# Maple integration test file: "6 Hyperbolic functions\6.5 Hyperbolic secant\6.5.7 (d hyper)^m (a+b (c sech)^n)^p.txt"

lst:=[

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

# n>0
[(a+b*sech(c+d*x)^2)*sinh(c+d*x)^4,x,5,3/8*(a-4*b)*x-1/8*(5*a-4*b)*cosh(c+d*x)*sinh(c+d*x)/d+1/4*a*cosh(c+d*x)^3*sinh(c+d*x)/d+b*tanh(c+d*x)/d],
[(a+b*sech(c+d*x)^2)*sinh(c+d*x)^3,x,3,-(a-b)*cosh(c+d*x)/d+1/3*a*cosh(c+d*x)^3/d+b*sech(c+d*x)/d],
[(a+b*sech(c+d*x)^2)*sinh(c+d*x)^2,x,4,-1/2*(a-2*b)*x+1/2*a*cosh(c+d*x)*sinh(c+d*x)/d-b*tanh(c+d*x)/d],
[(a+b*sech(c+d*x)^2)*sinh(c+d*x),x,3,a*cosh(c+d*x)/d-b*sech(c+d*x)/d],
[csch(c+d*x)*(a+b*sech(c+d*x)^2),x,3,-(a+b)*arctanh(cosh(c+d*x))/d+b*sech(c+d*x)/d],
[csch(c+d*x)^2*(a+b*sech(c+d*x)^2),x,3,-(a+b)*coth(c+d*x)/d-b*tanh(c+d*x)/d],
[csch(c+d*x)^3*(a+b*sech(c+d*x)^2),x,4,1/2*(a+3*b)*arctanh(cosh(c+d*x))/d-1/2*(a+b)*coth(c+d*x)*csch(c+d*x)/d-b*sech(c+d*x)/d],
[csch(c+d*x)^4*(a+b*sech(c+d*x)^2),x,3,(a+2*b)*coth(c+d*x)/d-1/3*(a+b)*coth(c+d*x)^3/d+b*tanh(c+d*x)/d],
[(a+b*sech(c+d*x)^2)^2*sinh(c+d*x)^4,x,6,1/8*(3*a^2-24*a*b+8*b^2)*x-1/8*a*(a-8*b)*cosh(c+d*x)*sinh(c+d*x)/d-1/4*(a^2-8*a*b+4*b^2)*tanh(c+d*x)/d+1/4*a^2*sinh(c+d*x)^4*tanh(c+d*x)/d-1/3*b^2*tanh(c+d*x)^3/d],
[(a+b*sech(c+d*x)^2)^2*sinh(c+d*x)^3,x,3,-a*(a-2*b)*cosh(c+d*x)/d+1/3*a^2*cosh(c+d*x)^3/d+(2*a-b)*b*sech(c+d*x)/d+1/3*b^2*sech(c+d*x)^3/d],
[(a+b*sech(c+d*x)^2)^2*sinh(c+d*x)^2,x,5,-1/2*a*(a-4*b)*x+1/2*a*(a-4*b)*tanh(c+d*x)/d+1/2*a^2*sinh(c+d*x)^2*tanh(c+d*x)/d+1/3*b^2*tanh(c+d*x)^3/d],
[(a+b*sech(c+d*x)^2)^2*sinh(c+d*x),x,3,a^2*cosh(c+d*x)/d-2*a*b*sech(c+d*x)/d-1/3*b^2*sech(c+d*x)^3/d],
[csch(c+d*x)*(a+b*sech(c+d*x)^2)^2,x,4,-(a+b)^2*arctanh(cosh(c+d*x))/d+b*(2*a+b)*sech(c+d*x)/d+1/3*b^2*sech(c+d*x)^3/d],
[csch(c+d*x)^2*(a+b*sech(c+d*x)^2)^2,x,3,-(a+b)^2*coth(c+d*x)/d-2*b*(a+b)*tanh(c+d*x)/d+1/3*b^2*tanh(c+d*x)^3/d],
[csch(c+d*x)^3*(a+b*sech(c+d*x)^2)^2,x,5,1/2*(a+b)*(a+5*b)*arctanh(cosh(c+d*x))/d-1/6*(3*a^2+6*a*b+5*b^2)*coth(c+d*x)*csch(c+d*x)/d-1/3*b*(6*a+5*b)*sech(c+d*x)/d+1/3*b^2*csch(c+d*x)^2*sech(c+d*x)^3/d],
[csch(c+d*x)^4*(a+b*sech(c+d*x)^2)^2,x,3,(a+b)*(a+3*b)*coth(c+d*x)/d-1/3*(a+b)^2*coth(c+d*x)^3/d+b*(2*a+3*b)*tanh(c+d*x)/d-1/3*b^2*tanh(c+d*x)^3/d],
[(a+b*sech(c+d*x)^2)^3*sinh(c+d*x)^4,x,6,3/8*a*(a^2-12*a*b+8*b^2)*x-3/8*a*(a^2-12*a*b+8*b^2)*tanh(c+d*x)/d+1/8*b*(6*a^2-23*a*b-8*b^2)*tanh(c+d*x)^3/d-3/40*(5*a-16*b)*b^2*tanh(c+d*x)^5/d-3/8*(a-2*b)*sinh(c+d*x)^2*tanh(c+d*x)*(a+b-b*tanh(c+d*x)^2)^2/d+1/4*cosh(c+d*x)*sinh(c+d*x)^3*(a+b-b*tanh(c+d*x)^2)^3/d],
[(a+b*sech(c+d*x)^2)^3*sinh(c+d*x)^3,x,3,-a^2*(a-3*b)*cosh(c+d*x)/d+1/3*a^3*cosh(c+d*x)^3/d+3*a*(a-b)*b*sech(c+d*x)/d+1/3*(3*a-b)*b^2*sech(c+d*x)^3/d+1/5*b^3*sech(c+d*x)^5/d],
[(a+b*sech(c+d*x)^2)^3*sinh(c+d*x)^2,x,6,-1/2*a^2*(a-6*b)*x+1/4*a^3/(d*(1-tanh(c+d*x)))-3*a^2*b*tanh(c+d*x)/d+1/3*b^2*(3*a+b)*tanh(c+d*x)^3/d-1/5*b^3*tanh(c+d*x)^5/d-1/4*a^3/(d*(1+tanh(c+d*x))),-1/2*a^2*(a-6*b)*x-1/30*b*(81*a^2-28*a*b-4*b^2)*tanh(c+d*x)/d-1/30*(33*a-2*b)*b*tanh(c+d*x)*(a+b-b*tanh(c+d*x)^2)/d-7/10*b*tanh(c+d*x)*(a+b-b*tanh(c+d*x)^2)^2/d+1/2*cosh(c+d*x)*sinh(c+d*x)*(a+b-b*tanh(c+d*x)^2)^3/d],
[(a+b*sech(c+d*x)^2)^3*sinh(c+d*x),x,3,a^3*cosh(c+d*x)/d-3*a^2*b*sech(c+d*x)/d-a*b^2*sech(c+d*x)^3/d-1/5*b^3*sech(c+d*x)^5/d],
[csch(c+d*x)*(a+b*sech(c+d*x)^2)^3,x,4,-(a+b)^3*arctanh(cosh(c+d*x))/d+b*(3*a^2+3*a*b+b^2)*sech(c+d*x)/d+1/3*b^2*(3*a+b)*sech(c+d*x)^3/d+1/5*b^3*sech(c+d*x)^5/d],
[csch(c+d*x)^2*(a+b*sech(c+d*x)^2)^3,x,3,-(a+b)^3*coth(c+d*x)/d-3*b*(a+b)^2*tanh(c+d*x)/d+b^2*(a+b)*tanh(c+d*x)^3/d-1/5*b^3*tanh(c+d*x)^5/d],
[csch(c+d*x)^3*(a+b*sech(c+d*x)^2)^3,x,5,1/2*(a+b)^2*(a+7*b)*arctanh(cosh(c+d*x))/d-1/2*(a+b)^2*(a+7*b)*sech(c+d*x)/d-1/6*b*(6*a^2+15*a*b+7*b^2)*sech(c+d*x)^3/d-1/10*b^2*(5*a+7*b)*sech(c+d*x)^5/d-1/2*(a+b)*(b+a*cosh(c+d*x)^2)^2*csch(c+d*x)^2*sech(c+d*x)^5/d],
[csch(c+d*x)^4*(a+b*sech(c+d*x)^2)^3,x,3,(a+b)^2*(a+4*b)*coth(c+d*x)/d-1/3*(a+b)^3*coth(c+d*x)^3/d+3*b*(a+b)*(a+2*b)*tanh(c+d*x)/d-1/3*b^2*(3*a+4*b)*tanh(c+d*x)^3/d+1/5*b^3*tanh(c+d*x)^5/d],

# n<0
[sinh(c+d*x)^4/(a+b*sech(c+d*x)^2),x,6,1/8*(3*a^2+12*a*b+8*b^2)*x/a^3-1/8*(5*a+4*b)*cosh(c+d*x)*sinh(c+d*x)/(a^2*d)+1/4*cosh(c+d*x)^3*sinh(c+d*x)/(a*d)-(a+b)^(3/2)*arctanh(sqrt(b)*tanh(c+d*x)/sqrt(a+b))*sqrt(b)/(a^3*d)],
[sinh(c+d*x)^3/(a+b*sech(c+d*x)^2),x,4,-(a+b)*cosh(c+d*x)/(a^2*d)+1/3*cosh(c+d*x)^3/(a*d)+(a+b)*arctan(cosh(c+d*x)*sqrt(a)/sqrt(b))*sqrt(b)/(a^(5/2)*d)],
[sinh(c+d*x)^2/(a+b*sech(c+d*x)^2),x,5,-1/2*(a+2*b)*x/a^2+1/2*cosh(c+d*x)*sinh(c+d*x)/(a*d)+arctanh(sqrt(b)*tanh(c+d*x)/sqrt(a+b))*sqrt(b)*sqrt(a+b)/(a^2*d)],
[sinh(c+d*x)/(a+b*sech(c+d*x)^2),x,3,cosh(c+d*x)/(a*d)-arctan(cosh(c+d*x)*sqrt(a)/sqrt(b))*sqrt(b)/(a^(3/2)*d)],
[csch(c+d*x)/(a+b*sech(c+d*x)^2),x,4,-arctanh(cosh(c+d*x))/((a+b)*d)+arctan(cosh(c+d*x)*sqrt(a)/sqrt(b))*sqrt(b)/((a+b)*d*sqrt(a))],
[csch(c+d*x)^2/(a+b*sech(c+d*x)^2),x,3,-coth(c+d*x)/((a+b)*d)+arctanh(sqrt(b)*tanh(c+d*x)/sqrt(a+b))*sqrt(b)/((a+b)^(3/2)*d)],
[csch(c+d*x)^3/(a+b*sech(c+d*x)^2),x,5,1/2*(a-b)*arctanh(cosh(c+d*x))/((a+b)^2*d)-1/2*coth(c+d*x)*csch(c+d*x)/((a+b)*d)-arctan(cosh(c+d*x)*sqrt(a)/sqrt(b))*sqrt(a)*sqrt(b)/((a+b)^2*d)],
[csch(c+d*x)^4/(a+b*sech(c+d*x)^2),x,4,a*coth(c+d*x)/((a+b)^2*d)-1/3*coth(c+d*x)^3/((a+b)*d)-a*arctanh(sqrt(b)*tanh(c+d*x)/sqrt(a+b))*sqrt(b)/((a+b)^(5/2)*d)],
[sinh(c+d*x)^4/(a+b*sech(c+d*x)^2)^2,x,7,3/8*(a^2+8*a*b+8*b^2)*x/a^4-3/2*(a+2*b)*arctanh(sqrt(b)*tanh(c+d*x)/sqrt(a+b))*sqrt(b)*sqrt(a+b)/(a^4*d)-1/8*(5*a+6*b)*cosh(c+d*x)*sinh(c+d*x)/(a^2*d*(a+b-b*tanh(c+d*x)^2))+1/4*cosh(c+d*x)^3*sinh(c+d*x)/(a*d*(a+b-b*tanh(c+d*x)^2))-3/8*b*(3*a+4*b)*tanh(c+d*x)/(a^3*d*(a+b-b*tanh(c+d*x)^2))],
[sinh(c+d*x)^3/(a+b*sech(c+d*x)^2)^2,x,5,-(a+2*b)*cosh(c+d*x)/(a^3*d)+1/3*cosh(c+d*x)^3/(a^2*d)-1/2*b*(a+b)*cosh(c+d*x)/(a^3*d*(b+a*cosh(c+d*x)^2))+1/2*(3*a+5*b)*arctan(cosh(c+d*x)*sqrt(a)/sqrt(b))*sqrt(b)/(a^(7/2)*d)],
[sinh(c+d*x)^2/(a+b*sech(c+d*x)^2)^2,x,6,-1/2*(a+4*b)*x/a^3+1/2*(3*a+4*b)*arctanh(sqrt(b)*tanh(c+d*x)/sqrt(a+b))*sqrt(b)/(a^3*d*sqrt(a+b))+1/2*cosh(c+d*x)*sinh(c+d*x)/(a*d*(a+b-b*tanh(c+d*x)^2))+b*tanh(c+d*x)/(a^2*d*(a+b-b*tanh(c+d*x)^2))],
[sinh(c+d*x)/(a+b*sech(c+d*x)^2)^2,x,4,3/2*cosh(c+d*x)/(a^2*d)-1/2*cosh(c+d*x)^3/(a*d*(b+a*cosh(c+d*x)^2))-3/2*arctan(cosh(c+d*x)*sqrt(a)/sqrt(b))*sqrt(b)/(a^(5/2)*d)],
[csch(c+d*x)/(a+b*sech(c+d*x)^2)^2,x,5,-arctanh(cosh(c+d*x))/((a+b)^2*d)-1/2*b*cosh(c+d*x)/(a*(a+b)*d*(b+a*cosh(c+d*x)^2))+1/2*(3*a+b)*arctan(cosh(c+d*x)*sqrt(a)/sqrt(b))*sqrt(b)/(a^(3/2)*(a+b)^2*d)],
[csch(c+d*x)^2/(a+b*sech(c+d*x)^2)^2,x,4,-3/2*coth(c+d*x)/((a+b)^2*d)+3/2*arctanh(sqrt(b)*tanh(c+d*x)/sqrt(a+b))*sqrt(b)/((a+b)^(5/2)*d)+1/2*coth(c+d*x)/((a+b)*d*(a+b-b*tanh(c+d*x)^2))],
[csch(c+d*x)^3/(a+b*sech(c+d*x)^2)^2,x,6,1/2*(a-3*b)*arctanh(cosh(c+d*x))/((a+b)^3*d)-1/2*(a-b)*cosh(c+d*x)/((a+b)^2*d*(b+a*cosh(c+d*x)^2))-1/2*coth(c+d*x)*csch(c+d*x)/((a+b)*d*(b+a*cosh(c+d*x)^2))-1/2*(3*a-b)*arctan(cosh(c+d*x)*sqrt(a)/sqrt(b))*sqrt(b)/((a+b)^3*d*sqrt(a))],
[csch(c+d*x)^4/(a+b*sech(c+d*x)^2)^2,x,5,(a-b)*coth(c+d*x)/((a+b)^3*d)-1/3*coth(c+d*x)^3/((a+b)^2*d)-1/2*(3*a-2*b)*arctanh(sqrt(b)*tanh(c+d*x)/sqrt(a+b))*sqrt(b)/((a+b)^(7/2)*d)-1/2*a*b*tanh(c+d*x)/((a+b)^3*d*(a+b-b*tanh(c+d*x)^2))],
[sinh(c+d*x)^4/(a+b*sech(c+d*x)^2)^3,x,8,3/8*(a^2+12*a*b+16*b^2)*x/a^5-3/8*(5*a^2+20*a*b+16*b^2)*arctanh(sqrt(b)*tanh(c+d*x)/sqrt(a+b))*sqrt(b)/(a^5*d*sqrt(a+b))-1/8*(5*a+8*b)*cosh(c+d*x)*sinh(c+d*x)/(a^2*d*(a+b-b*tanh(c+d*x)^2)^2)+1/4*cosh(c+d*x)^3*sinh(c+d*x)/(a*d*(a+b-b*tanh(c+d*x)^2)^2)-1/8*b*(7*a+12*b)*tanh(c+d*x)/(a^3*d*(a+b-b*tanh(c+d*x)^2)^2)-3/2*b*(a+2*b)*tanh(c+d*x)/(a^4*d*(a+b-b*tanh(c+d*x)^2))],
[sinh(c+d*x)^3/(a+b*sech(c+d*x)^2)^3,x,6,-(a+3*b)*cosh(c+d*x)/(a^4*d)+1/3*cosh(c+d*x)^3/(a^3*d)+1/4*b^2*(a+b)*cosh(c+d*x)/(a^4*d*(b+a*cosh(c+d*x)^2)^2)-1/8*b*(9*a+13*b)*cosh(c+d*x)/(a^4*d*(b+a*cosh(c+d*x)^2))+5/8*(3*a+7*b)*arctan(cosh(c+d*x)*sqrt(a)/sqrt(b))*sqrt(b)/(a^(9/2)*d)],
[sinh(c+d*x)^2/(a+b*sech(c+d*x)^2)^3,x,7,-1/2*(a+6*b)*x/a^4+1/8*(15*a^2+40*a*b+24*b^2)*arctanh(sqrt(b)*tanh(c+d*x)/sqrt(a+b))*sqrt(b)/(a^4*(a+b)^(3/2)*d)+1/2*cosh(c+d*x)*sinh(c+d*x)/(a*d*(a+b-b*tanh(c+d*x)^2)^2)+3/4*b*tanh(c+d*x)/(a^2*d*(a+b-b*tanh(c+d*x)^2)^2)+1/8*b*(11*a+12*b)*tanh(c+d*x)/(a^3*(a+b)*d*(a+b-b*tanh(c+d*x)^2))],
[sinh(c+d*x)/(a+b*sech(c+d*x)^2)^3,x,5,15/8*cosh(c+d*x)/(a^3*d)-1/4*cosh(c+d*x)^5/(a*d*(b+a*cosh(c+d*x)^2)^2)-5/8*cosh(c+d*x)^3/(a^2*d*(b+a*cosh(c+d*x)^2))-15/8*arctan(cosh(c+d*x)*sqrt(a)/sqrt(b))*sqrt(b)/(a^(7/2)*d)],
[csch(c+d*x)/(a+b*sech(c+d*x)^2)^3,x,6,-arctanh(cosh(c+d*x))/((a+b)^3*d)-1/4*b*cosh(c+d*x)^3/(a*(a+b)*d*(b+a*cosh(c+d*x)^2)^2)-1/8*b*(7*a+3*b)*cosh(c+d*x)/(a^2*(a+b)^2*d*(b+a*cosh(c+d*x)^2))+1/8*(15*a^2+10*a*b+3*b^2)*arctan(cosh(c+d*x)*sqrt(a)/sqrt(b))*sqrt(b)/(a^(5/2)*(a+b)^3*d)],
[csch(c+d*x)^2/(a+b*sech(c+d*x)^2)^3,x,5,-15/8*coth(c+d*x)/((a+b)^3*d)+15/8*arctanh(sqrt(b)*tanh(c+d*x)/sqrt(a+b))*sqrt(b)/((a+b)^(7/2)*d)+1/4*coth(c+d*x)/((a+b)*d*(a+b-b*tanh(c+d*x)^2)^2)+5/8*coth(c+d*x)/((a+b)^2*d*(a+b-b*tanh(c+d*x)^2))],
[csch(c+d*x)^3/(a+b*sech(c+d*x)^2)^3,x,7,1/2*(a-5*b)*arctanh(cosh(c+d*x))/((a+b)^4*d)+1/4*(2*a-b)*b*cosh(c+d*x)/(a*(a+b)^2*d*(b+a*cosh(c+d*x)^2)^2)-1/8*(4*a^2-9*a*b-b^2)*cosh(c+d*x)/(a*(a+b)^3*d*(b+a*cosh(c+d*x)^2))-1/2*cosh(c+d*x)*coth(c+d*x)^2/((a+b)*d*(b+a*cosh(c+d*x)^2)^2)-1/8*(15*a^2-10*a*b-b^2)*arctan(cosh(c+d*x)*sqrt(a)/sqrt(b))*sqrt(b)/(a^(3/2)*(a+b)^4*d)],
[csch(c+d*x)^4/(a+b*sech(c+d*x)^2)^3,x,6,(a-2*b)*coth(c+d*x)/((a+b)^4*d)-1/3*coth(c+d*x)^3/((a+b)^3*d)-5/8*(3*a-4*b)*arctanh(sqrt(b)*tanh(c+d*x)/sqrt(a+b))*sqrt(b)/((a+b)^(9/2)*d)-1/4*a*b*tanh(c+d*x)/((a+b)^3*d*(a+b-b*tanh(c+d*x)^2)^2)-1/8*(7*a-4*b)*b*tanh(c+d*x)/((a+b)^4*d*(a+b-b*tanh(c+d*x)^2))],

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

# n>0
[cosh(c+d*x)^4*(a+b*sech(c+d*x)^2),x,3,1/8*(3*a+4*b)*x+1/8*(3*a+4*b)*cosh(c+d*x)*sinh(c+d*x)/d+1/4*a*cosh(c+d*x)^3*sinh(c+d*x)/d],
[cosh(c+d*x)^3*(a+b*sech(c+d*x)^2),x,3,(a+b)*sinh(c+d*x)/d+1/3*a*sinh(c+d*x)^3/d],
[cosh(c+d*x)^2*(a+b*sech(c+d*x)^2),x,2,1/2*(a+2*b)*x+1/2*a*cosh(c+d*x)*sinh(c+d*x)/d],
[cosh(c+d*x)*(a+b*sech(c+d*x)^2),x,2,b*arctan(sinh(c+d*x))/d+a*sinh(c+d*x)/d],
[sech(c+d*x)*(a+b*sech(c+d*x)^2),x,2,1/2*(2*a+b)*arctan(sinh(c+d*x))/d+1/2*b*sech(c+d*x)*tanh(c+d*x)/d],
[sech(c+d*x)^2*(a+b*sech(c+d*x)^2),x,3,(a+b)*tanh(c+d*x)/d-1/3*b*tanh(c+d*x)^3/d,1/3*(3*a+2*b)*tanh(c+d*x)/d+1/3*b*sech(c+d*x)^2*tanh(c+d*x)/d],
[sech(c+d*x)^3*(a+b*sech(c+d*x)^2),x,3,1/8*(4*a+3*b)*arctan(sinh(c+d*x))/d+1/8*(4*a+3*b)*sech(c+d*x)*tanh(c+d*x)/d+1/4*b*sech(c+d*x)^3*tanh(c+d*x)/d],
[sech(c+d*x)^4*(a+b*sech(c+d*x)^2),x,3,(a+b)*tanh(c+d*x)/d-1/3*(a+2*b)*tanh(c+d*x)^3/d+1/5*b*tanh(c+d*x)^5/d,1/5*(5*a+4*b)*tanh(c+d*x)/d+1/5*b*sech(c+d*x)^4*tanh(c+d*x)/d-1/15*(5*a+4*b)*tanh(c+d*x)^3/d],
[cosh(c+d*x)^4*(a+b*sech(c+d*x)^2)^2,x,4,1/8*(3*a^2+8*a*b+8*b^2)*x+3/8*a*(a+2*b)*cosh(c+d*x)*sinh(c+d*x)/d+1/4*a*cosh(c+d*x)^3*sinh(c+d*x)*(a+b-b*tanh(c+d*x)^2)/d],
[cosh(c+d*x)^3*(a+b*sech(c+d*x)^2)^2,x,4,b^2*arctan(sinh(c+d*x))/d+a*(a+2*b)*sinh(c+d*x)/d+1/3*a^2*sinh(c+d*x)^3/d],
[cosh(c+d*x)^2*(a+b*sech(c+d*x)^2)^2,x,5,1/2*a*(a+4*b)*x+1/2*a^2*cosh(c+d*x)*sinh(c+d*x)/d+b^2*tanh(c+d*x)/d],
[cosh(c+d*x)*(a+b*sech(c+d*x)^2)^2,x,5,1/2*b*(4*a+b)*arctan(sinh(c+d*x))/d+a^2*sinh(c+d*x)/d+1/2*b^2*sech(c+d*x)*tanh(c+d*x)/d],
[sech(c+d*x)*(a+b*sech(c+d*x)^2)^2,x,4,1/8*(8*a^2+8*a*b+3*b^2)*arctan(sinh(c+d*x))/d+3/8*b*(2*a+b)*sech(c+d*x)*tanh(c+d*x)/d+1/4*b*sech(c+d*x)^3*(a+b+a*sinh(c+d*x)^2)*tanh(c+d*x)/d],
[sech(c+d*x)^2*(a+b*sech(c+d*x)^2)^2,x,3,(a+b)^2*tanh(c+d*x)/d-2/3*b*(a+b)*tanh(c+d*x)^3/d+1/5*b^2*tanh(c+d*x)^5/d],
[sech(c+d*x)^3*(a+b*sech(c+d*x)^2)^2,x,5,1/16*(8*a^2+12*a*b+5*b^2)*arctan(sinh(c+d*x))/d+1/16*(8*a^2+12*a*b+5*b^2)*sech(c+d*x)*tanh(c+d*x)/d+1/24*b*(8*a+5*b)*sech(c+d*x)^3*tanh(c+d*x)/d+1/6*b*sech(c+d*x)^5*(a+b+a*sinh(c+d*x)^2)*tanh(c+d*x)/d],
[sech(c+d*x)^4*(a+b*sech(c+d*x)^2)^2,x,3,(a+b)^2*tanh(c+d*x)/d-1/3*(a+b)*(a+3*b)*tanh(c+d*x)^3/d+1/5*b*(2*a+3*b)*tanh(c+d*x)^5/d-1/7*b^2*tanh(c+d*x)^7/d],
[cosh(c+d*x)^4*(a+b*sech(c+d*x)^2)^3,x,6,3/8*a*(a^2+4*a*b+8*b^2)*x+3/8*a^2*(a+4*b)*cosh(c+d*x)*sinh(c+d*x)/d+1/4*a^3*cosh(c+d*x)^3*sinh(c+d*x)/d+b^3*tanh(c+d*x)/d],
[cosh(c+d*x)^3*(a+b*sech(c+d*x)^2)^3,x,5,1/2*b^2*(6*a+b)*arctan(sinh(c+d*x))/d+a^2*(a+3*b)*sinh(c+d*x)/d+1/3*a^3*sinh(c+d*x)^3/d+1/2*b^3*sech(c+d*x)*tanh(c+d*x)/d],
[cosh(c+d*x)^2*(a+b*sech(c+d*x)^2)^3,x,5,1/2*a^2*(a+6*b)*x+1/2*a^3*cosh(c+d*x)*sinh(c+d*x)/d+b^2*(3*a+b)*tanh(c+d*x)/d-1/3*b^3*tanh(c+d*x)^3/d],
[cosh(c+d*x)*(a+b*sech(c+d*x)^2)^3,x,6,3/8*b*(8*a^2+4*a*b+b^2)*arctan(sinh(c+d*x))/d+a^3*sinh(c+d*x)/d+3/8*b^2*(4*a+b)*sech(c+d*x)*tanh(c+d*x)/d+1/4*b^3*sech(c+d*x)^3*tanh(c+d*x)/d],
[sech(c+d*x)*(a+b*sech(c+d*x)^2)^3,x,5,1/16*(2*a+b)*(8*a^2+8*a*b+5*b^2)*arctan(sinh(c+d*x))/d+1/48*b*(44*a^2+44*a*b+15*b^2)*sech(c+d*x)*tanh(c+d*x)/d+5/24*b*(2*a+b)*sech(c+d*x)^3*(a+b+a*sinh(c+d*x)^2)*tanh(c+d*x)/d+1/6*b*sech(c+d*x)^5*(a+b+a*sinh(c+d*x)^2)^2*tanh(c+d*x)/d],
[sech(c+d*x)^2*(a+b*sech(c+d*x)^2)^3,x,3,(a+b)^3*tanh(c+d*x)/d-b*(a+b)^2*tanh(c+d*x)^3/d+3/5*b^2*(a+b)*tanh(c+d*x)^5/d-1/7*b^3*tanh(c+d*x)^7/d],
[sech(c+d*x)^3*(a+b*sech(c+d*x)^2)^3,x,6,1/128*(64*a^3+144*a^2*b+120*a*b^2+35*b^3)*arctan(sinh(c+d*x))/d+1/128*(64*a^3+144*a^2*b+120*a*b^2+35*b^3)*sech(c+d*x)*tanh(c+d*x)/d+1/192*b*(72*a^2+92*a*b+35*b^2)*sech(c+d*x)^3*tanh(c+d*x)/d+1/48*b*(12*a+7*b)*sech(c+d*x)^5*(a+b+a*sinh(c+d*x)^2)*tanh(c+d*x)/d+1/8*b*sech(c+d*x)^7*(a+b+a*sinh(c+d*x)^2)^2*tanh(c+d*x)/d],
[sech(c+d*x)^4*(a+b*sech(c+d*x)^2)^3,x,3,(a+b)^3*tanh(c+d*x)/d-1/3*(a+b)^2*(a+4*b)*tanh(c+d*x)^3/d+3/5*b*(a+b)*(a+2*b)*tanh(c+d*x)^5/d-1/7*b^2*(3*a+4*b)*tanh(c+d*x)^7/d+1/9*b^3*tanh(c+d*x)^9/d],

# n<0
[cosh(c+d*x)^4/(a+b*sech(c+d*x)^2),x,6,1/8*(3*a^2-4*a*b+8*b^2)*x/a^3+1/8*(3*a-4*b)*cosh(c+d*x)*sinh(c+d*x)/(a^2*d)+1/4*cosh(c+d*x)^3*sinh(c+d*x)/(a*d)-b^(5/2)*arctanh(sqrt(b)*tanh(c+d*x)/sqrt(a+b))/(a^3*d*sqrt(a+b))],
[cosh(c+d*x)^3/(a+b*sech(c+d*x)^2),x,4,(a-b)*sinh(c+d*x)/(a^2*d)+1/3*sinh(c+d*x)^3/(a*d)+b^2*arctan(sinh(c+d*x)*sqrt(a)/sqrt(a+b))/(a^(5/2)*d*sqrt(a+b))],
[cosh(c+d*x)^2/(a+b*sech(c+d*x)^2),x,5,1/2*(a-2*b)*x/a^2+1/2*cosh(c+d*x)*sinh(c+d*x)/(a*d)+b^(3/2)*arctanh(sqrt(b)*tanh(c+d*x)/sqrt(a+b))/(a^2*d*sqrt(a+b))],
[cosh(c+d*x)/(a+b*sech(c+d*x)^2),x,3,sinh(c+d*x)/(a*d)-b*arctan(sinh(c+d*x)*sqrt(a)/sqrt(a+b))/(a^(3/2)*d*sqrt(a+b))],
[sech(c+d*x)/(a+b*sech(c+d*x)^2),x,2,arctan(sinh(c+d*x)*sqrt(a)/sqrt(a+b))/(d*sqrt(a)*sqrt(a+b))],
[sech(c+d*x)^2/(a+b*sech(c+d*x)^2),x,2,arctanh(sqrt(b)*tanh(c+d*x)/sqrt(a+b))/(d*sqrt(b)*sqrt(a+b))],
[sech(c+d*x)^3/(a+b*sech(c+d*x)^2),x,4,arctan(sinh(c+d*x))/(b*d)-arctan(sinh(c+d*x)*sqrt(a)/sqrt(a+b))*sqrt(a)/(b*d*sqrt(a+b))],
[sech(c+d*x)^4/(a+b*sech(c+d*x)^2),x,3,-a*arctanh(sqrt(b)*tanh(c+d*x)/sqrt(a+b))/(b^(3/2)*d*sqrt(a+b))+tanh(c+d*x)/(b*d)],
[sech(c+d*x)^5/(a+b*sech(c+d*x)^2),x,5,-1/2*(2*a-b)*arctan(sinh(c+d*x))/(b^2*d)+a^(3/2)*arctan(sinh(c+d*x)*sqrt(a)/sqrt(a+b))/(b^2*d*sqrt(a+b))+1/2*sech(c+d*x)*tanh(c+d*x)/(b*d)],
[sech(c+d*x)^6/(a+b*sech(c+d*x)^2),x,4,a^2*arctanh(sqrt(b)*tanh(c+d*x)/sqrt(a+b))/(b^(5/2)*d*sqrt(a+b))-(a-b)*tanh(c+d*x)/(b^2*d)-1/3*tanh(c+d*x)^3/(b*d)],
[cosh(c+d*x)^3/(a+b*sech(c+d*x)^2)^2,x,5,1/2*b^2*(6*a+5*b)*arctan(sinh(c+d*x)*sqrt(a)/sqrt(a+b))/(a^(7/2)*(a+b)^(3/2)*d)+(a-2*b)*sinh(c+d*x)/(a^3*d)+1/3*sinh(c+d*x)^3/(a^2*d)-1/2*b^3*sinh(c+d*x)/(a^3*(a+b)*d*(a+b+a*sinh(c+d*x)^2))],
[cosh(c+d*x)^2/(a+b*sech(c+d*x)^2)^2,x,6,1/2*(a-4*b)*x/a^3+1/2*b^(3/2)*(5*a+4*b)*arctanh(sqrt(b)*tanh(c+d*x)/sqrt(a+b))/(a^3*(a+b)^(3/2)*d)+1/2*cosh(c+d*x)*sinh(c+d*x)/(a*d*(a+b-b*tanh(c+d*x)^2))+1/2*b*(a+2*b)*tanh(c+d*x)/(a^2*(a+b)*d*(a+b-b*tanh(c+d*x)^2))],
[cosh(c+d*x)/(a+b*sech(c+d*x)^2)^2,x,5,-1/2*b*(4*a+3*b)*arctan(sinh(c+d*x)*sqrt(a)/sqrt(a+b))/(a^(5/2)*(a+b)^(3/2)*d)+sinh(c+d*x)/(a^2*d)+1/2*b^2*sinh(c+d*x)/(a^2*(a+b)*d*(a+b+a*sinh(c+d*x)^2))],
[sech(c+d*x)/(a+b*sech(c+d*x)^2)^2,x,3,1/2*(2*a+b)*arctan(sinh(c+d*x)*sqrt(a)/sqrt(a+b))/(a^(3/2)*(a+b)^(3/2)*d)-1/2*b*sinh(c+d*x)/(a*(a+b)*d*(a+b+a*sinh(c+d*x)^2))],
[sech(c+d*x)^2/(a+b*sech(c+d*x)^2)^2,x,3,1/2*arctanh(sqrt(b)*tanh(c+d*x)/sqrt(a+b))/((a+b)^(3/2)*d*sqrt(b))+1/2*tanh(c+d*x)/((a+b)*d*(a+b-b*tanh(c+d*x)^2))],
[sech(c+d*x)^3/(a+b*sech(c+d*x)^2)^2,x,3,1/2*sinh(c+d*x)/((a+b)*d*(a+b+a*sinh(c+d*x)^2))+1/2*arctan(sinh(c+d*x)*sqrt(a)/sqrt(a+b))/((a+b)^(3/2)*d*sqrt(a))],
[sech(c+d*x)^4/(a+b*sech(c+d*x)^2)^2,x,3,1/2*(a+2*b)*arctanh(sqrt(b)*tanh(c+d*x)/sqrt(a+b))/(b^(3/2)*(a+b)^(3/2)*d)-1/2*a*tanh(c+d*x)/(b*(a+b)*d*(a+b-b*tanh(c+d*x)^2))],
[sech(c+d*x)^5/(a+b*sech(c+d*x)^2)^2,x,5,arctan(sinh(c+d*x))/(b^2*d)-1/2*a*sinh(c+d*x)/(b*(a+b)*d*(a+b+a*sinh(c+d*x)^2))-1/2*(2*a+3*b)*arctan(sinh(c+d*x)*sqrt(a)/sqrt(a+b))*sqrt(a)/(b^2*(a+b)^(3/2)*d)],
[sech(c+d*x)^6/(a+b*sech(c+d*x)^2)^2,x,5,-1/2*a*(3*a+4*b)*arctanh(sqrt(b)*tanh(c+d*x)/sqrt(a+b))/(b^(5/2)*(a+b)^(3/2)*d)+tanh(c+d*x)/(b^2*d)+1/2*a^2*tanh(c+d*x)/(b^2*(a+b)*d*(a+b-b*tanh(c+d*x)^2))],
[sech(c+d*x)^7/(a+b*sech(c+d*x)^2)^2,x,6,-1/2*(4*a-b)*arctan(sinh(c+d*x))/(b^3*d)+1/2*a^(3/2)*(4*a+5*b)*arctan(sinh(c+d*x)*sqrt(a)/sqrt(a+b))/(b^3*(a+b)^(3/2)*d)+1/2*a*(2*a+b)*sinh(c+d*x)/(b^2*(a+b)*d*(a+b+a*sinh(c+d*x)^2))+1/2*sech(c+d*x)*tanh(c+d*x)/(b*d*(a+b+a*sinh(c+d*x)^2))],
[cosh(c+d*x)^2/(a+b*sech(c+d*x)^2)^3,x,7,1/2*(a-6*b)*x/a^4+1/8*b^(3/2)*(35*a^2+56*a*b+24*b^2)*arctanh(sqrt(b)*tanh(c+d*x)/sqrt(a+b))/(a^4*(a+b)^(5/2)*d)+1/2*cosh(c+d*x)*sinh(c+d*x)/(a*d*(a+b-b*tanh(c+d*x)^2)^2)+1/4*b*(2*a+3*b)*tanh(c+d*x)/(a^2*(a+b)*d*(a+b-b*tanh(c+d*x)^2)^2)+1/8*b*(4*a+3*b)*(a+4*b)*tanh(c+d*x)/(a^3*(a+b)^2*d*(a+b-b*tanh(c+d*x)^2))],
[cosh(c+d*x)/(a+b*sech(c+d*x)^2)^3,x,6,-3/8*b*(4*(a+b)^2+(2*a+b)^2)*arctan(sinh(c+d*x)*sqrt(a)/sqrt(a+b))/(a^(7/2)*(a+b)^(5/2)*d)+sinh(c+d*x)/(a^3*d)-1/4*b^3*sinh(c+d*x)/(a^3*(a+b)*d*(a+b+a*sinh(c+d*x)^2)^2)+3/8*b^2*(4*a+3*b)*sinh(c+d*x)/(a^3*(a+b)^2*d*(a+b+a*sinh(c+d*x)^2))],
[sech(c+d*x)/(a+b*sech(c+d*x)^2)^3,x,4,1/8*(8*a^2+8*a*b+3*b^2)*arctan(sinh(c+d*x)*sqrt(a)/sqrt(a+b))/(a^(5/2)*(a+b)^(5/2)*d)-1/4*b*cosh(c+d*x)^2*sinh(c+d*x)/(a*(a+b)*d*(a+b+a*sinh(c+d*x)^2)^2)-3/8*b*(2*a+b)*sinh(c+d*x)/(a^2*(a+b)^2*d*(a+b+a*sinh(c+d*x)^2))],
[sech(c+d*x)^2/(a+b*sech(c+d*x)^2)^3,x,4,3/8*arctanh(sqrt(b)*tanh(c+d*x)/sqrt(a+b))/((a+b)^(5/2)*d*sqrt(b))+1/4*tanh(c+d*x)/((a+b)*d*(a+b-b*tanh(c+d*x)^2)^2)+3/8*tanh(c+d*x)/((a+b)^2*d*(a+b-b*tanh(c+d*x)^2))],
[sech(c+d*x)^3/(a+b*sech(c+d*x)^2)^3,x,4,1/8*(4*a+b)*arctan(sinh(c+d*x)*sqrt(a)/sqrt(a+b))/(a^(3/2)*(a+b)^(5/2)*d)-1/4*b*sinh(c+d*x)/(a*(a+b)*d*(a+b+a*sinh(c+d*x)^2)^2)+1/8*(4*a+b)*sinh(c+d*x)/(a*(a+b)^2*d*(a+b+a*sinh(c+d*x)^2))],
[sech(c+d*x)^4/(a+b*sech(c+d*x)^2)^3,x,4,1/8*(a+4*b)*arctanh(sqrt(b)*tanh(c+d*x)/sqrt(a+b))/(b^(3/2)*(a+b)^(5/2)*d)-1/4*a*tanh(c+d*x)/(b*(a+b)*d*(a+b-b*tanh(c+d*x)^2)^2)+1/8*(a+4*b)*tanh(c+d*x)/(b*(a+b)^2*d*(a+b-b*tanh(c+d*x)^2))],
[sech(c+d*x)^5/(a+b*sech(c+d*x)^2)^3,x,4,1/4*sinh(c+d*x)/((a+b)*d*(a+b+a*sinh(c+d*x)^2)^2)+3/8*sinh(c+d*x)/((a+b)^2*d*(a+b+a*sinh(c+d*x)^2))+3/8*arctan(sinh(c+d*x)*sqrt(a)/sqrt(a+b))/((a+b)^(5/2)*d*sqrt(a))],
[sech(c+d*x)^6/(a+b*sech(c+d*x)^2)^3,x,4,1/8*(3*a^2+8*a*b+8*b^2)*arctanh(sqrt(b)*tanh(c+d*x)/sqrt(a+b))/(b^(5/2)*(a+b)^(5/2)*d)-1/4*a*sech(c+d*x)^2*tanh(c+d*x)/(b*(a+b)*d*(a+b-b*tanh(c+d*x)^2)^2)-3/8*a*(a+2*b)*tanh(c+d*x)/(b^2*(a+b)^2*d*(a+b-b*tanh(c+d*x)^2))],
[sech(c+d*x)^7/(a+b*sech(c+d*x)^2)^3,x,6,arctan(sinh(c+d*x))/(b^3*d)-1/4*a*sinh(c+d*x)/(b*(a+b)*d*(a+b+a*sinh(c+d*x)^2)^2)-1/8*a*(4*a+7*b)*sinh(c+d*x)/(b^2*(a+b)^2*d*(a+b+a*sinh(c+d*x)^2))-1/8*(8*a^2+20*a*b+15*b^2)*arctan(sinh(c+d*x)*sqrt(a)/sqrt(a+b))*sqrt(a)/(b^3*(a+b)^(5/2)*d)],

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

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

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

# p>0
[(a+b*sech(c+d*x)^2)*tanh(c+d*x)^4,x,4,a*x-a*tanh(c+d*x)/d-1/3*a*tanh(c+d*x)^3/d+1/5*b*tanh(c+d*x)^5/d],
[(a+b*sech(c+d*x)^2)*tanh(c+d*x)^3,x,4,a*log(cosh(c+d*x))/d+1/2*(a-b)*sech(c+d*x)^2/d+1/4*b*sech(c+d*x)^4/d],
[(a+b*sech(c+d*x)^2)*tanh(c+d*x)^2,x,4,a*x-a*tanh(c+d*x)/d+1/3*b*tanh(c+d*x)^3/d],
[(a+b*sech(c+d*x)^2)*tanh(c+d*x),x,3,a*log(cosh(c+d*x))/d-1/2*b*sech(c+d*x)^2/d],
[a+b*sech(c+d*x)^2,x,3,a*x+b*tanh(c+d*x)/d],
[coth(c+d*x)*(a+b*sech(c+d*x)^2),x,4,-b*log(cosh(c+d*x))/d+(a+b)*log(sinh(c+d*x))/d],
[coth(c+d*x)^2*(a+b*sech(c+d*x)^2),x,4,a*x-(a+b)*coth(c+d*x)/d],
[coth(c+d*x)^3*(a+b*sech(c+d*x)^2),x,4,-1/2*(a+b)*csch(c+d*x)^2/d+a*log(sinh(c+d*x))/d],
[coth(c+d*x)^4*(a+b*sech(c+d*x)^2),x,4,a*x-a*coth(c+d*x)/d-1/3*(a+b)*coth(c+d*x)^3/d],
[coth(c+d*x)^5*(a+b*sech(c+d*x)^2),x,4,-1/2*(2*a+b)*csch(c+d*x)^2/d-1/4*(a+b)*csch(c+d*x)^4/d+a*log(sinh(c+d*x))/d],
[(a+b*sech(c+d*x)^2)^2*tanh(c+d*x)^4,x,4,a^2*x-a^2*tanh(c+d*x)/d-1/3*a^2*tanh(c+d*x)^3/d+1/5*b*(2*a+b)*tanh(c+d*x)^5/d-1/7*b^2*tanh(c+d*x)^7/d],
[(a+b*sech(c+d*x)^2)^2*tanh(c+d*x)^3,x,4,a^2*log(cosh(c+d*x))/d+1/2*a*(a-2*b)*sech(c+d*x)^2/d+1/4*(2*a-b)*b*sech(c+d*x)^4/d+1/6*b^2*sech(c+d*x)^6/d],
[(a+b*sech(c+d*x)^2)^2*tanh(c+d*x)^2,x,4,a^2*x-a^2*tanh(c+d*x)/d+1/3*b*(2*a+b)*tanh(c+d*x)^3/d-1/5*b^2*tanh(c+d*x)^5/d],
[(a+b*sech(c+d*x)^2)^2*tanh(c+d*x),x,4,a^2*log(cosh(c+d*x))/d-a*b*sech(c+d*x)^2/d-1/4*b^2*sech(c+d*x)^4/d],
[(a+b*sech(c+d*x)^2)^2,x,4,a^2*x+b*(2*a+b)*tanh(c+d*x)/d-1/3*b^2*tanh(c+d*x)^3/d],
[coth(c+d*x)*(a+b*sech(c+d*x)^2)^2,x,4,-b*(2*a+b)*log(cosh(c+d*x))/d+(a+b)^2*log(sinh(c+d*x))/d+1/2*b^2*sech(c+d*x)^2/d],
[coth(c+d*x)^2*(a+b*sech(c+d*x)^2)^2,x,4,a^2*x-(a+b)^2*coth(c+d*x)/d-b^2*tanh(c+d*x)/d],
[coth(c+d*x)^3*(a+b*sech(c+d*x)^2)^2,x,4,-1/2*(a+b)^2*csch(c+d*x)^2/d+b^2*log(cosh(c+d*x))/d+(a^2-b^2)*log(sinh(c+d*x))/d],
[coth(c+d*x)^4*(a+b*sech(c+d*x)^2)^2,x,4,a^2*x-(a^2-b^2)*coth(c+d*x)/d-1/3*(a+b)^2*coth(c+d*x)^3/d],
[coth(c+d*x)^5*(a+b*sech(c+d*x)^2)^2,x,4,-a*(a+b)*csch(c+d*x)^2/d-1/4*(a+b)^2*csch(c+d*x)^4/d+a^2*log(sinh(c+d*x))/d],
[coth(c+d*x)^6*(a+b*sech(c+d*x)^2)^2,x,4,a^2*x-a^2*coth(c+d*x)/d-1/3*(a^2-b^2)*coth(c+d*x)^3/d-1/5*(a+b)^2*coth(c+d*x)^5/d],
[coth(c+d*x)^7*(a+b*sech(c+d*x)^2)^2,x,5,-a*(a+b)*csch(c+d*x)^2/d-1/4*(a+b)^2*csch(c+d*x)^4/d-1/6*(b+a*cosh(c+d*x)^2)^3*csch(c+d*x)^6/((a+b)*d)+a^2*log(sinh(c+d*x))/d],
[(a+b*sech(c+d*x)^2)^3*tanh(c+d*x)^4,x,4,a^3*x-a^3*tanh(c+d*x)/d-1/3*a^3*tanh(c+d*x)^3/d+1/5*b*(3*a^2+3*a*b+b^2)*tanh(c+d*x)^5/d-1/7*b^2*(3*a+2*b)*tanh(c+d*x)^7/d+1/9*b^3*tanh(c+d*x)^9/d],
[(a+b*sech(c+d*x)^2)^3*tanh(c+d*x)^3,x,5,a^3*log(cosh(c+d*x))/d-3/2*a^2*b*sech(c+d*x)^2/d-3/4*a*b^2*sech(c+d*x)^4/d-1/6*b^3*sech(c+d*x)^6/d+1/8*(b+a*cosh(c+d*x)^2)^4*sech(c+d*x)^8/(b*d)],
[(a+b*sech(c+d*x)^2)^3*tanh(c+d*x)^2,x,4,a^3*x-a^3*tanh(c+d*x)/d+1/3*b*(3*a^2+3*a*b+b^2)*tanh(c+d*x)^3/d-1/5*b^2*(3*a+2*b)*tanh(c+d*x)^5/d+1/7*b^3*tanh(c+d*x)^7/d],
[(a+b*sech(c+d*x)^2)^3*tanh(c+d*x),x,4,a^3*log(cosh(c+d*x))/d-3/2*a^2*b*sech(c+d*x)^2/d-3/4*a*b^2*sech(c+d*x)^4/d-1/6*b^3*sech(c+d*x)^6/d],
[(a+b*sech(c+d*x)^2)^3,x,4,a^3*x+b*(3*a^2+3*a*b+b^2)*tanh(c+d*x)/d-1/3*b^2*(3*a+2*b)*tanh(c+d*x)^3/d+1/5*b^3*tanh(c+d*x)^5/d],
[coth(c+d*x)*(a+b*sech(c+d*x)^2)^3,x,4,-b*(3*a^2+3*a*b+b^2)*log(cosh(c+d*x))/d+(a+b)^3*log(sinh(c+d*x))/d+1/2*b^2*(3*a+b)*sech(c+d*x)^2/d+1/4*b^3*sech(c+d*x)^4/d],
[coth(c+d*x)^2*(a+b*sech(c+d*x)^2)^3,x,4,a^3*x-(a+b)^3*coth(c+d*x)/d-b^2*(3*a+2*b)*tanh(c+d*x)/d+1/3*b^3*tanh(c+d*x)^3/d],
[coth(c+d*x)^3*(a+b*sech(c+d*x)^2)^3,x,4,-1/2*(a+b)^3*csch(c+d*x)^2/d+b^2*(3*a+2*b)*log(cosh(c+d*x))/d+(a-2*b)*(a+b)^2*log(sinh(c+d*x))/d-1/2*b^3*sech(c+d*x)^2/d],
[coth(c+d*x)^4*(a+b*sech(c+d*x)^2)^3,x,4,a^3*x-(a-2*b)*(a+b)^2*coth(c+d*x)/d-1/3*(a+b)^3*coth(c+d*x)^3/d+b^3*tanh(c+d*x)/d],
[coth(c+d*x)^5*(a+b*sech(c+d*x)^2)^3,x,4,-1/2*(2*a-b)*(a+b)^2*csch(c+d*x)^2/d-1/4*(a+b)^3*csch(c+d*x)^4/d-b^3*log(cosh(c+d*x))/d+(a^3+b^3)*log(sinh(c+d*x))/d],
[coth(c+d*x)^6*(a+b*sech(c+d*x)^2)^3,x,4,a^3*x-(a^3+b^3)*coth(c+d*x)/d-1/3*(a-2*b)*(a+b)^2*coth(c+d*x)^3/d-1/5*(a+b)^3*coth(c+d*x)^5/d],
[coth(c+d*x)^7*(a+b*sech(c+d*x)^2)^3,x,4,-3/2*a^2*(a+b)*csch(c+d*x)^2/d-3/4*a*(a+b)^2*csch(c+d*x)^4/d-1/6*(a+b)^3*csch(c+d*x)^6/d+a^3*log(sinh(c+d*x))/d],
[(a+b*sech(c+d*x)^2)^4,x,4,a^4*x+b*(2*a+b)*(2*a^2+2*a*b+b^2)*tanh(c+d*x)/d-1/3*b^2*(6*a^2+8*a*b+3*b^2)*tanh(c+d*x)^3/d+1/5*b^3*(4*a+3*b)*tanh(c+d*x)^5/d-1/7*b^4*tanh(c+d*x)^7/d],
[(a+b*sech(c+d*x)^2)^5,x,4,a^5*x+b*(5*a^4+10*a^3*b+10*a^2*b^2+5*a*b^3+b^4)*tanh(c+d*x)/d-1/3*b^2*(10*a^3+20*a^2*b+15*a*b^2+4*b^3)*tanh(c+d*x)^3/d+1/5*b^3*(10*a^2+15*a*b+6*b^2)*tanh(c+d*x)^5/d-1/7*b^4*(5*a+4*b)*tanh(c+d*x)^7/d+1/9*b^5*tanh(c+d*x)^9/d],

# p<0
[tanh(c+d*x)^5/(a+b*sech(c+d*x)^2),x,4,-(a+2*b)*log(cosh(c+d*x))/(b^2*d)+1/2*(a+b)^2*log(b+a*cosh(c+d*x)^2)/(a*b^2*d)-1/2*sech(c+d*x)^2/(b*d)],
[tanh(c+d*x)^4/(a+b*sech(c+d*x)^2),x,6,x/a-(a+b)^(3/2)*arctanh(sqrt(b)*tanh(c+d*x)/sqrt(a+b))/(a*b^(3/2)*d)+tanh(c+d*x)/(b*d)],
[tanh(c+d*x)^3/(a+b*sech(c+d*x)^2),x,4,-log(cosh(c+d*x))/(b*d)+1/2*(a+b)*log(b+a*cosh(c+d*x)^2)/(a*b*d)],
[tanh(c+d*x)^2/(a+b*sech(c+d*x)^2),x,5,x/a-arctanh(sqrt(b)*tanh(c+d*x)/sqrt(a+b))*sqrt(a+b)/(a*d*sqrt(b))],
[tanh(c+d*x)/(a+b*sech(c+d*x)^2),x,2,1/2*log(b+a*cosh(c+d*x)^2)/(a*d)],
[1/(a+b*sech(c+d*x)^2),x,3,x/a-arctanh(sqrt(b)*tanh(c+d*x)/sqrt(a+b))*sqrt(b)/(a*d*sqrt(a+b))],
[coth(c+d*x)/(a+b*sech(c+d*x)^2),x,4,1/2*b*log(b+a*cosh(c+d*x)^2)/(a*(a+b)*d)+log(sinh(c+d*x))/((a+b)*d)],
[coth(c+d*x)^2/(a+b*sech(c+d*x)^2),x,6,x/a-b^(3/2)*arctanh(sqrt(b)*tanh(c+d*x)/sqrt(a+b))/(a*(a+b)^(3/2)*d)-coth(c+d*x)/((a+b)*d)],
[coth(c+d*x)^3/(a+b*sech(c+d*x)^2),x,4,-1/2*csch(c+d*x)^2/((a+b)*d)+1/2*b^2*log(b+a*cosh(c+d*x)^2)/(a*(a+b)^2*d)+(a+2*b)*log(sinh(c+d*x))/((a+b)^2*d)],
[coth(c+d*x)^4/(a+b*sech(c+d*x)^2),x,7,x/a-b^(5/2)*arctanh(sqrt(b)*tanh(c+d*x)/sqrt(a+b))/(a*(a+b)^(5/2)*d)-(a+2*b)*coth(c+d*x)/((a+b)^2*d)-1/3*coth(c+d*x)^3/((a+b)*d)],
[tanh(c+d*x)^5/(a+b*sech(c+d*x)^2)^2,x,4,1/2*(a+b)^2/(a^2*b*d*(b+a*cosh(c+d*x)^2))+log(cosh(c+d*x))/(b^2*d)+1/2*(1/a^2+(-1)/b^2)*log(b+a*cosh(c+d*x)^2)/d],
[tanh(c+d*x)^4/(a+b*sech(c+d*x)^2)^2,x,6,x/a^2+1/2*(a-2*b)*arctanh(sqrt(b)*tanh(c+d*x)/sqrt(a+b))*sqrt(a+b)/(a^2*b^(3/2)*d)-1/2*(a+b)*tanh(c+d*x)/(a*b*d*(a+b-b*tanh(c+d*x)^2))],
[tanh(c+d*x)^3/(a+b*sech(c+d*x)^2)^2,x,4,1/2*(a+b)/(a^2*d*(b+a*cosh(c+d*x)^2))+1/2*log(b+a*cosh(c+d*x)^2)/(a^2*d)],
[tanh(c+d*x)^2/(a+b*sech(c+d*x)^2)^2,x,6,x/a^2-1/2*(a+2*b)*arctanh(sqrt(b)*tanh(c+d*x)/sqrt(a+b))/(a^2*d*sqrt(b)*sqrt(a+b))-1/2*tanh(c+d*x)/(a*d*(a+b-b*tanh(c+d*x)^2))],
[tanh(c+d*x)/(a+b*sech(c+d*x)^2)^2,x,4,1/2*b/(a^2*d*(b+a*cosh(c+d*x)^2))+1/2*log(b+a*cosh(c+d*x)^2)/(a^2*d)],
[1/(a+b*sech(c+d*x)^2)^2,x,5,x/a^2-1/2*(3*a+2*b)*arctanh(sqrt(b)*tanh(c+d*x)/sqrt(a+b))*sqrt(b)/(a^2*(a+b)^(3/2)*d)-1/2*b*tanh(c+d*x)/(a*(a+b)*d*(a+b-b*tanh(c+d*x)^2))],
[coth(c+d*x)/(a+b*sech(c+d*x)^2)^2,x,4,1/2*b^2/(a^2*(a+b)*d*(b+a*cosh(c+d*x)^2))+1/2*b*(2*a+b)*log(b+a*cosh(c+d*x)^2)/(a^2*(a+b)^2*d)+log(sinh(c+d*x))/((a+b)^2*d)],
[coth(c+d*x)^2/(a+b*sech(c+d*x)^2)^2,x,7,x/a^2-1/2*b^(3/2)*(5*a+2*b)*arctanh(sqrt(b)*tanh(c+d*x)/sqrt(a+b))/(a^2*(a+b)^(5/2)*d)-1/2*(2*a-b)*coth(c+d*x)/(a*(a+b)^2*d)-1/2*b*coth(c+d*x)/(a*(a+b)*d*(a+b-b*tanh(c+d*x)^2))],
[coth(c+d*x)^3/(a+b*sech(c+d*x)^2)^2,x,4,1/2*b^3/(a^2*(a+b)^2*d*(b+a*cosh(c+d*x)^2))-1/2*csch(c+d*x)^2/((a+b)^2*d)+1/2*b^2*(3*a+b)*log(b+a*cosh(c+d*x)^2)/(a^2*(a+b)^3*d)+(a+3*b)*log(sinh(c+d*x))/((a+b)^3*d)],
[coth(c+d*x)^4/(a+b*sech(c+d*x)^2)^2,x,8,x/a^2-1/2*b^(5/2)*(7*a+2*b)*arctanh(sqrt(b)*tanh(c+d*x)/sqrt(a+b))/(a^2*(a+b)^(7/2)*d)-1/2*(2*a^2+6*a*b-b^2)*coth(c+d*x)/(a*(a+b)^3*d)-1/6*(2*a-3*b)*coth(c+d*x)^3/(a*(a+b)^2*d)-1/2*b*coth(c+d*x)^3/(a*(a+b)*d*(a+b-b*tanh(c+d*x)^2))],
[tanh(c+d*x)^6/(a+b*sech(c+d*x)^2)^3,x,7,x/a^3-1/8*(3*a^2-4*a*b+8*b^2)*arctanh(sqrt(b)*tanh(c+d*x)/sqrt(a+b))*sqrt(a+b)/(a^3*b^(5/2)*d)-1/4*(a+b)*tanh(c+d*x)^3/(a*b*d*(a+b-b*tanh(c+d*x)^2)^2)+1/8*(3*a-4*b)*(a+b)*tanh(c+d*x)/(a^2*b^2*d*(a+b-b*tanh(c+d*x)^2))],
[tanh(c+d*x)^5/(a+b*sech(c+d*x)^2)^3,x,4,-1/4*(a+b)^2/(a^3*d*(b+a*cosh(c+d*x)^2)^2)+(a+b)/(a^3*d*(b+a*cosh(c+d*x)^2))+1/2*log(b+a*cosh(c+d*x)^2)/(a^3*d)],
[tanh(c+d*x)^4/(a+b*sech(c+d*x)^2)^3,x,7,x/a^3+1/8*(a^2-4*a*b-8*b^2)*arctanh(sqrt(b)*tanh(c+d*x)/sqrt(a+b))/(a^3*b^(3/2)*d*sqrt(a+b))-1/4*(a+b)*tanh(c+d*x)/(a*b*d*(a+b-b*tanh(c+d*x)^2)^2)+1/8*(a-4*b)*tanh(c+d*x)/(a^2*b*d*(a+b-b*tanh(c+d*x)^2))],
[tanh(c+d*x)^3/(a+b*sech(c+d*x)^2)^3,x,4,-1/4*b*(a+b)/(a^3*d*(b+a*cosh(c+d*x)^2)^2)+1/2*(a+2*b)/(a^3*d*(b+a*cosh(c+d*x)^2))+1/2*log(b+a*cosh(c+d*x)^2)/(a^3*d)],
[tanh(c+d*x)^2/(a+b*sech(c+d*x)^2)^3,x,7,x/a^3-1/8*(3*a^2+12*a*b+8*b^2)*arctanh(sqrt(b)*tanh(c+d*x)/sqrt(a+b))/(a^3*(a+b)^(3/2)*d*sqrt(b))-1/4*tanh(c+d*x)/(a*d*(a+b-b*tanh(c+d*x)^2)^2)-1/8*(3*a+4*b)*tanh(c+d*x)/(a^2*(a+b)*d*(a+b-b*tanh(c+d*x)^2))],
[tanh(c+d*x)/(a+b*sech(c+d*x)^2)^3,x,4,-1/4*b^2/(a^3*d*(b+a*cosh(c+d*x)^2)^2)+b/(a^3*d*(b+a*cosh(c+d*x)^2))+1/2*log(b+a*cosh(c+d*x)^2)/(a^3*d)],
[1/(a+b*sech(c+d*x)^2)^3,x,6,x/a^3-1/8*(15*a^2+20*a*b+8*b^2)*arctanh(sqrt(b)*tanh(c+d*x)/sqrt(a+b))*sqrt(b)/(a^3*(a+b)^(5/2)*d)-1/4*b*tanh(c+d*x)/(a*(a+b)*d*(a+b-b*tanh(c+d*x)^2)^2)-1/8*b*(7*a+4*b)*tanh(c+d*x)/(a^2*(a+b)^2*d*(a+b-b*tanh(c+d*x)^2))],
[coth(c+d*x)/(a+b*sech(c+d*x)^2)^3,x,4,-1/4*b^3/(a^3*(a+b)*d*(b+a*cosh(c+d*x)^2)^2)+1/2*b^2*(3*a+2*b)/(a^3*(a+b)^2*d*(b+a*cosh(c+d*x)^2))+1/2*b*(3*a^2+3*a*b+b^2)*log(b+a*cosh(c+d*x)^2)/(a^3*(a+b)^3*d)+log(sinh(c+d*x))/((a+b)^3*d)],
[coth(c+d*x)^2/(a+b*sech(c+d*x)^2)^3,x,8,x/a^3-1/8*b^(3/2)*(35*a^2+28*a*b+8*b^2)*arctanh(sqrt(b)*tanh(c+d*x)/sqrt(a+b))/(a^3*(a+b)^(7/2)*d)-1/8*(8*a^2-11*a*b-4*b^2)*coth(c+d*x)/(a^2*(a+b)^3*d)-1/4*b*coth(c+d*x)/(a*(a+b)*d*(a+b-b*tanh(c+d*x)^2)^2)-1/8*b*(9*a+4*b)*coth(c+d*x)/(a^2*(a+b)^2*d*(a+b-b*tanh(c+d*x)^2))],
[coth(c+d*x)^3/(a+b*sech(c+d*x)^2)^3,x,4,-1/4*b^4/(a^3*(a+b)^2*d*(b+a*cosh(c+d*x)^2)^2)+b^3*(2*a+b)/(a^3*(a+b)^3*d*(b+a*cosh(c+d*x)^2))-1/2*csch(c+d*x)^2/((a+b)^3*d)+1/2*b^2*(6*a^2+4*a*b+b^2)*log(b+a*cosh(c+d*x)^2)/(a^3*(a+b)^4*d)+(a+4*b)*log(sinh(c+d*x))/((a+b)^4*d)],
[coth(c+d*x)^4/(a+b*sech(c+d*x)^2)^3,x,9,x/a^3-1/8*b^(5/2)*(63*a^2+36*a*b+8*b^2)*arctanh(sqrt(b)*tanh(c+d*x)/sqrt(a+b))/(a^3*(a+b)^(9/2)*d)-1/8*(8*a^3+32*a^2*b-15*a*b^2-4*b^3)*coth(c+d*x)/(a^2*(a+b)^4*d)-1/24*(8*a^2-39*a*b-12*b^2)*coth(c+d*x)^3/(a^2*(a+b)^3*d)-1/4*b*coth(c+d*x)^3/(a*(a+b)*d*(a+b-b*tanh(c+d*x)^2)^2)-1/8*b*(11*a+4*b)*coth(c+d*x)^3/(a^2*(a+b)^2*d*(a+b-b*tanh(c+d*x)^2))],
[1/(a+b*sech(c+d*x)^2)^4,x,7,x/a^4-1/16*(35*a^3+70*a^2*b+56*a*b^2+16*b^3)*arctanh(sqrt(b)*tanh(c+d*x)/sqrt(a+b))*sqrt(b)/(a^4*(a+b)^(7/2)*d)-1/6*b*tanh(c+d*x)/(a*(a+b)*d*(a+b-b*tanh(c+d*x)^2)^3)-1/24*b*(11*a+6*b)*tanh(c+d*x)/(a^2*(a+b)^2*d*(a+b-b*tanh(c+d*x)^2)^2)-1/16*b*(19*a^2+22*a*b+8*b^2)*tanh(c+d*x)/(a^3*(a+b)^3*d*(a+b-b*tanh(c+d*x)^2))],

# Integrands of the form Tanh[e+f x]^m (a+b Sech[e+f x]^2)^(p/2) when a+b=0
[(1-sech(x)^2)^(3/2),x,4,coth(x)*log(cosh(x))*sqrt(tanh(x)^2)-1/2*coth(x)*(tanh(x)^2)^(3/2)],
[sqrt(1-sech(x)^2),x,3,coth(x)*log(cosh(x))*sqrt(tanh(x)^2)],
[1/sqrt(1-sech(x)^2),x,3,log(sinh(x))*tanh(x)/sqrt(tanh(x)^2)],
[(-1+sech(x)^2)^(3/2),x,4,-coth(x)*log(cosh(x))*sqrt(-tanh(x)^2)+1/2*sqrt(-tanh(x)^2)*tanh(x)],
[sqrt(-1+sech(x)^2),x,3,coth(x)*log(cosh(x))*sqrt(-tanh(x)^2)],
[1/sqrt(-1+sech(x)^2),x,3,log(sinh(x))*tanh(x)/sqrt(-tanh(x)^2)],

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

# p>0
[sqrt(a+b*sech(x)^2)*tanh(x)^5,x,7,1/3*(a+2*b)*(a+b*sech(x)^2)^(3/2)/b^2-1/5*(a+b*sech(x)^2)^(5/2)/b^2+arctanh(sqrt(a+b*sech(x)^2)/sqrt(a))*sqrt(a)-sqrt(a+b*sech(x)^2)],
[sqrt(a+b*sech(x)^2)*tanh(x)^4,x,9,-1/8*(a^2+6*a*b-3*b^2)*arctan(sqrt(b)*tanh(x)/sqrt(a+b-b*tanh(x)^2))/b^(3/2)+arctanh(sqrt(a)*tanh(x)/sqrt(a+b-b*tanh(x)^2))*sqrt(a)+1/8*(a-3*b)*sqrt(a+b-b*tanh(x)^2)*tanh(x)/b-1/4*sqrt(a+b-b*tanh(x)^2)*tanh(x)^3],
[sqrt(a+b*sech(x)^2)*tanh(x)^3,x,6,1/3*(a+b*sech(x)^2)^(3/2)/b+arctanh(sqrt(a+b*sech(x)^2)/sqrt(a))*sqrt(a)-sqrt(a+b*sech(x)^2)],
[sqrt(a+b*sech(x)^2)*tanh(x)^2,x,8,arctanh(sqrt(a)*tanh(x)/sqrt(a+b-b*tanh(x)^2))*sqrt(a)-1/2*(a-b)*arctan(sqrt(b)*tanh(x)/sqrt(a+b-b*tanh(x)^2))/sqrt(b)-1/2*sqrt(a+b-b*tanh(x)^2)*tanh(x)],
[sqrt(a+b*sech(x)^2)*tanh(x),x,5,arctanh(sqrt(a+b*sech(x)^2)/sqrt(a))*sqrt(a)-sqrt(a+b*sech(x)^2)],
[sqrt(a+b*sech(x)^2),x,6,arctanh(sqrt(a)*tanh(x)/sqrt(a+b-b*tanh(x)^2))*sqrt(a)+arctan(sqrt(b)*tanh(x)/sqrt(a+b-b*tanh(x)^2))*sqrt(b)],
[coth(x)*sqrt(a+b*sech(x)^2),x,7,arctanh(sqrt(a+b*sech(x)^2)/sqrt(a))*sqrt(a)-arctanh(sqrt(a+b*sech(x)^2)/sqrt(a+b))*sqrt(a+b)],
[coth(x)^2*sqrt(a+b*sech(x)^2),x,6,arctanh(sqrt(a)*tanh(x)/sqrt(a+b-b*tanh(x)^2))*sqrt(a)-coth(x)*sqrt(a+b-b*tanh(x)^2)],
[coth(x)^3*sqrt(a+b*sech(x)^2),x,8,arctanh(sqrt(a+b*sech(x)^2)/sqrt(a))*sqrt(a)-1/2*(2*a+b)*arctanh(sqrt(a+b*sech(x)^2)/sqrt(a+b))/sqrt(a+b)-1/2*coth(x)^2*sqrt(a+b*sech(x)^2)],
[coth(x)^4*sqrt(a+b*sech(x)^2),x,7,arctanh(sqrt(a)*tanh(x)/sqrt(a+b-b*tanh(x)^2))*sqrt(a)-1/3*(3*a+2*b)*coth(x)*sqrt(a+b-b*tanh(x)^2)/(a+b)-1/3*coth(x)^3*sqrt(a+b-b*tanh(x)^2)],
[coth(x)^5*sqrt(a+b*sech(x)^2),x,9,-1/8*(8*a^2+12*a*b+3*b^2)*arctanh(sqrt(a+b*sech(x)^2)/sqrt(a+b))/(a+b)^(3/2)+arctanh(sqrt(a+b*sech(x)^2)/sqrt(a))*sqrt(a)-1/8*(4*a+3*b)*coth(x)^2*sqrt(a+b*sech(x)^2)/(a+b)-1/4*coth(x)^4*sqrt(a+b*sech(x)^2)],
[(a+b*sech(x)^2)^(3/2)*tanh(x)^3,x,7,a^(3/2)*arctanh(sqrt(a+b*sech(x)^2)/sqrt(a))-1/3*(a+b*sech(x)^2)^(3/2)+1/5*(a+b*sech(x)^2)^(5/2)/b-a*sqrt(a+b*sech(x)^2)],
[(a+b*sech(x)^2)^(3/2)*tanh(x)^2,x,9,a^(3/2)*arctanh(sqrt(a)*tanh(x)/sqrt(a+b-b*tanh(x)^2))-1/8*(3*a^2-6*a*b-b^2)*arctan(sqrt(b)*tanh(x)/sqrt(a+b-b*tanh(x)^2))/sqrt(b)-1/8*(5*a+b)*sqrt(a+b-b*tanh(x)^2)*tanh(x)+1/4*b*sqrt(a+b-b*tanh(x)^2)*tanh(x)^3],
[(a+b*sech(x)^2)^(3/2)*tanh(x),x,6,a^(3/2)*arctanh(sqrt(a+b*sech(x)^2)/sqrt(a))-1/3*(a+b*sech(x)^2)^(3/2)-a*sqrt(a+b*sech(x)^2)],
[(a+b*sech(x)^2)^(3/2),x,7,a^(3/2)*arctanh(sqrt(a)*tanh(x)/sqrt(a+b-b*tanh(x)^2))+1/2*(3*a+b)*arctan(sqrt(b)*tanh(x)/sqrt(a+b-b*tanh(x)^2))*sqrt(b)+1/2*b*sqrt(a+b-b*tanh(x)^2)*tanh(x)],
[coth(x)*(a+b*sech(x)^2)^(3/2),x,8,a^(3/2)*arctanh(sqrt(a+b*sech(x)^2)/sqrt(a))-(a+b)^(3/2)*arctanh(sqrt(a+b*sech(x)^2)/sqrt(a+b))+b*sqrt(a+b*sech(x)^2)],
[coth(x)^2*(a+b*sech(x)^2)^(3/2),x,8,-b^(3/2)*arctan(sqrt(b)*tanh(x)/sqrt(a+b-b*tanh(x)^2))+a^(3/2)*arctanh(sqrt(a)*tanh(x)/sqrt(a+b-b*tanh(x)^2))-(a+b)*coth(x)*sqrt(a+b-b*tanh(x)^2)],
[(a+b*sech(c+d*x)^2)^(5/2),x,8,a^(5/2)*arctanh(sqrt(a)*tanh(c+d*x)/sqrt(a+b-b*tanh(c+d*x)^2))/d+1/8*(15*a^2+10*a*b+3*b^2)*arctan(sqrt(b)*tanh(c+d*x)/sqrt(a+b-b*tanh(c+d*x)^2))*sqrt(b)/d+1/8*b*(7*a+3*b)*sqrt(a+b-b*tanh(c+d*x)^2)*tanh(c+d*x)/d+1/4*b*tanh(c+d*x)*(a+b-b*tanh(c+d*x)^2)^(3/2)/d],

# p<0
[tanh(x)^5/sqrt(a+b*sech(x)^2),x,6,-1/3*(a+b*sech(x)^2)^(3/2)/b^2+arctanh(sqrt(a+b*sech(x)^2)/sqrt(a))/sqrt(a)+(a+2*b)*sqrt(a+b*sech(x)^2)/b^2],
[tanh(x)^4/sqrt(a+b*sech(x)^2),x,8,-1/2*(a+3*b)*arctan(sqrt(b)*tanh(x)/sqrt(a+b-b*tanh(x)^2))/b^(3/2)+arctanh(sqrt(a)*tanh(x)/sqrt(a+b-b*tanh(x)^2))/sqrt(a)+1/2*sqrt(a+b-b*tanh(x)^2)*tanh(x)/b],
[tanh(x)^3/sqrt(a+b*sech(x)^2),x,5,arctanh(sqrt(a+b*sech(x)^2)/sqrt(a))/sqrt(a)+sqrt(a+b*sech(x)^2)/b],
[tanh(x)^2/sqrt(a+b*sech(x)^2),x,7,arctanh(sqrt(a)*tanh(x)/sqrt(a+b-b*tanh(x)^2))/sqrt(a)-arctan(sqrt(b)*tanh(x)/sqrt(a+b-b*tanh(x)^2))/sqrt(b)],
[tanh(x)/sqrt(a+b*sech(x)^2),x,4,arctanh(sqrt(a+b*sech(x)^2)/sqrt(a))/sqrt(a)],
[1/sqrt(a+b*sech(x)^2),x,3,arctanh(sqrt(a)*tanh(x)/sqrt(a+b-b*tanh(x)^2))/sqrt(a)],
[coth(x)/sqrt(a+b*sech(x)^2),x,7,arctanh(sqrt(a+b*sech(x)^2)/sqrt(a))/sqrt(a)-arctanh(sqrt(a+b*sech(x)^2)/sqrt(a+b))/sqrt(a+b)],
[coth(x)^2/sqrt(a+b*sech(x)^2),x,6,arctanh(sqrt(a)*tanh(x)/sqrt(a+b-b*tanh(x)^2))/sqrt(a)-coth(x)*sqrt(a+b-b*tanh(x)^2)/(a+b)],
[coth(x)^3/sqrt(a+b*sech(x)^2),x,8,-1/2*(2*a+3*b)*arctanh(sqrt(a+b*sech(x)^2)/sqrt(a+b))/(a+b)^(3/2)+arctanh(sqrt(a+b*sech(x)^2)/sqrt(a))/sqrt(a)-1/2*coth(x)^2*sqrt(a+b*sech(x)^2)/(a+b)],
[tanh(x)^5/(a+b*sech(x)^2)^(3/2),x,6,arctanh(sqrt(a+b*sech(x)^2)/sqrt(a))/a^(3/2)-(a+b)^2/(a*b^2*sqrt(a+b*sech(x)^2))-sqrt(a+b*sech(x)^2)/b^2],
[tanh(x)^4/(a+b*sech(x)^2)^(3/2),x,8,arctan(sqrt(b)*tanh(x)/sqrt(a+b-b*tanh(x)^2))/b^(3/2)+arctanh(sqrt(a)*tanh(x)/sqrt(a+b-b*tanh(x)^2))/a^(3/2)-(a+b)*tanh(x)/(a*b*sqrt(a+b-b*tanh(x)^2))],
[tanh(x)^3/(a+b*sech(x)^2)^(3/2),x,5,arctanh(sqrt(a+b*sech(x)^2)/sqrt(a))/a^(3/2)+(-a-b)/(a*b*sqrt(a+b*sech(x)^2))],
[tanh(x)^2/(a+b*sech(x)^2)^(3/2),x,5,arctanh(sqrt(a)*tanh(x)/sqrt(a+b-b*tanh(x)^2))/a^(3/2)-tanh(x)/(a*sqrt(a+b-b*tanh(x)^2))],
[tanh(x)/(a+b*sech(x)^2)^(3/2),x,5,arctanh(sqrt(a+b*sech(x)^2)/sqrt(a))/a^(3/2)+(-1)/(a*sqrt(a+b*sech(x)^2))],
[1/(a+b*sech(x)^2)^(3/2),x,4,arctanh(sqrt(a)*tanh(x)/sqrt(a+b-b*tanh(x)^2))/a^(3/2)-b*tanh(x)/(a*(a+b)*sqrt(a+b-b*tanh(x)^2))],
[coth(x)/(a+b*sech(x)^2)^(3/2),x,8,arctanh(sqrt(a+b*sech(x)^2)/sqrt(a))/a^(3/2)-arctanh(sqrt(a+b*sech(x)^2)/sqrt(a+b))/(a+b)^(3/2)-b/(a*(a+b)*sqrt(a+b*sech(x)^2))],
[coth(x)^2/(a+b*sech(x)^2)^(3/2),x,7,arctanh(sqrt(a)*tanh(x)/sqrt(a+b-b*tanh(x)^2))/a^(3/2)-b*coth(x)/(a*(a+b)*sqrt(a+b-b*tanh(x)^2))-(a-b)*coth(x)*sqrt(a+b-b*tanh(x)^2)/(a*(a+b)^2)],
[tanh(x)^6/(a+b*sech(x)^2)^(5/2),x,9,-arctan(sqrt(b)*tanh(x)/sqrt(a+b-b*tanh(x)^2))/b^(5/2)+arctanh(sqrt(a)*tanh(x)/sqrt(a+b-b*tanh(x)^2))/a^(5/2)-(1/a^2+(-1)/b^2)*tanh(x)/sqrt(a+b-b*tanh(x)^2)-1/3*(a+b)*tanh(x)^3/(a*b*(a+b-b*tanh(x)^2)^(3/2))],
[tanh(x)^5/(a+b*sech(x)^2)^(5/2),x,6,arctanh(sqrt(a+b*sech(x)^2)/sqrt(a))/a^(5/2)-1/3*(a+b)^2/(a*b^2*(a+b*sech(x)^2)^(3/2))+((-1)/a^2+1/b^2)/sqrt(a+b*sech(x)^2)],
[tanh(x)^4/(a+b*sech(x)^2)^(5/2),x,7,arctanh(sqrt(a)*tanh(x)/sqrt(a+b-b*tanh(x)^2))/a^(5/2)+1/3*(a-3*b)*tanh(x)/(a^2*b*sqrt(a+b-b*tanh(x)^2))-1/3*(a+b)*tanh(x)/(a*b*(a+b-b*tanh(x)^2)^(3/2))],
[tanh(x)^3/(a+b*sech(x)^2)^(5/2),x,6,arctanh(sqrt(a+b*sech(x)^2)/sqrt(a))/a^(5/2)+1/3*(-a-b)/(a*b*(a+b*sech(x)^2)^(3/2))+(-1)/(a^2*sqrt(a+b*sech(x)^2))],
[tanh(x)^2/(a+b*sech(x)^2)^(5/2),x,7,arctanh(sqrt(a)*tanh(x)/sqrt(a+b-b*tanh(x)^2))/a^(5/2)-1/3*(2*a+3*b)*tanh(x)/(a^2*(a+b)*sqrt(a+b-b*tanh(x)^2))-1/3*tanh(x)/(a*(a+b-b*tanh(x)^2)^(3/2))],
[tanh(x)/(a+b*sech(x)^2)^(5/2),x,6,arctanh(sqrt(a+b*sech(x)^2)/sqrt(a))/a^(5/2)+(-1/3)/(a*(a+b*sech(x)^2)^(3/2))+(-1)/(a^2*sqrt(a+b*sech(x)^2))],
[1/(a+b*sech(x)^2)^(5/2),x,6,arctanh(sqrt(a)*tanh(x)/sqrt(a+b-b*tanh(x)^2))/a^(5/2)-1/3*b*(5*a+3*b)*tanh(x)/(a^2*(a+b)^2*sqrt(a+b-b*tanh(x)^2))-1/3*b*tanh(x)/(a*(a+b)*(a+b-b*tanh(x)^2)^(3/2))],
[coth(x)/(a+b*sech(x)^2)^(5/2),x,9,arctanh(sqrt(a+b*sech(x)^2)/sqrt(a))/a^(5/2)-arctanh(sqrt(a+b*sech(x)^2)/sqrt(a+b))/(a+b)^(5/2)-1/3*b/(a*(a+b)*(a+b*sech(x)^2)^(3/2))-b*(2*a+b)/(a^2*(a+b)^2*sqrt(a+b*sech(x)^2))],
[coth(x)^2/(a+b*sech(x)^2)^(5/2),x,8,arctanh(sqrt(a)*tanh(x)/sqrt(a+b-b*tanh(x)^2))/a^(5/2)-1/3*b*(7*a+3*b)*coth(x)/(a^2*(a+b)^2*sqrt(a+b-b*tanh(x)^2))-1/3*(a-3*b)*(3*a+b)*coth(x)*sqrt(a+b-b*tanh(x)^2)/(a^2*(a+b)^3)-1/3*b*coth(x)/(a*(a+b)*(a+b-b*tanh(x)^2)^(3/2))],
[1/(a+b*sech(c+d*x)^2)^(7/2),x,7,arctanh(sqrt(a)*tanh(c+d*x)/sqrt(a+b-b*tanh(c+d*x)^2))/(a^(7/2)*d)-1/15*b*(33*a^2+40*a*b+15*b^2)*tanh(c+d*x)/(a^3*(a+b)^3*d*sqrt(a+b-b*tanh(c+d*x)^2))-1/5*b*tanh(c+d*x)/(a*(a+b)*d*(a+b-b*tanh(c+d*x)^2)^(5/2))-1/15*b*(9*a+5*b)*tanh(c+d*x)/(a^2*(a+b)^2*d*(a+b-b*tanh(c+d*x)^2)^(3/2))]]:

# Integrands of the form Tanh[e+f x]^m (a+b Sech[e+f x]^2)^p when p symbolic

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

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

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