Optimal. Leaf size=35 \[ \frac{1}{2} i \text{PolyLog}\left (2,i e^{-x}\right )-\frac{1}{2} i \text{PolyLog}\left (2,-i e^{-x}\right ) \]
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Rubi [A] time = 0.0284399, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.75, 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 4849
Rule 2391
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*}
Mathematica [A] time = 0.0334939, size = 59, normalized size = 1.69 \[ x \cot ^{-1}\left (e^x\right )+\frac{1}{2} i \left (-\text{PolyLog}\left (2,-i e^x\right )+\text{PolyLog}\left (2,i e^x\right )+x \left (\log \left (1-i e^x\right )-\log \left (1+i e^x\right )\right )\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.033, size = 59, normalized size = 1.7 \begin{align*} \ln \left ({{\rm e}^{x}} \right ){\rm arccot} \left ({{\rm e}^{x}}\right )-{\frac{i}{2}}\ln \left ({{\rm e}^{x}} \right ) \ln \left ( 1+i{{\rm e}^{x}} \right ) +{\frac{i}{2}}\ln \left ({{\rm e}^{x}} \right ) \ln \left ( 1-i{{\rm e}^{x}} \right ) -{\frac{i}{2}}{\it dilog} \left ( 1+i{{\rm e}^{x}} \right ) +{\frac{i}{2}}{\it dilog} \left ( 1-i{{\rm e}^{x}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.61049, size = 46, normalized size = 1.31 \begin{align*} 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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.15755, size = 147, normalized size = 4.2 \begin{align*} 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 ) \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 \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 \operatorname{arccot}\left (e^{x}\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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