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 = 0.02, 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 = {2320, 4941,
2438} \begin {gather*} \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 {gather*}
Antiderivative was successfully verified.
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Rule 2320
Rule 2438
Rule 4941
Rubi steps
\begin {align*} \int \cot ^{-1}\left (e^x\right ) \, dx &=\text {Subst}\left (\int \frac {\cot ^{-1}(x)}{x} \, dx,x,e^x\right )\\ &=\frac {1}{2} i \text {Subst}\left (\int \frac {\log \left (1-\frac {i}{x}\right )}{x} \, dx,x,e^x\right )-\frac {1}{2} i \text {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 = 0.06, size = 59, normalized size = 1.69 \begin {gather*} x \cot ^{-1}\left (e^x\right )+\frac {1}{2} i \left (x \left (\log \left (1-i e^x\right )-\log \left (1+i e^x\right )\right )-\text {PolyLog}\left (2,-i e^x\right )+\text {PolyLog}\left (2,i e^x\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 58 vs. \(2 (25 ) = 50\).
time = 0.11, size = 59, normalized size = 1.69
method | result | size |
derivativedivides | \(\ln \left ({\mathrm e}^{x}\right ) \mathrm {arccot}\left ({\mathrm e}^{x}\right )-\frac {i \ln \left ({\mathrm e}^{x}\right ) \ln \left (1+i {\mathrm e}^{x}\right )}{2}+\frac {i \ln \left ({\mathrm e}^{x}\right ) \ln \left (1-i {\mathrm e}^{x}\right )}{2}-\frac {i \dilog \left (1+i {\mathrm e}^{x}\right )}{2}+\frac {i \dilog \left (1-i {\mathrm e}^{x}\right )}{2}\) | \(59\) |
default | \(\ln \left ({\mathrm e}^{x}\right ) \mathrm {arccot}\left ({\mathrm e}^{x}\right )-\frac {i \ln \left ({\mathrm e}^{x}\right ) \ln \left (1+i {\mathrm e}^{x}\right )}{2}+\frac {i \ln \left ({\mathrm e}^{x}\right ) \ln \left (1-i {\mathrm e}^{x}\right )}{2}-\frac {i \dilog \left (1+i {\mathrm e}^{x}\right )}{2}+\frac {i \dilog \left (1-i {\mathrm e}^{x}\right )}{2}\) | \(59\) |
risch | \(\frac {i x \ln \left (1+i {\mathrm e}^{x}\right )}{2}+\frac {\pi x}{2}+\frac {i \dilog \left (1-i {\mathrm e}^{x}\right )}{2}+\frac {i \ln \left (-i {\mathrm e}^{x}\right ) \ln \left (-i \left (-{\mathrm e}^{x}+i\right )\right )}{2}-\frac {i \ln \left (-i \left (-{\mathrm e}^{x}+i\right )\right ) x}{2}+\frac {i \dilog \left (-i {\mathrm e}^{x}\right )}{2}\) | \(73\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.53, size = 34, normalized size = 0.97 \begin {gather*} 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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 40 vs. \(2 (19) = 38\).
time = 2.99, size = 40, normalized size = 1.14 \begin {gather*} 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 {gather*}
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 \begin {gather*} \int \operatorname {acot}{\left (e^{x} \right )}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \mathrm {acot}\left ({\mathrm {e}}^x\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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