Integrand size = 6, antiderivative size = 58 \[ \int \coth ^{-1}(a x)^2 \, dx=\frac {\coth ^{-1}(a x)^2}{a}+x \coth ^{-1}(a x)^2-\frac {2 \coth ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{a}-\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{a} \]
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Time = 0.06 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {6022, 6132, 6056, 2449, 2352} \[ \int \coth ^{-1}(a x)^2 \, dx=-\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{a}+x \coth ^{-1}(a x)^2+\frac {\coth ^{-1}(a x)^2}{a}-\frac {2 \log \left (\frac {2}{1-a x}\right ) \coth ^{-1}(a x)}{a} \]
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Rule 2352
Rule 2449
Rule 6022
Rule 6056
Rule 6132
Rubi steps \begin{align*} \text {integral}& = x \coth ^{-1}(a x)^2-(2 a) \int \frac {x \coth ^{-1}(a x)}{1-a^2 x^2} \, dx \\ & = \frac {\coth ^{-1}(a x)^2}{a}+x \coth ^{-1}(a x)^2-2 \int \frac {\coth ^{-1}(a x)}{1-a x} \, dx \\ & = \frac {\coth ^{-1}(a x)^2}{a}+x \coth ^{-1}(a x)^2-\frac {2 \coth ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{a}+2 \int \frac {\log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx \\ & = \frac {\coth ^{-1}(a x)^2}{a}+x \coth ^{-1}(a x)^2-\frac {2 \coth ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{a}-\frac {2 \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-a x}\right )}{a} \\ & = \frac {\coth ^{-1}(a x)^2}{a}+x \coth ^{-1}(a x)^2-\frac {2 \coth ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{a}-\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{a} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.79 \[ \int \coth ^{-1}(a x)^2 \, dx=\frac {\coth ^{-1}(a x) \left ((-1+a x) \coth ^{-1}(a x)-2 \log \left (1-e^{-2 \coth ^{-1}(a x)}\right )\right )+\operatorname {PolyLog}\left (2,e^{-2 \coth ^{-1}(a x)}\right )}{a} \]
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Time = 0.50 (sec) , antiderivative size = 116, normalized size of antiderivative = 2.00
method | result | size |
derivativedivides | \(\frac {\operatorname {arccoth}\left (a x \right )^{2} \left (a x -1\right )+2 \operatorname {arccoth}\left (a x \right )^{2}-2 \,\operatorname {arccoth}\left (a x \right ) \ln \left (1-\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )-2 \operatorname {polylog}\left (2, \frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )-2 \,\operatorname {arccoth}\left (a x \right ) \ln \left (1+\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )-2 \operatorname {polylog}\left (2, -\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )}{a}\) | \(116\) |
default | \(\frac {\operatorname {arccoth}\left (a x \right )^{2} \left (a x -1\right )+2 \operatorname {arccoth}\left (a x \right )^{2}-2 \,\operatorname {arccoth}\left (a x \right ) \ln \left (1-\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )-2 \operatorname {polylog}\left (2, \frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )-2 \,\operatorname {arccoth}\left (a x \right ) \ln \left (1+\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )-2 \operatorname {polylog}\left (2, -\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )}{a}\) | \(116\) |
risch | \(\frac {\ln \left (a x -1\right )^{2} x}{4}-\frac {\ln \left (a x -1\right ) x}{2}-\frac {\ln \left (a x -1\right )^{2}}{4 a}+\frac {\ln \left (a x -1\right )}{2 a}+\frac {1}{a}+\frac {\ln \left (a x +1\right )^{2} x}{4}-\frac {x \ln \left (a x +1\right )}{2}+\frac {\ln \left (a x +1\right )^{2}}{4 a}+\frac {\ln \left (a x +1\right )}{2 a}-\frac {\left (-1+\ln \left (a x -1\right )\right ) \left (a x -1\right ) \ln \left (a x +1\right )}{2 a}+\frac {\ln \left (a x -1\right ) \left (a x -1\right )}{2 a}-\frac {\operatorname {dilog}\left (\frac {a x}{2}+\frac {1}{2}\right )}{a}-\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{a}\) | \(163\) |
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\[ \int \coth ^{-1}(a x)^2 \, dx=\int { \operatorname {arcoth}\left (a x\right )^{2} \,d x } \]
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\[ \int \coth ^{-1}(a x)^2 \, dx=\int \operatorname {acoth}^{2}{\left (a x \right )}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 135 vs. \(2 (55) = 110\).
Time = 0.21 (sec) , antiderivative size = 135, normalized size of antiderivative = 2.33 \[ \int \coth ^{-1}(a x)^2 \, dx=x \operatorname {arcoth}\left (a x\right )^{2} + \frac {1}{4} \, {\left (a {\left (\frac {\log \left (a x + 1\right )^{2} + 2 \, \log \left (a x + 1\right ) \log \left (a x - 1\right ) - \log \left (a x - 1\right )^{2}}{a^{3}} - \frac {4 \, {\left (\log \left (a x - 1\right ) \log \left (\frac {1}{2} \, a x + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, a x + \frac {1}{2}\right )\right )}}{a^{3}}\right )} - \frac {2 \, {\left (\frac {\log \left (a x + 1\right )}{a} - \frac {\log \left (a x - 1\right )}{a}\right )} \log \left (a^{2} x^{2} - 1\right )}{a}\right )} a + \frac {\operatorname {arcoth}\left (a x\right ) \log \left (a^{2} x^{2} - 1\right )}{a} \]
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\[ \int \coth ^{-1}(a x)^2 \, dx=\int { \operatorname {arcoth}\left (a x\right )^{2} \,d x } \]
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Timed out. \[ \int \coth ^{-1}(a x)^2 \, dx=\int {\mathrm {acoth}\left (a\,x\right )}^2 \,d x \]
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