\(\int \frac {(e+f x) \text {csch}^3(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx\) [500]

Optimal result

Integrand size = 34, antiderivative size = 1117 \[ \int \frac {(e+f x) \text {csch}^3(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx =\text {Too large to display} \] Output:

-3/2*b*f*x*arctan(sinh(d*x+c))/a^2/d+3/2*b*(f*x+e)*arctan(sinh(d*x+c))/a^2 
/d+b^6*(f*x+e)*ln(1+exp(2*d*x+2*c))/a^3/(a^2+b^2)^2/d-b^2*f*x*ln(tanh(d*x+ 
c))/a^3/d+b^2*(f*x+e)*ln(tanh(d*x+c))/a^3/d-f*polylog(2,exp(2*d*x+2*c))/a/ 
d^2+4*(f*x+e)*arctanh(exp(2*d*x+2*c))/a/d+f*polylog(2,-exp(2*d*x+2*c))/a/d 
^2+3/2*I*b*f*polylog(2,I*exp(d*x+c))/a^2/d^2+1/2*b^2*f*polylog(2,exp(2*d*x 
+2*c))/a^3/d^2+3*b*f*x*arctan(exp(d*x+c))/a^2/d-b^3*(f*x+e)*arctan(exp(d*x 
+c))/a^2/(a^2+b^2)/d+1/2*b^6*f*polylog(2,-exp(2*d*x+2*c))/a^3/(a^2+b^2)^2/ 
d^2-2*b^2*f*x*arctanh(exp(2*d*x+2*c))/a^3/d-2*b^5*(f*x+e)*arctan(exp(d*x+c 
))/a^2/(a^2+b^2)^2/d+1/2*b^4*f*tanh(d*x+c)/a^3/(a^2+b^2)/d^2+1/2*b*(f*x+e) 
*sech(d*x+c)*tanh(d*x+c)/a^2/d-f*csch(2*d*x+2*c)/a/d^2-1/2*b^2*f*polylog(2 
,-exp(2*d*x+2*c))/a^3/d^2+1/2*b^2*f*x/a^3/d-1/2*b^2*f*tanh(d*x+c)/a^3/d^2- 
1/2*b^2*(f*x+e)*tanh(d*x+c)^2/a^3/d+1/2*b*f*sech(d*x+c)/a^2/d^2-2*(f*x+e)* 
coth(2*d*x+2*c)*csch(2*d*x+2*c)/a/d+b*f*arctanh(cosh(d*x+c))/a^2/d^2+b*(f* 
x+e)*csch(d*x+c)/a^2/d-b^6*f*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/ 
a^3/(a^2+b^2)^2/d^2-b^6*f*polylog(2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a^3 
/(a^2+b^2)^2/d^2-b^6*(f*x+e)*ln(1+b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a^3/(a 
^2+b^2)^2/d-b^6*(f*x+e)*ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a^3/(a^2+b^ 
2)^2/d-1/2*b^3*(f*x+e)*sech(d*x+c)*tanh(d*x+c)/a^2/(a^2+b^2)/d-I*b^5*f*pol 
ylog(2,I*exp(d*x+c))/a^2/(a^2+b^2)^2/d^2-1/2*I*b^3*f*polylog(2,I*exp(d*x+c 
))/a^2/(a^2+b^2)/d^2+1/2*I*b^3*f*polylog(2,-I*exp(d*x+c))/a^2/(a^2+b^2)...
 

Mathematica [A] (warning: unable to verify)

Time = 11.10 (sec) , antiderivative size = 1675, normalized size of antiderivative = 1.50 \[ \int \frac {(e+f x) \text {csch}^3(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx =\text {Too large to display} \] Input:

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

Output:

8*(-1/16*(-(a*b*f*(c + d*x)) + 2*(a*b*f + b^2*(d*e - c*f) + a^2*(-2*d*e + 
2*c*f))*(c + d*x) + (a^2*(d*e + d*f*x)^2)/f - (b^2*(d*e + d*f*x)^2)/(2*f) 
+ 2*(2*a^2 - b^2)*f*(c + d*x)*Log[1 + E^(-c - d*x)] - 2*(a*b*f + b^2*(d*e 
- c*f) + a^2*(-2*d*e + 2*c*f))*Log[1 + E^(c + d*x)] - 2*(2*a^2 - b^2)*f*Po 
lyLog[2, -E^(-c - d*x)])/(a^3*d^2) - (a*b*f*(c + d*x) - 2*(a*b*f + 2*a^2*( 
d*e - c*f) + b^2*(-(d*e) + c*f))*(c + d*x) + (a^2*(d*e + d*f*x)^2)/f - (b^ 
2*(d*e + d*f*x)^2)/(2*f) + 2*(2*a^2 - b^2)*f*(c + d*x)*Log[1 - E^(-c - d*x 
)] + 2*(a*b*f + 2*a^2*(d*e - c*f) + b^2*(-(d*e) + c*f))*Log[1 - E^(c + d*x 
)] - 2*(2*a^2 - b^2)*f*PolyLog[2, E^(-c - d*x)])/(16*a^3*d^2) - (b^6*(-2*d 
*e*(c + d*x) + 2*c*f*(c + d*x) - f*(c + d*x)^2 + (4*a*Sqrt[a^2 + b^2]*d*e* 
ArcTan[(a + b*E^(c + d*x))/Sqrt[-a^2 - b^2]])/Sqrt[-(a^2 + b^2)^2] - (4*a* 
Sqrt[-(a^2 + b^2)^2]*d*e*ArcTanh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]])/(-a 
^2 - b^2)^(3/2) + 2*f*(c + d*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^ 
2])] + 2*f*(c + d*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] - 2*c* 
f*Log[b - 2*a*E^(c + d*x) - b*E^(2*(c + d*x))] + 2*d*e*Log[2*a*E^(c + d*x) 
 + b*(-1 + E^(2*(c + d*x)))] + 2*f*PolyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a 
^2 + b^2])] + 2*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))]))/( 
16*a^3*(a^2 + b^2)^2*d^2) + (-4*a^3*d*e*(c + d*x) - 6*a*b^2*d*e*(c + d*x) 
+ 4*a^3*c*f*(c + d*x) + 6*a*b^2*c*f*(c + d*x) - 2*a^3*f*(c + d*x)^2 - 3*a* 
b^2*f*(c + d*x)^2 + 6*a^2*b*d*e*ArcTan[E^(c + d*x)] + 10*b^3*d*e*ArcTan...
 

Rubi [A] (verified)

Time = 6.18 (sec) , antiderivative size = 943, normalized size of antiderivative = 0.84, number of steps used = 26, number of rules used = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.735, Rules used = {6123, 5984, 3042, 26, 4673, 26, 3042, 26, 4670, 2715, 2838, 6123, 5985, 2009, 6123, 5985, 2009, 6107, 6107, 6095, 2620, 2715, 2838, 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.

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

\(\Big \downarrow \) 6123

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

\(\Big \downarrow \) 5984

\(\displaystyle \frac {8 \int (e+f x) \text {csch}^3(2 c+2 d x)dx}{a}-\frac {b \int \frac {(e+f x) \text {csch}^2(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {b \int \frac {(e+f x) \text {csch}^2(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {8 \int -i (e+f x) \csc (2 i c+2 i d x)^3dx}{a}\)

\(\Big \downarrow \) 26

\(\displaystyle -\frac {b \int \frac {(e+f x) \text {csch}^2(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {8 i \int (e+f x) \csc (2 i c+2 i d x)^3dx}{a}\)

\(\Big \downarrow \) 4673

\(\displaystyle -\frac {b \int \frac {(e+f x) \text {csch}^2(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {8 i \left (\frac {1}{2} \int -i (e+f x) \text {csch}(2 c+2 d x)dx-\frac {i f \text {csch}(2 c+2 d x)}{8 d^2}-\frac {i (e+f x) \coth (2 c+2 d x) \text {csch}(2 c+2 d x)}{4 d}\right )}{a}\)

\(\Big \downarrow \) 26

\(\displaystyle -\frac {b \int \frac {(e+f x) \text {csch}^2(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {8 i \left (-\frac {1}{2} i \int (e+f x) \text {csch}(2 c+2 d x)dx-\frac {i f \text {csch}(2 c+2 d x)}{8 d^2}-\frac {i (e+f x) \coth (2 c+2 d x) \text {csch}(2 c+2 d x)}{4 d}\right )}{a}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {b \int \frac {(e+f x) \text {csch}^2(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {8 i \left (-\frac {1}{2} i \int i (e+f x) \csc (2 i c+2 i d x)dx-\frac {i f \text {csch}(2 c+2 d x)}{8 d^2}-\frac {i (e+f x) \coth (2 c+2 d x) \text {csch}(2 c+2 d x)}{4 d}\right )}{a}\)

\(\Big \downarrow \) 26

\(\displaystyle -\frac {b \int \frac {(e+f x) \text {csch}^2(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {8 i \left (\frac {1}{2} \int (e+f x) \csc (2 i c+2 i d x)dx-\frac {i f \text {csch}(2 c+2 d x)}{8 d^2}-\frac {i (e+f x) \coth (2 c+2 d x) \text {csch}(2 c+2 d x)}{4 d}\right )}{a}\)

\(\Big \downarrow \) 4670

\(\displaystyle -\frac {b \int \frac {(e+f x) \text {csch}^2(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {8 i \left (\frac {1}{2} \left (\frac {i f \int \log \left (1-e^{2 c+2 d x}\right )dx}{2 d}-\frac {i f \int \log \left (1+e^{2 c+2 d x}\right )dx}{2 d}+\frac {i (e+f x) \text {arctanh}\left (e^{2 c+2 d x}\right )}{d}\right )-\frac {i f \text {csch}(2 c+2 d x)}{8 d^2}-\frac {i (e+f x) \coth (2 c+2 d x) \text {csch}(2 c+2 d x)}{4 d}\right )}{a}\)

\(\Big \downarrow \) 2715

\(\displaystyle -\frac {b \int \frac {(e+f x) \text {csch}^2(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {8 i \left (\frac {1}{2} \left (\frac {i f \int e^{-2 c-2 d x} \log \left (1-e^{2 c+2 d x}\right )de^{2 c+2 d x}}{4 d^2}-\frac {i f \int e^{-2 c-2 d x} \log \left (1+e^{2 c+2 d x}\right )de^{2 c+2 d x}}{4 d^2}+\frac {i (e+f x) \text {arctanh}\left (e^{2 c+2 d x}\right )}{d}\right )-\frac {i f \text {csch}(2 c+2 d x)}{8 d^2}-\frac {i (e+f x) \coth (2 c+2 d x) \text {csch}(2 c+2 d x)}{4 d}\right )}{a}\)

\(\Big \downarrow \) 2838

\(\displaystyle -\frac {b \int \frac {(e+f x) \text {csch}^2(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {8 i \left (\frac {1}{2} \left (\frac {i (e+f x) \text {arctanh}\left (e^{2 c+2 d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{4 d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{4 d^2}\right )-\frac {i f \text {csch}(2 c+2 d x)}{8 d^2}-\frac {i (e+f x) \coth (2 c+2 d x) \text {csch}(2 c+2 d x)}{4 d}\right )}{a}\)

\(\Big \downarrow \) 6123

\(\displaystyle -\frac {b \left (\frac {\int (e+f x) \text {csch}^2(c+d x) \text {sech}^3(c+d x)dx}{a}-\frac {b \int \frac {(e+f x) \text {csch}(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}\right )}{a}-\frac {8 i \left (\frac {1}{2} \left (\frac {i (e+f x) \text {arctanh}\left (e^{2 c+2 d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{4 d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{4 d^2}\right )-\frac {i f \text {csch}(2 c+2 d x)}{8 d^2}-\frac {i (e+f x) \coth (2 c+2 d x) \text {csch}(2 c+2 d x)}{4 d}\right )}{a}\)

\(\Big \downarrow \) 5985

\(\displaystyle -\frac {b \left (\frac {-f \int \left (\frac {\text {csch}(c+d x) \text {sech}^2(c+d x)}{2 d}-\frac {3 \arctan (\sinh (c+d x))}{2 d}-\frac {3 \text {csch}(c+d x)}{2 d}\right )dx-\frac {3 (e+f x) \arctan (\sinh (c+d x))}{2 d}-\frac {3 (e+f x) \text {csch}(c+d x)}{2 d}+\frac {(e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{2 d}}{a}-\frac {b \int \frac {(e+f x) \text {csch}(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}\right )}{a}-\frac {8 i \left (\frac {1}{2} \left (\frac {i (e+f x) \text {arctanh}\left (e^{2 c+2 d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{4 d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{4 d^2}\right )-\frac {i f \text {csch}(2 c+2 d x)}{8 d^2}-\frac {i (e+f x) \coth (2 c+2 d x) \text {csch}(2 c+2 d x)}{4 d}\right )}{a}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {b \left (-\frac {b \int \frac {(e+f x) \text {csch}(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {-f \left (\frac {3 x \arctan \left (e^{c+d x}\right )}{d}-\frac {3 x \arctan (\sinh (c+d x))}{2 d}+\frac {\text {arctanh}(\cosh (c+d x))}{d^2}-\frac {3 i \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{2 d^2}+\frac {3 i \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{2 d^2}+\frac {\text {sech}(c+d x)}{2 d^2}\right )-\frac {3 (e+f x) \arctan (\sinh (c+d x))}{2 d}-\frac {3 (e+f x) \text {csch}(c+d x)}{2 d}+\frac {(e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{2 d}}{a}\right )}{a}-\frac {8 i \left (\frac {1}{2} \left (\frac {i (e+f x) \text {arctanh}\left (e^{2 c+2 d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{4 d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{4 d^2}\right )-\frac {i f \text {csch}(2 c+2 d x)}{8 d^2}-\frac {i (e+f x) \coth (2 c+2 d x) \text {csch}(2 c+2 d x)}{4 d}\right )}{a}\)

\(\Big \downarrow \) 6123

\(\displaystyle -\frac {b \left (-\frac {b \left (\frac {\int (e+f x) \text {csch}(c+d x) \text {sech}^3(c+d x)dx}{a}-\frac {b \int \frac {(e+f x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}\right )}{a}+\frac {-f \left (\frac {3 x \arctan \left (e^{c+d x}\right )}{d}-\frac {3 x \arctan (\sinh (c+d x))}{2 d}+\frac {\text {arctanh}(\cosh (c+d x))}{d^2}-\frac {3 i \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{2 d^2}+\frac {3 i \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{2 d^2}+\frac {\text {sech}(c+d x)}{2 d^2}\right )-\frac {3 (e+f x) \arctan (\sinh (c+d x))}{2 d}-\frac {3 (e+f x) \text {csch}(c+d x)}{2 d}+\frac {(e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{2 d}}{a}\right )}{a}-\frac {8 i \left (\frac {1}{2} \left (\frac {i (e+f x) \text {arctanh}\left (e^{2 c+2 d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{4 d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{4 d^2}\right )-\frac {i f \text {csch}(2 c+2 d x)}{8 d^2}-\frac {i (e+f x) \coth (2 c+2 d x) \text {csch}(2 c+2 d x)}{4 d}\right )}{a}\)

\(\Big \downarrow \) 5985

\(\displaystyle -\frac {b \left (-\frac {b \left (\frac {-f \int \left (\frac {\log (\tanh (c+d x))}{d}-\frac {\tanh ^2(c+d x)}{2 d}\right )dx-\frac {(e+f x) \tanh ^2(c+d x)}{2 d}+\frac {(e+f x) \log (\tanh (c+d x))}{d}}{a}-\frac {b \int \frac {(e+f x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}\right )}{a}+\frac {-f \left (\frac {3 x \arctan \left (e^{c+d x}\right )}{d}-\frac {3 x \arctan (\sinh (c+d x))}{2 d}+\frac {\text {arctanh}(\cosh (c+d x))}{d^2}-\frac {3 i \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{2 d^2}+\frac {3 i \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{2 d^2}+\frac {\text {sech}(c+d x)}{2 d^2}\right )-\frac {3 (e+f x) \arctan (\sinh (c+d x))}{2 d}-\frac {3 (e+f x) \text {csch}(c+d x)}{2 d}+\frac {(e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{2 d}}{a}\right )}{a}-\frac {8 i \left (\frac {1}{2} \left (\frac {i (e+f x) \text {arctanh}\left (e^{2 c+2 d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{4 d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{4 d^2}\right )-\frac {i f \text {csch}(2 c+2 d x)}{8 d^2}-\frac {i (e+f x) \coth (2 c+2 d x) \text {csch}(2 c+2 d x)}{4 d}\right )}{a}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {b \left (-\frac {b \left (\frac {-f \left (\frac {2 x \text {arctanh}\left (e^{2 c+2 d x}\right )}{d}+\frac {\operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 d^2}-\frac {\operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 d^2}+\frac {\tanh (c+d x)}{2 d^2}+\frac {x \log (\tanh (c+d x))}{d}-\frac {x}{2 d}\right )-\frac {(e+f x) \tanh ^2(c+d x)}{2 d}+\frac {(e+f x) \log (\tanh (c+d x))}{d}}{a}-\frac {b \int \frac {(e+f x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}\right )}{a}+\frac {-f \left (\frac {3 x \arctan \left (e^{c+d x}\right )}{d}-\frac {3 x \arctan (\sinh (c+d x))}{2 d}+\frac {\text {arctanh}(\cosh (c+d x))}{d^2}-\frac {3 i \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{2 d^2}+\frac {3 i \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{2 d^2}+\frac {\text {sech}(c+d x)}{2 d^2}\right )-\frac {3 (e+f x) \arctan (\sinh (c+d x))}{2 d}-\frac {3 (e+f x) \text {csch}(c+d x)}{2 d}+\frac {(e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{2 d}}{a}\right )}{a}-\frac {8 i \left (\frac {1}{2} \left (\frac {i (e+f x) \text {arctanh}\left (e^{2 c+2 d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{4 d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{4 d^2}\right )-\frac {i f \text {csch}(2 c+2 d x)}{8 d^2}-\frac {i (e+f x) \coth (2 c+2 d x) \text {csch}(2 c+2 d x)}{4 d}\right )}{a}\)

\(\Big \downarrow \) 6107

\(\displaystyle -\frac {b \left (-\frac {b \left (\frac {-f \left (\frac {2 x \text {arctanh}\left (e^{2 c+2 d x}\right )}{d}+\frac {\operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 d^2}-\frac {\operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 d^2}+\frac {\tanh (c+d x)}{2 d^2}+\frac {x \log (\tanh (c+d x))}{d}-\frac {x}{2 d}\right )-\frac {(e+f x) \tanh ^2(c+d x)}{2 d}+\frac {(e+f x) \log (\tanh (c+d x))}{d}}{a}-\frac {b \left (\frac {\int (e+f x) \text {sech}^3(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}+\frac {b^2 \int \frac {(e+f x) \text {sech}(c+d x)}{a+b \sinh (c+d x)}dx}{a^2+b^2}\right )}{a}\right )}{a}+\frac {-f \left (\frac {3 x \arctan \left (e^{c+d x}\right )}{d}-\frac {3 x \arctan (\sinh (c+d x))}{2 d}+\frac {\text {arctanh}(\cosh (c+d x))}{d^2}-\frac {3 i \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{2 d^2}+\frac {3 i \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{2 d^2}+\frac {\text {sech}(c+d x)}{2 d^2}\right )-\frac {3 (e+f x) \arctan (\sinh (c+d x))}{2 d}-\frac {3 (e+f x) \text {csch}(c+d x)}{2 d}+\frac {(e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{2 d}}{a}\right )}{a}-\frac {8 i \left (\frac {1}{2} \left (\frac {i (e+f x) \text {arctanh}\left (e^{2 c+2 d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{4 d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{4 d^2}\right )-\frac {i f \text {csch}(2 c+2 d x)}{8 d^2}-\frac {i (e+f x) \coth (2 c+2 d x) \text {csch}(2 c+2 d x)}{4 d}\right )}{a}\)

\(\Big \downarrow \) 6107

\(\displaystyle -\frac {b \left (-\frac {b \left (\frac {-f \left (\frac {2 x \text {arctanh}\left (e^{2 c+2 d x}\right )}{d}+\frac {\operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 d^2}-\frac {\operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 d^2}+\frac {\tanh (c+d x)}{2 d^2}+\frac {x \log (\tanh (c+d x))}{d}-\frac {x}{2 d}\right )-\frac {(e+f x) \tanh ^2(c+d x)}{2 d}+\frac {(e+f x) \log (\tanh (c+d x))}{d}}{a}-\frac {b \left (\frac {\int (e+f x) \text {sech}^3(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}+\frac {b^2 \left (\frac {b^2 \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{a^2+b^2}+\frac {\int (e+f x) \text {sech}(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{a^2+b^2}\right )}{a}\right )}{a}+\frac {-f \left (\frac {3 x \arctan \left (e^{c+d x}\right )}{d}-\frac {3 x \arctan (\sinh (c+d x))}{2 d}+\frac {\text {arctanh}(\cosh (c+d x))}{d^2}-\frac {3 i \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{2 d^2}+\frac {3 i \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{2 d^2}+\frac {\text {sech}(c+d x)}{2 d^2}\right )-\frac {3 (e+f x) \arctan (\sinh (c+d x))}{2 d}-\frac {3 (e+f x) \text {csch}(c+d x)}{2 d}+\frac {(e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{2 d}}{a}\right )}{a}-\frac {8 i \left (\frac {1}{2} \left (\frac {i (e+f x) \text {arctanh}\left (e^{2 c+2 d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{4 d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{4 d^2}\right )-\frac {i f \text {csch}(2 c+2 d x)}{8 d^2}-\frac {i (e+f x) \coth (2 c+2 d x) \text {csch}(2 c+2 d x)}{4 d}\right )}{a}\)

\(\Big \downarrow \) 6095

\(\displaystyle -\frac {b \left (-\frac {b \left (\frac {-f \left (\frac {2 x \text {arctanh}\left (e^{2 c+2 d x}\right )}{d}+\frac {\operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 d^2}-\frac {\operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 d^2}+\frac {\tanh (c+d x)}{2 d^2}+\frac {x \log (\tanh (c+d x))}{d}-\frac {x}{2 d}\right )-\frac {(e+f x) \tanh ^2(c+d x)}{2 d}+\frac {(e+f x) \log (\tanh (c+d x))}{d}}{a}-\frac {b \left (\frac {\int (e+f x) \text {sech}^3(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}+\frac {b^2 \left (\frac {b^2 \left (\int \frac {e^{c+d x} (e+f x)}{a+b e^{c+d x}-\sqrt {a^2+b^2}}dx+\int \frac {e^{c+d x} (e+f x)}{a+b e^{c+d x}+\sqrt {a^2+b^2}}dx-\frac {(e+f x)^2}{2 b f}\right )}{a^2+b^2}+\frac {\int (e+f x) \text {sech}(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{a^2+b^2}\right )}{a}\right )}{a}+\frac {-f \left (\frac {3 x \arctan \left (e^{c+d x}\right )}{d}-\frac {3 x \arctan (\sinh (c+d x))}{2 d}+\frac {\text {arctanh}(\cosh (c+d x))}{d^2}-\frac {3 i \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{2 d^2}+\frac {3 i \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{2 d^2}+\frac {\text {sech}(c+d x)}{2 d^2}\right )-\frac {3 (e+f x) \arctan (\sinh (c+d x))}{2 d}-\frac {3 (e+f x) \text {csch}(c+d x)}{2 d}+\frac {(e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{2 d}}{a}\right )}{a}-\frac {8 i \left (\frac {1}{2} \left (\frac {i (e+f x) \text {arctanh}\left (e^{2 c+2 d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{4 d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{4 d^2}\right )-\frac {i f \text {csch}(2 c+2 d x)}{8 d^2}-\frac {i (e+f x) \coth (2 c+2 d x) \text {csch}(2 c+2 d x)}{4 d}\right )}{a}\)

\(\Big \downarrow \) 2620

\(\displaystyle -\frac {b \left (-\frac {b \left (\frac {-f \left (\frac {2 x \text {arctanh}\left (e^{2 c+2 d x}\right )}{d}+\frac {\operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 d^2}-\frac {\operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 d^2}+\frac {\tanh (c+d x)}{2 d^2}+\frac {x \log (\tanh (c+d x))}{d}-\frac {x}{2 d}\right )-\frac {(e+f x) \tanh ^2(c+d x)}{2 d}+\frac {(e+f x) \log (\tanh (c+d x))}{d}}{a}-\frac {b \left (\frac {\int (e+f x) \text {sech}^3(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}+\frac {b^2 \left (\frac {b^2 \left (-\frac {f \int \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right )dx}{b d}-\frac {f \int \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right )dx}{b d}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {(e+f x)^2}{2 b f}\right )}{a^2+b^2}+\frac {\int (e+f x) \text {sech}(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{a^2+b^2}\right )}{a}\right )}{a}+\frac {-f \left (\frac {3 x \arctan \left (e^{c+d x}\right )}{d}-\frac {3 x \arctan (\sinh (c+d x))}{2 d}+\frac {\text {arctanh}(\cosh (c+d x))}{d^2}-\frac {3 i \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{2 d^2}+\frac {3 i \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{2 d^2}+\frac {\text {sech}(c+d x)}{2 d^2}\right )-\frac {3 (e+f x) \arctan (\sinh (c+d x))}{2 d}-\frac {3 (e+f x) \text {csch}(c+d x)}{2 d}+\frac {(e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{2 d}}{a}\right )}{a}-\frac {8 i \left (\frac {1}{2} \left (\frac {i (e+f x) \text {arctanh}\left (e^{2 c+2 d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{4 d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{4 d^2}\right )-\frac {i f \text {csch}(2 c+2 d x)}{8 d^2}-\frac {i (e+f x) \coth (2 c+2 d x) \text {csch}(2 c+2 d x)}{4 d}\right )}{a}\)

\(\Big \downarrow \) 2715

\(\displaystyle -\frac {b \left (-\frac {b \left (\frac {-f \left (\frac {2 x \text {arctanh}\left (e^{2 c+2 d x}\right )}{d}+\frac {\operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 d^2}-\frac {\operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 d^2}+\frac {\tanh (c+d x)}{2 d^2}+\frac {x \log (\tanh (c+d x))}{d}-\frac {x}{2 d}\right )-\frac {(e+f x) \tanh ^2(c+d x)}{2 d}+\frac {(e+f x) \log (\tanh (c+d x))}{d}}{a}-\frac {b \left (\frac {b^2 \left (\frac {b^2 \left (-\frac {f \int e^{-c-d x} \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right )de^{c+d x}}{b d^2}-\frac {f \int e^{-c-d x} \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right )de^{c+d x}}{b d^2}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {(e+f x)^2}{2 b f}\right )}{a^2+b^2}+\frac {\int (e+f x) \text {sech}(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{a^2+b^2}+\frac {\int (e+f x) \text {sech}^3(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{a}\right )}{a}+\frac {-f \left (\frac {3 x \arctan \left (e^{c+d x}\right )}{d}-\frac {3 x \arctan (\sinh (c+d x))}{2 d}+\frac {\text {arctanh}(\cosh (c+d x))}{d^2}-\frac {3 i \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{2 d^2}+\frac {3 i \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{2 d^2}+\frac {\text {sech}(c+d x)}{2 d^2}\right )-\frac {3 (e+f x) \arctan (\sinh (c+d x))}{2 d}-\frac {3 (e+f x) \text {csch}(c+d x)}{2 d}+\frac {(e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{2 d}}{a}\right )}{a}-\frac {8 i \left (\frac {1}{2} \left (\frac {i (e+f x) \text {arctanh}\left (e^{2 c+2 d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{4 d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{4 d^2}\right )-\frac {i f \text {csch}(2 c+2 d x)}{8 d^2}-\frac {i (e+f x) \coth (2 c+2 d x) \text {csch}(2 c+2 d x)}{4 d}\right )}{a}\)

\(\Big \downarrow \) 2838

\(\displaystyle -\frac {b \left (-\frac {b \left (\frac {-f \left (\frac {2 x \text {arctanh}\left (e^{2 c+2 d x}\right )}{d}+\frac {\operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 d^2}-\frac {\operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 d^2}+\frac {\tanh (c+d x)}{2 d^2}+\frac {x \log (\tanh (c+d x))}{d}-\frac {x}{2 d}\right )-\frac {(e+f x) \tanh ^2(c+d x)}{2 d}+\frac {(e+f x) \log (\tanh (c+d x))}{d}}{a}-\frac {b \left (\frac {b^2 \left (\frac {\int (e+f x) \text {sech}(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}+\frac {b^2 \left (\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b d^2}+\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b d^2}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {(e+f x)^2}{2 b f}\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {\int (e+f x) \text {sech}^3(c+d x) (a-b \sinh (c+d x))dx}{a^2+b^2}\right )}{a}\right )}{a}+\frac {-f \left (\frac {3 x \arctan \left (e^{c+d x}\right )}{d}-\frac {3 x \arctan (\sinh (c+d x))}{2 d}+\frac {\text {arctanh}(\cosh (c+d x))}{d^2}-\frac {3 i \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{2 d^2}+\frac {3 i \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{2 d^2}+\frac {\text {sech}(c+d x)}{2 d^2}\right )-\frac {3 (e+f x) \arctan (\sinh (c+d x))}{2 d}-\frac {3 (e+f x) \text {csch}(c+d x)}{2 d}+\frac {(e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{2 d}}{a}\right )}{a}-\frac {8 i \left (\frac {1}{2} \left (\frac {i (e+f x) \text {arctanh}\left (e^{2 c+2 d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{4 d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{4 d^2}\right )-\frac {i f \text {csch}(2 c+2 d x)}{8 d^2}-\frac {i (e+f x) \coth (2 c+2 d x) \text {csch}(2 c+2 d x)}{4 d}\right )}{a}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {b \left (-\frac {b \left (\frac {-f \left (\frac {2 x \text {arctanh}\left (e^{2 c+2 d x}\right )}{d}+\frac {\operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 d^2}-\frac {\operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 d^2}+\frac {\tanh (c+d x)}{2 d^2}+\frac {x \log (\tanh (c+d x))}{d}-\frac {x}{2 d}\right )-\frac {(e+f x) \tanh ^2(c+d x)}{2 d}+\frac {(e+f x) \log (\tanh (c+d x))}{d}}{a}-\frac {b \left (\frac {b^2 \left (\frac {\int (a (e+f x) \text {sech}(c+d x)-b (e+f x) \tanh (c+d x))dx}{a^2+b^2}+\frac {b^2 \left (\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b d^2}+\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b d^2}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {(e+f x)^2}{2 b f}\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {\int \left (a (e+f x) \text {sech}^3(c+d x)-b (e+f x) \text {sech}^2(c+d x) \tanh (c+d x)\right )dx}{a^2+b^2}\right )}{a}\right )}{a}+\frac {-f \left (\frac {3 x \arctan \left (e^{c+d x}\right )}{d}-\frac {3 x \arctan (\sinh (c+d x))}{2 d}+\frac {\text {arctanh}(\cosh (c+d x))}{d^2}-\frac {3 i \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{2 d^2}+\frac {3 i \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{2 d^2}+\frac {\text {sech}(c+d x)}{2 d^2}\right )-\frac {3 (e+f x) \arctan (\sinh (c+d x))}{2 d}-\frac {3 (e+f x) \text {csch}(c+d x)}{2 d}+\frac {(e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{2 d}}{a}\right )}{a}-\frac {8 i \left (\frac {1}{2} \left (\frac {i (e+f x) \text {arctanh}\left (e^{2 c+2 d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{4 d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{4 d^2}\right )-\frac {i f \text {csch}(2 c+2 d x)}{8 d^2}-\frac {i (e+f x) \coth (2 c+2 d x) \text {csch}(2 c+2 d x)}{4 d}\right )}{a}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {8 i \left (-\frac {i f \text {csch}(2 c+2 d x)}{8 d^2}-\frac {i (e+f x) \coth (2 c+2 d x) \text {csch}(2 c+2 d x)}{4 d}+\frac {1}{2} \left (\frac {i (e+f x) \text {arctanh}\left (e^{2 c+2 d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{4 d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{4 d^2}\right )\right )}{a}-\frac {b \left (\frac {\frac {(e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{2 d}-\frac {3 (e+f x) \arctan (\sinh (c+d x))}{2 d}-\frac {3 (e+f x) \text {csch}(c+d x)}{2 d}-f \left (\frac {3 x \arctan \left (e^{c+d x}\right )}{d}-\frac {3 x \arctan (\sinh (c+d x))}{2 d}+\frac {\text {arctanh}(\cosh (c+d x))}{d^2}-\frac {3 i \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{2 d^2}+\frac {3 i \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{2 d^2}+\frac {\text {sech}(c+d x)}{2 d^2}\right )}{a}-\frac {b \left (\frac {-\frac {(e+f x) \tanh ^2(c+d x)}{2 d}+\frac {(e+f x) \log (\tanh (c+d x))}{d}-f \left (\frac {2 \text {arctanh}\left (e^{2 c+2 d x}\right ) x}{d}+\frac {\log (\tanh (c+d x)) x}{d}-\frac {x}{2 d}+\frac {\operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 d^2}-\frac {\operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 d^2}+\frac {\tanh (c+d x)}{2 d^2}\right )}{a}-\frac {b \left (\frac {\left (\frac {\left (-\frac {(e+f x)^2}{2 b f}+\frac {\log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right ) (e+f x)}{b d}+\frac {\log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right ) (e+f x)}{b d}+\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b d^2}+\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b d^2}\right ) b^2}{a^2+b^2}+\frac {\frac {b (e+f x)^2}{2 f}+\frac {2 a \arctan \left (e^{c+d x}\right ) (e+f x)}{d}-\frac {b \log \left (1+e^{2 (c+d x)}\right ) (e+f x)}{d}-\frac {i a f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i a f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}-\frac {b f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d^2}}{a^2+b^2}\right ) b^2}{a^2+b^2}+\frac {\frac {b (e+f x) \text {sech}^2(c+d x)}{2 d}+\frac {a f \text {sech}(c+d x)}{2 d^2}+\frac {a (e+f x) \tanh (c+d x) \text {sech}(c+d x)}{2 d}+\frac {a (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i a f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{2 d^2}+\frac {i a f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{2 d^2}-\frac {b f \tanh (c+d x)}{2 d^2}}{a^2+b^2}\right )}{a}\right )}{a}\right )}{a}\)

Input:

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

Output:

((-8*I)*(((-1/8*I)*f*Csch[2*c + 2*d*x])/d^2 - ((I/4)*(e + f*x)*Coth[2*c + 
2*d*x]*Csch[2*c + 2*d*x])/d + ((I*(e + f*x)*ArcTanh[E^(2*c + 2*d*x)])/d + 
((I/4)*f*PolyLog[2, -E^(2*c + 2*d*x)])/d^2 - ((I/4)*f*PolyLog[2, E^(2*c + 
2*d*x)])/d^2)/2))/a - (b*(((-3*(e + f*x)*ArcTan[Sinh[c + d*x]])/(2*d) - (3 
*(e + f*x)*Csch[c + d*x])/(2*d) + ((e + f*x)*Csch[c + d*x]*Sech[c + d*x]^2 
)/(2*d) - f*((3*x*ArcTan[E^(c + d*x)])/d - (3*x*ArcTan[Sinh[c + d*x]])/(2* 
d) + ArcTanh[Cosh[c + d*x]]/d^2 - (((3*I)/2)*PolyLog[2, (-I)*E^(c + d*x)]) 
/d^2 + (((3*I)/2)*PolyLog[2, I*E^(c + d*x)])/d^2 + Sech[c + d*x]/(2*d^2))) 
/a - (b*((((e + f*x)*Log[Tanh[c + d*x]])/d - ((e + f*x)*Tanh[c + d*x]^2)/( 
2*d) - f*(-1/2*x/d + (2*x*ArcTanh[E^(2*c + 2*d*x)])/d + (x*Log[Tanh[c + d* 
x]])/d + PolyLog[2, -E^(2*c + 2*d*x)]/(2*d^2) - PolyLog[2, E^(2*c + 2*d*x) 
]/(2*d^2) + Tanh[c + d*x]/(2*d^2)))/a - (b*((b^2*((b^2*(-1/2*(e + f*x)^2/( 
b*f) + ((e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b*d) + 
((e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b*d) + (f*Poly 
Log[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b*d^2) + (f*PolyLog[2, 
-((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b*d^2)))/(a^2 + b^2) + ((b*(e 
+ f*x)^2)/(2*f) + (2*a*(e + f*x)*ArcTan[E^(c + d*x)])/d - (b*(e + f*x)*Log 
[1 + E^(2*(c + d*x))])/d - (I*a*f*PolyLog[2, (-I)*E^(c + d*x)])/d^2 + (I*a 
*f*PolyLog[2, I*E^(c + d*x)])/d^2 - (b*f*PolyLog[2, -E^(2*(c + d*x))])/(2* 
d^2))/(a^2 + b^2)))/(a^2 + b^2) + ((a*(e + f*x)*ArcTan[E^(c + d*x)])/d ...
 

Defintions of rubi rules used

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 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 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 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 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 4673

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_)), x_Symbol] :> 
 Simp[(-b^2)*(c + d*x)*Cot[e + f*x]*((b*Csc[e + f*x])^(n - 2)/(f*(n - 1))), 
 x] + (-Simp[b^2*d*((b*Csc[e + f*x])^(n - 2)/(f^2*(n - 1)*(n - 2))), x] + S 
imp[b^2*((n - 2)/(n - 1))   Int[(c + d*x)*(b*Csc[e + f*x])^(n - 2), x], x]) 
 /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2]
 

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 7293

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

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 3562 vs. \(2 (1042 ) = 2084\).

Time = 129.46 (sec) , antiderivative size = 3563, normalized size of antiderivative = 3.19

method	result	size
risch	\(\text {Expression too large to display}\)	\(3563\)

Input:

int((f*x+e)*csch(d*x+c)^3*sech(d*x+c)^3/(a+b*sinh(d*x+c)),x,method=_RETURN 
VERBOSE)
 

Output:

-12/d^2/(a^2+b^2)*c*a^2*f/(4*a^2+4*b^2)*b*arctan(exp(d*x+c))+12/d/(a^2+b^2 
)*b^2*f/(4*a^2+4*b^2)*ln(1+I*exp(d*x+c))*a*x+12/d^2/(a^2+b^2)*b^2*f/(4*a^2 
+4*b^2)*ln(1+I*exp(d*x+c))*a*c+12/d/(a^2+b^2)*b^2*f/(4*a^2+4*b^2)*ln(1-I*e 
xp(d*x+c))*a*x+12/d^2/(a^2+b^2)*b^2*f/(4*a^2+4*b^2)*ln(1-I*exp(d*x+c))*a*c 
+1/(a^2+b^2)/d^2/a^2*b^3*f*ln(exp(d*x+c)+1)-1/(a^2+b^2)/d^2/a^2*b^3*f*ln(e 
xp(d*x+c)-1)+1/(a^2+b^2)/d/a^3*b^4*e*ln(exp(d*x+c)-1)+1/(a^2+b^2)/d/a^3*b^ 
4*e*ln(exp(d*x+c)+1)-1/(a^2+b^2)/d^2/a^3*b^4*f*dilog(exp(d*x+c))+1/(a^2+b^ 
2)/d^2/a^3*b^4*f*dilog(exp(d*x+c)+1)-1/d^2*b^2*f/(a^2+b^2)^(3/2)*arctanh(1 
/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))-6*I/d^2/(a^2+b^2)*a^2*f/(4*a^2+4* 
b^2)*dilog(1+I*exp(d*x+c))*b+6*I/d^2/(a^2+b^2)*a^2*f/(4*a^2+4*b^2)*dilog(1 
-I*exp(d*x+c))*b-10*I/d/(a^2+b^2)*b^3*f/(4*a^2+4*b^2)*ln(1+I*exp(d*x+c))*x 
+10*I/d/(a^2+b^2)*b^3*f/(4*a^2+4*b^2)*ln(1-I*exp(d*x+c))*x+10*I/d^2/(a^2+b 
^2)*b^3*f/(4*a^2+4*b^2)*ln(1-I*exp(d*x+c))*c-10*I/d^2/(a^2+b^2)*b^3*f/(4*a 
^2+4*b^2)*ln(1+I*exp(d*x+c))*c+1/2/d/(a^2+b^2)^(3/2)*b^2*e*arctanh(1/2*(2* 
b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))+20/d/(a^2+b^2)*b^3*e/(4*a^2+4*b^2)*arct 
an(exp(d*x+c))-1/2/d/(a^2+b^2)^(5/2)*b^4*e*arctanh(1/2*(2*b*exp(d*x+c)+2*a 
)/(a^2+b^2)^(1/2))+8/d/(a^2+b^2)*a^3*e/(4*a^2+4*b^2)*ln(1+exp(2*d*x+2*c))+ 
8/d^2/(a^2+b^2)*a^3*f/(4*a^2+4*b^2)*dilog(1+I*exp(d*x+c))+8/d^2/(a^2+b^2)* 
a^3*f/(4*a^2+4*b^2)*dilog(1-I*exp(d*x+c))-1/d^2/a^2*b^4*f/(a^2+b^2)^(3/2)* 
arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))-2/(a^2+b^2)/d*ln(exp(...
 

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. 16848 vs. \(2 (1017) = 2034\).

Time = 0.55 (sec) , antiderivative size = 16848, normalized size of antiderivative = 15.08 \[ \int \frac {(e+f x) \text {csch}^3(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

-(b^6*log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/((a^7 + 2*a^5*b^2 + 
a^3*b^4)*d) + (3*a^2*b + 5*b^3)*arctan(e^(-d*x - c))/((a^4 + 2*a^2*b^2 + b 
^4)*d) - (2*a^3 + 3*a*b^2)*log(e^(-2*d*x - 2*c) + 1)/((a^4 + 2*a^2*b^2 + b 
^4)*d) + (4*a*b^2*e^(-4*d*x - 4*c) - (3*a^2*b + 2*b^3)*e^(-d*x - c) + 2*(2 
*a^3 + a*b^2)*e^(-2*d*x - 2*c) + (a^2*b - 2*b^3)*e^(-3*d*x - 3*c) - (a^2*b 
 - 2*b^3)*e^(-5*d*x - 5*c) + 2*(2*a^3 + a*b^2)*e^(-6*d*x - 6*c) + (3*a^2*b 
 + 2*b^3)*e^(-7*d*x - 7*c))/((a^4 + a^2*b^2 - 2*(a^4 + a^2*b^2)*e^(-4*d*x 
- 4*c) + (a^4 + a^2*b^2)*e^(-8*d*x - 8*c))*d) + (2*a^2 - b^2)*log(e^(-d*x 
- c) + 1)/(a^3*d) + (2*a^2 - b^2)*log(e^(-d*x - c) - 1)/(a^3*d))*e + (128* 
a^2*d*integrate(1/64*x/(a^3*d*e^(d*x + c) + a^3*d), x) - 64*b^2*d*integrat 
e(1/64*x/(a^3*d*e^(d*x + c) + a^3*d), x) - 128*a^2*d*integrate(1/64*x/(a^3 
*d*e^(d*x + c) - a^3*d), x) + 64*b^2*d*integrate(1/64*x/(a^3*d*e^(d*x + c) 
 - a^3*d), x) - a*b*((d*x + c)/(a^3*d^2) - log(e^(d*x + c) + 1)/(a^3*d^2)) 
 + a*b*((d*x + c)/(a^3*d^2) - log(e^(d*x + c) - 1)/(a^3*d^2)) + (a*b^2 + ( 
a^2*b*e^(7*c) + (3*a^2*b*d*e^(7*c) + 2*b^3*d*e^(7*c))*x)*e^(7*d*x) - (2*a^ 
3*e^(6*c) + a*b^2*e^(6*c) + 2*(2*a^3*d*e^(6*c) + a*b^2*d*e^(6*c))*x)*e^(6* 
d*x) - (a^2*b*e^(5*c) + (a^2*b*d*e^(5*c) - 2*b^3*d*e^(5*c))*x)*e^(5*d*x) - 
 (4*a*b^2*d*x*e^(4*c) + a*b^2*e^(4*c))*e^(4*d*x) - (a^2*b*e^(3*c) - (a^2*b 
*d*e^(3*c) - 2*b^3*d*e^(3*c))*x)*e^(3*d*x) + (2*a^3*e^(2*c) + a*b^2*e^(2*c 
) - 2*(2*a^3*d*e^(2*c) + a*b^2*d*e^(2*c))*x)*e^(2*d*x) + (a^2*b*e^c - (...
 

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(48*e**(8*c + 8*d*x)*atan(e**(c + d*x))*a**7*b*f + 21*e**(8*c + 8*d*x)*ata 
n(e**(c + d*x))*a**5*b**3*d*e + 96*e**(8*c + 8*d*x)*atan(e**(c + d*x))*a** 
5*b**3*f + 35*e**(8*c + 8*d*x)*atan(e**(c + d*x))*a**3*b**5*d*e + 48*e**(8 
*c + 8*d*x)*atan(e**(c + d*x))*a**3*b**5*f - 96*e**(4*c + 4*d*x)*atan(e**( 
c + d*x))*a**7*b*f - 42*e**(4*c + 4*d*x)*atan(e**(c + d*x))*a**5*b**3*d*e 
- 192*e**(4*c + 4*d*x)*atan(e**(c + d*x))*a**5*b**3*f - 70*e**(4*c + 4*d*x 
)*atan(e**(c + d*x))*a**3*b**5*d*e - 96*e**(4*c + 4*d*x)*atan(e**(c + d*x) 
)*a**3*b**5*f + 48*atan(e**(c + d*x))*a**7*b*f + 21*atan(e**(c + d*x))*a** 
5*b**3*d*e + 96*atan(e**(c + d*x))*a**5*b**3*f + 35*atan(e**(c + d*x))*a** 
3*b**5*d*e + 48*atan(e**(c + d*x))*a**3*b**5*f + 3584*e**(13*c + 8*d*x)*in 
t((e**(5*d*x)*x)/(e**(14*c + 14*d*x)*b + 2*e**(13*c + 13*d*x)*a - e**(12*c 
 + 12*d*x)*b - 3*e**(10*c + 10*d*x)*b - 6*e**(9*c + 9*d*x)*a + 3*e**(8*c + 
 8*d*x)*b + 3*e**(6*c + 6*d*x)*b + 6*e**(5*c + 5*d*x)*a - 3*e**(4*c + 4*d* 
x)*b - e**(2*c + 2*d*x)*b - 2*e**(c + d*x)*a + b),x)*a**9*d**2*f + 8064*e* 
*(13*c + 8*d*x)*int((e**(5*d*x)*x)/(e**(14*c + 14*d*x)*b + 2*e**(13*c + 13 
*d*x)*a - e**(12*c + 12*d*x)*b - 3*e**(10*c + 10*d*x)*b - 6*e**(9*c + 9*d* 
x)*a + 3*e**(8*c + 8*d*x)*b + 3*e**(6*c + 6*d*x)*b + 6*e**(5*c + 5*d*x)*a 
- 3*e**(4*c + 4*d*x)*b - e**(2*c + 2*d*x)*b - 2*e**(c + d*x)*a + b),x)*a** 
7*b**2*d**2*f + 5376*e**(13*c + 8*d*x)*int((e**(5*d*x)*x)/(e**(14*c + 14*d 
*x)*b + 2*e**(13*c + 13*d*x)*a - e**(12*c + 12*d*x)*b - 3*e**(10*c + 10...