# Maple integration test file: "4 Trig functions\4.5 Secant\4.5.2.3 (g sec)^p (a+b sec)^m (c+d sec)^n.txt"

lst:=[

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

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

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

# m>0
[sec(e+f*x)*(a+a*sec(e+f*x))*(c-c*sec(e+f*x))^4,x,12,7/8*a*c^4*arctanh(sin(e+f*x))/f-1/8*a*c^4*sec(e+f*x)*tan(e+f*x)/f-3/4*a*c^4*sec(e+f*x)^3*tan(e+f*x)/f+4/3*a*c^4*tan(e+f*x)^3/f+1/5*a*c^4*tan(e+f*x)^5/f],
[sec(e+f*x)*(a+a*sec(e+f*x))*(c-c*sec(e+f*x))^3,x,9,5/8*a*c^3*arctanh(sin(e+f*x))/f-3/8*a*c^3*sec(e+f*x)*tan(e+f*x)/f-1/4*a*c^3*sec(e+f*x)^3*tan(e+f*x)/f+2/3*a*c^3*tan(e+f*x)^3/f],
[sec(e+f*x)*(a+a*sec(e+f*x))*(c-c*sec(e+f*x))^2,x,6,1/2*a*c^2*arctanh(sin(e+f*x))/f-1/2*a*c^2*sec(e+f*x)*tan(e+f*x)/f+1/3*a*c^2*tan(e+f*x)^3/f],
[sec(e+f*x)*(a+a*sec(e+f*x))*(c-c*sec(e+f*x)),x,3,1/2*a*c*arctanh(sin(e+f*x))/f-1/2*a*c*sec(e+f*x)*tan(e+f*x)/f],
[sec(e+f*x)*(a+a*sec(e+f*x))/(c-c*sec(e+f*x)),x,2,-a*arctanh(sin(e+f*x))/(c*f)-2*a*tan(e+f*x)/(f*(c-c*sec(e+f*x)))],
[sec(e+f*x)*(a+a*sec(e+f*x))/(c-c*sec(e+f*x))^2,x,1,-1/3*(a+a*sec(e+f*x))*tan(e+f*x)/(f*(c-c*sec(e+f*x))^2)],
[sec(e+f*x)*(a+a*sec(e+f*x))/(c-c*sec(e+f*x))^3,x,2,-1/5*(a+a*sec(e+f*x))*tan(e+f*x)/(f*(c-c*sec(e+f*x))^3)-1/15*(a+a*sec(e+f*x))*tan(e+f*x)/(c*f*(c-c*sec(e+f*x))^2)],
[sec(e+f*x)*(a+a*sec(e+f*x))/(c-c*sec(e+f*x))^4,x,3,-1/7*(a+a*sec(e+f*x))*tan(e+f*x)/(f*(c-c*sec(e+f*x))^4)-2/35*(a+a*sec(e+f*x))*tan(e+f*x)/(c*f*(c-c*sec(e+f*x))^3)-2/105*(a+a*sec(e+f*x))*tan(e+f*x)/(f*(c^2-c^2*sec(e+f*x))^2)],
[sec(e+f*x)*(a+a*sec(e+f*x))/(c-c*sec(e+f*x))^5,x,4,-1/9*(a+a*sec(e+f*x))*tan(e+f*x)/(f*(c-c*sec(e+f*x))^5)-1/21*(a+a*sec(e+f*x))*tan(e+f*x)/(c*f*(c-c*sec(e+f*x))^4)-2/105*(a+a*sec(e+f*x))*tan(e+f*x)/(c^2*f*(c-c*sec(e+f*x))^3)-2/315*(a+a*sec(e+f*x))*tan(e+f*x)/(c*f*(c^2-c^2*sec(e+f*x))^2)],
[sec(e+f*x)*(a+a*sec(e+f*x))^2*(c-c*sec(e+f*x))^5,x,14,9/16*a^2*c^5*arctanh(sin(e+f*x))/f-3/16*a^2*c^5*sec(e+f*x)*tan(e+f*x)/f-3/8*a^2*c^5*sec(e+f*x)^3*tan(e+f*x)/f+1/4*a^2*c^5*sec(e+f*x)*tan(e+f*x)^3/f+1/2*a^2*c^5*sec(e+f*x)^3*tan(e+f*x)^3/f-4/5*a^2*c^5*tan(e+f*x)^5/f-1/7*a^2*c^5*tan(e+f*x)^7/f],
[sec(e+f*x)*(a+a*sec(e+f*x))^2*(c-c*sec(e+f*x))^4,x,11,7/16*a^2*c^4*arctanh(sin(e+f*x))/f-5/16*a^2*c^4*sec(e+f*x)*tan(e+f*x)/f-1/8*a^2*c^4*sec(e+f*x)^3*tan(e+f*x)/f+1/4*a^2*c^4*sec(e+f*x)*tan(e+f*x)^3/f+1/6*a^2*c^4*sec(e+f*x)^3*tan(e+f*x)^3/f-2/5*a^2*c^4*tan(e+f*x)^5/f],
[sec(e+f*x)*(a+a*sec(e+f*x))^2*(c-c*sec(e+f*x))^3,x,7,3/8*a^2*c^3*arctanh(sin(e+f*x))/f-3/8*a^2*c^3*sec(e+f*x)*tan(e+f*x)/f+1/4*a^2*c^3*sec(e+f*x)*tan(e+f*x)^3/f-1/5*a^2*c^3*tan(e+f*x)^5/f],
[sec(e+f*x)*(a+a*sec(e+f*x))^2*(c-c*sec(e+f*x))^2,x,4,3/8*a^2*c^2*arctanh(sin(e+f*x))/f-3/8*a^2*c^2*sec(e+f*x)*tan(e+f*x)/f+1/4*a^2*c^2*sec(e+f*x)*tan(e+f*x)^3/f],
[sec(e+f*x)*(a+a*sec(e+f*x))^2*(c-c*sec(e+f*x)),x,6,1/2*a^2*c*arctanh(sin(e+f*x))/f-1/2*a^2*c*sec(e+f*x)*tan(e+f*x)/f-1/3*a^2*c*tan(e+f*x)^3/f],
[sec(e+f*x)*(a+a*sec(e+f*x))^2/(c-c*sec(e+f*x)),x,5,-3*a^2*arctanh(sin(e+f*x))/(c*f)-3*a^2*tan(e+f*x)/(c*f)-2*(a^2+a^2*sec(e+f*x))*tan(e+f*x)/(f*(c-c*sec(e+f*x)))],
[sec(e+f*x)*(a+a*sec(e+f*x))^2/(c-c*sec(e+f*x))^2,x,3,a^2*arctanh(sin(e+f*x))/(c^2*f)-2/3*(a^2+a^2*sec(e+f*x))*tan(e+f*x)/(f*(c-c*sec(e+f*x))^2)+2*a^2*tan(e+f*x)/(f*(c^2-c^2*sec(e+f*x)))],
[sec(e+f*x)*(a+a*sec(e+f*x))^2/(c-c*sec(e+f*x))^3,x,1,-1/5*(a+a*sec(e+f*x))^2*tan(e+f*x)/(f*(c-c*sec(e+f*x))^3)],
[sec(e+f*x)*(a+a*sec(e+f*x))^2/(c-c*sec(e+f*x))^4,x,2,-1/7*(a+a*sec(e+f*x))^2*tan(e+f*x)/(f*(c-c*sec(e+f*x))^4)-1/35*(a+a*sec(e+f*x))^2*tan(e+f*x)/(c*f*(c-c*sec(e+f*x))^3)],
[sec(e+f*x)*(a+a*sec(e+f*x))^2/(c-c*sec(e+f*x))^5,x,3,-1/9*(a+a*sec(e+f*x))^2*tan(e+f*x)/(f*(c-c*sec(e+f*x))^5)-2/63*(a+a*sec(e+f*x))^2*tan(e+f*x)/(c*f*(c-c*sec(e+f*x))^4)-2/315*(a+a*sec(e+f*x))^2*tan(e+f*x)/(c^2*f*(c-c*sec(e+f*x))^3)],
[sec(e+f*x)*(a+a*sec(e+f*x))^2/(c-c*sec(e+f*x))^6,x,4,-1/11*(a+a*sec(e+f*x))^2*tan(e+f*x)/(f*(c-c*sec(e+f*x))^6)-1/33*(a+a*sec(e+f*x))^2*tan(e+f*x)/(c*f*(c-c*sec(e+f*x))^5)-2/231*(a+a*sec(e+f*x))^2*tan(e+f*x)/(c^2*f*(c-c*sec(e+f*x))^4)-2/1155*(a+a*sec(e+f*x))^2*tan(e+f*x)/(f*(c^2-c^2*sec(e+f*x))^3)],
[sec(e+f*x)*(a+a*sec(e+f*x))^3*(c-c*sec(e+f*x))^6,x,16,55/128*a^3*c^6*arctanh(sin(e+f*x))/f-25/128*a^3*c^6*sec(e+f*x)*tan(e+f*x)/f-15/64*a^3*c^6*sec(e+f*x)^3*tan(e+f*x)/f+5/24*a^3*c^6*sec(e+f*x)*tan(e+f*x)^3/f+5/16*a^3*c^6*sec(e+f*x)^3*tan(e+f*x)^3/f-1/6*a^3*c^6*sec(e+f*x)*tan(e+f*x)^5/f-3/8*a^3*c^6*sec(e+f*x)^3*tan(e+f*x)^5/f+4/7*a^3*c^6*tan(e+f*x)^7/f+1/9*a^3*c^6*tan(e+f*x)^9/f],
[sec(e+f*x)*(a+a*sec(e+f*x))^3*(c-c*sec(e+f*x))^5,x,13,45/128*a^3*c^5*arctanh(sin(e+f*x))/f-35/128*a^3*c^5*sec(e+f*x)*tan(e+f*x)/f-5/64*a^3*c^5*sec(e+f*x)^3*tan(e+f*x)/f+5/24*a^3*c^5*sec(e+f*x)*tan(e+f*x)^3/f+5/48*a^3*c^5*sec(e+f*x)^3*tan(e+f*x)^3/f-1/6*a^3*c^5*sec(e+f*x)*tan(e+f*x)^5/f-1/8*a^3*c^5*sec(e+f*x)^3*tan(e+f*x)^5/f+2/7*a^3*c^5*tan(e+f*x)^7/f],
[sec(e+f*x)*(a+a*sec(e+f*x))^3*(c-c*sec(e+f*x))^4,x,8,5/16*a^3*c^4*arctanh(sin(e+f*x))/f-5/16*a^3*c^4*sec(e+f*x)*tan(e+f*x)/f+5/24*a^3*c^4*sec(e+f*x)*tan(e+f*x)^3/f-1/6*a^3*c^4*sec(e+f*x)*tan(e+f*x)^5/f+1/7*a^3*c^4*tan(e+f*x)^7/f],
[sec(e+f*x)*(a+a*sec(e+f*x))^3*(c-c*sec(e+f*x))^3,x,5,5/16*a^3*c^3*arctanh(sin(e+f*x))/f-5/16*a^3*c^3*sec(e+f*x)*tan(e+f*x)/f+5/24*a^3*c^3*sec(e+f*x)*tan(e+f*x)^3/f-1/6*a^3*c^3*sec(e+f*x)*tan(e+f*x)^5/f],
[sec(e+f*x)*(a+a*sec(e+f*x))^3*(c-c*sec(e+f*x))^2,x,7,3/8*a^3*c^2*arctanh(sin(e+f*x))/f-3/8*a^3*c^2*sec(e+f*x)*tan(e+f*x)/f+1/4*a^3*c^2*sec(e+f*x)*tan(e+f*x)^3/f+1/5*a^3*c^2*tan(e+f*x)^5/f],
[sec(e+f*x)*(a+a*sec(e+f*x))^3*(c-c*sec(e+f*x)),x,9,5/8*a^3*c*arctanh(sin(e+f*x))/f-3/8*a^3*c*sec(e+f*x)*tan(e+f*x)/f-1/4*a^3*c*sec(e+f*x)^3*tan(e+f*x)/f-2/3*a^3*c*tan(e+f*x)^3/f],
[sec(e+f*x)*(a+a*sec(e+f*x))^3/(c-c*sec(e+f*x)),x,6,-15/2*a^3*arctanh(sin(e+f*x))/(c*f)-10*a^3*tan(e+f*x)/(c*f)-5/2*a^3*sec(e+f*x)*tan(e+f*x)/(c*f)-2*a*(a+a*sec(e+f*x))^2*tan(e+f*x)/(f*(c-c*sec(e+f*x)))],
[sec(e+f*x)*(a+a*sec(e+f*x))^3/(c-c*sec(e+f*x))^2,x,6,5*a^3*arctanh(sin(e+f*x))/(c^2*f)+5*a^3*tan(e+f*x)/(c^2*f)-2/3*a*(a+a*sec(e+f*x))^2*tan(e+f*x)/(f*(c-c*sec(e+f*x))^2)+10/3*(a^3+a^3*sec(e+f*x))*tan(e+f*x)/(f*(c^2-c^2*sec(e+f*x)))],
[sec(e+f*x)*(a+a*sec(e+f*x))^3/(c-c*sec(e+f*x))^3,x,4,-a^3*arctanh(sin(e+f*x))/(c^3*f)-2/5*a*(a+a*sec(e+f*x))^2*tan(e+f*x)/(f*(c-c*sec(e+f*x))^3)+2/3*(a^3+a^3*sec(e+f*x))*tan(e+f*x)/(c*f*(c-c*sec(e+f*x))^2)-2*a^3*tan(e+f*x)/(f*(c^3-c^3*sec(e+f*x)))],
[sec(e+f*x)*(a+a*sec(e+f*x))^3/(c-c*sec(e+f*x))^4,x,1,-1/7*(a+a*sec(e+f*x))^3*tan(e+f*x)/(f*(c-c*sec(e+f*x))^4)],
[sec(e+f*x)*(a+a*sec(e+f*x))^3/(c-c*sec(e+f*x))^5,x,2,-1/9*(a+a*sec(e+f*x))^3*tan(e+f*x)/(f*(c-c*sec(e+f*x))^5)-1/63*(a+a*sec(e+f*x))^3*tan(e+f*x)/(c*f*(c-c*sec(e+f*x))^4)],
[sec(e+f*x)*(a+a*sec(e+f*x))^3/(c-c*sec(e+f*x))^6,x,3,-1/11*(a+a*sec(e+f*x))^3*tan(e+f*x)/(f*(c-c*sec(e+f*x))^6)-2/99*(a+a*sec(e+f*x))^3*tan(e+f*x)/(c*f*(c-c*sec(e+f*x))^5)-2/693*(a+a*sec(e+f*x))^3*tan(e+f*x)/(c^2*f*(c-c*sec(e+f*x))^4)],
[sec(e+f*x)*(a+a*sec(e+f*x))^3/(c-c*sec(e+f*x))^7,x,4,-1/13*(a+a*sec(e+f*x))^3*tan(e+f*x)/(f*(c-c*sec(e+f*x))^7)-3/143*(a+a*sec(e+f*x))^3*tan(e+f*x)/(c*f*(c-c*sec(e+f*x))^6)-2/429*(a+a*sec(e+f*x))^3*tan(e+f*x)/(c^2*f*(c-c*sec(e+f*x))^5)-2/3003*(a+a*sec(e+f*x))^3*tan(e+f*x)/(c^3*f*(c-c*sec(e+f*x))^4)],

# m<0
[sec(e+f*x)*(c-c*sec(e+f*x))^4/(a+a*sec(e+f*x)),x,10,-35/2*c^4*arctanh(sin(e+f*x))/(a*f)+28*c^4*tan(e+f*x)/(a*f)-21/2*c^4*sec(e+f*x)*tan(e+f*x)/(a*f)+2*c*(c-c*sec(e+f*x))^3*tan(e+f*x)/(f*(a+a*sec(e+f*x)))+7/3*c^4*tan(e+f*x)^3/(a*f)],
[sec(e+f*x)*(c-c*sec(e+f*x))^3/(a+a*sec(e+f*x)),x,6,-15/2*c^3*arctanh(sin(e+f*x))/(a*f)+10*c^3*tan(e+f*x)/(a*f)-5/2*c^3*sec(e+f*x)*tan(e+f*x)/(a*f)+2*c*(c-c*sec(e+f*x))^2*tan(e+f*x)/(f*(a+a*sec(e+f*x)))],
[sec(e+f*x)*(c-c*sec(e+f*x))^2/(a+a*sec(e+f*x)),x,5,-3*c^2*arctanh(sin(e+f*x))/(a*f)+3*c^2*tan(e+f*x)/(a*f)+2*(c^2-c^2*sec(e+f*x))*tan(e+f*x)/(f*(a+a*sec(e+f*x)))],
[sec(e+f*x)*(c-c*sec(e+f*x))/(a+a*sec(e+f*x)),x,2,-c*arctanh(sin(e+f*x))/(a*f)+2*c*tan(e+f*x)/(f*(a+a*sec(e+f*x)))],
[sec(e+f*x)/((a+a*sec(e+f*x))*(c-c*sec(e+f*x))),x,3,csc(e+f*x)/(a*c*f)],
[sec(e+f*x)/((a+a*sec(e+f*x))*(c-c*sec(e+f*x))^2),x,6,-1/3*cot(e+f*x)^3/(a*c^2*f)+csc(e+f*x)/(a*c^2*f)-1/3*csc(e+f*x)^3/(a*c^2*f)],
[sec(e+f*x)/((a+a*sec(e+f*x))*(c-c*sec(e+f*x))^3),x,10,2/5*cot(e+f*x)^5/(a*c^3*f)+csc(e+f*x)/(a*c^3*f)-csc(e+f*x)^3/(a*c^3*f)+2/5*csc(e+f*x)^5/(a*c^3*f)],
[sec(e+f*x)/((a+a*sec(e+f*x))*(c-c*sec(e+f*x))^4),x,13,-1/5*cot(e+f*x)^5/(a*c^4*f)-4/7*cot(e+f*x)^7/(a*c^4*f)+csc(e+f*x)/(a*c^4*f)-2*csc(e+f*x)^3/(a*c^4*f)+9/5*csc(e+f*x)^5/(a*c^4*f)-4/7*csc(e+f*x)^7/(a*c^4*f)],
[sec(e+f*x)*(c-c*sec(e+f*x))^5/(a+a*sec(e+f*x))^2,x,11,105/2*c^5*arctanh(sin(e+f*x))/(a^2*f)-84*c^5*tan(e+f*x)/(a^2*f)+63/2*c^5*sec(e+f*x)*tan(e+f*x)/(a^2*f)-6*c^2*(c-c*sec(e+f*x))^3*tan(e+f*x)/(f*(a^2+a^2*sec(e+f*x)))+2/3*c*(c-c*sec(e+f*x))^4*tan(e+f*x)/(f*(a+a*sec(e+f*x))^2)-7*c^5*tan(e+f*x)^3/(a^2*f)],
[sec(e+f*x)*(c-c*sec(e+f*x))^4/(a+a*sec(e+f*x))^2,x,7,35/2*c^4*arctanh(sin(e+f*x))/(a^2*f)-70/3*c^4*tan(e+f*x)/(a^2*f)+35/6*c^4*sec(e+f*x)*tan(e+f*x)/(a^2*f)+2/3*c*(c-c*sec(e+f*x))^3*tan(e+f*x)/(f*(a+a*sec(e+f*x))^2)-14/3*(c^2-c^2*sec(e+f*x))^2*tan(e+f*x)/(f*(a^2+a^2*sec(e+f*x)))],
[sec(e+f*x)*(c-c*sec(e+f*x))^3/(a+a*sec(e+f*x))^2,x,6,5*c^3*arctanh(sin(e+f*x))/(a^2*f)-5*c^3*tan(e+f*x)/(a^2*f)+2/3*c*(c-c*sec(e+f*x))^2*tan(e+f*x)/(f*(a+a*sec(e+f*x))^2)-10/3*(c^3-c^3*sec(e+f*x))*tan(e+f*x)/(f*(a^2+a^2*sec(e+f*x)))],
[sec(e+f*x)*(c-c*sec(e+f*x))^2/(a+a*sec(e+f*x))^2,x,3,c^2*arctanh(sin(e+f*x))/(a^2*f)-2*c^2*tan(e+f*x)/(f*(a^2+a^2*sec(e+f*x)))+2/3*(c^2-c^2*sec(e+f*x))*tan(e+f*x)/(f*(a+a*sec(e+f*x))^2)],
[sec(e+f*x)*(c-c*sec(e+f*x))/(a+a*sec(e+f*x))^2,x,1,1/3*(c-c*sec(e+f*x))*tan(e+f*x)/(f*(a+a*sec(e+f*x))^2)],
[sec(e+f*x)/((a+a*sec(e+f*x))^2*(c-c*sec(e+f*x))),x,6,1/3*cot(e+f*x)^3/(a^2*c*f)+csc(e+f*x)/(a^2*c*f)-1/3*csc(e+f*x)^3/(a^2*c*f)],
[sec(e+f*x)/((a+a*sec(e+f*x))^2*(c-c*sec(e+f*x))^2),x,3,csc(e+f*x)/(a^2*c^2*f)-1/3*csc(e+f*x)^3/(a^2*c^2*f)],
[sec(e+f*x)/((a+a*sec(e+f*x))^2*(c-c*sec(e+f*x))^3),x,7,1/5*cot(e+f*x)^5/(a^2*c^3*f)+csc(e+f*x)/(a^2*c^3*f)-2/3*csc(e+f*x)^3/(a^2*c^3*f)+1/5*csc(e+f*x)^5/(a^2*c^3*f)],
[sec(e+f*x)/((a+a*sec(e+f*x))^2*(c-c*sec(e+f*x))^4),x,10,-2/7*cot(e+f*x)^7/(a^2*c^4*f)+csc(e+f*x)/(a^2*c^4*f)-4/3*csc(e+f*x)^3/(a^2*c^4*f)+csc(e+f*x)^5/(a^2*c^4*f)-2/7*csc(e+f*x)^7/(a^2*c^4*f)],
[sec(e+f*x)/((a+a*sec(e+f*x))^2*(c-c*sec(e+f*x))^5),x,13,1/7*cot(e+f*x)^7/(a^2*c^5*f)+4/9*cot(e+f*x)^9/(a^2*c^5*f)+csc(e+f*x)/(a^2*c^5*f)-7/3*csc(e+f*x)^3/(a^2*c^5*f)+3*csc(e+f*x)^5/(a^2*c^5*f)-13/7*csc(e+f*x)^7/(a^2*c^5*f)+4/9*csc(e+f*x)^9/(a^2*c^5*f)],
[sec(e+f*x)*(c-c*sec(e+f*x))^6/(a+a*sec(e+f*x))^3,x,12,-231/2*c^6*arctanh(sin(e+f*x))/(a^3*f)+924/5*c^6*tan(e+f*x)/(a^3*f)-693/10*c^6*sec(e+f*x)*tan(e+f*x)/(a^3*f)-22/15*c^2*(c-c*sec(e+f*x))^4*tan(e+f*x)/(a*f*(a+a*sec(e+f*x))^2)+2/5*c*(c-c*sec(e+f*x))^5*tan(e+f*x)/(f*(a+a*sec(e+f*x))^3)+66/5*(c^2-c^2*sec(e+f*x))^3*tan(e+f*x)/(f*(a^3+a^3*sec(e+f*x)))+77/5*c^6*tan(e+f*x)^3/(a^3*f)],
[sec(e+f*x)*(c-c*sec(e+f*x))^5/(a+a*sec(e+f*x))^3,x,8,-63/2*c^5*arctanh(sin(e+f*x))/(a^3*f)+42*c^5*tan(e+f*x)/(a^3*f)-21/2*c^5*sec(e+f*x)*tan(e+f*x)/(a^3*f)-6/5*c^2*(c-c*sec(e+f*x))^3*tan(e+f*x)/(a*f*(a+a*sec(e+f*x))^2)+2/5*c*(c-c*sec(e+f*x))^4*tan(e+f*x)/(f*(a+a*sec(e+f*x))^3)+42/5*c*(c^2-c^2*sec(e+f*x))^2*tan(e+f*x)/(f*(a^3+a^3*sec(e+f*x)))],
[sec(e+f*x)*(c-c*sec(e+f*x))^4/(a+a*sec(e+f*x))^3,x,7,-7*c^4*arctanh(sin(e+f*x))/(a^3*f)+7*c^4*tan(e+f*x)/(a^3*f)+2/5*c*(c-c*sec(e+f*x))^3*tan(e+f*x)/(f*(a+a*sec(e+f*x))^3)-14/15*(c^2-c^2*sec(e+f*x))^2*tan(e+f*x)/(a*f*(a+a*sec(e+f*x))^2)+14/3*(c^4-c^4*sec(e+f*x))*tan(e+f*x)/(f*(a^3+a^3*sec(e+f*x)))],
[sec(e+f*x)*(c-c*sec(e+f*x))^3/(a+a*sec(e+f*x))^3,x,4,-c^3*arctanh(sin(e+f*x))/(a^3*f)+2*c^3*tan(e+f*x)/(f*(a^3+a^3*sec(e+f*x)))+2/5*c*(c-c*sec(e+f*x))^2*tan(e+f*x)/(f*(a+a*sec(e+f*x))^3)-2/3*(c^3-c^3*sec(e+f*x))*tan(e+f*x)/(a*f*(a+a*sec(e+f*x))^2)],
[sec(e+f*x)*(c-c*sec(e+f*x))^2/(a+a*sec(e+f*x))^3,x,1,1/5*(c-c*sec(e+f*x))^2*tan(e+f*x)/(f*(a+a*sec(e+f*x))^3)],
[sec(e+f*x)*(c-c*sec(e+f*x))/(a+a*sec(e+f*x))^3,x,2,1/5*(c-c*sec(e+f*x))*tan(e+f*x)/(f*(a+a*sec(e+f*x))^3)+1/15*(c-c*sec(e+f*x))*tan(e+f*x)/(a*f*(a+a*sec(e+f*x))^2)],
[sec(e+f*x)/((a+a*sec(e+f*x))^3*(c-c*sec(e+f*x))),x,10,-2/5*cot(e+f*x)^5/(a^3*c*f)+csc(e+f*x)/(a^3*c*f)-csc(e+f*x)^3/(a^3*c*f)+2/5*csc(e+f*x)^5/(a^3*c*f)],
[sec(e+f*x)/((a+a*sec(e+f*x))^3*(c-c*sec(e+f*x))^2),x,7,-1/5*cot(e+f*x)^5/(a^3*c^2*f)+csc(e+f*x)/(a^3*c^2*f)-2/3*csc(e+f*x)^3/(a^3*c^2*f)+1/5*csc(e+f*x)^5/(a^3*c^2*f)],
[sec(e+f*x)/((a+a*sec(e+f*x))^3*(c-c*sec(e+f*x))^3),x,4,csc(e+f*x)/(a^3*c^3*f)-2/3*csc(e+f*x)^3/(a^3*c^3*f)+1/5*csc(e+f*x)^5/(a^3*c^3*f)],
[sec(e+f*x)/((a+a*sec(e+f*x))^3*(c-c*sec(e+f*x))^4),x,7,-1/7*cot(e+f*x)^7/(a^3*c^4*f)+csc(e+f*x)/(a^3*c^4*f)-csc(e+f*x)^3/(a^3*c^4*f)+3/5*csc(e+f*x)^5/(a^3*c^4*f)-1/7*csc(e+f*x)^7/(a^3*c^4*f)],
[sec(e+f*x)/((a+a*sec(e+f*x))^3*(c-c*sec(e+f*x))^5),x,10,2/9*cot(e+f*x)^9/(a^3*c^5*f)+csc(e+f*x)/(a^3*c^5*f)-5/3*csc(e+f*x)^3/(a^3*c^5*f)+9/5*csc(e+f*x)^5/(a^3*c^5*f)-csc(e+f*x)^7/(a^3*c^5*f)+2/9*csc(e+f*x)^9/(a^3*c^5*f)],
[sec(e+f*x)/((a+a*sec(e+f*x))^3*(c-c*sec(e+f*x))^6),x,13,-1/9*cot(e+f*x)^9/(a^3*c^6*f)-4/11*cot(e+f*x)^11/(a^3*c^6*f)+csc(e+f*x)/(a^3*c^6*f)-8/3*csc(e+f*x)^3/(a^3*c^6*f)+22/5*csc(e+f*x)^5/(a^3*c^6*f)-4*csc(e+f*x)^7/(a^3*c^6*f)+17/9*csc(e+f*x)^9/(a^3*c^6*f)-4/11*csc(e+f*x)^11/(a^3*c^6*f)],

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

# m>0
[sec(e+f*x)*(a+a*sec(e+f*x))*(c-c*sec(e+f*x))^(7/2),x,4,-8/21*c^2*(a+a*sec(e+f*x))*(c-c*sec(e+f*x))^(3/2)*tan(e+f*x)/f-2/9*c*(a+a*sec(e+f*x))*(c-c*sec(e+f*x))^(5/2)*tan(e+f*x)/f-256/315*c^4*(a+a*sec(e+f*x))*tan(e+f*x)/(f*sqrt(c-c*sec(e+f*x)))-64/105*c^3*(a+a*sec(e+f*x))*sqrt(c-c*sec(e+f*x))*tan(e+f*x)/f],
[sec(e+f*x)*(a+a*sec(e+f*x))*(c-c*sec(e+f*x))^(5/2),x,3,-2/7*c*(a+a*sec(e+f*x))*(c-c*sec(e+f*x))^(3/2)*tan(e+f*x)/f-64/105*c^3*(a+a*sec(e+f*x))*tan(e+f*x)/(f*sqrt(c-c*sec(e+f*x)))-16/35*c^2*(a+a*sec(e+f*x))*sqrt(c-c*sec(e+f*x))*tan(e+f*x)/f],
[sec(e+f*x)*(a+a*sec(e+f*x))*(c-c*sec(e+f*x))^(3/2),x,2,-8/15*c^2*(a+a*sec(e+f*x))*tan(e+f*x)/(f*sqrt(c-c*sec(e+f*x)))-2/5*c*(a+a*sec(e+f*x))*sqrt(c-c*sec(e+f*x))*tan(e+f*x)/f],
[sec(e+f*x)*(a+a*sec(e+f*x))*sqrt(c-c*sec(e+f*x)),x,1,-2/3*c*(a+a*sec(e+f*x))*tan(e+f*x)/(f*sqrt(c-c*sec(e+f*x)))],
[sec(e+f*x)*(a+a*sec(e+f*x))/sqrt(c-c*sec(e+f*x)),x,3,-2*a*arctan(sqrt(c)*tan(e+f*x)/(sqrt(2)*sqrt(c-c*sec(e+f*x))))*sqrt(2)/(f*sqrt(c))+2*a*tan(e+f*x)/(f*sqrt(c-c*sec(e+f*x)))],
[sec(e+f*x)*(a+a*sec(e+f*x))/(c-c*sec(e+f*x))^(3/2),x,3,a*arctan(sqrt(c)*tan(e+f*x)/(sqrt(2)*sqrt(c-c*sec(e+f*x))))/(c^(3/2)*f*sqrt(2))-a*tan(e+f*x)/(f*(c-c*sec(e+f*x))^(3/2))],
[sec(e+f*x)*(a+a*sec(e+f*x))/(c-c*sec(e+f*x))^(5/2),x,4,1/8*a*arctan(sqrt(c)*tan(e+f*x)/(sqrt(2)*sqrt(c-c*sec(e+f*x))))/(c^(5/2)*f*sqrt(2))-1/2*a*tan(e+f*x)/(f*(c-c*sec(e+f*x))^(5/2))+1/8*a*tan(e+f*x)/(c*f*(c-c*sec(e+f*x))^(3/2))],
[sec(e+f*x)*(a+a*sec(e+f*x))^2*(c-c*sec(e+f*x))^(7/2),x,4,-8/33*c^2*(a+a*sec(e+f*x))^2*(c-c*sec(e+f*x))^(3/2)*tan(e+f*x)/f-2/11*c*(a+a*sec(e+f*x))^2*(c-c*sec(e+f*x))^(5/2)*tan(e+f*x)/f-256/1155*c^4*(a+a*sec(e+f*x))^2*tan(e+f*x)/(f*sqrt(c-c*sec(e+f*x)))-64/231*c^3*(a+a*sec(e+f*x))^2*sqrt(c-c*sec(e+f*x))*tan(e+f*x)/f],
[sec(e+f*x)*(a+a*sec(e+f*x))^2*(c-c*sec(e+f*x))^(5/2),x,3,-2/9*c*(a+a*sec(e+f*x))^2*(c-c*sec(e+f*x))^(3/2)*tan(e+f*x)/f-64/315*c^3*(a+a*sec(e+f*x))^2*tan(e+f*x)/(f*sqrt(c-c*sec(e+f*x)))-16/63*c^2*(a+a*sec(e+f*x))^2*sqrt(c-c*sec(e+f*x))*tan(e+f*x)/f],
[sec(e+f*x)*(a+a*sec(e+f*x))^2*(c-c*sec(e+f*x))^(3/2),x,2,-8/35*c^2*(a+a*sec(e+f*x))^2*tan(e+f*x)/(f*sqrt(c-c*sec(e+f*x)))-2/7*c*(a+a*sec(e+f*x))^2*sqrt(c-c*sec(e+f*x))*tan(e+f*x)/f],
[sec(e+f*x)*(a+a*sec(e+f*x))^2*sqrt(c-c*sec(e+f*x)),x,1,-2/5*c*(a+a*sec(e+f*x))^2*tan(e+f*x)/(f*sqrt(c-c*sec(e+f*x)))],
[sec(e+f*x)*(a+a*sec(e+f*x))^2/sqrt(c-c*sec(e+f*x)),x,4,-4*a^2*arctan(sqrt(c)*tan(e+f*x)/(sqrt(2)*sqrt(c-c*sec(e+f*x))))*sqrt(2)/(f*sqrt(c))+16/3*a^2*tan(e+f*x)/(f*sqrt(c-c*sec(e+f*x)))-2/3*a^2*sqrt(c-c*sec(e+f*x))*tan(e+f*x)/(c*f),-4*a^2*arctan(sqrt(c)*tan(e+f*x)/(sqrt(2)*sqrt(c-c*sec(e+f*x))))*sqrt(2)/(f*sqrt(c))+4*a^2*tan(e+f*x)/(f*sqrt(c-c*sec(e+f*x)))+2/3*(a^2+a^2*sec(e+f*x))*tan(e+f*x)/(f*sqrt(c-c*sec(e+f*x)))],
[sec(e+f*x)*(a+a*sec(e+f*x))^2/(c-c*sec(e+f*x))^(3/2),x,4,3*a^2*arctan(sqrt(c)*tan(e+f*x)/(sqrt(2)*sqrt(c-c*sec(e+f*x))))*sqrt(2)/(c^(3/2)*f)-2*a^2*tan(e+f*x)/(f*(c-c*sec(e+f*x))^(3/2))-2*a^2*tan(e+f*x)/(c*f*sqrt(c-c*sec(e+f*x))),3*a^2*arctan(sqrt(c)*tan(e+f*x)/(sqrt(2)*sqrt(c-c*sec(e+f*x))))*sqrt(2)/(c^(3/2)*f)-(a^2+a^2*sec(e+f*x))*tan(e+f*x)/(f*(c-c*sec(e+f*x))^(3/2))-3*a^2*tan(e+f*x)/(c*f*sqrt(c-c*sec(e+f*x)))],
[sec(e+f*x)*(a+a*sec(e+f*x))^2/(c-c*sec(e+f*x))^(5/2),x,4,-3/4*a^2*arctan(sqrt(c)*tan(e+f*x)/(sqrt(2)*sqrt(c-c*sec(e+f*x))))/(c^(5/2)*f*sqrt(2))-a^2*tan(e+f*x)/(f*(c-c*sec(e+f*x))^(5/2))+5/4*a^2*tan(e+f*x)/(c*f*(c-c*sec(e+f*x))^(3/2)),-3/4*a^2*arctan(sqrt(c)*tan(e+f*x)/(sqrt(2)*sqrt(c-c*sec(e+f*x))))/(c^(5/2)*f*sqrt(2))-1/2*(a^2+a^2*sec(e+f*x))*tan(e+f*x)/(f*(c-c*sec(e+f*x))^(5/2))+3/4*a^2*tan(e+f*x)/(c*f*(c-c*sec(e+f*x))^(3/2))],
[sec(e+f*x)*(a+a*sec(e+f*x))^2/(c-c*sec(e+f*x))^(7/2),x,5,-1/16*a^2*arctan(sqrt(c)*tan(e+f*x)/(sqrt(2)*sqrt(c-c*sec(e+f*x))))/(c^(7/2)*f*sqrt(2))-1/3*(a^2+a^2*sec(e+f*x))*tan(e+f*x)/(f*(c-c*sec(e+f*x))^(7/2))+1/4*a^2*tan(e+f*x)/(c*f*(c-c*sec(e+f*x))^(5/2))-1/16*a^2*tan(e+f*x)/(c^2*f*(c-c*sec(e+f*x))^(3/2))],
[sec(e+f*x)*(a+a*sec(e+f*x))^3*(c-c*sec(e+f*x))^(7/2),x,4,-24/143*c^2*(a+a*sec(e+f*x))^3*(c-c*sec(e+f*x))^(3/2)*tan(e+f*x)/f-2/13*c*(a+a*sec(e+f*x))^3*(c-c*sec(e+f*x))^(5/2)*tan(e+f*x)/f-256/3003*c^4*(a+a*sec(e+f*x))^3*tan(e+f*x)/(f*sqrt(c-c*sec(e+f*x)))-64/429*c^3*(a+a*sec(e+f*x))^3*sqrt(c-c*sec(e+f*x))*tan(e+f*x)/f],
[sec(e+f*x)*(a+a*sec(e+f*x))^3*(c-c*sec(e+f*x))^(5/2),x,3,-2/11*c*(a+a*sec(e+f*x))^3*(c-c*sec(e+f*x))^(3/2)*tan(e+f*x)/f-64/693*c^3*(a+a*sec(e+f*x))^3*tan(e+f*x)/(f*sqrt(c-c*sec(e+f*x)))-16/99*c^2*(a+a*sec(e+f*x))^3*sqrt(c-c*sec(e+f*x))*tan(e+f*x)/f],
[sec(e+f*x)*(a+a*sec(e+f*x))^3*(c-c*sec(e+f*x))^(3/2),x,2,-8/63*c^2*(a+a*sec(e+f*x))^3*tan(e+f*x)/(f*sqrt(c-c*sec(e+f*x)))-2/9*c*(a+a*sec(e+f*x))^3*sqrt(c-c*sec(e+f*x))*tan(e+f*x)/f],
[sec(e+f*x)*(a+a*sec(e+f*x))^3*sqrt(c-c*sec(e+f*x)),x,1,-2/7*c*(a+a*sec(e+f*x))^3*tan(e+f*x)/(f*sqrt(c-c*sec(e+f*x)))],
[sec(e+f*x)*(a+a*sec(e+f*x))^3/sqrt(c-c*sec(e+f*x)),x,5,-8*a^3*arctan(sqrt(c)*tan(e+f*x)/(sqrt(2)*sqrt(c-c*sec(e+f*x))))*sqrt(2)/(f*sqrt(c))+8*a^3*tan(e+f*x)/(f*sqrt(c-c*sec(e+f*x)))+2/5*a*(a+a*sec(e+f*x))^2*tan(e+f*x)/(f*sqrt(c-c*sec(e+f*x)))+4/3*(a^3+a^3*sec(e+f*x))*tan(e+f*x)/(f*sqrt(c-c*sec(e+f*x)))],
[sec(e+f*x)*(a+a*sec(e+f*x))^3/(c-c*sec(e+f*x))^(3/2),x,5,10*a^3*arctan(sqrt(c)*tan(e+f*x)/(sqrt(2)*sqrt(c-c*sec(e+f*x))))*sqrt(2)/(c^(3/2)*f)-a*(a+a*sec(e+f*x))^2*tan(e+f*x)/(f*(c-c*sec(e+f*x))^(3/2))-10*a^3*tan(e+f*x)/(c*f*sqrt(c-c*sec(e+f*x)))-5/3*(a^3+a^3*sec(e+f*x))*tan(e+f*x)/(c*f*sqrt(c-c*sec(e+f*x)))],
[sec(e+f*x)*(a+a*sec(e+f*x))^3/(c-c*sec(e+f*x))^(5/2),x,5,-15/2*a^3*arctan(sqrt(c)*tan(e+f*x)/(sqrt(2)*sqrt(c-c*sec(e+f*x))))/(c^(5/2)*f*sqrt(2))-1/2*a*(a+a*sec(e+f*x))^2*tan(e+f*x)/(f*(c-c*sec(e+f*x))^(5/2))+5/4*(a^3+a^3*sec(e+f*x))*tan(e+f*x)/(c*f*(c-c*sec(e+f*x))^(3/2))+15/4*a^3*tan(e+f*x)/(c^2*f*sqrt(c-c*sec(e+f*x)))],

# m<0
[sec(e+f*x)*(c-c*sec(e+f*x))^(7/2)/(a+a*sec(e+f*x)),x,4,12/5*c^2*(c-c*sec(e+f*x))^(3/2)*tan(e+f*x)/(a*f)+2*c*(c-c*sec(e+f*x))^(5/2)*tan(e+f*x)/(f*(a+a*sec(e+f*x)))+128/5*c^4*tan(e+f*x)/(a*f*sqrt(c-c*sec(e+f*x)))+32/5*c^3*sqrt(c-c*sec(e+f*x))*tan(e+f*x)/(a*f)],
[sec(e+f*x)*(c-c*sec(e+f*x))^(5/2)/(a+a*sec(e+f*x)),x,3,2*c*(c-c*sec(e+f*x))^(3/2)*tan(e+f*x)/(f*(a+a*sec(e+f*x)))+32/3*c^3*tan(e+f*x)/(a*f*sqrt(c-c*sec(e+f*x)))+8/3*c^2*sqrt(c-c*sec(e+f*x))*tan(e+f*x)/(a*f)],
[sec(e+f*x)*(c-c*sec(e+f*x))^(3/2)/(a+a*sec(e+f*x)),x,2,4*c^2*tan(e+f*x)/(a*f*sqrt(c-c*sec(e+f*x)))+2*c*sqrt(c-c*sec(e+f*x))*tan(e+f*x)/(f*(a+a*sec(e+f*x)))],
[sec(e+f*x)*sqrt(c-c*sec(e+f*x))/(a+a*sec(e+f*x)),x,1,2*c*tan(e+f*x)/(f*(a+a*sec(e+f*x))*sqrt(c-c*sec(e+f*x)))],
[sec(e+f*x)/((a+a*sec(e+f*x))*sqrt(c-c*sec(e+f*x))),x,3,-arctan(sqrt(c)*tan(e+f*x)/(sqrt(2)*sqrt(c-c*sec(e+f*x))))/(a*f*sqrt(2)*sqrt(c))+tan(e+f*x)/(f*(a+a*sec(e+f*x))*sqrt(c-c*sec(e+f*x)))],
[sec(e+f*x)/((a+a*sec(e+f*x))*(c-c*sec(e+f*x))^(3/2)),x,4,-3/4*arctan(sqrt(c)*tan(e+f*x)/(sqrt(2)*sqrt(c-c*sec(e+f*x))))/(a*c^(3/2)*f*sqrt(2))-3/4*tan(e+f*x)/(a*f*(c-c*sec(e+f*x))^(3/2))+tan(e+f*x)/(f*(a+a*sec(e+f*x))*(c-c*sec(e+f*x))^(3/2))],
[sec(e+f*x)/((a+a*sec(e+f*x))*(c-c*sec(e+f*x))^(5/2)),x,5,-15/32*arctan(sqrt(c)*tan(e+f*x)/(sqrt(2)*sqrt(c-c*sec(e+f*x))))/(a*c^(5/2)*f*sqrt(2))-5/8*tan(e+f*x)/(a*f*(c-c*sec(e+f*x))^(5/2))+tan(e+f*x)/(f*(a+a*sec(e+f*x))*(c-c*sec(e+f*x))^(5/2))-15/32*tan(e+f*x)/(a*c*f*(c-c*sec(e+f*x))^(3/2))],
[sec(e+f*x)*(c-c*sec(e+f*x))^(7/2)/(a+a*sec(e+f*x))^2,x,4,-4*c^2*(c-c*sec(e+f*x))^(3/2)*tan(e+f*x)/(f*(a^2+a^2*sec(e+f*x)))+2/3*c*(c-c*sec(e+f*x))^(5/2)*tan(e+f*x)/(f*(a+a*sec(e+f*x))^2)-64/3*c^4*tan(e+f*x)/(a^2*f*sqrt(c-c*sec(e+f*x)))-16/3*c^3*sqrt(c-c*sec(e+f*x))*tan(e+f*x)/(a^2*f)],
[sec(e+f*x)*(c-c*sec(e+f*x))^(5/2)/(a+a*sec(e+f*x))^2,x,3,2/3*c*(c-c*sec(e+f*x))^(3/2)*tan(e+f*x)/(f*(a+a*sec(e+f*x))^2)-16/3*c^3*tan(e+f*x)/(a^2*f*sqrt(c-c*sec(e+f*x)))-8/3*c^2*sqrt(c-c*sec(e+f*x))*tan(e+f*x)/(f*(a^2+a^2*sec(e+f*x)))],
[sec(e+f*x)*(c-c*sec(e+f*x))^(3/2)/(a+a*sec(e+f*x))^2,x,2,-4/3*c^2*tan(e+f*x)/(f*(a^2+a^2*sec(e+f*x))*sqrt(c-c*sec(e+f*x)))+2/3*c*sqrt(c-c*sec(e+f*x))*tan(e+f*x)/(f*(a+a*sec(e+f*x))^2)],
[sec(e+f*x)*sqrt(c-c*sec(e+f*x))/(a+a*sec(e+f*x))^2,x,1,2/3*c*tan(e+f*x)/(f*(a+a*sec(e+f*x))^2*sqrt(c-c*sec(e+f*x)))],
[sec(e+f*x)/((a+a*sec(e+f*x))^2*sqrt(c-c*sec(e+f*x))),x,4,-1/2*arctan(sqrt(c)*tan(e+f*x)/(sqrt(2)*sqrt(c-c*sec(e+f*x))))/(a^2*f*sqrt(2)*sqrt(c))+1/3*tan(e+f*x)/(f*(a+a*sec(e+f*x))^2*sqrt(c-c*sec(e+f*x)))+1/2*tan(e+f*x)/(f*(a^2+a^2*sec(e+f*x))*sqrt(c-c*sec(e+f*x)))],
[sec(e+f*x)/((a+a*sec(e+f*x))^2*(c-c*sec(e+f*x))^(3/2)),x,5,-5/8*arctan(sqrt(c)*tan(e+f*x)/(sqrt(2)*sqrt(c-c*sec(e+f*x))))/(a^2*c^(3/2)*f*sqrt(2))-5/8*tan(e+f*x)/(a^2*f*(c-c*sec(e+f*x))^(3/2))+1/3*tan(e+f*x)/(f*(a+a*sec(e+f*x))^2*(c-c*sec(e+f*x))^(3/2))+5/6*tan(e+f*x)/(f*(a^2+a^2*sec(e+f*x))*(c-c*sec(e+f*x))^(3/2))],
[sec(e+f*x)/((a+a*sec(e+f*x))^2*(c-c*sec(e+f*x))^(5/2)),x,6,-35/64*arctan(sqrt(c)*tan(e+f*x)/(sqrt(2)*sqrt(c-c*sec(e+f*x))))/(a^2*c^(5/2)*f*sqrt(2))-35/48*tan(e+f*x)/(a^2*f*(c-c*sec(e+f*x))^(5/2))+1/3*tan(e+f*x)/(f*(a+a*sec(e+f*x))^2*(c-c*sec(e+f*x))^(5/2))+7/6*tan(e+f*x)/(f*(a^2+a^2*sec(e+f*x))*(c-c*sec(e+f*x))^(5/2))-35/64*tan(e+f*x)/(a^2*c*f*(c-c*sec(e+f*x))^(3/2))],
[sec(e+f*x)*(c-c*sec(e+f*x))^(7/2)/(a+a*sec(e+f*x))^3,x,4,-4/5*c^2*(c-c*sec(e+f*x))^(3/2)*tan(e+f*x)/(a*f*(a+a*sec(e+f*x))^2)+2/5*c*(c-c*sec(e+f*x))^(5/2)*tan(e+f*x)/(f*(a+a*sec(e+f*x))^3)+32/5*c^4*tan(e+f*x)/(a^3*f*sqrt(c-c*sec(e+f*x)))+16/5*c^3*sqrt(c-c*sec(e+f*x))*tan(e+f*x)/(f*(a^3+a^3*sec(e+f*x)))],
[sec(e+f*x)*(c-c*sec(e+f*x))^(5/2)/(a+a*sec(e+f*x))^3,x,3,2/5*c*(c-c*sec(e+f*x))^(3/2)*tan(e+f*x)/(f*(a+a*sec(e+f*x))^3)+16/15*c^3*tan(e+f*x)/(f*(a^3+a^3*sec(e+f*x))*sqrt(c-c*sec(e+f*x)))-8/15*c^2*sqrt(c-c*sec(e+f*x))*tan(e+f*x)/(a*f*(a+a*sec(e+f*x))^2)],
[sec(e+f*x)*(c-c*sec(e+f*x))^(3/2)/(a+a*sec(e+f*x))^3,x,2,-4/15*c^2*tan(e+f*x)/(a*f*(a+a*sec(e+f*x))^2*sqrt(c-c*sec(e+f*x)))+2/5*c*sqrt(c-c*sec(e+f*x))*tan(e+f*x)/(f*(a+a*sec(e+f*x))^3)],
[sec(e+f*x)*sqrt(c-c*sec(e+f*x))/(a+a*sec(e+f*x))^3,x,1,2/5*c*tan(e+f*x)/(f*(a+a*sec(e+f*x))^3*sqrt(c-c*sec(e+f*x)))],
[sec(e+f*x)/((a+a*sec(e+f*x))^3*sqrt(c-c*sec(e+f*x))),x,5,-1/4*arctan(sqrt(c)*tan(e+f*x)/(sqrt(2)*sqrt(c-c*sec(e+f*x))))/(a^3*f*sqrt(2)*sqrt(c))+1/5*tan(e+f*x)/(f*(a+a*sec(e+f*x))^3*sqrt(c-c*sec(e+f*x)))+1/6*tan(e+f*x)/(a*f*(a+a*sec(e+f*x))^2*sqrt(c-c*sec(e+f*x)))+1/4*tan(e+f*x)/(f*(a^3+a^3*sec(e+f*x))*sqrt(c-c*sec(e+f*x)))],
[sec(e+f*x)/((a+a*sec(e+f*x))^3*(c-c*sec(e+f*x))^(3/2)),x,6,-7/16*arctan(sqrt(c)*tan(e+f*x)/(sqrt(2)*sqrt(c-c*sec(e+f*x))))/(a^3*c^(3/2)*f*sqrt(2))-7/16*tan(e+f*x)/(a^3*f*(c-c*sec(e+f*x))^(3/2))+1/5*tan(e+f*x)/(f*(a+a*sec(e+f*x))^3*(c-c*sec(e+f*x))^(3/2))+7/30*tan(e+f*x)/(a*f*(a+a*sec(e+f*x))^2*(c-c*sec(e+f*x))^(3/2))+7/12*tan(e+f*x)/(f*(a^3+a^3*sec(e+f*x))*(c-c*sec(e+f*x))^(3/2))],
[sec(e+f*x)/((a+a*sec(e+f*x))^3*(c-c*sec(e+f*x))^(5/2)),x,7,-63/128*arctan(sqrt(c)*tan(e+f*x)/(sqrt(2)*sqrt(c-c*sec(e+f*x))))/(a^3*c^(5/2)*f*sqrt(2))-21/32*tan(e+f*x)/(a^3*f*(c-c*sec(e+f*x))^(5/2))+1/5*tan(e+f*x)/(f*(a+a*sec(e+f*x))^3*(c-c*sec(e+f*x))^(5/2))+3/10*tan(e+f*x)/(a*f*(a+a*sec(e+f*x))^2*(c-c*sec(e+f*x))^(5/2))+21/20*tan(e+f*x)/(f*(a^3+a^3*sec(e+f*x))*(c-c*sec(e+f*x))^(5/2))-63/128*tan(e+f*x)/(a^3*c*f*(c-c*sec(e+f*x))^(3/2))],

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

# m>0
[sec(e+f*x)*(c-c*sec(e+f*x))^(5/2)*sqrt(a+a*sec(e+f*x)),x,1,1/3*a*(c-c*sec(e+f*x))^(5/2)*tan(e+f*x)/(f*sqrt(a+a*sec(e+f*x)))],
[sec(e+f*x)*(c-c*sec(e+f*x))^(3/2)*sqrt(a+a*sec(e+f*x)),x,1,1/2*a*(c-c*sec(e+f*x))^(3/2)*tan(e+f*x)/(f*sqrt(a+a*sec(e+f*x)))],
[sec(e+f*x)*sqrt(a+a*sec(e+f*x))*sqrt(c-c*sec(e+f*x)),x,1,-c*sqrt(a+a*sec(e+f*x))*tan(e+f*x)/(f*sqrt(c-c*sec(e+f*x)))],
[sec(e+f*x)*sqrt(a+a*sec(e+f*x))/sqrt(c-c*sec(e+f*x)),x,1,a*log(1-sec(e+f*x))*tan(e+f*x)/(f*sqrt(a+a*sec(e+f*x))*sqrt(c-c*sec(e+f*x)))],
[sec(e+f*x)*sqrt(a+a*sec(e+f*x))/(c-c*sec(e+f*x))^(3/2),x,1,-1/2*sqrt(a+a*sec(e+f*x))*tan(e+f*x)/(f*(c-c*sec(e+f*x))^(3/2))],
[sec(e+f*x)*sqrt(a+a*sec(e+f*x))/(c-c*sec(e+f*x))^(5/2),x,1,-1/2*a*tan(e+f*x)/(f*(c-c*sec(e+f*x))^(5/2)*sqrt(a+a*sec(e+f*x)))],
[sec(e+f*x)*(a+a*sec(e+f*x))^(3/2)*(c-c*sec(e+f*x))^(7/2),x,2,1/10*a^2*(c-c*sec(e+f*x))^(7/2)*tan(e+f*x)/(f*sqrt(a+a*sec(e+f*x)))+1/5*a*(c-c*sec(e+f*x))^(7/2)*sqrt(a+a*sec(e+f*x))*tan(e+f*x)/f],
[sec(e+f*x)*(a+a*sec(e+f*x))^(3/2)*(c-c*sec(e+f*x))^(5/2),x,2,1/6*a^2*(c-c*sec(e+f*x))^(5/2)*tan(e+f*x)/(f*sqrt(a+a*sec(e+f*x)))+1/4*a*(c-c*sec(e+f*x))^(5/2)*sqrt(a+a*sec(e+f*x))*tan(e+f*x)/f],
[sec(e+f*x)*(a+a*sec(e+f*x))^(3/2)*(c-c*sec(e+f*x))^(3/2),x,2,-1/3*c^2*(a+a*sec(e+f*x))^(3/2)*tan(e+f*x)/(f*sqrt(c-c*sec(e+f*x)))-1/3*c*(a+a*sec(e+f*x))^(3/2)*sqrt(c-c*sec(e+f*x))*tan(e+f*x)/f],
[sec(e+f*x)*(a+a*sec(e+f*x))^(3/2)*sqrt(c-c*sec(e+f*x)),x,1,-1/2*c*(a+a*sec(e+f*x))^(3/2)*tan(e+f*x)/(f*sqrt(c-c*sec(e+f*x)))],
[sec(e+f*x)*(a+a*sec(e+f*x))^(3/2)/sqrt(c-c*sec(e+f*x)),x,2,2*a^2*log(1-sec(e+f*x))*tan(e+f*x)/(f*sqrt(a+a*sec(e+f*x))*sqrt(c-c*sec(e+f*x)))+a*sqrt(a+a*sec(e+f*x))*tan(e+f*x)/(f*sqrt(c-c*sec(e+f*x)))],
[sec(e+f*x)*(a+a*sec(e+f*x))^(3/2)/(c-c*sec(e+f*x))^(3/2),x,2,-a*sqrt(a+a*sec(e+f*x))*tan(e+f*x)/(f*(c-c*sec(e+f*x))^(3/2))-a^2*log(1-sec(e+f*x))*tan(e+f*x)/(c*f*sqrt(a+a*sec(e+f*x))*sqrt(c-c*sec(e+f*x)))],
[sec(e+f*x)*(a+a*sec(e+f*x))^(3/2)/(c-c*sec(e+f*x))^(5/2),x,1,-1/4*(a+a*sec(e+f*x))^(3/2)*tan(e+f*x)/(f*(c-c*sec(e+f*x))^(5/2))],
[sec(e+f*x)*(a+a*sec(e+f*x))^(3/2)/(c-c*sec(e+f*x))^(7/2),x,2,-1/6*(a+a*sec(e+f*x))^(3/2)*tan(e+f*x)/(f*(c-c*sec(e+f*x))^(7/2))-1/24*(a+a*sec(e+f*x))^(3/2)*tan(e+f*x)/(c*f*(c-c*sec(e+f*x))^(5/2))],
[sec(e+f*x)*(a+a*sec(e+f*x))^(3/2)/(c-c*sec(e+f*x))^(9/2),x,2,1/12*a^2*tan(e+f*x)/(c*f*(c-c*sec(e+f*x))^(7/2)*sqrt(a+a*sec(e+f*x)))-1/4*a*sqrt(a+a*sec(e+f*x))*tan(e+f*x)/(f*(c-c*sec(e+f*x))^(9/2))],
[sec(e+f*x)*(a+a*sec(e+f*x))^(3/2)/(c-c*sec(e+f*x))^(11/2),x,2,1/20*a^2*tan(e+f*x)/(c*f*(c-c*sec(e+f*x))^(9/2)*sqrt(a+a*sec(e+f*x)))-1/5*a*sqrt(a+a*sec(e+f*x))*tan(e+f*x)/(f*(c-c*sec(e+f*x))^(11/2))],
[sec(e+f*x)*(a+a*sec(e+f*x))^(5/2)*(c-c*sec(e+f*x))^(7/2),x,3,1/6*a*(a+a*sec(e+f*x))^(3/2)*(c-c*sec(e+f*x))^(7/2)*tan(e+f*x)/f+1/15*a^3*(c-c*sec(e+f*x))^(7/2)*tan(e+f*x)/(f*sqrt(a+a*sec(e+f*x)))+2/15*a^2*(c-c*sec(e+f*x))^(7/2)*sqrt(a+a*sec(e+f*x))*tan(e+f*x)/f],
[sec(e+f*x)*(a+a*sec(e+f*x))^(5/2)*(c-c*sec(e+f*x))^(5/2),x,3,-1/5*c*(a+a*sec(e+f*x))^(5/2)*(c-c*sec(e+f*x))^(3/2)*tan(e+f*x)/f-2/15*c^3*(a+a*sec(e+f*x))^(5/2)*tan(e+f*x)/(f*sqrt(c-c*sec(e+f*x)))-1/5*c^2*(a+a*sec(e+f*x))^(5/2)*sqrt(c-c*sec(e+f*x))*tan(e+f*x)/f],
[sec(e+f*x)*(a+a*sec(e+f*x))^(5/2)*(c-c*sec(e+f*x))^(3/2),x,2,-1/6*c^2*(a+a*sec(e+f*x))^(5/2)*tan(e+f*x)/(f*sqrt(c-c*sec(e+f*x)))-1/4*c*(a+a*sec(e+f*x))^(5/2)*sqrt(c-c*sec(e+f*x))*tan(e+f*x)/f],
[sec(e+f*x)*(a+a*sec(e+f*x))^(5/2)*sqrt(c-c*sec(e+f*x)),x,1,-1/3*c*(a+a*sec(e+f*x))^(5/2)*tan(e+f*x)/(f*sqrt(c-c*sec(e+f*x)))],
[sec(e+f*x)*(a+a*sec(e+f*x))^(5/2)/sqrt(c-c*sec(e+f*x)),x,3,1/2*a*(a+a*sec(e+f*x))^(3/2)*tan(e+f*x)/(f*sqrt(c-c*sec(e+f*x)))+4*a^3*log(1-sec(e+f*x))*tan(e+f*x)/(f*sqrt(a+a*sec(e+f*x))*sqrt(c-c*sec(e+f*x)))+2*a^2*sqrt(a+a*sec(e+f*x))*tan(e+f*x)/(f*sqrt(c-c*sec(e+f*x)))],
[sec(e+f*x)*(a+a*sec(e+f*x))^(5/2)/(c-c*sec(e+f*x))^(3/2),x,3,-a*(a+a*sec(e+f*x))^(3/2)*tan(e+f*x)/(f*(c-c*sec(e+f*x))^(3/2))-4*a^3*log(1-sec(e+f*x))*tan(e+f*x)/(c*f*sqrt(a+a*sec(e+f*x))*sqrt(c-c*sec(e+f*x)))-2*a^2*sqrt(a+a*sec(e+f*x))*tan(e+f*x)/(c*f*sqrt(c-c*sec(e+f*x)))],
[sec(e+f*x)*(a+a*sec(e+f*x))^(5/2)/(c-c*sec(e+f*x))^(5/2),x,3,-1/2*a*(a+a*sec(e+f*x))^(3/2)*tan(e+f*x)/(f*(c-c*sec(e+f*x))^(5/2))+a^2*sqrt(a+a*sec(e+f*x))*tan(e+f*x)/(c*f*(c-c*sec(e+f*x))^(3/2))+a^3*log(1-sec(e+f*x))*tan(e+f*x)/(c^2*f*sqrt(a+a*sec(e+f*x))*sqrt(c-c*sec(e+f*x)))],
[sec(e+f*x)*(a+a*sec(e+f*x))^(5/2)/(c-c*sec(e+f*x))^(7/2),x,1,-1/6*(a+a*sec(e+f*x))^(5/2)*tan(e+f*x)/(f*(c-c*sec(e+f*x))^(7/2))],
[sec(e+f*x)*(a+a*sec(e+f*x))^(5/2)/(c-c*sec(e+f*x))^(9/2),x,2,-1/8*(a+a*sec(e+f*x))^(5/2)*tan(e+f*x)/(f*(c-c*sec(e+f*x))^(9/2))-1/48*(a+a*sec(e+f*x))^(5/2)*tan(e+f*x)/(c*f*(c-c*sec(e+f*x))^(7/2))],
[sec(e+f*x)*(a+a*sec(e+f*x))^(5/2)/(c-c*sec(e+f*x))^(11/2),x,3,-1/10*(a+a*sec(e+f*x))^(5/2)*tan(e+f*x)/(f*(c-c*sec(e+f*x))^(11/2))-1/40*(a+a*sec(e+f*x))^(5/2)*tan(e+f*x)/(c*f*(c-c*sec(e+f*x))^(9/2))-1/240*(a+a*sec(e+f*x))^(5/2)*tan(e+f*x)/(c^2*f*(c-c*sec(e+f*x))^(7/2))],

# m<0
[sec(e+f*x)*(c-c*sec(e+f*x))^(5/2)/sqrt(a+a*sec(e+f*x)),x,3,-1/2*c*(c-c*sec(e+f*x))^(3/2)*tan(e+f*x)/(f*sqrt(a+a*sec(e+f*x)))-4*c^3*log(1+sec(e+f*x))*tan(e+f*x)/(f*sqrt(a+a*sec(e+f*x))*sqrt(c-c*sec(e+f*x)))-2*c^2*sqrt(c-c*sec(e+f*x))*tan(e+f*x)/(f*sqrt(a+a*sec(e+f*x)))],
[sec(e+f*x)*(c-c*sec(e+f*x))^(3/2)/sqrt(a+a*sec(e+f*x)),x,2,-2*c^2*log(1+sec(e+f*x))*tan(e+f*x)/(f*sqrt(a+a*sec(e+f*x))*sqrt(c-c*sec(e+f*x)))-c*sqrt(c-c*sec(e+f*x))*tan(e+f*x)/(f*sqrt(a+a*sec(e+f*x)))],
[sec(e+f*x)*sqrt(c-c*sec(e+f*x))/sqrt(a+a*sec(e+f*x)),x,1,-c*log(1+sec(e+f*x))*tan(e+f*x)/(f*sqrt(a+a*sec(e+f*x))*sqrt(c-c*sec(e+f*x)))],
[sec(e+f*x)/(sqrt(a+a*sec(e+f*x))*sqrt(c-c*sec(e+f*x))),x,2,-arctanh(cos(e+f*x))*tan(e+f*x)/(f*sqrt(a+a*sec(e+f*x))*sqrt(c-c*sec(e+f*x)))],
[sec(e+f*x)/((c-c*sec(e+f*x))^(3/2)*sqrt(a+a*sec(e+f*x))),x,3,-1/2*tan(e+f*x)/(f*(c-c*sec(e+f*x))^(3/2)*sqrt(a+a*sec(e+f*x)))-1/2*arctanh(cos(e+f*x))*tan(e+f*x)/(c*f*sqrt(a+a*sec(e+f*x))*sqrt(c-c*sec(e+f*x)))],
[sec(e+f*x)/((c-c*sec(e+f*x))^(5/2)*sqrt(a+a*sec(e+f*x))),x,4,-1/4*tan(e+f*x)/(f*(c-c*sec(e+f*x))^(5/2)*sqrt(a+a*sec(e+f*x)))-1/4*tan(e+f*x)/(c*f*(c-c*sec(e+f*x))^(3/2)*sqrt(a+a*sec(e+f*x)))-1/4*arctanh(cos(e+f*x))*tan(e+f*x)/(c^2*f*sqrt(a+a*sec(e+f*x))*sqrt(c-c*sec(e+f*x)))],
[sec(e+f*x)*(c-c*sec(e+f*x))^(5/2)/(a+a*sec(e+f*x))^(3/2),x,3,c*(c-c*sec(e+f*x))^(3/2)*tan(e+f*x)/(f*(a+a*sec(e+f*x))^(3/2))+4*c^3*log(1+sec(e+f*x))*tan(e+f*x)/(a*f*sqrt(a+a*sec(e+f*x))*sqrt(c-c*sec(e+f*x)))+2*c^2*sqrt(c-c*sec(e+f*x))*tan(e+f*x)/(a*f*sqrt(a+a*sec(e+f*x)))],
[sec(e+f*x)*(c-c*sec(e+f*x))^(3/2)/(a+a*sec(e+f*x))^(3/2),x,2,c^2*log(1+sec(e+f*x))*tan(e+f*x)/(a*f*sqrt(a+a*sec(e+f*x))*sqrt(c-c*sec(e+f*x)))+c*sqrt(c-c*sec(e+f*x))*tan(e+f*x)/(f*(a+a*sec(e+f*x))^(3/2))],
[sec(e+f*x)*sqrt(c-c*sec(e+f*x))/(a+a*sec(e+f*x))^(3/2),x,1,1/2*sqrt(c-c*sec(e+f*x))*tan(e+f*x)/(f*(a+a*sec(e+f*x))^(3/2))],
[sec(e+f*x)/((a+a*sec(e+f*x))^(3/2)*sqrt(c-c*sec(e+f*x))),x,3,1/2*tan(e+f*x)/(f*(a+a*sec(e+f*x))^(3/2)*sqrt(c-c*sec(e+f*x)))-1/2*arctanh(cos(e+f*x))*tan(e+f*x)/(a*f*sqrt(a+a*sec(e+f*x))*sqrt(c-c*sec(e+f*x)))],
[sec(e+f*x)/((a+a*sec(e+f*x))^(3/2)*(c-c*sec(e+f*x))^(3/2)),x,3,1/2*csc(e+f*x)/(a*c*f*sqrt(a+a*sec(e+f*x))*sqrt(c-c*sec(e+f*x)))-1/2*arctanh(cos(e+f*x))*tan(e+f*x)/(a*c*f*sqrt(a+a*sec(e+f*x))*sqrt(c-c*sec(e+f*x)))],
[sec(e+f*x)/((a+a*sec(e+f*x))^(3/2)*(c-c*sec(e+f*x))^(5/2)),x,4,3/8*csc(e+f*x)/(a*c^2*f*sqrt(a+a*sec(e+f*x))*sqrt(c-c*sec(e+f*x)))-1/4*tan(e+f*x)/(f*(a+a*sec(e+f*x))^(3/2)*(c-c*sec(e+f*x))^(5/2))-3/8*arctanh(cos(e+f*x))*tan(e+f*x)/(a*c^2*f*sqrt(a+a*sec(e+f*x))*sqrt(c-c*sec(e+f*x)))],
[sec(e+f*x)*(c-c*sec(e+f*x))^(5/2)/(a+a*sec(e+f*x))^(5/2),x,3,1/2*c*(c-c*sec(e+f*x))^(3/2)*tan(e+f*x)/(f*(a+a*sec(e+f*x))^(5/2))-c^3*log(1+sec(e+f*x))*tan(e+f*x)/(a^2*f*sqrt(a+a*sec(e+f*x))*sqrt(c-c*sec(e+f*x)))-c^2*sqrt(c-c*sec(e+f*x))*tan(e+f*x)/(a*f*(a+a*sec(e+f*x))^(3/2))],
[sec(e+f*x)*(c-c*sec(e+f*x))^(3/2)/(a+a*sec(e+f*x))^(5/2),x,1,1/4*(c-c*sec(e+f*x))^(3/2)*tan(e+f*x)/(f*(a+a*sec(e+f*x))^(5/2))],
[sec(e+f*x)*sqrt(c-c*sec(e+f*x))/(a+a*sec(e+f*x))^(5/2),x,1,1/2*c*tan(e+f*x)/(f*(a+a*sec(e+f*x))^(5/2)*sqrt(c-c*sec(e+f*x)))],
[sec(e+f*x)/((a+a*sec(e+f*x))^(5/2)*sqrt(c-c*sec(e+f*x))),x,4,1/4*tan(e+f*x)/(f*(a+a*sec(e+f*x))^(5/2)*sqrt(c-c*sec(e+f*x)))+1/4*tan(e+f*x)/(a*f*(a+a*sec(e+f*x))^(3/2)*sqrt(c-c*sec(e+f*x)))-1/4*arctanh(cos(e+f*x))*tan(e+f*x)/(a^2*f*sqrt(a+a*sec(e+f*x))*sqrt(c-c*sec(e+f*x)))],
[sec(e+f*x)/((a+a*sec(e+f*x))^(5/2)*(c-c*sec(e+f*x))^(3/2)),x,4,3/8*csc(e+f*x)/(a^2*c*f*sqrt(a+a*sec(e+f*x))*sqrt(c-c*sec(e+f*x)))+1/4*tan(e+f*x)/(f*(a+a*sec(e+f*x))^(5/2)*(c-c*sec(e+f*x))^(3/2))-3/8*arctanh(cos(e+f*x))*tan(e+f*x)/(a^2*c*f*sqrt(a+a*sec(e+f*x))*sqrt(c-c*sec(e+f*x)))],
[sec(e+f*x)/((a+a*sec(e+f*x))^(5/2)*(c-c*sec(e+f*x))^(5/2)),x,4,3/8*csc(e+f*x)/(a^2*c^2*f*sqrt(a+a*sec(e+f*x))*sqrt(c-c*sec(e+f*x)))-1/4*cot(e+f*x)^2*csc(e+f*x)/(a^2*c^2*f*sqrt(a+a*sec(e+f*x))*sqrt(c-c*sec(e+f*x)))-3/8*arctanh(cos(e+f*x))*tan(e+f*x)/(a^2*c^2*f*sqrt(a+a*sec(e+f*x))*sqrt(c-c*sec(e+f*x)))],

# Integrands of the form Sec[e+f x] (a+a Sec[e+f x])^m (c-c Sec[e+f x])^n with m and/or n symbolic
[sec(e+f*x)*(a+a*sec(e+f*x))^m*(c-c*sec(e+f*x))^n,x,3,-2^(1/2+n)*c*hypergeom([1/2+m,1/2-n],[3/2+m],1/2*(1+sec(e+f*x)))*(1-sec(e+f*x))^(1/2-n)*(a+a*sec(e+f*x))^m*(c-c*sec(e+f*x))^(-1+n)*tan(e+f*x)/(f*(1+2*m))],
[sec(e+f*x)*(a+a*sec(e+f*x))^m*(c-c*sec(e+f*x))^2,x,3,1/5*2^(1/2+m)*a*hypergeom([5/2,1/2-m],[7/2],1/2*(1-sec(e+f*x)))*(1+sec(e+f*x))^(1/2-m)*(a+a*sec(e+f*x))^(-1+m)*(c-c*sec(e+f*x))^2*tan(e+f*x)/f],
[sec(e+f*x)*(a+a*sec(e+f*x))^m*(c-c*sec(e+f*x)),x,3,1/3*2^(1/2+m)*a*hypergeom([3/2,1/2-m],[5/2],1/2*(1-sec(e+f*x)))*(1+sec(e+f*x))^(1/2-m)*(a+a*sec(e+f*x))^(-1+m)*(c-c*sec(e+f*x))*tan(e+f*x)/f],
[sec(e+f*x)*(a+a*sec(e+f*x))^m/(c-c*sec(e+f*x)),x,3,-2^(1/2+m)*a*hypergeom([-1/2,1/2-m],[1/2],1/2*(1-sec(e+f*x)))*(1+sec(e+f*x))^(1/2-m)*(a+a*sec(e+f*x))^(-1+m)*tan(e+f*x)/(f*(c-c*sec(e+f*x)))],
[sec(e+f*x)*(a+a*sec(e+f*x))^m/(c-c*sec(e+f*x))^2,x,3,-1/3*2^(1/2+m)*a*hypergeom([-3/2,1/2-m],[-1/2],1/2*(1-sec(e+f*x)))*(1+sec(e+f*x))^(1/2-m)*(a+a*sec(e+f*x))^(-1+m)*tan(e+f*x)/(f*(c-c*sec(e+f*x))^2)],
[sec(e+f*x)*(a+a*sec(e+f*x))^m*(c-c*sec(e+f*x))^(5/2),x,3,-2*c*(a+a*sec(e+f*x))^m*(c-c*sec(e+f*x))^(3/2)*tan(e+f*x)/(f*(5+2*m))-64*c^3*(a+a*sec(e+f*x))^m*tan(e+f*x)/(f*(5+2*m)*(3+8*m+4*m^2)*sqrt(c-c*sec(e+f*x)))-16*c^2*(a+a*sec(e+f*x))^m*sqrt(c-c*sec(e+f*x))*tan(e+f*x)/(f*(15+16*m+4*m^2))],
[sec(e+f*x)*(a+a*sec(e+f*x))^m*(c-c*sec(e+f*x))^(3/2),x,2,-8*c^2*(a+a*sec(e+f*x))^m*tan(e+f*x)/(f*(3+8*m+4*m^2)*sqrt(c-c*sec(e+f*x)))-2*c*(a+a*sec(e+f*x))^m*sqrt(c-c*sec(e+f*x))*tan(e+f*x)/(f*(3+2*m))],
[sec(e+f*x)*(a+a*sec(e+f*x))^m*(c-c*sec(e+f*x))^(1/2),x,1,-2*c*(a+a*sec(e+f*x))^m*tan(e+f*x)/(f*(1+2*m)*sqrt(c-c*sec(e+f*x)))],
[sec(e+f*x)*(a+a*sec(e+f*x))^m/(c-c*sec(e+f*x))^(1/2),x,2,-hypergeom([1,1/2+m],[3/2+m],1/2*(1+sec(e+f*x)))*(a+a*sec(e+f*x))^m*tan(e+f*x)/(f*(1+2*m)*sqrt(c-c*sec(e+f*x)))],
[sec(e+f*x)*(a+a*sec(e+f*x))^m/(c-c*sec(e+f*x))^(3/2),x,2,-1/2*hypergeom([2,1/2+m],[3/2+m],1/2*(1+sec(e+f*x)))*(a+a*sec(e+f*x))^m*tan(e+f*x)/(c*f*(1+2*m)*sqrt(c-c*sec(e+f*x)))],
[sec(e+f*x)*(a+a*sec(e+f*x))^m/(c-c*sec(e+f*x))^(5/2),x,2,-1/4*hypergeom([3,1/2+m],[3/2+m],1/2*(1+sec(e+f*x)))*(a+a*sec(e+f*x))^m*tan(e+f*x)/(c^2*f*(1+2*m)*sqrt(c-c*sec(e+f*x)))],
[sec(e+f*x)*(a+a*sec(e+f*x))^m*(c-c*sec(e+f*x))^(-3-m),x,3,-(a+a*sec(e+f*x))^m*(c-c*sec(e+f*x))^(-3-m)*tan(e+f*x)/(f*(1+2*m))+2*(a+a*sec(e+f*x))^(1+m)*(c-c*sec(e+f*x))^(-3-m)*tan(e+f*x)/(a*f*(3+8*m+4*m^2))-2*(a+a*sec(e+f*x))^(2+m)*(c-c*sec(e+f*x))^(-3-m)*tan(e+f*x)/(a^2*f*(1+2*m)*(15+16*m+4*m^2))],
[sec(e+f*x)*(a+a*sec(e+f*x))^m*(c-c*sec(e+f*x))^(-2-m),x,2,-(a+a*sec(e+f*x))^m*(c-c*sec(e+f*x))^(-2-m)*tan(e+f*x)/(f*(1+2*m))+(a+a*sec(e+f*x))^(1+m)*(c-c*sec(e+f*x))^(-2-m)*tan(e+f*x)/(a*f*(3+8*m+4*m^2))],
[sec(e+f*x)*(a+a*sec(e+f*x))^m*(c-c*sec(e+f*x))^(-1-m),x,1,-(a+a*sec(e+f*x))^m*(c-c*sec(e+f*x))^(-1-m)*tan(e+f*x)/(f*(1+2*m))],
[sec(e+f*x)*(a+a*sec(e+f*x))^m/(c-c*sec(e+f*x))^m,x,3,-2^(1/2-m)*c*hypergeom([1/2+m,1/2+m],[3/2+m],1/2*(1+sec(e+f*x)))*(1-sec(e+f*x))^(1/2+m)*(a+a*sec(e+f*x))^m*(c-c*sec(e+f*x))^(-1-m)*tan(e+f*x)/(f*(1+2*m))],
[sec(e+f*x)*(a+a*sec(e+f*x))^m*(c-c*sec(e+f*x))^(1-m),x,3,-2^(3/2-m)*c*hypergeom([-1/2+m,1/2+m],[3/2+m],1/2*(1+sec(e+f*x)))*(1-sec(e+f*x))^(-1/2+m)*(a+a*sec(e+f*x))^m*tan(e+f*x)/(f*(1+2*m)*(c-c*sec(e+f*x))^m)],
[sec(e+f*x)*(a+a*sec(e+f*x))^m*(c-c*sec(e+f*x))^(2-m),x,3,-2^(5/2-m)*c^2*hypergeom([-3/2+m,1/2+m],[3/2+m],1/2*(1+sec(e+f*x)))*(1-sec(e+f*x))^(-1/2+m)*(a+a*sec(e+f*x))^m*tan(e+f*x)/(f*(1+2*m)*(c-c*sec(e+f*x))^m)],

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

# Integrands of the form (g Sec[e+f x])^p (a+a Sec[e+f x])^m (c-c Sec[e+f x])
[sec(e+f*x)^2*(a+a*sec(e+f*x))^3*(c-c*sec(e+f*x)),x,10,1/4*a^3*c*arctanh(sin(e+f*x))/f+1/4*a^3*c*sec(e+f*x)*tan(e+f*x)/f-1/2*a^3*c*sec(e+f*x)^3*tan(e+f*x)/f-2/3*a^3*c*tan(e+f*x)^3/f-1/5*a^3*c*tan(e+f*x)^5/f],
[sec(e+f*x)^2*(a+a*sec(e+f*x))^2*(c-c*sec(e+f*x)),x,7,1/8*a^2*c*arctanh(sin(e+f*x))/f+1/8*a^2*c*sec(e+f*x)*tan(e+f*x)/f-1/4*a^2*c*sec(e+f*x)^3*tan(e+f*x)/f-1/3*a^2*c*tan(e+f*x)^3/f],
[sec(e+f*x)^2*(a+a*sec(e+f*x))*(c-c*sec(e+f*x)),x,3,-1/3*a*c*tan(e+f*x)^3/f],
[sec(e+f*x)^2*(c-c*sec(e+f*x))/(a+a*sec(e+f*x)),x,5,2*c*arctanh(sin(e+f*x))/(a*f)-c*tan(e+f*x)/(a*f)-2*c*tan(e+f*x)/(f*(a+a*sec(e+f*x)))],
[sec(e+f*x)^2*(c-c*sec(e+f*x))/(a+a*sec(e+f*x))^2,x,4,-c*arctanh(sin(e+f*x))/(a^2*f)+7/3*c*tan(e+f*x)/(a^2*f*(1+sec(e+f*x)))-2/3*c*tan(e+f*x)/(f*(a+a*sec(e+f*x))^2)],
[sec(e+f*x)^2*(c-c*sec(e+f*x))/(a+a*sec(e+f*x))^3,x,3,-2/5*c*tan(e+f*x)/(f*(a+a*sec(e+f*x))^3)+11/15*c*tan(e+f*x)/(a*f*(a+a*sec(e+f*x))^2)-4/15*c*tan(e+f*x)/(f*(a^3+a^3*sec(e+f*x)))],
[(g*sec(e+f*x))^p*(a+a*sec(e+f*x))^2*(c-c*sec(e+f*x)),x,5,-1/3*a^2*c*(cos(e+f*x)^2)^(1/2*(3+p))*hypergeom([3/2,1/2*(3+p)],[5/2],sin(e+f*x)^2)*(g*sec(e+f*x))^p*tan(e+f*x)^3/f-1/3*a^2*c*(cos(e+f*x)^2)^(1/2*(4+p))*hypergeom([3/2,1/2*(4+p)],[5/2],sin(e+f*x)^2)*(g*sec(e+f*x))^(1+p)*tan(e+f*x)^3/(f*g)],
[(g*sec(e+f*x))^p*(a+a*sec(e+f*x))*(c-c*sec(e+f*x)),x,2,-1/3*a*c*(cos(e+f*x)^2)^(1/2*(3+p))*hypergeom([3/2,1/2*(3+p)],[5/2],sin(e+f*x)^2)*(g*sec(e+f*x))^p*tan(e+f*x)^3/f],
[(g*sec(e+f*x))^p*(c-c*sec(e+f*x))/(a+a*sec(e+f*x)),x,6,-c*g*(1-2*p)*hypergeom([1/2,1/2*(1-p)],[1/2*(3-p)],cos(e+f*x)^2)*(g*sec(e+f*x))^(-1+p)*sin(e+f*x)/(a*f*(1-p)*sqrt(sin(e+f*x)^2))+2*c*hypergeom([1/2,-1/2*p],[1/2*(2-p)],cos(e+f*x)^2)*(g*sec(e+f*x))^p*sin(e+f*x)/(a*f*sqrt(sin(e+f*x)^2))-2*c*(g*sec(e+f*x))^p*tan(e+f*x)/(f*(a+a*sec(e+f*x)))],
[(g*sec(e+f*x))^p*(c-c*sec(e+f*x))/(a+a*sec(e+f*x))^2,x,7,-1/3*c*g*(3-4*p)*hypergeom([1/2,1/2*(1-p)],[1/2*(3-p)],cos(e+f*x)^2)*(g*sec(e+f*x))^(-1+p)*sin(e+f*x)/(a^2*f*sqrt(sin(e+f*x)^2))+1/3*c*(5-4*p)*hypergeom([1/2,-1/2*p],[1/2*(2-p)],cos(e+f*x)^2)*(g*sec(e+f*x))^p*sin(e+f*x)/(a^2*f*sqrt(sin(e+f*x)^2))-1/3*c*(5-4*p)*(g*sec(e+f*x))^p*tan(e+f*x)/(a^2*f*(1+sec(e+f*x)))-2/3*c*(g*sec(e+f*x))^p*tan(e+f*x)/(f*(a+a*sec(e+f*x))^2)],

# Integrands of the form (g Sec[e+f x])^p (a+a Sec[e+f x])^(m/2) / (c-c Sec[e+f x])

# m>0
[(g*sec(e+f*x))^(3/2)*sqrt(a+a*sec(e+f*x))/(c-c*sec(e+f*x)),x,5,-2*g^(3/2)*arctanh(sqrt(a)*sqrt(g)*tan(e+f*x)/(sqrt(g*sec(e+f*x))*sqrt(a+a*sec(e+f*x))))*sqrt(a)/(c*f)+2*g*cot(e+f*x)*sqrt(g*sec(e+f*x))*sqrt(a+a*sec(e+f*x))/(c*f),-2*a*g*sqrt(g*sec(e+f*x))*tan(e+f*x)/(f*(c-c*sec(e+f*x))*sqrt(a+a*sec(e+f*x)))+2*a*g^(3/2)*arctan(sqrt(c)*sqrt(g*sec(e+f*x))/(sqrt(g)*sqrt(c-c*sec(e+f*x))))*tan(e+f*x)/(f*sqrt(c)*sqrt(a+a*sec(e+f*x))*sqrt(c-c*sec(e+f*x)))],

# m<0
[sec(e+f*x)^2/((c-c*sec(e+f*x))*sqrt(a+a*sec(e+f*x))),x,4,-arctan(sqrt(a)*tan(e+f*x)/(sqrt(2)*sqrt(a+a*sec(e+f*x))))/(c*f*sqrt(2)*sqrt(a))+cot(e+f*x)*sqrt(a+a*sec(e+f*x))/(a*c*f),-tan(e+f*x)/(f*(c-c*sec(e+f*x))*sqrt(a+a*sec(e+f*x)))-arctanh(sqrt(c-c*sec(e+f*x))/(sqrt(2)*sqrt(c)))*tan(e+f*x)/(f*sqrt(2)*sqrt(c)*sqrt(a+a*sec(e+f*x))*sqrt(c-c*sec(e+f*x)))],
[sec(e+f*x)^(5/2)/((c-c*sec(e+f*x))*sqrt(a+a*sec(e+f*x))),x,8,-2*arcsinh(sqrt(a)*tan(e+f*x)/sqrt(a+a*sec(e+f*x)))/(c*f*sqrt(a))+arctanh(sin(e+f*x)*sqrt(a)*sqrt(sec(e+f*x))/(sqrt(2)*sqrt(a+a*sec(e+f*x))))/(c*f*sqrt(2)*sqrt(a))+csc(e+f*x)*sqrt(a+a*sec(e+f*x))/(a*c*f*sqrt(sec(e+f*x))),-sec(e+f*x)^(3/2)*sin(e+f*x)/(f*(c-c*sec(e+f*x))*sqrt(a+a*sec(e+f*x)))+2*arctan(sqrt(c)*sqrt(sec(e+f*x))/sqrt(c-c*sec(e+f*x)))*tan(e+f*x)/(f*sqrt(c)*sqrt(a+a*sec(e+f*x))*sqrt(c-c*sec(e+f*x)))-arctan(sqrt(2)*sqrt(c)*sqrt(sec(e+f*x))/sqrt(c-c*sec(e+f*x)))*tan(e+f*x)/(f*sqrt(2)*sqrt(c)*sqrt(a+a*sec(e+f*x))*sqrt(c-c*sec(e+f*x)))],
[(g*sec(e+f*x))^(3/2)/((c-c*sec(e+f*x))*sqrt(a+a*sec(e+f*x))),x,4,-g^(3/2)*arctanh(sqrt(a)*sqrt(g)*tan(e+f*x)/(sqrt(2)*sqrt(g*sec(e+f*x))*sqrt(a+a*sec(e+f*x))))/(c*f*sqrt(2)*sqrt(a))+g*cot(e+f*x)*sqrt(g*sec(e+f*x))*sqrt(a+a*sec(e+f*x))/(a*c*f),-g*sqrt(g*sec(e+f*x))*tan(e+f*x)/(f*(c-c*sec(e+f*x))*sqrt(a+a*sec(e+f*x)))+g^(3/2)*arctan(sqrt(2)*sqrt(c)*sqrt(g*sec(e+f*x))/(sqrt(g)*sqrt(c-c*sec(e+f*x))))*tan(e+f*x)/(f*sqrt(2)*sqrt(c)*sqrt(a+a*sec(e+f*x))*sqrt(c-c*sec(e+f*x)))],
[(g*sec(e+f*x))^(5/2)/((c-c*sec(e+f*x))*sqrt(a+a*sec(e+f*x))),x,8,-2*g^(5/2)*arctanh(sqrt(a)*sqrt(g)*tan(e+f*x)/(sqrt(g*sec(e+f*x))*sqrt(a+a*sec(e+f*x))))/(c*f*sqrt(a))+g^(5/2)*arctanh(sqrt(a)*sqrt(g)*tan(e+f*x)/(sqrt(2)*sqrt(g*sec(e+f*x))*sqrt(a+a*sec(e+f*x))))/(c*f*sqrt(2)*sqrt(a))+g^2*cot(e+f*x)*sqrt(g*sec(e+f*x))*sqrt(a+a*sec(e+f*x))/(a*c*f),-g^2*sqrt(g*sec(e+f*x))*tan(e+f*x)/(f*(c-c*sec(e+f*x))*sqrt(a+a*sec(e+f*x)))+2*g^(5/2)*arctan(sqrt(c)*sqrt(g*sec(e+f*x))/(sqrt(g)*sqrt(c-c*sec(e+f*x))))*tan(e+f*x)/(f*sqrt(c)*sqrt(a+a*sec(e+f*x))*sqrt(c-c*sec(e+f*x)))-g^(5/2)*arctan(sqrt(2)*sqrt(c)*sqrt(g*sec(e+f*x))/(sqrt(g)*sqrt(c-c*sec(e+f*x))))*tan(e+f*x)/(f*sqrt(2)*sqrt(c)*sqrt(a+a*sec(e+f*x))*sqrt(c-c*sec(e+f*x)))],

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

# m>0

# m<0
[sec(e+f*x)^2/(sqrt(a+a*sec(e+f*x))*sqrt(c-c*sec(e+f*x))),x,3,log(tan(e+f*x))*tan(e+f*x)/(f*sqrt(a+a*sec(e+f*x))*sqrt(c-c*sec(e+f*x)))],

# Integrands of the form (g Sec[e+f x])^p (a+a Sec[e+f x])^m (c+d Sec[e+f x])^n
[sec(e+f*x)*sqrt(a+a*sec(e+f*x))/(c-d*sec(e+f*x)),x,2,2*arctanh(sqrt(a)*sqrt(d)*tan(e+f*x)/(sqrt(c-d)*sqrt(a+a*sec(e+f*x))))*sqrt(a)/(f*sqrt(c-d)*sqrt(d))],

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

# m>0
[sec(e+f*x)*(a+a*sec(e+f*x))*(c+d*sec(e+f*x))^4,x,8,1/8*a*(8*c^4+16*c^3*d+24*c^2*d^2+12*c*d^3+3*d^4)*arctanh(sin(e+f*x))/f+1/30*a*(12*c^4+95*c^3*d+112*c^2*d^2+80*c*d^3+16*d^4)*tan(e+f*x)/f+1/120*a*d*(24*c^3+130*c^2*d+116*c*d^2+45*d^3)*sec(e+f*x)*tan(e+f*x)/f+1/60*a*(12*c^2+35*c*d+16*d^2)*(c+d*sec(e+f*x))^2*tan(e+f*x)/f+1/20*a*(4*c+5*d)*(c+d*sec(e+f*x))^3*tan(e+f*x)/f+1/5*a*(c+d*sec(e+f*x))^4*tan(e+f*x)/f],
[sec(e+f*x)*(a+a*sec(e+f*x))*(c+d*sec(e+f*x))^3,x,7,1/8*a*(8*c^3+12*c^2*d+12*c*d^2+3*d^3)*arctanh(sin(e+f*x))/f+1/6*a*(3*c^3+16*c^2*d+12*c*d^2+4*d^3)*tan(e+f*x)/f+1/24*a*d*(6*c^2+20*c*d+9*d^2)*sec(e+f*x)*tan(e+f*x)/f+1/12*a*(3*c+4*d)*(c+d*sec(e+f*x))^2*tan(e+f*x)/f+1/4*a*(c+d*sec(e+f*x))^3*tan(e+f*x)/f],
[sec(e+f*x)*(a+a*sec(e+f*x))*(c+d*sec(e+f*x))^2,x,6,1/2*a*(2*c^2+2*c*d+d^2)*arctanh(sin(e+f*x))/f+2/3*a*(c^2+3*c*d+d^2)*tan(e+f*x)/f+1/6*a*d*(2*c+3*d)*sec(e+f*x)*tan(e+f*x)/f+1/3*a*(c+d*sec(e+f*x))^2*tan(e+f*x)/f],
[sec(e+f*x)*(a+a*sec(e+f*x))*(c+d*sec(e+f*x)),x,5,1/2*a*(2*c+d)*arctanh(sin(e+f*x))/f+a*(c+d)*tan(e+f*x)/f+1/2*a*d*sec(e+f*x)*tan(e+f*x)/f],
[sec(e+f*x)*(a+a*sec(e+f*x))/(c+d*sec(e+f*x)),x,5,a*arctanh(sin(e+f*x))/(d*f)-2*a*arctanh(sqrt(c-d)*tan(1/2*(e+f*x))/sqrt(c+d))*sqrt(c-d)/(d*f*sqrt(c+d))],
[sec(e+f*x)*(a+a*sec(e+f*x))/(c+d*sec(e+f*x))^2,x,5,2*a*arctanh(sqrt(c-d)*tan(1/2*(e+f*x))/sqrt(c+d))/((c+d)^(3/2)*f*sqrt(c-d))+a*tan(e+f*x)/((c+d)*f*(c+d*sec(e+f*x)))],
[sec(e+f*x)*(a+a*sec(e+f*x))/(c+d*sec(e+f*x))^3,x,6,a*(2*c-d)*arctanh(sqrt(c-d)*tan(1/2*(e+f*x))/sqrt(c+d))/((c-d)^(3/2)*(c+d)^(5/2)*f)+1/2*a*tan(e+f*x)/((c+d)*f*(c+d*sec(e+f*x))^2)+1/2*a*(c-2*d)*tan(e+f*x)/((c-d)*(c+d)^2*f*(c+d*sec(e+f*x)))],
[sec(e+f*x)*(a+a*sec(e+f*x))/(c+d*sec(e+f*x))^4,x,7,a*(2*c^2-2*c*d+d^2)*arctanh(sqrt(c-d)*tan(1/2*(e+f*x))/sqrt(c+d))/((c-d)^(5/2)*(c+d)^(7/2)*f)+1/3*a*tan(e+f*x)/((c+d)*f*(c+d*sec(e+f*x))^3)+1/6*a*(2*c-3*d)*tan(e+f*x)/((c-d)*(c+d)^2*f*(c+d*sec(e+f*x))^2)+1/6*a*(c-4*d)*(2*c-d)*tan(e+f*x)/((c-d)^2*(c+d)^3*f*(c+d*sec(e+f*x)))],
[sec(e+f*x)*(a+a*sec(e+f*x))^2*(c+d*sec(e+f*x))^4,x,9,1/16*a^2*(24*c^4+64*c^3*d+84*c^2*d^2+48*c*d^3+11*d^4)*arctanh(sin(e+f*x))/f-1/60*a^2*(4*c^5-48*c^4*d-311*c^3*d^2-448*c^2*d^3-288*c*d^4-64*d^5)*tan(e+f*x)/(d*f)-1/240*a^2*(8*c^4-96*c^3*d-438*c^2*d^2-464*c*d^3-165*d^4)*sec(e+f*x)*tan(e+f*x)/f-1/120*a^2*(4*c^3-48*c^2*d-123*c*d^2-64*d^3)*(c+d*sec(e+f*x))^2*tan(e+f*x)/(d*f)-1/120*a^2*(4*c^2-48*c*d-55*d^2)*(c+d*sec(e+f*x))^3*tan(e+f*x)/(d*f)-1/30*a^2*(c-12*d)*(c+d*sec(e+f*x))^4*tan(e+f*x)/(d*f)+1/6*a^2*(c+d*sec(e+f*x))^5*tan(e+f*x)/(d*f),1/16*a^2*(24*c^4+64*c^3*d+84*c^2*d^2+48*c*d^3+11*d^4)*tan(e+f*x)/f+1/48*(24*c^4+64*c^3*d+84*c^2*d^2+48*c*d^3+11*d^4)*(a^2+a^2*sec(e+f*x))*tan(e+f*x)/f+1/30*d*(9*c+2*d)*(a+a*sec(e+f*x))^2*(c+d*sec(e+f*x))^2*tan(e+f*x)/f+1/6*d*(a+a*sec(e+f*x))^2*(c+d*sec(e+f*x))^3*tan(e+f*x)/f+1/120*d*(a+a*sec(e+f*x))^2*(2*(52*c^3+56*c^2*d+48*c*d^2+9*d^3)+d*(48*c^2+32*c*d+19*d^2)*sec(e+f*x))*tan(e+f*x)/f+1/8*a^3*(24*c^4+64*c^3*d+84*c^2*d^2+48*c*d^3+11*d^4)*arctan(sqrt(a-a*sec(e+f*x))/sqrt(a*(1+sec(e+f*x))))*tan(e+f*x)/(f*sqrt(a-a*sec(e+f*x))*sqrt(a+a*sec(e+f*x)))],
[sec(e+f*x)*(a+a*sec(e+f*x))^2*(c+d*sec(e+f*x))^3,x,8,3/8*a^2*(2*c+d)*(2*c^2+3*c*d+2*d^2)*arctanh(sin(e+f*x))/f-1/10*a^2*(c^4-10*c^3*d-44*c^2*d^2-40*c*d^3-12*d^4)*tan(e+f*x)/(d*f)-1/40*a^2*(2*c^3-20*c^2*d-57*c*d^2-30*d^3)*sec(e+f*x)*tan(e+f*x)/f-1/20*a^2*(c^2-10*c*d-12*d^2)*(c+d*sec(e+f*x))^2*tan(e+f*x)/(d*f)-1/20*a^2*(c-10*d)*(c+d*sec(e+f*x))^3*tan(e+f*x)/(d*f)+1/5*a^2*(c+d*sec(e+f*x))^4*tan(e+f*x)/(d*f),3/8*a^2*(2*c+d)*(2*c^2+3*c*d+2*d^2)*tan(e+f*x)/f+1/8*(2*c+d)*(2*c^2+3*c*d+2*d^2)*(a^2+a^2*sec(e+f*x))*tan(e+f*x)/f+1/5*d*(a+a*sec(e+f*x))^2*(c+d*sec(e+f*x))^2*tan(e+f*x)/f+1/20*d*(a+a*sec(e+f*x))^2*(2*(8*c^2+5*c*d+2*d^2)+d*(7*c+2*d)*sec(e+f*x))*tan(e+f*x)/f+3/4*a^3*(2*c+d)*(2*c^2+3*c*d+2*d^2)*arctan(sqrt(a-a*sec(e+f*x))/sqrt(a*(1+sec(e+f*x))))*tan(e+f*x)/(f*sqrt(a-a*sec(e+f*x))*sqrt(a+a*sec(e+f*x)))],
[sec(e+f*x)*(a+a*sec(e+f*x))^2*(c+d*sec(e+f*x))^2,x,8,1/8*a^2*(12*c^2+16*c*d+7*d^2)*arctanh(sin(e+f*x))/f-1/6*a^2*(c^3-8*c^2*d-20*c*d^2-8*d^3)*tan(e+f*x)/(d*f)-1/24*a^2*(2*c*(c-8*d)-21*d^2)*sec(e+f*x)*tan(e+f*x)/f-1/12*a^2*(c-8*d)*(c+d*sec(e+f*x))^2*tan(e+f*x)/(d*f)+1/4*a^2*(c+d*sec(e+f*x))^3*tan(e+f*x)/(d*f),1/8*a^2*(12*c^2+16*c*d+7*d^2)*tan(e+f*x)/f+1/12*d*(5*c+2*d)*(a+a*sec(e+f*x))^2*tan(e+f*x)/f+1/24*(12*c^2+16*c*d+7*d^2)*(a^2+a^2*sec(e+f*x))*tan(e+f*x)/f+1/4*d*(a+a*sec(e+f*x))^2*(c+d*sec(e+f*x))*tan(e+f*x)/f+1/4*a^3*(12*c^2+16*c*d+7*d^2)*arctan(sqrt(a-a*sec(e+f*x))/sqrt(a*(1+sec(e+f*x))))*tan(e+f*x)/(f*sqrt(a-a*sec(e+f*x))*sqrt(a+a*sec(e+f*x)))],
[sec(e+f*x)*(a+a*sec(e+f*x))^2*(c+d*sec(e+f*x)),x,6,1/2*a^2*(3*c+2*d)*arctanh(sin(e+f*x))/f+2/3*a^2*(3*c+2*d)*tan(e+f*x)/f+1/6*a^2*(3*c+2*d)*sec(e+f*x)*tan(e+f*x)/f+1/3*d*(a+a*sec(e+f*x))^2*tan(e+f*x)/f],
[sec(e+f*x)*(a+a*sec(e+f*x))^2/(c+d*sec(e+f*x)),x,8,-a^2*(c-2*d)*arctanh(sin(e+f*x))/(d^2*f)+2*a^2*(c-d)^(3/2)*arctanh(sqrt(c-d)*tan(1/2*(e+f*x))/sqrt(c+d))/(d^2*f*sqrt(c+d))+a^2*tan(e+f*x)/(d*f),a^2*tan(e+f*x)/(d*f)-2*a^3*(c-2*d)*arctan(sqrt(a-a*sec(e+f*x))/sqrt(a*(1+sec(e+f*x))))*tan(e+f*x)/(d^2*f*sqrt(a-a*sec(e+f*x))*sqrt(a+a*sec(e+f*x)))-2*a^3*(c-d)^(3/2)*arctan(sqrt(c+d)*sqrt(a+a*sec(e+f*x))/(sqrt(c-d)*sqrt(a-a*sec(e+f*x))))*tan(e+f*x)/(d^2*f*sqrt(c+d)*sqrt(a-a*sec(e+f*x))*sqrt(a+a*sec(e+f*x)))],
[sec(e+f*x)*(a+a*sec(e+f*x))^2/(c+d*sec(e+f*x))^2,x,8,a^2*arctanh(sin(e+f*x))/(d^2*f)-2*a^2*(c+2*d)*arctanh(sqrt(c-d)*tan(1/2*(e+f*x))/sqrt(c+d))*sqrt(c-d)/(d^2*(c+d)^(3/2)*f)-a^2*(c-d)*tan(e+f*x)/(d*(c+d)*f*(c+d*sec(e+f*x))),-a^2*(c-d)*tan(e+f*x)/(d*(c+d)*f*(c+d*sec(e+f*x)))+2*a^3*arctan(sqrt(a-a*sec(e+f*x))/sqrt(a*(1+sec(e+f*x))))*tan(e+f*x)/(d^2*f*sqrt(a-a*sec(e+f*x))*sqrt(a+a*sec(e+f*x)))+2*a^3*(c+2*d)*arctan(sqrt(c+d)*sqrt(a+a*sec(e+f*x))/(sqrt(c-d)*sqrt(a-a*sec(e+f*x))))*sqrt(c-d)*tan(e+f*x)/(d^2*(c+d)^(3/2)*f*sqrt(a-a*sec(e+f*x))*sqrt(a+a*sec(e+f*x)))],
[sec(e+f*x)*(a+a*sec(e+f*x))^2/(c+d*sec(e+f*x))^3,x,5,3*a^2*arctanh(sqrt(c-d)*tan(1/2*(e+f*x))/sqrt(c+d))/((c+d)^(5/2)*f*sqrt(c-d))+1/2*(a^2+a^2*sec(e+f*x))*tan(e+f*x)/((c+d)*f*(c+d*sec(e+f*x))^2)+3/2*a^2*tan(e+f*x)/((c+d)^2*f*(c+d*sec(e+f*x))),1/2*(a^2+a^2*sec(e+f*x))*tan(e+f*x)/((c+d)*f*(c+d*sec(e+f*x))^2)+3/2*a^2*tan(e+f*x)/((c+d)^2*f*(c+d*sec(e+f*x)))-3*a^3*arctan(sqrt(c+d)*sqrt(a+a*sec(e+f*x))/(sqrt(c-d)*sqrt(a-a*sec(e+f*x))))*tan(e+f*x)/((c+d)^(5/2)*f*sqrt(c-d)*sqrt(a-a*sec(e+f*x))*sqrt(a+a*sec(e+f*x)))],
[sec(e+f*x)*(a+a*sec(e+f*x))^2/(c+d*sec(e+f*x))^4,x,6,a^2*(3*c-2*d)*arctanh(sqrt(c-d)*tan(1/2*(e+f*x))/sqrt(c+d))/((c-d)^(3/2)*(c+d)^(7/2)*f)-1/3*d*(a+a*sec(e+f*x))^2*tan(e+f*x)/((c^2-d^2)*f*(c+d*sec(e+f*x))^3)+1/6*(3*c-2*d)*(a^2+a^2*sec(e+f*x))*tan(e+f*x)/((c-d)*(c+d)^2*f*(c+d*sec(e+f*x))^2)+1/2*a^2*(3*c-2*d)*tan(e+f*x)/((c-d)*(c+d)^3*f*(c+d*sec(e+f*x))),-1/3*d*(a+a*sec(e+f*x))^2*tan(e+f*x)/((c^2-d^2)*f*(c+d*sec(e+f*x))^3)+1/6*(3*c-2*d)*(a^2+a^2*sec(e+f*x))*tan(e+f*x)/((c-d)*(c+d)^2*f*(c+d*sec(e+f*x))^2)+1/2*a^2*(3*c-2*d)*tan(e+f*x)/((c-d)*(c+d)^3*f*(c+d*sec(e+f*x)))-a^3*(3*c-2*d)*arctan(sqrt(c+d)*sqrt(a+a*sec(e+f*x))/(sqrt(c-d)*sqrt(a-a*sec(e+f*x))))*tan(e+f*x)/((c-d)^(3/2)*(c+d)^(7/2)*f*sqrt(a-a*sec(e+f*x))*sqrt(a+a*sec(e+f*x)))],
[sec(e+f*x)*(a+a*sec(e+f*x))^2/(c+d*sec(e+f*x))^5,x,8,1/4*a^2*(12*c^2-16*c*d+7*d^2)*arctanh(sqrt(c-d)*tan(1/2*(e+f*x))/sqrt(c+d))/((c-d)^(5/2)*(c+d)^(9/2)*f)-1/4*a^2*(c-d)*tan(e+f*x)/(d*(c+d)*f*(c+d*sec(e+f*x))^4)+1/12*a^2*(c+8*d)*tan(e+f*x)/(d*(c+d)^2*f*(c+d*sec(e+f*x))^3)+1/24*a^2*(2*c^2+16*c*d-21*d^2)*tan(e+f*x)/((c-d)*d*(c+d)^3*f*(c+d*sec(e+f*x))^2)+1/24*a^2*(2*c^3+16*c^2*d-59*c*d^2+32*d^3)*tan(e+f*x)/((c-d)^2*d*(c+d)^4*f*(c+d*sec(e+f*x))),-1/4*a^2*(c-d)*tan(e+f*x)/(d*(c+d)*f*(c+d*sec(e+f*x))^4)+1/12*a^2*(c+8*d)*tan(e+f*x)/(d*(c+d)^2*f*(c+d*sec(e+f*x))^3)+1/24*a^2*(2*c^2+16*c*d-21*d^2)*tan(e+f*x)/((c-d)*d*(c+d)^3*f*(c+d*sec(e+f*x))^2)+1/24*a^2*(2*c^3+16*c^2*d-59*c*d^2+32*d^3)*tan(e+f*x)/((c-d)^2*d*(c+d)^4*f*(c+d*sec(e+f*x)))-1/4*a^3*(12*c^2-16*c*d+7*d^2)*arctan(sqrt(c+d)*sqrt(a+a*sec(e+f*x))/(sqrt(c-d)*sqrt(a-a*sec(e+f*x))))*tan(e+f*x)/((c-d)^(5/2)*(c+d)^(9/2)*f*sqrt(a-a*sec(e+f*x))*sqrt(a+a*sec(e+f*x)))],
[sec(e+f*x)*(a+a*sec(e+f*x))^3*(c+d*sec(e+f*x))^3,x,9,1/16*a^3*(40*c^3+90*c^2*d+78*c*d^2+23*d^3)*arctanh(sin(e+f*x))/f+1/16*a^3*(40*c^3+90*c^2*d+78*c*d^2+23*d^3)*tan(e+f*x)/f+1/48*(40*c^3+90*c^2*d+78*c*d^2+23*d^3)*(a^3+a^3*sec(e+f*x))*tan(e+f*x)/f+1/30*a*(3*c+8*d)*(a+a*sec(e+f*x))^2*(c+d*sec(e+f*x))^2*tan(e+f*x)/f+1/6*a*(a+a*sec(e+f*x))^2*(c+d*sec(e+f*x))^3*tan(e+f*x)/f+1/120*a*(a+a*sec(e+f*x))^2*(2*(4*c^3+74*c^2*d+66*c*d^2+21*d^3)+d*(6*c^2+62*c*d+31*d^2)*sec(e+f*x))*tan(e+f*x)/f,1/16*a^3*(40*c^3+90*c^2*d+78*c*d^2+23*d^3)*tan(e+f*x)/f+1/120*a*(40*c^3+90*c^2*d+78*c*d^2+23*d^3)*(a+a*sec(e+f*x))^2*tan(e+f*x)/f+1/48*(40*c^3+90*c^2*d+78*c*d^2+23*d^3)*(a^3+a^3*sec(e+f*x))*tan(e+f*x)/f+1/6*d*(a+a*sec(e+f*x))^3*(c+d*sec(e+f*x))^2*tan(e+f*x)/f+1/120*d*(a+a*sec(e+f*x))^3*(70*c^2+54*c*d+19*d^2+4*d*(8*c+3*d)*sec(e+f*x))*tan(e+f*x)/f+1/8*a^4*(40*c^3+90*c^2*d+78*c*d^2+23*d^3)*arctan(sqrt(a-a*sec(e+f*x))/sqrt(a*(1+sec(e+f*x))))*tan(e+f*x)/(f*sqrt(a-a*sec(e+f*x))*sqrt(a+a*sec(e+f*x)))],
[sec(e+f*x)*(a+a*sec(e+f*x))^3*(c+d*sec(e+f*x))^2,x,9,1/8*a^3*(20*c^2+30*c*d+13*d^2)*arctanh(sin(e+f*x))/f+1/30*a^3*(2*c^4-15*c^3*d+72*c^2*d^2+180*c*d^3+76*d^4)*tan(e+f*x)/(d^2*f)+1/120*a^3*(4*c^3-30*c^2*d+146*c*d^2+195*d^3)*sec(e+f*x)*tan(e+f*x)/(d*f)+1/60*a^3*(2*c^2-15*c*d+76*d^2)*(c+d*sec(e+f*x))^2*tan(e+f*x)/(d^2*f)-1/20*a^3*(2*c-11*d)*(c+d*sec(e+f*x))^3*tan(e+f*x)/(d^2*f)+1/5*(a^3+a^3*sec(e+f*x))*(c+d*sec(e+f*x))^3*tan(e+f*x)/(d*f),1/8*a^3*(20*c^2+30*c*d+13*d^2)*tan(e+f*x)/f+1/60*a*(20*c^2+30*c*d+13*d^2)*(a+a*sec(e+f*x))^2*tan(e+f*x)/f+3/20*d*(2*c+d)*(a+a*sec(e+f*x))^3*tan(e+f*x)/f+1/24*(20*c^2+30*c*d+13*d^2)*(a^3+a^3*sec(e+f*x))*tan(e+f*x)/f+1/5*d*(a+a*sec(e+f*x))^3*(c+d*sec(e+f*x))*tan(e+f*x)/f+1/4*a^4*(20*c^2+30*c*d+13*d^2)*arctan(sqrt(a-a*sec(e+f*x))/sqrt(a*(1+sec(e+f*x))))*tan(e+f*x)/(f*sqrt(a-a*sec(e+f*x))*sqrt(a+a*sec(e+f*x)))],
[sec(e+f*x)*(a+a*sec(e+f*x))^3*(c+d*sec(e+f*x)),x,10,5/8*a^3*(4*c+3*d)*arctanh(sin(e+f*x))/f+a^3*(4*c+3*d)*tan(e+f*x)/f+3/8*a^3*(4*c+3*d)*sec(e+f*x)*tan(e+f*x)/f+1/4*d*(a+a*sec(e+f*x))^3*tan(e+f*x)/f+1/12*a^3*(4*c+3*d)*tan(e+f*x)^3/f],
[sec(e+f*x)*(a+a*sec(e+f*x))^3/(c+d*sec(e+f*x)),x,9,1/2*a^3*arctanh(sin(e+f*x))/(d*f)+a^3*(c^2-3*c*d+3*d^2)*arctanh(sin(e+f*x))/(d^3*f)-2*a^3*(c-d)^(5/2)*arctanh(sqrt(c-d)*tan(1/2*(e+f*x))/sqrt(c+d))/(d^3*f*sqrt(c+d))-a^3*(c-3*d)*tan(e+f*x)/(d^2*f)+1/2*a^3*sec(e+f*x)*tan(e+f*x)/(d*f),-1/2*a^3*(2*c-5*d)*tan(e+f*x)/(d^2*f)+1/2*(a^3+a^3*sec(e+f*x))*tan(e+f*x)/(d*f)+a^4*(2*c^2-6*c*d+7*d^2)*arctan(sqrt(a-a*sec(e+f*x))/sqrt(a*(1+sec(e+f*x))))*tan(e+f*x)/(d^3*f*sqrt(a-a*sec(e+f*x))*sqrt(a+a*sec(e+f*x)))+2*a^4*(c-d)^(5/2)*arctan(sqrt(c+d)*sqrt(a+a*sec(e+f*x))/(sqrt(c-d)*sqrt(a-a*sec(e+f*x))))*tan(e+f*x)/(d^3*f*sqrt(c+d)*sqrt(a-a*sec(e+f*x))*sqrt(a+a*sec(e+f*x)))],
[sec(e+f*x)*(a+a*sec(e+f*x))^3/(c+d*sec(e+f*x))^2,x,9,-a^3*(2*c-3*d)*arctanh(sin(e+f*x))/(d^3*f)+2*a^3*(c-d)^(3/2)*(2*c+3*d)*arctanh(sqrt(c-d)*tan(1/2*(e+f*x))/sqrt(c+d))/(d^3*(c+d)^(3/2)*f)+2*a^3*c*tan(e+f*x)/(d^2*(c+d)*f)-(c-d)*(a^3+a^3*sec(e+f*x))*tan(e+f*x)/(d*(c+d)*f*(c+d*sec(e+f*x))),2*a^3*c*tan(e+f*x)/(d^2*(c+d)*f)-(c-d)*(a^3+a^3*sec(e+f*x))*tan(e+f*x)/(d*(c+d)*f*(c+d*sec(e+f*x)))-2*a^4*(2*c-3*d)*arctan(sqrt(a-a*sec(e+f*x))/sqrt(a*(1+sec(e+f*x))))*tan(e+f*x)/(d^3*f*sqrt(a-a*sec(e+f*x))*sqrt(a+a*sec(e+f*x)))-2*a^4*(c-d)^(3/2)*(2*c+3*d)*arctan(sqrt(c+d)*sqrt(a+a*sec(e+f*x))/(sqrt(c-d)*sqrt(a-a*sec(e+f*x))))*tan(e+f*x)/(d^3*(c+d)^(3/2)*f*sqrt(a-a*sec(e+f*x))*sqrt(a+a*sec(e+f*x)))],
[sec(e+f*x)*(a+a*sec(e+f*x))^3/(c+d*sec(e+f*x))^3,x,9,a^3*arctanh(sin(e+f*x))/(d^3*f)-a^3*(2*c^2+6*c*d+7*d^2)*arctanh(sqrt(c-d)*tan(1/2*(e+f*x))/sqrt(c+d))*sqrt(c-d)/(d^3*(c+d)^(5/2)*f)-1/2*(c-d)*(a^3+a^3*sec(e+f*x))*tan(e+f*x)/(d*(c+d)*f*(c+d*sec(e+f*x))^2)-1/2*a^3*(c-d)*(2*c+5*d)*tan(e+f*x)/(d^2*(c+d)^2*f*(c+d*sec(e+f*x))),-1/2*(c-d)*(a^3+a^3*sec(e+f*x))*tan(e+f*x)/(d*(c+d)*f*(c+d*sec(e+f*x))^2)-1/2*a^3*(c-d)*(2*c+5*d)*tan(e+f*x)/(d^2*(c+d)^2*f*(c+d*sec(e+f*x)))+2*a^4*arctan(sqrt(a-a*sec(e+f*x))/sqrt(a*(1+sec(e+f*x))))*tan(e+f*x)/(d^3*f*sqrt(a-a*sec(e+f*x))*sqrt(a+a*sec(e+f*x)))+a^4*(2*c^2+6*c*d+7*d^2)*arctan(sqrt(c+d)*sqrt(a+a*sec(e+f*x))/(sqrt(c-d)*sqrt(a-a*sec(e+f*x))))*sqrt(c-d)*tan(e+f*x)/(d^3*(c+d)^(5/2)*f*sqrt(a-a*sec(e+f*x))*sqrt(a+a*sec(e+f*x)))],
[sec(e+f*x)*(a+a*sec(e+f*x))^3/(c+d*sec(e+f*x))^4,x,6,5*a^3*arctanh(sqrt(c-d)*tan(1/2*(e+f*x))/sqrt(c+d))/((c+d)^(7/2)*f*sqrt(c-d))+1/3*a*(a+a*sec(e+f*x))^2*tan(e+f*x)/((c+d)*f*(c+d*sec(e+f*x))^3)-5/6*a^3*(c-d)*tan(e+f*x)/(d*(c+d)^2*f*(c+d*sec(e+f*x))^2)+5/6*a^3*(c+4*d)*tan(e+f*x)/(d*(c+d)^3*f*(c+d*sec(e+f*x))),1/3*a*(a+a*sec(e+f*x))^2*tan(e+f*x)/((c+d)*f*(c+d*sec(e+f*x))^3)+5/6*(a^3+a^3*sec(e+f*x))*tan(e+f*x)/((c+d)^2*f*(c+d*sec(e+f*x))^2)+5/2*a^3*tan(e+f*x)/((c+d)^3*f*(c+d*sec(e+f*x)))-5*a^4*arctan(sqrt(c+d)*sqrt(a+a*sec(e+f*x))/(sqrt(c-d)*sqrt(a-a*sec(e+f*x))))*tan(e+f*x)/((c+d)^(7/2)*f*sqrt(c-d)*sqrt(a-a*sec(e+f*x))*sqrt(a+a*sec(e+f*x)))],
[sec(e+f*x)*(a+a*sec(e+f*x))^3/(c+d*sec(e+f*x))^5,x,7,5/4*a^3*(4*c-3*d)*arctanh(sqrt(c-d)*tan(1/2*(e+f*x))/sqrt(c+d))/((c-d)^(3/2)*(c+d)^(9/2)*f)-1/4*d*(a+a*sec(e+f*x))^3*tan(e+f*x)/((c^2-d^2)*f*(c+d*sec(e+f*x))^4)+1/12*a*(4*c-3*d)*(a+a*sec(e+f*x))^2*tan(e+f*x)/((c-d)*(c+d)^2*f*(c+d*sec(e+f*x))^3)-5/24*a^3*(4*c-3*d)*tan(e+f*x)/(d*(c+d)^3*f*(c+d*sec(e+f*x))^2)+5/24*a^3*(4*c-3*d)*(c+4*d)*tan(e+f*x)/((c-d)*d*(c+d)^4*f*(c+d*sec(e+f*x))),-1/4*d*(a+a*sec(e+f*x))^3*tan(e+f*x)/((c^2-d^2)*f*(c+d*sec(e+f*x))^4)+1/12*a*(4*c-3*d)*(a+a*sec(e+f*x))^2*tan(e+f*x)/((c-d)*(c+d)^2*f*(c+d*sec(e+f*x))^3)+5/24*(4*c-3*d)*(a^3+a^3*sec(e+f*x))*tan(e+f*x)/((c-d)*(c+d)^3*f*(c+d*sec(e+f*x))^2)+5/8*a^3*(4*c-3*d)*tan(e+f*x)/((c-d)*(c+d)^4*f*(c+d*sec(e+f*x)))-5/4*a^4*(4*c-3*d)*arctan(sqrt(c+d)*sqrt(a+a*sec(e+f*x))/(sqrt(c-d)*sqrt(a-a*sec(e+f*x))))*tan(e+f*x)/((c-d)^(3/2)*(c+d)^(9/2)*f*sqrt(a-a*sec(e+f*x))*sqrt(a+a*sec(e+f*x)))],

# m<0
[sec(e+f*x)*(c+d*sec(e+f*x))^4/(a+a*sec(e+f*x)),x,7,1/2*d*(8*c^3-12*c^2*d+12*c*d^2-3*d^3)*arctanh(sin(e+f*x))/(a*f)-1/3*(3*c-4*d)*d*(c+d*sec(e+f*x))^2*tan(e+f*x)/(a*f)+(c-d)*(c+d*sec(e+f*x))^3*tan(e+f*x)/(f*(a+a*sec(e+f*x)))-1/6*d*(4*(3*c^3-16*c^2*d+12*c*d^2-4*d^3)+d*(6*c^2-20*c*d+9*d^2)*sec(e+f*x))*tan(e+f*x)/(a*f),-1/3*(3*c-4*d)*d*(c+d*sec(e+f*x))^2*tan(e+f*x)/(a*f)+(c-d)*(c+d*sec(e+f*x))^3*tan(e+f*x)/(f*(a+a*sec(e+f*x)))-1/6*d*(4*(3*c^3-16*c^2*d+12*c*d^2-4*d^3)+d*(6*c^2-20*c*d+9*d^2)*sec(e+f*x))*tan(e+f*x)/(a*f)+d*(8*c^3-12*c^2*d+12*c*d^2-3*d^3)*arctan(sqrt(a-a*sec(e+f*x))/sqrt(a*(1+sec(e+f*x))))*tan(e+f*x)/(f*sqrt(a-a*sec(e+f*x))*sqrt(a+a*sec(e+f*x)))],
[sec(e+f*x)*(c+d*sec(e+f*x))^3/(a+a*sec(e+f*x)),x,6,3/2*d*(2*c^2-2*c*d+d^2)*arctanh(sin(e+f*x))/(a*f)+(c-d)*(c+d*sec(e+f*x))^2*tan(e+f*x)/(f*(a+a*sec(e+f*x)))-1/2*d*(4*(c^2-3*c*d+d^2)+(2*c-3*d)*d*sec(e+f*x))*tan(e+f*x)/(a*f),(c-d)*(c+d*sec(e+f*x))^2*tan(e+f*x)/(f*(a+a*sec(e+f*x)))-1/2*d*(4*(c^2-3*c*d+d^2)+(2*c-3*d)*d*sec(e+f*x))*tan(e+f*x)/(a*f)+3*d*(2*c^2-2*c*d+d^2)*arctan(sqrt(a-a*sec(e+f*x))/sqrt(a*(1+sec(e+f*x))))*tan(e+f*x)/(f*sqrt(a-a*sec(e+f*x))*sqrt(a+a*sec(e+f*x)))],
[sec(e+f*x)*(c+d*sec(e+f*x))^2/(a+a*sec(e+f*x)),x,6,(2*c-d)*d*arctanh(sin(e+f*x))/(a*f)+d^2*tan(e+f*x)/(a*f)+(c-d)^2*tan(e+f*x)/(f*(a+a*sec(e+f*x))),d^2*tan(e+f*x)/(a*f)+(c-d)^2*tan(e+f*x)/(f*(a+a*sec(e+f*x)))+2*(2*c-d)*d*arctan(sqrt(a-a*sec(e+f*x))/sqrt(a*(1+sec(e+f*x))))*tan(e+f*x)/(f*sqrt(a-a*sec(e+f*x))*sqrt(a+a*sec(e+f*x)))],
[sec(e+f*x)*(c+d*sec(e+f*x))/(a+a*sec(e+f*x)),x,3,d*arctanh(sin(e+f*x))/(a*f)+(c-d)*tan(e+f*x)/(f*(a+a*sec(e+f*x)))],
[sec(e+f*x)/((a+a*sec(e+f*x))*(c+d*sec(e+f*x))),x,4,-2*d*arctanh(sqrt(c-d)*tan(1/2*(e+f*x))/sqrt(c+d))/(a*(c-d)^(3/2)*f*sqrt(c+d))+tan(e+f*x)/((c-d)*f*(a+a*sec(e+f*x))),tan(e+f*x)/((c-d)*f*(a+a*sec(e+f*x)))+2*d*arctan(sqrt(c+d)*sqrt(a+a*sec(e+f*x))/(sqrt(c-d)*sqrt(a-a*sec(e+f*x))))*tan(e+f*x)/((c-d)^(3/2)*f*sqrt(c+d)*sqrt(a-a*sec(e+f*x))*sqrt(a+a*sec(e+f*x)))],
[sec(e+f*x)/((a+a*sec(e+f*x))*(c+d*sec(e+f*x))^2),x,6,-2*d*(2*c+d)*arctanh(sqrt(c-d)*tan(1/2*(e+f*x))/sqrt(c+d))/(a*(c-d)^(5/2)*(c+d)^(3/2)*f)+(c+2*d)*tan(e+f*x)/((c-d)^2*(c+d)*f*(a+a*sec(e+f*x)))-d*tan(e+f*x)/((c^2-d^2)*f*(a+a*sec(e+f*x))*(c+d*sec(e+f*x))),(c+2*d)*tan(e+f*x)/((c-d)^2*(c+d)*f*(a+a*sec(e+f*x)))-d*tan(e+f*x)/((c^2-d^2)*f*(a+a*sec(e+f*x))*(c+d*sec(e+f*x)))+2*d*(2*c+d)*arctan(sqrt(c+d)*sqrt(a+a*sec(e+f*x))/(sqrt(c-d)*sqrt(a-a*sec(e+f*x))))*tan(e+f*x)/((c-d)^(5/2)*(c+d)^(3/2)*f*sqrt(a-a*sec(e+f*x))*sqrt(a+a*sec(e+f*x)))],
[sec(e+f*x)/((a+a*sec(e+f*x))*(c+d*sec(e+f*x))^3),x,7,-3*d*(2*c^2+2*c*d+d^2)*arctanh(sqrt(c-d)*tan(1/2*(e+f*x))/sqrt(c+d))/(a*(c-d)^(7/2)*(c+d)^(5/2)*f)+1/2*d*(2*c+3*d)*tan(e+f*x)/(a*(c-d)^2*(c+d)*f*(c+d*sec(e+f*x))^2)+tan(e+f*x)/((c-d)*f*(a+a*sec(e+f*x))*(c+d*sec(e+f*x))^2)+1/2*d*(2*c+d)*(c+4*d)*tan(e+f*x)/(a*(c-d)^3*(c+d)^2*f*(c+d*sec(e+f*x))),1/2*(2*c+d)*(c+4*d)*tan(e+f*x)/((c-d)^3*(c+d)^2*f*(a+a*sec(e+f*x)))-1/2*d*tan(e+f*x)/((c^2-d^2)*f*(a+a*sec(e+f*x))*(c+d*sec(e+f*x))^2)-1/2*d*(4*c+d)*tan(e+f*x)/((c^2-d^2)^2*f*(a+a*sec(e+f*x))*(c+d*sec(e+f*x)))+3*d*(2*c^2+2*c*d+d^2)*arctan(sqrt(c+d)*sqrt(a+a*sec(e+f*x))/(sqrt(c-d)*sqrt(a-a*sec(e+f*x))))*tan(e+f*x)/((c-d)^(7/2)*(c+d)^(5/2)*f*sqrt(a-a*sec(e+f*x))*sqrt(a+a*sec(e+f*x)))],
[sec(e+f*x)*(c+d*sec(e+f*x))^5/(a+a*sec(e+f*x))^2,x,8,5/2*(2*c-d)*d^2*(2*c^2-3*c*d+2*d^2)*arctanh(sin(e+f*x))/(a^2*f)-1/3*d*(c^2+10*c*d-12*d^2)*(c+d*sec(e+f*x))^2*tan(e+f*x)/(a^2*f)+1/3*(c-d)*(c+10*d)*(c+d*sec(e+f*x))^3*tan(e+f*x)/(f*(a^2+a^2*sec(e+f*x)))+1/3*(c-d)*(c+d*sec(e+f*x))^4*tan(e+f*x)/(f*(a+a*sec(e+f*x))^2)-1/6*d*(4*(c^4+10*c^3*d-44*c^2*d^2+40*c*d^3-12*d^4)+d*(2*c^3+20*c^2*d-57*c*d^2+30*d^3)*sec(e+f*x))*tan(e+f*x)/(a^2*f),-1/3*d*(c^2+10*c*d-12*d^2)*(c+d*sec(e+f*x))^2*tan(e+f*x)/(a^2*f)+1/3*(c-d)*(c+10*d)*(c+d*sec(e+f*x))^3*tan(e+f*x)/(f*(a^2+a^2*sec(e+f*x)))+1/3*(c-d)*(c+d*sec(e+f*x))^4*tan(e+f*x)/(f*(a+a*sec(e+f*x))^2)-1/6*d*(4*(c^4+10*c^3*d-44*c^2*d^2+40*c*d^3-12*d^4)+d*(2*c^3+20*c^2*d-57*c*d^2+30*d^3)*sec(e+f*x))*tan(e+f*x)/(a^2*f)+5*(2*c-d)*d^2*(2*c^2-3*c*d+2*d^2)*arctan(sqrt(a-a*sec(e+f*x))/sqrt(a*(1+sec(e+f*x))))*tan(e+f*x)/(a*f*sqrt(a-a*sec(e+f*x))*sqrt(a+a*sec(e+f*x)))],
[sec(e+f*x)*(c+d*sec(e+f*x))^4/(a+a*sec(e+f*x))^2,x,7,1/2*d^2*(12*c^2-16*c*d+7*d^2)*arctanh(sin(e+f*x))/(a^2*f)+1/3*(c-d)*(c+8*d)*(c+d*sec(e+f*x))^2*tan(e+f*x)/(f*(a^2+a^2*sec(e+f*x)))+1/3*(c-d)*(c+d*sec(e+f*x))^3*tan(e+f*x)/(f*(a+a*sec(e+f*x))^2)-1/6*d*(4*(c^3+8*c^2*d-20*c*d^2+8*d^3)+d*(2*c^2+16*c*d-21*d^2)*sec(e+f*x))*tan(e+f*x)/(a^2*f),1/3*(c-d)*(c+8*d)*(c+d*sec(e+f*x))^2*tan(e+f*x)/(f*(a^2+a^2*sec(e+f*x)))+1/3*(c-d)*(c+d*sec(e+f*x))^3*tan(e+f*x)/(f*(a+a*sec(e+f*x))^2)-1/6*d*(4*(c^3+8*c^2*d-20*c*d^2+8*d^3)+d*(2*c^2+16*c*d-21*d^2)*sec(e+f*x))*tan(e+f*x)/(a^2*f)+d^2*(12*c^2-16*c*d+7*d^2)*arctan(sqrt(a-a*sec(e+f*x))/sqrt(a*(1+sec(e+f*x))))*tan(e+f*x)/(a*f*sqrt(a-a*sec(e+f*x))*sqrt(a+a*sec(e+f*x)))],
[sec(e+f*x)*(c+d*sec(e+f*x))^3/(a+a*sec(e+f*x))^2,x,6,(3*c-2*d)*d^2*arctanh(sin(e+f*x))/(a^2*f)+1/3*(c-d)*(c+d*sec(e+f*x))^2*tan(e+f*x)/(f*(a+a*sec(e+f*x))^2)+1/3*(c^3+4*c^2*d-12*c*d^2+10*d^3-(c-4*d)*d^2*sec(e+f*x))*tan(e+f*x)/(f*(a^2+a^2*sec(e+f*x))),1/3*(c-d)*(c+d*sec(e+f*x))^2*tan(e+f*x)/(f*(a+a*sec(e+f*x))^2)+1/3*(c^3+4*c^2*d-12*c*d^2+10*d^3-(c-4*d)*d^2*sec(e+f*x))*tan(e+f*x)/(f*(a^2+a^2*sec(e+f*x)))+2*(3*c-2*d)*d^2*arctan(sqrt(a-a*sec(e+f*x))/sqrt(a*(1+sec(e+f*x))))*tan(e+f*x)/(a*f*sqrt(a-a*sec(e+f*x))*sqrt(a+a*sec(e+f*x)))],
[sec(e+f*x)*(c+d*sec(e+f*x))^2/(a+a*sec(e+f*x))^2,x,6,d^2*arctanh(sin(e+f*x))/(a^2*f)+1/3*(c-d)^2*tan(e+f*x)/(f*(a+a*sec(e+f*x))^2)+1/3*(c-d)*(c+5*d)*tan(e+f*x)/(f*(a^2+a^2*sec(e+f*x))),1/3*(c-d)^2*tan(e+f*x)/(f*(a+a*sec(e+f*x))^2)+1/3*(c-d)*(c+5*d)*tan(e+f*x)/(f*(a^2+a^2*sec(e+f*x)))+2*d^2*arctan(sqrt(a-a*sec(e+f*x))/sqrt(a*(1+sec(e+f*x))))*tan(e+f*x)/(a*f*sqrt(a-a*sec(e+f*x))*sqrt(a+a*sec(e+f*x)))],
[sec(e+f*x)*(c+d*sec(e+f*x))/(a+a*sec(e+f*x))^2,x,2,1/3*(c-d)*tan(e+f*x)/(f*(a+a*sec(e+f*x))^2)+1/3*(c+2*d)*tan(e+f*x)/(f*(a^2+a^2*sec(e+f*x)))],
[sec(e+f*x)/((a+a*sec(e+f*x))^2*(c+d*sec(e+f*x))),x,6,2*d^2*arctanh(sqrt(c-d)*tan(1/2*(e+f*x))/sqrt(c+d))/(a^2*(c-d)^(5/2)*f*sqrt(c+d))+1/3*tan(e+f*x)/((c-d)*f*(a+a*sec(e+f*x))^2)+1/3*(c-4*d)*tan(e+f*x)/((c-d)^2*f*(a^2+a^2*sec(e+f*x))),1/3*tan(e+f*x)/((c-d)*f*(a+a*sec(e+f*x))^2)+1/3*(c-4*d)*tan(e+f*x)/((c-d)^2*f*(a^2+a^2*sec(e+f*x)))-2*d^2*arctan(sqrt(c+d)*sqrt(a+a*sec(e+f*x))/(sqrt(c-d)*sqrt(a-a*sec(e+f*x))))*tan(e+f*x)/(a*(c-d)^(5/2)*f*sqrt(c+d)*sqrt(a-a*sec(e+f*x))*sqrt(a+a*sec(e+f*x)))],
[sec(e+f*x)/((a+a*sec(e+f*x))^2*(c+d*sec(e+f*x))^2),x,7,2*d^2*(3*c+2*d)*arctanh(sqrt(c-d)*tan(1/2*(e+f*x))/sqrt(c+d))/(a^2*(c-d)^(7/2)*(c+d)^(3/2)*f)+1/3*d*(c^2-6*c*d-10*d^2)*tan(e+f*x)/(a^2*(c-d)^3*(c+d)*f*(c+d*sec(e+f*x)))+1/3*(c-6*d)*tan(e+f*x)/(a^2*(c-d)^2*f*(1+sec(e+f*x))*(c+d*sec(e+f*x)))+1/3*tan(e+f*x)/((c-d)*f*(a+a*sec(e+f*x))^2*(c+d*sec(e+f*x))),1/3*(c+4*d)*tan(e+f*x)/((c-d)^2*(c+d)*f*(a+a*sec(e+f*x))^2)+1/3*(c^2-6*c*d-10*d^2)*tan(e+f*x)/((c-d)^3*(c+d)*f*(a^2+a^2*sec(e+f*x)))-d*tan(e+f*x)/((c^2-d^2)*f*(a+a*sec(e+f*x))^2*(c+d*sec(e+f*x)))-2*d^2*(3*c+2*d)*arctan(sqrt(c+d)*sqrt(a+a*sec(e+f*x))/(sqrt(c-d)*sqrt(a-a*sec(e+f*x))))*tan(e+f*x)/(a*(c-d)^(7/2)*(c+d)^(3/2)*f*sqrt(a-a*sec(e+f*x))*sqrt(a+a*sec(e+f*x)))],
[sec(e+f*x)/((a+a*sec(e+f*x))^2*(c+d*sec(e+f*x))^3),x,8,d^2*(12*c^2+16*c*d+7*d^2)*arctanh(sqrt(c-d)*tan(1/2*(e+f*x))/sqrt(c+d))/(a^2*(c-d)^(9/2)*(c+d)^(5/2)*f)+1/6*d*(2*c^2-16*c*d-21*d^2)*tan(e+f*x)/(a^2*(c-d)^3*(c+d)*f*(c+d*sec(e+f*x))^2)+1/3*(c-8*d)*tan(e+f*x)/(a^2*(c-d)^2*f*(1+sec(e+f*x))*(c+d*sec(e+f*x))^2)+1/3*tan(e+f*x)/((c-d)*f*(a+a*sec(e+f*x))^2*(c+d*sec(e+f*x))^2)+1/6*d*(2*c^3-16*c^2*d-59*c*d^2-32*d^3)*tan(e+f*x)/(a^2*(c-d)^4*(c+d)^2*f*(c+d*sec(e+f*x))),1/6*(2*c^2+22*c*d+11*d^2)*tan(e+f*x)/((c-d)^3*(c+d)^2*f*(a+a*sec(e+f*x))^2)+1/6*(2*c^3-16*c^2*d-59*c*d^2-32*d^3)*tan(e+f*x)/((c-d)^4*(c+d)^2*f*(a^2+a^2*sec(e+f*x)))-1/2*d*tan(e+f*x)/((c^2-d^2)*f*(a+a*sec(e+f*x))^2*(c+d*sec(e+f*x))^2)-1/2*d*(5*c+2*d)*tan(e+f*x)/((c^2-d^2)^2*f*(a+a*sec(e+f*x))^2*(c+d*sec(e+f*x)))-d^2*(12*c^2+16*c*d+7*d^2)*arctan(sqrt(c+d)*sqrt(a+a*sec(e+f*x))/(sqrt(c-d)*sqrt(a-a*sec(e+f*x))))*tan(e+f*x)/(a*(c-d)^(9/2)*(c+d)^(5/2)*f*sqrt(a-a*sec(e+f*x))*sqrt(a+a*sec(e+f*x)))],
[sec(e+f*x)*(c+d*sec(e+f*x))^6/(a+a*sec(e+f*x))^3,x,9,1/2*d^3*(40*c^3-90*c^2*d+78*c*d^2-23*d^3)*arctanh(sin(e+f*x))/(a^3*f)-2/15*d*(2*c^5+18*c^4*d+107*c^3*d^2-472*c^2*d^3+456*c*d^4-136*d^5)*tan(e+f*x)/(a^3*f)-1/30*d^2*(4*c^4+36*c^3*d+216*c^2*d^2-626*c*d^3+345*d^4)*sec(e+f*x)*tan(e+f*x)/(a^3*f)-1/15*d*(2*c^3+18*c^2*d+111*c*d^2-136*d^3)*(c+d*sec(e+f*x))^2*tan(e+f*x)/(a^3*f)+1/15*(c-d)*(2*c^2+18*c*d+115*d^2)*(c+d*sec(e+f*x))^3*tan(e+f*x)/(f*(a^3+a^3*sec(e+f*x)))+1/15*(c-d)*(2*c+13*d)*(c+d*sec(e+f*x))^4*tan(e+f*x)/(a*f*(a+a*sec(e+f*x))^2)+1/5*(c-d)*(c+d*sec(e+f*x))^5*tan(e+f*x)/(f*(a+a*sec(e+f*x))^3),-1/15*d*(2*c^3+18*c^2*d+111*c*d^2-136*d^3)*(c+d*sec(e+f*x))^2*tan(e+f*x)/(a^3*f)+1/15*(c-d)*(2*c^2+18*c*d+115*d^2)*(c+d*sec(e+f*x))^3*tan(e+f*x)/(f*(a^3+a^3*sec(e+f*x)))+1/15*(c-d)*(2*c+13*d)*(c+d*sec(e+f*x))^4*tan(e+f*x)/(a*f*(a+a*sec(e+f*x))^2)+1/5*(c-d)*(c+d*sec(e+f*x))^5*tan(e+f*x)/(f*(a+a*sec(e+f*x))^3)-1/30*d*(4*(2*c^5+18*c^4*d+107*c^3*d^2-472*c^2*d^3+456*c*d^4-136*d^5)+d*(4*c^4+36*c^3*d+216*c^2*d^2-626*c*d^3+345*d^4)*sec(e+f*x))*tan(e+f*x)/(a^3*f)+d^3*(40*c^3-90*c^2*d+78*c*d^2-23*d^3)*arctan(sqrt(a-a*sec(e+f*x))/sqrt(a*(1+sec(e+f*x))))*tan(e+f*x)/(a^2*f*sqrt(a-a*sec(e+f*x))*sqrt(a+a*sec(e+f*x)))],
[sec(e+f*x)*(c+d*sec(e+f*x))^5/(a+a*sec(e+f*x))^3,x,8,1/2*d^3*(20*c^2-30*c*d+13*d^2)*arctanh(sin(e+f*x))/(a^3*f)-2/15*d*(2*c^4+15*c^3*d+72*c^2*d^2-180*c*d^3+76*d^4)*tan(e+f*x)/(a^3*f)-1/30*d^2*(4*c^3+30*c^2*d+146*c*d^2-195*d^3)*sec(e+f*x)*tan(e+f*x)/(a^3*f)+1/15*(c-d)*(2*c^2+15*c*d+76*d^2)*(c+d*sec(e+f*x))^2*tan(e+f*x)/(f*(a^3+a^3*sec(e+f*x)))+1/15*(c-d)*(2*c+11*d)*(c+d*sec(e+f*x))^3*tan(e+f*x)/(a*f*(a+a*sec(e+f*x))^2)+1/5*(c-d)*(c+d*sec(e+f*x))^4*tan(e+f*x)/(f*(a+a*sec(e+f*x))^3),1/15*(c-d)*(2*c^2+15*c*d+76*d^2)*(c+d*sec(e+f*x))^2*tan(e+f*x)/(f*(a^3+a^3*sec(e+f*x)))+1/15*(c-d)*(2*c+11*d)*(c+d*sec(e+f*x))^3*tan(e+f*x)/(a*f*(a+a*sec(e+f*x))^2)+1/5*(c-d)*(c+d*sec(e+f*x))^4*tan(e+f*x)/(f*(a+a*sec(e+f*x))^3)-1/30*d*(4*(2*c^4+15*c^3*d+72*c^2*d^2-180*c*d^3+76*d^4)+d*(4*c^3+30*c^2*d+146*c*d^2-195*d^3)*sec(e+f*x))*tan(e+f*x)/(a^3*f)+d^3*(20*c^2-30*c*d+13*d^2)*arctan(sqrt(a-a*sec(e+f*x))/sqrt(a*(1+sec(e+f*x))))*tan(e+f*x)/(a^2*f*sqrt(a-a*sec(e+f*x))*sqrt(a+a*sec(e+f*x)))],
[sec(e+f*x)*(c+d*sec(e+f*x))^4/(a+a*sec(e+f*x))^3,x,7,(4*c-3*d)*d^3*arctanh(sin(e+f*x))/(a^3*f)+1/15*(c-d)*(2*c+9*d)*(c+d*sec(e+f*x))^2*tan(e+f*x)/(a*f*(a+a*sec(e+f*x))^2)+1/5*(c-d)*(c+d*sec(e+f*x))^3*tan(e+f*x)/(f*(a+a*sec(e+f*x))^3)+1/15*(2*c^4+8*c^3*d+21*c^2*d^2-88*c*d^3+72*d^4-d^2*(2*c^2+10*c*d-27*d^2)*sec(e+f*x))*tan(e+f*x)/(f*(a^3+a^3*sec(e+f*x))),1/15*(c-d)*(2*c+9*d)*(c+d*sec(e+f*x))^2*tan(e+f*x)/(a*f*(a+a*sec(e+f*x))^2)+1/5*(c-d)*(c+d*sec(e+f*x))^3*tan(e+f*x)/(f*(a+a*sec(e+f*x))^3)+1/15*(2*c^4+8*c^3*d+21*c^2*d^2-88*c*d^3+72*d^4-d^2*(2*c^2+10*c*d-27*d^2)*sec(e+f*x))*tan(e+f*x)/(f*(a^3+a^3*sec(e+f*x)))+2*(4*c-3*d)*d^3*arctan(sqrt(a-a*sec(e+f*x))/sqrt(a*(1+sec(e+f*x))))*tan(e+f*x)/(a^2*f*sqrt(a-a*sec(e+f*x))*sqrt(a+a*sec(e+f*x)))],
[sec(e+f*x)*(c+d*sec(e+f*x))^3/(a+a*sec(e+f*x))^3,x,6,d^3*arctanh(sin(e+f*x))/(a^3*f)+1/5*(c-d)*(c+d*sec(e+f*x))^2*tan(e+f*x)/(f*(a+a*sec(e+f*x))^3)+1/15*(c-d)*(2*(2*c^2+8*c*d+11*d^2)+(2*c^2+11*c*d+29*d^2)*sec(e+f*x))*tan(e+f*x)/(a*f*(a+a*sec(e+f*x))^2),1/5*(c-d)*(c+d*sec(e+f*x))^2*tan(e+f*x)/(f*(a+a*sec(e+f*x))^3)+1/15*(c-d)*(2*(2*c^2+8*c*d+11*d^2)+(2*c^2+11*c*d+29*d^2)*sec(e+f*x))*tan(e+f*x)/(a*f*(a+a*sec(e+f*x))^2)+2*d^3*arctan(sqrt(a-a*sec(e+f*x))/sqrt(a*(1+sec(e+f*x))))*tan(e+f*x)/(a^2*f*sqrt(a-a*sec(e+f*x))*sqrt(a+a*sec(e+f*x)))],
[sec(e+f*x)*(c+d*sec(e+f*x))^2/(a+a*sec(e+f*x))^3,x,4,1/5*(c-d)^2*tan(e+f*x)/(f*(a+a*sec(e+f*x))^3)+2/15*(c-d)*(c+4*d)*tan(e+f*x)/(a*f*(a+a*sec(e+f*x))^2)+1/15*(2*c^2+6*c*d+7*d^2)*tan(e+f*x)/(f*(a^3+a^3*sec(e+f*x)))],
[sec(e+f*x)*(c+d*sec(e+f*x))/(a+a*sec(e+f*x))^3,x,3,1/5*(c-d)*tan(e+f*x)/(f*(a+a*sec(e+f*x))^3)+1/15*(2*c+3*d)*tan(e+f*x)/(a*f*(a+a*sec(e+f*x))^2)+1/15*(2*c+3*d)*tan(e+f*x)/(f*(a^3+a^3*sec(e+f*x)))],
[sec(e+f*x)/((a+a*sec(e+f*x))^3*(c+d*sec(e+f*x))),x,7,-2*d^3*arctanh(sqrt(c-d)*tan(1/2*(e+f*x))/sqrt(c+d))/(a^3*(c-d)^(7/2)*f*sqrt(c+d))+1/5*tan(e+f*x)/((c-d)*f*(a+a*sec(e+f*x))^3)+1/15*(2*c-7*d)*tan(e+f*x)/(a*(c-d)^2*f*(a+a*sec(e+f*x))^2)+1/15*(2*c^2-9*c*d+22*d^2)*tan(e+f*x)/((c-d)^3*f*(a^3+a^3*sec(e+f*x))),1/5*tan(e+f*x)/((c-d)*f*(a+a*sec(e+f*x))^3)+1/15*(2*c-7*d)*tan(e+f*x)/(a*(c-d)^2*f*(a+a*sec(e+f*x))^2)+1/15*(2*c^2-9*c*d+22*d^2)*tan(e+f*x)/((c-d)^3*f*(a^3+a^3*sec(e+f*x)))+2*d^3*arctan(sqrt(c+d)*sqrt(a+a*sec(e+f*x))/(sqrt(c-d)*sqrt(a-a*sec(e+f*x))))*tan(e+f*x)/(a^2*(c-d)^(7/2)*f*sqrt(c+d)*sqrt(a-a*sec(e+f*x))*sqrt(a+a*sec(e+f*x)))],
[sec(e+f*x)/((a+a*sec(e+f*x))^3*(c+d*sec(e+f*x))^2),x,8,-2*d^3*(4*c+3*d)*arctanh(sqrt(c-d)*tan(1/2*(e+f*x))/sqrt(c+d))/(a^3*(c-d)^(9/2)*(c+d)^(3/2)*f)+1/15*d*(2*c^3-12*c^2*d+43*c*d^2+72*d^3)*tan(e+f*x)/(a^3*(c-d)^4*(c+d)*f*(c+d*sec(e+f*x)))+1/5*tan(e+f*x)/((c-d)*f*(a+a*sec(e+f*x))^3*(c+d*sec(e+f*x)))+1/15*(2*c-9*d)*tan(e+f*x)/(a*(c-d)^2*f*(a+a*sec(e+f*x))^2*(c+d*sec(e+f*x)))+1/15*(2*c^2-12*c*d+45*d^2)*tan(e+f*x)/((c-d)^3*f*(a^3+a^3*sec(e+f*x))*(c+d*sec(e+f*x))),1/5*(c+6*d)*tan(e+f*x)/((c-d)^2*(c+d)*f*(a+a*sec(e+f*x))^3)+1/15*(2*c^2-10*c*d-27*d^2)*tan(e+f*x)/(a*(c-d)^3*(c+d)*f*(a+a*sec(e+f*x))^2)+1/15*(2*c^3-12*c^2*d+43*c*d^2+72*d^3)*tan(e+f*x)/((c-d)^4*(c+d)*f*(a^3+a^3*sec(e+f*x)))-d*tan(e+f*x)/((c^2-d^2)*f*(a+a*sec(e+f*x))^3*(c+d*sec(e+f*x)))+2*d^3*(4*c+3*d)*arctan(sqrt(c+d)*sqrt(a+a*sec(e+f*x))/(sqrt(c-d)*sqrt(a-a*sec(e+f*x))))*tan(e+f*x)/(a^2*(c-d)^(9/2)*(c+d)^(3/2)*f*sqrt(a-a*sec(e+f*x))*sqrt(a+a*sec(e+f*x)))],
[sec(e+f*x)/((a+a*sec(e+f*x))^3*(c+d*sec(e+f*x))^3),x,9,-d^3*(20*c^2+30*c*d+13*d^2)*arctanh(sqrt(c-d)*tan(1/2*(e+f*x))/sqrt(c+d))/(a^3*(c-d)^(11/2)*(c+d)^(5/2)*f)+1/30*d*(4*c^3-30*c^2*d+146*c*d^2+195*d^3)*tan(e+f*x)/(a^3*(c-d)^4*(c+d)*f*(c+d*sec(e+f*x))^2)+1/5*tan(e+f*x)/((c-d)*f*(a+a*sec(e+f*x))^3*(c+d*sec(e+f*x))^2)+1/15*(2*c-11*d)*tan(e+f*x)/(a*(c-d)^2*f*(a+a*sec(e+f*x))^2*(c+d*sec(e+f*x))^2)+1/15*(2*c^2-15*c*d+76*d^2)*tan(e+f*x)/((c-d)^3*f*(a^3+a^3*sec(e+f*x))*(c+d*sec(e+f*x))^2)+1/30*d*(4*c^4-30*c^3*d+142*c^2*d^2+525*c*d^3+304*d^4)*tan(e+f*x)/(a^3*(c-d)^5*(c+d)^2*f*(c+d*sec(e+f*x))),1/10*(2*c^2+39*c*d+22*d^2)*tan(e+f*x)/((c-d)^3*(c+d)^2*f*(a+a*sec(e+f*x))^3)+1/30*(4*c^3-26*c^2*d-184*c*d^2-109*d^3)*tan(e+f*x)/(a*(c-d)^4*(c+d)^2*f*(a+a*sec(e+f*x))^2)+1/30*(4*c^4-30*c^3*d+142*c^2*d^2+525*c*d^3+304*d^4)*tan(e+f*x)/((c-d)^5*(c+d)^2*f*(a^3+a^3*sec(e+f*x)))-1/2*d*tan(e+f*x)/((c^2-d^2)*f*(a+a*sec(e+f*x))^3*(c+d*sec(e+f*x))^2)-3/2*d*(2*c+d)*tan(e+f*x)/((c^2-d^2)^2*f*(a+a*sec(e+f*x))^3*(c+d*sec(e+f*x)))+d^3*(20*c^2+30*c*d+13*d^2)*arctan(sqrt(c+d)*sqrt(a+a*sec(e+f*x))/(sqrt(c-d)*sqrt(a-a*sec(e+f*x))))*tan(e+f*x)/(a^2*(c-d)^(11/2)*(c+d)^(5/2)*f*sqrt(a-a*sec(e+f*x))*sqrt(a+a*sec(e+f*x)))],

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

# m>0
[sec(e+f*x)*sqrt(a+a*sec(e+f*x))/sqrt(c+d*sec(e+f*x)),x,2,2*arctanh(sqrt(a)*sqrt(d)*tan(e+f*x)/(sqrt(a+a*sec(e+f*x))*sqrt(c+d*sec(e+f*x))))*sqrt(a)/(f*sqrt(d))],

# m<0
[sec(e+f*x)*sqrt(c+d*sec(e+f*x))/sqrt(a+a*sec(e+f*x)),x,5,arctan(sqrt(a)*sqrt(c-d)*tan(e+f*x)/(sqrt(2)*sqrt(a+a*sec(e+f*x))*sqrt(c+d*sec(e+f*x))))*sqrt(2)*sqrt(c-d)/(f*sqrt(a))+2*arctanh(sqrt(a)*sqrt(d)*tan(e+f*x)/(sqrt(a+a*sec(e+f*x))*sqrt(c+d*sec(e+f*x))))*sqrt(d)/(f*sqrt(a))],
[sec(e+f*x)/(sqrt(a+a*sec(e+f*x))*sqrt(c+d*sec(e+f*x))),x,2,arctan(sqrt(a)*sqrt(c-d)*tan(e+f*x)/(sqrt(2)*sqrt(a+a*sec(e+f*x))*sqrt(c+d*sec(e+f*x))))*sqrt(2)/(f*sqrt(a)*sqrt(c-d))],

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

# m>0

# m<0
[sec(e+f*x)^2/(sqrt(a+a*sec(e+f*x))*sqrt(c+d*sec(e+f*x))),x,5,-arctan(sqrt(a)*sqrt(c-d)*tan(e+f*x)/(sqrt(2)*sqrt(a+a*sec(e+f*x))*sqrt(c+d*sec(e+f*x))))*sqrt(2)/(f*sqrt(a)*sqrt(c-d))+2*arctanh(sqrt(a)*sqrt(d)*tan(e+f*x)/(sqrt(a+a*sec(e+f*x))*sqrt(c+d*sec(e+f*x))))/(f*sqrt(a)*sqrt(d))],

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

# m>0
[sec(e+f*x)*sqrt(a+a*sec(e+f*x))/(c+d*sec(e+f*x)),x,2,2*arctan(sqrt(a)*sqrt(d)*tan(e+f*x)/(sqrt(c+d)*sqrt(a+a*sec(e+f*x))))*sqrt(a)/(f*sqrt(d)*sqrt(c+d))],
[(g*sec(e+f*x))^(3/2)*sqrt(a+a*sec(e+f*x))/(c+d*sec(e+f*x)),x,5,2*g^(3/2)*arctanh(sqrt(a)*sqrt(g)*tan(e+f*x)/(sqrt(g*sec(e+f*x))*sqrt(a+a*sec(e+f*x))))*sqrt(a)/(d*f)-2*g^(3/2)*arctanh(sqrt(a)*sqrt(c)*sqrt(g)*tan(e+f*x)/(sqrt(c+d)*sqrt(g*sec(e+f*x))*sqrt(a+a*sec(e+f*x))))*sqrt(a)*sqrt(c)/(d*f*sqrt(c+d))],

# m<0
[sec(e+f*x)/((c+d*sec(e+f*x))*sqrt(a+a*sec(e+f*x))),x,5,arctan(sqrt(a)*tan(e+f*x)/(sqrt(2)*sqrt(a+a*sec(e+f*x))))*sqrt(2)/((c-d)*f*sqrt(a))-2*arctan(sqrt(a)*sqrt(d)*tan(e+f*x)/(sqrt(c+d)*sqrt(a+a*sec(e+f*x))))*sqrt(d)/((c-d)*f*sqrt(a)*sqrt(c+d))],
[sec(e+f*x)^2/((c+d*sec(e+f*x))*sqrt(a+a*sec(e+f*x))),x,5,-arctan(sqrt(a)*tan(e+f*x)/(sqrt(2)*sqrt(a+a*sec(e+f*x))))*sqrt(2)/((c-d)*f*sqrt(a))+2*c*arctan(sqrt(a)*sqrt(d)*tan(e+f*x)/(sqrt(c+d)*sqrt(a+a*sec(e+f*x))))/((c-d)*f*sqrt(a)*sqrt(d)*sqrt(c+d))],
[(g*sec(e+f*x))^(3/2)/((c+d*sec(e+f*x))*sqrt(a+a*sec(e+f*x))),x,5,-g^(3/2)*arctanh(sqrt(a)*sqrt(g)*tan(e+f*x)/(sqrt(2)*sqrt(g*sec(e+f*x))*sqrt(a+a*sec(e+f*x))))*sqrt(2)/((c-d)*f*sqrt(a))+2*g^(3/2)*arctanh(sqrt(a)*sqrt(c)*sqrt(g)*tan(e+f*x)/(sqrt(c+d)*sqrt(g*sec(e+f*x))*sqrt(a+a*sec(e+f*x))))*sqrt(c)/((c-d)*f*sqrt(a)*sqrt(c+d))],
[(g*sec(e+f*x))^(5/2)/((c+d*sec(e+f*x))*sqrt(a+a*sec(e+f*x))),x,8,2*g^(5/2)*arctanh(sqrt(a)*sqrt(g)*tan(e+f*x)/(sqrt(g*sec(e+f*x))*sqrt(a+a*sec(e+f*x))))/(d*f*sqrt(a))+g^(5/2)*arctanh(sqrt(a)*sqrt(g)*tan(e+f*x)/(sqrt(2)*sqrt(g*sec(e+f*x))*sqrt(a+a*sec(e+f*x))))*sqrt(2)/((c-d)*f*sqrt(a))-2*c^(3/2)*g^(5/2)*arctanh(sqrt(a)*sqrt(c)*sqrt(g)*tan(e+f*x)/(sqrt(c+d)*sqrt(g*sec(e+f*x))*sqrt(a+a*sec(e+f*x))))/((c-d)*d*f*sqrt(a)*sqrt(c+d))],

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

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

# m>0
[sec(e+f*x)*(a+b*sec(e+f*x))*(c+d*sec(e+f*x))^4,x,8,1/8*(8*a*c^4+16*b*c^3*d+24*a*c^2*d^2+12*b*c*d^3+3*a*d^4)*arctanh(sin(e+f*x))/f+1/30*(12*b*c^4+95*a*c^3*d+112*b*c^2*d^2+80*a*c*d^3+16*b*d^4)*tan(e+f*x)/f+1/120*d*(24*b*c^3+130*a*c^2*d+116*b*c*d^2+45*a*d^3)*sec(e+f*x)*tan(e+f*x)/f+1/60*(12*b*c^2+35*a*c*d+16*b*d^2)*(c+d*sec(e+f*x))^2*tan(e+f*x)/f+1/20*(4*b*c+5*a*d)*(c+d*sec(e+f*x))^3*tan(e+f*x)/f+1/5*b*(c+d*sec(e+f*x))^4*tan(e+f*x)/f],
[sec(e+f*x)*(a+b*sec(e+f*x))*(c+d*sec(e+f*x))^3,x,7,1/8*(8*a*c^3+12*b*c^2*d+12*a*c*d^2+3*b*d^3)*arctanh(sin(e+f*x))/f+1/6*(4*a*d*(4*c^2+d^2)+3*b*(c^3+4*c*d^2))*tan(e+f*x)/f+1/24*d*(6*b*c^2+20*a*c*d+9*b*d^2)*sec(e+f*x)*tan(e+f*x)/f+1/12*(3*b*c+4*a*d)*(c+d*sec(e+f*x))^2*tan(e+f*x)/f+1/4*b*(c+d*sec(e+f*x))^3*tan(e+f*x)/f],
[sec(e+f*x)*(a+b*sec(e+f*x))*(c+d*sec(e+f*x))^2,x,6,1/2*(2*b*c*d+a*(2*c^2+d^2))*arctanh(sin(e+f*x))/f+2/3*(3*a*c*d+b*(c^2+d^2))*tan(e+f*x)/f+1/6*d*(2*b*c+3*a*d)*sec(e+f*x)*tan(e+f*x)/f+1/3*b*(c+d*sec(e+f*x))^2*tan(e+f*x)/f],
[sec(e+f*x)*(a+b*sec(e+f*x))*(c+d*sec(e+f*x)),x,5,1/2*(2*a*c+b*d)*arctanh(sin(e+f*x))/f+(b*c+a*d)*tan(e+f*x)/f+1/2*b*d*sec(e+f*x)*tan(e+f*x)/f],
[sec(e+f*x)*(a+b*sec(e+f*x))/(c+d*sec(e+f*x)),x,5,b*arctanh(sin(e+f*x))/(d*f)-2*(b*c-a*d)*arctanh(sqrt(c-d)*tan(1/2*(e+f*x))/sqrt(c+d))/(d*f*sqrt(c-d)*sqrt(c+d))],
[sec(e+f*x)*(a+b*sec(e+f*x))/(c+d*sec(e+f*x))^2,x,5,2*(a*c-b*d)*arctanh(sqrt(c-d)*tan(1/2*(e+f*x))/sqrt(c+d))/((c-d)^(3/2)*(c+d)^(3/2)*f)+(b*c-a*d)*tan(e+f*x)/((c^2-d^2)*f*(c+d*sec(e+f*x)))],
[sec(e+f*x)*(a+b*sec(e+f*x))/(c+d*sec(e+f*x))^3,x,6,-(3*b*c*d-a*(2*c^2+d^2))*arctanh(sqrt(c-d)*tan(1/2*(e+f*x))/sqrt(c+d))/((c-d)^(5/2)*(c+d)^(5/2)*f)+1/2*(b*c-a*d)*tan(e+f*x)/((c^2-d^2)*f*(c+d*sec(e+f*x))^2)-1/2*(3*a*c*d-b*(c^2+2*d^2))*tan(e+f*x)/((c^2-d^2)^2*f*(c+d*sec(e+f*x)))],
[sec(e+f*x)*(a+b*sec(e+f*x))/(c+d*sec(e+f*x))^4,x,7,(2*a*c^3-4*b*c^2*d+3*a*c*d^2-b*d^3)*arctanh(sqrt(c-d)*tan(1/2*(e+f*x))/sqrt(c+d))/((c-d)^(7/2)*(c+d)^(7/2)*f)+1/3*(b*c-a*d)*tan(e+f*x)/((c^2-d^2)*f*(c+d*sec(e+f*x))^3)+1/6*(2*b*c^2-5*a*c*d+3*b*d^2)*tan(e+f*x)/((c^2-d^2)^2*f*(c+d*sec(e+f*x))^2)+1/6*(2*b*c^3-11*a*c^2*d+13*b*c*d^2-4*a*d^3)*tan(e+f*x)/((c^2-d^2)^3*f*(c+d*sec(e+f*x)))],
#  {Sec[e + f*x]*(a + b*Sec[e + f*x])^2*(c + d*Sec[e + f*x])^4, x, 8, ((16*a*b*d*(4*c^3 + 3*c*d^2) + 2*a^2*(8*c^4 + 24*c^2*d^2 + 3*d^4) + b^2*(8*c^4 + 36*c^2*d^2 + 5*d^4))*ArcTanh[Sin[e + f*x]])/(16*f) + ((10*a^2*c*d^2*(19*c^2 + 16*d^2) + 16*a*b*d*(3*c^4 + 28*c^2*d^2 + 4*d^4) - b^2*(4*c^5 - 121*c^3*d^2 - 128*c*d^4))*Tan[e + f*x])/(60*d*f) + ((10*a^2*d^2*(26*c^2 + 9*d^2) + 16*a*b*d*(6*c^3 + 29*c*d^2) - b^2*(8*c^4 - 178*c^2*d^2 - 75*d^4))*Sec[e + f*x]*Tan[e + f*x])/(240*f) + ((70*a^2*c*d^2 + 16*a*b*d*(3*c^2 + 4*d^2) - b^2*(4*c^3 - 53*c*d^2))*(c + d*Sec[e + f*x])^2*Tan[e + f*x])/(120*d*f) + ((5*(6*a^2 + 5*b^2)*d^2 - 4*b*c*(b*c - 12*a*d))*(c + d*Sec[e + f*x])^3*Tan[e + f*x])/(120*d*f) - (b*(b*c - 12*a*d)*(c + d*Sec[e + f*x])^4*Tan[e + f*x])/(30*d*f) + (b^2*(c + d*Sec[e + f*x])^5*Tan[e + f*x])/(6*d*f)}
# {Sec[e + f*x]*(a + b*Sec[e + f*x])^2*(c + d*Sec[e + f*x])^3, x, 7, ((6*a*b*d*(4*c^2 + d^2) + b^2*c*(4*c^2 + 9*d^2) + 4*a^2*(2*c^3 + 3*c*d^2))*ArcTanh[Sin[e + f*x]])/(8*f) + ((20*a^2*d^2*(4*c^2 + d^2) + 30*a*b*c*d*(c^2 + 4*d^2) - b^2*(3*c^4 - 52*c^2*d^2 - 16*d^4))*Tan[e + f*x])/(30*d*f) + ((100*a^2*c*d^2 + 30*a*b*d*(2*c^2 + 3*d^2) - b^2*(6*c^3 - 71*c*d^2))*Sec[e + f*x]*Tan[e + f*x])/(120*f) + ((4*(5*a^2 + 4*b^2)*d^2 - 3*b*c*(b*c - 10*a*d))*(c + d*Sec[e + f*x])^2*Tan[e + f*x])/(60*d*f) - (b*(b*c - 10*a*d)*(c + d*Sec[e + f*x])^3*Tan[e + f*x])/(20*d*f) + (b^2*(c + d*Sec[e + f*x])^4*Tan[e + f*x])/(5*d*f)}
# {Sec[e + f*x]*(a + b*Sec[e + f*x])^2*(c + d*Sec[e + f*x])^2, x, 6, ((16*a*b*c*d + 4*a^2*(2*c^2 + d^2) + b^2*(4*c^2 + 3*d^2))*ArcTanh[Sin[e + f*x]])/(8*f) + ((8*a^2*b*c*d + 8*b^3*c*d - a^3*d^2 + 4*a*b^2*(3*c^2 + 2*d^2))*Tan[e + f*x])/(6*b*f) + ((2*a*d*(8*b*c - a*d) + 3*b^2*(4*c^2 + 3*d^2))*Sec[e + f*x]*Tan[e + f*x])/(24*f) + (d*(8*b*c - a*d)*(a + b*Sec[e + f*x])^2*Tan[e + f*x])/(12*b*f) + (d^2*(a + b*Sec[e + f*x])^3*Tan[e + f*x])/(4*b*f)}
# {Sec[e + f*x]*(a + b*Sec[e + f*x])^2*(c + d*Sec[e + f*x])^1, x, 5, ((2*a^2*c + b^2*c + 2*a*b*d)*ArcTanh[Sin[e + f*x]])/(2*f) + (2*(3*a*b*c + a^2*d + b^2*d)*Tan[e + f*x])/(3*f) + (b*(3*b*c + 2*a*d)*Sec[e + f*x]*Tan[e + f*x])/(6*f) + (d*(a + b*Sec[e + f*x])^2*Tan[e + f*x])/(3*f)}
# {(Sec[e + f*x]*(a + b*Sec[e + f*x])^2)/(c + d*Sec[e + f*x])^1, x, 6, -((b*(b*c - 2*a*d)*ArcTanh[Sin[e + f*x]])/(d^2*f)) + (2*(b*c - a*d)^2*ArcTanh[(Sqrt[c - d]*Tan[(e + f*x)/2])/Sqrt[c + d]])/(Sqrt[c - d]*d^2*Sqrt[c + d]*f) + (b^2*Tan[e + f*x])/(d*f)}
# {(Sec[e + f*x]*(a + b*Sec[e + f*x])^2)/(c + d*Sec[e + f*x])^2, x, 6, (b^2*ArcTanh[Sin[e + f*x]])/(d^2*f) - (2*(b*c - a*d)*(b*c^2 + a*c*d - 2*b*d^2)*ArcTanh[(Sqrt[c - d]*Tan[(e + f*x)/2])/Sqrt[c + d]])/((c - d)^(3/2)*d^2*(c + d)^(3/2)*f) - ((b*c - a*d)^2*Tan[e + f*x])/(d*(c^2 - d^2)*f*(c + d*Sec[e + f*x]))}
# {(Sec[e + f*x]*(a + b*Sec[e + f*x])^2)/(c + d*Sec[e + f*x])^3, x, 6, -(((6*a*b*c*d - a^2*(2*c^2 + d^2) - b^2*(c^2 + 2*d^2))*ArcTanh[(Sqrt[c - d]*Tan[(e + f*x)/2])/Sqrt[c + d]])/((c - d)^(5/2)*(c + d)^(5/2)*f)) - ((b*c - a*d)^2*Tan[e + f*x])/(2*d*(c^2 - d^2)*f*(c + d*Sec[e + f*x])^2) + ((b*c - a*d)*(3*a*c*d + b*(c^2 - 4*d^2))*Tan[e + f*x])/(2*(c - d)^2*d*(c + d)^2*f*(c + d*Sec[e + f*x]))}
# {(Sec[e + f*x]*(a + b*Sec[e + f*x])^2)/(c + d*Sec[e + f*x])^4, x, 7, -(((2*a*b*d*(4*c^2 + d^2) - b^2*c*(c^2 + 4*d^2) - a^2*(2*c^3 + 3*c*d^2))*ArcTanh[(Sqrt[c - d]*Tan[(e + f*x)/2])/Sqrt[c + d]])/((c - d)^(7/2)*(c + d)^(7/2)*f)) - ((b*c - a*d)^2*Tan[e + f*x])/(3*d*(c^2 - d^2)*f*(c + d*Sec[e + f*x])^3) + ((b*c - a*d)*(5*a*c*d + b*(c^2 - 6*d^2))*Tan[e + f*x])/(6*(c - d)^2*d*(c + d)^2*f*(c + d*Sec[e + f*x])^2) - ((a^2*d^2*(11*c^2 + 4*d^2) - a*b*(4*c^3*d + 26*c*d^3) - b^2*(c^4 - 10*c^2*d^2 - 6*d^4))*Tan[e + f*x])/(6*d*(c^2 - d^2)^3*f*(c + d*Sec[e + f*x]))}

# {(Sec[e + f*x]*(a + b*Sec[e + f*x])^2)/(c + d*Sec[e + f*x])^5, x, 8, -((10*a*b*d*(4*c^3 + 3*c*d^2) - a^2*(8*c^4 + 24*c^2*d^2 + 3*d^4) - b^2*(4*c^4 + 27*c^2*d^2 + 4*d^4))*ArcTanh[(Sqrt[c - d]*Tan[(e + f*x)/2])/Sqrt[c + d]])/(4*(c - d)^(9/2)*(c + d)^(9/2)*f) - ((b*c - a*d)^2*Tan[e + f*x])/(4*d*(c^2 - d^2)*f*(c + d*Sec[e + f*x])^4) + ((b*c - a*d)*(7*a*c*d + b*(c^2 - 8*d^2))*Tan[e + f*x])/(12*(c - d)^2*d*(c + d)^2*f*(c + d*Sec[e + f*x])^3) - ((a^2*d^2*(26*c^2 + 9*d^2) - 2*a*b*d*(6*c^3 + 29*c*d^2) - b^2*(2*c^4 - 25*c^2*d^2 - 12*d^4))*Tan[e + f*x])/(24*d*(c^2 - d^2)^3*f*(c + d*Sec[e + f*x])^2) - ((5*a^2*c*d^2*(10*c^2 + 11*d^2) - 2*a*b*d*(6*c^4 + 83*c^2*d^2 + 16*d^4) - b^2*(2*c^5 - 39*c^3*d^2 - 68*c*d^4))*Tan[e + f*x])/(24*d*(c^2 - d^2)^4*f*(c + d*Sec[e + f*x]))} 
#  {Sec[e + f*x]*(a + b*Sec[e + f*x])^3*(c + d*Sec[e + f*x])^3, x, 8, ((18*a^2*b*d*(4*c^2 + d^2) + b^3*d*(18*c^2 + 5*d^2) + 6*a*b^2*c*(4*c^2 + 9*d^2) + 8*a^3*(2*c^3 + 3*c*d^2))*ArcTanh[Sin[e + f*x]])/(16*f) + ((40*a^3*d^3*(4*c^2 + d^2) + 90*a^2*b*c*d^2*(c^2 + 4*d^2) - 6*a*b^2*d*(3*c^4 - 52*c^2*d^2 - 16*d^4) + b^3*(2*c^5 + 17*c^3*d^2 + 96*c*d^4))*Tan[e + f*x])/(60*d^2*f) + ((200*a^3*c*d^3 + 90*a^2*b*d^2*(2*c^2 + 3*d^2) - 6*a*b^2*d*(6*c^3 - 71*c*d^2) + b^3*(4*c^4 + 36*c^2*d^2 + 75*d^4))*Sec[e + f*x]*Tan[e + f*x])/(240*d*f) + ((90*a^2*b*c*d^2 + 40*a^3*d^3 + b^3*(2*c^3 + 21*c*d^2) - a*b^2*(18*c^2*d - 96*d^3))*(c + d*Sec[e + f*x])^2*Tan[e + f*x])/(120*d^2*f) - (b*(18*a*b*c*d - 90*a^2*d^2 - b^2*(2*c^2 + 25*d^2))*(c + d*Sec[e + f*x])^3*Tan[e + f*x])/(120*d^2*f) - (b^2*(2*b*c - 13*a*d)*(c + d*Sec[e + f*x])^4*Tan[e + f*x])/(30*d^2*f) + (b^2*(a + b*Sec[e + f*x])*(c + d*Sec[e + f*x])^4*Tan[e + f*x])/(6*d*f)}
# {Sec[e + f*x]*(a + b*Sec[e + f*x])^3*(c + d*Sec[e + f*x])^2, x, 7, ((24*a^2*b*c*d + 6*b^3*c*d + 4*a^3*(2*c^2 + d^2) + 3*a*b^2*(4*c^2 + 3*d^2))*ArcTanh[Sin[e + f*x]])/(8*f) + ((30*a^3*b*c*d + 120*a*b^3*c*d - 3*a^4*d^2 + 4*b^4*(5*c^2 + 4*d^2) + 4*a^2*b^2*(20*c^2 + 13*d^2))*Tan[e + f*x])/(30*b*f) + ((60*a^2*b*c*d + 90*b^3*c*d - 6*a^3*d^2 + a*b^2*(100*c^2 + 71*d^2))*Sec[e + f*x]*Tan[e + f*x])/(120*f) + ((3*a*d*(10*b*c - a*d) + 4*b^2*(5*c^2 + 4*d^2))*(a + b*Sec[e + f*x])^2*Tan[e + f*x])/(60*b*f) + (d*(10*b*c - a*d)*(a + b*Sec[e + f*x])^3*Tan[e + f*x])/(20*b*f) + (d^2*(a + b*Sec[e + f*x])^4*Tan[e + f*x])/(5*b*f)}
# {Sec[e + f*x]*(a + b*Sec[e + f*x])^3*(c + d*Sec[e + f*x])^1, x, 6, ((8*a^3*c + 12*a*b^2*c + 12*a^2*b*d + 3*b^3*d)*ArcTanh[Sin[e + f*x]])/(8*f) + ((16*a^2*b*c + 4*b^3*c + 3*a^3*d + 12*a*b^2*d)*Tan[e + f*x])/(6*f) + (b*(20*a*b*c + 6*a^2*d + 9*b^2*d)*Sec[e + f*x]*Tan[e + f*x])/(24*f) + ((4*b*c + 3*a*d)*(a + b*Sec[e + f*x])^2*Tan[e + f*x])/(12*f) + (d*(a + b*Sec[e + f*x])^3*Tan[e + f*x])/(4*f)}
# {(Sec[e + f*x]*(a + b*Sec[e + f*x])^3)/(c + d*Sec[e + f*x])^1, x, 7, -(b*(6*a*b*c*d - 6*a^2*d^2 - b^2*(2*c^2 + d^2))*ArcTanh[Sin[e + f*x]])/(2*d^3*f) - (2*(b*c - a*d)^3*ArcTanh[(Sqrt[c - d]*Tan[(e + f*x)/2])/Sqrt[c + d]])/(Sqrt[c - d]*d^3*Sqrt[c + d]*f) - (b^2*(2*b*c - 5*a*d)*Tan[e + f*x])/(2*d^2*f) + (b^2*(a + b*Sec[e + f*x])*Tan[e + f*x])/(2*d*f)}
# {(Sec[e + f*x]*(a + b*Sec[e + f*x])^3)/(c + d*Sec[e + f*x])^2, x, 7, -((b^2*(2*b*c - 3*a*d)*ArcTanh[Sin[e + f*x]])/(d^3*f)) + (2*(b*c - a*d)^2*(2*b*c^2 + a*c*d - 3*b*d^2)*ArcTanh[(Sqrt[c - d]*Tan[(e + f*x)/2])/Sqrt[c + d]])/((c - d)^(3/2)*d^3*(c + d)^(3/2)*f) - (b*(2*a*b*c*d - a^2*d^2 - b^2*(2*c^2 - d^2))*Tan[e + f*x])/(d^2*(c^2 - d^2)*f) - ((b*c - a*d)^2*(a + b*Sec[e + f*x])*Tan[e + f*x])/(d*(c^2 - d^2)*f*(c + d*Sec[e + f*x]))}
# {(Sec[e + f*x]*(a + b*Sec[e + f*x])^3)/(c + d*Sec[e + f*x])^3, x, 7, (b^3*ArcTanh[Sin[e + f*x]])/(d^3*f) - ((b*c - a*d)*(2*a*b*c*d*(c^2 - 4*d^2) + a^2*d^2*(2*c^2 + d^2) + b^2*(2*c^4 - 5*c^2*d^2 + 6*d^4))*ArcTanh[(Sqrt[c - d]*Tan[(e + f*x)/2])/Sqrt[c + d]])/((c - d)^(5/2)*d^3*(c + d)^(5/2)*f) - ((b*c - a*d)^2*(a + b*Sec[e + f*x])*Tan[e + f*x])/(2*d*(c^2 - d^2)*f*(c + d*Sec[e + f*x])^2) - ((b*c - a*d)^2*(2*b*c^2 + 3*a*c*d - 5*b*d^2)*Tan[e + f*x])/(2*d^2*(c^2 - d^2)^2*f*(c + d*Sec[e + f*x]))}
# {(Sec[e + f*x]*(a + b*Sec[e + f*x])^3)/(c + d*Sec[e + f*x])^4, x, 7, -(((a*c - b*d)*(10*a*b*c*d - b^2*(3*c^2 + 2*d^2) - a^2*(2*c^2 + 3*d^2))*ArcTanh[(Sqrt[c - d]*Tan[(e + f*x)/2])/Sqrt[c + d]])/((c - d)^(7/2)*(c + d)^(7/2)*f)) - ((b*c - a*d)^2*(a + b*Sec[e + f*x])*Tan[e + f*x])/(3*d*(c^2 - d^2)*f*(c + d*Sec[e + f*x])^3) - ((b*c - a*d)^2*(2*b*c^2 + 5*a*c*d - 7*b*d^2)*Tan[e + f*x])/(6*d^2*(c^2 - d^2)^2*f*(c + d*Sec[e + f*x])^2) + ((b*c - a*d)*(5*a*b*c*d*(c^2 - 7*d^2) + a^2*d^2*(11*c^2 + 4*d^2) + b^2*(2*c^4 - 5*c^2*d^2 + 18*d^4))*Tan[e + f*x])/(6*(c - d)^3*d^2*(c + d)^3*f*(c + d*Sec[e + f*x]))}

# {(Sec[e + f*x]*(a + b*Sec[e + f*x])^3)/(c + d*Sec[e + f*x])^5, x, 8, -((5*b^3*c*d*(3*c^2 + 4*d^2) + 15*a^2*b*d*(4*c^3 + 3*c*d^2) - a^3*(8*c^4 + 24*c^2*d^2 + 3*d^4) - 3*a*b^2*(4*c^4 + 27*c^2*d^2 + 4*d^4))*ArcTanh[(Sqrt[c - d]*Tan[(e + f*x)/2])/Sqrt[c + d]])/(4*(c - d)^(9/2)*(c + d)^(9/2)*f) - ((b*c - a*d)^2*(a + b*Sec[e + f*x])*Tan[e + f*x])/(4*d*(c^2 - d^2)*f*(c + d*Sec[e + f*x])^4) - ((b*c - a*d)^2*(2*b*c^2 + 7*a*c*d - 9*b*d^2)*Tan[e + f*x])/(12*d^2*(c^2 - d^2)^2*f*(c + d*Sec[e + f*x])^3) + ((b*c - a*d)*(a^2*d^2*(26*c^2 + 9*d^2) + a*b*(8*c^3*d - 78*c*d^3) + b^2*(2*c^4 - 3*c^2*d^2 + 36*d^4))*Tan[e + f*x])/(24*(c - d)^3*d^2*(c + d)^3*f*(c + d*Sec[e + f*x])^2) - ((5*a^3*c*d^3*(10*c^2 + 11*d^2) - 3*a*b^2*c*d*(2*c^4 - 39*c^2*d^2 - 68*d^4) - 3*a^2*b*d^2*(6*c^4 + 83*c^2*d^2 + 16*d^4) - b^3*(2*c^6 - 5*c^4*d^2 + 84*c^2*d^4 + 24*d^6))*Tan[e + f*x])/(24*d^2*(c^2 - d^2)^4*f*(c + d*Sec[e + f*x]))} 

# m<0
[sec(e+f*x)*(c+d*sec(e+f*x))^4/(a+b*sec(e+f*x)),x,12,1/2*d^3*(4*b*c-a*d)*arctanh(sin(e+f*x))/(b^2*f)+d*(2*b*c-a*d)*(2*b^2*c^2-2*a*b*c*d+a^2*d^2)*arctanh(sin(e+f*x))/(b^4*f)+2*(b*c-a*d)^4*arctanh(sqrt(a-b)*tan(1/2*(e+f*x))/sqrt(a+b))/(b^4*f*sqrt(a-b)*sqrt(a+b))+d^4*tan(e+f*x)/(b*f)+d^2*(6*b^2*c^2-4*a*b*c*d+a^2*d^2)*tan(e+f*x)/(b^3*f)+1/2*d^3*(4*b*c-a*d)*sec(e+f*x)*tan(e+f*x)/(b^2*f)+1/3*d^4*tan(e+f*x)^3/(b*f)],
[sec(e+f*x)*(c+d*sec(e+f*x))^3/(a+b*sec(e+f*x)),x,10,1/2*d^3*arctanh(sin(e+f*x))/(b*f)+d*(3*b^2*c^2-3*a*b*c*d+a^2*d^2)*arctanh(sin(e+f*x))/(b^3*f)+2*(b*c-a*d)^3*arctanh(sqrt(a-b)*tan(1/2*(e+f*x))/sqrt(a+b))/(b^3*f*sqrt(a-b)*sqrt(a+b))+d^2*(3*b*c-a*d)*tan(e+f*x)/(b^2*f)+1/2*d^3*sec(e+f*x)*tan(e+f*x)/(b*f)],
[sec(e+f*x)*(c+d*sec(e+f*x))^2/(a+b*sec(e+f*x)),x,8,d*(2*b*c-a*d)*arctanh(sin(e+f*x))/(b^2*f)+2*(b*c-a*d)^2*arctanh(sqrt(a-b)*tan(1/2*(e+f*x))/sqrt(a+b))/(b^2*f*sqrt(a-b)*sqrt(a+b))+d^2*tan(e+f*x)/(b*f)],
[sec(e+f*x)*(c+d*sec(e+f*x))/(a+b*sec(e+f*x)),x,5,d*arctanh(sin(e+f*x))/(b*f)+2*(b*c-a*d)*arctanh(sqrt(a-b)*tan(1/2*(e+f*x))/sqrt(a+b))/(b*f*sqrt(a-b)*sqrt(a+b))],
[sec(e+f*x)/((a+b*sec(e+f*x))*(c+d*sec(e+f*x))),x,6,2*b*arctanh(sqrt(a-b)*tan(1/2*(e+f*x))/sqrt(a+b))/((b*c-a*d)*f*sqrt(a-b)*sqrt(a+b))-2*d*arctanh(sqrt(c-d)*tan(1/2*(e+f*x))/sqrt(c+d))/((b*c-a*d)*f*sqrt(c-d)*sqrt(c+d))],
[sec(e+f*x)/((a+b*sec(e+f*x))*(c+d*sec(e+f*x))^2),x,7,-2*d*(2*b*c^2-a*c*d-b*d^2)*arctanh(sqrt(c-d)*tan(1/2*(e+f*x))/sqrt(c+d))/((c-d)^(3/2)*(c+d)^(3/2)*(b*c-a*d)^2*f)+d^2*sin(e+f*x)/((b*c-a*d)*(c^2-d^2)*f*(d+c*cos(e+f*x)))+2*b^2*arctanh(sqrt(a-b)*tan(1/2*(e+f*x))/sqrt(a+b))/((b*c-a*d)^2*f*sqrt(a-b)*sqrt(a+b))],

#  {Sec[e + f*x]/((a + b*Sec[e + f*x])*(c + d*Sec[e + f*x])^3), x, 9, (2*b^3*ArcTanh[(Sqrt[a - b]*Tan[(e + f*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*Sqrt[a + b]*(b*c - a*d)^3*f) + (d*(6*a*b*c^3*d - a^2*d^2*(2*c^2 + d^2) - b^2*(6*c^4 - 5*c^2*d^2 + 2*d^4))*ArcTanh[(Sqrt[c - d]*Tan[(e + f*x)/2])/Sqrt[c + d]])/((c - d)^(5/2)*(c + d)^(5/2)*(b*c - a*d)^3*f) + (d^2*Tan[e + f*x])/(2*(b*c - a*d)*(c^2 - d^2)*f*(c + d*Sec[e + f*x])^2) + (d^2*(5*b*c^2 - 3*a*c*d - 2*b*d^2)*Tan[e + f*x])/(2*(b*c - a*d)^2*(c^2 - d^2)^2*f*(c + d*Sec[e + f*x]))} 
[sec(e+f*x)*(c+d*sec(e+f*x))^5/(a+b*sec(e+f*x))^2,x,16,1/2*d^4*(5*b*c-2*a*d)*arctanh(sin(e+f*x))/(b^3*f)+d^2*(10*b^3*c^3-20*a*b^2*c^2*d+15*a^2*b*c*d^2-4*a^3*d^3)*arctanh(sin(e+f*x))/(b^5*f)+2*(b*c-a*d)^5*arctanh(sqrt(a-b)*tan(1/2*(e+f*x))/sqrt(a+b))/(a*(a-b)^(3/2)*b^3*(a+b)^(3/2)*f)-(b*c-a*d)^5*sin(e+f*x)/(b^4*(a^2-b^2)*f*(b+a*cos(e+f*x)))+2*(b*c-a*d)^4*(b*c+4*a*d)*arctanh(sqrt(a-b)*tan(1/2*(e+f*x))/sqrt(a+b))/(a*b^5*f*sqrt(a-b)*sqrt(a+b))+d^5*tan(e+f*x)/(b^2*f)+d^3*(10*b^2*c^2-10*a*b*c*d+3*a^2*d^2)*tan(e+f*x)/(b^4*f)+1/2*d^4*(5*b*c-2*a*d)*sec(e+f*x)*tan(e+f*x)/(b^3*f)+1/3*d^5*tan(e+f*x)^3/(b^2*f)],
[sec(e+f*x)*(c+d*sec(e+f*x))^4/(a+b*sec(e+f*x))^2,x,14,1/2*d^4*arctanh(sin(e+f*x))/(b^2*f)+d^2*(6*b^2*c^2-8*a*b*c*d+3*a^2*d^2)*arctanh(sin(e+f*x))/(b^4*f)+2*(b*c-a*d)^4*arctanh(sqrt(a-b)*tan(1/2*(e+f*x))/sqrt(a+b))/(a*(a-b)^(3/2)*b^2*(a+b)^(3/2)*f)-(b*c-a*d)^4*sin(e+f*x)/(b^3*(a^2-b^2)*f*(b+a*cos(e+f*x)))+2*(b*c-a*d)^3*(b*c+3*a*d)*arctanh(sqrt(a-b)*tan(1/2*(e+f*x))/sqrt(a+b))/(a*b^4*f*sqrt(a-b)*sqrt(a+b))+2*d^3*(2*b*c-a*d)*tan(e+f*x)/(b^3*f)+1/2*d^4*sec(e+f*x)*tan(e+f*x)/(b^2*f)],
[sec(e+f*x)*(c+d*sec(e+f*x))^3/(a+b*sec(e+f*x))^2,x,12,d^2*(3*b*c-2*a*d)*arctanh(sin(e+f*x))/(b^3*f)+2*(b*c-a*d)^3*arctanh(sqrt(a-b)*tan(1/2*(e+f*x))/sqrt(a+b))/(a*(a-b)^(3/2)*b*(a+b)^(3/2)*f)-(b*c-a*d)^3*sin(e+f*x)/(b^2*(a^2-b^2)*f*(b+a*cos(e+f*x)))+2*(b*c-a*d)^2*(b*c+2*a*d)*arctanh(sqrt(a-b)*tan(1/2*(e+f*x))/sqrt(a+b))/(a*b^3*f*sqrt(a-b)*sqrt(a+b))+d^3*tan(e+f*x)/(b^2*f)],
[sec(e+f*x)*(c+d*sec(e+f*x))^2/(a+b*sec(e+f*x))^2,x,10,d^2*arctanh(sin(e+f*x))/(b^2*f)+2*(b*c-a*d)^2*arctanh(sqrt(a-b)*tan(1/2*(e+f*x))/sqrt(a+b))/(a*(a-b)^(3/2)*(a+b)^(3/2)*f)-(b*c-a*d)^2*sin(e+f*x)/(b*(a^2-b^2)*f*(b+a*cos(e+f*x)))+2*(b^2*c^2-a^2*d^2)*arctanh(sqrt(a-b)*tan(1/2*(e+f*x))/sqrt(a+b))/(a*b^2*f*sqrt(a-b)*sqrt(a+b))],
[sec(e+f*x)*(c+d*sec(e+f*x))/(a+b*sec(e+f*x))^2,x,5,2*(a*c-b*d)*arctanh(sqrt(a-b)*tan(1/2*(e+f*x))/sqrt(a+b))/((a-b)^(3/2)*(a+b)^(3/2)*f)-(b*c-a*d)*tan(e+f*x)/((a^2-b^2)*f*(a+b*sec(e+f*x)))],
[sec(e+f*x)/((a+b*sec(e+f*x))^2*(c+d*sec(e+f*x))),x,7,2*b*(a*b*c-2*a^2*d+b^2*d)*arctanh(sqrt(a-b)*tan(1/2*(e+f*x))/sqrt(a+b))/((a-b)^(3/2)*(a+b)^(3/2)*(b*c-a*d)^2*f)-b^2*sin(e+f*x)/((a^2-b^2)*(b*c-a*d)*f*(b+a*cos(e+f*x)))+2*d^2*arctanh(sqrt(c-d)*tan(1/2*(e+f*x))/sqrt(c+d))/((b*c-a*d)^2*f*sqrt(c-d)*sqrt(c+d))],
#  {Sec[e + f*x]/((a + b*Sec[e + f*x])^2*(c + d*Sec[e + f*x])^2), x, 9, (2*b^2*(a*b*c - 3*a^2*d + 2*b^2*d)*ArcTanh[(Sqrt[a - b]*Tan[(e + f*x)/2])/Sqrt[a + b]])/((a - b)^(3/2)*(a + b)^(3/2)*(b*c - a*d)^3*f) + (2*d^2*(3*b*c^2 - a*c*d - 2*b*d^2)*ArcTanh[(Sqrt[c - d]*Tan[(e + f*x)/2])/Sqrt[c + d]])/((c - d)^(3/2)*(c + d)^(3/2)*(b*c - a*d)^3*f) - (d*(a^2*d^2 + b^2*(c^2 - 2*d^2))*Tan[e + f*x])/((a^2 - b^2)*(b*c - a*d)^2*(c^2 - d^2)*f*(c + d*Sec[e + f*x])) - (b^2*Tan[e + f*x])/((a^2 - b^2)*(b*c - a*d)*f*(a + b*Sec[e + f*x])*(c + d*Sec[e + f*x]))}

# {Sec[e + f*x]/((a + b*Sec[e + f*x])^2*(c + d*Sec[e + f*x])^3), x, 10, (2*b^3*(a*b*c - 4*a^2*d + 3*b^2*d)*ArcTanh[(Sqrt[a - b]*Tan[(e + f*x)/2])/Sqrt[a + b]])/((a - b)^(3/2)*(a + b)^(3/2)*(b*c - a*d)^4*f) - (d^2*(2*a*b*c*d*(4*c^2 - d^2) - a^2*d^2*(2*c^2 + d^2) - 3*b^2*(4*c^4 - 5*c^2*d^2 + 2*d^4))*ArcTanh[(Sqrt[c - d]*Tan[(e + f*x)/2])/Sqrt[c + d]])/((c - d)^(5/2)*(c + d)^(5/2)*(b*c - a*d)^4*f) - (d*(a^2*d^2 + b^2*(2*c^2 - 3*d^2))*Tan[e + f*x])/(2*(a^2 - b^2)*(b*c - a*d)^2*(c^2 - d^2)*f*(c + d*Sec[e + f*x])^2) - (b^2*Tan[e + f*x])/((a^2 - b^2)*(b*c - a*d)*f*(a + b*Sec[e + f*x])*(c + d*Sec[e + f*x])^2) + (d*(3*a^3*c*d^3 - 3*a*b^2*c*d^3 - a^2*b*d^2*(7*c^2 - 4*d^2) - b^3*(2*c^4 - 11*c^2*d^2 + 6*d^4))*Tan[e + f*x])/(2*(a^2 - b^2)*(b*c - a*d)^3*(c^2 - d^2)^2*f*(c + d*Sec[e + f*x]))} 
#  {(Sec[e + f*x]*(c + d*Sec[e + f*x])^6)/(a + b*Sec[e + f*x])^3, x, 10, (d^3*(72*a^2*b*c*d^2 - 20*a^3*d^3 - 3*a*b^2*d*(30*c^2 + d^2) + b^3*(40*c^3 + 6*c*d^2))*ArcTanh[Sin[e + f*x]])/(2*b^6*f) + ((b*c - a*d)^4*(8*a^3*b*c*d - 14*a*b^3*c*d + 20*a^4*d^2 + a^2*b^2*(2*c^2 - 47*d^2) + b^4*(c^2 + 30*d^2))*ArcTanh[(Sqrt[a - b]*Tan[(e + f*x)/2])/Sqrt[a + b]])/((a - b)^(5/2)*b^6*(a + b)^(5/2)*f) - (d*(216*a^5*b*c*d^4 - 60*a^6*d^5 - a^4*b^2*d^3*(250*c^2 - 101*d^2) + 6*a^3*b^3*c*d^2*(13*c^2 - 63*d^2) - 6*a*b^5*c*(3*c^4 + 37*c^2*d^2 - 18*d^4) + 2*a^2*b^4*d*(9*c^4 + 232*c^2*d^2 - 14*d^4) + b^6*d*(63*c^4 - 88*c^2*d^2 - 4*d^4))*Tan[e + f*x])/(6*b^5*(a^2 - b^2)^2*f) + (d^2*(68*a^4*b*c*d^3 - 30*a^5*d^4 - 2*b^5*c*d*(15*c^2 - 7*d^2) - 12*a^3*b^2*d^2*(3*c^2 - 4*d^2) - 2*a^2*b^3*c*d*(3*c^2 + 59*d^2) + 9*a*b^4*(c^4 + 10*c^2*d^2 - d^4))*Sec[e + f*x]*Tan[e + f*x])/(6*b^4*(a^2 - b^2)^2*f) - (d*(27*a^3*b*c*d^2 - 20*a^4*d^3 + 31*a^2*b^2*d^3 + b^4*d*(27*c^2 - 2*d^2) - 9*a*b^3*c*(c^2 + 6*d^2))*(c + d*Sec[e + f*x])^2*Tan[e + f*x])/(6*b^3*(a^2 - b^2)^2*f) - ((b*c - a*d)^2*(3*a*b*c + 5*a^2*d - 8*b^2*d)*(c + d*Sec[e + f*x])^3*Tan[e + f*x])/(2*b^2*(a^2 - b^2)^2*f*(a + b*Sec[e + f*x])) - ((b*c - a*d)^2*(c + d*Sec[e + f*x])^4*Tan[e + f*x])/(2*b*(a^2 - b^2)*f*(a + b*Sec[e + f*x])^2)}
# {(Sec[e + f*x]*(c + d*Sec[e + f*x])^5)/(a + b*Sec[e + f*x])^3, x, 9, -(d^3*(30*a*b*c*d - 12*a^2*d^2 - b^2*(20*c^2 + d^2))*ArcTanh[Sin[e + f*x]])/(2*b^5*f) + ((b*c - a*d)^3*(6*a^3*b*c*d - 12*a*b^3*c*d + 12*a^4*d^2 + a^2*b^2*(2*c^2 - 29*d^2) + b^4*(c^2 + 20*d^2))*ArcTanh[(Sqrt[a - b]*Tan[(e + f*x)/2])/Sqrt[a + b]])/((a - b)^(5/2)*b^5*(a + b)^(5/2)*f) + (d*(30*a^4*b*c*d^3 - 12*a^5*d^4 - a^3*b^2*d^2*(16*c^2 - 21*d^2) - b^5*c*d*(17*c^2 - 10*d^2) - a^2*b^3*c*d*(4*c^2 + 55*d^2) + a*b^4*(6*c^4 + 43*c^2*d^2 - 6*d^4))*Tan[e + f*x])/(2*b^4*(a^2 - b^2)^2*f) - (d^2*(7*a^3*b*c*d^2 - 6*a^4*d^3 + b^4*d*(8*c^2 - d^2) + a^2*b^2*d*(c^2 + 10*d^2) - a*b^3*c*(3*c^2 + 16*d^2))*Sec[e + f*x]*Tan[e + f*x])/(2*b^3*(a^2 - b^2)^2*f) - ((b*c - a*d)^2*(3*a*b*c + 4*a^2*d - 7*b^2*d)*(c + d*Sec[e + f*x])^2*Tan[e + f*x])/(2*b^2*(a^2 - b^2)^2*f*(a + b*Sec[e + f*x])) - ((b*c - a*d)^2*(c + d*Sec[e + f*x])^3*Tan[e + f*x])/(2*b*(a^2 - b^2)*f*(a + b*Sec[e + f*x])^2)}
# {(Sec[e + f*x]*(c + d*Sec[e + f*x])^4)/(a + b*Sec[e + f*x])^3, x, 8, (d^3*(4*b*c - 3*a*d)*ArcTanh[Sin[e + f*x]])/(b^4*f) + ((b*c - a*d)^2*(4*a^3*b*c*d - 10*a*b^3*c*d + 6*a^4*d^2 + a^2*b^2*(2*c^2 - 15*d^2) + b^4*(c^2 + 12*d^2))*ArcTanh[(Sqrt[a - b]*Tan[(e + f*x)/2])/Sqrt[a + b]])/((a - b)^(5/2)*b^4*(a + b)^(5/2)*f) - (d^2*(2*a*b*c*d - 3*a^2*d^2 - b^2*(c^2 - 2*d^2))*Tan[e + f*x])/(2*b^3*(a^2 - b^2)*f) - (3*(b*c - a*d)^3*(a*b*c + a^2*d - 2*b^2*d)*Tan[e + f*x])/(2*b^3*(a^2 - b^2)^2*f*(a + b*Sec[e + f*x])) - ((b*c - a*d)^2*(c + d*Sec[e + f*x])^2*Tan[e + f*x])/(2*b*(a^2 - b^2)*f*(a + b*Sec[e + f*x])^2)}
# {(Sec[e + f*x]*(c + d*Sec[e + f*x])^3)/(a + b*Sec[e + f*x])^3, x, 7, (d^3*ArcTanh[Sin[e + f*x]])/(b^3*f) + ((b*c - a*d)*(2*a^3*b*c*d - 8*a*b^3*c*d + 2*a^4*d^2 + a^2*b^2*(2*c^2 - 5*d^2) + b^4*(c^2 + 6*d^2))*ArcTanh[(Sqrt[a - b]*Tan[(e + f*x)/2])/Sqrt[a + b]])/((a - b)^(5/2)*b^3*(a + b)^(5/2)*f) - ((b*c - a*d)^2*(3*a*b*c + 2*a^2*d - 5*b^2*d)*Tan[e + f*x])/(2*b^2*(a^2 - b^2)^2*f*(a + b*Sec[e + f*x])) - ((b*c - a*d)^2*(c + d*Sec[e + f*x])*Tan[e + f*x])/(2*b*(a^2 - b^2)*f*(a + b*Sec[e + f*x])^2)}
# {(Sec[e + f*x]*(c + d*Sec[e + f*x])^2)/(a + b*Sec[e + f*x])^3, x, 6, -(((6*a*b*c*d - a^2*(2*c^2 + d^2) - b^2*(c^2 + 2*d^2))*ArcTanh[(Sqrt[a - b]*Tan[(e + f*x)/2])/Sqrt[a + b]])/((a - b)^(5/2)*(a + b)^(5/2)*f)) - ((b*c - a*d)^2*Tan[e + f*x])/(2*b*(a^2 - b^2)*f*(a + b*Sec[e + f*x])^2) - ((b*c - a*d)*(3*a*b*c + a^2*d - 4*b^2*d)*Tan[e + f*x])/(2*b*(a^2 - b^2)^2*f*(a + b*Sec[e + f*x]))}
# {(Sec[e + f*x]*(c + d*Sec[e + f*x])^1)/(a + b*Sec[e + f*x])^3, x, 6, ((2*a^2*c + b^2*c - 3*a*b*d)*ArcTanh[(Sqrt[a - b]*Tan[(e + f*x)/2])/Sqrt[a + b]])/((a - b)^(5/2)*(a + b)^(5/2)*f) - ((b*c - a*d)*Tan[e + f*x])/(2*(a^2 - b^2)*f*(a + b*Sec[e + f*x])^2) - ((3*a*b*c - a^2*d - 2*b^2*d)*Tan[e + f*x])/(2*(a^2 - b^2)^2*f*(a + b*Sec[e + f*x]))}
# {Sec[e + f*x]/((a + b*Sec[e + f*x])^3*(c + d*Sec[e + f*x])^1), x, 9, -((b*(6*a^3*b*c*d - 6*a^4*d^2 - a^2*b^2*(2*c^2 - 5*d^2) - b^4*(c^2 + 2*d^2))*ArcTanh[(Sqrt[a - b]*Tan[(e + f*x)/2])/Sqrt[a + b]])/((a - b)^(5/2)*(a + b)^(5/2)*(b*c - a*d)^3*f)) - (2*d^3*ArcTanh[(Sqrt[c - d]*Tan[(e + f*x)/2])/Sqrt[c + d]])/(Sqrt[c - d]*Sqrt[c + d]*(b*c - a*d)^3*f) - (b^2*Tan[e + f*x])/(2*(a^2 - b^2)*(b*c - a*d)*f*(a + b*Sec[e + f*x])^2) - (b^2*(3*a*b*c - 5*a^2*d + 2*b^2*d)*Tan[e + f*x])/(2*(a^2 - b^2)^2*(b*c - a*d)^2*f*(a + b*Sec[e + f*x]))}
# {Sec[e + f*x]/((a + b*Sec[e + f*x])^3*(c + d*Sec[e + f*x])^2), x, 10, -((b^2*(8*a^3*b*c*d - 2*a*b^3*c*d - 12*a^4*d^2 - a^2*b^2*(2*c^2 - 15*d^2) - b^4*(c^2 + 6*d^2))*ArcTanh[(Sqrt[a - b]*Tan[(e + f*x)/2])/Sqrt[a + b]])/((a - b)^(5/2)*(a + b)^(5/2)*(b*c - a*d)^4*f)) - (2*d^3*(4*b*c^2 - a*c*d - 3*b*d^2)*ArcTanh[(Sqrt[c - d]*Tan[(e + f*x)/2])/Sqrt[c + d]])/((c - d)^(3/2)*(c + d)^(3/2)*(b*c - a*d)^4*f) + (d*(2*a^4*d^3 + a^2*b^2*d*(7*c^2 - 11*d^2) - 2*b^4*d*(2*c^2 - 3*d^2) - 3*a*b^3*c*(c^2 - d^2))*Tan[e + f*x])/(2*(a^2 - b^2)^2*(b*c - a*d)^3*(c^2 - d^2)*f*(c + d*Sec[e + f*x])) - (b^2*Tan[e + f*x])/(2*(a^2 - b^2)*(b*c - a*d)*f*(a + b*Sec[e + f*x])^2*(c + d*Sec[e + f*x])) - (3*b^2*(a*b*c - 2*a^2*d + b^2*d)*Tan[e + f*x])/(2*(a^2 - b^2)^2*(b*c - a*d)^2*f*(a + b*Sec[e + f*x])*(c + d*Sec[e + f*x]))}

# {Sec[e + f*x]/((a + b*Sec[e + f*x])^3*(c + d*Sec[e + f*x])^3), x, 11, -((b^3*(10*a^3*b*c*d - 4*a*b^3*c*d - 20*a^4*d^2 - a^2*b^2*(2*c^2 - 29*d^2) - b^4*(c^2 + 12*d^2))*ArcTanh[(Sqrt[a - b]*Tan[(e + f*x)/2])/Sqrt[a + b]])/((a - b)^(5/2)*(a + b)^(5/2)*(b*c - a*d)^5*f)) - (d^3*(a^2*d^2*(2*c^2 + d^2) - a*b*(10*c^3*d - 4*c*d^3) + b^2*(20*c^4 - 29*c^2*d^2 + 12*d^4))*ArcTanh[(Sqrt[c - d]*Tan[(e + f*x)/2])/Sqrt[c + d]])/((c - d)^(5/2)*(c + d)^(5/2)*(b*c - a*d)^5*f) + (d*(a^4*d^3 - b^4*d*(5*c^2 - 6*d^2) + 2*a^2*b^2*d*(4*c^2 - 5*d^2) - 3*a*b^3*c*(c^2 - d^2))*Tan[e + f*x])/(2*(a^2 - b^2)^2*(b*c - a*d)^3*(c^2 - d^2)*f*(c + d*Sec[e + f*x])^2) - (b^2*Tan[e + f*x])/(2*(a^2 - b^2)*(b*c - a*d)*f*(a + b*Sec[e + f*x])^2*(c + d*Sec[e + f*x])^2) - (b^2*(3*a*b*c - 7*a^2*d + 4*b^2*d)*Tan[e + f*x])/(2*(a^2 - b^2)^2*(b*c - a*d)^2*f*(a + b*Sec[e + f*x])*(c + d*Sec[e + f*x])^2) - (3*d*(a^5*c*d^4 - 2*a^3*b^2*c*d^4 + a*b^4*c*(c^4 - 2*c^2*d^2 + 2*d^4) + b^5*d*(2*c^4 - 7*c^2*d^2 + 4*d^4) - a^2*b^3*d*(3*c^4 - 12*c^2*d^2 + 7*d^4) - a^4*b*(3*c^2*d^3 - 2*d^5))*Tan[e + f*x])/(2*(a^2 - b^2)^2*(b*c - a*d)^4*(c^2 - d^2)^2*f*(c + d*Sec[e + f*x]))} 

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

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

# m>0
[sec(e+f*x)*sqrt(a+b*sec(e+f*x))/sqrt(c+d*sec(e+f*x)),x,1,2*cot(e+f*x)*EllipticPi(sqrt((a+b)/(c+d))*sqrt(c+d*sec(e+f*x))/sqrt(a+b*sec(e+f*x)),b*(c+d)/((a+b)*d),sqrt((a-b)*(c+d)/((a+b)*(c-d))))*(a+b*sec(e+f*x))*sqrt(-(b*c-a*d)*(1-sec(e+f*x))/((c+d)*(a+b*sec(e+f*x))))*sqrt((b*c-a*d)*(1+sec(e+f*x))/((c-d)*(a+b*sec(e+f*x))))/(d*f*sqrt((a+b)/(c+d)))],

# m<0
[sec(e+f*x)/(sqrt(a+b*sec(e+f*x))*sqrt(c+d*sec(e+f*x))),x,1,2*cot(e+f*x)*EllipticF(sqrt(c+d)*sqrt(a+b*sec(e+f*x))/(sqrt(a+b)*sqrt(c+d*sec(e+f*x))),sqrt((a+b)*(c-d)/((a-b)*(c+d))))*(c+d*sec(e+f*x))*sqrt(a+b)*sqrt((b*c-a*d)*(1-sec(e+f*x))/((a+b)*(c+d*sec(e+f*x))))*sqrt(-(b*c-a*d)*(1+sec(e+f*x))/((a-b)*(c+d*sec(e+f*x))))/((b*c-a*d)*f*sqrt(c+d))],
[sec(e+f*x)/(sqrt(2+3*sec(e+f*x))*sqrt(-4+5*sec(e+f*x))),x,1,2*cot(e+f*x)*EllipticF(sqrt(2+3*sec(e+f*x))/(sqrt(5)*sqrt(-4+5*sec(e+f*x))),sqrt(45))*(4-5*sec(e+f*x))*sqrt((1-sec(e+f*x))/(4-5*sec(e+f*x)))*sqrt((1+sec(e+f*x))/(4-5*sec(e+f*x)))/f],
[sec(e+f*x)/(sqrt(4-5*sec(e+f*x))*sqrt(2+3*sec(e+f*x))),x,1,2/3*I*cot(e+f*x)*EllipticF(I*sqrt(5)*sqrt(4-5*sec(e+f*x))/sqrt(2+3*sec(e+f*x)),sqrt(1/45))*(2+3*sec(e+f*x))*sqrt((1-sec(e+f*x))/(2+3*sec(e+f*x)))*sqrt((1+sec(e+f*x))/(2+3*sec(e+f*x)))/(f*sqrt(5))],

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

# m>0

# m<0
[sec(e+f*x)^2/(sqrt(a+b*sec(e+f*x))*sqrt(c+d*sec(e+f*x))),x,3,2*cot(e+f*x)*EllipticPi(sqrt((a+b)/(c+d))*sqrt(c+d*sec(e+f*x))/sqrt(a+b*sec(e+f*x)),b*(c+d)/((a+b)*d),sqrt((a-b)*(c+d)/((a+b)*(c-d))))*(a+b*sec(e+f*x))*sqrt(-(b*c-a*d)*(1-sec(e+f*x))/((c+d)*(a+b*sec(e+f*x))))*sqrt((b*c-a*d)*(1+sec(e+f*x))/((c-d)*(a+b*sec(e+f*x))))/(b*d*f*sqrt((a+b)/(c+d)))-2*a*cot(e+f*x)*EllipticF(sqrt(c+d)*sqrt(a+b*sec(e+f*x))/(sqrt(a+b)*sqrt(c+d*sec(e+f*x))),sqrt((a+b)*(c-d)/((a-b)*(c+d))))*(c+d*sec(e+f*x))*sqrt(a+b)*sqrt((b*c-a*d)*(1-sec(e+f*x))/((a+b)*(c+d*sec(e+f*x))))*sqrt(-(b*c-a*d)*(1+sec(e+f*x))/((a-b)*(c+d*sec(e+f*x))))/(b*(b*c-a*d)*f*sqrt(c+d))],

# Integrands of the form (g Sec[e+f x])^(3/2) (a+b Sec[e+f x])^m (c+d Sec[e+f x])^(n/2)
#  {(g*Sec[e + f*x])^(3/2)*(c + d*Sec[e + f*x])^(5/2)/(a + b*Sec[e + f*x]), x, 0, 0}

# {(g*Sec[e + f*x])^(3/2)*(c + d*Sec[e + f*x])^(3/2)/(a + b*Sec[e + f*x]), x, 0, 0} 
[(g*sec(e+f*x))^(3/2)*(c+d*sec(e+f*x))^(1/2)/(a+b*sec(e+f*x)),x,7,2*d*g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2,sqrt(2*c/(c+d)))*sqrt((d+c*cos(e+f*x))/(c+d))*sqrt(g*sec(e+f*x))/(b*f*sqrt(c+d*sec(e+f*x)))+2*(b*c-a*d)*g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2*a/(a+b),sqrt(2*c/(c+d)))*sqrt((d+c*cos(e+f*x))/(c+d))*sqrt(g*sec(e+f*x))/(b*(a+b)*f*sqrt(c+d*sec(e+f*x)))],
[(g*sec(e+f*x))^(3/2)/((a+b*sec(e+f*x))*(c+d*sec(e+f*x))^(1/2)),x,3,2*g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2*a/(a+b),sqrt(2*c/(c+d)))*sqrt((d+c*cos(e+f*x))/(c+d))*sqrt(g*sec(e+f*x))/((a+b)*f*sqrt(c+d*sec(e+f*x)))],
#  {(g*Sec[e + f*x])^(3/2)/((c + d*Sec[e + f*x])^(3/2)*(a + b*Sec[e + f*x])), x, 0, 0}

# {(g*Sec[e + f*x])^(3/2)/((c + d*Sec[e + f*x])^(5/2)*(a + b*Sec[e + f*x])), x, 0, 0} 
[sqrt(g*sec(e+f*x))*sqrt(c+d*sec(e+f*x))/(a+b*cos(e+f*x)),x,8,2*d*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2,sqrt(2*c/(c+d)))*sqrt((d+c*cos(e+f*x))/(c+d))*sqrt(g*sec(e+f*x))/(a*f*sqrt(c+d*sec(e+f*x)))+2*(a*c-b*d)*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2*b/(a+b),sqrt(2*c/(c+d)))*sqrt((d+c*cos(e+f*x))/(c+d))*sqrt(g*sec(e+f*x))/(a*(a+b)*f*sqrt(c+d*sec(e+f*x)))],

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

# m>0
[sec(e+f*x)*sqrt(a+b*sec(e+f*x))/(c+c*sec(e+f*x)),x,1,EllipticE(tan(e+f*x)/(1+sec(e+f*x)),sqrt((a-b)/(a+b)))*sqrt(1/(1+sec(e+f*x)))*sqrt(a+b*sec(e+f*x))/(c*f*sqrt((a+b*sec(e+f*x))/((a+b)*(1+sec(e+f*x)))))],
[(g*sec(e+f*x))^(3/2)*sqrt(a+b*sec(e+f*x))/(c+c*sec(e+f*x)),x,11,-g*(b+a*cos(e+f*x))*sin(e+f*x)*sqrt(g*sec(e+f*x))/(f*(c+c*cos(e+f*x))*sqrt(a+b*sec(e+f*x)))+g*(b+a*cos(e+f*x))*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2*a/(a+b)))*sqrt(g*sec(e+f*x))/(c*f*sqrt((b+a*cos(e+f*x))/(a+b))*sqrt(a+b*sec(e+f*x)))+(a-b)*g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticF(sin(1/2*(e+f*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(e+f*x))/(a+b))*sqrt(g*sec(e+f*x))/(c*f*sqrt(a+b*sec(e+f*x)))+2*b*g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2,sqrt(2*a/(a+b)))*sqrt((b+a*cos(e+f*x))/(a+b))*sqrt(g*sec(e+f*x))/(c*f*sqrt(a+b*sec(e+f*x)))],

# m<0
[sec(e+f*x)/((c+c*sec(e+f*x))*sqrt(a+b*sec(e+f*x))),x,3,-2*cot(e+f*x)*EllipticF(sqrt(a+b*sec(e+f*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(e+f*x))/(a+b))*sqrt(-b*(1+sec(e+f*x))/(a-b))/((a-b)*c*f)+EllipticE(tan(e+f*x)/(1+sec(e+f*x)),sqrt((a-b)/(a+b)))*sqrt(1/(1+sec(e+f*x)))*sqrt(a+b*sec(e+f*x))/((a-b)*c*f*sqrt((a+b*sec(e+f*x))/((a+b)*(1+sec(e+f*x)))))],
[sec(e+f*x)^2/((c+c*sec(e+f*x))*sqrt(a+b*sec(e+f*x))),x,3,2*a*cot(e+f*x)*EllipticF(sqrt(a+b*sec(e+f*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(e+f*x))/(a+b))*sqrt(-b*(1+sec(e+f*x))/(a-b))/((a-b)*b*c*f)-EllipticE(tan(e+f*x)/(1+sec(e+f*x)),sqrt((a-b)/(a+b)))*sqrt(1/(1+sec(e+f*x)))*sqrt(a+b*sec(e+f*x))/((a-b)*c*f*sqrt((a+b*sec(e+f*x))/((a+b)*(1+sec(e+f*x)))))],
[(g*sec(e+f*x))^(3/2)/((c+c*sec(e+f*x))*sqrt(a+b*sec(e+f*x))),x,7,-g*(b+a*cos(e+f*x))*sin(e+f*x)*sqrt(g*sec(e+f*x))/((a-b)*f*(c+c*cos(e+f*x))*sqrt(a+b*sec(e+f*x)))+g*(b+a*cos(e+f*x))*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2*a/(a+b)))*sqrt(g*sec(e+f*x))/((a-b)*c*f*sqrt((b+a*cos(e+f*x))/(a+b))*sqrt(a+b*sec(e+f*x)))+g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticF(sin(1/2*(e+f*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(e+f*x))/(a+b))*sqrt(g*sec(e+f*x))/(c*f*sqrt(a+b*sec(e+f*x)))],
[(g*sec(e+f*x))^(5/2)/((c+c*sec(e+f*x))*sqrt(a+b*sec(e+f*x))),x,11,g^2*(b+a*cos(e+f*x))*sin(e+f*x)*sqrt(g*sec(e+f*x))/((a-b)*f*(c+c*cos(e+f*x))*sqrt(a+b*sec(e+f*x)))-g^2*(b+a*cos(e+f*x))*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticE(sin(1/2*(e+f*x)),sqrt(2*a/(a+b)))*sqrt(g*sec(e+f*x))/((a-b)*c*f*sqrt((b+a*cos(e+f*x))/(a+b))*sqrt(a+b*sec(e+f*x)))-g^2*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticF(sin(1/2*(e+f*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(e+f*x))/(a+b))*sqrt(g*sec(e+f*x))/(c*f*sqrt(a+b*sec(e+f*x)))+2*g^2*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2,sqrt(2*a/(a+b)))*sqrt((b+a*cos(e+f*x))/(a+b))*sqrt(g*sec(e+f*x))/(c*f*sqrt(a+b*sec(e+f*x)))],

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

# m>0
[sec(e+f*x)*sqrt(a+b*sec(e+f*x))/(c+d*sec(e+f*x)),x,3,2*cot(e+f*x)*EllipticF(sqrt(a+b*sec(e+f*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(e+f*x))/(a+b))*sqrt(-b*(1+sec(e+f*x))/(a-b))/(d*f)-2*(b*c-a*d)*EllipticPi(sqrt(1-sec(e+f*x))/sqrt(2),2*d/(c+d),sqrt(2*b/(a+b)))*sqrt((a+b*sec(e+f*x))/(a+b))*tan(e+f*x)/(d*(c+d)*f*sqrt(a+b*sec(e+f*x))*sqrt(-tan(e+f*x)^2))],
[(g*sec(e+f*x))^(3/2)*sqrt(a+b*sec(e+f*x))/(c+d*sec(e+f*x)),x,7,2*b*g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2,sqrt(2*a/(a+b)))*sqrt((b+a*cos(e+f*x))/(a+b))*sqrt(g*sec(e+f*x))/(d*f*sqrt(a+b*sec(e+f*x)))-2*(b*c-a*d)*g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2*c/(c+d),sqrt(2*a/(a+b)))*sqrt((b+a*cos(e+f*x))/(a+b))*sqrt(g*sec(e+f*x))/(d*(c+d)*f*sqrt(a+b*sec(e+f*x)))],

# m<0
[sec(e+f*x)/((c+d*sec(e+f*x))*sqrt(a+b*sec(e+f*x))),x,1,2*EllipticPi(sqrt(1-sec(e+f*x))/sqrt(2),2*d/(c+d),sqrt(2*b/(a+b)))*sqrt((a+b*sec(e+f*x))/(a+b))*tan(e+f*x)/((c+d)*f*sqrt(a+b*sec(e+f*x))*sqrt(-tan(e+f*x)^2))],
[sec(e+f*x)^2/((c+d*sec(e+f*x))*sqrt(a+b*sec(e+f*x))),x,3,2*cot(e+f*x)*EllipticF(sqrt(a+b*sec(e+f*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(e+f*x))/(a+b))*sqrt(-b*(1+sec(e+f*x))/(a-b))/(b*d*f)-2*c*EllipticPi(sqrt(1-sec(e+f*x))/sqrt(2),2*d/(c+d),sqrt(2*b/(a+b)))*sqrt((a+b*sec(e+f*x))/(a+b))*tan(e+f*x)/(d*(c+d)*f*sqrt(a+b*sec(e+f*x))*sqrt(-tan(e+f*x)^2))],
[(g*sec(e+f*x))^(3/2)/((c+d*sec(e+f*x))*sqrt(a+b*sec(e+f*x))),x,3,2*g*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2*c/(c+d),sqrt(2*a/(a+b)))*sqrt((b+a*cos(e+f*x))/(a+b))*sqrt(g*sec(e+f*x))/((c+d)*f*sqrt(a+b*sec(e+f*x)))],
[(g*sec(e+f*x))^(5/2)/((c+d*sec(e+f*x))*sqrt(a+b*sec(e+f*x))),x,7,2*g^2*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2,sqrt(2*a/(a+b)))*sqrt((b+a*cos(e+f*x))/(a+b))*sqrt(g*sec(e+f*x))/(d*f*sqrt(a+b*sec(e+f*x)))-2*c*g^2*sqrt(cos(1/2*(e+f*x))^2)/cos(1/2*(e+f*x))*EllipticPi(sin(1/2*(e+f*x)),2*c/(c+d),sqrt(2*a/(a+b)))*sqrt((b+a*cos(e+f*x))/(a+b))*sqrt(g*sec(e+f*x))/(d*(c+d)*f*sqrt(a+b*sec(e+f*x)))],

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

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

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

# n>0

# n<0
[sec(e+f*x)*tan(e+f*x)^4/(c-c*sec(e+f*x))^7,x,4,1/20*cot(1/2*(e+f*x))^5/(c^7*f)-1/14*cot(1/2*(e+f*x))^7/(c^7*f)+1/36*cot(1/2*(e+f*x))^9/(c^7*f)],
[sec(e+f*x)*tan(e+f*x)^4/(c-c*sec(e+f*x))^8,x,4,1/40*cot(1/2*(e+f*x))^5/(c^8*f)-3/56*cot(1/2*(e+f*x))^7/(c^8*f)+1/24*cot(1/2*(e+f*x))^9/(c^8*f)-1/88*cot(1/2*(e+f*x))^11/(c^8*f)]]:
