Optimal. Leaf size=103 \[ -\frac {\text {Li}_2\left (1-\frac {2}{1-a x}\right )}{3 a^3}-\frac {\tanh ^{-1}(a x)}{3 a^3}+\frac {\coth ^{-1}(a x)^2}{3 a^3}-\frac {2 \log \left (\frac {2}{1-a x}\right ) \coth ^{-1}(a x)}{3 a^3}+\frac {x}{3 a^2}+\frac {1}{3} x^3 \coth ^{-1}(a x)^2+\frac {x^2 \coth ^{-1}(a x)}{3 a} \]
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Rubi [A] time = 0.15, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {5917, 5981, 321, 206, 5985, 5919, 2402, 2315} \[ -\frac {\text {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{3 a^3}+\frac {x}{3 a^2}-\frac {\tanh ^{-1}(a x)}{3 a^3}+\frac {\coth ^{-1}(a x)^2}{3 a^3}-\frac {2 \log \left (\frac {2}{1-a x}\right ) \coth ^{-1}(a x)}{3 a^3}+\frac {1}{3} x^3 \coth ^{-1}(a x)^2+\frac {x^2 \coth ^{-1}(a x)}{3 a} \]
Antiderivative was successfully verified.
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Rule 206
Rule 321
Rule 2315
Rule 2402
Rule 5917
Rule 5919
Rule 5981
Rule 5985
Rubi steps
\begin {align*} \int x^2 \coth ^{-1}(a x)^2 \, dx &=\frac {1}{3} x^3 \coth ^{-1}(a x)^2-\frac {1}{3} (2 a) \int \frac {x^3 \coth ^{-1}(a x)}{1-a^2 x^2} \, dx\\ &=\frac {1}{3} x^3 \coth ^{-1}(a x)^2+\frac {2 \int x \coth ^{-1}(a x) \, dx}{3 a}-\frac {2 \int \frac {x \coth ^{-1}(a x)}{1-a^2 x^2} \, dx}{3 a}\\ &=\frac {x^2 \coth ^{-1}(a x)}{3 a}+\frac {\coth ^{-1}(a x)^2}{3 a^3}+\frac {1}{3} x^3 \coth ^{-1}(a x)^2-\frac {1}{3} \int \frac {x^2}{1-a^2 x^2} \, dx-\frac {2 \int \frac {\coth ^{-1}(a x)}{1-a x} \, dx}{3 a^2}\\ &=\frac {x}{3 a^2}+\frac {x^2 \coth ^{-1}(a x)}{3 a}+\frac {\coth ^{-1}(a x)^2}{3 a^3}+\frac {1}{3} x^3 \coth ^{-1}(a x)^2-\frac {2 \coth ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{3 a^3}-\frac {\int \frac {1}{1-a^2 x^2} \, dx}{3 a^2}+\frac {2 \int \frac {\log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx}{3 a^2}\\ &=\frac {x}{3 a^2}+\frac {x^2 \coth ^{-1}(a x)}{3 a}+\frac {\coth ^{-1}(a x)^2}{3 a^3}+\frac {1}{3} x^3 \coth ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)}{3 a^3}-\frac {2 \coth ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{3 a^3}-\frac {2 \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-a x}\right )}{3 a^3}\\ &=\frac {x}{3 a^2}+\frac {x^2 \coth ^{-1}(a x)}{3 a}+\frac {\coth ^{-1}(a x)^2}{3 a^3}+\frac {1}{3} x^3 \coth ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)}{3 a^3}-\frac {2 \coth ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{3 a^3}-\frac {\text {Li}_2\left (1-\frac {2}{1-a x}\right )}{3 a^3}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 66, normalized size = 0.64 \[ \frac {\left (a^3 x^3-1\right ) \coth ^{-1}(a x)^2+\coth ^{-1}(a x) \left (a^2 x^2-2 \log \left (1-e^{-2 \coth ^{-1}(a x)}\right )-1\right )+\text {Li}_2\left (e^{-2 \coth ^{-1}(a x)}\right )+a x}{3 a^3} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{2} \operatorname {arcoth}\left (a x\right )^{2}, x\right ) \]
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 \[ \int x^{2} \operatorname {arcoth}\left (a x\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 176, normalized size = 1.71 \[ \frac {x^{3} \mathrm {arccoth}\left (a x \right )^{2}}{3}+\frac {x^{2} \mathrm {arccoth}\left (a x \right )}{3 a}+\frac {\mathrm {arccoth}\left (a x \right ) \ln \left (a x -1\right )}{3 a^{3}}+\frac {\mathrm {arccoth}\left (a x \right ) \ln \left (a x +1\right )}{3 a^{3}}+\frac {x}{3 a^{2}}+\frac {\ln \left (a x -1\right )}{6 a^{3}}-\frac {\ln \left (a x +1\right )}{6 a^{3}}+\frac {\ln \left (a x -1\right )^{2}}{12 a^{3}}-\frac {\dilog \left (\frac {1}{2}+\frac {a x}{2}\right )}{3 a^{3}}-\frac {\ln \left (a x -1\right ) \ln \left (\frac {1}{2}+\frac {a x}{2}\right )}{6 a^{3}}-\frac {\ln \left (a x +1\right )^{2}}{12 a^{3}}-\frac {\ln \left (-\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (\frac {1}{2}+\frac {a x}{2}\right )}{6 a^{3}}+\frac {\ln \left (-\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (a x +1\right )}{6 a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 134, normalized size = 1.30 \[ \frac {1}{3} \, x^{3} \operatorname {arcoth}\left (a x\right )^{2} + \frac {1}{12} \, a^{2} {\left (\frac {4 \, a x - \log \left (a x + 1\right )^{2} + 2 \, \log \left (a x + 1\right ) \log \left (a x - 1\right ) + \log \left (a x - 1\right )^{2} + 2 \, \log \left (a x - 1\right )}{a^{5}} - \frac {4 \, {\left (\log \left (a x - 1\right ) \log \left (\frac {1}{2} \, a x + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, a x + \frac {1}{2}\right )\right )}}{a^{5}} - \frac {2 \, \log \left (a x + 1\right )}{a^{5}}\right )} + \frac {1}{3} \, a {\left (\frac {x^{2}}{a^{2}} + \frac {\log \left (a^{2} x^{2} - 1\right )}{a^{4}}\right )} \operatorname {arcoth}\left (a x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^2\,{\mathrm {acoth}\left (a\,x\right )}^2 \,d x \]
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 \[ \int x^{2} \operatorname {acoth}^{2}{\left (a x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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