# Maple integration test file: "4 Trig functions\4.5 Secant\4.5.1.4 (d tan)^n (a+b sec)^m.txt"

lst:=[

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

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

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

# n>0
[(a+a*sec(c+d*x))*tan(c+d*x)^9,x,3,-a*log(cos(c+d*x))/d+a*sec(c+d*x)/d-2*a*sec(c+d*x)^2/d-4/3*a*sec(c+d*x)^3/d+3/2*a*sec(c+d*x)^4/d+6/5*a*sec(c+d*x)^5/d-2/3*a*sec(c+d*x)^6/d-4/7*a*sec(c+d*x)^7/d+1/8*a*sec(c+d*x)^8/d+1/9*a*sec(c+d*x)^9/d],
[(a+a*sec(c+d*x))*tan(c+d*x)^7,x,3,a*log(cos(c+d*x))/d-a*sec(c+d*x)/d+3/2*a*sec(c+d*x)^2/d+a*sec(c+d*x)^3/d-3/4*a*sec(c+d*x)^4/d-3/5*a*sec(c+d*x)^5/d+1/6*a*sec(c+d*x)^6/d+1/7*a*sec(c+d*x)^7/d],
[(a+a*sec(c+d*x))*tan(c+d*x)^5,x,3,-a*log(cos(c+d*x))/d+a*sec(c+d*x)/d-a*sec(c+d*x)^2/d-2/3*a*sec(c+d*x)^3/d+1/4*a*sec(c+d*x)^4/d+1/5*a*sec(c+d*x)^5/d],
[(a+a*sec(c+d*x))*tan(c+d*x)^3,x,3,a*log(cos(c+d*x))/d-a*sec(c+d*x)/d+1/2*a*sec(c+d*x)^2/d+1/3*a*sec(c+d*x)^3/d],
[(a+a*sec(c+d*x))*tan(c+d*x),x,3,-a*log(cos(c+d*x))/d+a*sec(c+d*x)/d],
[cot(c+d*x)*(a+a*sec(c+d*x)),x,2,a*log(1-cos(c+d*x))/d],
[cot(c+d*x)^3*(a+a*sec(c+d*x)),x,3,-1/2*a/(d*(1-cos(c+d*x)))-3/4*a*log(1-cos(c+d*x))/d-1/4*a*log(1+cos(c+d*x))/d],
[cot(c+d*x)^5*(a+a*sec(c+d*x)),x,3,-1/8*a/(d*(1-cos(c+d*x))^2)+3/4*a/(d*(1-cos(c+d*x)))+1/8*a/(d*(1+cos(c+d*x)))+11/16*a*log(1-cos(c+d*x))/d+5/16*a*log(1+cos(c+d*x))/d],
[cot(c+d*x)^7*(a+a*sec(c+d*x)),x,3,-1/24*a/(d*(1-cos(c+d*x))^3)+9/32*a/(d*(1-cos(c+d*x))^2)-15/16*a/(d*(1-cos(c+d*x)))+1/32*a/(d*(1+cos(c+d*x))^2)-1/4*a/(d*(1+cos(c+d*x)))-21/32*a*log(1-cos(c+d*x))/d-11/32*a*log(1+cos(c+d*x))/d],
[(a+a*sec(c+d*x))*tan(c+d*x)^8,x,6,a*x+35/128*a*arctanh(sin(c+d*x))/d-1/128*(128*a+35*a*sec(c+d*x))*tan(c+d*x)/d+1/192*(64*a+35*a*sec(c+d*x))*tan(c+d*x)^3/d-1/240*(48*a+35*a*sec(c+d*x))*tan(c+d*x)^5/d+1/56*(8*a+7*a*sec(c+d*x))*tan(c+d*x)^7/d],
[(a+a*sec(c+d*x))*tan(c+d*x)^6,x,5,-a*x-5/16*a*arctanh(sin(c+d*x))/d+1/16*(16*a+5*a*sec(c+d*x))*tan(c+d*x)/d-1/24*(8*a+5*a*sec(c+d*x))*tan(c+d*x)^3/d+1/30*(6*a+5*a*sec(c+d*x))*tan(c+d*x)^5/d],
[(a+a*sec(c+d*x))*tan(c+d*x)^4,x,4,a*x+3/8*a*arctanh(sin(c+d*x))/d-1/8*(8*a+3*a*sec(c+d*x))*tan(c+d*x)/d+1/12*(4*a+3*a*sec(c+d*x))*tan(c+d*x)^3/d],
[(a+a*sec(c+d*x))*tan(c+d*x)^2,x,3,-a*x-1/2*a*arctanh(sin(c+d*x))/d+1/2*(2*a+a*sec(c+d*x))*tan(c+d*x)/d],
[cot(c+d*x)^2*(a+a*sec(c+d*x)),x,2,-a*x-cot(c+d*x)*(a+a*sec(c+d*x))/d],
[cot(c+d*x)^4*(a+a*sec(c+d*x)),x,3,a*x-1/3*cot(c+d*x)^3*(a+a*sec(c+d*x))/d+1/3*cot(c+d*x)*(3*a+2*a*sec(c+d*x))/d],
[cot(c+d*x)^6*(a+a*sec(c+d*x)),x,4,-a*x-1/5*cot(c+d*x)^5*(a+a*sec(c+d*x))/d+1/15*cot(c+d*x)^3*(5*a+4*a*sec(c+d*x))/d-1/15*cot(c+d*x)*(15*a+8*a*sec(c+d*x))/d],
[cot(c+d*x)^8*(a+a*sec(c+d*x)),x,5,a*x-1/7*cot(c+d*x)^7*(a+a*sec(c+d*x))/d+1/35*cot(c+d*x)^5*(7*a+6*a*sec(c+d*x))/d+1/35*cot(c+d*x)*(35*a+16*a*sec(c+d*x))/d-1/105*cot(c+d*x)^3*(35*a+24*a*sec(c+d*x))/d],
[cot(c+d*x)^10*(a+a*sec(c+d*x)),x,6,-a*x-1/9*cot(c+d*x)^9*(a+a*sec(c+d*x))/d+1/63*cot(c+d*x)^7*(9*a+8*a*sec(c+d*x))/d-1/105*cot(c+d*x)^5*(21*a+16*a*sec(c+d*x))/d+1/315*cot(c+d*x)^3*(105*a+64*a*sec(c+d*x))/d-1/315*cot(c+d*x)*(315*a+128*a*sec(c+d*x))/d],
[(a+a*sec(c+d*x))^2*tan(c+d*x)^9,x,3,-a^2*log(cos(c+d*x))/d+2*a^2*sec(c+d*x)/d-3/2*a^2*sec(c+d*x)^2/d-8/3*a^2*sec(c+d*x)^3/d+1/2*a^2*sec(c+d*x)^4/d+12/5*a^2*sec(c+d*x)^5/d+1/3*a^2*sec(c+d*x)^6/d-8/7*a^2*sec(c+d*x)^7/d-3/8*a^2*sec(c+d*x)^8/d+2/9*a^2*sec(c+d*x)^9/d+1/10*a^2*sec(c+d*x)^10/d],
[(a+a*sec(c+d*x))^2*tan(c+d*x)^7,x,3,a^2*log(cos(c+d*x))/d-2*a^2*sec(c+d*x)/d+a^2*sec(c+d*x)^2/d+2*a^2*sec(c+d*x)^3/d-6/5*a^2*sec(c+d*x)^5/d-1/3*a^2*sec(c+d*x)^6/d+2/7*a^2*sec(c+d*x)^7/d+1/8*a^2*sec(c+d*x)^8/d],
[(a+a*sec(c+d*x))^2*tan(c+d*x)^5,x,3,-a^2*log(cos(c+d*x))/d+2*a^2*sec(c+d*x)/d-1/2*a^2*sec(c+d*x)^2/d-4/3*a^2*sec(c+d*x)^3/d-1/4*a^2*sec(c+d*x)^4/d+2/5*a^2*sec(c+d*x)^5/d+1/6*a^2*sec(c+d*x)^6/d],
[(a+a*sec(c+d*x))^2*tan(c+d*x)^3,x,3,a^2*log(cos(c+d*x))/d-2*a^2*sec(c+d*x)/d+2/3*a^2*sec(c+d*x)^3/d+1/4*a^2*sec(c+d*x)^4/d],
[(a+a*sec(c+d*x))^2*tan(c+d*x),x,3,-a^2*log(cos(c+d*x))/d+2*a^2*sec(c+d*x)/d+1/2*a^2*sec(c+d*x)^2/d],
[cot(c+d*x)*(a+a*sec(c+d*x))^2,x,3,2*a^2*log(1-cos(c+d*x))/d-a^2*log(cos(c+d*x))/d],
[cot(c+d*x)^3*(a+a*sec(c+d*x))^2,x,3,-a^2/(d*(1-cos(c+d*x)))-a^2*log(1-cos(c+d*x))/d],
[cot(c+d*x)^5*(a+a*sec(c+d*x))^2,x,3,-1/4*a^2/(d*(1-cos(c+d*x))^2)+5/4*a^2/(d*(1-cos(c+d*x)))+7/8*a^2*log(1-cos(c+d*x))/d+1/8*a^2*log(1+cos(c+d*x))/d],
[cot(c+d*x)^7*(a+a*sec(c+d*x))^2,x,3,-1/12*a^2/(d*(1-cos(c+d*x))^3)+1/2*a^2/(d*(1-cos(c+d*x))^2)-23/16*a^2/(d*(1-cos(c+d*x)))-1/16*a^2/(d*(1+cos(c+d*x)))-13/16*a^2*log(1-cos(c+d*x))/d-3/16*a^2*log(1+cos(c+d*x))/d],
[cot(c+d*x)^9*(a+a*sec(c+d*x))^2,x,3,-1/32*a^2/(d*(1-cos(c+d*x))^4)+11/48*a^2/(d*(1-cos(c+d*x))^3)-3/4*a^2/(d*(1-cos(c+d*x))^2)+51/32*a^2/(d*(1-cos(c+d*x)))-1/64*a^2/(d*(1+cos(c+d*x))^2)+9/64*a^2/(d*(1+cos(c+d*x)))+99/128*a^2*log(1-cos(c+d*x))/d+29/128*a^2*log(1+cos(c+d*x))/d],
[(a+a*sec(c+d*x))^2*tan(c+d*x)^6,x,12,-a^2*x-5/8*a^2*arctanh(sin(c+d*x))/d+a^2*tan(c+d*x)/d+5/8*a^2*sec(c+d*x)*tan(c+d*x)/d-1/3*a^2*tan(c+d*x)^3/d-5/12*a^2*sec(c+d*x)*tan(c+d*x)^3/d+1/5*a^2*tan(c+d*x)^5/d+1/3*a^2*sec(c+d*x)*tan(c+d*x)^5/d+1/7*a^2*tan(c+d*x)^7/d],
[(a+a*sec(c+d*x))^2*tan(c+d*x)^4,x,10,a^2*x+3/4*a^2*arctanh(sin(c+d*x))/d-a^2*tan(c+d*x)/d-3/4*a^2*sec(c+d*x)*tan(c+d*x)/d+1/3*a^2*tan(c+d*x)^3/d+1/2*a^2*sec(c+d*x)*tan(c+d*x)^3/d+1/5*a^2*tan(c+d*x)^5/d],
[(a+a*sec(c+d*x))^2*tan(c+d*x)^2,x,8,-a^2*x-a^2*arctanh(sin(c+d*x))/d+a^2*tan(c+d*x)/d+a^2*sec(c+d*x)*tan(c+d*x)/d+1/3*a^2*tan(c+d*x)^3/d],
[cot(c+d*x)^2*(a+a*sec(c+d*x))^2,x,8,-a^2*x-2*a^2*cot(c+d*x)/d-2*a^2*csc(c+d*x)/d],
[cot(c+d*x)^4*(a+a*sec(c+d*x))^2,x,9,a^2*x+a^2*cot(c+d*x)/d-2/3*a^2*cot(c+d*x)^3/d+2*a^2*csc(c+d*x)/d-2/3*a^2*csc(c+d*x)^3/d],
[cot(c+d*x)^6*(a+a*sec(c+d*x))^2,x,11,-a^2*x-a^2*cot(c+d*x)/d+1/3*a^2*cot(c+d*x)^3/d-2/5*a^2*cot(c+d*x)^5/d-2*a^2*csc(c+d*x)/d+4/3*a^2*csc(c+d*x)^3/d-2/5*a^2*csc(c+d*x)^5/d],
[cot(c+d*x)^8*(a+a*sec(c+d*x))^2,x,12,a^2*x+a^2*cot(c+d*x)/d-1/3*a^2*cot(c+d*x)^3/d+1/5*a^2*cot(c+d*x)^5/d-2/7*a^2*cot(c+d*x)^7/d+2*a^2*csc(c+d*x)/d-2*a^2*csc(c+d*x)^3/d+6/5*a^2*csc(c+d*x)^5/d-2/7*a^2*csc(c+d*x)^7/d],
[cot(c+d*x)^10*(a+a*sec(c+d*x))^2,x,13,-a^2*x-a^2*cot(c+d*x)/d+1/3*a^2*cot(c+d*x)^3/d-1/5*a^2*cot(c+d*x)^5/d+1/7*a^2*cot(c+d*x)^7/d-2/9*a^2*cot(c+d*x)^9/d-2*a^2*csc(c+d*x)/d+8/3*a^2*csc(c+d*x)^3/d-12/5*a^2*csc(c+d*x)^5/d+8/7*a^2*csc(c+d*x)^7/d-2/9*a^2*csc(c+d*x)^9/d],
[(a+a*sec(c+d*x))^3*tan(c+d*x)^9,x,3,-a^3*log(cos(c+d*x))/d+3*a^3*sec(c+d*x)/d-1/2*a^3*sec(c+d*x)^2/d-11/3*a^3*sec(c+d*x)^3/d-3/2*a^3*sec(c+d*x)^4/d+14/5*a^3*sec(c+d*x)^5/d+7/3*a^3*sec(c+d*x)^6/d-6/7*a^3*sec(c+d*x)^7/d-11/8*a^3*sec(c+d*x)^8/d-1/9*a^3*sec(c+d*x)^9/d+3/10*a^3*sec(c+d*x)^10/d+1/11*a^3*sec(c+d*x)^11/d],
[(a+a*sec(c+d*x))^3*tan(c+d*x)^7,x,3,a^3*log(cos(c+d*x))/d-3*a^3*sec(c+d*x)/d+8/3*a^3*sec(c+d*x)^3/d+3/2*a^3*sec(c+d*x)^4/d-6/5*a^3*sec(c+d*x)^5/d-4/3*a^3*sec(c+d*x)^6/d+3/8*a^3*sec(c+d*x)^8/d+1/9*a^3*sec(c+d*x)^9/d],
[(a+a*sec(c+d*x))^3*tan(c+d*x)^5,x,3,-a^3*log(cos(c+d*x))/d+3*a^3*sec(c+d*x)/d+1/2*a^3*sec(c+d*x)^2/d-5/3*a^3*sec(c+d*x)^3/d-5/4*a^3*sec(c+d*x)^4/d+1/5*a^3*sec(c+d*x)^5/d+1/2*a^3*sec(c+d*x)^6/d+1/7*a^3*sec(c+d*x)^7/d],
[(a+a*sec(c+d*x))^3*tan(c+d*x)^3,x,3,a^3*log(cos(c+d*x))/d-3*a^3*sec(c+d*x)/d-a^3*sec(c+d*x)^2/d+2/3*a^3*sec(c+d*x)^3/d+3/4*a^3*sec(c+d*x)^4/d+1/5*a^3*sec(c+d*x)^5/d],
[(a+a*sec(c+d*x))^3*tan(c+d*x),x,3,-a^3*log(cos(c+d*x))/d+3*a^3*sec(c+d*x)/d+3/2*a^3*sec(c+d*x)^2/d+1/3*a^3*sec(c+d*x)^3/d],
[cot(c+d*x)*(a+a*sec(c+d*x))^3,x,3,4*a^3*log(1-cos(c+d*x))/d-3*a^3*log(cos(c+d*x))/d+a^3*sec(c+d*x)/d],
[cot(c+d*x)^3*(a+a*sec(c+d*x))^3,x,3,-2*a^3/(d*(1-cos(c+d*x)))-a^3*log(1-cos(c+d*x))/d],
[cot(c+d*x)^5*(a+a*sec(c+d*x))^3,x,3,-1/2*a^3/(d*(1-cos(c+d*x))^2)+2*a^3/(d*(1-cos(c+d*x)))+a^3*log(1-cos(c+d*x))/d],
[cot(c+d*x)^7*(a+a*sec(c+d*x))^3,x,3,-1/6*a^3/(d*(1-cos(c+d*x))^3)+7/8*a^3/(d*(1-cos(c+d*x))^2)-17/8*a^3/(d*(1-cos(c+d*x)))-15/16*a^3*log(1-cos(c+d*x))/d-1/16*a^3*log(1+cos(c+d*x))/d],
[cot(c+d*x)^9*(a+a*sec(c+d*x))^3,x,3,-1/16*a^3/(d*(1-cos(c+d*x))^4)+5/12*a^3/(d*(1-cos(c+d*x))^3)-39/32*a^3/(d*(1-cos(c+d*x))^2)+9/4*a^3/(d*(1-cos(c+d*x)))+1/32*a^3/(d*(1+cos(c+d*x)))+57/64*a^3*log(1-cos(c+d*x))/d+7/64*a^3*log(1+cos(c+d*x))/d],
[(a+a*sec(c+d*x))^3*tan(c+d*x)^6,x,17,-a^3*x-125/128*a^3*arctanh(sin(c+d*x))/d+a^3*tan(c+d*x)/d+115/128*a^3*sec(c+d*x)*tan(c+d*x)/d+5/64*a^3*sec(c+d*x)^3*tan(c+d*x)/d-1/3*a^3*tan(c+d*x)^3/d-5/8*a^3*sec(c+d*x)*tan(c+d*x)^3/d-5/48*a^3*sec(c+d*x)^3*tan(c+d*x)^3/d+1/5*a^3*tan(c+d*x)^5/d+1/2*a^3*sec(c+d*x)*tan(c+d*x)^5/d+1/8*a^3*sec(c+d*x)^3*tan(c+d*x)^5/d+3/7*a^3*tan(c+d*x)^7/d],
[(a+a*sec(c+d*x))^3*tan(c+d*x)^4,x,14,a^3*x+19/16*a^3*arctanh(sin(c+d*x))/d-a^3*tan(c+d*x)/d-17/16*a^3*sec(c+d*x)*tan(c+d*x)/d-1/8*a^3*sec(c+d*x)^3*tan(c+d*x)/d+1/3*a^3*tan(c+d*x)^3/d+3/4*a^3*sec(c+d*x)*tan(c+d*x)^3/d+1/6*a^3*sec(c+d*x)^3*tan(c+d*x)^3/d+3/5*a^3*tan(c+d*x)^5/d],
[(a+a*sec(c+d*x))^3*tan(c+d*x)^2,x,11,-a^3*x-13/8*a^3*arctanh(sin(c+d*x))/d+a^3*tan(c+d*x)/d+11/8*a^3*sec(c+d*x)*tan(c+d*x)/d+1/4*a^3*sec(c+d*x)^3*tan(c+d*x)/d+a^3*tan(c+d*x)^3/d],
[cot(c+d*x)^2*(a+a*sec(c+d*x))^3,x,11,-a^3*x+a^3*arctanh(sin(c+d*x))/d-4*a^3*cot(c+d*x)/d-4*a^3*csc(c+d*x)/d],
[cot(c+d*x)^4*(a+a*sec(c+d*x))^3,x,11,a^3*x+a^3*cot(c+d*x)/d-4/3*a^3*cot(c+d*x)^3/d+3*a^3*csc(c+d*x)/d-4/3*a^3*csc(c+d*x)^3/d],
[cot(c+d*x)^6*(a+a*sec(c+d*x))^3,x,14,-a^3*x-a^3*cot(c+d*x)/d+1/3*a^3*cot(c+d*x)^3/d-4/5*a^3*cot(c+d*x)^5/d-3*a^3*csc(c+d*x)/d+7/3*a^3*csc(c+d*x)^3/d-4/5*a^3*csc(c+d*x)^5/d],
[cot(c+d*x)^8*(a+a*sec(c+d*x))^3,x,15,a^3*x+a^3*cot(c+d*x)/d-1/3*a^3*cot(c+d*x)^3/d+1/5*a^3*cot(c+d*x)^5/d-4/7*a^3*cot(c+d*x)^7/d+3*a^3*csc(c+d*x)/d-10/3*a^3*csc(c+d*x)^3/d+11/5*a^3*csc(c+d*x)^5/d-4/7*a^3*csc(c+d*x)^7/d],
[cot(c+d*x)^10*(a+a*sec(c+d*x))^3,x,16,-a^3*x-a^3*cot(c+d*x)/d+1/3*a^3*cot(c+d*x)^3/d-1/5*a^3*cot(c+d*x)^5/d+1/7*a^3*cot(c+d*x)^7/d-4/9*a^3*cot(c+d*x)^9/d-3*a^3*csc(c+d*x)/d+13/3*a^3*csc(c+d*x)^3/d-21/5*a^3*csc(c+d*x)^5/d+15/7*a^3*csc(c+d*x)^7/d-4/9*a^3*csc(c+d*x)^9/d],
[cot(c+d*x)^12*(a+a*sec(c+d*x))^3,x,17,a^3*x+a^3*cot(c+d*x)/d-1/3*a^3*cot(c+d*x)^3/d+1/5*a^3*cot(c+d*x)^5/d-1/7*a^3*cot(c+d*x)^7/d+1/9*a^3*cot(c+d*x)^9/d-4/11*a^3*cot(c+d*x)^11/d+3*a^3*csc(c+d*x)/d-16/3*a^3*csc(c+d*x)^3/d+34/5*a^3*csc(c+d*x)^5/d-36/7*a^3*csc(c+d*x)^7/d+19/9*a^3*csc(c+d*x)^9/d-4/11*a^3*csc(c+d*x)^11/d],

# n<0
[tan(c+d*x)^9/(a+a*sec(c+d*x)),x,3,-log(cos(c+d*x))/(a*d)-sec(c+d*x)/(a*d)-3/2*sec(c+d*x)^2/(a*d)+sec(c+d*x)^3/(a*d)+3/4*sec(c+d*x)^4/(a*d)-3/5*sec(c+d*x)^5/(a*d)-1/6*sec(c+d*x)^6/(a*d)+1/7*sec(c+d*x)^7/(a*d)],
[tan(c+d*x)^7/(a+a*sec(c+d*x)),x,3,log(cos(c+d*x))/(a*d)+sec(c+d*x)/(a*d)+sec(c+d*x)^2/(a*d)-2/3*sec(c+d*x)^3/(a*d)-1/4*sec(c+d*x)^4/(a*d)+1/5*sec(c+d*x)^5/(a*d)],
[tan(c+d*x)^5/(a+a*sec(c+d*x)),x,3,-log(cos(c+d*x))/(a*d)-sec(c+d*x)/(a*d)-1/2*sec(c+d*x)^2/(a*d)+1/3*sec(c+d*x)^3/(a*d)],
[tan(c+d*x)^3/(a+a*sec(c+d*x)),x,3,log(cos(c+d*x))/(a*d)+sec(c+d*x)/(a*d)],
[tan(c+d*x)/(a+a*sec(c+d*x)),x,2,-log(1+cos(c+d*x))/(a*d)],
[cot(c+d*x)/(a+a*sec(c+d*x)),x,3,1/2/(a*d*(1+cos(c+d*x)))+1/4*log(1-cos(c+d*x))/(a*d)+3/4*log(1+cos(c+d*x))/(a*d)],
[cot(c+d*x)^3/(a+a*sec(c+d*x)),x,3,(-1/8)/(a*d*(1-cos(c+d*x)))+1/8/(a*d*(1+cos(c+d*x))^2)+(-3/4)/(a*d*(1+cos(c+d*x)))-5/16*log(1-cos(c+d*x))/(a*d)-11/16*log(1+cos(c+d*x))/(a*d)],
[cot(c+d*x)^5/(a+a*sec(c+d*x)),x,3,(-1/32)/(a*d*(1-cos(c+d*x))^2)+1/4/(a*d*(1-cos(c+d*x)))+1/24/(a*d*(1+cos(c+d*x))^3)+(-9/32)/(a*d*(1+cos(c+d*x))^2)+15/16/(a*d*(1+cos(c+d*x)))+11/32*log(1-cos(c+d*x))/(a*d)+21/32*log(1+cos(c+d*x))/(a*d)],
[tan(c+d*x)^8/(a+a*sec(c+d*x)),x,6,x/a-5/16*arctanh(sin(c+d*x))/(a*d)-1/16*(16-5*sec(c+d*x))*tan(c+d*x)/(a*d)+1/24*(8-5*sec(c+d*x))*tan(c+d*x)^3/(a*d)-1/30*(6-5*sec(c+d*x))*tan(c+d*x)^5/(a*d)],
[tan(c+d*x)^6/(a+a*sec(c+d*x)),x,5,-x/a+3/8*arctanh(sin(c+d*x))/(a*d)+1/8*(8-3*sec(c+d*x))*tan(c+d*x)/(a*d)-1/12*(4-3*sec(c+d*x))*tan(c+d*x)^3/(a*d)],
[tan(c+d*x)^4/(a+a*sec(c+d*x)),x,4,x/a-1/2*arctanh(sin(c+d*x))/(a*d)-1/2*(2-sec(c+d*x))*tan(c+d*x)/(a*d)],
[tan(c+d*x)^2/(a+a*sec(c+d*x)),x,3,-x/a+arctanh(sin(c+d*x))/(a*d)],
[cot(c+d*x)^2/(a+a*sec(c+d*x)),x,4,-x/a-1/3*cot(c+d*x)*(3-2*sec(c+d*x))/(a*d)+1/3*cot(c+d*x)^3*(1-sec(c+d*x))/(a*d)],
[cot(c+d*x)^4/(a+a*sec(c+d*x)),x,5,x/a+1/15*cot(c+d*x)*(15-8*sec(c+d*x))/(a*d)-1/15*cot(c+d*x)^3*(5-4*sec(c+d*x))/(a*d)+1/5*cot(c+d*x)^5*(1-sec(c+d*x))/(a*d)],
[cot(c+d*x)^6/(a+a*sec(c+d*x)),x,6,-x/a+1/105*cot(c+d*x)^3*(35-24*sec(c+d*x))/(a*d)-1/35*cot(c+d*x)*(35-16*sec(c+d*x))/(a*d)-1/35*cot(c+d*x)^5*(7-6*sec(c+d*x))/(a*d)+1/7*cot(c+d*x)^7*(1-sec(c+d*x))/(a*d)],
[tan(c+d*x)^9/(a+a*sec(c+d*x))^2,x,3,-log(cos(c+d*x))/(a^2*d)-2*sec(c+d*x)/(a^2*d)-1/2*sec(c+d*x)^2/(a^2*d)+4/3*sec(c+d*x)^3/(a^2*d)-1/4*sec(c+d*x)^4/(a^2*d)-2/5*sec(c+d*x)^5/(a^2*d)+1/6*sec(c+d*x)^6/(a^2*d)],
[tan(c+d*x)^7/(a+a*sec(c+d*x))^2,x,3,log(cos(c+d*x))/(a^2*d)+2*sec(c+d*x)/(a^2*d)-2/3*sec(c+d*x)^3/(a^2*d)+1/4*sec(c+d*x)^4/(a^2*d)],
[tan(c+d*x)^5/(a+a*sec(c+d*x))^2,x,3,-log(cos(c+d*x))/(a^2*d)-2*sec(c+d*x)/(a^2*d)+1/2*sec(c+d*x)^2/(a^2*d)],
[tan(c+d*x)^3/(a+a*sec(c+d*x))^2,x,3,-log(cos(c+d*x))/(a^2*d)+2*log(1+cos(c+d*x))/(a^2*d)],
[tan(c+d*x)/(a+a*sec(c+d*x))^2,x,3,(-1)/(a^2*d*(1+cos(c+d*x)))-log(1+cos(c+d*x))/(a^2*d)],
[cot(c+d*x)/(a+a*sec(c+d*x))^2,x,3,(-1/4)/(a^2*d*(1+cos(c+d*x))^2)+5/4/(a^2*d*(1+cos(c+d*x)))+1/8*log(1-cos(c+d*x))/(a^2*d)+7/8*log(1+cos(c+d*x))/(a^2*d)],
[cot(c+d*x)^3/(a+a*sec(c+d*x))^2,x,3,(-1/16)/(a^2*d*(1-cos(c+d*x)))+(-1/12)/(a^2*d*(1+cos(c+d*x))^3)+1/2/(a^2*d*(1+cos(c+d*x))^2)+(-23/16)/(a^2*d*(1+cos(c+d*x)))-3/16*log(1-cos(c+d*x))/(a^2*d)-13/16*log(1+cos(c+d*x))/(a^2*d)],
[cot(c+d*x)^5/(a+a*sec(c+d*x))^2,x,3,(-1/64)/(a^2*d*(1-cos(c+d*x))^2)+9/64/(a^2*d*(1-cos(c+d*x)))+(-1/32)/(a^2*d*(1+cos(c+d*x))^4)+11/48/(a^2*d*(1+cos(c+d*x))^3)+(-3/4)/(a^2*d*(1+cos(c+d*x))^2)+51/32/(a^2*d*(1+cos(c+d*x)))+29/128*log(1-cos(c+d*x))/(a^2*d)+99/128*log(1+cos(c+d*x))/(a^2*d)],
[tan(c+d*x)^8/(a+a*sec(c+d*x))^2,x,11,x/a^2-3/4*arctanh(sin(c+d*x))/(a^2*d)-tan(c+d*x)/(a^2*d)+3/4*sec(c+d*x)*tan(c+d*x)/(a^2*d)+1/3*tan(c+d*x)^3/(a^2*d)-1/2*sec(c+d*x)*tan(c+d*x)^3/(a^2*d)+1/5*tan(c+d*x)^5/(a^2*d)],
[tan(c+d*x)^6/(a+a*sec(c+d*x))^2,x,9,-x/a^2+arctanh(sin(c+d*x))/(a^2*d)+tan(c+d*x)/(a^2*d)-sec(c+d*x)*tan(c+d*x)/(a^2*d)+1/3*tan(c+d*x)^3/(a^2*d)],
[tan(c+d*x)^4/(a+a*sec(c+d*x))^2,x,5,x/a^2-2*arctanh(sin(c+d*x))/(a^2*d)+tan(c+d*x)/(a^2*d)],
[tan(c+d*x)^2/(a+a*sec(c+d*x))^2,x,9,-x/a^2+2*tan(c+d*x)/(a*d*(a+a*sec(c+d*x))),-x/a^2-2*cot(c+d*x)/(a^2*d)+2*csc(c+d*x)/(a^2*d)],
[cot(c+d*x)^2/(a+a*sec(c+d*x))^2,x,12,-x/a^2-cot(c+d*x)/(a^2*d)+1/3*cot(c+d*x)^3/(a^2*d)-2/5*cot(c+d*x)^5/(a^2*d)+2*csc(c+d*x)/(a^2*d)-4/3*csc(c+d*x)^3/(a^2*d)+2/5*csc(c+d*x)^5/(a^2*d)],
[cot(c+d*x)^4/(a+a*sec(c+d*x))^2,x,13,x/a^2+cot(c+d*x)/(a^2*d)-1/3*cot(c+d*x)^3/(a^2*d)+1/5*cot(c+d*x)^5/(a^2*d)-2/7*cot(c+d*x)^7/(a^2*d)-2*csc(c+d*x)/(a^2*d)+2*csc(c+d*x)^3/(a^2*d)-6/5*csc(c+d*x)^5/(a^2*d)+2/7*csc(c+d*x)^7/(a^2*d)],
[cot(c+d*x)^6/(a+a*sec(c+d*x))^2,x,14,-x/a^2-cot(c+d*x)/(a^2*d)+1/3*cot(c+d*x)^3/(a^2*d)-1/5*cot(c+d*x)^5/(a^2*d)+1/7*cot(c+d*x)^7/(a^2*d)-2/9*cot(c+d*x)^9/(a^2*d)+2*csc(c+d*x)/(a^2*d)-8/3*csc(c+d*x)^3/(a^2*d)+12/5*csc(c+d*x)^5/(a^2*d)-8/7*csc(c+d*x)^7/(a^2*d)+2/9*csc(c+d*x)^9/(a^2*d)],
[tan(c+d*x)^11/(a+a*sec(c+d*x))^3,x,3,log(cos(c+d*x))/(a^3*d)+3*sec(c+d*x)/(a^3*d)-1/2*sec(c+d*x)^2/(a^3*d)-5/3*sec(c+d*x)^3/(a^3*d)+5/4*sec(c+d*x)^4/(a^3*d)+1/5*sec(c+d*x)^5/(a^3*d)-1/2*sec(c+d*x)^6/(a^3*d)+1/7*sec(c+d*x)^7/(a^3*d)],
[tan(c+d*x)^9/(a+a*sec(c+d*x))^3,x,3,-log(cos(c+d*x))/(a^3*d)-3*sec(c+d*x)/(a^3*d)+sec(c+d*x)^2/(a^3*d)+2/3*sec(c+d*x)^3/(a^3*d)-3/4*sec(c+d*x)^4/(a^3*d)+1/5*sec(c+d*x)^5/(a^3*d)],
[tan(c+d*x)^7/(a+a*sec(c+d*x))^3,x,3,log(cos(c+d*x))/(a^3*d)+3*sec(c+d*x)/(a^3*d)-3/2*sec(c+d*x)^2/(a^3*d)+1/3*sec(c+d*x)^3/(a^3*d)],
[tan(c+d*x)^5/(a+a*sec(c+d*x))^3,x,3,3*log(cos(c+d*x))/(a^3*d)-4*log(1+cos(c+d*x))/(a^3*d)+sec(c+d*x)/(a^3*d)],
[tan(c+d*x)^3/(a+a*sec(c+d*x))^3,x,3,2/(a^3*d*(1+cos(c+d*x)))+log(1+cos(c+d*x))/(a^3*d)],
[tan(c+d*x)/(a+a*sec(c+d*x))^3,x,3,1/2/(a^3*d*(1+cos(c+d*x))^2)+(-2)/(a^3*d*(1+cos(c+d*x)))-log(1+cos(c+d*x))/(a^3*d)],
[cot(c+d*x)/(a+a*sec(c+d*x))^3,x,3,1/6/(a^3*d*(1+cos(c+d*x))^3)+(-7/8)/(a^3*d*(1+cos(c+d*x))^2)+17/8/(a^3*d*(1+cos(c+d*x)))+1/16*log(1-cos(c+d*x))/(a^3*d)+15/16*log(1+cos(c+d*x))/(a^3*d)],
[cot(c+d*x)^3/(a+a*sec(c+d*x))^3,x,3,(-1/32)/(a^3*d*(1-cos(c+d*x)))+1/16/(a^3*d*(1+cos(c+d*x))^4)+(-5/12)/(a^3*d*(1+cos(c+d*x))^3)+39/32/(a^3*d*(1+cos(c+d*x))^2)+(-9/4)/(a^3*d*(1+cos(c+d*x)))-7/64*log(1-cos(c+d*x))/(a^3*d)-57/64*log(1+cos(c+d*x))/(a^3*d)],
[cot(c+d*x)^5/(a+a*sec(c+d*x))^3,x,3,(-1/128)/(a^3*d*(1-cos(c+d*x))^2)+5/64/(a^3*d*(1-cos(c+d*x)))+1/40/(a^3*d*(1+cos(c+d*x))^5)+(-13/64)/(a^3*d*(1+cos(c+d*x))^4)+35/48/(a^3*d*(1+cos(c+d*x))^3)+(-99/64)/(a^3*d*(1+cos(c+d*x))^2)+303/128/(a^3*d*(1+cos(c+d*x)))+37/256*log(1-cos(c+d*x))/(a^3*d)+219/256*log(1+cos(c+d*x))/(a^3*d)],
[tan(c+d*x)^12/(a+a*sec(c+d*x))^3,x,18,x/a^3-125/128*arctanh(sin(c+d*x))/(a^3*d)-tan(c+d*x)/(a^3*d)+115/128*sec(c+d*x)*tan(c+d*x)/(a^3*d)+5/64*sec(c+d*x)^3*tan(c+d*x)/(a^3*d)+1/3*tan(c+d*x)^3/(a^3*d)-5/8*sec(c+d*x)*tan(c+d*x)^3/(a^3*d)-5/48*sec(c+d*x)^3*tan(c+d*x)^3/(a^3*d)-1/5*tan(c+d*x)^5/(a^3*d)+1/2*sec(c+d*x)*tan(c+d*x)^5/(a^3*d)+1/8*sec(c+d*x)^3*tan(c+d*x)^5/(a^3*d)-3/7*tan(c+d*x)^7/(a^3*d)],
[tan(c+d*x)^10/(a+a*sec(c+d*x))^3,x,15,-x/a^3+19/16*arctanh(sin(c+d*x))/(a^3*d)+tan(c+d*x)/(a^3*d)-17/16*sec(c+d*x)*tan(c+d*x)/(a^3*d)-1/8*sec(c+d*x)^3*tan(c+d*x)/(a^3*d)-1/3*tan(c+d*x)^3/(a^3*d)+3/4*sec(c+d*x)*tan(c+d*x)^3/(a^3*d)+1/6*sec(c+d*x)^3*tan(c+d*x)^3/(a^3*d)-3/5*tan(c+d*x)^5/(a^3*d)],
[tan(c+d*x)^8/(a+a*sec(c+d*x))^3,x,12,x/a^3-13/8*arctanh(sin(c+d*x))/(a^3*d)-tan(c+d*x)/(a^3*d)+11/8*sec(c+d*x)*tan(c+d*x)/(a^3*d)+1/4*sec(c+d*x)^3*tan(c+d*x)/(a^3*d)-tan(c+d*x)^3/(a^3*d)],
[tan(c+d*x)^6/(a+a*sec(c+d*x))^3,x,6,-x/a^3+7/2*arctanh(sin(c+d*x))/(a^3*d)-5/2*tan(c+d*x)/(a^3*d)-1/2*(1-sec(c+d*x))*tan(c+d*x)/(a^3*d)],
[tan(c+d*x)^4/(a+a*sec(c+d*x))^3,x,12,x/a^3+arctanh(sin(c+d*x))/(a^3*d)-4*tan(c+d*x)/(a^2*d*(a+a*sec(c+d*x))),x/a^3+arctanh(sin(c+d*x))/(a^3*d)+4*cot(c+d*x)/(a^3*d)-4*csc(c+d*x)/(a^3*d)],
[tan(c+d*x)^2/(a+a*sec(c+d*x))^3,x,12,-x/a^3+2*tan(c+d*x)/(a^2*d*(a+a*sec(c+d*x)))-1/3*tan(c+d*x)^3/(d*(a+a*sec(c+d*x))^3),-x/a^3-cot(c+d*x)/(a^3*d)+4/3*cot(c+d*x)^3/(a^3*d)+3*csc(c+d*x)/(a^3*d)-4/3*csc(c+d*x)^3/(a^3*d)],
[cot(c+d*x)^2/(a+a*sec(c+d*x))^3,x,16,-x/a^3-cot(c+d*x)/(a^3*d)+1/3*cot(c+d*x)^3/(a^3*d)-1/5*cot(c+d*x)^5/(a^3*d)+4/7*cot(c+d*x)^7/(a^3*d)+3*csc(c+d*x)/(a^3*d)-10/3*csc(c+d*x)^3/(a^3*d)+11/5*csc(c+d*x)^5/(a^3*d)-4/7*csc(c+d*x)^7/(a^3*d)],
[cot(c+d*x)^4/(a+a*sec(c+d*x))^3,x,17,x/a^3+cot(c+d*x)/(a^3*d)-1/3*cot(c+d*x)^3/(a^3*d)+1/5*cot(c+d*x)^5/(a^3*d)-1/7*cot(c+d*x)^7/(a^3*d)+4/9*cot(c+d*x)^9/(a^3*d)-3*csc(c+d*x)/(a^3*d)+13/3*csc(c+d*x)^3/(a^3*d)-21/5*csc(c+d*x)^5/(a^3*d)+15/7*csc(c+d*x)^7/(a^3*d)-4/9*csc(c+d*x)^9/(a^3*d)],
[cot(c+d*x)^6/(a+a*sec(c+d*x))^3,x,18,-x/a^3-cot(c+d*x)/(a^3*d)+1/3*cot(c+d*x)^3/(a^3*d)-1/5*cot(c+d*x)^5/(a^3*d)+1/7*cot(c+d*x)^7/(a^3*d)-1/9*cot(c+d*x)^9/(a^3*d)+4/11*cot(c+d*x)^11/(a^3*d)+3*csc(c+d*x)/(a^3*d)-16/3*csc(c+d*x)^3/(a^3*d)+34/5*csc(c+d*x)^5/(a^3*d)-36/7*csc(c+d*x)^7/(a^3*d)+19/9*csc(c+d*x)^9/(a^3*d)-4/11*csc(c+d*x)^11/(a^3*d)],

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

# n>0
[(a+a*sec(c+d*x))*(e*tan(c+d*x))^(5/2),x,17,a*e^(5/2)*arctan(1-sqrt(2)*sqrt(e*tan(c+d*x))/sqrt(e))/(d*sqrt(2))-a*e^(5/2)*arctan(1+sqrt(2)*sqrt(e*tan(c+d*x))/sqrt(e))/(d*sqrt(2))-1/2*a*e^(5/2)*log(sqrt(e)-sqrt(2)*sqrt(e*tan(c+d*x))+sqrt(e)*tan(c+d*x))/(d*sqrt(2))+1/2*a*e^(5/2)*log(sqrt(e)+sqrt(2)*sqrt(e*tan(c+d*x))+sqrt(e)*tan(c+d*x))/(d*sqrt(2))+6/5*a*e^2*cos(c+d*x)*sqrt(cos(c-1/4*Pi+d*x)^2)/cos(c-1/4*Pi+d*x)*EllipticE(sin(c-1/4*Pi+d*x),sqrt(2))*sqrt(e*tan(c+d*x))/(d*sqrt(sin(2*c+2*d*x)))-6/5*a*e*cos(c+d*x)*(e*tan(c+d*x))^(3/2)/d+2/15*e*(5*a+3*a*sec(c+d*x))*(e*tan(c+d*x))^(3/2)/d],
[(a+a*sec(c+d*x))*(e*tan(c+d*x))^(3/2),x,16,a*e^(3/2)*arctan(1-sqrt(2)*sqrt(e*tan(c+d*x))/sqrt(e))/(d*sqrt(2))-a*e^(3/2)*arctan(1+sqrt(2)*sqrt(e*tan(c+d*x))/sqrt(e))/(d*sqrt(2))+1/2*a*e^(3/2)*log(sqrt(e)-sqrt(2)*sqrt(e*tan(c+d*x))+sqrt(e)*tan(c+d*x))/(d*sqrt(2))-1/2*a*e^(3/2)*log(sqrt(e)+sqrt(2)*sqrt(e*tan(c+d*x))+sqrt(e)*tan(c+d*x))/(d*sqrt(2))-1/3*a*e^2*sqrt(cos(c-1/4*Pi+d*x)^2)/cos(c-1/4*Pi+d*x)*EllipticF(sin(c-1/4*Pi+d*x),sqrt(2))*sec(c+d*x)*sqrt(sin(2*c+2*d*x))/(d*sqrt(e*tan(c+d*x)))+2/3*e*(3*a+a*sec(c+d*x))*sqrt(e*tan(c+d*x))/d],
[(a+a*sec(c+d*x))*sqrt(e*tan(c+d*x)),x,16,-a*arctan(1-sqrt(2)*sqrt(e*tan(c+d*x))/sqrt(e))*sqrt(e)/(d*sqrt(2))+a*arctan(1+sqrt(2)*sqrt(e*tan(c+d*x))/sqrt(e))*sqrt(e)/(d*sqrt(2))+1/2*a*log(sqrt(e)-sqrt(2)*sqrt(e*tan(c+d*x))+sqrt(e)*tan(c+d*x))*sqrt(e)/(d*sqrt(2))-1/2*a*log(sqrt(e)+sqrt(2)*sqrt(e*tan(c+d*x))+sqrt(e)*tan(c+d*x))*sqrt(e)/(d*sqrt(2))-2*a*cos(c+d*x)*sqrt(cos(c-1/4*Pi+d*x)^2)/cos(c-1/4*Pi+d*x)*EllipticE(sin(c-1/4*Pi+d*x),sqrt(2))*sqrt(e*tan(c+d*x))/(d*sqrt(sin(2*c+2*d*x)))+2*a*cos(c+d*x)*(e*tan(c+d*x))^(3/2)/(d*e)],
[(a+a*sec(c+d*x))/sqrt(e*tan(c+d*x)),x,15,-a*arctan(1-sqrt(2)*sqrt(e*tan(c+d*x))/sqrt(e))/(d*sqrt(2)*sqrt(e))+a*arctan(1+sqrt(2)*sqrt(e*tan(c+d*x))/sqrt(e))/(d*sqrt(2)*sqrt(e))-1/2*a*log(sqrt(e)-sqrt(2)*sqrt(e*tan(c+d*x))+sqrt(e)*tan(c+d*x))/(d*sqrt(2)*sqrt(e))+1/2*a*log(sqrt(e)+sqrt(2)*sqrt(e*tan(c+d*x))+sqrt(e)*tan(c+d*x))/(d*sqrt(2)*sqrt(e))+a*sqrt(cos(c-1/4*Pi+d*x)^2)/cos(c-1/4*Pi+d*x)*EllipticF(sin(c-1/4*Pi+d*x),sqrt(2))*sec(c+d*x)*sqrt(sin(2*c+2*d*x))/(d*sqrt(e*tan(c+d*x)))],
[(a+a*sec(c+d*x))/(e*tan(c+d*x))^(3/2),x,17,a*arctan(1-sqrt(2)*sqrt(e*tan(c+d*x))/sqrt(e))/(d*e^(3/2)*sqrt(2))-a*arctan(1+sqrt(2)*sqrt(e*tan(c+d*x))/sqrt(e))/(d*e^(3/2)*sqrt(2))-1/2*a*log(sqrt(e)-sqrt(2)*sqrt(e*tan(c+d*x))+sqrt(e)*tan(c+d*x))/(d*e^(3/2)*sqrt(2))+1/2*a*log(sqrt(e)+sqrt(2)*sqrt(e*tan(c+d*x))+sqrt(e)*tan(c+d*x))/(d*e^(3/2)*sqrt(2))-2*(a+a*sec(c+d*x))/(d*e*sqrt(e*tan(c+d*x)))-2*a*cos(c+d*x)*sqrt(cos(c-1/4*Pi+d*x)^2)/cos(c-1/4*Pi+d*x)*EllipticE(sin(c-1/4*Pi+d*x),sqrt(2))*sqrt(e*tan(c+d*x))/(d*e^2*sqrt(sin(2*c+2*d*x)))+2*a*cos(c+d*x)*(e*tan(c+d*x))^(3/2)/(d*e^3)],
[(a+a*sec(c+d*x))/(e*tan(c+d*x))^(5/2),x,16,a*arctan(1-sqrt(2)*sqrt(e*tan(c+d*x))/sqrt(e))/(d*e^(5/2)*sqrt(2))-a*arctan(1+sqrt(2)*sqrt(e*tan(c+d*x))/sqrt(e))/(d*e^(5/2)*sqrt(2))+1/2*a*log(sqrt(e)-sqrt(2)*sqrt(e*tan(c+d*x))+sqrt(e)*tan(c+d*x))/(d*e^(5/2)*sqrt(2))-1/2*a*log(sqrt(e)+sqrt(2)*sqrt(e*tan(c+d*x))+sqrt(e)*tan(c+d*x))/(d*e^(5/2)*sqrt(2))-1/3*a*sqrt(cos(c-1/4*Pi+d*x)^2)/cos(c-1/4*Pi+d*x)*EllipticF(sin(c-1/4*Pi+d*x),sqrt(2))*sec(c+d*x)*sqrt(sin(2*c+2*d*x))/(d*e^2*sqrt(e*tan(c+d*x)))-2/3*(a+a*sec(c+d*x))/(d*e*(e*tan(c+d*x))^(3/2))],
[(a+a*sec(c+d*x))/(e*tan(c+d*x))^(7/2),x,18,-a*arctan(1-sqrt(2)*sqrt(e*tan(c+d*x))/sqrt(e))/(d*e^(7/2)*sqrt(2))+a*arctan(1+sqrt(2)*sqrt(e*tan(c+d*x))/sqrt(e))/(d*e^(7/2)*sqrt(2))+1/2*a*log(sqrt(e)-sqrt(2)*sqrt(e*tan(c+d*x))+sqrt(e)*tan(c+d*x))/(d*e^(7/2)*sqrt(2))-1/2*a*log(sqrt(e)+sqrt(2)*sqrt(e*tan(c+d*x))+sqrt(e)*tan(c+d*x))/(d*e^(7/2)*sqrt(2))+2/5*(5*a+3*a*sec(c+d*x))/(d*e^3*sqrt(e*tan(c+d*x)))+6/5*a*cos(c+d*x)*sqrt(cos(c-1/4*Pi+d*x)^2)/cos(c-1/4*Pi+d*x)*EllipticE(sin(c-1/4*Pi+d*x),sqrt(2))*sqrt(e*tan(c+d*x))/(d*e^4*sqrt(sin(2*c+2*d*x)))-2/5*(a+a*sec(c+d*x))/(d*e*(e*tan(c+d*x))^(5/2))-6/5*a*cos(c+d*x)*(e*tan(c+d*x))^(3/2)/(d*e^5)],
[(a+a*sec(c+d*x))^2*(e*tan(c+d*x))^(5/2),x,21,a^2*e^(5/2)*arctan(1-sqrt(2)*sqrt(e*tan(c+d*x))/sqrt(e))/(d*sqrt(2))-a^2*e^(5/2)*arctan(1+sqrt(2)*sqrt(e*tan(c+d*x))/sqrt(e))/(d*sqrt(2))-1/2*a^2*e^(5/2)*log(sqrt(e)-sqrt(2)*sqrt(e*tan(c+d*x))+sqrt(e)*tan(c+d*x))/(d*sqrt(2))+1/2*a^2*e^(5/2)*log(sqrt(e)+sqrt(2)*sqrt(e*tan(c+d*x))+sqrt(e)*tan(c+d*x))/(d*sqrt(2))+12/5*a^2*e^2*cos(c+d*x)*sqrt(cos(c-1/4*Pi+d*x)^2)/cos(c-1/4*Pi+d*x)*EllipticE(sin(c-1/4*Pi+d*x),sqrt(2))*sqrt(e*tan(c+d*x))/(d*sqrt(sin(2*c+2*d*x)))+2/3*a^2*e*(e*tan(c+d*x))^(3/2)/d-12/5*a^2*e*cos(c+d*x)*(e*tan(c+d*x))^(3/2)/d+4/5*a^2*e*sec(c+d*x)*(e*tan(c+d*x))^(3/2)/d+2/7*a^2*(e*tan(c+d*x))^(7/2)/(d*e)],
[(a+a*sec(c+d*x))^2*(e*tan(c+d*x))^(3/2),x,20,a^2*e^(3/2)*arctan(1-sqrt(2)*sqrt(e*tan(c+d*x))/sqrt(e))/(d*sqrt(2))-a^2*e^(3/2)*arctan(1+sqrt(2)*sqrt(e*tan(c+d*x))/sqrt(e))/(d*sqrt(2))+1/2*a^2*e^(3/2)*log(sqrt(e)-sqrt(2)*sqrt(e*tan(c+d*x))+sqrt(e)*tan(c+d*x))/(d*sqrt(2))-1/2*a^2*e^(3/2)*log(sqrt(e)+sqrt(2)*sqrt(e*tan(c+d*x))+sqrt(e)*tan(c+d*x))/(d*sqrt(2))-2/3*a^2*e^2*sqrt(cos(c-1/4*Pi+d*x)^2)/cos(c-1/4*Pi+d*x)*EllipticF(sin(c-1/4*Pi+d*x),sqrt(2))*sec(c+d*x)*sqrt(sin(2*c+2*d*x))/(d*sqrt(e*tan(c+d*x)))+2*a^2*e*sqrt(e*tan(c+d*x))/d+4/3*a^2*e*sec(c+d*x)*sqrt(e*tan(c+d*x))/d+2/5*a^2*(e*tan(c+d*x))^(5/2)/(d*e)],
[(a+a*sec(c+d*x))^2*sqrt(e*tan(c+d*x)),x,19,-a^2*arctan(1-sqrt(2)*sqrt(e*tan(c+d*x))/sqrt(e))*sqrt(e)/(d*sqrt(2))+a^2*arctan(1+sqrt(2)*sqrt(e*tan(c+d*x))/sqrt(e))*sqrt(e)/(d*sqrt(2))+1/2*a^2*log(sqrt(e)-sqrt(2)*sqrt(e*tan(c+d*x))+sqrt(e)*tan(c+d*x))*sqrt(e)/(d*sqrt(2))-1/2*a^2*log(sqrt(e)+sqrt(2)*sqrt(e*tan(c+d*x))+sqrt(e)*tan(c+d*x))*sqrt(e)/(d*sqrt(2))-4*a^2*cos(c+d*x)*sqrt(cos(c-1/4*Pi+d*x)^2)/cos(c-1/4*Pi+d*x)*EllipticE(sin(c-1/4*Pi+d*x),sqrt(2))*sqrt(e*tan(c+d*x))/(d*sqrt(sin(2*c+2*d*x)))+2/3*a^2*(e*tan(c+d*x))^(3/2)/(d*e)+4*a^2*cos(c+d*x)*(e*tan(c+d*x))^(3/2)/(d*e)],
[(a+a*sec(c+d*x))^2/sqrt(e*tan(c+d*x)),x,18,-a^2*arctan(1-sqrt(2)*sqrt(e*tan(c+d*x))/sqrt(e))/(d*sqrt(2)*sqrt(e))+a^2*arctan(1+sqrt(2)*sqrt(e*tan(c+d*x))/sqrt(e))/(d*sqrt(2)*sqrt(e))-1/2*a^2*log(sqrt(e)-sqrt(2)*sqrt(e*tan(c+d*x))+sqrt(e)*tan(c+d*x))/(d*sqrt(2)*sqrt(e))+1/2*a^2*log(sqrt(e)+sqrt(2)*sqrt(e*tan(c+d*x))+sqrt(e)*tan(c+d*x))/(d*sqrt(2)*sqrt(e))+2*a^2*sqrt(cos(c-1/4*Pi+d*x)^2)/cos(c-1/4*Pi+d*x)*EllipticF(sin(c-1/4*Pi+d*x),sqrt(2))*sec(c+d*x)*sqrt(sin(2*c+2*d*x))/(d*sqrt(e*tan(c+d*x)))+2*a^2*sqrt(e*tan(c+d*x))/(d*e)],
[(a+a*sec(c+d*x))^2/(e*tan(c+d*x))^(3/2),x,20,a^2*arctan(1-sqrt(2)*sqrt(e*tan(c+d*x))/sqrt(e))/(d*e^(3/2)*sqrt(2))-a^2*arctan(1+sqrt(2)*sqrt(e*tan(c+d*x))/sqrt(e))/(d*e^(3/2)*sqrt(2))-1/2*a^2*log(sqrt(e)-sqrt(2)*sqrt(e*tan(c+d*x))+sqrt(e)*tan(c+d*x))/(d*e^(3/2)*sqrt(2))+1/2*a^2*log(sqrt(e)+sqrt(2)*sqrt(e*tan(c+d*x))+sqrt(e)*tan(c+d*x))/(d*e^(3/2)*sqrt(2))-4*a^2/(d*e*sqrt(e*tan(c+d*x)))-4*a^2*cos(c+d*x)/(d*e*sqrt(e*tan(c+d*x)))-4*a^2*cos(c+d*x)*sqrt(cos(c-1/4*Pi+d*x)^2)/cos(c-1/4*Pi+d*x)*EllipticE(sin(c-1/4*Pi+d*x),sqrt(2))*sqrt(e*tan(c+d*x))/(d*e^2*sqrt(sin(2*c+2*d*x)))],
[(a+a*sec(c+d*x))^2/(e*tan(c+d*x))^(5/2),x,20,a^2*arctan(1-sqrt(2)*sqrt(e*tan(c+d*x))/sqrt(e))/(d*e^(5/2)*sqrt(2))-a^2*arctan(1+sqrt(2)*sqrt(e*tan(c+d*x))/sqrt(e))/(d*e^(5/2)*sqrt(2))+1/2*a^2*log(sqrt(e)-sqrt(2)*sqrt(e*tan(c+d*x))+sqrt(e)*tan(c+d*x))/(d*e^(5/2)*sqrt(2))-1/2*a^2*log(sqrt(e)+sqrt(2)*sqrt(e*tan(c+d*x))+sqrt(e)*tan(c+d*x))/(d*e^(5/2)*sqrt(2))-2/3*a^2*sqrt(cos(c-1/4*Pi+d*x)^2)/cos(c-1/4*Pi+d*x)*EllipticF(sin(c-1/4*Pi+d*x),sqrt(2))*sec(c+d*x)*sqrt(sin(2*c+2*d*x))/(d*e^2*sqrt(e*tan(c+d*x)))-4/3*a^2/(d*e*(e*tan(c+d*x))^(3/2))-4/3*a^2*sec(c+d*x)/(d*e*(e*tan(c+d*x))^(3/2))],
[(a+a*sec(c+d*x))^2/(e*tan(c+d*x))^(7/2),x,22,-a^2*arctan(1-sqrt(2)*sqrt(e*tan(c+d*x))/sqrt(e))/(d*e^(7/2)*sqrt(2))+a^2*arctan(1+sqrt(2)*sqrt(e*tan(c+d*x))/sqrt(e))/(d*e^(7/2)*sqrt(2))+1/2*a^2*log(sqrt(e)-sqrt(2)*sqrt(e*tan(c+d*x))+sqrt(e)*tan(c+d*x))/(d*e^(7/2)*sqrt(2))-1/2*a^2*log(sqrt(e)+sqrt(2)*sqrt(e*tan(c+d*x))+sqrt(e)*tan(c+d*x))/(d*e^(7/2)*sqrt(2))+2*a^2/(d*e^3*sqrt(e*tan(c+d*x)))+12/5*a^2*cos(c+d*x)/(d*e^3*sqrt(e*tan(c+d*x)))+12/5*a^2*cos(c+d*x)*sqrt(cos(c-1/4*Pi+d*x)^2)/cos(c-1/4*Pi+d*x)*EllipticE(sin(c-1/4*Pi+d*x),sqrt(2))*sqrt(e*tan(c+d*x))/(d*e^4*sqrt(sin(2*c+2*d*x)))-4/5*a^2/(d*e*(e*tan(c+d*x))^(5/2))-4/5*a^2*sec(c+d*x)/(d*e*(e*tan(c+d*x))^(5/2))],

# n<0
[(e*tan(c+d*x))^(11/2)/(a+a*sec(c+d*x)),x,18,e^(11/2)*arctan(1-sqrt(2)*sqrt(e*tan(c+d*x))/sqrt(e))/(a*d*sqrt(2))-e^(11/2)*arctan(1+sqrt(2)*sqrt(e*tan(c+d*x))/sqrt(e))/(a*d*sqrt(2))+1/2*e^(11/2)*log(sqrt(e)-sqrt(2)*sqrt(e*tan(c+d*x))+sqrt(e)*tan(c+d*x))/(a*d*sqrt(2))-1/2*e^(11/2)*log(sqrt(e)+sqrt(2)*sqrt(e*tan(c+d*x))+sqrt(e)*tan(c+d*x))/(a*d*sqrt(2))+5/21*e^6*sqrt(cos(c-1/4*Pi+d*x)^2)/cos(c-1/4*Pi+d*x)*EllipticF(sin(c-1/4*Pi+d*x),sqrt(2))*sec(c+d*x)*sqrt(sin(2*c+2*d*x))/(a*d*sqrt(e*tan(c+d*x)))+2/21*e^5*(21-5*sec(c+d*x))*sqrt(e*tan(c+d*x))/(a*d)-2/35*e^3*(7-5*sec(c+d*x))*(e*tan(c+d*x))^(5/2)/(a*d)],
[(e*tan(c+d*x))^(9/2)/(a+a*sec(c+d*x)),x,18,-e^(9/2)*arctan(1-sqrt(2)*sqrt(e*tan(c+d*x))/sqrt(e))/(a*d*sqrt(2))+e^(9/2)*arctan(1+sqrt(2)*sqrt(e*tan(c+d*x))/sqrt(e))/(a*d*sqrt(2))+1/2*e^(9/2)*log(sqrt(e)-sqrt(2)*sqrt(e*tan(c+d*x))+sqrt(e)*tan(c+d*x))/(a*d*sqrt(2))-1/2*e^(9/2)*log(sqrt(e)+sqrt(2)*sqrt(e*tan(c+d*x))+sqrt(e)*tan(c+d*x))/(a*d*sqrt(2))+6/5*e^4*cos(c+d*x)*sqrt(cos(c-1/4*Pi+d*x)^2)/cos(c-1/4*Pi+d*x)*EllipticE(sin(c-1/4*Pi+d*x),sqrt(2))*sqrt(e*tan(c+d*x))/(a*d*sqrt(sin(2*c+2*d*x)))-6/5*e^3*cos(c+d*x)*(e*tan(c+d*x))^(3/2)/(a*d)-2/15*e^3*(5-3*sec(c+d*x))*(e*tan(c+d*x))^(3/2)/(a*d)],
[(e*tan(c+d*x))^(7/2)/(a+a*sec(c+d*x)),x,17,-e^(7/2)*arctan(1-sqrt(2)*sqrt(e*tan(c+d*x))/sqrt(e))/(a*d*sqrt(2))+e^(7/2)*arctan(1+sqrt(2)*sqrt(e*tan(c+d*x))/sqrt(e))/(a*d*sqrt(2))-1/2*e^(7/2)*log(sqrt(e)-sqrt(2)*sqrt(e*tan(c+d*x))+sqrt(e)*tan(c+d*x))/(a*d*sqrt(2))+1/2*e^(7/2)*log(sqrt(e)+sqrt(2)*sqrt(e*tan(c+d*x))+sqrt(e)*tan(c+d*x))/(a*d*sqrt(2))-1/3*e^4*sqrt(cos(c-1/4*Pi+d*x)^2)/cos(c-1/4*Pi+d*x)*EllipticF(sin(c-1/4*Pi+d*x),sqrt(2))*sec(c+d*x)*sqrt(sin(2*c+2*d*x))/(a*d*sqrt(e*tan(c+d*x)))-2/3*e^3*(3-sec(c+d*x))*sqrt(e*tan(c+d*x))/(a*d)],
[(e*tan(c+d*x))^(5/2)/(a+a*sec(c+d*x)),x,17,e^(5/2)*arctan(1-sqrt(2)*sqrt(e*tan(c+d*x))/sqrt(e))/(a*d*sqrt(2))-e^(5/2)*arctan(1+sqrt(2)*sqrt(e*tan(c+d*x))/sqrt(e))/(a*d*sqrt(2))-1/2*e^(5/2)*log(sqrt(e)-sqrt(2)*sqrt(e*tan(c+d*x))+sqrt(e)*tan(c+d*x))/(a*d*sqrt(2))+1/2*e^(5/2)*log(sqrt(e)+sqrt(2)*sqrt(e*tan(c+d*x))+sqrt(e)*tan(c+d*x))/(a*d*sqrt(2))-2*e^2*cos(c+d*x)*sqrt(cos(c-1/4*Pi+d*x)^2)/cos(c-1/4*Pi+d*x)*EllipticE(sin(c-1/4*Pi+d*x),sqrt(2))*sqrt(e*tan(c+d*x))/(a*d*sqrt(sin(2*c+2*d*x)))+2*e*cos(c+d*x)*(e*tan(c+d*x))^(3/2)/(a*d)],
[(e*tan(c+d*x))^(3/2)/(a+a*sec(c+d*x)),x,16,e^(3/2)*arctan(1-sqrt(2)*sqrt(e*tan(c+d*x))/sqrt(e))/(a*d*sqrt(2))-e^(3/2)*arctan(1+sqrt(2)*sqrt(e*tan(c+d*x))/sqrt(e))/(a*d*sqrt(2))+1/2*e^(3/2)*log(sqrt(e)-sqrt(2)*sqrt(e*tan(c+d*x))+sqrt(e)*tan(c+d*x))/(a*d*sqrt(2))-1/2*e^(3/2)*log(sqrt(e)+sqrt(2)*sqrt(e*tan(c+d*x))+sqrt(e)*tan(c+d*x))/(a*d*sqrt(2))+e^2*sqrt(cos(c-1/4*Pi+d*x)^2)/cos(c-1/4*Pi+d*x)*EllipticF(sin(c-1/4*Pi+d*x),sqrt(2))*sec(c+d*x)*sqrt(sin(2*c+2*d*x))/(a*d*sqrt(e*tan(c+d*x)))],
[sqrt(e*tan(c+d*x))/(a+a*sec(c+d*x)),x,18,-arctan(1-sqrt(2)*sqrt(e*tan(c+d*x))/sqrt(e))*sqrt(e)/(a*d*sqrt(2))+arctan(1+sqrt(2)*sqrt(e*tan(c+d*x))/sqrt(e))*sqrt(e)/(a*d*sqrt(2))+1/2*log(sqrt(e)-sqrt(2)*sqrt(e*tan(c+d*x))+sqrt(e)*tan(c+d*x))*sqrt(e)/(a*d*sqrt(2))-1/2*log(sqrt(e)+sqrt(2)*sqrt(e*tan(c+d*x))+sqrt(e)*tan(c+d*x))*sqrt(e)/(a*d*sqrt(2))+2*e*(1-sec(c+d*x))/(a*d*sqrt(e*tan(c+d*x)))-2*cos(c+d*x)*sqrt(cos(c-1/4*Pi+d*x)^2)/cos(c-1/4*Pi+d*x)*EllipticE(sin(c-1/4*Pi+d*x),sqrt(2))*sqrt(e*tan(c+d*x))/(a*d*sqrt(sin(2*c+2*d*x)))+2*cos(c+d*x)*(e*tan(c+d*x))^(3/2)/(a*d*e)],
[1/((a+a*sec(c+d*x))*sqrt(e*tan(c+d*x))),x,17,-arctan(1-sqrt(2)*sqrt(e*tan(c+d*x))/sqrt(e))/(a*d*sqrt(2)*sqrt(e))+arctan(1+sqrt(2)*sqrt(e*tan(c+d*x))/sqrt(e))/(a*d*sqrt(2)*sqrt(e))-1/2*log(sqrt(e)-sqrt(2)*sqrt(e*tan(c+d*x))+sqrt(e)*tan(c+d*x))/(a*d*sqrt(2)*sqrt(e))+1/2*log(sqrt(e)+sqrt(2)*sqrt(e*tan(c+d*x))+sqrt(e)*tan(c+d*x))/(a*d*sqrt(2)*sqrt(e))-1/3*sqrt(cos(c-1/4*Pi+d*x)^2)/cos(c-1/4*Pi+d*x)*EllipticF(sin(c-1/4*Pi+d*x),sqrt(2))*sec(c+d*x)*sqrt(sin(2*c+2*d*x))/(a*d*sqrt(e*tan(c+d*x)))+2/3*e*(1-sec(c+d*x))/(a*d*(e*tan(c+d*x))^(3/2))],
[1/((a+a*sec(c+d*x))*(e*tan(c+d*x))^(3/2)),x,19,arctan(1-sqrt(2)*sqrt(e*tan(c+d*x))/sqrt(e))/(a*d*e^(3/2)*sqrt(2))-arctan(1+sqrt(2)*sqrt(e*tan(c+d*x))/sqrt(e))/(a*d*e^(3/2)*sqrt(2))-1/2*log(sqrt(e)-sqrt(2)*sqrt(e*tan(c+d*x))+sqrt(e)*tan(c+d*x))/(a*d*e^(3/2)*sqrt(2))+1/2*log(sqrt(e)+sqrt(2)*sqrt(e*tan(c+d*x))+sqrt(e)*tan(c+d*x))/(a*d*e^(3/2)*sqrt(2))-2/5*(5-3*sec(c+d*x))/(a*d*e*sqrt(e*tan(c+d*x)))+6/5*cos(c+d*x)*sqrt(cos(c-1/4*Pi+d*x)^2)/cos(c-1/4*Pi+d*x)*EllipticE(sin(c-1/4*Pi+d*x),sqrt(2))*sqrt(e*tan(c+d*x))/(a*d*e^2*sqrt(sin(2*c+2*d*x)))+2/5*e*(1-sec(c+d*x))/(a*d*(e*tan(c+d*x))^(5/2))-6/5*cos(c+d*x)*(e*tan(c+d*x))^(3/2)/(a*d*e^3)],
[1/((a+a*sec(c+d*x))*(e*tan(c+d*x))^(5/2)),x,18,arctan(1-sqrt(2)*sqrt(e*tan(c+d*x))/sqrt(e))/(a*d*e^(5/2)*sqrt(2))-arctan(1+sqrt(2)*sqrt(e*tan(c+d*x))/sqrt(e))/(a*d*e^(5/2)*sqrt(2))+1/2*log(sqrt(e)-sqrt(2)*sqrt(e*tan(c+d*x))+sqrt(e)*tan(c+d*x))/(a*d*e^(5/2)*sqrt(2))-1/2*log(sqrt(e)+sqrt(2)*sqrt(e*tan(c+d*x))+sqrt(e)*tan(c+d*x))/(a*d*e^(5/2)*sqrt(2))+5/21*sqrt(cos(c-1/4*Pi+d*x)^2)/cos(c-1/4*Pi+d*x)*EllipticF(sin(c-1/4*Pi+d*x),sqrt(2))*sec(c+d*x)*sqrt(sin(2*c+2*d*x))/(a*d*e^2*sqrt(e*tan(c+d*x)))+2/7*e*(1-sec(c+d*x))/(a*d*(e*tan(c+d*x))^(7/2))-2/21*(7-5*sec(c+d*x))/(a*d*e*(e*tan(c+d*x))^(3/2))],
[(e*tan(c+d*x))^(13/2)/(a+a*sec(c+d*x))^2,x,22,e^(13/2)*arctan(1-sqrt(2)*sqrt(e*tan(c+d*x))/sqrt(e))/(a^2*d*sqrt(2))-e^(13/2)*arctan(1+sqrt(2)*sqrt(e*tan(c+d*x))/sqrt(e))/(a^2*d*sqrt(2))-1/2*e^(13/2)*log(sqrt(e)-sqrt(2)*sqrt(e*tan(c+d*x))+sqrt(e)*tan(c+d*x))/(a^2*d*sqrt(2))+1/2*e^(13/2)*log(sqrt(e)+sqrt(2)*sqrt(e*tan(c+d*x))+sqrt(e)*tan(c+d*x))/(a^2*d*sqrt(2))-12/5*e^6*cos(c+d*x)*sqrt(cos(c-1/4*Pi+d*x)^2)/cos(c-1/4*Pi+d*x)*EllipticE(sin(c-1/4*Pi+d*x),sqrt(2))*sqrt(e*tan(c+d*x))/(a^2*d*sqrt(sin(2*c+2*d*x)))+2/3*e^5*(e*tan(c+d*x))^(3/2)/(a^2*d)+12/5*e^5*cos(c+d*x)*(e*tan(c+d*x))^(3/2)/(a^2*d)-4/5*e^5*sec(c+d*x)*(e*tan(c+d*x))^(3/2)/(a^2*d)+2/7*e^3*(e*tan(c+d*x))^(7/2)/(a^2*d)],
[(e*tan(c+d*x))^(11/2)/(a+a*sec(c+d*x))^2,x,21,e^(11/2)*arctan(1-sqrt(2)*sqrt(e*tan(c+d*x))/sqrt(e))/(a^2*d*sqrt(2))-e^(11/2)*arctan(1+sqrt(2)*sqrt(e*tan(c+d*x))/sqrt(e))/(a^2*d*sqrt(2))+1/2*e^(11/2)*log(sqrt(e)-sqrt(2)*sqrt(e*tan(c+d*x))+sqrt(e)*tan(c+d*x))/(a^2*d*sqrt(2))-1/2*e^(11/2)*log(sqrt(e)+sqrt(2)*sqrt(e*tan(c+d*x))+sqrt(e)*tan(c+d*x))/(a^2*d*sqrt(2))+2/3*e^6*sqrt(cos(c-1/4*Pi+d*x)^2)/cos(c-1/4*Pi+d*x)*EllipticF(sin(c-1/4*Pi+d*x),sqrt(2))*sec(c+d*x)*sqrt(sin(2*c+2*d*x))/(a^2*d*sqrt(e*tan(c+d*x)))+2*e^5*sqrt(e*tan(c+d*x))/(a^2*d)-4/3*e^5*sec(c+d*x)*sqrt(e*tan(c+d*x))/(a^2*d)+2/5*e^3*(e*tan(c+d*x))^(5/2)/(a^2*d)],
[(e*tan(c+d*x))^(9/2)/(a+a*sec(c+d*x))^2,x,20,-e^(9/2)*arctan(1-sqrt(2)*sqrt(e*tan(c+d*x))/sqrt(e))/(a^2*d*sqrt(2))+e^(9/2)*arctan(1+sqrt(2)*sqrt(e*tan(c+d*x))/sqrt(e))/(a^2*d*sqrt(2))+1/2*e^(9/2)*log(sqrt(e)-sqrt(2)*sqrt(e*tan(c+d*x))+sqrt(e)*tan(c+d*x))/(a^2*d*sqrt(2))-1/2*e^(9/2)*log(sqrt(e)+sqrt(2)*sqrt(e*tan(c+d*x))+sqrt(e)*tan(c+d*x))/(a^2*d*sqrt(2))+4*e^4*cos(c+d*x)*sqrt(cos(c-1/4*Pi+d*x)^2)/cos(c-1/4*Pi+d*x)*EllipticE(sin(c-1/4*Pi+d*x),sqrt(2))*sqrt(e*tan(c+d*x))/(a^2*d*sqrt(sin(2*c+2*d*x)))+2/3*e^3*(e*tan(c+d*x))^(3/2)/(a^2*d)-4*e^3*cos(c+d*x)*(e*tan(c+d*x))^(3/2)/(a^2*d)],
[(e*tan(c+d*x))^(7/2)/(a+a*sec(c+d*x))^2,x,19,-e^(7/2)*arctan(1-sqrt(2)*sqrt(e*tan(c+d*x))/sqrt(e))/(a^2*d*sqrt(2))+e^(7/2)*arctan(1+sqrt(2)*sqrt(e*tan(c+d*x))/sqrt(e))/(a^2*d*sqrt(2))-1/2*e^(7/2)*log(sqrt(e)-sqrt(2)*sqrt(e*tan(c+d*x))+sqrt(e)*tan(c+d*x))/(a^2*d*sqrt(2))+1/2*e^(7/2)*log(sqrt(e)+sqrt(2)*sqrt(e*tan(c+d*x))+sqrt(e)*tan(c+d*x))/(a^2*d*sqrt(2))-2*e^4*sqrt(cos(c-1/4*Pi+d*x)^2)/cos(c-1/4*Pi+d*x)*EllipticF(sin(c-1/4*Pi+d*x),sqrt(2))*sec(c+d*x)*sqrt(sin(2*c+2*d*x))/(a^2*d*sqrt(e*tan(c+d*x)))+2*e^3*sqrt(e*tan(c+d*x))/(a^2*d)],
[(e*tan(c+d*x))^(5/2)/(a+a*sec(c+d*x))^2,x,21,e^(5/2)*arctan(1-sqrt(2)*sqrt(e*tan(c+d*x))/sqrt(e))/(a^2*d*sqrt(2))-e^(5/2)*arctan(1+sqrt(2)*sqrt(e*tan(c+d*x))/sqrt(e))/(a^2*d*sqrt(2))-1/2*e^(5/2)*log(sqrt(e)-sqrt(2)*sqrt(e*tan(c+d*x))+sqrt(e)*tan(c+d*x))/(a^2*d*sqrt(2))+1/2*e^(5/2)*log(sqrt(e)+sqrt(2)*sqrt(e*tan(c+d*x))+sqrt(e)*tan(c+d*x))/(a^2*d*sqrt(2))-4*e^3/(a^2*d*sqrt(e*tan(c+d*x)))+4*e^3*cos(c+d*x)/(a^2*d*sqrt(e*tan(c+d*x)))+4*e^2*cos(c+d*x)*sqrt(cos(c-1/4*Pi+d*x)^2)/cos(c-1/4*Pi+d*x)*EllipticE(sin(c-1/4*Pi+d*x),sqrt(2))*sqrt(e*tan(c+d*x))/(a^2*d*sqrt(sin(2*c+2*d*x)))],
[(e*tan(c+d*x))^(3/2)/(a+a*sec(c+d*x))^2,x,21,e^(3/2)*arctan(1-sqrt(2)*sqrt(e*tan(c+d*x))/sqrt(e))/(a^2*d*sqrt(2))-e^(3/2)*arctan(1+sqrt(2)*sqrt(e*tan(c+d*x))/sqrt(e))/(a^2*d*sqrt(2))+1/2*e^(3/2)*log(sqrt(e)-sqrt(2)*sqrt(e*tan(c+d*x))+sqrt(e)*tan(c+d*x))/(a^2*d*sqrt(2))-1/2*e^(3/2)*log(sqrt(e)+sqrt(2)*sqrt(e*tan(c+d*x))+sqrt(e)*tan(c+d*x))/(a^2*d*sqrt(2))+2/3*e^2*sqrt(cos(c-1/4*Pi+d*x)^2)/cos(c-1/4*Pi+d*x)*EllipticF(sin(c-1/4*Pi+d*x),sqrt(2))*sec(c+d*x)*sqrt(sin(2*c+2*d*x))/(a^2*d*sqrt(e*tan(c+d*x)))-4/3*e^3/(a^2*d*(e*tan(c+d*x))^(3/2))+4/3*e^3*sec(c+d*x)/(a^2*d*(e*tan(c+d*x))^(3/2))],
[sqrt(e*tan(c+d*x))/(a+a*sec(c+d*x))^2,x,23,-arctan(1-sqrt(2)*sqrt(e*tan(c+d*x))/sqrt(e))*sqrt(e)/(a^2*d*sqrt(2))+arctan(1+sqrt(2)*sqrt(e*tan(c+d*x))/sqrt(e))*sqrt(e)/(a^2*d*sqrt(2))+1/2*log(sqrt(e)-sqrt(2)*sqrt(e*tan(c+d*x))+sqrt(e)*tan(c+d*x))*sqrt(e)/(a^2*d*sqrt(2))-1/2*log(sqrt(e)+sqrt(2)*sqrt(e*tan(c+d*x))+sqrt(e)*tan(c+d*x))*sqrt(e)/(a^2*d*sqrt(2))+2*e/(a^2*d*sqrt(e*tan(c+d*x)))-12/5*e*cos(c+d*x)/(a^2*d*sqrt(e*tan(c+d*x)))-12/5*cos(c+d*x)*sqrt(cos(c-1/4*Pi+d*x)^2)/cos(c-1/4*Pi+d*x)*EllipticE(sin(c-1/4*Pi+d*x),sqrt(2))*sqrt(e*tan(c+d*x))/(a^2*d*sqrt(sin(2*c+2*d*x)))-4/5*e^3/(a^2*d*(e*tan(c+d*x))^(5/2))+4/5*e^3*sec(c+d*x)/(a^2*d*(e*tan(c+d*x))^(5/2))],
[1/((a+a*sec(c+d*x))^2*sqrt(e*tan(c+d*x))),x,23,-arctan(1-sqrt(2)*sqrt(e*tan(c+d*x))/sqrt(e))/(a^2*d*sqrt(2)*sqrt(e))+arctan(1+sqrt(2)*sqrt(e*tan(c+d*x))/sqrt(e))/(a^2*d*sqrt(2)*sqrt(e))-1/2*log(sqrt(e)-sqrt(2)*sqrt(e*tan(c+d*x))+sqrt(e)*tan(c+d*x))/(a^2*d*sqrt(2)*sqrt(e))+1/2*log(sqrt(e)+sqrt(2)*sqrt(e*tan(c+d*x))+sqrt(e)*tan(c+d*x))/(a^2*d*sqrt(2)*sqrt(e))-10/21*sqrt(cos(c-1/4*Pi+d*x)^2)/cos(c-1/4*Pi+d*x)*EllipticF(sin(c-1/4*Pi+d*x),sqrt(2))*sec(c+d*x)*sqrt(sin(2*c+2*d*x))/(a^2*d*sqrt(e*tan(c+d*x)))-4/7*e^3/(a^2*d*(e*tan(c+d*x))^(7/2))+4/7*e^3*sec(c+d*x)/(a^2*d*(e*tan(c+d*x))^(7/2))+2/3*e/(a^2*d*(e*tan(c+d*x))^(3/2))-20/21*e*sec(c+d*x)/(a^2*d*(e*tan(c+d*x))^(3/2))],

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

# n>0
[sqrt(a+a*sec(c+d*x))*tan(c+d*x)^5,x,8,2/3*(a+a*sec(c+d*x))^(3/2)/(a*d)+2/5*(a+a*sec(c+d*x))^(5/2)/(a^2*d)-6/7*(a+a*sec(c+d*x))^(7/2)/(a^3*d)+2/9*(a+a*sec(c+d*x))^(9/2)/(a^4*d)-2*arctanh(sqrt(a+a*sec(c+d*x))/sqrt(a))*sqrt(a)/d+2*sqrt(a+a*sec(c+d*x))/d],
[sqrt(a+a*sec(c+d*x))*tan(c+d*x)^3,x,6,-2/3*(a+a*sec(c+d*x))^(3/2)/(a*d)+2/5*(a+a*sec(c+d*x))^(5/2)/(a^2*d)+2*arctanh(sqrt(a+a*sec(c+d*x))/sqrt(a))*sqrt(a)/d-2*sqrt(a+a*sec(c+d*x))/d],
[sqrt(a+a*sec(c+d*x))*tan(c+d*x),x,4,-2*arctanh(sqrt(a+a*sec(c+d*x))/sqrt(a))*sqrt(a)/d+2*sqrt(a+a*sec(c+d*x))/d],
[cot(c+d*x)*sqrt(a+a*sec(c+d*x)),x,6,2*arctanh(sqrt(a+a*sec(c+d*x))/sqrt(a))*sqrt(a)/d-arctanh(sqrt(a+a*sec(c+d*x))/(sqrt(2)*sqrt(a)))*sqrt(2)*sqrt(a)/d],
[cot(c+d*x)^3*sqrt(a+a*sec(c+d*x)),x,8,-2*arctanh(sqrt(a+a*sec(c+d*x))/sqrt(a))*sqrt(a)/d+7/4*arctanh(sqrt(a+a*sec(c+d*x))/(sqrt(2)*sqrt(a)))*sqrt(a)/(d*sqrt(2))+1/4*a/(d*sqrt(a+a*sec(c+d*x)))+1/2*a/(d*(1-sec(c+d*x))*sqrt(a+a*sec(c+d*x)))],
[cot(c+d*x)^5*sqrt(a+a*sec(c+d*x)),x,10,43/96*a^2/(d*(a+a*sec(c+d*x))^(3/2))-1/4*a^2/(d*(1-sec(c+d*x))^2*(a+a*sec(c+d*x))^(3/2))-15/16*a^2/(d*(1-sec(c+d*x))*(a+a*sec(c+d*x))^(3/2))+2*arctanh(sqrt(a+a*sec(c+d*x))/sqrt(a))*sqrt(a)/d-107/64*arctanh(sqrt(a+a*sec(c+d*x))/(sqrt(2)*sqrt(a)))*sqrt(a)/(d*sqrt(2))-21/64*a/(d*sqrt(a+a*sec(c+d*x)))],
[sqrt(a+a*sec(c+d*x))*tan(c+d*x)^6,x,4,-2*arctan(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))*sqrt(a)/d+2*a*tan(c+d*x)/(d*sqrt(a+a*sec(c+d*x)))-2/3*a^2*tan(c+d*x)^3/(d*(a+a*sec(c+d*x))^(3/2))+2/5*a^3*tan(c+d*x)^5/(d*(a+a*sec(c+d*x))^(5/2))+2*a^4*tan(c+d*x)^7/(d*(a+a*sec(c+d*x))^(7/2))+10/9*a^5*tan(c+d*x)^9/(d*(a+a*sec(c+d*x))^(9/2))+2/11*a^6*tan(c+d*x)^11/(d*(a+a*sec(c+d*x))^(11/2))],
[sqrt(a+a*sec(c+d*x))*tan(c+d*x)^4,x,4,2*arctan(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))*sqrt(a)/d-2*a*tan(c+d*x)/(d*sqrt(a+a*sec(c+d*x)))+2/3*a^2*tan(c+d*x)^3/(d*(a+a*sec(c+d*x))^(3/2))+6/5*a^3*tan(c+d*x)^5/(d*(a+a*sec(c+d*x))^(5/2))+2/7*a^4*tan(c+d*x)^7/(d*(a+a*sec(c+d*x))^(7/2))],
[sqrt(a+a*sec(c+d*x))*tan(c+d*x)^2,x,4,-2*arctan(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))*sqrt(a)/d+2*a*tan(c+d*x)/(d*sqrt(a+a*sec(c+d*x)))+2/3*a^2*tan(c+d*x)^3/(d*(a+a*sec(c+d*x))^(3/2))],
[cot(c+d*x)^2*sqrt(a+a*sec(c+d*x)),x,5,-2*arctan(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))*sqrt(a)/d+arctan(sqrt(a)*tan(c+d*x)/(sqrt(2)*sqrt(a+a*sec(c+d*x))))*sqrt(a)/(d*sqrt(2))-cot(c+d*x)*sqrt(a+a*sec(c+d*x))/d],
[cot(c+d*x)^4*sqrt(a+a*sec(c+d*x)),x,7,1/12*cot(c+d*x)^3*(a+a*sec(c+d*x))^(3/2)/(a*d)-1/4*cos(c+d*x)*cot(c+d*x)^3*sec(1/2*(c+d*x))^2*(a+a*sec(c+d*x))^(3/2)/(a*d)+2*arctan(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))*sqrt(a)/d-9/8*arctan(sqrt(a)*tan(c+d*x)/(sqrt(2)*sqrt(a+a*sec(c+d*x))))*sqrt(a)/(d*sqrt(2))+7/8*cot(c+d*x)*sqrt(a+a*sec(c+d*x))/d],
[cot(c+d*x)^6*sqrt(a+a*sec(c+d*x)),x,9,-23/192*cot(c+d*x)^3*(a+a*sec(c+d*x))^(3/2)/(a*d)+87/160*cot(c+d*x)^5*(a+a*sec(c+d*x))^(5/2)/(a^2*d)-17/32*cos(c+d*x)*cot(c+d*x)^5*sec(1/2*(c+d*x))^2*(a+a*sec(c+d*x))^(5/2)/(a^2*d)-1/16*cos(c+d*x)^2*cot(c+d*x)^5*sec(1/2*(c+d*x))^4*(a+a*sec(c+d*x))^(5/2)/(a^2*d)-2*arctan(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))*sqrt(a)/d+151/128*arctan(sqrt(a)*tan(c+d*x)/(sqrt(2)*sqrt(a+a*sec(c+d*x))))*sqrt(a)/(d*sqrt(2))-105/128*cot(c+d*x)*sqrt(a+a*sec(c+d*x))/d],
[(a+a*sec(c+d*x))^(3/2)*tan(c+d*x)^5,x,9,-2*a^(3/2)*arctanh(sqrt(a+a*sec(c+d*x))/sqrt(a))/d+2/3*(a+a*sec(c+d*x))^(3/2)/d+2/5*(a+a*sec(c+d*x))^(5/2)/(a*d)+2/7*(a+a*sec(c+d*x))^(7/2)/(a^2*d)-2/3*(a+a*sec(c+d*x))^(9/2)/(a^3*d)+2/11*(a+a*sec(c+d*x))^(11/2)/(a^4*d)+2*a*sqrt(a+a*sec(c+d*x))/d],
[(a+a*sec(c+d*x))^(3/2)*tan(c+d*x)^3,x,7,2*a^(3/2)*arctanh(sqrt(a+a*sec(c+d*x))/sqrt(a))/d-2/3*(a+a*sec(c+d*x))^(3/2)/d-2/5*(a+a*sec(c+d*x))^(5/2)/(a*d)+2/7*(a+a*sec(c+d*x))^(7/2)/(a^2*d)-2*a*sqrt(a+a*sec(c+d*x))/d],
[(a+a*sec(c+d*x))^(3/2)*tan(c+d*x),x,5,-2*a^(3/2)*arctanh(sqrt(a+a*sec(c+d*x))/sqrt(a))/d+2/3*(a+a*sec(c+d*x))^(3/2)/d+2*a*sqrt(a+a*sec(c+d*x))/d],
[cot(c+d*x)*(a+a*sec(c+d*x))^(3/2),x,6,2*a^(3/2)*arctanh(sqrt(a+a*sec(c+d*x))/sqrt(a))/d-2*a^(3/2)*arctanh(sqrt(a+a*sec(c+d*x))/(sqrt(2)*sqrt(a)))*sqrt(2)/d],
[cot(c+d*x)^3*(a+a*sec(c+d*x))^(3/2),x,7,-2*a^(3/2)*arctanh(sqrt(a+a*sec(c+d*x))/sqrt(a))/d+5/2*a^(3/2)*arctanh(sqrt(a+a*sec(c+d*x))/(sqrt(2)*sqrt(a)))/(d*sqrt(2))+1/2*a*sqrt(a+a*sec(c+d*x))/(d*(1-sec(c+d*x)))],
[cot(c+d*x)^5*(a+a*sec(c+d*x))^(3/2),x,9,2*a^(3/2)*arctanh(sqrt(a+a*sec(c+d*x))/sqrt(a))/d-71/32*a^(3/2)*arctanh(sqrt(a+a*sec(c+d*x))/(sqrt(2)*sqrt(a)))/(d*sqrt(2))+7/32*a^2/(d*sqrt(a+a*sec(c+d*x)))-1/4*a^2/(d*(1-sec(c+d*x))^2*sqrt(a+a*sec(c+d*x)))-13/16*a^2/(d*(1-sec(c+d*x))*sqrt(a+a*sec(c+d*x)))],
[(a+a*sec(c+d*x))^(3/2)*tan(c+d*x)^6,x,4,-2*a^(3/2)*arctan(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))/d+2*a^2*tan(c+d*x)/(d*sqrt(a+a*sec(c+d*x)))-2/3*a^3*tan(c+d*x)^3/(d*(a+a*sec(c+d*x))^(3/2))+2/5*a^4*tan(c+d*x)^5/(d*(a+a*sec(c+d*x))^(5/2))+30/7*a^5*tan(c+d*x)^7/(d*(a+a*sec(c+d*x))^(7/2))+34/9*a^6*tan(c+d*x)^9/(d*(a+a*sec(c+d*x))^(9/2))+14/11*a^7*tan(c+d*x)^11/(d*(a+a*sec(c+d*x))^(11/2))+2/13*a^8*tan(c+d*x)^13/(d*(a+a*sec(c+d*x))^(13/2))],
[(a+a*sec(c+d*x))^(3/2)*tan(c+d*x)^4,x,4,2*a^(3/2)*arctan(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))/d-2*a^2*tan(c+d*x)/(d*sqrt(a+a*sec(c+d*x)))+2/3*a^3*tan(c+d*x)^3/(d*(a+a*sec(c+d*x))^(3/2))+14/5*a^4*tan(c+d*x)^5/(d*(a+a*sec(c+d*x))^(5/2))+10/7*a^5*tan(c+d*x)^7/(d*(a+a*sec(c+d*x))^(7/2))+2/9*a^6*tan(c+d*x)^9/(d*(a+a*sec(c+d*x))^(9/2))],
[(a+a*sec(c+d*x))^(3/2)*tan(c+d*x)^2,x,4,-2*a^(3/2)*arctan(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))/d+2*a^2*tan(c+d*x)/(d*sqrt(a+a*sec(c+d*x)))+2*a^3*tan(c+d*x)^3/(d*(a+a*sec(c+d*x))^(3/2))+2/5*a^4*tan(c+d*x)^5/(d*(a+a*sec(c+d*x))^(5/2))],
[cot(c+d*x)^2*(a+a*sec(c+d*x))^(3/2),x,3,-2*a^(3/2)*arctan(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))/d-2*a*cot(c+d*x)*sqrt(a+a*sec(c+d*x))/d],
[cot(c+d*x)^4*(a+a*sec(c+d*x))^(3/2),x,6,2*a^(3/2)*arctan(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))/d-1/3*cot(c+d*x)^3*(a+a*sec(c+d*x))^(3/2)/d-1/2*a^(3/2)*arctan(sqrt(a)*tan(c+d*x)/(sqrt(2)*sqrt(a+a*sec(c+d*x))))/(d*sqrt(2))+3/2*a*cot(c+d*x)*sqrt(a+a*sec(c+d*x))/d],
[cot(c+d*x)^6*(a+a*sec(c+d*x))^(3/2),x,8,-2*a^(3/2)*arctan(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))/d+5/24*cot(c+d*x)^3*(a+a*sec(c+d*x))^(3/2)/d+3/20*cot(c+d*x)^5*(a+a*sec(c+d*x))^(5/2)/(a*d)-1/4*cos(c+d*x)*cot(c+d*x)^5*sec(1/2*(c+d*x))^2*(a+a*sec(c+d*x))^(5/2)/(a*d)+11/16*a^(3/2)*arctan(sqrt(a)*tan(c+d*x)/(sqrt(2)*sqrt(a+a*sec(c+d*x))))/(d*sqrt(2))-21/16*a*cot(c+d*x)*sqrt(a+a*sec(c+d*x))/d],
[(a+a*sec(c+d*x))^(5/2)*tan(c+d*x)^5,x,10,-2*a^(5/2)*arctanh(sqrt(a+a*sec(c+d*x))/sqrt(a))/d+2/3*a*(a+a*sec(c+d*x))^(3/2)/d+2/5*(a+a*sec(c+d*x))^(5/2)/d+2/7*(a+a*sec(c+d*x))^(7/2)/(a*d)+2/9*(a+a*sec(c+d*x))^(9/2)/(a^2*d)-6/11*(a+a*sec(c+d*x))^(11/2)/(a^3*d)+2/13*(a+a*sec(c+d*x))^(13/2)/(a^4*d)+2*a^2*sqrt(a+a*sec(c+d*x))/d],
[(a+a*sec(c+d*x))^(5/2)*tan(c+d*x)^3,x,8,2*a^(5/2)*arctanh(sqrt(a+a*sec(c+d*x))/sqrt(a))/d-2/3*a*(a+a*sec(c+d*x))^(3/2)/d-2/5*(a+a*sec(c+d*x))^(5/2)/d-2/7*(a+a*sec(c+d*x))^(7/2)/(a*d)+2/9*(a+a*sec(c+d*x))^(9/2)/(a^2*d)-2*a^2*sqrt(a+a*sec(c+d*x))/d],
[(a+a*sec(c+d*x))^(5/2)*tan(c+d*x),x,6,-2*a^(5/2)*arctanh(sqrt(a+a*sec(c+d*x))/sqrt(a))/d+2/3*a*(a+a*sec(c+d*x))^(3/2)/d+2/5*(a+a*sec(c+d*x))^(5/2)/d+2*a^2*sqrt(a+a*sec(c+d*x))/d],
[cot(c+d*x)*(a+a*sec(c+d*x))^(5/2),x,7,2*a^(5/2)*arctanh(sqrt(a+a*sec(c+d*x))/sqrt(a))/d-4*a^(5/2)*arctanh(sqrt(a+a*sec(c+d*x))/(sqrt(2)*sqrt(a)))*sqrt(2)/d+2*a^2*sqrt(a+a*sec(c+d*x))/d],
[cot(c+d*x)^3*(a+a*sec(c+d*x))^(5/2),x,7,-2*a^(5/2)*arctanh(sqrt(a+a*sec(c+d*x))/sqrt(a))/d+3*a^(5/2)*arctanh(sqrt(a+a*sec(c+d*x))/(sqrt(2)*sqrt(a)))/(d*sqrt(2))+a^2*sqrt(a+a*sec(c+d*x))/(d*(1-sec(c+d*x)))],
[cot(c+d*x)^5*(a+a*sec(c+d*x))^(5/2),x,8,2*a^(5/2)*arctanh(sqrt(a+a*sec(c+d*x))/sqrt(a))/d-43/16*a^(5/2)*arctanh(sqrt(a+a*sec(c+d*x))/(sqrt(2)*sqrt(a)))/(d*sqrt(2))-1/4*a^2*sqrt(a+a*sec(c+d*x))/(d*(1-sec(c+d*x))^2)-11/16*a^2*sqrt(a+a*sec(c+d*x))/(d*(1-sec(c+d*x)))],
[(a+a*sec(c+d*x))^(5/2)*tan(c+d*x)^6,x,4,-2*a^(5/2)*arctan(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))/d+2*a^3*tan(c+d*x)/(d*sqrt(a+a*sec(c+d*x)))-2/3*a^4*tan(c+d*x)^3/(d*(a+a*sec(c+d*x))^(3/2))+2/5*a^5*tan(c+d*x)^5/(d*(a+a*sec(c+d*x))^(5/2))+62/7*a^6*tan(c+d*x)^7/(d*(a+a*sec(c+d*x))^(7/2))+98/9*a^7*tan(c+d*x)^9/(d*(a+a*sec(c+d*x))^(9/2))+62/11*a^8*tan(c+d*x)^11/(d*(a+a*sec(c+d*x))^(11/2))+18/13*a^9*tan(c+d*x)^13/(d*(a+a*sec(c+d*x))^(13/2))+2/15*a^10*tan(c+d*x)^15/(d*(a+a*sec(c+d*x))^(15/2))],
[(a+a*sec(c+d*x))^(5/2)*tan(c+d*x)^4,x,4,2*a^(5/2)*arctan(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))/d-2*a^3*tan(c+d*x)/(d*sqrt(a+a*sec(c+d*x)))+2/3*a^4*tan(c+d*x)^3/(d*(a+a*sec(c+d*x))^(3/2))+6*a^5*tan(c+d*x)^5/(d*(a+a*sec(c+d*x))^(5/2))+34/7*a^6*tan(c+d*x)^7/(d*(a+a*sec(c+d*x))^(7/2))+14/9*a^7*tan(c+d*x)^9/(d*(a+a*sec(c+d*x))^(9/2))+2/11*a^8*tan(c+d*x)^11/(d*(a+a*sec(c+d*x))^(11/2))],
[(a+a*sec(c+d*x))^(5/2)*tan(c+d*x)^2,x,4,-2*a^(5/2)*arctan(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))/d+2*a^3*tan(c+d*x)/(d*sqrt(a+a*sec(c+d*x)))+14/3*a^4*tan(c+d*x)^3/(d*(a+a*sec(c+d*x))^(3/2))+2*a^5*tan(c+d*x)^5/(d*(a+a*sec(c+d*x))^(5/2))+2/7*a^6*tan(c+d*x)^7/(d*(a+a*sec(c+d*x))^(7/2))],
[cot(c+d*x)^2*(a+a*sec(c+d*x))^(5/2),x,3,-2*a^(5/2)*arctan(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))/d-4*a^2*cot(c+d*x)*sqrt(a+a*sec(c+d*x))/d],
[cot(c+d*x)^4*(a+a*sec(c+d*x))^(5/2),x,4,2*a^(5/2)*arctan(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))/d-2/3*a*cot(c+d*x)^3*(a+a*sec(c+d*x))^(3/2)/d+2*a^2*cot(c+d*x)*sqrt(a+a*sec(c+d*x))/d],
[cot(c+d*x)^6*(a+a*sec(c+d*x))^(5/2),x,7,-2*a^(5/2)*arctan(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))/d+1/2*a*cot(c+d*x)^3*(a+a*sec(c+d*x))^(3/2)/d-1/5*cot(c+d*x)^5*(a+a*sec(c+d*x))^(5/2)/d+1/4*a^(5/2)*arctan(sqrt(a)*tan(c+d*x)/(sqrt(2)*sqrt(a+a*sec(c+d*x))))/(d*sqrt(2))-7/4*a^2*cot(c+d*x)*sqrt(a+a*sec(c+d*x))/d],

# n<0
[tan(c+d*x)^5/sqrt(a+a*sec(c+d*x)),x,7,2/3*(a+a*sec(c+d*x))^(3/2)/(a^2*d)-6/5*(a+a*sec(c+d*x))^(5/2)/(a^3*d)+2/7*(a+a*sec(c+d*x))^(7/2)/(a^4*d)-2*arctanh(sqrt(a+a*sec(c+d*x))/sqrt(a))/(d*sqrt(a))+2*sqrt(a+a*sec(c+d*x))/(a*d)],
[tan(c+d*x)^3/sqrt(a+a*sec(c+d*x)),x,5,2/3*(a+a*sec(c+d*x))^(3/2)/(a^2*d)+2*arctanh(sqrt(a+a*sec(c+d*x))/sqrt(a))/(d*sqrt(a))-2*sqrt(a+a*sec(c+d*x))/(a*d)],
[tan(c+d*x)/sqrt(a+a*sec(c+d*x)),x,3,-2*arctanh(sqrt(a+a*sec(c+d*x))/sqrt(a))/(d*sqrt(a))],
[cot(c+d*x)/sqrt(a+a*sec(c+d*x)),x,7,2*arctanh(sqrt(a+a*sec(c+d*x))/sqrt(a))/(d*sqrt(a))-arctanh(sqrt(a+a*sec(c+d*x))/(sqrt(2)*sqrt(a)))/(d*sqrt(2)*sqrt(a))+(-1)/(d*sqrt(a+a*sec(c+d*x)))],
[cot(c+d*x)^3/sqrt(a+a*sec(c+d*x)),x,9,-1/12*a/(d*(a+a*sec(c+d*x))^(3/2))+1/2*a/(d*(1-sec(c+d*x))*(a+a*sec(c+d*x))^(3/2))-2*arctanh(sqrt(a+a*sec(c+d*x))/sqrt(a))/(d*sqrt(a))+9/8*arctanh(sqrt(a+a*sec(c+d*x))/(sqrt(2)*sqrt(a)))/(d*sqrt(2)*sqrt(a))+7/8/(d*sqrt(a+a*sec(c+d*x)))],
[cot(c+d*x)^5/sqrt(a+a*sec(c+d*x)),x,11,87/160*a^2/(d*(a+a*sec(c+d*x))^(5/2))-1/4*a^2/(d*(1-sec(c+d*x))^2*(a+a*sec(c+d*x))^(5/2))-17/16*a^2/(d*(1-sec(c+d*x))*(a+a*sec(c+d*x))^(5/2))+23/192*a/(d*(a+a*sec(c+d*x))^(3/2))+2*arctanh(sqrt(a+a*sec(c+d*x))/sqrt(a))/(d*sqrt(a))-151/128*arctanh(sqrt(a+a*sec(c+d*x))/(sqrt(2)*sqrt(a)))/(d*sqrt(2)*sqrt(a))+(-105/128)/(d*sqrt(a+a*sec(c+d*x)))],
[tan(c+d*x)^6/sqrt(a+a*sec(c+d*x)),x,4,-2*arctan(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))/(d*sqrt(a))+2*tan(c+d*x)/(d*sqrt(a+a*sec(c+d*x)))-2/3*a*tan(c+d*x)^3/(d*(a+a*sec(c+d*x))^(3/2))+2/5*a^2*tan(c+d*x)^5/(d*(a+a*sec(c+d*x))^(5/2))+6/7*a^3*tan(c+d*x)^7/(d*(a+a*sec(c+d*x))^(7/2))+2/9*a^4*tan(c+d*x)^9/(d*(a+a*sec(c+d*x))^(9/2))],
[tan(c+d*x)^4/sqrt(a+a*sec(c+d*x)),x,5,2*arctan(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))/(d*sqrt(a))-2*tan(c+d*x)/(d*sqrt(a+a*sec(c+d*x)))+2/3*a*tan(c+d*x)^3/(d*(a+a*sec(c+d*x))^(3/2))+2/5*a^2*tan(c+d*x)^5/(d*(a+a*sec(c+d*x))^(5/2))],
[tan(c+d*x)^2/sqrt(a+a*sec(c+d*x)),x,3,-2*arctan(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))/(d*sqrt(a))+2*tan(c+d*x)/(d*sqrt(a+a*sec(c+d*x)))],
[cot(c+d*x)^2/sqrt(a+a*sec(c+d*x)),x,6,-2*arctan(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))/(d*sqrt(a))+7/4*arctan(sqrt(a)*tan(c+d*x)/(sqrt(2)*sqrt(a+a*sec(c+d*x))))/(d*sqrt(2)*sqrt(a))-1/4*cot(c+d*x)*sqrt(a+a*sec(c+d*x))/(a*d)-1/4*cos(c+d*x)*cot(c+d*x)*sec(1/2*(c+d*x))^2*sqrt(a+a*sec(c+d*x))/(a*d)],
[cot(c+d*x)^4/sqrt(a+a*sec(c+d*x)),x,8,43/96*cot(c+d*x)^3*(a+a*sec(c+d*x))^(3/2)/(a^2*d)-15/32*cos(c+d*x)*cot(c+d*x)^3*sec(1/2*(c+d*x))^2*(a+a*sec(c+d*x))^(3/2)/(a^2*d)-1/16*cos(c+d*x)^2*cot(c+d*x)^3*sec(1/2*(c+d*x))^4*(a+a*sec(c+d*x))^(3/2)/(a^2*d)+2*arctan(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))/(d*sqrt(a))-107/64*arctan(sqrt(a)*tan(c+d*x)/(sqrt(2)*sqrt(a+a*sec(c+d*x))))/(d*sqrt(2)*sqrt(a))+21/64*cot(c+d*x)*sqrt(a+a*sec(c+d*x))/(a*d)],
[cot(c+d*x)^6/sqrt(a+a*sec(c+d*x)),x,10,-323/768*cot(c+d*x)^3*(a+a*sec(c+d*x))^(3/2)/(a^2*d)+579/640*cot(c+d*x)^5*(a+a*sec(c+d*x))^(5/2)/(a^3*d)-101/128*cos(c+d*x)*cot(c+d*x)^5*sec(1/2*(c+d*x))^2*(a+a*sec(c+d*x))^(5/2)/(a^3*d)-23/192*cos(c+d*x)^2*cot(c+d*x)^5*sec(1/2*(c+d*x))^4*(a+a*sec(c+d*x))^(5/2)/(a^3*d)-1/48*cos(c+d*x)^3*cot(c+d*x)^5*sec(1/2*(c+d*x))^6*(a+a*sec(c+d*x))^(5/2)/(a^3*d)-2*arctan(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))/(d*sqrt(a))+835/512*arctan(sqrt(a)*tan(c+d*x)/(sqrt(2)*sqrt(a+a*sec(c+d*x))))/(d*sqrt(2)*sqrt(a))-189/512*cot(c+d*x)*sqrt(a+a*sec(c+d*x))/(a*d)],
[tan(c+d*x)^5/(a+a*sec(c+d*x))^(3/2),x,6,-2*arctanh(sqrt(a+a*sec(c+d*x))/sqrt(a))/(a^(3/2)*d)-2*(a+a*sec(c+d*x))^(3/2)/(a^3*d)+2/5*(a+a*sec(c+d*x))^(5/2)/(a^4*d)+2*sqrt(a+a*sec(c+d*x))/(a^2*d)],
[tan(c+d*x)^3/(a+a*sec(c+d*x))^(3/2),x,4,2*arctanh(sqrt(a+a*sec(c+d*x))/sqrt(a))/(a^(3/2)*d)+2*sqrt(a+a*sec(c+d*x))/(a^2*d)],
[tan(c+d*x)/(a+a*sec(c+d*x))^(3/2),x,4,-2*arctanh(sqrt(a+a*sec(c+d*x))/sqrt(a))/(a^(3/2)*d)+2/(a*d*sqrt(a+a*sec(c+d*x)))],
[cot(c+d*x)/(a+a*sec(c+d*x))^(3/2),x,8,2*arctanh(sqrt(a+a*sec(c+d*x))/sqrt(a))/(a^(3/2)*d)+(-1/3)/(d*(a+a*sec(c+d*x))^(3/2))-1/2*arctanh(sqrt(a+a*sec(c+d*x))/(sqrt(2)*sqrt(a)))/(a^(3/2)*d*sqrt(2))+(-3/2)/(a*d*sqrt(a+a*sec(c+d*x)))],
[cot(c+d*x)^3/(a+a*sec(c+d*x))^(3/2),x,10,-2*arctanh(sqrt(a+a*sec(c+d*x))/sqrt(a))/(a^(3/2)*d)-3/20*a/(d*(a+a*sec(c+d*x))^(5/2))+1/2*a/(d*(1-sec(c+d*x))*(a+a*sec(c+d*x))^(5/2))+5/24/(d*(a+a*sec(c+d*x))^(3/2))+11/16*arctanh(sqrt(a+a*sec(c+d*x))/(sqrt(2)*sqrt(a)))/(a^(3/2)*d*sqrt(2))+21/16/(a*d*sqrt(a+a*sec(c+d*x)))],
[cot(c+d*x)^5/(a+a*sec(c+d*x))^(3/2),x,12,2*arctanh(sqrt(a+a*sec(c+d*x))/sqrt(a))/(a^(3/2)*d)+139/224*a^2/(d*(a+a*sec(c+d*x))^(7/2))-1/4*a^2/(d*(1-sec(c+d*x))^2*(a+a*sec(c+d*x))^(7/2))-19/16*a^2/(d*(1-sec(c+d*x))*(a+a*sec(c+d*x))^(7/2))+15/64*a/(d*(a+a*sec(c+d*x))^(5/2))+(-53/384)/(d*(a+a*sec(c+d*x))^(3/2))-203/256*arctanh(sqrt(a+a*sec(c+d*x))/(sqrt(2)*sqrt(a)))/(a^(3/2)*d*sqrt(2))+(-309/256)/(a*d*sqrt(a+a*sec(c+d*x)))],
[tan(c+d*x)^6/(a+a*sec(c+d*x))^(3/2),x,5,-2*arctan(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))/(a^(3/2)*d)+2*tan(c+d*x)/(a*d*sqrt(a+a*sec(c+d*x)))-2/3*tan(c+d*x)^3/(d*(a+a*sec(c+d*x))^(3/2))+2/5*a*tan(c+d*x)^5/(d*(a+a*sec(c+d*x))^(5/2))+2/7*a^2*tan(c+d*x)^7/(d*(a+a*sec(c+d*x))^(7/2))],
[tan(c+d*x)^4/(a+a*sec(c+d*x))^(3/2),x,4,2*arctan(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))/(a^(3/2)*d)-2*tan(c+d*x)/(a*d*sqrt(a+a*sec(c+d*x)))+2/3*tan(c+d*x)^3/(d*(a+a*sec(c+d*x))^(3/2))],
[tan(c+d*x)^2/(a+a*sec(c+d*x))^(3/2),x,4,-2*arctan(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))/(a^(3/2)*d)+2*arctan(sqrt(a)*tan(c+d*x)/(sqrt(2)*sqrt(a+a*sec(c+d*x))))*sqrt(2)/(a^(3/2)*d)],
[cot(c+d*x)^2/(a+a*sec(c+d*x))^(3/2),x,7,-2*arctan(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))/(a^(3/2)*d)+71/32*arctan(sqrt(a)*tan(c+d*x)/(sqrt(2)*sqrt(a+a*sec(c+d*x))))/(a^(3/2)*d*sqrt(2))+7/32*cot(c+d*x)*sqrt(a+a*sec(c+d*x))/(a^2*d)-13/32*cos(c+d*x)*cot(c+d*x)*sec(1/2*(c+d*x))^2*sqrt(a+a*sec(c+d*x))/(a^2*d)-1/16*cos(c+d*x)^2*cot(c+d*x)*sec(1/2*(c+d*x))^4*sqrt(a+a*sec(c+d*x))/(a^2*d)],
[cot(c+d*x)^4/(a+a*sec(c+d*x))^(3/2),x,9,2*arctan(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))/(a^(3/2)*d)+277/384*cot(c+d*x)^3*(a+a*sec(c+d*x))^(3/2)/(a^3*d)-81/128*cos(c+d*x)*cot(c+d*x)^3*sec(1/2*(c+d*x))^2*(a+a*sec(c+d*x))^(3/2)/(a^3*d)-7/64*cos(c+d*x)^2*cot(c+d*x)^3*sec(1/2*(c+d*x))^4*(a+a*sec(c+d*x))^(3/2)/(a^3*d)-1/48*cos(c+d*x)^3*cot(c+d*x)^3*sec(1/2*(c+d*x))^6*(a+a*sec(c+d*x))^(3/2)/(a^3*d)-533/256*arctan(sqrt(a)*tan(c+d*x)/(sqrt(2)*sqrt(a+a*sec(c+d*x))))/(a^(3/2)*d*sqrt(2))-21/256*cot(c+d*x)*sqrt(a+a*sec(c+d*x))/(a^2*d)],
[cot(c+d*x)^6/(a+a*sec(c+d*x))^(3/2),x,11,-2*arctan(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))/(a^(3/2)*d)-8171/12288*cot(c+d*x)^3*(a+a*sec(c+d*x))^(3/2)/(a^3*d)+12267/10240*cot(c+d*x)^5*(a+a*sec(c+d*x))^(5/2)/(a^4*d)-2045/2048*cos(c+d*x)*cot(c+d*x)^5*sec(1/2*(c+d*x))^2*(a+a*sec(c+d*x))^(5/2)/(a^4*d)-511/3072*cos(c+d*x)^2*cot(c+d*x)^5*sec(1/2*(c+d*x))^4*(a+a*sec(c+d*x))^(5/2)/(a^4*d)-29/768*cos(c+d*x)^3*cot(c+d*x)^5*sec(1/2*(c+d*x))^6*(a+a*sec(c+d*x))^(5/2)/(a^4*d)-1/128*cos(c+d*x)^4*cot(c+d*x)^5*sec(1/2*(c+d*x))^8*(a+a*sec(c+d*x))^(5/2)/(a^4*d)+16363/8192*arctan(sqrt(a)*tan(c+d*x)/(sqrt(2)*sqrt(a+a*sec(c+d*x))))/(a^(3/2)*d*sqrt(2))-21/8192*cot(c+d*x)*sqrt(a+a*sec(c+d*x))/(a^2*d)],
[tan(c+d*x)^5/(a+a*sec(c+d*x))^(5/2),x,5,-2*arctanh(sqrt(a+a*sec(c+d*x))/sqrt(a))/(a^(5/2)*d)+2/3*(a+a*sec(c+d*x))^(3/2)/(a^4*d)-6*sqrt(a+a*sec(c+d*x))/(a^3*d)],
[tan(c+d*x)^3/(a+a*sec(c+d*x))^(5/2),x,4,2*arctanh(sqrt(a+a*sec(c+d*x))/sqrt(a))/(a^(5/2)*d)+(-4)/(a^2*d*sqrt(a+a*sec(c+d*x)))],
[tan(c+d*x)/(a+a*sec(c+d*x))^(5/2),x,5,-2*arctanh(sqrt(a+a*sec(c+d*x))/sqrt(a))/(a^(5/2)*d)+2/3/(a*d*(a+a*sec(c+d*x))^(3/2))+2/(a^2*d*sqrt(a+a*sec(c+d*x)))],
[cot(c+d*x)/(a+a*sec(c+d*x))^(5/2),x,9,2*arctanh(sqrt(a+a*sec(c+d*x))/sqrt(a))/(a^(5/2)*d)+(-1/5)/(d*(a+a*sec(c+d*x))^(5/2))+(-1/2)/(a*d*(a+a*sec(c+d*x))^(3/2))-1/4*arctanh(sqrt(a+a*sec(c+d*x))/(sqrt(2)*sqrt(a)))/(a^(5/2)*d*sqrt(2))+(-7/4)/(a^2*d*sqrt(a+a*sec(c+d*x)))],
[cot(c+d*x)^3/(a+a*sec(c+d*x))^(5/2),x,11,-2*arctanh(sqrt(a+a*sec(c+d*x))/sqrt(a))/(a^(5/2)*d)-5/28*a/(d*(a+a*sec(c+d*x))^(7/2))+1/2*a/(d*(1-sec(c+d*x))*(a+a*sec(c+d*x))^(7/2))+3/40/(d*(a+a*sec(c+d*x))^(5/2))+19/48/(a*d*(a+a*sec(c+d*x))^(3/2))+13/32*arctanh(sqrt(a+a*sec(c+d*x))/(sqrt(2)*sqrt(a)))/(a^(5/2)*d*sqrt(2))+51/32/(a^2*d*sqrt(a+a*sec(c+d*x)))],
[cot(c+d*x)^5/(a+a*sec(c+d*x))^(5/2),x,13,2*arctanh(sqrt(a+a*sec(c+d*x))/sqrt(a))/(a^(5/2)*d)+199/288*a^2/(d*(a+a*sec(c+d*x))^(9/2))-1/4*a^2/(d*(1-sec(c+d*x))^2*(a+a*sec(c+d*x))^(9/2))-21/16*a^2/(d*(1-sec(c+d*x))*(a+a*sec(c+d*x))^(9/2))+135/448*a/(d*(a+a*sec(c+d*x))^(7/2))+7/640/(d*(a+a*sec(c+d*x))^(5/2))+(-83/256)/(a*d*(a+a*sec(c+d*x))^(3/2))-263/512*arctanh(sqrt(a+a*sec(c+d*x))/(sqrt(2)*sqrt(a)))/(a^(5/2)*d*sqrt(2))+(-761/512)/(a^2*d*sqrt(a+a*sec(c+d*x)))],
[tan(c+d*x)^6/(a+a*sec(c+d*x))^(5/2),x,4,-2*arctan(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))/(a^(5/2)*d)+2*tan(c+d*x)/(a^2*d*sqrt(a+a*sec(c+d*x)))-2/3*tan(c+d*x)^3/(a*d*(a+a*sec(c+d*x))^(3/2))+2/5*tan(c+d*x)^5/(d*(a+a*sec(c+d*x))^(5/2))],
[tan(c+d*x)^4/(a+a*sec(c+d*x))^(5/2),x,5,2*arctan(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))/(a^(5/2)*d)-4*arctan(sqrt(a)*tan(c+d*x)/(sqrt(2)*sqrt(a+a*sec(c+d*x))))*sqrt(2)/(a^(5/2)*d)+2*tan(c+d*x)/(a^2*d*sqrt(a+a*sec(c+d*x)))],
[tan(c+d*x)^2/(a+a*sec(c+d*x))^(5/2),x,5,-2*arctan(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))/(a^(5/2)*d)+3*arctan(sqrt(a)*tan(c+d*x)/(sqrt(2)*sqrt(a+a*sec(c+d*x))))/(a^(5/2)*d*sqrt(2))+1/2*sec(1/2*(c+d*x))^2*sin(c+d*x)/(a^2*d*sqrt(a+a*sec(c+d*x)))],
[cot(c+d*x)^2/(a+a*sec(c+d*x))^(5/2),x,8,-2*arctan(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))/(a^(5/2)*d)+319/128*arctan(sqrt(a)*tan(c+d*x)/(sqrt(2)*sqrt(a+a*sec(c+d*x))))/(a^(5/2)*d*sqrt(2))+63/128*cot(c+d*x)*sqrt(a+a*sec(c+d*x))/(a^3*d)-191/384*cos(c+d*x)*cot(c+d*x)*sec(1/2*(c+d*x))^2*sqrt(a+a*sec(c+d*x))/(a^3*d)-19/192*cos(c+d*x)^2*cot(c+d*x)*sec(1/2*(c+d*x))^4*sqrt(a+a*sec(c+d*x))/(a^3*d)-1/48*cos(c+d*x)^3*cot(c+d*x)*sec(1/2*(c+d*x))^6*sqrt(a+a*sec(c+d*x))/(a^3*d)],
[cot(c+d*x)^4/(a+a*sec(c+d*x))^(5/2),x,10,2*arctan(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))/(a^(5/2)*d)+5587/6144*cot(c+d*x)^3*(a+a*sec(c+d*x))^(3/2)/(a^4*d)-1527/2048*cos(c+d*x)*cot(c+d*x)^3*sec(1/2*(c+d*x))^2*(a+a*sec(c+d*x))^(3/2)/(a^4*d)-145/1024*cos(c+d*x)^2*cot(c+d*x)^3*sec(1/2*(c+d*x))^4*(a+a*sec(c+d*x))^(3/2)/(a^4*d)-9/256*cos(c+d*x)^3*cot(c+d*x)^3*sec(1/2*(c+d*x))^6*(a+a*sec(c+d*x))^(3/2)/(a^4*d)-1/128*cos(c+d*x)^4*cot(c+d*x)^3*sec(1/2*(c+d*x))^8*(a+a*sec(c+d*x))^(3/2)/(a^4*d)-9683/4096*arctan(sqrt(a)*tan(c+d*x)/(sqrt(2)*sqrt(a+a*sec(c+d*x))))/(a^(5/2)*d*sqrt(2))-1491/4096*cot(c+d*x)*sqrt(a+a*sec(c+d*x))/(a^3*d)],
[cot(c+d*x)^6/(a+a*sec(c+d*x))^(5/2),x,12,-2*arctan(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))/(a^(5/2)*d)-41693/49152*cot(c+d*x)^3*(a+a*sec(c+d*x))^(3/2)/(a^4*d)+58077/40960*cot(c+d*x)^5*(a+a*sec(c+d*x))^(5/2)/(a^5*d)-9467/8192*cos(c+d*x)*cot(c+d*x)^5*sec(1/2*(c+d*x))^2*(a+a*sec(c+d*x))^(5/2)/(a^5*d)-2473/12288*cos(c+d*x)^2*cot(c+d*x)^5*sec(1/2*(c+d*x))^4*(a+a*sec(c+d*x))^(5/2)/(a^5*d)-155/3072*cos(c+d*x)^3*cot(c+d*x)^5*sec(1/2*(c+d*x))^6*(a+a*sec(c+d*x))^(5/2)/(a^5*d)-7/512*cos(c+d*x)^4*cot(c+d*x)^5*sec(1/2*(c+d*x))^8*(a+a*sec(c+d*x))^(5/2)/(a^5*d)-1/320*cos(c+d*x)^5*cot(c+d*x)^5*sec(1/2*(c+d*x))^10*(a+a*sec(c+d*x))^(5/2)/(a^5*d)+74461/32768*arctan(sqrt(a)*tan(c+d*x)/(sqrt(2)*sqrt(a+a*sec(c+d*x))))/(a^(5/2)*d*sqrt(2))+8925/32768*cot(c+d*x)*sqrt(a+a*sec(c+d*x))/(a^3*d)],
[tan(e+f*x)^2/(a+a*sec(e+f*x))^(9/2),x,-7,-2*arctan(sqrt(a)*tan(e+f*x)/sqrt(a+a*sec(e+f*x)))/(a^(9/2)*f)+91/32*arctan(sqrt(a)*tan(e+f*x)/(sqrt(2)*sqrt(a+a*sec(e+f*x))))/(a^(9/2)*f*sqrt(2))+1/3*tan(e+f*x)/(a*f*(a+a*sec(e+f*x))^(7/2))+11/24*tan(e+f*x)/(a^2*f*(a+a*sec(e+f*x))^(5/2))+27/32*tan(e+f*x)/(a^3*f*(a+a*sec(e+f*x))^(3/2))],

# Integrands of the form (d Tan[e+f x])^n (a+a Sec[e+f x])^m with m and/or n symbolic
[(a+a*sec(c+d*x))^n*(e*tan(c+d*x))^m,x,1,2^(1+m+n)*AppellF1(1/2*(1+m),m+n,1,1/2*(3+m),(-a+a*sec(c+d*x))/(a+a*sec(c+d*x)),(a-a*sec(c+d*x))/(a+a*sec(c+d*x)))*(1/(1+sec(c+d*x)))^(1+m+n)*(a+a*sec(c+d*x))^n*(e*tan(c+d*x))^(1+m)/(d*e*(1+m))],
[(a+a*sec(c+d*x))^3*(e*tan(c+d*x))^m,x,8,3*a^3*(e*tan(c+d*x))^(1+m)/(d*e*(1+m))+a^3*hypergeom([1,1/2*(1+m)],[1/2*(3+m)],-tan(c+d*x)^2)*(e*tan(c+d*x))^(1+m)/(d*e*(1+m))+3*a^3*(cos(c+d*x)^2)^(1/2*(2+m))*hypergeom([1/2*(1+m),1/2*(2+m)],[1/2*(3+m)],sin(c+d*x)^2)*sec(c+d*x)*(e*tan(c+d*x))^(1+m)/(d*e*(1+m))+a^3*(cos(c+d*x)^2)^(1/2*(4+m))*hypergeom([1/2*(1+m),1/2*(4+m)],[1/2*(3+m)],sin(c+d*x)^2)*sec(c+d*x)^3*(e*tan(c+d*x))^(1+m)/(d*e*(1+m))],
[(a+a*sec(c+d*x))^2*(e*tan(c+d*x))^m,x,7,a^2*(e*tan(c+d*x))^(1+m)/(d*e*(1+m))+a^2*hypergeom([1,1/2*(1+m)],[1/2*(3+m)],-tan(c+d*x)^2)*(e*tan(c+d*x))^(1+m)/(d*e*(1+m))+2*a^2*(cos(c+d*x)^2)^(1/2*(2+m))*hypergeom([1/2*(1+m),1/2*(2+m)],[1/2*(3+m)],sin(c+d*x)^2)*sec(c+d*x)*(e*tan(c+d*x))^(1+m)/(d*e*(1+m))],
[(a+a*sec(c+d*x))*(e*tan(c+d*x))^m,x,4,a*hypergeom([1,1/2*(1+m)],[1/2*(3+m)],-tan(c+d*x)^2)*(e*tan(c+d*x))^(1+m)/(d*e*(1+m))+a*(cos(c+d*x)^2)^(1/2*(2+m))*hypergeom([1/2*(1+m),1/2*(2+m)],[1/2*(3+m)],sin(c+d*x)^2)*sec(c+d*x)*(e*tan(c+d*x))^(1+m)/(d*e*(1+m))],
[(e*tan(c+d*x))^m/(a+a*sec(c+d*x)),x,5,e*hypergeom([1,1/2*(-1+m)],[1/2*(1+m)],-tan(c+d*x)^2)*(e*tan(c+d*x))^(-1+m)/(a*d*(1-m))-e*(cos(c+d*x)^2)^(1/2*m)*hypergeom([1/2*(-1+m),1/2*m],[1/2*(1+m)],sin(c+d*x)^2)*sec(c+d*x)*(e*tan(c+d*x))^(-1+m)/(a*d*(1-m))],
[(e*tan(c+d*x))^m/(a+a*sec(c+d*x))^2,x,8,-e^3*(e*tan(c+d*x))^(-3+m)/(a^2*d*(3-m))-e^3*hypergeom([1,1/2*(-3+m)],[1/2*(-1+m)],-tan(c+d*x)^2)*(e*tan(c+d*x))^(-3+m)/(a^2*d*(3-m))+2*e^3*(cos(c+d*x)^2)^(1/2*(-2+m))*hypergeom([1/2*(-3+m),1/2*(-2+m)],[1/2*(-1+m)],sin(c+d*x)^2)*sec(c+d*x)*(e*tan(c+d*x))^(-3+m)/(a^2*d*(3-m))],
[(e*tan(c+d*x))^m/(a+a*sec(c+d*x))^3,x,9,3*e^5*(e*tan(c+d*x))^(-5+m)/(a^3*d*(5-m))+e^5*hypergeom([1,1/2*(-5+m)],[1/2*(-3+m)],-tan(c+d*x)^2)*(e*tan(c+d*x))^(-5+m)/(a^3*d*(5-m))-3*e^5*(cos(c+d*x)^2)^(1/2*(-4+m))*hypergeom([1/2*(-5+m),1/2*(-4+m)],[1/2*(-3+m)],sin(c+d*x)^2)*sec(c+d*x)*(e*tan(c+d*x))^(-5+m)/(a^3*d*(5-m))-e^5*(cos(c+d*x)^2)^(1/2*(-2+m))*hypergeom([1/2*(-5+m),1/2*(-2+m)],[1/2*(-3+m)],sin(c+d*x)^2)*sec(c+d*x)^3*(e*tan(c+d*x))^(-5+m)/(a^3*d*(5-m))],
[(a+a*sec(c+d*x))^(3/2)*(e*tan(c+d*x))^m,x,1,2^(5/2+m)*AppellF1(1/2*(1+m),3/2+m,1,1/2*(3+m),(-a+a*sec(c+d*x))/(a+a*sec(c+d*x)),(a-a*sec(c+d*x))/(a+a*sec(c+d*x)))*(1/(1+sec(c+d*x)))^(5/2+m)*(a+a*sec(c+d*x))^(3/2)*(e*tan(c+d*x))^(1+m)/(d*e*(1+m))],
[(a+a*sec(c+d*x))^(1/2)*(e*tan(c+d*x))^m,x,1,2^(3/2+m)*AppellF1(1/2*(1+m),1/2+m,1,1/2*(3+m),(-a+a*sec(c+d*x))/(a+a*sec(c+d*x)),(a-a*sec(c+d*x))/(a+a*sec(c+d*x)))*(1/(1+sec(c+d*x)))^(3/2+m)*sqrt(a+a*sec(c+d*x))*(e*tan(c+d*x))^(1+m)/(d*e*(1+m))],
[(e*tan(c+d*x))^m/(a+a*sec(c+d*x))^(1/2),x,1,2^(1/2+m)*AppellF1(1/2*(1+m),-1/2+m,1,1/2*(3+m),(-a+a*sec(c+d*x))/(a+a*sec(c+d*x)),(a-a*sec(c+d*x))/(a+a*sec(c+d*x)))*(1/(1+sec(c+d*x)))^(1/2+m)*(e*tan(c+d*x))^(1+m)/(d*e*(1+m)*sqrt(a+a*sec(c+d*x)))],
[(e*tan(c+d*x))^m/(a+a*sec(c+d*x))^(3/2),x,1,2^(-1/2+m)*AppellF1(1/2*(1+m),-3/2+m,1,1/2*(3+m),(-a+a*sec(c+d*x))/(a+a*sec(c+d*x)),(a-a*sec(c+d*x))/(a+a*sec(c+d*x)))*(1/(1+sec(c+d*x)))^(-1/2+m)*(e*tan(c+d*x))^(1+m)/(d*e*(1+m)*(a+a*sec(c+d*x))^(3/2))],
[(a+a*sec(c+d*x))^n*tan(c+d*x)^7,x,4,7*(a+a*sec(c+d*x))^(4+n)/(a^4*d*(4+n))+hypergeom([1,4+n],[5+n],1+sec(c+d*x))*(a+a*sec(c+d*x))^(4+n)/(a^4*d*(4+n))-5*(a+a*sec(c+d*x))^(5+n)/(a^5*d*(5+n))+(a+a*sec(c+d*x))^(6+n)/(a^6*d*(6+n))],
[(a+a*sec(c+d*x))^n*tan(c+d*x)^5,x,4,-3*(a+a*sec(c+d*x))^(3+n)/(a^3*d*(3+n))-hypergeom([1,3+n],[4+n],1+sec(c+d*x))*(a+a*sec(c+d*x))^(3+n)/(a^3*d*(3+n))+(a+a*sec(c+d*x))^(4+n)/(a^4*d*(4+n))],
[(a+a*sec(c+d*x))^n*tan(c+d*x)^3,x,3,(a+a*sec(c+d*x))^(2+n)/(a^2*d*(2+n))+hypergeom([1,2+n],[3+n],1+sec(c+d*x))*(a+a*sec(c+d*x))^(2+n)/(a^2*d*(2+n))],
[(a+a*sec(c+d*x))^n*tan(c+d*x),x,2,-hypergeom([1,1+n],[2+n],1+sec(c+d*x))*(a+a*sec(c+d*x))^(1+n)/(a*d*(1+n))],
[cot(c+d*x)*(a+a*sec(c+d*x))^n,x,4,-1/2*hypergeom([1,n],[1+n],1/2*(1+sec(c+d*x)))*(a+a*sec(c+d*x))^n/(d*n)+hypergeom([1,n],[1+n],1+sec(c+d*x))*(a+a*sec(c+d*x))^n/(d*n)],
[cot(c+d*x)^3*(a+a*sec(c+d*x))^n,x,5,-1/4*a*(4-n)*hypergeom([1,-1+n],[n],1/2*(1+sec(c+d*x)))*(a+a*sec(c+d*x))^(-1+n)/(d*(1-n))+a*hypergeom([1,-1+n],[n],1+sec(c+d*x))*(a+a*sec(c+d*x))^(-1+n)/(d*(1-n))+1/2*a*(a+a*sec(c+d*x))^(-1+n)/(d*(1-sec(c+d*x)))],
[(a+a*sec(c+d*x))^n*tan(c+d*x)^4,x,1,1/5*2^(5+n)*AppellF1(5/2,4+n,1,7/2,(-a+a*sec(c+d*x))/(a+a*sec(c+d*x)),(a-a*sec(c+d*x))/(a+a*sec(c+d*x)))*(1/(1+sec(c+d*x)))^(5+n)*(a+a*sec(c+d*x))^n*tan(c+d*x)^5/d],
[(a+a*sec(c+d*x))^n*tan(c+d*x)^2,x,1,1/3*2^(3+n)*AppellF1(3/2,2+n,1,5/2,(-a+a*sec(c+d*x))/(a+a*sec(c+d*x)),(a-a*sec(c+d*x))/(a+a*sec(c+d*x)))*(1/(1+sec(c+d*x)))^(3+n)*(a+a*sec(c+d*x))^n*tan(c+d*x)^3/d],
[cot(c+d*x)^2*(a+a*sec(c+d*x))^n,x,1,-2^(-1+n)*AppellF1(-1/2,-2+n,1,1/2,(-a+a*sec(c+d*x))/(a+a*sec(c+d*x)),(a-a*sec(c+d*x))/(a+a*sec(c+d*x)))*cot(c+d*x)*(1/(1+sec(c+d*x)))^(-1+n)*(a+a*sec(c+d*x))^n/d],
[cot(c+d*x)^4*(a+a*sec(c+d*x))^n,x,1,-1/3*2^(-3+n)*AppellF1(-3/2,-4+n,1,-1/2,(-a+a*sec(c+d*x))/(a+a*sec(c+d*x)),(a-a*sec(c+d*x))/(a+a*sec(c+d*x)))*cot(c+d*x)^3*(1/(1+sec(c+d*x)))^(-3+n)*(a+a*sec(c+d*x))^n/d],
[(a+a*sec(c+d*x))^n*tan(c+d*x)^(3/2),x,1,1/5*2^(7/2+n)*AppellF1(5/4,3/2+n,1,9/4,(-a+a*sec(c+d*x))/(a+a*sec(c+d*x)),(a-a*sec(c+d*x))/(a+a*sec(c+d*x)))*(1/(1+sec(c+d*x)))^(5/2+n)*(a+a*sec(c+d*x))^n*tan(c+d*x)^(5/2)/d],
[(a+a*sec(c+d*x))^n*tan(c+d*x)^(1/2),x,1,1/3*2^(5/2+n)*AppellF1(3/4,1/2+n,1,7/4,(-a+a*sec(c+d*x))/(a+a*sec(c+d*x)),(a-a*sec(c+d*x))/(a+a*sec(c+d*x)))*(1/(1+sec(c+d*x)))^(3/2+n)*(a+a*sec(c+d*x))^n*tan(c+d*x)^(3/2)/d],
[(a+a*sec(c+d*x))^n/tan(c+d*x)^(1/2),x,1,2^(3/2+n)*AppellF1(1/4,-1/2+n,1,5/4,(-a+a*sec(c+d*x))/(a+a*sec(c+d*x)),(a-a*sec(c+d*x))/(a+a*sec(c+d*x)))*(1/(1+sec(c+d*x)))^(1/2+n)*(a+a*sec(c+d*x))^n*sqrt(tan(c+d*x))/d],
[(a+a*sec(c+d*x))^n/tan(c+d*x)^(3/2),x,1,-2^(1/2+n)*AppellF1(-1/4,-3/2+n,1,3/4,(-a+a*sec(c+d*x))/(a+a*sec(c+d*x)),(a-a*sec(c+d*x))/(a+a*sec(c+d*x)))*(1/(1+sec(c+d*x)))^(-1/2+n)*(a+a*sec(c+d*x))^n/(d*sqrt(tan(c+d*x)))],

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

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

# n>0
[(e*cot(c+d*x))^(5/2)*(a+a*sec(c+d*x)),x,17,-2/3*(e*cot(c+d*x))^(5/2)*(a+a*sec(c+d*x))*tan(c+d*x)/d-1/3*a*(e*cot(c+d*x))^(5/2)*sqrt(cos(c-1/4*Pi+d*x)^2)/cos(c-1/4*Pi+d*x)*EllipticF(sin(c-1/4*Pi+d*x),sqrt(2))*sec(c+d*x)*sqrt(sin(2*c+2*d*x))*tan(c+d*x)^2/d+a*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))*(e*cot(c+d*x))^(5/2)*tan(c+d*x)^(5/2)/(d*sqrt(2))-a*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))*(e*cot(c+d*x))^(5/2)*tan(c+d*x)^(5/2)/(d*sqrt(2))+1/2*a*(e*cot(c+d*x))^(5/2)*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))*tan(c+d*x)^(5/2)/(d*sqrt(2))-1/2*a*(e*cot(c+d*x))^(5/2)*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))*tan(c+d*x)^(5/2)/(d*sqrt(2))],
[(e*cot(c+d*x))^(3/2)*(a+a*sec(c+d*x)),x,18,-2*(e*cot(c+d*x))^(3/2)*(a+a*sec(c+d*x))*tan(c+d*x)/d-2*a*(e*cot(c+d*x))^(3/2)*sqrt(cos(c-1/4*Pi+d*x)^2)/cos(c-1/4*Pi+d*x)*EllipticE(sin(c-1/4*Pi+d*x),sqrt(2))*sin(c+d*x)*tan(c+d*x)/(d*sqrt(sin(2*c+2*d*x)))+a*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))*(e*cot(c+d*x))^(3/2)*tan(c+d*x)^(3/2)/(d*sqrt(2))-a*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))*(e*cot(c+d*x))^(3/2)*tan(c+d*x)^(3/2)/(d*sqrt(2))-1/2*a*(e*cot(c+d*x))^(3/2)*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))*tan(c+d*x)^(3/2)/(d*sqrt(2))+1/2*a*(e*cot(c+d*x))^(3/2)*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))*tan(c+d*x)^(3/2)/(d*sqrt(2))+2*a*(e*cot(c+d*x))^(3/2)*sin(c+d*x)*tan(c+d*x)^2/d],
[(a+a*sec(c+d*x))*sqrt(e*cot(c+d*x)),x,16,a*sqrt(cos(c-1/4*Pi+d*x)^2)/cos(c-1/4*Pi+d*x)*EllipticF(sin(c-1/4*Pi+d*x),sqrt(2))*sec(c+d*x)*sqrt(e*cot(c+d*x))*sqrt(sin(2*c+2*d*x))/d-a*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))*sqrt(e*cot(c+d*x))*sqrt(tan(c+d*x))/(d*sqrt(2))+a*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))*sqrt(e*cot(c+d*x))*sqrt(tan(c+d*x))/(d*sqrt(2))-1/2*a*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))*sqrt(e*cot(c+d*x))*sqrt(tan(c+d*x))/(d*sqrt(2))+1/2*a*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))*sqrt(e*cot(c+d*x))*sqrt(tan(c+d*x))/(d*sqrt(2))],
[(a+a*sec(c+d*x))/sqrt(e*cot(c+d*x)),x,17,2*a*sin(c+d*x)/(d*sqrt(e*cot(c+d*x)))-2*a*cos(c+d*x)*sqrt(cos(c-1/4*Pi+d*x)^2)/cos(c-1/4*Pi+d*x)*EllipticE(sin(c-1/4*Pi+d*x),sqrt(2))/(d*sqrt(e*cot(c+d*x))*sqrt(sin(2*c+2*d*x)))-a*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2)*sqrt(e*cot(c+d*x))*sqrt(tan(c+d*x)))+a*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2)*sqrt(e*cot(c+d*x))*sqrt(tan(c+d*x)))+1/2*a*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2)*sqrt(e*cot(c+d*x))*sqrt(tan(c+d*x)))-1/2*a*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2)*sqrt(e*cot(c+d*x))*sqrt(tan(c+d*x)))],
[(a+a*sec(c+d*x))/(e*cot(c+d*x))^(3/2),x,17,2/3*cot(c+d*x)*(3*a+a*sec(c+d*x))/(d*(e*cot(c+d*x))^(3/2))-1/3*a*cot(c+d*x)*csc(c+d*x)*sqrt(cos(c-1/4*Pi+d*x)^2)/cos(c-1/4*Pi+d*x)*EllipticF(sin(c-1/4*Pi+d*x),sqrt(2))*sqrt(sin(2*c+2*d*x))/(d*(e*cot(c+d*x))^(3/2))+a*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(d*(e*cot(c+d*x))^(3/2)*sqrt(2)*tan(c+d*x)^(3/2))-a*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(d*(e*cot(c+d*x))^(3/2)*sqrt(2)*tan(c+d*x)^(3/2))+1/2*a*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*(e*cot(c+d*x))^(3/2)*sqrt(2)*tan(c+d*x)^(3/2))-1/2*a*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*(e*cot(c+d*x))^(3/2)*sqrt(2)*tan(c+d*x)^(3/2))],
[(e*cot(c+d*x))^(5/2)*(a+a*sec(c+d*x))^2,x,21,-4/3*a^2*(e*cot(c+d*x))^(5/2)*tan(c+d*x)/d-4/3*a^2*(e*cot(c+d*x))^(5/2)*sec(c+d*x)*tan(c+d*x)/d-2/3*a^2*(e*cot(c+d*x))^(5/2)*sqrt(cos(c-1/4*Pi+d*x)^2)/cos(c-1/4*Pi+d*x)*EllipticF(sin(c-1/4*Pi+d*x),sqrt(2))*sec(c+d*x)*sqrt(sin(2*c+2*d*x))*tan(c+d*x)^2/d+a^2*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))*(e*cot(c+d*x))^(5/2)*tan(c+d*x)^(5/2)/(d*sqrt(2))-a^2*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))*(e*cot(c+d*x))^(5/2)*tan(c+d*x)^(5/2)/(d*sqrt(2))+1/2*a^2*(e*cot(c+d*x))^(5/2)*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))*tan(c+d*x)^(5/2)/(d*sqrt(2))-1/2*a^2*(e*cot(c+d*x))^(5/2)*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))*tan(c+d*x)^(5/2)/(d*sqrt(2))],
[(e*cot(c+d*x))^(3/2)*(a+a*sec(c+d*x))^2,x,21,-4*a^2*(e*cot(c+d*x))^(3/2)*sin(c+d*x)/d-4*a^2*(e*cot(c+d*x))^(3/2)*tan(c+d*x)/d-4*a^2*(e*cot(c+d*x))^(3/2)*sqrt(cos(c-1/4*Pi+d*x)^2)/cos(c-1/4*Pi+d*x)*EllipticE(sin(c-1/4*Pi+d*x),sqrt(2))*sin(c+d*x)*tan(c+d*x)/(d*sqrt(sin(2*c+2*d*x)))+a^2*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))*(e*cot(c+d*x))^(3/2)*tan(c+d*x)^(3/2)/(d*sqrt(2))-a^2*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))*(e*cot(c+d*x))^(3/2)*tan(c+d*x)^(3/2)/(d*sqrt(2))-1/2*a^2*(e*cot(c+d*x))^(3/2)*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))*tan(c+d*x)^(3/2)/(d*sqrt(2))+1/2*a^2*(e*cot(c+d*x))^(3/2)*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))*tan(c+d*x)^(3/2)/(d*sqrt(2))],
[(a+a*sec(c+d*x))^2*sqrt(e*cot(c+d*x)),x,19,2*a^2*sqrt(cos(c-1/4*Pi+d*x)^2)/cos(c-1/4*Pi+d*x)*EllipticF(sin(c-1/4*Pi+d*x),sqrt(2))*sec(c+d*x)*sqrt(e*cot(c+d*x))*sqrt(sin(2*c+2*d*x))/d-a^2*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))*sqrt(e*cot(c+d*x))*sqrt(tan(c+d*x))/(d*sqrt(2))+a^2*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))*sqrt(e*cot(c+d*x))*sqrt(tan(c+d*x))/(d*sqrt(2))-1/2*a^2*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))*sqrt(e*cot(c+d*x))*sqrt(tan(c+d*x))/(d*sqrt(2))+1/2*a^2*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))*sqrt(e*cot(c+d*x))*sqrt(tan(c+d*x))/(d*sqrt(2))+2*a^2*sqrt(e*cot(c+d*x))*tan(c+d*x)/d],
[(a+a*sec(c+d*x))^2/sqrt(e*cot(c+d*x)),x,20,4*a^2*sin(c+d*x)/(d*sqrt(e*cot(c+d*x)))-4*a^2*cos(c+d*x)*sqrt(cos(c-1/4*Pi+d*x)^2)/cos(c-1/4*Pi+d*x)*EllipticE(sin(c-1/4*Pi+d*x),sqrt(2))/(d*sqrt(e*cot(c+d*x))*sqrt(sin(2*c+2*d*x)))-a^2*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2)*sqrt(e*cot(c+d*x))*sqrt(tan(c+d*x)))+a^2*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(d*sqrt(2)*sqrt(e*cot(c+d*x))*sqrt(tan(c+d*x)))+1/2*a^2*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2)*sqrt(e*cot(c+d*x))*sqrt(tan(c+d*x)))-1/2*a^2*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*sqrt(2)*sqrt(e*cot(c+d*x))*sqrt(tan(c+d*x)))+2/3*a^2*tan(c+d*x)/(d*sqrt(e*cot(c+d*x)))],
[(a+a*sec(c+d*x))^2/(e*cot(c+d*x))^(3/2),x,21,2*a^2*cot(c+d*x)/(d*(e*cot(c+d*x))^(3/2))+4/3*a^2*csc(c+d*x)/(d*(e*cot(c+d*x))^(3/2))-2/3*a^2*cot(c+d*x)*csc(c+d*x)*sqrt(cos(c-1/4*Pi+d*x)^2)/cos(c-1/4*Pi+d*x)*EllipticF(sin(c-1/4*Pi+d*x),sqrt(2))*sqrt(sin(2*c+2*d*x))/(d*(e*cot(c+d*x))^(3/2))+a^2*arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(d*(e*cot(c+d*x))^(3/2)*sqrt(2)*tan(c+d*x)^(3/2))-a^2*arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(d*(e*cot(c+d*x))^(3/2)*sqrt(2)*tan(c+d*x)^(3/2))+1/2*a^2*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*(e*cot(c+d*x))^(3/2)*sqrt(2)*tan(c+d*x)^(3/2))-1/2*a^2*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(d*(e*cot(c+d*x))^(3/2)*sqrt(2)*tan(c+d*x)^(3/2))+2/5*a^2*tan(c+d*x)/(d*(e*cot(c+d*x))^(3/2))],

# n<0
[(e*cot(c+d*x))^(3/2)/(a+a*sec(c+d*x)),x,20,2/5*cot(c+d*x)*(e*cot(c+d*x))^(3/2)*(1-sec(c+d*x))/(a*d)-2/5*(e*cot(c+d*x))^(3/2)*(5-3*sec(c+d*x))*tan(c+d*x)/(a*d)+6/5*(e*cot(c+d*x))^(3/2)*sqrt(cos(c-1/4*Pi+d*x)^2)/cos(c-1/4*Pi+d*x)*EllipticE(sin(c-1/4*Pi+d*x),sqrt(2))*sin(c+d*x)*tan(c+d*x)/(a*d*sqrt(sin(2*c+2*d*x)))+arctan(1-sqrt(2)*sqrt(tan(c+d*x)))*(e*cot(c+d*x))^(3/2)*tan(c+d*x)^(3/2)/(a*d*sqrt(2))-arctan(1+sqrt(2)*sqrt(tan(c+d*x)))*(e*cot(c+d*x))^(3/2)*tan(c+d*x)^(3/2)/(a*d*sqrt(2))-1/2*(e*cot(c+d*x))^(3/2)*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))*tan(c+d*x)^(3/2)/(a*d*sqrt(2))+1/2*(e*cot(c+d*x))^(3/2)*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))*tan(c+d*x)^(3/2)/(a*d*sqrt(2))-6/5*(e*cot(c+d*x))^(3/2)*sin(c+d*x)*tan(c+d*x)^2/(a*d)],
[sqrt(e*cot(c+d*x))/(a+a*sec(c+d*x)),x,18,2/3*cot(c+d*x)*(1-sec(c+d*x))*sqrt(e*cot(c+d*x))/(a*d)-1/3*sqrt(cos(c-1/4*Pi+d*x)^2)/cos(c-1/4*Pi+d*x)*EllipticF(sin(c-1/4*Pi+d*x),sqrt(2))*sec(c+d*x)*sqrt(e*cot(c+d*x))*sqrt(sin(2*c+2*d*x))/(a*d)-arctan(1-sqrt(2)*sqrt(tan(c+d*x)))*sqrt(e*cot(c+d*x))*sqrt(tan(c+d*x))/(a*d*sqrt(2))+arctan(1+sqrt(2)*sqrt(tan(c+d*x)))*sqrt(e*cot(c+d*x))*sqrt(tan(c+d*x))/(a*d*sqrt(2))-1/2*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))*sqrt(e*cot(c+d*x))*sqrt(tan(c+d*x))/(a*d*sqrt(2))+1/2*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))*sqrt(e*cot(c+d*x))*sqrt(tan(c+d*x))/(a*d*sqrt(2))],
[1/((a+a*sec(c+d*x))*sqrt(e*cot(c+d*x))),x,19,2*cot(c+d*x)*(1-sec(c+d*x))/(a*d*sqrt(e*cot(c+d*x)))+2*sin(c+d*x)/(a*d*sqrt(e*cot(c+d*x)))-2*cos(c+d*x)*sqrt(cos(c-1/4*Pi+d*x)^2)/cos(c-1/4*Pi+d*x)*EllipticE(sin(c-1/4*Pi+d*x),sqrt(2))/(a*d*sqrt(e*cot(c+d*x))*sqrt(sin(2*c+2*d*x)))-arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(a*d*sqrt(2)*sqrt(e*cot(c+d*x))*sqrt(tan(c+d*x)))+arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(a*d*sqrt(2)*sqrt(e*cot(c+d*x))*sqrt(tan(c+d*x)))+1/2*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(a*d*sqrt(2)*sqrt(e*cot(c+d*x))*sqrt(tan(c+d*x)))-1/2*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(a*d*sqrt(2)*sqrt(e*cot(c+d*x))*sqrt(tan(c+d*x)))],
[1/((e*cot(c+d*x))^(3/2)*(a+a*sec(c+d*x))),x,17,cot(c+d*x)*csc(c+d*x)*sqrt(cos(c-1/4*Pi+d*x)^2)/cos(c-1/4*Pi+d*x)*EllipticF(sin(c-1/4*Pi+d*x),sqrt(2))*sqrt(sin(2*c+2*d*x))/(a*d*(e*cot(c+d*x))^(3/2))+arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(a*d*(e*cot(c+d*x))^(3/2)*sqrt(2)*tan(c+d*x)^(3/2))-arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(a*d*(e*cot(c+d*x))^(3/2)*sqrt(2)*tan(c+d*x)^(3/2))+1/2*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(a*d*(e*cot(c+d*x))^(3/2)*sqrt(2)*tan(c+d*x)^(3/2))-1/2*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(a*d*(e*cot(c+d*x))^(3/2)*sqrt(2)*tan(c+d*x)^(3/2))],
[1/((e*cot(c+d*x))^(5/2)*(a+a*sec(c+d*x))),x,18,2*cos(c+d*x)*cot(c+d*x)/(a*d*(e*cot(c+d*x))^(5/2))-2*cos(c+d*x)*cot(c+d*x)^2*sqrt(cos(c-1/4*Pi+d*x)^2)/cos(c-1/4*Pi+d*x)*EllipticE(sin(c-1/4*Pi+d*x),sqrt(2))/(a*d*(e*cot(c+d*x))^(5/2)*sqrt(sin(2*c+2*d*x)))+arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(a*d*(e*cot(c+d*x))^(5/2)*sqrt(2)*tan(c+d*x)^(5/2))-arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(a*d*(e*cot(c+d*x))^(5/2)*sqrt(2)*tan(c+d*x)^(5/2))-1/2*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(a*d*(e*cot(c+d*x))^(5/2)*sqrt(2)*tan(c+d*x)^(5/2))+1/2*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(a*d*(e*cot(c+d*x))^(5/2)*sqrt(2)*tan(c+d*x)^(5/2))],
[1/((e*cot(c+d*x))^(7/2)*(a+a*sec(c+d*x))),x,18,-2/3*cot(c+d*x)^3*(3-sec(c+d*x))/(a*d*(e*cot(c+d*x))^(7/2))-1/3*cot(c+d*x)^3*csc(c+d*x)*sqrt(cos(c-1/4*Pi+d*x)^2)/cos(c-1/4*Pi+d*x)*EllipticF(sin(c-1/4*Pi+d*x),sqrt(2))*sqrt(sin(2*c+2*d*x))/(a*d*(e*cot(c+d*x))^(7/2))-arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(a*d*(e*cot(c+d*x))^(7/2)*sqrt(2)*tan(c+d*x)^(7/2))+arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(a*d*(e*cot(c+d*x))^(7/2)*sqrt(2)*tan(c+d*x)^(7/2))-1/2*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(a*d*(e*cot(c+d*x))^(7/2)*sqrt(2)*tan(c+d*x)^(7/2))+1/2*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(a*d*(e*cot(c+d*x))^(7/2)*sqrt(2)*tan(c+d*x)^(7/2))],
[1/((e*cot(c+d*x))^(9/2)*(a+a*sec(c+d*x))),x,19,-6/5*cos(c+d*x)*cot(c+d*x)^3/(a*d*(e*cot(c+d*x))^(9/2))-2/15*cot(c+d*x)^3*(5-3*sec(c+d*x))/(a*d*(e*cot(c+d*x))^(9/2))+6/5*cos(c+d*x)*cot(c+d*x)^4*sqrt(cos(c-1/4*Pi+d*x)^2)/cos(c-1/4*Pi+d*x)*EllipticE(sin(c-1/4*Pi+d*x),sqrt(2))/(a*d*(e*cot(c+d*x))^(9/2)*sqrt(sin(2*c+2*d*x)))-arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(a*d*(e*cot(c+d*x))^(9/2)*sqrt(2)*tan(c+d*x)^(9/2))+arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(a*d*(e*cot(c+d*x))^(9/2)*sqrt(2)*tan(c+d*x)^(9/2))+1/2*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(a*d*(e*cot(c+d*x))^(9/2)*sqrt(2)*tan(c+d*x)^(9/2))-1/2*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(a*d*(e*cot(c+d*x))^(9/2)*sqrt(2)*tan(c+d*x)^(9/2))],
[1/((a+a*sec(c+d*x))^2*sqrt(e*cot(c+d*x))),x,24,2*cot(c+d*x)/(a^2*d*sqrt(e*cot(c+d*x)))-12/5*cos(c+d*x)*cot(c+d*x)/(a^2*d*sqrt(e*cot(c+d*x)))-4/5*cot(c+d*x)^3/(a^2*d*sqrt(e*cot(c+d*x)))+4/5*cot(c+d*x)^2*csc(c+d*x)/(a^2*d*sqrt(e*cot(c+d*x)))-12/5*cos(c+d*x)*sqrt(cos(c-1/4*Pi+d*x)^2)/cos(c-1/4*Pi+d*x)*EllipticE(sin(c-1/4*Pi+d*x),sqrt(2))/(a^2*d*sqrt(e*cot(c+d*x))*sqrt(sin(2*c+2*d*x)))-arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(a^2*d*sqrt(2)*sqrt(e*cot(c+d*x))*sqrt(tan(c+d*x)))+arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(a^2*d*sqrt(2)*sqrt(e*cot(c+d*x))*sqrt(tan(c+d*x)))+1/2*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(a^2*d*sqrt(2)*sqrt(e*cot(c+d*x))*sqrt(tan(c+d*x)))-1/2*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(a^2*d*sqrt(2)*sqrt(e*cot(c+d*x))*sqrt(tan(c+d*x)))],
[1/((e*cot(c+d*x))^(3/2)*(a+a*sec(c+d*x))^2),x,22,-4/3*cot(c+d*x)^3/(a^2*d*(e*cot(c+d*x))^(3/2))+4/3*cot(c+d*x)^2*csc(c+d*x)/(a^2*d*(e*cot(c+d*x))^(3/2))+2/3*cot(c+d*x)*csc(c+d*x)*sqrt(cos(c-1/4*Pi+d*x)^2)/cos(c-1/4*Pi+d*x)*EllipticF(sin(c-1/4*Pi+d*x),sqrt(2))*sqrt(sin(2*c+2*d*x))/(a^2*d*(e*cot(c+d*x))^(3/2))+arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(a^2*d*(e*cot(c+d*x))^(3/2)*sqrt(2)*tan(c+d*x)^(3/2))-arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(a^2*d*(e*cot(c+d*x))^(3/2)*sqrt(2)*tan(c+d*x)^(3/2))+1/2*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(a^2*d*(e*cot(c+d*x))^(3/2)*sqrt(2)*tan(c+d*x)^(3/2))-1/2*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(a^2*d*(e*cot(c+d*x))^(3/2)*sqrt(2)*tan(c+d*x)^(3/2))],
[1/((e*cot(c+d*x))^(5/2)*(a+a*sec(c+d*x))^2),x,22,-4*cot(c+d*x)^3/(a^2*d*(e*cot(c+d*x))^(5/2))+4*cos(c+d*x)*cot(c+d*x)^3/(a^2*d*(e*cot(c+d*x))^(5/2))+4*cos(c+d*x)*cot(c+d*x)^2*sqrt(cos(c-1/4*Pi+d*x)^2)/cos(c-1/4*Pi+d*x)*EllipticE(sin(c-1/4*Pi+d*x),sqrt(2))/(a^2*d*(e*cot(c+d*x))^(5/2)*sqrt(sin(2*c+2*d*x)))+arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(a^2*d*(e*cot(c+d*x))^(5/2)*sqrt(2)*tan(c+d*x)^(5/2))-arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(a^2*d*(e*cot(c+d*x))^(5/2)*sqrt(2)*tan(c+d*x)^(5/2))-1/2*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(a^2*d*(e*cot(c+d*x))^(5/2)*sqrt(2)*tan(c+d*x)^(5/2))+1/2*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(a^2*d*(e*cot(c+d*x))^(5/2)*sqrt(2)*tan(c+d*x)^(5/2))],
[1/((e*cot(c+d*x))^(7/2)*(a+a*sec(c+d*x))^2),x,20,2*cot(c+d*x)^3/(a^2*d*(e*cot(c+d*x))^(7/2))-2*cot(c+d*x)^3*csc(c+d*x)*sqrt(cos(c-1/4*Pi+d*x)^2)/cos(c-1/4*Pi+d*x)*EllipticF(sin(c-1/4*Pi+d*x),sqrt(2))*sqrt(sin(2*c+2*d*x))/(a^2*d*(e*cot(c+d*x))^(7/2))-arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(a^2*d*(e*cot(c+d*x))^(7/2)*sqrt(2)*tan(c+d*x)^(7/2))+arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(a^2*d*(e*cot(c+d*x))^(7/2)*sqrt(2)*tan(c+d*x)^(7/2))-1/2*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(a^2*d*(e*cot(c+d*x))^(7/2)*sqrt(2)*tan(c+d*x)^(7/2))+1/2*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(a^2*d*(e*cot(c+d*x))^(7/2)*sqrt(2)*tan(c+d*x)^(7/2))],
[1/((e*cot(c+d*x))^(9/2)*(a+a*sec(c+d*x))^2),x,21,2/3*cot(c+d*x)^3/(a^2*d*(e*cot(c+d*x))^(9/2))-4*cos(c+d*x)*cot(c+d*x)^3/(a^2*d*(e*cot(c+d*x))^(9/2))+4*cos(c+d*x)*cot(c+d*x)^4*sqrt(cos(c-1/4*Pi+d*x)^2)/cos(c-1/4*Pi+d*x)*EllipticE(sin(c-1/4*Pi+d*x),sqrt(2))/(a^2*d*(e*cot(c+d*x))^(9/2)*sqrt(sin(2*c+2*d*x)))-arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(a^2*d*(e*cot(c+d*x))^(9/2)*sqrt(2)*tan(c+d*x)^(9/2))+arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(a^2*d*(e*cot(c+d*x))^(9/2)*sqrt(2)*tan(c+d*x)^(9/2))+1/2*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(a^2*d*(e*cot(c+d*x))^(9/2)*sqrt(2)*tan(c+d*x)^(9/2))-1/2*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(a^2*d*(e*cot(c+d*x))^(9/2)*sqrt(2)*tan(c+d*x)^(9/2))],
[1/((e*cot(c+d*x))^(11/2)*(a+a*sec(c+d*x))^2),x,22,2/5*cot(c+d*x)^3/(a^2*d*(e*cot(c+d*x))^(11/2))+2*cot(c+d*x)^5/(a^2*d*(e*cot(c+d*x))^(11/2))-4/3*cot(c+d*x)^4*csc(c+d*x)/(a^2*d*(e*cot(c+d*x))^(11/2))+2/3*cot(c+d*x)^5*csc(c+d*x)*sqrt(cos(c-1/4*Pi+d*x)^2)/cos(c-1/4*Pi+d*x)*EllipticF(sin(c-1/4*Pi+d*x),sqrt(2))*sqrt(sin(2*c+2*d*x))/(a^2*d*(e*cot(c+d*x))^(11/2))+arctan(1-sqrt(2)*sqrt(tan(c+d*x)))/(a^2*d*(e*cot(c+d*x))^(11/2)*sqrt(2)*tan(c+d*x)^(11/2))-arctan(1+sqrt(2)*sqrt(tan(c+d*x)))/(a^2*d*(e*cot(c+d*x))^(11/2)*sqrt(2)*tan(c+d*x)^(11/2))+1/2*log(1-sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(a^2*d*(e*cot(c+d*x))^(11/2)*sqrt(2)*tan(c+d*x)^(11/2))-1/2*log(1+sqrt(2)*sqrt(tan(c+d*x))+tan(c+d*x))/(a^2*d*(e*cot(c+d*x))^(11/2)*sqrt(2)*tan(c+d*x)^(11/2))],

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

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

# n>0
[(a+b*sec(c+d*x))*tan(c+d*x)^7,x,7,a*log(cos(c+d*x))/d-16/35*b*sec(c+d*x)/d+1/70*(35*a+16*b*sec(c+d*x))*tan(c+d*x)^2/d-1/140*(35*a+24*b*sec(c+d*x))*tan(c+d*x)^4/d+1/42*(7*a+6*b*sec(c+d*x))*tan(c+d*x)^6/d],
[(a+b*sec(c+d*x))*tan(c+d*x)^5,x,6,-a*log(cos(c+d*x))/d+8/15*b*sec(c+d*x)/d-1/30*(15*a+8*b*sec(c+d*x))*tan(c+d*x)^2/d+1/20*(5*a+4*b*sec(c+d*x))*tan(c+d*x)^4/d],
[(a+b*sec(c+d*x))*tan(c+d*x)^3,x,5,a*log(cos(c+d*x))/d-2/3*b*sec(c+d*x)/d+1/6*(3*a+2*b*sec(c+d*x))*tan(c+d*x)^2/d],
[(a+b*sec(c+d*x))*tan(c+d*x),x,4,-a*log(cos(c+d*x))/d+b*sec(c+d*x)/d],
[cot(c+d*x)*(a+b*sec(c+d*x)),x,5,1/2*(a+b)*log(1-cos(c+d*x))/d+1/2*(a-b)*log(1+cos(c+d*x))/d],
[cot(c+d*x)^3*(a+b*sec(c+d*x)),x,6,-1/4*(2*a+b)*log(1-cos(c+d*x))/d-1/4*(2*a-b)*log(1+cos(c+d*x))/d-1/2*cot(c+d*x)^2*(a+b*sec(c+d*x))/d],
[cot(c+d*x)^5*(a+b*sec(c+d*x)),x,7,1/16*(8*a+3*b)*log(1-cos(c+d*x))/d+1/16*(8*a-3*b)*log(1+cos(c+d*x))/d-1/4*cot(c+d*x)^4*(a+b*sec(c+d*x))/d+1/8*cot(c+d*x)^2*(4*a+3*b*sec(c+d*x))/d],
[cot(c+d*x)^7*(a+b*sec(c+d*x)),x,8,-1/32*(16*a+5*b)*log(1-cos(c+d*x))/d-1/32*(16*a-5*b)*log(1+cos(c+d*x))/d-1/6*cot(c+d*x)^6*(a+b*sec(c+d*x))/d+1/24*cot(c+d*x)^4*(6*a+5*b*sec(c+d*x))/d-1/16*cot(c+d*x)^2*(8*a+5*b*sec(c+d*x))/d],
[(a+b*sec(c+d*x))*tan(c+d*x)^6,x,5,-a*x-5/16*b*arctanh(sin(c+d*x))/d+1/16*(16*a+5*b*sec(c+d*x))*tan(c+d*x)/d-1/24*(8*a+5*b*sec(c+d*x))*tan(c+d*x)^3/d+1/30*(6*a+5*b*sec(c+d*x))*tan(c+d*x)^5/d],
[(a+b*sec(c+d*x))*tan(c+d*x)^4,x,4,a*x+3/8*b*arctanh(sin(c+d*x))/d-1/8*(8*a+3*b*sec(c+d*x))*tan(c+d*x)/d+1/12*(4*a+3*b*sec(c+d*x))*tan(c+d*x)^3/d],
[(a+b*sec(c+d*x))*tan(c+d*x)^2,x,3,-a*x-1/2*b*arctanh(sin(c+d*x))/d+1/2*(2*a+b*sec(c+d*x))*tan(c+d*x)/d],
[cot(c+d*x)^2*(a+b*sec(c+d*x)),x,2,-a*x-cot(c+d*x)*(a+b*sec(c+d*x))/d],
[cot(c+d*x)^4*(a+b*sec(c+d*x)),x,3,a*x-1/3*cot(c+d*x)^3*(a+b*sec(c+d*x))/d+1/3*cot(c+d*x)*(3*a+2*b*sec(c+d*x))/d],
[cot(c+d*x)^6*(a+b*sec(c+d*x)),x,4,-a*x-1/5*cot(c+d*x)^5*(a+b*sec(c+d*x))/d+1/15*cot(c+d*x)^3*(5*a+4*b*sec(c+d*x))/d-1/15*cot(c+d*x)*(15*a+8*b*sec(c+d*x))/d],
[cot(c+d*x)^8*(a+b*sec(c+d*x)),x,5,a*x-1/7*cot(c+d*x)^7*(a+b*sec(c+d*x))/d+1/35*cot(c+d*x)^5*(7*a+6*b*sec(c+d*x))/d+1/35*cot(c+d*x)*(35*a+16*b*sec(c+d*x))/d-1/105*cot(c+d*x)^3*(35*a+24*b*sec(c+d*x))/d],
[(a+b*sec(c+d*x))^2*tan(c+d*x)^9,x,3,-a^2*log(cos(c+d*x))/d+2*a*b*sec(c+d*x)/d-2*a^2*sec(c+d*x)^2/d-8/3*a*b*sec(c+d*x)^3/d+3/2*a^2*sec(c+d*x)^4/d+12/5*a*b*sec(c+d*x)^5/d-2/3*a^2*sec(c+d*x)^6/d-8/7*a*b*sec(c+d*x)^7/d+1/8*a^2*sec(c+d*x)^8/d+2/9*a*b*sec(c+d*x)^9/d+1/10*b^2*tan(c+d*x)^10/d,-a^2*log(cos(c+d*x))/d+2*a*b*sec(c+d*x)/d-1/2*(4*a^2-b^2)*sec(c+d*x)^2/d-8/3*a*b*sec(c+d*x)^3/d+1/2*(3*a^2-2*b^2)*sec(c+d*x)^4/d+12/5*a*b*sec(c+d*x)^5/d-1/3*(2*a^2-3*b^2)*sec(c+d*x)^6/d-8/7*a*b*sec(c+d*x)^7/d+1/8*(a^2-4*b^2)*sec(c+d*x)^8/d+2/9*a*b*sec(c+d*x)^9/d+1/10*b^2*sec(c+d*x)^10/d],
[(a+b*sec(c+d*x))^2*tan(c+d*x)^7,x,3,a^2*log(cos(c+d*x))/d-2*a*b*sec(c+d*x)/d+3/2*a^2*sec(c+d*x)^2/d+2*a*b*sec(c+d*x)^3/d-3/4*a^2*sec(c+d*x)^4/d-6/5*a*b*sec(c+d*x)^5/d+1/6*a^2*sec(c+d*x)^6/d+2/7*a*b*sec(c+d*x)^7/d+1/8*b^2*tan(c+d*x)^8/d,a^2*log(cos(c+d*x))/d-2*a*b*sec(c+d*x)/d+1/2*(3*a^2-b^2)*sec(c+d*x)^2/d+2*a*b*sec(c+d*x)^3/d-3/4*(a^2-b^2)*sec(c+d*x)^4/d-6/5*a*b*sec(c+d*x)^5/d+1/6*(a^2-3*b^2)*sec(c+d*x)^6/d+2/7*a*b*sec(c+d*x)^7/d+1/8*b^2*sec(c+d*x)^8/d],
[(a+b*sec(c+d*x))^2*tan(c+d*x)^5,x,3,-a^2*log(cos(c+d*x))/d+2*a*b*sec(c+d*x)/d-a^2*sec(c+d*x)^2/d-4/3*a*b*sec(c+d*x)^3/d+1/4*a^2*sec(c+d*x)^4/d+2/5*a*b*sec(c+d*x)^5/d+1/6*b^2*tan(c+d*x)^6/d,-a^2*log(cos(c+d*x))/d+2*a*b*sec(c+d*x)/d-1/2*(2*a^2-b^2)*sec(c+d*x)^2/d-4/3*a*b*sec(c+d*x)^3/d+1/4*(a^2-2*b^2)*sec(c+d*x)^4/d+2/5*a*b*sec(c+d*x)^5/d+1/6*b^2*sec(c+d*x)^6/d],
[(a+b*sec(c+d*x))^2*tan(c+d*x)^3,x,3,a^2*log(cos(c+d*x))/d-2*a*b*sec(c+d*x)/d+1/2*(a^2-b^2)*sec(c+d*x)^2/d+2/3*a*b*sec(c+d*x)^3/d+1/4*b^2*sec(c+d*x)^4/d],
[(a+b*sec(c+d*x))^2*tan(c+d*x),x,3,-a^2*log(cos(c+d*x))/d+2*a*b*sec(c+d*x)/d+1/2*b^2*sec(c+d*x)^2/d],
[cot(c+d*x)*(a+b*sec(c+d*x))^2,x,3,a^2*log(cos(c+d*x))/d+1/2*(a+b)^2*log(1-sec(c+d*x))/d+1/2*(a-b)^2*log(1+sec(c+d*x))/d],
[cot(c+d*x)^3*(a+b*sec(c+d*x))^2,x,4,-a^2*log(cos(c+d*x))/d-1/2*a*(a+b)*log(1-sec(c+d*x))/d-1/2*a*(a-b)*log(1+sec(c+d*x))/d-1/2*cot(c+d*x)^2*(a^2+b^2+2*a*b*sec(c+d*x))/d],
[cot(c+d*x)^5*(a+b*sec(c+d*x))^2,x,5,a^2*log(cos(c+d*x))/d+1/8*a*(4*a+3*b)*log(1-sec(c+d*x))/d+1/8*a*(4*a-3*b)*log(1+sec(c+d*x))/d+1/4*a*cot(c+d*x)^2*(2*a+3*b*sec(c+d*x))/d-1/4*cot(c+d*x)^4*(a^2+b^2+2*a*b*sec(c+d*x))/d],
[(a+b*sec(c+d*x))^2*tan(c+d*x)^6,x,12,-a^2*x-5/8*a*b*arctanh(sin(c+d*x))/d+a^2*tan(c+d*x)/d+5/8*a*b*sec(c+d*x)*tan(c+d*x)/d-1/3*a^2*tan(c+d*x)^3/d-5/12*a*b*sec(c+d*x)*tan(c+d*x)^3/d+1/5*a^2*tan(c+d*x)^5/d+1/3*a*b*sec(c+d*x)*tan(c+d*x)^5/d+1/7*b^2*tan(c+d*x)^7/d],
[(a+b*sec(c+d*x))^2*tan(c+d*x)^4,x,10,a^2*x+3/4*a*b*arctanh(sin(c+d*x))/d-a^2*tan(c+d*x)/d-3/4*a*b*sec(c+d*x)*tan(c+d*x)/d+1/3*a^2*tan(c+d*x)^3/d+1/2*a*b*sec(c+d*x)*tan(c+d*x)^3/d+1/5*b^2*tan(c+d*x)^5/d],
[(a+b*sec(c+d*x))^2*tan(c+d*x)^2,x,8,-a^2*x-a*b*arctanh(sin(c+d*x))/d+a^2*tan(c+d*x)/d+a*b*sec(c+d*x)*tan(c+d*x)/d+1/3*b^2*tan(c+d*x)^3/d],
[cot(c+d*x)^2*(a+b*sec(c+d*x))^2,x,8,-a^2*x-a^2*cot(c+d*x)/d-b^2*cot(c+d*x)/d-2*a*b*csc(c+d*x)/d],
[cot(c+d*x)^4*(a+b*sec(c+d*x))^2,x,9,a^2*x+a^2*cot(c+d*x)/d-1/3*a^2*cot(c+d*x)^3/d-1/3*b^2*cot(c+d*x)^3/d+2*a*b*csc(c+d*x)/d-2/3*a*b*csc(c+d*x)^3/d],
[cot(c+d*x)^6*(a+b*sec(c+d*x))^2,x,11,-a^2*x-a^2*cot(c+d*x)/d+1/3*a^2*cot(c+d*x)^3/d-1/5*a^2*cot(c+d*x)^5/d-1/5*b^2*cot(c+d*x)^5/d-2*a*b*csc(c+d*x)/d+4/3*a*b*csc(c+d*x)^3/d-2/5*a*b*csc(c+d*x)^5/d],
[cot(c+d*x)^8*(a+b*sec(c+d*x))^2,x,12,a^2*x+a^2*cot(c+d*x)/d-1/3*a^2*cot(c+d*x)^3/d+1/5*a^2*cot(c+d*x)^5/d-1/7*a^2*cot(c+d*x)^7/d-1/7*b^2*cot(c+d*x)^7/d+2*a*b*csc(c+d*x)/d-2*a*b*csc(c+d*x)^3/d+6/5*a*b*csc(c+d*x)^5/d-2/7*a*b*csc(c+d*x)^7/d],

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

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

# n>0

# n<0
[(e*tan(c+d*x))^(5/2)/(a+b*sec(c+d*x)),x,38,a*e^(5/2)*arctan(1-sqrt(2)*sqrt(e*tan(c+d*x))/sqrt(e))/(b^2*d*sqrt(2))-(a^2-b^2)*e^(5/2)*arctan(1-sqrt(2)*sqrt(e*tan(c+d*x))/sqrt(e))/(a*b^2*d*sqrt(2))-a*e^(5/2)*arctan(1+sqrt(2)*sqrt(e*tan(c+d*x))/sqrt(e))/(b^2*d*sqrt(2))+(a^2-b^2)*e^(5/2)*arctan(1+sqrt(2)*sqrt(e*tan(c+d*x))/sqrt(e))/(a*b^2*d*sqrt(2))-1/2*a*e^(5/2)*log(sqrt(e)-sqrt(2)*sqrt(e*tan(c+d*x))+sqrt(e)*tan(c+d*x))/(b^2*d*sqrt(2))+1/2*(a^2-b^2)*e^(5/2)*log(sqrt(e)-sqrt(2)*sqrt(e*tan(c+d*x))+sqrt(e)*tan(c+d*x))/(a*b^2*d*sqrt(2))+1/2*a*e^(5/2)*log(sqrt(e)+sqrt(2)*sqrt(e*tan(c+d*x))+sqrt(e)*tan(c+d*x))/(b^2*d*sqrt(2))-1/2*(a^2-b^2)*e^(5/2)*log(sqrt(e)+sqrt(2)*sqrt(e*tan(c+d*x))+sqrt(e)*tan(c+d*x))/(a*b^2*d*sqrt(2))+2*e^2*EllipticPi(sqrt(sin(c+d*x))/sqrt(1+cos(c+d*x)),-sqrt(a-b)/sqrt(a+b),I)*sqrt(2)*sqrt(a-b)*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(e*tan(c+d*x))/(a*b*d*sqrt(sin(c+d*x)))-2*e^2*EllipticPi(sqrt(sin(c+d*x))/sqrt(1+cos(c+d*x)),sqrt(a-b)/sqrt(a+b),I)*sqrt(2)*sqrt(a-b)*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(e*tan(c+d*x))/(a*b*d*sqrt(sin(c+d*x)))-2*e^2*cos(c+d*x)*sqrt(cos(c-1/4*Pi+d*x)^2)/cos(c-1/4*Pi+d*x)*EllipticE(sin(c-1/4*Pi+d*x),sqrt(2))*sqrt(e*tan(c+d*x))/(b*d*sqrt(sin(2*c+2*d*x)))+2*e*cos(c+d*x)*(e*tan(c+d*x))^(3/2)/(b*d)],
[(e*tan(c+d*x))^(3/2)/(a+b*sec(c+d*x)),x,35,a*e^(3/2)*arctan(1-sqrt(2)*sqrt(e*tan(c+d*x))/sqrt(e))/(b^2*d*sqrt(2))-(a^2-b^2)*e^(3/2)*arctan(1-sqrt(2)*sqrt(e*tan(c+d*x))/sqrt(e))/(a*b^2*d*sqrt(2))-a*e^(3/2)*arctan(1+sqrt(2)*sqrt(e*tan(c+d*x))/sqrt(e))/(b^2*d*sqrt(2))+(a^2-b^2)*e^(3/2)*arctan(1+sqrt(2)*sqrt(e*tan(c+d*x))/sqrt(e))/(a*b^2*d*sqrt(2))+1/2*a*e^(3/2)*log(sqrt(e)-sqrt(2)*sqrt(e*tan(c+d*x))+sqrt(e)*tan(c+d*x))/(b^2*d*sqrt(2))-1/2*(a^2-b^2)*e^(3/2)*log(sqrt(e)-sqrt(2)*sqrt(e*tan(c+d*x))+sqrt(e)*tan(c+d*x))/(a*b^2*d*sqrt(2))-1/2*a*e^(3/2)*log(sqrt(e)+sqrt(2)*sqrt(e*tan(c+d*x))+sqrt(e)*tan(c+d*x))/(b^2*d*sqrt(2))+1/2*(a^2-b^2)*e^(3/2)*log(sqrt(e)+sqrt(2)*sqrt(e*tan(c+d*x))+sqrt(e)*tan(c+d*x))/(a*b^2*d*sqrt(2))-2*e^2*EllipticPi(sqrt(-cos(c+d*x))/sqrt(1+sin(c+d*x)),b/(a-sqrt(a^2-b^2)),I)*sqrt(2)*sqrt(a^2-b^2)*sqrt(sin(c+d*x))/(a*b*d*sqrt(-cos(c+d*x))*sqrt(e*tan(c+d*x)))+2*e^2*EllipticPi(sqrt(-cos(c+d*x))/sqrt(1+sin(c+d*x)),b/(a+sqrt(a^2-b^2)),I)*sqrt(2)*sqrt(a^2-b^2)*sqrt(sin(c+d*x))/(a*b*d*sqrt(-cos(c+d*x))*sqrt(e*tan(c+d*x)))+e^2*sqrt(cos(c-1/4*Pi+d*x)^2)/cos(c-1/4*Pi+d*x)*EllipticF(sin(c-1/4*Pi+d*x),sqrt(2))*sec(c+d*x)*sqrt(sin(2*c+2*d*x))/(b*d*sqrt(e*tan(c+d*x)))],
[(e*tan(c+d*x))^(1/2)/(a+b*sec(c+d*x)),x,21,-arctan(1-sqrt(2)*sqrt(e*tan(c+d*x))/sqrt(e))*sqrt(e)/(a*d*sqrt(2))+arctan(1+sqrt(2)*sqrt(e*tan(c+d*x))/sqrt(e))*sqrt(e)/(a*d*sqrt(2))+1/2*log(sqrt(e)-sqrt(2)*sqrt(e*tan(c+d*x))+sqrt(e)*tan(c+d*x))*sqrt(e)/(a*d*sqrt(2))-1/2*log(sqrt(e)+sqrt(2)*sqrt(e*tan(c+d*x))+sqrt(e)*tan(c+d*x))*sqrt(e)/(a*d*sqrt(2))+2*b*EllipticPi(sqrt(sin(c+d*x))/sqrt(1+cos(c+d*x)),-sqrt(a-b)/sqrt(a+b),I)*sqrt(2)*sqrt(cos(c+d*x))*sqrt(e*tan(c+d*x))/(a*d*sqrt(a-b)*sqrt(a+b)*sqrt(sin(c+d*x)))-2*b*EllipticPi(sqrt(sin(c+d*x))/sqrt(1+cos(c+d*x)),sqrt(a-b)/sqrt(a+b),I)*sqrt(2)*sqrt(cos(c+d*x))*sqrt(e*tan(c+d*x))/(a*d*sqrt(a-b)*sqrt(a+b)*sqrt(sin(c+d*x)))],
[1/((a+b*sec(c+d*x))*(e*tan(c+d*x))^(1/2)),x,19,-arctan(1-sqrt(2)*sqrt(e*tan(c+d*x))/sqrt(e))/(a*d*sqrt(2)*sqrt(e))+arctan(1+sqrt(2)*sqrt(e*tan(c+d*x))/sqrt(e))/(a*d*sqrt(2)*sqrt(e))-1/2*log(sqrt(e)-sqrt(2)*sqrt(e*tan(c+d*x))+sqrt(e)*tan(c+d*x))/(a*d*sqrt(2)*sqrt(e))+1/2*log(sqrt(e)+sqrt(2)*sqrt(e*tan(c+d*x))+sqrt(e)*tan(c+d*x))/(a*d*sqrt(2)*sqrt(e))-2*b*EllipticPi(sqrt(-cos(c+d*x))/sqrt(1+sin(c+d*x)),b/(a-sqrt(a^2-b^2)),I)*sqrt(2)*sqrt(sin(c+d*x))/(a*d*sqrt(a^2-b^2)*sqrt(-cos(c+d*x))*sqrt(e*tan(c+d*x)))+2*b*EllipticPi(sqrt(-cos(c+d*x))/sqrt(1+sin(c+d*x)),b/(a+sqrt(a^2-b^2)),I)*sqrt(2)*sqrt(sin(c+d*x))/(a*d*sqrt(a^2-b^2)*sqrt(-cos(c+d*x))*sqrt(e*tan(c+d*x)))],
[1/((a+b*sec(c+d*x))*(e*tan(c+d*x))^(3/2)),x,39,a*arctan(1-sqrt(2)*sqrt(e*tan(c+d*x))/sqrt(e))/((a^2-b^2)*d*e^(3/2)*sqrt(2))-b^2*arctan(1-sqrt(2)*sqrt(e*tan(c+d*x))/sqrt(e))/(a*(a^2-b^2)*d*e^(3/2)*sqrt(2))-a*arctan(1+sqrt(2)*sqrt(e*tan(c+d*x))/sqrt(e))/((a^2-b^2)*d*e^(3/2)*sqrt(2))+b^2*arctan(1+sqrt(2)*sqrt(e*tan(c+d*x))/sqrt(e))/(a*(a^2-b^2)*d*e^(3/2)*sqrt(2))-1/2*a*log(sqrt(e)-sqrt(2)*sqrt(e*tan(c+d*x))+sqrt(e)*tan(c+d*x))/((a^2-b^2)*d*e^(3/2)*sqrt(2))+1/2*b^2*log(sqrt(e)-sqrt(2)*sqrt(e*tan(c+d*x))+sqrt(e)*tan(c+d*x))/(a*(a^2-b^2)*d*e^(3/2)*sqrt(2))+1/2*a*log(sqrt(e)+sqrt(2)*sqrt(e*tan(c+d*x))+sqrt(e)*tan(c+d*x))/((a^2-b^2)*d*e^(3/2)*sqrt(2))-1/2*b^2*log(sqrt(e)+sqrt(2)*sqrt(e*tan(c+d*x))+sqrt(e)*tan(c+d*x))/(a*(a^2-b^2)*d*e^(3/2)*sqrt(2))-2*(a-b*sec(c+d*x))/((a^2-b^2)*d*e*sqrt(e*tan(c+d*x)))+2*b^3*EllipticPi(sqrt(sin(c+d*x))/sqrt(1+cos(c+d*x)),-sqrt(a-b)/sqrt(a+b),I)*sqrt(2)*sqrt(cos(c+d*x))*sqrt(e*tan(c+d*x))/(a*(a-b)^(3/2)*(a+b)^(3/2)*d*e^2*sqrt(sin(c+d*x)))-2*b^3*EllipticPi(sqrt(sin(c+d*x))/sqrt(1+cos(c+d*x)),sqrt(a-b)/sqrt(a+b),I)*sqrt(2)*sqrt(cos(c+d*x))*sqrt(e*tan(c+d*x))/(a*(a-b)^(3/2)*(a+b)^(3/2)*d*e^2*sqrt(sin(c+d*x)))+2*b*cos(c+d*x)*sqrt(cos(c-1/4*Pi+d*x)^2)/cos(c-1/4*Pi+d*x)*EllipticE(sin(c-1/4*Pi+d*x),sqrt(2))*sqrt(e*tan(c+d*x))/((a^2-b^2)*d*e^2*sqrt(sin(2*c+2*d*x)))-2*b*cos(c+d*x)*(e*tan(c+d*x))^(3/2)/((a^2-b^2)*d*e^3)],
[1/((a+b*sec(c+d*x))*(e*tan(c+d*x))^(5/2)),x,36,a*arctan(1-sqrt(2)*sqrt(e*tan(c+d*x))/sqrt(e))/((a^2-b^2)*d*e^(5/2)*sqrt(2))-b^2*arctan(1-sqrt(2)*sqrt(e*tan(c+d*x))/sqrt(e))/(a*(a^2-b^2)*d*e^(5/2)*sqrt(2))-a*arctan(1+sqrt(2)*sqrt(e*tan(c+d*x))/sqrt(e))/((a^2-b^2)*d*e^(5/2)*sqrt(2))+b^2*arctan(1+sqrt(2)*sqrt(e*tan(c+d*x))/sqrt(e))/(a*(a^2-b^2)*d*e^(5/2)*sqrt(2))+1/2*a*log(sqrt(e)-sqrt(2)*sqrt(e*tan(c+d*x))+sqrt(e)*tan(c+d*x))/((a^2-b^2)*d*e^(5/2)*sqrt(2))-1/2*b^2*log(sqrt(e)-sqrt(2)*sqrt(e*tan(c+d*x))+sqrt(e)*tan(c+d*x))/(a*(a^2-b^2)*d*e^(5/2)*sqrt(2))-1/2*a*log(sqrt(e)+sqrt(2)*sqrt(e*tan(c+d*x))+sqrt(e)*tan(c+d*x))/((a^2-b^2)*d*e^(5/2)*sqrt(2))+1/2*b^2*log(sqrt(e)+sqrt(2)*sqrt(e*tan(c+d*x))+sqrt(e)*tan(c+d*x))/(a*(a^2-b^2)*d*e^(5/2)*sqrt(2))-2*b^3*EllipticPi(sqrt(-cos(c+d*x))/sqrt(1+sin(c+d*x)),b/(a-sqrt(a^2-b^2)),I)*sqrt(2)*sqrt(sin(c+d*x))/(a*(a^2-b^2)^(3/2)*d*e^2*sqrt(-cos(c+d*x))*sqrt(e*tan(c+d*x)))+2*b^3*EllipticPi(sqrt(-cos(c+d*x))/sqrt(1+sin(c+d*x)),b/(a+sqrt(a^2-b^2)),I)*sqrt(2)*sqrt(sin(c+d*x))/(a*(a^2-b^2)^(3/2)*d*e^2*sqrt(-cos(c+d*x))*sqrt(e*tan(c+d*x)))+1/3*b*sqrt(cos(c-1/4*Pi+d*x)^2)/cos(c-1/4*Pi+d*x)*EllipticF(sin(c-1/4*Pi+d*x),sqrt(2))*sec(c+d*x)*sqrt(sin(2*c+2*d*x))/((a^2-b^2)*d*e^2*sqrt(e*tan(c+d*x)))-2/3*(a-b*sec(c+d*x))/((a^2-b^2)*d*e*(e*tan(c+d*x))^(3/2))],

#  Mathematica indicates these have a closed-form antiderivative in terms of EllipticPi. 
#  {(e*Tan[c + d*x])^(5/2)/(a + b*Sec[c + d*x])^2, x, 21, 0}
# {(e*Tan[c + d*x])^(3/2)/(a + b*Sec[c + d*x])^2, x, 20, 0}
# {(e*Tan[c + d*x])^(1/2)/(a + b*Sec[c + d*x])^2, x, 9, 0}
# {1/(e*Tan[c + d*x])^(1/2)/(a + b*Sec[c + d*x])^2, x, 9, 0}
# {1/(e*Tan[c + d*x])^(3/2)/(a + b*Sec[c + d*x])^2, x, 22, 0}

# {1/(e*Tan[c + d*x])^(5/2)/(a + b*Sec[c + d*x])^2, x, 22, 0} 

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

# m>0
[sqrt(a+b*sec(c+d*x))*tan(c+d*x)^5,x,5,-2/3*a*(a^2-2*b^2)*(a+b*sec(c+d*x))^(3/2)/(b^4*d)+2/5*(3*a^2-2*b^2)*(a+b*sec(c+d*x))^(5/2)/(b^4*d)-6/7*a*(a+b*sec(c+d*x))^(7/2)/(b^4*d)+2/9*(a+b*sec(c+d*x))^(9/2)/(b^4*d)-2*arctanh(sqrt(a+b*sec(c+d*x))/sqrt(a))*sqrt(a)/d+2*sqrt(a+b*sec(c+d*x))/d],
[sqrt(a+b*sec(c+d*x))*tan(c+d*x)^3,x,5,-2/3*a*(a+b*sec(c+d*x))^(3/2)/(b^2*d)+2/5*(a+b*sec(c+d*x))^(5/2)/(b^2*d)+2*arctanh(sqrt(a+b*sec(c+d*x))/sqrt(a))*sqrt(a)/d-2*sqrt(a+b*sec(c+d*x))/d],
[sqrt(a+b*sec(c+d*x))*tan(c+d*x),x,4,-2*arctanh(sqrt(a+b*sec(c+d*x))/sqrt(a))*sqrt(a)/d+2*sqrt(a+b*sec(c+d*x))/d],
[cot(c+d*x)*sqrt(a+b*sec(c+d*x)),x,7,2*arctanh(sqrt(a+b*sec(c+d*x))/sqrt(a))*sqrt(a)/d-arctanh(sqrt(a+b*sec(c+d*x))/sqrt(a-b))*sqrt(a-b)/d-arctanh(sqrt(a+b*sec(c+d*x))/sqrt(a+b))*sqrt(a+b)/d],
[cot(c+d*x)^3*sqrt(a+b*sec(c+d*x)),x,13,-2*arctanh(sqrt(a+b*sec(c+d*x))/sqrt(a))*sqrt(a)/d+a*arctanh(sqrt(a+b*sec(c+d*x))/sqrt(a-b))/(d*sqrt(a-b))-3/4*b*arctanh(sqrt(a+b*sec(c+d*x))/sqrt(a-b))/(d*sqrt(a-b))+a*arctanh(sqrt(a+b*sec(c+d*x))/sqrt(a+b))/(d*sqrt(a+b))+3/4*b*arctanh(sqrt(a+b*sec(c+d*x))/sqrt(a+b))/(d*sqrt(a+b))-1/2*cot(c+d*x)^2*sqrt(a+b*sec(c+d*x))/d],
[sqrt(a+b*sec(c+d*x))*tan(c+d*x)^2,x,7,-2/3*a*(a-b)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b^2*d)-2/3*(a+2*b)*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b*d)+2*cot(c+d*x)*EllipticPi(sqrt(a+b*sec(c+d*x))/sqrt(a+b),(a+b)/a,sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/d+2/3*sqrt(a+b*sec(c+d*x))*tan(c+d*x)/d],
[sqrt(a+b*sec(c+d*x)),x,1,-2*cot(c+d*x)*EllipticPi(sqrt(a+b)/sqrt(a+b*sec(c+d*x)),a/(a+b),sqrt((a-b)/(a+b)))*(a+b*sec(c+d*x))*sqrt(-b*(1-sec(c+d*x))/(a+b*sec(c+d*x)))*sqrt(b*(1+sec(c+d*x))/(a+b*sec(c+d*x)))/(d*sqrt(a+b))],
[cot(c+d*x)^2*sqrt(a+b*sec(c+d*x)),x,5,cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/d+2*cot(c+d*x)*EllipticPi(sqrt(a+b)/sqrt(a+b*sec(c+d*x)),a/(a+b),sqrt((a-b)/(a+b)))*(a+b*sec(c+d*x))*sqrt(-b*(1-sec(c+d*x))/(a+b*sec(c+d*x)))*sqrt(b*(1+sec(c+d*x))/(a+b*sec(c+d*x)))/(d*sqrt(a+b))-cot(c+d*x)*sqrt(a+b*sec(c+d*x))/d],

# m<0
[tan(c+d*x)^5/sqrt(a+b*sec(c+d*x)),x,5,2/3*(3*a^2-2*b^2)*(a+b*sec(c+d*x))^(3/2)/(b^4*d)-6/5*a*(a+b*sec(c+d*x))^(5/2)/(b^4*d)+2/7*(a+b*sec(c+d*x))^(7/2)/(b^4*d)-2*arctanh(sqrt(a+b*sec(c+d*x))/sqrt(a))/(d*sqrt(a))-2*a*(a^2-2*b^2)*sqrt(a+b*sec(c+d*x))/(b^4*d)],
[tan(c+d*x)^3/sqrt(a+b*sec(c+d*x)),x,5,2/3*(a+b*sec(c+d*x))^(3/2)/(b^2*d)+2*arctanh(sqrt(a+b*sec(c+d*x))/sqrt(a))/(d*sqrt(a))-2*a*sqrt(a+b*sec(c+d*x))/(b^2*d)],
[tan(c+d*x)/sqrt(a+b*sec(c+d*x)),x,3,-2*arctanh(sqrt(a+b*sec(c+d*x))/sqrt(a))/(d*sqrt(a))],
[cot(c+d*x)/sqrt(a+b*sec(c+d*x)),x,7,2*arctanh(sqrt(a+b*sec(c+d*x))/sqrt(a))/(d*sqrt(a))-arctanh(sqrt(a+b*sec(c+d*x))/sqrt(a-b))/(d*sqrt(a-b))-arctanh(sqrt(a+b*sec(c+d*x))/sqrt(a+b))/(d*sqrt(a+b))],
[cot(c+d*x)^3/sqrt(a+b*sec(c+d*x)),x,11,-1/4*b*arctanh(sqrt(a+b*sec(c+d*x))/sqrt(a-b))/((a-b)^(3/2)*d)+1/4*b*arctanh(sqrt(a+b*sec(c+d*x))/sqrt(a+b))/((a+b)^(3/2)*d)-2*arctanh(sqrt(a+b*sec(c+d*x))/sqrt(a))/(d*sqrt(a))+arctanh(sqrt(a+b*sec(c+d*x))/sqrt(a-b))/(d*sqrt(a-b))+arctanh(sqrt(a+b*sec(c+d*x))/sqrt(a+b))/(d*sqrt(a+b))+1/4*sqrt(a+b*sec(c+d*x))/((a+b)*d*(1-sec(c+d*x)))+1/4*sqrt(a+b*sec(c+d*x))/((a-b)*d*(1+sec(c+d*x)))],
[tan(c+d*x)^4/sqrt(a+b*sec(c+d*x)),x,11,-2*cot(c+d*x)*EllipticPi(sqrt(a+b*sec(c+d*x))/sqrt(a+b),(a+b)/a,sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a*d)-2/15*(a-b)*(8*a^2-21*b^2)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(-b*(-1+sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b^4*d)+2/15*(-8*a^2+2*a*b+21*b^2)*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(-b*(-1+sec(c+d*x))/(a+b))*sqrt(b*(1+sec(c+d*x))/(-a+b))/(b^3*d)-8/15*a*sqrt(a+b*sec(c+d*x))*tan(c+d*x)/(b^2*d)+2/5*sec(c+d*x)*sqrt(a+b*sec(c+d*x))*tan(c+d*x)/(b*d),4*(a-b)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b^2*d)-2/15*(a-b)*(8*a^2+9*b^2)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b^4*d)+4*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b*d)-2/15*(8*a^2-2*a*b+9*b^2)*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b^3*d)-2*cot(c+d*x)*EllipticPi(sqrt(a+b*sec(c+d*x))/sqrt(a+b),(a+b)/a,sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a*d)-8/15*a*sqrt(a+b*sec(c+d*x))*tan(c+d*x)/(b^2*d)+2/5*sec(c+d*x)*sqrt(a+b*sec(c+d*x))*tan(c+d*x)/(b*d)],
[tan(c+d*x)^2/sqrt(a+b*sec(c+d*x)),x,6,-2*(a-b)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b^2*d)-2*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b*d)+2*cot(c+d*x)*EllipticPi(sqrt(a+b*sec(c+d*x))/sqrt(a+b),(a+b)/a,sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a*d)],
[1/sqrt(a+b*sec(c+d*x)),x,1,-2*cot(c+d*x)*EllipticPi(sqrt(a+b*sec(c+d*x))/sqrt(a+b),(a+b)/a,sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a*d)],
[cot(c+d*x)^2/sqrt(a+b*sec(c+d*x)),x,9,cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(d*sqrt(a+b))-cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(d*sqrt(a+b))+2*cot(c+d*x)*EllipticPi(sqrt(a+b*sec(c+d*x))/sqrt(a+b),(a+b)/a,sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a*d)-cot(c+d*x)/(d*sqrt(a+b*sec(c+d*x)))+b^2*tan(c+d*x)/((a^2-b^2)*d*sqrt(a+b*sec(c+d*x)))],
[tan(c+d*x)^5/(a+b*sec(c+d*x))^(3/2),x,5,-2*arctanh(sqrt(a+b*sec(c+d*x))/sqrt(a))/(a^(3/2)*d)-2*a*(a+b*sec(c+d*x))^(3/2)/(b^4*d)+2/5*(a+b*sec(c+d*x))^(5/2)/(b^4*d)+2*(a^2-b^2)^2/(a*b^4*d*sqrt(a+b*sec(c+d*x)))+2*(3*a^2-2*b^2)*sqrt(a+b*sec(c+d*x))/(b^4*d)],
[tan(c+d*x)^3/(a+b*sec(c+d*x))^(3/2),x,5,2*arctanh(sqrt(a+b*sec(c+d*x))/sqrt(a))/(a^(3/2)*d)+2*(a^2-b^2)/(a*b^2*d*sqrt(a+b*sec(c+d*x)))+2*sqrt(a+b*sec(c+d*x))/(b^2*d)],
[tan(c+d*x)/(a+b*sec(c+d*x))^(3/2),x,4,-2*arctanh(sqrt(a+b*sec(c+d*x))/sqrt(a))/(a^(3/2)*d)+2/(a*d*sqrt(a+b*sec(c+d*x)))],
[cot(c+d*x)/(a+b*sec(c+d*x))^(3/2),x,7,2*arctanh(sqrt(a+b*sec(c+d*x))/sqrt(a))/(a^(3/2)*d)-arctanh(sqrt(a+b*sec(c+d*x))/sqrt(a-b))/((a-b)^(3/2)*d)-arctanh(sqrt(a+b*sec(c+d*x))/sqrt(a+b))/((a+b)^(3/2)*d)+2*b^2/(a*(a^2-b^2)*d*sqrt(a+b*sec(c+d*x)))],
[cot(c+d*x)^3/(a+b*sec(c+d*x))^(3/2),x,11,-2*arctanh(sqrt(a+b*sec(c+d*x))/sqrt(a))/(a^(3/2)*d)+1/4*(4*a-7*b)*arctanh(sqrt(a+b*sec(c+d*x))/sqrt(a-b))/((a-b)^(5/2)*d)+1/4*(4*a+7*b)*arctanh(sqrt(a+b*sec(c+d*x))/sqrt(a+b))/((a+b)^(5/2)*d)+2*b^4/(a*(a^2-b^2)^2*d*sqrt(a+b*sec(c+d*x)))+1/4*sqrt(a+b*sec(c+d*x))/((a+b)^2*d*(1-sec(c+d*x)))+1/4*sqrt(a+b*sec(c+d*x))/((a-b)^2*d*(1+sec(c+d*x))),-2*arctanh(sqrt(a+b*sec(c+d*x))/sqrt(a))/(a^(3/2)*d)+1/2*(2*a-3*b)*arctanh(sqrt(a+b*sec(c+d*x))/sqrt(a-b))/((a-b)^(5/2)*d)-1/4*b*arctanh(sqrt(a+b*sec(c+d*x))/sqrt(a-b))/((a-b)^(5/2)*d)+1/4*b*arctanh(sqrt(a+b*sec(c+d*x))/sqrt(a+b))/((a+b)^(5/2)*d)+1/2*(2*a+3*b)*arctanh(sqrt(a+b*sec(c+d*x))/sqrt(a+b))/((a+b)^(5/2)*d)+2*b^4/(a*(a^2-b^2)^2*d*sqrt(a+b*sec(c+d*x)))+1/4*sqrt(a+b*sec(c+d*x))/((a+b)^2*d*(1-sec(c+d*x)))+1/4*sqrt(a+b*sec(c+d*x))/((a-b)^2*d*(1+sec(c+d*x)))],
[tan(c+d*x)^4/(a+b*sec(c+d*x))^(3/2),x,17,-2*cot(c+d*x)*EllipticPi(sqrt(a+b*sec(c+d*x))/sqrt(a+b),(a+b)/a,sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a^2*d)+2/3*(8*a^4-11*a^2*b^2+3*b^4)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(-b*(-1+sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a*b^4*d*sqrt(a+b))+2/3*(2*a+b)*(4*a^2+a*b-3*b^2)*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(-b*(-1+sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a*b^3*d*sqrt(a+b))-4*a*tan(c+d*x)/((a^2-b^2)*d*sqrt(a+b*sec(c+d*x)))+2*b^2*tan(c+d*x)/(a*(a^2-b^2)*d*sqrt(a+b*sec(c+d*x)))-2*a^2*sec(c+d*x)*tan(c+d*x)/(b*(a^2-b^2)*d*sqrt(a+b*sec(c+d*x)))+2/3*(4*a^2-b^2)*sqrt(a+b*sec(c+d*x))*tan(c+d*x)/(b^2*(a^2-b^2)*d),2*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a*d*sqrt(a+b))-4*a*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b^2*d*sqrt(a+b))+2/3*a*(8*a^2-5*b^2)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b^4*d*sqrt(a+b))-2*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a*d*sqrt(a+b))-4*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b*d*sqrt(a+b))+2/3*(2*a+b)*(4*a+b)*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b^3*d*sqrt(a+b))-2*cot(c+d*x)*EllipticPi(sqrt(a+b*sec(c+d*x))/sqrt(a+b),(a+b)/a,sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a^2*d)-4*a*tan(c+d*x)/((a^2-b^2)*d*sqrt(a+b*sec(c+d*x)))+2*b^2*tan(c+d*x)/(a*(a^2-b^2)*d*sqrt(a+b*sec(c+d*x)))-2*a^2*sec(c+d*x)*tan(c+d*x)/(b*(a^2-b^2)*d*sqrt(a+b*sec(c+d*x)))+2/3*(4*a^2-b^2)*sqrt(a+b*sec(c+d*x))*tan(c+d*x)/(b^2*(a^2-b^2)*d)],
[tan(c+d*x)^2/(a+b*sec(c+d*x))^(3/2),x,7,2*(a-b)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a*b^2*d)+2*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a*b*d)+2*cot(c+d*x)*EllipticPi(sqrt(a+b*sec(c+d*x))/sqrt(a+b),(a+b)/a,sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a^2*d)+2*tan(c+d*x)/(a*d*sqrt(a+b*sec(c+d*x)))],
[1/(a+b*sec(c+d*x))^(3/2),x,6,2*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a*d*sqrt(a+b))-2*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a*d*sqrt(a+b))-2*cot(c+d*x)*EllipticPi(sqrt(a+b*sec(c+d*x))/sqrt(a+b),(a+b)/a,sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a^2*d)+2*b^2*tan(c+d*x)/(a*(a^2-b^2)*d*sqrt(a+b*sec(c+d*x)))],
[cot(c+d*x)^2/(a+b*sec(c+d*x))^(3/2),x,14,-cot(c+d*x)/(d*(a+b*sec(c+d*x))^(3/2))+2*cot(c+d*x)*EllipticPi(sqrt(a+b*sec(c+d*x))/sqrt(a+b),(a+b)/a,sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a^2*d)+2*(a^2+b^2)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(-b*(-1+sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a*(a-b)*(a+b)^(3/2)*d)-(a^2-a*b+2*b^2)*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(-b*(-1+sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a*(a-b)*(a+b)^(3/2)*d)+b^2*tan(c+d*x)/((a^2-b^2)*d*(a+b*sec(c+d*x))^(3/2))+2*b^2*(a^2+b^2)*tan(c+d*x)/(a*(a^2-b^2)^2*d*sqrt(a+b*sec(c+d*x))),-cot(c+d*x)/(d*(a+b*sec(c+d*x))^(3/2))+4*a*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/((a-b)*(a+b)^(3/2)*d)-(3*a-b)*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/((a-b)*(a+b)^(3/2)*d)-2*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a*d*sqrt(a+b))+2*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a*d*sqrt(a+b))+2*cot(c+d*x)*EllipticPi(sqrt(a+b*sec(c+d*x))/sqrt(a+b),(a+b)/a,sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a^2*d)+b^2*tan(c+d*x)/((a^2-b^2)*d*(a+b*sec(c+d*x))^(3/2))+4*a*b^2*tan(c+d*x)/((a^2-b^2)^2*d*sqrt(a+b*sec(c+d*x)))-2*b^2*tan(c+d*x)/(a*(a^2-b^2)*d*sqrt(a+b*sec(c+d*x)))],

# Integrands of the form (d Tan[e+f x])^n (a+b Sec[e+f x])^m with n symbolic
[(a+b*sec(e+f*x))^3*(d*tan(e+f*x))^n,x,8,3*a*b^2*(d*tan(e+f*x))^(1+n)/(d*f*(1+n))+a^3*hypergeom([1,1/2*(1+n)],[1/2*(3+n)],-tan(e+f*x)^2)*(d*tan(e+f*x))^(1+n)/(d*f*(1+n))+3*a^2*b*(cos(e+f*x)^2)^(1/2*(2+n))*hypergeom([1/2*(1+n),1/2*(2+n)],[1/2*(3+n)],sin(e+f*x)^2)*sec(e+f*x)*(d*tan(e+f*x))^(1+n)/(d*f*(1+n))+b^3*(cos(e+f*x)^2)^(1/2*(4+n))*hypergeom([1/2*(1+n),1/2*(4+n)],[1/2*(3+n)],sin(e+f*x)^2)*sec(e+f*x)^3*(d*tan(e+f*x))^(1+n)/(d*f*(1+n))],
[(a+b*sec(e+f*x))^2*(d*tan(e+f*x))^n,x,7,b^2*(d*tan(e+f*x))^(1+n)/(d*f*(1+n))+a^2*hypergeom([1,1/2*(1+n)],[1/2*(3+n)],-tan(e+f*x)^2)*(d*tan(e+f*x))^(1+n)/(d*f*(1+n))+2*a*b*(cos(e+f*x)^2)^(1/2*(2+n))*hypergeom([1/2*(1+n),1/2*(2+n)],[1/2*(3+n)],sin(e+f*x)^2)*sec(e+f*x)*(d*tan(e+f*x))^(1+n)/(d*f*(1+n))],
[(a+b*sec(e+f*x))*(d*tan(e+f*x))^n,x,4,a*hypergeom([1,1/2*(1+n)],[1/2*(3+n)],-tan(e+f*x)^2)*(d*tan(e+f*x))^(1+n)/(d*f*(1+n))+b*(cos(e+f*x)^2)^(1/2*(2+n))*hypergeom([1/2*(1+n),1/2*(2+n)],[1/2*(3+n)],sin(e+f*x)^2)*sec(e+f*x)*(d*tan(e+f*x))^(1+n)/(d*f*(1+n))],
[(d*tan(e+f*x))^n/(a+b*sec(e+f*x)),x,-1,d*AppellF1(1-n,1/2*(1-n),1/2*(1-n),2-n,(a+b)/(a+b*sec(e+f*x)),(a-b)/(a+b*sec(e+f*x)))*(-b*(1-sec(e+f*x))/(a+b*sec(e+f*x)))^(1/2*(1-n))*(b*(1+sec(e+f*x))/(a+b*sec(e+f*x)))^(1/2*(1-n))*(d*tan(e+f*x))^(-1+n)*(-tan(e+f*x)^2)^(1/2*(1-n)+1/2*(-1+n))/(a*f*(1-n))-d*hypergeom([1,1/2*(1+n)],[1/2*(3+n)],-tan(e+f*x)^2)*(d*tan(e+f*x))^(-1+n)*(-tan(e+f*x)^2)^(1/2*(1-n)+1/2*(1+n))/(a*f*(1+n))],

#  {(d*Tan[e + f*x])^n/(a + b*Sec[e + f*x])^2, x, 0, 0} 
[(a+b*sec(c+d*x))^(3/2)*(e*tan(c+d*x))^m,x,0,Unintegrable((a+b*sec(c+d*x))^(3/2)*(e*tan(c+d*x))^m,x)],
[(a+b*sec(c+d*x))^(1/2)*(e*tan(c+d*x))^m,x,0,Unintegrable(sqrt(a+b*sec(c+d*x))*(e*tan(c+d*x))^m,x)],
[(e*tan(c+d*x))^m/(a+b*sec(c+d*x))^(1/2),x,0,Unintegrable((e*tan(c+d*x))^m/sqrt(a+b*sec(c+d*x)),x)],
[(e*tan(c+d*x))^m/(a+b*sec(c+d*x))^(3/2),x,0,Unintegrable((e*tan(c+d*x))^m/(a+b*sec(c+d*x))^(3/2),x)],

# Integrands of the form (d Tan[e+f x])^n (a+b Sec[e+f x])^m with m symbolic
[(a+b*sec(c+d*x))^n*(e*tan(c+d*x))^m,x,0,Unintegrable((a+b*sec(c+d*x))^n*(e*tan(c+d*x))^m,x)],
[(a+b*sec(c+d*x))^n*tan(c+d*x)^5,x,5,-a*(a^2-2*b^2)*(a+b*sec(c+d*x))^(1+n)/(b^4*d*(1+n))-hypergeom([1,1+n],[2+n],1+b*sec(c+d*x)/a)*(a+b*sec(c+d*x))^(1+n)/(a*d*(1+n))+(3*a^2-2*b^2)*(a+b*sec(c+d*x))^(2+n)/(b^4*d*(2+n))-3*a*(a+b*sec(c+d*x))^(3+n)/(b^4*d*(3+n))+(a+b*sec(c+d*x))^(4+n)/(b^4*d*(4+n))],
[(a+b*sec(c+d*x))^n*tan(c+d*x)^3,x,4,-a*(a+b*sec(c+d*x))^(1+n)/(b^2*d*(1+n))+hypergeom([1,1+n],[2+n],1+b*sec(c+d*x)/a)*(a+b*sec(c+d*x))^(1+n)/(a*d*(1+n))+(a+b*sec(c+d*x))^(2+n)/(b^2*d*(2+n))],
[(a+b*sec(c+d*x))^n*tan(c+d*x),x,2,-hypergeom([1,1+n],[2+n],1+b*sec(c+d*x)/a)*(a+b*sec(c+d*x))^(1+n)/(a*d*(1+n))],
[cot(c+d*x)*(a+b*sec(c+d*x))^n,x,8,-1/2*hypergeom([1,1+n],[2+n],(a+b*sec(c+d*x))/(a-b))*(a+b*sec(c+d*x))^(1+n)/((a-b)*d*(1+n))-1/2*hypergeom([1,1+n],[2+n],(a+b*sec(c+d*x))/(a+b))*(a+b*sec(c+d*x))^(1+n)/((a+b)*d*(1+n))+hypergeom([1,1+n],[2+n],1+b*sec(c+d*x)/a)*(a+b*sec(c+d*x))^(1+n)/(a*d*(1+n))],
[cot(c+d*x)^3*(a+b*sec(c+d*x))^n,x,10,1/2*hypergeom([1,1+n],[2+n],(a+b*sec(c+d*x))/(a-b))*(a+b*sec(c+d*x))^(1+n)/((a-b)*d*(1+n))+1/2*hypergeom([1,1+n],[2+n],(a+b*sec(c+d*x))/(a+b))*(a+b*sec(c+d*x))^(1+n)/((a+b)*d*(1+n))-hypergeom([1,1+n],[2+n],1+b*sec(c+d*x)/a)*(a+b*sec(c+d*x))^(1+n)/(a*d*(1+n))-1/4*b*hypergeom([2,1+n],[2+n],(a+b*sec(c+d*x))/(a-b))*(a+b*sec(c+d*x))^(1+n)/((a-b)^2*d*(1+n))+1/4*b*hypergeom([2,1+n],[2+n],(a+b*sec(c+d*x))/(a+b))*(a+b*sec(c+d*x))^(1+n)/((a+b)^2*d*(1+n))],
[(a+b*sec(c+d*x))^n*tan(c+d*x)^4,x,0,Unintegrable((a+b*sec(c+d*x))^n*tan(c+d*x)^4,x)],
[(a+b*sec(c+d*x))^n*tan(c+d*x)^2,x,9,(a+b)*AppellF1(1/2,1/2,-1-n,3/2,1/2*(1-sec(c+d*x)),b*(1-sec(c+d*x))/(a+b))*(a+b*sec(c+d*x))^n*sqrt(2)*tan(c+d*x)/(b*d*((a+b*sec(c+d*x))/(a+b))^n*sqrt(1+sec(c+d*x)))-a*AppellF1(1/2,1/2,-n,3/2,1/2*(1-sec(c+d*x)),b*(1-sec(c+d*x))/(a+b))*(a+b*sec(c+d*x))^n*sqrt(2)*tan(c+d*x)/(b*d*((a+b*sec(c+d*x))/(a+b))^n*sqrt(1+sec(c+d*x)))-Unintegrable((a+b*sec(c+d*x))^n,x)],
[cot(c+d*x)^2*(a+b*sec(c+d*x))^n,x,0,Unintegrable(cot(c+d*x)^2*(a+b*sec(c+d*x))^n,x)],
[cot(c+d*x)^4*(a+b*sec(c+d*x))^n,x,0,Unintegrable(cot(c+d*x)^4*(a+b*sec(c+d*x))^n,x)],
[(a+b*sec(c+d*x))^n*tan(c+d*x)^(3/2),x,0,Unintegrable((a+b*sec(c+d*x))^n*tan(c+d*x)^(3/2),x)],
[(a+b*sec(c+d*x))^n*tan(c+d*x)^(1/2),x,0,Unintegrable((a+b*sec(c+d*x))^n*sqrt(tan(c+d*x)),x)],
[(a+b*sec(c+d*x))^n/tan(c+d*x)^(1/2),x,0,Unintegrable((a+b*sec(c+d*x))^n/sqrt(tan(c+d*x)),x)],
[(a+b*sec(c+d*x))^n/tan(c+d*x)^(3/2),x,0,Unintegrable((a+b*sec(c+d*x))^n/tan(c+d*x)^(3/2),x)]]:

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