Integrand size = 7, antiderivative size = 46 \[ \int \log \left (a \coth ^n(x)\right ) \, dx=-2 n x \text {arctanh}\left (e^{2 x}\right )+x \log \left (a \coth ^n(x)\right )-\frac {1}{2} n \operatorname {PolyLog}\left (2,-e^{2 x}\right )+\frac {1}{2} n \operatorname {PolyLog}\left (2,e^{2 x}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 46, 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} \[ \int \log \left (a \coth ^n(x)\right ) \, dx=x \log \left (a \coth ^n(x)\right )-2 n x \text {arctanh}\left (e^{2 x}\right )-\frac {1}{2} n \operatorname {PolyLog}\left (2,-e^{2 x}\right )+\frac {1}{2} n \operatorname {PolyLog}\left (2,e^{2 x}\right ) \]
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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 ^n(x)\right )+\int n x \text {csch}(x) \text {sech}(x) \, dx \\ & = x \log \left (a \coth ^n(x)\right )+n \int x \text {csch}(x) \text {sech}(x) \, dx \\ & = x \log \left (a \coth ^n(x)\right )+(2 n) \int x \text {csch}(2 x) \, dx \\ & = -2 n x \tanh ^{-1}\left (e^{2 x}\right )+x \log \left (a \coth ^n(x)\right )-n \int \log \left (1-e^{2 x}\right ) \, dx+n \int \log \left (1+e^{2 x}\right ) \, dx \\ & = -2 n x \tanh ^{-1}\left (e^{2 x}\right )+x \log \left (a \coth ^n(x)\right )-\frac {1}{2} n \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 x}\right )+\frac {1}{2} n \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 x}\right ) \\ & = -2 n x \tanh ^{-1}\left (e^{2 x}\right )+x \log \left (a \coth ^n(x)\right )-\frac {1}{2} n \text {Li}_2\left (-e^{2 x}\right )+\frac {1}{2} n \text {Li}_2\left (e^{2 x}\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.20 \[ \int \log \left (a \coth ^n(x)\right ) \, dx=-\frac {1}{2} \log \left (a \coth ^n(x)\right ) \log (1-\tanh (x))+\frac {1}{2} \log \left (a \coth ^n(x)\right ) \log (1+\tanh (x))-\frac {1}{2} n \operatorname {PolyLog}(2,-\tanh (x))+\frac {1}{2} n \operatorname {PolyLog}(2,\tanh (x)) \]
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Time = 4.26 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.93
method | result | size |
default | \(x \left (\ln \left (a \left (\coth ^{n}\left (x \right )\right )\right )-n \ln \left (\coth \left (x \right )\right )\right )+n \left (\frac {\operatorname {dilog}\left (\coth \left (x \right )\right )}{2}+\frac {\operatorname {dilog}\left (\coth \left (x \right )+1\right )}{2}+\frac {\ln \left (\coth \left (x \right )\right ) \ln \left (\coth \left (x \right )+1\right )}{2}\right )\) | \(43\) |
risch | \(\text {Expression too large to display}\) | \(2220\) |
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Result contains complex when optimal does not.
Time = 0.33 (sec) , antiderivative size = 116, normalized size of antiderivative = 2.52 \[ \int \log \left (a \coth ^n(x)\right ) \, dx=n x \log \left (\frac {\cosh \left (x\right )}{\sinh \left (x\right )}\right ) + n x \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) - n x \log \left (i \, \cosh \left (x\right ) + i \, \sinh \left (x\right ) + 1\right ) - n x \log \left (-i \, \cosh \left (x\right ) - i \, \sinh \left (x\right ) + 1\right ) + n x \log \left (-\cosh \left (x\right ) - \sinh \left (x\right ) + 1\right ) + n {\rm Li}_2\left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) - n {\rm Li}_2\left (i \, \cosh \left (x\right ) + i \, \sinh \left (x\right )\right ) - n {\rm Li}_2\left (-i \, \cosh \left (x\right ) - i \, \sinh \left (x\right )\right ) + n {\rm Li}_2\left (-\cosh \left (x\right ) - \sinh \left (x\right )\right ) + x \log \left (a\right ) \]
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\[ \int \log \left (a \coth ^n(x)\right ) \, dx=\int \log {\left (a \coth ^{n}{\left (x \right )} \right )}\, dx \]
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none
Time = 0.28 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.33 \[ \int \log \left (a \coth ^n(x)\right ) \, dx=-\frac {1}{2} \, {\left (2 \, x \log \left (e^{\left (2 \, x\right )} + 1\right ) - 2 \, x \log \left (e^{x} + 1\right ) - 2 \, x \log \left (-e^{x} + 1\right ) + {\rm Li}_2\left (-e^{\left (2 \, x\right )}\right ) - 2 \, {\rm Li}_2\left (-e^{x}\right ) - 2 \, {\rm Li}_2\left (e^{x}\right )\right )} n + x \log \left (a \coth \left (x\right )^{n}\right ) \]
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\[ \int \log \left (a \coth ^n(x)\right ) \, dx=\int { \log \left (a \coth \left (x\right )^{n}\right ) \,d x } \]
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Timed out. \[ \int \log \left (a \coth ^n(x)\right ) \, dx=\int \ln \left (a\,{\mathrm {coth}\left (x\right )}^n\right ) \,d x \]
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