Optimal. Leaf size=130 \[ \frac {x^2}{4 b}+\frac {x^3}{3}-\frac {x^2 \log \left (1+e^{2 (a+b x)}\right )}{b}-\frac {x \text {PolyLog}\left (2,-e^{2 (a+b x)}\right )}{b^2}+\frac {\text {PolyLog}\left (3,-e^{2 (a+b x)}\right )}{2 b^3}-\frac {x \cosh (a+b x) \sinh (a+b x)}{2 b^2}+\frac {\sinh ^2(a+b x)}{4 b^3}+\frac {x^2 \sinh ^2(a+b x)}{2 b} \]
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Rubi [A]
time = 0.15, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 9, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5557, 5480,
3391, 30, 3799, 2221, 2611, 2320, 6724} \begin {gather*} \frac {\text {Li}_3\left (-e^{2 (a+b x)}\right )}{2 b^3}+\frac {\sinh ^2(a+b x)}{4 b^3}-\frac {x \text {Li}_2\left (-e^{2 (a+b x)}\right )}{b^2}-\frac {x \sinh (a+b x) \cosh (a+b x)}{2 b^2}-\frac {x^2 \log \left (e^{2 (a+b x)}+1\right )}{b}+\frac {x^2 \sinh ^2(a+b x)}{2 b}+\frac {x^2}{4 b}+\frac {x^3}{3} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2221
Rule 2320
Rule 2611
Rule 3391
Rule 3799
Rule 5480
Rule 5557
Rule 6724
Rubi steps
\begin {align*} \int x^2 \sinh ^2(a+b x) \tanh (a+b x) \, dx &=\int x^2 \cosh (a+b x) \sinh (a+b x) \, dx-\int x^2 \tanh (a+b x) \, dx\\ &=\frac {x^3}{3}+\frac {x^2 \sinh ^2(a+b x)}{2 b}-2 \int \frac {e^{2 (a+b x)} x^2}{1+e^{2 (a+b x)}} \, dx-\frac {\int x \sinh ^2(a+b x) \, dx}{b}\\ &=\frac {x^3}{3}-\frac {x^2 \log \left (1+e^{2 (a+b x)}\right )}{b}-\frac {x \cosh (a+b x) \sinh (a+b x)}{2 b^2}+\frac {\sinh ^2(a+b x)}{4 b^3}+\frac {x^2 \sinh ^2(a+b x)}{2 b}+\frac {\int x \, dx}{2 b}+\frac {2 \int x \log \left (1+e^{2 (a+b x)}\right ) \, dx}{b}\\ &=\frac {x^2}{4 b}+\frac {x^3}{3}-\frac {x^2 \log \left (1+e^{2 (a+b x)}\right )}{b}-\frac {x \text {Li}_2\left (-e^{2 (a+b x)}\right )}{b^2}-\frac {x \cosh (a+b x) \sinh (a+b x)}{2 b^2}+\frac {\sinh ^2(a+b x)}{4 b^3}+\frac {x^2 \sinh ^2(a+b x)}{2 b}+\frac {\int \text {Li}_2\left (-e^{2 (a+b x)}\right ) \, dx}{b^2}\\ &=\frac {x^2}{4 b}+\frac {x^3}{3}-\frac {x^2 \log \left (1+e^{2 (a+b x)}\right )}{b}-\frac {x \text {Li}_2\left (-e^{2 (a+b x)}\right )}{b^2}-\frac {x \cosh (a+b x) \sinh (a+b x)}{2 b^2}+\frac {\sinh ^2(a+b x)}{4 b^3}+\frac {x^2 \sinh ^2(a+b x)}{2 b}+\frac {\text {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{2 (a+b x)}\right )}{2 b^3}\\ &=\frac {x^2}{4 b}+\frac {x^3}{3}-\frac {x^2 \log \left (1+e^{2 (a+b x)}\right )}{b}-\frac {x \text {Li}_2\left (-e^{2 (a+b x)}\right )}{b^2}+\frac {\text {Li}_3\left (-e^{2 (a+b x)}\right )}{2 b^3}-\frac {x \cosh (a+b x) \sinh (a+b x)}{2 b^2}+\frac {\sinh ^2(a+b x)}{4 b^3}+\frac {x^2 \sinh ^2(a+b x)}{2 b}\\ \end {align*}
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Mathematica [A]
time = 1.96, size = 159, normalized size = 1.22 \begin {gather*} \frac {1}{24} \left (\frac {4 \left (2 b^2 x^2 \left (\frac {2 b e^{2 a} x}{1+e^{2 a}}-3 \log \left (1+e^{2 (a+b x)}\right )\right )-6 b x \text {PolyLog}\left (2,-e^{2 (a+b x)}\right )+3 \text {PolyLog}\left (3,-e^{2 (a+b x)}\right )\right )}{b^3}+\frac {3 \cosh (2 b x) \left (\left (1+2 b^2 x^2\right ) \cosh (2 a)-2 b x \sinh (2 a)\right )}{b^3}+\frac {3 \left (-2 b x \cosh (2 a)+\left (1+2 b^2 x^2\right ) \sinh (2 a)\right ) \sinh (2 b x)}{b^3}-8 x^3 \tanh (a)\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.44, size = 152, normalized size = 1.17
method | result | size |
risch | \(\frac {x^{3}}{3}+\frac {\left (2 b^{2} x^{2}-2 b x +1\right ) {\mathrm e}^{2 b x +2 a}}{16 b^{3}}+\frac {\left (2 b^{2} x^{2}+2 b x +1\right ) {\mathrm e}^{-2 b x -2 a}}{16 b^{3}}+\frac {2 a^{2} \ln \left ({\mathrm e}^{b x +a}\right )}{b^{3}}-\frac {2 a^{2} x}{b^{2}}-\frac {4 a^{3}}{3 b^{3}}-\frac {x^{2} \ln \left ({\mathrm e}^{2 b x +2 a}+1\right )}{b}-\frac {x \polylog \left (2, -{\mathrm e}^{2 b x +2 a}\right )}{b^{2}}+\frac {\polylog \left (3, -{\mathrm e}^{2 b x +2 a}\right )}{2 b^{3}}\) | \(152\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.32, size = 138, normalized size = 1.06 \begin {gather*} \frac {2}{3} \, x^{3} - \frac {{\left (16 \, b^{3} x^{3} e^{\left (2 \, a\right )} - 3 \, {\left (2 \, b^{2} x^{2} e^{\left (4 \, a\right )} - 2 \, b x e^{\left (4 \, a\right )} + e^{\left (4 \, a\right )}\right )} e^{\left (2 \, b x\right )} - 3 \, {\left (2 \, b^{2} x^{2} + 2 \, b x + 1\right )} e^{\left (-2 \, b x\right )}\right )} e^{\left (-2 \, a\right )}}{48 \, b^{3}} - \frac {2 \, b^{2} x^{2} \log \left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right ) + 2 \, b x {\rm Li}_2\left (-e^{\left (2 \, b x + 2 \, a\right )}\right ) - {\rm Li}_{3}(-e^{\left (2 \, b x + 2 \, a\right )})}{2 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains complex when optimal does not.
time = 0.40, size = 789, normalized size = 6.07 \begin {gather*} \frac {3 \, {\left (2 \, b^{2} x^{2} - 2 \, b x + 1\right )} \cosh \left (b x + a\right )^{4} + 12 \, {\left (2 \, b^{2} x^{2} - 2 \, b x + 1\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + 3 \, {\left (2 \, b^{2} x^{2} - 2 \, b x + 1\right )} \sinh \left (b x + a\right )^{4} + 6 \, b^{2} x^{2} + 16 \, {\left (b^{3} x^{3} + 2 \, a^{3}\right )} \cosh \left (b x + a\right )^{2} + 2 \, {\left (8 \, b^{3} x^{3} + 16 \, a^{3} + 9 \, {\left (2 \, b^{2} x^{2} - 2 \, b x + 1\right )} \cosh \left (b x + a\right )^{2}\right )} \sinh \left (b x + a\right )^{2} + 6 \, b x - 96 \, {\left (b x \cosh \left (b x + a\right )^{2} + 2 \, b x \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b x \sinh \left (b x + a\right )^{2}\right )} {\rm Li}_2\left (i \, \cosh \left (b x + a\right ) + i \, \sinh \left (b x + a\right )\right ) - 96 \, {\left (b x \cosh \left (b x + a\right )^{2} + 2 \, b x \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b x \sinh \left (b x + a\right )^{2}\right )} {\rm Li}_2\left (-i \, \cosh \left (b x + a\right ) - i \, \sinh \left (b x + a\right )\right ) - 48 \, {\left (a^{2} \cosh \left (b x + a\right )^{2} + 2 \, a^{2} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + a^{2} \sinh \left (b x + a\right )^{2}\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + i\right ) - 48 \, {\left (a^{2} \cosh \left (b x + a\right )^{2} + 2 \, a^{2} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + a^{2} \sinh \left (b x + a\right )^{2}\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - i\right ) - 48 \, {\left ({\left (b^{2} x^{2} - a^{2}\right )} \cosh \left (b x + a\right )^{2} + 2 \, {\left (b^{2} x^{2} - a^{2}\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + {\left (b^{2} x^{2} - a^{2}\right )} \sinh \left (b x + a\right )^{2}\right )} \log \left (i \, \cosh \left (b x + a\right ) + i \, \sinh \left (b x + a\right ) + 1\right ) - 48 \, {\left ({\left (b^{2} x^{2} - a^{2}\right )} \cosh \left (b x + a\right )^{2} + 2 \, {\left (b^{2} x^{2} - a^{2}\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + {\left (b^{2} x^{2} - a^{2}\right )} \sinh \left (b x + a\right )^{2}\right )} \log \left (-i \, \cosh \left (b x + a\right ) - i \, \sinh \left (b x + a\right ) + 1\right ) + 96 \, {\left (\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2}\right )} {\rm polylog}\left (3, i \, \cosh \left (b x + a\right ) + i \, \sinh \left (b x + a\right )\right ) + 96 \, {\left (\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2}\right )} {\rm polylog}\left (3, -i \, \cosh \left (b x + a\right ) - i \, \sinh \left (b x + a\right )\right ) + 4 \, {\left (3 \, {\left (2 \, b^{2} x^{2} - 2 \, b x + 1\right )} \cosh \left (b x + a\right )^{3} + 8 \, {\left (b^{3} x^{3} + 2 \, a^{3}\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 3}{48 \, {\left (b^{3} \cosh \left (b x + a\right )^{2} + 2 \, b^{3} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b^{3} \sinh \left (b x + a\right )^{2}\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*} \int x^{2} \sinh ^{3}{\left (a + b x \right )} \operatorname {sech}{\left (a + b x \right )}\, dx \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 = -0.01 \begin {gather*} \int \frac {x^2\,{\mathrm {sinh}\left (a+b\,x\right )}^3}{\mathrm {cosh}\left (a+b\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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