Optimal. Leaf size=35 \[ \frac {1}{2} i \text {Li}_2\left (i e^{-x}\right )-\frac {1}{2} i \text {Li}_2\left (-i e^{-x}\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {2282, 4849, 2391} \[ \frac {1}{2} i \text {PolyLog}\left (2,i e^{-x}\right )-\frac {1}{2} i \text {PolyLog}\left (2,-i e^{-x}\right ) \]
Antiderivative was successfully verified.
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Rule 2282
Rule 2391
Rule 4849
Rubi steps
\begin {align*} \int \cot ^{-1}\left (e^x\right ) \, dx &=\operatorname {Subst}\left (\int \frac {\cot ^{-1}(x)}{x} \, dx,x,e^x\right )\\ &=\frac {1}{2} i \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {i}{x}\right )}{x} \, dx,x,e^x\right )-\frac {1}{2} i \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {i}{x}\right )}{x} \, dx,x,e^x\right )\\ &=-\frac {1}{2} i \text {Li}_2\left (-i e^{-x}\right )+\frac {1}{2} i \text {Li}_2\left (i e^{-x}\right )\\ \end {align*}
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Mathematica [A] time = 0.08, size = 59, normalized size = 1.69 \[ x \cot ^{-1}\left (e^x\right )+\frac {1}{2} i \left (-\text {Li}_2\left (-i e^x\right )+\text {Li}_2\left (i e^x\right )+x \left (\log \left (1-i e^x\right )-\log \left (1+i e^x\right )\right )\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.49, size = 40, normalized size = 1.14 \[ x \operatorname {arccot}\left (e^{x}\right ) - \frac {1}{2} i \, x \log \left (i \, e^{x} + 1\right ) + \frac {1}{2} i \, x \log \left (-i \, e^{x} + 1\right ) + \frac {1}{2} i \, {\rm Li}_2\left (i \, e^{x}\right ) - \frac {1}{2} i \, {\rm Li}_2\left (-i \, e^{x}\right ) \]
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 \[ \int \operatorname {arccot}\left (e^{x}\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 59, normalized size = 1.69 \[ \ln \left ({\mathrm e}^{x}\right ) \mathrm {arccot}\left ({\mathrm e}^{x}\right )-\frac {i \ln \left ({\mathrm e}^{x}\right ) \ln \left (1+i {\mathrm e}^{x}\right )}{2}+\frac {i \ln \left ({\mathrm e}^{x}\right ) \ln \left (1-i {\mathrm e}^{x}\right )}{2}-\frac {i \dilog \left (1+i {\mathrm e}^{x}\right )}{2}+\frac {i \dilog \left (1-i {\mathrm e}^{x}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 34, normalized size = 0.97 \[ x \operatorname {arccot}\left (e^{x}\right ) + \frac {1}{4} \, \pi \log \left (e^{\left (2 \, x\right )} + 1\right ) + \frac {1}{2} i \, {\rm Li}_2\left (i \, e^{x} + 1\right ) - \frac {1}{2} i \, {\rm Li}_2\left (-i \, e^{x} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \mathrm {acot}\left ({\mathrm {e}}^x\right ) \,d x \]
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 \[ \int \operatorname {acot}{\left (e^{x} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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