Optimal. Leaf size=71 \[ -\frac {1}{2} i x \text {Li}_2\left (-i e^{-x}\right )+\frac {1}{2} i x \text {Li}_2\left (i e^{-x}\right )-\frac {1}{2} i \text {Li}_3\left (-i e^{-x}\right )+\frac {1}{2} i \text {Li}_3\left (i e^{-x}\right ) \]
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Rubi [A] time = 0.05, antiderivative size = 71, normalized size of antiderivative = 1.00, 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 2282
Rule 2531
Rule 5144
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*}
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Mathematica [A] time = 0.01, size = 58, normalized size = 0.82 \[ -\frac {1}{2} i \left (x \text {Li}_2\left (-i e^{-x}\right )-x \text {Li}_2\left (i e^{-x}\right )+\text {Li}_3\left (-i e^{-x}\right )-\text {Li}_3\left (i e^{-x}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [C] time = 0.67, size = 65, normalized size = 0.92 \[ \frac {1}{2} \, x^{2} \operatorname {arccot}\left (e^{x}\right ) - \frac {1}{4} i \, x^{2} \log \left (i \, e^{x} + 1\right ) + \frac {1}{4} i \, x^{2} \log \left (-i \, e^{x} + 1\right ) + \frac {1}{2} i \, x {\rm Li}_2\left (i \, e^{x}\right ) - \frac {1}{2} i \, x {\rm Li}_2\left (-i \, e^{x}\right ) - \frac {1}{2} i \, {\rm polylog}\left (3, i \, e^{x}\right ) + \frac {1}{2} i \, {\rm polylog}\left (3, -i \, e^{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \operatorname {arccot}\left (e^{x}\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.27, size = 50, normalized size = 0.70 \[ \frac {\pi \,x^{2}}{4}+\frac {i \polylog \left (2, i {\mathrm e}^{x}\right ) x}{2}-\frac {i \polylog \left (3, i {\mathrm e}^{x}\right )}{2}-\frac {i \polylog \left (2, -i {\mathrm e}^{x}\right ) x}{2}+\frac {i \polylog \left (3, -i {\mathrm e}^{x}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{2} \, x^{2} \arctan \left (e^{\left (-x\right )}\right ) + \int \frac {x^{2} e^{x}}{2 \, {\left (e^{\left (2 \, x\right )} + 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x\,\mathrm {acot}\left ({\mathrm {e}}^x\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \operatorname {acot}{\left (e^{x} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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