# Maple integration test file: "4 Trig functions\4.2 Cosine\4.2.4.1 (a+b cos)^m (A+B cos+C cos^2).txt"

lst:=[

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

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

# Integrands of the form Cos[e+f x]^m (A+C Cos[e+f x]^2)
[cos(c+d*x)^7*(A+C*cos(c+d*x)^2),x,3,(A+C)*sin(c+d*x)/d-1/3*(3*A+4*C)*sin(c+d*x)^3/d+3/5*(A+2*C)*sin(c+d*x)^5/d-1/7*(A+4*C)*sin(c+d*x)^7/d+1/9*C*sin(c+d*x)^9/d],
[cos(c+d*x)^5*(A+C*cos(c+d*x)^2),x,3,(A+C)*sin(c+d*x)/d-1/3*(2*A+3*C)*sin(c+d*x)^3/d+1/5*(A+3*C)*sin(c+d*x)^5/d-1/7*C*sin(c+d*x)^7/d],
[cos(c+d*x)^3*(A+C*cos(c+d*x)^2),x,3,(A+C)*sin(c+d*x)/d-1/3*(A+2*C)*sin(c+d*x)^3/d+1/5*C*sin(c+d*x)^5/d],
[cos(c+d*x)*(A+C*cos(c+d*x)^2),x,2,(A+C)*sin(c+d*x)/d-1/3*C*sin(c+d*x)^3/d],
[(A+C*cos(c+d*x)^2)*sec(c+d*x),x,2,A*arctanh(sin(c+d*x))/d+C*sin(c+d*x)/d],
[(A+C*cos(c+d*x)^2)*sec(c+d*x)^3,x,2,1/2*(A+2*C)*arctanh(sin(c+d*x))/d+1/2*A*sec(c+d*x)*tan(c+d*x)/d],
[(A+C*cos(c+d*x)^2)*sec(c+d*x)^5,x,3,1/8*(3*A+4*C)*arctanh(sin(c+d*x))/d+1/8*(3*A+4*C)*sec(c+d*x)*tan(c+d*x)/d+1/4*A*sec(c+d*x)^3*tan(c+d*x)/d],
[(A+C*cos(c+d*x)^2)*sec(c+d*x)^7,x,4,1/16*(5*A+6*C)*arctanh(sin(c+d*x))/d+1/16*(5*A+6*C)*sec(c+d*x)*tan(c+d*x)/d+1/24*(5*A+6*C)*sec(c+d*x)^3*tan(c+d*x)/d+1/6*A*sec(c+d*x)^5*tan(c+d*x)/d],
[cos(c+d*x)^6*(A+C*cos(c+d*x)^2),x,5,5/128*(8*A+7*C)*x+5/128*(8*A+7*C)*cos(c+d*x)*sin(c+d*x)/d+5/192*(8*A+7*C)*cos(c+d*x)^3*sin(c+d*x)/d+1/48*(8*A+7*C)*cos(c+d*x)^5*sin(c+d*x)/d+1/8*C*cos(c+d*x)^7*sin(c+d*x)/d],
[cos(c+d*x)^4*(A+C*cos(c+d*x)^2),x,4,1/16*(6*A+5*C)*x+1/16*(6*A+5*C)*cos(c+d*x)*sin(c+d*x)/d+1/24*(6*A+5*C)*cos(c+d*x)^3*sin(c+d*x)/d+1/6*C*cos(c+d*x)^5*sin(c+d*x)/d],
[cos(c+d*x)^2*(A+C*cos(c+d*x)^2),x,3,1/8*(4*A+3*C)*x+1/8*(4*A+3*C)*cos(c+d*x)*sin(c+d*x)/d+1/4*C*cos(c+d*x)^3*sin(c+d*x)/d],
[(A+C*cos(c+d*x)^2)*sec(c+d*x)^2,x,2,C*x+A*tan(c+d*x)/d],
[(A+C*cos(c+d*x)^2)*sec(c+d*x)^4,x,3,1/3*(2*A+3*C)*tan(c+d*x)/d+1/3*A*sec(c+d*x)^2*tan(c+d*x)/d],
[(A+C*cos(c+d*x)^2)*sec(c+d*x)^6,x,3,1/5*(4*A+5*C)*tan(c+d*x)/d+1/5*A*sec(c+d*x)^4*tan(c+d*x)/d+1/15*(4*A+5*C)*tan(c+d*x)^3/d],
[(A+C*cos(c+d*x)^2)*sec(c+d*x)^8,x,3,1/7*(6*A+7*C)*tan(c+d*x)/d+1/7*A*sec(c+d*x)^6*tan(c+d*x)/d+2/21*(6*A+7*C)*tan(c+d*x)^3/d+1/35*(6*A+7*C)*tan(c+d*x)^5/d],

# Integrands of the form (a Cos[e+f x])^(m/2) (A+C Cos[e+f x]^2)
[(b*cos(c+d*x))^(5/2)*(A+C*cos(c+d*x)^2),x,4,2/45*b*(9*A+7*C)*(b*cos(c+d*x))^(3/2)*sin(c+d*x)/d+2/9*C*(b*cos(c+d*x))^(7/2)*sin(c+d*x)/(b*d)+2/15*b^2*(9*A+7*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(b*cos(c+d*x))^(3/2)*(A+C*cos(c+d*x)^2),x,4,2/7*C*(b*cos(c+d*x))^(5/2)*sin(c+d*x)/(b*d)+2/21*b^2*(7*A+5*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))+2/21*b*(7*A+5*C)*sin(c+d*x)*sqrt(b*cos(c+d*x))/d],
[(b*cos(c+d*x))^(1/2)*(A+C*cos(c+d*x)^2),x,3,2/5*C*(b*cos(c+d*x))^(3/2)*sin(c+d*x)/(b*d)+2/5*(5*A+3*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(A+C*cos(c+d*x)^2)/(b*cos(c+d*x))^(1/2),x,3,2/3*(3*A+C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))+2/3*C*sin(c+d*x)*sqrt(b*cos(c+d*x))/(b*d)],
[(A+C*cos(c+d*x)^2)/(b*cos(c+d*x))^(3/2),x,3,2*A*sin(c+d*x)/(b*d*sqrt(b*cos(c+d*x)))-2*(A-C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(b^2*d*sqrt(cos(c+d*x)))],
[(A+C*cos(c+d*x)^2)/(b*cos(c+d*x))^(5/2),x,3,2/3*A*sin(c+d*x)/(b*d*(b*cos(c+d*x))^(3/2))+2/3*(A+3*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(b^2*d*sqrt(b*cos(c+d*x)))],
[(A+C*cos(c+d*x)^2)/(b*cos(c+d*x))^(7/2),x,4,2/5*A*sin(c+d*x)/(b*d*(b*cos(c+d*x))^(5/2))+2/5*(3*A+5*C)*sin(c+d*x)/(b^3*d*sqrt(b*cos(c+d*x)))-2/5*(3*A+5*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(b^4*d*sqrt(cos(c+d*x)))],
[(A+C*cos(c+d*x)^2)/(b*cos(c+d*x))^(9/2),x,4,2/7*A*sin(c+d*x)/(b*d*(b*cos(c+d*x))^(7/2))+2/21*(5*A+7*C)*sin(c+d*x)/(b^3*d*(b*cos(c+d*x))^(3/2))+2/21*(5*A+7*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(b^4*d*sqrt(b*cos(c+d*x)))],
[(3-5*cos(c+d*x)^2)*sqrt(cos(c+d*x)),x,1,-2*cos(c+d*x)^(3/2)*sin(c+d*x)/d],
[(1-3*cos(c+d*x)^2)/sqrt(cos(c+d*x)),x,1,-2*sin(c+d*x)*sqrt(cos(c+d*x))/d],

# Integrands of the form (a Sec[e+f x])^(m/2) (A+C Cos[e+f x]^2)
[(A+C*cos(c+d*x)^2)*(b*sec(c+d*x))^(9/2),x,5,2/21*b^3*(5*A+7*C)*(b*sec(c+d*x))^(3/2)*sin(c+d*x)/d+2/21*b^4*(5*A+7*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(b*sec(c+d*x))/d+2/7*A*b^2*(b*sec(c+d*x))^(5/2)*tan(c+d*x)/d],
[(A+C*cos(c+d*x)^2)*(b*sec(c+d*x))^(7/2),x,5,-2/5*b^4*(3*A+5*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(d*sqrt(cos(c+d*x))*sqrt(b*sec(c+d*x)))+2/5*b^3*(3*A+5*C)*sin(c+d*x)*sqrt(b*sec(c+d*x))/d+2/5*A*b^2*(b*sec(c+d*x))^(3/2)*tan(c+d*x)/d],
[(A+C*cos(c+d*x)^2)*(b*sec(c+d*x))^(5/2),x,4,2/3*b^2*(A+3*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(b*sec(c+d*x))/d+2/3*A*b^2*sqrt(b*sec(c+d*x))*tan(c+d*x)/d],
[(A+C*cos(c+d*x)^2)*(b*sec(c+d*x))^(3/2),x,4,-2*b^2*(A-C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(d*sqrt(cos(c+d*x))*sqrt(b*sec(c+d*x)))+2*A*b^2*tan(c+d*x)/(d*sqrt(b*sec(c+d*x)))],
[(A+C*cos(c+d*x)^2)*(b*sec(c+d*x))^(1/2),x,4,2/3*(3*A+C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(b*sec(c+d*x))/d+2/3*b^2*C*tan(c+d*x)/(d*(b*sec(c+d*x))^(3/2))],
[(A+C*cos(c+d*x)^2)/(b*sec(c+d*x))^(1/2),x,4,2/5*(5*A+3*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(d*sqrt(cos(c+d*x))*sqrt(b*sec(c+d*x)))+2/5*b^2*C*tan(c+d*x)/(d*(b*sec(c+d*x))^(5/2))],
[(A+C*cos(c+d*x)^2)/(b*sec(c+d*x))^(3/2),x,5,2/21*(7*A+5*C)*sin(c+d*x)/(b*d*sqrt(b*sec(c+d*x)))+2/21*(7*A+5*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(b*sec(c+d*x))/(b^2*d)+2/7*b^2*C*tan(c+d*x)/(d*(b*sec(c+d*x))^(7/2))],
[(A+C*cos(c+d*x)^2)/(b*sec(c+d*x))^(5/2),x,5,2/45*(9*A+7*C)*sin(c+d*x)/(b*d*(b*sec(c+d*x))^(3/2))+2/15*(9*A+7*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(b^2*d*sqrt(cos(c+d*x))*sqrt(b*sec(c+d*x)))+2/9*b^2*C*tan(c+d*x)/(d*(b*sec(c+d*x))^(9/2))],

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

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

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

# n>0
[cos(c+d*x)^2*(A+C*cos(c+d*x)^2)*sqrt(b*cos(c+d*x)),x,5,2/45*(9*A+7*C)*(b*cos(c+d*x))^(3/2)*sin(c+d*x)/(b*d)+2/9*C*(b*cos(c+d*x))^(7/2)*sin(c+d*x)/(b^3*d)+2/15*(9*A+7*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[cos(c+d*x)*(A+C*cos(c+d*x)^2)*sqrt(b*cos(c+d*x)),x,5,2/7*C*(b*cos(c+d*x))^(5/2)*sin(c+d*x)/(b^2*d)+2/21*b*(7*A+5*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))+2/21*(7*A+5*C)*sin(c+d*x)*sqrt(b*cos(c+d*x))/d],
[(A+C*cos(c+d*x)^2)*sqrt(b*cos(c+d*x)),x,3,2/5*C*(b*cos(c+d*x))^(3/2)*sin(c+d*x)/(b*d)+2/5*(5*A+3*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(A+C*cos(c+d*x)^2)*sec(c+d*x)*sqrt(b*cos(c+d*x)),x,4,2/3*b*(3*A+C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))+2/3*C*sin(c+d*x)*sqrt(b*cos(c+d*x))/d],
[(A+C*cos(c+d*x)^2)*sec(c+d*x)^2*sqrt(b*cos(c+d*x)),x,4,2*A*b*sin(c+d*x)/(d*sqrt(b*cos(c+d*x)))-2*(A-C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(A+C*cos(c+d*x)^2)*sec(c+d*x)^3*sqrt(b*cos(c+d*x)),x,4,2/3*A*b^2*sin(c+d*x)/(d*(b*cos(c+d*x))^(3/2))+2/3*b*(A+3*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))],
[(A+C*cos(c+d*x)^2)*sec(c+d*x)^4*sqrt(b*cos(c+d*x)),x,5,2/5*A*b^3*sin(c+d*x)/(d*(b*cos(c+d*x))^(5/2))+2/5*b*(3*A+5*C)*sin(c+d*x)/(d*sqrt(b*cos(c+d*x)))-2/5*(3*A+5*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(A+C*cos(c+d*x)^2)*sec(c+d*x)^5*sqrt(b*cos(c+d*x)),x,5,2/7*A*b^4*sin(c+d*x)/(d*(b*cos(c+d*x))^(7/2))+2/21*b^2*(5*A+7*C)*sin(c+d*x)/(d*(b*cos(c+d*x))^(3/2))+2/21*b*(5*A+7*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))],
[cos(c+d*x)*(b*cos(c+d*x))^(3/2)*(A+C*cos(c+d*x)^2),x,5,2/45*(9*A+7*C)*(b*cos(c+d*x))^(3/2)*sin(c+d*x)/d+2/9*C*(b*cos(c+d*x))^(7/2)*sin(c+d*x)/(b^2*d)+2/15*b*(9*A+7*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(b*cos(c+d*x))^(3/2)*(A+C*cos(c+d*x)^2),x,4,2/7*C*(b*cos(c+d*x))^(5/2)*sin(c+d*x)/(b*d)+2/21*b^2*(7*A+5*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))+2/21*b*(7*A+5*C)*sin(c+d*x)*sqrt(b*cos(c+d*x))/d],
[(b*cos(c+d*x))^(3/2)*(A+C*cos(c+d*x)^2)*sec(c+d*x),x,4,2/5*C*(b*cos(c+d*x))^(3/2)*sin(c+d*x)/d+2/5*b*(5*A+3*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(b*cos(c+d*x))^(3/2)*(A+C*cos(c+d*x)^2)*sec(c+d*x)^2,x,4,2/3*b^2*(3*A+C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))+2/3*b*C*sin(c+d*x)*sqrt(b*cos(c+d*x))/d],
[(b*cos(c+d*x))^(3/2)*(A+C*cos(c+d*x)^2)*sec(c+d*x)^3,x,4,2*A*b^2*sin(c+d*x)/(d*sqrt(b*cos(c+d*x)))-2*b*(A-C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(b*cos(c+d*x))^(3/2)*(A+C*cos(c+d*x)^2)*sec(c+d*x)^4,x,4,2/3*A*b^3*sin(c+d*x)/(d*(b*cos(c+d*x))^(3/2))+2/3*b^2*(A+3*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))],
[(b*cos(c+d*x))^(3/2)*(A+C*cos(c+d*x)^2)*sec(c+d*x)^5,x,5,2/5*A*b^4*sin(c+d*x)/(d*(b*cos(c+d*x))^(5/2))+2/5*b^2*(3*A+5*C)*sin(c+d*x)/(d*sqrt(b*cos(c+d*x)))-2/5*b*(3*A+5*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(b*cos(c+d*x))^(3/2)*(A+C*cos(c+d*x)^2)*sec(c+d*x)^6,x,5,2/7*A*b^5*sin(c+d*x)/(d*(b*cos(c+d*x))^(7/2))+2/21*b^3*(5*A+7*C)*sin(c+d*x)/(d*(b*cos(c+d*x))^(3/2))+2/21*b^2*(5*A+7*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))],
[(b*cos(c+d*x))^(5/2)*(A+C*cos(c+d*x)^2),x,4,2/45*b*(9*A+7*C)*(b*cos(c+d*x))^(3/2)*sin(c+d*x)/d+2/9*C*(b*cos(c+d*x))^(7/2)*sin(c+d*x)/(b*d)+2/15*b^2*(9*A+7*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(b*cos(c+d*x))^(5/2)*(A+C*cos(c+d*x)^2)*sec(c+d*x),x,5,2/7*C*(b*cos(c+d*x))^(5/2)*sin(c+d*x)/d+2/21*b^3*(7*A+5*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))+2/21*b^2*(7*A+5*C)*sin(c+d*x)*sqrt(b*cos(c+d*x))/d],
[(b*cos(c+d*x))^(5/2)*(A+C*cos(c+d*x)^2)*sec(c+d*x)^2,x,4,2/5*b*C*(b*cos(c+d*x))^(3/2)*sin(c+d*x)/d+2/5*b^2*(5*A+3*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(b*cos(c+d*x))^(5/2)*(A+C*cos(c+d*x)^2)*sec(c+d*x)^3,x,4,2/3*b^3*(3*A+C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))+2/3*b^2*C*sin(c+d*x)*sqrt(b*cos(c+d*x))/d],
[(b*cos(c+d*x))^(5/2)*(A+C*cos(c+d*x)^2)*sec(c+d*x)^4,x,4,2*A*b^3*sin(c+d*x)/(d*sqrt(b*cos(c+d*x)))-2*b^2*(A-C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(b*cos(c+d*x))^(5/2)*(A+C*cos(c+d*x)^2)*sec(c+d*x)^5,x,4,2/3*A*b^4*sin(c+d*x)/(d*(b*cos(c+d*x))^(3/2))+2/3*b^3*(A+3*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))],
[(b*cos(c+d*x))^(5/2)*(A+C*cos(c+d*x)^2)*sec(c+d*x)^6,x,5,2/5*A*b^5*sin(c+d*x)/(d*(b*cos(c+d*x))^(5/2))+2/5*b^3*(3*A+5*C)*sin(c+d*x)/(d*sqrt(b*cos(c+d*x)))-2/5*b^2*(3*A+5*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(b*cos(c+d*x))^(5/2)*(A+C*cos(c+d*x)^2)*sec(c+d*x)^7,x,5,2/7*A*b^6*sin(c+d*x)/(d*(b*cos(c+d*x))^(7/2))+2/21*b^4*(5*A+7*C)*sin(c+d*x)/(d*(b*cos(c+d*x))^(3/2))+2/21*b^3*(5*A+7*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))],

# n<0
[cos(c+d*x)^4*(A+C*cos(c+d*x)^2)/(b*cos(c+d*x))^(1/2),x,6,2/77*(11*A+9*C)*(b*cos(c+d*x))^(5/2)*sin(c+d*x)/(b^3*d)+2/11*C*(b*cos(c+d*x))^(9/2)*sin(c+d*x)/(b^5*d)+10/231*(11*A+9*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))+10/231*(11*A+9*C)*sin(c+d*x)*sqrt(b*cos(c+d*x))/(b*d)],
[cos(c+d*x)^3*(A+C*cos(c+d*x)^2)/(b*cos(c+d*x))^(1/2),x,5,2/45*(9*A+7*C)*(b*cos(c+d*x))^(3/2)*sin(c+d*x)/(b^2*d)+2/9*C*(b*cos(c+d*x))^(7/2)*sin(c+d*x)/(b^4*d)+2/15*(9*A+7*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(b*d*sqrt(cos(c+d*x)))],
[cos(c+d*x)^2*(A+C*cos(c+d*x)^2)/(b*cos(c+d*x))^(1/2),x,5,2/7*C*(b*cos(c+d*x))^(5/2)*sin(c+d*x)/(b^3*d)+2/21*(7*A+5*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))+2/21*(7*A+5*C)*sin(c+d*x)*sqrt(b*cos(c+d*x))/(b*d)],
[cos(c+d*x)*(A+C*cos(c+d*x)^2)/(b*cos(c+d*x))^(1/2),x,4,2/5*C*(b*cos(c+d*x))^(3/2)*sin(c+d*x)/(b^2*d)+2/5*(5*A+3*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(b*d*sqrt(cos(c+d*x)))],
[(A+C*cos(c+d*x)^2)/(b*cos(c+d*x))^(1/2),x,3,2/3*(3*A+C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))+2/3*C*sin(c+d*x)*sqrt(b*cos(c+d*x))/(b*d)],
[(A+C*cos(c+d*x)^2)*sec(c+d*x)/(b*cos(c+d*x))^(1/2),x,4,2*A*sin(c+d*x)/(d*sqrt(b*cos(c+d*x)))-2*(A-C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(b*d*sqrt(cos(c+d*x)))],
[(A+C*cos(c+d*x)^2)*sec(c+d*x)^2/(b*cos(c+d*x))^(1/2),x,4,2/3*A*b*sin(c+d*x)/(d*(b*cos(c+d*x))^(3/2))+2/3*(A+3*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))],
[(A+C*cos(c+d*x)^2)*sec(c+d*x)^3/(b*cos(c+d*x))^(1/2),x,5,2/5*A*b^2*sin(c+d*x)/(d*(b*cos(c+d*x))^(5/2))+2/5*(3*A+5*C)*sin(c+d*x)/(d*sqrt(b*cos(c+d*x)))-2/5*(3*A+5*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(b*d*sqrt(cos(c+d*x)))],
[(A+C*cos(c+d*x)^2)*sec(c+d*x)^4/(b*cos(c+d*x))^(1/2),x,5,2/7*A*b^3*sin(c+d*x)/(d*(b*cos(c+d*x))^(7/2))+2/21*b*(5*A+7*C)*sin(c+d*x)/(d*(b*cos(c+d*x))^(3/2))+2/21*(5*A+7*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))],
[(A+C*cos(c+d*x)^2)*sec(c+d*x)^5/(b*cos(c+d*x))^(1/2),x,6,2/9*A*b^4*sin(c+d*x)/(d*(b*cos(c+d*x))^(9/2))+2/45*b^2*(7*A+9*C)*sin(c+d*x)/(d*(b*cos(c+d*x))^(5/2))+2/15*(7*A+9*C)*sin(c+d*x)/(d*sqrt(b*cos(c+d*x)))-2/15*(7*A+9*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(b*d*sqrt(cos(c+d*x)))],
[cos(c+d*x)^4*(A+C*cos(c+d*x)^2)/(b*cos(c+d*x))^(3/2),x,5,2/45*(9*A+7*C)*(b*cos(c+d*x))^(3/2)*sin(c+d*x)/(b^3*d)+2/9*C*(b*cos(c+d*x))^(7/2)*sin(c+d*x)/(b^5*d)+2/15*(9*A+7*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(b^2*d*sqrt(cos(c+d*x)))],
[cos(c+d*x)^3*(A+C*cos(c+d*x)^2)/(b*cos(c+d*x))^(3/2),x,5,2/7*C*(b*cos(c+d*x))^(5/2)*sin(c+d*x)/(b^4*d)+2/21*(7*A+5*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(b*d*sqrt(b*cos(c+d*x)))+2/21*(7*A+5*C)*sin(c+d*x)*sqrt(b*cos(c+d*x))/(b^2*d)],
[cos(c+d*x)^2*(A+C*cos(c+d*x)^2)/(b*cos(c+d*x))^(3/2),x,4,2/5*C*(b*cos(c+d*x))^(3/2)*sin(c+d*x)/(b^3*d)+2/5*(5*A+3*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(b^2*d*sqrt(cos(c+d*x)))],
[cos(c+d*x)*(A+C*cos(c+d*x)^2)/(b*cos(c+d*x))^(3/2),x,4,2/3*(3*A+C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(b*d*sqrt(b*cos(c+d*x)))+2/3*C*sin(c+d*x)*sqrt(b*cos(c+d*x))/(b^2*d)],
[(A+C*cos(c+d*x)^2)/(b*cos(c+d*x))^(3/2),x,3,2*A*sin(c+d*x)/(b*d*sqrt(b*cos(c+d*x)))-2*(A-C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(b^2*d*sqrt(cos(c+d*x)))],
[(A+C*cos(c+d*x)^2)*sec(c+d*x)/(b*cos(c+d*x))^(3/2),x,4,2/3*A*sin(c+d*x)/(d*(b*cos(c+d*x))^(3/2))+2/3*(A+3*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(b*d*sqrt(b*cos(c+d*x)))],
[(A+C*cos(c+d*x)^2)*sec(c+d*x)^2/(b*cos(c+d*x))^(3/2),x,5,2/5*A*b*sin(c+d*x)/(d*(b*cos(c+d*x))^(5/2))+2/5*(3*A+5*C)*sin(c+d*x)/(b*d*sqrt(b*cos(c+d*x)))-2/5*(3*A+5*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(b^2*d*sqrt(cos(c+d*x)))],
[(A+C*cos(c+d*x)^2)*sec(c+d*x)^3/(b*cos(c+d*x))^(3/2),x,5,2/7*A*b^2*sin(c+d*x)/(d*(b*cos(c+d*x))^(7/2))+2/21*(5*A+7*C)*sin(c+d*x)/(d*(b*cos(c+d*x))^(3/2))+2/21*(5*A+7*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(b*d*sqrt(b*cos(c+d*x)))],
[cos(c+d*x)^5*(A+C*cos(c+d*x)^2)/(b*cos(c+d*x))^(5/2),x,5,2/45*(9*A+7*C)*(b*cos(c+d*x))^(3/2)*sin(c+d*x)/(b^4*d)+2/9*C*(b*cos(c+d*x))^(7/2)*sin(c+d*x)/(b^6*d)+2/15*(9*A+7*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(b^3*d*sqrt(cos(c+d*x)))],
[cos(c+d*x)^4*(A+C*cos(c+d*x)^2)/(b*cos(c+d*x))^(5/2),x,5,2/7*C*(b*cos(c+d*x))^(5/2)*sin(c+d*x)/(b^5*d)+2/21*(7*A+5*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(b^2*d*sqrt(b*cos(c+d*x)))+2/21*(7*A+5*C)*sin(c+d*x)*sqrt(b*cos(c+d*x))/(b^3*d)],
[cos(c+d*x)^3*(A+C*cos(c+d*x)^2)/(b*cos(c+d*x))^(5/2),x,4,2/5*C*(b*cos(c+d*x))^(3/2)*sin(c+d*x)/(b^4*d)+2/5*(5*A+3*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(b^3*d*sqrt(cos(c+d*x)))],
[cos(c+d*x)^2*(A+C*cos(c+d*x)^2)/(b*cos(c+d*x))^(5/2),x,4,2/3*(3*A+C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(b^2*d*sqrt(b*cos(c+d*x)))+2/3*C*sin(c+d*x)*sqrt(b*cos(c+d*x))/(b^3*d)],
[cos(c+d*x)*(A+C*cos(c+d*x)^2)/(b*cos(c+d*x))^(5/2),x,4,2*A*sin(c+d*x)/(b^2*d*sqrt(b*cos(c+d*x)))-2*(A-C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(b^3*d*sqrt(cos(c+d*x)))],
[(A+C*cos(c+d*x)^2)/(b*cos(c+d*x))^(5/2),x,3,2/3*A*sin(c+d*x)/(b*d*(b*cos(c+d*x))^(3/2))+2/3*(A+3*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(b^2*d*sqrt(b*cos(c+d*x)))],
[(A+C*cos(c+d*x)^2)*sec(c+d*x)/(b*cos(c+d*x))^(5/2),x,5,2/5*A*sin(c+d*x)/(d*(b*cos(c+d*x))^(5/2))+2/5*(3*A+5*C)*sin(c+d*x)/(b^2*d*sqrt(b*cos(c+d*x)))-2/5*(3*A+5*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(b^3*d*sqrt(cos(c+d*x)))],
[(A+C*cos(c+d*x)^2)*sec(c+d*x)^2/(b*cos(c+d*x))^(5/2),x,5,2/7*A*b*sin(c+d*x)/(d*(b*cos(c+d*x))^(7/2))+2/21*(5*A+7*C)*sin(c+d*x)/(b*d*(b*cos(c+d*x))^(3/2))+2/21*(5*A+7*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(b^2*d*sqrt(b*cos(c+d*x)))],
[(A+C*cos(c+d*x)^2)/(b*cos(c+d*x))^(7/2),x,4,2/5*A*sin(c+d*x)/(b*d*(b*cos(c+d*x))^(5/2))+2/5*(3*A+5*C)*sin(c+d*x)/(b^3*d*sqrt(b*cos(c+d*x)))-2/5*(3*A+5*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(b^4*d*sqrt(cos(c+d*x)))],
[(A+C*cos(c+d*x)^2)/(b*cos(c+d*x))^(9/2),x,4,2/7*A*sin(c+d*x)/(b*d*(b*cos(c+d*x))^(7/2))+2/21*(5*A+7*C)*sin(c+d*x)/(b^3*d*(b*cos(c+d*x))^(3/2))+2/21*(5*A+7*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(b^4*d*sqrt(b*cos(c+d*x)))],

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

# n>0
[cos(c+d*x)^(5/2)*(A+C*cos(c+d*x)^2)*sqrt(b*cos(c+d*x)),x,4,(A+C)*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))-1/3*(A+2*C)*sin(c+d*x)^3*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))+1/5*C*sin(c+d*x)^5*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[cos(c+d*x)^(3/2)*(A+C*cos(c+d*x)^2)*sqrt(b*cos(c+d*x)),x,4,1/4*C*cos(c+d*x)^(5/2)*sin(c+d*x)*sqrt(b*cos(c+d*x))/d+1/8*(4*A+3*C)*x*sqrt(b*cos(c+d*x))/sqrt(cos(c+d*x))+1/8*(4*A+3*C)*sin(c+d*x)*sqrt(cos(c+d*x))*sqrt(b*cos(c+d*x))/d],
[(A+C*cos(c+d*x)^2)*sqrt(cos(c+d*x))*sqrt(b*cos(c+d*x)),x,3,(A+C)*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))-1/3*C*sin(c+d*x)^3*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(A+C*cos(c+d*x)^2)*sqrt(b*cos(c+d*x))/sqrt(cos(c+d*x)),x,4,A*x*sqrt(b*cos(c+d*x))/sqrt(cos(c+d*x))+1/2*C*x*sqrt(b*cos(c+d*x))/sqrt(cos(c+d*x))+1/2*C*sin(c+d*x)*sqrt(cos(c+d*x))*sqrt(b*cos(c+d*x))/d],
[(A+C*cos(c+d*x)^2)*sqrt(b*cos(c+d*x))/cos(c+d*x)^(3/2),x,3,A*arctanh(sin(c+d*x))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))+C*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(A+C*cos(c+d*x)^2)*sqrt(b*cos(c+d*x))/cos(c+d*x)^(5/2),x,3,A*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*cos(c+d*x)^(3/2))+C*x*sqrt(b*cos(c+d*x))/sqrt(cos(c+d*x))],
[(A+C*cos(c+d*x)^2)*sqrt(b*cos(c+d*x))/cos(c+d*x)^(7/2),x,3,1/2*A*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*cos(c+d*x)^(5/2))+1/2*(A+2*C)*arctanh(sin(c+d*x))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(A+C*cos(c+d*x)^2)*sqrt(b*cos(c+d*x))/cos(c+d*x)^(9/2),x,4,1/3*A*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*cos(c+d*x)^(7/2))+1/3*(2*A+3*C)*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*cos(c+d*x)^(3/2))],
[(A+C*cos(c+d*x)^2)*sqrt(b*cos(c+d*x))/cos(c+d*x)^(11/2),x,4,1/4*A*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*cos(c+d*x)^(9/2))+1/8*(3*A+4*C)*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*cos(c+d*x)^(5/2))+1/8*(3*A+4*C)*arctanh(sin(c+d*x))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[cos(c+d*x)^(3/2)*(b*cos(c+d*x))^(3/2)*(A+C*cos(c+d*x)^2),x,4,b*(A+C)*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))-1/3*b*(A+2*C)*sin(c+d*x)^3*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))+1/5*b*C*sin(c+d*x)^5*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(b*cos(c+d*x))^(3/2)*(A+C*cos(c+d*x)^2)*sqrt(cos(c+d*x)),x,4,1/4*b*C*cos(c+d*x)^(5/2)*sin(c+d*x)*sqrt(b*cos(c+d*x))/d+1/8*b*(4*A+3*C)*x*sqrt(b*cos(c+d*x))/sqrt(cos(c+d*x))+1/8*b*(4*A+3*C)*sin(c+d*x)*sqrt(cos(c+d*x))*sqrt(b*cos(c+d*x))/d],
[(b*cos(c+d*x))^(3/2)*(A+C*cos(c+d*x)^2)/sqrt(cos(c+d*x)),x,3,b*(A+C)*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))-1/3*b*C*sin(c+d*x)^3*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(b*cos(c+d*x))^(3/2)*(A+C*cos(c+d*x)^2)/cos(c+d*x)^(3/2),x,4,A*b*x*sqrt(b*cos(c+d*x))/sqrt(cos(c+d*x))+1/2*b*C*x*sqrt(b*cos(c+d*x))/sqrt(cos(c+d*x))+1/2*b*C*sin(c+d*x)*sqrt(cos(c+d*x))*sqrt(b*cos(c+d*x))/d],
[(b*cos(c+d*x))^(3/2)*(A+C*cos(c+d*x)^2)/cos(c+d*x)^(5/2),x,3,A*b*arctanh(sin(c+d*x))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))+b*C*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(b*cos(c+d*x))^(3/2)*(A+C*cos(c+d*x)^2)/cos(c+d*x)^(7/2),x,3,A*b*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*cos(c+d*x)^(3/2))+b*C*x*sqrt(b*cos(c+d*x))/sqrt(cos(c+d*x))],
[(b*cos(c+d*x))^(3/2)*(A+C*cos(c+d*x)^2)/cos(c+d*x)^(9/2),x,3,1/2*A*b*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*cos(c+d*x)^(5/2))+1/2*b*(A+2*C)*arctanh(sin(c+d*x))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(b*cos(c+d*x))^(3/2)*(A+C*cos(c+d*x)^2)/cos(c+d*x)^(11/2),x,4,1/3*A*b*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*cos(c+d*x)^(7/2))+1/3*b*(2*A+3*C)*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*cos(c+d*x)^(3/2))],
[(b*cos(c+d*x))^(3/2)*(A+C*cos(c+d*x)^2)/cos(c+d*x)^(13/2),x,4,1/4*A*b*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*cos(c+d*x)^(9/2))+1/8*b*(3*A+4*C)*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*cos(c+d*x)^(5/2))+1/8*b*(3*A+4*C)*arctanh(sin(c+d*x))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(b*cos(c+d*x))^(5/2)*(A+C*cos(c+d*x)^2)*sqrt(cos(c+d*x)),x,4,b^2*(A+C)*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))-1/3*b^2*(A+2*C)*sin(c+d*x)^3*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))+1/5*b^2*C*sin(c+d*x)^5*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(b*cos(c+d*x))^(5/2)*(A+C*cos(c+d*x)^2)/sqrt(cos(c+d*x)),x,4,1/4*b^2*C*cos(c+d*x)^(5/2)*sin(c+d*x)*sqrt(b*cos(c+d*x))/d+1/8*b^2*(4*A+3*C)*x*sqrt(b*cos(c+d*x))/sqrt(cos(c+d*x))+1/8*b^2*(4*A+3*C)*sin(c+d*x)*sqrt(cos(c+d*x))*sqrt(b*cos(c+d*x))/d],
[(b*cos(c+d*x))^(5/2)*(A+C*cos(c+d*x)^2)/cos(c+d*x)^(3/2),x,3,b^2*(A+C)*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))-1/3*b^2*C*sin(c+d*x)^3*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(b*cos(c+d*x))^(5/2)*(A+C*cos(c+d*x)^2)/cos(c+d*x)^(5/2),x,4,A*b^2*x*sqrt(b*cos(c+d*x))/sqrt(cos(c+d*x))+1/2*b^2*C*x*sqrt(b*cos(c+d*x))/sqrt(cos(c+d*x))+1/2*b^2*C*sin(c+d*x)*sqrt(cos(c+d*x))*sqrt(b*cos(c+d*x))/d],
[(b*cos(c+d*x))^(5/2)*(A+C*cos(c+d*x)^2)/cos(c+d*x)^(7/2),x,3,A*b^2*arctanh(sin(c+d*x))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))+b^2*C*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(b*cos(c+d*x))^(5/2)*(A+C*cos(c+d*x)^2)/cos(c+d*x)^(9/2),x,3,A*b^2*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*cos(c+d*x)^(3/2))+b^2*C*x*sqrt(b*cos(c+d*x))/sqrt(cos(c+d*x))],
[(b*cos(c+d*x))^(5/2)*(A+C*cos(c+d*x)^2)/cos(c+d*x)^(11/2),x,3,1/2*A*b^2*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*cos(c+d*x)^(5/2))+1/2*b^2*(A+2*C)*arctanh(sin(c+d*x))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(b*cos(c+d*x))^(5/2)*(A+C*cos(c+d*x)^2)/cos(c+d*x)^(13/2),x,4,1/3*A*b^2*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*cos(c+d*x)^(7/2))+1/3*b^2*(2*A+3*C)*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*cos(c+d*x)^(3/2))],
[(b*cos(c+d*x))^(5/2)*(A+C*cos(c+d*x)^2)/cos(c+d*x)^(15/2),x,4,1/4*A*b^2*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*cos(c+d*x)^(9/2))+1/8*b^2*(3*A+4*C)*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*cos(c+d*x)^(5/2))+1/8*b^2*(3*A+4*C)*arctanh(sin(c+d*x))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],

# n<0
[cos(c+d*x)^(5/2)*(A+C*cos(c+d*x)^2)/sqrt(b*cos(c+d*x)),x,4,1/8*(4*A+3*C)*cos(c+d*x)^(3/2)*sin(c+d*x)/(d*sqrt(b*cos(c+d*x)))+1/4*C*cos(c+d*x)^(7/2)*sin(c+d*x)/(d*sqrt(b*cos(c+d*x)))+1/8*(4*A+3*C)*x*sqrt(cos(c+d*x))/sqrt(b*cos(c+d*x))],
[cos(c+d*x)^(3/2)*(A+C*cos(c+d*x)^2)/sqrt(b*cos(c+d*x)),x,3,(A+C)*sin(c+d*x)*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))-1/3*C*sin(c+d*x)^3*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))],
[(A+C*cos(c+d*x)^2)*sqrt(cos(c+d*x))/sqrt(b*cos(c+d*x)),x,4,1/2*C*cos(c+d*x)^(3/2)*sin(c+d*x)/(d*sqrt(b*cos(c+d*x)))+A*x*sqrt(cos(c+d*x))/sqrt(b*cos(c+d*x))+1/2*C*x*sqrt(cos(c+d*x))/sqrt(b*cos(c+d*x))],
[(A+C*cos(c+d*x)^2)/(sqrt(cos(c+d*x))*sqrt(b*cos(c+d*x))),x,3,A*arctanh(sin(c+d*x))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))+C*sin(c+d*x)*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))],
[(A+C*cos(c+d*x)^2)/(cos(c+d*x)^(3/2)*sqrt(b*cos(c+d*x))),x,3,A*sin(c+d*x)/(d*sqrt(cos(c+d*x))*sqrt(b*cos(c+d*x)))+C*x*sqrt(cos(c+d*x))/sqrt(b*cos(c+d*x))],
[(A+C*cos(c+d*x)^2)/(cos(c+d*x)^(5/2)*sqrt(b*cos(c+d*x))),x,3,1/2*A*sin(c+d*x)/(d*cos(c+d*x)^(3/2)*sqrt(b*cos(c+d*x)))+1/2*(A+2*C)*arctanh(sin(c+d*x))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))],
[(A+C*cos(c+d*x)^2)/(cos(c+d*x)^(7/2)*sqrt(b*cos(c+d*x))),x,4,1/3*A*sin(c+d*x)/(d*cos(c+d*x)^(5/2)*sqrt(b*cos(c+d*x)))+1/3*(2*A+3*C)*sin(c+d*x)/(d*sqrt(cos(c+d*x))*sqrt(b*cos(c+d*x)))],
[(A+C*cos(c+d*x)^2)/(cos(c+d*x)^(9/2)*sqrt(b*cos(c+d*x))),x,4,1/4*A*sin(c+d*x)/(d*cos(c+d*x)^(7/2)*sqrt(b*cos(c+d*x)))+1/8*(3*A+4*C)*sin(c+d*x)/(d*cos(c+d*x)^(3/2)*sqrt(b*cos(c+d*x)))+1/8*(3*A+4*C)*arctanh(sin(c+d*x))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))],
[cos(c+d*x)^(7/2)*(A+C*cos(c+d*x)^2)/(b*cos(c+d*x))^(3/2),x,4,1/8*(4*A+3*C)*cos(c+d*x)^(3/2)*sin(c+d*x)/(b*d*sqrt(b*cos(c+d*x)))+1/4*C*cos(c+d*x)^(7/2)*sin(c+d*x)/(b*d*sqrt(b*cos(c+d*x)))+1/8*(4*A+3*C)*x*sqrt(cos(c+d*x))/(b*sqrt(b*cos(c+d*x)))],
[cos(c+d*x)^(5/2)*(A+C*cos(c+d*x)^2)/(b*cos(c+d*x))^(3/2),x,3,(A+C)*sin(c+d*x)*sqrt(cos(c+d*x))/(b*d*sqrt(b*cos(c+d*x)))-1/3*C*sin(c+d*x)^3*sqrt(cos(c+d*x))/(b*d*sqrt(b*cos(c+d*x)))],
[cos(c+d*x)^(3/2)*(A+C*cos(c+d*x)^2)/(b*cos(c+d*x))^(3/2),x,4,1/2*C*cos(c+d*x)^(3/2)*sin(c+d*x)/(b*d*sqrt(b*cos(c+d*x)))+A*x*sqrt(cos(c+d*x))/(b*sqrt(b*cos(c+d*x)))+1/2*C*x*sqrt(cos(c+d*x))/(b*sqrt(b*cos(c+d*x)))],
[(A+C*cos(c+d*x)^2)*sqrt(cos(c+d*x))/(b*cos(c+d*x))^(3/2),x,3,A*arctanh(sin(c+d*x))*sqrt(cos(c+d*x))/(b*d*sqrt(b*cos(c+d*x)))+C*sin(c+d*x)*sqrt(cos(c+d*x))/(b*d*sqrt(b*cos(c+d*x)))],
[(A+C*cos(c+d*x)^2)/((b*cos(c+d*x))^(3/2)*sqrt(cos(c+d*x))),x,3,A*sin(c+d*x)/(b*d*sqrt(cos(c+d*x))*sqrt(b*cos(c+d*x)))+C*x*sqrt(cos(c+d*x))/(b*sqrt(b*cos(c+d*x)))],
[(A+C*cos(c+d*x)^2)/(cos(c+d*x)^(3/2)*(b*cos(c+d*x))^(3/2)),x,3,1/2*A*sin(c+d*x)/(b*d*cos(c+d*x)^(3/2)*sqrt(b*cos(c+d*x)))+1/2*(A+2*C)*arctanh(sin(c+d*x))*sqrt(cos(c+d*x))/(b*d*sqrt(b*cos(c+d*x)))],
[(A+C*cos(c+d*x)^2)/(cos(c+d*x)^(5/2)*(b*cos(c+d*x))^(3/2)),x,4,1/3*A*sin(c+d*x)/(b*d*cos(c+d*x)^(5/2)*sqrt(b*cos(c+d*x)))+1/3*(2*A+3*C)*sin(c+d*x)/(b*d*sqrt(cos(c+d*x))*sqrt(b*cos(c+d*x)))],
[(A+C*cos(c+d*x)^2)/(cos(c+d*x)^(7/2)*(b*cos(c+d*x))^(3/2)),x,4,1/4*A*sin(c+d*x)/(b*d*cos(c+d*x)^(7/2)*sqrt(b*cos(c+d*x)))+1/8*(3*A+4*C)*sin(c+d*x)/(b*d*cos(c+d*x)^(3/2)*sqrt(b*cos(c+d*x)))+1/8*(3*A+4*C)*arctanh(sin(c+d*x))*sqrt(cos(c+d*x))/(b*d*sqrt(b*cos(c+d*x)))],
[cos(c+d*x)^(9/2)*(A+C*cos(c+d*x)^2)/(b*cos(c+d*x))^(5/2),x,4,1/8*(4*A+3*C)*cos(c+d*x)^(3/2)*sin(c+d*x)/(b^2*d*sqrt(b*cos(c+d*x)))+1/4*C*cos(c+d*x)^(7/2)*sin(c+d*x)/(b^2*d*sqrt(b*cos(c+d*x)))+1/8*(4*A+3*C)*x*sqrt(cos(c+d*x))/(b^2*sqrt(b*cos(c+d*x)))],
[cos(c+d*x)^(7/2)*(A+C*cos(c+d*x)^2)/(b*cos(c+d*x))^(5/2),x,3,(A+C)*sin(c+d*x)*sqrt(cos(c+d*x))/(b^2*d*sqrt(b*cos(c+d*x)))-1/3*C*sin(c+d*x)^3*sqrt(cos(c+d*x))/(b^2*d*sqrt(b*cos(c+d*x)))],
[cos(c+d*x)^(5/2)*(A+C*cos(c+d*x)^2)/(b*cos(c+d*x))^(5/2),x,4,1/2*C*cos(c+d*x)^(3/2)*sin(c+d*x)/(b^2*d*sqrt(b*cos(c+d*x)))+A*x*sqrt(cos(c+d*x))/(b^2*sqrt(b*cos(c+d*x)))+1/2*C*x*sqrt(cos(c+d*x))/(b^2*sqrt(b*cos(c+d*x)))],
[cos(c+d*x)^(3/2)*(A+C*cos(c+d*x)^2)/(b*cos(c+d*x))^(5/2),x,3,A*arctanh(sin(c+d*x))*sqrt(cos(c+d*x))/(b^2*d*sqrt(b*cos(c+d*x)))+C*sin(c+d*x)*sqrt(cos(c+d*x))/(b^2*d*sqrt(b*cos(c+d*x)))],
[(A+C*cos(c+d*x)^2)*sqrt(cos(c+d*x))/(b*cos(c+d*x))^(5/2),x,3,A*sin(c+d*x)/(b^2*d*sqrt(cos(c+d*x))*sqrt(b*cos(c+d*x)))+C*x*sqrt(cos(c+d*x))/(b^2*sqrt(b*cos(c+d*x)))],
[(A+C*cos(c+d*x)^2)/((b*cos(c+d*x))^(5/2)*sqrt(cos(c+d*x))),x,3,1/2*A*sin(c+d*x)/(b^2*d*cos(c+d*x)^(3/2)*sqrt(b*cos(c+d*x)))+1/2*(A+2*C)*arctanh(sin(c+d*x))*sqrt(cos(c+d*x))/(b^2*d*sqrt(b*cos(c+d*x)))],
[(A+C*cos(c+d*x)^2)/(cos(c+d*x)^(3/2)*(b*cos(c+d*x))^(5/2)),x,4,1/3*A*sin(c+d*x)/(b^2*d*cos(c+d*x)^(5/2)*sqrt(b*cos(c+d*x)))+1/3*(2*A+3*C)*sin(c+d*x)/(b^2*d*sqrt(cos(c+d*x))*sqrt(b*cos(c+d*x)))],
[(A+C*cos(c+d*x)^2)/(cos(c+d*x)^(5/2)*(b*cos(c+d*x))^(5/2)),x,4,1/4*A*sin(c+d*x)/(b^2*d*cos(c+d*x)^(7/2)*sqrt(b*cos(c+d*x)))+1/8*(3*A+4*C)*sin(c+d*x)/(b^2*d*cos(c+d*x)^(3/2)*sqrt(b*cos(c+d*x)))+1/8*(3*A+4*C)*arctanh(sin(c+d*x))*sqrt(cos(c+d*x))/(b^2*d*sqrt(b*cos(c+d*x)))],

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

# n>0
[cos(c+d*x)^2*(b*cos(c+d*x))^(1/3)*(A+C*cos(c+d*x)^2),x,3,3/13*C*(b*cos(c+d*x))^(10/3)*sin(c+d*x)/(b^3*d)-3/130*(13*A+10*C)*(b*cos(c+d*x))^(10/3)*hypergeom([1/2,5/3],[8/3],cos(c+d*x)^2)*sin(c+d*x)/(b^3*d*sqrt(sin(c+d*x)^2))],
[cos(c+d*x)*(b*cos(c+d*x))^(1/3)*(A+C*cos(c+d*x)^2),x,3,3/10*C*(b*cos(c+d*x))^(7/3)*sin(c+d*x)/(b^2*d)-3/70*(10*A+7*C)*(b*cos(c+d*x))^(7/3)*hypergeom([1/2,7/6],[13/6],cos(c+d*x)^2)*sin(c+d*x)/(b^2*d*sqrt(sin(c+d*x)^2))],
[(b*cos(c+d*x))^(1/3)*(A+C*cos(c+d*x)^2),x,2,3/7*C*(b*cos(c+d*x))^(4/3)*sin(c+d*x)/(b*d)-3/28*(7*A+4*C)*(b*cos(c+d*x))^(4/3)*hypergeom([1/2,2/3],[5/3],cos(c+d*x)^2)*sin(c+d*x)/(b*d*sqrt(sin(c+d*x)^2))],
[(b*cos(c+d*x))^(1/3)*(A+C*cos(c+d*x)^2)*sec(c+d*x),x,3,3/4*C*(b*cos(c+d*x))^(1/3)*sin(c+d*x)/d-3/4*(4*A+C)*(b*cos(c+d*x))^(1/3)*hypergeom([1/6,1/2],[7/6],cos(c+d*x)^2)*sin(c+d*x)/(d*sqrt(sin(c+d*x)^2))],
[(b*cos(c+d*x))^(1/3)*(A+C*cos(c+d*x)^2)*sec(c+d*x)^2,x,3,3/2*A*b*sin(c+d*x)/(d*(b*cos(c+d*x))^(2/3))+3/8*(A-2*C)*(b*cos(c+d*x))^(4/3)*hypergeom([1/2,2/3],[5/3],cos(c+d*x)^2)*sin(c+d*x)/(b*d*sqrt(sin(c+d*x)^2))],
[(b*cos(c+d*x))^(1/3)*(A+C*cos(c+d*x)^2)*sec(c+d*x)^3,x,3,3/5*A*b^2*sin(c+d*x)/(d*(b*cos(c+d*x))^(5/3))-3/5*(2*A+5*C)*(b*cos(c+d*x))^(1/3)*hypergeom([1/6,1/2],[7/6],cos(c+d*x)^2)*sin(c+d*x)/(d*sqrt(sin(c+d*x)^2))],
[cos(c+d*x)^2*(b*cos(c+d*x))^(2/3)*(A+C*cos(c+d*x)^2),x,3,3/14*C*(b*cos(c+d*x))^(11/3)*sin(c+d*x)/(b^3*d)-3/154*(14*A+11*C)*(b*cos(c+d*x))^(11/3)*hypergeom([1/2,11/6],[17/6],cos(c+d*x)^2)*sin(c+d*x)/(b^3*d*sqrt(sin(c+d*x)^2))],
[cos(c+d*x)*(b*cos(c+d*x))^(2/3)*(A+C*cos(c+d*x)^2),x,3,3/11*C*(b*cos(c+d*x))^(8/3)*sin(c+d*x)/(b^2*d)-3/88*(11*A+8*C)*(b*cos(c+d*x))^(8/3)*hypergeom([1/2,4/3],[7/3],cos(c+d*x)^2)*sin(c+d*x)/(b^2*d*sqrt(sin(c+d*x)^2))],
[(b*cos(c+d*x))^(2/3)*(A+C*cos(c+d*x)^2),x,2,3/8*C*(b*cos(c+d*x))^(5/3)*sin(c+d*x)/(b*d)-3/40*(8*A+5*C)*(b*cos(c+d*x))^(5/3)*hypergeom([1/2,5/6],[11/6],cos(c+d*x)^2)*sin(c+d*x)/(b*d*sqrt(sin(c+d*x)^2))],
[(b*cos(c+d*x))^(2/3)*(A+C*cos(c+d*x)^2)*sec(c+d*x),x,3,3/5*C*(b*cos(c+d*x))^(2/3)*sin(c+d*x)/d-3/10*(5*A+2*C)*(b*cos(c+d*x))^(2/3)*hypergeom([1/3,1/2],[4/3],cos(c+d*x)^2)*sin(c+d*x)/(d*sqrt(sin(c+d*x)^2))],
[(b*cos(c+d*x))^(2/3)*(A+C*cos(c+d*x)^2)*sec(c+d*x)^2,x,3,3*A*b*sin(c+d*x)/(d*(b*cos(c+d*x))^(1/3))+3/5*(2*A-C)*(b*cos(c+d*x))^(5/3)*hypergeom([1/2,5/6],[11/6],cos(c+d*x)^2)*sin(c+d*x)/(b*d*sqrt(sin(c+d*x)^2))],
[(b*cos(c+d*x))^(2/3)*(A+C*cos(c+d*x)^2)*sec(c+d*x)^3,x,3,3/4*A*b^2*sin(c+d*x)/(d*(b*cos(c+d*x))^(4/3))-3/8*(A+4*C)*(b*cos(c+d*x))^(2/3)*hypergeom([1/3,1/2],[4/3],cos(c+d*x)^2)*sin(c+d*x)/(d*sqrt(sin(c+d*x)^2))],
[cos(c+d*x)^2*(b*cos(c+d*x))^(4/3)*(A+C*cos(c+d*x)^2),x,3,3/16*C*(b*cos(c+d*x))^(13/3)*sin(c+d*x)/(b^3*d)-3/208*(16*A+13*C)*(b*cos(c+d*x))^(13/3)*hypergeom([1/2,13/6],[19/6],cos(c+d*x)^2)*sin(c+d*x)/(b^3*d*sqrt(sin(c+d*x)^2))],
[cos(c+d*x)*(b*cos(c+d*x))^(4/3)*(A+C*cos(c+d*x)^2),x,3,3/13*C*(b*cos(c+d*x))^(10/3)*sin(c+d*x)/(b^2*d)-3/130*(13*A+10*C)*(b*cos(c+d*x))^(10/3)*hypergeom([1/2,5/3],[8/3],cos(c+d*x)^2)*sin(c+d*x)/(b^2*d*sqrt(sin(c+d*x)^2))],
[(b*cos(c+d*x))^(4/3)*(A+C*cos(c+d*x)^2),x,2,3/10*C*(b*cos(c+d*x))^(7/3)*sin(c+d*x)/(b*d)-3/70*(10*A+7*C)*(b*cos(c+d*x))^(7/3)*hypergeom([1/2,7/6],[13/6],cos(c+d*x)^2)*sin(c+d*x)/(b*d*sqrt(sin(c+d*x)^2))],
[(b*cos(c+d*x))^(4/3)*(A+C*cos(c+d*x)^2)*sec(c+d*x),x,3,3/7*C*(b*cos(c+d*x))^(4/3)*sin(c+d*x)/d-3/28*(7*A+4*C)*(b*cos(c+d*x))^(4/3)*hypergeom([1/2,2/3],[5/3],cos(c+d*x)^2)*sin(c+d*x)/(d*sqrt(sin(c+d*x)^2))],
[(b*cos(c+d*x))^(4/3)*(A+C*cos(c+d*x)^2)*sec(c+d*x)^2,x,3,3/4*b*C*(b*cos(c+d*x))^(1/3)*sin(c+d*x)/d-3/4*b*(4*A+C)*(b*cos(c+d*x))^(1/3)*hypergeom([1/6,1/2],[7/6],cos(c+d*x)^2)*sin(c+d*x)/(d*sqrt(sin(c+d*x)^2))],
[(b*cos(c+d*x))^(4/3)*(A+C*cos(c+d*x)^2)*sec(c+d*x)^3,x,3,3/2*A*b^2*sin(c+d*x)/(d*(b*cos(c+d*x))^(2/3))+3/8*(A-2*C)*(b*cos(c+d*x))^(4/3)*hypergeom([1/2,2/3],[5/3],cos(c+d*x)^2)*sin(c+d*x)/(d*sqrt(sin(c+d*x)^2))],

# n<0
[cos(c+d*x)^2*(A+C*cos(c+d*x)^2)/(b*cos(c+d*x))^(1/3),x,3,3/11*C*(b*cos(c+d*x))^(8/3)*sin(c+d*x)/(b^3*d)-3/88*(11*A+8*C)*(b*cos(c+d*x))^(8/3)*hypergeom([1/2,4/3],[7/3],cos(c+d*x)^2)*sin(c+d*x)/(b^3*d*sqrt(sin(c+d*x)^2))],
[cos(c+d*x)*(A+C*cos(c+d*x)^2)/(b*cos(c+d*x))^(1/3),x,3,3/8*C*(b*cos(c+d*x))^(5/3)*sin(c+d*x)/(b^2*d)-3/40*(8*A+5*C)*(b*cos(c+d*x))^(5/3)*hypergeom([1/2,5/6],[11/6],cos(c+d*x)^2)*sin(c+d*x)/(b^2*d*sqrt(sin(c+d*x)^2))],
[(A+C*cos(c+d*x)^2)/(b*cos(c+d*x))^(1/3),x,2,3/5*C*(b*cos(c+d*x))^(2/3)*sin(c+d*x)/(b*d)-3/10*(5*A+2*C)*(b*cos(c+d*x))^(2/3)*hypergeom([1/3,1/2],[4/3],cos(c+d*x)^2)*sin(c+d*x)/(b*d*sqrt(sin(c+d*x)^2))],
[(A+C*cos(c+d*x)^2)*sec(c+d*x)/(b*cos(c+d*x))^(1/3),x,3,3*A*sin(c+d*x)/(d*(b*cos(c+d*x))^(1/3))+3/5*(2*A-C)*(b*cos(c+d*x))^(5/3)*hypergeom([1/2,5/6],[11/6],cos(c+d*x)^2)*sin(c+d*x)/(b^2*d*sqrt(sin(c+d*x)^2))],
[(A+C*cos(c+d*x)^2)*sec(c+d*x)^2/(b*cos(c+d*x))^(1/3),x,3,3/4*A*b*sin(c+d*x)/(d*(b*cos(c+d*x))^(4/3))-3/8*(A+4*C)*(b*cos(c+d*x))^(2/3)*hypergeom([1/3,1/2],[4/3],cos(c+d*x)^2)*sin(c+d*x)/(b*d*sqrt(sin(c+d*x)^2))],
[(A+C*cos(c+d*x)^2)*sec(c+d*x)^3/(b*cos(c+d*x))^(1/3),x,3,3/7*A*b^2*sin(c+d*x)/(d*(b*cos(c+d*x))^(7/3))+3/7*(4*A+7*C)*hypergeom([-1/6,1/2],[5/6],cos(c+d*x)^2)*sin(c+d*x)/(d*(b*cos(c+d*x))^(1/3)*sqrt(sin(c+d*x)^2))],
[cos(c+d*x)^2*(A+C*cos(c+d*x)^2)/(b*cos(c+d*x))^(2/3),x,3,3/10*C*(b*cos(c+d*x))^(7/3)*sin(c+d*x)/(b^3*d)-3/70*(10*A+7*C)*(b*cos(c+d*x))^(7/3)*hypergeom([1/2,7/6],[13/6],cos(c+d*x)^2)*sin(c+d*x)/(b^3*d*sqrt(sin(c+d*x)^2))],
[cos(c+d*x)*(A+C*cos(c+d*x)^2)/(b*cos(c+d*x))^(2/3),x,3,3/7*C*(b*cos(c+d*x))^(4/3)*sin(c+d*x)/(b^2*d)-3/28*(7*A+4*C)*(b*cos(c+d*x))^(4/3)*hypergeom([1/2,2/3],[5/3],cos(c+d*x)^2)*sin(c+d*x)/(b^2*d*sqrt(sin(c+d*x)^2))],
[(A+C*cos(c+d*x)^2)/(b*cos(c+d*x))^(2/3),x,2,3/4*C*(b*cos(c+d*x))^(1/3)*sin(c+d*x)/(b*d)-3/4*(4*A+C)*(b*cos(c+d*x))^(1/3)*hypergeom([1/6,1/2],[7/6],cos(c+d*x)^2)*sin(c+d*x)/(b*d*sqrt(sin(c+d*x)^2))],
[(A+C*cos(c+d*x)^2)*sec(c+d*x)/(b*cos(c+d*x))^(2/3),x,3,3/2*A*sin(c+d*x)/(d*(b*cos(c+d*x))^(2/3))+3/8*(A-2*C)*(b*cos(c+d*x))^(4/3)*hypergeom([1/2,2/3],[5/3],cos(c+d*x)^2)*sin(c+d*x)/(b^2*d*sqrt(sin(c+d*x)^2))],
[(A+C*cos(c+d*x)^2)*sec(c+d*x)^2/(b*cos(c+d*x))^(2/3),x,3,3/5*A*b*sin(c+d*x)/(d*(b*cos(c+d*x))^(5/3))-3/5*(2*A+5*C)*(b*cos(c+d*x))^(1/3)*hypergeom([1/6,1/2],[7/6],cos(c+d*x)^2)*sin(c+d*x)/(b*d*sqrt(sin(c+d*x)^2))],
[(A+C*cos(c+d*x)^2)*sec(c+d*x)^3/(b*cos(c+d*x))^(2/3),x,3,3/8*A*b^2*sin(c+d*x)/(d*(b*cos(c+d*x))^(8/3))+3/16*(5*A+8*C)*hypergeom([-1/3,1/2],[2/3],cos(c+d*x)^2)*sin(c+d*x)/(d*(b*cos(c+d*x))^(2/3)*sqrt(sin(c+d*x)^2))],
[cos(c+d*x)^2*(A+C*cos(c+d*x)^2)/(b*cos(c+d*x))^(4/3),x,3,3/8*C*(b*cos(c+d*x))^(5/3)*sin(c+d*x)/(b^3*d)-3/40*(8*A+5*C)*(b*cos(c+d*x))^(5/3)*hypergeom([1/2,5/6],[11/6],cos(c+d*x)^2)*sin(c+d*x)/(b^3*d*sqrt(sin(c+d*x)^2))],
[cos(c+d*x)*(A+C*cos(c+d*x)^2)/(b*cos(c+d*x))^(4/3),x,3,3/5*C*(b*cos(c+d*x))^(2/3)*sin(c+d*x)/(b^2*d)-3/10*(5*A+2*C)*(b*cos(c+d*x))^(2/3)*hypergeom([1/3,1/2],[4/3],cos(c+d*x)^2)*sin(c+d*x)/(b^2*d*sqrt(sin(c+d*x)^2))],
[(A+C*cos(c+d*x)^2)/(b*cos(c+d*x))^(4/3),x,2,3*A*sin(c+d*x)/(b*d*(b*cos(c+d*x))^(1/3))+3/5*(2*A-C)*(b*cos(c+d*x))^(5/3)*hypergeom([1/2,5/6],[11/6],cos(c+d*x)^2)*sin(c+d*x)/(b^3*d*sqrt(sin(c+d*x)^2))],
[(A+C*cos(c+d*x)^2)*sec(c+d*x)/(b*cos(c+d*x))^(4/3),x,3,3/4*A*sin(c+d*x)/(d*(b*cos(c+d*x))^(4/3))-3/8*(A+4*C)*(b*cos(c+d*x))^(2/3)*hypergeom([1/3,1/2],[4/3],cos(c+d*x)^2)*sin(c+d*x)/(b^2*d*sqrt(sin(c+d*x)^2))],
[(A+C*cos(c+d*x)^2)*sec(c+d*x)^2/(b*cos(c+d*x))^(4/3),x,3,3/7*A*b*sin(c+d*x)/(d*(b*cos(c+d*x))^(7/3))+3/7*(4*A+7*C)*hypergeom([-1/6,1/2],[5/6],cos(c+d*x)^2)*sin(c+d*x)/(b*d*(b*cos(c+d*x))^(1/3)*sqrt(sin(c+d*x)^2))],
[(A+C*cos(c+d*x)^2)*sec(c+d*x)^3/(b*cos(c+d*x))^(4/3),x,3,3/10*A*b^2*sin(c+d*x)/(d*(b*cos(c+d*x))^(10/3))+3/40*(7*A+10*C)*hypergeom([-2/3,1/2],[1/3],cos(c+d*x)^2)*sin(c+d*x)/(d*(b*cos(c+d*x))^(4/3)*sqrt(sin(c+d*x)^2))],

# Integrands of the form (a Cos[c+d x])^m (b Cos[c+d x])^n (A+C Cos[c+d x]^2) with m symbolic
[cos(c+d*x)^m*(b*cos(c+d*x))^(4/3)*(A+C*cos(c+d*x)^2),x,3,3*b*C*cos(c+d*x)^(2+m)*(b*cos(c+d*x))^(1/3)*sin(c+d*x)/(d*(10+3*m))-3*b*(C*(7+3*m)+A*(10+3*m))*cos(c+d*x)^(2+m)*(b*cos(c+d*x))^(1/3)*hypergeom([1/2,1/6*(7+3*m)],[1/6*(13+3*m)],cos(c+d*x)^2)*sin(c+d*x)/(d*(70+51*m+9*m^2)*sqrt(sin(c+d*x)^2))],
[cos(c+d*x)^m*(b*cos(c+d*x))^(2/3)*(A+C*cos(c+d*x)^2),x,3,3*C*cos(c+d*x)^(1+m)*(b*cos(c+d*x))^(2/3)*sin(c+d*x)/(d*(8+3*m))-3*(C*(5+3*m)+A*(8+3*m))*cos(c+d*x)^(1+m)*(b*cos(c+d*x))^(2/3)*hypergeom([1/2,1/6*(5+3*m)],[1/6*(11+3*m)],cos(c+d*x)^2)*sin(c+d*x)/(d*(40+39*m+9*m^2)*sqrt(sin(c+d*x)^2))],
[cos(c+d*x)^m*(b*cos(c+d*x))^(1/3)*(A+C*cos(c+d*x)^2),x,3,3*C*cos(c+d*x)^(1+m)*(b*cos(c+d*x))^(1/3)*sin(c+d*x)/(d*(7+3*m))-3*(C*(4+3*m)+A*(7+3*m))*cos(c+d*x)^(1+m)*(b*cos(c+d*x))^(1/3)*hypergeom([1/2,1/6*(4+3*m)],[1/6*(10+3*m)],cos(c+d*x)^2)*sin(c+d*x)/(d*(28+33*m+9*m^2)*sqrt(sin(c+d*x)^2))],
[cos(c+d*x)^m*(A+C*cos(c+d*x)^2)/(b*cos(c+d*x))^(1/3),x,3,3*C*cos(c+d*x)^(1+m)*sin(c+d*x)/(d*(5+3*m)*(b*cos(c+d*x))^(1/3))-3*(C*(2+3*m)+A*(5+3*m))*cos(c+d*x)^(1+m)*hypergeom([1/2,1/6*(2+3*m)],[1/6*(8+3*m)],cos(c+d*x)^2)*sin(c+d*x)/(d*(10+21*m+9*m^2)*(b*cos(c+d*x))^(1/3)*sqrt(sin(c+d*x)^2))],
[cos(c+d*x)^m*(A+C*cos(c+d*x)^2)/(b*cos(c+d*x))^(2/3),x,3,3*C*cos(c+d*x)^(1+m)*sin(c+d*x)/(d*(4+3*m)*(b*cos(c+d*x))^(2/3))-3*(C+3*C*m+A*(4+3*m))*cos(c+d*x)^(1+m)*hypergeom([1/2,1/6*(1+3*m)],[1/6*(7+3*m)],cos(c+d*x)^2)*sin(c+d*x)/(d*(4+15*m+9*m^2)*(b*cos(c+d*x))^(2/3)*sqrt(sin(c+d*x)^2))],
[cos(c+d*x)^m*(A+C*cos(c+d*x)^2)/(b*cos(c+d*x))^(4/3),x,3,3*C*cos(c+d*x)^m*sin(c+d*x)/(b*d*(2+3*m)*(b*cos(c+d*x))^(1/3))-3*(C*(1-3*m)-A*(2+3*m))*cos(c+d*x)^m*hypergeom([1/2,1/6*(-1+3*m)],[1/6*(5+3*m)],cos(c+d*x)^2)*sin(c+d*x)/(b*d*(2-3*m-9*m^2)*(b*cos(c+d*x))^(1/3)*sqrt(sin(c+d*x)^2))],

# Integrands of the form (a Cos[c+d x])^m (b Cos[c+d x])^n (A+C Cos[c+d x]^2) with n symbolic
[(a*cos(c+d*x))^m*(b*cos(c+d*x))^n*(A+C*cos(c+d*x)^2),x,3,C*(a*cos(c+d*x))^(1+m)*(b*cos(c+d*x))^n*sin(c+d*x)/(a*d*(2+m+n))-(C*(1+m+n)+A*(2+m+n))*(a*cos(c+d*x))^(1+m)*(b*cos(c+d*x))^n*hypergeom([1/2,1/2*(1+m+n)],[1/2*(3+m+n)],cos(c+d*x)^2)*sin(c+d*x)/(a*d*(1+m+n)*(2+m+n)*sqrt(sin(c+d*x)^2))],
[cos(c+d*x)^2*(b*cos(c+d*x))^n*(A+C*cos(c+d*x)^2),x,3,C*(b*cos(c+d*x))^(3+n)*sin(c+d*x)/(b^3*d*(4+n))-(C*(3+n)+A*(4+n))*(b*cos(c+d*x))^(3+n)*hypergeom([1/2,1/2*(3+n)],[1/2*(5+n)],cos(c+d*x)^2)*sin(c+d*x)/(b^3*d*(3+n)*(4+n)*sqrt(sin(c+d*x)^2))],
[cos(c+d*x)*(b*cos(c+d*x))^n*(A+C*cos(c+d*x)^2),x,3,C*(b*cos(c+d*x))^(2+n)*sin(c+d*x)/(b^2*d*(3+n))-(C*(2+n)+A*(3+n))*(b*cos(c+d*x))^(2+n)*hypergeom([1/2,1/2*(2+n)],[1/2*(4+n)],cos(c+d*x)^2)*sin(c+d*x)/(b^2*d*(2+n)*(3+n)*sqrt(sin(c+d*x)^2))],
[(b*cos(c+d*x))^n*(A+C*cos(c+d*x)^2),x,2,C*(b*cos(c+d*x))^(1+n)*sin(c+d*x)/(b*d*(2+n))-(C*(1+n)+A*(2+n))*(b*cos(c+d*x))^(1+n)*hypergeom([1/2,1/2*(1+n)],[1/2*(3+n)],cos(c+d*x)^2)*sin(c+d*x)/(b*d*(1+n)*(2+n)*sqrt(sin(c+d*x)^2))],
[(b*cos(c+d*x))^n*(A+C*cos(c+d*x)^2)*sec(c+d*x),x,3,C*(b*cos(c+d*x))^n*sin(c+d*x)/(d*(1+n))-(A+A*n+C*n)*(b*cos(c+d*x))^n*hypergeom([1/2,1/2*n],[1/2*(2+n)],cos(c+d*x)^2)*sin(c+d*x)/(d*n*(1+n)*sqrt(sin(c+d*x)^2))],
[(b*cos(c+d*x))^n*(A+C*cos(c+d*x)^2)*sec(c+d*x)^2,x,3,b*C*(b*cos(c+d*x))^(-1+n)*sin(c+d*x)/(d*n)-b*(C*(1-n)-A*n)*(b*cos(c+d*x))^(-1+n)*hypergeom([1/2,1/2*(-1+n)],[1/2*(1+n)],cos(c+d*x)^2)*sin(c+d*x)/(d*(1-n)*n*sqrt(sin(c+d*x)^2))],
[(b*cos(c+d*x))^n*(A+C*cos(c+d*x)^2)*sec(c+d*x)^3,x,3,-b^2*C*(b*cos(c+d*x))^(-2+n)*sin(c+d*x)/(d*(1-n))+b^2*(A*(1-n)+C*(2-n))*(b*cos(c+d*x))^(-2+n)*hypergeom([1/2,1/2*(-2+n)],[1/2*n],cos(c+d*x)^2)*sin(c+d*x)/(d*(2-3*n+n^2)*sqrt(sin(c+d*x)^2))],
[(b*cos(c+d*x))^n*(A+C*cos(c+d*x)^2)*sec(c+d*x)^4,x,3,-b^3*C*(b*cos(c+d*x))^(-3+n)*sin(c+d*x)/(d*(2-n))+b^3*(A*(2-n)+C*(3-n))*(b*cos(c+d*x))^(-3+n)*hypergeom([1/2,1/2*(-3+n)],[1/2*(-1+n)],cos(c+d*x)^2)*sin(c+d*x)/(d*(6-5*n+n^2)*sqrt(sin(c+d*x)^2))],
[cos(c+d*x)^(5/2)*(b*cos(c+d*x))^n*(A+C*cos(c+d*x)^2),x,3,2*C*cos(c+d*x)^(7/2)*(b*cos(c+d*x))^n*sin(c+d*x)/(d*(9+2*n))-2*(C*(7+2*n)+A*(9+2*n))*cos(c+d*x)^(7/2)*(b*cos(c+d*x))^n*hypergeom([1/2,1/4*(7+2*n)],[1/4*(11+2*n)],cos(c+d*x)^2)*sin(c+d*x)/(d*(63+32*n+4*n^2)*sqrt(sin(c+d*x)^2))],
[cos(c+d*x)^(3/2)*(b*cos(c+d*x))^n*(A+C*cos(c+d*x)^2),x,3,2*C*cos(c+d*x)^(5/2)*(b*cos(c+d*x))^n*sin(c+d*x)/(d*(7+2*n))-2*(C*(5+2*n)+A*(7+2*n))*cos(c+d*x)^(5/2)*(b*cos(c+d*x))^n*hypergeom([1/2,1/4*(5+2*n)],[1/4*(9+2*n)],cos(c+d*x)^2)*sin(c+d*x)/(d*(35+24*n+4*n^2)*sqrt(sin(c+d*x)^2))],
[(b*cos(c+d*x))^n*(A+C*cos(c+d*x)^2)*sqrt(cos(c+d*x)),x,3,2*C*cos(c+d*x)^(3/2)*(b*cos(c+d*x))^n*sin(c+d*x)/(d*(5+2*n))-2*(C*(3+2*n)+A*(5+2*n))*cos(c+d*x)^(3/2)*(b*cos(c+d*x))^n*hypergeom([1/2,1/4*(3+2*n)],[1/4*(7+2*n)],cos(c+d*x)^2)*sin(c+d*x)/(d*(15+16*n+4*n^2)*sqrt(sin(c+d*x)^2))],
[(b*cos(c+d*x))^n*(A+C*cos(c+d*x)^2)/sqrt(cos(c+d*x)),x,3,2*C*(b*cos(c+d*x))^n*sin(c+d*x)*sqrt(cos(c+d*x))/(d*(3+2*n))-2*(C+2*C*n+A*(3+2*n))*(b*cos(c+d*x))^n*hypergeom([1/2,1/4*(1+2*n)],[1/4*(5+2*n)],cos(c+d*x)^2)*sin(c+d*x)*sqrt(cos(c+d*x))/(d*(3+8*n+4*n^2)*sqrt(sin(c+d*x)^2))],
[(b*cos(c+d*x))^n*(A+C*cos(c+d*x)^2)/cos(c+d*x)^(3/2),x,3,2*C*(b*cos(c+d*x))^n*sin(c+d*x)/(d*(1+2*n)*sqrt(cos(c+d*x)))+2*(A-C*(1-2*n)+2*A*n)*(b*cos(c+d*x))^n*hypergeom([1/2,1/4*(-1+2*n)],[1/4*(3+2*n)],cos(c+d*x)^2)*sin(c+d*x)/(d*(1-4*n^2)*sqrt(cos(c+d*x))*sqrt(sin(c+d*x)^2))],
[(b*cos(c+d*x))^n*(A+C*cos(c+d*x)^2)/cos(c+d*x)^(5/2),x,3,-2*C*(b*cos(c+d*x))^n*sin(c+d*x)/(d*(1-2*n)*cos(c+d*x)^(3/2))+2*(A+C*(3-2*n)-2*A*n)*(b*cos(c+d*x))^n*hypergeom([1/2,1/4*(-3+2*n)],[1/4*(1+2*n)],cos(c+d*x)^2)*sin(c+d*x)/(d*(3-8*n+4*n^2)*cos(c+d*x)^(3/2)*sqrt(sin(c+d*x)^2))],
[(b*cos(c+d*x))^n*(A+C*cos(c+d*x)^2)/cos(c+d*x)^(7/2),x,3,-2*C*(b*cos(c+d*x))^n*sin(c+d*x)/(d*(3-2*n)*cos(c+d*x)^(5/2))+2*(A*(3-2*n)+C*(5-2*n))*(b*cos(c+d*x))^n*hypergeom([1/2,1/4*(-5+2*n)],[1/4*(-1+2*n)],cos(c+d*x)^2)*sin(c+d*x)/(d*(15-16*n+4*n^2)*cos(c+d*x)^(5/2)*sqrt(sin(c+d*x)^2))],
[(b*cos(c+d*x))^n*(A+C*cos(c+d*x)^2)/cos(c+d*x)^(9/2),x,3,-2*C*(b*cos(c+d*x))^n*sin(c+d*x)/(d*(5-2*n)*cos(c+d*x)^(7/2))+2*(A*(5-2*n)+C*(7-2*n))*(b*cos(c+d*x))^n*hypergeom([1/2,1/4*(-7+2*n)],[1/4*(-3+2*n)],cos(c+d*x)^2)*sin(c+d*x)/(d*(35-24*n+4*n^2)*cos(c+d*x)^(7/2)*sqrt(sin(c+d*x)^2))],

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

# Integrands of the form (a+a Cos[e+f x])^m (A+C Cos[e+f x]^2)
[(a+a*cos(e+f*x))^m*(A+C*cos(e+f*x)^2),x,4,-C*(a+a*cos(e+f*x))^m*sin(e+f*x)/(f*(2+3*m+m^2))+C*(a+a*cos(e+f*x))^(1+m)*sin(e+f*x)/(a*f*(2+m))+2^(1/2+m)*(C*(1+m+m^2)+A*(2+3*m+m^2))*(1+cos(e+f*x))^(-1/2-m)*(a+a*cos(e+f*x))^m*hypergeom([1/2,1/2-m],[3/2],1/2*(1-cos(e+f*x)))*sin(e+f*x)/(f*(2+3*m+m^2))],

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

# n>0
[(a+a*cos(c+d*x))^(2/3)*(A+C*cos(c+d*x)^2),x,4,-9/40*C*(a+a*cos(c+d*x))^(2/3)*sin(c+d*x)/d+3/8*C*(a+a*cos(c+d*x))^(5/3)*sin(c+d*x)/(a*d)+1/10*(40*A+19*C)*(a+a*cos(c+d*x))^(2/3)*hypergeom([-1/6,1/2],[3/2],1/2*(1-cos(c+d*x)))*sin(c+d*x)/(2^(5/6)*d*(1+cos(c+d*x))^(7/6))],
[(a+a*cos(c+d*x))^(1/3)*(A+C*cos(c+d*x)^2),x,4,-9/28*C*(a+a*cos(c+d*x))^(1/3)*sin(c+d*x)/d+3/7*C*(a+a*cos(c+d*x))^(4/3)*sin(c+d*x)/(a*d)+1/14*(28*A+13*C)*(a+a*cos(c+d*x))^(1/3)*hypergeom([1/6,1/2],[3/2],1/2*(1-cos(c+d*x)))*sin(c+d*x)/(2^(1/6)*d*(1+cos(c+d*x))^(5/6))],

# n<0
[(A+C*cos(c+d*x)^2)/(a+a*cos(c+d*x))^(1/3),x,4,-9/10*C*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(1/3))+3/5*C*(a+a*cos(c+d*x))^(2/3)*sin(c+d*x)/(a*d)+1/5*(10*A+7*C)*hypergeom([1/2,5/6],[3/2],1/2*(1-cos(c+d*x)))*sin(c+d*x)/(2^(5/6)*d*(1+cos(c+d*x))^(1/6)*(a+a*cos(c+d*x))^(1/3))],
[(A+C*cos(c+d*x)^2)/(a+a*cos(c+d*x))^(2/3),x,4,3*(A+C)*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(2/3))+3/4*C*(a+a*cos(c+d*x))^(1/3)*sin(c+d*x)/(a*d)-1/2*(4*A+7*C)*(a+a*cos(c+d*x))^(1/3)*hypergeom([1/6,1/2],[3/2],1/2*(1-cos(c+d*x)))*sin(c+d*x)/(2^(1/6)*a*d*(1+cos(c+d*x))^(5/6))],

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

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

# n>0
[(a+b*cos(c+d*x))^(2/3)*(A+C*cos(c+d*x)^2),x,8,3/8*C*(a+b*cos(c+d*x))^(5/3)*sin(c+d*x)/(b*d)-3/4*a*(a+b)*C*AppellF1(1/2,1/2,-5/3,3/2,1/2*(1-cos(c+d*x)),b*(1-cos(c+d*x))/(a+b))*(a+b*cos(c+d*x))^(2/3)*sin(c+d*x)/(b^2*d*((a+b*cos(c+d*x))/(a+b))^(2/3)*sqrt(2)*sqrt(1+cos(c+d*x)))+1/4*(3*a^2*C+b^2*(8*A+5*C))*AppellF1(1/2,1/2,-2/3,3/2,1/2*(1-cos(c+d*x)),b*(1-cos(c+d*x))/(a+b))*(a+b*cos(c+d*x))^(2/3)*sin(c+d*x)/(b^2*d*((a+b*cos(c+d*x))/(a+b))^(2/3)*sqrt(2)*sqrt(1+cos(c+d*x)))],
[(a+b*cos(c+d*x))^(1/3)*(A+C*cos(c+d*x)^2),x,8,3/7*C*(a+b*cos(c+d*x))^(4/3)*sin(c+d*x)/(b*d)-3/7*a*(a+b)*C*AppellF1(1/2,1/2,-4/3,3/2,1/2*(1-cos(c+d*x)),b*(1-cos(c+d*x))/(a+b))*(a+b*cos(c+d*x))^(1/3)*sin(c+d*x)*sqrt(2)/(b^2*d*((a+b*cos(c+d*x))/(a+b))^(1/3)*sqrt(1+cos(c+d*x)))+1/7*(3*a^2*C+b^2*(7*A+4*C))*AppellF1(1/2,1/2,-1/3,3/2,1/2*(1-cos(c+d*x)),b*(1-cos(c+d*x))/(a+b))*(a+b*cos(c+d*x))^(1/3)*sin(c+d*x)*sqrt(2)/(b^2*d*((a+b*cos(c+d*x))/(a+b))^(1/3)*sqrt(1+cos(c+d*x)))],

# n<0
[(A+C*cos(c+d*x)^2)/(a+b*cos(c+d*x))^(1/3),x,8,3/5*C*(a+b*cos(c+d*x))^(2/3)*sin(c+d*x)/(b*d)-3/5*a*C*AppellF1(1/2,1/2,-2/3,3/2,1/2*(1-cos(c+d*x)),b*(1-cos(c+d*x))/(a+b))*(a+b*cos(c+d*x))^(2/3)*sin(c+d*x)*sqrt(2)/(b^2*d*((a+b*cos(c+d*x))/(a+b))^(2/3)*sqrt(1+cos(c+d*x)))+1/5*(3*a^2*C+b^2*(5*A+2*C))*AppellF1(1/2,1/2,1/3,3/2,1/2*(1-cos(c+d*x)),b*(1-cos(c+d*x))/(a+b))*((a+b*cos(c+d*x))/(a+b))^(1/3)*sin(c+d*x)*sqrt(2)/(b^2*d*(a+b*cos(c+d*x))^(1/3)*sqrt(1+cos(c+d*x)))],
[(A+C*cos(c+d*x)^2)/(a+b*cos(c+d*x))^(2/3),x,8,3/4*C*(a+b*cos(c+d*x))^(1/3)*sin(c+d*x)/(b*d)-3/2*a*C*AppellF1(1/2,1/2,-1/3,3/2,1/2*(1-cos(c+d*x)),b*(1-cos(c+d*x))/(a+b))*(a+b*cos(c+d*x))^(1/3)*sin(c+d*x)/(b^2*d*((a+b*cos(c+d*x))/(a+b))^(1/3)*sqrt(2)*sqrt(1+cos(c+d*x)))+1/2*(3*a^2*C+b^2*(4*A+C))*AppellF1(1/2,1/2,2/3,3/2,1/2*(1-cos(c+d*x)),b*(1-cos(c+d*x))/(a+b))*((a+b*cos(c+d*x))/(a+b))^(2/3)*sin(c+d*x)/(b^2*d*(a+b*cos(c+d*x))^(2/3)*sqrt(2)*sqrt(1+cos(c+d*x)))],

# Integrands of the form (a+b Cos[e+f x])^m (A+C Cos[e+f x]^2) with m symbolic
[(a+b*cos(e+f*x))^m*(A-A*cos(e+f*x)^2),x,7,-4*A*AppellF1(1/2,-3/2,-m,3/2,1/2*(1-cos(e+f*x)),b*(1-cos(e+f*x))/(a+b))*(a+b*cos(e+f*x))^m*sin(e+f*x)*sqrt(2)/(f*((a+b*cos(e+f*x))/(a+b))^m*sqrt(1+cos(e+f*x)))+4*A*AppellF1(1/2,-1/2,-m,3/2,1/2*(1-cos(e+f*x)),b*(1-cos(e+f*x))/(a+b))*(a+b*cos(e+f*x))^m*sin(e+f*x)*sqrt(2)/(f*((a+b*cos(e+f*x))/(a+b))^m*sqrt(1+cos(e+f*x)))],
[(a+b*cos(e+f*x))^m*(A+C*cos(e+f*x)^2),x,8,C*(a+b*cos(e+f*x))^(1+m)*sin(e+f*x)/(b*f*(2+m))-a*(a+b)*C*AppellF1(1/2,1/2,-1-m,3/2,1/2*(1-cos(e+f*x)),b*(1-cos(e+f*x))/(a+b))*(a+b*cos(e+f*x))^m*sin(e+f*x)*sqrt(2)/(b^2*f*(2+m)*((a+b*cos(e+f*x))/(a+b))^m*sqrt(1+cos(e+f*x)))+(a^2*C+b^2*(C*(1+m)+A*(2+m)))*AppellF1(1/2,1/2,-m,3/2,1/2*(1-cos(e+f*x)),b*(1-cos(e+f*x))/(a+b))*(a+b*cos(e+f*x))^m*sin(e+f*x)*sqrt(2)/(b^2*f*(2+m)*((a+b*cos(e+f*x))/(a+b))^m*sqrt(1+cos(e+f*x)))],

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

# Integrands of the form (a Cos[e+f x])^m (B Cos[e+f x]+C Cos[e+f x]^2)
[(a*cos(e+f*x))^m*(B*cos(e+f*x)+C*cos(e+f*x)^2),x,4,-B*(a*cos(e+f*x))^(2+m)*hypergeom([1/2,1/2*(2+m)],[1/2*(4+m)],cos(e+f*x)^2)*sin(e+f*x)/(a^2*f*(2+m)*sqrt(sin(e+f*x)^2))-C*(a*cos(e+f*x))^(3+m)*hypergeom([1/2,1/2*(3+m)],[1/2*(5+m)],cos(e+f*x)^2)*sin(e+f*x)/(a^3*f*(3+m)*sqrt(sin(e+f*x)^2))],

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

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

# n>0

# n<0

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

# n>0

# n<0

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

# n>0
[cos(c+d*x)^m*(b*cos(c+d*x))^(1/3)*(B*cos(c+d*x)+C*cos(c+d*x)^2),x,5,-3*B*cos(c+d*x)^(2+m)*(b*cos(c+d*x))^(1/3)*hypergeom([1/2,1/6*(7+3*m)],[1/6*(13+3*m)],cos(c+d*x)^2)*sin(c+d*x)/(d*(7+3*m)*sqrt(sin(c+d*x)^2))-3*C*cos(c+d*x)^(3+m)*(b*cos(c+d*x))^(1/3)*hypergeom([1/2,1/6*(10+3*m)],[1/6*(16+3*m)],cos(c+d*x)^2)*sin(c+d*x)/(d*(10+3*m)*sqrt(sin(c+d*x)^2))],
[cos(c+d*x)^m*(b*cos(c+d*x))^(2/3)*(B*cos(c+d*x)+C*cos(c+d*x)^2),x,5,-3*B*cos(c+d*x)^(2+m)*(b*cos(c+d*x))^(2/3)*hypergeom([1/2,1/6*(8+3*m)],[1/6*(14+3*m)],cos(c+d*x)^2)*sin(c+d*x)/(d*(8+3*m)*sqrt(sin(c+d*x)^2))-3*C*cos(c+d*x)^(3+m)*(b*cos(c+d*x))^(2/3)*hypergeom([1/2,1/6*(11+3*m)],[1/6*(17+3*m)],cos(c+d*x)^2)*sin(c+d*x)/(d*(11+3*m)*sqrt(sin(c+d*x)^2))],
[cos(c+d*x)^m*(b*cos(c+d*x))^(4/3)*(B*cos(c+d*x)+C*cos(c+d*x)^2),x,5,-3*b*B*cos(c+d*x)^(3+m)*(b*cos(c+d*x))^(1/3)*hypergeom([1/2,1/6*(10+3*m)],[1/6*(16+3*m)],cos(c+d*x)^2)*sin(c+d*x)/(d*(10+3*m)*sqrt(sin(c+d*x)^2))-3*b*C*cos(c+d*x)^(4+m)*(b*cos(c+d*x))^(1/3)*hypergeom([1/2,1/6*(13+3*m)],[1/6*(19+3*m)],cos(c+d*x)^2)*sin(c+d*x)/(d*(13+3*m)*sqrt(sin(c+d*x)^2))],

# n<0
[cos(c+d*x)^m*(B*cos(c+d*x)+C*cos(c+d*x)^2)/(b*cos(c+d*x))^(1/3),x,5,-3*B*cos(c+d*x)^(2+m)*hypergeom([1/2,1/6*(5+3*m)],[1/6*(11+3*m)],cos(c+d*x)^2)*sin(c+d*x)/(d*(5+3*m)*(b*cos(c+d*x))^(1/3)*sqrt(sin(c+d*x)^2))-3*C*cos(c+d*x)^(3+m)*hypergeom([1/2,1/6*(8+3*m)],[1/6*(14+3*m)],cos(c+d*x)^2)*sin(c+d*x)/(d*(8+3*m)*(b*cos(c+d*x))^(1/3)*sqrt(sin(c+d*x)^2))],
[cos(c+d*x)^m*(B*cos(c+d*x)+C*cos(c+d*x)^2)/(b*cos(c+d*x))^(2/3),x,5,-3*B*cos(c+d*x)^(2+m)*hypergeom([1/2,1/6*(4+3*m)],[1/6*(10+3*m)],cos(c+d*x)^2)*sin(c+d*x)/(d*(4+3*m)*(b*cos(c+d*x))^(2/3)*sqrt(sin(c+d*x)^2))-3*C*cos(c+d*x)^(3+m)*hypergeom([1/2,1/6*(7+3*m)],[1/6*(13+3*m)],cos(c+d*x)^2)*sin(c+d*x)/(d*(7+3*m)*(b*cos(c+d*x))^(2/3)*sqrt(sin(c+d*x)^2))],
[cos(c+d*x)^m*(B*cos(c+d*x)+C*cos(c+d*x)^2)/(b*cos(c+d*x))^(4/3),x,5,-3*B*cos(c+d*x)^(1+m)*hypergeom([1/2,1/6*(2+3*m)],[1/6*(8+3*m)],cos(c+d*x)^2)*sin(c+d*x)/(b*d*(2+3*m)*(b*cos(c+d*x))^(1/3)*sqrt(sin(c+d*x)^2))-3*C*cos(c+d*x)^(2+m)*hypergeom([1/2,1/6*(5+3*m)],[1/6*(11+3*m)],cos(c+d*x)^2)*sin(c+d*x)/(b*d*(5+3*m)*(b*cos(c+d*x))^(1/3)*sqrt(sin(c+d*x)^2))],

# Integrands of the form (a Cos[e+f x])^m (b Cos[e+f x])^n (B Cos[e+f x]+C Cos[e+f x]^2) with n symbolic
[(a*cos(c+d*x))^m*(b*cos(c+d*x))^n*(B*cos(c+d*x)+C*cos(c+d*x)^2),x,5,-B*(a*cos(c+d*x))^(2+m)*(b*cos(c+d*x))^n*hypergeom([1/2,1/2*(2+m+n)],[1/2*(4+m+n)],cos(c+d*x)^2)*sin(c+d*x)/(a^2*d*(2+m+n)*sqrt(sin(c+d*x)^2))-C*(a*cos(c+d*x))^(3+m)*(b*cos(c+d*x))^n*hypergeom([1/2,1/2*(3+m+n)],[1/2*(5+m+n)],cos(c+d*x)^2)*sin(c+d*x)/(a^3*d*(3+m+n)*sqrt(sin(c+d*x)^2))],
[cos(c+d*x)^2*(b*cos(c+d*x))^n*(B*cos(c+d*x)+C*cos(c+d*x)^2),x,5,-B*(b*cos(c+d*x))^(4+n)*hypergeom([1/2,1/2*(4+n)],[1/2*(6+n)],cos(c+d*x)^2)*sin(c+d*x)/(b^4*d*(4+n)*sqrt(sin(c+d*x)^2))-C*(b*cos(c+d*x))^(5+n)*hypergeom([1/2,1/2*(5+n)],[1/2*(7+n)],cos(c+d*x)^2)*sin(c+d*x)/(b^5*d*(5+n)*sqrt(sin(c+d*x)^2))],
[cos(c+d*x)*(b*cos(c+d*x))^n*(B*cos(c+d*x)+C*cos(c+d*x)^2),x,5,-B*(b*cos(c+d*x))^(3+n)*hypergeom([1/2,1/2*(3+n)],[1/2*(5+n)],cos(c+d*x)^2)*sin(c+d*x)/(b^3*d*(3+n)*sqrt(sin(c+d*x)^2))-C*(b*cos(c+d*x))^(4+n)*hypergeom([1/2,1/2*(4+n)],[1/2*(6+n)],cos(c+d*x)^2)*sin(c+d*x)/(b^4*d*(4+n)*sqrt(sin(c+d*x)^2))],
[(b*cos(c+d*x))^n*(B*cos(c+d*x)+C*cos(c+d*x)^2),x,4,-B*(b*cos(c+d*x))^(2+n)*hypergeom([1/2,1/2*(2+n)],[1/2*(4+n)],cos(c+d*x)^2)*sin(c+d*x)/(b^2*d*(2+n)*sqrt(sin(c+d*x)^2))-C*(b*cos(c+d*x))^(3+n)*hypergeom([1/2,1/2*(3+n)],[1/2*(5+n)],cos(c+d*x)^2)*sin(c+d*x)/(b^3*d*(3+n)*sqrt(sin(c+d*x)^2))],
[(b*cos(c+d*x))^n*(B*cos(c+d*x)+C*cos(c+d*x)^2)*sec(c+d*x),x,5,-B*(b*cos(c+d*x))^(1+n)*hypergeom([1/2,1/2*(1+n)],[1/2*(3+n)],cos(c+d*x)^2)*sin(c+d*x)/(b*d*(1+n)*sqrt(sin(c+d*x)^2))-C*(b*cos(c+d*x))^(2+n)*hypergeom([1/2,1/2*(2+n)],[1/2*(4+n)],cos(c+d*x)^2)*sin(c+d*x)/(b^2*d*(2+n)*sqrt(sin(c+d*x)^2))],
[(b*cos(c+d*x))^n*(B*cos(c+d*x)+C*cos(c+d*x)^2)*sec(c+d*x)^2,x,5,-B*(b*cos(c+d*x))^n*hypergeom([1/2,1/2*n],[1/2*(2+n)],cos(c+d*x)^2)*sin(c+d*x)/(d*n*sqrt(sin(c+d*x)^2))-C*(b*cos(c+d*x))^(1+n)*hypergeom([1/2,1/2*(1+n)],[1/2*(3+n)],cos(c+d*x)^2)*sin(c+d*x)/(b*d*(1+n)*sqrt(sin(c+d*x)^2))],
[(b*cos(c+d*x))^n*(B*cos(c+d*x)+C*cos(c+d*x)^2)*sec(c+d*x)^3,x,5,b*B*(b*cos(c+d*x))^(-1+n)*hypergeom([1/2,1/2*(-1+n)],[1/2*(1+n)],cos(c+d*x)^2)*sin(c+d*x)/(d*(1-n)*sqrt(sin(c+d*x)^2))-C*(b*cos(c+d*x))^n*hypergeom([1/2,1/2*n],[1/2*(2+n)],cos(c+d*x)^2)*sin(c+d*x)/(d*n*sqrt(sin(c+d*x)^2))],
[(b*cos(c+d*x))^n*(B*cos(c+d*x)+C*cos(c+d*x)^2)*sec(c+d*x)^4,x,5,b^2*B*(b*cos(c+d*x))^(-2+n)*hypergeom([1/2,1/2*(-2+n)],[1/2*n],cos(c+d*x)^2)*sin(c+d*x)/(d*(2-n)*sqrt(sin(c+d*x)^2))+b*C*(b*cos(c+d*x))^(-1+n)*hypergeom([1/2,1/2*(-1+n)],[1/2*(1+n)],cos(c+d*x)^2)*sin(c+d*x)/(d*(1-n)*sqrt(sin(c+d*x)^2))],
[cos(c+d*x)^(5/2)*(b*cos(c+d*x))^n*(B*cos(c+d*x)+C*cos(c+d*x)^2),x,5,-2*B*cos(c+d*x)^(9/2)*(b*cos(c+d*x))^n*hypergeom([1/2,1/4*(9+2*n)],[1/4*(13+2*n)],cos(c+d*x)^2)*sin(c+d*x)/(d*(9+2*n)*sqrt(sin(c+d*x)^2))-2*C*cos(c+d*x)^(11/2)*(b*cos(c+d*x))^n*hypergeom([1/2,1/4*(11+2*n)],[1/4*(15+2*n)],cos(c+d*x)^2)*sin(c+d*x)/(d*(11+2*n)*sqrt(sin(c+d*x)^2))],
[cos(c+d*x)^(3/2)*(b*cos(c+d*x))^n*(B*cos(c+d*x)+C*cos(c+d*x)^2),x,5,-2*B*cos(c+d*x)^(7/2)*(b*cos(c+d*x))^n*hypergeom([1/2,1/4*(7+2*n)],[1/4*(11+2*n)],cos(c+d*x)^2)*sin(c+d*x)/(d*(7+2*n)*sqrt(sin(c+d*x)^2))-2*C*cos(c+d*x)^(9/2)*(b*cos(c+d*x))^n*hypergeom([1/2,1/4*(9+2*n)],[1/4*(13+2*n)],cos(c+d*x)^2)*sin(c+d*x)/(d*(9+2*n)*sqrt(sin(c+d*x)^2))],
[(b*cos(c+d*x))^n*(B*cos(c+d*x)+C*cos(c+d*x)^2)*sqrt(cos(c+d*x)),x,5,-2*B*cos(c+d*x)^(5/2)*(b*cos(c+d*x))^n*hypergeom([1/2,1/4*(5+2*n)],[1/4*(9+2*n)],cos(c+d*x)^2)*sin(c+d*x)/(d*(5+2*n)*sqrt(sin(c+d*x)^2))-2*C*cos(c+d*x)^(7/2)*(b*cos(c+d*x))^n*hypergeom([1/2,1/4*(7+2*n)],[1/4*(11+2*n)],cos(c+d*x)^2)*sin(c+d*x)/(d*(7+2*n)*sqrt(sin(c+d*x)^2))],
[(b*cos(c+d*x))^n*(B*cos(c+d*x)+C*cos(c+d*x)^2)/sqrt(cos(c+d*x)),x,5,-2*B*cos(c+d*x)^(3/2)*(b*cos(c+d*x))^n*hypergeom([1/2,1/4*(3+2*n)],[1/4*(7+2*n)],cos(c+d*x)^2)*sin(c+d*x)/(d*(3+2*n)*sqrt(sin(c+d*x)^2))-2*C*cos(c+d*x)^(5/2)*(b*cos(c+d*x))^n*hypergeom([1/2,1/4*(5+2*n)],[1/4*(9+2*n)],cos(c+d*x)^2)*sin(c+d*x)/(d*(5+2*n)*sqrt(sin(c+d*x)^2))],
[(b*cos(c+d*x))^n*(B*cos(c+d*x)+C*cos(c+d*x)^2)/cos(c+d*x)^(3/2),x,5,-2*C*cos(c+d*x)^(3/2)*(b*cos(c+d*x))^n*hypergeom([1/2,1/4*(3+2*n)],[1/4*(7+2*n)],cos(c+d*x)^2)*sin(c+d*x)/(d*(3+2*n)*sqrt(sin(c+d*x)^2))-2*B*(b*cos(c+d*x))^n*hypergeom([1/2,1/4*(1+2*n)],[1/4*(5+2*n)],cos(c+d*x)^2)*sin(c+d*x)*sqrt(cos(c+d*x))/(d*(1+2*n)*sqrt(sin(c+d*x)^2))],
[(b*cos(c+d*x))^n*(B*cos(c+d*x)+C*cos(c+d*x)^2)/cos(c+d*x)^(5/2),x,5,2*B*(b*cos(c+d*x))^n*hypergeom([1/2,1/4*(-1+2*n)],[1/4*(3+2*n)],cos(c+d*x)^2)*sin(c+d*x)/(d*(1-2*n)*sqrt(cos(c+d*x))*sqrt(sin(c+d*x)^2))-2*C*(b*cos(c+d*x))^n*hypergeom([1/2,1/4*(1+2*n)],[1/4*(5+2*n)],cos(c+d*x)^2)*sin(c+d*x)*sqrt(cos(c+d*x))/(d*(1+2*n)*sqrt(sin(c+d*x)^2))],
[(b*cos(c+d*x))^n*(B*cos(c+d*x)+C*cos(c+d*x)^2)/cos(c+d*x)^(7/2),x,5,2*B*(b*cos(c+d*x))^n*hypergeom([1/2,1/4*(-3+2*n)],[1/4*(1+2*n)],cos(c+d*x)^2)*sin(c+d*x)/(d*(3-2*n)*cos(c+d*x)^(3/2)*sqrt(sin(c+d*x)^2))+2*C*(b*cos(c+d*x))^n*hypergeom([1/2,1/4*(-1+2*n)],[1/4*(3+2*n)],cos(c+d*x)^2)*sin(c+d*x)/(d*(1-2*n)*sqrt(cos(c+d*x))*sqrt(sin(c+d*x)^2))],
[(b*cos(c+d*x))^n*(B*cos(c+d*x)+C*cos(c+d*x)^2)/cos(c+d*x)^(9/2),x,5,2*B*(b*cos(c+d*x))^n*hypergeom([1/2,1/4*(-5+2*n)],[1/4*(-1+2*n)],cos(c+d*x)^2)*sin(c+d*x)/(d*(5-2*n)*cos(c+d*x)^(5/2)*sqrt(sin(c+d*x)^2))+2*C*(b*cos(c+d*x))^n*hypergeom([1/2,1/4*(-3+2*n)],[1/4*(1+2*n)],cos(c+d*x)^2)*sin(c+d*x)/(d*(3-2*n)*cos(c+d*x)^(3/2)*sqrt(sin(c+d*x)^2))],

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

# Integrands of the form (a+a Cos[e+f x])^m (B Cos[e+f x]+C Cos[e+f x]^2)
[(a+a*cos(e+f*x))^m*(B*cos(e+f*x)+C*cos(e+f*x)^2),x,4,-(C-B*(2+m))*(a+a*cos(e+f*x))^m*sin(e+f*x)/(f*(1+m)*(2+m))+C*(a+a*cos(e+f*x))^(1+m)*sin(e+f*x)/(a*f*(2+m))+2^(1/2+m)*(B*m*(2+m)+C*(1+m+m^2))*(1+cos(e+f*x))^(-1/2-m)*(a+a*cos(e+f*x))^m*hypergeom([1/2,1/2-m],[3/2],1/2*(1-cos(e+f*x)))*sin(e+f*x)/(f*(2+3*m+m^2))],

# Integrands of the form (a+b Cos[e+f x])^m (B Cos[e+f x]+C Cos[e+f x]^2)
[(a+b*cos(e+f*x))^m*(B*cos(e+f*x)+C*cos(e+f*x)^2),x,8,C*(a+b*cos(e+f*x))^(1+m)*sin(e+f*x)/(b*f*(2+m))-(a+b)*(a*C-b*B*(2+m))*AppellF1(1/2,1/2,-1-m,3/2,1/2*(1-cos(e+f*x)),b*(1-cos(e+f*x))/(a+b))*(a+b*cos(e+f*x))^m*sin(e+f*x)*sqrt(2)/(b^2*f*(2+m)*((a+b*cos(e+f*x))/(a+b))^m*sqrt(1+cos(e+f*x)))+(a^2*C+b^2*C*(1+m)-a*b*B*(2+m))*AppellF1(1/2,1/2,-m,3/2,1/2*(1-cos(e+f*x)),b*(1-cos(e+f*x))/(a+b))*(a+b*cos(e+f*x))^m*sin(e+f*x)*sqrt(2)/(b^2*f*(2+m)*((a+b*cos(e+f*x))/(a+b))^m*sqrt(1+cos(e+f*x)))],

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

# n>0
[(a+b*cos(c+d*x))^(2/3)*(B*cos(c+d*x)+C*cos(c+d*x)^2),x,8,3/8*C*(a+b*cos(c+d*x))^(5/3)*sin(c+d*x)/(b*d)+1/4*(a+b)*(8*b*B-3*a*C)*AppellF1(1/2,1/2,-5/3,3/2,1/2*(1-cos(c+d*x)),b*(1-cos(c+d*x))/(a+b))*(a+b*cos(c+d*x))^(2/3)*sin(c+d*x)/(b^2*d*((a+b*cos(c+d*x))/(a+b))^(2/3)*sqrt(2)*sqrt(1+cos(c+d*x)))-1/4*(8*a*b*B-3*a^2*C-5*b^2*C)*AppellF1(1/2,1/2,-2/3,3/2,1/2*(1-cos(c+d*x)),b*(1-cos(c+d*x))/(a+b))*(a+b*cos(c+d*x))^(2/3)*sin(c+d*x)/(b^2*d*((a+b*cos(c+d*x))/(a+b))^(2/3)*sqrt(2)*sqrt(1+cos(c+d*x)))],
[(a+b*cos(c+d*x))^(1/3)*(B*cos(c+d*x)+C*cos(c+d*x)^2),x,8,3/7*C*(a+b*cos(c+d*x))^(4/3)*sin(c+d*x)/(b*d)+1/7*(a+b)*(7*b*B-3*a*C)*AppellF1(1/2,1/2,-4/3,3/2,1/2*(1-cos(c+d*x)),b*(1-cos(c+d*x))/(a+b))*(a+b*cos(c+d*x))^(1/3)*sin(c+d*x)*sqrt(2)/(b^2*d*((a+b*cos(c+d*x))/(a+b))^(1/3)*sqrt(1+cos(c+d*x)))-1/7*(7*a*b*B-3*a^2*C-4*b^2*C)*AppellF1(1/2,1/2,-1/3,3/2,1/2*(1-cos(c+d*x)),b*(1-cos(c+d*x))/(a+b))*(a+b*cos(c+d*x))^(1/3)*sin(c+d*x)*sqrt(2)/(b^2*d*((a+b*cos(c+d*x))/(a+b))^(1/3)*sqrt(1+cos(c+d*x)))],

# n<0
[(B*cos(c+d*x)+C*cos(c+d*x)^2)/(a+b*cos(c+d*x))^(1/3),x,8,3/5*C*(a+b*cos(c+d*x))^(2/3)*sin(c+d*x)/(b*d)+1/5*(5*b*B-3*a*C)*AppellF1(1/2,1/2,-2/3,3/2,1/2*(1-cos(c+d*x)),b*(1-cos(c+d*x))/(a+b))*(a+b*cos(c+d*x))^(2/3)*sin(c+d*x)*sqrt(2)/(b^2*d*((a+b*cos(c+d*x))/(a+b))^(2/3)*sqrt(1+cos(c+d*x)))-1/5*(5*a*b*B-3*a^2*C-2*b^2*C)*AppellF1(1/2,1/2,1/3,3/2,1/2*(1-cos(c+d*x)),b*(1-cos(c+d*x))/(a+b))*((a+b*cos(c+d*x))/(a+b))^(1/3)*sin(c+d*x)*sqrt(2)/(b^2*d*(a+b*cos(c+d*x))^(1/3)*sqrt(1+cos(c+d*x)))],
[(B*cos(c+d*x)+C*cos(c+d*x)^2)/(a+b*cos(c+d*x))^(2/3),x,8,3/4*C*(a+b*cos(c+d*x))^(1/3)*sin(c+d*x)/(b*d)+1/2*(4*b*B-3*a*C)*AppellF1(1/2,1/2,-1/3,3/2,1/2*(1-cos(c+d*x)),b*(1-cos(c+d*x))/(a+b))*(a+b*cos(c+d*x))^(1/3)*sin(c+d*x)/(b^2*d*((a+b*cos(c+d*x))/(a+b))^(1/3)*sqrt(2)*sqrt(1+cos(c+d*x)))-1/2*(4*a*b*B-3*a^2*C-b^2*C)*AppellF1(1/2,1/2,2/3,3/2,1/2*(1-cos(c+d*x)),b*(1-cos(c+d*x))/(a+b))*((a+b*cos(c+d*x))/(a+b))^(2/3)*sin(c+d*x)/(b^2*d*(a+b*cos(c+d*x))^(2/3)*sqrt(2)*sqrt(1+cos(c+d*x)))],

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

# Integrands of the form (a Cos[e+f x])^m (A+B Cos[e+f x]+C Cos[e+f x]^2)
[(a*cos(e+f*x))^m*(A+B*cos(e+f*x)+C*cos(e+f*x)^2),x,4,C*(a*cos(e+f*x))^(1+m)*sin(e+f*x)/(a*f*(2+m))-(C*(1+m)+A*(2+m))*(a*cos(e+f*x))^(1+m)*hypergeom([1/2,1/2*(1+m)],[1/2*(3+m)],cos(e+f*x)^2)*sin(e+f*x)/(a*f*(1+m)*(2+m)*sqrt(sin(e+f*x)^2))-B*(a*cos(e+f*x))^(2+m)*hypergeom([1/2,1/2*(2+m)],[1/2*(4+m)],cos(e+f*x)^2)*sin(e+f*x)/(a^2*f*(2+m)*sqrt(sin(e+f*x)^2))],

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

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

# n>0
[cos(c+d*x)^2*(A+B*cos(c+d*x)+C*cos(c+d*x)^2)*sqrt(b*cos(c+d*x)),x,10,2/45*(9*A+7*C)*(b*cos(c+d*x))^(3/2)*sin(c+d*x)/(b*d)+2/7*B*(b*cos(c+d*x))^(5/2)*sin(c+d*x)/(b^2*d)+2/9*C*(b*cos(c+d*x))^(7/2)*sin(c+d*x)/(b^3*d)+10/21*b*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))+10/21*B*sin(c+d*x)*sqrt(b*cos(c+d*x))/d+2/15*(9*A+7*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[cos(c+d*x)*(A+B*cos(c+d*x)+C*cos(c+d*x)^2)*sqrt(b*cos(c+d*x)),x,9,2/5*B*(b*cos(c+d*x))^(3/2)*sin(c+d*x)/(b*d)+2/7*C*(b*cos(c+d*x))^(5/2)*sin(c+d*x)/(b^2*d)+2/21*b*(7*A+5*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))+2/21*(7*A+5*C)*sin(c+d*x)*sqrt(b*cos(c+d*x))/d+6/5*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(A+B*cos(c+d*x)+C*cos(c+d*x)^2)*sqrt(b*cos(c+d*x)),x,7,2/5*C*(b*cos(c+d*x))^(3/2)*sin(c+d*x)/(b*d)+2/3*b*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))+2/3*B*sin(c+d*x)*sqrt(b*cos(c+d*x))/d+2/5*(5*A+3*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(A+B*cos(c+d*x)+C*cos(c+d*x)^2)*sec(c+d*x)*sqrt(b*cos(c+d*x)),x,7,2/3*b*(3*A+C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))+2/3*C*sin(c+d*x)*sqrt(b*cos(c+d*x))/d+2*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(A+B*cos(c+d*x)+C*cos(c+d*x)^2)*sec(c+d*x)^2*sqrt(b*cos(c+d*x)),x,7,2*A*b*sin(c+d*x)/(d*sqrt(b*cos(c+d*x)))+2*b*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))-2*(A-C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(A+B*cos(c+d*x)+C*cos(c+d*x)^2)*sec(c+d*x)^3*sqrt(b*cos(c+d*x)),x,8,2/3*A*b^2*sin(c+d*x)/(d*(b*cos(c+d*x))^(3/2))+2*b*B*sin(c+d*x)/(d*sqrt(b*cos(c+d*x)))+2/3*b*(A+3*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))-2*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(A+B*cos(c+d*x)+C*cos(c+d*x)^2)*sec(c+d*x)^4*sqrt(b*cos(c+d*x)),x,9,2/5*A*b^3*sin(c+d*x)/(d*(b*cos(c+d*x))^(5/2))+2/3*b^2*B*sin(c+d*x)/(d*(b*cos(c+d*x))^(3/2))+2/5*b*(3*A+5*C)*sin(c+d*x)/(d*sqrt(b*cos(c+d*x)))+2/3*b*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))-2/5*(3*A+5*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(A+B*cos(c+d*x)+C*cos(c+d*x)^2)*sec(c+d*x)^5*sqrt(b*cos(c+d*x)),x,10,2/7*A*b^4*sin(c+d*x)/(d*(b*cos(c+d*x))^(7/2))+2/5*b^3*B*sin(c+d*x)/(d*(b*cos(c+d*x))^(5/2))+2/21*b^2*(5*A+7*C)*sin(c+d*x)/(d*(b*cos(c+d*x))^(3/2))+6/5*b*B*sin(c+d*x)/(d*sqrt(b*cos(c+d*x)))+2/21*b*(5*A+7*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))-6/5*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[cos(c+d*x)*(b*cos(c+d*x))^(3/2)*(A+B*cos(c+d*x)+C*cos(c+d*x)^2),x,10,2/45*(9*A+7*C)*(b*cos(c+d*x))^(3/2)*sin(c+d*x)/d+2/7*B*(b*cos(c+d*x))^(5/2)*sin(c+d*x)/(b*d)+2/9*C*(b*cos(c+d*x))^(7/2)*sin(c+d*x)/(b^2*d)+10/21*b^2*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))+10/21*b*B*sin(c+d*x)*sqrt(b*cos(c+d*x))/d+2/15*b*(9*A+7*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(b*cos(c+d*x))^(3/2)*(A+B*cos(c+d*x)+C*cos(c+d*x)^2),x,8,2/5*B*(b*cos(c+d*x))^(3/2)*sin(c+d*x)/d+2/7*C*(b*cos(c+d*x))^(5/2)*sin(c+d*x)/(b*d)+2/21*b^2*(7*A+5*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))+2/21*b*(7*A+5*C)*sin(c+d*x)*sqrt(b*cos(c+d*x))/d+6/5*b*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(b*cos(c+d*x))^(3/2)*(A+B*cos(c+d*x)+C*cos(c+d*x)^2)*sec(c+d*x),x,8,2/5*C*(b*cos(c+d*x))^(3/2)*sin(c+d*x)/d+2/3*b^2*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))+2/3*b*B*sin(c+d*x)*sqrt(b*cos(c+d*x))/d+2/5*b*(5*A+3*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(b*cos(c+d*x))^(3/2)*(A+B*cos(c+d*x)+C*cos(c+d*x)^2)*sec(c+d*x)^2,x,7,2/3*b^2*(3*A+C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))+2/3*b*C*sin(c+d*x)*sqrt(b*cos(c+d*x))/d+2*b*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(b*cos(c+d*x))^(3/2)*(A+B*cos(c+d*x)+C*cos(c+d*x)^2)*sec(c+d*x)^3,x,7,2*A*b^2*sin(c+d*x)/(d*sqrt(b*cos(c+d*x)))+2*b^2*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))-2*b*(A-C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(b*cos(c+d*x))^(3/2)*(A+B*cos(c+d*x)+C*cos(c+d*x)^2)*sec(c+d*x)^4,x,8,2/3*A*b^3*sin(c+d*x)/(d*(b*cos(c+d*x))^(3/2))+2*b^2*B*sin(c+d*x)/(d*sqrt(b*cos(c+d*x)))+2/3*b^2*(A+3*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))-2*b*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(b*cos(c+d*x))^(3/2)*(A+B*cos(c+d*x)+C*cos(c+d*x)^2)*sec(c+d*x)^5,x,9,2/5*A*b^4*sin(c+d*x)/(d*(b*cos(c+d*x))^(5/2))+2/3*b^3*B*sin(c+d*x)/(d*(b*cos(c+d*x))^(3/2))+2/5*b^2*(3*A+5*C)*sin(c+d*x)/(d*sqrt(b*cos(c+d*x)))+2/3*b^2*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))-2/5*b*(3*A+5*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(b*cos(c+d*x))^(3/2)*(A+B*cos(c+d*x)+C*cos(c+d*x)^2)*sec(c+d*x)^6,x,10,2/7*A*b^5*sin(c+d*x)/(d*(b*cos(c+d*x))^(7/2))+2/5*b^4*B*sin(c+d*x)/(d*(b*cos(c+d*x))^(5/2))+2/21*b^3*(5*A+7*C)*sin(c+d*x)/(d*(b*cos(c+d*x))^(3/2))+6/5*b^2*B*sin(c+d*x)/(d*sqrt(b*cos(c+d*x)))+2/21*b^2*(5*A+7*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))-6/5*b*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(b*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x)+C*cos(c+d*x)^2),x,9,2/45*b*(9*A+7*C)*(b*cos(c+d*x))^(3/2)*sin(c+d*x)/d+2/7*B*(b*cos(c+d*x))^(5/2)*sin(c+d*x)/d+2/9*C*(b*cos(c+d*x))^(7/2)*sin(c+d*x)/(b*d)+10/21*b^3*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))+10/21*b^2*B*sin(c+d*x)*sqrt(b*cos(c+d*x))/d+2/15*b^2*(9*A+7*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(b*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x)+C*cos(c+d*x)^2)*sec(c+d*x),x,9,2/5*b*B*(b*cos(c+d*x))^(3/2)*sin(c+d*x)/d+2/7*C*(b*cos(c+d*x))^(5/2)*sin(c+d*x)/d+2/21*b^3*(7*A+5*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))+2/21*b^2*(7*A+5*C)*sin(c+d*x)*sqrt(b*cos(c+d*x))/d+6/5*b^2*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(b*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x)+C*cos(c+d*x)^2)*sec(c+d*x)^2,x,8,2/5*b*C*(b*cos(c+d*x))^(3/2)*sin(c+d*x)/d+2/3*b^3*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))+2/3*b^2*B*sin(c+d*x)*sqrt(b*cos(c+d*x))/d+2/5*b^2*(5*A+3*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(b*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x)+C*cos(c+d*x)^2)*sec(c+d*x)^3,x,7,2/3*b^3*(3*A+C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))+2/3*b^2*C*sin(c+d*x)*sqrt(b*cos(c+d*x))/d+2*b^2*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(b*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x)+C*cos(c+d*x)^2)*sec(c+d*x)^4,x,7,2*A*b^3*sin(c+d*x)/(d*sqrt(b*cos(c+d*x)))+2*b^3*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))-2*b^2*(A-C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(b*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x)+C*cos(c+d*x)^2)*sec(c+d*x)^5,x,8,2/3*A*b^4*sin(c+d*x)/(d*(b*cos(c+d*x))^(3/2))+2*b^3*B*sin(c+d*x)/(d*sqrt(b*cos(c+d*x)))+2/3*b^3*(A+3*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))-2*b^2*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(b*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x)+C*cos(c+d*x)^2)*sec(c+d*x)^6,x,9,2/5*A*b^5*sin(c+d*x)/(d*(b*cos(c+d*x))^(5/2))+2/3*b^4*B*sin(c+d*x)/(d*(b*cos(c+d*x))^(3/2))+2/5*b^3*(3*A+5*C)*sin(c+d*x)/(d*sqrt(b*cos(c+d*x)))+2/3*b^3*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))-2/5*b^2*(3*A+5*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(b*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x)+C*cos(c+d*x)^2)*sec(c+d*x)^7,x,10,2/7*A*b^6*sin(c+d*x)/(d*(b*cos(c+d*x))^(7/2))+2/5*b^5*B*sin(c+d*x)/(d*(b*cos(c+d*x))^(5/2))+2/21*b^4*(5*A+7*C)*sin(c+d*x)/(d*(b*cos(c+d*x))^(3/2))+6/5*b^3*B*sin(c+d*x)/(d*sqrt(b*cos(c+d*x)))+2/21*b^3*(5*A+7*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))-6/5*b^2*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],

# n<0
[cos(c+d*x)^3*(A+B*cos(c+d*x)+C*cos(c+d*x)^2)/sqrt(b*cos(c+d*x)),x,10,2/45*(9*A+7*C)*(b*cos(c+d*x))^(3/2)*sin(c+d*x)/(b^2*d)+2/7*B*(b*cos(c+d*x))^(5/2)*sin(c+d*x)/(b^3*d)+2/9*C*(b*cos(c+d*x))^(7/2)*sin(c+d*x)/(b^4*d)+10/21*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))+10/21*B*sin(c+d*x)*sqrt(b*cos(c+d*x))/(b*d)+2/15*(9*A+7*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(b*d*sqrt(cos(c+d*x)))],
[cos(c+d*x)^2*(A+B*cos(c+d*x)+C*cos(c+d*x)^2)/sqrt(b*cos(c+d*x)),x,9,2/5*B*(b*cos(c+d*x))^(3/2)*sin(c+d*x)/(b^2*d)+2/7*C*(b*cos(c+d*x))^(5/2)*sin(c+d*x)/(b^3*d)+2/21*(7*A+5*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))+2/21*(7*A+5*C)*sin(c+d*x)*sqrt(b*cos(c+d*x))/(b*d)+6/5*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(b*d*sqrt(cos(c+d*x)))],
[cos(c+d*x)*(A+B*cos(c+d*x)+C*cos(c+d*x)^2)/sqrt(b*cos(c+d*x)),x,8,2/5*C*(b*cos(c+d*x))^(3/2)*sin(c+d*x)/(b^2*d)+2/3*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))+2/3*B*sin(c+d*x)*sqrt(b*cos(c+d*x))/(b*d)+2/5*(5*A+3*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(b*d*sqrt(cos(c+d*x)))],
[(A+B*cos(c+d*x)+C*cos(c+d*x)^2)/sqrt(b*cos(c+d*x)),x,6,2/3*(3*A+C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))+2/3*C*sin(c+d*x)*sqrt(b*cos(c+d*x))/(b*d)+2*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(b*d*sqrt(cos(c+d*x)))],
[(A+B*cos(c+d*x)+C*cos(c+d*x)^2)*sec(c+d*x)/sqrt(b*cos(c+d*x)),x,7,2*A*sin(c+d*x)/(d*sqrt(b*cos(c+d*x)))+2*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))-2*(A-C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(b*d*sqrt(cos(c+d*x)))],
[(A+B*cos(c+d*x)+C*cos(c+d*x)^2)*sec(c+d*x)^2/sqrt(b*cos(c+d*x)),x,8,2/3*A*b*sin(c+d*x)/(d*(b*cos(c+d*x))^(3/2))+2*B*sin(c+d*x)/(d*sqrt(b*cos(c+d*x)))+2/3*(A+3*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))-2*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(b*d*sqrt(cos(c+d*x)))],
[(A+B*cos(c+d*x)+C*cos(c+d*x)^2)*sec(c+d*x)^3/sqrt(b*cos(c+d*x)),x,9,2/5*A*b^2*sin(c+d*x)/(d*(b*cos(c+d*x))^(5/2))+2/3*b*B*sin(c+d*x)/(d*(b*cos(c+d*x))^(3/2))+2/5*(3*A+5*C)*sin(c+d*x)/(d*sqrt(b*cos(c+d*x)))+2/3*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))-2/5*(3*A+5*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(b*d*sqrt(cos(c+d*x)))],
[(A+B*cos(c+d*x)+C*cos(c+d*x)^2)*sec(c+d*x)^4/sqrt(b*cos(c+d*x)),x,10,2/7*A*b^3*sin(c+d*x)/(d*(b*cos(c+d*x))^(7/2))+2/5*b^2*B*sin(c+d*x)/(d*(b*cos(c+d*x))^(5/2))+2/21*b*(5*A+7*C)*sin(c+d*x)/(d*(b*cos(c+d*x))^(3/2))+6/5*B*sin(c+d*x)/(d*sqrt(b*cos(c+d*x)))+2/21*(5*A+7*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))-6/5*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(b*d*sqrt(cos(c+d*x)))],
[cos(c+d*x)^4*(A+B*cos(c+d*x)+C*cos(c+d*x)^2)/(b*cos(c+d*x))^(3/2),x,10,2/45*(9*A+7*C)*(b*cos(c+d*x))^(3/2)*sin(c+d*x)/(b^3*d)+2/7*B*(b*cos(c+d*x))^(5/2)*sin(c+d*x)/(b^4*d)+2/9*C*(b*cos(c+d*x))^(7/2)*sin(c+d*x)/(b^5*d)+10/21*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(b*d*sqrt(b*cos(c+d*x)))+10/21*B*sin(c+d*x)*sqrt(b*cos(c+d*x))/(b^2*d)+2/15*(9*A+7*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(b^2*d*sqrt(cos(c+d*x)))],
[cos(c+d*x)^3*(A+B*cos(c+d*x)+C*cos(c+d*x)^2)/(b*cos(c+d*x))^(3/2),x,9,2/5*B*(b*cos(c+d*x))^(3/2)*sin(c+d*x)/(b^3*d)+2/7*C*(b*cos(c+d*x))^(5/2)*sin(c+d*x)/(b^4*d)+2/21*(7*A+5*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(b*d*sqrt(b*cos(c+d*x)))+2/21*(7*A+5*C)*sin(c+d*x)*sqrt(b*cos(c+d*x))/(b^2*d)+6/5*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(b^2*d*sqrt(cos(c+d*x)))],
[cos(c+d*x)^2*(A+B*cos(c+d*x)+C*cos(c+d*x)^2)/(b*cos(c+d*x))^(3/2),x,8,2/5*C*(b*cos(c+d*x))^(3/2)*sin(c+d*x)/(b^3*d)+2/3*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(b*d*sqrt(b*cos(c+d*x)))+2/3*B*sin(c+d*x)*sqrt(b*cos(c+d*x))/(b^2*d)+2/5*(5*A+3*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(b^2*d*sqrt(cos(c+d*x)))],
[cos(c+d*x)*(A+B*cos(c+d*x)+C*cos(c+d*x)^2)/(b*cos(c+d*x))^(3/2),x,7,2/3*(3*A+C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(b*d*sqrt(b*cos(c+d*x)))+2/3*C*sin(c+d*x)*sqrt(b*cos(c+d*x))/(b^2*d)+2*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(b^2*d*sqrt(cos(c+d*x)))],
[(A+B*cos(c+d*x)+C*cos(c+d*x)^2)/(b*cos(c+d*x))^(3/2),x,6,2*A*sin(c+d*x)/(b*d*sqrt(b*cos(c+d*x)))+2*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(b*d*sqrt(b*cos(c+d*x)))-2*(A-C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(b^2*d*sqrt(cos(c+d*x)))],
[(A+B*cos(c+d*x)+C*cos(c+d*x)^2)*sec(c+d*x)/(b*cos(c+d*x))^(3/2),x,8,2/3*A*sin(c+d*x)/(d*(b*cos(c+d*x))^(3/2))+2*B*sin(c+d*x)/(b*d*sqrt(b*cos(c+d*x)))+2/3*(A+3*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(b*d*sqrt(b*cos(c+d*x)))-2*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(b^2*d*sqrt(cos(c+d*x)))],
[(A+B*cos(c+d*x)+C*cos(c+d*x)^2)*sec(c+d*x)^2/(b*cos(c+d*x))^(3/2),x,9,2/5*A*b*sin(c+d*x)/(d*(b*cos(c+d*x))^(5/2))+2/3*B*sin(c+d*x)/(d*(b*cos(c+d*x))^(3/2))+2/5*(3*A+5*C)*sin(c+d*x)/(b*d*sqrt(b*cos(c+d*x)))+2/3*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(b*d*sqrt(b*cos(c+d*x)))-2/5*(3*A+5*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(b^2*d*sqrt(cos(c+d*x)))],
[(A+B*cos(c+d*x)+C*cos(c+d*x)^2)*sec(c+d*x)^3/(b*cos(c+d*x))^(3/2),x,10,2/7*A*b^2*sin(c+d*x)/(d*(b*cos(c+d*x))^(7/2))+2/5*b*B*sin(c+d*x)/(d*(b*cos(c+d*x))^(5/2))+2/21*(5*A+7*C)*sin(c+d*x)/(d*(b*cos(c+d*x))^(3/2))+6/5*B*sin(c+d*x)/(b*d*sqrt(b*cos(c+d*x)))+2/21*(5*A+7*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(b*d*sqrt(b*cos(c+d*x)))-6/5*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(b^2*d*sqrt(cos(c+d*x)))],
[cos(c+d*x)^5*(A+B*cos(c+d*x)+C*cos(c+d*x)^2)/(b*cos(c+d*x))^(5/2),x,10,2/45*(9*A+7*C)*(b*cos(c+d*x))^(3/2)*sin(c+d*x)/(b^4*d)+2/7*B*(b*cos(c+d*x))^(5/2)*sin(c+d*x)/(b^5*d)+2/9*C*(b*cos(c+d*x))^(7/2)*sin(c+d*x)/(b^6*d)+10/21*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(b^2*d*sqrt(b*cos(c+d*x)))+10/21*B*sin(c+d*x)*sqrt(b*cos(c+d*x))/(b^3*d)+2/15*(9*A+7*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(b^3*d*sqrt(cos(c+d*x)))],
[cos(c+d*x)^4*(A+B*cos(c+d*x)+C*cos(c+d*x)^2)/(b*cos(c+d*x))^(5/2),x,9,2/5*B*(b*cos(c+d*x))^(3/2)*sin(c+d*x)/(b^4*d)+2/7*C*(b*cos(c+d*x))^(5/2)*sin(c+d*x)/(b^5*d)+2/21*(7*A+5*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(b^2*d*sqrt(b*cos(c+d*x)))+2/21*(7*A+5*C)*sin(c+d*x)*sqrt(b*cos(c+d*x))/(b^3*d)+6/5*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(b^3*d*sqrt(cos(c+d*x)))],
[cos(c+d*x)^3*(A+B*cos(c+d*x)+C*cos(c+d*x)^2)/(b*cos(c+d*x))^(5/2),x,8,2/5*C*(b*cos(c+d*x))^(3/2)*sin(c+d*x)/(b^4*d)+2/3*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(b^2*d*sqrt(b*cos(c+d*x)))+2/3*B*sin(c+d*x)*sqrt(b*cos(c+d*x))/(b^3*d)+2/5*(5*A+3*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(b^3*d*sqrt(cos(c+d*x)))],
[cos(c+d*x)^2*(A+B*cos(c+d*x)+C*cos(c+d*x)^2)/(b*cos(c+d*x))^(5/2),x,7,2/3*(3*A+C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(b^2*d*sqrt(b*cos(c+d*x)))+2/3*C*sin(c+d*x)*sqrt(b*cos(c+d*x))/(b^3*d)+2*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(b^3*d*sqrt(cos(c+d*x)))],
[cos(c+d*x)*(A+B*cos(c+d*x)+C*cos(c+d*x)^2)/(b*cos(c+d*x))^(5/2),x,7,2*A*sin(c+d*x)/(b^2*d*sqrt(b*cos(c+d*x)))+2*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(b^2*d*sqrt(b*cos(c+d*x)))-2*(A-C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(b^3*d*sqrt(cos(c+d*x)))],
[(A+B*cos(c+d*x)+C*cos(c+d*x)^2)/(b*cos(c+d*x))^(5/2),x,7,2/3*A*sin(c+d*x)/(b*d*(b*cos(c+d*x))^(3/2))+2*B*sin(c+d*x)/(b^2*d*sqrt(b*cos(c+d*x)))+2/3*(A+3*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(b^2*d*sqrt(b*cos(c+d*x)))-2*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(b^3*d*sqrt(cos(c+d*x)))],
[(A+B*cos(c+d*x)+C*cos(c+d*x)^2)*sec(c+d*x)/(b*cos(c+d*x))^(5/2),x,9,2/5*A*sin(c+d*x)/(d*(b*cos(c+d*x))^(5/2))+2/3*B*sin(c+d*x)/(b*d*(b*cos(c+d*x))^(3/2))+2/5*(3*A+5*C)*sin(c+d*x)/(b^2*d*sqrt(b*cos(c+d*x)))+2/3*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(b^2*d*sqrt(b*cos(c+d*x)))-2/5*(3*A+5*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(b^3*d*sqrt(cos(c+d*x)))],
[(A+B*cos(c+d*x)+C*cos(c+d*x)^2)*sec(c+d*x)^2/(b*cos(c+d*x))^(5/2),x,10,2/7*A*b*sin(c+d*x)/(d*(b*cos(c+d*x))^(7/2))+2/5*B*sin(c+d*x)/(d*(b*cos(c+d*x))^(5/2))+2/21*(5*A+7*C)*sin(c+d*x)/(b*d*(b*cos(c+d*x))^(3/2))+6/5*B*sin(c+d*x)/(b^2*d*sqrt(b*cos(c+d*x)))+2/21*(5*A+7*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(b^2*d*sqrt(b*cos(c+d*x)))-6/5*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(b^3*d*sqrt(cos(c+d*x)))],
[(A+B*cos(c+d*x)+C*cos(c+d*x)^2)/(b*cos(c+d*x))^(7/2),x,8,2/5*A*sin(c+d*x)/(b*d*(b*cos(c+d*x))^(5/2))+2/3*B*sin(c+d*x)/(b^2*d*(b*cos(c+d*x))^(3/2))+2/5*(3*A+5*C)*sin(c+d*x)/(b^3*d*sqrt(b*cos(c+d*x)))+2/3*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(b^3*d*sqrt(b*cos(c+d*x)))-2/5*(3*A+5*C)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(b^4*d*sqrt(cos(c+d*x)))],

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

# n>0
[cos(c+d*x)^(5/2)*(A+B*cos(c+d*x)+C*cos(c+d*x)^2)*sqrt(b*cos(c+d*x)),x,8,1/4*B*cos(c+d*x)^(5/2)*sin(c+d*x)*sqrt(b*cos(c+d*x))/d+1/5*C*cos(c+d*x)^(7/2)*sin(c+d*x)*sqrt(b*cos(c+d*x))/d+3/8*B*x*sqrt(b*cos(c+d*x))/sqrt(cos(c+d*x))+1/5*(5*A+4*C)*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))-1/15*(5*A+4*C)*sin(c+d*x)^3*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))+3/8*B*sin(c+d*x)*sqrt(cos(c+d*x))*sqrt(b*cos(c+d*x))/d],
[cos(c+d*x)^(3/2)*(A+B*cos(c+d*x)+C*cos(c+d*x)^2)*sqrt(b*cos(c+d*x)),x,7,1/4*C*cos(c+d*x)^(5/2)*sin(c+d*x)*sqrt(b*cos(c+d*x))/d+1/8*(4*A+3*C)*x*sqrt(b*cos(c+d*x))/sqrt(cos(c+d*x))+B*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))-1/3*B*sin(c+d*x)^3*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))+1/8*(4*A+3*C)*sin(c+d*x)*sqrt(cos(c+d*x))*sqrt(b*cos(c+d*x))/d],
[(A+B*cos(c+d*x)+C*cos(c+d*x)^2)*sqrt(cos(c+d*x))*sqrt(b*cos(c+d*x)),x,3,1/3*C*cos(c+d*x)^(3/2)*sin(c+d*x)*sqrt(b*cos(c+d*x))/d+1/2*B*x*sqrt(b*cos(c+d*x))/sqrt(cos(c+d*x))+1/3*(3*A+2*C)*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))+1/2*B*sin(c+d*x)*sqrt(cos(c+d*x))*sqrt(b*cos(c+d*x))/d],
[(A+B*cos(c+d*x)+C*cos(c+d*x)^2)*sqrt(b*cos(c+d*x))/sqrt(cos(c+d*x)),x,5,A*x*sqrt(b*cos(c+d*x))/sqrt(cos(c+d*x))+1/2*C*x*sqrt(b*cos(c+d*x))/sqrt(cos(c+d*x))+B*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))+1/2*C*sin(c+d*x)*sqrt(cos(c+d*x))*sqrt(b*cos(c+d*x))/d],
[(A+B*cos(c+d*x)+C*cos(c+d*x)^2)*sqrt(b*cos(c+d*x))/cos(c+d*x)^(3/2),x,4,B*x*sqrt(b*cos(c+d*x))/sqrt(cos(c+d*x))+A*arctanh(sin(c+d*x))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))+C*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(A+B*cos(c+d*x)+C*cos(c+d*x)^2)*sqrt(b*cos(c+d*x))/cos(c+d*x)^(5/2),x,4,A*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*cos(c+d*x)^(3/2))+C*x*sqrt(b*cos(c+d*x))/sqrt(cos(c+d*x))+B*arctanh(sin(c+d*x))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(A+B*cos(c+d*x)+C*cos(c+d*x)^2)*sqrt(b*cos(c+d*x))/cos(c+d*x)^(7/2),x,6,1/2*A*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*cos(c+d*x)^(5/2))+B*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*cos(c+d*x)^(3/2))+1/2*(A+2*C)*arctanh(sin(c+d*x))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(A+B*cos(c+d*x)+C*cos(c+d*x)^2)*sqrt(b*cos(c+d*x))/cos(c+d*x)^(9/2),x,7,1/3*A*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*cos(c+d*x)^(7/2))+1/2*B*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*cos(c+d*x)^(5/2))+1/3*(2*A+3*C)*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*cos(c+d*x)^(3/2))+1/2*B*arctanh(sin(c+d*x))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(A+B*cos(c+d*x)+C*cos(c+d*x)^2)*sqrt(b*cos(c+d*x))/cos(c+d*x)^(11/2),x,7,1/4*A*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*cos(c+d*x)^(9/2))+1/8*(3*A+4*C)*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*cos(c+d*x)^(5/2))+B*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*cos(c+d*x)^(3/2))+1/3*B*sin(c+d*x)^3*sqrt(b*cos(c+d*x))/(d*cos(c+d*x)^(7/2))+1/8*(3*A+4*C)*arctanh(sin(c+d*x))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[cos(c+d*x)^(3/2)*(b*cos(c+d*x))^(3/2)*(A+B*cos(c+d*x)+C*cos(c+d*x)^2),x,8,1/4*b*B*cos(c+d*x)^(5/2)*sin(c+d*x)*sqrt(b*cos(c+d*x))/d+1/5*b*C*cos(c+d*x)^(7/2)*sin(c+d*x)*sqrt(b*cos(c+d*x))/d+3/8*b*B*x*sqrt(b*cos(c+d*x))/sqrt(cos(c+d*x))+1/5*b*(5*A+4*C)*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))-1/15*b*(5*A+4*C)*sin(c+d*x)^3*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))+3/8*b*B*sin(c+d*x)*sqrt(cos(c+d*x))*sqrt(b*cos(c+d*x))/d],
[(b*cos(c+d*x))^(3/2)*(A+B*cos(c+d*x)+C*cos(c+d*x)^2)*sqrt(cos(c+d*x)),x,7,1/4*b*C*cos(c+d*x)^(5/2)*sin(c+d*x)*sqrt(b*cos(c+d*x))/d+1/8*b*(4*A+3*C)*x*sqrt(b*cos(c+d*x))/sqrt(cos(c+d*x))+b*B*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))-1/3*b*B*sin(c+d*x)^3*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))+1/8*b*(4*A+3*C)*sin(c+d*x)*sqrt(cos(c+d*x))*sqrt(b*cos(c+d*x))/d],
[(b*cos(c+d*x))^(3/2)*(A+B*cos(c+d*x)+C*cos(c+d*x)^2)/sqrt(cos(c+d*x)),x,3,1/3*b*C*cos(c+d*x)^(3/2)*sin(c+d*x)*sqrt(b*cos(c+d*x))/d+1/2*b*B*x*sqrt(b*cos(c+d*x))/sqrt(cos(c+d*x))+1/3*b*(3*A+2*C)*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))+1/2*b*B*sin(c+d*x)*sqrt(cos(c+d*x))*sqrt(b*cos(c+d*x))/d],
[(b*cos(c+d*x))^(3/2)*(A+B*cos(c+d*x)+C*cos(c+d*x)^2)/cos(c+d*x)^(3/2),x,5,A*b*x*sqrt(b*cos(c+d*x))/sqrt(cos(c+d*x))+1/2*b*C*x*sqrt(b*cos(c+d*x))/sqrt(cos(c+d*x))+b*B*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))+1/2*b*C*sin(c+d*x)*sqrt(cos(c+d*x))*sqrt(b*cos(c+d*x))/d],
[(b*cos(c+d*x))^(3/2)*(A+B*cos(c+d*x)+C*cos(c+d*x)^2)/cos(c+d*x)^(5/2),x,4,b*B*x*sqrt(b*cos(c+d*x))/sqrt(cos(c+d*x))+A*b*arctanh(sin(c+d*x))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))+b*C*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(b*cos(c+d*x))^(3/2)*(A+B*cos(c+d*x)+C*cos(c+d*x)^2)/cos(c+d*x)^(7/2),x,4,A*b*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*cos(c+d*x)^(3/2))+b*C*x*sqrt(b*cos(c+d*x))/sqrt(cos(c+d*x))+b*B*arctanh(sin(c+d*x))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(b*cos(c+d*x))^(3/2)*(A+B*cos(c+d*x)+C*cos(c+d*x)^2)/cos(c+d*x)^(9/2),x,6,1/2*A*b*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*cos(c+d*x)^(5/2))+b*B*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*cos(c+d*x)^(3/2))+1/2*b*(A+2*C)*arctanh(sin(c+d*x))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(b*cos(c+d*x))^(3/2)*(A+B*cos(c+d*x)+C*cos(c+d*x)^2)/cos(c+d*x)^(11/2),x,7,1/3*A*b*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*cos(c+d*x)^(7/2))+1/2*b*B*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*cos(c+d*x)^(5/2))+1/3*b*(2*A+3*C)*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*cos(c+d*x)^(3/2))+1/2*b*B*arctanh(sin(c+d*x))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(b*cos(c+d*x))^(3/2)*(A+B*cos(c+d*x)+C*cos(c+d*x)^2)/cos(c+d*x)^(13/2),x,7,1/4*A*b*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*cos(c+d*x)^(9/2))+1/8*b*(3*A+4*C)*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*cos(c+d*x)^(5/2))+b*B*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*cos(c+d*x)^(3/2))+1/3*b*B*sin(c+d*x)^3*sqrt(b*cos(c+d*x))/(d*cos(c+d*x)^(7/2))+1/8*b*(3*A+4*C)*arctanh(sin(c+d*x))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(b*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x)+C*cos(c+d*x)^2)*sqrt(cos(c+d*x)),x,8,1/4*b^2*B*cos(c+d*x)^(5/2)*sin(c+d*x)*sqrt(b*cos(c+d*x))/d+1/5*b^2*C*cos(c+d*x)^(7/2)*sin(c+d*x)*sqrt(b*cos(c+d*x))/d+3/8*b^2*B*x*sqrt(b*cos(c+d*x))/sqrt(cos(c+d*x))+1/5*b^2*(5*A+4*C)*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))-1/15*b^2*(5*A+4*C)*sin(c+d*x)^3*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))+3/8*b^2*B*sin(c+d*x)*sqrt(cos(c+d*x))*sqrt(b*cos(c+d*x))/d],
[(b*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x)+C*cos(c+d*x)^2)/sqrt(cos(c+d*x)),x,7,1/4*b^2*C*cos(c+d*x)^(5/2)*sin(c+d*x)*sqrt(b*cos(c+d*x))/d+1/8*b^2*(4*A+3*C)*x*sqrt(b*cos(c+d*x))/sqrt(cos(c+d*x))+b^2*B*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))-1/3*b^2*B*sin(c+d*x)^3*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))+1/8*b^2*(4*A+3*C)*sin(c+d*x)*sqrt(cos(c+d*x))*sqrt(b*cos(c+d*x))/d],
[(b*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x)+C*cos(c+d*x)^2)/cos(c+d*x)^(3/2),x,3,1/3*b^2*C*cos(c+d*x)^(3/2)*sin(c+d*x)*sqrt(b*cos(c+d*x))/d+1/2*b^2*B*x*sqrt(b*cos(c+d*x))/sqrt(cos(c+d*x))+1/3*b^2*(3*A+2*C)*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))+1/2*b^2*B*sin(c+d*x)*sqrt(cos(c+d*x))*sqrt(b*cos(c+d*x))/d],
[(b*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x)+C*cos(c+d*x)^2)/cos(c+d*x)^(5/2),x,5,A*b^2*x*sqrt(b*cos(c+d*x))/sqrt(cos(c+d*x))+1/2*b^2*C*x*sqrt(b*cos(c+d*x))/sqrt(cos(c+d*x))+b^2*B*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))+1/2*b^2*C*sin(c+d*x)*sqrt(cos(c+d*x))*sqrt(b*cos(c+d*x))/d],
[(b*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x)+C*cos(c+d*x)^2)/cos(c+d*x)^(7/2),x,4,b^2*B*x*sqrt(b*cos(c+d*x))/sqrt(cos(c+d*x))+A*b^2*arctanh(sin(c+d*x))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))+b^2*C*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(b*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x)+C*cos(c+d*x)^2)/cos(c+d*x)^(9/2),x,4,A*b^2*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*cos(c+d*x)^(3/2))+b^2*C*x*sqrt(b*cos(c+d*x))/sqrt(cos(c+d*x))+b^2*B*arctanh(sin(c+d*x))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(b*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x)+C*cos(c+d*x)^2)/cos(c+d*x)^(11/2),x,6,1/2*A*b^2*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*cos(c+d*x)^(5/2))+b^2*B*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*cos(c+d*x)^(3/2))+1/2*b^2*(A+2*C)*arctanh(sin(c+d*x))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(b*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x)+C*cos(c+d*x)^2)/cos(c+d*x)^(13/2),x,7,1/3*A*b^2*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*cos(c+d*x)^(7/2))+1/2*b^2*B*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*cos(c+d*x)^(5/2))+1/3*b^2*(2*A+3*C)*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*cos(c+d*x)^(3/2))+1/2*b^2*B*arctanh(sin(c+d*x))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(b*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x)+C*cos(c+d*x)^2)/cos(c+d*x)^(15/2),x,7,1/4*A*b^2*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*cos(c+d*x)^(9/2))+1/8*b^2*(3*A+4*C)*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*cos(c+d*x)^(5/2))+b^2*B*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*cos(c+d*x)^(3/2))+1/3*b^2*B*sin(c+d*x)^3*sqrt(b*cos(c+d*x))/(d*cos(c+d*x)^(7/2))+1/8*b^2*(3*A+4*C)*arctanh(sin(c+d*x))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],

# n<0
[cos(c+d*x)^(5/2)*(A+B*cos(c+d*x)+C*cos(c+d*x)^2)/sqrt(b*cos(c+d*x)),x,7,1/8*(4*A+3*C)*cos(c+d*x)^(3/2)*sin(c+d*x)/(d*sqrt(b*cos(c+d*x)))+1/4*C*cos(c+d*x)^(7/2)*sin(c+d*x)/(d*sqrt(b*cos(c+d*x)))+1/8*(4*A+3*C)*x*sqrt(cos(c+d*x))/sqrt(b*cos(c+d*x))+B*sin(c+d*x)*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))-1/3*B*sin(c+d*x)^3*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))],
[cos(c+d*x)^(3/2)*(A+B*cos(c+d*x)+C*cos(c+d*x)^2)/sqrt(b*cos(c+d*x)),x,3,1/2*B*cos(c+d*x)^(3/2)*sin(c+d*x)/(d*sqrt(b*cos(c+d*x)))+1/3*C*cos(c+d*x)^(5/2)*sin(c+d*x)/(d*sqrt(b*cos(c+d*x)))+1/2*B*x*sqrt(cos(c+d*x))/sqrt(b*cos(c+d*x))+1/3*(3*A+2*C)*sin(c+d*x)*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))],
[(A+B*cos(c+d*x)+C*cos(c+d*x)^2)*sqrt(cos(c+d*x))/sqrt(b*cos(c+d*x)),x,5,1/2*C*cos(c+d*x)^(3/2)*sin(c+d*x)/(d*sqrt(b*cos(c+d*x)))+A*x*sqrt(cos(c+d*x))/sqrt(b*cos(c+d*x))+1/2*C*x*sqrt(cos(c+d*x))/sqrt(b*cos(c+d*x))+B*sin(c+d*x)*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))],
[(A+B*cos(c+d*x)+C*cos(c+d*x)^2)/(sqrt(cos(c+d*x))*sqrt(b*cos(c+d*x))),x,4,B*x*sqrt(cos(c+d*x))/sqrt(b*cos(c+d*x))+A*arctanh(sin(c+d*x))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))+C*sin(c+d*x)*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))],
[(A+B*cos(c+d*x)+C*cos(c+d*x)^2)/(cos(c+d*x)^(3/2)*sqrt(b*cos(c+d*x))),x,4,A*sin(c+d*x)/(d*sqrt(cos(c+d*x))*sqrt(b*cos(c+d*x)))+C*x*sqrt(cos(c+d*x))/sqrt(b*cos(c+d*x))+B*arctanh(sin(c+d*x))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))],
[(A+B*cos(c+d*x)+C*cos(c+d*x)^2)/(cos(c+d*x)^(5/2)*sqrt(b*cos(c+d*x))),x,6,1/2*A*sin(c+d*x)/(d*cos(c+d*x)^(3/2)*sqrt(b*cos(c+d*x)))+B*sin(c+d*x)/(d*sqrt(cos(c+d*x))*sqrt(b*cos(c+d*x)))+1/2*(A+2*C)*arctanh(sin(c+d*x))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))],
[(A+B*cos(c+d*x)+C*cos(c+d*x)^2)/(cos(c+d*x)^(7/2)*sqrt(b*cos(c+d*x))),x,7,1/3*A*sin(c+d*x)/(d*cos(c+d*x)^(5/2)*sqrt(b*cos(c+d*x)))+1/2*B*sin(c+d*x)/(d*cos(c+d*x)^(3/2)*sqrt(b*cos(c+d*x)))+1/3*(2*A+3*C)*sin(c+d*x)/(d*sqrt(cos(c+d*x))*sqrt(b*cos(c+d*x)))+1/2*B*arctanh(sin(c+d*x))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))],
[(A+B*cos(c+d*x)+C*cos(c+d*x)^2)/(cos(c+d*x)^(9/2)*sqrt(b*cos(c+d*x))),x,7,1/4*A*sin(c+d*x)/(d*cos(c+d*x)^(7/2)*sqrt(b*cos(c+d*x)))+1/8*(3*A+4*C)*sin(c+d*x)/(d*cos(c+d*x)^(3/2)*sqrt(b*cos(c+d*x)))+1/3*B*sin(c+d*x)^3/(d*cos(c+d*x)^(5/2)*sqrt(b*cos(c+d*x)))+B*sin(c+d*x)/(d*sqrt(cos(c+d*x))*sqrt(b*cos(c+d*x)))+1/8*(3*A+4*C)*arctanh(sin(c+d*x))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))],
[cos(c+d*x)^(7/2)*(A+B*cos(c+d*x)+C*cos(c+d*x)^2)/(b*cos(c+d*x))^(3/2),x,7,1/8*(4*A+3*C)*cos(c+d*x)^(3/2)*sin(c+d*x)/(b*d*sqrt(b*cos(c+d*x)))+1/4*C*cos(c+d*x)^(7/2)*sin(c+d*x)/(b*d*sqrt(b*cos(c+d*x)))+1/8*(4*A+3*C)*x*sqrt(cos(c+d*x))/(b*sqrt(b*cos(c+d*x)))+B*sin(c+d*x)*sqrt(cos(c+d*x))/(b*d*sqrt(b*cos(c+d*x)))-1/3*B*sin(c+d*x)^3*sqrt(cos(c+d*x))/(b*d*sqrt(b*cos(c+d*x)))],
[cos(c+d*x)^(5/2)*(A+B*cos(c+d*x)+C*cos(c+d*x)^2)/(b*cos(c+d*x))^(3/2),x,3,1/2*B*cos(c+d*x)^(3/2)*sin(c+d*x)/(b*d*sqrt(b*cos(c+d*x)))+1/3*C*cos(c+d*x)^(5/2)*sin(c+d*x)/(b*d*sqrt(b*cos(c+d*x)))+1/2*B*x*sqrt(cos(c+d*x))/(b*sqrt(b*cos(c+d*x)))+1/3*(3*A+2*C)*sin(c+d*x)*sqrt(cos(c+d*x))/(b*d*sqrt(b*cos(c+d*x)))],
[cos(c+d*x)^(3/2)*(A+B*cos(c+d*x)+C*cos(c+d*x)^2)/(b*cos(c+d*x))^(3/2),x,5,1/2*C*cos(c+d*x)^(3/2)*sin(c+d*x)/(b*d*sqrt(b*cos(c+d*x)))+A*x*sqrt(cos(c+d*x))/(b*sqrt(b*cos(c+d*x)))+1/2*C*x*sqrt(cos(c+d*x))/(b*sqrt(b*cos(c+d*x)))+B*sin(c+d*x)*sqrt(cos(c+d*x))/(b*d*sqrt(b*cos(c+d*x)))],
[(A+B*cos(c+d*x)+C*cos(c+d*x)^2)*sqrt(cos(c+d*x))/(b*cos(c+d*x))^(3/2),x,4,B*x*sqrt(cos(c+d*x))/(b*sqrt(b*cos(c+d*x)))+A*arctanh(sin(c+d*x))*sqrt(cos(c+d*x))/(b*d*sqrt(b*cos(c+d*x)))+C*sin(c+d*x)*sqrt(cos(c+d*x))/(b*d*sqrt(b*cos(c+d*x)))],
[(A+B*cos(c+d*x)+C*cos(c+d*x)^2)/((b*cos(c+d*x))^(3/2)*sqrt(cos(c+d*x))),x,4,A*sin(c+d*x)/(b*d*sqrt(cos(c+d*x))*sqrt(b*cos(c+d*x)))+C*x*sqrt(cos(c+d*x))/(b*sqrt(b*cos(c+d*x)))+B*arctanh(sin(c+d*x))*sqrt(cos(c+d*x))/(b*d*sqrt(b*cos(c+d*x)))],
[(A+B*cos(c+d*x)+C*cos(c+d*x)^2)/(cos(c+d*x)^(3/2)*(b*cos(c+d*x))^(3/2)),x,6,1/2*A*sin(c+d*x)/(b*d*cos(c+d*x)^(3/2)*sqrt(b*cos(c+d*x)))+B*sin(c+d*x)/(b*d*sqrt(cos(c+d*x))*sqrt(b*cos(c+d*x)))+1/2*(A+2*C)*arctanh(sin(c+d*x))*sqrt(cos(c+d*x))/(b*d*sqrt(b*cos(c+d*x)))],
[(A+B*cos(c+d*x)+C*cos(c+d*x)^2)/(cos(c+d*x)^(5/2)*(b*cos(c+d*x))^(3/2)),x,7,1/3*A*sin(c+d*x)/(b*d*cos(c+d*x)^(5/2)*sqrt(b*cos(c+d*x)))+1/2*B*sin(c+d*x)/(b*d*cos(c+d*x)^(3/2)*sqrt(b*cos(c+d*x)))+1/3*(2*A+3*C)*sin(c+d*x)/(b*d*sqrt(cos(c+d*x))*sqrt(b*cos(c+d*x)))+1/2*B*arctanh(sin(c+d*x))*sqrt(cos(c+d*x))/(b*d*sqrt(b*cos(c+d*x)))],
[(A+B*cos(c+d*x)+C*cos(c+d*x)^2)/(cos(c+d*x)^(7/2)*(b*cos(c+d*x))^(3/2)),x,7,1/4*A*sin(c+d*x)/(b*d*cos(c+d*x)^(7/2)*sqrt(b*cos(c+d*x)))+1/8*(3*A+4*C)*sin(c+d*x)/(b*d*cos(c+d*x)^(3/2)*sqrt(b*cos(c+d*x)))+1/3*B*sin(c+d*x)^3/(b*d*cos(c+d*x)^(5/2)*sqrt(b*cos(c+d*x)))+B*sin(c+d*x)/(b*d*sqrt(cos(c+d*x))*sqrt(b*cos(c+d*x)))+1/8*(3*A+4*C)*arctanh(sin(c+d*x))*sqrt(cos(c+d*x))/(b*d*sqrt(b*cos(c+d*x)))],
[cos(c+d*x)^(9/2)*(A+B*cos(c+d*x)+C*cos(c+d*x)^2)/(b*cos(c+d*x))^(5/2),x,7,1/8*(4*A+3*C)*cos(c+d*x)^(3/2)*sin(c+d*x)/(b^2*d*sqrt(b*cos(c+d*x)))+1/4*C*cos(c+d*x)^(7/2)*sin(c+d*x)/(b^2*d*sqrt(b*cos(c+d*x)))+1/8*(4*A+3*C)*x*sqrt(cos(c+d*x))/(b^2*sqrt(b*cos(c+d*x)))+B*sin(c+d*x)*sqrt(cos(c+d*x))/(b^2*d*sqrt(b*cos(c+d*x)))-1/3*B*sin(c+d*x)^3*sqrt(cos(c+d*x))/(b^2*d*sqrt(b*cos(c+d*x)))],
[cos(c+d*x)^(7/2)*(A+B*cos(c+d*x)+C*cos(c+d*x)^2)/(b*cos(c+d*x))^(5/2),x,3,1/2*B*cos(c+d*x)^(3/2)*sin(c+d*x)/(b^2*d*sqrt(b*cos(c+d*x)))+1/3*C*cos(c+d*x)^(5/2)*sin(c+d*x)/(b^2*d*sqrt(b*cos(c+d*x)))+1/2*B*x*sqrt(cos(c+d*x))/(b^2*sqrt(b*cos(c+d*x)))+1/3*(3*A+2*C)*sin(c+d*x)*sqrt(cos(c+d*x))/(b^2*d*sqrt(b*cos(c+d*x)))],
[cos(c+d*x)^(5/2)*(A+B*cos(c+d*x)+C*cos(c+d*x)^2)/(b*cos(c+d*x))^(5/2),x,5,1/2*C*cos(c+d*x)^(3/2)*sin(c+d*x)/(b^2*d*sqrt(b*cos(c+d*x)))+A*x*sqrt(cos(c+d*x))/(b^2*sqrt(b*cos(c+d*x)))+1/2*C*x*sqrt(cos(c+d*x))/(b^2*sqrt(b*cos(c+d*x)))+B*sin(c+d*x)*sqrt(cos(c+d*x))/(b^2*d*sqrt(b*cos(c+d*x)))],
[cos(c+d*x)^(3/2)*(A+B*cos(c+d*x)+C*cos(c+d*x)^2)/(b*cos(c+d*x))^(5/2),x,4,B*x*sqrt(cos(c+d*x))/(b^2*sqrt(b*cos(c+d*x)))+A*arctanh(sin(c+d*x))*sqrt(cos(c+d*x))/(b^2*d*sqrt(b*cos(c+d*x)))+C*sin(c+d*x)*sqrt(cos(c+d*x))/(b^2*d*sqrt(b*cos(c+d*x)))],
[(A+B*cos(c+d*x)+C*cos(c+d*x)^2)*sqrt(cos(c+d*x))/(b*cos(c+d*x))^(5/2),x,4,A*sin(c+d*x)/(b^2*d*sqrt(cos(c+d*x))*sqrt(b*cos(c+d*x)))+C*x*sqrt(cos(c+d*x))/(b^2*sqrt(b*cos(c+d*x)))+B*arctanh(sin(c+d*x))*sqrt(cos(c+d*x))/(b^2*d*sqrt(b*cos(c+d*x)))],
[(A+B*cos(c+d*x)+C*cos(c+d*x)^2)/((b*cos(c+d*x))^(5/2)*sqrt(cos(c+d*x))),x,6,1/2*A*sin(c+d*x)/(b^2*d*cos(c+d*x)^(3/2)*sqrt(b*cos(c+d*x)))+B*sin(c+d*x)/(b^2*d*sqrt(cos(c+d*x))*sqrt(b*cos(c+d*x)))+1/2*(A+2*C)*arctanh(sin(c+d*x))*sqrt(cos(c+d*x))/(b^2*d*sqrt(b*cos(c+d*x)))],
[(A+B*cos(c+d*x)+C*cos(c+d*x)^2)/(cos(c+d*x)^(3/2)*(b*cos(c+d*x))^(5/2)),x,7,1/3*A*sin(c+d*x)/(b^2*d*cos(c+d*x)^(5/2)*sqrt(b*cos(c+d*x)))+1/2*B*sin(c+d*x)/(b^2*d*cos(c+d*x)^(3/2)*sqrt(b*cos(c+d*x)))+1/3*(2*A+3*C)*sin(c+d*x)/(b^2*d*sqrt(cos(c+d*x))*sqrt(b*cos(c+d*x)))+1/2*B*arctanh(sin(c+d*x))*sqrt(cos(c+d*x))/(b^2*d*sqrt(b*cos(c+d*x)))],
[(A+B*cos(c+d*x)+C*cos(c+d*x)^2)/(cos(c+d*x)^(5/2)*(b*cos(c+d*x))^(5/2)),x,7,1/4*A*sin(c+d*x)/(b^2*d*cos(c+d*x)^(7/2)*sqrt(b*cos(c+d*x)))+1/8*(3*A+4*C)*sin(c+d*x)/(b^2*d*cos(c+d*x)^(3/2)*sqrt(b*cos(c+d*x)))+1/3*B*sin(c+d*x)^3/(b^2*d*cos(c+d*x)^(5/2)*sqrt(b*cos(c+d*x)))+B*sin(c+d*x)/(b^2*d*sqrt(cos(c+d*x))*sqrt(b*cos(c+d*x)))+1/8*(3*A+4*C)*arctanh(sin(c+d*x))*sqrt(cos(c+d*x))/(b^2*d*sqrt(b*cos(c+d*x)))],

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

# n>0
[cos(c+d*x)*(b*cos(c+d*x))^(2/3)*(A+B*cos(c+d*x)+C*cos(c+d*x)^2),x,5,3/11*C*(b*cos(c+d*x))^(8/3)*sin(c+d*x)/(b^2*d)-3/88*(11*A+8*C)*(b*cos(c+d*x))^(8/3)*hypergeom([1/2,4/3],[7/3],cos(c+d*x)^2)*sin(c+d*x)/(b^2*d*sqrt(sin(c+d*x)^2))-3/11*B*(b*cos(c+d*x))^(11/3)*hypergeom([1/2,11/6],[17/6],cos(c+d*x)^2)*sin(c+d*x)/(b^3*d*sqrt(sin(c+d*x)^2))],
[(b*cos(c+d*x))^(2/3)*(A+B*cos(c+d*x)+C*cos(c+d*x)^2),x,4,3/8*C*(b*cos(c+d*x))^(5/3)*sin(c+d*x)/(b*d)-3/40*(8*A+5*C)*(b*cos(c+d*x))^(5/3)*hypergeom([1/2,5/6],[11/6],cos(c+d*x)^2)*sin(c+d*x)/(b*d*sqrt(sin(c+d*x)^2))-3/8*B*(b*cos(c+d*x))^(8/3)*hypergeom([1/2,4/3],[7/3],cos(c+d*x)^2)*sin(c+d*x)/(b^2*d*sqrt(sin(c+d*x)^2))],
[(b*cos(c+d*x))^(2/3)*(A+B*cos(c+d*x)+C*cos(c+d*x)^2)*sec(c+d*x),x,5,3/5*C*(b*cos(c+d*x))^(2/3)*sin(c+d*x)/d-3/10*(5*A+2*C)*(b*cos(c+d*x))^(2/3)*hypergeom([1/3,1/2],[4/3],cos(c+d*x)^2)*sin(c+d*x)/(d*sqrt(sin(c+d*x)^2))-3/5*B*(b*cos(c+d*x))^(5/3)*hypergeom([1/2,5/6],[11/6],cos(c+d*x)^2)*sin(c+d*x)/(b*d*sqrt(sin(c+d*x)^2))],
[(b*cos(c+d*x))^(2/3)*(A+B*cos(c+d*x)+C*cos(c+d*x)^2)*sec(c+d*x)^2,x,5,3*A*b*sin(c+d*x)/(d*(b*cos(c+d*x))^(1/3))-3/2*B*(b*cos(c+d*x))^(2/3)*hypergeom([1/3,1/2],[4/3],cos(c+d*x)^2)*sin(c+d*x)/(d*sqrt(sin(c+d*x)^2))+3/5*(2*A-C)*(b*cos(c+d*x))^(5/3)*hypergeom([1/2,5/6],[11/6],cos(c+d*x)^2)*sin(c+d*x)/(b*d*sqrt(sin(c+d*x)^2))],
[(b*cos(c+d*x))^(2/3)*(A+B*cos(c+d*x)+C*cos(c+d*x)^2)*sec(c+d*x)^3,x,5,3/4*A*b^2*sin(c+d*x)/(d*(b*cos(c+d*x))^(4/3))+3*b*B*hypergeom([-1/6,1/2],[5/6],cos(c+d*x)^2)*sin(c+d*x)/(d*(b*cos(c+d*x))^(1/3)*sqrt(sin(c+d*x)^2))-3/8*(A+4*C)*(b*cos(c+d*x))^(2/3)*hypergeom([1/3,1/2],[4/3],cos(c+d*x)^2)*sin(c+d*x)/(d*sqrt(sin(c+d*x)^2))],
[(b*cos(c+d*x))^(2/3)*(A+B*cos(c+d*x)+C*cos(c+d*x)^2)*sec(c+d*x)^4,x,5,3/7*A*b^3*sin(c+d*x)/(d*(b*cos(c+d*x))^(7/3))+3/4*b^2*B*hypergeom([-2/3,1/2],[1/3],cos(c+d*x)^2)*sin(c+d*x)/(d*(b*cos(c+d*x))^(4/3)*sqrt(sin(c+d*x)^2))+3/7*b*(4*A+7*C)*hypergeom([-1/6,1/2],[5/6],cos(c+d*x)^2)*sin(c+d*x)/(d*(b*cos(c+d*x))^(1/3)*sqrt(sin(c+d*x)^2))],
[cos(c+d*x)*(b*cos(c+d*x))^(4/3)*(A+B*cos(c+d*x)+C*cos(c+d*x)^2),x,5,3/13*C*(b*cos(c+d*x))^(10/3)*sin(c+d*x)/(b^2*d)-3/130*(13*A+10*C)*(b*cos(c+d*x))^(10/3)*hypergeom([1/2,5/3],[8/3],cos(c+d*x)^2)*sin(c+d*x)/(b^2*d*sqrt(sin(c+d*x)^2))-3/13*B*(b*cos(c+d*x))^(13/3)*hypergeom([1/2,13/6],[19/6],cos(c+d*x)^2)*sin(c+d*x)/(b^3*d*sqrt(sin(c+d*x)^2))],
[(b*cos(c+d*x))^(4/3)*(A+B*cos(c+d*x)+C*cos(c+d*x)^2),x,4,3/10*C*(b*cos(c+d*x))^(7/3)*sin(c+d*x)/(b*d)-3/70*(10*A+7*C)*(b*cos(c+d*x))^(7/3)*hypergeom([1/2,7/6],[13/6],cos(c+d*x)^2)*sin(c+d*x)/(b*d*sqrt(sin(c+d*x)^2))-3/10*B*(b*cos(c+d*x))^(10/3)*hypergeom([1/2,5/3],[8/3],cos(c+d*x)^2)*sin(c+d*x)/(b^2*d*sqrt(sin(c+d*x)^2))],
[(b*cos(c+d*x))^(4/3)*(A+B*cos(c+d*x)+C*cos(c+d*x)^2)*sec(c+d*x),x,5,3/7*C*(b*cos(c+d*x))^(4/3)*sin(c+d*x)/d-3/28*(7*A+4*C)*(b*cos(c+d*x))^(4/3)*hypergeom([1/2,2/3],[5/3],cos(c+d*x)^2)*sin(c+d*x)/(d*sqrt(sin(c+d*x)^2))-3/7*B*(b*cos(c+d*x))^(7/3)*hypergeom([1/2,7/6],[13/6],cos(c+d*x)^2)*sin(c+d*x)/(b*d*sqrt(sin(c+d*x)^2))],
[(b*cos(c+d*x))^(4/3)*(A+B*cos(c+d*x)+C*cos(c+d*x)^2)*sec(c+d*x)^2,x,5,3/4*b*C*(b*cos(c+d*x))^(1/3)*sin(c+d*x)/d-3/4*b*(4*A+C)*(b*cos(c+d*x))^(1/3)*hypergeom([1/6,1/2],[7/6],cos(c+d*x)^2)*sin(c+d*x)/(d*sqrt(sin(c+d*x)^2))-3/4*B*(b*cos(c+d*x))^(4/3)*hypergeom([1/2,2/3],[5/3],cos(c+d*x)^2)*sin(c+d*x)/(d*sqrt(sin(c+d*x)^2))],
[(b*cos(c+d*x))^(4/3)*(A+B*cos(c+d*x)+C*cos(c+d*x)^2)*sec(c+d*x)^3,x,5,3/2*A*b^2*sin(c+d*x)/(d*(b*cos(c+d*x))^(2/3))-3*b*B*(b*cos(c+d*x))^(1/3)*hypergeom([1/6,1/2],[7/6],cos(c+d*x)^2)*sin(c+d*x)/(d*sqrt(sin(c+d*x)^2))+3/8*(A-2*C)*(b*cos(c+d*x))^(4/3)*hypergeom([1/2,2/3],[5/3],cos(c+d*x)^2)*sin(c+d*x)/(d*sqrt(sin(c+d*x)^2))],
[(b*cos(c+d*x))^(4/3)*(A+B*cos(c+d*x)+C*cos(c+d*x)^2)*sec(c+d*x)^4,x,5,3/5*A*b^3*sin(c+d*x)/(d*(b*cos(c+d*x))^(5/3))+3/2*b^2*B*hypergeom([-1/3,1/2],[2/3],cos(c+d*x)^2)*sin(c+d*x)/(d*(b*cos(c+d*x))^(2/3)*sqrt(sin(c+d*x)^2))-3/5*b*(2*A+5*C)*(b*cos(c+d*x))^(1/3)*hypergeom([1/6,1/2],[7/6],cos(c+d*x)^2)*sin(c+d*x)/(d*sqrt(sin(c+d*x)^2))],

# n<0
[cos(c+d*x)^2*(A+B*cos(c+d*x)+C*cos(c+d*x)^2)/(b*cos(c+d*x))^(1/3),x,5,3/11*C*(b*cos(c+d*x))^(8/3)*sin(c+d*x)/(b^3*d)-3/88*(11*A+8*C)*(b*cos(c+d*x))^(8/3)*hypergeom([1/2,4/3],[7/3],cos(c+d*x)^2)*sin(c+d*x)/(b^3*d*sqrt(sin(c+d*x)^2))-3/11*B*(b*cos(c+d*x))^(11/3)*hypergeom([1/2,11/6],[17/6],cos(c+d*x)^2)*sin(c+d*x)/(b^4*d*sqrt(sin(c+d*x)^2))],
[cos(c+d*x)*(A+B*cos(c+d*x)+C*cos(c+d*x)^2)/(b*cos(c+d*x))^(1/3),x,5,3/8*C*(b*cos(c+d*x))^(5/3)*sin(c+d*x)/(b^2*d)-3/40*(8*A+5*C)*(b*cos(c+d*x))^(5/3)*hypergeom([1/2,5/6],[11/6],cos(c+d*x)^2)*sin(c+d*x)/(b^2*d*sqrt(sin(c+d*x)^2))-3/8*B*(b*cos(c+d*x))^(8/3)*hypergeom([1/2,4/3],[7/3],cos(c+d*x)^2)*sin(c+d*x)/(b^3*d*sqrt(sin(c+d*x)^2))],
[(A+B*cos(c+d*x)+C*cos(c+d*x)^2)/(b*cos(c+d*x))^(1/3),x,4,3/5*C*(b*cos(c+d*x))^(2/3)*sin(c+d*x)/(b*d)-3/10*(5*A+2*C)*(b*cos(c+d*x))^(2/3)*hypergeom([1/3,1/2],[4/3],cos(c+d*x)^2)*sin(c+d*x)/(b*d*sqrt(sin(c+d*x)^2))-3/5*B*(b*cos(c+d*x))^(5/3)*hypergeom([1/2,5/6],[11/6],cos(c+d*x)^2)*sin(c+d*x)/(b^2*d*sqrt(sin(c+d*x)^2))],
[(A+B*cos(c+d*x)+C*cos(c+d*x)^2)*sec(c+d*x)/(b*cos(c+d*x))^(1/3),x,5,3*A*sin(c+d*x)/(d*(b*cos(c+d*x))^(1/3))-3/2*B*(b*cos(c+d*x))^(2/3)*hypergeom([1/3,1/2],[4/3],cos(c+d*x)^2)*sin(c+d*x)/(b*d*sqrt(sin(c+d*x)^2))+3/5*(2*A-C)*(b*cos(c+d*x))^(5/3)*hypergeom([1/2,5/6],[11/6],cos(c+d*x)^2)*sin(c+d*x)/(b^2*d*sqrt(sin(c+d*x)^2))],
[(A+B*cos(c+d*x)+C*cos(c+d*x)^2)*sec(c+d*x)^2/(b*cos(c+d*x))^(1/3),x,5,3/4*A*b*sin(c+d*x)/(d*(b*cos(c+d*x))^(4/3))+3*B*hypergeom([-1/6,1/2],[5/6],cos(c+d*x)^2)*sin(c+d*x)/(d*(b*cos(c+d*x))^(1/3)*sqrt(sin(c+d*x)^2))-3/8*(A+4*C)*(b*cos(c+d*x))^(2/3)*hypergeom([1/3,1/2],[4/3],cos(c+d*x)^2)*sin(c+d*x)/(b*d*sqrt(sin(c+d*x)^2))],
[(A+B*cos(c+d*x)+C*cos(c+d*x)^2)*sec(c+d*x)^3/(b*cos(c+d*x))^(1/3),x,5,3/7*A*b^2*sin(c+d*x)/(d*(b*cos(c+d*x))^(7/3))+3/4*b*B*hypergeom([-2/3,1/2],[1/3],cos(c+d*x)^2)*sin(c+d*x)/(d*(b*cos(c+d*x))^(4/3)*sqrt(sin(c+d*x)^2))+3/7*(4*A+7*C)*hypergeom([-1/6,1/2],[5/6],cos(c+d*x)^2)*sin(c+d*x)/(d*(b*cos(c+d*x))^(1/3)*sqrt(sin(c+d*x)^2))],
[cos(c+d*x)^3*(A+B*cos(c+d*x)+C*cos(c+d*x)^2)/(b*cos(c+d*x))^(4/3),x,5,3/11*C*(b*cos(c+d*x))^(8/3)*sin(c+d*x)/(b^4*d)-3/88*(11*A+8*C)*(b*cos(c+d*x))^(8/3)*hypergeom([1/2,4/3],[7/3],cos(c+d*x)^2)*sin(c+d*x)/(b^4*d*sqrt(sin(c+d*x)^2))-3/11*B*(b*cos(c+d*x))^(11/3)*hypergeom([1/2,11/6],[17/6],cos(c+d*x)^2)*sin(c+d*x)/(b^5*d*sqrt(sin(c+d*x)^2))],
[cos(c+d*x)^2*(A+B*cos(c+d*x)+C*cos(c+d*x)^2)/(b*cos(c+d*x))^(4/3),x,5,3/8*C*(b*cos(c+d*x))^(5/3)*sin(c+d*x)/(b^3*d)-3/40*(8*A+5*C)*(b*cos(c+d*x))^(5/3)*hypergeom([1/2,5/6],[11/6],cos(c+d*x)^2)*sin(c+d*x)/(b^3*d*sqrt(sin(c+d*x)^2))-3/8*B*(b*cos(c+d*x))^(8/3)*hypergeom([1/2,4/3],[7/3],cos(c+d*x)^2)*sin(c+d*x)/(b^4*d*sqrt(sin(c+d*x)^2))],
[cos(c+d*x)*(A+B*cos(c+d*x)+C*cos(c+d*x)^2)/(b*cos(c+d*x))^(4/3),x,5,3/5*C*(b*cos(c+d*x))^(2/3)*sin(c+d*x)/(b^2*d)-3/10*(5*A+2*C)*(b*cos(c+d*x))^(2/3)*hypergeom([1/3,1/2],[4/3],cos(c+d*x)^2)*sin(c+d*x)/(b^2*d*sqrt(sin(c+d*x)^2))-3/5*B*(b*cos(c+d*x))^(5/3)*hypergeom([1/2,5/6],[11/6],cos(c+d*x)^2)*sin(c+d*x)/(b^3*d*sqrt(sin(c+d*x)^2))],
[(A+B*cos(c+d*x)+C*cos(c+d*x)^2)/(b*cos(c+d*x))^(4/3),x,4,3*A*sin(c+d*x)/(b*d*(b*cos(c+d*x))^(1/3))-3/2*B*(b*cos(c+d*x))^(2/3)*hypergeom([1/3,1/2],[4/3],cos(c+d*x)^2)*sin(c+d*x)/(b^2*d*sqrt(sin(c+d*x)^2))+3/5*(2*A-C)*(b*cos(c+d*x))^(5/3)*hypergeom([1/2,5/6],[11/6],cos(c+d*x)^2)*sin(c+d*x)/(b^3*d*sqrt(sin(c+d*x)^2))],
[(A+B*cos(c+d*x)+C*cos(c+d*x)^2)*sec(c+d*x)/(b*cos(c+d*x))^(4/3),x,5,3/4*A*sin(c+d*x)/(d*(b*cos(c+d*x))^(4/3))+3*B*hypergeom([-1/6,1/2],[5/6],cos(c+d*x)^2)*sin(c+d*x)/(b*d*(b*cos(c+d*x))^(1/3)*sqrt(sin(c+d*x)^2))-3/8*(A+4*C)*(b*cos(c+d*x))^(2/3)*hypergeom([1/3,1/2],[4/3],cos(c+d*x)^2)*sin(c+d*x)/(b^2*d*sqrt(sin(c+d*x)^2))],
[(A+B*cos(c+d*x)+C*cos(c+d*x)^2)*sec(c+d*x)^2/(b*cos(c+d*x))^(4/3),x,5,3/7*A*b*sin(c+d*x)/(d*(b*cos(c+d*x))^(7/3))+3/4*B*hypergeom([-2/3,1/2],[1/3],cos(c+d*x)^2)*sin(c+d*x)/(d*(b*cos(c+d*x))^(4/3)*sqrt(sin(c+d*x)^2))+3/7*(4*A+7*C)*hypergeom([-1/6,1/2],[5/6],cos(c+d*x)^2)*sin(c+d*x)/(b*d*(b*cos(c+d*x))^(1/3)*sqrt(sin(c+d*x)^2))],

# Integrands of the form (a Cos[e+f x])^m (b Cos[e+f x])^n (A+B Cos[e+f x]+C Cos[e+f x]^2) with m symbolic
[cos(c+d*x)^m*(b*cos(c+d*x))^(4/3)*(A+B*cos(c+d*x)+C*cos(c+d*x)^2),x,5,3*b*C*cos(c+d*x)^(2+m)*(b*cos(c+d*x))^(1/3)*sin(c+d*x)/(d*(10+3*m))-3*b*(C*(7+3*m)+A*(10+3*m))*cos(c+d*x)^(2+m)*(b*cos(c+d*x))^(1/3)*hypergeom([1/2,1/6*(7+3*m)],[1/6*(13+3*m)],cos(c+d*x)^2)*sin(c+d*x)/(d*(70+51*m+9*m^2)*sqrt(sin(c+d*x)^2))-3*b*B*cos(c+d*x)^(3+m)*(b*cos(c+d*x))^(1/3)*hypergeom([1/2,1/6*(10+3*m)],[1/6*(16+3*m)],cos(c+d*x)^2)*sin(c+d*x)/(d*(10+3*m)*sqrt(sin(c+d*x)^2))],
[cos(c+d*x)^m*(b*cos(c+d*x))^(2/3)*(A+B*cos(c+d*x)+C*cos(c+d*x)^2),x,5,3*C*cos(c+d*x)^(1+m)*(b*cos(c+d*x))^(2/3)*sin(c+d*x)/(d*(8+3*m))-3*(C*(5+3*m)+A*(8+3*m))*cos(c+d*x)^(1+m)*(b*cos(c+d*x))^(2/3)*hypergeom([1/2,1/6*(5+3*m)],[1/6*(11+3*m)],cos(c+d*x)^2)*sin(c+d*x)/(d*(40+39*m+9*m^2)*sqrt(sin(c+d*x)^2))-3*B*cos(c+d*x)^(2+m)*(b*cos(c+d*x))^(2/3)*hypergeom([1/2,1/6*(8+3*m)],[1/6*(14+3*m)],cos(c+d*x)^2)*sin(c+d*x)/(d*(8+3*m)*sqrt(sin(c+d*x)^2))],
[cos(c+d*x)^m*(b*cos(c+d*x))^(1/3)*(A+B*cos(c+d*x)+C*cos(c+d*x)^2),x,5,3*C*cos(c+d*x)^(1+m)*(b*cos(c+d*x))^(1/3)*sin(c+d*x)/(d*(7+3*m))-3*(C*(4+3*m)+A*(7+3*m))*cos(c+d*x)^(1+m)*(b*cos(c+d*x))^(1/3)*hypergeom([1/2,1/6*(4+3*m)],[1/6*(10+3*m)],cos(c+d*x)^2)*sin(c+d*x)/(d*(28+33*m+9*m^2)*sqrt(sin(c+d*x)^2))-3*B*cos(c+d*x)^(2+m)*(b*cos(c+d*x))^(1/3)*hypergeom([1/2,1/6*(7+3*m)],[1/6*(13+3*m)],cos(c+d*x)^2)*sin(c+d*x)/(d*(7+3*m)*sqrt(sin(c+d*x)^2))],
[cos(c+d*x)^m*(A+B*cos(c+d*x)+C*cos(c+d*x)^2)/(b*cos(c+d*x))^(1/3),x,5,3*C*cos(c+d*x)^(1+m)*sin(c+d*x)/(d*(5+3*m)*(b*cos(c+d*x))^(1/3))-3*(C*(2+3*m)+A*(5+3*m))*cos(c+d*x)^(1+m)*hypergeom([1/2,1/6*(2+3*m)],[1/6*(8+3*m)],cos(c+d*x)^2)*sin(c+d*x)/(d*(10+21*m+9*m^2)*(b*cos(c+d*x))^(1/3)*sqrt(sin(c+d*x)^2))-3*B*cos(c+d*x)^(2+m)*hypergeom([1/2,1/6*(5+3*m)],[1/6*(11+3*m)],cos(c+d*x)^2)*sin(c+d*x)/(d*(5+3*m)*(b*cos(c+d*x))^(1/3)*sqrt(sin(c+d*x)^2))],
[cos(c+d*x)^m*(A+B*cos(c+d*x)+C*cos(c+d*x)^2)/(b*cos(c+d*x))^(2/3),x,5,3*C*cos(c+d*x)^(1+m)*sin(c+d*x)/(d*(4+3*m)*(b*cos(c+d*x))^(2/3))-3*(C+3*C*m+A*(4+3*m))*cos(c+d*x)^(1+m)*hypergeom([1/2,1/6*(1+3*m)],[1/6*(7+3*m)],cos(c+d*x)^2)*sin(c+d*x)/(d*(4+15*m+9*m^2)*(b*cos(c+d*x))^(2/3)*sqrt(sin(c+d*x)^2))-3*B*cos(c+d*x)^(2+m)*hypergeom([1/2,1/6*(4+3*m)],[1/6*(10+3*m)],cos(c+d*x)^2)*sin(c+d*x)/(d*(4+3*m)*(b*cos(c+d*x))^(2/3)*sqrt(sin(c+d*x)^2))],
[cos(c+d*x)^m*(A+B*cos(c+d*x)+C*cos(c+d*x)^2)/(b*cos(c+d*x))^(4/3),x,5,3*C*cos(c+d*x)^m*sin(c+d*x)/(b*d*(2+3*m)*(b*cos(c+d*x))^(1/3))-3*(C*(1-3*m)-A*(2+3*m))*cos(c+d*x)^m*hypergeom([1/2,1/6*(-1+3*m)],[1/6*(5+3*m)],cos(c+d*x)^2)*sin(c+d*x)/(b*d*(2-3*m-9*m^2)*(b*cos(c+d*x))^(1/3)*sqrt(sin(c+d*x)^2))-3*B*cos(c+d*x)^(1+m)*hypergeom([1/2,1/6*(2+3*m)],[1/6*(8+3*m)],cos(c+d*x)^2)*sin(c+d*x)/(b*d*(2+3*m)*(b*cos(c+d*x))^(1/3)*sqrt(sin(c+d*x)^2))],

# Integrands of the form (a Cos[e+f x])^m (b Cos[e+f x])^n (A+B Cos[e+f x]+C Cos[e+f x]^2) with n symbolic
[(a*cos(c+d*x))^m*(b*cos(c+d*x))^n*(A+B*cos(c+d*x)+C*cos(c+d*x)^2),x,5,C*(a*cos(c+d*x))^(1+m)*(b*cos(c+d*x))^n*sin(c+d*x)/(a*d*(2+m+n))-(C*(1+m+n)+A*(2+m+n))*(a*cos(c+d*x))^(1+m)*(b*cos(c+d*x))^n*hypergeom([1/2,1/2*(1+m+n)],[1/2*(3+m+n)],cos(c+d*x)^2)*sin(c+d*x)/(a*d*(1+m+n)*(2+m+n)*sqrt(sin(c+d*x)^2))-B*(a*cos(c+d*x))^(2+m)*(b*cos(c+d*x))^n*hypergeom([1/2,1/2*(2+m+n)],[1/2*(4+m+n)],cos(c+d*x)^2)*sin(c+d*x)/(a^2*d*(2+m+n)*sqrt(sin(c+d*x)^2))],
[cos(c+d*x)^2*(b*cos(c+d*x))^n*(A+B*cos(c+d*x)+C*cos(c+d*x)^2),x,5,C*(b*cos(c+d*x))^(3+n)*sin(c+d*x)/(b^3*d*(4+n))-(C*(3+n)+A*(4+n))*(b*cos(c+d*x))^(3+n)*hypergeom([1/2,1/2*(3+n)],[1/2*(5+n)],cos(c+d*x)^2)*sin(c+d*x)/(b^3*d*(3+n)*(4+n)*sqrt(sin(c+d*x)^2))-B*(b*cos(c+d*x))^(4+n)*hypergeom([1/2,1/2*(4+n)],[1/2*(6+n)],cos(c+d*x)^2)*sin(c+d*x)/(b^4*d*(4+n)*sqrt(sin(c+d*x)^2))],
[cos(c+d*x)*(b*cos(c+d*x))^n*(A+B*cos(c+d*x)+C*cos(c+d*x)^2),x,5,C*(b*cos(c+d*x))^(2+n)*sin(c+d*x)/(b^2*d*(3+n))-(C*(2+n)+A*(3+n))*(b*cos(c+d*x))^(2+n)*hypergeom([1/2,1/2*(2+n)],[1/2*(4+n)],cos(c+d*x)^2)*sin(c+d*x)/(b^2*d*(2+n)*(3+n)*sqrt(sin(c+d*x)^2))-B*(b*cos(c+d*x))^(3+n)*hypergeom([1/2,1/2*(3+n)],[1/2*(5+n)],cos(c+d*x)^2)*sin(c+d*x)/(b^3*d*(3+n)*sqrt(sin(c+d*x)^2))],
[(b*cos(c+d*x))^n*(A+B*cos(c+d*x)+C*cos(c+d*x)^2),x,4,C*(b*cos(c+d*x))^(1+n)*sin(c+d*x)/(b*d*(2+n))-(C*(1+n)+A*(2+n))*(b*cos(c+d*x))^(1+n)*hypergeom([1/2,1/2*(1+n)],[1/2*(3+n)],cos(c+d*x)^2)*sin(c+d*x)/(b*d*(1+n)*(2+n)*sqrt(sin(c+d*x)^2))-B*(b*cos(c+d*x))^(2+n)*hypergeom([1/2,1/2*(2+n)],[1/2*(4+n)],cos(c+d*x)^2)*sin(c+d*x)/(b^2*d*(2+n)*sqrt(sin(c+d*x)^2))],
[(b*cos(c+d*x))^n*(A+B*cos(c+d*x)+C*cos(c+d*x)^2)*sec(c+d*x),x,5,C*(b*cos(c+d*x))^n*sin(c+d*x)/(d*(1+n))-(A+A*n+C*n)*(b*cos(c+d*x))^n*hypergeom([1/2,1/2*n],[1/2*(2+n)],cos(c+d*x)^2)*sin(c+d*x)/(d*n*(1+n)*sqrt(sin(c+d*x)^2))-B*(b*cos(c+d*x))^(1+n)*hypergeom([1/2,1/2*(1+n)],[1/2*(3+n)],cos(c+d*x)^2)*sin(c+d*x)/(b*d*(1+n)*sqrt(sin(c+d*x)^2))],
[(b*cos(c+d*x))^n*(A+B*cos(c+d*x)+C*cos(c+d*x)^2)*sec(c+d*x)^2,x,5,b*C*(b*cos(c+d*x))^(-1+n)*sin(c+d*x)/(d*n)-b*(C*(1-n)-A*n)*(b*cos(c+d*x))^(-1+n)*hypergeom([1/2,1/2*(-1+n)],[1/2*(1+n)],cos(c+d*x)^2)*sin(c+d*x)/(d*(1-n)*n*sqrt(sin(c+d*x)^2))-B*(b*cos(c+d*x))^n*hypergeom([1/2,1/2*n],[1/2*(2+n)],cos(c+d*x)^2)*sin(c+d*x)/(d*n*sqrt(sin(c+d*x)^2))],
[(b*cos(c+d*x))^n*(A+B*cos(c+d*x)+C*cos(c+d*x)^2)*sec(c+d*x)^3,x,5,-b^2*C*(b*cos(c+d*x))^(-2+n)*sin(c+d*x)/(d*(1-n))+b^2*(A*(1-n)+C*(2-n))*(b*cos(c+d*x))^(-2+n)*hypergeom([1/2,1/2*(-2+n)],[1/2*n],cos(c+d*x)^2)*sin(c+d*x)/(d*(2-3*n+n^2)*sqrt(sin(c+d*x)^2))+b*B*(b*cos(c+d*x))^(-1+n)*hypergeom([1/2,1/2*(-1+n)],[1/2*(1+n)],cos(c+d*x)^2)*sin(c+d*x)/(d*(1-n)*sqrt(sin(c+d*x)^2))],
[(b*cos(c+d*x))^n*(A+B*cos(c+d*x)+C*cos(c+d*x)^2)*sec(c+d*x)^4,x,5,-b^3*C*(b*cos(c+d*x))^(-3+n)*sin(c+d*x)/(d*(2-n))+b^3*(A*(2-n)+C*(3-n))*(b*cos(c+d*x))^(-3+n)*hypergeom([1/2,1/2*(-3+n)],[1/2*(-1+n)],cos(c+d*x)^2)*sin(c+d*x)/(d*(6-5*n+n^2)*sqrt(sin(c+d*x)^2))+b^2*B*(b*cos(c+d*x))^(-2+n)*hypergeom([1/2,1/2*(-2+n)],[1/2*n],cos(c+d*x)^2)*sin(c+d*x)/(d*(2-n)*sqrt(sin(c+d*x)^2))],
[cos(c+d*x)^(3/2)*(b*cos(c+d*x))^n*(A+B*cos(c+d*x)+C*cos(c+d*x)^2),x,5,2*C*cos(c+d*x)^(5/2)*(b*cos(c+d*x))^n*sin(c+d*x)/(d*(7+2*n))-2*(C*(5+2*n)+A*(7+2*n))*cos(c+d*x)^(5/2)*(b*cos(c+d*x))^n*hypergeom([1/2,1/4*(5+2*n)],[1/4*(9+2*n)],cos(c+d*x)^2)*sin(c+d*x)/(d*(35+24*n+4*n^2)*sqrt(sin(c+d*x)^2))-2*B*cos(c+d*x)^(7/2)*(b*cos(c+d*x))^n*hypergeom([1/2,1/4*(7+2*n)],[1/4*(11+2*n)],cos(c+d*x)^2)*sin(c+d*x)/(d*(7+2*n)*sqrt(sin(c+d*x)^2))],
[(b*cos(c+d*x))^n*(A+B*cos(c+d*x)+C*cos(c+d*x)^2)*sqrt(cos(c+d*x)),x,5,2*C*cos(c+d*x)^(3/2)*(b*cos(c+d*x))^n*sin(c+d*x)/(d*(5+2*n))-2*(C*(3+2*n)+A*(5+2*n))*cos(c+d*x)^(3/2)*(b*cos(c+d*x))^n*hypergeom([1/2,1/4*(3+2*n)],[1/4*(7+2*n)],cos(c+d*x)^2)*sin(c+d*x)/(d*(15+16*n+4*n^2)*sqrt(sin(c+d*x)^2))-2*B*cos(c+d*x)^(5/2)*(b*cos(c+d*x))^n*hypergeom([1/2,1/4*(5+2*n)],[1/4*(9+2*n)],cos(c+d*x)^2)*sin(c+d*x)/(d*(5+2*n)*sqrt(sin(c+d*x)^2))],
[(b*cos(c+d*x))^n*(A+B*cos(c+d*x)+C*cos(c+d*x)^2)/sqrt(cos(c+d*x)),x,5,2*C*(b*cos(c+d*x))^n*sin(c+d*x)*sqrt(cos(c+d*x))/(d*(3+2*n))-2*B*cos(c+d*x)^(3/2)*(b*cos(c+d*x))^n*hypergeom([1/2,1/4*(3+2*n)],[1/4*(7+2*n)],cos(c+d*x)^2)*sin(c+d*x)/(d*(3+2*n)*sqrt(sin(c+d*x)^2))-2*(C+2*C*n+A*(3+2*n))*(b*cos(c+d*x))^n*hypergeom([1/2,1/4*(1+2*n)],[1/4*(5+2*n)],cos(c+d*x)^2)*sin(c+d*x)*sqrt(cos(c+d*x))/(d*(3+8*n+4*n^2)*sqrt(sin(c+d*x)^2))],
[(b*cos(c+d*x))^n*(A+B*cos(c+d*x)+C*cos(c+d*x)^2)/cos(c+d*x)^(3/2),x,5,2*C*(b*cos(c+d*x))^n*sin(c+d*x)/(d*(1+2*n)*sqrt(cos(c+d*x)))+2*(A-C*(1-2*n)+2*A*n)*(b*cos(c+d*x))^n*hypergeom([1/2,1/4*(-1+2*n)],[1/4*(3+2*n)],cos(c+d*x)^2)*sin(c+d*x)/(d*(1-4*n^2)*sqrt(cos(c+d*x))*sqrt(sin(c+d*x)^2))-2*B*(b*cos(c+d*x))^n*hypergeom([1/2,1/4*(1+2*n)],[1/4*(5+2*n)],cos(c+d*x)^2)*sin(c+d*x)*sqrt(cos(c+d*x))/(d*(1+2*n)*sqrt(sin(c+d*x)^2))],
[(b*cos(c+d*x))^n*(A+B*cos(c+d*x)+C*cos(c+d*x)^2)/cos(c+d*x)^(5/2),x,5,-2*C*(b*cos(c+d*x))^n*sin(c+d*x)/(d*(1-2*n)*cos(c+d*x)^(3/2))+2*(A+C*(3-2*n)-2*A*n)*(b*cos(c+d*x))^n*hypergeom([1/2,1/4*(-3+2*n)],[1/4*(1+2*n)],cos(c+d*x)^2)*sin(c+d*x)/(d*(3-8*n+4*n^2)*cos(c+d*x)^(3/2)*sqrt(sin(c+d*x)^2))+2*B*(b*cos(c+d*x))^n*hypergeom([1/2,1/4*(-1+2*n)],[1/4*(3+2*n)],cos(c+d*x)^2)*sin(c+d*x)/(d*(1-2*n)*sqrt(cos(c+d*x))*sqrt(sin(c+d*x)^2))],
[(b*cos(c+d*x))^n*(A+B*cos(c+d*x)+C*cos(c+d*x)^2)/cos(c+d*x)^(7/2),x,5,-2*C*(b*cos(c+d*x))^n*sin(c+d*x)/(d*(3-2*n)*cos(c+d*x)^(5/2))+2*(A*(3-2*n)+C*(5-2*n))*(b*cos(c+d*x))^n*hypergeom([1/2,1/4*(-5+2*n)],[1/4*(-1+2*n)],cos(c+d*x)^2)*sin(c+d*x)/(d*(15-16*n+4*n^2)*cos(c+d*x)^(5/2)*sqrt(sin(c+d*x)^2))+2*B*(b*cos(c+d*x))^n*hypergeom([1/2,1/4*(-3+2*n)],[1/4*(1+2*n)],cos(c+d*x)^2)*sin(c+d*x)/(d*(3-2*n)*cos(c+d*x)^(3/2)*sqrt(sin(c+d*x)^2))],

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

# Integrands of the form (a+a Cos[e+f x])^m (A+B Cos[e+f x]+C Cos[e+f x]^2)
[(a+a*cos(e+f*x))^m*(A+B*cos(e+f*x)+C*cos(e+f*x)^2),x,4,-(C-B*(2+m))*(a+a*cos(e+f*x))^m*sin(e+f*x)/(f*(1+m)*(2+m))+C*(a+a*cos(e+f*x))^(1+m)*sin(e+f*x)/(a*f*(2+m))+2^(1/2+m)*(B*m*(2+m)+C*(1+m+m^2)+A*(2+3*m+m^2))*(1+cos(e+f*x))^(-1/2-m)*(a+a*cos(e+f*x))^m*hypergeom([1/2,1/2-m],[3/2],1/2*(1-cos(e+f*x)))*sin(e+f*x)/(f*(2+3*m+m^2))],

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

# n>0
[(a+a*cos(c+d*x))^(2/3)*(A+B*cos(c+d*x)+C*cos(c+d*x)^2),x,4,3/40*(8*B-3*C)*(a+a*cos(c+d*x))^(2/3)*sin(c+d*x)/d+3/8*C*(a+a*cos(c+d*x))^(5/3)*sin(c+d*x)/(a*d)+1/10*(40*A+16*B+19*C)*(a+a*cos(c+d*x))^(2/3)*hypergeom([-1/6,1/2],[3/2],1/2*(1-cos(c+d*x)))*sin(c+d*x)/(2^(5/6)*d*(1+cos(c+d*x))^(7/6))],
[(a+a*cos(c+d*x))^(1/3)*(A+B*cos(c+d*x)+C*cos(c+d*x)^2),x,4,3/28*(7*B-3*C)*(a+a*cos(c+d*x))^(1/3)*sin(c+d*x)/d+3/7*C*(a+a*cos(c+d*x))^(4/3)*sin(c+d*x)/(a*d)+1/14*(28*A+7*B+13*C)*(a+a*cos(c+d*x))^(1/3)*hypergeom([1/6,1/2],[3/2],1/2*(1-cos(c+d*x)))*sin(c+d*x)/(2^(1/6)*d*(1+cos(c+d*x))^(5/6))],

# n<0
[(A+B*cos(c+d*x)+C*cos(c+d*x)^2)/(a+a*cos(c+d*x))^(1/3),x,4,3/10*(5*B-3*C)*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(1/3))+3/5*C*(a+a*cos(c+d*x))^(2/3)*sin(c+d*x)/(a*d)+1/5*(10*A-5*B+7*C)*hypergeom([1/2,5/6],[3/2],1/2*(1-cos(c+d*x)))*sin(c+d*x)/(2^(5/6)*d*(1+cos(c+d*x))^(1/6)*(a+a*cos(c+d*x))^(1/3))],
[(A+B*cos(c+d*x)+C*cos(c+d*x)^2)/(a+a*cos(c+d*x))^(2/3),x,4,3*(A-B+C)*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(2/3))+3/4*C*(a+a*cos(c+d*x))^(1/3)*sin(c+d*x)/(a*d)-1/2*(4*A-8*B+7*C)*(a+a*cos(c+d*x))^(1/3)*hypergeom([1/6,1/2],[3/2],1/2*(1-cos(c+d*x)))*sin(c+d*x)/(2^(1/6)*a*d*(1+cos(c+d*x))^(5/6))],

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

# Integrands of the form (a+b Cos[e+f x])^(n/3) (A+B Cos[e+f x]+C Cos[e+f x]^2)

# n>0
[(a+b*cos(c+d*x))^(2/3)*(A+B*cos(c+d*x)+C*cos(c+d*x)^2),x,8,3/8*C*(a+b*cos(c+d*x))^(5/3)*sin(c+d*x)/(b*d)+1/4*(a+b)*(8*b*B-3*a*C)*AppellF1(1/2,1/2,-5/3,3/2,1/2*(1-cos(c+d*x)),b*(1-cos(c+d*x))/(a+b))*(a+b*cos(c+d*x))^(2/3)*sin(c+d*x)/(b^2*d*((a+b*cos(c+d*x))/(a+b))^(2/3)*sqrt(2)*sqrt(1+cos(c+d*x)))+1/4*(8*A*b^2-8*a*b*B+3*a^2*C+5*b^2*C)*AppellF1(1/2,1/2,-2/3,3/2,1/2*(1-cos(c+d*x)),b*(1-cos(c+d*x))/(a+b))*(a+b*cos(c+d*x))^(2/3)*sin(c+d*x)/(b^2*d*((a+b*cos(c+d*x))/(a+b))^(2/3)*sqrt(2)*sqrt(1+cos(c+d*x)))],
[(a+b*cos(c+d*x))^(1/3)*(A+B*cos(c+d*x)+C*cos(c+d*x)^2),x,8,3/7*C*(a+b*cos(c+d*x))^(4/3)*sin(c+d*x)/(b*d)+1/7*(a+b)*(7*b*B-3*a*C)*AppellF1(1/2,1/2,-4/3,3/2,1/2*(1-cos(c+d*x)),b*(1-cos(c+d*x))/(a+b))*(a+b*cos(c+d*x))^(1/3)*sin(c+d*x)*sqrt(2)/(b^2*d*((a+b*cos(c+d*x))/(a+b))^(1/3)*sqrt(1+cos(c+d*x)))+1/7*(7*A*b^2-7*a*b*B+3*a^2*C+4*b^2*C)*AppellF1(1/2,1/2,-1/3,3/2,1/2*(1-cos(c+d*x)),b*(1-cos(c+d*x))/(a+b))*(a+b*cos(c+d*x))^(1/3)*sin(c+d*x)*sqrt(2)/(b^2*d*((a+b*cos(c+d*x))/(a+b))^(1/3)*sqrt(1+cos(c+d*x)))],

# n<0
[(A+B*cos(c+d*x)+C*cos(c+d*x)^2)/(a+b*cos(c+d*x))^(1/3),x,8,3/5*C*(a+b*cos(c+d*x))^(2/3)*sin(c+d*x)/(b*d)+1/5*(5*b*B-3*a*C)*AppellF1(1/2,1/2,-2/3,3/2,1/2*(1-cos(c+d*x)),b*(1-cos(c+d*x))/(a+b))*(a+b*cos(c+d*x))^(2/3)*sin(c+d*x)*sqrt(2)/(b^2*d*((a+b*cos(c+d*x))/(a+b))^(2/3)*sqrt(1+cos(c+d*x)))+1/5*(5*A*b^2-5*a*b*B+3*a^2*C+2*b^2*C)*AppellF1(1/2,1/2,1/3,3/2,1/2*(1-cos(c+d*x)),b*(1-cos(c+d*x))/(a+b))*((a+b*cos(c+d*x))/(a+b))^(1/3)*sin(c+d*x)*sqrt(2)/(b^2*d*(a+b*cos(c+d*x))^(1/3)*sqrt(1+cos(c+d*x)))],
[(A+B*cos(c+d*x)+C*cos(c+d*x)^2)/(a+b*cos(c+d*x))^(2/3),x,8,3/4*C*(a+b*cos(c+d*x))^(1/3)*sin(c+d*x)/(b*d)+1/2*(4*b*B-3*a*C)*AppellF1(1/2,1/2,-1/3,3/2,1/2*(1-cos(c+d*x)),b*(1-cos(c+d*x))/(a+b))*(a+b*cos(c+d*x))^(1/3)*sin(c+d*x)/(b^2*d*((a+b*cos(c+d*x))/(a+b))^(1/3)*sqrt(2)*sqrt(1+cos(c+d*x)))+1/2*(4*A*b^2-4*a*b*B+3*a^2*C+b^2*C)*AppellF1(1/2,1/2,2/3,3/2,1/2*(1-cos(c+d*x)),b*(1-cos(c+d*x))/(a+b))*((a+b*cos(c+d*x))/(a+b))^(2/3)*sin(c+d*x)/(b^2*d*(a+b*cos(c+d*x))^(2/3)*sqrt(2)*sqrt(1+cos(c+d*x)))],

# Integrands of the form (a+b Cos[e+f x])^m (A+B Cos[e+f x]+C Cos[e+f x]^2) with m symbolic
[(a+b*cos(e+f*x))^m*(A+(A+C)*cos(e+f*x)+C*cos(e+f*x)^2),x,7,4*C*AppellF1(1/2,-3/2,-m,3/2,1/2*(1-cos(e+f*x)),b*(1-cos(e+f*x))/(a+b))*(a+b*cos(e+f*x))^m*sin(e+f*x)*sqrt(2)/(f*((a+b*cos(e+f*x))/(a+b))^m*sqrt(1+cos(e+f*x)))+2*(A-C)*AppellF1(1/2,-1/2,-m,3/2,1/2*(1-cos(e+f*x)),b*(1-cos(e+f*x))/(a+b))*(a+b*cos(e+f*x))^m*sin(e+f*x)*sqrt(2)/(f*((a+b*cos(e+f*x))/(a+b))^m*sqrt(1+cos(e+f*x)))],
[(a+b*cos(e+f*x))^m*(A+B*cos(e+f*x)+C*cos(e+f*x)^2),x,8,C*(a+b*cos(e+f*x))^(1+m)*sin(e+f*x)/(b*f*(2+m))-(a+b)*(a*C-b*B*(2+m))*AppellF1(1/2,1/2,-1-m,3/2,1/2*(1-cos(e+f*x)),b*(1-cos(e+f*x))/(a+b))*(a+b*cos(e+f*x))^m*sin(e+f*x)*sqrt(2)/(b^2*f*(2+m)*((a+b*cos(e+f*x))/(a+b))^m*sqrt(1+cos(e+f*x)))+(a^2*C+b^2*C*(1+m)+A*b^2*(2+m)-a*b*B*(2+m))*AppellF1(1/2,1/2,-m,3/2,1/2*(1-cos(e+f*x)),b*(1-cos(e+f*x))/(a+b))*(a+b*cos(e+f*x))^m*sin(e+f*x)*sqrt(2)/(b^2*f*(2+m)*((a+b*cos(e+f*x))/(a+b))^m*sqrt(1+cos(e+f*x)))]]:
