Optimal. Leaf size=40 \[ -\frac {1}{3} \sqrt {1-x^2}+\frac {1}{9} \left (1-x^2\right )^{3/2}+\frac {1}{3} x^3 \cos ^{-1}(x) \]
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Rubi [A]
time = 0.01, 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 = {4724, 272, 45}
\begin {gather*} \frac {1}{3} x^3 \text {ArcCos}(x)+\frac {1}{9} \left (1-x^2\right )^{3/2}-\frac {\sqrt {1-x^2}}{3} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rule 4724
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} \text {Subst}\left (\int \frac {x}{\sqrt {1-x}} \, dx,x,x^2\right )\\ &=\frac {1}{3} x^3 \cos ^{-1}(x)+\frac {1}{6} \text {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 \begin {gather*} -\frac {1}{9} \sqrt {1-x^2} \left (2+x^2\right )+\frac {1}{3} x^3 \cos ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 34, normalized size = 0.85
method | result | size |
default | \(\frac {x^{3} \arccos \left (x \right )}{3}-\frac {x^{2} \sqrt {-x^{2}+1}}{9}-\frac {2 \sqrt {-x^{2}+1}}{9}\) | \(34\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.91, size = 33, normalized size = 0.82 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.06, size = 24, normalized size = 0.60 \begin {gather*} \frac {1}{3} \, x^{3} \arccos \left (x\right ) - \frac {1}{9} \, {\left (x^{2} + 2\right )} \sqrt {-x^{2} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.09, size = 32, normalized size = 0.80 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.46, size = 33, normalized size = 0.82 \begin {gather*} \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 {gather*}
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 \begin {gather*} \frac {x^3\,\mathrm {acos}\left (x\right )}{3}-\frac {\sqrt {1-x^2}\,\left (x^2+2\right )}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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