Optimal. Leaf size=27 \[ -\frac{x^2}{6}+\frac{1}{6} \log \left (x^2+1\right )+\frac{1}{3} x^3 \tan ^{-1}(x) \]
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Rubi [A] time = 0.0162793, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4852, 266, 43} \[ -\frac{x^2}{6}+\frac{1}{6} \log \left (x^2+1\right )+\frac{1}{3} x^3 \tan ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 4852
Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^2 \tan ^{-1}(x) \, dx &=\frac{1}{3} x^3 \tan ^{-1}(x)-\frac{1}{3} \int \frac{x^3}{1+x^2} \, dx\\ &=\frac{1}{3} x^3 \tan ^{-1}(x)-\frac{1}{6} \operatorname{Subst}\left (\int \frac{x}{1+x} \, dx,x,x^2\right )\\ &=\frac{1}{3} x^3 \tan ^{-1}(x)-\frac{1}{6} \operatorname{Subst}\left (\int \left (1+\frac{1}{-1-x}\right ) \, dx,x,x^2\right )\\ &=-\frac{x^2}{6}+\frac{1}{3} x^3 \tan ^{-1}(x)+\frac{1}{6} \log \left (1+x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0042958, size = 23, normalized size = 0.85 \[ \frac{1}{6} \left (-x^2+\log \left (x^2+1\right )+2 x^3 \tan ^{-1}(x)\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 22, normalized size = 0.8 \begin{align*} -{\frac{{x}^{2}}{6}}+{\frac{{x}^{3}\arctan \left ( x \right ) }{3}}+{\frac{\ln \left ({x}^{2}+1 \right ) }{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.920129, size = 28, normalized size = 1.04 \begin{align*} \frac{1}{3} \, x^{3} \arctan \left (x\right ) - \frac{1}{6} \, x^{2} + \frac{1}{6} \, \log \left (x^{2} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.03142, size = 65, normalized size = 2.41 \begin{align*} \frac{1}{3} \, x^{3} \arctan \left (x\right ) - \frac{1}{6} \, x^{2} + \frac{1}{6} \, \log \left (x^{2} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.325198, size = 20, normalized size = 0.74 \begin{align*} \frac{x^{3} \operatorname{atan}{\left (x \right )}}{3} - \frac{x^{2}}{6} + \frac{\log{\left (x^{2} + 1 \right )}}{6} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06491, size = 28, normalized size = 1.04 \begin{align*} \frac{1}{3} \, x^{3} \arctan \left (x\right ) - \frac{1}{6} \, x^{2} + \frac{1}{6} \, \log \left (x^{2} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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