Optimal. Leaf size=393 \[ \frac {3 i e f^2 x}{4 a d^2}+\frac {3 i f^3 x^2}{8 a d^2}-\frac {i (e+f x)^3}{a d}+\frac {3 i (e+f x)^4}{8 a f}+\frac {6 f^2 (e+f x) \cosh (c+d x)}{a d^3}+\frac {(e+f x)^3 \cosh (c+d x)}{a d}+\frac {6 i f (e+f x)^2 \log \left (1+i e^{c+d x}\right )}{a d^2}+\frac {12 i f^2 (e+f x) \text {PolyLog}\left (2,-i e^{c+d x}\right )}{a d^3}-\frac {12 i f^3 \text {PolyLog}\left (3,-i e^{c+d x}\right )}{a d^4}-\frac {6 f^3 \sinh (c+d x)}{a d^4}-\frac {3 f (e+f x)^2 \sinh (c+d x)}{a d^2}-\frac {3 i f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 a d^3}-\frac {i (e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 a d}+\frac {3 i f^3 \sinh ^2(c+d x)}{8 a d^4}+\frac {3 i f (e+f x)^2 \sinh ^2(c+d x)}{4 a d^2}-\frac {i (e+f x)^3 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d} \]
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Rubi [A]
time = 0.50, 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 = {5676, 3392, 32, 3391, 3377, 2717, 3399, 4269, 3797, 2221, 2611, 2320, 6724}
\begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 32
Rule 2221
Rule 2320
Rule 2611
Rule 2717
Rule 3377
Rule 3391
Rule 3392
Rule 3399
Rule 3797
Rule 4269
Rule 5676
Rule 6724
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 ) \text {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 = 6.98, size = 397, normalized size = 1.01 \begin {gather*} \frac {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+\frac {96 f^2 (e+f x) \cosh (c+d x)}{d^3}+\frac {16 (e+f x)^3 \cosh (c+d x)}{d}+\frac {3 i f^3 \cosh (2 (c+d x))}{d^4}+\frac {6 i f (e+f x)^2 \cosh (2 (c+d x))}{d^2}+\frac {32 f \left (-i d e^c x \left (3 e^2+3 e f x+f^2 x^2\right )+3 \left (1+i e^c\right ) (e+f x)^2 \log \left (1+i e^{c+d x}\right )\right )}{d^2 \left (-i+e^c\right )}+\frac {192 i f^2 (e+f x) \text {PolyLog}\left (2,-i e^{c+d x}\right )}{d^3}-\frac {192 i f^3 \text {PolyLog}\left (3,-i e^{c+d x}\right )}{d^4}-\frac {32 i (e+f x)^3 \sinh \left (\frac {d x}{2}\right )}{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 )}-\frac {96 f^3 \sinh (c+d x)}{d^4}-\frac {48 f (e+f x)^2 \sinh (c+d x)}{d^2}-\frac {6 i f^2 (e+f x) \sinh (2 (c+d x))}{d^3}-\frac {4 i (e+f x)^3 \sinh (2 (c+d x))}{d}}{16 a} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 939 vs. \(2 (354 ) = 708\).
time = 2.79, size = 940, normalized size = 2.39
method | result | size |
risch | \(-\frac {12 i f^{3} \polylog \left (3, -i {\mathrm e}^{d x +c}\right )}{a \,d^{4}}-\frac {6 i e \,f^{2} c^{2}}{a \,d^{3}}-\frac {6 i f^{3} c^{2} \ln \left (1+i {\mathrm e}^{d x +c}\right )}{a \,d^{4}}+\frac {3 i e^{4}}{8 a f}+\frac {3 i f^{3} x^{4}}{8 a}+\frac {3 i e^{3} x}{2 a}+\frac {2 f^{3} x^{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 d^{3} x^{3} f^{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 {\left (d^{3} x^{3} f^{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 {12 i e \,f^{2} \ln \left (1+i {\mathrm e}^{d x +c}\right ) c}{a \,d^{3}}-\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 ) x}{a \,d^{2}}+\frac {\left (d^{3} x^{3} f^{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 {6 i \ln \left ({\mathrm e}^{d x +c}\right ) e^{2} f}{a \,d^{2}}+\frac {6 i f^{3} c^{2} \ln \left ({\mathrm e}^{d x +c}-i\right )}{a \,d^{4}}+\frac {6 i f^{3} c^{2} x}{a \,d^{3}}-\frac {6 i f^{3} c^{2} \ln \left ({\mathrm e}^{d x +c}\right )}{a \,d^{4}}+\frac {6 i \ln \left ({\mathrm e}^{d x +c}-i\right ) e^{2} f}{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}\right )}{a \,d^{3}}-\frac {6 i e \,f^{2} x^{2}}{a d}+\frac {6 i f^{3} \ln \left (1+i {\mathrm e}^{d x +c}\right ) x^{2}}{a \,d^{2}}+\frac {12 i e \,f^{2} \polylog \left (2, -i {\mathrm e}^{d x +c}\right )}{a \,d^{3}}+\frac {12 i f^{3} \polylog \left (2, -i {\mathrm e}^{d x +c}\right ) x}{a \,d^{3}}+\frac {9 i f \,e^{2} x^{2}}{4 a}-\frac {2 i f^{3} x^{3}}{a d}+\frac {i \left (4 d^{3} x^{3} f^{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 {3 i f^{2} e \,x^{3}}{2 a}+\frac {4 i f^{3} c^{3}}{a \,d^{4}}\) | \(940\) |
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 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 1045 vs. \(2 (349) = 698\).
time = 0.37, size = 1045, normalized size = 2.66 \begin {gather*} \frac {4 \, d^{3} f^{3} x^{3} + 6 \, d^{2} f^{3} x^{2} + 6 \, d f^{3} x + 4 \, d^{3} e^{3} + 3 \, f^{3} - 384 \, {\left ({\left (-i \, d f^{3} x - i \, d f^{2} e\right )} e^{\left (3 \, d x + 3 \, c\right )} - {\left (d f^{3} x + d f^{2} e\right )} e^{\left (2 \, d x + 2 \, c\right )}\right )} {\rm Li}_2\left (-i \, e^{\left (d x + c\right )}\right ) + 6 \, {\left (2 \, d^{3} f x + d^{2} f\right )} e^{2} + 6 \, {\left (2 \, d^{3} f^{2} x^{2} + 2 \, d^{2} f^{2} x + d f^{2}\right )} e + {\left (-4 i \, d^{3} f^{3} x^{3} + 6 i \, d^{2} f^{3} x^{2} - 6 i \, d f^{3} x - 4 i \, d^{3} e^{3} + 3 i \, f^{3} - 6 \, {\left (2 i \, d^{3} f x - i \, d^{2} f\right )} e^{2} - 6 \, {\left (2 i \, d^{3} f^{2} x^{2} - 2 i \, d^{2} f^{2} x + i \, d f^{2}\right )} e\right )} e^{\left (5 \, d x + 5 \, c\right )} + 3 \, {\left (4 \, d^{3} f^{3} x^{3} - 14 \, d^{2} f^{3} x^{2} + 30 \, d f^{3} x + 4 \, d^{3} e^{3} - 31 \, f^{3} + 2 \, {\left (6 \, d^{3} f x - 7 \, d^{2} f\right )} e^{2} + 2 \, {\left (6 \, d^{3} f^{2} x^{2} - 14 \, d^{2} f^{2} x + 15 \, d f^{2}\right )} e\right )} e^{\left (4 \, d x + 4 \, c\right )} - 4 \, {\left (-3 i \, d^{4} f^{3} x^{4} + 20 i \, d^{3} f^{3} x^{3} - 12 i \, d^{2} f^{3} x^{2} + 24 i \, d f^{3} x + 8 \, {\left (2 i \, c^{3} - 3 i\right )} f^{3} + 4 \, {\left (-3 i \, d^{4} x + i \, d^{3}\right )} e^{3} + 6 \, {\left (-3 i \, d^{4} f x^{2} + 10 i \, d^{3} f x + 2 \, {\left (4 i \, c - i\right )} d^{2} f\right )} e^{2} + 12 \, {\left (-i \, d^{4} f^{2} x^{3} + 5 i \, d^{3} f^{2} x^{2} - 2 i \, d^{2} f^{2} x + 2 \, {\left (-2 i \, c^{2} + i\right )} d f^{2}\right )} e\right )} e^{\left (3 \, d x + 3 \, c\right )} + 4 \, {\left (3 \, d^{4} f^{3} x^{4} + 4 \, d^{3} f^{3} x^{3} + 12 \, d^{2} f^{3} x^{2} + 24 \, d f^{3} x - 8 \, {\left (2 \, c^{3} - 3\right )} f^{3} + 4 \, {\left (3 \, d^{4} x + 5 \, d^{3}\right )} e^{3} + 6 \, {\left (3 \, d^{4} f x^{2} + 2 \, d^{3} f x - 2 \, {\left (4 \, c - 1\right )} d^{2} f\right )} e^{2} + 12 \, {\left (d^{4} f^{2} x^{3} + d^{3} f^{2} x^{2} + 2 \, d^{2} f^{2} x + 2 \, {\left (2 \, c^{2} + 1\right )} d f^{2}\right )} e\right )} e^{\left (2 \, d x + 2 \, c\right )} - 3 \, {\left (4 i \, d^{3} f^{3} x^{3} + 14 i \, d^{2} f^{3} x^{2} + 30 i \, d f^{3} x + 4 i \, d^{3} e^{3} + 31 i \, f^{3} + 2 \, {\left (6 i \, d^{3} f x + 7 i \, d^{2} f\right )} e^{2} + 2 \, {\left (6 i \, d^{3} f^{2} x^{2} + 14 i \, d^{2} f^{2} x + 15 i \, d f^{2}\right )} e\right )} e^{\left (d x + c\right )} - 192 \, {\left ({\left (-i \, c^{2} f^{3} + 2 i \, c d f^{2} e - i \, d^{2} f e^{2}\right )} e^{\left (3 \, d x + 3 \, c\right )} - {\left (c^{2} f^{3} - 2 \, c d f^{2} e + d^{2} f e^{2}\right )} e^{\left (2 \, d x + 2 \, c\right )}\right )} \log \left (e^{\left (d x + c\right )} - i\right ) - 192 \, {\left ({\left (-i \, d^{2} f^{3} x^{2} + i \, c^{2} f^{3} + 2 \, {\left (-i \, d^{2} f^{2} x - i \, c d f^{2}\right )} e\right )} e^{\left (3 \, d x + 3 \, c\right )} - {\left (d^{2} f^{3} x^{2} - c^{2} f^{3} + 2 \, {\left (d^{2} f^{2} x + c d f^{2}\right )} e\right )} e^{\left (2 \, d x + 2 \, c\right )}\right )} \log \left (i \, e^{\left (d x + c\right )} + 1\right ) - 384 \, {\left (i \, f^{3} e^{\left (3 \, d x + 3 \, c\right )} + f^{3} e^{\left (2 \, d x + 2 \, c\right )}\right )} {\rm polylog}\left (3, -i \, e^{\left (d x + c\right )}\right )}{32 \, {\left (a d^{4} e^{\left (3 \, d x + 3 \, c\right )} - i \, a d^{4} e^{\left (2 \, d x + 2 \, c\right )}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {2 e^{3} + 6 e^{2} f x + 6 e f^{2} x^{2} + 2 f^{3} x^{3}}{a d e^{c} e^{d x} - i a d} - \frac {i \left (\int \left (- \frac {i d e^{3}}{e^{c} e^{3 d x} - i e^{2 d x}}\right )\, dx + \int \left (- \frac {i d f^{3} x^{3}}{e^{c} e^{3 d x} - i e^{2 d x}}\right )\, dx + \int \left (- \frac {d e^{3} e^{c} e^{d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\right )\, dx + \int \left (- \frac {4 d e^{3} e^{3 c} e^{3 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\right )\, dx + \int \frac {d e^{3} e^{5 c} e^{5 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\, dx + \int \left (- \frac {3 i d e f^{2} x^{2}}{e^{c} e^{3 d x} - i e^{2 d x}}\right )\, dx + \int \left (- \frac {3 i d e^{2} f x}{e^{c} e^{3 d x} - i e^{2 d x}}\right )\, dx + \int \frac {4 i d e^{3} e^{2 c} e^{2 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\, dx + \int \frac {i d e^{3} e^{4 c} e^{4 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\, dx + \int \left (- \frac {24 i e^{2} f e^{2 c} e^{2 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\right )\, dx + \int \left (- \frac {24 i f^{3} x^{2} 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^{3} x^{3} e^{c} e^{d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\right )\, dx + \int \left (- \frac {4 d f^{3} x^{3} e^{3 c} e^{3 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\right )\, dx + \int \frac {d f^{3} x^{3} 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^{3} x^{3} e^{2 c} e^{2 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\, dx + \int \frac {i d f^{3} x^{3} e^{4 c} e^{4 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\, dx + \int \left (- \frac {48 i e 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 {3 d e 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 {12 d e 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 {3 d e f^{2} x^{2} e^{5 c} e^{5 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\, dx + \int \left (- \frac {3 d e^{2} f x e^{c} e^{d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\right )\, dx + \int \left (- \frac {12 d e^{2} f x e^{3 c} e^{3 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\right )\, dx + \int \frac {3 d e^{2} f x e^{5 c} e^{5 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\, dx + \int \frac {12 i d e 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 {3 i d e f^{2} x^{2} e^{4 c} e^{4 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\, dx + \int \frac {12 i d e^{2} f x e^{2 c} e^{2 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\, dx + \int \frac {3 i d e^{2} 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} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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