Optimal. Leaf size=58 \[ -\frac {\text {Li}_2\left (1-\frac {2}{1-a x}\right )}{a}+x \coth ^{-1}(a x)^2+\frac {\coth ^{-1}(a x)^2}{a}-\frac {2 \log \left (\frac {2}{1-a x}\right ) \coth ^{-1}(a x)}{a} \]
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Rubi [A] time = 0.08, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {5911, 5985, 5919, 2402, 2315} \[ -\frac {\text {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{a}+x \coth ^{-1}(a x)^2+\frac {\coth ^{-1}(a x)^2}{a}-\frac {2 \log \left (\frac {2}{1-a x}\right ) \coth ^{-1}(a x)}{a} \]
Antiderivative was successfully verified.
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Rule 2315
Rule 2402
Rule 5911
Rule 5919
Rule 5985
Rubi steps
\begin {align*} \int \coth ^{-1}(a x)^2 \, dx &=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 \operatorname {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 {\text {Li}_2\left (1-\frac {2}{1-a x}\right )}{a}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 46, normalized size = 0.79 \[ \frac {\text {Li}_2\left (e^{-2 \coth ^{-1}(a x)}\right )+\coth ^{-1}(a x) \left ((a x-1) \coth ^{-1}(a x)-2 \log \left (1-e^{-2 \coth ^{-1}(a x)}\right )\right )}{a} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\operatorname {arcoth}\left (a x\right )^{2}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {arcoth}\left (a x\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.30, size = 122, normalized size = 2.10 \[ x \mathrm {arccoth}\left (a x \right )^{2}+\frac {\mathrm {arccoth}\left (a x \right )^{2}}{a}-\frac {2 \,\mathrm {arccoth}\left (a x \right ) \ln \left (1-\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )}{a}-\frac {2 \,\mathrm {arccoth}\left (a x \right ) \ln \left (1+\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )}{a}-\frac {2 \polylog \left (2, \frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )}{a}-\frac {2 \polylog \left (2, -\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.32, size = 135, normalized size = 2.33 \[ x \operatorname {arcoth}\left (a x\right )^{2} + \frac {1}{4} \, {\left (a {\left (\frac {\log \left (a x + 1\right )^{2} + 2 \, \log \left (a x + 1\right ) \log \left (a x - 1\right ) - \log \left (a x - 1\right )^{2}}{a^{3}} - \frac {4 \, {\left (\log \left (a x - 1\right ) \log \left (\frac {1}{2} \, a x + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, a x + \frac {1}{2}\right )\right )}}{a^{3}}\right )} - \frac {2 \, {\left (\frac {\log \left (a x + 1\right )}{a} - \frac {\log \left (a x - 1\right )}{a}\right )} \log \left (a^{2} x^{2} - 1\right )}{a}\right )} a + \frac {\operatorname {arcoth}\left (a x\right ) \log \left (a^{2} x^{2} - 1\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int {\mathrm {acoth}\left (a\,x\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {acoth}^{2}{\left (a x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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