Integrand size = 4, antiderivative size = 25 \[ \int \coth ^{-1}\left (e^x\right ) \, dx=\frac {\operatorname {PolyLog}\left (2,-e^{-x}\right )}{2}-\frac {\operatorname {PolyLog}\left (2,e^{-x}\right )}{2} \]
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Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2320, 6032} \[ \int \coth ^{-1}\left (e^x\right ) \, dx=\frac {\operatorname {PolyLog}\left (2,-e^{-x}\right )}{2}-\frac {\operatorname {PolyLog}\left (2,e^{-x}\right )}{2} \]
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Rule 2320
Rule 6032
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {\coth ^{-1}(x)}{x} \, dx,x,e^x\right ) \\ & = \frac {\operatorname {PolyLog}\left (2,-e^{-x}\right )}{2}-\frac {\operatorname {PolyLog}\left (2,e^{-x}\right )}{2} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(51\) vs. \(2(25)=50\).
Time = 0.00 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.04 \[ \int \coth ^{-1}\left (e^x\right ) \, dx=x \coth ^{-1}\left (e^x\right )+\frac {1}{2} x \log \left (1-e^x\right )-\frac {1}{2} x \log \left (1+e^x\right )-\frac {\operatorname {PolyLog}\left (2,-e^x\right )}{2}+\frac {\operatorname {PolyLog}\left (2,e^x\right )}{2} \]
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Time = 0.10 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88
method | result | size |
risch | \(-\frac {x \ln \left ({\mathrm e}^{x}-1\right )}{2}-\frac {\operatorname {dilog}\left ({\mathrm e}^{x}\right )}{2}-\frac {\operatorname {dilog}\left (1+{\mathrm e}^{x}\right )}{2}\) | \(22\) |
derivativedivides | \(\ln \left ({\mathrm e}^{x}\right ) \operatorname {arccoth}\left ({\mathrm e}^{x}\right )-\frac {\operatorname {dilog}\left ({\mathrm e}^{x}\right )}{2}-\frac {\operatorname {dilog}\left (1+{\mathrm e}^{x}\right )}{2}-\frac {\ln \left ({\mathrm e}^{x}\right ) \ln \left (1+{\mathrm e}^{x}\right )}{2}\) | \(31\) |
default | \(\ln \left ({\mathrm e}^{x}\right ) \operatorname {arccoth}\left ({\mathrm e}^{x}\right )-\frac {\operatorname {dilog}\left ({\mathrm e}^{x}\right )}{2}-\frac {\operatorname {dilog}\left (1+{\mathrm e}^{x}\right )}{2}-\frac {\ln \left ({\mathrm e}^{x}\right ) \ln \left (1+{\mathrm e}^{x}\right )}{2}\) | \(31\) |
parts | \(x \,\operatorname {arccoth}\left ({\mathrm e}^{x}\right )+\frac {x \ln \left (1-{\mathrm e}^{x}\right )}{2}+\frac {\operatorname {polylog}\left (2, {\mathrm e}^{x}\right )}{2}-\frac {x \ln \left (1+{\mathrm e}^{x}\right )}{2}-\frac {\operatorname {polylog}\left (2, -{\mathrm e}^{x}\right )}{2}\) | \(39\) |
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Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (17) = 34\).
Time = 0.26 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.56 \[ \int \coth ^{-1}\left (e^x\right ) \, dx=\frac {1}{2} \, x \log \left (\frac {\cosh \left (x\right ) + \sinh \left (x\right ) + 1}{\cosh \left (x\right ) + \sinh \left (x\right ) - 1}\right ) - \frac {1}{2} \, x \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) + \frac {1}{2} \, x \log \left (-\cosh \left (x\right ) - \sinh \left (x\right ) + 1\right ) + \frac {1}{2} \, {\rm Li}_2\left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) - \frac {1}{2} \, {\rm Li}_2\left (-\cosh \left (x\right ) - \sinh \left (x\right )\right ) \]
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\[ \int \coth ^{-1}\left (e^x\right ) \, dx=\int \operatorname {acoth}{\left (e^{x} \right )}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (17) = 34\).
Time = 0.26 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.32 \[ \int \coth ^{-1}\left (e^x\right ) \, dx=-\frac {1}{2} \, x {\left (\log \left (e^{x} + 1\right ) - \log \left (e^{x} - 1\right )\right )} + x \operatorname {arcoth}\left (e^{x}\right ) + \frac {1}{2} \, \log \left (-e^{x}\right ) \log \left (e^{x} + 1\right ) - \frac {1}{2} \, x \log \left (e^{x} - 1\right ) + \frac {1}{2} \, {\rm Li}_2\left (e^{x} + 1\right ) - \frac {1}{2} \, {\rm Li}_2\left (-e^{x} + 1\right ) \]
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\[ \int \coth ^{-1}\left (e^x\right ) \, dx=\int { \operatorname {arcoth}\left (e^{x}\right ) \,d x } \]
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Timed out. \[ \int \coth ^{-1}\left (e^x\right ) \, dx=\int \mathrm {acoth}\left ({\mathrm {e}}^x\right ) \,d x \]
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