# Maple integration test file: "4 Trig functions\4.5 Secant\4.5.3.1 (a+b sec)^m (d sec)^n (A+B sec).txt"

lst:=[

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

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

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

# n>0
[(b*sec(c+d*x))^(5/2)*(A+B*sec(c+d*x)),x,8,2/3*A*b*(b*sec(c+d*x))^(3/2)*sin(c+d*x)/d+2/5*B*(b*sec(c+d*x))^(5/2)*sin(c+d*x)/d-6/5*b^3*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(d*sqrt(cos(c+d*x))*sqrt(b*sec(c+d*x)))+6/5*b^2*B*sin(c+d*x)*sqrt(b*sec(c+d*x))/d+2/3*A*b^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(b*sec(c+d*x))/d],
[(b*sec(c+d*x))^(3/2)*(A+B*sec(c+d*x)),x,7,2/3*B*(b*sec(c+d*x))^(3/2)*sin(c+d*x)/d-2*A*b^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(d*sqrt(cos(c+d*x))*sqrt(b*sec(c+d*x)))+2*A*b*sin(c+d*x)*sqrt(b*sec(c+d*x))/d+2/3*b*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(b*sec(c+d*x))/d],
[(b*sec(c+d*x))^(1/2)*(A+B*sec(c+d*x)),x,6,-2*b*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(d*sqrt(cos(c+d*x))*sqrt(b*sec(c+d*x)))+2*B*sin(c+d*x)*sqrt(b*sec(c+d*x))/d+2*A*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(b*sec(c+d*x))/d],

# n<0
[(A+B*sec(c+d*x))/(b*sec(c+d*x))^(1/2),x,5,2*A*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(d*sqrt(cos(c+d*x))*sqrt(b*sec(c+d*x)))+2*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(b*sec(c+d*x))/(b*d)],
[(A+B*sec(c+d*x))/(b*sec(c+d*x))^(3/2),x,6,2/3*A*sin(c+d*x)/(b*d*sqrt(b*sec(c+d*x)))+2*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(b*d*sqrt(cos(c+d*x))*sqrt(b*sec(c+d*x)))+2/3*A*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(b*sec(c+d*x))/(b^2*d)],
[(A+B*sec(c+d*x))/(b*sec(c+d*x))^(5/2),x,7,2/5*A*sin(c+d*x)/(b*d*(b*sec(c+d*x))^(3/2))+2/3*B*sin(c+d*x)/(b^2*d*sqrt(b*sec(c+d*x)))+6/5*A*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(b^2*d*sqrt(cos(c+d*x))*sqrt(b*sec(c+d*x)))+2/3*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(b*sec(c+d*x))/(b^3*d)],

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

# Integrands of the form Sec[c+d x]^m (A+B Sec[c+d x]) (b Sec[c+d x])^(n/3)

# n>0
[sec(c+d*x)^2*(b*sec(c+d*x))^(2/3)*(A+B*sec(c+d*x)),x,6,3/5*A*hypergeom([-5/6,1/2],[1/6],cos(c+d*x)^2)*(b*sec(c+d*x))^(5/3)*sin(c+d*x)/(b*d*sqrt(sin(c+d*x)^2))+3/8*B*hypergeom([-4/3,1/2],[-1/3],cos(c+d*x)^2)*(b*sec(c+d*x))^(8/3)*sin(c+d*x)/(b^2*d*sqrt(sin(c+d*x)^2))],
[sec(c+d*x)*(b*sec(c+d*x))^(2/3)*(A+B*sec(c+d*x)),x,6,3/2*A*hypergeom([-1/3,1/2],[2/3],cos(c+d*x)^2)*(b*sec(c+d*x))^(2/3)*sin(c+d*x)/(d*sqrt(sin(c+d*x)^2))+3/5*B*hypergeom([-5/6,1/2],[1/6],cos(c+d*x)^2)*(b*sec(c+d*x))^(5/3)*sin(c+d*x)/(b*d*sqrt(sin(c+d*x)^2))],
[(b*sec(c+d*x))^(2/3)*(A+B*sec(c+d*x)),x,5,-3*A*b*hypergeom([1/6,1/2],[7/6],cos(c+d*x)^2)*sin(c+d*x)/(d*(b*sec(c+d*x))^(1/3)*sqrt(sin(c+d*x)^2))+3/2*B*hypergeom([-1/3,1/2],[2/3],cos(c+d*x)^2)*(b*sec(c+d*x))^(2/3)*sin(c+d*x)/(d*sqrt(sin(c+d*x)^2))],
[cos(c+d*x)*(b*sec(c+d*x))^(2/3)*(A+B*sec(c+d*x)),x,6,-3/4*A*b^2*hypergeom([1/2,2/3],[5/3],cos(c+d*x)^2)*sin(c+d*x)/(d*(b*sec(c+d*x))^(4/3)*sqrt(sin(c+d*x)^2))-3*b*B*hypergeom([1/6,1/2],[7/6],cos(c+d*x)^2)*sin(c+d*x)/(d*(b*sec(c+d*x))^(1/3)*sqrt(sin(c+d*x)^2))],
[cos(c+d*x)^2*(b*sec(c+d*x))^(2/3)*(A+B*sec(c+d*x)),x,6,-3/7*A*b^3*hypergeom([1/2,7/6],[13/6],cos(c+d*x)^2)*sin(c+d*x)/(d*(b*sec(c+d*x))^(7/3)*sqrt(sin(c+d*x)^2))-3/4*b^2*B*hypergeom([1/2,2/3],[5/3],cos(c+d*x)^2)*sin(c+d*x)/(d*(b*sec(c+d*x))^(4/3)*sqrt(sin(c+d*x)^2))],
[sec(c+d*x)^2*(b*sec(c+d*x))^(4/3)*(A+B*sec(c+d*x)),x,6,3/7*A*hypergeom([-7/6,1/2],[-1/6],cos(c+d*x)^2)*(b*sec(c+d*x))^(7/3)*sin(c+d*x)/(b*d*sqrt(sin(c+d*x)^2))+3/10*B*hypergeom([-5/3,1/2],[-2/3],cos(c+d*x)^2)*(b*sec(c+d*x))^(10/3)*sin(c+d*x)/(b^2*d*sqrt(sin(c+d*x)^2))],
[sec(c+d*x)*(b*sec(c+d*x))^(4/3)*(A+B*sec(c+d*x)),x,6,3/4*A*hypergeom([-2/3,1/2],[1/3],cos(c+d*x)^2)*(b*sec(c+d*x))^(4/3)*sin(c+d*x)/(d*sqrt(sin(c+d*x)^2))+3/7*B*hypergeom([-7/6,1/2],[-1/6],cos(c+d*x)^2)*(b*sec(c+d*x))^(7/3)*sin(c+d*x)/(b*d*sqrt(sin(c+d*x)^2))],
[(b*sec(c+d*x))^(4/3)*(A+B*sec(c+d*x)),x,5,3*A*b*hypergeom([-1/6,1/2],[5/6],cos(c+d*x)^2)*(b*sec(c+d*x))^(1/3)*sin(c+d*x)/(d*sqrt(sin(c+d*x)^2))+3/4*B*hypergeom([-2/3,1/2],[1/3],cos(c+d*x)^2)*(b*sec(c+d*x))^(4/3)*sin(c+d*x)/(d*sqrt(sin(c+d*x)^2))],
[cos(c+d*x)*(b*sec(c+d*x))^(4/3)*(A+B*sec(c+d*x)),x,6,-3/2*A*b^2*hypergeom([1/3,1/2],[4/3],cos(c+d*x)^2)*sin(c+d*x)/(d*(b*sec(c+d*x))^(2/3)*sqrt(sin(c+d*x)^2))+3*b*B*hypergeom([-1/6,1/2],[5/6],cos(c+d*x)^2)*(b*sec(c+d*x))^(1/3)*sin(c+d*x)/(d*sqrt(sin(c+d*x)^2))],
[cos(c+d*x)^2*(b*sec(c+d*x))^(4/3)*(A+B*sec(c+d*x)),x,6,-3/5*A*b^3*hypergeom([1/2,5/6],[11/6],cos(c+d*x)^2)*sin(c+d*x)/(d*(b*sec(c+d*x))^(5/3)*sqrt(sin(c+d*x)^2))-3/2*b^2*B*hypergeom([1/3,1/2],[4/3],cos(c+d*x)^2)*sin(c+d*x)/(d*(b*sec(c+d*x))^(2/3)*sqrt(sin(c+d*x)^2))],

# n<0
[sec(c+d*x)^2*(A+B*sec(c+d*x))/(b*sec(c+d*x))^(2/3),x,6,3*A*hypergeom([-1/6,1/2],[5/6],cos(c+d*x)^2)*(b*sec(c+d*x))^(1/3)*sin(c+d*x)/(b*d*sqrt(sin(c+d*x)^2))+3/4*B*hypergeom([-2/3,1/2],[1/3],cos(c+d*x)^2)*(b*sec(c+d*x))^(4/3)*sin(c+d*x)/(b^2*d*sqrt(sin(c+d*x)^2))],
[sec(c+d*x)*(A+B*sec(c+d*x))/(b*sec(c+d*x))^(2/3),x,6,-3/2*A*hypergeom([1/3,1/2],[4/3],cos(c+d*x)^2)*sin(c+d*x)/(d*(b*sec(c+d*x))^(2/3)*sqrt(sin(c+d*x)^2))+3*B*hypergeom([-1/6,1/2],[5/6],cos(c+d*x)^2)*(b*sec(c+d*x))^(1/3)*sin(c+d*x)/(b*d*sqrt(sin(c+d*x)^2))],
[(A+B*sec(c+d*x))/(b*sec(c+d*x))^(2/3),x,5,-3/5*A*b*hypergeom([1/2,5/6],[11/6],cos(c+d*x)^2)*sin(c+d*x)/(d*(b*sec(c+d*x))^(5/3)*sqrt(sin(c+d*x)^2))-3/2*B*hypergeom([1/3,1/2],[4/3],cos(c+d*x)^2)*sin(c+d*x)/(d*(b*sec(c+d*x))^(2/3)*sqrt(sin(c+d*x)^2))],
[sec(c+d*x)*(A+B*sec(c+d*x))/(b*sec(c+d*x))^(2/3),x,6,-3/2*A*hypergeom([1/3,1/2],[4/3],cos(c+d*x)^2)*sin(c+d*x)/(d*(b*sec(c+d*x))^(2/3)*sqrt(sin(c+d*x)^2))+3*B*hypergeom([-1/6,1/2],[5/6],cos(c+d*x)^2)*(b*sec(c+d*x))^(1/3)*sin(c+d*x)/(b*d*sqrt(sin(c+d*x)^2))],
[sec(c+d*x)^2*(A+B*sec(c+d*x))/(b*sec(c+d*x))^(2/3),x,6,3*A*hypergeom([-1/6,1/2],[5/6],cos(c+d*x)^2)*(b*sec(c+d*x))^(1/3)*sin(c+d*x)/(b*d*sqrt(sin(c+d*x)^2))+3/4*B*hypergeom([-2/3,1/2],[1/3],cos(c+d*x)^2)*(b*sec(c+d*x))^(4/3)*sin(c+d*x)/(b^2*d*sqrt(sin(c+d*x)^2))],
[sec(c+d*x)^2*(A+B*sec(c+d*x))/(b*sec(c+d*x))^(4/3),x,6,-3*A*hypergeom([1/6,1/2],[7/6],cos(c+d*x)^2)*sin(c+d*x)/(b*d*(b*sec(c+d*x))^(1/3)*sqrt(sin(c+d*x)^2))+3/2*B*hypergeom([-1/3,1/2],[2/3],cos(c+d*x)^2)*(b*sec(c+d*x))^(2/3)*sin(c+d*x)/(b^2*d*sqrt(sin(c+d*x)^2))],
[sec(c+d*x)*(A+B*sec(c+d*x))/(b*sec(c+d*x))^(4/3),x,6,-3/4*A*hypergeom([1/2,2/3],[5/3],cos(c+d*x)^2)*sin(c+d*x)/(d*(b*sec(c+d*x))^(4/3)*sqrt(sin(c+d*x)^2))-3*B*hypergeom([1/6,1/2],[7/6],cos(c+d*x)^2)*sin(c+d*x)/(b*d*(b*sec(c+d*x))^(1/3)*sqrt(sin(c+d*x)^2))],
[(A+B*sec(c+d*x))/(b*sec(c+d*x))^(4/3),x,5,-3/7*A*b*hypergeom([1/2,7/6],[13/6],cos(c+d*x)^2)*sin(c+d*x)/(d*(b*sec(c+d*x))^(7/3)*sqrt(sin(c+d*x)^2))-3/4*B*hypergeom([1/2,2/3],[5/3],cos(c+d*x)^2)*sin(c+d*x)/(d*(b*sec(c+d*x))^(4/3)*sqrt(sin(c+d*x)^2))],
[sec(c+d*x)*(A+B*sec(c+d*x))/(b*sec(c+d*x))^(4/3),x,6,-3/4*A*hypergeom([1/2,2/3],[5/3],cos(c+d*x)^2)*sin(c+d*x)/(d*(b*sec(c+d*x))^(4/3)*sqrt(sin(c+d*x)^2))-3*B*hypergeom([1/6,1/2],[7/6],cos(c+d*x)^2)*sin(c+d*x)/(b*d*(b*sec(c+d*x))^(1/3)*sqrt(sin(c+d*x)^2))],
[sec(c+d*x)^2*(A+B*sec(c+d*x))/(b*sec(c+d*x))^(4/3),x,6,-3*A*hypergeom([1/6,1/2],[7/6],cos(c+d*x)^2)*sin(c+d*x)/(b*d*(b*sec(c+d*x))^(1/3)*sqrt(sin(c+d*x)^2))+3/2*B*hypergeom([-1/3,1/2],[2/3],cos(c+d*x)^2)*(b*sec(c+d*x))^(2/3)*sin(c+d*x)/(b^2*d*sqrt(sin(c+d*x)^2))],

# Integrands of the form Sec[c+d x]^m (A+B Sec[c+d x]) (b Sec[c+d x])^n with m symbolic
[sec(c+d*x)^m*(b*sec(c+d*x))^(4/3)*(A+B*sec(c+d*x)),x,6,3*A*b*hypergeom([1/2,1/6*(-1-3*m)],[1/6*(5-3*m)],cos(c+d*x)^2)*sec(c+d*x)^m*(b*sec(c+d*x))^(1/3)*sin(c+d*x)/(d*(1+3*m)*sqrt(sin(c+d*x)^2))+3*b*B*hypergeom([1/2,1/6*(-4-3*m)],[1/6*(2-3*m)],cos(c+d*x)^2)*sec(c+d*x)^(1+m)*(b*sec(c+d*x))^(1/3)*sin(c+d*x)/(d*(4+3*m)*sqrt(sin(c+d*x)^2))],
[sec(c+d*x)^m*(b*sec(c+d*x))^(2/3)*(A+B*sec(c+d*x)),x,6,-3*A*hypergeom([1/2,1/6*(1-3*m)],[1/6*(7-3*m)],cos(c+d*x)^2)*sec(c+d*x)^(-1+m)*(b*sec(c+d*x))^(2/3)*sin(c+d*x)/(d*(1-3*m)*sqrt(sin(c+d*x)^2))+3*B*hypergeom([1/2,1/6*(-2-3*m)],[1/6*(4-3*m)],cos(c+d*x)^2)*sec(c+d*x)^m*(b*sec(c+d*x))^(2/3)*sin(c+d*x)/(d*(2+3*m)*sqrt(sin(c+d*x)^2))],
[sec(c+d*x)^m*(b*sec(c+d*x))^(1/3)*(A+B*sec(c+d*x)),x,6,-3*A*hypergeom([1/2,1/6*(2-3*m)],[1/6*(8-3*m)],cos(c+d*x)^2)*sec(c+d*x)^(-1+m)*(b*sec(c+d*x))^(1/3)*sin(c+d*x)/(d*(2-3*m)*sqrt(sin(c+d*x)^2))+3*B*hypergeom([1/2,1/6*(-1-3*m)],[1/6*(5-3*m)],cos(c+d*x)^2)*sec(c+d*x)^m*(b*sec(c+d*x))^(1/3)*sin(c+d*x)/(d*(1+3*m)*sqrt(sin(c+d*x)^2))],
[sec(c+d*x)^m*(A+B*sec(c+d*x))/(b*sec(c+d*x))^(1/3),x,6,-3*A*hypergeom([1/2,1/6*(4-3*m)],[1/6*(10-3*m)],cos(c+d*x)^2)*sec(c+d*x)^(-1+m)*sin(c+d*x)/(d*(4-3*m)*(b*sec(c+d*x))^(1/3)*sqrt(sin(c+d*x)^2))-3*B*hypergeom([1/2,1/6*(1-3*m)],[1/6*(7-3*m)],cos(c+d*x)^2)*sec(c+d*x)^m*sin(c+d*x)/(d*(1-3*m)*(b*sec(c+d*x))^(1/3)*sqrt(sin(c+d*x)^2))],
[sec(c+d*x)^m*(A+B*sec(c+d*x))/(b*sec(c+d*x))^(2/3),x,6,-3*A*hypergeom([1/2,1/6*(5-3*m)],[1/6*(11-3*m)],cos(c+d*x)^2)*sec(c+d*x)^(-1+m)*sin(c+d*x)/(d*(5-3*m)*(b*sec(c+d*x))^(2/3)*sqrt(sin(c+d*x)^2))-3*B*hypergeom([1/2,1/6*(2-3*m)],[1/6*(8-3*m)],cos(c+d*x)^2)*sec(c+d*x)^m*sin(c+d*x)/(d*(2-3*m)*(b*sec(c+d*x))^(2/3)*sqrt(sin(c+d*x)^2))],
[sec(c+d*x)^m*(A+B*sec(c+d*x))/(b*sec(c+d*x))^(4/3),x,6,-3*A*hypergeom([1/2,1/6*(7-3*m)],[1/6*(13-3*m)],cos(c+d*x)^2)*sec(c+d*x)^(-2+m)*sin(c+d*x)/(b*d*(7-3*m)*(b*sec(c+d*x))^(1/3)*sqrt(sin(c+d*x)^2))-3*B*hypergeom([1/2,1/6*(4-3*m)],[1/6*(10-3*m)],cos(c+d*x)^2)*sec(c+d*x)^(-1+m)*sin(c+d*x)/(b*d*(4-3*m)*(b*sec(c+d*x))^(1/3)*sqrt(sin(c+d*x)^2))],

# Integrands of the form Sec[c+d x]^m (A+B Sec[c+d x]) (b Sec[c+d x])^n with n symbolic
[sec(c+d*x)^m*(b*sec(c+d*x))^n*(A+B*sec(c+d*x)),x,6,-A*hypergeom([1/2,1/2*(1-m-n)],[1/2*(3-m-n)],cos(c+d*x)^2)*sec(c+d*x)^(-1+m)*(b*sec(c+d*x))^n*sin(c+d*x)/(d*(1-m-n)*sqrt(sin(c+d*x)^2))+B*hypergeom([1/2,1/2*(-m-n)],[1/2*(2-m-n)],cos(c+d*x)^2)*sec(c+d*x)^m*(b*sec(c+d*x))^n*sin(c+d*x)/(d*(m+n)*sqrt(sin(c+d*x)^2))],
[sec(c+d*x)^2*(b*sec(c+d*x))^n*(A+B*sec(c+d*x)),x,6,A*hypergeom([1/2,1/2*(-1-n)],[1/2*(1-n)],cos(c+d*x)^2)*(b*sec(c+d*x))^(1+n)*sin(c+d*x)/(b*d*(1+n)*sqrt(sin(c+d*x)^2))+B*hypergeom([1/2,1/2*(-2-n)],[-1/2*n],cos(c+d*x)^2)*(b*sec(c+d*x))^(2+n)*sin(c+d*x)/(b^2*d*(2+n)*sqrt(sin(c+d*x)^2))],
[sec(c+d*x)*(b*sec(c+d*x))^n*(A+B*sec(c+d*x)),x,6,A*hypergeom([1/2,-1/2*n],[1/2*(2-n)],cos(c+d*x)^2)*(b*sec(c+d*x))^n*sin(c+d*x)/(d*n*sqrt(sin(c+d*x)^2))+B*hypergeom([1/2,1/2*(-1-n)],[1/2*(1-n)],cos(c+d*x)^2)*(b*sec(c+d*x))^(1+n)*sin(c+d*x)/(b*d*(1+n)*sqrt(sin(c+d*x)^2))],
[(b*sec(c+d*x))^n*(A+B*sec(c+d*x)),x,5,-A*b*hypergeom([1/2,1/2*(1-n)],[1/2*(3-n)],cos(c+d*x)^2)*(b*sec(c+d*x))^(-1+n)*sin(c+d*x)/(d*(1-n)*sqrt(sin(c+d*x)^2))+B*hypergeom([1/2,-1/2*n],[1/2*(2-n)],cos(c+d*x)^2)*(b*sec(c+d*x))^n*sin(c+d*x)/(d*n*sqrt(sin(c+d*x)^2))],
[cos(c+d*x)*(b*sec(c+d*x))^n*(A+B*sec(c+d*x)),x,6,-A*b^2*hypergeom([1/2,1/2*(2-n)],[1/2*(4-n)],cos(c+d*x)^2)*(b*sec(c+d*x))^(-2+n)*sin(c+d*x)/(d*(2-n)*sqrt(sin(c+d*x)^2))-b*B*hypergeom([1/2,1/2*(1-n)],[1/2*(3-n)],cos(c+d*x)^2)*(b*sec(c+d*x))^(-1+n)*sin(c+d*x)/(d*(1-n)*sqrt(sin(c+d*x)^2))],
[cos(c+d*x)^2*(b*sec(c+d*x))^n*(A+B*sec(c+d*x)),x,6,-A*b^3*hypergeom([1/2,1/2*(3-n)],[1/2*(5-n)],cos(c+d*x)^2)*(b*sec(c+d*x))^(-3+n)*sin(c+d*x)/(d*(3-n)*sqrt(sin(c+d*x)^2))-b^2*B*hypergeom([1/2,1/2*(2-n)],[1/2*(4-n)],cos(c+d*x)^2)*(b*sec(c+d*x))^(-2+n)*sin(c+d*x)/(d*(2-n)*sqrt(sin(c+d*x)^2))],
[sec(c+d*x)^(3/2)*(b*sec(c+d*x))^n*(A+B*sec(c+d*x)),x,6,2*B*hypergeom([1/2,1/4*(-3-2*n)],[1/4*(1-2*n)],cos(c+d*x)^2)*sec(c+d*x)^(3/2)*(b*sec(c+d*x))^n*sin(c+d*x)/(d*(3+2*n)*sqrt(sin(c+d*x)^2))+2*A*hypergeom([1/2,1/4*(-1-2*n)],[1/4*(3-2*n)],cos(c+d*x)^2)*(b*sec(c+d*x))^n*sin(c+d*x)*sqrt(sec(c+d*x))/(d*(1+2*n)*sqrt(sin(c+d*x)^2))],
[(b*sec(c+d*x))^n*(A+B*sec(c+d*x))*sqrt(sec(c+d*x)),x,6,-2*A*hypergeom([1/2,1/4*(1-2*n)],[1/4*(5-2*n)],cos(c+d*x)^2)*(b*sec(c+d*x))^n*sin(c+d*x)/(d*(1-2*n)*sqrt(sec(c+d*x))*sqrt(sin(c+d*x)^2))+2*B*hypergeom([1/2,1/4*(-1-2*n)],[1/4*(3-2*n)],cos(c+d*x)^2)*(b*sec(c+d*x))^n*sin(c+d*x)*sqrt(sec(c+d*x))/(d*(1+2*n)*sqrt(sin(c+d*x)^2))],
[(b*sec(c+d*x))^n*(A+B*sec(c+d*x))/sqrt(sec(c+d*x)),x,6,-2*A*hypergeom([1/2,1/4*(3-2*n)],[1/4*(7-2*n)],cos(c+d*x)^2)*(b*sec(c+d*x))^n*sin(c+d*x)/(d*(3-2*n)*sec(c+d*x)^(3/2)*sqrt(sin(c+d*x)^2))-2*B*hypergeom([1/2,1/4*(1-2*n)],[1/4*(5-2*n)],cos(c+d*x)^2)*(b*sec(c+d*x))^n*sin(c+d*x)/(d*(1-2*n)*sqrt(sec(c+d*x))*sqrt(sin(c+d*x)^2))],
[(b*sec(c+d*x))^n*(A+B*sec(c+d*x))/sec(c+d*x)^(3/2),x,6,-2*A*hypergeom([1/2,1/4*(5-2*n)],[1/4*(9-2*n)],cos(c+d*x)^2)*(b*sec(c+d*x))^n*sin(c+d*x)/(d*(5-2*n)*sec(c+d*x)^(5/2)*sqrt(sin(c+d*x)^2))-2*B*hypergeom([1/2,1/4*(3-2*n)],[1/4*(7-2*n)],cos(c+d*x)^2)*(b*sec(c+d*x))^n*sin(c+d*x)/(d*(3-2*n)*sec(c+d*x)^(3/2)*sqrt(sin(c+d*x)^2))],

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

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

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

# n>0
[sec(c+d*x)^4*(a+a*sec(c+d*x))*(A+B*sec(c+d*x)),x,7,3/8*a*(A+B)*arctanh(sin(c+d*x))/d+1/5*a*(5*A+4*B)*tan(c+d*x)/d+3/8*a*(A+B)*sec(c+d*x)*tan(c+d*x)/d+1/4*a*(A+B)*sec(c+d*x)^3*tan(c+d*x)/d+1/5*a*B*sec(c+d*x)^4*tan(c+d*x)/d+1/15*a*(5*A+4*B)*tan(c+d*x)^3/d],
[sec(c+d*x)^3*(a+a*sec(c+d*x))*(A+B*sec(c+d*x)),x,6,1/8*a*(4*A+3*B)*arctanh(sin(c+d*x))/d+a*(A+B)*tan(c+d*x)/d+1/8*a*(4*A+3*B)*sec(c+d*x)*tan(c+d*x)/d+1/4*a*B*sec(c+d*x)^3*tan(c+d*x)/d+1/3*a*(A+B)*tan(c+d*x)^3/d],
[sec(c+d*x)^2*(a+a*sec(c+d*x))*(A+B*sec(c+d*x)),x,6,1/2*a*(A+B)*arctanh(sin(c+d*x))/d+1/3*a*(3*A+2*B)*tan(c+d*x)/d+1/2*a*(A+B)*sec(c+d*x)*tan(c+d*x)/d+1/3*a*B*sec(c+d*x)^2*tan(c+d*x)/d],
[sec(c+d*x)*(a+a*sec(c+d*x))*(A+B*sec(c+d*x)),x,5,1/2*a*(2*A+B)*arctanh(sin(c+d*x))/d+a*(A+B)*tan(c+d*x)/d+1/2*a*B*sec(c+d*x)*tan(c+d*x)/d],
[(a+a*sec(c+d*x))*(A+B*sec(c+d*x)),x,4,a*A*x+a*(A+B)*arctanh(sin(c+d*x))/d+a*B*tan(c+d*x)/d],
[cos(c+d*x)*(a+a*sec(c+d*x))*(A+B*sec(c+d*x)),x,3,a*(A+B)*x+a*B*arctanh(sin(c+d*x))/d+a*A*sin(c+d*x)/d],
[cos(c+d*x)^2*(a+a*sec(c+d*x))*(A+B*sec(c+d*x)),x,4,1/2*a*(A+2*B)*x+a*(A+B)*sin(c+d*x)/d+1/2*a*A*cos(c+d*x)*sin(c+d*x)/d],
[cos(c+d*x)^3*(a+a*sec(c+d*x))*(A+B*sec(c+d*x)),x,5,1/2*a*(A+B)*x+1/3*a*(2*A+3*B)*sin(c+d*x)/d+1/2*a*(A+B)*cos(c+d*x)*sin(c+d*x)/d+1/3*a*A*cos(c+d*x)^2*sin(c+d*x)/d],
[cos(c+d*x)^4*(a+a*sec(c+d*x))*(A+B*sec(c+d*x)),x,6,1/8*a*(3*A+4*B)*x+a*(A+B)*sin(c+d*x)/d+1/8*a*(3*A+4*B)*cos(c+d*x)*sin(c+d*x)/d+1/4*a*A*cos(c+d*x)^3*sin(c+d*x)/d-1/3*a*(A+B)*sin(c+d*x)^3/d],
[cos(c+d*x)^5*(a+a*sec(c+d*x))*(A+B*sec(c+d*x)),x,7,3/8*a*(A+B)*x+1/5*a*(4*A+5*B)*sin(c+d*x)/d+3/8*a*(A+B)*cos(c+d*x)*sin(c+d*x)/d+1/4*a*(A+B)*cos(c+d*x)^3*sin(c+d*x)/d+1/5*a*A*cos(c+d*x)^4*sin(c+d*x)/d-1/15*a*(4*A+5*B)*sin(c+d*x)^3/d],
[sec(c+d*x)^3*(a+a*sec(c+d*x))^2*(A+B*sec(c+d*x)),x,7,1/8*a^2*(7*A+6*B)*arctanh(sin(c+d*x))/d+1/5*a^2*(10*A+9*B)*tan(c+d*x)/d+1/8*a^2*(7*A+6*B)*sec(c+d*x)*tan(c+d*x)/d+1/20*a^2*(5*A+6*B)*sec(c+d*x)^3*tan(c+d*x)/d+1/5*B*sec(c+d*x)^3*(a^2+a^2*sec(c+d*x))*tan(c+d*x)/d+1/15*a^2*(10*A+9*B)*tan(c+d*x)^3/d],
[sec(c+d*x)^2*(a+a*sec(c+d*x))^2*(A+B*sec(c+d*x)),x,7,1/8*a^2*(8*A+7*B)*arctanh(sin(c+d*x))/d+1/6*a^2*(8*A+7*B)*tan(c+d*x)/d+1/24*a^2*(8*A+7*B)*sec(c+d*x)*tan(c+d*x)/d+1/12*(4*A-B)*(a+a*sec(c+d*x))^2*tan(c+d*x)/d+1/4*B*(a+a*sec(c+d*x))^3*tan(c+d*x)/(a*d)],
[sec(c+d*x)*(a+a*sec(c+d*x))^2*(A+B*sec(c+d*x)),x,6,1/2*a^2*(3*A+2*B)*arctanh(sin(c+d*x))/d+2/3*a^2*(3*A+2*B)*tan(c+d*x)/d+1/6*a^2*(3*A+2*B)*sec(c+d*x)*tan(c+d*x)/d+1/3*B*(a+a*sec(c+d*x))^2*tan(c+d*x)/d],
[(a+a*sec(c+d*x))^2*(A+B*sec(c+d*x)),x,5,a^2*A*x+1/2*a^2*(4*A+3*B)*arctanh(sin(c+d*x))/d+1/2*a^2*(2*A+3*B)*tan(c+d*x)/d+1/2*B*(a^2+a^2*sec(c+d*x))*tan(c+d*x)/d],
[cos(c+d*x)*(a+a*sec(c+d*x))^2*(A+B*sec(c+d*x)),x,4,a^2*(2*A+B)*x+a^2*(A+2*B)*arctanh(sin(c+d*x))/d+a^2*(A-B)*sin(c+d*x)/d+B*(a^2+a^2*sec(c+d*x))*sin(c+d*x)/d],
[cos(c+d*x)^2*(a+a*sec(c+d*x))^2*(A+B*sec(c+d*x)),x,4,1/2*a^2*(3*A+4*B)*x+a^2*B*arctanh(sin(c+d*x))/d+1/2*a^2*(3*A+2*B)*sin(c+d*x)/d+1/2*A*cos(c+d*x)*(a^2+a^2*sec(c+d*x))*sin(c+d*x)/d],
[cos(c+d*x)^3*(a+a*sec(c+d*x))^2*(A+B*sec(c+d*x)),x,5,1/2*a^2*(2*A+3*B)*x+2/3*a^2*(2*A+3*B)*sin(c+d*x)/d+1/6*a^2*(2*A+3*B)*cos(c+d*x)*sin(c+d*x)/d+1/3*A*cos(c+d*x)^2*(a+a*sec(c+d*x))^2*sin(c+d*x)/d],
[cos(c+d*x)^4*(a+a*sec(c+d*x))^2*(A+B*sec(c+d*x)),x,6,1/8*a^2*(7*A+8*B)*x+1/3*a^2*(4*A+5*B)*sin(c+d*x)/d+1/8*a^2*(7*A+8*B)*cos(c+d*x)*sin(c+d*x)/d+1/12*a^2*(5*A+4*B)*cos(c+d*x)^2*sin(c+d*x)/d+1/4*A*cos(c+d*x)^3*(a^2+a^2*sec(c+d*x))*sin(c+d*x)/d],
[cos(c+d*x)^5*(a+a*sec(c+d*x))^2*(A+B*sec(c+d*x)),x,7,1/8*a^2*(6*A+7*B)*x+1/5*a^2*(9*A+10*B)*sin(c+d*x)/d+1/8*a^2*(6*A+7*B)*cos(c+d*x)*sin(c+d*x)/d+1/20*a^2*(6*A+5*B)*cos(c+d*x)^3*sin(c+d*x)/d+1/5*A*cos(c+d*x)^4*(a^2+a^2*sec(c+d*x))*sin(c+d*x)/d-1/15*a^2*(9*A+10*B)*sin(c+d*x)^3/d],
[sec(c+d*x)^3*(a+a*sec(c+d*x))^3*(A+B*sec(c+d*x)),x,8,1/16*a^3*(26*A+23*B)*arctanh(sin(c+d*x))/d+1/5*a^3*(19*A+17*B)*tan(c+d*x)/d+1/16*a^3*(26*A+23*B)*sec(c+d*x)*tan(c+d*x)/d+1/40*a^3*(22*A+21*B)*sec(c+d*x)^3*tan(c+d*x)/d+1/6*a*B*sec(c+d*x)^3*(a+a*sec(c+d*x))^2*tan(c+d*x)/d+1/15*(3*A+4*B)*sec(c+d*x)^3*(a^3+a^3*sec(c+d*x))*tan(c+d*x)/d+1/15*a^3*(19*A+17*B)*tan(c+d*x)^3/d],
[sec(c+d*x)^2*(a+a*sec(c+d*x))^3*(A+B*sec(c+d*x)),x,11,1/8*a^3*(15*A+13*B)*arctanh(sin(c+d*x))/d+1/5*a^3*(15*A+13*B)*tan(c+d*x)/d+3/40*a^3*(15*A+13*B)*sec(c+d*x)*tan(c+d*x)/d+1/20*(5*A-B)*(a+a*sec(c+d*x))^3*tan(c+d*x)/d+1/5*B*(a+a*sec(c+d*x))^4*tan(c+d*x)/(a*d)+1/60*a^3*(15*A+13*B)*tan(c+d*x)^3/d],
[sec(c+d*x)*(a+a*sec(c+d*x))^3*(A+B*sec(c+d*x)),x,10,5/8*a^3*(4*A+3*B)*arctanh(sin(c+d*x))/d+a^3*(4*A+3*B)*tan(c+d*x)/d+3/8*a^3*(4*A+3*B)*sec(c+d*x)*tan(c+d*x)/d+1/4*B*(a+a*sec(c+d*x))^3*tan(c+d*x)/d+1/12*a^3*(4*A+3*B)*tan(c+d*x)^3/d],
[(a+a*sec(c+d*x))^3*(A+B*sec(c+d*x)),x,6,a^3*A*x+1/2*a^3*(7*A+5*B)*arctanh(sin(c+d*x))/d+5/2*a^3*(A+B)*tan(c+d*x)/d+1/3*a*B*(a+a*sec(c+d*x))^2*tan(c+d*x)/d+1/6*(3*A+5*B)*(a^3+a^3*sec(c+d*x))*tan(c+d*x)/d],
[cos(c+d*x)*(a+a*sec(c+d*x))^3*(A+B*sec(c+d*x)),x,5,a^3*(3*A+B)*x+1/2*a^3*(6*A+7*B)*arctanh(sin(c+d*x))/d-5/2*a^3*B*sin(c+d*x)/d+1/2*a*B*(a+a*sec(c+d*x))^2*sin(c+d*x)/d+(A+2*B)*(a^3+a^3*sec(c+d*x))*sin(c+d*x)/d],
[cos(c+d*x)^2*(a+a*sec(c+d*x))^3*(A+B*sec(c+d*x)),x,5,1/2*a^3*(7*A+6*B)*x+a^3*(A+3*B)*arctanh(sin(c+d*x))/d+5/2*a^3*A*sin(c+d*x)/d+1/2*a*A*cos(c+d*x)*(a+a*sec(c+d*x))^2*sin(c+d*x)/d-1/2*(A-2*B)*(a^3+a^3*sec(c+d*x))*sin(c+d*x)/d],
[cos(c+d*x)^3*(a+a*sec(c+d*x))^3*(A+B*sec(c+d*x)),x,5,1/2*a^3*(5*A+7*B)*x+a^3*B*arctanh(sin(c+d*x))/d+5/2*a^3*(A+B)*sin(c+d*x)/d+1/3*a*A*cos(c+d*x)^2*(a+a*sec(c+d*x))^2*sin(c+d*x)/d+1/6*(5*A+3*B)*cos(c+d*x)*(a^3+a^3*sec(c+d*x))*sin(c+d*x)/d],
[cos(c+d*x)^4*(a+a*sec(c+d*x))^3*(A+B*sec(c+d*x)),x,8,5/8*a^3*(3*A+4*B)*x+a^3*(3*A+4*B)*sin(c+d*x)/d+3/8*a^3*(3*A+4*B)*cos(c+d*x)*sin(c+d*x)/d+1/4*A*cos(c+d*x)^3*(a+a*sec(c+d*x))^3*sin(c+d*x)/d-1/12*a^3*(3*A+4*B)*sin(c+d*x)^3/d],
[cos(c+d*x)^5*(a+a*sec(c+d*x))^3*(A+B*sec(c+d*x)),x,7,1/8*a^3*(13*A+15*B)*x+1/15*a^3*(38*A+45*B)*sin(c+d*x)/d+1/8*a^3*(13*A+15*B)*cos(c+d*x)*sin(c+d*x)/d+1/60*a^3*(43*A+45*B)*cos(c+d*x)^2*sin(c+d*x)/d+1/5*a*A*cos(c+d*x)^4*(a+a*sec(c+d*x))^2*sin(c+d*x)/d+1/20*(7*A+5*B)*cos(c+d*x)^3*(a^3+a^3*sec(c+d*x))*sin(c+d*x)/d],
[cos(c+d*x)^6*(a+a*sec(c+d*x))^3*(A+B*sec(c+d*x)),x,8,1/16*a^3*(23*A+26*B)*x+1/5*a^3*(17*A+19*B)*sin(c+d*x)/d+1/16*a^3*(23*A+26*B)*cos(c+d*x)*sin(c+d*x)/d+1/40*a^3*(21*A+22*B)*cos(c+d*x)^3*sin(c+d*x)/d+1/6*a*A*cos(c+d*x)^5*(a+a*sec(c+d*x))^2*sin(c+d*x)/d+1/15*(4*A+3*B)*cos(c+d*x)^4*(a^3+a^3*sec(c+d*x))*sin(c+d*x)/d-1/15*a^3*(17*A+19*B)*sin(c+d*x)^3/d],
[sec(c+d*x)^2*(a+a*sec(c+d*x))^4*(A+B*sec(c+d*x)),x,14,7/16*a^4*(8*A+7*B)*arctanh(sin(c+d*x))/d+4/5*a^4*(8*A+7*B)*tan(c+d*x)/d+27/80*a^4*(8*A+7*B)*sec(c+d*x)*tan(c+d*x)/d+1/40*a^4*(8*A+7*B)*sec(c+d*x)^3*tan(c+d*x)/d+1/30*(6*A-B)*(a+a*sec(c+d*x))^4*tan(c+d*x)/d+1/6*B*(a+a*sec(c+d*x))^5*tan(c+d*x)/(a*d)+2/15*a^4*(8*A+7*B)*tan(c+d*x)^3/d],
[sec(c+d*x)*(a+a*sec(c+d*x))^4*(A+B*sec(c+d*x)),x,13,7/8*a^4*(5*A+4*B)*arctanh(sin(c+d*x))/d+8/5*a^4*(5*A+4*B)*tan(c+d*x)/d+27/40*a^4*(5*A+4*B)*sec(c+d*x)*tan(c+d*x)/d+1/20*a^4*(5*A+4*B)*sec(c+d*x)^3*tan(c+d*x)/d+1/5*B*(a+a*sec(c+d*x))^4*tan(c+d*x)/d+4/15*a^4*(5*A+4*B)*tan(c+d*x)^3/d],
[(a+a*sec(c+d*x))^4*(A+B*sec(c+d*x)),x,7,a^4*A*x+1/8*a^4*(48*A+35*B)*arctanh(sin(c+d*x))/d+5/8*a^4*(8*A+7*B)*tan(c+d*x)/d+1/4*a*B*(a+a*sec(c+d*x))^3*tan(c+d*x)/d+1/12*(4*A+7*B)*(a^2+a^2*sec(c+d*x))^2*tan(c+d*x)/d+1/24*(32*A+35*B)*(a^4+a^4*sec(c+d*x))*tan(c+d*x)/d],
[cos(c+d*x)*(a+a*sec(c+d*x))^4*(A+B*sec(c+d*x)),x,6,a^4*(4*A+B)*x+1/2*a^4*(13*A+12*B)*arctanh(sin(c+d*x))/d-5/2*a^4*(A+2*B)*sin(c+d*x)/d+1/3*a*B*(a+a*sec(c+d*x))^3*sin(c+d*x)/d+1/2*(A+2*B)*(a^2+a^2*sec(c+d*x))^2*sin(c+d*x)/d+1/3*(9*A+11*B)*(a^4+a^4*sec(c+d*x))*sin(c+d*x)/d],
[cos(c+d*x)^2*(a+a*sec(c+d*x))^4*(A+B*sec(c+d*x)),x,6,1/2*a^4*(13*A+8*B)*x+1/2*a^4*(8*A+13*B)*arctanh(sin(c+d*x))/d+5/2*a^4*(A-B)*sin(c+d*x)/d+1/2*a*A*cos(c+d*x)*(a+a*sec(c+d*x))^3*sin(c+d*x)/d-1/2*(A-B)*(a^2+a^2*sec(c+d*x))^2*sin(c+d*x)/d+1/2*(A+6*B)*(a^4+a^4*sec(c+d*x))*sin(c+d*x)/d],
[cos(c+d*x)^3*(a+a*sec(c+d*x))^4*(A+B*sec(c+d*x)),x,6,1/2*a^4*(12*A+13*B)*x+a^4*(A+4*B)*arctanh(sin(c+d*x))/d+5/2*a^4*(2*A+B)*sin(c+d*x)/d+1/3*a*A*cos(c+d*x)^2*(a+a*sec(c+d*x))^3*sin(c+d*x)/d+1/2*(2*A+B)*cos(c+d*x)*(a^2+a^2*sec(c+d*x))^2*sin(c+d*x)/d-1/6*(8*A-3*B)*(a^4+a^4*sec(c+d*x))*sin(c+d*x)/d],
[cos(c+d*x)^4*(a+a*sec(c+d*x))^4*(A+B*sec(c+d*x)),x,6,1/8*a^4*(35*A+48*B)*x+a^4*B*arctanh(sin(c+d*x))/d+5/8*a^4*(7*A+8*B)*sin(c+d*x)/d+1/4*a*A*cos(c+d*x)^3*(a+a*sec(c+d*x))^3*sin(c+d*x)/d+1/12*(7*A+4*B)*cos(c+d*x)^2*(a^2+a^2*sec(c+d*x))^2*sin(c+d*x)/d+1/24*(35*A+32*B)*cos(c+d*x)*(a^4+a^4*sec(c+d*x))*sin(c+d*x)/d],
[cos(c+d*x)^5*(a+a*sec(c+d*x))^4*(A+B*sec(c+d*x)),x,11,7/8*a^4*(4*A+5*B)*x+8/5*a^4*(4*A+5*B)*sin(c+d*x)/d+27/40*a^4*(4*A+5*B)*cos(c+d*x)*sin(c+d*x)/d+1/20*a^4*(4*A+5*B)*cos(c+d*x)^3*sin(c+d*x)/d+1/5*A*cos(c+d*x)^4*(a+a*sec(c+d*x))^4*sin(c+d*x)/d-4/15*a^4*(4*A+5*B)*sin(c+d*x)^3/d],
[cos(c+d*x)^6*(a+a*sec(c+d*x))^4*(A+B*sec(c+d*x)),x,8,7/16*a^4*(7*A+8*B)*x+1/15*a^4*(72*A+83*B)*sin(c+d*x)/d+7/16*a^4*(7*A+8*B)*cos(c+d*x)*sin(c+d*x)/d+1/120*a^4*(159*A+176*B)*cos(c+d*x)^2*sin(c+d*x)/d+1/6*a*A*cos(c+d*x)^5*(a+a*sec(c+d*x))^3*sin(c+d*x)/d+1/10*(3*A+2*B)*cos(c+d*x)^4*(a^2+a^2*sec(c+d*x))^2*sin(c+d*x)/d+1/120*(73*A+72*B)*cos(c+d*x)^3*(a^4+a^4*sec(c+d*x))*sin(c+d*x)/d],
[cos(c+d*x)^7*(a+a*sec(c+d*x))^4*(A+B*sec(c+d*x)),x,9,1/16*a^4*(44*A+49*B)*x+1/35*a^4*(227*A+252*B)*sin(c+d*x)/d+1/16*a^4*(44*A+49*B)*cos(c+d*x)*sin(c+d*x)/d+1/280*a^4*(276*A+301*B)*cos(c+d*x)^3*sin(c+d*x)/d+1/7*a*A*cos(c+d*x)^6*(a+a*sec(c+d*x))^3*sin(c+d*x)/d+1/42*(10*A+7*B)*cos(c+d*x)^5*(a^2+a^2*sec(c+d*x))^2*sin(c+d*x)/d+7/15*(A+B)*cos(c+d*x)^4*(a^4+a^4*sec(c+d*x))*sin(c+d*x)/d-1/105*a^4*(227*A+252*B)*sin(c+d*x)^3/d],

# n<0
[sec(c+d*x)^4*(A+B*sec(c+d*x))/(a+a*sec(c+d*x)),x,6,3/2*(A-B)*arctanh(sin(c+d*x))/(a*d)-(3*A-4*B)*tan(c+d*x)/(a*d)+3/2*(A-B)*sec(c+d*x)*tan(c+d*x)/(a*d)+(A-B)*sec(c+d*x)^3*tan(c+d*x)/(d*(a+a*sec(c+d*x)))-1/3*(3*A-4*B)*tan(c+d*x)^3/(a*d)],
[sec(c+d*x)^3*(A+B*sec(c+d*x))/(a+a*sec(c+d*x)),x,6,-1/2*(2*A-3*B)*arctanh(sin(c+d*x))/(a*d)+2*(A-B)*tan(c+d*x)/(a*d)-1/2*(2*A-3*B)*sec(c+d*x)*tan(c+d*x)/(a*d)+(A-B)*sec(c+d*x)^2*tan(c+d*x)/(d*(a+a*sec(c+d*x)))],
[sec(c+d*x)^2*(A+B*sec(c+d*x))/(a+a*sec(c+d*x)),x,5,(A-B)*arctanh(sin(c+d*x))/(a*d)+B*tan(c+d*x)/(a*d)-(A-B)*tan(c+d*x)/(d*(a+a*sec(c+d*x)))],
[sec(c+d*x)*(A+B*sec(c+d*x))/(a+a*sec(c+d*x)),x,3,B*arctanh(sin(c+d*x))/(a*d)+(A-B)*tan(c+d*x)/(d*(a+a*sec(c+d*x)))],
[(A+B*sec(c+d*x))/(a+a*sec(c+d*x)),x,2,A*x/a-(A-B)*tan(c+d*x)/(d*(a+a*sec(c+d*x)))],
[cos(c+d*x)*(A+B*sec(c+d*x))/(a+a*sec(c+d*x)),x,4,-(A-B)*x/a+(2*A-B)*sin(c+d*x)/(a*d)-(A-B)*sin(c+d*x)/(d*(a+a*sec(c+d*x)))],
[cos(c+d*x)^2*(A+B*sec(c+d*x))/(a+a*sec(c+d*x)),x,5,1/2*(3*A-2*B)*x/a-2*(A-B)*sin(c+d*x)/(a*d)+1/2*(3*A-2*B)*cos(c+d*x)*sin(c+d*x)/(a*d)-(A-B)*cos(c+d*x)*sin(c+d*x)/(d*(a+a*sec(c+d*x)))],
[cos(c+d*x)^3*(A+B*sec(c+d*x))/(a+a*sec(c+d*x)),x,6,-3/2*(A-B)*x/a+(4*A-3*B)*sin(c+d*x)/(a*d)-3/2*(A-B)*cos(c+d*x)*sin(c+d*x)/(a*d)-(A-B)*cos(c+d*x)^2*sin(c+d*x)/(d*(a+a*sec(c+d*x)))-1/3*(4*A-3*B)*sin(c+d*x)^3/(a*d)],
[sec(c+d*x)^5*(A+B*sec(c+d*x))/(a+a*sec(c+d*x))^2,x,7,1/2*(7*A-10*B)*arctanh(sin(c+d*x))/(a^2*d)-4*(2*A-3*B)*tan(c+d*x)/(a^2*d)+1/2*(7*A-10*B)*sec(c+d*x)*tan(c+d*x)/(a^2*d)+1/3*(7*A-10*B)*sec(c+d*x)^3*tan(c+d*x)/(a^2*d*(1+sec(c+d*x)))+1/3*(A-B)*sec(c+d*x)^4*tan(c+d*x)/(d*(a+a*sec(c+d*x))^2)-4/3*(2*A-3*B)*tan(c+d*x)^3/(a^2*d)],
[sec(c+d*x)^4*(A+B*sec(c+d*x))/(a+a*sec(c+d*x))^2,x,7,-1/2*(4*A-7*B)*arctanh(sin(c+d*x))/(a^2*d)+2/3*(5*A-8*B)*tan(c+d*x)/(a^2*d)-1/2*(4*A-7*B)*sec(c+d*x)*tan(c+d*x)/(a^2*d)+1/3*(5*A-8*B)*sec(c+d*x)^2*tan(c+d*x)/(a^2*d*(1+sec(c+d*x)))+1/3*(A-B)*sec(c+d*x)^3*tan(c+d*x)/(d*(a+a*sec(c+d*x))^2)],
[sec(c+d*x)^3*(A+B*sec(c+d*x))/(a+a*sec(c+d*x))^2,x,6,(A-2*B)*arctanh(sin(c+d*x))/(a^2*d)-1/3*(A-4*B)*tan(c+d*x)/(a^2*d)-(A-2*B)*tan(c+d*x)/(a^2*d*(1+sec(c+d*x)))+1/3*(A-B)*sec(c+d*x)^2*tan(c+d*x)/(d*(a+a*sec(c+d*x))^2)],
[sec(c+d*x)^2*(A+B*sec(c+d*x))/(a+a*sec(c+d*x))^2,x,4,B*arctanh(sin(c+d*x))/(a^2*d)+1/3*(2*A-5*B)*tan(c+d*x)/(a^2*d*(1+sec(c+d*x)))-1/3*(A-B)*tan(c+d*x)/(d*(a+a*sec(c+d*x))^2)],
[sec(c+d*x)*(A+B*sec(c+d*x))/(a+a*sec(c+d*x))^2,x,2,1/3*(A-B)*tan(c+d*x)/(d*(a+a*sec(c+d*x))^2)+1/3*(A+2*B)*tan(c+d*x)/(d*(a^2+a^2*sec(c+d*x)))],
[(A+B*sec(c+d*x))/(a+a*sec(c+d*x))^2,x,3,A*x/a^2-1/3*(4*A-B)*tan(c+d*x)/(a^2*d*(1+sec(c+d*x)))-1/3*(A-B)*tan(c+d*x)/(d*(a+a*sec(c+d*x))^2)],
[cos(c+d*x)*(A+B*sec(c+d*x))/(a+a*sec(c+d*x))^2,x,5,-(2*A-B)*x/a^2+2/3*(5*A-2*B)*sin(c+d*x)/(a^2*d)-(2*A-B)*sin(c+d*x)/(a^2*d*(1+sec(c+d*x)))-1/3*(A-B)*sin(c+d*x)/(d*(a+a*sec(c+d*x))^2)],
[cos(c+d*x)^2*(A+B*sec(c+d*x))/(a+a*sec(c+d*x))^2,x,6,1/2*(7*A-4*B)*x/a^2-2/3*(8*A-5*B)*sin(c+d*x)/(a^2*d)+1/2*(7*A-4*B)*cos(c+d*x)*sin(c+d*x)/(a^2*d)-1/3*(8*A-5*B)*cos(c+d*x)*sin(c+d*x)/(a^2*d*(1+sec(c+d*x)))-1/3*(A-B)*cos(c+d*x)*sin(c+d*x)/(d*(a+a*sec(c+d*x))^2)],
[cos(c+d*x)^3*(A+B*sec(c+d*x))/(a+a*sec(c+d*x))^2,x,7,-1/2*(10*A-7*B)*x/a^2+4*(3*A-2*B)*sin(c+d*x)/(a^2*d)-1/2*(10*A-7*B)*cos(c+d*x)*sin(c+d*x)/(a^2*d)-1/3*(10*A-7*B)*cos(c+d*x)^2*sin(c+d*x)/(a^2*d*(1+sec(c+d*x)))-1/3*(A-B)*cos(c+d*x)^2*sin(c+d*x)/(d*(a+a*sec(c+d*x))^2)-4/3*(3*A-2*B)*sin(c+d*x)^3/(a^2*d)],
[sec(c+d*x)^5*(A+B*sec(c+d*x))/(a+a*sec(c+d*x))^3,x,8,-1/2*(6*A-13*B)*arctanh(sin(c+d*x))/(a^3*d)+8/15*(9*A-19*B)*tan(c+d*x)/(a^3*d)-1/2*(6*A-13*B)*sec(c+d*x)*tan(c+d*x)/(a^3*d)+1/5*(A-B)*sec(c+d*x)^4*tan(c+d*x)/(d*(a+a*sec(c+d*x))^3)+1/15*(6*A-11*B)*sec(c+d*x)^3*tan(c+d*x)/(a*d*(a+a*sec(c+d*x))^2)+4/15*(9*A-19*B)*sec(c+d*x)^2*tan(c+d*x)/(d*(a^3+a^3*sec(c+d*x)))],
[sec(c+d*x)^4*(A+B*sec(c+d*x))/(a+a*sec(c+d*x))^3,x,7,(A-3*B)*arctanh(sin(c+d*x))/(a^3*d)-1/15*(7*A-27*B)*tan(c+d*x)/(a^3*d)+1/5*(A-B)*sec(c+d*x)^3*tan(c+d*x)/(d*(a+a*sec(c+d*x))^3)+1/15*(4*A-9*B)*sec(c+d*x)^2*tan(c+d*x)/(a*d*(a+a*sec(c+d*x))^2)-(A-3*B)*tan(c+d*x)/(d*(a^3+a^3*sec(c+d*x)))],
[sec(c+d*x)^3*(A+B*sec(c+d*x))/(a+a*sec(c+d*x))^3,x,5,B*arctanh(sin(c+d*x))/(a^3*d)+1/5*(A-B)*sec(c+d*x)^2*tan(c+d*x)/(d*(a+a*sec(c+d*x))^3)-1/15*(2*A-7*B)*tan(c+d*x)/(a*d*(a+a*sec(c+d*x))^2)+1/15*(4*A-29*B)*tan(c+d*x)/(d*(a^3+a^3*sec(c+d*x)))],
[sec(c+d*x)^2*(A+B*sec(c+d*x))/(a+a*sec(c+d*x))^3,x,3,-1/5*(A-B)*tan(c+d*x)/(d*(a+a*sec(c+d*x))^3)+1/15*(3*A-8*B)*tan(c+d*x)/(a*d*(a+a*sec(c+d*x))^2)+1/15*(3*A+7*B)*tan(c+d*x)/(d*(a^3+a^3*sec(c+d*x)))],
[sec(c+d*x)*(A+B*sec(c+d*x))/(a+a*sec(c+d*x))^3,x,3,1/5*(A-B)*tan(c+d*x)/(d*(a+a*sec(c+d*x))^3)+1/15*(2*A+3*B)*tan(c+d*x)/(a*d*(a+a*sec(c+d*x))^2)+1/15*(2*A+3*B)*tan(c+d*x)/(d*(a^3+a^3*sec(c+d*x)))],
[(A+B*sec(c+d*x))/(a+a*sec(c+d*x))^3,x,4,A*x/a^3-1/5*(A-B)*tan(c+d*x)/(d*(a+a*sec(c+d*x))^3)-1/15*(7*A-2*B)*tan(c+d*x)/(a*d*(a+a*sec(c+d*x))^2)-2/15*(11*A-B)*tan(c+d*x)/(d*(a^3+a^3*sec(c+d*x)))],
[cos(c+d*x)*(A+B*sec(c+d*x))/(a+a*sec(c+d*x))^3,x,6,-(3*A-B)*x/a^3+2/15*(36*A-11*B)*sin(c+d*x)/(a^3*d)-1/5*(A-B)*sin(c+d*x)/(d*(a+a*sec(c+d*x))^3)-1/15*(9*A-4*B)*sin(c+d*x)/(a*d*(a+a*sec(c+d*x))^2)-(3*A-B)*sin(c+d*x)/(d*(a^3+a^3*sec(c+d*x)))],
[cos(c+d*x)^2*(A+B*sec(c+d*x))/(a+a*sec(c+d*x))^3,x,7,1/2*(13*A-6*B)*x/a^3-8/15*(19*A-9*B)*sin(c+d*x)/(a^3*d)+1/2*(13*A-6*B)*cos(c+d*x)*sin(c+d*x)/(a^3*d)-1/5*(A-B)*cos(c+d*x)*sin(c+d*x)/(d*(a+a*sec(c+d*x))^3)-1/15*(11*A-6*B)*cos(c+d*x)*sin(c+d*x)/(a*d*(a+a*sec(c+d*x))^2)-4/15*(19*A-9*B)*cos(c+d*x)*sin(c+d*x)/(d*(a^3+a^3*sec(c+d*x)))],
[cos(c+d*x)^3*(A+B*sec(c+d*x))/(a+a*sec(c+d*x))^3,x,8,-1/2*(23*A-13*B)*x/a^3+4/5*(34*A-19*B)*sin(c+d*x)/(a^3*d)-1/2*(23*A-13*B)*cos(c+d*x)*sin(c+d*x)/(a^3*d)-1/5*(A-B)*cos(c+d*x)^2*sin(c+d*x)/(d*(a+a*sec(c+d*x))^3)-1/15*(13*A-8*B)*cos(c+d*x)^2*sin(c+d*x)/(a*d*(a+a*sec(c+d*x))^2)-1/3*(23*A-13*B)*cos(c+d*x)^2*sin(c+d*x)/(d*(a^3+a^3*sec(c+d*x)))-4/15*(34*A-19*B)*sin(c+d*x)^3/(a^3*d)],
[sec(c+d*x)^6*(A+B*sec(c+d*x))/(a+a*sec(c+d*x))^4,x,9,-1/2*(8*A-21*B)*arctanh(sin(c+d*x))/(a^4*d)+8/105*(83*A-216*B)*tan(c+d*x)/(a^4*d)-1/2*(8*A-21*B)*sec(c+d*x)*tan(c+d*x)/(a^4*d)+1/105*(52*A-129*B)*sec(c+d*x)^3*tan(c+d*x)/(a^4*d*(1+sec(c+d*x))^2)+4/105*(83*A-216*B)*sec(c+d*x)^2*tan(c+d*x)/(a^4*d*(1+sec(c+d*x)))+1/7*(A-B)*sec(c+d*x)^5*tan(c+d*x)/(d*(a+a*sec(c+d*x))^4)+1/5*(A-2*B)*sec(c+d*x)^4*tan(c+d*x)/(a*d*(a+a*sec(c+d*x))^3)],
[sec(c+d*x)^5*(A+B*sec(c+d*x))/(a+a*sec(c+d*x))^4,x,8,(A-4*B)*arctanh(sin(c+d*x))/(a^4*d)-1/105*(55*A-244*B)*tan(c+d*x)/(a^4*d)+1/105*(25*A-88*B)*sec(c+d*x)^2*tan(c+d*x)/(a^4*d*(1+sec(c+d*x))^2)-(A-4*B)*tan(c+d*x)/(a^4*d*(1+sec(c+d*x)))+1/7*(A-B)*sec(c+d*x)^4*tan(c+d*x)/(d*(a+a*sec(c+d*x))^4)+1/35*(5*A-12*B)*sec(c+d*x)^3*tan(c+d*x)/(a*d*(a+a*sec(c+d*x))^3)],
[sec(c+d*x)^4*(A+B*sec(c+d*x))/(a+a*sec(c+d*x))^4,x,6,B*arctanh(sin(c+d*x))/(a^4*d)-1/105*(6*A-55*B)*tan(c+d*x)/(a^4*d*(1+sec(c+d*x))^2)+1/105*(12*A-215*B)*tan(c+d*x)/(a^4*d*(1+sec(c+d*x)))+1/7*(A-B)*sec(c+d*x)^3*tan(c+d*x)/(d*(a+a*sec(c+d*x))^4)+1/35*(3*A-10*B)*sec(c+d*x)^2*tan(c+d*x)/(a*d*(a+a*sec(c+d*x))^3)],
[sec(c+d*x)^3*(A+B*sec(c+d*x))/(a+a*sec(c+d*x))^4,x,4,-1/7*(A-B)*sec(c+d*x)^3*tan(c+d*x)/(d*(a+a*sec(c+d*x))^4)+1/35*(4*A+3*B)*tan(c+d*x)/(a*d*(a+a*sec(c+d*x))^3)-8/105*(4*A+3*B)*tan(c+d*x)/(d*(a^2+a^2*sec(c+d*x))^2)+1/15*(4*A+3*B)*tan(c+d*x)/(d*(a^4+a^4*sec(c+d*x)))],
[sec(c+d*x)^2*(A+B*sec(c+d*x))/(a+a*sec(c+d*x))^4,x,4,-1/7*(A-B)*tan(c+d*x)/(d*(a+a*sec(c+d*x))^4)+1/35*(4*A-11*B)*tan(c+d*x)/(a*d*(a+a*sec(c+d*x))^3)+1/105*(8*A+13*B)*tan(c+d*x)/(d*(a^2+a^2*sec(c+d*x))^2)+1/105*(8*A+13*B)*tan(c+d*x)/(d*(a^4+a^4*sec(c+d*x)))],
[sec(c+d*x)*(A+B*sec(c+d*x))/(a+a*sec(c+d*x))^4,x,4,1/7*(A-B)*tan(c+d*x)/(d*(a+a*sec(c+d*x))^4)+1/35*(3*A+4*B)*tan(c+d*x)/(a*d*(a+a*sec(c+d*x))^3)+2/105*(3*A+4*B)*tan(c+d*x)/(d*(a^2+a^2*sec(c+d*x))^2)+2/105*(3*A+4*B)*tan(c+d*x)/(d*(a^4+a^4*sec(c+d*x)))],
[(A+B*sec(c+d*x))/(a+a*sec(c+d*x))^4,x,5,A*x/a^4-1/105*(55*A-6*B)*tan(c+d*x)/(a^4*d*(1+sec(c+d*x))^2)-2/105*(80*A-3*B)*tan(c+d*x)/(a^4*d*(1+sec(c+d*x)))-1/7*(A-B)*tan(c+d*x)/(d*(a+a*sec(c+d*x))^4)-1/35*(10*A-3*B)*tan(c+d*x)/(a*d*(a+a*sec(c+d*x))^3)],
[cos(c+d*x)*(A+B*sec(c+d*x))/(a+a*sec(c+d*x))^4,x,7,-(4*A-B)*x/a^4+8/105*(83*A-20*B)*sin(c+d*x)/(a^4*d)-1/105*(88*A-25*B)*sin(c+d*x)/(a^4*d*(1+sec(c+d*x))^2)-(4*A-B)*sin(c+d*x)/(a^4*d*(1+sec(c+d*x)))-1/7*(A-B)*sin(c+d*x)/(d*(a+a*sec(c+d*x))^4)-1/35*(12*A-5*B)*sin(c+d*x)/(a*d*(a+a*sec(c+d*x))^3)],
[cos(c+d*x)^2*(A+B*sec(c+d*x))/(a+a*sec(c+d*x))^4,x,8,1/2*(21*A-8*B)*x/a^4-8/105*(216*A-83*B)*sin(c+d*x)/(a^4*d)+1/2*(21*A-8*B)*cos(c+d*x)*sin(c+d*x)/(a^4*d)-1/105*(129*A-52*B)*cos(c+d*x)*sin(c+d*x)/(a^4*d*(1+sec(c+d*x))^2)-4/105*(216*A-83*B)*cos(c+d*x)*sin(c+d*x)/(a^4*d*(1+sec(c+d*x)))-1/7*(A-B)*cos(c+d*x)*sin(c+d*x)/(d*(a+a*sec(c+d*x))^4)-1/5*(2*A-B)*cos(c+d*x)*sin(c+d*x)/(a*d*(a+a*sec(c+d*x))^3)],
[cos(c+d*x)^3*(A+B*sec(c+d*x))/(a+a*sec(c+d*x))^4,x,9,-1/2*(44*A-21*B)*x/a^4+8/35*(227*A-108*B)*sin(c+d*x)/(a^4*d)-1/2*(44*A-21*B)*cos(c+d*x)*sin(c+d*x)/(a^4*d)-1/105*(178*A-87*B)*cos(c+d*x)^2*sin(c+d*x)/(a^4*d*(1+sec(c+d*x))^2)-1/3*(44*A-21*B)*cos(c+d*x)^2*sin(c+d*x)/(a^4*d*(1+sec(c+d*x)))-1/7*(A-B)*cos(c+d*x)^2*sin(c+d*x)/(d*(a+a*sec(c+d*x))^4)-1/35*(16*A-9*B)*cos(c+d*x)^2*sin(c+d*x)/(a*d*(a+a*sec(c+d*x))^3)-8/105*(227*A-108*B)*sin(c+d*x)^3/(a^4*d)],

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

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

# n<0
[sec(c+d*x)^4*(A+B*sec(c+d*x))/(a+a*sec(c+d*x))^(1/2),x,6,-(A-B)*arctan(sqrt(a)*tan(c+d*x)/(sqrt(2)*sqrt(a+a*sec(c+d*x))))*sqrt(2)/(d*sqrt(a))+4/105*(49*A-37*B)*tan(c+d*x)/(d*sqrt(a+a*sec(c+d*x)))+2/35*(7*A-B)*sec(c+d*x)^2*tan(c+d*x)/(d*sqrt(a+a*sec(c+d*x)))+2/7*B*sec(c+d*x)^3*tan(c+d*x)/(d*sqrt(a+a*sec(c+d*x)))-2/105*(7*A-31*B)*sqrt(a+a*sec(c+d*x))*tan(c+d*x)/(a*d)],
[sec(c+d*x)^3*(A+B*sec(c+d*x))/(a+a*sec(c+d*x))^(1/2),x,5,(A-B)*arctan(sqrt(a)*tan(c+d*x)/(sqrt(2)*sqrt(a+a*sec(c+d*x))))*sqrt(2)/(d*sqrt(a))-4/15*(5*A-7*B)*tan(c+d*x)/(d*sqrt(a+a*sec(c+d*x)))+2/5*B*sec(c+d*x)^2*tan(c+d*x)/(d*sqrt(a+a*sec(c+d*x)))+2/15*(5*A-B)*sqrt(a+a*sec(c+d*x))*tan(c+d*x)/(a*d)],
[sec(c+d*x)^2*(A+B*sec(c+d*x))/(a+a*sec(c+d*x))^(1/2),x,4,-(A-B)*arctan(sqrt(a)*tan(c+d*x)/(sqrt(2)*sqrt(a+a*sec(c+d*x))))*sqrt(2)/(d*sqrt(a))+2/3*(3*A-2*B)*tan(c+d*x)/(d*sqrt(a+a*sec(c+d*x)))+2/3*B*sqrt(a+a*sec(c+d*x))*tan(c+d*x)/(a*d)],
[sec(c+d*x)*(A+B*sec(c+d*x))/(a+a*sec(c+d*x))^(1/2),x,3,(A-B)*arctan(sqrt(a)*tan(c+d*x)/(sqrt(2)*sqrt(a+a*sec(c+d*x))))*sqrt(2)/(d*sqrt(a))+2*B*tan(c+d*x)/(d*sqrt(a+a*sec(c+d*x)))],
[(A+B*sec(c+d*x))/(a+a*sec(c+d*x))^(1/2),x,5,2*A*arctan(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))/(d*sqrt(a))-(A-B)*arctan(sqrt(a)*tan(c+d*x)/(sqrt(2)*sqrt(a+a*sec(c+d*x))))*sqrt(2)/(d*sqrt(a))],
[cos(c+d*x)*(A+B*sec(c+d*x))/(a+a*sec(c+d*x))^(1/2),x,6,-(A-2*B)*arctan(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))/(d*sqrt(a))+(A-B)*arctan(sqrt(a)*tan(c+d*x)/(sqrt(2)*sqrt(a+a*sec(c+d*x))))*sqrt(2)/(d*sqrt(a))+A*sin(c+d*x)/(d*sqrt(a+a*sec(c+d*x)))],
[cos(c+d*x)^2*(A+B*sec(c+d*x))/(a+a*sec(c+d*x))^(1/2),x,7,1/4*(7*A-4*B)*arctan(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))/(d*sqrt(a))-(A-B)*arctan(sqrt(a)*tan(c+d*x)/(sqrt(2)*sqrt(a+a*sec(c+d*x))))*sqrt(2)/(d*sqrt(a))-1/4*(A-4*B)*sin(c+d*x)/(d*sqrt(a+a*sec(c+d*x)))+1/2*A*cos(c+d*x)*sin(c+d*x)/(d*sqrt(a+a*sec(c+d*x)))],
[cos(c+d*x)^3*(A+B*sec(c+d*x))/(a+a*sec(c+d*x))^(1/2),x,8,-1/8*(9*A-14*B)*arctan(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))/(d*sqrt(a))+(A-B)*arctan(sqrt(a)*tan(c+d*x)/(sqrt(2)*sqrt(a+a*sec(c+d*x))))*sqrt(2)/(d*sqrt(a))+1/8*(7*A-2*B)*sin(c+d*x)/(d*sqrt(a+a*sec(c+d*x)))-1/12*(A-6*B)*cos(c+d*x)*sin(c+d*x)/(d*sqrt(a+a*sec(c+d*x)))+1/3*A*cos(c+d*x)^2*sin(c+d*x)/(d*sqrt(a+a*sec(c+d*x)))],
[sec(c+d*x)^4*(A+B*sec(c+d*x))/(a+a*sec(c+d*x))^(3/2),x,6,1/2*(11*A-15*B)*arctan(sqrt(a)*tan(c+d*x)/(sqrt(2)*sqrt(a+a*sec(c+d*x))))/(a^(3/2)*d*sqrt(2))+1/2*(A-B)*sec(c+d*x)^3*tan(c+d*x)/(d*(a+a*sec(c+d*x))^(3/2))-1/15*(65*A-93*B)*tan(c+d*x)/(a*d*sqrt(a+a*sec(c+d*x)))-1/10*(5*A-9*B)*sec(c+d*x)^2*tan(c+d*x)/(a*d*sqrt(a+a*sec(c+d*x)))+1/30*(35*A-39*B)*sqrt(a+a*sec(c+d*x))*tan(c+d*x)/(a^2*d)],
[sec(c+d*x)^3*(A+B*sec(c+d*x))/(a+a*sec(c+d*x))^(3/2),x,5,-1/2*(7*A-11*B)*arctan(sqrt(a)*tan(c+d*x)/(sqrt(2)*sqrt(a+a*sec(c+d*x))))/(a^(3/2)*d*sqrt(2))+1/2*(A-B)*sec(c+d*x)^2*tan(c+d*x)/(d*(a+a*sec(c+d*x))^(3/2))+1/3*(9*A-13*B)*tan(c+d*x)/(a*d*sqrt(a+a*sec(c+d*x)))-1/6*(3*A-7*B)*sqrt(a+a*sec(c+d*x))*tan(c+d*x)/(a^2*d)],
[sec(c+d*x)^2*(A+B*sec(c+d*x))/(a+a*sec(c+d*x))^(3/2),x,4,1/2*(3*A-7*B)*arctan(sqrt(a)*tan(c+d*x)/(sqrt(2)*sqrt(a+a*sec(c+d*x))))/(a^(3/2)*d*sqrt(2))-1/2*(A-B)*tan(c+d*x)/(d*(a+a*sec(c+d*x))^(3/2))+2*B*tan(c+d*x)/(a*d*sqrt(a+a*sec(c+d*x)))],
[sec(c+d*x)*(A+B*sec(c+d*x))/(a+a*sec(c+d*x))^(3/2),x,3,1/2*(A+3*B)*arctan(sqrt(a)*tan(c+d*x)/(sqrt(2)*sqrt(a+a*sec(c+d*x))))/(a^(3/2)*d*sqrt(2))+1/2*(A-B)*tan(c+d*x)/(d*(a+a*sec(c+d*x))^(3/2))],
[(A+B*sec(c+d*x))/(a+a*sec(c+d*x))^(3/2),x,6,2*A*arctan(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))/(a^(3/2)*d)-1/2*(5*A-B)*arctan(sqrt(a)*tan(c+d*x)/(sqrt(2)*sqrt(a+a*sec(c+d*x))))/(a^(3/2)*d*sqrt(2))-1/2*(A-B)*tan(c+d*x)/(d*(a+a*sec(c+d*x))^(3/2))],
[cos(c+d*x)*(A+B*sec(c+d*x))/(a+a*sec(c+d*x))^(3/2),x,7,-(3*A-2*B)*arctan(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))/(a^(3/2)*d)-1/2*(A-B)*sin(c+d*x)/(d*(a+a*sec(c+d*x))^(3/2))+1/2*(9*A-5*B)*arctan(sqrt(a)*tan(c+d*x)/(sqrt(2)*sqrt(a+a*sec(c+d*x))))/(a^(3/2)*d*sqrt(2))+1/2*(3*A-B)*sin(c+d*x)/(a*d*sqrt(a+a*sec(c+d*x)))],
[cos(c+d*x)^2*(A+B*sec(c+d*x))/(a+a*sec(c+d*x))^(3/2),x,8,1/4*(19*A-12*B)*arctan(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))/(a^(3/2)*d)-1/2*(A-B)*cos(c+d*x)*sin(c+d*x)/(d*(a+a*sec(c+d*x))^(3/2))-1/2*(13*A-9*B)*arctan(sqrt(a)*tan(c+d*x)/(sqrt(2)*sqrt(a+a*sec(c+d*x))))/(a^(3/2)*d*sqrt(2))-1/4*(7*A-6*B)*sin(c+d*x)/(a*d*sqrt(a+a*sec(c+d*x)))+1/2*(2*A-B)*cos(c+d*x)*sin(c+d*x)/(a*d*sqrt(a+a*sec(c+d*x)))],
[cos(c+d*x)^3*(A+B*sec(c+d*x))/(a+a*sec(c+d*x))^(3/2),x,9,-1/8*(47*A-38*B)*arctan(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))/(a^(3/2)*d)-1/2*(A-B)*cos(c+d*x)^2*sin(c+d*x)/(d*(a+a*sec(c+d*x))^(3/2))+1/2*(17*A-13*B)*arctan(sqrt(a)*tan(c+d*x)/(sqrt(2)*sqrt(a+a*sec(c+d*x))))/(a^(3/2)*d*sqrt(2))+7/8*(3*A-2*B)*sin(c+d*x)/(a*d*sqrt(a+a*sec(c+d*x)))-1/12*(13*A-12*B)*cos(c+d*x)*sin(c+d*x)/(a*d*sqrt(a+a*sec(c+d*x)))+1/6*(5*A-3*B)*cos(c+d*x)^2*sin(c+d*x)/(a*d*sqrt(a+a*sec(c+d*x)))],
[sec(c+d*x)^4*(A+B*sec(c+d*x))/(a+a*sec(c+d*x))^(5/2),x,6,-1/16*(75*A-163*B)*arctan(sqrt(a)*tan(c+d*x)/(sqrt(2)*sqrt(a+a*sec(c+d*x))))/(a^(5/2)*d*sqrt(2))+1/4*(A-B)*sec(c+d*x)^3*tan(c+d*x)/(d*(a+a*sec(c+d*x))^(5/2))+1/16*(9*A-17*B)*sec(c+d*x)^2*tan(c+d*x)/(a*d*(a+a*sec(c+d*x))^(3/2))+1/24*(93*A-197*B)*tan(c+d*x)/(a^2*d*sqrt(a+a*sec(c+d*x)))-1/48*(39*A-95*B)*sqrt(a+a*sec(c+d*x))*tan(c+d*x)/(a^3*d)],
[sec(c+d*x)^3*(A+B*sec(c+d*x))/(a+a*sec(c+d*x))^(5/2),x,5,1/16*(19*A-75*B)*arctan(sqrt(a)*tan(c+d*x)/(sqrt(2)*sqrt(a+a*sec(c+d*x))))/(a^(5/2)*d*sqrt(2))+1/4*(A-B)*sec(c+d*x)^2*tan(c+d*x)/(d*(a+a*sec(c+d*x))^(5/2))-1/16*(5*A-13*B)*tan(c+d*x)/(a*d*(a+a*sec(c+d*x))^(3/2))-1/4*(A-9*B)*tan(c+d*x)/(a^2*d*sqrt(a+a*sec(c+d*x)))],
[sec(c+d*x)^2*(A+B*sec(c+d*x))/(a+a*sec(c+d*x))^(5/2),x,4,1/16*(5*A+19*B)*arctan(sqrt(a)*tan(c+d*x)/(sqrt(2)*sqrt(a+a*sec(c+d*x))))/(a^(5/2)*d*sqrt(2))-1/4*(A-B)*tan(c+d*x)/(d*(a+a*sec(c+d*x))^(5/2))+1/16*(5*A-13*B)*tan(c+d*x)/(a*d*(a+a*sec(c+d*x))^(3/2))],
[sec(c+d*x)*(A+B*sec(c+d*x))/(a+a*sec(c+d*x))^(5/2),x,4,1/16*(3*A+5*B)*arctan(sqrt(a)*tan(c+d*x)/(sqrt(2)*sqrt(a+a*sec(c+d*x))))/(a^(5/2)*d*sqrt(2))+1/4*(A-B)*tan(c+d*x)/(d*(a+a*sec(c+d*x))^(5/2))+1/16*(3*A+5*B)*tan(c+d*x)/(a*d*(a+a*sec(c+d*x))^(3/2))],
[(A+B*sec(c+d*x))/(a+a*sec(c+d*x))^(5/2),x,7,2*A*arctan(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))/(a^(5/2)*d)-1/16*(43*A-3*B)*arctan(sqrt(a)*tan(c+d*x)/(sqrt(2)*sqrt(a+a*sec(c+d*x))))/(a^(5/2)*d*sqrt(2))-1/4*(A-B)*tan(c+d*x)/(d*(a+a*sec(c+d*x))^(5/2))-1/16*(11*A-3*B)*tan(c+d*x)/(a*d*(a+a*sec(c+d*x))^(3/2))],
[cos(c+d*x)*(A+B*sec(c+d*x))/(a+a*sec(c+d*x))^(5/2),x,8,-(5*A-2*B)*arctan(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))/(a^(5/2)*d)-1/4*(A-B)*sin(c+d*x)/(d*(a+a*sec(c+d*x))^(5/2))-1/16*(15*A-7*B)*sin(c+d*x)/(a*d*(a+a*sec(c+d*x))^(3/2))+1/16*(115*A-43*B)*arctan(sqrt(a)*tan(c+d*x)/(sqrt(2)*sqrt(a+a*sec(c+d*x))))/(a^(5/2)*d*sqrt(2))+1/16*(35*A-11*B)*sin(c+d*x)/(a^2*d*sqrt(a+a*sec(c+d*x)))],
[cos(c+d*x)^2*(A+B*sec(c+d*x))/(a+a*sec(c+d*x))^(5/2),x,9,1/4*(39*A-20*B)*arctan(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))/(a^(5/2)*d)-1/4*(A-B)*cos(c+d*x)*sin(c+d*x)/(d*(a+a*sec(c+d*x))^(5/2))-1/16*(19*A-11*B)*cos(c+d*x)*sin(c+d*x)/(a*d*(a+a*sec(c+d*x))^(3/2))-1/16*(219*A-115*B)*arctan(sqrt(a)*tan(c+d*x)/(sqrt(2)*sqrt(a+a*sec(c+d*x))))/(a^(5/2)*d*sqrt(2))-7/16*(9*A-5*B)*sin(c+d*x)/(a^2*d*sqrt(a+a*sec(c+d*x)))+1/16*(31*A-15*B)*cos(c+d*x)*sin(c+d*x)/(a^2*d*sqrt(a+a*sec(c+d*x)))],

# Integrands of the form Sec[e+f x]^n (a - a Sec[e+f x])^(m/2) (A+A Sec[e+f x])
[(A+A*sec(c+d*x))/(a-a*sec(c+d*x))^(1/2),x,5,2*A*arctan(sqrt(a)*tan(c+d*x)/sqrt(a-a*sec(c+d*x)))/(d*sqrt(a))-2*A*arctan(sqrt(a)*tan(c+d*x)/(sqrt(2)*sqrt(a-a*sec(c+d*x))))*sqrt(2)/(d*sqrt(a))],
[cos(c+d*x)*(A+A*sec(c+d*x))/(a-a*sec(c+d*x))^(1/2),x,6,3*A*arctan(sqrt(a)*tan(c+d*x)/sqrt(a-a*sec(c+d*x)))/(d*sqrt(a))-2*A*arctan(sqrt(a)*tan(c+d*x)/(sqrt(2)*sqrt(a-a*sec(c+d*x))))*sqrt(2)/(d*sqrt(a))+A*sin(c+d*x)/(d*sqrt(a-a*sec(c+d*x)))],
[cos(c+d*x)^2*(A+A*sec(c+d*x))/(a-a*sec(c+d*x))^(1/2),x,7,11/4*A*arctan(sqrt(a)*tan(c+d*x)/sqrt(a-a*sec(c+d*x)))/(d*sqrt(a))-2*A*arctan(sqrt(a)*tan(c+d*x)/(sqrt(2)*sqrt(a-a*sec(c+d*x))))*sqrt(2)/(d*sqrt(a))+5/4*A*sin(c+d*x)/(d*sqrt(a-a*sec(c+d*x)))+1/2*A*cos(c+d*x)*sin(c+d*x)/(d*sqrt(a-a*sec(c+d*x)))],
[cos(c+d*x)^3*(A+A*sec(c+d*x))/(a-a*sec(c+d*x))^(1/2),x,8,23/8*A*arctan(sqrt(a)*tan(c+d*x)/sqrt(a-a*sec(c+d*x)))/(d*sqrt(a))-2*A*arctan(sqrt(a)*tan(c+d*x)/(sqrt(2)*sqrt(a-a*sec(c+d*x))))*sqrt(2)/(d*sqrt(a))+9/8*A*sin(c+d*x)/(d*sqrt(a-a*sec(c+d*x)))+7/12*A*cos(c+d*x)*sin(c+d*x)/(d*sqrt(a-a*sec(c+d*x)))+1/3*A*cos(c+d*x)^2*sin(c+d*x)/(d*sqrt(a-a*sec(c+d*x)))],
[(A+A*sec(c+d*x))/(a-a*sec(c+d*x))^(3/2),x,6,2*A*arctan(sqrt(a)*tan(c+d*x)/sqrt(a-a*sec(c+d*x)))/(a^(3/2)*d)-3*A*arctan(sqrt(a)*tan(c+d*x)/(sqrt(2)*sqrt(a-a*sec(c+d*x))))/(a^(3/2)*d*sqrt(2))-A*tan(c+d*x)/(d*(a-a*sec(c+d*x))^(3/2)),2*A*arctan(sqrt(a)*tan(c+d*x)/sqrt(a-a*sec(c+d*x)))/(a^(3/2)*d)-3*A*arctan(sqrt(a)*tan(c+d*x)/(sqrt(2)*sqrt(a-a*sec(c+d*x))))/(a^(3/2)*d*sqrt(2))+1/2*A*csc(1/2*(c+d*x))^2*sin(c+d*x)/(a*d*sqrt(a-a*sec(c+d*x)))],
[cos(c+d*x)*(A+A*sec(c+d*x))/(a-a*sec(c+d*x))^(3/2),x,7,5*A*arctan(sqrt(a)*tan(c+d*x)/sqrt(a-a*sec(c+d*x)))/(a^(3/2)*d)-A*sin(c+d*x)/(d*(a-a*sec(c+d*x))^(3/2))-7*A*arctan(sqrt(a)*tan(c+d*x)/(sqrt(2)*sqrt(a-a*sec(c+d*x))))/(a^(3/2)*d*sqrt(2))+2*A*sin(c+d*x)/(a*d*sqrt(a-a*sec(c+d*x)))],
[cos(c+d*x)^2*(A+A*sec(c+d*x))/(a-a*sec(c+d*x))^(3/2),x,8,31/4*A*arctan(sqrt(a)*tan(c+d*x)/sqrt(a-a*sec(c+d*x)))/(a^(3/2)*d)-A*cos(c+d*x)*sin(c+d*x)/(d*(a-a*sec(c+d*x))^(3/2))-11*A*arctan(sqrt(a)*tan(c+d*x)/(sqrt(2)*sqrt(a-a*sec(c+d*x))))/(a^(3/2)*d*sqrt(2))+13/4*A*sin(c+d*x)/(a*d*sqrt(a-a*sec(c+d*x)))+3/2*A*cos(c+d*x)*sin(c+d*x)/(a*d*sqrt(a-a*sec(c+d*x)))],
[cos(c+d*x)^3*(A+A*sec(c+d*x))/(a-a*sec(c+d*x))^(3/2),x,9,85/8*A*arctan(sqrt(a)*tan(c+d*x)/sqrt(a-a*sec(c+d*x)))/(a^(3/2)*d)-A*cos(c+d*x)^2*sin(c+d*x)/(d*(a-a*sec(c+d*x))^(3/2))-15*A*arctan(sqrt(a)*tan(c+d*x)/(sqrt(2)*sqrt(a-a*sec(c+d*x))))/(a^(3/2)*d*sqrt(2))+35/8*A*sin(c+d*x)/(a*d*sqrt(a-a*sec(c+d*x)))+25/12*A*cos(c+d*x)*sin(c+d*x)/(a*d*sqrt(a-a*sec(c+d*x)))+4/3*A*cos(c+d*x)^2*sin(c+d*x)/(a*d*sqrt(a-a*sec(c+d*x)))],
[(A+A*sec(c+d*x))/(a-a*sec(c+d*x))^(5/2),x,7,2*A*arctan(sqrt(a)*tan(c+d*x)/sqrt(a-a*sec(c+d*x)))/(a^(5/2)*d)-23/8*A*arctan(sqrt(a)*tan(c+d*x)/(sqrt(2)*sqrt(a-a*sec(c+d*x))))/(a^(5/2)*d*sqrt(2))-1/2*A*tan(c+d*x)/(d*(a-a*sec(c+d*x))^(5/2))-7/8*A*tan(c+d*x)/(a*d*(a-a*sec(c+d*x))^(3/2)),2*A*arctan(sqrt(a)*tan(c+d*x)/sqrt(a-a*sec(c+d*x)))/(a^(5/2)*d)-23/8*A*arctan(sqrt(a)*tan(c+d*x)/(sqrt(2)*sqrt(a-a*sec(c+d*x))))/(a^(5/2)*d*sqrt(2))+7/16*A*csc(1/2*(c+d*x))^2*sin(c+d*x)/(a^2*d*sqrt(a-a*sec(c+d*x)))-1/8*A*cos(c+d*x)*csc(1/2*(c+d*x))^4*sin(c+d*x)/(a^2*d*sqrt(a-a*sec(c+d*x)))],
[cos(c+d*x)*(A+A*sec(c+d*x))/(a-a*sec(c+d*x))^(5/2),x,8,7*A*arctan(sqrt(a)*tan(c+d*x)/sqrt(a-a*sec(c+d*x)))/(a^(5/2)*d)-1/2*A*sin(c+d*x)/(d*(a-a*sec(c+d*x))^(5/2))-11/8*A*sin(c+d*x)/(a*d*(a-a*sec(c+d*x))^(3/2))-79/8*A*arctan(sqrt(a)*tan(c+d*x)/(sqrt(2)*sqrt(a-a*sec(c+d*x))))/(a^(5/2)*d*sqrt(2))+23/8*A*sin(c+d*x)/(a^2*d*sqrt(a-a*sec(c+d*x)))],
[cos(c+d*x)^2*(A+A*sec(c+d*x))/(a-a*sec(c+d*x))^(5/2),x,9,59/4*A*arctan(sqrt(a)*tan(c+d*x)/sqrt(a-a*sec(c+d*x)))/(a^(5/2)*d)-1/2*A*cos(c+d*x)*sin(c+d*x)/(d*(a-a*sec(c+d*x))^(5/2))-15/8*A*cos(c+d*x)*sin(c+d*x)/(a*d*(a-a*sec(c+d*x))^(3/2))-167/8*A*arctan(sqrt(a)*tan(c+d*x)/(sqrt(2)*sqrt(a-a*sec(c+d*x))))/(a^(5/2)*d*sqrt(2))+49/8*A*sin(c+d*x)/(a^2*d*sqrt(a-a*sec(c+d*x)))+23/8*A*cos(c+d*x)*sin(c+d*x)/(a^2*d*sqrt(a-a*sec(c+d*x)))],
[cos(c+d*x)^3*(A+A*sec(c+d*x))/(a-a*sec(c+d*x))^(5/2),x,10,203/8*A*arctan(sqrt(a)*tan(c+d*x)/sqrt(a-a*sec(c+d*x)))/(a^(5/2)*d)-1/2*A*cos(c+d*x)^2*sin(c+d*x)/(d*(a-a*sec(c+d*x))^(5/2))-19/8*A*cos(c+d*x)^2*sin(c+d*x)/(a*d*(a-a*sec(c+d*x))^(3/2))-287/8*A*arctan(sqrt(a)*tan(c+d*x)/(sqrt(2)*sqrt(a-a*sec(c+d*x))))/(a^(5/2)*d*sqrt(2))+21/2*A*sin(c+d*x)/(a^2*d*sqrt(a-a*sec(c+d*x)))+119/24*A*cos(c+d*x)*sin(c+d*x)/(a^2*d*sqrt(a-a*sec(c+d*x)))+77/24*A*cos(c+d*x)^2*sin(c+d*x)/(a^2*d*sqrt(a-a*sec(c+d*x)))],

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

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

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

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

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

# n<0
[sec(c+d*x)^(5/2)*(A+B*sec(c+d*x))/sqrt(a+a*sec(c+d*x)),x,7,-1/4*(4*A-7*B)*arcsinh(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))/(d*sqrt(a))+(A-B)*arctanh(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*sec(c+d*x))))*sqrt(2)/(d*sqrt(a))+1/4*(4*A-B)*sec(c+d*x)^(3/2)*sin(c+d*x)/(d*sqrt(a+a*sec(c+d*x)))+1/2*B*sec(c+d*x)^(5/2)*sin(c+d*x)/(d*sqrt(a+a*sec(c+d*x)))],
[sec(c+d*x)^(3/2)*(A+B*sec(c+d*x))/sqrt(a+a*sec(c+d*x)),x,6,(2*A-B)*arcsinh(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))/(d*sqrt(a))-(A-B)*arctanh(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*sec(c+d*x))))*sqrt(2)/(d*sqrt(a))+B*sec(c+d*x)^(3/2)*sin(c+d*x)/(d*sqrt(a+a*sec(c+d*x)))],
[(A+B*sec(c+d*x))*sqrt(sec(c+d*x))/sqrt(a+a*sec(c+d*x)),x,5,2*B*arcsinh(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))/(d*sqrt(a))+(A-B)*arctanh(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*sec(c+d*x))))*sqrt(2)/(d*sqrt(a))],
[(A+B*sec(c+d*x))/(sqrt(sec(c+d*x))*sqrt(a+a*sec(c+d*x))),x,3,-(A-B)*arctanh(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*sec(c+d*x))))*sqrt(2)/(d*sqrt(a))+2*A*sin(c+d*x)*sqrt(sec(c+d*x))/(d*sqrt(a+a*sec(c+d*x)))],
[(A+B*sec(c+d*x))/(sec(c+d*x)^(3/2)*sqrt(a+a*sec(c+d*x))),x,4,(A-B)*arctanh(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*sec(c+d*x))))*sqrt(2)/(d*sqrt(a))+2/3*A*sin(c+d*x)/(d*sqrt(sec(c+d*x))*sqrt(a+a*sec(c+d*x)))-2/3*(A-3*B)*sin(c+d*x)*sqrt(sec(c+d*x))/(d*sqrt(a+a*sec(c+d*x)))],
[(A+B*sec(c+d*x))/(sec(c+d*x)^(5/2)*sqrt(a+a*sec(c+d*x))),x,5,-(A-B)*arctanh(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*sec(c+d*x))))*sqrt(2)/(d*sqrt(a))+2/5*A*sin(c+d*x)/(d*sec(c+d*x)^(3/2)*sqrt(a+a*sec(c+d*x)))-2/15*(A-5*B)*sin(c+d*x)/(d*sqrt(sec(c+d*x))*sqrt(a+a*sec(c+d*x)))+2/15*(13*A-5*B)*sin(c+d*x)*sqrt(sec(c+d*x))/(d*sqrt(a+a*sec(c+d*x)))],
[(A+B*sec(c+d*x))/(sec(c+d*x)^(7/2)*sqrt(a+a*sec(c+d*x))),x,6,(A-B)*arctanh(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*sec(c+d*x))))*sqrt(2)/(d*sqrt(a))+2/7*A*sin(c+d*x)/(d*sec(c+d*x)^(5/2)*sqrt(a+a*sec(c+d*x)))-2/35*(A-7*B)*sin(c+d*x)/(d*sec(c+d*x)^(3/2)*sqrt(a+a*sec(c+d*x)))+2/105*(31*A-7*B)*sin(c+d*x)/(d*sqrt(sec(c+d*x))*sqrt(a+a*sec(c+d*x)))-2/105*(43*A-91*B)*sin(c+d*x)*sqrt(sec(c+d*x))/(d*sqrt(a+a*sec(c+d*x)))],
[sec(c+d*x)^(7/2)*(A+B*sec(c+d*x))/(a+a*sec(c+d*x))^(3/2),x,8,-1/4*(12*A-19*B)*arcsinh(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))/(a^(3/2)*d)+1/2*(A-B)*sec(c+d*x)^(7/2)*sin(c+d*x)/(d*(a+a*sec(c+d*x))^(3/2))+1/2*(9*A-13*B)*arctanh(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*sec(c+d*x))))/(a^(3/2)*d*sqrt(2))+1/4*(6*A-7*B)*sec(c+d*x)^(3/2)*sin(c+d*x)/(a*d*sqrt(a+a*sec(c+d*x)))-1/2*(A-2*B)*sec(c+d*x)^(5/2)*sin(c+d*x)/(a*d*sqrt(a+a*sec(c+d*x)))],
[sec(c+d*x)^(5/2)*(A+B*sec(c+d*x))/(a+a*sec(c+d*x))^(3/2),x,7,(2*A-3*B)*arcsinh(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))/(a^(3/2)*d)+1/2*(A-B)*sec(c+d*x)^(5/2)*sin(c+d*x)/(d*(a+a*sec(c+d*x))^(3/2))-1/2*(5*A-9*B)*arctanh(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*sec(c+d*x))))/(a^(3/2)*d*sqrt(2))-1/2*(A-3*B)*sec(c+d*x)^(3/2)*sin(c+d*x)/(a*d*sqrt(a+a*sec(c+d*x)))],
[sec(c+d*x)^(3/2)*(A+B*sec(c+d*x))/(a+a*sec(c+d*x))^(3/2),x,6,2*B*arcsinh(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))/(a^(3/2)*d)+1/2*(A-B)*sec(c+d*x)^(3/2)*sin(c+d*x)/(d*(a+a*sec(c+d*x))^(3/2))+1/2*(A-5*B)*arctanh(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*sec(c+d*x))))/(a^(3/2)*d*sqrt(2))],
[(A+B*sec(c+d*x))*sqrt(sec(c+d*x))/(a+a*sec(c+d*x))^(3/2),x,3,-1/2*(A-B)*sec(c+d*x)^(3/2)*sin(c+d*x)/(d*(a+a*sec(c+d*x))^(3/2))+1/2*(3*A+B)*arctanh(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*sec(c+d*x))))/(a^(3/2)*d*sqrt(2))],
[(A+B*sec(c+d*x))/((a+a*sec(c+d*x))^(3/2)*sqrt(sec(c+d*x))),x,4,-1/2*(7*A-3*B)*arctanh(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*sec(c+d*x))))/(a^(3/2)*d*sqrt(2))-1/2*(A-B)*sin(c+d*x)*sqrt(sec(c+d*x))/(d*(a+a*sec(c+d*x))^(3/2))+1/2*(5*A-B)*sin(c+d*x)*sqrt(sec(c+d*x))/(a*d*sqrt(a+a*sec(c+d*x)))],
[(A+B*sec(c+d*x))/(sec(c+d*x)^(3/2)*(a+a*sec(c+d*x))^(3/2)),x,5,1/2*(11*A-7*B)*arctanh(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*sec(c+d*x))))/(a^(3/2)*d*sqrt(2))-1/2*(A-B)*sin(c+d*x)/(d*(a+a*sec(c+d*x))^(3/2)*sqrt(sec(c+d*x)))+1/6*(7*A-3*B)*sin(c+d*x)/(a*d*sqrt(sec(c+d*x))*sqrt(a+a*sec(c+d*x)))-1/6*(19*A-15*B)*sin(c+d*x)*sqrt(sec(c+d*x))/(a*d*sqrt(a+a*sec(c+d*x)))],
[(A+B*sec(c+d*x))/(sec(c+d*x)^(5/2)*(a+a*sec(c+d*x))^(3/2)),x,6,-1/2*(A-B)*sin(c+d*x)/(d*sec(c+d*x)^(3/2)*(a+a*sec(c+d*x))^(3/2))-1/2*(15*A-11*B)*arctanh(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*sec(c+d*x))))/(a^(3/2)*d*sqrt(2))+1/10*(9*A-5*B)*sin(c+d*x)/(a*d*sec(c+d*x)^(3/2)*sqrt(a+a*sec(c+d*x)))-1/30*(39*A-35*B)*sin(c+d*x)/(a*d*sqrt(sec(c+d*x))*sqrt(a+a*sec(c+d*x)))+1/30*(147*A-95*B)*sin(c+d*x)*sqrt(sec(c+d*x))/(a*d*sqrt(a+a*sec(c+d*x)))],
[sec(c+d*x)^(7/2)*(A+B*sec(c+d*x))/(a+a*sec(c+d*x))^(5/2),x,8,(2*A-5*B)*arcsinh(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))/(a^(5/2)*d)+1/4*(A-B)*sec(c+d*x)^(7/2)*sin(c+d*x)/(d*(a+a*sec(c+d*x))^(5/2))+1/16*(7*A-15*B)*sec(c+d*x)^(5/2)*sin(c+d*x)/(a*d*(a+a*sec(c+d*x))^(3/2))-1/16*(43*A-115*B)*arctanh(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*sec(c+d*x))))/(a^(5/2)*d*sqrt(2))-1/16*(11*A-35*B)*sec(c+d*x)^(3/2)*sin(c+d*x)/(a^2*d*sqrt(a+a*sec(c+d*x)))],
[sec(c+d*x)^(5/2)*(A+B*sec(c+d*x))/(a+a*sec(c+d*x))^(5/2),x,7,2*B*arcsinh(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))/(a^(5/2)*d)+1/4*(A-B)*sec(c+d*x)^(5/2)*sin(c+d*x)/(d*(a+a*sec(c+d*x))^(5/2))+1/16*(3*A-11*B)*sec(c+d*x)^(3/2)*sin(c+d*x)/(a*d*(a+a*sec(c+d*x))^(3/2))+1/16*(3*A-43*B)*arctanh(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*sec(c+d*x))))/(a^(5/2)*d*sqrt(2))],
[sec(c+d*x)^(3/2)*(A+B*sec(c+d*x))/(a+a*sec(c+d*x))^(5/2),x,4,-1/4*(A-B)*sec(c+d*x)^(5/2)*sin(c+d*x)/(d*(a+a*sec(c+d*x))^(5/2))+1/16*(5*A+3*B)*sec(c+d*x)^(3/2)*sin(c+d*x)/(a*d*(a+a*sec(c+d*x))^(3/2))+1/16*(5*A+3*B)*arctanh(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*sec(c+d*x))))/(a^(5/2)*d*sqrt(2))],
[(A+B*sec(c+d*x))*sqrt(sec(c+d*x))/(a+a*sec(c+d*x))^(5/2),x,5,-1/4*(A-B)*sec(c+d*x)^(3/2)*sin(c+d*x)/(d*(a+a*sec(c+d*x))^(5/2))-1/16*(9*A-B)*sec(c+d*x)^(3/2)*sin(c+d*x)/(a*d*(a+a*sec(c+d*x))^(3/2))+1/16*(19*A+5*B)*arctanh(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*sec(c+d*x))))/(a^(5/2)*d*sqrt(2)),1/16*(19*A+5*B)*arctanh(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*sec(c+d*x))))/(a^(5/2)*d*sqrt(2))+1/4*(A-B)*sin(c+d*x)*sqrt(sec(c+d*x))/(d*(a+a*sec(c+d*x))^(5/2))+1/16*(5*A+3*B)*sin(c+d*x)*sqrt(sec(c+d*x))/(a*d*(a+a*sec(c+d*x))^(3/2))-1/16*(9*A-B)*sin(c+d*x)*sqrt(sec(c+d*x))/(a^2*d*sqrt(a+a*sec(c+d*x)))],
[(A+B*sec(c+d*x))/((a+a*sec(c+d*x))^(5/2)*sqrt(sec(c+d*x))),x,5,-1/16*(75*A-19*B)*arctanh(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*sec(c+d*x))))/(a^(5/2)*d*sqrt(2))-1/4*(A-B)*sin(c+d*x)*sqrt(sec(c+d*x))/(d*(a+a*sec(c+d*x))^(5/2))-1/16*(13*A-5*B)*sin(c+d*x)*sqrt(sec(c+d*x))/(a*d*(a+a*sec(c+d*x))^(3/2))+1/16*(49*A-9*B)*sin(c+d*x)*sqrt(sec(c+d*x))/(a^2*d*sqrt(a+a*sec(c+d*x)))],
[(A+B*sec(c+d*x))/(sec(c+d*x)^(3/2)*(a+a*sec(c+d*x))^(5/2)),x,6,1/16*(163*A-75*B)*arctanh(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*sec(c+d*x))))/(a^(5/2)*d*sqrt(2))-1/4*(A-B)*sin(c+d*x)/(d*(a+a*sec(c+d*x))^(5/2)*sqrt(sec(c+d*x)))-1/16*(17*A-9*B)*sin(c+d*x)/(a*d*(a+a*sec(c+d*x))^(3/2)*sqrt(sec(c+d*x)))+1/48*(95*A-39*B)*sin(c+d*x)/(a^2*d*sqrt(sec(c+d*x))*sqrt(a+a*sec(c+d*x)))-1/48*(299*A-147*B)*sin(c+d*x)*sqrt(sec(c+d*x))/(a^2*d*sqrt(a+a*sec(c+d*x)))],
[(A+B*sec(c+d*x))/(sec(c+d*x)^(5/2)*(a+a*sec(c+d*x))^(5/2)),x,7,-1/4*(A-B)*sin(c+d*x)/(d*sec(c+d*x)^(3/2)*(a+a*sec(c+d*x))^(5/2))-1/16*(21*A-13*B)*sin(c+d*x)/(a*d*sec(c+d*x)^(3/2)*(a+a*sec(c+d*x))^(3/2))-1/16*(283*A-163*B)*arctanh(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*sec(c+d*x))))/(a^(5/2)*d*sqrt(2))+1/80*(157*A-85*B)*sin(c+d*x)/(a^2*d*sec(c+d*x)^(3/2)*sqrt(a+a*sec(c+d*x)))-1/240*(787*A-475*B)*sin(c+d*x)/(a^2*d*sqrt(sec(c+d*x))*sqrt(a+a*sec(c+d*x)))+1/240*(2671*A-1495*B)*sin(c+d*x)*sqrt(sec(c+d*x))/(a^2*d*sqrt(a+a*sec(c+d*x)))],

# Integrands of the form Sec[e+f x]^n (a+a Sec[e+f x])^(m/3) (A+B Sec[e+f x])
[(a+a*sec(c+d*x))^(2/3)*(A+B*sec(c+d*x)),x,9,3/2*B*(a+a*sec(c+d*x))^(2/3)*tan(c+d*x)/(d*(1+sec(c+d*x)))+3/7*A*AppellF1(7/6,1/2,1,13/6,1/2*(1+sec(c+d*x)),1+sec(c+d*x))*(a+a*sec(c+d*x))^(2/3)*sqrt(2)*tan(c+d*x)/(d*sqrt(1-sec(c+d*x)))-1/2*3^(3/4)*B*sqrt(cos(arccos((2^(1/3)-(1+sec(c+d*x))^(1/3)*(1-sqrt(3)))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))))^2)/cos(arccos((2^(1/3)-(1+sec(c+d*x))^(1/3)*(1-sqrt(3)))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))))*EllipticF(sin(arccos((2^(1/3)-(1+sec(c+d*x))^(1/3)*(1-sqrt(3)))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3))))),sqrt(1/4*(2+sqrt(3))))*(a+a*sec(c+d*x))^(2/3)*(2^(1/3)-(1+sec(c+d*x))^(1/3))*sqrt((2^(2/3)+2^(1/3)*(1+sec(c+d*x))^(1/3)+(1+sec(c+d*x))^(2/3))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))^2)*tan(c+d*x)/(2^(1/3)*d*(1-sec(c+d*x))*(1+sec(c+d*x))*sqrt(-(1+sec(c+d*x))^(1/3)*(2^(1/3)-(1+sec(c+d*x))^(1/3))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))^2))],
[(A+B*sec(c+d*x))/(a+a*sec(c+d*x))^(1/3),x,8,3*A*AppellF1(1/6,1/2,1,7/6,1/2*(1+sec(c+d*x)),1+sec(c+d*x))*sqrt(2)*tan(c+d*x)/(d*(a+a*sec(c+d*x))^(1/3)*sqrt(1-sec(c+d*x)))-3^(3/4)*B*sqrt(cos(arccos((2^(1/3)-(1+sec(c+d*x))^(1/3)*(1-sqrt(3)))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))))^2)/cos(arccos((2^(1/3)-(1+sec(c+d*x))^(1/3)*(1-sqrt(3)))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))))*EllipticF(sin(arccos((2^(1/3)-(1+sec(c+d*x))^(1/3)*(1-sqrt(3)))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3))))),sqrt(1/4*(2+sqrt(3))))*(2^(1/3)-(1+sec(c+d*x))^(1/3))*sqrt((2^(2/3)+2^(1/3)*(1+sec(c+d*x))^(1/3)+(1+sec(c+d*x))^(2/3))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))^2)*tan(c+d*x)/(2^(1/3)*d*(1-sec(c+d*x))*(a+a*sec(c+d*x))^(1/3)*sqrt(-(1+sec(c+d*x))^(1/3)*(2^(1/3)-(1+sec(c+d*x))^(1/3))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))^2))],
[(A+B*sec(c+d*x))/(a+a*sec(c+d*x))^(4/3),x,9,3/5*B*tan(c+d*x)/(a*d*(1+sec(c+d*x))*(a+a*sec(c+d*x))^(1/3))-3/5*A*AppellF1(-5/6,1/2,1,1/6,1/2*(1+sec(c+d*x)),1+sec(c+d*x))*sqrt(2)*tan(c+d*x)/(a*d*(1+sec(c+d*x))*(a+a*sec(c+d*x))^(1/3)*sqrt(1-sec(c+d*x)))-1/5*3^(3/4)*B*sqrt(cos(arccos((2^(1/3)-(1+sec(c+d*x))^(1/3)*(1-sqrt(3)))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))))^2)/cos(arccos((2^(1/3)-(1+sec(c+d*x))^(1/3)*(1-sqrt(3)))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))))*EllipticF(sin(arccos((2^(1/3)-(1+sec(c+d*x))^(1/3)*(1-sqrt(3)))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3))))),sqrt(1/4*(2+sqrt(3))))*(2^(1/3)-(1+sec(c+d*x))^(1/3))*sqrt((2^(2/3)+2^(1/3)*(1+sec(c+d*x))^(1/3)+(1+sec(c+d*x))^(2/3))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))^2)*tan(c+d*x)/(2^(1/3)*a*d*(1-sec(c+d*x))*(a+a*sec(c+d*x))^(1/3)*sqrt(-(1+sec(c+d*x))^(1/3)*(2^(1/3)-(1+sec(c+d*x))^(1/3))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))^2))],
[(a+a*sec(c+d*x))^(4/3)*(A+B*sec(c+d*x)),x,11,3/4*a*B*(a+a*sec(c+d*x))^(1/3)*tan(c+d*x)/d-15/4*a*B*(a+a*sec(c+d*x))^(1/3)*(1+sqrt(3))*tan(c+d*x)/(d*(1+sec(c+d*x))^(2/3)*(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3))))+3/11*a*A*AppellF1(11/6,1/2,1,17/6,1/2*(1+sec(c+d*x)),1+sec(c+d*x))*(1+sec(c+d*x))*(a+a*sec(c+d*x))^(1/3)*sqrt(2)*tan(c+d*x)/(d*sqrt(1-sec(c+d*x)))+15/2*3^(1/4)*a*B*sqrt(cos(arccos((2^(1/3)-(1+sec(c+d*x))^(1/3)*(1-sqrt(3)))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))))^2)/cos(arccos((2^(1/3)-(1+sec(c+d*x))^(1/3)*(1-sqrt(3)))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))))*EllipticE(sin(arccos((2^(1/3)-(1+sec(c+d*x))^(1/3)*(1-sqrt(3)))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3))))),sqrt(1/4*(2+sqrt(3))))*(a+a*sec(c+d*x))^(1/3)*(2^(1/3)-(1+sec(c+d*x))^(1/3))*sqrt((2^(2/3)+2^(1/3)*(1+sec(c+d*x))^(1/3)+(1+sec(c+d*x))^(2/3))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))^2)*tan(c+d*x)/(2^(2/3)*d*(1-sec(c+d*x))*(1+sec(c+d*x))^(2/3)*sqrt(-(1+sec(c+d*x))^(1/3)*(2^(1/3)-(1+sec(c+d*x))^(1/3))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))^2))+5/4*3^(3/4)*a*B*sqrt(cos(arccos((2^(1/3)-(1+sec(c+d*x))^(1/3)*(1-sqrt(3)))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))))^2)/cos(arccos((2^(1/3)-(1+sec(c+d*x))^(1/3)*(1-sqrt(3)))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))))*EllipticF(sin(arccos((2^(1/3)-(1+sec(c+d*x))^(1/3)*(1-sqrt(3)))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3))))),sqrt(1/4*(2+sqrt(3))))*(a+a*sec(c+d*x))^(1/3)*(2^(1/3)-(1+sec(c+d*x))^(1/3))*(1-sqrt(3))*sqrt((2^(2/3)+2^(1/3)*(1+sec(c+d*x))^(1/3)+(1+sec(c+d*x))^(2/3))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))^2)*tan(c+d*x)/(2^(2/3)*d*(1-sec(c+d*x))*(1+sec(c+d*x))^(2/3)*sqrt(-(1+sec(c+d*x))^(1/3)*(2^(1/3)-(1+sec(c+d*x))^(1/3))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))^2))],
[(a+a*sec(c+d*x))^(1/3)*(A+B*sec(c+d*x)),x,10,-3*B*(a+a*sec(c+d*x))^(1/3)*(1+sqrt(3))*tan(c+d*x)/(d*(1+sec(c+d*x))^(2/3)*(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3))))+3/5*A*AppellF1(5/6,1/2,1,11/6,1/2*(1+sec(c+d*x)),1+sec(c+d*x))*(a+a*sec(c+d*x))^(1/3)*sqrt(2)*tan(c+d*x)/(d*sqrt(1-sec(c+d*x)))+3*2^(1/3)*3^(1/4)*B*sqrt(cos(arccos((2^(1/3)-(1+sec(c+d*x))^(1/3)*(1-sqrt(3)))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))))^2)/cos(arccos((2^(1/3)-(1+sec(c+d*x))^(1/3)*(1-sqrt(3)))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))))*EllipticE(sin(arccos((2^(1/3)-(1+sec(c+d*x))^(1/3)*(1-sqrt(3)))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3))))),sqrt(1/4*(2+sqrt(3))))*(a+a*sec(c+d*x))^(1/3)*(2^(1/3)-(1+sec(c+d*x))^(1/3))*sqrt((2^(2/3)+2^(1/3)*(1+sec(c+d*x))^(1/3)+(1+sec(c+d*x))^(2/3))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))^2)*tan(c+d*x)/(d*(1-sec(c+d*x))*(1+sec(c+d*x))^(2/3)*sqrt(-(1+sec(c+d*x))^(1/3)*(2^(1/3)-(1+sec(c+d*x))^(1/3))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))^2))+3^(3/4)*B*sqrt(cos(arccos((2^(1/3)-(1+sec(c+d*x))^(1/3)*(1-sqrt(3)))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))))^2)/cos(arccos((2^(1/3)-(1+sec(c+d*x))^(1/3)*(1-sqrt(3)))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))))*EllipticF(sin(arccos((2^(1/3)-(1+sec(c+d*x))^(1/3)*(1-sqrt(3)))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3))))),sqrt(1/4*(2+sqrt(3))))*(a+a*sec(c+d*x))^(1/3)*(2^(1/3)-(1+sec(c+d*x))^(1/3))*(1-sqrt(3))*sqrt((2^(2/3)+2^(1/3)*(1+sec(c+d*x))^(1/3)+(1+sec(c+d*x))^(2/3))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))^2)*tan(c+d*x)/(2^(2/3)*d*(1-sec(c+d*x))*(1+sec(c+d*x))^(2/3)*sqrt(-(1+sec(c+d*x))^(1/3)*(2^(1/3)-(1+sec(c+d*x))^(1/3))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))^2))],
[(A+B*sec(c+d*x))/(a+a*sec(c+d*x))^(2/3),x,11,3*B*tan(c+d*x)/(d*(a+a*sec(c+d*x))^(2/3))+3*B*(1+sec(c+d*x))^(1/3)*(1+sqrt(3))*tan(c+d*x)/(d*(a+a*sec(c+d*x))^(2/3)*(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3))))-3*A*AppellF1(-1/6,1/2,1,5/6,1/2*(1+sec(c+d*x)),1+sec(c+d*x))*sqrt(2)*tan(c+d*x)/(d*(a+a*sec(c+d*x))^(2/3)*sqrt(1-sec(c+d*x)))-3*2^(1/3)*3^(1/4)*B*sqrt(cos(arccos((2^(1/3)-(1+sec(c+d*x))^(1/3)*(1-sqrt(3)))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))))^2)/cos(arccos((2^(1/3)-(1+sec(c+d*x))^(1/3)*(1-sqrt(3)))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))))*EllipticE(sin(arccos((2^(1/3)-(1+sec(c+d*x))^(1/3)*(1-sqrt(3)))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3))))),sqrt(1/4*(2+sqrt(3))))*(1+sec(c+d*x))^(1/3)*(2^(1/3)-(1+sec(c+d*x))^(1/3))*sqrt((2^(2/3)+2^(1/3)*(1+sec(c+d*x))^(1/3)+(1+sec(c+d*x))^(2/3))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))^2)*tan(c+d*x)/(d*(1-sec(c+d*x))*(a+a*sec(c+d*x))^(2/3)*sqrt(-(1+sec(c+d*x))^(1/3)*(2^(1/3)-(1+sec(c+d*x))^(1/3))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))^2))-3^(3/4)*B*sqrt(cos(arccos((2^(1/3)-(1+sec(c+d*x))^(1/3)*(1-sqrt(3)))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))))^2)/cos(arccos((2^(1/3)-(1+sec(c+d*x))^(1/3)*(1-sqrt(3)))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))))*EllipticF(sin(arccos((2^(1/3)-(1+sec(c+d*x))^(1/3)*(1-sqrt(3)))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3))))),sqrt(1/4*(2+sqrt(3))))*(1+sec(c+d*x))^(1/3)*(2^(1/3)-(1+sec(c+d*x))^(1/3))*(1-sqrt(3))*sqrt((2^(2/3)+2^(1/3)*(1+sec(c+d*x))^(1/3)+(1+sec(c+d*x))^(2/3))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))^2)*tan(c+d*x)/(2^(2/3)*d*(1-sec(c+d*x))*(a+a*sec(c+d*x))^(2/3)*sqrt(-(1+sec(c+d*x))^(1/3)*(2^(1/3)-(1+sec(c+d*x))^(1/3))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))^2))],

# Integrands of the form (d Sec[e+f x])^n (a+a Sec[e+f x])^m (A+B Sec[e+f x]) with m and/or n symbolic
[(c*sec(e+f*x))^n*(a+a*sec(e+f*x))^m*(A+B*sec(e+f*x)),x,7,-B*AppellF1(n,1/2,-1/2-m,1+n,sec(e+f*x),-sec(e+f*x))*(c*sec(e+f*x))^n*(1+sec(e+f*x))^(-1/2-m)*(a+a*sec(e+f*x))^m*tan(e+f*x)/(f*n*sqrt(1-sec(e+f*x)))-(A-B)*AppellF1(n,1/2,1/2-m,1+n,sec(e+f*x),-sec(e+f*x))*(c*sec(e+f*x))^n*(1+sec(e+f*x))^(-1/2-m)*(a+a*sec(e+f*x))^m*tan(e+f*x)/(f*n*sqrt(1-sec(e+f*x)))],
[sec(c+d*x)^(-1-n)*(a+a*sec(c+d*x))^n*(A+B*sec(c+d*x)),x,4,A*(a+a*sec(c+d*x))^n*sin(c+d*x)/(d*(1+n)*sec(c+d*x)^n)+(B+A*n+B*n)*hypergeom([1/2-n,-n],[1-n],-2*sec(c+d*x)/(1-sec(c+d*x)))*sec(c+d*x)^(1-n)*((1+sec(c+d*x))/(1-sec(c+d*x)))^(1/2-n)*(a+a*sec(c+d*x))^n*sin(c+d*x)/(d*n*(1+n)*(1+sec(c+d*x)))],

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

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

# m>0
[sec(c+d*x)^3*(a+b*sec(c+d*x))*(A+B*sec(c+d*x)),x,6,1/8*(4*a*A+3*b*B)*arctanh(sin(c+d*x))/d+(A*b+a*B)*tan(c+d*x)/d+1/8*(4*a*A+3*b*B)*sec(c+d*x)*tan(c+d*x)/d+1/4*b*B*sec(c+d*x)^3*tan(c+d*x)/d+1/3*(A*b+a*B)*tan(c+d*x)^3/d],
[sec(c+d*x)^2*(a+b*sec(c+d*x))*(A+B*sec(c+d*x)),x,6,1/2*(A*b+a*B)*arctanh(sin(c+d*x))/d+1/3*(3*a*A+2*b*B)*tan(c+d*x)/d+1/2*(A*b+a*B)*sec(c+d*x)*tan(c+d*x)/d+1/3*b*B*sec(c+d*x)^2*tan(c+d*x)/d],
[sec(c+d*x)*(a+b*sec(c+d*x))*(A+B*sec(c+d*x)),x,5,1/2*(2*a*A+b*B)*arctanh(sin(c+d*x))/d+(A*b+a*B)*tan(c+d*x)/d+1/2*b*B*sec(c+d*x)*tan(c+d*x)/d],
[(a+b*sec(c+d*x))*(A+B*sec(c+d*x)),x,4,a*A*x+(A*b+a*B)*arctanh(sin(c+d*x))/d+b*B*tan(c+d*x)/d],
[cos(c+d*x)*(a+b*sec(c+d*x))*(A+B*sec(c+d*x)),x,3,(A*b+a*B)*x+b*B*arctanh(sin(c+d*x))/d+a*A*sin(c+d*x)/d],
[cos(c+d*x)^2*(a+b*sec(c+d*x))*(A+B*sec(c+d*x)),x,4,1/2*(a*A+2*b*B)*x+(A*b+a*B)*sin(c+d*x)/d+1/2*a*A*cos(c+d*x)*sin(c+d*x)/d],
[cos(c+d*x)^3*(a+b*sec(c+d*x))*(A+B*sec(c+d*x)),x,5,1/2*(A*b+a*B)*x+1/3*(2*a*A+3*b*B)*sin(c+d*x)/d+1/2*(A*b+a*B)*cos(c+d*x)*sin(c+d*x)/d+1/3*a*A*cos(c+d*x)^2*sin(c+d*x)/d],
[cos(c+d*x)^4*(a+b*sec(c+d*x))*(A+B*sec(c+d*x)),x,6,1/8*(3*a*A+4*b*B)*x+(A*b+a*B)*sin(c+d*x)/d+1/8*(3*a*A+4*b*B)*cos(c+d*x)*sin(c+d*x)/d+1/4*a*A*cos(c+d*x)^3*sin(c+d*x)/d-1/3*(A*b+a*B)*sin(c+d*x)^3/d],
[sec(c+d*x)^3*(a+b*sec(c+d*x))^2*(A+B*sec(c+d*x)),x,7,1/8*(4*a^2*A+3*A*b^2+6*a*b*B)*arctanh(sin(c+d*x))/d+1/5*(4*b^2*B+5*a*(2*A*b+a*B))*tan(c+d*x)/d+1/8*(4*a^2*A+3*A*b^2+6*a*b*B)*sec(c+d*x)*tan(c+d*x)/d+1/20*b*(5*A*b+6*a*B)*sec(c+d*x)^3*tan(c+d*x)/d+1/5*b*B*sec(c+d*x)^3*(a+b*sec(c+d*x))*tan(c+d*x)/d+1/15*(4*b^2*B+5*a*(2*A*b+a*B))*tan(c+d*x)^3/d],
[sec(c+d*x)^2*(a+b*sec(c+d*x))^2*(A+B*sec(c+d*x)),x,7,1/8*(8*a*A*b+4*a^2*B+3*b^2*B)*arctanh(sin(c+d*x))/d+1/6*(4*a^2*A*b+4*A*b^3-a^3*B+8*a*b^2*B)*tan(c+d*x)/(b*d)+1/24*(8*a*A*b-2*a^2*B+9*b^2*B)*sec(c+d*x)*tan(c+d*x)/d+1/12*(4*A*b-a*B)*(a+b*sec(c+d*x))^2*tan(c+d*x)/(b*d)+1/4*B*(a+b*sec(c+d*x))^3*tan(c+d*x)/(b*d)],
[sec(c+d*x)*(a+b*sec(c+d*x))^2*(A+B*sec(c+d*x)),x,6,1/2*(2*a^2*A+A*b^2+2*a*b*B)*arctanh(sin(c+d*x))/d+2/3*(3*a*A*b+a^2*B+b^2*B)*tan(c+d*x)/d+1/6*b*(3*A*b+2*a*B)*sec(c+d*x)*tan(c+d*x)/d+1/3*B*(a+b*sec(c+d*x))^2*tan(c+d*x)/d],
[(a+b*sec(c+d*x))^2*(A+B*sec(c+d*x)),x,5,a^2*A*x+1/2*(4*a*A*b+2*a^2*B+b^2*B)*arctanh(sin(c+d*x))/d+1/2*b*(2*A*b+3*a*B)*tan(c+d*x)/d+1/2*b*B*(a+b*sec(c+d*x))*tan(c+d*x)/d],
[cos(c+d*x)*(a+b*sec(c+d*x))^2*(A+B*sec(c+d*x)),x,5,a*(2*A*b+a*B)*x+b*(A*b+2*a*B)*arctanh(sin(c+d*x))/d+a^2*A*sin(c+d*x)/d+b^2*B*tan(c+d*x)/d],
[cos(c+d*x)^2*(a+b*sec(c+d*x))^2*(A+B*sec(c+d*x)),x,5,1/2*(a^2*A+2*A*b^2+4*a*b*B)*x+b^2*B*arctanh(sin(c+d*x))/d+a*(2*A*b+a*B)*sin(c+d*x)/d+1/2*a^2*A*cos(c+d*x)*sin(c+d*x)/d],
[cos(c+d*x)^3*(a+b*sec(c+d*x))^2*(A+B*sec(c+d*x)),x,5,1/2*(2*a*A*b+a^2*B+2*b^2*B)*x+1/3*(2*a^2*A+3*A*b^2+6*a*b*B)*sin(c+d*x)/d+1/2*a*(2*A*b+a*B)*cos(c+d*x)*sin(c+d*x)/d+1/3*a^2*A*cos(c+d*x)^2*sin(c+d*x)/d],
[cos(c+d*x)^4*(a+b*sec(c+d*x))^2*(A+B*sec(c+d*x)),x,7,1/8*(3*a^2*A+4*A*b^2+8*a*b*B)*x+(2*a*A*b+a^2*B+b^2*B)*sin(c+d*x)/d+1/8*(3*a^2*A+4*A*b^2+8*a*b*B)*cos(c+d*x)*sin(c+d*x)/d+1/4*a^2*A*cos(c+d*x)^3*sin(c+d*x)/d-1/3*a*(2*A*b+a*B)*sin(c+d*x)^3/d],
[cos(c+d*x)^5*(a+b*sec(c+d*x))^2*(A+B*sec(c+d*x)),x,7,1/8*(6*a*A*b+3*a^2*B+4*b^2*B)*x+1/5*(4*a^2*A+5*A*b^2+10*a*b*B)*sin(c+d*x)/d+1/8*(6*a*A*b+3*a^2*B+4*b^2*B)*cos(c+d*x)*sin(c+d*x)/d+1/4*a*(2*A*b+a*B)*cos(c+d*x)^3*sin(c+d*x)/d+1/5*a^2*A*cos(c+d*x)^4*sin(c+d*x)/d-1/15*(4*a^2*A+5*A*b^2+10*a*b*B)*sin(c+d*x)^3/d],
[sec(c+d*x)^2*(a+b*sec(c+d*x))^3*(A+B*sec(c+d*x)),x,8,1/8*(12*a^2*A*b+3*A*b^3+4*a^3*B+9*a*b^2*B)*arctanh(sin(c+d*x))/d+1/30*(15*a^3*A*b+60*a*A*b^3-3*a^4*B+52*a^2*b^2*B+16*b^4*B)*tan(c+d*x)/(b*d)+1/120*(30*a^2*A*b+45*A*b^3-6*a^3*B+71*a*b^2*B)*sec(c+d*x)*tan(c+d*x)/d+1/60*(15*a*A*b-3*a^2*B+16*b^2*B)*(a+b*sec(c+d*x))^2*tan(c+d*x)/(b*d)+1/20*(5*A*b-a*B)*(a+b*sec(c+d*x))^3*tan(c+d*x)/(b*d)+1/5*B*(a+b*sec(c+d*x))^4*tan(c+d*x)/(b*d)],
[sec(c+d*x)*(a+b*sec(c+d*x))^3*(A+B*sec(c+d*x)),x,7,1/8*(8*a^3*A+12*a*A*b^2+12*a^2*b*B+3*b^3*B)*arctanh(sin(c+d*x))/d+1/6*(16*a^2*A*b+4*A*b^3+3*a^3*B+12*a*b^2*B)*tan(c+d*x)/d+1/24*b*(20*a*A*b+6*a^2*B+9*b^2*B)*sec(c+d*x)*tan(c+d*x)/d+1/12*(4*A*b+3*a*B)*(a+b*sec(c+d*x))^2*tan(c+d*x)/d+1/4*B*(a+b*sec(c+d*x))^3*tan(c+d*x)/d],
[(a+b*sec(c+d*x))^3*(A+B*sec(c+d*x)),x,6,a^3*A*x+1/2*(6*a^2*A*b+A*b^3+2*a^3*B+3*a*b^2*B)*arctanh(sin(c+d*x))/d+1/3*b*(9*a*A*b+8*a^2*B+2*b^2*B)*tan(c+d*x)/d+1/6*b^2*(3*A*b+5*a*B)*sec(c+d*x)*tan(c+d*x)/d+1/3*b*B*(a+b*sec(c+d*x))^2*tan(c+d*x)/d],
[cos(c+d*x)*(a+b*sec(c+d*x))^3*(A+B*sec(c+d*x)),x,6,a^2*(3*A*b+a*B)*x+1/2*b*(6*a*A*b+6*a^2*B+b^2*B)*arctanh(sin(c+d*x))/d+1/2*a^2*(2*a*A-b*B)*sin(c+d*x)/d+1/2*b*B*(a+b*sec(c+d*x))^2*sin(c+d*x)/d+b^2*(A*b+2*a*B)*tan(c+d*x)/d,a^2*(3*A*b+a*B)*x+1/2*b*(6*a*A*b+6*a^2*B+b^2*B)*arctanh(sin(c+d*x))/d+a*A*(a+b*sec(c+d*x))^2*sin(c+d*x)/d-b*(2*a^2*A-A*b^2-3*a*b*B)*tan(c+d*x)/d-1/2*b^2*(2*a*A-b*B)*sec(c+d*x)*tan(c+d*x)/d],
[cos(c+d*x)^2*(a+b*sec(c+d*x))^3*(A+B*sec(c+d*x)),x,6,1/2*a*(a^2*A+6*A*b^2+6*a*b*B)*x+b^2*(A*b+3*a*B)*arctanh(sin(c+d*x))/d+a^2*(2*A*b+a*B)*sin(c+d*x)/d+1/2*a*A*cos(c+d*x)*(a+b*sec(c+d*x))^2*sin(c+d*x)/d-1/2*b^2*(a*A-2*b*B)*tan(c+d*x)/d],
[cos(c+d*x)^3*(a+b*sec(c+d*x))^3*(A+B*sec(c+d*x)),x,6,1/2*(3*a^2*A*b+2*A*b^3+a^3*B+6*a*b^2*B)*x+b^3*B*arctanh(sin(c+d*x))/d+1/3*a*(2*a^2*A+8*A*b^2+9*a*b*B)*sin(c+d*x)/d+1/6*a^2*(5*A*b+3*a*B)*cos(c+d*x)*sin(c+d*x)/d+1/3*a*A*cos(c+d*x)^2*(a+b*sec(c+d*x))^2*sin(c+d*x)/d],
[cos(c+d*x)^4*(a+b*sec(c+d*x))^3*(A+B*sec(c+d*x)),x,6,1/8*(3*a^3*A+12*a*A*b^2+12*a^2*b*B+8*b^3*B)*x+1/3*(6*a^2*A*b+3*A*b^3+2*a^3*B+9*a*b^2*B)*sin(c+d*x)/d+1/8*a*(3*a^2*A+10*A*b^2+12*a*b*B)*cos(c+d*x)*sin(c+d*x)/d+1/6*a^2*(3*A*b+2*a*B)*cos(c+d*x)^2*sin(c+d*x)/d+1/4*a*A*cos(c+d*x)^3*(a+b*sec(c+d*x))^2*sin(c+d*x)/d],
[cos(c+d*x)^5*(a+b*sec(c+d*x))^3*(A+B*sec(c+d*x)),x,8,1/8*(9*a^2*A*b+4*A*b^3+3*a^3*B+12*a*b^2*B)*x+1/5*(4*a^3*A+14*a*A*b^2+15*a^2*b*B+5*b^3*B)*sin(c+d*x)/d+1/8*(9*a^2*A*b+4*A*b^3+3*a^3*B+12*a*b^2*B)*cos(c+d*x)*sin(c+d*x)/d+1/20*a^2*(7*A*b+5*a*B)*cos(c+d*x)^3*sin(c+d*x)/d+1/5*a*A*cos(c+d*x)^4*(a+b*sec(c+d*x))^2*sin(c+d*x)/d-1/15*a*(4*a^2*A+12*A*b^2+15*a*b*B)*sin(c+d*x)^3/d],
[sec(c+d*x)^2*(a+b*sec(c+d*x))^4*(A+B*sec(c+d*x)),x,9,1/16*(32*a^3*A*b+24*a*A*b^3+8*a^4*B+36*a^2*b^2*B+5*b^4*B)*arctanh(sin(c+d*x))/d+1/60*(24*a^4*A*b+224*a^2*A*b^3+32*A*b^5-4*a^5*B+121*a^3*b^2*B+128*a*b^4*B)*tan(c+d*x)/(b*d)+1/240*(48*a^3*A*b+232*a*A*b^3-8*a^4*B+178*a^2*b^2*B+75*b^4*B)*sec(c+d*x)*tan(c+d*x)/d+1/120*(24*a^2*A*b+32*A*b^3-4*a^3*B+53*a*b^2*B)*(a+b*sec(c+d*x))^2*tan(c+d*x)/(b*d)+1/120*(24*a*A*b-4*a^2*B+25*b^2*B)*(a+b*sec(c+d*x))^3*tan(c+d*x)/(b*d)+1/30*(6*A*b-a*B)*(a+b*sec(c+d*x))^4*tan(c+d*x)/(b*d)+1/6*B*(a+b*sec(c+d*x))^5*tan(c+d*x)/(b*d)],
[sec(c+d*x)*(a+b*sec(c+d*x))^4*(A+B*sec(c+d*x)),x,8,1/8*(8*a^4*A+24*a^2*A*b^2+3*A*b^4+16*a^3*b*B+12*a*b^3*B)*arctanh(sin(c+d*x))/d+1/30*(95*a^3*A*b+80*a*A*b^3+12*a^4*B+112*a^2*b^2*B+16*b^4*B)*tan(c+d*x)/d+1/120*b*(130*a^2*A*b+45*A*b^3+24*a^3*B+116*a*b^2*B)*sec(c+d*x)*tan(c+d*x)/d+1/60*(35*a*A*b+12*a^2*B+16*b^2*B)*(a+b*sec(c+d*x))^2*tan(c+d*x)/d+1/20*(5*A*b+4*a*B)*(a+b*sec(c+d*x))^3*tan(c+d*x)/d+1/5*B*(a+b*sec(c+d*x))^4*tan(c+d*x)/d],
[(a+b*sec(c+d*x))^4*(A+B*sec(c+d*x)),x,7,a^4*A*x+1/8*(32*a^3*A*b+16*a*A*b^3+8*a^4*B+24*a^2*b^2*B+3*b^4*B)*arctanh(sin(c+d*x))/d+1/6*b*(34*a^2*A*b+4*A*b^3+19*a^3*B+16*a*b^2*B)*tan(c+d*x)/d+1/24*b^2*(32*a*A*b+26*a^2*B+9*b^2*B)*sec(c+d*x)*tan(c+d*x)/d+1/12*b*(4*A*b+7*a*B)*(a+b*sec(c+d*x))^2*tan(c+d*x)/d+1/4*b*B*(a+b*sec(c+d*x))^3*tan(c+d*x)/d],
[cos(c+d*x)*(a+b*sec(c+d*x))^4*(A+B*sec(c+d*x)),x,7,a^3*(4*A*b+a*B)*x+1/2*b*(12*a^2*A*b+A*b^3+8*a^3*B+4*a*b^2*B)*arctanh(sin(c+d*x))/d+a*A*(a+b*sec(c+d*x))^3*sin(c+d*x)/d-1/3*b*(6*a^3*A-12*a*A*b^2-17*a^2*b*B-2*b^3*B)*tan(c+d*x)/d-1/6*b^2*(6*a^2*A-3*A*b^2-8*a*b*B)*sec(c+d*x)*tan(c+d*x)/d-1/3*b*(3*a*A-b*B)*(a+b*sec(c+d*x))^2*tan(c+d*x)/d],
[cos(c+d*x)^2*(a+b*sec(c+d*x))^4*(A+B*sec(c+d*x)),x,7,1/2*a^2*(a^2*A+12*A*b^2+8*a*b*B)*x+1/2*b^2*(8*a*A*b+12*a^2*B+b^2*B)*arctanh(sin(c+d*x))/d+1/2*a*(5*A*b+2*a*B)*(a+b*sec(c+d*x))^2*sin(c+d*x)/d+1/2*a*A*cos(c+d*x)*(a+b*sec(c+d*x))^3*sin(c+d*x)/d-1/2*b*(13*a^2*A*b-2*A*b^3+4*a^3*B-8*a*b^2*B)*tan(c+d*x)/d-1/2*b^2*(6*a*A*b+2*a^2*B-b^2*B)*sec(c+d*x)*tan(c+d*x)/d],
[cos(c+d*x)^3*(a+b*sec(c+d*x))^4*(A+B*sec(c+d*x)),x,7,1/2*a*(4*a^2*A*b+8*A*b^3+a^3*B+12*a*b^2*B)*x+b^3*(A*b+4*a*B)*arctanh(sin(c+d*x))/d+1/3*a^2*(2*a^2*A+9*A*b^2+9*a*b*B)*sin(c+d*x)/d+1/2*a*(2*A*b+a*B)*cos(c+d*x)*(a+b*sec(c+d*x))^2*sin(c+d*x)/d+1/3*a*A*cos(c+d*x)^2*(a+b*sec(c+d*x))^3*sin(c+d*x)/d-1/6*b^2*(8*a*A*b+3*a^2*B-6*b^2*B)*tan(c+d*x)/d],
[cos(c+d*x)^4*(a+b*sec(c+d*x))^4*(A+B*sec(c+d*x)),x,7,1/8*(3*a^4*A+24*a^2*A*b^2+8*A*b^4+16*a^3*b*B+32*a*b^3*B)*x+b^4*B*arctanh(sin(c+d*x))/d+1/6*a*(16*a^2*A*b+19*A*b^3+4*a^3*B+34*a*b^2*B)*sin(c+d*x)/d+1/24*a^2*(9*a^2*A+26*A*b^2+32*a*b*B)*cos(c+d*x)*sin(c+d*x)/d+1/12*a*(7*A*b+4*a*B)*cos(c+d*x)^2*(a+b*sec(c+d*x))^2*sin(c+d*x)/d+1/4*a*A*cos(c+d*x)^3*(a+b*sec(c+d*x))^3*sin(c+d*x)/d],
[cos(c+d*x)^5*(a+b*sec(c+d*x))^4*(A+B*sec(c+d*x)),x,7,1/8*(12*a^3*A*b+16*a*A*b^3+3*a^4*B+24*a^2*b^2*B+8*b^4*B)*x+1/15*(8*a^4*A+60*a^2*A*b^2+15*A*b^4+40*a^3*b*B+60*a*b^3*B)*sin(c+d*x)/d+1/40*a*(60*a^2*A*b+56*A*b^3+15*a^3*B+110*a*b^2*B)*cos(c+d*x)*sin(c+d*x)/d+1/30*a^2*(8*a^2*A+18*A*b^2+25*a*b*B)*cos(c+d*x)^2*sin(c+d*x)/d+1/20*a*(8*A*b+5*a*B)*cos(c+d*x)^3*(a+b*sec(c+d*x))^2*sin(c+d*x)/d+1/5*a*A*cos(c+d*x)^4*(a+b*sec(c+d*x))^3*sin(c+d*x)/d],
[cos(c+d*x)^6*(a+b*sec(c+d*x))^4*(A+B*sec(c+d*x)),x,9,1/16*(5*a^4*A+36*a^2*A*b^2+8*A*b^4+24*a^3*b*B+32*a*b^3*B)*x+1/15*(48*a^3*A*b+53*a*A*b^3+12*a^4*B+87*a^2*b^2*B+15*b^4*B)*sin(c+d*x)/d+1/16*(5*a^4*A+36*a^2*A*b^2+8*A*b^4+24*a^3*b*B+32*a*b^3*B)*cos(c+d*x)*sin(c+d*x)/d+1/120*a^2*(25*a^2*A+48*A*b^2+72*a*b*B)*cos(c+d*x)^3*sin(c+d*x)/d+1/10*a*(3*A*b+2*a*B)*cos(c+d*x)^4*(a+b*sec(c+d*x))^2*sin(c+d*x)/d+1/6*a*A*cos(c+d*x)^5*(a+b*sec(c+d*x))^3*sin(c+d*x)/d-1/15*a*(16*a^2*A*b+13*A*b^3+4*a^3*B+27*a*b^2*B)*sin(c+d*x)^3/d],

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

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

# n>0
[sec(c+d*x)^4*(A+B*sec(c+d*x))*sqrt(a+b*sec(c+d*x)),x,7,-2/315*(a-b)*(24*a^3*A*b+57*a*A*b^3-16*a^4*B-24*a^2*b^2*B+147*b^4*B)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b^5*d)-2/315*(a-b)*(3*b^3*(25*A-49*B)+18*a*b^2*(A-2*B)+12*a^2*b*(2*A-B)-16*a^3*B)*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b^4*d)-2/315*(12*a^2*A*b-75*A*b^3-8*a^3*B-13*a*b^2*B)*sqrt(a+b*sec(c+d*x))*tan(c+d*x)/(b^3*d)+2/315*(9*a*A*b-6*a^2*B+49*b^2*B)*sec(c+d*x)*sqrt(a+b*sec(c+d*x))*tan(c+d*x)/(b^2*d)+2/63*(9*A*b+a*B)*sec(c+d*x)^2*sqrt(a+b*sec(c+d*x))*tan(c+d*x)/(b*d)+2/9*B*sec(c+d*x)^3*sqrt(a+b*sec(c+d*x))*tan(c+d*x)/d],
[sec(c+d*x)^3*(A+B*sec(c+d*x))*sqrt(a+b*sec(c+d*x)),x,6,2/105*(a-b)*(14*a^2*A*b-63*A*b^3-8*a^3*B-19*a*b^2*B)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b^4*d)+2/105*(a-b)*(b^2*(63*A-25*B)+2*a*b*(7*A-3*B)-8*a^2*B)*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b^3*d)+2/105*(7*a*A*b-4*a^2*B+25*b^2*B)*sqrt(a+b*sec(c+d*x))*tan(c+d*x)/(b^2*d)+2/35*(7*A*b+a*B)*sec(c+d*x)*sqrt(a+b*sec(c+d*x))*tan(c+d*x)/(b*d)+2/7*B*sec(c+d*x)^2*sqrt(a+b*sec(c+d*x))*tan(c+d*x)/d],
[sec(c+d*x)^2*(A+B*sec(c+d*x))*sqrt(a+b*sec(c+d*x)),x,5,-2/15*(a-b)*(5*a*A*b-2*a^2*B+9*b^2*B)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b^3*d)-2/15*(a-b)*(5*A*b-2*a*B-9*b*B)*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b^2*d)+2/5*B*(a+b*sec(c+d*x))^(3/2)*tan(c+d*x)/(b*d)+2/15*(5*A*b-2*a*B)*sqrt(a+b*sec(c+d*x))*tan(c+d*x)/(b*d)],
[sec(c+d*x)*(A+B*sec(c+d*x))*sqrt(a+b*sec(c+d*x)),x,4,-2/3*(a-b)*(3*A*b+a*B)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b^2*d)+2/3*(a-b)*(3*A-B)*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b*d)+2/3*B*sqrt(a+b*sec(c+d*x))*tan(c+d*x)/d],
[(A+B*sec(c+d*x))*sqrt(a+b*sec(c+d*x)),x,5,-2*(a-b)*B*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b*d)+2*(A*b+(a-b)*B)*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b*d)-2*A*cot(c+d*x)*EllipticPi(sqrt(a+b*sec(c+d*x))/sqrt(a+b),(a+b)/a,sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/d],
[cos(c+d*x)*(A+B*sec(c+d*x))*sqrt(a+b*sec(c+d*x)),x,6,A*(a-b)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b*d)+(A+2*B)*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/d-(A*b+2*a*B)*cot(c+d*x)*EllipticPi(sqrt(a+b*sec(c+d*x))/sqrt(a+b),(a+b)/a,sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a*d)+A*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/d],
[cos(c+d*x)^2*(A+B*sec(c+d*x))*sqrt(a+b*sec(c+d*x)),x,7,1/4*(a-b)*(A*b+4*a*B)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a*b*d)+1/4*(A*b+2*a*(A+2*B))*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a*d)-1/4*(4*a^2*A-A*b^2+4*a*b*B)*cot(c+d*x)*EllipticPi(sqrt(a+b*sec(c+d*x))/sqrt(a+b),(a+b)/a,sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a^2*d)+1/4*(A*b+4*a*B)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(a*d)+1/2*A*cos(c+d*x)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/d],
[cos(c+d*x)^3*(A+B*sec(c+d*x))*sqrt(a+b*sec(c+d*x)),x,8,1/24*(a-b)*(16*a^2*A-3*A*b^2+6*a*b*B)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a^2*b*d)+1/24*(2*a+b)*(8*a*A-3*A*b+6*a*B)*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a^2*d)-1/8*(4*a^2*A*b+A*b^3+8*a^3*B-2*a*b^2*B)*cot(c+d*x)*EllipticPi(sqrt(a+b*sec(c+d*x))/sqrt(a+b),(a+b)/a,sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a^3*d)+1/24*(16*a^2*A-3*A*b^2+6*a*b*B)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(a^2*d)+1/12*(A*b+6*a*B)*cos(c+d*x)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(a*d)+1/3*A*cos(c+d*x)^2*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/d],
[sec(c+d*x)^3*(a+b*sec(c+d*x))^(3/2)*(A+B*sec(c+d*x)),x,7,2/315*(a-b)*(18*a^3*A*b-246*a*A*b^3-8*a^4*B-33*a^2*b^2*B-147*b^4*B)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b^4*d)-2/315*(a-b)*(3*b^3*(25*A-49*B)-3*a*b^2*(57*A-13*B)-6*a^2*b*(3*A-B)+8*a^3*B)*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b^3*d)-2/315*(18*a*A*b-8*a^2*B-49*b^2*B)*(a+b*sec(c+d*x))^(3/2)*tan(c+d*x)/(b^2*d)+2/63*(9*A*b-4*a*B)*(a+b*sec(c+d*x))^(5/2)*tan(c+d*x)/(b^2*d)+2/9*B*sec(c+d*x)*(a+b*sec(c+d*x))^(5/2)*tan(c+d*x)/(b*d)-2/315*(18*a^2*A*b-75*A*b^3-8*a^3*B-39*a*b^2*B)*sqrt(a+b*sec(c+d*x))*tan(c+d*x)/(b^2*d)],
[sec(c+d*x)^2*(a+b*sec(c+d*x))^(3/2)*(A+B*sec(c+d*x)),x,6,-2/105*(a-b)*(21*a^2*A*b+63*A*b^3-6*a^3*B+82*a*b^2*B)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b^3*d)+2/105*(a-b)*(b^2*(63*A-25*B)+6*a^2*B-a*(21*A*b-57*b*B))*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b^2*d)+2/35*(7*A*b-2*a*B)*(a+b*sec(c+d*x))^(3/2)*tan(c+d*x)/(b*d)+2/7*B*(a+b*sec(c+d*x))^(5/2)*tan(c+d*x)/(b*d)+2/105*(21*a*A*b-6*a^2*B+25*b^2*B)*sqrt(a+b*sec(c+d*x))*tan(c+d*x)/(b*d)],
[sec(c+d*x)*(a+b*sec(c+d*x))^(3/2)*(A+B*sec(c+d*x)),x,5,-2/15*(a-b)*(20*a*A*b+3*a^2*B+9*b^2*B)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b^2*d)+2/15*(a-b)*(15*a*A-5*A*b-3*a*B+9*b*B)*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b*d)+2/5*B*(a+b*sec(c+d*x))^(3/2)*tan(c+d*x)/d+2/15*(5*A*b+3*a*B)*sqrt(a+b*sec(c+d*x))*tan(c+d*x)/d],
[(a+b*sec(c+d*x))^(3/2)*(A+B*sec(c+d*x)),x,6,-2/3*(a-b)*(3*A*b+4*a*B)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b*d)-2/3*(b^2*(3*A-B)-3*a^2*B-a*(6*A*b-4*b*B))*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b*d)-2*a*A*cot(c+d*x)*EllipticPi(sqrt(a+b*sec(c+d*x))/sqrt(a+b),(a+b)/a,sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/d+2/3*b*B*sqrt(a+b*sec(c+d*x))*tan(c+d*x)/d],
[cos(c+d*x)*(a+b*sec(c+d*x))^(3/2)*(A+B*sec(c+d*x)),x,6,(a-b)*(a*A-2*b*B)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b*d)+(2*b*(A-B)+a*(A+4*B))*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/d-(3*A*b+2*a*B)*cot(c+d*x)*EllipticPi(sqrt(a+b*sec(c+d*x))/sqrt(a+b),(a+b)/a,sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/d+a*A*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/d],
[cos(c+d*x)^2*(a+b*sec(c+d*x))^(3/2)*(A+B*sec(c+d*x)),x,7,1/4*(a-b)*(5*A*b+4*a*B)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b*d)+1/4*(2*a*A+5*A*b+4*a*B+8*b*B)*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/d-1/4*(4*a^2*A+3*A*b^2+12*a*b*B)*cot(c+d*x)*EllipticPi(sqrt(a+b*sec(c+d*x))/sqrt(a+b),(a+b)/a,sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a*d)+1/4*(5*A*b+4*a*B)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/d+1/2*a*A*cos(c+d*x)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/d],
[cos(c+d*x)^3*(a+b*sec(c+d*x))^(3/2)*(A+B*sec(c+d*x)),x,8,1/24*(a-b)*(16*a^2*A+3*A*b^2+30*a*b*B)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a*b*d)+1/24*(16*a^2*A+14*a*A*b+3*A*b^2+12*a^2*B+30*a*b*B)*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a*d)-1/8*(12*a^2*A*b-A*b^3+8*a^3*B+6*a*b^2*B)*cot(c+d*x)*EllipticPi(sqrt(a+b*sec(c+d*x))/sqrt(a+b),(a+b)/a,sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a^2*d)+1/24*(16*a^2*A+3*A*b^2+30*a*b*B)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(a*d)+1/12*(7*A*b+6*a*B)*cos(c+d*x)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/d+1/3*a*A*cos(c+d*x)^2*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/d],
[sec(c+d*x)^3*(a+b*sec(c+d*x))^(5/2)*(A+B*sec(c+d*x)),x,8,2/3465*(a-b)*(110*a^4*A*b-3069*a^2*A*b^3-1617*A*b^5-40*a^5*B-255*a^3*b^2*B-3705*a*b^4*B)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b^4*d)-2/3465*(a-b)*(6*a*b^3*(209*A-505*B)-3*b^4*(539*A-225*B)-15*a^2*b^2*(121*A-19*B)+40*a^4*B-a^3*(110*A*b-30*b*B))*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b^3*d)-2/3465*(110*a^2*A*b-539*A*b^3-40*a^3*B-335*a*b^2*B)*(a+b*sec(c+d*x))^(3/2)*tan(c+d*x)/(b^2*d)-2/693*(22*a*A*b-8*a^2*B-81*b^2*B)*(a+b*sec(c+d*x))^(5/2)*tan(c+d*x)/(b^2*d)+2/99*(11*A*b-4*a*B)*(a+b*sec(c+d*x))^(7/2)*tan(c+d*x)/(b^2*d)+2/11*B*sec(c+d*x)*(a+b*sec(c+d*x))^(7/2)*tan(c+d*x)/(b*d)-2/3465*(110*a^3*A*b-1254*a*A*b^3-40*a^4*B-285*a^2*b^2*B-675*b^4*B)*sqrt(a+b*sec(c+d*x))*tan(c+d*x)/(b^2*d)],
[sec(c+d*x)^2*(a+b*sec(c+d*x))^(5/2)*(A+B*sec(c+d*x)),x,7,-2/315*(a-b)*(45*a^3*A*b+435*a*A*b^3-10*a^4*B+279*a^2*b^2*B+147*b^4*B)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b^3*d)-2/315*(a-b)*(3*b^3*(25*A-49*B)-6*a*b^2*(60*A-19*B)+15*a^2*b*(3*A-11*B)-10*a^3*B)*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b^2*d)+2/315*(45*a*A*b-10*a^2*B+49*b^2*B)*(a+b*sec(c+d*x))^(3/2)*tan(c+d*x)/(b*d)+2/63*(9*A*b-2*a*B)*(a+b*sec(c+d*x))^(5/2)*tan(c+d*x)/(b*d)+2/9*B*(a+b*sec(c+d*x))^(7/2)*tan(c+d*x)/(b*d)+2/315*(45*a^2*A*b+75*A*b^3-10*a^3*B+114*a*b^2*B)*sqrt(a+b*sec(c+d*x))*tan(c+d*x)/(b*d)],
[sec(c+d*x)*(a+b*sec(c+d*x))^(5/2)*(A+B*sec(c+d*x)),x,6,-2/105*(a-b)*(161*a^2*A*b+63*A*b^3+15*a^3*B+145*a*b^2*B)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b^2*d)+2/105*(a-b)*(b^2*(63*A-25*B)-8*a*b*(7*A-15*B)+15*a^2*(7*A-B))*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b*d)+2/35*(7*A*b+5*a*B)*(a+b*sec(c+d*x))^(3/2)*tan(c+d*x)/d+2/7*B*(a+b*sec(c+d*x))^(5/2)*tan(c+d*x)/d+2/105*(56*a*A*b+15*a^2*B+25*b^2*B)*sqrt(a+b*sec(c+d*x))*tan(c+d*x)/d],
[(a+b*sec(c+d*x))^(5/2)*(A+B*sec(c+d*x)),x,7,-2/15*(a-b)*(35*a*A*b+23*a^2*B+9*b^2*B)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b*d)+2/15*(a^2*b*(45*A-23*B)-a*b^2*(35*A-17*B)+b^3*(5*A-9*B)+15*a^3*B)*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b*d)-2*a^2*A*cot(c+d*x)*EllipticPi(sqrt(a+b*sec(c+d*x))/sqrt(a+b),(a+b)/a,sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/d+2/5*b*B*(a+b*sec(c+d*x))^(3/2)*tan(c+d*x)/d+2/15*b*(5*A*b+8*a*B)*sqrt(a+b*sec(c+d*x))*tan(c+d*x)/d],
[cos(c+d*x)*(a+b*sec(c+d*x))^(5/2)*(A+B*sec(c+d*x)),x,7,a*A*(a+b*sec(c+d*x))^(3/2)*sin(c+d*x)/d+1/3*(a-b)*(3*a^2*A-6*A*b^2-14*a*b*B)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b*d)+1/3*(2*a*b*(9*A-7*B)-2*b^2*(3*A-B)+3*a^2*(A+6*B))*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/d-a*(5*A*b+2*a*B)*cot(c+d*x)*EllipticPi(sqrt(a+b*sec(c+d*x))/sqrt(a+b),(a+b)/a,sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/d-1/3*b*(3*a*A-2*b*B)*sqrt(a+b*sec(c+d*x))*tan(c+d*x)/d],
[cos(c+d*x)^2*(a+b*sec(c+d*x))^(5/2)*(A+B*sec(c+d*x)),x,7,1/2*a*A*cos(c+d*x)*(a+b*sec(c+d*x))^(3/2)*sin(c+d*x)/d+1/4*(a-b)*(9*a*A*b+4*a^2*B-8*b^2*B)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b*d)+1/4*(8*b^2*(A-B)+2*a^2*(A+2*B)+3*a*b*(3*A+8*B))*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/d-1/4*(4*a^2*A+15*A*b^2+20*a*b*B)*cot(c+d*x)*EllipticPi(sqrt(a+b*sec(c+d*x))/sqrt(a+b),(a+b)/a,sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/d+1/4*a*(7*A*b+4*a*B)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/d],
[cos(c+d*x)^3*(a+b*sec(c+d*x))^(5/2)*(A+B*sec(c+d*x)),x,8,1/3*a*A*cos(c+d*x)^2*(a+b*sec(c+d*x))^(3/2)*sin(c+d*x)/d+1/24*(a-b)*(16*a^2*A+33*A*b^2+54*a*b*B)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b*d)+1/24*(16*a^2*A+26*a*A*b+33*A*b^2+12*a^2*B+54*a*b*B+48*b^2*B)*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/d-1/8*(20*a^2*A*b+5*A*b^3+8*a^3*B+30*a*b^2*B)*cot(c+d*x)*EllipticPi(sqrt(a+b*sec(c+d*x))/sqrt(a+b),(a+b)/a,sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a*d)+1/24*(16*a^2*A+33*A*b^2+54*a*b*B)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/d+1/4*a*(3*A*b+2*a*B)*cos(c+d*x)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/d],
[cos(c+d*x)^4*(a+b*sec(c+d*x))^(5/2)*(A+B*sec(c+d*x)),x,9,1/4*a*A*cos(c+d*x)^3*(a+b*sec(c+d*x))^(3/2)*sin(c+d*x)/d+1/192*(a-b)*(284*a^2*A*b+15*A*b^3+128*a^3*B+264*a*b^2*B)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a*b*d)+1/192*(15*A*b^3+8*a^3*(9*A+16*B)+4*a^2*b*(71*A+52*B)+2*a*b^2*(59*A+132*B))*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a*d)-1/64*(48*a^4*A+120*a^2*A*b^2-5*A*b^4+160*a^3*b*B+40*a*b^3*B)*cot(c+d*x)*EllipticPi(sqrt(a+b*sec(c+d*x))/sqrt(a+b),(a+b)/a,sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a^2*d)+1/192*(284*a^2*A*b+15*A*b^3+128*a^3*B+264*a*b^2*B)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(a*d)+1/96*(36*a^2*A+59*A*b^2+104*a*b*B)*cos(c+d*x)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/d+1/24*a*(11*A*b+8*a*B)*cos(c+d*x)^2*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/d],

# n<0
[sec(c+d*x)^3*(A+B*sec(c+d*x))/sqrt(a+b*sec(c+d*x)),x,5,2/15*(a-b)*(10*a*A*b-8*a^2*B-9*b^2*B)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b^4*d)+2/15*(b^2*(5*A-9*B)-8*a^2*B+2*a*b*(5*A+B))*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b^3*d)+2/15*(5*A*b-4*a*B)*sqrt(a+b*sec(c+d*x))*tan(c+d*x)/(b^2*d)+2/5*B*sec(c+d*x)*sqrt(a+b*sec(c+d*x))*tan(c+d*x)/(b*d)],
[sec(c+d*x)^2*(A+B*sec(c+d*x))/sqrt(a+b*sec(c+d*x)),x,4,-2/3*(a-b)*(3*A*b-2*a*B)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b^3*d)-2/3*(3*A*b-(2*a+b)*B)*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b^2*d)+2/3*B*sqrt(a+b*sec(c+d*x))*tan(c+d*x)/(b*d)],
[sec(c+d*x)*(A+B*sec(c+d*x))/sqrt(a+b*sec(c+d*x)),x,3,-2*(a-b)*B*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b^2*d)+2*(A-B)*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b*d)],
[(A+B*sec(c+d*x))/sqrt(a+b*sec(c+d*x)),x,3,2*B*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b*d)-2*A*cot(c+d*x)*EllipticPi(sqrt(a+b*sec(c+d*x))/sqrt(a+b),(a+b)/a,sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a*d)],
[cos(c+d*x)*(A+B*sec(c+d*x))/sqrt(a+b*sec(c+d*x)),x,6,A*(a-b)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a*b*d)+A*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a*d)+(A*b-2*a*B)*cot(c+d*x)*EllipticPi(sqrt(a+b*sec(c+d*x))/sqrt(a+b),(a+b)/a,sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a^2*d)+A*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(a*d)],
[cos(c+d*x)^2*(A+B*sec(c+d*x))/sqrt(a+b*sec(c+d*x)),x,7,-1/4*(a-b)*(3*A*b-4*a*B)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a^2*b*d)-1/4*(3*A*b-2*a*(A+2*B))*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a^2*d)-1/4*(4*a^2*A+3*A*b^2-4*a*b*B)*cot(c+d*x)*EllipticPi(sqrt(a+b*sec(c+d*x))/sqrt(a+b),(a+b)/a,sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a^3*d)-1/4*(3*A*b-4*a*B)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(a^2*d)+1/2*A*cos(c+d*x)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(a*d)],
[cos(c+d*x)^3*(A+B*sec(c+d*x))/sqrt(a+b*sec(c+d*x)),x,8,1/24*(a-b)*(16*a^2*A+15*A*b^2-18*a*b*B)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a^3*b*d)+1/24*(16*a^2*A-10*a*A*b+15*A*b^2+12*a^2*B-18*a*b*B)*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a^3*d)+1/8*(4*a^2*A*b+5*A*b^3-8*a^3*B-6*a*b^2*B)*cot(c+d*x)*EllipticPi(sqrt(a+b*sec(c+d*x))/sqrt(a+b),(a+b)/a,sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a^4*d)+1/24*(16*a^2*A+15*A*b^2-18*a*b*B)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(a^3*d)-1/12*(5*A*b-6*a*B)*cos(c+d*x)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(a^2*d)+1/3*A*cos(c+d*x)^2*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(a*d)],
[sec(c+d*x)^3*(A+B*sec(c+d*x))/(a+b*sec(c+d*x))^(3/2),x,5,-2/3*(6*a^2*A*b-3*A*b^3-8*a^3*B+5*a*b^2*B)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b^4*d*sqrt(a+b))-2/3*(2*a+b)*(3*A*b-(4*a+b)*B)*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b^3*d*sqrt(a+b))-2*a^2*(A*b-a*B)*tan(c+d*x)/(b^2*(a^2-b^2)*d*sqrt(a+b*sec(c+d*x)))+2/3*B*sqrt(a+b*sec(c+d*x))*tan(c+d*x)/(b^2*d)],
[sec(c+d*x)^2*(A+B*sec(c+d*x))/(a+b*sec(c+d*x))^(3/2),x,4,2*(a*A*b-2*a^2*B+b^2*B)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b^3*d*sqrt(a+b))+2*(A*b-(2*a+b)*B)*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b^2*d*sqrt(a+b))+2*a*(A*b-a*B)*tan(c+d*x)/(b*(a^2-b^2)*d*sqrt(a+b*sec(c+d*x)))],
[sec(c+d*x)*(A+B*sec(c+d*x))/(a+b*sec(c+d*x))^(3/2),x,4,-2*(A*b-a*B)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b^2*d*sqrt(a+b))+2*(A+B)*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b*d*sqrt(a+b))-2*(A*b-a*B)*tan(c+d*x)/((a^2-b^2)*d*sqrt(a+b*sec(c+d*x)))],
[(A+B*sec(c+d*x))/(a+b*sec(c+d*x))^(3/2),x,6,2*(A*b-a*B)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a*b*d*sqrt(a+b))-2*(A*b-a*B)*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a*b*d*sqrt(a+b))-2*A*cot(c+d*x)*EllipticPi(sqrt(a+b*sec(c+d*x))/sqrt(a+b),(a+b)/a,sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a^2*d)+2*b*(A*b-a*B)*tan(c+d*x)/(a*(a^2-b^2)*d*sqrt(a+b*sec(c+d*x)))],
[cos(c+d*x)*(A+B*sec(c+d*x))/(a+b*sec(c+d*x))^(3/2),x,7,(a^2*A-3*A*b^2+2*a*b*B)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a^2*b*d*sqrt(a+b))+(3*A*b+a*(A-2*B))*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a^2*d*sqrt(a+b))+(3*A*b-2*a*B)*cot(c+d*x)*EllipticPi(sqrt(a+b*sec(c+d*x))/sqrt(a+b),(a+b)/a,sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a^3*d)+A*sin(c+d*x)/(a*d*sqrt(a+b*sec(c+d*x)))+b*(a^2*A-3*A*b^2+2*a*b*B)*tan(c+d*x)/(a^2*(a^2-b^2)*d*sqrt(a+b*sec(c+d*x)))],
[cos(c+d*x)^2*(A+B*sec(c+d*x))/(a+b*sec(c+d*x))^(3/2),x,8,-1/4*(7*a^2*A*b-15*A*b^3-4*a^3*B+12*a*b^2*B)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a^3*b*d*sqrt(a+b))-1/4*(15*A*b^2+a*b*(5*A-12*B)-2*a^2*(A+2*B))*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a^3*d*sqrt(a+b))-1/4*(4*a^2*A+15*A*b^2-12*a*b*B)*cot(c+d*x)*EllipticPi(sqrt(a+b*sec(c+d*x))/sqrt(a+b),(a+b)/a,sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a^4*d)-1/4*(5*A*b-4*a*B)*sin(c+d*x)/(a^2*d*sqrt(a+b*sec(c+d*x)))+1/2*A*cos(c+d*x)*sin(c+d*x)/(a*d*sqrt(a+b*sec(c+d*x)))-1/4*b*(7*a^2*A*b-15*A*b^3-4*a^3*B+12*a*b^2*B)*tan(c+d*x)/(a^3*(a^2-b^2)*d*sqrt(a+b*sec(c+d*x)))],
[cos(c+d*x)^3*(A+B*sec(c+d*x))/(a+b*sec(c+d*x))^(3/2),x,9,1/24*(16*a^4*A+41*a^2*A*b^2-105*A*b^4-42*a^3*b*B+90*a*b^3*B)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a^4*b*d*sqrt(a+b))+1/24*(105*A*b^3+5*a*b^2*(7*A-18*B)+4*a^3*(4*A+3*B)-6*a^2*b*(A+5*B))*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a^4*d*sqrt(a+b))+1/8*(12*a^2*A*b+35*A*b^3-8*a^3*B-30*a*b^2*B)*cot(c+d*x)*EllipticPi(sqrt(a+b*sec(c+d*x))/sqrt(a+b),(a+b)/a,sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a^5*d)+1/24*(16*a^2*A+35*A*b^2-30*a*b*B)*sin(c+d*x)/(a^3*d*sqrt(a+b*sec(c+d*x)))-1/12*(7*A*b-6*a*B)*cos(c+d*x)*sin(c+d*x)/(a^2*d*sqrt(a+b*sec(c+d*x)))+1/3*A*cos(c+d*x)^2*sin(c+d*x)/(a*d*sqrt(a+b*sec(c+d*x)))+1/24*b*(16*a^4*A+41*a^2*A*b^2-105*A*b^4-42*a^3*b*B+90*a*b^3*B)*tan(c+d*x)/(a^4*(a^2-b^2)*d*sqrt(a+b*sec(c+d*x)))],
[sec(c+d*x)^4*(A+B*sec(c+d*x))/(a+b*sec(c+d*x))^(5/2),x,6,-2/3*(8*a^4*A*b-15*a^2*A*b^3+3*A*b^5-16*a^5*B+28*a^3*b^2*B-8*a*b^4*B)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/((a-b)*b^5*(a+b)^(3/2)*d)+2/3*(9*a*b^3*(A-B)+b^4*(3*A-B)+16*a^4*B-2*a^2*b^2*(3*A+8*B)-a^3*(8*A*b-12*b*B))*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b^4*(a^2-b^2)*d*sqrt(a+b))+2/3*a*(A*b-a*B)*sec(c+d*x)^2*tan(c+d*x)/(b*(a^2-b^2)*d*(a+b*sec(c+d*x))^(3/2))-2/3*a^2*(3*a^2*A*b-7*A*b^3-6*a^3*B+10*a*b^2*B)*tan(c+d*x)/(b^3*(a^2-b^2)^2*d*sqrt(a+b*sec(c+d*x)))-2/3*(a*A*b-2*a^2*B+b^2*B)*sqrt(a+b*sec(c+d*x))*tan(c+d*x)/(b^3*(a^2-b^2)*d)],
[sec(c+d*x)^3*(A+B*sec(c+d*x))/(a+b*sec(c+d*x))^(5/2),x,5,2/3*(2*a^3*A*b-6*a*A*b^3-8*a^4*B+15*a^2*b^2*B-3*b^4*B)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/((a-b)*b^4*(a+b)^(3/2)*d)+2/3*(2*a^2*b*(A-3*B)-3*b^3*(A-B)-8*a^3*B+3*a*b^2*(A+3*B))*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b^3*(a^2-b^2)*d*sqrt(a+b))-2/3*a^2*(A*b-a*B)*tan(c+d*x)/(b^2*(a^2-b^2)*d*(a+b*sec(c+d*x))^(3/2))+2/3*a*(2*a^2*A*b-6*A*b^3-5*a^3*B+9*a*b^2*B)*tan(c+d*x)/(b^2*(a^2-b^2)^2*d*sqrt(a+b*sec(c+d*x)))],
[sec(c+d*x)^2*(A+B*sec(c+d*x))/(a+b*sec(c+d*x))^(5/2),x,5,2/3*(a^2*A*b+3*A*b^3+2*a^3*B-6*a*b^2*B)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/((a-b)*b^3*(a+b)^(3/2)*d)+2/3*(2*a^2*B-3*b^2*(A+B)+a*b*(A+3*B))*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b^2*(a^2-b^2)*d*sqrt(a+b))+2/3*a*(A*b-a*B)*tan(c+d*x)/(b*(a^2-b^2)*d*(a+b*sec(c+d*x))^(3/2))+2/3*(a^2*A*b+3*A*b^3+2*a^3*B-6*a*b^2*B)*tan(c+d*x)/(b*(a^2-b^2)^2*d*sqrt(a+b*sec(c+d*x)))],
[sec(c+d*x)*(A+B*sec(c+d*x))/(a+b*sec(c+d*x))^(5/2),x,5,-2/3*(4*a*A*b-a^2*B-3*b^2*B)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/((a-b)*b^2*(a+b)^(3/2)*d)+2/3*(3*a*A-A*b+a*B-3*b*B)*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/((a-b)*b*(a+b)^(3/2)*d)-2/3*(A*b-a*B)*tan(c+d*x)/((a^2-b^2)*d*(a+b*sec(c+d*x))^(3/2))-2/3*(4*a*A*b-a^2*B-3*b^2*B)*tan(c+d*x)/((a^2-b^2)^2*d*sqrt(a+b*sec(c+d*x)))],
[(A+B*sec(c+d*x))/(a+b*sec(c+d*x))^(5/2),x,7,2/3*(7*a^2*A*b-3*A*b^3-4*a^3*B)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a^2*(a-b)*b*(a+b)^(3/2)*d)-2/3*(6*a^2*A*b-a*A*b^2-3*A*b^3-3*a^3*B+a^2*b*B)*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a^2*(a-b)*b*(a+b)^(3/2)*d)-2*A*cot(c+d*x)*EllipticPi(sqrt(a+b*sec(c+d*x))/sqrt(a+b),(a+b)/a,sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a^3*d)+2/3*b*(A*b-a*B)*tan(c+d*x)/(a*(a^2-b^2)*d*(a+b*sec(c+d*x))^(3/2))+2/3*b*(7*a^2*A*b-3*A*b^3-4*a^3*B)*tan(c+d*x)/(a^2*(a^2-b^2)^2*d*sqrt(a+b*sec(c+d*x)))],
[cos(c+d*x)*(A+B*sec(c+d*x))/(a+b*sec(c+d*x))^(5/2),x,8,A*sin(c+d*x)/(a*d*(a+b*sec(c+d*x))^(3/2))+1/3*(3*a^4*A-26*a^2*A*b^2+15*A*b^4+14*a^3*b*B-6*a*b^3*B)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a^3*(a-b)*b*(a+b)^(3/2)*d)-1/3*(15*A*b^3+a*b^2*(5*A-6*B)-3*a^3*(A-4*B)-a^2*b*(21*A+2*B))*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a^3*(a^2-b^2)*d*sqrt(a+b))+(5*A*b-2*a*B)*cot(c+d*x)*EllipticPi(sqrt(a+b*sec(c+d*x))/sqrt(a+b),(a+b)/a,sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a^4*d)+1/3*b*(3*a^2*A-5*A*b^2+2*a*b*B)*tan(c+d*x)/(a^2*(a^2-b^2)*d*(a+b*sec(c+d*x))^(3/2))+1/3*b*(3*a^4*A-26*a^2*A*b^2+15*A*b^4+14*a^3*b*B-6*a*b^3*B)*tan(c+d*x)/(a^3*(a^2-b^2)^2*d*sqrt(a+b*sec(c+d*x)))],
[cos(c+d*x)^2*(A+B*sec(c+d*x))/(a+b*sec(c+d*x))^(5/2),x,9,-1/4*(7*A*b-4*a*B)*sin(c+d*x)/(a^2*d*(a+b*sec(c+d*x))^(3/2))+1/2*A*cos(c+d*x)*sin(c+d*x)/(a*d*(a+b*sec(c+d*x))^(3/2))-1/12*(33*a^4*A*b-170*a^2*A*b^3+105*A*b^5-12*a^5*B+104*a^3*b^2*B-60*a*b^4*B)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a^4*(a-b)*b*(a+b)^(3/2)*d)+1/12*(105*A*b^4+5*a*b^3*(7*A-12*B)+6*a^4*(A+2*B)-5*a^2*b^2*(27*A+4*B)-a^3*(27*A*b-84*b*B))*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a^4*(a^2-b^2)*d*sqrt(a+b))-1/4*(4*a^2*A+35*A*b^2-20*a*b*B)*cot(c+d*x)*EllipticPi(sqrt(a+b*sec(c+d*x))/sqrt(a+b),(a+b)/a,sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a^5*d)-1/12*b*(27*a^2*A*b-35*A*b^3-12*a^3*B+20*a*b^2*B)*tan(c+d*x)/(a^3*(a^2-b^2)*d*(a+b*sec(c+d*x))^(3/2))-1/12*b*(33*a^4*A*b-170*a^2*A*b^3+105*A*b^5-12*a^5*B+104*a^3*b^2*B-60*a*b^4*B)*tan(c+d*x)/(a^4*(a^2-b^2)^2*d*sqrt(a+b*sec(c+d*x)))],
[sec(e+f*x)*(A+A*sec(e+f*x))/sqrt(a+b*sec(e+f*x)),x,1,-2*A*(a-b)*cot(e+f*x)*EllipticE(sqrt(a+b*sec(e+f*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(e+f*x))/(a+b))*sqrt(-b*(1+sec(e+f*x))/(a-b))/(b^2*f)],
[sec(e+f*x)*(A-A*sec(e+f*x))/sqrt(a+b*sec(e+f*x)),x,1,2*A*(a+b)*cot(e+f*x)*EllipticE(sqrt(a+b*sec(e+f*x))/sqrt(a-b),sqrt((a-b)/(a+b)))*sqrt(a-b)*sqrt(b*(1-sec(e+f*x))/(a+b))*sqrt(-b*(1+sec(e+f*x))/(a-b))/(b^2*f)],

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

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

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

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

# n>0
[sec(c+d*x)^(3/2)*(A+B*sec(c+d*x))*sqrt(a+b*sec(c+d*x)),x,13,1/4*(4*A*b+3*a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(d*sqrt(a+b*sec(c+d*x)))+1/4*(4*a*A*b-a^2*B+4*b^2*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(b*d*sqrt(a+b*sec(c+d*x)))+1/2*B*sec(c+d*x)^(3/2)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/d-1/4*(4*A*b+a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt(a+b*sec(c+d*x))/(b*d*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x)))+1/4*(4*A*b+a*B)*sin(c+d*x)*sqrt(sec(c+d*x))*sqrt(a+b*sec(c+d*x))/(b*d)],
[(A+B*sec(c+d*x))*sqrt(sec(c+d*x))*sqrt(a+b*sec(c+d*x)),x,12,(2*a*A+b*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(d*sqrt(a+b*sec(c+d*x)))+(2*A*b+a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(d*sqrt(a+b*sec(c+d*x)))-B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt(a+b*sec(c+d*x))/(d*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x)))+B*sin(c+d*x)*sqrt(sec(c+d*x))*sqrt(a+b*sec(c+d*x))/d],
[(A+B*sec(c+d*x))*sqrt(a+b*sec(c+d*x))/sqrt(sec(c+d*x)),x,11,2*a*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(d*sqrt(a+b*sec(c+d*x)))+2*b*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(d*sqrt(a+b*sec(c+d*x)))+2*A*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt(a+b*sec(c+d*x))/(d*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x)))],
[(A+B*sec(c+d*x))*sqrt(a+b*sec(c+d*x))/sec(c+d*x)^(3/2),x,8,2/3*A*(a^2-b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(a*d*sqrt(a+b*sec(c+d*x)))+2/3*A*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(d*sqrt(sec(c+d*x)))+2/3*(A*b+3*a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt(a+b*sec(c+d*x))/(a*d*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x)))],
[(A+B*sec(c+d*x))*sqrt(a+b*sec(c+d*x))/sec(c+d*x)^(5/2),x,9,-2/15*(a^2-b^2)*(2*A*b-5*a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(a^2*d*sqrt(a+b*sec(c+d*x)))+2/5*A*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(d*sec(c+d*x)^(3/2))+2/15*(A*b+5*a*B)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(a*d*sqrt(sec(c+d*x)))+2/15*(9*a^2*A-2*A*b^2+5*a*b*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt(a+b*sec(c+d*x))/(a^2*d*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x)))],
[(A+B*sec(c+d*x))*sqrt(a+b*sec(c+d*x))/sec(c+d*x)^(7/2),x,10,2/105*(a^2-b^2)*(25*a^2*A+8*A*b^2-14*a*b*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(a^3*d*sqrt(a+b*sec(c+d*x)))+2/7*A*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(d*sec(c+d*x)^(5/2))+2/35*(A*b+7*a*B)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(a*d*sec(c+d*x)^(3/2))+2/105*(25*a^2*A-4*A*b^2+7*a*b*B)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(a^2*d*sqrt(sec(c+d*x)))+2/105*(19*a^2*A*b+8*A*b^3+63*a^3*B-14*a*b^2*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt(a+b*sec(c+d*x))/(a^3*d*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x)))],
[sec(c+d*x)^(3/2)*(a+b*sec(c+d*x))^(3/2)*(A+B*sec(c+d*x)),x,14,1/24*(42*a*A*b+17*a^2*B+16*b^2*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(d*sqrt(a+b*sec(c+d*x)))+1/8*(6*a^2*A*b+8*A*b^3-a^3*B+12*a*b^2*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(b*d*sqrt(a+b*sec(c+d*x)))+1/12*(6*A*b+7*a*B)*sec(c+d*x)^(3/2)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/d+1/3*b*B*sec(c+d*x)^(5/2)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/d-1/24*(30*a*A*b+3*a^2*B+16*b^2*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt(a+b*sec(c+d*x))/(b*d*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x)))+1/24*(30*a*A*b+3*a^2*B+16*b^2*B)*sin(c+d*x)*sqrt(sec(c+d*x))*sqrt(a+b*sec(c+d*x))/(b*d)],
[(a+b*sec(c+d*x))^(3/2)*(A+B*sec(c+d*x))*sqrt(sec(c+d*x)),x,13,1/4*(8*a^2*A+4*A*b^2+7*a*b*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(d*sqrt(a+b*sec(c+d*x)))+1/4*(12*a*A*b+3*a^2*B+4*b^2*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(d*sqrt(a+b*sec(c+d*x)))+1/2*b*B*sec(c+d*x)^(3/2)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/d-1/4*(4*A*b+5*a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt(a+b*sec(c+d*x))/(d*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x)))+1/4*(4*A*b+5*a*B)*sin(c+d*x)*sqrt(sec(c+d*x))*sqrt(a+b*sec(c+d*x))/d],
[(a+b*sec(c+d*x))^(3/2)*(A+B*sec(c+d*x))/sqrt(sec(c+d*x)),x,12,(2*a*A*b+2*a^2*B+b^2*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(d*sqrt(a+b*sec(c+d*x)))+b*(2*A*b+3*a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(d*sqrt(a+b*sec(c+d*x)))+(2*a*A-b*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt(a+b*sec(c+d*x))/(d*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x)))+b*B*sin(c+d*x)*sqrt(sec(c+d*x))*sqrt(a+b*sec(c+d*x))/d],
[(a+b*sec(c+d*x))^(3/2)*(A+B*sec(c+d*x))/sec(c+d*x)^(3/2),x,12,2/3*(a^2*A-A*b^2+3*a*b*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(d*sqrt(a+b*sec(c+d*x)))+2*b^2*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(d*sqrt(a+b*sec(c+d*x)))+2/3*a*A*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(d*sqrt(sec(c+d*x)))+2/3*(4*A*b+3*a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt(a+b*sec(c+d*x))/(d*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x)))],
[(a+b*sec(c+d*x))^(3/2)*(A+B*sec(c+d*x))/sec(c+d*x)^(5/2),x,9,2/15*(a^2-b^2)*(3*A*b+5*a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(a*d*sqrt(a+b*sec(c+d*x)))+2/5*a*A*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(d*sec(c+d*x)^(3/2))+2/15*(6*A*b+5*a*B)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(d*sqrt(sec(c+d*x)))+2/15*(9*a^2*A+3*A*b^2+20*a*b*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt(a+b*sec(c+d*x))/(a*d*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x)))],
[(a+b*sec(c+d*x))^(3/2)*(A+B*sec(c+d*x))/sec(c+d*x)^(7/2),x,10,2/105*(a^2-b^2)*(25*a^2*A-6*A*b^2+21*a*b*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(a^2*d*sqrt(a+b*sec(c+d*x)))+2/7*a*A*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(d*sec(c+d*x)^(5/2))+2/35*(8*A*b+7*a*B)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(d*sec(c+d*x)^(3/2))+2/105*(25*a^2*A+3*A*b^2+42*a*b*B)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(a*d*sqrt(sec(c+d*x)))+2/105*(82*a^2*A*b-6*A*b^3+63*a^3*B+21*a*b^2*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt(a+b*sec(c+d*x))/(a^2*d*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x)))],
[(a+b*sec(c+d*x))^(3/2)*(A+B*sec(c+d*x))/sec(c+d*x)^(9/2),x,11,2/315*(a^2-b^2)*(39*a^2*A*b+8*A*b^3+75*a^3*B-18*a*b^2*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(a^3*d*sqrt(a+b*sec(c+d*x)))+2/9*a*A*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(d*sec(c+d*x)^(7/2))+2/63*(10*A*b+9*a*B)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(d*sec(c+d*x)^(5/2))+2/315*(49*a^2*A+3*A*b^2+72*a*b*B)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(a*d*sec(c+d*x)^(3/2))+2/315*(88*a^2*A*b-4*A*b^3+75*a^3*B+9*a*b^2*B)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(a^2*d*sqrt(sec(c+d*x)))+2/315*(147*a^4*A+33*a^2*A*b^2+8*A*b^4+246*a^3*b*B-18*a*b^3*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt(a+b*sec(c+d*x))/(a^3*d*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x)))],
[sec(c+d*x)^(3/2)*(a+b*sec(c+d*x))^(5/2)*(A+B*sec(c+d*x)),x,15,1/4*b*B*sec(c+d*x)^(5/2)*(a+b*sec(c+d*x))^(3/2)*sin(c+d*x)/d+1/192*(472*a^2*A*b+128*A*b^3+133*a^3*B+356*a*b^2*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(d*sqrt(a+b*sec(c+d*x)))+1/64*(40*a^3*A*b+160*a*A*b^3-5*a^4*B+120*a^2*b^2*B+48*b^4*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(b*d*sqrt(a+b*sec(c+d*x)))+1/96*(104*a*A*b+59*a^2*B+36*b^2*B)*sec(c+d*x)^(3/2)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/d+1/24*b*(8*A*b+11*a*B)*sec(c+d*x)^(5/2)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/d-1/192*(264*a^2*A*b+128*A*b^3+15*a^3*B+284*a*b^2*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt(a+b*sec(c+d*x))/(b*d*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x)))+1/192*(264*a^2*A*b+128*A*b^3+15*a^3*B+284*a*b^2*B)*sin(c+d*x)*sqrt(sec(c+d*x))*sqrt(a+b*sec(c+d*x))/(b*d)],
[(a+b*sec(c+d*x))^(5/2)*(A+B*sec(c+d*x))*sqrt(sec(c+d*x)),x,14,1/3*b*B*sec(c+d*x)^(3/2)*(a+b*sec(c+d*x))^(3/2)*sin(c+d*x)/d+1/24*(48*a^3*A+66*a*A*b^2+59*a^2*b*B+16*b^3*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(d*sqrt(a+b*sec(c+d*x)))+1/8*(30*a^2*A*b+8*A*b^3+5*a^3*B+20*a*b^2*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(d*sqrt(a+b*sec(c+d*x)))+1/4*b*(2*A*b+3*a*B)*sec(c+d*x)^(3/2)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/d-1/24*(54*a*A*b+33*a^2*B+16*b^2*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt(a+b*sec(c+d*x))/(d*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x)))+1/24*(54*a*A*b+33*a^2*B+16*b^2*B)*sin(c+d*x)*sqrt(sec(c+d*x))*sqrt(a+b*sec(c+d*x))/d],
[(a+b*sec(c+d*x))^(5/2)*(A+B*sec(c+d*x))/sqrt(sec(c+d*x)),x,13,1/2*b*B*(a+b*sec(c+d*x))^(3/2)*sin(c+d*x)*sqrt(sec(c+d*x))/d+1/4*(16*a^2*A*b+4*A*b^3+8*a^3*B+11*a*b^2*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(d*sqrt(a+b*sec(c+d*x)))+1/4*b*(20*a*A*b+15*a^2*B+4*b^2*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(d*sqrt(a+b*sec(c+d*x)))+1/4*(8*a^2*A-4*A*b^2-9*a*b*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt(a+b*sec(c+d*x))/(d*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x)))+1/4*b*(4*A*b+7*a*B)*sin(c+d*x)*sqrt(sec(c+d*x))*sqrt(a+b*sec(c+d*x))/d],
[(a+b*sec(c+d*x))^(5/2)*(A+B*sec(c+d*x))/sec(c+d*x)^(3/2),x,13,2/3*a*A*(a+b*sec(c+d*x))^(3/2)*sin(c+d*x)/(d*sqrt(sec(c+d*x)))+1/3*(2*a^3*A+4*a*A*b^2+12*a^2*b*B+3*b^3*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(d*sqrt(a+b*sec(c+d*x)))+b^2*(2*A*b+5*a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(d*sqrt(a+b*sec(c+d*x)))+1/3*(14*a*A*b+6*a^2*B-3*b^2*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt(a+b*sec(c+d*x))/(d*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x)))-1/3*b*(2*a*A-3*b*B)*sin(c+d*x)*sqrt(sec(c+d*x))*sqrt(a+b*sec(c+d*x))/d],
[(a+b*sec(c+d*x))^(5/2)*(A+B*sec(c+d*x))/sec(c+d*x)^(5/2),x,13,2/5*a*A*(a+b*sec(c+d*x))^(3/2)*sin(c+d*x)/(d*sec(c+d*x)^(3/2))+2/15*(8*a^2*A*b-8*A*b^3+5*a^3*B+10*a*b^2*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(d*sqrt(a+b*sec(c+d*x)))+2*b^3*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(d*sqrt(a+b*sec(c+d*x)))+2/15*a*(8*A*b+5*a*B)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(d*sqrt(sec(c+d*x)))+2/15*(9*a^2*A+23*A*b^2+35*a*b*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt(a+b*sec(c+d*x))/(d*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x)))],
[(a+b*sec(c+d*x))^(5/2)*(A+B*sec(c+d*x))/sec(c+d*x)^(7/2),x,10,2/7*a*A*(a+b*sec(c+d*x))^(3/2)*sin(c+d*x)/(d*sec(c+d*x)^(5/2))+2/105*(a^2-b^2)*(25*a^2*A+15*A*b^2+56*a*b*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(a*d*sqrt(a+b*sec(c+d*x)))+2/35*a*(10*A*b+7*a*B)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(d*sec(c+d*x)^(3/2))+2/105*(25*a^2*A+45*A*b^2+77*a*b*B)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(d*sqrt(sec(c+d*x)))+2/105*(145*a^2*A*b+15*A*b^3+63*a^3*B+161*a*b^2*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt(a+b*sec(c+d*x))/(a*d*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x)))],
[(a+b*sec(c+d*x))^(5/2)*(A+B*sec(c+d*x))/sec(c+d*x)^(9/2),x,11,2/9*a*A*(a+b*sec(c+d*x))^(3/2)*sin(c+d*x)/(d*sec(c+d*x)^(7/2))+2/315*(a^2-b^2)*(114*a^2*A*b-10*A*b^3+75*a^3*B+45*a*b^2*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(a^2*d*sqrt(a+b*sec(c+d*x)))+2/21*a*(4*A*b+3*a*B)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(d*sec(c+d*x)^(5/2))+2/315*(49*a^2*A+75*A*b^2+135*a*b*B)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(d*sec(c+d*x)^(3/2))+2/315*(163*a^2*A*b+5*A*b^3+75*a^3*B+135*a*b^2*B)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(a*d*sqrt(sec(c+d*x)))+2/315*(147*a^4*A+279*a^2*A*b^2-10*A*b^4+435*a^3*b*B+45*a*b^3*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt(a+b*sec(c+d*x))/(a^2*d*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x)))],
[(a+b*sec(c+d*x))^(5/2)*(A+B*sec(c+d*x))/sec(c+d*x)^(11/2),x,12,2/11*a*A*(a+b*sec(c+d*x))^(3/2)*sin(c+d*x)/(d*sec(c+d*x)^(9/2))+2/3465*(a^2-b^2)*(675*a^4*A+285*a^2*A*b^2+40*A*b^4+1254*a^3*b*B-110*a*b^3*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(a^3*d*sqrt(a+b*sec(c+d*x)))+2/99*a*(14*A*b+11*a*B)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(d*sec(c+d*x)^(7/2))+2/693*(81*a^2*A+113*A*b^2+209*a*b*B)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(d*sec(c+d*x)^(5/2))+2/3465*(1145*a^2*A*b+15*A*b^3+539*a^3*B+825*a*b^2*B)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(a*d*sec(c+d*x)^(3/2))+2/3465*(675*a^4*A+1025*a^2*A*b^2-20*A*b^4+1793*a^3*b*B+55*a*b^3*B)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(a^2*d*sqrt(sec(c+d*x)))+2/3465*(3705*a^4*A*b+255*a^2*A*b^3+40*A*b^5+1617*a^5*B+3069*a^3*b^2*B-110*a*b^4*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt(a+b*sec(c+d*x))/(a^3*d*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x)))],

# n<0
[sec(c+d*x)^(5/2)*(A+B*sec(c+d*x))/sqrt(a+b*sec(c+d*x)),x,13,1/4*(4*A*b-a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(b*d*sqrt(a+b*sec(c+d*x)))-1/4*(4*a*A*b-3*a^2*B-4*b^2*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(b^2*d*sqrt(a+b*sec(c+d*x)))+1/2*B*sec(c+d*x)^(3/2)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(b*d)-1/4*(4*A*b-3*a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt(a+b*sec(c+d*x))/(b^2*d*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x)))+1/4*(4*A*b-3*a*B)*sin(c+d*x)*sqrt(sec(c+d*x))*sqrt(a+b*sec(c+d*x))/(b^2*d)],
[sec(c+d*x)^(3/2)*(A+B*sec(c+d*x))/sqrt(a+b*sec(c+d*x)),x,12,B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(d*sqrt(a+b*sec(c+d*x)))+(2*A*b-a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(b*d*sqrt(a+b*sec(c+d*x)))-B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt(a+b*sec(c+d*x))/(b*d*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x)))+B*sin(c+d*x)*sqrt(sec(c+d*x))*sqrt(a+b*sec(c+d*x))/(b*d)],
[(A+B*sec(c+d*x))*sqrt(sec(c+d*x))/sqrt(a+b*sec(c+d*x)),x,7,2*A*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(d*sqrt(a+b*sec(c+d*x)))+2*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(d*sqrt(a+b*sec(c+d*x)))],
[(A+B*sec(c+d*x))/(sqrt(sec(c+d*x))*sqrt(a+b*sec(c+d*x))),x,7,-2*(A*b-a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(a*d*sqrt(a+b*sec(c+d*x)))+2*A*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt(a+b*sec(c+d*x))/(a*d*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x)))],
[(A+B*sec(c+d*x))/(sec(c+d*x)^(3/2)*sqrt(a+b*sec(c+d*x))),x,8,2/3*(a^2*A+2*A*b^2-3*a*b*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(a^2*d*sqrt(a+b*sec(c+d*x)))+2/3*A*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(a*d*sqrt(sec(c+d*x)))-2/3*(2*A*b-3*a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt(a+b*sec(c+d*x))/(a^2*d*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x)))],
[(A+B*sec(c+d*x))/(sec(c+d*x)^(5/2)*sqrt(a+b*sec(c+d*x))),x,9,-2/15*(7*a^2*A*b+8*A*b^3-5*a^3*B-10*a*b^2*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(a^3*d*sqrt(a+b*sec(c+d*x)))+2/5*A*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(a*d*sec(c+d*x)^(3/2))-2/15*(4*A*b-5*a*B)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(a^2*d*sqrt(sec(c+d*x)))+2/15*(9*a^2*A+8*A*b^2-10*a*b*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt(a+b*sec(c+d*x))/(a^3*d*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x)))],
[sec(c+d*x)^(5/2)*(A+B*sec(c+d*x))/(a+b*sec(c+d*x))^(3/2),x,13,2*a*(A*b-a*B)*sec(c+d*x)^(3/2)*sin(c+d*x)/(b*(a^2-b^2)*d*sqrt(a+b*sec(c+d*x)))+B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(b*d*sqrt(a+b*sec(c+d*x)))+(2*A*b-3*a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(b^2*d*sqrt(a+b*sec(c+d*x)))+(2*a*A*b-3*a^2*B+b^2*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt(a+b*sec(c+d*x))/(b^2*(a^2-b^2)*d*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x)))-(2*a*A*b-3*a^2*B+b^2*B)*sin(c+d*x)*sqrt(sec(c+d*x))*sqrt(a+b*sec(c+d*x))/(b^2*(a^2-b^2)*d)],
[sec(c+d*x)^(3/2)*(A+B*sec(c+d*x))/(a+b*sec(c+d*x))^(3/2),x,9,2*a*(A*b-a*B)*sin(c+d*x)*sqrt(sec(c+d*x))/(b*(a^2-b^2)*d*sqrt(a+b*sec(c+d*x)))+2*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(b*d*sqrt(a+b*sec(c+d*x)))-2*(A*b-a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt(a+b*sec(c+d*x))/(b*(a^2-b^2)*d*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x)))],
[(A+B*sec(c+d*x))*sqrt(sec(c+d*x))/(a+b*sec(c+d*x))^(3/2),x,8,-2*(A*b-a*B)*sin(c+d*x)*sqrt(sec(c+d*x))/((a^2-b^2)*d*sqrt(a+b*sec(c+d*x)))+2*A*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(a*d*sqrt(a+b*sec(c+d*x)))+2*(A*b-a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt(a+b*sec(c+d*x))/(a*(a^2-b^2)*d*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x)))],
[(A+B*sec(c+d*x))/((a+b*sec(c+d*x))^(3/2)*sqrt(sec(c+d*x))),x,8,2*b*(A*b-a*B)*sin(c+d*x)*sqrt(sec(c+d*x))/(a*(a^2-b^2)*d*sqrt(a+b*sec(c+d*x)))-2*(2*A*b-a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(a^2*d*sqrt(a+b*sec(c+d*x)))+2*(a^2*A-2*A*b^2+a*b*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt(a+b*sec(c+d*x))/(a^2*(a^2-b^2)*d*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x)))],
[(A+B*sec(c+d*x))/(sec(c+d*x)^(3/2)*(a+b*sec(c+d*x))^(3/2)),x,9,2*b*(A*b-a*B)*sin(c+d*x)/(a*(a^2-b^2)*d*sqrt(sec(c+d*x))*sqrt(a+b*sec(c+d*x)))+2/3*(a^2*A+8*A*b^2-6*a*b*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(a^3*d*sqrt(a+b*sec(c+d*x)))+2/3*(a^2*A-4*A*b^2+3*a*b*B)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(a^2*(a^2-b^2)*d*sqrt(sec(c+d*x)))-2/3*(5*a^2*A*b-8*A*b^3-3*a^3*B+6*a*b^2*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt(a+b*sec(c+d*x))/(a^3*(a^2-b^2)*d*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x)))],
[(A+B*sec(c+d*x))/(sec(c+d*x)^(5/2)*(a+b*sec(c+d*x))^(3/2)),x,10,2*b*(A*b-a*B)*sin(c+d*x)/(a*(a^2-b^2)*d*sec(c+d*x)^(3/2)*sqrt(a+b*sec(c+d*x)))-2/15*(12*a^2*A*b+48*A*b^3-5*a^3*B-40*a*b^2*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(a^4*d*sqrt(a+b*sec(c+d*x)))+2/5*(a^2*A-6*A*b^2+5*a*b*B)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(a^2*(a^2-b^2)*d*sec(c+d*x)^(3/2))-2/15*(9*a^2*A*b-24*A*b^3-5*a^3*B+20*a*b^2*B)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(a^3*(a^2-b^2)*d*sqrt(sec(c+d*x)))+2/15*(9*a^4*A+24*a^2*A*b^2-48*A*b^4-25*a^3*b*B+40*a*b^3*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt(a+b*sec(c+d*x))/(a^4*(a^2-b^2)*d*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x)))],
[sec(c+d*x)^(5/2)*(A+B*sec(c+d*x))/(a+b*sec(c+d*x))^(5/2),x,13,2/3*a*(A*b-a*B)*sec(c+d*x)^(3/2)*sin(c+d*x)/(b*(a^2-b^2)*d*(a+b*sec(c+d*x))^(3/2))-2/3*a*(4*A*b^3+3*a^3*B-7*a*b^2*B)*sin(c+d*x)*sqrt(sec(c+d*x))/(b^2*(a^2-b^2)^2*d*sqrt(a+b*sec(c+d*x)))+2/3*(A*b-a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(b*(a^2-b^2)*d*sqrt(a+b*sec(c+d*x)))+2*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(b^2*d*sqrt(a+b*sec(c+d*x)))+2/3*(4*A*b^3+3*a^3*B-7*a*b^2*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt(a+b*sec(c+d*x))/(b^2*(a^2-b^2)^2*d*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x)))],
[sec(c+d*x)^(3/2)*(A+B*sec(c+d*x))/(a+b*sec(c+d*x))^(5/2),x,9,2/3*a*(A*b-a*B)*sin(c+d*x)*sqrt(sec(c+d*x))/(b*(a^2-b^2)*d*(a+b*sec(c+d*x))^(3/2))+2/3*(2*a^2*A*b+2*A*b^3+a^3*B-5*a*b^2*B)*sin(c+d*x)*sqrt(sec(c+d*x))/(b*(a^2-b^2)^2*d*sqrt(a+b*sec(c+d*x)))-2/3*(A*b-a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(a*(a^2-b^2)*d*sqrt(a+b*sec(c+d*x)))-2/3*(3*a^2*A+A*b^2-4*a*b*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt(a+b*sec(c+d*x))/(a*(a^2-b^2)^2*d*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x)))],
[(A+B*sec(c+d*x))*sqrt(sec(c+d*x))/(a+b*sec(c+d*x))^(5/2),x,9,-2/3*(A*b-a*B)*sin(c+d*x)*sqrt(sec(c+d*x))/((a^2-b^2)*d*(a+b*sec(c+d*x))^(3/2))-2/3*(5*a^2*A*b-A*b^3-2*a^3*B-2*a*b^2*B)*sin(c+d*x)*sqrt(sec(c+d*x))/(a*(a^2-b^2)^2*d*sqrt(a+b*sec(c+d*x)))+2/3*(3*a^2*A-2*A*b^2-a*b*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(a^2*(a^2-b^2)*d*sqrt(a+b*sec(c+d*x)))+2/3*(6*a^2*A*b-2*A*b^3-3*a^3*B-a*b^2*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt(a+b*sec(c+d*x))/(a^2*(a^2-b^2)^2*d*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x)))],
[(A+B*sec(c+d*x))/((a+b*sec(c+d*x))^(5/2)*sqrt(sec(c+d*x))),x,9,2/3*b*(A*b-a*B)*sin(c+d*x)*sqrt(sec(c+d*x))/(a*(a^2-b^2)*d*(a+b*sec(c+d*x))^(3/2))+2/3*b*(8*a^2*A*b-4*A*b^3-5*a^3*B+a*b^2*B)*sin(c+d*x)*sqrt(sec(c+d*x))/(a^2*(a^2-b^2)^2*d*sqrt(a+b*sec(c+d*x)))-2/3*(9*a^2*A*b-8*A*b^3-3*a^3*B+2*a*b^2*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(a^3*(a^2-b^2)*d*sqrt(a+b*sec(c+d*x)))+2/3*(3*a^4*A-15*a^2*A*b^2+8*A*b^4+6*a^3*b*B-2*a*b^3*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt(a+b*sec(c+d*x))/(a^3*(a^2-b^2)^2*d*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x)))],
[(A+B*sec(c+d*x))/(sec(c+d*x)^(3/2)*(a+b*sec(c+d*x))^(5/2)),x,10,2/3*b*(A*b-a*B)*sin(c+d*x)/(a*(a^2-b^2)*d*(a+b*sec(c+d*x))^(3/2)*sqrt(sec(c+d*x)))+2/3*b*(10*a^2*A*b-6*A*b^3-7*a^3*B+3*a*b^2*B)*sin(c+d*x)/(a^2*(a^2-b^2)^2*d*sqrt(sec(c+d*x))*sqrt(a+b*sec(c+d*x)))+2/3*(a^4*A+16*a^2*A*b^2-16*A*b^4-9*a^3*b*B+8*a*b^3*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(a^4*(a^2-b^2)*d*sqrt(a+b*sec(c+d*x)))+2/3*(a^4*A-13*a^2*A*b^2+8*A*b^4+8*a^3*b*B-4*a*b^3*B)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(a^3*(a^2-b^2)^2*d*sqrt(sec(c+d*x)))-2/3*(8*a^4*A*b-28*a^2*A*b^3+16*A*b^5-3*a^5*B+15*a^3*b^2*B-8*a*b^4*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt(a+b*sec(c+d*x))/(a^4*(a^2-b^2)^2*d*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x)))],
[(A+B*sec(c+d*x))/(sec(c+d*x)^(5/2)*(a+b*sec(c+d*x))^(5/2)),x,11,2/3*b*(A*b-a*B)*sin(c+d*x)/(a*(a^2-b^2)*d*sec(c+d*x)^(3/2)*(a+b*sec(c+d*x))^(3/2))+2/3*b*(12*a^2*A*b-8*A*b^3-9*a^3*B+5*a*b^2*B)*sin(c+d*x)/(a^2*(a^2-b^2)^2*d*sec(c+d*x)^(3/2)*sqrt(a+b*sec(c+d*x)))-2/15*(17*a^4*A*b+116*a^2*A*b^3-128*A*b^5-5*a^5*B-80*a^3*b^2*B+80*a*b^4*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(a^5*(a^2-b^2)*d*sqrt(a+b*sec(c+d*x)))+2/15*(3*a^4*A-71*a^2*A*b^2+48*A*b^4+50*a^3*b*B-30*a*b^3*B)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(a^3*(a^2-b^2)^2*d*sec(c+d*x)^(3/2))-2/15*(14*a^4*A*b-98*a^2*A*b^3+64*A*b^5-5*a^5*B+65*a^3*b^2*B-40*a*b^4*B)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(a^4*(a^2-b^2)^2*d*sqrt(sec(c+d*x)))+2/15*(9*a^6*A+55*a^4*A*b^2-212*a^2*A*b^4+128*A*b^6-40*a^5*b*B+140*a^3*b^3*B-80*a*b^5*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt(a+b*sec(c+d*x))/(a^5*(a^2-b^2)^2*d*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x)))],

# Integrands of the form Sec[e+f x]^n (a+b Sec[e+f x])^(m/3) (A+B Sec[e+f x])
[(a+b*sec(c+d*x))^(2/3)*(A+B*sec(c+d*x)),x,4,B*AppellF1(1/2,1/2,-2/3,3/2,1/2*(1-sec(c+d*x)),b*(1-sec(c+d*x))/(a+b))*(a+b*sec(c+d*x))^(2/3)*sqrt(2)*tan(c+d*x)/(d*((a+b*sec(c+d*x))/(a+b))^(2/3)*sqrt(1+sec(c+d*x)))+A*Unintegrable((a+b*sec(c+d*x))^(2/3),x)],
[(a+b*sec(c+d*x))^(1/3)*(A+B*sec(c+d*x)),x,4,B*AppellF1(1/2,1/2,-1/3,3/2,1/2*(1-sec(c+d*x)),b*(1-sec(c+d*x))/(a+b))*(a+b*sec(c+d*x))^(1/3)*sqrt(2)*tan(c+d*x)/(d*((a+b*sec(c+d*x))/(a+b))^(1/3)*sqrt(1+sec(c+d*x)))+A*Unintegrable((a+b*sec(c+d*x))^(1/3),x)],
[(A+B*sec(c+d*x))/(a+b*sec(c+d*x))^(1/3),x,4,B*AppellF1(1/2,1/2,1/3,3/2,1/2*(1-sec(c+d*x)),b*(1-sec(c+d*x))/(a+b))*((a+b*sec(c+d*x))/(a+b))^(1/3)*sqrt(2)*tan(c+d*x)/(d*(a+b*sec(c+d*x))^(1/3)*sqrt(1+sec(c+d*x)))+A*Unintegrable(1/(a+b*sec(c+d*x))^(1/3),x)],
[(A+B*sec(c+d*x))/(a+b*sec(c+d*x))^(2/3),x,4,B*AppellF1(1/2,1/2,2/3,3/2,1/2*(1-sec(c+d*x)),b*(1-sec(c+d*x))/(a+b))*((a+b*sec(c+d*x))/(a+b))^(2/3)*sqrt(2)*tan(c+d*x)/(d*(a+b*sec(c+d*x))^(2/3)*sqrt(1+sec(c+d*x)))+A*Unintegrable(1/(a+b*sec(c+d*x))^(2/3),x)],

# Integrands of the form (d Sec[e+f x])^n (a+b Sec[e+f x])^m (A+B Sec[e+f x]) with m and/or n symbolic
[(c*sec(e+f*x))^n*(a+b*sec(e+f*x))^m*(A+B*sec(e+f*x)),x,0,Unintegrable((c*sec(e+f*x))^n*(a+b*sec(e+f*x))^m*(A+B*sec(e+f*x)),x)],
[sec(c+d*x)^m*(a+b*sec(c+d*x))^4*(A+B*sec(c+d*x)),x,9,b*(A*b^3*(8+6*m+m^2)+4*a*b^2*B*(8+6*m+m^2)+2*a^3*B*(19+8*m+m^2)+a^2*A*b*(68+37*m+5*m^2))*sec(c+d*x)^(1+m)*sin(c+d*x)/(d*(4+m)*(3+4*m+m^2))+b^2*(b^2*B*(3+m)^2+2*a*A*b*(4+m)^2+a^2*B*(26+9*m+m^2))*sec(c+d*x)^(2+m)*sin(c+d*x)/(d*(4+m)*(6+5*m+m^2))+b*(A*b*(4+m)+a*B*(7+m))*sec(c+d*x)^(1+m)*(a+b*sec(c+d*x))^2*sin(c+d*x)/(d*(3+m)*(4+m))+b*B*sec(c+d*x)^(1+m)*(a+b*sec(c+d*x))^3*sin(c+d*x)/(d*(4+m))-(A*b^4*m*(2+m)+4*a*b^3*B*m*(2+m)+6*a^2*A*b^2*m*(3+m)+4*a^3*b*B*m*(3+m)+a^4*A*(3+4*m+m^2))*hypergeom([1/2,1/2*(1-m)],[1/2*(3-m)],cos(c+d*x)^2)*sec(c+d*x)^(-1+m)*sin(c+d*x)/(d*(3+m)*(1-m^2)*sqrt(sin(c+d*x)^2))+(b^4*B*(3+4*m+m^2)+4*a*A*b^3*(4+5*m+m^2)+6*a^2*b^2*B*(4+5*m+m^2)+4*a^3*A*b*(8+6*m+m^2)+a^4*B*(8+6*m+m^2))*hypergeom([1/2,-1/2*m],[1/2*(2-m)],cos(c+d*x)^2)*sec(c+d*x)^m*sin(c+d*x)/(d*m*(2+m)*(4+m)*sqrt(sin(c+d*x)^2))],
[sec(c+d*x)^m*(a+b*sec(c+d*x))^3*(A+B*sec(c+d*x)),x,8,b*(b^2*B*(2+m)+3*a*A*b*(3+m)+2*a^2*B*(4+m))*sec(c+d*x)^(1+m)*sin(c+d*x)/(d*(1+m)*(3+m))+b^2*(A*b*(3+m)+a*B*(5+m))*sec(c+d*x)^(2+m)*sin(c+d*x)/(d*(2+m)*(3+m))+b*B*sec(c+d*x)^(1+m)*(a+b*sec(c+d*x))^2*sin(c+d*x)/(d*(3+m))-(b^3*B*m*(2+m)+3*a*A*b^2*m*(3+m)+3*a^2*b*B*m*(3+m)+a^3*A*(3+4*m+m^2))*hypergeom([1/2,1/2*(1-m)],[1/2*(3-m)],cos(c+d*x)^2)*sec(c+d*x)^(-1+m)*sin(c+d*x)/(d*(3+m)*(1-m^2)*sqrt(sin(c+d*x)^2))+(A*b^3*(1+m)+3*a*b^2*B*(1+m)+3*a^2*A*b*(2+m)+a^3*B*(2+m))*hypergeom([1/2,-1/2*m],[1/2*(2-m)],cos(c+d*x)^2)*sec(c+d*x)^m*sin(c+d*x)/(d*m*(2+m)*sqrt(sin(c+d*x)^2))],
[sec(c+d*x)^m*(a+b*sec(c+d*x))^2*(A+B*sec(c+d*x)),x,7,b*(A*b*(2+m)+a*B*(3+m))*sec(c+d*x)^(1+m)*sin(c+d*x)/(d*(1+m)*(2+m))+b*B*sec(c+d*x)^(1+m)*(a+b*sec(c+d*x))*sin(c+d*x)/(d*(2+m))-(A*b^2*m+2*a*b*B*m+a^2*A*(1+m))*hypergeom([1/2,1/2*(1-m)],[1/2*(3-m)],cos(c+d*x)^2)*sec(c+d*x)^(-1+m)*sin(c+d*x)/(d*(1-m^2)*sqrt(sin(c+d*x)^2))+(b^2*B*(1+m)+a*(2*A*b+a*B)*(2+m))*hypergeom([1/2,-1/2*m],[1/2*(2-m)],cos(c+d*x)^2)*sec(c+d*x)^m*sin(c+d*x)/(d*m*(2+m)*sqrt(sin(c+d*x)^2))],
[sec(c+d*x)^m*(a+b*sec(c+d*x))*(A+B*sec(c+d*x)),x,6,b*B*sec(c+d*x)^(1+m)*sin(c+d*x)/(d*(1+m))-(b*B*m+a*A*(1+m))*hypergeom([1/2,1/2*(1-m)],[1/2*(3-m)],cos(c+d*x)^2)*sec(c+d*x)^(-1+m)*sin(c+d*x)/(d*(1-m^2)*sqrt(sin(c+d*x)^2))+(A*b+a*B)*hypergeom([1/2,-1/2*m],[1/2*(2-m)],cos(c+d*x)^2)*sec(c+d*x)^m*sin(c+d*x)/(d*m*sqrt(sin(c+d*x)^2))],

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

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

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

# n>0
[cos(c+d*x)^(7/2)*(a+a*sec(c+d*x))*(A+B*sec(c+d*x)),x,8,6/5*a*(A+B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+2/21*a*(5*A+7*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+2/5*a*(A+B)*cos(c+d*x)^(3/2)*sin(c+d*x)/d+2/7*a*A*cos(c+d*x)^(5/2)*sin(c+d*x)/d+2/21*a*(5*A+7*B)*sin(c+d*x)*sqrt(cos(c+d*x))/d],
[cos(c+d*x)^(5/2)*(a+a*sec(c+d*x))*(A+B*sec(c+d*x)),x,7,2/5*a*(3*A+5*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+2/3*a*(A+B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+2/5*a*A*cos(c+d*x)^(3/2)*sin(c+d*x)/d+2/3*a*(A+B)*sin(c+d*x)*sqrt(cos(c+d*x))/d],
[cos(c+d*x)^(3/2)*(a+a*sec(c+d*x))*(A+B*sec(c+d*x)),x,6,2*a*(A+B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+2/3*a*(A+3*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+2/3*a*A*sin(c+d*x)*sqrt(cos(c+d*x))/d],
[(a+a*sec(c+d*x))*(A+B*sec(c+d*x))*sqrt(cos(c+d*x)),x,6,2*a*(A-B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+2*a*(A+B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+2*a*B*sin(c+d*x)/(d*sqrt(cos(c+d*x)))],
[(a+a*sec(c+d*x))*(A+B*sec(c+d*x))/sqrt(cos(c+d*x)),x,7,-2*a*(A+B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+2/3*a*(3*A+B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+2/3*a*B*sin(c+d*x)/(d*cos(c+d*x)^(3/2))+2*a*(A+B)*sin(c+d*x)/(d*sqrt(cos(c+d*x)))],
[(a+a*sec(c+d*x))*(A+B*sec(c+d*x))/cos(c+d*x)^(3/2),x,8,-2/5*a*(5*A+3*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+2/3*a*(A+B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+2/5*a*B*sin(c+d*x)/(d*cos(c+d*x)^(5/2))+2/3*a*(A+B)*sin(c+d*x)/(d*cos(c+d*x)^(3/2))+2/5*a*(5*A+3*B)*sin(c+d*x)/(d*sqrt(cos(c+d*x)))],
[cos(c+d*x)^(9/2)*(a+a*sec(c+d*x))^2*(A+B*sec(c+d*x)),x,9,4/15*a^2*(8*A+9*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+4/21*a^2*(5*A+6*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+4/45*a^2*(8*A+9*B)*cos(c+d*x)^(3/2)*sin(c+d*x)/d+2/63*a^2*(11*A+9*B)*cos(c+d*x)^(5/2)*sin(c+d*x)/d+2/9*A*cos(c+d*x)^(5/2)*(a^2+a^2*cos(c+d*x))*sin(c+d*x)/d+4/21*a^2*(5*A+6*B)*sin(c+d*x)*sqrt(cos(c+d*x))/d],
[cos(c+d*x)^(7/2)*(a+a*sec(c+d*x))^2*(A+B*sec(c+d*x)),x,8,4/5*a^2*(3*A+4*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+4/21*a^2*(6*A+7*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+2/35*a^2*(9*A+7*B)*cos(c+d*x)^(3/2)*sin(c+d*x)/d+2/7*A*cos(c+d*x)^(3/2)*(a^2+a^2*cos(c+d*x))*sin(c+d*x)/d+4/21*a^2*(6*A+7*B)*sin(c+d*x)*sqrt(cos(c+d*x))/d],
[cos(c+d*x)^(5/2)*(a+a*sec(c+d*x))^2*(A+B*sec(c+d*x)),x,7,4/5*a^2*(4*A+5*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+4/3*a^2*(A+2*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+2/15*a^2*(7*A+5*B)*sin(c+d*x)*sqrt(cos(c+d*x))/d+2/5*A*(a^2+a^2*cos(c+d*x))*sin(c+d*x)*sqrt(cos(c+d*x))/d],
[cos(c+d*x)^(3/2)*(a+a*sec(c+d*x))^2*(A+B*sec(c+d*x)),x,7,4*a^2*A*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+4/3*a^2*(2*A+3*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+2*B*(a^2+a^2*cos(c+d*x))*sin(c+d*x)/(d*sqrt(cos(c+d*x)))+2/3*a^2*(A-3*B)*sin(c+d*x)*sqrt(cos(c+d*x))/d],
[(a+a*sec(c+d*x))^2*(A+B*sec(c+d*x))*sqrt(cos(c+d*x)),x,7,-4*a^2*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+4/3*a^2*(3*A+2*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+2/3*B*(a^2+a^2*cos(c+d*x))*sin(c+d*x)/(d*cos(c+d*x)^(3/2))+2/3*a^2*(3*A+5*B)*sin(c+d*x)/(d*sqrt(cos(c+d*x)))],
[(a+a*sec(c+d*x))^2*(A+B*sec(c+d*x))/sqrt(cos(c+d*x)),x,8,-4/5*a^2*(5*A+4*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+4/3*a^2*(2*A+B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+2/15*a^2*(5*A+7*B)*sin(c+d*x)/(d*cos(c+d*x)^(3/2))+2/5*B*(a^2+a^2*cos(c+d*x))*sin(c+d*x)/(d*cos(c+d*x)^(5/2))+4/5*a^2*(5*A+4*B)*sin(c+d*x)/(d*sqrt(cos(c+d*x)))],
[(a+a*sec(c+d*x))^2*(A+B*sec(c+d*x))/cos(c+d*x)^(3/2),x,9,-4/5*a^2*(4*A+3*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+4/21*a^2*(7*A+6*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+2/35*a^2*(7*A+9*B)*sin(c+d*x)/(d*cos(c+d*x)^(5/2))+4/21*a^2*(7*A+6*B)*sin(c+d*x)/(d*cos(c+d*x)^(3/2))+2/7*B*(a^2+a^2*cos(c+d*x))*sin(c+d*x)/(d*cos(c+d*x)^(7/2))+4/5*a^2*(4*A+3*B)*sin(c+d*x)/(d*sqrt(cos(c+d*x)))],

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

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

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

# n<0
[cos(c+d*x)^(7/2)*(A+B*sec(c+d*x))/sqrt(a+a*sec(c+d*x)),x,7,(A-B)*arctanh(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*sec(c+d*x))))*sqrt(2)*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(d*sqrt(a))-2/35*(A-7*B)*cos(c+d*x)^(3/2)*sin(c+d*x)/(d*sqrt(a+a*sec(c+d*x)))+2/7*A*cos(c+d*x)^(5/2)*sin(c+d*x)/(d*sqrt(a+a*sec(c+d*x)))-2/105*(43*A-91*B)*sin(c+d*x)/(d*sqrt(cos(c+d*x))*sqrt(a+a*sec(c+d*x)))+2/105*(31*A-7*B)*sin(c+d*x)*sqrt(cos(c+d*x))/(d*sqrt(a+a*sec(c+d*x)))],
[cos(c+d*x)^(5/2)*(A+B*sec(c+d*x))/sqrt(a+a*sec(c+d*x)),x,6,-(A-B)*arctanh(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*sec(c+d*x))))*sqrt(2)*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(d*sqrt(a))+2/5*A*cos(c+d*x)^(3/2)*sin(c+d*x)/(d*sqrt(a+a*sec(c+d*x)))+2/15*(13*A-5*B)*sin(c+d*x)/(d*sqrt(cos(c+d*x))*sqrt(a+a*sec(c+d*x)))-2/15*(A-5*B)*sin(c+d*x)*sqrt(cos(c+d*x))/(d*sqrt(a+a*sec(c+d*x)))],
[cos(c+d*x)^(3/2)*(A+B*sec(c+d*x))/sqrt(a+a*sec(c+d*x)),x,5,(A-B)*arctanh(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*sec(c+d*x))))*sqrt(2)*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(d*sqrt(a))-2/3*(A-3*B)*sin(c+d*x)/(d*sqrt(cos(c+d*x))*sqrt(a+a*sec(c+d*x)))+2/3*A*sin(c+d*x)*sqrt(cos(c+d*x))/(d*sqrt(a+a*sec(c+d*x)))],
[(A+B*sec(c+d*x))*sqrt(cos(c+d*x))/sqrt(a+a*sec(c+d*x)),x,4,-(A-B)*arctanh(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*sec(c+d*x))))*sqrt(2)*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(d*sqrt(a))+2*A*sin(c+d*x)/(d*sqrt(cos(c+d*x))*sqrt(a+a*sec(c+d*x)))],
[(A+B*sec(c+d*x))/(sqrt(cos(c+d*x))*sqrt(a+a*sec(c+d*x))),x,6,2*B*arcsinh(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(d*sqrt(a))+(A-B)*arctanh(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*sec(c+d*x))))*sqrt(2)*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(d*sqrt(a))],
[(A+B*sec(c+d*x))/(cos(c+d*x)^(3/2)*sqrt(a+a*sec(c+d*x))),x,7,(2*A-B)*arcsinh(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(d*sqrt(a))-(A-B)*arctanh(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*sec(c+d*x))))*sqrt(2)*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(d*sqrt(a))+B*sin(c+d*x)/(d*cos(c+d*x)^(3/2)*sqrt(a+a*sec(c+d*x)))],
[(A+B*sec(c+d*x))/(cos(c+d*x)^(5/2)*sqrt(a+a*sec(c+d*x))),x,8,-1/4*(4*A-7*B)*arcsinh(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(d*sqrt(a))+(A-B)*arctanh(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*sec(c+d*x))))*sqrt(2)*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(d*sqrt(a))+1/2*B*sin(c+d*x)/(d*cos(c+d*x)^(5/2)*sqrt(a+a*sec(c+d*x)))+1/4*(4*A-B)*sin(c+d*x)/(d*cos(c+d*x)^(3/2)*sqrt(a+a*sec(c+d*x)))],
[cos(c+d*x)^(5/2)*(A+B*sec(c+d*x))/(a+a*sec(c+d*x))^(3/2),x,7,-1/2*(A-B)*cos(c+d*x)^(3/2)*sin(c+d*x)/(d*(a+a*sec(c+d*x))^(3/2))-1/2*(15*A-11*B)*arctanh(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*sec(c+d*x))))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(3/2)*d*sqrt(2))+1/10*(9*A-5*B)*cos(c+d*x)^(3/2)*sin(c+d*x)/(a*d*sqrt(a+a*sec(c+d*x)))+1/30*(147*A-95*B)*sin(c+d*x)/(a*d*sqrt(cos(c+d*x))*sqrt(a+a*sec(c+d*x)))-1/30*(39*A-35*B)*sin(c+d*x)*sqrt(cos(c+d*x))/(a*d*sqrt(a+a*sec(c+d*x)))],
[cos(c+d*x)^(3/2)*(A+B*sec(c+d*x))/(a+a*sec(c+d*x))^(3/2),x,6,-1/2*(A-B)*sin(c+d*x)*sqrt(cos(c+d*x))/(d*(a+a*sec(c+d*x))^(3/2))+1/2*(11*A-7*B)*arctanh(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*sec(c+d*x))))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(3/2)*d*sqrt(2))-1/6*(19*A-15*B)*sin(c+d*x)/(a*d*sqrt(cos(c+d*x))*sqrt(a+a*sec(c+d*x)))+1/6*(7*A-3*B)*sin(c+d*x)*sqrt(cos(c+d*x))/(a*d*sqrt(a+a*sec(c+d*x)))],
[(A+B*sec(c+d*x))*sqrt(cos(c+d*x))/(a+a*sec(c+d*x))^(3/2),x,5,-1/2*(A-B)*sin(c+d*x)/(d*(a+a*sec(c+d*x))^(3/2)*sqrt(cos(c+d*x)))-1/2*(7*A-3*B)*arctanh(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*sec(c+d*x))))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(3/2)*d*sqrt(2))+1/2*(5*A-B)*sin(c+d*x)/(a*d*sqrt(cos(c+d*x))*sqrt(a+a*sec(c+d*x)))],
[(A+B*sec(c+d*x))/((a+a*sec(c+d*x))^(3/2)*sqrt(cos(c+d*x))),x,4,-1/2*(A-B)*sin(c+d*x)/(d*cos(c+d*x)^(3/2)*(a+a*sec(c+d*x))^(3/2))+1/2*(3*A+B)*arctanh(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*sec(c+d*x))))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(3/2)*d*sqrt(2))],
[(A+B*sec(c+d*x))/(cos(c+d*x)^(3/2)*(a+a*sec(c+d*x))^(3/2)),x,7,1/2*(A-B)*sin(c+d*x)/(d*cos(c+d*x)^(3/2)*(a+a*sec(c+d*x))^(3/2))+2*B*arcsinh(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(3/2)*d)+1/2*(A-5*B)*arctanh(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*sec(c+d*x))))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(3/2)*d*sqrt(2))],
[(A+B*sec(c+d*x))/(cos(c+d*x)^(5/2)*(a+a*sec(c+d*x))^(3/2)),x,8,1/2*(A-B)*sin(c+d*x)/(d*cos(c+d*x)^(5/2)*(a+a*sec(c+d*x))^(3/2))+(2*A-3*B)*arcsinh(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(3/2)*d)-1/2*(5*A-9*B)*arctanh(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*sec(c+d*x))))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(3/2)*d*sqrt(2))-1/2*(A-3*B)*sin(c+d*x)/(a*d*cos(c+d*x)^(3/2)*sqrt(a+a*sec(c+d*x)))],
[(A+B*sec(c+d*x))/(cos(c+d*x)^(7/2)*(a+a*sec(c+d*x))^(3/2)),x,9,1/2*(A-B)*sin(c+d*x)/(d*cos(c+d*x)^(7/2)*(a+a*sec(c+d*x))^(3/2))-1/4*(12*A-19*B)*arcsinh(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(3/2)*d)+1/2*(9*A-13*B)*arctanh(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*sec(c+d*x))))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(3/2)*d*sqrt(2))-1/2*(A-2*B)*sin(c+d*x)/(a*d*cos(c+d*x)^(5/2)*sqrt(a+a*sec(c+d*x)))+1/4*(6*A-7*B)*sin(c+d*x)/(a*d*cos(c+d*x)^(3/2)*sqrt(a+a*sec(c+d*x)))],
[cos(c+d*x)^(5/2)*(A+B*sec(c+d*x))/(a+a*sec(c+d*x))^(5/2),x,8,-1/4*(A-B)*cos(c+d*x)^(3/2)*sin(c+d*x)/(d*(a+a*sec(c+d*x))^(5/2))-1/16*(21*A-13*B)*cos(c+d*x)^(3/2)*sin(c+d*x)/(a*d*(a+a*sec(c+d*x))^(3/2))-1/16*(283*A-163*B)*arctanh(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*sec(c+d*x))))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(5/2)*d*sqrt(2))+1/80*(157*A-85*B)*cos(c+d*x)^(3/2)*sin(c+d*x)/(a^2*d*sqrt(a+a*sec(c+d*x)))+1/240*(2671*A-1495*B)*sin(c+d*x)/(a^2*d*sqrt(cos(c+d*x))*sqrt(a+a*sec(c+d*x)))-1/240*(787*A-475*B)*sin(c+d*x)*sqrt(cos(c+d*x))/(a^2*d*sqrt(a+a*sec(c+d*x)))],
[cos(c+d*x)^(3/2)*(A+B*sec(c+d*x))/(a+a*sec(c+d*x))^(5/2),x,7,-1/4*(A-B)*sin(c+d*x)*sqrt(cos(c+d*x))/(d*(a+a*sec(c+d*x))^(5/2))-1/16*(17*A-9*B)*sin(c+d*x)*sqrt(cos(c+d*x))/(a*d*(a+a*sec(c+d*x))^(3/2))+1/16*(163*A-75*B)*arctanh(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*sec(c+d*x))))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(5/2)*d*sqrt(2))-1/48*(299*A-147*B)*sin(c+d*x)/(a^2*d*sqrt(cos(c+d*x))*sqrt(a+a*sec(c+d*x)))+1/48*(95*A-39*B)*sin(c+d*x)*sqrt(cos(c+d*x))/(a^2*d*sqrt(a+a*sec(c+d*x)))],
[(A+B*sec(c+d*x))*sqrt(cos(c+d*x))/(a+a*sec(c+d*x))^(5/2),x,6,-1/4*(A-B)*sin(c+d*x)/(d*(a+a*sec(c+d*x))^(5/2)*sqrt(cos(c+d*x)))-1/16*(13*A-5*B)*sin(c+d*x)/(a*d*(a+a*sec(c+d*x))^(3/2)*sqrt(cos(c+d*x)))-1/16*(75*A-19*B)*arctanh(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*sec(c+d*x))))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(5/2)*d*sqrt(2))+1/16*(49*A-9*B)*sin(c+d*x)/(a^2*d*sqrt(cos(c+d*x))*sqrt(a+a*sec(c+d*x)))],
[(A+B*sec(c+d*x))/((a+a*sec(c+d*x))^(5/2)*sqrt(cos(c+d*x))),x,6,1/4*(A-B)*sin(c+d*x)/(d*(a+a*sec(c+d*x))^(5/2)*sqrt(cos(c+d*x)))+1/16*(5*A+3*B)*sin(c+d*x)/(a*d*(a+a*sec(c+d*x))^(3/2)*sqrt(cos(c+d*x)))+1/16*(19*A+5*B)*arctanh(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*sec(c+d*x))))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(5/2)*d*sqrt(2))-1/16*(9*A-B)*sin(c+d*x)/(a^2*d*sqrt(cos(c+d*x))*sqrt(a+a*sec(c+d*x)))],
[(A+B*sec(c+d*x))/(cos(c+d*x)^(3/2)*(a+a*sec(c+d*x))^(5/2)),x,5,-1/4*(A-B)*sin(c+d*x)/(d*cos(c+d*x)^(5/2)*(a+a*sec(c+d*x))^(5/2))+1/16*(5*A+3*B)*sin(c+d*x)/(a*d*cos(c+d*x)^(3/2)*(a+a*sec(c+d*x))^(3/2))+1/16*(5*A+3*B)*arctanh(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*sec(c+d*x))))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(5/2)*d*sqrt(2))],
[(A+B*sec(c+d*x))/(cos(c+d*x)^(5/2)*(a+a*sec(c+d*x))^(5/2)),x,8,1/4*(A-B)*sin(c+d*x)/(d*cos(c+d*x)^(5/2)*(a+a*sec(c+d*x))^(5/2))+1/16*(3*A-11*B)*sin(c+d*x)/(a*d*cos(c+d*x)^(3/2)*(a+a*sec(c+d*x))^(3/2))+2*B*arcsinh(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(5/2)*d)+1/16*(3*A-43*B)*arctanh(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*sec(c+d*x))))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(5/2)*d*sqrt(2))],
[(A+B*sec(c+d*x))/(cos(c+d*x)^(7/2)*(a+a*sec(c+d*x))^(5/2)),x,9,1/4*(A-B)*sin(c+d*x)/(d*cos(c+d*x)^(7/2)*(a+a*sec(c+d*x))^(5/2))+1/16*(7*A-15*B)*sin(c+d*x)/(a*d*cos(c+d*x)^(5/2)*(a+a*sec(c+d*x))^(3/2))+(2*A-5*B)*arcsinh(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(5/2)*d)-1/16*(43*A-115*B)*arctanh(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*sec(c+d*x))))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(5/2)*d*sqrt(2))-1/16*(11*A-35*B)*sin(c+d*x)/(a^2*d*cos(c+d*x)^(3/2)*sqrt(a+a*sec(c+d*x)))],

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

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

# n>0
[cos(c+d*x)^(7/2)*(a+b*sec(c+d*x))*(A+B*sec(c+d*x)),x,8,6/5*(A*b+a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+2/21*(5*a*A+7*b*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+2/5*(A*b+a*B)*cos(c+d*x)^(3/2)*sin(c+d*x)/d+2/7*a*A*cos(c+d*x)^(5/2)*sin(c+d*x)/d+2/21*(5*a*A+7*b*B)*sin(c+d*x)*sqrt(cos(c+d*x))/d],
[cos(c+d*x)^(5/2)*(a+b*sec(c+d*x))*(A+B*sec(c+d*x)),x,7,2/5*(3*a*A+5*b*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+2/3*(A*b+a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+2/5*a*A*cos(c+d*x)^(3/2)*sin(c+d*x)/d+2/3*(A*b+a*B)*sin(c+d*x)*sqrt(cos(c+d*x))/d],
[cos(c+d*x)^(3/2)*(a+b*sec(c+d*x))*(A+B*sec(c+d*x)),x,6,2*(A*b+a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+2/3*(a*A+3*b*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+2/3*a*A*sin(c+d*x)*sqrt(cos(c+d*x))/d],
[cos(c+d*x)^(1/2)*(a+b*sec(c+d*x))*(A+B*sec(c+d*x)),x,6,2*(a*A-b*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+2*(A*b+a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+2*b*B*sin(c+d*x)/(d*sqrt(cos(c+d*x)))],
[(a+b*sec(c+d*x))*(A+B*sec(c+d*x))/cos(c+d*x)^(1/2),x,7,-2*(A*b+a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+2/3*(3*a*A+b*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+2/3*b*B*sin(c+d*x)/(d*cos(c+d*x)^(3/2))+2*(A*b+a*B)*sin(c+d*x)/(d*sqrt(cos(c+d*x)))],
[(a+b*sec(c+d*x))*(A+B*sec(c+d*x))/cos(c+d*x)^(3/2),x,8,-2/5*(5*a*A+3*b*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+2/3*(A*b+a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+2/5*b*B*sin(c+d*x)/(d*cos(c+d*x)^(5/2))+2/3*(A*b+a*B)*sin(c+d*x)/(d*cos(c+d*x)^(3/2))+2/5*(5*a*A+3*b*B)*sin(c+d*x)/(d*sqrt(cos(c+d*x)))],
[cos(c+d*x)^(7/2)*(a+b*sec(c+d*x))^2*(A+B*sec(c+d*x)),x,7,2/5*(6*a*A*b+3*a^2*B+5*b^2*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+2/21*(5*a^2*A+7*b*(A*b+2*a*B))*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+2/35*a*(9*A*b+7*a*B)*cos(c+d*x)^(3/2)*sin(c+d*x)/d+2/7*a*A*cos(c+d*x)^(3/2)*(b+a*cos(c+d*x))*sin(c+d*x)/d+2/21*(5*a^2*A+7*b*(A*b+2*a*B))*sin(c+d*x)*sqrt(cos(c+d*x))/d],
[cos(c+d*x)^(5/2)*(a+b*sec(c+d*x))^2*(A+B*sec(c+d*x)),x,6,2/5*(3*a^2*A+5*b*(A*b+2*a*B))*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+2/3*(2*a*A*b+a^2*B+3*b^2*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+2/15*a*(7*A*b+5*a*B)*sin(c+d*x)*sqrt(cos(c+d*x))/d+2/5*a*A*(b+a*cos(c+d*x))*sin(c+d*x)*sqrt(cos(c+d*x))/d],
[cos(c+d*x)^(3/2)*(a+b*sec(c+d*x))^2*(A+B*sec(c+d*x)),x,6,2*(2*a*A*b+a^2*B-b^2*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+2/3*(a^2*A+3*A*b^2+6*a*b*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+2*b^2*B*sin(c+d*x)/(d*sqrt(cos(c+d*x)))+2/3*a^2*A*sin(c+d*x)*sqrt(cos(c+d*x))/d],
[cos(c+d*x)^(1/2)*(a+b*sec(c+d*x))^2*(A+B*sec(c+d*x)),x,6,2*(a^2*A-A*b^2-2*a*b*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+2/3*(6*a*A*b+3*a^2*B+b^2*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+2/3*b^2*B*sin(c+d*x)/(d*cos(c+d*x)^(3/2))+2*b*(A*b+2*a*B)*sin(c+d*x)/(d*sqrt(cos(c+d*x)))],
[(a+b*sec(c+d*x))^2*(A+B*sec(c+d*x))/cos(c+d*x)^(1/2),x,7,-2/5*(10*a*A*b+5*a^2*B+3*b^2*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+2/3*(3*a^2*A+A*b^2+2*a*b*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+2/5*b^2*B*sin(c+d*x)/(d*cos(c+d*x)^(5/2))+2/3*b*(A*b+2*a*B)*sin(c+d*x)/(d*cos(c+d*x)^(3/2))+2/5*(10*a*A*b+5*a^2*B+3*b^2*B)*sin(c+d*x)/(d*sqrt(cos(c+d*x)))],
[(a+b*sec(c+d*x))^2*(A+B*sec(c+d*x))/cos(c+d*x)^(3/2),x,8,-2/5*(5*a^2*A+3*A*b^2+6*a*b*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+2/21*(14*a*A*b+7*a^2*B+5*b^2*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+2/7*b^2*B*sin(c+d*x)/(d*cos(c+d*x)^(7/2))+2/5*b*(A*b+2*a*B)*sin(c+d*x)/(d*cos(c+d*x)^(5/2))+2/21*(14*a*A*b+7*a^2*B+5*b^2*B)*sin(c+d*x)/(d*cos(c+d*x)^(3/2))+2/5*(5*a^2*A+3*A*b^2+6*a*b*B)*sin(c+d*x)/(d*sqrt(cos(c+d*x)))],

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

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

# n>0
[cos(c+d*x)^(7/2)*(A+B*sec(c+d*x))*sqrt(a+b*sec(c+d*x)),x,11,2/105*(a^2-b^2)*(25*a^2*A+8*A*b^2-14*a*b*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(a^3*d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+2/35*(A*b+7*a*B)*cos(c+d*x)^(3/2)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(a*d)+2/7*A*cos(c+d*x)^(5/2)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/d+2/105*(25*a^2*A-4*A*b^2+7*a*b*B)*sin(c+d*x)*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(a^2*d)+2/105*(19*a^2*A*b+8*A*b^3+63*a^3*B-14*a*b^2*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(a^3*d*sqrt((b+a*cos(c+d*x))/(a+b)))],
[cos(c+d*x)^(5/2)*(A+B*sec(c+d*x))*sqrt(a+b*sec(c+d*x)),x,10,-2/15*(a^2-b^2)*(2*A*b-5*a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(a^2*d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+2/5*A*cos(c+d*x)^(3/2)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/d+2/15*(A*b+5*a*B)*sin(c+d*x)*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(a*d)+2/15*(9*a^2*A-2*A*b^2+5*a*b*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(a^2*d*sqrt((b+a*cos(c+d*x))/(a+b)))],
[cos(c+d*x)^(3/2)*(A+B*sec(c+d*x))*sqrt(a+b*sec(c+d*x)),x,9,2/3*A*(a^2-b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(a*d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+2/3*A*sin(c+d*x)*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/d+2/3*(A*b+3*a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(a*d*sqrt((b+a*cos(c+d*x))/(a+b)))],
[(A+B*sec(c+d*x))*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)),x,12,2*a*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+2*b*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+2*A*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(d*sqrt((b+a*cos(c+d*x))/(a+b)))],
[(A+B*sec(c+d*x))*sqrt(a+b*sec(c+d*x))/sqrt(cos(c+d*x)),x,13,(2*a*A+b*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+(2*A*b+a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+B*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(d*sqrt(cos(c+d*x)))-B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(d*sqrt((b+a*cos(c+d*x))/(a+b)))],
[(A+B*sec(c+d*x))*sqrt(a+b*sec(c+d*x))/cos(c+d*x)^(3/2),x,14,1/4*(4*A*b+3*a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+1/4*(4*a*A*b-a^2*B+4*b^2*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(b*d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+1/2*B*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(d*cos(c+d*x)^(3/2))+1/4*(4*A*b+a*B)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(b*d*sqrt(cos(c+d*x)))-1/4*(4*A*b+a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(b*d*sqrt((b+a*cos(c+d*x))/(a+b)))],
[cos(c+d*x)^(9/2)*(a+b*sec(c+d*x))^(3/2)*(A+B*sec(c+d*x)),x,12,2/315*(a^2-b^2)*(39*a^2*A*b+8*A*b^3+75*a^3*B-18*a*b^2*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(a^3*d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+2/315*(49*a^2*A+3*A*b^2+72*a*b*B)*cos(c+d*x)^(3/2)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(a*d)+2/63*(10*A*b+9*a*B)*cos(c+d*x)^(5/2)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/d+2/9*a*A*cos(c+d*x)^(7/2)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/d+2/315*(88*a^2*A*b-4*A*b^3+75*a^3*B+9*a*b^2*B)*sin(c+d*x)*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(a^2*d)+2/315*(147*a^4*A+33*a^2*A*b^2+8*A*b^4+246*a^3*b*B-18*a*b^3*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(a^3*d*sqrt((b+a*cos(c+d*x))/(a+b)))],
[cos(c+d*x)^(7/2)*(a+b*sec(c+d*x))^(3/2)*(A+B*sec(c+d*x)),x,11,2/105*(a^2-b^2)*(25*a^2*A-6*A*b^2+21*a*b*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(a^2*d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+2/35*(8*A*b+7*a*B)*cos(c+d*x)^(3/2)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/d+2/7*a*A*cos(c+d*x)^(5/2)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/d+2/105*(25*a^2*A+3*A*b^2+42*a*b*B)*sin(c+d*x)*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(a*d)+2/105*(82*a^2*A*b-6*A*b^3+63*a^3*B+21*a*b^2*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(a^2*d*sqrt((b+a*cos(c+d*x))/(a+b)))],
[cos(c+d*x)^(5/2)*(a+b*sec(c+d*x))^(3/2)*(A+B*sec(c+d*x)),x,10,2/15*(a^2-b^2)*(3*A*b+5*a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(a*d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+2/5*a*A*cos(c+d*x)^(3/2)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/d+2/15*(6*A*b+5*a*B)*sin(c+d*x)*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/d+2/15*(9*a^2*A+3*A*b^2+20*a*b*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(a*d*sqrt((b+a*cos(c+d*x))/(a+b)))],
[cos(c+d*x)^(3/2)*(a+b*sec(c+d*x))^(3/2)*(A+B*sec(c+d*x)),x,13,2/3*(a^2*A-A*b^2+3*a*b*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+2*b^2*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+2/3*a*A*sin(c+d*x)*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/d+2/3*(4*A*b+3*a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(d*sqrt((b+a*cos(c+d*x))/(a+b)))],
[(a+b*sec(c+d*x))^(3/2)*(A+B*sec(c+d*x))*sqrt(cos(c+d*x)),x,13,(2*a*A*b+2*a^2*B+b^2*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+b*(2*A*b+3*a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+b*B*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(d*sqrt(cos(c+d*x)))+(2*a*A-b*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(d*sqrt((b+a*cos(c+d*x))/(a+b)))],
[(a+b*sec(c+d*x))^(3/2)*(A+B*sec(c+d*x))/sqrt(cos(c+d*x)),x,14,1/4*(8*a^2*A+4*A*b^2+7*a*b*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+1/4*(12*a*A*b+3*a^2*B+4*b^2*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+1/2*b*B*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(d*cos(c+d*x)^(3/2))+1/4*(4*A*b+5*a*B)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(d*sqrt(cos(c+d*x)))-1/4*(4*A*b+5*a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(d*sqrt((b+a*cos(c+d*x))/(a+b)))],
[(a+b*sec(c+d*x))^(3/2)*(A+B*sec(c+d*x))/cos(c+d*x)^(3/2),x,15,1/24*(42*a*A*b+17*a^2*B+16*b^2*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+1/8*(6*a^2*A*b+8*A*b^3-a^3*B+12*a*b^2*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(b*d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+1/3*b*B*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(d*cos(c+d*x)^(5/2))+1/12*(6*A*b+7*a*B)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(d*cos(c+d*x)^(3/2))+1/24*(30*a*A*b+3*a^2*B+16*b^2*B)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(b*d*sqrt(cos(c+d*x)))-1/24*(30*a*A*b+3*a^2*B+16*b^2*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(b*d*sqrt((b+a*cos(c+d*x))/(a+b)))],
[cos(c+d*x)^(11/2)*(a+b*sec(c+d*x))^(5/2)*(A+B*sec(c+d*x)),x,13,2/11*a*A*cos(c+d*x)^(9/2)*(a+b*sec(c+d*x))^(3/2)*sin(c+d*x)/d+2/3465*(a^2-b^2)*(675*a^4*A+285*a^2*A*b^2+40*A*b^4+1254*a^3*b*B-110*a*b^3*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(a^3*d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+2/3465*(1145*a^2*A*b+15*A*b^3+539*a^3*B+825*a*b^2*B)*cos(c+d*x)^(3/2)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(a*d)+2/693*(81*a^2*A+113*A*b^2+209*a*b*B)*cos(c+d*x)^(5/2)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/d+2/99*a*(14*A*b+11*a*B)*cos(c+d*x)^(7/2)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/d+2/3465*(675*a^4*A+1025*a^2*A*b^2-20*A*b^4+1793*a^3*b*B+55*a*b^3*B)*sin(c+d*x)*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(a^2*d)+2/3465*(3705*a^4*A*b+255*a^2*A*b^3+40*A*b^5+1617*a^5*B+3069*a^3*b^2*B-110*a*b^4*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(a^3*d*sqrt((b+a*cos(c+d*x))/(a+b)))],
[cos(c+d*x)^(9/2)*(a+b*sec(c+d*x))^(5/2)*(A+B*sec(c+d*x)),x,12,2/9*a*A*cos(c+d*x)^(7/2)*(a+b*sec(c+d*x))^(3/2)*sin(c+d*x)/d+2/315*(a^2-b^2)*(114*a^2*A*b-10*A*b^3+75*a^3*B+45*a*b^2*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(a^2*d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+2/315*(49*a^2*A+75*A*b^2+135*a*b*B)*cos(c+d*x)^(3/2)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/d+2/21*a*(4*A*b+3*a*B)*cos(c+d*x)^(5/2)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/d+2/315*(163*a^2*A*b+5*A*b^3+75*a^3*B+135*a*b^2*B)*sin(c+d*x)*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(a*d)+2/315*(147*a^4*A+279*a^2*A*b^2-10*A*b^4+435*a^3*b*B+45*a*b^3*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(a^2*d*sqrt((b+a*cos(c+d*x))/(a+b)))],
[cos(c+d*x)^(7/2)*(a+b*sec(c+d*x))^(5/2)*(A+B*sec(c+d*x)),x,11,2/7*a*A*cos(c+d*x)^(5/2)*(a+b*sec(c+d*x))^(3/2)*sin(c+d*x)/d+2/105*(a^2-b^2)*(25*a^2*A+15*A*b^2+56*a*b*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(a*d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+2/35*a*(10*A*b+7*a*B)*cos(c+d*x)^(3/2)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/d+2/105*(25*a^2*A+45*A*b^2+77*a*b*B)*sin(c+d*x)*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/d+2/105*(145*a^2*A*b+15*A*b^3+63*a^3*B+161*a*b^2*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(a*d*sqrt((b+a*cos(c+d*x))/(a+b)))],
[cos(c+d*x)^(5/2)*(a+b*sec(c+d*x))^(5/2)*(A+B*sec(c+d*x)),x,14,2/5*a*A*cos(c+d*x)^(3/2)*(a+b*sec(c+d*x))^(3/2)*sin(c+d*x)/d+2/15*(8*a^2*A*b-8*A*b^3+5*a^3*B+10*a*b^2*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+2*b^3*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+2/15*a*(8*A*b+5*a*B)*sin(c+d*x)*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/d+2/15*(9*a^2*A+23*A*b^2+35*a*b*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(d*sqrt((b+a*cos(c+d*x))/(a+b)))],
[cos(c+d*x)^(3/2)*(a+b*sec(c+d*x))^(5/2)*(A+B*sec(c+d*x)),x,14,2/3*a*A*(a+b*sec(c+d*x))^(3/2)*sin(c+d*x)*sqrt(cos(c+d*x))/d+1/3*(2*a^3*A+4*a*A*b^2+12*a^2*b*B+3*b^3*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+b^2*(2*A*b+5*a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))-1/3*b*(2*a*A-3*b*B)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(d*sqrt(cos(c+d*x)))+1/3*(14*a*A*b+6*a^2*B-3*b^2*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(d*sqrt((b+a*cos(c+d*x))/(a+b)))],
[(a+b*sec(c+d*x))^(5/2)*(A+B*sec(c+d*x))*sqrt(cos(c+d*x)),x,14,1/2*b*B*(a+b*sec(c+d*x))^(3/2)*sin(c+d*x)/(d*sqrt(cos(c+d*x)))+1/4*(16*a^2*A*b+4*A*b^3+8*a^3*B+11*a*b^2*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+1/4*b*(20*a*A*b+15*a^2*B+4*b^2*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+1/4*b*(4*A*b+7*a*B)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(d*sqrt(cos(c+d*x)))+1/4*(8*a^2*A-4*A*b^2-9*a*b*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(d*sqrt((b+a*cos(c+d*x))/(a+b)))],
[(a+b*sec(c+d*x))^(5/2)*(A+B*sec(c+d*x))/sqrt(cos(c+d*x)),x,15,1/3*b*B*(a+b*sec(c+d*x))^(3/2)*sin(c+d*x)/(d*cos(c+d*x)^(3/2))+1/24*(48*a^3*A+66*a*A*b^2+59*a^2*b*B+16*b^3*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+1/8*(30*a^2*A*b+8*A*b^3+5*a^3*B+20*a*b^2*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+1/4*b*(2*A*b+3*a*B)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(d*cos(c+d*x)^(3/2))+1/24*(54*a*A*b+33*a^2*B+16*b^2*B)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(d*sqrt(cos(c+d*x)))-1/24*(54*a*A*b+33*a^2*B+16*b^2*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(d*sqrt((b+a*cos(c+d*x))/(a+b)))],
[(a+b*sec(c+d*x))^(5/2)*(A+B*sec(c+d*x))/cos(c+d*x)^(3/2),x,16,1/4*b*B*(a+b*sec(c+d*x))^(3/2)*sin(c+d*x)/(d*cos(c+d*x)^(5/2))+1/192*(472*a^2*A*b+128*A*b^3+133*a^3*B+356*a*b^2*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+1/64*(40*a^3*A*b+160*a*A*b^3-5*a^4*B+120*a^2*b^2*B+48*b^4*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(b*d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+1/24*b*(8*A*b+11*a*B)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(d*cos(c+d*x)^(5/2))+1/96*(104*a*A*b+59*a^2*B+36*b^2*B)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(d*cos(c+d*x)^(3/2))+1/192*(264*a^2*A*b+128*A*b^3+15*a^3*B+284*a*b^2*B)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(b*d*sqrt(cos(c+d*x)))-1/192*(264*a^2*A*b+128*A*b^3+15*a^3*B+284*a*b^2*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(b*d*sqrt((b+a*cos(c+d*x))/(a+b)))],

# n<0
[cos(c+d*x)^(5/2)*(A+B*sec(c+d*x))/sqrt(a+b*sec(c+d*x)),x,10,-2/15*(7*a^2*A*b+8*A*b^3-5*a^3*B-10*a*b^2*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(a^3*d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+2/5*A*cos(c+d*x)^(3/2)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(a*d)-2/15*(4*A*b-5*a*B)*sin(c+d*x)*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(a^2*d)+2/15*(9*a^2*A+8*A*b^2-10*a*b*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(a^3*d*sqrt((b+a*cos(c+d*x))/(a+b)))],
[cos(c+d*x)^(3/2)*(A+B*sec(c+d*x))/sqrt(a+b*sec(c+d*x)),x,9,2/3*(a^2*A+2*A*b^2-3*a*b*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(a^2*d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+2/3*A*sin(c+d*x)*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(a*d)-2/3*(2*A*b-3*a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(a^2*d*sqrt((b+a*cos(c+d*x))/(a+b)))],
[(A+B*sec(c+d*x))*sqrt(cos(c+d*x))/sqrt(a+b*sec(c+d*x)),x,8,-2*(A*b-a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(a*d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+2*A*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(a*d*sqrt((b+a*cos(c+d*x))/(a+b)))],
[(A+B*sec(c+d*x))/(sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))),x,8,2*A*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+2*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))],
[(A+B*sec(c+d*x))/(cos(c+d*x)^(3/2)*sqrt(a+b*sec(c+d*x))),x,13,B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+(2*A*b-a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(b*d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+B*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(b*d*sqrt(cos(c+d*x)))-B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(b*d*sqrt((b+a*cos(c+d*x))/(a+b)))],
[(A+B*sec(c+d*x))/(cos(c+d*x)^(5/2)*sqrt(a+b*sec(c+d*x))),x,14,1/4*(4*A*b-a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(b*d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))-1/4*(4*a*A*b-3*a^2*B-4*b^2*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(b^2*d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+1/2*B*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(b*d*cos(c+d*x)^(3/2))+1/4*(4*A*b-3*a*B)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(b^2*d*sqrt(cos(c+d*x)))-1/4*(4*A*b-3*a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(b^2*d*sqrt((b+a*cos(c+d*x))/(a+b)))],
[cos(c+d*x)^(5/2)*(A+B*sec(c+d*x))/(a+b*sec(c+d*x))^(3/2),x,11,2*b*(A*b-a*B)*cos(c+d*x)^(3/2)*sin(c+d*x)/(a*(a^2-b^2)*d*sqrt(a+b*sec(c+d*x)))-2/15*(12*a^2*A*b+48*A*b^3-5*a^3*B-40*a*b^2*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(a^4*d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+2/5*(a^2*A-6*A*b^2+5*a*b*B)*cos(c+d*x)^(3/2)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(a^2*(a^2-b^2)*d)-2/15*(9*a^2*A*b-24*A*b^3-5*a^3*B+20*a*b^2*B)*sin(c+d*x)*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(a^3*(a^2-b^2)*d)+2/15*(9*a^4*A+24*a^2*A*b^2-48*A*b^4-25*a^3*b*B+40*a*b^3*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(a^4*(a^2-b^2)*d*sqrt((b+a*cos(c+d*x))/(a+b)))],
[cos(c+d*x)^(3/2)*(A+B*sec(c+d*x))/(a+b*sec(c+d*x))^(3/2),x,10,2*b*(A*b-a*B)*sin(c+d*x)*sqrt(cos(c+d*x))/(a*(a^2-b^2)*d*sqrt(a+b*sec(c+d*x)))+2/3*(a^2*A+8*A*b^2-6*a*b*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(a^3*d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+2/3*(a^2*A-4*A*b^2+3*a*b*B)*sin(c+d*x)*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(a^2*(a^2-b^2)*d)-2/3*(5*a^2*A*b-8*A*b^3-3*a^3*B+6*a*b^2*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(a^3*(a^2-b^2)*d*sqrt((b+a*cos(c+d*x))/(a+b)))],
[(A+B*sec(c+d*x))*sqrt(cos(c+d*x))/(a+b*sec(c+d*x))^(3/2),x,9,2*b*(A*b-a*B)*sin(c+d*x)/(a*(a^2-b^2)*d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))-2*(2*A*b-a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(a^2*d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+2*(a^2*A-2*A*b^2+a*b*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(a^2*(a^2-b^2)*d*sqrt((b+a*cos(c+d*x))/(a+b)))],
[(A+B*sec(c+d*x))/((a+b*sec(c+d*x))^(3/2)*sqrt(cos(c+d*x))),x,9,-2*(A*b-a*B)*sin(c+d*x)/((a^2-b^2)*d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+2*A*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(a*d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+2*(A*b-a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(a*(a^2-b^2)*d*sqrt((b+a*cos(c+d*x))/(a+b)))],
[(A+B*sec(c+d*x))/(cos(c+d*x)^(3/2)*(a+b*sec(c+d*x))^(3/2)),x,10,2*a*(A*b-a*B)*sin(c+d*x)/(b*(a^2-b^2)*d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+2*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(b*d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))-2*(A*b-a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(b*(a^2-b^2)*d*sqrt((b+a*cos(c+d*x))/(a+b)))],
[(A+B*sec(c+d*x))/(cos(c+d*x)^(5/2)*(a+b*sec(c+d*x))^(3/2)),x,14,2*a*(A*b-a*B)*sin(c+d*x)/(b*(a^2-b^2)*d*cos(c+d*x)^(3/2)*sqrt(a+b*sec(c+d*x)))+B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(b*d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+(2*A*b-3*a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(b^2*d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))-(2*a*A*b-3*a^2*B+b^2*B)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(b^2*(a^2-b^2)*d*sqrt(cos(c+d*x)))+(2*a*A*b-3*a^2*B+b^2*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(b^2*(a^2-b^2)*d*sqrt((b+a*cos(c+d*x))/(a+b)))],
[(A+B*sec(c+d*x))/(cos(c+d*x)^(7/2)*(a+b*sec(c+d*x))^(3/2)),x,15,2*a*(A*b-a*B)*sin(c+d*x)/(b*(a^2-b^2)*d*cos(c+d*x)^(5/2)*sqrt(a+b*sec(c+d*x)))+1/4*(4*A*b-5*a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(b^2*d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))-1/4*(12*a*A*b-15*a^2*B-4*b^2*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(b^3*d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))-1/2*(4*a*A*b-5*a^2*B+b^2*B)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(b^2*(a^2-b^2)*d*cos(c+d*x)^(3/2))+1/4*(12*a^2*A*b-4*A*b^3-15*a^3*B+7*a*b^2*B)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(b^3*(a^2-b^2)*d*sqrt(cos(c+d*x)))-1/4*(12*a^2*A*b-4*A*b^3-15*a^3*B+7*a*b^2*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(b^3*(a^2-b^2)*d*sqrt((b+a*cos(c+d*x))/(a+b)))],
[cos(c+d*x)^(5/2)*(A+B*sec(c+d*x))/(a+b*sec(c+d*x))^(5/2),x,12,2/3*b*(A*b-a*B)*cos(c+d*x)^(3/2)*sin(c+d*x)/(a*(a^2-b^2)*d*(a+b*sec(c+d*x))^(3/2))+2/3*b*(12*a^2*A*b-8*A*b^3-9*a^3*B+5*a*b^2*B)*cos(c+d*x)^(3/2)*sin(c+d*x)/(a^2*(a^2-b^2)^2*d*sqrt(a+b*sec(c+d*x)))-2/15*(17*a^4*A*b+116*a^2*A*b^3-128*A*b^5-5*a^5*B-80*a^3*b^2*B+80*a*b^4*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(a^5*(a^2-b^2)*d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+2/15*(3*a^4*A-71*a^2*A*b^2+48*A*b^4+50*a^3*b*B-30*a*b^3*B)*cos(c+d*x)^(3/2)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(a^3*(a^2-b^2)^2*d)-2/15*(14*a^4*A*b-98*a^2*A*b^3+64*A*b^5-5*a^5*B+65*a^3*b^2*B-40*a*b^4*B)*sin(c+d*x)*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(a^4*(a^2-b^2)^2*d)+2/15*(9*a^6*A+55*a^4*A*b^2-212*a^2*A*b^4+128*A*b^6-40*a^5*b*B+140*a^3*b^3*B-80*a*b^5*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(a^5*(a^2-b^2)^2*d*sqrt((b+a*cos(c+d*x))/(a+b)))],
[cos(c+d*x)^(3/2)*(A+B*sec(c+d*x))/(a+b*sec(c+d*x))^(5/2),x,11,2/3*b*(A*b-a*B)*sin(c+d*x)*sqrt(cos(c+d*x))/(a*(a^2-b^2)*d*(a+b*sec(c+d*x))^(3/2))+2/3*b*(10*a^2*A*b-6*A*b^3-7*a^3*B+3*a*b^2*B)*sin(c+d*x)*sqrt(cos(c+d*x))/(a^2*(a^2-b^2)^2*d*sqrt(a+b*sec(c+d*x)))+2/3*(a^4*A+16*a^2*A*b^2-16*A*b^4-9*a^3*b*B+8*a*b^3*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(a^4*(a^2-b^2)*d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+2/3*(a^4*A-13*a^2*A*b^2+8*A*b^4+8*a^3*b*B-4*a*b^3*B)*sin(c+d*x)*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(a^3*(a^2-b^2)^2*d)-2/3*(8*a^4*A*b-28*a^2*A*b^3+16*A*b^5-3*a^5*B+15*a^3*b^2*B-8*a*b^4*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(a^4*(a^2-b^2)^2*d*sqrt((b+a*cos(c+d*x))/(a+b)))],
[(A+B*sec(c+d*x))*sqrt(cos(c+d*x))/(a+b*sec(c+d*x))^(5/2),x,10,2/3*b*(A*b-a*B)*sin(c+d*x)/(a*(a^2-b^2)*d*(a+b*sec(c+d*x))^(3/2)*sqrt(cos(c+d*x)))+2/3*b*(8*a^2*A*b-4*A*b^3-5*a^3*B+a*b^2*B)*sin(c+d*x)/(a^2*(a^2-b^2)^2*d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))-2/3*(9*a^2*A*b-8*A*b^3-3*a^3*B+2*a*b^2*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(a^3*(a^2-b^2)*d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+2/3*(3*a^4*A-15*a^2*A*b^2+8*A*b^4+6*a^3*b*B-2*a*b^3*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(a^3*(a^2-b^2)^2*d*sqrt((b+a*cos(c+d*x))/(a+b)))],
[(A+B*sec(c+d*x))/((a+b*sec(c+d*x))^(5/2)*sqrt(cos(c+d*x))),x,10,-2/3*(A*b-a*B)*sin(c+d*x)/((a^2-b^2)*d*(a+b*sec(c+d*x))^(3/2)*sqrt(cos(c+d*x)))-2/3*(5*a^2*A*b-A*b^3-2*a^3*B-2*a*b^2*B)*sin(c+d*x)/(a*(a^2-b^2)^2*d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+2/3*(3*a^2*A-2*A*b^2-a*b*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(a^2*(a^2-b^2)*d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+2/3*(6*a^2*A*b-2*A*b^3-3*a^3*B-a*b^2*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(a^2*(a^2-b^2)^2*d*sqrt((b+a*cos(c+d*x))/(a+b)))],
[(A+B*sec(c+d*x))/(cos(c+d*x)^(3/2)*(a+b*sec(c+d*x))^(5/2)),x,10,2/3*a*(A*b-a*B)*sin(c+d*x)/(b*(a^2-b^2)*d*(a+b*sec(c+d*x))^(3/2)*sqrt(cos(c+d*x)))+2/3*(2*a^2*A*b+2*A*b^3+a^3*B-5*a*b^2*B)*sin(c+d*x)/(b*(a^2-b^2)^2*d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))-2/3*(A*b-a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(a*(a^2-b^2)*d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))-2/3*(3*a^2*A+A*b^2-4*a*b*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(a*(a^2-b^2)^2*d*sqrt((b+a*cos(c+d*x))/(a+b)))],
[(A+B*sec(c+d*x))/(cos(c+d*x)^(5/2)*(a+b*sec(c+d*x))^(5/2)),x,14,2/3*a*(A*b-a*B)*sin(c+d*x)/(b*(a^2-b^2)*d*cos(c+d*x)^(3/2)*(a+b*sec(c+d*x))^(3/2))-2/3*a*(4*A*b^3+3*a^3*B-7*a*b^2*B)*sin(c+d*x)/(b^2*(a^2-b^2)^2*d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+2/3*(A*b-a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(b*(a^2-b^2)*d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+2*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(b^2*d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+2/3*(4*A*b^3+3*a^3*B-7*a*b^2*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(b^2*(a^2-b^2)^2*d*sqrt((b+a*cos(c+d*x))/(a+b)))],
[(A+B*sec(c+d*x))/(cos(c+d*x)^(7/2)*(a+b*sec(c+d*x))^(5/2)),x,15,2/3*a*(A*b-a*B)*sin(c+d*x)/(b*(a^2-b^2)*d*cos(c+d*x)^(5/2)*(a+b*sec(c+d*x))^(3/2))+2/3*a*(2*a^2*A*b-6*A*b^3-5*a^3*B+9*a*b^2*B)*sin(c+d*x)/(b^2*(a^2-b^2)^2*d*cos(c+d*x)^(3/2)*sqrt(a+b*sec(c+d*x)))-1/3*(2*a*A*b-5*a^2*B+3*b^2*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(b^2*(a^2-b^2)*d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+(2*A*b-5*a*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(b^3*d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))-1/3*(6*a^3*A*b-14*a*A*b^3-15*a^4*B+26*a^2*b^2*B-3*b^4*B)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(b^3*(a^2-b^2)^2*d*sqrt(cos(c+d*x)))+1/3*(6*a^3*A*b-14*a*A*b^3-15*a^4*B+26*a^2*b^2*B-3*b^4*B)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(b^3*(a^2-b^2)^2*d*sqrt((b+a*cos(c+d*x))/(a+b)))]]:
