Optimal. Leaf size=115 \[ \frac {2 \text {Li}_3\left (-e^{a+b x}\right )}{b^3}-\frac {2 \text {Li}_3\left (e^{a+b x}\right )}{b^3}+\frac {2 \cosh (a+b x)}{b^3}-\frac {2 x \text {Li}_2\left (-e^{a+b x}\right )}{b^2}+\frac {2 x \text {Li}_2\left (e^{a+b x}\right )}{b^2}-\frac {2 x \sinh (a+b x)}{b^2}+\frac {x^2 \cosh (a+b x)}{b}-\frac {2 x^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b} \]
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Rubi [A] time = 0.12, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {5450, 3296, 2638, 4182, 2531, 2282, 6589} \[ -\frac {2 x \text {PolyLog}\left (2,-e^{a+b x}\right )}{b^2}+\frac {2 x \text {PolyLog}\left (2,e^{a+b x}\right )}{b^2}+\frac {2 \text {PolyLog}\left (3,-e^{a+b x}\right )}{b^3}-\frac {2 \text {PolyLog}\left (3,e^{a+b x}\right )}{b^3}-\frac {2 x \sinh (a+b x)}{b^2}+\frac {2 \cosh (a+b x)}{b^3}+\frac {x^2 \cosh (a+b x)}{b}-\frac {2 x^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 2531
Rule 2638
Rule 3296
Rule 4182
Rule 5450
Rule 6589
Rubi steps
\begin {align*} \int x^2 \cosh (a+b x) \coth (a+b x) \, dx &=\int x^2 \text {csch}(a+b x) \, dx+\int x^2 \sinh (a+b x) \, dx\\ &=-\frac {2 x^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b}+\frac {x^2 \cosh (a+b x)}{b}-\frac {2 \int x \cosh (a+b x) \, dx}{b}-\frac {2 \int x \log \left (1-e^{a+b x}\right ) \, dx}{b}+\frac {2 \int x \log \left (1+e^{a+b x}\right ) \, dx}{b}\\ &=-\frac {2 x^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b}+\frac {x^2 \cosh (a+b x)}{b}-\frac {2 x \text {Li}_2\left (-e^{a+b x}\right )}{b^2}+\frac {2 x \text {Li}_2\left (e^{a+b x}\right )}{b^2}-\frac {2 x \sinh (a+b x)}{b^2}+\frac {2 \int \text {Li}_2\left (-e^{a+b x}\right ) \, dx}{b^2}-\frac {2 \int \text {Li}_2\left (e^{a+b x}\right ) \, dx}{b^2}+\frac {2 \int \sinh (a+b x) \, dx}{b^2}\\ &=-\frac {2 x^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b}+\frac {2 \cosh (a+b x)}{b^3}+\frac {x^2 \cosh (a+b x)}{b}-\frac {2 x \text {Li}_2\left (-e^{a+b x}\right )}{b^2}+\frac {2 x \text {Li}_2\left (e^{a+b x}\right )}{b^2}-\frac {2 x \sinh (a+b x)}{b^2}+\frac {2 \operatorname {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{a+b x}\right )}{b^3}-\frac {2 \operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{a+b x}\right )}{b^3}\\ &=-\frac {2 x^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b}+\frac {2 \cosh (a+b x)}{b^3}+\frac {x^2 \cosh (a+b x)}{b}-\frac {2 x \text {Li}_2\left (-e^{a+b x}\right )}{b^2}+\frac {2 x \text {Li}_2\left (e^{a+b x}\right )}{b^2}+\frac {2 \text {Li}_3\left (-e^{a+b x}\right )}{b^3}-\frac {2 \text {Li}_3\left (e^{a+b x}\right )}{b^3}-\frac {2 x \sinh (a+b x)}{b^2}\\ \end {align*}
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Mathematica [A] time = 4.08, size = 138, normalized size = 1.20 \[ \frac {b^2 x^2 \cosh (a+b x)-2 b^2 x^2 \tanh ^{-1}(\sinh (a+b x)+\cosh (a+b x))-2 b x \text {Li}_2(-\cosh (a+b x)-\sinh (a+b x))+2 b x \text {Li}_2(\cosh (a+b x)+\sinh (a+b x))+2 \text {Li}_3(-\cosh (a+b x)-\sinh (a+b x))-2 \text {Li}_3(\cosh (a+b x)+\sinh (a+b x))-2 b x \sinh (a+b x)+2 \cosh (a+b x)}{b^3} \]
Warning: Unable to verify antiderivative.
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fricas [C] time = 0.55, size = 391, normalized size = 3.40 \[ \frac {b^{2} x^{2} + {\left (b^{2} x^{2} - 2 \, b x + 2\right )} \cosh \left (b x + a\right )^{2} + 2 \, {\left (b^{2} x^{2} - 2 \, b x + 2\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + {\left (b^{2} x^{2} - 2 \, b x + 2\right )} \sinh \left (b x + a\right )^{2} + 2 \, b x + 4 \, {\left (b x \cosh \left (b x + a\right ) + b x \sinh \left (b x + a\right )\right )} {\rm Li}_2\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) - 4 \, {\left (b x \cosh \left (b x + a\right ) + b x \sinh \left (b x + a\right )\right )} {\rm Li}_2\left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right ) - 2 \, {\left (b^{2} x^{2} \cosh \left (b x + a\right ) + b^{2} x^{2} \sinh \left (b x + a\right )\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) + 2 \, {\left (a^{2} \cosh \left (b x + a\right ) + a^{2} \sinh \left (b x + a\right )\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) + 2 \, {\left ({\left (b^{2} x^{2} - a^{2}\right )} \cosh \left (b x + a\right ) + {\left (b^{2} x^{2} - a^{2}\right )} \sinh \left (b x + a\right )\right )} \log \left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right ) + 1\right ) - 4 \, {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} {\rm polylog}\left (3, \cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) + 4 \, {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} {\rm polylog}\left (3, -\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right ) + 2}{2 \, {\left (b^{3} \cosh \left (b x + a\right ) + b^{3} \sinh \left (b x + a\right )\right )}} \]
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 \[ \int x^{2} \cosh \left (b x + a\right )^{2} \operatorname {csch}\left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.51, size = 196, normalized size = 1.70 \[ \frac {\left (x^{2} b^{2}-2 b x +2\right ) {\mathrm e}^{b x +a}}{2 b^{3}}+\frac {\left (x^{2} b^{2}+2 b x +2\right ) {\mathrm e}^{-b x -a}}{2 b^{3}}-\frac {2 a^{2} \arctanh \left ({\mathrm e}^{b x +a}\right )}{b^{3}}-\frac {\ln \left (1+{\mathrm e}^{b x +a}\right ) x^{2}}{b}+\frac {\ln \left (1+{\mathrm e}^{b x +a}\right ) a^{2}}{b^{3}}-\frac {2 \polylog \left (2, -{\mathrm e}^{b x +a}\right ) x}{b^{2}}+\frac {2 \polylog \left (3, -{\mathrm e}^{b x +a}\right )}{b^{3}}+\frac {\ln \left (1-{\mathrm e}^{b x +a}\right ) x^{2}}{b}-\frac {\ln \left (1-{\mathrm e}^{b x +a}\right ) a^{2}}{b^{3}}+\frac {2 \polylog \left (2, {\mathrm e}^{b x +a}\right ) x}{b^{2}}-\frac {2 \polylog \left (3, {\mathrm e}^{b x +a}\right )}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 152, normalized size = 1.32 \[ \frac {{\left ({\left (b^{2} x^{2} e^{\left (2 \, a\right )} - 2 \, b x e^{\left (2 \, a\right )} + 2 \, e^{\left (2 \, a\right )}\right )} e^{\left (b x\right )} + {\left (b^{2} x^{2} + 2 \, b x + 2\right )} e^{\left (-b x\right )}\right )} e^{\left (-a\right )}}{2 \, b^{3}} - \frac {b^{2} x^{2} \log \left (e^{\left (b x + a\right )} + 1\right ) + 2 \, b x {\rm Li}_2\left (-e^{\left (b x + a\right )}\right ) - 2 \, {\rm Li}_{3}(-e^{\left (b x + a\right )})}{b^{3}} + \frac {b^{2} x^{2} \log \left (-e^{\left (b x + a\right )} + 1\right ) + 2 \, b x {\rm Li}_2\left (e^{\left (b x + a\right )}\right ) - 2 \, {\rm Li}_{3}(e^{\left (b x + a\right )})}{b^{3}} \]
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 \[ \int \frac {x^2\,{\mathrm {cosh}\left (a+b\,x\right )}^2}{\mathrm {sinh}\left (a+b\,x\right )} \,d x \]
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 \[ \int x^{2} \cosh ^{2}{\left (a + b x \right )} \operatorname {csch}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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