Optimal. Leaf size=103 \[ \frac {x}{3 a^2}+\frac {x^2 \coth ^{-1}(a x)}{3 a}+\frac {\coth ^{-1}(a x)^2}{3 a^3}+\frac {1}{3} x^3 \coth ^{-1}(a x)^2-\frac {\tanh ^{-1}(a x)}{3 a^3}-\frac {2 \coth ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{3 a^3}-\frac {\text {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{3 a^3} \]
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Rubi [A]
time = 0.11, 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 = {6038, 6128,
327, 212, 6132, 6056, 2449, 2352} \begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 327
Rule 2352
Rule 2449
Rule 6038
Rule 6056
Rule 6128
Rule 6132
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 \text {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 = 0.18, size = 66, normalized size = 0.64 \begin {gather*} \frac {a x+\left (-1+a^3 x^3\right ) \coth ^{-1}(a x)^2+\coth ^{-1}(a x) \left (-1+a^2 x^2-2 \log \left (1-e^{-2 \coth ^{-1}(a x)}\right )\right )+\text {PolyLog}\left (2,e^{-2 \coth ^{-1}(a x)}\right )}{3 a^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.24, size = 145, normalized size = 1.41
method | result | size |
derivativedivides | \(\frac {\frac {a^{3} x^{3} \mathrm {arccoth}\left (a x \right )^{2}}{3}+\frac {\mathrm {arccoth}\left (a x \right ) a^{2} x^{2}}{3}+\frac {\mathrm {arccoth}\left (a x \right ) \ln \left (a x -1\right )}{3}+\frac {\mathrm {arccoth}\left (a x \right ) \ln \left (a x +1\right )}{3}+\frac {a x}{3}+\frac {\ln \left (a x -1\right )}{6}-\frac {\ln \left (a x +1\right )}{6}-\frac {\dilog \left (\frac {a x}{2}+\frac {1}{2}\right )}{3}-\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{6}+\frac {\ln \left (a x -1\right )^{2}}{12}-\frac {\ln \left (a x +1\right )^{2}}{12}+\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{6}}{a^{3}}\) | \(145\) |
default | \(\frac {\frac {a^{3} x^{3} \mathrm {arccoth}\left (a x \right )^{2}}{3}+\frac {\mathrm {arccoth}\left (a x \right ) a^{2} x^{2}}{3}+\frac {\mathrm {arccoth}\left (a x \right ) \ln \left (a x -1\right )}{3}+\frac {\mathrm {arccoth}\left (a x \right ) \ln \left (a x +1\right )}{3}+\frac {a x}{3}+\frac {\ln \left (a x -1\right )}{6}-\frac {\ln \left (a x +1\right )}{6}-\frac {\dilog \left (\frac {a x}{2}+\frac {1}{2}\right )}{3}-\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{6}+\frac {\ln \left (a x -1\right )^{2}}{12}-\frac {\ln \left (a x +1\right )^{2}}{12}+\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{6}}{a^{3}}\) | \(145\) |
risch | \(\frac {\ln \left (a x -1\right )^{2} x^{3}}{12}-\frac {\ln \left (a x -1\right )^{2}}{12 a^{3}}-\frac {\ln \left (a x -1\right ) x^{3}}{18}-\frac {\ln \left (a x -1\right ) x^{2}}{12 a}-\frac {\ln \left (a x -1\right ) x}{6 a^{2}}+\frac {11 \ln \left (a x -1\right )}{36 a^{3}}+\frac {x}{3 a^{2}}+\frac {\ln \left (a x +1\right )^{2} x^{3}}{12}+\frac {\ln \left (a x +1\right )^{2}}{12 a^{3}}-\frac {x^{3} \ln \left (a x +1\right )}{18}+\frac {\ln \left (a x +1\right ) x^{2}}{12 a}-\frac {\ln \left (a x +1\right ) x}{6 a^{2}}+\frac {5 \ln \left (a x +1\right )}{36 a^{3}}-\frac {\left (\left (-\frac {1}{9}+\frac {\ln \left (a x -1\right )}{3}\right ) \left (a x -1\right )^{3}+\left (-\frac {1}{2}+\ln \left (a x -1\right )\right ) \left (a x -1\right )^{2}+\left (-1+\ln \left (a x -1\right )\right ) \left (a x -1\right )\right ) \ln \left (a x +1\right )}{2 a^{3}}+\frac {\ln \left (a x -1\right ) \left (a x -1\right )}{6 a^{3}}-\frac {\dilog \left (\frac {a x}{2}+\frac {1}{2}\right )}{3 a^{3}}-\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{3 a^{3}}+\frac {\ln \left (a x -1\right ) \left (a x -1\right )^{2}}{12 a^{3}}+\frac {17}{54 a^{3}}+\frac {\left (a x -1\right )^{3} \ln \left (a x -1\right )}{18 a^{3}}\) | \(303\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 134, normalized size = 1.30 \begin {gather*} \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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int x^{2} \operatorname {acoth}^{2}{\left (a x \right )}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int x^2\,{\mathrm {acoth}\left (a\,x\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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