Optimal. Leaf size=51 \[ -x^2 \tanh ^{-1}\left (e^x\right )+\frac {1}{2} x^2 \tanh ^{-1}(\cosh (x))-x \text {PolyLog}\left (2,-e^x\right )+x \text {PolyLog}\left (2,e^x\right )+\text {PolyLog}\left (3,-e^x\right )-\text {PolyLog}\left (3,e^x\right ) \]
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Rubi [A]
time = 0.04, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 5, integrand size = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {6408, 4267,
2611, 2320, 6724} \begin {gather*} -x \text {Li}_2\left (-e^x\right )+x \text {Li}_2\left (e^x\right )+\text {Li}_3\left (-e^x\right )-\text {Li}_3\left (e^x\right )+x^2 \left (-\tanh ^{-1}\left (e^x\right )\right )+\frac {1}{2} x^2 \tanh ^{-1}(\cosh (x)) \end {gather*}
Antiderivative was successfully verified.
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Rule 2320
Rule 2611
Rule 4267
Rule 6408
Rule 6724
Rubi steps
\begin {align*} \int x \tanh ^{-1}(\cosh (x)) \, dx &=\frac {1}{2} x^2 \tanh ^{-1}(\cosh (x))+\frac {1}{2} \int x^2 \text {csch}(x) \, dx\\ &=-x^2 \tanh ^{-1}\left (e^x\right )+\frac {1}{2} x^2 \tanh ^{-1}(\cosh (x))-\int x \log \left (1-e^x\right ) \, dx+\int x \log \left (1+e^x\right ) \, dx\\ &=-x^2 \tanh ^{-1}\left (e^x\right )+\frac {1}{2} x^2 \tanh ^{-1}(\cosh (x))-x \text {Li}_2\left (-e^x\right )+x \text {Li}_2\left (e^x\right )+\int \text {Li}_2\left (-e^x\right ) \, dx-\int \text {Li}_2\left (e^x\right ) \, dx\\ &=-x^2 \tanh ^{-1}\left (e^x\right )+\frac {1}{2} x^2 \tanh ^{-1}(\cosh (x))-x \text {Li}_2\left (-e^x\right )+x \text {Li}_2\left (e^x\right )+\text {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^x\right )-\text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^x\right )\\ &=-x^2 \tanh ^{-1}\left (e^x\right )+\frac {1}{2} x^2 \tanh ^{-1}(\cosh (x))-x \text {Li}_2\left (-e^x\right )+x \text {Li}_2\left (e^x\right )+\text {Li}_3\left (-e^x\right )-\text {Li}_3\left (e^x\right )\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 81, normalized size = 1.59 \begin {gather*} \frac {1}{2} \left (x^2 \tanh ^{-1}(\cosh (x))+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 [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.17, size = 479, normalized size = 9.39
method | result | size |
risch | \(\frac {i \pi \mathrm {csgn}\left (i \left ({\mathrm e}^{x}-1\right )\right )^{2} \mathrm {csgn}\left (i \left ({\mathrm e}^{x}-1\right )^{2}\right ) x^{2}}{8}-\frac {i \pi \mathrm {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{x}+1\right )^{2}\right )^{3} x^{2}}{8}-\frac {i \pi \,\mathrm {csgn}\left (i \left ({\mathrm e}^{x}-1\right )\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{x}-1\right )^{2}\right )^{2} x^{2}}{4}+\frac {i \pi \,\mathrm {csgn}\left (i \left ({\mathrm e}^{x}-1\right )^{2}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{x}-1\right )^{2}\right ) x^{2}}{8}+\frac {i \pi \,\mathrm {csgn}\left (i \left ({\mathrm e}^{x}+1\right )\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{x}+1\right )^{2}\right )^{2} x^{2}}{4}+x \polylog \left (2, {\mathrm e}^{x}\right )-x \polylog \left (2, -{\mathrm e}^{x}\right )+\polylog \left (3, -{\mathrm e}^{x}\right )-\polylog \left (3, {\mathrm e}^{x}\right )+\frac {x^{2} \ln \left (1-{\mathrm e}^{x}\right )}{2}-\frac {x^{2} \ln \left ({\mathrm e}^{x}-1\right )}{2}+\frac {i \pi \mathrm {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{x}-1\right )^{2}\right )^{2} x^{2}}{4}-\frac {i \pi \,\mathrm {csgn}\left (i \left ({\mathrm e}^{x}+1\right )^{2}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{x}+1\right )^{2}\right ) x^{2}}{8}-\frac {i \pi \mathrm {csgn}\left (i \left ({\mathrm e}^{x}+1\right )\right )^{2} \mathrm {csgn}\left (i \left ({\mathrm e}^{x}+1\right )^{2}\right ) x^{2}}{8}-\frac {i \pi \,\mathrm {csgn}\left (i {\mathrm e}^{-x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{x}-1\right )^{2}\right )^{2} x^{2}}{8}-\frac {i \pi \mathrm {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{x}-1\right )^{2}\right )^{3} x^{2}}{8}-\frac {i \pi \,x^{2}}{4}-\frac {i \pi \mathrm {csgn}\left (i \left ({\mathrm e}^{x}+1\right )^{2}\right )^{3} x^{2}}{8}+\frac {i \pi \,\mathrm {csgn}\left (i \left ({\mathrm e}^{x}+1\right )^{2}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{x}+1\right )^{2}\right )^{2} x^{2}}{8}-\frac {i \pi \,\mathrm {csgn}\left (i \left ({\mathrm e}^{x}-1\right )^{2}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{x}-1\right )^{2}\right )^{2} x^{2}}{8}+\frac {i \pi \mathrm {csgn}\left (i \left ({\mathrm e}^{x}-1\right )^{2}\right )^{3} x^{2}}{8}+\frac {i \pi \,\mathrm {csgn}\left (i {\mathrm e}^{-x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{x}+1\right )^{2}\right )^{2} x^{2}}{8}\) | \(479\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 56, normalized size = 1.10 \begin {gather*} \frac {1}{2} \, x^{2} \operatorname {artanh}\left (\cosh \left (x\right )\right ) - \frac {1}{2} \, x^{2} \log \left (e^{x} + 1\right ) + \frac {1}{2} \, x^{2} \log \left (-e^{x} + 1\right ) - x {\rm Li}_2\left (-e^{x}\right ) + x {\rm Li}_2\left (e^{x}\right ) + {\rm Li}_{3}(-e^{x}) - {\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. 88 vs.
\(2 (42) = 84\).
time = 0.42, size = 88, normalized size = 1.73 \begin {gather*} \frac {1}{4} \, x^{2} \log \left (-\frac {\cosh \left (x\right ) + 1}{\cosh \left (x\right ) - 1}\right ) - \frac {1}{2} \, x^{2} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) + \frac {1}{2} \, x^{2} \log \left (-\cosh \left (x\right ) - \sinh \left (x\right ) + 1\right ) + x {\rm Li}_2\left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) - x {\rm Li}_2\left (-\cosh \left (x\right ) - \sinh \left (x\right )\right ) - {\rm polylog}\left (3, \cosh \left (x\right ) + \sinh \left (x\right )\right ) + {\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 {atanh}{\left (\cosh {\left (x \right )} \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 {atanh}\left (\mathrm {cosh}\left (x\right )\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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