\(\int \frac {(e+f x) \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)} \, dx\) [432]

Optimal result

Integrand size = 32, antiderivative size = 322 \[ \int \frac {(e+f x) \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {(e+f x)^2}{2 a f}+\frac {\left (a^2+b^2\right ) (e+f x)^2}{2 a b^2 f}-\frac {f \cosh (c+d x)}{b d^2}-\frac {\left (a^2+b^2\right ) (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a b^2 d}-\frac {\left (a^2+b^2\right ) (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a b^2 d}+\frac {(e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a d}-\frac {\left (a^2+b^2\right ) f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a b^2 d^2}-\frac {\left (a^2+b^2\right ) f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a b^2 d^2}+\frac {f \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{2 a d^2}+\frac {(e+f x) \sinh (c+d x)}{b d} \] Output:

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

Mathematica [A] (warning: unable to verify)

Time = 3.56 (sec) , antiderivative size = 487, normalized size of antiderivative = 1.51 \[ \int \frac {(e+f x) \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {-\frac {2 f \cosh (c+d x)}{b}+\frac {d \left (2 c e+2 d e x+d f x^2+2 (e+f x) \log \left (1-e^{-c-d x}\right )+2 (e+f x) \log \left (1+e^{-c-d x}\right )\right )-2 f \operatorname {PolyLog}\left (2,-e^{-c-d x}\right )-2 f \operatorname {PolyLog}\left (2,e^{-c-d x}\right )}{a}-\frac {\left (a^2+b^2\right ) \left (-2 d e (c+d x)+2 c f (c+d x)-f (c+d x)^2+\frac {4 a \sqrt {a^2+b^2} d e \arctan \left (\frac {a+b e^{c+d x}}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-\left (a^2+b^2\right )^2}}-\frac {4 a \sqrt {-\left (a^2+b^2\right )^2} d e \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )}{\left (-a^2-b^2\right )^{3/2}}+2 f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )+2 f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-2 c f \log \left (b-2 a e^{c+d x}-b e^{2 (c+d x)}\right )+2 d e \log \left (2 a e^{c+d x}+b \left (-1+e^{2 (c+d x)}\right )\right )+2 f \operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )+2 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right )}{a b^2}+\frac {2 d (e+f x) \sinh (c+d x)}{b}}{2 d^2} \] Input:

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

Output:

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

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 2.97 (sec) , antiderivative size = 422, normalized size of antiderivative = 1.31, number of steps used = 30, number of rules used = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.906, Rules used = {6119, 5973, 3042, 26, 4201, 2620, 2715, 2838, 5969, 3042, 25, 3115, 24, 6099, 3042, 3777, 26, 3042, 26, 3118, 5969, 3042, 25, 3115, 24, 6095, 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.

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

\(\Big \downarrow \) 6119

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

\(\Big \downarrow \) 5973

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

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {b \int \frac {(e+f x) \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\int (e+f x) \cosh (c+d x) \sinh (c+d x)dx+\int -i (e+f x) \tan \left (i c+i d x+\frac {\pi }{2}\right )dx}{a}\)

\(\Big \downarrow \) 26

\(\displaystyle -\frac {b \int \frac {(e+f x) \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\int (e+f x) \cosh (c+d x) \sinh (c+d x)dx-i \int (e+f x) \tan \left (\frac {1}{2} (2 i c+\pi )+i d x\right )dx}{a}\)

\(\Big \downarrow \) 4201

\(\displaystyle -\frac {b \int \frac {(e+f x) \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\int (e+f x) \cosh (c+d x) \sinh (c+d x)dx-i \left (2 i \int \frac {e^{2 c+2 d x-i \pi } (e+f x)}{1+e^{2 c+2 d x-i \pi }}dx-\frac {i (e+f x)^2}{2 f}\right )}{a}\)

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

\(\displaystyle -\frac {b \int \frac {(e+f x) \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\int (e+f x) \cosh (c+d x) \sinh (c+d x)dx-i \left (2 i \left (\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \int e^{-2 c-2 d x+i \pi } \log \left (1+e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}\right )-\frac {i (e+f x)^2}{2 f}\right )}{a}\)

\(\Big \downarrow \) 2838

\(\displaystyle -\frac {b \int \frac {(e+f x) \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\int (e+f x) \cosh (c+d x) \sinh (c+d x)dx-i \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{a}\)

\(\Big \downarrow \) 5969

\(\displaystyle -\frac {b \int \frac {(e+f x) \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {-\frac {f \int \sinh ^2(c+d x)dx}{2 d}-i \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )+\frac {(e+f x) \sinh ^2(c+d x)}{2 d}}{a}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {b \int \frac {(e+f x) \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {-\frac {f \int -\sin (i c+i d x)^2dx}{2 d}-i \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )+\frac {(e+f x) \sinh ^2(c+d x)}{2 d}}{a}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {b \int \frac {(e+f x) \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\frac {f \int \sin (i c+i d x)^2dx}{2 d}-i \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )+\frac {(e+f x) \sinh ^2(c+d x)}{2 d}}{a}\)

\(\Big \downarrow \) 3115

\(\displaystyle -\frac {b \int \frac {(e+f x) \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {\frac {f \left (\frac {\int 1dx}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d}-i \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )+\frac {(e+f x) \sinh ^2(c+d x)}{2 d}}{a}\)

\(\Big \downarrow \) 24

\(\displaystyle -\frac {b \int \frac {(e+f x) \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {-i \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )+\frac {(e+f x) \sinh ^2(c+d x)}{2 d}+\frac {f \left (\frac {x}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d}}{a}\)

\(\Big \downarrow \) 6099

\(\displaystyle -\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}-\frac {a \int (e+f x) \cosh (c+d x)dx}{b^2}+\frac {\int (e+f x) \cosh (c+d x) \sinh (c+d x)dx}{b}\right )}{a}+\frac {-i \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )+\frac {(e+f x) \sinh ^2(c+d x)}{2 d}+\frac {f \left (\frac {x}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d}}{a}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-i \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )+\frac {(e+f x) \sinh ^2(c+d x)}{2 d}+\frac {f \left (\frac {x}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d}}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}-\frac {a \int (e+f x) \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{b^2}+\frac {\int (e+f x) \cosh (c+d x) \sinh (c+d x)dx}{b}\right )}{a}\)

\(\Big \downarrow \) 3777

\(\displaystyle \frac {-i \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )+\frac {(e+f x) \sinh ^2(c+d x)}{2 d}+\frac {f \left (\frac {x}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d}}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}-\frac {a \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {i f \int -i \sinh (c+d x)dx}{d}\right )}{b^2}+\frac {\int (e+f x) \cosh (c+d x) \sinh (c+d x)dx}{b}\right )}{a}\)

\(\Big \downarrow \) 26

\(\displaystyle -\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}-\frac {a \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \int \sinh (c+d x)dx}{d}\right )}{b^2}+\frac {\int (e+f x) \cosh (c+d x) \sinh (c+d x)dx}{b}\right )}{a}+\frac {-i \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )+\frac {(e+f x) \sinh ^2(c+d x)}{2 d}+\frac {f \left (\frac {x}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d}}{a}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-i \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )+\frac {(e+f x) \sinh ^2(c+d x)}{2 d}+\frac {f \left (\frac {x}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d}}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}-\frac {a \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \int -i \sin (i c+i d x)dx}{d}\right )}{b^2}+\frac {\int (e+f x) \cosh (c+d x) \sinh (c+d x)dx}{b}\right )}{a}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {-i \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )+\frac {(e+f x) \sinh ^2(c+d x)}{2 d}+\frac {f \left (\frac {x}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d}}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}-\frac {a \left (\frac {(e+f x) \sinh (c+d x)}{d}+\frac {i f \int \sin (i c+i d x)dx}{d}\right )}{b^2}+\frac {\int (e+f x) \cosh (c+d x) \sinh (c+d x)dx}{b}\right )}{a}\)

\(\Big \downarrow \) 3118

\(\displaystyle -\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}+\frac {\int (e+f x) \cosh (c+d x) \sinh (c+d x)dx}{b}-\frac {a \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{b^2}\right )}{a}+\frac {-i \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )+\frac {(e+f x) \sinh ^2(c+d x)}{2 d}+\frac {f \left (\frac {x}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d}}{a}\)

\(\Big \downarrow \) 5969

\(\displaystyle -\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}+\frac {\frac {(e+f x) \sinh ^2(c+d x)}{2 d}-\frac {f \int \sinh ^2(c+d x)dx}{2 d}}{b}-\frac {a \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{b^2}\right )}{a}+\frac {-i \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )+\frac {(e+f x) \sinh ^2(c+d x)}{2 d}+\frac {f \left (\frac {x}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d}}{a}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-i \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )+\frac {(e+f x) \sinh ^2(c+d x)}{2 d}+\frac {f \left (\frac {x}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d}}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}+\frac {\frac {(e+f x) \sinh ^2(c+d x)}{2 d}-\frac {f \int -\sin (i c+i d x)^2dx}{2 d}}{b}-\frac {a \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{b^2}\right )}{a}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {-i \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )+\frac {(e+f x) \sinh ^2(c+d x)}{2 d}+\frac {f \left (\frac {x}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d}}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}+\frac {\frac {(e+f x) \sinh ^2(c+d x)}{2 d}+\frac {f \int \sin (i c+i d x)^2dx}{2 d}}{b}-\frac {a \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{b^2}\right )}{a}\)

\(\Big \downarrow \) 3115

\(\displaystyle -\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}+\frac {\frac {f \left (\frac {\int 1dx}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d}+\frac {(e+f x) \sinh ^2(c+d x)}{2 d}}{b}-\frac {a \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{b^2}\right )}{a}+\frac {-i \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )+\frac {(e+f x) \sinh ^2(c+d x)}{2 d}+\frac {f \left (\frac {x}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d}}{a}\)

\(\Big \downarrow \) 24

\(\displaystyle -\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}-\frac {a \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{b^2}+\frac {\frac {(e+f x) \sinh ^2(c+d x)}{2 d}+\frac {f \left (\frac {x}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d}}{b}\right )}{a}+\frac {-i \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )+\frac {(e+f x) \sinh ^2(c+d x)}{2 d}+\frac {f \left (\frac {x}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d}}{a}\)

\(\Big \downarrow \) 6095

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

\(\displaystyle -\frac {b \left (\frac {\left (a^2+b^2\right ) \left (-\frac {f \int e^{-c-d x} \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right )de^{c+d x}}{b d^2}-\frac {f \int e^{-c-d x} \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right )de^{c+d x}}{b d^2}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {(e+f x)^2}{2 b f}\right )}{b^2}-\frac {a \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{b^2}+\frac {\frac {(e+f x) \sinh ^2(c+d x)}{2 d}+\frac {f \left (\frac {x}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d}}{b}\right )}{a}+\frac {-i \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )+\frac {(e+f x) \sinh ^2(c+d x)}{2 d}+\frac {f \left (\frac {x}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d}}{a}\)

\(\Big \downarrow \) 2838

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

Input:

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

Output:

((-I)*(((-1/2*I)*(e + f*x)^2)/f + (2*I)*(((e + f*x)*Log[1 + E^(2*c - I*Pi 
+ 2*d*x)])/(2*d) + (f*PolyLog[2, -E^(2*c - I*Pi + 2*d*x)])/(4*d^2))) + ((e 
 + f*x)*Sinh[c + d*x]^2)/(2*d) + (f*(x/2 - (Cosh[c + d*x]*Sinh[c + d*x])/( 
2*d)))/(2*d))/a - (b*(((a^2 + b^2)*(-1/2*(e + f*x)^2/(b*f) + ((e + f*x)*Lo 
g[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)))/b^2 - (a*(-((f*Cosh[c + d*x])/d^2) + ((e 
+ f*x)*Sinh[c + d*x])/d))/b^2 + (((e + f*x)*Sinh[c + d*x]^2)/(2*d) + (f*(x 
/2 - (Cosh[c + d*x]*Sinh[c + d*x])/(2*d)))/(2*d))/b))/a
 

Defintions of rubi rules used

rule 24

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
 

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 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 3115

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* 
x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n)   Int[(b*Sin 
[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 
2*n]
 

rule 3118

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Cos[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 4201

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (Complex[0, fz_])*(f_.)*(x_)], x 
_Symbol] :> Simp[(-I)*((c + d*x)^(m + 1)/(d*(m + 1))), x] + Simp[2*I   Int[ 
(c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*fz*x)))), x], x] 
 /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
 

rule 5969

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

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 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 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. \(931\) vs. \(2(304)=608\).

Time = 2.97 (sec) , antiderivative size = 932, normalized size of antiderivative = 2.89

method	result	size
risch	\(-\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}{d \,b^{2}}-\frac {2 a c f \ln \left ({\mathrm e}^{d x +c}\right )}{d^{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 )}{d^{2} b^{2}}-\frac {c f \ln \left ({\mathrm e}^{d x +c}-1\right )}{d^{2} a}+\frac {c f \ln \left (b \,{\mathrm e}^{2 d x +2 c}+2 a \,{\mathrm e}^{d x +c}-b \right )}{d^{2} a}-\frac {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}-\frac {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}+\frac {f \ln \left ({\mathrm e}^{d x +c}+1\right ) x}{d a}-\frac {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}-\frac {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}+\frac {a f \,x^{2}}{2 b^{2}}-\frac {a e x}{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 )}{d^{2} b^{2}}+\frac {2 a e \ln \left ({\mathrm e}^{d x +c}\right )}{d \,b^{2}}-\frac {a e \ln \left (b \,{\mathrm e}^{2 d x +2 c}+2 a \,{\mathrm e}^{d x +c}-b \right )}{d \,b^{2}}+\frac {a f \,c^{2}}{d^{2} b^{2}}+\frac {e \ln \left ({\mathrm e}^{d x +c}-1\right )}{d a}-\frac {e \ln \left (b \,{\mathrm e}^{2 d x +2 c}+2 a \,{\mathrm e}^{d x +c}-b \right )}{d a}+\frac {e \ln \left ({\mathrm e}^{d x +c}+1\right )}{d a}+\frac {f \operatorname {dilog}\left ({\mathrm e}^{d x +c}+1\right )}{d^{2} a}-\frac {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}-\frac {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}-\frac {f \operatorname {dilog}\left ({\mathrm e}^{d x +c}\right )}{d^{2} a}+\frac {\left (d x f +d e -f \right ) {\mathrm e}^{d x +c}}{2 b \,d^{2}}-\frac {\left (d x f +d e +f \right ) {\mathrm e}^{-d x -c}}{2 b \,d^{2}}+\frac {2 a f c x}{d \,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}{d^{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}{d^{2} b^{2}}+\frac {c a f \ln \left (b \,{\mathrm e}^{2 d x +2 c}+2 a \,{\mathrm e}^{d x +c}-b \right )}{d^{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}{d \,b^{2}}\)	\(932\)

Input:

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

Output:

-1/d*a/b^2*f*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/b^2*c*f*ln(exp(d*x+c))-1/d^2*a/b^2*f*dilog((-b*exp(d*x+c)+(a^2+b^2 
)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))-1/d^2*c*f/a*ln(exp(d*x+c)-1)+1/d^2*c*f/a* 
ln(b*exp(2*d*x+2*c)+2*a*exp(d*x+c)-b)-1/d^2*f/a*ln((-b*exp(d*x+c)+(a^2+b^2 
)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*c-1/d^2*f/a*ln((b*exp(d*x+c)+(a^2+b^2)^(1 
/2)+a)/(a+(a^2+b^2)^(1/2)))*c+1/d*f/a*ln(exp(d*x+c)+1)*x-1/d*f/a*ln((-b*ex 
p(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*x-1/d*f/a*ln((b*exp(d*x+ 
c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*x+1/2*a/b^2*f*x^2-a/b^2*e*x-1/d 
^2/b^2*a*f*dilog((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))+2/d 
/b^2*a*e*ln(exp(d*x+c))-1/d/b^2*a*e*ln(b*exp(2*d*x+2*c)+2*a*exp(d*x+c)-b)+ 
1/d^2/b^2*a*f*c^2+1/d*e/a*ln(exp(d*x+c)-1)-1/d*e/a*ln(b*exp(2*d*x+2*c)+2*a 
*exp(d*x+c)-b)+1/d*e/a*ln(exp(d*x+c)+1)+1/d^2*f/a*dilog(exp(d*x+c)+1)-1/d^ 
2*f/a*dilog((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))-1/d^2* 
f/a*dilog((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))-1/d^2*f*di 
log(exp(d*x+c))/a+1/2*(d*f*x+d*e-f)/b/d^2*exp(d*x+c)-1/2*(d*f*x+d*e+f)/b/d 
^2*exp(-d*x-c)+2/d/b^2*a*f*c*x-1/d^2/b^2*a*f*ln((-b*exp(d*x+c)+(a^2+b^2)^( 
1/2)-a)/(-a+(a^2+b^2)^(1/2)))*c-1/d^2/b^2*a*f*ln((b*exp(d*x+c)+(a^2+b^2)^( 
1/2)+a)/(a+(a^2+b^2)^(1/2)))*c+1/d^2/b^2*c*a*f*ln(b*exp(2*d*x+2*c)+2*a*exp 
(d*x+c)-b)-1/d/b^2*a*f*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1 
/2)))*x
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1108 vs. \(2 (301) = 602\).

Time = 0.12 (sec) , antiderivative size = 1108, normalized size of antiderivative = 3.44 \[ \int \frac {(e+f x) \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/2*(a*b*d*f*x + a*b*d*e + a*b*f - (a*b*d*f*x + a*b*d*e - a*b*f)*cosh(d*x 
 + c)^2 - (a*b*d*f*x + a*b*d*e - a*b*f)*sinh(d*x + c)^2 - (a^2*d^2*f*x^2 + 
 2*a^2*d^2*e*x + 4*a^2*c*d*e - 2*a^2*c^2*f)*cosh(d*x + c) + 2*((a^2 + b^2) 
*f*cosh(d*x + c) + (a^2 + b^2)*f*sinh(d*x + c))*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*((a^2 + b^2)*f*cosh(d*x + c) + (a^2 + b^2)*f*sinh(d*x + c 
))*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) + b^2*f 
*sinh(d*x + c))*dilog(cosh(d*x + c) + sinh(d*x + c)) - 2*(b^2*f*cosh(d*x + 
 c) + b^2*f*sinh(d*x + c))*dilog(-cosh(d*x + c) - sinh(d*x + c)) + 2*(((a^ 
2 + b^2)*d*e - (a^2 + b^2)*c*f)*cosh(d*x + c) + ((a^2 + b^2)*d*e - (a^2 + 
b^2)*c*f)*sinh(d*x + c))*log(2*b*cosh(d*x + c) + 2*b*sinh(d*x + c) + 2*b*s 
qrt((a^2 + b^2)/b^2) + 2*a) + 2*(((a^2 + b^2)*d*e - (a^2 + b^2)*c*f)*cosh( 
d*x + c) + ((a^2 + b^2)*d*e - (a^2 + b^2)*c*f)*sinh(d*x + c))*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*(((a^ 
2 + b^2)*d*f*x + (a^2 + b^2)*c*f)*cosh(d*x + c) + ((a^2 + b^2)*d*f*x + (a^ 
2 + b^2)*c*f)*sinh(d*x + c))*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*(((a^2 
+ b^2)*d*f*x + (a^2 + b^2)*c*f)*cosh(d*x + c) + ((a^2 + b^2)*d*f*x + (a^2 
+ b^2)*c*f)*sinh(d*x + c))*log(-(a*cosh(d*x + c) + a*sinh(d*x + c) - (b...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

-1/2*e*(2*(d*x + c)*a/(b^2*d) - e^(d*x + c)/(b*d) + e^(-d*x - c)/(b*d) - 2 
*log(e^(-d*x - c) + 1)/(a*d) - 2*log(e^(-d*x - c) - 1)/(a*d) + 2*(a^2 + b^ 
2)*log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/(a*b^2*d)) - 1/4*f*(2*( 
a*d^2*x^2*e^c - (b*d*x*e^(2*c) - b*e^(2*c))*e^(d*x) + (b*d*x + b)*e^(-d*x) 
)*e^(-c)/(b^2*d^2) - integrate(8*((a^3*e^c + a*b^2*e^c)*x*e^(d*x) - (a^2*b 
 + b^3)*x)/(a*b^3*e^(2*d*x + 2*c) + 2*a^2*b^2*e^(d*x + c) - a*b^3), x) + 4 
*integrate(x/(a*e^(d*x + c) + a), x) - 4*integrate(x/(a*e^(d*x + c) - a), 
x))
 

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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