Optimal. Leaf size=38 \[ -\frac {x^2}{2}+x \log \left (1+e^{2 x}\right )+x \log (a \text {sech}(x))+\frac {1}{2} \text {Li}_2\left (-e^{2 x}\right ) \]
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Rubi [A]
time = 0.04, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {2628, 3799,
2221, 2317, 2438} \begin {gather*} \frac {1}{2} \text {PolyLog}\left (2,-e^{2 x}\right )+x \log (a \text {sech}(x))-\frac {x^2}{2}+x \log \left (e^{2 x}+1\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 2221
Rule 2317
Rule 2438
Rule 2628
Rule 3799
Rubi steps
\begin {align*} \int \log (a \text {sech}(x)) \, dx &=x \log (a \text {sech}(x))+\int x \tanh (x) \, dx\\ &=-\frac {x^2}{2}+x \log (a \text {sech}(x))+2 \int \frac {e^{2 x} x}{1+e^{2 x}} \, dx\\ &=-\frac {x^2}{2}+x \log \left (1+e^{2 x}\right )+x \log (a \text {sech}(x))-\int \log \left (1+e^{2 x}\right ) \, dx\\ &=-\frac {x^2}{2}+x \log \left (1+e^{2 x}\right )+x \log (a \text {sech}(x))-\frac {1}{2} \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 x}\right )\\ &=-\frac {x^2}{2}+x \log \left (1+e^{2 x}\right )+x \log (a \text {sech}(x))+\frac {1}{2} \text {Li}_2\left (-e^{2 x}\right )\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 37, normalized size = 0.97 \begin {gather*} \frac {1}{2} \left (x \left (x+2 \log \left (1+e^{-2 x}\right )+2 \log (a \text {sech}(x))\right )-\text {Li}_2\left (-e^{-2 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 = 0.10, size = 314, normalized size = 8.26
method | result | size |
risch | \(x \ln \left ({\mathrm e}^{x}\right )+\frac {i x \pi \,\mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{1+{\mathrm e}^{2 x}}\right ) \mathrm {csgn}\left (\frac {i a \,{\mathrm e}^{x}}{1+{\mathrm e}^{2 x}}\right )^{2}}{2}-\frac {i x \pi \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{1+{\mathrm e}^{2 x}}\right )^{3}}{2}-\frac {i x \pi \mathrm {csgn}\left (\frac {i a \,{\mathrm e}^{x}}{1+{\mathrm e}^{2 x}}\right )^{3}}{2}-\frac {i x \pi \,\mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (\frac {i}{1+{\mathrm e}^{2 x}}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{1+{\mathrm e}^{2 x}}\right )}{2}+\frac {i x \pi \,\mathrm {csgn}\left (i a \right ) \mathrm {csgn}\left (\frac {i a \,{\mathrm e}^{x}}{1+{\mathrm e}^{2 x}}\right )^{2}}{2}-\frac {i x \pi \,\mathrm {csgn}\left (i a \right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{1+{\mathrm e}^{2 x}}\right ) \mathrm {csgn}\left (\frac {i a \,{\mathrm e}^{x}}{1+{\mathrm e}^{2 x}}\right )}{2}+x \ln \left (2\right )+x \ln \left (a \right )-\frac {x^{2}}{2}+\frac {i x \pi \,\mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{1+{\mathrm e}^{2 x}}\right )^{2}}{2}+\frac {i x \pi \,\mathrm {csgn}\left (\frac {i}{1+{\mathrm e}^{2 x}}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{1+{\mathrm e}^{2 x}}\right )^{2}}{2}-\ln \left ({\mathrm e}^{x}\right ) \ln \left (1+{\mathrm e}^{2 x}\right )+\ln \left ({\mathrm e}^{x}\right ) \ln \left (1+i {\mathrm e}^{x}\right )+\ln \left ({\mathrm e}^{x}\right ) \ln \left (1-i {\mathrm e}^{x}\right )+\dilog \left (1+i {\mathrm e}^{x}\right )+\dilog \left (1-i {\mathrm e}^{x}\right )\) | \(314\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.54, size = 31, normalized size = 0.82 \begin {gather*} -\frac {1}{2} \, x^{2} + x \log \left (a \operatorname {sech}\left (x\right )\right ) + x \log \left (e^{\left (2 \, x\right )} + 1\right ) + \frac {1}{2} \, {\rm Li}_2\left (-e^{\left (2 \, x\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains complex when optimal does not.
time = 0.40, size = 84, normalized size = 2.21 \begin {gather*} -\frac {1}{2} \, x^{2} + x \log \left (\frac {2 \, {\left (a \cosh \left (x\right ) + a \sinh \left (x\right )\right )}}{\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 1}\right ) + x \log \left (i \, \cosh \left (x\right ) + i \, \sinh \left (x\right ) + 1\right ) + x \log \left (-i \, \cosh \left (x\right ) - i \, \sinh \left (x\right ) + 1\right ) + {\rm Li}_2\left (i \, \cosh \left (x\right ) + i \, \sinh \left (x\right )\right ) + {\rm Li}_2\left (-i \, \cosh \left (x\right ) - i \, \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 \log {\left (a \operatorname {sech}{\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 = -0.03 \begin {gather*} -\int \ln \left (\mathrm {cosh}\left (x\right )\right )-\ln \left (a\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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