\(\int \frac {(e+f x)^2 \text {csch}^3(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx\) [492]

Optimal result

Integrand size = 34, antiderivative size = 1210 \[ \int \frac {(e+f x)^2 \text {csch}^3(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx =\text {Too large to display} \] Output:

-b^2*f*(f*x+e)*polylog(2,-exp(2*d*x+2*c))/a^3/d^2+b^4*(f*x+e)^2*ln(1+exp(2 
*d*x+2*c))/a^3/(a^2+b^2)/d+1/2*b^2*f^2*polylog(3,-exp(2*d*x+2*c))/a^3/d^3- 
2*b^2*(f*x+e)^2*arctanh(exp(2*d*x+2*c))/a^3/d+2*b*(f*x+e)^2*arctan(exp(d*x 
+c))/a^2/d+1/2*f^2*polylog(3,exp(2*d*x+2*c))/a/d^3+2*(f*x+e)^2*arctanh(exp 
(2*d*x+2*c))/a/d-1/2*f^2*polylog(3,-exp(2*d*x+2*c))/a/d^3+2*I*b^3*f*(f*x+e 
)*polylog(2,-I*exp(d*x+c))/a^2/(a^2+b^2)/d^2+2*I*b*f^2*polylog(3,-I*exp(d* 
x+c))/a^2/d^3+2*I*b^3*f^2*polylog(3,I*exp(d*x+c))/a^2/(a^2+b^2)/d^3+2*I*b* 
f*(f*x+e)*polylog(2,I*exp(d*x+c))/a^2/d^2+b^4*f*(f*x+e)*polylog(2,-exp(2*d 
*x+2*c))/a^3/(a^2+b^2)/d^2-f*(f*x+e)*polylog(2,exp(2*d*x+2*c))/a/d^2-2*I*b 
^3*f*(f*x+e)*polylog(2,I*exp(d*x+c))/a^2/(a^2+b^2)/d^2+4*b*f*(f*x+e)*arcta 
nh(exp(d*x+c))/a^2/d^2+2*b*f^2*polylog(2,-exp(d*x+c))/a^2/d^3-2*b*f^2*poly 
log(2,exp(d*x+c))/a^2/d^3-1/2*b^2*f^2*polylog(3,exp(2*d*x+2*c))/a^3/d^3+f^ 
2*ln(sinh(d*x+c))/a/d^3-1/2*b^4*f^2*polylog(3,-exp(2*d*x+2*c))/a^3/(a^2+b^ 
2)/d^3-2*I*b*f^2*polylog(3,I*exp(d*x+c))/a^2/d^3+f*(f*x+e)*polylog(2,-exp( 
2*d*x+2*c))/a/d^2-f*(f*x+e)*coth(d*x+c)/a/d^2+b*(f*x+e)^2*csch(d*x+c)/a^2/ 
d+b^2*f*(f*x+e)*polylog(2,exp(2*d*x+2*c))/a^3/d^2-1/2*(f*x+e)^2*coth(d*x+c 
)^2/a/d-2*b^3*(f*x+e)^2*arctan(exp(d*x+c))/a^2/(a^2+b^2)/d-2*I*b^3*f^2*pol 
ylog(3,-I*exp(d*x+c))/a^2/(a^2+b^2)/d^3-2*I*b*f*(f*x+e)*polylog(2,-I*exp(d 
*x+c))/a^2/d^2+1/2*(f*x+e)^2/a/d-2*b^4*f*(f*x+e)*polylog(2,-b*exp(d*x+c)/( 
a+(a^2+b^2)^(1/2)))/a^3/(a^2+b^2)/d^2-2*b^4*f*(f*x+e)*polylog(2,-b*exp(...
 

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(2784\) vs. \(2(1210)=2420\).

Time = 10.20 (sec) , antiderivative size = 2784, normalized size of antiderivative = 2.30 \[ \int \frac {(e+f x)^2 \text {csch}^3(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Result too large to show} \] Input:

Integrate[((e + f*x)^2*Csch[c + d*x]^3*Sech[c + d*x])/(a + b*Sinh[c + d*x] 
),x]
 

Output:

(b*(e + f*x)^2*Csch[c])/(a^2*d) - ((e + f*x)^2*Csch[(c + d*x)/2]^2)/(8*a*d 
) + (-12*a*d^3*e^2*E^(2*c)*x + 12*a*d^3*e^2*(1 + E^(2*c))*x + 12*a*d^3*e*f 
*x^2 + 4*a*d^3*f^2*x^3 + 12*b*d^2*e^2*(1 + E^(2*c))*ArcTan[E^(c + d*x)] - 
6*a*d^2*e^2*(1 + E^(2*c))*(2*d*x - Log[1 + E^(2*(c + d*x))]) + (12*I)*b*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*a*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)*b*(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*P 
olyLog[2, I*E^(c + d*x)] + 2*PolyLog[3, (-I)*E^(c + d*x)] - 2*PolyLog[3, I 
*E^(c + d*x)]) - a*(1 + E^(2*c))*f^2*(2*d^2*x^2*(2*d*x - 3*Log[1 + E^(2*(c 
 + d*x))]) - 6*d*x*PolyLog[2, -E^(2*(c + d*x))] + 3*PolyLog[3, -E^(2*(c + 
d*x))]))/(6*(a^2 + b^2)*d^3*(1 + E^(2*c))) + (12*a^2*d^3*e^2*E^(2*c)*x - 1 
2*b^2*d^3*e^2*E^(2*c)*x - 12*a^2*d*E^(2*c)*f^2*x + 12*a^2*d^3*e*E^(2*c)*f* 
x^2 - 12*b^2*d^3*e*E^(2*c)*f*x^2 + 4*a^2*d^3*E^(2*c)*f^2*x^3 - 4*b^2*d^3*E 
^(2*c)*f^2*x^3 - 24*a*b*d*e*f*ArcTanh[E^(c + d*x)] + 24*a*b*d*e*E^(2*c)*f* 
ArcTanh[E^(c + d*x)] + 12*a*b*d*f^2*x*Log[1 - E^(c + d*x)] - 12*a*b*d*E^(2 
*c)*f^2*x*Log[1 - E^(c + d*x)] - 12*a*b*d*f^2*x*Log[1 + E^(c + d*x)] + 12* 
a*b*d*E^(2*c)*f^2*x*Log[1 + E^(c + d*x)] + 6*a^2*d^2*e^2*Log[1 - E^(2*(c + 
 d*x))] - 6*b^2*d^2*e^2*Log[1 - E^(2*(c + d*x))] - 6*a^2*d^2*e^2*E^(2*c...
 

Rubi [A] (verified)

Time = 6.34 (sec) , antiderivative size = 1118, normalized size of antiderivative = 0.92, number of steps used = 24, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.676, Rules used = {6123, 5985, 27, 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.

\(\displaystyle \int \frac {(e+f x)^2 \text {csch}^3(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx\)

\(\Big \downarrow \) 6123

\(\displaystyle \frac {\int (e+f x)^2 \text {csch}^3(c+d x) \text {sech}(c+d x)dx}{a}-\frac {b \int \frac {(e+f x)^2 \text {csch}^2(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)}dx}{a}\)

\(\Big \downarrow \) 5985

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6123

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

\(\Big \downarrow \) 5985

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 6123

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

\(\Big \downarrow \) 5984

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4670

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 6107

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

\(\Big \downarrow \) 6095

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

\(\Big \downarrow \) 7292

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {-\frac {\coth ^2(c+d x) (e+f x)^2}{2 d}-\frac {\log (\tanh (c+d x)) (e+f x)^2}{d}+\frac {f \left (\frac {2 \text {arctanh}\left (e^{2 c+2 d x}\right ) (e+f x)^2}{f}+\frac {\log (\tanh (c+d x)) (e+f x)^2}{f}+\frac {(e+f x)^2}{2 f}-\frac {\coth (c+d x) (e+f x)}{d}+\frac {\operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right ) (e+f x)}{d}-\frac {\operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right ) (e+f x)}{d}+\frac {f \log (\sinh (c+d x))}{d^2}-\frac {f \operatorname {PolyLog}\left (3,-e^{2 c+2 d x}\right )}{2 d^2}+\frac {f \operatorname {PolyLog}\left (3,e^{2 c+2 d x}\right )}{2 d^2}\right )}{d}}{a}-\frac {b \left (\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}\right )}{a}\)

Input:

Int[((e + f*x)^2*Csch[c + d*x]^3*Sech[c + d*x])/(a + b*Sinh[c + d*x]),x]
 

Output:

(-1/2*((e + f*x)^2*Coth[c + d*x]^2)/d - ((e + f*x)^2*Log[Tanh[c + d*x]])/d 
 + (f*((e + f*x)^2/(2*f) + (2*(e + f*x)^2*ArcTanh[E^(2*c + 2*d*x)])/f - (( 
e + f*x)*Coth[c + d*x])/d + (f*Log[Sinh[c + d*x]])/d^2 + ((e + f*x)^2*Log[ 
Tanh[c + d*x]])/f + ((e + f*x)*PolyLog[2, -E^(2*c + 2*d*x)])/d - ((e + f*x 
)*PolyLog[2, E^(2*c + 2*d*x)])/d - (f*PolyLog[3, -E^(2*c + 2*d*x)])/(2*d^2 
) + (f*PolyLog[3, E^(2*c + 2*d*x)])/(2*d^2)))/d)/a - (b*((-(((e + f*x)^2*A 
rcTan[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))])...
 

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]

\[\int \frac {\left (f x +e \right )^{2} \operatorname {csch}\left (d x +c \right )^{3} \operatorname {sech}\left (d x +c \right )}{a +b \sinh \left (d x +c \right )}d x\]

Input:

int((f*x+e)^2*csch(d*x+c)^3*sech(d*x+c)/(a+b*sinh(d*x+c)),x)
 

Output:

int((f*x+e)^2*csch(d*x+c)^3*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. 13309 vs. \(2 (1110) = 2220\).

Time = 0.43 (sec) , antiderivative size = 13309, normalized size of antiderivative = 11.00 \[ \int \frac {(e+f x)^2 \text {csch}^3(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \] Input:

integrate((f*x+e)^2*csch(d*x+c)^3*sech(d*x+c)/(a+b*sinh(d*x+c)),x, algorit 
hm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(e+f x)^2 \text {csch}^3(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \] Input:

integrate((f*x+e)**2*csch(d*x+c)**3*sech(d*x+c)/(a+b*sinh(d*x+c)),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {(e+f x)^2 \text {csch}^3(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{2} \operatorname {csch}\left (d x + c\right )^{3} \operatorname {sech}\left (d x + c\right )}{b \sinh \left (d x + c\right ) + a} \,d x } \] Input:

integrate((f*x+e)^2*csch(d*x+c)^3*sech(d*x+c)/(a+b*sinh(d*x+c)),x, algorit 
hm="maxima")
 

Output:

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

Giac [F(-1)]

Timed out. \[ \int \frac {(e+f x)^2 \text {csch}^3(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \] Input:

integrate((f*x+e)^2*csch(d*x+c)^3*sech(d*x+c)/(a+b*sinh(d*x+c)),x, algorit 
hm="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(e+f x)^2 \text {csch}^3(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {{\left (e+f\,x\right )}^2}{\mathrm {cosh}\left (c+d\,x\right )\,{\mathrm {sinh}\left (c+d\,x\right )}^3\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \] Input:

int((e + f*x)^2/(cosh(c + d*x)*sinh(c + d*x)^3*(a + b*sinh(c + d*x))),x)
 

Output:

int((e + f*x)^2/(cosh(c + d*x)*sinh(c + d*x)^3*(a + b*sinh(c + d*x))), x)
                                                                                    
                                                                                    
 

Reduce [F]

\[ \int \frac {(e+f x)^2 \text {csch}^3(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {too large to display} \] Input:

int((f*x+e)^2*csch(d*x+c)^3*sech(d*x+c)/(a+b*sinh(d*x+c)),x)
 

Output:

(2*e**(4*c + 4*d*x)*atan(e**(c + d*x))*a**3*b*e**2 - 4*e**(2*c + 2*d*x)*at 
an(e**(c + d*x))*a**3*b*e**2 + 2*atan(e**(c + d*x))*a**3*b*e**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 - 3*e**(8*c + 8*d*x)*b - 4*e**(7*c + 7*d*x)*a + 2*e**(6*c + 6*d*x)*b 
+ 2*e**(4*c + 4*d*x)*b + 4*e**(3*c + 3*d*x)*a - 3*e**(2*c + 2*d*x)*b - 2*e 
**(c + d*x)*a + b),x)*a**5*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 - 3*e**(8*c + 8*d*x)*b - 
4*e**(7*c + 7*d*x)*a + 2*e**(6*c + 6*d*x)*b + 2*e**(4*c + 4*d*x)*b + 4*e** 
(3*c + 3*d*x)*a - 3*e**(2*c + 2*d*x)*b - 2*e**(c + d*x)*a + b),x)*a**3*b** 
2*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 - 3*e**(8*c + 8*d*x)*b - 4*e**(7*c + 7*d*x)*a + 2*e** 
(6*c + 6*d*x)*b + 2*e**(4*c + 4*d*x)*b + 4*e**(3*c + 3*d*x)*a - 3*e**(2*c 
+ 2*d*x)*b - 2*e**(c + d*x)*a + b),x)*a**5*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 - 3*e**(8*c + 
 8*d*x)*b - 4*e**(7*c + 7*d*x)*a + 2*e**(6*c + 6*d*x)*b + 2*e**(4*c + 4*d* 
x)*b + 4*e**(3*c + 3*d*x)*a - 3*e**(2*c + 2*d*x)*b - 2*e**(c + d*x)*a + b) 
,x)*a**3*b**2*d*e*f + e**(4*c + 4*d*x)*log(e**(2*c + 2*d*x) + 1)*a**4*e**2 
 - e**(4*c + 4*d*x)*log(e**(c + d*x) - 1)*a**4*e**2 + e**(4*c + 4*d*x)*log 
(e**(c + d*x) - 1)*b**4*e**2 - e**(4*c + 4*d*x)*log(e**(c + d*x) + 1)*a**4 
*e**2 + e**(4*c + 4*d*x)*log(e**(c + d*x) + 1)*b**4*e**2 - e**(4*c + 4*...