Optimal. Leaf size=27 \[ \frac {1}{3} x^3 \tan ^{-1}(x)-\frac {x^2}{6}+\frac {1}{6} \log \left (x^2+1\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, 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 43
Rule 266
Rule 4852
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*}
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Mathematica [A] time = 0.01, size = 23, normalized size = 0.85 \[ \frac {1}{6} \left (2 x^3 \tan ^{-1}(x)-x^2+\log \left (x^2+1\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 21, normalized size = 0.78 \[ \frac {1}{3} \, x^{3} \arctan \relax (x) - \frac {1}{6} \, x^{2} + \frac {1}{6} \, \log \left (x^{2} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.90, size = 21, normalized size = 0.78 \[ \frac {1}{3} \, x^{3} \arctan \relax (x) - \frac {1}{6} \, x^{2} + \frac {1}{6} \, \log \left (x^{2} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 22, normalized size = 0.81 \[ \frac {x^{3} \arctan \relax (x )}{3}-\frac {x^{2}}{6}+\frac {\ln \left (x^{2}+1\right )}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 21, normalized size = 0.78 \[ \frac {1}{3} \, x^{3} \arctan \relax (x) - \frac {1}{6} \, x^{2} + \frac {1}{6} \, \log \left (x^{2} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.17, size = 21, normalized size = 0.78 \[ \frac {\ln \left (x^2+1\right )}{6}+\frac {x^3\,\mathrm {atan}\relax (x)}{3}-\frac {x^2}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.35, size = 20, normalized size = 0.74 \[ \frac {x^{3} \operatorname {atan}{\relax (x )}}{3} - \frac {x^{2}}{6} + \frac {\log {\left (x^{2} + 1 \right )}}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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