Optimal. Leaf size=393 \[ -\frac {12 i f^3 \text {Li}_3\left (-i e^{c+d x}\right )}{a d^4}+\frac {3 i f^3 \sinh ^2(c+d x)}{8 a d^4}-\frac {6 f^3 \sinh (c+d x)}{a d^4}+\frac {12 i f^2 (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{a d^3}+\frac {6 f^2 (e+f x) \cosh (c+d x)}{a d^3}-\frac {3 i f^2 (e+f x) \sinh (c+d x) \cosh (c+d x)}{4 a d^3}+\frac {6 i f (e+f x)^2 \log \left (1+i e^{c+d x}\right )}{a d^2}+\frac {3 i f (e+f x)^2 \sinh ^2(c+d x)}{4 a d^2}-\frac {3 f (e+f x)^2 \sinh (c+d x)}{a d^2}+\frac {(e+f x)^3 \cosh (c+d x)}{a d}-\frac {i (e+f x)^3 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{a d}-\frac {i (e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 a d}+\frac {3 i e f^2 x}{4 a d^2}+\frac {3 i f^3 x^2}{8 a d^2}-\frac {i (e+f x)^3}{a d}+\frac {3 i (e+f x)^4}{8 a f} \]
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Rubi [A] time = 0.70, antiderivative size = 393, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 13, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.419, Rules used = {5557, 3311, 32, 3310, 3296, 2637, 3318, 4184, 3716, 2190, 2531, 2282, 6589} \[ \frac {12 i f^2 (e+f x) \text {PolyLog}\left (2,-i e^{c+d x}\right )}{a d^3}-\frac {12 i f^3 \text {PolyLog}\left (3,-i e^{c+d x}\right )}{a d^4}+\frac {6 f^2 (e+f x) \cosh (c+d x)}{a d^3}-\frac {3 i f^2 (e+f x) \sinh (c+d x) \cosh (c+d x)}{4 a d^3}+\frac {6 i f (e+f x)^2 \log \left (1+i e^{c+d x}\right )}{a d^2}+\frac {3 i f (e+f x)^2 \sinh ^2(c+d x)}{4 a d^2}-\frac {3 f (e+f x)^2 \sinh (c+d x)}{a d^2}+\frac {3 i f^3 \sinh ^2(c+d x)}{8 a d^4}-\frac {6 f^3 \sinh (c+d x)}{a d^4}+\frac {(e+f x)^3 \cosh (c+d x)}{a d}-\frac {i (e+f x)^3 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{a d}-\frac {i (e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 a d}+\frac {3 i e f^2 x}{4 a d^2}+\frac {3 i f^3 x^2}{8 a d^2}-\frac {i (e+f x)^3}{a d}+\frac {3 i (e+f x)^4}{8 a f} \]
Antiderivative was successfully verified.
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Rule 32
Rule 2190
Rule 2282
Rule 2531
Rule 2637
Rule 3296
Rule 3310
Rule 3311
Rule 3318
Rule 3716
Rule 4184
Rule 5557
Rule 6589
Rubi steps
\begin {align*} \int \frac {(e+f x)^3 \sinh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx &=i \int \frac {(e+f x)^3 \sinh ^2(c+d x)}{a+i a \sinh (c+d x)} \, dx-\frac {i \int (e+f x)^3 \sinh ^2(c+d x) \, dx}{a}\\ &=-\frac {i (e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 a d}+\frac {3 i f (e+f x)^2 \sinh ^2(c+d x)}{4 a d^2}+\frac {i \int (e+f x)^3 \, dx}{2 a}+\frac {\int (e+f x)^3 \sinh (c+d x) \, dx}{a}-\frac {\left (3 i f^2\right ) \int (e+f x) \sinh ^2(c+d x) \, dx}{2 a d^2}-\int \frac {(e+f x)^3 \sinh (c+d x)}{a+i a \sinh (c+d x)} \, dx\\ &=\frac {i (e+f x)^4}{8 a f}+\frac {(e+f x)^3 \cosh (c+d x)}{a d}-\frac {3 i f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 a d^3}-\frac {i (e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 a d}+\frac {3 i f^3 \sinh ^2(c+d x)}{8 a d^4}+\frac {3 i f (e+f x)^2 \sinh ^2(c+d x)}{4 a d^2}-i \int \frac {(e+f x)^3}{a+i a \sinh (c+d x)} \, dx+\frac {i \int (e+f x)^3 \, dx}{a}-\frac {(3 f) \int (e+f x)^2 \cosh (c+d x) \, dx}{a d}+\frac {\left (3 i f^2\right ) \int (e+f x) \, dx}{4 a d^2}\\ &=\frac {3 i e f^2 x}{4 a d^2}+\frac {3 i f^3 x^2}{8 a d^2}+\frac {3 i (e+f x)^4}{8 a f}+\frac {(e+f x)^3 \cosh (c+d x)}{a d}-\frac {3 f (e+f x)^2 \sinh (c+d x)}{a d^2}-\frac {3 i f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 a d^3}-\frac {i (e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 a d}+\frac {3 i f^3 \sinh ^2(c+d x)}{8 a d^4}+\frac {3 i f (e+f x)^2 \sinh ^2(c+d x)}{4 a d^2}-\frac {i \int (e+f x)^3 \csc ^2\left (\frac {1}{2} \left (i c+\frac {\pi }{2}\right )+\frac {i d x}{2}\right ) \, dx}{2 a}+\frac {\left (6 f^2\right ) \int (e+f x) \sinh (c+d x) \, dx}{a d^2}\\ &=\frac {3 i e f^2 x}{4 a d^2}+\frac {3 i f^3 x^2}{8 a d^2}+\frac {3 i (e+f x)^4}{8 a f}+\frac {6 f^2 (e+f x) \cosh (c+d x)}{a d^3}+\frac {(e+f x)^3 \cosh (c+d x)}{a d}-\frac {3 f (e+f x)^2 \sinh (c+d x)}{a d^2}-\frac {3 i f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 a d^3}-\frac {i (e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 a d}+\frac {3 i f^3 \sinh ^2(c+d x)}{8 a d^4}+\frac {3 i f (e+f x)^2 \sinh ^2(c+d x)}{4 a d^2}-\frac {i (e+f x)^3 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d}+\frac {(3 i f) \int (e+f x)^2 \coth \left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right ) \, dx}{a d}-\frac {\left (6 f^3\right ) \int \cosh (c+d x) \, dx}{a d^3}\\ &=\frac {3 i e f^2 x}{4 a d^2}+\frac {3 i f^3 x^2}{8 a d^2}-\frac {i (e+f x)^3}{a d}+\frac {3 i (e+f x)^4}{8 a f}+\frac {6 f^2 (e+f x) \cosh (c+d x)}{a d^3}+\frac {(e+f x)^3 \cosh (c+d x)}{a d}-\frac {6 f^3 \sinh (c+d x)}{a d^4}-\frac {3 f (e+f x)^2 \sinh (c+d x)}{a d^2}-\frac {3 i f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 a d^3}-\frac {i (e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 a d}+\frac {3 i f^3 \sinh ^2(c+d x)}{8 a d^4}+\frac {3 i f (e+f x)^2 \sinh ^2(c+d x)}{4 a d^2}-\frac {i (e+f x)^3 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d}-\frac {(6 f) \int \frac {e^{2 \left (\frac {c}{2}+\frac {d x}{2}\right )} (e+f x)^2}{1+i e^{2 \left (\frac {c}{2}+\frac {d x}{2}\right )}} \, dx}{a d}\\ &=\frac {3 i e f^2 x}{4 a d^2}+\frac {3 i f^3 x^2}{8 a d^2}-\frac {i (e+f x)^3}{a d}+\frac {3 i (e+f x)^4}{8 a f}+\frac {6 f^2 (e+f x) \cosh (c+d x)}{a d^3}+\frac {(e+f x)^3 \cosh (c+d x)}{a d}+\frac {6 i f (e+f x)^2 \log \left (1+i e^{c+d x}\right )}{a d^2}-\frac {6 f^3 \sinh (c+d x)}{a d^4}-\frac {3 f (e+f x)^2 \sinh (c+d x)}{a d^2}-\frac {3 i f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 a d^3}-\frac {i (e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 a d}+\frac {3 i f^3 \sinh ^2(c+d x)}{8 a d^4}+\frac {3 i f (e+f x)^2 \sinh ^2(c+d x)}{4 a d^2}-\frac {i (e+f x)^3 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d}-\frac {\left (12 i f^2\right ) \int (e+f x) \log \left (1+i e^{2 \left (\frac {c}{2}+\frac {d x}{2}\right )}\right ) \, dx}{a d^2}\\ &=\frac {3 i e f^2 x}{4 a d^2}+\frac {3 i f^3 x^2}{8 a d^2}-\frac {i (e+f x)^3}{a d}+\frac {3 i (e+f x)^4}{8 a f}+\frac {6 f^2 (e+f x) \cosh (c+d x)}{a d^3}+\frac {(e+f x)^3 \cosh (c+d x)}{a d}+\frac {6 i f (e+f x)^2 \log \left (1+i e^{c+d x}\right )}{a d^2}+\frac {12 i f^2 (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{a d^3}-\frac {6 f^3 \sinh (c+d x)}{a d^4}-\frac {3 f (e+f x)^2 \sinh (c+d x)}{a d^2}-\frac {3 i f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 a d^3}-\frac {i (e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 a d}+\frac {3 i f^3 \sinh ^2(c+d x)}{8 a d^4}+\frac {3 i f (e+f x)^2 \sinh ^2(c+d x)}{4 a d^2}-\frac {i (e+f x)^3 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d}-\frac {\left (12 i f^3\right ) \int \text {Li}_2\left (-i e^{2 \left (\frac {c}{2}+\frac {d x}{2}\right )}\right ) \, dx}{a d^3}\\ &=\frac {3 i e f^2 x}{4 a d^2}+\frac {3 i f^3 x^2}{8 a d^2}-\frac {i (e+f x)^3}{a d}+\frac {3 i (e+f x)^4}{8 a f}+\frac {6 f^2 (e+f x) \cosh (c+d x)}{a d^3}+\frac {(e+f x)^3 \cosh (c+d x)}{a d}+\frac {6 i f (e+f x)^2 \log \left (1+i e^{c+d x}\right )}{a d^2}+\frac {12 i f^2 (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{a d^3}-\frac {6 f^3 \sinh (c+d x)}{a d^4}-\frac {3 f (e+f x)^2 \sinh (c+d x)}{a d^2}-\frac {3 i f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 a d^3}-\frac {i (e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 a d}+\frac {3 i f^3 \sinh ^2(c+d x)}{8 a d^4}+\frac {3 i f (e+f x)^2 \sinh ^2(c+d x)}{4 a d^2}-\frac {i (e+f x)^3 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d}-\frac {\left (12 i f^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{2 \left (\frac {c}{2}+\frac {d x}{2}\right )}\right )}{a d^4}\\ &=\frac {3 i e f^2 x}{4 a d^2}+\frac {3 i f^3 x^2}{8 a d^2}-\frac {i (e+f x)^3}{a d}+\frac {3 i (e+f x)^4}{8 a f}+\frac {6 f^2 (e+f x) \cosh (c+d x)}{a d^3}+\frac {(e+f x)^3 \cosh (c+d x)}{a d}+\frac {6 i f (e+f x)^2 \log \left (1+i e^{c+d x}\right )}{a d^2}+\frac {12 i f^2 (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{a d^3}-\frac {12 i f^3 \text {Li}_3\left (-i e^{c+d x}\right )}{a d^4}-\frac {6 f^3 \sinh (c+d x)}{a d^4}-\frac {3 f (e+f x)^2 \sinh (c+d x)}{a d^2}-\frac {3 i f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 a d^3}-\frac {i (e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 a d}+\frac {3 i f^3 \sinh ^2(c+d x)}{8 a d^4}+\frac {3 i f (e+f x)^2 \sinh ^2(c+d x)}{4 a d^2}-\frac {i (e+f x)^3 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d}\\ \end {align*}
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Mathematica [A] time = 7.32, size = 376, normalized size = 0.96 \[ \frac {-\frac {192 i f^2 \left (d (e+f x) \text {Li}_2\left (i e^{-c-d x}\right )+f \text {Li}_3\left (i e^{-c-d x}\right )\right )}{d^4}-\frac {96 f^3 \sinh (c+d x)}{d^4}+\frac {3 i f^3 \cosh (2 (c+d x))}{d^4}-\frac {6 i f^2 (e+f x) \sinh (2 (c+d x))}{d^3}+\frac {96 f^2 (e+f x) \cosh (c+d x)}{d^3}+\frac {96 i f (e+f x)^2 \log \left (1-i e^{-c-d x}\right )}{d^2}-\frac {48 f (e+f x)^2 \sinh (c+d x)}{d^2}+\frac {6 i f (e+f x)^2 \cosh (2 (c+d x))}{d^2}+\frac {32 (e+f x)^3}{\left (e^c-i\right ) d}-\frac {4 i (e+f x)^3 \sinh (2 (c+d x))}{d}+\frac {16 (e+f x)^3 \cosh (c+d x)}{d}-\frac {32 i \sinh \left (\frac {d x}{2}\right ) (e+f x)^3}{d \left (\cosh \left (\frac {c}{2}\right )+i \sinh \left (\frac {c}{2}\right )\right ) \left (\cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right )}+24 i e^3 x+36 i e^2 f x^2+24 i e f^2 x^3+6 i f^3 x^4}{16 a} \]
Antiderivative was successfully verified.
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fricas [C] time = 0.90, size = 1029, normalized size = 2.62 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (f x + e\right )}^{3} \sinh \left (d x + c\right )^{3}}{i \, a \sinh \left (d x + c\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.27, size = 928, normalized size = 2.36 \[ \frac {12 i e \,f^{2} c \ln \left ({\mathrm e}^{d x +c}\right )}{a \,d^{3}}+\frac {12 i e \,f^{2} \ln \left (1+i {\mathrm e}^{d x +c}\right ) x}{a \,d^{2}}-\frac {12 i e \,f^{2} c x}{a \,d^{2}}-\frac {12 i e \,f^{2} c \ln \left ({\mathrm e}^{d x +c}-i\right )}{a \,d^{3}}+\frac {12 i e \,f^{2} \ln \left (1+i {\mathrm e}^{d x +c}\right ) c}{a \,d^{3}}+\frac {3 i e^{3} x}{2 a}+\frac {3 i x^{4} f^{3}}{8 a}+\frac {6 i f^{3} c^{2} x}{a \,d^{3}}-\frac {6 i e \,f^{2} c^{2}}{a \,d^{3}}-\frac {6 i e \,f^{2} x^{2}}{a d}+\frac {12 i e \,f^{2} \polylog \left (2, -i {\mathrm e}^{d x +c}\right )}{a \,d^{3}}+\frac {6 i f^{3} c^{2} \ln \left ({\mathrm e}^{d x +c}-i\right )}{a \,d^{4}}-\frac {6 i \ln \left ({\mathrm e}^{d x +c}\right ) e^{2} f}{a \,d^{2}}+\frac {6 i f^{3} \ln \left (1+i {\mathrm e}^{d x +c}\right ) x^{2}}{a \,d^{2}}-\frac {6 i f^{3} \ln \left (1+i {\mathrm e}^{d x +c}\right ) c^{2}}{a \,d^{4}}+\frac {6 i \ln \left ({\mathrm e}^{d x +c}-i\right ) e^{2} f}{a \,d^{2}}-\frac {6 i f^{3} c^{2} \ln \left ({\mathrm e}^{d x +c}\right )}{a \,d^{4}}+\frac {12 i f^{3} \polylog \left (2, -i {\mathrm e}^{d x +c}\right ) x}{a \,d^{3}}+\frac {\left (f^{3} x^{3} d^{3}+3 d^{3} e \,f^{2} x^{2}+3 d^{3} e^{2} f x -3 d^{2} f^{3} x^{2}+d^{3} e^{3}-6 d^{2} e \,f^{2} x -3 d^{2} e^{2} f +6 d \,f^{3} x +6 d e \,f^{2}-6 f^{3}\right ) {\mathrm e}^{d x +c}}{2 a \,d^{4}}+\frac {\left (f^{3} x^{3} d^{3}+3 d^{3} e \,f^{2} x^{2}+3 d^{3} e^{2} f x +3 d^{2} f^{3} x^{2}+d^{3} e^{3}+6 d^{2} e \,f^{2} x +3 d^{2} e^{2} f +6 d \,f^{3} x +6 d e \,f^{2}+6 f^{3}\right ) {\mathrm e}^{-d x -c}}{2 a \,d^{4}}+\frac {2 x^{3} f^{3}+6 e \,f^{2} x^{2}+6 e^{2} f x +2 e^{3}}{d a \left ({\mathrm e}^{d x +c}-i\right )}+\frac {i \left (4 f^{3} x^{3} d^{3}+12 d^{3} e \,f^{2} x^{2}+12 d^{3} e^{2} f x +6 d^{2} f^{3} x^{2}+4 d^{3} e^{3}+12 d^{2} e \,f^{2} x +6 d^{2} e^{2} f +6 d \,f^{3} x +6 d e \,f^{2}+3 f^{3}\right ) {\mathrm e}^{-2 d x -2 c}}{32 a \,d^{4}}-\frac {i \left (4 f^{3} x^{3} d^{3}+12 d^{3} e \,f^{2} x^{2}+12 d^{3} e^{2} f x -6 d^{2} f^{3} x^{2}+4 d^{3} e^{3}-12 d^{2} e \,f^{2} x -6 d^{2} e^{2} f +6 d \,f^{3} x +6 d e \,f^{2}-3 f^{3}\right ) {\mathrm e}^{2 d x +2 c}}{32 a \,d^{4}}-\frac {12 i f^{3} \polylog \left (3, -i {\mathrm e}^{d x +c}\right )}{a \,d^{4}}+\frac {9 i e^{2} f \,x^{2}}{4 a}+\frac {3 i e \,f^{2} x^{3}}{2 a}-\frac {2 i f^{3} x^{3}}{a d}+\frac {4 i f^{3} c^{3}}{a \,d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\mathrm {sinh}\left (c+d\,x\right )}^3\,{\left (e+f\,x\right )}^3}{a+a\,\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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