\(\int \frac {(e+f x)^2 \text {csch}^2(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx\) [469]

Optimal result

Integrand size = 36, antiderivative size = 914 \[ \int \frac {(e+f x)^2 \text {csch}^2(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx =\text {Too large to display} \] Output:

b^2*(f*x+e)^2/a/(a^2+b^2)/d+2*b*(f*x+e)^2*arctanh(exp(d*x+c))/a^2/d-2*b*f^ 
2*polylog(3,-exp(d*x+c))/a^2/d^3+2*b*f^2*polylog(3,exp(d*x+c))/a^2/d^3-b^2 
*f^2*polylog(2,-exp(2*d*x+2*c))/a/(a^2+b^2)/d^3+b^2*(f*x+e)^2*tanh(d*x+c)/ 
a/(a^2+b^2)/d+b^3*(f*x+e)^2*sech(d*x+c)/a^2/(a^2+b^2)/d-b*(f*x+e)^2*sech(d 
*x+c)/a^2/d+2*I*b^3*f^2*polylog(2,-I*exp(d*x+c))/a^2/(a^2+b^2)/d^3+2*I*b*f 
^2*polylog(2,I*exp(d*x+c))/a^2/d^3+2*b^4*f^2*polylog(3,-b*exp(d*x+c)/(a+(a 
^2+b^2)^(1/2)))/a^2/(a^2+b^2)^(3/2)/d^3-2*b^4*f^2*polylog(3,-b*exp(d*x+c)/ 
(a-(a^2+b^2)^(1/2)))/a^2/(a^2+b^2)^(3/2)/d^3-2*b^2*f*(f*x+e)*ln(1+exp(2*d* 
x+2*c))/a/(a^2+b^2)/d^2-4*b^3*f*(f*x+e)*arctan(exp(d*x+c))/a^2/(a^2+b^2)/d 
^2-2*I*b^3*f^2*polylog(2,I*exp(d*x+c))/a^2/(a^2+b^2)/d^3-b^4*(f*x+e)^2*ln( 
1+b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a^2/(a^2+b^2)^(3/2)/d+b^4*(f*x+e)^2*ln 
(1+b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a^2/(a^2+b^2)^(3/2)/d+2*b*f*(f*x+e)*p 
olylog(2,-exp(d*x+c))/a^2/d^2-2*b*f*(f*x+e)*polylog(2,exp(d*x+c))/a^2/d^2+ 
2*f*(f*x+e)*ln(1-exp(4*d*x+4*c))/a/d^2+1/2*f^2*polylog(2,exp(4*d*x+4*c))/a 
/d^3-2*(f*x+e)^2*coth(2*d*x+2*c)/a/d-2*(f*x+e)^2/a/d-2*b^4*f*(f*x+e)*polyl 
og(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a^2/(a^2+b^2)^(3/2)/d^2+2*b^4*f*(f 
*x+e)*polylog(2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a^2/(a^2+b^2)^(3/2)/d^2 
+4*b*f*(f*x+e)*arctan(exp(d*x+c))/a^2/d^2-2*I*b*f^2*polylog(2,-I*exp(d*x+c 
))/a^2/d^3
 

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1829\) vs. \(2(914)=1828\).

Time = 9.45 (sec) , antiderivative size = 1829, normalized size of antiderivative = 2.00 \[ \int \frac {(e+f x)^2 \text {csch}^2(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx =\text {Too large to display} \] Input:

Integrate[((e + f*x)^2*Csch[c + d*x]^2*Sech[c + d*x]^2)/(a + b*Sinh[c + d* 
x]),x]
 

Output:

4*(-1/4*(f*(4*a*d^2*e*E^(2*c)*x + 2*a*d^2*E^(2*c)*f*x^2 - 4*b*d*e*ArcTan[E 
^(c + d*x)] - 4*b*d*e*E^(2*c)*ArcTan[E^(c + d*x)] - (2*I)*b*d*f*x*Log[1 - 
I*E^(c + d*x)] - (2*I)*b*d*E^(2*c)*f*x*Log[1 - I*E^(c + d*x)] + (2*I)*b*d* 
f*x*Log[1 + I*E^(c + d*x)] + (2*I)*b*d*E^(2*c)*f*x*Log[1 + I*E^(c + d*x)] 
- 2*a*d*e*Log[1 + E^(2*(c + d*x))] - 2*a*d*e*E^(2*c)*Log[1 + E^(2*(c + d*x 
))] - 2*a*d*f*x*Log[1 + E^(2*(c + d*x))] - 2*a*d*E^(2*c)*f*x*Log[1 + E^(2* 
(c + d*x))] + (2*I)*b*(1 + E^(2*c))*f*PolyLog[2, (-I)*E^(c + d*x)] - (2*I) 
*b*(1 + E^(2*c))*f*PolyLog[2, I*E^(c + d*x)] - a*f*PolyLog[2, -E^(2*(c + d 
*x))] - a*E^(2*c)*f*PolyLog[2, -E^(2*(c + d*x))]))/((a^2 + b^2)*d^3*(1 + E 
^(2*c))) + (d*(d*x*(3*a*f*(-2*e*E^c + f*x) + b*d*(-3*e^2*E^c + 3*e*f*x + f 
^2*x^2)) + 3*(1 + E^c)*f*x*(2*a*f + b*d*(2*e + f*x))*Log[1 + E^(-c - d*x)] 
 + 3*e*(1 + E^c)*(b*d*e + 2*a*f)*Log[1 + E^(c + d*x)]) - 6*(1 + E^c)*f*(a* 
f + b*d*(e + f*x))*PolyLog[2, -E^(-c - d*x)] - 6*b*(1 + E^c)*f^2*PolyLog[3 
, -E^(-c - d*x)])/(12*a^2*d^3*(1 + E^c)) + (d*(d*x*(-3*a*f*(2*e*E^c + f*x) 
 + b*d*(3*e^2*E^c + 3*e*f*x + f^2*x^2)) - 3*(-1 + E^c)*f*x*(-2*a*f + b*d*( 
2*e + f*x))*Log[1 - E^(-c - d*x)] - 3*e*(-1 + E^c)*(b*d*e - 2*a*f)*Log[1 - 
 E^(c + d*x)]) + 6*(-1 + E^c)*f*(-(a*f) + b*d*(e + f*x))*PolyLog[2, E^(-c 
- d*x)] + 6*b*(-1 + E^c)*f^2*PolyLog[3, E^(-c - d*x)])/(12*a^2*d^3*(-1 + E 
^c)) + (b^4*(-2*d^2*e^2*ArcTanh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]] + 2*d 
^2*e*f*x*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] + d^2*f^2*x^2*L...
 

Rubi [A] (verified)

Time = 6.36 (sec) , antiderivative size = 818, normalized size of antiderivative = 0.89, number of steps used = 30, number of rules used = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.806, Rules used = {6123, 5984, 3042, 25, 4672, 26, 3042, 26, 4201, 2620, 2715, 2838, 6123, 5985, 25, 6107, 3042, 3803, 25, 2694, 27, 2620, 3011, 2720, 7143, 7292, 27, 7293, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {(e+f x)^2 \text {csch}^2(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx\)

\(\Big \downarrow \) 6123

\(\displaystyle \frac {\int (e+f x)^2 \text {csch}^2(c+d x) \text {sech}^2(c+d x)dx}{a}-\frac {b \int \frac {(e+f x)^2 \text {csch}(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}\)

\(\Big \downarrow \) 5984

\(\displaystyle \frac {4 \int (e+f x)^2 \text {csch}^2(2 c+2 d x)dx}{a}-\frac {b \int \frac {(e+f x)^2 \text {csch}(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {b \int \frac {(e+f x)^2 \text {csch}(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {4 \int -(e+f x)^2 \csc (2 i c+2 i d x)^2dx}{a}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {b \int \frac {(e+f x)^2 \text {csch}(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {4 \int (e+f x)^2 \csc (2 i c+2 i d x)^2dx}{a}\)

\(\Big \downarrow \) 4672

\(\displaystyle -\frac {b \int \frac {(e+f x)^2 \text {csch}(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {4 \left (\frac {(e+f x)^2 \coth (2 c+2 d x)}{2 d}-\frac {i f \int -i (e+f x) \coth (2 c+2 d x)dx}{d}\right )}{a}\)

\(\Big \downarrow \) 26

\(\displaystyle -\frac {b \int \frac {(e+f x)^2 \text {csch}(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {4 \left (\frac {(e+f x)^2 \coth (2 c+2 d x)}{2 d}-\frac {f \int (e+f x) \coth (2 c+2 d x)dx}{d}\right )}{a}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {b \int \frac {(e+f x)^2 \text {csch}(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {4 \left (\frac {(e+f x)^2 \coth (2 c+2 d x)}{2 d}-\frac {f \int -i (e+f x) \tan \left (2 i c+2 i d x+\frac {\pi }{2}\right )dx}{d}\right )}{a}\)

\(\Big \downarrow \) 26

\(\displaystyle -\frac {b \int \frac {(e+f x)^2 \text {csch}(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {4 \left (\frac {(e+f x)^2 \coth (2 c+2 d x)}{2 d}+\frac {i f \int (e+f x) \tan \left (\frac {1}{2} (4 i c+\pi )+2 i d x\right )dx}{d}\right )}{a}\)

\(\Big \downarrow \) 4201

\(\displaystyle -\frac {b \int \frac {(e+f x)^2 \text {csch}(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {4 \left (\frac {(e+f x)^2 \coth (2 c+2 d x)}{2 d}+\frac {i f \left (2 i \int \frac {e^{4 c+4 d x-i \pi } (e+f x)}{1+e^{4 c+4 d x-i \pi }}dx-\frac {i (e+f x)^2}{2 f}\right )}{d}\right )}{a}\)

\(\Big \downarrow \) 2620

\(\displaystyle -\frac {b \int \frac {(e+f x)^2 \text {csch}(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {4 \left (\frac {(e+f x)^2 \coth (2 c+2 d x)}{2 d}+\frac {i f \left (2 i \left (\frac {(e+f x) \log \left (1+e^{4 c+4 d x-i \pi }\right )}{4 d}-\frac {f \int \log \left (1+e^{4 c+4 d x-i \pi }\right )dx}{4 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}\right )}{a}\)

\(\Big \downarrow \) 2715

\(\displaystyle -\frac {b \int \frac {(e+f x)^2 \text {csch}(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {4 \left (\frac {(e+f x)^2 \coth (2 c+2 d x)}{2 d}+\frac {i f \left (2 i \left (\frac {(e+f x) \log \left (1+e^{4 c+4 d x-i \pi }\right )}{4 d}-\frac {f \int e^{-4 c-4 d x+i \pi } \log \left (1+e^{4 c+4 d x-i \pi }\right )de^{4 c+4 d x-i \pi }}{16 d^2}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}\right )}{a}\)

\(\Big \downarrow \) 2838

\(\displaystyle -\frac {b \int \frac {(e+f x)^2 \text {csch}(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {4 \left (\frac {(e+f x)^2 \coth (2 c+2 d x)}{2 d}+\frac {i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{4 c+4 d x-i \pi }\right )}{16 d^2}+\frac {(e+f x) \log \left (1+e^{4 c+4 d x-i \pi }\right )}{4 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}\right )}{a}\)

\(\Big \downarrow \) 6123

\(\displaystyle -\frac {b \left (\frac {\int (e+f x)^2 \text {csch}(c+d x) \text {sech}^2(c+d x)dx}{a}-\frac {b \int \frac {(e+f x)^2 \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}\right )}{a}-\frac {4 \left (\frac {(e+f x)^2 \coth (2 c+2 d x)}{2 d}+\frac {i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{4 c+4 d x-i \pi }\right )}{16 d^2}+\frac {(e+f x) \log \left (1+e^{4 c+4 d x-i \pi }\right )}{4 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}\right )}{a}\)

\(\Big \downarrow \) 5985

\(\displaystyle -\frac {b \left (\frac {-2 f \int -\left ((e+f x) \left (\frac {\text {arctanh}(\cosh (c+d x))}{d}-\frac {\text {sech}(c+d x)}{d}\right )\right )dx-\frac {(e+f x)^2 \text {arctanh}(\cosh (c+d x))}{d}+\frac {(e+f x)^2 \text {sech}(c+d x)}{d}}{a}-\frac {b \int \frac {(e+f x)^2 \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}\right )}{a}-\frac {4 \left (\frac {(e+f x)^2 \coth (2 c+2 d x)}{2 d}+\frac {i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{4 c+4 d x-i \pi }\right )}{16 d^2}+\frac {(e+f x) \log \left (1+e^{4 c+4 d x-i \pi }\right )}{4 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}\right )}{a}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {b \left (\frac {2 f \int (e+f x) \left (\frac {\text {arctanh}(\cosh (c+d x))}{d}-\frac {\text {sech}(c+d x)}{d}\right )dx-\frac {(e+f x)^2 \text {arctanh}(\cosh (c+d x))}{d}+\frac {(e+f x)^2 \text {sech}(c+d x)}{d}}{a}-\frac {b \int \frac {(e+f x)^2 \text {sech}^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}\right )}{a}-\frac {4 \left (\frac {(e+f x)^2 \coth (2 c+2 d x)}{2 d}+\frac {i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{4 c+4 d x-i \pi }\right )}{16 d^2}+\frac {(e+f x) \log \left (1+e^{4 c+4 d x-i \pi }\right )}{4 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}\right )}{a}\)

\(\Big \downarrow \) 6107

\(\displaystyle -\frac {b \left (\frac {2 f \int (e+f x) \left (\frac {\text {arctanh}(\cosh (c+d x))}{d}-\frac {\text {sech}(c+d x)}{d}\right )dx-\frac {(e+f x)^2 \text {arctanh}(\cosh (c+d x))}{d}+\frac {(e+f x)^2 \text {sech}(c+d x)}{d}}{a}-\frac {b \left (\frac {b^2 \int \frac {(e+f x)^2}{a+b \sinh (c+d x)}dx}{a^2+b^2}+\frac {\int (e+f x)^2 \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{a}\right )}{a}-\frac {4 \left (\frac {(e+f x)^2 \coth (2 c+2 d x)}{2 d}+\frac {i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{4 c+4 d x-i \pi }\right )}{16 d^2}+\frac {(e+f x) \log \left (1+e^{4 c+4 d x-i \pi }\right )}{4 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}\right )}{a}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {b \left (\frac {2 f \int (e+f x) \left (\frac {\text {arctanh}(\cosh (c+d x))}{d}-\frac {\text {sech}(c+d x)}{d}\right )dx-\frac {(e+f x)^2 \text {arctanh}(\cosh (c+d x))}{d}+\frac {(e+f x)^2 \text {sech}(c+d x)}{d}}{a}-\frac {b \left (\frac {\int (e+f x)^2 \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}+\frac {b^2 \int \frac {(e+f x)^2}{a-i b \sin (i c+i d x)}dx}{a^2+b^2}\right )}{a}\right )}{a}-\frac {4 \left (\frac {(e+f x)^2 \coth (2 c+2 d x)}{2 d}+\frac {i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{4 c+4 d x-i \pi }\right )}{16 d^2}+\frac {(e+f x) \log \left (1+e^{4 c+4 d x-i \pi }\right )}{4 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}\right )}{a}\)

\(\Big \downarrow \) 3803

\(\displaystyle -\frac {b \left (\frac {2 f \int (e+f x) \left (\frac {\text {arctanh}(\cosh (c+d x))}{d}-\frac {\text {sech}(c+d x)}{d}\right )dx-\frac {(e+f x)^2 \text {arctanh}(\cosh (c+d x))}{d}+\frac {(e+f x)^2 \text {sech}(c+d x)}{d}}{a}-\frac {b \left (\frac {2 b^2 \int -\frac {e^{c+d x} (e+f x)^2}{-2 e^{c+d x} a-b e^{2 (c+d x)}+b}dx}{a^2+b^2}+\frac {\int (e+f x)^2 \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{a}\right )}{a}-\frac {4 \left (\frac {(e+f x)^2 \coth (2 c+2 d x)}{2 d}+\frac {i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{4 c+4 d x-i \pi }\right )}{16 d^2}+\frac {(e+f x) \log \left (1+e^{4 c+4 d x-i \pi }\right )}{4 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}\right )}{a}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {b \left (\frac {2 f \int (e+f x) \left (\frac {\text {arctanh}(\cosh (c+d x))}{d}-\frac {\text {sech}(c+d x)}{d}\right )dx-\frac {(e+f x)^2 \text {arctanh}(\cosh (c+d x))}{d}+\frac {(e+f x)^2 \text {sech}(c+d x)}{d}}{a}-\frac {b \left (\frac {\int (e+f x)^2 \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}-\frac {2 b^2 \int \frac {e^{c+d x} (e+f x)^2}{-2 e^{c+d x} a-b e^{2 (c+d x)}+b}dx}{a^2+b^2}\right )}{a}\right )}{a}-\frac {4 \left (\frac {(e+f x)^2 \coth (2 c+2 d x)}{2 d}+\frac {i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{4 c+4 d x-i \pi }\right )}{16 d^2}+\frac {(e+f x) \log \left (1+e^{4 c+4 d x-i \pi }\right )}{4 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}\right )}{a}\)

\(\Big \downarrow \) 2694

\(\displaystyle -\frac {b \left (\frac {2 f \int (e+f x) \left (\frac {\text {arctanh}(\cosh (c+d x))}{d}-\frac {\text {sech}(c+d x)}{d}\right )dx-\frac {(e+f x)^2 \text {arctanh}(\cosh (c+d x))}{d}+\frac {(e+f x)^2 \text {sech}(c+d x)}{d}}{a}-\frac {b \left (\frac {\int (e+f x)^2 \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}-\frac {2 b^2 \left (\frac {b \int -\frac {e^{c+d x} (e+f x)^2}{2 \left (a+b e^{c+d x}-\sqrt {a^2+b^2}\right )}dx}{\sqrt {a^2+b^2}}-\frac {b \int -\frac {e^{c+d x} (e+f x)^2}{2 \left (a+b e^{c+d x}+\sqrt {a^2+b^2}\right )}dx}{\sqrt {a^2+b^2}}\right )}{a^2+b^2}\right )}{a}\right )}{a}-\frac {4 \left (\frac {(e+f x)^2 \coth (2 c+2 d x)}{2 d}+\frac {i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{4 c+4 d x-i \pi }\right )}{16 d^2}+\frac {(e+f x) \log \left (1+e^{4 c+4 d x-i \pi }\right )}{4 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}\right )}{a}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {b \left (\frac {2 f \int (e+f x) \left (\frac {\text {arctanh}(\cosh (c+d x))}{d}-\frac {\text {sech}(c+d x)}{d}\right )dx-\frac {(e+f x)^2 \text {arctanh}(\cosh (c+d x))}{d}+\frac {(e+f x)^2 \text {sech}(c+d x)}{d}}{a}-\frac {b \left (\frac {\int (e+f x)^2 \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}-\frac {2 b^2 \left (\frac {b \int \frac {e^{c+d x} (e+f x)^2}{a+b e^{c+d x}+\sqrt {a^2+b^2}}dx}{2 \sqrt {a^2+b^2}}-\frac {b \int \frac {e^{c+d x} (e+f x)^2}{a+b e^{c+d x}-\sqrt {a^2+b^2}}dx}{2 \sqrt {a^2+b^2}}\right )}{a^2+b^2}\right )}{a}\right )}{a}-\frac {4 \left (\frac {(e+f x)^2 \coth (2 c+2 d x)}{2 d}+\frac {i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{4 c+4 d x-i \pi }\right )}{16 d^2}+\frac {(e+f x) \log \left (1+e^{4 c+4 d x-i \pi }\right )}{4 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}\right )}{a}\)

\(\Big \downarrow \) 2620

\(\displaystyle -\frac {b \left (\frac {2 f \int (e+f x) \left (\frac {\text {arctanh}(\cosh (c+d x))}{d}-\frac {\text {sech}(c+d x)}{d}\right )dx-\frac {(e+f x)^2 \text {arctanh}(\cosh (c+d x))}{d}+\frac {(e+f x)^2 \text {sech}(c+d x)}{d}}{a}-\frac {b \left (\frac {\int (e+f x)^2 \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}-\frac {2 b^2 \left (\frac {b \left (\frac {(e+f x)^2 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {2 f \int (e+f x) \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right )dx}{b d}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {(e+f x)^2 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}-\frac {2 f \int (e+f x) \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right )dx}{b d}\right )}{2 \sqrt {a^2+b^2}}\right )}{a^2+b^2}\right )}{a}\right )}{a}-\frac {4 \left (\frac {(e+f x)^2 \coth (2 c+2 d x)}{2 d}+\frac {i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{4 c+4 d x-i \pi }\right )}{16 d^2}+\frac {(e+f x) \log \left (1+e^{4 c+4 d x-i \pi }\right )}{4 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}\right )}{a}\)

\(\Big \downarrow \) 3011

\(\displaystyle -\frac {b \left (\frac {2 f \int (e+f x) \left (\frac {\text {arctanh}(\cosh (c+d x))}{d}-\frac {\text {sech}(c+d x)}{d}\right )dx-\frac {(e+f x)^2 \text {arctanh}(\cosh (c+d x))}{d}+\frac {(e+f x)^2 \text {sech}(c+d x)}{d}}{a}-\frac {b \left (\frac {\int (e+f x)^2 \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}-\frac {2 b^2 \left (\frac {b \left (\frac {(e+f x)^2 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {2 f \left (\frac {f \int \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )dx}{d}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {(e+f x)^2 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}-\frac {2 f \left (\frac {f \int \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )dx}{d}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}\right )}{a^2+b^2}\right )}{a}\right )}{a}-\frac {4 \left (\frac {(e+f x)^2 \coth (2 c+2 d x)}{2 d}+\frac {i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{4 c+4 d x-i \pi }\right )}{16 d^2}+\frac {(e+f x) \log \left (1+e^{4 c+4 d x-i \pi }\right )}{4 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}\right )}{a}\)

\(\Big \downarrow \) 2720

\(\displaystyle -\frac {b \left (\frac {2 f \int (e+f x) \left (\frac {\text {arctanh}(\cosh (c+d x))}{d}-\frac {\text {sech}(c+d x)}{d}\right )dx-\frac {(e+f x)^2 \text {arctanh}(\cosh (c+d x))}{d}+\frac {(e+f x)^2 \text {sech}(c+d x)}{d}}{a}-\frac {b \left (\frac {\int (e+f x)^2 \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}-\frac {2 b^2 \left (\frac {b \left (\frac {(e+f x)^2 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {2 f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {(e+f x)^2 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}-\frac {2 f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}\right )}{a^2+b^2}\right )}{a}\right )}{a}-\frac {4 \left (\frac {(e+f x)^2 \coth (2 c+2 d x)}{2 d}+\frac {i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{4 c+4 d x-i \pi }\right )}{16 d^2}+\frac {(e+f x) \log \left (1+e^{4 c+4 d x-i \pi }\right )}{4 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}\right )}{a}\)

\(\Big \downarrow \) 7143

\(\displaystyle -\frac {b \left (\frac {2 f \int (e+f x) \left (\frac {\text {arctanh}(\cosh (c+d x))}{d}-\frac {\text {sech}(c+d x)}{d}\right )dx-\frac {(e+f x)^2 \text {arctanh}(\cosh (c+d x))}{d}+\frac {(e+f x)^2 \text {sech}(c+d x)}{d}}{a}-\frac {b \left (\frac {\int (e+f x)^2 \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}-\frac {2 b^2 \left (\frac {b \left (\frac {(e+f x)^2 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {2 f \left (\frac {f \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {(e+f x)^2 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}-\frac {2 f \left (\frac {f \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}\right )}{a^2+b^2}\right )}{a}\right )}{a}-\frac {4 \left (\frac {(e+f x)^2 \coth (2 c+2 d x)}{2 d}+\frac {i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{4 c+4 d x-i \pi }\right )}{16 d^2}+\frac {(e+f x) \log \left (1+e^{4 c+4 d x-i \pi }\right )}{4 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}\right )}{a}\)

\(\Big \downarrow \) 7292

\(\displaystyle -\frac {b \left (\frac {2 f \int \frac {(e+f x) (\text {arctanh}(\cosh (c+d x))-\text {sech}(c+d x))}{d}dx-\frac {(e+f x)^2 \text {arctanh}(\cosh (c+d x))}{d}+\frac {(e+f x)^2 \text {sech}(c+d x)}{d}}{a}-\frac {b \left (\frac {\int (e+f x)^2 \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}-\frac {2 b^2 \left (\frac {b \left (\frac {(e+f x)^2 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {2 f \left (\frac {f \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {(e+f x)^2 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}-\frac {2 f \left (\frac {f \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}\right )}{a^2+b^2}\right )}{a}\right )}{a}-\frac {4 \left (\frac {(e+f x)^2 \coth (2 c+2 d x)}{2 d}+\frac {i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{4 c+4 d x-i \pi }\right )}{16 d^2}+\frac {(e+f x) \log \left (1+e^{4 c+4 d x-i \pi }\right )}{4 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}\right )}{a}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {b \left (\frac {\frac {2 f \int (e+f x) (\text {arctanh}(\cosh (c+d x))-\text {sech}(c+d x))dx}{d}-\frac {(e+f x)^2 \text {arctanh}(\cosh (c+d x))}{d}+\frac {(e+f x)^2 \text {sech}(c+d x)}{d}}{a}-\frac {b \left (\frac {\int (e+f x)^2 \text {sech}^2(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}-\frac {2 b^2 \left (\frac {b \left (\frac {(e+f x)^2 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {2 f \left (\frac {f \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {(e+f x)^2 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}-\frac {2 f \left (\frac {f \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}\right )}{a^2+b^2}\right )}{a}\right )}{a}-\frac {4 \left (\frac {(e+f x)^2 \coth (2 c+2 d x)}{2 d}+\frac {i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{4 c+4 d x-i \pi }\right )}{16 d^2}+\frac {(e+f x) \log \left (1+e^{4 c+4 d x-i \pi }\right )}{4 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}\right )}{a}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {b \left (\frac {\frac {2 f \int ((e+f x) \text {arctanh}(\cosh (c+d x))-(e+f x) \text {sech}(c+d x))dx}{d}-\frac {(e+f x)^2 \text {arctanh}(\cosh (c+d x))}{d}+\frac {(e+f x)^2 \text {sech}(c+d x)}{d}}{a}-\frac {b \left (\frac {\int \left (a (e+f x)^2 \text {sech}^2(c+d x)-b (e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)\right )dx}{a^2+b^2}-\frac {2 b^2 \left (\frac {b \left (\frac {(e+f x)^2 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {2 f \left (\frac {f \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {(e+f x)^2 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}-\frac {2 f \left (\frac {f \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}\right )}{a^2+b^2}\right )}{a}\right )}{a}-\frac {4 \left (\frac {(e+f x)^2 \coth (2 c+2 d x)}{2 d}+\frac {i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{4 c+4 d x-i \pi }\right )}{16 d^2}+\frac {(e+f x) \log \left (1+e^{4 c+4 d x-i \pi }\right )}{4 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}\right )}{a}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {4 \left (\frac {\coth (2 c+2 d x) (e+f x)^2}{2 d}+\frac {i f \left (2 i \left (\frac {(e+f x) \log \left (1+e^{4 c+4 d x-i \pi }\right )}{4 d}+\frac {f \operatorname {PolyLog}\left (2,-e^{4 c+4 d x-i \pi }\right )}{16 d^2}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}\right )}{a}-\frac {b \left (\frac {-\frac {\text {arctanh}(\cosh (c+d x)) (e+f x)^2}{d}+\frac {\text {sech}(c+d x) (e+f x)^2}{d}+\frac {2 f \left (-\frac {\text {arctanh}\left (e^{c+d x}\right ) (e+f x)^2}{f}+\frac {\text {arctanh}(\cosh (c+d x)) (e+f x)^2}{2 f}-\frac {2 \arctan \left (e^{c+d x}\right ) (e+f x)}{d}-\frac {\operatorname {PolyLog}\left (2,-e^{c+d x}\right ) (e+f x)}{d}+\frac {\operatorname {PolyLog}\left (2,e^{c+d x}\right ) (e+f x)}{d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}+\frac {f \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{d^2}-\frac {f \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{d^2}\right )}{d}}{a}-\frac {b \left (\frac {\frac {2 i b \operatorname {PolyLog}\left (2,-i e^{c+d x}\right ) f^2}{d^3}-\frac {2 i b \operatorname {PolyLog}\left (2,i e^{c+d x}\right ) f^2}{d^3}-\frac {a \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right ) f^2}{d^3}-\frac {4 b (e+f x) \arctan \left (e^{c+d x}\right ) f}{d^2}-\frac {2 a (e+f x) \log \left (1+e^{2 (c+d x)}\right ) f}{d^2}+\frac {a (e+f x)^2}{d}+\frac {b (e+f x)^2 \text {sech}(c+d x)}{d}+\frac {a (e+f x)^2 \tanh (c+d x)}{d}}{a^2+b^2}-\frac {2 b^2 \left (\frac {b \left (\frac {(e+f x)^2 \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right )}{b d}-\frac {2 f \left (\frac {f \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {(e+f x)^2 \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right )}{b d}-\frac {2 f \left (\frac {f \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}\right )}{a^2+b^2}\right )}{a}\right )}{a}\)

Input:

Int[((e + f*x)^2*Csch[c + d*x]^2*Sech[c + d*x]^2)/(a + b*Sinh[c + d*x]),x]
 

Output:

(-4*(((e + f*x)^2*Coth[2*c + 2*d*x])/(2*d) + (I*f*(((-1/2*I)*(e + f*x)^2)/ 
f + (2*I)*(((e + f*x)*Log[1 + E^(4*c - I*Pi + 4*d*x)])/(4*d) + (f*PolyLog[ 
2, -E^(4*c - I*Pi + 4*d*x)])/(16*d^2))))/d))/a - (b*((-(((e + f*x)^2*ArcTa 
nh[Cosh[c + d*x]])/d) + (2*f*((-2*(e + f*x)*ArcTan[E^(c + d*x)])/d - ((e + 
 f*x)^2*ArcTanh[E^(c + d*x)])/f + ((e + f*x)^2*ArcTanh[Cosh[c + d*x]])/(2* 
f) - ((e + f*x)*PolyLog[2, -E^(c + d*x)])/d + (I*f*PolyLog[2, (-I)*E^(c + 
d*x)])/d^2 - (I*f*PolyLog[2, I*E^(c + d*x)])/d^2 + ((e + f*x)*PolyLog[2, E 
^(c + d*x)])/d + (f*PolyLog[3, -E^(c + d*x)])/d^2 - (f*PolyLog[3, E^(c + d 
*x)])/d^2))/d + ((e + f*x)^2*Sech[c + d*x])/d)/a - (b*((-2*b^2*(-1/2*(b*(( 
(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b*d) - (2*f*( 
-(((e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/d) + (f 
*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/d^2))/(b*d)))/Sqrt[ 
a^2 + b^2] + (b*(((e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2] 
)])/(b*d) - (2*f*(-(((e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 
+ b^2]))])/d) + (f*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/d 
^2))/(b*d)))/(2*Sqrt[a^2 + b^2])))/(a^2 + b^2) + ((a*(e + f*x)^2)/d - (4*b 
*f*(e + f*x)*ArcTan[E^(c + d*x)])/d^2 - (2*a*f*(e + f*x)*Log[1 + E^(2*(c + 
 d*x))])/d^2 + ((2*I)*b*f^2*PolyLog[2, (-I)*E^(c + d*x)])/d^3 - ((2*I)*b*f 
^2*PolyLog[2, I*E^(c + d*x)])/d^3 - (a*f^2*PolyLog[2, -E^(2*(c + d*x))])/d 
^3 + (b*(e + f*x)^2*Sech[c + d*x])/d + (a*(e + f*x)^2*Tanh[c + d*x])/d)...
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2620

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ 
((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp 
[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si 
mp[d*(m/(b*f*g*n*Log[F]))   Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x 
)))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
 

rule 2694

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.) 
*(F_)^(v_)), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[2*(c/q)   Int 
[(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Simp[2*(c/q)   Int[(f + g*x) 
^m*(F^u/(b + q + 2*c*F^u)), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[ 
v, 2*u] && LinearQ[u, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[m, 0]
 

rule 2715

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[1/(d*e*n*Log[F])   Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) 
))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
 

rule 2720

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] 
   Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct 
ionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ 
[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) 
*(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
 

rule 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

rule 3011

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) 
*(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + 
b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F]))   Int[(f + g*x)^( 
m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e 
, f, g, n}, x] && GtQ[m, 0]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3803

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (Complex[0, fz_])* 
(f_.)*(x_)]), x_Symbol] :> Simp[2   Int[(c + d*x)^m*(E^((-I)*e + f*fz*x)/(( 
-I)*b + 2*a*E^((-I)*e + f*fz*x) + I*b*E^(2*((-I)*e + f*fz*x)))), x], x] /; 
FreeQ[{a, b, c, d, e, f, fz}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]
 

rule 4201

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (Complex[0, fz_])*(f_.)*(x_)], x 
_Symbol] :> Simp[(-I)*((c + d*x)^(m + 1)/(d*(m + 1))), x] + Simp[2*I   Int[ 
(c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*fz*x)))), x], x] 
 /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
 

rule 4672

Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp 
[(-(c + d*x)^m)*(Cot[e + f*x]/f), x] + Simp[d*(m/f)   Int[(c + d*x)^(m - 1) 
*Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
 

rule 5984

Int[Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + 
(b_.)*(x_)]^(n_.), x_Symbol] :> Simp[2^n   Int[(c + d*x)^m*Csch[2*a + 2*b*x 
]^n, x], x] /; FreeQ[{a, b, c, d}, x] && RationalQ[m] && IntegerQ[n]
 

rule 5985

Int[Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + 
(b_.)*(x_)]^(p_.), x_Symbol] :> With[{u = IntHide[Csch[a + b*x]^n*Sech[a + 
b*x]^p, x]}, Simp[(c + d*x)^m   u, x] - Simp[d*m   Int[(c + d*x)^(m - 1)*u, 
 x], x]] /; FreeQ[{a, b, c, d}, x] && IntegersQ[n, p] && GtQ[m, 0] && NeQ[n 
, p]
 

rule 6107

Int[(((e_.) + (f_.)*(x_))^(m_.)*Sech[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_ 
.)*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[b^2/(a^2 + b^2)   Int[(e + 
f*x)^m*(Sech[c + d*x]^(n - 2)/(a + b*Sinh[c + d*x])), x], x] + Simp[1/(a^2 
+ b^2)   Int[(e + f*x)^m*Sech[c + d*x]^n*(a - b*Sinh[c + d*x]), x], x] /; F 
reeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[a^2 + b^2, 0] && IGtQ[n, 0 
]
 

rule 6123

Int[(Csch[(c_.) + (d_.)*(x_)]^(n_.)*((e_.) + (f_.)*(x_))^(m_.)*Sech[(c_.) + 
 (d_.)*(x_)]^(p_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :> S 
imp[1/a   Int[(e + f*x)^m*Sech[c + d*x]^p*Csch[c + d*x]^n, x], x] - Simp[b/ 
a   Int[(e + f*x)^m*Sech[c + d*x]^p*(Csch[c + d*x]^(n - 1)/(a + b*Sinh[c + 
d*x])), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] 
&& IGtQ[p, 0]
 

rule 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d 
, e, n, p}, x] && EqQ[b*d, a*e]
 

rule 7292

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =! 
= u]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

Maple [F]

\[\int \frac {\left (f x +e \right )^{2} \operatorname {csch}\left (d x +c \right )^{2} \operatorname {sech}\left (d x +c \right )^{2}}{a +b \sinh \left (d x +c \right )}d x\]

Input:

int((f*x+e)^2*csch(d*x+c)^2*sech(d*x+c)^2/(a+b*sinh(d*x+c)),x)
 

Output:

int((f*x+e)^2*csch(d*x+c)^2*sech(d*x+c)^2/(a+b*sinh(d*x+c)),x)
 

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 10432 vs. \(2 (845) = 1690\).

Time = 0.33 (sec) , antiderivative size = 10432, normalized size of antiderivative = 11.41 \[ \int \frac {(e+f x)^2 \text {csch}^2(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \] Input:

integrate((f*x+e)^2*csch(d*x+c)^2*sech(d*x+c)^2/(a+b*sinh(d*x+c)),x, algor 
ithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(e+f x)^2 \text {csch}^2(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \] Input:

integrate((f*x+e)**2*csch(d*x+c)**2*sech(d*x+c)**2/(a+b*sinh(d*x+c)),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {(e+f x)^2 \text {csch}^2(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{2} \operatorname {csch}\left (d x + c\right )^{2} \operatorname {sech}\left (d x + c\right )^{2}}{b \sinh \left (d x + c\right ) + a} \,d x } \] Input:

integrate((f*x+e)^2*csch(d*x+c)^2*sech(d*x+c)^2/(a+b*sinh(d*x+c)),x, algor 
ithm="maxima")
 

Output:

-2*a*e*f*(2*(d*x + c)/((a^2 + b^2)*d^2) - log(e^(2*d*x + 2*c) + 1)/((a^2 + 
 b^2)*d^2)) + 4*b*f^2*integrate(x*e^(d*x + c)/(a^2*d*e^(2*d*x + 2*c) + b^2 
*d*e^(2*d*x + 2*c) + a^2*d + b^2*d), x) - 4*a*f^2*integrate(x/(a^2*d*e^(2* 
d*x + 2*c) + b^2*d*e^(2*d*x + 2*c) + a^2*d + b^2*d), x) + (b^4*log((b*e^(- 
d*x - c) - a - sqrt(a^2 + b^2))/(b*e^(-d*x - c) - a + sqrt(a^2 + b^2)))/(( 
a^4 + a^2*b^2)*sqrt(a^2 + b^2)*d) - 2*(a*b*e^(-d*x - c) + b^2*e^(-2*d*x - 
2*c) - a*b*e^(-3*d*x - 3*c) + 2*a^2 + b^2)/((a^3 + a*b^2 - (a^3 + a*b^2)*e 
^(-4*d*x - 4*c))*d) + b*log(e^(-d*x - c) + 1)/(a^2*d) - b*log(e^(-d*x - c) 
 - 1)/(a^2*d))*e^2 - 4*e*f*x/(a*d) + 4*b*e*f*arctan(e^(d*x + c))/((a^2 + b 
^2)*d^2) + 2*((2*a^2*f^2 + b^2*f^2)*x^2 + 2*(2*a^2*e*f + b^2*e*f)*x + (a*b 
*f^2*x^2*e^(3*c) + 2*a*b*e*f*x*e^(3*c))*e^(3*d*x) + (b^2*f^2*x^2*e^(2*c) + 
 2*b^2*e*f*x*e^(2*c))*e^(2*d*x) - (a*b*f^2*x^2*e^c + 2*a*b*e*f*x*e^c)*e^(d 
*x))/(a^3*d + a*b^2*d - (a^3*d*e^(4*c) + a*b^2*d*e^(4*c))*e^(4*d*x)) + 2*e 
*f*log(e^(d*x + c) + 1)/(a*d^2) + 2*e*f*log(e^(d*x + c) - 1)/(a*d^2) + (d^ 
2*x^2*log(e^(d*x + c) + 1) + 2*d*x*dilog(-e^(d*x + c)) - 2*polylog(3, -e^( 
d*x + c)))*b*f^2/(a^2*d^3) - (d^2*x^2*log(-e^(d*x + c) + 1) + 2*d*x*dilog( 
e^(d*x + c)) - 2*polylog(3, e^(d*x + c)))*b*f^2/(a^2*d^3) + 2*(b*d*e*f + a 
*f^2)*(d*x*log(e^(d*x + c) + 1) + dilog(-e^(d*x + c)))/(a^2*d^3) - 2*(b*d* 
e*f - a*f^2)*(d*x*log(-e^(d*x + c) + 1) + dilog(e^(d*x + c)))/(a^2*d^3) - 
1/3*(b*d^3*f^2*x^3 + 3*(b*d*e*f + a*f^2)*d^2*x^2)/(a^2*d^3) + 1/3*(b*d^...
 

Giac [F(-1)]

Timed out. \[ \int \frac {(e+f x)^2 \text {csch}^2(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \] Input:

integrate((f*x+e)^2*csch(d*x+c)^2*sech(d*x+c)^2/(a+b*sinh(d*x+c)),x, algor 
ithm="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(e+f x)^2 \text {csch}^2(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {{\left (e+f\,x\right )}^2}{{\mathrm {cosh}\left (c+d\,x\right )}^2\,{\mathrm {sinh}\left (c+d\,x\right )}^2\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \] Input:

int((e + f*x)^2/(cosh(c + d*x)^2*sinh(c + d*x)^2*(a + b*sinh(c + d*x))),x)
 

Output:

int((e + f*x)^2/(cosh(c + d*x)^2*sinh(c + d*x)^2*(a + b*sinh(c + d*x))), x 
)
 

Reduce [F]

\[ \int \frac {(e+f x)^2 \text {csch}^2(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {too large to display} \] Input:

int((f*x+e)^2*csch(d*x+c)^2*sech(d*x+c)^2/(a+b*sinh(d*x+c)),x)
 

Output:

(2*e**(4*c + 4*d*x)*sqrt(a**2 + b**2)*atan((e**(c + d*x)*b*i + a*i)/sqrt(a 
**2 + b**2))*b**4*e**2*i - 2*sqrt(a**2 + b**2)*atan((e**(c + d*x)*b*i + a* 
i)/sqrt(a**2 + b**2))*b**4*e**2*i + 32*e**(9*c + 4*d*x)*int((e**(5*d*x)*x* 
*2)/(e**(10*c + 10*d*x)*b + 2*e**(9*c + 9*d*x)*a - e**(8*c + 8*d*x)*b - 2* 
e**(6*c + 6*d*x)*b - 4*e**(5*c + 5*d*x)*a + 2*e**(4*c + 4*d*x)*b + e**(2*c 
 + 2*d*x)*b + 2*e**(c + d*x)*a - b),x)*a**6*d*f**2 + 64*e**(9*c + 4*d*x)*i 
nt((e**(5*d*x)*x**2)/(e**(10*c + 10*d*x)*b + 2*e**(9*c + 9*d*x)*a - e**(8* 
c + 8*d*x)*b - 2*e**(6*c + 6*d*x)*b - 4*e**(5*c + 5*d*x)*a + 2*e**(4*c + 4 
*d*x)*b + e**(2*c + 2*d*x)*b + 2*e**(c + d*x)*a - b),x)*a**4*b**2*d*f**2 + 
 32*e**(9*c + 4*d*x)*int((e**(5*d*x)*x**2)/(e**(10*c + 10*d*x)*b + 2*e**(9 
*c + 9*d*x)*a - e**(8*c + 8*d*x)*b - 2*e**(6*c + 6*d*x)*b - 4*e**(5*c + 5* 
d*x)*a + 2*e**(4*c + 4*d*x)*b + e**(2*c + 2*d*x)*b + 2*e**(c + d*x)*a - b) 
,x)*a**2*b**4*d*f**2 + 64*e**(9*c + 4*d*x)*int((e**(5*d*x)*x)/(e**(10*c + 
10*d*x)*b + 2*e**(9*c + 9*d*x)*a - e**(8*c + 8*d*x)*b - 2*e**(6*c + 6*d*x) 
*b - 4*e**(5*c + 5*d*x)*a + 2*e**(4*c + 4*d*x)*b + e**(2*c + 2*d*x)*b + 2* 
e**(c + d*x)*a - b),x)*a**6*d*e*f + 128*e**(9*c + 4*d*x)*int((e**(5*d*x)*x 
)/(e**(10*c + 10*d*x)*b + 2*e**(9*c + 9*d*x)*a - e**(8*c + 8*d*x)*b - 2*e* 
*(6*c + 6*d*x)*b - 4*e**(5*c + 5*d*x)*a + 2*e**(4*c + 4*d*x)*b + e**(2*c + 
 2*d*x)*b + 2*e**(c + d*x)*a - b),x)*a**4*b**2*d*e*f + 64*e**(9*c + 4*d*x) 
*int((e**(5*d*x)*x)/(e**(10*c + 10*d*x)*b + 2*e**(9*c + 9*d*x)*a - e**(...