Optimal. Leaf size=9 \[ \frac {1}{2} \log ^2(\cosh (x)) \]
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Rubi [A]
time = 0.01, antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 3, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3556, 4426,
2338} \begin {gather*} \frac {1}{2} \log ^2(\cosh (x)) \end {gather*}
Antiderivative was successfully verified.
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Rule 2338
Rule 3556
Rule 4426
Rubi steps
\begin {align*} \int \log (\cosh (x)) \tanh (x) \, dx &=\text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,\cosh (x)\right )\\ &=\frac {1}{2} \log ^2(\cosh (x))\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 9, normalized size = 1.00 \begin {gather*} \frac {1}{2} \log ^2(\cosh (x)) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 8, normalized size = 0.89
method | result | size |
derivativedivides | \(\frac {\ln \left (\cosh \left (x \right )\right )^{2}}{2}\) | \(8\) |
default | \(\frac {\ln \left (\cosh \left (x \right )\right )^{2}}{2}\) | \(8\) |
risch | \(\left (x -\ln \left (1+{\mathrm e}^{2 x}\right )\right ) \ln \left ({\mathrm e}^{x}\right )+\frac {\ln \left (1+{\mathrm e}^{2 x}\right )^{2}}{2}-\ln \left (1+{\mathrm e}^{2 x}\right ) \ln \left (2\right )+\frac {i \ln \left (1+{\mathrm e}^{2 x}\right ) \pi \,\mathrm {csgn}\left (i \left (1+{\mathrm e}^{2 x}\right )\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x} \left (1+{\mathrm e}^{2 x}\right )\right )^{2}}{2}-\frac {i \pi \,\mathrm {csgn}\left (i {\mathrm e}^{-x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x} \left (1+{\mathrm e}^{2 x}\right )\right )^{2} x}{2}-\frac {i \ln \left (1+{\mathrm e}^{2 x}\right ) \pi \mathrm {csgn}\left (i {\mathrm e}^{-x} \left (1+{\mathrm e}^{2 x}\right )\right )^{3}}{2}+x \ln \left (2\right )-\frac {x^{2}}{2}+\frac {i \pi \mathrm {csgn}\left (i {\mathrm e}^{-x} \left (1+{\mathrm e}^{2 x}\right )\right )^{3} x}{2}+\frac {i \pi \,\mathrm {csgn}\left (i {\mathrm e}^{-x}\right ) \mathrm {csgn}\left (i \left (1+{\mathrm e}^{2 x}\right )\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x} \left (1+{\mathrm e}^{2 x}\right )\right ) x}{2}+\frac {i \ln \left (1+{\mathrm e}^{2 x}\right ) \pi \,\mathrm {csgn}\left (i {\mathrm e}^{-x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x} \left (1+{\mathrm e}^{2 x}\right )\right )^{2}}{2}-\frac {i \pi \,\mathrm {csgn}\left (i \left (1+{\mathrm e}^{2 x}\right )\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x} \left (1+{\mathrm e}^{2 x}\right )\right )^{2} x}{2}-\frac {i \ln \left (1+{\mathrm e}^{2 x}\right ) \pi \,\mathrm {csgn}\left (i {\mathrm e}^{-x}\right ) \mathrm {csgn}\left (i \left (1+{\mathrm e}^{2 x}\right )\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x} \left (1+{\mathrm e}^{2 x}\right )\right )}{2}\) | \(308\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.78, size = 7, normalized size = 0.78 \begin {gather*} \frac {1}{2} \, \log \left (\cosh \left (x\right )\right )^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.66, size = 7, normalized size = 0.78 \begin {gather*} \frac {1}{2} \, \log \left (\cosh \left (x\right )\right )^{2} \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 \log {\left (\cosh {\left (x \right )} \right )} \tanh {\left (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 [B]
time = 0.44, size = 16, normalized size = 1.78 \begin {gather*} \frac {{\ln \left (\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}\right )}^2}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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