Integrand size = 5, antiderivative size = 38 \[ \int \log (a \text {csch}(x)) \, dx=-\frac {x^2}{2}+x \log \left (1-e^{2 x}\right )+x \log (a \text {csch}(x))+\frac {\operatorname {PolyLog}\left (2,e^{2 x}\right )}{2} \]
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Time = 0.04 (sec) , antiderivative size = 38, 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, 3797, 2221, 2317, 2438} \[ \int \log (a \text {csch}(x)) \, dx=x \log (a \text {csch}(x))+\frac {\operatorname {PolyLog}\left (2,e^{2 x}\right )}{2}-\frac {x^2}{2}+x \log \left (1-e^{2 x}\right ) \]
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Rule 2221
Rule 2317
Rule 2438
Rule 2628
Rule 3797
Rubi steps \begin{align*} \text {integral}& = x \log (a \text {csch}(x))+\int x \coth (x) \, dx \\ & = -\frac {x^2}{2}+x \log (a \text {csch}(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 {csch}(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 {csch}(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 {csch}(x))+\frac {\text {Li}_2\left (e^{2 x}\right )}{2} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00 \[ \int \log (a \text {csch}(x)) \, dx=\frac {x^2}{2}+x \log \left (1-e^{-2 x}\right )+x \log (a \text {csch}(x))-\frac {1}{2} \operatorname {PolyLog}\left (2,e^{-2 x}\right ) \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.81 (sec) , antiderivative size = 293, normalized size of antiderivative = 7.71
method | result | size |
risch | \(x \ln \left ({\mathrm e}^{x}\right )+\frac {i \pi \,\operatorname {csgn}\left (\frac {i {\mathrm e}^{x}}{-1+{\mathrm e}^{2 x}}\right ) \operatorname {csgn}\left (\frac {i a \,{\mathrm e}^{x}}{-1+{\mathrm e}^{2 x}}\right )^{2} x}{2}+\frac {i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{x}\right ) \operatorname {csgn}\left (\frac {i {\mathrm e}^{x}}{-1+{\mathrm e}^{2 x}}\right )^{2} x}{2}-\frac {i \pi \,\operatorname {csgn}\left (\frac {i {\mathrm e}^{x}}{-1+{\mathrm e}^{2 x}}\right ) \operatorname {csgn}\left (\frac {i a \,{\mathrm e}^{x}}{-1+{\mathrm e}^{2 x}}\right ) \operatorname {csgn}\left (i a \right ) x}{2}+\frac {i \pi \operatorname {csgn}\left (\frac {i a \,{\mathrm e}^{x}}{-1+{\mathrm e}^{2 x}}\right )^{2} \operatorname {csgn}\left (i a \right ) x}{2}-\frac {i \pi \operatorname {csgn}\left (\frac {i a \,{\mathrm e}^{x}}{-1+{\mathrm e}^{2 x}}\right )^{3} x}{2}-\frac {i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{x}\right ) \operatorname {csgn}\left (\frac {i}{-1+{\mathrm e}^{2 x}}\right ) \operatorname {csgn}\left (\frac {i {\mathrm e}^{x}}{-1+{\mathrm e}^{2 x}}\right ) x}{2}+x \ln \left (2\right )+\ln \left (a \right ) x -\frac {x^{2}}{2}+\frac {i \pi \,\operatorname {csgn}\left (\frac {i}{-1+{\mathrm e}^{2 x}}\right ) \operatorname {csgn}\left (\frac {i {\mathrm e}^{x}}{-1+{\mathrm e}^{2 x}}\right )^{2} x}{2}-\frac {i \pi \operatorname {csgn}\left (\frac {i {\mathrm e}^{x}}{-1+{\mathrm e}^{2 x}}\right )^{3} x}{2}-\ln \left ({\mathrm e}^{x}\right ) \ln \left (-1+{\mathrm e}^{2 x}\right )+\operatorname {dilog}\left (1+{\mathrm e}^{x}\right )+\ln \left ({\mathrm e}^{x}\right ) \ln \left (1+{\mathrm e}^{x}\right )-\operatorname {dilog}\left ({\mathrm e}^{x}\right )\) | \(293\) |
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Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (31) = 62\).
Time = 0.35 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.00 \[ \int \log (a \text {csch}(x)) \, dx=-\frac {1}{2} \, x^{2} + x \log \left (\frac {2 \, {\left (a \cosh \left (x\right ) + a \sinh \left (x\right )\right )}}{\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 1}\right ) + x \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) + x \log \left (-\cosh \left (x\right ) - \sinh \left (x\right ) + 1\right ) + {\rm Li}_2\left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) + {\rm Li}_2\left (-\cosh \left (x\right ) - \sinh \left (x\right )\right ) \]
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\[ \int \log (a \text {csch}(x)) \, dx=\int \log {\left (a \operatorname {csch}{\left (x \right )} \right )}\, dx \]
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none
Time = 0.25 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.97 \[ \int \log (a \text {csch}(x)) \, dx=-\frac {1}{2} \, x^{2} + x \log \left (a \operatorname {csch}\left (x\right )\right ) + x \log \left (e^{x} + 1\right ) + x \log \left (-e^{x} + 1\right ) + {\rm Li}_2\left (-e^{x}\right ) + {\rm Li}_2\left (e^{x}\right ) \]
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\[ \int \log (a \text {csch}(x)) \, dx=\int { \log \left (a \operatorname {csch}\left (x\right )\right ) \,d x } \]
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Timed out. \[ \int \log (a \text {csch}(x)) \, dx=\int \ln \left (\frac {a}{\mathrm {sinh}\left (x\right )}\right ) \,d x \]
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