Optimal. Leaf size=104 \[ \frac {2 i \text {Li}_2\left (-i e^{a+b x}\right )}{b^3}-\frac {2 i \text {Li}_2\left (i e^{a+b x}\right )}{b^3}+\frac {2 \cosh (a+b x)}{b^3}-\frac {4 x \tan ^{-1}\left (e^{a+b x}\right )}{b^2}-\frac {2 x \sinh (a+b x)}{b^2}+\frac {x^2 \cosh (a+b x)}{b}+\frac {x^2 \text {sech}(a+b x)}{b} \]
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Rubi [A] time = 0.12, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {5449, 3296, 2638, 5418, 4180, 2279, 2391} \[ \frac {2 i \text {PolyLog}\left (2,-i e^{a+b x}\right )}{b^3}-\frac {2 i \text {PolyLog}\left (2,i e^{a+b x}\right )}{b^3}-\frac {4 x \tan ^{-1}\left (e^{a+b x}\right )}{b^2}-\frac {2 x \sinh (a+b x)}{b^2}+\frac {2 \cosh (a+b x)}{b^3}+\frac {x^2 \cosh (a+b x)}{b}+\frac {x^2 \text {sech}(a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 2279
Rule 2391
Rule 2638
Rule 3296
Rule 4180
Rule 5418
Rule 5449
Rubi steps
\begin {align*} \int x^2 \sinh (a+b x) \tanh ^2(a+b x) \, dx &=\int x^2 \sinh (a+b x) \, dx-\int x^2 \text {sech}(a+b x) \tanh (a+b x) \, dx\\ &=\frac {x^2 \cosh (a+b x)}{b}+\frac {x^2 \text {sech}(a+b x)}{b}-\frac {2 \int x \cosh (a+b x) \, dx}{b}-\frac {2 \int x \text {sech}(a+b x) \, dx}{b}\\ &=-\frac {4 x \tan ^{-1}\left (e^{a+b x}\right )}{b^2}+\frac {x^2 \cosh (a+b x)}{b}+\frac {x^2 \text {sech}(a+b x)}{b}-\frac {2 x \sinh (a+b x)}{b^2}+\frac {(2 i) \int \log \left (1-i e^{a+b x}\right ) \, dx}{b^2}-\frac {(2 i) \int \log \left (1+i e^{a+b x}\right ) \, dx}{b^2}+\frac {2 \int \sinh (a+b x) \, dx}{b^2}\\ &=-\frac {4 x \tan ^{-1}\left (e^{a+b x}\right )}{b^2}+\frac {2 \cosh (a+b x)}{b^3}+\frac {x^2 \cosh (a+b x)}{b}+\frac {x^2 \text {sech}(a+b x)}{b}-\frac {2 x \sinh (a+b x)}{b^2}+\frac {(2 i) \operatorname {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{a+b x}\right )}{b^3}-\frac {(2 i) \operatorname {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{a+b x}\right )}{b^3}\\ &=-\frac {4 x \tan ^{-1}\left (e^{a+b x}\right )}{b^2}+\frac {2 \cosh (a+b x)}{b^3}+\frac {x^2 \cosh (a+b x)}{b}+\frac {2 i \text {Li}_2\left (-i e^{a+b x}\right )}{b^3}-\frac {2 i \text {Li}_2\left (i e^{a+b x}\right )}{b^3}+\frac {x^2 \text {sech}(a+b x)}{b}-\frac {2 x \sinh (a+b x)}{b^2}\\ \end {align*}
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Mathematica [A] time = 0.51, size = 172, normalized size = 1.65 \[ \frac {b^2 x^2 \text {sech}(a+b x)+\cosh (b x) \left (\cosh (a) \left (b^2 x^2+2\right )-2 b x \sinh (a)\right )+\sinh (b x) \left (\sinh (a) \left (b^2 x^2+2\right )-2 b x \cosh (a)\right )+2 i \left (\text {Li}_2\left (-i e^{a+b x}\right )-\text {Li}_2\left (i e^{a+b x}\right )\right )+(-2 i a-2 i b x+\pi ) \left (\log \left (1-i e^{a+b x}\right )-\log \left (1+i e^{a+b x}\right )\right )-(\pi -2 i a) \log \left (\cot \left (\frac {1}{4} (2 i a+2 i b x+\pi )\right )\right )}{b^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.73, size = 869, normalized size = 8.36 \[ \text {result too large to display} \]
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 \[ \int x^{2} \operatorname {sech}\left (b x + a\right )^{2} \sinh \left (b x + a\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.48, size = 205, normalized size = 1.97 \[ \frac {\left (x^{2} b^{2}-2 b x +2\right ) {\mathrm e}^{b x +a}}{2 b^{3}}+\frac {\left (x^{2} b^{2}+2 b x +2\right ) {\mathrm e}^{-b x -a}}{2 b^{3}}+\frac {2 x^{2} {\mathrm e}^{b x +a}}{b \left (1+{\mathrm e}^{2 b x +2 a}\right )}+\frac {2 i \ln \left (1+i {\mathrm e}^{b x +a}\right ) x}{b^{2}}+\frac {2 i \ln \left (1+i {\mathrm e}^{b x +a}\right ) a}{b^{3}}-\frac {2 i \ln \left (1-i {\mathrm e}^{b x +a}\right ) x}{b^{2}}-\frac {2 i \ln \left (1-i {\mathrm e}^{b x +a}\right ) a}{b^{3}}+\frac {2 i \dilog \left (1+i {\mathrm e}^{b x +a}\right )}{b^{3}}-\frac {2 i \dilog \left (1-i {\mathrm e}^{b x +a}\right )}{b^{3}}+\frac {4 a \arctan \left ({\mathrm e}^{b x +a}\right )}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {{\left (b^{2} x^{2} e^{\left (4 \, a\right )} - 2 \, b x e^{\left (4 \, a\right )} + 2 \, e^{\left (4 \, a\right )}\right )} e^{\left (3 \, b x\right )} + 2 \, {\left (3 \, b^{2} x^{2} e^{\left (2 \, a\right )} + 2 \, e^{\left (2 \, a\right )}\right )} e^{\left (b x\right )} + {\left (b^{2} x^{2} + 2 \, b x + 2\right )} e^{\left (-b x\right )}}{2 \, {\left (b^{3} e^{\left (2 \, b x + 3 \, a\right )} + b^{3} e^{a}\right )}} - 4 \, \int \frac {x e^{\left (b x + a\right )}}{b e^{\left (2 \, b x + 2 \, a\right )} + b}\,{d x} \]
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 \[ \int \frac {x^2\,{\mathrm {sinh}\left (a+b\,x\right )}^3}{{\mathrm {cosh}\left (a+b\,x\right )}^2} \,d x \]
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 \[ \int x^{2} \sinh ^{3}{\left (a + b x \right )} \operatorname {sech}^{2}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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