Optimal. Leaf size=71 \[ -\frac{1}{2} i x \text{PolyLog}\left (2,-i e^{-x}\right )+\frac{1}{2} i x \text{PolyLog}\left (2,i e^{-x}\right )-\frac{1}{2} i \text{PolyLog}\left (3,-i e^{-x}\right )+\frac{1}{2} i \text{PolyLog}\left (3,i e^{-x}\right ) \]
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Rubi [A] time = 0.0465794, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used = {5144, 2531, 2282, 6589} \[ -\frac{1}{2} i x \text{PolyLog}\left (2,-i e^{-x}\right )+\frac{1}{2} i x \text{PolyLog}\left (2,i e^{-x}\right )-\frac{1}{2} i \text{PolyLog}\left (3,-i e^{-x}\right )+\frac{1}{2} i \text{PolyLog}\left (3,i e^{-x}\right ) \]
Antiderivative was successfully verified.
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Rule 5144
Rule 2531
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int x \cot ^{-1}\left (e^x\right ) \, dx &=\frac{1}{2} i \int x \log \left (1-i e^{-x}\right ) \, dx-\frac{1}{2} i \int x \log \left (1+i e^{-x}\right ) \, dx\\ &=-\frac{1}{2} i x \text{Li}_2\left (-i e^{-x}\right )+\frac{1}{2} i x \text{Li}_2\left (i e^{-x}\right )+\frac{1}{2} i \int \text{Li}_2\left (-i e^{-x}\right ) \, dx-\frac{1}{2} i \int \text{Li}_2\left (i e^{-x}\right ) \, dx\\ &=-\frac{1}{2} i x \text{Li}_2\left (-i e^{-x}\right )+\frac{1}{2} i x \text{Li}_2\left (i e^{-x}\right )-\frac{1}{2} i \operatorname{Subst}\left (\int \frac{\text{Li}_2(-i x)}{x} \, dx,x,e^{-x}\right )+\frac{1}{2} i \operatorname{Subst}\left (\int \frac{\text{Li}_2(i x)}{x} \, dx,x,e^{-x}\right )\\ &=-\frac{1}{2} i x \text{Li}_2\left (-i e^{-x}\right )+\frac{1}{2} i x \text{Li}_2\left (i e^{-x}\right )-\frac{1}{2} i \text{Li}_3\left (-i e^{-x}\right )+\frac{1}{2} i \text{Li}_3\left (i e^{-x}\right )\\ \end{align*}
Mathematica [A] time = 0.0104373, size = 58, normalized size = 0.82 \[ -\frac{1}{2} i \left (x \text{PolyLog}\left (2,-i e^{-x}\right )-x \text{PolyLog}\left (2,i e^{-x}\right )+\text{PolyLog}\left (3,-i e^{-x}\right )-\text{PolyLog}\left (3,i e^{-x}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.185, size = 50, normalized size = 0.7 \begin{align*}{\frac{\pi \,{x}^{2}}{4}}+{\frac{i}{2}}{\it polylog} \left ( 2,i{{\rm e}^{x}} \right ) x-{\frac{i}{2}}{\it polylog} \left ( 3,i{{\rm e}^{x}} \right ) -{\frac{i}{2}}x{\it polylog} \left ( 2,-i{{\rm e}^{x}} \right ) +{\frac{i}{2}}{\it polylog} \left ( 3,-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}{2} \, x^{2} \arctan \left (e^{\left (-x\right )}\right ) + \int \frac{x^{2} e^{x}}{2 \,{\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 = 2.19322, size = 238, normalized size = 3.35 \begin{align*} \frac{1}{2} \, x^{2} \operatorname{arccot}\left (e^{x}\right ) - \frac{1}{4} i \, x^{2} \log \left (i \, e^{x} + 1\right ) + \frac{1}{4} i \, x^{2} \log \left (-i \, e^{x} + 1\right ) + \frac{1}{2} i \, x{\rm Li}_2\left (i \, e^{x}\right ) - \frac{1}{2} i \, x{\rm Li}_2\left (-i \, e^{x}\right ) - \frac{1}{2} i \,{\rm polylog}\left (3, i \, e^{x}\right ) + \frac{1}{2} i \,{\rm polylog}\left (3, -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 \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 \operatorname{arccot}\left (e^{x}\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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