrt(2))-1/2*b*d^(5/2)*log(sqrt(g)+cot(e+f*x)*sqrt(g)-sqrt(2)*sqrt(d)*sqrt(g*cos(e+f*x))/sqrt(d*sin(e+f*x)))/((a^2-b^2)*f*g^(3/2)*sqrt(2))-1/2*a^2*d^(5/2)*log(sqrt(g)+cot(e+f*x)*sqrt(g)+sqrt(2)*sqrt(d)*sqrt(g*cos(e+f*x))/sqrt(d*sin(e+f*x)))/(b*(a^2-b^2)*f*g^(3/2)*sqrt(2))+1/2*b*d^(5/2)*log(sqrt(g)+cot(e+f*x)*sqrt(g)+sqrt(2)*sqrt(d)*sqrt(g*cos(e+f*x))/sqrt(d*sin(e+f*x)))/((a^2-b^2)*f*g^(3/2)*sqrt(2))+2*a*d*(d*sin(e+f*x))^(3/2)/((a^2-b^2)*f*g*sqrt(g*cos(e+f*x)))-2*a^3*d^3*EllipticPi(sqrt(g*cos(e+f*x))/(sqrt(g)*sqrt(1+sin(e+f*x))),-sqrt(-a+b)/sqrt(a+b),I)*sqrt(2)*sqrt(sin(e+f*x))/(b*(-a+b)^(3/2)*(a+b)^(3/2)*f*g^(3/2)*sqrt(d*sin(e+f*x)))+2*a^3*d^3*EllipticPi(sqrt(g*cos(e+f*x))/(sqrt(g)*sqrt(1+sin(e+f*x))),sqrt(-a+b)/sqrt(a+b),I)*sqrt(2)*sqrt(sin(e+f*x))/(b*(-a+b)^(3/2)*(a+b)^(3/2)*f*g^(3/2)*sqrt(d*sin(e+f*x)))-2*b*d^2*sqrt(d*sin(e+f*x))/((a^2-b^2)*f*g*sqrt(g*cos(e+f*x)))-2*a*d^2*sqrt(cos(e-1/4*Pi+f*x)^2)/cos(e-1/4*Pi+f*x)*EllipticE(sin(e-1/4*Pi+f*x),sqrt(2))*sqrt(g*cos(e+f*x))*sqrt(d*sin(e+f*x))/((a^2-b^2)*f*g^2*sqrt(sin(2*e+2*f*x)))],
[(d*sin(e+f*x))^(3/2)/((g*cos(e+f*x))^(3/2)*(a+b*sin(e+f*x))),x,10,-2*b*(d*sin(e+f*x))^(3/2)/((a^2-b^2)*f*g*sqrt(g*cos(e+f*x)))+2*a^2*d^2*EllipticPi(sqrt(g*cos(e+f*x))/(sqrt(g)*sqrt(1+sin(e+f*x))),-sqrt(-a+b)/sqrt(a+b),I)*sqrt(2)*sqrt(sin(e+f*x))/((-a+b)^(3/2)*(a+b)^(3/2)*f*g^(3/2)*sqrt(d*sin(e+f*x)))-2*a^2*d^2*EllipticPi(sqrt(g*cos(e+f*x))/(sqrt(g)*sqrt(1+sin(e+f*x))),sqrt(-a+b)/sqrt(a+b),I)*sqrt(2)*sqrt(sin(e+f*x))/((-a+b)^(3/2)*(a+b)^(3/2)*f*g^(3/2)*sqrt(d*sin(e+f*x)))+2*a*d*sqrt(d*sin(e+f*x))/((a^2-b^2)*f*g*sqrt(g*cos(e+f*x)))+2*b*d*sqrt(cos(e-1/4*Pi+f*x)^2)/cos(e-1/4*Pi+f*x)*EllipticE(sin(e-1/4*Pi+f*x),sqrt(2))*sqrt(g*cos(e+f*x))*sqrt(d*sin(e+f*x))/((a^2-b^2)*f*g^2*sqrt(sin(2*e+2*f*x)))],
[(d*sin(e+f*x))^(1/2)/((g*cos(e+f*x))^(3/2)*(a+b*sin(e+f*x))),x,11,2*a*(d*sin(e+f*x))^(3/2)/((a^2-b^2)*d*f*g*sqrt(g*cos(e+f*x)))-2*a*b*d*EllipticPi(sqrt(g*cos(e+f*x))/(sqrt(g)*sqrt(1+sin(e+f*x))),-sqrt(-a+b)/sqrt(a+b),I)*sqrt(2)*sqrt(sin(e+f*x))/((-a+b)^(3/2)*(a+b)^(3/2)*f*g^(3/2)*sqrt(d*sin(e+f*x)))+2*a*b*d*EllipticPi(sqrt(g*cos(e+f*x))/(sqrt(g)*sqrt(1+sin(e+f*x))),sqrt(-a+b)/sqrt(a+b),I)*sqrt(2)*sqrt(sin(e+f*x))/((-a+b)^(3/2)*(a+b)^(3/2)*f*g^(3/2)*sqrt(d*sin(e+f*x)))-2*b*sqrt(d*sin(e+f*x))/((a^2-b^2)*f*g*sqrt(g*cos(e+f*x)))-2*a*sqrt(cos(e-1/4*Pi+f*x)^2)/cos(e-1/4*Pi+f*x)*EllipticE(sin(e-1/4*Pi+f*x),sqrt(2))*sqrt(g*cos(e+f*x))*sqrt(d*sin(e+f*x))/((a^2-b^2)*f*g^2*sqrt(sin(2*e+2*f*x)))],
[1/((g*cos(e+f*x))^(3/2)*(d*sin(e+f*x))^(1/2)*(a+b*sin(e+f*x))),x,11,-2*b*(d*sin(e+f*x))^(3/2)/((a^2-b^2)*d^2*f*g*sqrt(g*cos(e+f*x)))+2*b^2*EllipticPi(sqrt(g*cos(e+f*x))/(sqrt(g)*sqrt(1+sin(e+f*x))),-sqrt(-a+b)/sqrt(a+b),I)*sqrt(2)*sqrt(sin(e+f*x))/((-a+b)^(3/2)*(a+b)^(3/2)*f*g^(3/2)*sqrt(d*sin(e+f*x)))-2*b^2*EllipticPi(sqrt(g*cos(e+f*x))/(sqrt(g)*sqrt(1+sin(e+f*x))),sqrt(-a+b)/sqrt(a+b),I)*sqrt(2)*sqrt(sin(e+f*x))/((-a+b)^(3/2)*(a+b)^(3/2)*f*g^(3/2)*sqrt(d*sin(e+f*x)))+2*a*sqrt(d*sin(e+f*x))/((a^2-b^2)*d*f*g*sqrt(g*cos(e+f*x)))+2*b*sqrt(cos(e-1/4*Pi+f*x)^2)/cos(e-1/4*Pi+f*x)*EllipticE(sin(e-1/4*Pi+f*x),sqrt(2))*sqrt(g*cos(e+f*x))*sqrt(d*sin(e+f*x))/((a^2-b^2)*d*f*g^2*sqrt(sin(2*e+2*f*x)))],
[1/((g*cos(e+f*x))^(3/2)*(d*sin(e+f*x))^(3/2)*(a+b*sin(e+f*x))),x,16,4*a*(d*sin(e+f*x))^(3/2)/((a^2-b^2)*d^3*f*g*sqrt(g*cos(e+f*x)))+2*b^2*(g*cos(e+f*x))^(3/2)/(a*(a^2-b^2)*d*f*g^3*sqrt(d*sin(e+f*x)))-2*a/((a^2-b^2)*d*f*g*sqrt(g*cos(e+f*x))*sqrt(d*sin(e+f*x)))-2*b^3*EllipticPi(sqrt(g*cos(e+f*x))/(sqrt(g)*sqrt(1+sin(e+f*x))),-sqrt(-a+b)/sqrt(a+b),I)*sqrt(2)*sqrt(sin(e+f*x))/(a*(-a+b)^(3/2)*(a+b)^(3/2)*d*f*g^(3/2)*sqrt(d*sin(e+f*x)))+2*b^3*EllipticPi(sqrt(g*cos(e+f*x))/(sqrt(g)*sqrt(1+sin(e+f*x))),sqrt(-a+b)/sqrt(a+b),I)*sqrt(2)*sqrt(sin(e+f*x))/(a*(-a+b)^(3/2)*(a+b)^(3/2)*d*f*g^(3/2)*sqrt(d*sin(e+f*x)))-2*b*sqrt(d*sin(e+f*x))/((a^2-b^2)*d^2*f*g*sqrt(g*cos(e+f*x)))-4*a*sqrt(cos(e-1/4*Pi+f*x)^2)/cos(e-1/4*Pi+f*x)*EllipticE(sin(e-1/4*Pi+f*x),sqrt(2))*sqrt(g*cos(e+f*x))*sqrt(d*sin(e+f*x))/((a^2-b^2)*d^2*f*g^2*sqrt(sin(2*e+2*f*x)))+2*b^2*sqrt(cos(e-1/4*Pi+f*x)^2)/cos(e-1/4*Pi+f*x)*EllipticE(sin(e-1/4*Pi+f*x),sqrt(2))*sqrt(g*cos(e+f*x))*sqrt(d*sin(e+f*x))/(a*(a^2-b^2)*d^2*f*g^2*sqrt(sin(2*e+2*f*x)))],
[1/((g*cos(e+f*x))^(3/2)*(d*sin(e+f*x))^(5/2)*(a+b*sin(e+f*x))),x,19,2/3*b^2*(g*cos(e+f*x))^(3/2)/(a*(a^2-b^2)*d*f*g^3*(d*sin(e+f*x))^(3/2))-2/3*a/((a^2-b^2)*d*f*g*(d*sin(e+f*x))^(3/2)*sqrt(g*cos(e+f*x)))-4*b*(d*sin(e+f*x))^(3/2)/((a^2-b^2)*d^4*f*g*sqrt(g*cos(e+f*x)))-2*b^3*(g*cos(e+f*x))^(3/2)/(a^2*(a^2-b^2)*d^2*f*g^3*sqrt(d*sin(e+f*x)))+2*b/((a^2-b^2)*d^2*f*g*sqrt(g*cos(e+f*x))*sqrt(d*sin(e+f*x)))+2*b^4*EllipticPi(sqrt(g*cos(e+f*x))/(sqrt(g)*sqrt(1+sin(e+f*x))),-sqrt(-a+b)/sqrt(a+b),I)*sqrt(2)*sqrt(sin(e+f*x))/(a^2*(-a+b)^(3/2)*(a+b)^(3/2)*d^2*f*g^(3/2)*sqrt(d*sin(e+f*x)))-2*b^4*EllipticPi(sqrt(g*cos(e+f*x))/(sqrt(g)*sqrt(1+sin(e+f*x))),sqrt(-a+b)/sqrt(a+b),I)*sqrt(2)*sqrt(sin(e+f*x))/(a^2*(-a+b)^(3/2)*(a+b)^(3/2)*d^2*f*g^(3/2)*sqrt(d*sin(e+f*x)))+8/3*a*sqrt(d*sin(e+f*x))/((a^2-b^2)*d^3*f*g*sqrt(g*cos(e+f*x)))+4*b*sqrt(cos(e-1/4*Pi+f*x)^2)/cos(e-1/4*Pi+f*x)*EllipticE(sin(e-1/4*Pi+f*x),sqrt(2))*sqrt(g*cos(e+f*x))*sqrt(d*sin(e+f*x))/((a^2-b^2)*d^3*f*g^2*sqrt(sin(2*e+2*f*x)))-2*b^3*sqrt(cos(e-1/4*Pi+f*x)^2)/cos(e-1/4*Pi+f*x)*EllipticE(sin(e-1/4*Pi+f*x),sqrt(2))*sqrt(g*cos(e+f*x))*sqrt(d*sin(e+f*x))/(a^2*(a^2-b^2)*d^3*f*g^2*sqrt(sin(2*e+2*f*x)))],

# Integrands of the form (g Cos[e+f x])^p (d Sin[e+f x])^n / (a+b Sin[e+f x])^2

# Integrands of the form (g Cos[e+f x])^(p/2) (d Sin[e+f x])^(n/2) / (a+b Sin[e+f x])^2

# p>0
[(g*cos(e+f*x))^(3/2)/((a+b*sin(e+f*x))^2*sqrt(d*sin(e+f*x))),x,8,b*g^2*EllipticPi(sqrt(d*sin(e+f*x))/(sqrt(d)*sqrt(1+cos(e+f*x))),-a/(b-sqrt(-a^2+b^2)),I)*sqrt(2)*sqrt(cos(e+f*x))/(a^2*f*sqrt(-a^2+b^2)*sqrt(d)*sqrt(g*cos(e+f*x)))-b*g^2*EllipticPi(sqrt(d*sin(e+f*x))/(sqrt(d)*sqrt(1+cos(e+f*x))),-a/(b+sqrt(-a^2+b^2)),I)*sqrt(2)*sqrt(cos(e+f*x))/(a^2*f*sqrt(-a^2+b^2)*sqrt(d)*sqrt(g*cos(e+f*x)))+g*sqrt(g*cos(e+f*x))*sqrt(d*sin(e+f*x))/(a*d*f*(a+b*sin(e+f*x)))+1/2*g^2*sqrt(cos(e-1/4*Pi+f*x)^2)/cos(e-1/4*Pi+f*x)*EllipticF(sin(e-1/4*Pi+f*x),sqrt(2))*sqrt(sin(2*e+2*f*x))/(a^2*f*sqrt(g*cos(e+f*x))*sqrt(d*sin(e+f*x)))],

# p<0

# Integrands of the form (g Sec[e+f x])^p (d Sin[e+f x])^n (a+b Sin[e+f x])^m

# Integrands of the form Sec[e+f x]^1 (d Sin[e+f x])^n (a+b Sin[e+f x])^m

# Integrands of the form Sec[e+f x]^2 (d Sin[e+f x])^n (a+b Sin[e+f x])^m

# Integrands of the form Sec[e+f x]^2 Sin[e+f x]^n (a+b Sin[e+f x])^m

# m>0
[sec(c+d*x)^2*sin(c+d*x)^4*(a+b*sin(c+d*x)),x,8,-3/2*a*x+2*b*cos(c+d*x)/d-1/3*b*cos(c+d*x)^3/d+b*sec(c+d*x)/d+3/2*a*tan(c+d*x)/d-1/2*a*sin(c+d*x)^2*tan(c+d*x)/d],
[sec(c+d*x)^2*sin(c+d*x)^3*(a+b*sin(c+d*x)),x,8,-3/2*b*x+a*cos(c+d*x)/d+a*sec(c+d*x)/d+3/2*b*tan(c+d*x)/d-1/2*b*sin(c+d*x)^2*tan(c+d*x)/d],
[sec(c+d*x)^2*sin(c+d*x)^2*(a+b*sin(c+d*x)),x,7,-a*x+b*cos(c+d*x)/d+b*sec(c+d*x)/d+a*tan(c+d*x)/d],
[sec(c+d*x)^2*sin(c+d*x)*(a+b*sin(c+d*x)),x,5,-b*x+a*sec(c+d*x)/d+b*tan(c+d*x)/d],
[csc(c+d*x)*sec(c+d*x)^2*(a+b*sin(c+d*x)),x,6,-a*arctanh(cos(c+d*x))/d+a*sec(c+d*x)/d+b*tan(c+d*x)/d],
[csc(c+d*x)^2*sec(c+d*x)^2*(a+b*sin(c+d*x)),x,7,-b*arctanh(cos(c+d*x))/d-a*cot(c+d*x)/d+b*sec(c+d*x)/d+a*tan(c+d*x)/d],
[csc(c+d*x)^3*sec(c+d*x)^2*(a+b*sin(c+d*x)),x,8,-3/2*a*arctanh(cos(c+d*x))/d-b*cot(c+d*x)/d+3/2*a*sec(c+d*x)/d-1/2*a*csc(c+d*x)^2*sec(c+d*x)/d+b*tan(c+d*x)/d],
[sec(c+d*x)^2*sin(c+d*x)^3*(a+b*sin(c+d*x))^2,x,8,-3*a*b*x+(a^2+2*b^2)*cos(c+d*x)/d-1/3*b^2*cos(c+d*x)^3/d+(a^2+b^2)*sec(c+d*x)/d+3*a*b*tan(c+d*x)/d-a*b*sin(c+d*x)^2*tan(c+d*x)/d],
[sec(c+d*x)^2*sin(c+d*x)^2*(a+b*sin(c+d*x))^2,x,11,-a^2*x-3/2*b^2*x+2*a*b*cos(c+d*x)/d+2*a*b*sec(c+d*x)/d+a^2*tan(c+d*x)/d+3/2*b^2*tan(c+d*x)/d-1/2*b^2*sin(c+d*x)^2*tan(c+d*x)/d],
[sec(c+d*x)^2*sin(c+d*x)*(a+b*sin(c+d*x))^2,x,4,-2*a*b*x+2*b^2*cos(c+d*x)/d+sec(c+d*x)*(a+b*sin(c+d*x))^2/d],
[csc(c+d*x)*sec(c+d*x)^2*(a+b*sin(c+d*x))^2,x,8,-a^2*arctanh(cos(c+d*x))/d+(a^2+b^2)*sec(c+d*x)/d+2*a*b*tan(c+d*x)/d,(a^2+b^2)*sec(c+d*x)/d-a^2*arctanh(sqrt(cos(c+d*x)^2))*sec(c+d*x)*sqrt(cos(c+d*x)^2)/d+2*a*b*tan(c+d*x)/d],
[csc(c+d*x)^2*sec(c+d*x)^2*(a+b*sin(c+d*x))^2,x,7,-2*a*b*arctanh(cos(c+d*x))/d-a^2*cot(c+d*x)/d+2*a*b*sec(c+d*x)/d+(a^2+b^2)*tan(c+d*x)/d],
[csc(c+d*x)^3*sec(c+d*x)^2*(a+b*sin(c+d*x))^2,x,10,-1/2*(3*a^2+2*b^2)*arctanh(cos(c+d*x))/d-2*a*b*cot(c+d*x)/d+1/2*(3*a^2+2*b^2)*sec(c+d*x)/d-1/2*a^2*csc(c+d*x)^2*sec(c+d*x)/d+2*a*b*tan(c+d*x)/d,-2*a*b*cot(c+d*x)/d+1/2*(3*a^2+2*b^2)*sec(c+d*x)/d-1/2*a^2*csc(c+d*x)^2*sec(c+d*x)/d-1/2*(3*a^2+2*b^2)*arctanh(sqrt(cos(c+d*x)^2))*sec(c+d*x)*sqrt(cos(c+d*x)^2)/d+2*a*b*tan(c+d*x)/d],
[csc(c+d*x)^4*sec(c+d*x)^2*(a+b*sin(c+d*x))^2,x,8,-3*a*b*arctanh(cos(c+d*x))/d-(2*a^2+b^2)*cot(c+d*x)/d-1/3*a^2*cot(c+d*x)^3/d+3*a*b*sec(c+d*x)/d-a*b*csc(c+d*x)^2*sec(c+d*x)/d+(a^2+b^2)*tan(c+d*x)/d],
[sec(c+d*x)^2*sin(c+d*x)^3*(a+b*sin(c+d*x))^3,x,17,-9/2*a^2*b*x-15/8*b^3*x+a^3*cos(c+d*x)/d+6*a*b^2*cos(c+d*x)/d-a*b^2*cos(c+d*x)^3/d+a^3*sec(c+d*x)/d+3*a*b^2*sec(c+d*x)/d+9/2*a^2*b*tan(c+d*x)/d+15/8*b^3*tan(c+d*x)/d-3/2*a^2*b*sin(c+d*x)^2*tan(c+d*x)/d-5/8*b^3*sin(c+d*x)^2*tan(c+d*x)/d-1/4*b^3*sin(c+d*x)^4*tan(c+d*x)/d],
[sec(c+d*x)^2*sin(c+d*x)^2*(a+b*sin(c+d*x))^3,x,14,-a^3*x-9/2*a*b^2*x+3*a^2*b*cos(c+d*x)/d+2*b^3*cos(c+d*x)/d-1/3*b^3*cos(c+d*x)^3/d+3*a^2*b*sec(c+d*x)/d+b^3*sec(c+d*x)/d+a^3*tan(c+d*x)/d+9/2*a*b^2*tan(c+d*x)/d-3/2*a*b^2*sin(c+d*x)^2*tan(c+d*x)/d],
[sec(c+d*x)^2*sin(c+d*x)*(a+b*sin(c+d*x))^3,x,3,-3/2*b*(2*a^2+b^2)*x+6*a*b^2*cos(c+d*x)/d+3/2*b^3*cos(c+d*x)*sin(c+d*x)/d+sec(c+d*x)*(a+b*sin(c+d*x))^3/d],
[csc(c+d*x)*sec(c+d*x)^2*(a+b*sin(c+d*x))^3,x,11,-b^3*x-a^3*arctanh(cos(c+d*x))/d+a^3*sec(c+d*x)/d+3*a*b^2*sec(c+d*x)/d+3*a^2*b*tan(c+d*x)/d+b^3*tan(c+d*x)/d],
[csc(c+d*x)^2*sec(c+d*x)^2*(a+b*sin(c+d*x))^3,x,12,-3*a^2*b*arctanh(cos(c+d*x))/d-a^3*cot(c+d*x)/d+3*a^2*b*sec(c+d*x)/d+b^3*sec(c+d*x)/d+a^3*tan(c+d*x)/d+3*a*b^2*tan(c+d*x)/d],
[csc(c+d*x)^3*sec(c+d*x)^2*(a+b*sin(c+d*x))^3,x,14,-3/2*a^3*arctanh(cos(c+d*x))/d-3*a*b^2*arctanh(cos(c+d*x))/d-3*a^2*b*cot(c+d*x)/d+3/2*a^3*sec(c+d*x)/d+3*a*b^2*sec(c+d*x)/d-1/2*a^3*csc(c+d*x)^2*sec(c+d*x)/d+3*a^2*b*tan(c+d*x)/d+b^3*tan(c+d*x)/d],
[csc(c+d*x)^4*sec(c+d*x)^2*(a+b*sin(c+d*x))^3,x,15,-9/2*a^2*b*arctanh(cos(c+d*x))/d-b^3*arctanh(cos(c+d*x))/d-2*a^3*cot(c+d*x)/d-3*a*b^2*cot(c+d*x)/d-1/3*a^3*cot(c+d*x)^3/d+9/2*a^2*b*sec(c+d*x)/d+b^3*sec(c+d*x)/d-3/2*a^2*b*csc(c+d*x)^2*sec(c+d*x)/d+a^3*tan(c+d*x)/d+3*a*b^2*tan(c+d*x)/d],

# m<0
[sec(c+d*x)^2*sin(c+d*x)^4/(a+b*sin(c+d*x))^2,x,12,-x/b^2-2*a^5*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(b^2*(a^2-b^2)^(5/2)*d)+4*a^3*(a^2-2*b^2)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(b^2*(a^2-b^2)^(5/2)*d)+1/2*cos(c+d*x)/((a+b)^2*d*(1-sin(c+d*x)))-1/2*cos(c+d*x)/((a-b)^2*d*(1+sin(c+d*x)))-a^4*cos(c+d*x)/(b*(a^2-b^2)^2*d*(a+b*sin(c+d*x)))],
[sec(c+d*x)^2*sin(c+d*x)^3/(a+b*sin(c+d*x))^2,x,12,2*a^4*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(b*(a^2-b^2)^(5/2)*d)-2*a^2*(a^2-3*b^2)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(b*(a^2-b^2)^(5/2)*d)+1/2*cos(c+d*x)/((a+b)^2*d*(1-sin(c+d*x)))+1/2*cos(c+d*x)/((a-b)^2*d*(1+sin(c+d*x)))+a^3*cos(c+d*x)/((a^2-b^2)^2*d*(a+b*sin(c+d*x)))],
[sec(c+d*x)^2*sin(c+d*x)^2/(a+b*sin(c+d*x))^2,x,12,-2*a^3*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/((a^2-b^2)^(5/2)*d)-4*a*b^2*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/((a^2-b^2)^(5/2)*d)+1/2*cos(c+d*x)/((a+b)^2*d*(1-sin(c+d*x)))-1/2*cos(c+d*x)/((a-b)^2*d*(1+sin(c+d*x)))-a^2*b*cos(c+d*x)/((a^2-b^2)^2*d*(a+b*sin(c+d*x)))],
[sec(c+d*x)^2*sin(c+d*x)/(a+b*sin(c+d*x))^2,x,6,2*b*(2*a^2+b^2)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/((a^2-b^2)^(5/2)*d)-a*sec(c+d*x)/((a^2-b^2)*d*(a+b*sin(c+d*x)))+sec(c+d*x)*(2*a^2+b^2-3*a*b*sin(c+d*x))/((a^2-b^2)^2*d)],
[csc(c+d*x)*sec(c+d*x)^2/(a+b*sin(c+d*x))^2,x,13,2*b^3*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/((a^2-b^2)^(5/2)*d)+2*b^3*(3*a^2-b^2)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(a^2*(a^2-b^2)^(5/2)*d)-arctanh(cos(c+d*x))/(a^2*d)+1/2*cos(c+d*x)/((a+b)^2*d*(1-sin(c+d*x)))+1/2*cos(c+d*x)/((a-b)^2*d*(1+sin(c+d*x)))+b^4*cos(c+d*x)/(a*(a^2-b^2)^2*d*(a+b*sin(c+d*x)))],
[csc(c+d*x)^2*sec(c+d*x)^2/(a+b*sin(c+d*x))^2,x,15,-2*b^4*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(a*(a^2-b^2)^(5/2)*d)-4*b^4*(2*a^2-b^2)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(a^3*(a^2-b^2)^(5/2)*d)+2*b*arctanh(cos(c+d*x))/(a^3*d)-cot(c+d*x)/(a^2*d)+1/2*cos(c+d*x)/((a+b)^2*d*(1-sin(c+d*x)))-1/2*cos(c+d*x)/((a-b)^2*d*(1+sin(c+d*x)))-b^5*cos(c+d*x)/(a^2*(a^2-b^2)^2*d*(a+b*sin(c+d*x)))],
[csc(c+d*x)^3*sec(c+d*x)^2/(a+b*sin(c+d*x))^2,x,17,2*b^5*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(a^2*(a^2-b^2)^(5/2)*d)+2*b^5*(5*a^2-3*b^2)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(a^4*(a^2-b^2)^(5/2)*d)-1/2*arctanh(cos(c+d*x))/(a^2*d)-(a^2+3*b^2)*arctanh(cos(c+d*x))/(a^4*d)+2*b*cot(c+d*x)/(a^3*d)-1/2*cot(c+d*x)*csc(c+d*x)/(a^2*d)+1/2*cos(c+d*x)/((a+b)^2*d*(1-sin(c+d*x)))+1/2*cos(c+d*x)/((a-b)^2*d*(1+sin(c+d*x)))+b^6*cos(c+d*x)/(a^3*(a^2-b^2)^2*d*(a+b*sin(c+d*x)))],
[sec(c+d*x)^2*sin(c+d*x)^4/(a+b*sin(c+d*x))^3,x,18,4*a^4*(a^2-2*b^2)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(b^2*(a^2-b^2)^(7/2)*d)-a^4*(2*a^2+b^2)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(b^2*(a^2-b^2)^(7/2)*d)-2*a^2*(a^4-3*a^2*b^2+6*b^4)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(b^2*(a^2-b^2)^(7/2)*d)+1/2*cos(c+d*x)/((a+b)^3*d*(1-sin(c+d*x)))-1/2*cos(c+d*x)/((a-b)^3*d*(1+sin(c+d*x)))-1/2*a^4*cos(c+d*x)/(b*(a^2-b^2)^2*d*(a+b*sin(c+d*x))^2)-3/2*a^5*cos(c+d*x)/(b*(a^2-b^2)^3*d*(a+b*sin(c+d*x)))+2*a^3*(a^2-2*b^2)*cos(c+d*x)/(b*(a^2-b^2)^3*d*(a+b*sin(c+d*x)))],
[sec(c+d*x)^2*sin(c+d*x)^3/(a+b*sin(c+d*x))^3,x,18,-2*a^3*(a^2-3*b^2)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(b*(a^2-b^2)^(7/2)*d)+a^3*(2*a^2+b^2)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(b*(a^2-b^2)^(7/2)*d)+2*a*b*(a^2+3*b^2)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/((a^2-b^2)^(7/2)*d)+1/2*cos(c+d*x)/((a+b)^3*d*(1-sin(c+d*x)))+1/2*cos(c+d*x)/((a-b)^3*d*(1+sin(c+d*x)))+1/2*a^3*cos(c+d*x)/((a^2-b^2)^2*d*(a+b*sin(c+d*x))^2)+3/2*a^4*cos(c+d*x)/((a^2-b^2)^3*d*(a+b*sin(c+d*x)))-a^2*(a^2-3*b^2)*cos(c+d*x)/((a^2-b^2)^3*d*(a+b*sin(c+d*x)))],
[sec(c+d*x)^2*sin(c+d*x)^2/(a+b*sin(c+d*x))^3,x,18,-4*a^2*b^2*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/((a^2-b^2)^(7/2)*d)-a^2*(2*a^2+b^2)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/((a^2-b^2)^(7/2)*d)-2*b^2*(3*a^2+b^2)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/((a^2-b^2)^(7/2)*d)+1/2*cos(c+d*x)/((a+b)^3*d*(1-sin(c+d*x)))-1/2*cos(c+d*x)/((a-b)^3*d*(1+sin(c+d*x)))-1/2*a^2*b*cos(c+d*x)/((a^2-b^2)^2*d*(a+b*sin(c+d*x))^2)-3/2*a^3*b*cos(c+d*x)/((a^2-b^2)^3*d*(a+b*sin(c+d*x)))-2*a*b^3*cos(c+d*x)/((a^2-b^2)^3*d*(a+b*sin(c+d*x)))],
[sec(c+d*x)^2*sin(c+d*x)/(a+b*sin(c+d*x))^3,x,7,3*a*b*(2*a^2+3*b^2)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/((a^2-b^2)^(7/2)*d)-1/2*a*sec(c+d*x)/((a^2-b^2)*d*(a+b*sin(c+d*x))^2)-1/2*(3*a^2+2*b^2)*sec(c+d*x)/((a^2-b^2)^2*d*(a+b*sin(c+d*x)))+1/2*sec(c+d*x)*(3*a*(2*a^2+3*b^2)-b*(11*a^2+4*b^2)*sin(c+d*x))/((a^2-b^2)^3*d)],
[csc(c+d*x)*sec(c+d*x)^2/(a+b*sin(c+d*x))^3,x,19,2*b^3*(3*a^2-b^2)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(a*(a^2-b^2)^(7/2)*d)+b^3*(2*a^2+b^2)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(a*(a^2-b^2)^(7/2)*d)+2*b^3*(6*a^4-3*a^2*b^2+b^4)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(a^3*(a^2-b^2)^(7/2)*d)-arctanh(cos(c+d*x))/(a^3*d)+1/2*cos(c+d*x)/((a+b)^3*d*(1-sin(c+d*x)))+1/2*cos(c+d*x)/((a-b)^3*d*(1+sin(c+d*x)))+1/2*b^4*cos(c+d*x)/(a*(a^2-b^2)^2*d*(a+b*sin(c+d*x))^2)+3/2*b^4*cos(c+d*x)/((a^2-b^2)^3*d*(a+b*sin(c+d*x)))+b^4*(3*a^2-b^2)*cos(c+d*x)/(a^2*(a^2-b^2)^3*d*(a+b*sin(c+d*x)))],
[csc(c+d*x)^2*sec(c+d*x)^2/(a+b*sin(c+d*x))^3,x,21,-4*b^4*(2*a^2-b^2)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(a^2*(a^2-b^2)^(7/2)*d)-b^4*(2*a^2+b^2)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(a^2*(a^2-b^2)^(7/2)*d)-2*b^4*(10*a^4-9*a^2*b^2+3*b^4)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(a^4*(a^2-b^2)^(7/2)*d)+3*b*arctanh(cos(c+d*x))/(a^4*d)-cot(c+d*x)/(a^3*d)+1/2*cos(c+d*x)/((a+b)^3*d*(1-sin(c+d*x)))-1/2*cos(c+d*x)/((a-b)^3*d*(1+sin(c+d*x)))-1/2*b^5*cos(c+d*x)/(a^2*(a^2-b^2)^2*d*(a+b*sin(c+d*x))^2)-3/2*b^5*cos(c+d*x)/(a*(a^2-b^2)^3*d*(a+b*sin(c+d*x)))-2*b^5*(2*a^2-b^2)*cos(c+d*x)/(a^3*(a^2-b^2)^3*d*(a+b*sin(c+d*x)))],
[csc(c+d*x)^3*sec(c+d*x)^2/(a+b*sin(c+d*x))^3,x,23,2*b^5*(5*a^2-3*b^2)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(a^3*(a^2-b^2)^(7/2)*d)+b^5*(2*a^2+b^2)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(a^3*(a^2-b^2)^(7/2)*d)+2*b^5*(15*a^4-17*a^2*b^2+6*b^4)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(a^5*(a^2-b^2)^(7/2)*d)-1/2*arctanh(cos(c+d*x))/(a^3*d)-(a^2+6*b^2)*arctanh(cos(c+d*x))/(a^5*d)+3*b*cot(c+d*x)/(a^4*d)-1/2*cot(c+d*x)*csc(c+d*x)/(a^3*d)+1/2*cos(c+d*x)/((a+b)^3*d*(1-sin(c+d*x)))+1/2*cos(c+d*x)/((a-b)^3*d*(1+sin(c+d*x)))+1/2*b^6*cos(c+d*x)/(a^3*(a^2-b^2)^2*d*(a+b*sin(c+d*x))^2)+3/2*b^6*cos(c+d*x)/(a^2*(a^2-b^2)^3*d*(a+b*sin(c+d*x)))+b^6*(5*a^2-3*b^2)*cos(c+d*x)/(a^4*(a^2-b^2)^3*d*(a+b*sin(c+d*x)))],

# Integrands of the form Sec[e+f x]^2 Sin[e+f x]^n (a+b Sin[e+f x])^(m/2)

# Integrands of the form Sec[e+f x]^2 (d Sin[e+f x])^(n/2) (a+b Sin[e+f x])^(m/2)
[sec(e+f*x)^2*(a+b*sin(e+f*x))^(1/2)/sqrt(d*sin(e+f*x)),x,2,sec(e+f*x)*sqrt(d*sin(e+f*x))*sqrt(a+b*sin(e+f*x))/(d*f)-EllipticF(sqrt(d)*sqrt(a+b*sin(e+f*x))/(sqrt(a+b)*sqrt(d*sin(e+f*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-csc(e+f*x))/(a+b))*sqrt(a*(1+csc(e+f*x))/(a-b))*tan(e+f*x)/(f*sqrt(d))],
[sec(e+f*x)^2*(a+b*sin(e+f*x))^(3/2)/sqrt(d*sin(e+f*x)),x,-1,sec(e+f*x)*(b+a*sin(e+f*x))*sqrt(a+b*sin(e+f*x))/(f*sqrt(d*sin(e+f*x)))-(a+b)^(3/2)*EllipticF(sqrt(d)*sqrt(a+b*sin(e+f*x))/(sqrt(a+b)*sqrt(d*sin(e+f*x))),sqrt((-a-b)/(a-b)))*sqrt(-a*(-1+csc(e+f*x))/(a+b))*sqrt(a*(1+csc(e+f*x))/(a-b))*tan(e+f*x)/(f*sqrt(d))-b*(a+b)*EllipticE(sqrt((-b-a*csc(e+f*x))/(a-b)),sqrt((-a+b)/(a+b)))*(1+sin(e+f*x))*sqrt(-a*(-1+csc(e+f*x))/(a+b))*sqrt((b+a*csc(e+f*x))/(-a+b))*tan(e+f*x)/(f*sqrt(a*(1+csc(e+f*x))/(a-b))*sqrt(d*sin(e+f*x))*sqrt(a+b*sin(e+f*x)))],

# Integrands of the form Sec[e+f x]^3 (d Sin[e+f x])^n (a+b Sin[e+f x])^m

# Integrands of the form Sec[e+f x]^4 (d Sin[e+f x])^n (a+b Sin[e+f x])^m

# Integrands of the form Sec[e+f x]^4 Sin[e+f x]^n (a+b Sin[e+f x])^m

# Integrands of the form Sec[e+f x]^4 Sin[e+f x]^n (a+b Sin[e+f x])^(m/2)

# Integrands of the form Sec[e+f x]^4 (d Sin[e+f x])^(n/2) (a+b Sin[e+f x])^(m/2)
[sec(e+f*x)^4*(a+b*sin(e+f*x))^(5/2)/sqrt(d*sin(e+f*x)),x,-1,1/3*sec(e+f*x)^3*(a+b*sin(e+f*x))^(5/2)*sqrt(d*sin(e+f*x))/(d*f)+5/6*a*sec(e+f*x)*(b+a*sin(e+f*x))*sqrt(a+b*sin(e+f*x))/(f*sqrt(d*sin(e+f*x)))-5/6*a*(a+b)^(3/2)*EllipticF(sqrt(d)*sqrt(a+b*sin(e+f*x))/(sqrt(a+b)*sqrt(d*sin(e+f*x))),sqrt((-a-b)/(a-b)))*sqrt(-a*(-1+csc(e+f*x))/(a+b))*sqrt(a*(1+csc(e+f*x))/(a-b))*tan(e+f*x)/(f*sqrt(d))-5/6*a*b*(a+b)*EllipticE(sqrt((-b-a*csc(e+f*x))/(a-b)),sqrt((-a+b)/(a+b)))*(1+sin(e+f*x))*sqrt(-a*(-1+csc(e+f*x))/(a+b))*sqrt((b+a*csc(e+f*x))/(-a+b))*tan(e+f*x)/(f*sqrt(a*(1+csc(e+f*x))/(a-b))*sqrt(d*sin(e+f*x))*sqrt(a+b*sin(e+f*x)))],

# Integrands of the form Sec[e+f x]^5 (d Sin[e+f x])^n (a+b Sin[e+f x])^m

# Integrands of the form Sec[e+f x]^5 Sin[e+f x]^n (a+b Sin[e+f x])^m

# m>0
[sec(c+d*x)^5*sin(c+d*x)^7*(a+b*sin(c+d*x)),x,11,35/8*b*arctanh(sin(c+d*x))/d+1/2*a*cos(c+d*x)^2/d-3*a*log(cos(c+d*x))/d-3/2*a*sec(c+d*x)^2/d+1/4*a*sec(c+d*x)^4/d-35/8*b*sin(c+d*x)/d-35/24*b*sin(c+d*x)^3/d-7/8*b*sin(c+d*x)^3*tan(c+d*x)^2/d+1/4*b*sin(c+d*x)^3*tan(c+d*x)^4/d],
[sec(c+d*x)^5*sin(c+d*x)^6*(a+b*sin(c+d*x)),x,10,15/8*a*arctanh(sin(c+d*x))/d+1/2*b*cos(c+d*x)^2/d-3*b*log(cos(c+d*x))/d-3/2*b*sec(c+d*x)^2/d+1/4*b*sec(c+d*x)^4/d-15/8*a*sin(c+d*x)/d-5/8*a*sin(c+d*x)*tan(c+d*x)^2/d+1/4*a*sin(c+d*x)*tan(c+d*x)^4/d],
[sec(c+d*x)^5*sin(c+d*x)^5*(a+b*sin(c+d*x)),x,7,-1/16*(8*a+15*b)*log(1-sin(c+d*x))/d-1/16*(8*a-15*b)*log(1+sin(c+d*x))/d-15/8*b*sin(c+d*x)/d-1/8*(4*a+5*b*sin(c+d*x))*tan(c+d*x)^2/d+1/4*(a+b*sin(c+d*x))*tan(c+d*x)^4/d],
[sec(c+d*x)^5*sin(c+d*x)^4*(a+b*sin(c+d*x)),x,7,3/8*a*arctanh(sin(c+d*x))/d-b*log(cos(c+d*x))/d-3/8*a*sec(c+d*x)*tan(c+d*x)/d-1/2*b*tan(c+d*x)^2/d+1/4*a*sec(c+d*x)*tan(c+d*x)^3/d+1/4*b*tan(c+d*x)^4/d],
[sec(c+d*x)^5*sin(c+d*x)^3*(a+b*sin(c+d*x)),x,6,3/8*b*arctanh(sin(c+d*x))/d-3/8*b*sec(c+d*x)*tan(c+d*x)/d+1/4*b*sec(c+d*x)*tan(c+d*x)^3/d+1/4*a*tan(c+d*x)^4/d],
[sec(c+d*x)^5*sin(c+d*x)^2*(a+b*sin(c+d*x)),x,6,-1/8*a*arctanh(sin(c+d*x))/d-1/8*a*sec(c+d*x)*tan(c+d*x)/d+1/4*a*sec(c+d*x)^3*tan(c+d*x)/d+1/4*b*tan(c+d*x)^4/d],
[sec(c+d*x)^5*sin(c+d*x)*(a+b*sin(c+d*x)),x,6,-1/8*b*arctanh(sin(c+d*x))/d+1/4*a*sec(c+d*x)^4/d-1/8*b*sec(c+d*x)*tan(c+d*x)/d+1/4*b*sec(c+d*x)^3*tan(c+d*x)/d],
[csc(c+d*x)*sec(c+d*x)^5*(a+b*sin(c+d*x)),x,8,3/8*b*arctanh(sin(c+d*x))/d+a*log(tan(c+d*x))/d+3/8*b*sec(c+d*x)*tan(c+d*x)/d+1/4*b*sec(c+d*x)^3*tan(c+d*x)/d+a*tan(c+d*x)^2/d+1/4*a*tan(c+d*x)^4/d],
[csc(c+d*x)^2*sec(c+d*x)^5*(a+b*sin(c+d*x)),x,10,15/8*a*arctanh(sin(c+d*x))/d-15/8*a*csc(c+d*x)/d+b*log(tan(c+d*x))/d+5/8*a*csc(c+d*x)*sec(c+d*x)^2/d+1/4*a*csc(c+d*x)*sec(c+d*x)^4/d+b*tan(c+d*x)^2/d+1/4*b*tan(c+d*x)^4/d],
[csc(c+d*x)^3*sec(c+d*x)^5*(a+b*sin(c+d*x)),x,10,15/8*b*arctanh(sin(c+d*x))/d-1/2*a*cot(c+d*x)^2/d-15/8*b*csc(c+d*x)/d+3*a*log(tan(c+d*x))/d+5/8*b*csc(c+d*x)*sec(c+d*x)^2/d+1/4*b*csc(c+d*x)*sec(c+d*x)^4/d+3/2*a*tan(c+d*x)^2/d+1/4*a*tan(c+d*x)^4/d],
[csc(c+d*x)^4*sec(c+d*x)^5*(a+b*sin(c+d*x)),x,11,35/8*a*arctanh(sin(c+d*x))/d-1/2*b*cot(c+d*x)^2/d-35/8*a*csc(c+d*x)/d-35/24*a*csc(c+d*x)^3/d+3*b*log(tan(c+d*x))/d+7/8*a*csc(c+d*x)^3*sec(c+d*x)^2/d+1/4*a*csc(c+d*x)^3*sec(c+d*x)^4/d+3/2*b*tan(c+d*x)^2/d+1/4*b*tan(c+d*x)^4/d],
[sec(c+d*x)^5*sin(c+d*x)^6*(a+b*sin(c+d*x))^2,x,9,-1/16*(15*a^2+48*a*b+35*b^2)*log(1-sin(c+d*x))/d+1/16*(15*a^2-48*a*b+35*b^2)*log(1+sin(c+d*x))/d-(a^2+3*b^2)*sin(c+d*x)/d-a*b*sin(c+d*x)^2/d-1/3*b^2*sin(c+d*x)^3/d-1/8*sec(c+d*x)^2*(11*b+9*a*sin(c+d*x))*(a+b*sin(c+d*x))/d+1/4*sec(c+d*x)^3*(a+b*sin(c+d*x))^2*tan(c+d*x)/d],
[sec(c+d*x)^5*sin(c+d*x)^5*(a+b*sin(c+d*x))^2,x,8,-1/8*(4*a^2+15*a*b+12*b^2)*log(1-sin(c+d*x))/d+1/8*(15*a*b-4*(a^2+3*b^2))*log(1+sin(c+d*x))/d-2*a*b*sin(c+d*x)/d-1/2*b^2*sin(c+d*x)^2/d+1/4*sec(c+d*x)^4*(a+b*sin(c+d*x))^2/d-1/4*sec(c+d*x)^2*(a+b*sin(c+d*x))*(4*a+5*b*sin(c+d*x))/d],
[sec(c+d*x)^5*sin(c+d*x)^4*(a+b*sin(c+d*x))^2,x,9,-1/16*(3*a^2+16*a*b+15*b^2)*log(1-sin(c+d*x))/d+1/16*(3*a^2-16*a*b+15*b^2)*log(1+sin(c+d*x))/d-b^2*sin(c+d*x)/d-1/8*sec(c+d*x)^2*(7*b+5*a*sin(c+d*x))*(a+b*sin(c+d*x))/d+1/4*sec(c+d*x)^3*(a+b*sin(c+d*x))^2*tan(c+d*x)/d],
[sec(c+d*x)^5*sin(c+d*x)^3*(a+b*sin(c+d*x))^2,x,7,-1/8*b*(3*a+4*b)*log(1-sin(c+d*x))/d+1/8*(3*a-4*b)*b*log(1+sin(c+d*x))/d+1/4*sec(c+d*x)^4*(a+b*sin(c+d*x))^2/d-1/4*sec(c+d*x)^2*(a+b*sin(c+d*x))*(2*a+3*b*sin(c+d*x))/d],
[sec(c+d*x)^5*sin(c+d*x)^2*(a+b*sin(c+d*x))^2,x,5,-1/8*(a^2-3*b^2)*arctanh(sin(c+d*x))/d-1/8*sec(c+d*x)^2*(4*a*b+(a^2+3*b^2)*sin(c+d*x))/d+1/4*sec(c+d*x)^3*(a+b*sin(c+d*x))^2*tan(c+d*x)/d],
[sec(c+d*x)^5*sin(c+d*x)*(a+b*sin(c+d*x))^2,x,6,-1/4*a*b*arctanh(sin(c+d*x))/d+1/4*sec(c+d*x)^4*(a+b*sin(c+d*x))^2/d-1/4*sec(c+d*x)^2*(b^2+a*b*sin(c+d*x))/d],
[csc(c+d*x)*sec(c+d*x)^5*(a+b*sin(c+d*x))^2,x,6,-1/8*a*(4*a+3*b)*log(1-sin(c+d*x))/d+a^2*log(sin(c+d*x))/d-1/8*a*(4*a-3*b)*log(1+sin(c+d*x))/d+1/4*a*sec(c+d*x)^2*(2*a+3*b*sin(c+d*x))/d+1/4*sec(c+d*x)^4*(a^2+b^2+2*a*b*sin(c+d*x))/d],
[csc(c+d*x)^2*sec(c+d*x)^5*(a+b*sin(c+d*x))^2,x,6,-a^2*csc(c+d*x)/d-1/16*(15*a^2+16*a*b+3*b^2)*log(1-sin(c+d*x))/d+2*a*b*log(sin(c+d*x))/d+1/16*(15*a^2-16*a*b+3*b^2)*log(1+sin(c+d*x))/d+1/8*b*sec(c+d*x)^2*(8*a+(3+7*a^2/b^2)*b*sin(c+d*x))/d+1/4*b*sec(c+d*x)^4*(2*a+(a^2+b^2)*sin(c+d*x)/b)/d],
[csc(c+d*x)^3*sec(c+d*x)^5*(a+b*sin(c+d*x))^2,x,6,-2*a*b*csc(c+d*x)/d-1/2*a^2*csc(c+d*x)^2/d-1/8*(12*a^2+15*a*b+4*b^2)*log(1-sin(c+d*x))/d+(3*a^2+b^2)*log(sin(c+d*x))/d-1/8*(12*a^2-15*a*b+4*b^2)*log(1+sin(c+d*x))/d+1/4*sec(c+d*x)^4*(a^2+b^2+2*a*b*sin(c+d*x))/d+1/4*sec(c+d*x)^2*(2*(2*a^2+b^2)+7*a*b*sin(c+d*x))/d],
[sec(c+d*x)^5*sin(c+d*x)^5*(a+b*sin(c+d*x))^3,x,8,-1/16*(a+b)*(8*a^2+37*a*b+35*b^2)*log(1-sin(c+d*x))/d-1/16*(a-b)*(8*a^2-37*a*b+35*b^2)*log(1+sin(c+d*x))/d-1/8*b*(24*a^2+35*b^2)*sin(c+d*x)/d-3/2*a*b^2*sin(c+d*x)^2/d-1/3*b^3*sin(c+d*x)^3/d+1/4*sec(c+d*x)^4*(a+b*sin(c+d*x))^3/d-1/8*sec(c+d*x)^2*(a+b*sin(c+d*x))^2*(8*a+11*b*sin(c+d*x))/d],
[sec(c+d*x)^5*sin(c+d*x)^4*(a+b*sin(c+d*x))^3,x,9,-3/16*(a+b)*(a^2+7*a*b+8*b^2)*log(1-sin(c+d*x))/d+3/16*(a-b)*(a^2-7*a*b+8*b^2)*log(1+sin(c+d*x))/d-29/8*a*b^2*sin(c+d*x)/d-1/2*b^3*sin(c+d*x)^2/d-1/8*sec(c+d*x)^2*(8*b+5*a*sin(c+d*x))*(a+b*sin(c+d*x))^2/d+1/4*sec(c+d*x)^3*(a+b*sin(c+d*x))^3*tan(c+d*x)/d],
[sec(c+d*x)^5*sin(c+d*x)^3*(a+b*sin(c+d*x))^3,x,8,-3/16*b*(a+b)*(3*a+5*b)*log(1-sin(c+d*x))/d+3/16*(3*a-5*b)*(a-b)*b*log(1+sin(c+d*x))/d-15/8*b^3*sin(c+d*x)/d+1/4*sec(c+d*x)^4*(a+b*sin(c+d*x))^3/d-1/8*sec(c+d*x)^2*(a+b*sin(c+d*x))^2*(4*a+7*b*sin(c+d*x))/d],
[sec(c+d*x)^5*sin(c+d*x)^2*(a+b*sin(c+d*x))^3,x,7,1/16*(a^3-9*a*b^2-8*b^3)*log(1-sin(c+d*x))/d-1/16*(a^3-9*a*b^2+8*b^3)*log(1+sin(c+d*x))/d-1/8*sec(c+d*x)^2*(a+b*sin(c+d*x))*(5*a*b+(a^2+4*b^2)*sin(c+d*x))/d+1/4*sec(c+d*x)^3*(a+b*sin(c+d*x))^3*tan(c+d*x)/d],
[sec(c+d*x)^5*sin(c+d*x)*(a+b*sin(c+d*x))^3,x,5,-3/8*b*(a^2-b^2)*arctanh(sin(c+d*x))/d+1/4*sec(c+d*x)^4*(a+b*sin(c+d*x))^3/d-3/8*sec(c+d*x)^2*(a+b*sin(c+d*x))*(b^2+a*b*sin(c+d*x))/d],
[csc(c+d*x)*sec(c+d*x)^5*(a+b*sin(c+d*x))^3,x,6,-1/16*(8*a^3+9*a^2*b-b^3)*log(1-sin(c+d*x))/d+a^3*log(sin(c+d*x))/d-1/16*(8*a^3-9*a^2*b+b^3)*log(1+sin(c+d*x))/d+1/8*sec(c+d*x)^2*(4*a^3+b*(9*a^2-b^2)*sin(c+d*x))/d+1/4*sec(c+d*x)^4*(a*(a^2+3*b^2)+b*(3*a^2+b^2)*sin(c+d*x))/d],
[csc(c+d*x)^2*sec(c+d*x)^5*(a+b*sin(c+d*x))^3,x,6,-a^3*csc(c+d*x)/d-3/16*a*(a+b)*(5*a+3*b)*log(1-sin(c+d*x))/d+3*a^2*b*log(sin(c+d*x))/d+3/16*a*(5*a-3*b)*(a-b)*log(1+sin(c+d*x))/d+1/4*b*sec(c+d*x)^4*(3*a^2+b^2+a*(3+a^2/b^2)*b*sin(c+d*x))/d+1/8*a*b*sec(c+d*x)^2*(12*a+(9+7*a^2/b^2)*b*sin(c+d*x))/d],
[csc(c+d*x)^3*sec(c+d*x)^5*(a+b*sin(c+d*x))^3,x,6,-3*a^2*b*csc(c+d*x)/d-1/2*a^3*csc(c+d*x)^2/d-3/16*(a+b)*(8*a^2+7*a*b+b^2)*log(1-sin(c+d*x))/d+3*a*(a^2+b^2)*log(sin(c+d*x))/d-3/16*(a-b)*(8*a^2-7*a*b+b^2)*log(1+sin(c+d*x))/d+1/4*b^2*sec(c+d*x)^4*(a*(3+a^2/b^2)+(1+3*a^2/b^2)*b*sin(c+d*x))/d+1/8*b^2*sec(c+d*x)^2*(4*a*(3+2*a^2/b^2)+3*(1+7*a^2/b^2)*b*sin(c+d*x))/d],

# m<0

# Integrands of the form Sec[e+f x]^5 Sin[e+f x]^n (a+b Sin[e+f x])^(m/2)

# Integrands of the form Sec[e+f x]^5 Sin[e+f x]^n (a+b Sin[e+f x])^m with n symbolic
[sec(c+d*x)^5*sin(c+d*x)^n*(a+b*sin(c+d*x))^4,x,6,-1/8*(6*a^2*b^2*(1-n^2)-a^4*(3-4*n+n^2)-b^4*(3+4*n+n^2))*hypergeom([1,1/2*(1+n)],[1/2*(3+n)],sin(c+d*x)^2)*sin(c+d*x)^(1+n)/(d*(1+n))-1/2*a*b*n*(a^2*(2-n)-b^2*(2+n))*hypergeom([1,1/2*(2+n)],[1/2*(4+n)],sin(c+d*x)^2)*sin(c+d*x)^(2+n)/(d*(2+n))+1/4*sec(c+d*x)^4*sin(c+d*x)^(1+n)*(a^4+6*a^2*b^2+b^4+4*a*b*(a^2+b^2)*sin(c+d*x))/d+1/8*sec(c+d*x)^2*sin(c+d*x)^(1+n)*(a^4*(3-n)-6*a^2*b^2*(1+n)-b^4*(5+n)+4*a*b*(a^2*(2-n)-b^2*(2+n))*sin(c+d*x))/d],
[sec(c+d*x)^5*sin(c+d*x)^n*(a+b*sin(c+d*x))^3,x,5,1/4*a*(a^2*(3-n)-3*b^2*(1+n))*hypergeom([2,1/2*(1+n)],[1/2*(3+n)],sin(c+d*x)^2)*sin(c+d*x)^(1+n)/(d*(1+n))+1/4*b*(3*a^2*(2-n)-b^2*(2+n))*hypergeom([2,1/2*(2+n)],[1/2*(4+n)],sin(c+d*x)^2)*sin(c+d*x)^(2+n)/(d*(2+n))+1/4*sec(c+d*x)^4*sin(c+d*x)^(1+n)*(a*(a^2+3*b^2)+b*(3*a^2+b^2)*sin(c+d*x))/d],
[sec(c+d*x)^5*sin(c+d*x)^n*(a+b*sin(c+d*x))^2,x,5,1/4*(a^2*(3-n)-b^2*(1+n))*hypergeom([2,1/2*(1+n)],[1/2*(3+n)],sin(c+d*x)^2)*sin(c+d*x)^(1+n)/(d*(1+n))+1/2*a*b*(2-n)*hypergeom([2,1/2*(2+n)],[1/2*(4+n)],sin(c+d*x)^2)*sin(c+d*x)^(2+n)/(d*(2+n))+1/4*sec(c+d*x)^4*sin(c+d*x)^(1+n)*(a^2+b^2+2*a*b*sin(c+d*x))/d],
[sec(c+d*x)^5*sin(c+d*x)^n*(a+b*sin(c+d*x)),x,4,a*hypergeom([3,1/2*(1+n)],[1/2*(3+n)],sin(c+d*x)^2)*sin(c+d*x)^(1+n)/(d*(1+n))+b*hypergeom([3,1/2*(2+n)],[1/2*(4+n)],sin(c+d*x)^2)*sin(c+d*x)^(2+n)/(d*(2+n))],
[sec(c+d*x)^5*sin(c+d*x)^n/(a+b*sin(c+d*x)),x,10,1/16*(3*a^2-9*a*b+8*b^2)*hypergeom([1,1+n],[2+n],-sin(c+d*x))*sin(c+d*x)^(1+n)/((a-b)^3*d*(1+n))+1/16*(3*a^2+9*a*b+8*b^2)*hypergeom([1,1+n],[2+n],sin(c+d*x))*sin(c+d*x)^(1+n)/((a+b)^3*d*(1+n))-b^6*hypergeom([1,1+n],[2+n],-b*sin(c+d*x)/a)*sin(c+d*x)^(1+n)/(a*(a^2-b^2)^3*d*(1+n))+1/16*(3*a-5*b)*hypergeom([2,1+n],[2+n],-sin(c+d*x))*sin(c+d*x)^(1+n)/((a-b)^2*d*(1+n))+1/16*(3*a+5*b)*hypergeom([2,1+n],[2+n],sin(c+d*x))*sin(c+d*x)^(1+n)/((a+b)^2*d*(1+n))+1/8*hypergeom([3,1+n],[2+n],-sin(c+d*x))*sin(c+d*x)^(1+n)/((a-b)*d*(1+n))+1/8*hypergeom([3,1+n],[2+n],sin(c+d*x))*sin(c+d*x)^(1+n)/((a+b)*d*(1+n))],

# Integrands of the form Sec[e+f x]^5 Sin[e+f x]^n (a+b Sin[e+f x])^m with m symbolic
[sec(c+d*x)^5*sin(c+d*x)^n*(a+b*sin(c+d*x))^p,x,17,3/16*AppellF1(1+n,-p,1,2+n,-b*sin(c+d*x)/a,-sin(c+d*x))*sin(c+d*x)^(1+n)*(a+b*sin(c+d*x))^p/(d*(1+n)*(1+b*sin(c+d*x)/a)^p)+3/16*AppellF1(1+n,-p,1,2+n,-b*sin(c+d*x)/a,sin(c+d*x))*sin(c+d*x)^(1+n)*(a+b*sin(c+d*x))^p/(d*(1+n)*(1+b*sin(c+d*x)/a)^p)+3/16*AppellF1(1+n,-p,2,2+n,-b*sin(c+d*x)/a,-sin(c+d*x))*sin(c+d*x)^(1+n)*(a+b*sin(c+d*x))^p/(d*(1+n)*(1+b*sin(c+d*x)/a)^p)+3/16*AppellF1(1+n,-p,2,2+n,-b*sin(c+d*x)/a,sin(c+d*x))*sin(c+d*x)^(1+n)*(a+b*sin(c+d*x))^p/(d*(1+n)*(1+b*sin(c+d*x)/a)^p)+1/8*AppellF1(1+n,-p,3,2+n,-b*sin(c+d*x)/a,-sin(c+d*x))*sin(c+d*x)^(1+n)*(a+b*sin(c+d*x))^p/(d*(1+n)*(1+b*sin(c+d*x)/a)^p)+1/8*AppellF1(1+n,-p,3,2+n,-b*sin(c+d*x)/a,sin(c+d*x))*sin(c+d*x)^(1+n)*(a+b*sin(c+d*x))^p/(d*(1+n)*(1+b*sin(c+d*x)/a)^p)],

# Integrands of the form Sec[e+f x]^6 (d Sin[e+f x])^n (a+b Sin[e+f x])^m

# Integrands of the form Sec[e+f x]^6 Sin[e+f x]^n (a+b Sin[e+f x])^m

# Integrands of the form Sec[e+f x]^6 Sin[e+f x]^n (a+b Sin[e+f x])^(m/2)

# Integrands of the form Sec[e+f x]^6 (d Sin[e+f x])^(n/2) (a+b Sin[e+f x])^(m/2)
[sec(e+f*x)^6*(a+b*sin(e+f*x))^(9/2)/sqrt(d*sin(e+f*x)),x,-1,1/5*sec(e+f*x)^5*(a+b*sin(e+f*x))^(9/2)*sqrt(d*sin(e+f*x))/(d*f)-3/5*a*b*(-2*a^2+b^2)*cos(e+f*x)*sqrt(a+b*sin(e+f*x))/(f*sqrt(d*sin(e+f*x)))-3/20*a*sec(e+f*x)^3*(-a*(7*a^2+b^2)+2*b*(-7*a^2+b^2)*sin(e+f*x)+5*a*(a^2-b^2)*sin(e+f*x)^2+(8*a^2*b-4*b^3)*sin(e+f*x)^3)*sqrt(d*sin(e+f*x))*sqrt(a+b*sin(e+f*x))/(d*f)-3/20*a*(a+b)^(3/2)*(5*a^2+3*a*b-4*b^2)*EllipticF(sqrt(d)*sqrt(a+b*sin(e+f*x))/(sqrt(a+b)*sqrt(d*sin(e+f*x))),sqrt((-a-b)/(a-b)))*sqrt(-a*(-1+csc(e+f*x))/(a+b))*sqrt(a*(1+csc(e+f*x))/(a-b))*tan(e+f*x)/(f*sqrt(d))-3/5*b*(2*a^4-3*a^2*b^2+b^4)*EllipticE(sqrt((-b-a*csc(e+f*x))/(a-b)),sqrt(1-2*a/(a+b)))*sqrt(-a*(-1+csc(e+f*x))/(a+b))*sqrt(d*sin(e+f*x))*sqrt(-a*csc(e+f*x)^2*(1+sin(e+f*x))*(a+b*sin(e+f*x))/(a-b)^2)*tan(e+f*x)/(d*f*sqrt(a+b*sin(e+f*x)))],

# Integrands of the form (g Cos[e+f x])^p (a+b Sin[e+f x])^m (c+d Sin[e+f x])^n

# Integrands of the form Cos[e+f x]^2 (a+b Sin[e+f x])^m (c+d Sin[e+f x])^n

# Integrands of the form Cos[e+f x]^2 (a+b Sin[e+f x])^m (c+d Sin[e+f x])^(n/3)
[cos(e+f*x)^2*(a+b*sin(e+f*x))^2*(c+d*sin(e+f*x))^(4/3),x,11,-9/2080*(64*a*b*c*d-26*a^2*d^2-b^2*(18*c^2-13*d^2))*cos(e+f*x)*(c+d*sin(e+f*x))^(7/3)/(d^3*f)-9/208*b*(3*b*c-2*a*d)*cos(e+f*x)*sin(e+f*x)*(c+d*sin(e+f*x))^(7/3)/(d^2*f)+3/16*cos(e+f*x)*(a+b*sin(e+f*x))^2*(c+d*sin(e+f*x))^(7/3)/(d*f)-3/1040*(c+d)^2*(208*a^2*c*d^2-64*a*b*d*(3*c^2-5*d^2)+b^2*c*(54*c^2+d^2))*AppellF1(1/2,1/2,-7/3,3/2,1/2*(1-sin(e+f*x)),d*(1-sin(e+f*x))/(c+d))*cos(e+f*x)*(c+d*sin(e+f*x))^(1/3)/(d^4*f*((c+d*sin(e+f*x))/(c+d))^(1/3)*sqrt(2)*sqrt(1+sin(e+f*x)))-3/1040*(c-d)*(c+d)^2*(192*a*b*c*d-208*a^2*d^2-b^2*(54*c^2+91*d^2))*AppellF1(1/2,1/2,-4/3,3/2,1/2*(1-sin(e+f*x)),d*(1-sin(e+f*x))/(c+d))*cos(e+f*x)*(c+d*sin(e+f*x))^(1/3)/(d^4*f*((c+d*sin(e+f*x))/(c+d))^(1/3)*sqrt(2)*sqrt(1+sin(e+f*x)))],
[cos(e+f*x)^2*(a+b*sin(e+f*x))*(c+d*sin(e+f*x))^(4/3),x,10,-3/130*(6*b*c-13*a*d)*cos(e+f*x)*(c+d*sin(e+f*x))^(7/3)/(d^2*f)+3/13*b*cos(e+f*x)*sin(e+f*x)*(c+d*sin(e+f*x))^(7/3)/(d*f)+3/65*(c+d)^2*(6*b*c^2-13*a*c*d-10*b*d^2)*AppellF1(1/2,1/2,-7/3,3/2,1/2*(1-sin(e+f*x)),d*(1-sin(e+f*x))/(c+d))*cos(e+f*x)*(c+d*sin(e+f*x))^(1/3)/(d^3*f*((c+d*sin(e+f*x))/(c+d))^(1/3)*sqrt(2)*sqrt(1+sin(e+f*x)))-3/65*(c-d)*(c+d)^2*(6*b*c-13*a*d)*AppellF1(1/2,1/2,-4/3,3/2,1/2*(1-sin(e+f*x)),d*(1-sin(e+f*x))/(c+d))*cos(e+f*x)*(c+d*sin(e+f*x))^(1/3)/(d^3*f*((c+d*sin(e+f*x))/(c+d))^(1/3)*sqrt(2)*sqrt(1+sin(e+f*x)))],
[cos(e+f*x)^2*(c+d*sin(e+f*x))^(4/3),x,2,3/7*AppellF1(7/3,-1/2,-1/2,10/3,(c+d*sin(e+f*x))/(c-d),(c+d*sin(e+f*x))/(c+d))*cos(e+f*x)*(c+d*sin(e+f*x))^(7/3)/(d*f*sqrt(1+(-c-d*sin(e+f*x))/(c-d))*sqrt(1+(-c-d*sin(e+f*x))/(c+d)))],
[cos(e+f*x)^2*(c+d*sin(e+f*x))^(4/3)/(a+b*sin(e+f*x)),x,0,Unintegrable(cos(e+f*x)^2*(c+d*sin(e+f*x))^(4/3)/(a+b*sin(e+f*x)),x)],
[cos(e+f*x)^2*(c+d*sin(e+f*x))^(4/3)/(a+b*sin(e+f*x))^2,x,0,Unintegrable(cos(e+f*x)^2*(c+d*sin(e+f*x))^(4/3)/(a+b*sin(e+f*x))^2,x)],

# Integrands of the form Cos[e+f x]^2 (a+b Sin[e+f x])^m (c+d Sin[e+f x])^n with m and/or n symbolic
[cos(e+f*x)^2*(a+b*sin(e+f*x))^m*(c+d*sin(e+f*x))^n,x,0,Unintegrable(cos(e+f*x)^2*(a+b*sin(e+f*x))^m*(c+d*sin(e+f*x))^n,x)],
[cos(e+f*x)^2*(a+b*sin(e+f*x))^m*(c+d*sin(e+f*x))^(4/3),x,0,Unintegrable(cos(e+f*x)^2*(a+b*sin(e+f*x))^m*(c+d*sin(e+f*x))^(4/3),x)],
[cos(e+f*x)^2*(a+b*sin(e+f*x))^2*(c+d*sin(e+f*x))^n,x,11,(2*a^2*d^2*(3+n)-4*a*b*c*d*(4+n)+b^2*(6*c^2-d^2*(3+n)))*cos(e+f*x)*(c+d*sin(e+f*x))^(1+n)/(d^3*f*(2+n)*(3+n)*(4+n))-b*(3*b*c-2*a*d)*cos(e+f*x)*sin(e+f*x)*(c+d*sin(e+f*x))^(1+n)/(d^2*f*(3+n)*(4+n))+cos(e+f*x)*(a+b*sin(e+f*x))^2*(c+d*sin(e+f*x))^(1+n)/(d*f*(4+n))-(c+d)*(a^2*c*d^2*(12+7*n+n^2)-2*a*b*d*(4+n)*(2*c^2-d^2*(2+n))+b^2*c*(6*c^2-d^2*(3-n-n^2)))*AppellF1(1/2,1/2,-1-n,3/2,1/2*(1-sin(e+f*x)),d*(1-sin(e+f*x))/(c+d))*cos(e+f*x)*(c+d*sin(e+f*x))^n*sqrt(2)/(d^4*f*(2+n)*(3+n)*(4+n)*((c+d*sin(e+f*x))/(c+d))^n*sqrt(1+sin(e+f*x)))-(c^2-d^2)*(4*a*b*c*d*(4+n)-a^2*d^2*(12+7*n+n^2)-b^2*(6*c^2+d^2*(3+4*n+n^2)))*AppellF1(1/2,1/2,-n,3/2,1/2*(1-sin(e+f*x)),d*(1-sin(e+f*x))/(c+d))*cos(e+f*x)*(c+d*sin(e+f*x))^n*sqrt(2)/(d^4*f*(2+n)*(3+n)*(4+n)*((c+d*sin(e+f*x))/(c+d))^n*sqrt(1+sin(e+f*x)))],
[cos(e+f*x)^2*(a+b*sin(e+f*x))*(c+d*sin(e+f*x))^n,x,10,-(2*b*c-a*d*(3+n))*cos(e+f*x)*(c+d*sin(e+f*x))^(1+n)/(d^2*f*(2+n)*(3+n))+b*cos(e+f*x)*sin(e+f*x)*(c+d*sin(e+f*x))^(1+n)/(d*f*(3+n))-(c+d)*(a*c*d*(3+n)-b*(2*c^2-d^2*(2+n)))*AppellF1(1/2,1/2,-1-n,3/2,1/2*(1-sin(e+f*x)),d*(1-sin(e+f*x))/(c+d))*cos(e+f*x)*(c+d*sin(e+f*x))^n*sqrt(2)/(d^3*f*(2+n)*(3+n)*((c+d*sin(e+f*x))/(c+d))^n*sqrt(1+sin(e+f*x)))-(c^2-d^2)*(2*b*c-a*d*(3+n))*AppellF1(1/2,1/2,-n,3/2,1/2*(1-sin(e+f*x)),d*(1-sin(e+f*x))/(c+d))*cos(e+f*x)*(c+d*sin(e+f*x))^n*sqrt(2)/(d^3*f*(2+n)*(3+n)*((c+d*sin(e+f*x))/(c+d))^n*sqrt(1+sin(e+f*x)))],
[cos(e+f*x)^2*(c+d*sin(e+f*x))^n,x,2,AppellF1(1+n,-1/2,-1/2,2+n,(c+d*sin(e+f*x))/(c-d),(c+d*sin(e+f*x))/(c+d))*cos(e+f*x)*(c+d*sin(e+f*x))^(1+n)/(d*f*(1+n)*sqrt(1+(-c-d*sin(e+f*x))/(c-d))*sqrt(1+(-c-d*sin(e+f*x))/(c+d)))],
[cos(e+f*x)^2*(c+d*sin(e+f*x))^n/(a+b*sin(e+f*x)),x,0,Unintegrable(cos(e+f*x)^2*(c+d*sin(e+f*x))^n/(a+b*sin(e+f*x)),x)],
[cos(e+f*x)^2*(c+d*sin(e+f*x))^n/(a+b*sin(e+f*x))^2,x,0,Unintegrable(cos(e+f*x)^2*(c+d*sin(e+f*x))^n/(a+b*sin(e+f*x))^2,x)],

# Integrands of the form (g Cos[e+f x])^p (a+b Sin[e+f x])^m (c+d Sin[e+f x])

# Integrands of the form (g Cos[e+f x])^m (a+b Sin[e+f x])^n (A+B Sin[e+f x])

# n>0
[cos(c+d*x)^7*(a+b*sin(c+d*x))*(A+B*sin(c+d*x)),x,3,a*A*sin(c+d*x)/d+1/2*(A*b+a*B)*sin(c+d*x)^2/d-1/3*(3*a*A-b*B)*sin(c+d*x)^3/d-3/4*(A*b+a*B)*sin(c+d*x)^4/d+3/5*(a*A-b*B)*sin(c+d*x)^5/d+1/2*(A*b+a*B)*sin(c+d*x)^6/d-1/7*(a*A-3*b*B)*sin(c+d*x)^7/d-1/8*(A*b+a*B)*sin(c+d*x)^8/d-1/9*b*B*sin(c+d*x)^9/d],
[cos(c+d*x)^5*(a+b*sin(c+d*x))*(A+B*sin(c+d*x)),x,3,a*A*sin(c+d*x)/d+1/2*(A*b+a*B)*sin(c+d*x)^2/d-1/3*(2*a*A-b*B)*sin(c+d*x)^3/d-1/2*(A*b+a*B)*sin(c+d*x)^4/d+1/5*(a*A-2*b*B)*sin(c+d*x)^5/d+1/6*(A*b+a*B)*sin(c+d*x)^6/d+1/7*b*B*sin(c+d*x)^7/d],
[cos(c+d*x)^3*(a+b*sin(c+d*x))*(A+B*sin(c+d*x)),x,3,a*A*sin(c+d*x)/d+1/2*(A*b+a*B)*sin(c+d*x)^2/d-1/3*(a*A-b*B)*sin(c+d*x)^3/d-1/4*(A*b+a*B)*sin(c+d*x)^4/d-1/5*b*B*sin(c+d*x)^5/d],
[cos(c+d*x)*(a+b*sin(c+d*x))*(A+B*sin(c+d*x)),x,3,a*A*sin(c+d*x)/d+1/2*(A*b+a*B)*sin(c+d*x)^2/d+1/3*b*B*sin(c+d*x)^3/d],
[sec(c+d*x)*(a+b*sin(c+d*x))*(A+B*sin(c+d*x)),x,5,-1/2*(a+b)*(A+B)*log(1-sin(c+d*x))/d+1/2*(a-b)*(A-B)*log(1+sin(c+d*x))/d-b*B*sin(c+d*x)/d],
[sec(c+d*x)^3*(a+b*sin(c+d*x))*(A+B*sin(c+d*x)),x,3,1/2*(a*A-b*B)*arctanh(sin(c+d*x))/d+1/2*sec(c+d*x)^2*(A*b+a*B+(a*A+b*B)*sin(c+d*x))/d],
[sec(c+d*x)^5*(a+b*sin(c+d*x))*(A+B*sin(c+d*x)),x,4,1/8*(3*a*A-b*B)*arctanh(sin(c+d*x))/d+1/4*sec(c+d*x)^4*(A*b+a*B+(a*A+b*B)*sin(c+d*x))/d+1/8*(3*a*A-b*B)*sec(c+d*x)*tan(c+d*x)/d],
[sec(c+d*x)^7*(a+b*sin(c+d*x))*(A+B*sin(c+d*x)),x,5,1/16*(5*a*A-b*B)*arctanh(sin(c+d*x))/d+1/6*sec(c+d*x)^6*(A*b+a*B+(a*A+b*B)*sin(c+d*x))/d+1/16*(5*a*A-b*B)*sec(c+d*x)*tan(c+d*x)/d+1/24*(5*a*A-b*B)*sec(c+d*x)^3*tan(c+d*x)/d],
[cos(c+d*x)^7*(a+b*sin(c+d*x))^2*(A+B*sin(c+d*x)),x,3,-1/3*(a^2-b^2)^3*(A*b-a*B)*(a+b*sin(c+d*x))^3/(b^8*d)+1/4*(a^2-b^2)^2*(6*a*A*b-7*a^2*B+b^2*B)*(a+b*sin(c+d*x))^4/(b^8*d)-3/5*(a^2-b^2)*(5*a^2*A*b-A*b^3-7*a^3*B+3*a*b^2*B)*(a+b*sin(c+d*x))^5/(b^8*d)+1/6*(20*a^3*A*b-12*a*A*b^3-35*a^4*B+30*a^2*b^2*B-3*b^4*B)*(a+b*sin(c+d*x))^6/(b^8*d)-1/7*(15*a^2*A*b-3*A*b^3-35*a^3*B+15*a*b^2*B)*(a+b*sin(c+d*x))^7/(b^8*d)+3/8*(2*a*A*b-7*a^2*B+b^2*B)*(a+b*sin(c+d*x))^8/(b^8*d)-1/9*(A*b-7*a*B)*(a+b*sin(c+d*x))^9/(b^8*d)-1/10*B*(a+b*sin(c+d*x))^10/(b^8*d)],
[cos(c+d*x)^5*(a+b*sin(c+d*x))^2*(A+B*sin(c+d*x)),x,3,1/3*(a^2-b^2)^2*(A*b-a*B)*(a+b*sin(c+d*x))^3/(b^6*d)-1/4*(a^2-b^2)*(4*a*A*b-5*a^2*B+b^2*B)*(a+b*sin(c+d*x))^4/(b^6*d)+2/5*(3*a^2*A*b-A*b^3-5*a^3*B+3*a*b^2*B)*(a+b*sin(c+d*x))^5/(b^6*d)-1/3*(2*a*A*b-5*a^2*B+b^2*B)*(a+b*sin(c+d*x))^6/(b^6*d)+1/7*(A*b-5*a*B)*(a+b*sin(c+d*x))^7/(b^6*d)+1/8*B*(a+b*sin(c+d*x))^8/(b^6*d)],
[cos(c+d*x)^3*(a+b*sin(c+d*x))^2*(A+B*sin(c+d*x)),x,3,-1/3*(a^2-b^2)*(A*b-a*B)*(a+b*sin(c+d*x))^3/(b^4*d)+1/4*(2*a*A*b-3*a^2*B+b^2*B)*(a+b*sin(c+d*x))^4/(b^4*d)-1/5*(A*b-3*a*B)*(a+b*sin(c+d*x))^5/(b^4*d)-1/6*B*(a+b*sin(c+d*x))^6/(b^4*d)],
[cos(c+d*x)*(a+b*sin(c+d*x))^2*(A+B*sin(c+d*x)),x,3,1/3*(A*b-a*B)*(a+b*sin(c+d*x))^3/(b^2*d)+1/4*B*(a+b*sin(c+d*x))^4/(b^2*d)],
[sec(c+d*x)*(a+b*sin(c+d*x))^2*(A+B*sin(c+d*x)),x,6,-1/2*(a+b)^2*(A+B)*log(1-sin(c+d*x))/d+1/2*(a-b)^2*(A-B)*log(1+sin(c+d*x))/d-b*(A*b+2*a*B)*sin(c+d*x)/d-1/2*b^2*B*sin(c+d*x)^2/d],
[sec(c+d*x)^3*(a+b*sin(c+d*x))^2*(A+B*sin(c+d*x)),x,5,-1/4*(a+b)*(a*A-b*(A+2*B))*log(1-sin(c+d*x))/d+1/4*(a-b)*(a*A+b*(A-2*B))*log(1+sin(c+d*x))/d+1/2*sec(c+d*x)^2*(a+b*sin(c+d*x))*(A*b+a*B+(a*A+b*B)*sin(c+d*x))/d],
[sec(c+d*x)^5*(a+b*sin(c+d*x))^2*(A+B*sin(c+d*x)),x,4,1/8*(3*a^2*A-A*b^2-2*a*b*B)*arctanh(sin(c+d*x))/d+1/4*sec(c+d*x)^4*(B+A*sin(c+d*x))*(a+b*sin(c+d*x))^2/d+1/8*sec(c+d*x)^2*(2*b*(2*a*A-b*B)+(3*a^2*A+A*b^2-2*a*b*B)*sin(c+d*x))/d],
[sec(c+d*x)^7*(a+b*sin(c+d*x))^2*(A+B*sin(c+d*x)),x,5,1/16*(5*a^2*A-A*b^2-2*a*b*B)*arctanh(sin(c+d*x))/d+1/6*sec(c+d*x)^6*(B+A*sin(c+d*x))*(a+b*sin(c+d*x))^2/d+1/24*sec(c+d*x)^4*(2*b*(4*a*A-b*B)+(5*a^2*A+3*A*b^2-2*a*b*B)*sin(c+d*x))/d+1/16*(5*a^2*A-A*b^2-2*a*b*B)*sec(c+d*x)*tan(c+d*x)/d],

# n<0
[cos(c+d*x)^7*(A+B*sin(c+d*x))/(a+b*sin(c+d*x)),x,3,-(a^2-b^2)^3*(A*b-a*B)*log(a+b*sin(c+d*x))/(b^8*d)+(a^5*A*b-3*a^3*A*b^3+3*a*A*b^5-a^6*B+3*a^4*b^2*B-3*a^2*b^4*B+b^6*B)*sin(c+d*x)/(b^7*d)-1/2*(a^4-3*a^2*b^2+3*b^4)*(A*b-a*B)*sin(c+d*x)^2/(b^6*d)+1/3*(a^3*A*b-3*a*A*b^3-a^4*B+3*a^2*b^2*B-3*b^4*B)*sin(c+d*x)^3/(b^5*d)-1/4*(a^2-3*b^2)*(A*b-a*B)*sin(c+d*x)^4/(b^4*d)+1/5*(a*A*b-a^2*B+3*b^2*B)*sin(c+d*x)^5/(b^3*d)-1/6*(A*b-a*B)*sin(c+d*x)^6/(b^2*d)-1/7*B*sin(c+d*x)^7/(b*d)],
[cos(c+d*x)^5*(A+B*sin(c+d*x))/(a+b*sin(c+d*x)),x,3,(a^2-b^2)^2*(A*b-a*B)*log(a+b*sin(c+d*x))/(b^6*d)-(a^3*A*b-2*a*A*b^3-a^4*B+2*a^2*b^2*B-b^4*B)*sin(c+d*x)/(b^5*d)+1/2*(a^2-2*b^2)*(A*b-a*B)*sin(c+d*x)^2/(b^4*d)-1/3*(a*A*b-a^2*B+2*b^2*B)*sin(c+d*x)^3/(b^3*d)+1/4*(A*b-a*B)*sin(c+d*x)^4/(b^2*d)+1/5*B*sin(c+d*x)^5/(b*d)],
[cos(c+d*x)^3*(A+B*sin(c+d*x))/(a+b*sin(c+d*x)),x,3,-(a^2-b^2)*(A*b-a*B)*log(a+b*sin(c+d*x))/(b^4*d)+(a*A*b-a^2*B+b^2*B)*sin(c+d*x)/(b^3*d)-1/2*(A*b-a*B)*sin(c+d*x)^2/(b^2*d)-1/3*B*sin(c+d*x)^3/(b*d)],
[cos(c+d*x)*(A+B*sin(c+d*x))/(a+b*sin(c+d*x)),x,3,(A*b-a*B)*log(a+b*sin(c+d*x))/(b^2*d)+B*sin(c+d*x)/(b*d)],
[sec(c+d*x)*(A+B*sin(c+d*x))/(a+b*sin(c+d*x)),x,3,-1/2*(A+B)*log(1-sin(c+d*x))/((a+b)*d)+1/2*(A-B)*log(1+sin(c+d*x))/((a-b)*d)-(A*b-a*B)*log(a+b*sin(c+d*x))/((a^2-b^2)*d)],
[sec(c+d*x)^3*(A+B*sin(c+d*x))/(a+b*sin(c+d*x)),x,4,-1/4*(a*A+b*(2*A+B))*log(1-sin(c+d*x))/((a+b)^2*d)+1/4*(a*A-b*(2*A-B))*log(1+sin(c+d*x))/((a-b)^2*d)+b^2*(A*b-a*B)*log(a+b*sin(c+d*x))/((a^2-b^2)^2*d)-1/2*sec(c+d*x)^2*(A*b-a*B-(a*A-b*B)*sin(c+d*x))/((a^2-b^2)*d)],
[sec(c+d*x)^5*(A+B*sin(c+d*x))/(a+b*sin(c+d*x)),x,5,-1/16*(3*a^2*A+a*b*(9*A+B)+b^2*(8*A+3*B))*log(1-sin(c+d*x))/((a+b)^3*d)+1/16*(3*a^2*A+b^2*(8*A-3*B)-a*b*(9*A-B))*log(1+sin(c+d*x))/((a-b)^3*d)-b^4*(A*b-a*B)*log(a+b*sin(c+d*x))/((a^2-b^2)^3*d)-1/4*sec(c+d*x)^4*(A*b-a*B-(a*A-b*B)*sin(c+d*x))/((a^2-b^2)*d)+1/8*sec(c+d*x)^2*(4*b^2*(A*b-a*B)+(3*a^3*A-7*a*A*b^2+a^2*b*B+3*b^3*B)*sin(c+d*x))/((a^2-b^2)^2*d)],
[sec(c+d*x)^7*(A+B*sin(c+d*x))/(a+b*sin(c+d*x)),x,6,-1/32*(5*a^3*A+a^2*b*(20*A+B)+a*b^2*(29*A+4*B)+b^3*(16*A+5*B))*log(1-sin(c+d*x))/((a+b)^4*d)+1/32*(5*a^3*A-b^3*(16*A-5*B)+a*b^2*(29*A-4*B)-a^2*b*(20*A-B))*log(1+sin(c+d*x))/((a-b)^4*d)+b^6*(A*b-a*B)*log(a+b*sin(c+d*x))/((a^2-b^2)^4*d)-1/6*sec(c+d*x)^6*(A*b-a*B-(a*A-b*B)*sin(c+d*x))/((a^2-b^2)*d)+1/24*sec(c+d*x)^4*(6*b^2*(A*b-a*B)+(5*a^3*A-11*a*A*b^2+a^2*b*B+5*b^3*B)*sin(c+d*x))/((a^2-b^2)^2*d)-1/16*sec(c+d*x)^2*(8*b^4*(A*b-a*B)-(5*a^5*A-16*a^3*A*b^2+19*a*A*b^4+a^4*b*B-4*a^2*b^3*B-5*b^5*B)*sin(c+d*x))/((a^2-b^2)^3*d)],
[cos(c+d*x)^7*(A+B*sin(c+d*x))/(a+b*sin(c+d*x))^2,x,3,(a^2-b^2)^2*(6*a*A*b-7*a^2*B+b^2*B)*log(a+b*sin(c+d*x))/(b^8*d)-(5*a^4*A*b-9*a^2*A*b^3+3*A*b^5-6*a^5*B+12*a^3*b^2*B-6*a*b^4*B)*sin(c+d*x)/(b^7*d)+1/2*(4*a^3*A*b-6*a*A*b^3-5*a^4*B+9*a^2*b^2*B-3*b^4*B)*sin(c+d*x)^2/(b^6*d)-1/3*(3*a^2*A*b-3*A*b^3-4*a^3*B+6*a*b^2*B)*sin(c+d*x)^3/(b^5*d)+1/4*(2*a*A*b-3*a^2*B+3*b^2*B)*sin(c+d*x)^4/(b^4*d)-1/5*(A*b-2*a*B)*sin(c+d*x)^5/(b^3*d)-1/6*B*sin(c+d*x)^6/(b^2*d)+(a^2-b^2)^3*(A*b-a*B)/(b^8*d*(a+b*sin(c+d*x)))],
[cos(c+d*x)^5*(A+B*sin(c+d*x))/(a+b*sin(c+d*x))^2,x,3,-(a^2-b^2)*(4*a*A*b-5*a^2*B+b^2*B)*log(a+b*sin(c+d*x))/(b^6*d)+(3*a^2*A*b-2*A*b^3-4*a^3*B+4*a*b^2*B)*sin(c+d*x)/(b^5*d)-1/2*(2*a*A*b-3*a^2*B+2*b^2*B)*sin(c+d*x)^2/(b^4*d)+1/3*(A*b-2*a*B)*sin(c+d*x)^3/(b^3*d)+1/4*B*sin(c+d*x)^4/(b^2*d)-(a^2-b^2)^2*(A*b-a*B)/(b^6*d*(a+b*sin(c+d*x)))],
[cos(c+d*x)^3*(A+B*sin(c+d*x))/(a+b*sin(c+d*x))^2,x,3,(2*a*A*b-3*a^2*B+b^2*B)*log(a+b*sin(c+d*x))/(b^4*d)-(A*b-2*a*B)*sin(c+d*x)/(b^3*d)-1/2*B*sin(c+d*x)^2/(b^2*d)+(a^2-b^2)*(A*b-a*B)/(b^4*d*(a+b*sin(c+d*x)))],
[cos(c+d*x)*(A+B*sin(c+d*x))/(a+b*sin(c+d*x))^2,x,3,B*log(a+b*sin(c+d*x))/(b^2*d)+(-A*b+a*B)/(b^2*d*(a+b*sin(c+d*x)))],
[sec(c+d*x)*(A+B*sin(c+d*x))/(a+b*sin(c+d*x))^2,x,3,-1/2*(A+B)*log(1-sin(c+d*x))/((a+b)^2*d)+1/2*(A-B)*log(1+sin(c+d*x))/((a-b)^2*d)-(2*a*A*b-a^2*B-b^2*B)*log(a+b*sin(c+d*x))/((a^2-b^2)^2*d)+(A*b-a*B)/((a^2-b^2)*d*(a+b*sin(c+d*x)))],
[sec(c+d*x)^3*(A+B*sin(c+d*x))/(a+b*sin(c+d*x))^2,x,4,-1/4*(a*A+3*A*b+2*b*B)*log(1-sin(c+d*x))/((a+b)^3*d)+1/4*(a*A-3*A*b+2*b*B)*log(1+sin(c+d*x))/((a-b)^3*d)+b^2*(4*a*A*b-3*a^2*B-b^2*B)*log(a+b*sin(c+d*x))/((a^2-b^2)^3*d)-1/2*b*(a^2*A+3*A*b^2-4*a*b*B)/((a^2-b^2)^2*d*(a+b*sin(c+d*x)))-1/2*sec(c+d*x)^2*(A*b-a*B-(a*A-b*B)*sin(c+d*x))/((a^2-b^2)*d*(a+b*sin(c+d*x)))],
[sec(c+d*x)^5*(A+B*sin(c+d*x))/(a+b*sin(c+d*x))^2,x,5,-1/16*(3*a^2*A+2*a*b*(6*A+B)+b^2*(15*A+8*B))*log(1-sin(c+d*x))/((a+b)^4*d)+1/16*(3*a^2*A+b^2*(15*A-8*B)-2*a*b*(6*A-B))*log(1+sin(c+d*x))/((a-b)^4*d)-b^4*(6*a*A*b-5*a^2*B-b^2*B)*log(a+b*sin(c+d*x))/((a^2-b^2)^4*d)-1/8*b*(3*a^4*A-12*a^2*A*b^2-15*A*b^4+2*a^3*b*B+22*a*b^3*B)/((a^2-b^2)^3*d*(a+b*sin(c+d*x)))-1/4*sec(c+d*x)^4*(A*b-a*B-(a*A-b*B)*sin(c+d*x))/((a^2-b^2)*d*(a+b*sin(c+d*x)))+1/8*sec(c+d*x)^2*(b*(a^2*A+5*A*b^2-6*a*b*B)+(3*a^3*A-9*a*A*b^2+2*a^2*b*B+4*b^3*B)*sin(c+d*x))/((a^2-b^2)^2*d*(a+b*sin(c+d*x)))],
[sec(c+d*x)^7*(A+B*sin(c+d*x))/(a+b*sin(c+d*x))^2,x,6,-1/32*(5*a^3*A+a^2*b*(25*A+2*B)+a*b^2*(47*A+10*B)+b^3*(35*A+16*B))*log(1-sin(c+d*x))/((a+b)^5*d)+1/32*(5*a^3*A-b^3*(35*A-16*B)+a*b^2*(47*A-10*B)-a^2*(25*A*b-2*b*B))*log(1+sin(c+d*x))/((a-b)^5*d)+b^6*(8*a*A*b-7*a^2*B-b^2*B)*log(a+b*sin(c+d*x))/((a^2-b^2)^5*d)-1/16*b*(5*a^6*A-23*a^4*A*b^2+47*a^2*A*b^4+35*A*b^6+2*a^5*b*B-12*a^3*b^3*B-54*a*b^5*B)/((a^2-b^2)^4*d*(a+b*sin(c+d*x)))-1/6*sec(c+d*x)^6*(A*b-a*B-(a*A-b*B)*sin(c+d*x))/((a^2-b^2)*d*(a+b*sin(c+d*x)))+1/24*sec(c+d*x)^4*(b*(a^2*A+7*A*b^2-8*a*b*B)+(5*a^3*A-13*a*A*b^2+2*a^2*b*B+6*b^3*B)*sin(c+d*x))/((a^2-b^2)^2*d*(a+b*sin(c+d*x)))+1/48*sec(c+d*x)^2*(b*(5*a^4*A-18*a^2*A*b^2-35*A*b^4+2*a^3*b*B+46*a*b^3*B)+3*(5*a^5*A-18*a^3*A*b^2+29*a*A*b^4+2*a^4*b*B-10*a^2*b^3*B-8*b^5*B)*sin(c+d*x))/((a^2-b^2)^3*d*(a+b*sin(c+d*x)))],

# Integrands of the form (g Cos[e+f x])^m (a+b Sin[e+f x])^(n/2) (A+B Sin[e+f x])

# Integrands of the form (g Cos[e+f x])^(m/2) (a+b Sin[e+f x])^n (A+B Sin[e+f x])

# Integrands of the form (g Cos[e+f x])^(m/2) (a+b Sin[e+f x])^(n/2) (A+B Sin[e+f x])

# Integrands of the form (g Cos[e+f x])^m (a+b Sin[e+f x])^n (A+B Sin[e+f x]) with m and/or n symbolic
[(g*cos(e+f*x))^(-1-m)*(a+b*sin(e+f*x))^m*(A+B*sin(e+f*x)),x,0,Unintegrable((g*cos(e+f*x))^(-1-m)*(a+b*sin(e+f*x))^m*(A+B*sin(e+f*x)),x)],

# Integrands of the form (g Cos[e+f x])^p (a+b Sin[e+f x])^m (c+d Sin[e+f x])^n

# Integrands of the form (g Cos[e+f x])^p (a+b Sin[e+f x])^m (c+d Sin[e+f x])^n with p symbolic
[(g*cos(e+f*x))^p/((a+b*sin(e+f*x))*(c+d*sin(e+f*x))),x,4,-g*AppellF1(1-p,1/2*(1-p),1/2*(1-p),2-p,(a+b)/(a+b*sin(e+f*x)),(a-b)/(a+b*sin(e+f*x)))*(g*cos(e+f*x))^(-1+p)*(-b*(1-sin(e+f*x))/(a+b*sin(e+f*x)))^(1/2*(1-p))*(b*(1+sin(e+f*x))/(a+b*sin(e+f*x)))^(1/2*(1-p))/((b*c-a*d)*f*(1-p))+g*AppellF1(1-p,1/2*(1-p),1/2*(1-p),2-p,(c+d)/(c+d*sin(e+f*x)),(c-d)/(c+d*sin(e+f*x)))*(g*cos(e+f*x))^(-1+p)*(-d*(1-sin(e+f*x))/(c+d*sin(e+f*x)))^(1/2*(1-p))*(d*(1+sin(e+f*x))/(c+d*sin(e+f*x)))^(1/2*(1-p))/((b*c-a*d)*f*(1-p))],
[(g*cos(e+f*x))^p/((a+b*sin(e+f*x))*(c+d*sin(e+f*x))^2),x,5,-b*g*AppellF1(1-p,1/2*(1-p),1/2*(1-p),2-p,(a+b)/(a+b*sin(e+f*x)),(a-b)/(a+b*sin(e+f*x)))*(g*cos(e+f*x))^(-1+p)*(-b*(1-sin(e+f*x))/(a+b*sin(e+f*x)))^(1/2*(1-p))*(b*(1+sin(e+f*x))/(a+b*sin(e+f*x)))^(1/2*(1-p))/((b*c-a*d)^2*f*(1-p))+b*g*AppellF1(1-p,1/2*(1-p),1/2*(1-p),2-p,(c+d)/(c+d*sin(e+f*x)),(c-d)/(c+d*sin(e+f*x)))*(g*cos(e+f*x))^(-1+p)*(-d*(1-sin(e+f*x))/(c+d*sin(e+f*x)))^(1/2*(1-p))*(d*(1+sin(e+f*x))/(c+d*sin(e+f*x)))^(1/2*(1-p))/((b*c-a*d)^2*f*(1-p))+g*AppellF1(2-p,1/2*(1-p),1/2*(1-p),3-p,(c+d)/(c+d*sin(e+f*x)),(c-d)/(c+d*sin(e+f*x)))*(g*cos(e+f*x))^(-1+p)*(-d*(1-sin(e+f*x))/(c+d*sin(e+f*x)))^(1/2*(1-p))*(d*(1+sin(e+f*x))/(c+d*sin(e+f*x)))^(1/2*(1-p))/((b*c-a*d)*f*(2-p)*(c+d*sin(e+f*x)))],

# Integrands of the form (g Sec[e+f x])^p (a+b Sin[e+f x])^m (c+d Sin[e+f x])^n

# Integrands of the form (g Sec[e+f x])^p (a+b Sin[e+f x])^m (c+d Sin[e+f x])^n

# Integrands of the form (g Sec[e+f x])^p (a+b Sin[e+f x])^m (c+d Sin[e+f x])^n with p symbolic
[(g*sec(e+f*x))^p/((a+b*sin(e+f*x))*(c+d*sin(e+f*x))),x,5,-AppellF1(1+p,1/2*(1+p),1/2*(1+p),2+p,(a+b)/(a+b*sin(e+f*x)),(a-b)/(a+b*sin(e+f*x)))*sec(e+f*x)*(g*sec(e+f*x))^p*(-b*(1-sin(e+f*x))/(a+b*sin(e+f*x)))^(1/2*(1+p))*(b*(1+sin(e+f*x))/(a+b*sin(e+f*x)))^(1/2*(1+p))/((b*c-a*d)*f*(1+p))+AppellF1(1+p,1/2*(1+p),1/2*(1+p),2+p,(c+d)/(c+d*sin(e+f*x)),(c-d)/(c+d*sin(e+f*x)))*sec(e+f*x)*(g*sec(e+f*x))^p*(-d*(1-sin(e+f*x))/(c+d*sin(e+f*x)))^(1/2*(1+p))*(d*(1+sin(e+f*x))/(c+d*sin(e+f*x)))^(1/2*(1+p))/((b*c-a*d)*f*(1+p))]]:
