# Maple integration test file: "4 Trig functions\4.5 Secant\4.5.7 (d trig)^m (a+b (c sec)^n)^p.txt"

lst:=[

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

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

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

# p>0
[(a+b*sec(e+f*x)^2)*sin(e+f*x)^7,x,3,-(a-3*b)*cos(e+f*x)/f+(a-b)*cos(e+f*x)^3/f-1/5*(3*a-b)*cos(e+f*x)^5/f+1/7*a*cos(e+f*x)^7/f+b*sec(e+f*x)/f],
[(a+b*sec(e+f*x)^2)*sin(e+f*x)^5,x,3,-(a-2*b)*cos(e+f*x)/f+1/3*(2*a-b)*cos(e+f*x)^3/f-1/5*a*cos(e+f*x)^5/f+b*sec(e+f*x)/f],
[(a+b*sec(e+f*x)^2)*sin(e+f*x)^3,x,3,-(a-b)*cos(e+f*x)/f+1/3*a*cos(e+f*x)^3/f+b*sec(e+f*x)/f],
[(a+b*sec(e+f*x)^2)*sin(e+f*x),x,3,-a*cos(e+f*x)/f+b*sec(e+f*x)/f],
[csc(e+f*x)*(a+b*sec(e+f*x)^2),x,3,-(a+b)*arctanh(cos(e+f*x))/f+b*sec(e+f*x)/f],
[csc(e+f*x)^3*(a+b*sec(e+f*x)^2),x,4,-1/2*(a+3*b)*arctanh(cos(e+f*x))/f-1/2*(a+b)*cot(e+f*x)*csc(e+f*x)/f+b*sec(e+f*x)/f],
[csc(e+f*x)^5*(a+b*sec(e+f*x)^2),x,5,-3/8*(a+5*b)*arctanh(cos(e+f*x))/f-1/8*(3*a+7*b)*cot(e+f*x)*csc(e+f*x)/f-1/4*(a+b)*cot(e+f*x)*csc(e+f*x)^3/f+b*sec(e+f*x)/f],
[(a+b*sec(e+f*x)^2)*sin(e+f*x)^6,x,6,5/16*(a-6*b)*x-1/16*(11*a-18*b)*cos(e+f*x)*sin(e+f*x)/f+1/24*(13*a-6*b)*cos(e+f*x)^3*sin(e+f*x)/f-1/6*a*cos(e+f*x)^5*sin(e+f*x)/f+b*tan(e+f*x)/f],
[(a+b*sec(e+f*x)^2)*sin(e+f*x)^4,x,5,3/8*(a-4*b)*x-1/8*(5*a-4*b)*cos(e+f*x)*sin(e+f*x)/f+1/4*a*cos(e+f*x)^3*sin(e+f*x)/f+b*tan(e+f*x)/f],
[(a+b*sec(e+f*x)^2)*sin(e+f*x)^2,x,4,1/2*(a-2*b)*x-1/2*a*cos(e+f*x)*sin(e+f*x)/f+b*tan(e+f*x)/f],
[a+b*sec(e+f*x)^2,x,3,a*x+b*tan(e+f*x)/f],
[csc(e+f*x)^2*(a+b*sec(e+f*x)^2),x,3,-(a+b)*cot(e+f*x)/f+b*tan(e+f*x)/f],
[csc(e+f*x)^4*(a+b*sec(e+f*x)^2),x,3,-(a+2*b)*cot(e+f*x)/f-1/3*(a+b)*cot(e+f*x)^3/f+b*tan(e+f*x)/f],
[csc(e+f*x)^6*(a+b*sec(e+f*x)^2),x,3,-(a+3*b)*cot(e+f*x)/f-1/3*(2*a+3*b)*cot(e+f*x)^3/f-1/5*(a+b)*cot(e+f*x)^5/f+b*tan(e+f*x)/f],
[(a+b*sec(e+f*x)^2)^2*sin(e+f*x)^5,x,3,-(a^2-4*a*b+b^2)*cos(e+f*x)/f+2/3*a*(a-b)*cos(e+f*x)^3/f-1/5*a^2*cos(e+f*x)^5/f+2*(a-b)*b*sec(e+f*x)/f+1/3*b^2*sec(e+f*x)^3/f],
[(a+b*sec(e+f*x)^2)^2*sin(e+f*x)^3,x,3,-a*(a-2*b)*cos(e+f*x)/f+1/3*a^2*cos(e+f*x)^3/f+(2*a-b)*b*sec(e+f*x)/f+1/3*b^2*sec(e+f*x)^3/f],
[(a+b*sec(e+f*x)^2)^2*sin(e+f*x),x,3,-a^2*cos(e+f*x)/f+2*a*b*sec(e+f*x)/f+1/3*b^2*sec(e+f*x)^3/f],
[csc(e+f*x)*(a+b*sec(e+f*x)^2)^2,x,4,-(a+b)^2*arctanh(cos(e+f*x))/f+b*(2*a+b)*sec(e+f*x)/f+1/3*b^2*sec(e+f*x)^3/f],
[csc(e+f*x)^3*(a+b*sec(e+f*x)^2)^2,x,5,-1/2*(a+b)*(a+5*b)*arctanh(cos(e+f*x))/f-1/6*(3*a^2+6*a*b+5*b^2)*cot(e+f*x)*csc(e+f*x)/f+1/3*b*(6*a+5*b)*sec(e+f*x)/f+1/3*b^2*csc(e+f*x)^2*sec(e+f*x)^3/f],
[csc(e+f*x)^5*(a+b*sec(e+f*x)^2)^2,x,6,-1/8*(3*a^2+30*a*b+35*b^2)*arctanh(cos(e+f*x))/f-1/24*(3*a+7*b)^2*cot(e+f*x)*csc(e+f*x)/f-1/12*(3*a^2+6*a*b+7*b^2)*cot(e+f*x)*csc(e+f*x)^3/f+1/3*b*(6*a+7*b)*sec(e+f*x)/f+1/3*b^2*csc(e+f*x)^4*sec(e+f*x)^3/f],
[(a+b*sec(e+f*x)^2)^2*sin(e+f*x)^6,x,7,5/16*(a^2-12*a*b+8*b^2)*x-1/16*(3*a^2-36*a*b+8*b^2)*cos(e+f*x)*sin(e+f*x)/f+1/24*a*(a-12*b)*cos(e+f*x)^3*sin(e+f*x)/f-1/6*(a^2-12*a*b+12*b^2)*tan(e+f*x)/f+1/6*a^2*sin(e+f*x)^6*tan(e+f*x)/f+1/3*b^2*tan(e+f*x)^3/f],
[(a+b*sec(e+f*x)^2)^2*sin(e+f*x)^4,x,6,1/8*(3*a^2-24*a*b+8*b^2)*x-1/8*a*(a-8*b)*cos(e+f*x)*sin(e+f*x)/f-1/4*(a^2-8*a*b+4*b^2)*tan(e+f*x)/f+1/4*a^2*sin(e+f*x)^4*tan(e+f*x)/f+1/3*b^2*tan(e+f*x)^3/f],
[(a+b*sec(e+f*x)^2)^2*sin(e+f*x)^2,x,5,1/2*a*(a-4*b)*x-1/2*a*(a-4*b)*tan(e+f*x)/f+1/2*a^2*sin(e+f*x)^2*tan(e+f*x)/f+1/3*b^2*tan(e+f*x)^3/f],
[(a+b*sec(e+f*x)^2)^2,x,4,a^2*x+b*(2*a+b)*tan(e+f*x)/f+1/3*b^2*tan(e+f*x)^3/f],
[csc(e+f*x)^2*(a+b*sec(e+f*x)^2)^2,x,3,-(a+b)^2*cot(e+f*x)/f+2*b*(a+b)*tan(e+f*x)/f+1/3*b^2*tan(e+f*x)^3/f],
[csc(e+f*x)^4*(a+b*sec(e+f*x)^2)^2,x,3,-(a+b)*(a+3*b)*cot(e+f*x)/f-1/3*(a+b)^2*cot(e+f*x)^3/f+b*(2*a+3*b)*tan(e+f*x)/f+1/3*b^2*tan(e+f*x)^3/f],
[csc(e+f*x)^6*(a+b*sec(e+f*x)^2)^2,x,3,-(a^2+6*a*b+6*b^2)*cot(e+f*x)/f-2/3*(a+b)*(a+2*b)*cot(e+f*x)^3/f-1/5*(a+b)^2*cot(e+f*x)^5/f+2*b*(a+2*b)*tan(e+f*x)/f+1/3*b^2*tan(e+f*x)^3/f],

# p<0
[sin(e+f*x)^5/(a+b*sec(e+f*x)^2),x,4,-(a+b)^2*cos(e+f*x)/(a^3*f)+1/3*(2*a+b)*cos(e+f*x)^3/(a^2*f)-1/5*cos(e+f*x)^5/(a*f)+(a+b)^2*arctan(cos(e+f*x)*sqrt(a)/sqrt(b))*sqrt(b)/(a^(7/2)*f)],
[sin(e+f*x)^3/(a+b*sec(e+f*x)^2),x,4,-(a+b)*cos(e+f*x)/(a^2*f)+1/3*cos(e+f*x)^3/(a*f)+(a+b)*arctan(cos(e+f*x)*sqrt(a)/sqrt(b))*sqrt(b)/(a^(5/2)*f)],
[sin(e+f*x)/(a+b*sec(e+f*x)^2),x,3,-cos(e+f*x)/(a*f)+arctan(cos(e+f*x)*sqrt(a)/sqrt(b))*sqrt(b)/(a^(3/2)*f)],
[csc(e+f*x)/(a+b*sec(e+f*x)^2),x,4,-arctanh(cos(e+f*x))/((a+b)*f)+arctan(cos(e+f*x)*sqrt(a)/sqrt(b))*sqrt(b)/((a+b)*f*sqrt(a))],
[csc(e+f*x)^3/(a+b*sec(e+f*x)^2),x,5,-1/2*(a-b)*arctanh(cos(e+f*x))/((a+b)^2*f)-1/2*cot(e+f*x)*csc(e+f*x)/((a+b)*f)+arctan(cos(e+f*x)*sqrt(a)/sqrt(b))*sqrt(a)*sqrt(b)/((a+b)^2*f)],
[csc(e+f*x)^5/(a+b*sec(e+f*x)^2),x,6,-1/8*(3*a^2-6*a*b-b^2)*arctanh(cos(e+f*x))/((a+b)^3*f)-1/8*(3*a-b)*cot(e+f*x)*csc(e+f*x)/((a+b)^2*f)-1/4*cot(e+f*x)*csc(e+f*x)^3/((a+b)*f)+a^(3/2)*arctan(cos(e+f*x)*sqrt(a)/sqrt(b))*sqrt(b)/((a+b)^3*f)],
[sin(e+f*x)^6/(a+b*sec(e+f*x)^2),x,7,1/16*(5*a^3+30*a^2*b+40*a*b^2+16*b^3)*x/a^4-1/16*(11*a^2+18*a*b+8*b^2)*cos(e+f*x)*sin(e+f*x)/(a^3*f)+1/8*(3*a+2*b)*cos(e+f*x)^3*sin(e+f*x)/(a^2*f)+1/6*cos(e+f*x)^3*sin(e+f*x)^3/(a*f)-(a+b)^(5/2)*arctan(sqrt(b)*tan(e+f*x)/sqrt(a+b))*sqrt(b)/(a^4*f)],
[sin(e+f*x)^4/(a+b*sec(e+f*x)^2),x,6,1/8*(3*a^2+12*a*b+8*b^2)*x/a^3-1/8*(5*a+4*b)*cos(e+f*x)*sin(e+f*x)/(a^2*f)+1/4*cos(e+f*x)^3*sin(e+f*x)/(a*f)-(a+b)^(3/2)*arctan(sqrt(b)*tan(e+f*x)/sqrt(a+b))*sqrt(b)/(a^3*f)],
[sin(e+f*x)^2/(a+b*sec(e+f*x)^2),x,5,1/2*(a+2*b)*x/a^2-1/2*cos(e+f*x)*sin(e+f*x)/(a*f)-arctan(sqrt(b)*tan(e+f*x)/sqrt(a+b))*sqrt(b)*sqrt(a+b)/(a^2*f)],
[1/(a+b*sec(e+f*x)^2),x,3,x/a+arctan(cot(e+f*x)*sqrt(a+b)/sqrt(b))*sqrt(b)/(a*f*sqrt(a+b))],
[csc(e+f*x)^2/(a+b*sec(e+f*x)^2),x,3,-cot(e+f*x)/((a+b)*f)-arctan(sqrt(b)*tan(e+f*x)/sqrt(a+b))*sqrt(b)/((a+b)^(3/2)*f)],
[csc(e+f*x)^4/(a+b*sec(e+f*x)^2),x,4,-a*cot(e+f*x)/((a+b)^2*f)-1/3*cot(e+f*x)^3/((a+b)*f)-a*arctan(sqrt(b)*tan(e+f*x)/sqrt(a+b))*sqrt(b)/((a+b)^(5/2)*f)],
[csc(e+f*x)^6/(a+b*sec(e+f*x)^2),x,4,-a^2*cot(e+f*x)/((a+b)^3*f)-1/3*(2*a+b)*cot(e+f*x)^3/((a+b)^2*f)-1/5*cot(e+f*x)^5/((a+b)*f)-a^2*arctan(sqrt(b)*tan(e+f*x)/sqrt(a+b))*sqrt(b)/((a+b)^(7/2)*f)],
[sin(e+f*x)^5/(a+b*sec(e+f*x)^2)^2,x,6,-1/2*(a+b)*(3*a+7*b)*cos(e+f*x)/(a^4*f)+1/6*(a+b)*(3*a+7*b)*cos(e+f*x)^3/(a^3*b*f)-1/5*cos(e+f*x)^5/(a^2*f)-1/2*(a+b)^2*cos(e+f*x)^5/(a^2*b*f*(b+a*cos(e+f*x)^2))+1/2*(a+b)*(3*a+7*b)*arctan(cos(e+f*x)*sqrt(a)/sqrt(b))*sqrt(b)/(a^(9/2)*f)],
[sin(e+f*x)^3/(a+b*sec(e+f*x)^2)^2,x,5,-(a+2*b)*cos(e+f*x)/(a^3*f)+1/3*cos(e+f*x)^3/(a^2*f)-1/2*b*(a+b)*cos(e+f*x)/(a^3*f*(b+a*cos(e+f*x)^2))+1/2*(3*a+5*b)*arctan(cos(e+f*x)*sqrt(a)/sqrt(b))*sqrt(b)/(a^(7/2)*f)],
[sin(e+f*x)/(a+b*sec(e+f*x)^2)^2,x,4,-3/2*cos(e+f*x)/(a^2*f)+1/2*cos(e+f*x)^3/(a*f*(b+a*cos(e+f*x)^2))+3/2*arctan(cos(e+f*x)*sqrt(a)/sqrt(b))*sqrt(b)/(a^(5/2)*f)],
[csc(e+f*x)/(a+b*sec(e+f*x)^2)^2,x,5,-arctanh(cos(e+f*x))/((a+b)^2*f)-1/2*b*cos(e+f*x)/(a*(a+b)*f*(b+a*cos(e+f*x)^2))+1/2*(3*a+b)*arctan(cos(e+f*x)*sqrt(a)/sqrt(b))*sqrt(b)/(a^(3/2)*(a+b)^2*f)],
[csc(e+f*x)^3/(a+b*sec(e+f*x)^2)^2,x,6,-1/2*(a-3*b)*arctanh(cos(e+f*x))/((a+b)^3*f)+1/2*(a-b)*cos(e+f*x)/((a+b)^2*f*(b+a*cos(e+f*x)^2))-1/2*cot(e+f*x)*csc(e+f*x)/((a+b)*f*(b+a*cos(e+f*x)^2))+1/2*(3*a-b)*arctan(cos(e+f*x)*sqrt(a)/sqrt(b))*sqrt(b)/((a+b)^3*f*sqrt(a))],
[csc(e+f*x)^5/(a+b*sec(e+f*x)^2)^2,x,7,-3/8*(a^2-6*a*b+b^2)*arctanh(cos(e+f*x))/((a+b)^4*f)+3/8*a*(a-3*b)*cos(e+f*x)/((a+b)^3*f*(b+a*cos(e+f*x)^2))-1/8*(a-5*b)*cot(e+f*x)*csc(e+f*x)/((a+b)^2*f*(b+a*cos(e+f*x)^2))-1/4*cot(e+f*x)*csc(e+f*x)^3/((a+b)*f*(b+a*cos(e+f*x)^2))+3/2*(a-b)*arctan(cos(e+f*x)*sqrt(a)/sqrt(b))*sqrt(a)*sqrt(b)/((a+b)^4*f)],
[sin(e+f*x)^6/(a+b*sec(e+f*x)^2)^2,x,8,1/16*(5*a^3+60*a^2*b+120*a*b^2+64*b^3)*x/a^5-1/2*(a+b)^(3/2)*(3*a+8*b)*arctan(sqrt(b)*tan(e+f*x)/sqrt(a+b))*sqrt(b)/(a^5*f)-1/48*(33*a^2+82*a*b+48*b^2)*cos(e+f*x)*sin(e+f*x)/(a^3*f*(a+b+b*tan(e+f*x)^2))+1/24*(9*a+8*b)*cos(e+f*x)^3*sin(e+f*x)/(a^2*f*(a+b+b*tan(e+f*x)^2))+1/6*cos(e+f*x)^3*sin(e+f*x)^3/(a*f*(a+b+b*tan(e+f*x)^2))-1/16*b*(19*a^2+52*a*b+32*b^2)*tan(e+f*x)/(a^4*f*(a+b+b*tan(e+f*x)^2))],
[sin(e+f*x)^4/(a+b*sec(e+f*x)^2)^2,x,7,3/8*(a^2+8*a*b+8*b^2)*x/a^4-3/2*(a+2*b)*arctan(sqrt(b)*tan(e+f*x)/sqrt(a+b))*sqrt(b)*sqrt(a+b)/(a^4*f)-1/8*(5*a+6*b)*cos(e+f*x)*sin(e+f*x)/(a^2*f*(a+b+b*tan(e+f*x)^2))+1/4*cos(e+f*x)^3*sin(e+f*x)/(a*f*(a+b+b*tan(e+f*x)^2))-3/8*b*(3*a+4*b)*tan(e+f*x)/(a^3*f*(a+b+b*tan(e+f*x)^2))],
[sin(e+f*x)^2/(a+b*sec(e+f*x)^2)^2,x,6,1/2*(a+4*b)*x/a^3-1/2*(3*a+4*b)*arctan(sqrt(b)*tan(e+f*x)/sqrt(a+b))*sqrt(b)/(a^3*f*sqrt(a+b))-1/2*cos(e+f*x)*sin(e+f*x)/(a*f*(a+b+b*tan(e+f*x)^2))-b*tan(e+f*x)/(a^2*f*(a+b+b*tan(e+f*x)^2))],
[1/(a+b*sec(e+f*x)^2)^2,x,5,x/a^2-1/2*(3*a+2*b)*arctan(sqrt(b)*tan(e+f*x)/sqrt(a+b))*sqrt(b)/(a^2*(a+b)^(3/2)*f)-1/2*b*tan(e+f*x)/(a*(a+b)*f*(a+b+b*tan(e+f*x)^2))],
[csc(e+f*x)^2/(a+b*sec(e+f*x)^2)^2,x,4,-3/2*cot(e+f*x)/((a+b)^2*f)-3/2*arctan(sqrt(b)*tan(e+f*x)/sqrt(a+b))*sqrt(b)/((a+b)^(5/2)*f)+1/2*cot(e+f*x)/((a+b)*f*(a+b+b*tan(e+f*x)^2))],
[csc(e+f*x)^4/(a+b*sec(e+f*x)^2)^2,x,5,-(a-b)*cot(e+f*x)/((a+b)^3*f)-1/3*cot(e+f*x)^3/((a+b)^2*f)-1/2*(3*a-2*b)*arctan(sqrt(b)*tan(e+f*x)/sqrt(a+b))*sqrt(b)/((a+b)^(7/2)*f)-1/2*a*b*tan(e+f*x)/((a+b)^3*f*(a+b+b*tan(e+f*x)^2))],
[csc(e+f*x)^6/(a+b*sec(e+f*x)^2)^2,x,6,-1/5*(5*a^2-10*a*b-b^2)*cot(e+f*x)/((a+b)^4*f)-1/15*(10*a+3*b)*cot(e+f*x)^3/((a+b)^3*f)-1/2*a*(3*a-4*b)*arctan(sqrt(b)*tan(e+f*x)/sqrt(a+b))*sqrt(b)/((a+b)^(9/2)*f)-1/5*cot(e+f*x)^5/((a+b)*f*(a+b+b*tan(e+f*x)^2))-1/10*b*(5*a^2+2*b^2)*tan(e+f*x)/((a+b)^4*f*(a+b+b*tan(e+f*x)^2))],
[sin(e+f*x)^5/(a+b*sec(e+f*x)^2)^3,x,6,-1/2*(3*a^2+14*a*b+13*b^2)*cos(e+f*x)/(a^5*f)+1/12*(a+3*b)*(3*a+5*b)*cos(e+f*x)^3/(a^4*b*f)-1/5*cos(e+f*x)^5/(a^3*f)-1/4*(a+b)^2*cos(e+f*x)^7/(a^2*b*f*(b+a*cos(e+f*x)^2)^2)-1/8*b*(a+b)*(3*a+11*b)*cos(e+f*x)/(a^5*f*(b+a*cos(e+f*x)^2))+1/8*(15*a^2+70*a*b+63*b^2)*arctan(cos(e+f*x)*sqrt(a)/sqrt(b))*sqrt(b)/(a^(11/2)*f)],
[sin(e+f*x)^3/(a+b*sec(e+f*x)^2)^3,x,6,-(a+3*b)*cos(e+f*x)/(a^4*f)+1/3*cos(e+f*x)^3/(a^3*f)+1/4*b^2*(a+b)*cos(e+f*x)/(a^4*f*(b+a*cos(e+f*x)^2)^2)-1/8*b*(9*a+13*b)*cos(e+f*x)/(a^4*f*(b+a*cos(e+f*x)^2))+5/8*(3*a+7*b)*arctan(cos(e+f*x)*sqrt(a)/sqrt(b))*sqrt(b)/(a^(9/2)*f)],
[sin(e+f*x)/(a+b*sec(e+f*x)^2)^3,x,5,-15/8*cos(e+f*x)/(a^3*f)+1/4*cos(e+f*x)^5/(a*f*(b+a*cos(e+f*x)^2)^2)+5/8*cos(e+f*x)^3/(a^2*f*(b+a*cos(e+f*x)^2))+15/8*arctan(cos(e+f*x)*sqrt(a)/sqrt(b))*sqrt(b)/(a^(7/2)*f)],
[csc(e+f*x)/(a+b*sec(e+f*x)^2)^3,x,6,-arctanh(cos(e+f*x))/((a+b)^3*f)-1/4*b*cos(e+f*x)^3/(a*(a+b)*f*(b+a*cos(e+f*x)^2)^2)-1/8*b*(7*a+3*b)*cos(e+f*x)/(a^2*(a+b)^2*f*(b+a*cos(e+f*x)^2))+1/8*(15*a^2+10*a*b+3*b^2)*arctan(cos(e+f*x)*sqrt(a)/sqrt(b))*sqrt(b)/(a^(5/2)*(a+b)^3*f)],
[csc(e+f*x)^3/(a+b*sec(e+f*x)^2)^3,x,7,-1/2*(a-5*b)*arctanh(cos(e+f*x))/((a+b)^4*f)-1/4*(2*a-b)*b*cos(e+f*x)/(a*(a+b)^2*f*(b+a*cos(e+f*x)^2)^2)+1/8*(4*a^2-9*a*b-b^2)*cos(e+f*x)/(a*(a+b)^3*f*(b+a*cos(e+f*x)^2))-1/2*cos(e+f*x)*cot(e+f*x)^2/((a+b)*f*(b+a*cos(e+f*x)^2)^2)+1/8*(15*a^2-10*a*b-b^2)*arctan(cos(e+f*x)*sqrt(a)/sqrt(b))*sqrt(b)/(a^(3/2)*(a+b)^4*f)],
[csc(e+f*x)^5/(a+b*sec(e+f*x)^2)^3,x,8,-3/8*(a^2-10*a*b+5*b^2)*arctanh(cos(e+f*x))/((a+b)^5*f)+1/8*(a^2-9*a*b+2*b^2)*cos(e+f*x)/((a+b)^3*f*(b+a*cos(e+f*x)^2)^2)+3/8*(a^2-6*a*b+b^2)*cos(e+f*x)/((a+b)^4*f*(b+a*cos(e+f*x)^2))-1/8*(a-7*b)*cot(e+f*x)*csc(e+f*x)/((a+b)^2*f*(b+a*cos(e+f*x)^2)^2)-1/4*cot(e+f*x)^3*csc(e+f*x)/((a+b)*f*(b+a*cos(e+f*x)^2)^2)+3/8*(5*a^2-10*a*b+b^2)*arctan(cos(e+f*x)*sqrt(a)/sqrt(b))*sqrt(b)/((a+b)^5*f*sqrt(a))],
[sin(e+f*x)^6/(a+b*sec(e+f*x)^2)^3,x,9,5/16*(a+2*b)*(a^2+16*a*b+16*b^2)*x/a^6-5/8*(a+4*b)*(3*a+4*b)*arctan(sqrt(b)*tan(e+f*x)/sqrt(a+b))*sqrt(b)*sqrt(a+b)/(a^6*f)-1/48*(33*a^2+110*a*b+80*b^2)*cos(e+f*x)*sin(e+f*x)/(a^3*f*(a+b+b*tan(e+f*x)^2)^2)+1/24*(9*a+10*b)*cos(e+f*x)^3*sin(e+f*x)/(a^2*f*(a+b+b*tan(e+f*x)^2)^2)+1/6*cos(e+f*x)^3*sin(e+f*x)^3/(a*f*(a+b+b*tan(e+f*x)^2)^2)-5/48*b*(9*a^2+32*a*b+24*b^2)*tan(e+f*x)/(a^4*f*(a+b+b*tan(e+f*x)^2)^2)-5/16*b*(5*a^2+20*a*b+16*b^2)*tan(e+f*x)/(a^5*f*(a+b+b*tan(e+f*x)^2))],
[sin(e+f*x)^4/(a+b*sec(e+f*x)^2)^3,x,8,3/8*(a^2+12*a*b+16*b^2)*x/a^5-3/8*(5*a^2+20*a*b+16*b^2)*arctan(sqrt(b)*tan(e+f*x)/sqrt(a+b))*sqrt(b)/(a^5*f*sqrt(a+b))-1/8*(5*a+8*b)*cos(e+f*x)*sin(e+f*x)/(a^2*f*(a+b+b*tan(e+f*x)^2)^2)+1/4*cos(e+f*x)^3*sin(e+f*x)/(a*f*(a+b+b*tan(e+f*x)^2)^2)-1/8*b*(7*a+12*b)*tan(e+f*x)/(a^3*f*(a+b+b*tan(e+f*x)^2)^2)-3/2*b*(a+2*b)*tan(e+f*x)/(a^4*f*(a+b+b*tan(e+f*x)^2))],
[sin(e+f*x)^2/(a+b*sec(e+f*x)^2)^3,x,7,1/2*(a+6*b)*x/a^4-1/8*(15*a^2+40*a*b+24*b^2)*arctan(sqrt(b)*tan(e+f*x)/sqrt(a+b))*sqrt(b)/(a^4*(a+b)^(3/2)*f)-1/2*cos(e+f*x)*sin(e+f*x)/(a*f*(a+b+b*tan(e+f*x)^2)^2)-3/4*b*tan(e+f*x)/(a^2*f*(a+b+b*tan(e+f*x)^2)^2)-1/8*b*(11*a+12*b)*tan(e+f*x)/(a^3*(a+b)*f*(a+b+b*tan(e+f*x)^2))],
[1/(a+b*sec(e+f*x)^2)^3,x,6,x/a^3-1/8*(15*a^2+20*a*b+8*b^2)*arctan(sqrt(b)*tan(e+f*x)/sqrt(a+b))*sqrt(b)/(a^3*(a+b)^(5/2)*f)-1/4*b*tan(e+f*x)/(a*(a+b)*f*(a+b+b*tan(e+f*x)^2)^2)-1/8*b*(7*a+4*b)*tan(e+f*x)/(a^2*(a+b)^2*f*(a+b+b*tan(e+f*x)^2))],
[csc(e+f*x)^2/(a+b*sec(e+f*x)^2)^3,x,5,-15/8*cot(e+f*x)/((a+b)^3*f)-15/8*arctan(sqrt(b)*tan(e+f*x)/sqrt(a+b))*sqrt(b)/((a+b)^(7/2)*f)+1/4*cot(e+f*x)/((a+b)*f*(a+b+b*tan(e+f*x)^2)^2)+5/8*cot(e+f*x)/((a+b)^2*f*(a+b+b*tan(e+f*x)^2))],
[csc(e+f*x)^4/(a+b*sec(e+f*x)^2)^3,x,6,-(a-2*b)*cot(e+f*x)/((a+b)^4*f)-1/3*cot(e+f*x)^3/((a+b)^3*f)-5/8*(3*a-4*b)*arctan(sqrt(b)*tan(e+f*x)/sqrt(a+b))*sqrt(b)/((a+b)^(9/2)*f)-1/4*a*b*tan(e+f*x)/((a+b)^3*f*(a+b+b*tan(e+f*x)^2)^2)-1/8*(7*a-4*b)*b*tan(e+f*x)/((a+b)^4*f*(a+b+b*tan(e+f*x)^2))],
[csc(e+f*x)^6/(a+b*sec(e+f*x)^2)^3,x,7,-1/5*(5*a^2-20*a*b+2*b^2)*cot(e+f*x)/((a+b)^5*f)-1/15*(10*a+b)*cot(e+f*x)^3/((a+b)^4*f)-1/8*(15*a^2-40*a*b+8*b^2)*arctan(sqrt(b)*tan(e+f*x)/sqrt(a+b))*sqrt(b)/((a+b)^(11/2)*f)-1/5*cot(e+f*x)^5/((a+b)*f*(a+b+b*tan(e+f*x)^2)^2)-1/20*b*(5*a^2+4*b^2)*tan(e+f*x)/((a+b)^4*f*(a+b+b*tan(e+f*x)^2)^2)-1/40*b*(35*a^2-40*a*b+24*b^2)*tan(e+f*x)/((a+b)^5*f*(a+b+b*tan(e+f*x)^2))],

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

# p>0
[sin(e+f*x)^5*sqrt(a+b*sec(e+f*x)^2),x,6,2/15*(5*a+b)*cos(e+f*x)^3*(a+b*sec(e+f*x)^2)^(3/2)/(a^2*f)-1/5*cos(e+f*x)^5*(a+b*sec(e+f*x)^2)^(3/2)/(a*f)+arctanh(sec(e+f*x)*sqrt(b)/sqrt(a+b*sec(e+f*x)^2))*sqrt(b)/f-cos(e+f*x)*sqrt(a+b*sec(e+f*x)^2)/f],
[sin(e+f*x)^3*sqrt(a+b*sec(e+f*x)^2),x,5,1/3*cos(e+f*x)^3*(a+b*sec(e+f*x)^2)^(3/2)/(a*f)+arctanh(sec(e+f*x)*sqrt(b)/sqrt(a+b*sec(e+f*x)^2))*sqrt(b)/f-cos(e+f*x)*sqrt(a+b*sec(e+f*x)^2)/f],
[sin(e+f*x)*sqrt(a+b*sec(e+f*x)^2),x,4,arctanh(sec(e+f*x)*sqrt(b)/sqrt(a+b*sec(e+f*x)^2))*sqrt(b)/f-cos(e+f*x)*sqrt(a+b*sec(e+f*x)^2)/f],
[csc(e+f*x)*sqrt(a+b*sec(e+f*x)^2),x,6,arctanh(sec(e+f*x)*sqrt(b)/sqrt(a+b*sec(e+f*x)^2))*sqrt(b)/f-arctanh(sec(e+f*x)*sqrt(a+b)/sqrt(a+b*sec(e+f*x)^2))*sqrt(a+b)/f],
[csc(e+f*x)^3*sqrt(a+b*sec(e+f*x)^2),x,7,arctanh(sec(e+f*x)*sqrt(b)/sqrt(a+b*sec(e+f*x)^2))*sqrt(b)/f-1/2*(a+2*b)*arctanh(sec(e+f*x)*sqrt(a+b)/sqrt(a+b*sec(e+f*x)^2))/(f*sqrt(a+b))-1/2*cot(e+f*x)*csc(e+f*x)*sqrt(a+b*sec(e+f*x)^2)/f],
[csc(e+f*x)^5*sqrt(a+b*sec(e+f*x)^2),x,8,-1/8*(3*a^2+12*a*b+8*b^2)*arctanh(sec(e+f*x)*sqrt(a+b)/sqrt(a+b*sec(e+f*x)^2))/((a+b)^(3/2)*f)+arctanh(sec(e+f*x)*sqrt(b)/sqrt(a+b*sec(e+f*x)^2))*sqrt(b)/f-1/8*(3*a+4*b)*cot(e+f*x)*csc(e+f*x)*sqrt(a+b*sec(e+f*x)^2)/((a+b)*f)-1/4*cot(e+f*x)*csc(e+f*x)^3*sqrt(a+b*sec(e+f*x)^2)/f],
[sin(e+f*x)^6*sqrt(a+b*sec(e+f*x)^2),x,9,1/16*(5*a^3-15*a^2*b-5*a*b^2-b^3)*arctan(sqrt(a)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/(a^(5/2)*f)+arctanh(sqrt(b)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))*sqrt(b)/f-1/16*(a-b)*(5*a+b)*cos(e+f*x)*sin(e+f*x)*sqrt(a+b+b*tan(e+f*x)^2)/(a^2*f)-1/24*(5*a-b)*cos(e+f*x)*sin(e+f*x)^3*sqrt(a+b+b*tan(e+f*x)^2)/(a*f)-1/6*cos(e+f*x)*sin(e+f*x)^5*sqrt(a+b+b*tan(e+f*x)^2)/f],
[sin(e+f*x)^4*sqrt(a+b*sec(e+f*x)^2),x,8,1/8*(3*a^2-6*a*b-b^2)*arctan(sqrt(a)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/(a^(3/2)*f)+arctanh(sqrt(b)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))*sqrt(b)/f-1/8*(3*a-b)*cos(e+f*x)*sin(e+f*x)*sqrt(a+b+b*tan(e+f*x)^2)/(a*f)-1/4*cos(e+f*x)*sin(e+f*x)^3*sqrt(a+b+b*tan(e+f*x)^2)/f],
[sin(e+f*x)^2*sqrt(a+b*sec(e+f*x)^2),x,7,1/2*(a-b)*arctan(sqrt(a)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/(f*sqrt(a))+arctanh(sqrt(b)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))*sqrt(b)/f-1/2*cos(e+f*x)*sin(e+f*x)*sqrt(a+b+b*tan(e+f*x)^2)/f],
[sqrt(a+b*sec(e+f*x)^2),x,6,arctan(sqrt(a)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))*sqrt(a)/f+arctanh(sqrt(b)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))*sqrt(b)/f],
[csc(e+f*x)^2*sqrt(a+b*sec(e+f*x)^2),x,4,arctanh(sqrt(b)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))*sqrt(b)/f-cot(e+f*x)*sqrt(a+b+b*tan(e+f*x)^2)/f],
[csc(e+f*x)^4*sqrt(a+b*sec(e+f*x)^2),x,5,arctanh(sqrt(b)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))*sqrt(b)/f-cot(e+f*x)*sqrt(a+b+b*tan(e+f*x)^2)/f-1/3*cot(e+f*x)^3*(a+b+b*tan(e+f*x)^2)^(3/2)/((a+b)*f)],
[csc(e+f*x)^6*sqrt(a+b*sec(e+f*x)^2),x,6,arctanh(sqrt(b)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))*sqrt(b)/f-cot(e+f*x)*sqrt(a+b+b*tan(e+f*x)^2)/f-2/15*(5*a+4*b)*cot(e+f*x)^3*(a+b+b*tan(e+f*x)^2)^(3/2)/((a+b)^2*f)-1/5*cot(e+f*x)^5*(a+b+b*tan(e+f*x)^2)^(3/2)/((a+b)*f)],
[(a+b*sec(e+f*x)^2)^(3/2)*sin(e+f*x)^5,x,7,-1/3*(3*a-4*b)*cos(e+f*x)*(a+b*sec(e+f*x)^2)^(3/2)/(a*f)+2/3*cos(e+f*x)^3*(a+b*sec(e+f*x)^2)^(5/2)/(a*f)-1/5*cos(e+f*x)^5*(a+b*sec(e+f*x)^2)^(5/2)/(a*f)+1/2*(3*a-4*b)*arctanh(sec(e+f*x)*sqrt(b)/sqrt(a+b*sec(e+f*x)^2))*sqrt(b)/f+1/2*(3*a-4*b)*b*sec(e+f*x)*sqrt(a+b*sec(e+f*x)^2)/(a*f)],
[(a+b*sec(e+f*x)^2)^(3/2)*sin(e+f*x)^3,x,6,-1/3*(3*a-2*b)*cos(e+f*x)*(a+b*sec(e+f*x)^2)^(3/2)/(a*f)+1/3*cos(e+f*x)^3*(a+b*sec(e+f*x)^2)^(5/2)/(a*f)+1/2*(3*a-2*b)*arctanh(sec(e+f*x)*sqrt(b)/sqrt(a+b*sec(e+f*x)^2))*sqrt(b)/f+1/2*(3*a-2*b)*b*sec(e+f*x)*sqrt(a+b*sec(e+f*x)^2)/(a*f)],
[(a+b*sec(e+f*x)^2)^(3/2)*sin(e+f*x),x,5,-cos(e+f*x)*(a+b*sec(e+f*x)^2)^(3/2)/f+3/2*a*arctanh(sec(e+f*x)*sqrt(b)/sqrt(a+b*sec(e+f*x)^2))*sqrt(b)/f+3/2*b*sec(e+f*x)*sqrt(a+b*sec(e+f*x)^2)/f],
[csc(e+f*x)*(a+b*sec(e+f*x)^2)^(3/2),x,7,-(a+b)^(3/2)*arctanh(sec(e+f*x)*sqrt(a+b)/sqrt(a+b*sec(e+f*x)^2))/f+1/2*(3*a+2*b)*arctanh(sec(e+f*x)*sqrt(b)/sqrt(a+b*sec(e+f*x)^2))*sqrt(b)/f+1/2*b*sec(e+f*x)*sqrt(a+b*sec(e+f*x)^2)/f],
[csc(e+f*x)^3*(a+b*sec(e+f*x)^2)^(3/2),x,8,-1/2*cot(e+f*x)*csc(e+f*x)*(a+b*sec(e+f*x)^2)^(3/2)/f+1/2*(3*a+4*b)*arctanh(sec(e+f*x)*sqrt(b)/sqrt(a+b*sec(e+f*x)^2))*sqrt(b)/f-1/2*(a+4*b)*arctanh(sec(e+f*x)*sqrt(a+b)/sqrt(a+b*sec(e+f*x)^2))*sqrt(a+b)/f+b*sec(e+f*x)*sqrt(a+b*sec(e+f*x)^2)/f],
[csc(e+f*x)^5*(a+b*sec(e+f*x)^2)^(3/2),x,9,-1/4*cot(e+f*x)*csc(e+f*x)^3*(a+b*sec(e+f*x)^2)^(3/2)/f+3/2*(a+2*b)*arctanh(sec(e+f*x)*sqrt(b)/sqrt(a+b*sec(e+f*x)^2))*sqrt(b)/f-3/8*(a^2+8*a*b+8*b^2)*arctanh(sec(e+f*x)*sqrt(a+b)/sqrt(a+b*sec(e+f*x)^2))/(f*sqrt(a+b))+3/8*(a+4*b)*sec(e+f*x)*sqrt(a+b*sec(e+f*x)^2)/f-3/8*(a+2*b)*csc(e+f*x)^2*sec(e+f*x)*sqrt(a+b*sec(e+f*x)^2)/f],
[(a+b*sec(e+f*x)^2)^(3/2)*sin(e+f*x)^6,x,10,1/16*(5*a^3-45*a^2*b+15*a*b^2+b^3)*arctan(sqrt(a)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/(a^(3/2)*f)+1/2*(3*a-5*b)*arctanh(sqrt(b)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))*sqrt(b)/f-1/16*(5*a^2-26*a*b+b^2)*sqrt(a+b+b*tan(e+f*x)^2)*tan(e+f*x)/(a*f)+1/48*(5*a^2-40*a*b+3*b^2)*sin(e+f*x)^2*sqrt(a+b+b*tan(e+f*x)^2)*tan(e+f*x)/(a*f)+1/24*(5*a-3*b)*sin(e+f*x)^4*sqrt(a+b+b*tan(e+f*x)^2)*tan(e+f*x)/f-1/6*cos(e+f*x)*sin(e+f*x)^5*(a+b+b*tan(e+f*x)^2)^(3/2)/f],
[(a+b*sec(e+f*x)^2)^(3/2)*sin(e+f*x)^4,x,9,3/8*(a^2-6*a*b+b^2)*arctan(sqrt(a)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/(f*sqrt(a))+3/2*(a-b)*arctanh(sqrt(b)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))*sqrt(b)/f-3/8*(a-3*b)*sqrt(a+b+b*tan(e+f*x)^2)*tan(e+f*x)/f+3/8*(a-b)*sin(e+f*x)^2*sqrt(a+b+b*tan(e+f*x)^2)*tan(e+f*x)/f-1/4*cos(e+f*x)*sin(e+f*x)^3*(a+b+b*tan(e+f*x)^2)^(3/2)/f],
[(a+b*sec(e+f*x)^2)^(3/2)*sin(e+f*x)^2,x,8,1/2*(a-3*b)*arctan(sqrt(a)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))*sqrt(a)/f+1/2*(3*a-b)*arctanh(sqrt(b)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))*sqrt(b)/f+b*sqrt(a+b+b*tan(e+f*x)^2)*tan(e+f*x)/f-1/2*cos(e+f*x)*sin(e+f*x)*(a+b+b*tan(e+f*x)^2)^(3/2)/f],
[(a+b*sec(e+f*x)^2)^(3/2),x,7,a^(3/2)*arctan(sqrt(a)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/f+1/2*(3*a+b)*arctanh(sqrt(b)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))*sqrt(b)/f+1/2*b*sqrt(a+b+b*tan(e+f*x)^2)*tan(e+f*x)/f],
[csc(e+f*x)^2*(a+b*sec(e+f*x)^2)^(3/2),x,5,3/2*(a+b)*arctanh(sqrt(b)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))*sqrt(b)/f+3/2*b*sqrt(a+b+b*tan(e+f*x)^2)*tan(e+f*x)/f-cot(e+f*x)*(a+b+b*tan(e+f*x)^2)^(3/2)/f],
[csc(e+f*x)^4*(a+b*sec(e+f*x)^2)^(3/2),x,6,1/2*(3*a+5*b)*arctanh(sqrt(b)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))*sqrt(b)/f+1/2*b*(3*a+5*b)*sqrt(a+b+b*tan(e+f*x)^2)*tan(e+f*x)/((a+b)*f)-1/3*(3*a+5*b)*cot(e+f*x)*(a+b+b*tan(e+f*x)^2)^(3/2)/((a+b)*f)-1/3*cot(e+f*x)^3*(a+b+b*tan(e+f*x)^2)^(5/2)/((a+b)*f)],
[csc(e+f*x)^6*(a+b*sec(e+f*x)^2)^(3/2),x,7,1/2*(3*a+7*b)*arctanh(sqrt(b)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))*sqrt(b)/f+1/2*b*(3*a+7*b)*sqrt(a+b+b*tan(e+f*x)^2)*tan(e+f*x)/((a+b)*f)-1/3*(3*a+7*b)*cot(e+f*x)*(a+b+b*tan(e+f*x)^2)^(3/2)/((a+b)*f)-2/3*cot(e+f*x)^3*(a+b+b*tan(e+f*x)^2)^(5/2)/((a+b)*f)-1/5*cot(e+f*x)^5*(a+b+b*tan(e+f*x)^2)^(5/2)/((a+b)*f)],

# p<0
[sin(e+f*x)^5/sqrt(a+b*sec(e+f*x)^2),x,4,-1/15*(15*a^2+20*a*b+8*b^2)*cos(e+f*x)*sqrt(a+b*sec(e+f*x)^2)/(a^3*f)+2/15*(5*a+2*b)*cos(e+f*x)^3*sqrt(a+b*sec(e+f*x)^2)/(a^2*f)-1/5*cos(e+f*x)^5*sqrt(a+b*sec(e+f*x)^2)/(a*f)],
[sin(e+f*x)^3/sqrt(a+b*sec(e+f*x)^2),x,3,-1/3*(3*a+2*b)*cos(e+f*x)*sqrt(a+b*sec(e+f*x)^2)/(a^2*f)+1/3*cos(e+f*x)^3*sqrt(a+b*sec(e+f*x)^2)/(a*f)],
[sin(e+f*x)/sqrt(a+b*sec(e+f*x)^2),x,2,-cos(e+f*x)*sqrt(a+b*sec(e+f*x)^2)/(a*f)],
[csc(e+f*x)/sqrt(a+b*sec(e+f*x)^2),x,3,-arctanh(sec(e+f*x)*sqrt(a+b)/sqrt(a+b*sec(e+f*x)^2))/(f*sqrt(a+b))],
[csc(e+f*x)^3/sqrt(a+b*sec(e+f*x)^2),x,5,-1/2*a*arctanh(sec(e+f*x)*sqrt(a+b)/sqrt(a+b*sec(e+f*x)^2))/((a+b)^(3/2)*f)-1/2*cot(e+f*x)*csc(e+f*x)*sqrt(a+b*sec(e+f*x)^2)/((a+b)*f)],
[csc(e+f*x)^5/sqrt(a+b*sec(e+f*x)^2),x,6,-3/8*a^2*arctanh(sec(e+f*x)*sqrt(a+b)/sqrt(a+b*sec(e+f*x)^2))/((a+b)^(5/2)*f)-1/8*(5*a+2*b)*cot(e+f*x)*csc(e+f*x)*sqrt(a+b*sec(e+f*x)^2)/((a+b)^2*f)-1/4*cot(e+f*x)^3*csc(e+f*x)*sqrt(a+b*sec(e+f*x)^2)/((a+b)*f)],
[sin(e+f*x)^6/sqrt(a+b*sec(e+f*x)^2),x,7,5/16*(a+b)^3*arctan(sqrt(a)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/(a^(7/2)*f)-1/48*(33*a^2+40*a*b+15*b^2)*cos(e+f*x)*sin(e+f*x)*sqrt(a+b+b*tan(e+f*x)^2)/(a^3*f)+1/24*(9*a+5*b)*cos(e+f*x)^3*sin(e+f*x)*sqrt(a+b+b*tan(e+f*x)^2)/(a^2*f)+1/6*cos(e+f*x)^3*sin(e+f*x)^3*sqrt(a+b+b*tan(e+f*x)^2)/(a*f)],
[sin(e+f*x)^4/sqrt(a+b*sec(e+f*x)^2),x,6,3/8*(a+b)^2*arctan(sqrt(a)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/(a^(5/2)*f)-1/8*(5*a+3*b)*cos(e+f*x)*sin(e+f*x)*sqrt(a+b+b*tan(e+f*x)^2)/(a^2*f)+1/4*cos(e+f*x)^3*sin(e+f*x)*sqrt(a+b+b*tan(e+f*x)^2)/(a*f)],
[sin(e+f*x)^2/sqrt(a+b*sec(e+f*x)^2),x,5,1/2*(a+b)*arctan(sqrt(a)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/(a^(3/2)*f)-1/2*cos(e+f*x)*sin(e+f*x)*sqrt(a+b+b*tan(e+f*x)^2)/(a*f)],
[1/sqrt(a+b*sec(e+f*x)^2),x,3,arctan(sqrt(a)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/(f*sqrt(a))],
[csc(e+f*x)^2/sqrt(a+b*sec(e+f*x)^2),x,2,-cot(e+f*x)*sqrt(a+b+b*tan(e+f*x)^2)/((a+b)*f)],
[csc(e+f*x)^4/sqrt(a+b*sec(e+f*x)^2),x,3,-1/3*(3*a+b)*cot(e+f*x)*sqrt(a+b+b*tan(e+f*x)^2)/((a+b)^2*f)-1/3*cot(e+f*x)^3*sqrt(a+b+b*tan(e+f*x)^2)/((a+b)*f)],
[csc(e+f*x)^6/sqrt(a+b*sec(e+f*x)^2),x,4,-1/15*(15*a^2+10*a*b+3*b^2)*cot(e+f*x)*sqrt(a+b+b*tan(e+f*x)^2)/((a+b)^3*f)-2/15*(5*a+3*b)*cot(e+f*x)^3*sqrt(a+b+b*tan(e+f*x)^2)/((a+b)^2*f)-1/5*cot(e+f*x)^5*sqrt(a+b+b*tan(e+f*x)^2)/((a+b)*f)],
[sin(e+f*x)^5/(a+b*sec(e+f*x)^2)^(3/2),x,5,-1/15*(15*a^2+40*a*b+24*b^2)*cos(e+f*x)/(a^3*f*sqrt(a+b*sec(e+f*x)^2))+2/15*(5*a+3*b)*cos(e+f*x)^3/(a^2*f*sqrt(a+b*sec(e+f*x)^2))-1/5*cos(e+f*x)^5/(a*f*sqrt(a+b*sec(e+f*x)^2))-2/15*b*(15*a^2+40*a*b+24*b^2)*sec(e+f*x)/(a^4*f*sqrt(a+b*sec(e+f*x)^2))],
[sin(e+f*x)^3/(a+b*sec(e+f*x)^2)^(3/2),x,4,-1/3*(3*a+4*b)*cos(e+f*x)/(a^2*f*sqrt(a+b*sec(e+f*x)^2))+1/3*cos(e+f*x)^3/(a*f*sqrt(a+b*sec(e+f*x)^2))-2/3*b*(3*a+4*b)*sec(e+f*x)/(a^3*f*sqrt(a+b*sec(e+f*x)^2))],
[sin(e+f*x)/(a+b*sec(e+f*x)^2)^(3/2),x,3,-cos(e+f*x)/(a*f*sqrt(a+b*sec(e+f*x)^2))-2*b*sec(e+f*x)/(a^2*f*sqrt(a+b*sec(e+f*x)^2))],
[csc(e+f*x)/(a+b*sec(e+f*x)^2)^(3/2),x,4,-arctanh(sec(e+f*x)*sqrt(a+b)/sqrt(a+b*sec(e+f*x)^2))/((a+b)^(3/2)*f)-b*sec(e+f*x)/(a*(a+b)*f*sqrt(a+b*sec(e+f*x)^2))],
[csc(e+f*x)^3/(a+b*sec(e+f*x)^2)^(3/2),x,6,-1/2*(a-2*b)*arctanh(sec(e+f*x)*sqrt(a+b)/sqrt(a+b*sec(e+f*x)^2))/((a+b)^(5/2)*f)-1/2*cot(e+f*x)*csc(e+f*x)/((a+b)*f*sqrt(a+b*sec(e+f*x)^2))-3/2*b*sec(e+f*x)/((a+b)^2*f*sqrt(a+b*sec(e+f*x)^2))],
[csc(e+f*x)^5/(a+b*sec(e+f*x)^2)^(3/2),x,7,-3/8*a*(a-4*b)*arctanh(sec(e+f*x)*sqrt(a+b)/sqrt(a+b*sec(e+f*x)^2))/((a+b)^(7/2)*f)-5/8*a*cot(e+f*x)*csc(e+f*x)/((a+b)^2*f*sqrt(a+b*sec(e+f*x)^2))-1/4*cot(e+f*x)^3*csc(e+f*x)/((a+b)*f*sqrt(a+b*sec(e+f*x)^2))-1/8*(13*a-2*b)*b*sec(e+f*x)/((a+b)^3*f*sqrt(a+b*sec(e+f*x)^2))],
[sin(e+f*x)^6/(a+b*sec(e+f*x)^2)^(3/2),x,8,5/16*(a+b)^2*(a+7*b)*arctan(sqrt(a)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/(a^(9/2)*f)-1/48*(a+b)*(33*a+35*b)*cos(e+f*x)*sin(e+f*x)/(a^3*f*sqrt(a+b+b*tan(e+f*x)^2))+1/24*(9*a+7*b)*cos(e+f*x)^3*sin(e+f*x)/(a^2*f*sqrt(a+b+b*tan(e+f*x)^2))+1/6*cos(e+f*x)^3*sin(e+f*x)^3/(a*f*sqrt(a+b+b*tan(e+f*x)^2))-1/48*b*(81*a^2+190*a*b+105*b^2)*tan(e+f*x)/(a^4*f*sqrt(a+b+b*tan(e+f*x)^2))],
[sin(e+f*x)^4/(a+b*sec(e+f*x)^2)^(3/2),x,7,3/8*(a+b)*(a+5*b)*arctan(sqrt(a)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/(a^(7/2)*f)-5/8*(a+b)*cos(e+f*x)*sin(e+f*x)/(a^2*f*sqrt(a+b+b*tan(e+f*x)^2))+1/4*cos(e+f*x)^3*sin(e+f*x)/(a*f*sqrt(a+b+b*tan(e+f*x)^2))-1/8*b*(13*a+15*b)*tan(e+f*x)/(a^3*f*sqrt(a+b+b*tan(e+f*x)^2))],
[sin(e+f*x)^2/(a+b*sec(e+f*x)^2)^(3/2),x,6,1/2*(a+3*b)*arctan(sqrt(a)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/(a^(5/2)*f)-1/2*cos(e+f*x)*sin(e+f*x)/(a*f*sqrt(a+b+b*tan(e+f*x)^2))-3/2*b*tan(e+f*x)/(a^2*f*sqrt(a+b+b*tan(e+f*x)^2))],
[1/(a+b*sec(e+f*x)^2)^(3/2),x,4,arctan(sqrt(a)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/(a^(3/2)*f)-b*tan(e+f*x)/(a*(a+b)*f*sqrt(a+b+b*tan(e+f*x)^2))],
[csc(e+f*x)^2/(a+b*sec(e+f*x)^2)^(3/2),x,3,-cot(e+f*x)/((a+b)*f*sqrt(a+b+b*tan(e+f*x)^2))-2*b*tan(e+f*x)/((a+b)^2*f*sqrt(a+b+b*tan(e+f*x)^2))],
[csc(e+f*x)^4/(a+b*sec(e+f*x)^2)^(3/2),x,4,-1/3*(3*a-b)*cot(e+f*x)/((a+b)^2*f*sqrt(a+b+b*tan(e+f*x)^2))-1/3*cot(e+f*x)^3/((a+b)*f*sqrt(a+b+b*tan(e+f*x)^2))-2/3*(3*a-b)*b*tan(e+f*x)/((a+b)^3*f*sqrt(a+b+b*tan(e+f*x)^2))],
[csc(e+f*x)^6/(a+b*sec(e+f*x)^2)^(3/2),x,5,-1/15*(15*a^2-10*a*b-b^2)*cot(e+f*x)/((a+b)^3*f*sqrt(a+b+b*tan(e+f*x)^2))-2/15*(5*a+2*b)*cot(e+f*x)^3/((a+b)^2*f*sqrt(a+b+b*tan(e+f*x)^2))-1/5*cot(e+f*x)^5/((a+b)*f*sqrt(a+b+b*tan(e+f*x)^2))-2/15*b*(15*a^2-10*a*b-b^2)*tan(e+f*x)/((a+b)^4*f*sqrt(a+b+b*tan(e+f*x)^2))],
[sin(e+f*x)^5/(a+b*sec(e+f*x)^2)^(5/2),x,6,-1/5*(5*a^2+20*a*b+16*b^2)*cos(e+f*x)/(a^3*f*(a+b*sec(e+f*x)^2)^(3/2))+2/15*(5*a+4*b)*cos(e+f*x)^3/(a^2*f*(a+b*sec(e+f*x)^2)^(3/2))-1/5*cos(e+f*x)^5/(a*f*(a+b*sec(e+f*x)^2)^(3/2))-4/15*b*(5*a^2+20*a*b+16*b^2)*sec(e+f*x)/(a^4*f*(a+b*sec(e+f*x)^2)^(3/2))-8/15*b*(5*a^2+20*a*b+16*b^2)*sec(e+f*x)/(a^5*f*sqrt(a+b*sec(e+f*x)^2))],
[sin(e+f*x)^3/(a+b*sec(e+f*x)^2)^(5/2),x,5,-(a+2*b)*cos(e+f*x)/(a^2*f*(a+b*sec(e+f*x)^2)^(3/2))+1/3*cos(e+f*x)^3/(a*f*(a+b*sec(e+f*x)^2)^(3/2))-4/3*b*(a+2*b)*sec(e+f*x)/(a^3*f*(a+b*sec(e+f*x)^2)^(3/2))-8/3*b*(a+2*b)*sec(e+f*x)/(a^4*f*sqrt(a+b*sec(e+f*x)^2))],
[sin(e+f*x)/(a+b*sec(e+f*x)^2)^(5/2),x,4,-cos(e+f*x)/(a*f*(a+b*sec(e+f*x)^2)^(3/2))-4/3*b*sec(e+f*x)/(a^2*f*(a+b*sec(e+f*x)^2)^(3/2))-8/3*b*sec(e+f*x)/(a^3*f*sqrt(a+b*sec(e+f*x)^2))],
[csc(e+f*x)/(a+b*sec(e+f*x)^2)^(5/2),x,6,-arctanh(sec(e+f*x)*sqrt(a+b)/sqrt(a+b*sec(e+f*x)^2))/((a+b)^(5/2)*f)-1/3*b*sec(e+f*x)/(a*(a+b)*f*(a+b*sec(e+f*x)^2)^(3/2))-1/3*b*(5*a+2*b)*sec(e+f*x)/(a^2*(a+b)^2*f*sqrt(a+b*sec(e+f*x)^2))],
[csc(e+f*x)^3/(a+b*sec(e+f*x)^2)^(5/2),x,7,-1/2*(a-4*b)*arctanh(sec(e+f*x)*sqrt(a+b)/sqrt(a+b*sec(e+f*x)^2))/((a+b)^(7/2)*f)-1/2*cot(e+f*x)*csc(e+f*x)/((a+b)*f*(a+b*sec(e+f*x)^2)^(3/2))-5/6*b*sec(e+f*x)/((a+b)^2*f*(a+b*sec(e+f*x)^2)^(3/2))-1/6*(13*a-2*b)*b*sec(e+f*x)/(a*(a+b)^3*f*sqrt(a+b*sec(e+f*x)^2))],
[csc(e+f*x)^5/(a+b*sec(e+f*x)^2)^(5/2),x,8,-1/8*(3*a^2-24*a*b+8*b^2)*arctanh(sec(e+f*x)*sqrt(a+b)/sqrt(a+b*sec(e+f*x)^2))/((a+b)^(9/2)*f)-1/8*(5*a-2*b)*cot(e+f*x)*csc(e+f*x)/((a+b)^2*f*(a+b*sec(e+f*x)^2)^(3/2))-1/4*cot(e+f*x)^3*csc(e+f*x)/((a+b)*f*(a+b*sec(e+f*x)^2)^(3/2))-1/24*(23*a-12*b)*b*sec(e+f*x)/((a+b)^3*f*(a+b*sec(e+f*x)^2)^(3/2))-5/24*(11*a-10*b)*b*sec(e+f*x)/((a+b)^4*f*sqrt(a+b*sec(e+f*x)^2))],
[sin(e+f*x)^6/(a+b*sec(e+f*x)^2)^(5/2),x,9,5/16*(a+b)*(a^2+14*a*b+21*b^2)*arctan(sqrt(a)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/(a^(11/2)*f)-1/48*b*(113*a^2+420*a*b+315*b^2)*tan(e+f*x)/(a^5*f*sqrt(a+b+b*tan(e+f*x)^2))-1/16*(a+b)*(11*a+21*b)*cos(e+f*x)*sin(e+f*x)/(a^3*f*(a+b+b*tan(e+f*x)^2)^(3/2))+3/8*(a+b)*cos(e+f*x)^3*sin(e+f*x)/(a^2*f*(a+b+b*tan(e+f*x)^2)^(3/2))+1/6*cos(e+f*x)^3*sin(e+f*x)^3/(a*f*(a+b+b*tan(e+f*x)^2)^(3/2))-7/48*b*(a+b)*(7*a+15*b)*tan(e+f*x)/(a^4*f*(a+b+b*tan(e+f*x)^2)^(3/2))],
[sin(e+f*x)^4/(a+b*sec(e+f*x)^2)^(5/2),x,8,1/8*(3*a^2+30*a*b+35*b^2)*arctan(sqrt(a)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/(a^(9/2)*f)-5/24*b*(11*a+21*b)*tan(e+f*x)/(a^4*f*sqrt(a+b+b*tan(e+f*x)^2))-1/8*(5*a+7*b)*cos(e+f*x)*sin(e+f*x)/(a^2*f*(a+b+b*tan(e+f*x)^2)^(3/2))+1/4*cos(e+f*x)^3*sin(e+f*x)/(a*f*(a+b+b*tan(e+f*x)^2)^(3/2))-1/24*b*(23*a+35*b)*tan(e+f*x)/(a^3*f*(a+b+b*tan(e+f*x)^2)^(3/2))],
[sin(e+f*x)^2/(a+b*sec(e+f*x)^2)^(5/2),x,7,1/2*(a+5*b)*arctan(sqrt(a)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/(a^(7/2)*f)-1/6*b*(13*a+15*b)*tan(e+f*x)/(a^3*(a+b)*f*sqrt(a+b+b*tan(e+f*x)^2))-1/2*cos(e+f*x)*sin(e+f*x)/(a*f*(a+b+b*tan(e+f*x)^2)^(3/2))-5/6*b*tan(e+f*x)/(a^2*f*(a+b+b*tan(e+f*x)^2)^(3/2))],
[1/(a+b*sec(e+f*x)^2)^(5/2),x,6,arctan(sqrt(a)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/(a^(5/2)*f)-1/3*b*(5*a+3*b)*tan(e+f*x)/(a^2*(a+b)^2*f*sqrt(a+b+b*tan(e+f*x)^2))-1/3*b*tan(e+f*x)/(a*(a+b)*f*(a+b+b*tan(e+f*x)^2)^(3/2))],
[csc(e+f*x)^2/(a+b*sec(e+f*x)^2)^(5/2),x,4,-8/3*b*tan(e+f*x)/((a+b)^3*f*sqrt(a+b+b*tan(e+f*x)^2))-cot(e+f*x)/((a+b)*f*(a+b+b*tan(e+f*x)^2)^(3/2))-4/3*b*tan(e+f*x)/((a+b)^2*f*(a+b+b*tan(e+f*x)^2)^(3/2))],
[csc(e+f*x)^4/(a+b*sec(e+f*x)^2)^(5/2),x,5,-8/3*(a-b)*b*tan(e+f*x)/((a+b)^4*f*sqrt(a+b+b*tan(e+f*x)^2))-(a-b)*cot(e+f*x)/((a+b)^2*f*(a+b+b*tan(e+f*x)^2)^(3/2))-1/3*cot(e+f*x)^3/((a+b)*f*(a+b+b*tan(e+f*x)^2)^(3/2))-4/3*(a-b)*b*tan(e+f*x)/((a+b)^3*f*(a+b+b*tan(e+f*x)^2)^(3/2))],
[csc(e+f*x)^6/(a+b*sec(e+f*x)^2)^(5/2),x,6,-8/15*b*(5*a^2-10*a*b+b^2)*tan(e+f*x)/((a+b)^5*f*sqrt(a+b+b*tan(e+f*x)^2))-1/5*(5*a^2-10*a*b+b^2)*cot(e+f*x)/((a+b)^3*f*(a+b+b*tan(e+f*x)^2)^(3/2))-2/15*(5*a+b)*cot(e+f*x)^3/((a+b)^2*f*(a+b+b*tan(e+f*x)^2)^(3/2))-1/5*cot(e+f*x)^5/((a+b)*f*(a+b+b*tan(e+f*x)^2)^(3/2))-4/15*b*(5*a^2-10*a*b+b^2)*tan(e+f*x)/((a+b)^4*f*(a+b+b*tan(e+f*x)^2)^(3/2))],

# Integrands of the form Sin[e+f x]^m (a+b Sec[e+f x]^2)^p when p symbolic
[(a+b*sec(e+f*x)^2)^p*(d*sin(e+f*x))^m,x,-1,AppellF1(1/2*(1+m),1/2+p,-p,1/2*(3+m),sin(e+f*x)^2,a*sin(e+f*x)^2/(a+b))*(cos(e+f*x)^2)^(1/2+p)*(a+b*sec(e+f*x)^2)^p*(d*sin(e+f*x))^m*tan(e+f*x)/(f*(1+m)*((a+b-a*sin(e+f*x)^2)/(a+b))^p)],
[(a+b*sec(e+f*x)^2)^p*sin(e+f*x)^5,x,5,1/15*(10*a+b*(3-2*p))*cos(e+f*x)^3*(a+b*sec(e+f*x)^2)^(1+p)/(a^2*f)-1/5*cos(e+f*x)^5*(a+b*sec(e+f*x)^2)^(1+p)/(a*f)-1/15*(15*a^2+10*a*b*(1-2*p)+b^2*(3-8*p+4*p^2))*cos(e+f*x)*hypergeom([-1/2,-p],[1/2],-b*sec(e+f*x)^2/a)*(a+b*sec(e+f*x)^2)^p/(a^2*f*(1+b*sec(e+f*x)^2/a)^p)],
[(a+b*sec(e+f*x)^2)^p*sin(e+f*x)^3,x,4,1/3*cos(e+f*x)^3*(a+b*sec(e+f*x)^2)^(1+p)/(a*f)-1/3*(3*a+b-2*b*p)*cos(e+f*x)*hypergeom([-1/2,-p],[1/2],-b*sec(e+f*x)^2/a)*(a+b*sec(e+f*x)^2)^p/(a*f*(1+b*sec(e+f*x)^2/a)^p)],
[(a+b*sec(e+f*x)^2)^p*sin(e+f*x),x,3,-cos(e+f*x)*hypergeom([-1/2,-p],[1/2],-b*sec(e+f*x)^2/a)*(a+b*sec(e+f*x)^2)^p/(f*(1+b*sec(e+f*x)^2/a)^p)],
[csc(e+f*x)*(a+b*sec(e+f*x)^2)^p,x,3,-AppellF1(1/2,1,-p,3/2,sec(e+f*x)^2,-b*sec(e+f*x)^2/a)*sec(e+f*x)*(a+b*sec(e+f*x)^2)^p/(f*(1+b*sec(e+f*x)^2/a)^p)],
[csc(e+f*x)^3*(a+b*sec(e+f*x)^2)^p,x,3,1/3*AppellF1(3/2,2,-p,5/2,sec(e+f*x)^2,-b*sec(e+f*x)^2/a)*sec(e+f*x)^3*(a+b*sec(e+f*x)^2)^p/(f*(1+b*sec(e+f*x)^2/a)^p)],
[(a+b*sec(e+f*x)^2)^p*sin(e+f*x)^4,x,3,1/5*AppellF1(5/2,3,-p,7/2,-tan(e+f*x)^2,-b*tan(e+f*x)^2/(a+b))*tan(e+f*x)^5*(a+b+b*tan(e+f*x)^2)^p/(f*(1+b*tan(e+f*x)^2/(a+b))^p)],
[(a+b*sec(e+f*x)^2)^p*sin(e+f*x)^2,x,3,1/3*AppellF1(3/2,2,-p,5/2,-tan(e+f*x)^2,-b*tan(e+f*x)^2/(a+b))*tan(e+f*x)^3*(a+b+b*tan(e+f*x)^2)^p/(f*(1+b*tan(e+f*x)^2/(a+b))^p)],
[(a+b*sec(e+f*x)^2)^p,x,3,AppellF1(1/2,1,-p,3/2,-tan(e+f*x)^2,-b*tan(e+f*x)^2/(a+b))*tan(e+f*x)*(a+b+b*tan(e+f*x)^2)^p/(f*(1+b*tan(e+f*x)^2/(a+b))^p)],
[csc(e+f*x)^2*(a+b*sec(e+f*x)^2)^p,x,3,-cot(e+f*x)*hypergeom([-1/2,-p],[1/2],-b*tan(e+f*x)^2/(a+b))*(a+b+b*tan(e+f*x)^2)^p/(f*(1+b*tan(e+f*x)^2/(a+b))^p)],
[csc(e+f*x)^4*(a+b*sec(e+f*x)^2)^p,x,4,-1/3*cot(e+f*x)^3*(a+b+b*tan(e+f*x)^2)^(1+p)/((a+b)*f)-1/3*(3*a+2*b*(1+p))*cot(e+f*x)*hypergeom([-1/2,-p],[1/2],-b*tan(e+f*x)^2/(a+b))*(a+b+b*tan(e+f*x)^2)^p/((a+b)*f*(1+b*tan(e+f*x)^2/(a+b))^p)],
[csc(e+f*x)^6*(a+b*sec(e+f*x)^2)^p,x,5,-1/15*(10*a+b*(7+2*p))*cot(e+f*x)^3*(a+b+b*tan(e+f*x)^2)^(1+p)/((a+b)^2*f)-1/5*cot(e+f*x)^5*(a+b+b*tan(e+f*x)^2)^(1+p)/((a+b)*f)-1/15*(15*a^2+20*a*b*(1+p)+4*b^2*(2+3*p+p^2))*cot(e+f*x)*hypergeom([-1/2,-p],[1/2],-b*tan(e+f*x)^2/(a+b))*(a+b+b*tan(e+f*x)^2)^p/((a+b)^2*f*(1+b*tan(e+f*x)^2/(a+b))^p)],

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

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

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

# Integrands of the form Sec[e+f x]^m (a+b Sec[e+f x]^2)^p when a+b=0
[(a-a*sec(c+d*x)^2)^4,x,6,a^4*x-a^4*tan(c+d*x)/d+1/3*a^4*tan(c+d*x)^3/d-1/5*a^4*tan(c+d*x)^5/d+1/7*a^4*tan(c+d*x)^7/d],
[(a-a*sec(c+d*x)^2)^3,x,5,a^3*x-a^3*tan(c+d*x)/d+1/3*a^3*tan(c+d*x)^3/d-1/5*a^3*tan(c+d*x)^5/d],
[(a-a*sec(c+d*x)^2)^2,x,4,a^2*x-a^2*tan(c+d*x)/d+1/3*a^2*tan(c+d*x)^3/d],
[a-a*sec(c+d*x)^2,x,3,a*x-a*tan(c+d*x)/d],
[1/(a-a*sec(c+d*x)^2),x,3,x/a+cot(c+d*x)/(a*d)],
[1/(a-a*sec(c+d*x)^2)^2,x,4,x/a^2+cot(c+d*x)/(a^2*d)-1/3*cot(c+d*x)^3/(a^2*d)],
[1/(a-a*sec(c+d*x)^2)^3,x,5,x/a^3+cot(c+d*x)/(a^3*d)-1/3*cot(c+d*x)^3/(a^3*d)+1/5*cot(c+d*x)^5/(a^3*d)],
[1/(a-a*sec(c+d*x)^2)^4,x,6,x/a^4+cot(c+d*x)/(a^4*d)-1/3*cot(c+d*x)^3/(a^4*d)+1/5*cot(c+d*x)^5/(a^4*d)-1/7*cot(c+d*x)^7/(a^4*d)],

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

# p>0
[sec(e+f*x)^5*(a+b*sec(e+f*x)^2),x,4,1/16*(6*a+5*b)*arctanh(sin(e+f*x))/f+1/16*(6*a+5*b)*sec(e+f*x)*tan(e+f*x)/f+1/24*(6*a+5*b)*sec(e+f*x)^3*tan(e+f*x)/f+1/6*b*sec(e+f*x)^5*tan(e+f*x)/f],
[sec(e+f*x)^3*(a+b*sec(e+f*x)^2),x,3,1/8*(4*a+3*b)*arctanh(sin(e+f*x))/f+1/8*(4*a+3*b)*sec(e+f*x)*tan(e+f*x)/f+1/4*b*sec(e+f*x)^3*tan(e+f*x)/f],
[sec(e+f*x)*(a+b*sec(e+f*x)^2),x,2,1/2*(2*a+b)*arctanh(sin(e+f*x))/f+1/2*b*sec(e+f*x)*tan(e+f*x)/f],
[cos(e+f*x)*(a+b*sec(e+f*x)^2),x,2,b*arctanh(sin(e+f*x))/f+a*sin(e+f*x)/f],
[cos(e+f*x)^3*(a+b*sec(e+f*x)^2),x,3,(a+b)*sin(e+f*x)/f-1/3*a*sin(e+f*x)^3/f],
[cos(e+f*x)^5*(a+b*sec(e+f*x)^2),x,4,(a+b)*sin(e+f*x)/f-1/3*(2*a+b)*sin(e+f*x)^3/f+1/5*a*sin(e+f*x)^5/f],
[sec(e+f*x)^6*(a+b*sec(e+f*x)^2),x,3,1/7*(7*a+6*b)*tan(e+f*x)/f+1/7*b*sec(e+f*x)^6*tan(e+f*x)/f+2/21*(7*a+6*b)*tan(e+f*x)^3/f+1/35*(7*a+6*b)*tan(e+f*x)^5/f],
[sec(e+f*x)^4*(a+b*sec(e+f*x)^2),x,3,1/5*(5*a+4*b)*tan(e+f*x)/f+1/5*b*sec(e+f*x)^4*tan(e+f*x)/f+1/15*(5*a+4*b)*tan(e+f*x)^3/f],
[sec(e+f*x)^2*(a+b*sec(e+f*x)^2),x,3,1/3*(3*a+2*b)*tan(e+f*x)/f+1/3*b*sec(e+f*x)^2*tan(e+f*x)/f],
[a+b*sec(e+f*x)^2,x,3,a*x+b*tan(e+f*x)/f],
[cos(e+f*x)^2*(a+b*sec(e+f*x)^2),x,2,1/2*(a+2*b)*x+1/2*a*cos(e+f*x)*sin(e+f*x)/f],
[cos(e+f*x)^4*(a+b*sec(e+f*x)^2),x,3,1/8*(3*a+4*b)*x+1/8*(3*a+4*b)*cos(e+f*x)*sin(e+f*x)/f+1/4*a*cos(e+f*x)^3*sin(e+f*x)/f],
[cos(e+f*x)^6*(a+b*sec(e+f*x)^2),x,4,1/16*(5*a+6*b)*x+1/16*(5*a+6*b)*cos(e+f*x)*sin(e+f*x)/f+1/24*(5*a+6*b)*cos(e+f*x)^3*sin(e+f*x)/f+1/6*a*cos(e+f*x)^5*sin(e+f*x)/f],
[sec(e+f*x)^5*(a+b*sec(e+f*x)^2)^2,x,6,1/128*(48*a^2+80*a*b+35*b^2)*arctanh(sin(e+f*x))/f+1/128*(48*a^2+80*a*b+35*b^2)*sec(e+f*x)*tan(e+f*x)/f+1/192*(48*a^2+80*a*b+35*b^2)*sec(e+f*x)^3*tan(e+f*x)/f+1/48*b*(10*a+7*b)*sec(e+f*x)^5*tan(e+f*x)/f+1/8*b*sec(e+f*x)^7*(a+b-a*sin(e+f*x)^2)*tan(e+f*x)/f],
[sec(e+f*x)^3*(a+b*sec(e+f*x)^2)^2,x,5,1/16*(8*a^2+12*a*b+5*b^2)*arctanh(sin(e+f*x))/f+1/16*(8*a^2+12*a*b+5*b^2)*sec(e+f*x)*tan(e+f*x)/f+1/24*b*(8*a+5*b)*sec(e+f*x)^3*tan(e+f*x)/f+1/6*b*sec(e+f*x)^5*(a+b-a*sin(e+f*x)^2)*tan(e+f*x)/f],
[sec(e+f*x)*(a+b*sec(e+f*x)^2)^2,x,4,1/8*(8*a^2+8*a*b+3*b^2)*arctanh(sin(e+f*x))/f+3/8*b*(2*a+b)*sec(e+f*x)*tan(e+f*x)/f+1/4*b*sec(e+f*x)^3*(a+b-a*sin(e+f*x)^2)*tan(e+f*x)/f],
[cos(e+f*x)*(a+b*sec(e+f*x)^2)^2,x,5,1/2*b*(4*a+b)*arctanh(sin(e+f*x))/f+a^2*sin(e+f*x)/f+1/2*b^2*sec(e+f*x)*tan(e+f*x)/f],
[cos(e+f*x)^3*(a+b*sec(e+f*x)^2)^2,x,4,b^2*arctanh(sin(e+f*x))/f+a*(a+2*b)*sin(e+f*x)/f-1/3*a^2*sin(e+f*x)^3/f],
[cos(e+f*x)^5*(a+b*sec(e+f*x)^2)^2,x,3,(a+b)^2*sin(e+f*x)/f-2/3*a*(a+b)*sin(e+f*x)^3/f+1/5*a^2*sin(e+f*x)^5/f],
[sec(e+f*x)^6*(a+b*sec(e+f*x)^2)^2,x,3,(a+b)^2*tan(e+f*x)/f+2/3*(a+b)*(a+2*b)*tan(e+f*x)^3/f+1/5*(a^2+6*a*b+6*b^2)*tan(e+f*x)^5/f+2/7*b*(a+2*b)*tan(e+f*x)^7/f+1/9*b^2*tan(e+f*x)^9/f],
[sec(e+f*x)^4*(a+b*sec(e+f*x)^2)^2,x,3,(a+b)^2*tan(e+f*x)/f+1/3*(a+b)*(a+3*b)*tan(e+f*x)^3/f+1/5*b*(2*a+3*b)*tan(e+f*x)^5/f+1/7*b^2*tan(e+f*x)^7/f],
[sec(e+f*x)^2*(a+b*sec(e+f*x)^2)^2,x,3,(a+b)^2*tan(e+f*x)/f+2/3*b*(a+b)*tan(e+f*x)^3/f+1/5*b^2*tan(e+f*x)^5/f],
[(a+b*sec(e+f*x)^2)^2,x,4,a^2*x+b*(2*a+b)*tan(e+f*x)/f+1/3*b^2*tan(e+f*x)^3/f],
[cos(e+f*x)^2*(a+b*sec(e+f*x)^2)^2,x,5,1/2*a*(a+4*b)*x+1/2*a^2*cos(e+f*x)*sin(e+f*x)/f+b^2*tan(e+f*x)/f],
[cos(e+f*x)^4*(a+b*sec(e+f*x)^2)^2,x,4,1/8*(3*a^2+8*a*b+8*b^2)*x+3/8*a*(a+2*b)*cos(e+f*x)*sin(e+f*x)/f+1/4*a*cos(e+f*x)^3*sin(e+f*x)*(a+b+b*tan(e+f*x)^2)/f],
[cos(e+f*x)^6*(a+b*sec(e+f*x)^2)^2,x,5,1/16*(5*a^2+12*a*b+8*b^2)*x+1/16*(5*a^2+12*a*b+8*b^2)*cos(e+f*x)*sin(e+f*x)/f+1/24*a*(5*a+8*b)*cos(e+f*x)^3*sin(e+f*x)/f+1/6*a*cos(e+f*x)^5*sin(e+f*x)*(a+b+b*tan(e+f*x)^2)/f],
[(a+b*sec(c+d*x)^2)^3,x,4,a^3*x+b*(3*a^2+3*a*b+b^2)*tan(c+d*x)/d+1/3*b^2*(3*a+2*b)*tan(c+d*x)^3/d+1/5*b^3*tan(c+d*x)^5/d],
[(a+b*sec(c+d*x)^2)^4,x,4,a^4*x+b*(2*a+b)*(2*a^2+2*a*b+b^2)*tan(c+d*x)/d+1/3*b^2*(6*a^2+8*a*b+3*b^2)*tan(c+d*x)^3/d+1/5*b^3*(4*a+3*b)*tan(c+d*x)^5/d+1/7*b^4*tan(c+d*x)^7/d],

# p<0
[sec(e+f*x)^5/(a+b*sec(e+f*x)^2),x,5,-1/2*(2*a-b)*arctanh(sin(e+f*x))/(b^2*f)+a^(3/2)*arctanh(sin(e+f*x)*sqrt(a)/sqrt(a+b))/(b^2*f*sqrt(a+b))+1/2*sec(e+f*x)*tan(e+f*x)/(b*f)],
[sec(e+f*x)^3/(a+b*sec(e+f*x)^2),x,4,arctanh(sin(e+f*x))/(b*f)-arctanh(sin(e+f*x)*sqrt(a)/sqrt(a+b))*sqrt(a)/(b*f*sqrt(a+b))],
[sec(e+f*x)/(a+b*sec(e+f*x)^2),x,2,arctanh(sin(e+f*x)*sqrt(a)/sqrt(a+b))/(f*sqrt(a)*sqrt(a+b))],
[cos(e+f*x)/(a+b*sec(e+f*x)^2),x,3,sin(e+f*x)/(a*f)-b*arctanh(sin(e+f*x)*sqrt(a)/sqrt(a+b))/(a^(3/2)*f*sqrt(a+b))],
[cos(e+f*x)^3/(a+b*sec(e+f*x)^2),x,4,(a-b)*sin(e+f*x)/(a^2*f)-1/3*sin(e+f*x)^3/(a*f)+b^2*arctanh(sin(e+f*x)*sqrt(a)/sqrt(a+b))/(a^(5/2)*f*sqrt(a+b))],
[cos(e+f*x)^5/(a+b*sec(e+f*x)^2),x,4,(a^2-a*b+b^2)*sin(e+f*x)/(a^3*f)-1/3*(2*a-b)*sin(e+f*x)^3/(a^2*f)+1/5*sin(e+f*x)^5/(a*f)-b^3*arctanh(sin(e+f*x)*sqrt(a)/sqrt(a+b))/(a^(7/2)*f*sqrt(a+b))],
[sec(e+f*x)^6/(a+b*sec(e+f*x)^2),x,4,a^2*arctan(sqrt(b)*tan(e+f*x)/sqrt(a+b))/(b^(5/2)*f*sqrt(a+b))-(a-b)*tan(e+f*x)/(b^2*f)+1/3*tan(e+f*x)^3/(b*f)],
[sec(e+f*x)^4/(a+b*sec(e+f*x)^2),x,3,-a*arctan(sqrt(b)*tan(e+f*x)/sqrt(a+b))/(b^(3/2)*f*sqrt(a+b))+tan(e+f*x)/(b*f)],
[sec(e+f*x)^2/(a+b*sec(e+f*x)^2),x,2,arctan(sqrt(b)*tan(e+f*x)/sqrt(a+b))/(f*sqrt(b)*sqrt(a+b))],
[1/(a+b*sec(e+f*x)^2),x,3,x/a+arctan(cot(e+f*x)*sqrt(a+b)/sqrt(b))*sqrt(b)/(a*f*sqrt(a+b))],
[cos(e+f*x)^2/(a+b*sec(e+f*x)^2),x,5,1/2*(a-2*b)*x/a^2+1/2*cos(e+f*x)*sin(e+f*x)/(a*f)+b^(3/2)*arctan(sqrt(b)*tan(e+f*x)/sqrt(a+b))/(a^2*f*sqrt(a+b))],
[cos(e+f*x)^4/(a+b*sec(e+f*x)^2),x,6,1/8*(3*a^2-4*a*b+8*b^2)*x/a^3+1/8*(3*a-4*b)*cos(e+f*x)*sin(e+f*x)/(a^2*f)+1/4*cos(e+f*x)^3*sin(e+f*x)/(a*f)-b^(5/2)*arctan(sqrt(b)*tan(e+f*x)/sqrt(a+b))/(a^3*f*sqrt(a+b))],
[cos(e+f*x)^6/(a+b*sec(e+f*x)^2),x,7,1/16*(5*a^3-6*a^2*b+8*a*b^2-16*b^3)*x/a^4+1/16*(5*a^2-6*a*b+8*b^2)*cos(e+f*x)*sin(e+f*x)/(a^3*f)+1/24*(5*a-6*b)*cos(e+f*x)^3*sin(e+f*x)/(a^2*f)+1/6*cos(e+f*x)^5*sin(e+f*x)/(a*f)+b^(7/2)*arctan(sqrt(b)*tan(e+f*x)/sqrt(a+b))/(a^4*f*sqrt(a+b))],
[sec(e+f*x)^5/(a+b*sec(e+f*x)^2)^2,x,5,arctanh(sin(e+f*x))/(b^2*f)-1/2*a*sin(e+f*x)/(b*(a+b)*f*(a+b-a*sin(e+f*x)^2))-1/2*(2*a+3*b)*arctanh(sin(e+f*x)*sqrt(a)/sqrt(a+b))*sqrt(a)/(b^2*(a+b)^(3/2)*f)],
[sec(e+f*x)^3/(a+b*sec(e+f*x)^2)^2,x,3,1/2*sin(e+f*x)/((a+b)*f*(a+b-a*sin(e+f*x)^2))+1/2*arctanh(sin(e+f*x)*sqrt(a)/sqrt(a+b))/((a+b)^(3/2)*f*sqrt(a))],
[sec(e+f*x)/(a+b*sec(e+f*x)^2)^2,x,3,1/2*(2*a+b)*arctanh(sin(e+f*x)*sqrt(a)/sqrt(a+b))/(a^(3/2)*(a+b)^(3/2)*f)-1/2*b*sin(e+f*x)/(a*(a+b)*f*(a+b-a*sin(e+f*x)^2))],
[cos(e+f*x)/(a+b*sec(e+f*x)^2)^2,x,5,-1/2*b*(4*a+3*b)*arctanh(sin(e+f*x)*sqrt(a)/sqrt(a+b))/(a^(5/2)*(a+b)^(3/2)*f)+sin(e+f*x)/(a^2*f)+1/2*b^2*sin(e+f*x)/(a^2*(a+b)*f*(a+b-a*sin(e+f*x)^2))],
[cos(e+f*x)^3/(a+b*sec(e+f*x)^2)^2,x,5,1/2*b^2*(6*a+5*b)*arctanh(sin(e+f*x)*sqrt(a)/sqrt(a+b))/(a^(7/2)*(a+b)^(3/2)*f)+(a-2*b)*sin(e+f*x)/(a^3*f)-1/3*sin(e+f*x)^3/(a^2*f)-1/2*b^3*sin(e+f*x)/(a^3*(a+b)*f*(a+b-a*sin(e+f*x)^2))],
[cos(e+f*x)^5/(a+b*sec(e+f*x)^2)^2,x,5,-1/2*b^3*(8*a+7*b)*arctanh(sin(e+f*x)*sqrt(a)/sqrt(a+b))/(a^(9/2)*(a+b)^(3/2)*f)+(a^2-2*a*b+3*b^2)*sin(e+f*x)/(a^4*f)-2/3*(a-b)*sin(e+f*x)^3/(a^3*f)+1/5*sin(e+f*x)^5/(a^2*f)+1/2*b^4*sin(e+f*x)/(a^4*(a+b)*f*(a+b-a*sin(e+f*x)^2))],
[sec(e+f*x)^6/(a+b*sec(e+f*x)^2)^2,x,5,-1/2*a*(3*a+4*b)*arctan(sqrt(b)*tan(e+f*x)/sqrt(a+b))/(b^(5/2)*(a+b)^(3/2)*f)+tan(e+f*x)/(b^2*f)+1/2*a^2*tan(e+f*x)/(b^2*(a+b)*f*(a+b+b*tan(e+f*x)^2))],
[sec(e+f*x)^4/(a+b*sec(e+f*x)^2)^2,x,3,1/2*(a+2*b)*arctan(sqrt(b)*tan(e+f*x)/sqrt(a+b))/(b^(3/2)*(a+b)^(3/2)*f)-1/2*a*tan(e+f*x)/(b*(a+b)*f*(a+b+b*tan(e+f*x)^2))],
[sec(e+f*x)^2/(a+b*sec(e+f*x)^2)^2,x,3,1/2*arctan(sqrt(b)*tan(e+f*x)/sqrt(a+b))/((a+b)^(3/2)*f*sqrt(b))+1/2*tan(e+f*x)/((a+b)*f*(a+b+b*tan(e+f*x)^2))],
[1/(a+b*sec(e+f*x)^2)^2,x,5,x/a^2-1/2*(3*a+2*b)*arctan(sqrt(b)*tan(e+f*x)/sqrt(a+b))*sqrt(b)/(a^2*(a+b)^(3/2)*f)-1/2*b*tan(e+f*x)/(a*(a+b)*f*(a+b+b*tan(e+f*x)^2))],
[cos(e+f*x)^2/(a+b*sec(e+f*x)^2)^2,x,6,1/2*(a-4*b)*x/a^3+1/2*b^(3/2)*(5*a+4*b)*arctan(sqrt(b)*tan(e+f*x)/sqrt(a+b))/(a^3*(a+b)^(3/2)*f)+1/2*cos(e+f*x)*sin(e+f*x)/(a*f*(a+b+b*tan(e+f*x)^2))+1/2*b*(a+2*b)*tan(e+f*x)/(a^2*(a+b)*f*(a+b+b*tan(e+f*x)^2))],
[cos(e+f*x)^4/(a+b*sec(e+f*x)^2)^2,x,7,1/8*(3*a^2-8*a*b+24*b^2)*x/a^4-1/2*b^(5/2)*(7*a+6*b)*arctan(sqrt(b)*tan(e+f*x)/sqrt(a+b))/(a^4*(a+b)^(3/2)*f)+3/8*(a-2*b)*cos(e+f*x)*sin(e+f*x)/(a^2*f*(a+b+b*tan(e+f*x)^2))+1/4*cos(e+f*x)^3*sin(e+f*x)/(a*f*(a+b+b*tan(e+f*x)^2))+1/8*(a-3*b)*b*(3*a+4*b)*tan(e+f*x)/(a^3*(a+b)*f*(a+b+b*tan(e+f*x)^2))],
[cos(e+f*x)^6/(a+b*sec(e+f*x)^2)^2,x,8,1/16*(5*a^3-12*a^2*b+24*a*b^2-64*b^3)*x/a^5+1/2*b^(7/2)*(9*a+8*b)*arctan(sqrt(b)*tan(e+f*x)/sqrt(a+b))/(a^5*(a+b)^(3/2)*f)+1/48*(15*a^2-26*a*b+48*b^2)*cos(e+f*x)*sin(e+f*x)/(a^3*f*(a+b+b*tan(e+f*x)^2))+1/24*(5*a-8*b)*cos(e+f*x)^3*sin(e+f*x)/(a^2*f*(a+b+b*tan(e+f*x)^2))+1/6*cos(e+f*x)^5*sin(e+f*x)/(a*f*(a+b+b*tan(e+f*x)^2))+1/16*b*(5*a^3-7*a^2*b+12*a*b^2+32*b^3)*tan(e+f*x)/(a^4*(a+b)*f*(a+b+b*tan(e+f*x)^2))],
[sec(e+f*x)^5/(a+b*sec(e+f*x)^2)^3,x,4,1/4*sin(e+f*x)/((a+b)*f*(a+b-a*sin(e+f*x)^2)^2)+3/8*sin(e+f*x)/((a+b)^2*f*(a+b-a*sin(e+f*x)^2))+3/8*arctanh(sin(e+f*x)*sqrt(a)/sqrt(a+b))/((a+b)^(5/2)*f*sqrt(a))],
[sec(e+f*x)^3/(a+b*sec(e+f*x)^2)^3,x,4,1/8*(4*a+b)*arctanh(sin(e+f*x)*sqrt(a)/sqrt(a+b))/(a^(3/2)*(a+b)^(5/2)*f)-1/4*b*sin(e+f*x)/(a*(a+b)*f*(a+b-a*sin(e+f*x)^2)^2)+1/8*(4*a+b)*sin(e+f*x)/(a*(a+b)^2*f*(a+b-a*sin(e+f*x)^2))],
[sec(e+f*x)/(a+b*sec(e+f*x)^2)^3,x,4,1/8*(8*a^2+8*a*b+3*b^2)*arctanh(sin(e+f*x)*sqrt(a)/sqrt(a+b))/(a^(5/2)*(a+b)^(5/2)*f)-1/4*b*cos(e+f*x)^2*sin(e+f*x)/(a*(a+b)*f*(a+b-a*sin(e+f*x)^2)^2)-3/8*b*(2*a+b)*sin(e+f*x)/(a^2*(a+b)^2*f*(a+b-a*sin(e+f*x)^2))],
[cos(e+f*x)/(a+b*sec(e+f*x)^2)^3,x,6,-3/8*b*(4*(a+b)^2+(2*a+b)^2)*arctanh(sin(e+f*x)*sqrt(a)/sqrt(a+b))/(a^(7/2)*(a+b)^(5/2)*f)+sin(e+f*x)/(a^3*f)-1/4*b^3*sin(e+f*x)/(a^3*(a+b)*f*(a+b-a*sin(e+f*x)^2)^2)+3/8*b^2*(4*a+3*b)*sin(e+f*x)/(a^3*(a+b)^2*f*(a+b-a*sin(e+f*x)^2))],
[cos(e+f*x)^3/(a+b*sec(e+f*x)^2)^3,x,6,1/8*b^2*(48*a^2+80*a*b+35*b^2)*arctanh(sin(e+f*x)*sqrt(a)/sqrt(a+b))/(a^(9/2)*(a+b)^(5/2)*f)+(a-3*b)*sin(e+f*x)/(a^4*f)-1/3*sin(e+f*x)^3/(a^3*f)+1/4*b^4*sin(e+f*x)/(a^4*(a+b)*f*(a+b-a*sin(e+f*x)^2)^2)-1/8*b^3*(16*a+13*b)*sin(e+f*x)/(a^4*(a+b)^2*f*(a+b-a*sin(e+f*x)^2))],
[cos(e+f*x)^5/(a+b*sec(e+f*x)^2)^3,x,6,-1/8*b^3*(80*a^2+140*a*b+63*b^2)*arctanh(sin(e+f*x)*sqrt(a)/sqrt(a+b))/(a^(11/2)*(a+b)^(5/2)*f)+(a^2-3*a*b+6*b^2)*sin(e+f*x)/(a^5*f)-1/3*(2*a-3*b)*sin(e+f*x)^3/(a^4*f)+1/5*sin(e+f*x)^5/(a^3*f)-1/4*b^5*sin(e+f*x)/(a^5*(a+b)*f*(a+b-a*sin(e+f*x)^2)^2)+1/8*b^4*(20*a+17*b)*sin(e+f*x)/(a^5*(a+b)^2*f*(a+b-a*sin(e+f*x)^2))],
[sec(e+f*x)^6/(a+b*sec(e+f*x)^2)^3,x,4,1/8*(3*a^2+8*a*b+8*b^2)*arctan(sqrt(b)*tan(e+f*x)/sqrt(a+b))/(b^(5/2)*(a+b)^(5/2)*f)-1/4*a*sec(e+f*x)^2*tan(e+f*x)/(b*(a+b)*f*(a+b+b*tan(e+f*x)^2)^2)-3/8*a*(a+2*b)*tan(e+f*x)/(b^2*(a+b)^2*f*(a+b+b*tan(e+f*x)^2))],
[sec(e+f*x)^4/(a+b*sec(e+f*x)^2)^3,x,4,1/8*(a+4*b)*arctan(sqrt(b)*tan(e+f*x)/sqrt(a+b))/(b^(3/2)*(a+b)^(5/2)*f)-1/4*a*tan(e+f*x)/(b*(a+b)*f*(a+b+b*tan(e+f*x)^2)^2)+1/8*(a+4*b)*tan(e+f*x)/(b*(a+b)^2*f*(a+b+b*tan(e+f*x)^2))],
[sec(e+f*x)^2/(a+b*sec(e+f*x)^2)^3,x,4,3/8*arctan(sqrt(b)*tan(e+f*x)/sqrt(a+b))/((a+b)^(5/2)*f*sqrt(b))+1/4*tan(e+f*x)/((a+b)*f*(a+b+b*tan(e+f*x)^2)^2)+3/8*tan(e+f*x)/((a+b)^2*f*(a+b+b*tan(e+f*x)^2))],
[1/(a+b*sec(e+f*x)^2)^3,x,6,x/a^3-1/8*(15*a^2+20*a*b+8*b^2)*arctan(sqrt(b)*tan(e+f*x)/sqrt(a+b))*sqrt(b)/(a^3*(a+b)^(5/2)*f)-1/4*b*tan(e+f*x)/(a*(a+b)*f*(a+b+b*tan(e+f*x)^2)^2)-1/8*b*(7*a+4*b)*tan(e+f*x)/(a^2*(a+b)^2*f*(a+b+b*tan(e+f*x)^2))],
[cos(e+f*x)^2/(a+b*sec(e+f*x)^2)^3,x,7,1/2*(a-6*b)*x/a^4+1/8*b^(3/2)*(35*a^2+56*a*b+24*b^2)*arctan(sqrt(b)*tan(e+f*x)/sqrt(a+b))/(a^4*(a+b)^(5/2)*f)+1/2*cos(e+f*x)*sin(e+f*x)/(a*f*(a+b+b*tan(e+f*x)^2)^2)+1/4*b*(2*a+3*b)*tan(e+f*x)/(a^2*(a+b)*f*(a+b+b*tan(e+f*x)^2)^2)+1/8*b*(4*a+3*b)*(a+4*b)*tan(e+f*x)/(a^3*(a+b)^2*f*(a+b+b*tan(e+f*x)^2))],
[cos(e+f*x)^4/(a+b*sec(e+f*x)^2)^3,x,8,3/8*(a^2-4*a*b+16*b^2)*x/a^5-3/8*b^(5/2)*(21*a^2+36*a*b+16*b^2)*arctan(sqrt(b)*tan(e+f*x)/sqrt(a+b))/(a^5*(a+b)^(5/2)*f)+1/8*(3*a-8*b)*cos(e+f*x)*sin(e+f*x)/(a^2*f*(a+b+b*tan(e+f*x)^2)^2)+1/4*cos(e+f*x)^3*sin(e+f*x)/(a*f*(a+b+b*tan(e+f*x)^2)^2)+1/8*b*(3*a^2-7*a*b-12*b^2)*tan(e+f*x)/(a^3*(a+b)*f*(a+b+b*tan(e+f*x)^2)^2)+3/8*b*(a+2*b)*(a^2-4*a*b-4*b^2)*tan(e+f*x)/(a^4*(a+b)^2*f*(a+b+b*tan(e+f*x)^2))],
[cos(e+f*x)^6/(a+b*sec(e+f*x)^2)^3,x,9,1/16*(5*a^3-18*a^2*b+48*a*b^2-160*b^3)*x/a^6+1/8*b^(7/2)*(99*a^2+176*a*b+80*b^2)*arctan(sqrt(b)*tan(e+f*x)/sqrt(a+b))/(a^6*(a+b)^(5/2)*f)+1/48*(15*a^2-34*a*b+80*b^2)*cos(e+f*x)*sin(e+f*x)/(a^3*f*(a+b+b*tan(e+f*x)^2)^2)+5/24*(a-2*b)*cos(e+f*x)^3*sin(e+f*x)/(a^2*f*(a+b+b*tan(e+f*x)^2)^2)+1/6*cos(e+f*x)^5*sin(e+f*x)/(a*f*(a+b+b*tan(e+f*x)^2)^2)+1/48*b*(15*a^3-29*a^2*b+64*a*b^2+120*b^3)*tan(e+f*x)/(a^4*(a+b)*f*(a+b+b*tan(e+f*x)^2)^2)+1/16*b*(5*a^4-8*a^3*b+17*a^2*b^2+116*a*b^3+80*b^4)*tan(e+f*x)/(a^5*(a+b)^2*f*(a+b+b*tan(e+f*x)^2))],
[1/(a+b*sec(c+d*x)^2)^4,x,7,x/a^4-1/16*(35*a^3+70*a^2*b+56*a*b^2+16*b^3)*arctan(sqrt(b)*tan(c+d*x)/sqrt(a+b))*sqrt(b)/(a^4*(a+b)^(7/2)*d)-1/6*b*tan(c+d*x)/(a*(a+b)*d*(a+b+b*tan(c+d*x)^2)^3)-1/24*b*(11*a+6*b)*tan(c+d*x)/(a^2*(a+b)^2*d*(a+b+b*tan(c+d*x)^2)^2)-1/16*b*(19*a^2+22*a*b+8*b^2)*tan(c+d*x)/(a^3*(a+b)^3*d*(a+b+b*tan(c+d*x)^2))],

# Integrands of the form Sec[e+f x]^m (a+b Sec[e+f x]^2)^(p/2) when a+b=0
[(a-a*sec(c+d*x)^2)^(7/2),x,6,-a^3*cot(c+d*x)*log(cos(c+d*x))*sqrt(-a*tan(c+d*x)^2)/d-1/2*a^3*sqrt(-a*tan(c+d*x)^2)*tan(c+d*x)/d+1/4*a^3*sqrt(-a*tan(c+d*x)^2)*tan(c+d*x)^3/d-1/6*a^3*sqrt(-a*tan(c+d*x)^2)*tan(c+d*x)^5/d],
[(a-a*sec(c+d*x)^2)^(5/2),x,5,-a^2*cot(c+d*x)*log(cos(c+d*x))*sqrt(-a*tan(c+d*x)^2)/d-1/2*a^2*sqrt(-a*tan(c+d*x)^2)*tan(c+d*x)/d+1/4*a^2*sqrt(-a*tan(c+d*x)^2)*tan(c+d*x)^3/d],
[(a-a*sec(c+d*x)^2)^(3/2),x,4,-a*cot(c+d*x)*log(cos(c+d*x))*sqrt(-a*tan(c+d*x)^2)/d-1/2*a*sqrt(-a*tan(c+d*x)^2)*tan(c+d*x)/d],
[(a-a*sec(c+d*x)^2)^(1/2),x,3,-cot(c+d*x)*log(cos(c+d*x))*sqrt(-a*tan(c+d*x)^2)/d],
[1/(a-a*sec(c+d*x)^2)^(1/2),x,3,log(sin(c+d*x))*tan(c+d*x)/(d*sqrt(-a*tan(c+d*x)^2))],
[1/(a-a*sec(c+d*x)^2)^(3/2),x,4,1/2*cot(c+d*x)/(a*d*sqrt(-a*tan(c+d*x)^2))+log(sin(c+d*x))*tan(c+d*x)/(a*d*sqrt(-a*tan(c+d*x)^2))],
[1/(a-a*sec(c+d*x)^2)^(5/2),x,5,1/2*cot(c+d*x)/(a^2*d*sqrt(-a*tan(c+d*x)^2))-1/4*cot(c+d*x)^3/(a^2*d*sqrt(-a*tan(c+d*x)^2))+log(sin(c+d*x))*tan(c+d*x)/(a^2*d*sqrt(-a*tan(c+d*x)^2))],
[1/(a-a*sec(c+d*x)^2)^(7/2),x,6,1/2*cot(c+d*x)/(a^3*d*sqrt(-a*tan(c+d*x)^2))-1/4*cot(c+d*x)^3/(a^3*d*sqrt(-a*tan(c+d*x)^2))+1/6*cot(c+d*x)^5/(a^3*d*sqrt(-a*tan(c+d*x)^2))+log(sin(c+d*x))*tan(c+d*x)/(a^3*d*sqrt(-a*tan(c+d*x)^2))],

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

# p>0
[sec(e+f*x)^5*sqrt(a+b*sec(e+f*x)^2),x,11,-1/15*(2*a^2-3*a*b-8*b^2)*sin(e+f*x)*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2))/(b^2*f)+1/15*(2*a^2-3*a*b-8*b^2)*EllipticE(sin(e+f*x),sqrt(a/(a+b)))*sqrt(cos(e+f*x)^2)*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2))/(b^2*f*sqrt(1-a*sin(e+f*x)^2/(a+b)))-1/15*(a-8*b)*(a+b)*EllipticF(sin(e+f*x),sqrt(a/(a+b)))*sqrt(cos(e+f*x)^2)*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2))*sqrt(1-a*sin(e+f*x)^2/(a+b))/(b*f*(a+b-a*sin(e+f*x)^2))+1/15*(a+4*b)*sec(e+f*x)*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2))*tan(e+f*x)/(b*f)+1/5*sec(e+f*x)^3*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2))*tan(e+f*x)/f],
[sec(e+f*x)^3*sqrt(a+b*sec(e+f*x)^2),x,10,1/3*(a+2*b)*sin(e+f*x)*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2))/(b*f)-1/3*(a+2*b)*EllipticE(sin(e+f*x),sqrt(a/(a+b)))*sqrt(cos(e+f*x)^2)*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2))/(b*f*sqrt(1-a*sin(e+f*x)^2/(a+b)))+2/3*(a+b)*EllipticF(sin(e+f*x),sqrt(a/(a+b)))*sqrt(cos(e+f*x)^2)*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2))*sqrt(1-a*sin(e+f*x)^2/(a+b))/(f*(a+b-a*sin(e+f*x)^2))+1/3*sec(e+f*x)*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2))*tan(e+f*x)/f],
[sec(e+f*x)*sqrt(a+b*sec(e+f*x)^2),x,10,sin(e+f*x)*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2))/f-EllipticE(sin(e+f*x),sqrt(a/(a+b)))*sqrt(cos(e+f*x)^2)*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2))/(f*sqrt(1-a*sin(e+f*x)^2/(a+b)))+(a+b)*EllipticF(sin(e+f*x),sqrt(a/(a+b)))*sqrt(cos(e+f*x)^2)*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2))*sqrt(1-a*sin(e+f*x)^2/(a+b))/(f*(a+b-a*sin(e+f*x)^2))],
[cos(e+f*x)*sqrt(a+b*sec(e+f*x)^2),x,5,EllipticE(sin(e+f*x),sqrt(a/(a+b)))*sqrt(cos(e+f*x)^2)*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2))/(f*sqrt(1-a*sin(e+f*x)^2/(a+b)))],
[cos(e+f*x)^3*sqrt(a+b*sec(e+f*x)^2),x,9,1/3*cos(e+f*x)^2*sin(e+f*x)*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2))/f+1/3*(2*a+b)*EllipticE(sin(e+f*x),sqrt(a/(a+b)))*sqrt(cos(e+f*x)^2)*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2))/(a*f*sqrt(1-a*sin(e+f*x)^2/(a+b)))-1/3*b*(a+b)*EllipticF(sin(e+f*x),sqrt(a/(a+b)))*sqrt(cos(e+f*x)^2)*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2))*sqrt(1-a*sin(e+f*x)^2/(a+b))/(a*f*(a+b-a*sin(e+f*x)^2))],
[cos(e+f*x)^5*sqrt(a+b*sec(e+f*x)^2),x,10,2/15*(2*a-b)*cos(e+f*x)^2*sin(e+f*x)*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2))/(a*f)+1/5*cos(e+f*x)^2*sin(e+f*x)*(a+b-a*sin(e+f*x)^2)*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2))/(a*f)+1/15*(8*a^2+3*a*b-2*b^2)*EllipticE(sin(e+f*x),sqrt(a/(a+b)))*sqrt(cos(e+f*x)^2)*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2))/(a^2*f*sqrt(1-a*sin(e+f*x)^2/(a+b)))-2/15*(2*a-b)*b*(a+b)*EllipticF(sin(e+f*x),sqrt(a/(a+b)))*sqrt(cos(e+f*x)^2)*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2))*sqrt(1-a*sin(e+f*x)^2/(a+b))/(a^2*f*(a+b-a*sin(e+f*x)^2))],
[sec(e+f*x)^6*sqrt(a+b*sec(e+f*x)^2),x,6,1/16*(a+b)*(a^2-2*a*b+5*b^2)*arctanh(sqrt(b)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/(b^(5/2)*f)+1/16*(a^2-2*a*b+5*b^2)*sqrt(a+b+b*tan(e+f*x)^2)*tan(e+f*x)/(b^2*f)-1/24*(3*a-5*b)*tan(e+f*x)*(a+b+b*tan(e+f*x)^2)^(3/2)/(b^2*f)+1/6*sec(e+f*x)^2*tan(e+f*x)*(a+b+b*tan(e+f*x)^2)^(3/2)/(b*f)],
[sec(e+f*x)^4*sqrt(a+b*sec(e+f*x)^2),x,5,-1/8*(a-3*b)*(a+b)*arctanh(sqrt(b)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/(b^(3/2)*f)-1/8*(a-3*b)*sqrt(a+b+b*tan(e+f*x)^2)*tan(e+f*x)/(b*f)+1/4*tan(e+f*x)*(a+b+b*tan(e+f*x)^2)^(3/2)/(b*f)],
[sec(e+f*x)^2*sqrt(a+b*sec(e+f*x)^2),x,4,1/2*(a+b)*arctanh(sqrt(b)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/(f*sqrt(b))+1/2*sqrt(a+b+b*tan(e+f*x)^2)*tan(e+f*x)/f],
[sqrt(a+b*sec(e+f*x)^2),x,6,arctan(sqrt(a)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))*sqrt(a)/f+arctanh(sqrt(b)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))*sqrt(b)/f],
[cos(e+f*x)^2*sqrt(a+b*sec(e+f*x)^2),x,4,1/2*(a+b)*arctan(sqrt(a)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/(f*sqrt(a))+1/2*cos(e+f*x)*sin(e+f*x)*sqrt(a+b+b*tan(e+f*x)^2)/f],
[cos(e+f*x)^4*sqrt(a+b*sec(e+f*x)^2),x,5,1/8*(3*a-b)*(a+b)*arctan(sqrt(a)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/(a^(3/2)*f)+1/8*(3*a-b)*cos(e+f*x)*sin(e+f*x)*sqrt(a+b+b*tan(e+f*x)^2)/(a*f)+1/4*cos(e+f*x)^3*sin(e+f*x)*(a+b+b*tan(e+f*x)^2)^(3/2)/(a*f)],
[cos(e+f*x)^6*sqrt(a+b*sec(e+f*x)^2),x,7,1/16*(a+b)*(5*a^2-2*a*b+b^2)*arctan(sqrt(a)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/(a^(5/2)*f)+1/48*(3*a-b)*(5*a+3*b)*cos(e+f*x)*sin(e+f*x)*sqrt(a+b+b*tan(e+f*x)^2)/(a^2*f)+1/24*(5*a+b)*cos(e+f*x)^3*sin(e+f*x)*sqrt(a+b+b*tan(e+f*x)^2)/(a*f)+1/6*cos(e+f*x)^5*sin(e+f*x)*sqrt(a+b+b*tan(e+f*x)^2)/f],
[sec(e+f*x)^5*(a+b*sec(e+f*x)^2)^(3/2),x,12,-2/35*(a+2*b)*(a^2-4*a*b-4*b^2)*sin(e+f*x)*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2))/(b^2*f)+2/35*(a+2*b)*(a^2-4*a*b-4*b^2)*EllipticE(sin(e+f*x),sqrt(a/(a+b)))*sqrt(cos(e+f*x)^2)*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2))/(b^2*f*sqrt(1-a*sin(e+f*x)^2/(a+b)))-1/35*(a+b)*(a^2-16*a*b-16*b^2)*EllipticF(sin(e+f*x),sqrt(a/(a+b)))*sqrt(cos(e+f*x)^2)*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2))*sqrt(1-a*sin(e+f*x)^2/(a+b))/(b*f*(a+b-a*sin(e+f*x)^2))+1/35*(a^2+11*a*b+8*b^2)*sec(e+f*x)*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2))*tan(e+f*x)/(b*f)+2/35*(4*a+3*b)*sec(e+f*x)^3*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2))*tan(e+f*x)/f+1/7*b*sec(e+f*x)^5*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2))*tan(e+f*x)/f],
[sec(e+f*x)^3*(a+b*sec(e+f*x)^2)^(3/2),x,11,1/15*(3*a^2+13*a*b+8*b^2)*sin(e+f*x)*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2))/(b*f)-1/15*(3*a^2+13*a*b+8*b^2)*EllipticE(sin(e+f*x),sqrt(a/(a+b)))*sqrt(cos(e+f*x)^2)*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2))/(b*f*sqrt(1-a*sin(e+f*x)^2/(a+b)))+1/15*(a+b)*(9*a+8*b)*EllipticF(sin(e+f*x),sqrt(a/(a+b)))*sqrt(cos(e+f*x)^2)*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2))*sqrt(1-a*sin(e+f*x)^2/(a+b))/(f*(a+b-a*sin(e+f*x)^2))+2/15*(3*a+2*b)*sec(e+f*x)*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2))*tan(e+f*x)/f+1/5*b*sec(e+f*x)^3*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2))*tan(e+f*x)/f],
[sec(e+f*x)*(a+b*sec(e+f*x)^2)^(3/2),x,10,2/3*(2*a+b)*sin(e+f*x)*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2))/f-2/3*(2*a+b)*EllipticE(sin(e+f*x),sqrt(a/(a+b)))*sqrt(cos(e+f*x)^2)*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2))/(f*sqrt(1-a*sin(e+f*x)^2/(a+b)))+1/3*(a+b)*(3*a+2*b)*EllipticF(sin(e+f*x),sqrt(a/(a+b)))*sqrt(cos(e+f*x)^2)*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2))*sqrt(1-a*sin(e+f*x)^2/(a+b))/(f*(a+b-a*sin(e+f*x)^2))+1/3*b*sec(e+f*x)*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2))*tan(e+f*x)/f],
[cos(e+f*x)*(a+b*sec(e+f*x)^2)^(3/2),x,9,b*sin(e+f*x)*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2))/f+(a-b)*EllipticE(sin(e+f*x),sqrt(a/(a+b)))*sqrt(cos(e+f*x)^2)*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2))/(f*sqrt(1-a*sin(e+f*x)^2/(a+b)))+b*(a+b)*EllipticF(sin(e+f*x),sqrt(a/(a+b)))*sqrt(cos(e+f*x)^2)*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2))*sqrt(1-a*sin(e+f*x)^2/(a+b))/(f*(a+b-a*sin(e+f*x)^2))],
[cos(e+f*x)^3*(a+b*sec(e+f*x)^2)^(3/2),x,9,1/3*a*cos(e+f*x)^2*sin(e+f*x)*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2))/f+2/3*(a+2*b)*EllipticE(sin(e+f*x),sqrt(a/(a+b)))*sqrt(cos(e+f*x)^2)*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2))/(f*sqrt(1-a*sin(e+f*x)^2/(a+b)))-1/3*b*(a+b)*EllipticF(sin(e+f*x),sqrt(a/(a+b)))*sqrt(cos(e+f*x)^2)*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2))*sqrt(1-a*sin(e+f*x)^2/(a+b))/(f*(a+b-a*sin(e+f*x)^2))],
[cos(e+f*x)^5*(a+b*sec(e+f*x)^2)^(3/2),x,10,-2/15*(a-3*(a+b))*cos(e+f*x)^2*sin(e+f*x)*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2))/f+1/5*a*cos(e+f*x)^4*sin(e+f*x)*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2))/f+1/15*(8*a^2+13*a*b+3*b^2)*EllipticE(sin(e+f*x),sqrt(a/(a+b)))*sqrt(cos(e+f*x)^2)*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2))/(a*f*sqrt(1-a*sin(e+f*x)^2/(a+b)))-1/15*b*(a+b)*(4*a+3*b)*EllipticF(sin(e+f*x),sqrt(a/(a+b)))*sqrt(cos(e+f*x)^2)*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2))*sqrt(1-a*sin(e+f*x)^2/(a+b))/(a*f*(a+b-a*sin(e+f*x)^2))],
[sec(e+f*x)^6*(a+b*sec(e+f*x)^2)^(3/2),x,7,1/128*(a+b)^2*(3*a^2-10*a*b+35*b^2)*arctanh(sqrt(b)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/(b^(5/2)*f)+1/128*(a+b)*(3*a^2-10*a*b+35*b^2)*sqrt(a+b+b*tan(e+f*x)^2)*tan(e+f*x)/(b^2*f)+1/192*(3*a^2-10*a*b+35*b^2)*tan(e+f*x)*(a+b+b*tan(e+f*x)^2)^(3/2)/(b^2*f)-1/48*(3*a-7*b)*tan(e+f*x)*(a+b+b*tan(e+f*x)^2)^(5/2)/(b^2*f)+1/8*sec(e+f*x)^2*tan(e+f*x)*(a+b+b*tan(e+f*x)^2)^(5/2)/(b*f)],
[sec(e+f*x)^4*(a+b*sec(e+f*x)^2)^(3/2),x,6,-1/16*(a-5*b)*(a+b)^2*arctanh(sqrt(b)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/(b^(3/2)*f)-1/16*(a-5*b)*(a+b)*sqrt(a+b+b*tan(e+f*x)^2)*tan(e+f*x)/(b*f)-1/24*(a-5*b)*tan(e+f*x)*(a+b+b*tan(e+f*x)^2)^(3/2)/(b*f)+1/6*tan(e+f*x)*(a+b+b*tan(e+f*x)^2)^(5/2)/(b*f)],
[sec(e+f*x)^2*(a+b*sec(e+f*x)^2)^(3/2),x,5,3/8*(a+b)^2*arctanh(sqrt(b)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/(f*sqrt(b))+3/8*(a+b)*sqrt(a+b+b*tan(e+f*x)^2)*tan(e+f*x)/f+1/4*tan(e+f*x)*(a+b+b*tan(e+f*x)^2)^(3/2)/f],
[(a+b*sec(e+f*x)^2)^(3/2),x,7,a^(3/2)*arctan(sqrt(a)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/f+1/2*(3*a+b)*arctanh(sqrt(b)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))*sqrt(b)/f+1/2*b*sqrt(a+b+b*tan(e+f*x)^2)*tan(e+f*x)/f],
[cos(e+f*x)^2*(a+b*sec(e+f*x)^2)^(3/2),x,7,b^(3/2)*arctanh(sqrt(b)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/f+1/2*(a+3*b)*arctan(sqrt(a)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))*sqrt(a)/f+1/2*a*cos(e+f*x)*sin(e+f*x)*sqrt(a+b+b*tan(e+f*x)^2)/f],
[cos(e+f*x)^4*(a+b*sec(e+f*x)^2)^(3/2),x,5,3/8*(a+b)^2*arctan(sqrt(a)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/(f*sqrt(a))+3/8*(a+b)*cos(e+f*x)*sin(e+f*x)*sqrt(a+b+b*tan(e+f*x)^2)/f+1/4*cos(e+f*x)^3*sin(e+f*x)*(a+b+b*tan(e+f*x)^2)^(3/2)/f],
[cos(e+f*x)^6*(a+b*sec(e+f*x)^2)^(3/2),x,6,1/16*(5*a-b)*(a+b)^2*arctan(sqrt(a)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/(a^(3/2)*f)+1/16*(5*a-b)*(a+b)*cos(e+f*x)*sin(e+f*x)*sqrt(a+b+b*tan(e+f*x)^2)/(a*f)+1/24*(5*a-b)*cos(e+f*x)^3*sin(e+f*x)*(a+b+b*tan(e+f*x)^2)^(3/2)/(a*f)+1/6*cos(e+f*x)^5*sin(e+f*x)*(a+b+b*tan(e+f*x)^2)^(5/2)/(a*f)],
[(a+b*sec(c+d*x)^2)^(5/2),x,8,a^(5/2)*arctan(sqrt(a)*tan(c+d*x)/sqrt(a+b+b*tan(c+d*x)^2))/d+1/8*(15*a^2+10*a*b+3*b^2)*arctanh(sqrt(b)*tan(c+d*x)/sqrt(a+b+b*tan(c+d*x)^2))*sqrt(b)/d+1/8*b*(7*a+3*b)*sqrt(a+b+b*tan(c+d*x)^2)*tan(c+d*x)/d+1/4*b*tan(c+d*x)*(a+b+b*tan(c+d*x)^2)^(3/2)/d],
[(1+sec(x)^2)^(3/2),x,6,2*arcsinh(tan(x)/sqrt(2))+arctan(tan(x)/sqrt(2+tan(x)^2))+1/2*sqrt(2+tan(x)^2)*tan(x)],
[sqrt(1+sec(x)^2),x,5,arcsinh(tan(x)/sqrt(2))+arctan(tan(x)/sqrt(2+tan(x)^2))],

# p<0
[sec(e+f*x)^5/sqrt(a+b*sec(e+f*x)^2),x,10,2/3*(a-b)*EllipticE(sin(e+f*x),sqrt(a/(a+b)))*(a+b-a*sin(e+f*x)^2)/(b^2*f*sqrt(cos(e+f*x)^2)*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2))*sqrt(1-a*sin(e+f*x)^2/(a+b)))-1/3*(a-2*b)*EllipticF(sin(e+f*x),sqrt(a/(a+b)))*sqrt(1-a*sin(e+f*x)^2/(a+b))/(b*f*sqrt(cos(e+f*x)^2)*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2)))-2/3*(a-b)*sec(e+f*x)*(a+b-a*sin(e+f*x)^2)*tan(e+f*x)/(b^2*f*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2)))+1/3*sec(e+f*x)^3*(a+b-a*sin(e+f*x)^2)*tan(e+f*x)/(b*f*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2)))],
[sec(e+f*x)^3/sqrt(a+b*sec(e+f*x)^2),x,7,-EllipticE(sin(e+f*x)*sqrt(a)/sqrt(a+b),sqrt((a+b)/a))*sqrt(a)*sqrt(a+b)*sqrt(1-a*sin(e+f*x)^2/(a+b))/(b*f*sqrt(cos(e+f*x)^2)*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2)))+sec(e+f*x)*(a+b-a*sin(e+f*x)^2)*tan(e+f*x)/(b*f*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2)))],
[sec(e+f*x)/sqrt(a+b*sec(e+f*x)^2),x,5,EllipticF(sin(e+f*x),sqrt(a/(a+b)))*sqrt(1-a*sin(e+f*x)^2/(a+b))/(f*sqrt(cos(e+f*x)^2)*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2)))],
[cos(e+f*x)/sqrt(a+b*sec(e+f*x)^2),x,5,EllipticE(sin(e+f*x)*sqrt(a)/sqrt(a+b),sqrt((a+b)/a))*sqrt(a+b)*sqrt(1-a*sin(e+f*x)^2/(a+b))/(f*sqrt(a)*sqrt(cos(e+f*x)^2)*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2)))],
[cos(e+f*x)^3/sqrt(a+b*sec(e+f*x)^2),x,9,1/3*sin(e+f*x)*(a+b-a*sin(e+f*x)^2)/(a*f*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2)))+2/3*(a-b)*EllipticE(sin(e+f*x),sqrt(a/(a+b)))*(a+b-a*sin(e+f*x)^2)/(a^2*f*sqrt(cos(e+f*x)^2)*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2))*sqrt(1-a*sin(e+f*x)^2/(a+b)))-1/3*(a-2*b)*b*EllipticF(sin(e+f*x),sqrt(a/(a+b)))*sqrt(1-a*sin(e+f*x)^2/(a+b))/(a^2*f*sqrt(cos(e+f*x)^2)*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2)))],
[cos(e+f*x)^5/sqrt(a+b*sec(e+f*x)^2),x,10,4/15*(a-b)*sin(e+f*x)*(a+b-a*sin(e+f*x)^2)/(a^2*f*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2)))+1/5*cos(e+f*x)^2*sin(e+f*x)*(a+b-a*sin(e+f*x)^2)/(a*f*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2)))+1/15*(8*a^2-7*a*b+8*b^2)*EllipticE(sin(e+f*x),sqrt(a/(a+b)))*(a+b-a*sin(e+f*x)^2)/(a^3*f*sqrt(cos(e+f*x)^2)*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2))*sqrt(1-a*sin(e+f*x)^2/(a+b)))-1/15*b*(4*a^2-3*a*b+8*b^2)*EllipticF(sin(e+f*x),sqrt(a/(a+b)))*sqrt(1-a*sin(e+f*x)^2/(a+b))/(a^3*f*sqrt(cos(e+f*x)^2)*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2)))],
[sec(e+f*x)^6/sqrt(a+b*sec(e+f*x)^2),x,5,1/8*(3*a^2-2*a*b+3*b^2)*arctanh(sqrt(b)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/(b^(5/2)*f)-3/8*(a-b)*sqrt(a+b+b*tan(e+f*x)^2)*tan(e+f*x)/(b^2*f)+1/4*sec(e+f*x)^2*sqrt(a+b+b*tan(e+f*x)^2)*tan(e+f*x)/(b*f)],
[sec(e+f*x)^4/sqrt(a+b*sec(e+f*x)^2),x,4,-1/2*(a-b)*arctanh(sqrt(b)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/(b^(3/2)*f)+1/2*sqrt(a+b+b*tan(e+f*x)^2)*tan(e+f*x)/(b*f)],
[sec(e+f*x)^2/sqrt(a+b*sec(e+f*x)^2),x,3,arctanh(sqrt(b)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/(f*sqrt(b))],
[1/sqrt(a+b*sec(e+f*x)^2),x,3,arctan(sqrt(a)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/(f*sqrt(a))],
[cos(e+f*x)^2/sqrt(a+b*sec(e+f*x)^2),x,4,1/2*(a-b)*arctan(sqrt(a)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/(a^(3/2)*f)+1/2*cos(e+f*x)*sin(e+f*x)*sqrt(a+b+b*tan(e+f*x)^2)/(a*f)],
[cos(e+f*x)^4/sqrt(a+b*sec(e+f*x)^2),x,6,1/8*(3*a^2-2*a*b+3*b^2)*arctan(sqrt(a)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/(a^(5/2)*f)+3/8*(a-b)*cos(e+f*x)*sin(e+f*x)*sqrt(a+b+b*tan(e+f*x)^2)/(a^2*f)+1/4*cos(e+f*x)^3*sin(e+f*x)*sqrt(a+b+b*tan(e+f*x)^2)/(a*f)],
[cos(e+f*x)^6/sqrt(a+b*sec(e+f*x)^2),x,7,1/16*(a-b)*(5*a^2+2*a*b+5*b^2)*arctan(sqrt(a)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/(a^(7/2)*f)+1/48*(15*a^2-14*a*b+15*b^2)*cos(e+f*x)*sin(e+f*x)*sqrt(a+b+b*tan(e+f*x)^2)/(a^3*f)+5/24*(a-b)*cos(e+f*x)^3*sin(e+f*x)*sqrt(a+b+b*tan(e+f*x)^2)/(a^2*f)+1/6*cos(e+f*x)^5*sin(e+f*x)*sqrt(a+b+b*tan(e+f*x)^2)/(a*f)],
[sec(e+f*x)^5/(a+b*sec(e+f*x)^2)^(3/2),x,10,a*(2*a+b)*sin(e+f*x)/(b^2*(a+b)*f*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2)))-(2*a+b)*EllipticE(sin(e+f*x),sqrt(a/(a+b)))*(a+b-a*sin(e+f*x)^2)/(b^2*(a+b)*f*sqrt(cos(e+f*x)^2)*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2))*sqrt(1-a*sin(e+f*x)^2/(a+b)))+EllipticF(sin(e+f*x),sqrt(a/(a+b)))*sqrt(1-a*sin(e+f*x)^2/(a+b))/(b*f*sqrt(cos(e+f*x)^2)*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2)))+sec(e+f*x)*tan(e+f*x)/(b*f*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2)))],
[sec(e+f*x)^3/(a+b*sec(e+f*x)^2)^(3/2),x,7,-a*sin(e+f*x)/(b*(a+b)*f*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2)))+EllipticE(sin(e+f*x),sqrt(a/(a+b)))*(a+b-a*sin(e+f*x)^2)/(b*(a+b)*f*sqrt(cos(e+f*x)^2)*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2))*sqrt(1-a*sin(e+f*x)^2/(a+b)))],
[sec(e+f*x)/(a+b*sec(e+f*x)^2)^(3/2),x,9,sin(e+f*x)/((a+b)*f*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2)))-EllipticE(sin(e+f*x),sqrt(a/(a+b)))*(a+b-a*sin(e+f*x)^2)/(a*(a+b)*f*sqrt(cos(e+f*x)^2)*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2))*sqrt(1-a*sin(e+f*x)^2/(a+b)))+EllipticF(sin(e+f*x),sqrt(a/(a+b)))*sqrt(1-a*sin(e+f*x)^2/(a+b))/(a*f*sqrt(cos(e+f*x)^2)*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2)))],
[cos(e+f*x)/(a+b*sec(e+f*x)^2)^(3/2),x,9,-b*sin(e+f*x)/(a*(a+b)*f*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2)))+(a+2*b)*EllipticE(sin(e+f*x),sqrt(a/(a+b)))*(a+b-a*sin(e+f*x)^2)/(a^2*(a+b)*f*sqrt(cos(e+f*x)^2)*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2))*sqrt(1-a*sin(e+f*x)^2/(a+b)))-2*b*EllipticF(sin(e+f*x),sqrt(a/(a+b)))*sqrt(1-a*sin(e+f*x)^2/(a+b))/(a^2*f*sqrt(cos(e+f*x)^2)*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2)))],
[cos(e+f*x)^3/(a+b*sec(e+f*x)^2)^(3/2),x,10,-b*cos(e+f*x)^2*sin(e+f*x)/(a*(a+b)*f*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2)))+1/3*(a+4*b)*sin(e+f*x)*(a+b-a*sin(e+f*x)^2)/(a^2*(a+b)*f*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2)))+1/3*(2*a^2-3*a*b-8*b^2)*EllipticE(sin(e+f*x),sqrt(a/(a+b)))*(a+b-a*sin(e+f*x)^2)/(a^3*(a+b)*f*sqrt(cos(e+f*x)^2)*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2))*sqrt(1-a*sin(e+f*x)^2/(a+b)))-1/3*(a-8*b)*b*EllipticF(sin(e+f*x),sqrt(a/(a+b)))*sqrt(1-a*sin(e+f*x)^2/(a+b))/(a^3*f*sqrt(cos(e+f*x)^2)*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2)))],
[cos(e+f*x)^5/(a+b*sec(e+f*x)^2)^(3/2),x,11,-b*cos(e+f*x)^4*sin(e+f*x)/(a*(a+b)*f*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2)))+1/15*(4*a^2-5*a*b-24*b^2)*sin(e+f*x)*(a+b-a*sin(e+f*x)^2)/(a^3*(a+b)*f*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2)))+1/5*(a+6*b)*cos(e+f*x)^2*sin(e+f*x)*(a+b-a*sin(e+f*x)^2)/(a^2*(a+b)*f*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2)))+1/15*(8*a^3-9*a^2*b+16*a*b^2+48*b^3)*EllipticE(sin(e+f*x),sqrt(a/(a+b)))*(a+b-a*sin(e+f*x)^2)/(a^4*(a+b)*f*sqrt(cos(e+f*x)^2)*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2))*sqrt(1-a*sin(e+f*x)^2/(a+b)))-4/15*b*(a^2-2*a*b+12*b^2)*EllipticF(sin(e+f*x),sqrt(a/(a+b)))*sqrt(1-a*sin(e+f*x)^2/(a+b))/(a^4*f*sqrt(cos(e+f*x)^2)*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2)))],
[sec(e+f*x)^6/(a+b*sec(e+f*x)^2)^(3/2),x,5,-1/2*(3*a-b)*arctanh(sqrt(b)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/(b^(5/2)*f)-a*sec(e+f*x)^2*tan(e+f*x)/(b*(a+b)*f*sqrt(a+b+b*tan(e+f*x)^2))+1/2*(3*a+b)*sqrt(a+b+b*tan(e+f*x)^2)*tan(e+f*x)/(b^2*(a+b)*f)],
[sec(e+f*x)^4/(a+b*sec(e+f*x)^2)^(3/2),x,4,arctanh(sqrt(b)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/(b^(3/2)*f)-a*tan(e+f*x)/(b*(a+b)*f*sqrt(a+b+b*tan(e+f*x)^2))],
[sec(e+f*x)^2/(a+b*sec(e+f*x)^2)^(3/2),x,2,tan(e+f*x)/((a+b)*f*sqrt(a+b+b*tan(e+f*x)^2))],
[1/(a+b*sec(e+f*x)^2)^(3/2),x,4,arctan(sqrt(a)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/(a^(3/2)*f)-b*tan(e+f*x)/(a*(a+b)*f*sqrt(a+b+b*tan(e+f*x)^2))],
[cos(e+f*x)^2/(a+b*sec(e+f*x)^2)^(3/2),x,6,1/2*(a-3*b)*arctan(sqrt(a)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/(a^(5/2)*f)+1/2*cos(e+f*x)*sin(e+f*x)/(a*f*sqrt(a+b+b*tan(e+f*x)^2))+1/2*b*(a+3*b)*tan(e+f*x)/(a^2*(a+b)*f*sqrt(a+b+b*tan(e+f*x)^2))],
[cos(e+f*x)^4/(a+b*sec(e+f*x)^2)^(3/2),x,7,3/8*(a^2-2*a*b+5*b^2)*arctan(sqrt(a)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/(a^(7/2)*f)+1/8*(3*a-5*b)*cos(e+f*x)*sin(e+f*x)/(a^2*f*sqrt(a+b+b*tan(e+f*x)^2))+1/4*cos(e+f*x)^3*sin(e+f*x)/(a*f*sqrt(a+b+b*tan(e+f*x)^2))+1/8*(a-3*b)*b*(3*a+5*b)*tan(e+f*x)/(a^3*(a+b)*f*sqrt(a+b+b*tan(e+f*x)^2))],
[cos(e+f*x)^6/(a+b*sec(e+f*x)^2)^(3/2),x,8,1/16*(5*a^3-9*a^2*b+15*a*b^2-35*b^3)*arctan(sqrt(a)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/(a^(9/2)*f)+1/48*(15*a^2-22*a*b+35*b^2)*cos(e+f*x)*sin(e+f*x)/(a^3*f*sqrt(a+b+b*tan(e+f*x)^2))+1/24*(5*a-7*b)*cos(e+f*x)^3*sin(e+f*x)/(a^2*f*sqrt(a+b+b*tan(e+f*x)^2))+1/6*cos(e+f*x)^5*sin(e+f*x)/(a*f*sqrt(a+b+b*tan(e+f*x)^2))+1/48*b*(15*a^3-17*a^2*b+25*a*b^2+105*b^3)*tan(e+f*x)/(a^4*(a+b)*f*sqrt(a+b+b*tan(e+f*x)^2))],
[sec(e+f*x)^5/(a+b*sec(e+f*x)^2)^(5/2),x,10,-2/3*a*(a+2*b)*sin(e+f*x)/(b^2*(a+b)^2*f*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2)))-1/3*a*sin(e+f*x)/(b*(a+b)*f*(a+b-a*sin(e+f*x)^2)*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2)))+2/3*(a+2*b)*EllipticE(sin(e+f*x),sqrt(a/(a+b)))*(a+b-a*sin(e+f*x)^2)/(b^2*(a+b)^2*f*sqrt(cos(e+f*x)^2)*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2))*sqrt(1-a*sin(e+f*x)^2/(a+b)))-1/3*EllipticF(sin(e+f*x),sqrt(a/(a+b)))*sqrt(1-a*sin(e+f*x)^2/(a+b))/(b*(a+b)*f*sqrt(cos(e+f*x)^2)*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2)))],
[sec(e+f*x)^3/(a+b*sec(e+f*x)^2)^(5/2),x,10,-1/3*(a-b)*sin(e+f*x)/(b*(a+b)^2*f*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2)))+1/3*sin(e+f*x)/((a+b)*f*(a+b-a*sin(e+f*x)^2)*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2)))+1/3*(a-b)*EllipticE(sin(e+f*x),sqrt(a/(a+b)))*(a+b-a*sin(e+f*x)^2)/(a*b*(a+b)^2*f*sqrt(cos(e+f*x)^2)*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2))*sqrt(1-a*sin(e+f*x)^2/(a+b)))+1/3*EllipticF(sin(e+f*x),sqrt(a/(a+b)))*sqrt(1-a*sin(e+f*x)^2/(a+b))/(a*(a+b)*f*sqrt(cos(e+f*x)^2)*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2)))],
[sec(e+f*x)/(a+b*sec(e+f*x)^2)^(5/2),x,10,2/3*(2*a+b)*sin(e+f*x)/(a*(a+b)^2*f*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2)))-1/3*b*sin(e+f*x)/(a*(a+b)*f*(a+b-a*sin(e+f*x)^2)*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2)))-2/3*(2*a+b)*EllipticE(sin(e+f*x),sqrt(a/(a+b)))*(a+b-a*sin(e+f*x)^2)/(a^2*(a+b)^2*f*sqrt(cos(e+f*x)^2)*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2))*sqrt(1-a*sin(e+f*x)^2/(a+b)))+1/3*(3*a+2*b)*EllipticF(sin(e+f*x),sqrt(a/(a+b)))*sqrt(1-a*sin(e+f*x)^2/(a+b))/(a^2*(a+b)*f*sqrt(cos(e+f*x)^2)*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2)))],
[cos(e+f*x)/(a+b*sec(e+f*x)^2)^(5/2),x,10,-2/3*b*(3*a+2*b)*sin(e+f*x)/(a^2*(a+b)^2*f*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2)))-1/3*b*cos(e+f*x)^2*sin(e+f*x)/(a*(a+b)*f*(a+b-a*sin(e+f*x)^2)*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2)))+1/3*(3*a^2+13*a*b+8*b^2)*EllipticE(sin(e+f*x),sqrt(a/(a+b)))*(a+b-a*sin(e+f*x)^2)/(a^3*(a+b)^2*f*sqrt(cos(e+f*x)^2)*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2))*sqrt(1-a*sin(e+f*x)^2/(a+b)))-1/3*b*(9*a+8*b)*EllipticF(sin(e+f*x),sqrt(a/(a+b)))*sqrt(1-a*sin(e+f*x)^2/(a+b))/(a^3*(a+b)*f*sqrt(cos(e+f*x)^2)*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2)))],
[cos(e+f*x)^3/(a+b*sec(e+f*x)^2)^(5/2),x,11,-2/3*b*(4*a+3*b)*cos(e+f*x)^2*sin(e+f*x)/(a^2*(a+b)^2*f*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2)))-1/3*b*cos(e+f*x)^4*sin(e+f*x)/(a*(a+b)*f*(a+b-a*sin(e+f*x)^2)*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2)))+1/3*(a^2+11*a*b+8*b^2)*sin(e+f*x)*(a+b-a*sin(e+f*x)^2)/(a^3*(a+b)^2*f*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2)))+2/3*(a+2*b)*(a^2-4*a*b-4*b^2)*EllipticE(sin(e+f*x),sqrt(a/(a+b)))*(a+b-a*sin(e+f*x)^2)/(a^4*(a+b)^2*f*sqrt(cos(e+f*x)^2)*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2))*sqrt(1-a*sin(e+f*x)^2/(a+b)))-1/3*b*(a^2-16*a*b-16*b^2)*EllipticF(sin(e+f*x),sqrt(a/(a+b)))*sqrt(1-a*sin(e+f*x)^2/(a+b))/(a^4*(a+b)*f*sqrt(cos(e+f*x)^2)*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2)))],
[cos(e+f*x)^5/(a+b*sec(e+f*x)^2)^(5/2),x,12,-2/3*b*(5*a+4*b)*cos(e+f*x)^4*sin(e+f*x)/(a^2*(a+b)^2*f*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2)))-1/3*b*cos(e+f*x)^6*sin(e+f*x)/(a*(a+b)*f*(a+b-a*sin(e+f*x)^2)*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2)))+2/15*(2*a^3-3*a^2*b-42*a*b^2-32*b^3)*sin(e+f*x)*(a+b-a*sin(e+f*x)^2)/(a^4*(a+b)^2*f*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2)))+1/15*(3*a^2+61*a*b+48*b^2)*cos(e+f*x)^2*sin(e+f*x)*(a+b-a*sin(e+f*x)^2)/(a^3*(a+b)^2*f*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2)))+1/15*(8*a^4-11*a^3*b+27*a^2*b^2+184*a*b^3+128*b^4)*EllipticE(sin(e+f*x),sqrt(a/(a+b)))*(a+b-a*sin(e+f*x)^2)/(a^5*(a+b)^2*f*sqrt(cos(e+f*x)^2)*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2))*sqrt(1-a*sin(e+f*x)^2/(a+b)))-1/15*b*(4*a^3-9*a^2*b+120*a*b^2+128*b^3)*EllipticF(sin(e+f*x),sqrt(a/(a+b)))*sqrt(1-a*sin(e+f*x)^2/(a+b))/(a^5*(a+b)*f*sqrt(cos(e+f*x)^2)*sqrt(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2)))],
[sec(e+f*x)^6/(a+b*sec(e+f*x)^2)^(5/2),x,5,arctanh(sqrt(b)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/(b^(5/2)*f)-1/3*a*(3*a+5*b)*tan(e+f*x)/(b^2*(a+b)^2*f*sqrt(a+b+b*tan(e+f*x)^2))-1/3*a*sec(e+f*x)^2*tan(e+f*x)/(b*(a+b)*f*(a+b+b*tan(e+f*x)^2)^(3/2))],
[sec(e+f*x)^4/(a+b*sec(e+f*x)^2)^(5/2),x,3,2/3*tan(e+f*x)/((a+b)^2*f*sqrt(a+b+b*tan(e+f*x)^2))+1/3*sec(e+f*x)^2*tan(e+f*x)/((a+b)*f*(a+b+b*tan(e+f*x)^2)^(3/2))],
[sec(e+f*x)^2/(a+b*sec(e+f*x)^2)^(5/2),x,3,2/3*tan(e+f*x)/((a+b)^2*f*sqrt(a+b+b*tan(e+f*x)^2))+1/3*tan(e+f*x)/((a+b)*f*(a+b+b*tan(e+f*x)^2)^(3/2))],
[1/(a+b*sec(e+f*x)^2)^(5/2),x,6,arctan(sqrt(a)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/(a^(5/2)*f)-1/3*b*(5*a+3*b)*tan(e+f*x)/(a^2*(a+b)^2*f*sqrt(a+b+b*tan(e+f*x)^2))-1/3*b*tan(e+f*x)/(a*(a+b)*f*(a+b+b*tan(e+f*x)^2)^(3/2))],
[cos(e+f*x)^2/(a+b*sec(e+f*x)^2)^(5/2),x,7,1/2*(a-5*b)*arctan(sqrt(a)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/(a^(7/2)*f)+1/6*b*(3*a^2+22*a*b+15*b^2)*tan(e+f*x)/(a^3*(a+b)^2*f*sqrt(a+b+b*tan(e+f*x)^2))+1/2*cos(e+f*x)*sin(e+f*x)/(a*f*(a+b+b*tan(e+f*x)^2)^(3/2))+1/6*b*(3*a+5*b)*tan(e+f*x)/(a^2*(a+b)*f*(a+b+b*tan(e+f*x)^2)^(3/2))],
[cos(e+f*x)^4/(a+b*sec(e+f*x)^2)^(5/2),x,8,1/8*(3*a^2-10*a*b+35*b^2)*arctan(sqrt(a)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/(a^(9/2)*f)+1/24*b*(9*a^3-15*a^2*b-145*a*b^2-105*b^3)*tan(e+f*x)/(a^4*(a+b)^2*f*sqrt(a+b+b*tan(e+f*x)^2))+1/8*(3*a-7*b)*cos(e+f*x)*sin(e+f*x)/(a^2*f*(a+b+b*tan(e+f*x)^2)^(3/2))+1/4*cos(e+f*x)^3*sin(e+f*x)/(a*f*(a+b+b*tan(e+f*x)^2)^(3/2))+1/24*b*(9*a^2-18*a*b-35*b^2)*tan(e+f*x)/(a^3*(a+b)*f*(a+b+b*tan(e+f*x)^2)^(3/2))],
[cos(e+f*x)^6/(a+b*sec(e+f*x)^2)^(5/2),x,9,5/16*(a-3*b)*(a^2+7*b^2)*arctan(sqrt(a)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/(a^(11/2)*f)+1/48*b*(15*a^4-20*a^3*b+38*a^2*b^2+420*a*b^3+315*b^4)*tan(e+f*x)/(a^5*(a+b)^2*f*sqrt(a+b+b*tan(e+f*x)^2))+1/16*(5*a^2-10*a*b+21*b^2)*cos(e+f*x)*sin(e+f*x)/(a^3*f*(a+b+b*tan(e+f*x)^2)^(3/2))+1/24*(5*a-9*b)*cos(e+f*x)^3*sin(e+f*x)/(a^2*f*(a+b+b*tan(e+f*x)^2)^(3/2))+1/6*cos(e+f*x)^5*sin(e+f*x)/(a*f*(a+b+b*tan(e+f*x)^2)^(3/2))+1/48*b*(15*a^3-25*a^2*b+49*a*b^2+105*b^3)*tan(e+f*x)/(a^4*(a+b)*f*(a+b+b*tan(e+f*x)^2)^(3/2))],
[1/(a+b*sec(c+d*x)^2)^(7/2),x,7,arctan(sqrt(a)*tan(c+d*x)/sqrt(a+b+b*tan(c+d*x)^2))/(a^(7/2)*d)-1/15*b*(33*a^2+40*a*b+15*b^2)*tan(c+d*x)/(a^3*(a+b)^3*d*sqrt(a+b+b*tan(c+d*x)^2))-1/5*b*tan(c+d*x)/(a*(a+b)*d*(a+b+b*tan(c+d*x)^2)^(5/2))-1/15*b*(9*a+5*b)*tan(c+d*x)/(a^2*(a+b)^2*d*(a+b+b*tan(c+d*x)^2)^(3/2))],
[1/sqrt(1+sec(x)^2),x,3,arctan(tan(x)/sqrt(2+tan(x)^2))],

# Integrands of the form Sec[e+f x]^m (a+b Sec[e+f x]^2)^p when p symbolic
[(d*sec(e+f*x))^m*(a+b*sec(e+f*x)^2)^p,x,-1,AppellF1(1/2*m,1/2,-p,1/2*(2+m),sec(e+f*x)^2,-b*sec(e+f*x)^2/a)*cos(e+f*x)*(d*sec(e+f*x))^m*(a+b*sec(e+f*x)^2)^p*sqrt(-tan(e+f*x)^2)/(f*m*(1+b*sec(e+f*x)^2/a)^p*sin(e+f*x))],
[sec(e+f*x)^3*(a+b*sec(e+f*x)^2)^p,x,5,AppellF1(1/2,2+p,-p,3/2,sin(e+f*x)^2,a*sin(e+f*x)^2/(a+b))*(cos(e+f*x)^2)^p*sin(e+f*x)*(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2))^p/(f*(1-a*sin(e+f*x)^2/(a+b))^p)],
[sec(e+f*x)*(a+b*sec(e+f*x)^2)^p,x,5,AppellF1(1/2,1+p,-p,3/2,sin(e+f*x)^2,a*sin(e+f*x)^2/(a+b))*(cos(e+f*x)^2)^p*sin(e+f*x)*(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2))^p/(f*(1-a*sin(e+f*x)^2/(a+b))^p)],
[cos(e+f*x)*(a+b*sec(e+f*x)^2)^p,x,5,AppellF1(1/2,p,-p,3/2,sin(e+f*x)^2,a*sin(e+f*x)^2/(a+b))*(cos(e+f*x)^2)^p*sin(e+f*x)*(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2))^p/(f*(1-a*sin(e+f*x)^2/(a+b))^p)],
[cos(e+f*x)^3*(a+b*sec(e+f*x)^2)^p,x,5,AppellF1(1/2,-1+p,-p,3/2,sin(e+f*x)^2,a*sin(e+f*x)^2/(a+b))*(cos(e+f*x)^2)^p*sin(e+f*x)*(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2))^p/(f*(1-a*sin(e+f*x)^2/(a+b))^p)],
[cos(e+f*x)^5*(a+b*sec(e+f*x)^2)^p,x,5,AppellF1(1/2,-2+p,-p,3/2,sin(e+f*x)^2,a*sin(e+f*x)^2/(a+b))*(cos(e+f*x)^2)^p*sin(e+f*x)*(sec(e+f*x)^2*(a+b-a*sin(e+f*x)^2))^p/(f*(1-a*sin(e+f*x)^2/(a+b))^p)],
[sec(e+f*x)^6*(a+b*sec(e+f*x)^2)^p,x,5,-(3*a-2*b*(2+p))*tan(e+f*x)*(a+b+b*tan(e+f*x)^2)^(1+p)/(b^2*f*(15+16*p+4*p^2))+sec(e+f*x)^2*tan(e+f*x)*(a+b+b*tan(e+f*x)^2)^(1+p)/(b*f*(5+2*p))+(3*a^2-4*a*b*(1+p)+4*b^2*(2+3*p+p^2))*hypergeom([1/2,-p],[3/2],-b*tan(e+f*x)^2/(a+b))*tan(e+f*x)*(a+b+b*tan(e+f*x)^2)^p/(b^2*f*(15+16*p+4*p^2)*(1+b*tan(e+f*x)^2/(a+b))^p)],
[sec(e+f*x)^4*(a+b*sec(e+f*x)^2)^p,x,4,tan(e+f*x)*(a+b+b*tan(e+f*x)^2)^(1+p)/(b*f*(3+2*p))-(a-2*b*(1+p))*hypergeom([1/2,-p],[3/2],-b*tan(e+f*x)^2/(a+b))*tan(e+f*x)*(a+b+b*tan(e+f*x)^2)^p/(b*f*(3+2*p)*(1+b*tan(e+f*x)^2/(a+b))^p)],
[sec(e+f*x)^2*(a+b*sec(e+f*x)^2)^p,x,3,hypergeom([1/2,-p],[3/2],-b*tan(e+f*x)^2/(a+b))*tan(e+f*x)*(a+b+b*tan(e+f*x)^2)^p/(f*(1+b*tan(e+f*x)^2/(a+b))^p)],
[(a+b*sec(e+f*x)^2)^p,x,3,AppellF1(1/2,1,-p,3/2,-tan(e+f*x)^2,-b*tan(e+f*x)^2/(a+b))*tan(e+f*x)*(a+b+b*tan(e+f*x)^2)^p/(f*(1+b*tan(e+f*x)^2/(a+b))^p)],
[cos(e+f*x)^2*(a+b*sec(e+f*x)^2)^p,x,3,AppellF1(1/2,2,-p,3/2,-tan(e+f*x)^2,-b*tan(e+f*x)^2/(a+b))*tan(e+f*x)*(a+b+b*tan(e+f*x)^2)^p/(f*(1+b*tan(e+f*x)^2/(a+b))^p)],
[cos(e+f*x)^4*(a+b*sec(e+f*x)^2)^p,x,3,AppellF1(1/2,3,-p,3/2,-tan(e+f*x)^2,-b*tan(e+f*x)^2/(a+b))*tan(e+f*x)*(a+b+b*tan(e+f*x)^2)^p/(f*(1+b*tan(e+f*x)^2/(a+b))^p)],
[cos(e+f*x)^6*(a+b*sec(e+f*x)^2)^p,x,3,AppellF1(1/2,4,-p,3/2,-tan(e+f*x)^2,-b*tan(e+f*x)^2/(a+b))*tan(e+f*x)*(a+b+b*tan(e+f*x)^2)^p/(f*(1+b*tan(e+f*x)^2/(a+b))^p)],

# Integrands of the form Sec[e+f x]^m (a+b Sec[e+f x]^3)^p

# Integrands of the form Sec[e+f x]^m (a+b Sec[e+f x]^4)^p

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

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

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

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

# p>0
[(a+b*sec(e+f*x)^2)*tan(e+f*x)^5,x,4,-a*log(cos(e+f*x))/f-1/2*(2*a-b)*sec(e+f*x)^2/f+1/4*(a-2*b)*sec(e+f*x)^4/f+1/6*b*sec(e+f*x)^6/f],
[(a+b*sec(e+f*x)^2)*tan(e+f*x)^3,x,4,a*log(cos(e+f*x))/f+1/2*(a-b)*sec(e+f*x)^2/f+1/4*b*sec(e+f*x)^4/f],
[(a+b*sec(e+f*x)^2)*tan(e+f*x),x,3,-a*log(cos(e+f*x))/f+1/2*b*sec(e+f*x)^2/f],
[cot(e+f*x)*(a+b*sec(e+f*x)^2),x,4,-b*log(cos(e+f*x))/f+(a+b)*log(sin(e+f*x))/f],
[cot(e+f*x)^3*(a+b*sec(e+f*x)^2),x,4,-1/2*(a+b)*csc(e+f*x)^2/f-a*log(sin(e+f*x))/f],
[cot(e+f*x)^5*(a+b*sec(e+f*x)^2),x,4,1/2*(2*a+b)*csc(e+f*x)^2/f-1/4*(a+b)*csc(e+f*x)^4/f+a*log(sin(e+f*x))/f],
[(a+b*sec(e+f*x)^2)*tan(e+f*x)^6,x,4,-a*x+a*tan(e+f*x)/f-1/3*a*tan(e+f*x)^3/f+1/5*a*tan(e+f*x)^5/f+1/7*b*tan(e+f*x)^7/f],
[(a+b*sec(e+f*x)^2)*tan(e+f*x)^4,x,4,a*x-a*tan(e+f*x)/f+1/3*a*tan(e+f*x)^3/f+1/5*b*tan(e+f*x)^5/f],
[(a+b*sec(e+f*x)^2)*tan(e+f*x)^2,x,4,-a*x+a*tan(e+f*x)/f+1/3*b*tan(e+f*x)^3/f],
[a+b*sec(e+f*x)^2,x,3,a*x+b*tan(e+f*x)/f],
[cot(e+f*x)^2*(a+b*sec(e+f*x)^2),x,4,-a*x-(a+b)*cot(e+f*x)/f],
[cot(e+f*x)^4*(a+b*sec(e+f*x)^2),x,4,a*x+a*cot(e+f*x)/f-1/3*(a+b)*cot(e+f*x)^3/f],
[cot(e+f*x)^6*(a+b*sec(e+f*x)^2),x,4,-a*x-a*cot(e+f*x)/f+1/3*a*cot(e+f*x)^3/f-1/5*(a+b)*cot(e+f*x)^5/f],
[(a+b*sec(e+f*x)^2)^2*tan(e+f*x)^5,x,4,-a^2*log(cos(e+f*x))/f-a*(a-b)*sec(e+f*x)^2/f+1/4*(a^2-4*a*b+b^2)*sec(e+f*x)^4/f+1/3*(a-b)*b*sec(e+f*x)^6/f+1/8*b^2*sec(e+f*x)^8/f],
[(a+b*sec(e+f*x)^2)^2*tan(e+f*x)^3,x,4,a^2*log(cos(e+f*x))/f+1/2*a*(a-2*b)*sec(e+f*x)^2/f+1/4*(2*a-b)*b*sec(e+f*x)^4/f+1/6*b^2*sec(e+f*x)^6/f],
[(a+b*sec(e+f*x)^2)^2*tan(e+f*x),x,4,-a^2*log(cos(e+f*x))/f+a*b*sec(e+f*x)^2/f+1/4*b^2*sec(e+f*x)^4/f],
[cot(e+f*x)*(a+b*sec(e+f*x)^2)^2,x,4,-b*(2*a+b)*log(cos(e+f*x))/f+(a+b)^2*log(sin(e+f*x))/f+1/2*b^2*sec(e+f*x)^2/f],
[cot(e+f*x)^3*(a+b*sec(e+f*x)^2)^2,x,4,-1/2*(a+b)^2*csc(e+f*x)^2/f-b^2*log(cos(e+f*x))/f-(a^2-b^2)*log(sin(e+f*x))/f],
[cot(e+f*x)^5*(a+b*sec(e+f*x)^2)^2,x,4,a*(a+b)*csc(e+f*x)^2/f-1/4*(a+b)^2*csc(e+f*x)^4/f+a^2*log(sin(e+f*x))/f],
[(a+b*sec(e+f*x)^2)^2*tan(e+f*x)^6,x,4,-a^2*x+a^2*tan(e+f*x)/f-1/3*a^2*tan(e+f*x)^3/f+1/5*a^2*tan(e+f*x)^5/f+1/7*b*(2*a+b)*tan(e+f*x)^7/f+1/9*b^2*tan(e+f*x)^9/f],
[(a+b*sec(e+f*x)^2)^2*tan(e+f*x)^4,x,4,a^2*x-a^2*tan(e+f*x)/f+1/3*a^2*tan(e+f*x)^3/f+1/5*b*(2*a+b)*tan(e+f*x)^5/f+1/7*b^2*tan(e+f*x)^7/f],
[(a+b*sec(e+f*x)^2)^2*tan(e+f*x)^2,x,4,-a^2*x+a^2*tan(e+f*x)/f+1/3*b*(2*a+b)*tan(e+f*x)^3/f+1/5*b^2*tan(e+f*x)^5/f],
[(a+b*sec(e+f*x)^2)^2,x,4,a^2*x+b*(2*a+b)*tan(e+f*x)/f+1/3*b^2*tan(e+f*x)^3/f],
[cot(e+f*x)^2*(a+b*sec(e+f*x)^2)^2,x,4,-a^2*x-(a+b)^2*cot(e+f*x)/f+b^2*tan(e+f*x)/f],
[cot(e+f*x)^4*(a+b*sec(e+f*x)^2)^2,x,4,a^2*x+(a^2-b^2)*cot(e+f*x)/f-1/3*(a+b)^2*cot(e+f*x)^3/f],
[cot(e+f*x)^6*(a+b*sec(e+f*x)^2)^2,x,4,-a^2*x-a^2*cot(e+f*x)/f+1/3*(a^2-b^2)*cot(e+f*x)^3/f-1/5*(a+b)^2*cot(e+f*x)^5/f],

# p<0
[tan(e+f*x)^5/(a+b*sec(e+f*x)^2),x,4,(a+2*b)*log(cos(e+f*x))/(b^2*f)-1/2*(a+b)^2*log(b+a*cos(e+f*x)^2)/(a*b^2*f)+1/2*sec(e+f*x)^2/(b*f)],
[tan(e+f*x)^3/(a+b*sec(e+f*x)^2),x,4,-log(cos(e+f*x))/(b*f)+1/2*(a+b)*log(b+a*cos(e+f*x)^2)/(a*b*f)],
[tan(e+f*x)/(a+b*sec(e+f*x)^2),x,2,-1/2*log(b+a*cos(e+f*x)^2)/(a*f)],
[cot(e+f*x)/(a+b*sec(e+f*x)^2),x,4,1/2*b*log(b+a*cos(e+f*x)^2)/(a*(a+b)*f)+log(sin(e+f*x))/((a+b)*f)],
[cot(e+f*x)^3/(a+b*sec(e+f*x)^2),x,4,-1/2*csc(e+f*x)^2/((a+b)*f)-1/2*b^2*log(b+a*cos(e+f*x)^2)/(a*(a+b)^2*f)-(a+2*b)*log(sin(e+f*x))/((a+b)^2*f)],
[cot(e+f*x)^5/(a+b*sec(e+f*x)^2),x,4,1/2*(2*a+3*b)*csc(e+f*x)^2/((a+b)^2*f)-1/4*csc(e+f*x)^4/((a+b)*f)+1/2*b^3*log(b+a*cos(e+f*x)^2)/(a*(a+b)^3*f)+(a^2+3*a*b+3*b^2)*log(sin(e+f*x))/((a+b)^3*f)],
[tan(e+f*x)^6/(a+b*sec(e+f*x)^2),x,7,-x/a+(a+b)^(5/2)*arctan(sqrt(b)*tan(e+f*x)/sqrt(a+b))/(a*b^(5/2)*f)-(a+2*b)*tan(e+f*x)/(b^2*f)+1/3*tan(e+f*x)^3/(b*f)],
[tan(e+f*x)^4/(a+b*sec(e+f*x)^2),x,6,x/a-(a+b)^(3/2)*arctan(sqrt(b)*tan(e+f*x)/sqrt(a+b))/(a*b^(3/2)*f)+tan(e+f*x)/(b*f)],
[tan(e+f*x)^2/(a+b*sec(e+f*x)^2),x,5,-x/a+arctan(sqrt(b)*tan(e+f*x)/sqrt(a+b))*sqrt(a+b)/(a*f*sqrt(b))],
[1/(a+b*sec(e+f*x)^2),x,3,x/a+arctan(cot(e+f*x)*sqrt(a+b)/sqrt(b))*sqrt(b)/(a*f*sqrt(a+b))],
[cot(e+f*x)^2/(a+b*sec(e+f*x)^2),x,6,-x/a+b^(3/2)*arctan(sqrt(b)*tan(e+f*x)/sqrt(a+b))/(a*(a+b)^(3/2)*f)-cot(e+f*x)/((a+b)*f)],
[cot(e+f*x)^4/(a+b*sec(e+f*x)^2),x,7,x/a-b^(5/2)*arctan(sqrt(b)*tan(e+f*x)/sqrt(a+b))/(a*(a+b)^(5/2)*f)+(a+2*b)*cot(e+f*x)/((a+b)^2*f)-1/3*cot(e+f*x)^3/((a+b)*f)],
[cot(e+f*x)^6/(a+b*sec(e+f*x)^2),x,8,-x/a+b^(7/2)*arctan(sqrt(b)*tan(e+f*x)/sqrt(a+b))/(a*(a+b)^(7/2)*f)-(a^2+3*a*b+3*b^2)*cot(e+f*x)/((a+b)^3*f)+1/3*(a+2*b)*cot(e+f*x)^3/((a+b)^2*f)-1/5*cot(e+f*x)^5/((a+b)*f)],
[tan(e+f*x)^5/(a+b*sec(e+f*x)^2)^2,x,4,-1/2*(a+b)^2/(a^2*b*f*(b+a*cos(e+f*x)^2))-log(cos(e+f*x))/(b^2*f)-1/2*(1/a^2+(-1)/b^2)*log(b+a*cos(e+f*x)^2)/f],
[tan(e+f*x)^3/(a+b*sec(e+f*x)^2)^2,x,4,1/2*(a+b)/(a^2*f*(b+a*cos(e+f*x)^2))+1/2*log(b+a*cos(e+f*x)^2)/(a^2*f)],
[tan(e+f*x)/(a+b*sec(e+f*x)^2)^2,x,4,-1/2*b/(a^2*f*(b+a*cos(e+f*x)^2))-1/2*log(b+a*cos(e+f*x)^2)/(a^2*f)],
[cot(e+f*x)/(a+b*sec(e+f*x)^2)^2,x,4,1/2*b^2/(a^2*(a+b)*f*(b+a*cos(e+f*x)^2))+1/2*b*(2*a+b)*log(b+a*cos(e+f*x)^2)/(a^2*(a+b)^2*f)+log(sin(e+f*x))/((a+b)^2*f)],
[cot(e+f*x)^3/(a+b*sec(e+f*x)^2)^2,x,4,-1/2*b^3/(a^2*(a+b)^2*f*(b+a*cos(e+f*x)^2))-1/2*csc(e+f*x)^2/((a+b)^2*f)-1/2*b^2*(3*a+b)*log(b+a*cos(e+f*x)^2)/(a^2*(a+b)^3*f)-(a+3*b)*log(sin(e+f*x))/((a+b)^3*f)],
[cot(e+f*x)^5/(a+b*sec(e+f*x)^2)^2,x,4,1/2*b^4/(a^2*(a+b)^3*f*(b+a*cos(e+f*x)^2))+(a+2*b)*csc(e+f*x)^2/((a+b)^3*f)-1/4*csc(e+f*x)^4/((a+b)^2*f)+1/2*b^3*(4*a+b)*log(b+a*cos(e+f*x)^2)/(a^2*(a+b)^4*f)+(a^2+4*a*b+6*b^2)*log(sin(e+f*x))/((a+b)^4*f)],
[tan(e+f*x)^6/(a+b*sec(e+f*x)^2)^2,x,7,-x/a^2-1/2*(3*a-2*b)*(a+b)^(3/2)*arctan(sqrt(b)*tan(e+f*x)/sqrt(a+b))/(a^2*b^(5/2)*f)+1/2*(3*a+b)*tan(e+f*x)/(a*b^2*f)-1/2*(a+b)*tan(e+f*x)^3/(a*b*f*(a+b+b*tan(e+f*x)^2))],
[tan(e+f*x)^4/(a+b*sec(e+f*x)^2)^2,x,6,x/a^2+1/2*(a-2*b)*arctan(sqrt(b)*tan(e+f*x)/sqrt(a+b))*sqrt(a+b)/(a^2*b^(3/2)*f)-1/2*(a+b)*tan(e+f*x)/(a*b*f*(a+b+b*tan(e+f*x)^2))],
[tan(e+f*x)^2/(a+b*sec(e+f*x)^2)^2,x,6,-x/a^2+1/2*(a+2*b)*arctan(sqrt(b)*tan(e+f*x)/sqrt(a+b))/(a^2*f*sqrt(b)*sqrt(a+b))+1/2*tan(e+f*x)/(a*f*(a+b+b*tan(e+f*x)^2))],
[1/(a+b*sec(e+f*x)^2)^2,x,5,x/a^2-1/2*(3*a+2*b)*arctan(sqrt(b)*tan(e+f*x)/sqrt(a+b))*sqrt(b)/(a^2*(a+b)^(3/2)*f)-1/2*b*tan(e+f*x)/(a*(a+b)*f*(a+b+b*tan(e+f*x)^2))],
[cot(e+f*x)^2/(a+b*sec(e+f*x)^2)^2,x,7,-x/a^2+1/2*b^(3/2)*(5*a+2*b)*arctan(sqrt(b)*tan(e+f*x)/sqrt(a+b))/(a^2*(a+b)^(5/2)*f)-1/2*(2*a-b)*cot(e+f*x)/(a*(a+b)^2*f)-1/2*b*cot(e+f*x)/(a*(a+b)*f*(a+b+b*tan(e+f*x)^2))],
[cot(e+f*x)^4/(a+b*sec(e+f*x)^2)^2,x,8,x/a^2-1/2*b^(5/2)*(7*a+2*b)*arctan(sqrt(b)*tan(e+f*x)/sqrt(a+b))/(a^2*(a+b)^(7/2)*f)+1/2*(2*a^2+6*a*b-b^2)*cot(e+f*x)/(a*(a+b)^3*f)-1/6*(2*a-3*b)*cot(e+f*x)^3/(a*(a+b)^2*f)-1/2*b*cot(e+f*x)^3/(a*(a+b)*f*(a+b+b*tan(e+f*x)^2))],
[cot(e+f*x)^6/(a+b*sec(e+f*x)^2)^2,x,9,-x/a^2+1/2*b^(7/2)*(9*a+2*b)*arctan(sqrt(b)*tan(e+f*x)/sqrt(a+b))/(a^2*(a+b)^(9/2)*f)-1/2*(2*a^3+8*a^2*b+12*a*b^2-b^3)*cot(e+f*x)/(a*(a+b)^4*f)+1/6*(2*a^2+6*a*b-3*b^2)*cot(e+f*x)^3/(a*(a+b)^3*f)-1/10*(2*a-5*b)*cot(e+f*x)^5/(a*(a+b)^2*f)-1/2*b*cot(e+f*x)^5/(a*(a+b)*f*(a+b+b*tan(e+f*x)^2))],
[tan(e+f*x)^5/(a+b*sec(e+f*x)^2)^3,x,4,1/4*(a+b)^2/(a^3*f*(b+a*cos(e+f*x)^2)^2)+(-a-b)/(a^3*f*(b+a*cos(e+f*x)^2))-1/2*log(b+a*cos(e+f*x)^2)/(a^3*f)],
[tan(e+f*x)^3/(a+b*sec(e+f*x)^2)^3,x,4,-1/4*b*(a+b)/(a^3*f*(b+a*cos(e+f*x)^2)^2)+1/2*(a+2*b)/(a^3*f*(b+a*cos(e+f*x)^2))+1/2*log(b+a*cos(e+f*x)^2)/(a^3*f)],
[tan(e+f*x)/(a+b*sec(e+f*x)^2)^3,x,4,1/4*b^2/(a^3*f*(b+a*cos(e+f*x)^2)^2)-b/(a^3*f*(b+a*cos(e+f*x)^2))-1/2*log(b+a*cos(e+f*x)^2)/(a^3*f)],
[cot(e+f*x)/(a+b*sec(e+f*x)^2)^3,x,4,-1/4*b^3/(a^3*(a+b)*f*(b+a*cos(e+f*x)^2)^2)+1/2*b^2*(3*a+2*b)/(a^3*(a+b)^2*f*(b+a*cos(e+f*x)^2))+1/2*b*(3*a^2+3*a*b+b^2)*log(b+a*cos(e+f*x)^2)/(a^3*(a+b)^3*f)+log(sin(e+f*x))/((a+b)^3*f)],
[cot(e+f*x)^3/(a+b*sec(e+f*x)^2)^3,x,4,1/4*b^4/(a^3*(a+b)^2*f*(b+a*cos(e+f*x)^2)^2)-b^3*(2*a+b)/(a^3*(a+b)^3*f*(b+a*cos(e+f*x)^2))-1/2*csc(e+f*x)^2/((a+b)^3*f)-1/2*b^2*(6*a^2+4*a*b+b^2)*log(b+a*cos(e+f*x)^2)/(a^3*(a+b)^4*f)-(a+4*b)*log(sin(e+f*x))/((a+b)^4*f)],
[cot(e+f*x)^5/(a+b*sec(e+f*x)^2)^3,x,4,-1/4*b^5/(a^3*(a+b)^3*f*(b+a*cos(e+f*x)^2)^2)+1/2*b^4*(5*a+2*b)/(a^3*(a+b)^4*f*(b+a*cos(e+f*x)^2))+1/2*(2*a+5*b)*csc(e+f*x)^2/((a+b)^4*f)-1/4*csc(e+f*x)^4/((a+b)^3*f)+1/2*b^3*(10*a^2+5*a*b+b^2)*log(b+a*cos(e+f*x)^2)/(a^3*(a+b)^5*f)+(a^2+5*a*b+10*b^2)*log(sin(e+f*x))/((a+b)^5*f)],
[tan(e+f*x)^6/(a+b*sec(e+f*x)^2)^3,x,7,-x/a^3+1/8*(3*a^2-4*a*b+8*b^2)*arctan(sqrt(b)*tan(e+f*x)/sqrt(a+b))*sqrt(a+b)/(a^3*b^(5/2)*f)-1/4*(a+b)*tan(e+f*x)^3/(a*b*f*(a+b+b*tan(e+f*x)^2)^2)-1/8*(3*a-4*b)*(a+b)*tan(e+f*x)/(a^2*b^2*f*(a+b+b*tan(e+f*x)^2))],
[tan(e+f*x)^4/(a+b*sec(e+f*x)^2)^3,x,7,x/a^3+1/8*(a^2-4*a*b-8*b^2)*arctan(sqrt(b)*tan(e+f*x)/sqrt(a+b))/(a^3*b^(3/2)*f*sqrt(a+b))-1/4*(a+b)*tan(e+f*x)/(a*b*f*(a+b+b*tan(e+f*x)^2)^2)+1/8*(a-4*b)*tan(e+f*x)/(a^2*b*f*(a+b+b*tan(e+f*x)^2))],
[tan(e+f*x)^2/(a+b*sec(e+f*x)^2)^3,x,7,-x/a^3+1/8*(3*a^2+12*a*b+8*b^2)*arctan(sqrt(b)*tan(e+f*x)/sqrt(a+b))/(a^3*(a+b)^(3/2)*f*sqrt(b))+1/4*tan(e+f*x)/(a*f*(a+b+b*tan(e+f*x)^2)^2)+1/8*(3*a+4*b)*tan(e+f*x)/(a^2*(a+b)*f*(a+b+b*tan(e+f*x)^2))],
[1/(a+b*sec(e+f*x)^2)^3,x,6,x/a^3-1/8*(15*a^2+20*a*b+8*b^2)*arctan(sqrt(b)*tan(e+f*x)/sqrt(a+b))*sqrt(b)/(a^3*(a+b)^(5/2)*f)-1/4*b*tan(e+f*x)/(a*(a+b)*f*(a+b+b*tan(e+f*x)^2)^2)-1/8*b*(7*a+4*b)*tan(e+f*x)/(a^2*(a+b)^2*f*(a+b+b*tan(e+f*x)^2))],
[cot(e+f*x)^2/(a+b*sec(e+f*x)^2)^3,x,8,-x/a^3+1/8*b^(3/2)*(35*a^2+28*a*b+8*b^2)*arctan(sqrt(b)*tan(e+f*x)/sqrt(a+b))/(a^3*(a+b)^(7/2)*f)-1/8*(8*a^2-11*a*b-4*b^2)*cot(e+f*x)/(a^2*(a+b)^3*f)-1/4*b*cot(e+f*x)/(a*(a+b)*f*(a+b+b*tan(e+f*x)^2)^2)-1/8*b*(9*a+4*b)*cot(e+f*x)/(a^2*(a+b)^2*f*(a+b+b*tan(e+f*x)^2))],
[cot(e+f*x)^4/(a+b*sec(e+f*x)^2)^3,x,9,x/a^3-1/8*b^(5/2)*(63*a^2+36*a*b+8*b^2)*arctan(sqrt(b)*tan(e+f*x)/sqrt(a+b))/(a^3*(a+b)^(9/2)*f)+1/8*(8*a^3+32*a^2*b-15*a*b^2-4*b^3)*cot(e+f*x)/(a^2*(a+b)^4*f)-1/24*(8*a^2-39*a*b-12*b^2)*cot(e+f*x)^3/(a^2*(a+b)^3*f)-1/4*b*cot(e+f*x)^3/(a*(a+b)*f*(a+b+b*tan(e+f*x)^2)^2)-1/8*b*(11*a+4*b)*cot(e+f*x)^3/(a^2*(a+b)^2*f*(a+b+b*tan(e+f*x)^2))],
[cot(e+f*x)^6/(a+b*sec(e+f*x)^2)^3,x,10,-x/a^3+1/8*b^(7/2)*(99*a^2+44*a*b+8*b^2)*arctan(sqrt(b)*tan(e+f*x)/sqrt(a+b))/(a^3*(a+b)^(11/2)*f)-1/8*(8*a^4+40*a^3*b+80*a^2*b^2-19*a*b^3-4*b^4)*cot(e+f*x)/(a^2*(a+b)^5*f)+1/24*(8*a^3+32*a^2*b-51*a*b^2-12*b^3)*cot(e+f*x)^3/(a^2*(a+b)^4*f)-1/40*(8*a^2-75*a*b-20*b^2)*cot(e+f*x)^5/(a^2*(a+b)^3*f)-1/4*b*cot(e+f*x)^5/(a*(a+b)*f*(a+b+b*tan(e+f*x)^2)^2)-1/8*b*(13*a+4*b)*cot(e+f*x)^5/(a^2*(a+b)^2*f*(a+b+b*tan(e+f*x)^2))],

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

# p>0
[sqrt(a+b*sec(e+f*x)^2)*tan(e+f*x)^5,x,7,-1/3*(a+2*b)*(a+b*sec(e+f*x)^2)^(3/2)/(b^2*f)+1/5*(a+b*sec(e+f*x)^2)^(5/2)/(b^2*f)-arctanh(sqrt(a+b*sec(e+f*x)^2)/sqrt(a))*sqrt(a)/f+sqrt(a+b*sec(e+f*x)^2)/f],
[sqrt(a+b*sec(e+f*x)^2)*tan(e+f*x)^3,x,6,1/3*(a+b*sec(e+f*x)^2)^(3/2)/(b*f)+arctanh(sqrt(a+b*sec(e+f*x)^2)/sqrt(a))*sqrt(a)/f-sqrt(a+b*sec(e+f*x)^2)/f],
[sqrt(a+b*sec(e+f*x)^2)*tan(e+f*x),x,5,-arctanh(sqrt(a+b*sec(e+f*x)^2)/sqrt(a))*sqrt(a)/f+sqrt(a+b*sec(e+f*x)^2)/f],
[cot(e+f*x)*sqrt(a+b*sec(e+f*x)^2),x,7,arctanh(sqrt(a+b*sec(e+f*x)^2)/sqrt(a))*sqrt(a)/f-arctanh(sqrt(a+b*sec(e+f*x)^2)/sqrt(a+b))*sqrt(a+b)/f],
[cot(e+f*x)^3*sqrt(a+b*sec(e+f*x)^2),x,8,-arctanh(sqrt(a+b*sec(e+f*x)^2)/sqrt(a))*sqrt(a)/f+1/2*(2*a+b)*arctanh(sqrt(a+b*sec(e+f*x)^2)/sqrt(a+b))/(f*sqrt(a+b))-1/2*cot(e+f*x)^2*sqrt(a+b*sec(e+f*x)^2)/f],
[cot(e+f*x)^5*sqrt(a+b*sec(e+f*x)^2),x,9,-1/8*(8*a^2+12*a*b+3*b^2)*arctanh(sqrt(a+b*sec(e+f*x)^2)/sqrt(a+b))/((a+b)^(3/2)*f)+arctanh(sqrt(a+b*sec(e+f*x)^2)/sqrt(a))*sqrt(a)/f+1/8*(4*a+3*b)*cot(e+f*x)^2*sqrt(a+b*sec(e+f*x)^2)/((a+b)*f)-1/4*cot(e+f*x)^4*sqrt(a+b*sec(e+f*x)^2)/f],
[sqrt(a+b*sec(e+f*x)^2)*tan(e+f*x)^6,x,10,1/16*(a^3+5*a^2*b+15*a*b^2-5*b^3)*arctanh(sqrt(b)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/(b^(5/2)*f)-arctan(sqrt(a)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))*sqrt(a)/f-1/16*(a-b)*(a+5*b)*sqrt(a+b+b*tan(e+f*x)^2)*tan(e+f*x)/(b^2*f)+1/24*(a-5*b)*sqrt(a+b+b*tan(e+f*x)^2)*tan(e+f*x)^3/(b*f)+1/6*sqrt(a+b+b*tan(e+f*x)^2)*tan(e+f*x)^5/f],
[sqrt(a+b*sec(e+f*x)^2)*tan(e+f*x)^4,x,9,-1/8*(a^2+6*a*b-3*b^2)*arctanh(sqrt(b)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/(b^(3/2)*f)+arctan(sqrt(a)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))*sqrt(a)/f+1/8*(a-3*b)*sqrt(a+b+b*tan(e+f*x)^2)*tan(e+f*x)/(b*f)+1/4*sqrt(a+b+b*tan(e+f*x)^2)*tan(e+f*x)^3/f],
[sqrt(a+b*sec(e+f*x)^2)*tan(e+f*x)^2,x,8,-arctan(sqrt(a)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))*sqrt(a)/f+1/2*(a-b)*arctanh(sqrt(b)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/(f*sqrt(b))+1/2*sqrt(a+b+b*tan(e+f*x)^2)*tan(e+f*x)/f],
[sqrt(a+b*sec(e+f*x)^2),x,6,arctan(sqrt(a)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))*sqrt(a)/f+arctanh(sqrt(b)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))*sqrt(b)/f],
[cot(e+f*x)^2*sqrt(a+b*sec(e+f*x)^2),x,6,-arctan(sqrt(a)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))*sqrt(a)/f-cot(e+f*x)*sqrt(a+b+b*tan(e+f*x)^2)/f],
[cot(e+f*x)^4*sqrt(a+b*sec(e+f*x)^2),x,7,arctan(sqrt(a)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))*sqrt(a)/f+1/3*(3*a+2*b)*cot(e+f*x)*sqrt(a+b+b*tan(e+f*x)^2)/((a+b)*f)-1/3*cot(e+f*x)^3*sqrt(a+b+b*tan(e+f*x)^2)/f],
[cot(e+f*x)^6*sqrt(a+b*sec(e+f*x)^2),x,8,-arctan(sqrt(a)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))*sqrt(a)/f-1/15*(15*a^2+25*a*b+8*b^2)*cot(e+f*x)*sqrt(a+b+b*tan(e+f*x)^2)/((a+b)^2*f)-1/15*(b-5*(a+b))*cot(e+f*x)^3*sqrt(a+b+b*tan(e+f*x)^2)/((a+b)*f)-1/5*cot(e+f*x)^5*sqrt(a+b+b*tan(e+f*x)^2)/f],
[(a+b*sec(e+f*x)^2)^(3/2)*tan(e+f*x)^5,x,8,-a^(3/2)*arctanh(sqrt(a+b*sec(e+f*x)^2)/sqrt(a))/f+1/3*(a+b*sec(e+f*x)^2)^(3/2)/f-1/5*(a+2*b)*(a+b*sec(e+f*x)^2)^(5/2)/(b^2*f)+1/7*(a+b*sec(e+f*x)^2)^(7/2)/(b^2*f)+a*sqrt(a+b*sec(e+f*x)^2)/f],
[(a+b*sec(e+f*x)^2)^(3/2)*tan(e+f*x)^3,x,7,a^(3/2)*arctanh(sqrt(a+b*sec(e+f*x)^2)/sqrt(a))/f-1/3*(a+b*sec(e+f*x)^2)^(3/2)/f+1/5*(a+b*sec(e+f*x)^2)^(5/2)/(b*f)-a*sqrt(a+b*sec(e+f*x)^2)/f],
[(a+b*sec(e+f*x)^2)^(3/2)*tan(e+f*x),x,6,-a^(3/2)*arctanh(sqrt(a+b*sec(e+f*x)^2)/sqrt(a))/f+1/3*(a+b*sec(e+f*x)^2)^(3/2)/f+a*sqrt(a+b*sec(e+f*x)^2)/f],
[cot(e+f*x)*(a+b*sec(e+f*x)^2)^(3/2),x,8,a^(3/2)*arctanh(sqrt(a+b*sec(e+f*x)^2)/sqrt(a))/f-(a+b)^(3/2)*arctanh(sqrt(a+b*sec(e+f*x)^2)/sqrt(a+b))/f+b*sqrt(a+b*sec(e+f*x)^2)/f],
[cot(e+f*x)^3*(a+b*sec(e+f*x)^2)^(3/2),x,8,-a^(3/2)*arctanh(sqrt(a+b*sec(e+f*x)^2)/sqrt(a))/f+1/2*(2*a-b)*arctanh(sqrt(a+b*sec(e+f*x)^2)/sqrt(a+b))*sqrt(a+b)/f-1/2*(a+b)*cot(e+f*x)^2*sqrt(a+b*sec(e+f*x)^2)/f],
[cot(e+f*x)^5*(a+b*sec(e+f*x)^2)^(3/2),x,9,a^(3/2)*arctanh(sqrt(a+b*sec(e+f*x)^2)/sqrt(a))/f-1/8*(8*a^2+4*a*b-b^2)*arctanh(sqrt(a+b*sec(e+f*x)^2)/sqrt(a+b))/(f*sqrt(a+b))+1/8*(4*a-b)*cot(e+f*x)^2*sqrt(a+b*sec(e+f*x)^2)/f-1/4*(a+b)*cot(e+f*x)^4*sqrt(a+b*sec(e+f*x)^2)/f],
[(a+b*sec(e+f*x)^2)^(3/2)*tan(e+f*x)^6,x,11,-a^(3/2)*arctan(sqrt(a)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/f+1/128*(3*a^4+20*a^3*b+90*a^2*b^2-60*a*b^3-5*b^4)*arctanh(sqrt(b)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/(b^(5/2)*f)-1/128*(3*a^3+17*a^2*b-55*a*b^2-5*b^3)*sqrt(a+b+b*tan(e+f*x)^2)*tan(e+f*x)/(b^2*f)+1/192*(3*a^2-50*a*b-5*b^2)*sqrt(a+b+b*tan(e+f*x)^2)*tan(e+f*x)^3/(b*f)+1/48*(9*a+b)*sqrt(a+b+b*tan(e+f*x)^2)*tan(e+f*x)^5/f+1/8*b*sqrt(a+b+b*tan(e+f*x)^2)*tan(e+f*x)^7/f],
[(a+b*sec(e+f*x)^2)^(3/2)*tan(e+f*x)^4,x,10,a^(3/2)*arctan(sqrt(a)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/f-1/16*(a-b)*(a^2+10*a*b+b^2)*arctanh(sqrt(b)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/(b^(3/2)*f)+1/16*(a^2-8*a*b-b^2)*sqrt(a+b+b*tan(e+f*x)^2)*tan(e+f*x)/(b*f)+1/24*(7*a+b)*sqrt(a+b+b*tan(e+f*x)^2)*tan(e+f*x)^3/f+1/6*b*sqrt(a+b+b*tan(e+f*x)^2)*tan(e+f*x)^5/f],
[(a+b*sec(e+f*x)^2)^(3/2)*tan(e+f*x)^2,x,9,-a^(3/2)*arctan(sqrt(a)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/f+1/8*(3*a^2-6*a*b-b^2)*arctanh(sqrt(b)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/(f*sqrt(b))+1/8*(5*a+b)*sqrt(a+b+b*tan(e+f*x)^2)*tan(e+f*x)/f+1/4*b*sqrt(a+b+b*tan(e+f*x)^2)*tan(e+f*x)^3/f],
[(a+b*sec(e+f*x)^2)^(3/2),x,7,a^(3/2)*arctan(sqrt(a)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/f+1/2*(3*a+b)*arctanh(sqrt(b)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))*sqrt(b)/f+1/2*b*sqrt(a+b+b*tan(e+f*x)^2)*tan(e+f*x)/f],
[cot(e+f*x)^2*(a+b*sec(e+f*x)^2)^(3/2),x,8,-a^(3/2)*arctan(sqrt(a)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/f+b^(3/2)*arctanh(sqrt(b)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/f-(a+b)*cot(e+f*x)*sqrt(a+b+b*tan(e+f*x)^2)/f],
[cot(e+f*x)^4*(a+b*sec(e+f*x)^2)^(3/2),x,7,a^(3/2)*arctan(sqrt(a)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/f+1/3*(3*a-b)*cot(e+f*x)*sqrt(a+b+b*tan(e+f*x)^2)/f-1/3*(a+b)*cot(e+f*x)^3*sqrt(a+b+b*tan(e+f*x)^2)/f],
[cot(e+f*x)^6*(a+b*sec(e+f*x)^2)^(3/2),x,8,-a^(3/2)*arctan(sqrt(a)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/f-1/15*(15*a^2+10*a*b-2*b^2)*cot(e+f*x)*sqrt(a+b+b*tan(e+f*x)^2)/((a+b)*f)+1/15*(5*a-b)*cot(e+f*x)^3*sqrt(a+b+b*tan(e+f*x)^2)/f-1/5*(a+b)*cot(e+f*x)^5*sqrt(a+b+b*tan(e+f*x)^2)/f],

# p<0
[tan(e+f*x)^5/sqrt(a+b*sec(e+f*x)^2),x,6,1/3*(a+b*sec(e+f*x)^2)^(3/2)/(b^2*f)-arctanh(sqrt(a+b*sec(e+f*x)^2)/sqrt(a))/(f*sqrt(a))-(a+2*b)*sqrt(a+b*sec(e+f*x)^2)/(b^2*f)],
[tan(e+f*x)^3/sqrt(a+b*sec(e+f*x)^2),x,5,arctanh(sqrt(a+b*sec(e+f*x)^2)/sqrt(a))/(f*sqrt(a))+sqrt(a+b*sec(e+f*x)^2)/(b*f)],
[tan(e+f*x)/sqrt(a+b*sec(e+f*x)^2),x,4,-arctanh(sqrt(a+b*sec(e+f*x)^2)/sqrt(a))/(f*sqrt(a))],
[cot(e+f*x)/sqrt(a+b*sec(e+f*x)^2),x,7,arctanh(sqrt(a+b*sec(e+f*x)^2)/sqrt(a))/(f*sqrt(a))-arctanh(sqrt(a+b*sec(e+f*x)^2)/sqrt(a+b))/(f*sqrt(a+b))],
[cot(e+f*x)^3/sqrt(a+b*sec(e+f*x)^2),x,8,1/2*(2*a+3*b)*arctanh(sqrt(a+b*sec(e+f*x)^2)/sqrt(a+b))/((a+b)^(3/2)*f)-arctanh(sqrt(a+b*sec(e+f*x)^2)/sqrt(a))/(f*sqrt(a))-1/2*cot(e+f*x)^2*sqrt(a+b*sec(e+f*x)^2)/((a+b)*f)],
[cot(e+f*x)^5/sqrt(a+b*sec(e+f*x)^2),x,9,-1/8*(8*a^2+20*a*b+15*b^2)*arctanh(sqrt(a+b*sec(e+f*x)^2)/sqrt(a+b))/((a+b)^(5/2)*f)+arctanh(sqrt(a+b*sec(e+f*x)^2)/sqrt(a))/(f*sqrt(a))+1/8*(4*a+7*b)*cot(e+f*x)^2*sqrt(a+b*sec(e+f*x)^2)/((a+b)^2*f)-1/4*cot(e+f*x)^4*sqrt(a+b*sec(e+f*x)^2)/((a+b)*f)],
[tan(e+f*x)^6/sqrt(a+b*sec(e+f*x)^2),x,9,1/8*(3*a^2+10*a*b+15*b^2)*arctanh(sqrt(b)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/(b^(5/2)*f)-arctan(sqrt(a)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/(f*sqrt(a))-1/8*(3*a+7*b)*sqrt(a+b+b*tan(e+f*x)^2)*tan(e+f*x)/(b^2*f)+1/4*sqrt(a+b+b*tan(e+f*x)^2)*tan(e+f*x)^3/(b*f)],
[tan(e+f*x)^4/sqrt(a+b*sec(e+f*x)^2),x,8,-1/2*(a+3*b)*arctanh(sqrt(b)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/(b^(3/2)*f)+arctan(sqrt(a)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/(f*sqrt(a))+1/2*sqrt(a+b+b*tan(e+f*x)^2)*tan(e+f*x)/(b*f)],
[tan(e+f*x)^2/sqrt(a+b*sec(e+f*x)^2),x,7,-arctan(sqrt(a)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/(f*sqrt(a))+arctanh(sqrt(b)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/(f*sqrt(b))],
[1/sqrt(a+b*sec(e+f*x)^2),x,3,arctan(sqrt(a)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/(f*sqrt(a))],
[cot(e+f*x)^2/sqrt(a+b*sec(e+f*x)^2),x,6,-arctan(sqrt(a)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/(f*sqrt(a))-cot(e+f*x)*sqrt(a+b+b*tan(e+f*x)^2)/((a+b)*f)],
[cot(e+f*x)^4/sqrt(a+b*sec(e+f*x)^2),x,7,arctan(sqrt(a)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/(f*sqrt(a))+1/3*(3*a+5*b)*cot(e+f*x)*sqrt(a+b+b*tan(e+f*x)^2)/((a+b)^2*f)-1/3*cot(e+f*x)^3*sqrt(a+b+b*tan(e+f*x)^2)/((a+b)*f)],
[cot(e+f*x)^6/sqrt(a+b*sec(e+f*x)^2),x,8,-arctan(sqrt(a)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/(f*sqrt(a))-1/15*(15*a^2+40*a*b+33*b^2)*cot(e+f*x)*sqrt(a+b+b*tan(e+f*x)^2)/((a+b)^3*f)+1/15*(5*a+9*b)*cot(e+f*x)^3*sqrt(a+b+b*tan(e+f*x)^2)/((a+b)^2*f)-1/5*cot(e+f*x)^5*sqrt(a+b+b*tan(e+f*x)^2)/((a+b)*f)],
[tan(e+f*x)^5/(a+b*sec(e+f*x)^2)^(3/2),x,6,-arctanh(sqrt(a+b*sec(e+f*x)^2)/sqrt(a))/(a^(3/2)*f)+(a+b)^2/(a*b^2*f*sqrt(a+b*sec(e+f*x)^2))+sqrt(a+b*sec(e+f*x)^2)/(b^2*f)],
[tan(e+f*x)^3/(a+b*sec(e+f*x)^2)^(3/2),x,5,arctanh(sqrt(a+b*sec(e+f*x)^2)/sqrt(a))/(a^(3/2)*f)+(-a-b)/(a*b*f*sqrt(a+b*sec(e+f*x)^2))],
[tan(e+f*x)/(a+b*sec(e+f*x)^2)^(3/2),x,5,-arctanh(sqrt(a+b*sec(e+f*x)^2)/sqrt(a))/(a^(3/2)*f)+1/(a*f*sqrt(a+b*sec(e+f*x)^2))],
[cot(e+f*x)/(a+b*sec(e+f*x)^2)^(3/2),x,8,arctanh(sqrt(a+b*sec(e+f*x)^2)/sqrt(a))/(a^(3/2)*f)-arctanh(sqrt(a+b*sec(e+f*x)^2)/sqrt(a+b))/((a+b)^(3/2)*f)-b/(a*(a+b)*f*sqrt(a+b*sec(e+f*x)^2))],
[cot(e+f*x)^3/(a+b*sec(e+f*x)^2)^(3/2),x,9,-arctanh(sqrt(a+b*sec(e+f*x)^2)/sqrt(a))/(a^(3/2)*f)+1/2*(2*a+5*b)*arctanh(sqrt(a+b*sec(e+f*x)^2)/sqrt(a+b))/((a+b)^(5/2)*f)-1/2*(a-2*b)*b/(a*(a+b)^2*f*sqrt(a+b*sec(e+f*x)^2))-1/2*cot(e+f*x)^2/((a+b)*f*sqrt(a+b*sec(e+f*x)^2))],
[cot(e+f*x)^5/(a+b*sec(e+f*x)^2)^(3/2),x,10,arctanh(sqrt(a+b*sec(e+f*x)^2)/sqrt(a))/(a^(3/2)*f)-1/8*(8*a^2+28*a*b+35*b^2)*arctanh(sqrt(a+b*sec(e+f*x)^2)/sqrt(a+b))/((a+b)^(7/2)*f)+1/8*b*(4*a^2+11*a*b-8*b^2)/(a*(a+b)^3*f*sqrt(a+b*sec(e+f*x)^2))+1/8*(4*a+9*b)*cot(e+f*x)^2/((a+b)^2*f*sqrt(a+b*sec(e+f*x)^2))-1/4*cot(e+f*x)^4/((a+b)*f*sqrt(a+b*sec(e+f*x)^2))],
[tan(e+f*x)^6/(a+b*sec(e+f*x)^2)^(3/2),x,9,-arctan(sqrt(a)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/(a^(3/2)*f)-1/2*(3*a+5*b)*arctanh(sqrt(b)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/(b^(5/2)*f)+1/2*(3*a+2*b)*sqrt(a+b+b*tan(e+f*x)^2)*tan(e+f*x)/(a*b^2*f)-(a+b)*tan(e+f*x)^3/(a*b*f*sqrt(a+b+b*tan(e+f*x)^2))],
[tan(e+f*x)^4/(a+b*sec(e+f*x)^2)^(3/2),x,8,arctan(sqrt(a)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/(a^(3/2)*f)+arctanh(sqrt(b)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/(b^(3/2)*f)-(a+b)*tan(e+f*x)/(a*b*f*sqrt(a+b+b*tan(e+f*x)^2))],
[tan(e+f*x)^2/(a+b*sec(e+f*x)^2)^(3/2),x,5,-arctan(sqrt(a)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/(a^(3/2)*f)+tan(e+f*x)/(a*f*sqrt(a+b+b*tan(e+f*x)^2))],
[1/(a+b*sec(e+f*x)^2)^(3/2),x,4,arctan(sqrt(a)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/(a^(3/2)*f)-b*tan(e+f*x)/(a*(a+b)*f*sqrt(a+b+b*tan(e+f*x)^2))],
[cot(e+f*x)^2/(a+b*sec(e+f*x)^2)^(3/2),x,7,-arctan(sqrt(a)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/(a^(3/2)*f)-b*cot(e+f*x)/(a*(a+b)*f*sqrt(a+b+b*tan(e+f*x)^2))-(a-b)*cot(e+f*x)*sqrt(a+b+b*tan(e+f*x)^2)/(a*(a+b)^2*f)],
[cot(e+f*x)^4/(a+b*sec(e+f*x)^2)^(3/2),x,8,arctan(sqrt(a)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/(a^(3/2)*f)-b*cot(e+f*x)^3/(a*(a+b)*f*sqrt(a+b+b*tan(e+f*x)^2))+1/3*(3*a-b)*(a+3*b)*cot(e+f*x)*sqrt(a+b+b*tan(e+f*x)^2)/(a*(a+b)^3*f)-1/3*(a-3*b)*cot(e+f*x)^3*sqrt(a+b+b*tan(e+f*x)^2)/(a*(a+b)^2*f)],
[cot(e+f*x)^6/(a+b*sec(e+f*x)^2)^(3/2),x,9,-arctan(sqrt(a)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/(a^(3/2)*f)-b*cot(e+f*x)^5/(a*(a+b)*f*sqrt(a+b+b*tan(e+f*x)^2))-1/15*(15*a^3+55*a^2*b+73*a*b^2-15*b^3)*cot(e+f*x)*sqrt(a+b+b*tan(e+f*x)^2)/(a*(a+b)^4*f)+1/15*(5*a^2+14*a*b-15*b^2)*cot(e+f*x)^3*sqrt(a+b+b*tan(e+f*x)^2)/(a*(a+b)^3*f)-1/5*(a-5*b)*cot(e+f*x)^5*sqrt(a+b+b*tan(e+f*x)^2)/(a*(a+b)^2*f)],
[tan(e+f*x)^5/(a+b*sec(e+f*x)^2)^(5/2),x,6,-arctanh(sqrt(a+b*sec(e+f*x)^2)/sqrt(a))/(a^(5/2)*f)+1/3*(a+b)^2/(a*b^2*f*(a+b*sec(e+f*x)^2)^(3/2))+(1/a^2+(-1)/b^2)/(f*sqrt(a+b*sec(e+f*x)^2))],
[tan(e+f*x)^3/(a+b*sec(e+f*x)^2)^(5/2),x,6,arctanh(sqrt(a+b*sec(e+f*x)^2)/sqrt(a))/(a^(5/2)*f)+1/3*(-a-b)/(a*b*f*(a+b*sec(e+f*x)^2)^(3/2))+(-1)/(a^2*f*sqrt(a+b*sec(e+f*x)^2))],
[tan(e+f*x)/(a+b*sec(e+f*x)^2)^(5/2),x,6,-arctanh(sqrt(a+b*sec(e+f*x)^2)/sqrt(a))/(a^(5/2)*f)+1/3/(a*f*(a+b*sec(e+f*x)^2)^(3/2))+1/(a^2*f*sqrt(a+b*sec(e+f*x)^2))],
[cot(e+f*x)/(a+b*sec(e+f*x)^2)^(5/2),x,9,arctanh(sqrt(a+b*sec(e+f*x)^2)/sqrt(a))/(a^(5/2)*f)-arctanh(sqrt(a+b*sec(e+f*x)^2)/sqrt(a+b))/((a+b)^(5/2)*f)-1/3*b/(a*(a+b)*f*(a+b*sec(e+f*x)^2)^(3/2))-b*(2*a+b)/(a^2*(a+b)^2*f*sqrt(a+b*sec(e+f*x)^2))],
[cot(e+f*x)^3/(a+b*sec(e+f*x)^2)^(5/2),x,10,-arctanh(sqrt(a+b*sec(e+f*x)^2)/sqrt(a))/(a^(5/2)*f)+1/2*(2*a+7*b)*arctanh(sqrt(a+b*sec(e+f*x)^2)/sqrt(a+b))/((a+b)^(7/2)*f)-1/6*(3*a-2*b)*b/(a*(a+b)^2*f*(a+b*sec(e+f*x)^2)^(3/2))-1/2*cot(e+f*x)^2/((a+b)*f*(a+b*sec(e+f*x)^2)^(3/2))-1/2*b*(a^2-6*a*b-2*b^2)/(a^2*(a+b)^3*f*sqrt(a+b*sec(e+f*x)^2))],
[cot(e+f*x)^5/(a+b*sec(e+f*x)^2)^(5/2),x,11,arctanh(sqrt(a+b*sec(e+f*x)^2)/sqrt(a))/(a^(5/2)*f)-1/8*(8*a^2+36*a*b+63*b^2)*arctanh(sqrt(a+b*sec(e+f*x)^2)/sqrt(a+b))/((a+b)^(9/2)*f)+1/24*b*(12*a^2+39*a*b-8*b^2)/(a*(a+b)^3*f*(a+b*sec(e+f*x)^2)^(3/2))+1/8*(4*a+11*b)*cot(e+f*x)^2/((a+b)^2*f*(a+b*sec(e+f*x)^2)^(3/2))-1/4*cot(e+f*x)^4/((a+b)*f*(a+b*sec(e+f*x)^2)^(3/2))+1/8*b*(4*a^3+15*a^2*b-32*a*b^2-8*b^3)/(a^2*(a+b)^4*f*sqrt(a+b*sec(e+f*x)^2))],
[tan(e+f*x)^6/(a+b*sec(e+f*x)^2)^(5/2),x,9,-arctan(sqrt(a)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/(a^(5/2)*f)+arctanh(sqrt(b)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/(b^(5/2)*f)+(1/a^2+(-1)/b^2)*tan(e+f*x)/(f*sqrt(a+b+b*tan(e+f*x)^2))-1/3*(a+b)*tan(e+f*x)^3/(a*b*f*(a+b+b*tan(e+f*x)^2)^(3/2))],
[tan(e+f*x)^4/(a+b*sec(e+f*x)^2)^(5/2),x,7,arctan(sqrt(a)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/(a^(5/2)*f)+1/3*(a-3*b)*tan(e+f*x)/(a^2*b*f*sqrt(a+b+b*tan(e+f*x)^2))-1/3*(a+b)*tan(e+f*x)/(a*b*f*(a+b+b*tan(e+f*x)^2)^(3/2))],
[tan(e+f*x)^2/(a+b*sec(e+f*x)^2)^(5/2),x,7,-arctan(sqrt(a)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/(a^(5/2)*f)+1/3*(2*a+3*b)*tan(e+f*x)/(a^2*(a+b)*f*sqrt(a+b+b*tan(e+f*x)^2))+1/3*tan(e+f*x)/(a*f*(a+b+b*tan(e+f*x)^2)^(3/2))],
[1/(a+b*sec(e+f*x)^2)^(5/2),x,6,arctan(sqrt(a)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/(a^(5/2)*f)-1/3*b*(5*a+3*b)*tan(e+f*x)/(a^2*(a+b)^2*f*sqrt(a+b+b*tan(e+f*x)^2))-1/3*b*tan(e+f*x)/(a*(a+b)*f*(a+b+b*tan(e+f*x)^2)^(3/2))],
[cot(e+f*x)^2/(a+b*sec(e+f*x)^2)^(5/2),x,8,-arctan(sqrt(a)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/(a^(5/2)*f)-1/3*b*(7*a+3*b)*cot(e+f*x)/(a^2*(a+b)^2*f*sqrt(a+b+b*tan(e+f*x)^2))-1/3*(a-3*b)*(3*a+b)*cot(e+f*x)*sqrt(a+b+b*tan(e+f*x)^2)/(a^2*(a+b)^3*f)-1/3*b*cot(e+f*x)/(a*(a+b)*f*(a+b+b*tan(e+f*x)^2)^(3/2))],
[cot(e+f*x)^4/(a+b*sec(e+f*x)^2)^(5/2),x,9,arctan(sqrt(a)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/(a^(5/2)*f)-b*(3*a+b)*cot(e+f*x)^3/(a^2*(a+b)^2*f*sqrt(a+b+b*tan(e+f*x)^2))+1/3*(a-b)*(3*a^2+14*a*b+3*b^2)*cot(e+f*x)*sqrt(a+b+b*tan(e+f*x)^2)/(a^2*(a+b)^4*f)-1/3*(a^2-10*a*b-3*b^2)*cot(e+f*x)^3*sqrt(a+b+b*tan(e+f*x)^2)/(a^2*(a+b)^3*f)-1/3*b*cot(e+f*x)^3/(a*(a+b)*f*(a+b+b*tan(e+f*x)^2)^(3/2))],
[cot(e+f*x)^6/(a+b*sec(e+f*x)^2)^(5/2),x,10,-arctan(sqrt(a)*tan(e+f*x)/sqrt(a+b+b*tan(e+f*x)^2))/(a^(5/2)*f)-1/3*b*(11*a+3*b)*cot(e+f*x)^5/(a^2*(a+b)^2*f*sqrt(a+b+b*tan(e+f*x)^2))-1/15*(15*a^4+70*a^3*b+128*a^2*b^2-70*a*b^3-15*b^4)*cot(e+f*x)*sqrt(a+b+b*tan(e+f*x)^2)/(a^2*(a+b)^5*f)+1/15*(5*a^3+19*a^2*b-65*a*b^2-15*b^3)*cot(e+f*x)^3*sqrt(a+b+b*tan(e+f*x)^2)/(a^2*(a+b)^4*f)-1/5*(a^2-20*a*b-5*b^2)*cot(e+f*x)^5*sqrt(a+b+b*tan(e+f*x)^2)/(a^2*(a+b)^3*f)-1/3*b*cot(e+f*x)^5/(a*(a+b)*f*(a+b+b*tan(e+f*x)^2)^(3/2))],

# Integrands of the form Tan[e+f x]^m (a+b Sec[e+f x]^2)^p when p symbolic
[(a+b*sec(e+f*x)^2)^p*(d*tan(e+f*x))^m,x,4,AppellF1(1/2*(1+m),1,-p,1/2*(3+m),-tan(e+f*x)^2,-b*tan(e+f*x)^2/(a+b))*(d*tan(e+f*x))^(1+m)*(a+b+b*tan(e+f*x)^2)^p/(d*f*(1+m)*(1+b*tan(e+f*x)^2/(a+b))^p)],
[(a+b*sec(e+f*x)^2)^p*tan(e+f*x)^5,x,5,-1/2*(a+2*b)*(a+b*sec(e+f*x)^2)^(1+p)/(b^2*f*(1+p))-1/2*hypergeom([1,1+p],[2+p],1+b*sec(e+f*x)^2/a)*(a+b*sec(e+f*x)^2)^(1+p)/(a*f*(1+p))+1/2*(a+b*sec(e+f*x)^2)^(2+p)/(b^2*f*(2+p))],
[(a+b*sec(e+f*x)^2)^p*tan(e+f*x)^3,x,4,1/2*(a+b*sec(e+f*x)^2)^(1+p)/(b*f*(1+p))+1/2*hypergeom([1,1+p],[2+p],1+b*sec(e+f*x)^2/a)*(a+b*sec(e+f*x)^2)^(1+p)/(a*f*(1+p))],
[(a+b*sec(e+f*x)^2)^p*tan(e+f*x),x,3,-1/2*hypergeom([1,1+p],[2+p],1+b*sec(e+f*x)^2/a)*(a+b*sec(e+f*x)^2)^(1+p)/(a*f*(1+p))],
[cot(e+f*x)*(a+b*sec(e+f*x)^2)^p,x,5,-1/2*hypergeom([1,1+p],[2+p],(a+b*sec(e+f*x)^2)/(a+b))*(a+b*sec(e+f*x)^2)^(1+p)/((a+b)*f*(1+p))+1/2*hypergeom([1,1+p],[2+p],1+b*sec(e+f*x)^2/a)*(a+b*sec(e+f*x)^2)^(1+p)/(a*f*(1+p))],
[cot(e+f*x)^3*(a+b*sec(e+f*x)^2)^p,x,6,-1/2*cot(e+f*x)^2*(a+b*sec(e+f*x)^2)^(1+p)/((a+b)*f)+1/2*(a+b-b*p)*hypergeom([1,1+p],[2+p],(a+b*sec(e+f*x)^2)/(a+b))*(a+b*sec(e+f*x)^2)^(1+p)/((a+b)^2*f*(1+p))-1/2*hypergeom([1,1+p],[2+p],1+b*sec(e+f*x)^2/a)*(a+b*sec(e+f*x)^2)^(1+p)/(a*f*(1+p))],
[(a+b*sec(e+f*x)^2)^p*tan(e+f*x)^4,x,4,1/5*AppellF1(5/2,1,-p,7/2,-tan(e+f*x)^2,-b*tan(e+f*x)^2/(a+b))*tan(e+f*x)^5*(a+b+b*tan(e+f*x)^2)^p/(f*(1+b*tan(e+f*x)^2/(a+b))^p)],
[(a+b*sec(e+f*x)^2)^p*tan(e+f*x)^2,x,4,1/3*AppellF1(3/2,1,-p,5/2,-tan(e+f*x)^2,-b*tan(e+f*x)^2/(a+b))*tan(e+f*x)^3*(a+b+b*tan(e+f*x)^2)^p/(f*(1+b*tan(e+f*x)^2/(a+b))^p)],
[(a+b*sec(e+f*x)^2)^p,x,3,AppellF1(1/2,1,-p,3/2,-tan(e+f*x)^2,-b*tan(e+f*x)^2/(a+b))*tan(e+f*x)*(a+b+b*tan(e+f*x)^2)^p/(f*(1+b*tan(e+f*x)^2/(a+b))^p)],
[cot(e+f*x)^2*(a+b*sec(e+f*x)^2)^p,x,4,-AppellF1(-1/2,1,-p,1/2,-tan(e+f*x)^2,-b*tan(e+f*x)^2/(a+b))*cot(e+f*x)*(a+b+b*tan(e+f*x)^2)^p/(f*(1+b*tan(e+f*x)^2/(a+b))^p)],
[cot(e+f*x)^4*(a+b*sec(e+f*x)^2)^p,x,4,-1/3*AppellF1(-3/2,1,-p,-1/2,-tan(e+f*x)^2,-b*tan(e+f*x)^2/(a+b))*cot(e+f*x)^3*(a+b+b*tan(e+f*x)^2)^p/(f*(1+b*tan(e+f*x)^2/(a+b))^p)],

# Integrands of the form Tan[e+f x]^m (a+b Sec[e+f x]^3)^p

# Integrands of the form Tan[e+f x]^m (a+b Sec[e+f x]^3)^p

# p>0
[(a+b*sec(e+f*x)^3)*tan(e+f*x)^5,x,3,-a*log(cos(e+f*x))/f-a*sec(e+f*x)^2/f+1/3*b*sec(e+f*x)^3/f+1/4*a*sec(e+f*x)^4/f-2/5*b*sec(e+f*x)^5/f+1/7*b*sec(e+f*x)^7/f],
[(a+b*sec(e+f*x)^3)*tan(e+f*x)^3,x,3,a*log(cos(e+f*x))/f+1/2*a*sec(e+f*x)^2/f-1/3*b*sec(e+f*x)^3/f+1/5*b*sec(e+f*x)^5/f],
[(a+b*sec(e+f*x)^3)*tan(e+f*x),x,3,-a*log(cos(e+f*x))/f+1/3*b*sec(e+f*x)^3/f],
[cot(e+f*x)*(a+b*sec(e+f*x)^3),x,3,1/2*(a+b)*log(1-cos(e+f*x))/f+1/2*(a-b)*log(1+cos(e+f*x))/f+b*sec(e+f*x)/f],
[cot(e+f*x)^3*(a+b*sec(e+f*x)^3),x,5,-1/2*(a+b*cos(e+f*x))*csc(e+f*x)^2/f-1/4*(2*a-b)*log(1-cos(e+f*x))/f-1/4*(2*a+b)*log(1+cos(e+f*x))/f],

# p<0
[tan(e+f*x)^5/(a+b*sec(e+f*x)^3),x,11,-1/3*(a^(2/3)-2*b^(2/3))*log(b^(1/3)+a^(1/3)*cos(e+f*x))/(a^(1/3)*b^(4/3)*f)+1/6*(a^(2/3)-2*b^(2/3))*log(b^(2/3)-a^(1/3)*b^(1/3)*cos(e+f*x)+a^(2/3)*cos(e+f*x)^2)/(a^(1/3)*b^(4/3)*f)-1/3*log(b+a*cos(e+f*x)^3)/(a*f)+sec(e+f*x)/(b*f)-(a^(2/3)+2*b^(2/3))*arctan((b^(1/3)-2*a^(1/3)*cos(e+f*x))/(b^(1/3)*sqrt(3)))/(a^(1/3)*b^(4/3)*f*sqrt(3))],
[tan(e+f*x)^3/(a+b*sec(e+f*x)^3),x,9,-1/3*log(b^(1/3)+a^(1/3)*cos(e+f*x))/(a^(1/3)*b^(2/3)*f)+1/6*log(b^(2/3)-a^(1/3)*b^(1/3)*cos(e+f*x)+a^(2/3)*cos(e+f*x)^2)/(a^(1/3)*b^(2/3)*f)+1/3*log(b+a*cos(e+f*x)^3)/(a*f)+arctan((b^(1/3)-2*a^(1/3)*cos(e+f*x))/(b^(1/3)*sqrt(3)))/(a^(1/3)*b^(2/3)*f*sqrt(3))],
[tan(e+f*x)/(a+b*sec(e+f*x)^3),x,2,-1/3*log(b+a*cos(e+f*x)^3)/(a*f)],
[cot(e+f*x)/(a+b*sec(e+f*x)^3),x,11,1/2*log(1-cos(e+f*x))/((a+b)*f)+1/2*log(1+cos(e+f*x))/((a-b)*f)-1/3*(a^(2/3)+b^(2/3))*b^(2/3)*log(b^(1/3)+a^(1/3)*cos(e+f*x))/(a^(1/3)*(a^2-b^2)*f)+1/6*(a^(2/3)+b^(2/3))*b^(2/3)*log(b^(2/3)-a^(1/3)*b^(1/3)*cos(e+f*x)+a^(2/3)*cos(e+f*x)^2)/(a^(1/3)*(a^2-b^2)*f)-1/3*b^2*log(b+a*cos(e+f*x)^3)/(a*(a^2-b^2)*f)-b^(2/3)*arctan((b^(1/3)-2*a^(1/3)*cos(e+f*x))/(b^(1/3)*sqrt(3)))/(a^(1/3)*(a^(4/3)+a^(2/3)*b^(2/3)+b^(4/3))*f*sqrt(3))],
[cot(e+f*x)^3/(a+b*sec(e+f*x)^3),x,11,(-1/4)/((a+b)*f*(1-cos(e+f*x)))+(-1/4)/((a-b)*f*(1+cos(e+f*x)))-1/4*(2*a+5*b)*log(1-cos(e+f*x))/((a+b)^2*f)-1/4*(2*a-5*b)*log(1+cos(e+f*x))/((a-b)^2*f)-1/3*b^(4/3)*(a^2+3*a^(2/3)*b^(4/3)+2*b^2)*log(b^(1/3)+a^(1/3)*cos(e+f*x))/(a^(1/3)*(a^2-b^2)^2*f)+1/6*b^(4/3)*(a^2+3*a^(2/3)*b^(4/3)+2*b^2)*log(b^(2/3)-a^(1/3)*b^(1/3)*cos(e+f*x)+a^(2/3)*cos(e+f*x)^2)/(a^(1/3)*(a^2-b^2)^2*f)-1/3*b^2*(2*a^2+b^2)*log(b+a*cos(e+f*x)^3)/(a*(a^2-b^2)^2*f)+b^(4/3)*(a^2-3*a^(2/3)*b^(4/3)+2*b^2)*arctan((b^(1/3)-2*a^(1/3)*cos(e+f*x))/(b^(1/3)*sqrt(3)))/(a^(1/3)*(a^2-b^2)^2*f*sqrt(3))],

# Integrands of the form Tan[e+f x]^m (a+b Sec[e+f x]^4)^p

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

# Integrands of the form Tan[e+f x]^m (a+b Sec[e+f x]^n)^p when p symbolic
[(a+b*(c*sec(e+f*x))^n)^p*(d*tan(e+f*x))^m,x,0,Unintegrable((a+b*(c*sec(e+f*x))^n)^p*(d*tan(e+f*x))^m,x)],
[(a+b*(c*sec(e+f*x))^n)^p*tan(e+f*x)^5,x,15,-hypergeom([1,1+p],[2+p],1+b*(c*sec(e+f*x))^n/a)*(a+b*(c*sec(e+f*x))^n)^(1+p)/(a*f*n*(1+p))-hypergeom([2/n,-p],[(2+n)/n],-b*(c*sec(e+f*x))^n/a)*sec(e+f*x)^2*(a+b*(c*sec(e+f*x))^n)^p/(f*(1+b*(c*sec(e+f*x))^n/a)^p)+1/4*hypergeom([4/n,-p],[(4+n)/n],-b*(c*sec(e+f*x))^n/a)*sec(e+f*x)^4*(a+b*(c*sec(e+f*x))^n)^p/(f*(1+b*(c*sec(e+f*x))^n/a)^p)],
[(a+b*(c*sec(e+f*x))^n)^p*tan(e+f*x)^3,x,11,hypergeom([1,1+p],[2+p],1+b*(c*sec(e+f*x))^n/a)*(a+b*(c*sec(e+f*x))^n)^(1+p)/(a*f*n*(1+p))+1/2*hypergeom([2/n,-p],[(2+n)/n],-b*(c*sec(e+f*x))^n/a)*sec(e+f*x)^2*(a+b*(c*sec(e+f*x))^n)^p/(f*(1+b*(c*sec(e+f*x))^n/a)^p)],
[(a+b*(c*sec(e+f*x))^n)^p*tan(e+f*x),x,5,-hypergeom([1,1+p],[2+p],1+b*(c*sec(e+f*x))^n/a)*(a+b*(c*sec(e+f*x))^n)^(1+p)/(a*f*n*(1+p))],
[cot(e+f*x)*(a+b*(c*sec(e+f*x))^n)^p,x,0,Unintegrable(cot(e+f*x)*(a+b*(c*sec(e+f*x))^n)^p,x)],
[cot(e+f*x)^3*(a+b*(c*sec(e+f*x))^n)^p,x,0,Unintegrable(cot(e+f*x)^3*(a+b*(c*sec(e+f*x))^n)^p,x)],
[(a+b*(c*sec(e+f*x))^n)^p*tan(e+f*x)^2,x,0,Unintegrable((a+b*(c*sec(e+f*x))^n)^p*tan(e+f*x)^2,x)],
[(a+b*(c*sec(e+f*x))^n)^p,x,0,Unintegrable((a+b*(c*sec(e+f*x))^n)^p,x)],
[cot(e+f*x)^2*(a+b*(c*sec(e+f*x))^n)^p,x,0,Unintegrable(cot(e+f*x)^2*(a+b*(c*sec(e+f*x))^n)^p,x)]]:
