Optimal. Leaf size=70 \[ \frac{1}{2} x^2 \text{PolyLog}\left (2,-e^{-x}\right )-\frac{1}{2} x^2 \text{PolyLog}\left (2,e^{-x}\right )+x \text{PolyLog}\left (3,-e^{-x}\right )-x \text{PolyLog}\left (3,e^{-x}\right )+\text{PolyLog}\left (4,-e^{-x}\right )-\text{PolyLog}\left (4,e^{-x}\right ) \]
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Rubi [A] time = 0.0730378, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.625, Rules used = {6214, 2531, 6609, 2282, 6589} \[ \frac{1}{2} x^2 \text{PolyLog}\left (2,-e^{-x}\right )-\frac{1}{2} x^2 \text{PolyLog}\left (2,e^{-x}\right )+x \text{PolyLog}\left (3,-e^{-x}\right )-x \text{PolyLog}\left (3,e^{-x}\right )+\text{PolyLog}\left (4,-e^{-x}\right )-\text{PolyLog}\left (4,e^{-x}\right ) \]
Antiderivative was successfully verified.
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Rule 6214
Rule 2531
Rule 6609
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int x^2 \coth ^{-1}\left (e^x\right ) \, dx &=-\left (\frac{1}{2} \int x^2 \log \left (1-e^{-x}\right ) \, dx\right )+\frac{1}{2} \int x^2 \log \left (1+e^{-x}\right ) \, dx\\ &=\frac{1}{2} x^2 \text{Li}_2\left (-e^{-x}\right )-\frac{1}{2} x^2 \text{Li}_2\left (e^{-x}\right )-\int x \text{Li}_2\left (-e^{-x}\right ) \, dx+\int x \text{Li}_2\left (e^{-x}\right ) \, dx\\ &=\frac{1}{2} x^2 \text{Li}_2\left (-e^{-x}\right )-\frac{1}{2} x^2 \text{Li}_2\left (e^{-x}\right )+x \text{Li}_3\left (-e^{-x}\right )-x \text{Li}_3\left (e^{-x}\right )-\int \text{Li}_3\left (-e^{-x}\right ) \, dx+\int \text{Li}_3\left (e^{-x}\right ) \, dx\\ &=\frac{1}{2} x^2 \text{Li}_2\left (-e^{-x}\right )-\frac{1}{2} x^2 \text{Li}_2\left (e^{-x}\right )+x \text{Li}_3\left (-e^{-x}\right )-x \text{Li}_3\left (e^{-x}\right )+\operatorname{Subst}\left (\int \frac{\text{Li}_3(-x)}{x} \, dx,x,e^{-x}\right )-\operatorname{Subst}\left (\int \frac{\text{Li}_3(x)}{x} \, dx,x,e^{-x}\right )\\ &=\frac{1}{2} x^2 \text{Li}_2\left (-e^{-x}\right )-\frac{1}{2} x^2 \text{Li}_2\left (e^{-x}\right )+x \text{Li}_3\left (-e^{-x}\right )-x \text{Li}_3\left (e^{-x}\right )+\text{Li}_4\left (-e^{-x}\right )-\text{Li}_4\left (e^{-x}\right )\\ \end{align*}
Mathematica [A] time = 0.0193149, size = 93, normalized size = 1.33 \[ \frac{1}{6} \left (-3 x^2 \text{PolyLog}\left (2,-e^x\right )+3 x^2 \text{PolyLog}\left (2,e^x\right )+6 x \text{PolyLog}\left (3,-e^x\right )-6 x \text{PolyLog}\left (3,e^x\right )-6 \text{PolyLog}\left (4,-e^x\right )+6 \text{PolyLog}\left (4,e^x\right )+x^3 \log \left (1-e^x\right )-x^3 \log \left (e^x+1\right )+2 x^3 \coth ^{-1}\left (e^x\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 79, normalized size = 1.1 \begin{align*}{\frac{{x}^{3}{\rm arccoth} \left ({{\rm e}^{x}}\right )}{3}}-{\frac{{x}^{3}\ln \left ({{\rm e}^{x}}+1 \right ) }{6}}-{\frac{{x}^{2}{\it polylog} \left ( 2,-{{\rm e}^{x}} \right ) }{2}}+x{\it polylog} \left ( 3,-{{\rm e}^{x}} \right ) -{\it polylog} \left ( 4,-{{\rm e}^{x}} \right ) +{\frac{{x}^{3}\ln \left ( 1-{{\rm e}^{x}} \right ) }{6}}+{\frac{{x}^{2}{\it polylog} \left ( 2,{{\rm e}^{x}} \right ) }{2}}-x{\it polylog} \left ( 3,{{\rm e}^{x}} \right ) +{\it polylog} \left ( 4,{{\rm e}^{x}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.10916, size = 103, normalized size = 1.47 \begin{align*} \frac{1}{3} \, x^{3} \operatorname{arcoth}\left (e^{x}\right ) - \frac{1}{6} \, x^{3} \log \left (e^{x} + 1\right ) + \frac{1}{6} \, x^{3} \log \left (-e^{x} + 1\right ) - \frac{1}{2} \, x^{2}{\rm Li}_2\left (-e^{x}\right ) + \frac{1}{2} \, x^{2}{\rm Li}_2\left (e^{x}\right ) + x{\rm Li}_{3}(-e^{x}) - x{\rm Li}_{3}(e^{x}) -{\rm Li}_{4}(-e^{x}) +{\rm Li}_{4}(e^{x}) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 1.77765, size = 462, normalized size = 6.6 \begin{align*} \frac{1}{6} \, x^{3} \log \left (\frac{\cosh \left (x\right ) + \sinh \left (x\right ) + 1}{\cosh \left (x\right ) + \sinh \left (x\right ) - 1}\right ) - \frac{1}{6} \, x^{3} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) + \frac{1}{6} \, x^{3} \log \left (-\cosh \left (x\right ) - \sinh \left (x\right ) + 1\right ) + \frac{1}{2} \, x^{2}{\rm Li}_2\left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) - \frac{1}{2} \, x^{2}{\rm Li}_2\left (-\cosh \left (x\right ) - \sinh \left (x\right )\right ) - x{\rm polylog}\left (3, \cosh \left (x\right ) + \sinh \left (x\right )\right ) + x{\rm polylog}\left (3, -\cosh \left (x\right ) - \sinh \left (x\right )\right ) +{\rm polylog}\left (4, \cosh \left (x\right ) + \sinh \left (x\right )\right ) -{\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 (e^{x} \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 (e^{x}\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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