# Maple integration test file: "4 Trig functions\4.5 Secant\4.5.4.1 (a+b sec)^m (A+B sec+C sec^2).txt"

lst:=[

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

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

# Integrands of the form Sec[e+f x]^m (A+C Sec[e+f x]^2)
[sec(c+d*x)^6*(A+C*sec(c+d*x)^2),x,3,1/7*(7*A+6*C)*tan(c+d*x)/d+1/7*C*sec(c+d*x)^6*tan(c+d*x)/d+2/21*(7*A+6*C)*tan(c+d*x)^3/d+1/35*(7*A+6*C)*tan(c+d*x)^5/d],
[sec(c+d*x)^5*(A+C*sec(c+d*x)^2),x,4,1/16*(6*A+5*C)*arctanh(sin(c+d*x))/d+1/16*(6*A+5*C)*sec(c+d*x)*tan(c+d*x)/d+1/24*(6*A+5*C)*sec(c+d*x)^3*tan(c+d*x)/d+1/6*C*sec(c+d*x)^5*tan(c+d*x)/d],
[sec(c+d*x)^4*(A+C*sec(c+d*x)^2),x,3,1/5*(5*A+4*C)*tan(c+d*x)/d+1/5*C*sec(c+d*x)^4*tan(c+d*x)/d+1/15*(5*A+4*C)*tan(c+d*x)^3/d],
[sec(c+d*x)^3*(A+C*sec(c+d*x)^2),x,3,1/8*(4*A+3*C)*arctanh(sin(c+d*x))/d+1/8*(4*A+3*C)*sec(c+d*x)*tan(c+d*x)/d+1/4*C*sec(c+d*x)^3*tan(c+d*x)/d],
[sec(c+d*x)^2*(A+C*sec(c+d*x)^2),x,3,1/3*(3*A+2*C)*tan(c+d*x)/d+1/3*C*sec(c+d*x)^2*tan(c+d*x)/d],
[sec(c+d*x)*(A+C*sec(c+d*x)^2),x,2,1/2*(2*A+C)*arctanh(sin(c+d*x))/d+1/2*C*sec(c+d*x)*tan(c+d*x)/d],
[A+C*sec(c+d*x)^2,x,3,A*x+C*tan(c+d*x)/d],
[cos(c+d*x)*(A+C*sec(c+d*x)^2),x,2,C*arctanh(sin(c+d*x))/d+A*sin(c+d*x)/d],
[cos(c+d*x)^2*(A+C*sec(c+d*x)^2),x,2,1/2*(A+2*C)*x+1/2*A*cos(c+d*x)*sin(c+d*x)/d],
[cos(c+d*x)^3*(A+C*sec(c+d*x)^2),x,3,(A+C)*sin(c+d*x)/d-1/3*A*sin(c+d*x)^3/d],
[cos(c+d*x)^4*(A+C*sec(c+d*x)^2),x,3,1/8*(3*A+4*C)*x+1/8*(3*A+4*C)*cos(c+d*x)*sin(c+d*x)/d+1/4*A*cos(c+d*x)^3*sin(c+d*x)/d],
[cos(c+d*x)^5*(A+C*sec(c+d*x)^2),x,4,(A+C)*sin(c+d*x)/d-1/3*(2*A+C)*sin(c+d*x)^3/d+1/5*A*sin(c+d*x)^5/d],
[cos(c+d*x)^6*(A+C*sec(c+d*x)^2),x,4,1/16*(5*A+6*C)*x+1/16*(5*A+6*C)*cos(c+d*x)*sin(c+d*x)/d+1/24*(5*A+6*C)*cos(c+d*x)^3*sin(c+d*x)/d+1/6*A*cos(c+d*x)^5*sin(c+d*x)/d],
[sec(c+d*x)^m*(-C*m/(1+m)+C*sec(c+d*x)^2),x,1,C*sec(c+d*x)^(1+m)*sin(c+d*x)/(d*(1+m))],
[sec(c+d*x)^m*(A-A*(1+m)*sec(c+d*x)^2/m),x,1,-A*sec(c+d*x)^(1+m)*sin(c+d*x)/(d*m)],

# Integrands of the form (b Sec[e+f x])^(m/2) (A+C Sec[e+f x]^2)
[(b*sec(c+d*x))^(5/2)*(A+C*sec(c+d*x)^2),x,4,2/21*b*(7*A+5*C)*(b*sec(c+d*x))^(3/2)*sin(c+d*x)/d+2/21*b^2*(7*A+5*C)*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+2/7*C*(b*sec(c+d*x))^(5/2)*tan(c+d*x)/d],
[(b*sec(c+d*x))^(3/2)*(A+C*sec(c+d*x)^2),x,4,-2/5*b^2*(5*A+3*C)*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)))+2/5*b*(5*A+3*C)*sin(c+d*x)*sqrt(b*sec(c+d*x))/d+2/5*C*(b*sec(c+d*x))^(3/2)*tan(c+d*x)/d],
[(b*sec(c+d*x))^(1/2)*(A+C*sec(c+d*x)^2),x,3,2/3*(3*A+C)*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+2/3*C*sqrt(b*sec(c+d*x))*tan(c+d*x)/d],
[(A+C*sec(c+d*x)^2)/(b*sec(c+d*x))^(1/2),x,3,2*(A-C)*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*C*tan(c+d*x)/(d*sqrt(b*sec(c+d*x)))],
[(A+C*sec(c+d*x)^2)/(b*sec(c+d*x))^(3/2),x,3,2/3*(A+3*C)*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)+2/3*A*tan(c+d*x)/(d*(b*sec(c+d*x))^(3/2))],
[(A+C*sec(c+d*x)^2)/(b*sec(c+d*x))^(5/2),x,3,2/5*(3*A+5*C)*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)))+2/5*A*tan(c+d*x)/(d*(b*sec(c+d*x))^(5/2))],
[(A+C*sec(c+d*x)^2)/(b*sec(c+d*x))^(7/2),x,4,2/21*(5*A+7*C)*sin(c+d*x)/(b^3*d*sqrt(b*sec(c+d*x)))+2/21*(5*A+7*C)*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^4*d)+2/7*A*tan(c+d*x)/(d*(b*sec(c+d*x))^(7/2))],
[(A+C*sec(c+d*x)^2)/(b*sec(c+d*x))^(9/2),x,4,2/45*(7*A+9*C)*sin(c+d*x)/(b^3*d*(b*sec(c+d*x))^(3/2))+2/15*(7*A+9*C)*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*d*sqrt(cos(c+d*x))*sqrt(b*sec(c+d*x)))+2/9*A*tan(c+d*x)/(d*(b*sec(c+d*x))^(9/2))],
[(3+3*sec(c+d*x)^2)/sec(c+d*x)^(1/2),x,1,6*sin(c+d*x)*sqrt(sec(c+d*x))/d],

# Integrands of the form Sec[e+f x]^m (A+C Sec[e+f x]^2) when A (m+1)+C m=0
[sec(e+f*x)^m*(m-(1+m)*sec(e+f*x)^2),x,1,-sec(e+f*x)^(1+m)*sin(e+f*x)/f],
[sec(e+f*x)^5*(5-6*sec(e+f*x)^2),x,1,-sec(e+f*x)^5*tan(e+f*x)/f],
[sec(e+f*x)^4*(4-5*sec(e+f*x)^2),x,1,-sec(e+f*x)^4*tan(e+f*x)/f],
[sec(e+f*x)^3*(3-4*sec(e+f*x)^2),x,1,-sec(e+f*x)^3*tan(e+f*x)/f],
[sec(e+f*x)^2*(2-3*sec(e+f*x)^2),x,1,-sec(e+f*x)^2*tan(e+f*x)/f],
[sec(e+f*x)*(1-2*sec(e+f*x)^2),x,1,-sec(e+f*x)*tan(e+f*x)/f],
[-sec(e+f*x)^2,x,2,-tan(e+f*x)/f],
[-cos(e+f*x),x,1,-sin(e+f*x)/f],
[cos(e+f*x)^2*(-2+sec(e+f*x)^2),x,1,-cos(e+f*x)*sin(e+f*x)/f],
[cos(e+f*x)^3*(-3+2*sec(e+f*x)^2),x,1,-cos(e+f*x)^2*sin(e+f*x)/f],
[cos(e+f*x)^4*(-4+3*sec(e+f*x)^2),x,1,-cos(e+f*x)^3*sin(e+f*x)/f],
[cos(e+f*x)^5*(-5+4*sec(e+f*x)^2),x,1,-cos(e+f*x)^4*sin(e+f*x)/f],

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

# Integrands of the form Sec[e+f x]^m (B Sec[e+f x]+C Sec[e+f x]^2)
[sec(c+d*x)^3*(B*sec(c+d*x)+C*sec(c+d*x)^2),x,7,3/8*C*arctanh(sin(c+d*x))/d+B*tan(c+d*x)/d+3/8*C*sec(c+d*x)*tan(c+d*x)/d+1/4*C*sec(c+d*x)^3*tan(c+d*x)/d+1/3*B*tan(c+d*x)^3/d],
[sec(c+d*x)^2*(B*sec(c+d*x)+C*sec(c+d*x)^2),x,6,1/2*B*arctanh(sin(c+d*x))/d+C*tan(c+d*x)/d+1/2*B*sec(c+d*x)*tan(c+d*x)/d+1/3*C*tan(c+d*x)^3/d],
[sec(c+d*x)*(B*sec(c+d*x)+C*sec(c+d*x)^2),x,6,1/2*C*arctanh(sin(c+d*x))/d+B*tan(c+d*x)/d+1/2*C*sec(c+d*x)*tan(c+d*x)/d],
[B*sec(c+d*x)+C*sec(c+d*x)^2,x,4,B*arctanh(sin(c+d*x))/d+C*tan(c+d*x)/d],
[cos(c+d*x)*(B*sec(c+d*x)+C*sec(c+d*x)^2),x,4,B*x+C*arctanh(sin(c+d*x))/d],
[cos(c+d*x)^2*(B*sec(c+d*x)+C*sec(c+d*x)^2),x,3,C*x+B*sin(c+d*x)/d],
[cos(c+d*x)^3*(B*sec(c+d*x)+C*sec(c+d*x)^2),x,5,1/2*B*x+C*sin(c+d*x)/d+1/2*B*cos(c+d*x)*sin(c+d*x)/d],
[cos(c+d*x)^4*(B*sec(c+d*x)+C*sec(c+d*x)^2),x,6,1/2*C*x+B*sin(c+d*x)/d+1/2*C*cos(c+d*x)*sin(c+d*x)/d-1/3*B*sin(c+d*x)^3/d],
[cos(c+d*x)^5*(B*sec(c+d*x)+C*sec(c+d*x)^2),x,7,3/8*B*x+C*sin(c+d*x)/d+3/8*B*cos(c+d*x)*sin(c+d*x)/d+1/4*B*cos(c+d*x)^3*sin(c+d*x)/d-1/3*C*sin(c+d*x)^3/d],
[cos(c+d*x)^6*(B*sec(c+d*x)+C*sec(c+d*x)^2),x,7,3/8*C*x+B*sin(c+d*x)/d+3/8*C*cos(c+d*x)*sin(c+d*x)/d+1/4*C*cos(c+d*x)^3*sin(c+d*x)/d-2/3*B*sin(c+d*x)^3/d+1/5*B*sin(c+d*x)^5/d],

# Integrands of the form (b Sec[e+f x])^(m/2) (B Sec[e+f x]+C Sec[e+f x]^2)
[(b*sec(c+d*x))^(3/2)*(B*sec(c+d*x)+C*sec(c+d*x)^2),x,10,2/3*B*(b*sec(c+d*x))^(3/2)*sin(c+d*x)/d+2/5*C*(b*sec(c+d*x))^(5/2)*sin(c+d*x)/(b*d)-6/5*b^2*C*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*C*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)*(B*sec(c+d*x)+C*sec(c+d*x)^2),x,9,2/3*C*(b*sec(c+d*x))^(3/2)*sin(c+d*x)/(b*d)-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+2/3*C*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)+C*sec(c+d*x)^2)/(b*sec(c+d*x))^(1/2),x,8,-2*C*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*C*sin(c+d*x)*sqrt(b*sec(c+d*x))/(b*d)+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)],
[(B*sec(c+d*x)+C*sec(c+d*x)^2)/(b*sec(c+d*x))^(3/2),x,7,2*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*C*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)],
[(B*sec(c+d*x)+C*sec(c+d*x)^2)/(b*sec(c+d*x))^(5/2),x,8,2/3*B*sin(c+d*x)/(b^2*d*sqrt(b*sec(c+d*x)))+2*C*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)],
[(B*sec(c+d*x)+C*sec(c+d*x)^2)/(b*sec(c+d*x))^(7/2),x,9,2/5*B*sin(c+d*x)/(b^2*d*(b*sec(c+d*x))^(3/2))+2/3*C*sin(c+d*x)/(b^3*d*sqrt(b*sec(c+d*x)))+6/5*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(b^3*d*sqrt(cos(c+d*x))*sqrt(b*sec(c+d*x)))+2/3*C*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^4*d)],
[(B*sec(c+d*x)+C*sec(c+d*x)^2)/(b*sec(c+d*x))^(9/2),x,10,2/7*B*sin(c+d*x)/(b^2*d*(b*sec(c+d*x))^(5/2))+2/5*C*sin(c+d*x)/(b^3*d*(b*sec(c+d*x))^(3/2))+10/21*B*sin(c+d*x)/(b^4*d*sqrt(b*sec(c+d*x)))+6/5*C*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*d*sqrt(cos(c+d*x))*sqrt(b*sec(c+d*x)))+10/21*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(b*sec(c+d*x))/(b^5*d)],

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

# Integrands of the form Sec[e+f x]^m (A+B Sec[e+f x]+C Sec[e+f x]^2)
[sec(c+d*x)^4*(A+B*sec(c+d*x)+C*sec(c+d*x)^2),x,7,3/8*B*arctanh(sin(c+d*x))/d+1/5*(5*A+4*C)*tan(c+d*x)/d+3/8*B*sec(c+d*x)*tan(c+d*x)/d+1/4*B*sec(c+d*x)^3*tan(c+d*x)/d+1/5*C*sec(c+d*x)^4*tan(c+d*x)/d+1/15*(5*A+4*C)*tan(c+d*x)^3/d],
[sec(c+d*x)^3*(A+B*sec(c+d*x)+C*sec(c+d*x)^2),x,6,1/8*(4*A+3*C)*arctanh(sin(c+d*x))/d+B*tan(c+d*x)/d+1/8*(4*A+3*C)*sec(c+d*x)*tan(c+d*x)/d+1/4*C*sec(c+d*x)^3*tan(c+d*x)/d+1/3*B*tan(c+d*x)^3/d],
[sec(c+d*x)^2*(A+B*sec(c+d*x)+C*sec(c+d*x)^2),x,6,1/2*B*arctanh(sin(c+d*x))/d+1/3*(3*A+2*C)*tan(c+d*x)/d+1/2*B*sec(c+d*x)*tan(c+d*x)/d+1/3*C*sec(c+d*x)^2*tan(c+d*x)/d],
[sec(c+d*x)*(A+B*sec(c+d*x)+C*sec(c+d*x)^2),x,5,1/2*(2*A+C)*arctanh(sin(c+d*x))/d+B*tan(c+d*x)/d+1/2*C*sec(c+d*x)*tan(c+d*x)/d],
[A+B*sec(c+d*x)+C*sec(c+d*x)^2,x,4,A*x+B*arctanh(sin(c+d*x))/d+C*tan(c+d*x)/d],
[cos(c+d*x)*(A+B*sec(c+d*x)+C*sec(c+d*x)^2),x,4,B*x+C*arctanh(sin(c+d*x))/d+A*sin(c+d*x)/d],
[cos(c+d*x)^2*(A+B*sec(c+d*x)+C*sec(c+d*x)^2),x,4,1/2*(A+2*C)*x+B*sin(c+d*x)/d+1/2*A*cos(c+d*x)*sin(c+d*x)/d],
[cos(c+d*x)^3*(A+B*sec(c+d*x)+C*sec(c+d*x)^2),x,6,1/2*B*x+(A+C)*sin(c+d*x)/d+1/2*B*cos(c+d*x)*sin(c+d*x)/d-1/3*A*sin(c+d*x)^3/d],
[cos(c+d*x)^4*(A+B*sec(c+d*x)+C*sec(c+d*x)^2),x,6,1/8*(3*A+4*C)*x+B*sin(c+d*x)/d+1/8*(3*A+4*C)*cos(c+d*x)*sin(c+d*x)/d+1/4*A*cos(c+d*x)^3*sin(c+d*x)/d-1/3*B*sin(c+d*x)^3/d],
[cos(c+d*x)^5*(A+B*sec(c+d*x)+C*sec(c+d*x)^2),x,8,3/8*B*x+(A+C)*sin(c+d*x)/d+3/8*B*cos(c+d*x)*sin(c+d*x)/d+1/4*B*cos(c+d*x)^3*sin(c+d*x)/d-1/3*(2*A+C)*sin(c+d*x)^3/d+1/5*A*sin(c+d*x)^5/d],
[cos(c+d*x)^6*(A+B*sec(c+d*x)+C*sec(c+d*x)^2),x,7,1/16*(5*A+6*C)*x+B*sin(c+d*x)/d+1/16*(5*A+6*C)*cos(c+d*x)*sin(c+d*x)/d+1/24*(5*A+6*C)*cos(c+d*x)^3*sin(c+d*x)/d+1/6*A*cos(c+d*x)^5*sin(c+d*x)/d-2/3*B*sin(c+d*x)^3/d+1/5*B*sin(c+d*x)^5/d],

# Integrands of the form (b Sec[e+f x])^(m/2) (A+B Sec[e+f x]+C Sec[e+f x]^2)
[(b*sec(c+d*x))^(3/2)*(A+B*sec(c+d*x)+C*sec(c+d*x)^2),x,8,2/3*B*(b*sec(c+d*x))^(3/2)*sin(c+d*x)/d-2/5*b^2*(5*A+3*C)*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)))+2/5*b*(5*A+3*C)*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+2/5*C*(b*sec(c+d*x))^(3/2)*tan(c+d*x)/d],
[(b*sec(c+d*x))^(1/2)*(A+B*sec(c+d*x)+C*sec(c+d*x)^2),x,7,-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+2/3*(3*A+C)*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+2/3*C*sqrt(b*sec(c+d*x))*tan(c+d*x)/d],
[(A+B*sec(c+d*x)+C*sec(c+d*x)^2)/(b*sec(c+d*x))^(1/2),x,6,2*(A-C)*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)+2*C*tan(c+d*x)/(d*sqrt(b*sec(c+d*x)))],
[(A+B*sec(c+d*x)+C*sec(c+d*x)^2)/(b*sec(c+d*x))^(3/2),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*sqrt(cos(c+d*x))*sqrt(b*sec(c+d*x)))+2/3*(A+3*C)*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)+2/3*A*tan(c+d*x)/(d*(b*sec(c+d*x))^(3/2))],
[(A+B*sec(c+d*x)+C*sec(c+d*x)^2)/(b*sec(c+d*x))^(5/2),x,7,2/3*B*sin(c+d*x)/(b^2*d*sqrt(b*sec(c+d*x)))+2/5*(3*A+5*C)*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)+2/5*A*tan(c+d*x)/(d*(b*sec(c+d*x))^(5/2))],
[(A+B*sec(c+d*x)+C*sec(c+d*x)^2)/(b*sec(c+d*x))^(7/2),x,8,2/5*B*sin(c+d*x)/(b^2*d*(b*sec(c+d*x))^(3/2))+2/21*(5*A+7*C)*sin(c+d*x)/(b^3*d*sqrt(b*sec(c+d*x)))+6/5*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(b^3*d*sqrt(cos(c+d*x))*sqrt(b*sec(c+d*x)))+2/21*(5*A+7*C)*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^4*d)+2/7*A*tan(c+d*x)/(d*(b*sec(c+d*x))^(7/2))]]:

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