Integrand size = 5, antiderivative size = 51 \[ \int x \coth ^{-1}(\cosh (x)) \, dx=\frac {1}{2} x^2 \coth ^{-1}(\cosh (x))-x^2 \text {arctanh}\left (e^x\right )-x \operatorname {PolyLog}\left (2,-e^x\right )+x \operatorname {PolyLog}\left (2,e^x\right )+\operatorname {PolyLog}\left (3,-e^x\right )-\operatorname {PolyLog}\left (3,e^x\right ) \]
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Time = 0.05 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {6409, 4267, 2611, 2320, 6724} \[ \int x \coth ^{-1}(\cosh (x)) \, dx=-x^2 \text {arctanh}\left (e^x\right )-x \operatorname {PolyLog}\left (2,-e^x\right )+x \operatorname {PolyLog}\left (2,e^x\right )+\operatorname {PolyLog}\left (3,-e^x\right )-\operatorname {PolyLog}\left (3,e^x\right )+\frac {1}{2} x^2 \coth ^{-1}(\cosh (x)) \]
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Rule 2320
Rule 2611
Rule 4267
Rule 6409
Rule 6724
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} x^2 \coth ^{-1}(\cosh (x))+\frac {1}{2} \int x^2 \text {csch}(x) \, dx \\ & = \frac {1}{2} x^2 \coth ^{-1}(\cosh (x))-x^2 \text {arctanh}\left (e^x\right )-\int x \log \left (1-e^x\right ) \, dx+\int x \log \left (1+e^x\right ) \, dx \\ & = \frac {1}{2} x^2 \coth ^{-1}(\cosh (x))-x^2 \text {arctanh}\left (e^x\right )-x \operatorname {PolyLog}\left (2,-e^x\right )+x \operatorname {PolyLog}\left (2,e^x\right )+\int \operatorname {PolyLog}\left (2,-e^x\right ) \, dx-\int \operatorname {PolyLog}\left (2,e^x\right ) \, dx \\ & = \frac {1}{2} x^2 \coth ^{-1}(\cosh (x))-x^2 \text {arctanh}\left (e^x\right )-x \operatorname {PolyLog}\left (2,-e^x\right )+x \operatorname {PolyLog}\left (2,e^x\right )+\text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-x)}{x} \, dx,x,e^x\right )-\text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^x\right ) \\ & = \frac {1}{2} x^2 \coth ^{-1}(\cosh (x))-x^2 \text {arctanh}\left (e^x\right )-x \operatorname {PolyLog}\left (2,-e^x\right )+x \operatorname {PolyLog}\left (2,e^x\right )+\operatorname {PolyLog}\left (3,-e^x\right )-\operatorname {PolyLog}\left (3,e^x\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.37 \[ \int x \coth ^{-1}(\cosh (x)) \, dx=\frac {1}{2} x^2 \coth ^{-1}(\cosh (x))+\frac {1}{2} x^2 \log \left (1-e^x\right )-\frac {1}{2} x^2 \log \left (1+e^x\right )-x \operatorname {PolyLog}\left (2,-e^x\right )+x \operatorname {PolyLog}\left (2,e^x\right )+\operatorname {PolyLog}\left (3,-e^x\right )-\operatorname {PolyLog}\left (3,e^x\right ) \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.32 (sec) , antiderivative size = 379, normalized size of antiderivative = 7.43
method | result | size |
risch | \(-\frac {x^{2} \ln \left ({\mathrm e}^{x}-1\right )}{2}-x \operatorname {polylog}\left (2, -{\mathrm e}^{x}\right )+\operatorname {polylog}\left (3, -{\mathrm e}^{x}\right )-\frac {i \pi \left ({\operatorname {csgn}\left (i \left (1+{\mathrm e}^{x}\right )\right )}^{2} \operatorname {csgn}\left (i \left (1+{\mathrm e}^{x}\right )^{2}\right )-2 \,\operatorname {csgn}\left (i \left (1+{\mathrm e}^{x}\right )\right ) {\operatorname {csgn}\left (i \left (1+{\mathrm e}^{x}\right )^{2}\right )}^{2}+{\operatorname {csgn}\left (i \left (1+{\mathrm e}^{x}\right )^{2}\right )}^{3}+\operatorname {csgn}\left (i \left (1+{\mathrm e}^{x}\right )^{2}\right ) \operatorname {csgn}\left (i {\mathrm e}^{-x}\right ) \operatorname {csgn}\left (i {\mathrm e}^{-x} \left (1+{\mathrm e}^{x}\right )^{2}\right )-\operatorname {csgn}\left (i \left (1+{\mathrm e}^{x}\right )^{2}\right ) \operatorname {csgn}\left (i {\mathrm e}^{-x} \left (1+{\mathrm e}^{x}\right )^{2}\right )^{2}-{\operatorname {csgn}\left (i \left ({\mathrm e}^{x}-1\right )\right )}^{2} \operatorname {csgn}\left (i \left ({\mathrm e}^{x}-1\right )^{2}\right )+2 \,\operatorname {csgn}\left (i \left ({\mathrm e}^{x}-1\right )\right ) {\operatorname {csgn}\left (i \left ({\mathrm e}^{x}-1\right )^{2}\right )}^{2}-{\operatorname {csgn}\left (i \left ({\mathrm e}^{x}-1\right )^{2}\right )}^{3}-\operatorname {csgn}\left (i \left ({\mathrm e}^{x}-1\right )^{2}\right ) \operatorname {csgn}\left (i {\mathrm e}^{-x}\right ) \operatorname {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{x}-1\right )^{2}\right )+\operatorname {csgn}\left (i \left ({\mathrm e}^{x}-1\right )^{2}\right ) \operatorname {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{x}-1\right )^{2}\right )^{2}-\operatorname {csgn}\left (i {\mathrm e}^{-x}\right ) \operatorname {csgn}\left (i {\mathrm e}^{-x} \left (1+{\mathrm e}^{x}\right )^{2}\right )^{2}+\operatorname {csgn}\left (i {\mathrm e}^{-x}\right ) \operatorname {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{x}-1\right )^{2}\right )^{2}+\operatorname {csgn}\left (i {\mathrm e}^{-x} \left (1+{\mathrm e}^{x}\right )^{2}\right )^{3}-\operatorname {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{x}-1\right )^{2}\right )^{3}\right ) x^{2}}{8}+\frac {x^{2} \ln \left (1-{\mathrm e}^{x}\right )}{2}+x \operatorname {polylog}\left (2, {\mathrm e}^{x}\right )-\operatorname {polylog}\left (3, {\mathrm e}^{x}\right )\) | \(379\) |
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Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (42) = 84\).
Time = 0.25 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.71 \[ \int x \coth ^{-1}(\cosh (x)) \, dx=\frac {1}{4} \, x^{2} \log \left (\frac {\cosh \left (x\right ) + 1}{\cosh \left (x\right ) - 1}\right ) - \frac {1}{2} \, x^{2} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) + \frac {1}{2} \, x^{2} \log \left (-\cosh \left (x\right ) - \sinh \left (x\right ) + 1\right ) + x {\rm Li}_2\left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) - x {\rm Li}_2\left (-\cosh \left (x\right ) - \sinh \left (x\right )\right ) - {\rm polylog}\left (3, \cosh \left (x\right ) + \sinh \left (x\right )\right ) + {\rm polylog}\left (3, -\cosh \left (x\right ) - \sinh \left (x\right )\right ) \]
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\[ \int x \coth ^{-1}(\cosh (x)) \, dx=\int x \operatorname {acoth}{\left (\cosh {\left (x \right )} \right )}\, dx \]
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none
Time = 0.25 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.10 \[ \int x \coth ^{-1}(\cosh (x)) \, dx=\frac {1}{2} \, x^{2} \operatorname {arcoth}\left (\cosh \left (x\right )\right ) - \frac {1}{2} \, x^{2} \log \left (e^{x} + 1\right ) + \frac {1}{2} \, x^{2} \log \left (-e^{x} + 1\right ) - x {\rm Li}_2\left (-e^{x}\right ) + x {\rm Li}_2\left (e^{x}\right ) + {\rm Li}_{3}(-e^{x}) - {\rm Li}_{3}(e^{x}) \]
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\[ \int x \coth ^{-1}(\cosh (x)) \, dx=\int { x \operatorname {arcoth}\left (\cosh \left (x\right )\right ) \,d x } \]
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Timed out. \[ \int x \coth ^{-1}(\cosh (x)) \, dx=\int x\,\mathrm {acoth}\left (\mathrm {cosh}\left (x\right )\right ) \,d x \]
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