Optimal. Leaf size=38 \[ x \log (a \text {sech}(x))+\frac {1}{2} \text {Li}_2\left (-e^{2 x}\right )-\frac {x^2}{2}+x \log \left (e^{2 x}+1\right ) \]
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Rubi [A] time = 0.05, 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 = {2548, 3718, 2190, 2279, 2391} \[ \frac {1}{2} \text {PolyLog}\left (2,-e^{2 x}\right )+x \log (a \text {sech}(x))-\frac {x^2}{2}+x \log \left (e^{2 x}+1\right ) \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2279
Rule 2391
Rule 2548
Rule 3718
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} \operatorname {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 = 0.02, size = 37, normalized size = 0.97 \[ \frac {1}{2} \left (x \left (2 \log (a \text {sech}(x))+x+2 \log \left (e^{-2 x}+1\right )\right )-\text {Li}_2\left (-e^{-2 x}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [C] time = 0.50, size = 84, normalized size = 2.21 \[ -\frac {1}{2} \, x^{2} + x \log \left (\frac {2 \, {\left (a \cosh \relax (x) + a \sinh \relax (x)\right )}}{\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} + 1}\right ) + x \log \left (i \, \cosh \relax (x) + i \, \sinh \relax (x) + 1\right ) + x \log \left (-i \, \cosh \relax (x) - i \, \sinh \relax (x) + 1\right ) + {\rm Li}_2\left (i \, \cosh \relax (x) + i \, \sinh \relax (x)\right ) + {\rm Li}_2\left (-i \, \cosh \relax (x) - i \, \sinh \relax (x)\right ) \]
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 \[ \int \log \left (a \operatorname {sech}\relax (x)\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.15, size = 314, normalized size = 8.26 \[ -\frac {i \pi x \,\mathrm {csgn}\left (i a \right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{{\mathrm e}^{2 x}+1}\right ) \mathrm {csgn}\left (\frac {i a \,{\mathrm e}^{x}}{{\mathrm e}^{2 x}+1}\right )}{2}+\frac {i \pi x \,\mathrm {csgn}\left (i a \right ) \mathrm {csgn}\left (\frac {i a \,{\mathrm e}^{x}}{{\mathrm e}^{2 x}+1}\right )^{2}}{2}-\frac {i \pi x \,\mathrm {csgn}\left (\frac {i}{{\mathrm e}^{2 x}+1}\right ) \mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{{\mathrm e}^{2 x}+1}\right )}{2}+\frac {i \pi x \,\mathrm {csgn}\left (\frac {i}{{\mathrm e}^{2 x}+1}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{{\mathrm e}^{2 x}+1}\right )^{2}}{2}+\frac {i \pi x \,\mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{{\mathrm e}^{2 x}+1}\right )^{2}}{2}-\frac {i \pi x \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{{\mathrm e}^{2 x}+1}\right )^{3}}{2}+\frac {i \pi x \,\mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{{\mathrm e}^{2 x}+1}\right ) \mathrm {csgn}\left (\frac {i a \,{\mathrm e}^{x}}{{\mathrm e}^{2 x}+1}\right )^{2}}{2}-\frac {i \pi x \mathrm {csgn}\left (\frac {i a \,{\mathrm e}^{x}}{{\mathrm e}^{2 x}+1}\right )^{3}}{2}-\frac {x^{2}}{2}+x \ln \relax (a )+x \ln \left ({\mathrm e}^{x}\right )+\ln \left (-i {\mathrm e}^{x}+1\right ) \ln \left ({\mathrm e}^{x}\right )+\ln \left (i {\mathrm e}^{x}+1\right ) \ln \left ({\mathrm e}^{x}\right )-\ln \left ({\mathrm e}^{2 x}+1\right ) \ln \left ({\mathrm e}^{x}\right )+\ln \relax (2) x +\dilog \left (-i {\mathrm e}^{x}+1\right )+\dilog \left (i {\mathrm e}^{x}+1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.15, size = 31, normalized size = 0.82 \[ -\frac {1}{2} \, x^{2} + x \log \left (a \operatorname {sech}\relax (x)\right ) + x \log \left (e^{\left (2 \, x\right )} + 1\right ) + \frac {1}{2} \, {\rm Li}_2\left (-e^{\left (2 \, x\right )}\right ) \]
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 \[ -\int \ln \left (\mathrm {cosh}\relax (x)\right )-\ln \relax (a) \,d x \]
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 \[ \int \log {\left (a \operatorname {sech}{\relax (x )} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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