Optimal. Leaf size=25 \[ \frac{1}{2} \text{PolyLog}\left (2,-e^{-x}\right )-\frac{1}{2} \text{PolyLog}\left (2,e^{-x}\right ) \]
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Rubi [A] time = 0.0121473, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {2282, 5913} \[ \frac{1}{2} \text{PolyLog}\left (2,-e^{-x}\right )-\frac{1}{2} \text{PolyLog}\left (2,e^{-x}\right ) \]
Antiderivative was successfully verified.
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Rule 2282
Rule 5913
Rubi steps
\begin{align*} \int \coth ^{-1}\left (e^x\right ) \, dx &=\operatorname{Subst}\left (\int \frac{\coth ^{-1}(x)}{x} \, dx,x,e^x\right )\\ &=\frac{\text{Li}_2\left (-e^{-x}\right )}{2}-\frac{\text{Li}_2\left (e^{-x}\right )}{2}\\ \end{align*}
Mathematica [B] time = 0.0343158, size = 51, normalized size = 2.04 \[ -\frac{1}{2} \text{PolyLog}\left (2,-e^x\right )+\frac{1}{2} \text{PolyLog}\left (2,e^x\right )+\frac{1}{2} x \log \left (1-e^x\right )-\frac{1}{2} x \log \left (e^x+1\right )+x \coth ^{-1}\left (e^x\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.052, size = 31, normalized size = 1.2 \begin{align*} \ln \left ({{\rm e}^{x}} \right ){\rm arccoth} \left ({{\rm e}^{x}}\right )-{\frac{{\it dilog} \left ({{\rm e}^{x}} \right ) }{2}}-{\frac{{\it dilog} \left ({{\rm e}^{x}}+1 \right ) }{2}}-{\frac{\ln \left ({{\rm e}^{x}} \right ) \ln \left ({{\rm e}^{x}}+1 \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.11888, size = 78, normalized size = 3.12 \begin{align*} -\frac{1}{2} \, x{\left (\log \left (e^{x} + 1\right ) - \log \left (e^{x} - 1\right )\right )} + x \operatorname{arcoth}\left (e^{x}\right ) + \frac{1}{2} \, \log \left (-e^{x}\right ) \log \left (e^{x} + 1\right ) - \frac{1}{2} \, x \log \left (e^{x} - 1\right ) + \frac{1}{2} \,{\rm Li}_2\left (e^{x} + 1\right ) - \frac{1}{2} \,{\rm Li}_2\left (-e^{x} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.77649, size = 262, normalized size = 10.48 \begin{align*} \frac{1}{2} \, x \log \left (\frac{\cosh \left (x\right ) + \sinh \left (x\right ) + 1}{\cosh \left (x\right ) + \sinh \left (x\right ) - 1}\right ) - \frac{1}{2} \, x \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) + \frac{1}{2} \, x \log \left (-\cosh \left (x\right ) - \sinh \left (x\right ) + 1\right ) + \frac{1}{2} \,{\rm Li}_2\left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) - \frac{1}{2} \,{\rm Li}_2\left (-\cosh \left (x\right ) - \sinh \left (x\right )\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{acoth}{\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{arcoth}\left (e^{x}\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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