Optimal. Leaf size=51 \[ \frac {1}{2} x \text {PolyLog}\left (2,-e^{-x}\right )-\frac {1}{2} x \text {PolyLog}\left (2,e^{-x}\right )+\frac {1}{2} \text {PolyLog}\left (3,-e^{-x}\right )-\frac {1}{2} \text {PolyLog}\left (3,e^{-x}\right ) \]
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Rubi [A]
time = 0.03, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 4, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {6349, 2611,
2320, 6724} \begin {gather*} \frac {1}{2} x \text {Li}_2\left (-e^{-x}\right )-\frac {1}{2} x \text {Li}_2\left (e^{-x}\right )+\frac {\text {Li}_3\left (-e^{-x}\right )}{2}-\frac {\text {Li}_3\left (e^{-x}\right )}{2} \end {gather*}
Antiderivative was successfully verified.
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Rule 2320
Rule 2611
Rule 6349
Rule 6724
Rubi steps
\begin {align*} \int x \coth ^{-1}\left (e^x\right ) \, dx &=-\left (\frac {1}{2} \int x \log \left (1-e^{-x}\right ) \, dx\right )+\frac {1}{2} \int x \log \left (1+e^{-x}\right ) \, dx\\ &=\frac {1}{2} x \text {Li}_2\left (-e^{-x}\right )-\frac {1}{2} x \text {Li}_2\left (e^{-x}\right )-\frac {1}{2} \int \text {Li}_2\left (-e^{-x}\right ) \, dx+\frac {1}{2} \int \text {Li}_2\left (e^{-x}\right ) \, dx\\ &=\frac {1}{2} x \text {Li}_2\left (-e^{-x}\right )-\frac {1}{2} x \text {Li}_2\left (e^{-x}\right )+\frac {1}{2} \text {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{-x}\right )-\frac {1}{2} \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{-x}\right )\\ &=\frac {1}{2} x \text {Li}_2\left (-e^{-x}\right )-\frac {1}{2} x \text {Li}_2\left (e^{-x}\right )+\frac {\text {Li}_3\left (-e^{-x}\right )}{2}-\frac {\text {Li}_3\left (e^{-x}\right )}{2}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 71, normalized size = 1.39 \begin {gather*} \frac {1}{4} \left (2 x^2 \coth ^{-1}\left (e^x\right )+x^2 \log \left (1-e^x\right )-x^2 \log \left (1+e^x\right )-2 x \text {PolyLog}\left (2,-e^x\right )+2 x \text {PolyLog}\left (2,e^x\right )+2 \text {PolyLog}\left (3,-e^x\right )-2 \text {PolyLog}\left (3,e^x\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 62, normalized size = 1.22
method | result | size |
risch | \(-\frac {x^{2} \ln \left ({\mathrm e}^{x}-1\right )}{4}+\frac {x^{2} \ln \left (1-{\mathrm e}^{x}\right )}{4}+\frac {x \polylog \left (2, {\mathrm e}^{x}\right )}{2}-\frac {\polylog \left (3, {\mathrm e}^{x}\right )}{2}-\frac {x \polylog \left (2, -{\mathrm e}^{x}\right )}{2}+\frac {\polylog \left (3, -{\mathrm e}^{x}\right )}{2}\) | \(54\) |
default | \(\frac {x^{2} \mathrm {arccoth}\left ({\mathrm e}^{x}\right )}{2}-\frac {x^{2} \ln \left ({\mathrm e}^{x}+1\right )}{4}-\frac {x \polylog \left (2, -{\mathrm e}^{x}\right )}{2}+\frac {\polylog \left (3, -{\mathrm e}^{x}\right )}{2}+\frac {x^{2} \ln \left (1-{\mathrm e}^{x}\right )}{4}+\frac {x \polylog \left (2, {\mathrm e}^{x}\right )}{2}-\frac {\polylog \left (3, {\mathrm e}^{x}\right )}{2}\) | \(62\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 59, normalized size = 1.16 \begin {gather*} \frac {1}{2} \, x^{2} \operatorname {arcoth}\left (e^{x}\right ) - \frac {1}{4} \, x^{2} \log \left (e^{x} + 1\right ) + \frac {1}{4} \, x^{2} \log \left (-e^{x} + 1\right ) - \frac {1}{2} \, x {\rm Li}_2\left (-e^{x}\right ) + \frac {1}{2} \, x {\rm Li}_2\left (e^{x}\right ) + \frac {1}{2} \, {\rm Li}_{3}(-e^{x}) - \frac {1}{2} \, {\rm Li}_{3}(e^{x}) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 94 vs.
\(2 (37) = 74\).
time = 0.34, size = 94, normalized size = 1.84 \begin {gather*} \frac {1}{4} \, x^{2} \log \left (\frac {\cosh \left (x\right ) + \sinh \left (x\right ) + 1}{\cosh \left (x\right ) + \sinh \left (x\right ) - 1}\right ) - \frac {1}{4} \, x^{2} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) + \frac {1}{4} \, x^{2} \log \left (-\cosh \left (x\right ) - \sinh \left (x\right ) + 1\right ) + \frac {1}{2} \, x {\rm Li}_2\left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) - \frac {1}{2} \, x {\rm Li}_2\left (-\cosh \left (x\right ) - \sinh \left (x\right )\right ) - \frac {1}{2} \, {\rm polylog}\left (3, \cosh \left (x\right ) + \sinh \left (x\right )\right ) + \frac {1}{2} \, {\rm polylog}\left (3, -\cosh \left (x\right ) - \sinh \left (x\right )\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 x \operatorname {acoth}{\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 = -0.02 \begin {gather*} \int x\,\mathrm {acoth}\left ({\mathrm {e}}^x\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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