\(\int \frac {(e+f x)^2 \text {csch}^2(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx\) [465]

Optimal result

Integrand size = 34, antiderivative size = 982 \[ \int \frac {(e+f x)^2 \text {csch}^2(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx =\text {Too large to display} \] Output:

-2*(f*x+e)^2*arctan(exp(d*x+c))/a/d+2*I*f*(f*x+e)*polylog(2,-I*exp(d*x+c)) 
/a/d^2+2*I*f^2*polylog(3,I*exp(d*x+c))/a/d^3+b*f*(f*x+e)*polylog(2,-exp(2* 
d*x+2*c))/a^2/d^2-b^3*(f*x+e)^2*ln(1+exp(2*d*x+2*c))/a^2/(a^2+b^2)/d+2*I*b 
^2*f*(f*x+e)*polylog(2,I*exp(d*x+c))/a/(a^2+b^2)/d^2+2*I*b^2*f^2*polylog(3 
,-I*exp(d*x+c))/a/(a^2+b^2)/d^3-b^3*f*(f*x+e)*polylog(2,-exp(2*d*x+2*c))/a 
^2/(a^2+b^2)/d^2+b^3*(f*x+e)^2*ln(1+b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a^2/ 
(a^2+b^2)/d+b^3*(f*x+e)^2*ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a^2/(a^2+ 
b^2)/d-2*b^3*f^2*polylog(3,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a^2/(a^2+b^2 
)/d^3-2*b^3*f^2*polylog(3,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a^2/(a^2+b^2) 
/d^3-(f*x+e)^2*csch(d*x+c)/a/d-2*I*b^2*f*(f*x+e)*polylog(2,-I*exp(d*x+c))/ 
a/(a^2+b^2)/d^2-2*f^2*polylog(2,-exp(d*x+c))/a/d^3+2*f^2*polylog(2,exp(d*x 
+c))/a/d^3-2*I*b^2*f^2*polylog(3,I*exp(d*x+c))/a/(a^2+b^2)/d^3-4*f*(f*x+e) 
*arctanh(exp(d*x+c))/a/d^2+1/2*b*f^2*polylog(3,exp(2*d*x+2*c))/a^2/d^3-b*f 
*(f*x+e)*polylog(2,exp(2*d*x+2*c))/a^2/d^2-1/2*b*f^2*polylog(3,-exp(2*d*x+ 
2*c))/a^2/d^3+2*b*(f*x+e)^2*arctanh(exp(2*d*x+2*c))/a^2/d-2*I*f^2*polylog( 
3,-I*exp(d*x+c))/a/d^3+2*b^3*f*(f*x+e)*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2 
)^(1/2)))/a^2/(a^2+b^2)/d^2+2*b^3*f*(f*x+e)*polylog(2,-b*exp(d*x+c)/(a-(a^ 
2+b^2)^(1/2)))/a^2/(a^2+b^2)/d^2+1/2*b^3*f^2*polylog(3,-exp(2*d*x+2*c))/a^ 
2/(a^2+b^2)/d^3+2*b^2*(f*x+e)^2*arctan(exp(d*x+c))/a/(a^2+b^2)/d-2*I*f*(f* 
x+e)*polylog(2,I*exp(d*x+c))/a/d^2
 

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(2323\) vs. \(2(982)=1964\).

Time = 10.30 (sec) , antiderivative size = 2323, normalized size of antiderivative = 2.37 \[ \int \frac {(e+f x)^2 \text {csch}^2(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Result too large to show} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 4.87 (sec) , antiderivative size = 889, normalized size of antiderivative = 0.91, number of steps used = 21, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.588, Rules used = {6123, 5985, 25, 6123, 5984, 3042, 26, 4670, 3011, 2720, 6107, 6095, 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.

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

\(\Big \downarrow \) 6123

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

\(\Big \downarrow \) 5985

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 6123

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

\(\Big \downarrow \) 5984

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4670

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

\(\Big \downarrow \) 3011

\(\displaystyle \frac {2 f \int (e+f x) \left (\frac {\arctan (\sinh (c+d x))}{d}+\frac {\text {csch}(c+d x)}{d}\right )dx-\frac {(e+f x)^2 \arctan (\sinh (c+d x))}{d}-\frac {(e+f x)^2 \text {csch}(c+d x)}{d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^2 \text {sech}(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {2 i \left (-\frac {i f \left (\frac {f \int \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )dx}{2 d}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 d}\right )}{d}+\frac {i f \left (\frac {f \int \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )dx}{2 d}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 d}\right )}{d}+\frac {i (e+f x)^2 \text {arctanh}\left (e^{2 c+2 d x}\right )}{d}\right )}{a}\right )}{a}\)

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 6107

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

\(\Big \downarrow \) 6095

\(\displaystyle \frac {2 f \int (e+f x) \left (\frac {\arctan (\sinh (c+d x))}{d}+\frac {\text {csch}(c+d x)}{d}\right )dx-\frac {(e+f x)^2 \arctan (\sinh (c+d x))}{d}-\frac {(e+f x)^2 \text {csch}(c+d x)}{d}}{a}-\frac {b \left (-\frac {b \left (\frac {b^2 \left (\int \frac {e^{c+d x} (e+f x)^2}{a+b e^{c+d x}-\sqrt {a^2+b^2}}dx+\int \frac {e^{c+d x} (e+f x)^2}{a+b e^{c+d x}+\sqrt {a^2+b^2}}dx-\frac {(e+f x)^3}{3 b f}\right )}{a^2+b^2}+\frac {\int (e+f x)^2 \text {sech}(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{a}+\frac {2 i \left (-\frac {i f \left (\frac {f \int e^{-2 c-2 d x} \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )de^{2 c+2 d x}}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 d}\right )}{d}+\frac {i f \left (\frac {f \int e^{-2 c-2 d x} \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )de^{2 c+2 d x}}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 d}\right )}{d}+\frac {i (e+f x)^2 \text {arctanh}\left (e^{2 c+2 d x}\right )}{d}\right )}{a}\right )}{a}\)

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

\(\Big \downarrow \) 7292

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(-(((e + f*x)^2*ArcTan[Sinh[c + d*x]])/d) - ((e + f*x)^2*Csch[c + d*x])/d 
+ (2*f*(-(((e + f*x)^2*ArcTan[E^(c + d*x)])/f) + ((e + f*x)^2*ArcTan[Sinh[ 
c + d*x]])/(2*f) - (2*(e + f*x)*ArcTanh[E^(c + d*x)])/d - (f*PolyLog[2, -E 
^(c + d*x)])/d^2 + (I*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/d - (I*(e + 
f*x)*PolyLog[2, I*E^(c + d*x)])/d + (f*PolyLog[2, E^(c + d*x)])/d^2 - (I*f 
*PolyLog[3, (-I)*E^(c + d*x)])/d^2 + (I*f*PolyLog[3, I*E^(c + d*x)])/d^2)) 
/d)/a - (b*(-((b*((b^2*(-1/3*(e + f*x)^3/(b*f) + ((e + f*x)^2*Log[1 + (b*E 
^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b*d) + ((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*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)))/(a^2 + b^2) + ((b*(e + f*x)^3)/(3 
*f) + (2*a*(e + f*x)^2*ArcTan[E^(c + d*x)])/d - (b*(e + f*x)^2*Log[1 + E^( 
2*(c + d*x))])/d - ((2*I)*a*f*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/d^2 
+ ((2*I)*a*f*(e + f*x)*PolyLog[2, I*E^(c + d*x)])/d^2 - (b*f*(e + f*x)*Pol 
yLog[2, -E^(2*(c + d*x))])/d^2 + ((2*I)*a*f^2*PolyLog[3, (-I)*E^(c + d*x)] 
)/d^3 - ((2*I)*a*f^2*PolyLog[3, I*E^(c + d*x)])/d^3 + (b*f^2*PolyLog[3, -E 
^(2*(c + d*x))])/(2*d^3))/(a^2 + b^2)))/a) + ((2*I)*((I*(e + f*x)^2*ArcTan 
h[E^(2*c + 2*d*x)])/d - (I*f*(-1/2*((e + f*x)*PolyLog[2, -E^(2*c + 2*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 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 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 4670

Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x 
_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] 
 + (-Simp[d*(m/(f*fz*I))   Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*fz*x 
)], x], x] + Simp[d*(m/(f*fz*I))   Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e 
+ f*fz*x)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[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 6095

Int[(Cosh[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sin 
h[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[-(e + f*x)^(m + 1)/(b*f*(m + 1)), 
 x] + (Int[(e + f*x)^m*(E^(c + d*x)/(a - Rt[a^2 + b^2, 2] + b*E^(c + d*x))) 
, x] + Int[(e + f*x)^m*(E^(c + d*x)/(a + Rt[a^2 + b^2, 2] + b*E^(c + d*x))) 
, x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[a^2 + b^2, 0]
 

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]

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

Input:

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

Output:

int((f*x+e)^2*csch(d*x+c)^2*sech(d*x+c)/(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. 5664 vs. \(2 (897) = 1794\).

Time = 0.22 (sec) , antiderivative size = 5664, normalized size of antiderivative = 5.77 \[ \int \frac {(e+f x)^2 \text {csch}^2(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

(b^3*log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/((a^4 + a^2*b^2)*d) + 
 2*a*arctan(e^(-d*x - c))/((a^2 + b^2)*d) + b*log(e^(-2*d*x - 2*c) + 1)/(( 
a^2 + b^2)*d) + 2*e^(-d*x - c)/((a*e^(-2*d*x - 2*c) - a)*d) - b*log(e^(-d* 
x - c) + 1)/(a^2*d) - b*log(e^(-d*x - c) - 1)/(a^2*d))*e^2 - 2*(f^2*x^2*e^ 
c + 2*e*f*x*e^c)*e^(d*x)/(a*d*e^(2*d*x + 2*c) - a*d) - 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^3*f^2*x^3 + 3*(b*d* 
e*f - a*f^2)*d^2*x^2)/(a^2*d^3) - integrate(2*(b^4*f^2*x^2 + 2*b^4*e*f*x - 
 (a*b^3*f^2*x^2*e^c + 2*a*b^3*e*f*x*e^c)*e^(d*x))/(a^4*b + a^2*b^3 - (a^4* 
b*e^(2*c) + a^2*b^3*e^(2*c))*e^(2*d*x) - 2*(a^5*e^c + a^3*b^2*e^c)*e^(d*x) 
), x) - integrate(2*(b*f^2*x^2 + 2*b*e*f*x + (a*f^2*x^2*e^c + 2*a*e*f*x*e^ 
c)*e^(d*x))/(a^2 + b^2 + (a^2*e^(2*c) + b^2*e^(2*c))*e^(2*d*x)), x)
 

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - 2*e**(2*c + 2*d*x)*atan(e**(c + d*x))*a**3*e**2 + 2*atan(e**(c + d*x)) 
*a**3*e**2 + 16*e**(6*c + 2*d*x)*int((e**(4*d*x)*x**2)/(e**(8*c + 8*d*x)*b 
 + 2*e**(7*c + 7*d*x)*a - 2*e**(6*c + 6*d*x)*b - 2*e**(5*c + 5*d*x)*a - 2* 
e**(3*c + 3*d*x)*a + 2*e**(2*c + 2*d*x)*b + 2*e**(c + d*x)*a - b),x)*a**4* 
d*f**2 + 16*e**(6*c + 2*d*x)*int((e**(4*d*x)*x**2)/(e**(8*c + 8*d*x)*b + 2 
*e**(7*c + 7*d*x)*a - 2*e**(6*c + 6*d*x)*b - 2*e**(5*c + 5*d*x)*a - 2*e**( 
3*c + 3*d*x)*a + 2*e**(2*c + 2*d*x)*b + 2*e**(c + d*x)*a - b),x)*a**2*b**2 
*d*f**2 + 32*e**(6*c + 2*d*x)*int((e**(4*d*x)*x)/(e**(8*c + 8*d*x)*b + 2*e 
**(7*c + 7*d*x)*a - 2*e**(6*c + 6*d*x)*b - 2*e**(5*c + 5*d*x)*a - 2*e**(3* 
c + 3*d*x)*a + 2*e**(2*c + 2*d*x)*b + 2*e**(c + d*x)*a - b),x)*a**4*d*e*f 
+ 32*e**(6*c + 2*d*x)*int((e**(4*d*x)*x)/(e**(8*c + 8*d*x)*b + 2*e**(7*c + 
 7*d*x)*a - 2*e**(6*c + 6*d*x)*b - 2*e**(5*c + 5*d*x)*a - 2*e**(3*c + 3*d* 
x)*a + 2*e**(2*c + 2*d*x)*b + 2*e**(c + d*x)*a - b),x)*a**2*b**2*d*e*f + e 
**(2*c + 2*d*x)*log(e**(2*c + 2*d*x) + 1)*a**2*b*e**2 - e**(2*c + 2*d*x)*l 
og(e**(c + d*x) - 1)*a**2*b*e**2 - e**(2*c + 2*d*x)*log(e**(c + d*x) - 1)* 
b**3*e**2 - e**(2*c + 2*d*x)*log(e**(c + d*x) + 1)*a**2*b*e**2 - e**(2*c + 
 2*d*x)*log(e**(c + d*x) + 1)*b**3*e**2 + e**(2*c + 2*d*x)*log(e**(2*c + 2 
*d*x)*b + 2*e**(c + d*x)*a - b)*b**3*e**2 - 2*e**(c + d*x)*a**3*e**2 - 2*e 
**(c + d*x)*a*b**2*e**2 - 16*e**(4*c)*int((e**(4*d*x)*x**2)/(e**(8*c + 8*d 
*x)*b + 2*e**(7*c + 7*d*x)*a - 2*e**(6*c + 6*d*x)*b - 2*e**(5*c + 5*d*x...