Optimal. Leaf size=103 \[ -\frac{1}{2} i x^2 \text{PolyLog}\left (2,-i e^{-x}\right )+\frac{1}{2} i x^2 \text{PolyLog}\left (2,i e^{-x}\right )-i x \text{PolyLog}\left (3,-i e^{-x}\right )+i x \text{PolyLog}\left (3,i e^{-x}\right )-i \text{PolyLog}\left (4,-i e^{-x}\right )+i \text{PolyLog}\left (4,i e^{-x}\right ) \]
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Rubi [A] time = 0.0700694, antiderivative size = 103, 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 = {5144, 2531, 6609, 2282, 6589} \[ -\frac{1}{2} i x^2 \text{PolyLog}\left (2,-i e^{-x}\right )+\frac{1}{2} i x^2 \text{PolyLog}\left (2,i e^{-x}\right )-i x \text{PolyLog}\left (3,-i e^{-x}\right )+i x \text{PolyLog}\left (3,i e^{-x}\right )-i \text{PolyLog}\left (4,-i e^{-x}\right )+i \text{PolyLog}\left (4,i e^{-x}\right ) \]
Antiderivative was successfully verified.
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Rule 5144
Rule 2531
Rule 6609
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int x^2 \cot ^{-1}\left (e^x\right ) \, dx &=\frac{1}{2} i \int x^2 \log \left (1-i e^{-x}\right ) \, dx-\frac{1}{2} i \int x^2 \log \left (1+i e^{-x}\right ) \, dx\\ &=-\frac{1}{2} i x^2 \text{Li}_2\left (-i e^{-x}\right )+\frac{1}{2} i x^2 \text{Li}_2\left (i e^{-x}\right )+i \int x \text{Li}_2\left (-i e^{-x}\right ) \, dx-i \int x \text{Li}_2\left (i e^{-x}\right ) \, dx\\ &=-\frac{1}{2} i x^2 \text{Li}_2\left (-i e^{-x}\right )+\frac{1}{2} i x^2 \text{Li}_2\left (i e^{-x}\right )-i x \text{Li}_3\left (-i e^{-x}\right )+i x \text{Li}_3\left (i e^{-x}\right )+i \int \text{Li}_3\left (-i e^{-x}\right ) \, dx-i \int \text{Li}_3\left (i e^{-x}\right ) \, dx\\ &=-\frac{1}{2} i x^2 \text{Li}_2\left (-i e^{-x}\right )+\frac{1}{2} i x^2 \text{Li}_2\left (i e^{-x}\right )-i x \text{Li}_3\left (-i e^{-x}\right )+i x \text{Li}_3\left (i e^{-x}\right )-i \operatorname{Subst}\left (\int \frac{\text{Li}_3(-i x)}{x} \, dx,x,e^{-x}\right )+i \operatorname{Subst}\left (\int \frac{\text{Li}_3(i x)}{x} \, dx,x,e^{-x}\right )\\ &=-\frac{1}{2} i x^2 \text{Li}_2\left (-i e^{-x}\right )+\frac{1}{2} i x^2 \text{Li}_2\left (i e^{-x}\right )-i x \text{Li}_3\left (-i e^{-x}\right )+i x \text{Li}_3\left (i e^{-x}\right )-i \text{Li}_4\left (-i e^{-x}\right )+i \text{Li}_4\left (i e^{-x}\right )\\ \end{align*}
Mathematica [A] time = 0.0082515, size = 103, normalized size = 1. \[ -\frac{1}{2} i x^2 \text{PolyLog}\left (2,-i e^{-x}\right )+\frac{1}{2} i x^2 \text{PolyLog}\left (2,i e^{-x}\right )-i x \text{PolyLog}\left (3,-i e^{-x}\right )+i x \text{PolyLog}\left (3,i e^{-x}\right )-i \text{PolyLog}\left (4,-i e^{-x}\right )+i \text{PolyLog}\left (4,i e^{-x}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.154, size = 76, normalized size = 0.7 \begin{align*}{\frac{\pi \,{x}^{3}}{6}}+{\frac{i}{2}}{\it polylog} \left ( 2,i{{\rm e}^{x}} \right ){x}^{2}-ix{\it polylog} \left ( 3,i{{\rm e}^{x}} \right ) +i{\it polylog} \left ( 4,i{{\rm e}^{x}} \right ) -{\frac{i}{2}}{x}^{2}{\it polylog} \left ( 2,-i{{\rm e}^{x}} \right ) +i{\it polylog} \left ( 3,-i{{\rm e}^{x}} \right ) x-i{\it polylog} \left ( 4,-i{{\rm e}^{x}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{3} \, x^{3} \arctan \left (e^{\left (-x\right )}\right ) + \int \frac{x^{3} e^{x}}{3 \,{\left (e^{\left (2 \, x\right )} + 1\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 2.2362, size = 298, normalized size = 2.89 \begin{align*} \frac{1}{3} \, x^{3} \operatorname{arccot}\left (e^{x}\right ) - \frac{1}{6} i \, x^{3} \log \left (i \, e^{x} + 1\right ) + \frac{1}{6} i \, x^{3} \log \left (-i \, e^{x} + 1\right ) + \frac{1}{2} i \, x^{2}{\rm Li}_2\left (i \, e^{x}\right ) - \frac{1}{2} i \, x^{2}{\rm Li}_2\left (-i \, e^{x}\right ) - i \, x{\rm polylog}\left (3, i \, e^{x}\right ) + i \, x{\rm polylog}\left (3, -i \, e^{x}\right ) + i \,{\rm polylog}\left (4, i \, e^{x}\right ) - i \,{\rm polylog}\left (4, -i \, e^{x}\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{acot}{\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{arccot}\left (e^{x}\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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