Integrand size = 31, antiderivative size = 287 \[ \int \frac {(e+f x)^2 \sinh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {i f^2 x}{4 a d^2}-\frac {i (e+f x)^2}{a d}+\frac {i (e+f x)^3}{2 a f}+\frac {2 f^2 \cosh (c+d x)}{a d^3}+\frac {(e+f x)^2 \cosh (c+d x)}{a d}+\frac {4 i f (e+f x) \log \left (1+i e^{c+d x}\right )}{a d^2}+\frac {4 i f^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a d^3}-\frac {2 f (e+f x) \sinh (c+d x)}{a d^2}-\frac {i f^2 \cosh (c+d x) \sinh (c+d x)}{4 a d^3}-\frac {i (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{2 a d}+\frac {i f (e+f x) \sinh ^2(c+d x)}{2 a d^2}-\frac {i (e+f x)^2 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d} \]
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Time = 0.38 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.419, Rules used = {5676, 3392, 32, 2715, 8, 3377, 2718, 3399, 4269, 3797, 2221, 2317, 2438} \[ \int \frac {(e+f x)^2 \sinh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {4 i f^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a d^3}+\frac {2 f^2 \cosh (c+d x)}{a d^3}-\frac {i f^2 \sinh (c+d x) \cosh (c+d x)}{4 a d^3}+\frac {4 i f (e+f x) \log \left (1+i e^{c+d x}\right )}{a d^2}+\frac {i f (e+f x) \sinh ^2(c+d x)}{2 a d^2}-\frac {2 f (e+f x) \sinh (c+d x)}{a d^2}+\frac {(e+f x)^2 \cosh (c+d x)}{a d}-\frac {i (e+f x)^2 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{a d}-\frac {i (e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 a d}+\frac {i f^2 x}{4 a d^2}-\frac {i (e+f x)^2}{a d}+\frac {i (e+f x)^3}{2 a f} \]
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Rule 8
Rule 32
Rule 2221
Rule 2317
Rule 2438
Rule 2715
Rule 2718
Rule 3377
Rule 3392
Rule 3399
Rule 3797
Rule 4269
Rule 5676
Rubi steps \begin{align*} \text {integral}& = i \int \frac {(e+f x)^2 \sinh ^2(c+d x)}{a+i a \sinh (c+d x)} \, dx-\frac {i \int (e+f x)^2 \sinh ^2(c+d x) \, dx}{a} \\ & = -\frac {i (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{2 a d}+\frac {i f (e+f x) \sinh ^2(c+d x)}{2 a d^2}+\frac {i \int (e+f x)^2 \, dx}{2 a}+\frac {\int (e+f x)^2 \sinh (c+d x) \, dx}{a}-\frac {\left (i f^2\right ) \int \sinh ^2(c+d x) \, dx}{2 a d^2}-\int \frac {(e+f x)^2 \sinh (c+d x)}{a+i a \sinh (c+d x)} \, dx \\ & = \frac {i (e+f x)^3}{6 a f}+\frac {(e+f x)^2 \cosh (c+d x)}{a d}-\frac {i f^2 \cosh (c+d x) \sinh (c+d x)}{4 a d^3}-\frac {i (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{2 a d}+\frac {i f (e+f x) \sinh ^2(c+d x)}{2 a d^2}-i \int \frac {(e+f x)^2}{a+i a \sinh (c+d x)} \, dx+\frac {i \int (e+f x)^2 \, dx}{a}-\frac {(2 f) \int (e+f x) \cosh (c+d x) \, dx}{a d}+\frac {\left (i f^2\right ) \int 1 \, dx}{4 a d^2} \\ & = \frac {i f^2 x}{4 a d^2}+\frac {i (e+f x)^3}{2 a f}+\frac {(e+f x)^2 \cosh (c+d x)}{a d}-\frac {2 f (e+f x) \sinh (c+d x)}{a d^2}-\frac {i f^2 \cosh (c+d x) \sinh (c+d x)}{4 a d^3}-\frac {i (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{2 a d}+\frac {i f (e+f x) \sinh ^2(c+d x)}{2 a d^2}-\frac {i \int (e+f x)^2 \csc ^2\left (\frac {1}{2} \left (i c+\frac {\pi }{2}\right )+\frac {i d x}{2}\right ) \, dx}{2 a}+\frac {\left (2 f^2\right ) \int \sinh (c+d x) \, dx}{a d^2} \\ & = \frac {i f^2 x}{4 a d^2}+\frac {i (e+f x)^3}{2 a f}+\frac {2 f^2 \cosh (c+d x)}{a d^3}+\frac {(e+f x)^2 \cosh (c+d x)}{a d}-\frac {2 f (e+f x) \sinh (c+d x)}{a d^2}-\frac {i f^2 \cosh (c+d x) \sinh (c+d x)}{4 a d^3}-\frac {i (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{2 a d}+\frac {i f (e+f x) \sinh ^2(c+d x)}{2 a d^2}-\frac {i (e+f x)^2 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d}+\frac {(2 i f) \int (e+f x) \coth \left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right ) \, dx}{a d} \\ & = \frac {i f^2 x}{4 a d^2}-\frac {i (e+f x)^2}{a d}+\frac {i (e+f x)^3}{2 a f}+\frac {2 f^2 \cosh (c+d x)}{a d^3}+\frac {(e+f x)^2 \cosh (c+d x)}{a d}-\frac {2 f (e+f x) \sinh (c+d x)}{a d^2}-\frac {i f^2 \cosh (c+d x) \sinh (c+d x)}{4 a d^3}-\frac {i (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{2 a d}+\frac {i f (e+f x) \sinh ^2(c+d x)}{2 a d^2}-\frac {i (e+f x)^2 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d}-\frac {(4 f) \int \frac {e^{2 \left (\frac {c}{2}+\frac {d x}{2}\right )} (e+f x)}{1+i e^{2 \left (\frac {c}{2}+\frac {d x}{2}\right )}} \, dx}{a d} \\ & = \frac {i f^2 x}{4 a d^2}-\frac {i (e+f x)^2}{a d}+\frac {i (e+f x)^3}{2 a f}+\frac {2 f^2 \cosh (c+d x)}{a d^3}+\frac {(e+f x)^2 \cosh (c+d x)}{a d}+\frac {4 i f (e+f x) \log \left (1+i e^{c+d x}\right )}{a d^2}-\frac {2 f (e+f x) \sinh (c+d x)}{a d^2}-\frac {i f^2 \cosh (c+d x) \sinh (c+d x)}{4 a d^3}-\frac {i (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{2 a d}+\frac {i f (e+f x) \sinh ^2(c+d x)}{2 a d^2}-\frac {i (e+f x)^2 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d}-\frac {\left (4 i f^2\right ) \int \log \left (1+i e^{2 \left (\frac {c}{2}+\frac {d x}{2}\right )}\right ) \, dx}{a d^2} \\ & = \frac {i f^2 x}{4 a d^2}-\frac {i (e+f x)^2}{a d}+\frac {i (e+f x)^3}{2 a f}+\frac {2 f^2 \cosh (c+d x)}{a d^3}+\frac {(e+f x)^2 \cosh (c+d x)}{a d}+\frac {4 i f (e+f x) \log \left (1+i e^{c+d x}\right )}{a d^2}-\frac {2 f (e+f x) \sinh (c+d x)}{a d^2}-\frac {i f^2 \cosh (c+d x) \sinh (c+d x)}{4 a d^3}-\frac {i (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{2 a d}+\frac {i f (e+f x) \sinh ^2(c+d x)}{2 a d^2}-\frac {i (e+f x)^2 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d}-\frac {\left (4 i f^2\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{2 \left (\frac {c}{2}+\frac {d x}{2}\right )}\right )}{a d^3} \\ & = \frac {i f^2 x}{4 a d^2}-\frac {i (e+f x)^2}{a d}+\frac {i (e+f x)^3}{2 a f}+\frac {2 f^2 \cosh (c+d x)}{a d^3}+\frac {(e+f x)^2 \cosh (c+d x)}{a d}+\frac {4 i f (e+f x) \log \left (1+i e^{c+d x}\right )}{a d^2}+\frac {4 i f^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a d^3}-\frac {2 f (e+f x) \sinh (c+d x)}{a d^2}-\frac {i f^2 \cosh (c+d x) \sinh (c+d x)}{4 a d^3}-\frac {i (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{2 a d}+\frac {i f (e+f x) \sinh ^2(c+d x)}{2 a d^2}-\frac {i (e+f x)^2 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d} \\ \end{align*}
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1661\) vs. \(2(287)=574\).
Time = 3.72 (sec) , antiderivative size = 1661, normalized size of antiderivative = 5.79 \[ \int \frac {(e+f x)^2 \sinh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {e^{-2 c} \left (-6 i d^2 e^2 e^c \cosh \left (\frac {3 d x}{2}\right )+6 d^2 e^2 e^{4 c} \cosh \left (\frac {3 d x}{2}\right )-14 i d e e^c f \cosh \left (\frac {3 d x}{2}\right )-14 d e e^{4 c} f \cosh \left (\frac {3 d x}{2}\right )-15 i e^c f^2 \cosh \left (\frac {3 d x}{2}\right )+15 e^{4 c} f^2 \cosh \left (\frac {3 d x}{2}\right )-12 i d^2 e e^c f x \cosh \left (\frac {3 d x}{2}\right )+12 d^2 e e^{4 c} f x \cosh \left (\frac {3 d x}{2}\right )-14 i d e^c f^2 x \cosh \left (\frac {3 d x}{2}\right )-14 d e^{4 c} f^2 x \cosh \left (\frac {3 d x}{2}\right )-6 i d^2 e^c f^2 x^2 \cosh \left (\frac {3 d x}{2}\right )+6 d^2 e^{4 c} f^2 x^2 \cosh \left (\frac {3 d x}{2}\right )+2 d^2 e^2 \cosh \left (\frac {5 d x}{2}\right )-2 i d^2 e^2 e^{5 c} \cosh \left (\frac {5 d x}{2}\right )+2 d e f \cosh \left (\frac {5 d x}{2}\right )+2 i d e e^{5 c} f \cosh \left (\frac {5 d x}{2}\right )+f^2 \cosh \left (\frac {5 d x}{2}\right )-i e^{5 c} f^2 \cosh \left (\frac {5 d x}{2}\right )+4 d^2 e f x \cosh \left (\frac {5 d x}{2}\right )-4 i d^2 e e^{5 c} f x \cosh \left (\frac {5 d x}{2}\right )+2 d f^2 x \cosh \left (\frac {5 d x}{2}\right )+2 i d e^{5 c} f^2 x \cosh \left (\frac {5 d x}{2}\right )+2 d^2 f^2 x^2 \cosh \left (\frac {5 d x}{2}\right )-2 i d^2 e^{5 c} f^2 x^2 \cosh \left (\frac {5 d x}{2}\right )+8 e^{2 c} \cosh \left (\frac {d x}{2}\right ) \left (2 \left (1-i e^c\right ) f^2+2 d \left (1+i e^c\right ) f (e+f x)+d^2 \left (5-i e^c\right ) (e+f x)^2+d^3 \left (1+i e^c\right ) x \left (3 e^2+3 e f x+f^2 x^2\right )+8 d \left (1+i e^c\right ) f (e+f x) \log \left (1-i e^{-c-d x}\right )\right )-40 d^2 e^2 e^{2 c} \sinh \left (\frac {d x}{2}\right )-8 i d^2 e^2 e^{3 c} \sinh \left (\frac {d x}{2}\right )-16 d e e^{2 c} f \sinh \left (\frac {d x}{2}\right )+16 i d e e^{3 c} f \sinh \left (\frac {d x}{2}\right )-16 e^{2 c} f^2 \sinh \left (\frac {d x}{2}\right )-16 i e^{3 c} f^2 \sinh \left (\frac {d x}{2}\right )-24 d^3 e^2 e^{2 c} x \sinh \left (\frac {d x}{2}\right )+24 i d^3 e^2 e^{3 c} x \sinh \left (\frac {d x}{2}\right )-80 d^2 e e^{2 c} f x \sinh \left (\frac {d x}{2}\right )-16 i d^2 e e^{3 c} f x \sinh \left (\frac {d x}{2}\right )-16 d e^{2 c} f^2 x \sinh \left (\frac {d x}{2}\right )+16 i d e^{3 c} f^2 x \sinh \left (\frac {d x}{2}\right )-24 d^3 e e^{2 c} f x^2 \sinh \left (\frac {d x}{2}\right )+24 i d^3 e e^{3 c} f x^2 \sinh \left (\frac {d x}{2}\right )-40 d^2 e^{2 c} f^2 x^2 \sinh \left (\frac {d x}{2}\right )-8 i d^2 e^{3 c} f^2 x^2 \sinh \left (\frac {d x}{2}\right )-8 d^3 e^{2 c} f^2 x^3 \sinh \left (\frac {d x}{2}\right )+8 i d^3 e^{3 c} f^2 x^3 \sinh \left (\frac {d x}{2}\right )-64 d e e^{2 c} f \log \left (1-i e^{-c-d x}\right ) \sinh \left (\frac {d x}{2}\right )+64 i d e e^{3 c} f \log \left (1-i e^{-c-d x}\right ) \sinh \left (\frac {d x}{2}\right )-64 d e^{2 c} f^2 x \log \left (1-i e^{-c-d x}\right ) \sinh \left (\frac {d x}{2}\right )+64 i d e^{3 c} f^2 x \log \left (1-i e^{-c-d x}\right ) \sinh \left (\frac {d x}{2}\right )+64 e^{2 c} f^2 \operatorname {PolyLog}\left (2,i e^{-c-d x}\right ) \left (\left (-1-i e^c\right ) \cosh \left (\frac {d x}{2}\right )+\left (1-i e^c\right ) \sinh \left (\frac {d x}{2}\right )\right )+6 i d^2 e^2 e^c \sinh \left (\frac {3 d x}{2}\right )+6 d^2 e^2 e^{4 c} \sinh \left (\frac {3 d x}{2}\right )+14 i d e e^c f \sinh \left (\frac {3 d x}{2}\right )-14 d e e^{4 c} f \sinh \left (\frac {3 d x}{2}\right )+15 i e^c f^2 \sinh \left (\frac {3 d x}{2}\right )+15 e^{4 c} f^2 \sinh \left (\frac {3 d x}{2}\right )+12 i d^2 e e^c f x \sinh \left (\frac {3 d x}{2}\right )+12 d^2 e e^{4 c} f x \sinh \left (\frac {3 d x}{2}\right )+14 i d e^c f^2 x \sinh \left (\frac {3 d x}{2}\right )-14 d e^{4 c} f^2 x \sinh \left (\frac {3 d x}{2}\right )+6 i d^2 e^c f^2 x^2 \sinh \left (\frac {3 d x}{2}\right )+6 d^2 e^{4 c} f^2 x^2 \sinh \left (\frac {3 d x}{2}\right )-2 d^2 e^2 \sinh \left (\frac {5 d x}{2}\right )-2 i d^2 e^2 e^{5 c} \sinh \left (\frac {5 d x}{2}\right )-2 d e f \sinh \left (\frac {5 d x}{2}\right )+2 i d e e^{5 c} f \sinh \left (\frac {5 d x}{2}\right )-f^2 \sinh \left (\frac {5 d x}{2}\right )-i e^{5 c} f^2 \sinh \left (\frac {5 d x}{2}\right )-4 d^2 e f x \sinh \left (\frac {5 d x}{2}\right )-4 i d^2 e e^{5 c} f x \sinh \left (\frac {5 d x}{2}\right )-2 d f^2 x \sinh \left (\frac {5 d x}{2}\right )+2 i d e^{5 c} f^2 x \sinh \left (\frac {5 d x}{2}\right )-2 d^2 f^2 x^2 \sinh \left (\frac {5 d x}{2}\right )-2 i d^2 e^{5 c} f^2 x^2 \sinh \left (\frac {5 d x}{2}\right )\right )}{16 a d^3 \left (\left (-i+e^c\right ) \cosh \left (\frac {d x}{2}\right )+\left (i+e^c\right ) \sinh \left (\frac {d x}{2}\right )\right )} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 559 vs. \(2 (257 ) = 514\).
Time = 2.42 (sec) , antiderivative size = 560, normalized size of antiderivative = 1.95
method | result | size |
risch | \(-\frac {2 i f^{2} x^{2}}{a d}-\frac {4 i f^{2} c x}{a \,d^{2}}+\frac {i \left (2 d^{2} x^{2} f^{2}+4 d^{2} e f x +2 d^{2} e^{2}+2 x d \,f^{2}+2 d e f +f^{2}\right ) {\mathrm e}^{-2 d x -2 c}}{16 d^{3} a}+\frac {2 i e f \ln \left (1+{\mathrm e}^{2 d x +2 c}\right )}{a \,d^{2}}-\frac {2 i c \,f^{2} \ln \left (1+{\mathrm e}^{2 d x +2 c}\right )}{a \,d^{3}}+\frac {\left (d^{2} x^{2} f^{2}+2 d^{2} e f x +d^{2} e^{2}-2 x d \,f^{2}-2 d e f +2 f^{2}\right ) {\mathrm e}^{d x +c}}{2 d^{3} a}+\frac {\left (d^{2} x^{2} f^{2}+2 d^{2} e f x +d^{2} e^{2}+2 x d \,f^{2}+2 d e f +2 f^{2}\right ) {\mathrm e}^{-d x -c}}{2 d^{3} a}+\frac {4 i f^{2} \ln \left (1+i {\mathrm e}^{d x +c}\right ) x}{a \,d^{2}}+\frac {2 x^{2} f^{2}+4 e f x +2 e^{2}}{d a \left ({\mathrm e}^{d x +c}-i\right )}+\frac {4 i c \,f^{2} \ln \left ({\mathrm e}^{d x +c}\right )}{a \,d^{3}}-\frac {4 e f \arctan \left ({\mathrm e}^{d x +c}\right )}{a \,d^{2}}+\frac {4 i f^{2} \operatorname {polylog}\left (2, -i {\mathrm e}^{d x +c}\right )}{a \,d^{3}}+\frac {4 i f^{2} \ln \left (1+i {\mathrm e}^{d x +c}\right ) c}{a \,d^{3}}+\frac {3 i f e \,x^{2}}{2 a}+\frac {i e^{3}}{2 a f}-\frac {i \left (2 d^{2} x^{2} f^{2}+4 d^{2} e f x +2 d^{2} e^{2}-2 x d \,f^{2}-2 d e f +f^{2}\right ) {\mathrm e}^{2 d x +2 c}}{16 d^{3} a}-\frac {2 i f^{2} c^{2}}{a \,d^{3}}+\frac {3 i e^{2} x}{2 a}-\frac {4 i e f \ln \left ({\mathrm e}^{d x +c}\right )}{a \,d^{2}}+\frac {4 c \,f^{2} \arctan \left ({\mathrm e}^{d x +c}\right )}{a \,d^{3}}+\frac {i f^{2} x^{3}}{2 a}\) | \(560\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 595 vs. \(2 (244) = 488\).
Time = 0.26 (sec) , antiderivative size = 595, normalized size of antiderivative = 2.07 \[ \int \frac {(e+f x)^2 \sinh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {2 \, d^{2} f^{2} x^{2} + 2 \, d^{2} e^{2} + 2 \, d e f + f^{2} + 2 \, {\left (2 \, d^{2} e f + d f^{2}\right )} x - 64 \, {\left (-i \, f^{2} e^{\left (3 \, d x + 3 \, c\right )} - f^{2} e^{\left (2 \, d x + 2 \, c\right )}\right )} {\rm Li}_2\left (-i \, e^{\left (d x + c\right )}\right ) + {\left (-2 i \, d^{2} f^{2} x^{2} - 2 i \, d^{2} e^{2} + 2 i \, d e f - i \, f^{2} - 2 \, {\left (2 i \, d^{2} e f - i \, d f^{2}\right )} x\right )} e^{\left (5 \, d x + 5 \, c\right )} + {\left (6 \, d^{2} f^{2} x^{2} + 6 \, d^{2} e^{2} - 14 \, d e f + 15 \, f^{2} + 2 \, {\left (6 \, d^{2} e f - 7 \, d f^{2}\right )} x\right )} e^{\left (4 \, d x + 4 \, c\right )} - 8 \, {\left (-i \, d^{3} f^{2} x^{3} + i \, d^{2} e^{2} + 2 \, {\left (4 i \, c - i\right )} d e f + 2 \, {\left (-2 i \, c^{2} + i\right )} f^{2} + {\left (-3 i \, d^{3} e f + 5 i \, d^{2} f^{2}\right )} x^{2} + {\left (-3 i \, d^{3} e^{2} + 10 i \, d^{2} e f - 2 i \, d f^{2}\right )} x\right )} e^{\left (3 \, d x + 3 \, c\right )} + 8 \, {\left (d^{3} f^{2} x^{3} + 5 \, d^{2} e^{2} - 2 \, {\left (4 \, c - 1\right )} d e f + 2 \, {\left (2 \, c^{2} + 1\right )} f^{2} + {\left (3 \, d^{3} e f + d^{2} f^{2}\right )} x^{2} + {\left (3 \, d^{3} e^{2} + 2 \, d^{2} e f + 2 \, d f^{2}\right )} x\right )} e^{\left (2 \, d x + 2 \, c\right )} + {\left (-6 i \, d^{2} f^{2} x^{2} - 6 i \, d^{2} e^{2} - 14 i \, d e f - 15 i \, f^{2} - 2 \, {\left (6 i \, d^{2} e f + 7 i \, d f^{2}\right )} x\right )} e^{\left (d x + c\right )} - 64 \, {\left ({\left (-i \, d e f + i \, c f^{2}\right )} e^{\left (3 \, d x + 3 \, c\right )} - {\left (d e f - c f^{2}\right )} e^{\left (2 \, d x + 2 \, c\right )}\right )} \log \left (e^{\left (d x + c\right )} - i\right ) - 64 \, {\left ({\left (-i \, d f^{2} x - i \, c f^{2}\right )} e^{\left (3 \, d x + 3 \, c\right )} - {\left (d f^{2} x + c f^{2}\right )} e^{\left (2 \, d x + 2 \, c\right )}\right )} \log \left (i \, e^{\left (d x + c\right )} + 1\right )}{16 \, {\left (a d^{3} e^{\left (3 \, d x + 3 \, c\right )} - i \, a d^{3} e^{\left (2 \, d x + 2 \, c\right )}\right )}} \]
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\[ \int \frac {(e+f x)^2 \sinh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {2 e^{2} + 4 e f x + 2 f^{2} x^{2}}{a d e^{c} e^{d x} - i a d} - \frac {i \left (\int \left (- \frac {i d e^{2}}{e^{c} e^{3 d x} - i e^{2 d x}}\right )\, dx + \int \left (- \frac {i d f^{2} x^{2}}{e^{c} e^{3 d x} - i e^{2 d x}}\right )\, dx + \int \left (- \frac {d e^{2} e^{c} e^{d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\right )\, dx + \int \left (- \frac {4 d e^{2} e^{3 c} e^{3 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\right )\, dx + \int \frac {d e^{2} e^{5 c} e^{5 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\, dx + \int \left (- \frac {2 i d e f x}{e^{c} e^{3 d x} - i e^{2 d x}}\right )\, dx + \int \frac {4 i d e^{2} e^{2 c} e^{2 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\, dx + \int \frac {i d e^{2} e^{4 c} e^{4 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\, dx + \int \left (- \frac {16 i e f e^{2 c} e^{2 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\right )\, dx + \int \left (- \frac {16 i f^{2} x e^{2 c} e^{2 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\right )\, dx + \int \left (- \frac {d f^{2} x^{2} e^{c} e^{d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\right )\, dx + \int \left (- \frac {4 d f^{2} x^{2} e^{3 c} e^{3 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\right )\, dx + \int \frac {d f^{2} x^{2} e^{5 c} e^{5 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\, dx + \int \frac {4 i d f^{2} x^{2} e^{2 c} e^{2 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\, dx + \int \frac {i d f^{2} x^{2} e^{4 c} e^{4 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\, dx + \int \left (- \frac {2 d e f x e^{c} e^{d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\right )\, dx + \int \left (- \frac {8 d e f x e^{3 c} e^{3 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\right )\, dx + \int \frac {2 d e f x e^{5 c} e^{5 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\, dx + \int \frac {8 i d e f x e^{2 c} e^{2 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\, dx + \int \frac {2 i d e f x e^{4 c} e^{4 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\, dx\right ) e^{- 2 c}}{4 a d} \]
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Exception generated. \[ \int \frac {(e+f x)^2 \sinh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\text {Exception raised: RuntimeError} \]
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\[ \int \frac {(e+f x)^2 \sinh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{2} \sinh \left (d x + c\right )^{3}}{i \, a \sinh \left (d x + c\right ) + a} \,d x } \]
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Timed out. \[ \int \frac {(e+f x)^2 \sinh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\int \frac {{\mathrm {sinh}\left (c+d\,x\right )}^3\,{\left (e+f\,x\right )}^2}{a+a\,\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}} \,d x \]
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