Optimal. Leaf size=77 \[ -x^2 \text{PolyLog}\left (2,-e^x\right )+x^2 \text{PolyLog}\left (2,e^x\right )+2 x \text{PolyLog}\left (3,-e^x\right )-2 x \text{PolyLog}\left (3,e^x\right )-2 \text{PolyLog}\left (4,-e^x\right )+2 \text{PolyLog}\left (4,e^x\right )-\frac{2}{3} x^3 \tanh ^{-1}\left (e^x\right )+\frac{1}{3} x^3 \coth ^{-1}(\cosh (x)) \]
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Rubi [A] time = 0.0900436, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.857, Rules used = {6274, 4182, 2531, 6609, 2282, 6589} \[ -x^2 \text{PolyLog}\left (2,-e^x\right )+x^2 \text{PolyLog}\left (2,e^x\right )+2 x \text{PolyLog}\left (3,-e^x\right )-2 x \text{PolyLog}\left (3,e^x\right )-2 \text{PolyLog}\left (4,-e^x\right )+2 \text{PolyLog}\left (4,e^x\right )-\frac{2}{3} x^3 \tanh ^{-1}\left (e^x\right )+\frac{1}{3} x^3 \coth ^{-1}(\cosh (x)) \]
Antiderivative was successfully verified.
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Rule 6274
Rule 4182
Rule 2531
Rule 6609
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int x^2 \coth ^{-1}(\cosh (x)) \, dx &=\frac{1}{3} x^3 \coth ^{-1}(\cosh (x))+\frac{1}{3} \int x^3 \text{csch}(x) \, dx\\ &=\frac{1}{3} x^3 \coth ^{-1}(\cosh (x))-\frac{2}{3} x^3 \tanh ^{-1}\left (e^x\right )-\int x^2 \log \left (1-e^x\right ) \, dx+\int x^2 \log \left (1+e^x\right ) \, dx\\ &=\frac{1}{3} x^3 \coth ^{-1}(\cosh (x))-\frac{2}{3} x^3 \tanh ^{-1}\left (e^x\right )-x^2 \text{Li}_2\left (-e^x\right )+x^2 \text{Li}_2\left (e^x\right )+2 \int x \text{Li}_2\left (-e^x\right ) \, dx-2 \int x \text{Li}_2\left (e^x\right ) \, dx\\ &=\frac{1}{3} x^3 \coth ^{-1}(\cosh (x))-\frac{2}{3} x^3 \tanh ^{-1}\left (e^x\right )-x^2 \text{Li}_2\left (-e^x\right )+x^2 \text{Li}_2\left (e^x\right )+2 x \text{Li}_3\left (-e^x\right )-2 x \text{Li}_3\left (e^x\right )-2 \int \text{Li}_3\left (-e^x\right ) \, dx+2 \int \text{Li}_3\left (e^x\right ) \, dx\\ &=\frac{1}{3} x^3 \coth ^{-1}(\cosh (x))-\frac{2}{3} x^3 \tanh ^{-1}\left (e^x\right )-x^2 \text{Li}_2\left (-e^x\right )+x^2 \text{Li}_2\left (e^x\right )+2 x \text{Li}_3\left (-e^x\right )-2 x \text{Li}_3\left (e^x\right )-2 \operatorname{Subst}\left (\int \frac{\text{Li}_3(-x)}{x} \, dx,x,e^x\right )+2 \operatorname{Subst}\left (\int \frac{\text{Li}_3(x)}{x} \, dx,x,e^x\right )\\ &=\frac{1}{3} x^3 \coth ^{-1}(\cosh (x))-\frac{2}{3} x^3 \tanh ^{-1}\left (e^x\right )-x^2 \text{Li}_2\left (-e^x\right )+x^2 \text{Li}_2\left (e^x\right )+2 x \text{Li}_3\left (-e^x\right )-2 x \text{Li}_3\left (e^x\right )-2 \text{Li}_4\left (-e^x\right )+2 \text{Li}_4\left (e^x\right )\\ \end{align*}
Mathematica [A] time = 0.0286605, size = 109, normalized size = 1.42 \[ \frac{1}{24} \left (24 x^2 \text{PolyLog}\left (2,-e^{-x}\right )+24 x^2 \text{PolyLog}\left (2,e^x\right )+48 x \text{PolyLog}\left (3,-e^{-x}\right )-48 x \text{PolyLog}\left (3,e^x\right )+48 \text{PolyLog}\left (4,-e^{-x}\right )+48 \text{PolyLog}\left (4,e^x\right )-2 x^4-8 x^3 \log \left (e^{-x}+1\right )+8 x^3 \log \left (1-e^x\right )+8 x^3 \coth ^{-1}(\cosh (x))+\pi ^4\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.141, size = 471, normalized size = 6.1 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.17831, size = 105, normalized size = 1.36 \begin{align*} \frac{1}{3} \, x^{3} \operatorname{arcoth}\left (\cosh \left (x\right )\right ) - \frac{1}{3} \, x^{3} \log \left (e^{x} + 1\right ) + \frac{1}{3} \, x^{3} \log \left (-e^{x} + 1\right ) - x^{2}{\rm Li}_2\left (-e^{x}\right ) + x^{2}{\rm Li}_2\left (e^{x}\right ) + 2 \, x{\rm Li}_{3}(-e^{x}) - 2 \, x{\rm Li}_{3}(e^{x}) - 2 \,{\rm Li}_{4}(-e^{x}) + 2 \,{\rm Li}_{4}(e^{x}) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 1.77078, size = 435, normalized size = 5.65 \begin{align*} \frac{1}{6} \, x^{3} \log \left (\frac{\cosh \left (x\right ) + 1}{\cosh \left (x\right ) - 1}\right ) - \frac{1}{3} \, x^{3} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) + \frac{1}{3} \, x^{3} \log \left (-\cosh \left (x\right ) - \sinh \left (x\right ) + 1\right ) + x^{2}{\rm Li}_2\left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) - x^{2}{\rm Li}_2\left (-\cosh \left (x\right ) - \sinh \left (x\right )\right ) - 2 \, x{\rm polylog}\left (3, \cosh \left (x\right ) + \sinh \left (x\right )\right ) + 2 \, x{\rm polylog}\left (3, -\cosh \left (x\right ) - \sinh \left (x\right )\right ) + 2 \,{\rm polylog}\left (4, \cosh \left (x\right ) + \sinh \left (x\right )\right ) - 2 \,{\rm polylog}\left (4, -\cosh \left (x\right ) - \sinh \left (x\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \operatorname{acoth}{\left (\cosh{\left (x \right )} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \operatorname{arcoth}\left (\cosh \left (x\right )\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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