# Maple integration test file: "4 Trig functions\4.2 Cosine\4.2.3.1 (a+b cos)^m (c+d cos)^n (A+B cos).txt"

lst:=[

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

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

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

# n>0
[cos(c+d*x)^3*(a+a*cos(c+d*x))*(A+B*cos(c+d*x)),x,8,3/8*a*(A+B)*x+1/5*a*(5*A+4*B)*sin(c+d*x)/d+3/8*a*(A+B)*cos(c+d*x)*sin(c+d*x)/d+1/4*a*(A+B)*cos(c+d*x)^3*sin(c+d*x)/d+1/5*a*B*cos(c+d*x)^4*sin(c+d*x)/d-1/15*a*(5*A+4*B)*sin(c+d*x)^3/d],
[cos(c+d*x)^2*(a+a*cos(c+d*x))*(A+B*cos(c+d*x)),x,7,1/8*a*(4*A+3*B)*x+a*(A+B)*sin(c+d*x)/d+1/8*a*(4*A+3*B)*cos(c+d*x)*sin(c+d*x)/d+1/4*a*B*cos(c+d*x)^3*sin(c+d*x)/d-1/3*a*(A+B)*sin(c+d*x)^3/d],
[cos(c+d*x)*(a+a*cos(c+d*x))*(A+B*cos(c+d*x)),x,3,1/2*a*(A+B)*x+1/3*a*(3*A+2*B)*sin(c+d*x)/d+1/2*a*(A+B)*cos(c+d*x)*sin(c+d*x)/d+1/3*a*B*cos(c+d*x)^2*sin(c+d*x)/d],
[(a+a*cos(c+d*x))*(A+B*cos(c+d*x)),x,1,1/2*a*(2*A+B)*x+a*(A+B)*sin(c+d*x)/d+1/2*a*B*cos(c+d*x)*sin(c+d*x)/d],
[(a+a*cos(c+d*x))*(A+B*cos(c+d*x))*sec(c+d*x),x,4,a*(A+B)*x+a*A*arctanh(sin(c+d*x))/d+a*B*sin(c+d*x)/d],
[(a+a*cos(c+d*x))*(A+B*cos(c+d*x))*sec(c+d*x)^2,x,4,a*B*x+a*(A+B)*arctanh(sin(c+d*x))/d+a*A*tan(c+d*x)/d],
[(a+a*cos(c+d*x))*(A+B*cos(c+d*x))*sec(c+d*x)^3,x,6,1/2*a*(A+2*B)*arctanh(sin(c+d*x))/d+a*(A+B)*tan(c+d*x)/d+1/2*a*A*sec(c+d*x)*tan(c+d*x)/d],
[(a+a*cos(c+d*x))*(A+B*cos(c+d*x))*sec(c+d*x)^4,x,7,1/2*a*(A+B)*arctanh(sin(c+d*x))/d+1/3*a*(2*A+3*B)*tan(c+d*x)/d+1/2*a*(A+B)*sec(c+d*x)*tan(c+d*x)/d+1/3*a*A*sec(c+d*x)^2*tan(c+d*x)/d],
[(a+a*cos(c+d*x))*(A+B*cos(c+d*x))*sec(c+d*x)^5,x,7,1/8*a*(3*A+4*B)*arctanh(sin(c+d*x))/d+a*(A+B)*tan(c+d*x)/d+1/8*a*(3*A+4*B)*sec(c+d*x)*tan(c+d*x)/d+1/4*a*A*sec(c+d*x)^3*tan(c+d*x)/d+1/3*a*(A+B)*tan(c+d*x)^3/d],
[cos(c+d*x)^3*(a+a*cos(c+d*x))^2*(A+B*cos(c+d*x)),x,9,1/16*a^2*(12*A+11*B)*x+1/5*a^2*(9*A+8*B)*sin(c+d*x)/d+1/16*a^2*(12*A+11*B)*cos(c+d*x)*sin(c+d*x)/d+1/24*a^2*(12*A+11*B)*cos(c+d*x)^3*sin(c+d*x)/d+1/30*a^2*(6*A+7*B)*cos(c+d*x)^4*sin(c+d*x)/d+1/6*B*cos(c+d*x)^4*(a^2+a^2*cos(c+d*x))*sin(c+d*x)/d-1/15*a^2*(9*A+8*B)*sin(c+d*x)^3/d],
[cos(c+d*x)^2*(a+a*cos(c+d*x))^2*(A+B*cos(c+d*x)),x,8,1/8*a^2*(7*A+6*B)*x+1/5*a^2*(10*A+9*B)*sin(c+d*x)/d+1/8*a^2*(7*A+6*B)*cos(c+d*x)*sin(c+d*x)/d+1/20*a^2*(5*A+6*B)*cos(c+d*x)^3*sin(c+d*x)/d+1/5*B*cos(c+d*x)^3*(a^2+a^2*cos(c+d*x))*sin(c+d*x)/d-1/15*a^2*(10*A+9*B)*sin(c+d*x)^3/d],
[cos(c+d*x)*(a+a*cos(c+d*x))^2*(A+B*cos(c+d*x)),x,4,1/8*a^2*(8*A+7*B)*x+1/6*a^2*(8*A+7*B)*sin(c+d*x)/d+1/24*a^2*(8*A+7*B)*cos(c+d*x)*sin(c+d*x)/d+1/12*(4*A-B)*(a+a*cos(c+d*x))^2*sin(c+d*x)/d+1/4*B*(a+a*cos(c+d*x))^3*sin(c+d*x)/(a*d)],
[(a+a*cos(c+d*x))^2*(A+B*cos(c+d*x)),x,2,1/2*a^2*(3*A+2*B)*x+2/3*a^2*(3*A+2*B)*sin(c+d*x)/d+1/6*a^2*(3*A+2*B)*cos(c+d*x)*sin(c+d*x)/d+1/3*B*(a+a*cos(c+d*x))^2*sin(c+d*x)/d],
[(a+a*cos(c+d*x))^2*(A+B*cos(c+d*x))*sec(c+d*x),x,5,1/2*a^2*(4*A+3*B)*x+a^2*A*arctanh(sin(c+d*x))/d+1/2*a^2*(2*A+3*B)*sin(c+d*x)/d+1/2*B*(a^2+a^2*cos(c+d*x))*sin(c+d*x)/d],
[(a+a*cos(c+d*x))^2*(A+B*cos(c+d*x))*sec(c+d*x)^2,x,5,a^2*(A+2*B)*x+a^2*(2*A+B)*arctanh(sin(c+d*x))/d-a^2*(A-B)*sin(c+d*x)/d+A*(a^2+a^2*cos(c+d*x))*tan(c+d*x)/d],
[(a+a*cos(c+d*x))^2*(A+B*cos(c+d*x))*sec(c+d*x)^3,x,5,a^2*B*x+1/2*a^2*(3*A+4*B)*arctanh(sin(c+d*x))/d+1/2*a^2*(3*A+2*B)*tan(c+d*x)/d+1/2*A*(a^2+a^2*cos(c+d*x))*sec(c+d*x)*tan(c+d*x)/d],
[(a+a*cos(c+d*x))^2*(A+B*cos(c+d*x))*sec(c+d*x)^4,x,7,1/2*a^2*(2*A+3*B)*arctanh(sin(c+d*x))/d+1/3*a^2*(5*A+6*B)*tan(c+d*x)/d+1/6*a^2*(4*A+3*B)*sec(c+d*x)*tan(c+d*x)/d+1/3*A*(a^2+a^2*cos(c+d*x))*sec(c+d*x)^2*tan(c+d*x)/d],
[(a+a*cos(c+d*x))^2*(A+B*cos(c+d*x))*sec(c+d*x)^5,x,8,1/8*a^2*(7*A+8*B)*arctanh(sin(c+d*x))/d+1/3*a^2*(4*A+5*B)*tan(c+d*x)/d+1/8*a^2*(7*A+8*B)*sec(c+d*x)*tan(c+d*x)/d+1/12*a^2*(5*A+4*B)*sec(c+d*x)^2*tan(c+d*x)/d+1/4*A*(a^2+a^2*cos(c+d*x))*sec(c+d*x)^3*tan(c+d*x)/d],
[cos(c+d*x)^2*(a+a*cos(c+d*x))^3*(A+B*cos(c+d*x)),x,9,1/16*a^3*(26*A+23*B)*x+1/5*a^3*(19*A+17*B)*sin(c+d*x)/d+1/16*a^3*(26*A+23*B)*cos(c+d*x)*sin(c+d*x)/d+1/40*a^3*(22*A+21*B)*cos(c+d*x)^3*sin(c+d*x)/d+1/6*a*B*cos(c+d*x)^3*(a+a*cos(c+d*x))^2*sin(c+d*x)/d+1/15*(3*A+4*B)*cos(c+d*x)^3*(a^3+a^3*cos(c+d*x))*sin(c+d*x)/d-1/15*a^3*(19*A+17*B)*sin(c+d*x)^3/d],
[cos(c+d*x)*(a+a*cos(c+d*x))^3*(A+B*cos(c+d*x)),x,10,1/8*a^3*(15*A+13*B)*x+1/5*a^3*(15*A+13*B)*sin(c+d*x)/d+3/40*a^3*(15*A+13*B)*cos(c+d*x)*sin(c+d*x)/d+1/20*(5*A-B)*(a+a*cos(c+d*x))^3*sin(c+d*x)/d+1/5*B*(a+a*cos(c+d*x))^4*sin(c+d*x)/(a*d)-1/60*a^3*(15*A+13*B)*sin(c+d*x)^3/d],
[(a+a*cos(c+d*x))^3*(A+B*cos(c+d*x)),x,8,5/8*a^3*(4*A+3*B)*x+a^3*(4*A+3*B)*sin(c+d*x)/d+3/8*a^3*(4*A+3*B)*cos(c+d*x)*sin(c+d*x)/d+1/4*B*(a+a*cos(c+d*x))^3*sin(c+d*x)/d-1/12*a^3*(4*A+3*B)*sin(c+d*x)^3/d],
[(a+a*cos(c+d*x))^3*(A+B*cos(c+d*x))*sec(c+d*x),x,6,1/2*a^3*(7*A+5*B)*x+a^3*A*arctanh(sin(c+d*x))/d+5/2*a^3*(A+B)*sin(c+d*x)/d+1/3*a*B*(a+a*cos(c+d*x))^2*sin(c+d*x)/d+1/6*(3*A+5*B)*(a^3+a^3*cos(c+d*x))*sin(c+d*x)/d],
[(a+a*cos(c+d*x))^3*(A+B*cos(c+d*x))*sec(c+d*x)^2,x,6,1/2*a^3*(6*A+7*B)*x+a^3*(3*A+B)*arctanh(sin(c+d*x))/d+5/2*a^3*B*sin(c+d*x)/d-1/2*(2*A-B)*(a^3+a^3*cos(c+d*x))*sin(c+d*x)/d+a*A*(a+a*cos(c+d*x))^2*tan(c+d*x)/d],
[(a+a*cos(c+d*x))^3*(A+B*cos(c+d*x))*sec(c+d*x)^3,x,6,a^3*(A+3*B)*x+1/2*a^3*(7*A+6*B)*arctanh(sin(c+d*x))/d-5/2*a^3*A*sin(c+d*x)/d+(2*A+B)*(a^3+a^3*cos(c+d*x))*tan(c+d*x)/d+1/2*a*A*(a+a*cos(c+d*x))^2*sec(c+d*x)*tan(c+d*x)/d],
[(a+a*cos(c+d*x))^3*(A+B*cos(c+d*x))*sec(c+d*x)^4,x,6,a^3*B*x+1/2*a^3*(5*A+7*B)*arctanh(sin(c+d*x))/d+5/2*a^3*(A+B)*tan(c+d*x)/d+1/6*(5*A+3*B)*(a^3+a^3*cos(c+d*x))*sec(c+d*x)*tan(c+d*x)/d+1/3*a*A*(a+a*cos(c+d*x))^2*sec(c+d*x)^2*tan(c+d*x)/d],
[(a+a*cos(c+d*x))^3*(A+B*cos(c+d*x))*sec(c+d*x)^5,x,8,5/8*a^3*(3*A+4*B)*arctanh(sin(c+d*x))/d+1/3*a^3*(9*A+11*B)*tan(c+d*x)/d+1/24*a^3*(27*A+28*B)*sec(c+d*x)*tan(c+d*x)/d+1/6*(3*A+2*B)*(a^3+a^3*cos(c+d*x))*sec(c+d*x)^2*tan(c+d*x)/d+1/4*a*A*(a+a*cos(c+d*x))^2*sec(c+d*x)^3*tan(c+d*x)/d],
[(a+a*cos(c+d*x))^3*(A+B*cos(c+d*x))*sec(c+d*x)^6,x,9,1/8*a^3*(13*A+15*B)*arctanh(sin(c+d*x))/d+1/15*a^3*(38*A+45*B)*tan(c+d*x)/d+1/8*a^3*(13*A+15*B)*sec(c+d*x)*tan(c+d*x)/d+1/60*a^3*(43*A+45*B)*sec(c+d*x)^2*tan(c+d*x)/d+1/20*(7*A+5*B)*(a^3+a^3*cos(c+d*x))*sec(c+d*x)^3*tan(c+d*x)/d+1/5*a*A*(a+a*cos(c+d*x))^2*sec(c+d*x)^4*tan(c+d*x)/d],
[cos(c+d*x)^2*(a+a*cos(c+d*x))^4*(A+B*cos(c+d*x)),x,10,1/16*a^4*(49*A+44*B)*x+1/35*a^4*(252*A+227*B)*sin(c+d*x)/d+1/16*a^4*(49*A+44*B)*cos(c+d*x)*sin(c+d*x)/d+1/280*a^4*(301*A+276*B)*cos(c+d*x)^3*sin(c+d*x)/d+1/7*a*B*cos(c+d*x)^3*(a+a*cos(c+d*x))^3*sin(c+d*x)/d+1/42*(7*A+10*B)*cos(c+d*x)^3*(a^2+a^2*cos(c+d*x))^2*sin(c+d*x)/d+7/15*(A+B)*cos(c+d*x)^3*(a^4+a^4*cos(c+d*x))*sin(c+d*x)/d-1/105*a^4*(252*A+227*B)*sin(c+d*x)^3/d],
[cos(c+d*x)*(a+a*cos(c+d*x))^4*(A+B*cos(c+d*x)),x,13,7/16*a^4*(8*A+7*B)*x+4/5*a^4*(8*A+7*B)*sin(c+d*x)/d+27/80*a^4*(8*A+7*B)*cos(c+d*x)*sin(c+d*x)/d+1/40*a^4*(8*A+7*B)*cos(c+d*x)^3*sin(c+d*x)/d+1/30*(6*A-B)*(a+a*cos(c+d*x))^4*sin(c+d*x)/d+1/6*B*(a+a*cos(c+d*x))^5*sin(c+d*x)/(a*d)-2/15*a^4*(8*A+7*B)*sin(c+d*x)^3/d],
[(a+a*cos(c+d*x))^4*(A+B*cos(c+d*x)),x,11,7/8*a^4*(5*A+4*B)*x+8/5*a^4*(5*A+4*B)*sin(c+d*x)/d+27/40*a^4*(5*A+4*B)*cos(c+d*x)*sin(c+d*x)/d+1/20*a^4*(5*A+4*B)*cos(c+d*x)^3*sin(c+d*x)/d+1/5*B*(a+a*cos(c+d*x))^4*sin(c+d*x)/d-4/15*a^4*(5*A+4*B)*sin(c+d*x)^3/d],
[(a+a*cos(c+d*x))^4*(A+B*cos(c+d*x))*sec(c+d*x),x,7,1/8*a^4*(48*A+35*B)*x+a^4*A*arctanh(sin(c+d*x))/d+5/8*a^4*(8*A+7*B)*sin(c+d*x)/d+1/4*a*B*(a+a*cos(c+d*x))^3*sin(c+d*x)/d+1/12*(4*A+7*B)*(a^2+a^2*cos(c+d*x))^2*sin(c+d*x)/d+1/24*(32*A+35*B)*(a^4+a^4*cos(c+d*x))*sin(c+d*x)/d],
[(a+a*cos(c+d*x))^4*(A+B*cos(c+d*x))*sec(c+d*x)^2,x,7,1/2*a^4*(13*A+12*B)*x+a^4*(4*A+B)*arctanh(sin(c+d*x))/d+5/2*a^4*(A+2*B)*sin(c+d*x)/d-1/3*(3*A-B)*(a^2+a^2*cos(c+d*x))^2*sin(c+d*x)/d-1/6*(3*A-8*B)*(a^4+a^4*cos(c+d*x))*sin(c+d*x)/d+a*A*(a+a*cos(c+d*x))^3*tan(c+d*x)/d],
[(a+a*cos(c+d*x))^4*(A+B*cos(c+d*x))*sec(c+d*x)^3,x,7,1/2*a^4*(8*A+13*B)*x+1/2*a^4*(13*A+8*B)*arctanh(sin(c+d*x))/d-5/2*a^4*(A-B)*sin(c+d*x)/d-1/2*(6*A+B)*(a^4+a^4*cos(c+d*x))*sin(c+d*x)/d+1/2*(5*A+2*B)*(a^2+a^2*cos(c+d*x))^2*tan(c+d*x)/d+1/2*a*A*(a+a*cos(c+d*x))^3*sec(c+d*x)*tan(c+d*x)/d],
[(a+a*cos(c+d*x))^4*(A+B*cos(c+d*x))*sec(c+d*x)^4,x,7,a^4*(A+4*B)*x+1/2*a^4*(12*A+13*B)*arctanh(sin(c+d*x))/d-5/2*a^4*(2*A+B)*sin(c+d*x)/d+1/3*(11*A+9*B)*(a^4+a^4*cos(c+d*x))*tan(c+d*x)/d+1/2*(2*A+B)*(a^2+a^2*cos(c+d*x))^2*sec(c+d*x)*tan(c+d*x)/d+1/3*a*A*(a+a*cos(c+d*x))^3*sec(c+d*x)^2*tan(c+d*x)/d],
[(a+a*cos(c+d*x))^4*(A+B*cos(c+d*x))*sec(c+d*x)^5,x,7,a^4*B*x+1/8*a^4*(35*A+48*B)*arctanh(sin(c+d*x))/d+5/8*a^4*(7*A+8*B)*tan(c+d*x)/d+1/24*(35*A+32*B)*(a^4+a^4*cos(c+d*x))*sec(c+d*x)*tan(c+d*x)/d+1/12*(7*A+4*B)*(a^2+a^2*cos(c+d*x))^2*sec(c+d*x)^2*tan(c+d*x)/d+1/4*a*A*(a+a*cos(c+d*x))^3*sec(c+d*x)^3*tan(c+d*x)/d],
[(a+a*cos(c+d*x))^4*(A+B*cos(c+d*x))*sec(c+d*x)^6,x,9,7/8*a^4*(4*A+5*B)*arctanh(sin(c+d*x))/d+1/15*a^4*(83*A+100*B)*tan(c+d*x)/d+1/120*a^4*(244*A+275*B)*sec(c+d*x)*tan(c+d*x)/d+1/30*(26*A+25*B)*(a^4+a^4*cos(c+d*x))*sec(c+d*x)^2*tan(c+d*x)/d+1/20*(8*A+5*B)*(a^2+a^2*cos(c+d*x))^2*sec(c+d*x)^3*tan(c+d*x)/d+1/5*a*A*(a+a*cos(c+d*x))^3*sec(c+d*x)^4*tan(c+d*x)/d],
[(a+a*cos(c+d*x))^4*(A+B*cos(c+d*x))*sec(c+d*x)^7,x,10,7/16*a^4*(7*A+8*B)*arctanh(sin(c+d*x))/d+1/15*a^4*(72*A+83*B)*tan(c+d*x)/d+7/16*a^4*(7*A+8*B)*sec(c+d*x)*tan(c+d*x)/d+1/120*a^4*(159*A+176*B)*sec(c+d*x)^2*tan(c+d*x)/d+1/120*(73*A+72*B)*(a^4+a^4*cos(c+d*x))*sec(c+d*x)^3*tan(c+d*x)/d+1/10*(3*A+2*B)*(a^2+a^2*cos(c+d*x))^2*sec(c+d*x)^4*tan(c+d*x)/d+1/6*a*A*(a+a*cos(c+d*x))^3*sec(c+d*x)^5*tan(c+d*x)/d],

# n<0
[cos(c+d*x)^4*(A+B*cos(c+d*x))/(a+a*cos(c+d*x)),x,7,-3/8*(4*A-5*B)*x/a+4*(A-B)*sin(c+d*x)/(a*d)-3/8*(4*A-5*B)*cos(c+d*x)*sin(c+d*x)/(a*d)-1/4*(4*A-5*B)*cos(c+d*x)^3*sin(c+d*x)/(a*d)+(A-B)*cos(c+d*x)^4*sin(c+d*x)/(d*(a+a*cos(c+d*x)))-4/3*(A-B)*sin(c+d*x)^3/(a*d)],
[cos(c+d*x)^3*(A+B*cos(c+d*x))/(a+a*cos(c+d*x)),x,6,3/2*(A-B)*x/a-(3*A-4*B)*sin(c+d*x)/(a*d)+3/2*(A-B)*cos(c+d*x)*sin(c+d*x)/(a*d)+(A-B)*cos(c+d*x)^3*sin(c+d*x)/(d*(a+a*cos(c+d*x)))+1/3*(3*A-4*B)*sin(c+d*x)^3/(a*d)],
[cos(c+d*x)^2*(A+B*cos(c+d*x))/(a+a*cos(c+d*x)),x,2,-(A-B)*x/a+1/2*B*x/a+(A-B)*sin(c+d*x)/(a*d)+1/2*B*cos(c+d*x)*sin(c+d*x)/(a*d)+(A-B)*sin(c+d*x)/(a*d*(1+cos(c+d*x))),-1/2*(2*A-3*B)*x/a+2*(A-B)*sin(c+d*x)/(a*d)-1/2*(2*A-3*B)*cos(c+d*x)*sin(c+d*x)/(a*d)+(A-B)*cos(c+d*x)^2*sin(c+d*x)/(d*(a+a*cos(c+d*x)))],
[cos(c+d*x)*(A+B*cos(c+d*x))/(a+a*cos(c+d*x)),x,5,(A-B)*x/a+B*sin(c+d*x)/(a*d)-(A-B)*sin(c+d*x)/(a*d*(1+cos(c+d*x)))],
[(A+B*cos(c+d*x))/(a+a*cos(c+d*x)),x,2,B*x/a+(A-B)*sin(c+d*x)/(d*(a+a*cos(c+d*x)))],
[(A+B*cos(c+d*x))*sec(c+d*x)/(a+a*cos(c+d*x)),x,3,A*arctanh(sin(c+d*x))/(a*d)-(A-B)*sin(c+d*x)/(d*(a+a*cos(c+d*x)))],
[(A+B*cos(c+d*x))*sec(c+d*x)^2/(a+a*cos(c+d*x)),x,5,-(A-B)*arctanh(sin(c+d*x))/(a*d)+(2*A-B)*tan(c+d*x)/(a*d)-(A-B)*tan(c+d*x)/(d*(a+a*cos(c+d*x)))],
[(A+B*cos(c+d*x))*sec(c+d*x)^3/(a+a*cos(c+d*x)),x,6,1/2*(3*A-2*B)*arctanh(sin(c+d*x))/(a*d)-2*(A-B)*tan(c+d*x)/(a*d)+1/2*(3*A-2*B)*sec(c+d*x)*tan(c+d*x)/(a*d)-(A-B)*sec(c+d*x)*tan(c+d*x)/(d*(a+a*cos(c+d*x)))],
[(A+B*cos(c+d*x))*sec(c+d*x)^4/(a+a*cos(c+d*x)),x,6,-3/2*(A-B)*arctanh(sin(c+d*x))/(a*d)+(4*A-3*B)*tan(c+d*x)/(a*d)-3/2*(A-B)*sec(c+d*x)*tan(c+d*x)/(a*d)-(A-B)*sec(c+d*x)^2*tan(c+d*x)/(d*(a+a*cos(c+d*x)))+1/3*(4*A-3*B)*tan(c+d*x)^3/(a*d)],
[cos(c+d*x)^4*(A+B*cos(c+d*x))/(a+a*cos(c+d*x))^2,x,7,1/2*(7*A-10*B)*x/a^2-4*(2*A-3*B)*sin(c+d*x)/(a^2*d)+1/2*(7*A-10*B)*cos(c+d*x)*sin(c+d*x)/(a^2*d)+1/3*(7*A-10*B)*cos(c+d*x)^3*sin(c+d*x)/(a^2*d*(1+cos(c+d*x)))+1/3*(A-B)*cos(c+d*x)^4*sin(c+d*x)/(d*(a+a*cos(c+d*x))^2)+4/3*(2*A-3*B)*sin(c+d*x)^3/(a^2*d)],
[cos(c+d*x)^3*(A+B*cos(c+d*x))/(a+a*cos(c+d*x))^2,x,3,-1/2*(4*A-7*B)*x/a^2+2/3*(5*A-8*B)*sin(c+d*x)/(a^2*d)-1/2*(4*A-7*B)*cos(c+d*x)*sin(c+d*x)/(a^2*d)+1/3*(5*A-8*B)*cos(c+d*x)^2*sin(c+d*x)/(a^2*d*(1+cos(c+d*x)))+1/3*(A-B)*cos(c+d*x)^3*sin(c+d*x)/(d*(a+a*cos(c+d*x))^2)],
[cos(c+d*x)^2*(A+B*cos(c+d*x))/(a+a*cos(c+d*x))^2,x,6,(A-2*B)*x/a^2-1/3*(A-4*B)*sin(c+d*x)/(a^2*d)-(A-2*B)*sin(c+d*x)/(a^2*d*(1+cos(c+d*x)))+1/3*(A-B)*cos(c+d*x)^2*sin(c+d*x)/(d*(a+a*cos(c+d*x))^2)],
[cos(c+d*x)*(A+B*cos(c+d*x))/(a+a*cos(c+d*x))^2,x,4,B*x/a^2+1/3*(2*A-5*B)*sin(c+d*x)/(a^2*d*(1+cos(c+d*x)))-1/3*(A-B)*sin(c+d*x)/(d*(a+a*cos(c+d*x))^2)],
[(A+B*cos(c+d*x))/(a+a*cos(c+d*x))^2,x,2,1/3*(A-B)*sin(c+d*x)/(d*(a+a*cos(c+d*x))^2)+1/3*(A+2*B)*sin(c+d*x)/(d*(a^2+a^2*cos(c+d*x)))],
[(A+B*cos(c+d*x))*sec(c+d*x)/(a+a*cos(c+d*x))^2,x,4,A*arctanh(sin(c+d*x))/(a^2*d)-1/3*(4*A-B)*sin(c+d*x)/(a^2*d*(1+cos(c+d*x)))-1/3*(A-B)*sin(c+d*x)/(d*(a+a*cos(c+d*x))^2)],
[(A+B*cos(c+d*x))*sec(c+d*x)^2/(a+a*cos(c+d*x))^2,x,6,-(2*A-B)*arctanh(sin(c+d*x))/(a^2*d)+2/3*(5*A-2*B)*tan(c+d*x)/(a^2*d)-(2*A-B)*tan(c+d*x)/(a^2*d*(1+cos(c+d*x)))-1/3*(A-B)*tan(c+d*x)/(d*(a+a*cos(c+d*x))^2)],
[(A+B*cos(c+d*x))*sec(c+d*x)^3/(a+a*cos(c+d*x))^2,x,7,1/2*(7*A-4*B)*arctanh(sin(c+d*x))/(a^2*d)-2/3*(8*A-5*B)*tan(c+d*x)/(a^2*d)+1/2*(7*A-4*B)*sec(c+d*x)*tan(c+d*x)/(a^2*d)-1/3*(8*A-5*B)*sec(c+d*x)*tan(c+d*x)/(a^2*d*(1+cos(c+d*x)))-1/3*(A-B)*sec(c+d*x)*tan(c+d*x)/(d*(a+a*cos(c+d*x))^2)],
[(A+B*cos(c+d*x))*sec(c+d*x)^4/(a+a*cos(c+d*x))^2,x,7,-1/2*(10*A-7*B)*arctanh(sin(c+d*x))/(a^2*d)+4*(3*A-2*B)*tan(c+d*x)/(a^2*d)-1/2*(10*A-7*B)*sec(c+d*x)*tan(c+d*x)/(a^2*d)-1/3*(10*A-7*B)*sec(c+d*x)^2*tan(c+d*x)/(a^2*d*(1+cos(c+d*x)))-1/3*(A-B)*sec(c+d*x)^2*tan(c+d*x)/(d*(a+a*cos(c+d*x))^2)+4/3*(3*A-2*B)*tan(c+d*x)^3/(a^2*d)],
[cos(c+d*x)^5*(A+B*cos(c+d*x))/(a+a*cos(c+d*x))^3,x,8,1/2*(13*A-23*B)*x/a^3-4/5*(19*A-34*B)*sin(c+d*x)/(a^3*d)+1/2*(13*A-23*B)*cos(c+d*x)*sin(c+d*x)/(a^3*d)+1/5*(A-B)*cos(c+d*x)^5*sin(c+d*x)/(d*(a+a*cos(c+d*x))^3)+1/15*(8*A-13*B)*cos(c+d*x)^4*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^2)+1/3*(13*A-23*B)*cos(c+d*x)^3*sin(c+d*x)/(d*(a^3+a^3*cos(c+d*x)))+4/15*(19*A-34*B)*sin(c+d*x)^3/(a^3*d)],
[cos(c+d*x)^4*(A+B*cos(c+d*x))/(a+a*cos(c+d*x))^3,x,4,-1/2*(6*A-13*B)*x/a^3+8/15*(9*A-19*B)*sin(c+d*x)/(a^3*d)-1/2*(6*A-13*B)*cos(c+d*x)*sin(c+d*x)/(a^3*d)+1/5*(A-B)*cos(c+d*x)^4*sin(c+d*x)/(d*(a+a*cos(c+d*x))^3)+1/15*(6*A-11*B)*cos(c+d*x)^3*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^2)+4/15*(9*A-19*B)*cos(c+d*x)^2*sin(c+d*x)/(d*(a^3+a^3*cos(c+d*x)))],
[cos(c+d*x)^3*(A+B*cos(c+d*x))/(a+a*cos(c+d*x))^3,x,7,(A-3*B)*x/a^3-1/15*(7*A-27*B)*sin(c+d*x)/(a^3*d)+1/5*(A-B)*cos(c+d*x)^3*sin(c+d*x)/(d*(a+a*cos(c+d*x))^3)+1/15*(4*A-9*B)*cos(c+d*x)^2*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^2)-(A-3*B)*sin(c+d*x)/(d*(a^3+a^3*cos(c+d*x)))],
[cos(c+d*x)^2*(A+B*cos(c+d*x))/(a+a*cos(c+d*x))^3,x,5,B*x/a^3+1/5*(A-B)*cos(c+d*x)^2*sin(c+d*x)/(d*(a+a*cos(c+d*x))^3)-1/15*(2*A-7*B)*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^2)+1/15*(4*A-29*B)*sin(c+d*x)/(d*(a^3+a^3*cos(c+d*x)))],
[cos(c+d*x)*(A+B*cos(c+d*x))/(a+a*cos(c+d*x))^3,x,4,-1/5*(A-B)*sin(c+d*x)/(d*(a+a*cos(c+d*x))^3)+1/15*(3*A-8*B)*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^2)+1/15*(3*A+7*B)*sin(c+d*x)/(d*(a^3+a^3*cos(c+d*x)))],
[(A+B*cos(c+d*x))/(a+a*cos(c+d*x))^3,x,3,1/5*(A-B)*sin(c+d*x)/(d*(a+a*cos(c+d*x))^3)+1/15*(2*A+3*B)*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^2)+1/15*(2*A+3*B)*sin(c+d*x)/(d*(a^3+a^3*cos(c+d*x)))],
[(A+B*cos(c+d*x))*sec(c+d*x)/(a+a*cos(c+d*x))^3,x,5,A*arctanh(sin(c+d*x))/(a^3*d)-1/5*(A-B)*sin(c+d*x)/(d*(a+a*cos(c+d*x))^3)-1/15*(7*A-2*B)*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^2)-2/15*(11*A-B)*sin(c+d*x)/(d*(a^3+a^3*cos(c+d*x)))],
[(A+B*cos(c+d*x))*sec(c+d*x)^2/(a+a*cos(c+d*x))^3,x,7,-(3*A-B)*arctanh(sin(c+d*x))/(a^3*d)+2/15*(36*A-11*B)*tan(c+d*x)/(a^3*d)-1/5*(A-B)*tan(c+d*x)/(d*(a+a*cos(c+d*x))^3)-1/15*(9*A-4*B)*tan(c+d*x)/(a*d*(a+a*cos(c+d*x))^2)-(3*A-B)*tan(c+d*x)/(d*(a^3+a^3*cos(c+d*x)))],
[(A+B*cos(c+d*x))*sec(c+d*x)^3/(a+a*cos(c+d*x))^3,x,8,1/2*(13*A-6*B)*arctanh(sin(c+d*x))/(a^3*d)-8/15*(19*A-9*B)*tan(c+d*x)/(a^3*d)+1/2*(13*A-6*B)*sec(c+d*x)*tan(c+d*x)/(a^3*d)-1/5*(A-B)*sec(c+d*x)*tan(c+d*x)/(d*(a+a*cos(c+d*x))^3)-1/15*(11*A-6*B)*sec(c+d*x)*tan(c+d*x)/(a*d*(a+a*cos(c+d*x))^2)-4/15*(19*A-9*B)*sec(c+d*x)*tan(c+d*x)/(d*(a^3+a^3*cos(c+d*x)))],
[cos(c+d*x)^5*(A+B*cos(c+d*x))/(a+a*cos(c+d*x))^4,x,5,-1/2*(8*A-21*B)*x/a^4+8/105*(83*A-216*B)*sin(c+d*x)/(a^4*d)-1/2*(8*A-21*B)*cos(c+d*x)*sin(c+d*x)/(a^4*d)+1/105*(52*A-129*B)*cos(c+d*x)^3*sin(c+d*x)/(a^4*d*(1+cos(c+d*x))^2)+4/105*(83*A-216*B)*cos(c+d*x)^2*sin(c+d*x)/(a^4*d*(1+cos(c+d*x)))+1/7*(A-B)*cos(c+d*x)^5*sin(c+d*x)/(d*(a+a*cos(c+d*x))^4)+1/5*(A-2*B)*cos(c+d*x)^4*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^3)],
[cos(c+d*x)^4*(A+B*cos(c+d*x))/(a+a*cos(c+d*x))^4,x,8,(A-4*B)*x/a^4-1/105*(55*A-244*B)*sin(c+d*x)/(a^4*d)+1/105*(25*A-88*B)*cos(c+d*x)^2*sin(c+d*x)/(a^4*d*(1+cos(c+d*x))^2)-(A-4*B)*sin(c+d*x)/(a^4*d*(1+cos(c+d*x)))+1/7*(A-B)*cos(c+d*x)^4*sin(c+d*x)/(d*(a+a*cos(c+d*x))^4)+1/35*(5*A-12*B)*cos(c+d*x)^3*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^3)],
[cos(c+d*x)^3*(A+B*cos(c+d*x))/(a+a*cos(c+d*x))^4,x,6,B*x/a^4-1/105*(6*A-55*B)*sin(c+d*x)/(a^4*d*(1+cos(c+d*x))^2)+1/105*(12*A-215*B)*sin(c+d*x)/(a^4*d*(1+cos(c+d*x)))+1/7*(A-B)*cos(c+d*x)^3*sin(c+d*x)/(d*(a+a*cos(c+d*x))^4)+1/35*(3*A-10*B)*cos(c+d*x)^2*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^3)],
[cos(c+d*x)^2*(A+B*cos(c+d*x))/(a+a*cos(c+d*x))^4,x,5,-2/105*(A+27*B)*sin(c+d*x)/(a^4*d*(1+cos(c+d*x))^2)+1/105*(13*A+36*B)*sin(c+d*x)/(a^4*d*(1+cos(c+d*x)))+1/7*(A-B)*cos(c+d*x)^2*sin(c+d*x)/(d*(a+a*cos(c+d*x))^4)-1/35*(A-8*B)*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^3)],
[cos(c+d*x)*(A+B*cos(c+d*x))/(a+a*cos(c+d*x))^4,x,5,-1/7*(A-B)*sin(c+d*x)/(d*(a+a*cos(c+d*x))^4)+1/35*(4*A-11*B)*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^3)+1/105*(8*A+13*B)*sin(c+d*x)/(d*(a^2+a^2*cos(c+d*x))^2)+1/105*(8*A+13*B)*sin(c+d*x)/(d*(a^4+a^4*cos(c+d*x)))],
[(A+B*cos(c+d*x))/(a+a*cos(c+d*x))^4,x,4,1/7*(A-B)*sin(c+d*x)/(d*(a+a*cos(c+d*x))^4)+1/35*(3*A+4*B)*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^3)+2/105*(3*A+4*B)*sin(c+d*x)/(d*(a^2+a^2*cos(c+d*x))^2)+2/105*(3*A+4*B)*sin(c+d*x)/(d*(a^4+a^4*cos(c+d*x)))],
[(A+B*cos(c+d*x))*sec(c+d*x)/(a+a*cos(c+d*x))^4,x,6,A*arctanh(sin(c+d*x))/(a^4*d)-1/105*(55*A-6*B)*sin(c+d*x)/(a^4*d*(1+cos(c+d*x))^2)-2/105*(80*A-3*B)*sin(c+d*x)/(a^4*d*(1+cos(c+d*x)))-1/7*(A-B)*sin(c+d*x)/(d*(a+a*cos(c+d*x))^4)-1/35*(10*A-3*B)*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^3)],
[(A+B*cos(c+d*x))*sec(c+d*x)^2/(a+a*cos(c+d*x))^4,x,8,-(4*A-B)*arctanh(sin(c+d*x))/(a^4*d)+8/105*(83*A-20*B)*tan(c+d*x)/(a^4*d)-1/105*(88*A-25*B)*tan(c+d*x)/(a^4*d*(1+cos(c+d*x))^2)-(4*A-B)*tan(c+d*x)/(a^4*d*(1+cos(c+d*x)))-1/7*(A-B)*tan(c+d*x)/(d*(a+a*cos(c+d*x))^4)-1/35*(12*A-5*B)*tan(c+d*x)/(a*d*(a+a*cos(c+d*x))^3)],
[(A+B*cos(c+d*x))*sec(c+d*x)^3/(a+a*cos(c+d*x))^4,x,9,1/2*(21*A-8*B)*arctanh(sin(c+d*x))/(a^4*d)-8/105*(216*A-83*B)*tan(c+d*x)/(a^4*d)+1/2*(21*A-8*B)*sec(c+d*x)*tan(c+d*x)/(a^4*d)-1/105*(129*A-52*B)*sec(c+d*x)*tan(c+d*x)/(a^4*d*(1+cos(c+d*x))^2)-4/105*(216*A-83*B)*sec(c+d*x)*tan(c+d*x)/(a^4*d*(1+cos(c+d*x)))-1/7*(A-B)*sec(c+d*x)*tan(c+d*x)/(d*(a+a*cos(c+d*x))^4)-1/5*(2*A-B)*sec(c+d*x)*tan(c+d*x)/(a*d*(a+a*cos(c+d*x))^3)],

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

# n>0
[cos(c+d*x)^3*(a+a*cos(c+d*x))^(1/2)*(A+B*cos(c+d*x)),x,5,4/105*(9*A+8*B)*(a+a*cos(c+d*x))^(3/2)*sin(c+d*x)/(a*d)+4/45*a*(9*A+8*B)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+2/63*a*(9*A+8*B)*cos(c+d*x)^3*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+2/9*a*B*cos(c+d*x)^4*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))-8/315*(9*A+8*B)*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/d],
[cos(c+d*x)^2*(a+a*cos(c+d*x))^(1/2)*(A+B*cos(c+d*x)),x,4,2/35*(7*A+6*B)*(a+a*cos(c+d*x))^(3/2)*sin(c+d*x)/(a*d)+2/15*a*(7*A+6*B)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+2/7*a*B*cos(c+d*x)^3*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))-4/105*(7*A+6*B)*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/d],
[cos(c+d*x)*(a+a*cos(c+d*x))^(1/2)*(A+B*cos(c+d*x)),x,4,2/5*B*(a+a*cos(c+d*x))^(3/2)*sin(c+d*x)/(a*d)+2/15*a*(5*A+7*B)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+2/15*(5*A-2*B)*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/d],
[(a+a*cos(c+d*x))^(1/2)*(A+B*cos(c+d*x)),x,2,2/3*a*(3*A+B)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+2/3*B*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/d],
[(a+a*cos(c+d*x))^(1/2)*(A+B*cos(c+d*x))*sec(c+d*x),x,3,2*A*arctanh(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))*sqrt(a)/d+2*a*B*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))],
[(a+a*cos(c+d*x))^(1/2)*(A+B*cos(c+d*x))*sec(c+d*x)^2,x,3,(A+2*B)*arctanh(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))*sqrt(a)/d+a*A*tan(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))],
[(a+a*cos(c+d*x))^(1/2)*(A+B*cos(c+d*x))*sec(c+d*x)^3,x,4,1/4*(3*A+4*B)*arctanh(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))*sqrt(a)/d+1/4*a*(3*A+4*B)*tan(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+1/2*a*A*sec(c+d*x)*tan(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))],
[(a+a*cos(c+d*x))^(1/2)*(A+B*cos(c+d*x))*sec(c+d*x)^4,x,5,1/8*(5*A+6*B)*arctanh(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))*sqrt(a)/d+1/8*a*(5*A+6*B)*tan(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+1/12*a*(5*A+6*B)*sec(c+d*x)*tan(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+1/3*a*A*sec(c+d*x)^2*tan(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))],
[cos(c+d*x)^3*(a+a*cos(c+d*x))^(3/2)*(A+B*cos(c+d*x)),x,6,4/1155*(187*A+168*B)*(a+a*cos(c+d*x))^(3/2)*sin(c+d*x)/d+4/495*a^2*(187*A+168*B)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+2/693*a^2*(187*A+168*B)*cos(c+d*x)^3*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+2/99*a^2*(11*A+12*B)*cos(c+d*x)^4*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))-8/3465*a*(187*A+168*B)*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/d+2/11*a*B*cos(c+d*x)^4*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/d],
[cos(c+d*x)^2*(a+a*cos(c+d*x))^(3/2)*(A+B*cos(c+d*x)),x,5,2/105*(39*A+34*B)*(a+a*cos(c+d*x))^(3/2)*sin(c+d*x)/d+2/45*a^2*(39*A+34*B)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+2/63*a^2*(9*A+10*B)*cos(c+d*x)^3*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))-4/315*a*(39*A+34*B)*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/d+2/9*a*B*cos(c+d*x)^3*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/d],
[cos(c+d*x)*(a+a*cos(c+d*x))^(3/2)*(A+B*cos(c+d*x)),x,5,2/35*(7*A-2*B)*(a+a*cos(c+d*x))^(3/2)*sin(c+d*x)/d+2/7*B*(a+a*cos(c+d*x))^(5/2)*sin(c+d*x)/(a*d)+8/105*a^2*(21*A+19*B)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+2/105*a*(21*A+19*B)*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/d],
[(a+a*cos(c+d*x))^(3/2)*(A+B*cos(c+d*x)),x,3,2/5*B*(a+a*cos(c+d*x))^(3/2)*sin(c+d*x)/d+8/15*a^2*(5*A+3*B)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+2/15*a*(5*A+3*B)*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/d],
[(a+a*cos(c+d*x))^(3/2)*(A+B*cos(c+d*x))*sec(c+d*x),x,4,2*a^(3/2)*A*arctanh(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))/d+2/3*a^2*(3*A+4*B)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+2/3*a*B*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/d],
[(a+a*cos(c+d*x))^(3/2)*(A+B*cos(c+d*x))*sec(c+d*x)^2,x,4,a^(3/2)*(3*A+2*B)*arctanh(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))/d-a^2*(A-2*B)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+a*A*sqrt(a+a*cos(c+d*x))*tan(c+d*x)/d],
[(a+a*cos(c+d*x))^(3/2)*(A+B*cos(c+d*x))*sec(c+d*x)^3,x,4,1/4*a^(3/2)*(7*A+12*B)*arctanh(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))/d+1/4*a^2*(5*A+4*B)*tan(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+1/2*a*A*sec(c+d*x)*sqrt(a+a*cos(c+d*x))*tan(c+d*x)/d],
[(a+a*cos(c+d*x))^(3/2)*(A+B*cos(c+d*x))*sec(c+d*x)^4,x,5,1/8*a^(3/2)*(11*A+14*B)*arctanh(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))/d+1/8*a^2*(11*A+14*B)*tan(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+1/12*a^2*(7*A+6*B)*sec(c+d*x)*tan(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+1/3*a*A*sec(c+d*x)^2*sqrt(a+a*cos(c+d*x))*tan(c+d*x)/d],
[(a+a*cos(c+d*x))^(3/2)*(A+B*cos(c+d*x))*sec(c+d*x)^5,x,6,1/64*a^(3/2)*(75*A+88*B)*arctanh(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))/d+1/64*a^2*(75*A+88*B)*tan(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+1/96*a^2*(75*A+88*B)*sec(c+d*x)*tan(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+1/24*a^2*(9*A+8*B)*sec(c+d*x)^2*tan(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+1/4*a*A*sec(c+d*x)^3*sqrt(a+a*cos(c+d*x))*tan(c+d*x)/d],
[cos(c+d*x)^2*(a+a*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x)),x,6,2/1155*a*(803*A+710*B)*(a+a*cos(c+d*x))^(3/2)*sin(c+d*x)/d+2/11*a*B*cos(c+d*x)^3*(a+a*cos(c+d*x))^(3/2)*sin(c+d*x)/d+2/495*a^3*(803*A+710*B)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+2/693*a^3*(209*A+194*B)*cos(c+d*x)^3*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))-4/3465*a^2*(803*A+710*B)*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/d+2/99*a^2*(11*A+14*B)*cos(c+d*x)^3*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/d],
[cos(c+d*x)*(a+a*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x)),x,6,2/105*a*(15*A+13*B)*(a+a*cos(c+d*x))^(3/2)*sin(c+d*x)/d+2/63*(9*A-2*B)*(a+a*cos(c+d*x))^(5/2)*sin(c+d*x)/d+2/9*B*(a+a*cos(c+d*x))^(7/2)*sin(c+d*x)/(a*d)+64/315*a^3*(15*A+13*B)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+16/315*a^2*(15*A+13*B)*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/d],
[(a+a*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x)),x,4,2/35*a*(7*A+5*B)*(a+a*cos(c+d*x))^(3/2)*sin(c+d*x)/d+2/7*B*(a+a*cos(c+d*x))^(5/2)*sin(c+d*x)/d+64/105*a^3*(7*A+5*B)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+16/105*a^2*(7*A+5*B)*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/d],
[(a+a*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x))*sec(c+d*x),x,5,2*a^(5/2)*A*arctanh(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))/d+2/5*a*B*(a+a*cos(c+d*x))^(3/2)*sin(c+d*x)/d+2/15*a^3*(35*A+32*B)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+2/15*a^2*(5*A+8*B)*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/d],
[(a+a*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x))*sec(c+d*x)^2,x,5,a^(5/2)*(5*A+2*B)*arctanh(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))/d+1/3*a^3*(3*A+14*B)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))-1/3*a^2*(3*A-2*B)*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/d+a*A*(a+a*cos(c+d*x))^(3/2)*tan(c+d*x)/d],
[(a+a*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x))*sec(c+d*x)^3,x,5,1/4*a^(5/2)*(19*A+20*B)*arctanh(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))/d-1/4*a^3*(9*A-4*B)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+1/2*a*A*(a+a*cos(c+d*x))^(3/2)*sec(c+d*x)*tan(c+d*x)/d+1/4*a^2*(7*A+4*B)*sqrt(a+a*cos(c+d*x))*tan(c+d*x)/d],
[(a+a*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x))*sec(c+d*x)^4,x,5,1/8*a^(5/2)*(25*A+38*B)*arctanh(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))/d+1/3*a*A*(a+a*cos(c+d*x))^(3/2)*sec(c+d*x)^2*tan(c+d*x)/d+1/24*a^3*(49*A+54*B)*tan(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+1/4*a^2*(3*A+2*B)*sec(c+d*x)*sqrt(a+a*cos(c+d*x))*tan(c+d*x)/d],
[(a+a*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x))*sec(c+d*x)^5,x,6,1/64*a^(5/2)*(163*A+200*B)*arctanh(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))/d+1/4*a*A*(a+a*cos(c+d*x))^(3/2)*sec(c+d*x)^3*tan(c+d*x)/d+1/64*a^3*(163*A+200*B)*tan(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+1/96*a^3*(95*A+104*B)*sec(c+d*x)*tan(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+1/24*a^2*(11*A+8*B)*sec(c+d*x)^2*sqrt(a+a*cos(c+d*x))*tan(c+d*x)/d],
[(a+a*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x))*sec(c+d*x)^6,x,7,1/128*a^(5/2)*(283*A+326*B)*arctanh(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))/d+1/5*a*A*(a+a*cos(c+d*x))^(3/2)*sec(c+d*x)^4*tan(c+d*x)/d+1/128*a^3*(283*A+326*B)*tan(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+1/192*a^3*(283*A+326*B)*sec(c+d*x)*tan(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+1/240*a^3*(157*A+170*B)*sec(c+d*x)^2*tan(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+1/40*a^2*(13*A+10*B)*sec(c+d*x)^3*sqrt(a+a*cos(c+d*x))*tan(c+d*x)/d],

# n<0
[cos(c+d*x)^3*(A+B*cos(c+d*x))/(a+a*cos(c+d*x))^(1/2),x,7,-(A-B)*arctanh(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*cos(c+d*x))))*sqrt(2)/(d*sqrt(a))+4/105*(49*A-37*B)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+2/35*(7*A-B)*cos(c+d*x)^2*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+2/7*B*cos(c+d*x)^3*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))-2/105*(7*A-31*B)*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/(a*d)],
[cos(c+d*x)^2*(A+B*cos(c+d*x))/(a+a*cos(c+d*x))^(1/2),x,6,(A-B)*arctanh(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*cos(c+d*x))))*sqrt(2)/(d*sqrt(a))-4/15*(5*A-7*B)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+2/5*B*cos(c+d*x)^2*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+2/15*(5*A-B)*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/(a*d)],
[cos(c+d*x)*(A+B*cos(c+d*x))/(a+a*cos(c+d*x))^(1/2),x,5,-(A-B)*arctanh(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*cos(c+d*x))))*sqrt(2)/(d*sqrt(a))+2/3*(3*A-2*B)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+2/3*B*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/(a*d)],
[(A+B*cos(c+d*x))/(a+a*cos(c+d*x))^(1/2),x,3,(A-B)*arctanh(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*cos(c+d*x))))*sqrt(2)/(d*sqrt(a))+2*B*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))],
[(A+B*cos(c+d*x))*sec(c+d*x)/(a+a*cos(c+d*x))^(1/2),x,5,2*A*arctanh(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))/(d*sqrt(a))-(A-B)*arctanh(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*cos(c+d*x))))*sqrt(2)/(d*sqrt(a))],
[(A+B*cos(c+d*x))*sec(c+d*x)^2/(a+a*cos(c+d*x))^(1/2),x,6,-(A-2*B)*arctanh(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))/(d*sqrt(a))+(A-B)*arctanh(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*cos(c+d*x))))*sqrt(2)/(d*sqrt(a))+A*tan(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))],
[(A+B*cos(c+d*x))*sec(c+d*x)^3/(a+a*cos(c+d*x))^(1/2),x,7,1/4*(7*A-4*B)*arctanh(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))/(d*sqrt(a))-(A-B)*arctanh(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*cos(c+d*x))))*sqrt(2)/(d*sqrt(a))-1/4*(A-4*B)*tan(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+1/2*A*sec(c+d*x)*tan(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))],
[cos(c+d*x)^4*(A+B*cos(c+d*x))/(a+a*cos(c+d*x))^(3/2),x,8,1/2*(A-B)*cos(c+d*x)^4*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(3/2))-1/2*(15*A-19*B)*arctanh(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*cos(c+d*x))))/(a^(3/2)*d*sqrt(2))+1/105*(651*A-799*B)*sin(c+d*x)/(a*d*sqrt(a+a*cos(c+d*x)))+1/70*(63*A-67*B)*cos(c+d*x)^2*sin(c+d*x)/(a*d*sqrt(a+a*cos(c+d*x)))-1/14*(7*A-11*B)*cos(c+d*x)^3*sin(c+d*x)/(a*d*sqrt(a+a*cos(c+d*x)))-1/210*(273*A-397*B)*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/(a^2*d)],
[cos(c+d*x)^3*(A+B*cos(c+d*x))/(a+a*cos(c+d*x))^(3/2),x,7,1/2*(A-B)*cos(c+d*x)^3*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(3/2))+1/2*(11*A-15*B)*arctanh(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*cos(c+d*x))))/(a^(3/2)*d*sqrt(2))-1/15*(65*A-93*B)*sin(c+d*x)/(a*d*sqrt(a+a*cos(c+d*x)))-1/10*(5*A-9*B)*cos(c+d*x)^2*sin(c+d*x)/(a*d*sqrt(a+a*cos(c+d*x)))+1/30*(35*A-39*B)*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/(a^2*d)],
[cos(c+d*x)^2*(A+B*cos(c+d*x))/(a+a*cos(c+d*x))^(3/2),x,6,1/2*(A-B)*cos(c+d*x)^2*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(3/2))-1/2*(7*A-11*B)*arctanh(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*cos(c+d*x))))/(a^(3/2)*d*sqrt(2))+1/3*(9*A-13*B)*sin(c+d*x)/(a*d*sqrt(a+a*cos(c+d*x)))-1/6*(3*A-7*B)*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/(a^2*d)],
[cos(c+d*x)*(A+B*cos(c+d*x))/(a+a*cos(c+d*x))^(3/2),x,5,-1/2*(A-B)*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(3/2))+1/2*(3*A-7*B)*arctanh(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*cos(c+d*x))))/(a^(3/2)*d*sqrt(2))+2*B*sin(c+d*x)/(a*d*sqrt(a+a*cos(c+d*x)))],
[(A+B*cos(c+d*x))/(a+a*cos(c+d*x))^(3/2),x,3,1/2*(A-B)*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(3/2))+1/2*(A+3*B)*arctanh(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*cos(c+d*x))))/(a^(3/2)*d*sqrt(2))],
[(A+B*cos(c+d*x))*sec(c+d*x)/(a+a*cos(c+d*x))^(3/2),x,6,2*A*arctanh(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))/(a^(3/2)*d)-1/2*(A-B)*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(3/2))-1/2*(5*A-B)*arctanh(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*cos(c+d*x))))/(a^(3/2)*d*sqrt(2))],
[(A+B*cos(c+d*x))*sec(c+d*x)^2/(a+a*cos(c+d*x))^(3/2),x,7,-(3*A-2*B)*arctanh(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))/(a^(3/2)*d)+1/2*(9*A-5*B)*arctanh(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*cos(c+d*x))))/(a^(3/2)*d*sqrt(2))-1/2*(A-B)*tan(c+d*x)/(d*(a+a*cos(c+d*x))^(3/2))+1/2*(3*A-B)*tan(c+d*x)/(a*d*sqrt(a+a*cos(c+d*x)))],
[(A+B*cos(c+d*x))*sec(c+d*x)^3/(a+a*cos(c+d*x))^(3/2),x,8,1/4*(19*A-12*B)*arctanh(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))/(a^(3/2)*d)-1/2*(13*A-9*B)*arctanh(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*cos(c+d*x))))/(a^(3/2)*d*sqrt(2))-1/2*(A-B)*sec(c+d*x)*tan(c+d*x)/(d*(a+a*cos(c+d*x))^(3/2))-1/4*(7*A-6*B)*tan(c+d*x)/(a*d*sqrt(a+a*cos(c+d*x)))+1/2*(2*A-B)*sec(c+d*x)*tan(c+d*x)/(a*d*sqrt(a+a*cos(c+d*x)))],
[cos(c+d*x)^4*(A+B*cos(c+d*x))/(a+a*cos(c+d*x))^(5/2),x,8,1/4*(A-B)*cos(c+d*x)^4*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(5/2))+1/16*(13*A-21*B)*cos(c+d*x)^3*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^(3/2))+1/16*(163*A-283*B)*arctanh(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*cos(c+d*x))))/(a^(5/2)*d*sqrt(2))-1/120*(985*A-1729*B)*sin(c+d*x)/(a^2*d*sqrt(a+a*cos(c+d*x)))-1/80*(85*A-157*B)*cos(c+d*x)^2*sin(c+d*x)/(a^2*d*sqrt(a+a*cos(c+d*x)))+1/240*(475*A-787*B)*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/(a^3*d)],
[cos(c+d*x)^3*(A+B*cos(c+d*x))/(a+a*cos(c+d*x))^(5/2),x,7,1/4*(A-B)*cos(c+d*x)^3*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(5/2))+1/16*(9*A-17*B)*cos(c+d*x)^2*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^(3/2))-1/16*(75*A-163*B)*arctanh(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*cos(c+d*x))))/(a^(5/2)*d*sqrt(2))+1/24*(93*A-197*B)*sin(c+d*x)/(a^2*d*sqrt(a+a*cos(c+d*x)))-1/48*(39*A-95*B)*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/(a^3*d)],
[cos(c+d*x)^2*(A+B*cos(c+d*x))/(a+a*cos(c+d*x))^(5/2),x,6,1/4*(A-B)*cos(c+d*x)^2*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(5/2))-1/16*(5*A-13*B)*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^(3/2))+1/16*(19*A-75*B)*arctanh(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*cos(c+d*x))))/(a^(5/2)*d*sqrt(2))-1/4*(A-9*B)*sin(c+d*x)/(a^2*d*sqrt(a+a*cos(c+d*x)))],
[cos(c+d*x)*(A+B*cos(c+d*x))/(a+a*cos(c+d*x))^(5/2),x,5,-1/4*(A-B)*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(5/2))+1/16*(5*A-13*B)*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^(3/2))+1/16*(5*A+19*B)*arctanh(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*cos(c+d*x))))/(a^(5/2)*d*sqrt(2))],
[(A+B*cos(c+d*x))/(a+a*cos(c+d*x))^(5/2),x,4,1/4*(A-B)*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(5/2))+1/16*(3*A+5*B)*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^(3/2))+1/16*(3*A+5*B)*arctanh(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*cos(c+d*x))))/(a^(5/2)*d*sqrt(2))],
[(A+B*cos(c+d*x))*sec(c+d*x)/(a+a*cos(c+d*x))^(5/2),x,7,2*A*arctanh(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))/(a^(5/2)*d)-1/4*(A-B)*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(5/2))-1/16*(11*A-3*B)*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^(3/2))-1/16*(43*A-3*B)*arctanh(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*cos(c+d*x))))/(a^(5/2)*d*sqrt(2))],
[(A+B*cos(c+d*x))*sec(c+d*x)^2/(a+a*cos(c+d*x))^(5/2),x,8,-(5*A-2*B)*arctanh(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))/(a^(5/2)*d)+1/16*(115*A-43*B)*arctanh(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*cos(c+d*x))))/(a^(5/2)*d*sqrt(2))-1/4*(A-B)*tan(c+d*x)/(d*(a+a*cos(c+d*x))^(5/2))-1/16*(15*A-7*B)*tan(c+d*x)/(a*d*(a+a*cos(c+d*x))^(3/2))+1/16*(35*A-11*B)*tan(c+d*x)/(a^2*d*sqrt(a+a*cos(c+d*x)))],
[(A+B*cos(c+d*x))*sec(c+d*x)^3/(a+a*cos(c+d*x))^(5/2),x,9,1/4*(39*A-20*B)*arctanh(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))/(a^(5/2)*d)-1/16*(219*A-115*B)*arctanh(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*cos(c+d*x))))/(a^(5/2)*d*sqrt(2))-1/4*(A-B)*sec(c+d*x)*tan(c+d*x)/(d*(a+a*cos(c+d*x))^(5/2))-1/16*(19*A-11*B)*sec(c+d*x)*tan(c+d*x)/(a*d*(a+a*cos(c+d*x))^(3/2))-7/16*(9*A-5*B)*tan(c+d*x)/(a^2*d*sqrt(a+a*cos(c+d*x)))+1/16*(31*A-15*B)*sec(c+d*x)*tan(c+d*x)/(a^2*d*sqrt(a+a*cos(c+d*x)))],

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

# n>0
[cos(c+d*x)^(5/2)*(a+a*cos(c+d*x))*(A+B*cos(c+d*x)),x,8,2/15*a*(9*A+7*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+10/21*a*(A+B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+2/45*a*(9*A+7*B)*cos(c+d*x)^(3/2)*sin(c+d*x)/d+2/7*a*(A+B)*cos(c+d*x)^(5/2)*sin(c+d*x)/d+2/9*a*B*cos(c+d*x)^(7/2)*sin(c+d*x)/d+10/21*a*(A+B)*sin(c+d*x)*sqrt(cos(c+d*x))/d],
[cos(c+d*x)^(3/2)*(a+a*cos(c+d*x))*(A+B*cos(c+d*x)),x,7,6/5*a*(A+B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+2/21*a*(7*A+5*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+2/5*a*(A+B)*cos(c+d*x)^(3/2)*sin(c+d*x)/d+2/7*a*B*cos(c+d*x)^(5/2)*sin(c+d*x)/d+2/21*a*(7*A+5*B)*sin(c+d*x)*sqrt(cos(c+d*x))/d],
[(a+a*cos(c+d*x))*(A+B*cos(c+d*x))*sqrt(cos(c+d*x)),x,6,2/5*a*(5*A+3*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+2/3*a*(A+B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+2/5*a*B*cos(c+d*x)^(3/2)*sin(c+d*x)/d+2/3*a*(A+B)*sin(c+d*x)*sqrt(cos(c+d*x))/d],
[(a+a*cos(c+d*x))*(A+B*cos(c+d*x))/sqrt(cos(c+d*x)),x,5,2*a*(A+B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+2/3*a*(3*A+B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+2/3*a*B*sin(c+d*x)*sqrt(cos(c+d*x))/d],
[(a+a*cos(c+d*x))*(A+B*cos(c+d*x))/cos(c+d*x)^(3/2),x,5,-2*a*(A-B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+2*a*(A+B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+2*a*A*sin(c+d*x)/(d*sqrt(cos(c+d*x)))],
[(a+a*cos(c+d*x))*(A+B*cos(c+d*x))/cos(c+d*x)^(5/2),x,6,-2*a*(A+B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+2/3*a*(A+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))/d+2/3*a*A*sin(c+d*x)/(d*cos(c+d*x)^(3/2))+2*a*(A+B)*sin(c+d*x)/(d*sqrt(cos(c+d*x)))],
[(a+a*cos(c+d*x))*(A+B*cos(c+d*x))/cos(c+d*x)^(7/2),x,7,-2/5*a*(3*A+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))/d+2/3*a*(A+B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+2/5*a*A*sin(c+d*x)/(d*cos(c+d*x)^(5/2))+2/3*a*(A+B)*sin(c+d*x)/(d*cos(c+d*x)^(3/2))+2/5*a*(3*A+5*B)*sin(c+d*x)/(d*sqrt(cos(c+d*x)))],
[cos(c+d*x)^(3/2)*(a+a*cos(c+d*x))^2*(A+B*cos(c+d*x)),x,8,4/15*a^2*(9*A+8*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+4/21*a^2*(6*A+5*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+4/45*a^2*(9*A+8*B)*cos(c+d*x)^(3/2)*sin(c+d*x)/d+2/63*a^2*(9*A+11*B)*cos(c+d*x)^(5/2)*sin(c+d*x)/d+2/9*B*cos(c+d*x)^(5/2)*(a^2+a^2*cos(c+d*x))*sin(c+d*x)/d+4/21*a^2*(6*A+5*B)*sin(c+d*x)*sqrt(cos(c+d*x))/d],
[(a+a*cos(c+d*x))^2*(A+B*cos(c+d*x))*sqrt(cos(c+d*x)),x,7,4/5*a^2*(4*A+3*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+4/21*a^2*(7*A+6*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+2/35*a^2*(7*A+9*B)*cos(c+d*x)^(3/2)*sin(c+d*x)/d+2/7*B*cos(c+d*x)^(3/2)*(a^2+a^2*cos(c+d*x))*sin(c+d*x)/d+4/21*a^2*(7*A+6*B)*sin(c+d*x)*sqrt(cos(c+d*x))/d],
[(a+a*cos(c+d*x))^2*(A+B*cos(c+d*x))/sqrt(cos(c+d*x)),x,6,4/5*a^2*(5*A+4*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+4/3*a^2*(2*A+B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+2/15*a^2*(5*A+7*B)*sin(c+d*x)*sqrt(cos(c+d*x))/d+2/5*B*(a^2+a^2*cos(c+d*x))*sin(c+d*x)*sqrt(cos(c+d*x))/d],
[(a+a*cos(c+d*x))^2*(A+B*cos(c+d*x))/cos(c+d*x)^(3/2),x,6,4*a^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))/d+4/3*a^2*(3*A+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))/d+2*A*(a^2+a^2*cos(c+d*x))*sin(c+d*x)/(d*sqrt(cos(c+d*x)))-2/3*a^2*(3*A-B)*sin(c+d*x)*sqrt(cos(c+d*x))/d],
[(a+a*cos(c+d*x))^2*(A+B*cos(c+d*x))/cos(c+d*x)^(5/2),x,6,-4*a^2*A*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+4/3*a^2*(2*A+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))/d+2/3*A*(a^2+a^2*cos(c+d*x))*sin(c+d*x)/(d*cos(c+d*x)^(3/2))+2/3*a^2*(5*A+3*B)*sin(c+d*x)/(d*sqrt(cos(c+d*x)))],
[(a+a*cos(c+d*x))^2*(A+B*cos(c+d*x))/cos(c+d*x)^(7/2),x,7,-4/5*a^2*(4*A+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))/d+4/3*a^2*(A+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))/d+2/15*a^2*(7*A+5*B)*sin(c+d*x)/(d*cos(c+d*x)^(3/2))+2/5*A*(a^2+a^2*cos(c+d*x))*sin(c+d*x)/(d*cos(c+d*x)^(5/2))+4/5*a^2*(4*A+5*B)*sin(c+d*x)/(d*sqrt(cos(c+d*x)))],
[(a+a*cos(c+d*x))^2*(A+B*cos(c+d*x))/cos(c+d*x)^(9/2),x,8,-4/5*a^2*(3*A+4*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+4/21*a^2*(6*A+7*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+2/35*a^2*(9*A+7*B)*sin(c+d*x)/(d*cos(c+d*x)^(5/2))+4/21*a^2*(6*A+7*B)*sin(c+d*x)/(d*cos(c+d*x)^(3/2))+2/7*A*(a^2+a^2*cos(c+d*x))*sin(c+d*x)/(d*cos(c+d*x)^(7/2))+4/5*a^2*(3*A+4*B)*sin(c+d*x)/(d*sqrt(cos(c+d*x)))],
[cos(c+d*x)^(3/2)*(a+a*cos(c+d*x))^3*(A+B*cos(c+d*x)),x,9,4/15*a^3*(17*A+15*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+4/231*a^3*(121*A+105*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+4/45*a^3*(17*A+15*B)*cos(c+d*x)^(3/2)*sin(c+d*x)/d+20/693*a^3*(22*A+21*B)*cos(c+d*x)^(5/2)*sin(c+d*x)/d+2/11*a*B*cos(c+d*x)^(5/2)*(a+a*cos(c+d*x))^2*sin(c+d*x)/d+2/99*(11*A+15*B)*cos(c+d*x)^(5/2)*(a^3+a^3*cos(c+d*x))*sin(c+d*x)/d+4/231*a^3*(121*A+105*B)*sin(c+d*x)*sqrt(cos(c+d*x))/d],
[(a+a*cos(c+d*x))^3*(A+B*cos(c+d*x))*sqrt(cos(c+d*x)),x,8,4/15*a^3*(21*A+17*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+4/21*a^3*(13*A+11*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+4/105*a^3*(24*A+23*B)*cos(c+d*x)^(3/2)*sin(c+d*x)/d+2/9*a*B*cos(c+d*x)^(3/2)*(a+a*cos(c+d*x))^2*sin(c+d*x)/d+2/63*(9*A+13*B)*cos(c+d*x)^(3/2)*(a^3+a^3*cos(c+d*x))*sin(c+d*x)/d+4/21*a^3*(13*A+11*B)*sin(c+d*x)*sqrt(cos(c+d*x))/d],
[(a+a*cos(c+d*x))^3*(A+B*cos(c+d*x))/sqrt(cos(c+d*x)),x,7,4/5*a^3*(9*A+7*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+4/21*a^3*(21*A+13*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+4/105*a^3*(42*A+41*B)*sin(c+d*x)*sqrt(cos(c+d*x))/d+2/7*a*B*(a+a*cos(c+d*x))^2*sin(c+d*x)*sqrt(cos(c+d*x))/d+2/35*(7*A+11*B)*(a^3+a^3*cos(c+d*x))*sin(c+d*x)*sqrt(cos(c+d*x))/d],
[(a+a*cos(c+d*x))^3*(A+B*cos(c+d*x))/cos(c+d*x)^(3/2),x,7,4/5*a^3*(5*A+9*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+4/3*a^3*(5*A+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))/d+2*a*A*(a+a*cos(c+d*x))^2*sin(c+d*x)/(d*sqrt(cos(c+d*x)))-4/15*a^3*(5*A-6*B)*sin(c+d*x)*sqrt(cos(c+d*x))/d-2/5*(5*A-B)*(a^3+a^3*cos(c+d*x))*sin(c+d*x)*sqrt(cos(c+d*x))/d],
[(a+a*cos(c+d*x))^3*(A+B*cos(c+d*x))/cos(c+d*x)^(5/2),x,7,-4*a^3*(A-B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+20/3*a^3*(A+B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+2/3*a*A*(a+a*cos(c+d*x))^2*sin(c+d*x)/(d*cos(c+d*x)^(3/2))+2/3*(7*A+3*B)*(a^3+a^3*cos(c+d*x))*sin(c+d*x)/(d*sqrt(cos(c+d*x)))-4/3*a^3*(4*A+B)*sin(c+d*x)*sqrt(cos(c+d*x))/d],
[(a+a*cos(c+d*x))^3*(A+B*cos(c+d*x))/cos(c+d*x)^(7/2),x,7,-4/5*a^3*(9*A+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))/d+4/3*a^3*(3*A+5*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+2/5*a*A*(a+a*cos(c+d*x))^2*sin(c+d*x)/(d*cos(c+d*x)^(5/2))+2/15*(9*A+5*B)*(a^3+a^3*cos(c+d*x))*sin(c+d*x)/(d*cos(c+d*x)^(3/2))+4/15*a^3*(21*A+20*B)*sin(c+d*x)/(d*sqrt(cos(c+d*x)))],
[(a+a*cos(c+d*x))^3*(A+B*cos(c+d*x))/cos(c+d*x)^(9/2),x,8,-4/5*a^3*(7*A+9*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+4/21*a^3*(13*A+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))/d+4/105*a^3*(41*A+42*B)*sin(c+d*x)/(d*cos(c+d*x)^(3/2))+2/7*a*A*(a+a*cos(c+d*x))^2*sin(c+d*x)/(d*cos(c+d*x)^(7/2))+2/35*(11*A+7*B)*(a^3+a^3*cos(c+d*x))*sin(c+d*x)/(d*cos(c+d*x)^(5/2))+4/5*a^3*(7*A+9*B)*sin(c+d*x)/(d*sqrt(cos(c+d*x)))],
[(a+a*cos(c+d*x))^3*(A+B*cos(c+d*x))/cos(c+d*x)^(11/2),x,9,-4/15*a^3*(17*A+21*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+4/21*a^3*(11*A+13*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+4/105*a^3*(23*A+24*B)*sin(c+d*x)/(d*cos(c+d*x)^(5/2))+4/21*a^3*(11*A+13*B)*sin(c+d*x)/(d*cos(c+d*x)^(3/2))+2/9*a*A*(a+a*cos(c+d*x))^2*sin(c+d*x)/(d*cos(c+d*x)^(9/2))+2/63*(13*A+9*B)*(a^3+a^3*cos(c+d*x))*sin(c+d*x)/(d*cos(c+d*x)^(7/2))+4/15*a^3*(17*A+21*B)*sin(c+d*x)/(d*sqrt(cos(c+d*x)))],

# n<0
[cos(c+d*x)^(5/2)*(A+B*cos(c+d*x))/(a+a*cos(c+d*x)),x,6,-3/5*(5*A-7*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(a*d)+5/3*(A-B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/(a*d)-1/5*(5*A-7*B)*cos(c+d*x)^(3/2)*sin(c+d*x)/(a*d)+(A-B)*cos(c+d*x)^(5/2)*sin(c+d*x)/(d*(a+a*cos(c+d*x)))+5/3*(A-B)*sin(c+d*x)*sqrt(cos(c+d*x))/(a*d)],
[cos(c+d*x)^(3/2)*(A+B*cos(c+d*x))/(a+a*cos(c+d*x)),x,5,3*(A-B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(a*d)-1/3*(3*A-5*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/(a*d)+(A-B)*cos(c+d*x)^(3/2)*sin(c+d*x)/(d*(a+a*cos(c+d*x)))-1/3*(3*A-5*B)*sin(c+d*x)*sqrt(cos(c+d*x))/(a*d)],
[(A+B*cos(c+d*x))*sqrt(cos(c+d*x))/(a+a*cos(c+d*x)),x,4,-(A-3*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(a*d)+(A-B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/(a*d)+(A-B)*sin(c+d*x)*sqrt(cos(c+d*x))/(d*(a+a*cos(c+d*x)))],
[(A+B*cos(c+d*x))/((a+a*cos(c+d*x))*sqrt(cos(c+d*x))),x,4,(A-B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(a*d)+(A+B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/(a*d)-(A-B)*sin(c+d*x)*sqrt(cos(c+d*x))/(d*(a+a*cos(c+d*x)))],
[(A+B*cos(c+d*x))/(cos(c+d*x)^(3/2)*(a+a*cos(c+d*x))),x,5,-(3*A-B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(a*d)-(A-B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/(a*d)+(3*A-B)*sin(c+d*x)/(a*d*sqrt(cos(c+d*x)))-(A-B)*sin(c+d*x)/(d*(a+a*cos(c+d*x))*sqrt(cos(c+d*x)))],
[(A+B*cos(c+d*x))/(cos(c+d*x)^(5/2)*(a+a*cos(c+d*x))),x,6,3*(A-B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(a*d)+1/3*(5*A-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))/(a*d)+1/3*(5*A-3*B)*sin(c+d*x)/(a*d*cos(c+d*x)^(3/2))-(A-B)*sin(c+d*x)/(d*cos(c+d*x)^(3/2)*(a+a*cos(c+d*x)))-3*(A-B)*sin(c+d*x)/(a*d*sqrt(cos(c+d*x)))],
[cos(c+d*x)^(7/2)*(A+B*cos(c+d*x))/(a+a*cos(c+d*x))^2,x,7,-7/5*(5*A-8*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(a^2*d)+5/3*(2*A-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))/(a^2*d)-7/15*(5*A-8*B)*cos(c+d*x)^(3/2)*sin(c+d*x)/(a^2*d)+(2*A-3*B)*cos(c+d*x)^(5/2)*sin(c+d*x)/(a^2*d*(1+cos(c+d*x)))+1/3*(A-B)*cos(c+d*x)^(7/2)*sin(c+d*x)/(d*(a+a*cos(c+d*x))^2)+5/3*(2*A-3*B)*sin(c+d*x)*sqrt(cos(c+d*x))/(a^2*d)],
[cos(c+d*x)^(5/2)*(A+B*cos(c+d*x))/(a+a*cos(c+d*x))^2,x,6,(4*A-7*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(a^2*d)-5/3*(A-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))/(a^2*d)+1/3*(4*A-7*B)*cos(c+d*x)^(3/2)*sin(c+d*x)/(a^2*d*(1+cos(c+d*x)))+1/3*(A-B)*cos(c+d*x)^(5/2)*sin(c+d*x)/(d*(a+a*cos(c+d*x))^2)-5/3*(A-2*B)*sin(c+d*x)*sqrt(cos(c+d*x))/(a^2*d)],
[cos(c+d*x)^(3/2)*(A+B*cos(c+d*x))/(a+a*cos(c+d*x))^2,x,5,-(A-4*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(a^2*d)+1/3*(2*A-5*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/(a^2*d)+1/3*(A-B)*cos(c+d*x)^(3/2)*sin(c+d*x)/(d*(a+a*cos(c+d*x))^2)+1/3*(2*A-5*B)*sin(c+d*x)*sqrt(cos(c+d*x))/(a^2*d*(1+cos(c+d*x)))],
[(A+B*cos(c+d*x))*sqrt(cos(c+d*x))/(a+a*cos(c+d*x))^2,x,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))/(a^2*d)+1/3*(A+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))/(a^2*d)+B*sin(c+d*x)*sqrt(cos(c+d*x))/(a^2*d*(1+cos(c+d*x)))+1/3*(A-B)*sin(c+d*x)*sqrt(cos(c+d*x))/(d*(a+a*cos(c+d*x))^2)],
[(A+B*cos(c+d*x))/((a+a*cos(c+d*x))^2*sqrt(cos(c+d*x))),x,5,A*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(a^2*d)+1/3*(2*A+B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/(a^2*d)-A*sin(c+d*x)*sqrt(cos(c+d*x))/(a^2*d*(1+cos(c+d*x)))-1/3*(A-B)*sin(c+d*x)*sqrt(cos(c+d*x))/(d*(a+a*cos(c+d*x))^2)],
[(A+B*cos(c+d*x))/(cos(c+d*x)^(3/2)*(a+a*cos(c+d*x))^2),x,6,-(4*A-B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(a^2*d)-1/3*(5*A-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))/(a^2*d)+(4*A-B)*sin(c+d*x)/(a^2*d*sqrt(cos(c+d*x)))-1/3*(5*A-2*B)*sin(c+d*x)/(a^2*d*(1+cos(c+d*x))*sqrt(cos(c+d*x)))-1/3*(A-B)*sin(c+d*x)/(d*(a+a*cos(c+d*x))^2*sqrt(cos(c+d*x)))],
[(A+B*cos(c+d*x))/(cos(c+d*x)^(5/2)*(a+a*cos(c+d*x))^2),x,7,(7*A-4*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(a^2*d)+5/3*(2*A-B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/(a^2*d)+5/3*(2*A-B)*sin(c+d*x)/(a^2*d*cos(c+d*x)^(3/2))-1/3*(7*A-4*B)*sin(c+d*x)/(a^2*d*cos(c+d*x)^(3/2)*(1+cos(c+d*x)))-1/3*(A-B)*sin(c+d*x)/(d*cos(c+d*x)^(3/2)*(a+a*cos(c+d*x))^2)-(7*A-4*B)*sin(c+d*x)/(a^2*d*sqrt(cos(c+d*x)))],
[cos(c+d*x)^(9/2)*(A+B*cos(c+d*x))/(a+a*cos(c+d*x))^3,x,8,-7/10*(17*A-33*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(a^3*d)+1/2*(11*A-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))/(a^3*d)-7/30*(17*A-33*B)*cos(c+d*x)^(3/2)*sin(c+d*x)/(a^3*d)+1/5*(A-B)*cos(c+d*x)^(9/2)*sin(c+d*x)/(d*(a+a*cos(c+d*x))^3)+1/15*(7*A-12*B)*cos(c+d*x)^(7/2)*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^2)+3/10*(11*A-21*B)*cos(c+d*x)^(5/2)*sin(c+d*x)/(d*(a^3+a^3*cos(c+d*x)))+1/2*(11*A-21*B)*sin(c+d*x)*sqrt(cos(c+d*x))/(a^3*d)],
[cos(c+d*x)^(7/2)*(A+B*cos(c+d*x))/(a+a*cos(c+d*x))^3,x,7,7/10*(7*A-17*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(a^3*d)-1/6*(13*A-33*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/(a^3*d)+1/5*(A-B)*cos(c+d*x)^(7/2)*sin(c+d*x)/(d*(a+a*cos(c+d*x))^3)+1/3*(A-2*B)*cos(c+d*x)^(5/2)*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^2)+7/30*(7*A-17*B)*cos(c+d*x)^(3/2)*sin(c+d*x)/(d*(a^3+a^3*cos(c+d*x)))-1/6*(13*A-33*B)*sin(c+d*x)*sqrt(cos(c+d*x))/(a^3*d)],
[cos(c+d*x)^(5/2)*(A+B*cos(c+d*x))/(a+a*cos(c+d*x))^3,x,6,-1/10*(9*A-49*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(a^3*d)+1/6*(3*A-13*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/(a^3*d)+1/5*(A-B)*cos(c+d*x)^(5/2)*sin(c+d*x)/(d*(a+a*cos(c+d*x))^3)+1/15*(3*A-8*B)*cos(c+d*x)^(3/2)*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^2)+1/6*(3*A-13*B)*sin(c+d*x)*sqrt(cos(c+d*x))/(d*(a^3+a^3*cos(c+d*x)))],
[cos(c+d*x)^(3/2)*(A+B*cos(c+d*x))/(a+a*cos(c+d*x))^3,x,6,-1/10*(A+9*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(a^3*d)+1/6*(A+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))/(a^3*d)+1/5*(A-B)*cos(c+d*x)^(3/2)*sin(c+d*x)/(d*(a+a*cos(c+d*x))^3)+1/15*(A-6*B)*sin(c+d*x)*sqrt(cos(c+d*x))/(a*d*(a+a*cos(c+d*x))^2)+1/10*(A+9*B)*sin(c+d*x)*sqrt(cos(c+d*x))/(d*(a^3+a^3*cos(c+d*x)))],
[(A+B*cos(c+d*x))*sqrt(cos(c+d*x))/(a+a*cos(c+d*x))^3,x,6,1/10*(A-B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(a^3*d)+1/6*(A+B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/(a^3*d)+1/5*(A-B)*sin(c+d*x)*sqrt(cos(c+d*x))/(d*(a+a*cos(c+d*x))^3)+1/15*(A+4*B)*sin(c+d*x)*sqrt(cos(c+d*x))/(a*d*(a+a*cos(c+d*x))^2)-1/10*(A-B)*sin(c+d*x)*sqrt(cos(c+d*x))/(d*(a^3+a^3*cos(c+d*x)))],
[(A+B*cos(c+d*x))/((a+a*cos(c+d*x))^3*sqrt(cos(c+d*x))),x,6,1/10*(9*A+B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(a^3*d)+1/6*(3*A+B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/(a^3*d)-1/5*(A-B)*sin(c+d*x)*sqrt(cos(c+d*x))/(d*(a+a*cos(c+d*x))^3)-1/15*(6*A-B)*sin(c+d*x)*sqrt(cos(c+d*x))/(a*d*(a+a*cos(c+d*x))^2)-1/10*(9*A+B)*sin(c+d*x)*sqrt(cos(c+d*x))/(d*(a^3+a^3*cos(c+d*x)))],
[(A+B*cos(c+d*x))/(cos(c+d*x)^(3/2)*(a+a*cos(c+d*x))^3),x,7,-1/10*(49*A-9*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(a^3*d)-1/6*(13*A-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))/(a^3*d)+1/10*(49*A-9*B)*sin(c+d*x)/(a^3*d*sqrt(cos(c+d*x)))-1/5*(A-B)*sin(c+d*x)/(d*(a+a*cos(c+d*x))^3*sqrt(cos(c+d*x)))-1/15*(8*A-3*B)*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^2*sqrt(cos(c+d*x)))-1/6*(13*A-3*B)*sin(c+d*x)/(d*(a^3+a^3*cos(c+d*x))*sqrt(cos(c+d*x)))],
[(A+B*cos(c+d*x))/(cos(c+d*x)^(5/2)*(a+a*cos(c+d*x))^3),x,8,7/10*(17*A-7*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(a^3*d)+1/6*(33*A-13*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/(a^3*d)+1/6*(33*A-13*B)*sin(c+d*x)/(a^3*d*cos(c+d*x)^(3/2))-1/5*(A-B)*sin(c+d*x)/(d*cos(c+d*x)^(3/2)*(a+a*cos(c+d*x))^3)-1/3*(2*A-B)*sin(c+d*x)/(a*d*cos(c+d*x)^(3/2)*(a+a*cos(c+d*x))^2)-7/30*(17*A-7*B)*sin(c+d*x)/(d*cos(c+d*x)^(3/2)*(a^3+a^3*cos(c+d*x)))-7/10*(17*A-7*B)*sin(c+d*x)/(a^3*d*sqrt(cos(c+d*x)))],

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

# n>0
[cos(c+d*x)^(5/2)*(a+a*cos(c+d*x))^(1/2)*(A+B*cos(c+d*x)),x,6,5/64*(8*A+7*B)*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))*sqrt(a)/d+5/96*a*(8*A+7*B)*cos(c+d*x)^(3/2)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+1/24*a*(8*A+7*B)*cos(c+d*x)^(5/2)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+1/4*a*B*cos(c+d*x)^(7/2)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+5/64*a*(8*A+7*B)*sin(c+d*x)*sqrt(cos(c+d*x))/(d*sqrt(a+a*cos(c+d*x)))],
[cos(c+d*x)^(3/2)*(a+a*cos(c+d*x))^(1/2)*(A+B*cos(c+d*x)),x,5,1/8*(6*A+5*B)*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))*sqrt(a)/d+1/12*a*(6*A+5*B)*cos(c+d*x)^(3/2)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+1/3*a*B*cos(c+d*x)^(5/2)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+1/8*a*(6*A+5*B)*sin(c+d*x)*sqrt(cos(c+d*x))/(d*sqrt(a+a*cos(c+d*x)))],
[cos(c+d*x)^(1/2)*(a+a*cos(c+d*x))^(1/2)*(A+B*cos(c+d*x)),x,4,1/4*(4*A+3*B)*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))*sqrt(a)/d+1/2*a*B*cos(c+d*x)^(3/2)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+1/4*a*(4*A+3*B)*sin(c+d*x)*sqrt(cos(c+d*x))/(d*sqrt(a+a*cos(c+d*x)))],
[(a+a*cos(c+d*x))^(1/2)*(A+B*cos(c+d*x))/cos(c+d*x)^(1/2),x,3,(2*A+B)*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))*sqrt(a)/d+a*B*sin(c+d*x)*sqrt(cos(c+d*x))/(d*sqrt(a+a*cos(c+d*x)))],
[(a+a*cos(c+d*x))^(1/2)*(A+B*cos(c+d*x))/cos(c+d*x)^(3/2),x,3,2*B*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))*sqrt(a)/d+2*a*A*sin(c+d*x)/(d*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x)))],
[(a+a*cos(c+d*x))^(1/2)*(A+B*cos(c+d*x))/cos(c+d*x)^(5/2),x,2,2/3*a*A*sin(c+d*x)/(d*cos(c+d*x)^(3/2)*sqrt(a+a*cos(c+d*x)))+2/3*a*(2*A+3*B)*sin(c+d*x)/(d*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x)))],
[(a+a*cos(c+d*x))^(1/2)*(A+B*cos(c+d*x))/cos(c+d*x)^(7/2),x,3,2/5*a*A*sin(c+d*x)/(d*cos(c+d*x)^(5/2)*sqrt(a+a*cos(c+d*x)))+2/15*a*(4*A+5*B)*sin(c+d*x)/(d*cos(c+d*x)^(3/2)*sqrt(a+a*cos(c+d*x)))+4/15*a*(4*A+5*B)*sin(c+d*x)/(d*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x)))],
[(a+a*cos(c+d*x))^(1/2)*(A+B*cos(c+d*x))/cos(c+d*x)^(9/2),x,4,2/7*a*A*sin(c+d*x)/(d*cos(c+d*x)^(7/2)*sqrt(a+a*cos(c+d*x)))+2/35*a*(6*A+7*B)*sin(c+d*x)/(d*cos(c+d*x)^(5/2)*sqrt(a+a*cos(c+d*x)))+8/105*a*(6*A+7*B)*sin(c+d*x)/(d*cos(c+d*x)^(3/2)*sqrt(a+a*cos(c+d*x)))+16/105*a*(6*A+7*B)*sin(c+d*x)/(d*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x)))],
[cos(c+d*x)^(3/2)*(a+a*cos(c+d*x))^(3/2)*(A+B*cos(c+d*x)),x,6,1/64*a^(3/2)*(88*A+75*B)*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))/d+1/96*a^2*(88*A+75*B)*cos(c+d*x)^(3/2)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+1/24*a^2*(8*A+9*B)*cos(c+d*x)^(5/2)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+1/64*a^2*(88*A+75*B)*sin(c+d*x)*sqrt(cos(c+d*x))/(d*sqrt(a+a*cos(c+d*x)))+1/4*a*B*cos(c+d*x)^(5/2)*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/d],
[cos(c+d*x)^(1/2)*(a+a*cos(c+d*x))^(3/2)*(A+B*cos(c+d*x)),x,5,1/8*a^(3/2)*(14*A+11*B)*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))/d+1/12*a^2*(6*A+7*B)*cos(c+d*x)^(3/2)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+1/8*a^2*(14*A+11*B)*sin(c+d*x)*sqrt(cos(c+d*x))/(d*sqrt(a+a*cos(c+d*x)))+1/3*a*B*cos(c+d*x)^(3/2)*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/d],
[(a+a*cos(c+d*x))^(3/2)*(A+B*cos(c+d*x))/cos(c+d*x)^(1/2),x,4,1/4*a^(3/2)*(12*A+7*B)*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))/d+1/4*a^2*(4*A+5*B)*sin(c+d*x)*sqrt(cos(c+d*x))/(d*sqrt(a+a*cos(c+d*x)))+1/2*a*B*sin(c+d*x)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))/d],
[(a+a*cos(c+d*x))^(3/2)*(A+B*cos(c+d*x))/cos(c+d*x)^(3/2),x,4,a^(3/2)*(2*A+3*B)*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))/d-a^2*(2*A-B)*sin(c+d*x)*sqrt(cos(c+d*x))/(d*sqrt(a+a*cos(c+d*x)))+2*a*A*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(a+a*cos(c+d*x))^(3/2)*(A+B*cos(c+d*x))/cos(c+d*x)^(5/2),x,4,2*a^(3/2)*B*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))/d+2/3*a^2*(4*A+3*B)*sin(c+d*x)/(d*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x)))+2/3*a*A*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/(d*cos(c+d*x)^(3/2))],
[(a+a*cos(c+d*x))^(3/2)*(A+B*cos(c+d*x))/cos(c+d*x)^(7/2),x,3,2/15*a^2*(6*A+5*B)*sin(c+d*x)/(d*cos(c+d*x)^(3/2)*sqrt(a+a*cos(c+d*x)))+2/15*a^2*(18*A+25*B)*sin(c+d*x)/(d*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x)))+2/5*a*A*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/(d*cos(c+d*x)^(5/2))],
[(a+a*cos(c+d*x))^(3/2)*(A+B*cos(c+d*x))/cos(c+d*x)^(9/2),x,4,2/35*a^2*(8*A+7*B)*sin(c+d*x)/(d*cos(c+d*x)^(5/2)*sqrt(a+a*cos(c+d*x)))+2/105*a^2*(52*A+63*B)*sin(c+d*x)/(d*cos(c+d*x)^(3/2)*sqrt(a+a*cos(c+d*x)))+4/105*a^2*(52*A+63*B)*sin(c+d*x)/(d*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x)))+2/7*a*A*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/(d*cos(c+d*x)^(7/2))],
[(a+a*cos(c+d*x))^(3/2)*(A+B*cos(c+d*x))/cos(c+d*x)^(11/2),x,5,2/63*a^2*(10*A+9*B)*sin(c+d*x)/(d*cos(c+d*x)^(7/2)*sqrt(a+a*cos(c+d*x)))+2/105*a^2*(34*A+39*B)*sin(c+d*x)/(d*cos(c+d*x)^(5/2)*sqrt(a+a*cos(c+d*x)))+8/315*a^2*(34*A+39*B)*sin(c+d*x)/(d*cos(c+d*x)^(3/2)*sqrt(a+a*cos(c+d*x)))+16/315*a^2*(34*A+39*B)*sin(c+d*x)/(d*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x)))+2/9*a*A*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/(d*cos(c+d*x)^(9/2))],
[cos(c+d*x)^(3/2)*(a+a*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x)),x,7,1/128*a^(5/2)*(326*A+283*B)*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))/d+1/5*a*B*cos(c+d*x)^(5/2)*(a+a*cos(c+d*x))^(3/2)*sin(c+d*x)/d+1/192*a^3*(326*A+283*B)*cos(c+d*x)^(3/2)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+1/240*a^3*(170*A+157*B)*cos(c+d*x)^(5/2)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+1/128*a^3*(326*A+283*B)*sin(c+d*x)*sqrt(cos(c+d*x))/(d*sqrt(a+a*cos(c+d*x)))+1/40*a^2*(10*A+13*B)*cos(c+d*x)^(5/2)*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/d],
[cos(c+d*x)^(1/2)*(a+a*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x)),x,6,1/64*a^(5/2)*(200*A+163*B)*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))/d+1/4*a*B*cos(c+d*x)^(3/2)*(a+a*cos(c+d*x))^(3/2)*sin(c+d*x)/d+1/96*a^3*(104*A+95*B)*cos(c+d*x)^(3/2)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+1/64*a^3*(200*A+163*B)*sin(c+d*x)*sqrt(cos(c+d*x))/(d*sqrt(a+a*cos(c+d*x)))+1/24*a^2*(8*A+11*B)*cos(c+d*x)^(3/2)*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/d],
[(a+a*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x))/cos(c+d*x)^(1/2),x,5,1/8*a^(5/2)*(38*A+25*B)*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))/d+1/3*a*B*(a+a*cos(c+d*x))^(3/2)*sin(c+d*x)*sqrt(cos(c+d*x))/d+1/24*a^3*(54*A+49*B)*sin(c+d*x)*sqrt(cos(c+d*x))/(d*sqrt(a+a*cos(c+d*x)))+1/4*a^2*(2*A+3*B)*sin(c+d*x)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))/d],
[(a+a*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x))/cos(c+d*x)^(3/2),x,5,1/4*a^(5/2)*(20*A+19*B)*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))/d+2*a*A*(a+a*cos(c+d*x))^(3/2)*sin(c+d*x)/(d*sqrt(cos(c+d*x)))-1/4*a^3*(4*A-9*B)*sin(c+d*x)*sqrt(cos(c+d*x))/(d*sqrt(a+a*cos(c+d*x)))-1/2*a^2*(4*A-B)*sin(c+d*x)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))/d],
[(a+a*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x))/cos(c+d*x)^(5/2),x,5,a^(5/2)*(2*A+5*B)*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))/d+2/3*a*A*(a+a*cos(c+d*x))^(3/2)*sin(c+d*x)/(d*cos(c+d*x)^(3/2))-1/3*a^3*(14*A+3*B)*sin(c+d*x)*sqrt(cos(c+d*x))/(d*sqrt(a+a*cos(c+d*x)))+2*a^2*(2*A+B)*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(a+a*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x))/cos(c+d*x)^(7/2),x,5,2*a^(5/2)*B*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))/d+2/5*a*A*(a+a*cos(c+d*x))^(3/2)*sin(c+d*x)/(d*cos(c+d*x)^(5/2))+2/15*a^3*(32*A+35*B)*sin(c+d*x)/(d*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x)))+2/15*a^2*(8*A+5*B)*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/(d*cos(c+d*x)^(3/2))],
[(a+a*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x))/cos(c+d*x)^(9/2),x,4,2/7*a*A*(a+a*cos(c+d*x))^(3/2)*sin(c+d*x)/(d*cos(c+d*x)^(7/2))+2/15*a^3*(10*A+11*B)*sin(c+d*x)/(d*cos(c+d*x)^(3/2)*sqrt(a+a*cos(c+d*x)))+2/105*a^3*(230*A+301*B)*sin(c+d*x)/(d*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x)))+2/35*a^2*(10*A+7*B)*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/(d*cos(c+d*x)^(5/2))],
[(a+a*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x))/cos(c+d*x)^(11/2),x,5,2/9*a*A*(a+a*cos(c+d*x))^(3/2)*sin(c+d*x)/(d*cos(c+d*x)^(9/2))+2/315*a^3*(124*A+135*B)*sin(c+d*x)/(d*cos(c+d*x)^(5/2)*sqrt(a+a*cos(c+d*x)))+2/315*a^3*(292*A+345*B)*sin(c+d*x)/(d*cos(c+d*x)^(3/2)*sqrt(a+a*cos(c+d*x)))+4/315*a^3*(292*A+345*B)*sin(c+d*x)/(d*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x)))+2/21*a^2*(4*A+3*B)*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/(d*cos(c+d*x)^(7/2))],
[(a+a*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x))/cos(c+d*x)^(13/2),x,6,2/11*a*A*(a+a*cos(c+d*x))^(3/2)*sin(c+d*x)/(d*cos(c+d*x)^(11/2))+2/693*a^3*(194*A+209*B)*sin(c+d*x)/(d*cos(c+d*x)^(7/2)*sqrt(a+a*cos(c+d*x)))+2/1155*a^3*(710*A+803*B)*sin(c+d*x)/(d*cos(c+d*x)^(5/2)*sqrt(a+a*cos(c+d*x)))+8/3465*a^3*(710*A+803*B)*sin(c+d*x)/(d*cos(c+d*x)^(3/2)*sqrt(a+a*cos(c+d*x)))+16/3465*a^3*(710*A+803*B)*sin(c+d*x)/(d*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x)))+2/99*a^2*(14*A+11*B)*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/(d*cos(c+d*x)^(9/2))],

# n<0
[cos(c+d*x)^(3/2)*(A+B*cos(c+d*x))/(a+a*cos(c+d*x))^(1/2),x,7,-1/4*(4*A-7*B)*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))/(d*sqrt(a))+(A-B)*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))*sqrt(2)/(d*sqrt(a))+1/2*B*cos(c+d*x)^(3/2)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+1/4*(4*A-B)*sin(c+d*x)*sqrt(cos(c+d*x))/(d*sqrt(a+a*cos(c+d*x)))],
[cos(c+d*x)^(1/2)*(A+B*cos(c+d*x))/(a+a*cos(c+d*x))^(1/2),x,6,(2*A-B)*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))/(d*sqrt(a))-(A-B)*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))*sqrt(2)/(d*sqrt(a))+B*sin(c+d*x)*sqrt(cos(c+d*x))/(d*sqrt(a+a*cos(c+d*x)))],
[(A+B*cos(c+d*x))/(cos(c+d*x)^(1/2)*(a+a*cos(c+d*x))^(1/2)),x,5,2*B*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))/(d*sqrt(a))+(A-B)*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))*sqrt(2)/(d*sqrt(a))],
[(A+B*cos(c+d*x))/(cos(c+d*x)^(3/2)*(a+a*cos(c+d*x))^(1/2)),x,4,-(A-B)*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))*sqrt(2)/(d*sqrt(a))+2*A*sin(c+d*x)/(d*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x)))],
[(A+B*cos(c+d*x))/(cos(c+d*x)^(5/2)*(a+a*cos(c+d*x))^(1/2)),x,5,(A-B)*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))*sqrt(2)/(d*sqrt(a))+2/3*A*sin(c+d*x)/(d*cos(c+d*x)^(3/2)*sqrt(a+a*cos(c+d*x)))-2/3*(A-3*B)*sin(c+d*x)/(d*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x)))],
[(A+B*cos(c+d*x))/(cos(c+d*x)^(7/2)*(a+a*cos(c+d*x))^(1/2)),x,6,-(A-B)*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))*sqrt(2)/(d*sqrt(a))+2/5*A*sin(c+d*x)/(d*cos(c+d*x)^(5/2)*sqrt(a+a*cos(c+d*x)))-2/15*(A-5*B)*sin(c+d*x)/(d*cos(c+d*x)^(3/2)*sqrt(a+a*cos(c+d*x)))+2/15*(13*A-5*B)*sin(c+d*x)/(d*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x)))],
[cos(c+d*x)^(3/2)*(A+B*cos(c+d*x))/(a+a*cos(c+d*x))^(3/2),x,7,(2*A-3*B)*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))/(a^(3/2)*d)+1/2*(A-B)*cos(c+d*x)^(3/2)*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(3/2))-1/2*(5*A-9*B)*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))/(a^(3/2)*d*sqrt(2))-1/2*(A-3*B)*sin(c+d*x)*sqrt(cos(c+d*x))/(a*d*sqrt(a+a*cos(c+d*x)))],
[cos(c+d*x)^(1/2)*(A+B*cos(c+d*x))/(a+a*cos(c+d*x))^(3/2),x,6,2*B*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))/(a^(3/2)*d)+1/2*(A-5*B)*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))/(a^(3/2)*d*sqrt(2))+1/2*(A-B)*sin(c+d*x)*sqrt(cos(c+d*x))/(d*(a+a*cos(c+d*x))^(3/2))],
[(A+B*cos(c+d*x))/(cos(c+d*x)^(1/2)*(a+a*cos(c+d*x))^(3/2)),x,4,1/2*(3*A+B)*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))/(a^(3/2)*d*sqrt(2))-1/2*(A-B)*sin(c+d*x)*sqrt(cos(c+d*x))/(d*(a+a*cos(c+d*x))^(3/2))],
[(A+B*cos(c+d*x))/(cos(c+d*x)^(3/2)*(a+a*cos(c+d*x))^(3/2)),x,5,-1/2*(7*A-3*B)*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))/(a^(3/2)*d*sqrt(2))-1/2*(A-B)*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(3/2)*sqrt(cos(c+d*x)))+1/2*(5*A-B)*sin(c+d*x)/(a*d*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x)))],
[(A+B*cos(c+d*x))/(cos(c+d*x)^(5/2)*(a+a*cos(c+d*x))^(3/2)),x,6,-1/2*(A-B)*sin(c+d*x)/(d*cos(c+d*x)^(3/2)*(a+a*cos(c+d*x))^(3/2))+1/2*(11*A-7*B)*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))/(a^(3/2)*d*sqrt(2))+1/6*(7*A-3*B)*sin(c+d*x)/(a*d*cos(c+d*x)^(3/2)*sqrt(a+a*cos(c+d*x)))-1/6*(19*A-15*B)*sin(c+d*x)/(a*d*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x)))],
[cos(c+d*x)^(5/2)*(A+B*cos(c+d*x))/(a+a*cos(c+d*x))^(5/2),x,8,(2*A-5*B)*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))/(a^(5/2)*d)+1/4*(A-B)*cos(c+d*x)^(5/2)*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(5/2))+1/16*(7*A-15*B)*cos(c+d*x)^(3/2)*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^(3/2))-1/16*(43*A-115*B)*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))/(a^(5/2)*d*sqrt(2))-1/16*(11*A-35*B)*sin(c+d*x)*sqrt(cos(c+d*x))/(a^2*d*sqrt(a+a*cos(c+d*x)))],
[cos(c+d*x)^(3/2)*(A+B*cos(c+d*x))/(a+a*cos(c+d*x))^(5/2),x,7,2*B*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))/(a^(5/2)*d)+1/4*(A-B)*cos(c+d*x)^(3/2)*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(5/2))+1/16*(3*A-43*B)*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))/(a^(5/2)*d*sqrt(2))+1/16*(3*A-11*B)*sin(c+d*x)*sqrt(cos(c+d*x))/(a*d*(a+a*cos(c+d*x))^(3/2))],
[cos(c+d*x)^(1/2)*(A+B*cos(c+d*x))/(a+a*cos(c+d*x))^(5/2),x,5,1/16*(5*A+3*B)*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))/(a^(5/2)*d*sqrt(2))+1/4*(A-B)*sin(c+d*x)*sqrt(cos(c+d*x))/(d*(a+a*cos(c+d*x))^(5/2))+1/16*(A+7*B)*sin(c+d*x)*sqrt(cos(c+d*x))/(a*d*(a+a*cos(c+d*x))^(3/2))],
[(A+B*cos(c+d*x))/(cos(c+d*x)^(1/2)*(a+a*cos(c+d*x))^(5/2)),x,5,1/16*(19*A+5*B)*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))/(a^(5/2)*d*sqrt(2))-1/4*(A-B)*sin(c+d*x)*sqrt(cos(c+d*x))/(d*(a+a*cos(c+d*x))^(5/2))-1/16*(9*A-B)*sin(c+d*x)*sqrt(cos(c+d*x))/(a*d*(a+a*cos(c+d*x))^(3/2))],
[(A+B*cos(c+d*x))/(cos(c+d*x)^(3/2)*(a+a*cos(c+d*x))^(5/2)),x,6,-1/16*(75*A-19*B)*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))/(a^(5/2)*d*sqrt(2))-1/4*(A-B)*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(5/2)*sqrt(cos(c+d*x)))-1/16*(13*A-5*B)*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^(3/2)*sqrt(cos(c+d*x)))+1/16*(49*A-9*B)*sin(c+d*x)/(a^2*d*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x)))],
[(A+B*cos(c+d*x))/(cos(c+d*x)^(5/2)*(a+a*cos(c+d*x))^(5/2)),x,7,-1/4*(A-B)*sin(c+d*x)/(d*cos(c+d*x)^(3/2)*(a+a*cos(c+d*x))^(5/2))-1/16*(17*A-9*B)*sin(c+d*x)/(a*d*cos(c+d*x)^(3/2)*(a+a*cos(c+d*x))^(3/2))+1/16*(163*A-75*B)*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))/(a^(5/2)*d*sqrt(2))+1/48*(95*A-39*B)*sin(c+d*x)/(a^2*d*cos(c+d*x)^(3/2)*sqrt(a+a*cos(c+d*x)))-1/48*(299*A-147*B)*sin(c+d*x)/(a^2*d*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x)))],
[cos(c+d*x)^(7/2)*(A+B*cos(c+d*x))/(a+a*cos(c+d*x))^(7/2),x,9,(2*A-7*B)*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))/(a^(7/2)*d)+1/6*(A-B)*cos(c+d*x)^(7/2)*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(7/2))+1/16*(3*A-7*B)*cos(c+d*x)^(5/2)*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^(5/2))+1/192*(79*A-259*B)*cos(c+d*x)^(3/2)*sin(c+d*x)/(a^2*d*(a+a*cos(c+d*x))^(3/2))-1/64*(177*A-637*B)*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))/(a^(7/2)*d*sqrt(2))-7/64*(7*A-27*B)*sin(c+d*x)*sqrt(cos(c+d*x))/(a^3*d*sqrt(a+a*cos(c+d*x)))],
[cos(c+d*x)^(5/2)*(A+B*cos(c+d*x))/(a+a*cos(c+d*x))^(7/2),x,8,2*B*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))/(a^(7/2)*d)+1/6*(A-B)*cos(c+d*x)^(5/2)*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(7/2))+1/48*(5*A-17*B)*cos(c+d*x)^(3/2)*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^(5/2))+1/64*(5*A-177*B)*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))/(a^(7/2)*d*sqrt(2))+1/64*(5*A-49*B)*sin(c+d*x)*sqrt(cos(c+d*x))/(a^2*d*(a+a*cos(c+d*x))^(3/2))],
[cos(c+d*x)^(3/2)*(A+B*cos(c+d*x))/(a+a*cos(c+d*x))^(7/2),x,6,1/6*(A-B)*cos(c+d*x)^(3/2)*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(7/2))+1/64*(7*A+5*B)*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))/(a^(7/2)*d*sqrt(2))+1/48*(A-13*B)*sin(c+d*x)*sqrt(cos(c+d*x))/(a*d*(a+a*cos(c+d*x))^(5/2))+1/192*(17*A+67*B)*sin(c+d*x)*sqrt(cos(c+d*x))/(a^2*d*(a+a*cos(c+d*x))^(3/2))],
[cos(c+d*x)^(1/2)*(A+B*cos(c+d*x))/(a+a*cos(c+d*x))^(7/2),x,6,1/64*(13*A+7*B)*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))/(a^(7/2)*d*sqrt(2))+1/6*(A-B)*sin(c+d*x)*sqrt(cos(c+d*x))/(d*(a+a*cos(c+d*x))^(7/2))+1/16*(A+3*B)*sin(c+d*x)*sqrt(cos(c+d*x))/(a*d*(a+a*cos(c+d*x))^(5/2))-1/192*(5*A-17*B)*sin(c+d*x)*sqrt(cos(c+d*x))/(a^2*d*(a+a*cos(c+d*x))^(3/2))],
[(A+B*cos(c+d*x))/(cos(c+d*x)^(1/2)*(a+a*cos(c+d*x))^(7/2)),x,6,1/64*(63*A+13*B)*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))/(a^(7/2)*d*sqrt(2))-1/6*(A-B)*sin(c+d*x)*sqrt(cos(c+d*x))/(d*(a+a*cos(c+d*x))^(7/2))-1/16*(5*A-B)*sin(c+d*x)*sqrt(cos(c+d*x))/(a*d*(a+a*cos(c+d*x))^(5/2))-1/192*(103*A+5*B)*sin(c+d*x)*sqrt(cos(c+d*x))/(a^2*d*(a+a*cos(c+d*x))^(3/2))],
[(A+B*cos(c+d*x))/(cos(c+d*x)^(3/2)*(a+a*cos(c+d*x))^(7/2)),x,7,-3/64*(121*A-21*B)*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))/(a^(7/2)*d*sqrt(2))-1/6*(A-B)*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(7/2)*sqrt(cos(c+d*x)))-1/48*(19*A-7*B)*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^(5/2)*sqrt(cos(c+d*x)))-1/192*(199*A-43*B)*sin(c+d*x)/(a^2*d*(a+a*cos(c+d*x))^(3/2)*sqrt(cos(c+d*x)))+1/192*(691*A-103*B)*sin(c+d*x)/(a^3*d*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x)))],
[(A+B*cos(c+d*x))/(cos(c+d*x)^(5/2)*(a+a*cos(c+d*x))^(7/2)),x,8,-1/6*(A-B)*sin(c+d*x)/(d*cos(c+d*x)^(3/2)*(a+a*cos(c+d*x))^(7/2))-1/48*(23*A-11*B)*sin(c+d*x)/(a*d*cos(c+d*x)^(3/2)*(a+a*cos(c+d*x))^(5/2))-1/64*(109*A-41*B)*sin(c+d*x)/(a^2*d*cos(c+d*x)^(3/2)*(a+a*cos(c+d*x))^(3/2))+1/64*(1015*A-363*B)*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))/(a^(7/2)*d*sqrt(2))+1/192*(579*A-199*B)*sin(c+d*x)/(a^3*d*cos(c+d*x)^(3/2)*sqrt(a+a*cos(c+d*x)))-1/192*(1887*A-691*B)*sin(c+d*x)/(a^3*d*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x)))],

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

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

# n>0
[cos(c+d*x)^2*(a+b*cos(c+d*x))*(A+B*cos(c+d*x)),x,7,1/8*(4*a*A+3*b*B)*x+(A*b+a*B)*sin(c+d*x)/d+1/8*(4*a*A+3*b*B)*cos(c+d*x)*sin(c+d*x)/d+1/4*b*B*cos(c+d*x)^3*sin(c+d*x)/d-1/3*(A*b+a*B)*sin(c+d*x)^3/d],
[cos(c+d*x)*(a+b*cos(c+d*x))*(A+B*cos(c+d*x)),x,3,1/2*(A*b+a*B)*x+1/3*(3*a*A+2*b*B)*sin(c+d*x)/d+1/2*(A*b+a*B)*cos(c+d*x)*sin(c+d*x)/d+1/3*b*B*cos(c+d*x)^2*sin(c+d*x)/d],
[(a+b*cos(c+d*x))*(A+B*cos(c+d*x)),x,1,1/2*(2*a*A+b*B)*x+(A*b+a*B)*sin(c+d*x)/d+1/2*b*B*cos(c+d*x)*sin(c+d*x)/d],
[(a+b*cos(c+d*x))*(A+B*cos(c+d*x))*sec(c+d*x),x,4,(A*b+a*B)*x+a*A*arctanh(sin(c+d*x))/d+b*B*sin(c+d*x)/d],
[(a+b*cos(c+d*x))*(A+B*cos(c+d*x))*sec(c+d*x)^2,x,4,b*B*x+(A*b+a*B)*arctanh(sin(c+d*x))/d+a*A*tan(c+d*x)/d],
[(a+b*cos(c+d*x))*(A+B*cos(c+d*x))*sec(c+d*x)^3,x,6,1/2*(a*A+2*b*B)*arctanh(sin(c+d*x))/d+(A*b+a*B)*tan(c+d*x)/d+1/2*a*A*sec(c+d*x)*tan(c+d*x)/d],
[(a+b*cos(c+d*x))*(A+B*cos(c+d*x))*sec(c+d*x)^4,x,7,1/2*(A*b+a*B)*arctanh(sin(c+d*x))/d+1/3*(2*a*A+3*b*B)*tan(c+d*x)/d+1/2*(A*b+a*B)*sec(c+d*x)*tan(c+d*x)/d+1/3*a*A*sec(c+d*x)^2*tan(c+d*x)/d],
[(a+b*cos(c+d*x))*(A+B*cos(c+d*x))*sec(c+d*x)^5,x,7,1/8*(3*a*A+4*b*B)*arctanh(sin(c+d*x))/d+(A*b+a*B)*tan(c+d*x)/d+1/8*(3*a*A+4*b*B)*sec(c+d*x)*tan(c+d*x)/d+1/4*a*A*sec(c+d*x)^3*tan(c+d*x)/d+1/3*(A*b+a*B)*tan(c+d*x)^3/d],
[cos(c+d*x)^2*(a+b*cos(c+d*x))^2*(A+B*cos(c+d*x)),x,7,1/8*(4*a^2*A+3*A*b^2+6*a*b*B)*x+1/5*(4*b^2*B+5*a*(2*A*b+a*B))*sin(c+d*x)/d+1/8*(4*a^2*A+3*A*b^2+6*a*b*B)*cos(c+d*x)*sin(c+d*x)/d+1/20*b*(5*A*b+6*a*B)*cos(c+d*x)^3*sin(c+d*x)/d+1/5*b*B*cos(c+d*x)^3*(a+b*cos(c+d*x))*sin(c+d*x)/d-1/15*(4*b^2*B+5*a*(2*A*b+a*B))*sin(c+d*x)^3/d],
[cos(c+d*x)*(a+b*cos(c+d*x))^2*(A+B*cos(c+d*x)),x,4,1/8*(8*a*A*b+4*a^2*B+3*b^2*B)*x+1/6*(4*a^2*A*b+4*A*b^3-a^3*B+8*a*b^2*B)*sin(c+d*x)/(b*d)+1/24*(8*a*A*b-2*a^2*B+9*b^2*B)*cos(c+d*x)*sin(c+d*x)/d+1/12*(4*A*b-a*B)*(a+b*cos(c+d*x))^2*sin(c+d*x)/(b*d)+1/4*B*(a+b*cos(c+d*x))^3*sin(c+d*x)/(b*d)],
[(a+b*cos(c+d*x))^2*(A+B*cos(c+d*x)),x,2,1/2*(2*a^2*A+A*b^2+2*a*b*B)*x+2/3*(3*a*A*b+a^2*B+b^2*B)*sin(c+d*x)/d+1/6*b*(3*A*b+2*a*B)*cos(c+d*x)*sin(c+d*x)/d+1/3*B*(a+b*cos(c+d*x))^2*sin(c+d*x)/d],
[(a+b*cos(c+d*x))^2*(A+B*cos(c+d*x))*sec(c+d*x),x,4,1/2*(4*a*A*b+2*a^2*B+b^2*B)*x+a^2*A*arctanh(sin(c+d*x))/d+1/2*b*(2*A*b+3*a*B)*sin(c+d*x)/d+1/2*b*B*(a+b*cos(c+d*x))*sin(c+d*x)/d],
[(a+b*cos(c+d*x))^2*(A+B*cos(c+d*x))*sec(c+d*x)^2,x,4,b*(A*b+2*a*B)*x+a*(2*A*b+a*B)*arctanh(sin(c+d*x))/d+b^2*B*sin(c+d*x)/d+a^2*A*tan(c+d*x)/d],
[(a+b*cos(c+d*x))^2*(A+B*cos(c+d*x))*sec(c+d*x)^3,x,4,b^2*B*x+1/2*(a^2*A+2*A*b^2+4*a*b*B)*arctanh(sin(c+d*x))/d+a*(2*A*b+a*B)*tan(c+d*x)/d+1/2*a^2*A*sec(c+d*x)*tan(c+d*x)/d],
[(a+b*cos(c+d*x))^2*(A+B*cos(c+d*x))*sec(c+d*x)^4,x,6,1/2*(2*a*A*b+a^2*B+2*b^2*B)*arctanh(sin(c+d*x))/d+1/3*(2*a^2*A+3*A*b^2+6*a*b*B)*tan(c+d*x)/d+1/2*a*(2*A*b+a*B)*sec(c+d*x)*tan(c+d*x)/d+1/3*a^2*A*sec(c+d*x)^2*tan(c+d*x)/d],
[(a+b*cos(c+d*x))^2*(A+B*cos(c+d*x))*sec(c+d*x)^5,x,7,1/8*(3*a^2*A+4*A*b^2+8*a*b*B)*arctanh(sin(c+d*x))/d+1/3*(4*a*A*b+2*a^2*B+3*b^2*B)*tan(c+d*x)/d+1/8*(3*a^2*A+4*A*b^2+8*a*b*B)*sec(c+d*x)*tan(c+d*x)/d+1/3*a*(2*A*b+a*B)*sec(c+d*x)^2*tan(c+d*x)/d+1/4*a^2*A*sec(c+d*x)^3*tan(c+d*x)/d],
[cos(c+d*x)^2*(a+b*cos(c+d*x))^3*(A+B*cos(c+d*x)),x,8,1/16*(8*a^3*A+18*a*A*b^2+18*a^2*b*B+5*b^3*B)*x+1/5*(15*a^2*A*b+4*A*b^3+5*a^3*B+12*a*b^2*B)*sin(c+d*x)/d+1/16*(8*a^3*A+18*a*A*b^2+18*a^2*b*B+5*b^3*B)*cos(c+d*x)*sin(c+d*x)/d+1/24*b*(18*a*A*b+14*a^2*B+5*b^2*B)*cos(c+d*x)^3*sin(c+d*x)/d+1/15*b^2*(3*A*b+4*a*B)*cos(c+d*x)^4*sin(c+d*x)/d+1/6*b*B*cos(c+d*x)^3*(a+b*cos(c+d*x))^2*sin(c+d*x)/d-1/15*(15*a^2*A*b+4*A*b^3+5*a^3*B+12*a*b^2*B)*sin(c+d*x)^3/d],
[cos(c+d*x)*(a+b*cos(c+d*x))^3*(A+B*cos(c+d*x)),x,5,1/8*(12*a^2*A*b+3*A*b^3+4*a^3*B+9*a*b^2*B)*x+1/30*(15*a^3*A*b+60*a*A*b^3-3*a^4*B+52*a^2*b^2*B+16*b^4*B)*sin(c+d*x)/(b*d)+1/120*(30*a^2*A*b+45*A*b^3-6*a^3*B+71*a*b^2*B)*cos(c+d*x)*sin(c+d*x)/d+1/60*(15*a*A*b-3*a^2*B+16*b^2*B)*(a+b*cos(c+d*x))^2*sin(c+d*x)/(b*d)+1/20*(5*A*b-a*B)*(a+b*cos(c+d*x))^3*sin(c+d*x)/(b*d)+1/5*B*(a+b*cos(c+d*x))^4*sin(c+d*x)/(b*d)],
[(a+b*cos(c+d*x))^3*(A+B*cos(c+d*x)),x,3,1/8*(8*a^3*A+12*a*A*b^2+12*a^2*b*B+3*b^3*B)*x+1/6*(16*a^2*A*b+4*A*b^3+3*a^3*B+12*a*b^2*B)*sin(c+d*x)/d+1/24*b*(20*a*A*b+6*a^2*B+9*b^2*B)*cos(c+d*x)*sin(c+d*x)/d+1/12*(4*A*b+3*a*B)*(a+b*cos(c+d*x))^2*sin(c+d*x)/d+1/4*B*(a+b*cos(c+d*x))^3*sin(c+d*x)/d],
[(a+b*cos(c+d*x))^3*(A+B*cos(c+d*x))*sec(c+d*x),x,5,1/2*(6*a^2*A*b+A*b^3+2*a^3*B+3*a*b^2*B)*x+a^3*A*arctanh(sin(c+d*x))/d+1/3*b*(9*a*A*b+8*a^2*B+2*b^2*B)*sin(c+d*x)/d+1/6*b^2*(3*A*b+5*a*B)*cos(c+d*x)*sin(c+d*x)/d+1/3*b*B*(a+b*cos(c+d*x))^2*sin(c+d*x)/d],
[(a+b*cos(c+d*x))^3*(A+B*cos(c+d*x))*sec(c+d*x)^2,x,5,1/2*b*(6*a*A*b+6*a^2*B+b^2*B)*x+a^2*(3*A*b+a*B)*arctanh(sin(c+d*x))/d-b*(2*a^2*A-A*b^2-3*a*b*B)*sin(c+d*x)/d-1/2*b^2*(2*a*A-b*B)*cos(c+d*x)*sin(c+d*x)/d+a*A*(a+b*cos(c+d*x))^2*tan(c+d*x)/d],
[(a+b*cos(c+d*x))^3*(A+B*cos(c+d*x))*sec(c+d*x)^3,x,5,b^2*(A*b+3*a*B)*x+1/2*a*(a^2*A+6*A*b^2+6*a*b*B)*arctanh(sin(c+d*x))/d-1/2*b^2*(a*A-2*b*B)*sin(c+d*x)/d+a^2*(2*A*b+a*B)*tan(c+d*x)/d+1/2*a*A*(a+b*cos(c+d*x))^2*sec(c+d*x)*tan(c+d*x)/d],
[(a+b*cos(c+d*x))^3*(A+B*cos(c+d*x))*sec(c+d*x)^4,x,5,b^3*B*x+1/2*(3*a^2*A*b+2*A*b^3+a^3*B+6*a*b^2*B)*arctanh(sin(c+d*x))/d+1/3*a*(2*a^2*A+8*A*b^2+9*a*b*B)*tan(c+d*x)/d+1/6*a^2*(5*A*b+3*a*B)*sec(c+d*x)*tan(c+d*x)/d+1/3*a*A*(a+b*cos(c+d*x))^2*sec(c+d*x)^2*tan(c+d*x)/d],
[(a+b*cos(c+d*x))^3*(A+B*cos(c+d*x))*sec(c+d*x)^5,x,7,1/8*(3*a^3*A+12*a*A*b^2+12*a^2*b*B+8*b^3*B)*arctanh(sin(c+d*x))/d+1/3*(6*a^2*A*b+3*A*b^3+2*a^3*B+9*a*b^2*B)*tan(c+d*x)/d+1/8*a*(3*a^2*A+10*A*b^2+12*a*b*B)*sec(c+d*x)*tan(c+d*x)/d+1/6*a^2*(3*A*b+2*a*B)*sec(c+d*x)^2*tan(c+d*x)/d+1/4*a*A*(a+b*cos(c+d*x))^2*sec(c+d*x)^3*tan(c+d*x)/d],
[(a+b*cos(c+d*x))^3*(A+B*cos(c+d*x))*sec(c+d*x)^6,x,8,1/8*(9*a^2*A*b+4*A*b^3+3*a^3*B+12*a*b^2*B)*arctanh(sin(c+d*x))/d+1/15*(8*a^3*A+30*a*A*b^2+30*a^2*b*B+15*b^3*B)*tan(c+d*x)/d+1/8*(9*a^2*A*b+4*A*b^3+3*a^3*B+12*a*b^2*B)*sec(c+d*x)*tan(c+d*x)/d+1/15*a*(4*a^2*A+12*A*b^2+15*a*b*B)*sec(c+d*x)^2*tan(c+d*x)/d+1/20*a^2*(7*A*b+5*a*B)*sec(c+d*x)^3*tan(c+d*x)/d+1/5*a*A*(a+b*cos(c+d*x))^2*sec(c+d*x)^4*tan(c+d*x)/d],
[cos(c+d*x)^2*(a+b*cos(c+d*x))^4*(A+B*cos(c+d*x)),x,9,1/16*(8*a^4*A+36*a^2*A*b^2+5*A*b^4+24*a^3*b*B+20*a*b^3*B)*x+1/35*(140*a^3*A*b+112*a*A*b^3+35*a^4*B+168*a^2*b^2*B+24*b^4*B)*sin(c+d*x)/d+1/16*(8*a^4*A+36*a^2*A*b^2+5*A*b^4+24*a^3*b*B+20*a*b^3*B)*cos(c+d*x)*sin(c+d*x)/d+1/168*b*(224*a^2*A*b+35*A*b^3+104*a^3*B+140*a*b^2*B)*cos(c+d*x)^3*sin(c+d*x)/d+1/105*b^2*(49*a*A*b+31*a^2*B+18*b^2*B)*cos(c+d*x)^4*sin(c+d*x)/d+1/42*b*(7*A*b+10*a*B)*cos(c+d*x)^3*(a+b*cos(c+d*x))^2*sin(c+d*x)/d+1/7*b*B*cos(c+d*x)^3*(a+b*cos(c+d*x))^3*sin(c+d*x)/d-1/105*(140*a^3*A*b+112*a*A*b^3+35*a^4*B+168*a^2*b^2*B+24*b^4*B)*sin(c+d*x)^3/d],
[cos(c+d*x)*(a+b*cos(c+d*x))^4*(A+B*cos(c+d*x)),x,6,1/16*(32*a^3*A*b+24*a*A*b^3+8*a^4*B+36*a^2*b^2*B+5*b^4*B)*x+1/60*(24*a^4*A*b+224*a^2*A*b^3+32*A*b^5-4*a^5*B+121*a^3*b^2*B+128*a*b^4*B)*sin(c+d*x)/(b*d)+1/240*(48*a^3*A*b+232*a*A*b^3-8*a^4*B+178*a^2*b^2*B+75*b^4*B)*cos(c+d*x)*sin(c+d*x)/d+1/120*(24*a^2*A*b+32*A*b^3-4*a^3*B+53*a*b^2*B)*(a+b*cos(c+d*x))^2*sin(c+d*x)/(b*d)+1/120*(24*a*A*b-4*a^2*B+25*b^2*B)*(a+b*cos(c+d*x))^3*sin(c+d*x)/(b*d)+1/30*(6*A*b-a*B)*(a+b*cos(c+d*x))^4*sin(c+d*x)/(b*d)+1/6*B*(a+b*cos(c+d*x))^5*sin(c+d*x)/(b*d)],
[(a+b*cos(c+d*x))^4*(A+B*cos(c+d*x)),x,4,1/8*(8*a^4*A+24*a^2*A*b^2+3*A*b^4+16*a^3*b*B+12*a*b^3*B)*x+1/30*(95*a^3*A*b+80*a*A*b^3+12*a^4*B+112*a^2*b^2*B+16*b^4*B)*sin(c+d*x)/d+1/120*b*(130*a^2*A*b+45*A*b^3+24*a^3*B+116*a*b^2*B)*cos(c+d*x)*sin(c+d*x)/d+1/60*(35*a*A*b+12*a^2*B+16*b^2*B)*(a+b*cos(c+d*x))^2*sin(c+d*x)/d+1/20*(5*A*b+4*a*B)*(a+b*cos(c+d*x))^3*sin(c+d*x)/d+1/5*B*(a+b*cos(c+d*x))^4*sin(c+d*x)/d],
[(a+b*cos(c+d*x))^4*(A+B*cos(c+d*x))*sec(c+d*x),x,6,1/8*(32*a^3*A*b+16*a*A*b^3+8*a^4*B+24*a^2*b^2*B+3*b^4*B)*x+a^4*A*arctanh(sin(c+d*x))/d+1/6*b*(34*a^2*A*b+4*A*b^3+19*a^3*B+16*a*b^2*B)*sin(c+d*x)/d+1/24*b^2*(32*a*A*b+26*a^2*B+9*b^2*B)*cos(c+d*x)*sin(c+d*x)/d+1/12*b*(4*A*b+7*a*B)*(a+b*cos(c+d*x))^2*sin(c+d*x)/d+1/4*b*B*(a+b*cos(c+d*x))^3*sin(c+d*x)/d],
[(a+b*cos(c+d*x))^4*(A+B*cos(c+d*x))*sec(c+d*x)^2,x,6,1/2*b*(12*a^2*A*b+A*b^3+8*a^3*B+4*a*b^2*B)*x+a^3*(4*A*b+a*B)*arctanh(sin(c+d*x))/d-1/3*b*(6*a^3*A-12*a*A*b^2-17*a^2*b*B-2*b^3*B)*sin(c+d*x)/d-1/6*b^2*(6*a^2*A-3*A*b^2-8*a*b*B)*cos(c+d*x)*sin(c+d*x)/d-1/3*b*(3*a*A-b*B)*(a+b*cos(c+d*x))^2*sin(c+d*x)/d+a*A*(a+b*cos(c+d*x))^3*tan(c+d*x)/d],
[(a+b*cos(c+d*x))^4*(A+B*cos(c+d*x))*sec(c+d*x)^3,x,6,1/2*b^2*(8*a*A*b+12*a^2*B+b^2*B)*x+1/2*a^2*(a^2*A+12*A*b^2+8*a*b*B)*arctanh(sin(c+d*x))/d-1/2*b*(13*a^2*A*b-2*A*b^3+4*a^3*B-8*a*b^2*B)*sin(c+d*x)/d-1/2*b^2*(6*a*A*b+2*a^2*B-b^2*B)*cos(c+d*x)*sin(c+d*x)/d+1/2*a*(5*A*b+2*a*B)*(a+b*cos(c+d*x))^2*tan(c+d*x)/d+1/2*a*A*(a+b*cos(c+d*x))^3*sec(c+d*x)*tan(c+d*x)/d],
[(a+b*cos(c+d*x))^4*(A+B*cos(c+d*x))*sec(c+d*x)^4,x,6,b^3*(A*b+4*a*B)*x+1/2*a*(4*a^2*A*b+8*A*b^3+a^3*B+12*a*b^2*B)*arctanh(sin(c+d*x))/d-1/6*b^2*(8*a*A*b+3*a^2*B-6*b^2*B)*sin(c+d*x)/d+1/3*a^2*(2*a^2*A+9*A*b^2+9*a*b*B)*tan(c+d*x)/d+1/2*a*(2*A*b+a*B)*(a+b*cos(c+d*x))^2*sec(c+d*x)*tan(c+d*x)/d+1/3*a*A*(a+b*cos(c+d*x))^3*sec(c+d*x)^2*tan(c+d*x)/d],
[(a+b*cos(c+d*x))^4*(A+B*cos(c+d*x))*sec(c+d*x)^5,x,6,b^4*B*x+1/8*(3*a^4*A+24*a^2*A*b^2+8*A*b^4+16*a^3*b*B+32*a*b^3*B)*arctanh(sin(c+d*x))/d+1/6*a*(16*a^2*A*b+19*A*b^3+4*a^3*B+34*a*b^2*B)*tan(c+d*x)/d+1/24*a^2*(9*a^2*A+26*A*b^2+32*a*b*B)*sec(c+d*x)*tan(c+d*x)/d+1/12*a*(7*A*b+4*a*B)*(a+b*cos(c+d*x))^2*sec(c+d*x)^2*tan(c+d*x)/d+1/4*a*A*(a+b*cos(c+d*x))^3*sec(c+d*x)^3*tan(c+d*x)/d],
[(a+b*cos(c+d*x))^4*(A+B*cos(c+d*x))*sec(c+d*x)^6,x,8,1/8*(12*a^3*A*b+16*a*A*b^3+3*a^4*B+24*a^2*b^2*B+8*b^4*B)*arctanh(sin(c+d*x))/d+1/15*(8*a^4*A+60*a^2*A*b^2+15*A*b^4+40*a^3*b*B+60*a*b^3*B)*tan(c+d*x)/d+1/40*a*(60*a^2*A*b+56*A*b^3+15*a^3*B+110*a*b^2*B)*sec(c+d*x)*tan(c+d*x)/d+1/30*a^2*(8*a^2*A+18*A*b^2+25*a*b*B)*sec(c+d*x)^2*tan(c+d*x)/d+1/20*a*(8*A*b+5*a*B)*(a+b*cos(c+d*x))^2*sec(c+d*x)^3*tan(c+d*x)/d+1/5*a*A*(a+b*cos(c+d*x))^3*sec(c+d*x)^4*tan(c+d*x)/d],
[(a+b*cos(c+d*x))^4*(A+B*cos(c+d*x))*sec(c+d*x)^7,x,9,1/16*(5*a^4*A+36*a^2*A*b^2+8*A*b^4+24*a^3*b*B+32*a*b^3*B)*arctanh(sin(c+d*x))/d+1/15*(32*a^3*A*b+40*a*A*b^3+8*a^4*B+60*a^2*b^2*B+15*b^4*B)*tan(c+d*x)/d+1/16*(5*a^4*A+36*a^2*A*b^2+8*A*b^4+24*a^3*b*B+32*a*b^3*B)*sec(c+d*x)*tan(c+d*x)/d+1/15*a*(16*a^2*A*b+13*A*b^3+4*a^3*B+27*a*b^2*B)*sec(c+d*x)^2*tan(c+d*x)/d+1/120*a^2*(25*a^2*A+48*A*b^2+72*a*b*B)*sec(c+d*x)^3*tan(c+d*x)/d+1/10*a*(3*A*b+2*a*B)*(a+b*cos(c+d*x))^2*sec(c+d*x)^4*tan(c+d*x)/d+1/6*a*A*(a+b*cos(c+d*x))^3*sec(c+d*x)^5*tan(c+d*x)/d],

# n<0
[cos(c+d*x)^3*(A+B*cos(c+d*x))/(a+b*cos(c+d*x)),x,6,1/2*(2*a^2+b^2)*(A*b-a*B)*x/b^4-1/3*(3*a*A*b-3*a^2*B-2*b^2*B)*sin(c+d*x)/(b^3*d)+1/2*(A*b-a*B)*cos(c+d*x)*sin(c+d*x)/(b^2*d)+1/3*B*cos(c+d*x)^2*sin(c+d*x)/(b*d)-2*a^3*(A*b-a*B)*arctan(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/(b^4*d*sqrt(a-b)*sqrt(a+b))],
[cos(c+d*x)^2*(A+B*cos(c+d*x))/(a+b*cos(c+d*x)),x,5,-1/2*(2*a*A*b-2*a^2*B-b^2*B)*x/b^3+(A*b-a*B)*sin(c+d*x)/(b^2*d)+1/2*B*cos(c+d*x)*sin(c+d*x)/(b*d)+2*a^2*(A*b-a*B)*arctan(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/(b^3*d*sqrt(a-b)*sqrt(a+b))],
[cos(c+d*x)*(A+B*cos(c+d*x))/(a+b*cos(c+d*x)),x,6,(A*b-a*B)*x/b^2+B*sin(c+d*x)/(b*d)-2*a*(A*b-a*B)*arctan(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/(b^2*d*sqrt(a-b)*sqrt(a+b))],
[(A+B*cos(c+d*x))/(a+b*cos(c+d*x)),x,3,B*x/b+2*(A*b-a*B)*arctan(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/(b*d*sqrt(a-b)*sqrt(a+b))],
[(A+B*cos(c+d*x))*sec(c+d*x)/(a+b*cos(c+d*x)),x,4,A*arctanh(sin(c+d*x))/(a*d)-2*(A*b-a*B)*arctan(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/(a*d*sqrt(a-b)*sqrt(a+b))],
[(A+B*cos(c+d*x))*sec(c+d*x)^2/(a+b*cos(c+d*x)),x,6,-(A*b-a*B)*arctanh(sin(c+d*x))/(a^2*d)+2*b*(A*b-a*B)*arctan(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/(a^2*d*sqrt(a-b)*sqrt(a+b))+A*tan(c+d*x)/(a*d)],
[(A+B*cos(c+d*x))*sec(c+d*x)^3/(a+b*cos(c+d*x)),x,6,1/2*(a^2*A+2*A*b^2-2*a*b*B)*arctanh(sin(c+d*x))/(a^3*d)-2*b^2*(A*b-a*B)*arctan(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/(a^3*d*sqrt(a-b)*sqrt(a+b))-(A*b-a*B)*tan(c+d*x)/(a^2*d)+1/2*A*sec(c+d*x)*tan(c+d*x)/(a*d)],
[(A+B*cos(c+d*x))*sec(c+d*x)^4/(a+b*cos(c+d*x)),x,7,-1/2*(a^2+2*b^2)*(A*b-a*B)*arctanh(sin(c+d*x))/(a^4*d)+2*b^3*(A*b-a*B)*arctan(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/(a^4*d*sqrt(a-b)*sqrt(a+b))+1/3*(2*a^2*A+3*A*b^2-3*a*b*B)*tan(c+d*x)/(a^3*d)-1/2*(A*b-a*B)*sec(c+d*x)*tan(c+d*x)/(a^2*d)+1/3*A*sec(c+d*x)^2*tan(c+d*x)/(a*d)],
[cos(c+d*x)^3*(A+B*cos(c+d*x))/(a+b*cos(c+d*x))^2,x,6,-1/2*(4*a*A*b-6*a^2*B-b^2*B)*x/b^4+2*a^2*(2*a^2*A*b-3*A*b^3-3*a^3*B+4*a*b^2*B)*arctan(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/((a-b)^(3/2)*b^4*(a+b)^(3/2)*d)+(2*a^2*A*b-A*b^3-3*a^3*B+2*a*b^2*B)*sin(c+d*x)/(b^3*(a^2-b^2)*d)-1/2*(2*a*A*b-3*a^2*B+b^2*B)*cos(c+d*x)*sin(c+d*x)/(b^2*(a^2-b^2)*d)+a*(A*b-a*B)*cos(c+d*x)^2*sin(c+d*x)/(b*(a^2-b^2)*d*(a+b*cos(c+d*x)))],
[cos(c+d*x)^2*(A+B*cos(c+d*x))/(a+b*cos(c+d*x))^2,x,5,(A*b-2*a*B)*x/b^3-2*a*(a^2*A*b-2*A*b^3-2*a^3*B+3*a*b^2*B)*arctan(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/((a-b)^(3/2)*b^3*(a+b)^(3/2)*d)+B*sin(c+d*x)/(b^2*d)-a^2*(A*b-a*B)*sin(c+d*x)/(b^2*(a^2-b^2)*d*(a+b*cos(c+d*x)))],
[cos(c+d*x)*(A+B*cos(c+d*x))/(a+b*cos(c+d*x))^2,x,5,B*x/b^2-2*(A*b^3+a^3*B-2*a*b^2*B)*arctan(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/((a-b)^(3/2)*b^2*(a+b)^(3/2)*d)+a*(A*b-a*B)*sin(c+d*x)/(b*(a^2-b^2)*d*(a+b*cos(c+d*x)))],
[(A+B*cos(c+d*x))/(a+b*cos(c+d*x))^2,x,4,2*(a*A-b*B)*arctan(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/((a-b)^(3/2)*(a+b)^(3/2)*d)-(A*b-a*B)*sin(c+d*x)/((a^2-b^2)*d*(a+b*cos(c+d*x)))],
[(A+B*cos(c+d*x))*sec(c+d*x)/(a+b*cos(c+d*x))^2,x,5,-2*(2*a^2*A*b-A*b^3-a^3*B)*arctan(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/(a^2*(a-b)^(3/2)*(a+b)^(3/2)*d)+A*arctanh(sin(c+d*x))/(a^2*d)+b*(A*b-a*B)*sin(c+d*x)/(a*(a^2-b^2)*d*(a+b*cos(c+d*x)))],
[(A+B*cos(c+d*x))*sec(c+d*x)^2/(a+b*cos(c+d*x))^2,x,6,2*b*(3*a^2*A*b-2*A*b^3-2*a^3*B+a*b^2*B)*arctan(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/(a^3*(a-b)^(3/2)*(a+b)^(3/2)*d)-(2*A*b-a*B)*arctanh(sin(c+d*x))/(a^3*d)+(a^2*A-2*A*b^2+a*b*B)*tan(c+d*x)/(a^2*(a^2-b^2)*d)+b*(A*b-a*B)*tan(c+d*x)/(a*(a^2-b^2)*d*(a+b*cos(c+d*x)))],
[(A+B*cos(c+d*x))*sec(c+d*x)^3/(a+b*cos(c+d*x))^2,x,7,-2*b^2*(4*a^2*A*b-3*A*b^3-3*a^3*B+2*a*b^2*B)*arctan(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/(a^4*(a-b)^(3/2)*(a+b)^(3/2)*d)+1/2*(a^2*A+6*A*b^2-4*a*b*B)*arctanh(sin(c+d*x))/(a^4*d)-(2*a^2*A*b-3*A*b^3-a^3*B+2*a*b^2*B)*tan(c+d*x)/(a^3*(a^2-b^2)*d)+1/2*(a^2*A-3*A*b^2+2*a*b*B)*sec(c+d*x)*tan(c+d*x)/(a^2*(a^2-b^2)*d)+b*(A*b-a*B)*sec(c+d*x)*tan(c+d*x)/(a*(a^2-b^2)*d*(a+b*cos(c+d*x)))],
[cos(c+d*x)^4*(A+B*cos(c+d*x))/(a+b*cos(c+d*x))^3,x,7,-1/2*(6*a*A*b-12*a^2*B-b^2*B)*x/b^5+a^2*(6*a^4*A*b-15*a^2*A*b^3+12*A*b^5-12*a^5*B+29*a^3*b^2*B-20*a*b^4*B)*arctan(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/((a-b)^(5/2)*b^5*(a+b)^(5/2)*d)+1/2*(6*a^4*A*b-11*a^2*A*b^3+2*A*b^5-12*a^5*B+21*a^3*b^2*B-6*a*b^4*B)*sin(c+d*x)/(b^4*(a^2-b^2)^2*d)-1/2*(3*a^3*A*b-6*a*A*b^3-6*a^4*B+10*a^2*b^2*B-b^4*B)*cos(c+d*x)*sin(c+d*x)/(b^3*(a^2-b^2)^2*d)+1/2*a*(A*b-a*B)*cos(c+d*x)^3*sin(c+d*x)/(b*(a^2-b^2)*d*(a+b*cos(c+d*x))^2)+1/2*a*(2*a^2*A*b-5*A*b^3-4*a^3*B+7*a*b^2*B)*cos(c+d*x)^2*sin(c+d*x)/(b^2*(a^2-b^2)^2*d*(a+b*cos(c+d*x)))],
[cos(c+d*x)^3*(A+B*cos(c+d*x))/(a+b*cos(c+d*x))^3,x,6,(A*b-3*a*B)*x/b^4-a*(2*a^4*A*b-5*a^2*A*b^3+6*A*b^5-6*a^5*B+15*a^3*b^2*B-12*a*b^4*B)*arctan(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/((a-b)^(5/2)*b^4*(a+b)^(5/2)*d)-1/2*(a*A*b-3*a^2*B+2*b^2*B)*sin(c+d*x)/(b^3*(a^2-b^2)*d)+1/2*a*(A*b-a*B)*cos(c+d*x)^2*sin(c+d*x)/(b*(a^2-b^2)*d*(a+b*cos(c+d*x))^2)-1/2*a^2*(a^2*A*b-4*A*b^3-3*a^3*B+6*a*b^2*B)*sin(c+d*x)/(b^3*(a^2-b^2)^2*d*(a+b*cos(c+d*x)))],
[cos(c+d*x)^2*(A+B*cos(c+d*x))/(a+b*cos(c+d*x))^3,x,5,B*x/b^3+(a^2*A*b^3+2*A*b^5-2*a^5*B+5*a^3*b^2*B-6*a*b^4*B)*arctan(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/((a-b)^(5/2)*b^3*(a+b)^(5/2)*d)-1/2*a^2*(A*b-a*B)*sin(c+d*x)/(b^2*(a^2-b^2)*d*(a+b*cos(c+d*x))^2)+1/2*a*(a^2*A*b-4*A*b^3-3*a^3*B+6*a*b^2*B)*sin(c+d*x)/(b^2*(a^2-b^2)^2*d*(a+b*cos(c+d*x)))],
[cos(c+d*x)*(A+B*cos(c+d*x))/(a+b*cos(c+d*x))^3,x,6,-(3*a*A*b-a^2*B-2*b^2*B)*arctan(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/((a-b)^(5/2)*(a+b)^(5/2)*d)+1/2*a*(A*b-a*B)*sin(c+d*x)/(b*(a^2-b^2)*d*(a+b*cos(c+d*x))^2)+1/2*(a^2*A*b+2*A*b^3+a^3*B-4*a*b^2*B)*sin(c+d*x)/(b*(a^2-b^2)^2*d*(a+b*cos(c+d*x)))],
[(A+B*cos(c+d*x))/(a+b*cos(c+d*x))^3,x,5,(2*a^2*A+A*b^2-3*a*b*B)*arctan(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/((a-b)^(5/2)*(a+b)^(5/2)*d)-1/2*(A*b-a*B)*sin(c+d*x)/((a^2-b^2)*d*(a+b*cos(c+d*x))^2)-1/2*(3*a*A*b-a^2*B-2*b^2*B)*sin(c+d*x)/((a^2-b^2)^2*d*(a+b*cos(c+d*x)))],
[(A+B*cos(c+d*x))*sec(c+d*x)/(a+b*cos(c+d*x))^3,x,6,-(6*a^4*A*b-5*a^2*A*b^3+2*A*b^5-2*a^5*B-a^3*b^2*B)*arctan(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/(a^3*(a-b)^(5/2)*(a+b)^(5/2)*d)+A*arctanh(sin(c+d*x))/(a^3*d)+1/2*b*(A*b-a*B)*sin(c+d*x)/(a*(a^2-b^2)*d*(a+b*cos(c+d*x))^2)+1/2*b*(5*a^2*A*b-2*A*b^3-3*a^3*B)*sin(c+d*x)/(a^2*(a^2-b^2)^2*d*(a+b*cos(c+d*x)))],
[(A+B*cos(c+d*x))*sec(c+d*x)^2/(a+b*cos(c+d*x))^3,x,7,b*(12*a^4*A*b-15*a^2*A*b^3+6*A*b^5-6*a^5*B+5*a^3*b^2*B-2*a*b^4*B)*arctan(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/(a^4*(a-b)^(5/2)*(a+b)^(5/2)*d)-(3*A*b-a*B)*arctanh(sin(c+d*x))/(a^4*d)+1/2*(2*a^4*A-11*a^2*A*b^2+6*A*b^4+5*a^3*b*B-2*a*b^3*B)*tan(c+d*x)/(a^3*(a^2-b^2)^2*d)+1/2*b*(A*b-a*B)*tan(c+d*x)/(a*(a^2-b^2)*d*(a+b*cos(c+d*x))^2)+1/2*b*(6*a^2*A*b-3*A*b^3-4*a^3*B+a*b^2*B)*tan(c+d*x)/(a^2*(a^2-b^2)^2*d*(a+b*cos(c+d*x)))],
[(A+B*cos(c+d*x))*sec(c+d*x)^3/(a+b*cos(c+d*x))^3,x,8,-b^2*(20*a^4*A*b-29*a^2*A*b^3+12*A*b^5-12*a^5*B+15*a^3*b^2*B-6*a*b^4*B)*arctan(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/(a^5*(a-b)^(5/2)*(a+b)^(5/2)*d)+1/2*(a^2*A+12*A*b^2-6*a*b*B)*arctanh(sin(c+d*x))/(a^5*d)-1/2*(6*a^4*A*b-21*a^2*A*b^3+12*A*b^5-2*a^5*B+11*a^3*b^2*B-6*a*b^4*B)*tan(c+d*x)/(a^4*(a^2-b^2)^2*d)+1/2*(a^4*A-10*a^2*A*b^2+6*A*b^4+6*a^3*b*B-3*a*b^3*B)*sec(c+d*x)*tan(c+d*x)/(a^3*(a^2-b^2)^2*d)+1/2*b*(A*b-a*B)*sec(c+d*x)*tan(c+d*x)/(a*(a^2-b^2)*d*(a+b*cos(c+d*x))^2)+1/2*b*(7*a^2*A*b-4*A*b^3-5*a^3*B+2*a*b^2*B)*sec(c+d*x)*tan(c+d*x)/(a^2*(a^2-b^2)^2*d*(a+b*cos(c+d*x)))],
[cos(c+d*x)^4*(A+B*cos(c+d*x))/(a+b*cos(c+d*x))^4,x,7,(A*b-4*a*B)*x/b^5-a*(2*a^6*A*b-7*a^4*A*b^3+8*a^2*A*b^5-8*A*b^7-8*a^7*B+28*a^5*b^2*B-35*a^3*b^4*B+20*a*b^6*B)*arctan(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/((a-b)^(7/2)*b^5*(a+b)^(7/2)*d)-1/6*(3*a^3*A*b-8*a*A*b^3-12*a^4*B+23*a^2*b^2*B-6*b^4*B)*sin(c+d*x)/(b^4*(a^2-b^2)^2*d)+1/3*a*(A*b-a*B)*cos(c+d*x)^3*sin(c+d*x)/(b*(a^2-b^2)*d*(a+b*cos(c+d*x))^3)+1/6*a*(a^2*A*b-6*A*b^3-4*a^3*B+9*a*b^2*B)*cos(c+d*x)^2*sin(c+d*x)/(b^2*(a^2-b^2)^2*d*(a+b*cos(c+d*x))^2)-1/2*a^2*(a^4*A*b-2*a^2*A*b^3+6*A*b^5-4*a^5*B+11*a^3*b^2*B-12*a*b^4*B)*sin(c+d*x)/(b^4*(a^2-b^2)^3*d*(a+b*cos(c+d*x)))],
[cos(c+d*x)^3*(A+B*cos(c+d*x))/(a+b*cos(c+d*x))^4,x,6,B*x/b^4-(3*a^2*A*b^5+2*A*b^7+2*a^7*B-7*a^5*b^2*B+8*a^3*b^4*B-8*a*b^6*B)*arctan(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/((a-b)^(7/2)*b^4*(a+b)^(7/2)*d)+1/3*a*(A*b-a*B)*cos(c+d*x)^2*sin(c+d*x)/(b*(a^2-b^2)*d*(a+b*cos(c+d*x))^3)+1/6*a^2*(5*A*b^3+3*a^3*B-8*a*b^2*B)*sin(c+d*x)/(b^3*(a^2-b^2)^2*d*(a+b*cos(c+d*x))^2)-1/6*a*(a^2*A*b^3-16*A*b^5+9*a^5*B-28*a^3*b^2*B+34*a*b^4*B)*sin(c+d*x)/(b^3*(a^2-b^2)^3*d*(a+b*cos(c+d*x)))],
[cos(c+d*x)^2*(A+B*cos(c+d*x))/(a+b*cos(c+d*x))^4,x,6,(a^3*A+4*a*A*b^2-3*a^2*b*B-2*b^3*B)*arctan(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/((a-b)^(7/2)*(a+b)^(7/2)*d)-1/3*a^2*(A*b-a*B)*sin(c+d*x)/(b^2*(a^2-b^2)*d*(a+b*cos(c+d*x))^3)+1/6*a*(a^2*A*b-6*A*b^3-4*a^3*B+9*a*b^2*B)*sin(c+d*x)/(b^2*(a^2-b^2)^2*d*(a+b*cos(c+d*x))^2)+1/6*(a^4*A*b-10*a^2*A*b^3-6*A*b^5+2*a^5*B-5*a^3*b^2*B+18*a*b^4*B)*sin(c+d*x)/(b^2*(a^2-b^2)^3*d*(a+b*cos(c+d*x)))],
[cos(c+d*x)*(A+B*cos(c+d*x))/(a+b*cos(c+d*x))^4,x,7,-(4*a^2*A*b+A*b^3-a^3*B-4*a*b^2*B)*arctan(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/((a-b)^(7/2)*(a+b)^(7/2)*d)+1/3*a*(A*b-a*B)*sin(c+d*x)/(b*(a^2-b^2)*d*(a+b*cos(c+d*x))^3)+1/6*(2*a^2*A*b+3*A*b^3+a^3*B-6*a*b^2*B)*sin(c+d*x)/(b*(a^2-b^2)^2*d*(a+b*cos(c+d*x))^2)+1/6*(2*a^3*A*b+13*a*A*b^3+a^4*B-10*a^2*b^2*B-6*b^4*B)*sin(c+d*x)/(b*(a^2-b^2)^3*d*(a+b*cos(c+d*x)))],
[(A+B*cos(c+d*x))/(a+b*cos(c+d*x))^4,x,6,(2*a^3*A+3*a*A*b^2-4*a^2*b*B-b^3*B)*arctan(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/((a-b)^(7/2)*(a+b)^(7/2)*d)-1/3*(A*b-a*B)*sin(c+d*x)/((a^2-b^2)*d*(a+b*cos(c+d*x))^3)-1/6*(5*a*A*b-2*a^2*B-3*b^2*B)*sin(c+d*x)/((a^2-b^2)^2*d*(a+b*cos(c+d*x))^2)-1/6*(11*a^2*A*b+4*A*b^3-2*a^3*B-13*a*b^2*B)*sin(c+d*x)/((a^2-b^2)^3*d*(a+b*cos(c+d*x)))],
[(A+B*cos(c+d*x))*sec(c+d*x)/(a+b*cos(c+d*x))^4,x,7,-(8*a^6*A*b-8*a^4*A*b^3+7*a^2*A*b^5-2*A*b^7-2*a^7*B-3*a^5*b^2*B)*arctan(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/(a^4*(a-b)^(7/2)*(a+b)^(7/2)*d)+A*arctanh(sin(c+d*x))/(a^4*d)+1/3*b*(A*b-a*B)*sin(c+d*x)/(a*(a^2-b^2)*d*(a+b*cos(c+d*x))^3)+1/6*b*(8*a^2*A*b-3*A*b^3-5*a^3*B)*sin(c+d*x)/(a^2*(a^2-b^2)^2*d*(a+b*cos(c+d*x))^2)+1/6*b*(26*a^4*A*b-17*a^2*A*b^3+6*A*b^5-11*a^5*B-4*a^3*b^2*B)*sin(c+d*x)/(a^3*(a^2-b^2)^3*d*(a+b*cos(c+d*x)))],
[(A+B*cos(c+d*x))*sec(c+d*x)^2/(a+b*cos(c+d*x))^4,x,8,b*(20*a^6*A*b-35*a^4*A*b^3+28*a^2*A*b^5-8*A*b^7-8*a^7*B+8*a^5*b^2*B-7*a^3*b^4*B+2*a*b^6*B)*arctan(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/(a^5*(a-b)^(7/2)*(a+b)^(7/2)*d)-(4*A*b-a*B)*arctanh(sin(c+d*x))/(a^5*d)+1/6*(6*a^6*A-65*a^4*A*b^2+68*a^2*A*b^4-24*A*b^6+26*a^5*b*B-17*a^3*b^3*B+6*a*b^5*B)*tan(c+d*x)/(a^4*(a^2-b^2)^3*d)+1/3*b*(A*b-a*B)*tan(c+d*x)/(a*(a^2-b^2)*d*(a+b*cos(c+d*x))^3)+1/6*b*(9*a^2*A*b-4*A*b^3-6*a^3*B+a*b^2*B)*tan(c+d*x)/(a^2*(a^2-b^2)^2*d*(a+b*cos(c+d*x))^2)+1/2*b*(12*a^4*A*b-11*a^2*A*b^3+4*A*b^5-6*a^5*B+2*a^3*b^2*B-a*b^4*B)*tan(c+d*x)/(a^3*(a^2-b^2)^3*d*(a+b*cos(c+d*x)))],
[(A+B*cos(c+d*x))*sec(c+d*x)^3/(a+b*cos(c+d*x))^4,x,9,-b^2*(40*a^6*A*b-84*a^4*A*b^3+69*a^2*A*b^5-20*A*b^7-20*a^7*B+35*a^5*b^2*B-28*a^3*b^4*B+8*a*b^6*B)*arctan(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/(a^6*(a-b)^(7/2)*(a+b)^(7/2)*d)+1/2*(a^2*A+20*A*b^2-8*a*b*B)*arctanh(sin(c+d*x))/(a^6*d)-1/6*(24*a^6*A*b-146*a^4*A*b^3+167*a^2*A*b^5-60*A*b^7-6*a^7*B+65*a^5*b^2*B-68*a^3*b^4*B+24*a*b^6*B)*tan(c+d*x)/(a^5*(a^2-b^2)^3*d)+1/2*(a^6*A-23*a^4*A*b^2+27*a^2*A*b^4-10*A*b^6+12*a^5*b*B-11*a^3*b^3*B+4*a*b^5*B)*sec(c+d*x)*tan(c+d*x)/(a^4*(a^2-b^2)^3*d)+1/3*b*(A*b-a*B)*sec(c+d*x)*tan(c+d*x)/(a*(a^2-b^2)*d*(a+b*cos(c+d*x))^3)+1/6*b*(10*a^2*A*b-5*A*b^3-7*a^3*B+2*a*b^2*B)*sec(c+d*x)*tan(c+d*x)/(a^2*(a^2-b^2)^2*d*(a+b*cos(c+d*x))^2)+1/6*b*(48*a^4*A*b-53*a^2*A*b^3+20*A*b^5-27*a^5*B+20*a^3*b^2*B-8*a*b^4*B)*sec(c+d*x)*tan(c+d*x)/(a^3*(a^2-b^2)^3*d*(a+b*cos(c+d*x)))],
[cos(c+d*x)^3*(a*B+b*B*cos(c+d*x))/(a+b*cos(c+d*x)),x,3,B*sin(c+d*x)/d-1/3*B*sin(c+d*x)^3/d],
[cos(c+d*x)^2*(a*B+b*B*cos(c+d*x))/(a+b*cos(c+d*x)),x,3,1/2*B*x+1/2*B*cos(c+d*x)*sin(c+d*x)/d],
[cos(c+d*x)*(a*B+b*B*cos(c+d*x))/(a+b*cos(c+d*x)),x,2,B*sin(c+d*x)/d],
[(a*B+b*B*cos(c+d*x))/(a+b*cos(c+d*x)),x,2,B*x],
[(a*B+b*B*cos(c+d*x))*sec(c+d*x)/(a+b*cos(c+d*x)),x,2,B*arctanh(sin(c+d*x))/d],
[(a*B+b*B*cos(c+d*x))*sec(c+d*x)^2/(a+b*cos(c+d*x)),x,3,B*tan(c+d*x)/d],
[(a*B+b*B*cos(c+d*x))*sec(c+d*x)^3/(a+b*cos(c+d*x)),x,3,1/2*B*arctanh(sin(c+d*x))/d+1/2*B*sec(c+d*x)*tan(c+d*x)/d],
[(a*B+b*B*cos(c+d*x))*sec(c+d*x)^4/(a+b*cos(c+d*x)),x,3,B*tan(c+d*x)/d+1/3*B*tan(c+d*x)^3/d],
[cos(c+d*x)^3*(a*B+b*B*cos(c+d*x))/(a+b*cos(c+d*x))^2,x,6,1/2*(2*a^2+b^2)*B*x/b^3-a*B*sin(c+d*x)/(b^2*d)+1/2*B*cos(c+d*x)*sin(c+d*x)/(b*d)-2*a^3*B*arctan(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/(b^3*d*sqrt(a-b)*sqrt(a+b))],
[cos(c+d*x)^2*(a*B+b*B*cos(c+d*x))/(a+b*cos(c+d*x))^2,x,6,-a*B*x/b^2+B*sin(c+d*x)/(b*d)+2*a^2*B*arctan(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/(b^2*d*sqrt(a-b)*sqrt(a+b))],
[cos(c+d*x)*(a*B+b*B*cos(c+d*x))/(a+b*cos(c+d*x))^2,x,4,B*x/b-2*a*B*arctan(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/(b*d*sqrt(a-b)*sqrt(a+b))],
[(a*B+b*B*cos(c+d*x))/(a+b*cos(c+d*x))^2,x,3,2*B*arctan(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/(d*sqrt(a-b)*sqrt(a+b))],
[(a*B+b*B*cos(c+d*x))*sec(c+d*x)/(a+b*cos(c+d*x))^2,x,5,B*arctanh(sin(c+d*x))/(a*d)-2*b*B*arctan(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/(a*d*sqrt(a-b)*sqrt(a+b))],
[(a*B+b*B*cos(c+d*x))*sec(c+d*x)^2/(a+b*cos(c+d*x))^2,x,7,-b*B*arctanh(sin(c+d*x))/(a^2*d)+2*b^2*B*arctan(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/(a^2*d*sqrt(a-b)*sqrt(a+b))+B*tan(c+d*x)/(a*d)],
[(a*B+b*B*cos(c+d*x))*sec(c+d*x)^3/(a+b*cos(c+d*x))^2,x,7,1/2*(a^2+2*b^2)*B*arctanh(sin(c+d*x))/(a^3*d)-2*b^3*B*arctan(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/(a^3*d*sqrt(a-b)*sqrt(a+b))-b*B*tan(c+d*x)/(a^2*d)+1/2*B*sec(c+d*x)*tan(c+d*x)/(a*d)],

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

# n>0
[cos(c+d*x)^3*(a+b*cos(c+d*x))^(1/2)*(A+B*cos(c+d*x)),x,9,-2/315*(36*a*A*b-24*a^2*B-49*b^2*B)*(a+b*cos(c+d*x))^(3/2)*sin(c+d*x)/(b^3*d)+2/21*(3*A*b-2*a*B)*cos(c+d*x)*(a+b*cos(c+d*x))^(3/2)*sin(c+d*x)/(b^2*d)+2/9*B*cos(c+d*x)^2*(a+b*cos(c+d*x))^(3/2)*sin(c+d*x)/(b*d)+2/315*(24*a^2*A*b+75*A*b^3-16*a^3*B-36*a*b^2*B)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(b^3*d)+2/315*(24*a^3*A*b+57*a*A*b^3-16*a^4*B-24*a^2*b^2*B+147*b^4*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(b^4*d*sqrt((a+b*cos(c+d*x))/(a+b)))-2/315*(a^2-b^2)*(24*a^2*A*b+75*A*b^3-16*a^3*B-36*a*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*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(b^4*d*sqrt(a+b*cos(c+d*x)))],
[cos(c+d*x)^2*(a+b*cos(c+d*x))^(1/2)*(A+B*cos(c+d*x)),x,8,2/35*(7*A*b-4*a*B)*(a+b*cos(c+d*x))^(3/2)*sin(c+d*x)/(b^2*d)+2/7*B*cos(c+d*x)*(a+b*cos(c+d*x))^(3/2)*sin(c+d*x)/(b*d)-2/105*(14*a*A*b-8*a^2*B-25*b^2*B)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(b^2*d)-2/105*(14*a^2*A*b-63*A*b^3-8*a^3*B-19*a*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*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(b^3*d*sqrt((a+b*cos(c+d*x))/(a+b)))+2/105*(a^2-b^2)*(14*a*A*b-8*a^2*B-25*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*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(b^3*d*sqrt(a+b*cos(c+d*x)))],
[cos(c+d*x)*(a+b*cos(c+d*x))^(1/2)*(A+B*cos(c+d*x)),x,8,2/5*B*(a+b*cos(c+d*x))^(3/2)*sin(c+d*x)/(b*d)+2/15*(5*A*b-2*a*B)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(b*d)+2/15*(5*a*A*b-2*a^2*B+9*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*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(b^2*d*sqrt((a+b*cos(c+d*x))/(a+b)))-2/15*(a^2-b^2)*(5*A*b-2*a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(b^2*d*sqrt(a+b*cos(c+d*x)))],
[(a+b*cos(c+d*x))^(1/2)*(A+B*cos(c+d*x)),x,6,2/3*B*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/d+2/3*(3*A*b+a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(b*d*sqrt((a+b*cos(c+d*x))/(a+b)))-2/3*(a^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*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(b*d*sqrt(a+b*cos(c+d*x)))],
[(a+b*cos(c+d*x))^(1/2)*(A+B*cos(c+d*x))*sec(c+d*x),x,8,2*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(d*sqrt((a+b*cos(c+d*x))/(a+b)))+2*A*b*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(d*sqrt(a+b*cos(c+d*x)))+2*a*A*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(d*sqrt(a+b*cos(c+d*x)))],
[(a+b*cos(c+d*x))^(1/2)*(A+B*cos(c+d*x))*sec(c+d*x)^2,x,9,-A*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(d*sqrt((a+b*cos(c+d*x))/(a+b)))+(a*A+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*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(d*sqrt(a+b*cos(c+d*x)))+(A*b+2*a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(d*sqrt(a+b*cos(c+d*x)))+A*sqrt(a+b*cos(c+d*x))*tan(c+d*x)/d],
[(a+b*cos(c+d*x))^(1/2)*(A+B*cos(c+d*x))*sec(c+d*x)^3,x,10,-1/4*(A*b+4*a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(a*d*sqrt((a+b*cos(c+d*x))/(a+b)))+1/4*(3*A*b+4*a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(d*sqrt(a+b*cos(c+d*x)))+1/4*(4*a^2*A-A*b^2+4*a*b*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(a*d*sqrt(a+b*cos(c+d*x)))+1/4*(A*b+4*a*B)*sqrt(a+b*cos(c+d*x))*tan(c+d*x)/(a*d)+1/2*A*sec(c+d*x)*sqrt(a+b*cos(c+d*x))*tan(c+d*x)/d],
[(a+b*cos(c+d*x))^(1/2)*(A+B*cos(c+d*x))*sec(c+d*x)^4,x,11,-1/24*(16*a^2*A-3*A*b^2+6*a*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*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(a^2*d*sqrt((a+b*cos(c+d*x))/(a+b)))+1/24*(16*a^2*A-A*b^2+18*a*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*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(a*d*sqrt(a+b*cos(c+d*x)))+1/8*(4*a^2*A*b+A*b^3+8*a^3*B-2*a*b^2*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(a^2*d*sqrt(a+b*cos(c+d*x)))+1/24*(16*a^2*A-3*A*b^2+6*a*b*B)*sqrt(a+b*cos(c+d*x))*tan(c+d*x)/(a^2*d)+1/12*(A*b+6*a*B)*sec(c+d*x)*sqrt(a+b*cos(c+d*x))*tan(c+d*x)/(a*d)+1/3*A*sec(c+d*x)^2*sqrt(a+b*cos(c+d*x))*tan(c+d*x)/d],
[cos(c+d*x)^2*(a+b*cos(c+d*x))^(3/2)*(A+B*cos(c+d*x)),x,9,-2/315*(18*a*A*b-8*a^2*B-49*b^2*B)*(a+b*cos(c+d*x))^(3/2)*sin(c+d*x)/(b^2*d)+2/63*(9*A*b-4*a*B)*(a+b*cos(c+d*x))^(5/2)*sin(c+d*x)/(b^2*d)+2/9*B*cos(c+d*x)*(a+b*cos(c+d*x))^(5/2)*sin(c+d*x)/(b*d)-2/315*(18*a^2*A*b-75*A*b^3-8*a^3*B-39*a*b^2*B)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(b^2*d)-2/315*(18*a^3*A*b-246*a*A*b^3-8*a^4*B-33*a^2*b^2*B-147*b^4*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(b^3*d*sqrt((a+b*cos(c+d*x))/(a+b)))+2/315*(a^2-b^2)*(18*a^2*A*b-75*A*b^3-8*a^3*B-39*a*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*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(b^3*d*sqrt(a+b*cos(c+d*x)))],
[cos(c+d*x)*(a+b*cos(c+d*x))^(3/2)*(A+B*cos(c+d*x)),x,9,2/35*(7*A*b-2*a*B)*(a+b*cos(c+d*x))^(3/2)*sin(c+d*x)/(b*d)+2/7*B*(a+b*cos(c+d*x))^(5/2)*sin(c+d*x)/(b*d)+2/105*(21*a*A*b-6*a^2*B+25*b^2*B)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(b*d)+2/105*(21*a^2*A*b+63*A*b^3-6*a^3*B+82*a*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*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(b^2*d*sqrt((a+b*cos(c+d*x))/(a+b)))-2/105*(a^2-b^2)*(21*a*A*b-6*a^2*B+25*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*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(b^2*d*sqrt(a+b*cos(c+d*x)))],
[(a+b*cos(c+d*x))^(3/2)*(A+B*cos(c+d*x)),x,7,2/5*B*(a+b*cos(c+d*x))^(3/2)*sin(c+d*x)/d+2/15*(5*A*b+3*a*B)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/d+2/15*(20*a*A*b+3*a^2*B+9*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*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(b*d*sqrt((a+b*cos(c+d*x))/(a+b)))-2/15*(a^2-b^2)*(5*A*b+3*a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(b*d*sqrt(a+b*cos(c+d*x)))],
[(a+b*cos(c+d*x))^(3/2)*(A+B*cos(c+d*x))*sec(c+d*x),x,9,2/3*b*B*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/d+2/3*(3*A*b+4*a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(d*sqrt((a+b*cos(c+d*x))/(a+b)))+2/3*(3*a*A*b-a^2*B+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*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(d*sqrt(a+b*cos(c+d*x)))+2*a^2*A*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(d*sqrt(a+b*cos(c+d*x)))],
[(a+b*cos(c+d*x))^(3/2)*(A+B*cos(c+d*x))*sec(c+d*x)^2,x,9,-(a*A-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*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(d*sqrt((a+b*cos(c+d*x))/(a+b)))+(a^2*A+2*A*b^2+2*a*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*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(d*sqrt(a+b*cos(c+d*x)))+a*(3*A*b+2*a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(d*sqrt(a+b*cos(c+d*x)))+a*A*sqrt(a+b*cos(c+d*x))*tan(c+d*x)/d],
[(a+b*cos(c+d*x))^(3/2)*(A+B*cos(c+d*x))*sec(c+d*x)^3,x,10,-1/4*(5*A*b+4*a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(d*sqrt((a+b*cos(c+d*x))/(a+b)))+1/4*(7*a*A*b+4*a^2*B+8*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*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(d*sqrt(a+b*cos(c+d*x)))+1/4*(4*a^2*A+3*A*b^2+12*a*b*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(d*sqrt(a+b*cos(c+d*x)))+1/4*(5*A*b+4*a*B)*sqrt(a+b*cos(c+d*x))*tan(c+d*x)/d+1/2*a*A*sec(c+d*x)*sqrt(a+b*cos(c+d*x))*tan(c+d*x)/d],
[(a+b*cos(c+d*x))^(3/2)*(A+B*cos(c+d*x))*sec(c+d*x)^4,x,11,-1/24*(16*a^2*A+3*A*b^2+30*a*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*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(a*d*sqrt((a+b*cos(c+d*x))/(a+b)))+1/24*(16*a^2*A+17*A*b^2+42*a*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*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(d*sqrt(a+b*cos(c+d*x)))+1/8*(12*a^2*A*b-A*b^3+8*a^3*B+6*a*b^2*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(a*d*sqrt(a+b*cos(c+d*x)))+1/24*(16*a^2*A+3*A*b^2+30*a*b*B)*sqrt(a+b*cos(c+d*x))*tan(c+d*x)/(a*d)+1/12*(7*A*b+6*a*B)*sec(c+d*x)*sqrt(a+b*cos(c+d*x))*tan(c+d*x)/d+1/3*a*A*sec(c+d*x)^2*sqrt(a+b*cos(c+d*x))*tan(c+d*x)/d],
[cos(c+d*x)^2*(a+b*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x)),x,10,-2/3465*(110*a^2*A*b-539*A*b^3-40*a^3*B-335*a*b^2*B)*(a+b*cos(c+d*x))^(3/2)*sin(c+d*x)/(b^2*d)-2/693*(22*a*A*b-8*a^2*B-81*b^2*B)*(a+b*cos(c+d*x))^(5/2)*sin(c+d*x)/(b^2*d)+2/99*(11*A*b-4*a*B)*(a+b*cos(c+d*x))^(7/2)*sin(c+d*x)/(b^2*d)+2/11*B*cos(c+d*x)*(a+b*cos(c+d*x))^(7/2)*sin(c+d*x)/(b*d)-2/3465*(110*a^3*A*b-1254*a*A*b^3-40*a^4*B-285*a^2*b^2*B-675*b^4*B)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(b^2*d)-2/3465*(110*a^4*A*b-3069*a^2*A*b^3-1617*A*b^5-40*a^5*B-255*a^3*b^2*B-3705*a*b^4*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(b^3*d*sqrt((a+b*cos(c+d*x))/(a+b)))+2/3465*(a^2-b^2)*(110*a^3*A*b-1254*a*A*b^3-40*a^4*B-285*a^2*b^2*B-675*b^4*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(b^3*d*sqrt(a+b*cos(c+d*x)))],
[cos(c+d*x)*(a+b*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x)),x,10,2/315*(45*a*A*b-10*a^2*B+49*b^2*B)*(a+b*cos(c+d*x))^(3/2)*sin(c+d*x)/(b*d)+2/63*(9*A*b-2*a*B)*(a+b*cos(c+d*x))^(5/2)*sin(c+d*x)/(b*d)+2/9*B*(a+b*cos(c+d*x))^(7/2)*sin(c+d*x)/(b*d)+2/315*(45*a^2*A*b+75*A*b^3-10*a^3*B+114*a*b^2*B)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(b*d)+2/315*(45*a^3*A*b+435*a*A*b^3-10*a^4*B+279*a^2*b^2*B+147*b^4*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(b^2*d*sqrt((a+b*cos(c+d*x))/(a+b)))-2/315*(a^2-b^2)*(45*a^2*A*b+75*A*b^3-10*a^3*B+114*a*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*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(b^2*d*sqrt(a+b*cos(c+d*x)))],
[(a+b*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x)),x,8,2/35*(7*A*b+5*a*B)*(a+b*cos(c+d*x))^(3/2)*sin(c+d*x)/d+2/7*B*(a+b*cos(c+d*x))^(5/2)*sin(c+d*x)/d+2/105*(56*a*A*b+15*a^2*B+25*b^2*B)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/d+2/105*(161*a^2*A*b+63*A*b^3+15*a^3*B+145*a*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*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(b*d*sqrt((a+b*cos(c+d*x))/(a+b)))-2/105*(a^2-b^2)*(56*a*A*b+15*a^2*B+25*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*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(b*d*sqrt(a+b*cos(c+d*x)))],
[(a+b*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x))*sec(c+d*x),x,10,2/5*b*B*(a+b*cos(c+d*x))^(3/2)*sin(c+d*x)/d+2/15*b*(5*A*b+8*a*B)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/d+2/15*(35*a*A*b+23*a^2*B+9*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*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(d*sqrt((a+b*cos(c+d*x))/(a+b)))+2/15*(10*a^2*A*b+5*A*b^3-8*a^3*B+8*a*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*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(d*sqrt(a+b*cos(c+d*x)))+2*a^3*A*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(d*sqrt(a+b*cos(c+d*x)))],
[(a+b*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x))*sec(c+d*x)^2,x,10,-1/3*b*(3*a*A-2*b*B)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/d-1/3*(3*a^2*A-6*A*b^2-14*a*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*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(d*sqrt((a+b*cos(c+d*x))/(a+b)))+1/3*(3*a^3*A+12*a*A*b^2+4*a^2*b*B+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*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(d*sqrt(a+b*cos(c+d*x)))+a^2*(5*A*b+2*a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(d*sqrt(a+b*cos(c+d*x)))+a*A*(a+b*cos(c+d*x))^(3/2)*tan(c+d*x)/d],
[(a+b*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x))*sec(c+d*x)^3,x,10,-1/4*(9*a*A*b+4*a^2*B-8*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*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(d*sqrt((a+b*cos(c+d*x))/(a+b)))+1/4*(11*a^2*A*b+8*A*b^3+4*a^3*B+16*a*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*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(d*sqrt(a+b*cos(c+d*x)))+1/4*a*(4*a^2*A+15*A*b^2+20*a*b*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(d*sqrt(a+b*cos(c+d*x)))+1/2*a*A*(a+b*cos(c+d*x))^(3/2)*sec(c+d*x)*tan(c+d*x)/d+1/4*a*(7*A*b+4*a*B)*sqrt(a+b*cos(c+d*x))*tan(c+d*x)/d],
[(a+b*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x))*sec(c+d*x)^4,x,11,-1/24*(16*a^2*A+33*A*b^2+54*a*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*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(d*sqrt((a+b*cos(c+d*x))/(a+b)))+1/24*(16*a^3*A+59*a*A*b^2+66*a^2*b*B+48*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*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(d*sqrt(a+b*cos(c+d*x)))+1/8*(20*a^2*A*b+5*A*b^3+8*a^3*B+30*a*b^2*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(d*sqrt(a+b*cos(c+d*x)))+1/3*a*A*(a+b*cos(c+d*x))^(3/2)*sec(c+d*x)^2*tan(c+d*x)/d+1/24*(16*a^2*A+33*A*b^2+54*a*b*B)*sqrt(a+b*cos(c+d*x))*tan(c+d*x)/d+1/4*a*(3*A*b+2*a*B)*sec(c+d*x)*sqrt(a+b*cos(c+d*x))*tan(c+d*x)/d],
[(a+b*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x))*sec(c+d*x)^5,x,12,-1/192*(284*a^2*A*b+15*A*b^3+128*a^3*B+264*a*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*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(a*d*sqrt((a+b*cos(c+d*x))/(a+b)))+1/192*(356*a^2*A*b+133*A*b^3+128*a^3*B+472*a*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*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(d*sqrt(a+b*cos(c+d*x)))+1/64*(48*a^4*A+120*a^2*A*b^2-5*A*b^4+160*a^3*b*B+40*a*b^3*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(a*d*sqrt(a+b*cos(c+d*x)))+1/4*a*A*(a+b*cos(c+d*x))^(3/2)*sec(c+d*x)^3*tan(c+d*x)/d+1/192*(284*a^2*A*b+15*A*b^3+128*a^3*B+264*a*b^2*B)*sqrt(a+b*cos(c+d*x))*tan(c+d*x)/(a*d)+1/96*(36*a^2*A+59*A*b^2+104*a*b*B)*sec(c+d*x)*sqrt(a+b*cos(c+d*x))*tan(c+d*x)/d+1/24*a*(11*A*b+8*a*B)*sec(c+d*x)^2*sqrt(a+b*cos(c+d*x))*tan(c+d*x)/d],

# n<0
[cos(c+d*x)^3*(A+B*cos(c+d*x))/(a+b*cos(c+d*x))^(1/2),x,8,-2/105*(28*a*A*b-24*a^2*B-25*b^2*B)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(b^3*d)+2/35*(7*A*b-6*a*B)*cos(c+d*x)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(b^2*d)+2/7*B*cos(c+d*x)^2*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(b*d)+2/105*(56*a^2*A*b+63*A*b^3-48*a^3*B-44*a*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*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(b^4*d*sqrt((a+b*cos(c+d*x))/(a+b)))-2/105*(56*a^3*A*b+49*a*A*b^3-48*a^4*B-32*a^2*b^2*B-25*b^4*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(b^4*d*sqrt(a+b*cos(c+d*x)))],
[cos(c+d*x)^2*(A+B*cos(c+d*x))/(a+b*cos(c+d*x))^(1/2),x,7,2/15*(5*A*b-4*a*B)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(b^2*d)+2/5*B*cos(c+d*x)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(b*d)-2/15*(10*a*A*b-8*a^2*B-9*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*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(b^3*d*sqrt((a+b*cos(c+d*x))/(a+b)))+2/15*(10*a^2*A*b+5*A*b^3-8*a^3*B-7*a*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*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(b^3*d*sqrt(a+b*cos(c+d*x)))],
[cos(c+d*x)*(A+B*cos(c+d*x))/(a+b*cos(c+d*x))^(1/2),x,7,2/3*B*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(b*d)+2/3*(3*A*b-2*a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(b^2*d*sqrt((a+b*cos(c+d*x))/(a+b)))-2/3*(3*a*A*b-2*a^2*B-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*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(b^2*d*sqrt(a+b*cos(c+d*x)))],
[(A+B*cos(c+d*x))/(a+b*cos(c+d*x))^(1/2),x,5,2*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(b*d*sqrt((a+b*cos(c+d*x))/(a+b)))+2*(A*b-a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(b*d*sqrt(a+b*cos(c+d*x)))],
[(A+B*cos(c+d*x))*sec(c+d*x)/(a+b*cos(c+d*x))^(1/2),x,5,2*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(d*sqrt(a+b*cos(c+d*x)))+2*A*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(d*sqrt(a+b*cos(c+d*x)))],
[(A+B*cos(c+d*x))*sec(c+d*x)^2/(a+b*cos(c+d*x))^(1/2),x,9,-A*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(a*d*sqrt((a+b*cos(c+d*x))/(a+b)))+A*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(d*sqrt(a+b*cos(c+d*x)))-(A*b-2*a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(a*d*sqrt(a+b*cos(c+d*x)))+A*sqrt(a+b*cos(c+d*x))*tan(c+d*x)/(a*d)],
[(A+B*cos(c+d*x))*sec(c+d*x)^3/(a+b*cos(c+d*x))^(1/2),x,10,1/4*(3*A*b-4*a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(a^2*d*sqrt((a+b*cos(c+d*x))/(a+b)))-1/4*(A*b-4*a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(a*d*sqrt(a+b*cos(c+d*x)))+1/4*(4*a^2*A+3*A*b^2-4*a*b*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(a^2*d*sqrt(a+b*cos(c+d*x)))-1/4*(3*A*b-4*a*B)*sqrt(a+b*cos(c+d*x))*tan(c+d*x)/(a^2*d)+1/2*A*sec(c+d*x)*sqrt(a+b*cos(c+d*x))*tan(c+d*x)/(a*d)],
[cos(c+d*x)^3*(A+B*cos(c+d*x))/(a+b*cos(c+d*x))^(3/2),x,8,2*a*(A*b-a*B)*cos(c+d*x)^2*sin(c+d*x)/(b*(a^2-b^2)*d*sqrt(a+b*cos(c+d*x)))+2/15*(20*a^2*A*b-5*A*b^3-24*a^3*B+9*a*b^2*B)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(b^3*(a^2-b^2)*d)-2/5*(5*a*A*b-6*a^2*B+b^2*B)*cos(c+d*x)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(b^2*(a^2-b^2)*d)-2/15*(40*a^3*A*b-25*a*A*b^3-48*a^4*B+24*a^2*b^2*B+9*b^4*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(b^4*(a^2-b^2)*d*sqrt((a+b*cos(c+d*x))/(a+b)))+2/15*(40*a^2*A*b+5*A*b^3-48*a^3*B-12*a*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*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(b^4*d*sqrt(a+b*cos(c+d*x)))],
[cos(c+d*x)^2*(A+B*cos(c+d*x))/(a+b*cos(c+d*x))^(3/2),x,7,-2*a^2*(A*b-a*B)*sin(c+d*x)/(b^2*(a^2-b^2)*d*sqrt(a+b*cos(c+d*x)))+2/3*B*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(b^2*d)+2/3*(6*a^2*A*b-3*A*b^3-8*a^3*B+5*a*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*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(b^3*(a^2-b^2)*d*sqrt((a+b*cos(c+d*x))/(a+b)))-2/3*(6*a*A*b-8*a^2*B-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*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(b^3*d*sqrt(a+b*cos(c+d*x)))],
[cos(c+d*x)*(A+B*cos(c+d*x))/(a+b*cos(c+d*x))^(3/2),x,7,2*a*(A*b-a*B)*sin(c+d*x)/(b*(a^2-b^2)*d*sqrt(a+b*cos(c+d*x)))-2*(a*A*b-2*a^2*B+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*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(b^2*(a^2-b^2)*d*sqrt((a+b*cos(c+d*x))/(a+b)))+2*(A*b-2*a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(b^2*d*sqrt(a+b*cos(c+d*x)))],
[(A+B*cos(c+d*x))/(a+b*cos(c+d*x))^(3/2),x,6,-2*(A*b-a*B)*sin(c+d*x)/((a^2-b^2)*d*sqrt(a+b*cos(c+d*x)))+2*(A*b-a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(b*(a^2-b^2)*d*sqrt((a+b*cos(c+d*x))/(a+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*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(b*d*sqrt(a+b*cos(c+d*x)))],
[(A+B*cos(c+d*x))*sec(c+d*x)/(a+b*cos(c+d*x))^(3/2),x,7,2*b*(A*b-a*B)*sin(c+d*x)/(a*(a^2-b^2)*d*sqrt(a+b*cos(c+d*x)))-2*(A*b-a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(a*(a^2-b^2)*d*sqrt((a+b*cos(c+d*x))/(a+b)))+2*A*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(a*d*sqrt(a+b*cos(c+d*x)))],
[(A+B*cos(c+d*x))*sec(c+d*x)^2/(a+b*cos(c+d*x))^(3/2),x,10,b*(a^2*A-3*A*b^2+2*a*b*B)*sin(c+d*x)/(a^2*(a^2-b^2)*d*sqrt(a+b*cos(c+d*x)))-(a^2*A-3*A*b^2+2*a*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*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(a^2*(a^2-b^2)*d*sqrt((a+b*cos(c+d*x))/(a+b)))+A*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(a*d*sqrt(a+b*cos(c+d*x)))-(3*A*b-2*a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(a^2*d*sqrt(a+b*cos(c+d*x)))+A*tan(c+d*x)/(a*d*sqrt(a+b*cos(c+d*x)))],
[(A+B*cos(c+d*x))*sec(c+d*x)^3/(a+b*cos(c+d*x))^(3/2),x,11,-1/4*b*(7*a^2*A*b-15*A*b^3-4*a^3*B+12*a*b^2*B)*sin(c+d*x)/(a^3*(a^2-b^2)*d*sqrt(a+b*cos(c+d*x)))+1/4*(7*a^2*A*b-15*A*b^3-4*a^3*B+12*a*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*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(a^3*(a^2-b^2)*d*sqrt((a+b*cos(c+d*x))/(a+b)))-1/4*(5*A*b-4*a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(a^2*d*sqrt(a+b*cos(c+d*x)))+1/4*(4*a^2*A+15*A*b^2-12*a*b*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(a^3*d*sqrt(a+b*cos(c+d*x)))-1/4*(5*A*b-4*a*B)*tan(c+d*x)/(a^2*d*sqrt(a+b*cos(c+d*x)))+1/2*A*sec(c+d*x)*tan(c+d*x)/(a*d*sqrt(a+b*cos(c+d*x)))],
[cos(c+d*x)^4*(A+B*cos(c+d*x))/(a+b*cos(c+d*x))^(5/2),x,9,2/3*a*(A*b-a*B)*cos(c+d*x)^3*sin(c+d*x)/(b*(a^2-b^2)*d*(a+b*cos(c+d*x))^(3/2))+2/3*a*(5*a^2*A*b-9*A*b^3-8*a^3*B+12*a*b^2*B)*cos(c+d*x)^2*sin(c+d*x)/(b^2*(a^2-b^2)^2*d*sqrt(a+b*cos(c+d*x)))+2/15*(40*a^4*A*b-65*a^2*A*b^3+5*A*b^5-64*a^5*B+98*a^3*b^2*B-14*a*b^4*B)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(b^4*(a^2-b^2)^2*d)-2/15*(30*a^3*A*b-50*a*A*b^3-48*a^4*B+71*a^2*b^2*B-3*b^4*B)*cos(c+d*x)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(b^3*(a^2-b^2)^2*d)-2/15*(80*a^5*A*b-140*a^3*A*b^3+40*a*A*b^5-128*a^6*B+212*a^4*b^2*B-55*a^2*b^4*B-9*b^6*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(b^5*(a^2-b^2)^2*d*sqrt((a+b*cos(c+d*x))/(a+b)))+2/15*(80*a^4*A*b-80*a^2*A*b^3-5*A*b^5-128*a^5*B+116*a^3*b^2*B+17*a*b^4*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(b^5*(a^2-b^2)*d*sqrt(a+b*cos(c+d*x)))],
[cos(c+d*x)^3*(A+B*cos(c+d*x))/(a+b*cos(c+d*x))^(5/2),x,8,2/3*a*(A*b-a*B)*cos(c+d*x)^2*sin(c+d*x)/(b*(a^2-b^2)*d*(a+b*cos(c+d*x))^(3/2))-2/3*a^2*(3*a^2*A*b-7*A*b^3-6*a^3*B+10*a*b^2*B)*sin(c+d*x)/(b^3*(a^2-b^2)^2*d*sqrt(a+b*cos(c+d*x)))-2/3*(a*A*b-2*a^2*B+b^2*B)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(b^3*(a^2-b^2)*d)+2/3*(8*a^4*A*b-15*a^2*A*b^3+3*A*b^5-16*a^5*B+28*a^3*b^2*B-8*a*b^4*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(b^4*(a^2-b^2)^2*d*sqrt((a+b*cos(c+d*x))/(a+b)))-2/3*(8*a^3*A*b-9*a*A*b^3-16*a^4*B+16*a^2*b^2*B+b^4*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(b^4*(a^2-b^2)*d*sqrt(a+b*cos(c+d*x)))],
[cos(c+d*x)^2*(A+B*cos(c+d*x))/(a+b*cos(c+d*x))^(5/2),x,7,-2/3*a^2*(A*b-a*B)*sin(c+d*x)/(b^2*(a^2-b^2)*d*(a+b*cos(c+d*x))^(3/2))+2/3*a*(2*a^2*A*b-6*A*b^3-5*a^3*B+9*a*b^2*B)*sin(c+d*x)/(b^2*(a^2-b^2)^2*d*sqrt(a+b*cos(c+d*x)))-2/3*(2*a^3*A*b-6*a*A*b^3-8*a^4*B+15*a^2*b^2*B-3*b^4*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(b^3*(a^2-b^2)^2*d*sqrt((a+b*cos(c+d*x))/(a+b)))+2/3*(2*a^2*A*b-3*A*b^3-8*a^3*B+9*a*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*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(b^3*(a^2-b^2)*d*sqrt(a+b*cos(c+d*x)))],
[cos(c+d*x)*(A+B*cos(c+d*x))/(a+b*cos(c+d*x))^(5/2),x,8,2/3*a*(A*b-a*B)*sin(c+d*x)/(b*(a^2-b^2)*d*(a+b*cos(c+d*x))^(3/2))+2/3*(a^2*A*b+3*A*b^3+2*a^3*B-6*a*b^2*B)*sin(c+d*x)/(b*(a^2-b^2)^2*d*sqrt(a+b*cos(c+d*x)))-2/3*(a^2*A*b+3*A*b^3+2*a^3*B-6*a*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*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(b^2*(a^2-b^2)^2*d*sqrt((a+b*cos(c+d*x))/(a+b)))+2/3*(a*A*b+2*a^2*B-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*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(b^2*(a^2-b^2)*d*sqrt(a+b*cos(c+d*x)))],
[(A+B*cos(c+d*x))/(a+b*cos(c+d*x))^(5/2),x,7,-2/3*(A*b-a*B)*sin(c+d*x)/((a^2-b^2)*d*(a+b*cos(c+d*x))^(3/2))-2/3*(4*a*A*b-a^2*B-3*b^2*B)*sin(c+d*x)/((a^2-b^2)^2*d*sqrt(a+b*cos(c+d*x)))+2/3*(4*a*A*b-a^2*B-3*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*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(b*(a^2-b^2)^2*d*sqrt((a+b*cos(c+d*x))/(a+b)))-2/3*(A*b-a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(b*(a^2-b^2)*d*sqrt(a+b*cos(c+d*x)))],
[(A+B*cos(c+d*x))*sec(c+d*x)/(a+b*cos(c+d*x))^(5/2),x,10,2/3*b*(A*b-a*B)*sin(c+d*x)/(a*(a^2-b^2)*d*(a+b*cos(c+d*x))^(3/2))+2/3*b*(7*a^2*A*b-3*A*b^3-4*a^3*B)*sin(c+d*x)/(a^2*(a^2-b^2)^2*d*sqrt(a+b*cos(c+d*x)))-2/3*(7*a^2*A*b-3*A*b^3-4*a^3*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(a^2*(a^2-b^2)^2*d*sqrt((a+b*cos(c+d*x))/(a+b)))+2/3*(A*b-a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(a*(a^2-b^2)*d*sqrt(a+b*cos(c+d*x)))+2*A*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(a^2*d*sqrt(a+b*cos(c+d*x)))],
[(A+B*cos(c+d*x))*sec(c+d*x)^2/(a+b*cos(c+d*x))^(5/2),x,11,1/3*b*(3*a^2*A-5*A*b^2+2*a*b*B)*sin(c+d*x)/(a^2*(a^2-b^2)*d*(a+b*cos(c+d*x))^(3/2))+1/3*b*(3*a^4*A-26*a^2*A*b^2+15*A*b^4+14*a^3*b*B-6*a*b^3*B)*sin(c+d*x)/(a^3*(a^2-b^2)^2*d*sqrt(a+b*cos(c+d*x)))-1/3*(3*a^4*A-26*a^2*A*b^2+15*A*b^4+14*a^3*b*B-6*a*b^3*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(a^3*(a^2-b^2)^2*d*sqrt((a+b*cos(c+d*x))/(a+b)))+1/3*(3*a^2*A-5*A*b^2+2*a*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*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(a^2*(a^2-b^2)*d*sqrt(a+b*cos(c+d*x)))-(5*A*b-2*a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(a^3*d*sqrt(a+b*cos(c+d*x)))+A*tan(c+d*x)/(a*d*(a+b*cos(c+d*x))^(3/2))],
[(A+B*cos(c+d*x))*sec(c+d*x)^3/(a+b*cos(c+d*x))^(5/2),x,12,-1/12*b*(27*a^2*A*b-35*A*b^3-12*a^3*B+20*a*b^2*B)*sin(c+d*x)/(a^3*(a^2-b^2)*d*(a+b*cos(c+d*x))^(3/2))-1/12*b*(33*a^4*A*b-170*a^2*A*b^3+105*A*b^5-12*a^5*B+104*a^3*b^2*B-60*a*b^4*B)*sin(c+d*x)/(a^4*(a^2-b^2)^2*d*sqrt(a+b*cos(c+d*x)))+1/12*(33*a^4*A*b-170*a^2*A*b^3+105*A*b^5-12*a^5*B+104*a^3*b^2*B-60*a*b^4*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(a^4*(a^2-b^2)^2*d*sqrt((a+b*cos(c+d*x))/(a+b)))-1/12*(27*a^2*A*b-35*A*b^3-12*a^3*B+20*a*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*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(a^3*(a^2-b^2)*d*sqrt(a+b*cos(c+d*x)))+1/4*(4*a^2*A+35*A*b^2-20*a*b*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(a^4*d*sqrt(a+b*cos(c+d*x)))-1/4*(7*A*b-4*a*B)*tan(c+d*x)/(a^2*d*(a+b*cos(c+d*x))^(3/2))+1/2*A*sec(c+d*x)*tan(c+d*x)/(a*d*(a+b*cos(c+d*x))^(3/2))],
[(a*B+b*B*cos(c+d*x))/(a+b*cos(c+d*x))^(3/2),x,3,2*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(d*sqrt(a+b*cos(c+d*x)))],
[(a*B+b*B*cos(c+d*x))*sec(c+d*x)/(a+b*cos(c+d*x))^(3/2),x,3,2*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(d*sqrt(a+b*cos(c+d*x)))],
[(a*B+b*B*cos(c+d*x))/(a+b*cos(c+d*x))^(5/2),x,5,-2*b*B*sin(c+d*x)/((a^2-b^2)*d*sqrt(a+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*b/(a+b)))*sqrt(a+b*cos(c+d*x))/((a^2-b^2)*d*sqrt((a+b*cos(c+d*x))/(a+b)))],
[(a*B+b*B*cos(c+d*x))*sec(c+d*x)/(a+b*cos(c+d*x))^(5/2),x,8,2*b^2*B*sin(c+d*x)/(a*(a^2-b^2)*d*sqrt(a+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*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(a*(a^2-b^2)*d*sqrt((a+b*cos(c+d*x))/(a+b)))+2*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(a*d*sqrt(a+b*cos(c+d*x)))],

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

# n>0
[cos(c+d*x)^(5/2)*(a+b*cos(c+d*x))*(A+B*cos(c+d*x)),x,8,2/15*(9*a*A+7*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))/d+10/21*(A*b+a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+2/45*(9*a*A+7*b*B)*cos(c+d*x)^(3/2)*sin(c+d*x)/d+2/7*(A*b+a*B)*cos(c+d*x)^(5/2)*sin(c+d*x)/d+2/9*b*B*cos(c+d*x)^(7/2)*sin(c+d*x)/d+10/21*(A*b+a*B)*sin(c+d*x)*sqrt(cos(c+d*x))/d],
[cos(c+d*x)^(3/2)*(a+b*cos(c+d*x))*(A+B*cos(c+d*x)),x,7,6/5*(A*b+a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+2/21*(7*a*A+5*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))/d+2/5*(A*b+a*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+2/21*(7*a*A+5*b*B)*sin(c+d*x)*sqrt(cos(c+d*x))/d],
[cos(c+d*x)^(1/2)*(a+b*cos(c+d*x))*(A+B*cos(c+d*x)),x,6,2/5*(5*a*A+3*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))/d+2/3*(A*b+a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+2/5*b*B*cos(c+d*x)^(3/2)*sin(c+d*x)/d+2/3*(A*b+a*B)*sin(c+d*x)*sqrt(cos(c+d*x))/d],
[(a+b*cos(c+d*x))*(A+B*cos(c+d*x))/cos(c+d*x)^(1/2),x,5,2*(A*b+a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+2/3*(3*a*A+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))/d+2/3*b*B*sin(c+d*x)*sqrt(cos(c+d*x))/d],
[(a+b*cos(c+d*x))*(A+B*cos(c+d*x))/cos(c+d*x)^(3/2),x,5,-2*(a*A-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))/d+2*(A*b+a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+2*a*A*sin(c+d*x)/(d*sqrt(cos(c+d*x)))],
[(a+b*cos(c+d*x))*(A+B*cos(c+d*x))/cos(c+d*x)^(5/2),x,6,-2*(A*b+a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+2/3*(a*A+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))/d+2/3*a*A*sin(c+d*x)/(d*cos(c+d*x)^(3/2))+2*(A*b+a*B)*sin(c+d*x)/(d*sqrt(cos(c+d*x)))],
[(a+b*cos(c+d*x))*(A+B*cos(c+d*x))/cos(c+d*x)^(7/2),x,7,-2/5*(3*a*A+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))/d+2/3*(A*b+a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+2/5*a*A*sin(c+d*x)/(d*cos(c+d*x)^(5/2))+2/3*(A*b+a*B)*sin(c+d*x)/(d*cos(c+d*x)^(3/2))+2/5*(3*a*A+5*b*B)*sin(c+d*x)/(d*sqrt(cos(c+d*x)))],
[cos(c+d*x)^(5/2)*(a+b*cos(c+d*x))^2*(A+B*cos(c+d*x)),x,8,2/15*(9*a^2*A+7*A*b^2+14*a*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))/d+10/231*(9*b^2*B+11*a*(2*A*b+a*B))*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+2/45*(9*a^2*A+7*A*b^2+14*a*b*B)*cos(c+d*x)^(3/2)*sin(c+d*x)/d+2/77*(9*b^2*B+11*a*(2*A*b+a*B))*cos(c+d*x)^(5/2)*sin(c+d*x)/d+2/99*b*(11*A*b+13*a*B)*cos(c+d*x)^(7/2)*sin(c+d*x)/d+2/11*b*B*cos(c+d*x)^(7/2)*(a+b*cos(c+d*x))*sin(c+d*x)/d+10/231*(9*b^2*B+11*a*(2*A*b+a*B))*sin(c+d*x)*sqrt(cos(c+d*x))/d],
[cos(c+d*x)^(3/2)*(a+b*cos(c+d*x))^2*(A+B*cos(c+d*x)),x,7,2/15*(7*b^2*B+9*a*(2*A*b+a*B))*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+2/21*(7*a^2*A+5*A*b^2+10*a*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))/d+2/45*(7*b^2*B+9*a*(2*A*b+a*B))*cos(c+d*x)^(3/2)*sin(c+d*x)/d+2/63*b*(9*A*b+11*a*B)*cos(c+d*x)^(5/2)*sin(c+d*x)/d+2/9*b*B*cos(c+d*x)^(5/2)*(a+b*cos(c+d*x))*sin(c+d*x)/d+2/21*(7*a^2*A+5*A*b^2+10*a*b*B)*sin(c+d*x)*sqrt(cos(c+d*x))/d],
[cos(c+d*x)^(1/2)*(a+b*cos(c+d*x))^2*(A+B*cos(c+d*x)),x,6,2/5*(5*a^2*A+3*A*b^2+6*a*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))/d+2/21*(5*b^2*B+7*a*(2*A*b+a*B))*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+2/35*b*(7*A*b+9*a*B)*cos(c+d*x)^(3/2)*sin(c+d*x)/d+2/7*b*B*cos(c+d*x)^(3/2)*(a+b*cos(c+d*x))*sin(c+d*x)/d+2/21*(5*b^2*B+7*a*(2*A*b+a*B))*sin(c+d*x)*sqrt(cos(c+d*x))/d],
[(a+b*cos(c+d*x))^2*(A+B*cos(c+d*x))/cos(c+d*x)^(1/2),x,5,2/5*(3*b^2*B+5*a*(2*A*b+a*B))*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+2/3*(3*a^2*A+A*b^2+2*a*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))/d+2/15*b*(5*A*b+7*a*B)*sin(c+d*x)*sqrt(cos(c+d*x))/d+2/5*b*B*(a+b*cos(c+d*x))*sin(c+d*x)*sqrt(cos(c+d*x))/d],
[(a+b*cos(c+d*x))^2*(A+B*cos(c+d*x))/cos(c+d*x)^(3/2),x,5,-2*(a^2*A-A*b^2-2*a*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))/d+2/3*(6*a*A*b+3*a^2*B+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))/d+2*a^2*A*sin(c+d*x)/(d*sqrt(cos(c+d*x)))+2/3*b^2*B*sin(c+d*x)*sqrt(cos(c+d*x))/d],
[(a+b*cos(c+d*x))^2*(A+B*cos(c+d*x))/cos(c+d*x)^(5/2),x,5,-2*(2*a*A*b+a^2*B-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))/d+2/3*(a^2*A+3*A*b^2+6*a*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))/d+2/3*a^2*A*sin(c+d*x)/(d*cos(c+d*x)^(3/2))+2*a*(2*A*b+a*B)*sin(c+d*x)/(d*sqrt(cos(c+d*x)))],
[(a+b*cos(c+d*x))^2*(A+B*cos(c+d*x))/cos(c+d*x)^(7/2),x,6,-2/5*(3*a^2*A+5*A*b^2+10*a*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))/d+2/3*(2*a*A*b+a^2*B+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))/d+2/5*a^2*A*sin(c+d*x)/(d*cos(c+d*x)^(5/2))+2/3*a*(2*A*b+a*B)*sin(c+d*x)/(d*cos(c+d*x)^(3/2))+2/5*(3*a^2*A+5*A*b^2+10*a*b*B)*sin(c+d*x)/(d*sqrt(cos(c+d*x)))],
[cos(c+d*x)^(3/2)*(a+b*cos(c+d*x))^3*(A+B*cos(c+d*x)),x,8,2/15*(27*a^2*A*b+7*A*b^3+9*a^3*B+21*a*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))/d+2/231*(77*a^3*A+165*a*A*b^2+165*a^2*b*B+45*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))/d+2/45*(27*a^2*A*b+7*A*b^3+9*a^3*B+21*a*b^2*B)*cos(c+d*x)^(3/2)*sin(c+d*x)/d+2/77*b*(33*a*A*b+26*a^2*B+9*b^2*B)*cos(c+d*x)^(5/2)*sin(c+d*x)/d+2/99*b^2*(11*A*b+15*a*B)*cos(c+d*x)^(7/2)*sin(c+d*x)/d+2/11*b*B*cos(c+d*x)^(5/2)*(a+b*cos(c+d*x))^2*sin(c+d*x)/d+2/231*(77*a^3*A+165*a*A*b^2+165*a^2*b*B+45*b^3*B)*sin(c+d*x)*sqrt(cos(c+d*x))/d],
[cos(c+d*x)^(1/2)*(a+b*cos(c+d*x))^3*(A+B*cos(c+d*x)),x,7,2/15*(15*a^3*A+27*a*A*b^2+27*a^2*b*B+7*b^3*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+2/21*(21*a^2*A*b+5*A*b^3+7*a^3*B+15*a*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))/d+2/45*b*(27*a*A*b+22*a^2*B+7*b^2*B)*cos(c+d*x)^(3/2)*sin(c+d*x)/d+2/63*b^2*(9*A*b+13*a*B)*cos(c+d*x)^(5/2)*sin(c+d*x)/d+2/9*b*B*cos(c+d*x)^(3/2)*(a+b*cos(c+d*x))^2*sin(c+d*x)/d+2/21*(21*a^2*A*b+5*A*b^3+7*a^3*B+15*a*b^2*B)*sin(c+d*x)*sqrt(cos(c+d*x))/d],
[(a+b*cos(c+d*x))^3*(A+B*cos(c+d*x))/cos(c+d*x)^(1/2),x,6,2/5*(15*a^2*A*b+3*A*b^3+5*a^3*B+9*a*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))/d+2/21*(21*a^3*A+21*a*A*b^2+21*a^2*b*B+5*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))/d+2/35*b^2*(7*A*b+11*a*B)*cos(c+d*x)^(3/2)*sin(c+d*x)/d+2/21*b*(21*a*A*b+18*a^2*B+5*b^2*B)*sin(c+d*x)*sqrt(cos(c+d*x))/d+2/7*b*B*(a+b*cos(c+d*x))^2*sin(c+d*x)*sqrt(cos(c+d*x))/d],
[(a+b*cos(c+d*x))^3*(A+B*cos(c+d*x))/cos(c+d*x)^(3/2),x,6,-2/5*(5*a^3*A-15*a*A*b^2-15*a^2*b*B-3*b^3*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+2/3*(9*a^2*A*b+A*b^3+3*a^3*B+3*a*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))/d-2/5*b^2*(5*a*A-b*B)*cos(c+d*x)^(3/2)*sin(c+d*x)/d+2*a*A*(a+b*cos(c+d*x))^2*sin(c+d*x)/(d*sqrt(cos(c+d*x)))-2/3*b*(6*a^2*A-A*b^2-3*a*b*B)*sin(c+d*x)*sqrt(cos(c+d*x))/d],
[(a+b*cos(c+d*x))^3*(A+B*cos(c+d*x))/cos(c+d*x)^(5/2),x,6,-2*(3*a^2*A*b-A*b^3+a^3*B-3*a*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))/d+2/3*(a^3*A+9*a*A*b^2+9*a^2*b*B+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))/d+2/3*a*A*(a+b*cos(c+d*x))^2*sin(c+d*x)/(d*cos(c+d*x)^(3/2))+2/3*a^2*(7*A*b+3*a*B)*sin(c+d*x)/(d*sqrt(cos(c+d*x)))-2/3*b^2*(a*A-b*B)*sin(c+d*x)*sqrt(cos(c+d*x))/d],
[(a+b*cos(c+d*x))^3*(A+B*cos(c+d*x))/cos(c+d*x)^(7/2),x,6,-2/5*(3*a^3*A+15*a*A*b^2+15*a^2*b*B-5*b^3*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+2/3*(3*a^2*A*b+3*A*b^3+a^3*B+9*a*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))/d+2/15*a^2*(9*A*b+5*a*B)*sin(c+d*x)/(d*cos(c+d*x)^(3/2))+2/5*a*A*(a+b*cos(c+d*x))^2*sin(c+d*x)/(d*cos(c+d*x)^(5/2))+2/5*a*(3*a^2*A+14*A*b^2+15*a*b*B)*sin(c+d*x)/(d*sqrt(cos(c+d*x)))],

# n<0
[cos(c+d*x)^(5/2)*(A+B*cos(c+d*x))/(a+b*cos(c+d*x)),x,7,-2/5*(5*a*A*b-5*a^2*B-3*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))/(b^3*d)+2/3*(3*a^2+b^2)*(A*b-a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/(b^4*d)-2*a^3*(A*b-a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(a+b),sqrt(2))/(b^4*(a+b)*d)+2/5*B*cos(c+d*x)^(3/2)*sin(c+d*x)/(b*d)+2/3*(A*b-a*B)*sin(c+d*x)*sqrt(cos(c+d*x))/(b^2*d)],
[cos(c+d*x)^(3/2)*(A+B*cos(c+d*x))/(a+b*cos(c+d*x)),x,6,2*(A*b-a*B)*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)-2/3*(3*a*A*b-3*a^2*B-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))/(b^3*d)+2*a^2*(A*b-a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(a+b),sqrt(2))/(b^3*(a+b)*d)+2/3*B*sin(c+d*x)*sqrt(cos(c+d*x))/(b*d)],
[cos(c+d*x)^(1/2)*(A+B*cos(c+d*x))/(a+b*cos(c+d*x)),x,5,2*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(b*d)+2*(A*b-a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/(b^2*d)-2*a*(A*b-a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(a+b),sqrt(2))/(b^2*(a+b)*d)],
[(A+B*cos(c+d*x))/(cos(c+d*x)^(1/2)*(a+b*cos(c+d*x))),x,3,2*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/(b*d)+2*(A*b-a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(a+b),sqrt(2))/(b*(a+b)*d)],
[(A+B*cos(c+d*x))/(cos(c+d*x)^(3/2)*(a+b*cos(c+d*x))),x,5,-2*A*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(a*d)-2*(A*b-a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(a+b),sqrt(2))/(a*(a+b)*d)+2*A*sin(c+d*x)/(a*d*sqrt(cos(c+d*x)))],
[(A+B*cos(c+d*x))/(cos(c+d*x)^(5/2)*(a+b*cos(c+d*x))),x,7,2*(A*b-a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(a^2*d)+2/3*A*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/(a*d)+2*b*(A*b-a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(a+b),sqrt(2))/(a^2*(a+b)*d)+2/3*A*sin(c+d*x)/(a*d*cos(c+d*x)^(3/2))-2*(A*b-a*B)*sin(c+d*x)/(a^2*d*sqrt(cos(c+d*x)))],
[cos(c+d*x)^(5/2)*(A+B*cos(c+d*x))/(a+b*cos(c+d*x))^2,x,7,(3*a^2*A*b-2*A*b^3-5*a^3*B+4*a*b^2*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(b^3*(a^2-b^2)*d)-1/3*(9*a^3*A*b-12*a*A*b^3-15*a^4*B+16*a^2*b^2*B+2*b^4*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/(b^4*(a^2-b^2)*d)+a^2*(3*a^2*A*b-5*A*b^3-5*a^3*B+7*a*b^2*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(a+b),sqrt(2))/((a-b)*b^4*(a+b)^2*d)+a*(A*b-a*B)*cos(c+d*x)^(3/2)*sin(c+d*x)/(b*(a^2-b^2)*d*(a+b*cos(c+d*x)))-1/3*(3*a*A*b-5*a^2*B+2*b^2*B)*sin(c+d*x)*sqrt(cos(c+d*x))/(b^2*(a^2-b^2)*d)],
[cos(c+d*x)^(3/2)*(A+B*cos(c+d*x))/(a+b*cos(c+d*x))^2,x,6,-(a*A*b-3*a^2*B+2*b^2*B)*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*(a^2-b^2)*d)+(a^2*A*b-2*A*b^3-3*a^3*B+4*a*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))/(b^3*(a^2-b^2)*d)-a*(a^2*A*b-3*A*b^3-3*a^3*B+5*a*b^2*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(a+b),sqrt(2))/((a-b)*b^3*(a+b)^2*d)+a*(A*b-a*B)*sin(c+d*x)*sqrt(cos(c+d*x))/(b*(a^2-b^2)*d*(a+b*cos(c+d*x)))],
[cos(c+d*x)^(1/2)*(A+B*cos(c+d*x))/(a+b*cos(c+d*x))^2,x,6,(A*b-a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(b*(a^2-b^2)*d)+(a*A*b+a^2*B-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))/(b^2*(a^2-b^2)*d)-(a^2*A*b+A*b^3+a^3*B-3*a*b^2*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(a+b),sqrt(2))/((a-b)*b^2*(a+b)^2*d)-(A*b-a*B)*sin(c+d*x)*sqrt(cos(c+d*x))/((a^2-b^2)*d*(a+b*cos(c+d*x)))],
[(A+B*cos(c+d*x))/(cos(c+d*x)^(1/2)*(a+b*cos(c+d*x))^2),x,6,-(A*b-a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(a*(a^2-b^2)*d)-(A*b-a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/(b*(a^2-b^2)*d)+(3*a^2*A*b-A*b^3-a^3*B-a*b^2*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(a+b),sqrt(2))/(a*(a-b)*b*(a+b)^2*d)+b*(A*b-a*B)*sin(c+d*x)*sqrt(cos(c+d*x))/(a*(a^2-b^2)*d*(a+b*cos(c+d*x)))],
[(A+B*cos(c+d*x))/(cos(c+d*x)^(3/2)*(a+b*cos(c+d*x))^2),x,7,-(2*a^2*A-3*A*b^2+a*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))/(a^2*(a^2-b^2)*d)+(A*b-a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/(a*(a^2-b^2)*d)-(5*a^2*A*b-3*A*b^3-3*a^3*B+a*b^2*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(a+b),sqrt(2))/(a^2*(a-b)*(a+b)^2*d)+(2*a^2*A-3*A*b^2+a*b*B)*sin(c+d*x)/(a^2*(a^2-b^2)*d*sqrt(cos(c+d*x)))+b*(A*b-a*B)*sin(c+d*x)/(a*(a^2-b^2)*d*(a+b*cos(c+d*x))*sqrt(cos(c+d*x)))],
[(A+B*cos(c+d*x))/(cos(c+d*x)^(5/2)*(a+b*cos(c+d*x))^2),x,8,(4*a^2*A*b-5*A*b^3-2*a^3*B+3*a*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))/(a^3*(a^2-b^2)*d)+1/3*(2*a^2*A-5*A*b^2+3*a*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))/(a^2*(a^2-b^2)*d)+b*(7*a^2*A*b-5*A*b^3-5*a^3*B+3*a*b^2*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(a+b),sqrt(2))/(a^3*(a-b)*(a+b)^2*d)+1/3*(2*a^2*A-5*A*b^2+3*a*b*B)*sin(c+d*x)/(a^2*(a^2-b^2)*d*cos(c+d*x)^(3/2))+b*(A*b-a*B)*sin(c+d*x)/(a*(a^2-b^2)*d*cos(c+d*x)^(3/2)*(a+b*cos(c+d*x)))-(4*a^2*A*b-5*A*b^3-2*a^3*B+3*a*b^2*B)*sin(c+d*x)/(a^3*(a^2-b^2)*d*sqrt(cos(c+d*x)))],
[cos(c+d*x)^(5/2)*(A+B*cos(c+d*x))/(a+b*cos(c+d*x))^3,x,7,-1/4*(3*a^3*A*b-9*a*A*b^3-15*a^4*B+29*a^2*b^2*B-8*b^4*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(b^3*(a^2-b^2)^2*d)+1/4*(3*a^4*A*b-5*a^2*A*b^3+8*A*b^5-15*a^5*B+33*a^3*b^2*B-24*a*b^4*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/(b^4*(a^2-b^2)^2*d)-1/4*a*(3*a^4*A*b-6*a^2*A*b^3+15*A*b^5-15*a^5*B+38*a^3*b^2*B-35*a*b^4*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(a+b),sqrt(2))/((a-b)^2*b^4*(a+b)^3*d)+1/2*a*(A*b-a*B)*cos(c+d*x)^(3/2)*sin(c+d*x)/(b*(a^2-b^2)*d*(a+b*cos(c+d*x))^2)+1/4*a*(a^2*A*b-7*A*b^3-5*a^3*B+11*a*b^2*B)*sin(c+d*x)*sqrt(cos(c+d*x))/(b^2*(a^2-b^2)^2*d*(a+b*cos(c+d*x)))],
[cos(c+d*x)^(3/2)*(A+B*cos(c+d*x))/(a+b*cos(c+d*x))^3,x,7,-1/4*(a^2*A*b+5*A*b^3+3*a^3*B-9*a*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))/(b^2*(a^2-b^2)^2*d)+1/4*(a^3*A*b-7*a*A*b^3+3*a^4*B-5*a^2*b^2*B+8*b^4*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/(b^3*(a^2-b^2)^2*d)-1/4*(a^4*A*b-10*a^2*A*b^3-3*A*b^5+3*a^5*B-6*a^3*b^2*B+15*a*b^4*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(a+b),sqrt(2))/((a-b)^2*b^3*(a+b)^3*d)+1/2*a*(A*b-a*B)*sin(c+d*x)*sqrt(cos(c+d*x))/(b*(a^2-b^2)*d*(a+b*cos(c+d*x))^2)+1/4*(a^2*A*b+5*A*b^3+3*a^3*B-9*a*b^2*B)*sin(c+d*x)*sqrt(cos(c+d*x))/(b*(a^2-b^2)^2*d*(a+b*cos(c+d*x)))],
[cos(c+d*x)^(1/2)*(A+B*cos(c+d*x))/(a+b*cos(c+d*x))^3,x,7,1/4*(5*a^2*A*b+A*b^3-a^3*B-5*a*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))/(a*b*(a^2-b^2)^2*d)+1/4*(3*a^2*A*b+3*A*b^3+a^3*B-7*a*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))/(b^2*(a^2-b^2)^2*d)-1/4*(3*a^4*A*b+10*a^2*A*b^3-A*b^5+a^5*B-10*a^3*b^2*B-3*a*b^4*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(a+b),sqrt(2))/(a*(a-b)^2*b^2*(a+b)^3*d)-1/2*(A*b-a*B)*sin(c+d*x)*sqrt(cos(c+d*x))/((a^2-b^2)*d*(a+b*cos(c+d*x))^2)-1/4*(5*a^2*A*b+A*b^3-a^3*B-5*a*b^2*B)*sin(c+d*x)*sqrt(cos(c+d*x))/(a*(a^2-b^2)^2*d*(a+b*cos(c+d*x)))],
[(A+B*cos(c+d*x))/(cos(c+d*x)^(1/2)*(a+b*cos(c+d*x))^3),x,7,-1/4*(9*a^2*A*b-3*A*b^3-5*a^3*B-a*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))/(a^2*(a^2-b^2)^2*d)-1/4*(7*a^2*A*b-A*b^3-3*a^3*B-3*a*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))/(a*b*(a^2-b^2)^2*d)+1/4*(15*a^4*A*b-6*a^2*A*b^3+3*A*b^5-3*a^5*B-10*a^3*b^2*B+a*b^4*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(a+b),sqrt(2))/(a^2*(a-b)^2*b*(a+b)^3*d)+1/2*b*(A*b-a*B)*sin(c+d*x)*sqrt(cos(c+d*x))/(a*(a^2-b^2)*d*(a+b*cos(c+d*x))^2)+1/4*b*(9*a^2*A*b-3*A*b^3-5*a^3*B-a*b^2*B)*sin(c+d*x)*sqrt(cos(c+d*x))/(a^2*(a^2-b^2)^2*d*(a+b*cos(c+d*x)))],
[(A+B*cos(c+d*x))/(cos(c+d*x)^(3/2)*(a+b*cos(c+d*x))^3),x,8,-1/4*(8*a^4*A-29*a^2*A*b^2+15*A*b^4+9*a^3*b*B-3*a*b^3*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(a^3*(a^2-b^2)^2*d)+1/4*(11*a^2*A*b-5*A*b^3-7*a^3*B+a*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))/(a^2*(a^2-b^2)^2*d)-1/4*(35*a^4*A*b-38*a^2*A*b^3+15*A*b^5-15*a^5*B+6*a^3*b^2*B-3*a*b^4*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(a+b),sqrt(2))/(a^3*(a-b)^2*(a+b)^3*d)+1/4*(8*a^4*A-29*a^2*A*b^2+15*A*b^4+9*a^3*b*B-3*a*b^3*B)*sin(c+d*x)/(a^3*(a^2-b^2)^2*d*sqrt(cos(c+d*x)))+1/2*b*(A*b-a*B)*sin(c+d*x)/(a*(a^2-b^2)*d*(a+b*cos(c+d*x))^2*sqrt(cos(c+d*x)))+1/4*b*(11*a^2*A*b-5*A*b^3-7*a^3*B+a*b^2*B)*sin(c+d*x)/(a^2*(a^2-b^2)^2*d*(a+b*cos(c+d*x))*sqrt(cos(c+d*x)))],
[(A+B*cos(c+d*x))/(cos(c+d*x)^(5/2)*(a+b*cos(c+d*x))^3),x,9,1/4*(24*a^4*A*b-65*a^2*A*b^3+35*A*b^5-8*a^5*B+29*a^3*b^2*B-15*a*b^4*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(a^4*(a^2-b^2)^2*d)+1/12*(8*a^4*A-61*a^2*A*b^2+35*A*b^4+33*a^3*b*B-15*a*b^3*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/(a^3*(a^2-b^2)^2*d)+1/4*b*(63*a^4*A*b-86*a^2*A*b^3+35*A*b^5-35*a^5*B+38*a^3*b^2*B-15*a*b^4*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(a+b),sqrt(2))/(a^4*(a-b)^2*(a+b)^3*d)+1/12*(8*a^4*A-61*a^2*A*b^2+35*A*b^4+33*a^3*b*B-15*a*b^3*B)*sin(c+d*x)/(a^3*(a^2-b^2)^2*d*cos(c+d*x)^(3/2))+1/2*b*(A*b-a*B)*sin(c+d*x)/(a*(a^2-b^2)*d*cos(c+d*x)^(3/2)*(a+b*cos(c+d*x))^2)+1/4*b*(13*a^2*A*b-7*A*b^3-9*a^3*B+3*a*b^2*B)*sin(c+d*x)/(a^2*(a^2-b^2)^2*d*cos(c+d*x)^(3/2)*(a+b*cos(c+d*x)))-1/4*(24*a^4*A*b-65*a^2*A*b^3+35*A*b^5-8*a^5*B+29*a^3*b^2*B-15*a*b^4*B)*sin(c+d*x)/(a^4*(a^2-b^2)^2*d*sqrt(cos(c+d*x)))],
[cos(c+d*x)^(5/2)*(a*B+b*B*cos(c+d*x))/(a+b*cos(c+d*x)),x,3,6/5*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+2/5*B*cos(c+d*x)^(3/2)*sin(c+d*x)/d],
[cos(c+d*x)^(3/2)*(a*B+b*B*cos(c+d*x))/(a+b*cos(c+d*x)),x,3,2/3*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+2/3*B*sin(c+d*x)*sqrt(cos(c+d*x))/d],
[cos(c+d*x)^(1/2)*(a*B+b*B*cos(c+d*x))/(a+b*cos(c+d*x)),x,2,2*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d],
[(a*B+b*B*cos(c+d*x))/(cos(c+d*x)^(1/2)*(a+b*cos(c+d*x))),x,2,2*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d],
[(a*B+b*B*cos(c+d*x))/(cos(c+d*x)^(3/2)*(a+b*cos(c+d*x))),x,3,-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))/d+2*B*sin(c+d*x)/(d*sqrt(cos(c+d*x)))],
[(a*B+b*B*cos(c+d*x))/(cos(c+d*x)^(5/2)*(a+b*cos(c+d*x))),x,3,2/3*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+2/3*B*sin(c+d*x)/(d*cos(c+d*x)^(3/2))],
[cos(c+d*x)^(5/2)*(a*B+b*B*cos(c+d*x))/(a+b*cos(c+d*x))^2,x,7,-2*a*B*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)+2/3*(3*a^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))/(b^3*d)-2*a^3*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(a+b),sqrt(2))/(b^3*(a+b)*d)+2/3*B*sin(c+d*x)*sqrt(cos(c+d*x))/(b*d)],
[cos(c+d*x)^(3/2)*(a*B+b*B*cos(c+d*x))/(a+b*cos(c+d*x))^2,x,6,2*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(b*d)-2*a*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/(b^2*d)+2*a^2*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(a+b),sqrt(2))/(b^2*(a+b)*d)],
[cos(c+d*x)^(1/2)*(a*B+b*B*cos(c+d*x))/(a+b*cos(c+d*x))^2,x,4,2*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/(b*d)-2*a*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(a+b),sqrt(2))/(b*(a+b)*d)],
[(a*B+b*B*cos(c+d*x))/(cos(c+d*x)^(1/2)*(a+b*cos(c+d*x))^2),x,2,2*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(a+b),sqrt(2))/((a+b)*d)],
[(a*B+b*B*cos(c+d*x))/(cos(c+d*x)^(3/2)*(a+b*cos(c+d*x))^2),x,6,-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))/(a*d)-2*b*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(a+b),sqrt(2))/(a*(a+b)*d)+2*B*sin(c+d*x)/(a*d*sqrt(cos(c+d*x)))],
[(a*B+b*B*cos(c+d*x))/(cos(c+d*x)^(5/2)*(a+b*cos(c+d*x))^2),x,8,2*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))/(a^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))/(a*d)+2*b^2*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(a+b),sqrt(2))/(a^2*(a+b)*d)+2/3*B*sin(c+d*x)/(a*d*cos(c+d*x)^(3/2))-2*b*B*sin(c+d*x)/(a^2*d*sqrt(cos(c+d*x)))],

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

# n>0
[cos(c+d*x)^(3/2)*(a+b*cos(c+d*x))^(1/2)*(A+B*cos(c+d*x)),x,8,1/3*B*(a+b*cos(c+d*x))^(3/2)*sin(c+d*x)*sqrt(cos(c+d*x))/(b*d)+1/24*(6*a*A*b-3*a^2*B+16*b^2*B)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(b^2*d*sqrt(cos(c+d*x)))+1/4*(2*A*b-a*B)*sin(c+d*x)*sqrt(cos(c+d*x))*sqrt(a+b*cos(c+d*x))/(b*d)-1/24*(a-b)*(6*a*A*b-3*a^2*B+16*b^2*B)*cot(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*b^2*d)+1/24*(a+2*b)*(6*A*b-3*a*B+8*b*B)*cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b^2*d)+1/8*(2*a^2*A*b-8*A*b^3-a^3*B-4*a*b^2*B)*cot(c+d*x)*EllipticPi(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),(a+b)/b,sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b^3*d)],
[cos(c+d*x)^(1/2)*(a+b*cos(c+d*x))^(1/2)*(A+B*cos(c+d*x)),x,7,1/4*(4*A*b+a*B)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(b*d*sqrt(cos(c+d*x)))+1/2*B*sin(c+d*x)*sqrt(cos(c+d*x))*sqrt(a+b*cos(c+d*x))/d-1/4*(a-b)*(4*A*b+a*B)*cot(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*b*d)+1/4*(4*A*b+(a+2*b)*B)*cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b*d)-1/4*(4*a*A*b-a^2*B+4*b^2*B)*cot(c+d*x)*EllipticPi(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),(a+b)/b,sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b^2*d)],
[(a+b*cos(c+d*x))^(1/2)*(A+B*cos(c+d*x))/cos(c+d*x)^(1/2),x,6,B*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))-(a-b)*B*cot(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*d)+(2*A+B)*cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/d-(2*A*b+a*B)*cot(c+d*x)*EllipticPi(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),(a+b)/b,sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b*d)],
[(a+b*cos(c+d*x))^(1/2)*(A+B*cos(c+d*x))/cos(c+d*x)^(3/2),x,5,2*A*(a-b)*cot(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*d)+2*(A*b-a*(A-B))*cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*d)-2*B*cot(c+d*x)*EllipticPi(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),(a+b)/b,sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/d],
[(a+b*cos(c+d*x))^(1/2)*(A+B*cos(c+d*x))/cos(c+d*x)^(5/2),x,4,2/3*A*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(d*cos(c+d*x)^(3/2))+2/3*(a-b)*(A*b+3*a*B)*cot(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^2*d)+2/3*(a-b)*(A-3*B)*cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*d)],
[(a+b*cos(c+d*x))^(1/2)*(A+B*cos(c+d*x))/cos(c+d*x)^(7/2),x,5,2/5*A*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(d*cos(c+d*x)^(5/2))+2/15*(A*b+5*a*B)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(a*d*cos(c+d*x)^(3/2))+2/15*(a-b)*(9*a^2*A-2*A*b^2+5*a*b*B)*cot(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^3*d)-2/15*(a-b)*(9*a*A+2*A*b-5*a*B)*cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^2*d)],
[(a+b*cos(c+d*x))^(1/2)*(A+B*cos(c+d*x))/cos(c+d*x)^(9/2),x,6,2/7*A*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(d*cos(c+d*x)^(7/2))+2/35*(A*b+7*a*B)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(a*d*cos(c+d*x)^(5/2))+2/105*(25*a^2*A-4*A*b^2+7*a*b*B)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(a^2*d*cos(c+d*x)^(3/2))+2/105*(a-b)*(19*a^2*A*b+8*A*b^3+63*a^3*B-14*a*b^2*B)*cot(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^4*d)+2/105*(a-b)*(8*A*b^2+a^2*(25*A-63*B)+2*a*b*(3*A-7*B))*cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^3*d)],
[cos(c+d*x)^(3/2)*(a+b*cos(c+d*x))^(3/2)*(A+B*cos(c+d*x)),x,9,1/24*(8*A*b-3*a*B)*(a+b*cos(c+d*x))^(3/2)*sin(c+d*x)*sqrt(cos(c+d*x))/(b*d)+1/4*B*(a+b*cos(c+d*x))^(5/2)*sin(c+d*x)*sqrt(cos(c+d*x))/(b*d)+1/192*(24*a^2*A*b+128*A*b^3-9*a^3*B+156*a*b^2*B)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(b^2*d*sqrt(cos(c+d*x)))+1/32*(8*a*A*b-3*a^2*B+12*b^2*B)*sin(c+d*x)*sqrt(cos(c+d*x))*sqrt(a+b*cos(c+d*x))/(b*d)-1/192*(a-b)*(24*a^2*A*b+128*A*b^3-9*a^3*B+156*a*b^2*B)*cot(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*b^2*d)-1/192*(9*a^3*B-6*a^2*b*(4*A+B)-8*b^3*(16*A+9*B)-4*a*b^2*(28*A+39*B))*cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b^2*d)+1/64*(8*a^3*A*b-96*a*A*b^3-3*a^4*B-24*a^2*b^2*B-48*b^4*B)*cot(c+d*x)*EllipticPi(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),(a+b)/b,sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b^3*d)],
[cos(c+d*x)^(1/2)*(a+b*cos(c+d*x))^(3/2)*(A+B*cos(c+d*x)),x,8,1/3*b*B*cos(c+d*x)^(3/2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/d+1/24*(30*a*A*b+3*a^2*B+16*b^2*B)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(b*d*sqrt(cos(c+d*x)))+1/12*(6*A*b+7*a*B)*sin(c+d*x)*sqrt(cos(c+d*x))*sqrt(a+b*cos(c+d*x))/d-1/24*(a-b)*(30*a*A*b+3*a^2*B+16*b^2*B)*cot(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*b*d)+1/24*(30*a*A*b+12*A*b^2+3*a^2*B+14*a*b*B+16*b^2*B)*cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b*d)-1/8*(6*a^2*A*b+8*A*b^3-a^3*B+12*a*b^2*B)*cot(c+d*x)*EllipticPi(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),(a+b)/b,sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b^2*d)],
[(a+b*cos(c+d*x))^(3/2)*(A+B*cos(c+d*x))/cos(c+d*x)^(1/2),x,7,1/4*(4*A*b+5*a*B)*sin(c+d*x)*sqrt(a+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(a+b*cos(c+d*x))/d-1/4*(a-b)*(4*A*b+5*a*B)*cot(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*d)+1/4*(8*a*A+4*A*b+5*a*B+2*b*B)*cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/d-1/4*(12*a*A*b+3*a^2*B+4*b^2*B)*cot(c+d*x)*EllipticPi(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),(a+b)/b,sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b*d)],
[(a+b*cos(c+d*x))^(3/2)*(A+B*cos(c+d*x))/cos(c+d*x)^(3/2),x,7,2*a*A*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))-(2*a*A-b*B)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))+(a-b)*(2*a*A-b*B)*cot(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*d)-(2*a*(A-B)-b*(4*A+B))*cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/d-(2*A*b+3*a*B)*cot(c+d*x)*EllipticPi(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),(a+b)/b,sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/d],
[(a+b*cos(c+d*x))^(3/2)*(A+B*cos(c+d*x))/cos(c+d*x)^(5/2),x,6,2/3*a*A*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(d*cos(c+d*x)^(3/2))+2/3*(a-b)*(4*A*b+3*a*B)*cot(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*d)+2/3*(3*A*b^2+a^2*(A-3*B)-a*(4*A*b-6*b*B))*cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*d)-2*b*B*cot(c+d*x)*EllipticPi(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),(a+b)/b,sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/d],
[(a+b*cos(c+d*x))^(3/2)*(A+B*cos(c+d*x))/cos(c+d*x)^(7/2),x,5,2/5*a*A*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(d*cos(c+d*x)^(5/2))+2/15*(6*A*b+5*a*B)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(d*cos(c+d*x)^(3/2))+2/15*(a-b)*(9*a^2*A+3*A*b^2+20*a*b*B)*cot(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^2*d)-2/15*(a-b)*(9*a*A-3*A*b-5*a*B+15*b*B)*cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*d)],
[(a+b*cos(c+d*x))^(3/2)*(A+B*cos(c+d*x))/cos(c+d*x)^(9/2),x,6,2/7*a*A*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(d*cos(c+d*x)^(7/2))+2/35*(8*A*b+7*a*B)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(d*cos(c+d*x)^(5/2))+2/105*(25*a^2*A+3*A*b^2+42*a*b*B)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(a*d*cos(c+d*x)^(3/2))+2/105*(a-b)*(82*a^2*A*b-6*A*b^3+63*a^3*B+21*a*b^2*B)*cot(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^3*d)-2/105*(a-b)*(6*A*b^2-a^2*(25*A-63*B)+3*a*b*(19*A-7*B))*cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^2*d)],
[(a+b*cos(c+d*x))^(3/2)*(A+B*cos(c+d*x))/cos(c+d*x)^(11/2),x,7,2/9*a*A*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(d*cos(c+d*x)^(9/2))+2/63*(10*A*b+9*a*B)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(d*cos(c+d*x)^(7/2))+2/315*(49*a^2*A+3*A*b^2+72*a*b*B)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(a*d*cos(c+d*x)^(5/2))+2/315*(88*a^2*A*b-4*A*b^3+75*a^3*B+9*a*b^2*B)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(a^2*d*cos(c+d*x)^(3/2))+2/315*(a-b)*(147*a^4*A+33*a^2*A*b^2+8*A*b^4+246*a^3*b*B-18*a*b^3*B)*cot(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^4*d)+2/315*(a-b)*(8*A*b^3-a^3*(147*A-75*B)+3*a^2*b*(13*A-57*B)+6*a*b^2*(A-3*B))*cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^3*d)],
[cos(c+d*x)^(3/2)*(a+b*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x)),x,10,1/240*(50*a*A*b-15*a^2*B+64*b^2*B)*(a+b*cos(c+d*x))^(3/2)*sin(c+d*x)*sqrt(cos(c+d*x))/(b*d)+1/40*(10*A*b-3*a*B)*(a+b*cos(c+d*x))^(5/2)*sin(c+d*x)*sqrt(cos(c+d*x))/(b*d)+1/5*B*(a+b*cos(c+d*x))^(7/2)*sin(c+d*x)*sqrt(cos(c+d*x))/(b*d)+1/1920*(150*a^3*A*b+2840*a*A*b^3-45*a^4*B+1692*a^2*b^2*B+1024*b^4*B)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(b^2*d*sqrt(cos(c+d*x)))+1/320*(50*a^2*A*b+120*A*b^3-15*a^3*B+172*a*b^2*B)*sin(c+d*x)*sqrt(cos(c+d*x))*sqrt(a+b*cos(c+d*x))/(b*d)-1/1920*(a-b)*(150*a^3*A*b+2840*a*A*b^3-45*a^4*B+1692*a^2*b^2*B+1024*b^4*B)*cot(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*b^2*d)-1/1920*(45*a^4*B-30*a^3*b*(5*A+B)-16*b^4*(45*A+64*B)-8*a*b^3*(355*A+193*B)-4*a^2*b^2*(295*A+423*B))*cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b^2*d)+1/128*(10*a^4*A*b-240*a^2*A*b^3-96*A*b^5-3*a^5*B-40*a^3*b^2*B-240*a*b^4*B)*cot(c+d*x)*EllipticPi(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),(a+b)/b,sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b^3*d)],
[cos(c+d*x)^(1/2)*(a+b*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x)),x,9,1/4*b*B*cos(c+d*x)^(3/2)*(a+b*cos(c+d*x))^(3/2)*sin(c+d*x)/d+1/24*(8*A*b+11*a*B)*(a+b*cos(c+d*x))^(3/2)*sin(c+d*x)*sqrt(cos(c+d*x))/d+1/192*(264*a^2*A*b+128*A*b^3+15*a^3*B+284*a*b^2*B)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(b*d*sqrt(cos(c+d*x)))+1/32*(24*a*A*b+5*a^2*B+12*b^2*B)*sin(c+d*x)*sqrt(cos(c+d*x))*sqrt(a+b*cos(c+d*x))/d-1/192*(a-b)*(264*a^2*A*b+128*A*b^3+15*a^3*B+284*a*b^2*B)*cot(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*b*d)+1/192*(15*a^3*B+8*b^3*(16*A+9*B)+2*a^2*b*(132*A+59*B)+4*a*b^2*(52*A+71*B))*cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b*d)-1/64*(40*a^3*A*b+160*a*A*b^3-5*a^4*B+120*a^2*b^2*B+48*b^4*B)*cot(c+d*x)*EllipticPi(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),(a+b)/b,sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b^2*d)],
[(a+b*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x))/cos(c+d*x)^(1/2),x,8,1/3*b*B*(a+b*cos(c+d*x))^(3/2)*sin(c+d*x)*sqrt(cos(c+d*x))/d+1/24*(54*a*A*b+33*a^2*B+16*b^2*B)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))+1/4*b*(2*A*b+3*a*B)*sin(c+d*x)*sqrt(cos(c+d*x))*sqrt(a+b*cos(c+d*x))/d-1/24*(a-b)*(54*a*A*b+33*a^2*B+16*b^2*B)*cot(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*d)+1/24*(4*b^2*(3*A+4*B)+a^2*(48*A+33*B)+a*(54*A*b+26*b*B))*cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/d-1/8*(30*a^2*A*b+8*A*b^3+5*a^3*B+20*a*b^2*B)*cot(c+d*x)*EllipticPi(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),(a+b)/b,sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b*d)],
[(a+b*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x))/cos(c+d*x)^(3/2),x,8,2*a*A*(a+b*cos(c+d*x))^(3/2)*sin(c+d*x)/(d*sqrt(cos(c+d*x)))-1/4*(8*a^2*A-4*A*b^2-9*a*b*B)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))-1/2*b*(4*a*A-b*B)*sin(c+d*x)*sqrt(cos(c+d*x))*sqrt(a+b*cos(c+d*x))/d+1/4*(a-b)*(8*a^2*A-4*A*b^2-9*a*b*B)*cot(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*d)-1/4*(8*a^2*(A-B)-2*b^2*(2*A+B)-3*a*b*(8*A+3*B))*cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/d-1/4*(20*a*A*b+15*a^2*B+4*b^2*B)*cot(c+d*x)*EllipticPi(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),(a+b)/b,sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/d],
[(a+b*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x))/cos(c+d*x)^(5/2),x,8,2/3*a*A*(a+b*cos(c+d*x))^(3/2)*sin(c+d*x)/(d*cos(c+d*x)^(3/2))+2*a*(2*A*b+a*B)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))-1/3*(14*a*A*b+6*a^2*B-3*b^2*B)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))+1/3*(a-b)*(14*a*A*b+6*a^2*B-3*b^2*B)*cot(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*d)-1/3*(2*a*b*(7*A-9*B)-2*a^2*(A-3*B)-3*b^2*(6*A+B))*cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/d-b*(2*A*b+5*a*B)*cot(c+d*x)*EllipticPi(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),(a+b)/b,sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/d],
[(a+b*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x))/cos(c+d*x)^(7/2),x,7,2/5*a*A*(a+b*cos(c+d*x))^(3/2)*sin(c+d*x)/(d*cos(c+d*x)^(5/2))+2/15*a*(8*A*b+5*a*B)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(d*cos(c+d*x)^(3/2))+2/15*(a-b)*(9*a^2*A+23*A*b^2+35*a*b*B)*cot(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*d)+2/15*(15*A*b^3-a*b^2*(23*A-45*B)+a^2*b*(17*A-35*B)-a^3*(9*A-5*B))*cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*d)-2*b^2*B*cot(c+d*x)*EllipticPi(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),(a+b)/b,sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/d],
[(a+b*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x))/cos(c+d*x)^(9/2),x,6,2/7*a*A*(a+b*cos(c+d*x))^(3/2)*sin(c+d*x)/(d*cos(c+d*x)^(7/2))+2/35*a*(10*A*b+7*a*B)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(d*cos(c+d*x)^(5/2))+2/105*(25*a^2*A+45*A*b^2+77*a*b*B)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(d*cos(c+d*x)^(3/2))+2/105*(a-b)*(145*a^2*A*b+15*A*b^3+63*a^3*B+161*a*b^2*B)*cot(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^2*d)+2/105*(a-b)*(a^2*(25*A-63*B)+15*b^2*(A-7*B)-8*a*b*(15*A-7*B))*cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*d)],
[(a+b*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x))/cos(c+d*x)^(11/2),x,7,2/9*a*A*(a+b*cos(c+d*x))^(3/2)*sin(c+d*x)/(d*cos(c+d*x)^(9/2))+2/21*a*(4*A*b+3*a*B)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(d*cos(c+d*x)^(7/2))+2/315*(49*a^2*A+75*A*b^2+135*a*b*B)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(d*cos(c+d*x)^(5/2))+2/315*(163*a^2*A*b+5*A*b^3+75*a^3*B+135*a*b^2*B)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(a*d*cos(c+d*x)^(3/2))+2/315*(a-b)*(147*a^4*A+279*a^2*A*b^2-10*A*b^4+435*a^3*b*B+45*a*b^3*B)*cot(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^3*d)-2/315*(a-b)*(10*A*b^3-6*a^2*b*(19*A-60*B)+3*a^3*(49*A-25*B)+15*a*b^2*(11*A-3*B))*cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^2*d)],
[(a+b*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x))/cos(c+d*x)^(13/2),x,8,2/11*a*A*(a+b*cos(c+d*x))^(3/2)*sin(c+d*x)/(d*cos(c+d*x)^(11/2))+2/99*a*(14*A*b+11*a*B)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(d*cos(c+d*x)^(9/2))+2/693*(81*a^2*A+113*A*b^2+209*a*b*B)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(d*cos(c+d*x)^(7/2))+2/3465*(1145*a^2*A*b+15*A*b^3+539*a^3*B+825*a*b^2*B)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(a*d*cos(c+d*x)^(5/2))+2/3465*(675*a^4*A+1025*a^2*A*b^2-20*A*b^4+1793*a^3*b*B+55*a*b^3*B)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(a^2*d*cos(c+d*x)^(3/2))+2/3465*(a-b)*(3705*a^4*A*b+255*a^2*A*b^3+40*A*b^5+1617*a^5*B+3069*a^3*b^2*B-110*a*b^4*B)*cot(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^4*d)+2/3465*(a-b)*(40*A*b^4+3*a^4*(225*A-539*B)-6*a^3*b*(505*A-209*B)+15*a^2*b^2*(19*A-121*B)+10*a*b^3*(3*A-11*B))*cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^3*d)],
[(a+b*cos(c+d*x))^(5/2)*(3/2*b*B/a+B*cos(c+d*x))/cos(c+d*x)^(5/2),x,6,b*B*(a+b*cos(c+d*x))^(3/2)*sin(c+d*x)/(d*cos(c+d*x)^(3/2))+2*(a-b)*(a^2+3*b^2)*B*cot(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*d)-(a-3*b)*(2*a^2-a*b+3*b^2)*B*cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*d)-b*(5*a+3*b^2/a)*B*cot(c+d*x)*EllipticPi(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),(a+b)/b,sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/d],

# n<0
[cos(c+d*x)^(3/2)*(A+B*cos(c+d*x))/(a+b*cos(c+d*x))^(1/2),x,7,1/4*(4*A*b-3*a*B)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(b^2*d*sqrt(cos(c+d*x)))+1/2*B*sin(c+d*x)*sqrt(cos(c+d*x))*sqrt(a+b*cos(c+d*x))/(b*d)-1/4*(a-b)*(4*A*b-3*a*B)*cot(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*b^2*d)+1/4*(4*A*b-3*a*B+2*b*B)*cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b^2*d)+1/4*(4*a*A*b-3*a^2*B-4*b^2*B)*cot(c+d*x)*EllipticPi(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),(a+b)/b,sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b^3*d)],
[cos(c+d*x)^(1/2)*(A+B*cos(c+d*x))/(a+b*cos(c+d*x))^(1/2),x,7,a*B*sin(c+d*x)/(b*d*sqrt(cos(c+d*x))*sqrt(a+b*cos(c+d*x)))+B*sin(c+d*x)*sqrt(cos(c+d*x))/(d*sqrt(a+b*cos(c+d*x)))-(a-b)*B*cot(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*b*d)+B*cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b*d)-(2*A*b-a*B)*cot(c+d*x)*EllipticPi(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),(a+b)/b,sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b^2*d)],
[(A+B*cos(c+d*x))/(cos(c+d*x)^(1/2)*(a+b*cos(c+d*x))^(1/2)),x,3,2*A*cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*d)-2*B*cot(c+d*x)*EllipticPi(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),(a+b)/b,sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b*d)],
[(A+B*cos(c+d*x))/(cos(c+d*x)^(3/2)*(a+b*cos(c+d*x))^(1/2)),x,3,2*A*(a-b)*cot(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^2*d)-2*(A-B)*cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*d)],
[(A+B*cos(c+d*x))/(cos(c+d*x)^(5/2)*(a+b*cos(c+d*x))^(1/2)),x,4,2/3*A*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(a*d*cos(c+d*x)^(3/2))-2/3*(a-b)*(2*A*b-3*a*B)*cot(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^3*d)+2/3*(2*A*b+a*(A-3*B))*cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^2*d)],
[(A+B*cos(c+d*x))/(cos(c+d*x)^(7/2)*(a+b*cos(c+d*x))^(1/2)),x,5,2/5*A*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(a*d*cos(c+d*x)^(5/2))-2/15*(4*A*b-5*a*B)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(a^2*d*cos(c+d*x)^(3/2))+2/15*(a-b)*(9*a^2*A+8*A*b^2-10*a*b*B)*cot(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^4*d)-2/15*(8*A*b^2+a^2*(9*A-5*B)-2*a*b*(A+5*B))*cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^3*d)],
[cos(c+d*x)^(3/2)*(A+B*cos(c+d*x))/(a+b*cos(c+d*x))^(3/2),x,7,2*a*(A*b-a*B)*sin(c+d*x)*sqrt(cos(c+d*x))/(b*(a^2-b^2)*d*sqrt(a+b*cos(c+d*x)))-(2*a*A*b-3*a^2*B+b^2*B)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(b^2*(a^2-b^2)*d*sqrt(cos(c+d*x)))+(2*a*A*b-3*a^2*B+b^2*B)*cot(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*b^2*d*sqrt(a+b))-(2*A*b-(3*a+b)*B)*cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b^2*d*sqrt(a+b))-(2*A*b-3*a*B)*cot(c+d*x)*EllipticPi(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),(a+b)/b,sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b^3*d)],
[cos(c+d*x)^(1/2)*(A+B*cos(c+d*x))/(a+b*cos(c+d*x))^(3/2),x,6,2*a*(A*b-a*B)*sin(c+d*x)/(b*(a^2-b^2)*d*sqrt(cos(c+d*x))*sqrt(a+b*cos(c+d*x)))-2*(A*b-a*B)*cot(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*b*d*sqrt(a+b))+2*(A*b-a*B)*cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*b*d*sqrt(a+b))-2*B*cot(c+d*x)*EllipticPi(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),(a+b)/b,sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b^2*d)],
[(A+B*cos(c+d*x))/(cos(c+d*x)^(1/2)*(a+b*cos(c+d*x))^(3/2)),x,4,-2*(A*b-a*B)*sin(c+d*x)/((a^2-b^2)*d*sqrt(cos(c+d*x))*sqrt(a+b*cos(c+d*x)))+2*(A*b-a*B)*cot(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^2*d*sqrt(a+b))+2*(A+B)*cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*d*sqrt(a+b))],
[(A+B*cos(c+d*x))/(cos(c+d*x)^(3/2)*(a+b*cos(c+d*x))^(3/2)),x,4,2*b*(A*b-a*B)*sin(c+d*x)/(a*(a^2-b^2)*d*sqrt(cos(c+d*x))*sqrt(a+b*cos(c+d*x)))+2*(a^2*A-2*A*b^2+a*b*B)*cot(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^3*d*sqrt(a+b))-2*(2*A*b+a*(A-B))*cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^2*d*sqrt(a+b))],
[(A+B*cos(c+d*x))/(cos(c+d*x)^(5/2)*(a+b*cos(c+d*x))^(3/2)),x,5,2*b*(A*b-a*B)*sin(c+d*x)/(a*(a^2-b^2)*d*cos(c+d*x)^(3/2)*sqrt(a+b*cos(c+d*x)))+2/3*(a^2*A-4*A*b^2+3*a*b*B)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(a^2*(a^2-b^2)*d*cos(c+d*x)^(3/2))-2/3*(5*a^2*A*b-8*A*b^3-3*a^3*B+6*a*b^2*B)*cot(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^4*d*sqrt(a+b))+2/3*(a+2*b)*(4*A*b+a*(A-3*B))*cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^3*d*sqrt(a+b))],
[cos(c+d*x)^(5/2)*(A+B*cos(c+d*x))/(a+b*cos(c+d*x))^(5/2),x,8,2/3*a*(A*b-a*B)*cos(c+d*x)^(3/2)*sin(c+d*x)/(b*(a^2-b^2)*d*(a+b*cos(c+d*x))^(3/2))+2/3*a*(2*a^2*A*b-6*A*b^3-5*a^3*B+9*a*b^2*B)*sin(c+d*x)*sqrt(cos(c+d*x))/(b^2*(a^2-b^2)^2*d*sqrt(a+b*cos(c+d*x)))-1/3*(6*a^3*A*b-14*a*A*b^3-15*a^4*B+26*a^2*b^2*B-3*b^4*B)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(b^3*(a^2-b^2)^2*d*sqrt(cos(c+d*x)))+1/3*(6*a^3*A*b-14*a*A*b^3-15*a^4*B+26*a^2*b^2*B-3*b^4*B)*cot(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*(a-b)*b^3*(a+b)^(3/2)*d)-1/3*(6*a^2*A*b+2*a*A*b^2-12*A*b^3-15*a^3*B-5*a^2*b*B+21*a*b^2*B+3*b^3*B)*cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/((a-b)*b^3*(a+b)^(3/2)*d)-(2*A*b-5*a*B)*cot(c+d*x)*EllipticPi(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),(a+b)/b,sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b^4*d)],
[cos(c+d*x)^(3/2)*(A+B*cos(c+d*x))/(a+b*cos(c+d*x))^(5/2),x,7,2/3*a*(A*b-a*B)*sin(c+d*x)*sqrt(cos(c+d*x))/(b*(a^2-b^2)*d*(a+b*cos(c+d*x))^(3/2))-2/3*a*(4*A*b^3+3*a^3*B-7*a*b^2*B)*sin(c+d*x)/(b^2*(a^2-b^2)^2*d*sqrt(cos(c+d*x))*sqrt(a+b*cos(c+d*x)))+2/3*(4*A*b^3+3*a^3*B-7*a*b^2*B)*cot(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*(a-b)*b^2*(a+b)^(3/2)*d)+2/3*(a*A*b^2-3*A*b^3-3*a^3*B-a^2*b*B+6*a*b^2*B)*cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*(a-b)*b^2*(a+b)^(3/2)*d)-2*B*cot(c+d*x)*EllipticPi(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),(a+b)/b,sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b^3*d)],
[cos(c+d*x)^(1/2)*(A+B*cos(c+d*x))/(a+b*cos(c+d*x))^(5/2),x,5,-2/3*(A*b-a*B)*sin(c+d*x)*sqrt(cos(c+d*x))/((a^2-b^2)*d*(a+b*cos(c+d*x))^(3/2))+2/3*(3*a^2*A+A*b^2-4*a*b*B)*sin(c+d*x)/((a^2-b^2)^2*d*sqrt(cos(c+d*x))*sqrt(a+b*cos(c+d*x)))-2/3*(3*a^2*A+A*b^2-4*a*b*B)*cot(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^2*(a-b)*(a+b)^(3/2)*d)+2/3*(3*a*A-A*b+a*B-3*b*B)*cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*(a-b)*(a+b)^(3/2)*d)],
[(A+B*cos(c+d*x))/(cos(c+d*x)^(1/2)*(a+b*cos(c+d*x))^(5/2)),x,5,2/3*b*(A*b-a*B)*sin(c+d*x)*sqrt(cos(c+d*x))/(a*(a^2-b^2)*d*(a+b*cos(c+d*x))^(3/2))-2/3*(6*a^2*A*b-2*A*b^3-3*a^3*B-a*b^2*B)*sin(c+d*x)/(a*(a^2-b^2)^2*d*sqrt(cos(c+d*x))*sqrt(a+b*cos(c+d*x)))+2/3*(6*a^2*A*b-2*A*b^3-3*a^3*B-a*b^2*B)*cot(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^3*(a-b)*(a+b)^(3/2)*d)-2/3*(2*A*b^2-3*a^2*(A+B)+a*b*(3*A+B))*cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^2*(a^2-b^2)*d*sqrt(a+b))],
[(A+B*cos(c+d*x))/(cos(c+d*x)^(3/2)*(a+b*cos(c+d*x))^(5/2)),x,5,2/3*b*(A*b-a*B)*sin(c+d*x)/(a*(a^2-b^2)*d*(a+b*cos(c+d*x))^(3/2)*sqrt(cos(c+d*x)))+2/3*b*(8*a^2*A*b-4*A*b^3-5*a^3*B+a*b^2*B)*sin(c+d*x)/(a^2*(a^2-b^2)^2*d*sqrt(cos(c+d*x))*sqrt(a+b*cos(c+d*x)))+2/3*(3*a^4*A-15*a^2*A*b^2+8*A*b^4+6*a^3*b*B-2*a*b^3*B)*cot(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^4*(a-b)*(a+b)^(3/2)*d)+2/3*(8*A*b^3-3*a^3*(A-B)+2*a*b^2*(3*A-B)-3*a^2*b*(3*A+B))*cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^3*(a^2-b^2)*d*sqrt(a+b))],
[(A+B*cos(c+d*x))/(cos(c+d*x)^(5/2)*(a+b*cos(c+d*x))^(5/2)),x,6,2/3*b*(A*b-a*B)*sin(c+d*x)/(a*(a^2-b^2)*d*cos(c+d*x)^(3/2)*(a+b*cos(c+d*x))^(3/2))+2/3*b*(10*a^2*A*b-6*A*b^3-7*a^3*B+3*a*b^2*B)*sin(c+d*x)/(a^2*(a^2-b^2)^2*d*cos(c+d*x)^(3/2)*sqrt(a+b*cos(c+d*x)))+2/3*(a^4*A-13*a^2*A*b^2+8*A*b^4+8*a^3*b*B-4*a*b^3*B)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(a^3*(a^2-b^2)^2*d*cos(c+d*x)^(3/2))-2/3*(8*a^4*A*b-28*a^2*A*b^3+16*A*b^5-3*a^5*B+15*a^3*b^2*B-8*a*b^4*B)*cot(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^5*(a-b)*(a+b)^(3/2)*d)-2/3*(16*A*b^4-a^4*(A-3*B)+4*a*b^3*(3*A-2*B)-9*a^3*b*(A-B)-2*a^2*b^2*(8*A+3*B))*cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^4*(a^2-b^2)*d*sqrt(a+b))],
[cos(c+d*x)^(3/2)*(a*B+b*B*cos(c+d*x))/(a+b*cos(c+d*x))^(3/2),x,9,a*B*sin(c+d*x)/(b*d*sqrt(cos(c+d*x))*sqrt(a+b*cos(c+d*x)))+B*sin(c+d*x)*sqrt(cos(c+d*x))/(d*sqrt(a+b*cos(c+d*x)))-(a-b)*B*cot(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*b*d)+B*cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b*d)+a*B*cot(c+d*x)*EllipticPi(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),(a+b)/b,sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b^2*d)],
[cos(c+d*x)^(1/2)*(a*B+b*B*cos(c+d*x))/(a+b*cos(c+d*x))^(3/2),x,2,-2*B*cot(c+d*x)*EllipticPi(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),(a+b)/b,sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b*d)],
[(a*B+b*B*cos(c+d*x))/(cos(c+d*x)^(1/2)*(a+b*cos(c+d*x))^(3/2)),x,2,2*B*cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*d)],
[(a*B+b*B*cos(c+d*x))/(cos(c+d*x)^(3/2)*(a+b*cos(c+d*x))^(3/2)),x,4,2*(a-b)*B*cot(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^2*d)-2*B*cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*d)],
[(1+cos(c+d*x))/(cos(c+d*x)^(3/2)*sqrt(2+3*cos(c+d*x))),x,1,-cot(c+d*x)*EllipticE(sqrt(2+3*cos(c+d*x))/(sqrt(5)*sqrt(cos(c+d*x))),sqrt(5))*sqrt(-1-sec(c+d*x))*sqrt(1-sec(c+d*x))/d],
[(1+cos(c+d*x))/(cos(c+d*x)^(3/2)*sqrt(-2+3*cos(c+d*x))),x,1,-cot(c+d*x)*EllipticE(sqrt(-2+3*cos(c+d*x))/sqrt(cos(c+d*x)),sqrt(1/5))*sqrt(5)*sqrt(-1+sec(c+d*x))*sqrt(1+sec(c+d*x))/d],
[(1+cos(c+d*x))/(cos(c+d*x)^(3/2)*sqrt(2-3*cos(c+d*x))),x,2,csc(c+d*x)*EllipticE(sqrt(2-3*cos(c+d*x))/sqrt(-cos(c+d*x)),sqrt(1/5))*sqrt(5)*sqrt(-cos(c+d*x))*sqrt(cos(c+d*x))*sqrt(-1+sec(c+d*x))*sqrt(1+sec(c+d*x))/d],
[(1+cos(c+d*x))/(cos(c+d*x)^(3/2)*sqrt(-2-3*cos(c+d*x))),x,2,csc(c+d*x)*EllipticE(sqrt(-2-3*cos(c+d*x))/(sqrt(5)*sqrt(-cos(c+d*x))),sqrt(5))*sqrt(-cos(c+d*x))*sqrt(cos(c+d*x))*sqrt(-1-sec(c+d*x))*sqrt(1-sec(c+d*x))/d],
[(1+cos(c+d*x))/(cos(c+d*x)^(3/2)*sqrt(3+2*cos(c+d*x))),x,1,2/3*cot(c+d*x)*EllipticE(sqrt(3+2*cos(c+d*x))/(sqrt(5)*sqrt(cos(c+d*x))),sqrt(-5))*sqrt(1-sec(c+d*x))*sqrt(1+sec(c+d*x))/d],
[(1+cos(c+d*x))/(cos(c+d*x)^(3/2)*sqrt(3-2*cos(c+d*x))),x,1,2/3*cot(c+d*x)*EllipticE(sqrt(3-2*cos(c+d*x))/sqrt(cos(c+d*x)),sqrt(-1/5))*sqrt(5)*sqrt(1-sec(c+d*x))*sqrt(1+sec(c+d*x))/d],
[(1+cos(c+d*x))/(cos(c+d*x)^(3/2)*sqrt(-3+2*cos(c+d*x))),x,2,-2/3*csc(c+d*x)*EllipticE(sqrt(-3+2*cos(c+d*x))/sqrt(-cos(c+d*x)),sqrt(-1/5))*sqrt(5)*sqrt(-cos(c+d*x))*sqrt(cos(c+d*x))*sqrt(1-sec(c+d*x))*sqrt(1+sec(c+d*x))/d],
[(1+cos(c+d*x))/(cos(c+d*x)^(3/2)*sqrt(-3-2*cos(c+d*x))),x,2,-2/3*csc(c+d*x)*EllipticE(sqrt(-3-2*cos(c+d*x))/(sqrt(5)*sqrt(-cos(c+d*x))),sqrt(-5))*sqrt(-cos(c+d*x))*sqrt(cos(c+d*x))*sqrt(1-sec(c+d*x))*sqrt(1+sec(c+d*x))/d],

# Integrands of the form (c Cos[e+f x])^m (a+b Cos[e+f x])^n (A+B Cos[e+f x]) with m and/or n symbolic
[(c*cos(e+f*x))^m*(a+b*cos(e+f*x))^n*(A+B*cos(e+f*x)),x,0,Unintegrable((c*cos(e+f*x))^m*(a+b*cos(e+f*x))^n*(A+B*cos(e+f*x)),x)],
[(c*cos(e+f*x))^m*(a+b*cos(e+f*x))^4*(A+B*cos(e+f*x)),x,7,b*(A*b^3*(15+8*m+m^2)+4*a*b^2*B*(15+8*m+m^2)+2*a^3*B*(28+10*m+m^2)+a^2*A*b*(110+47*m+5*m^2))*(c*cos(e+f*x))^(1+m)*sin(e+f*x)/(c*f*(5+m)*(8+6*m+m^2))+b^2*(b^2*B*(4+m)^2+2*a*A*b*(5+m)^2+a^2*B*(36+11*m+m^2))*cos(e+f*x)*(c*cos(e+f*x))^(1+m)*sin(e+f*x)/(c*f*(3+m)*(4+m)*(5+m))+b*(A*b*(5+m)+a*B*(8+m))*(c*cos(e+f*x))^(1+m)*(a+b*cos(e+f*x))^2*sin(e+f*x)/(c*f*(4+m)*(5+m))+b*B*(c*cos(e+f*x))^(1+m)*(a+b*cos(e+f*x))^3*sin(e+f*x)/(c*f*(5+m))-(A*b^4*(3+4*m+m^2)+4*a*b^3*B*(3+4*m+m^2)+6*a^2*A*b^2*(4+5*m+m^2)+4*a^3*b*B*(4+5*m+m^2)+a^4*A*(8+6*m+m^2))*(c*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)/(c*f*(4+m)*(2+3*m+m^2)*sqrt(sin(e+f*x)^2))-(b^4*B*(8+6*m+m^2)+4*a*A*b^3*(10+7*m+m^2)+6*a^2*b^2*B*(10+7*m+m^2)+4*a^3*A*b*(15+8*m+m^2)+a^4*B*(15+8*m+m^2))*(c*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)/(c^2*f*(2+m)*(3+m)*(5+m)*sqrt(sin(e+f*x)^2))],
[(c*cos(e+f*x))^m*(a+b*cos(e+f*x))^3*(A+B*cos(e+f*x)),x,6,b*(b^2*B*(3+m)+3*a*A*b*(4+m)+2*a^2*B*(5+m))*(c*cos(e+f*x))^(1+m)*sin(e+f*x)/(c*f*(2+m)*(4+m))+b^2*(A*b*(4+m)+a*B*(6+m))*cos(e+f*x)*(c*cos(e+f*x))^(1+m)*sin(e+f*x)/(c*f*(3+m)*(4+m))+b*B*(c*cos(e+f*x))^(1+m)*(a+b*cos(e+f*x))^2*sin(e+f*x)/(c*f*(4+m))-(a^2*(2+m)*(b*B*(1+m)+a*A*(4+m))+b*(1+m)*(b^2*B*(3+m)+3*a*A*b*(4+m)+2*a^2*B*(5+m)))*(c*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)/(c*f*(1+m)*(2+m)*(4+m)*sqrt(sin(e+f*x)^2))-(A*b^3*(2+m)+3*a*b^2*B*(2+m)+3*a^2*A*b*(3+m)+a^3*B*(3+m))*(c*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)/(c^2*f*(2+m)*(3+m)*sqrt(sin(e+f*x)^2))],
[(c*cos(e+f*x))^m*(a+b*cos(e+f*x))^2*(A+B*cos(e+f*x)),x,5,b*(A*b*(3+m)+a*B*(4+m))*(c*cos(e+f*x))^(1+m)*sin(e+f*x)/(c*f*(2+m)*(3+m))+b*B*(c*cos(e+f*x))^(1+m)*(a+b*cos(e+f*x))*sin(e+f*x)/(c*f*(3+m))-(A*b^2*(1+m)+2*a*b*B*(1+m)+a^2*A*(2+m))*(c*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)/(c*f*(1+m)*(2+m)*sqrt(sin(e+f*x)^2))-(b^2*B*(2+m)+a*(2*A*b+a*B)*(3+m))*(c*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)/(c^2*f*(2+m)*(3+m)*sqrt(sin(e+f*x)^2))],
[(c*cos(e+f*x))^m*(a+b*cos(e+f*x))*(A+B*cos(e+f*x)),x,5,b*B*(c*cos(e+f*x))^(1+m)*sin(e+f*x)/(c*f*(2+m))-(b*B*(1+m)+a*A*(2+m))*(c*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)/(c*f*(1+m)*(2+m)*sqrt(sin(e+f*x)^2))-(A*b+a*B)*(c*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)/(c^2*f*(2+m)*sqrt(sin(e+f*x)^2))],
[(c*cos(e+f*x))^m*(A+B*cos(e+f*x))/(a+b*cos(e+f*x)),x,7,a*(A*b-a*B)*c*AppellF1(1/2,1/2*(1-m),1,3/2,sin(e+f*x)^2,-b^2*sin(e+f*x)^2/(a^2-b^2))*(c*cos(e+f*x))^(-1+m)*(cos(e+f*x)^2)^(1/2*(1-m))*sin(e+f*x)/(b*(a^2-b^2)*f)-(A*b-a*B)*AppellF1(1/2,-1/2*m,1,3/2,sin(e+f*x)^2,-b^2*sin(e+f*x)^2/(a^2-b^2))*(c*cos(e+f*x))^m*sin(e+f*x)/((a^2-b^2)*f*(cos(e+f*x)^2)^(1/2*m))-B*(c*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)/(b*c*f*(1+m)*sqrt(sin(e+f*x)^2))],
[(c*cos(e+f*x))^m*(a+b*cos(e+f*x))^(3/2)*(A+B*cos(e+f*x)),x,1,2*b*B*(c*cos(e+f*x))^(1+m)*sin(e+f*x)*sqrt(a+b*cos(e+f*x))/(c*f*(5+2*m))+2*Unintegrable((c*cos(e+f*x))^m*(1/2*a*c*(2*b*B*(1+m)+2*a*A*(5/2+m))+1/2*c*(b^2*B*(3+2*m)+a*(2*A*b+a*B)*(5+2*m))*cos(e+f*x)+1/2*b*c*(2*a*B*(3+m)+A*b*(5+2*m))*cos(e+f*x)^2)/sqrt(a+b*cos(e+f*x)),x)/(c*(5+2*m))],
[(c*cos(e+f*x))^m*(a+b*cos(e+f*x))^(1/2)*(A+B*cos(e+f*x)),x,0,Unintegrable((c*cos(e+f*x))^m*(A+B*cos(e+f*x))*sqrt(a+b*cos(e+f*x)),x)],
[(c*cos(e+f*x))^m*(A+B*cos(e+f*x))/(a+b*cos(e+f*x))^(1/2),x,0,Unintegrable((c*cos(e+f*x))^m*(A+B*cos(e+f*x))/sqrt(a+b*cos(e+f*x)),x)],
[(c*cos(e+f*x))^m*(A+B*cos(e+f*x))/(a+b*cos(e+f*x))^(3/2),x,1,2*b*(A*b-a*B)*(c*cos(e+f*x))^(1+m)*sin(e+f*x)/(a*(a^2-b^2)*c*f*sqrt(a+b*cos(e+f*x)))+2*Unintegrable((c*cos(e+f*x))^m*(1/2*c*(a*(a*A-b*B)+2*b*(A*b-a*B)*(1/2+m))-1/2*a*(A*b-a*B)*c*cos(e+f*x)-1/2*b*(A*b-a*B)*c*(3+2*m)*cos(e+f*x)^2)/sqrt(a+b*cos(e+f*x)),x)/(a*(a^2-b^2)*c)],

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

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

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

# n>0
[(a+a*cos(c+d*x))*(A+B*cos(c+d*x))*sec(c+d*x)^(7/2),x,9,2/3*a*(A+B)*sec(c+d*x)^(3/2)*sin(c+d*x)/d+2/5*a*A*sec(c+d*x)^(5/2)*sin(c+d*x)/d+2/5*a*(3*A+5*B)*sin(c+d*x)*sqrt(sec(c+d*x))/d-2/5*a*(3*A+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(cos(c+d*x))*sqrt(sec(c+d*x))/d+2/3*a*(A+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))*sqrt(sec(c+d*x))/d],
[(a+a*cos(c+d*x))*(A+B*cos(c+d*x))*sec(c+d*x)^(5/2),x,8,2/3*a*A*sec(c+d*x)^(3/2)*sin(c+d*x)/d+2*a*(A+B)*sin(c+d*x)*sqrt(sec(c+d*x))/d-2*a*(A+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(cos(c+d*x))*sqrt(sec(c+d*x))/d+2/3*a*(A+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))*sqrt(sec(c+d*x))/d],
[(a+a*cos(c+d*x))*(A+B*cos(c+d*x))*sec(c+d*x)^(3/2),x,7,2*a*A*sin(c+d*x)*sqrt(sec(c+d*x))/d-2*a*(A-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(cos(c+d*x))*sqrt(sec(c+d*x))/d+2*a*(A+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))*sqrt(sec(c+d*x))/d],
[(a+a*cos(c+d*x))*(A+B*cos(c+d*x))*sqrt(sec(c+d*x)),x,7,2/3*a*B*sin(c+d*x)/(d*sqrt(sec(c+d*x)))+2*a*(A+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(cos(c+d*x))*sqrt(sec(c+d*x))/d+2/3*a*(3*A+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))*sqrt(sec(c+d*x))/d],
[(a+a*cos(c+d*x))*(A+B*cos(c+d*x))/sqrt(sec(c+d*x)),x,8,2/5*a*B*sin(c+d*x)/(d*sec(c+d*x)^(3/2))+2/3*a*(A+B)*sin(c+d*x)/(d*sqrt(sec(c+d*x)))+2/5*a*(5*A+3*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(cos(c+d*x))*sqrt(sec(c+d*x))/d+2/3*a*(A+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))*sqrt(sec(c+d*x))/d],
[(a+a*cos(c+d*x))*(A+B*cos(c+d*x))/sec(c+d*x)^(3/2),x,9,2/7*a*B*sin(c+d*x)/(d*sec(c+d*x)^(5/2))+2/5*a*(A+B)*sin(c+d*x)/(d*sec(c+d*x)^(3/2))+2/21*a*(7*A+5*B)*sin(c+d*x)/(d*sqrt(sec(c+d*x)))+6/5*a*(A+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(cos(c+d*x))*sqrt(sec(c+d*x))/d+2/21*a*(7*A+5*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))*sqrt(sec(c+d*x))/d],
[(a+a*cos(c+d*x))^2*(A+B*cos(c+d*x))*sec(c+d*x)^(7/2),x,9,2/15*a^2*(7*A+5*B)*sec(c+d*x)^(3/2)*sin(c+d*x)/d+2/5*A*sec(c+d*x)^(3/2)*(a^2+a^2*sec(c+d*x))*sin(c+d*x)/d+4/5*a^2*(4*A+5*B)*sin(c+d*x)*sqrt(sec(c+d*x))/d-4/5*a^2*(4*A+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(cos(c+d*x))*sqrt(sec(c+d*x))/d+4/3*a^2*(A+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))*sqrt(sec(c+d*x))/d],
[(a+a*cos(c+d*x))^2*(A+B*cos(c+d*x))*sec(c+d*x)^(5/2),x,8,2/3*a^2*(5*A+3*B)*sin(c+d*x)*sqrt(sec(c+d*x))/d+2/3*A*(a^2+a^2*sec(c+d*x))*sin(c+d*x)*sqrt(sec(c+d*x))/d-4*a^2*A*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d+4/3*a^2*(2*A+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))*sqrt(sec(c+d*x))/d],
[(a+a*cos(c+d*x))^2*(A+B*cos(c+d*x))*sec(c+d*x)^(3/2),x,8,2/3*B*(a^2+a^2*sec(c+d*x))*sin(c+d*x)/(d*sqrt(sec(c+d*x)))+2/3*a^2*(3*A-B)*sin(c+d*x)*sqrt(sec(c+d*x))/d+4*a^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(cos(c+d*x))*sqrt(sec(c+d*x))/d+4/3*a^2*(3*A+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))*sqrt(sec(c+d*x))/d],
[(a+a*cos(c+d*x))^2*(A+B*cos(c+d*x))*sqrt(sec(c+d*x)),x,8,2/5*B*(a^2+a^2*sec(c+d*x))*sin(c+d*x)/(d*sec(c+d*x)^(3/2))+2/15*a^2*(5*A+7*B)*sin(c+d*x)/(d*sqrt(sec(c+d*x)))+4/5*a^2*(5*A+4*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(cos(c+d*x))*sqrt(sec(c+d*x))/d+4/3*a^2*(2*A+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))*sqrt(sec(c+d*x))/d],
[(a+a*cos(c+d*x))^2*(A+B*cos(c+d*x))/sqrt(sec(c+d*x)),x,9,2/35*a^2*(7*A+9*B)*sin(c+d*x)/(d*sec(c+d*x)^(3/2))+2/7*B*(a^2+a^2*sec(c+d*x))*sin(c+d*x)/(d*sec(c+d*x)^(5/2))+4/21*a^2*(7*A+6*B)*sin(c+d*x)/(d*sqrt(sec(c+d*x)))+4/5*a^2*(4*A+3*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(cos(c+d*x))*sqrt(sec(c+d*x))/d+4/21*a^2*(7*A+6*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))*sqrt(sec(c+d*x))/d],
[(a+a*cos(c+d*x))^3*(A+B*cos(c+d*x))*sec(c+d*x)^(9/2),x,10,4/105*a^3*(41*A+42*B)*sec(c+d*x)^(3/2)*sin(c+d*x)/d+2/7*a*A*sec(c+d*x)^(3/2)*(a+a*sec(c+d*x))^2*sin(c+d*x)/d+2/35*(11*A+7*B)*sec(c+d*x)^(3/2)*(a^3+a^3*sec(c+d*x))*sin(c+d*x)/d+4/5*a^3*(7*A+9*B)*sin(c+d*x)*sqrt(sec(c+d*x))/d-4/5*a^3*(7*A+9*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(cos(c+d*x))*sqrt(sec(c+d*x))/d+4/21*a^3*(13*A+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))*sqrt(sec(c+d*x))/d],
[(a+a*cos(c+d*x))^3*(A+B*cos(c+d*x))*sec(c+d*x)^(7/2),x,9,4/15*a^3*(21*A+20*B)*sin(c+d*x)*sqrt(sec(c+d*x))/d+2/5*a*A*(a+a*sec(c+d*x))^2*sin(c+d*x)*sqrt(sec(c+d*x))/d+2/15*(9*A+5*B)*(a^3+a^3*sec(c+d*x))*sin(c+d*x)*sqrt(sec(c+d*x))/d-4/5*a^3*(9*A+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(cos(c+d*x))*sqrt(sec(c+d*x))/d+4/3*a^3*(3*A+5*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))*sqrt(sec(c+d*x))/d],
[(a+a*cos(c+d*x))^3*(A+B*cos(c+d*x))*sec(c+d*x)^(5/2),x,9,2/3*a*B*(a+a*sec(c+d*x))^2*sin(c+d*x)/(d*sqrt(sec(c+d*x)))+4/3*a^3*(4*A+B)*sin(c+d*x)*sqrt(sec(c+d*x))/d+2/3*(A-B)*(a^3+a^3*sec(c+d*x))*sin(c+d*x)*sqrt(sec(c+d*x))/d-4*a^3*(A-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(cos(c+d*x))*sqrt(sec(c+d*x))/d+20/3*a^3*(A+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))*sqrt(sec(c+d*x))/d],
[(a+a*cos(c+d*x))^3*(A+B*cos(c+d*x))*sec(c+d*x)^(3/2),x,9,2/5*a*B*(a+a*sec(c+d*x))^2*sin(c+d*x)/(d*sec(c+d*x)^(3/2))+2/15*(5*A+9*B)*(a^3+a^3*sec(c+d*x))*sin(c+d*x)/(d*sqrt(sec(c+d*x)))+4/15*a^3*(5*A-6*B)*sin(c+d*x)*sqrt(sec(c+d*x))/d+4/5*a^3*(5*A+9*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(cos(c+d*x))*sqrt(sec(c+d*x))/d+4/3*a^3*(5*A+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))*sqrt(sec(c+d*x))/d],
[(a+a*cos(c+d*x))^3*(A+B*cos(c+d*x))*sqrt(sec(c+d*x)),x,9,2/7*a*B*(a+a*sec(c+d*x))^2*sin(c+d*x)/(d*sec(c+d*x)^(5/2))+2/35*(7*A+11*B)*(a^3+a^3*sec(c+d*x))*sin(c+d*x)/(d*sec(c+d*x)^(3/2))+4/105*a^3*(42*A+41*B)*sin(c+d*x)/(d*sqrt(sec(c+d*x)))+4/5*a^3*(9*A+7*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(cos(c+d*x))*sqrt(sec(c+d*x))/d+4/21*a^3*(21*A+13*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))*sqrt(sec(c+d*x))/d],
[(a+a*cos(c+d*x))^3*(A+B*cos(c+d*x))/sqrt(sec(c+d*x)),x,10,4/105*a^3*(24*A+23*B)*sin(c+d*x)/(d*sec(c+d*x)^(3/2))+2/9*a*B*(a+a*sec(c+d*x))^2*sin(c+d*x)/(d*sec(c+d*x)^(7/2))+2/63*(9*A+13*B)*(a^3+a^3*sec(c+d*x))*sin(c+d*x)/(d*sec(c+d*x)^(5/2))+4/21*a^3*(13*A+11*B)*sin(c+d*x)/(d*sqrt(sec(c+d*x)))+4/15*a^3*(21*A+17*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(cos(c+d*x))*sqrt(sec(c+d*x))/d+4/21*a^3*(13*A+11*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))*sqrt(sec(c+d*x))/d],

# n<0
[(A+B*cos(c+d*x))*sec(c+d*x)^(5/2)/(a+a*cos(c+d*x)),x,9,1/3*(5*A-3*B)*sec(c+d*x)^(3/2)*sin(c+d*x)/(a*d)-(A-B)*sec(c+d*x)^(5/2)*sin(c+d*x)/(d*(a+a*sec(c+d*x)))-3*(A-B)*sin(c+d*x)*sqrt(sec(c+d*x))/(a*d)+3*(A-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(cos(c+d*x))*sqrt(sec(c+d*x))/(a*d)+1/3*(5*A-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))*sqrt(sec(c+d*x))/(a*d)],
[(A+B*cos(c+d*x))*sec(c+d*x)^(3/2)/(a+a*cos(c+d*x)),x,8,-(A-B)*sec(c+d*x)^(3/2)*sin(c+d*x)/(d*(a+a*sec(c+d*x)))+(3*A-B)*sin(c+d*x)*sqrt(sec(c+d*x))/(a*d)-(3*A-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(cos(c+d*x))*sqrt(sec(c+d*x))/(a*d)-(A-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))*sqrt(sec(c+d*x))/(a*d)],
[(A+B*cos(c+d*x))*sqrt(sec(c+d*x))/(a+a*cos(c+d*x)),x,7,-(A-B)*sin(c+d*x)*sqrt(sec(c+d*x))/(d*(a+a*sec(c+d*x)))+(A-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(cos(c+d*x))*sqrt(sec(c+d*x))/(a*d)+(A+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))*sqrt(sec(c+d*x))/(a*d)],
[(A+B*cos(c+d*x))/((a+a*cos(c+d*x))*sqrt(sec(c+d*x))),x,7,(A-B)*sin(c+d*x)*sqrt(sec(c+d*x))/(d*(a+a*sec(c+d*x)))-(A-3*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(cos(c+d*x))*sqrt(sec(c+d*x))/(a*d)+(A-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))*sqrt(sec(c+d*x))/(a*d)],
[(A+B*cos(c+d*x))/((a+a*cos(c+d*x))*sec(c+d*x)^(3/2)),x,8,-1/3*(3*A-5*B)*sin(c+d*x)/(a*d*sqrt(sec(c+d*x)))+(A-B)*sin(c+d*x)/(d*(a+a*sec(c+d*x))*sqrt(sec(c+d*x)))+3*(A-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(cos(c+d*x))*sqrt(sec(c+d*x))/(a*d)-1/3*(3*A-5*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))*sqrt(sec(c+d*x))/(a*d)],
[(A+B*cos(c+d*x))/((a+a*cos(c+d*x))*sec(c+d*x)^(5/2)),x,9,-1/5*(5*A-7*B)*sin(c+d*x)/(a*d*sec(c+d*x)^(3/2))+(A-B)*sin(c+d*x)/(d*sec(c+d*x)^(3/2)*(a+a*sec(c+d*x)))+5/3*(A-B)*sin(c+d*x)/(a*d*sqrt(sec(c+d*x)))-3/5*(5*A-7*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(cos(c+d*x))*sqrt(sec(c+d*x))/(a*d)+5/3*(A-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))*sqrt(sec(c+d*x))/(a*d)],
[(A+B*cos(c+d*x))*sec(c+d*x)^(3/2)/(a+a*cos(c+d*x))^2,x,9,-1/3*(5*A-2*B)*sec(c+d*x)^(3/2)*sin(c+d*x)/(a^2*d*(1+sec(c+d*x)))-1/3*(A-B)*sec(c+d*x)^(5/2)*sin(c+d*x)/(d*(a+a*sec(c+d*x))^2)+(4*A-B)*sin(c+d*x)*sqrt(sec(c+d*x))/(a^2*d)-(4*A-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(cos(c+d*x))*sqrt(sec(c+d*x))/(a^2*d)-1/3*(5*A-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))*sqrt(sec(c+d*x))/(a^2*d)],
[(A+B*cos(c+d*x))*sqrt(sec(c+d*x))/(a+a*cos(c+d*x))^2,x,8,-1/3*(A-B)*sec(c+d*x)^(3/2)*sin(c+d*x)/(d*(a+a*sec(c+d*x))^2)-A*sin(c+d*x)*sqrt(sec(c+d*x))/(a^2*d*(1+sec(c+d*x)))+A*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^2*d)+1/3*(2*A+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))*sqrt(sec(c+d*x))/(a^2*d)],
[(A+B*cos(c+d*x))/((a+a*cos(c+d*x))^2*sqrt(sec(c+d*x))),x,8,1/3*(A+2*B)*sin(c+d*x)*sqrt(sec(c+d*x))/(a^2*d*(1+sec(c+d*x)))-1/3*(A-B)*sin(c+d*x)*sqrt(sec(c+d*x))/(d*(a+a*sec(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(cos(c+d*x))*sqrt(sec(c+d*x))/(a^2*d)+1/3*(A+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))*sqrt(sec(c+d*x))/(a^2*d)],
[(A+B*cos(c+d*x))/((a+a*cos(c+d*x))^2*sec(c+d*x)^(3/2)),x,8,1/3*(2*A-5*B)*sin(c+d*x)*sqrt(sec(c+d*x))/(a^2*d*(1+sec(c+d*x)))+1/3*(A-B)*sin(c+d*x)*sqrt(sec(c+d*x))/(d*(a+a*sec(c+d*x))^2)-(A-4*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(cos(c+d*x))*sqrt(sec(c+d*x))/(a^2*d)+1/3*(2*A-5*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))*sqrt(sec(c+d*x))/(a^2*d)],
[(A+B*cos(c+d*x))/((a+a*cos(c+d*x))^2*sec(c+d*x)^(5/2)),x,9,-5/3*(A-2*B)*sin(c+d*x)/(a^2*d*sqrt(sec(c+d*x)))+1/3*(4*A-7*B)*sin(c+d*x)/(a^2*d*(1+sec(c+d*x))*sqrt(sec(c+d*x)))+1/3*(A-B)*sin(c+d*x)/(d*(a+a*sec(c+d*x))^2*sqrt(sec(c+d*x)))+(4*A-7*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(cos(c+d*x))*sqrt(sec(c+d*x))/(a^2*d)-5/3*(A-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))*sqrt(sec(c+d*x))/(a^2*d)],
[(A+B*cos(c+d*x))*sec(c+d*x)^(3/2)/(a+a*cos(c+d*x))^3,x,10,-1/5*(A-B)*sec(c+d*x)^(7/2)*sin(c+d*x)/(d*(a+a*sec(c+d*x))^3)-1/15*(8*A-3*B)*sec(c+d*x)^(5/2)*sin(c+d*x)/(a*d*(a+a*sec(c+d*x))^2)-1/6*(13*A-3*B)*sec(c+d*x)^(3/2)*sin(c+d*x)/(d*(a^3+a^3*sec(c+d*x)))+1/10*(49*A-9*B)*sin(c+d*x)*sqrt(sec(c+d*x))/(a^3*d)-1/10*(49*A-9*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(cos(c+d*x))*sqrt(sec(c+d*x))/(a^3*d)-1/6*(13*A-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))*sqrt(sec(c+d*x))/(a^3*d)],
[(A+B*cos(c+d*x))*sqrt(sec(c+d*x))/(a+a*cos(c+d*x))^3,x,9,-1/5*(A-B)*sec(c+d*x)^(5/2)*sin(c+d*x)/(d*(a+a*sec(c+d*x))^3)-1/15*(6*A-B)*sec(c+d*x)^(3/2)*sin(c+d*x)/(a*d*(a+a*sec(c+d*x))^2)-1/10*(9*A+B)*sin(c+d*x)*sqrt(sec(c+d*x))/(d*(a^3+a^3*sec(c+d*x)))+1/10*(9*A+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(cos(c+d*x))*sqrt(sec(c+d*x))/(a^3*d)+1/6*(3*A+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))*sqrt(sec(c+d*x))/(a^3*d)],
[(A+B*cos(c+d*x))/((a+a*cos(c+d*x))^3*sqrt(sec(c+d*x))),x,9,-1/5*(A-B)*sec(c+d*x)^(3/2)*sin(c+d*x)/(d*(a+a*sec(c+d*x))^3)-1/15*(4*A+B)*sin(c+d*x)*sqrt(sec(c+d*x))/(a*d*(a+a*sec(c+d*x))^2)+1/6*(A+B)*sin(c+d*x)*sqrt(sec(c+d*x))/(d*(a^3+a^3*sec(c+d*x)))+1/10*(A-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(cos(c+d*x))*sqrt(sec(c+d*x))/(a^3*d)+1/6*(A+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))*sqrt(sec(c+d*x))/(a^3*d)],
[(A+B*cos(c+d*x))/((a+a*cos(c+d*x))^3*sec(c+d*x)^(3/2)),x,9,-1/5*(A-B)*sin(c+d*x)*sqrt(sec(c+d*x))/(d*(a+a*sec(c+d*x))^3)+1/15*(2*A+3*B)*sin(c+d*x)*sqrt(sec(c+d*x))/(a*d*(a+a*sec(c+d*x))^2)+1/6*(A+3*B)*sin(c+d*x)*sqrt(sec(c+d*x))/(d*(a^3+a^3*sec(c+d*x)))-1/10*(A+9*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(cos(c+d*x))*sqrt(sec(c+d*x))/(a^3*d)+1/6*(A+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))*sqrt(sec(c+d*x))/(a^3*d)],
[(A+B*cos(c+d*x))/((a+a*cos(c+d*x))^3*sec(c+d*x)^(5/2)),x,9,1/5*(A-B)*sin(c+d*x)*sqrt(sec(c+d*x))/(d*(a+a*sec(c+d*x))^3)+1/15*(3*A-8*B)*sin(c+d*x)*sqrt(sec(c+d*x))/(a*d*(a+a*sec(c+d*x))^2)+1/6*(3*A-13*B)*sin(c+d*x)*sqrt(sec(c+d*x))/(d*(a^3+a^3*sec(c+d*x)))-1/10*(9*A-49*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(cos(c+d*x))*sqrt(sec(c+d*x))/(a^3*d)+1/6*(3*A-13*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))*sqrt(sec(c+d*x))/(a^3*d)],
[(A+B*cos(c+d*x))/((a+a*cos(c+d*x))^3*sec(c+d*x)^(7/2)),x,10,-1/6*(13*A-33*B)*sin(c+d*x)/(a^3*d*sqrt(sec(c+d*x)))+1/5*(A-B)*sin(c+d*x)/(d*(a+a*sec(c+d*x))^3*sqrt(sec(c+d*x)))+1/3*(A-2*B)*sin(c+d*x)/(a*d*(a+a*sec(c+d*x))^2*sqrt(sec(c+d*x)))+7/30*(7*A-17*B)*sin(c+d*x)/(d*(a^3+a^3*sec(c+d*x))*sqrt(sec(c+d*x)))+7/10*(7*A-17*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(cos(c+d*x))*sqrt(sec(c+d*x))/(a^3*d)-1/6*(13*A-33*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))*sqrt(sec(c+d*x))/(a^3*d)],

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

# n>0
[(A+B*cos(c+d*x))*sec(c+d*x)^(11/2)*sqrt(a+a*cos(c+d*x)),x,6,16/315*a*(8*A+9*B)*sec(c+d*x)^(3/2)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+4/105*a*(8*A+9*B)*sec(c+d*x)^(5/2)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+2/63*a*(8*A+9*B)*sec(c+d*x)^(7/2)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+2/9*a*A*sec(c+d*x)^(9/2)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+32/315*a*(8*A+9*B)*sin(c+d*x)*sqrt(sec(c+d*x))/(d*sqrt(a+a*cos(c+d*x)))],
[(A+B*cos(c+d*x))*sec(c+d*x)^(9/2)*sqrt(a+a*cos(c+d*x)),x,5,8/105*a*(6*A+7*B)*sec(c+d*x)^(3/2)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+2/35*a*(6*A+7*B)*sec(c+d*x)^(5/2)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+2/7*a*A*sec(c+d*x)^(7/2)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+16/105*a*(6*A+7*B)*sin(c+d*x)*sqrt(sec(c+d*x))/(d*sqrt(a+a*cos(c+d*x)))],
[(A+B*cos(c+d*x))*sec(c+d*x)^(7/2)*sqrt(a+a*cos(c+d*x)),x,4,2/15*a*(4*A+5*B)*sec(c+d*x)^(3/2)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+2/5*a*A*sec(c+d*x)^(5/2)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+4/15*a*(4*A+5*B)*sin(c+d*x)*sqrt(sec(c+d*x))/(d*sqrt(a+a*cos(c+d*x)))],
[(A+B*cos(c+d*x))*sec(c+d*x)^(5/2)*sqrt(a+a*cos(c+d*x)),x,3,2/3*a*A*sec(c+d*x)^(3/2)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+2/3*a*(2*A+3*B)*sin(c+d*x)*sqrt(sec(c+d*x))/(d*sqrt(a+a*cos(c+d*x)))],
[(A+B*cos(c+d*x))*sec(c+d*x)^(3/2)*sqrt(a+a*cos(c+d*x)),x,4,2*B*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))*sqrt(a)*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d+2*a*A*sin(c+d*x)*sqrt(sec(c+d*x))/(d*sqrt(a+a*cos(c+d*x)))],
[(A+B*cos(c+d*x))*sqrt(a+a*cos(c+d*x))*sqrt(sec(c+d*x)),x,4,a*B*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x))*sqrt(sec(c+d*x)))+(2*A+B)*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))*sqrt(a)*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d],
[(A+B*cos(c+d*x))*sqrt(a+a*cos(c+d*x))/sqrt(sec(c+d*x)),x,5,1/2*a*B*sin(c+d*x)/(d*sec(c+d*x)^(3/2)*sqrt(a+a*cos(c+d*x)))+1/4*a*(4*A+3*B)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x))*sqrt(sec(c+d*x)))+1/4*(4*A+3*B)*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))*sqrt(a)*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d],
[(A+B*cos(c+d*x))*sqrt(a+a*cos(c+d*x))/sec(c+d*x)^(3/2),x,6,1/3*a*B*sin(c+d*x)/(d*sec(c+d*x)^(5/2)*sqrt(a+a*cos(c+d*x)))+1/12*a*(6*A+5*B)*sin(c+d*x)/(d*sec(c+d*x)^(3/2)*sqrt(a+a*cos(c+d*x)))+1/8*a*(6*A+5*B)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x))*sqrt(sec(c+d*x)))+1/8*(6*A+5*B)*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))*sqrt(a)*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d],
[(a+a*cos(c+d*x))^(3/2)*(A+B*cos(c+d*x))*sec(c+d*x)^(13/2),x,7,16/3465*a^2*(168*A+187*B)*sec(c+d*x)^(3/2)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+4/1155*a^2*(168*A+187*B)*sec(c+d*x)^(5/2)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+2/693*a^2*(168*A+187*B)*sec(c+d*x)^(7/2)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+2/99*a^2*(12*A+11*B)*sec(c+d*x)^(9/2)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+2/11*a*A*sec(c+d*x)^(11/2)*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/d+32/3465*a^2*(168*A+187*B)*sin(c+d*x)*sqrt(sec(c+d*x))/(d*sqrt(a+a*cos(c+d*x)))],
[(a+a*cos(c+d*x))^(3/2)*(A+B*cos(c+d*x))*sec(c+d*x)^(11/2),x,6,8/315*a^2*(34*A+39*B)*sec(c+d*x)^(3/2)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+2/105*a^2*(34*A+39*B)*sec(c+d*x)^(5/2)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+2/63*a^2*(10*A+9*B)*sec(c+d*x)^(7/2)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+2/9*a*A*sec(c+d*x)^(9/2)*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/d+16/315*a^2*(34*A+39*B)*sin(c+d*x)*sqrt(sec(c+d*x))/(d*sqrt(a+a*cos(c+d*x)))],
[(a+a*cos(c+d*x))^(3/2)*(A+B*cos(c+d*x))*sec(c+d*x)^(9/2),x,5,2/105*a^2*(52*A+63*B)*sec(c+d*x)^(3/2)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+2/35*a^2*(8*A+7*B)*sec(c+d*x)^(5/2)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+2/7*a*A*sec(c+d*x)^(7/2)*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/d+4/105*a^2*(52*A+63*B)*sin(c+d*x)*sqrt(sec(c+d*x))/(d*sqrt(a+a*cos(c+d*x)))],
[(a+a*cos(c+d*x))^(3/2)*(A+B*cos(c+d*x))*sec(c+d*x)^(7/2),x,4,2/15*a^2*(6*A+5*B)*sec(c+d*x)^(3/2)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+2/5*a*A*sec(c+d*x)^(5/2)*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/d+2/15*a^2*(18*A+25*B)*sin(c+d*x)*sqrt(sec(c+d*x))/(d*sqrt(a+a*cos(c+d*x)))],
[(a+a*cos(c+d*x))^(3/2)*(A+B*cos(c+d*x))*sec(c+d*x)^(5/2),x,5,2/3*a*A*sec(c+d*x)^(3/2)*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/d+2*a^(3/2)*B*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d+2/3*a^2*(4*A+3*B)*sin(c+d*x)*sqrt(sec(c+d*x))/(d*sqrt(a+a*cos(c+d*x)))],
[(a+a*cos(c+d*x))^(3/2)*(A+B*cos(c+d*x))*sec(c+d*x)^(3/2),x,5,-a^2*(2*A-B)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x))*sqrt(sec(c+d*x)))+a^(3/2)*(2*A+3*B)*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d+2*a*A*sin(c+d*x)*sqrt(a+a*cos(c+d*x))*sqrt(sec(c+d*x))/d],
[(a+a*cos(c+d*x))^(3/2)*(A+B*cos(c+d*x))*sqrt(sec(c+d*x)),x,5,1/4*a^2*(4*A+5*B)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x))*sqrt(sec(c+d*x)))+1/2*a*B*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/(d*sqrt(sec(c+d*x)))+1/4*a^(3/2)*(12*A+7*B)*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d],
[(a+a*cos(c+d*x))^(3/2)*(A+B*cos(c+d*x))/sqrt(sec(c+d*x)),x,6,1/12*a^2*(6*A+7*B)*sin(c+d*x)/(d*sec(c+d*x)^(3/2)*sqrt(a+a*cos(c+d*x)))+1/3*a*B*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/(d*sec(c+d*x)^(3/2))+1/8*a^2*(14*A+11*B)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x))*sqrt(sec(c+d*x)))+1/8*a^(3/2)*(14*A+11*B)*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d],
[(a+a*cos(c+d*x))^(3/2)*(A+B*cos(c+d*x))/sec(c+d*x)^(3/2),x,7,1/24*a^2*(8*A+9*B)*sin(c+d*x)/(d*sec(c+d*x)^(5/2)*sqrt(a+a*cos(c+d*x)))+1/96*a^2*(88*A+75*B)*sin(c+d*x)/(d*sec(c+d*x)^(3/2)*sqrt(a+a*cos(c+d*x)))+1/4*a*B*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/(d*sec(c+d*x)^(5/2))+1/64*a^2*(88*A+75*B)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x))*sqrt(sec(c+d*x)))+1/64*a^(3/2)*(88*A+75*B)*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d],
[(a+a*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x))*sec(c+d*x)^(15/2),x,8,2/13*a*A*(a+a*cos(c+d*x))^(3/2)*sec(c+d*x)^(13/2)*sin(c+d*x)/d+16/45045*a^3*(4184*A+4615*B)*sec(c+d*x)^(3/2)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+4/15015*a^3*(4184*A+4615*B)*sec(c+d*x)^(5/2)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+2/9009*a^3*(4184*A+4615*B)*sec(c+d*x)^(7/2)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+2/1287*a^3*(280*A+299*B)*sec(c+d*x)^(9/2)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+2/143*a^2*(16*A+13*B)*sec(c+d*x)^(11/2)*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/d+32/45045*a^3*(4184*A+4615*B)*sin(c+d*x)*sqrt(sec(c+d*x))/(d*sqrt(a+a*cos(c+d*x)))],
[(a+a*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x))*sec(c+d*x)^(13/2),x,7,2/11*a*A*(a+a*cos(c+d*x))^(3/2)*sec(c+d*x)^(11/2)*sin(c+d*x)/d+8/3465*a^3*(710*A+803*B)*sec(c+d*x)^(3/2)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+2/1155*a^3*(710*A+803*B)*sec(c+d*x)^(5/2)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+2/693*a^3*(194*A+209*B)*sec(c+d*x)^(7/2)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+2/99*a^2*(14*A+11*B)*sec(c+d*x)^(9/2)*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/d+16/3465*a^3*(710*A+803*B)*sin(c+d*x)*sqrt(sec(c+d*x))/(d*sqrt(a+a*cos(c+d*x)))],
[(a+a*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x))*sec(c+d*x)^(11/2),x,6,2/9*a*A*(a+a*cos(c+d*x))^(3/2)*sec(c+d*x)^(9/2)*sin(c+d*x)/d+2/315*a^3*(292*A+345*B)*sec(c+d*x)^(3/2)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+2/315*a^3*(124*A+135*B)*sec(c+d*x)^(5/2)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+2/21*a^2*(4*A+3*B)*sec(c+d*x)^(7/2)*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/d+4/315*a^3*(292*A+345*B)*sin(c+d*x)*sqrt(sec(c+d*x))/(d*sqrt(a+a*cos(c+d*x)))],
[(a+a*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x))*sec(c+d*x)^(9/2),x,5,2/7*a*A*(a+a*cos(c+d*x))^(3/2)*sec(c+d*x)^(7/2)*sin(c+d*x)/d+2/15*a^3*(10*A+11*B)*sec(c+d*x)^(3/2)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+2/35*a^2*(10*A+7*B)*sec(c+d*x)^(5/2)*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/d+2/105*a^3*(230*A+301*B)*sin(c+d*x)*sqrt(sec(c+d*x))/(d*sqrt(a+a*cos(c+d*x)))],
[(a+a*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x))*sec(c+d*x)^(7/2),x,6,2/5*a*A*(a+a*cos(c+d*x))^(3/2)*sec(c+d*x)^(5/2)*sin(c+d*x)/d+2/15*a^2*(8*A+5*B)*sec(c+d*x)^(3/2)*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/d+2*a^(5/2)*B*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d+2/15*a^3*(32*A+35*B)*sin(c+d*x)*sqrt(sec(c+d*x))/(d*sqrt(a+a*cos(c+d*x)))],
[(a+a*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x))*sec(c+d*x)^(5/2),x,6,2/3*a*A*(a+a*cos(c+d*x))^(3/2)*sec(c+d*x)^(3/2)*sin(c+d*x)/d-1/3*a^3*(14*A+3*B)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x))*sqrt(sec(c+d*x)))+a^(5/2)*(2*A+5*B)*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d+2*a^2*(2*A+B)*sin(c+d*x)*sqrt(a+a*cos(c+d*x))*sqrt(sec(c+d*x))/d],
[(a+a*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x))*sec(c+d*x)^(3/2),x,6,-1/4*a^3*(4*A-9*B)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x))*sqrt(sec(c+d*x)))-1/2*a^2*(4*A-B)*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/(d*sqrt(sec(c+d*x)))+2*a*A*(a+a*cos(c+d*x))^(3/2)*sin(c+d*x)*sqrt(sec(c+d*x))/d+1/4*a^(5/2)*(20*A+19*B)*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d],
[(a+a*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x))*sqrt(sec(c+d*x)),x,6,1/3*a*B*(a+a*cos(c+d*x))^(3/2)*sin(c+d*x)/(d*sqrt(sec(c+d*x)))+1/24*a^3*(54*A+49*B)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x))*sqrt(sec(c+d*x)))+1/4*a^2*(2*A+3*B)*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/(d*sqrt(sec(c+d*x)))+1/8*a^(5/2)*(38*A+25*B)*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d],
[(a+a*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x))/sqrt(sec(c+d*x)),x,7,1/4*a*B*(a+a*cos(c+d*x))^(3/2)*sin(c+d*x)/(d*sec(c+d*x)^(3/2))+1/96*a^3*(104*A+95*B)*sin(c+d*x)/(d*sec(c+d*x)^(3/2)*sqrt(a+a*cos(c+d*x)))+1/24*a^2*(8*A+11*B)*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/(d*sec(c+d*x)^(3/2))+1/64*a^3*(200*A+163*B)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x))*sqrt(sec(c+d*x)))+1/64*a^(5/2)*(200*A+163*B)*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d],
[(a+a*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x))/sec(c+d*x)^(3/2),x,8,1/5*a*B*(a+a*cos(c+d*x))^(3/2)*sin(c+d*x)/(d*sec(c+d*x)^(5/2))+1/240*a^3*(170*A+157*B)*sin(c+d*x)/(d*sec(c+d*x)^(5/2)*sqrt(a+a*cos(c+d*x)))+1/192*a^3*(326*A+283*B)*sin(c+d*x)/(d*sec(c+d*x)^(3/2)*sqrt(a+a*cos(c+d*x)))+1/40*a^2*(10*A+13*B)*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/(d*sec(c+d*x)^(5/2))+1/128*a^3*(326*A+283*B)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x))*sqrt(sec(c+d*x)))+1/128*a^(5/2)*(326*A+283*B)*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d],

# n<0
[(A+B*cos(c+d*x))*sec(c+d*x)^(11/2)/sqrt(a+a*cos(c+d*x)),x,9,-2/315*(29*A-93*B)*sec(c+d*x)^(3/2)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+2/105*(19*A-3*B)*sec(c+d*x)^(5/2)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))-2/63*(A-9*B)*sec(c+d*x)^(7/2)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+2/9*A*sec(c+d*x)^(9/2)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))-(A-B)*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))*sqrt(2)*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(d*sqrt(a))+2/315*(257*A-129*B)*sin(c+d*x)*sqrt(sec(c+d*x))/(d*sqrt(a+a*cos(c+d*x)))],
[(A+B*cos(c+d*x))*sec(c+d*x)^(9/2)/sqrt(a+a*cos(c+d*x)),x,8,2/105*(31*A-7*B)*sec(c+d*x)^(3/2)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))-2/35*(A-7*B)*sec(c+d*x)^(5/2)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+2/7*A*sec(c+d*x)^(7/2)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+(A-B)*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))*sqrt(2)*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(d*sqrt(a))-2/105*(43*A-91*B)*sin(c+d*x)*sqrt(sec(c+d*x))/(d*sqrt(a+a*cos(c+d*x)))],
[(A+B*cos(c+d*x))*sec(c+d*x)^(7/2)/sqrt(a+a*cos(c+d*x)),x,7,-2/15*(A-5*B)*sec(c+d*x)^(3/2)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+2/5*A*sec(c+d*x)^(5/2)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))-(A-B)*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))*sqrt(2)*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(d*sqrt(a))+2/15*(13*A-5*B)*sin(c+d*x)*sqrt(sec(c+d*x))/(d*sqrt(a+a*cos(c+d*x)))],
[(A+B*cos(c+d*x))*sec(c+d*x)^(5/2)/sqrt(a+a*cos(c+d*x)),x,6,2/3*A*sec(c+d*x)^(3/2)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+(A-B)*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))*sqrt(2)*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(d*sqrt(a))-2/3*(A-3*B)*sin(c+d*x)*sqrt(sec(c+d*x))/(d*sqrt(a+a*cos(c+d*x)))],
[(A+B*cos(c+d*x))*sec(c+d*x)^(3/2)/sqrt(a+a*cos(c+d*x)),x,5,-(A-B)*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))*sqrt(2)*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(d*sqrt(a))+2*A*sin(c+d*x)*sqrt(sec(c+d*x))/(d*sqrt(a+a*cos(c+d*x)))],
[(A+B*cos(c+d*x))*sec(c+d*x)^(1/2)/sqrt(a+a*cos(c+d*x)),x,6,2*B*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(d*sqrt(a))+(A-B)*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))*sqrt(2)*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(d*sqrt(a))],
[(A+B*cos(c+d*x))/(sec(c+d*x)^(1/2)*sqrt(a+a*cos(c+d*x))),x,7,B*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x))*sqrt(sec(c+d*x)))+(2*A-B)*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(d*sqrt(a))-(A-B)*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))*sqrt(2)*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(d*sqrt(a))],
[(A+B*cos(c+d*x))/(sec(c+d*x)^(3/2)*sqrt(a+a*cos(c+d*x))),x,8,1/2*B*sin(c+d*x)/(d*sec(c+d*x)^(3/2)*sqrt(a+a*cos(c+d*x)))+1/4*(4*A-B)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x))*sqrt(sec(c+d*x)))-1/4*(4*A-7*B)*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(d*sqrt(a))+(A-B)*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))*sqrt(2)*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(d*sqrt(a))],
[(a*A+(A*b+a*B)*cos(c+d*x)+b*B*cos(c+d*x)^2)*sqrt(sec(c+d*x))/sqrt(a+a*cos(c+d*x)),x,7,b*B*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x))*sqrt(sec(c+d*x)))+(2*A*b+2*a*B-b*B)*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(d*sqrt(a))+(a-b)*(A-B)*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))*sqrt(2)*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(d*sqrt(a))],
[(A+B*cos(c+d*x))*sec(c+d*x)^(9/2)/(a+a*cos(c+d*x))^(3/2),x,9,-1/2*(A-B)*sec(c+d*x)^(7/2)*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(3/2))+1/210*(397*A-273*B)*sec(c+d*x)^(3/2)*sin(c+d*x)/(a*d*sqrt(a+a*cos(c+d*x)))-1/70*(67*A-63*B)*sec(c+d*x)^(5/2)*sin(c+d*x)/(a*d*sqrt(a+a*cos(c+d*x)))+1/14*(11*A-7*B)*sec(c+d*x)^(7/2)*sin(c+d*x)/(a*d*sqrt(a+a*cos(c+d*x)))+1/2*(19*A-15*B)*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(3/2)*d*sqrt(2))-1/210*(1201*A-1029*B)*sin(c+d*x)*sqrt(sec(c+d*x))/(a*d*sqrt(a+a*cos(c+d*x)))],
[(A+B*cos(c+d*x))*sec(c+d*x)^(7/2)/(a+a*cos(c+d*x))^(3/2),x,8,-1/2*(A-B)*sec(c+d*x)^(5/2)*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(3/2))-1/30*(39*A-35*B)*sec(c+d*x)^(3/2)*sin(c+d*x)/(a*d*sqrt(a+a*cos(c+d*x)))+1/10*(9*A-5*B)*sec(c+d*x)^(5/2)*sin(c+d*x)/(a*d*sqrt(a+a*cos(c+d*x)))-1/2*(15*A-11*B)*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(3/2)*d*sqrt(2))+1/30*(147*A-95*B)*sin(c+d*x)*sqrt(sec(c+d*x))/(a*d*sqrt(a+a*cos(c+d*x)))],
[(A+B*cos(c+d*x))*sec(c+d*x)^(5/2)/(a+a*cos(c+d*x))^(3/2),x,7,-1/2*(A-B)*sec(c+d*x)^(3/2)*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(3/2))+1/6*(7*A-3*B)*sec(c+d*x)^(3/2)*sin(c+d*x)/(a*d*sqrt(a+a*cos(c+d*x)))+1/2*(11*A-7*B)*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(3/2)*d*sqrt(2))-1/6*(19*A-15*B)*sin(c+d*x)*sqrt(sec(c+d*x))/(a*d*sqrt(a+a*cos(c+d*x)))],
[(A+B*cos(c+d*x))*sec(c+d*x)^(3/2)/(a+a*cos(c+d*x))^(3/2),x,6,-1/2*(A-B)*sin(c+d*x)*sqrt(sec(c+d*x))/(d*(a+a*cos(c+d*x))^(3/2))-1/2*(7*A-3*B)*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(3/2)*d*sqrt(2))+1/2*(5*A-B)*sin(c+d*x)*sqrt(sec(c+d*x))/(a*d*sqrt(a+a*cos(c+d*x)))],
[(A+B*cos(c+d*x))*sec(c+d*x)^(1/2)/(a+a*cos(c+d*x))^(3/2),x,5,-1/2*(A-B)*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(3/2)*sqrt(sec(c+d*x)))+1/2*(3*A+B)*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(3/2)*d*sqrt(2))],
[(A+B*cos(c+d*x))/((a+a*cos(c+d*x))^(3/2)*sec(c+d*x)^(1/2)),x,7,1/2*(A-B)*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(3/2)*sqrt(sec(c+d*x)))+2*B*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(3/2)*d)+1/2*(A-5*B)*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(3/2)*d*sqrt(2))],
[(A+B*cos(c+d*x))/((a+a*cos(c+d*x))^(3/2)*sec(c+d*x)^(3/2)),x,8,1/2*(A-B)*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(3/2)*sec(c+d*x)^(3/2))-1/2*(A-3*B)*sin(c+d*x)/(a*d*sqrt(a+a*cos(c+d*x))*sqrt(sec(c+d*x)))+(2*A-3*B)*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(3/2)*d)-1/2*(5*A-9*B)*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(3/2)*d*sqrt(2))],
[(A+B*cos(c+d*x))*sec(c+d*x)^(7/2)/(a+a*cos(c+d*x))^(5/2),x,9,-1/4*(A-B)*sec(c+d*x)^(5/2)*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(5/2))-1/16*(21*A-13*B)*sec(c+d*x)^(5/2)*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^(3/2))-1/240*(787*A-475*B)*sec(c+d*x)^(3/2)*sin(c+d*x)/(a^2*d*sqrt(a+a*cos(c+d*x)))+1/80*(157*A-85*B)*sec(c+d*x)^(5/2)*sin(c+d*x)/(a^2*d*sqrt(a+a*cos(c+d*x)))-1/16*(283*A-163*B)*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(5/2)*d*sqrt(2))+1/240*(2671*A-1495*B)*sin(c+d*x)*sqrt(sec(c+d*x))/(a^2*d*sqrt(a+a*cos(c+d*x)))],
[(A+B*cos(c+d*x))*sec(c+d*x)^(5/2)/(a+a*cos(c+d*x))^(5/2),x,8,-1/4*(A-B)*sec(c+d*x)^(3/2)*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(5/2))-1/16*(17*A-9*B)*sec(c+d*x)^(3/2)*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^(3/2))+1/48*(95*A-39*B)*sec(c+d*x)^(3/2)*sin(c+d*x)/(a^2*d*sqrt(a+a*cos(c+d*x)))+1/16*(163*A-75*B)*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(5/2)*d*sqrt(2))-1/48*(299*A-147*B)*sin(c+d*x)*sqrt(sec(c+d*x))/(a^2*d*sqrt(a+a*cos(c+d*x)))],
[(A+B*cos(c+d*x))*sec(c+d*x)^(3/2)/(a+a*cos(c+d*x))^(5/2),x,7,-1/4*(A-B)*sin(c+d*x)*sqrt(sec(c+d*x))/(d*(a+a*cos(c+d*x))^(5/2))-1/16*(13*A-5*B)*sin(c+d*x)*sqrt(sec(c+d*x))/(a*d*(a+a*cos(c+d*x))^(3/2))-1/16*(75*A-19*B)*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(5/2)*d*sqrt(2))+1/16*(49*A-9*B)*sin(c+d*x)*sqrt(sec(c+d*x))/(a^2*d*sqrt(a+a*cos(c+d*x)))],
[(A+B*cos(c+d*x))*sec(c+d*x)^(1/2)/(a+a*cos(c+d*x))^(5/2),x,6,-1/4*(A-B)*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(5/2)*sqrt(sec(c+d*x)))-1/16*(9*A-B)*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^(3/2)*sqrt(sec(c+d*x)))+1/16*(19*A+5*B)*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(5/2)*d*sqrt(2))],
[(A+B*cos(c+d*x))/((a+a*cos(c+d*x))^(5/2)*sec(c+d*x)^(1/2)),x,6,1/4*(A-B)*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(5/2)*sqrt(sec(c+d*x)))+1/16*(A+7*B)*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^(3/2)*sqrt(sec(c+d*x)))+1/16*(5*A+3*B)*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(5/2)*d*sqrt(2))],
[(A+B*cos(c+d*x))/((a+a*cos(c+d*x))^(5/2)*sec(c+d*x)^(3/2)),x,8,1/4*(A-B)*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(5/2)*sec(c+d*x)^(3/2))+1/16*(3*A-11*B)*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^(3/2)*sqrt(sec(c+d*x)))+2*B*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(5/2)*d)+1/16*(3*A-43*B)*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(5/2)*d*sqrt(2))],
[(A+B*cos(c+d*x))/((a+a*cos(c+d*x))^(5/2)*sec(c+d*x)^(5/2)),x,9,1/4*(A-B)*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(5/2)*sec(c+d*x)^(5/2))+1/16*(7*A-15*B)*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^(3/2)*sec(c+d*x)^(3/2))-1/16*(11*A-35*B)*sin(c+d*x)/(a^2*d*sqrt(a+a*cos(c+d*x))*sqrt(sec(c+d*x)))+(2*A-5*B)*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(5/2)*d)-1/16*(43*A-115*B)*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(5/2)*d*sqrt(2))],
[(A+B*cos(c+d*x))*sec(c+d*x)^(5/2)/(a+a*cos(c+d*x))^(7/2),x,9,-1/6*(A-B)*sec(c+d*x)^(3/2)*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(7/2))-1/48*(23*A-11*B)*sec(c+d*x)^(3/2)*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^(5/2))-1/64*(109*A-41*B)*sec(c+d*x)^(3/2)*sin(c+d*x)/(a^2*d*(a+a*cos(c+d*x))^(3/2))+1/192*(579*A-199*B)*sec(c+d*x)^(3/2)*sin(c+d*x)/(a^3*d*sqrt(a+a*cos(c+d*x)))+1/64*(1015*A-363*B)*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(7/2)*d*sqrt(2))-1/192*(1887*A-691*B)*sin(c+d*x)*sqrt(sec(c+d*x))/(a^3*d*sqrt(a+a*cos(c+d*x)))],
[(A+B*cos(c+d*x))*sec(c+d*x)^(3/2)/(a+a*cos(c+d*x))^(7/2),x,8,-1/6*(A-B)*sin(c+d*x)*sqrt(sec(c+d*x))/(d*(a+a*cos(c+d*x))^(7/2))-1/48*(19*A-7*B)*sin(c+d*x)*sqrt(sec(c+d*x))/(a*d*(a+a*cos(c+d*x))^(5/2))-1/192*(199*A-43*B)*sin(c+d*x)*sqrt(sec(c+d*x))/(a^2*d*(a+a*cos(c+d*x))^(3/2))-3/64*(121*A-21*B)*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(7/2)*d*sqrt(2))+1/192*(691*A-103*B)*sin(c+d*x)*sqrt(sec(c+d*x))/(a^3*d*sqrt(a+a*cos(c+d*x)))],
[(A+B*cos(c+d*x))*sec(c+d*x)^(1/2)/(a+a*cos(c+d*x))^(7/2),x,7,-1/6*(A-B)*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(7/2)*sqrt(sec(c+d*x)))-1/16*(5*A-B)*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^(5/2)*sqrt(sec(c+d*x)))-1/192*(103*A+5*B)*sin(c+d*x)/(a^2*d*(a+a*cos(c+d*x))^(3/2)*sqrt(sec(c+d*x)))+1/64*(63*A+13*B)*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(7/2)*d*sqrt(2))],
[(A+B*cos(c+d*x))/((a+a*cos(c+d*x))^(7/2)*sec(c+d*x)^(1/2)),x,7,1/6*(A-B)*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(7/2)*sqrt(sec(c+d*x)))+1/16*(A+3*B)*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^(5/2)*sqrt(sec(c+d*x)))-1/192*(5*A-17*B)*sin(c+d*x)/(a^2*d*(a+a*cos(c+d*x))^(3/2)*sqrt(sec(c+d*x)))+1/64*(13*A+7*B)*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(7/2)*d*sqrt(2))],
[(A+B*cos(c+d*x))/((a+a*cos(c+d*x))^(7/2)*sec(c+d*x)^(3/2)),x,7,1/6*(A-B)*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(7/2)*sec(c+d*x)^(3/2))+1/48*(A-13*B)*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^(5/2)*sqrt(sec(c+d*x)))+1/192*(17*A+67*B)*sin(c+d*x)/(a^2*d*(a+a*cos(c+d*x))^(3/2)*sqrt(sec(c+d*x)))+1/64*(7*A+5*B)*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(7/2)*d*sqrt(2))],
[(A+B*cos(c+d*x))/((a+a*cos(c+d*x))^(7/2)*sec(c+d*x)^(5/2)),x,9,1/6*(A-B)*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(7/2)*sec(c+d*x)^(5/2))+1/48*(5*A-17*B)*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^(5/2)*sec(c+d*x)^(3/2))+1/64*(5*A-49*B)*sin(c+d*x)/(a^2*d*(a+a*cos(c+d*x))^(3/2)*sqrt(sec(c+d*x)))+2*B*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(7/2)*d)+1/64*(5*A-177*B)*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(7/2)*d*sqrt(2))],
[(A+B*cos(c+d*x))/((a+a*cos(c+d*x))^(7/2)*sec(c+d*x)^(7/2)),x,10,1/6*(A-B)*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(7/2)*sec(c+d*x)^(7/2))+1/16*(3*A-7*B)*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^(5/2)*sec(c+d*x)^(5/2))+1/192*(79*A-259*B)*sin(c+d*x)/(a^2*d*(a+a*cos(c+d*x))^(3/2)*sec(c+d*x)^(3/2))-7/64*(7*A-27*B)*sin(c+d*x)/(a^3*d*sqrt(a+a*cos(c+d*x))*sqrt(sec(c+d*x)))+(2*A-7*B)*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(7/2)*d)-1/64*(177*A-637*B)*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(7/2)*d*sqrt(2))],

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

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

# n>0
[(a+b*cos(c+d*x))*(A+B*cos(c+d*x))*sec(c+d*x)^(7/2),x,9,2/3*(A*b+a*B)*sec(c+d*x)^(3/2)*sin(c+d*x)/d+2/5*a*A*sec(c+d*x)^(5/2)*sin(c+d*x)/d+2/5*(3*a*A+5*b*B)*sin(c+d*x)*sqrt(sec(c+d*x))/d-2/5*(3*a*A+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(cos(c+d*x))*sqrt(sec(c+d*x))/d+2/3*(A*b+a*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))*sqrt(sec(c+d*x))/d],
[(a+b*cos(c+d*x))*(A+B*cos(c+d*x))*sec(c+d*x)^(5/2),x,8,2/3*a*A*sec(c+d*x)^(3/2)*sin(c+d*x)/d+2*(A*b+a*B)*sin(c+d*x)*sqrt(sec(c+d*x))/d-2*(A*b+a*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(cos(c+d*x))*sqrt(sec(c+d*x))/d+2/3*(a*A+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))*sqrt(sec(c+d*x))/d],
[(a+b*cos(c+d*x))*(A+B*cos(c+d*x))*sec(c+d*x)^(3/2),x,7,2*a*A*sin(c+d*x)*sqrt(sec(c+d*x))/d-2*(a*A-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(cos(c+d*x))*sqrt(sec(c+d*x))/d+2*(A*b+a*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))*sqrt(sec(c+d*x))/d],
[(a+b*cos(c+d*x))*(A+B*cos(c+d*x))*sqrt(sec(c+d*x)),x,7,2/3*b*B*sin(c+d*x)/(d*sqrt(sec(c+d*x)))+2*(A*b+a*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(cos(c+d*x))*sqrt(sec(c+d*x))/d+2/3*(3*a*A+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))*sqrt(sec(c+d*x))/d],
[(a+b*cos(c+d*x))*(A+B*cos(c+d*x))/sqrt(sec(c+d*x)),x,8,2/5*b*B*sin(c+d*x)/(d*sec(c+d*x)^(3/2))+2/3*(A*b+a*B)*sin(c+d*x)/(d*sqrt(sec(c+d*x)))+2/5*(5*a*A+3*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(cos(c+d*x))*sqrt(sec(c+d*x))/d+2/3*(A*b+a*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))*sqrt(sec(c+d*x))/d],
[(a+b*cos(c+d*x))*(A+B*cos(c+d*x))/sec(c+d*x)^(3/2),x,9,2/7*b*B*sin(c+d*x)/(d*sec(c+d*x)^(5/2))+2/5*(A*b+a*B)*sin(c+d*x)/(d*sec(c+d*x)^(3/2))+2/21*(7*a*A+5*b*B)*sin(c+d*x)/(d*sqrt(sec(c+d*x)))+6/5*(A*b+a*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(cos(c+d*x))*sqrt(sec(c+d*x))/d+2/21*(7*a*A+5*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))*sqrt(sec(c+d*x))/d],
[(a+b*cos(c+d*x))^2*(A+B*cos(c+d*x))*sec(c+d*x)^(7/2),x,9,2/15*a*(7*A*b+5*a*B)*sec(c+d*x)^(3/2)*sin(c+d*x)/d+2/5*a*A*sec(c+d*x)^(3/2)*(b+a*sec(c+d*x))*sin(c+d*x)/d+2/5*(3*a^2*A+5*b*(A*b+2*a*B))*sin(c+d*x)*sqrt(sec(c+d*x))/d-2/5*(3*a^2*A+5*b*(A*b+2*a*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(cos(c+d*x))*sqrt(sec(c+d*x))/d+2/3*(2*a*A*b+a^2*B+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))*sqrt(sec(c+d*x))/d],
[(a+b*cos(c+d*x))^2*(A+B*cos(c+d*x))*sec(c+d*x)^(5/2),x,8,2/3*a*(5*A*b+3*a*B)*sin(c+d*x)*sqrt(sec(c+d*x))/d+2/3*a*A*(b+a*sec(c+d*x))*sin(c+d*x)*sqrt(sec(c+d*x))/d-2*(2*a*A*b+a^2*B-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(cos(c+d*x))*sqrt(sec(c+d*x))/d+2/3*(a^2*A+3*A*b^2+6*a*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))*sqrt(sec(c+d*x))/d],
[(a+b*cos(c+d*x))^2*(A+B*cos(c+d*x))*sec(c+d*x)^(3/2),x,8,2/3*b^2*B*sin(c+d*x)/(d*sqrt(sec(c+d*x)))+2*a^2*A*sin(c+d*x)*sqrt(sec(c+d*x))/d-2*(a^2*A-b*(A*b+2*a*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(cos(c+d*x))*sqrt(sec(c+d*x))/d+2/3*(6*a*A*b+3*a^2*B+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))*sqrt(sec(c+d*x))/d],
[(a+b*cos(c+d*x))^2*(A+B*cos(c+d*x))*sqrt(sec(c+d*x)),x,8,2/5*b^2*B*sin(c+d*x)/(d*sec(c+d*x)^(3/2))+2/3*b*(A*b+2*a*B)*sin(c+d*x)/(d*sqrt(sec(c+d*x)))+2/5*(10*a*A*b+5*a^2*B+3*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(cos(c+d*x))*sqrt(sec(c+d*x))/d+2/3*(3*a^2*A+A*b^2+2*a*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))*sqrt(sec(c+d*x))/d],
[(a+b*cos(c+d*x))^2*(A+B*cos(c+d*x))/sqrt(sec(c+d*x)),x,9,2/7*b^2*B*sin(c+d*x)/(d*sec(c+d*x)^(5/2))+2/5*b*(A*b+2*a*B)*sin(c+d*x)/(d*sec(c+d*x)^(3/2))+2/21*(14*a*A*b+7*a^2*B+5*b^2*B)*sin(c+d*x)/(d*sqrt(sec(c+d*x)))+2/5*(5*a^2*A+3*A*b^2+6*a*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(cos(c+d*x))*sqrt(sec(c+d*x))/d+2/21*(14*a*A*b+7*a^2*B+5*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))*sqrt(sec(c+d*x))/d],
[(a+b*cos(c+d*x))^3*(A+B*cos(c+d*x))*sec(c+d*x)^(9/2),x,10,2/21*a*(5*a^2*A+18*A*b^2+21*a*b*B)*sec(c+d*x)^(3/2)*sin(c+d*x)/d+2/35*a^2*(11*A*b+7*a*B)*sec(c+d*x)^(5/2)*sin(c+d*x)/d+2/7*a*A*sec(c+d*x)^(3/2)*(b+a*sec(c+d*x))^2*sin(c+d*x)/d+2/5*(9*a^2*A*b+5*A*b^3+3*a^3*B+15*a*b^2*B)*sin(c+d*x)*sqrt(sec(c+d*x))/d-2/5*(9*a^2*A*b+5*A*b^3+3*a^3*B+15*a*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(cos(c+d*x))*sqrt(sec(c+d*x))/d+2/21*(5*a^3*A+21*a*A*b^2+21*a^2*b*B+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))*sqrt(sec(c+d*x))/d],
[(a+b*cos(c+d*x))^3*(A+B*cos(c+d*x))*sec(c+d*x)^(7/2),x,9,2/15*a^2*(9*A*b+5*a*B)*sec(c+d*x)^(3/2)*sin(c+d*x)/d+2/5*a*(3*a^2*A+14*A*b^2+15*a*b*B)*sin(c+d*x)*sqrt(sec(c+d*x))/d+2/5*a*A*(b+a*sec(c+d*x))^2*sin(c+d*x)*sqrt(sec(c+d*x))/d-2/5*(3*a^3*A+15*a*A*b^2+15*a^2*b*B-5*b^3*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(cos(c+d*x))*sqrt(sec(c+d*x))/d+2/3*(3*a^2*A*b+3*A*b^3+a^3*B+9*a*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))*sqrt(sec(c+d*x))/d],
[(a+b*cos(c+d*x))^3*(A+B*cos(c+d*x))*sec(c+d*x)^(5/2),x,9,2/3*a^2*(a*A-b*B)*sec(c+d*x)^(3/2)*sin(c+d*x)/d+2/3*b*B*(b+a*sec(c+d*x))^2*sin(c+d*x)/(d*sqrt(sec(c+d*x)))+2/3*a*(9*a*A*b+3*a^2*B-2*b^2*B)*sin(c+d*x)*sqrt(sec(c+d*x))/d-2*(3*a^2*A*b-A*b^3+a^3*B-3*a*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(cos(c+d*x))*sqrt(sec(c+d*x))/d+2/3*(a^3*A+9*a*A*b^2+9*a^2*b*B+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))*sqrt(sec(c+d*x))/d],
[(a+b*cos(c+d*x))^3*(A+B*cos(c+d*x))*sec(c+d*x)^(3/2),x,9,2/5*b*B*(b+a*sec(c+d*x))^2*sin(c+d*x)/(d*sec(c+d*x)^(3/2))+2/15*b^2*(5*A*b+9*a*B)*sin(c+d*x)/(d*sqrt(sec(c+d*x)))+2/5*a^2*(5*a*A-b*B)*sin(c+d*x)*sqrt(sec(c+d*x))/d-2/5*(5*a^3*A-15*a*A*b^2-15*a^2*b*B-3*b^3*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(cos(c+d*x))*sqrt(sec(c+d*x))/d+2/3*(9*a^2*A*b+A*b^3+3*a^3*B+3*a*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))*sqrt(sec(c+d*x))/d],
[(a+b*cos(c+d*x))^3*(A+B*cos(c+d*x))*sqrt(sec(c+d*x)),x,9,2/35*b^2*(7*A*b+11*a*B)*sin(c+d*x)/(d*sec(c+d*x)^(3/2))+2/7*b*B*(b+a*sec(c+d*x))^2*sin(c+d*x)/(d*sec(c+d*x)^(5/2))+2/21*b*(21*a*A*b+18*a^2*B+5*b^2*B)*sin(c+d*x)/(d*sqrt(sec(c+d*x)))+2/5*(15*a^2*A*b+3*A*b^3+5*a^3*B+9*a*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(cos(c+d*x))*sqrt(sec(c+d*x))/d+2/21*(21*a^3*A+21*a*A*b^2+21*a^2*b*B+5*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))*sqrt(sec(c+d*x))/d],
[(a+b*cos(c+d*x))^3*(A+B*cos(c+d*x))/sqrt(sec(c+d*x)),x,10,2/63*b^2*(9*A*b+13*a*B)*sin(c+d*x)/(d*sec(c+d*x)^(5/2))+2/45*b*(27*a*A*b+22*a^2*B+7*b^2*B)*sin(c+d*x)/(d*sec(c+d*x)^(3/2))+2/9*b*B*(b+a*sec(c+d*x))^2*sin(c+d*x)/(d*sec(c+d*x)^(7/2))+2/21*(21*a^2*A*b+5*A*b^3+7*a^3*B+15*a*b^2*B)*sin(c+d*x)/(d*sqrt(sec(c+d*x)))+2/15*(15*a^3*A+27*a*A*b^2+27*a^2*b*B+7*b^3*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(cos(c+d*x))*sqrt(sec(c+d*x))/d+2/21*(21*a^2*A*b+5*A*b^3+7*a^3*B+15*a*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))*sqrt(sec(c+d*x))/d],

# n<0
[(A+B*cos(c+d*x))*sec(c+d*x)^(5/2)/(a+b*cos(c+d*x)),x,11,2/3*A*sec(c+d*x)^(3/2)*sin(c+d*x)/(a*d)-2*(A*b-a*B)*sin(c+d*x)*sqrt(sec(c+d*x))/(a^2*d)+2*(A*b-a*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(cos(c+d*x))*sqrt(sec(c+d*x))/(a^2*d)+2/3*A*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(sec(c+d*x))/(a*d)+2*b*(A*b-a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(a+b),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^2*(a+b)*d)],
[(A+B*cos(c+d*x))*sec(c+d*x)^(3/2)/(a+b*cos(c+d*x)),x,8,2*A*sin(c+d*x)*sqrt(sec(c+d*x))/(a*d)-2*A*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a*d)-2*(A*b-a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(a+b),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a*(a+b)*d)],
[(A+B*cos(c+d*x))*sqrt(sec(c+d*x))/(a+b*cos(c+d*x)),x,6,2*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))*sqrt(sec(c+d*x))/(b*d)+2*(A*b-a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(a+b),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(b*(a+b)*d)],
[(A+B*cos(c+d*x))/((a+b*cos(c+d*x))*sqrt(sec(c+d*x))),x,8,2*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(cos(c+d*x))*sqrt(sec(c+d*x))/(b*d)+2*(A*b-a*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))*sqrt(sec(c+d*x))/(b^2*d)-2*a*(A*b-a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(a+b),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(b^2*(a+b)*d)],
[(A+B*cos(c+d*x))/((a+b*cos(c+d*x))*sec(c+d*x)^(3/2)),x,10,2/3*B*sin(c+d*x)/(b*d*sqrt(sec(c+d*x)))+2*(A*b-a*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(cos(c+d*x))*sqrt(sec(c+d*x))/(b^2*d)-2/3*(3*a*A*b-3*a^2*B-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))*sqrt(sec(c+d*x))/(b^3*d)+2*a^2*(A*b-a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(a+b),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(b^3*(a+b)*d)],
[(A+B*cos(c+d*x))*sec(c+d*x)^(5/2)/(a+b*cos(c+d*x))^2,x,12,1/3*(2*a^2*A-5*A*b^2+3*a*b*B)*sec(c+d*x)^(3/2)*sin(c+d*x)/(a^2*(a^2-b^2)*d)+b*(A*b-a*B)*sec(c+d*x)^(5/2)*sin(c+d*x)/(a*(a^2-b^2)*d*(b+a*sec(c+d*x)))-(4*a^2*A*b-5*A*b^3-2*a^3*B+3*a*b^2*B)*sin(c+d*x)*sqrt(sec(c+d*x))/(a^3*(a^2-b^2)*d)+(4*a^2*A*b-5*A*b^3-2*a^3*B+3*a*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(cos(c+d*x))*sqrt(sec(c+d*x))/(a^3*(a^2-b^2)*d)+1/3*(2*a^2*A-5*A*b^2+3*a*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))*sqrt(sec(c+d*x))/(a^2*(a^2-b^2)*d)+b*(7*a^2*A*b-5*A*b^3-5*a^3*B+3*a*b^2*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(a+b),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^3*(a-b)*(a+b)^2*d)],
[(A+B*cos(c+d*x))*sec(c+d*x)^(3/2)/(a+b*cos(c+d*x))^2,x,11,b*(A*b-a*B)*sec(c+d*x)^(3/2)*sin(c+d*x)/(a*(a^2-b^2)*d*(b+a*sec(c+d*x)))+(2*a^2*A-3*A*b^2+a*b*B)*sin(c+d*x)*sqrt(sec(c+d*x))/(a^2*(a^2-b^2)*d)-(2*a^2*A-3*A*b^2+a*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(cos(c+d*x))*sqrt(sec(c+d*x))/(a^2*(a^2-b^2)*d)+(A*b-a*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))*sqrt(sec(c+d*x))/(a*(a^2-b^2)*d)-(5*a^2*A*b-3*A*b^3-3*a^3*B+a*b^2*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(a+b),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^2*(a-b)*(a+b)^2*d)],
[(A+B*cos(c+d*x))*sqrt(sec(c+d*x))/(a+b*cos(c+d*x))^2,x,10,b*(A*b-a*B)*sin(c+d*x)*sqrt(sec(c+d*x))/(a*(a^2-b^2)*d*(b+a*sec(c+d*x)))-(A*b-a*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(cos(c+d*x))*sqrt(sec(c+d*x))/(a*(a^2-b^2)*d)-(A*b-a*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))*sqrt(sec(c+d*x))/(b*(a^2-b^2)*d)+(3*a^2*A*b-A*b^3-a^3*B-a*b^2*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(a+b),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a*(a-b)*b*(a+b)^2*d)],
[(A+B*cos(c+d*x))/((a+b*cos(c+d*x))^2*sqrt(sec(c+d*x))),x,10,-(A*b-a*B)*sin(c+d*x)*sqrt(sec(c+d*x))/((a^2-b^2)*d*(b+a*sec(c+d*x)))+(A*b-a*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(cos(c+d*x))*sqrt(sec(c+d*x))/(b*(a^2-b^2)*d)+(a*A*b+a^2*B-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))*sqrt(sec(c+d*x))/(b^2*(a^2-b^2)*d)-(a^2*A*b+A*b^3+a^3*B-3*a*b^2*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(a+b),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/((a-b)*b^2*(a+b)^2*d)],
[(A+B*cos(c+d*x))/((a+b*cos(c+d*x))^2*sec(c+d*x)^(3/2)),x,10,a*(A*b-a*B)*sin(c+d*x)*sqrt(sec(c+d*x))/(b*(a^2-b^2)*d*(b+a*sec(c+d*x)))-(a*A*b-3*a^2*B+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(cos(c+d*x))*sqrt(sec(c+d*x))/(b^2*(a^2-b^2)*d)+(a^2*A*b-2*A*b^3-3*a^3*B+4*a*b^2*B)*sqrt(cos(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(sec(c+d*x))/(b^3*(a^2-b^2)*d)-a*(a^2*A*b-3*A*b^3-3*a^3*B+5*a*b^2*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(a+b),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/((a-b)*b^3*(a+b)^2*d)],
[(A+B*cos(c+d*x))/((a+b*cos(c+d*x))^2*sec(c+d*x)^(5/2)),x,11,-1/3*(3*a*A*b-5*a^2*B+2*b^2*B)*sin(c+d*x)/(b^2*(a^2-b^2)*d*sqrt(sec(c+d*x)))+a*(A*b-a*B)*sin(c+d*x)/(b*(a^2-b^2)*d*(b+a*sec(c+d*x))*sqrt(sec(c+d*x)))+(3*a^2*A*b-2*A*b^3-5*a^3*B+4*a*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(cos(c+d*x))*sqrt(sec(c+d*x))/(b^3*(a^2-b^2)*d)-1/3*(9*a^3*A*b-12*a*A*b^3-15*a^4*B+16*a^2*b^2*B+2*b^4*B)*sqrt(cos(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(sec(c+d*x))/(b^4*(a^2-b^2)*d)+a^2*(3*a^2*A*b-5*A*b^3-5*a^3*B+7*a*b^2*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(a+b),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/((a-b)*b^4*(a+b)^2*d)],
[(A+B*cos(c+d*x))*sec(c+d*x)^(3/2)/(a+b*cos(c+d*x))^3,x,12,1/2*b*(A*b-a*B)*sec(c+d*x)^(5/2)*sin(c+d*x)/(a*(a^2-b^2)*d*(b+a*sec(c+d*x))^2)+1/4*b*(11*a^2*A*b-5*A*b^3-7*a^3*B+a*b^2*B)*sec(c+d*x)^(3/2)*sin(c+d*x)/(a^2*(a^2-b^2)^2*d*(b+a*sec(c+d*x)))+1/4*(8*a^4*A-29*a^2*A*b^2+15*A*b^4+9*a^3*b*B-3*a*b^3*B)*sin(c+d*x)*sqrt(sec(c+d*x))/(a^3*(a^2-b^2)^2*d)-1/4*(8*a^4*A-29*a^2*A*b^2+15*A*b^4+9*a^3*b*B-3*a*b^3*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^3*(a^2-b^2)^2*d)+1/4*(11*a^2*A*b-5*A*b^3-7*a^3*B+a*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))*sqrt(sec(c+d*x))/(a^2*(a^2-b^2)^2*d)-1/4*(35*a^4*A*b-38*a^2*A*b^3+15*A*b^5-15*a^5*B+6*a^3*b^2*B-3*a*b^4*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(a+b),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^3*(a-b)^2*(a+b)^3*d)],
[(A+B*cos(c+d*x))*sqrt(sec(c+d*x))/(a+b*cos(c+d*x))^3,x,11,1/2*b*(A*b-a*B)*sec(c+d*x)^(3/2)*sin(c+d*x)/(a*(a^2-b^2)*d*(b+a*sec(c+d*x))^2)+1/4*b*(9*a^2*A*b-3*A*b^3-5*a^3*B-a*b^2*B)*sin(c+d*x)*sqrt(sec(c+d*x))/(a^2*(a^2-b^2)^2*d*(b+a*sec(c+d*x)))-1/4*(9*a^2*A*b-3*A*b^3-5*a^3*B-a*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(cos(c+d*x))*sqrt(sec(c+d*x))/(a^2*(a^2-b^2)^2*d)-1/4*(7*a^2*A*b-A*b^3-3*a^3*B-3*a*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))*sqrt(sec(c+d*x))/(a*b*(a^2-b^2)^2*d)+1/4*(15*a^4*A*b-6*a^2*A*b^3+3*A*b^5-3*a^5*B-10*a^3*b^2*B+a*b^4*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(a+b),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^2*(a-b)^2*b*(a+b)^3*d)],
[(A+B*cos(c+d*x))/((a+b*cos(c+d*x))^3*sqrt(sec(c+d*x))),x,11,1/2*b*(A*b-a*B)*sin(c+d*x)*sqrt(sec(c+d*x))/(a*(a^2-b^2)*d*(b+a*sec(c+d*x))^2)-1/4*(7*a^2*A*b-A*b^3-3*a^3*B-3*a*b^2*B)*sin(c+d*x)*sqrt(sec(c+d*x))/(a*(a^2-b^2)^2*d*(b+a*sec(c+d*x)))+1/4*(5*a^2*A*b+A*b^3-a^3*B-5*a*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(cos(c+d*x))*sqrt(sec(c+d*x))/(a*b*(a^2-b^2)^2*d)+1/4*(3*a^2*A*b+3*A*b^3+a^3*B-7*a*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))*sqrt(sec(c+d*x))/(b^2*(a^2-b^2)^2*d)-1/4*(3*a^4*A*b+10*a^2*A*b^3-A*b^5+a^5*B-10*a^3*b^2*B-3*a*b^4*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(a+b),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a*(a-b)^2*b^2*(a+b)^3*d)],
[(A+B*cos(c+d*x))/((a+b*cos(c+d*x))^3*sec(c+d*x)^(3/2)),x,11,-1/2*(A*b-a*B)*sin(c+d*x)*sqrt(sec(c+d*x))/((a^2-b^2)*d*(b+a*sec(c+d*x))^2)+1/4*(3*a^2*A*b+3*A*b^3+a^3*B-7*a*b^2*B)*sin(c+d*x)*sqrt(sec(c+d*x))/(b*(a^2-b^2)^2*d*(b+a*sec(c+d*x)))-1/4*(a^2*A*b+5*A*b^3+3*a^3*B-9*a*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(cos(c+d*x))*sqrt(sec(c+d*x))/(b^2*(a^2-b^2)^2*d)+1/4*(a^3*A*b-7*a*A*b^3+3*a^4*B-5*a^2*b^2*B+8*b^4*B)*sqrt(cos(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(sec(c+d*x))/(b^3*(a^2-b^2)^2*d)-1/4*(a^4*A*b-10*a^2*A*b^3-3*A*b^5+3*a^5*B-6*a^3*b^2*B+15*a*b^4*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(a+b),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/((a-b)^2*b^3*(a+b)^3*d)],
[(A+B*cos(c+d*x))/((a+b*cos(c+d*x))^3*sec(c+d*x)^(5/2)),x,11,1/2*a*(A*b-a*B)*sin(c+d*x)*sqrt(sec(c+d*x))/(b*(a^2-b^2)*d*(b+a*sec(c+d*x))^2)+1/4*a*(a^2*A*b-7*A*b^3-5*a^3*B+11*a*b^2*B)*sin(c+d*x)*sqrt(sec(c+d*x))/(b^2*(a^2-b^2)^2*d*(b+a*sec(c+d*x)))-1/4*(3*a^3*A*b-9*a*A*b^3-15*a^4*B+29*a^2*b^2*B-8*b^4*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(b^3*(a^2-b^2)^2*d)+1/4*(3*a^4*A*b-5*a^2*A*b^3+8*A*b^5-15*a^5*B+33*a^3*b^2*B-24*a*b^4*B)*sqrt(cos(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(sec(c+d*x))/(b^4*(a^2-b^2)^2*d)-1/4*a*(3*a^4*A*b-6*a^2*A*b^3+15*A*b^5-15*a^5*B+38*a^3*b^2*B-35*a*b^4*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(a+b),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/((a-b)^2*b^4*(a+b)^3*d)],
[(A+B*cos(c+d*x))/((a+b*cos(c+d*x))^3*sec(c+d*x)^(7/2)),x,12,-1/12*(15*a^3*A*b-33*a*A*b^3-35*a^4*B+61*a^2*b^2*B-8*b^4*B)*sin(c+d*x)/(b^3*(a^2-b^2)^2*d*sqrt(sec(c+d*x)))+1/2*a*(A*b-a*B)*sin(c+d*x)/(b*(a^2-b^2)*d*(b+a*sec(c+d*x))^2*sqrt(sec(c+d*x)))+1/4*a*(3*a^2*A*b-9*A*b^3-7*a^3*B+13*a*b^2*B)*sin(c+d*x)/(b^2*(a^2-b^2)^2*d*(b+a*sec(c+d*x))*sqrt(sec(c+d*x)))+1/4*(15*a^4*A*b-29*a^2*A*b^3+8*A*b^5-35*a^5*B+65*a^3*b^2*B-24*a*b^4*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(b^4*(a^2-b^2)^2*d)-1/12*(45*a^5*A*b-99*a^3*A*b^3+72*a*A*b^5-105*a^6*B+223*a^4*b^2*B-128*a^2*b^4*B-8*b^6*B)*sqrt(cos(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(sec(c+d*x))/(b^5*(a^2-b^2)^2*d)+1/4*a^2*(15*a^4*A*b-38*a^2*A*b^3+35*A*b^5-35*a^5*B+86*a^3*b^2*B-63*a*b^4*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(a+b),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/((a-b)^2*b^5*(a+b)^3*d)],
[(a*B+b*B*cos(c+d*x))*sec(c+d*x)^(5/2)/(a+b*cos(c+d*x)),x,4,2/3*B*sec(c+d*x)^(3/2)*sin(c+d*x)/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))*sqrt(sec(c+d*x))/d],
[(a*B+b*B*cos(c+d*x))*sec(c+d*x)^(3/2)/(a+b*cos(c+d*x)),x,4,2*B*sin(c+d*x)*sqrt(sec(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(cos(c+d*x))*sqrt(sec(c+d*x))/d],
[(a*B+b*B*cos(c+d*x))*sqrt(sec(c+d*x))/(a+b*cos(c+d*x)),x,3,2*B*sqrt(cos(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(sec(c+d*x))/d],
[(a*B+b*B*cos(c+d*x))/((a+b*cos(c+d*x))*sqrt(sec(c+d*x))),x,3,2*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d],
[(a*B+b*B*cos(c+d*x))/((a+b*cos(c+d*x))*sec(c+d*x)^(3/2)),x,4,2/3*B*sin(c+d*x)/(d*sqrt(sec(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))*sqrt(sec(c+d*x))/d],
[(a*B+b*B*cos(c+d*x))/((a+b*cos(c+d*x))*sec(c+d*x)^(5/2)),x,4,2/5*B*sin(c+d*x)/(d*sec(c+d*x)^(3/2))+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(cos(c+d*x))*sqrt(sec(c+d*x))/d],

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

# n>0
[(A+B*cos(c+d*x))*sec(c+d*x)^(9/2)*sqrt(a+b*cos(c+d*x)),x,7,2/105*(25*a^2*A-4*A*b^2+7*a*b*B)*sec(c+d*x)^(3/2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(a^2*d)+2/35*(A*b+7*a*B)*sec(c+d*x)^(5/2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(a*d)+2/7*A*sec(c+d*x)^(7/2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/d+2/105*(a-b)*(19*a^2*A*b+8*A*b^3+63*a^3*B-14*a*b^2*B)*csc(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^4*d*sqrt(sec(c+d*x)))+2/105*(a-b)*(8*A*b^2+a^2*(25*A-63*B)+2*a*b*(3*A-7*B))*csc(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^3*d*sqrt(sec(c+d*x)))],
[(A+B*cos(c+d*x))*sec(c+d*x)^(7/2)*sqrt(a+b*cos(c+d*x)),x,6,2/15*(A*b+5*a*B)*sec(c+d*x)^(3/2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(a*d)+2/5*A*sec(c+d*x)^(5/2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/d+2/15*(a-b)*(9*a^2*A-2*A*b^2+5*a*b*B)*csc(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^3*d*sqrt(sec(c+d*x)))-2/15*(a-b)*(9*a*A+2*A*b-5*a*B)*csc(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^2*d*sqrt(sec(c+d*x)))],
[(A+B*cos(c+d*x))*sec(c+d*x)^(5/2)*sqrt(a+b*cos(c+d*x)),x,5,2/3*A*sec(c+d*x)^(3/2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/d+2/3*(a-b)*(A*b+3*a*B)*csc(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^2*d*sqrt(sec(c+d*x)))+2/3*(a-b)*(A-3*B)*csc(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*d*sqrt(sec(c+d*x)))],
[(A+B*cos(c+d*x))*sec(c+d*x)^(3/2)*sqrt(a+b*cos(c+d*x)),x,6,2*A*(a-b)*csc(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*d*sqrt(sec(c+d*x)))+2*(A*b-a*(A-B))*csc(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*d*sqrt(sec(c+d*x)))-2*B*csc(c+d*x)*EllipticPi(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),(a+b)/b,sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(d*sqrt(sec(c+d*x)))],
[(A+B*cos(c+d*x))*sec(c+d*x)^(1/2)*sqrt(a+b*cos(c+d*x)),x,7,B*sin(c+d*x)*sqrt(a+b*cos(c+d*x))*sqrt(sec(c+d*x))/d-(a-b)*B*csc(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*d*sqrt(sec(c+d*x)))+(2*A+B)*csc(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(d*sqrt(sec(c+d*x)))-(2*A*b+a*B)*csc(c+d*x)*EllipticPi(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),(a+b)/b,sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b*d*sqrt(sec(c+d*x)))],
[(A+B*cos(c+d*x))*sqrt(a+b*cos(c+d*x))/sec(c+d*x)^(1/2),x,8,1/2*B*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(d*sqrt(sec(c+d*x)))+1/4*(4*A*b+a*B)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))*sqrt(sec(c+d*x))/(b*d)-1/4*(a-b)*(4*A*b+a*B)*csc(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*b*d*sqrt(sec(c+d*x)))+1/4*(4*A*b+(a+2*b)*B)*csc(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b*d*sqrt(sec(c+d*x)))-1/4*(4*a*A*b-a^2*B+4*b^2*B)*csc(c+d*x)*EllipticPi(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),(a+b)/b,sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b^2*d*sqrt(sec(c+d*x)))],
[(A+B*cos(c+d*x))*sqrt(a+b*cos(c+d*x))/sec(c+d*x)^(3/2),x,9,1/3*B*(a+b*cos(c+d*x))^(3/2)*sin(c+d*x)/(b*d*sqrt(sec(c+d*x)))+1/4*(2*A*b-a*B)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(b*d*sqrt(sec(c+d*x)))+1/24*(6*a*A*b-3*a^2*B+16*b^2*B)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))*sqrt(sec(c+d*x))/(b^2*d)-1/24*(a-b)*(6*a*A*b-3*a^2*B+16*b^2*B)*csc(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*b^2*d*sqrt(sec(c+d*x)))+1/24*(a+2*b)*(6*A*b-3*a*B+8*b*B)*csc(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b^2*d*sqrt(sec(c+d*x)))+1/8*(2*a^2*A*b-8*A*b^3-a^3*B-4*a*b^2*B)*csc(c+d*x)*EllipticPi(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),(a+b)/b,sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b^3*d*sqrt(sec(c+d*x)))],
[(a+b*cos(c+d*x))^(3/2)*(A+B*cos(c+d*x))*sec(c+d*x)^(11/2),x,8,2/315*(88*a^2*A*b-4*A*b^3+75*a^3*B+9*a*b^2*B)*sec(c+d*x)^(3/2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(a^2*d)+2/315*(49*a^2*A+3*A*b^2+72*a*b*B)*sec(c+d*x)^(5/2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(a*d)+2/63*(10*A*b+9*a*B)*sec(c+d*x)^(7/2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/d+2/9*a*A*sec(c+d*x)^(9/2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/d+2/315*(a-b)*(147*a^4*A+33*a^2*A*b^2+8*A*b^4+246*a^3*b*B-18*a*b^3*B)*csc(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^4*d*sqrt(sec(c+d*x)))+2/315*(a-b)*(8*A*b^3-a^3*(147*A-75*B)+3*a^2*b*(13*A-57*B)+6*a*b^2*(A-3*B))*csc(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^3*d*sqrt(sec(c+d*x)))],
[(a+b*cos(c+d*x))^(3/2)*(A+B*cos(c+d*x))*sec(c+d*x)^(9/2),x,7,2/105*(25*a^2*A+3*A*b^2+42*a*b*B)*sec(c+d*x)^(3/2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(a*d)+2/35*(8*A*b+7*a*B)*sec(c+d*x)^(5/2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/d+2/7*a*A*sec(c+d*x)^(7/2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/d+2/105*(a-b)*(82*a^2*A*b-6*A*b^3+63*a^3*B+21*a*b^2*B)*csc(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^3*d*sqrt(sec(c+d*x)))-2/105*(a-b)*(6*A*b^2-a^2*(25*A-63*B)+3*a*b*(19*A-7*B))*csc(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^2*d*sqrt(sec(c+d*x)))],
[(a+b*cos(c+d*x))^(3/2)*(A+B*cos(c+d*x))*sec(c+d*x)^(7/2),x,6,2/15*(6*A*b+5*a*B)*sec(c+d*x)^(3/2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/d+2/5*a*A*sec(c+d*x)^(5/2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/d+2/15*(a-b)*(9*a^2*A+3*A*b^2+20*a*b*B)*csc(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^2*d*sqrt(sec(c+d*x)))-2/15*(a-b)*(9*a*A-3*A*b-5*a*B+15*b*B)*csc(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*d*sqrt(sec(c+d*x)))],
[(a+b*cos(c+d*x))^(3/2)*(A+B*cos(c+d*x))*sec(c+d*x)^(5/2),x,7,2/3*a*A*sec(c+d*x)^(3/2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/d+2/3*(a-b)*(4*A*b+3*a*B)*csc(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*d*sqrt(sec(c+d*x)))+2/3*(3*A*b^2+a^2*(A-3*B)-a*(4*A*b-6*b*B))*csc(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*d*sqrt(sec(c+d*x)))-2*b*B*csc(c+d*x)*EllipticPi(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),(a+b)/b,sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(d*sqrt(sec(c+d*x)))],
[(a+b*cos(c+d*x))^(3/2)*(A+B*cos(c+d*x))*sec(c+d*x)^(3/2),x,8,2*a*A*sin(c+d*x)*sqrt(a+b*cos(c+d*x))*sqrt(sec(c+d*x))/d-(2*a*A-b*B)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))*sqrt(sec(c+d*x))/d+(a-b)*(2*a*A-b*B)*csc(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*d*sqrt(sec(c+d*x)))-(2*a*(A-B)-b*(4*A+B))*csc(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(d*sqrt(sec(c+d*x)))-(2*A*b+3*a*B)*csc(c+d*x)*EllipticPi(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),(a+b)/b,sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(d*sqrt(sec(c+d*x)))],
[(a+b*cos(c+d*x))^(3/2)*(A+B*cos(c+d*x))*sec(c+d*x)^(1/2),x,8,1/2*b*B*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(d*sqrt(sec(c+d*x)))+1/4*(4*A*b+5*a*B)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))*sqrt(sec(c+d*x))/d-1/4*(a-b)*(4*A*b+5*a*B)*csc(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*d*sqrt(sec(c+d*x)))+1/4*(8*a*A+4*A*b+5*a*B+2*b*B)*csc(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(d*sqrt(sec(c+d*x)))-1/4*(12*a*A*b+3*a^2*B+4*b^2*B)*csc(c+d*x)*EllipticPi(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),(a+b)/b,sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b*d*sqrt(sec(c+d*x)))],
[(a+b*cos(c+d*x))^(3/2)*(A+B*cos(c+d*x))/sec(c+d*x)^(1/2),x,9,1/3*b*B*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(d*sec(c+d*x)^(3/2))+1/12*(6*A*b+7*a*B)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(d*sqrt(sec(c+d*x)))+1/24*(30*a*A*b+3*a^2*B+16*b^2*B)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))*sqrt(sec(c+d*x))/(b*d)-1/24*(a-b)*(30*a*A*b+3*a^2*B+16*b^2*B)*csc(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*b*d*sqrt(sec(c+d*x)))+1/24*(30*a*A*b+12*A*b^2+3*a^2*B+14*a*b*B+16*b^2*B)*csc(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b*d*sqrt(sec(c+d*x)))-1/8*(6*a^2*A*b+8*A*b^3-a^3*B+12*a*b^2*B)*csc(c+d*x)*EllipticPi(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),(a+b)/b,sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b^2*d*sqrt(sec(c+d*x)))],
[(a+b*cos(c+d*x))^(3/2)*(A+B*cos(c+d*x))/sec(c+d*x)^(3/2),x,10,1/24*(8*A*b-3*a*B)*(a+b*cos(c+d*x))^(3/2)*sin(c+d*x)/(b*d*sqrt(sec(c+d*x)))+1/4*B*(a+b*cos(c+d*x))^(5/2)*sin(c+d*x)/(b*d*sqrt(sec(c+d*x)))+1/32*(8*a*A*b-3*a^2*B+12*b^2*B)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(b*d*sqrt(sec(c+d*x)))+1/192*(24*a^2*A*b+128*A*b^3-9*a^3*B+156*a*b^2*B)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))*sqrt(sec(c+d*x))/(b^2*d)-1/192*(a-b)*(24*a^2*A*b+128*A*b^3-9*a^3*B+156*a*b^2*B)*csc(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*b^2*d*sqrt(sec(c+d*x)))-1/192*(9*a^3*B-6*a^2*b*(4*A+B)-8*b^3*(16*A+9*B)-4*a*b^2*(28*A+39*B))*csc(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b^2*d*sqrt(sec(c+d*x)))+1/64*(8*a^3*A*b-96*a*A*b^3-3*a^4*B-24*a^2*b^2*B-48*b^4*B)*csc(c+d*x)*EllipticPi(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),(a+b)/b,sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b^3*d*sqrt(sec(c+d*x)))],
[(a+b*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x))*sec(c+d*x)^(13/2),x,9,2/11*a*A*(a+b*cos(c+d*x))^(3/2)*sec(c+d*x)^(11/2)*sin(c+d*x)/d+2/3465*(675*a^4*A+1025*a^2*A*b^2-20*A*b^4+1793*a^3*b*B+55*a*b^3*B)*sec(c+d*x)^(3/2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(a^2*d)+2/3465*(1145*a^2*A*b+15*A*b^3+539*a^3*B+825*a*b^2*B)*sec(c+d*x)^(5/2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(a*d)+2/693*(81*a^2*A+113*A*b^2+209*a*b*B)*sec(c+d*x)^(7/2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/d+2/99*a*(14*A*b+11*a*B)*sec(c+d*x)^(9/2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/d+2/3465*(a-b)*(3705*a^4*A*b+255*a^2*A*b^3+40*A*b^5+1617*a^5*B+3069*a^3*b^2*B-110*a*b^4*B)*csc(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^4*d*sqrt(sec(c+d*x)))+2/3465*(a-b)*(40*A*b^4+3*a^4*(225*A-539*B)-6*a^3*b*(505*A-209*B)+15*a^2*b^2*(19*A-121*B)+10*a*b^3*(3*A-11*B))*csc(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^3*d*sqrt(sec(c+d*x)))],
[(a+b*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x))*sec(c+d*x)^(11/2),x,8,2/9*a*A*(a+b*cos(c+d*x))^(3/2)*sec(c+d*x)^(9/2)*sin(c+d*x)/d+2/315*(163*a^2*A*b+5*A*b^3+75*a^3*B+135*a*b^2*B)*sec(c+d*x)^(3/2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(a*d)+2/315*(49*a^2*A+75*A*b^2+135*a*b*B)*sec(c+d*x)^(5/2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/d+2/21*a*(4*A*b+3*a*B)*sec(c+d*x)^(7/2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/d+2/315*(a-b)*(147*a^4*A+279*a^2*A*b^2-10*A*b^4+435*a^3*b*B+45*a*b^3*B)*csc(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^3*d*sqrt(sec(c+d*x)))-2/315*(a-b)*(10*A*b^3-6*a^2*b*(19*A-60*B)+3*a^3*(49*A-25*B)+15*a*b^2*(11*A-3*B))*csc(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^2*d*sqrt(sec(c+d*x)))],
[(a+b*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x))*sec(c+d*x)^(9/2),x,7,2/7*a*A*(a+b*cos(c+d*x))^(3/2)*sec(c+d*x)^(7/2)*sin(c+d*x)/d+2/105*(25*a^2*A+45*A*b^2+77*a*b*B)*sec(c+d*x)^(3/2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/d+2/35*a*(10*A*b+7*a*B)*sec(c+d*x)^(5/2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/d+2/105*(a-b)*(145*a^2*A*b+15*A*b^3+63*a^3*B+161*a*b^2*B)*csc(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^2*d*sqrt(sec(c+d*x)))+2/105*(a-b)*(a^2*(25*A-63*B)+15*b^2*(A-7*B)-8*a*b*(15*A-7*B))*csc(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*d*sqrt(sec(c+d*x)))],
[(a+b*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x))*sec(c+d*x)^(7/2),x,8,2/5*a*A*(a+b*cos(c+d*x))^(3/2)*sec(c+d*x)^(5/2)*sin(c+d*x)/d+2/15*a*(8*A*b+5*a*B)*sec(c+d*x)^(3/2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/d+2/15*(a-b)*(9*a^2*A+23*A*b^2+35*a*b*B)*csc(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*d*sqrt(sec(c+d*x)))+2/15*(15*A*b^3-a*b^2*(23*A-45*B)+a^2*b*(17*A-35*B)-a^3*(9*A-5*B))*csc(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*d*sqrt(sec(c+d*x)))-2*b^2*B*csc(c+d*x)*EllipticPi(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),(a+b)/b,sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(d*sqrt(sec(c+d*x)))],
[(a+b*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x))*sec(c+d*x)^(5/2),x,9,2/3*a*A*(a+b*cos(c+d*x))^(3/2)*sec(c+d*x)^(3/2)*sin(c+d*x)/d+2*a*(2*A*b+a*B)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))*sqrt(sec(c+d*x))/d-1/3*(14*a*A*b+6*a^2*B-3*b^2*B)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))*sqrt(sec(c+d*x))/d+1/3*(a-b)*(14*a*A*b+6*a^2*B-3*b^2*B)*csc(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*d*sqrt(sec(c+d*x)))-1/3*(2*a*b*(7*A-9*B)-2*a^2*(A-3*B)-3*b^2*(6*A+B))*csc(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(d*sqrt(sec(c+d*x)))-b*(2*A*b+5*a*B)*csc(c+d*x)*EllipticPi(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),(a+b)/b,sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(d*sqrt(sec(c+d*x)))],
[(a+b*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x))*sec(c+d*x)^(3/2),x,9,-1/2*b*(4*a*A-b*B)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(d*sqrt(sec(c+d*x)))+2*a*A*(a+b*cos(c+d*x))^(3/2)*sin(c+d*x)*sqrt(sec(c+d*x))/d-1/4*(8*a^2*A-4*A*b^2-9*a*b*B)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))*sqrt(sec(c+d*x))/d+1/4*(a-b)*(8*a^2*A-4*A*b^2-9*a*b*B)*csc(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*d*sqrt(sec(c+d*x)))-1/4*(8*a^2*(A-B)-2*b^2*(2*A+B)-3*a*b*(8*A+3*B))*csc(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(d*sqrt(sec(c+d*x)))-1/4*(20*a*A*b+15*a^2*B+4*b^2*B)*csc(c+d*x)*EllipticPi(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),(a+b)/b,sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(d*sqrt(sec(c+d*x)))],
[(a+b*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x))*sec(c+d*x)^(1/2),x,9,1/3*b*B*(a+b*cos(c+d*x))^(3/2)*sin(c+d*x)/(d*sqrt(sec(c+d*x)))+1/4*b*(2*A*b+3*a*B)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(d*sqrt(sec(c+d*x)))+1/24*(54*a*A*b+33*a^2*B+16*b^2*B)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))*sqrt(sec(c+d*x))/d-1/24*(a-b)*(54*a*A*b+33*a^2*B+16*b^2*B)*csc(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*d*sqrt(sec(c+d*x)))+1/24*(4*b^2*(3*A+4*B)+a^2*(48*A+33*B)+a*(54*A*b+26*b*B))*csc(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(d*sqrt(sec(c+d*x)))-1/8*(30*a^2*A*b+8*A*b^3+5*a^3*B+20*a*b^2*B)*csc(c+d*x)*EllipticPi(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),(a+b)/b,sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b*d*sqrt(sec(c+d*x)))],
[(a+b*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x))/sec(c+d*x)^(1/2),x,10,1/4*b*B*(a+b*cos(c+d*x))^(3/2)*sin(c+d*x)/(d*sec(c+d*x)^(3/2))+1/24*(8*A*b+11*a*B)*(a+b*cos(c+d*x))^(3/2)*sin(c+d*x)/(d*sqrt(sec(c+d*x)))+1/32*(24*a*A*b+5*a^2*B+12*b^2*B)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(d*sqrt(sec(c+d*x)))+1/192*(264*a^2*A*b+128*A*b^3+15*a^3*B+284*a*b^2*B)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))*sqrt(sec(c+d*x))/(b*d)-1/192*(a-b)*(264*a^2*A*b+128*A*b^3+15*a^3*B+284*a*b^2*B)*csc(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*b*d*sqrt(sec(c+d*x)))+1/192*(15*a^3*B+8*b^3*(16*A+9*B)+2*a^2*b*(132*A+59*B)+4*a*b^2*(52*A+71*B))*csc(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b*d*sqrt(sec(c+d*x)))-1/64*(40*a^3*A*b+160*a*A*b^3-5*a^4*B+120*a^2*b^2*B+48*b^4*B)*csc(c+d*x)*EllipticPi(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),(a+b)/b,sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b^2*d*sqrt(sec(c+d*x)))],
[(a+b*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x))/sec(c+d*x)^(3/2),x,11,1/240*(50*a*A*b-15*a^2*B+64*b^2*B)*(a+b*cos(c+d*x))^(3/2)*sin(c+d*x)/(b*d*sqrt(sec(c+d*x)))+1/40*(10*A*b-3*a*B)*(a+b*cos(c+d*x))^(5/2)*sin(c+d*x)/(b*d*sqrt(sec(c+d*x)))+1/5*B*(a+b*cos(c+d*x))^(7/2)*sin(c+d*x)/(b*d*sqrt(sec(c+d*x)))+1/320*(50*a^2*A*b+120*A*b^3-15*a^3*B+172*a*b^2*B)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(b*d*sqrt(sec(c+d*x)))+1/1920*(150*a^3*A*b+2840*a*A*b^3-45*a^4*B+1692*a^2*b^2*B+1024*b^4*B)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))*sqrt(sec(c+d*x))/(b^2*d)-1/1920*(a-b)*(150*a^3*A*b+2840*a*A*b^3-45*a^4*B+1692*a^2*b^2*B+1024*b^4*B)*csc(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*b^2*d*sqrt(sec(c+d*x)))-1/1920*(45*a^4*B-30*a^3*b*(5*A+B)-16*b^4*(45*A+64*B)-8*a*b^3*(355*A+193*B)-4*a^2*b^2*(295*A+423*B))*csc(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b^2*d*sqrt(sec(c+d*x)))+1/128*(10*a^4*A*b-240*a^2*A*b^3-96*A*b^5-3*a^5*B-40*a^3*b^2*B-240*a*b^4*B)*csc(c+d*x)*EllipticPi(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),(a+b)/b,sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b^3*d*sqrt(sec(c+d*x)))],

# n<0
[(A+B*cos(c+d*x))*sec(c+d*x)^(7/2)/sqrt(a+b*cos(c+d*x)),x,6,-2/15*(4*A*b-5*a*B)*sec(c+d*x)^(3/2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(a^2*d)+2/5*A*sec(c+d*x)^(5/2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(a*d)+2/15*(a-b)*(9*a^2*A+8*A*b^2-10*a*b*B)*csc(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^4*d*sqrt(sec(c+d*x)))-2/15*(8*A*b^2+a^2*(9*A-5*B)-2*a*b*(A+5*B))*csc(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^3*d*sqrt(sec(c+d*x)))],
[(A+B*cos(c+d*x))*sec(c+d*x)^(5/2)/sqrt(a+b*cos(c+d*x)),x,5,2/3*A*sec(c+d*x)^(3/2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(a*d)-2/3*(a-b)*(2*A*b-3*a*B)*csc(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^3*d*sqrt(sec(c+d*x)))+2/3*(2*A*b+a*(A-3*B))*csc(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^2*d*sqrt(sec(c+d*x)))],
[(A+B*cos(c+d*x))*sec(c+d*x)^(3/2)/sqrt(a+b*cos(c+d*x)),x,4,2*A*(a-b)*csc(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^2*d*sqrt(sec(c+d*x)))-2*(A-B)*csc(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*d*sqrt(sec(c+d*x)))],
[(A+B*cos(c+d*x))*sec(c+d*x)^(1/2)/sqrt(a+b*cos(c+d*x)),x,4,2*A*csc(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*d*sqrt(sec(c+d*x)))-2*B*csc(c+d*x)*EllipticPi(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),(a+b)/b,sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b*d*sqrt(sec(c+d*x)))],
[(A+B*cos(c+d*x))/(sec(c+d*x)^(1/2)*sqrt(a+b*cos(c+d*x))),x,8,B*sin(c+d*x)/(d*sqrt(a+b*cos(c+d*x))*sqrt(sec(c+d*x)))+a*B*sin(c+d*x)*sqrt(sec(c+d*x))/(b*d*sqrt(a+b*cos(c+d*x)))-(a-b)*B*csc(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*b*d*sqrt(sec(c+d*x)))+B*csc(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b*d*sqrt(sec(c+d*x)))-(2*A*b-a*B)*csc(c+d*x)*EllipticPi(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),(a+b)/b,sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b^2*d*sqrt(sec(c+d*x)))],
[(A+B*cos(c+d*x))/(sec(c+d*x)^(3/2)*sqrt(a+b*cos(c+d*x))),x,8,1/2*B*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(b*d*sqrt(sec(c+d*x)))+1/4*(4*A*b-3*a*B)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))*sqrt(sec(c+d*x))/(b^2*d)-1/4*(a-b)*(4*A*b-3*a*B)*csc(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*b^2*d*sqrt(sec(c+d*x)))+1/4*(4*A*b-3*a*B+2*b*B)*csc(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b^2*d*sqrt(sec(c+d*x)))+1/4*(4*a*A*b-3*a^2*B-4*b^2*B)*csc(c+d*x)*EllipticPi(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),(a+b)/b,sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b^3*d*sqrt(sec(c+d*x)))],
[(A+B*cos(c+d*x))*sec(c+d*x)^(5/2)/(a+b*cos(c+d*x))^(3/2),x,6,2*b*(A*b-a*B)*sec(c+d*x)^(3/2)*sin(c+d*x)/(a*(a^2-b^2)*d*sqrt(a+b*cos(c+d*x)))+2/3*(a^2*A-4*A*b^2+3*a*b*B)*sec(c+d*x)^(3/2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(a^2*(a^2-b^2)*d)-2/3*(5*a^2*A*b-8*A*b^3-3*a^3*B+6*a*b^2*B)*csc(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^4*d*sqrt(a+b)*sqrt(sec(c+d*x)))+2/3*(a+2*b)*(4*A*b+a*(A-3*B))*csc(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^3*d*sqrt(a+b)*sqrt(sec(c+d*x)))],
[(A+B*cos(c+d*x))*sec(c+d*x)^(3/2)/(a+b*cos(c+d*x))^(3/2),x,5,2*b*(A*b-a*B)*sin(c+d*x)*sqrt(sec(c+d*x))/(a*(a^2-b^2)*d*sqrt(a+b*cos(c+d*x)))+2*(a^2*A-2*A*b^2+a*b*B)*csc(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^3*d*sqrt(a+b)*sqrt(sec(c+d*x)))-2*(2*A*b+a*(A-B))*csc(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^2*d*sqrt(a+b)*sqrt(sec(c+d*x)))],
[(A+B*cos(c+d*x))*sec(c+d*x)^(1/2)/(a+b*cos(c+d*x))^(3/2),x,5,-2*(A*b-a*B)*sin(c+d*x)*sqrt(sec(c+d*x))/((a^2-b^2)*d*sqrt(a+b*cos(c+d*x)))+2*(A*b-a*B)*csc(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^2*d*sqrt(a+b)*sqrt(sec(c+d*x)))+2*(A+B)*csc(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*d*sqrt(a+b)*sqrt(sec(c+d*x)))],
[(A+B*cos(c+d*x))/((a+b*cos(c+d*x))^(3/2)*sec(c+d*x)^(1/2)),x,7,2*a*(A*b-a*B)*sin(c+d*x)*sqrt(sec(c+d*x))/(b*(a^2-b^2)*d*sqrt(a+b*cos(c+d*x)))-2*(A*b-a*B)*csc(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*b*d*sqrt(a+b)*sqrt(sec(c+d*x)))+2*(A*b-a*B)*csc(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*b*d*sqrt(a+b)*sqrt(sec(c+d*x)))-2*B*csc(c+d*x)*EllipticPi(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),(a+b)/b,sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b^2*d*sqrt(sec(c+d*x)))],
[(A+B*cos(c+d*x))/((a+b*cos(c+d*x))^(3/2)*sec(c+d*x)^(3/2)),x,8,2*a*(A*b-a*B)*sin(c+d*x)/(b*(a^2-b^2)*d*sqrt(a+b*cos(c+d*x))*sqrt(sec(c+d*x)))-(2*a*A*b-3*a^2*B+b^2*B)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))*sqrt(sec(c+d*x))/(b^2*(a^2-b^2)*d)+(2*a*A*b-3*a^2*B+b^2*B)*csc(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*b^2*d*sqrt(a+b)*sqrt(sec(c+d*x)))-(2*A*b-(3*a+b)*B)*csc(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b^2*d*sqrt(a+b)*sqrt(sec(c+d*x)))-(2*A*b-3*a*B)*csc(c+d*x)*EllipticPi(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),(a+b)/b,sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b^3*d*sqrt(sec(c+d*x)))],
[(A+B*cos(c+d*x))*sec(c+d*x)^(5/2)/(a+b*cos(c+d*x))^(5/2),x,7,2/3*b*(A*b-a*B)*sec(c+d*x)^(3/2)*sin(c+d*x)/(a*(a^2-b^2)*d*(a+b*cos(c+d*x))^(3/2))+2/3*b*(10*a^2*A*b-6*A*b^3-7*a^3*B+3*a*b^2*B)*sec(c+d*x)^(3/2)*sin(c+d*x)/(a^2*(a^2-b^2)^2*d*sqrt(a+b*cos(c+d*x)))+2/3*(a^4*A-13*a^2*A*b^2+8*A*b^4+8*a^3*b*B-4*a*b^3*B)*sec(c+d*x)^(3/2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(a^3*(a^2-b^2)^2*d)-2/3*(8*a^4*A*b-28*a^2*A*b^3+16*A*b^5-3*a^5*B+15*a^3*b^2*B-8*a*b^4*B)*csc(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^5*(a-b)*(a+b)^(3/2)*d*sqrt(sec(c+d*x)))-2/3*(16*A*b^4-a^4*(A-3*B)+4*a*b^3*(3*A-2*B)-9*a^3*b*(A-B)-2*a^2*b^2*(8*A+3*B))*csc(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^4*(a^2-b^2)*d*sqrt(a+b)*sqrt(sec(c+d*x)))],
[(A+B*cos(c+d*x))*sec(c+d*x)^(3/2)/(a+b*cos(c+d*x))^(5/2),x,6,2/3*b*(A*b-a*B)*sin(c+d*x)*sqrt(sec(c+d*x))/(a*(a^2-b^2)*d*(a+b*cos(c+d*x))^(3/2))+2/3*b*(8*a^2*A*b-4*A*b^3-5*a^3*B+a*b^2*B)*sin(c+d*x)*sqrt(sec(c+d*x))/(a^2*(a^2-b^2)^2*d*sqrt(a+b*cos(c+d*x)))+2/3*(3*a^4*A-15*a^2*A*b^2+8*A*b^4+6*a^3*b*B-2*a*b^3*B)*csc(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^4*(a-b)*(a+b)^(3/2)*d*sqrt(sec(c+d*x)))+2/3*(8*A*b^3-3*a^3*(A-B)+2*a*b^2*(3*A-B)-3*a^2*b*(3*A+B))*csc(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^3*(a^2-b^2)*d*sqrt(a+b)*sqrt(sec(c+d*x)))],
[(A+B*cos(c+d*x))*sec(c+d*x)^(1/2)/(a+b*cos(c+d*x))^(5/2),x,6,2/3*b*(A*b-a*B)*sin(c+d*x)/(a*(a^2-b^2)*d*(a+b*cos(c+d*x))^(3/2)*sqrt(sec(c+d*x)))-2/3*(6*a^2*A*b-2*A*b^3-3*a^3*B-a*b^2*B)*sin(c+d*x)*sqrt(sec(c+d*x))/(a*(a^2-b^2)^2*d*sqrt(a+b*cos(c+d*x)))+2/3*(6*a^2*A*b-2*A*b^3-3*a^3*B-a*b^2*B)*csc(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^3*(a-b)*(a+b)^(3/2)*d*sqrt(sec(c+d*x)))-2/3*(2*A*b^2-3*a^2*(A+B)+a*b*(3*A+B))*csc(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^2*(a^2-b^2)*d*sqrt(a+b)*sqrt(sec(c+d*x)))],
[(A+B*cos(c+d*x))/((a+b*cos(c+d*x))^(5/2)*sec(c+d*x)^(1/2)),x,6,-2/3*(A*b-a*B)*sin(c+d*x)/((a^2-b^2)*d*(a+b*cos(c+d*x))^(3/2)*sqrt(sec(c+d*x)))+2/3*(3*a^2*A+A*b^2-4*a*b*B)*sin(c+d*x)*sqrt(sec(c+d*x))/((a^2-b^2)^2*d*sqrt(a+b*cos(c+d*x)))-2/3*(3*a^2*A+A*b^2-4*a*b*B)*csc(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^2*(a-b)*(a+b)^(3/2)*d*sqrt(sec(c+d*x)))+2/3*(a*(3*A+B)-b*(A+3*B))*csc(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*(a-b)*(a+b)^(3/2)*d*sqrt(sec(c+d*x)))],
[(A+B*cos(c+d*x))/((a+b*cos(c+d*x))^(5/2)*sec(c+d*x)^(3/2)),x,8,2/3*a*(A*b-a*B)*sin(c+d*x)/(b*(a^2-b^2)*d*(a+b*cos(c+d*x))^(3/2)*sqrt(sec(c+d*x)))-2/3*a*(4*A*b^3+3*a^3*B-7*a*b^2*B)*sin(c+d*x)*sqrt(sec(c+d*x))/(b^2*(a^2-b^2)^2*d*sqrt(a+b*cos(c+d*x)))+2/3*(4*A*b^3+3*a^3*B-7*a*b^2*B)*csc(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*(a-b)*b^2*(a+b)^(3/2)*d*sqrt(sec(c+d*x)))-2/3*(3*A*b^3+3*a^3*B+a^2*b*B-a*b^2*(A+6*B))*csc(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*(a-b)*b^2*(a+b)^(3/2)*d*sqrt(sec(c+d*x)))-2*B*csc(c+d*x)*EllipticPi(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),(a+b)/b,sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b^3*d*sqrt(sec(c+d*x)))],
[(A+B*cos(c+d*x))/((a+b*cos(c+d*x))^(5/2)*sec(c+d*x)^(5/2)),x,9,2/3*a*(A*b-a*B)*sin(c+d*x)/(b*(a^2-b^2)*d*(a+b*cos(c+d*x))^(3/2)*sec(c+d*x)^(3/2))+2/3*a*(2*a^2*A*b-6*A*b^3-5*a^3*B+9*a*b^2*B)*sin(c+d*x)/(b^2*(a^2-b^2)^2*d*sqrt(a+b*cos(c+d*x))*sqrt(sec(c+d*x)))-1/3*(6*a^3*A*b-14*a*A*b^3-15*a^4*B+26*a^2*b^2*B-3*b^4*B)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))*sqrt(sec(c+d*x))/(b^3*(a^2-b^2)^2*d)+1/3*(6*a^3*A*b-14*a*A*b^3-15*a^4*B+26*a^2*b^2*B-3*b^4*B)*csc(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*(a-b)*b^3*(a+b)^(3/2)*d*sqrt(sec(c+d*x)))+1/3*(3*b^3*(4*A-B)+15*a^3*B-a*b^2*(2*A+21*B)-a^2*(6*A*b-5*b*B))*csc(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/((a-b)*b^3*(a+b)^(3/2)*d*sqrt(sec(c+d*x)))-(2*A*b-5*a*B)*csc(c+d*x)*EllipticPi(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),(a+b)/b,sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b^4*d*sqrt(sec(c+d*x)))],
[(a*B+b*B*cos(c+d*x))*sec(c+d*x)^(3/2)/(a+b*cos(c+d*x))^(3/2),x,5,2*(a-b)*B*csc(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^2*d*sqrt(sec(c+d*x)))-2*B*csc(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*d*sqrt(sec(c+d*x)))],
[(a*B+b*B*cos(c+d*x))*sec(c+d*x)^(1/2)/(a+b*cos(c+d*x))^(3/2),x,3,2*B*csc(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*d*sqrt(sec(c+d*x)))],
[(a*B+b*B*cos(c+d*x))/((a+b*cos(c+d*x))^(3/2)*sec(c+d*x)^(1/2)),x,3,-2*B*csc(c+d*x)*EllipticPi(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),(a+b)/b,sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b*d*sqrt(sec(c+d*x)))],
[(a*B+b*B*cos(c+d*x))/((a+b*cos(c+d*x))^(3/2)*sec(c+d*x)^(3/2)),x,10,B*sin(c+d*x)/(d*sqrt(a+b*cos(c+d*x))*sqrt(sec(c+d*x)))+a*B*sin(c+d*x)*sqrt(sec(c+d*x))/(b*d*sqrt(a+b*cos(c+d*x)))-(a-b)*B*csc(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*b*d*sqrt(sec(c+d*x)))+B*csc(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b*d*sqrt(sec(c+d*x)))+a*B*csc(c+d*x)*EllipticPi(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),(a+b)/b,sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b^2*d*sqrt(sec(c+d*x)))],

# Integrands of the form (c Sec[e+f x])^m (a+b Cos[e+f x])^n (A+B Cos[e+f x]) with m and/or n symbolic
[(a+b*cos(e+f*x))^n*(A+B*cos(e+f*x))*(c*sec(e+f*x))^m,x,1,(c*cos(e+f*x))^m*(c*sec(e+f*x))^m*Unintegrable((a+b*cos(e+f*x))^n*(A+B*cos(e+f*x))/(c*cos(e+f*x))^m,x)],
[(a+b*cos(e+f*x))^4*(A+B*cos(e+f*x))*(c*sec(e+f*x))^m,x,10,-c^6*(4*a^3*A*b*(15-8*m+m^2)+a^4*B*(15-8*m+m^2)+4*a*A*b^3*(10-7*m+m^2)+6*a^2*b^2*B*(10-7*m+m^2)+b^4*B*(8-6*m+m^2))*hypergeom([1/2,1/2*(6-m)],[1/2*(8-m)],cos(e+f*x)^2)*(c*sec(e+f*x))^(-6+m)*sin(e+f*x)/(f*(48-44*m+12*m^2-m^3)*sqrt(sin(e+f*x)^2))-c^5*(a^4*A*(8-6*m+m^2)+6*a^2*A*b^2*(4-5*m+m^2)+4*a^3*b*B*(4-5*m+m^2)+A*b^4*(3-4*m+m^2)+4*a*b^3*B*(3-4*m+m^2))*hypergeom([1/2,1/2*(5-m)],[1/2*(7-m)],cos(e+f*x)^2)*(c*sec(e+f*x))^(-5+m)*sin(e+f*x)/(f*(1-m)*(15-8*m+m^2)*sqrt(sin(e+f*x)^2))-a*c^5*(4*a^2*A*b*(3-4*m+m^2)+a^3*B*(3-4*m+m^2)+2*A*b^3*(4-2*m+m^2)+a*b^2*B*(8-13*m+5*m^2))*(c*sec(e+f*x))^(-5+m)*tan(e+f*x)/(f*(1-m)*(8-6*m+m^2))-a^2*c^5*(2*a*b*B*(1-m)^2+a^2*A*(2-m)^2+A*b^2*(6-m+m^2))*sec(e+f*x)*(c*sec(e+f*x))^(-5+m)*tan(e+f*x)/(f*(6-11*m+6*m^2-m^3))-a*c^5*(a*B*(1-m)-A*b*(2+m))*(c*sec(e+f*x))^(-5+m)*(b+a*sec(e+f*x))^2*tan(e+f*x)/(f*(2-3*m+m^2))-a*A*c^5*(c*sec(e+f*x))^(-5+m)*(b+a*sec(e+f*x))^3*tan(e+f*x)/(f*(1-m))],
[(a+b*cos(e+f*x))^3*(A+B*cos(e+f*x))*(c*sec(e+f*x))^m,x,9,-c^5*(a^3*A*(8-6*m+m^2)+3*a*A*b^2*(4-5*m+m^2)+3*a^2*b*B*(4-5*m+m^2)+b^3*B*(3-4*m+m^2))*hypergeom([1/2,1/2*(5-m)],[1/2*(7-m)],cos(e+f*x)^2)*(c*sec(e+f*x))^(-5+m)*sin(e+f*x)/(f*(1-m)*(15-8*m+m^2)*sqrt(sin(e+f*x)^2))-c^4*(A*b^3*(2-m)+3*a*b^2*B*(2-m)+3*a^2*A*b*(3-m)+a^3*B*(3-m))*hypergeom([1/2,1/2*(4-m)],[1/2*(6-m)],cos(e+f*x)^2)*(c*sec(e+f*x))^(-4+m)*sin(e+f*x)/(f*(8-6*m+m^2)*sqrt(sin(e+f*x)^2))-a*c^4*(3*a*b*B*(1-m)+a^2*A*(2-m)-2*A*b^2*m)*(c*sec(e+f*x))^(-4+m)*tan(e+f*x)/(f*(3-4*m+m^2))-a^2*c^4*(a*B*(1-m)-A*b*(1+m))*sec(e+f*x)*(c*sec(e+f*x))^(-4+m)*tan(e+f*x)/(f*(2-3*m+m^2))-a*A*c^4*(c*sec(e+f*x))^(-4+m)*(b+a*sec(e+f*x))^2*tan(e+f*x)/(f*(1-m))],
[(a+b*cos(e+f*x))^2*(A+B*cos(e+f*x))*(c*sec(e+f*x))^m,x,8,-c^4*(b^2*B*(2-m)+2*a*A*b*(3-m)+a^2*B*(3-m))*hypergeom([1/2,1/2*(4-m)],[1/2*(6-m)],cos(e+f*x)^2)*(c*sec(e+f*x))^(-4+m)*sin(e+f*x)/(f*(8-6*m+m^2)*sqrt(sin(e+f*x)^2))-c^3*(A*b^2*(1-m)+2*a*b*B*(1-m)+a^2*A*(2-m))*hypergeom([1/2,1/2*(3-m)],[1/2*(5-m)],cos(e+f*x)^2)*(c*sec(e+f*x))^(-3+m)*sin(e+f*x)/(f*(3-4*m+m^2)*sqrt(sin(e+f*x)^2))-a*c^3*(a*B*(1-m)-A*b*m)*(c*sec(e+f*x))^(-3+m)*tan(e+f*x)/(f*(2-3*m+m^2))-a*A*c^3*(c*sec(e+f*x))^(-3+m)*(b+a*sec(e+f*x))*tan(e+f*x)/(f*(1-m))],
[(a+b*cos(e+f*x))*(A+B*cos(e+f*x))*(c*sec(e+f*x))^m,x,7,-c^3*(b*B*(1-m)+a*A*(2-m))*hypergeom([1/2,1/2*(3-m)],[1/2*(5-m)],cos(e+f*x)^2)*(c*sec(e+f*x))^(-3+m)*sin(e+f*x)/(f*(3-4*m+m^2)*sqrt(sin(e+f*x)^2))-(A*b+a*B)*c^2*hypergeom([1/2,1/2*(2-m)],[1/2*(4-m)],cos(e+f*x)^2)*(c*sec(e+f*x))^(-2+m)*sin(e+f*x)/(f*(2-m)*sqrt(sin(e+f*x)^2))-a*A*c^2*(c*sec(e+f*x))^(-2+m)*tan(e+f*x)/(f*(1-m))],
[(A+B*cos(e+f*x))*(c*sec(e+f*x))^m/(a+b*cos(e+f*x)),x,10,-(A*b-a*B)*AppellF1(1/2,1/2*m,1,3/2,sin(e+f*x)^2,-b^2*sin(e+f*x)^2/(a^2-b^2))*cos(e+f*x)*(cos(e+f*x)^2)^(1/2*m)*(c*sec(e+f*x))^(1+m)*sin(e+f*x)/((a^2-b^2)*c*f)+a*(A*b-a*B)*AppellF1(1/2,1/2*(1+m),1,3/2,sin(e+f*x)^2,-b^2*sin(e+f*x)^2/(a^2-b^2))*(cos(e+f*x)^2)^(1/2*(1+m))*(c*sec(e+f*x))^(1+m)*sin(e+f*x)/(b*(a^2-b^2)*c*f)-B*c*hypergeom([1/2,1/2*(1-m)],[1/2*(3-m)],cos(e+f*x)^2)*(c*sec(e+f*x))^(-1+m)*sin(e+f*x)/(b*f*(1-m)*sqrt(sin(e+f*x)^2))],
[(a+b*cos(e+f*x))^(3/2)*(A+B*cos(e+f*x))*(c*sec(e+f*x))^m,x,2,2*b*B*cos(e+f*x)*(c*sec(e+f*x))^m*sin(e+f*x)*sqrt(a+b*cos(e+f*x))/(f*(5-2*m))+2*(c*cos(e+f*x))^m*(c*sec(e+f*x))^m*Unintegrable((1/2*a*c*(2*b*B*(1-m)+2*a*A*(5/2-m))+1/2*c*(b^2*B*(3-2*m)+a*(2*A*b+a*B)*(5-2*m))*cos(e+f*x)+1/2*b*c*(A*b*(5-2*m)+2*a*B*(3-m))*cos(e+f*x)^2)/((c*cos(e+f*x))^m*sqrt(a+b*cos(e+f*x))),x)/(c*(5-2*m))],
[(a+b*cos(e+f*x))^(1/2)*(A+B*cos(e+f*x))*(c*sec(e+f*x))^m,x,1,(c*cos(e+f*x))^m*(c*sec(e+f*x))^m*Unintegrable((A+B*cos(e+f*x))*sqrt(a+b*cos(e+f*x))/(c*cos(e+f*x))^m,x)],
[(A+B*cos(e+f*x))*(c*sec(e+f*x))^m/(a+b*cos(e+f*x))^(1/2),x,1,(c*cos(e+f*x))^m*(c*sec(e+f*x))^m*Unintegrable((A+B*cos(e+f*x))/((c*cos(e+f*x))^m*sqrt(a+b*cos(e+f*x))),x)],
[(A+B*cos(e+f*x))*(c*sec(e+f*x))^m/(a+b*cos(e+f*x))^(3/2),x,2,2*b*(A*b-a*B)*cos(e+f*x)*(c*sec(e+f*x))^m*sin(e+f*x)/(a*(a^2-b^2)*f*sqrt(a+b*cos(e+f*x)))+2*(c*cos(e+f*x))^m*(c*sec(e+f*x))^m*Unintegrable((1/2*c*(a^2*A+A*b^2*(1-2*m)-2*a*b*B*(1-m))-1/2*a*(A*b-a*B)*c*cos(e+f*x)-1/2*b*(A*b-a*B)*c*(3-2*m)*cos(e+f*x)^2)/((c*cos(e+f*x))^m*sqrt(a+b*cos(e+f*x))),x)/(a*(a^2-b^2)*c)]]:

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

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

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

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