\(\int \frac {(e+f x) \text {csch}^3(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx\) [493]

Optimal result

Integrand size = 32, antiderivative size = 762 \[ \int \frac {(e+f x) \text {csch}^3(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {f x}{2 a d}+\frac {2 b f x \arctan \left (e^{c+d x}\right )}{a^2 d}-\frac {2 b^3 (e+f x) \arctan \left (e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d}-\frac {b f x \arctan (\sinh (c+d x))}{a^2 d}+\frac {b (e+f x) \arctan (\sinh (c+d x))}{a^2 d}+\frac {2 f x \text {arctanh}\left (e^{2 c+2 d x}\right )}{a d}-\frac {2 b^2 (e+f x) \text {arctanh}\left (e^{2 c+2 d x}\right )}{a^3 d}+\frac {b f \text {arctanh}(\cosh (c+d x))}{a^2 d^2}-\frac {f \coth (c+d x)}{2 a d^2}-\frac {(e+f x) \coth ^2(c+d x)}{2 a d}+\frac {b (e+f x) \text {csch}(c+d x)}{a^2 d}-\frac {b^4 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d}-\frac {b^4 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d}+\frac {b^4 (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{a^3 \left (a^2+b^2\right ) d}+\frac {f x \log (\tanh (c+d x))}{a d}-\frac {(e+f x) \log (\tanh (c+d x))}{a d}-\frac {i b f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a^2 d^2}+\frac {i b^3 f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^2}+\frac {i b f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{a^2 d^2}-\frac {i b^3 f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac {b^4 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^2}-\frac {b^4 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^2}+\frac {b^4 f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 a^3 \left (a^2+b^2\right ) d^2}+\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 a d^2}-\frac {b^2 f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 a^3 d^2}-\frac {f \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 a d^2}+\frac {b^2 f \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 a^3 d^2} \] Output:

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

Mathematica [A] (warning: unable to verify)

Time = 9.15 (sec) , antiderivative size = 1009, normalized size of antiderivative = 1.32 \[ \int \frac {(e+f x) \text {csch}^3(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx =\text {Too large to display} \] Input:

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

Output:

((2*b*d*e*Cosh[(c + d*x)/2] - a*f*Cosh[(c + d*x)/2] - 2*b*c*f*Cosh[(c + d* 
x)/2] + 2*b*f*(c + d*x)*Cosh[(c + d*x)/2])*Csch[(c + d*x)/2])/(4*a^2*d^2) 
+ ((-(d*e) + c*f - f*(c + d*x))*Csch[(c + d*x)/2]^2)/(8*a*d^2) + (-1/2*((a 
^2 - b^2)*(d*e + d*f*x)^2)/f - (a*b*f + a^2*(d*e + d*f*x) - b^2*(d*e + d*f 
*x))*Log[1 - E^(-c - d*x)] + (a*b*f - a^2*(d*e + d*f*x) + b^2*(d*e + d*f*x 
))*Log[1 + E^(-c - d*x)] + (a^2 - b^2)*f*PolyLog[2, -E^(-c - d*x)] + (a^2 
- b^2)*f*PolyLog[2, E^(-c - d*x)])/(a^3*d^2) - (b^4*(-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]))]))/(2*a^3*(a^2 + b^2) 
*d^2) + (-(a*d*e*(c + d*x)) + a*c*f*(c + d*x) - (a*f*(c + d*x)^2)/2 + 2*b* 
d*e*ArcTan[E^(c + d*x)] - 2*b*c*f*ArcTan[E^(c + d*x)] + I*b*f*(c + d*x)*Lo 
g[1 - I*E^(c + d*x)] - I*b*f*(c + d*x)*Log[1 + I*E^(c + d*x)] + a*d*e*Log[ 
1 + E^(2*(c + d*x))] - a*c*f*Log[1 + E^(2*(c + d*x))] + a*f*(c + d*x)*Log[ 
1 + E^(2*(c + d*x))] - I*b*f*PolyLog[2, (-I)*E^(c + d*x)] + I*b*f*PolyL...
 

Rubi [A] (verified)

Time = 4.26 (sec) , antiderivative size = 673, normalized size of antiderivative = 0.88, number of steps used = 21, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {6123, 5985, 2009, 6123, 5985, 2009, 6123, 5984, 3042, 26, 4670, 2715, 2838, 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.

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

\(\Big \downarrow \) 6123

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

\(\Big \downarrow \) 5985

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 6123

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

\(\Big \downarrow \) 5985

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 6123

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

\(\Big \downarrow \) 5984

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4670

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

\(\Big \downarrow \) 6107

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

\(\Big \downarrow \) 6095

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(-1/2*((e + f*x)*Coth[c + d*x]^2)/d - ((e + f*x)*Log[Tanh[c + d*x]])/d - f 
*(-1/2*x/d - (2*x*ArcTanh[E^(2*c + 2*d*x)])/d + Coth[c + d*x]/(2*d^2) - (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)))/a - (b*((-(((e + f*x)*ArcTan[Sinh[c + d*x]])/d 
) - ((e + f*x)*Csch[c + d*x])/d - f*((2*x*ArcTan[E^(c + d*x)])/d - (x*ArcT 
an[Sinh[c + d*x]])/d + ArcTanh[Cosh[c + d*x]]/d^2 - (I*PolyLog[2, (-I)*E^( 
c + d*x)])/d^2 + (I*PolyLog[2, I*E^(c + d*x)])/d^2))/a - (b*(-((b*((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*PolyLog[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*I)*((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*P 
olyLog[2, E^(2*c + 2*d*x)])/d^2))/a))/a))/a
 

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 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. 1477 vs. \(2 (714 ) = 1428\).

Time = 13.31 (sec) , antiderivative size = 1478, normalized size of antiderivative = 1.94

method	result	size
risch	\(\text {Expression too large to display}\)	\(1478\)

Input:

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

Output:

8/d*e/(4*a^2+4*b^2)*b*arctan(exp(d*x+c))+4/d*a*e/(4*a^2+4*b^2)*ln(1+exp(2* 
d*x+2*c))+4/d^2*a*f/(4*a^2+4*b^2)*dilog(1+I*exp(d*x+c))+4/d^2*a*f/(4*a^2+4 
*b^2)*dilog(1-I*exp(d*x+c))+1/d^2*b^2*f/(a^2+b^2)^(3/2)*arctanh(1/2*(2*b*e 
xp(d*x+c)+2*a)/(a^2+b^2)^(1/2))+1/d^2*b^2/a^3*f*dilog(exp(d*x+c)+1)+1/d*b^ 
2/a^3*e*ln(exp(d*x+c)-1)+1/d*b^2/a^3*e*ln(exp(d*x+c)+1)-1/d^2*b/a^2*f*ln(e 
xp(d*x+c)-1)+1/d^2*b/a^2*f*ln(exp(d*x+c)+1)-1/d^2*b^2/a^3*f*dilog(exp(d*x+ 
c))-1/d^2/a^3*b^4*f/(a^2+b^2)*dilog((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+ 
(a^2+b^2)^(1/2)))-1/d^2/a^3*b^4*f/(a^2+b^2)*dilog((b*exp(d*x+c)+(a^2+b^2)^ 
(1/2)+a)/(a+(a^2+b^2)^(1/2)))+4/d*a*f/(4*a^2+4*b^2)*ln(1+I*exp(d*x+c))*x+4 
/d*a*f/(4*a^2+4*b^2)*ln(1-I*exp(d*x+c))*x-8/d^2*c*f/(4*a^2+4*b^2)*b*arctan 
(exp(d*x+c))+4*I/d^2*f/(4*a^2+4*b^2)*dilog(1-I*exp(d*x+c))*b-4*I/d^2*f/(4* 
a^2+4*b^2)*dilog(1+I*exp(d*x+c))*b-4/d^2*a*c*f/(4*a^2+4*b^2)*ln(1+exp(2*d* 
x+2*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))-1/d/a^3*b^4*e/(a^2+b^2)*ln(b*exp(2*d*x+2*c)+2*a*exp(d*x+c) 
-b)+4/d^2*a*f/(4*a^2+4*b^2)*ln(1+I*exp(d*x+c))*c+4/d^2*a*f/(4*a^2+4*b^2)*l 
n(1-I*exp(d*x+c))*c-1/d^2/a^2*b^2*f/(a^2+b^2)^(1/2)*arctanh(1/2*(2*b*exp(d 
*x+c)+2*a)/(a^2+b^2)^(1/2))+1/d^2*c*f/a*ln(exp(d*x+c)-1)+1/d^2/a*f*dilog(e 
xp(d*x+c))-1/d/a*e*ln(exp(d*x+c)+1)-1/d/a*e*ln(exp(d*x+c)-1)-1/d^2*b^2/a^3 
*c*f*ln(exp(d*x+c)-1)+1/d*b^2/a^3*f*ln(exp(d*x+c)+1)*x-1/d/a*f*ln(exp(d*x+ 
c)+1)*x-1/d^2*f/a*dilog(exp(d*x+c)+1)-(-2*b*d*f*x*exp(3*d*x+3*c)+2*a*d*...
 

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. 5731 vs. \(2 (695) = 1390\).

Time = 0.27 (sec) , antiderivative size = 5731, normalized size of antiderivative = 7.52 \[ \int \frac {(e+f x) \text {csch}^3(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

integrate((f*x+e)*csch(d*x+c)^3*sech(d*x+c)/(a+b*sinh(d*x+c)),x, algorithm 
="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 + (16*a^2*d*integrate(1/16*x/(a^3*d*e^(d*x + c) + a^3*d), x) - 16*b^2 
*d*integrate(1/16*x/(a^3*d*e^(d*x + c) + a^3*d), x) - 16*a^2*d*integrate(1 
/16*x/(a^3*d*e^(d*x + c) - a^3*d), x) + 16*b^2*d*integrate(1/16*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)) + 
(2*b*d*x*e^(3*d*x + 3*c) - 2*b*d*x*e^(d*x + c) - (2*a*d*x*e^(2*c) + a*e^(2 
*c))*e^(2*d*x) + a)/(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) + 16*integrate(-1/8*(a*b^4*x*e^(d*x + c) - b^5*x)/(a^5*b + a^3*b 
^3 - (a^5*b*e^(2*c) + a^3*b^3*e^(2*c))*e^(2*d*x) - 2*(a^6*e^c + a^4*b^2*e^ 
c)*e^(d*x)), x) + 16*integrate(1/8*(b*x*e^(d*x + c) - a*x)/(a^2 + b^2 + (a 
^2*e^(2*c) + b^2*e^(2*c))*e^(2*d*x)), x))*f
 

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(2*e**(4*c + 4*d*x)*atan(e**(c + d*x))*a**3*b*e - 4*e**(2*c + 2*d*x)*atan( 
e**(c + d*x))*a**3*b*e + 2*atan(e**(c + d*x))*a**3*b*e + 32*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*f + 32*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*f + e**(4*c + 
4*d*x)*log(e**(2*c + 2*d*x) + 1)*a**4*e - e**(4*c + 4*d*x)*log(e**(c + d*x 
) - 1)*a**4*e + e**(4*c + 4*d*x)*log(e**(c + d*x) - 1)*b**4*e - e**(4*c + 
4*d*x)*log(e**(c + d*x) + 1)*a**4*e + e**(4*c + 4*d*x)*log(e**(c + d*x) + 
1)*b**4*e - e**(4*c + 4*d*x)*log(e**(2*c + 2*d*x)*b + 2*e**(c + d*x)*a - b 
)*b**4*e - e**(4*c + 4*d*x)*a**4*e - e**(4*c + 4*d*x)*a**2*b**2*e + 2*e**( 
3*c + 3*d*x)*a**3*b*e + 2*e**(3*c + 3*d*x)*a*b**3*e - 64*e**(7*c + 2*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*f - 64*e**(7*c + 2*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)...