Optimal. Leaf size=40 \[ \frac{1}{9} \left (1-x^2\right )^{3/2}-\frac{\sqrt{1-x^2}}{3}+\frac{1}{3} x^3 \cos ^{-1}(x) \]
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Rubi [A] time = 0.0215891, antiderivative size = 40, 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 = {4628, 266, 43} \[ \frac{1}{3} x^3 \cos ^{-1}(x)+\frac{1}{9} \left (1-x^2\right )^{3/2}-\frac{\sqrt{1-x^2}}{3} \]
Antiderivative was successfully verified.
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Rule 4628
Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^2 \cos ^{-1}(x) \, dx &=\frac{1}{3} x^3 \cos ^{-1}(x)+\frac{1}{3} \int \frac{x^3}{\sqrt{1-x^2}} \, dx\\ &=\frac{1}{3} x^3 \cos ^{-1}(x)+\frac{1}{6} \operatorname{Subst}\left (\int \frac{x}{\sqrt{1-x}} \, dx,x,x^2\right )\\ &=\frac{1}{3} x^3 \cos ^{-1}(x)+\frac{1}{6} \operatorname{Subst}\left (\int \left (\frac{1}{\sqrt{1-x}}-\sqrt{1-x}\right ) \, dx,x,x^2\right )\\ &=-\frac{1}{3} \sqrt{1-x^2}+\frac{1}{9} \left (1-x^2\right )^{3/2}+\frac{1}{3} x^3 \cos ^{-1}(x)\\ \end{align*}
Mathematica [A] time = 0.0103854, size = 30, normalized size = 0.75 \[ \frac{1}{3} x^3 \cos ^{-1}(x)-\frac{1}{9} \sqrt{1-x^2} \left (x^2+2\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 34, normalized size = 0.9 \begin{align*}{\frac{{x}^{3}\arccos \left ( x \right ) }{3}}-{\frac{{x}^{2}}{9}\sqrt{-{x}^{2}+1}}-{\frac{2}{9}\sqrt{-{x}^{2}+1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.45287, size = 45, normalized size = 1.12 \begin{align*} \frac{1}{3} \, x^{3} \arccos \left (x\right ) - \frac{1}{9} \, \sqrt{-x^{2} + 1} x^{2} - \frac{2}{9} \, \sqrt{-x^{2} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.548079, size = 68, normalized size = 1.7 \begin{align*} \frac{1}{3} \, x^{3} \arccos \left (x\right ) - \frac{1}{9} \,{\left (x^{2} + 2\right )} \sqrt{-x^{2} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.330808, size = 32, normalized size = 0.8 \begin{align*} \frac{x^{3} \operatorname{acos}{\left (x \right )}}{3} - \frac{x^{2} \sqrt{1 - x^{2}}}{9} - \frac{2 \sqrt{1 - x^{2}}}{9} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09278, size = 45, normalized size = 1.12 \begin{align*} \frac{1}{3} \, x^{3} \arccos \left (x\right ) - \frac{1}{9} \, \sqrt{-x^{2} + 1} x^{2} - \frac{2}{9} \, \sqrt{-x^{2} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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