Optimal. Leaf size=185 \[ -\frac {3 \text {Li}_4\left (-e^{2 (a+b x)}\right )}{4 b^4}-\frac {3 \sinh (a+b x) \cosh (a+b x)}{8 b^4}+\frac {3 x \text {Li}_3\left (-e^{2 (a+b x)}\right )}{2 b^3}+\frac {3 x \sinh ^2(a+b x)}{4 b^3}-\frac {3 x^2 \text {Li}_2\left (-e^{2 (a+b x)}\right )}{2 b^2}-\frac {3 x^2 \sinh (a+b x) \cosh (a+b x)}{4 b^2}-\frac {x^3 \log \left (e^{2 (a+b x)}+1\right )}{b}+\frac {x^3 \sinh ^2(a+b x)}{2 b}+\frac {3 x}{8 b^3}+\frac {x^3}{4 b}+\frac {x^4}{4} \]
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Rubi [A] time = 0.25, antiderivative size = 185, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 12, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {5449, 5372, 3311, 30, 2635, 8, 3718, 2190, 2531, 6609, 2282, 6589} \[ -\frac {3 x^2 \text {PolyLog}\left (2,-e^{2 (a+b x)}\right )}{2 b^2}+\frac {3 x \text {PolyLog}\left (3,-e^{2 (a+b x)}\right )}{2 b^3}-\frac {3 \text {PolyLog}\left (4,-e^{2 (a+b x)}\right )}{4 b^4}-\frac {3 x^2 \sinh (a+b x) \cosh (a+b x)}{4 b^2}+\frac {3 x \sinh ^2(a+b x)}{4 b^3}-\frac {3 \sinh (a+b x) \cosh (a+b x)}{8 b^4}-\frac {x^3 \log \left (e^{2 (a+b x)}+1\right )}{b}+\frac {x^3 \sinh ^2(a+b x)}{2 b}+\frac {3 x}{8 b^3}+\frac {x^3}{4 b}+\frac {x^4}{4} \]
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 2190
Rule 2282
Rule 2531
Rule 2635
Rule 3311
Rule 3718
Rule 5372
Rule 5449
Rule 6589
Rule 6609
Rubi steps
\begin {align*} \int x^3 \sinh ^2(a+b x) \tanh (a+b x) \, dx &=\int x^3 \cosh (a+b x) \sinh (a+b x) \, dx-\int x^3 \tanh (a+b x) \, dx\\ &=\frac {x^4}{4}+\frac {x^3 \sinh ^2(a+b x)}{2 b}-2 \int \frac {e^{2 (a+b x)} x^3}{1+e^{2 (a+b x)}} \, dx-\frac {3 \int x^2 \sinh ^2(a+b x) \, dx}{2 b}\\ &=\frac {x^4}{4}-\frac {x^3 \log \left (1+e^{2 (a+b x)}\right )}{b}-\frac {3 x^2 \cosh (a+b x) \sinh (a+b x)}{4 b^2}+\frac {3 x \sinh ^2(a+b x)}{4 b^3}+\frac {x^3 \sinh ^2(a+b x)}{2 b}-\frac {3 \int \sinh ^2(a+b x) \, dx}{4 b^3}+\frac {3 \int x^2 \, dx}{4 b}+\frac {3 \int x^2 \log \left (1+e^{2 (a+b x)}\right ) \, dx}{b}\\ &=\frac {x^3}{4 b}+\frac {x^4}{4}-\frac {x^3 \log \left (1+e^{2 (a+b x)}\right )}{b}-\frac {3 x^2 \text {Li}_2\left (-e^{2 (a+b x)}\right )}{2 b^2}-\frac {3 \cosh (a+b x) \sinh (a+b x)}{8 b^4}-\frac {3 x^2 \cosh (a+b x) \sinh (a+b x)}{4 b^2}+\frac {3 x \sinh ^2(a+b x)}{4 b^3}+\frac {x^3 \sinh ^2(a+b x)}{2 b}+\frac {3 \int 1 \, dx}{8 b^3}+\frac {3 \int x \text {Li}_2\left (-e^{2 (a+b x)}\right ) \, dx}{b^2}\\ &=\frac {3 x}{8 b^3}+\frac {x^3}{4 b}+\frac {x^4}{4}-\frac {x^3 \log \left (1+e^{2 (a+b x)}\right )}{b}-\frac {3 x^2 \text {Li}_2\left (-e^{2 (a+b x)}\right )}{2 b^2}+\frac {3 x \text {Li}_3\left (-e^{2 (a+b x)}\right )}{2 b^3}-\frac {3 \cosh (a+b x) \sinh (a+b x)}{8 b^4}-\frac {3 x^2 \cosh (a+b x) \sinh (a+b x)}{4 b^2}+\frac {3 x \sinh ^2(a+b x)}{4 b^3}+\frac {x^3 \sinh ^2(a+b x)}{2 b}-\frac {3 \int \text {Li}_3\left (-e^{2 (a+b x)}\right ) \, dx}{2 b^3}\\ &=\frac {3 x}{8 b^3}+\frac {x^3}{4 b}+\frac {x^4}{4}-\frac {x^3 \log \left (1+e^{2 (a+b x)}\right )}{b}-\frac {3 x^2 \text {Li}_2\left (-e^{2 (a+b x)}\right )}{2 b^2}+\frac {3 x \text {Li}_3\left (-e^{2 (a+b x)}\right )}{2 b^3}-\frac {3 \cosh (a+b x) \sinh (a+b x)}{8 b^4}-\frac {3 x^2 \cosh (a+b x) \sinh (a+b x)}{4 b^2}+\frac {3 x \sinh ^2(a+b x)}{4 b^3}+\frac {x^3 \sinh ^2(a+b x)}{2 b}-\frac {3 \operatorname {Subst}\left (\int \frac {\text {Li}_3(-x)}{x} \, dx,x,e^{2 (a+b x)}\right )}{4 b^4}\\ &=\frac {3 x}{8 b^3}+\frac {x^3}{4 b}+\frac {x^4}{4}-\frac {x^3 \log \left (1+e^{2 (a+b x)}\right )}{b}-\frac {3 x^2 \text {Li}_2\left (-e^{2 (a+b x)}\right )}{2 b^2}+\frac {3 x \text {Li}_3\left (-e^{2 (a+b x)}\right )}{2 b^3}-\frac {3 \text {Li}_4\left (-e^{2 (a+b x)}\right )}{4 b^4}-\frac {3 \cosh (a+b x) \sinh (a+b x)}{8 b^4}-\frac {3 x^2 \cosh (a+b x) \sinh (a+b x)}{4 b^2}+\frac {3 x \sinh ^2(a+b x)}{4 b^3}+\frac {x^3 \sinh ^2(a+b x)}{2 b}\\ \end {align*}
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Mathematica [A] time = 2.99, size = 191, normalized size = 1.03 \[ \frac {1}{16} \left (\frac {12 \left (2 b^2 x^2 \text {Li}_2\left (-e^{-2 (a+b x)}\right )+2 b x \text {Li}_3\left (-e^{-2 (a+b x)}\right )+\text {Li}_4\left (-e^{-2 (a+b x)}\right )\right )}{b^4}+\frac {\cosh (2 b x) \left (2 b x \cosh (2 a) \left (2 b^2 x^2+3\right )-3 \sinh (2 a) \left (2 b^2 x^2+1\right )\right )}{b^4}+\frac {\sinh (2 b x) \left (2 b x \sinh (2 a) \left (2 b^2 x^2+3\right )-3 \cosh (2 a) \left (2 b^2 x^2+1\right )\right )}{b^4}-\frac {16 x^3 \log \left (e^{-2 (a+b x)}+1\right )}{b}-\frac {8 x^4}{e^{2 a}+1}-4 x^4 \tanh (a)\right ) \]
Antiderivative was successfully verified.
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fricas [C] time = 0.60, size = 966, normalized size = 5.22 \[ \text {result too large to display} \]
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 \[ \int x^{3} \operatorname {sech}\left (b x + a\right ) \sinh \left (b x + a\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.49, size = 189, normalized size = 1.02 \[ \frac {x^{4}}{4}+\frac {\left (4 x^{3} b^{3}-6 x^{2} b^{2}+6 b x -3\right ) {\mathrm e}^{2 b x +2 a}}{32 b^{4}}+\frac {\left (4 x^{3} b^{3}+6 x^{2} b^{2}+6 b x +3\right ) {\mathrm e}^{-2 b x -2 a}}{32 b^{4}}+\frac {2 a^{3} x}{b^{3}}+\frac {3 a^{4}}{2 b^{4}}-\frac {x^{3} \ln \left (1+{\mathrm e}^{2 b x +2 a}\right )}{b}-\frac {3 x^{2} \polylog \left (2, -{\mathrm e}^{2 b x +2 a}\right )}{2 b^{2}}+\frac {3 x \polylog \left (3, -{\mathrm e}^{2 b x +2 a}\right )}{2 b^{3}}-\frac {3 \polylog \left (4, -{\mathrm e}^{2 b x +2 a}\right )}{4 b^{4}}-\frac {2 a^{3} \ln \left ({\mathrm e}^{b x +a}\right )}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 181, normalized size = 0.98 \[ \frac {1}{2} \, x^{4} - \frac {{\left (8 \, b^{4} x^{4} e^{\left (2 \, a\right )} - {\left (4 \, b^{3} x^{3} e^{\left (4 \, a\right )} - 6 \, b^{2} x^{2} e^{\left (4 \, a\right )} + 6 \, b x e^{\left (4 \, a\right )} - 3 \, e^{\left (4 \, a\right )}\right )} e^{\left (2 \, b x\right )} - {\left (4 \, b^{3} x^{3} + 6 \, b^{2} x^{2} + 6 \, b x + 3\right )} e^{\left (-2 \, b x\right )}\right )} e^{\left (-2 \, a\right )}}{32 \, b^{4}} - \frac {4 \, b^{3} x^{3} \log \left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right ) + 6 \, b^{2} x^{2} {\rm Li}_2\left (-e^{\left (2 \, b x + 2 \, a\right )}\right ) - 6 \, b x {\rm Li}_{3}(-e^{\left (2 \, b x + 2 \, a\right )}) + 3 \, {\rm Li}_{4}(-e^{\left (2 \, b x + 2 \, a\right )})}{3 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^3\,{\mathrm {sinh}\left (a+b\,x\right )}^3}{\mathrm {cosh}\left (a+b\,x\right )} \,d x \]
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 \[ \int x^{3} \sinh ^{3}{\left (a + b x \right )} \operatorname {sech}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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