Optimal. Leaf size=241 \[ -\frac {12 f^3 \text {Li}_3\left (-i e^{c+d x}\right )}{a d^4}+\frac {6 i f^3 \sinh (c+d x)}{a d^4}+\frac {12 f^2 (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{a d^3}-\frac {6 i f^2 (e+f x) \cosh (c+d x)}{a d^3}+\frac {6 f (e+f x)^2 \log \left (1+i e^{c+d x}\right )}{a d^2}+\frac {3 i f (e+f x)^2 \sinh (c+d x)}{a d^2}-\frac {i (e+f x)^3 \cosh (c+d x)}{a d}-\frac {(e+f x)^3 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{a d}-\frac {(e+f x)^3}{a d}+\frac {(e+f x)^4}{4 a f} \]
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Rubi [A] time = 0.53, antiderivative size = 241, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 11, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.355, Rules used = {5557, 3296, 2637, 32, 3318, 4184, 3716, 2190, 2531, 2282, 6589} \[ \frac {12 f^2 (e+f x) \text {PolyLog}\left (2,-i e^{c+d x}\right )}{a d^3}-\frac {12 f^3 \text {PolyLog}\left (3,-i e^{c+d x}\right )}{a d^4}-\frac {6 i f^2 (e+f x) \cosh (c+d x)}{a d^3}+\frac {6 f (e+f x)^2 \log \left (1+i e^{c+d x}\right )}{a d^2}+\frac {3 i f (e+f x)^2 \sinh (c+d x)}{a d^2}+\frac {6 i f^3 \sinh (c+d x)}{a d^4}-\frac {i (e+f x)^3 \cosh (c+d x)}{a d}-\frac {(e+f x)^3 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{a d}-\frac {(e+f x)^3}{a d}+\frac {(e+f x)^4}{4 a f} \]
Antiderivative was successfully verified.
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Rule 32
Rule 2190
Rule 2282
Rule 2531
Rule 2637
Rule 3296
Rule 3318
Rule 3716
Rule 4184
Rule 5557
Rule 6589
Rubi steps
\begin {align*} \int \frac {(e+f x)^3 \sinh ^2(c+d x)}{a+i a \sinh (c+d x)} \, dx &=i \int \frac {(e+f x)^3 \sinh (c+d x)}{a+i a \sinh (c+d x)} \, dx-\frac {i \int (e+f x)^3 \sinh (c+d x) \, dx}{a}\\ &=-\frac {i (e+f x)^3 \cosh (c+d x)}{a d}+\frac {\int (e+f x)^3 \, dx}{a}+\frac {(3 i f) \int (e+f x)^2 \cosh (c+d x) \, dx}{a d}-\int \frac {(e+f x)^3}{a+i a \sinh (c+d x)} \, dx\\ &=\frac {(e+f x)^4}{4 a f}-\frac {i (e+f x)^3 \cosh (c+d x)}{a d}+\frac {3 i f (e+f x)^2 \sinh (c+d x)}{a d^2}-\frac {\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 i f^2\right ) \int (e+f x) \sinh (c+d x) \, dx}{a d^2}\\ &=\frac {(e+f x)^4}{4 a f}-\frac {6 i f^2 (e+f x) \cosh (c+d x)}{a d^3}-\frac {i (e+f x)^3 \cosh (c+d x)}{a d}+\frac {3 i f (e+f x)^2 \sinh (c+d x)}{a d^2}-\frac {(e+f x)^3 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d}+\frac {(3 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 i f^3\right ) \int \cosh (c+d x) \, dx}{a d^3}\\ &=-\frac {(e+f x)^3}{a d}+\frac {(e+f x)^4}{4 a f}-\frac {6 i f^2 (e+f x) \cosh (c+d x)}{a d^3}-\frac {i (e+f x)^3 \cosh (c+d x)}{a d}+\frac {6 i f^3 \sinh (c+d x)}{a d^4}+\frac {3 i f (e+f x)^2 \sinh (c+d x)}{a d^2}-\frac {(e+f x)^3 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d}+\frac {(6 i 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 {(e+f x)^3}{a d}+\frac {(e+f x)^4}{4 a f}-\frac {6 i f^2 (e+f x) \cosh (c+d x)}{a d^3}-\frac {i (e+f x)^3 \cosh (c+d x)}{a d}+\frac {6 f (e+f x)^2 \log \left (1+i e^{c+d x}\right )}{a d^2}+\frac {6 i f^3 \sinh (c+d x)}{a d^4}+\frac {3 i f (e+f x)^2 \sinh (c+d x)}{a d^2}-\frac {(e+f x)^3 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d}-\frac {\left (12 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 {(e+f x)^3}{a d}+\frac {(e+f x)^4}{4 a f}-\frac {6 i f^2 (e+f x) \cosh (c+d x)}{a d^3}-\frac {i (e+f x)^3 \cosh (c+d x)}{a d}+\frac {6 f (e+f x)^2 \log \left (1+i e^{c+d x}\right )}{a d^2}+\frac {12 f^2 (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{a d^3}+\frac {6 i f^3 \sinh (c+d x)}{a d^4}+\frac {3 i f (e+f x)^2 \sinh (c+d x)}{a d^2}-\frac {(e+f x)^3 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d}-\frac {\left (12 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 {(e+f x)^3}{a d}+\frac {(e+f x)^4}{4 a f}-\frac {6 i f^2 (e+f x) \cosh (c+d x)}{a d^3}-\frac {i (e+f x)^3 \cosh (c+d x)}{a d}+\frac {6 f (e+f x)^2 \log \left (1+i e^{c+d x}\right )}{a d^2}+\frac {12 f^2 (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{a d^3}+\frac {6 i f^3 \sinh (c+d x)}{a d^4}+\frac {3 i f (e+f x)^2 \sinh (c+d x)}{a d^2}-\frac {(e+f x)^3 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d}-\frac {\left (12 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 {(e+f x)^3}{a d}+\frac {(e+f x)^4}{4 a f}-\frac {6 i f^2 (e+f x) \cosh (c+d x)}{a d^3}-\frac {i (e+f x)^3 \cosh (c+d x)}{a d}+\frac {6 f (e+f x)^2 \log \left (1+i e^{c+d x}\right )}{a d^2}+\frac {12 f^2 (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{a d^3}-\frac {12 f^3 \text {Li}_3\left (-i e^{c+d x}\right )}{a d^4}+\frac {6 i f^3 \sinh (c+d x)}{a d^4}+\frac {3 i f (e+f x)^2 \sinh (c+d x)}{a d^2}-\frac {(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 [B] time = 6.53, size = 857, normalized size = 3.56 \[ \frac {\frac {i f^3 x^4 \sinh \left (c+\frac {d x}{2}\right ) d^4+4 i e f^2 x^3 \sinh \left (c+\frac {d x}{2}\right ) d^4+6 i e^2 f x^2 \sinh \left (c+\frac {d x}{2}\right ) d^4+4 i e^3 x \sinh \left (c+\frac {d x}{2}\right ) d^4-10 e^3 \sinh \left (\frac {d x}{2}\right ) d^3-10 f^3 x^3 \sinh \left (\frac {d x}{2}\right ) d^3-30 e f^2 x^2 \sinh \left (\frac {d x}{2}\right ) d^3-30 e^2 f x \sinh \left (\frac {d x}{2}\right ) d^3+2 e^3 \sinh \left (2 c+\frac {3 d x}{2}\right ) d^3+2 f^3 x^3 \sinh \left (2 c+\frac {3 d x}{2}\right ) d^3+6 e f^2 x^2 \sinh \left (2 c+\frac {3 d x}{2}\right ) d^3+6 e^2 f x \sinh \left (2 c+\frac {3 d x}{2}\right ) d^3-6 f^3 x^2 \cosh \left (2 c+\frac {3 d x}{2}\right ) d^2-6 e^2 f \cosh \left (2 c+\frac {3 d x}{2}\right ) d^2-12 e f^2 x \cosh \left (2 c+\frac {3 d x}{2}\right ) d^2+6 i f^3 x^2 \sinh \left (c+\frac {d x}{2}\right ) d^2+6 i e^2 f \sinh \left (c+\frac {d x}{2}\right ) d^2+12 i e f^2 x \sinh \left (c+\frac {d x}{2}\right ) d^2+6 i f^3 x^2 \sinh \left (c+\frac {3 d x}{2}\right ) d^2+6 i e^2 f \sinh \left (c+\frac {3 d x}{2}\right ) d^2+12 i e f^2 x \sinh \left (c+\frac {3 d x}{2}\right ) d^2-2 i (e+f x) \left (6 f^2+d^2 (e+f x)^2\right ) \cosh \left (c+\frac {d x}{2}\right ) d-2 i (e+f x) \left (6 f^2+d^2 (e+f x)^2\right ) \cosh \left (c+\frac {3 d x}{2}\right ) d-12 e f^2 \sinh \left (\frac {d x}{2}\right ) d-12 f^3 x \sinh \left (\frac {d x}{2}\right ) d+12 e f^2 \sinh \left (2 c+\frac {3 d x}{2}\right ) d+12 f^3 x \sinh \left (2 c+\frac {3 d x}{2}\right ) d+\left (x \left (4 e^3+6 f x e^2+4 f^2 x^2 e+f^3 x^3\right ) d^4+6 f (e+f x)^2 d^2+12 f^3\right ) \cosh \left (\frac {d x}{2}\right )-12 f^3 \cosh \left (2 c+\frac {3 d x}{2}\right )+12 i f^3 \sinh \left (c+\frac {d x}{2}\right )+12 i f^3 \sinh \left (c+\frac {3 d x}{2}\right )}{\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 {8 i \left (d^3 (e+f x)^3+3 d^2 \left (1+i e^c\right ) f \log \left (1-i e^{-c-d x}\right ) (e+f x)^2+6 i \left (i-e^c\right ) 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 )\right )}{-i+e^c}}{4 a d^4} \]
Antiderivative was successfully verified.
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fricas [C] time = 0.52, size = 816, normalized size = 3.39 \[ -\frac {2 \, d^{3} f^{3} x^{3} + 2 \, d^{3} e^{3} + 6 \, d^{2} e^{2} f + 12 \, d e f^{2} + 12 \, f^{3} + 6 \, {\left (d^{3} e f^{2} + d^{2} f^{3}\right )} x^{2} + 6 \, {\left (d^{3} e^{2} f + 2 \, d^{2} e f^{2} + 2 \, d f^{3}\right )} x - {\left (48 \, {\left (d f^{3} x + d e f^{2}\right )} e^{\left (2 \, d x + 2 \, c\right )} + {\left (-48 i \, d f^{3} x - 48 i \, d e f^{2}\right )} e^{\left (d x + c\right )}\right )} {\rm Li}_2\left (-i \, e^{\left (d x + c\right )}\right ) - {\left (-2 i \, d^{3} f^{3} x^{3} - 2 i \, d^{3} e^{3} + 6 i \, d^{2} e^{2} f - 12 i \, d e f^{2} + 12 i \, f^{3} + {\left (-6 i \, d^{3} e f^{2} + 6 i \, d^{2} f^{3}\right )} x^{2} + {\left (-6 i \, d^{3} e^{2} f + 12 i \, d^{2} e f^{2} - 12 i \, d f^{3}\right )} x\right )} e^{\left (3 \, d x + 3 \, c\right )} - {\left (d^{4} f^{3} x^{4} - 2 \, d^{3} e^{3} - 6 \, {\left (4 \, c - 1\right )} d^{2} e^{2} f + 12 \, {\left (2 \, c^{2} - 1\right )} d e f^{2} - 4 \, {\left (2 \, c^{3} - 3\right )} f^{3} + 2 \, {\left (2 \, d^{4} e f^{2} - 5 \, d^{3} f^{3}\right )} x^{3} + 6 \, {\left (d^{4} e^{2} f - 5 \, d^{3} e f^{2} + d^{2} f^{3}\right )} x^{2} + 2 \, {\left (2 \, d^{4} e^{3} - 15 \, d^{3} e^{2} f + 6 \, d^{2} e f^{2} - 6 \, d f^{3}\right )} x\right )} e^{\left (2 \, d x + 2 \, c\right )} - {\left (-i \, d^{4} f^{3} x^{4} - 10 i \, d^{3} e^{3} + {\left (24 i \, c - 6 i\right )} d^{2} e^{2} f + {\left (-24 i \, c^{2} - 12 i\right )} d e f^{2} + {\left (8 i \, c^{3} - 12 i\right )} f^{3} + {\left (-4 i \, d^{4} e f^{2} - 2 i \, d^{3} f^{3}\right )} x^{3} + {\left (-6 i \, d^{4} e^{2} f - 6 i \, d^{3} e f^{2} - 6 i \, d^{2} f^{3}\right )} x^{2} + {\left (-4 i \, d^{4} e^{3} - 6 i \, d^{3} e^{2} f - 12 i \, d^{2} e f^{2} - 12 i \, d f^{3}\right )} x\right )} e^{\left (d x + c\right )} - {\left (24 \, {\left (d^{2} e^{2} f - 2 \, c d e f^{2} + c^{2} f^{3}\right )} e^{\left (2 \, d x + 2 \, c\right )} + {\left (-24 i \, d^{2} e^{2} f + 48 i \, c d e f^{2} - 24 i \, c^{2} f^{3}\right )} e^{\left (d x + c\right )}\right )} \log \left (e^{\left (d x + c\right )} - i\right ) - {\left (24 \, {\left (d^{2} f^{3} x^{2} + 2 \, d^{2} e f^{2} x + 2 \, c d e f^{2} - c^{2} f^{3}\right )} e^{\left (2 \, d x + 2 \, c\right )} + {\left (-24 i \, d^{2} f^{3} x^{2} - 48 i \, d^{2} e f^{2} x - 48 i \, c d e f^{2} + 24 i \, c^{2} f^{3}\right )} e^{\left (d x + c\right )}\right )} \log \left (i \, e^{\left (d x + c\right )} + 1\right ) + {\left (48 \, f^{3} e^{\left (2 \, d x + 2 \, c\right )} - 48 i \, f^{3} e^{\left (d x + c\right )}\right )} {\rm polylog}\left (3, -i \, e^{\left (d x + c\right )}\right )}{4 \, a d^{4} e^{\left (2 \, d x + 2 \, c\right )} - 4 i \, a d^{4} e^{\left (d x + c\right )}} \]
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 \[ \int \frac {{\left (f x + e\right )}^{3} \sinh \left (d x + c\right )^{2}}{i \, a \sinh \left (d x + c\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.29, size = 688, normalized size = 2.85 \[ \frac {12 f^{2} e \ln \left (1+i {\mathrm e}^{d x +c}\right ) x}{a \,d^{2}}+\frac {12 f^{2} e \ln \left (1+i {\mathrm e}^{d x +c}\right ) c}{a \,d^{3}}-\frac {12 f^{2} e c \ln \left ({\mathrm e}^{d x +c}-i\right )}{a \,d^{3}}+\frac {12 f^{2} e c \ln \left ({\mathrm e}^{d x +c}\right )}{a \,d^{3}}-\frac {12 f^{2} e c x}{a \,d^{2}}-\frac {2 f^{3} x^{3}}{a d}+\frac {4 f^{3} c^{3}}{a \,d^{4}}-\frac {12 f^{3} \polylog \left (3, -i {\mathrm e}^{d x +c}\right )}{a \,d^{4}}+\frac {e \,f^{2} x^{3}}{a}+\frac {3 e^{2} f \,x^{2}}{2 a}-\frac {6 f \ln \left ({\mathrm e}^{d x +c}\right ) e^{2}}{a \,d^{2}}+\frac {6 f^{3} c^{2} x}{a \,d^{3}}-\frac {i \left (f^{3} x^{3} d^{3}+3 d^{3} e \,f^{2} x^{2}+3 d^{3} e^{2} f x -3 d^{2} f^{3} x^{2}+d^{3} e^{3}-6 d^{2} e \,f^{2} x -3 d^{2} e^{2} f +6 d \,f^{3} x +6 d e \,f^{2}-6 f^{3}\right ) {\mathrm e}^{d x +c}}{2 a \,d^{4}}-\frac {2 i \left (x^{3} f^{3}+3 e \,f^{2} x^{2}+3 e^{2} f x +e^{3}\right )}{d a \left ({\mathrm e}^{d x +c}-i\right )}-\frac {i \left (f^{3} x^{3} d^{3}+3 d^{3} e \,f^{2} x^{2}+3 d^{3} e^{2} f x +3 d^{2} f^{3} x^{2}+d^{3} e^{3}+6 d^{2} e \,f^{2} x +3 d^{2} e^{2} f +6 d \,f^{3} x +6 d e \,f^{2}+6 f^{3}\right ) {\mathrm e}^{-d x -c}}{2 a \,d^{4}}+\frac {6 f^{3} c^{2} \ln \left ({\mathrm e}^{d x +c}-i\right )}{a \,d^{4}}+\frac {6 f \ln \left ({\mathrm e}^{d x +c}-i\right ) e^{2}}{a \,d^{2}}+\frac {12 f^{3} \polylog \left (2, -i {\mathrm e}^{d x +c}\right ) x}{a \,d^{3}}+\frac {12 f^{2} e \polylog \left (2, -i {\mathrm e}^{d x +c}\right )}{a \,d^{3}}+\frac {6 f^{3} \ln \left (1+i {\mathrm e}^{d x +c}\right ) x^{2}}{a \,d^{2}}-\frac {6 f^{3} \ln \left (1+i {\mathrm e}^{d x +c}\right ) c^{2}}{a \,d^{4}}-\frac {6 f^{2} e \,c^{2}}{a \,d^{3}}-\frac {6 f^{2} e \,x^{2}}{a d}-\frac {6 f^{3} c^{2} \ln \left ({\mathrm e}^{d x +c}\right )}{a \,d^{4}}+\frac {x^{4} f^{3}}{4 a}+\frac {e^{3} x}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.71, size = 670, normalized size = 2.78 \[ -\frac {3}{4} \, e^{2} f {\left (\frac {4 \, x e^{\left (d x + c\right )}}{a d e^{\left (d x + c\right )} - i \, a d} - \frac {-2 i \, d^{2} x^{2} e^{c} - 2 i \, d x e^{c} - {\left (2 i \, d x e^{\left (3 \, c\right )} - 2 i \, e^{\left (3 \, c\right )}\right )} e^{\left (2 \, d x\right )} + 2 \, {\left (d^{2} x^{2} e^{\left (2 \, c\right )} - 3 \, d x e^{\left (2 \, c\right )} + e^{\left (2 \, c\right )}\right )} e^{\left (d x\right )} - 2 \, {\left (d x + 1\right )} e^{\left (-d x\right )} - 2 i \, e^{c}}{a d^{2} e^{\left (d x + 2 \, c\right )} - i \, a d^{2} e^{c}} - \frac {8 \, \log \left ({\left (e^{\left (d x + c\right )} - i\right )} e^{\left (-c\right )}\right )}{a d^{2}}\right )} + \frac {1}{2} \, e^{3} {\left (\frac {2 \, {\left (d x + c\right )}}{a d} + \frac {-5 i \, e^{\left (-d x - c\right )} + 1}{{\left (i \, a e^{\left (-d x - c\right )} + a e^{\left (-2 \, d x - 2 \, c\right )}\right )} d} - \frac {i \, e^{\left (-d x - c\right )}}{a d}\right )} + \frac {-i \, d^{4} f^{3} x^{4} - 12 i \, d e f^{2} - {\left (4 i \, d^{4} e f^{2} + 10 i \, d^{3} f^{3}\right )} x^{3} - 12 i \, f^{3} - {\left (30 i \, d^{3} e f^{2} + 6 i \, d^{2} f^{3}\right )} x^{2} - {\left (12 i \, d^{2} e f^{2} + 12 i \, d f^{3}\right )} x - {\left (2 i \, d^{3} f^{3} x^{3} e^{\left (2 \, c\right )} + {\left (6 i \, d^{3} e f^{2} - 6 i \, d^{2} f^{3}\right )} x^{2} e^{\left (2 \, c\right )} + {\left (-12 i \, d^{2} e f^{2} + 12 i \, d f^{3}\right )} x e^{\left (2 \, c\right )} + {\left (12 i \, d e f^{2} - 12 i \, f^{3}\right )} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )} + {\left (d^{4} f^{3} x^{4} e^{c} + 2 \, {\left (2 \, d^{4} e f^{2} - d^{3} f^{3}\right )} x^{3} e^{c} - 6 \, {\left (d^{3} e f^{2} - d^{2} f^{3}\right )} x^{2} e^{c} + 12 \, {\left (d^{2} e f^{2} - d f^{3}\right )} x e^{c} - 12 \, {\left (d e f^{2} - f^{3}\right )} e^{c}\right )} e^{\left (d x\right )}}{4 \, {\left (a d^{4} e^{\left (d x + c\right )} - i \, a d^{4}\right )}} + \frac {12 \, {\left (d x \log \left (i \, e^{\left (d x + c\right )} + 1\right ) + {\rm Li}_2\left (-i \, e^{\left (d x + c\right )}\right )\right )} e f^{2}}{a d^{3}} + \frac {6 \, {\left (d^{2} x^{2} \log \left (i \, e^{\left (d x + c\right )} + 1\right ) + 2 \, d x {\rm Li}_2\left (-i \, e^{\left (d x + c\right )}\right ) - 2 \, {\rm Li}_{3}(-i \, e^{\left (d x + c\right )})\right )} f^{3}}{a d^{4}} - \frac {2 \, {\left (d^{3} f^{3} x^{3} + 3 \, d^{3} e f^{2} x^{2}\right )}}{a d^{4}} \]
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 \[ \int \frac {{\mathrm {sinh}\left (c+d\,x\right )}^2\,{\left (e+f\,x\right )}^3}{a+a\,\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {2 i e^{3} e^{c} + 6 i e^{2} f x e^{c} + 6 i e f^{2} x^{2} e^{c} + 2 i f^{3} x^{3} e^{c}}{- i a d e^{c} - a d e^{- d x}} - \frac {i \left (\int \frac {i d e^{3}}{e^{c} e^{2 d x} - i e^{d x}}\, dx + \int \frac {i d f^{3} x^{3}}{e^{c} e^{2 d x} - i e^{d x}}\, dx + \int \frac {d e^{3} e^{c} e^{d x}}{e^{c} e^{2 d x} - i e^{d x}}\, dx + \int \frac {d e^{3} e^{3 c} e^{3 d x}}{e^{c} e^{2 d x} - i e^{d x}}\, dx + \int \frac {3 i d e f^{2} x^{2}}{e^{c} e^{2 d x} - i e^{d x}}\, dx + \int \frac {3 i d e^{2} f x}{e^{c} e^{2 d x} - i e^{d x}}\, dx + \int \frac {i d e^{3} e^{2 c} e^{2 d x}}{e^{c} e^{2 d x} - i e^{d x}}\, dx + \int \frac {12 i e^{2} f e^{2 c} e^{2 d x}}{e^{c} e^{2 d x} - i e^{d x}}\, dx + \int \frac {12 i f^{3} x^{2} e^{2 c} e^{2 d x}}{e^{c} e^{2 d x} - i e^{d x}}\, dx + \int \frac {d f^{3} x^{3} e^{c} e^{d x}}{e^{c} e^{2 d x} - i e^{d x}}\, dx + \int \frac {d f^{3} x^{3} e^{3 c} e^{3 d x}}{e^{c} e^{2 d x} - i e^{d x}}\, dx + \int \frac {i d f^{3} x^{3} e^{2 c} e^{2 d x}}{e^{c} e^{2 d x} - i e^{d x}}\, dx + \int \frac {24 i e f^{2} x e^{2 c} e^{2 d x}}{e^{c} e^{2 d x} - i e^{d x}}\, dx + \int \frac {3 d e f^{2} x^{2} e^{c} e^{d x}}{e^{c} e^{2 d x} - i e^{d x}}\, dx + \int \frac {3 d e f^{2} x^{2} e^{3 c} e^{3 d x}}{e^{c} e^{2 d x} - i e^{d x}}\, dx + \int \frac {3 d e^{2} f x e^{c} e^{d x}}{e^{c} e^{2 d x} - i e^{d x}}\, dx + \int \frac {3 d e^{2} f x e^{3 c} e^{3 d x}}{e^{c} e^{2 d x} - i e^{d x}}\, dx + \int \frac {3 i d e f^{2} x^{2} e^{2 c} e^{2 d x}}{e^{c} e^{2 d x} - i e^{d x}}\, dx + \int \frac {3 i d e^{2} f x e^{2 c} e^{2 d x}}{e^{c} e^{2 d x} - i e^{d x}}\, dx\right ) e^{- c}}{2 a d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
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