Integrand size = 5, antiderivative size = 39 \[ \int \log (a \cosh (x)) \, dx=\frac {x^2}{2}-x \log \left (1+e^{2 x}\right )+x \log (a \cosh (x))-\frac {1}{2} \operatorname {PolyLog}\left (2,-e^{2 x}\right ) \]
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Time = 0.04 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {2628, 3799, 2221, 2317, 2438} \[ \int \log (a \cosh (x)) \, dx=x \log (a \cosh (x))-\frac {1}{2} \operatorname {PolyLog}\left (2,-e^{2 x}\right )+\frac {x^2}{2}-x \log \left (e^{2 x}+1\right ) \]
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Rule 2221
Rule 2317
Rule 2438
Rule 2628
Rule 3799
Rubi steps \begin{align*} \text {integral}& = x \log (a \cosh (x))-\int x \tanh (x) \, dx \\ & = \frac {x^2}{2}+x \log (a \cosh (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 \cosh (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 \cosh (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 \cosh (x))-\frac {1}{2} \text {Li}_2\left (-e^{2 x}\right ) \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.95 \[ \int \log (a \cosh (x)) \, dx=x \log (a \cosh (x))+\frac {1}{2} \left (-x \left (x+2 \log \left (1+e^{-2 x}\right )\right )+\operatorname {PolyLog}\left (2,-e^{-2 x}\right )\right ) \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 2.76 (sec) , antiderivative size = 321, normalized size of antiderivative = 8.23
method | result | size |
risch | \(-x \ln \left ({\mathrm e}^{x}\right )+\frac {i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{-x} \left (1+{\mathrm e}^{2 x}\right )\right ) {\operatorname {csgn}\left (i a \left (1+{\mathrm e}^{2 x}\right ) {\mathrm e}^{-x}\right )}^{2} x}{2}+\frac {i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{-x}\right ) \operatorname {csgn}\left (i {\mathrm e}^{-x} \left (1+{\mathrm e}^{2 x}\right )\right )^{2} x}{2}-\frac {i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{-x}\right ) \operatorname {csgn}\left (i \left (1+{\mathrm e}^{2 x}\right )\right ) \operatorname {csgn}\left (i {\mathrm e}^{-x} \left (1+{\mathrm e}^{2 x}\right )\right ) x}{2}+\frac {i \pi {\operatorname {csgn}\left (i a \left (1+{\mathrm e}^{2 x}\right ) {\mathrm e}^{-x}\right )}^{2} \operatorname {csgn}\left (i a \right ) x}{2}+\frac {i \pi \,\operatorname {csgn}\left (i \left (1+{\mathrm e}^{2 x}\right )\right ) \operatorname {csgn}\left (i {\mathrm e}^{-x} \left (1+{\mathrm e}^{2 x}\right )\right )^{2} x}{2}-\frac {i \pi {\operatorname {csgn}\left (i a \left (1+{\mathrm e}^{2 x}\right ) {\mathrm e}^{-x}\right )}^{3} x}{2}-x \ln \left (2\right )+\ln \left (a \right ) x +\frac {x^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{-x} \left (1+{\mathrm e}^{2 x}\right )\right ) \operatorname {csgn}\left (i a \left (1+{\mathrm e}^{2 x}\right ) {\mathrm e}^{-x}\right ) \operatorname {csgn}\left (i a \right ) x}{2}-\frac {i \pi \operatorname {csgn}\left (i {\mathrm e}^{-x} \left (1+{\mathrm e}^{2 x}\right )\right )^{3} x}{2}+\ln \left ({\mathrm e}^{x}\right ) \ln \left (1+{\mathrm e}^{2 x}\right )-\ln \left ({\mathrm e}^{x}\right ) \ln \left (1+i {\mathrm e}^{x}\right )-\ln \left ({\mathrm e}^{x}\right ) \ln \left (1-i {\mathrm e}^{x}\right )-\operatorname {dilog}\left (1+i {\mathrm e}^{x}\right )-\operatorname {dilog}\left (1-i {\mathrm e}^{x}\right )\) | \(321\) |
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Result contains complex when optimal does not.
Time = 0.32 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.67 \[ \int \log (a \cosh (x)) \, dx=\frac {1}{2} \, x^{2} + x \log \left (a \cosh \left (x\right )\right ) - x \log \left (i \, \cosh \left (x\right ) + i \, \sinh \left (x\right ) + 1\right ) - x \log \left (-i \, \cosh \left (x\right ) - i \, \sinh \left (x\right ) + 1\right ) - {\rm Li}_2\left (i \, \cosh \left (x\right ) + i \, \sinh \left (x\right )\right ) - {\rm Li}_2\left (-i \, \cosh \left (x\right ) - i \, \sinh \left (x\right )\right ) \]
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\[ \int \log (a \cosh (x)) \, dx=\int \log {\left (a \cosh {\left (x \right )} \right )}\, dx \]
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none
Time = 0.31 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.82 \[ \int \log (a \cosh (x)) \, dx=\frac {1}{2} \, x^{2} + x \log \left (a \cosh \left (x\right )\right ) - x \log \left (e^{\left (2 \, x\right )} + 1\right ) - \frac {1}{2} \, {\rm Li}_2\left (-e^{\left (2 \, x\right )}\right ) \]
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\[ \int \log (a \cosh (x)) \, dx=\int { \log \left (a \cosh \left (x\right )\right ) \,d x } \]
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Timed out. \[ \int \log (a \cosh (x)) \, dx=\int \ln \left (a\,\mathrm {cosh}\left (x\right )\right ) \,d x \]
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