\(\int \frac {(e+f x) \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)} \, dx\) [427]

Optimal result

Integrand size = 30, antiderivative size = 286 \[ \int \frac {(e+f x) \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {(e+f x)^2}{2 b f}-\frac {2 (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{a d}-\frac {\sqrt {a^2+b^2} (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a b d}+\frac {\sqrt {a^2+b^2} (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a b d}-\frac {f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a d^2}+\frac {f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a d^2}-\frac {\sqrt {a^2+b^2} f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a b d^2}+\frac {\sqrt {a^2+b^2} f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a b d^2} \] Output:

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

Mathematica [A] (verified)

Time = 1.30 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.04 \[ \int \frac {(e+f x) \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {-a (c+d x) (c f-d (2 e+f x))+2 b \left (d (e+f x) \left (\log \left (1-e^{c+d x}\right )-\log \left (1+e^{c+d x}\right )\right )-f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )+f \operatorname {PolyLog}\left (2,e^{c+d x}\right )\right )+2 \sqrt {a^2+b^2} \left (2 d e \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )-2 c f \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )-f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )+f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-f \operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )+f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right )}{2 a b d^2} \] Input:

Integrate[((e + f*x)*Cosh[c + d*x]*Coth[c + d*x])/(a + b*Sinh[c + d*x]),x]
 

Output:

(-(a*(c + d*x)*(c*f - d*(2*e + f*x))) + 2*b*(d*(e + f*x)*(Log[1 - E^(c + d 
*x)] - Log[1 + E^(c + d*x)]) - f*PolyLog[2, -E^(c + d*x)] + f*PolyLog[2, E 
^(c + d*x)]) + 2*Sqrt[a^2 + b^2]*(2*d*e*ArcTanh[(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*PolyLog[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])) 
]))/(2*a*b*d^2)
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 2.35 (sec) , antiderivative size = 368, normalized size of antiderivative = 1.29, number of steps used = 25, number of rules used = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {6119, 5973, 3042, 26, 3777, 3042, 3117, 4670, 2715, 2838, 6099, 17, 3042, 26, 3777, 3042, 3117, 3803, 25, 2694, 27, 2620, 2715, 2838}

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.

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

\(\Big \downarrow \) 6119

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

\(\Big \downarrow \) 5973

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {b \int \frac {(e+f x) \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \int (e+f x) \csc (i c+i d x)dx-i \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \int \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}\right )}{a}\)

\(\Big \downarrow \) 3117

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

\(\Big \downarrow \) 4670

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

\(\Big \downarrow \) 6099

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

\(\Big \downarrow \) 17

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3117

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

\(\Big \downarrow \) 3803

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2694

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

(I*(((2*I)*(e + f*x)*ArcTanh[E^(c + d*x)])/d + (I*f*PolyLog[2, -E^(c + d*x 
)])/d^2 - (I*f*PolyLog[2, E^(c + d*x)])/d^2) - I*((I*(e + f*x)*Cosh[c + d* 
x])/d - (I*f*Sinh[c + d*x])/d^2))/a - (b*(-1/2*(a*(e + f*x)^2)/(b^2*f) - ( 
2*(a^2 + b^2)*(-1/2*(b*(((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)))/Sqrt[a^2 + b^2] + (b*(((e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sq 
rt[a^2 + b^2])])/(b*d) + (f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b 
^2]))])/(b*d^2)))/(2*Sqrt[a^2 + b^2])))/b^2 - (I*((I*(e + f*x)*Cosh[c + d* 
x])/d - (I*f*Sinh[c + d*x])/d^2))/b))/a
 

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 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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 2694

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.) 
*(F_)^(v_)), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[2*(c/q)   Int 
[(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Simp[2*(c/q)   Int[(f + g*x) 
^m*(F^u/(b + q + 2*c*F^u)), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[ 
v, 2*u] && LinearQ[u, x] && NeQ[b^2 - 4*a*c, 0] && 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 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 3803

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (Complex[0, fz_])* 
(f_.)*(x_)]), x_Symbol] :> Simp[2   Int[(c + d*x)^m*(E^((-I)*e + f*fz*x)/(( 
-I)*b + 2*a*E^((-I)*e + f*fz*x) + I*b*E^(2*((-I)*e + f*fz*x)))), x], x] /; 
FreeQ[{a, b, c, d, e, f, fz}, x] && NeQ[a^2 - b^2, 0] && IGtQ[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 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 6099

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(969\) vs. \(2(261)=522\).

Time = 1.23 (sec) , antiderivative size = 970, normalized size of antiderivative = 3.39

method	result	size
risch	\(\frac {f \,x^{2}}{2 b}+\frac {e x}{b}-\frac {a f \ln \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right ) c}{b \,d^{2} \sqrt {a^{2}+b^{2}}}-\frac {a f \ln \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right ) x}{b d \sqrt {a^{2}+b^{2}}}+\frac {a f \ln \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right ) x}{b d \sqrt {a^{2}+b^{2}}}-\frac {a f \operatorname {dilog}\left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right )}{b \,d^{2} \sqrt {a^{2}+b^{2}}}+\frac {a f \operatorname {dilog}\left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right )}{b \,d^{2} \sqrt {a^{2}+b^{2}}}-\frac {f \ln \left ({\mathrm e}^{d x +c}+1\right ) x}{d a}-\frac {f \operatorname {dilog}\left ({\mathrm e}^{d x +c}+1\right )}{d^{2} a}-\frac {f \operatorname {dilog}\left ({\mathrm e}^{d x +c}\right )}{d^{2} a}+\frac {2 a e \,\operatorname {arctanh}\left (\frac {2 b \,{\mathrm e}^{d x +c}+2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{b d \sqrt {a^{2}+b^{2}}}+\frac {2 b e \,\operatorname {arctanh}\left (\frac {2 b \,{\mathrm e}^{d x +c}+2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{d a \sqrt {a^{2}+b^{2}}}-\frac {2 b c f \,\operatorname {arctanh}\left (\frac {2 b \,{\mathrm e}^{d x +c}+2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{d^{2} a \sqrt {a^{2}+b^{2}}}-\frac {b f \ln \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right ) c}{d^{2} a \sqrt {a^{2}+b^{2}}}+\frac {b f \ln \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right ) c}{d^{2} a \sqrt {a^{2}+b^{2}}}+\frac {a f \ln \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right ) c}{b \,d^{2} \sqrt {a^{2}+b^{2}}}-\frac {c f \ln \left ({\mathrm e}^{d x +c}-1\right )}{d^{2} a}-\frac {b f \ln \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right ) x}{d a \sqrt {a^{2}+b^{2}}}+\frac {b f \ln \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right ) x}{d a \sqrt {a^{2}+b^{2}}}-\frac {b f \operatorname {dilog}\left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right )}{d^{2} a \sqrt {a^{2}+b^{2}}}+\frac {b f \operatorname {dilog}\left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right )}{d^{2} a \sqrt {a^{2}+b^{2}}}-\frac {e \ln \left ({\mathrm e}^{d x +c}+1\right )}{d a}+\frac {e \ln \left ({\mathrm e}^{d x +c}-1\right )}{d a}-\frac {2 c a f \,\operatorname {arctanh}\left (\frac {2 b \,{\mathrm e}^{d x +c}+2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{b \,d^{2} \sqrt {a^{2}+b^{2}}}\)	\(970\)

Input:

int((f*x+e)*cosh(d*x+c)*coth(d*x+c)/(a+b*sinh(d*x+c)),x,method=_RETURNVERB 
OSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 598 vs. \(2 (257) = 514\).

Time = 0.14 (sec) , antiderivative size = 598, normalized size of antiderivative = 2.09 \[ \int \frac {(e+f x) \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)} \, dx =\text {Too large to display} \] Input:

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

Output:

1/2*(a*d^2*f*x^2 + 2*a*d^2*e*x - 2*b*f*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*f*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*f*dilog(cosh(d*x + c) + sinh(d*x + c)) - 2*b*f* 
dilog(-cosh(d*x + c) - sinh(d*x + c)) + 2*(b*d*e - b*c*f)*sqrt((a^2 + b^2) 
/b^2)*log(2*b*cosh(d*x + c) + 2*b*sinh(d*x + c) + 2*b*sqrt((a^2 + b^2)/b^2 
) + 2*a) - 2*(b*d*e - b*c*f)*sqrt((a^2 + b^2)/b^2)*log(2*b*cosh(d*x + c) + 
 2*b*sinh(d*x + c) - 2*b*sqrt((a^2 + b^2)/b^2) + 2*a) - 2*(b*d*f*x + b*c*f 
)*sqrt((a^2 + b^2)/b^2)*log(-(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) + 2*(b*d*f*x + b 
*c*f)*sqrt((a^2 + b^2)/b^2)*log(-(a*cosh(d*x + c) + a*sinh(d*x + c) - (b*c 
osh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b) - 2*(b*d*f*x 
 + b*d*e)*log(cosh(d*x + c) + sinh(d*x + c) + 1) + 2*(b*d*e - b*c*f)*log(c 
osh(d*x + c) + sinh(d*x + c) - 1) + 2*(b*d*f*x + b*c*f)*log(-cosh(d*x + c) 
 - sinh(d*x + c) + 1))/(a*b*d^2)
 

Sympy [F]

\[ \int \frac {(e+f x) \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\left (e + f x\right ) \cosh {\left (c + d x \right )} \coth {\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {(e+f x) \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )} \cosh \left (d x + c\right ) \coth \left (d x + c\right )}{b \sinh \left (d x + c\right ) + a} \,d x } \] Input:

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

Output:

-1/2*(4*(a^2*e^c + b^2*e^c)*integrate(x*e^(d*x)/(a*b^2*e^(2*d*x + 2*c) + 2 
*a^2*b*e^(d*x + c) - a*b^2), x) - x^2/b - 2*integrate(x/(a*e^(d*x + c) + a 
), x) - 2*integrate(x/(a*e^(d*x + c) - a), x))*f + e*((d*x + c)/(b*d) - lo 
g(e^(-d*x - c) + 1)/(a*d) + log(e^(-d*x - c) - 1)/(a*d) - sqrt(a^2 + b^2)* 
log((b*e^(-d*x - c) - a - sqrt(a^2 + b^2))/(b*e^(-d*x - c) - a + sqrt(a^2 
+ b^2)))/(a*b*d))
 

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(e+f x) \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\left (f x +e \right ) \cosh \left (d x +c \right ) \coth \left (d x +c \right )}{a +b \sinh \left (d x +c \right )}d x \] Input:

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

Output:

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