Optimal. Leaf size=39 \[ -\frac{\log \left (a^2 x^2+1\right )}{6 a^3}+\frac{x^2}{6 a}+\frac{1}{3} x^3 \cot ^{-1}(a x) \]
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Rubi [A] time = 0.0260955, antiderivative size = 39, normalized size of antiderivative = 1., 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 = {4853, 266, 43} \[ -\frac{\log \left (a^2 x^2+1\right )}{6 a^3}+\frac{x^2}{6 a}+\frac{1}{3} x^3 \cot ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 4853
Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^2 \cot ^{-1}(a x) \, dx &=\frac{1}{3} x^3 \cot ^{-1}(a x)+\frac{1}{3} a \int \frac{x^3}{1+a^2 x^2} \, dx\\ &=\frac{1}{3} x^3 \cot ^{-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 \cot ^{-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 \cot ^{-1}(a x)-\frac{\log \left (1+a^2 x^2\right )}{6 a^3}\\ \end{align*}
Mathematica [A] time = 0.0104125, size = 39, normalized size = 1. \[ -\frac{\log \left (a^2 x^2+1\right )}{6 a^3}+\frac{x^2}{6 a}+\frac{1}{3} x^3 \cot ^{-1}(a x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 34, normalized size = 0.9 \begin{align*}{\frac{{x}^{2}}{6\,a}}+{\frac{{x}^{3}{\rm arccot} \left (ax\right )}{3}}-{\frac{\ln \left ({a}^{2}{x}^{2}+1 \right ) }{6\,{a}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.988707, size = 49, normalized size = 1.26 \begin{align*} \frac{1}{3} \, x^{3} \operatorname{arccot}\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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.92215, size = 84, normalized size = 2.15 \begin{align*} \frac{2 \, a^{3} x^{3} \operatorname{arccot}\left (a x\right ) + a^{2} x^{2} - \log \left (a^{2} x^{2} + 1\right )}{6 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.637841, size = 37, normalized size = 0.95 \begin{align*} \begin{cases} \frac{x^{3} \operatorname{acot}{\left (a x \right )}}{3} + \frac{x^{2}}{6 a} - \frac{\log{\left (a^{2} x^{2} + 1 \right )}}{6 a^{3}} & \text{for}\: a \neq 0 \\\frac{\pi x^{3}}{6} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13748, size = 54, normalized size = 1.38 \begin{align*} \frac{1}{3} \, x^{3} \arctan \left (\frac{1}{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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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