3.5.83 \(\int \frac {(e+f x) \coth ^2(c+d x) \text {csch}(c+d x)}{a+b \sinh (c+d x)} \, dx\) [483]

3.5.83.1 Optimal result

Integrand size = 32, antiderivative size = 413 \[ \int \frac {(e+f x) \coth ^2(c+d x) \text {csch}(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {(e+f x) \text {arctanh}\left (e^{c+d x}\right )}{a d}-\frac {2 b^2 (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{a^3 d}+\frac {b (e+f x) \coth (c+d x)}{a^2 d}-\frac {f \text {csch}(c+d x)}{2 a d^2}-\frac {(e+f x) \coth (c+d x) \text {csch}(c+d x)}{2 a d}-\frac {b \sqrt {a^2+b^2} (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d}+\frac {b \sqrt {a^2+b^2} (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d}-\frac {b f \log (\sinh (c+d x))}{a^2 d^2}-\frac {f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{2 a d^2}-\frac {b^2 f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a^3 d^2}+\frac {f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{2 a d^2}+\frac {b^2 f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a^3 d^2}-\frac {b \sqrt {a^2+b^2} f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d^2}+\frac {b \sqrt {a^2+b^2} f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d^2} \]

output

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

3.5.83.2 Mathematica [A] (verified)

Time = 8.83 (sec) , antiderivative size = 615, normalized size of antiderivative = 1.49 \[ \int \frac {(e+f x) \coth ^2(c+d x) \text {csch}(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {\left (2 b d e \cosh \left (\frac {1}{2} (c+d x)\right )-a f \cosh \left (\frac {1}{2} (c+d x)\right )-2 b c f \cosh \left (\frac {1}{2} (c+d x)\right )+2 b f (c+d x) \cosh \left (\frac {1}{2} (c+d x)\right )\right ) \text {csch}\left (\frac {1}{2} (c+d x)\right )}{4 a^2 d^2}+\frac {(-d e+c f-f (c+d x)) \text {csch}^2\left (\frac {1}{2} (c+d x)\right )}{8 a d^2}+\frac {-2 a b f (c+d x)+\left (-2 a b f+a^2 (d e+d f x)+2 b^2 (d e+d f x)\right ) \log \left (1-e^{-c-d x}\right )-\left (2 a b f+a^2 (d e+d f x)+2 b^2 (d e+d f x)\right ) \log \left (1+e^{-c-d x}\right )+\left (a^2+2 b^2\right ) f \operatorname {PolyLog}\left (2,-e^{-c-d x}\right )-\left (a^2+2 b^2\right ) f \operatorname {PolyLog}\left (2,e^{-c-d x}\right )}{2 a^3 d^2}-\frac {b \sqrt {a^2+b^2} \left (-2 d e \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )+2 c f \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )+f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )-f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+f \operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )-f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right )}{a^3 d^2}+\frac {(-d e+c f-f (c+d x)) \text {sech}^2\left (\frac {1}{2} (c+d x)\right )}{8 a d^2}+\frac {\text {sech}\left (\frac {1}{2} (c+d x)\right ) \left (2 b d e \sinh \left (\frac {1}{2} (c+d x)\right )+a f \sinh \left (\frac {1}{2} (c+d x)\right )-2 b c f \sinh \left (\frac {1}{2} (c+d x)\right )+2 b f (c+d x) \sinh \left (\frac {1}{2} (c+d x)\right )\right )}{4 a^2 d^2} \]

input

Integrate[((e + f*x)*Coth[c + d*x]^2*Csch[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) + (-2*a*b*f 
*(c + d*x) + (-2*a*b*f + a^2*(d*e + d*f*x) + 2*b^2*(d*e + d*f*x))*Log[1 - 
E^(-c - d*x)] - (2*a*b*f + a^2*(d*e + d*f*x) + 2*b^2*(d*e + d*f*x))*Log[1 
+ E^(-c - d*x)] + (a^2 + 2*b^2)*f*PolyLog[2, -E^(-c - d*x)] - (a^2 + 2*b^2 
)*f*PolyLog[2, E^(-c - d*x)])/(2*a^3*d^2) - (b*Sqrt[a^2 + b^2]*(-2*d*e*Arc 
Tanh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]] + 2*c*f*ArcTanh[(a + b*E^(c + d* 
x))/Sqrt[a^2 + b^2]] + f*(c + d*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + 
 b^2])] - f*(c + d*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] + f*P 
olyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] - f*PolyLog[2, -((b*E^(c 
 + d*x))/(a + Sqrt[a^2 + b^2]))]))/(a^3*d^2) + ((-(d*e) + c*f - f*(c + d*x 
))*Sech[(c + d*x)/2]^2)/(8*a*d^2) + (Sech[(c + d*x)/2]*(2*b*d*e*Sinh[(c + 
d*x)/2] + a*f*Sinh[(c + d*x)/2] - 2*b*c*f*Sinh[(c + d*x)/2] + 2*b*f*(c + d 
*x)*Sinh[(c + d*x)/2]))/(4*a^2*d^2)
 

3.5.83.3 Rubi [F]

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.

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

\(\Big \downarrow \) 6121

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

\(\Big \downarrow \) 5980

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4670

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

\(\Big \downarrow \) 4673

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4670

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

\(\Big \downarrow \) 6103

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 4203

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

\(\Big \downarrow \) 17

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3956

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

\(\Big \downarrow \) 6119

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

\(\Big \downarrow \) 5973

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3117

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

input

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

output

$Aborted
 

3.5.83.3.1 Defintions of rubi rules used

rule 17

Int[(c_.)*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[c*((a + b*x)^(m + 1 
)/(b*(m + 1))), x] /; FreeQ[{a, b, c, m}, x] && NeQ[m, -1]
 

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 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 3117

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; 
 FreeQ[{c, d}, x]
 

rule 3777

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

rule 3956

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d 
*x], x]]/d, x] /; FreeQ[{c, d}, x]
 

rule 4203

Int[((c_.) + (d_.)*(x_))^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symb 
ol] :> Simp[b*(c + d*x)^m*((b*Tan[e + f*x])^(n - 1)/(f*(n - 1))), x] + (-Si 
mp[b*d*(m/(f*(n - 1)))   Int[(c + d*x)^(m - 1)*(b*Tan[e + f*x])^(n - 1), x] 
, x] - Simp[b^2   Int[(c + d*x)^m*(b*Tan[e + f*x])^(n - 2), x], x]) /; Free 
Q[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 0]
 

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 5973

Int[Cosh[(a_.) + (b_.)*(x_)]^(n_.)*Coth[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + 
(d_.)*(x_))^(m_.), x_Symbol] :> Int[(c + d*x)^m*Cosh[a + b*x]^n*Coth[a + b* 
x]^(p - 2), x] + Int[(c + d*x)^m*Cosh[a + b*x]^(n - 2)*Coth[a + b*x]^p, x] 
/; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IGtQ[p, 0]
 

rule 5980

Int[Coth[(a_.) + (b_.)*(x_)]^(p_)*Csch[(a_.) + (b_.)*(x_)]*((c_.) + (d_.)*( 
x_))^(m_.), x_Symbol] :> Int[(c + d*x)^m*Csch[a + b*x]*Coth[a + b*x]^(p - 2 
), x] + Int[(c + d*x)^m*Csch[a + b*x]^3*Coth[a + b*x]^(p - 2), x] /; FreeQ[ 
{a, b, c, d, m}, x] && IGtQ[p/2, 0]
 

rule 6103

Int[(Coth[(c_.) + (d_.)*(x_)]^(n_.)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_ 
.)*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[1/a   Int[(e + f*x)^m*Coth[ 
c + d*x]^n, x], x] - Simp[b/a   Int[(e + f*x)^m*Cosh[c + d*x]*(Coth[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]
 

rule 6119

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

rule 6121

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

3.5.83.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1283\) vs. \(2(379)=758\).

Time = 1.96 (sec) , antiderivative size = 1284, normalized size of antiderivative = 3.11

method	result	size
risch	\(\text {Expression too large to display}\)	\(1284\)

input

int((f*x+e)*coth(d*x+c)^2*csch(d*x+c)/(a+b*sinh(d*x+c)),x,method=_RETURNVE 
RBOSE)
 

output

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

3.5.83.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3585 vs. \(2 (373) = 746\).

Time = 0.34 (sec) , antiderivative size = 3585, normalized size of antiderivative = 8.68 \[ \int \frac {(e+f x) \coth ^2(c+d x) \text {csch}(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \]

input

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

output

1/2*(4*(a*b*d*f*x + a*b*c*f)*cosh(d*x + c)^4 + 4*(a*b*d*f*x + a*b*c*f)*sin 
h(d*x + c)^4 - 4*a*b*d*e + 4*a*b*c*f - 2*(a^2*d*f*x + a^2*d*e + a^2*f)*cos 
h(d*x + c)^3 - 2*(a^2*d*f*x + a^2*d*e + a^2*f - 8*(a*b*d*f*x + a*b*c*f)*co 
sh(d*x + c))*sinh(d*x + c)^3 - 4*(a*b*d*f*x - a*b*d*e + 2*a*b*c*f)*cosh(d* 
x + c)^2 - 2*(2*a*b*d*f*x - 2*a*b*d*e + 4*a*b*c*f - 12*(a*b*d*f*x + a*b*c* 
f)*cosh(d*x + c)^2 + 3*(a^2*d*f*x + a^2*d*e + a^2*f)*cosh(d*x + c))*sinh(d 
*x + c)^2 - 2*(b^2*f*cosh(d*x + c)^4 + 4*b^2*f*cosh(d*x + c)*sinh(d*x + c) 
^3 + b^2*f*sinh(d*x + c)^4 - 2*b^2*f*cosh(d*x + c)^2 + b^2*f + 2*(3*b^2*f* 
cosh(d*x + c)^2 - b^2*f)*sinh(d*x + c)^2 + 4*(b^2*f*cosh(d*x + c)^3 - b^2* 
f*cosh(d*x + c))*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2)*dilog((a*cosh(d*x + 
c) + a*sinh(d*x + c) + (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2 
)/b^2) - b)/b + 1) + 2*(b^2*f*cosh(d*x + c)^4 + 4*b^2*f*cosh(d*x + c)*sinh 
(d*x + c)^3 + b^2*f*sinh(d*x + c)^4 - 2*b^2*f*cosh(d*x + c)^2 + b^2*f + 2* 
(3*b^2*f*cosh(d*x + c)^2 - b^2*f)*sinh(d*x + c)^2 + 4*(b^2*f*cosh(d*x + c) 
^3 - b^2*f*cosh(d*x + c))*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2)*dilog((a*co 
sh(d*x + c) + a*sinh(d*x + c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt(( 
a^2 + b^2)/b^2) - b)/b + 1) + 2*((b^2*d*e - b^2*c*f)*cosh(d*x + c)^4 + 4*( 
b^2*d*e - b^2*c*f)*cosh(d*x + c)*sinh(d*x + c)^3 + (b^2*d*e - b^2*c*f)*sin 
h(d*x + c)^4 + b^2*d*e - b^2*c*f - 2*(b^2*d*e - b^2*c*f)*cosh(d*x + c)^2 - 
 2*(b^2*d*e - b^2*c*f - 3*(b^2*d*e - b^2*c*f)*cosh(d*x + c)^2)*sinh(d*x...
 

3.5.83.6 Sympy [F]

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

input

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

output

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

3.5.83.7 Maxima [F]

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

input

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

output

(2*a^2*d*integrate(1/4*x/(a^3*d*e^(d*x + c) + a^3*d), x) + 4*b^2*d*integra 
te(1/4*x/(a^3*d*e^(d*x + c) + a^3*d), x) + 2*a^2*d*integrate(1/4*x/(a^3*d* 
e^(d*x + c) - a^3*d), x) + 4*b^2*d*integrate(1/4*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*(a^2*b*e^c + 
b^3*e^c)*integrate(x*e^(d*x)/(a^3*b*e^(2*d*x + 2*c) + 2*a^4*e^(d*x + c) - 
a^3*b), x) + (2*b*d*x*e^(2*d*x + 2*c) - 2*b*d*x - (a*d*x*e^(3*c) + a*e^(3* 
c))*e^(3*d*x) - (a*d*x*e^c - a*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))*f + 1/2*e*(2*(a*e^(-d*x - c) + 2*b*e^( 
-2*d*x - 2*c) + a*e^(-3*d*x - 3*c) - 2*b)/((2*a^2*e^(-2*d*x - 2*c) - a^2*e 
^(-4*d*x - 4*c) - a^2)*d) - (a^2 + 2*b^2)*log(e^(-d*x - c) + 1)/(a^3*d) + 
(a^2 + 2*b^2)*log(e^(-d*x - c) - 1)/(a^3*d) - 2*(a^2*b + b^3)*log((b*e^(-d 
*x - c) - a - sqrt(a^2 + b^2))/(b*e^(-d*x - c) - a + sqrt(a^2 + b^2)))/(sq 
rt(a^2 + b^2)*a^3*d))
 

3.5.83.8 Giac [F(-1)]

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

input

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

output

Timed out
 

3.5.83.9 Mupad [F(-1)]

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

input

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

output

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