# Maple integration test file: "6 Hyperbolic functions\6.5 Hyperbolic secant\6.5.3 Hyperbolic secant functions.txt"

lst:=[

# Integration Problems Involving Hyperbolic Secants

# Integrands of the form (c Sech[a+b x])^n

# Integrands of the form Sech[a+b x]^n
[sech(a+b*x),x,1,arctan(sinh(a+b*x))/b],
[sech(a+b*x)^2,x,2,tanh(a+b*x)/b],
[sech(a+b*x)^3,x,2,1/2*arctan(sinh(a+b*x))/b+1/2*sech(a+b*x)*tanh(a+b*x)/b],
[sech(a+b*x)^4,x,2,tanh(a+b*x)/b-1/3*tanh(a+b*x)^3/b],
[sech(a+b*x)^5,x,3,3/8*arctan(sinh(a+b*x))/b+3/8*sech(a+b*x)*tanh(a+b*x)/b+1/4*sech(a+b*x)^3*tanh(a+b*x)/b],
[sech(a+b*x)^6,x,2,tanh(a+b*x)/b-2/3*tanh(a+b*x)^3/b+1/5*tanh(a+b*x)^5/b],
[sech(7*x)^4,x,2,1/7*tanh(7*x)-1/21*tanh(7*x)^3],
[sech(Pi*x)^6,x,2,tanh(Pi*x)/Pi-2/3*tanh(Pi*x)^3/Pi+1/5*tanh(Pi*x)^5/Pi],

# Integrands of the form (c Sech[a+b x])^(n/2)
[sech(a+b*x)^(5/2),x,3,2/3*sech(a+b*x)^(3/2)*sinh(a+b*x)/b-2/3*I*sqrt(cos(1/2*I*(a+b*x))^2)/cos(1/2*I*(a+b*x))*EllipticF(sin(1/2*I*(a+b*x)),sqrt(2))*sqrt(cosh(a+b*x))*sqrt(sech(a+b*x))/b],
[sech(a+b*x)^(3/2),x,3,2*sinh(a+b*x)*sqrt(sech(a+b*x))/b+2*I*sqrt(cos(1/2*I*(a+b*x))^2)/cos(1/2*I*(a+b*x))*EllipticE(sin(1/2*I*(a+b*x)),sqrt(2))*sqrt(cosh(a+b*x))*sqrt(sech(a+b*x))/b],
[sech(a+b*x)^(1/2),x,2,-2*I*sqrt(cos(1/2*I*(a+b*x))^2)/cos(1/2*I*(a+b*x))*EllipticF(sin(1/2*I*(a+b*x)),sqrt(2))*sqrt(cosh(a+b*x))*sqrt(sech(a+b*x))/b],
[1/sech(a+b*x)^(1/2),x,2,-2*I*sqrt(cos(1/2*I*(a+b*x))^2)/cos(1/2*I*(a+b*x))*EllipticE(sin(1/2*I*(a+b*x)),sqrt(2))*sqrt(cosh(a+b*x))*sqrt(sech(a+b*x))/b],
[1/sech(a+b*x)^(3/2),x,3,2/3*sinh(a+b*x)/(b*sqrt(sech(a+b*x)))-2/3*I*sqrt(cos(1/2*I*(a+b*x))^2)/cos(1/2*I*(a+b*x))*EllipticF(sin(1/2*I*(a+b*x)),sqrt(2))*sqrt(cosh(a+b*x))*sqrt(sech(a+b*x))/b],
[1/sech(a+b*x)^(5/2),x,3,2/5*sinh(a+b*x)/(b*sech(a+b*x)^(3/2))-6/5*I*sqrt(cos(1/2*I*(a+b*x))^2)/cos(1/2*I*(a+b*x))*EllipticE(sin(1/2*I*(a+b*x)),sqrt(2))*sqrt(cosh(a+b*x))*sqrt(sech(a+b*x))/b],
[(b*sech(c+d*x))^(7/2),x,4,2/5*b*(b*sech(c+d*x))^(5/2)*sinh(c+d*x)/d+6/5*I*b^4*sqrt(cos(1/2*I*(c+d*x))^2)/cos(1/2*I*(c+d*x))*EllipticE(sin(1/2*I*(c+d*x)),sqrt(2))/(d*sqrt(cosh(c+d*x))*sqrt(b*sech(c+d*x)))+6/5*b^3*sinh(c+d*x)*sqrt(b*sech(c+d*x))/d],
[(b*sech(c+d*x))^(5/2),x,3,2/3*b*(b*sech(c+d*x))^(3/2)*sinh(c+d*x)/d-2/3*I*b^2*sqrt(cos(1/2*I*(c+d*x))^2)/cos(1/2*I*(c+d*x))*EllipticF(sin(1/2*I*(c+d*x)),sqrt(2))*sqrt(cosh(c+d*x))*sqrt(b*sech(c+d*x))/d],
[(b*sech(c+d*x))^(3/2),x,3,2*I*b^2*sqrt(cos(1/2*I*(c+d*x))^2)/cos(1/2*I*(c+d*x))*EllipticE(sin(1/2*I*(c+d*x)),sqrt(2))/(d*sqrt(cosh(c+d*x))*sqrt(b*sech(c+d*x)))+2*b*sinh(c+d*x)*sqrt(b*sech(c+d*x))/d],
[(b*sech(c+d*x))^(1/2),x,2,-2*I*sqrt(cos(1/2*I*(c+d*x))^2)/cos(1/2*I*(c+d*x))*EllipticF(sin(1/2*I*(c+d*x)),sqrt(2))*sqrt(cosh(c+d*x))*sqrt(b*sech(c+d*x))/d],
[1/(b*sech(c+d*x))^(1/2),x,2,-2*I*sqrt(cos(1/2*I*(c+d*x))^2)/cos(1/2*I*(c+d*x))*EllipticE(sin(1/2*I*(c+d*x)),sqrt(2))/(d*sqrt(cosh(c+d*x))*sqrt(b*sech(c+d*x)))],
[1/(b*sech(c+d*x))^(3/2),x,3,2/3*sinh(c+d*x)/(b*d*sqrt(b*sech(c+d*x)))-2/3*I*sqrt(cos(1/2*I*(c+d*x))^2)/cos(1/2*I*(c+d*x))*EllipticF(sin(1/2*I*(c+d*x)),sqrt(2))*sqrt(cosh(c+d*x))*sqrt(b*sech(c+d*x))/(b^2*d)],
[1/(b*sech(c+d*x))^(5/2),x,3,2/5*sinh(c+d*x)/(b*d*(b*sech(c+d*x))^(3/2))-6/5*I*sqrt(cos(1/2*I*(c+d*x))^2)/cos(1/2*I*(c+d*x))*EllipticE(sin(1/2*I*(c+d*x)),sqrt(2))/(b^2*d*sqrt(cosh(c+d*x))*sqrt(b*sech(c+d*x)))],
[1/(b*sech(c+d*x))^(7/2),x,4,2/7*sinh(c+d*x)/(b*d*(b*sech(c+d*x))^(5/2))+10/21*sinh(c+d*x)/(b^3*d*sqrt(b*sech(c+d*x)))-10/21*I*sqrt(cos(1/2*I*(c+d*x))^2)/cos(1/2*I*(c+d*x))*EllipticF(sin(1/2*I*(c+d*x)),sqrt(2))*sqrt(cosh(c+d*x))*sqrt(b*sech(c+d*x))/(b^4*d)],

# Integrands of the form (c Sech[a+b x])^n with n symbolic
[(b*sech(c+d*x))^n,x,2,-b*hypergeom([1/2,1/2*(1-n)],[1/2*(3-n)],cosh(c+d*x)^2)*(b*sech(c+d*x))^(-1+n)*sinh(c+d*x)/(d*(1-n)*sqrt(-sinh(c+d*x)^2))],

# Integrands of the form (c Sech[a+b x]^m)^n

# Integrands of the form (c Sech[a+b x]^2)^n
[(sech(a+b*x)^2)^(7/2),x,5,5/16*arcsin(tanh(a+b*x))/b+5/24*(sech(a+b*x)^2)^(3/2)*tanh(a+b*x)/b+1/6*(sech(a+b*x)^2)^(5/2)*tanh(a+b*x)/b+5/16*sqrt(sech(a+b*x)^2)*tanh(a+b*x)/b],
[(sech(a+b*x)^2)^(5/2),x,4,3/8*arcsin(tanh(a+b*x))/b+1/4*(sech(a+b*x)^2)^(3/2)*tanh(a+b*x)/b+3/8*sqrt(sech(a+b*x)^2)*tanh(a+b*x)/b],
[(sech(a+b*x)^2)^(3/2),x,3,1/2*arcsin(tanh(a+b*x))/b+1/2*sqrt(sech(a+b*x)^2)*tanh(a+b*x)/b],
[(sech(a+b*x)^2)^(1/2),x,2,arcsin(tanh(a+b*x))/b],
[1/(sech(a+b*x)^2)^(1/2),x,2,tanh(a+b*x)/(b*sqrt(sech(a+b*x)^2))],
[1/(sech(a+b*x)^2)^(3/2),x,3,1/3*tanh(a+b*x)/(b*(sech(a+b*x)^2)^(3/2))+2/3*tanh(a+b*x)/(b*sqrt(sech(a+b*x)^2))],
[1/(sech(a+b*x)^2)^(5/2),x,4,1/5*tanh(a+b*x)/(b*(sech(a+b*x)^2)^(5/2))+4/15*tanh(a+b*x)/(b*(sech(a+b*x)^2)^(3/2))+8/15*tanh(a+b*x)/(b*sqrt(sech(a+b*x)^2))],
[1/(sech(a+b*x)^2)^(7/2),x,5,1/7*tanh(a+b*x)/(b*(sech(a+b*x)^2)^(7/2))+6/35*tanh(a+b*x)/(b*(sech(a+b*x)^2)^(5/2))+8/35*tanh(a+b*x)/(b*(sech(a+b*x)^2)^(3/2))+16/35*tanh(a+b*x)/(b*sqrt(sech(a+b*x)^2))],
[(a*sech(x)^2)^(5/2),x,5,3/8*a^(5/2)*arctan(sqrt(a)*tanh(x)/sqrt(a*sech(x)^2))+1/4*a*(a*sech(x)^2)^(3/2)*tanh(x)+3/8*a^2*sqrt(a*sech(x)^2)*tanh(x)],
[(a*sech(x)^2)^(3/2),x,4,1/2*a^(3/2)*arctan(sqrt(a)*tanh(x)/sqrt(a*sech(x)^2))+1/2*a*sqrt(a*sech(x)^2)*tanh(x)],
[(a*sech(x)^2)^(1/2),x,3,arctan(sqrt(a)*tanh(x)/sqrt(a*sech(x)^2))*sqrt(a)],
[1/(a*sech(x)^2)^(1/2),x,2,tanh(x)/sqrt(a*sech(x)^2)],
[1/(a*sech(x)^2)^(3/2),x,3,1/3*tanh(x)/(a*sech(x)^2)^(3/2)+2/3*tanh(x)/(a*sqrt(a*sech(x)^2))],
[1/(a*sech(x)^2)^(5/2),x,4,1/5*tanh(x)/(a*sech(x)^2)^(5/2)+4/15*tanh(x)/(a*(a*sech(x)^2)^(3/2))+8/15*tanh(x)/(a^2*sqrt(a*sech(x)^2))],
[1/(a*sech(x)^2)^(7/2),x,5,1/7*tanh(x)/(a*sech(x)^2)^(7/2)+6/35*tanh(x)/(a*(a*sech(x)^2)^(5/2))+8/35*tanh(x)/(a^2*(a*sech(x)^2)^(3/2))+16/35*tanh(x)/(a^3*sqrt(a*sech(x)^2))],

# Integrands of the form (c Sech[a+b x]^3)^n
[(a*sech(x)^3)^(5/2),x,7,154/195*I*a^2*cosh(x)^(3/2)*sqrt(cos(1/2*I*x)^2)/cos(1/2*I*x)*EllipticE(sin(1/2*I*x),sqrt(2))*sqrt(a*sech(x)^3)+154/195*a^2*cosh(x)*sinh(x)*sqrt(a*sech(x)^3)+154/585*a^2*sqrt(a*sech(x)^3)*tanh(x)+22/117*a^2*sech(x)^2*sqrt(a*sech(x)^3)*tanh(x)+2/13*a^2*sech(x)^4*sqrt(a*sech(x)^3)*tanh(x)],
[(a*sech(x)^3)^(3/2),x,5,-10/21*I*a*cosh(x)^(3/2)*sqrt(cos(1/2*I*x)^2)/cos(1/2*I*x)*EllipticF(sin(1/2*I*x),sqrt(2))*sqrt(a*sech(x)^3)+10/21*a*sinh(x)*sqrt(a*sech(x)^3)+2/7*a*sech(x)*sqrt(a*sech(x)^3)*tanh(x)],
[(a*sech(x)^3)^(1/2),x,4,2*I*cosh(x)^(3/2)*sqrt(cos(1/2*I*x)^2)/cos(1/2*I*x)*EllipticE(sin(1/2*I*x),sqrt(2))*sqrt(a*sech(x)^3)+2*cosh(x)*sinh(x)*sqrt(a*sech(x)^3)],
[1/(a*sech(x)^3)^(1/2),x,4,-2/3*I*sqrt(cos(1/2*I*x)^2)/cos(1/2*I*x)*EllipticF(sin(1/2*I*x),sqrt(2))/(cosh(x)^(3/2)*sqrt(a*sech(x)^3))+2/3*tanh(x)/sqrt(a*sech(x)^3)],
[1/(a*sech(x)^3)^(3/2),x,5,-14/15*I*sqrt(cos(1/2*I*x)^2)/cos(1/2*I*x)*EllipticE(sin(1/2*I*x),sqrt(2))/(a*cosh(x)^(3/2)*sqrt(a*sech(x)^3))+14/45*sinh(x)/(a*sqrt(a*sech(x)^3))+2/9*cosh(x)^2*sinh(x)/(a*sqrt(a*sech(x)^3))],
[1/(a*sech(x)^3)^(5/2),x,7,-26/77*I*sqrt(cos(1/2*I*x)^2)/cos(1/2*I*x)*EllipticF(sin(1/2*I*x),sqrt(2))/(a^2*cosh(x)^(3/2)*sqrt(a*sech(x)^3))+78/385*cosh(x)*sinh(x)/(a^2*sqrt(a*sech(x)^3))+26/165*cosh(x)^3*sinh(x)/(a^2*sqrt(a*sech(x)^3))+2/15*cosh(x)^5*sinh(x)/(a^2*sqrt(a*sech(x)^3))+26/77*tanh(x)/(a^2*sqrt(a*sech(x)^3))],

# Integrands of the form (c Sech[a+b x]^4)^n
[(a*sech(x)^4)^(7/2),x,3,a^3*cosh(x)*sinh(x)*sqrt(a*sech(x)^4)-2*a^3*sinh(x)^2*sqrt(a*sech(x)^4)*tanh(x)+3*a^3*sinh(x)^2*sqrt(a*sech(x)^4)*tanh(x)^3-20/7*a^3*sinh(x)^2*sqrt(a*sech(x)^4)*tanh(x)^5+5/3*a^3*sinh(x)^2*sqrt(a*sech(x)^4)*tanh(x)^7-6/11*a^3*sinh(x)^2*sqrt(a*sech(x)^4)*tanh(x)^9+1/13*a^3*sinh(x)^2*sqrt(a*sech(x)^4)*tanh(x)^11],
[(a*sech(x)^4)^(5/2),x,3,a^2*cosh(x)*sinh(x)*sqrt(a*sech(x)^4)-4/3*a^2*sinh(x)^2*sqrt(a*sech(x)^4)*tanh(x)+6/5*a^2*sinh(x)^2*sqrt(a*sech(x)^4)*tanh(x)^3-4/7*a^2*sinh(x)^2*sqrt(a*sech(x)^4)*tanh(x)^5+1/9*a^2*sinh(x)^2*sqrt(a*sech(x)^4)*tanh(x)^7],
[(a*sech(x)^4)^(3/2),x,3,a*cosh(x)*sinh(x)*sqrt(a*sech(x)^4)-2/3*a*sinh(x)^2*sqrt(a*sech(x)^4)*tanh(x)+1/5*a*sinh(x)^2*sqrt(a*sech(x)^4)*tanh(x)^3],
[(a*sech(x)^4)^(1/2),x,3,cosh(x)*sinh(x)*sqrt(a*sech(x)^4)],
[1/(a*sech(x)^4)^(1/2),x,3,1/2*x*sech(x)^2/sqrt(a*sech(x)^4)+1/2*tanh(x)/sqrt(a*sech(x)^4)],
[1/(a*sech(x)^4)^(3/2),x,5,5/16*x*sech(x)^2/(a*sqrt(a*sech(x)^4))+5/24*cosh(x)*sinh(x)/(a*sqrt(a*sech(x)^4))+1/6*cosh(x)^3*sinh(x)/(a*sqrt(a*sech(x)^4))+5/16*tanh(x)/(a*sqrt(a*sech(x)^4))],
[1/(a*sech(x)^4)^(5/2),x,7,63/256*x*sech(x)^2/(a^2*sqrt(a*sech(x)^4))+21/128*cosh(x)*sinh(x)/(a^2*sqrt(a*sech(x)^4))+21/160*cosh(x)^3*sinh(x)/(a^2*sqrt(a*sech(x)^4))+9/80*cosh(x)^5*sinh(x)/(a^2*sqrt(a*sech(x)^4))+1/10*cosh(x)^7*sinh(x)/(a^2*sqrt(a*sech(x)^4))+63/256*tanh(x)/(a^2*sqrt(a*sech(x)^4))],

# Integrands of the form (c Sech[a+b x]^m)^n

# Integrands of the form Hyper[c+d x]^m (a+b Sech[c+d x])^n

# Integrands of the form Sinh[c+d x]^m (a+b Sech[c+d x])^n

# a^2-b^2=0
[sinh(x)^4/(a+a*sech(x)),x,7,-1/8*x/a-1/8*cosh(x)*sinh(x)/a+1/4*cosh(x)^3*sinh(x)/a-1/3*sinh(x)^3/a],
[sinh(x)^3/(a+a*sech(x)),x,6,1/3*cosh(x)^3/a-1/2*sinh(x)^2/a],
[sinh(x)^2/(a+a*sech(x)),x,5,1/2*x/a-sinh(x)/a+1/2*cosh(x)*sinh(x)/a],
[sinh(x)/(a+a*sech(x)),x,5,cosh(x)/a-log(1+cosh(x))/a],
[csch(x)/(a+a*sech(x)),x,6,-1/2*arctanh(cosh(x))/a-1/2*coth(x)*csch(x)/a+1/2*csch(x)^2/a],
[csch(x)^2/(a+a*sech(x)),x,6,-1/3*coth(x)^3/a+1/3*csch(x)^3/a],
[csch(x)^3/(a+a*sech(x)),x,7,1/8*arctanh(cosh(x))/a-1/8*coth(x)*csch(x)/a-1/4*coth(x)*csch(x)^3/a+1/4*csch(x)^4/a],
[csch(x)^4/(a+a*sech(x)),x,7,1/3*coth(x)^3/a-1/5*coth(x)^5/a+1/5*csch(x)^5/a],

# a^2-b^2!=0
[sinh(x)^4/(a+b*sech(x)),x,6,1/8*(3*a^4-12*a^2*b^2+8*b^4)*x/a^5-2*(a-b)^(3/2)*b*(a+b)^(3/2)*arctan(sqrt(a-b)*tanh(1/2*x)/sqrt(a+b))/a^5+1/8*(8*b*(a^2-b^2)-a*(3*a^2-4*b^2)*cosh(x))*sinh(x)/a^4-1/12*(4*b-3*a*cosh(x))*sinh(x)^3/a^2],
[sinh(x)^3/(a+b*sech(x)),x,5,-(a^2-b^2)*cosh(x)/a^3-1/2*b*cosh(x)^2/a^2+1/3*cosh(x)^3/a+b*(a^2-b^2)*log(b+a*cosh(x))/a^4],
[sinh(x)^2/(a+b*sech(x)),x,5,-1/2*(a^2-2*b^2)*x/a^3-1/2*(2*b-a*cosh(x))*sinh(x)/a^2+2*b*arctan(sqrt(a-b)*tanh(1/2*x)/sqrt(a+b))*sqrt(a-b)*sqrt(a+b)/a^3],
[sinh(x)/(a+b*sech(x)),x,5,cosh(x)/a-b*log(b+a*cosh(x))/a^2],
[csch(x)/(a+b*sech(x)),x,4,1/2*log(1-cosh(x))/(a+b)-1/2*log(1+cosh(x))/(a-b)+b*log(b+a*cosh(x))/(a^2-b^2)],
[csch(x)^2/(a+b*sech(x)),x,5,2*a*b*arctan(sqrt(a-b)*tanh(1/2*x)/sqrt(a+b))/((a-b)^(3/2)*(a+b)^(3/2))+(b-a*cosh(x))*csch(x)/(a^2-b^2)],
[csch(x)^3/(a+b*sech(x)),x,6,1/2*(b-a*cosh(x))*csch(x)^2/(a^2-b^2)-1/4*a*log(1-cosh(x))/(a+b)^2+1/4*a*log(1+cosh(x))/(a-b)^2-a^2*b*log(b+a*cosh(x))/(a^2-b^2)^2],
[csch(x)^4/(a+b*sech(x)),x,6,-2*a^3*b*arctan(sqrt(a-b)*tanh(1/2*x)/sqrt(a+b))/((a-b)^(5/2)*(a+b)^(5/2))-1/3*(3*a^2*b-a*(2*a^2+b^2)*cosh(x))*csch(x)/(a^2-b^2)^2+1/3*(b-a*cosh(x))*csch(x)^3/(a^2-b^2)],

# Integrands of the form Cosh[c+d x]^m (a+b Sech[c+d x])^n

# a^2-b^2=0
[cosh(x)^4/(a+a*sech(x)),x,7,15/8*x/a-4*sinh(x)/a+15/8*cosh(x)*sinh(x)/a+5/4*cosh(x)^3*sinh(x)/a-cosh(x)^3*sinh(x)/(a+a*sech(x))-4/3*sinh(x)^3/a],
[cosh(x)^3/(a+a*sech(x)),x,6,-3/2*x/a+4*sinh(x)/a-3/2*cosh(x)*sinh(x)/a-cosh(x)^2*sinh(x)/(a+a*sech(x))+4/3*sinh(x)^3/a],
[cosh(x)^2/(a+a*sech(x)),x,5,3/2*x/a-2*sinh(x)/a+3/2*cosh(x)*sinh(x)/a-cosh(x)*sinh(x)/(a+a*sech(x))],
[cosh(x)/(a+a*sech(x)),x,4,-x/a+2*sinh(x)/a-sinh(x)/(a+a*sech(x))],
[sech(x)/(a+a*sech(x)),x,1,tanh(x)/(a+a*sech(x))],
[sech(x)^2/(a+a*sech(x)),x,3,arctan(sinh(x))/a-tanh(x)/(a+a*sech(x))],
[sech(x)^3/(a+a*sech(x)),x,4,-arctan(sinh(x))/a+tanh(x)/a+tanh(x)/(a+a*sech(x))],
[sech(x)^4/(a+a*sech(x)),x,6,3/2*arctan(sinh(x))/a-2*tanh(x)/a+3/2*sech(x)*tanh(x)/a-sech(x)^2*tanh(x)/(a+a*sech(x))],
[1/(a+a*sech(c+d*x)),x,2,x/a-tanh(c+d*x)/(d*(a+a*sech(c+d*x)))],
[1/(a-a*sech(c+d*x)),x,2,x/a-tanh(c+d*x)/(d*(a-a*sech(c+d*x)))],
[(a+a*sech(c+d*x))^(5/2),x,5,2*a^(5/2)*arctanh(sqrt(a)*tanh(c+d*x)/sqrt(a+a*sech(c+d*x)))/d+14/3*a^3*tanh(c+d*x)/(d*sqrt(a+a*sech(c+d*x)))+2/3*a^2*sqrt(a+a*sech(c+d*x))*tanh(c+d*x)/d],
[(a+a*sech(c+d*x))^(3/2),x,4,2*a^(3/2)*arctanh(sqrt(a)*tanh(c+d*x)/sqrt(a+a*sech(c+d*x)))/d+2*a^2*tanh(c+d*x)/(d*sqrt(a+a*sech(c+d*x)))],
[(a+a*sech(c+d*x))^(1/2),x,2,2*arctanh(sqrt(a)*tanh(c+d*x)/sqrt(a+a*sech(c+d*x)))*sqrt(a)/d],
[1/(a+a*sech(c+d*x))^(1/2),x,5,2*arctanh(sqrt(a)*tanh(c+d*x)/sqrt(a+a*sech(c+d*x)))/(d*sqrt(a))-arctanh(sqrt(a)*tanh(c+d*x)/(sqrt(2)*sqrt(a+a*sech(c+d*x))))*sqrt(2)/(d*sqrt(a))],
[1/(a+a*sech(c+d*x))^(3/2),x,6,2*arctanh(sqrt(a)*tanh(c+d*x)/sqrt(a+a*sech(c+d*x)))/(a^(3/2)*d)-5/2*arctanh(sqrt(a)*tanh(c+d*x)/(sqrt(2)*sqrt(a+a*sech(c+d*x))))/(a^(3/2)*d*sqrt(2))-1/2*tanh(c+d*x)/(d*(a+a*sech(c+d*x))^(3/2))],
[sqrt(a-a*sech(c+d*x)),x,2,2*arctanh(sqrt(a)*tanh(c+d*x)/sqrt(a-a*sech(c+d*x)))*sqrt(a)/d],
[1/sqrt(a-a*sech(c+d*x)),x,5,2*arctanh(sqrt(a)*tanh(c+d*x)/sqrt(a-a*sech(c+d*x)))/(d*sqrt(a))-arctanh(sqrt(a)*tanh(c+d*x)/(sqrt(2)*sqrt(a-a*sech(c+d*x))))*sqrt(2)/(d*sqrt(a))],
[sqrt(3+3*sech(x)),x,2,2*arctanh(tanh(x)/sqrt(1+sech(x)))*sqrt(3)],
[sqrt(3-3*sech(x)),x,2,2*arctanh(tanh(x)/sqrt(1-sech(x)))*sqrt(3)],

# a^2-b^2!=0
[(a+b*sech(c+d*x))^4,x,6,a^4*x+2*a*b*(2*a^2+b^2)*arctan(sinh(c+d*x))/d+1/3*b^2*(17*a^2+2*b^2)*tanh(c+d*x)/d+4/3*a*b^3*sech(c+d*x)*tanh(c+d*x)/d+1/3*b^2*(a+b*sech(c+d*x))^2*tanh(c+d*x)/d],
[(a+b*sech(c+d*x))^3,x,5,a^3*x+1/2*b*(6*a^2+b^2)*arctan(sinh(c+d*x))/d+5/2*a*b^2*tanh(c+d*x)/d+1/2*b^2*(a+b*sech(c+d*x))*tanh(c+d*x)/d],
[(a+b*sech(c+d*x))^2,x,4,a^2*x+2*a*b*arctan(sinh(c+d*x))/d+b^2*tanh(c+d*x)/d],
[a+b*sech(c+d*x),x,2,a*x+b*arctan(sinh(c+d*x))/d],
[1/(a+b*sech(c+d*x)),x,3,x/a-2*b*arctan(sqrt(a-b)*tanh(1/2*(c+d*x))/sqrt(a+b))/(a*d*sqrt(a-b)*sqrt(a+b))],
[1/(a+b*sech(c+d*x))^2,x,5,x/a^2-2*b*(2*a^2-b^2)*arctan(sqrt(a-b)*tanh(1/2*(c+d*x))/sqrt(a+b))/(a^2*(a-b)^(3/2)*(a+b)^(3/2)*d)+b^2*tanh(c+d*x)/(a*(a^2-b^2)*d*(a+b*sech(c+d*x)))],
[1/(a+b*sech(c+d*x))^3,x,6,x/a^3-b*(6*a^4-5*a^2*b^2+2*b^4)*arctan(sqrt(a-b)*tanh(1/2*(c+d*x))/sqrt(a+b))/(a^3*(a-b)^(5/2)*(a+b)^(5/2)*d)+1/2*b^2*tanh(c+d*x)/(a*(a^2-b^2)*d*(a+b*sech(c+d*x))^2)+1/2*b^2*(5*a^2-2*b^2)*tanh(c+d*x)/(a^2*(a^2-b^2)^2*d*(a+b*sech(c+d*x)))],
[1/sqrt(a+b*sech(c+d*x)),x,1,2*coth(c+d*x)*EllipticPi(sqrt(a+b*sech(c+d*x))/sqrt(a+b),(a+b)/a,sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sech(c+d*x))/(a+b))*sqrt(-b*(1+sech(c+d*x))/(a-b))/(a*d)],
[cosh(x)^4/(a+b*sech(x)),x,8,1/8*(3*a^4+4*a^2*b^2+8*b^4)*x/a^5-1/3*b*(2*a^2+3*b^2)*sinh(x)/a^4+1/8*(3*a^2+4*b^2)*cosh(x)*sinh(x)/a^3-1/3*b*cosh(x)^2*sinh(x)/a^2+1/4*cosh(x)^3*sinh(x)/a-2*b^5*arctan(sqrt(a-b)*tanh(1/2*x)/sqrt(a+b))/(a^5*sqrt(a-b)*sqrt(a+b))],
[cosh(x)^3/(a+b*sech(x)),x,7,-1/2*b*(a^2+2*b^2)*x/a^4+1/3*(2*a^2+3*b^2)*sinh(x)/a^3-1/2*b*cosh(x)*sinh(x)/a^2+1/3*cosh(x)^2*sinh(x)/a+2*b^4*arctan(sqrt(a-b)*tanh(1/2*x)/sqrt(a+b))/(a^4*sqrt(a-b)*sqrt(a+b))],
[cosh(x)^2/(a+b*sech(x)),x,6,1/2*(a^2+2*b^2)*x/a^3-b*sinh(x)/a^2+1/2*cosh(x)*sinh(x)/a-2*b^3*arctan(sqrt(a-b)*tanh(1/2*x)/sqrt(a+b))/(a^3*sqrt(a-b)*sqrt(a+b))],
[cosh(x)/(a+b*sech(x)),x,5,-b*x/a^2+sinh(x)/a+2*b^2*arctan(sqrt(a-b)*tanh(1/2*x)/sqrt(a+b))/(a^2*sqrt(a-b)*sqrt(a+b))],
[sech(x)/(a+b*sech(x)),x,3,2*arctan(sqrt(a-b)*tanh(1/2*x)/sqrt(a+b))/(sqrt(a-b)*sqrt(a+b))],
[sech(x)^2/(a+b*sech(x)),x,5,arctan(sinh(x))/b-2*a*arctan(sqrt(a-b)*tanh(1/2*x)/sqrt(a+b))/(b*sqrt(a-b)*sqrt(a+b))],
[sech(x)^3/(a+b*sech(x)),x,6,-a*arctan(sinh(x))/b^2+2*a^2*arctan(sqrt(a-b)*tanh(1/2*x)/sqrt(a+b))/(b^2*sqrt(a-b)*sqrt(a+b))+tanh(x)/b],
[sech(x)^4/(a+b*sech(x)),x,7,1/2*(2*a^2+b^2)*arctan(sinh(x))/b^3-2*a^3*arctan(sqrt(a-b)*tanh(1/2*x)/sqrt(a+b))/(b^3*sqrt(a-b)*sqrt(a+b))-a*tanh(x)/b^2+1/2*sech(x)*tanh(x)/b],

# Integrands of the form Tanh[c+d x]^m (a+b Sech[c+d x])^n when a^2-b^2=0

# n>0

# n<0
[tanh(x)^6/(a+a*sech(x)),x,5,x/a-3/8*arctan(sinh(x))/a-1/8*(8-3*sech(x))*tanh(x)/a-1/12*(4-3*sech(x))*tanh(x)^3/a],
[tanh(x)^5/(a+a*sech(x)),x,3,log(cosh(x))/a+sech(x)/a+1/2*sech(x)^2/a-1/3*sech(x)^3/a],
[tanh(x)^4/(a+a*sech(x)),x,4,x/a-1/2*arctan(sinh(x))/a-1/2*(2-sech(x))*tanh(x)/a],
[tanh(x)^3/(a+a*sech(x)),x,3,log(cosh(x))/a+sech(x)/a],
[tanh(x)^2/(a+a*sech(x)),x,3,x/a-arctan(sinh(x))/a],
[tanh(x)/(a+a*sech(x)),x,2,log(1+cosh(x))/a],
[coth(x)/(a+a*sech(x)),x,3,1/2/(a*(1+cosh(x)))+1/4*log(1-cosh(x))/a+3/4*log(1+cosh(x))/a],
[coth(x)^2/(a+a*sech(x)),x,4,x/a-1/3*coth(x)*(3-2*sech(x))/a-1/3*coth(x)^3*(1-sech(x))/a],
[coth(x)^3/(a+a*sech(x)),x,3,1/8/(a*(1-cosh(x)))+(-1/8)/(a*(1+cosh(x))^2)+3/4/(a*(1+cosh(x)))+5/16*log(1-cosh(x))/a+11/16*log(1+cosh(x))/a],
[coth(x)^4/(a+a*sech(x)),x,5,x/a-1/15*coth(x)*(15-8*sech(x))/a-1/15*coth(x)^3*(5-4*sech(x))/a-1/5*coth(x)^5*(1-sech(x))/a],

# Integrands of the form Tanh[c+d x]^m (a+b Sech[c+d x])^n

# n>0

# n<0
[tanh(x)^7/(a+b*sech(x)),x,3,log(cosh(x))/a-(a^2-b^2)^3*log(a+b*sech(x))/(a*b^6)+(a^4-3*a^2*b^2+3*b^4)*sech(x)/b^5-1/2*a*(a^2-3*b^2)*sech(x)^2/b^4+1/3*(a^2-3*b^2)*sech(x)^3/b^3-1/4*a*sech(x)^4/b^2+1/5*sech(x)^5/b],
[tanh(x)^6/(a+b*sech(x)),x,15,x/a-3/8*arctan(sinh(x))/b-1/2*(a^2-3*b^2)*arctan(sinh(x))/b^3-(a^4-3*a^2*b^2+3*b^4)*arctan(sinh(x))/b^5+2*(a-b)^(5/2)*(a+b)^(5/2)*arctan(sqrt(a-b)*tanh(1/2*x)/sqrt(a+b))/(a*b^5)+a*tanh(x)/b^2+a*(a^2-3*b^2)*tanh(x)/b^4-3/8*sech(x)*tanh(x)/b-1/2*(a^2-3*b^2)*sech(x)*tanh(x)/b^3-1/4*sech(x)^3*tanh(x)/b-1/3*a*tanh(x)^3/b^2],
[tanh(x)^5/(a+b*sech(x)),x,3,log(cosh(x))/a+(a^2-b^2)^2*log(a+b*sech(x))/(a*b^4)-(a^2-2*b^2)*sech(x)/b^3+1/2*a*sech(x)^2/b^2-1/3*sech(x)^3/b],
[tanh(x)^4/(a+b*sech(x)),x,6,x/a+1/2*(2*a^2-3*b^2)*arctan(sinh(x))/b^3-2*(a-b)^(3/2)*(a+b)^(3/2)*arctan(sqrt(a-b)*tanh(1/2*x)/sqrt(a+b))/(a*b^3)-a*tanh(x)/b^2+1/2*sech(x)*tanh(x)/b],
[tanh(x)^3/(a+b*sech(x)),x,3,log(cosh(x))/a+(1-a^2/b^2)*log(a+b*sech(x))/a+sech(x)/b],
[tanh(x)^2/(a+b*sech(x)),x,7,x/a-arctan(sinh(x))/b+2*arctan(sqrt(a-b)*tanh(1/2*x)/sqrt(a+b))*sqrt(a-b)*sqrt(a+b)/(a*b)],
[tanh(x)/(a+b*sech(x)),x,4,log(cosh(x))/a+log(a+b*sech(x))/a],
[coth(x)/(a+b*sech(x)),x,3,log(cosh(x))/a+1/2*log(1-sech(x))/(a+b)+1/2*log(1+sech(x))/(a-b)-b^2*log(a+b*sech(x))/(a*(a^2-b^2))],
[coth(x)^2/(a+b*sech(x)),x,9,a*x/(a^2-b^2)-b^2*x/(a*(a^2-b^2))+2*b^3*arctan(sqrt(a-b)*tanh(1/2*x)/sqrt(a+b))/(a*(a-b)^(3/2)*(a+b)^(3/2))-a*coth(x)/(a^2-b^2)+b*csch(x)/(a^2-b^2)],
[coth(x)^3/(a+b*sech(x)),x,3,log(cosh(x))/a+1/4*(2*a+3*b)*log(1-sech(x))/(a+b)^2+1/4*(2*a-3*b)*log(1+sech(x))/(a-b)^2+b^4*log(a+b*sech(x))/(a*(a^2-b^2)^2)+(-1/4)/((a+b)*(1-sech(x)))+(-1/4)/((a-b)*(1+sech(x)))],
[coth(x)^4/(a+b*sech(x)),x,15,-a*b^2*x/(a^2-b^2)^2+b^4*x/(a*(a^2-b^2)^2)+a*x/(a^2-b^2)-2*b^5*arctan(sqrt(a-b)*tanh(1/2*x)/sqrt(a+b))/(a*(a-b)^(5/2)*(a+b)^(5/2))+a*b^2*coth(x)/(a^2-b^2)^2-a*coth(x)/(a^2-b^2)-1/3*a*coth(x)^3/(a^2-b^2)-b^3*csch(x)/(a^2-b^2)^2+b*csch(x)/(a^2-b^2)+1/3*b*csch(x)^3/(a^2-b^2)],
[coth(x)^5/(a+b*sech(x)),x,3,log(cosh(x))/a+1/16*(8*a^2+21*a*b+15*b^2)*log(1-sech(x))/(a+b)^3+1/16*(8*a^2-21*a*b+15*b^2)*log(1+sech(x))/(a-b)^3-b^6*log(a+b*sech(x))/(a*(a^2-b^2)^3)+(-1/16)/((a+b)*(1-sech(x))^2)+1/16*(-5*a-7*b)/((a+b)^2*(1-sech(x)))+(-1/16)/((a-b)*(1+sech(x))^2)+1/16*(-5*a+7*b)/((a-b)^2*(1+sech(x)))],

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

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

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

# Integrands of the form E^(a+b x) Sech[c+d x]^m

# Integrands of the form E^(c (a+b x)) (Sech[a c+b c x]^2)^(m/2)
[exp(c*(a+b*x))*(sech(a*c+b*c*x)^2)^(7/2),x,6,32/3*cosh(a*c+b*c*x)*sqrt(sech(a*c+b*c*x)^2)/(b*c*(1+exp(2*c*(a+b*x)))^6)-192/5*cosh(a*c+b*c*x)*sqrt(sech(a*c+b*c*x)^2)/(b*c*(1+exp(2*c*(a+b*x)))^5)+48*cosh(a*c+b*c*x)*sqrt(sech(a*c+b*c*x)^2)/(b*c*(1+exp(2*c*(a+b*x)))^4)-64/3*cosh(a*c+b*c*x)*sqrt(sech(a*c+b*c*x)^2)/(b*c*(1+exp(2*c*(a+b*x)))^3)],
[exp(c*(a+b*x))*(sech(a*c+b*c*x)^2)^(5/2),x,6,-4*cosh(a*c+b*c*x)*sqrt(sech(a*c+b*c*x)^2)/(b*c*(1+exp(2*c*(a+b*x)))^4)+32/3*cosh(a*c+b*c*x)*sqrt(sech(a*c+b*c*x)^2)/(b*c*(1+exp(2*c*(a+b*x)))^3)-8*cosh(a*c+b*c*x)*sqrt(sech(a*c+b*c*x)^2)/(b*c*(1+exp(2*c*(a+b*x)))^2)],
[exp(c*(a+b*x))*(sech(a*c+b*c*x)^2)^(3/2),x,4,2*exp(4*c*(a+b*x))*cosh(a*c+b*c*x)*sqrt(sech(a*c+b*c*x)^2)/(b*c*(1+exp(2*c*(a+b*x)))^2)],
[exp(c*(a+b*x))*sqrt(sech(a*c+b*c*x)^2),x,4,cosh(a*c+b*c*x)*log(1+exp(2*c*(a+b*x)))*sqrt(sech(a*c+b*c*x)^2)/(b*c)],
[exp(c*(a+b*x))/sqrt(sech(a*c+b*c*x)^2),x,5,1/4*exp(2*c*(a+b*x))*sech(a*c+b*c*x)/(b*c*sqrt(sech(a*c+b*c*x)^2))+1/2*x*sech(a*c+b*c*x)/sqrt(sech(a*c+b*c*x)^2)],
[exp(c*(a+b*x))/(sech(a*c+b*c*x)^2)^(3/2),x,6,-1/16*sech(a*c+b*c*x)/(b*c*exp(2*c*(a+b*x))*sqrt(sech(a*c+b*c*x)^2))+3/16*exp(2*c*(a+b*x))*sech(a*c+b*c*x)/(b*c*sqrt(sech(a*c+b*c*x)^2))+1/32*exp(4*c*(a+b*x))*sech(a*c+b*c*x)/(b*c*sqrt(sech(a*c+b*c*x)^2))+3/8*x*sech(a*c+b*c*x)/sqrt(sech(a*c+b*c*x)^2)],
[exp(c*(a+b*x))/(sech(a*c+b*c*x)^2)^(5/2),x,6,-1/128*sech(a*c+b*c*x)/(b*c*exp(4*c*(a+b*x))*sqrt(sech(a*c+b*c*x)^2))-5/64*sech(a*c+b*c*x)/(b*c*exp(2*c*(a+b*x))*sqrt(sech(a*c+b*c*x)^2))+5/32*exp(2*c*(a+b*x))*sech(a*c+b*c*x)/(b*c*sqrt(sech(a*c+b*c*x)^2))+5/128*exp(4*c*(a+b*x))*sech(a*c+b*c*x)/(b*c*sqrt(sech(a*c+b*c*x)^2))+1/192*exp(6*c*(a+b*x))*sech(a*c+b*c*x)/(b*c*sqrt(sech(a*c+b*c*x)^2))+5/16*x*sech(a*c+b*c*x)/sqrt(sech(a*c+b*c*x)^2)],

# Integrands of the form x^m Sech[a+b Log[c x^n]]^p

# Integrands of the form x^m Sech[b Log[c x^n]]^(p/2)
[x^5/sech(2*log(c*x))^(1/2),x,6,2/21*x^2/(c^4*sqrt(sech(2*log(c*x))))+1/7*x^6/sqrt(sech(2*log(c*x)))+1/21*(c^2+1/x^2)*sqrt(cos(2*arccot(c*x))^2)/cos(2*arccot(c*x))*EllipticF(sin(2*arccot(c*x)),sqrt(1/2))*sqrt((c^4+1/x^4)/(c^2+1/x^2)^2)/(c^5*(c^4+1/x^4)*x*sqrt(sech(2*log(c*x))))],
[x^4/sech(2*log(c*x))^(1/2),x,3,1/6*(c^4+1/x^4)*x^5/(c^4*sqrt(sech(2*log(c*x))))],
[x^3/sech(2*log(c*x))^(1/2),x,8,2/5/(c^4*sqrt(sech(2*log(c*x))))+(-2/5)/(c^4*(c^2+1/x^2)*x^2*sqrt(sech(2*log(c*x))))+1/5*x^4/sqrt(sech(2*log(c*x)))+2/5*(c^2+1/x^2)*sqrt(cos(2*arccot(c*x))^2)/cos(2*arccot(c*x))*EllipticE(sin(2*arccot(c*x)),sqrt(1/2))*sqrt((c^4+1/x^4)/(c^2+1/x^2)^2)/(c^3*(c^4+1/x^4)*x*sqrt(sech(2*log(c*x))))-1/5*(c^2+1/x^2)*sqrt(cos(2*arccot(c*x))^2)/cos(2*arccot(c*x))*EllipticF(sin(2*arccot(c*x)),sqrt(1/2))*sqrt((c^4+1/x^4)/(c^2+1/x^2)^2)/(c^3*(c^4+1/x^4)*x*sqrt(sech(2*log(c*x))))],
[x^2/sech(2*log(c*x))^(1/2),x,6,1/4*x^3/sqrt(sech(2*log(c*x)))+1/4*arctanh(sqrt(1+1/(c^4*x^4)))/(c^4*x*sqrt(1+1/(c^4*x^4))*sqrt(sech(2*log(c*x))))],
[x/sech(2*log(c*x))^(1/2),x,5,1/3*x^2/sqrt(sech(2*log(c*x)))-1/3*(c^2+1/x^2)*sqrt(cos(2*arccot(c*x))^2)/cos(2*arccot(c*x))*EllipticF(sin(2*arccot(c*x)),sqrt(1/2))*sqrt((c^4+1/x^4)/(c^2+1/x^2)^2)/(c*(c^4+1/x^4)*x*sqrt(sech(2*log(c*x))))],
[1/sech(2*log(c*x))^(1/2),x,6,1/2*x/sqrt(sech(2*log(c*x)))-1/2*arccsch(c^2*x^2)/(c^2*x*sqrt(1+1/(c^4*x^4))*sqrt(sech(2*log(c*x))))],
[sech(2*log(c*x))^(1/2)/x,x,3,-I*sqrt(cos(I*log(c*x))^2)/cos(I*log(c*x))*EllipticF(sin(I*log(c*x)),sqrt(2))*sqrt(cosh(2*log(c*x)))*sqrt(sech(2*log(c*x)))],
[sech(2*log(c*x))^(1/2)/x^2,x,5,-1/2*c^2*x*arccsch(c^2*x^2)*sqrt(1+1/(c^4*x^4))*sqrt(sech(2*log(c*x)))],
[sech(2*log(c*x))^(1/2)/x^3,x,6,-(c^4+1/x^4)*sqrt(sech(2*log(c*x)))/(c^2+1/x^2)+c*(c^2+1/x^2)*x*sqrt(cos(2*arccot(c*x))^2)/cos(2*arccot(c*x))*EllipticE(sin(2*arccot(c*x)),sqrt(1/2))*sqrt((c^4+1/x^4)/(c^2+1/x^2)^2)*sqrt(sech(2*log(c*x)))-1/2*c*(c^2+1/x^2)*x*sqrt(cos(2*arccot(c*x))^2)/cos(2*arccot(c*x))*EllipticF(sin(2*arccot(c*x)),sqrt(1/2))*sqrt((c^4+1/x^4)/(c^2+1/x^2)^2)*sqrt(sech(2*log(c*x)))],
[sech(2*log(c*x))^(1/2)/x^4,x,3,-1/2*(c^4+1/x^4)*x*sqrt(sech(2*log(c*x)))],
[sech(2*log(c*x))^(1/2)/x^5,x,5,-1/3*(c^4+1/x^4)*sqrt(sech(2*log(c*x)))+1/6*c^3*(c^2+1/x^2)*x*sqrt(cos(2*arccot(c*x))^2)/cos(2*arccot(c*x))*EllipticF(sin(2*arccot(c*x)),sqrt(1/2))*sqrt((c^4+1/x^4)/(c^2+1/x^2)^2)*sqrt(sech(2*log(c*x)))],
[x^8/sech(2*log(c*x))^(3/2),x,8,1/32*x/(c^4*(c^4+1/x^4)*sech(2*log(c*x))^(3/2))+1/16*x^5/((c^4+1/x^4)*sech(2*log(c*x))^(3/2))+1/12*x^9/sech(2*log(c*x))^(3/2)-1/32*arctanh(sqrt(1+1/(c^4*x^4)))/(c^12*(1+1/(c^4*x^4))^(3/2)*x^3*sech(2*log(c*x))^(3/2))],
[x^7/sech(2*log(c*x))^(3/2),x,7,4/77/(c^4*(c^4+1/x^4)*sech(2*log(c*x))^(3/2))+6/77*x^4/((c^4+1/x^4)*sech(2*log(c*x))^(3/2))+1/11*x^8/sech(2*log(c*x))^(3/2)+2/77*(c^2+1/x^2)*sqrt(cos(2*arccot(c*x))^2)/cos(2*arccot(c*x))*EllipticF(sin(2*arccot(c*x)),sqrt(1/2))*sqrt((c^4+1/x^4)/(c^2+1/x^2)^2)/(c^5*(c^4+1/x^4)^2*x^3*sech(2*log(c*x))^(3/2))],
[x^6/sech(2*log(c*x))^(3/2),x,3,1/10*(c^4+1/x^4)*x^7/(c^4*sech(2*log(c*x))^(3/2))],
[x^5/sech(2*log(c*x))^(3/2),x,9,(-4/15)/(c^4*(c^4+1/x^4)*(c^2+1/x^2)*x^4*sech(2*log(c*x))^(3/2))+4/15/(c^4*(c^4+1/x^4)*x^2*sech(2*log(c*x))^(3/2))+2/15*x^2/((c^4+1/x^4)*sech(2*log(c*x))^(3/2))+1/9*x^6/sech(2*log(c*x))^(3/2)+4/15*(c^2+1/x^2)*sqrt(cos(2*arccot(c*x))^2)/cos(2*arccot(c*x))*EllipticE(sin(2*arccot(c*x)),sqrt(1/2))*sqrt((c^4+1/x^4)/(c^2+1/x^2)^2)/(c^3*(c^4+1/x^4)^2*x^3*sech(2*log(c*x))^(3/2))-2/15*(c^2+1/x^2)*sqrt(cos(2*arccot(c*x))^2)/cos(2*arccot(c*x))*EllipticF(sin(2*arccot(c*x)),sqrt(1/2))*sqrt((c^4+1/x^4)/(c^2+1/x^2)^2)/(c^3*(c^4+1/x^4)^2*x^3*sech(2*log(c*x))^(3/2))],
[x^4/sech(2*log(c*x))^(3/2),x,7,3/16*x/((c^4+1/x^4)*sech(2*log(c*x))^(3/2))+1/8*x^5/sech(2*log(c*x))^(3/2)+3/16*arctanh(sqrt(1+1/(c^4*x^4)))/(c^8*(1+1/(c^4*x^4))^(3/2)*x^3*sech(2*log(c*x))^(3/2))],
[x^3/sech(2*log(c*x))^(3/2),x,6,2/7/((c^4+1/x^4)*sech(2*log(c*x))^(3/2))+1/7*x^4/sech(2*log(c*x))^(3/2)-2/7*(c^2+1/x^2)*sqrt(cos(2*arccot(c*x))^2)/cos(2*arccot(c*x))*EllipticF(sin(2*arccot(c*x)),sqrt(1/2))*sqrt((c^4+1/x^4)/(c^2+1/x^2)^2)/(c*(c^4+1/x^4)^2*x^3*sech(2*log(c*x))^(3/2))],
[x^2/sech(2*log(c*x))^(3/2),x,7,1/2/((c^4+1/x^4)*x*sech(2*log(c*x))^(3/2))+1/6*x^3/sech(2*log(c*x))^(3/2)-1/2*arccsch(c^2*x^2)/(c^6*(1+1/(c^4*x^4))^(3/2)*x^3*sech(2*log(c*x))^(3/2))],
[x/sech(2*log(c*x))^(3/2),x,8,(-12/5)/((c^4+1/x^4)*(c^2+1/x^2)*x^4*sech(2*log(c*x))^(3/2))+6/5/((c^4+1/x^4)*x^2*sech(2*log(c*x))^(3/2))+1/5*x^2/sech(2*log(c*x))^(3/2)+12/5*c*(c^2+1/x^2)*sqrt(cos(2*arccot(c*x))^2)/cos(2*arccot(c*x))*EllipticE(sin(2*arccot(c*x)),sqrt(1/2))*sqrt((c^4+1/x^4)/(c^2+1/x^2)^2)/((c^4+1/x^4)^2*x^3*sech(2*log(c*x))^(3/2))-6/5*c*(c^2+1/x^2)*sqrt(cos(2*arccot(c*x))^2)/cos(2*arccot(c*x))*EllipticF(sin(2*arccot(c*x)),sqrt(1/2))*sqrt((c^4+1/x^4)/(c^2+1/x^2)^2)/((c^4+1/x^4)^2*x^3*sech(2*log(c*x))^(3/2))],
[1/sech(2*log(c*x))^(3/2),x,7,(-3/4)/((c^4+1/x^4)*x^3*sech(2*log(c*x))^(3/2))+1/4*x/sech(2*log(c*x))^(3/2)+3/4*arctanh(sqrt(1+1/(c^4*x^4)))/(c^4*(1+1/(c^4*x^4))^(3/2)*x^3*sech(2*log(c*x))^(3/2))],
[sech(2*log(c*x))^(3/2)/x,x,4,sinh(2*log(c*x))*sqrt(sech(2*log(c*x)))+I*sqrt(cos(I*log(c*x))^2)/cos(I*log(c*x))*EllipticE(sin(I*log(c*x)),sqrt(2))*sqrt(cosh(2*log(c*x)))*sqrt(sech(2*log(c*x)))],
[sech(2*log(c*x))^(3/2)/x^2,x,3,1/2*(c^4+1/x^4)*x^3*sech(2*log(c*x))^(3/2)],
[sech(2*log(c*x))^(3/2)/x^3,x,5,1/2*(c^4+1/x^4)*x^2*sech(2*log(c*x))^(3/2)-1/4*(c^4+1/x^4)*(c^2+1/x^2)*x^3*sqrt(cos(2*arccot(c*x))^2)/cos(2*arccot(c*x))*EllipticF(sin(2*arccot(c*x)),sqrt(1/2))*sech(2*log(c*x))^(3/2)*sqrt((c^4+1/x^4)/(c^2+1/x^2)^2)/c],
[sech(2*log(c*x))^(3/2)/x^4,x,6,1/2*(c^4+1/x^4)*x*sech(2*log(c*x))^(3/2)-1/2*c^6*(1+1/(c^4*x^4))^(3/2)*x^3*arccsch(c^2*x^2)*sech(2*log(c*x))^(3/2)],

# Integrands of the form x^m Sech[a+b Log[c x^n]]^p
[sech(a+b*log(c*x^n)),x,4,2*exp(a)*x*(c*x^n)^b*hypergeom([1,1/2*(b+1/n)/b],[1/2*(3+1/(b*n))],-exp(2*a)*(c*x^n)^(2*b))/(1+b*n)],
[sech(a+b*log(c*x^n))^2,x,4,4*exp(2*a)*x*(c*x^n)^(2*b)*hypergeom([2,1/2*(2+1/(b*n))],[1/2*(4+1/(b*n))],-exp(2*a)*(c*x^n)^(2*b))/(1+2*b*n)],
[sech(a+b*log(c*x^n))^3,x,4,8*exp(3*a)*x*(c*x^n)^(3*b)*hypergeom([3,1/2*(3*b+1/n)/b],[1/2*(5+1/(b*n))],-exp(2*a)*(c*x^n)^(2*b))/(1+3*b*n)],
[sech(a+b*log(c*x^n))^4,x,4,16*exp(4*a)*x*(c*x^n)^(4*b)*hypergeom([4,1/2*(4+1/(b*n))],[1/2*(6+1/(b*n))],-exp(2*a)*(c*x^n)^(2*b))/(1+4*b*n)],
[(1-b^2*n^2)*sech(a+b*log(c*x^n))+2*b^2*n^2*sech(a+b*log(c*x^n))^3,x,-9,x*sech(a+b*log(c*x^n))+b*n*x*sech(a+b*log(c*x^n))*tanh(a+b*log(c*x^n))],
[sech(a+2*log(c*x^(1/2)))^3,x,3,2*c^6/(exp(a)*(c^4+1/(exp(2*a)*x^2))^2)],
[sech(a+2*log(c/x^(1/2)))^3,x,4,2*c^2/(exp(3*a)*(exp(-2*a)+c^4/x^2)^2)],
[sech(a+log(c*x^n)/(n*(-2+p)))^p,x,3,1/2*exp(2*a)*(2-p)*x*(1+(c*x^n)^(2/(n*(2-p)))/exp(2*a))*sech(a-log(c*x^n)/(n*(2-p)))^p/((1-p)*(c*x^n)^(2/(n*(2-p))))],
[sech(a-log(c*x^n)/(n*(-2+p)))^p,x,3,1/2*(2-p)*x*(1+1/(exp(2*a)*(c*x^n)^(2/(n*(2-p)))))*sech(a+log(c*x^n)/(n*(2-p)))^p/(1-p)],
[sech(a+b*log(c*x^n))/x,x,2,arctan(sinh(a+b*log(c*x^n)))/(b*n)],
[sech(a+b*log(c*x^n))^2/x,x,3,tanh(a+b*log(c*x^n))/(b*n)],
[sech(a+b*log(c*x^n))^3/x,x,3,1/2*arctan(sinh(a+b*log(c*x^n)))/(b*n)+1/2*sech(a+b*log(c*x^n))*tanh(a+b*log(c*x^n))/(b*n)],
[sech(a+b*log(c*x^n))^4/x,x,3,tanh(a+b*log(c*x^n))/(b*n)-1/3*tanh(a+b*log(c*x^n))^3/(b*n)],
[sech(a+b*log(c*x^n))^5/x,x,4,3/8*arctan(sinh(a+b*log(c*x^n)))/(b*n)+3/8*sech(a+b*log(c*x^n))*tanh(a+b*log(c*x^n))/(b*n)+1/4*sech(a+b*log(c*x^n))^3*tanh(a+b*log(c*x^n))/(b*n)],

# Integrands of the form x^m Sech[a+b Log[c x^n]]^(p/2)
[sech(a+b*log(c*x^n))^(5/2)/x,x,4,2/3*sech(a+b*log(c*x^n))^(3/2)*sinh(a+b*log(c*x^n))/(b*n)-2/3*I*sqrt(cos(1/2*I*(a+b*log(c*x^n)))^2)/cos(1/2*I*(a+b*log(c*x^n)))*EllipticF(sin(1/2*I*(a+b*log(c*x^n))),sqrt(2))*sqrt(cosh(a+b*log(c*x^n)))*sqrt(sech(a+b*log(c*x^n)))/(b*n)],
[sech(a+b*log(c*x^n))^(3/2)/x,x,4,2*sinh(a+b*log(c*x^n))*sqrt(sech(a+b*log(c*x^n)))/(b*n)+2*I*sqrt(cos(1/2*I*(a+b*log(c*x^n)))^2)/cos(1/2*I*(a+b*log(c*x^n)))*EllipticE(sin(1/2*I*(a+b*log(c*x^n))),sqrt(2))*sqrt(cosh(a+b*log(c*x^n)))*sqrt(sech(a+b*log(c*x^n)))/(b*n)],
[sqrt(sech(a+b*log(c*x^n)))/x,x,3,-2*I*sqrt(cos(1/2*I*(a+b*log(c*x^n)))^2)/cos(1/2*I*(a+b*log(c*x^n)))*EllipticF(sin(1/2*I*(a+b*log(c*x^n))),sqrt(2))*sqrt(cosh(a+b*log(c*x^n)))*sqrt(sech(a+b*log(c*x^n)))/(b*n)],
[1/(x*sqrt(sech(a+b*log(c*x^n)))),x,3,-2*I*sqrt(cos(1/2*I*(a+b*log(c*x^n)))^2)/cos(1/2*I*(a+b*log(c*x^n)))*EllipticE(sin(1/2*I*(a+b*log(c*x^n))),sqrt(2))*sqrt(cosh(a+b*log(c*x^n)))*sqrt(sech(a+b*log(c*x^n)))/(b*n)],
[1/(x*sech(a+b*log(c*x^n))^(3/2)),x,4,2/3*sinh(a+b*log(c*x^n))/(b*n*sqrt(sech(a+b*log(c*x^n))))-2/3*I*sqrt(cos(1/2*I*(a+b*log(c*x^n)))^2)/cos(1/2*I*(a+b*log(c*x^n)))*EllipticF(sin(1/2*I*(a+b*log(c*x^n))),sqrt(2))*sqrt(cosh(a+b*log(c*x^n)))*sqrt(sech(a+b*log(c*x^n)))/(b*n)],
[1/(x*sech(a+b*log(c*x^n))^(5/2)),x,4,2/5*sinh(a+b*log(c*x^n))/(b*n*sech(a+b*log(c*x^n))^(3/2))-6/5*I*sqrt(cos(1/2*I*(a+b*log(c*x^n)))^2)/cos(1/2*I*(a+b*log(c*x^n)))*EllipticE(sin(1/2*I*(a+b*log(c*x^n))),sqrt(2))*sqrt(cosh(a+b*log(c*x^n)))*sqrt(sech(a+b*log(c*x^n)))/(b*n)]]:
