# Maple integration test file: "6 Hyperbolic functions\6.1 Hyperbolic sine\6.1.7 hyper^m (a+b sinh^n)^p.txt"

lst:=[

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

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

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

# p>0
[sinh(c+d*x)^4*(a+b*sinh(c+d*x)^2),x,4,1/16*(6*a-5*b)*x-1/16*(6*a-5*b)*cosh(c+d*x)*sinh(c+d*x)/d+1/24*(6*a-5*b)*cosh(c+d*x)*sinh(c+d*x)^3/d+1/6*b*cosh(c+d*x)*sinh(c+d*x)^5/d],
[sinh(c+d*x)^3*(a+b*sinh(c+d*x)^2),x,3,-(a-b)*cosh(c+d*x)/d+1/3*(a-2*b)*cosh(c+d*x)^3/d+1/5*b*cosh(c+d*x)^5/d],
[sinh(c+d*x)^2*(a+b*sinh(c+d*x)^2),x,3,-1/8*(4*a-3*b)*x+1/8*(4*a-3*b)*cosh(c+d*x)*sinh(c+d*x)/d+1/4*b*cosh(c+d*x)*sinh(c+d*x)^3/d],
[sinh(c+d*x)*(a+b*sinh(c+d*x)^2),x,2,(a-b)*cosh(c+d*x)/d+1/3*b*cosh(c+d*x)^3/d],
[a+b*sinh(c+d*x)^2,x,3,a*x-1/2*b*x+1/2*b*cosh(c+d*x)*sinh(c+d*x)/d],
[csch(c+d*x)*(a+b*sinh(c+d*x)^2),x,2,-a*arctanh(cosh(c+d*x))/d+b*cosh(c+d*x)/d],
[csch(c+d*x)^2*(a+b*sinh(c+d*x)^2),x,2,b*x-a*coth(c+d*x)/d],
[csch(c+d*x)^3*(a+b*sinh(c+d*x)^2),x,2,1/2*(a-2*b)*arctanh(cosh(c+d*x))/d-1/2*a*coth(c+d*x)*csch(c+d*x)/d],
[csch(c+d*x)^4*(a+b*sinh(c+d*x)^2),x,3,1/3*(2*a-3*b)*coth(c+d*x)/d-1/3*a*coth(c+d*x)*csch(c+d*x)^2/d],
[sinh(c+d*x)^4*(a+b*sinh(c+d*x)^2)^2,x,6,1/128*(48*a^2-80*a*b+35*b^2)*x-1/128*(80*a^2-176*a*b+93*b^2)*cosh(c+d*x)*sinh(c+d*x)/d+1/192*(48*a^2-208*a*b+139*b^2)*cosh(c+d*x)^3*sinh(c+d*x)/d+1/48*(16*a-13*b)*b*cosh(c+d*x)^5*sinh(c+d*x)/d+1/8*b^2*cosh(c+d*x)^3*sinh(c+d*x)^5/d],
[sinh(c+d*x)^3*(a+b*sinh(c+d*x)^2)^2,x,3,-(a-b)^2*cosh(c+d*x)/d+1/3*(a-3*b)*(a-b)*cosh(c+d*x)^3/d+1/5*(2*a-3*b)*b*cosh(c+d*x)^5/d+1/7*b^2*cosh(c+d*x)^7/d],
[sinh(c+d*x)^2*(a+b*sinh(c+d*x)^2)^2,x,2,-1/16*(8*a^2-12*a*b+5*b^2)*x+1/16*(8*a^2-20*a*b+11*b^2)*cosh(c+d*x)*sinh(c+d*x)/d+1/8*(4*a-3*b)*b*cosh(c+d*x)^3*sinh(c+d*x)/d+1/6*b^2*cosh(c+d*x)^3*sinh(c+d*x)^3/d,-1/16*(8*a^2-12*a*b+5*b^2)*x+1/48*(16*a^2-36*a*b+15*b^2)*cosh(c+d*x)*sinh(c+d*x)/d+1/24*(4*a-5*b)*b*cosh(c+d*x)*sinh(c+d*x)^3/d+1/6*cosh(c+d*x)*sinh(c+d*x)*(a+b*sinh(c+d*x)^2)^2/d],
[sinh(c+d*x)*(a+b*sinh(c+d*x)^2)^2,x,3,(a-b)^2*cosh(c+d*x)/d+2/3*(a-b)*b*cosh(c+d*x)^3/d+1/5*b^2*cosh(c+d*x)^5/d],
[(a+b*sinh(c+d*x)^2)^2,x,1,1/8*(8*a^2-8*a*b+3*b^2)*x+1/8*(8*a-3*b)*b*cosh(c+d*x)*sinh(c+d*x)/d+1/4*b^2*cosh(c+d*x)*sinh(c+d*x)^3/d],
[csch(c+d*x)*(a+b*sinh(c+d*x)^2)^2,x,4,-a^2*arctanh(cosh(c+d*x))/d+(2*a-b)*b*cosh(c+d*x)/d+1/3*b^2*cosh(c+d*x)^3/d],
[csch(c+d*x)^2*(a+b*sinh(c+d*x)^2)^2,x,4,1/2*(4*a-b)*b*x-a^2*coth(c+d*x)/d+1/2*b^2*cosh(c+d*x)*sinh(c+d*x)/d,1/2*(4*a-b)*b*x-a^2*cosh(c+d*x)^2*coth(c+d*x)/d+1/2*(2*a^2+b^2)*cosh(c+d*x)*sinh(c+d*x)/d],
[csch(c+d*x)^3*(a+b*sinh(c+d*x)^2)^2,x,5,1/2*a*(a-4*b)*arctanh(cosh(c+d*x))/d+b^2*cosh(c+d*x)/d-1/2*a^2*coth(c+d*x)*csch(c+d*x)/d],
[csch(c+d*x)^4*(a+b*sinh(c+d*x)^2)^2,x,4,b^2*x+a*(a-2*b)*coth(c+d*x)/d-1/3*a^2*coth(c+d*x)^3/d],
[sinh(c+d*x)^4*(a+b*sinh(c+d*x)^2)^3,x,7,3/256*(4*a-3*b)*(8*a^2-14*a*b+7*b^2)*x-1/1280*(576*a^3-1744*a^2*b+1678*a*b^2-525*b^3)*cosh(c+d*x)*sinh(c+d*x)/d+1/640*(48*a^3-272*a^2*b+314*a*b^2-105*b^3)*cosh(c+d*x)^3*sinh(c+d*x)/d+3/80*(2*a-3*b)*cosh(c+d*x)^5*sinh(c+d*x)^3*(a-(a-b)*tanh(c+d*x)^2)^2/d+1/10*cosh(c+d*x)^7*sinh(c+d*x)^3*(a-(a-b)*tanh(c+d*x)^2)^3/d-1/160*b*cosh(c+d*x)^3*sinh(c+d*x)^3*(a*(14*a-9*b)-(22*a-21*b)*(a-b)*tanh(c+d*x)^2)/d],
[sinh(c+d*x)^3*(a+b*sinh(c+d*x)^2)^3,x,3,-(a-b)^3*cosh(c+d*x)/d+1/3*(a-4*b)*(a-b)^2*cosh(c+d*x)^3/d+3/5*(a-2*b)*(a-b)*b*cosh(c+d*x)^5/d+1/7*(3*a-4*b)*b^2*cosh(c+d*x)^7/d+1/9*b^3*cosh(c+d*x)^9/d],
[sinh(c+d*x)^2*(a+b*sinh(c+d*x)^2)^3,x,3,-1/128*(64*a^3-144*a^2*b+120*a*b^2-35*b^3)*x+1/384*(96*a^3-376*a^2*b+360*a*b^2-105*b^3)*cosh(c+d*x)*sinh(c+d*x)/d+1/192*b*(24*a^2-64*a*b+35*b^2)*cosh(c+d*x)*sinh(c+d*x)^3/d+1/48*(6*a-7*b)*cosh(c+d*x)*sinh(c+d*x)*(a+b*sinh(c+d*x)^2)^2/d+1/8*cosh(c+d*x)*sinh(c+d*x)*(a+b*sinh(c+d*x)^2)^3/d],
[sinh(c+d*x)*(a+b*sinh(c+d*x)^2)^3,x,3,(a-b)^3*cosh(c+d*x)/d+(a-b)^2*b*cosh(c+d*x)^3/d+3/5*(a-b)*b^2*cosh(c+d*x)^5/d+1/7*b^3*cosh(c+d*x)^7/d],
[(a+b*sinh(c+d*x)^2)^3,x,2,1/16*(2*a-b)*(8*a^2-8*a*b+5*b^2)*x+1/48*b*(64*a^2-54*a*b+15*b^2)*cosh(c+d*x)*sinh(c+d*x)/d+5/24*(2*a-b)*b^2*cosh(c+d*x)*sinh(c+d*x)^3/d+1/6*b*cosh(c+d*x)*sinh(c+d*x)*(a+b*sinh(c+d*x)^2)^2/d],
[csch(c+d*x)*(a+b*sinh(c+d*x)^2)^3,x,4,-a^3*arctanh(cosh(c+d*x))/d+b*(3*a^2-3*a*b+b^2)*cosh(c+d*x)/d+1/3*(3*a-2*b)*b^2*cosh(c+d*x)^3/d+1/5*b^3*cosh(c+d*x)^5/d],
[csch(c+d*x)^2*(a+b*sinh(c+d*x)^2)^3,x,5,3/8*b*(8*a^2-4*a*b+b^2)*x-1/8*a*(2*a+b)*(4*a+b)*coth(c+d*x)/d+1/4*b*cosh(c+d*x)^4*coth(c+d*x)*(a-(a-b)*tanh(c+d*x)^2)^2/d+1/8*b*cosh(c+d*x)^2*coth(c+d*x)*(a*(4*a+b)-(4*a-3*b)*(a-b)*tanh(c+d*x)^2)/d],
[csch(c+d*x)^3*(a+b*sinh(c+d*x)^2)^3,x,5,1/2*a^2*(a-6*b)*arctanh(cosh(c+d*x))/d+(3*a-b)*b^2*cosh(c+d*x)/d+1/3*b^3*cosh(c+d*x)^3/d-1/2*a^3*coth(c+d*x)*csch(c+d*x)/d],
[csch(c+d*x)^4*(a+b*sinh(c+d*x)^2)^3,x,5,1/2*(6*a-b)*b^2*x+1/2*a*(2*a^2-5*a*b-2*b^2)*coth(c+d*x)/d-1/6*a^2*(2*a+3*b)*coth(c+d*x)^3/d+1/2*b*cosh(c+d*x)^2*coth(c+d*x)^3*(a-(a-b)*tanh(c+d*x)^2)^2/d],

# p<0
[sinh(c+d*x)^7/(a+b*sinh(c+d*x)^2),x,4,(a^2+a*b+b^2)*cosh(c+d*x)/(b^3*d)-1/3*(a+2*b)*cosh(c+d*x)^3/(b^2*d)+1/5*cosh(c+d*x)^5/(b*d)-a^3*arctan(cosh(c+d*x)*sqrt(b)/sqrt(a-b))/(b^(7/2)*d*sqrt(a-b))],
[sinh(c+d*x)^6/(a+b*sinh(c+d*x)^2),x,6,1/8*(8*a^2+4*a*b+3*b^2)*x/b^3-1/8*(4*a+3*b)*cosh(c+d*x)*sinh(c+d*x)/(b^2*d)+1/4*cosh(c+d*x)*sinh(c+d*x)^3/(b*d)-a^(5/2)*arctanh(sqrt(a-b)*tanh(c+d*x)/sqrt(a))/(b^3*d*sqrt(a-b))],
[sinh(c+d*x)^5/(a+b*sinh(c+d*x)^2),x,4,-(a+b)*cosh(c+d*x)/(b^2*d)+1/3*cosh(c+d*x)^3/(b*d)+a^2*arctan(cosh(c+d*x)*sqrt(b)/sqrt(a-b))/(b^(5/2)*d*sqrt(a-b))],
[sinh(c+d*x)^4/(a+b*sinh(c+d*x)^2),x,5,-1/2*(2*a+b)*x/b^2+1/2*cosh(c+d*x)*sinh(c+d*x)/(b*d)+a^(3/2)*arctanh(sqrt(a-b)*tanh(c+d*x)/sqrt(a))/(b^2*d*sqrt(a-b))],
[sinh(c+d*x)^3/(a+b*sinh(c+d*x)^2),x,3,cosh(c+d*x)/(b*d)-a*arctan(cosh(c+d*x)*sqrt(b)/sqrt(a-b))/(b^(3/2)*d*sqrt(a-b))],
[sinh(c+d*x)^2/(a+b*sinh(c+d*x)^2),x,3,x/b-arctanh(sqrt(a-b)*tanh(c+d*x)/sqrt(a))*sqrt(a)/(b*d*sqrt(a-b))],
[sinh(c+d*x)/(a+b*sinh(c+d*x)^2),x,2,arctan(cosh(c+d*x)*sqrt(b)/sqrt(a-b))/(d*sqrt(a-b)*sqrt(b))],
[1/(a+b*sinh(c+d*x)^2),x,2,arctanh(sqrt(a-b)*tanh(c+d*x)/sqrt(a))/(d*sqrt(a)*sqrt(a-b))],
[csch(c+d*x)/(a+b*sinh(c+d*x)^2),x,4,-arctanh(cosh(c+d*x))/(a*d)-arctan(cosh(c+d*x)*sqrt(b)/sqrt(a-b))*sqrt(b)/(a*d*sqrt(a-b))],
[csch(c+d*x)^2/(a+b*sinh(c+d*x)^2),x,3,-coth(c+d*x)/(a*d)-b*arctanh(sqrt(a-b)*tanh(c+d*x)/sqrt(a))/(a^(3/2)*d*sqrt(a-b))],
[csch(c+d*x)^3/(a+b*sinh(c+d*x)^2),x,5,1/2*(a+2*b)*arctanh(cosh(c+d*x))/(a^2*d)-1/2*coth(c+d*x)*csch(c+d*x)/(a*d)+b^(3/2)*arctan(cosh(c+d*x)*sqrt(b)/sqrt(a-b))/(a^2*d*sqrt(a-b))],
[csch(c+d*x)^4/(a+b*sinh(c+d*x)^2),x,4,(a+b)*coth(c+d*x)/(a^2*d)-1/3*coth(c+d*x)^3/(a*d)+b^2*arctanh(sqrt(a-b)*tanh(c+d*x)/sqrt(a))/(a^(5/2)*d*sqrt(a-b))],
[csch(c+d*x)^5/(a+b*sinh(c+d*x)^2),x,6,-1/8*(3*a^2+4*a*b+8*b^2)*arctanh(cosh(c+d*x))/(a^3*d)+1/8*(3*a+4*b)*coth(c+d*x)*csch(c+d*x)/(a^2*d)-1/4*coth(c+d*x)*csch(c+d*x)^3/(a*d)-b^(5/2)*arctan(cosh(c+d*x)*sqrt(b)/sqrt(a-b))/(a^3*d*sqrt(a-b))],
[csch(c+d*x)^6/(a+b*sinh(c+d*x)^2),x,4,-(a^2+a*b+b^2)*coth(c+d*x)/(a^3*d)+1/3*(2*a+b)*coth(c+d*x)^3/(a^2*d)-1/5*coth(c+d*x)^5/(a*d)-b^3*arctanh(sqrt(a-b)*tanh(c+d*x)/sqrt(a))/(a^(7/2)*d*sqrt(a-b))],
[sinh(c+d*x)^4/(a+b*sinh(c+d*x)^2)^2,x,5,x/b^2-1/2*(2*a-3*b)*arctanh(sqrt(a-b)*tanh(c+d*x)/sqrt(a))*sqrt(a)/((a-b)^(3/2)*b^2*d)-1/2*a*tanh(c+d*x)/((a-b)*b*d*(a-(a-b)*tanh(c+d*x)^2))],
[sinh(c+d*x)^3/(a+b*sinh(c+d*x)^2)^2,x,3,1/2*(a-2*b)*arctan(cosh(c+d*x)*sqrt(b)/sqrt(a-b))/((a-b)^(3/2)*b^(3/2)*d)-1/2*a*cosh(c+d*x)/((a-b)*b*d*(a-b+b*cosh(c+d*x)^2))],
[sinh(c+d*x)^2/(a+b*sinh(c+d*x)^2)^2,x,4,1/2*cosh(c+d*x)*sinh(c+d*x)/((a-b)*d*(a+b*sinh(c+d*x)^2))-1/2*arctanh(sqrt(a-b)*tanh(c+d*x)/sqrt(a))/((a-b)^(3/2)*d*sqrt(a))],
[sinh(c+d*x)/(a+b*sinh(c+d*x)^2)^2,x,3,1/2*cosh(c+d*x)/((a-b)*d*(a-b+b*cosh(c+d*x)^2))+1/2*arctan(cosh(c+d*x)*sqrt(b)/sqrt(a-b))/((a-b)^(3/2)*d*sqrt(b))],
[1/(a+b*sinh(c+d*x)^2)^2,x,4,1/2*(2*a-b)*arctanh(sqrt(a-b)*tanh(c+d*x)/sqrt(a))/(a^(3/2)*(a-b)^(3/2)*d)-1/2*b*cosh(c+d*x)*sinh(c+d*x)/(a*(a-b)*d*(a+b*sinh(c+d*x)^2))],
[csch(c+d*x)/(a+b*sinh(c+d*x)^2)^2,x,5,-arctanh(cosh(c+d*x))/(a^2*d)-1/2*b*cosh(c+d*x)/(a*(a-b)*d*(a-b+b*cosh(c+d*x)^2))-1/2*(3*a-2*b)*arctan(cosh(c+d*x)*sqrt(b)/sqrt(a-b))*sqrt(b)/(a^2*(a-b)^(3/2)*d)],
[csch(c+d*x)^2/(a+b*sinh(c+d*x)^2)^2,x,4,-1/2*(4*a-3*b)*b*arctanh(sqrt(a-b)*tanh(c+d*x)/sqrt(a))/(a^(5/2)*(a-b)^(3/2)*d)-coth(c+d*x)/(a*d*(a-(a-b)*tanh(c+d*x)^2))+1/2*(2*a^2-4*a*b+3*b^2)*tanh(c+d*x)/(a^2*(a-b)*d*(a-(a-b)*tanh(c+d*x)^2))],
[csch(c+d*x)^3/(a+b*sinh(c+d*x)^2)^2,x,6,1/2*(5*a-4*b)*b^(3/2)*arctan(cosh(c+d*x)*sqrt(b)/sqrt(a-b))/(a^3*(a-b)^(3/2)*d)+1/2*(a+4*b)*arctanh(cosh(c+d*x))/(a^3*d)-1/2*(a-2*b)*b*cosh(c+d*x)/(a^2*(a-b)*d*(a-b+b*cosh(c+d*x)^2))-1/2*coth(c+d*x)*csch(c+d*x)/(a*d*(a-b+b*cosh(c+d*x)^2))],
[csch(c+d*x)^4/(a+b*sinh(c+d*x)^2)^2,x,5,1/2*(6*a-5*b)*b^2*arctanh(sqrt(a-b)*tanh(c+d*x)/sqrt(a))/(a^(7/2)*(a-b)^(3/2)*d)+1/2*(2*a^2+a*b-5*b^2)*coth(c+d*x)/(a^3*(a-b)*d)-1/6*(2*a-5*b)*coth(c+d*x)^3/(a^2*(a-b)*d)-1/2*b*csch(c+d*x)^3*sech(c+d*x)/(a*(a-b)*d*(a-(a-b)*tanh(c+d*x)^2))],
[sinh(c+d*x)^4/(a+b*sinh(c+d*x)^2)^3,x,4,3/8*arctanh(sqrt(a-b)*tanh(c+d*x)/sqrt(a))/((a-b)^(5/2)*d*sqrt(a))+1/4*tanh(c+d*x)^3/((a-b)*d*(a-(a-b)*tanh(c+d*x)^2)^2)-3/8*tanh(c+d*x)/((a-b)^2*d*(a-(a-b)*tanh(c+d*x)^2))],
[sinh(c+d*x)^3/(a+b*sinh(c+d*x)^2)^3,x,4,1/8*(a-4*b)*arctan(cosh(c+d*x)*sqrt(b)/sqrt(a-b))/((a-b)^(5/2)*b^(3/2)*d)-1/4*a*cosh(c+d*x)/((a-b)*b*d*(a-b+b*cosh(c+d*x)^2)^2)+1/8*(a-4*b)*cosh(c+d*x)/((a-b)^2*b*d*(a-b+b*cosh(c+d*x)^2))],
[sinh(c+d*x)^2/(a+b*sinh(c+d*x)^2)^3,x,5,-1/8*(4*a-b)*arctanh(sqrt(a-b)*tanh(c+d*x)/sqrt(a))/(a^(3/2)*(a-b)^(5/2)*d)+1/4*cosh(c+d*x)*sinh(c+d*x)/((a-b)*d*(a+b*sinh(c+d*x)^2)^2)+1/8*(2*a+b)*cosh(c+d*x)*sinh(c+d*x)/(a*(a-b)^2*d*(a+b*sinh(c+d*x)^2))],
[sinh(c+d*x)/(a+b*sinh(c+d*x)^2)^3,x,4,1/4*cosh(c+d*x)/((a-b)*d*(a-b+b*cosh(c+d*x)^2)^2)+3/8*cosh(c+d*x)/((a-b)^2*d*(a-b+b*cosh(c+d*x)^2))+3/8*arctan(cosh(c+d*x)*sqrt(b)/sqrt(a-b))/((a-b)^(5/2)*d*sqrt(b))],
[1/(a+b*sinh(c+d*x)^2)^3,x,5,1/8*(8*a^2-8*a*b+3*b^2)*arctanh(sqrt(a-b)*tanh(c+d*x)/sqrt(a))/(a^(5/2)*(a-b)^(5/2)*d)-1/4*b*cosh(c+d*x)*sinh(c+d*x)/(a*(a-b)*d*(a+b*sinh(c+d*x)^2)^2)-3/8*(2*a-b)*b*cosh(c+d*x)*sinh(c+d*x)/(a^2*(a-b)^2*d*(a+b*sinh(c+d*x)^2))],
[csch(c+d*x)/(a+b*sinh(c+d*x)^2)^3,x,6,-arctanh(cosh(c+d*x))/(a^3*d)-1/4*b*cosh(c+d*x)/(a*(a-b)*d*(a-b+b*cosh(c+d*x)^2)^2)-1/8*(7*a-4*b)*b*cosh(c+d*x)/(a^2*(a-b)^2*d*(a-b+b*cosh(c+d*x)^2))-1/8*(15*a^2-20*a*b+8*b^2)*arctan(cosh(c+d*x)*sqrt(b)/sqrt(a-b))*sqrt(b)/(a^3*(a-b)^(5/2)*d)],
[csch(c+d*x)^2/(a+b*sinh(c+d*x)^2)^3,x,5,-3/8*b*(8*a^2-12*a*b+5*b^2)*arctanh(sqrt(a-b)*tanh(c+d*x)/sqrt(a))/(a^(7/2)*(a-b)^(5/2)*d)-1/8*(4*a-5*b)*(2*a-3*b)*coth(c+d*x)/(a^3*(a-b)^2*d)-1/4*b*csch(c+d*x)*sech(c+d*x)^3/(a*(a-b)*d*(a-(a-b)*tanh(c+d*x)^2)^2)-1/8*b*coth(c+d*x)*(4*a-5*b-(4*a-b)*tanh(c+d*x)^2)/(a^2*(a-b)^2*d*(a-(a-b)*tanh(c+d*x)^2))],
[csch(c+d*x)^3/(a+b*sinh(c+d*x)^2)^3,x,7,1/8*b^(3/2)*(35*a^2-56*a*b+24*b^2)*arctan(cosh(c+d*x)*sqrt(b)/sqrt(a-b))/(a^4*(a-b)^(5/2)*d)+1/2*(a+6*b)*arctanh(cosh(c+d*x))/(a^4*d)-1/4*(2*a-3*b)*b*cosh(c+d*x)/(a^2*(a-b)*d*(a-b+b*cosh(c+d*x)^2)^2)-1/8*(a-4*b)*(4*a-3*b)*b*cosh(c+d*x)/(a^3*(a-b)^2*d*(a-b+b*cosh(c+d*x)^2))-1/2*coth(c+d*x)*csch(c+d*x)/(a*d*(a-b+b*cosh(c+d*x)^2)^2)],
[csch(c+d*x)^4/(a+b*sinh(c+d*x)^2)^3,x,6,1/8*b^2*(48*a^2-80*a*b+35*b^2)*arctanh(sqrt(a-b)*tanh(c+d*x)/sqrt(a))/(a^(9/2)*(a-b)^(5/2)*d)+1/8*(8*a^3-4*a^2*b-45*a*b^2+35*b^3)*coth(c+d*x)/(a^4*(a-b)^2*d)-1/24*(8*a^2-52*a*b+35*b^2)*coth(c+d*x)^3/(a^3*(a-b)^2*d)-1/4*b*csch(c+d*x)^3*sech(c+d*x)^3/(a*(a-b)*d*(a-(a-b)*tanh(c+d*x)^2)^2)-1/8*(10*a-7*b)*b*csch(c+d*x)^3*sech(c+d*x)/(a^2*(a-b)^2*d*(a-(a-b)*tanh(c+d*x)^2))],
[1/(1+sinh(x)^2),x,3,tanh(x)],
[1/(1+sinh(x)^2)^2,x,3,tanh(x)-1/3*tanh(x)^3],
[1/(1+sinh(x)^2)^3,x,3,tanh(x)-2/3*tanh(x)^3+1/5*tanh(x)^5],
[1/(1-sinh(x)^2),x,2,arctanh(sqrt(2)*tanh(x))/sqrt(2)],
[1/(1-sinh(x)^2)^2,x,4,1/4*cosh(x)*sinh(x)/(1-sinh(x)^2)+3/4*arctanh(sqrt(2)*tanh(x))/sqrt(2)],
[1/(1-sinh(x)^2)^3,x,5,1/8*cosh(x)*sinh(x)/(1-sinh(x)^2)^2+9/32*cosh(x)*sinh(x)/(1-sinh(x)^2)+19/32*arctanh(sqrt(2)*tanh(x))/sqrt(2)],

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

# p>0
[sinh(e+f*x)^3*sqrt(a+b*sinh(e+f*x)^2),x,5,-1/8*(a-b)*(a+3*b)*arctanh(cosh(e+f*x)*sqrt(b)/sqrt(a-b+b*cosh(e+f*x)^2))/(b^(3/2)*f)+1/4*cosh(e+f*x)*(a-b+b*cosh(e+f*x)^2)^(3/2)/(b*f)-1/8*(a+3*b)*cosh(e+f*x)*sqrt(a-b+b*cosh(e+f*x)^2)/(b*f)],
[sinh(e+f*x)*sqrt(a+b*sinh(e+f*x)^2),x,4,1/2*(a-b)*arctanh(cosh(e+f*x)*sqrt(b)/sqrt(a-b+b*cosh(e+f*x)^2))/(f*sqrt(b))+1/2*cosh(e+f*x)*sqrt(a-b+b*cosh(e+f*x)^2)/f],
[csch(e+f*x)*sqrt(a+b*sinh(e+f*x)^2),x,6,-arctanh(cosh(e+f*x)*sqrt(a)/sqrt(a-b+b*cosh(e+f*x)^2))*sqrt(a)/f+arctanh(cosh(e+f*x)*sqrt(b)/sqrt(a-b+b*cosh(e+f*x)^2))*sqrt(b)/f],
[csch(e+f*x)^3*sqrt(a+b*sinh(e+f*x)^2),x,4,1/2*(a-b)*arctanh(cosh(e+f*x)*sqrt(a)/sqrt(a-b+b*cosh(e+f*x)^2))/(f*sqrt(a))-1/2*coth(e+f*x)*csch(e+f*x)*sqrt(a-b+b*cosh(e+f*x)^2)/f],
[csch(e+f*x)^5*sqrt(a+b*sinh(e+f*x)^2),x,5,-1/8*(a-b)*(3*a+b)*arctanh(cosh(e+f*x)*sqrt(a)/sqrt(a-b+b*cosh(e+f*x)^2))/(a^(3/2)*f)-1/4*(a-b+b*cosh(e+f*x)^2)^(3/2)*coth(e+f*x)*csch(e+f*x)^3/(a*f)+1/8*(3*a+b)*coth(e+f*x)*csch(e+f*x)*sqrt(a-b+b*cosh(e+f*x)^2)/(a*f)],
[sinh(e+f*x)^4*sqrt(a+b*sinh(e+f*x)^2),x,7,1/15*(a-4*b)*cosh(e+f*x)*sinh(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(b*f)+1/5*cosh(e+f*x)*sinh(e+f*x)^3*sqrt(a+b*sinh(e+f*x)^2)/f+1/15*(2*a^2+3*a*b-8*b^2)*sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticE(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(b^2*f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))-1/15*(a-4*b)*sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticF(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(b*f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))-1/15*(2*a^2+3*a*b-8*b^2)*sqrt(a+b*sinh(e+f*x)^2)*tanh(e+f*x)/(b^2*f)],
[sinh(e+f*x)^2*sqrt(a+b*sinh(e+f*x)^2),x,6,1/3*cosh(e+f*x)*sinh(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/f-1/3*I*(a-2*b)*sqrt(cos(I*e+I*f*x)^2)/cos(I*e+I*f*x)*EllipticE(sin(I*e+I*f*x),sqrt(b/a))*sqrt(a+b*sinh(e+f*x)^2)/(b*f*sqrt(1+b*sinh(e+f*x)^2/a))+1/3*I*a*(a-b)*sqrt(cos(I*e+I*f*x)^2)/cos(I*e+I*f*x)*EllipticF(sin(I*e+I*f*x),sqrt(b/a))*sqrt(1+b*sinh(e+f*x)^2/a)/(b*f*sqrt(a+b*sinh(e+f*x)^2))],
[sqrt(a+b*sinh(e+f*x)^2),x,2,-I*sqrt(cos(I*e+I*f*x)^2)/cos(I*e+I*f*x)*EllipticE(sin(I*e+I*f*x),sqrt(b/a))*sqrt(a+b*sinh(e+f*x)^2)/(f*sqrt(1+b*sinh(e+f*x)^2/a))],
[csch(e+f*x)^2*sqrt(a+b*sinh(e+f*x)^2),x,7,-coth(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/f-sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticE(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))+b*sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticF(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(a*f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))+sqrt(a+b*sinh(e+f*x)^2)*tanh(e+f*x)/f],
[csch(e+f*x)^4*sqrt(a+b*sinh(e+f*x)^2),x,7,1/3*(2*a-b)*coth(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(a*f)-1/3*coth(e+f*x)*csch(e+f*x)^2*sqrt(a+b*sinh(e+f*x)^2)/f+1/3*(2*a-b)*sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticE(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(a*f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))-1/3*b*sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticF(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(a*f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))-1/3*(2*a-b)*sqrt(a+b*sinh(e+f*x)^2)*tanh(e+f*x)/(a*f)],
[sinh(e+f*x)^3*(a+b*sinh(e+f*x)^2)^(3/2),x,6,-1/16*(a-b)^2*(a+5*b)*arctanh(cosh(e+f*x)*sqrt(b)/sqrt(a-b+b*cosh(e+f*x)^2))/(b^(3/2)*f)-1/24*(a+5*b)*cosh(e+f*x)*(a-b+b*cosh(e+f*x)^2)^(3/2)/(b*f)+1/6*cosh(e+f*x)*(a-b+b*cosh(e+f*x)^2)^(5/2)/(b*f)-1/16*(a-b)*(a+5*b)*cosh(e+f*x)*sqrt(a-b+b*cosh(e+f*x)^2)/(b*f)],
[sinh(e+f*x)*(a+b*sinh(e+f*x)^2)^(3/2),x,5,1/4*cosh(e+f*x)*(a-b+b*cosh(e+f*x)^2)^(3/2)/f+3/8*(a-b)^2*arctanh(cosh(e+f*x)*sqrt(b)/sqrt(a-b+b*cosh(e+f*x)^2))/(f*sqrt(b))+3/8*(a-b)*cosh(e+f*x)*sqrt(a-b+b*cosh(e+f*x)^2)/f],
[csch(e+f*x)*(a+b*sinh(e+f*x)^2)^(3/2),x,7,-a^(3/2)*arctanh(cosh(e+f*x)*sqrt(a)/sqrt(a-b+b*cosh(e+f*x)^2))/f+1/2*(3*a-b)*arctanh(cosh(e+f*x)*sqrt(b)/sqrt(a-b+b*cosh(e+f*x)^2))*sqrt(b)/f+1/2*b*cosh(e+f*x)*sqrt(a-b+b*cosh(e+f*x)^2)/f],
[csch(e+f*x)^3*(a+b*sinh(e+f*x)^2)^(3/2),x,7,b^(3/2)*arctanh(cosh(e+f*x)*sqrt(b)/sqrt(a-b+b*cosh(e+f*x)^2))/f+1/2*(a-3*b)*arctanh(cosh(e+f*x)*sqrt(a)/sqrt(a-b+b*cosh(e+f*x)^2))*sqrt(a)/f-1/2*a*coth(e+f*x)*csch(e+f*x)*sqrt(a-b+b*cosh(e+f*x)^2)/f],
[csch(e+f*x)^5*(a+b*sinh(e+f*x)^2)^(3/2),x,5,-1/4*(a-b+b*cosh(e+f*x)^2)^(3/2)*coth(e+f*x)*csch(e+f*x)^3/f-3/8*(a-b)^2*arctanh(cosh(e+f*x)*sqrt(a)/sqrt(a-b+b*cosh(e+f*x)^2))/(f*sqrt(a))+3/8*(a-b)*coth(e+f*x)*csch(e+f*x)*sqrt(a-b+b*cosh(e+f*x)^2)/f],
[csch(e+f*x)^7*(a+b*sinh(e+f*x)^2)^(3/2),x,6,1/16*(a-b)^2*(5*a+b)*arctanh(cosh(e+f*x)*sqrt(a)/sqrt(a-b+b*cosh(e+f*x)^2))/(a^(3/2)*f)+1/24*(5*a+b)*(a-b+b*cosh(e+f*x)^2)^(3/2)*coth(e+f*x)*csch(e+f*x)^3/(a*f)-1/6*(a-b+b*cosh(e+f*x)^2)^(5/2)*coth(e+f*x)*csch(e+f*x)^5/(a*f)-1/16*(a-b)*(5*a+b)*coth(e+f*x)*csch(e+f*x)*sqrt(a-b+b*cosh(e+f*x)^2)/(a*f)],
[sinh(e+f*x)^4*(a+b*sinh(e+f*x)^2)^(3/2),x,8,1/35*(a^2-11*a*b+8*b^2)*cosh(e+f*x)*sinh(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(b*f)+2/35*(4*a-3*b)*cosh(e+f*x)*sinh(e+f*x)^3*sqrt(a+b*sinh(e+f*x)^2)/f+1/7*b*cosh(e+f*x)*sinh(e+f*x)^5*sqrt(a+b*sinh(e+f*x)^2)/f+2/35*(a-2*b)*(a^2+4*a*b-4*b^2)*sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticE(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(b^2*f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))-1/35*(a^2-11*a*b+8*b^2)*sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticF(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(b*f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))-2/35*(a-2*b)*(a^2+4*a*b-4*b^2)*sqrt(a+b*sinh(e+f*x)^2)*tanh(e+f*x)/(b^2*f)],
[sinh(e+f*x)^2*(a+b*sinh(e+f*x)^2)^(3/2),x,7,1/5*cosh(e+f*x)*sinh(e+f*x)*(a+b*sinh(e+f*x)^2)^(3/2)/f+1/15*(3*a-4*b)*cosh(e+f*x)*sinh(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/f-1/15*I*(3*a^2-13*a*b+8*b^2)*sqrt(cos(I*e+I*f*x)^2)/cos(I*e+I*f*x)*EllipticE(sin(I*e+I*f*x),sqrt(b/a))*sqrt(a+b*sinh(e+f*x)^2)/(b*f*sqrt(1+b*sinh(e+f*x)^2/a))+1/15*I*a*(3*a-4*b)*(a-b)*sqrt(cos(I*e+I*f*x)^2)/cos(I*e+I*f*x)*EllipticF(sin(I*e+I*f*x),sqrt(b/a))*sqrt(1+b*sinh(e+f*x)^2/a)/(b*f*sqrt(a+b*sinh(e+f*x)^2))],
[(a+b*sinh(e+f*x)^2)^(3/2),x,6,1/3*b*cosh(e+f*x)*sinh(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/f-2/3*I*(2*a-b)*sqrt(cos(I*e+I*f*x)^2)/cos(I*e+I*f*x)*EllipticE(sin(I*e+I*f*x),sqrt(b/a))*sqrt(a+b*sinh(e+f*x)^2)/(f*sqrt(1+b*sinh(e+f*x)^2/a))+1/3*I*a*(a-b)*sqrt(cos(I*e+I*f*x)^2)/cos(I*e+I*f*x)*EllipticF(sin(I*e+I*f*x),sqrt(b/a))*sqrt(1+b*sinh(e+f*x)^2/a)/(f*sqrt(a+b*sinh(e+f*x)^2))],
[csch(e+f*x)^2*(a+b*sinh(e+f*x)^2)^(3/2),x,6,-a*coth(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/f-(a+b)*sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticE(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))+2*b*sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticF(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))+(a+b)*sqrt(a+b*sinh(e+f*x)^2)*tanh(e+f*x)/f],
[csch(e+f*x)^4*(a+b*sinh(e+f*x)^2)^(3/2),x,7,2/3*(a-2*b)*coth(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/f-1/3*a*coth(e+f*x)*csch(e+f*x)^2*sqrt(a+b*sinh(e+f*x)^2)/f+2/3*(a-2*b)*sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticE(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))-1/3*(a-3*b)*b*sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticF(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(a*f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))-2/3*(a-2*b)*sqrt(a+b*sinh(e+f*x)^2)*tanh(e+f*x)/f],
[(a+b*sinh(c+d*x)^2)^(5/2),x,7,1/5*b*cosh(c+d*x)*sinh(c+d*x)*(a+b*sinh(c+d*x)^2)^(3/2)/d+4/15*(2*a-b)*b*cosh(c+d*x)*sinh(c+d*x)*sqrt(a+b*sinh(c+d*x)^2)/d-1/15*I*(23*a^2-23*a*b+8*b^2)*sqrt(cos(I*c+I*d*x)^2)/cos(I*c+I*d*x)*EllipticE(sin(I*c+I*d*x),sqrt(b/a))*sqrt(a+b*sinh(c+d*x)^2)/(d*sqrt(1+b*sinh(c+d*x)^2/a))+4/15*I*a*(a-b)*(2*a-b)*sqrt(cos(I*c+I*d*x)^2)/cos(I*c+I*d*x)*EllipticF(sin(I*c+I*d*x),sqrt(b/a))*sqrt(1+b*sinh(c+d*x)^2/a)/(d*sqrt(a+b*sinh(c+d*x)^2))],
[sqrt(1+sinh(x)^2),x,3,sqrt(cosh(x)^2)*tanh(x)],
[sqrt(-1-sinh(x)^2),x,3,sqrt(-cosh(x)^2)*tanh(x)],
[sqrt(1-sinh(x)^2),x,1,-I*sqrt(cos(I*x)^2)/cos(I*x)*EllipticE(sin(I*x),I)],
[sqrt(-1+sinh(x)^2),x,2,-I*sqrt(cos(I*x)^2)/cos(I*x)*EllipticE(sin(I*x),I)*sqrt(-1+sinh(x)^2)/sqrt(1-sinh(x)^2)],
[sqrt(a+b*sinh(x)^2),x,2,-I*sqrt(cos(I*x)^2)/cos(I*x)*EllipticE(sin(I*x),sqrt(b/a))*sqrt(a+b*sinh(x)^2)/sqrt(1+b*sinh(x)^2/a)],
[(1+sinh(x)^2)^(3/2),x,4,1/3*(cosh(x)^2)^(3/2)*tanh(x)+2/3*sqrt(cosh(x)^2)*tanh(x)],
[(-1-sinh(x)^2)^(3/2),x,4,1/3*(-cosh(x)^2)^(3/2)*tanh(x)-2/3*sqrt(-cosh(x)^2)*tanh(x)],
[(1-sinh(x)^2)^(3/2),x,4,-2*I*sqrt(cos(I*x)^2)/cos(I*x)*EllipticE(sin(I*x),I)+2/3*I*sqrt(cos(I*x)^2)/cos(I*x)*EllipticF(sin(I*x),I)-1/3*cosh(x)*sinh(x)*sqrt(1-sinh(x)^2)],
[(-1+sinh(x)^2)^(3/2),x,6,2/3*I*sqrt(cos(I*x)^2)/cos(I*x)*EllipticF(sin(I*x),I)*sqrt(1-sinh(x)^2)/sqrt(-1+sinh(x)^2)+1/3*cosh(x)*sinh(x)*sqrt(-1+sinh(x)^2)+2*I*sqrt(cos(I*x)^2)/cos(I*x)*EllipticE(sin(I*x),I)*sqrt(-1+sinh(x)^2)/sqrt(1-sinh(x)^2)],
[(a+b*sinh(x)^2)^(3/2),x,6,1/3*b*cosh(x)*sinh(x)*sqrt(a+b*sinh(x)^2)-2/3*I*(2*a-b)*sqrt(cos(I*x)^2)/cos(I*x)*EllipticE(sin(I*x),sqrt(b/a))*sqrt(a+b*sinh(x)^2)/sqrt(1+b*sinh(x)^2/a)+1/3*I*a*(a-b)*sqrt(cos(I*x)^2)/cos(I*x)*EllipticF(sin(I*x),sqrt(b/a))*sqrt(1+b*sinh(x)^2/a)/sqrt(a+b*sinh(x)^2)],

# p<0
[sinh(e+f*x)^3/sqrt(a+b*sinh(e+f*x)^2),x,4,-1/2*(a+b)*arctanh(cosh(e+f*x)*sqrt(b)/sqrt(a-b+b*cosh(e+f*x)^2))/(b^(3/2)*f)+1/2*cosh(e+f*x)*sqrt(a-b+b*cosh(e+f*x)^2)/(b*f)],
[sinh(e+f*x)/sqrt(a+b*sinh(e+f*x)^2),x,3,arctanh(cosh(e+f*x)*sqrt(b)/sqrt(a-b+b*cosh(e+f*x)^2))/(f*sqrt(b))],
[csch(e+f*x)/sqrt(a+b*sinh(e+f*x)^2),x,3,-arctanh(cosh(e+f*x)*sqrt(a)/sqrt(a-b+b*cosh(e+f*x)^2))/(f*sqrt(a))],
[csch(e+f*x)^3/sqrt(a+b*sinh(e+f*x)^2),x,4,1/2*(a+b)*arctanh(cosh(e+f*x)*sqrt(a)/sqrt(a-b+b*cosh(e+f*x)^2))/(a^(3/2)*f)-1/2*coth(e+f*x)*csch(e+f*x)*sqrt(a-b+b*cosh(e+f*x)^2)/(a*f)],
[sinh(e+f*x)^4/sqrt(a+b*sinh(e+f*x)^2),x,6,1/3*cosh(e+f*x)*sinh(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(b*f)+2/3*(a+b)*sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticE(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(b^2*f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))-1/3*sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticF(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(b*f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))-2/3*(a+b)*sqrt(a+b*sinh(e+f*x)^2)*tanh(e+f*x)/(b^2*f)],
[sinh(e+f*x)^2/sqrt(a+b*sinh(e+f*x)^2),x,5,-I*sqrt(cos(I*e+I*f*x)^2)/cos(I*e+I*f*x)*EllipticE(sin(I*e+I*f*x),sqrt(b/a))*sqrt(a+b*sinh(e+f*x)^2)/(b*f*sqrt(1+b*sinh(e+f*x)^2/a))+I*a*sqrt(cos(I*e+I*f*x)^2)/cos(I*e+I*f*x)*EllipticF(sin(I*e+I*f*x),sqrt(b/a))*sqrt(1+b*sinh(e+f*x)^2/a)/(b*f*sqrt(a+b*sinh(e+f*x)^2))],
[1/sqrt(a+b*sinh(e+f*x)^2),x,2,-I*sqrt(cos(I*e+I*f*x)^2)/cos(I*e+I*f*x)*EllipticF(sin(I*e+I*f*x),sqrt(b/a))*sqrt(1+b*sinh(e+f*x)^2/a)/(f*sqrt(a+b*sinh(e+f*x)^2))],
[csch(e+f*x)^2/sqrt(a+b*sinh(e+f*x)^2),x,5,-coth(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(a*f)-sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticE(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(a*f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))+sqrt(a+b*sinh(e+f*x)^2)*tanh(e+f*x)/(a*f)],
[csch(e+f*x)^4/sqrt(a+b*sinh(e+f*x)^2),x,7,2/3*(a+b)*coth(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(a^2*f)-1/3*coth(e+f*x)*csch(e+f*x)^2*sqrt(a+b*sinh(e+f*x)^2)/(a*f)+2/3*(a+b)*sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticE(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(a^2*f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))-1/3*b*sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticF(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(a^2*f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))-2/3*(a+b)*sqrt(a+b*sinh(e+f*x)^2)*tanh(e+f*x)/(a^2*f)],
[sinh(e+f*x)^3/(a+b*sinh(e+f*x)^2)^(3/2),x,4,arctanh(cosh(e+f*x)*sqrt(b)/sqrt(a-b+b*cosh(e+f*x)^2))/(b^(3/2)*f)-a*cosh(e+f*x)/((a-b)*b*f*sqrt(a-b+b*cosh(e+f*x)^2))],
[sinh(e+f*x)/(a+b*sinh(e+f*x)^2)^(3/2),x,2,cosh(e+f*x)/((a-b)*f*sqrt(a-b+b*cosh(e+f*x)^2))],
[csch(e+f*x)/(a+b*sinh(e+f*x)^2)^(3/2),x,4,-arctanh(cosh(e+f*x)*sqrt(a)/sqrt(a-b+b*cosh(e+f*x)^2))/(a^(3/2)*f)-b*cosh(e+f*x)/(a*(a-b)*f*sqrt(a-b+b*cosh(e+f*x)^2))],
[csch(e+f*x)^3/(a+b*sinh(e+f*x)^2)^(3/2),x,6,1/2*(a+3*b)*arctanh(cosh(e+f*x)*sqrt(a)/sqrt(a-b+b*cosh(e+f*x)^2))/(a^(5/2)*f)-1/2*(a-3*b)*b*cosh(e+f*x)/(a^2*(a-b)*f*sqrt(a-b+b*cosh(e+f*x)^2))-1/2*coth(e+f*x)*csch(e+f*x)/(a*f*sqrt(a-b+b*cosh(e+f*x)^2))],
[sinh(e+f*x)^6/(a+b*sinh(e+f*x)^2)^(3/2),x,7,-a*cosh(e+f*x)*sinh(e+f*x)^3/((a-b)*b*f*sqrt(a+b*sinh(e+f*x)^2))+1/3*(4*a-b)*cosh(e+f*x)*sinh(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/((a-b)*b^2*f)+1/3*(8*a^2-3*a*b-2*b^2)*sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticE(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/((a-b)*b^3*f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))-1/3*(4*a-b)*sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticF(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/((a-b)*b^2*f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))-1/3*(8*a^2-3*a*b-2*b^2)*sqrt(a+b*sinh(e+f*x)^2)*tanh(e+f*x)/((a-b)*b^3*f)],
[sinh(e+f*x)^4/(a+b*sinh(e+f*x)^2)^(3/2),x,6,-a*cosh(e+f*x)*sinh(e+f*x)/((a-b)*b*f*sqrt(a+b*sinh(e+f*x)^2))-(2*a-b)*sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticE(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/((a-b)*b^2*f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))+sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticF(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/((a-b)*b*f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))+(2*a-b)*sqrt(a+b*sinh(e+f*x)^2)*tanh(e+f*x)/((a-b)*b^2*f)],
[sinh(e+f*x)^2/(a+b*sinh(e+f*x)^2)^(3/2),x,6,cosh(e+f*x)*sinh(e+f*x)/((a-b)*f*sqrt(a+b*sinh(e+f*x)^2))+I*sqrt(cos(I*e+I*f*x)^2)/cos(I*e+I*f*x)*EllipticE(sin(I*e+I*f*x),sqrt(b/a))*sqrt(a+b*sinh(e+f*x)^2)/((a-b)*b*f*sqrt(1+b*sinh(e+f*x)^2/a))-I*sqrt(cos(I*e+I*f*x)^2)/cos(I*e+I*f*x)*EllipticF(sin(I*e+I*f*x),sqrt(b/a))*sqrt(1+b*sinh(e+f*x)^2/a)/(b*f*sqrt(a+b*sinh(e+f*x)^2))],
[1/(a+b*sinh(e+f*x)^2)^(3/2),x,4,-b*cosh(e+f*x)*sinh(e+f*x)/(a*(a-b)*f*sqrt(a+b*sinh(e+f*x)^2))-I*sqrt(cos(I*e+I*f*x)^2)/cos(I*e+I*f*x)*EllipticE(sin(I*e+I*f*x),sqrt(b/a))*sqrt(a+b*sinh(e+f*x)^2)/(a*(a-b)*f*sqrt(1+b*sinh(e+f*x)^2/a))],
[csch(e+f*x)^2/(a+b*sinh(e+f*x)^2)^(3/2),x,7,-b*coth(e+f*x)/(a*(a-b)*f*sqrt(a+b*sinh(e+f*x)^2))-(a-2*b)*coth(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(a^2*(a-b)*f)-(a-2*b)*sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticE(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(a^2*(a-b)*f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))-b*sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticF(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(a^2*(a-b)*f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))+(a-2*b)*sqrt(a+b*sinh(e+f*x)^2)*tanh(e+f*x)/(a^2*(a-b)*f)],
[sinh(e+f*x)^5/(a+b*sinh(e+f*x)^2)^(5/2),x,5,arctanh(cosh(e+f*x)*sqrt(b)/sqrt(a-b+b*cosh(e+f*x)^2))/(b^(5/2)*f)-1/3*a*cosh(e+f*x)*sinh(e+f*x)^2/((a-b)*b*f*(a-b+b*cosh(e+f*x)^2)^(3/2))-1/3*a*(3*a-5*b)*cosh(e+f*x)/((a-b)^2*b^2*f*sqrt(a-b+b*cosh(e+f*x)^2))],
[sinh(e+f*x)^3/(a+b*sinh(e+f*x)^2)^(5/2),x,3,1/3*cosh(e+f*x)*sinh(e+f*x)^2/((a-b)*f*(a-b+b*cosh(e+f*x)^2)^(3/2))-2/3*cosh(e+f*x)/((a-b)^2*f*sqrt(a-b+b*cosh(e+f*x)^2))],
[sinh(e+f*x)/(a+b*sinh(e+f*x)^2)^(5/2),x,3,1/3*cosh(e+f*x)/((a-b)*f*(a-b+b*cosh(e+f*x)^2)^(3/2))+2/3*cosh(e+f*x)/((a-b)^2*f*sqrt(a-b+b*cosh(e+f*x)^2))],
[csch(e+f*x)/(a+b*sinh(e+f*x)^2)^(5/2),x,6,-arctanh(cosh(e+f*x)*sqrt(a)/sqrt(a-b+b*cosh(e+f*x)^2))/(a^(5/2)*f)-1/3*b*cosh(e+f*x)/(a*(a-b)*f*(a-b+b*cosh(e+f*x)^2)^(3/2))-1/3*(5*a-3*b)*b*cosh(e+f*x)/(a^2*(a-b)^2*f*sqrt(a-b+b*cosh(e+f*x)^2))],
[sinh(e+f*x)^6/(a+b*sinh(e+f*x)^2)^(5/2),x,7,-1/3*a*cosh(e+f*x)*sinh(e+f*x)^3/((a-b)*b*f*(a+b*sinh(e+f*x)^2)^(3/2))-2/3*a*(2*a-3*b)*cosh(e+f*x)*sinh(e+f*x)/((a-b)^2*b^2*f*sqrt(a+b*sinh(e+f*x)^2))-1/3*(8*a^2-13*a*b+3*b^2)*sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticE(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/((a-b)^2*b^3*f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))+2/3*(2*a-3*b)*sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticF(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/((a-b)^2*b^2*f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))+1/3*(8*a^2-13*a*b+3*b^2)*sqrt(a+b*sinh(e+f*x)^2)*tanh(e+f*x)/((a-b)^2*b^3*f)],
[sinh(e+f*x)^4/(a+b*sinh(e+f*x)^2)^(5/2),x,5,-1/3*a*cosh(e+f*x)*sinh(e+f*x)/((a-b)*b*f*(a+b*sinh(e+f*x)^2)^(3/2))+2/3*(a-2*b)*cosh(e+f*x)*sqrt(cos(arctan(sinh(e+f*x)*sqrt(b)/sqrt(a)))^2)/cos(arctan(sinh(e+f*x)*sqrt(b)/sqrt(a)))*EllipticE(sin(arctan(sinh(e+f*x)*sqrt(b)/sqrt(a))),sqrt(1-a/b))*sqrt(a)/((a-b)^2*b^(3/2)*f*sqrt(a*cosh(e+f*x)^2/(a+b*sinh(e+f*x)^2))*sqrt(a+b*sinh(e+f*x)^2))-1/3*(a-3*b)*sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticF(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(a*(a-b)^2*b*f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))],
[sinh(e+f*x)^2/(a+b*sinh(e+f*x)^2)^(5/2),x,7,1/3*cosh(e+f*x)*sinh(e+f*x)/((a-b)*f*(a+b*sinh(e+f*x)^2)^(3/2))+1/3*(a+b)*cosh(e+f*x)*sinh(e+f*x)/(a*(a-b)^2*f*sqrt(a+b*sinh(e+f*x)^2))+1/3*I*(a+b)*sqrt(cos(I*e+I*f*x)^2)/cos(I*e+I*f*x)*EllipticE(sin(I*e+I*f*x),sqrt(b/a))*sqrt(a+b*sinh(e+f*x)^2)/(a*(a-b)^2*b*f*sqrt(1+b*sinh(e+f*x)^2/a))-1/3*I*sqrt(cos(I*e+I*f*x)^2)/cos(I*e+I*f*x)*EllipticF(sin(I*e+I*f*x),sqrt(b/a))*sqrt(1+b*sinh(e+f*x)^2/a)/((a-b)*b*f*sqrt(a+b*sinh(e+f*x)^2))],
[1/(a+b*sinh(e+f*x)^2)^(5/2),x,7,-1/3*b*cosh(e+f*x)*sinh(e+f*x)/(a*(a-b)*f*(a+b*sinh(e+f*x)^2)^(3/2))-2/3*(2*a-b)*b*cosh(e+f*x)*sinh(e+f*x)/(a^2*(a-b)^2*f*sqrt(a+b*sinh(e+f*x)^2))-2/3*I*(2*a-b)*sqrt(cos(I*e+I*f*x)^2)/cos(I*e+I*f*x)*EllipticE(sin(I*e+I*f*x),sqrt(b/a))*sqrt(a+b*sinh(e+f*x)^2)/(a^2*(a-b)^2*f*sqrt(1+b*sinh(e+f*x)^2/a))+1/3*I*sqrt(cos(I*e+I*f*x)^2)/cos(I*e+I*f*x)*EllipticF(sin(I*e+I*f*x),sqrt(b/a))*sqrt(1+b*sinh(e+f*x)^2/a)/(a*(a-b)*f*sqrt(a+b*sinh(e+f*x)^2))],
[csch(e+f*x)^2/(a+b*sinh(e+f*x)^2)^(5/2),x,8,-1/3*b*coth(e+f*x)/(a*(a-b)*f*(a+b*sinh(e+f*x)^2)^(3/2))-2/3*(3*a-2*b)*b*coth(e+f*x)/(a^2*(a-b)^2*f*sqrt(a+b*sinh(e+f*x)^2))-1/3*(3*a^2-13*a*b+8*b^2)*coth(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(a^3*(a-b)^2*f)-1/3*(3*a^2-13*a*b+8*b^2)*sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticE(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(a^3*(a-b)^2*f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))-2/3*(3*a-2*b)*b*sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticF(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(a^3*(a-b)^2*f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))+1/3*(3*a^2-13*a*b+8*b^2)*sqrt(a+b*sinh(e+f*x)^2)*tanh(e+f*x)/(a^3*(a-b)^2*f)],
[1/sqrt(1+sinh(x)^2),x,3,arctan(sinh(x))*cosh(x)/sqrt(cosh(x)^2)],
[1/sqrt(1-sinh(x)^2),x,1,-I*sqrt(cos(I*x)^2)/cos(I*x)*EllipticF(sin(I*x),I)],
[1/sqrt(-1+sinh(x)^2),x,2,-I*sqrt(cos(I*x)^2)/cos(I*x)*EllipticF(sin(I*x),I)*sqrt(1-sinh(x)^2)/sqrt(-1+sinh(x)^2)],
[1/sqrt(-1-sinh(x)^2),x,3,arctan(sinh(x))*cosh(x)/sqrt(-cosh(x)^2)],
[1/sqrt(a+b*sinh(x)^2),x,2,-I*sqrt(cos(I*x)^2)/cos(I*x)*EllipticF(sin(I*x),sqrt(b/a))*sqrt(1+b*sinh(x)^2/a)/sqrt(a+b*sinh(x)^2)],

# Integrands of the form (d Sinh[e+f x]) (a+b Sinh[e+f x]^2)^p when p symbolic
[(d*sinh(e+f*x))^m*(a+b*sinh(e+f*x)^2)^p,x,3,d*AppellF1(1/2,1/2*(1-m),-p,3/2,cosh(e+f*x)^2,-b*cosh(e+f*x)^2/(a-b))*cosh(e+f*x)*(a-b+b*cosh(e+f*x)^2)^p*(d*sinh(e+f*x))^(-1+m)*(-sinh(e+f*x)^2)^(1/2*(1-m))/(f*(1+b*cosh(e+f*x)^2/(a-b))^p)],
[sinh(e+f*x)^5*(a+b*sinh(e+f*x)^2)^p,x,5,-(3*a+2*b*(2+p))*cosh(e+f*x)*(a-b+b*cosh(e+f*x)^2)^(1+p)/(b^2*f*(15+16*p+4*p^2))+(3*a^2+4*a*b*(1+p)+4*b^2*(2+3*p+p^2))*cosh(e+f*x)*(a-b+b*cosh(e+f*x)^2)^p*hypergeom([1/2,-p],[3/2],-b*cosh(e+f*x)^2/(a-b))/(b^2*f*(15+16*p+4*p^2)*(1+b*cosh(e+f*x)^2/(a-b))^p)+cosh(e+f*x)*(a-b+b*cosh(e+f*x)^2)^(1+p)*sinh(e+f*x)^2/(b*f*(5+2*p))],
[sinh(e+f*x)^3*(a+b*sinh(e+f*x)^2)^p,x,4,cosh(e+f*x)*(a-b+b*cosh(e+f*x)^2)^(1+p)/(b*f*(3+2*p))-(a+2*b*(1+p))*cosh(e+f*x)*(a-b+b*cosh(e+f*x)^2)^p*hypergeom([1/2,-p],[3/2],-b*cosh(e+f*x)^2/(a-b))/(b*f*(3+2*p)*(1+b*cosh(e+f*x)^2/(a-b))^p)],
[sinh(e+f*x)*(a+b*sinh(e+f*x)^2)^p,x,3,cosh(e+f*x)*(a-b+b*cosh(e+f*x)^2)^p*hypergeom([1/2,-p],[3/2],-b*cosh(e+f*x)^2/(a-b))/(f*(1+b*cosh(e+f*x)^2/(a-b))^p)],
[csch(e+f*x)*(a+b*sinh(e+f*x)^2)^p,x,3,-AppellF1(1/2,1,-p,3/2,cosh(e+f*x)^2,-b*cosh(e+f*x)^2/(a-b))*cosh(e+f*x)*(a-b+b*cosh(e+f*x)^2)^p/(f*(1+b*cosh(e+f*x)^2/(a-b))^p)],
[csch(e+f*x)^3*(a+b*sinh(e+f*x)^2)^p,x,3,AppellF1(1/2,2,-p,3/2,cosh(e+f*x)^2,-b*cosh(e+f*x)^2/(a-b))*cosh(e+f*x)*(a-b+b*cosh(e+f*x)^2)^p/(f*(1+b*cosh(e+f*x)^2/(a-b))^p)],
[csch(e+f*x)^5*(a+b*sinh(e+f*x)^2)^p,x,3,-AppellF1(1/2,3,-p,3/2,cosh(e+f*x)^2,-b*cosh(e+f*x)^2/(a-b))*cosh(e+f*x)*(a-b+b*cosh(e+f*x)^2)^p/(f*(1+b*cosh(e+f*x)^2/(a-b))^p)],
[sinh(e+f*x)^4*(a+b*sinh(e+f*x)^2)^p,x,3,1/5*AppellF1(5/2,1/2,-p,7/2,-sinh(e+f*x)^2,-b*sinh(e+f*x)^2/a)*sinh(e+f*x)^4*(a+b*sinh(e+f*x)^2)^p*sqrt(cosh(e+f*x)^2)*tanh(e+f*x)/(f*(1+b*sinh(e+f*x)^2/a)^p)],
[sinh(e+f*x)^2*(a+b*sinh(e+f*x)^2)^p,x,3,1/3*AppellF1(3/2,2+p,-p,5/2,tanh(e+f*x)^2,(a-b)*tanh(e+f*x)^2/a)*(sech(e+f*x)^2)^p*(a+b*sinh(e+f*x)^2)^p*tanh(e+f*x)^3/(f*(1-(a-b)*tanh(e+f*x)^2/a)^p)],
[csch(e+f*x)^2*(a+b*sinh(e+f*x)^2)^p,x,3,-AppellF1(-1/2,1/2,-p,1/2,-sinh(e+f*x)^2,-b*sinh(e+f*x)^2/a)*csch(e+f*x)*sech(e+f*x)*(a+b*sinh(e+f*x)^2)^p*sqrt(cosh(e+f*x)^2)/(f*(1+b*sinh(e+f*x)^2/a)^p)],
[csch(e+f*x)^4*(a+b*sinh(e+f*x)^2)^p,x,3,-1/3*AppellF1(-3/2,1/2,-p,-1/2,-sinh(e+f*x)^2,-b*sinh(e+f*x)^2/a)*csch(e+f*x)^3*sech(e+f*x)*(a+b*sinh(e+f*x)^2)^p*sqrt(cosh(e+f*x)^2)/(f*(1+b*sinh(e+f*x)^2/a)^p)],

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

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

# p>0
[sinh(c+d*x)^4*(a+b*sinh(c+d*x)^3),x,7,3/8*a*x-b*cosh(c+d*x)/d+b*cosh(c+d*x)^3/d-3/5*b*cosh(c+d*x)^5/d+1/7*b*cosh(c+d*x)^7/d-3/8*a*cosh(c+d*x)*sinh(c+d*x)/d+1/4*a*cosh(c+d*x)*sinh(c+d*x)^3/d],
[sinh(c+d*x)^3*(a+b*sinh(c+d*x)^3),x,8,-5/16*b*x-a*cosh(c+d*x)/d+1/3*a*cosh(c+d*x)^3/d+5/16*b*cosh(c+d*x)*sinh(c+d*x)/d-5/24*b*cosh(c+d*x)*sinh(c+d*x)^3/d+1/6*b*cosh(c+d*x)*sinh(c+d*x)^5/d],
[sinh(c+d*x)^2*(a+b*sinh(c+d*x)^3),x,6,-1/2*a*x+b*cosh(c+d*x)/d-2/3*b*cosh(c+d*x)^3/d+1/5*b*cosh(c+d*x)^5/d+1/2*a*cosh(c+d*x)*sinh(c+d*x)/d],
[sinh(c+d*x)*(a+b*sinh(c+d*x)^3),x,6,3/8*b*x+a*cosh(c+d*x)/d-3/8*b*cosh(c+d*x)*sinh(c+d*x)/d+1/4*b*cosh(c+d*x)*sinh(c+d*x)^3/d],
[a+b*sinh(c+d*x)^3,x,3,a*x-b*cosh(c+d*x)/d+1/3*b*cosh(c+d*x)^3/d],
[csch(c+d*x)*(a+b*sinh(c+d*x)^3),x,5,-1/2*b*x-a*arctanh(cosh(c+d*x))/d+1/2*b*cosh(c+d*x)*sinh(c+d*x)/d],
[csch(c+d*x)^2*(a+b*sinh(c+d*x)^3),x,5,b*cosh(c+d*x)/d-a*coth(c+d*x)/d],
[csch(c+d*x)^3*(a+b*sinh(c+d*x)^3),x,4,b*x+1/2*a*arctanh(cosh(c+d*x))/d-1/2*a*coth(c+d*x)*csch(c+d*x)/d],
[csch(c+d*x)^4*(a+b*sinh(c+d*x)^3),x,5,-b*arctanh(cosh(c+d*x))/d+a*coth(c+d*x)/d-1/3*a*coth(c+d*x)^3/d],
[sinh(c+d*x)^3*(a+b*sinh(c+d*x)^3)^2,x,10,-5/8*a*b*x-a^2*cosh(c+d*x)/d+b^2*cosh(c+d*x)/d+1/3*a^2*cosh(c+d*x)^3/d-4/3*b^2*cosh(c+d*x)^3/d+6/5*b^2*cosh(c+d*x)^5/d-4/7*b^2*cosh(c+d*x)^7/d+1/9*b^2*cosh(c+d*x)^9/d+5/8*a*b*cosh(c+d*x)*sinh(c+d*x)/d-5/12*a*b*cosh(c+d*x)*sinh(c+d*x)^3/d+1/3*a*b*cosh(c+d*x)*sinh(c+d*x)^5/d],
[sinh(c+d*x)^2*(a+b*sinh(c+d*x)^3)^2,x,11,-1/2*a^2*x+35/128*b^2*x+2*a*b*cosh(c+d*x)/d-4/3*a*b*cosh(c+d*x)^3/d+2/5*a*b*cosh(c+d*x)^5/d+1/2*a^2*cosh(c+d*x)*sinh(c+d*x)/d-35/128*b^2*cosh(c+d*x)*sinh(c+d*x)/d+35/192*b^2*cosh(c+d*x)*sinh(c+d*x)^3/d-7/48*b^2*cosh(c+d*x)*sinh(c+d*x)^5/d+1/8*b^2*cosh(c+d*x)*sinh(c+d*x)^7/d],
[sinh(c+d*x)*(a+b*sinh(c+d*x)^3)^2,x,8,3/4*a*b*x+a^2*cosh(c+d*x)/d-b^2*cosh(c+d*x)/d+b^2*cosh(c+d*x)^3/d-3/5*b^2*cosh(c+d*x)^5/d+1/7*b^2*cosh(c+d*x)^7/d-3/4*a*b*cosh(c+d*x)*sinh(c+d*x)/d+1/2*a*b*cosh(c+d*x)*sinh(c+d*x)^3/d],
[(a+b*sinh(c+d*x)^3)^2,x,8,a^2*x-5/16*b^2*x-2*a*b*cosh(c+d*x)/d+2/3*a*b*cosh(c+d*x)^3/d+5/16*b^2*cosh(c+d*x)*sinh(c+d*x)/d-5/24*b^2*cosh(c+d*x)*sinh(c+d*x)^3/d+1/6*b^2*cosh(c+d*x)*sinh(c+d*x)^5/d],
[csch(c+d*x)*(a+b*sinh(c+d*x)^3)^2,x,7,-a*b*x-a^2*arctanh(cosh(c+d*x))/d+b^2*cosh(c+d*x)/d-2/3*b^2*cosh(c+d*x)^3/d+1/5*b^2*cosh(c+d*x)^5/d+a*b*cosh(c+d*x)*sinh(c+d*x)/d],
[csch(c+d*x)^2*(a+b*sinh(c+d*x)^3)^2,x,8,3/8*b^2*x+2*a*b*cosh(c+d*x)/d-a^2*coth(c+d*x)/d-3/8*b^2*cosh(c+d*x)*sinh(c+d*x)/d+1/4*b^2*cosh(c+d*x)*sinh(c+d*x)^3/d],
[csch(c+d*x)^3*(a+b*sinh(c+d*x)^3)^2,x,6,2*a*b*x+1/2*a^2*arctanh(cosh(c+d*x))/d-b^2*cosh(c+d*x)/d+1/3*b^2*cosh(c+d*x)^3/d-1/2*a^2*coth(c+d*x)*csch(c+d*x)/d],
[csch(c+d*x)^4*(a+b*sinh(c+d*x)^3)^2,x,7,-1/2*b^2*x-2*a*b*arctanh(cosh(c+d*x))/d+a^2*coth(c+d*x)/d-1/3*a^2*coth(c+d*x)^3/d+1/2*b^2*cosh(c+d*x)*sinh(c+d*x)/d],
[csch(c+d*x)^5*(a+b*sinh(c+d*x)^3)^2,x,8,-3/8*a^2*arctanh(cosh(c+d*x))/d+b^2*cosh(c+d*x)/d-2*a*b*coth(c+d*x)/d+3/8*a^2*coth(c+d*x)*csch(c+d*x)/d-1/4*a^2*coth(c+d*x)*csch(c+d*x)^3/d],
[csch(c+d*x)^6*(a+b*sinh(c+d*x)^3)^2,x,6,b^2*x+a*b*arctanh(cosh(c+d*x))/d-a^2*coth(c+d*x)/d+2/3*a^2*coth(c+d*x)^3/d-1/5*a^2*coth(c+d*x)^5/d-a*b*coth(c+d*x)*csch(c+d*x)/d],
[csch(c+d*x)^7*(a+b*sinh(c+d*x)^3)^2,x,9,5/16*a^2*arctanh(cosh(c+d*x))/d-b^2*arctanh(cosh(c+d*x))/d+2*a*b*coth(c+d*x)/d-2/3*a*b*coth(c+d*x)^3/d-5/16*a^2*coth(c+d*x)*csch(c+d*x)/d+5/24*a^2*coth(c+d*x)*csch(c+d*x)^3/d-1/6*a^2*coth(c+d*x)*csch(c+d*x)^5/d],
[sinh(c+d*x)^2*(a+b*sinh(c+d*x)^3)^3,x,13,-1/2*a^3*x+105/128*a*b^2*x+3*a^2*b*cosh(c+d*x)/d-b^3*cosh(c+d*x)/d-2*a^2*b*cosh(c+d*x)^3/d+5/3*b^3*cosh(c+d*x)^3/d+3/5*a^2*b*cosh(c+d*x)^5/d-2*b^3*cosh(c+d*x)^5/d+10/7*b^3*cosh(c+d*x)^7/d-5/9*b^3*cosh(c+d*x)^9/d+1/11*b^3*cosh(c+d*x)^11/d+1/2*a^3*cosh(c+d*x)*sinh(c+d*x)/d-105/128*a*b^2*cosh(c+d*x)*sinh(c+d*x)/d+35/64*a*b^2*cosh(c+d*x)*sinh(c+d*x)^3/d-7/16*a*b^2*cosh(c+d*x)*sinh(c+d*x)^5/d+3/8*a*b^2*cosh(c+d*x)*sinh(c+d*x)^7/d],
[sinh(c+d*x)*(a+b*sinh(c+d*x)^3)^3,x,14,9/8*a^2*b*x-63/256*b^3*x+a^3*cosh(c+d*x)/d-3*a*b^2*cosh(c+d*x)/d+3*a*b^2*cosh(c+d*x)^3/d-9/5*a*b^2*cosh(c+d*x)^5/d+3/7*a*b^2*cosh(c+d*x)^7/d-9/8*a^2*b*cosh(c+d*x)*sinh(c+d*x)/d+63/256*b^3*cosh(c+d*x)*sinh(c+d*x)/d+3/4*a^2*b*cosh(c+d*x)*sinh(c+d*x)^3/d-21/128*b^3*cosh(c+d*x)*sinh(c+d*x)^3/d+21/160*b^3*cosh(c+d*x)*sinh(c+d*x)^5/d-9/80*b^3*cosh(c+d*x)*sinh(c+d*x)^7/d+1/10*b^3*cosh(c+d*x)*sinh(c+d*x)^9/d],
[(a+b*sinh(c+d*x)^3)^3,x,10,a^3*x-15/16*a*b^2*x-3*a^2*b*cosh(c+d*x)/d+b^3*cosh(c+d*x)/d+a^2*b*cosh(c+d*x)^3/d-4/3*b^3*cosh(c+d*x)^3/d+6/5*b^3*cosh(c+d*x)^5/d-4/7*b^3*cosh(c+d*x)^7/d+1/9*b^3*cosh(c+d*x)^9/d+15/16*a*b^2*cosh(c+d*x)*sinh(c+d*x)/d-5/8*a*b^2*cosh(c+d*x)*sinh(c+d*x)^3/d+1/2*a*b^2*cosh(c+d*x)*sinh(c+d*x)^5/d],
[csch(c+d*x)*(a+b*sinh(c+d*x)^3)^3,x,12,-3/2*a^2*b*x+35/128*b^3*x-a^3*arctanh(cosh(c+d*x))/d+3*a*b^2*cosh(c+d*x)/d-2*a*b^2*cosh(c+d*x)^3/d+3/5*a*b^2*cosh(c+d*x)^5/d+3/2*a^2*b*cosh(c+d*x)*sinh(c+d*x)/d-35/128*b^3*cosh(c+d*x)*sinh(c+d*x)/d+35/192*b^3*cosh(c+d*x)*sinh(c+d*x)^3/d-7/48*b^3*cosh(c+d*x)*sinh(c+d*x)^5/d+1/8*b^3*cosh(c+d*x)*sinh(c+d*x)^7/d],
[csch(c+d*x)^2*(a+b*sinh(c+d*x)^3)^3,x,10,9/8*a*b^2*x+3*a^2*b*cosh(c+d*x)/d-b^3*cosh(c+d*x)/d+b^3*cosh(c+d*x)^3/d-3/5*b^3*cosh(c+d*x)^5/d+1/7*b^3*cosh(c+d*x)^7/d-a^3*coth(c+d*x)/d-9/8*a*b^2*cosh(c+d*x)*sinh(c+d*x)/d+3/4*a*b^2*cosh(c+d*x)*sinh(c+d*x)^3/d],
[csch(c+d*x)^3*(a+b*sinh(c+d*x)^3)^3,x,10,3*a^2*b*x-5/16*b^3*x+1/2*a^3*arctanh(cosh(c+d*x))/d-3*a*b^2*cosh(c+d*x)/d+a*b^2*cosh(c+d*x)^3/d-1/2*a^3*coth(c+d*x)*csch(c+d*x)/d+5/16*b^3*cosh(c+d*x)*sinh(c+d*x)/d-5/24*b^3*cosh(c+d*x)*sinh(c+d*x)^3/d+1/6*b^3*cosh(c+d*x)*sinh(c+d*x)^5/d],
[csch(c+d*x)^4*(a+b*sinh(c+d*x)^3)^3,x,9,-3/2*a*b^2*x-3*a^2*b*arctanh(cosh(c+d*x))/d+b^3*cosh(c+d*x)/d-2/3*b^3*cosh(c+d*x)^3/d+1/5*b^3*cosh(c+d*x)^5/d+a^3*coth(c+d*x)/d-1/3*a^3*coth(c+d*x)^3/d+3/2*a*b^2*cosh(c+d*x)*sinh(c+d*x)/d],
[csch(c+d*x)^5*(a+b*sinh(c+d*x)^3)^3,x,11,3/8*b^3*x-3/8*a^3*arctanh(cosh(c+d*x))/d+3*a*b^2*cosh(c+d*x)/d-3*a^2*b*coth(c+d*x)/d+3/8*a^3*coth(c+d*x)*csch(c+d*x)/d-1/4*a^3*coth(c+d*x)*csch(c+d*x)^3/d-3/8*b^3*cosh(c+d*x)*sinh(c+d*x)/d+1/4*b^3*cosh(c+d*x)*sinh(c+d*x)^3/d],
[csch(c+d*x)^6*(a+b*sinh(c+d*x)^3)^3,x,8,3*a*b^2*x+3/2*a^2*b*arctanh(cosh(c+d*x))/d-b^3*cosh(c+d*x)/d+1/3*b^3*cosh(c+d*x)^3/d-a^3*coth(c+d*x)/d+2/3*a^3*coth(c+d*x)^3/d-1/5*a^3*coth(c+d*x)^5/d-3/2*a^2*b*coth(c+d*x)*csch(c+d*x)/d],
[csch(c+d*x)^7*(a+b*sinh(c+d*x)^3)^3,x,11,-1/2*b^3*x+5/16*a^3*arctanh(cosh(c+d*x))/d-3*a*b^2*arctanh(cosh(c+d*x))/d+3*a^2*b*coth(c+d*x)/d-a^2*b*coth(c+d*x)^3/d-5/16*a^3*coth(c+d*x)*csch(c+d*x)/d+5/24*a^3*coth(c+d*x)*csch(c+d*x)^3/d-1/6*a^3*coth(c+d*x)*csch(c+d*x)^5/d+1/2*b^3*cosh(c+d*x)*sinh(c+d*x)/d],

# p<0
[sinh(c+d*x)^6/(a+b*sinh(c+d*x)^3),x,15,-a*x/b^2-cosh(c+d*x)/(b*d)+1/3*cosh(c+d*x)^3/(b*d)-2/3*(-1)^(2/3)*a^(4/3)*arctan((-1)^(1/6)*((-1)^(5/6)*b^(1/3)+I*a^(1/3)*tanh(1/2*(c+d*x)))/sqrt((-1)^(1/3)*a^(2/3)-b^(2/3)))/(b^2*d*sqrt((-1)^(1/3)*a^(2/3)-b^(2/3)))-2/3*a^(4/3)*arctanh((b^(1/3)-a^(1/3)*tanh(1/2*(c+d*x)))/sqrt(a^(2/3)+b^(2/3)))/(b^2*d*sqrt(a^(2/3)+b^(2/3)))-2/3*(-1)^(2/3)*a^(4/3)*arctan((-1)^(1/6)*((-1)^(1/6)*b^(1/3)+I*a^(1/3)*tanh(1/2*(c+d*x)))/sqrt((-1)^(1/3)*a^(2/3)-(-1)^(2/3)*b^(2/3)))/(b^2*d*sqrt((-1)^(1/3)*a^(2/3)-(-1)^(2/3)*b^(2/3)))],
[sinh(c+d*x)^5/(a+b*sinh(c+d*x)^3),x,15,-1/2*x/b+1/2*cosh(c+d*x)*sinh(c+d*x)/(b*d)+2/3*a*arctan((-1)^(1/6)*((-1)^(5/6)*b^(1/3)+I*a^(1/3)*tanh(1/2*(c+d*x)))/sqrt((-1)^(1/3)*a^(2/3)-b^(2/3)))/(b^(5/3)*d*sqrt((-1)^(1/3)*a^(2/3)-b^(2/3)))+2/3*a*arctan((-1)^(5/6)*((-1)^(1/6)*b^(1/3)+I*a^(1/3)*tanh(1/2*(c+d*x)))/sqrt(-(-1)^(2/3)*a^(2/3)-b^(2/3)))/(b^(5/3)*d*sqrt(-(-1)^(2/3)*a^(2/3)-b^(2/3)))+2/3*a*arctanh((b^(1/3)-a^(1/3)*tanh(1/2*(c+d*x)))/sqrt(a^(2/3)+b^(2/3)))/(b^(5/3)*d*sqrt(a^(2/3)+b^(2/3)))],
[sinh(c+d*x)^4/(a+b*sinh(c+d*x)^3),x,14,cosh(c+d*x)/(b*d)+2/3*(-1)^(1/3)*a^(2/3)*arctan((-1)^(1/6)*((-1)^(5/6)*b^(1/3)+I*a^(1/3)*tanh(1/2*(c+d*x)))/sqrt((-1)^(1/3)*a^(2/3)-b^(2/3)))/(b^(4/3)*d*sqrt((-1)^(1/3)*a^(2/3)-b^(2/3)))-2/3*a^(2/3)*arctanh((b^(1/3)-a^(1/3)*tanh(1/2*(c+d*x)))/sqrt(a^(2/3)+b^(2/3)))/(b^(4/3)*d*sqrt(a^(2/3)+b^(2/3)))-2/3*a^(2/3)*arctan((-1)^(1/6)*((-1)^(1/6)*b^(1/3)+I*a^(1/3)*tanh(1/2*(c+d*x)))/sqrt((-1)^(1/3)*a^(2/3)-(-1)^(2/3)*b^(2/3)))/(b^(4/3)*d*sqrt((-1)^(1/3)*a^(2/3)-(-1)^(2/3)*b^(2/3)))],
[sinh(c+d*x)^3/(a+b*sinh(c+d*x)^3),x,13,x/b+2/3*(-1)^(2/3)*a^(1/3)*arctan((-1)^(1/6)*((-1)^(5/6)*b^(1/3)+I*a^(1/3)*tanh(1/2*(c+d*x)))/sqrt((-1)^(1/3)*a^(2/3)-b^(2/3)))/(b*d*sqrt((-1)^(1/3)*a^(2/3)-b^(2/3)))+2/3*a^(1/3)*arctanh((b^(1/3)-a^(1/3)*tanh(1/2*(c+d*x)))/sqrt(a^(2/3)+b^(2/3)))/(b*d*sqrt(a^(2/3)+b^(2/3)))+2/3*(-1)^(2/3)*a^(1/3)*arctan((-1)^(1/6)*((-1)^(1/6)*b^(1/3)+I*a^(1/3)*tanh(1/2*(c+d*x)))/sqrt((-1)^(1/3)*a^(2/3)-(-1)^(2/3)*b^(2/3)))/(b*d*sqrt((-1)^(1/3)*a^(2/3)-(-1)^(2/3)*b^(2/3)))],
[sinh(c+d*x)^2/(a+b*sinh(c+d*x)^3),x,11,-2/3*arctan((-1)^(1/6)*((-1)^(5/6)*b^(1/3)+I*a^(1/3)*tanh(1/2*(c+d*x)))/sqrt((-1)^(1/3)*a^(2/3)-b^(2/3)))/(b^(2/3)*d*sqrt((-1)^(1/3)*a^(2/3)-b^(2/3)))-2/3*arctan((-1)^(5/6)*((-1)^(1/6)*b^(1/3)+I*a^(1/3)*tanh(1/2*(c+d*x)))/sqrt(-(-1)^(2/3)*a^(2/3)-b^(2/3)))/(b^(2/3)*d*sqrt(-(-1)^(2/3)*a^(2/3)-b^(2/3)))-2/3*arctanh((b^(1/3)-a^(1/3)*tanh(1/2*(c+d*x)))/sqrt(a^(2/3)+b^(2/3)))/(b^(2/3)*d*sqrt(a^(2/3)+b^(2/3)))],
[sinh(c+d*x)/(a+b*sinh(c+d*x)^3),x,11,-2/3*(-1)^(1/3)*arctan((-1)^(1/6)*((-1)^(5/6)*b^(1/3)+I*a^(1/3)*tanh(1/2*(c+d*x)))/sqrt((-1)^(1/3)*a^(2/3)-b^(2/3)))/(a^(1/3)*b^(1/3)*d*sqrt((-1)^(1/3)*a^(2/3)-b^(2/3)))+2/3*arctanh((b^(1/3)-a^(1/3)*tanh(1/2*(c+d*x)))/sqrt(a^(2/3)+b^(2/3)))/(a^(1/3)*b^(1/3)*d*sqrt(a^(2/3)+b^(2/3)))+2/3*arctan((-1)^(1/6)*((-1)^(1/6)*b^(1/3)+I*a^(1/3)*tanh(1/2*(c+d*x)))/sqrt((-1)^(1/3)*a^(2/3)-(-1)^(2/3)*b^(2/3)))/(a^(1/3)*b^(1/3)*d*sqrt((-1)^(1/3)*a^(2/3)-(-1)^(2/3)*b^(2/3)))],
[1/(a+b*sinh(c+d*x)^3),x,11,-2/3*(-1)^(2/3)*arctan((-1)^(1/6)*((-1)^(5/6)*b^(1/3)+I*a^(1/3)*tanh(1/2*(c+d*x)))/sqrt((-1)^(1/3)*a^(2/3)-b^(2/3)))/(a^(2/3)*d*sqrt((-1)^(1/3)*a^(2/3)-b^(2/3)))-2/3*arctanh((b^(1/3)-a^(1/3)*tanh(1/2*(c+d*x)))/sqrt(a^(2/3)+b^(2/3)))/(a^(2/3)*d*sqrt(a^(2/3)+b^(2/3)))-2/3*(-1)^(2/3)*arctan((-1)^(1/6)*((-1)^(1/6)*b^(1/3)+I*a^(1/3)*tanh(1/2*(c+d*x)))/sqrt((-1)^(1/3)*a^(2/3)-(-1)^(2/3)*b^(2/3)))/(a^(2/3)*d*sqrt((-1)^(1/3)*a^(2/3)-(-1)^(2/3)*b^(2/3)))],
[csch(c+d*x)/(a+b*sinh(c+d*x)^3),x,14,-arctanh(cosh(c+d*x))/(a*d)+2/3*b^(1/3)*arctan((-1)^(1/6)*((-1)^(5/6)*b^(1/3)+I*a^(1/3)*tanh(1/2*(c+d*x)))/sqrt((-1)^(1/3)*a^(2/3)-b^(2/3)))/(a*d*sqrt((-1)^(1/3)*a^(2/3)-b^(2/3)))+2/3*b^(1/3)*arctan((-1)^(5/6)*((-1)^(1/6)*b^(1/3)+I*a^(1/3)*tanh(1/2*(c+d*x)))/sqrt(-(-1)^(2/3)*a^(2/3)-b^(2/3)))/(a*d*sqrt(-(-1)^(2/3)*a^(2/3)-b^(2/3)))+2/3*b^(1/3)*arctanh((b^(1/3)-a^(1/3)*tanh(1/2*(c+d*x)))/sqrt(a^(2/3)+b^(2/3)))/(a*d*sqrt(a^(2/3)+b^(2/3)))],
[csch(c+d*x)^2/(a+b*sinh(c+d*x)^3),x,15,-coth(c+d*x)/(a*d)+2/3*(-1)^(1/3)*b^(2/3)*arctan((-1)^(1/6)*((-1)^(5/6)*b^(1/3)+I*a^(1/3)*tanh(1/2*(c+d*x)))/sqrt((-1)^(1/3)*a^(2/3)-b^(2/3)))/(a^(4/3)*d*sqrt((-1)^(1/3)*a^(2/3)-b^(2/3)))-2/3*b^(2/3)*arctanh((b^(1/3)-a^(1/3)*tanh(1/2*(c+d*x)))/sqrt(a^(2/3)+b^(2/3)))/(a^(4/3)*d*sqrt(a^(2/3)+b^(2/3)))-2/3*b^(2/3)*arctan((-1)^(1/6)*((-1)^(1/6)*b^(1/3)+I*a^(1/3)*tanh(1/2*(c+d*x)))/sqrt((-1)^(1/3)*a^(2/3)-(-1)^(2/3)*b^(2/3)))/(a^(4/3)*d*sqrt((-1)^(1/3)*a^(2/3)-(-1)^(2/3)*b^(2/3)))],
[csch(c+d*x)^3/(a+b*sinh(c+d*x)^3),x,15,1/2*arctanh(cosh(c+d*x))/(a*d)-1/2*coth(c+d*x)*csch(c+d*x)/(a*d)+2/3*(-1)^(2/3)*b*arctan((-1)^(1/6)*((-1)^(5/6)*b^(1/3)+I*a^(1/3)*tanh(1/2*(c+d*x)))/sqrt((-1)^(1/3)*a^(2/3)-b^(2/3)))/(a^(5/3)*d*sqrt((-1)^(1/3)*a^(2/3)-b^(2/3)))+2/3*b*arctanh((b^(1/3)-a^(1/3)*tanh(1/2*(c+d*x)))/sqrt(a^(2/3)+b^(2/3)))/(a^(5/3)*d*sqrt(a^(2/3)+b^(2/3)))+2/3*(-1)^(2/3)*b*arctan((-1)^(1/6)*((-1)^(1/6)*b^(1/3)+I*a^(1/3)*tanh(1/2*(c+d*x)))/sqrt((-1)^(1/3)*a^(2/3)-(-1)^(2/3)*b^(2/3)))/(a^(5/3)*d*sqrt((-1)^(1/3)*a^(2/3)-(-1)^(2/3)*b^(2/3)))],
[csch(c+d*x)^4/(a+b*sinh(c+d*x)^3),x,16,b*arctanh(cosh(c+d*x))/(a^2*d)+coth(c+d*x)/(a*d)-1/3*coth(c+d*x)^3/(a*d)-2/3*b^(4/3)*arctan((-1)^(1/6)*((-1)^(5/6)*b^(1/3)+I*a^(1/3)*tanh(1/2*(c+d*x)))/sqrt((-1)^(1/3)*a^(2/3)-b^(2/3)))/(a^2*d*sqrt((-1)^(1/3)*a^(2/3)-b^(2/3)))-2/3*b^(4/3)*arctan((-1)^(5/6)*((-1)^(1/6)*b^(1/3)+I*a^(1/3)*tanh(1/2*(c+d*x)))/sqrt(-(-1)^(2/3)*a^(2/3)-b^(2/3)))/(a^2*d*sqrt(-(-1)^(2/3)*a^(2/3)-b^(2/3)))-2/3*b^(4/3)*arctanh((b^(1/3)-a^(1/3)*tanh(1/2*(c+d*x)))/sqrt(a^(2/3)+b^(2/3)))/(a^2*d*sqrt(a^(2/3)+b^(2/3)))],
[1/(1+sinh(x)^3),x,12,-1/3*(-1)^(1/6)*log(1+(-1)^(5/6)-(-1)^(1/6)*tanh(1/2*x))+1/3*(-1)^(1/6)*log(1+(-1)^(1/6)+(-1)^(1/3)*tanh(1/2*x))-1/3*arctanh((1-tanh(1/2*x))/sqrt(2))*sqrt(2)-2/3*(-1)^(1/6)*arctan((I+(-1)^(1/6)*tanh(1/2*x))/sqrt(1-(-1)^(1/3)))/sqrt(1-(-1)^(1/3))],
[1/(1-sinh(x)^3),x,12,-1/3*(-1)^(5/6)*log(1+(-1)^(5/6)+(-1)^(2/3)*tanh(1/2*x))+1/3*(-1)^(5/6)*log(1+(-1)^(1/6)+(-1)^(5/6)*tanh(1/2*x))+1/3*arctanh((1+tanh(1/2*x))/sqrt(2))*sqrt(2)+2/3*(-1)^(5/6)*arctan((I-(-1)^(5/6)*tanh(1/2*x))/sqrt(1+(-1)^(2/3)))/sqrt(1+(-1)^(2/3))],

# Integrands of the form Sinh[e+f x]^m (a+b Sinh[e+f x]^3)^(p/2)

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

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

# p>0
[sinh(c+d*x)^4*(a+b*sinh(c+d*x)^4),x,6,1/128*(48*a+35*b)*x-1/128*(80*a+93*b)*cosh(c+d*x)*sinh(c+d*x)/d+1/192*(48*a+163*b)*cosh(c+d*x)^3*sinh(c+d*x)/d-25/48*b*cosh(c+d*x)^5*sinh(c+d*x)/d+1/8*b*cosh(c+d*x)^7*sinh(c+d*x)/d],
[sinh(c+d*x)^3*(a+b*sinh(c+d*x)^4),x,3,-(a+b)*cosh(c+d*x)/d+1/3*(a+3*b)*cosh(c+d*x)^3/d-3/5*b*cosh(c+d*x)^5/d+1/7*b*cosh(c+d*x)^7/d],
[sinh(c+d*x)^2*(a+b*sinh(c+d*x)^4),x,5,-1/16*(8*a+5*b)*x+1/16*(8*a+11*b)*cosh(c+d*x)*sinh(c+d*x)/d-13/24*b*cosh(c+d*x)^3*sinh(c+d*x)/d+1/6*b*cosh(c+d*x)^5*sinh(c+d*x)/d],
[sinh(c+d*x)*(a+b*sinh(c+d*x)^4),x,2,(a+b)*cosh(c+d*x)/d-2/3*b*cosh(c+d*x)^3/d+1/5*b*cosh(c+d*x)^5/d],
[a+b*sinh(c+d*x)^4,x,4,a*x+3/8*b*x-3/8*b*cosh(c+d*x)*sinh(c+d*x)/d+1/4*b*cosh(c+d*x)*sinh(c+d*x)^3/d],
[csch(c+d*x)*(a+b*sinh(c+d*x)^4),x,4,-a*arctanh(cosh(c+d*x))/d-b*cosh(c+d*x)/d+1/3*b*cosh(c+d*x)^3/d],
[csch(c+d*x)^2*(a+b*sinh(c+d*x)^4),x,4,-1/2*b*x-a*coth(c+d*x)/d+1/2*b*cosh(c+d*x)*sinh(c+d*x)/d],
[csch(c+d*x)^3*(a+b*sinh(c+d*x)^4),x,4,1/2*a*arctanh(cosh(c+d*x))/d+b*cosh(c+d*x)/d-1/2*a*coth(c+d*x)*csch(c+d*x)/d],
[csch(c+d*x)^4*(a+b*sinh(c+d*x)^4),x,4,b*x+a*coth(c+d*x)/d-1/3*a*coth(c+d*x)^3/d],
[csch(c+d*x)^5*(a+b*sinh(c+d*x)^4),x,4,-1/8*(3*a+8*b)*arctanh(cosh(c+d*x))/d+3/8*a*coth(c+d*x)*csch(c+d*x)/d-1/4*a*coth(c+d*x)*csch(c+d*x)^3/d],
[csch(c+d*x)^6*(a+b*sinh(c+d*x)^4),x,3,-(a+b)*coth(c+d*x)/d+2/3*a*coth(c+d*x)^3/d-1/5*a*coth(c+d*x)^5/d],
[csch(c+d*x)^7*(a+b*sinh(c+d*x)^4),x,5,1/16*(5*a+8*b)*arctanh(cosh(c+d*x))/d-1/16*(5*a+8*b)*coth(c+d*x)*csch(c+d*x)/d+5/24*a*coth(c+d*x)*csch(c+d*x)^3/d-1/6*a*coth(c+d*x)*csch(c+d*x)^5/d],
[sinh(c+d*x)^3*(a+b*sinh(c+d*x)^4)^2,x,3,-(a+b)^2*cosh(c+d*x)/d+1/3*(a+b)*(a+5*b)*cosh(c+d*x)^3/d-2/5*b*(3*a+5*b)*cosh(c+d*x)^5/d+2/7*b*(a+5*b)*cosh(c+d*x)^7/d-5/9*b^2*cosh(c+d*x)^9/d+1/11*b^2*cosh(c+d*x)^11/d],
[sinh(c+d*x)^2*(a+b*sinh(c+d*x)^4)^2,x,7,-1/256*(128*a^2+160*a*b+63*b^2)*x+1/256*(128*a^2+352*a*b+193*b^2)*cosh(c+d*x)*sinh(c+d*x)/d-1/384*b*(416*a+447*b)*cosh(c+d*x)^3*sinh(c+d*x)/d+1/480*b*(160*a+513*b)*cosh(c+d*x)^5*sinh(c+d*x)/d-41/80*b^2*cosh(c+d*x)^7*sinh(c+d*x)/d+1/10*b^2*cosh(c+d*x)^9*sinh(c+d*x)/d],
[sinh(c+d*x)*(a+b*sinh(c+d*x)^4)^2,x,3,(a+b)^2*cosh(c+d*x)/d-4/3*b*(a+b)*cosh(c+d*x)^3/d+2/5*b*(a+3*b)*cosh(c+d*x)^5/d-4/7*b^2*cosh(c+d*x)^7/d+1/9*b^2*cosh(c+d*x)^9/d],
[(a+b*sinh(c+d*x)^4)^2,x,6,1/128*(128*a^2+96*a*b+35*b^2)*x-1/128*b*(160*a+93*b)*cosh(c+d*x)*sinh(c+d*x)/d+1/192*b*(96*a+163*b)*cosh(c+d*x)^3*sinh(c+d*x)/d-25/48*b^2*cosh(c+d*x)^5*sinh(c+d*x)/d+1/8*b^2*cosh(c+d*x)^7*sinh(c+d*x)/d],
[csch(c+d*x)*(a+b*sinh(c+d*x)^4)^2,x,4,-a^2*arctanh(cosh(c+d*x))/d-b*(2*a+b)*cosh(c+d*x)/d+1/3*b*(2*a+3*b)*cosh(c+d*x)^3/d-3/5*b^2*cosh(c+d*x)^5/d+1/7*b^2*cosh(c+d*x)^7/d],
[csch(c+d*x)^2*(a+b*sinh(c+d*x)^4)^2,x,6,-1/16*b*(16*a+5*b)*x-a^2*coth(c+d*x)/d+1/16*b*(16*a+11*b)*cosh(c+d*x)*sinh(c+d*x)/d-13/24*b^2*cosh(c+d*x)^3*sinh(c+d*x)/d+1/6*b^2*cosh(c+d*x)^5*sinh(c+d*x)/d],
[csch(c+d*x)^3*(a+b*sinh(c+d*x)^4)^2,x,5,1/2*a^2*arctanh(cosh(c+d*x))/d+b*(2*a+b)*cosh(c+d*x)/d-2/3*b^2*cosh(c+d*x)^3/d+1/5*b^2*cosh(c+d*x)^5/d-1/2*a^2*coth(c+d*x)*csch(c+d*x)/d],
[csch(c+d*x)^4*(a+b*sinh(c+d*x)^4)^2,x,6,1/8*b*(16*a+3*b)*x+a^2*coth(c+d*x)/d-1/3*a^2*coth(c+d*x)^3/d-5/8*b^2*cosh(c+d*x)*sinh(c+d*x)/d+1/4*b^2*cosh(c+d*x)^3*sinh(c+d*x)/d],
[csch(c+d*x)^5*(a+b*sinh(c+d*x)^4)^2,x,6,-1/8*a*(3*a+16*b)*arctanh(cosh(c+d*x))/d-b^2*cosh(c+d*x)/d+1/3*b^2*cosh(c+d*x)^3/d+3/8*a^2*coth(c+d*x)*csch(c+d*x)/d-1/4*a^2*coth(c+d*x)*csch(c+d*x)^3/d],
[csch(c+d*x)^6*(a+b*sinh(c+d*x)^4)^2,x,5,-1/2*b^2*x-a*(a+2*b)*coth(c+d*x)/d+2/3*a^2*coth(c+d*x)^3/d-1/5*a^2*coth(c+d*x)^5/d+1/2*b^2*cosh(c+d*x)*sinh(c+d*x)/d],
[csch(c+d*x)^7*(a+b*sinh(c+d*x)^4)^2,x,6,1/16*a*(5*a+16*b)*arctanh(cosh(c+d*x))/d+b^2*cosh(c+d*x)/d-1/16*a*(5*a+16*b)*coth(c+d*x)*csch(c+d*x)/d+5/24*a^2*coth(c+d*x)*csch(c+d*x)^3/d-1/6*a^2*coth(c+d*x)*csch(c+d*x)^5/d],
[sinh(c+d*x)^5*(a+b*sinh(c+d*x)^4)^3,x,3,(a+b)^3*cosh(c+d*x)/d-2/3*(a+b)^2*(a+4*b)*cosh(c+d*x)^3/d+1/5*(a+b)*(a^2+17*a*b+28*b^2)*cosh(c+d*x)^5/d-4/7*b*(3*a^2+15*a*b+14*b^2)*cosh(c+d*x)^7/d+1/9*b*(3*a^2+45*a*b+70*b^2)*cosh(c+d*x)^9/d-2/11*b^2*(9*a+28*b)*cosh(c+d*x)^11/d+1/13*b^2*(3*a+28*b)*cosh(c+d*x)^13/d-8/15*b^3*cosh(c+d*x)^15/d+1/17*b^3*cosh(c+d*x)^17/d],
[sinh(c+d*x)^3*(a+b*sinh(c+d*x)^4)^3,x,3,-(a+b)^3*cosh(c+d*x)/d+1/3*(a+b)^2*(a+7*b)*cosh(c+d*x)^3/d-3/5*b*(a+b)*(3*a+7*b)*cosh(c+d*x)^5/d+1/7*b*(3*a^2+30*a*b+35*b^2)*cosh(c+d*x)^7/d-5/9*b^2*(3*a+7*b)*cosh(c+d*x)^9/d+3/11*b^2*(a+7*b)*cosh(c+d*x)^11/d-7/13*b^3*cosh(c+d*x)^13/d+1/15*b^3*cosh(c+d*x)^15/d],
[sinh(c+d*x)*(a+b*sinh(c+d*x)^4)^3,x,3,(a+b)^3*cosh(c+d*x)/d-2*b*(a+b)^2*cosh(c+d*x)^3/d+3/5*b*(a+b)*(a+5*b)*cosh(c+d*x)^5/d-4/7*b^2*(3*a+5*b)*cosh(c+d*x)^7/d+1/3*b^2*(a+5*b)*cosh(c+d*x)^9/d-6/11*b^3*cosh(c+d*x)^11/d+1/13*b^3*cosh(c+d*x)^13/d],
[csch(c+d*x)*(a+b*sinh(c+d*x)^4)^3,x,4,-a^3*arctanh(cosh(c+d*x))/d-b*(3*a^2+3*a*b+b^2)*cosh(c+d*x)/d+1/3*b*(3*a^2+9*a*b+5*b^2)*cosh(c+d*x)^3/d-1/5*b^2*(9*a+10*b)*cosh(c+d*x)^5/d+1/7*b^2*(3*a+10*b)*cosh(c+d*x)^7/d-5/9*b^3*cosh(c+d*x)^9/d+1/11*b^3*cosh(c+d*x)^11/d],
[csch(c+d*x)^3*(a+b*sinh(c+d*x)^4)^3,x,5,1/2*a^3*arctanh(cosh(c+d*x))/d+b*(3*a^2+3*a*b+b^2)*cosh(c+d*x)/d-2/3*b^2*(3*a+2*b)*cosh(c+d*x)^3/d+3/5*b^2*(a+2*b)*cosh(c+d*x)^5/d-4/7*b^3*cosh(c+d*x)^7/d+1/9*b^3*cosh(c+d*x)^9/d-1/2*a^3*coth(c+d*x)*csch(c+d*x)/d],
[csch(c+d*x)^5*(a+b*sinh(c+d*x)^4)^3,x,6,-3/8*a^2*(a+8*b)*arctanh(cosh(c+d*x))/d-b^2*(3*a+b)*cosh(c+d*x)/d+b^2*(a+b)*cosh(c+d*x)^3/d-3/5*b^3*cosh(c+d*x)^5/d+1/7*b^3*cosh(c+d*x)^7/d+3/8*a^3*coth(c+d*x)*csch(c+d*x)/d-1/4*a^3*coth(c+d*x)*csch(c+d*x)^3/d],
[csch(c+d*x)^7*(a+b*sinh(c+d*x)^4)^3,x,7,1/16*a^2*(5*a+24*b)*arctanh(cosh(c+d*x))/d+b^2*(3*a+b)*cosh(c+d*x)/d-2/3*b^3*cosh(c+d*x)^3/d+1/5*b^3*cosh(c+d*x)^5/d-1/16*a^2*(5*a+24*b)*coth(c+d*x)*csch(c+d*x)/d+5/24*a^3*coth(c+d*x)*csch(c+d*x)^3/d-1/6*a^3*coth(c+d*x)*csch(c+d*x)^5/d],
[csch(c+d*x)^9*(a+b*sinh(c+d*x)^4)^3,x,8,-1/128*a*(35*a^2+144*a*b+384*b^2)*arctanh(cosh(c+d*x))/d-b^3*cosh(c+d*x)/d+1/3*b^3*cosh(c+d*x)^3/d+1/128*a^2*(35*a+144*b)*coth(c+d*x)*csch(c+d*x)/d-1/192*a^2*(35*a+144*b)*coth(c+d*x)*csch(c+d*x)^3/d+7/48*a^3*coth(c+d*x)*csch(c+d*x)^5/d-1/8*a^3*coth(c+d*x)*csch(c+d*x)^7/d],
[csch(c+d*x)^11*(a+b*sinh(c+d*x)^4)^3,x,8,3/256*a*(21*a^2+80*a*b+128*b^2)*arctanh(cosh(c+d*x))/d+b^3*cosh(c+d*x)/d-3/256*a*(21*a^2+80*a*b+128*b^2)*coth(c+d*x)*csch(c+d*x)/d+1/128*a^2*(21*a+80*b)*coth(c+d*x)*csch(c+d*x)^3/d-1/160*a^2*(21*a+80*b)*coth(c+d*x)*csch(c+d*x)^5/d+9/80*a^3*coth(c+d*x)*csch(c+d*x)^7/d-1/10*a^3*coth(c+d*x)*csch(c+d*x)^9/d],
[csch(c+d*x)^13*(a+b*sinh(c+d*x)^4)^3,x,8,-1/1024*(231*a^3+840*a^2*b+1152*a*b^2+1024*b^3)*arctanh(cosh(c+d*x))/d+3/1024*a*(77*a^2+280*a*b+384*b^2)*coth(c+d*x)*csch(c+d*x)/d-1/512*a*(77*a^2+280*a*b+384*b^2)*coth(c+d*x)*csch(c+d*x)^3/d+7/640*a^2*(11*a+40*b)*coth(c+d*x)*csch(c+d*x)^5/d-3/320*a^2*(11*a+40*b)*coth(c+d*x)*csch(c+d*x)^7/d+11/120*a^3*coth(c+d*x)*csch(c+d*x)^9/d-1/12*a^3*coth(c+d*x)*csch(c+d*x)^11/d],
[sinh(c+d*x)^2*(a+b*sinh(c+d*x)^4)^3,x,9,-1/2048*(1024*a^3+1920*a^2*b+1512*a*b^2+429*b^3)*x+1/2048*(1024*a^3+4224*a^2*b+4632*a*b^2+1619*b^3)*cosh(c+d*x)*sinh(c+d*x)/d-1/3072*b*(4992*a^2+10728*a*b+5549*b^2)*cosh(c+d*x)^3*sinh(c+d*x)/d+1/3840*b*(1920*a^2+12312*a*b+10579*b^2)*cosh(c+d*x)^5*sinh(c+d*x)/d-1/4480*b^2*(6888*a+11821*b)*cosh(c+d*x)^7*sinh(c+d*x)/d+1/1680*b^2*(504*a+2593*b)*cosh(c+d*x)^9*sinh(c+d*x)/d-85/168*b^3*cosh(c+d*x)^11*sinh(c+d*x)/d+1/14*b^3*cosh(c+d*x)^13*sinh(c+d*x)/d],
[(a+b*sinh(c+d*x)^4)^3,x,8,1/1024*(1024*a^3+1152*a^2*b+840*a*b^2+231*b^3)*x-1/1024*b*(1920*a^2+2232*a*b+793*b^2)*cosh(c+d*x)*sinh(c+d*x)/d+1/1536*b*(1152*a^2+3912*a*b+2279*b^2)*cosh(c+d*x)^3*sinh(c+d*x)/d-1/1920*b^2*(3000*a+3481*b)*cosh(c+d*x)^5*sinh(c+d*x)/d+3/320*b^2*(40*a+139*b)*cosh(c+d*x)^7*sinh(c+d*x)/d-61/120*b^3*cosh(c+d*x)^9*sinh(c+d*x)/d+1/12*b^3*cosh(c+d*x)^11*sinh(c+d*x)/d],
[csch(c+d*x)^2*(a+b*sinh(c+d*x)^4)^3,x,8,-3/256*b*(128*a^2+80*a*b+21*b^2)*x-a^3*coth(c+d*x)/d+1/256*b*(384*a^2+528*a*b+193*b^2)*cosh(c+d*x)*sinh(c+d*x)/d-1/128*b^2*(208*a+149*b)*cosh(c+d*x)^3*sinh(c+d*x)/d+1/160*b^2*(80*a+171*b)*cosh(c+d*x)^5*sinh(c+d*x)/d-41/80*b^3*cosh(c+d*x)^7*sinh(c+d*x)/d+1/10*b^3*cosh(c+d*x)^9*sinh(c+d*x)/d],
[csch(c+d*x)^4*(a+b*sinh(c+d*x)^4)^3,x,8,1/128*b*(384*a^2+144*a*b+35*b^2)*x+a^3*coth(c+d*x)/d-1/3*a^3*coth(c+d*x)^3/d-3/128*b^2*(80*a+31*b)*cosh(c+d*x)*sinh(c+d*x)/d+1/192*b^2*(144*a+163*b)*cosh(c+d*x)^3*sinh(c+d*x)/d-25/48*b^3*cosh(c+d*x)^5*sinh(c+d*x)/d+1/8*b^3*cosh(c+d*x)^7*sinh(c+d*x)/d],
[csch(c+d*x)^6*(a+b*sinh(c+d*x)^4)^3,x,7,-1/16*b^2*(24*a+5*b)*x-a^2*(a+3*b)*coth(c+d*x)/d+2/3*a^3*coth(c+d*x)^3/d-1/5*a^3*coth(c+d*x)^5/d+1/16*b^2*(24*a+11*b)*cosh(c+d*x)*sinh(c+d*x)/d-13/24*b^3*cosh(c+d*x)^3*sinh(c+d*x)/d+1/6*b^3*cosh(c+d*x)^5*sinh(c+d*x)/d],
[csch(c+d*x)^8*(a+b*sinh(c+d*x)^4)^3,x,6,3/8*b^2*(8*a+b)*x+a^2*(a+3*b)*coth(c+d*x)/d-a^2*(a+b)*coth(c+d*x)^3/d+3/5*a^3*coth(c+d*x)^5/d-1/7*a^3*coth(c+d*x)^7/d-5/8*b^3*cosh(c+d*x)*sinh(c+d*x)/d+1/4*b^3*cosh(c+d*x)^3*sinh(c+d*x)/d],
[csch(c+d*x)^10*(a+b*sinh(c+d*x)^4)^3,x,5,-1/2*b^3*x-a*(a^2+3*a*b+3*b^2)*coth(c+d*x)/d+2/3*a^2*(2*a+3*b)*coth(c+d*x)^3/d-3/5*a^2*(2*a+b)*coth(c+d*x)^5/d+4/7*a^3*coth(c+d*x)^7/d-1/9*a^3*coth(c+d*x)^9/d+1/2*b^3*cosh(c+d*x)*sinh(c+d*x)/d],
[csch(c+d*x)^12*(a+b*sinh(c+d*x)^4)^3,x,4,b^3*x+a*(a^2+3*a*b+3*b^2)*coth(c+d*x)/d-1/3*a*(5*a^2+9*a*b+3*b^2)*coth(c+d*x)^3/d+1/5*a^2*(10*a+9*b)*coth(c+d*x)^5/d-1/7*a^2*(10*a+3*b)*coth(c+d*x)^7/d+5/9*a^3*coth(c+d*x)^9/d-1/11*a^3*coth(c+d*x)^11/d],
[csch(c+d*x)^14*(a+b*sinh(c+d*x)^4)^3,x,3,-(a+b)^3*coth(c+d*x)/d+2*a*(a+b)^2*coth(c+d*x)^3/d-3/5*a*(a+b)*(5*a+b)*coth(c+d*x)^5/d+4/7*a^2*(5*a+3*b)*coth(c+d*x)^7/d-1/3*a^2*(5*a+b)*coth(c+d*x)^9/d+6/11*a^3*coth(c+d*x)^11/d-1/13*a^3*coth(c+d*x)^13/d],
[csch(c+d*x)^16*(a+b*sinh(c+d*x)^4)^3,x,3,(a+b)^3*coth(c+d*x)/d-1/3*(a+b)^2*(7*a+b)*coth(c+d*x)^3/d+3/5*a*(a+b)*(7*a+3*b)*coth(c+d*x)^5/d-1/7*a*(35*a^2+30*a*b+3*b^2)*coth(c+d*x)^7/d+5/9*a^2*(7*a+3*b)*coth(c+d*x)^9/d-3/11*a^2*(7*a+b)*coth(c+d*x)^11/d+7/13*a^3*coth(c+d*x)^13/d-1/15*a^3*coth(c+d*x)^15/d],
[csch(c+d*x)^18*(a+b*sinh(c+d*x)^4)^3,x,3,-(a+b)^3*coth(c+d*x)/d+2/3*(a+b)^2*(4*a+b)*coth(c+d*x)^3/d-1/5*(a+b)*(28*a^2+17*a*b+b^2)*coth(c+d*x)^5/d+4/7*a*(14*a^2+15*a*b+3*b^2)*coth(c+d*x)^7/d-1/9*a*(70*a^2+45*a*b+3*b^2)*coth(c+d*x)^9/d+2/11*a^2*(28*a+9*b)*coth(c+d*x)^11/d-1/13*a^2*(28*a+3*b)*coth(c+d*x)^13/d+8/15*a^3*coth(c+d*x)^15/d-1/17*a^3*coth(c+d*x)^17/d],
[csch(c+d*x)^20*(a+b*sinh(c+d*x)^4)^3,x,3,(a+b)^3*coth(c+d*x)/d-(a+b)^2*(3*a+b)*coth(c+d*x)^3/d+3/5*(a+b)*(12*a^2+9*a*b+b^2)*coth(c+d*x)^5/d-1/7*(84*a^3+105*a^2*b+30*a*b^2+b^3)*coth(c+d*x)^7/d+1/3*a*(42*a^2+35*a*b+5*b^2)*coth(c+d*x)^9/d-3/11*a*(42*a^2+21*a*b+b^2)*coth(c+d*x)^11/d+21/13*a^2*(4*a+b)*coth(c+d*x)^13/d-1/5*a^2*(12*a+b)*coth(c+d*x)^15/d+9/17*a^3*coth(c+d*x)^17/d-1/19*a^3*coth(c+d*x)^19/d],

# p<0
[sinh(c+d*x)^7/(a-b*sinh(c+d*x)^4),x,6,cosh(c+d*x)/(b*d)-1/3*cosh(c+d*x)^3/(b*d)-1/2*a*arctan(b^(1/4)*cosh(c+d*x)/sqrt(sqrt(a)-sqrt(b)))/(b^(7/4)*d*sqrt(sqrt(a)-sqrt(b)))+1/2*a*arctanh(b^(1/4)*cosh(c+d*x)/sqrt(sqrt(a)+sqrt(b)))/(b^(7/4)*d*sqrt(sqrt(a)+sqrt(b)))],
[sinh(c+d*x)^5/(a-b*sinh(c+d*x)^4),x,6,-cosh(c+d*x)/(b*d)+1/2*arctan(b^(1/4)*cosh(c+d*x)/sqrt(sqrt(a)-sqrt(b)))*sqrt(a)/(b^(5/4)*d*sqrt(sqrt(a)-sqrt(b)))+1/2*arctanh(b^(1/4)*cosh(c+d*x)/sqrt(sqrt(a)+sqrt(b)))*sqrt(a)/(b^(5/4)*d*sqrt(sqrt(a)+sqrt(b)))],
[sinh(c+d*x)^3/(a-b*sinh(c+d*x)^4),x,4,-1/2*arctan(b^(1/4)*cosh(c+d*x)/sqrt(sqrt(a)-sqrt(b)))/(b^(3/4)*d*sqrt(sqrt(a)-sqrt(b)))+1/2*arctanh(b^(1/4)*cosh(c+d*x)/sqrt(sqrt(a)+sqrt(b)))/(b^(3/4)*d*sqrt(sqrt(a)+sqrt(b)))],
[sinh(c+d*x)/(a-b*sinh(c+d*x)^4),x,4,1/2*arctan(b^(1/4)*cosh(c+d*x)/sqrt(sqrt(a)-sqrt(b)))/(b^(1/4)*d*sqrt(a)*sqrt(sqrt(a)-sqrt(b)))+1/2*arctanh(b^(1/4)*cosh(c+d*x)/sqrt(sqrt(a)+sqrt(b)))/(b^(1/4)*d*sqrt(a)*sqrt(sqrt(a)+sqrt(b)))],
[csch(c+d*x)/(a-b*sinh(c+d*x)^4),x,7,-arctanh(cosh(c+d*x))/(a*d)-1/2*b^(1/4)*arctan(b^(1/4)*cosh(c+d*x)/sqrt(sqrt(a)-sqrt(b)))/(a*d*sqrt(sqrt(a)-sqrt(b)))+1/2*b^(1/4)*arctanh(b^(1/4)*cosh(c+d*x)/sqrt(sqrt(a)+sqrt(b)))/(a*d*sqrt(sqrt(a)+sqrt(b)))],
[csch(c+d*x)^3/(a-b*sinh(c+d*x)^4),x,7,1/2*arctanh(cosh(c+d*x))/(a*d)+1/4/(a*d*(1-cosh(c+d*x)))+(-1/4)/(a*d*(1+cosh(c+d*x)))+1/2*b^(3/4)*arctan(b^(1/4)*cosh(c+d*x)/sqrt(sqrt(a)-sqrt(b)))/(a^(3/2)*d*sqrt(sqrt(a)-sqrt(b)))+1/2*b^(3/4)*arctanh(b^(1/4)*cosh(c+d*x)/sqrt(sqrt(a)+sqrt(b)))/(a^(3/2)*d*sqrt(sqrt(a)+sqrt(b)))],
[sinh(c+d*x)^6/(a-b*sinh(c+d*x)^4),x,7,1/2*x/b-1/2*a^(3/4)*arctanh(sqrt(sqrt(a)-sqrt(b))*tanh(c+d*x)/a^(1/4))/(b^(3/2)*d*sqrt(sqrt(a)-sqrt(b)))+1/2*a^(3/4)*arctanh(sqrt(sqrt(a)+sqrt(b))*tanh(c+d*x)/a^(1/4))/(b^(3/2)*d*sqrt(sqrt(a)+sqrt(b)))+(-1/4)/(b*d*(1-tanh(c+d*x)))+1/4/(b*d*(1+tanh(c+d*x)))],
[sinh(c+d*x)^4/(a-b*sinh(c+d*x)^4),x,7,-x/b+1/2*a^(1/4)*arctanh(sqrt(sqrt(a)-sqrt(b))*tanh(c+d*x)/a^(1/4))/(b*d*sqrt(sqrt(a)-sqrt(b)))+1/2*a^(1/4)*arctanh(sqrt(sqrt(a)+sqrt(b))*tanh(c+d*x)/a^(1/4))/(b*d*sqrt(sqrt(a)+sqrt(b)))],
[sinh(c+d*x)^2/(a-b*sinh(c+d*x)^4),x,4,-1/2*arctanh(sqrt(sqrt(a)-sqrt(b))*tanh(c+d*x)/a^(1/4))/(a^(1/4)*d*sqrt(b)*sqrt(sqrt(a)-sqrt(b)))+1/2*arctanh(sqrt(sqrt(a)+sqrt(b))*tanh(c+d*x)/a^(1/4))/(a^(1/4)*d*sqrt(b)*sqrt(sqrt(a)+sqrt(b)))],
[1/(a-b*sinh(c+d*x)^4),x,4,1/2*arctanh(sqrt(sqrt(a)-sqrt(b))*tanh(c+d*x)/a^(1/4))/(a^(3/4)*d*sqrt(sqrt(a)-sqrt(b)))+1/2*arctanh(sqrt(sqrt(a)+sqrt(b))*tanh(c+d*x)/a^(1/4))/(a^(3/4)*d*sqrt(sqrt(a)+sqrt(b)))],
[csch(c+d*x)^2/(a-b*sinh(c+d*x)^4),x,6,-coth(c+d*x)/(a*d)-1/2*arctanh(sqrt(sqrt(a)-sqrt(b))*tanh(c+d*x)/a^(1/4))*sqrt(b)/(a^(5/4)*d*sqrt(sqrt(a)-sqrt(b)))+1/2*arctanh(sqrt(sqrt(a)+sqrt(b))*tanh(c+d*x)/a^(1/4))*sqrt(b)/(a^(5/4)*d*sqrt(sqrt(a)+sqrt(b)))],
[csch(c+d*x)^4/(a-b*sinh(c+d*x)^4),x,6,coth(c+d*x)/(a*d)-1/3*coth(c+d*x)^3/(a*d)+1/2*b*arctanh(sqrt(sqrt(a)-sqrt(b))*tanh(c+d*x)/a^(1/4))/(a^(7/4)*d*sqrt(sqrt(a)-sqrt(b)))+1/2*b*arctanh(sqrt(sqrt(a)+sqrt(b))*tanh(c+d*x)/a^(1/4))/(a^(7/4)*d*sqrt(sqrt(a)+sqrt(b)))],
[sinh(c+d*x)^9/(a-b*sinh(c+d*x)^4)^2,x,7,cosh(c+d*x)/(b^2*d)+1/4*a*cosh(c+d*x)*(a+b-b*cosh(c+d*x)^2)/((a-b)*b^2*d*(a-b+2*b*cosh(c+d*x)^2-b*cosh(c+d*x)^4))-1/8*arctan(b^(1/4)*cosh(c+d*x)/sqrt(sqrt(a)-sqrt(b)))*sqrt(a)*(5*sqrt(a)-6*sqrt(b))/(b^(9/4)*d*(sqrt(a)-sqrt(b))^(3/2))-1/8*arctanh(b^(1/4)*cosh(c+d*x)/sqrt(sqrt(a)+sqrt(b)))*sqrt(a)*(5*sqrt(a)+6*sqrt(b))/(b^(9/4)*d*(sqrt(a)+sqrt(b))^(3/2))],
[sinh(c+d*x)^7/(a-b*sinh(c+d*x)^4)^2,x,5,-1/4*a*cosh(c+d*x)*(2-cosh(c+d*x)^2)/((a-b)*b*d*(a-b+2*b*cosh(c+d*x)^2-b*cosh(c+d*x)^4))+1/8*arctan(b^(1/4)*cosh(c+d*x)/sqrt(sqrt(a)-sqrt(b)))*(3*sqrt(a)-4*sqrt(b))/(b^(7/4)*d*(sqrt(a)-sqrt(b))^(3/2))-1/8*arctanh(b^(1/4)*cosh(c+d*x)/sqrt(sqrt(a)+sqrt(b)))*(3*sqrt(a)+4*sqrt(b))/(b^(7/4)*d*(sqrt(a)+sqrt(b))^(3/2))],
[sinh(c+d*x)^5/(a-b*sinh(c+d*x)^4)^2,x,5,1/4*cosh(c+d*x)*(a+b-b*cosh(c+d*x)^2)/((a-b)*b*d*(a-b+2*b*cosh(c+d*x)^2-b*cosh(c+d*x)^4))-1/8*arctan(b^(1/4)*cosh(c+d*x)/sqrt(sqrt(a)-sqrt(b)))*(sqrt(a)-2*sqrt(b))/(b^(5/4)*d*sqrt(a)*(sqrt(a)-sqrt(b))^(3/2))-1/8*arctanh(b^(1/4)*cosh(c+d*x)/sqrt(sqrt(a)+sqrt(b)))*(sqrt(a)+2*sqrt(b))/(b^(5/4)*d*sqrt(a)*(sqrt(a)+sqrt(b))^(3/2))],
[sinh(c+d*x)^3/(a-b*sinh(c+d*x)^4)^2,x,5,-1/4*cosh(c+d*x)*(2-cosh(c+d*x)^2)/((a-b)*d*(a-b+2*b*cosh(c+d*x)^2-b*cosh(c+d*x)^4))-1/8*arctan(b^(1/4)*cosh(c+d*x)/sqrt(sqrt(a)-sqrt(b)))/(b^(3/4)*d*sqrt(a)*(sqrt(a)-sqrt(b))^(3/2))+1/8*arctanh(b^(1/4)*cosh(c+d*x)/sqrt(sqrt(a)+sqrt(b)))/(b^(3/4)*d*sqrt(a)*(sqrt(a)+sqrt(b))^(3/2))],
[sinh(c+d*x)/(a-b*sinh(c+d*x)^4)^2,x,5,1/4*cosh(c+d*x)*(a+b-b*cosh(c+d*x)^2)/(a*(a-b)*d*(a-b+2*b*cosh(c+d*x)^2-b*cosh(c+d*x)^4))+1/8*arctan(b^(1/4)*cosh(c+d*x)/sqrt(sqrt(a)-sqrt(b)))*(3*sqrt(a)-2*sqrt(b))/(a^(3/2)*b^(1/4)*d*(sqrt(a)-sqrt(b))^(3/2))+1/8*arctanh(b^(1/4)*cosh(c+d*x)/sqrt(sqrt(a)+sqrt(b)))*(3*sqrt(a)+2*sqrt(b))/(a^(3/2)*b^(1/4)*d*(sqrt(a)+sqrt(b))^(3/2))],
[csch(c+d*x)/(a-b*sinh(c+d*x)^4)^2,x,11,-arctanh(cosh(c+d*x))/(a^2*d)-1/4*b*cosh(c+d*x)*(2-cosh(c+d*x)^2)/(a*(a-b)*d*(a-b+2*b*cosh(c+d*x)^2-b*cosh(c+d*x)^4))-1/8*b^(1/4)*arctan(b^(1/4)*cosh(c+d*x)/sqrt(sqrt(a)-sqrt(b)))/(a^(3/2)*d*(sqrt(a)-sqrt(b))^(3/2))+1/8*b^(1/4)*arctanh(b^(1/4)*cosh(c+d*x)/sqrt(sqrt(a)+sqrt(b)))/(a^(3/2)*d*(sqrt(a)+sqrt(b))^(3/2))-1/2*b^(1/4)*arctan(b^(1/4)*cosh(c+d*x)/sqrt(sqrt(a)-sqrt(b)))/(a^2*d*sqrt(sqrt(a)-sqrt(b)))+1/2*b^(1/4)*arctanh(b^(1/4)*cosh(c+d*x)/sqrt(sqrt(a)+sqrt(b)))/(a^2*d*sqrt(sqrt(a)+sqrt(b)))],
[sinh(c+d*x)^8/(a-b*sinh(c+d*x)^4)^2,x,14,x/b^2+1/8*a^(1/4)*arctanh(sqrt(sqrt(a)-sqrt(b))*tanh(c+d*x)/a^(1/4))/(b^(3/2)*d*(sqrt(a)-sqrt(b))^(3/2))-1/8*a^(1/4)*arctanh(sqrt(sqrt(a)+sqrt(b))*tanh(c+d*x)/a^(1/4))/(b^(3/2)*d*(sqrt(a)+sqrt(b))^(3/2))-1/2*a^(1/4)*arctanh(sqrt(sqrt(a)-sqrt(b))*tanh(c+d*x)/a^(1/4))/(b^2*d*sqrt(sqrt(a)-sqrt(b)))-1/2*a^(1/4)*arctanh(sqrt(sqrt(a)+sqrt(b))*tanh(c+d*x)/a^(1/4))/(b^2*d*sqrt(sqrt(a)+sqrt(b)))-1/4*tanh(c+d*x)/((a-b)*b*d)+1/4*tanh(c+d*x)^5/(b*d*(a-2*a*tanh(c+d*x)^2+(a-b)*tanh(c+d*x)^4))],
[sinh(c+d*x)^6/(a-b*sinh(c+d*x)^4)^2,x,6,1/8*arctanh(sqrt(sqrt(a)-sqrt(b))*tanh(c+d*x)/a^(1/4))*(2*sqrt(a)-3*sqrt(b))/(a^(1/4)*b^(3/2)*d*(sqrt(a)-sqrt(b))^(3/2))-1/8*arctanh(sqrt(sqrt(a)+sqrt(b))*tanh(c+d*x)/a^(1/4))*(2*sqrt(a)+3*sqrt(b))/(a^(1/4)*b^(3/2)*d*(sqrt(a)+sqrt(b))^(3/2))+1/4*tanh(c+d*x)/((a-b)*b*d)+1/4*sech(c+d*x)^2*tanh(c+d*x)^3/(b*d*(a-2*a*tanh(c+d*x)^2+(a-b)*tanh(c+d*x)^4))],
[sinh(c+d*x)^4/(a-b*sinh(c+d*x)^4)^2,x,7,1/8*arctanh(sqrt(sqrt(a)-sqrt(b))*tanh(c+d*x)/a^(1/4))/(a^(3/4)*d*(sqrt(a)-sqrt(b))^(3/2)*sqrt(b))-1/8*arctanh(sqrt(sqrt(a)+sqrt(b))*tanh(c+d*x)/a^(1/4))/(a^(3/4)*d*sqrt(b)*(sqrt(a)+sqrt(b))^(3/2))-1/4*tanh(c+d*x)/(a*(a-b)*d)+1/4*tanh(c+d*x)^5/(a*d*(a-2*a*tanh(c+d*x)^2+(a-b)*tanh(c+d*x)^4))],
[sinh(c+d*x)^2/(a-b*sinh(c+d*x)^4)^2,x,5,-1/8*arctanh(sqrt(sqrt(a)-sqrt(b))*tanh(c+d*x)/a^(1/4))*(2*sqrt(a)-sqrt(b))/(a^(5/4)*d*(sqrt(a)-sqrt(b))^(3/2)*sqrt(b))+1/8*arctanh(sqrt(sqrt(a)+sqrt(b))*tanh(c+d*x)/a^(1/4))*(2*sqrt(a)+sqrt(b))/(a^(5/4)*d*sqrt(b)*(sqrt(a)+sqrt(b))^(3/2))+1/4*tanh(c+d*x)*(a-(a+b)*tanh(c+d*x)^2)/(a*(a-b)*d*(a-2*a*tanh(c+d*x)^2+(a-b)*tanh(c+d*x)^4))],
[1/(a-b*sinh(c+d*x)^4)^2,x,5,1/8*arctanh(sqrt(sqrt(a)-sqrt(b))*tanh(c+d*x)/a^(1/4))*(4*sqrt(a)-3*sqrt(b))/(a^(7/4)*d*(sqrt(a)-sqrt(b))^(3/2))+1/8*arctanh(sqrt(sqrt(a)+sqrt(b))*tanh(c+d*x)/a^(1/4))*(4*sqrt(a)+3*sqrt(b))/(a^(7/4)*d*(sqrt(a)+sqrt(b))^(3/2))-1/4*b*tanh(c+d*x)*(1-2*tanh(c+d*x)^2)/(a*(a-b)*d*(a-2*a*tanh(c+d*x)^2+(a-b)*tanh(c+d*x)^4))],
[csch(c+d*x)^2/(a-b*sinh(c+d*x)^4)^2,x,7,-coth(c+d*x)/(a^2*d)-1/8*arctanh(sqrt(sqrt(a)-sqrt(b))*tanh(c+d*x)/a^(1/4))*(6*sqrt(a)-5*sqrt(b))*sqrt(b)/(a^(9/4)*d*(sqrt(a)-sqrt(b))^(3/2))+1/8*arctanh(sqrt(sqrt(a)+sqrt(b))*tanh(c+d*x)/a^(1/4))*sqrt(b)*(6*sqrt(a)+5*sqrt(b))/(a^(9/4)*d*(sqrt(a)+sqrt(b))^(3/2))+1/4*b*tanh(c+d*x)*(a-(a+b)*tanh(c+d*x)^2)/(a^2*(a-b)*d*(a-2*a*tanh(c+d*x)^2+(a-b)*tanh(c+d*x)^4))],
[sinh(c+d*x)^9/(a-b*sinh(c+d*x)^4)^3,x,6,1/8*a*cosh(c+d*x)*(a+b-b*cosh(c+d*x)^2)/((a-b)*b^2*d*(a-b+2*b*cosh(c+d*x)^2-b*cosh(c+d*x)^4)^2)-1/32*cosh(c+d*x)*(9*a^2-11*a*b-10*b^2-2*(2*a-5*b)*b*cosh(c+d*x)^2)/((a-b)^2*b^2*d*(a-b+2*b*cosh(c+d*x)^2-b*cosh(c+d*x)^4))+1/64*arctan(b^(1/4)*cosh(c+d*x)/sqrt(sqrt(a)-sqrt(b)))*(5*a+12*b-14*sqrt(a)*sqrt(b))/(b^(9/4)*d*sqrt(a)*(sqrt(a)-sqrt(b))^(5/2))+1/64*arctanh(b^(1/4)*cosh(c+d*x)/sqrt(sqrt(a)+sqrt(b)))*(5*a+12*b+14*sqrt(a)*sqrt(b))/(b^(9/4)*d*sqrt(a)*(sqrt(a)+sqrt(b))^(5/2))],
[sinh(c+d*x)^7/(a-b*sinh(c+d*x)^4)^3,x,6,-1/8*a*cosh(c+d*x)*(2-cosh(c+d*x)^2)/((a-b)*b*d*(a-b+2*b*cosh(c+d*x)^2-b*cosh(c+d*x)^4)^2)+1/32*cosh(c+d*x)*(5*a-17*b-3*(a-3*b)*cosh(c+d*x)^2)/((a-b)^2*b*d*(a-b+2*b*cosh(c+d*x)^2-b*cosh(c+d*x)^4))+3/64*arctan(b^(1/4)*cosh(c+d*x)/sqrt(sqrt(a)-sqrt(b)))*(sqrt(a)-2*sqrt(b))/(b^(7/4)*d*sqrt(a)*(sqrt(a)-sqrt(b))^(5/2))-3/64*arctanh(b^(1/4)*cosh(c+d*x)/sqrt(sqrt(a)+sqrt(b)))*(sqrt(a)+2*sqrt(b))/(b^(7/4)*d*sqrt(a)*(sqrt(a)+sqrt(b))^(5/2))],
[sinh(c+d*x)^5/(a-b*sinh(c+d*x)^4)^3,x,6,1/8*cosh(c+d*x)*(a+b-b*cosh(c+d*x)^2)/((a-b)*b*d*(a-b+2*b*cosh(c+d*x)^2-b*cosh(c+d*x)^4)^2)-1/32*cosh(c+d*x)*(a^2-11*a*b-2*b^2+2*b*(2*a+b)*cosh(c+d*x)^2)/(a*(a-b)^2*b*d*(a-b+2*b*cosh(c+d*x)^2-b*cosh(c+d*x)^4))-1/64*arctan(b^(1/4)*cosh(c+d*x)/sqrt(sqrt(a)-sqrt(b)))*(3*a+4*b-10*sqrt(a)*sqrt(b))/(a^(3/2)*b^(5/4)*d*(sqrt(a)-sqrt(b))^(5/2))-1/64*arctanh(b^(1/4)*cosh(c+d*x)/sqrt(sqrt(a)+sqrt(b)))*(3*a+4*b+10*sqrt(a)*sqrt(b))/(a^(3/2)*b^(5/4)*d*(sqrt(a)+sqrt(b))^(5/2))],
[sinh(c+d*x)^3/(a-b*sinh(c+d*x)^4)^3,x,6,-1/8*cosh(c+d*x)*(2-cosh(c+d*x)^2)/((a-b)*d*(a-b+2*b*cosh(c+d*x)^2-b*cosh(c+d*x)^4)^2)-1/32*cosh(c+d*x)*(11*a+b-(5*a+b)*cosh(c+d*x)^2)/(a*(a-b)^2*d*(a-b+2*b*cosh(c+d*x)^2-b*cosh(c+d*x)^4))-1/64*arctan(b^(1/4)*cosh(c+d*x)/sqrt(sqrt(a)-sqrt(b)))*(5*sqrt(a)-2*sqrt(b))/(a^(3/2)*b^(3/4)*d*(sqrt(a)-sqrt(b))^(5/2))+1/64*arctanh(b^(1/4)*cosh(c+d*x)/sqrt(sqrt(a)+sqrt(b)))*(5*sqrt(a)+2*sqrt(b))/(a^(3/2)*b^(3/4)*d*(sqrt(a)+sqrt(b))^(5/2))],
[sinh(c+d*x)/(a-b*sinh(c+d*x)^4)^3,x,6,1/8*cosh(c+d*x)*(a+b-b*cosh(c+d*x)^2)/(a*(a-b)*d*(a-b+2*b*cosh(c+d*x)^2-b*cosh(c+d*x)^4)^2)+1/32*cosh(c+d*x)*((7*a-3*b)*(a+2*b)-6*(2*a-b)*b*cosh(c+d*x)^2)/(a^2*(a-b)^2*d*(a-b+2*b*cosh(c+d*x)^2-b*cosh(c+d*x)^4))+3/64*arctan(b^(1/4)*cosh(c+d*x)/sqrt(sqrt(a)-sqrt(b)))*(7*a+4*b-10*sqrt(a)*sqrt(b))/(a^(5/2)*b^(1/4)*d*(sqrt(a)-sqrt(b))^(5/2))+3/64*arctanh(b^(1/4)*cosh(c+d*x)/sqrt(sqrt(a)+sqrt(b)))*(7*a+4*b+10*sqrt(a)*sqrt(b))/(a^(5/2)*b^(1/4)*d*(sqrt(a)+sqrt(b))^(5/2))],
[csch(c+d*x)/(a-b*sinh(c+d*x)^4)^3,x,16,-arctanh(cosh(c+d*x))/(a^3*d)-1/8*b*cosh(c+d*x)*(2-cosh(c+d*x)^2)/(a*(a-b)*d*(a-b+2*b*cosh(c+d*x)^2-b*cosh(c+d*x)^4)^2)-1/4*b*cosh(c+d*x)*(2-cosh(c+d*x)^2)/(a^2*(a-b)*d*(a-b+2*b*cosh(c+d*x)^2-b*cosh(c+d*x)^4))-1/32*b*cosh(c+d*x)*(11*a+b-(5*a+b)*cosh(c+d*x)^2)/(a^2*(a-b)^2*d*(a-b+2*b*cosh(c+d*x)^2-b*cosh(c+d*x)^4))-1/64*b^(1/4)*arctan(b^(1/4)*cosh(c+d*x)/sqrt(sqrt(a)-sqrt(b)))*(5*sqrt(a)-2*sqrt(b))/(a^(5/2)*d*(sqrt(a)-sqrt(b))^(5/2))-1/8*b^(1/4)*arctan(b^(1/4)*cosh(c+d*x)/sqrt(sqrt(a)-sqrt(b)))/(a^(5/2)*d*(sqrt(a)-sqrt(b))^(3/2))+1/8*b^(1/4)*arctanh(b^(1/4)*cosh(c+d*x)/sqrt(sqrt(a)+sqrt(b)))/(a^(5/2)*d*(sqrt(a)+sqrt(b))^(3/2))+1/64*b^(1/4)*arctanh(b^(1/4)*cosh(c+d*x)/sqrt(sqrt(a)+sqrt(b)))*(5*sqrt(a)+2*sqrt(b))/(a^(5/2)*d*(sqrt(a)+sqrt(b))^(5/2))-1/2*b^(1/4)*arctan(b^(1/4)*cosh(c+d*x)/sqrt(sqrt(a)-sqrt(b)))/(a^3*d*sqrt(sqrt(a)-sqrt(b)))+1/2*b^(1/4)*arctanh(b^(1/4)*cosh(c+d*x)/sqrt(sqrt(a)+sqrt(b)))/(a^3*d*sqrt(sqrt(a)+sqrt(b)))],
[sinh(c+d*x)^8/(a-b*sinh(c+d*x)^4)^3,x,9,-1/64*arctanh(sqrt(sqrt(a)-sqrt(b))*tanh(c+d*x)/a^(1/4))*(2*sqrt(a)-5*sqrt(b))/(a^(3/4)*b^(3/2)*d*(sqrt(a)-sqrt(b))^(5/2))+1/64*arctanh(sqrt(sqrt(a)+sqrt(b))*tanh(c+d*x)/a^(1/4))*(2*sqrt(a)+5*sqrt(b))/(a^(3/4)*b^(3/2)*d*(sqrt(a)+sqrt(b))^(5/2))-1/32*(a+5*b)*tanh(c+d*x)/(a*(a-b)^2*b*d)-1/32*tanh(c+d*x)^3/(a*(a-b)*b*d)+1/8*tanh(c+d*x)^9/(a*d*(a-2*a*tanh(c+d*x)^2+(a-b)*tanh(c+d*x)^4)^2)-1/32*sech(c+d*x)^2*tanh(c+d*x)^5/(a*b*d*(a-2*a*tanh(c+d*x)^2+(a-b)*tanh(c+d*x)^4))],
[sinh(c+d*x)^6/(a-b*sinh(c+d*x)^4)^3,x,6,1/64*arctanh(sqrt(sqrt(a)-sqrt(b))*tanh(c+d*x)/a^(1/4))*(4*a+3*b-10*sqrt(a)*sqrt(b))/(a^(5/4)*b^(3/2)*d*(sqrt(a)-sqrt(b))^(5/2))-1/64*arctanh(sqrt(sqrt(a)+sqrt(b))*tanh(c+d*x)/a^(1/4))*(4*a+3*b+10*sqrt(a)*sqrt(b))/(a^(5/4)*b^(3/2)*d*(sqrt(a)+sqrt(b))^(5/2))+1/8*tanh(c+d*x)*(a*(a+3*b)-(a^2+6*a*b+b^2)*tanh(c+d*x)^2)/((a-b)^3*d*(a-2*a*tanh(c+d*x)^2+(a-b)*tanh(c+d*x)^4)^2)+1/32*tanh(c+d*x)*(2*a*(a^2-a*b-8*b^2)/(a-b)^3-(2*a^2+15*a*b+3*b^2)*tanh(c+d*x)^2/(a-b)^2)/(a*b*d*(a-2*a*tanh(c+d*x)^2+(a-b)*tanh(c+d*x)^4))],
[sinh(c+d*x)^4/(a-b*sinh(c+d*x)^4)^3,x,6,3/64*arctanh(sqrt(sqrt(a)-sqrt(b))*tanh(c+d*x)/a^(1/4))*(2*sqrt(a)-sqrt(b))/(a^(7/4)*d*(sqrt(a)-sqrt(b))^(5/2)*sqrt(b))-3/64*arctanh(sqrt(sqrt(a)+sqrt(b))*tanh(c+d*x)/a^(1/4))*(2*sqrt(a)+sqrt(b))/(a^(7/4)*d*sqrt(b)*(sqrt(a)+sqrt(b))^(5/2))-1/8*b*tanh(c+d*x)*(3*a+b-4*(a+b)*tanh(c+d*x)^2)/((a-b)^3*d*(a-2*a*tanh(c+d*x)^2+(a-b)*tanh(c+d*x)^4)^2)-1/32*tanh(c+d*x)*((9*a^2-24*a*b-b^2)/(a-b)^3-(17*a+3*b)*tanh(c+d*x)^2/(a-b)^2)/(a*d*(a-2*a*tanh(c+d*x)^2+(a-b)*tanh(c+d*x)^4))],
[sinh(c+d*x)^2/(a-b*sinh(c+d*x)^4)^3,x,6,-1/64*arctanh(sqrt(sqrt(a)-sqrt(b))*tanh(c+d*x)/a^(1/4))*(12*a+5*b-14*sqrt(a)*sqrt(b))/(a^(9/4)*d*(sqrt(a)-sqrt(b))^(5/2)*sqrt(b))+1/64*arctanh(sqrt(sqrt(a)+sqrt(b))*tanh(c+d*x)/a^(1/4))*(12*a+5*b+14*sqrt(a)*sqrt(b))/(a^(9/4)*d*sqrt(b)*(sqrt(a)+sqrt(b))^(5/2))+1/8*b*tanh(c+d*x)*(a*(a+3*b)-(a^2+6*a*b+b^2)*tanh(c+d*x)^2)/(a*(a-b)^3*d*(a-2*a*tanh(c+d*x)^2+(a-b)*tanh(c+d*x)^4)^2)+1/32*tanh(c+d*x)*(2*a*(5*a^2-9*a*b-4*b^2)/(a-b)^3-5*(2*a^2+3*a*b-b^2)*tanh(c+d*x)^2/(a-b)^2)/(a^2*d*(a-2*a*tanh(c+d*x)^2+(a-b)*tanh(c+d*x)^4))],
[1/(a-b*sinh(c+d*x)^4)^3,x,6,1/64*arctanh(sqrt(sqrt(a)-sqrt(b))*tanh(c+d*x)/a^(1/4))*(32*a+21*b-50*sqrt(a)*sqrt(b))/(a^(11/4)*d*(sqrt(a)-sqrt(b))^(5/2))+1/64*arctanh(sqrt(sqrt(a)+sqrt(b))*tanh(c+d*x)/a^(1/4))*(32*a+21*b+50*sqrt(a)*sqrt(b))/(a^(11/4)*d*(sqrt(a)+sqrt(b))^(5/2))-1/8*b^2*tanh(c+d*x)*(3*a+b-4*(a+b)*tanh(c+d*x)^2)/(a*(a-b)^3*d*(a-2*a*tanh(c+d*x)^2+(a-b)*tanh(c+d*x)^4)^2)-1/32*b*tanh(c+d*x)*((17*a^2-40*a*b+7*b^2)/(a-b)^3-(33*a-13*b)*tanh(c+d*x)^2/(a-b)^2)/(a^2*d*(a-2*a*tanh(c+d*x)^2+(a-b)*tanh(c+d*x)^4))],
[csch(c+d*x)^2/(a-b*sinh(c+d*x)^4)^3,x,8,-coth(c+d*x)/(a^3*d)-3/64*arctanh(sqrt(sqrt(a)-sqrt(b))*tanh(c+d*x)/a^(1/4))*sqrt(b)*(20*a+15*b-34*sqrt(a)*sqrt(b))/(a^(13/4)*d*(sqrt(a)-sqrt(b))^(5/2))+3/64*arctanh(sqrt(sqrt(a)+sqrt(b))*tanh(c+d*x)/a^(1/4))*sqrt(b)*(20*a+15*b+34*sqrt(a)*sqrt(b))/(a^(13/4)*d*(sqrt(a)+sqrt(b))^(5/2))+1/8*b^2*tanh(c+d*x)*(a*(a+3*b)-(a^2+6*a*b+b^2)*tanh(c+d*x)^2)/(a^2*(a-b)^3*d*(a-2*a*tanh(c+d*x)^2+(a-b)*tanh(c+d*x)^4)^2)+1/32*b*tanh(c+d*x)*(2*a^2*(9*a-17*b)/(a-b)^3-(18*a^2+15*a*b-13*b^2)*tanh(c+d*x)^2/(a-b)^2)/(a^3*d*(a-2*a*tanh(c+d*x)^2+(a-b)*tanh(c+d*x)^4))],
[1/(1-sinh(x)^4),x,3,1/2*arctanh(sqrt(2)*tanh(x))/sqrt(2)+1/2*tanh(x)],
[1/(1+sinh(x)^4),x,10,-1/4*arctan((sqrt(1+sqrt(2))-2*tanh(x))/sqrt(-1+sqrt(2)))/sqrt(1+sqrt(2))+1/4*arctan((sqrt(1+sqrt(2))+2*tanh(x))/sqrt(-1+sqrt(2)))/sqrt(1+sqrt(2))-1/8*log(sqrt(2)-2*sqrt(1+sqrt(2))*tanh(x)+2*tanh(x)^2)*sqrt(1+sqrt(2))+1/8*log(1+sqrt(2*(1+sqrt(2)))*tanh(x)+sqrt(2)*tanh(x)^2)*sqrt(1+sqrt(2))],

# Integrands of the form Sinh[e+f x]^m (a+b Sinh[e+f x]^4)^(p/2)

# Integrands of the form Sinh[e+f x]^m (a+b Sinh[e+f x]^n)^p
[1/(a+b*sinh(x)^5),x,17,-2/5*(-1)^(9/10)*arctanh((I*b^(1/5)-(-1)^(9/10)*a^(1/5)*tanh(1/2*x))/sqrt(-(-1)^(4/5)*a^(2/5)-b^(2/5)))/(a^(4/5)*sqrt(-(-1)^(4/5)*a^(2/5)-b^(2/5)))-2/5*arctanh((b^(1/5)-a^(1/5)*tanh(1/2*x))/sqrt(a^(2/5)+b^(2/5)))/(a^(4/5)*sqrt(a^(2/5)+b^(2/5)))+2/5*(-1)^(1/5)*arctanh((b^(1/5)+(-1)^(1/5)*a^(1/5)*tanh(1/2*x))/sqrt((-1)^(2/5)*a^(2/5)+b^(2/5)))/(a^(4/5)*sqrt((-1)^(2/5)*a^(2/5)+b^(2/5)))+2/5*(-1)^(9/10)*arctanh((-1)^(9/10)*((-1)^(1/5)*b^(1/5)+a^(1/5)*tanh(1/2*x))/sqrt(-(-1)^(4/5)*a^(2/5)+(-1)^(1/5)*b^(2/5)))/(a^(4/5)*sqrt(-(-1)^(4/5)*a^(2/5)+(-1)^(1/5)*b^(2/5)))+2/5*(-1)^(9/10)*arctanh((-1)^(3/10)*(b^(1/5)+(-1)^(3/5)*a^(1/5)*tanh(1/2*x))/sqrt(-(-1)^(4/5)*a^(2/5)+(-1)^(3/5)*b^(2/5)))/(a^(4/5)*sqrt(-(-1)^(4/5)*a^(2/5)+(-1)^(3/5)*b^(2/5)))],
[1/(a+b*sinh(x)^6),x,7,1/3*arctanh(sqrt(a^(1/3)-b^(1/3))*tanh(x)/a^(1/6))/(a^(5/6)*sqrt(a^(1/3)-b^(1/3)))+1/3*arctanh(sqrt(a^(1/3)+(-1)^(1/3)*b^(1/3))*tanh(x)/a^(1/6))/(a^(5/6)*sqrt(a^(1/3)+(-1)^(1/3)*b^(1/3)))+1/3*arctanh(sqrt(a^(1/3)-(-1)^(2/3)*b^(1/3))*tanh(x)/a^(1/6))/(a^(5/6)*sqrt(a^(1/3)-(-1)^(2/3)*b^(1/3)))],
[1/(a+b*sinh(x)^8),x,9,-1/4*arctanh(sqrt((-a)^(1/4)-b^(1/4))*tanh(x)/(-a)^(1/8))/((-a)^(7/8)*sqrt((-a)^(1/4)-b^(1/4)))-1/4*arctanh(sqrt((-a)^(1/4)-I*b^(1/4))*tanh(x)/(-a)^(1/8))/((-a)^(7/8)*sqrt((-a)^(1/4)-I*b^(1/4)))-1/4*arctanh(sqrt((-a)^(1/4)+I*b^(1/4))*tanh(x)/(-a)^(1/8))/((-a)^(7/8)*sqrt((-a)^(1/4)+I*b^(1/4)))-1/4*arctanh(sqrt((-a)^(1/4)+b^(1/4))*tanh(x)/(-a)^(1/8))/((-a)^(7/8)*sqrt((-a)^(1/4)+b^(1/4)))],
[1/(1+sinh(x)^5),x,17,-1/5*arctanh((1-tanh(1/2*x))/sqrt(2))*sqrt(2)-2/5*(-1)^(3/5)*arctan((1+(-1)^(3/5)*tanh(1/2*x))/sqrt(-1+(-1)^(1/5)))/sqrt(-1+(-1)^(1/5))+2/5*(-1)^(9/10)*arctanh((-1)^(7/10)*(1+(-1)^(1/5)*tanh(1/2*x))/sqrt(-(-1)^(2/5)*(1+(-1)^(2/5))))/sqrt(-(-1)^(2/5)*(1+(-1)^(2/5)))-2/5*(-1)^(4/5)*arctanh((1-(-1)^(4/5)*tanh(1/2*x))/sqrt(1-(-1)^(3/5)))/sqrt(1-(-1)^(3/5))+2/5*(-1)^(9/10)*arctan((I-(-1)^(9/10)*tanh(1/2*x))/sqrt(1+(-1)^(4/5)))/sqrt(1+(-1)^(4/5))],
[1/(1+sinh(x)^6),x,8,1/3*arctanh(sqrt(1+(-1)^(1/3))*tanh(x))/sqrt(1+(-1)^(1/3))+1/3*arctanh(sqrt(1-(-1)^(2/3))*tanh(x))/sqrt(1-(-1)^(2/3))+1/3*tanh(x)],
[1/(1+sinh(x)^8),x,9,1/4*arctanh(sqrt(1-(-1)^(1/4))*tanh(x))/sqrt(1-(-1)^(1/4))+1/4*arctanh(sqrt(1+(-1)^(1/4))*tanh(x))/sqrt(1+(-1)^(1/4))+1/4*arctanh(sqrt(1-(-1)^(3/4))*tanh(x))/sqrt(1-(-1)^(3/4))+1/4*arctanh(sqrt(1+(-1)^(3/4))*tanh(x))/sqrt(1+(-1)^(3/4))],
[1/(1-sinh(x)^5),x,17,1/5*arctanh((1+tanh(1/2*x))/sqrt(2))*sqrt(2)-2/5*(-1)^(1/10)*arctan((I+(-1)^(1/10)*tanh(1/2*x))/sqrt(1-(-1)^(1/5)))/sqrt(1-(-1)^(1/5))-2/5*arctanh(((-1)^(3/5)-tanh(1/2*x))/sqrt(1-(-1)^(1/5)))/sqrt(1-(-1)^(1/5))+2/5*arctanh(((-1)^(4/5)+tanh(1/2*x))/sqrt(1-(-1)^(3/5)))/sqrt(1-(-1)^(3/5))-2/5*(-1)^(1/10)*arctanh((-1)^(3/10)*(1+(-1)^(4/5)*tanh(1/2*x))/sqrt((-1)^(1/5)+(-1)^(3/5)))/sqrt((-1)^(1/5)+(-1)^(3/5))],
[1/(1-sinh(x)^6),x,7,1/3*arctanh(sqrt(2)*tanh(x))/sqrt(2)+1/3*arctanh(sqrt(1-(-1)^(1/3))*tanh(x))/sqrt(1-(-1)^(1/3))+1/3*arctanh(sqrt(1+(-1)^(2/3))*tanh(x))/sqrt(1+(-1)^(2/3))],
[1/(1-sinh(x)^8),x,10,1/4*arctanh(sqrt(1-I)*tanh(x))/sqrt(1-I)+1/4*arctanh(sqrt(1+I)*tanh(x))/sqrt(1+I)+1/4*arctanh(sqrt(2)*tanh(x))/sqrt(2)+1/4*tanh(x)],

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

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

# Integrands of the form Cosh[e+f x]^m (a+b Sinh[e+f x]^2)^p with a-b=0

# p>0

# p<0
[cosh(x)^5/(a+a*sinh(x)^2),x,3,sinh(x)/a+1/3*sinh(x)^3/a],
[cosh(x)^4/(a+a*sinh(x)^2),x,3,1/2*x/a+1/2*cosh(x)*sinh(x)/a],
[cosh(x)^3/(a+a*sinh(x)^2),x,2,sinh(x)/a],
[cosh(x)^2/(a+a*sinh(x)^2),x,2,x/a],
[cosh(x)/(a+a*sinh(x)^2),x,2,arctan(sinh(x))/a],
[sech(x)/(a+a*sinh(x)^2),x,3,1/2*arctan(sinh(x))/a+1/2*sech(x)*tanh(x)/a],
[sech(x)^3/(a+a*sinh(x)^2),x,4,3/8*arctan(sinh(x))/a+3/8*sech(x)*tanh(x)/a+1/4*sech(x)^3*tanh(x)/a],

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

# p>0
[cosh(c+d*x)^4*(a+b*sinh(c+d*x)^2),x,5,1/16*(6*a-b)*x+1/16*(6*a-b)*cosh(c+d*x)*sinh(c+d*x)/d+1/24*(6*a-b)*cosh(c+d*x)^3*sinh(c+d*x)/d+1/6*b*cosh(c+d*x)^5*sinh(c+d*x)/d],
[cosh(c+d*x)^3*(a+b*sinh(c+d*x)^2),x,3,a*sinh(c+d*x)/d+1/3*(a+b)*sinh(c+d*x)^3/d+1/5*b*sinh(c+d*x)^5/d],
[cosh(c+d*x)^2*(a+b*sinh(c+d*x)^2),x,4,1/8*(4*a-b)*x+1/8*(4*a-b)*cosh(c+d*x)*sinh(c+d*x)/d+1/4*b*cosh(c+d*x)^3*sinh(c+d*x)/d],
[cosh(c+d*x)*(a+b*sinh(c+d*x)^2),x,2,a*sinh(c+d*x)/d+1/3*b*sinh(c+d*x)^3/d],
[sech(c+d*x)*(a+b*sinh(c+d*x)^2),x,3,(a-b)*arctan(sinh(c+d*x))/d+b*sinh(c+d*x)/d],
[sech(c+d*x)^2*(a+b*sinh(c+d*x)^2),x,3,b*x+(a-b)*tanh(c+d*x)/d],
[sech(c+d*x)^3*(a+b*sinh(c+d*x)^2),x,3,1/2*(a+b)*arctan(sinh(c+d*x))/d+1/2*(a-b)*sech(c+d*x)*tanh(c+d*x)/d],
[sech(c+d*x)^4*(a+b*sinh(c+d*x)^2),x,2,a*tanh(c+d*x)/d-1/3*(a-b)*tanh(c+d*x)^3/d],
[sech(c+d*x)^5*(a+b*sinh(c+d*x)^2),x,4,1/8*(3*a+b)*arctan(sinh(c+d*x))/d+1/8*(3*a+b)*sech(c+d*x)*tanh(c+d*x)/d+1/4*(a-b)*sech(c+d*x)^3*tanh(c+d*x)/d],
[sech(c+d*x)^6*(a+b*sinh(c+d*x)^2),x,3,a*tanh(c+d*x)/d-1/3*(2*a-b)*tanh(c+d*x)^3/d+1/5*(a-b)*tanh(c+d*x)^5/d],
[cosh(c+d*x)^4*(a+b*sinh(c+d*x)^2)^2,x,6,1/128*(48*a^2-16*a*b+3*b^2)*x+1/128*(48*a^2-16*a*b+3*b^2)*cosh(c+d*x)*sinh(c+d*x)/d+1/192*(48*a^2-16*a*b+3*b^2)*cosh(c+d*x)^3*sinh(c+d*x)/d+1/48*(10*a-3*b)*b*cosh(c+d*x)^5*sinh(c+d*x)/d+1/8*b*cosh(c+d*x)^7*sinh(c+d*x)*(a-(a-b)*tanh(c+d*x)^2)/d],
[cosh(c+d*x)^3*(a+b*sinh(c+d*x)^2)^2,x,3,a^2*sinh(c+d*x)/d+1/3*a*(a+2*b)*sinh(c+d*x)^3/d+1/5*b*(2*a+b)*sinh(c+d*x)^5/d+1/7*b^2*sinh(c+d*x)^7/d],
[cosh(c+d*x)^2*(a+b*sinh(c+d*x)^2)^2,x,5,1/16*(8*a^2-4*a*b+b^2)*x+1/16*(8*a^2-4*a*b+b^2)*cosh(c+d*x)*sinh(c+d*x)/d+1/24*(8*a-3*b)*b*cosh(c+d*x)^3*sinh(c+d*x)/d+1/6*b*cosh(c+d*x)^5*sinh(c+d*x)*(a-(a-b)*tanh(c+d*x)^2)/d],
[cosh(c+d*x)*(a+b*sinh(c+d*x)^2)^2,x,3,a^2*sinh(c+d*x)/d+2/3*a*b*sinh(c+d*x)^3/d+1/5*b^2*sinh(c+d*x)^5/d],
[sech(c+d*x)*(a+b*sinh(c+d*x)^2)^2,x,4,(a-b)^2*arctan(sinh(c+d*x))/d+(2*a-b)*b*sinh(c+d*x)/d+1/3*b^2*sinh(c+d*x)^3/d],
[sech(c+d*x)^2*(a+b*sinh(c+d*x)^2)^2,x,5,1/2*(4*a-3*b)*b*x+1/2*b^2*cosh(c+d*x)*sinh(c+d*x)/d+(a-b)^2*tanh(c+d*x)/d],
[sech(c+d*x)^3*(a+b*sinh(c+d*x)^2)^2,x,5,1/2*(a-b)*(a+3*b)*arctan(sinh(c+d*x))/d+b^2*sinh(c+d*x)/d+1/2*(a-b)^2*sech(c+d*x)*tanh(c+d*x)/d],
[sech(c+d*x)^4*(a+b*sinh(c+d*x)^2)^2,x,4,b^2*x+(a^2-b^2)*tanh(c+d*x)/d-1/3*(a-b)^2*tanh(c+d*x)^3/d],
[sech(c+d*x)^5*(a+b*sinh(c+d*x)^2)^2,x,4,1/8*(3*a^2+2*a*b+3*b^2)*arctan(sinh(c+d*x))/d+3/8*(a^2-b^2)*sech(c+d*x)*tanh(c+d*x)/d+1/4*(a-b)*sech(c+d*x)^3*(a+b*sinh(c+d*x)^2)*tanh(c+d*x)/d],
[sech(c+d*x)^6*(a+b*sinh(c+d*x)^2)^2,x,3,a^2*tanh(c+d*x)/d-2/3*a*(a-b)*tanh(c+d*x)^3/d+1/5*(a-b)^2*tanh(c+d*x)^5/d],
[sech(c+d*x)^7*(a+b*sinh(c+d*x)^2)^2,x,5,1/16*(5*a^2+2*a*b+b^2)*arctan(sinh(c+d*x))/d+1/16*(5*a^2+2*a*b+b^2)*sech(c+d*x)*tanh(c+d*x)/d+1/24*(a-b)*(5*a+3*b)*sech(c+d*x)^3*tanh(c+d*x)/d+1/6*(a-b)*sech(c+d*x)^5*(a+b*sinh(c+d*x)^2)*tanh(c+d*x)/d],
[cosh(c+d*x)^4*(a+b*sinh(c+d*x)^2)^3,x,7,3/256*(4*a-b)*(8*a^2-2*a*b+b^2)*x+3/256*(4*a-b)*(8*a^2-2*a*b+b^2)*cosh(c+d*x)*sinh(c+d*x)/d+1/128*(4*a-b)*(8*a^2-2*a*b+b^2)*cosh(c+d*x)^3*sinh(c+d*x)/d+1/160*b*(44*a^2-28*a*b+5*b^2)*cosh(c+d*x)^5*sinh(c+d*x)/d+1/10*b*cosh(c+d*x)^9*sinh(c+d*x)*(a-(a-b)*tanh(c+d*x)^2)^2/d+1/80*b*cosh(c+d*x)^7*sinh(c+d*x)*(a*(10*a-b)-5*(a-b)*(2*a-b)*tanh(c+d*x)^2)/d],
[cosh(c+d*x)^3*(a+b*sinh(c+d*x)^2)^3,x,3,a^3*sinh(c+d*x)/d+1/3*a^2*(a+3*b)*sinh(c+d*x)^3/d+3/5*a*b*(a+b)*sinh(c+d*x)^5/d+1/7*b^2*(3*a+b)*sinh(c+d*x)^7/d+1/9*b^3*sinh(c+d*x)^9/d],
[cosh(c+d*x)^2*(a+b*sinh(c+d*x)^2)^3,x,6,1/128*(64*a^3-48*a^2*b+24*a*b^2-5*b^3)*x+1/128*(64*a^3-48*a^2*b+24*a*b^2-5*b^3)*cosh(c+d*x)*sinh(c+d*x)/d+1/192*b*(88*a^2-68*a*b+15*b^2)*cosh(c+d*x)^3*sinh(c+d*x)/d+1/8*b*cosh(c+d*x)^7*sinh(c+d*x)*(a-(a-b)*tanh(c+d*x)^2)^2/d+1/48*b*cosh(c+d*x)^5*sinh(c+d*x)*(a*(8*a-b)-(8*a-5*b)*(a-b)*tanh(c+d*x)^2)/d],
[cosh(c+d*x)*(a+b*sinh(c+d*x)^2)^3,x,3,a^3*sinh(c+d*x)/d+a^2*b*sinh(c+d*x)^3/d+3/5*a*b^2*sinh(c+d*x)^5/d+1/7*b^3*sinh(c+d*x)^7/d],
[sech(c+d*x)*(a+b*sinh(c+d*x)^2)^3,x,4,(a-b)^3*arctan(sinh(c+d*x))/d+b*(3*a^2-3*a*b+b^2)*sinh(c+d*x)/d+1/3*(3*a-b)*b^2*sinh(c+d*x)^3/d+1/5*b^3*sinh(c+d*x)^5/d],
[sech(c+d*x)^2*(a+b*sinh(c+d*x)^2)^3,x,6,3/8*b*(8*a^2-12*a*b+5*b^2)*x+3/8*(4*a-3*b)*b^2*cosh(c+d*x)*sinh(c+d*x)/d+1/4*b^3*cosh(c+d*x)^3*sinh(c+d*x)/d+(a-b)^3*tanh(c+d*x)/d],
[sech(c+d*x)^3*(a+b*sinh(c+d*x)^2)^3,x,5,1/2*(a-b)^2*(a+5*b)*arctan(sinh(c+d*x))/d+(3*a-2*b)*b^2*sinh(c+d*x)/d+1/3*b^3*sinh(c+d*x)^3/d+1/2*(a-b)^3*sech(c+d*x)*tanh(c+d*x)/d],
[sech(c+d*x)^4*(a+b*sinh(c+d*x)^2)^3,x,5,1/2*(6*a-5*b)*b^2*x+1/2*b^3*cosh(c+d*x)*sinh(c+d*x)/d+(a-b)^2*(a+2*b)*tanh(c+d*x)/d-1/3*(a-b)^3*tanh(c+d*x)^3/d],
[sech(c+d*x)^5*(a+b*sinh(c+d*x)^2)^3,x,6,3/8*(a-b)*(4*b^2+(a+b)^2)*arctan(sinh(c+d*x))/d+b^3*sinh(c+d*x)/d+3/8*(a-b)^2*(a+3*b)*sech(c+d*x)*tanh(c+d*x)/d+1/4*(a-b)^3*sech(c+d*x)^3*tanh(c+d*x)/d],
[sech(c+d*x)^6*(a+b*sinh(c+d*x)^2)^3,x,4,b^3*x+(a^3-b^3)*tanh(c+d*x)/d-1/3*(a-b)^2*(2*a+b)*tanh(c+d*x)^3/d+1/5*(a-b)^3*tanh(c+d*x)^5/d],
[sech(c+d*x)^7*(a+b*sinh(c+d*x)^2)^3,x,5,1/16*(a+b)*(5*a^2-2*a*b+5*b^2)*arctan(sinh(c+d*x))/d+1/48*(a-b)*(15*a^2+14*a*b+15*b^2)*sech(c+d*x)*tanh(c+d*x)/d+5/24*(a^2-b^2)*sech(c+d*x)^3*(a+b*sinh(c+d*x)^2)*tanh(c+d*x)/d+1/6*(a-b)*sech(c+d*x)^5*(a+b*sinh(c+d*x)^2)^2*tanh(c+d*x)/d],
[sech(c+d*x)^8*(a+b*sinh(c+d*x)^2)^3,x,3,a^3*tanh(c+d*x)/d-a^2*(a-b)*tanh(c+d*x)^3/d+3/5*a*(a-b)^2*tanh(c+d*x)^5/d-1/7*(a-b)^3*tanh(c+d*x)^7/d],

# p<0
[cosh(c+d*x)^7/(a+b*sinh(c+d*x)^2),x,4,(a^2-3*a*b+3*b^2)*sinh(c+d*x)/(b^3*d)-1/3*(a-3*b)*sinh(c+d*x)^3/(b^2*d)+1/5*sinh(c+d*x)^5/(b*d)-(a-b)^3*arctan(sinh(c+d*x)*sqrt(b)/sqrt(a))/(b^(7/2)*d*sqrt(a))],
[cosh(c+d*x)^6/(a+b*sinh(c+d*x)^2),x,6,1/8*(8*a^2-20*a*b+15*b^2)*x/b^3-1/8*(4*a-7*b)*cosh(c+d*x)*sinh(c+d*x)/(b^2*d)+1/4*cosh(c+d*x)^3*sinh(c+d*x)/(b*d)-(a-b)^(5/2)*arctanh(sqrt(a-b)*tanh(c+d*x)/sqrt(a))/(b^3*d*sqrt(a))],
[cosh(c+d*x)^5/(a+b*sinh(c+d*x)^2),x,4,-(a-2*b)*sinh(c+d*x)/(b^2*d)+1/3*sinh(c+d*x)^3/(b*d)+(a-b)^2*arctan(sinh(c+d*x)*sqrt(b)/sqrt(a))/(b^(5/2)*d*sqrt(a))],
[cosh(c+d*x)^4/(a+b*sinh(c+d*x)^2),x,5,-1/2*(2*a-3*b)*x/b^2+1/2*cosh(c+d*x)*sinh(c+d*x)/(b*d)+(a-b)^(3/2)*arctanh(sqrt(a-b)*tanh(c+d*x)/sqrt(a))/(b^2*d*sqrt(a))],
[cosh(c+d*x)^3/(a+b*sinh(c+d*x)^2),x,3,sinh(c+d*x)/(b*d)-(a-b)*arctan(sinh(c+d*x)*sqrt(b)/sqrt(a))/(b^(3/2)*d*sqrt(a))],
[cosh(c+d*x)^2/(a+b*sinh(c+d*x)^2),x,4,x/b-arctanh(sqrt(a-b)*tanh(c+d*x)/sqrt(a))*sqrt(a-b)/(b*d*sqrt(a))],
[cosh(c+d*x)/(a+b*sinh(c+d*x)^2),x,2,arctan(sinh(c+d*x)*sqrt(b)/sqrt(a))/(d*sqrt(a)*sqrt(b))],
[sech(c+d*x)/(a+b*sinh(c+d*x)^2),x,4,arctan(sinh(c+d*x))/((a-b)*d)-arctan(sinh(c+d*x)*sqrt(b)/sqrt(a))*sqrt(b)/((a-b)*d*sqrt(a))],
[sech(c+d*x)^2/(a+b*sinh(c+d*x)^2),x,3,-b*arctanh(sqrt(a-b)*tanh(c+d*x)/sqrt(a))/((a-b)^(3/2)*d*sqrt(a))+tanh(c+d*x)/((a-b)*d)],
[sech(c+d*x)^3/(a+b*sinh(c+d*x)^2),x,5,1/2*(a-3*b)*arctan(sinh(c+d*x))/((a-b)^2*d)+b^(3/2)*arctan(sinh(c+d*x)*sqrt(b)/sqrt(a))/((a-b)^2*d*sqrt(a))+1/2*sech(c+d*x)*tanh(c+d*x)/((a-b)*d)],
[sech(c+d*x)^4/(a+b*sinh(c+d*x)^2),x,4,b^2*arctanh(sqrt(a-b)*tanh(c+d*x)/sqrt(a))/((a-b)^(5/2)*d*sqrt(a))+(a-2*b)*tanh(c+d*x)/((a-b)^2*d)-1/3*tanh(c+d*x)^3/((a-b)*d)],
[sech(c+d*x)^5/(a+b*sinh(c+d*x)^2),x,6,1/8*(3*a^2-10*a*b+15*b^2)*arctan(sinh(c+d*x))/((a-b)^3*d)-b^(5/2)*arctan(sinh(c+d*x)*sqrt(b)/sqrt(a))/((a-b)^3*d*sqrt(a))+1/8*(3*a-7*b)*sech(c+d*x)*tanh(c+d*x)/((a-b)^2*d)+1/4*sech(c+d*x)^3*tanh(c+d*x)/((a-b)*d)],
[sech(c+d*x)^6/(a+b*sinh(c+d*x)^2),x,4,-b^3*arctanh(sqrt(a-b)*tanh(c+d*x)/sqrt(a))/((a-b)^(7/2)*d*sqrt(a))+(a^2-3*a*b+3*b^2)*tanh(c+d*x)/((a-b)^3*d)-1/3*(2*a-3*b)*tanh(c+d*x)^3/((a-b)^2*d)+1/5*tanh(c+d*x)^5/((a-b)*d)],
[cosh(c+d*x)^6/(a+b*sinh(c+d*x)^2)^2,x,6,-1/2*(4*a-5*b)*x/b^3+1/2*(a-b)^(3/2)*(4*a+b)*arctanh(sqrt(a-b)*tanh(c+d*x)/sqrt(a))/(a^(3/2)*b^3*d)+1/2*cosh(c+d*x)*sinh(c+d*x)/(b*d*(a-(a-b)*tanh(c+d*x)^2))+1/2*(a-b)*(2*a-b)*tanh(c+d*x)/(a*b^2*d*(a-(a-b)*tanh(c+d*x)^2))],
[cosh(c+d*x)^5/(a+b*sinh(c+d*x)^2)^2,x,5,-1/2*(3*a^2-2*a*b-b^2)*arctan(sinh(c+d*x)*sqrt(b)/sqrt(a))/(a^(3/2)*b^(5/2)*d)+sinh(c+d*x)/(b^2*d)+1/2*(a-b)^2*sinh(c+d*x)/(a*b^2*d*(a+b*sinh(c+d*x)^2))],
[cosh(c+d*x)^4/(a+b*sinh(c+d*x)^2)^2,x,5,x/b^2-1/2*(2*a+b)*arctanh(sqrt(a-b)*tanh(c+d*x)/sqrt(a))*sqrt(a-b)/(a^(3/2)*b^2*d)-1/2*(a-b)*tanh(c+d*x)/(a*b*d*(a-(a-b)*tanh(c+d*x)^2))],
[cosh(c+d*x)^3/(a+b*sinh(c+d*x)^2)^2,x,3,1/2*(a+b)*arctan(sinh(c+d*x)*sqrt(b)/sqrt(a))/(a^(3/2)*b^(3/2)*d)-1/2*(a-b)*sinh(c+d*x)/(a*b*d*(a+b*sinh(c+d*x)^2))],
[cosh(c+d*x)^2/(a+b*sinh(c+d*x)^2)^2,x,3,1/2*arctanh(sqrt(a-b)*tanh(c+d*x)/sqrt(a))/(a^(3/2)*d*sqrt(a-b))+1/2*tanh(c+d*x)/(a*d*(a-(a-b)*tanh(c+d*x)^2))],
[cosh(c+d*x)/(a+b*sinh(c+d*x)^2)^2,x,3,1/2*sinh(c+d*x)/(a*d*(a+b*sinh(c+d*x)^2))+1/2*arctan(sinh(c+d*x)*sqrt(b)/sqrt(a))/(a^(3/2)*d*sqrt(b))],
[sech(c+d*x)/(a+b*sinh(c+d*x)^2)^2,x,5,arctan(sinh(c+d*x))/((a-b)^2*d)-1/2*b*sinh(c+d*x)/(a*(a-b)*d*(a+b*sinh(c+d*x)^2))-1/2*(3*a-b)*arctan(sinh(c+d*x)*sqrt(b)/sqrt(a))*sqrt(b)/(a^(3/2)*(a-b)^2*d)],
[sech(c+d*x)^2/(a+b*sinh(c+d*x)^2)^2,x,5,-1/2*(4*a-b)*b*arctanh(sqrt(a-b)*tanh(c+d*x)/sqrt(a))/(a^(3/2)*(a-b)^(5/2)*d)+tanh(c+d*x)/((a-b)^2*d)+1/2*b^2*tanh(c+d*x)/(a*(a-b)^2*d*(a-(a-b)*tanh(c+d*x)^2))],
[sech(c+d*x)^3/(a+b*sinh(c+d*x)^2)^2,x,6,1/2*(a-5*b)*arctan(sinh(c+d*x))/((a-b)^3*d)+1/2*(5*a-b)*b^(3/2)*arctan(sinh(c+d*x)*sqrt(b)/sqrt(a))/(a^(3/2)*(a-b)^3*d)+1/2*b*(a+b)*sinh(c+d*x)/(a*(a-b)^2*d*(a+b*sinh(c+d*x)^2))+1/2*sech(c+d*x)*tanh(c+d*x)/((a-b)*d*(a+b*sinh(c+d*x)^2))],
[sech(c+d*x)^4/(a+b*sinh(c+d*x)^2)^2,x,5,1/2*(6*a-b)*b^2*arctanh(sqrt(a-b)*tanh(c+d*x)/sqrt(a))/(a^(3/2)*(a-b)^(7/2)*d)+(a-3*b)*tanh(c+d*x)/((a-b)^3*d)-1/3*tanh(c+d*x)^3/((a-b)^2*d)-1/2*b^3*tanh(c+d*x)/(a*(a-b)^3*d*(a-(a-b)*tanh(c+d*x)^2))],
[cosh(c+d*x)^6/(a+b*sinh(c+d*x)^2)^3,x,6,x/b^3-1/8*(8*a^2+4*a*b+3*b^2)*arctanh(sqrt(a-b)*tanh(c+d*x)/sqrt(a))*sqrt(a-b)/(a^(5/2)*b^3*d)-1/4*(a-b)*tanh(c+d*x)/(a*b*d*(a-(a-b)*tanh(c+d*x)^2)^2)-1/8*(a-b)*(4*a+3*b)*tanh(c+d*x)/(a^2*b^2*d*(a-(a-b)*tanh(c+d*x)^2))],
[cosh(c+d*x)^5/(a+b*sinh(c+d*x)^2)^3,x,4,1/8*(3*a^2+2*a*b+3*b^2)*arctan(sinh(c+d*x)*sqrt(b)/sqrt(a))/(a^(5/2)*b^(5/2)*d)-1/4*(a-b)*cosh(c+d*x)^2*sinh(c+d*x)/(a*b*d*(a+b*sinh(c+d*x)^2)^2)+3/8*(1/a^2+(-1)/b^2)*sinh(c+d*x)/(d*(a+b*sinh(c+d*x)^2))],
[cosh(c+d*x)^4/(a+b*sinh(c+d*x)^2)^3,x,4,3/8*arctanh(sqrt(a-b)*tanh(c+d*x)/sqrt(a))/(a^(5/2)*d*sqrt(a-b))+1/4*tanh(c+d*x)/(a*d*(a-(a-b)*tanh(c+d*x)^2)^2)+3/8*tanh(c+d*x)/(a^2*d*(a-(a-b)*tanh(c+d*x)^2))],
[cosh(c+d*x)^3/(a+b*sinh(c+d*x)^2)^3,x,4,1/8*(a+3*b)*arctan(sinh(c+d*x)*sqrt(b)/sqrt(a))/(a^(5/2)*b^(3/2)*d)-1/4*(a-b)*sinh(c+d*x)/(a*b*d*(a+b*sinh(c+d*x)^2)^2)+1/8*(a+3*b)*sinh(c+d*x)/(a^2*b*d*(a+b*sinh(c+d*x)^2))],
[cosh(c+d*x)^2/(a+b*sinh(c+d*x)^2)^3,x,4,1/8*(4*a-3*b)*arctanh(sqrt(a-b)*tanh(c+d*x)/sqrt(a))/(a^(5/2)*(a-b)^(3/2)*d)-1/4*b*tanh(c+d*x)/(a*(a-b)*d*(a-(a-b)*tanh(c+d*x)^2)^2)+1/8*(4*a-3*b)*tanh(c+d*x)/(a^2*(a-b)*d*(a-(a-b)*tanh(c+d*x)^2))],
[cosh(c+d*x)/(a+b*sinh(c+d*x)^2)^3,x,4,1/4*sinh(c+d*x)/(a*d*(a+b*sinh(c+d*x)^2)^2)+3/8*sinh(c+d*x)/(a^2*d*(a+b*sinh(c+d*x)^2))+3/8*arctan(sinh(c+d*x)*sqrt(b)/sqrt(a))/(a^(5/2)*d*sqrt(b))],
[sech(c+d*x)/(a+b*sinh(c+d*x)^2)^3,x,6,arctan(sinh(c+d*x))/((a-b)^3*d)-1/4*b*sinh(c+d*x)/(a*(a-b)*d*(a+b*sinh(c+d*x)^2)^2)-1/8*(7*a-3*b)*b*sinh(c+d*x)/(a^2*(a-b)^2*d*(a+b*sinh(c+d*x)^2))-1/8*(15*a^2-10*a*b+3*b^2)*arctan(sinh(c+d*x)*sqrt(b)/sqrt(a))*sqrt(b)/(a^(5/2)*(a-b)^3*d)],
[sech(c+d*x)^2/(a+b*sinh(c+d*x)^2)^3,x,6,-3/8*b*(8*a^2-4*a*b+b^2)*arctanh(sqrt(a-b)*tanh(c+d*x)/sqrt(a))/(a^(5/2)*(a-b)^(7/2)*d)+tanh(c+d*x)/((a-b)^3*d)-1/4*b^3*tanh(c+d*x)/(a*(a-b)^3*d*(a-(a-b)*tanh(c+d*x)^2)^2)+3/8*(4*a-b)*b^2*tanh(c+d*x)/(a^2*(a-b)^3*d*(a-(a-b)*tanh(c+d*x)^2))],
[sech(c+d*x)^3/(a+b*sinh(c+d*x)^2)^3,x,7,1/2*(a-7*b)*arctan(sinh(c+d*x))/((a-b)^4*d)+1/8*b^(3/2)*(35*a^2-14*a*b+3*b^2)*arctan(sinh(c+d*x)*sqrt(b)/sqrt(a))/(a^(5/2)*(a-b)^4*d)+1/4*b*(2*a+b)*sinh(c+d*x)/(a*(a-b)^2*d*(a+b*sinh(c+d*x)^2)^2)+1/8*(4*a-b)*b*(a+3*b)*sinh(c+d*x)/(a^2*(a-b)^3*d*(a+b*sinh(c+d*x)^2))+1/2*sech(c+d*x)*tanh(c+d*x)/((a-b)*d*(a+b*sinh(c+d*x)^2)^2)],
[sech(c+d*x)^4/(a+b*sinh(c+d*x)^2)^3,x,6,1/8*b^2*(48*a^2-16*a*b+3*b^2)*arctanh(sqrt(a-b)*tanh(c+d*x)/sqrt(a))/(a^(5/2)*(a-b)^(9/2)*d)+(a-4*b)*tanh(c+d*x)/((a-b)^4*d)-1/3*tanh(c+d*x)^3/((a-b)^3*d)+1/4*b^4*tanh(c+d*x)/(a*(a-b)^4*d*(a-(a-b)*tanh(c+d*x)^2)^2)-1/8*(16*a-3*b)*b^3*tanh(c+d*x)/(a^2*(a-b)^4*d*(a-(a-b)*tanh(c+d*x)^2))],
[cosh(x)^2/(1-sinh(x)^2),x,4,-x+arctanh(sqrt(2)*tanh(x))*sqrt(2)],
[cosh(x)^3/(1-sinh(x)^2),x,3,2*arctanh(sinh(x))-sinh(x)],
[cosh(x)^4/(1-sinh(x)^2),x,5,-5/2*x-1/2*cosh(x)*sinh(x)+2*arctanh(sqrt(2)*tanh(x))*sqrt(2)],

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

# p>0
[cosh(e+f*x)^3*sqrt(a+b*sinh(e+f*x)^2),x,5,-1/8*a*(a-4*b)*arctanh(sinh(e+f*x)*sqrt(b)/sqrt(a+b*sinh(e+f*x)^2))/(b^(3/2)*f)+1/4*sinh(e+f*x)*(a+b*sinh(e+f*x)^2)^(3/2)/(b*f)-1/8*(a-4*b)*sinh(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(b*f)],
[cosh(e+f*x)*sqrt(a+b*sinh(e+f*x)^2),x,4,1/2*a*arctanh(sinh(e+f*x)*sqrt(b)/sqrt(a+b*sinh(e+f*x)^2))/(f*sqrt(b))+1/2*sinh(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/f],
[sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2),x,6,arctan(sinh(e+f*x)*sqrt(a-b)/sqrt(a+b*sinh(e+f*x)^2))*sqrt(a-b)/f+arctanh(sinh(e+f*x)*sqrt(b)/sqrt(a+b*sinh(e+f*x)^2))*sqrt(b)/f],
[sech(e+f*x)^3*sqrt(a+b*sinh(e+f*x)^2),x,4,1/2*a*arctan(sinh(e+f*x)*sqrt(a-b)/sqrt(a+b*sinh(e+f*x)^2))/(f*sqrt(a-b))+1/2*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)*tanh(e+f*x)/f],
[sech(e+f*x)^5*sqrt(a+b*sinh(e+f*x)^2),x,5,1/8*a*(3*a-4*b)*arctan(sinh(e+f*x)*sqrt(a-b)/sqrt(a+b*sinh(e+f*x)^2))/((a-b)^(3/2)*f)+1/4*sech(e+f*x)^3*(a+b*sinh(e+f*x)^2)^(3/2)*tanh(e+f*x)/((a-b)*f)+1/8*(3*a-4*b)*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)*tanh(e+f*x)/((a-b)*f)],
[cosh(e+f*x)^4*sqrt(a+b*sinh(e+f*x)^2),x,7,1/5*cosh(e+f*x)*sinh(e+f*x)*(a+b*sinh(e+f*x)^2)^(3/2)/(b*f)-2/15*(a-3*b)*cosh(e+f*x)*sinh(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(b*f)+1/15*(2*a^2-7*a*b-3*b^2)*sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticE(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(b^2*f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))-1/15*(a-9*b)*sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticF(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(b*f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))-1/15*(2*a^2-7*a*b-3*b^2)*sqrt(a+b*sinh(e+f*x)^2)*tanh(e+f*x)/(b^2*f)],
[cosh(e+f*x)^2*sqrt(a+b*sinh(e+f*x)^2),x,6,1/3*cosh(e+f*x)*sinh(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/f-1/3*(a+b)*sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticE(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(b*f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))+2/3*sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticF(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))+1/3*(a+b)*sqrt(a+b*sinh(e+f*x)^2)*tanh(e+f*x)/(b*f)],
[sqrt(a+b*sinh(e+f*x)^2),x,2,-I*sqrt(cos(I*e+I*f*x)^2)/cos(I*e+I*f*x)*EllipticE(sin(I*e+I*f*x),sqrt(b/a))*sqrt(a+b*sinh(e+f*x)^2)/(f*sqrt(1+b*sinh(e+f*x)^2/a))],
[sech(e+f*x)^2*sqrt(a+b*sinh(e+f*x)^2),x,2,sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticE(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))],
[sech(e+f*x)^4*sqrt(a+b*sinh(e+f*x)^2),x,5,1/3*(2*a-b)*sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticE(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/((a-b)*f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))-1/3*b*sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticF(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/((a-b)*f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))+1/3*sech(e+f*x)^2*sqrt(a+b*sinh(e+f*x)^2)*tanh(e+f*x)/f],
[cosh(e+f*x)^3*(a+b*sinh(e+f*x)^2)^(3/2),x,6,-1/16*a^2*(a-6*b)*arctanh(sinh(e+f*x)*sqrt(b)/sqrt(a+b*sinh(e+f*x)^2))/(b^(3/2)*f)-1/24*(a-6*b)*sinh(e+f*x)*(a+b*sinh(e+f*x)^2)^(3/2)/(b*f)+1/6*sinh(e+f*x)*(a+b*sinh(e+f*x)^2)^(5/2)/(b*f)-1/16*a*(a-6*b)*sinh(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(b*f)],
[cosh(e+f*x)*(a+b*sinh(e+f*x)^2)^(3/2),x,5,1/4*sinh(e+f*x)*(a+b*sinh(e+f*x)^2)^(3/2)/f+3/8*a^2*arctanh(sinh(e+f*x)*sqrt(b)/sqrt(a+b*sinh(e+f*x)^2))/(f*sqrt(b))+3/8*a*sinh(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/f],
[sech(e+f*x)*(a+b*sinh(e+f*x)^2)^(3/2),x,7,(a-b)^(3/2)*arctan(sinh(e+f*x)*sqrt(a-b)/sqrt(a+b*sinh(e+f*x)^2))/f+1/2*(3*a-2*b)*arctanh(sinh(e+f*x)*sqrt(b)/sqrt(a+b*sinh(e+f*x)^2))*sqrt(b)/f+1/2*b*sinh(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/f],
[sech(e+f*x)^3*(a+b*sinh(e+f*x)^2)^(3/2),x,7,b^(3/2)*arctanh(sinh(e+f*x)*sqrt(b)/sqrt(a+b*sinh(e+f*x)^2))/f+1/2*(a+2*b)*arctan(sinh(e+f*x)*sqrt(a-b)/sqrt(a+b*sinh(e+f*x)^2))*sqrt(a-b)/f+1/2*(a-b)*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)*tanh(e+f*x)/f],
[sech(e+f*x)^5*(a+b*sinh(e+f*x)^2)^(3/2),x,5,3/8*a^2*arctan(sinh(e+f*x)*sqrt(a-b)/sqrt(a+b*sinh(e+f*x)^2))/(f*sqrt(a-b))+1/4*sech(e+f*x)^3*(a+b*sinh(e+f*x)^2)^(3/2)*tanh(e+f*x)/f+3/8*a*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)*tanh(e+f*x)/f],
[sech(e+f*x)^7*(a+b*sinh(e+f*x)^2)^(3/2),x,6,1/16*a^2*(5*a-6*b)*arctan(sinh(e+f*x)*sqrt(a-b)/sqrt(a+b*sinh(e+f*x)^2))/((a-b)^(3/2)*f)+1/24*(5*a-6*b)*sech(e+f*x)^3*(a+b*sinh(e+f*x)^2)^(3/2)*tanh(e+f*x)/((a-b)*f)+1/6*sech(e+f*x)^5*(a+b*sinh(e+f*x)^2)^(5/2)*tanh(e+f*x)/((a-b)*f)+1/16*a*(5*a-6*b)*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)*tanh(e+f*x)/((a-b)*f)],
[cosh(e+f*x)^4*(a+b*sinh(e+f*x)^2)^(3/2),x,8,1/35*(a^2+9*a*b-2*b^2)*cosh(e+f*x)*sinh(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(b*f)+2/35*(4*a-b)*cosh(e+f*x)^3*sinh(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/f+1/7*b*cosh(e+f*x)^5*sinh(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/f+2/35*(a+b)*(a^2-6*a*b+b^2)*sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticE(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(b^2*f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))-1/35*(a^2-18*a*b+b^2)*sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticF(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(b*f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))-2/35*(a+b)*(a^2-6*a*b+b^2)*sqrt(a+b*sinh(e+f*x)^2)*tanh(e+f*x)/(b^2*f)],
[cosh(e+f*x)^2*(a+b*sinh(e+f*x)^2)^(3/2),x,7,2/15*(3*a-b)*cosh(e+f*x)*sinh(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/f+1/5*b*cosh(e+f*x)^3*sinh(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/f-1/15*(3*a^2+7*a*b-2*b^2)*sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticE(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(b*f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))+1/15*(9*a-b)*sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticF(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))+1/15*(3*a^2+7*a*b-2*b^2)*sqrt(a+b*sinh(e+f*x)^2)*tanh(e+f*x)/(b*f)],
[(a+b*sinh(e+f*x)^2)^(3/2),x,6,1/3*b*cosh(e+f*x)*sinh(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/f-2/3*I*(2*a-b)*sqrt(cos(I*e+I*f*x)^2)/cos(I*e+I*f*x)*EllipticE(sin(I*e+I*f*x),sqrt(b/a))*sqrt(a+b*sinh(e+f*x)^2)/(f*sqrt(1+b*sinh(e+f*x)^2/a))+1/3*I*a*(a-b)*sqrt(cos(I*e+I*f*x)^2)/cos(I*e+I*f*x)*EllipticF(sin(I*e+I*f*x),sqrt(b/a))*sqrt(1+b*sinh(e+f*x)^2/a)/(f*sqrt(a+b*sinh(e+f*x)^2))],
[sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)^(3/2),x,6,(a-2*b)*sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticE(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))+b*sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticF(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))-(a-2*b)*sqrt(a+b*sinh(e+f*x)^2)*tanh(e+f*x)/f+(a-b)*sqrt(a+b*sinh(e+f*x)^2)*tanh(e+f*x)/f],
[sech(e+f*x)^4*(a+b*sinh(e+f*x)^2)^(3/2),x,5,2/3*(a+b)*sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticE(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))-1/3*b*sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticF(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))+1/3*(a-b)*sech(e+f*x)^2*sqrt(a+b*sinh(e+f*x)^2)*tanh(e+f*x)/f],

# p<0
[cosh(e+f*x)^3/sqrt(a+b*sinh(e+f*x)^2),x,4,-1/2*(a-2*b)*arctanh(sinh(e+f*x)*sqrt(b)/sqrt(a+b*sinh(e+f*x)^2))/(b^(3/2)*f)+1/2*sinh(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(b*f)],
[cosh(e+f*x)/sqrt(a+b*sinh(e+f*x)^2),x,3,arctanh(sinh(e+f*x)*sqrt(b)/sqrt(a+b*sinh(e+f*x)^2))/(f*sqrt(b))],
[sech(e+f*x)/sqrt(a+b*sinh(e+f*x)^2),x,3,arctan(sinh(e+f*x)*sqrt(a-b)/sqrt(a+b*sinh(e+f*x)^2))/(f*sqrt(a-b))],
[sech(e+f*x)^3/sqrt(a+b*sinh(e+f*x)^2),x,4,1/2*(a-2*b)*arctan(sinh(e+f*x)*sqrt(a-b)/sqrt(a+b*sinh(e+f*x)^2))/((a-b)^(3/2)*f)+1/2*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)*tanh(e+f*x)/((a-b)*f)],
[cosh(e+f*x)^4/sqrt(a+b*sinh(e+f*x)^2),x,6,1/3*cosh(e+f*x)*sinh(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(b*f)+2/3*(a-2*b)*sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticE(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(b^2*f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))-1/3*(a-3*b)*sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticF(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(a*b*f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))-2/3*(a-2*b)*sqrt(a+b*sinh(e+f*x)^2)*tanh(e+f*x)/(b^2*f)],
[cosh(e+f*x)^2/sqrt(a+b*sinh(e+f*x)^2),x,5,-sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticE(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(b*f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))+sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticF(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(a*f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))+sqrt(a+b*sinh(e+f*x)^2)*tanh(e+f*x)/(b*f)],
[1/sqrt(a+b*sinh(e+f*x)^2),x,2,-I*sqrt(cos(I*e+I*f*x)^2)/cos(I*e+I*f*x)*EllipticF(sin(I*e+I*f*x),sqrt(b/a))*sqrt(1+b*sinh(e+f*x)^2/a)/(f*sqrt(a+b*sinh(e+f*x)^2))],
[sech(e+f*x)^2/sqrt(a+b*sinh(e+f*x)^2),x,7,sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticE(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/((a-b)*f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))-b*sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticF(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(a*(a-b)*f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))],
[sech(e+f*x)^4/sqrt(a+b*sinh(e+f*x)^2),x,5,2/3*(a-2*b)*sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticE(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/((a-b)^2*f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))-1/3*(a-3*b)*b*sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticF(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(a*(a-b)^2*f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))+1/3*sech(e+f*x)^2*sqrt(a+b*sinh(e+f*x)^2)*tanh(e+f*x)/((a-b)*f)],
[cosh(e+f*x)^3/(a+b*sinh(e+f*x)^2)^(3/2),x,4,arctanh(sinh(e+f*x)*sqrt(b)/sqrt(a+b*sinh(e+f*x)^2))/(b^(3/2)*f)-(a-b)*sinh(e+f*x)/(a*b*f*sqrt(a+b*sinh(e+f*x)^2))],
[cosh(e+f*x)/(a+b*sinh(e+f*x)^2)^(3/2),x,2,sinh(e+f*x)/(a*f*sqrt(a+b*sinh(e+f*x)^2))],
[sech(e+f*x)/(a+b*sinh(e+f*x)^2)^(3/2),x,4,arctan(sinh(e+f*x)*sqrt(a-b)/sqrt(a+b*sinh(e+f*x)^2))/((a-b)^(3/2)*f)-b*sinh(e+f*x)/(a*(a-b)*f*sqrt(a+b*sinh(e+f*x)^2))],
[sech(e+f*x)^3/(a+b*sinh(e+f*x)^2)^(3/2),x,6,1/2*(a-4*b)*arctan(sinh(e+f*x)*sqrt(a-b)/sqrt(a+b*sinh(e+f*x)^2))/((a-b)^(5/2)*f)+1/2*b*(a+2*b)*sinh(e+f*x)/(a*(a-b)^2*f*sqrt(a+b*sinh(e+f*x)^2))+1/2*sech(e+f*x)*tanh(e+f*x)/((a-b)*f*sqrt(a+b*sinh(e+f*x)^2))],
[cosh(e+f*x)^6/(a+b*sinh(e+f*x)^2)^(3/2),x,7,-(a-b)*cosh(e+f*x)^3*sinh(e+f*x)/(a*b*f*sqrt(a+b*sinh(e+f*x)^2))+1/3*(4*a-3*b)*cosh(e+f*x)*sinh(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(a*b^2*f)+1/3*(8*a^2-13*a*b+3*b^2)*sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticE(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(a*b^3*f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))-2/3*(2*a-3*b)*sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticF(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(a*b^2*f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))-1/3*(8*a^2-13*a*b+3*b^2)*sqrt(a+b*sinh(e+f*x)^2)*tanh(e+f*x)/(a*b^3*f)],
[cosh(e+f*x)^4/(a+b*sinh(e+f*x)^2)^(3/2),x,6,-(a-b)*cosh(e+f*x)*sinh(e+f*x)/(a*b*f*sqrt(a+b*sinh(e+f*x)^2))-(2*a-b)*sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticE(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(a*b^2*f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))+sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticF(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(a*b*f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))+(2*a-b)*sqrt(a+b*sinh(e+f*x)^2)*tanh(e+f*x)/(a*b^2*f)],
[cosh(e+f*x)^2/(a+b*sinh(e+f*x)^2)^(3/2),x,2,cosh(e+f*x)*sqrt(cos(arctan(sinh(e+f*x)*sqrt(b)/sqrt(a)))^2)/cos(arctan(sinh(e+f*x)*sqrt(b)/sqrt(a)))*EllipticE(sin(arctan(sinh(e+f*x)*sqrt(b)/sqrt(a))),sqrt(1-a/b))/(f*sqrt(a)*sqrt(b)*sqrt(a*cosh(e+f*x)^2/(a+b*sinh(e+f*x)^2))*sqrt(a+b*sinh(e+f*x)^2))],
[1/(a+b*sinh(e+f*x)^2)^(3/2),x,4,-b*cosh(e+f*x)*sinh(e+f*x)/(a*(a-b)*f*sqrt(a+b*sinh(e+f*x)^2))-I*sqrt(cos(I*e+I*f*x)^2)/cos(I*e+I*f*x)*EllipticE(sin(I*e+I*f*x),sqrt(b/a))*sqrt(a+b*sinh(e+f*x)^2)/(a*(a-b)*f*sqrt(1+b*sinh(e+f*x)^2/a))],
[sech(e+f*x)^2/(a+b*sinh(e+f*x)^2)^(3/2),x,5,(a+b)*cosh(e+f*x)*sqrt(cos(arctan(sinh(e+f*x)*sqrt(b)/sqrt(a)))^2)/cos(arctan(sinh(e+f*x)*sqrt(b)/sqrt(a)))*EllipticE(sin(arctan(sinh(e+f*x)*sqrt(b)/sqrt(a))),sqrt(1-a/b))*sqrt(b)/((a-b)^2*f*sqrt(a)*sqrt(a*cosh(e+f*x)^2/(a+b*sinh(e+f*x)^2))*sqrt(a+b*sinh(e+f*x)^2))-2*b*sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticF(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(a*(a-b)^2*f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))+tanh(e+f*x)/((a-b)*f*sqrt(a+b*sinh(e+f*x)^2))],
[cosh(e+f*x)^5/(a+b*sinh(e+f*x)^2)^(5/2),x,5,arctanh(sinh(e+f*x)*sqrt(b)/sqrt(a+b*sinh(e+f*x)^2))/(b^(5/2)*f)-1/3*(a-b)*cosh(e+f*x)^2*sinh(e+f*x)/(a*b*f*(a+b*sinh(e+f*x)^2)^(3/2))-1/3*(a-b)*(3*a+2*b)*sinh(e+f*x)/(a^2*b^2*f*sqrt(a+b*sinh(e+f*x)^2))],
[cosh(e+f*x)^3/(a+b*sinh(e+f*x)^2)^(5/2),x,3,1/3*cosh(e+f*x)^2*sinh(e+f*x)/(a*f*(a+b*sinh(e+f*x)^2)^(3/2))+2/3*sinh(e+f*x)/(a^2*f*sqrt(a+b*sinh(e+f*x)^2))],
[cosh(e+f*x)/(a+b*sinh(e+f*x)^2)^(5/2),x,3,1/3*sinh(e+f*x)/(a*f*(a+b*sinh(e+f*x)^2)^(3/2))+2/3*sinh(e+f*x)/(a^2*f*sqrt(a+b*sinh(e+f*x)^2))],
[sech(e+f*x)/(a+b*sinh(e+f*x)^2)^(5/2),x,6,arctan(sinh(e+f*x)*sqrt(a-b)/sqrt(a+b*sinh(e+f*x)^2))/((a-b)^(5/2)*f)-1/3*b*sinh(e+f*x)/(a*(a-b)*f*(a+b*sinh(e+f*x)^2)^(3/2))-1/3*(5*a-2*b)*b*sinh(e+f*x)/(a^2*(a-b)^2*f*sqrt(a+b*sinh(e+f*x)^2))],
[cosh(e+f*x)^6/(a+b*sinh(e+f*x)^2)^(5/2),x,7,-1/3*(a-b)*cosh(e+f*x)^3*sinh(e+f*x)/(a*b*f*(a+b*sinh(e+f*x)^2)^(3/2))-2/3*(a-b)*(2*a+b)*cosh(e+f*x)*sinh(e+f*x)/(a^2*b^2*f*sqrt(a+b*sinh(e+f*x)^2))-1/3*(8*a^2-3*a*b-2*b^2)*sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticE(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(a^2*b^3*f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))+1/3*(4*a-b)*sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticF(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(a^2*b^2*f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))+1/3*(8*a^2-3*a*b-2*b^2)*sqrt(a+b*sinh(e+f*x)^2)*tanh(e+f*x)/(a^2*b^3*f)],
[cosh(e+f*x)^4/(a+b*sinh(e+f*x)^2)^(5/2),x,5,-1/3*(a-b)*cosh(e+f*x)*sinh(e+f*x)/(a*b*f*(a+b*sinh(e+f*x)^2)^(3/2))+2/3*(a+b)*cosh(e+f*x)*sqrt(cos(arctan(sinh(e+f*x)*sqrt(b)/sqrt(a)))^2)/cos(arctan(sinh(e+f*x)*sqrt(b)/sqrt(a)))*EllipticE(sin(arctan(sinh(e+f*x)*sqrt(b)/sqrt(a))),sqrt(1-a/b))/(a^(3/2)*b^(3/2)*f*sqrt(a*cosh(e+f*x)^2/(a+b*sinh(e+f*x)^2))*sqrt(a+b*sinh(e+f*x)^2))-1/3*sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticF(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(a^2*b*f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))],
[cosh(e+f*x)^2/(a+b*sinh(e+f*x)^2)^(5/2),x,5,1/3*cosh(e+f*x)*sinh(e+f*x)/(a*f*(a+b*sinh(e+f*x)^2)^(3/2))+1/3*(a-2*b)*cosh(e+f*x)*sqrt(cos(arctan(sinh(e+f*x)*sqrt(b)/sqrt(a)))^2)/cos(arctan(sinh(e+f*x)*sqrt(b)/sqrt(a)))*EllipticE(sin(arctan(sinh(e+f*x)*sqrt(b)/sqrt(a))),sqrt(1-a/b))/(a^(3/2)*(a-b)*f*sqrt(b)*sqrt(a*cosh(e+f*x)^2/(a+b*sinh(e+f*x)^2))*sqrt(a+b*sinh(e+f*x)^2))+1/3*sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticF(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(a^2*(a-b)*f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))],
[1/(a+b*sinh(e+f*x)^2)^(5/2),x,7,-1/3*b*cosh(e+f*x)*sinh(e+f*x)/(a*(a-b)*f*(a+b*sinh(e+f*x)^2)^(3/2))-2/3*(2*a-b)*b*cosh(e+f*x)*sinh(e+f*x)/(a^2*(a-b)^2*f*sqrt(a+b*sinh(e+f*x)^2))-2/3*I*(2*a-b)*sqrt(cos(I*e+I*f*x)^2)/cos(I*e+I*f*x)*EllipticE(sin(I*e+I*f*x),sqrt(b/a))*sqrt(a+b*sinh(e+f*x)^2)/(a^2*(a-b)^2*f*sqrt(1+b*sinh(e+f*x)^2/a))+1/3*I*sqrt(cos(I*e+I*f*x)^2)/cos(I*e+I*f*x)*EllipticF(sin(I*e+I*f*x),sqrt(b/a))*sqrt(1+b*sinh(e+f*x)^2/a)/(a*(a-b)*f*sqrt(a+b*sinh(e+f*x)^2))],
[sech(e+f*x)^2/(a+b*sinh(e+f*x)^2)^(5/2),x,6,1/3*b*(3*a+b)*cosh(e+f*x)*sinh(e+f*x)/(a*(a-b)^2*f*(a+b*sinh(e+f*x)^2)^(3/2))+1/3*(3*a^2+7*a*b-2*b^2)*cosh(e+f*x)*sqrt(cos(arctan(sinh(e+f*x)*sqrt(b)/sqrt(a)))^2)/cos(arctan(sinh(e+f*x)*sqrt(b)/sqrt(a)))*EllipticE(sin(arctan(sinh(e+f*x)*sqrt(b)/sqrt(a))),sqrt(1-a/b))*sqrt(b)/(a^(3/2)*(a-b)^3*f*sqrt(a*cosh(e+f*x)^2/(a+b*sinh(e+f*x)^2))*sqrt(a+b*sinh(e+f*x)^2))-1/3*(9*a-b)*b*sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticF(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(a^2*(a-b)^3*f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))+tanh(e+f*x)/((a-b)*f*(a+b*sinh(e+f*x)^2)^(3/2))],

# Integrands of the form (d Cosh[e+f x])^m (a+b Sinh[e+f x]^2)^p when p symbolic
[(d*cosh(e+f*x))^m*(a+b*sinh(e+f*x)^2)^p,x,3,d*AppellF1(1/2,1/2*(1-m),-p,3/2,-sinh(e+f*x)^2,-b*sinh(e+f*x)^2/a)*(d*cosh(e+f*x))^(-1+m)*(cosh(e+f*x)^2)^(1/2*(1-m))*sinh(e+f*x)*(a+b*sinh(e+f*x)^2)^p/(f*(1+b*sinh(e+f*x)^2/a)^p)],
[cosh(e+f*x)^5*(a+b*sinh(e+f*x)^2)^p,x,5,-(3*a-b*(7+2*p))*sinh(e+f*x)*(a+b*sinh(e+f*x)^2)^(1+p)/(b^2*f*(15+16*p+4*p^2))+cosh(e+f*x)^2*sinh(e+f*x)*(a+b*sinh(e+f*x)^2)^(1+p)/(b*f*(5+2*p))+(3*a^2-2*a*b*(5+2*p)+b^2*(15+16*p+4*p^2))*hypergeom([1/2,-p],[3/2],-b*sinh(e+f*x)^2/a)*sinh(e+f*x)*(a+b*sinh(e+f*x)^2)^p/(b^2*f*(15+16*p+4*p^2)*(1+b*sinh(e+f*x)^2/a)^p)],
[cosh(e+f*x)^3*(a+b*sinh(e+f*x)^2)^p,x,4,sinh(e+f*x)*(a+b*sinh(e+f*x)^2)^(1+p)/(b*f*(3+2*p))-(a-b*(3+2*p))*hypergeom([1/2,-p],[3/2],-b*sinh(e+f*x)^2/a)*sinh(e+f*x)*(a+b*sinh(e+f*x)^2)^p/(b*f*(3+2*p)*(1+b*sinh(e+f*x)^2/a)^p)],
[cosh(e+f*x)*(a+b*sinh(e+f*x)^2)^p,x,3,hypergeom([1/2,-p],[3/2],-b*sinh(e+f*x)^2/a)*sinh(e+f*x)*(a+b*sinh(e+f*x)^2)^p/(f*(1+b*sinh(e+f*x)^2/a)^p)],
[sech(e+f*x)*(a+b*sinh(e+f*x)^2)^p,x,3,AppellF1(1/2,1,-p,3/2,-sinh(e+f*x)^2,-b*sinh(e+f*x)^2/a)*sinh(e+f*x)*(a+b*sinh(e+f*x)^2)^p/(f*(1+b*sinh(e+f*x)^2/a)^p)],
[sech(e+f*x)^3*(a+b*sinh(e+f*x)^2)^p,x,3,AppellF1(1/2,2,-p,3/2,-sinh(e+f*x)^2,-b*sinh(e+f*x)^2/a)*sinh(e+f*x)*(a+b*sinh(e+f*x)^2)^p/(f*(1+b*sinh(e+f*x)^2/a)^p)],
[cosh(e+f*x)^4*(a+b*sinh(e+f*x)^2)^p,x,3,AppellF1(1/2,-3/2,-p,3/2,-sinh(e+f*x)^2,-b*sinh(e+f*x)^2/a)*(a+b*sinh(e+f*x)^2)^p*sqrt(cosh(e+f*x)^2)*tanh(e+f*x)/(f*(1+b*sinh(e+f*x)^2/a)^p)],
[cosh(e+f*x)^2*(a+b*sinh(e+f*x)^2)^p,x,3,AppellF1(1/2,-1/2,-p,3/2,-sinh(e+f*x)^2,-b*sinh(e+f*x)^2/a)*(a+b*sinh(e+f*x)^2)^p*sqrt(cosh(e+f*x)^2)*tanh(e+f*x)/(f*(1+b*sinh(e+f*x)^2/a)^p)],
[(a+b*sinh(e+f*x)^2)^p,x,3,AppellF1(1/2,1/2,-p,3/2,-sinh(e+f*x)^2,-b*sinh(e+f*x)^2/a)*(a+b*sinh(e+f*x)^2)^p*sqrt(cosh(e+f*x)^2)*tanh(e+f*x)/(f*(1+b*sinh(e+f*x)^2/a)^p)],
[sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)^p,x,3,AppellF1(1/2,3/2,-p,3/2,-sinh(e+f*x)^2,-b*sinh(e+f*x)^2/a)*(a+b*sinh(e+f*x)^2)^p*sqrt(cosh(e+f*x)^2)*tanh(e+f*x)/(f*(1+b*sinh(e+f*x)^2/a)^p)],
[sech(e+f*x)^4*(a+b*sinh(e+f*x)^2)^p,x,3,AppellF1(1/2,5/2,-p,3/2,-sinh(e+f*x)^2,-b*sinh(e+f*x)^2/a)*(a+b*sinh(e+f*x)^2)^p*sqrt(cosh(e+f*x)^2)*tanh(e+f*x)/(f*(1+b*sinh(e+f*x)^2/a)^p)],

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

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

# Integrands of the form Cosh[e+f x]^m (a+b Sinh[e+f x]^n)^p
[cosh(c+d*x)^5/(a+b*sqrt(sinh(c+d*x))),x,4,-2*a*(a^4+b^4)^2*log(a+b*sqrt(sinh(c+d*x)))/(b^10*d)-a^3*(a^4+2*b^4)*sinh(c+d*x)/(b^8*d)+2/3*a^2*(a^4+2*b^4)*sinh(c+d*x)^(3/2)/(b^7*d)-1/2*a*(a^4+2*b^4)*sinh(c+d*x)^2/(b^6*d)+2/5*(a^4+2*b^4)*sinh(c+d*x)^(5/2)/(b^5*d)-1/3*a^3*sinh(c+d*x)^3/(b^4*d)+2/7*a^2*sinh(c+d*x)^(7/2)/(b^3*d)-1/4*a*sinh(c+d*x)^4/(b^2*d)+2/9*sinh(c+d*x)^(9/2)/(b*d)+2*(a^4+b^4)^2*sqrt(sinh(c+d*x))/(b^9*d)],
[cosh(c+d*x)^3/(a+b*sqrt(sinh(c+d*x))),x,4,-2*a*(a^4+b^4)*log(a+b*sqrt(sinh(c+d*x)))/(b^6*d)-a^3*sinh(c+d*x)/(b^4*d)+2/3*a^2*sinh(c+d*x)^(3/2)/(b^3*d)-1/2*a*sinh(c+d*x)^2/(b^2*d)+2/5*sinh(c+d*x)^(5/2)/(b*d)+2*(a^4+b^4)*sqrt(sinh(c+d*x))/(b^5*d)],
[cosh(c+d*x)/(a+b*sqrt(sinh(c+d*x))),x,4,-2*a*log(a+b*sqrt(sinh(c+d*x)))/(b^2*d)+2*sqrt(sinh(c+d*x))/(b*d)],
[sech(c+d*x)/(a+b*sqrt(sinh(c+d*x))),x,19,a^3*arctan(sinh(c+d*x))/((a^4+b^4)*d)+a*b^2*log(cosh(c+d*x))/((a^4+b^4)*d)-2*a*b^2*log(a+b*sqrt(sinh(c+d*x)))/((a^4+b^4)*d)+b*(a^2-b^2)*arctan(1-sqrt(2)*sqrt(sinh(c+d*x)))/((a^4+b^4)*d*sqrt(2))-b*(a^2-b^2)*arctan(1+sqrt(2)*sqrt(sinh(c+d*x)))/((a^4+b^4)*d*sqrt(2))-1/2*b*(a^2+b^2)*log(1+sinh(c+d*x)-sqrt(2)*sqrt(sinh(c+d*x)))/((a^4+b^4)*d*sqrt(2))+1/2*b*(a^2+b^2)*log(1+sinh(c+d*x)+sqrt(2)*sqrt(sinh(c+d*x)))/((a^4+b^4)*d*sqrt(2))],

#  {Sech[c + d*x]^3/(a + b*Sqrt[Sinh[c + d*x]]), x, 00, (a^2*b^3*(a^2 + b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Sinh[c + d*x]]])/(Sqrt[2]*(a^4 + b^4)^2*d) + (b*(a^2 - 3*b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Sinh[c + d*x]]])/(4*Sqrt[2]*(a^4 + b^4)*d) - (a^2*b^3*(a^2 + b^2)*ArcTan[1 + Sqrt[2]*Sqrt[Sinh[c + d*x]]])/(Sqrt[2]*(a^4 + b^4)^2*d) - (b*(a^2 - 3*b^2)*ArcTan[1 + Sqrt[2]*Sqrt[Sinh[c + d*x]]])/(4*Sqrt[2]*(a^4 + b^4)*d) + (a^3*b^4*ArcTan[Sinh[c + d*x]])/((a^4 + b^4)^2*d) + (a^3*ArcTan[Sinh[c + d*x]])/(2*(a^4 + b^4)*d) - (a^2*b^3*(a^2 - b^2)*ArcTanh[(Sqrt[2]*Sqrt[Sinh[c + d*x]])/(1 + Sinh[c + d*x])])/(Sqrt[2]*(a^4 + b^4)^2*d) + (b*(a^2 + 3*b^2)*ArcTanh[(Sqrt[2]*Sqrt[Sinh[c + d*x]])/(1 + Sinh[c + d*x])])/(4*Sqrt[2]*(a^4 + b^4)*d) - (2*a*b^6*Log[a + b*Sqrt[Sinh[c + d*x]]])/((a^4 + b^4)^2*d) + (a*b^6*Log[1 + Sinh[c + d*x]^2])/(2*(a^4 + b^4)^2*d) - (a*(b^2 - a^2*Sinh[c + d*x]))/(2*(a^4 + b^4)*d*(1 + Sinh[c + d*x]^2)) + (b*Sqrt[Sinh[c + d*x]]*(b^2 - a^2*Sinh[c + d*x]))/(2*(a^4 + b^4)*d*(1 + Sinh[c + d*x]^2))} 
[cosh(c+d*x)^5/(a+b*sqrt(sinh(c+d*x)))^2,x,4,2*(a^4+b^4)*(9*a^4+b^4)*log(a+b*sqrt(sinh(c+d*x)))/(b^10*d)+a^2*(7*a^4+6*b^4)*sinh(c+d*x)/(b^8*d)-4/3*a*(3*a^4+2*b^4)*sinh(c+d*x)^(3/2)/(b^7*d)+1/2*(5*a^4+2*b^4)*sinh(c+d*x)^2/(b^6*d)-8/5*a^3*sinh(c+d*x)^(5/2)/(b^5*d)+a^2*sinh(c+d*x)^3/(b^4*d)-4/7*a*sinh(c+d*x)^(7/2)/(b^3*d)+1/4*sinh(c+d*x)^4/(b^2*d)-16*a^3*(a^4+b^4)*sqrt(sinh(c+d*x))/(b^9*d)+2*a*(a^4+b^4)^2/(b^10*d*(a+b*sqrt(sinh(c+d*x))))],
[cosh(c+d*x)^3/(a+b*sqrt(sinh(c+d*x)))^2,x,4,2*(5*a^4+b^4)*log(a+b*sqrt(sinh(c+d*x)))/(b^6*d)+3*a^2*sinh(c+d*x)/(b^4*d)-4/3*a*sinh(c+d*x)^(3/2)/(b^3*d)+1/2*sinh(c+d*x)^2/(b^2*d)-8*a^3*sqrt(sinh(c+d*x))/(b^5*d)+2*a*(a^4+b^4)/(b^6*d*(a+b*sqrt(sinh(c+d*x))))],
[cosh(c+d*x)/(a+b*sqrt(sinh(c+d*x)))^2,x,4,2*log(a+b*sqrt(sinh(c+d*x)))/(b^2*d)+2*a/(b^2*d*(a+b*sqrt(sinh(c+d*x))))],
[sech(c+d*x)/(a+b*sqrt(sinh(c+d*x)))^2,x,19,a^2*(a^4-3*b^4)*arctan(sinh(c+d*x))/((a^4+b^4)^2*d)+b^2*(3*a^4-b^4)*log(cosh(c+d*x))/((a^4+b^4)^2*d)-2*b^2*(3*a^4-b^4)*log(a+b*sqrt(sinh(c+d*x)))/((a^4+b^4)^2*d)-a*b*(a^4+2*a^2*b^2-b^4)*log(1+sinh(c+d*x)-sqrt(2)*sqrt(sinh(c+d*x)))/((a^4+b^4)^2*d*sqrt(2))+a*b*(a^4+2*a^2*b^2-b^4)*log(1+sinh(c+d*x)+sqrt(2)*sqrt(sinh(c+d*x)))/((a^4+b^4)^2*d*sqrt(2))+a*b*(a^4-2*a^2*b^2-b^4)*arctan(1-sqrt(2)*sqrt(sinh(c+d*x)))*sqrt(2)/((a^4+b^4)^2*d)-a*b*(a^4-2*a^2*b^2-b^4)*arctan(1+sqrt(2)*sqrt(sinh(c+d*x)))*sqrt(2)/((a^4+b^4)^2*d)+2*a*b^2/((a^4+b^4)*d*(a+b*sqrt(sinh(c+d*x))))],

#  {Sech[c + d*x]^3/(a + b*Sqrt[Sinh[c + d*x]])^2, x, 00, (a*b*(a^4 - 6*a^2*b^2 - b^4)*ArcTan[1 - Sqrt[2]*Sqrt[Sinh[c + d*x]]])/(2*Sqrt[2]*(a^4 + b^4)^2*d) + (Sqrt[2]*a*b^3*(2*a^6 + 3*a^4*b^2 - 2*a^2*b^4 - b^6)*ArcTan[1 - Sqrt[2]*Sqrt[Sinh[c + d*x]]])/((a^4 + b^4)^3*d) - (a*b*(a^4 - 6*a^2*b^2 - b^4)*ArcTan[1 + Sqrt[2]*Sqrt[Sinh[c + d*x]]])/(2*Sqrt[2]*(a^4 + b^4)^2*d) - (Sqrt[2]*a*b^3*(2*a^6 + 3*a^4*b^2 - 2*a^2*b^4 - b^6)*ArcTan[1 + Sqrt[2]*Sqrt[Sinh[c + d*x]]])/((a^4 + b^4)^3*d) + (a^2*b^4*(5*a^4 - 3*b^4)*ArcTan[Sinh[c + d*x]])/((a^4 + b^4)^3*d) + (a^2*(a^4 - 3*b^4)*ArcTan[Sinh[c + d*x]])/(2*(a^4 + b^4)^2*d) + (a*b*(a^4 + 6*a^2*b^2 - b^4)*ArcTanh[(Sqrt[2]*Sqrt[Sinh[c + d*x]])/(1 + Sinh[c + d*x])])/(2*Sqrt[2]*(a^4 + b^4)^2*d) - (Sqrt[2]*a*b^3*(2*a^6 - 3*a^4*b^2 - 2*a^2*b^4 + b^6)*ArcTanh[(Sqrt[2]*Sqrt[Sinh[c + d*x]])/(1 + Sinh[c + d*x])])/((a^4 + b^4)^3*d) - (2*b^6*(7*a^4 - b^4)*Log[a + b*Sqrt[Sinh[c + d*x]]])/((a^4 + b^4)^3*d) + (b^6*(7*a^4 - b^4)*Log[1 + Sinh[c + d*x]^2])/(2*(a^4 + b^4)^3*d) + (2*a*b^6)/((a^4 + b^4)^2*d*(a + b*Sqrt[Sinh[c + d*x]])) - (b^2*(3*a^4 - b^4) - a^2*(a^4 - 3*b^4)*Sinh[c + d*x])/(2*(a^4 + b^4)^2*d*(1 + Sinh[c + d*x]^2)) + (a*b*Sqrt[Sinh[c + d*x]]*(2*a^2*b^2 - (a^4 - b^4)*Sinh[c + d*x]))/((a^4 + b^4)^2*d*(1 + Sinh[c + d*x]^2))} 
[cosh(c+d*x)^5/(a+b*sinh(c+d*x)^n),x,6,hypergeom([1,1/n],[1+1/n],-b*sinh(c+d*x)^n/a)*sinh(c+d*x)/(a*d)+2/3*hypergeom([1,3/n],[(3+n)/n],-b*sinh(c+d*x)^n/a)*sinh(c+d*x)^3/(a*d)+1/5*hypergeom([1,5/n],[(5+n)/n],-b*sinh(c+d*x)^n/a)*sinh(c+d*x)^5/(a*d)],
[cosh(c+d*x)^3/(a+b*sinh(c+d*x)^n),x,5,hypergeom([1,1/n],[1+1/n],-b*sinh(c+d*x)^n/a)*sinh(c+d*x)/(a*d)+1/3*hypergeom([1,3/n],[(3+n)/n],-b*sinh(c+d*x)^n/a)*sinh(c+d*x)^3/(a*d)],
[cosh(c+d*x)/(a+b*sinh(c+d*x)^n),x,2,hypergeom([1,1/n],[1+1/n],-b*sinh(c+d*x)^n/a)*sinh(c+d*x)/(a*d)],
[cosh(c+d*x)^5/(a+b*sinh(c+d*x)^n)^2,x,6,hypergeom([2,1/n],[1+1/n],-b*sinh(c+d*x)^n/a)*sinh(c+d*x)/(a^2*d)+2/3*hypergeom([2,3/n],[(3+n)/n],-b*sinh(c+d*x)^n/a)*sinh(c+d*x)^3/(a^2*d)+1/5*hypergeom([2,5/n],[(5+n)/n],-b*sinh(c+d*x)^n/a)*sinh(c+d*x)^5/(a^2*d)],
[cosh(c+d*x)^3/(a+b*sinh(c+d*x)^n)^2,x,5,hypergeom([2,1/n],[1+1/n],-b*sinh(c+d*x)^n/a)*sinh(c+d*x)/(a^2*d)+1/3*hypergeom([2,3/n],[(3+n)/n],-b*sinh(c+d*x)^n/a)*sinh(c+d*x)^3/(a^2*d)],
[cosh(c+d*x)/(a+b*sinh(c+d*x)^n)^2,x,2,hypergeom([2,1/n],[1+1/n],-b*sinh(c+d*x)^n/a)*sinh(c+d*x)/(a^2*d)],

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

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

# Integrands of the form Tanh[e+f x]^m (a+b Sinh[e+f x]^2)^p
[coth(x)/(1-sinh(x)^2),x,4,log(sinh(x))-1/2*log(1-sinh(x)^2)],

# Integrands of the form Tanh[e+f x]^m (a+b Sinh[e+f x]^2)^(p/2) when a-b=0

# p>0
[sqrt(a+a*sinh(e+f*x)^2)*tanh(e+f*x)^5,x,5,-1/3*a^2/(f*(a*cosh(e+f*x)^2)^(3/2))+2*a/(f*sqrt(a*cosh(e+f*x)^2))+sqrt(a*cosh(e+f*x)^2)/f],
[sqrt(a+a*sinh(e+f*x)^2)*tanh(e+f*x)^3,x,5,a/(f*sqrt(a*cosh(e+f*x)^2))+sqrt(a*cosh(e+f*x)^2)/f],
[sqrt(a+a*sinh(e+f*x)^2)*tanh(e+f*x),x,4,sqrt(a*cosh(e+f*x)^2)/f],
[coth(e+f*x)*sqrt(a+a*sinh(e+f*x)^2),x,5,-arctanh(sqrt(a*cosh(e+f*x)^2)/sqrt(a))*sqrt(a)/f+sqrt(a*cosh(e+f*x)^2)/f],
[coth(e+f*x)^3*sqrt(a+a*sinh(e+f*x)^2),x,7,-1/2*(a*cosh(e+f*x)^2)^(3/2)*csch(e+f*x)^2/(a*f)-3/2*arctanh(sqrt(a*cosh(e+f*x)^2)/sqrt(a))*sqrt(a)/f+3/2*sqrt(a*cosh(e+f*x)^2)/f],
[sqrt(a+a*sinh(e+f*x)^2)*tanh(e+f*x)^6,x,7,-15/8*arctan(sinh(e+f*x))*sech(e+f*x)*sqrt(a*cosh(e+f*x)^2)/f+15/8*sqrt(a*cosh(e+f*x)^2)*tanh(e+f*x)/f-5/8*sqrt(a*cosh(e+f*x)^2)*tanh(e+f*x)^3/f-1/4*sqrt(a*cosh(e+f*x)^2)*tanh(e+f*x)^5/f],
[sqrt(a+a*sinh(e+f*x)^2)*tanh(e+f*x)^4,x,6,-3/2*arctan(sinh(e+f*x))*sech(e+f*x)*sqrt(a*cosh(e+f*x)^2)/f+3/2*sqrt(a*cosh(e+f*x)^2)*tanh(e+f*x)/f-1/2*sqrt(a*cosh(e+f*x)^2)*tanh(e+f*x)^3/f],
[sqrt(a+a*sinh(e+f*x)^2)*tanh(e+f*x)^2,x,5,-arctan(sinh(e+f*x))*sech(e+f*x)*sqrt(a*cosh(e+f*x)^2)/f+sqrt(a*cosh(e+f*x)^2)*tanh(e+f*x)/f],
[coth(e+f*x)^2*sqrt(a+a*sinh(e+f*x)^2),x,5,-csch(e+f*x)*sech(e+f*x)*sqrt(a*cosh(e+f*x)^2)/f+sqrt(a*cosh(e+f*x)^2)*tanh(e+f*x)/f],
[coth(e+f*x)^4*sqrt(a+a*sinh(e+f*x)^2),x,5,-2*csch(e+f*x)*sech(e+f*x)*sqrt(a*cosh(e+f*x)^2)/f-1/3*csch(e+f*x)^3*sech(e+f*x)*sqrt(a*cosh(e+f*x)^2)/f+sqrt(a*cosh(e+f*x)^2)*tanh(e+f*x)/f],
[coth(e+f*x)^6*sqrt(a+a*sinh(e+f*x)^2),x,5,-3*csch(e+f*x)*sech(e+f*x)*sqrt(a*cosh(e+f*x)^2)/f-csch(e+f*x)^3*sech(e+f*x)*sqrt(a*cosh(e+f*x)^2)/f-1/5*csch(e+f*x)^5*sech(e+f*x)*sqrt(a*cosh(e+f*x)^2)/f+sqrt(a*cosh(e+f*x)^2)*tanh(e+f*x)/f],

# p<0
[tanh(e+f*x)^5/sqrt(a+a*sinh(e+f*x)^2),x,5,-1/5*a^2/(f*(a*cosh(e+f*x)^2)^(5/2))+2/3*a/(f*(a*cosh(e+f*x)^2)^(3/2))+(-1)/(f*sqrt(a*cosh(e+f*x)^2))],
[tanh(e+f*x)^3/sqrt(a+a*sinh(e+f*x)^2),x,5,1/3*a/(f*(a*cosh(e+f*x)^2)^(3/2))+(-1)/(f*sqrt(a*cosh(e+f*x)^2))],
[tanh(e+f*x)/sqrt(a+a*sinh(e+f*x)^2),x,4,(-1)/(f*sqrt(a*cosh(e+f*x)^2))],
[coth(e+f*x)/sqrt(a+a*sinh(e+f*x)^2),x,4,-arctanh(sqrt(a*cosh(e+f*x)^2)/sqrt(a))/(f*sqrt(a))],
[coth(e+f*x)^3/sqrt(a+a*sinh(e+f*x)^2),x,6,-1/2*arctanh(sqrt(a*cosh(e+f*x)^2)/sqrt(a))/(f*sqrt(a))-1/2*csch(e+f*x)^2*sqrt(a*cosh(e+f*x)^2)/(a*f)],
[tanh(e+f*x)^4/sqrt(a+a*sinh(e+f*x)^2),x,5,3/8*arctan(sinh(e+f*x))*cosh(e+f*x)/(f*sqrt(a*cosh(e+f*x)^2))-3/8*tanh(e+f*x)/(f*sqrt(a*cosh(e+f*x)^2))-1/4*tanh(e+f*x)^3/(f*sqrt(a*cosh(e+f*x)^2))],
[tanh(e+f*x)^2/sqrt(a+a*sinh(e+f*x)^2),x,4,1/2*arctan(sinh(e+f*x))*cosh(e+f*x)/(f*sqrt(a*cosh(e+f*x)^2))-1/2*tanh(e+f*x)/(f*sqrt(a*cosh(e+f*x)^2))],
[coth(e+f*x)^2/sqrt(a+a*sinh(e+f*x)^2),x,4,-coth(e+f*x)/(f*sqrt(a*cosh(e+f*x)^2))],
[coth(e+f*x)^4/sqrt(a+a*sinh(e+f*x)^2),x,4,-coth(e+f*x)/(f*sqrt(a*cosh(e+f*x)^2))-1/3*coth(e+f*x)*csch(e+f*x)^2/(f*sqrt(a*cosh(e+f*x)^2))],
[coth(e+f*x)^6/sqrt(a+a*sinh(e+f*x)^2),x,5,-coth(e+f*x)/(f*sqrt(a*cosh(e+f*x)^2))-2/3*coth(e+f*x)*csch(e+f*x)^2/(f*sqrt(a*cosh(e+f*x)^2))-1/5*coth(e+f*x)*csch(e+f*x)^4/(f*sqrt(a*cosh(e+f*x)^2))],
[tanh(e+f*x)^5/(a+a*sinh(e+f*x)^2)^(3/2),x,5,-1/7*a^2/(f*(a*cosh(e+f*x)^2)^(7/2))+2/5*a/(f*(a*cosh(e+f*x)^2)^(5/2))+(-1/3)/(f*(a*cosh(e+f*x)^2)^(3/2))],
[tanh(e+f*x)^3/(a+a*sinh(e+f*x)^2)^(3/2),x,5,1/5*a/(f*(a*cosh(e+f*x)^2)^(5/2))+(-1/3)/(f*(a*cosh(e+f*x)^2)^(3/2))],
[tanh(e+f*x)/(a+a*sinh(e+f*x)^2)^(3/2),x,4,(-1/3)/(f*(a*cosh(e+f*x)^2)^(3/2))],
[coth(e+f*x)/(a+a*sinh(e+f*x)^2)^(3/2),x,5,-arctanh(sqrt(a*cosh(e+f*x)^2)/sqrt(a))/(a^(3/2)*f)+1/(a*f*sqrt(a*cosh(e+f*x)^2))],
[coth(e+f*x)^3/(a+a*sinh(e+f*x)^2)^(3/2),x,6,1/2*arctanh(sqrt(a*cosh(e+f*x)^2)/sqrt(a))/(a^(3/2)*f)-1/2*csch(e+f*x)^2*sqrt(a*cosh(e+f*x)^2)/(a^2*f)],
[tanh(e+f*x)^2/(a+a*sinh(e+f*x)^2)^(3/2),x,5,1/8*arctan(sinh(e+f*x))*cosh(e+f*x)/(a*f*sqrt(a*cosh(e+f*x)^2))+1/8*tanh(e+f*x)/(a*f*sqrt(a*cosh(e+f*x)^2))-1/4*sech(e+f*x)^2*tanh(e+f*x)/(a*f*sqrt(a*cosh(e+f*x)^2))],
[coth(e+f*x)^2/(a+a*sinh(e+f*x)^2)^(3/2),x,5,-arctan(sinh(e+f*x))*cosh(e+f*x)/(a*f*sqrt(a*cosh(e+f*x)^2))-coth(e+f*x)/(a*f*sqrt(a*cosh(e+f*x)^2))],
[coth(e+f*x)^4/(a+a*sinh(e+f*x)^2)^(3/2),x,4,-1/3*coth(e+f*x)*csch(e+f*x)^2/(a*f*sqrt(a*cosh(e+f*x)^2))],
[coth(e+f*x)^6/(a+a*sinh(e+f*x)^2)^(3/2),x,5,-1/3*coth(e+f*x)*csch(e+f*x)^2/(a*f*sqrt(a*cosh(e+f*x)^2))-1/5*coth(e+f*x)*csch(e+f*x)^4/(a*f*sqrt(a*cosh(e+f*x)^2))],
[coth(e+f*x)^8/(a+a*sinh(e+f*x)^2)^(3/2),x,5,-1/3*coth(e+f*x)*csch(e+f*x)^2/(a*f*sqrt(a*cosh(e+f*x)^2))-2/5*coth(e+f*x)*csch(e+f*x)^4/(a*f*sqrt(a*cosh(e+f*x)^2))-1/7*coth(e+f*x)*csch(e+f*x)^6/(a*f*sqrt(a*cosh(e+f*x)^2))],

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

# p>0
[sqrt(a+b*sinh(e+f*x)^2)*tanh(e+f*x)^5,x,6,-1/8*(8*a^2-24*a*b+15*b^2)*arctanh(sqrt(a+b*sinh(e+f*x)^2)/sqrt(a-b))/((a-b)^(3/2)*f)+1/8*(8*a-7*b)*sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)^(3/2)/((a-b)^2*f)-1/4*sech(e+f*x)^4*(a+b*sinh(e+f*x)^2)^(3/2)/((a-b)*f)+1/8*(8*a^2-24*a*b+15*b^2)*sqrt(a+b*sinh(e+f*x)^2)/((a-b)^2*f)],
[sqrt(a+b*sinh(e+f*x)^2)*tanh(e+f*x)^3,x,5,1/2*sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)^(3/2)/((a-b)*f)-1/2*(2*a-3*b)*arctanh(sqrt(a+b*sinh(e+f*x)^2)/sqrt(a-b))/(f*sqrt(a-b))+1/2*(2*a-3*b)*sqrt(a+b*sinh(e+f*x)^2)/((a-b)*f)],
[sqrt(a+b*sinh(e+f*x)^2)*tanh(e+f*x),x,4,-arctanh(sqrt(a+b*sinh(e+f*x)^2)/sqrt(a-b))*sqrt(a-b)/f+sqrt(a+b*sinh(e+f*x)^2)/f],
[coth(e+f*x)*sqrt(a+b*sinh(e+f*x)^2),x,4,-arctanh(sqrt(a+b*sinh(e+f*x)^2)/sqrt(a))*sqrt(a)/f+sqrt(a+b*sinh(e+f*x)^2)/f],
[coth(e+f*x)^3*sqrt(a+b*sinh(e+f*x)^2),x,5,-1/2*csch(e+f*x)^2*(a+b*sinh(e+f*x)^2)^(3/2)/(a*f)-1/2*(2*a+b)*arctanh(sqrt(a+b*sinh(e+f*x)^2)/sqrt(a))/(f*sqrt(a))+1/2*(2*a+b)*sqrt(a+b*sinh(e+f*x)^2)/(a*f)],
[coth(e+f*x)^5*sqrt(a+b*sinh(e+f*x)^2),x,6,-1/8*(8*a^2+8*a*b-b^2)*arctanh(sqrt(a+b*sinh(e+f*x)^2)/sqrt(a))/(a^(3/2)*f)-1/8*(8*a-b)*csch(e+f*x)^2*(a+b*sinh(e+f*x)^2)^(3/2)/(a^2*f)-1/4*csch(e+f*x)^4*(a+b*sinh(e+f*x)^2)^(3/2)/(a*f)+1/8*(8*a^2+8*a*b-b^2)*sqrt(a+b*sinh(e+f*x)^2)/(a^2*f)],
[sqrt(a+b*sinh(e+f*x)^2)*tanh(e+f*x)^4,x,7,-1/3*(7*a-8*b)*sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticE(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/((a-b)*f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))+1/3*(3*a-4*b)*sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticF(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/((a-b)*f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))+1/3*(7*a-8*b)*sqrt(a+b*sinh(e+f*x)^2)*tanh(e+f*x)/((a-b)*f)-1/3*(3*a-4*b)*sqrt(a+b*sinh(e+f*x)^2)*tanh(e+f*x)/((a-b)*f)-1/3*sqrt(a+b*sinh(e+f*x)^2)*tanh(e+f*x)^3/f],
[sqrt(a+b*sinh(e+f*x)^2)*tanh(e+f*x)^2,x,6,-2*sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticE(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))+sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticF(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))+sqrt(a+b*sinh(e+f*x)^2)*tanh(e+f*x)/f],
[sqrt(a+b*sinh(e+f*x)^2),x,2,-I*sqrt(cos(I*e+I*f*x)^2)/cos(I*e+I*f*x)*EllipticE(sin(I*e+I*f*x),sqrt(b/a))*sqrt(a+b*sinh(e+f*x)^2)/(f*sqrt(1+b*sinh(e+f*x)^2/a))],
[coth(e+f*x)^2*sqrt(a+b*sinh(e+f*x)^2),x,6,-coth(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/f-2*sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticE(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))+(a+b)*sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticF(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(a*f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))+2*sqrt(a+b*sinh(e+f*x)^2)*tanh(e+f*x)/f],
[coth(e+f*x)^4*sqrt(a+b*sinh(e+f*x)^2),x,7,-1/3*(3*a+b)*coth(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(a*f)-1/3*coth(e+f*x)^3*sqrt(a+b*sinh(e+f*x)^2)/f-1/3*(7*a+b)*sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticE(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(a*f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))+1/3*(3*a+5*b)*sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticF(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(a*f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))+1/3*(7*a+b)*sqrt(a+b*sinh(e+f*x)^2)*tanh(e+f*x)/(a*f)],
[(a+b*sinh(e+f*x)^2)^(3/2)*tanh(e+f*x)^5,x,7,1/24*(8*a^2-40*a*b+35*b^2)*(a+b*sinh(e+f*x)^2)^(3/2)/((a-b)^2*f)+1/8*(8*a-9*b)*sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)^(5/2)/((a-b)^2*f)-1/4*sech(e+f*x)^4*(a+b*sinh(e+f*x)^2)^(5/2)/((a-b)*f)-1/8*(8*a^2-40*a*b+35*b^2)*arctanh(sqrt(a+b*sinh(e+f*x)^2)/sqrt(a-b))/(f*sqrt(a-b))+1/8*(8*a^2-40*a*b+35*b^2)*sqrt(a+b*sinh(e+f*x)^2)/((a-b)*f)],
[(a+b*sinh(e+f*x)^2)^(3/2)*tanh(e+f*x)^3,x,6,1/6*(2*a-5*b)*(a+b*sinh(e+f*x)^2)^(3/2)/((a-b)*f)+1/2*sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)^(5/2)/((a-b)*f)-1/2*(2*a-5*b)*arctanh(sqrt(a+b*sinh(e+f*x)^2)/sqrt(a-b))*sqrt(a-b)/f+1/2*(2*a-5*b)*sqrt(a+b*sinh(e+f*x)^2)/f],
[(a+b*sinh(e+f*x)^2)^(3/2)*tanh(e+f*x),x,5,-(a-b)^(3/2)*arctanh(sqrt(a+b*sinh(e+f*x)^2)/sqrt(a-b))/f+1/3*(a+b*sinh(e+f*x)^2)^(3/2)/f+(a-b)*sqrt(a+b*sinh(e+f*x)^2)/f],
[coth(e+f*x)*(a+b*sinh(e+f*x)^2)^(3/2),x,5,-a^(3/2)*arctanh(sqrt(a+b*sinh(e+f*x)^2)/sqrt(a))/f+1/3*(a+b*sinh(e+f*x)^2)^(3/2)/f+a*sqrt(a+b*sinh(e+f*x)^2)/f],
[coth(e+f*x)^3*(a+b*sinh(e+f*x)^2)^(3/2),x,6,1/6*(2*a+3*b)*(a+b*sinh(e+f*x)^2)^(3/2)/(a*f)-1/2*csch(e+f*x)^2*(a+b*sinh(e+f*x)^2)^(5/2)/(a*f)-1/2*(2*a+3*b)*arctanh(sqrt(a+b*sinh(e+f*x)^2)/sqrt(a))*sqrt(a)/f+1/2*(2*a+3*b)*sqrt(a+b*sinh(e+f*x)^2)/f],
[coth(e+f*x)^5*(a+b*sinh(e+f*x)^2)^(3/2),x,7,1/24*(8*a^2+3*b*(8*a+b))*(a+b*sinh(e+f*x)^2)^(3/2)/(a^2*f)-1/8*(8*a+b)*csch(e+f*x)^2*(a+b*sinh(e+f*x)^2)^(5/2)/(a^2*f)-1/4*csch(e+f*x)^4*(a+b*sinh(e+f*x)^2)^(5/2)/(a*f)-1/8*(8*a^2+3*b*(8*a+b))*arctanh(sqrt(a+b*sinh(e+f*x)^2)/sqrt(a))/(f*sqrt(a))+1/8*(8*a^2+3*b*(8*a+b))*sqrt(a+b*sinh(e+f*x)^2)/(a*f)],
[(a+b*sinh(e+f*x)^2)^(3/2)*tanh(e+f*x)^4,x,8,-1/3*(3*a-8*b)*cosh(e+f*x)*sinh(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/f-8/3*(a-2*b)*sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticE(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))+1/3*(3*a-8*b)*sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticF(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))+8/3*(a-2*b)*sqrt(a+b*sinh(e+f*x)^2)*tanh(e+f*x)/f+(a-2*b)*sinh(e+f*x)^2*sqrt(a+b*sinh(e+f*x)^2)*tanh(e+f*x)/f-1/3*(a+b*sinh(e+f*x)^2)^(3/2)*tanh(e+f*x)^3/f],
[(a+b*sinh(e+f*x)^2)^(3/2)*tanh(e+f*x)^2,x,7,4/3*b*cosh(e+f*x)*sinh(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/f-1/3*(7*a-8*b)*sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticE(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))+1/3*(3*a-4*b)*sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticF(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))-(a+b*sinh(e+f*x)^2)^(3/2)*tanh(e+f*x)/f+1/3*(7*a-8*b)*sqrt(a+b*sinh(e+f*x)^2)*tanh(e+f*x)/f],
[(a+b*sinh(e+f*x)^2)^(3/2),x,6,1/3*b*cosh(e+f*x)*sinh(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/f-2/3*I*(2*a-b)*sqrt(cos(I*e+I*f*x)^2)/cos(I*e+I*f*x)*EllipticE(sin(I*e+I*f*x),sqrt(b/a))*sqrt(a+b*sinh(e+f*x)^2)/(f*sqrt(1+b*sinh(e+f*x)^2/a))+1/3*I*a*(a-b)*sqrt(cos(I*e+I*f*x)^2)/cos(I*e+I*f*x)*EllipticF(sin(I*e+I*f*x),sqrt(b/a))*sqrt(1+b*sinh(e+f*x)^2/a)/(f*sqrt(a+b*sinh(e+f*x)^2))],
[coth(e+f*x)^2*(a+b*sinh(e+f*x)^2)^(3/2),x,7,-coth(e+f*x)*(a+b*sinh(e+f*x)^2)^(3/2)/f+4/3*b*cosh(e+f*x)*sinh(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/f-1/3*(7*a+b)*sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticE(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))+1/3*(3*a+5*b)*sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticF(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))+1/3*(7*a+b)*sqrt(a+b*sinh(e+f*x)^2)*tanh(e+f*x)/f],
[coth(e+f*x)^4*(a+b*sinh(e+f*x)^2)^(3/2),x,8,-1/3*coth(e+f*x)^3*(a+b*sinh(e+f*x)^2)^(3/2)/f-(a+b)*cosh(e+f*x)^2*coth(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/f+1/3*(3*a+5*b)*cosh(e+f*x)*sinh(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/f-8/3*(a+b)*sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticE(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))+1/3*(3*a+b)*(a+3*b)*sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticF(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(a*f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))+8/3*(a+b)*sqrt(a+b*sinh(e+f*x)^2)*tanh(e+f*x)/f],

# p<0
[tanh(e+f*x)^5/sqrt(a+b*sinh(e+f*x)^2),x,5,-1/8*(8*a^2-8*a*b+3*b^2)*arctanh(sqrt(a+b*sinh(e+f*x)^2)/sqrt(a-b))/((a-b)^(5/2)*f)+1/8*(8*a-5*b)*sech(e+f*x)^2*sqrt(a+b*sinh(e+f*x)^2)/((a-b)^2*f)-1/4*sech(e+f*x)^4*sqrt(a+b*sinh(e+f*x)^2)/((a-b)*f)],
[tanh(e+f*x)^3/sqrt(a+b*sinh(e+f*x)^2),x,4,-1/2*(2*a-b)*arctanh(sqrt(a+b*sinh(e+f*x)^2)/sqrt(a-b))/((a-b)^(3/2)*f)+1/2*sech(e+f*x)^2*sqrt(a+b*sinh(e+f*x)^2)/((a-b)*f)],
[tanh(e+f*x)/sqrt(a+b*sinh(e+f*x)^2),x,3,-arctanh(sqrt(a+b*sinh(e+f*x)^2)/sqrt(a-b))/(f*sqrt(a-b))],
[coth(e+f*x)/sqrt(a+b*sinh(e+f*x)^2),x,3,-arctanh(sqrt(a+b*sinh(e+f*x)^2)/sqrt(a))/(f*sqrt(a))],
[coth(e+f*x)^3/sqrt(a+b*sinh(e+f*x)^2),x,4,-1/2*(2*a-b)*arctanh(sqrt(a+b*sinh(e+f*x)^2)/sqrt(a))/(a^(3/2)*f)-1/2*csch(e+f*x)^2*sqrt(a+b*sinh(e+f*x)^2)/(a*f)],
[coth(e+f*x)^5/sqrt(a+b*sinh(e+f*x)^2),x,5,-1/8*(8*a^2-8*a*b+3*b^2)*arctanh(sqrt(a+b*sinh(e+f*x)^2)/sqrt(a))/(a^(5/2)*f)-1/8*(8*a-3*b)*csch(e+f*x)^2*sqrt(a+b*sinh(e+f*x)^2)/(a^2*f)-1/4*csch(e+f*x)^4*sqrt(a+b*sinh(e+f*x)^2)/(a*f)],
[tanh(e+f*x)^4/sqrt(a+b*sinh(e+f*x)^2),x,5,-2/3*(2*a-b)*sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticE(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/((a-b)^2*f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))+1/3*(3*a-b)*sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticF(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/((a-b)^2*f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))+1/3*sech(e+f*x)^2*sqrt(a+b*sinh(e+f*x)^2)*tanh(e+f*x)/((a-b)*f)],
[tanh(e+f*x)^2/sqrt(a+b*sinh(e+f*x)^2),x,6,-sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticE(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/((a-b)*f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))+sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticF(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/((a-b)*f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))],
[1/sqrt(a+b*sinh(e+f*x)^2),x,2,-I*sqrt(cos(I*e+I*f*x)^2)/cos(I*e+I*f*x)*EllipticF(sin(I*e+I*f*x),sqrt(b/a))*sqrt(1+b*sinh(e+f*x)^2/a)/(f*sqrt(a+b*sinh(e+f*x)^2))],
[coth(e+f*x)^2/sqrt(a+b*sinh(e+f*x)^2),x,6,-coth(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(a*f)-sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticE(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(a*f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))+sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticF(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(a*f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))+sqrt(a+b*sinh(e+f*x)^2)*tanh(e+f*x)/(a*f)],
[coth(e+f*x)^4/sqrt(a+b*sinh(e+f*x)^2),x,7,-2/3*(2*a-b)*coth(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(a^2*f)-1/3*coth(e+f*x)*csch(e+f*x)^2*sqrt(a+b*sinh(e+f*x)^2)/(a*f)-2/3*(2*a-b)*sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticE(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(a^2*f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))+1/3*(3*a-b)*sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticF(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(a^2*f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))+2/3*(2*a-b)*sqrt(a+b*sinh(e+f*x)^2)*tanh(e+f*x)/(a^2*f)],
[tanh(e+f*x)^5/(a+b*sinh(e+f*x)^2)^(3/2),x,6,-1/8*(8*a^2+8*a*b-b^2)*arctanh(sqrt(a+b*sinh(e+f*x)^2)/sqrt(a-b))/((a-b)^(7/2)*f)+1/8*(8*a^2+8*a*b-b^2)/((a-b)^3*f*sqrt(a+b*sinh(e+f*x)^2))+1/8*(8*a-3*b)*sech(e+f*x)^2/((a-b)^2*f*sqrt(a+b*sinh(e+f*x)^2))-1/4*sech(e+f*x)^4/((a-b)*f*sqrt(a+b*sinh(e+f*x)^2))],
[tanh(e+f*x)^3/(a+b*sinh(e+f*x)^2)^(3/2),x,5,-1/2*(2*a+b)*arctanh(sqrt(a+b*sinh(e+f*x)^2)/sqrt(a-b))/((a-b)^(5/2)*f)+1/2*(2*a+b)/((a-b)^2*f*sqrt(a+b*sinh(e+f*x)^2))+1/2*sech(e+f*x)^2/((a-b)*f*sqrt(a+b*sinh(e+f*x)^2))],
[tanh(e+f*x)/(a+b*sinh(e+f*x)^2)^(3/2),x,4,-arctanh(sqrt(a+b*sinh(e+f*x)^2)/sqrt(a-b))/((a-b)^(3/2)*f)+1/((a-b)*f*sqrt(a+b*sinh(e+f*x)^2))],
[coth(e+f*x)/(a+b*sinh(e+f*x)^2)^(3/2),x,4,-arctanh(sqrt(a+b*sinh(e+f*x)^2)/sqrt(a))/(a^(3/2)*f)+1/(a*f*sqrt(a+b*sinh(e+f*x)^2))],
[coth(e+f*x)^3/(a+b*sinh(e+f*x)^2)^(3/2),x,5,-1/2*(2*a-3*b)*arctanh(sqrt(a+b*sinh(e+f*x)^2)/sqrt(a))/(a^(5/2)*f)+1/2*(2*a-3*b)/(a^2*f*sqrt(a+b*sinh(e+f*x)^2))-1/2*csch(e+f*x)^2/(a*f*sqrt(a+b*sinh(e+f*x)^2))],
[coth(e+f*x)^5/(a+b*sinh(e+f*x)^2)^(3/2),x,6,-1/8*(8*a^2-24*a*b+15*b^2)*arctanh(sqrt(a+b*sinh(e+f*x)^2)/sqrt(a))/(a^(7/2)*f)+1/8*(8*a^2-24*a*b+15*b^2)/(a^3*f*sqrt(a+b*sinh(e+f*x)^2))-1/8*(8*a-5*b)*csch(e+f*x)^2/(a^2*f*sqrt(a+b*sinh(e+f*x)^2))-1/4*csch(e+f*x)^4/(a*f*sqrt(a+b*sinh(e+f*x)^2))],
[tanh(e+f*x)^4/(a+b*sinh(e+f*x)^2)^(3/2),x,6,-1/3*(7*a+b)*cosh(e+f*x)*sqrt(cos(arctan(sinh(e+f*x)*sqrt(b)/sqrt(a)))^2)/cos(arctan(sinh(e+f*x)*sqrt(b)/sqrt(a)))*EllipticE(sin(arctan(sinh(e+f*x)*sqrt(b)/sqrt(a))),sqrt(1-a/b))*sqrt(a)*sqrt(b)/((a-b)^3*f*sqrt(a*cosh(e+f*x)^2/(a+b*sinh(e+f*x)^2))*sqrt(a+b*sinh(e+f*x)^2))+1/3*(3*a+5*b)*sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticF(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/((a-b)^3*f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))-4/3*a*tanh(e+f*x)/((a-b)^2*f*sqrt(a+b*sinh(e+f*x)^2))+1/3*sech(e+f*x)^2*tanh(e+f*x)/((a-b)*f*sqrt(a+b*sinh(e+f*x)^2))],
[tanh(e+f*x)^2/(a+b*sinh(e+f*x)^2)^(3/2),x,5,-2*cosh(e+f*x)*sqrt(cos(arctan(sinh(e+f*x)*sqrt(b)/sqrt(a)))^2)/cos(arctan(sinh(e+f*x)*sqrt(b)/sqrt(a)))*EllipticE(sin(arctan(sinh(e+f*x)*sqrt(b)/sqrt(a))),sqrt(1-a/b))*sqrt(a)*sqrt(b)/((a-b)^2*f*sqrt(a*cosh(e+f*x)^2/(a+b*sinh(e+f*x)^2))*sqrt(a+b*sinh(e+f*x)^2))+(a+b)*sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticF(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(a*(a-b)^2*f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))-tanh(e+f*x)/((a-b)*f*sqrt(a+b*sinh(e+f*x)^2))],
[1/(a+b*sinh(e+f*x)^2)^(3/2),x,4,-b*cosh(e+f*x)*sinh(e+f*x)/(a*(a-b)*f*sqrt(a+b*sinh(e+f*x)^2))-I*sqrt(cos(I*e+I*f*x)^2)/cos(I*e+I*f*x)*EllipticE(sin(I*e+I*f*x),sqrt(b/a))*sqrt(a+b*sinh(e+f*x)^2)/(a*(a-b)*f*sqrt(1+b*sinh(e+f*x)^2/a))],
[coth(e+f*x)^2/(a+b*sinh(e+f*x)^2)^(3/2),x,7,coth(e+f*x)/(a*f*sqrt(a+b*sinh(e+f*x)^2))-2*coth(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(a^2*f)-2*sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticE(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(a^2*f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))+sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticF(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(a^2*f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))+2*sqrt(a+b*sinh(e+f*x)^2)*tanh(e+f*x)/(a^2*f)],
[coth(e+f*x)^4/(a+b*sinh(e+f*x)^2)^(3/2),x,8,-(a-b)*coth(e+f*x)*csch(e+f*x)^2/(a*b*f*sqrt(a+b*sinh(e+f*x)^2))-1/3*(7*a-8*b)*coth(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(a^3*f)+1/3*(3*a-4*b)*coth(e+f*x)*csch(e+f*x)^2*sqrt(a+b*sinh(e+f*x)^2)/(a^2*b*f)-1/3*(7*a-8*b)*sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticE(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(a^3*f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))+1/3*(3*a-4*b)*sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticF(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(a^3*f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))+1/3*(7*a-8*b)*sqrt(a+b*sinh(e+f*x)^2)*tanh(e+f*x)/(a^3*f)],
[tanh(e+f*x)^5/(a+b*sinh(e+f*x)^2)^(5/2),x,7,-1/8*(8*a^2+24*a*b+3*b^2)*arctanh(sqrt(a+b*sinh(e+f*x)^2)/sqrt(a-b))/((a-b)^(9/2)*f)+1/24*(8*a^2+24*a*b+3*b^2)/((a-b)^3*f*(a+b*sinh(e+f*x)^2)^(3/2))+1/8*(8*a-b)*sech(e+f*x)^2/((a-b)^2*f*(a+b*sinh(e+f*x)^2)^(3/2))-1/4*sech(e+f*x)^4/((a-b)*f*(a+b*sinh(e+f*x)^2)^(3/2))+1/8*(8*a^2+24*a*b+3*b^2)/((a-b)^4*f*sqrt(a+b*sinh(e+f*x)^2))],
[tanh(e+f*x)^3/(a+b*sinh(e+f*x)^2)^(5/2),x,6,-1/2*(2*a+3*b)*arctanh(sqrt(a+b*sinh(e+f*x)^2)/sqrt(a-b))/((a-b)^(7/2)*f)+1/6*(2*a+3*b)/((a-b)^2*f*(a+b*sinh(e+f*x)^2)^(3/2))+1/2*sech(e+f*x)^2/((a-b)*f*(a+b*sinh(e+f*x)^2)^(3/2))+1/2*(2*a+3*b)/((a-b)^3*f*sqrt(a+b*sinh(e+f*x)^2))],
[tanh(e+f*x)/(a+b*sinh(e+f*x)^2)^(5/2),x,5,-arctanh(sqrt(a+b*sinh(e+f*x)^2)/sqrt(a-b))/((a-b)^(5/2)*f)+1/3/((a-b)*f*(a+b*sinh(e+f*x)^2)^(3/2))+1/((a-b)^2*f*sqrt(a+b*sinh(e+f*x)^2))],
[coth(e+f*x)/(a+b*sinh(e+f*x)^2)^(5/2),x,5,-arctanh(sqrt(a+b*sinh(e+f*x)^2)/sqrt(a))/(a^(5/2)*f)+1/3/(a*f*(a+b*sinh(e+f*x)^2)^(3/2))+1/(a^2*f*sqrt(a+b*sinh(e+f*x)^2))],
[coth(e+f*x)^3/(a+b*sinh(e+f*x)^2)^(5/2),x,6,-1/2*(2*a-5*b)*arctanh(sqrt(a+b*sinh(e+f*x)^2)/sqrt(a))/(a^(7/2)*f)+1/6*(2*a-5*b)/(a^2*f*(a+b*sinh(e+f*x)^2)^(3/2))-1/2*csch(e+f*x)^2/(a*f*(a+b*sinh(e+f*x)^2)^(3/2))+1/2*(2*a-5*b)/(a^3*f*sqrt(a+b*sinh(e+f*x)^2))],
[coth(e+f*x)^5/(a+b*sinh(e+f*x)^2)^(5/2),x,7,-1/8*(8*a^2-40*a*b+35*b^2)*arctanh(sqrt(a+b*sinh(e+f*x)^2)/sqrt(a))/(a^(9/2)*f)+1/24*(8*a^2-40*a*b+35*b^2)/(a^3*f*(a+b*sinh(e+f*x)^2)^(3/2))-1/8*(8*a-7*b)*csch(e+f*x)^2/(a^2*f*(a+b*sinh(e+f*x)^2)^(3/2))-1/4*csch(e+f*x)^4/(a*f*(a+b*sinh(e+f*x)^2)^(3/2))+1/8*(8*a^2-40*a*b+35*b^2)/(a^4*f*sqrt(a+b*sinh(e+f*x)^2))],
[tanh(e+f*x)^4/(a+b*sinh(e+f*x)^2)^(5/2),x,7,-1/3*b*(5*a+3*b)*cosh(e+f*x)*sinh(e+f*x)/((a-b)^3*f*(a+b*sinh(e+f*x)^2)^(3/2))-8/3*(a+b)*cosh(e+f*x)*sqrt(cos(arctan(sinh(e+f*x)*sqrt(b)/sqrt(a)))^2)/cos(arctan(sinh(e+f*x)*sqrt(b)/sqrt(a)))*EllipticE(sin(arctan(sinh(e+f*x)*sqrt(b)/sqrt(a))),sqrt(1-a/b))*sqrt(a)*sqrt(b)/((a-b)^4*f*sqrt(a*cosh(e+f*x)^2/(a+b*sinh(e+f*x)^2))*sqrt(a+b*sinh(e+f*x)^2))+1/3*(3*a+b)*(a+3*b)*sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticF(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(a*(a-b)^4*f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))-2/3*(2*a+b)*tanh(e+f*x)/((a-b)^2*f*(a+b*sinh(e+f*x)^2)^(3/2))+1/3*sech(e+f*x)^2*tanh(e+f*x)/((a-b)*f*(a+b*sinh(e+f*x)^2)^(3/2))],
[tanh(e+f*x)^2/(a+b*sinh(e+f*x)^2)^(5/2),x,6,-4/3*b*cosh(e+f*x)*sinh(e+f*x)/((a-b)^2*f*(a+b*sinh(e+f*x)^2)^(3/2))-1/3*(7*a+b)*cosh(e+f*x)*sqrt(cos(arctan(sinh(e+f*x)*sqrt(b)/sqrt(a)))^2)/cos(arctan(sinh(e+f*x)*sqrt(b)/sqrt(a)))*EllipticE(sin(arctan(sinh(e+f*x)*sqrt(b)/sqrt(a))),sqrt(1-a/b))*sqrt(b)/((a-b)^3*f*sqrt(a)*sqrt(a*cosh(e+f*x)^2/(a+b*sinh(e+f*x)^2))*sqrt(a+b*sinh(e+f*x)^2))+1/3*(3*a+5*b)*sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticF(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(a*(a-b)^3*f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))-tanh(e+f*x)/((a-b)*f*(a+b*sinh(e+f*x)^2)^(3/2))],
[1/(a+b*sinh(e+f*x)^2)^(5/2),x,7,-1/3*b*cosh(e+f*x)*sinh(e+f*x)/(a*(a-b)*f*(a+b*sinh(e+f*x)^2)^(3/2))-2/3*(2*a-b)*b*cosh(e+f*x)*sinh(e+f*x)/(a^2*(a-b)^2*f*sqrt(a+b*sinh(e+f*x)^2))-2/3*I*(2*a-b)*sqrt(cos(I*e+I*f*x)^2)/cos(I*e+I*f*x)*EllipticE(sin(I*e+I*f*x),sqrt(b/a))*sqrt(a+b*sinh(e+f*x)^2)/(a^2*(a-b)^2*f*sqrt(1+b*sinh(e+f*x)^2/a))+1/3*I*sqrt(cos(I*e+I*f*x)^2)/cos(I*e+I*f*x)*EllipticF(sin(I*e+I*f*x),sqrt(b/a))*sqrt(1+b*sinh(e+f*x)^2/a)/(a*(a-b)*f*sqrt(a+b*sinh(e+f*x)^2))],
[coth(e+f*x)^2/(a+b*sinh(e+f*x)^2)^(5/2),x,8,1/3*coth(e+f*x)/(a*f*(a+b*sinh(e+f*x)^2)^(3/2))+1/3*(3*a-4*b)*coth(e+f*x)/(a^2*(a-b)*f*sqrt(a+b*sinh(e+f*x)^2))-1/3*(7*a-8*b)*coth(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(a^3*(a-b)*f)-1/3*(7*a-8*b)*sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticE(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(a^3*(a-b)*f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))+1/3*(3*a-4*b)*sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticF(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(a^3*(a-b)*f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))+1/3*(7*a-8*b)*sqrt(a+b*sinh(e+f*x)^2)*tanh(e+f*x)/(a^3*(a-b)*f)],
[coth(e+f*x)^4/(a+b*sinh(e+f*x)^2)^(5/2),x,9,-1/3*(a-b)*coth(e+f*x)*csch(e+f*x)^2/(a*b*f*(a+b*sinh(e+f*x)^2)^(3/2))-2/3*(a-3*b)*coth(e+f*x)*csch(e+f*x)^2/(a^2*b*f*sqrt(a+b*sinh(e+f*x)^2))-8/3*(a-2*b)*coth(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(a^4*f)+1/3*(3*a-8*b)*coth(e+f*x)*csch(e+f*x)^2*sqrt(a+b*sinh(e+f*x)^2)/(a^3*b*f)-8/3*(a-2*b)*sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticE(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(a^4*f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))+1/3*(3*a-8*b)*sqrt(cos(arctan(sinh(e+f*x)))^2)/cos(arctan(sinh(e+f*x)))*EllipticF(sin(arctan(sinh(e+f*x))),sqrt(1-b/a))*sech(e+f*x)*sqrt(a+b*sinh(e+f*x)^2)/(a^4*f*sqrt(sech(e+f*x)^2*(a+b*sinh(e+f*x)^2)/a))+8/3*(a-2*b)*sqrt(a+b*sinh(e+f*x)^2)*tanh(e+f*x)/(a^4*f)],

# Integrands of the form (d Tanh[e+f x])^m (a+b Sinh[e+f x]^2)^p when p symbolic
[(a+b*sinh(e+f*x)^2)^p*(d*tanh(e+f*x))^m,x,3,AppellF1(1/2*(1+m),1/2*(1+m),-p,1/2*(3+m),-sinh(e+f*x)^2,-b*sinh(e+f*x)^2/a)*(cosh(e+f*x)^2)^(1/2*(1+m))*(a+b*sinh(e+f*x)^2)^p*(d*tanh(e+f*x))^(1+m)/(d*f*(1+m)*(1+b*sinh(e+f*x)^2/a)^p)],
[(a+b*sinh(c+d*x)^2)^p*tanh(c+d*x)^3,x,3,-1/2*(a-b*(1+p))*hypergeom([1,1+p],[2+p],(a+b*sinh(c+d*x)^2)/(a-b))*(a+b*sinh(c+d*x)^2)^(1+p)/((a-b)^2*d*(1+p))+1/2*sech(c+d*x)^2*(a+b*sinh(c+d*x)^2)^(1+p)/((a-b)*d)],
[(a+b*sinh(c+d*x)^2)^p*tanh(c+d*x),x,2,-1/2*hypergeom([1,1+p],[2+p],(a+b*sinh(c+d*x)^2)/(a-b))*(a+b*sinh(c+d*x)^2)^(1+p)/((a-b)*d*(1+p))],
[coth(c+d*x)*(a+b*sinh(c+d*x)^2)^p,x,2,-1/2*hypergeom([1,1+p],[2+p],1+b*sinh(c+d*x)^2/a)*(a+b*sinh(c+d*x)^2)^(1+p)/(a*d*(1+p))],
[coth(c+d*x)^3*(a+b*sinh(c+d*x)^2)^p,x,3,-1/2*csch(c+d*x)^2*(a+b*sinh(c+d*x)^2)^(1+p)/(a*d)-1/2*(a+b*p)*hypergeom([1,1+p],[2+p],1+b*sinh(c+d*x)^2/a)*(a+b*sinh(c+d*x)^2)^(1+p)/(a^2*d*(1+p))],
[(a+b*sinh(c+d*x)^2)^p*tanh(c+d*x)^4,x,3,1/5*AppellF1(5/2,5/2,-p,7/2,-sinh(c+d*x)^2,-b*sinh(c+d*x)^2/a)*sinh(c+d*x)^4*(a+b*sinh(c+d*x)^2)^p*sqrt(cosh(c+d*x)^2)*tanh(c+d*x)/(d*(1+b*sinh(c+d*x)^2/a)^p)],
[(a+b*sinh(c+d*x)^2)^p*tanh(c+d*x)^2,x,3,1/3*AppellF1(3/2,3/2,-p,5/2,-sinh(c+d*x)^2,-b*sinh(c+d*x)^2/a)*sinh(c+d*x)^2*(a+b*sinh(c+d*x)^2)^p*sqrt(cosh(c+d*x)^2)*tanh(c+d*x)/(d*(1+b*sinh(c+d*x)^2/a)^p)],
[coth(c+d*x)^2*(a+b*sinh(c+d*x)^2)^p,x,3,-AppellF1(-1/2,-1/2,-p,1/2,-sinh(c+d*x)^2,-b*sinh(c+d*x)^2/a)*csch(c+d*x)*sech(c+d*x)*(a+b*sinh(c+d*x)^2)^p*sqrt(cosh(c+d*x)^2)/(d*(1+b*sinh(c+d*x)^2/a)^p)],
[coth(c+d*x)^4*(a+b*sinh(c+d*x)^2)^p,x,3,-1/3*AppellF1(-3/2,-3/2,-p,-1/2,-sinh(c+d*x)^2,-b*sinh(c+d*x)^2/a)*csch(c+d*x)^3*sech(c+d*x)*(a+b*sinh(c+d*x)^2)^p*sqrt(cosh(c+d*x)^2)/(d*(1+b*sinh(c+d*x)^2/a)^p)],

# Integrands of the form Tanh[e+f x]^m (a+b Sinh[e+f x]^3)^p
[coth(x)^3/(a+b*sinh(x)^3),x,12,-1/2*csch(x)^2/a+log(sinh(x))/a-1/3*b^(2/3)*log(a^(1/3)+b^(1/3)*sinh(x))/a^(5/3)+1/6*b^(2/3)*log(a^(2/3)-a^(1/3)*b^(1/3)*sinh(x)+b^(2/3)*sinh(x)^2)/a^(5/3)-1/3*log(a+b*sinh(x)^3)/a+b^(2/3)*arctan((a^(1/3)-2*b^(1/3)*sinh(x))/(a^(1/3)*sqrt(3)))/(a^(5/3)*sqrt(3))],
[coth(x)/sqrt(a+b*sinh(x)^3),x,4,-2/3*arctanh(sqrt(a+b*sinh(x)^3)/sqrt(a))/sqrt(a)],
[coth(x)*sqrt(a+b*sinh(x)^3),x,5,-2/3*arctanh(sqrt(a+b*sinh(x)^3)/sqrt(a))*sqrt(a)+2/3*sqrt(a+b*sinh(x)^3)],

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

# Integrands of the form Tanh[e+f x]^m (a+b Sinh[e+f x]^n)^p
[coth(x)/sqrt(a+b*sinh(x)^n),x,4,-2*arctanh(sqrt(a+b*sinh(x)^n)/sqrt(a))/(n*sqrt(a))],
[coth(x)*sqrt(a+b*sinh(x)^n),x,5,-2*arctanh(sqrt(a+b*sinh(x)^n)/sqrt(a))*sqrt(a)/n+2*sqrt(a+b*sinh(x)^n)/n]]:
