Integrand size = 30, antiderivative size = 286 \[ \int \frac {(e+f x) \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {e x}{b}+\frac {f x^2}{2 b}-\frac {2 (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{a d}-\frac {\sqrt {a^2+b^2} (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a b d}+\frac {\sqrt {a^2+b^2} (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a b d}-\frac {f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a d^2}+\frac {f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a d^2}-\frac {\sqrt {a^2+b^2} f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a b d^2}+\frac {\sqrt {a^2+b^2} f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a b d^2} \]
e*x/b+1/2*f*x^2/b-2*(f*x+e)*arctanh(exp(d*x+c))/a/d-f*polylog(2,-exp(d*x+c ))/a/d^2+f*polylog(2,exp(d*x+c))/a/d^2-(f*x+e)*ln(1+b*exp(d*x+c)/(a-(a^2+b ^2)^(1/2)))*(a^2+b^2)^(1/2)/a/b/d+(f*x+e)*ln(1+b*exp(d*x+c)/(a+(a^2+b^2)^( 1/2)))*(a^2+b^2)^(1/2)/a/b/d-f*polylog(2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)) )*(a^2+b^2)^(1/2)/a/b/d^2+f*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))*( a^2+b^2)^(1/2)/a/b/d^2
Time = 1.36 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.04 \[ \int \frac {(e+f x) \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {-a (c+d x) (c f-d (2 e+f x))+2 b \left (d (e+f x) \left (\log \left (1-e^{c+d x}\right )-\log \left (1+e^{c+d x}\right )\right )-f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )+f \operatorname {PolyLog}\left (2,e^{c+d x}\right )\right )+2 \sqrt {a^2+b^2} \left (2 d e \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )-2 c f \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )-f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )+f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-f \operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )+f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right )}{2 a b d^2} \]
(-(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)
Result contains complex when optimal does not.
Time = 2.19 (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 |
\(\displaystyle \frac {i \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )-i \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{a}-\frac {b \left (-\frac {2 \left (a^2+b^2\right ) \left (\frac {b \left (\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b d^2}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b d^2}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}\right )}{b^2}-\frac {a (e+f x)^2}{2 b^2 f}-\frac {i \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{b}\right )}{a}\) |
(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
3.5.27.3.1 Defintions of rubi rules used
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]
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
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]
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]
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]
Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]
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]
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]
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]
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]
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]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(969\) vs. \(2(261)=522\).
Time = 1.95 (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 {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 {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 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 {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}}}-\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 {f \operatorname {dilog}\left ({\mathrm e}^{d x +c}\right )}{d^{2} a}-\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 {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 {c f \ln \left ({\mathrm e}^{d x +c}-1\right )}{d^{2} a}+\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 ) c}{b \,d^{2} \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 \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 {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}}}\) | \(970\) |
1/2*f*x^2/b+e*x/b+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))+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*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-2/b/d^2*c*a*f/(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-1/d^2*f/a*dilog(exp(d*x+c))-1/d*f/a*ln(exp (d*x+c)+1)*x-1/d^2*f/a*dilog(exp(d*x+c)+1)+2*b/d*e/a/(a^2+b^2)^(1/2)*arcta nh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))-1/d^2*c*f/a*ln(exp(d*x+c)-1)+ 1/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^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+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/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)*dil...
Leaf count of result is larger than twice the leaf count of optimal. 598 vs. \(2 (257) = 514\).
Time = 0.31 (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=\frac {a d^{2} f x^{2} + 2 \, a d^{2} e x - 2 \, b f \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} {\rm Li}_2\left (\frac {a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right ) + {\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )\right )} \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} - b}{b} + 1\right ) + 2 \, b f \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} {\rm Li}_2\left (\frac {a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right ) - {\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )\right )} \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} - b}{b} + 1\right ) + 2 \, b f {\rm Li}_2\left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right ) - 2 \, b f {\rm Li}_2\left (-\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )\right ) + 2 \, {\left (b d e - b c f\right )} \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} \log \left (2 \, b \cosh \left (d x + c\right ) + 2 \, b \sinh \left (d x + c\right ) + 2 \, b \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} + 2 \, a\right ) - 2 \, {\left (b d e - b c f\right )} \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} \log \left (2 \, b \cosh \left (d x + c\right ) + 2 \, b \sinh \left (d x + c\right ) - 2 \, b \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} + 2 \, a\right ) - 2 \, {\left (b d f x + b c f\right )} \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} \log \left (-\frac {a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right ) + {\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )\right )} \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} - b}{b}\right ) + 2 \, {\left (b d f x + b c f\right )} \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} \log \left (-\frac {a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right ) - {\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )\right )} \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} - b}{b}\right ) - 2 \, {\left (b d f x + b d e\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) + 1\right ) + 2 \, {\left (b d e - b c f\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) - 1\right ) + 2 \, {\left (b d f x + b c f\right )} \log \left (-\cosh \left (d x + c\right ) - \sinh \left (d x + c\right ) + 1\right )}{2 \, a b d^{2}} \]
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)
\[ \int \frac {(e+f x) \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\left (e + f x\right ) \cosh {\left (c + d x \right )} \coth {\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \]
\[ \int \frac {(e+f x) \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )} \cosh \left (d x + c\right ) \coth \left (d x + c\right )}{b \sinh \left (d x + c\right ) + a} \,d x } \]
-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))
Timed out. \[ \int \frac {(e+f x) \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {(e+f x) \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\mathrm {cosh}\left (c+d\,x\right )\,\mathrm {coth}\left (c+d\,x\right )\,\left (e+f\,x\right )}{a+b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x \]