Optimal. Leaf size=51 \[ -x \text{PolyLog}\left (2,-e^x\right )+x \text{PolyLog}\left (2,e^x\right )+\text{PolyLog}\left (3,-e^x\right )-\text{PolyLog}\left (3,e^x\right )+x^2 \left (-\tanh ^{-1}\left (e^x\right )\right )+\frac{1}{2} x^2 \tanh ^{-1}(\cosh (x)) \]
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Rubi [A] time = 0.0609738, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 5, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1., Rules used = {6273, 4182, 2531, 2282, 6589} \[ -x \text{PolyLog}\left (2,-e^x\right )+x \text{PolyLog}\left (2,e^x\right )+\text{PolyLog}\left (3,-e^x\right )-\text{PolyLog}\left (3,e^x\right )+x^2 \left (-\tanh ^{-1}\left (e^x\right )\right )+\frac{1}{2} x^2 \tanh ^{-1}(\cosh (x)) \]
Antiderivative was successfully verified.
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Rule 6273
Rule 4182
Rule 2531
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int x \tanh ^{-1}(\cosh (x)) \, dx &=\frac{1}{2} x^2 \tanh ^{-1}(\cosh (x))+\frac{1}{2} \int x^2 \text{csch}(x) \, dx\\ &=-x^2 \tanh ^{-1}\left (e^x\right )+\frac{1}{2} x^2 \tanh ^{-1}(\cosh (x))-\int x \log \left (1-e^x\right ) \, dx+\int x \log \left (1+e^x\right ) \, dx\\ &=-x^2 \tanh ^{-1}\left (e^x\right )+\frac{1}{2} x^2 \tanh ^{-1}(\cosh (x))-x \text{Li}_2\left (-e^x\right )+x \text{Li}_2\left (e^x\right )+\int \text{Li}_2\left (-e^x\right ) \, dx-\int \text{Li}_2\left (e^x\right ) \, dx\\ &=-x^2 \tanh ^{-1}\left (e^x\right )+\frac{1}{2} x^2 \tanh ^{-1}(\cosh (x))-x \text{Li}_2\left (-e^x\right )+x \text{Li}_2\left (e^x\right )+\operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^x\right )-\operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^x\right )\\ &=-x^2 \tanh ^{-1}\left (e^x\right )+\frac{1}{2} x^2 \tanh ^{-1}(\cosh (x))-x \text{Li}_2\left (-e^x\right )+x \text{Li}_2\left (e^x\right )+\text{Li}_3\left (-e^x\right )-\text{Li}_3\left (e^x\right )\\ \end{align*}
Mathematica [A] time = 0.0149095, size = 81, normalized size = 1.59 \[ \frac{1}{2} \left (2 x \text{PolyLog}\left (2,-e^{-x}\right )-2 x \text{PolyLog}\left (2,e^{-x}\right )+2 \text{PolyLog}\left (3,-e^{-x}\right )-2 \text{PolyLog}\left (3,e^{-x}\right )+x^2 \log \left (1-e^{-x}\right )-x^2 \log \left (e^{-x}+1\right )+x^2 \tanh ^{-1}(\cosh (x))\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.23, size = 479, normalized size = 9.4 \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 = 1.16409, size = 76, normalized size = 1.49 \begin{align*} \frac{1}{2} \, x^{2} \operatorname{artanh}\left (\cosh \left (x\right )\right ) - \frac{1}{2} \, x^{2} \log \left (e^{x} + 1\right ) + \frac{1}{2} \, x^{2} \log \left (-e^{x} + 1\right ) - x{\rm Li}_2\left (-e^{x}\right ) + x{\rm Li}_2\left (e^{x}\right ) +{\rm Li}_{3}(-e^{x}) -{\rm Li}_{3}(e^{x}) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 2.14733, size = 327, normalized size = 6.41 \begin{align*} \frac{1}{4} \, x^{2} \log \left (-\frac{\cosh \left (x\right ) + 1}{\cosh \left (x\right ) - 1}\right ) - \frac{1}{2} \, x^{2} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) + \frac{1}{2} \, x^{2} \log \left (-\cosh \left (x\right ) - \sinh \left (x\right ) + 1\right ) + x{\rm Li}_2\left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) - x{\rm Li}_2\left (-\cosh \left (x\right ) - \sinh \left (x\right )\right ) -{\rm polylog}\left (3, \cosh \left (x\right ) + \sinh \left (x\right )\right ) +{\rm polylog}\left (3, -\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 \operatorname{atanh}{\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 \operatorname{artanh}\left (\cosh \left (x\right )\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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