# Maple integration test file: "4 Trig functions\4.1 Sine\4.1.1.2 (g cos)^p (a+b sin)^m.txt"

lst:=[

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

# Integrands of the form (g Cos[e+f x])^p (a+b Sin[e+f x])^m when a^2-b^2=0

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

# m>0
[cos(c+d*x)^7*(a+a*sin(c+d*x)),x,3,8/5*(a+a*sin(c+d*x))^5/(a^4*d)-2*(a+a*sin(c+d*x))^6/(a^5*d)+6/7*(a+a*sin(c+d*x))^7/(a^6*d)-1/8*(a+a*sin(c+d*x))^8/(a^7*d)],
[cos(c+d*x)^6*(a+a*sin(c+d*x)),x,5,5/16*a*x-1/7*a*cos(c+d*x)^7/d+5/16*a*cos(c+d*x)*sin(c+d*x)/d+5/24*a*cos(c+d*x)^3*sin(c+d*x)/d+1/6*a*cos(c+d*x)^5*sin(c+d*x)/d],
[cos(c+d*x)^5*(a+a*sin(c+d*x)),x,3,(a+a*sin(c+d*x))^4/(a^3*d)-4/5*(a+a*sin(c+d*x))^5/(a^4*d)+1/6*(a+a*sin(c+d*x))^6/(a^5*d)],
[cos(c+d*x)^4*(a+a*sin(c+d*x)),x,4,3/8*a*x-1/5*a*cos(c+d*x)^5/d+3/8*a*cos(c+d*x)*sin(c+d*x)/d+1/4*a*cos(c+d*x)^3*sin(c+d*x)/d],
[cos(c+d*x)^3*(a+a*sin(c+d*x)),x,3,2/3*(a+a*sin(c+d*x))^3/(a^2*d)-1/4*(a+a*sin(c+d*x))^4/(a^3*d)],
[cos(c+d*x)^2*(a+a*sin(c+d*x)),x,3,1/2*a*x-1/3*a*cos(c+d*x)^3/d+1/2*a*cos(c+d*x)*sin(c+d*x)/d],
[cos(c+d*x)*(a+a*sin(c+d*x)),x,2,1/2*(a+a*sin(c+d*x))^2/(a*d),a*sin(c+d*x)/d+1/2*a*sin(c+d*x)^2/d],
[sec(c+d*x)*(a+a*sin(c+d*x)),x,2,-a*log(1-sin(c+d*x))/d],
[sec(c+d*x)^2*(a+a*sin(c+d*x)),x,3,a*sec(c+d*x)/d+a*tan(c+d*x)/d],
[sec(c+d*x)^3*(a+a*sin(c+d*x)),x,4,1/2*a*arctanh(sin(c+d*x))/d+1/2*a^2/(d*(a-a*sin(c+d*x)))],
[sec(c+d*x)^4*(a+a*sin(c+d*x)),x,3,1/3*a*sec(c+d*x)^3/d+a*tan(c+d*x)/d+1/3*a*tan(c+d*x)^3/d],
[sec(c+d*x)^5*(a+a*sin(c+d*x)),x,4,3/8*a*arctanh(sin(c+d*x))/d+1/8*a^3/(d*(a-a*sin(c+d*x))^2)+1/4*a^2/(d*(a-a*sin(c+d*x)))-1/8*a^2/(d*(a+a*sin(c+d*x)))],
[cos(c+d*x)^6*(a+a*sin(c+d*x))^2,x,6,45/128*a^2*x-9/56*a^2*cos(c+d*x)^7/d+45/128*a^2*cos(c+d*x)*sin(c+d*x)/d+15/64*a^2*cos(c+d*x)^3*sin(c+d*x)/d+3/16*a^2*cos(c+d*x)^5*sin(c+d*x)/d-1/8*cos(c+d*x)^7*(a^2+a^2*sin(c+d*x))/d],
[cos(c+d*x)^5*(a+a*sin(c+d*x))^2,x,3,4/5*(a+a*sin(c+d*x))^5/(a^3*d)-2/3*(a+a*sin(c+d*x))^6/(a^4*d)+1/7*(a+a*sin(c+d*x))^7/(a^5*d)],
[cos(c+d*x)^4*(a+a*sin(c+d*x))^2,x,5,7/16*a^2*x-7/30*a^2*cos(c+d*x)^5/d+7/16*a^2*cos(c+d*x)*sin(c+d*x)/d+7/24*a^2*cos(c+d*x)^3*sin(c+d*x)/d-1/6*cos(c+d*x)^5*(a^2+a^2*sin(c+d*x))/d],
[cos(c+d*x)^3*(a+a*sin(c+d*x))^2,x,3,1/2*(a+a*sin(c+d*x))^4/(a^2*d)-1/5*(a+a*sin(c+d*x))^5/(a^3*d)],
[cos(c+d*x)^2*(a+a*sin(c+d*x))^2,x,4,5/8*a^2*x-5/12*a^2*cos(c+d*x)^3/d+5/8*a^2*cos(c+d*x)*sin(c+d*x)/d-1/4*cos(c+d*x)^3*(a^2+a^2*sin(c+d*x))/d],
[cos(c+d*x)*(a+a*sin(c+d*x))^2,x,2,1/3*(a+a*sin(c+d*x))^3/(a*d)],
[sec(c+d*x)*(a+a*sin(c+d*x))^2,x,3,-2*a^2*log(1-sin(c+d*x))/d-a^2*sin(c+d*x)/d],
[sec(c+d*x)^2*(a+a*sin(c+d*x))^2,x,3,-a^2*x+2*a^4*cos(c+d*x)/(d*(a^2-a^2*sin(c+d*x)))],
[sec(c+d*x)^3*(a+a*sin(c+d*x))^2,x,2,a^3/(d*(a-a*sin(c+d*x)))],
[sec(c+d*x)^4*(a+a*sin(c+d*x))^2,x,3,1/3*a^4*cos(c+d*x)/(d*(a-a*sin(c+d*x))^2)+1/3*a^4*cos(c+d*x)/(d*(a^2-a^2*sin(c+d*x)))],
[sec(c+d*x)^5*(a+a*sin(c+d*x))^2,x,4,1/4*a^2*arctanh(sin(c+d*x))/d+1/4*a^4/(d*(a-a*sin(c+d*x))^2)+1/4*a^3/(d*(a-a*sin(c+d*x)))],
[sec(c+d*x)^6*(a+a*sin(c+d*x))^2,x,3,2/5*sec(c+d*x)^5*(a^2+a^2*sin(c+d*x))/d+3/5*a^2*tan(c+d*x)/d+1/5*a^2*tan(c+d*x)^3/d],
[sec(c+d*x)^7*(a+a*sin(c+d*x))^2,x,4,1/4*a^2*arctanh(sin(c+d*x))/d+1/12*a^5/(d*(a-a*sin(c+d*x))^3)+1/8*a^4/(d*(a-a*sin(c+d*x))^2)+3/16*a^3/(d*(a-a*sin(c+d*x)))-1/16*a^3/(d*(a+a*sin(c+d*x)))],
[sec(c+d*x)^8*(a+a*sin(c+d*x))^2,x,3,2/7*sec(c+d*x)^7*(a^2+a^2*sin(c+d*x))/d+5/7*a^2*tan(c+d*x)/d+10/21*a^2*tan(c+d*x)^3/d+1/7*a^2*tan(c+d*x)^5/d],
[cos(c+d*x)^6*(a+a*sin(c+d*x))^3,x,7,55/128*a^3*x-11/56*a^3*cos(c+d*x)^7/d+55/128*a^3*cos(c+d*x)*sin(c+d*x)/d+55/192*a^3*cos(c+d*x)^3*sin(c+d*x)/d+11/48*a^3*cos(c+d*x)^5*sin(c+d*x)/d-1/9*a*cos(c+d*x)^7*(a+a*sin(c+d*x))^2/d-11/72*cos(c+d*x)^7*(a^3+a^3*sin(c+d*x))/d],
[cos(c+d*x)^5*(a+a*sin(c+d*x))^3,x,3,2/3*(a+a*sin(c+d*x))^6/(a^3*d)-4/7*(a+a*sin(c+d*x))^7/(a^4*d)+1/8*(a+a*sin(c+d*x))^8/(a^5*d)],
[cos(c+d*x)^4*(a+a*sin(c+d*x))^3,x,6,9/16*a^3*x-3/10*a^3*cos(c+d*x)^5/d+9/16*a^3*cos(c+d*x)*sin(c+d*x)/d+3/8*a^3*cos(c+d*x)^3*sin(c+d*x)/d-1/7*a*cos(c+d*x)^5*(a+a*sin(c+d*x))^2/d-3/14*cos(c+d*x)^5*(a^3+a^3*sin(c+d*x))/d],
[cos(c+d*x)^3*(a+a*sin(c+d*x))^3,x,3,2/5*(a+a*sin(c+d*x))^5/(a^2*d)-1/6*(a+a*sin(c+d*x))^6/(a^3*d)],
[cos(c+d*x)^2*(a+a*sin(c+d*x))^3,x,5,7/8*a^3*x-7/12*a^3*cos(c+d*x)^3/d+7/8*a^3*cos(c+d*x)*sin(c+d*x)/d-1/5*a*cos(c+d*x)^3*(a+a*sin(c+d*x))^2/d-7/20*cos(c+d*x)^3*(a^3+a^3*sin(c+d*x))/d],
[cos(c+d*x)*(a+a*sin(c+d*x))^3,x,2,1/4*(a+a*sin(c+d*x))^4/(a*d)],
[sec(c+d*x)*(a+a*sin(c+d*x))^3,x,3,-4*a^3*log(1-sin(c+d*x))/d-3*a^3*sin(c+d*x)/d-1/2*a^3*sin(c+d*x)^2/d],
[sec(c+d*x)^2*(a+a*sin(c+d*x))^3,x,4,-3*a^3*x+3*a^3*cos(c+d*x)/d+2*a^5*cos(c+d*x)^3/(d*(a-a*sin(c+d*x))^2)],
[sec(c+d*x)^3*(a+a*sin(c+d*x))^3,x,3,a^3*log(1-sin(c+d*x))/d+2*a^4/(d*(a-a*sin(c+d*x)))],
[sec(c+d*x)^4*(a+a*sin(c+d*x))^3,x,2,1/3*a^6*cos(c+d*x)^3/(d*(a-a*sin(c+d*x))^3)],
[sec(c+d*x)^5*(a+a*sin(c+d*x))^3,x,2,1/2*a^5/(d*(a-a*sin(c+d*x))^2)],
[sec(c+d*x)^6*(a+a*sin(c+d*x))^3,x,4,1/5*a^6*cos(c+d*x)/(d*(a-a*sin(c+d*x))^3)+2/15*a^5*cos(c+d*x)/(d*(a-a*sin(c+d*x))^2)+2/15*a^6*cos(c+d*x)/(d*(a^3-a^3*sin(c+d*x)))],
[sec(c+d*x)^7*(a+a*sin(c+d*x))^3,x,4,1/8*a^3*arctanh(sin(c+d*x))/d+1/6*a^6/(d*(a-a*sin(c+d*x))^3)+1/8*a^5/(d*(a-a*sin(c+d*x))^2)+1/8*a^4/(d*(a-a*sin(c+d*x)))],
[sec(c+d*x)^8*(a+a*sin(c+d*x))^3,x,4,3/35*a^3*sec(c+d*x)^5/d+2/7*a*sec(c+d*x)^7*(a+a*sin(c+d*x))^2/d+3/7*a^3*tan(c+d*x)/d+2/7*a^3*tan(c+d*x)^3/d+3/35*a^3*tan(c+d*x)^5/d],
[cos(c+d*x)^5*(a+a*sin(c+d*x))^8,x,3,4/11*(a+a*sin(c+d*x))^11/(a^3*d)-1/3*(a+a*sin(c+d*x))^12/(a^4*d)+1/13*(a+a*sin(c+d*x))^13/(a^5*d)],
[cos(c+d*x)^4*(a+a*sin(c+d*x))^8,x,11,4199/1024*a^8*x-4199/1920*a^8*cos(c+d*x)^5/d+4199/1024*a^8*cos(c+d*x)*sin(c+d*x)/d+4199/1536*a^8*cos(c+d*x)^3*sin(c+d*x)/d-323/1320*a^3*cos(c+d*x)^5*(a+a*sin(c+d*x))^5/d-19/132*a^2*cos(c+d*x)^5*(a+a*sin(c+d*x))^6/d-1/12*a*cos(c+d*x)^5*(a+a*sin(c+d*x))^7/d-4199/6336*a^2*cos(c+d*x)^5*(a^2+a^2*sin(c+d*x))^3/d-323/792*cos(c+d*x)^5*(a^2+a^2*sin(c+d*x))^4/d-4199/4032*cos(c+d*x)^5*(a^4+a^4*sin(c+d*x))^2/d-4199/2688*cos(c+d*x)^5*(a^8+a^8*sin(c+d*x))/d],
[cos(c+d*x)^3*(a+a*sin(c+d*x))^8,x,3,1/5*(a+a*sin(c+d*x))^10/(a^2*d)-1/11*(a+a*sin(c+d*x))^11/(a^3*d)],
[cos(c+d*x)^2*(a+a*sin(c+d*x))^8,x,10,2431/256*a^8*x-2431/384*a^8*cos(c+d*x)^3/d+2431/256*a^8*cos(c+d*x)*sin(c+d*x)/d-17/48*a^3*cos(c+d*x)^3*(a+a*sin(c+d*x))^5/d-17/90*a^2*cos(c+d*x)^3*(a+a*sin(c+d*x))^6/d-1/10*a*cos(c+d*x)^3*(a+a*sin(c+d*x))^7/d-2431/2016*a^2*cos(c+d*x)^3*(a^2+a^2*sin(c+d*x))^3/d-221/336*cos(c+d*x)^3*(a^2+a^2*sin(c+d*x))^4/d-2431/1120*cos(c+d*x)^3*(a^4+a^4*sin(c+d*x))^2/d-2431/640*cos(c+d*x)^3*(a^8+a^8*sin(c+d*x))/d],
[cos(c+d*x)*(a+a*sin(c+d*x))^8,x,2,1/9*(a+a*sin(c+d*x))^9/(a*d)],
[sec(c+d*x)*(a+a*sin(c+d*x))^8,x,3,-128*a^8*log(1-sin(c+d*x))/d-64*a^8*sin(c+d*x)/d-16/3*a^5*(a+a*sin(c+d*x))^3/d-4/5*a^3*(a+a*sin(c+d*x))^5/d-1/3*a^2*(a+a*sin(c+d*x))^6/d-1/7*a*(a+a*sin(c+d*x))^7/d-2*(a^2+a^2*sin(c+d*x))^4/d-16*(a^4+a^4*sin(c+d*x))^2/d],
[sec(c+d*x)^2*(a+a*sin(c+d*x))^8,x,9,-3003/16*a^8*x+1001/10*a^8*cos(c+d*x)^5/d-3003/16*a^8*cos(c+d*x)*sin(c+d*x)/d-1001/8*a^8*cos(c+d*x)^3*sin(c+d*x)/d+2*a^15*cos(c+d*x)^13/(d*(a-a*sin(c+d*x))^7)+26*a^13*cos(c+d*x)^11/(d*(a-a*sin(c+d*x))^5)+286/3*a^14*cos(c+d*x)^9/(d*(a^2-a^2*sin(c+d*x))^3)+143/2*a^16*cos(c+d*x)^7/(d*(a^8-a^8*sin(c+d*x)))],
[sec(c+d*x)^3*(a+a*sin(c+d*x))^8,x,3,192*a^8*log(1-sin(c+d*x))/d+129*a^8*sin(c+d*x)/d+36*a^8*sin(c+d*x)^2/d+10*a^8*sin(c+d*x)^3/d+2*a^8*sin(c+d*x)^4/d+1/5*a^8*sin(c+d*x)^5/d+64*a^9/(d*(a-a*sin(c+d*x)))],
[sec(c+d*x)^4*(a+a*sin(c+d*x))^8,x,8,1155/8*a^8*x-385/4*a^8*cos(c+d*x)^3/d+1155/8*a^8*cos(c+d*x)*sin(c+d*x)/d+2/3*a^15*cos(c+d*x)^11/(d*(a-a*sin(c+d*x))^7)-22/3*a^13*cos(c+d*x)^9/(d*(a-a*sin(c+d*x))^5)-66*a^14*cos(c+d*x)^7/(d*(a^2-a^2*sin(c+d*x))^3)-231/4*a^16*cos(c+d*x)^5/(d*(a^8-a^8*sin(c+d*x)))],
[sec(c+d*x)^5*(a+a*sin(c+d*x))^8,x,3,-80*a^8*log(1-sin(c+d*x))/d-31*a^8*sin(c+d*x)/d-4*a^8*sin(c+d*x)^2/d-1/3*a^8*sin(c+d*x)^3/d+16*a^10/(d*(a-a*sin(c+d*x))^2)-80*a^9/(d*(a-a*sin(c+d*x)))],

# m<0
[cos(c+d*x)^6/(a+a*sin(c+d*x)),x,4,3/8*x/a+1/5*cos(c+d*x)^5/(a*d)+3/8*cos(c+d*x)*sin(c+d*x)/(a*d)+1/4*cos(c+d*x)^3*sin(c+d*x)/(a*d)],
[cos(c+d*x)^5/(a+a*sin(c+d*x)),x,3,-2/3*(a-a*sin(c+d*x))^3/(a^4*d)+1/4*(a-a*sin(c+d*x))^4/(a^5*d)],
[cos(c+d*x)^4/(a+a*sin(c+d*x)),x,3,1/2*x/a+1/3*cos(c+d*x)^3/(a*d)+1/2*cos(c+d*x)*sin(c+d*x)/(a*d)],
[cos(c+d*x)^3/(a+a*sin(c+d*x)),x,2,sin(c+d*x)/(a*d)-1/2*sin(c+d*x)^2/(a*d)],
[cos(c+d*x)^2/(a+a*sin(c+d*x)),x,2,x/a+cos(c+d*x)/(a*d)],
[cos(c+d*x)/(a+a*sin(c+d*x)),x,2,log(1+sin(c+d*x))/(a*d)],
[sec(c+d*x)/(a+a*sin(c+d*x)),x,4,1/2*arctanh(sin(c+d*x))/(a*d)+(-1/2)/(d*(a+a*sin(c+d*x)))],
[sec(c+d*x)^2/(a+a*sin(c+d*x)),x,3,-1/3*sec(c+d*x)/(d*(a+a*sin(c+d*x)))+2/3*tan(c+d*x)/(a*d)],
[sec(c+d*x)^3/(a+a*sin(c+d*x)),x,4,3/8*arctanh(sin(c+d*x))/(a*d)+1/8/(d*(a-a*sin(c+d*x)))-1/8*a/(d*(a+a*sin(c+d*x))^2)+(-1/4)/(d*(a+a*sin(c+d*x)))],
[sec(c+d*x)^4/(a+a*sin(c+d*x)),x,3,-1/5*sec(c+d*x)^3/(d*(a+a*sin(c+d*x)))+4/5*tan(c+d*x)/(a*d)+4/15*tan(c+d*x)^3/(a*d)],
[sec(c+d*x)^5/(a+a*sin(c+d*x)),x,4,5/16*arctanh(sin(c+d*x))/(a*d)+1/32*a/(d*(a-a*sin(c+d*x))^2)+1/8/(d*(a-a*sin(c+d*x)))-1/24*a^2/(d*(a+a*sin(c+d*x))^3)-3/32*a/(d*(a+a*sin(c+d*x))^2)+(-3/16)/(d*(a+a*sin(c+d*x)))],
[cos(c+d*x)^8/(a+a*sin(c+d*x))^2,x,5,7/16*x/a^2+7/30*cos(c+d*x)^5/(a^2*d)+7/16*cos(c+d*x)*sin(c+d*x)/(a^2*d)+7/24*cos(c+d*x)^3*sin(c+d*x)/(a^2*d)+1/6*cos(c+d*x)^7/(d*(a^2+a^2*sin(c+d*x)))],
[cos(c+d*x)^7/(a+a*sin(c+d*x))^2,x,3,-1/2*(a-a*sin(c+d*x))^4/(a^6*d)+1/5*(a-a*sin(c+d*x))^5/(a^7*d)],
[cos(c+d*x)^6/(a+a*sin(c+d*x))^2,x,4,5/8*x/a^2+5/12*cos(c+d*x)^3/(a^2*d)+5/8*cos(c+d*x)*sin(c+d*x)/(a^2*d)+1/4*cos(c+d*x)^5/(d*(a^2+a^2*sin(c+d*x)))],
[cos(c+d*x)^5/(a+a*sin(c+d*x))^2,x,2,-1/3*(a-a*sin(c+d*x))^3/(a^5*d)],
[cos(c+d*x)^4/(a+a*sin(c+d*x))^2,x,3,3/2*x/a^2+3/2*cos(c+d*x)/(a^2*d)+1/2*cos(c+d*x)^3/(d*(a^2+a^2*sin(c+d*x)))],
[cos(c+d*x)^3/(a+a*sin(c+d*x))^2,x,3,2*log(1+sin(c+d*x))/(a^2*d)-sin(c+d*x)/(a^2*d)],
[cos(c+d*x)^2/(a+a*sin(c+d*x))^2,x,2,-x/a^2-2*cos(c+d*x)/(d*(a^2+a^2*sin(c+d*x)))],
[cos(c+d*x)/(a+a*sin(c+d*x))^2,x,2,(-1)/(d*(a^2+a^2*sin(c+d*x)))],
[sec(c+d*x)/(a+a*sin(c+d*x))^2,x,4,1/4*arctanh(sin(c+d*x))/(a^2*d)+(-1/4)/(d*(a+a*sin(c+d*x))^2)+(-1/4)/(d*(a^2+a^2*sin(c+d*x)))],
[sec(c+d*x)^2/(a+a*sin(c+d*x))^2,x,4,-1/5*sec(c+d*x)/(d*(a+a*sin(c+d*x))^2)-1/5*sec(c+d*x)/(d*(a^2+a^2*sin(c+d*x)))+2/5*tan(c+d*x)/(a^2*d)],
[sec(c+d*x)^3/(a+a*sin(c+d*x))^2,x,4,1/4*arctanh(sin(c+d*x))/(a^2*d)-1/12*a/(d*(a+a*sin(c+d*x))^3)+(-1/8)/(d*(a+a*sin(c+d*x))^2)+1/16/(d*(a^2-a^2*sin(c+d*x)))+(-3/16)/(d*(a^2+a^2*sin(c+d*x)))],
[sec(c+d*x)^4/(a+a*sin(c+d*x))^2,x,4,-1/7*sec(c+d*x)^3/(d*(a+a*sin(c+d*x))^2)-1/7*sec(c+d*x)^3/(d*(a^2+a^2*sin(c+d*x)))+4/7*tan(c+d*x)/(a^2*d)+4/21*tan(c+d*x)^3/(a^2*d)],
[sec(c+d*x)^5/(a+a*sin(c+d*x))^2,x,4,15/64*arctanh(sin(c+d*x))/(a^2*d)+1/64/(d*(a-a*sin(c+d*x))^2)-1/32*a^2/(d*(a+a*sin(c+d*x))^4)-1/16*a/(d*(a+a*sin(c+d*x))^3)+(-3/32)/(d*(a+a*sin(c+d*x))^2)+5/64/(d*(a^2-a^2*sin(c+d*x)))+(-5/32)/(d*(a^2+a^2*sin(c+d*x)))],
[cos(c+d*x)^8/(a+a*sin(c+d*x))^3,x,5,7/8*x/a^3+7/15*cos(c+d*x)^5/(a^3*d)+7/8*cos(c+d*x)*sin(c+d*x)/(a^3*d)+7/12*cos(c+d*x)^3*sin(c+d*x)/(a^3*d)+2/3*cos(c+d*x)^7/(a*d*(a+a*sin(c+d*x))^2)],
[cos(c+d*x)^7/(a+a*sin(c+d*x))^3,x,2,-1/4*(a-a*sin(c+d*x))^4/(a^7*d)],
[cos(c+d*x)^6/(a+a*sin(c+d*x))^3,x,4,5/2*x/a^3+5/3*cos(c+d*x)^3/(a^3*d)+5/2*cos(c+d*x)*sin(c+d*x)/(a^3*d)+2*cos(c+d*x)^5/(a*d*(a+a*sin(c+d*x))^2)],
[cos(c+d*x)^5/(a+a*sin(c+d*x))^3,x,3,4*log(1+sin(c+d*x))/(a^3*d)-3*sin(c+d*x)/(a^3*d)+1/2*sin(c+d*x)^2/(a^3*d)],
[cos(c+d*x)^4/(a+a*sin(c+d*x))^3,x,3,-3*x/a^3-3*cos(c+d*x)/(a^3*d)-2*cos(c+d*x)^3/(a*d*(a+a*sin(c+d*x))^2)],
[cos(c+d*x)^3/(a+a*sin(c+d*x))^3,x,3,-log(1+sin(c+d*x))/(a^3*d)+(-2)/(d*(a^3+a^3*sin(c+d*x)))],
[cos(c+d*x)^2/(a+a*sin(c+d*x))^3,x,1,-1/3*cos(c+d*x)^3/(d*(a+a*sin(c+d*x))^3)],
[cos(c+d*x)/(a+a*sin(c+d*x))^3,x,2,(-1/2)/(a*d*(a+a*sin(c+d*x))^2)],
[sec(c+d*x)/(a+a*sin(c+d*x))^3,x,4,1/8*arctanh(sin(c+d*x))/(a^3*d)+(-1/6)/(d*(a+a*sin(c+d*x))^3)+(-1/8)/(a*d*(a+a*sin(c+d*x))^2)+(-1/8)/(d*(a^3+a^3*sin(c+d*x)))],
[sec(c+d*x)^2/(a+a*sin(c+d*x))^3,x,5,-1/7*sec(c+d*x)/(d*(a+a*sin(c+d*x))^3)-4/35*sec(c+d*x)/(a*d*(a+a*sin(c+d*x))^2)-4/35*sec(c+d*x)/(d*(a^3+a^3*sin(c+d*x)))+8/35*tan(c+d*x)/(a^3*d)],
[sec(c+d*x)^3/(a+a*sin(c+d*x))^3,x,4,5/32*arctanh(sin(c+d*x))/(a^3*d)-1/16*a/(d*(a+a*sin(c+d*x))^4)+(-1/12)/(d*(a+a*sin(c+d*x))^3)+(-3/32)/(a*d*(a+a*sin(c+d*x))^2)+1/32/(d*(a^3-a^3*sin(c+d*x)))+(-1/8)/(d*(a^3+a^3*sin(c+d*x)))],
[sec(c+d*x)^4/(a+a*sin(c+d*x))^3,x,5,-1/9*sec(c+d*x)^3/(d*(a+a*sin(c+d*x))^3)-2/21*sec(c+d*x)^3/(a*d*(a+a*sin(c+d*x))^2)-2/21*sec(c+d*x)^3/(d*(a^3+a^3*sin(c+d*x)))+8/21*tan(c+d*x)/(a^3*d)+8/63*tan(c+d*x)^3/(a^3*d)],
[sec(c+d*x)^5/(a+a*sin(c+d*x))^3,x,4,21/128*arctanh(sin(c+d*x))/(a^3*d)+1/128/(a*d*(a-a*sin(c+d*x))^2)-1/40*a^2/(d*(a+a*sin(c+d*x))^5)-3/64*a/(d*(a+a*sin(c+d*x))^4)+(-1/16)/(d*(a+a*sin(c+d*x))^3)+(-5/64)/(a*d*(a+a*sin(c+d*x))^2)+3/64/(d*(a^3-a^3*sin(c+d*x)))+(-15/128)/(d*(a^3+a^3*sin(c+d*x)))],
[cos(c+d*x)^8/(a+a*sin(c+d*x))^8,x,5,x/a^8-2/7*cos(c+d*x)^7/(a*d*(a+a*sin(c+d*x))^7)+2/5*cos(c+d*x)^5/(a^3*d*(a+a*sin(c+d*x))^5)-2/3*cos(c+d*x)^3/(a^2*d*(a^2+a^2*sin(c+d*x))^3)+2*cos(c+d*x)/(d*(a^8+a^8*sin(c+d*x)))],
[cos(c+d*x)^7/(a+a*sin(c+d*x))^8,x,2,-1/8*(a-a*sin(c+d*x))^4/(d*(a^3+a^3*sin(c+d*x))^4)],
[cos(c+d*x)^6/(a+a*sin(c+d*x))^8,x,2,-1/9*cos(c+d*x)^7/(d*(a+a*sin(c+d*x))^8)-1/63*cos(c+d*x)^7/(a*d*(a+a*sin(c+d*x))^7)],
[cos(c+d*x)^5/(a+a*sin(c+d*x))^8,x,3,(-4/5)/(a^3*d*(a+a*sin(c+d*x))^5)+(-1/3)/(a^5*d*(a+a*sin(c+d*x))^3)+1/(d*(a^2+a^2*sin(c+d*x))^4)],
[cos(c+d*x)^4/(a+a*sin(c+d*x))^8,x,4,-1/11*cos(c+d*x)^5/(d*(a+a*sin(c+d*x))^8)-1/33*cos(c+d*x)^5/(a*d*(a+a*sin(c+d*x))^7)-2/231*cos(c+d*x)^5/(a^2*d*(a+a*sin(c+d*x))^6)-2/1155*cos(c+d*x)^5/(a^3*d*(a+a*sin(c+d*x))^5)],
[cos(c+d*x)^3/(a+a*sin(c+d*x))^8,x,3,(-1/3)/(a^2*d*(a+a*sin(c+d*x))^6)+1/5/(a^3*d*(a+a*sin(c+d*x))^5)],
[cos(c+d*x)^2/(a+a*sin(c+d*x))^8,x,6,-1/13*cos(c+d*x)^3/(d*(a+a*sin(c+d*x))^8)-5/143*cos(c+d*x)^3/(a*d*(a+a*sin(c+d*x))^7)-20/1287*cos(c+d*x)^3/(a^2*d*(a+a*sin(c+d*x))^6)-20/3003*cos(c+d*x)^3/(a^3*d*(a+a*sin(c+d*x))^5)-8/3003*cos(c+d*x)^3/(d*(a^2+a^2*sin(c+d*x))^4)-8/9009*cos(c+d*x)^3/(a^2*d*(a^2+a^2*sin(c+d*x))^3)],
[cos(c+d*x)/(a+a*sin(c+d*x))^8,x,2,(-1/7)/(a*d*(a+a*sin(c+d*x))^7)],
[sec(c+d*x)/(a+a*sin(c+d*x))^8,x,4,1/256*arctanh(sin(c+d*x))/(a^8*d)+(-1/16)/(d*(a+a*sin(c+d*x))^8)+(-1/28)/(a*d*(a+a*sin(c+d*x))^7)+(-1/48)/(a^2*d*(a+a*sin(c+d*x))^6)+(-1/80)/(a^3*d*(a+a*sin(c+d*x))^5)+(-1/192)/(a^5*d*(a+a*sin(c+d*x))^3)+(-1/128)/(d*(a^2+a^2*sin(c+d*x))^4)+(-1/256)/(d*(a^4+a^4*sin(c+d*x))^2)+(-1/256)/(d*(a^8+a^8*sin(c+d*x)))],
[sec(c+d*x)^2/(a+a*sin(c+d*x))^8,x,10,-1/17*sec(c+d*x)/(d*(a+a*sin(c+d*x))^8)-3/85*sec(c+d*x)/(a*d*(a+a*sin(c+d*x))^7)-24/1105*sec(c+d*x)/(a^2*d*(a+a*sin(c+d*x))^6)-168/12155*sec(c+d*x)/(a^3*d*(a+a*sin(c+d*x))^5)-112/12155*sec(c+d*x)/(d*(a^2+a^2*sin(c+d*x))^4)-16/2431*sec(c+d*x)/(a^2*d*(a^2+a^2*sin(c+d*x))^3)-64/12155*sec(c+d*x)/(d*(a^4+a^4*sin(c+d*x))^2)-64/12155*sec(c+d*x)/(d*(a^8+a^8*sin(c+d*x)))+128/12155*tan(c+d*x)/(a^8*d)],
[sec(c+d*x)^3/(a+a*sin(c+d*x))^8,x,4,5/512*arctanh(sin(c+d*x))/(a^8*d)-1/36*a/(d*(a+a*sin(c+d*x))^9)+(-1/32)/(d*(a+a*sin(c+d*x))^8)+(-3/112)/(a*d*(a+a*sin(c+d*x))^7)+(-1/48)/(a^2*d*(a+a*sin(c+d*x))^6)+(-1/64)/(a^3*d*(a+a*sin(c+d*x))^5)+(-7/768)/(a^5*d*(a+a*sin(c+d*x))^3)+(-3/256)/(d*(a^2+a^2*sin(c+d*x))^4)+(-1/128)/(d*(a^4+a^4*sin(c+d*x))^2)+1/1024/(d*(a^8-a^8*sin(c+d*x)))+(-9/1024)/(d*(a^8+a^8*sin(c+d*x)))],
[sec(c+d*x)^4/(a+a*sin(c+d*x))^8,x,10,-1/19*sec(c+d*x)^3/(d*(a+a*sin(c+d*x))^8)-11/323*sec(c+d*x)^3/(a*d*(a+a*sin(c+d*x))^7)-22/969*sec(c+d*x)^3/(a^2*d*(a+a*sin(c+d*x))^6)-66/4199*sec(c+d*x)^3/(a^3*d*(a+a*sin(c+d*x))^5)-48/4199*sec(c+d*x)^3/(d*(a^2+a^2*sin(c+d*x))^4)-112/12597*sec(c+d*x)^3/(a^2*d*(a^2+a^2*sin(c+d*x))^3)-32/4199*sec(c+d*x)^3/(d*(a^4+a^4*sin(c+d*x))^2)-32/4199*sec(c+d*x)^3/(d*(a^8+a^8*sin(c+d*x)))+128/4199*tan(c+d*x)/(a^8*d)+128/12597*tan(c+d*x)^3/(a^8*d)],
[sec(c+d*x)^5/(a+a*sin(c+d*x))^8,x,4,33/2048*arctanh(sin(c+d*x))/(a^8*d)-1/80*a^2/(d*(a+a*sin(c+d*x))^10)-1/48*a/(d*(a+a*sin(c+d*x))^9)+(-3/128)/(d*(a+a*sin(c+d*x))^8)+(-5/224)/(a*d*(a+a*sin(c+d*x))^7)+(-5/256)/(a^2*d*(a+a*sin(c+d*x))^6)+(-21/1280)/(a^3*d*(a+a*sin(c+d*x))^5)+(-3/256)/(a^5*d*(a+a*sin(c+d*x))^3)+(-7/512)/(d*(a^2+a^2*sin(c+d*x))^4)+1/4096/(d*(a^4-a^4*sin(c+d*x))^2)+(-45/4096)/(d*(a^4+a^4*sin(c+d*x))^2)+11/4096/(d*(a^8-a^8*sin(c+d*x)))+(-55/4096)/(d*(a^8+a^8*sin(c+d*x)))],

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

# m>0
[cos(c+d*x)^7*sqrt(a+a*sin(c+d*x)),x,3,16/9*(a+a*sin(c+d*x))^(9/2)/(a^4*d)-24/11*(a+a*sin(c+d*x))^(11/2)/(a^5*d)+12/13*(a+a*sin(c+d*x))^(13/2)/(a^6*d)-2/15*(a+a*sin(c+d*x))^(15/2)/(a^7*d)],
[cos(c+d*x)^6*sqrt(a+a*sin(c+d*x)),x,4,-256/3003*a^4*cos(c+d*x)^7/(d*(a+a*sin(c+d*x))^(7/2))-64/429*a^3*cos(c+d*x)^7/(d*(a+a*sin(c+d*x))^(5/2))-24/143*a^2*cos(c+d*x)^7/(d*(a+a*sin(c+d*x))^(3/2))-2/13*a*cos(c+d*x)^7/(d*sqrt(a+a*sin(c+d*x)))],
[cos(c+d*x)^5*sqrt(a+a*sin(c+d*x)),x,3,8/7*(a+a*sin(c+d*x))^(7/2)/(a^3*d)-8/9*(a+a*sin(c+d*x))^(9/2)/(a^4*d)+2/11*(a+a*sin(c+d*x))^(11/2)/(a^5*d)],
[cos(c+d*x)^4*sqrt(a+a*sin(c+d*x)),x,3,-64/315*a^3*cos(c+d*x)^5/(d*(a+a*sin(c+d*x))^(5/2))-16/63*a^2*cos(c+d*x)^5/(d*(a+a*sin(c+d*x))^(3/2))-2/9*a*cos(c+d*x)^5/(d*sqrt(a+a*sin(c+d*x)))],
[cos(c+d*x)^3*sqrt(a+a*sin(c+d*x)),x,3,4/5*(a+a*sin(c+d*x))^(5/2)/(a^2*d)-2/7*(a+a*sin(c+d*x))^(7/2)/(a^3*d)],
[cos(c+d*x)^2*sqrt(a+a*sin(c+d*x)),x,2,-8/15*a^2*cos(c+d*x)^3/(d*(a+a*sin(c+d*x))^(3/2))-2/5*a*cos(c+d*x)^3/(d*sqrt(a+a*sin(c+d*x)))],
[cos(c+d*x)*sqrt(a+a*sin(c+d*x)),x,2,2/3*(a+a*sin(c+d*x))^(3/2)/(a*d)],
[sec(c+d*x)*sqrt(a+a*sin(c+d*x)),x,3,arctanh(sqrt(a+a*sin(c+d*x))/(sqrt(2)*sqrt(a)))*sqrt(2)*sqrt(a)/d],
[sec(c+d*x)^2*sqrt(a+a*sin(c+d*x)),x,3,-arctanh(cos(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(c+d*x))))*sqrt(a)/(d*sqrt(2))+sec(c+d*x)*sqrt(a+a*sin(c+d*x))/d],
[sec(c+d*x)^3*sqrt(a+a*sin(c+d*x)),x,5,3/4*arctanh(sqrt(a+a*sin(c+d*x))/(sqrt(2)*sqrt(a)))*sqrt(a)/(d*sqrt(2))-3/4*a/(d*sqrt(a+a*sin(c+d*x)))+1/2*sec(c+d*x)^2*sqrt(a+a*sin(c+d*x))/d],
[sec(c+d*x)^4*sqrt(a+a*sin(c+d*x)),x,5,-5/8*a^2*cos(c+d*x)/(d*(a+a*sin(c+d*x))^(3/2))-5/8*arctanh(cos(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(c+d*x))))*sqrt(a)/(d*sqrt(2))+5/6*a*sec(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))+1/3*sec(c+d*x)^3*sqrt(a+a*sin(c+d*x))/d],
[sec(c+d*x)^5*sqrt(a+a*sin(c+d*x)),x,7,-35/96*a^2/(d*(a+a*sin(c+d*x))^(3/2))+35/64*arctanh(sqrt(a+a*sin(c+d*x))/(sqrt(2)*sqrt(a)))*sqrt(a)/(d*sqrt(2))-35/64*a/(d*sqrt(a+a*sin(c+d*x)))+7/16*a*sec(c+d*x)^2/(d*sqrt(a+a*sin(c+d*x)))+1/4*sec(c+d*x)^4*sqrt(a+a*sin(c+d*x))/d],
[sec(c+d*x)^6*sqrt(a+a*sin(c+d*x)),x,7,-63/128*a^2*cos(c+d*x)/(d*(a+a*sin(c+d*x))^(3/2))-21/80*a^2*sec(c+d*x)/(d*(a+a*sin(c+d*x))^(3/2))-63/128*arctanh(cos(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(c+d*x))))*sqrt(a)/(d*sqrt(2))+21/32*a*sec(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))+3/10*a*sec(c+d*x)^3/(d*sqrt(a+a*sin(c+d*x)))+1/5*sec(c+d*x)^5*sqrt(a+a*sin(c+d*x))/d],
[cos(c+d*x)^7*(a+a*sin(c+d*x))^(3/2),x,3,16/11*(a+a*sin(c+d*x))^(11/2)/(a^4*d)-24/13*(a+a*sin(c+d*x))^(13/2)/(a^5*d)+4/5*(a+a*sin(c+d*x))^(15/2)/(a^6*d)-2/17*(a+a*sin(c+d*x))^(17/2)/(a^7*d)],
[cos(c+d*x)^6*(a+a*sin(c+d*x))^(3/2),x,5,-4096/45045*a^5*cos(c+d*x)^7/(d*(a+a*sin(c+d*x))^(7/2))-1024/6435*a^4*cos(c+d*x)^7/(d*(a+a*sin(c+d*x))^(5/2))-128/715*a^3*cos(c+d*x)^7/(d*(a+a*sin(c+d*x))^(3/2))-32/195*a^2*cos(c+d*x)^7/(d*sqrt(a+a*sin(c+d*x)))-2/15*a*cos(c+d*x)^7*sqrt(a+a*sin(c+d*x))/d],
[cos(c+d*x)^5*(a+a*sin(c+d*x))^(3/2),x,3,8/9*(a+a*sin(c+d*x))^(9/2)/(a^3*d)-8/11*(a+a*sin(c+d*x))^(11/2)/(a^4*d)+2/13*(a+a*sin(c+d*x))^(13/2)/(a^5*d)],
[cos(c+d*x)^4*(a+a*sin(c+d*x))^(3/2),x,4,-256/1155*a^4*cos(c+d*x)^5/(d*(a+a*sin(c+d*x))^(5/2))-64/231*a^3*cos(c+d*x)^5/(d*(a+a*sin(c+d*x))^(3/2))-8/33*a^2*cos(c+d*x)^5/(d*sqrt(a+a*sin(c+d*x)))-2/11*a*cos(c+d*x)^5*sqrt(a+a*sin(c+d*x))/d],
[cos(c+d*x)^3*(a+a*sin(c+d*x))^(3/2),x,3,4/7*(a+a*sin(c+d*x))^(7/2)/(a^2*d)-2/9*(a+a*sin(c+d*x))^(9/2)/(a^3*d)],
[cos(c+d*x)^2*(a+a*sin(c+d*x))^(3/2),x,3,-64/105*a^3*cos(c+d*x)^3/(d*(a+a*sin(c+d*x))^(3/2))-16/35*a^2*cos(c+d*x)^3/(d*sqrt(a+a*sin(c+d*x)))-2/7*a*cos(c+d*x)^3*sqrt(a+a*sin(c+d*x))/d],
[cos(c+d*x)*(a+a*sin(c+d*x))^(3/2),x,2,2/5*(a+a*sin(c+d*x))^(5/2)/(a*d)],
[sec(c+d*x)*(a+a*sin(c+d*x))^(3/2),x,4,2*a^(3/2)*arctanh(sqrt(a+a*sin(c+d*x))/(sqrt(2)*sqrt(a)))*sqrt(2)/d-2*a*sqrt(a+a*sin(c+d*x))/d],
[sec(c+d*x)^2*(a+a*sin(c+d*x))^(3/2),x,1,2*a*sec(c+d*x)*sqrt(a+a*sin(c+d*x))/d],
[sec(c+d*x)^3*(a+a*sin(c+d*x))^(3/2),x,4,1/2*sec(c+d*x)^2*(a+a*sin(c+d*x))^(3/2)/d+1/2*a^(3/2)*arctanh(sqrt(a+a*sin(c+d*x))/(sqrt(2)*sqrt(a)))/(d*sqrt(2))],
[sec(c+d*x)^4*(a+a*sin(c+d*x))^(3/2),x,4,1/3*sec(c+d*x)^3*(a+a*sin(c+d*x))^(3/2)/d-1/2*a^(3/2)*arctanh(cos(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(c+d*x))))/(d*sqrt(2))+1/2*a*sec(c+d*x)*sqrt(a+a*sin(c+d*x))/d],
[sec(c+d*x)^5*(a+a*sin(c+d*x))^(3/2),x,6,1/4*sec(c+d*x)^4*(a+a*sin(c+d*x))^(3/2)/d+15/32*a^(3/2)*arctanh(sqrt(a+a*sin(c+d*x))/(sqrt(2)*sqrt(a)))/(d*sqrt(2))-15/32*a^2/(d*sqrt(a+a*sin(c+d*x)))+5/16*a*sec(c+d*x)^2*sqrt(a+a*sin(c+d*x))/d],
[sec(c+d*x)^6*(a+a*sin(c+d*x))^(3/2),x,6,-7/16*a^3*cos(c+d*x)/(d*(a+a*sin(c+d*x))^(3/2))+1/5*sec(c+d*x)^5*(a+a*sin(c+d*x))^(3/2)/d-7/16*a^(3/2)*arctanh(cos(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(c+d*x))))/(d*sqrt(2))+7/12*a^2*sec(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))+7/30*a*sec(c+d*x)^3*sqrt(a+a*sin(c+d*x))/d],
[cos(c+d*x)^5*(a+a*sin(c+d*x))^(5/2),x,3,8/11*(a+a*sin(c+d*x))^(11/2)/(a^3*d)-8/13*(a+a*sin(c+d*x))^(13/2)/(a^4*d)+2/15*(a+a*sin(c+d*x))^(15/2)/(a^5*d)],
[cos(c+d*x)^4*(a+a*sin(c+d*x))^(5/2),x,5,-4096/15015*a^5*cos(c+d*x)^5/(d*(a+a*sin(c+d*x))^(5/2))-1024/3003*a^4*cos(c+d*x)^5/(d*(a+a*sin(c+d*x))^(3/2))-2/13*a*cos(c+d*x)^5*(a+a*sin(c+d*x))^(3/2)/d-128/429*a^3*cos(c+d*x)^5/(d*sqrt(a+a*sin(c+d*x)))-32/143*a^2*cos(c+d*x)^5*sqrt(a+a*sin(c+d*x))/d],
[cos(c+d*x)^3*(a+a*sin(c+d*x))^(5/2),x,3,4/9*(a+a*sin(c+d*x))^(9/2)/(a^2*d)-2/11*(a+a*sin(c+d*x))^(11/2)/(a^3*d)],
[cos(c+d*x)^2*(a+a*sin(c+d*x))^(5/2),x,4,-256/315*a^4*cos(c+d*x)^3/(d*(a+a*sin(c+d*x))^(3/2))-2/9*a*cos(c+d*x)^3*(a+a*sin(c+d*x))^(3/2)/d-64/105*a^3*cos(c+d*x)^3/(d*sqrt(a+a*sin(c+d*x)))-8/21*a^2*cos(c+d*x)^3*sqrt(a+a*sin(c+d*x))/d],
[cos(c+d*x)*(a+a*sin(c+d*x))^(5/2),x,2,2/7*(a+a*sin(c+d*x))^(7/2)/(a*d)],
[sec(c+d*x)*(a+a*sin(c+d*x))^(5/2),x,5,-2/3*a*(a+a*sin(c+d*x))^(3/2)/d+4*a^(5/2)*arctanh(sqrt(a+a*sin(c+d*x))/(sqrt(2)*sqrt(a)))*sqrt(2)/d-4*a^2*sqrt(a+a*sin(c+d*x))/d],
[sec(c+d*x)^2*(a+a*sin(c+d*x))^(5/2),x,2,-2*a*sec(c+d*x)*(a+a*sin(c+d*x))^(3/2)/d+8*a^2*sec(c+d*x)*sqrt(a+a*sin(c+d*x))/d],
[sec(c+d*x)^3*(a+a*sin(c+d*x))^(5/2),x,4,a*sec(c+d*x)^2*(a+a*sin(c+d*x))^(3/2)/d-a^(5/2)*arctanh(sqrt(a+a*sin(c+d*x))/(sqrt(2)*sqrt(a)))/(d*sqrt(2))],
[sec(c+d*x)^4*(a+a*sin(c+d*x))^(5/2),x,1,2/3*a*sec(c+d*x)^3*(a+a*sin(c+d*x))^(3/2)/d],
[sec(c+d*x)^5*(a+a*sin(c+d*x))^(5/2),x,5,3/16*a*sec(c+d*x)^2*(a+a*sin(c+d*x))^(3/2)/d+1/4*sec(c+d*x)^4*(a+a*sin(c+d*x))^(5/2)/d+3/16*a^(5/2)*arctanh(sqrt(a+a*sin(c+d*x))/(sqrt(2)*sqrt(a)))/(d*sqrt(2))],
[sec(c+d*x)^6*(a+a*sin(c+d*x))^(5/2),x,5,1/6*a*sec(c+d*x)^3*(a+a*sin(c+d*x))^(3/2)/d+1/5*sec(c+d*x)^5*(a+a*sin(c+d*x))^(5/2)/d-1/4*a^(5/2)*arctanh(cos(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(c+d*x))))/(d*sqrt(2))+1/4*a^2*sec(c+d*x)*sqrt(a+a*sin(c+d*x))/d],
[sec(c+d*x)^7*(a+a*sin(c+d*x))^(5/2),x,7,7/48*a*sec(c+d*x)^4*(a+a*sin(c+d*x))^(3/2)/d+1/6*sec(c+d*x)^6*(a+a*sin(c+d*x))^(5/2)/d+35/128*a^(5/2)*arctanh(sqrt(a+a*sin(c+d*x))/(sqrt(2)*sqrt(a)))/(d*sqrt(2))-35/128*a^3/(d*sqrt(a+a*sin(c+d*x)))+35/192*a^2*sec(c+d*x)^2*sqrt(a+a*sin(c+d*x))/d],
[cos(c+d*x)^7*(a+a*sin(c+d*x))^(7/2),x,3,16/15*(a+a*sin(c+d*x))^(15/2)/(a^4*d)-24/17*(a+a*sin(c+d*x))^(17/2)/(a^5*d)+12/19*(a+a*sin(c+d*x))^(19/2)/(a^6*d)-2/21*(a+a*sin(c+d*x))^(21/2)/(a^7*d)],
[cos(c+d*x)^6*(a+a*sin(c+d*x))^(7/2),x,7,-131072/969969*a^7*cos(c+d*x)^7/(d*(a+a*sin(c+d*x))^(7/2))-32768/138567*a^6*cos(c+d*x)^7/(d*(a+a*sin(c+d*x))^(5/2))-12288/46189*a^5*cos(c+d*x)^7/(d*(a+a*sin(c+d*x))^(3/2))-48/323*a^2*cos(c+d*x)^7*(a+a*sin(c+d*x))^(3/2)/d-2/19*a*cos(c+d*x)^7*(a+a*sin(c+d*x))^(5/2)/d-1024/4199*a^4*cos(c+d*x)^7/(d*sqrt(a+a*sin(c+d*x)))-64/323*a^3*cos(c+d*x)^7*sqrt(a+a*sin(c+d*x))/d],
[cos(c+d*x)^5*(a+a*sin(c+d*x))^(7/2),x,3,8/13*(a+a*sin(c+d*x))^(13/2)/(a^3*d)-8/15*(a+a*sin(c+d*x))^(15/2)/(a^4*d)+2/17*(a+a*sin(c+d*x))^(17/2)/(a^5*d)],
[cos(c+d*x)^4*(a+a*sin(c+d*x))^(7/2),x,6,-16384/45045*a^6*cos(c+d*x)^5/(d*(a+a*sin(c+d*x))^(5/2))-4096/9009*a^5*cos(c+d*x)^5/(d*(a+a*sin(c+d*x))^(3/2))-8/39*a^2*cos(c+d*x)^5*(a+a*sin(c+d*x))^(3/2)/d-2/15*a*cos(c+d*x)^5*(a+a*sin(c+d*x))^(5/2)/d-512/1287*a^4*cos(c+d*x)^5/(d*sqrt(a+a*sin(c+d*x)))-128/429*a^3*cos(c+d*x)^5*sqrt(a+a*sin(c+d*x))/d],
[cos(c+d*x)^3*(a+a*sin(c+d*x))^(7/2),x,3,4/11*(a+a*sin(c+d*x))^(11/2)/(a^2*d)-2/13*(a+a*sin(c+d*x))^(13/2)/(a^3*d)],
[cos(c+d*x)^2*(a+a*sin(c+d*x))^(7/2),x,5,-4096/3465*a^5*cos(c+d*x)^3/(d*(a+a*sin(c+d*x))^(3/2))-32/99*a^2*cos(c+d*x)^3*(a+a*sin(c+d*x))^(3/2)/d-2/11*a*cos(c+d*x)^3*(a+a*sin(c+d*x))^(5/2)/d-1024/1155*a^4*cos(c+d*x)^3/(d*sqrt(a+a*sin(c+d*x)))-128/231*a^3*cos(c+d*x)^3*sqrt(a+a*sin(c+d*x))/d],
[cos(c+d*x)*(a+a*sin(c+d*x))^(7/2),x,2,2/9*(a+a*sin(c+d*x))^(9/2)/(a*d)],
[sec(c+d*x)*(a+a*sin(c+d*x))^(7/2),x,6,-4/3*a^2*(a+a*sin(c+d*x))^(3/2)/d-2/5*a*(a+a*sin(c+d*x))^(5/2)/d+8*a^(7/2)*arctanh(sqrt(a+a*sin(c+d*x))/(sqrt(2)*sqrt(a)))*sqrt(2)/d-8*a^3*sqrt(a+a*sin(c+d*x))/d],
[sec(c+d*x)^2*(a+a*sin(c+d*x))^(7/2),x,3,-16/3*a^2*sec(c+d*x)*(a+a*sin(c+d*x))^(3/2)/d-2/3*a*sec(c+d*x)*(a+a*sin(c+d*x))^(5/2)/d+64/3*a^3*sec(c+d*x)*sqrt(a+a*sin(c+d*x))/d],
[sec(c+d*x)^3*(a+a*sin(c+d*x))^(7/2),x,5,a*sec(c+d*x)^2*(a+a*sin(c+d*x))^(5/2)/d-3*a^(7/2)*arctanh(sqrt(a+a*sin(c+d*x))/(sqrt(2)*sqrt(a)))*sqrt(2)/d+3*a^3*sqrt(a+a*sin(c+d*x))/d],
[sec(c+d*x)^4*(a+a*sin(c+d*x))^(7/2),x,2,-8/3*a^2*sec(c+d*x)^3*(a+a*sin(c+d*x))^(3/2)/d+2*a*sec(c+d*x)^3*(a+a*sin(c+d*x))^(5/2)/d],
[sec(c+d*x)^5*(a+a*sin(c+d*x))^(7/2),x,5,-1/8*a^2*sec(c+d*x)^2*(a+a*sin(c+d*x))^(3/2)/d+1/2*a*sec(c+d*x)^4*(a+a*sin(c+d*x))^(5/2)/d-1/8*a^(7/2)*arctanh(sqrt(a+a*sin(c+d*x))/(sqrt(2)*sqrt(a)))/(d*sqrt(2))],
[sec(c+d*x)^6*(a+a*sin(c+d*x))^(7/2),x,1,2/5*a*sec(c+d*x)^5*(a+a*sin(c+d*x))^(5/2)/d],
[sec(c+d*x)^7*(a+a*sin(c+d*x))^(7/2),x,6,5/64*a^2*sec(c+d*x)^2*(a+a*sin(c+d*x))^(3/2)/d+5/48*a*sec(c+d*x)^4*(a+a*sin(c+d*x))^(5/2)/d+1/6*sec(c+d*x)^6*(a+a*sin(c+d*x))^(7/2)/d+5/64*a^(7/2)*arctanh(sqrt(a+a*sin(c+d*x))/(sqrt(2)*sqrt(a)))/(d*sqrt(2))],
[sec(c+d*x)^8*(a+a*sin(c+d*x))^(7/2),x,6,1/12*a^2*sec(c+d*x)^3*(a+a*sin(c+d*x))^(3/2)/d+1/10*a*sec(c+d*x)^5*(a+a*sin(c+d*x))^(5/2)/d+1/7*sec(c+d*x)^7*(a+a*sin(c+d*x))^(7/2)/d-1/8*a^(7/2)*arctanh(cos(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(c+d*x))))/(d*sqrt(2))+1/8*a^3*sec(c+d*x)*sqrt(a+a*sin(c+d*x))/d],
[sec(c+d*x)^9*(a+a*sin(c+d*x))^(7/2),x,8,21/256*a^2*sec(c+d*x)^4*(a+a*sin(c+d*x))^(3/2)/d+3/32*a*sec(c+d*x)^6*(a+a*sin(c+d*x))^(5/2)/d+1/8*sec(c+d*x)^8*(a+a*sin(c+d*x))^(7/2)/d+315/2048*a^(7/2)*arctanh(sqrt(a+a*sin(c+d*x))/(sqrt(2)*sqrt(a)))/(d*sqrt(2))-315/2048*a^4/(d*sqrt(a+a*sin(c+d*x)))+105/1024*a^3*sec(c+d*x)^2*sqrt(a+a*sin(c+d*x))/d],
[sec(c+d*x)^10*(a+a*sin(c+d*x))^(7/2),x,8,-11/64*a^5*cos(c+d*x)/(d*(a+a*sin(c+d*x))^(3/2))+11/140*a^2*sec(c+d*x)^5*(a+a*sin(c+d*x))^(3/2)/d+11/126*a*sec(c+d*x)^7*(a+a*sin(c+d*x))^(5/2)/d+1/9*sec(c+d*x)^9*(a+a*sin(c+d*x))^(7/2)/d-11/64*a^(7/2)*arctanh(cos(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(c+d*x))))/(d*sqrt(2))+11/48*a^4*sec(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))+11/120*a^3*sec(c+d*x)^3*sqrt(a+a*sin(c+d*x))/d],

# m<0
[cos(c+d*x)^7/sqrt(a+a*sin(c+d*x)),x,3,16/7*(a+a*sin(c+d*x))^(7/2)/(a^4*d)-8/3*(a+a*sin(c+d*x))^(9/2)/(a^5*d)+12/11*(a+a*sin(c+d*x))^(11/2)/(a^6*d)-2/13*(a+a*sin(c+d*x))^(13/2)/(a^7*d)],
[cos(c+d*x)^6/sqrt(a+a*sin(c+d*x)),x,3,-64/693*a^3*cos(c+d*x)^7/(d*(a+a*sin(c+d*x))^(7/2))-16/99*a^2*cos(c+d*x)^7/(d*(a+a*sin(c+d*x))^(5/2))-2/11*a*cos(c+d*x)^7/(d*(a+a*sin(c+d*x))^(3/2))],
[cos(c+d*x)^5/sqrt(a+a*sin(c+d*x)),x,3,8/5*(a+a*sin(c+d*x))^(5/2)/(a^3*d)-8/7*(a+a*sin(c+d*x))^(7/2)/(a^4*d)+2/9*(a+a*sin(c+d*x))^(9/2)/(a^5*d)],
[cos(c+d*x)^4/sqrt(a+a*sin(c+d*x)),x,2,-8/35*a^2*cos(c+d*x)^5/(d*(a+a*sin(c+d*x))^(5/2))-2/7*a*cos(c+d*x)^5/(d*(a+a*sin(c+d*x))^(3/2))],
[cos(c+d*x)^3/sqrt(a+a*sin(c+d*x)),x,3,4/3*(a+a*sin(c+d*x))^(3/2)/(a^2*d)-2/5*(a+a*sin(c+d*x))^(5/2)/(a^3*d)],
[cos(c+d*x)^2/sqrt(a+a*sin(c+d*x)),x,1,-2/3*a*cos(c+d*x)^3/(d*(a+a*sin(c+d*x))^(3/2))],
[cos(c+d*x)/sqrt(a+a*sin(c+d*x)),x,2,2*sqrt(a+a*sin(c+d*x))/(a*d)],
[sec(c+d*x)/sqrt(a+a*sin(c+d*x)),x,4,arctanh(sqrt(a+a*sin(c+d*x))/(sqrt(2)*sqrt(a)))/(d*sqrt(2)*sqrt(a))+(-1)/(d*sqrt(a+a*sin(c+d*x)))],
[sec(c+d*x)^2/sqrt(a+a*sin(c+d*x)),x,4,-3/4*a*cos(c+d*x)/(d*(a+a*sin(c+d*x))^(3/2))-3/4*arctanh(cos(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(c+d*x))))/(d*sqrt(2)*sqrt(a))+sec(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))],
[sec(c+d*x)^3/sqrt(a+a*sin(c+d*x)),x,6,-5/12*a/(d*(a+a*sin(c+d*x))^(3/2))+5/8*arctanh(sqrt(a+a*sin(c+d*x))/(sqrt(2)*sqrt(a)))/(d*sqrt(2)*sqrt(a))+(-5/8)/(d*sqrt(a+a*sin(c+d*x)))+1/2*sec(c+d*x)^2/(d*sqrt(a+a*sin(c+d*x)))],
[sec(c+d*x)^4/sqrt(a+a*sin(c+d*x)),x,6,-35/64*a*cos(c+d*x)/(d*(a+a*sin(c+d*x))^(3/2))-7/24*a*sec(c+d*x)/(d*(a+a*sin(c+d*x))^(3/2))-35/64*arctanh(cos(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(c+d*x))))/(d*sqrt(2)*sqrt(a))+35/48*sec(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))+1/3*sec(c+d*x)^3/(d*sqrt(a+a*sin(c+d*x)))],
[sec(c+d*x)^5/sqrt(a+a*sin(c+d*x)),x,8,-21/64*a/(d*(a+a*sin(c+d*x))^(3/2))-9/40*a*sec(c+d*x)^2/(d*(a+a*sin(c+d*x))^(3/2))+63/128*arctanh(sqrt(a+a*sin(c+d*x))/(sqrt(2)*sqrt(a)))/(d*sqrt(2)*sqrt(a))+(-63/128)/(d*sqrt(a+a*sin(c+d*x)))+63/160*sec(c+d*x)^2/(d*sqrt(a+a*sin(c+d*x)))+1/4*sec(c+d*x)^4/(d*sqrt(a+a*sin(c+d*x)))],
[sec(c+d*x)^6/sqrt(a+a*sin(c+d*x)),x,8,-231/512*a*cos(c+d*x)/(d*(a+a*sin(c+d*x))^(3/2))-77/320*a*sec(c+d*x)/(d*(a+a*sin(c+d*x))^(3/2))-11/60*a*sec(c+d*x)^3/(d*(a+a*sin(c+d*x))^(3/2))-231/512*arctanh(cos(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(c+d*x))))/(d*sqrt(2)*sqrt(a))+77/128*sec(c+d*x)/(d*sqrt(a+a*sin(c+d*x)))+11/40*sec(c+d*x)^3/(d*sqrt(a+a*sin(c+d*x)))+1/5*sec(c+d*x)^5/(d*sqrt(a+a*sin(c+d*x)))],
[cos(c+d*x)^7/(a+a*sin(c+d*x))^(3/2),x,3,16/5*(a+a*sin(c+d*x))^(5/2)/(a^4*d)-24/7*(a+a*sin(c+d*x))^(7/2)/(a^5*d)+4/3*(a+a*sin(c+d*x))^(9/2)/(a^6*d)-2/11*(a+a*sin(c+d*x))^(11/2)/(a^7*d)],
[cos(c+d*x)^6/(a+a*sin(c+d*x))^(3/2),x,2,-8/63*a^2*cos(c+d*x)^7/(d*(a+a*sin(c+d*x))^(7/2))-2/9*a*cos(c+d*x)^7/(d*(a+a*sin(c+d*x))^(5/2))],
[cos(c+d*x)^5/(a+a*sin(c+d*x))^(3/2),x,3,8/3*(a+a*sin(c+d*x))^(3/2)/(a^3*d)-8/5*(a+a*sin(c+d*x))^(5/2)/(a^4*d)+2/7*(a+a*sin(c+d*x))^(7/2)/(a^5*d)],
[cos(c+d*x)^4/(a+a*sin(c+d*x))^(3/2),x,1,-2/5*a*cos(c+d*x)^5/(d*(a+a*sin(c+d*x))^(5/2))],
[cos(c+d*x)^3/(a+a*sin(c+d*x))^(3/2),x,3,-2/3*(a+a*sin(c+d*x))^(3/2)/(a^3*d)+4*sqrt(a+a*sin(c+d*x))/(a^2*d)],
[cos(c+d*x)^2/(a+a*sin(c+d*x))^(3/2),x,3,-2*arctanh(cos(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(c+d*x))))*sqrt(2)/(a^(3/2)*d)+2*cos(c+d*x)/(a*d*sqrt(a+a*sin(c+d*x)))],
[cos(c+d*x)/(a+a*sin(c+d*x))^(3/2),x,2,(-2)/(a*d*sqrt(a+a*sin(c+d*x)))],
[sec(c+d*x)/(a+a*sin(c+d*x))^(3/2),x,5,(-1/3)/(d*(a+a*sin(c+d*x))^(3/2))+1/2*arctanh(sqrt(a+a*sin(c+d*x))/(sqrt(2)*sqrt(a)))/(a^(3/2)*d*sqrt(2))+(-1/2)/(a*d*sqrt(a+a*sin(c+d*x)))],
[sec(c+d*x)^2/(a+a*sin(c+d*x))^(3/2),x,5,-15/32*cos(c+d*x)/(d*(a+a*sin(c+d*x))^(3/2))-1/4*sec(c+d*x)/(d*(a+a*sin(c+d*x))^(3/2))-15/32*arctanh(cos(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(c+d*x))))/(a^(3/2)*d*sqrt(2))+5/8*sec(c+d*x)/(a*d*sqrt(a+a*sin(c+d*x)))],
[sec(c+d*x)^3/(a+a*sin(c+d*x))^(3/2),x,7,(-7/24)/(d*(a+a*sin(c+d*x))^(3/2))-1/5*sec(c+d*x)^2/(d*(a+a*sin(c+d*x))^(3/2))+7/16*arctanh(sqrt(a+a*sin(c+d*x))/(sqrt(2)*sqrt(a)))/(a^(3/2)*d*sqrt(2))+(-7/16)/(a*d*sqrt(a+a*sin(c+d*x)))+7/20*sec(c+d*x)^2/(a*d*sqrt(a+a*sin(c+d*x)))],
[sec(c+d*x)^4/(a+a*sin(c+d*x))^(3/2),x,7,-105/256*cos(c+d*x)/(d*(a+a*sin(c+d*x))^(3/2))-7/32*sec(c+d*x)/(d*(a+a*sin(c+d*x))^(3/2))-1/6*sec(c+d*x)^3/(d*(a+a*sin(c+d*x))^(3/2))-105/256*arctanh(cos(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(c+d*x))))/(a^(3/2)*d*sqrt(2))+35/64*sec(c+d*x)/(a*d*sqrt(a+a*sin(c+d*x)))+1/4*sec(c+d*x)^3/(a*d*sqrt(a+a*sin(c+d*x)))],
[sec(c+d*x)^5/(a+a*sin(c+d*x))^(3/2),x,9,(-33/128)/(d*(a+a*sin(c+d*x))^(3/2))-99/560*sec(c+d*x)^2/(d*(a+a*sin(c+d*x))^(3/2))-1/7*sec(c+d*x)^4/(d*(a+a*sin(c+d*x))^(3/2))+99/256*arctanh(sqrt(a+a*sin(c+d*x))/(sqrt(2)*sqrt(a)))/(a^(3/2)*d*sqrt(2))+(-99/256)/(a*d*sqrt(a+a*sin(c+d*x)))+99/320*sec(c+d*x)^2/(a*d*sqrt(a+a*sin(c+d*x)))+11/56*sec(c+d*x)^4/(a*d*sqrt(a+a*sin(c+d*x)))],
[sec(c+d*x)^6/(a+a*sin(c+d*x))^(3/2),x,9,-3003/8192*cos(c+d*x)/(d*(a+a*sin(c+d*x))^(3/2))-1001/5120*sec(c+d*x)/(d*(a+a*sin(c+d*x))^(3/2))-143/960*sec(c+d*x)^3/(d*(a+a*sin(c+d*x))^(3/2))-1/8*sec(c+d*x)^5/(d*(a+a*sin(c+d*x))^(3/2))-3003/8192*arctanh(cos(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(c+d*x))))/(a^(3/2)*d*sqrt(2))+1001/2048*sec(c+d*x)/(a*d*sqrt(a+a*sin(c+d*x)))+143/640*sec(c+d*x)^3/(a*d*sqrt(a+a*sin(c+d*x)))+13/80*sec(c+d*x)^5/(a*d*sqrt(a+a*sin(c+d*x)))],
[cos(c+d*x)^10/(a+a*sin(c+d*x))^(5/2),x,3,-64/2145*a^3*cos(c+d*x)^11/(d*(a+a*sin(c+d*x))^(11/2))-16/195*a^2*cos(c+d*x)^11/(d*(a+a*sin(c+d*x))^(9/2))-2/15*a*cos(c+d*x)^11/(d*(a+a*sin(c+d*x))^(7/2))],
[cos(c+d*x)^9/(a+a*sin(c+d*x))^(5/2),x,3,32/5*(a+a*sin(c+d*x))^(5/2)/(a^5*d)-64/7*(a+a*sin(c+d*x))^(7/2)/(a^6*d)+16/3*(a+a*sin(c+d*x))^(9/2)/(a^7*d)-16/11*(a+a*sin(c+d*x))^(11/2)/(a^8*d)+2/13*(a+a*sin(c+d*x))^(13/2)/(a^9*d)],
[cos(c+d*x)^8/(a+a*sin(c+d*x))^(5/2),x,2,-8/99*a^2*cos(c+d*x)^9/(d*(a+a*sin(c+d*x))^(9/2))-2/11*a*cos(c+d*x)^9/(d*(a+a*sin(c+d*x))^(7/2))],
[cos(c+d*x)^7/(a+a*sin(c+d*x))^(5/2),x,3,16/3*(a+a*sin(c+d*x))^(3/2)/(a^4*d)-24/5*(a+a*sin(c+d*x))^(5/2)/(a^5*d)+12/7*(a+a*sin(c+d*x))^(7/2)/(a^6*d)-2/9*(a+a*sin(c+d*x))^(9/2)/(a^7*d)],
[cos(c+d*x)^6/(a+a*sin(c+d*x))^(5/2),x,1,-2/7*a*cos(c+d*x)^7/(d*(a+a*sin(c+d*x))^(7/2))],
[cos(c+d*x)^5/(a+a*sin(c+d*x))^(5/2),x,3,-8/3*(a+a*sin(c+d*x))^(3/2)/(a^4*d)+2/5*(a+a*sin(c+d*x))^(5/2)/(a^5*d)+8*sqrt(a+a*sin(c+d*x))/(a^3*d)],
[cos(c+d*x)^4/(a+a*sin(c+d*x))^(5/2),x,4,2/3*cos(c+d*x)^3/(a*d*(a+a*sin(c+d*x))^(3/2))-4*arctanh(cos(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(c+d*x))))*sqrt(2)/(a^(5/2)*d)+4*cos(c+d*x)/(a^2*d*sqrt(a+a*sin(c+d*x)))],
[cos(c+d*x)^3/(a+a*sin(c+d*x))^(5/2),x,3,(-4)/(a^2*d*sqrt(a+a*sin(c+d*x)))-2*sqrt(a+a*sin(c+d*x))/(a^3*d)],
[cos(c+d*x)^2/(a+a*sin(c+d*x))^(5/2),x,3,-cos(c+d*x)/(a*d*(a+a*sin(c+d*x))^(3/2))+arctanh(cos(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(c+d*x))))/(a^(5/2)*d*sqrt(2))],
[cos(c+d*x)/(a+a*sin(c+d*x))^(5/2),x,2,(-2/3)/(a*d*(a+a*sin(c+d*x))^(3/2))],
[sec(c+d*x)/(a+a*sin(c+d*x))^(5/2),x,6,(-1/5)/(d*(a+a*sin(c+d*x))^(5/2))+(-1/6)/(a*d*(a+a*sin(c+d*x))^(3/2))+1/4*arctanh(sqrt(a+a*sin(c+d*x))/(sqrt(2)*sqrt(a)))/(a^(5/2)*d*sqrt(2))+(-1/4)/(a^2*d*sqrt(a+a*sin(c+d*x)))],
[sec(c+d*x)^2/(a+a*sin(c+d*x))^(5/2),x,6,-1/6*sec(c+d*x)/(d*(a+a*sin(c+d*x))^(5/2))-35/128*cos(c+d*x)/(a*d*(a+a*sin(c+d*x))^(3/2))-7/48*sec(c+d*x)/(a*d*(a+a*sin(c+d*x))^(3/2))-35/128*arctanh(cos(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(c+d*x))))/(a^(5/2)*d*sqrt(2))+35/96*sec(c+d*x)/(a^2*d*sqrt(a+a*sin(c+d*x)))],
[sec(c+d*x)^3/(a+a*sin(c+d*x))^(5/2),x,8,-1/7*sec(c+d*x)^2/(d*(a+a*sin(c+d*x))^(5/2))+(-3/16)/(a*d*(a+a*sin(c+d*x))^(3/2))-9/70*sec(c+d*x)^2/(a*d*(a+a*sin(c+d*x))^(3/2))+9/32*arctanh(sqrt(a+a*sin(c+d*x))/(sqrt(2)*sqrt(a)))/(a^(5/2)*d*sqrt(2))+(-9/32)/(a^2*d*sqrt(a+a*sin(c+d*x)))+9/40*sec(c+d*x)^2/(a^2*d*sqrt(a+a*sin(c+d*x)))],
[sec(c+d*x)^4/(a+a*sin(c+d*x))^(5/2),x,8,-1/8*sec(c+d*x)^3/(d*(a+a*sin(c+d*x))^(5/2))-1155/4096*cos(c+d*x)/(a*d*(a+a*sin(c+d*x))^(3/2))-77/512*sec(c+d*x)/(a*d*(a+a*sin(c+d*x))^(3/2))-11/96*sec(c+d*x)^3/(a*d*(a+a*sin(c+d*x))^(3/2))-1155/4096*arctanh(cos(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*sin(c+d*x))))/(a^(5/2)*d*sqrt(2))+385/1024*sec(c+d*x)/(a^2*d*sqrt(a+a*sin(c+d*x)))+11/64*sec(c+d*x)^3/(a^2*d*sqrt(a+a*sin(c+d*x)))],

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

# m>0
[(e*cos(c+d*x))^(7/2)*(a+a*sin(c+d*x)),x,5,-2/9*a*(e*cos(c+d*x))^(9/2)/(d*e)+2/7*a*e*(e*cos(c+d*x))^(5/2)*sin(c+d*x)/d+10/21*a*e^4*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(e*cos(c+d*x)))+10/21*a*e^3*sin(c+d*x)*sqrt(e*cos(c+d*x))/d],
[(e*cos(c+d*x))^(5/2)*(a+a*sin(c+d*x)),x,4,-2/7*a*(e*cos(c+d*x))^(7/2)/(d*e)+2/5*a*e*(e*cos(c+d*x))^(3/2)*sin(c+d*x)/d+6/5*a*e^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(e*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(e*cos(c+d*x))^(3/2)*(a+a*sin(c+d*x)),x,4,-2/5*a*(e*cos(c+d*x))^(5/2)/(d*e)+2/3*a*e^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(e*cos(c+d*x)))+2/3*a*e*sin(c+d*x)*sqrt(e*cos(c+d*x))/d],
[(a+a*sin(c+d*x))*sqrt(e*cos(c+d*x)),x,3,-2/3*a*(e*cos(c+d*x))^(3/2)/(d*e)+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(e*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(a+a*sin(c+d*x))/sqrt(e*cos(c+d*x)),x,3,2*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))/(d*sqrt(e*cos(c+d*x)))-2*a*sqrt(e*cos(c+d*x))/(d*e)],
[(a+a*sin(c+d*x))/(e*cos(c+d*x))^(3/2),x,4,2*a/(d*e*sqrt(e*cos(c+d*x)))+2*a*sin(c+d*x)/(d*e*sqrt(e*cos(c+d*x)))-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(e*cos(c+d*x))/(d*e^2*sqrt(cos(c+d*x)))],
[(a+a*sin(c+d*x))/(e*cos(c+d*x))^(5/2),x,4,2/3*a/(d*e*(e*cos(c+d*x))^(3/2))+2/3*a*sin(c+d*x)/(d*e*(e*cos(c+d*x))^(3/2))+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))/(d*e^2*sqrt(e*cos(c+d*x)))],
[(a+a*sin(c+d*x))/(e*cos(c+d*x))^(7/2),x,5,2/5*a/(d*e*(e*cos(c+d*x))^(5/2))+2/5*a*sin(c+d*x)/(d*e*(e*cos(c+d*x))^(5/2))+6/5*a*sin(c+d*x)/(d*e^3*sqrt(e*cos(c+d*x)))-6/5*a*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(e*cos(c+d*x))/(d*e^4*sqrt(cos(c+d*x)))],
[(e*cos(c+d*x))^(7/2)*(a+a*sin(c+d*x))^2,x,6,-26/99*a^2*(e*cos(c+d*x))^(9/2)/(d*e)+26/77*a^2*e*(e*cos(c+d*x))^(5/2)*sin(c+d*x)/d-2/11*(e*cos(c+d*x))^(9/2)*(a^2+a^2*sin(c+d*x))/(d*e)+130/231*a^2*e^4*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(e*cos(c+d*x)))+130/231*a^2*e^3*sin(c+d*x)*sqrt(e*cos(c+d*x))/d],
[(e*cos(c+d*x))^(5/2)*(a+a*sin(c+d*x))^2,x,5,-22/63*a^2*(e*cos(c+d*x))^(7/2)/(d*e)+22/45*a^2*e*(e*cos(c+d*x))^(3/2)*sin(c+d*x)/d-2/9*(e*cos(c+d*x))^(7/2)*(a^2+a^2*sin(c+d*x))/(d*e)+22/15*a^2*e^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(e*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(e*cos(c+d*x))^(3/2)*(a+a*sin(c+d*x))^2,x,5,-18/35*a^2*(e*cos(c+d*x))^(5/2)/(d*e)-2/7*(e*cos(c+d*x))^(5/2)*(a^2+a^2*sin(c+d*x))/(d*e)+6/7*a^2*e^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(e*cos(c+d*x)))+6/7*a^2*e*sin(c+d*x)*sqrt(e*cos(c+d*x))/d],
[(a+a*sin(c+d*x))^2*sqrt(e*cos(c+d*x)),x,4,-14/15*a^2*(e*cos(c+d*x))^(3/2)/(d*e)-2/5*(e*cos(c+d*x))^(3/2)*(a^2+a^2*sin(c+d*x))/(d*e)+14/5*a^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(e*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(a+a*sin(c+d*x))^2/sqrt(e*cos(c+d*x)),x,4,10/3*a^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(e*cos(c+d*x)))-10/3*a^2*sqrt(e*cos(c+d*x))/(d*e)-2/3*(a^2+a^2*sin(c+d*x))*sqrt(e*cos(c+d*x))/(d*e)],
[(a+a*sin(c+d*x))^2/(e*cos(c+d*x))^(3/2),x,4,4*a^4*(e*cos(c+d*x))^(3/2)/(d*e^3*(a^2-a^2*sin(c+d*x)))-6*a^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(e*cos(c+d*x))/(d*e^2*sqrt(cos(c+d*x)))],
[(a+a*sin(c+d*x))^2/(e*cos(c+d*x))^(5/2),x,4,-2/3*a^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*e^2*sqrt(e*cos(c+d*x)))+4/3*a^4*sqrt(e*cos(c+d*x))/(d*e^3*(a^2-a^2*sin(c+d*x)))],
[(a+a*sin(c+d*x))^2/(e*cos(c+d*x))^(7/2),x,5,2/5*a^4*(e*cos(c+d*x))^(3/2)/(d*e^5*(a-a*sin(c+d*x))^2)+2/5*a^4*(e*cos(c+d*x))^(3/2)/(d*e^5*(a^2-a^2*sin(c+d*x)))-2/5*a^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(e*cos(c+d*x))/(d*e^4*sqrt(cos(c+d*x)))],
[(a+a*sin(c+d*x))^2/(e*cos(c+d*x))^(9/2),x,4,2/7*a^2*sin(c+d*x)/(d*e^3*(e*cos(c+d*x))^(3/2))+4/7*(a^2+a^2*sin(c+d*x))/(d*e*(e*cos(c+d*x))^(7/2))+2/7*a^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*e^4*sqrt(e*cos(c+d*x)))],
[(a+a*sin(c+d*x))^2/(e*cos(c+d*x))^(11/2),x,5,2/9*a^2*sin(c+d*x)/(d*e^3*(e*cos(c+d*x))^(5/2))+4/9*(a^2+a^2*sin(c+d*x))/(d*e*(e*cos(c+d*x))^(9/2))+2/3*a^2*sin(c+d*x)/(d*e^5*sqrt(e*cos(c+d*x)))-2/3*a^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(e*cos(c+d*x))/(d*e^6*sqrt(cos(c+d*x)))],
[(e*cos(c+d*x))^(7/2)*(a+a*sin(c+d*x))^3,x,7,-34/99*a^3*(e*cos(c+d*x))^(9/2)/(d*e)+34/77*a^3*e*(e*cos(c+d*x))^(5/2)*sin(c+d*x)/d-2/13*a*(e*cos(c+d*x))^(9/2)*(a+a*sin(c+d*x))^2/(d*e)-34/143*(e*cos(c+d*x))^(9/2)*(a^3+a^3*sin(c+d*x))/(d*e)+170/231*a^3*e^4*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(e*cos(c+d*x)))+170/231*a^3*e^3*sin(c+d*x)*sqrt(e*cos(c+d*x))/d],
[(e*cos(c+d*x))^(5/2)*(a+a*sin(c+d*x))^3,x,6,-10/21*a^3*(e*cos(c+d*x))^(7/2)/(d*e)+2/3*a^3*e*(e*cos(c+d*x))^(3/2)*sin(c+d*x)/d-2/11*a*(e*cos(c+d*x))^(7/2)*(a+a*sin(c+d*x))^2/(d*e)-10/33*(e*cos(c+d*x))^(7/2)*(a^3+a^3*sin(c+d*x))/(d*e)+2*a^3*e^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(e*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(e*cos(c+d*x))^(3/2)*(a+a*sin(c+d*x))^3,x,6,-26/35*a^3*(e*cos(c+d*x))^(5/2)/(d*e)-2/9*a*(e*cos(c+d*x))^(5/2)*(a+a*sin(c+d*x))^2/(d*e)-26/63*(e*cos(c+d*x))^(5/2)*(a^3+a^3*sin(c+d*x))/(d*e)+26/21*a^3*e^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(e*cos(c+d*x)))+26/21*a^3*e*sin(c+d*x)*sqrt(e*cos(c+d*x))/d],
[(a+a*sin(c+d*x))^3*sqrt(e*cos(c+d*x)),x,5,-22/15*a^3*(e*cos(c+d*x))^(3/2)/(d*e)-2/7*a*(e*cos(c+d*x))^(3/2)*(a+a*sin(c+d*x))^2/(d*e)-22/35*(e*cos(c+d*x))^(3/2)*(a^3+a^3*sin(c+d*x))/(d*e)+22/5*a^3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(e*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(a+a*sin(c+d*x))^3/sqrt(e*cos(c+d*x)),x,5,6*a^3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(e*cos(c+d*x)))-6*a^3*sqrt(e*cos(c+d*x))/(d*e)-2/5*a*(a+a*sin(c+d*x))^2*sqrt(e*cos(c+d*x))/(d*e)-6/5*(a^3+a^3*sin(c+d*x))*sqrt(e*cos(c+d*x))/(d*e)],
[(a+a*sin(c+d*x))^3/(e*cos(c+d*x))^(3/2),x,5,14/3*a^3*(e*cos(c+d*x))^(3/2)/(d*e^3)+4*a^5*(e*cos(c+d*x))^(7/2)/(d*e^5*(a-a*sin(c+d*x))^2)-14*a^3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(e*cos(c+d*x))/(d*e^2*sqrt(cos(c+d*x)))],
[(a+a*sin(c+d*x))^3/(e*cos(c+d*x))^(5/2),x,5,4/3*a^5*(e*cos(c+d*x))^(5/2)/(d*e^5*(a-a*sin(c+d*x))^2)-10/3*a^3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*e^2*sqrt(e*cos(c+d*x)))+10/3*a^3*sqrt(e*cos(c+d*x))/(d*e^3)],
[(a+a*sin(c+d*x))^3/(e*cos(c+d*x))^(7/2),x,5,4/5*a^5*(e*cos(c+d*x))^(3/2)/(d*e^5*(a-a*sin(c+d*x))^2)-6/5*a^6*(e*cos(c+d*x))^(3/2)/(d*e^5*(a^3-a^3*sin(c+d*x)))+6/5*a^3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(e*cos(c+d*x))/(d*e^4*sqrt(cos(c+d*x)))],
[(a+a*sin(c+d*x))^3/(e*cos(c+d*x))^(9/2),x,5,-2/21*a^3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*e^4*sqrt(e*cos(c+d*x)))+4/7*a^5*sqrt(e*cos(c+d*x))/(d*e^5*(a-a*sin(c+d*x))^2)-2/21*a^6*sqrt(e*cos(c+d*x))/(d*e^5*(a^3-a^3*sin(c+d*x)))],
[(a+a*sin(c+d*x))^3/(e*cos(c+d*x))^(11/2),x,6,2/9*a^6*(e*cos(c+d*x))^(3/2)/(d*e^7*(a-a*sin(c+d*x))^3)+2/15*a^5*(e*cos(c+d*x))^(3/2)/(d*e^7*(a-a*sin(c+d*x))^2)+2/15*a^6*(e*cos(c+d*x))^(3/2)/(d*e^7*(a^3-a^3*sin(c+d*x)))-2/15*a^3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(e*cos(c+d*x))/(d*e^6*sqrt(cos(c+d*x)))],
[(e*cos(c+d*x))^(3/2)*(a+a*sin(c+d*x))^4,x,7,-442/385*a^4*(e*cos(c+d*x))^(5/2)/(d*e)-2/11*a*(e*cos(c+d*x))^(5/2)*(a+a*sin(c+d*x))^3/(d*e)-34/99*(e*cos(c+d*x))^(5/2)*(a^2+a^2*sin(c+d*x))^2/(d*e)-442/693*(e*cos(c+d*x))^(5/2)*(a^4+a^4*sin(c+d*x))/(d*e)+442/231*a^4*e^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(e*cos(c+d*x)))+442/231*a^4*e*sin(c+d*x)*sqrt(e*cos(c+d*x))/d],
[(a+a*sin(c+d*x))^4*sqrt(e*cos(c+d*x)),x,6,-22/9*a^4*(e*cos(c+d*x))^(3/2)/(d*e)-2/9*a*(e*cos(c+d*x))^(3/2)*(a+a*sin(c+d*x))^3/(d*e)-10/21*(e*cos(c+d*x))^(3/2)*(a^2+a^2*sin(c+d*x))^2/(d*e)-22/21*(e*cos(c+d*x))^(3/2)*(a^4+a^4*sin(c+d*x))/(d*e)+22/3*a^4*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(e*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(a+a*sin(c+d*x))^4/sqrt(e*cos(c+d*x)),x,6,78/7*a^4*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(e*cos(c+d*x)))-78/7*a^4*sqrt(e*cos(c+d*x))/(d*e)-2/7*a*(a+a*sin(c+d*x))^3*sqrt(e*cos(c+d*x))/(d*e)-26/35*(a^2+a^2*sin(c+d*x))^2*sqrt(e*cos(c+d*x))/(d*e)-78/35*(a^4+a^4*sin(c+d*x))*sqrt(e*cos(c+d*x))/(d*e)],
[(a+a*sin(c+d*x))^4/(e*cos(c+d*x))^(3/2),x,6,-154/15*a^4*(e*cos(c+d*x))^(3/2)*sin(c+d*x)/(d*e^3)+4*a^7*(e*cos(c+d*x))^(11/2)/(d*e^7*(a-a*sin(c+d*x))^3)+44/3*a^8*(e*cos(c+d*x))^(7/2)/(d*e^5*(a^4-a^4*sin(c+d*x)))-154/5*a^4*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(e*cos(c+d*x))/(d*e^2*sqrt(cos(c+d*x)))],
[(a+a*sin(c+d*x))^4/(e*cos(c+d*x))^(5/2),x,6,4/3*a^7*(e*cos(c+d*x))^(9/2)/(d*e^7*(a-a*sin(c+d*x))^3)+12*a^8*(e*cos(c+d*x))^(5/2)/(d*e^5*(a^4-a^4*sin(c+d*x)))-10*a^4*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*e^2*sqrt(e*cos(c+d*x)))-10*a^4*sin(c+d*x)*sqrt(e*cos(c+d*x))/(d*e^3)],
[(a+a*sin(c+d*x))^4/(e*cos(c+d*x))^(7/2),x,5,4/5*a^7*(e*cos(c+d*x))^(7/2)/(d*e^7*(a-a*sin(c+d*x))^3)-28/5*a^8*(e*cos(c+d*x))^(3/2)/(d*e^5*(a^4-a^4*sin(c+d*x)))+42/5*a^4*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(e*cos(c+d*x))/(d*e^4*sqrt(cos(c+d*x)))],
[(a+a*sin(c+d*x))^4/(e*cos(c+d*x))^(9/2),x,5,4/7*a^7*(e*cos(c+d*x))^(5/2)/(d*e^7*(a-a*sin(c+d*x))^3)+10/21*a^4*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*e^4*sqrt(e*cos(c+d*x)))-20/21*a^8*sqrt(e*cos(c+d*x))/(d*e^5*(a^4-a^4*sin(c+d*x)))],
[(a+a*sin(c+d*x))^4/(e*cos(c+d*x))^(11/2),x,6,4/9*a^7*(e*cos(c+d*x))^(3/2)/(d*e^7*(a-a*sin(c+d*x))^3)-2/15*a^8*(e*cos(c+d*x))^(3/2)/(d*e^7*(a^2-a^2*sin(c+d*x))^2)-2/15*a^8*(e*cos(c+d*x))^(3/2)/(d*e^7*(a^4-a^4*sin(c+d*x)))+2/15*a^4*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(e*cos(c+d*x))/(d*e^6*sqrt(cos(c+d*x)))],
[(a+a*sin(c+d*x))^4/(e*cos(c+d*x))^(13/2),x,6,-2/77*a^4*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*e^6*sqrt(e*cos(c+d*x)))+4/11*a^7*sqrt(e*cos(c+d*x))/(d*e^7*(a-a*sin(c+d*x))^3)-2/77*a^8*sqrt(e*cos(c+d*x))/(d*e^7*(a^2-a^2*sin(c+d*x))^2)-2/77*a^8*sqrt(e*cos(c+d*x))/(d*e^7*(a^4-a^4*sin(c+d*x)))],

# m<0
[(e*cos(c+d*x))^(11/2)/(a+a*sin(c+d*x)),x,5,2/9*e*(e*cos(c+d*x))^(9/2)/(a*d)+2/7*e^3*(e*cos(c+d*x))^(5/2)*sin(c+d*x)/(a*d)+10/21*e^6*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))/(a*d*sqrt(e*cos(c+d*x)))+10/21*e^5*sin(c+d*x)*sqrt(e*cos(c+d*x))/(a*d)],
[(e*cos(c+d*x))^(9/2)/(a+a*sin(c+d*x)),x,4,2/7*e*(e*cos(c+d*x))^(7/2)/(a*d)+2/5*e^3*(e*cos(c+d*x))^(3/2)*sin(c+d*x)/(a*d)+6/5*e^4*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(e*cos(c+d*x))/(a*d*sqrt(cos(c+d*x)))],
[(e*cos(c+d*x))^(7/2)/(a+a*sin(c+d*x)),x,4,2/5*e*(e*cos(c+d*x))^(5/2)/(a*d)+2/3*e^4*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))/(a*d*sqrt(e*cos(c+d*x)))+2/3*e^3*sin(c+d*x)*sqrt(e*cos(c+d*x))/(a*d)],
[(e*cos(c+d*x))^(5/2)/(a+a*sin(c+d*x)),x,3,2/3*e*(e*cos(c+d*x))^(3/2)/(a*d)+2*e^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(e*cos(c+d*x))/(a*d*sqrt(cos(c+d*x)))],
[(e*cos(c+d*x))^(3/2)/(a+a*sin(c+d*x)),x,3,2*e^2*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))/(a*d*sqrt(e*cos(c+d*x)))+2*e*sqrt(e*cos(c+d*x))/(a*d)],
[sqrt(e*cos(c+d*x))/(a+a*sin(c+d*x)),x,3,-2*(e*cos(c+d*x))^(3/2)/(d*e*(a+a*sin(c+d*x)))-2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(e*cos(c+d*x))/(a*d*sqrt(cos(c+d*x)))],
[1/((a+a*sin(c+d*x))*sqrt(e*cos(c+d*x))),x,3,2/3*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))/(a*d*sqrt(e*cos(c+d*x)))-2/3*sqrt(e*cos(c+d*x))/(d*e*(a+a*sin(c+d*x)))],
[1/((e*cos(c+d*x))^(3/2)*(a+a*sin(c+d*x))),x,4,6/5*sin(c+d*x)/(a*d*e*sqrt(e*cos(c+d*x)))+(-2/5)/(d*e*(a+a*sin(c+d*x))*sqrt(e*cos(c+d*x)))-6/5*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(e*cos(c+d*x))/(a*d*e^2*sqrt(cos(c+d*x)))],
[1/((e*cos(c+d*x))^(5/2)*(a+a*sin(c+d*x))),x,4,10/21*sin(c+d*x)/(a*d*e*(e*cos(c+d*x))^(3/2))+(-2/7)/(d*e*(e*cos(c+d*x))^(3/2)*(a+a*sin(c+d*x)))+10/21*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))/(a*d*e^2*sqrt(e*cos(c+d*x)))],
[1/((e*cos(c+d*x))^(7/2)*(a+a*sin(c+d*x))),x,5,14/45*sin(c+d*x)/(a*d*e*(e*cos(c+d*x))^(5/2))+(-2/9)/(d*e*(e*cos(c+d*x))^(5/2)*(a+a*sin(c+d*x)))+14/15*sin(c+d*x)/(a*d*e^3*sqrt(e*cos(c+d*x)))-14/15*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(e*cos(c+d*x))/(a*d*e^4*sqrt(cos(c+d*x)))],
[(e*cos(c+d*x))^(11/2)/(a+a*sin(c+d*x))^2,x,5,18/35*e^3*(e*cos(c+d*x))^(5/2)*sin(c+d*x)/(a^2*d)+4/5*e*(e*cos(c+d*x))^(9/2)/(d*(a^2+a^2*sin(c+d*x)))+6/7*e^6*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))/(a^2*d*sqrt(e*cos(c+d*x)))+6/7*e^5*sin(c+d*x)*sqrt(e*cos(c+d*x))/(a^2*d)],
[(e*cos(c+d*x))^(9/2)/(a+a*sin(c+d*x))^2,x,4,14/15*e^3*(e*cos(c+d*x))^(3/2)*sin(c+d*x)/(a^2*d)+4/3*e*(e*cos(c+d*x))^(7/2)/(d*(a^2+a^2*sin(c+d*x)))+14/5*e^4*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(e*cos(c+d*x))/(a^2*d*sqrt(cos(c+d*x)))],
[(e*cos(c+d*x))^(7/2)/(a+a*sin(c+d*x))^2,x,4,4*e*(e*cos(c+d*x))^(5/2)/(d*(a^2+a^2*sin(c+d*x)))+10/3*e^4*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))/(a^2*d*sqrt(e*cos(c+d*x)))+10/3*e^3*sin(c+d*x)*sqrt(e*cos(c+d*x))/(a^2*d)],
[(e*cos(c+d*x))^(5/2)/(a+a*sin(c+d*x))^2,x,3,-4*e*(e*cos(c+d*x))^(3/2)/(d*(a^2+a^2*sin(c+d*x)))-6*e^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(e*cos(c+d*x))/(a^2*d*sqrt(cos(c+d*x)))],
[(e*cos(c+d*x))^(3/2)/(a+a*sin(c+d*x))^2,x,3,-2/3*e^2*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))/(a^2*d*sqrt(e*cos(c+d*x)))-4/3*e*sqrt(e*cos(c+d*x))/(d*(a^2+a^2*sin(c+d*x)))],
[sqrt(e*cos(c+d*x))/(a+a*sin(c+d*x))^2,x,4,-2/5*(e*cos(c+d*x))^(3/2)/(d*e*(a+a*sin(c+d*x))^2)-2/5*(e*cos(c+d*x))^(3/2)/(d*e*(a^2+a^2*sin(c+d*x)))-2/5*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(e*cos(c+d*x))/(a^2*d*sqrt(cos(c+d*x)))],
[1/((a+a*sin(c+d*x))^2*sqrt(e*cos(c+d*x))),x,4,2/7*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))/(a^2*d*sqrt(e*cos(c+d*x)))-2/7*sqrt(e*cos(c+d*x))/(d*e*(a+a*sin(c+d*x))^2)-2/7*sqrt(e*cos(c+d*x))/(d*e*(a^2+a^2*sin(c+d*x)))],
[1/((e*cos(c+d*x))^(3/2)*(a+a*sin(c+d*x))^2),x,5,2/3*sin(c+d*x)/(a^2*d*e*sqrt(e*cos(c+d*x)))+(-2/9)/(d*e*(a+a*sin(c+d*x))^2*sqrt(e*cos(c+d*x)))+(-2/9)/(d*e*(a^2+a^2*sin(c+d*x))*sqrt(e*cos(c+d*x)))-2/3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(e*cos(c+d*x))/(a^2*d*e^2*sqrt(cos(c+d*x)))],
[1/((e*cos(c+d*x))^(5/2)*(a+a*sin(c+d*x))^2),x,5,10/33*sin(c+d*x)/(a^2*d*e*(e*cos(c+d*x))^(3/2))+(-2/11)/(d*e*(e*cos(c+d*x))^(3/2)*(a+a*sin(c+d*x))^2)+(-2/11)/(d*e*(e*cos(c+d*x))^(3/2)*(a^2+a^2*sin(c+d*x)))+10/33*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))/(a^2*d*e^2*sqrt(e*cos(c+d*x)))],
[1/((e*cos(c+d*x))^(7/2)*(a+a*sin(c+d*x))^2),x,6,14/65*sin(c+d*x)/(a^2*d*e*(e*cos(c+d*x))^(5/2))+(-2/13)/(d*e*(e*cos(c+d*x))^(5/2)*(a+a*sin(c+d*x))^2)+(-2/13)/(d*e*(e*cos(c+d*x))^(5/2)*(a^2+a^2*sin(c+d*x)))+42/65*sin(c+d*x)/(a^2*d*e^3*sqrt(e*cos(c+d*x)))-42/65*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(e*cos(c+d*x))/(a^2*d*e^4*sqrt(cos(c+d*x)))],
[(e*cos(c+d*x))^(15/2)/(a+a*sin(c+d*x))^3,x,6,26/45*e^3*(e*cos(c+d*x))^(9/2)/(a^3*d)+26/35*e^5*(e*cos(c+d*x))^(5/2)*sin(c+d*x)/(a^3*d)+4/5*e*(e*cos(c+d*x))^(13/2)/(a*d*(a+a*sin(c+d*x))^2)+26/21*e^8*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))/(a^3*d*sqrt(e*cos(c+d*x)))+26/21*e^7*sin(c+d*x)*sqrt(e*cos(c+d*x))/(a^3*d)],
[(e*cos(c+d*x))^(13/2)/(a+a*sin(c+d*x))^3,x,5,22/21*e^3*(e*cos(c+d*x))^(7/2)/(a^3*d)+22/15*e^5*(e*cos(c+d*x))^(3/2)*sin(c+d*x)/(a^3*d)+4/3*e*(e*cos(c+d*x))^(11/2)/(a*d*(a+a*sin(c+d*x))^2)+22/5*e^6*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(e*cos(c+d*x))/(a^3*d*sqrt(cos(c+d*x)))],
[(e*cos(c+d*x))^(11/2)/(a+a*sin(c+d*x))^3,x,5,18/5*e^3*(e*cos(c+d*x))^(5/2)/(a^3*d)+4*e*(e*cos(c+d*x))^(9/2)/(a*d*(a+a*sin(c+d*x))^2)+6*e^6*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))/(a^3*d*sqrt(e*cos(c+d*x)))+6*e^5*sin(c+d*x)*sqrt(e*cos(c+d*x))/(a^3*d)],
[(e*cos(c+d*x))^(9/2)/(a+a*sin(c+d*x))^3,x,4,-14/3*e^3*(e*cos(c+d*x))^(3/2)/(a^3*d)-4*e*(e*cos(c+d*x))^(7/2)/(a*d*(a+a*sin(c+d*x))^2)-14*e^4*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(e*cos(c+d*x))/(a^3*d*sqrt(cos(c+d*x)))],
[(e*cos(c+d*x))^(7/2)/(a+a*sin(c+d*x))^3,x,4,-4/3*e*(e*cos(c+d*x))^(5/2)/(a*d*(a+a*sin(c+d*x))^2)-10/3*e^4*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))/(a^3*d*sqrt(e*cos(c+d*x)))-10/3*e^3*sqrt(e*cos(c+d*x))/(a^3*d)],
[(e*cos(c+d*x))^(5/2)/(a+a*sin(c+d*x))^3,x,4,-4/5*e*(e*cos(c+d*x))^(3/2)/(a*d*(a+a*sin(c+d*x))^2)+6/5*e*(e*cos(c+d*x))^(3/2)/(d*(a^3+a^3*sin(c+d*x)))+6/5*e^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(e*cos(c+d*x))/(a^3*d*sqrt(cos(c+d*x)))],
[(e*cos(c+d*x))^(3/2)/(a+a*sin(c+d*x))^3,x,4,-2/21*e^2*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))/(a^3*d*sqrt(e*cos(c+d*x)))-4/7*e*sqrt(e*cos(c+d*x))/(a*d*(a+a*sin(c+d*x))^2)+2/21*e*sqrt(e*cos(c+d*x))/(d*(a^3+a^3*sin(c+d*x)))],
[sqrt(e*cos(c+d*x))/(a+a*sin(c+d*x))^3,x,5,-2/9*(e*cos(c+d*x))^(3/2)/(d*e*(a+a*sin(c+d*x))^3)-2/15*(e*cos(c+d*x))^(3/2)/(a*d*e*(a+a*sin(c+d*x))^2)-2/15*(e*cos(c+d*x))^(3/2)/(d*e*(a^3+a^3*sin(c+d*x)))-2/15*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(e*cos(c+d*x))/(a^3*d*sqrt(cos(c+d*x)))],
[1/((a+a*sin(c+d*x))^3*sqrt(e*cos(c+d*x))),x,5,10/77*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))/(a^3*d*sqrt(e*cos(c+d*x)))-2/11*sqrt(e*cos(c+d*x))/(d*e*(a+a*sin(c+d*x))^3)-10/77*sqrt(e*cos(c+d*x))/(a*d*e*(a+a*sin(c+d*x))^2)-10/77*sqrt(e*cos(c+d*x))/(d*e*(a^3+a^3*sin(c+d*x)))],
[1/((e*cos(c+d*x))^(3/2)*(a+a*sin(c+d*x))^3),x,6,14/39*sin(c+d*x)/(a^3*d*e*sqrt(e*cos(c+d*x)))+(-2/13)/(d*e*(a+a*sin(c+d*x))^3*sqrt(e*cos(c+d*x)))+(-14/117)/(a*d*e*(a+a*sin(c+d*x))^2*sqrt(e*cos(c+d*x)))+(-14/117)/(d*e*(a^3+a^3*sin(c+d*x))*sqrt(e*cos(c+d*x)))-14/39*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(e*cos(c+d*x))/(a^3*d*e^2*sqrt(cos(c+d*x)))],
[(e*cos(c+d*x))^(15/2)/(a+a*sin(c+d*x))^4,x,6,234/35*e^5*(e*cos(c+d*x))^(5/2)*sin(c+d*x)/(a^4*d)+4*e*(e*cos(c+d*x))^(13/2)/(a*d*(a+a*sin(c+d*x))^3)+52/5*e^3*(e*cos(c+d*x))^(9/2)/(d*(a^4+a^4*sin(c+d*x)))+78/7*e^8*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))/(a^4*d*sqrt(e*cos(c+d*x)))+78/7*e^7*sin(c+d*x)*sqrt(e*cos(c+d*x))/(a^4*d)],
[(e*cos(c+d*x))^(13/2)/(a+a*sin(c+d*x))^4,x,5,-154/15*e^5*(e*cos(c+d*x))^(3/2)*sin(c+d*x)/(a^4*d)-4*e*(e*cos(c+d*x))^(11/2)/(a*d*(a+a*sin(c+d*x))^3)-44/3*e^3*(e*cos(c+d*x))^(7/2)/(d*(a^4+a^4*sin(c+d*x)))-154/5*e^6*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(e*cos(c+d*x))/(a^4*d*sqrt(cos(c+d*x)))],
[(e*cos(c+d*x))^(11/2)/(a+a*sin(c+d*x))^4,x,5,-4/3*e*(e*cos(c+d*x))^(9/2)/(a*d*(a+a*sin(c+d*x))^3)-12*e^3*(e*cos(c+d*x))^(5/2)/(d*(a^4+a^4*sin(c+d*x)))-10*e^6*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))/(a^4*d*sqrt(e*cos(c+d*x)))-10*e^5*sin(c+d*x)*sqrt(e*cos(c+d*x))/(a^4*d)],
[(e*cos(c+d*x))^(9/2)/(a+a*sin(c+d*x))^4,x,4,-4/5*e*(e*cos(c+d*x))^(7/2)/(a*d*(a+a*sin(c+d*x))^3)+28/5*e^3*(e*cos(c+d*x))^(3/2)/(d*(a^4+a^4*sin(c+d*x)))+42/5*e^4*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(e*cos(c+d*x))/(a^4*d*sqrt(cos(c+d*x)))],
[(e*cos(c+d*x))^(7/2)/(a+a*sin(c+d*x))^4,x,4,-4/7*e*(e*cos(c+d*x))^(5/2)/(a*d*(a+a*sin(c+d*x))^3)+10/21*e^4*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))/(a^4*d*sqrt(e*cos(c+d*x)))+20/21*e^3*sqrt(e*cos(c+d*x))/(d*(a^4+a^4*sin(c+d*x)))],
[(e*cos(c+d*x))^(5/2)/(a+a*sin(c+d*x))^4,x,5,-4/9*e*(e*cos(c+d*x))^(3/2)/(a*d*(a+a*sin(c+d*x))^3)+2/15*e*(e*cos(c+d*x))^(3/2)/(d*(a^2+a^2*sin(c+d*x))^2)+2/15*e*(e*cos(c+d*x))^(3/2)/(d*(a^4+a^4*sin(c+d*x)))+2/15*e^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(e*cos(c+d*x))/(a^4*d*sqrt(cos(c+d*x)))],
[(e*cos(c+d*x))^(3/2)/(a+a*sin(c+d*x))^4,x,5,-2/77*e^2*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))/(a^4*d*sqrt(e*cos(c+d*x)))-4/11*e*sqrt(e*cos(c+d*x))/(a*d*(a+a*sin(c+d*x))^3)+2/77*e*sqrt(e*cos(c+d*x))/(d*(a^2+a^2*sin(c+d*x))^2)+2/77*e*sqrt(e*cos(c+d*x))/(d*(a^4+a^4*sin(c+d*x)))],
[sqrt(e*cos(c+d*x))/(a+a*sin(c+d*x))^4,x,6,-2/13*(e*cos(c+d*x))^(3/2)/(d*e*(a+a*sin(c+d*x))^4)-10/117*(e*cos(c+d*x))^(3/2)/(a*d*e*(a+a*sin(c+d*x))^3)-2/39*(e*cos(c+d*x))^(3/2)/(d*e*(a^2+a^2*sin(c+d*x))^2)-2/39*(e*cos(c+d*x))^(3/2)/(d*e*(a^4+a^4*sin(c+d*x)))-2/39*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(e*cos(c+d*x))/(a^4*d*sqrt(cos(c+d*x)))],
[1/((a+a*sin(c+d*x))^4*sqrt(e*cos(c+d*x))),x,6,2/33*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))/(a^4*d*sqrt(e*cos(c+d*x)))-2/15*sqrt(e*cos(c+d*x))/(d*e*(a+a*sin(c+d*x))^4)-14/165*sqrt(e*cos(c+d*x))/(a*d*e*(a+a*sin(c+d*x))^3)-2/33*sqrt(e*cos(c+d*x))/(d*e*(a^2+a^2*sin(c+d*x))^2)-2/33*sqrt(e*cos(c+d*x))/(d*e*(a^4+a^4*sin(c+d*x)))],
[1/((e*cos(c+d*x))^(3/2)*(a+a*sin(c+d*x))^4),x,7,42/221*sin(c+d*x)/(a^4*d*e*sqrt(e*cos(c+d*x)))+(-2/17)/(d*e*(a+a*sin(c+d*x))^4*sqrt(e*cos(c+d*x)))+(-18/221)/(a*d*e*(a+a*sin(c+d*x))^3*sqrt(e*cos(c+d*x)))+(-14/221)/(d*e*(a^2+a^2*sin(c+d*x))^2*sqrt(e*cos(c+d*x)))+(-14/221)/(d*e*(a^4+a^4*sin(c+d*x))*sqrt(e*cos(c+d*x)))-42/221*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(e*cos(c+d*x))/(a^4*d*e^2*sqrt(cos(c+d*x)))],

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

# m>0
[(e*cos(c+d*x))^(3/2)*sqrt(a+a*sin(c+d*x)),x,8,-1/2*a*(e*cos(c+d*x))^(5/2)/(d*e*sqrt(a+a*sin(c+d*x)))+3/4*e*sqrt(e*cos(c+d*x))*sqrt(a+a*sin(c+d*x))/d-3/4*e^(3/2)*arcsinh(sqrt(e*cos(c+d*x))/sqrt(e))*sqrt(1+cos(c+d*x))*sqrt(a+a*sin(c+d*x))/(d*(1+cos(c+d*x)+sin(c+d*x)))+3/4*e^(3/2)*arctan(sin(c+d*x)*sqrt(e)/(sqrt(e*cos(c+d*x))*sqrt(1+cos(c+d*x))))*sqrt(1+cos(c+d*x))*sqrt(a+a*sin(c+d*x))/(d*(1+cos(c+d*x)+sin(c+d*x)))],
[sqrt(e*cos(c+d*x))*sqrt(a+a*sin(c+d*x)),x,7,-a*(e*cos(c+d*x))^(3/2)/(d*e*sqrt(a+a*sin(c+d*x)))+arcsinh(sqrt(e*cos(c+d*x))/sqrt(e))*sqrt(e)*sqrt(1+cos(c+d*x))*sqrt(a+a*sin(c+d*x))/(d*(1+cos(c+d*x)+sin(c+d*x)))+arctan(sin(c+d*x)*sqrt(e)/(sqrt(e*cos(c+d*x))*sqrt(1+cos(c+d*x))))*sqrt(e)*sqrt(1+cos(c+d*x))*sqrt(a+a*sin(c+d*x))/(d*(1+cos(c+d*x)+sin(c+d*x)))],
[sqrt(a+a*sin(c+d*x))/sqrt(e*cos(c+d*x)),x,6,-2*arcsinh(sqrt(e*cos(c+d*x))/sqrt(e))*sqrt(1+cos(c+d*x))*sqrt(a+a*sin(c+d*x))/(d*(1+cos(c+d*x)+sin(c+d*x))*sqrt(e))+2*arctan(sin(c+d*x)*sqrt(e)/(sqrt(e*cos(c+d*x))*sqrt(1+cos(c+d*x))))*sqrt(1+cos(c+d*x))*sqrt(a+a*sin(c+d*x))/(d*(1+cos(c+d*x)+sin(c+d*x))*sqrt(e))],
[sqrt(a+a*sin(c+d*x))/(e*cos(c+d*x))^(3/2),x,1,2*sqrt(a+a*sin(c+d*x))/(d*e*sqrt(e*cos(c+d*x)))],
[sqrt(a+a*sin(c+d*x))/(e*cos(c+d*x))^(5/2),x,2,4/3*(a+a*sin(c+d*x))^(3/2)/(a*d*e*(e*cos(c+d*x))^(3/2))-2*sqrt(a+a*sin(c+d*x))/(d*e*(e*cos(c+d*x))^(3/2))],
[sqrt(a+a*sin(c+d*x))/(e*cos(c+d*x))^(7/2),x,3,8/3*(a+a*sin(c+d*x))^(3/2)/(a*d*e*(e*cos(c+d*x))^(5/2))-16/15*(a+a*sin(c+d*x))^(5/2)/(a^2*d*e*(e*cos(c+d*x))^(5/2))-2/3*sqrt(a+a*sin(c+d*x))/(d*e*(e*cos(c+d*x))^(5/2))],
[sqrt(a+a*sin(c+d*x))/(e*cos(c+d*x))^(9/2),x,4,-12/5*(a+a*sin(c+d*x))^(3/2)/(a*d*e*(e*cos(c+d*x))^(7/2))+16/5*(a+a*sin(c+d*x))^(5/2)/(a^2*d*e*(e*cos(c+d*x))^(7/2))-32/35*(a+a*sin(c+d*x))^(7/2)/(a^3*d*e*(e*cos(c+d*x))^(7/2))-2/5*sqrt(a+a*sin(c+d*x))/(d*e*(e*cos(c+d*x))^(7/2))],
[(e*cos(c+d*x))^(5/2)*(a+a*sin(c+d*x))^(3/2),x,10,-15/32*a^3*(e*cos(c+d*x))^(7/2)/(d*e*(a+a*sin(c+d*x))^(3/2))+15/64*a^2*e*(e*cos(c+d*x))^(3/2)/(d*sqrt(a+a*sin(c+d*x)))-3/8*a^2*(e*cos(c+d*x))^(7/2)/(d*e*sqrt(a+a*sin(c+d*x)))-1/4*a*(e*cos(c+d*x))^(7/2)*sqrt(a+a*sin(c+d*x))/(d*e)+45/64*a*e^(5/2)*arcsinh(sqrt(e*cos(c+d*x))/sqrt(e))*sqrt(1+cos(c+d*x))*sqrt(a+a*sin(c+d*x))/(d*(1+cos(c+d*x)+sin(c+d*x)))+45/64*a*e^(5/2)*arctan(sin(c+d*x)*sqrt(e)/(sqrt(e*cos(c+d*x))*sqrt(1+cos(c+d*x))))*sqrt(1+cos(c+d*x))*sqrt(a+a*sin(c+d*x))/(d*(1+cos(c+d*x)+sin(c+d*x)))],
[(e*cos(c+d*x))^(3/2)*(a+a*sin(c+d*x))^(3/2),x,9,-7/12*a^2*(e*cos(c+d*x))^(5/2)/(d*e*sqrt(a+a*sin(c+d*x)))-1/3*a*(e*cos(c+d*x))^(5/2)*sqrt(a+a*sin(c+d*x))/(d*e)+7/8*a*e*sqrt(e*cos(c+d*x))*sqrt(a+a*sin(c+d*x))/d-7/8*a*e^(3/2)*arcsinh(sqrt(e*cos(c+d*x))/sqrt(e))*sqrt(1+cos(c+d*x))*sqrt(a+a*sin(c+d*x))/(d*(1+cos(c+d*x)+sin(c+d*x)))+7/8*a*e^(3/2)*arctan(sin(c+d*x)*sqrt(e)/(sqrt(e*cos(c+d*x))*sqrt(1+cos(c+d*x))))*sqrt(1+cos(c+d*x))*sqrt(a+a*sin(c+d*x))/(d*(1+cos(c+d*x)+sin(c+d*x)))],
[(a+a*sin(c+d*x))^(3/2)*sqrt(e*cos(c+d*x)),x,8,-5/4*a^2*(e*cos(c+d*x))^(3/2)/(d*e*sqrt(a+a*sin(c+d*x)))-1/2*a*(e*cos(c+d*x))^(3/2)*sqrt(a+a*sin(c+d*x))/(d*e)+5/4*a*arcsinh(sqrt(e*cos(c+d*x))/sqrt(e))*sqrt(e)*sqrt(1+cos(c+d*x))*sqrt(a+a*sin(c+d*x))/(d*(1+cos(c+d*x)+sin(c+d*x)))+5/4*a*arctan(sin(c+d*x)*sqrt(e)/(sqrt(e*cos(c+d*x))*sqrt(1+cos(c+d*x))))*sqrt(e)*sqrt(1+cos(c+d*x))*sqrt(a+a*sin(c+d*x))/(d*(1+cos(c+d*x)+sin(c+d*x)))],
[(a+a*sin(c+d*x))^(3/2)/sqrt(e*cos(c+d*x)),x,7,-a*sqrt(e*cos(c+d*x))*sqrt(a+a*sin(c+d*x))/(d*e)-3*a*arcsinh(sqrt(e*cos(c+d*x))/sqrt(e))*sqrt(1+cos(c+d*x))*sqrt(a+a*sin(c+d*x))/(d*(1+cos(c+d*x)+sin(c+d*x))*sqrt(e))+3*a*arctan(sin(c+d*x)*sqrt(e)/(sqrt(e*cos(c+d*x))*sqrt(1+cos(c+d*x))))*sqrt(1+cos(c+d*x))*sqrt(a+a*sin(c+d*x))/(d*(1+cos(c+d*x)+sin(c+d*x))*sqrt(e))],
[(a+a*sin(c+d*x))^(3/2)/(e*cos(c+d*x))^(3/2),x,7,4*a*sqrt(a+a*sin(c+d*x))/(d*e*sqrt(e*cos(c+d*x)))-2*a^2*arcsinh(sqrt(e*cos(c+d*x))/sqrt(e))*sqrt(1+cos(c+d*x))*sqrt(a+a*sin(c+d*x))/(d*e^(3/2)*(a+a*cos(c+d*x)+a*sin(c+d*x)))-2*a^2*arctan(sin(c+d*x)*sqrt(e)/(sqrt(e*cos(c+d*x))*sqrt(1+cos(c+d*x))))*sqrt(1+cos(c+d*x))*sqrt(a+a*sin(c+d*x))/(d*e^(3/2)*(a+a*cos(c+d*x)+a*sin(c+d*x)))],
[(a+a*sin(c+d*x))^(3/2)/(e*cos(c+d*x))^(5/2),x,1,2/3*(a+a*sin(c+d*x))^(3/2)/(d*e*(e*cos(c+d*x))^(3/2))],
[(a+a*sin(c+d*x))^(3/2)/(e*cos(c+d*x))^(7/2),x,2,2*(a+a*sin(c+d*x))^(3/2)/(d*e*(e*cos(c+d*x))^(5/2))-4/5*(a+a*sin(c+d*x))^(5/2)/(a*d*e*(e*cos(c+d*x))^(5/2))],
[(a+a*sin(c+d*x))^(3/2)/(e*cos(c+d*x))^(9/2),x,3,-2*(a+a*sin(c+d*x))^(3/2)/(d*e*(e*cos(c+d*x))^(7/2))+8/3*(a+a*sin(c+d*x))^(5/2)/(a*d*e*(e*cos(c+d*x))^(7/2))-16/21*(a+a*sin(c+d*x))^(7/2)/(a^2*d*e*(e*cos(c+d*x))^(7/2))],
[(a+a*sin(c+d*x))^(3/2)/(e*cos(c+d*x))^(11/2),x,4,-2/3*(a+a*sin(c+d*x))^(3/2)/(d*e*(e*cos(c+d*x))^(9/2))+4*(a+a*sin(c+d*x))^(5/2)/(a*d*e*(e*cos(c+d*x))^(9/2))-16/5*(a+a*sin(c+d*x))^(7/2)/(a^2*d*e*(e*cos(c+d*x))^(9/2))+32/45*(a+a*sin(c+d*x))^(9/2)/(a^3*d*e*(e*cos(c+d*x))^(9/2))],
[(e*cos(c+d*x))^(3/2)*(a+a*sin(c+d*x))^(5/2),x,10,-1/4*a*(e*cos(c+d*x))^(5/2)*(a+a*sin(c+d*x))^(3/2)/(d*e)-77/96*a^3*(e*cos(c+d*x))^(5/2)/(d*e*sqrt(a+a*sin(c+d*x)))-11/24*a^2*(e*cos(c+d*x))^(5/2)*sqrt(a+a*sin(c+d*x))/(d*e)+77/64*a^2*e*sqrt(e*cos(c+d*x))*sqrt(a+a*sin(c+d*x))/d-77/64*a^2*e^(3/2)*arcsinh(sqrt(e*cos(c+d*x))/sqrt(e))*sqrt(1+cos(c+d*x))*sqrt(a+a*sin(c+d*x))/(d*(1+cos(c+d*x)+sin(c+d*x)))+77/64*a^2*e^(3/2)*arctan(sin(c+d*x)*sqrt(e)/(sqrt(e*cos(c+d*x))*sqrt(1+cos(c+d*x))))*sqrt(1+cos(c+d*x))*sqrt(a+a*sin(c+d*x))/(d*(1+cos(c+d*x)+sin(c+d*x)))],
[(a+a*sin(c+d*x))^(5/2)*sqrt(e*cos(c+d*x)),x,9,-1/3*a*(e*cos(c+d*x))^(3/2)*(a+a*sin(c+d*x))^(3/2)/(d*e)-15/8*a^3*(e*cos(c+d*x))^(3/2)/(d*e*sqrt(a+a*sin(c+d*x)))-3/4*a^2*(e*cos(c+d*x))^(3/2)*sqrt(a+a*sin(c+d*x))/(d*e)+15/8*a^2*arcsinh(sqrt(e*cos(c+d*x))/sqrt(e))*sqrt(e)*sqrt(1+cos(c+d*x))*sqrt(a+a*sin(c+d*x))/(d*(1+cos(c+d*x)+sin(c+d*x)))+15/8*a^2*arctan(sin(c+d*x)*sqrt(e)/(sqrt(e*cos(c+d*x))*sqrt(1+cos(c+d*x))))*sqrt(e)*sqrt(1+cos(c+d*x))*sqrt(a+a*sin(c+d*x))/(d*(1+cos(c+d*x)+sin(c+d*x)))],
[(a+a*sin(c+d*x))^(5/2)/sqrt(e*cos(c+d*x)),x,8,-1/2*a*(a+a*sin(c+d*x))^(3/2)*sqrt(e*cos(c+d*x))/(d*e)-7/4*a^2*sqrt(e*cos(c+d*x))*sqrt(a+a*sin(c+d*x))/(d*e)-21/4*a^2*arcsinh(sqrt(e*cos(c+d*x))/sqrt(e))*sqrt(1+cos(c+d*x))*sqrt(a+a*sin(c+d*x))/(d*(1+cos(c+d*x)+sin(c+d*x))*sqrt(e))+21/4*a^2*arctan(sin(c+d*x)*sqrt(e)/(sqrt(e*cos(c+d*x))*sqrt(1+cos(c+d*x))))*sqrt(1+cos(c+d*x))*sqrt(a+a*sin(c+d*x))/(d*(1+cos(c+d*x)+sin(c+d*x))*sqrt(e))],
[(a+a*sin(c+d*x))^(5/2)/(e*cos(c+d*x))^(3/2),x,8,4*a*(a+a*sin(c+d*x))^(3/2)/(d*e*sqrt(e*cos(c+d*x)))+5*a^3*(e*cos(c+d*x))^(3/2)/(d*e^3*sqrt(a+a*sin(c+d*x)))-5*a^2*arcsinh(sqrt(e*cos(c+d*x))/sqrt(e))*sqrt(1+cos(c+d*x))*sqrt(a+a*sin(c+d*x))/(d*e^(3/2)*(1+cos(c+d*x)+sin(c+d*x)))-5*a^2*arctan(sin(c+d*x)*sqrt(e)/(sqrt(e*cos(c+d*x))*sqrt(1+cos(c+d*x))))*sqrt(1+cos(c+d*x))*sqrt(a+a*sin(c+d*x))/(d*e^(3/2)*(1+cos(c+d*x)+sin(c+d*x)))],
[(a+a*sin(c+d*x))^(5/2)/(e*cos(c+d*x))^(5/2),x,7,4/3*a*(a+a*sin(c+d*x))^(3/2)/(d*e*(e*cos(c+d*x))^(3/2))+2*a^2*arcsinh(sqrt(e*cos(c+d*x))/sqrt(e))*sqrt(1+cos(c+d*x))*sqrt(a+a*sin(c+d*x))/(d*e^(5/2)*(1+cos(c+d*x)+sin(c+d*x)))-2*a^2*arctan(sin(c+d*x)*sqrt(e)/(sqrt(e*cos(c+d*x))*sqrt(1+cos(c+d*x))))*sqrt(1+cos(c+d*x))*sqrt(a+a*sin(c+d*x))/(d*e^(5/2)*(1+cos(c+d*x)+sin(c+d*x)))],
[(a+a*sin(c+d*x))^(5/2)/(e*cos(c+d*x))^(7/2),x,1,2/5*(a+a*sin(c+d*x))^(5/2)/(d*e*(e*cos(c+d*x))^(5/2))],
[(a+a*sin(c+d*x))^(5/2)/(e*cos(c+d*x))^(9/2),x,2,2/3*(a+a*sin(c+d*x))^(5/2)/(d*e*(e*cos(c+d*x))^(7/2))-4/21*(a+a*sin(c+d*x))^(7/2)/(a*d*e*(e*cos(c+d*x))^(7/2))],
[(a+a*sin(c+d*x))^(5/2)/(e*cos(c+d*x))^(11/2),x,3,2*(a+a*sin(c+d*x))^(5/2)/(d*e*(e*cos(c+d*x))^(9/2))-8/5*(a+a*sin(c+d*x))^(7/2)/(a*d*e*(e*cos(c+d*x))^(9/2))+16/45*(a+a*sin(c+d*x))^(9/2)/(a^2*d*e*(e*cos(c+d*x))^(9/2))],
[(a+a*sin(c+d*x))^(5/2)/(e*cos(c+d*x))^(13/2),x,4,-2*(a+a*sin(c+d*x))^(5/2)/(d*e*(e*cos(c+d*x))^(11/2))+4*(a+a*sin(c+d*x))^(7/2)/(a*d*e*(e*cos(c+d*x))^(11/2))-16/7*(a+a*sin(c+d*x))^(9/2)/(a^2*d*e*(e*cos(c+d*x))^(11/2))+32/77*(a+a*sin(c+d*x))^(11/2)/(a^3*d*e*(e*cos(c+d*x))^(11/2))],

# m<0
[(e*cos(c+d*x))^(5/2)/sqrt(a+a*sin(c+d*x)),x,8,-1/2*a*(e*cos(c+d*x))^(7/2)/(d*e*(a+a*sin(c+d*x))^(3/2))+1/4*e*(e*cos(c+d*x))^(3/2)/(d*sqrt(a+a*sin(c+d*x)))+3/4*e^(5/2)*arcsinh(sqrt(e*cos(c+d*x))/sqrt(e))*sqrt(1+cos(c+d*x))*sqrt(a+a*sin(c+d*x))/(d*(a+a*cos(c+d*x)+a*sin(c+d*x)))+3/4*e^(5/2)*arctan(sin(c+d*x)*sqrt(e)/(sqrt(e*cos(c+d*x))*sqrt(1+cos(c+d*x))))*sqrt(1+cos(c+d*x))*sqrt(a+a*sin(c+d*x))/(d*(a+a*cos(c+d*x)+a*sin(c+d*x)))],
[(e*cos(c+d*x))^(3/2)/sqrt(a+a*sin(c+d*x)),x,7,e*sqrt(e*cos(c+d*x))*sqrt(a+a*sin(c+d*x))/(a*d)-e^(3/2)*arcsinh(sqrt(e*cos(c+d*x))/sqrt(e))*sqrt(1+cos(c+d*x))*sqrt(a+a*sin(c+d*x))/(a*d*(1+cos(c+d*x)+sin(c+d*x)))+e^(3/2)*arctan(sin(c+d*x)*sqrt(e)/(sqrt(e*cos(c+d*x))*sqrt(1+cos(c+d*x))))*sqrt(1+cos(c+d*x))*sqrt(a+a*sin(c+d*x))/(a*d*(1+cos(c+d*x)+sin(c+d*x)))],
[sqrt(e*cos(c+d*x))/sqrt(a+a*sin(c+d*x)),x,6,2*arcsinh(sqrt(e*cos(c+d*x))/sqrt(e))*sqrt(e)*sqrt(1+cos(c+d*x))*sqrt(a+a*sin(c+d*x))/(d*(a+a*cos(c+d*x)+a*sin(c+d*x)))+2*arctan(sin(c+d*x)*sqrt(e)/(sqrt(e*cos(c+d*x))*sqrt(1+cos(c+d*x))))*sqrt(e)*sqrt(1+cos(c+d*x))*sqrt(a+a*sin(c+d*x))/(d*(a+a*cos(c+d*x)+a*sin(c+d*x)))],
[1/(sqrt(e*cos(c+d*x))*sqrt(a+a*sin(c+d*x))),x,1,-2*sqrt(e*cos(c+d*x))/(d*e*sqrt(a+a*sin(c+d*x)))],
[1/((e*cos(c+d*x))^(3/2)*sqrt(a+a*sin(c+d*x))),x,2,(-2/3)/(d*e*sqrt(e*cos(c+d*x))*sqrt(a+a*sin(c+d*x)))+4/3*sqrt(a+a*sin(c+d*x))/(a*d*e*sqrt(e*cos(c+d*x)))],
[1/((e*cos(c+d*x))^(5/2)*sqrt(a+a*sin(c+d*x))),x,3,16/15*(a+a*sin(c+d*x))^(3/2)/(a^2*d*e*(e*cos(c+d*x))^(3/2))+(-2/5)/(d*e*(e*cos(c+d*x))^(3/2)*sqrt(a+a*sin(c+d*x)))-8/5*sqrt(a+a*sin(c+d*x))/(a*d*e*(e*cos(c+d*x))^(3/2))],
[1/((e*cos(c+d*x))^(7/2)*sqrt(a+a*sin(c+d*x))),x,4,16/7*(a+a*sin(c+d*x))^(3/2)/(a^2*d*e*(e*cos(c+d*x))^(5/2))-32/35*(a+a*sin(c+d*x))^(5/2)/(a^3*d*e*(e*cos(c+d*x))^(5/2))+(-2/7)/(d*e*(e*cos(c+d*x))^(5/2)*sqrt(a+a*sin(c+d*x)))-4/7*sqrt(a+a*sin(c+d*x))/(a*d*e*(e*cos(c+d*x))^(5/2))],
[(e*cos(c+d*x))^(7/2)/(a+a*sin(c+d*x))^(3/2),x,8,1/2*e*(e*cos(c+d*x))^(5/2)/(a*d*sqrt(a+a*sin(c+d*x)))+5/4*e^3*sqrt(e*cos(c+d*x))*sqrt(a+a*sin(c+d*x))/(a^2*d)-5/4*e^(7/2)*arcsinh(sqrt(e*cos(c+d*x))/sqrt(e))*sqrt(1+cos(c+d*x))*sqrt(a+a*sin(c+d*x))/(a^2*d*(1+cos(c+d*x)+sin(c+d*x)))+5/4*e^(7/2)*arctan(sin(c+d*x)*sqrt(e)/(sqrt(e*cos(c+d*x))*sqrt(1+cos(c+d*x))))*sqrt(1+cos(c+d*x))*sqrt(a+a*sin(c+d*x))/(a^2*d*(1+cos(c+d*x)+sin(c+d*x)))],
[(e*cos(c+d*x))^(5/2)/(a+a*sin(c+d*x))^(3/2),x,7,e*(e*cos(c+d*x))^(3/2)/(a*d*sqrt(a+a*sin(c+d*x)))+3*e^(5/2)*arcsinh(sqrt(e*cos(c+d*x))/sqrt(e))*sqrt(1+cos(c+d*x))*sqrt(a+a*sin(c+d*x))/(d*(a^2+a^2*cos(c+d*x)+a^2*sin(c+d*x)))+3*e^(5/2)*arctan(sin(c+d*x)*sqrt(e)/(sqrt(e*cos(c+d*x))*sqrt(1+cos(c+d*x))))*sqrt(1+cos(c+d*x))*sqrt(a+a*sin(c+d*x))/(d*(a^2+a^2*cos(c+d*x)+a^2*sin(c+d*x)))],
[(e*cos(c+d*x))^(3/2)/(a+a*sin(c+d*x))^(3/2),x,8,-2*(e*cos(c+d*x))^(5/2)/(d*e*(a+a*sin(c+d*x))^(3/2))-2*e*sqrt(e*cos(c+d*x))*sqrt(a+a*sin(c+d*x))/(a^2*d)+2*e^(3/2)*arcsinh(sqrt(e*cos(c+d*x))/sqrt(e))*sqrt(1+cos(c+d*x))*sqrt(a+a*sin(c+d*x))/(a^2*d*(1+cos(c+d*x)+sin(c+d*x)))-2*e^(3/2)*arctan(sin(c+d*x)*sqrt(e)/(sqrt(e*cos(c+d*x))*sqrt(1+cos(c+d*x))))*sqrt(1+cos(c+d*x))*sqrt(a+a*sin(c+d*x))/(a^2*d*(1+cos(c+d*x)+sin(c+d*x)))],
[sqrt(e*cos(c+d*x))/(a+a*sin(c+d*x))^(3/2),x,1,-2/3*(e*cos(c+d*x))^(3/2)/(d*e*(a+a*sin(c+d*x))^(3/2))],
[1/((a+a*sin(c+d*x))^(3/2)*sqrt(e*cos(c+d*x))),x,2,-2/5*sqrt(e*cos(c+d*x))/(d*e*(a+a*sin(c+d*x))^(3/2))-4/5*sqrt(e*cos(c+d*x))/(a*d*e*sqrt(a+a*sin(c+d*x)))],
[1/((e*cos(c+d*x))^(3/2)*(a+a*sin(c+d*x))^(3/2)),x,3,(-2/7)/(d*e*(a+a*sin(c+d*x))^(3/2)*sqrt(e*cos(c+d*x)))+(-8/21)/(a*d*e*sqrt(e*cos(c+d*x))*sqrt(a+a*sin(c+d*x)))+16/21*sqrt(a+a*sin(c+d*x))/(a^2*d*e*sqrt(e*cos(c+d*x)))],
[1/((e*cos(c+d*x))^(5/2)*(a+a*sin(c+d*x))^(3/2)),x,4,(-2/9)/(d*e*(e*cos(c+d*x))^(3/2)*(a+a*sin(c+d*x))^(3/2))+32/45*(a+a*sin(c+d*x))^(3/2)/(a^3*d*e*(e*cos(c+d*x))^(3/2))+(-4/15)/(a*d*e*(e*cos(c+d*x))^(3/2)*sqrt(a+a*sin(c+d*x)))-16/15*sqrt(a+a*sin(c+d*x))/(a^2*d*e*(e*cos(c+d*x))^(3/2))],
[1/((e*cos(c+d*x))^(7/2)*(a+a*sin(c+d*x))^(3/2)),x,5,(-2/11)/(d*e*(e*cos(c+d*x))^(5/2)*(a+a*sin(c+d*x))^(3/2))+128/77*(a+a*sin(c+d*x))^(3/2)/(a^3*d*e*(e*cos(c+d*x))^(5/2))-256/385*(a+a*sin(c+d*x))^(5/2)/(a^4*d*e*(e*cos(c+d*x))^(5/2))+(-16/77)/(a*d*e*(e*cos(c+d*x))^(5/2)*sqrt(a+a*sin(c+d*x)))-32/77*sqrt(a+a*sin(c+d*x))/(a^2*d*e*(e*cos(c+d*x))^(5/2))],
[(e*cos(c+d*x))^(9/2)/(a+a*sin(c+d*x))^(5/2),x,9,1/2*e*(e*cos(c+d*x))^(7/2)/(a*d*(a+a*sin(c+d*x))^(3/2))+7/4*e^3*(e*cos(c+d*x))^(3/2)/(a^2*d*sqrt(a+a*sin(c+d*x)))+21/4*e^(9/2)*arcsinh(sqrt(e*cos(c+d*x))/sqrt(e))*sqrt(1+cos(c+d*x))*sqrt(a+a*sin(c+d*x))/(d*(a^3+a^3*cos(c+d*x)+a^3*sin(c+d*x)))+21/4*e^(9/2)*arctan(sin(c+d*x)*sqrt(e)/(sqrt(e*cos(c+d*x))*sqrt(1+cos(c+d*x))))*sqrt(1+cos(c+d*x))*sqrt(a+a*sin(c+d*x))/(d*(a^3+a^3*cos(c+d*x)+a^3*sin(c+d*x)))],
[(e*cos(c+d*x))^(7/2)/(a+a*sin(c+d*x))^(5/2),x,8,-4*e*(e*cos(c+d*x))^(5/2)/(a*d*(a+a*sin(c+d*x))^(3/2))-5*e^3*sqrt(e*cos(c+d*x))*sqrt(a+a*sin(c+d*x))/(a^3*d)+5*e^(7/2)*arcsinh(sqrt(e*cos(c+d*x))/sqrt(e))*sqrt(1+cos(c+d*x))*sqrt(a+a*sin(c+d*x))/(a^3*d*(1+cos(c+d*x)+sin(c+d*x)))-5*e^(7/2)*arctan(sin(c+d*x)*sqrt(e)/(sqrt(e*cos(c+d*x))*sqrt(1+cos(c+d*x))))*sqrt(1+cos(c+d*x))*sqrt(a+a*sin(c+d*x))/(a^3*d*(1+cos(c+d*x)+sin(c+d*x)))],
[(e*cos(c+d*x))^(5/2)/(a+a*sin(c+d*x))^(5/2),x,7,-4/3*e*(e*cos(c+d*x))^(3/2)/(a*d*(a+a*sin(c+d*x))^(3/2))-2*e^(5/2)*arcsinh(sqrt(e*cos(c+d*x))/sqrt(e))*sqrt(1+cos(c+d*x))*sqrt(a+a*sin(c+d*x))/(d*(a^3+a^3*cos(c+d*x)+a^3*sin(c+d*x)))-2*e^(5/2)*arctan(sin(c+d*x)*sqrt(e)/(sqrt(e*cos(c+d*x))*sqrt(1+cos(c+d*x))))*sqrt(1+cos(c+d*x))*sqrt(a+a*sin(c+d*x))/(d*(a^3+a^3*cos(c+d*x)+a^3*sin(c+d*x)))],
[(e*cos(c+d*x))^(3/2)/(a+a*sin(c+d*x))^(5/2),x,1,-2/5*(e*cos(c+d*x))^(5/2)/(d*e*(a+a*sin(c+d*x))^(5/2))],
[sqrt(e*cos(c+d*x))/(a+a*sin(c+d*x))^(5/2),x,2,-2/7*(e*cos(c+d*x))^(3/2)/(d*e*(a+a*sin(c+d*x))^(5/2))-4/21*(e*cos(c+d*x))^(3/2)/(a*d*e*(a+a*sin(c+d*x))^(3/2))],
[1/((a+a*sin(c+d*x))^(5/2)*sqrt(e*cos(c+d*x))),x,3,-2/9*sqrt(e*cos(c+d*x))/(d*e*(a+a*sin(c+d*x))^(5/2))-8/45*sqrt(e*cos(c+d*x))/(a*d*e*(a+a*sin(c+d*x))^(3/2))-16/45*sqrt(e*cos(c+d*x))/(a^2*d*e*sqrt(a+a*sin(c+d*x)))],
[1/((e*cos(c+d*x))^(3/2)*(a+a*sin(c+d*x))^(5/2)),x,4,(-2/11)/(d*e*(a+a*sin(c+d*x))^(5/2)*sqrt(e*cos(c+d*x)))+(-12/77)/(a*d*e*(a+a*sin(c+d*x))^(3/2)*sqrt(e*cos(c+d*x)))+(-16/77)/(a^2*d*e*sqrt(e*cos(c+d*x))*sqrt(a+a*sin(c+d*x)))+32/77*sqrt(a+a*sin(c+d*x))/(a^3*d*e*sqrt(e*cos(c+d*x)))],
[1/((e*cos(c+d*x))^(5/2)*(a+a*sin(c+d*x))^(5/2)),x,5,(-2/13)/(d*e*(e*cos(c+d*x))^(3/2)*(a+a*sin(c+d*x))^(5/2))+(-16/117)/(a*d*e*(e*cos(c+d*x))^(3/2)*(a+a*sin(c+d*x))^(3/2))+256/585*(a+a*sin(c+d*x))^(3/2)/(a^4*d*e*(e*cos(c+d*x))^(3/2))+(-32/195)/(a^2*d*e*(e*cos(c+d*x))^(3/2)*sqrt(a+a*sin(c+d*x)))-128/195*sqrt(a+a*sin(c+d*x))/(a^3*d*e*(e*cos(c+d*x))^(3/2))],

# Integrands of the form (d Cos[e+f x])^(p/3) (a+a Sin[e+f x])^(m/2)
[(e*cos(c+d*x))^(7/3)/sqrt(a+a*sin(c+d*x)),x,3,-3/5*2^(1/6)*a*(e*cos(c+d*x))^(10/3)*hypergeom([-1/6,5/3],[8/3],1/2*(1-sin(c+d*x)))/(d*e*(1+sin(c+d*x))^(1/6)*(a+a*sin(c+d*x))^(3/2))],
[(e*cos(c+d*x))^(5/3)/sqrt(a+a*sin(c+d*x)),x,3,-3/4*a*(e*cos(c+d*x))^(8/3)*hypergeom([1/6,4/3],[7/3],1/2*(1-sin(c+d*x)))*(1+sin(c+d*x))^(1/6)/(2^(1/6)*d*e*(a+a*sin(c+d*x))^(3/2))],
[(e*cos(c+d*x))^(2/3)/sqrt(a+a*sin(c+d*x)),x,3,-3/5*2^(1/3)*a*(e*cos(c+d*x))^(5/3)*hypergeom([2/3,5/6],[11/6],1/2*(1-sin(c+d*x)))*(1+sin(c+d*x))^(2/3)/(d*e*(a+a*sin(c+d*x))^(3/2))],
[(e*cos(c+d*x))^(1/3)/sqrt(a+a*sin(c+d*x)),x,3,-3/2*a*(e*cos(c+d*x))^(4/3)*hypergeom([2/3,5/6],[5/3],1/2*(1-sin(c+d*x)))*(1+sin(c+d*x))^(5/6)/(2^(5/6)*d*e*(a+a*sin(c+d*x))^(3/2))],
[1/((e*cos(c+d*x))^(1/3)*sqrt(a+a*sin(c+d*x))),x,3,-3/2*(e*cos(c+d*x))^(2/3)*hypergeom([1/3,7/6],[4/3],1/2*(1-sin(c+d*x)))*(1+sin(c+d*x))^(1/6)/(2^(1/6)*d*e*sqrt(a+a*sin(c+d*x)))],
[1/((e*cos(c+d*x))^(4/3)*sqrt(a+a*sin(c+d*x))),x,3,3*hypergeom([-1/6,5/3],[5/6],1/2*(1-sin(c+d*x)))*(1+sin(c+d*x))^(2/3)/(2^(2/3)*d*e*(e*cos(c+d*x))^(1/3)*sqrt(a+a*sin(c+d*x)))],

# Integrands of the form (d Cos[e+f x])^p (a+a Sin[e+f x])^m when p symbolic
[(e*cos(c+d*x))^p*(a+a*sin(c+d*x))^8,x,2,-2^(17/2+1/2*p)*a^8*(e*cos(c+d*x))^(1+p)*hypergeom([1/2*(-15-p),1/2*(1+p)],[1/2*(3+p)],1/2*(1-sin(c+d*x)))*(1+sin(c+d*x))^(1/2*(-1-p))/(d*e*(1+p))],
[(e*cos(c+d*x))^p*(a+a*sin(c+d*x))^3,x,2,-2^(7/2+1/2*p)*a^3*(e*cos(c+d*x))^(1+p)*hypergeom([1/2*(-5-p),1/2*(1+p)],[1/2*(3+p)],1/2*(1-sin(c+d*x)))*(1+sin(c+d*x))^(1/2*(-1-p))/(d*e*(1+p))],
[(e*cos(c+d*x))^p*(a+a*sin(c+d*x))^2,x,2,-2^(5/2+1/2*p)*a^2*(e*cos(c+d*x))^(1+p)*hypergeom([1/2*(-3-p),1/2*(1+p)],[1/2*(3+p)],1/2*(1-sin(c+d*x)))*(1+sin(c+d*x))^(1/2*(-1-p))/(d*e*(1+p))],
[(e*cos(c+d*x))^p*(a+a*sin(c+d*x)),x,2,-2^(3/2+1/2*p)*a*(e*cos(c+d*x))^(1+p)*hypergeom([1/2*(-1-p),1/2*(1+p)],[1/2*(3+p)],1/2*(1-sin(c+d*x)))*(1+sin(c+d*x))^(1/2*(-1-p))/(d*e*(1+p))],
[(e*cos(c+d*x))^p/(a+a*sin(c+d*x)),x,2,-2^(-1/2+1/2*p)*(e*cos(c+d*x))^(1+p)*hypergeom([1/2*(3-p),1/2*(1+p)],[1/2*(3+p)],1/2*(1-sin(c+d*x)))*(1+sin(c+d*x))^(1/2*(-1-p))/(a*d*e*(1+p))],
[(e*cos(c+d*x))^p/(a+a*sin(c+d*x))^2,x,2,-2^(1/2*(-3+p))*(e*cos(c+d*x))^(1+p)*hypergeom([1/2*(5-p),1/2*(1+p)],[1/2*(3+p)],1/2*(1-sin(c+d*x)))*(1+sin(c+d*x))^(1/2*(-1-p))/(a^2*d*e*(1+p))],
[(e*cos(c+d*x))^p/(a+a*sin(c+d*x))^3,x,2,-2^(1/2*(-5+p))*(e*cos(c+d*x))^(1+p)*hypergeom([1/2*(7-p),1/2*(1+p)],[1/2*(3+p)],1/2*(1-sin(c+d*x)))*(1+sin(c+d*x))^(1/2*(-1-p))/(a^3*d*e*(1+p))],
[(e*cos(c+d*x))^p/(a+a*sin(c+d*x))^8,x,2,-2^(1/2*(-15+p))*(e*cos(c+d*x))^(1+p)*hypergeom([1/2*(17-p),1/2*(1+p)],[1/2*(3+p)],1/2*(1-sin(c+d*x)))*(1+sin(c+d*x))^(1/2*(-1-p))/(a^8*d*e*(1+p))],
[(e*cos(c+d*x))^p*(a+a*sin(c+d*x))^(7/2),x,3,-2^(4+1/2*p)*a^4*(e*cos(c+d*x))^(1+p)*hypergeom([1/2*(-6-p),1/2*(1+p)],[1/2*(3+p)],1/2*(1-sin(c+d*x)))/(d*e*(1+p)*(1+sin(c+d*x))^(1/2*p)*sqrt(a+a*sin(c+d*x)))],
[(e*cos(c+d*x))^p*(a+a*sin(c+d*x))^(5/2),x,3,-2^(3+1/2*p)*a^3*(e*cos(c+d*x))^(1+p)*hypergeom([1/2*(-4-p),1/2*(1+p)],[1/2*(3+p)],1/2*(1-sin(c+d*x)))/(d*e*(1+p)*(1+sin(c+d*x))^(1/2*p)*sqrt(a+a*sin(c+d*x)))],
[(e*cos(c+d*x))^p*(a+a*sin(c+d*x))^(3/2),x,3,-2^(2+1/2*p)*a^2*(e*cos(c+d*x))^(1+p)*hypergeom([1/2*(-2-p),1/2*(1+p)],[1/2*(3+p)],1/2*(1-sin(c+d*x)))/(d*e*(1+p)*(1+sin(c+d*x))^(1/2*p)*sqrt(a+a*sin(c+d*x)))],
[(e*cos(c+d*x))^p*(a+a*sin(c+d*x))^(1/2),x,3,-2^(1+1/2*p)*a*(e*cos(c+d*x))^(1+p)*hypergeom([-1/2*p,1/2*(1+p)],[1/2*(3+p)],1/2*(1-sin(c+d*x)))/(d*e*(1+p)*(1+sin(c+d*x))^(1/2*p)*sqrt(a+a*sin(c+d*x)))],
[(e*cos(c+d*x))^p/(a+a*sin(c+d*x))^(1/2),x,3,-2^(1/2*p)*a*(e*cos(c+d*x))^(1+p)*hypergeom([1/2*(2-p),1/2*(1+p)],[1/2*(3+p)],1/2*(1-sin(c+d*x)))*(1+sin(c+d*x))^(1-1/2*p)/(d*e*(1+p)*(a+a*sin(c+d*x))^(3/2))],
[(e*cos(c+d*x))^p/(a+a*sin(c+d*x))^(3/2),x,3,-2^(-1+1/2*p)*(e*cos(c+d*x))^(1+p)*hypergeom([1/2*(4-p),1/2*(1+p)],[1/2*(3+p)],1/2*(1-sin(c+d*x)))*(1+sin(c+d*x))^(1-1/2*p)/(d*e*(1+p)*(a+a*sin(c+d*x))^(3/2))],
[(e*cos(c+d*x))^p/(a+a*sin(c+d*x))^(5/2),x,3,-2^(-2+1/2*p)*(e*cos(c+d*x))^(1+p)*hypergeom([1/2*(6-p),1/2*(1+p)],[1/2*(3+p)],1/2*(1-sin(c+d*x)))*(1+sin(c+d*x))^(1-1/2*p)/(a*d*e*(1+p)*(a+a*sin(c+d*x))^(3/2))],

# Integrands of the form (d Cos[e+f x])^p (a+a Sin[e+f x])^m when m symbolic
[(e*cos(c+d*x))^p*(a+a*sin(c+d*x))^m,x,3,-2^(1/2+m+1/2*p)*a*(e*cos(c+d*x))^(1+p)*hypergeom([1/2*(1-2*m-p),1/2*(1+p)],[1/2*(3+p)],1/2*(1-sin(c+d*x)))*(1+sin(c+d*x))^(1/2*(1-2*m-p))*(a+a*sin(c+d*x))^(-1+m)/(d*e*(1+p))],
[cos(c+d*x)^7*(a+a*sin(c+d*x))^m,x,3,8*(a+a*sin(c+d*x))^(4+m)/(a^4*d*(4+m))-12*(a+a*sin(c+d*x))^(5+m)/(a^5*d*(5+m))+6*(a+a*sin(c+d*x))^(6+m)/(a^6*d*(6+m))-(a+a*sin(c+d*x))^(7+m)/(a^7*d*(7+m))],
[cos(c+d*x)^5*(a+a*sin(c+d*x))^m,x,3,4*(a+a*sin(c+d*x))^(3+m)/(a^3*d*(3+m))-4*(a+a*sin(c+d*x))^(4+m)/(a^4*d*(4+m))+(a+a*sin(c+d*x))^(5+m)/(a^5*d*(5+m))],
[cos(c+d*x)^3*(a+a*sin(c+d*x))^m,x,3,2*(a+a*sin(c+d*x))^(2+m)/(a^2*d*(2+m))-(a+a*sin(c+d*x))^(3+m)/(a^3*d*(3+m))],
[cos(c+d*x)*(a+a*sin(c+d*x))^m,x,2,(a+a*sin(c+d*x))^(1+m)/(a*d*(1+m))],
[sec(c+d*x)*(a+a*sin(c+d*x))^m,x,2,1/2*hypergeom([1,m],[1+m],1/2*(1+sin(c+d*x)))*(a+a*sin(c+d*x))^m/(d*m)],
[sec(c+d*x)^3*(a+a*sin(c+d*x))^m,x,2,-1/4*a*hypergeom([2,-1+m],[m],1/2*(1+sin(c+d*x)))*(a+a*sin(c+d*x))^(-1+m)/(d*(1-m))],
[sec(c+d*x)^5*(a+a*sin(c+d*x))^m,x,2,-1/8*a^2*hypergeom([3,-2+m],[-1+m],1/2*(1+sin(c+d*x)))*(a+a*sin(c+d*x))^(-2+m)/(d*(2-m))],
[cos(c+d*x)^4*(a+a*sin(c+d*x))^m,x,3,-1/5*2^(5/2+m)*a^2*cos(c+d*x)^5*hypergeom([5/2,-3/2-m],[7/2],1/2*(1-sin(c+d*x)))*(1+sin(c+d*x))^(-1/2-m)*(a+a*sin(c+d*x))^(-2+m)/d],
[cos(c+d*x)^2*(a+a*sin(c+d*x))^m,x,3,-1/3*2^(3/2+m)*a*cos(c+d*x)^3*hypergeom([3/2,-1/2-m],[5/2],1/2*(1-sin(c+d*x)))*(1+sin(c+d*x))^(-1/2-m)*(a+a*sin(c+d*x))^(-1+m)/d],
[sec(c+d*x)^2*(a+a*sin(c+d*x))^m,x,3,2^(-1/2+m)*hypergeom([-1/2,3/2-m],[1/2],1/2*(1-sin(c+d*x)))*sec(c+d*x)*(1+sin(c+d*x))^(1/2-m)*(a+a*sin(c+d*x))^m/d],
[sec(c+d*x)^4*(a+a*sin(c+d*x))^m,x,3,1/3*2^(-3/2+m)*hypergeom([-3/2,5/2-m],[-1/2],1/2*(1-sin(c+d*x)))*sec(c+d*x)^3*(1+sin(c+d*x))^(1/2-m)*(a+a*sin(c+d*x))^(1+m)/(a*d)],
[(e*cos(c+d*x))^(5/2)*(a+a*sin(c+d*x))^m,x,3,-1/7*2^(11/4+m)*a*(e*cos(c+d*x))^(7/2)*hypergeom([7/4,-3/4-m],[11/4],1/2*(1-sin(c+d*x)))*(1+sin(c+d*x))^(-3/4-m)*(a+a*sin(c+d*x))^(-1+m)/(d*e)],
[(e*cos(c+d*x))^(3/2)*(a+a*sin(c+d*x))^m,x,3,-1/5*2^(9/4+m)*a*(e*cos(c+d*x))^(5/2)*hypergeom([5/4,-1/4-m],[9/4],1/2*(1-sin(c+d*x)))*(1+sin(c+d*x))^(-1/4-m)*(a+a*sin(c+d*x))^(-1+m)/(d*e)],
[(e*cos(c+d*x))^(1/2)*(a+a*sin(c+d*x))^m,x,3,-1/3*2^(7/4+m)*a*(e*cos(c+d*x))^(3/2)*hypergeom([3/4,1/4-m],[7/4],1/2*(1-sin(c+d*x)))*(1+sin(c+d*x))^(1/4-m)*(a+a*sin(c+d*x))^(-1+m)/(d*e)],
[(a+a*sin(c+d*x))^m/(e*cos(c+d*x))^(1/2),x,3,-2^(5/4+m)*a*hypergeom([1/4,3/4-m],[5/4],1/2*(1-sin(c+d*x)))*(1+sin(c+d*x))^(3/4-m)*(a+a*sin(c+d*x))^(-1+m)*sqrt(e*cos(c+d*x))/(d*e)],
[(a+a*sin(c+d*x))^m/(e*cos(c+d*x))^(3/2),x,3,2^(3/4+m)*hypergeom([-1/4,5/4-m],[3/4],1/2*(1-sin(c+d*x)))*(1+sin(c+d*x))^(1/4-m)*(a+a*sin(c+d*x))^m/(d*e*sqrt(e*cos(c+d*x)))],
[(a+a*sin(c+d*x))^m/(e*cos(c+d*x))^(5/2),x,3,1/3*2^(1/4+m)*hypergeom([-3/4,7/4-m],[1/4],1/2*(1-sin(c+d*x)))*(1+sin(c+d*x))^(3/4-m)*(a+a*sin(c+d*x))^m/(d*e*(e*cos(c+d*x))^(3/2))],
[(e*cos(c+d*x))^(-4-m)*(a+a*sin(c+d*x))^m,x,4,-(e*cos(c+d*x))^(-3-m)*(a+a*sin(c+d*x))^m/(d*e*(3-m))-3*(e*cos(c+d*x))^(-3-m)*(a+a*sin(c+d*x))^(1+m)/(a*d*e*(1-m)*(3-m))+6*(e*cos(c+d*x))^(-3-m)*(a+a*sin(c+d*x))^(2+m)/(a^2*d*e*(3-m)*(1-m^2))-6*(e*cos(c+d*x))^(-3-m)*(a+a*sin(c+d*x))^(3+m)/(a^3*d*e*(9-10*m^2+m^4))],
[(e*cos(c+d*x))^(-3-m)*(a+a*sin(c+d*x))^m,x,3,-(e*cos(c+d*x))^(-2-m)*(a+a*sin(c+d*x))^m/(d*e*(2-m))+2*(e*cos(c+d*x))^(-2-m)*(a+a*sin(c+d*x))^(1+m)/(a*d*e*(2-m)*m)-2*(e*cos(c+d*x))^(-2-m)*(a+a*sin(c+d*x))^(2+m)/(a^2*d*e*m*(4-m^2))],
[(e*cos(c+d*x))^(-2-m)*(a+a*sin(c+d*x))^m,x,2,-(e*cos(c+d*x))^(-1-m)*(a+a*sin(c+d*x))^m/(d*e*(1-m))+(e*cos(c+d*x))^(-1-m)*(a+a*sin(c+d*x))^(1+m)/(a*d*e*(1-m^2))],
[(e*cos(c+d*x))^(-1-m)*(a+a*sin(c+d*x))^m,x,1,(a+a*sin(c+d*x))^m/(d*e*m*(e*cos(c+d*x))^m)],
[(a+a*sin(c+d*x))^m/(e*cos(c+d*x))^m,x,3,-2^(1/2+1/2*m)*a*(e*cos(c+d*x))^(1-m)*hypergeom([1/2*(1-m),1/2*(1-m)],[1/2*(3-m)],1/2*(1-sin(c+d*x)))*(1+sin(c+d*x))^(1/2*(1-m))*(a+a*sin(c+d*x))^(-1+m)/(d*e*(1-m))],
[(e*cos(c+d*x))^(1-m)*(a+a*sin(c+d*x))^m,x,3,2^(1-1/2*m)*(e*cos(c+d*x))^(2-m)*hypergeom([1/2*m,1/2*(2+m)],[1/2*(4+m)],1/2*(1+sin(c+d*x)))*(1-sin(c+d*x))^(-1+1/2*m)*(a+a*sin(c+d*x))^m/(d*e*(2+m))],
[(e*cos(c+d*x))^(2-m)*(a+a*sin(c+d*x))^m,x,3,-2^(3/2+1/2*m)*a*(e*cos(c+d*x))^(3-m)*hypergeom([1/2*(-1-m),1/2*(3-m)],[1/2*(5-m)],1/2*(1-sin(c+d*x)))*(1+sin(c+d*x))^(1/2*(-1-m))*(a+a*sin(c+d*x))^(-1+m)/(d*e*(3-m))],
[(e*cos(c+d*x))^(5-2*m)*(a+a*sin(c+d*x))^m,x,3,-8*a^3*(e*cos(c+d*x))^(6-2*m)*(a+a*sin(c+d*x))^(-3+m)/(d*e*(60-47*m+12*m^2-m^3))-4*a^2*(e*cos(c+d*x))^(6-2*m)*(a+a*sin(c+d*x))^(-2+m)/(d*e*(4-m)*(5-m))-a*(e*cos(c+d*x))^(6-2*m)*(a+a*sin(c+d*x))^(-1+m)/(d*e*(5-m))],
[(e*cos(c+d*x))^(3-2*m)*(a+a*sin(c+d*x))^m,x,2,-2*a^2*(e*cos(c+d*x))^(4-2*m)*(a+a*sin(c+d*x))^(-2+m)/(d*e*(2-m)*(3-m))-a*(e*cos(c+d*x))^(4-2*m)*(a+a*sin(c+d*x))^(-1+m)/(d*e*(3-m))],
[(e*cos(c+d*x))^(1-2*m)*(a+a*sin(c+d*x))^m,x,1,-a*(e*cos(c+d*x))^(2-2*m)*(a+a*sin(c+d*x))^(-1+m)/(d*e*(1-m))],
[(e*cos(c+d*x))^(-1-2*m)*(a+a*sin(c+d*x))^m,x,3,1/2*hypergeom([1,-m],[1-m],1/2*(1-sin(c+d*x)))*(a+a*sin(c+d*x))^m/(d*e*m*(e*cos(c+d*x))^(2*m))],
[(e*cos(c+d*x))^(-3-2*m)*(a+a*sin(c+d*x))^m,x,3,1/4*hypergeom([2,-1-m],[-m],1/2*(1-sin(c+d*x)))*(a+a*sin(c+d*x))^(1+m)/(a*d*e*(1+m)*(e*cos(c+d*x))^(2*(1+m)))],
[(e*cos(c+d*x))^(4-2*m)*(a+a*sin(c+d*x))^m,x,4,1/5*2^(5/2-m)*(e*cos(c+d*x))^(5-2*m)*hypergeom([5/2,1/2*(-3+2*m)],[7/2],1/2*(1+sin(c+d*x)))*(1-sin(c+d*x))^(-5/2+m)*(a+a*sin(c+d*x))^m/(d*e)],
[(e*cos(c+d*x))^(2-2*m)*(a+a*sin(c+d*x))^m,x,4,1/3*2^(3/2-m)*(e*cos(c+d*x))^(3-2*m)*hypergeom([3/2,1/2*(-1+2*m)],[5/2],1/2*(1+sin(c+d*x)))*(1-sin(c+d*x))^(-3/2+m)*(a+a*sin(c+d*x))^m/(d*e)],
[(a+a*sin(c+d*x))^m/(e*cos(c+d*x))^(2*m),x,4,2^(1/2-m)*(e*cos(c+d*x))^(1-2*m)*hypergeom([1/2,1/2*(1+2*m)],[3/2],1/2*(1+sin(c+d*x)))*(1-sin(c+d*x))^(-1/2+m)*(a+a*sin(c+d*x))^m/(d*e)],
[(e*cos(c+d*x))^(-2-2*m)*(a+a*sin(c+d*x))^m,x,4,-2^(-1/2-m)*(e*cos(c+d*x))^(-1-2*m)*hypergeom([-1/2,1/2*(3+2*m)],[1/2],1/2*(1+sin(c+d*x)))*(1-sin(c+d*x))^(1/2+m)*(a+a*sin(c+d*x))^m/(d*e)],

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

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

# m>0
[cos(c+d*x)^5*(a+b*sin(c+d*x)),x,4,-1/6*b*cos(c+d*x)^6/d+a*sin(c+d*x)/d-2/3*a*sin(c+d*x)^3/d+1/5*a*sin(c+d*x)^5/d],
[cos(c+d*x)^3*(a+b*sin(c+d*x)),x,3,-1/4*b*cos(c+d*x)^4/d+a*sin(c+d*x)/d-1/3*a*sin(c+d*x)^3/d],
[cos(c+d*x)*(a+b*sin(c+d*x)),x,2,1/2*(a+b*sin(c+d*x))^2/(b*d),a*sin(c+d*x)/d+1/2*b*sin(c+d*x)^2/d],
[sec(c+d*x)*(a+b*sin(c+d*x)),x,4,-1/2*(a+b)*log(1-sin(c+d*x))/d+1/2*(a-b)*log(1+sin(c+d*x))/d],
[sec(c+d*x)^3*(a+b*sin(c+d*x)),x,3,1/2*a*arctanh(sin(c+d*x))/d+1/2*sec(c+d*x)^2*(b+a*sin(c+d*x))/d],
[sec(c+d*x)^5*(a+b*sin(c+d*x)),x,4,3/8*a*arctanh(sin(c+d*x))/d+1/4*sec(c+d*x)^4*(b+a*sin(c+d*x))/d+3/8*a*sec(c+d*x)*tan(c+d*x)/d],
[cos(c+d*x)^4*(a+b*sin(c+d*x)),x,4,3/8*a*x-1/5*b*cos(c+d*x)^5/d+3/8*a*cos(c+d*x)*sin(c+d*x)/d+1/4*a*cos(c+d*x)^3*sin(c+d*x)/d],
[cos(c+d*x)^2*(a+b*sin(c+d*x)),x,3,1/2*a*x-1/3*b*cos(c+d*x)^3/d+1/2*a*cos(c+d*x)*sin(c+d*x)/d],
[sec(c+d*x)^2*(a+b*sin(c+d*x)),x,3,b*sec(c+d*x)/d+a*tan(c+d*x)/d],
[sec(c+d*x)^4*(a+b*sin(c+d*x)),x,3,1/3*b*sec(c+d*x)^3/d+a*tan(c+d*x)/d+1/3*a*tan(c+d*x)^3/d],
[sec(c+d*x)^6*(a+b*sin(c+d*x)),x,3,1/5*b*sec(c+d*x)^5/d+a*tan(c+d*x)/d+2/3*a*tan(c+d*x)^3/d+1/5*a*tan(c+d*x)^5/d],
[cos(c+d*x)^5*(a+b*sin(c+d*x))^2,x,4,-1/3*a*b*cos(c+d*x)^6/d+a^2*sin(c+d*x)/d-1/3*(2*a^2-b^2)*sin(c+d*x)^3/d+1/5*(a^2-2*b^2)*sin(c+d*x)^5/d+1/7*b^2*sin(c+d*x)^7/d],
[cos(c+d*x)^3*(a+b*sin(c+d*x))^2,x,3,-1/3*(a^2-b^2)*(a+b*sin(c+d*x))^3/(b^3*d)+1/2*a*(a+b*sin(c+d*x))^4/(b^3*d)-1/5*(a+b*sin(c+d*x))^5/(b^3*d)],
[cos(c+d*x)*(a+b*sin(c+d*x))^2,x,2,1/3*(a+b*sin(c+d*x))^3/(b*d)],
[sec(c+d*x)*(a+b*sin(c+d*x))^2,x,6,-1/2*(a+b)^2*log(1-sin(c+d*x))/d+1/2*(a-b)^2*log(1+sin(c+d*x))/d-b^2*sin(c+d*x)/d],
[sec(c+d*x)^3*(a+b*sin(c+d*x))^2,x,3,1/2*(a^2-b^2)*arctanh(sin(c+d*x))/d+1/2*sec(c+d*x)^2*(b+a*sin(c+d*x))*(a+b*sin(c+d*x))/d],
[sec(c+d*x)^5*(a+b*sin(c+d*x))^2,x,4,1/8*(3*a^2-b^2)*arctanh(sin(c+d*x))/d+1/4*sec(c+d*x)^4*(b+a*sin(c+d*x))*(a+b*sin(c+d*x))/d+1/8*sec(c+d*x)^2*(2*a*b+(3*a^2-b^2)*sin(c+d*x))/d],
[cos(c+d*x)^6*(a+b*sin(c+d*x))^2,x,6,5/128*(8*a^2+b^2)*x-9/56*a*b*cos(c+d*x)^7/d+5/128*(8*a^2+b^2)*cos(c+d*x)*sin(c+d*x)/d+5/192*(8*a^2+b^2)*cos(c+d*x)^3*sin(c+d*x)/d+1/48*(8*a^2+b^2)*cos(c+d*x)^5*sin(c+d*x)/d-1/8*b*cos(c+d*x)^7*(a+b*sin(c+d*x))/d],
[cos(c+d*x)^4*(a+b*sin(c+d*x))^2,x,5,1/16*(6*a^2+b^2)*x-7/30*a*b*cos(c+d*x)^5/d+1/16*(6*a^2+b^2)*cos(c+d*x)*sin(c+d*x)/d+1/24*(6*a^2+b^2)*cos(c+d*x)^3*sin(c+d*x)/d-1/6*b*cos(c+d*x)^5*(a+b*sin(c+d*x))/d],
[cos(c+d*x)^2*(a+b*sin(c+d*x))^2,x,4,1/8*(4*a^2+b^2)*x-5/12*a*b*cos(c+d*x)^3/d+1/8*(4*a^2+b^2)*cos(c+d*x)*sin(c+d*x)/d-1/4*b*cos(c+d*x)^3*(a+b*sin(c+d*x))/d],
[sec(c+d*x)^2*(a+b*sin(c+d*x))^2,x,3,-b^2*x+a*b*cos(c+d*x)/d+sec(c+d*x)*(b+a*sin(c+d*x))*(a+b*sin(c+d*x))/d],
[sec(c+d*x)^4*(a+b*sin(c+d*x))^2,x,4,1/3*a*b*sec(c+d*x)/d+1/3*sec(c+d*x)^3*(b+a*sin(c+d*x))*(a+b*sin(c+d*x))/d+1/3*(2*a^2-b^2)*tan(c+d*x)/d],
[sec(c+d*x)^6*(a+b*sin(c+d*x))^2,x,4,1/5*a*b*sec(c+d*x)^3/d+1/5*sec(c+d*x)^5*(b+a*sin(c+d*x))*(a+b*sin(c+d*x))/d+1/5*(4*a^2-b^2)*tan(c+d*x)/d+1/15*(4*a^2-b^2)*tan(c+d*x)^3/d],
[sec(c+d*x)^8*(a+b*sin(c+d*x))^2,x,4,1/7*a*b*sec(c+d*x)^5/d+1/7*sec(c+d*x)^7*(b+a*sin(c+d*x))*(a+b*sin(c+d*x))/d+1/7*(6*a^2-b^2)*tan(c+d*x)/d+2/21*(6*a^2-b^2)*tan(c+d*x)^3/d+1/35*(6*a^2-b^2)*tan(c+d*x)^5/d],
[cos(c+d*x)^5*(a+b*sin(c+d*x))^3,x,3,1/4*(a^2-b^2)^2*(a+b*sin(c+d*x))^4/(b^5*d)-4/5*a*(a^2-b^2)*(a+b*sin(c+d*x))^5/(b^5*d)+1/3*(3*a^2-b^2)*(a+b*sin(c+d*x))^6/(b^5*d)-4/7*a*(a+b*sin(c+d*x))^7/(b^5*d)+1/8*(a+b*sin(c+d*x))^8/(b^5*d)],
[cos(c+d*x)^3*(a+b*sin(c+d*x))^3,x,3,-1/4*(a^2-b^2)*(a+b*sin(c+d*x))^4/(b^3*d)+2/5*a*(a+b*sin(c+d*x))^5/(b^3*d)-1/6*(a+b*sin(c+d*x))^6/(b^3*d)],
[cos(c+d*x)*(a+b*sin(c+d*x))^3,x,2,1/4*(a+b*sin(c+d*x))^4/(b*d)],
[sec(c+d*x)*(a+b*sin(c+d*x))^3,x,6,-1/2*(a+b)^3*log(1-sin(c+d*x))/d+1/2*(a-b)^3*log(1+sin(c+d*x))/d-3*a*b^2*sin(c+d*x)/d-1/2*b^3*sin(c+d*x)^2/d],
[sec(c+d*x)^3*(a+b*sin(c+d*x))^3,x,6,-1/4*(a-2*b)*(a+b)^2*log(1-sin(c+d*x))/d+1/4*(a-b)^2*(a+2*b)*log(1+sin(c+d*x))/d+1/2*a*b^2*sin(c+d*x)/d+1/2*sec(c+d*x)^2*(b+a*sin(c+d*x))*(a+b*sin(c+d*x))^2/d],
[sec(c+d*x)^5*(a+b*sin(c+d*x))^3,x,4,3/8*a*(a^2-b^2)*arctanh(sin(c+d*x))/d+3/8*a*sec(c+d*x)^2*(b+a*sin(c+d*x))*(a+b*sin(c+d*x))/d+1/4*sec(c+d*x)^3*(a+b*sin(c+d*x))^3*tan(c+d*x)/d],
[cos(c+d*x)^4*(a+b*sin(c+d*x))^3,x,6,3/16*a*(2*a^2+b^2)*x-1/70*b*(17*a^2+4*b^2)*cos(c+d*x)^5/d+3/16*a*(2*a^2+b^2)*cos(c+d*x)*sin(c+d*x)/d+1/8*a*(2*a^2+b^2)*cos(c+d*x)^3*sin(c+d*x)/d-3/14*a*b*cos(c+d*x)^5*(a+b*sin(c+d*x))/d-1/7*b*cos(c+d*x)^5*(a+b*sin(c+d*x))^2/d],
[cos(c+d*x)^2*(a+b*sin(c+d*x))^3,x,5,1/8*a*(4*a^2+3*b^2)*x-1/60*b*(27*a^2+8*b^2)*cos(c+d*x)^3/d+1/8*a*(4*a^2+3*b^2)*cos(c+d*x)*sin(c+d*x)/d-7/20*a*b*cos(c+d*x)^3*(a+b*sin(c+d*x))/d-1/5*b*cos(c+d*x)^3*(a+b*sin(c+d*x))^2/d],
[sec(c+d*x)^2*(a+b*sin(c+d*x))^3,x,2,-3*a*b^2*x+2*b*(a^2+b^2)*cos(c+d*x)/d+a*b^2*cos(c+d*x)*sin(c+d*x)/d+sec(c+d*x)*(b+a*sin(c+d*x))*(a+b*sin(c+d*x))^2/d],
[sec(c+d*x)^4*(a+b*sin(c+d*x))^3,x,5,2/3*b*(a^2-b^2)*sec(c+d*x)/d+1/3*sec(c+d*x)^3*(b+a*sin(c+d*x))*(a+b*sin(c+d*x))^2/d+2/3*a*(a^2-b^2)*tan(c+d*x)/d],
[sec(c+d*x)^6*(a+b*sin(c+d*x))^3,x,5,2/15*b*(2*a^2-b^2)*sec(c+d*x)/d+1/5*sec(c+d*x)^5*(b+a*sin(c+d*x))*(a+b*sin(c+d*x))^2/d+2/15*sec(c+d*x)^3*(a+b*sin(c+d*x))*(a*b+(2*a^2-b^2)*sin(c+d*x))/d+2/15*a*(4*a^2-3*b^2)*tan(c+d*x)/d],
[sec(c+d*x)^8*(a+b*sin(c+d*x))^3,x,5,2/35*b*(3*a^2-b^2)*sec(c+d*x)^3/d+1/7*sec(c+d*x)^7*(b+a*sin(c+d*x))*(a+b*sin(c+d*x))^2/d+2/35*sec(c+d*x)^5*(a+b*sin(c+d*x))*(2*a*b+(3*a^2-b^2)*sin(c+d*x))/d+12/35*a*(2*a^2-b^2)*tan(c+d*x)/d+4/35*a*(2*a^2-b^2)*tan(c+d*x)^3/d],
[sec(c+d*x)^10*(a+b*sin(c+d*x))^3,x,5,2/63*b*(4*a^2-b^2)*sec(c+d*x)^5/d+1/9*sec(c+d*x)^9*(b+a*sin(c+d*x))*(a+b*sin(c+d*x))^2/d+2/63*sec(c+d*x)^7*(a+b*sin(c+d*x))*(3*a*b+(4*a^2-b^2)*sin(c+d*x))/d+2/21*a*(8*a^2-3*b^2)*tan(c+d*x)/d+4/63*a*(8*a^2-3*b^2)*tan(c+d*x)^3/d+2/105*a*(8*a^2-3*b^2)*tan(c+d*x)^5/d],
[cos(c+d*x)^5*(a+b*sin(c+d*x))^8,x,3,1/9*(a^2-b^2)^2*(a+b*sin(c+d*x))^9/(b^5*d)-2/5*a*(a^2-b^2)*(a+b*sin(c+d*x))^10/(b^5*d)+2/11*(3*a^2-b^2)*(a+b*sin(c+d*x))^11/(b^5*d)-1/3*a*(a+b*sin(c+d*x))^12/(b^5*d)+1/13*(a+b*sin(c+d*x))^13/(b^5*d)],
[cos(c+d*x)^3*(a+b*sin(c+d*x))^8,x,3,-1/9*(a^2-b^2)*(a+b*sin(c+d*x))^9/(b^3*d)+1/5*a*(a+b*sin(c+d*x))^10/(b^3*d)-1/11*(a+b*sin(c+d*x))^11/(b^3*d)],
[cos(c+d*x)*(a+b*sin(c+d*x))^8,x,2,1/9*(a+b*sin(c+d*x))^9/(b*d)],
[sec(c+d*x)*(a+b*sin(c+d*x))^8,x,6,-1/2*(a+b)^8*log(1-sin(c+d*x))/d+1/2*(a-b)^8*log(1+sin(c+d*x))/d-b^2*(28*a^6+70*a^4*b^2+28*a^2*b^4+b^6)*sin(c+d*x)/d-4*a*b^3*(7*a^4+7*a^2*b^2+b^4)*sin(c+d*x)^2/d-1/3*b^4*(70*a^4+28*a^2*b^2+b^4)*sin(c+d*x)^3/d-2*a*b^5*(7*a^2+b^2)*sin(c+d*x)^4/d-1/5*b^6*(28*a^2+b^2)*sin(c+d*x)^5/d-4/3*a*b^7*sin(c+d*x)^6/d-1/7*b^8*sin(c+d*x)^7/d],
[sec(c+d*x)^3*(a+b*sin(c+d*x))^8,x,7,-1/4*(a-7*b)*(a+b)^7*log(1-sin(c+d*x))/d+1/4*(a-b)^7*(a+7*b)*log(1+sin(c+d*x))/d+7/2*b^2*(3*a^6+30*a^4*b^2+20*a^2*b^4+b^6)*sin(c+d*x)/d+1/2*a*b^3*(35*a^4+112*a^2*b^2+24*b^4)*sin(c+d*x)^2/d+7/6*b^4*(15*a^4+20*a^2*b^2+b^4)*sin(c+d*x)^3/d+3/2*a*b^5*(7*a^2+4*b^2)*sin(c+d*x)^4/d+7/10*b^6*(5*a^2+b^2)*sin(c+d*x)^5/d+1/2*a*b^7*sin(c+d*x)^6/d+1/2*sec(c+d*x)^2*(b+a*sin(c+d*x))*(a+b*sin(c+d*x))^7/d],
[sec(c+d*x)^5*(a+b*sin(c+d*x))^8,x,8,-1/16*(a+b)^6*(3*a^2-18*a*b+35*b^2)*log(1-sin(c+d*x))/d+1/16*(a-b)^6*(3*a^2+18*a*b+35*b^2)*log(1+sin(c+d*x))/d+5/8*b^2*(6*a^6-35*a^4*b^2-84*a^2*b^4-7*b^6)*sin(c+d*x)/d+1/4*a*b^3*(15*a^4-77*a^2*b^2-48*b^4)*sin(c+d*x)^2/d+5/24*b^4*(9*a^4-42*a^2*b^2-7*b^4)*sin(c+d*x)^3/d-1/8*a*(13-3*a^2/b^2)*b^7*sin(c+d*x)^4/d+1/4*sec(c+d*x)^4*(b+a*sin(c+d*x))*(a+b*sin(c+d*x))^7/d-1/8*sec(c+d*x)^2*(a+b*sin(c+d*x))^5*(b*(a^2+7*b^2)-a*(3*a^2-11*b^2)*sin(c+d*x))/d],
[cos(c+d*x)^2*(a+b*sin(c+d*x))^8,x,10,1/256*(128*a^8+896*a^6*b^2+1120*a^4*b^4+280*a^2*b^6+7*b^8)*x-11/40320*a*b*(1792*a^6+10536*a^4*b^2+9588*a^2*b^4+1289*b^6)*cos(c+d*x)^3/d+1/256*(128*a^8+896*a^6*b^2+1120*a^4*b^4+280*a^2*b^6+7*b^8)*cos(c+d*x)*sin(c+d*x)/d-1/13440*b*(6272*a^6+28088*a^4*b^2+15956*a^2*b^4+735*b^6)*cos(c+d*x)^3*(a+b*sin(c+d*x))/d-13/3360*a*b*(112*a^4+348*a^2*b^2+101*b^4)*cos(c+d*x)^3*(a+b*sin(c+d*x))^2/d-1/2016*b*(784*a^4+1500*a^2*b^2+147*b^4)*cos(c+d*x)^3*(a+b*sin(c+d*x))^3/d-1/336*a*b*(112*a^2+109*b^2)*cos(c+d*x)^3*(a+b*sin(c+d*x))^4/d-1/240*b*(64*a^2+21*b^2)*cos(c+d*x)^3*(a+b*sin(c+d*x))^5/d-17/90*a*b*cos(c+d*x)^3*(a+b*sin(c+d*x))^6/d-1/10*b*cos(c+d*x)^3*(a+b*sin(c+d*x))^7/d],
[sec(c+d*x)^2*(a+b*sin(c+d*x))^8,x,7,-7/16*b^2*(64*a^6+240*a^4*b^2+120*a^2*b^4+5*b^6)*x+1/20*a*b*(40*a^6+1664*a^4*b^2+2789*a^2*b^4+512*b^6)*cos(c+d*x)/d+1/80*b^2*(80*a^6+2248*a^4*b^2+2502*a^2*b^4+175*b^6)*cos(c+d*x)*sin(c+d*x)/d+1/40*a*b*(40*a^4+624*a^2*b^2+337*b^4)*cos(c+d*x)*(a+b*sin(c+d*x))^2/d+1/120*b*(120*a^4+992*a^2*b^2+175*b^4)*cos(c+d*x)*(a+b*sin(c+d*x))^3/d+1/30*a*b*(30*a^2+113*b^2)*cos(c+d*x)*(a+b*sin(c+d*x))^4/d+1/6*b*(6*a^2+7*b^2)*cos(c+d*x)*(a+b*sin(c+d*x))^5/d+a*b*cos(c+d*x)*(a+b*sin(c+d*x))^6/d+sec(c+d*x)*(b+a*sin(c+d*x))*(a+b*sin(c+d*x))^7/d],
[sec(c+d*x)^4*(a+b*sin(c+d*x))^8,x,7,35/8*b^4*(16*a^4+16*a^2*b^2+b^4)*x+1/6*a*b*(8*a^6-104*a^4*b^2-803*a^2*b^4-256*b^6)*cos(c+d*x)/d+1/24*b^2*(16*a^6-200*a^4*b^2-866*a^2*b^4-105*b^6)*cos(c+d*x)*sin(c+d*x)/d+1/12*a*b*(8*a^4-88*a^2*b^2-151*b^4)*cos(c+d*x)*(a+b*sin(c+d*x))^2/d+1/12*b*(8*a^4-72*a^2*b^2-35*b^4)*cos(c+d*x)*(a+b*sin(c+d*x))^3/d+1/3*a*b*(2*a^2-13*b^2)*cos(c+d*x)*(a+b*sin(c+d*x))^4/d+1/3*b*(2*a^2-7*b^2)*cos(c+d*x)*(a+b*sin(c+d*x))^5/d+1/3*sec(c+d*x)^3*(b+a*sin(c+d*x))*(a+b*sin(c+d*x))^7/d-1/3*sec(c+d*x)*(a+b*sin(c+d*x))^6*(5*a*b-(2*a^2-7*b^2)*sin(c+d*x))/d],
[sec(c+d*x)^6*(a+b*sin(c+d*x))^8,x,7,-7/2*b^6*(8*a^2+b^2)*x+2/15*a*b*(8*a^6-48*a^4*b^2+163*a^2*b^4+192*b^6)*cos(c+d*x)/d+1/30*b^2*(16*a^6-88*a^4*b^2+282*a^2*b^4+105*b^6)*cos(c+d*x)*sin(c+d*x)/d+1/15*a*b*(8*a^4-32*a^2*b^2+87*b^4)*cos(c+d*x)*(a+b*sin(c+d*x))^2/d+1/15*b*(8*a^4-16*a^2*b^2+35*b^4)*cos(c+d*x)*(a+b*sin(c+d*x))^3/d+4/15*a*b*(2*a^2+b^2)*cos(c+d*x)*(a+b*sin(c+d*x))^4/d+1/5*sec(c+d*x)^5*(b+a*sin(c+d*x))*(a+b*sin(c+d*x))^7/d-1/15*sec(c+d*x)^3*(a+b*sin(c+d*x))^6*(3*a*b-(4*a^2-7*b^2)*sin(c+d*x))/d-4/15*sec(c+d*x)*(a+b*sin(c+d*x))^5*(b*(4*a^2-7*b^2)-a*(2*a^2+b^2)*sin(c+d*x))/d],
[sec(c+d*x)^8*(a+b*sin(c+d*x))^8,x,7,b^8*x+4/105*a*b*(24*a^6-88*a^4*b^2+125*a^2*b^4-96*b^6)*cos(c+d*x)/d+1/105*b^2*(48*a^6-152*a^4*b^2+174*a^2*b^4-105*b^6)*cos(c+d*x)*sin(c+d*x)/d+2/105*a*b*(24*a^4-40*a^2*b^2+9*b^4)*cos(c+d*x)*(a+b*sin(c+d*x))^2/d+2/105*b*(24*a^4+8*a^2*b^2-35*b^4)*cos(c+d*x)*(a+b*sin(c+d*x))^3/d+1/7*sec(c+d*x)^7*(b+a*sin(c+d*x))*(a+b*sin(c+d*x))^7/d-2/105*sec(c+d*x)^3*(a+b*sin(c+d*x))^5*(b*(6*a^2-7*b^2)-a*(12*a^2-11*b^2)*sin(c+d*x))/d-1/35*sec(c+d*x)^5*(a+b*sin(c+d*x))^6*(a*b-(6*a^2-7*b^2)*sin(c+d*x))/d-2/105*sec(c+d*x)*(a+b*sin(c+d*x))^4*(3*a*b*(12*a^2-11*b^2)-(24*a^4+8*a^2*b^2-35*b^4)*sin(c+d*x))/d],
[sec(c+d*x)^10*(a+b*sin(c+d*x))^8,x,10,128/315*a*b*(a^2-b^2)^3*sec(c+d*x)/d+64/315*a*(a^2-b^2)^2*sec(c+d*x)^3*(b+a*sin(c+d*x))*(a+b*sin(c+d*x))^2/d+16/105*a*(a^2-b^2)*sec(c+d*x)^5*(b+a*sin(c+d*x))*(a+b*sin(c+d*x))^4/d+1/9*sec(c+d*x)^9*(b+a*sin(c+d*x))*(a+b*sin(c+d*x))^7/d+1/63*sec(c+d*x)^7*(a+b*sin(c+d*x))^6*(a*b+(8*a^2-7*b^2)*sin(c+d*x))/d+128/315*a^2*(a^2-b^2)^3*tan(c+d*x)/d],

# m<0
[cos(c+d*x)^5/(a+b*sin(c+d*x)),x,3,(a^2-b^2)^2*log(a+b*sin(c+d*x))/(b^5*d)-a*(a^2-2*b^2)*sin(c+d*x)/(b^4*d)+1/2*(a^2-2*b^2)*sin(c+d*x)^2/(b^3*d)-1/3*a*sin(c+d*x)^3/(b^2*d)+1/4*sin(c+d*x)^4/(b*d)],
[cos(c+d*x)^3/(a+b*sin(c+d*x)),x,3,-(a^2-b^2)*log(a+b*sin(c+d*x))/(b^3*d)+a*sin(c+d*x)/(b^2*d)-1/2*sin(c+d*x)^2/(b*d)],
[cos(c+d*x)/(a+b*sin(c+d*x)),x,2,log(a+b*sin(c+d*x))/(b*d)],
[sec(c+d*x)/(a+b*sin(c+d*x)),x,6,-1/2*log(1-sin(c+d*x))/((a+b)*d)+1/2*log(1+sin(c+d*x))/((a-b)*d)-b*log(a+b*sin(c+d*x))/((a^2-b^2)*d)],
[sec(c+d*x)^3/(a+b*sin(c+d*x)),x,4,-1/4*(a+2*b)*log(1-sin(c+d*x))/((a+b)^2*d)+1/4*(a-2*b)*log(1+sin(c+d*x))/((a-b)^2*d)+b^3*log(a+b*sin(c+d*x))/((a^2-b^2)^2*d)-1/2*sec(c+d*x)^2*(b-a*sin(c+d*x))/((a^2-b^2)*d)],
[sec(c+d*x)^5/(a+b*sin(c+d*x)),x,5,-1/16*(3*a^2+9*a*b+8*b^2)*log(1-sin(c+d*x))/((a+b)^3*d)+1/16*(3*a^2-9*a*b+8*b^2)*log(1+sin(c+d*x))/((a-b)^3*d)-b^5*log(a+b*sin(c+d*x))/((a^2-b^2)^3*d)-1/4*sec(c+d*x)^4*(b-a*sin(c+d*x))/((a^2-b^2)*d)+1/8*sec(c+d*x)^2*(4*b^3+a*(3*a^2-7*b^2)*sin(c+d*x))/((a^2-b^2)^2*d)],
[cos(c+d*x)^6/(a+b*sin(c+d*x)),x,7,1/8*a*(8*a^4-20*a^2*b^2+15*b^4)*x/b^6-2*(a^2-b^2)^(5/2)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(b^6*d)+1/5*cos(c+d*x)^5/(b*d)-1/12*cos(c+d*x)^3*(4*(a^2-b^2)-3*a*b*sin(c+d*x))/(b^3*d)+1/8*cos(c+d*x)*(8*(a^2-b^2)^2-a*b*(4*a^2-7*b^2)*sin(c+d*x))/(b^5*d)],
[cos(c+d*x)^4/(a+b*sin(c+d*x)),x,6,-1/2*a*(2*a^2-3*b^2)*x/b^4+2*(a^2-b^2)^(3/2)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(b^4*d)+1/3*cos(c+d*x)^3/(b*d)-1/2*cos(c+d*x)*(2*(a^2-b^2)-a*b*sin(c+d*x))/(b^3*d)],
[cos(c+d*x)^2/(a+b*sin(c+d*x)),x,5,a*x/b^2+cos(c+d*x)/(b*d)-2*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))*sqrt(a^2-b^2)/(b^2*d)],
[sec(c+d*x)^2/(a+b*sin(c+d*x)),x,5,-2*b^2*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/((a^2-b^2)^(3/2)*d)-sec(c+d*x)*(b-a*sin(c+d*x))/((a^2-b^2)*d)],
[sec(c+d*x)^4/(a+b*sin(c+d*x)),x,6,2*b^4*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/((a^2-b^2)^(5/2)*d)-1/3*sec(c+d*x)^3*(b-a*sin(c+d*x))/((a^2-b^2)*d)+1/3*sec(c+d*x)*(3*b^3+a*(2*a^2-5*b^2)*sin(c+d*x))/((a^2-b^2)^2*d)],
[sec(c+d*x)^6/(a+b*sin(c+d*x)),x,7,-2*b^6*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/((a^2-b^2)^(7/2)*d)-1/5*sec(c+d*x)^5*(b-a*sin(c+d*x))/((a^2-b^2)*d)+1/15*sec(c+d*x)^3*(5*b^3+a*(4*a^2-9*b^2)*sin(c+d*x))/((a^2-b^2)^2*d)-1/15*sec(c+d*x)*(15*b^5-a*(8*a^4-26*a^2*b^2+33*b^4)*sin(c+d*x))/((a^2-b^2)^3*d)],
[cos(c+d*x)^7/(a+b*sin(c+d*x))^2,x,3,6*a*(a^2-b^2)^2*log(a+b*sin(c+d*x))/(b^7*d)-(5*a^4-9*a^2*b^2+3*b^4)*sin(c+d*x)/(b^6*d)+a*(2*a^2-3*b^2)*sin(c+d*x)^2/(b^5*d)-(a^2-b^2)*sin(c+d*x)^3/(b^4*d)+1/2*a*sin(c+d*x)^4/(b^3*d)-1/5*sin(c+d*x)^5/(b^2*d)+(a^2-b^2)^3/(b^7*d*(a+b*sin(c+d*x)))],
[cos(c+d*x)^5/(a+b*sin(c+d*x))^2,x,3,-4*a*(a^2-b^2)*log(a+b*sin(c+d*x))/(b^5*d)+(3*a^2-2*b^2)*sin(c+d*x)/(b^4*d)-a*sin(c+d*x)^2/(b^3*d)+1/3*sin(c+d*x)^3/(b^2*d)-(a^2-b^2)^2/(b^5*d*(a+b*sin(c+d*x)))],
[cos(c+d*x)^3/(a+b*sin(c+d*x))^2,x,3,2*a*log(a+b*sin(c+d*x))/(b^3*d)-sin(c+d*x)/(b^2*d)+(a^2-b^2)/(b^3*d*(a+b*sin(c+d*x)))],
[cos(c+d*x)/(a+b*sin(c+d*x))^2,x,2,(-1)/(b*d*(a+b*sin(c+d*x)))],
[sec(c+d*x)/(a+b*sin(c+d*x))^2,x,4,-1/2*log(1-sin(c+d*x))/((a+b)^2*d)+1/2*log(1+sin(c+d*x))/((a-b)^2*d)-2*a*b*log(a+b*sin(c+d*x))/((a^2-b^2)^2*d)+b/((a^2-b^2)*d*(a+b*sin(c+d*x)))],
[sec(c+d*x)^3/(a+b*sin(c+d*x))^2,x,4,-1/4*(a+3*b)*log(1-sin(c+d*x))/((a+b)^3*d)+1/4*(a-3*b)*log(1+sin(c+d*x))/((a-b)^3*d)+4*a*b^3*log(a+b*sin(c+d*x))/((a^2-b^2)^3*d)-1/2*b*(a^2+3*b^2)/((a^2-b^2)^2*d*(a+b*sin(c+d*x)))-1/2*sec(c+d*x)^2*(b-a*sin(c+d*x))/((a^2-b^2)*d*(a+b*sin(c+d*x)))],
[sec(c+d*x)^5/(a+b*sin(c+d*x))^2,x,5,-3/16*(a^2+4*a*b+5*b^2)*log(1-sin(c+d*x))/((a+b)^4*d)+3/16*(a^2-4*a*b+5*b^2)*log(1+sin(c+d*x))/((a-b)^4*d)-6*a*b^5*log(a+b*sin(c+d*x))/((a^2-b^2)^4*d)-3/8*b*(a^4-4*a^2*b^2-5*b^4)/((a^2-b^2)^3*d*(a+b*sin(c+d*x)))-1/4*sec(c+d*x)^4*(b-a*sin(c+d*x))/((a^2-b^2)*d*(a+b*sin(c+d*x)))+1/8*sec(c+d*x)^2*(b*(a^2+5*b^2)+3*a*(a^2-3*b^2)*sin(c+d*x))/((a^2-b^2)^2*d*(a+b*sin(c+d*x)))],
[cos(c+d*x)^6/(a+b*sin(c+d*x))^2,x,7,-5/8*(8*a^4-12*a^2*b^2+3*b^4)*x/b^6+10*a*(a^2-b^2)^(3/2)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(b^6*d)+5/12*cos(c+d*x)^3*(4*a-3*b*sin(c+d*x))/(b^3*d)-cos(c+d*x)^5/(b*d*(a+b*sin(c+d*x)))-5/8*cos(c+d*x)*(8*a*(a^2-b^2)-b*(4*a^2-3*b^2)*sin(c+d*x))/(b^5*d)],
[cos(c+d*x)^4/(a+b*sin(c+d*x))^2,x,6,3/2*(2*a^2-b^2)*x/b^4+3/2*cos(c+d*x)*(2*a-b*sin(c+d*x))/(b^3*d)-cos(c+d*x)^3/(b*d*(a+b*sin(c+d*x)))-6*a*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))*sqrt(a^2-b^2)/(b^4*d)],
[cos(c+d*x)^2/(a+b*sin(c+d*x))^2,x,5,-x/b^2-cos(c+d*x)/(b*d*(a+b*sin(c+d*x)))+2*a*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(b^2*d*sqrt(a^2-b^2))],
[sec(c+d*x)^2/(a+b*sin(c+d*x))^2,x,6,-6*a*b^2*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/((a^2-b^2)^(5/2)*d)+b*sec(c+d*x)/((a^2-b^2)*d*(a+b*sin(c+d*x)))-sec(c+d*x)*(3*a*b-(a^2+2*b^2)*sin(c+d*x))/((a^2-b^2)^2*d)],
[sec(c+d*x)^4/(a+b*sin(c+d*x))^2,x,7,10*a*b^4*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/((a^2-b^2)^(7/2)*d)+b*sec(c+d*x)^3/((a^2-b^2)*d*(a+b*sin(c+d*x)))-1/3*sec(c+d*x)^3*(5*a*b-(a^2+4*b^2)*sin(c+d*x))/((a^2-b^2)^2*d)+1/3*sec(c+d*x)*(15*a*b^3+(2*a^4-9*a^2*b^2-8*b^4)*sin(c+d*x))/((a^2-b^2)^3*d)],
[cos(c+d*x)^7/(a+b*sin(c+d*x))^3,x,3,-3*(5*a^4-6*a^2*b^2+b^4)*log(a+b*sin(c+d*x))/(b^7*d)+a*(10*a^2-9*b^2)*sin(c+d*x)/(b^6*d)-3/2*(2*a^2-b^2)*sin(c+d*x)^2/(b^5*d)+a*sin(c+d*x)^3/(b^4*d)-1/4*sin(c+d*x)^4/(b^3*d)+1/2*(a^2-b^2)^3/(b^7*d*(a+b*sin(c+d*x))^2)-6*a*(a^2-b^2)^2/(b^7*d*(a+b*sin(c+d*x)))],
[cos(c+d*x)^5/(a+b*sin(c+d*x))^3,x,3,2*(3*a^2-b^2)*log(a+b*sin(c+d*x))/(b^5*d)-3*a*sin(c+d*x)/(b^4*d)+1/2*sin(c+d*x)^2/(b^3*d)-1/2*(a^2-b^2)^2/(b^5*d*(a+b*sin(c+d*x))^2)+4*a*(a^2-b^2)/(b^5*d*(a+b*sin(c+d*x)))],
[cos(c+d*x)^3/(a+b*sin(c+d*x))^3,x,3,-log(a+b*sin(c+d*x))/(b^3*d)+1/2*(a^2-b^2)/(b^3*d*(a+b*sin(c+d*x))^2)-2*a/(b^3*d*(a+b*sin(c+d*x)))],
[cos(c+d*x)/(a+b*sin(c+d*x))^3,x,2,(-1/2)/(b*d*(a+b*sin(c+d*x))^2)],
[sec(c+d*x)/(a+b*sin(c+d*x))^3,x,4,-1/2*log(1-sin(c+d*x))/((a+b)^3*d)+1/2*log(1+sin(c+d*x))/((a-b)^3*d)-b*(3*a^2+b^2)*log(a+b*sin(c+d*x))/((a^2-b^2)^3*d)+1/2*b/((a^2-b^2)*d*(a+b*sin(c+d*x))^2)+2*a*b/((a^2-b^2)^2*d*(a+b*sin(c+d*x)))],
[sec(c+d*x)^3/(a+b*sin(c+d*x))^3,x,4,-1/4*(a+4*b)*log(1-sin(c+d*x))/((a+b)^4*d)+1/4*(a-4*b)*log(1+sin(c+d*x))/((a-b)^4*d)+2*b^3*(5*a^2+b^2)*log(a+b*sin(c+d*x))/((a^2-b^2)^4*d)-1/2*b*(a^2+2*b^2)/((a^2-b^2)^2*d*(a+b*sin(c+d*x))^2)-1/2*sec(c+d*x)^2*(b-a*sin(c+d*x))/((a^2-b^2)*d*(a+b*sin(c+d*x))^2)-1/2*a*b*(a^2+11*b^2)/((a^2-b^2)^3*d*(a+b*sin(c+d*x)))],
[sec(c+d*x)^5/(a+b*sin(c+d*x))^3,x,5,-3/16*(a^2+5*a*b+8*b^2)*log(1-sin(c+d*x))/((a+b)^5*d)+3/16*(a^2-5*a*b+8*b^2)*log(1+sin(c+d*x))/((a-b)^5*d)-3*b^5*(7*a^2+b^2)*log(a+b*sin(c+d*x))/((a^2-b^2)^5*d)-3/8*b*(a^4-5*a^2*b^2-4*b^4)/((a^2-b^2)^3*d*(a+b*sin(c+d*x))^2)-1/4*sec(c+d*x)^4*(b-a*sin(c+d*x))/((a^2-b^2)*d*(a+b*sin(c+d*x))^2)-3/8*a*b*(a^4-6*a^2*b^2-27*b^4)/((a^2-b^2)^4*d*(a+b*sin(c+d*x)))+1/8*sec(c+d*x)^2*(2*b*(a^2+3*b^2)+a*(3*a^2-11*b^2)*sin(c+d*x))/((a^2-b^2)^2*d*(a+b*sin(c+d*x))^2)],
[cos(c+d*x)^6/(a+b*sin(c+d*x))^3,x,7,5/2*a*(4*a^2-3*b^2)*x/b^6-1/2*cos(c+d*x)^5/(b*d*(a+b*sin(c+d*x))^2)-5/6*cos(c+d*x)^3*(4*a+b*sin(c+d*x))/(b^3*d*(a+b*sin(c+d*x)))+5/2*cos(c+d*x)*(4*a^2-b^2-2*a*b*sin(c+d*x))/(b^5*d)-5*(4*a^4-5*a^2*b^2+b^4)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(b^6*d*sqrt(a^2-b^2))],
[cos(c+d*x)^4/(a+b*sin(c+d*x))^3,x,6,-3*a*x/b^4-1/2*cos(c+d*x)^3/(b*d*(a+b*sin(c+d*x))^2)-3/2*cos(c+d*x)*(2*a+b*sin(c+d*x))/(b^3*d*(a+b*sin(c+d*x)))+3*(2*a^2-b^2)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(b^4*d*sqrt(a^2-b^2))],
[cos(c+d*x)^2/(a+b*sin(c+d*x))^3,x,6,arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/((a^2-b^2)^(3/2)*d)-1/2*cos(c+d*x)/(b*d*(a+b*sin(c+d*x))^2)+1/2*a*cos(c+d*x)/(b*(a^2-b^2)*d*(a+b*sin(c+d*x)))],
[sec(c+d*x)^2/(a+b*sin(c+d*x))^3,x,7,-3*b^2*(4*a^2+b^2)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/((a^2-b^2)^(7/2)*d)+1/2*b*sec(c+d*x)/((a^2-b^2)*d*(a+b*sin(c+d*x))^2)+5/2*a*b*sec(c+d*x)/((a^2-b^2)^2*d*(a+b*sin(c+d*x)))-1/2*sec(c+d*x)*(3*b*(4*a^2+b^2)-a*(2*a^2+13*b^2)*sin(c+d*x))/((a^2-b^2)^3*d)],
[sec(c+d*x)^4/(a+b*sin(c+d*x))^3,x,8,5*b^4*(6*a^2+b^2)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/((a^2-b^2)^(9/2)*d)+1/2*b*sec(c+d*x)^3/((a^2-b^2)*d*(a+b*sin(c+d*x))^2)+7/2*a*b*sec(c+d*x)^3/((a^2-b^2)^2*d*(a+b*sin(c+d*x)))-1/6*sec(c+d*x)^3*(5*b*(6*a^2+b^2)-a*(2*a^2+33*b^2)*sin(c+d*x))/((a^2-b^2)^3*d)+1/6*sec(c+d*x)*(15*b^3*(6*a^2+b^2)+a*(4*a^4-28*a^2*b^2-81*b^4)*sin(c+d*x))/((a^2-b^2)^4*d)],
[cos(c+d*x)^7/(a+b*sin(c+d*x))^8,x,3,1/7*(a^2-b^2)^3/(b^7*d*(a+b*sin(c+d*x))^7)-a*(a^2-b^2)^2/(b^7*d*(a+b*sin(c+d*x))^6)+3/5*(5*a^4-6*a^2*b^2+b^4)/(b^7*d*(a+b*sin(c+d*x))^5)-a*(5*a^2-3*b^2)/(b^7*d*(a+b*sin(c+d*x))^4)+(5*a^2-b^2)/(b^7*d*(a+b*sin(c+d*x))^3)-3*a/(b^7*d*(a+b*sin(c+d*x))^2)+1/(b^7*d*(a+b*sin(c+d*x)))],
[cos(c+d*x)^5/(a+b*sin(c+d*x))^8,x,3,-1/7*(a^2-b^2)^2/(b^5*d*(a+b*sin(c+d*x))^7)+2/3*a*(a^2-b^2)/(b^5*d*(a+b*sin(c+d*x))^6)-2/5*(3*a^2-b^2)/(b^5*d*(a+b*sin(c+d*x))^5)+a/(b^5*d*(a+b*sin(c+d*x))^4)+(-1/3)/(b^5*d*(a+b*sin(c+d*x))^3)],
[cos(c+d*x)^3/(a+b*sin(c+d*x))^8,x,3,1/7*(a^2-b^2)/(b^3*d*(a+b*sin(c+d*x))^7)-1/3*a/(b^3*d*(a+b*sin(c+d*x))^6)+1/5/(b^3*d*(a+b*sin(c+d*x))^5)],
[cos(c+d*x)/(a+b*sin(c+d*x))^8,x,2,(-1/7)/(b*d*(a+b*sin(c+d*x))^7)],
[sec(c+d*x)/(a+b*sin(c+d*x))^8,x,4,-1/2*log(1-sin(c+d*x))/((a+b)^8*d)+1/2*log(1+sin(c+d*x))/((a-b)^8*d)-8*a*b*(a^2+b^2)*(a^4+6*a^2*b^2+b^4)*log(a+b*sin(c+d*x))/((a^2-b^2)^8*d)+1/7*b/((a^2-b^2)*d*(a+b*sin(c+d*x))^7)+1/3*a*b/((a^2-b^2)^2*d*(a+b*sin(c+d*x))^6)+1/5*b*(3*a^2+b^2)/((a^2-b^2)^3*d*(a+b*sin(c+d*x))^5)+a*b*(a^2+b^2)/((a^2-b^2)^4*d*(a+b*sin(c+d*x))^4)+1/3*b*(5*a^4+10*a^2*b^2+b^4)/((a^2-b^2)^5*d*(a+b*sin(c+d*x))^3)+a*b*(3*a^4+10*a^2*b^2+3*b^4)/((a^2-b^2)^6*d*(a+b*sin(c+d*x))^2)+b*(7*a^6+35*a^4*b^2+21*a^2*b^4+b^6)/((a^2-b^2)^7*d*(a+b*sin(c+d*x)))],
[sec(c+d*x)^3/(a+b*sin(c+d*x))^8,x,4,-1/4*(a+9*b)*log(1-sin(c+d*x))/((a+b)^9*d)+1/4*(a-9*b)*log(1+sin(c+d*x))/((a-b)^9*d)+8*a*b^3*(15*a^6+63*a^4*b^2+45*a^2*b^4+5*b^6)*log(a+b*sin(c+d*x))/((a^2-b^2)^9*d)-1/14*b*(7*a^2+9*b^2)/((a^2-b^2)^2*d*(a+b*sin(c+d*x))^7)-1/2*sec(c+d*x)^2*(b-a*sin(c+d*x))/((a^2-b^2)*d*(a+b*sin(c+d*x))^7)-1/6*a*b*(3*a^2+13*b^2)/((a^2-b^2)^3*d*(a+b*sin(c+d*x))^6)-1/10*b*(5*a^4+50*a^2*b^2+9*b^4)/((a^2-b^2)^4*d*(a+b*sin(c+d*x))^5)-1/2*a*b*(a^4+20*a^2*b^2+11*b^4)/((a^2-b^2)^5*d*(a+b*sin(c+d*x))^4)-1/6*b*(3*a^6+115*a^4*b^2+129*a^2*b^4+9*b^6)/((a^2-b^2)^6*d*(a+b*sin(c+d*x))^3)-1/2*a*b*(a^6+77*a^4*b^2+147*a^2*b^4+31*b^6)/((a^2-b^2)^7*d*(a+b*sin(c+d*x))^2)-1/2*b*(a^8+196*a^6*b^2+574*a^4*b^4+244*a^2*b^6+9*b^8)/((a^2-b^2)^8*d*(a+b*sin(c+d*x)))],
[cos(c+d*x)^8/(a+b*sin(c+d*x))^8,x,11,x/b^8-1/8*a*(16*a^6-56*a^4*b^2+70*a^2*b^4-35*b^6)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/(b^8*(a^2-b^2)^(7/2)*d)-1/7*cos(c+d*x)^7/(b*d*(a+b*sin(c+d*x))^7)+1/6*a*cos(c+d*x)^7/(b*(a^2-b^2)*d*(a+b*sin(c+d*x))^6)-1/24*a*(6*a^2-11*b^2)*cos(c+d*x)^5/(b^3*(a^2-b^2)^2*d*(a+b*sin(c+d*x))^4)+1/16*a*(8*a^4-22*a^2*b^2+19*b^4)*cos(c+d*x)^3/(b^5*(a^2-b^2)^3*d*(a+b*sin(c+d*x))^2)+1/30*cos(c+d*x)^5*(6*(a^2-b^2)+5*a*b*sin(c+d*x))/(b^3*(a^2-b^2)*d*(a+b*sin(c+d*x))^5)-1/24*cos(c+d*x)^3*(8*(a^2-b^2)^2+a*b*(6*a^2-11*b^2)*sin(c+d*x))/(b^5*(a^2-b^2)^2*d*(a+b*sin(c+d*x))^3)+1/16*cos(c+d*x)*(16*(a^2-b^2)^3+a*b*(8*a^4-22*a^2*b^2+19*b^4)*sin(c+d*x))/(b^7*(a^2-b^2)^3*d*(a+b*sin(c+d*x)))],
[cos(c+d*x)^6/(a+b*sin(c+d*x))^8,x,11,5/8*a*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/((a^2-b^2)^(9/2)*d)-1/7*cos(c+d*x)^5/(b*d*(a+b*sin(c+d*x))^7)+1/168*a*(4*a^2-b^2)*cos(c+d*x)/(b^5*(a^2-b^2)*d*(a+b*sin(c+d*x))^4)+1/168*(4*a^4-9*a^2*b^2+12*b^4)*cos(c+d*x)/(b^5*(a^2-b^2)^2*d*(a+b*sin(c+d*x))^3)+1/336*a*(8*a^4-30*a^2*b^2+57*b^4)*cos(c+d*x)/(b^5*(a^2-b^2)^3*d*(a+b*sin(c+d*x))^2)+1/336*(8*a^6-38*a^4*b^2+87*a^2*b^4+48*b^6)*cos(c+d*x)/(b^5*(a^2-b^2)^4*d*(a+b*sin(c+d*x)))+5/42*cos(c+d*x)^3*(2*a+3*b*sin(c+d*x))/(b^3*d*(a+b*sin(c+d*x))^6)-1/42*cos(c+d*x)*(4*a^2+9*b^2+10*a*b*sin(c+d*x))/(b^5*d*(a+b*sin(c+d*x))^5)],
[cos(c+d*x)^4/(a+b*sin(c+d*x))^8,x,11,3/8*a*(2*a^2+b^2)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/((a^2-b^2)^(11/2)*d)-1/7*cos(c+d*x)^3/(b*d*(a+b*sin(c+d*x))^7)-1/140*(a^2-3*b^2)*cos(c+d*x)/(b^3*(a^2-b^2)*d*(a+b*sin(c+d*x))^5)-1/280*a*(2*a^2-11*b^2)*cos(c+d*x)/(b^3*(a^2-b^2)^2*d*(a+b*sin(c+d*x))^4)-1/280*(2*a^4-15*a^2*b^2-8*b^4)*cos(c+d*x)/(b^3*(a^2-b^2)^3*d*(a+b*sin(c+d*x))^3)-1/560*a*(4*a^4-36*a^2*b^2-73*b^4)*cos(c+d*x)/(b^3*(a^2-b^2)^4*d*(a+b*sin(c+d*x))^2)-1/560*(4*a^6-40*a^4*b^2-247*a^2*b^4-32*b^6)*cos(c+d*x)/(b^3*(a^2-b^2)^5*d*(a+b*sin(c+d*x)))+1/28*cos(c+d*x)*(a+3*b*sin(c+d*x))/(b^3*d*(a+b*sin(c+d*x))^6)],
[cos(c+d*x)^2/(a+b*sin(c+d*x))^8,x,11,1/8*a*(8*a^4+20*a^2*b^2+5*b^4)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/((a^2-b^2)^(13/2)*d)-1/7*cos(c+d*x)/(b*d*(a+b*sin(c+d*x))^7)+1/42*a*cos(c+d*x)/(b*(a^2-b^2)*d*(a+b*sin(c+d*x))^6)+1/210*(5*a^2+6*b^2)*cos(c+d*x)/(b*(a^2-b^2)^2*d*(a+b*sin(c+d*x))^5)+1/840*a*(20*a^2+79*b^2)*cos(c+d*x)/(b*(a^2-b^2)^3*d*(a+b*sin(c+d*x))^4)+1/840*(20*a^4+179*a^2*b^2+32*b^4)*cos(c+d*x)/(b*(a^2-b^2)^4*d*(a+b*sin(c+d*x))^3)+1/1680*a*(40*a^4+718*a^2*b^2+397*b^4)*cos(c+d*x)/(b*(a^2-b^2)^5*d*(a+b*sin(c+d*x))^2)+1/1680*(40*a^6+1518*a^4*b^2+1779*a^2*b^4+128*b^6)*cos(c+d*x)/(b*(a^2-b^2)^6*d*(a+b*sin(c+d*x)))],
[sec(c+d*x)^2/(a+b*sin(c+d*x))^8,x,12,-9/8*a*b^2*(64*a^6+336*a^4*b^2+280*a^2*b^4+35*b^6)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/((a^2-b^2)^(17/2)*d)+1/7*b*sec(c+d*x)/((a^2-b^2)*d*(a+b*sin(c+d*x))^7)+5/14*a*b*sec(c+d*x)/((a^2-b^2)^2*d*(a+b*sin(c+d*x))^6)+1/70*b*(49*a^2+16*b^2)*sec(c+d*x)/((a^2-b^2)^3*d*(a+b*sin(c+d*x))^5)+13/280*a*b*(28*a^2+27*b^2)*sec(c+d*x)/((a^2-b^2)^4*d*(a+b*sin(c+d*x))^4)+1/280*b*(700*a^4+1317*a^2*b^2+128*b^4)*sec(c+d*x)/((a^2-b^2)^5*d*(a+b*sin(c+d*x))^3)+11/560*a*b*(280*a^4+844*a^2*b^2+241*b^4)*sec(c+d*x)/((a^2-b^2)^6*d*(a+b*sin(c+d*x))^2)+1/560*b*(9800*a^6+41484*a^4*b^2+22767*a^2*b^4+1024*b^6)*sec(c+d*x)/((a^2-b^2)^7*d*(a+b*sin(c+d*x)))-1/560*sec(c+d*x)*(315*a*b*(64*a^6+336*a^4*b^2+280*a^2*b^4+35*b^6)-(560*a^8+42472*a^6*b^2+125634*a^4*b^4+54511*a^2*b^6+2048*b^8)*sin(c+d*x))/((a^2-b^2)^8*d)],
[sec(c+d*x)^4/(a+b*sin(c+d*x))^8,x,13,165/8*a*b^4*(32*a^6+112*a^4*b^2+70*a^2*b^4+7*b^6)*arctan((b+a*tan(1/2*(c+d*x)))/sqrt(a^2-b^2))/((a^2-b^2)^(19/2)*d)+1/7*b*sec(c+d*x)^3/((a^2-b^2)*d*(a+b*sin(c+d*x))^7)+17/42*a*b*sec(c+d*x)^3/((a^2-b^2)^2*d*(a+b*sin(c+d*x))^6)+1/14*b*(13*a^2+4*b^2)*sec(c+d*x)^3/((a^2-b^2)^3*d*(a+b*sin(c+d*x))^5)+1/56*a*b*(118*a^2+103*b^2)*sec(c+d*x)^3/((a^2-b^2)^4*d*(a+b*sin(c+d*x))^4)+1/168*b*(882*a^4+1421*a^2*b^2+128*b^4)*sec(c+d*x)^3/((a^2-b^2)^5*d*(a+b*sin(c+d*x))^3)+13/112*a*b*(140*a^4+336*a^2*b^2+85*b^4)*sec(c+d*x)^3/((a^2-b^2)^6*d*(a+b*sin(c+d*x))^2)+1/112*b*(9212*a^6+28420*a^4*b^2+12907*a^2*b^4+512*b^6)*sec(c+d*x)^3/((a^2-b^2)^7*d*(a+b*sin(c+d*x)))-1/336*sec(c+d*x)^3*(1155*a*b*(32*a^6+112*a^4*b^2+70*a^2*b^4+7*b^6)-(112*a^8+52528*a^6*b^2+142902*a^4*b^4+57665*a^2*b^6+2048*b^8)*sin(c+d*x))/((a^2-b^2)^8*d)+1/336*sec(c+d*x)*(3465*a*b^3*(32*a^6+112*a^4*b^2+70*a^2*b^4+7*b^6)+(224*a^10-6048*a^8*b^2-207332*a^6*b^4-413024*a^4*b^6-135489*a^2*b^8-4096*b^10)*sin(c+d*x))/((a^2-b^2)^9*d)],

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

# m>0
[cos(c+d*x)^5*sqrt(a+b*sin(c+d*x)),x,3,2/3*(a^2-b^2)^2*(a+b*sin(c+d*x))^(3/2)/(b^5*d)-8/5*a*(a^2-b^2)*(a+b*sin(c+d*x))^(5/2)/(b^5*d)+4/7*(3*a^2-b^2)*(a+b*sin(c+d*x))^(7/2)/(b^5*d)-8/9*a*(a+b*sin(c+d*x))^(9/2)/(b^5*d)+2/11*(a+b*sin(c+d*x))^(11/2)/(b^5*d)],
[cos(c+d*x)^3*sqrt(a+b*sin(c+d*x)),x,3,-2/3*(a^2-b^2)*(a+b*sin(c+d*x))^(3/2)/(b^3*d)+4/5*a*(a+b*sin(c+d*x))^(5/2)/(b^3*d)-2/7*(a+b*sin(c+d*x))^(7/2)/(b^3*d)],
[cos(c+d*x)*sqrt(a+b*sin(c+d*x)),x,2,2/3*(a+b*sin(c+d*x))^(3/2)/(b*d)],
[sec(c+d*x)*sqrt(a+b*sin(c+d*x)),x,5,-arctanh(sqrt(a+b*sin(c+d*x))/sqrt(a-b))*sqrt(a-b)/d+arctanh(sqrt(a+b*sin(c+d*x))/sqrt(a+b))*sqrt(a+b)/d],
[sec(c+d*x)^3*sqrt(a+b*sin(c+d*x)),x,6,-1/4*(2*a-b)*arctanh(sqrt(a+b*sin(c+d*x))/sqrt(a-b))/(d*sqrt(a-b))+1/4*(2*a+b)*arctanh(sqrt(a+b*sin(c+d*x))/sqrt(a+b))/(d*sqrt(a+b))+1/2*sec(c+d*x)*sqrt(a+b*sin(c+d*x))*tan(c+d*x)/d],
[sec(c+d*x)^5*sqrt(a+b*sin(c+d*x)),x,7,-1/32*(12*a^2-18*a*b+5*b^2)*arctanh(sqrt(a+b*sin(c+d*x))/sqrt(a-b))/((a-b)^(3/2)*d)+1/32*(12*a^2+18*a*b+5*b^2)*arctanh(sqrt(a+b*sin(c+d*x))/sqrt(a+b))/((a+b)^(3/2)*d)-1/16*sec(c+d*x)^2*(a*b-(6*a^2-5*b^2)*sin(c+d*x))*sqrt(a+b*sin(c+d*x))/((a^2-b^2)*d)+1/4*sec(c+d*x)^3*sqrt(a+b*sin(c+d*x))*tan(c+d*x)/d],
[cos(c+d*x)^4*sqrt(a+b*sin(c+d*x)),x,8,2/9*cos(c+d*x)^3*(a+b*sin(c+d*x))^(3/2)/(b*d)-4/21*a*cos(c+d*x)^3*sqrt(a+b*sin(c+d*x))/(b*d)-4/315*cos(c+d*x)*(4*a*(a^2-3*b^2)-3*b*(a^2+7*b^2)*sin(c+d*x))*sqrt(a+b*sin(c+d*x))/(b^3*d)-8/315*(4*a^4-15*a^2*b^2-21*b^4)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*sin(c+d*x))/(b^4*d*sqrt((a+b*sin(c+d*x))/(a+b)))+32/315*a*(a^4-4*a^2*b^2+3*b^4)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(b^4*d*sqrt(a+b*sin(c+d*x)))],
[cos(c+d*x)^2*sqrt(a+b*sin(c+d*x)),x,7,2/5*cos(c+d*x)*(a+b*sin(c+d*x))^(3/2)/(b*d)-4/15*a*cos(c+d*x)*sqrt(a+b*sin(c+d*x))/(b*d)+4/15*(a^2+3*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*sin(c+d*x))/(b^2*d*sqrt((a+b*sin(c+d*x))/(a+b)))-4/15*a*(a^2-b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(b^2*d*sqrt(a+b*sin(c+d*x)))],
[sec(c+d*x)^2*sqrt(a+b*sin(c+d*x)),x,7,-sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*sin(c+d*x))/(d*sqrt((a+b*sin(c+d*x))/(a+b)))+a*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(d*sqrt(a+b*sin(c+d*x)))+sqrt(a+b*sin(c+d*x))*tan(c+d*x)/d],
[sec(c+d*x)^4*sqrt(a+b*sin(c+d*x)),x,7,-1/6*sec(c+d*x)*(a*b-(4*a^2-3*b^2)*sin(c+d*x))*sqrt(a+b*sin(c+d*x))/((a^2-b^2)*d)-1/6*(4*a^2-3*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*sin(c+d*x))/((a^2-b^2)*d*sqrt((a+b*sin(c+d*x))/(a+b)))+2/3*a*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(d*sqrt(a+b*sin(c+d*x)))+1/3*sec(c+d*x)^2*sqrt(a+b*sin(c+d*x))*tan(c+d*x)/d],
[cos(c+d*x)^5*(a+b*sin(c+d*x))^(3/2),x,3,2/5*(a^2-b^2)^2*(a+b*sin(c+d*x))^(5/2)/(b^5*d)-8/7*a*(a^2-b^2)*(a+b*sin(c+d*x))^(7/2)/(b^5*d)+4/9*(3*a^2-b^2)*(a+b*sin(c+d*x))^(9/2)/(b^5*d)-8/11*a*(a+b*sin(c+d*x))^(11/2)/(b^5*d)+2/13*(a+b*sin(c+d*x))^(13/2)/(b^5*d)],
[cos(c+d*x)^3*(a+b*sin(c+d*x))^(3/2),x,3,-2/5*(a^2-b^2)*(a+b*sin(c+d*x))^(5/2)/(b^3*d)+4/7*a*(a+b*sin(c+d*x))^(7/2)/(b^3*d)-2/9*(a+b*sin(c+d*x))^(9/2)/(b^3*d)],
[cos(c+d*x)*(a+b*sin(c+d*x))^(3/2),x,2,2/5*(a+b*sin(c+d*x))^(5/2)/(b*d)],
[sec(c+d*x)*(a+b*sin(c+d*x))^(3/2),x,6,-(a-b)^(3/2)*arctanh(sqrt(a+b*sin(c+d*x))/sqrt(a-b))/d+(a+b)^(3/2)*arctanh(sqrt(a+b*sin(c+d*x))/sqrt(a+b))/d-2*b*sqrt(a+b*sin(c+d*x))/d],
[sec(c+d*x)^3*(a+b*sin(c+d*x))^(3/2),x,6,-1/4*(2*a+b)*arctanh(sqrt(a+b*sin(c+d*x))/sqrt(a-b))*sqrt(a-b)/d+1/4*(2*a-b)*arctanh(sqrt(a+b*sin(c+d*x))/sqrt(a+b))*sqrt(a+b)/d+1/2*sec(c+d*x)^2*(b+a*sin(c+d*x))*sqrt(a+b*sin(c+d*x))/d],
[sec(c+d*x)^5*(a+b*sin(c+d*x))^(3/2),x,7,-3/32*(4*a^2-2*a*b-b^2)*arctanh(sqrt(a+b*sin(c+d*x))/sqrt(a-b))/(d*sqrt(a-b))+3/32*(4*a^2+2*a*b-b^2)*arctanh(sqrt(a+b*sin(c+d*x))/sqrt(a+b))/(d*sqrt(a+b))-1/16*sec(c+d*x)^2*(b-6*a*sin(c+d*x))*sqrt(a+b*sin(c+d*x))/d+1/4*sec(c+d*x)^4*(b+a*sin(c+d*x))*sqrt(a+b*sin(c+d*x))/d],
[cos(c+d*x)^4*(a+b*sin(c+d*x))^(3/2),x,8,-2/11*b*cos(c+d*x)^5*sqrt(a+b*sin(c+d*x))/d+2/231*cos(c+d*x)^3*(a^2+3*b^2+28*a*b*sin(c+d*x))*sqrt(a+b*sin(c+d*x))/(b*d)-4/1155*cos(c+d*x)*(4*a^4-21*a^2*b^2-15*b^4-3*a*b*(a^2+31*b^2)*sin(c+d*x))*sqrt(a+b*sin(c+d*x))/(b^3*d)-32/1155*a*(a^4-6*a^2*b^2-27*b^4)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*sin(c+d*x))/(b^4*d*sqrt((a+b*sin(c+d*x))/(a+b)))+8/1155*(4*a^6-25*a^4*b^2+6*a^2*b^4+15*b^6)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(b^4*d*sqrt(a+b*sin(c+d*x)))],
[cos(c+d*x)^2*(a+b*sin(c+d*x))^(3/2),x,7,-2/7*b*cos(c+d*x)^3*sqrt(a+b*sin(c+d*x))/d+2/105*cos(c+d*x)*(3*a^2+5*b^2+24*a*b*sin(c+d*x))*sqrt(a+b*sin(c+d*x))/(b*d)+4/105*a*(3*a^2+29*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*sin(c+d*x))/(b^2*d*sqrt((a+b*sin(c+d*x))/(a+b)))-4/105*(3*a^4+2*a^2*b^2-5*b^4)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(b^2*d*sqrt(a+b*sin(c+d*x)))],
[sec(c+d*x)^2*(a+b*sin(c+d*x))^(3/2),x,6,sec(c+d*x)*(b+a*sin(c+d*x))*sqrt(a+b*sin(c+d*x))/d-a*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*sin(c+d*x))/(d*sqrt((a+b*sin(c+d*x))/(a+b)))+(a^2-b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(d*sqrt(a+b*sin(c+d*x)))],
[sec(c+d*x)^4*(a+b*sin(c+d*x))^(3/2),x,7,-1/6*sec(c+d*x)*(b-4*a*sin(c+d*x))*sqrt(a+b*sin(c+d*x))/d+1/3*sec(c+d*x)^3*(b+a*sin(c+d*x))*sqrt(a+b*sin(c+d*x))/d-2/3*a*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*sin(c+d*x))/(d*sqrt((a+b*sin(c+d*x))/(a+b)))+1/6*(4*a^2-b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(d*sqrt(a+b*sin(c+d*x)))],
[sec(c+d*x)^6*(a+b*sin(c+d*x))^(3/2),x,8,-1/30*sec(c+d*x)^3*(b-8*a*sin(c+d*x))*sqrt(a+b*sin(c+d*x))/d+1/5*sec(c+d*x)^5*(b+a*sin(c+d*x))*sqrt(a+b*sin(c+d*x))/d-1/60*sec(c+d*x)*(b*(8*a^4-13*a^2*b^2+5*b^4)-a*(32*a^4-61*a^2*b^2+29*b^4)*sin(c+d*x))*sqrt(a+b*sin(c+d*x))/((a^2-b^2)^2*d)-1/60*a*(32*a^2-29*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*sin(c+d*x))/((a^2-b^2)*d*sqrt((a+b*sin(c+d*x))/(a+b)))+1/60*(32*a^2-5*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(d*sqrt(a+b*sin(c+d*x)))],
[cos(c+d*x)^5*(a+b*sin(c+d*x))^(5/2),x,3,2/7*(a^2-b^2)^2*(a+b*sin(c+d*x))^(7/2)/(b^5*d)-8/9*a*(a^2-b^2)*(a+b*sin(c+d*x))^(9/2)/(b^5*d)+4/11*(3*a^2-b^2)*(a+b*sin(c+d*x))^(11/2)/(b^5*d)-8/13*a*(a+b*sin(c+d*x))^(13/2)/(b^5*d)+2/15*(a+b*sin(c+d*x))^(15/2)/(b^5*d)],
[cos(c+d*x)^3*(a+b*sin(c+d*x))^(5/2),x,3,-2/7*(a^2-b^2)*(a+b*sin(c+d*x))^(7/2)/(b^3*d)+4/9*a*(a+b*sin(c+d*x))^(9/2)/(b^3*d)-2/11*(a+b*sin(c+d*x))^(11/2)/(b^3*d)],
[cos(c+d*x)*(a+b*sin(c+d*x))^(5/2),x,2,2/7*(a+b*sin(c+d*x))^(7/2)/(b*d)],
[sec(c+d*x)*(a+b*sin(c+d*x))^(5/2),x,7,-(a-b)^(5/2)*arctanh(sqrt(a+b*sin(c+d*x))/sqrt(a-b))/d+(a+b)^(5/2)*arctanh(sqrt(a+b*sin(c+d*x))/sqrt(a+b))/d-2/3*b*(a+b*sin(c+d*x))^(3/2)/d-4*a*b*sqrt(a+b*sin(c+d*x))/d],
[sec(c+d*x)^3*(a+b*sin(c+d*x))^(5/2),x,7,-1/4*(a-b)^(3/2)*(2*a+3*b)*arctanh(sqrt(a+b*sin(c+d*x))/sqrt(a-b))/d+1/4*(2*a-3*b)*(a+b)^(3/2)*arctanh(sqrt(a+b*sin(c+d*x))/sqrt(a+b))/d+1/2*sec(c+d*x)^2*(b+a*sin(c+d*x))*(a+b*sin(c+d*x))^(3/2)/d+1/2*a*b*sqrt(a+b*sin(c+d*x))/d],
[sec(c+d*x)^5*(a+b*sin(c+d*x))^(5/2),x,7,1/4*sec(c+d*x)^4*(b+a*sin(c+d*x))*(a+b*sin(c+d*x))^(3/2)/d-3/32*(4*a^2+2*a*b-b^2)*arctanh(sqrt(a+b*sin(c+d*x))/sqrt(a-b))*sqrt(a-b)/d+3/32*(4*a^2-2*a*b-b^2)*arctanh(sqrt(a+b*sin(c+d*x))/sqrt(a+b))*sqrt(a+b)/d+3/16*sec(c+d*x)^2*(a*b+(2*a^2-b^2)*sin(c+d*x))*sqrt(a+b*sin(c+d*x))/d],
[cos(c+d*x)^4*(a+b*sin(c+d*x))^(5/2),x,9,-2/13*b*cos(c+d*x)^5*(a+b*sin(c+d*x))^(3/2)/d-32/143*a*b*cos(c+d*x)^5*sqrt(a+b*sin(c+d*x))/d+2/3003*cos(c+d*x)^3*(a*(5*a^2+59*b^2)+7*b*(53*a^2+11*b^2)*sin(c+d*x))*sqrt(a+b*sin(c+d*x))/(b*d)-4/15015*cos(c+d*x)*(4*a*(5*a^4-40*a^2*b^2-93*b^4)-3*b*(5*a^4+430*a^2*b^2+77*b^4)*sin(c+d*x))*sqrt(a+b*sin(c+d*x))/(b^3*d)-8/15015*(20*a^6-175*a^4*b^2-1662*a^2*b^4-231*b^6)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*sin(c+d*x))/(b^4*d*sqrt((a+b*sin(c+d*x))/(a+b)))+32/15015*a*(5*a^6-45*a^4*b^2-53*a^2*b^4+93*b^6)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(b^4*d*sqrt(a+b*sin(c+d*x)))],
[cos(c+d*x)^2*(a+b*sin(c+d*x))^(5/2),x,8,-2/9*b*cos(c+d*x)^3*(a+b*sin(c+d*x))^(3/2)/d-8/21*a*b*cos(c+d*x)^3*sqrt(a+b*sin(c+d*x))/d+2/315*cos(c+d*x)*(a*(5*a^2+27*b^2)+3*b*(25*a^2+7*b^2)*sin(c+d*x))*sqrt(a+b*sin(c+d*x))/(b*d)+4/315*(5*a^4+102*a^2*b^2+21*b^4)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*sin(c+d*x))/(b^2*d*sqrt((a+b*sin(c+d*x))/(a+b)))-4/315*a*(5*a^4+22*a^2*b^2-27*b^4)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(b^2*d*sqrt(a+b*sin(c+d*x)))],
[sec(c+d*x)^2*(a+b*sin(c+d*x))^(5/2),x,7,sec(c+d*x)*(b+a*sin(c+d*x))*(a+b*sin(c+d*x))^(3/2)/d+a*b*cos(c+d*x)*sqrt(a+b*sin(c+d*x))/d-(a^2+3*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*sin(c+d*x))/(d*sqrt((a+b*sin(c+d*x))/(a+b)))+a*(a^2-b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(d*sqrt(a+b*sin(c+d*x)))],
[sec(c+d*x)^4*(a+b*sin(c+d*x))^(5/2),x,7,1/3*sec(c+d*x)^3*(b+a*sin(c+d*x))*(a+b*sin(c+d*x))^(3/2)/d+1/6*sec(c+d*x)*(a*b+(4*a^2-3*b^2)*sin(c+d*x))*sqrt(a+b*sin(c+d*x))/d-1/6*(4*a^2-3*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*sin(c+d*x))/(d*sqrt((a+b*sin(c+d*x))/(a+b)))+2/3*a*(a^2-b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(d*sqrt(a+b*sin(c+d*x)))],
[sec(c+d*x)^6*(a+b*sin(c+d*x))^(5/2),x,8,1/5*sec(c+d*x)^5*(b+a*sin(c+d*x))*(a+b*sin(c+d*x))^(3/2)/d+1/30*sec(c+d*x)^3*(5*a*b+(8*a^2-3*b^2)*sin(c+d*x))*sqrt(a+b*sin(c+d*x))/d-1/60*sec(c+d*x)*(8*a*b*(a^2-b^2)-(32*a^4-41*a^2*b^2+9*b^4)*sin(c+d*x))*sqrt(a+b*sin(c+d*x))/((a^2-b^2)*d)-1/60*(32*a^2-9*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*sin(c+d*x))/(d*sqrt((a+b*sin(c+d*x))/(a+b)))+1/60*a*(32*a^2-17*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(d*sqrt(a+b*sin(c+d*x)))],
[sec(c+d*x)^8*(a+b*sin(c+d*x))^(5/2),x,9,1/7*sec(c+d*x)^7*(b+a*sin(c+d*x))*(a+b*sin(c+d*x))^(3/2)/d+3/70*sec(c+d*x)^5*(3*a*b+(4*a^2-b^2)*sin(c+d*x))*sqrt(a+b*sin(c+d*x))/d-1/140*sec(c+d*x)^3*(4*a*b*(a^2-b^2)-(32*a^4-39*a^2*b^2+7*b^4)*sin(c+d*x))*sqrt(a+b*sin(c+d*x))/((a^2-b^2)*d)-1/280*sec(c+d*x)*(a*b*(32*a^4-59*a^2*b^2+27*b^4)-(128*a^6-272*a^4*b^2+165*a^2*b^4-21*b^6)*sin(c+d*x))*sqrt(a+b*sin(c+d*x))/((a^2-b^2)^2*d)-1/280*(128*a^4-144*a^2*b^2+21*b^4)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*sin(c+d*x))/((a^2-b^2)*d*sqrt((a+b*sin(c+d*x))/(a+b)))+2/35*a*(8*a^2-3*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(d*sqrt(a+b*sin(c+d*x)))],

# m<0
[cos(c+d*x)^5/sqrt(a+b*sin(c+d*x)),x,3,-8/3*a*(a^2-b^2)*(a+b*sin(c+d*x))^(3/2)/(b^5*d)+4/5*(3*a^2-b^2)*(a+b*sin(c+d*x))^(5/2)/(b^5*d)-8/7*a*(a+b*sin(c+d*x))^(7/2)/(b^5*d)+2/9*(a+b*sin(c+d*x))^(9/2)/(b^5*d)+2*(a^2-b^2)^2*sqrt(a+b*sin(c+d*x))/(b^5*d)],
[cos(c+d*x)^3/sqrt(a+b*sin(c+d*x)),x,3,4/3*a*(a+b*sin(c+d*x))^(3/2)/(b^3*d)-2/5*(a+b*sin(c+d*x))^(5/2)/(b^3*d)-2*(a^2-b^2)*sqrt(a+b*sin(c+d*x))/(b^3*d)],
[cos(c+d*x)/sqrt(a+b*sin(c+d*x)),x,2,2*sqrt(a+b*sin(c+d*x))/(b*d)],
[sec(c+d*x)/sqrt(a+b*sin(c+d*x)),x,5,-arctanh(sqrt(a+b*sin(c+d*x))/sqrt(a-b))/(d*sqrt(a-b))+arctanh(sqrt(a+b*sin(c+d*x))/sqrt(a+b))/(d*sqrt(a+b))],
[sec(c+d*x)^3/sqrt(a+b*sin(c+d*x)),x,6,-1/4*(2*a-3*b)*arctanh(sqrt(a+b*sin(c+d*x))/sqrt(a-b))/((a-b)^(3/2)*d)+1/4*(2*a+3*b)*arctanh(sqrt(a+b*sin(c+d*x))/sqrt(a+b))/((a+b)^(3/2)*d)-1/2*sec(c+d*x)^2*(b-a*sin(c+d*x))*sqrt(a+b*sin(c+d*x))/((a^2-b^2)*d)],
[sec(c+d*x)^5/sqrt(a+b*sin(c+d*x)),x,7,-3/32*(4*a^2-10*a*b+7*b^2)*arctanh(sqrt(a+b*sin(c+d*x))/sqrt(a-b))/((a-b)^(5/2)*d)+3/32*(4*a^2+10*a*b+7*b^2)*arctanh(sqrt(a+b*sin(c+d*x))/sqrt(a+b))/((a+b)^(5/2)*d)-1/4*sec(c+d*x)^4*(b-a*sin(c+d*x))*sqrt(a+b*sin(c+d*x))/((a^2-b^2)*d)-1/16*sec(c+d*x)^2*(b*(a^2-7*b^2)-6*a*(a^2-2*b^2)*sin(c+d*x))*sqrt(a+b*sin(c+d*x))/((a^2-b^2)^2*d)],
[cos(c+d*x)^4/sqrt(a+b*sin(c+d*x)),x,7,2/7*cos(c+d*x)^3*sqrt(a+b*sin(c+d*x))/(b*d)-4/35*cos(c+d*x)*(4*a^2-5*b^2-3*a*b*sin(c+d*x))*sqrt(a+b*sin(c+d*x))/(b^3*d)-32/35*a*(a^2-2*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*sin(c+d*x))/(b^4*d*sqrt((a+b*sin(c+d*x))/(a+b)))+8/35*(4*a^4-9*a^2*b^2+5*b^4)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(b^4*d*sqrt(a+b*sin(c+d*x)))],
[cos(c+d*x)^2/sqrt(a+b*sin(c+d*x)),x,6,2/3*cos(c+d*x)*sqrt(a+b*sin(c+d*x))/(b*d)+4/3*a*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*sin(c+d*x))/(b^2*d*sqrt((a+b*sin(c+d*x))/(a+b)))-4/3*(a^2-b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(b^2*d*sqrt(a+b*sin(c+d*x)))],
[sec(c+d*x)^2/sqrt(a+b*sin(c+d*x)),x,6,-sec(c+d*x)*(b-a*sin(c+d*x))*sqrt(a+b*sin(c+d*x))/((a^2-b^2)*d)-a*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*sin(c+d*x))/((a^2-b^2)*d*sqrt((a+b*sin(c+d*x))/(a+b)))+sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(d*sqrt(a+b*sin(c+d*x)))],
[sec(c+d*x)^4/sqrt(a+b*sin(c+d*x)),x,7,-1/3*sec(c+d*x)^3*(b-a*sin(c+d*x))*sqrt(a+b*sin(c+d*x))/((a^2-b^2)*d)-1/6*sec(c+d*x)*(b*(a^2-5*b^2)-4*a*(a^2-2*b^2)*sin(c+d*x))*sqrt(a+b*sin(c+d*x))/((a^2-b^2)^2*d)-2/3*a*(a^2-2*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*sin(c+d*x))/((a^2-b^2)^2*d*sqrt((a+b*sin(c+d*x))/(a+b)))+1/6*(4*a^2-5*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/((a^2-b^2)*d*sqrt(a+b*sin(c+d*x)))],
[cos(c+d*x)^5/(a+b*sin(c+d*x))^(3/2),x,3,4/3*(3*a^2-b^2)*(a+b*sin(c+d*x))^(3/2)/(b^5*d)-8/5*a*(a+b*sin(c+d*x))^(5/2)/(b^5*d)+2/7*(a+b*sin(c+d*x))^(7/2)/(b^5*d)-2*(a^2-b^2)^2/(b^5*d*sqrt(a+b*sin(c+d*x)))-8*a*(a^2-b^2)*sqrt(a+b*sin(c+d*x))/(b^5*d)],
[cos(c+d*x)^3/(a+b*sin(c+d*x))^(3/2),x,3,-2/3*(a+b*sin(c+d*x))^(3/2)/(b^3*d)+2*(a^2-b^2)/(b^3*d*sqrt(a+b*sin(c+d*x)))+4*a*sqrt(a+b*sin(c+d*x))/(b^3*d)],
[cos(c+d*x)/(a+b*sin(c+d*x))^(3/2),x,2,(-2)/(b*d*sqrt(a+b*sin(c+d*x)))],
[sec(c+d*x)/(a+b*sin(c+d*x))^(3/2),x,6,-arctanh(sqrt(a+b*sin(c+d*x))/sqrt(a-b))/((a-b)^(3/2)*d)+arctanh(sqrt(a+b*sin(c+d*x))/sqrt(a+b))/((a+b)^(3/2)*d)+2*b/((a^2-b^2)*d*sqrt(a+b*sin(c+d*x)))],
[sec(c+d*x)^3/(a+b*sin(c+d*x))^(3/2),x,7,-1/4*(2*a-5*b)*arctanh(sqrt(a+b*sin(c+d*x))/sqrt(a-b))/((a-b)^(5/2)*d)+1/4*(2*a+5*b)*arctanh(sqrt(a+b*sin(c+d*x))/sqrt(a+b))/((a+b)^(5/2)*d)-1/2*b*(a^2+5*b^2)/((a^2-b^2)^2*d*sqrt(a+b*sin(c+d*x)))-1/2*sec(c+d*x)^2*(b-a*sin(c+d*x))/((a^2-b^2)*d*sqrt(a+b*sin(c+d*x)))],
[sec(c+d*x)^5/(a+b*sin(c+d*x))^(3/2),x,8,-3/32*(4*a^2-14*a*b+15*b^2)*arctanh(sqrt(a+b*sin(c+d*x))/sqrt(a-b))/((a-b)^(7/2)*d)+3/32*(4*a^2+14*a*b+15*b^2)*arctanh(sqrt(a+b*sin(c+d*x))/sqrt(a+b))/((a+b)^(7/2)*d)-3/16*b*(2*a^4-7*a^2*b^2-15*b^4)/((a^2-b^2)^3*d*sqrt(a+b*sin(c+d*x)))-1/4*sec(c+d*x)^4*(b-a*sin(c+d*x))/((a^2-b^2)*d*sqrt(a+b*sin(c+d*x)))+1/16*sec(c+d*x)^2*(b*(a^2+9*b^2)+2*a*(3*a^2-8*b^2)*sin(c+d*x))/((a^2-b^2)^2*d*sqrt(a+b*sin(c+d*x)))],
[cos(c+d*x)^6/(a+b*sin(c+d*x))^(3/2),x,8,-2*cos(c+d*x)^5/(b*d*sqrt(a+b*sin(c+d*x)))+20/63*cos(c+d*x)^3*(8*a-7*b*sin(c+d*x))*sqrt(a+b*sin(c+d*x))/(b^3*d)-8/63*cos(c+d*x)*(a*(32*a^2-33*b^2)-3*b*(8*a^2-7*b^2)*sin(c+d*x))*sqrt(a+b*sin(c+d*x))/(b^5*d)-16/63*(32*a^4-57*a^2*b^2+21*b^4)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*sin(c+d*x))/(b^6*d*sqrt((a+b*sin(c+d*x))/(a+b)))+16/63*a*(32*a^4-65*a^2*b^2+33*b^4)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(b^6*d*sqrt(a+b*sin(c+d*x)))],
[cos(c+d*x)^4/(a+b*sin(c+d*x))^(3/2),x,7,-2*cos(c+d*x)^3/(b*d*sqrt(a+b*sin(c+d*x)))+4/5*cos(c+d*x)*(4*a-3*b*sin(c+d*x))*sqrt(a+b*sin(c+d*x))/(b^3*d)+8/5*(4*a^2-3*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*sin(c+d*x))/(b^4*d*sqrt((a+b*sin(c+d*x))/(a+b)))-32/5*a*(a^2-b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(b^4*d*sqrt(a+b*sin(c+d*x)))],
[cos(c+d*x)^2/(a+b*sin(c+d*x))^(3/2),x,6,-2*cos(c+d*x)/(b*d*sqrt(a+b*sin(c+d*x)))-4*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*sin(c+d*x))/(b^2*d*sqrt((a+b*sin(c+d*x))/(a+b)))+4*a*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(b^2*d*sqrt(a+b*sin(c+d*x)))],
[sec(c+d*x)^2/(a+b*sin(c+d*x))^(3/2),x,7,2*b*sec(c+d*x)/((a^2-b^2)*d*sqrt(a+b*sin(c+d*x)))-sec(c+d*x)*(4*a*b-(a^2+3*b^2)*sin(c+d*x))*sqrt(a+b*sin(c+d*x))/((a^2-b^2)^2*d)-(a^2+3*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*sin(c+d*x))/((a^2-b^2)^2*d*sqrt((a+b*sin(c+d*x))/(a+b)))+a*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/((a^2-b^2)*d*sqrt(a+b*sin(c+d*x)))],
[sec(c+d*x)^4/(a+b*sin(c+d*x))^(3/2),x,8,2*b*sec(c+d*x)^3/((a^2-b^2)*d*sqrt(a+b*sin(c+d*x)))-1/3*sec(c+d*x)^3*(8*a*b-(a^2+7*b^2)*sin(c+d*x))*sqrt(a+b*sin(c+d*x))/((a^2-b^2)^2*d)-1/6*sec(c+d*x)*(a*b*(a^2-33*b^2)-(4*a^4-15*a^2*b^2-21*b^4)*sin(c+d*x))*sqrt(a+b*sin(c+d*x))/((a^2-b^2)^3*d)-1/6*(4*a^4-15*a^2*b^2-21*b^4)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*sin(c+d*x))/((a^2-b^2)^3*d*sqrt((a+b*sin(c+d*x))/(a+b)))+2/3*a*(a^2-3*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/((a^2-b^2)^2*d*sqrt(a+b*sin(c+d*x)))],
[cos(c+d*x)^5/(a+b*sin(c+d*x))^(5/2),x,3,-2/3*(a^2-b^2)^2/(b^5*d*(a+b*sin(c+d*x))^(3/2))-8/3*a*(a+b*sin(c+d*x))^(3/2)/(b^5*d)+2/5*(a+b*sin(c+d*x))^(5/2)/(b^5*d)+8*a*(a^2-b^2)/(b^5*d*sqrt(a+b*sin(c+d*x)))+4*(3*a^2-b^2)*sqrt(a+b*sin(c+d*x))/(b^5*d)],
[cos(c+d*x)^3/(a+b*sin(c+d*x))^(5/2),x,3,2/3*(a^2-b^2)/(b^3*d*(a+b*sin(c+d*x))^(3/2))-4*a/(b^3*d*sqrt(a+b*sin(c+d*x)))-2*sqrt(a+b*sin(c+d*x))/(b^3*d)],
[cos(c+d*x)/(a+b*sin(c+d*x))^(5/2),x,2,(-2/3)/(b*d*(a+b*sin(c+d*x))^(3/2))],
[sec(c+d*x)/(a+b*sin(c+d*x))^(5/2),x,7,-arctanh(sqrt(a+b*sin(c+d*x))/sqrt(a-b))/((a-b)^(5/2)*d)+arctanh(sqrt(a+b*sin(c+d*x))/sqrt(a+b))/((a+b)^(5/2)*d)+2/3*b/((a^2-b^2)*d*(a+b*sin(c+d*x))^(3/2))+4*a*b/((a^2-b^2)^2*d*sqrt(a+b*sin(c+d*x)))],
[sec(c+d*x)^3/(a+b*sin(c+d*x))^(5/2),x,8,-1/4*(2*a-7*b)*arctanh(sqrt(a+b*sin(c+d*x))/sqrt(a-b))/((a-b)^(7/2)*d)+1/4*(2*a+7*b)*arctanh(sqrt(a+b*sin(c+d*x))/sqrt(a+b))/((a+b)^(7/2)*d)-1/6*b*(3*a^2+7*b^2)/((a^2-b^2)^2*d*(a+b*sin(c+d*x))^(3/2))-1/2*sec(c+d*x)^2*(b-a*sin(c+d*x))/((a^2-b^2)*d*(a+b*sin(c+d*x))^(3/2))-1/2*a*b*(a^2+19*b^2)/((a^2-b^2)^3*d*sqrt(a+b*sin(c+d*x)))],
[sec(c+d*x)^5/(a+b*sin(c+d*x))^(5/2),x,9,-1/32*(12*a^2-54*a*b+77*b^2)*arctanh(sqrt(a+b*sin(c+d*x))/sqrt(a-b))/((a-b)^(9/2)*d)+1/32*(12*a^2+54*a*b+77*b^2)*arctanh(sqrt(a+b*sin(c+d*x))/sqrt(a+b))/((a+b)^(9/2)*d)-1/48*b*(18*a^4-81*a^2*b^2-77*b^4)/((a^2-b^2)^3*d*(a+b*sin(c+d*x))^(3/2))-1/4*sec(c+d*x)^4*(b-a*sin(c+d*x))/((a^2-b^2)*d*(a+b*sin(c+d*x))^(3/2))+1/16*sec(c+d*x)^2*(b*(3*a^2+11*b^2)+2*a*(3*a^2-10*b^2)*sin(c+d*x))/((a^2-b^2)^2*d*(a+b*sin(c+d*x))^(3/2))-1/8*a*b*(3*a^4-16*a^2*b^2-127*b^4)/((a^2-b^2)^4*d*sqrt(a+b*sin(c+d*x)))],
[cos(c+d*x)^8/(a+b*sin(c+d*x))^(5/2),x,9,-2/3*cos(c+d*x)^7/(b*d*(a+b*sin(c+d*x))^(3/2))-28/33*cos(c+d*x)^5*(12*a+b*sin(c+d*x))/(b^3*d*sqrt(a+b*sin(c+d*x)))+40/99*cos(c+d*x)^3*(32*a^2-3*b^2-28*a*b*sin(c+d*x))*sqrt(a+b*sin(c+d*x))/(b^5*d)-16/99*cos(c+d*x)*(128*a^4-144*a^2*b^2+15*b^4-3*a*b*(32*a^2-31*b^2)*sin(c+d*x))*sqrt(a+b*sin(c+d*x))/(b^7*d)-128/99*a*(8*a^2-9*b^2)*(4*a^2-3*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*sin(c+d*x))/(b^8*d*sqrt((a+b*sin(c+d*x))/(a+b)))+32/99*(128*a^6-272*a^4*b^2+159*a^2*b^4-15*b^6)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(b^8*d*sqrt(a+b*sin(c+d*x)))],
[cos(c+d*x)^6/(a+b*sin(c+d*x))^(5/2),x,8,-2/3*cos(c+d*x)^5/(b*d*(a+b*sin(c+d*x))^(3/2))-20/21*cos(c+d*x)^3*(8*a+b*sin(c+d*x))/(b^3*d*sqrt(a+b*sin(c+d*x)))+8/21*cos(c+d*x)*(32*a^2-5*b^2-24*a*b*sin(c+d*x))*sqrt(a+b*sin(c+d*x))/(b^5*d)+16/21*a*(32*a^2-29*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*sin(c+d*x))/(b^6*d*sqrt((a+b*sin(c+d*x))/(a+b)))-16/21*(32*a^4-37*a^2*b^2+5*b^4)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(b^6*d*sqrt(a+b*sin(c+d*x)))],
[cos(c+d*x)^4/(a+b*sin(c+d*x))^(5/2),x,7,-2/3*cos(c+d*x)^3/(b*d*(a+b*sin(c+d*x))^(3/2))-4/3*cos(c+d*x)*(4*a+b*sin(c+d*x))/(b^3*d*sqrt(a+b*sin(c+d*x)))-32/3*a*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*sin(c+d*x))/(b^4*d*sqrt((a+b*sin(c+d*x))/(a+b)))+8/3*(4*a^2-b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(b^4*d*sqrt(a+b*sin(c+d*x)))],
[cos(c+d*x)^2/(a+b*sin(c+d*x))^(5/2),x,7,-2/3*cos(c+d*x)/(b*d*(a+b*sin(c+d*x))^(3/2))+4/3*a*cos(c+d*x)/(b*(a^2-b^2)*d*sqrt(a+b*sin(c+d*x)))+4/3*a*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*sin(c+d*x))/(b^2*(a^2-b^2)*d*sqrt((a+b*sin(c+d*x))/(a+b)))-4/3*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/(b^2*d*sqrt(a+b*sin(c+d*x)))],
[sec(c+d*x)^2/(a+b*sin(c+d*x))^(5/2),x,8,2/3*b*sec(c+d*x)/((a^2-b^2)*d*(a+b*sin(c+d*x))^(3/2))+16/3*a*b*sec(c+d*x)/((a^2-b^2)^2*d*sqrt(a+b*sin(c+d*x)))-1/3*sec(c+d*x)*(b*(27*a^2+5*b^2)-a*(3*a^2+29*b^2)*sin(c+d*x))*sqrt(a+b*sin(c+d*x))/((a^2-b^2)^3*d)-1/3*a*(3*a^2+29*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*sin(c+d*x))/((a^2-b^2)^3*d*sqrt((a+b*sin(c+d*x))/(a+b)))+1/3*(3*a^2+5*b^2)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/((a^2-b^2)^2*d*sqrt(a+b*sin(c+d*x)))],
[sec(c+d*x)^4/(a+b*sin(c+d*x))^(5/2),x,9,2/3*b*sec(c+d*x)^3/((a^2-b^2)*d*(a+b*sin(c+d*x))^(3/2))+8*a*b*sec(c+d*x)^3/((a^2-b^2)^2*d*sqrt(a+b*sin(c+d*x)))-1/3*sec(c+d*x)^3*(b*(29*a^2+3*b^2)-a*(a^2+31*b^2)*sin(c+d*x))*sqrt(a+b*sin(c+d*x))/((a^2-b^2)^3*d)-1/6*sec(c+d*x)*(b*(a^4-114*a^2*b^2-15*b^4)-4*a*(a^4-6*a^2*b^2-27*b^4)*sin(c+d*x))*sqrt(a+b*sin(c+d*x))/((a^2-b^2)^4*d)-2/3*a*(a^4-6*a^2*b^2-27*b^4)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticE(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*sin(c+d*x))/((a^2-b^2)^4*d*sqrt((a+b*sin(c+d*x))/(a+b)))+1/6*(4*a^4-21*a^2*b^2-15*b^4)*sqrt(cos(1/2*(c-1/2*Pi+d*x))^2)/cos(1/2*(c-1/2*Pi+d*x))*EllipticF(sin(1/2*(c-1/2*Pi+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*sin(c+d*x))/(a+b))/((a^2-b^2)^3*d*sqrt(a+b*sin(c+d*x)))],

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

# n>0
[(e*cos(c+d*x))^(7/2)*(a+b*sin(c+d*x)),x,5,-2/9*b*(e*cos(c+d*x))^(9/2)/(d*e)+2/7*a*e*(e*cos(c+d*x))^(5/2)*sin(c+d*x)/d+10/21*a*e^4*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(e*cos(c+d*x)))+10/21*a*e^3*sin(c+d*x)*sqrt(e*cos(c+d*x))/d],
[(e*cos(c+d*x))^(5/2)*(a+b*sin(c+d*x)),x,4,-2/7*b*(e*cos(c+d*x))^(7/2)/(d*e)+2/5*a*e*(e*cos(c+d*x))^(3/2)*sin(c+d*x)/d+6/5*a*e^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(e*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(e*cos(c+d*x))^(3/2)*(a+b*sin(c+d*x)),x,4,-2/5*b*(e*cos(c+d*x))^(5/2)/(d*e)+2/3*a*e^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(e*cos(c+d*x)))+2/3*a*e*sin(c+d*x)*sqrt(e*cos(c+d*x))/d],
[(a+b*sin(c+d*x))*sqrt(e*cos(c+d*x)),x,3,-2/3*b*(e*cos(c+d*x))^(3/2)/(d*e)+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(e*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(a+b*sin(c+d*x))/sqrt(e*cos(c+d*x)),x,3,2*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))/(d*sqrt(e*cos(c+d*x)))-2*b*sqrt(e*cos(c+d*x))/(d*e)],
[(a+b*sin(c+d*x))/(e*cos(c+d*x))^(3/2),x,4,2*b/(d*e*sqrt(e*cos(c+d*x)))+2*a*sin(c+d*x)/(d*e*sqrt(e*cos(c+d*x)))-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(e*cos(c+d*x))/(d*e^2*sqrt(cos(c+d*x)))],
[(a+b*sin(c+d*x))/(e*cos(c+d*x))^(5/2),x,4,2/3*b/(d*e*(e*cos(c+d*x))^(3/2))+2/3*a*sin(c+d*x)/(d*e*(e*cos(c+d*x))^(3/2))+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))/(d*e^2*sqrt(e*cos(c+d*x)))],
[(a+b*sin(c+d*x))/(e*cos(c+d*x))^(7/2),x,5,2/5*b/(d*e*(e*cos(c+d*x))^(5/2))+2/5*a*sin(c+d*x)/(d*e*(e*cos(c+d*x))^(5/2))+6/5*a*sin(c+d*x)/(d*e^3*sqrt(e*cos(c+d*x)))-6/5*a*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(e*cos(c+d*x))/(d*e^4*sqrt(cos(c+d*x)))],
[(e*cos(c+d*x))^(7/2)*(a+b*sin(c+d*x))^2,x,6,-26/99*a*b*(e*cos(c+d*x))^(9/2)/(d*e)+2/77*(11*a^2+2*b^2)*e*(e*cos(c+d*x))^(5/2)*sin(c+d*x)/d-2/11*b*(e*cos(c+d*x))^(9/2)*(a+b*sin(c+d*x))/(d*e)+10/231*(11*a^2+2*b^2)*e^4*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(e*cos(c+d*x)))+10/231*(11*a^2+2*b^2)*e^3*sin(c+d*x)*sqrt(e*cos(c+d*x))/d],
[(e*cos(c+d*x))^(5/2)*(a+b*sin(c+d*x))^2,x,5,-22/63*a*b*(e*cos(c+d*x))^(7/2)/(d*e)+2/45*(9*a^2+2*b^2)*e*(e*cos(c+d*x))^(3/2)*sin(c+d*x)/d-2/9*b*(e*cos(c+d*x))^(7/2)*(a+b*sin(c+d*x))/(d*e)+2/15*(9*a^2+2*b^2)*e^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(e*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(e*cos(c+d*x))^(3/2)*(a+b*sin(c+d*x))^2,x,5,-18/35*a*b*(e*cos(c+d*x))^(5/2)/(d*e)-2/7*b*(e*cos(c+d*x))^(5/2)*(a+b*sin(c+d*x))/(d*e)+2/21*(7*a^2+2*b^2)*e^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(e*cos(c+d*x)))+2/21*(7*a^2+2*b^2)*e*sin(c+d*x)*sqrt(e*cos(c+d*x))/d],
[(a+b*sin(c+d*x))^2*sqrt(e*cos(c+d*x)),x,4,-14/15*a*b*(e*cos(c+d*x))^(3/2)/(d*e)-2/5*b*(e*cos(c+d*x))^(3/2)*(a+b*sin(c+d*x))/(d*e)+2/5*(5*a^2+2*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(e*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(a+b*sin(c+d*x))^2/sqrt(e*cos(c+d*x)),x,4,2/3*(3*a^2+2*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(e*cos(c+d*x)))-10/3*a*b*sqrt(e*cos(c+d*x))/(d*e)-2/3*b*(a+b*sin(c+d*x))*sqrt(e*cos(c+d*x))/(d*e)],
[(a+b*sin(c+d*x))^2/(e*cos(c+d*x))^(3/2),x,4,2*a*b*(e*cos(c+d*x))^(3/2)/(d*e^3)+2*(b+a*sin(c+d*x))*(a+b*sin(c+d*x))/(d*e*sqrt(e*cos(c+d*x)))-2*(a^2+2*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(e*cos(c+d*x))/(d*e^2*sqrt(cos(c+d*x)))],
[(a+b*sin(c+d*x))^2/(e*cos(c+d*x))^(5/2),x,4,2/3*(b+a*sin(c+d*x))*(a+b*sin(c+d*x))/(d*e*(e*cos(c+d*x))^(3/2))+2/3*(a^2-2*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*e^2*sqrt(e*cos(c+d*x)))+2/3*a*b*sqrt(e*cos(c+d*x))/(d*e^3)],
[(a+b*sin(c+d*x))^2/(e*cos(c+d*x))^(7/2),x,5,2/5*(b+a*sin(c+d*x))*(a+b*sin(c+d*x))/(d*e*(e*cos(c+d*x))^(5/2))+2/5*a*b/(d*e^3*sqrt(e*cos(c+d*x)))+2/5*(3*a^2-2*b^2)*sin(c+d*x)/(d*e^3*sqrt(e*cos(c+d*x)))-2/5*(3*a^2-2*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(e*cos(c+d*x))/(d*e^4*sqrt(cos(c+d*x)))],
[(e*cos(c+d*x))^(7/2)*(a+b*sin(c+d*x))^3,x,7,-2/1287*b*(177*a^2+44*b^2)*(e*cos(c+d*x))^(9/2)/(d*e)+2/77*a*(11*a^2+6*b^2)*e*(e*cos(c+d*x))^(5/2)*sin(c+d*x)/d-34/143*a*b*(e*cos(c+d*x))^(9/2)*(a+b*sin(c+d*x))/(d*e)-2/13*b*(e*cos(c+d*x))^(9/2)*(a+b*sin(c+d*x))^2/(d*e)+10/231*a*(11*a^2+6*b^2)*e^4*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(e*cos(c+d*x)))+10/231*a*(11*a^2+6*b^2)*e^3*sin(c+d*x)*sqrt(e*cos(c+d*x))/d],
[(e*cos(c+d*x))^(5/2)*(a+b*sin(c+d*x))^3,x,6,-2/231*b*(43*a^2+12*b^2)*(e*cos(c+d*x))^(7/2)/(d*e)+2/15*a*(3*a^2+2*b^2)*e*(e*cos(c+d*x))^(3/2)*sin(c+d*x)/d-10/33*a*b*(e*cos(c+d*x))^(7/2)*(a+b*sin(c+d*x))/(d*e)-2/11*b*(e*cos(c+d*x))^(7/2)*(a+b*sin(c+d*x))^2/(d*e)+2/5*a*(3*a^2+2*b^2)*e^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(e*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(e*cos(c+d*x))^(3/2)*(a+b*sin(c+d*x))^3,x,6,-2/315*b*(89*a^2+28*b^2)*(e*cos(c+d*x))^(5/2)/(d*e)-26/63*a*b*(e*cos(c+d*x))^(5/2)*(a+b*sin(c+d*x))/(d*e)-2/9*b*(e*cos(c+d*x))^(5/2)*(a+b*sin(c+d*x))^2/(d*e)+2/21*a*(7*a^2+6*b^2)*e^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(e*cos(c+d*x)))+2/21*a*(7*a^2+6*b^2)*e*sin(c+d*x)*sqrt(e*cos(c+d*x))/d],
[(a+b*sin(c+d*x))^3*sqrt(e*cos(c+d*x)),x,5,-2/105*b*(57*a^2+20*b^2)*(e*cos(c+d*x))^(3/2)/(d*e)-22/35*a*b*(e*cos(c+d*x))^(3/2)*(a+b*sin(c+d*x))/(d*e)-2/7*b*(e*cos(c+d*x))^(3/2)*(a+b*sin(c+d*x))^2/(d*e)+2/5*a*(5*a^2+6*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(e*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(a+b*sin(c+d*x))^3/sqrt(e*cos(c+d*x)),x,5,2*a*(a^2+2*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(e*cos(c+d*x)))-2/5*b*(11*a^2+4*b^2)*sqrt(e*cos(c+d*x))/(d*e)-6/5*a*b*(a+b*sin(c+d*x))*sqrt(e*cos(c+d*x))/(d*e)-2/5*b*(a+b*sin(c+d*x))^2*sqrt(e*cos(c+d*x))/(d*e)],
[(a+b*sin(c+d*x))^3/(e*cos(c+d*x))^(3/2),x,5,2/3*b*(3*a^2+4*b^2)*(e*cos(c+d*x))^(3/2)/(d*e^3)+2*a*b*(e*cos(c+d*x))^(3/2)*(a+b*sin(c+d*x))/(d*e^3)+2*(b+a*sin(c+d*x))*(a+b*sin(c+d*x))^2/(d*e*sqrt(e*cos(c+d*x)))-2*a*(a^2+6*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(e*cos(c+d*x))/(d*e^2*sqrt(cos(c+d*x)))],
[(a+b*sin(c+d*x))^3/(e*cos(c+d*x))^(5/2),x,5,2/3*(b+a*sin(c+d*x))*(a+b*sin(c+d*x))^2/(d*e*(e*cos(c+d*x))^(3/2))+2/3*a*(a^2-6*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*e^2*sqrt(e*cos(c+d*x)))+2/3*b*(a^2+4*b^2)*sqrt(e*cos(c+d*x))/(d*e^3)+2/3*a*b*(a+b*sin(c+d*x))*sqrt(e*cos(c+d*x))/(d*e^3)],
[(a+b*sin(c+d*x))^3/(e*cos(c+d*x))^(7/2),x,5,2/5*b*(3*a^2-4*b^2)*(e*cos(c+d*x))^(3/2)/(d*e^5)+2/5*(b+a*sin(c+d*x))*(a+b*sin(c+d*x))^2/(d*e*(e*cos(c+d*x))^(5/2))-2/5*(a+b*sin(c+d*x))*(a*b-(3*a^2-4*b^2)*sin(c+d*x))/(d*e^3*sqrt(e*cos(c+d*x)))-6/5*a*(a^2-2*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(e*cos(c+d*x))/(d*e^4*sqrt(cos(c+d*x)))],
[(a+b*sin(c+d*x))^3/(e*cos(c+d*x))^(9/2),x,5,2/7*(b+a*sin(c+d*x))*(a+b*sin(c+d*x))^2/(d*e*(e*cos(c+d*x))^(7/2))+2/21*(a+b*sin(c+d*x))*(a*b+(5*a^2-4*b^2)*sin(c+d*x))/(d*e^3*(e*cos(c+d*x))^(3/2))+2/21*a*(5*a^2-6*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*e^4*sqrt(e*cos(c+d*x)))+2/21*b*(5*a^2-4*b^2)*sqrt(e*cos(c+d*x))/(d*e^5)],
[(e*cos(c+d*x))^(7/2)*(a+b*sin(c+d*x))^4,x,8,-34/6435*a*b*(53*a^2+38*b^2)*(e*cos(c+d*x))^(9/2)/(d*e)+2/385*(55*a^4+60*a^2*b^2+4*b^4)*e*(e*cos(c+d*x))^(5/2)*sin(c+d*x)/d-2/715*b*(93*a^2+26*b^2)*(e*cos(c+d*x))^(9/2)*(a+b*sin(c+d*x))/(d*e)-14/65*a*b*(e*cos(c+d*x))^(9/2)*(a+b*sin(c+d*x))^2/(d*e)-2/15*b*(e*cos(c+d*x))^(9/2)*(a+b*sin(c+d*x))^3/(d*e)+2/231*(55*a^4+60*a^2*b^2+4*b^4)*e^4*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(e*cos(c+d*x)))+2/231*(55*a^4+60*a^2*b^2+4*b^4)*e^3*sin(c+d*x)*sqrt(e*cos(c+d*x))/d],
[(e*cos(c+d*x))^(5/2)*(a+b*sin(c+d*x))^4,x,7,-10/3003*a*b*(115*a^2+94*b^2)*(e*cos(c+d*x))^(7/2)/(d*e)+2/195*(39*a^4+52*a^2*b^2+4*b^4)*e*(e*cos(c+d*x))^(3/2)*sin(c+d*x)/d-2/429*b*(73*a^2+22*b^2)*(e*cos(c+d*x))^(7/2)*(a+b*sin(c+d*x))/(d*e)-38/143*a*b*(e*cos(c+d*x))^(7/2)*(a+b*sin(c+d*x))^2/(d*e)-2/13*b*(e*cos(c+d*x))^(7/2)*(a+b*sin(c+d*x))^3/(d*e)+2/65*(39*a^4+52*a^2*b^2+4*b^4)*e^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(e*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(e*cos(c+d*x))^(3/2)*(a+b*sin(c+d*x))^4,x,7,-26/3465*a*b*(79*a^2+74*b^2)*(e*cos(c+d*x))^(5/2)/(d*e)-2/693*b*(167*a^2+54*b^2)*(e*cos(c+d*x))^(5/2)*(a+b*sin(c+d*x))/(d*e)-34/99*a*b*(e*cos(c+d*x))^(5/2)*(a+b*sin(c+d*x))^2/(d*e)-2/11*b*(e*cos(c+d*x))^(5/2)*(a+b*sin(c+d*x))^3/(d*e)+2/231*(77*a^4+132*a^2*b^2+12*b^4)*e^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(e*cos(c+d*x)))+2/231*(77*a^4+132*a^2*b^2+12*b^4)*e*sin(c+d*x)*sqrt(e*cos(c+d*x))/d],
[(a+b*sin(c+d*x))^4*sqrt(e*cos(c+d*x)),x,6,-22/315*a*b*(17*a^2+18*b^2)*(e*cos(c+d*x))^(3/2)/(d*e)-2/105*b*(41*a^2+14*b^2)*(e*cos(c+d*x))^(3/2)*(a+b*sin(c+d*x))/(d*e)-10/21*a*b*(e*cos(c+d*x))^(3/2)*(a+b*sin(c+d*x))^2/(d*e)-2/9*b*(e*cos(c+d*x))^(3/2)*(a+b*sin(c+d*x))^3/(d*e)+2/15*(15*a^4+36*a^2*b^2+4*b^4)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(e*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(a+b*sin(c+d*x))^4/sqrt(e*cos(c+d*x)),x,6,2/7*(7*a^4+28*a^2*b^2+4*b^4)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(e*cos(c+d*x)))-6/35*a*b*(31*a^2+34*b^2)*sqrt(e*cos(c+d*x))/(d*e)-2/35*b*(29*a^2+10*b^2)*(a+b*sin(c+d*x))*sqrt(e*cos(c+d*x))/(d*e)-26/35*a*b*(a+b*sin(c+d*x))^2*sqrt(e*cos(c+d*x))/(d*e)-2/7*b*(a+b*sin(c+d*x))^3*sqrt(e*cos(c+d*x))/(d*e)],
[(a+b*sin(c+d*x))^4/(e*cos(c+d*x))^(3/2),x,6,2/15*a*b*(15*a^2+62*b^2)*(e*cos(c+d*x))^(3/2)/(d*e^3)+2/5*b*(5*a^2+6*b^2)*(e*cos(c+d*x))^(3/2)*(a+b*sin(c+d*x))/(d*e^3)+2*a*b*(e*cos(c+d*x))^(3/2)*(a+b*sin(c+d*x))^2/(d*e^3)+2*(b+a*sin(c+d*x))*(a+b*sin(c+d*x))^3/(d*e*sqrt(e*cos(c+d*x)))-2/5*(5*a^4+60*a^2*b^2+12*b^4)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(e*cos(c+d*x))/(d*e^2*sqrt(cos(c+d*x)))],
[(a+b*sin(c+d*x))^4/(e*cos(c+d*x))^(5/2),x,6,2/3*(b+a*sin(c+d*x))*(a+b*sin(c+d*x))^3/(d*e*(e*cos(c+d*x))^(3/2))+2/3*(a^4-12*a^2*b^2-4*b^4)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*e^2*sqrt(e*cos(c+d*x)))+2/3*a*b*(a^2+14*b^2)*sqrt(e*cos(c+d*x))/(d*e^3)+2/3*b*(a^2+2*b^2)*(a+b*sin(c+d*x))*sqrt(e*cos(c+d*x))/(d*e^3)+2/3*a*b*(a+b*sin(c+d*x))^2*sqrt(e*cos(c+d*x))/(d*e^3)],
[(a+b*sin(c+d*x))^4/(e*cos(c+d*x))^(7/2),x,6,2/5*a*b*(3*a^2-10*b^2)*(e*cos(c+d*x))^(3/2)/(d*e^5)+6/5*b*(a^2-2*b^2)*(e*cos(c+d*x))^(3/2)*(a+b*sin(c+d*x))/(d*e^5)+2/5*(b+a*sin(c+d*x))*(a+b*sin(c+d*x))^3/(d*e*(e*cos(c+d*x))^(5/2))-6/5*(a+b*sin(c+d*x))^2*(a*b-(a^2-2*b^2)*sin(c+d*x))/(d*e^3*sqrt(e*cos(c+d*x)))-6/5*(a^4-4*a^2*b^2-4*b^4)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(e*cos(c+d*x))/(d*e^4*sqrt(cos(c+d*x)))],
[(a+b*sin(c+d*x))^4/(e*cos(c+d*x))^(9/2),x,6,2/7*(b+a*sin(c+d*x))*(a+b*sin(c+d*x))^3/(d*e*(e*cos(c+d*x))^(7/2))-2/21*(a+b*sin(c+d*x))^2*(a*b-(5*a^2-6*b^2)*sin(c+d*x))/(d*e^3*(e*cos(c+d*x))^(3/2))+2/21*(5*a^4-12*a^2*b^2+12*b^4)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*e^4*sqrt(e*cos(c+d*x)))+10/21*a*b*(a^2-2*b^2)*sqrt(e*cos(c+d*x))/(d*e^5)+2/21*b*(5*a^2-6*b^2)*(a+b*sin(c+d*x))*sqrt(e*cos(c+d*x))/(d*e^5)],
[(a+b*sin(c+d*x))^4/(e*cos(c+d*x))^(11/2),x,6,2/45*a*b*(21*a^2-22*b^2)*(e*cos(c+d*x))^(3/2)/(d*e^7)+2/9*(b+a*sin(c+d*x))*(a+b*sin(c+d*x))^3/(d*e*(e*cos(c+d*x))^(9/2))+2/45*(a+b*sin(c+d*x))^2*(a*b+(7*a^2-6*b^2)*sin(c+d*x))/(d*e^3*(e*cos(c+d*x))^(5/2))-2/45*(a+b*sin(c+d*x))*(b*(7*a^2-6*b^2)-a*(21*a^2-22*b^2)*sin(c+d*x))/(d*e^5*sqrt(e*cos(c+d*x)))-2/15*(7*a^4-12*a^2*b^2+4*b^4)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(e*cos(c+d*x))/(d*e^6*sqrt(cos(c+d*x)))],

# n<0
[(e*cos(c+d*x))^(11/2)/(a+b*sin(c+d*x)),x,15,-(-a^2+b^2)^(9/4)*e^(11/2)*arctan(sqrt(b)*sqrt(e*cos(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/(b^(11/2)*d)-(-a^2+b^2)^(9/4)*e^(11/2)*arctanh(sqrt(b)*sqrt(e*cos(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/(b^(11/2)*d)+2/9*e*(e*cos(c+d*x))^(9/2)/(b*d)-2/35*e^3*(e*cos(c+d*x))^(5/2)*(7*(a^2-b^2)-5*a*b*sin(c+d*x))/(b^3*d)+2/21*a*(21*a^4-49*a^2*b^2+33*b^4)*e^6*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(b^6*d*sqrt(e*cos(c+d*x)))-a*(a^2-b^2)^3*e^6*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(c+d*x))/(b^6*d*(a^2-b*(b-sqrt(-a^2+b^2)))*sqrt(e*cos(c+d*x)))-a*(a^2-b^2)^3*e^6*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(c+d*x))/(b^6*d*(a^2-b*(b+sqrt(-a^2+b^2)))*sqrt(e*cos(c+d*x)))+2/21*e^5*(21*(a^2-b^2)^2-a*b*(7*a^2-12*b^2)*sin(c+d*x))*sqrt(e*cos(c+d*x))/(b^5*d)],
[(e*cos(c+d*x))^(9/2)/(a+b*sin(c+d*x)),x,14,(-a^2+b^2)^(7/4)*e^(9/2)*arctan(sqrt(b)*sqrt(e*cos(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/(b^(9/2)*d)-(-a^2+b^2)^(7/4)*e^(9/2)*arctanh(sqrt(b)*sqrt(e*cos(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/(b^(9/2)*d)+2/7*e*(e*cos(c+d*x))^(7/2)/(b*d)-2/15*e^3*(e*cos(c+d*x))^(3/2)*(5*(a^2-b^2)-3*a*b*sin(c+d*x))/(b^3*d)+a*(a^2-b^2)^2*e^5*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(c+d*x))/(b^5*d*(b-sqrt(-a^2+b^2))*sqrt(e*cos(c+d*x)))+a*(a^2-b^2)^2*e^5*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(c+d*x))/(b^5*d*(b+sqrt(-a^2+b^2))*sqrt(e*cos(c+d*x)))-2/5*a*(5*a^2-8*b^2)*e^4*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(e*cos(c+d*x))/(b^4*d*sqrt(cos(c+d*x)))],
[(e*cos(c+d*x))^(7/2)/(a+b*sin(c+d*x)),x,14,-(-a^2+b^2)^(5/4)*e^(7/2)*arctan(sqrt(b)*sqrt(e*cos(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/(b^(7/2)*d)-(-a^2+b^2)^(5/4)*e^(7/2)*arctanh(sqrt(b)*sqrt(e*cos(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/(b^(7/2)*d)+2/5*e*(e*cos(c+d*x))^(5/2)/(b*d)-2/3*a*(3*a^2-4*b^2)*e^4*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(b^4*d*sqrt(e*cos(c+d*x)))+a*(a^2-b^2)^2*e^4*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(c+d*x))/(b^4*d*(a^2-b*(b-sqrt(-a^2+b^2)))*sqrt(e*cos(c+d*x)))+a*(a^2-b^2)^2*e^4*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(c+d*x))/(b^4*d*(a^2-b*(b+sqrt(-a^2+b^2)))*sqrt(e*cos(c+d*x)))-2/3*e^3*(3*(a^2-b^2)-a*b*sin(c+d*x))*sqrt(e*cos(c+d*x))/(b^3*d)],
[(e*cos(c+d*x))^(5/2)/(a+b*sin(c+d*x)),x,13,(-a^2+b^2)^(3/4)*e^(5/2)*arctan(sqrt(b)*sqrt(e*cos(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/(b^(5/2)*d)-(-a^2+b^2)^(3/4)*e^(5/2)*arctanh(sqrt(b)*sqrt(e*cos(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/(b^(5/2)*d)+2/3*e*(e*cos(c+d*x))^(3/2)/(b*d)-a*(a^2-b^2)*e^3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(c+d*x))/(b^3*d*(b-sqrt(-a^2+b^2))*sqrt(e*cos(c+d*x)))-a*(a^2-b^2)*e^3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(c+d*x))/(b^3*d*(b+sqrt(-a^2+b^2))*sqrt(e*cos(c+d*x)))+2*a*e^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(e*cos(c+d*x))/(b^2*d*sqrt(cos(c+d*x)))],
[(e*cos(c+d*x))^(3/2)/(a+b*sin(c+d*x)),x,13,-(-a^2+b^2)^(1/4)*e^(3/2)*arctan(sqrt(b)*sqrt(e*cos(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/(b^(3/2)*d)-(-a^2+b^2)^(1/4)*e^(3/2)*arctanh(sqrt(b)*sqrt(e*cos(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/(b^(3/2)*d)+2*a*e^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(b^2*d*sqrt(e*cos(c+d*x)))-a*(a^2-b^2)*e^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(c+d*x))/(b^2*d*(a^2-b*(b-sqrt(-a^2+b^2)))*sqrt(e*cos(c+d*x)))-a*(a^2-b^2)*e^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(c+d*x))/(b^2*d*(a^2-b*(b+sqrt(-a^2+b^2)))*sqrt(e*cos(c+d*x)))+2*e*sqrt(e*cos(c+d*x))/(b*d)],
[sqrt(e*cos(c+d*x))/(a+b*sin(c+d*x)),x,9,arctan(sqrt(b)*sqrt(e*cos(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))*sqrt(e)/((-a^2+b^2)^(1/4)*d*sqrt(b))-arctanh(sqrt(b)*sqrt(e*cos(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))*sqrt(e)/((-a^2+b^2)^(1/4)*d*sqrt(b))+a*e*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(c+d*x))/(b*d*(b-sqrt(-a^2+b^2))*sqrt(e*cos(c+d*x)))+a*e*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(c+d*x))/(b*d*(b+sqrt(-a^2+b^2))*sqrt(e*cos(c+d*x)))],
[1/((a+b*sin(c+d*x))*sqrt(e*cos(c+d*x))),x,9,-arctan(sqrt(b)*sqrt(e*cos(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))*sqrt(b)/((-a^2+b^2)^(3/4)*d*sqrt(e))-arctanh(sqrt(b)*sqrt(e*cos(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))*sqrt(b)/((-a^2+b^2)^(3/4)*d*sqrt(e))+a*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(c+d*x))/(d*(a^2-b*(b-sqrt(-a^2+b^2)))*sqrt(e*cos(c+d*x)))+a*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(c+d*x))/(d*(a^2-b*(b+sqrt(-a^2+b^2)))*sqrt(e*cos(c+d*x)))],
[1/((e*cos(c+d*x))^(3/2)*(a+b*sin(c+d*x))),x,13,b^(3/2)*arctan(sqrt(b)*sqrt(e*cos(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/((-a^2+b^2)^(5/4)*d*e^(3/2))-b^(3/2)*arctanh(sqrt(b)*sqrt(e*cos(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/((-a^2+b^2)^(5/4)*d*e^(3/2))-2*(b-a*sin(c+d*x))/((a^2-b^2)*d*e*sqrt(e*cos(c+d*x)))-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/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(c+d*x))/((a^2-b^2)*d*e*(b-sqrt(-a^2+b^2))*sqrt(e*cos(c+d*x)))-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/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(c+d*x))/((a^2-b^2)*d*e*(b+sqrt(-a^2+b^2))*sqrt(e*cos(c+d*x)))-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(e*cos(c+d*x))/((a^2-b^2)*d*e^2*sqrt(cos(c+d*x)))],
[1/((e*cos(c+d*x))^(5/2)*(a+b*sin(c+d*x))),x,13,-b^(5/2)*arctan(sqrt(b)*sqrt(e*cos(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/((-a^2+b^2)^(7/4)*d*e^(5/2))-b^(5/2)*arctanh(sqrt(b)*sqrt(e*cos(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/((-a^2+b^2)^(7/4)*d*e^(5/2))-2/3*(b-a*sin(c+d*x))/((a^2-b^2)*d*e*(e*cos(c+d*x))^(3/2))+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))/((a^2-b^2)*d*e^2*sqrt(e*cos(c+d*x)))-a*b^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(c+d*x))/((a^2-b^2)*d*e^2*(a^2-b*(b-sqrt(-a^2+b^2)))*sqrt(e*cos(c+d*x)))-a*b^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(c+d*x))/((a^2-b^2)*d*e^2*(a^2-b*(b+sqrt(-a^2+b^2)))*sqrt(e*cos(c+d*x)))],
[1/((e*cos(c+d*x))^(7/2)*(a+b*sin(c+d*x))),x,14,b^(7/2)*arctan(sqrt(b)*sqrt(e*cos(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/((-a^2+b^2)^(9/4)*d*e^(7/2))-b^(7/2)*arctanh(sqrt(b)*sqrt(e*cos(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/((-a^2+b^2)^(9/4)*d*e^(7/2))-2/5*(b-a*sin(c+d*x))/((a^2-b^2)*d*e*(e*cos(c+d*x))^(5/2))+2/5*(5*b^3+a*(3*a^2-8*b^2)*sin(c+d*x))/((a^2-b^2)^2*d*e^3*sqrt(e*cos(c+d*x)))+a*b^3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(c+d*x))/((a^2-b^2)^2*d*e^3*(b-sqrt(-a^2+b^2))*sqrt(e*cos(c+d*x)))+a*b^3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(c+d*x))/((a^2-b^2)^2*d*e^3*(b+sqrt(-a^2+b^2))*sqrt(e*cos(c+d*x)))-2/5*a*(3*a^2-8*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(e*cos(c+d*x))/((a^2-b^2)^2*d*e^4*sqrt(cos(c+d*x)))],
[(e*cos(c+d*x))^(11/2)/(a+b*sin(c+d*x))^2,x,15,-9/2*a*(-a^2+b^2)^(5/4)*e^(11/2)*arctan(sqrt(b)*sqrt(e*cos(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/(b^(11/2)*d)-9/2*a*(-a^2+b^2)^(5/4)*e^(11/2)*arctanh(sqrt(b)*sqrt(e*cos(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/(b^(11/2)*d)+9/35*e^3*(e*cos(c+d*x))^(5/2)*(7*a-5*b*sin(c+d*x))/(b^3*d)-e*(e*cos(c+d*x))^(9/2)/(b*d*(a+b*sin(c+d*x)))-3/7*(21*a^4-28*a^2*b^2+5*b^4)*e^6*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(b^6*d*sqrt(e*cos(c+d*x)))+9/2*a^2*(a^2-b^2)^2*e^6*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(c+d*x))/(b^6*d*(a^2-b*(b-sqrt(-a^2+b^2)))*sqrt(e*cos(c+d*x)))+9/2*a^2*(a^2-b^2)^2*e^6*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(c+d*x))/(b^6*d*(a^2-b*(b+sqrt(-a^2+b^2)))*sqrt(e*cos(c+d*x)))-3/7*e^5*(21*a*(a^2-b^2)-b*(7*a^2-5*b^2)*sin(c+d*x))*sqrt(e*cos(c+d*x))/(b^5*d)],
[(e*cos(c+d*x))^(9/2)/(a+b*sin(c+d*x))^2,x,14,7/2*a*(-a^2+b^2)^(3/4)*e^(9/2)*arctan(sqrt(b)*sqrt(e*cos(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/(b^(9/2)*d)-7/2*a*(-a^2+b^2)^(3/4)*e^(9/2)*arctanh(sqrt(b)*sqrt(e*cos(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/(b^(9/2)*d)+7/15*e^3*(e*cos(c+d*x))^(3/2)*(5*a-3*b*sin(c+d*x))/(b^3*d)-e*(e*cos(c+d*x))^(7/2)/(b*d*(a+b*sin(c+d*x)))-7/2*a^2*(a^2-b^2)*e^5*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(c+d*x))/(b^5*d*(b-sqrt(-a^2+b^2))*sqrt(e*cos(c+d*x)))-7/2*a^2*(a^2-b^2)*e^5*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(c+d*x))/(b^5*d*(b+sqrt(-a^2+b^2))*sqrt(e*cos(c+d*x)))+7/5*(5*a^2-3*b^2)*e^4*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(e*cos(c+d*x))/(b^4*d*sqrt(cos(c+d*x)))],
[(e*cos(c+d*x))^(7/2)/(a+b*sin(c+d*x))^2,x,14,-5/2*a*(-a^2+b^2)^(1/4)*e^(7/2)*arctan(sqrt(b)*sqrt(e*cos(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/(b^(7/2)*d)-5/2*a*(-a^2+b^2)^(1/4)*e^(7/2)*arctanh(sqrt(b)*sqrt(e*cos(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/(b^(7/2)*d)-e*(e*cos(c+d*x))^(5/2)/(b*d*(a+b*sin(c+d*x)))+5/3*(3*a^2-b^2)*e^4*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(b^4*d*sqrt(e*cos(c+d*x)))-5/2*a^2*(a^2-b^2)*e^4*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(c+d*x))/(b^4*d*(a^2-b*(b-sqrt(-a^2+b^2)))*sqrt(e*cos(c+d*x)))-5/2*a^2*(a^2-b^2)*e^4*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(c+d*x))/(b^4*d*(a^2-b*(b+sqrt(-a^2+b^2)))*sqrt(e*cos(c+d*x)))+5/3*e^3*(3*a-b*sin(c+d*x))*sqrt(e*cos(c+d*x))/(b^3*d)],
[(e*cos(c+d*x))^(5/2)/(a+b*sin(c+d*x))^2,x,13,3/2*a*e^(5/2)*arctan(sqrt(b)*sqrt(e*cos(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/(b^(5/2)*(-a^2+b^2)^(1/4)*d)-3/2*a*e^(5/2)*arctanh(sqrt(b)*sqrt(e*cos(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/(b^(5/2)*(-a^2+b^2)^(1/4)*d)-e*(e*cos(c+d*x))^(3/2)/(b*d*(a+b*sin(c+d*x)))+3/2*a^2*e^3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(c+d*x))/(b^3*d*(b-sqrt(-a^2+b^2))*sqrt(e*cos(c+d*x)))+3/2*a^2*e^3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(c+d*x))/(b^3*d*(b+sqrt(-a^2+b^2))*sqrt(e*cos(c+d*x)))-3*e^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(e*cos(c+d*x))/(b^2*d*sqrt(cos(c+d*x)))],
[(e*cos(c+d*x))^(3/2)/(a+b*sin(c+d*x))^2,x,13,-1/2*a*e^(3/2)*arctan(sqrt(b)*sqrt(e*cos(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/(b^(3/2)*(-a^2+b^2)^(3/4)*d)-1/2*a*e^(3/2)*arctanh(sqrt(b)*sqrt(e*cos(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/(b^(3/2)*(-a^2+b^2)^(3/4)*d)-e^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(b^2*d*sqrt(e*cos(c+d*x)))+1/2*a^2*e^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(c+d*x))/(b^2*d*(a^2-b*(b-sqrt(-a^2+b^2)))*sqrt(e*cos(c+d*x)))+1/2*a^2*e^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(c+d*x))/(b^2*d*(a^2-b*(b+sqrt(-a^2+b^2)))*sqrt(e*cos(c+d*x)))-e*sqrt(e*cos(c+d*x))/(b*d*(a+b*sin(c+d*x)))],
[sqrt(e*cos(c+d*x))/(a+b*sin(c+d*x))^2,x,13,b*(e*cos(c+d*x))^(3/2)/((a^2-b^2)*d*e*(a+b*sin(c+d*x)))-1/2*a*arctan(sqrt(b)*sqrt(e*cos(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))*sqrt(e)/((-a^2+b^2)^(5/4)*d*sqrt(b))+1/2*a*arctanh(sqrt(b)*sqrt(e*cos(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))*sqrt(e)/((-a^2+b^2)^(5/4)*d*sqrt(b))+1/2*a^2*e*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(c+d*x))/(b*(a^2-b^2)*d*(b-sqrt(-a^2+b^2))*sqrt(e*cos(c+d*x)))+1/2*a^2*e*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(c+d*x))/(b*(a^2-b^2)*d*(b+sqrt(-a^2+b^2))*sqrt(e*cos(c+d*x)))+sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(e*cos(c+d*x))/((a^2-b^2)*d*sqrt(cos(c+d*x)))],
[1/((a+b*sin(c+d*x))^2*sqrt(e*cos(c+d*x))),x,13,3/2*a*arctan(sqrt(b)*sqrt(e*cos(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))*sqrt(b)/((-a^2+b^2)^(7/4)*d*sqrt(e))+3/2*a*arctanh(sqrt(b)*sqrt(e*cos(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))*sqrt(b)/((-a^2+b^2)^(7/4)*d*sqrt(e))-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))/((a^2-b^2)*d*sqrt(e*cos(c+d*x)))+3/2*a^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(c+d*x))/((a^2-b^2)*d*(a^2-b*(b-sqrt(-a^2+b^2)))*sqrt(e*cos(c+d*x)))+3/2*a^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(c+d*x))/((a^2-b^2)*d*(a^2-b*(b+sqrt(-a^2+b^2)))*sqrt(e*cos(c+d*x)))+b*sqrt(e*cos(c+d*x))/((a^2-b^2)*d*e*(a+b*sin(c+d*x)))],
[1/((e*cos(c+d*x))^(3/2)*(a+b*sin(c+d*x))^2),x,14,-5/2*a*b^(3/2)*arctan(sqrt(b)*sqrt(e*cos(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/((-a^2+b^2)^(9/4)*d*e^(3/2))+5/2*a*b^(3/2)*arctanh(sqrt(b)*sqrt(e*cos(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/((-a^2+b^2)^(9/4)*d*e^(3/2))+b/((a^2-b^2)*d*e*(a+b*sin(c+d*x))*sqrt(e*cos(c+d*x)))+(-5*a*b+(2*a^2+3*b^2)*sin(c+d*x))/((a^2-b^2)^2*d*e*sqrt(e*cos(c+d*x)))-5/2*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/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(c+d*x))/((a^2-b^2)^2*d*e*(b-sqrt(-a^2+b^2))*sqrt(e*cos(c+d*x)))-5/2*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/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(c+d*x))/((a^2-b^2)^2*d*e*(b+sqrt(-a^2+b^2))*sqrt(e*cos(c+d*x)))-(2*a^2+3*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(e*cos(c+d*x))/((a^2-b^2)^2*d*e^2*sqrt(cos(c+d*x)))],
[1/((e*cos(c+d*x))^(5/2)*(a+b*sin(c+d*x))^2),x,14,7/2*a*b^(5/2)*arctan(sqrt(b)*sqrt(e*cos(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/((-a^2+b^2)^(11/4)*d*e^(5/2))+7/2*a*b^(5/2)*arctanh(sqrt(b)*sqrt(e*cos(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/((-a^2+b^2)^(11/4)*d*e^(5/2))+b/((a^2-b^2)*d*e*(e*cos(c+d*x))^(3/2)*(a+b*sin(c+d*x)))+1/3*(-7*a*b+(2*a^2+5*b^2)*sin(c+d*x))/((a^2-b^2)^2*d*e*(e*cos(c+d*x))^(3/2))+1/3*(2*a^2+5*b^2)*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))/((a^2-b^2)^2*d*e^2*sqrt(e*cos(c+d*x)))-7/2*a^2*b^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(c+d*x))/((a^2-b^2)^2*d*e^2*(a^2-b*(b-sqrt(-a^2+b^2)))*sqrt(e*cos(c+d*x)))-7/2*a^2*b^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(c+d*x))/((a^2-b^2)^2*d*e^2*(a^2-b*(b+sqrt(-a^2+b^2)))*sqrt(e*cos(c+d*x)))],
[1/((e*cos(c+d*x))^(7/2)*(a+b*sin(c+d*x))^2),x,15,-9/2*a*b^(7/2)*arctan(sqrt(b)*sqrt(e*cos(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/((-a^2+b^2)^(13/4)*d*e^(7/2))+9/2*a*b^(7/2)*arctanh(sqrt(b)*sqrt(e*cos(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/((-a^2+b^2)^(13/4)*d*e^(7/2))+b/((a^2-b^2)*d*e*(e*cos(c+d*x))^(5/2)*(a+b*sin(c+d*x)))+1/5*(-9*a*b+(2*a^2+7*b^2)*sin(c+d*x))/((a^2-b^2)^2*d*e*(e*cos(c+d*x))^(5/2))+3/5*(15*a*b^3+(2*a^4-10*a^2*b^2-7*b^4)*sin(c+d*x))/((a^2-b^2)^3*d*e^3*sqrt(e*cos(c+d*x)))+9/2*a^2*b^3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(c+d*x))/((a^2-b^2)^3*d*e^3*(b-sqrt(-a^2+b^2))*sqrt(e*cos(c+d*x)))+9/2*a^2*b^3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(c+d*x))/((a^2-b^2)^3*d*e^3*(b+sqrt(-a^2+b^2))*sqrt(e*cos(c+d*x)))-3/5*(2*a^4-10*a^2*b^2-7*b^4)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(e*cos(c+d*x))/((a^2-b^2)^3*d*e^4*sqrt(cos(c+d*x)))],
[(e*cos(c+d*x))^(13/2)/(a+b*sin(c+d*x))^3,x,15,-11/8*(9*a^4-11*a^2*b^2+2*b^4)*e^(13/2)*arctan(sqrt(b)*sqrt(e*cos(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/(b^(13/2)*(-a^2+b^2)^(1/4)*d)+11/8*(9*a^4-11*a^2*b^2+2*b^4)*e^(13/2)*arctanh(sqrt(b)*sqrt(e*cos(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/(b^(13/2)*(-a^2+b^2)^(1/4)*d)-1/2*e*(e*cos(c+d*x))^(11/2)/(b*d*(a+b*sin(c+d*x))^2)-11/28*e^3*(e*cos(c+d*x))^(7/2)*(9*a+2*b*sin(c+d*x))/(b^3*d*(a+b*sin(c+d*x)))+11/60*e^5*(e*cos(c+d*x))^(3/2)*(5*(9*a^2-2*b^2)-27*a*b*sin(c+d*x))/(b^5*d)-11/8*a*(9*a^4-11*a^2*b^2+2*b^4)*e^7*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(c+d*x))/(b^7*d*(b-sqrt(-a^2+b^2))*sqrt(e*cos(c+d*x)))-11/8*a*(9*a^4-11*a^2*b^2+2*b^4)*e^7*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(c+d*x))/(b^7*d*(b+sqrt(-a^2+b^2))*sqrt(e*cos(c+d*x)))+11/20*a*(45*a^2-37*b^2)*e^6*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(e*cos(c+d*x))/(b^6*d*sqrt(cos(c+d*x)))],
[(e*cos(c+d*x))^(11/2)/(a+b*sin(c+d*x))^3,x,15,9/8*(7*a^4-9*a^2*b^2+2*b^4)*e^(11/2)*arctan(sqrt(b)*sqrt(e*cos(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/(b^(11/2)*(-a^2+b^2)^(3/4)*d)+9/8*(7*a^4-9*a^2*b^2+2*b^4)*e^(11/2)*arctanh(sqrt(b)*sqrt(e*cos(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/(b^(11/2)*(-a^2+b^2)^(3/4)*d)-1/2*e*(e*cos(c+d*x))^(9/2)/(b*d*(a+b*sin(c+d*x))^2)-9/20*e^3*(e*cos(c+d*x))^(5/2)*(7*a+2*b*sin(c+d*x))/(b^3*d*(a+b*sin(c+d*x)))+3/4*a*(21*a^2-13*b^2)*e^6*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(b^6*d*sqrt(e*cos(c+d*x)))-9/8*a*(7*a^4-9*a^2*b^2+2*b^4)*e^6*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(c+d*x))/(b^6*d*(a^2-b*(b-sqrt(-a^2+b^2)))*sqrt(e*cos(c+d*x)))-9/8*a*(7*a^4-9*a^2*b^2+2*b^4)*e^6*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(c+d*x))/(b^6*d*(a^2-b*(b+sqrt(-a^2+b^2)))*sqrt(e*cos(c+d*x)))+3/4*e^5*(3*(7*a^2-2*b^2)-7*a*b*sin(c+d*x))*sqrt(e*cos(c+d*x))/(b^5*d)],
[(e*cos(c+d*x))^(9/2)/(a+b*sin(c+d*x))^3,x,14,7/8*(5*a^2-2*b^2)*e^(9/2)*arctan(sqrt(b)*sqrt(e*cos(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/(b^(9/2)*(-a^2+b^2)^(1/4)*d)-7/8*(5*a^2-2*b^2)*e^(9/2)*arctanh(sqrt(b)*sqrt(e*cos(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/(b^(9/2)*(-a^2+b^2)^(1/4)*d)-1/2*e*(e*cos(c+d*x))^(7/2)/(b*d*(a+b*sin(c+d*x))^2)-7/12*e^3*(e*cos(c+d*x))^(3/2)*(5*a+2*b*sin(c+d*x))/(b^3*d*(a+b*sin(c+d*x)))+7/8*a*(5*a^2-2*b^2)*e^5*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(c+d*x))/(b^5*d*(b-sqrt(-a^2+b^2))*sqrt(e*cos(c+d*x)))+7/8*a*(5*a^2-2*b^2)*e^5*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(c+d*x))/(b^5*d*(b+sqrt(-a^2+b^2))*sqrt(e*cos(c+d*x)))-35/4*a*e^4*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(e*cos(c+d*x))/(b^4*d*sqrt(cos(c+d*x)))],
[(e*cos(c+d*x))^(7/2)/(a+b*sin(c+d*x))^3,x,14,-5/8*(3*a^2-2*b^2)*e^(7/2)*arctan(sqrt(b)*sqrt(e*cos(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/(b^(7/2)*(-a^2+b^2)^(3/4)*d)-5/8*(3*a^2-2*b^2)*e^(7/2)*arctanh(sqrt(b)*sqrt(e*cos(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/(b^(7/2)*(-a^2+b^2)^(3/4)*d)-1/2*e*(e*cos(c+d*x))^(5/2)/(b*d*(a+b*sin(c+d*x))^2)-15/4*a*e^4*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(b^4*d*sqrt(e*cos(c+d*x)))+5/8*a*(3*a^2-2*b^2)*e^4*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(c+d*x))/(b^4*d*(a^2-b*(b-sqrt(-a^2+b^2)))*sqrt(e*cos(c+d*x)))+5/8*a*(3*a^2-2*b^2)*e^4*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(c+d*x))/(b^4*d*(a^2-b*(b+sqrt(-a^2+b^2)))*sqrt(e*cos(c+d*x)))-5/4*e^3*(3*a+2*b*sin(c+d*x))*sqrt(e*cos(c+d*x))/(b^3*d*(a+b*sin(c+d*x)))],
[(e*cos(c+d*x))^(5/2)/(a+b*sin(c+d*x))^3,x,14,3/8*(a^2-2*b^2)*e^(5/2)*arctan(sqrt(b)*sqrt(e*cos(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/(b^(5/2)*(-a^2+b^2)^(5/4)*d)-3/8*(a^2-2*b^2)*e^(5/2)*arctanh(sqrt(b)*sqrt(e*cos(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/(b^(5/2)*(-a^2+b^2)^(5/4)*d)-1/2*e*(e*cos(c+d*x))^(3/2)/(b*d*(a+b*sin(c+d*x))^2)+3/4*a*e*(e*cos(c+d*x))^(3/2)/(b*(a^2-b^2)*d*(a+b*sin(c+d*x)))-3/8*a*(a^2-2*b^2)*e^3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(c+d*x))/(b^3*(a^2-b^2)*d*(b-sqrt(-a^2+b^2))*sqrt(e*cos(c+d*x)))-3/8*a*(a^2-2*b^2)*e^3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(c+d*x))/(b^3*(a^2-b^2)*d*(b+sqrt(-a^2+b^2))*sqrt(e*cos(c+d*x)))+3/4*a*e^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(e*cos(c+d*x))/(b^2*(a^2-b^2)*d*sqrt(cos(c+d*x)))],
[(e*cos(c+d*x))^(3/2)/(a+b*sin(c+d*x))^3,x,14,1/8*(a^2+2*b^2)*e^(3/2)*arctan(sqrt(b)*sqrt(e*cos(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/(b^(3/2)*(-a^2+b^2)^(7/4)*d)+1/8*(a^2+2*b^2)*e^(3/2)*arctanh(sqrt(b)*sqrt(e*cos(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/(b^(3/2)*(-a^2+b^2)^(7/4)*d)-1/4*a*e^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(b^2*(a^2-b^2)*d*sqrt(e*cos(c+d*x)))+1/8*a*(a^2+2*b^2)*e^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(c+d*x))/(b^2*(a^2-b^2)*d*(a^2-b*(b-sqrt(-a^2+b^2)))*sqrt(e*cos(c+d*x)))+1/8*a*(a^2+2*b^2)*e^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(c+d*x))/(b^2*(a^2-b^2)*d*(a^2-b*(b+sqrt(-a^2+b^2)))*sqrt(e*cos(c+d*x)))-1/2*e*sqrt(e*cos(c+d*x))/(b*d*(a+b*sin(c+d*x))^2)+1/4*a*e*sqrt(e*cos(c+d*x))/(b*(a^2-b^2)*d*(a+b*sin(c+d*x)))],
[sqrt(e*cos(c+d*x))/(a+b*sin(c+d*x))^3,x,14,1/2*b*(e*cos(c+d*x))^(3/2)/((a^2-b^2)*d*e*(a+b*sin(c+d*x))^2)+5/4*a*b*(e*cos(c+d*x))^(3/2)/((a^2-b^2)^2*d*e*(a+b*sin(c+d*x)))+1/8*(3*a^2+2*b^2)*arctan(sqrt(b)*sqrt(e*cos(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))*sqrt(e)/((-a^2+b^2)^(9/4)*d*sqrt(b))-1/8*(3*a^2+2*b^2)*arctanh(sqrt(b)*sqrt(e*cos(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))*sqrt(e)/((-a^2+b^2)^(9/4)*d*sqrt(b))+1/8*a*(3*a^2+2*b^2)*e*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(c+d*x))/(b*(a^2-b^2)^2*d*(b-sqrt(-a^2+b^2))*sqrt(e*cos(c+d*x)))+1/8*a*(3*a^2+2*b^2)*e*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(c+d*x))/(b*(a^2-b^2)^2*d*(b+sqrt(-a^2+b^2))*sqrt(e*cos(c+d*x)))+5/4*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(e*cos(c+d*x))/((a^2-b^2)^2*d*sqrt(cos(c+d*x)))],
[1/((a+b*sin(c+d*x))^3*sqrt(e*cos(c+d*x))),x,14,-3/8*(5*a^2+2*b^2)*arctan(sqrt(b)*sqrt(e*cos(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))*sqrt(b)/((-a^2+b^2)^(11/4)*d*sqrt(e))-3/8*(5*a^2+2*b^2)*arctanh(sqrt(b)*sqrt(e*cos(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))*sqrt(b)/((-a^2+b^2)^(11/4)*d*sqrt(e))-7/4*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))/((a^2-b^2)^2*d*sqrt(e*cos(c+d*x)))+3/8*a*(5*a^2+2*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(c+d*x))/((a^2-b^2)^2*d*(a^2-b*(b-sqrt(-a^2+b^2)))*sqrt(e*cos(c+d*x)))+3/8*a*(5*a^2+2*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(c+d*x))/((a^2-b^2)^2*d*(a^2-b*(b+sqrt(-a^2+b^2)))*sqrt(e*cos(c+d*x)))+1/2*b*sqrt(e*cos(c+d*x))/((a^2-b^2)*d*e*(a+b*sin(c+d*x))^2)+7/4*a*b*sqrt(e*cos(c+d*x))/((a^2-b^2)^2*d*e*(a+b*sin(c+d*x)))],
[1/((e*cos(c+d*x))^(3/2)*(a+b*sin(c+d*x))^3),x,15,5/8*b^(3/2)*(7*a^2+2*b^2)*arctan(sqrt(b)*sqrt(e*cos(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/((-a^2+b^2)^(13/4)*d*e^(3/2))-5/8*b^(3/2)*(7*a^2+2*b^2)*arctanh(sqrt(b)*sqrt(e*cos(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/((-a^2+b^2)^(13/4)*d*e^(3/2))+1/2*b/((a^2-b^2)*d*e*(a+b*sin(c+d*x))^2*sqrt(e*cos(c+d*x)))+9/4*a*b/((a^2-b^2)^2*d*e*(a+b*sin(c+d*x))*sqrt(e*cos(c+d*x)))+1/4*(-5*b*(7*a^2+2*b^2)+a*(8*a^2+37*b^2)*sin(c+d*x))/((a^2-b^2)^3*d*e*sqrt(e*cos(c+d*x)))-5/8*a*b*(7*a^2+2*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(c+d*x))/((a^2-b^2)^3*d*e*(b-sqrt(-a^2+b^2))*sqrt(e*cos(c+d*x)))-5/8*a*b*(7*a^2+2*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(c+d*x))/((a^2-b^2)^3*d*e*(b+sqrt(-a^2+b^2))*sqrt(e*cos(c+d*x)))-1/4*a*(8*a^2+37*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(e*cos(c+d*x))/((a^2-b^2)^3*d*e^2*sqrt(cos(c+d*x)))],
[1/((e*cos(c+d*x))^(5/2)*(a+b*sin(c+d*x))^3),x,15,-7/8*b^(5/2)*(9*a^2+2*b^2)*arctan(sqrt(b)*sqrt(e*cos(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/((-a^2+b^2)^(15/4)*d*e^(5/2))-7/8*b^(5/2)*(9*a^2+2*b^2)*arctanh(sqrt(b)*sqrt(e*cos(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/((-a^2+b^2)^(15/4)*d*e^(5/2))+1/2*b/((a^2-b^2)*d*e*(e*cos(c+d*x))^(3/2)*(a+b*sin(c+d*x))^2)+11/4*a*b/((a^2-b^2)^2*d*e*(e*cos(c+d*x))^(3/2)*(a+b*sin(c+d*x)))+1/12*(-7*b*(9*a^2+2*b^2)+a*(8*a^2+69*b^2)*sin(c+d*x))/((a^2-b^2)^3*d*e*(e*cos(c+d*x))^(3/2))+1/12*a*(8*a^2+69*b^2)*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))/((a^2-b^2)^3*d*e^2*sqrt(e*cos(c+d*x)))-7/8*a*b^2*(9*a^2+2*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(c+d*x))/((a^2-b^2)^3*d*e^2*(a^2-b*(b-sqrt(-a^2+b^2)))*sqrt(e*cos(c+d*x)))-7/8*a*b^2*(9*a^2+2*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(c+d*x))/((a^2-b^2)^3*d*e^2*(a^2-b*(b+sqrt(-a^2+b^2)))*sqrt(e*cos(c+d*x)))],
[1/((e*cos(c+d*x))^(7/2)*(a+b*sin(c+d*x))^3),x,16,9/8*b^(7/2)*(11*a^2+2*b^2)*arctan(sqrt(b)*sqrt(e*cos(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/((-a^2+b^2)^(17/4)*d*e^(7/2))-9/8*b^(7/2)*(11*a^2+2*b^2)*arctanh(sqrt(b)*sqrt(e*cos(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/((-a^2+b^2)^(17/4)*d*e^(7/2))+1/2*b/((a^2-b^2)*d*e*(e*cos(c+d*x))^(5/2)*(a+b*sin(c+d*x))^2)+13/4*a*b/((a^2-b^2)^2*d*e*(e*cos(c+d*x))^(5/2)*(a+b*sin(c+d*x)))+1/20*(-9*b*(11*a^2+2*b^2)+a*(8*a^2+109*b^2)*sin(c+d*x))/((a^2-b^2)^3*d*e*(e*cos(c+d*x))^(5/2))+3/20*(15*b^3*(11*a^2+2*b^2)+a*(8*a^4-64*a^2*b^2-139*b^4)*sin(c+d*x))/((a^2-b^2)^4*d*e^3*sqrt(e*cos(c+d*x)))+9/8*a*b^3*(11*a^2+2*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(c+d*x))/((a^2-b^2)^4*d*e^3*(b-sqrt(-a^2+b^2))*sqrt(e*cos(c+d*x)))+9/8*a*b^3*(11*a^2+2*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(c+d*x))/((a^2-b^2)^4*d*e^3*(b+sqrt(-a^2+b^2))*sqrt(e*cos(c+d*x)))-3/20*a*(8*a^4-64*a^2*b^2-139*b^4)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(e*cos(c+d*x))/((a^2-b^2)^4*d*e^4*sqrt(cos(c+d*x)))],
[(e*cos(c+d*x))^(15/2)/(a+b*sin(c+d*x))^4,x,16,39/16*a*(11*a^4-17*a^2*b^2+6*b^4)*e^(15/2)*arctan(sqrt(b)*sqrt(e*cos(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/(b^(15/2)*(-a^2+b^2)^(3/4)*d)+39/16*a*(11*a^4-17*a^2*b^2+6*b^4)*e^(15/2)*arctanh(sqrt(b)*sqrt(e*cos(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/(b^(15/2)*(-a^2+b^2)^(3/4)*d)-1/3*e*(e*cos(c+d*x))^(13/2)/(b*d*(a+b*sin(c+d*x))^3)-13/84*e^3*(e*cos(c+d*x))^(9/2)*(11*a+4*b*sin(c+d*x))/(b^3*d*(a+b*sin(c+d*x))^2)-39/280*e^5*(e*cos(c+d*x))^(5/2)*(77*a^2-20*b^2+22*a*b*sin(c+d*x))/(b^5*d*(a+b*sin(c+d*x)))+13/56*(231*a^4-203*a^2*b^2+20*b^4)*e^8*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(b^8*d*sqrt(e*cos(c+d*x)))-39/16*a^2*(11*a^4-17*a^2*b^2+6*b^4)*e^8*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(c+d*x))/(b^8*d*(a^2-b*(b-sqrt(-a^2+b^2)))*sqrt(e*cos(c+d*x)))-39/16*a^2*(11*a^4-17*a^2*b^2+6*b^4)*e^8*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(c+d*x))/(b^8*d*(a^2-b*(b+sqrt(-a^2+b^2)))*sqrt(e*cos(c+d*x)))+13/56*e^7*(21*a*(11*a^2-6*b^2)-b*(77*a^2-20*b^2)*sin(c+d*x))*sqrt(e*cos(c+d*x))/(b^7*d)],
[(e*cos(c+d*x))^(13/2)/(a+b*sin(c+d*x))^4,x,15,77/16*a*(3*a^2-2*b^2)*e^(13/2)*arctan(sqrt(b)*sqrt(e*cos(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/(b^(13/2)*(-a^2+b^2)^(1/4)*d)-77/16*a*(3*a^2-2*b^2)*e^(13/2)*arctanh(sqrt(b)*sqrt(e*cos(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/(b^(13/2)*(-a^2+b^2)^(1/4)*d)-1/3*e*(e*cos(c+d*x))^(11/2)/(b*d*(a+b*sin(c+d*x))^3)-11/60*e^3*(e*cos(c+d*x))^(7/2)*(9*a+4*b*sin(c+d*x))/(b^3*d*(a+b*sin(c+d*x))^2)-77/120*e^5*(e*cos(c+d*x))^(3/2)*(15*a^2-4*b^2+6*a*b*sin(c+d*x))/(b^5*d*(a+b*sin(c+d*x)))+77/16*a^2*(3*a^2-2*b^2)*e^7*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(c+d*x))/(b^7*d*(b-sqrt(-a^2+b^2))*sqrt(e*cos(c+d*x)))+77/16*a^2*(3*a^2-2*b^2)*e^7*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(c+d*x))/(b^7*d*(b+sqrt(-a^2+b^2))*sqrt(e*cos(c+d*x)))-77/40*(15*a^2-4*b^2)*e^6*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(e*cos(c+d*x))/(b^6*d*sqrt(cos(c+d*x)))],
[(e*cos(c+d*x))^(11/2)/(a+b*sin(c+d*x))^4,x,15,-15/16*a*(7*a^2-6*b^2)*e^(11/2)*arctan(sqrt(b)*sqrt(e*cos(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/(b^(11/2)*(-a^2+b^2)^(3/4)*d)-15/16*a*(7*a^2-6*b^2)*e^(11/2)*arctanh(sqrt(b)*sqrt(e*cos(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/(b^(11/2)*(-a^2+b^2)^(3/4)*d)-1/3*e*(e*cos(c+d*x))^(9/2)/(b*d*(a+b*sin(c+d*x))^3)-1/4*e^3*(e*cos(c+d*x))^(5/2)*(7*a+4*b*sin(c+d*x))/(b^3*d*(a+b*sin(c+d*x))^2)-5/8*(21*a^2-4*b^2)*e^6*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(b^6*d*sqrt(e*cos(c+d*x)))+15/16*a^2*(7*a^2-6*b^2)*e^6*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(c+d*x))/(b^6*d*(a^2-b*(b-sqrt(-a^2+b^2)))*sqrt(e*cos(c+d*x)))+15/16*a^2*(7*a^2-6*b^2)*e^6*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(c+d*x))/(b^6*d*(a^2-b*(b+sqrt(-a^2+b^2)))*sqrt(e*cos(c+d*x)))-5/8*e^5*(21*a^2-4*b^2+14*a*b*sin(c+d*x))*sqrt(e*cos(c+d*x))/(b^5*d*(a+b*sin(c+d*x)))],
[(e*cos(c+d*x))^(9/2)/(a+b*sin(c+d*x))^4,x,15,7/16*a*(5*a^2-6*b^2)*e^(9/2)*arctan(sqrt(b)*sqrt(e*cos(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/(b^(9/2)*(-a^2+b^2)^(5/4)*d)-7/16*a*(5*a^2-6*b^2)*e^(9/2)*arctanh(sqrt(b)*sqrt(e*cos(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/(b^(9/2)*(-a^2+b^2)^(5/4)*d)-1/3*e*(e*cos(c+d*x))^(7/2)/(b*d*(a+b*sin(c+d*x))^3)+7/8*(5*a^2-4*b^2)*e^3*(e*cos(c+d*x))^(3/2)/(b^3*(a^2-b^2)*d*(a+b*sin(c+d*x)))-7/12*e^3*(e*cos(c+d*x))^(3/2)*(5*a+4*b*sin(c+d*x))/(b^3*d*(a+b*sin(c+d*x))^2)-7/16*a^2*(5*a^2-6*b^2)*e^5*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(c+d*x))/(b^5*(a^2-b^2)*d*(b-sqrt(-a^2+b^2))*sqrt(e*cos(c+d*x)))-7/16*a^2*(5*a^2-6*b^2)*e^5*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(c+d*x))/(b^5*(a^2-b^2)*d*(b+sqrt(-a^2+b^2))*sqrt(e*cos(c+d*x)))+7/8*(5*a^2-4*b^2)*e^4*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(e*cos(c+d*x))/(b^4*(a^2-b^2)*d*sqrt(cos(c+d*x)))],
[(e*cos(c+d*x))^(7/2)/(a+b*sin(c+d*x))^4,x,15,-5/16*a*(a^2-2*b^2)*e^(7/2)*arctan(sqrt(b)*sqrt(e*cos(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/(b^(7/2)*(-a^2+b^2)^(7/4)*d)-5/16*a*(a^2-2*b^2)*e^(7/2)*arctanh(sqrt(b)*sqrt(e*cos(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/(b^(7/2)*(-a^2+b^2)^(7/4)*d)-1/3*e*(e*cos(c+d*x))^(5/2)/(b*d*(a+b*sin(c+d*x))^3)+5/24*(3*a^2-4*b^2)*e^4*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(b^4*(a^2-b^2)*d*sqrt(e*cos(c+d*x)))-5/16*a^2*(a^2-2*b^2)*e^4*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(c+d*x))/(b^4*(a^2-b^2)*d*(a^2-b*(b-sqrt(-a^2+b^2)))*sqrt(e*cos(c+d*x)))-5/16*a^2*(a^2-2*b^2)*e^4*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(c+d*x))/(b^4*(a^2-b^2)*d*(a^2-b*(b+sqrt(-a^2+b^2)))*sqrt(e*cos(c+d*x)))-5/24*(3*a^2-4*b^2)*e^3*sqrt(e*cos(c+d*x))/(b^3*(a^2-b^2)*d*(a+b*sin(c+d*x)))+5/12*e^3*(3*a+4*b*sin(c+d*x))*sqrt(e*cos(c+d*x))/(b^3*d*(a+b*sin(c+d*x))^2)],
[(e*cos(c+d*x))^(5/2)/(a+b*sin(c+d*x))^4,x,15,-1/16*a*(a^2-6*b^2)*e^(5/2)*arctan(sqrt(b)*sqrt(e*cos(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/(b^(5/2)*(-a^2+b^2)^(9/4)*d)+1/16*a*(a^2-6*b^2)*e^(5/2)*arctanh(sqrt(b)*sqrt(e*cos(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/(b^(5/2)*(-a^2+b^2)^(9/4)*d)-1/3*e*(e*cos(c+d*x))^(3/2)/(b*d*(a+b*sin(c+d*x))^3)+1/4*a*e*(e*cos(c+d*x))^(3/2)/(b*(a^2-b^2)*d*(a+b*sin(c+d*x))^2)+1/8*(a^2+4*b^2)*e*(e*cos(c+d*x))^(3/2)/(b*(a^2-b^2)^2*d*(a+b*sin(c+d*x)))-1/16*a^2*(a^2-6*b^2)*e^3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(c+d*x))/(b^3*(a^2-b^2)^2*d*(b-sqrt(-a^2+b^2))*sqrt(e*cos(c+d*x)))-1/16*a^2*(a^2-6*b^2)*e^3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(c+d*x))/(b^3*(a^2-b^2)^2*d*(b+sqrt(-a^2+b^2))*sqrt(e*cos(c+d*x)))+1/8*(a^2+4*b^2)*e^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(e*cos(c+d*x))/(b^2*(a^2-b^2)^2*d*sqrt(cos(c+d*x)))],
[(e*cos(c+d*x))^(3/2)/(a+b*sin(c+d*x))^4,x,15,-1/16*a*(a^2+6*b^2)*e^(3/2)*arctan(sqrt(b)*sqrt(e*cos(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/(b^(3/2)*(-a^2+b^2)^(11/4)*d)-1/16*a*(a^2+6*b^2)*e^(3/2)*arctanh(sqrt(b)*sqrt(e*cos(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/(b^(3/2)*(-a^2+b^2)^(11/4)*d)-1/24*(3*a^2+4*b^2)*e^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(b^2*(a^2-b^2)^2*d*sqrt(e*cos(c+d*x)))+1/16*a^2*(a^2+6*b^2)*e^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(c+d*x))/(b^2*(a^2-b^2)^2*d*(a^2-b*(b-sqrt(-a^2+b^2)))*sqrt(e*cos(c+d*x)))+1/16*a^2*(a^2+6*b^2)*e^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(c+d*x))/(b^2*(a^2-b^2)^2*d*(a^2-b*(b+sqrt(-a^2+b^2)))*sqrt(e*cos(c+d*x)))-1/3*e*sqrt(e*cos(c+d*x))/(b*d*(a+b*sin(c+d*x))^3)+1/12*a*e*sqrt(e*cos(c+d*x))/(b*(a^2-b^2)*d*(a+b*sin(c+d*x))^2)+1/24*(3*a^2+4*b^2)*e*sqrt(e*cos(c+d*x))/(b*(a^2-b^2)^2*d*(a+b*sin(c+d*x)))],
[sqrt(e*cos(c+d*x))/(a+b*sin(c+d*x))^4,x,15,1/3*b*(e*cos(c+d*x))^(3/2)/((a^2-b^2)*d*e*(a+b*sin(c+d*x))^3)+3/4*a*b*(e*cos(c+d*x))^(3/2)/((a^2-b^2)^2*d*e*(a+b*sin(c+d*x))^2)+1/8*b*(11*a^2+4*b^2)*(e*cos(c+d*x))^(3/2)/((a^2-b^2)^3*d*e*(a+b*sin(c+d*x)))-5/16*a*(a^2+2*b^2)*arctan(sqrt(b)*sqrt(e*cos(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))*sqrt(e)/((-a^2+b^2)^(13/4)*d*sqrt(b))+5/16*a*(a^2+2*b^2)*arctanh(sqrt(b)*sqrt(e*cos(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))*sqrt(e)/((-a^2+b^2)^(13/4)*d*sqrt(b))+5/16*a^2*(a^2+2*b^2)*e*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(c+d*x))/(b*(a^2-b^2)^3*d*(b-sqrt(-a^2+b^2))*sqrt(e*cos(c+d*x)))+5/16*a^2*(a^2+2*b^2)*e*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(c+d*x))/(b*(a^2-b^2)^3*d*(b+sqrt(-a^2+b^2))*sqrt(e*cos(c+d*x)))+1/8*(11*a^2+4*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(e*cos(c+d*x))/((a^2-b^2)^3*d*sqrt(cos(c+d*x)))],
[1/((a+b*sin(c+d*x))^4*sqrt(e*cos(c+d*x))),x,15,7/16*a*(5*a^2+6*b^2)*arctan(sqrt(b)*sqrt(e*cos(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))*sqrt(b)/((-a^2+b^2)^(15/4)*d*sqrt(e))+7/16*a*(5*a^2+6*b^2)*arctanh(sqrt(b)*sqrt(e*cos(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))*sqrt(b)/((-a^2+b^2)^(15/4)*d*sqrt(e))-1/24*(57*a^2+20*b^2)*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))/((a^2-b^2)^3*d*sqrt(e*cos(c+d*x)))+7/16*a^2*(5*a^2+6*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(c+d*x))/((a^2-b^2)^3*d*(a^2-b*(b-sqrt(-a^2+b^2)))*sqrt(e*cos(c+d*x)))+7/16*a^2*(5*a^2+6*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(c+d*x))/((a^2-b^2)^3*d*(a^2-b*(b+sqrt(-a^2+b^2)))*sqrt(e*cos(c+d*x)))+1/3*b*sqrt(e*cos(c+d*x))/((a^2-b^2)*d*e*(a+b*sin(c+d*x))^3)+11/12*a*b*sqrt(e*cos(c+d*x))/((a^2-b^2)^2*d*e*(a+b*sin(c+d*x))^2)+1/24*b*(57*a^2+20*b^2)*sqrt(e*cos(c+d*x))/((a^2-b^2)^3*d*e*(a+b*sin(c+d*x)))],
[1/((e*cos(c+d*x))^(3/2)*(a+b*sin(c+d*x))^4),x,16,-15/16*a*b^(3/2)*(7*a^2+6*b^2)*arctan(sqrt(b)*sqrt(e*cos(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/((-a^2+b^2)^(17/4)*d*e^(3/2))+15/16*a*b^(3/2)*(7*a^2+6*b^2)*arctanh(sqrt(b)*sqrt(e*cos(c+d*x))/((-a^2+b^2)^(1/4)*sqrt(e)))/((-a^2+b^2)^(17/4)*d*e^(3/2))+1/3*b/((a^2-b^2)*d*e*(a+b*sin(c+d*x))^3*sqrt(e*cos(c+d*x)))+13/12*a*b/((a^2-b^2)^2*d*e*(a+b*sin(c+d*x))^2*sqrt(e*cos(c+d*x)))+1/24*b*(89*a^2+28*b^2)/((a^2-b^2)^3*d*e*(a+b*sin(c+d*x))*sqrt(e*cos(c+d*x)))+1/8*(-15*a*b*(7*a^2+6*b^2)+(16*a^4+151*a^2*b^2+28*b^4)*sin(c+d*x))/((a^2-b^2)^4*d*e*sqrt(e*cos(c+d*x)))-15/16*a^2*b*(7*a^2+6*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(b-sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(c+d*x))/((a^2-b^2)^4*d*e*(b-sqrt(-a^2+b^2))*sqrt(e*cos(c+d*x)))-15/16*a^2*b*(7*a^2+6*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(b+sqrt(-a^2+b^2)),sqrt(2))*sqrt(cos(c+d*x))/((a^2-b^2)^4*d*e*(b+sqrt(-a^2+b^2))*sqrt(e*cos(c+d*x)))-1/8*(16*a^4+151*a^2*b^2+28*b^4)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(e*cos(c+d*x))/((a^2-b^2)^4*d*e^2*sqrt(cos(c+d*x)))],

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

# n>0
#  {(e*Cos[c + d*x])^(3/2)*Sqrt[a + b*Sin[c + d*x]], x, 1, (2*e*AppellF1[3/2, -(1/4), -(1/4), 5/2, (a + b*Sin[c + d*x])/(a + b), (a + b*Sin[c + d*x])/(a - b)]*Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x])^(3/2))/(3*b*d*(1 - (a + b*Sin[c + d*x])/(a - b))^(1/4)*(1 - (a + b*Sin[c + d*x])/(a + b))^(1/4))}
# {Sqrt[e*Cos[c + d*x]]*Sqrt[a + b*Sin[c + d*x]], x, 1, (1/(3*b*d*Sqrt[e*Cos[c + d*x]]))*(2*e*AppellF1[3/2, 1/4, 1/4, 5/2, (a + b*Sin[c + d*x])/(a + b), (a + b*Sin[c + d*x])/(a - b)]*(a + b*Sin[c + d*x])^(3/2)*(1 - (a + b*Sin[c + d*x])/(a - b))^(1/4)*(1 - (a + b*Sin[c + d*x])/(a + b))^(1/4))}
# {Sqrt[a + b*Sin[c + d*x]]/Sqrt[e*Cos[c + d*x]], x, 1, (2*e*AppellF1[3/2, 3/4, 3/4, 5/2, (a + b*Sin[c + d*x])/(a + b), (a + b*Sin[c + d*x])/(a - b)]*(a + b*Sin[c + d*x])^(3/2)*(1 - (a + b*Sin[c + d*x])/(a - b))^(3/4)*(1 - (a + b*Sin[c + d*x])/(a + b))^(3/4))/(3*b*d*(e*Cos[c + d*x])^(3/2))}
# {Sqrt[a + b*Sin[c + d*x]]/(e*Cos[c + d*x])^(3/2), x, 1, 0}
# {Sqrt[a + b*Sin[c + d*x]]/(e*Cos[c + d*x])^(5/2), x, 2, 0}
# {Sqrt[a + b*Sin[c + d*x]]/(e*Cos[c + d*x])^(7/2), x, 3, 0}

# {Sqrt[a + b*Sin[c + d*x]]/(e*Cos[c + d*x])^(9/2), x, 4, 0} 
#  {(e*Cos[c + d*x])^(5/2)*(a + b*Sin[c + d*x])^(3/2), x, 1, (2*e*AppellF1[5/2, -(3/4), -(3/4), 7/2, (a + b*Sin[c + d*x])/(a + b), (a + b*Sin[c + d*x])/(a - b)]*(e*Cos[c + d*x])^(3/2)*(a + b*Sin[c + d*x])^(5/2))/(5*b*d*(1 - (a + b*Sin[c + d*x])/(a - b))^(3/4)*(1 - (a + b*Sin[c + d*x])/(a + b))^(3/4))}
# {(e*Cos[c + d*x])^(3/2)*(a + b*Sin[c + d*x])^(3/2), x, 1, (2*e*AppellF1[5/2, -(1/4), -(1/4), 7/2, (a + b*Sin[c + d*x])/(a + b), (a + b*Sin[c + d*x])/(a - b)]*Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x])^(5/2))/(5*b*d*(1 - (a + b*Sin[c + d*x])/(a - b))^(1/4)*(1 - (a + b*Sin[c + d*x])/(a + b))^(1/4))}
# {Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x])^(3/2), x, 1, (1/(5*b*d*Sqrt[e*Cos[c + d*x]]))*(2*e*AppellF1[5/2, 1/4, 1/4, 7/2, (a + b*Sin[c + d*x])/(a + b), (a + b*Sin[c + d*x])/(a - b)]*(a + b*Sin[c + d*x])^(5/2)*(1 - (a + b*Sin[c + d*x])/(a - b))^(1/4)*(1 - (a + b*Sin[c + d*x])/(a + b))^(1/4))}
# {(a + b*Sin[c + d*x])^(3/2)/Sqrt[e*Cos[c + d*x]], x, 1, (1/(5*b*d*(e*Cos[c + d*x])^(3/2)))*(2*e*AppellF1[5/2, 3/4, 3/4, 7/2, (a + b*Sin[c + d*x])/(a + b), (a + b*Sin[c + d*x])/(a - b)]*(a + b*Sin[c + d*x])^(5/2)*(1 - (a + b*Sin[c + d*x])/(a - b))^(3/4)*(1 - (a + b*Sin[c + d*x])/(a + b))^(3/4))}
# {(a + b*Sin[c + d*x])^(3/2)/(e*Cos[c + d*x])^(3/2), x, 8, 0}
# {(a + b*Sin[c + d*x])^(3/2)/(e*Cos[c + d*x])^(5/2), x, 1, 0}
# {(a + b*Sin[c + d*x])^(3/2)/(e*Cos[c + d*x])^(7/2), x, 3, 0}
# {(a + b*Sin[c + d*x])^(3/2)/(e*Cos[c + d*x])^(9/2), x, 4, 0}

# {(a + b*Sin[c + d*x])^(3/2)/(e*Cos[c + d*x])^(11/2), x, 5, 0} 
#  {(e*Cos[c + d*x])^(3/2)*(a + b*Sin[c + d*x])^(5/2), x, 1, (2*e*AppellF1[7/2, -(1/4), -(1/4), 9/2, (a + b*Sin[c + d*x])/(a + b), (a + b*Sin[c + d*x])/(a - b)]*Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x])^(7/2))/(7*b*d*(1 - (a + b*Sin[c + d*x])/(a - b))^(1/4)*(1 - (a + b*Sin[c + d*x])/(a + b))^(1/4))}
# {Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x])^(5/2), x, 1, (2*e*AppellF1[7/2, 1/4, 1/4, 9/2, (a + b*Sin[c + d*x])/(a + b), (a + b*Sin[c + d*x])/(a - b)]*(a + b*Sin[c + d*x])^(7/2)*(1 - (a + b*Sin[c + d*x])/(a - b))^(1/4)*(1 - (a + b*Sin[c + d*x])/(a + b))^(1/4))/(7*b*d*Sqrt[e*Cos[c + d*x]])}
# {(a + b*Sin[c + d*x])^(5/2)/Sqrt[e*Cos[c + d*x]], x, 1, (2*e*AppellF1[7/2, 3/4, 3/4, 9/2, (a + b*Sin[c + d*x])/(a + b), (a + b*Sin[c + d*x])/(a - b)]*(a + b*Sin[c + d*x])^(7/2)*(1 - (a + b*Sin[c + d*x])/(a - b))^(3/4)*(1 - (a + b*Sin[c + d*x])/(a + b))^(3/4))/(7*b*d*(e*Cos[c + d*x])^(3/2))}
# {(a + b*Sin[c + d*x])^(5/2)/(e*Cos[c + d*x])^(3/2), x, 9, 0}
# {(a + b*Sin[c + d*x])^(5/2)/(e*Cos[c + d*x])^(5/2), x, 8, 0}
# {(a + b*Sin[c + d*x])^(5/2)/(e*Cos[c + d*x])^(7/2), x, 1, 0}
# {(a + b*Sin[c + d*x])^(5/2)/(e*Cos[c + d*x])^(9/2), x, 3, 0}
# {(a + b*Sin[c + d*x])^(5/2)/(e*Cos[c + d*x])^(11/2), x, 4, 0}

# {(a + b*Sin[c + d*x])^(5/2)/(e*Cos[c + d*x])^(13/2), x, 5, 0} 

# n<0
[1/(sqrt(c*cos(e+f*x))*sqrt(a+b*sin(e+f*x))),x,2,2*(-a+b)^(1/4)*EllipticF((a+b)^(1/4)*sqrt((1+cos(e+f*x)+sin(e+f*x))/(1+cos(e+f*x)-sin(e+f*x)))/(-a+b)^(1/4),I)*sqrt(2)*sqrt(c*cos(e+f*x))*sqrt((a+b*sin(e+f*x))/((a-b)*(1-sin(e+f*x))))/((a+b)^(1/4)*c*f*sqrt((1+cos(e+f*x)+sin(e+f*x))/(1+cos(e+f*x)-sin(e+f*x)))*sqrt(a+b*sin(e+f*x))),(a-b)^(1/4)*sqrt(cos(2*arctan((a+b)^(1/4)*sqrt((1+cos(e+f*x)+sin(e+f*x))/(1+cos(e+f*x)-sin(e+f*x)))/(a-b)^(1/4)))^2)/cos(2*arctan((a+b)^(1/4)*sqrt((1+cos(e+f*x)+sin(e+f*x))/(1+cos(e+f*x)-sin(e+f*x)))/(a-b)^(1/4)))*EllipticF(sin(2*arctan((a+b)^(1/4)*sqrt((1+cos(e+f*x)+sin(e+f*x))/(1+cos(e+f*x)-sin(e+f*x)))/(a-b)^(1/4))),sqrt(1/2))*sqrt(2)*(1+(1+cos(e+f*x)+sin(e+f*x))*sqrt(a+b)/((1+cos(e+f*x)-sin(e+f*x))*sqrt(a-b)))*sqrt(c*cos(e+f*x))*sqrt((a+b*sin(e+f*x))/((a-b)*(1-sin(e+f*x))))*sqrt((a+b*sin(e+f*x))/((a-b)*(1-cos(f*x)*sin(e)-cos(e)*sin(f*x))*(1+(1+cos(e+f*x)+sin(e+f*x))*sqrt(a+b)/((1+cos(e+f*x)-sin(e+f*x))*sqrt(a-b)))^2))/((a+b)^(1/4)*c*f*sqrt((1+cos(e+f*x)+sin(e+f*x))/(1+cos(e+f*x)-sin(e+f*x)))*sqrt(a+b*sin(e+f*x))*sqrt((a+b*sin(e+f*x))/((a-b)*(1-cos(f*x)*sin(e)-cos(e)*sin(f*x)))))],
#  {(e*Cos[c + d*x])^(5/2)/Sqrt[a + b*Sin[c + d*x]], x, 1, (2*e*AppellF1[1/2, -(3/4), -(3/4), 3/2, (a + b*Sin[c + d*x])/(a + b), (a + b*Sin[c + d*x])/(a - b)]*(e*Cos[c + d*x])^(3/2)*Sqrt[a + b*Sin[c + d*x]])/(b*d*(1 - (a + b*Sin[c + d*x])/(a - b))^(3/4)*(1 - (a + b*Sin[c + d*x])/(a + b))^(3/4))}
# {(e*Cos[c + d*x])^(3/2)/Sqrt[a + b*Sin[c + d*x]], x, 1, (2*e*AppellF1[1/2, -(1/4), -(1/4), 3/2, (a + b*Sin[c + d*x])/(a + b), (a + b*Sin[c + d*x])/(a - b)]*Sqrt[e*Cos[c + d*x]]*Sqrt[a + b*Sin[c + d*x]])/(b*d*(1 - (a + b*Sin[c + d*x])/(a - b))^(1/4)*(1 - (a + b*Sin[c + d*x])/(a + b))^(1/4))}
# {Sqrt[e*Cos[c + d*x]]/Sqrt[a + b*Sin[c + d*x]], x, 1, (2*e*AppellF1[1/2, 1/4, 1/4, 3/2, (a + b*Sin[c + d*x])/(a + b), (a + b*Sin[c + d*x])/(a - b)]*Sqrt[a + b*Sin[c + d*x]]*(1 - (a + b*Sin[c + d*x])/(a - b))^(1/4)*(1 - (a + b*Sin[c + d*x])/(a + b))^(1/4))/(b*d*Sqrt[e*Cos[c + d*x]])}
# {1/(Sqrt[e*Cos[c + d*x]]*Sqrt[a + b*Sin[c + d*x]]), x, 1, 0}
# {1/((e*Cos[c + d*x])^(3/2)*Sqrt[a + b*Sin[c + d*x]]), x, 2, 0}
# {1/((e*Cos[c + d*x])^(5/2)*Sqrt[a + b*Sin[c + d*x]]), x, 3, 0}

# {1/((e*Cos[c + d*x])^(7/2)*Sqrt[a + b*Sin[c + d*x]]), x, 4, 0} 
#  {(e*Cos[c + d*x])^(7/2)/(a + b*Sin[c + d*x])^(3/2), x, 1, -((2*e*AppellF1[-(1/2), -(5/4), -(5/4), 1/2, (a + b*Sin[c + d*x])/(a + b), (a + b*Sin[c + d*x])/(a - b)]*(e*Cos[c + d*x])^(5/2))/(b*d*Sqrt[a + b*Sin[c + d*x]]*(1 - (a + b*Sin[c + d*x])/(a - b))^(5/4)*(1 - (a + b*Sin[c + d*x])/(a + b))^(5/4)))}
# {(e*Cos[c + d*x])^(5/2)/(a + b*Sin[c + d*x])^(3/2), x, 1, -((2*e*AppellF1[-(1/2), -(3/4), -(3/4), 1/2, (a + b*Sin[c + d*x])/(a + b), (a + b*Sin[c + d*x])/(a - b)]*(e*Cos[c + d*x])^(3/2))/(b*d*Sqrt[a + b*Sin[c + d*x]]*(1 - (a + b*Sin[c + d*x])/(a - b))^(3/4)*(1 - (a + b*Sin[c + d*x])/(a + b))^(3/4)))}
# {(e*Cos[c + d*x])^(3/2)/(a + b*Sin[c + d*x])^(3/2), x, 1, -((2*e*AppellF1[-(1/2), -(1/4), -(1/4), 1/2, (a + b*Sin[c + d*x])/(a + b), (a + b*Sin[c + d*x])/(a - b)]*Sqrt[e*Cos[c + d*x]])/(b*d*Sqrt[a + b*Sin[c + d*x]]*(1 - (a + b*Sin[c + d*x])/(a - b))^(1/4)*(1 - (a + b*Sin[c + d*x])/(a + b))^(1/4)))}
# {Sqrt[e*Cos[c + d*x]]/(a + b*Sin[c + d*x])^(3/2), x, 1, -((2*2^(3/4)*e*Hypergeometric2F1[1/4, 3/4, 7/4, ((a - b)*(1 - Sin[c + d*x]))/(2*(a + b*Sin[c + d*x]))]*(1 - Sin[c + d*x])*(((a + b)*(1 + Sin[c + d*x]))/(a + b*Sin[c + d*x]))^(1/4))/(3*(a + b)*d*Sqrt[e*Cos[c + d*x]]*Sqrt[a + b*Sin[c + d*x]]))}
# {1/(Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x])^(3/2)), x, 2, 0}
# {1/((e*Cos[c + d*x])^(3/2)*(a + b*Sin[c + d*x])^(3/2)), x, 3, 0}
# {1/((e*Cos[c + d*x])^(5/2)*(a + b*Sin[c + d*x])^(3/2)), x, 4, 0}

# {1/((e*Cos[c + d*x])^(7/2)*(a + b*Sin[c + d*x])^(3/2)), x, 5, 0} 
#  {(e*Cos[c + d*x])^(9/2)/(a + b*Sin[c + d*x])^(5/2), x, 1, -((2*e*AppellF1[-(3/2), -(7/4), -(7/4), -(1/2), (a + b*Sin[c + d*x])/(a + b), (a + b*Sin[c + d*x])/(a - b)]*(e*Cos[c + d*x])^(7/2))/(3*b*d*(a + b*Sin[c + d*x])^(3/2)*(1 - (a + b*Sin[c + d*x])/(a - b))^(7/4)*(1 - (a + b*Sin[c + d*x])/(a + b))^(7/4)))}
# {(e*Cos[c + d*x])^(7/2)/(a + b*Sin[c + d*x])^(5/2), x, 1, -((2*e*AppellF1[-(3/2), -(5/4), -(5/4), -(1/2), (a + b*Sin[c + d*x])/(a + b), (a + b*Sin[c + d*x])/(a - b)]*(e*Cos[c + d*x])^(5/2))/(3*b*d*(a + b*Sin[c + d*x])^(3/2)*(1 - (a + b*Sin[c + d*x])/(a - b))^(5/4)*(1 - (a + b*Sin[c + d*x])/(a + b))^(5/4)))}
# {(e*Cos[c + d*x])^(5/2)/(a + b*Sin[c + d*x])^(5/2), x, 1, -((2*e*AppellF1[-(3/2), -(3/4), -(3/4), -(1/2), (a + b*Sin[c + d*x])/(a + b), (a + b*Sin[c + d*x])/(a - b)]*(e*Cos[c + d*x])^(3/2))/(3*b*d*(a + b*Sin[c + d*x])^(3/2)*(1 - (a + b*Sin[c + d*x])/(a - b))^(3/4)*(1 - (a + b*Sin[c + d*x])/(a + b))^(3/4)))}
# {(e*Cos[c + d*x])^(3/2)/(a + b*Sin[c + d*x])^(5/2), x, 1, 0}
# {Sqrt[e*Cos[c + d*x]]/(a + b*Sin[c + d*x])^(5/2), x, 2, 0}
# {1/(Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x])^(5/2)), x, 3, 0}
# {1/((e*Cos[c + d*x])^(3/2)*(a + b*Sin[c + d*x])^(5/2)), x, 4, 0}

# {1/((e*Cos[c + d*x])^(5/2)*(a + b*Sin[c + d*x])^(5/2)), x, 5, 0} 

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

#  {(e*Cos[c + d*x])^p*(a + b*Sin[c + d*x])^8, x, 10, -((a*b*(9 + m)*(3*b^6*(5053 + 4443*m + 1163*m^2 + 93*m^3) + a^2*b^4*(101978 + 91503*m + 28373*m^2 + 3753*m^3 + 185*m^4) + a^4*b^2*(95912 + 88506*m + 33953*m^2 + 6723*m^3 + 671*m^4 + 27*m^5) + a^6*(12176 + 11772*m + 6168*m^2 + 1809*m^3 + 303*m^4 + 27*m^5 + m^6))*(e*Cos[c + d*x])^(1 + m))/(d*e*(3 + m)*(4 + m)*(5 + m)*(6 + m)*(7 + m)*(8 + m)*(2 + 3*m + m^2))) + (1/(d*e*(6 + m)*(8 + m)*(8 + 6*m + m^2)))*((105*b^8 + 420*a^2*b^6*(8 + m) + 210*a^4*b^4*(48 + 14*m + m^2) + 28*a^6*b^2*(192 + 104*m + 18*m^2 + m^3) + a^8*(384 + 400*m + 140*m^2 + 20*m^3 + m^4))*(e*Cos[c + d*x])^(1 + m)*(Cos[c + d*x]^2)^((1/2)*(-1 - m))*Hypergeometric2F1[1/2, (1 - m)/2, 3/2, Sin[c + d*x]^2]*Sin[c + d*x]) - (1/(d*e*(4 + m)*(5 + m)*(6 + m)*(7 + m)*(8 + m)*(6 + 5*m + m^2)))*(b*(105*b^6*(105 + 71*m + 15*m^2 + m^3) + 3*a^2*b^4*(75798 + 51665*m + 12355*m^2 + 1255*m^3 + 47*m^4) + a^4*b^2*(367992 + 253870*m + 72195*m^2 + 10705*m^3 + 813*m^4 + 25*m^5) + a^6*(69264 + 48860*m + 18424*m^2 + 4025*m^3 + 511*m^4 + 35*m^5 + m^6))*(e*Cos[c + d*x])^(1 + m)*(a + b*Sin[c + d*x])) - (a*b*(11 + m)*(3*b^4*(783 + 308*m + 29*m^2) + 2*a^2*b^2*(3852 + 1529*m + 208*m^2 + 11*m^3) + a^4*(2232 + 902*m + 203*m^2 + 22*m^3 + m^4))*(e*Cos[c + d*x])^(1 + m)*(a + b*Sin[c + d*x])^2)/(d*e*(3 + m)*(4 + m)*(5 + m)*(6 + m)*(7 + m)*(8 + m)) - (b*(35*b^4*(35 + 12*m + m^2) + 2*a^2*b^2*(6068 + 2091*m + 232*m^2 + 9*m^3) + a^4*(5944 + 2070*m + 355*m^2 + 30*m^3 + m^4))*(e*Cos[c + d*x])^(1 + m)*(a + b*Sin[c + d*x])^3)/(d*e*(4 + m)*(5 + m)*(6 + m)*(7 + m)*(8 + m)) - (a*b*(13 + m)*(b^2*(83 + 13*m) + a^2*(82 + 13*m + m^2))*(e*Cos[c + d*x])^(1 + m)*(a + b*Sin[c + d*x])^4)/(d*e*(5 + m)*(6 + m)*(7 + m)*(8 + m)) - (b*(7*b^2*(7 + m) + a^2*(146 + 21*m + m^2))*(e*Cos[c + d*x])^(1 + m)*(a + b*Sin[c + d*x])^5)/(d*e*(6 + m)*(7 + m)*(8 + m)) - (a*b*(15 + m)*(e*Cos[c + d*x])^(1 + m)*(a + b*Sin[c + d*x])^6)/(d*e*(7 + m)*(8 + m)) - (b*(e*Cos[c + d*x])^(1 + m)*(a + b*Sin[c + d*x])^7)/(d*e*(8 + m))} 
[(e*cos(c+d*x))^p*(a+b*sin(c+d*x))^3,x,4,-b*(2*b^2*(2+p)+a^2*(11+6*p+p^2))*(e*cos(c+d*x))^(1+p)/(d*e*(3+p)*(2+3*p+p^2))-a*b*(5+p)*(e*cos(c+d*x))^(1+p)*(a+b*sin(c+d*x))/(d*e*(2+p)*(3+p))-b*(e*cos(c+d*x))^(1+p)*(a+b*sin(c+d*x))^2/(d*e*(3+p))-a*(3*b^2+a^2*(2+p))*(e*cos(c+d*x))^(1+p)*hypergeom([1/2,1/2*(1+p)],[1/2*(3+p)],cos(c+d*x)^2)*sin(c+d*x)/(d*e*(1+p)*(2+p)*sqrt(sin(c+d*x)^2))],
[(e*cos(c+d*x))^p*(a+b*sin(c+d*x))^2,x,3,-a*b*(3+p)*(e*cos(c+d*x))^(1+p)/(d*e*(1+p)*(2+p))-b*(e*cos(c+d*x))^(1+p)*(a+b*sin(c+d*x))/(d*e*(2+p))-(b^2+a^2*(2+p))*(e*cos(c+d*x))^(1+p)*hypergeom([1/2,1/2*(1+p)],[1/2*(3+p)],cos(c+d*x)^2)*sin(c+d*x)/(d*e*(1+p)*(2+p)*sqrt(sin(c+d*x)^2))],
[(e*cos(c+d*x))^p*(a+b*sin(c+d*x)),x,2,-b*(e*cos(c+d*x))^(1+p)/(d*e*(1+p))-a*(e*cos(c+d*x))^(1+p)*hypergeom([1/2,1/2*(1+p)],[1/2*(3+p)],cos(c+d*x)^2)*sin(c+d*x)/(d*e*(1+p)*sqrt(sin(c+d*x)^2))],
[(e*cos(c+d*x))^p/(a+b*sin(c+d*x)),x,1,-e*AppellF1(1-p,1/2*(1-p),1/2*(1-p),2-p,(a+b)/(a+b*sin(c+d*x)),(a-b)/(a+b*sin(c+d*x)))*(e*cos(c+d*x))^(-1+p)*(-b*(1-sin(c+d*x))/(a+b*sin(c+d*x)))^(1/2*(1-p))*(b*(1+sin(c+d*x))/(a+b*sin(c+d*x)))^(1/2*(1-p))/(b*d*(1-p))],
[(e*cos(c+d*x))^p/(a+b*sin(c+d*x))^2,x,1,-e*AppellF1(2-p,1/2*(1-p),1/2*(1-p),3-p,(a+b)/(a+b*sin(c+d*x)),(a-b)/(a+b*sin(c+d*x)))*(e*cos(c+d*x))^(-1+p)*(-b*(1-sin(c+d*x))/(a+b*sin(c+d*x)))^(1/2*(1-p))*(b*(1+sin(c+d*x))/(a+b*sin(c+d*x)))^(1/2*(1-p))/(b*d*(2-p)*(a+b*sin(c+d*x)))],
[(e*cos(c+d*x))^p/(a+b*sin(c+d*x))^3,x,1,-e*AppellF1(3-p,1/2*(1-p),1/2*(1-p),4-p,(a+b)/(a+b*sin(c+d*x)),(a-b)/(a+b*sin(c+d*x)))*(e*cos(c+d*x))^(-1+p)*(-b*(1-sin(c+d*x))/(a+b*sin(c+d*x)))^(1/2*(1-p))*(b*(1+sin(c+d*x))/(a+b*sin(c+d*x)))^(1/2*(1-p))/(b*d*(3-p)*(a+b*sin(c+d*x))^2)],
[(e*cos(c+d*x))^p/(a+b*sin(c+d*x))^8,x,1,-e*AppellF1(8-p,1/2*(1-p),1/2*(1-p),9-p,(a+b)/(a+b*sin(c+d*x)),(a-b)/(a+b*sin(c+d*x)))*(e*cos(c+d*x))^(-1+p)*(-b*(1-sin(c+d*x))/(a+b*sin(c+d*x)))^(1/2*(1-p))*(b*(1+sin(c+d*x))/(a+b*sin(c+d*x)))^(1/2*(1-p))/(b*d*(8-p)*(a+b*sin(c+d*x))^7)],
[(e*cos(c+d*x))^p*(a+b*sin(c+d*x))^(5/2),x,2,2/7*e*AppellF1(7/2,1/2*(1-p),1/2*(1-p),9/2,(a+b*sin(c+d*x))/(a-b),(a+b*sin(c+d*x))/(a+b))*(e*cos(c+d*x))^(-1+p)*(a+b*sin(c+d*x))^(7/2)*(1+(-a-b*sin(c+d*x))/(a-b))^(1/2*(1-p))*(1+(-a-b*sin(c+d*x))/(a+b))^(1/2*(1-p))/(b*d)],
[(e*cos(c+d*x))^p*(a+b*sin(c+d*x))^(3/2),x,2,2/5*e*AppellF1(5/2,1/2*(1-p),1/2*(1-p),7/2,(a+b*sin(c+d*x))/(a-b),(a+b*sin(c+d*x))/(a+b))*(e*cos(c+d*x))^(-1+p)*(a+b*sin(c+d*x))^(5/2)*(1+(-a-b*sin(c+d*x))/(a-b))^(1/2*(1-p))*(1+(-a-b*sin(c+d*x))/(a+b))^(1/2*(1-p))/(b*d)],
[(e*cos(c+d*x))^p*sqrt(a+b*sin(c+d*x)),x,2,2/3*e*AppellF1(3/2,1/2*(1-p),1/2*(1-p),5/2,(a+b*sin(c+d*x))/(a-b),(a+b*sin(c+d*x))/(a+b))*(e*cos(c+d*x))^(-1+p)*(a+b*sin(c+d*x))^(3/2)*(1+(-a-b*sin(c+d*x))/(a-b))^(1/2*(1-p))*(1+(-a-b*sin(c+d*x))/(a+b))^(1/2*(1-p))/(b*d)],
[(e*cos(c+d*x))^p/sqrt(a+b*sin(c+d*x)),x,2,2*e*AppellF1(1/2,1/2*(1-p),1/2*(1-p),3/2,(a+b*sin(c+d*x))/(a-b),(a+b*sin(c+d*x))/(a+b))*(e*cos(c+d*x))^(-1+p)*(1+(-a-b*sin(c+d*x))/(a-b))^(1/2*(1-p))*(1+(-a-b*sin(c+d*x))/(a+b))^(1/2*(1-p))*sqrt(a+b*sin(c+d*x))/(b*d)],
[(e*cos(c+d*x))^p/(a+b*sin(c+d*x))^(3/2),x,2,-2*e*AppellF1(-1/2,1/2*(1-p),1/2*(1-p),1/2,(a+b*sin(c+d*x))/(a-b),(a+b*sin(c+d*x))/(a+b))*(e*cos(c+d*x))^(-1+p)*(1+(-a-b*sin(c+d*x))/(a-b))^(1/2*(1-p))*(1+(-a-b*sin(c+d*x))/(a+b))^(1/2*(1-p))/(b*d*sqrt(a+b*sin(c+d*x)))],
[(e*cos(c+d*x))^p/(a+b*sin(c+d*x))^(5/2),x,2,-2/3*e*AppellF1(-3/2,1/2*(1-p),1/2*(1-p),-1/2,(a+b*sin(c+d*x))/(a-b),(a+b*sin(c+d*x))/(a+b))*(e*cos(c+d*x))^(-1+p)*(1+(-a-b*sin(c+d*x))/(a-b))^(1/2*(1-p))*(1+(-a-b*sin(c+d*x))/(a+b))^(1/2*(1-p))/(b*d*(a+b*sin(c+d*x))^(3/2))],

# Integrands of the form (d Cos[e+f x])^p (a+b Sin[e+f x])^m when m symbolic
[(e*cos(c+d*x))^p*(a+b*sin(c+d*x))^m,x,2,e*AppellF1(1+m,1/2*(1-p),1/2*(1-p),2+m,(a+b*sin(c+d*x))/(a-b),(a+b*sin(c+d*x))/(a+b))*(e*cos(c+d*x))^(-1+p)*(a+b*sin(c+d*x))^(1+m)*(1+(-a-b*sin(c+d*x))/(a-b))^(1/2*(1-p))*(1+(-a-b*sin(c+d*x))/(a+b))^(1/2*(1-p))/(b*d*(1+m))],
[cos(c+d*x)^7*(a+b*sin(c+d*x))^m,x,3,-(a^2-b^2)^3*(a+b*sin(c+d*x))^(1+m)/(b^7*d*(1+m))+6*a*(a^2-b^2)^2*(a+b*sin(c+d*x))^(2+m)/(b^7*d*(2+m))-3*(5*a^4-6*a^2*b^2+b^4)*(a+b*sin(c+d*x))^(3+m)/(b^7*d*(3+m))+4*a*(5*a^2-3*b^2)*(a+b*sin(c+d*x))^(4+m)/(b^7*d*(4+m))-3*(5*a^2-b^2)*(a+b*sin(c+d*x))^(5+m)/(b^7*d*(5+m))+6*a*(a+b*sin(c+d*x))^(6+m)/(b^7*d*(6+m))-(a+b*sin(c+d*x))^(7+m)/(b^7*d*(7+m))],
[cos(c+d*x)^5*(a+b*sin(c+d*x))^m,x,3,(a^2-b^2)^2*(a+b*sin(c+d*x))^(1+m)/(b^5*d*(1+m))-4*a*(a^2-b^2)*(a+b*sin(c+d*x))^(2+m)/(b^5*d*(2+m))+2*(3*a^2-b^2)*(a+b*sin(c+d*x))^(3+m)/(b^5*d*(3+m))-4*a*(a+b*sin(c+d*x))^(4+m)/(b^5*d*(4+m))+(a+b*sin(c+d*x))^(5+m)/(b^5*d*(5+m))],
[cos(c+d*x)^3*(a+b*sin(c+d*x))^m,x,3,-(a^2-b^2)*(a+b*sin(c+d*x))^(1+m)/(b^3*d*(1+m))+2*a*(a+b*sin(c+d*x))^(2+m)/(b^3*d*(2+m))-(a+b*sin(c+d*x))^(3+m)/(b^3*d*(3+m))],
[cos(c+d*x)*(a+b*sin(c+d*x))^m,x,2,(a+b*sin(c+d*x))^(1+m)/(b*d*(1+m))],
[sec(c+d*x)*(a+b*sin(c+d*x))^m,x,5,-1/2*hypergeom([1,1+m],[2+m],(a+b*sin(c+d*x))/(a-b))*(a+b*sin(c+d*x))^(1+m)/((a-b)*d*(1+m))+1/2*hypergeom([1,1+m],[2+m],(a+b*sin(c+d*x))/(a+b))*(a+b*sin(c+d*x))^(1+m)/((a+b)*d*(1+m))],
[sec(c+d*x)^3*(a+b*sin(c+d*x))^m,x,6,-1/4*(a-b*(1-m))*hypergeom([1,1+m],[2+m],(a+b*sin(c+d*x))/(a-b))*(a+b*sin(c+d*x))^(1+m)/((a-b)^2*d*(1+m))+1/4*(a+b-b*m)*hypergeom([1,1+m],[2+m],(a+b*sin(c+d*x))/(a+b))*(a+b*sin(c+d*x))^(1+m)/((a+b)^2*d*(1+m))-1/2*sec(c+d*x)^2*(b-a*sin(c+d*x))*(a+b*sin(c+d*x))^(1+m)/((a^2-b^2)*d)],
[sec(c+d*x)^5*(a+b*sin(c+d*x))^m,x,7,-1/16*(3*a^2-3*a*b*(2-m)+b^2*(3-4*m+m^2))*hypergeom([1,1+m],[2+m],(a+b*sin(c+d*x))/(a-b))*(a+b*sin(c+d*x))^(1+m)/((a-b)^3*d*(1+m))+1/16*(3*a^2+3*a*b*(2-m)+b^2*(3-4*m+m^2))*hypergeom([1,1+m],[2+m],(a+b*sin(c+d*x))/(a+b))*(a+b*sin(c+d*x))^(1+m)/((a+b)^3*d*(1+m))-1/4*sec(c+d*x)^4*(b-a*sin(c+d*x))*(a+b*sin(c+d*x))^(1+m)/((a^2-b^2)*d)+1/8*sec(c+d*x)^2*(a+b*sin(c+d*x))^(1+m)*(b*(b^2*(3-m)-a^2*(1+m))+a*(3*a^2-b^2*(5-2*m))*sin(c+d*x))/((a^2-b^2)^2*d)],
[cos(c+d*x)^4*(a+b*sin(c+d*x))^m,x,2,AppellF1(1+m,-3/2,-3/2,2+m,(a+b*sin(c+d*x))/(a-b),(a+b*sin(c+d*x))/(a+b))*cos(c+d*x)^3*(a+b*sin(c+d*x))^(1+m)/(b*d*(1+m)*(1+(-a-b*sin(c+d*x))/(a-b))^(3/2)*(1+(-a-b*sin(c+d*x))/(a+b))^(3/2))],
[cos(c+d*x)^2*(a+b*sin(c+d*x))^m,x,2,AppellF1(1+m,-1/2,-1/2,2+m,(a+b*sin(c+d*x))/(a-b),(a+b*sin(c+d*x))/(a+b))*cos(c+d*x)*(a+b*sin(c+d*x))^(1+m)/(b*d*(1+m)*sqrt(1+(-a-b*sin(c+d*x))/(a-b))*sqrt(1+(-a-b*sin(c+d*x))/(a+b)))],
[sec(c+d*x)^2*(a+b*sin(c+d*x))^m,x,2,AppellF1(1+m,3/2,3/2,2+m,(a+b*sin(c+d*x))/(a-b),(a+b*sin(c+d*x))/(a+b))*sec(c+d*x)^3*(a+b*sin(c+d*x))^(1+m)*(1+(-a-b*sin(c+d*x))/(a-b))^(3/2)*(1+(-a-b*sin(c+d*x))/(a+b))^(3/2)/(b*d*(1+m))],
[sec(c+d*x)^4*(a+b*sin(c+d*x))^m,x,2,AppellF1(1+m,5/2,5/2,2+m,(a+b*sin(c+d*x))/(a-b),(a+b*sin(c+d*x))/(a+b))*sec(c+d*x)^5*(a+b*sin(c+d*x))^(1+m)*(1+(-a-b*sin(c+d*x))/(a-b))^(5/2)*(1+(-a-b*sin(c+d*x))/(a+b))^(5/2)/(b*d*(1+m))],
[(e*cos(c+d*x))^(5/2)*(a+b*sin(c+d*x))^m,x,2,e*AppellF1(1+m,-3/4,-3/4,2+m,(a+b*sin(c+d*x))/(a-b),(a+b*sin(c+d*x))/(a+b))*(e*cos(c+d*x))^(3/2)*(a+b*sin(c+d*x))^(1+m)/(b*d*(1+m)*(1+(-a-b*sin(c+d*x))/(a-b))^(3/4)*(1+(-a-b*sin(c+d*x))/(a+b))^(3/4))],
[(e*cos(c+d*x))^(3/2)*(a+b*sin(c+d*x))^m,x,2,e*AppellF1(1+m,-1/4,-1/4,2+m,(a+b*sin(c+d*x))/(a-b),(a+b*sin(c+d*x))/(a+b))*(a+b*sin(c+d*x))^(1+m)*sqrt(e*cos(c+d*x))/(b*d*(1+m)*(1+(-a-b*sin(c+d*x))/(a-b))^(1/4)*(1+(-a-b*sin(c+d*x))/(a+b))^(1/4))],
[(e*cos(c+d*x))^(1/2)*(a+b*sin(c+d*x))^m,x,2,e*AppellF1(1+m,1/4,1/4,2+m,(a+b*sin(c+d*x))/(a-b),(a+b*sin(c+d*x))/(a+b))*(a+b*sin(c+d*x))^(1+m)*(1+(-a-b*sin(c+d*x))/(a-b))^(1/4)*(1+(-a-b*sin(c+d*x))/(a+b))^(1/4)/(b*d*(1+m)*sqrt(e*cos(c+d*x)))],
[(a+b*sin(c+d*x))^m/(e*cos(c+d*x))^(1/2),x,2,e*AppellF1(1+m,3/4,3/4,2+m,(a+b*sin(c+d*x))/(a-b),(a+b*sin(c+d*x))/(a+b))*(a+b*sin(c+d*x))^(1+m)*(1+(-a-b*sin(c+d*x))/(a-b))^(3/4)*(1+(-a-b*sin(c+d*x))/(a+b))^(3/4)/(b*d*(1+m)*(e*cos(c+d*x))^(3/2))],
[(a+b*sin(c+d*x))^m/(e*cos(c+d*x))^(3/2),x,2,e*AppellF1(1+m,5/4,5/4,2+m,(a+b*sin(c+d*x))/(a-b),(a+b*sin(c+d*x))/(a+b))*(a+b*sin(c+d*x))^(1+m)*(1+(-a-b*sin(c+d*x))/(a-b))^(5/4)*(1+(-a-b*sin(c+d*x))/(a+b))^(5/4)/(b*d*(1+m)*(e*cos(c+d*x))^(5/2))],
[(a+b*sin(c+d*x))^m/(e*cos(c+d*x))^(5/2),x,2,e*AppellF1(1+m,7/4,7/4,2+m,(a+b*sin(c+d*x))/(a-b),(a+b*sin(c+d*x))/(a+b))*(a+b*sin(c+d*x))^(1+m)*(1+(-a-b*sin(c+d*x))/(a-b))^(7/4)*(1+(-a-b*sin(c+d*x))/(a+b))^(7/4)/(b*d*(1+m)*(e*cos(c+d*x))^(7/2))],
[(e*cos(c+d*x))^(-4-m)*(a+b*sin(c+d*x))^m,x,9,-(e*cos(c+d*x))^(-3-m)*(a+b*sin(c+d*x))^(1+m)/((a-b)*d*e*(3+m))+2*b*(e*cos(c+d*x))^(-1-m)*(a+b*sin(c+d*x))^(1+m)/((a-b)^2*d*e^3*(1+m)*(3+m))+a*(e*cos(c+d*x))^(-3-m)*(1+sin(c+d*x))*(a+b*sin(c+d*x))^(1+m)/((a^2-b^2)*d*e*(3+m))+a*(3*b+a*(2+m))*(e*cos(c+d*x))^(-3-m)*(1-sin(c+d*x))*(1+sin(c+d*x))*(a+b*sin(c+d*x))^(1+m)/((a-b)*(a+b)^2*d*e*(1+m)*(3+m))-2^(3/2-1/2*m)*a*b*(e*cos(c+d*x))^(-1-m)*hypergeom([1/2*(-1-m),1/2*(1+m)],[1/2*(1-m)],1/2*(a-b)*(1-sin(c+d*x))/(a+b*sin(c+d*x)))*((a+b)*(1+sin(c+d*x))/(a+b*sin(c+d*x)))^(1/2*(1+m))*(a+b*sin(c+d*x))^(1+m)/((a-b)^2*(a+b)*d*e^3*(3+4*m+m^2))-2^(-1/2-1/2*m)*a*(2*a*b-b^2+a^2*(2+m))*(e*cos(c+d*x))^(-3-m)*hypergeom([1/2*(1-m),1/2*(3+m)],[1/2*(3-m)],1/2*(a-b)*(1-sin(c+d*x))/(a+b*sin(c+d*x)))*(1-sin(c+d*x))^2*((a+b)*(1+sin(c+d*x))/(a+b*sin(c+d*x)))^(1/2*(3+m))*(a+b*sin(c+d*x))^(1+m)/((a-b)*(a+b)^3*d*e*(1-m)*(3+m))],
[(e*cos(c+d*x))^(-3-m)*(a+b*sin(c+d*x))^m,x,-5,sec(c+d*x)^4*(-1+sin(c+d*x))*(1+sin(c+d*x))*(a+b*sin(c+d*x))^(1+m)/((a-b)*d*e^3*(2+m)*(e*cos(c+d*x))^m)+(-2*b+a*(2+m))*sec(c+d*x)^4*(-1+sin(c+d*x))*(1+sin(c+d*x))^2*(a+b*sin(c+d*x))^(1+m)/((a-b)^2*d*e^3*m*(2+m)*(e*cos(c+d*x))^m)-(-b^2+a^2*(1+m))*hypergeom([1/2*m,1+m],[2+m],-2*(a+b*sin(c+d*x))/((a-b)*(-1+sin(c+d*x))))*sec(c+d*x)^4*(1+sin(c+d*x))^3*((a+b)*(1+sin(c+d*x))/((a-b)*(-1+sin(c+d*x))))^(1/2*(-2+m))*(a+b*sin(c+d*x))^(1+m)/((a-b)^3*d*e^3*m*(1+m)*(e*cos(c+d*x))^m)],
[(e*cos(c+d*x))^(-2-m)*(a+b*sin(c+d*x))^m,x,3,-(e*cos(c+d*x))^(-1-m)*(a+b*sin(c+d*x))^(1+m)/((a-b)*d*e*(1+m))+2^(1/2-1/2*m)*a*(e*cos(c+d*x))^(-1-m)*hypergeom([1/2*(-1-m),1/2*(1+m)],[1/2*(1-m)],1/2*(a-b)*(1-sin(c+d*x))/(a+b*sin(c+d*x)))*((a+b)*(1+sin(c+d*x))/(a+b*sin(c+d*x)))^(1/2*(1+m))*(a+b*sin(c+d*x))^(1+m)/((a^2-b^2)*d*e*(1+m))],
[(e*cos(c+d*x))^(-1-m)*(a+b*sin(c+d*x))^m,x,1,e*(e*cos(c+d*x))^(-2-m)*hypergeom([1+m,1/2*(2+m)],[2+m],2*(a+b*sin(c+d*x))/((a+b)*(1+sin(c+d*x))))*(1-sin(c+d*x))*(-(a-b)*(1-sin(c+d*x))/((a+b)*(1+sin(c+d*x))))^(1/2*m)*(a+b*sin(c+d*x))^(1+m)/((a+b)*d*(1+m))],
[(a+b*sin(c+d*x))^m/(e*cos(c+d*x))^m,x,2,e*AppellF1(1+m,1/2*(1+m),1/2*(1+m),2+m,(a+b*sin(c+d*x))/(a-b),(a+b*sin(c+d*x))/(a+b))*(e*cos(c+d*x))^(-1-m)*(a+b*sin(c+d*x))^(1+m)*(1+(-a-b*sin(c+d*x))/(a-b))^(1/2*(1+m))*(1+(-a-b*sin(c+d*x))/(a+b))^(1/2*(1+m))/(b*d*(1+m))],
[(e*cos(c+d*x))^(1-m)*(a+b*sin(c+d*x))^m,x,2,e*AppellF1(1+m,1/2*m,1/2*m,2+m,(a+b*sin(c+d*x))/(a-b),(a+b*sin(c+d*x))/(a+b))*(a+b*sin(c+d*x))^(1+m)*(1+(-a-b*sin(c+d*x))/(a-b))^(1/2*m)*(1+(-a-b*sin(c+d*x))/(a+b))^(1/2*m)/(b*d*(1+m)*(e*cos(c+d*x))^m)],
[(e*cos(c+d*x))^(2-m)*(a+b*sin(c+d*x))^m,x,2,e*AppellF1(1+m,1/2*(-1+m),1/2*(-1+m),2+m,(a+b*sin(c+d*x))/(a-b),(a+b*sin(c+d*x))/(a+b))*(e*cos(c+d*x))^(1-m)*(a+b*sin(c+d*x))^(1+m)*(1+(-a-b*sin(c+d*x))/(a-b))^(1/2*(-1+m))*(1+(-a-b*sin(c+d*x))/(a+b))^(1/2*(-1+m))/(b*d*(1+m))]]:

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

# Integrands of the form (g Sec[e+f x])^p (a+b Sin[e+f x])^m when a^2-b^2=0

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