Optimal. Leaf size=40 \[ \frac {1}{3} x^3 \cos ^{-1}(x)+\frac {1}{9} \left (1-x^2\right )^{3/2}-\frac {\sqrt {1-x^2}}{3} \]
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Rubi [A] time = 0.02, antiderivative size = 40, 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 = {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 43
Rule 266
Rule 4628
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*}
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Mathematica [A] time = 0.01, 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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fricas [A] time = 0.43, size = 24, normalized size = 0.60 \[ \frac {1}{3} \, x^{3} \arccos \relax (x) - \frac {1}{9} \, {\left (x^{2} + 2\right )} \sqrt {-x^{2} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.02, size = 33, normalized size = 0.82 \[ \frac {1}{3} \, x^{3} \arccos \relax (x) - \frac {1}{9} \, \sqrt {-x^{2} + 1} x^{2} - \frac {2}{9} \, \sqrt {-x^{2} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 34, normalized size = 0.85 \[ \frac {x^{3} \arccos \relax (x )}{3}-\frac {\sqrt {-x^{2}+1}\, x^{2}}{9}-\frac {2 \sqrt {-x^{2}+1}}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.27, size = 33, normalized size = 0.82 \[ \frac {1}{3} \, x^{3} \arccos \relax (x) - \frac {1}{9} \, \sqrt {-x^{2} + 1} x^{2} - \frac {2}{9} \, \sqrt {-x^{2} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 24, normalized size = 0.60 \[ \frac {x^3\,\mathrm {acos}\relax (x)}{3}-\frac {\sqrt {1-x^2}\,\left (x^2+2\right )}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.37, size = 32, normalized size = 0.80 \[ \frac {x^{3} \operatorname {acos}{\relax (x )}}{3} - \frac {x^{2} \sqrt {1 - x^{2}}}{9} - \frac {2 \sqrt {1 - x^{2}}}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
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