Optimal. Leaf size=40 \[ \frac {\log \left (1-a^2 x^2\right )}{6 a^3}+\frac {1}{3} x^3 \coth ^{-1}(a x)+\frac {x^2}{6 a} \]
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Rubi [A] time = 0.03, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {5917, 266, 43} \[ \frac {\log \left (1-a^2 x^2\right )}{6 a^3}+\frac {x^2}{6 a}+\frac {1}{3} x^3 \coth ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 5917
Rubi steps
\begin {align*} \int x^2 \coth ^{-1}(a x) \, dx &=\frac {1}{3} x^3 \coth ^{-1}(a x)-\frac {1}{3} a \int \frac {x^3}{1-a^2 x^2} \, dx\\ &=\frac {1}{3} x^3 \coth ^{-1}(a x)-\frac {1}{6} a \operatorname {Subst}\left (\int \frac {x}{1-a^2 x} \, dx,x,x^2\right )\\ &=\frac {1}{3} x^3 \coth ^{-1}(a x)-\frac {1}{6} a \operatorname {Subst}\left (\int \left (-\frac {1}{a^2}-\frac {1}{a^2 \left (-1+a^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=\frac {x^2}{6 a}+\frac {1}{3} x^3 \coth ^{-1}(a x)+\frac {\log \left (1-a^2 x^2\right )}{6 a^3}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 40, normalized size = 1.00 \[ \frac {\log \left (1-a^2 x^2\right )}{6 a^3}+\frac {1}{3} x^3 \coth ^{-1}(a x)+\frac {x^2}{6 a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 44, normalized size = 1.10 \[ \frac {a^{3} x^{3} \log \left (\frac {a x + 1}{a x - 1}\right ) + a^{2} x^{2} + \log \left (a^{2} x^{2} - 1\right )}{6 \, a^{3}} \]
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 \[ \int x^{2} \operatorname {arcoth}\left (a x\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 41, normalized size = 1.02 \[ \frac {x^{3} \mathrm {arccoth}\left (a x \right )}{3}+\frac {x^{2}}{6 a}+\frac {\ln \left (a x -1\right )}{6 a^{3}}+\frac {\ln \left (a x +1\right )}{6 a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 35, normalized size = 0.88 \[ \frac {1}{3} \, x^{3} \operatorname {arcoth}\left (a x\right ) + \frac {1}{6} \, a {\left (\frac {x^{2}}{a^{2}} + \frac {\log \left (a^{2} x^{2} - 1\right )}{a^{4}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.22, size = 35, normalized size = 0.88 \[ \frac {\frac {\ln \left (a^2\,x^2-1\right )}{6}+\frac {a^2\,x^2}{6}}{a^3}+\frac {x^3\,\mathrm {acoth}\left (a\,x\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.67, size = 46, normalized size = 1.15 \[ \begin {cases} \frac {x^{3} \operatorname {acoth}{\left (a x \right )}}{3} + \frac {x^{2}}{6 a} + \frac {\log {\left (a x + 1 \right )}}{3 a^{3}} - \frac {\operatorname {acoth}{\left (a x \right )}}{3 a^{3}} & \text {for}\: a \neq 0 \\\frac {i \pi x^{3}}{6} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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