Integrand size = 7, antiderivative size = 37 \[ \int \log \left (a \coth ^2(x)\right ) \, dx=-4 x \text {arctanh}\left (e^{2 x}\right )+x \log \left (a \coth ^2(x)\right )-\operatorname {PolyLog}\left (2,-e^{2 x}\right )+\operatorname {PolyLog}\left (2,e^{2 x}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {2628, 12, 5569, 4267, 2317, 2438} \[ \int \log \left (a \coth ^2(x)\right ) \, dx=x \log \left (a \coth ^2(x)\right )-4 x \text {arctanh}\left (e^{2 x}\right )-\operatorname {PolyLog}\left (2,-e^{2 x}\right )+\operatorname {PolyLog}\left (2,e^{2 x}\right ) \]
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Rule 12
Rule 2317
Rule 2438
Rule 2628
Rule 4267
Rule 5569
Rubi steps \begin{align*} \text {integral}& = x \log \left (a \coth ^2(x)\right )-\int -2 x \text {csch}(x) \text {sech}(x) \, dx \\ & = x \log \left (a \coth ^2(x)\right )+2 \int x \text {csch}(x) \text {sech}(x) \, dx \\ & = x \log \left (a \coth ^2(x)\right )+4 \int x \text {csch}(2 x) \, dx \\ & = -4 x \tanh ^{-1}\left (e^{2 x}\right )+x \log \left (a \coth ^2(x)\right )-2 \int \log \left (1-e^{2 x}\right ) \, dx+2 \int \log \left (1+e^{2 x}\right ) \, dx \\ & = -4 x \tanh ^{-1}\left (e^{2 x}\right )+x \log \left (a \coth ^2(x)\right )-\text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 x}\right )+\text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 x}\right ) \\ & = -4 x \tanh ^{-1}\left (e^{2 x}\right )+x \log \left (a \coth ^2(x)\right )-\text {Li}_2\left (-e^{2 x}\right )+\text {Li}_2\left (e^{2 x}\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.27 \[ \int \log \left (a \coth ^2(x)\right ) \, dx=-\frac {1}{2} \log \left (a \coth ^2(x)\right ) \log (1-\tanh (x))+\frac {1}{2} \log \left (a \coth ^2(x)\right ) \log (1+\tanh (x))-\operatorname {PolyLog}(2,-\tanh (x))+\operatorname {PolyLog}(2,\tanh (x)) \]
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Time = 1.14 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.27
method | result | size |
derivativedivides | \(-\frac {\ln \left (\coth \left (x \right )-1\right ) \ln \left (a \left (\coth ^{2}\left (x \right )\right )\right )}{2}+\operatorname {dilog}\left (\coth \left (x \right )\right )+\ln \left (\coth \left (x \right )-1\right ) \ln \left (\coth \left (x \right )\right )+\frac {\ln \left (\coth \left (x \right )+1\right ) \ln \left (a \left (\coth ^{2}\left (x \right )\right )\right )}{2}+\operatorname {dilog}\left (\coth \left (x \right )+1\right )\) | \(47\) |
default | \(-\frac {\ln \left (\coth \left (x \right )-1\right ) \ln \left (a \left (\coth ^{2}\left (x \right )\right )\right )}{2}+\operatorname {dilog}\left (\coth \left (x \right )\right )+\ln \left (\coth \left (x \right )-1\right ) \ln \left (\coth \left (x \right )\right )+\frac {\ln \left (\coth \left (x \right )+1\right ) \ln \left (a \left (\coth ^{2}\left (x \right )\right )\right )}{2}+\operatorname {dilog}\left (\coth \left (x \right )+1\right )\) | \(47\) |
risch | \(-2 x \ln \left (-1+{\mathrm e}^{2 x}\right )-\frac {i \pi {\operatorname {csgn}\left (i \left (1+{\mathrm e}^{2 x}\right )\right )}^{2} \operatorname {csgn}\left (i \left (1+{\mathrm e}^{2 x}\right )^{2}\right ) x}{2}-\frac {i \pi {\operatorname {csgn}\left (i \left (1+{\mathrm e}^{2 x}\right )^{2}\right )}^{3} x}{2}+\frac {i \pi \,\operatorname {csgn}\left (i \left (1+{\mathrm e}^{2 x}\right )^{2}\right ) {\operatorname {csgn}\left (\frac {i \left (1+{\mathrm e}^{2 x}\right )^{2}}{\left (-1+{\mathrm e}^{2 x}\right )^{2}}\right )}^{2} x}{2}+\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (1+{\mathrm e}^{2 x}\right )^{2}}{\left (-1+{\mathrm e}^{2 x}\right )^{2}}\right ) {\operatorname {csgn}\left (\frac {i a \left (1+{\mathrm e}^{2 x}\right )^{2}}{\left (-1+{\mathrm e}^{2 x}\right )^{2}}\right )}^{2} x}{2}+\frac {i \pi {\operatorname {csgn}\left (\frac {i a \left (1+{\mathrm e}^{2 x}\right )^{2}}{\left (-1+{\mathrm e}^{2 x}\right )^{2}}\right )}^{2} \operatorname {csgn}\left (i a \right ) x}{2}+\frac {i \pi {\operatorname {csgn}\left (i \left (-1+{\mathrm e}^{2 x}\right )^{2}\right )}^{3} x}{2}+\ln \left (a \right ) x +2 \operatorname {dilog}\left (1+{\mathrm e}^{x}\right )+2 x \ln \left (1+{\mathrm e}^{x}\right )-2 \operatorname {dilog}\left ({\mathrm e}^{x}\right )+\frac {i \pi \,\operatorname {csgn}\left (\frac {i}{\left (-1+{\mathrm e}^{2 x}\right )^{2}}\right ) {\operatorname {csgn}\left (\frac {i \left (1+{\mathrm e}^{2 x}\right )^{2}}{\left (-1+{\mathrm e}^{2 x}\right )^{2}}\right )}^{2} x}{2}-\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (1+{\mathrm e}^{2 x}\right )^{2}}{\left (-1+{\mathrm e}^{2 x}\right )^{2}}\right ) \operatorname {csgn}\left (\frac {i a \left (1+{\mathrm e}^{2 x}\right )^{2}}{\left (-1+{\mathrm e}^{2 x}\right )^{2}}\right ) \operatorname {csgn}\left (i a \right ) x}{2}-\frac {i \pi {\operatorname {csgn}\left (\frac {i \left (1+{\mathrm e}^{2 x}\right )^{2}}{\left (-1+{\mathrm e}^{2 x}\right )^{2}}\right )}^{3} x}{2}-\frac {i \pi \,\operatorname {csgn}\left (i \left (1+{\mathrm e}^{2 x}\right )^{2}\right ) \operatorname {csgn}\left (\frac {i}{\left (-1+{\mathrm e}^{2 x}\right )^{2}}\right ) \operatorname {csgn}\left (\frac {i \left (1+{\mathrm e}^{2 x}\right )^{2}}{\left (-1+{\mathrm e}^{2 x}\right )^{2}}\right ) x}{2}-\frac {i \pi {\operatorname {csgn}\left (\frac {i a \left (1+{\mathrm e}^{2 x}\right )^{2}}{\left (-1+{\mathrm e}^{2 x}\right )^{2}}\right )}^{3} x}{2}+\frac {i \pi {\operatorname {csgn}\left (i \left (-1+{\mathrm e}^{2 x}\right )\right )}^{2} \operatorname {csgn}\left (i \left (-1+{\mathrm e}^{2 x}\right )^{2}\right ) x}{2}+i \pi \,\operatorname {csgn}\left (i \left (1+{\mathrm e}^{2 x}\right )\right ) {\operatorname {csgn}\left (i \left (1+{\mathrm e}^{2 x}\right )^{2}\right )}^{2} x -i \pi \,\operatorname {csgn}\left (i \left (-1+{\mathrm e}^{2 x}\right )\right ) {\operatorname {csgn}\left (i \left (-1+{\mathrm e}^{2 x}\right )^{2}\right )}^{2} x +2 \ln \left ({\mathrm e}^{x}\right ) \ln \left (1+{\mathrm e}^{2 x}\right )-2 \ln \left ({\mathrm e}^{x}\right ) \ln \left (1+i {\mathrm e}^{x}\right )-2 \ln \left ({\mathrm e}^{x}\right ) \ln \left (1-i {\mathrm e}^{x}\right )-2 \operatorname {dilog}\left (1+i {\mathrm e}^{x}\right )-2 \operatorname {dilog}\left (1-i {\mathrm e}^{x}\right )\) | \(562\) |
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Result contains complex when optimal does not.
Time = 0.33 (sec) , antiderivative size = 127, normalized size of antiderivative = 3.43 \[ \int \log \left (a \coth ^2(x)\right ) \, dx=x \log \left (\frac {a \cosh \left (x\right )^{2} + a \sinh \left (x\right )^{2} + a}{\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2} - 1}\right ) + 2 \, x \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) - 2 \, x \log \left (i \, \cosh \left (x\right ) + i \, \sinh \left (x\right ) + 1\right ) - 2 \, x \log \left (-i \, \cosh \left (x\right ) - i \, \sinh \left (x\right ) + 1\right ) + 2 \, x \log \left (-\cosh \left (x\right ) - \sinh \left (x\right ) + 1\right ) + 2 \, {\rm Li}_2\left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) - 2 \, {\rm Li}_2\left (i \, \cosh \left (x\right ) + i \, \sinh \left (x\right )\right ) - 2 \, {\rm Li}_2\left (-i \, \cosh \left (x\right ) - i \, \sinh \left (x\right )\right ) + 2 \, {\rm Li}_2\left (-\cosh \left (x\right ) - \sinh \left (x\right )\right ) \]
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\[ \int \log \left (a \coth ^2(x)\right ) \, dx=\int \log {\left (a \coth ^{2}{\left (x \right )} \right )}\, dx \]
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none
Time = 0.28 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.59 \[ \int \log \left (a \coth ^2(x)\right ) \, dx=x \log \left (a \coth \left (x\right )^{2}\right ) - 2 \, x \log \left (e^{\left (2 \, x\right )} + 1\right ) + 2 \, x \log \left (e^{x} + 1\right ) + 2 \, x \log \left (-e^{x} + 1\right ) - {\rm Li}_2\left (-e^{\left (2 \, x\right )}\right ) + 2 \, {\rm Li}_2\left (-e^{x}\right ) + 2 \, {\rm Li}_2\left (e^{x}\right ) \]
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\[ \int \log \left (a \coth ^2(x)\right ) \, dx=\int { \log \left (a \coth \left (x\right )^{2}\right ) \,d x } \]
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Timed out. \[ \int \log \left (a \coth ^2(x)\right ) \, dx=\int \ln \left (a\,{\mathrm {coth}\left (x\right )}^2\right ) \,d x \]
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