Optimal. Leaf size=25 \[ \frac {\text {Li}_2\left (-e^{-x}\right )}{2}-\frac {\text {Li}_2\left (e^{-x}\right )}{2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, 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.
[In]
[Out]
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 = 0.04, size = 51, normalized size = 2.04 \[ -\frac {\text {Li}_2\left (-e^x\right )}{2}+\frac {\text {Li}_2\left (e^x\right )}{2}+\frac {1}{2} x \log \left (1-e^x\right )-\frac {1}{2} x \log \left (e^x+1\right )+x \coth ^{-1}\left (e^x\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.58, size = 64, normalized size = 2.56 \[ \frac {1}{2} \, x \log \left (\frac {\cosh \relax (x) + \sinh \relax (x) + 1}{\cosh \relax (x) + \sinh \relax (x) - 1}\right ) - \frac {1}{2} \, x \log \left (\cosh \relax (x) + \sinh \relax (x) + 1\right ) + \frac {1}{2} \, x \log \left (-\cosh \relax (x) - \sinh \relax (x) + 1\right ) + \frac {1}{2} \, {\rm Li}_2\left (\cosh \relax (x) + \sinh \relax (x)\right ) - \frac {1}{2} \, {\rm Li}_2\left (-\cosh \relax (x) - \sinh \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {arcoth}\left (e^{x}\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.10, size = 31, normalized size = 1.24 \[ \ln \left ({\mathrm e}^{x}\right ) \mathrm {arccoth}\left ({\mathrm e}^{x}\right )-\frac {\dilog \left ({\mathrm e}^{x}\right )}{2}-\frac {\dilog \left ({\mathrm e}^{x}+1\right )}{2}-\frac {\ln \left ({\mathrm e}^{x}\right ) \ln \left ({\mathrm e}^{x}+1\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.32, size = 58, normalized size = 2.32 \[ -\frac {1}{2} \, x {\left (\log \left (e^{x} + 1\right ) - \log \left (e^{x} - 1\right )\right )} + x \operatorname {arcoth}\left (e^{x}\right ) + \frac {1}{2} \, \log \left (-e^{x}\right ) \log \left (e^{x} + 1\right ) - \frac {1}{2} \, x \log \left (e^{x} - 1\right ) + \frac {1}{2} \, {\rm Li}_2\left (e^{x} + 1\right ) - \frac {1}{2} \, {\rm Li}_2\left (-e^{x} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.04 \[ \int \mathrm {acoth}\left ({\mathrm {e}}^x\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {acoth}{\left (e^{x} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________