Optimal. Leaf size=66 \[ \frac {2}{9} x^3 \cos ^{-1}(x)+\frac {2}{27} \left (1-x^2\right )^{3/2}+\frac {4 \sqrt {1-x^2}}{9}-\frac {1}{3} \left (1-x^2\right )^{3/2} \cos ^{-1}(x)^2-\frac {2}{3} x \cos ^{-1}(x) \]
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Rubi [A] time = 0.07, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {4678, 4646, 444, 43} \[ \frac {2}{27} \left (1-x^2\right )^{3/2}+\frac {4 \sqrt {1-x^2}}{9}+\frac {2}{9} x^3 \cos ^{-1}(x)-\frac {1}{3} \left (1-x^2\right )^{3/2} \cos ^{-1}(x)^2-\frac {2}{3} x \cos ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 43
Rule 444
Rule 4646
Rule 4678
Rubi steps
\begin {align*} \int x \sqrt {1-x^2} \cos ^{-1}(x)^2 \, dx &=-\frac {1}{3} \left (1-x^2\right )^{3/2} \cos ^{-1}(x)^2-\frac {2}{3} \int \left (1-x^2\right ) \cos ^{-1}(x) \, dx\\ &=-\frac {2}{3} x \cos ^{-1}(x)+\frac {2}{9} x^3 \cos ^{-1}(x)-\frac {1}{3} \left (1-x^2\right )^{3/2} \cos ^{-1}(x)^2-\frac {2}{3} \int \frac {x \left (1-\frac {x^2}{3}\right )}{\sqrt {1-x^2}} \, dx\\ &=-\frac {2}{3} x \cos ^{-1}(x)+\frac {2}{9} x^3 \cos ^{-1}(x)-\frac {1}{3} \left (1-x^2\right )^{3/2} \cos ^{-1}(x)^2-\frac {1}{3} \operatorname {Subst}\left (\int \frac {1-\frac {x}{3}}{\sqrt {1-x}} \, dx,x,x^2\right )\\ &=-\frac {2}{3} x \cos ^{-1}(x)+\frac {2}{9} x^3 \cos ^{-1}(x)-\frac {1}{3} \left (1-x^2\right )^{3/2} \cos ^{-1}(x)^2-\frac {1}{3} \operatorname {Subst}\left (\int \left (\frac {2}{3 \sqrt {1-x}}+\frac {\sqrt {1-x}}{3}\right ) \, dx,x,x^2\right )\\ &=\frac {4 \sqrt {1-x^2}}{9}+\frac {2}{27} \left (1-x^2\right )^{3/2}-\frac {2}{3} x \cos ^{-1}(x)+\frac {2}{9} x^3 \cos ^{-1}(x)-\frac {1}{3} \left (1-x^2\right )^{3/2} \cos ^{-1}(x)^2\\ \end {align*}
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Mathematica [A] time = 0.06, size = 50, normalized size = 0.76 \[ \frac {1}{27} \left (-2 \sqrt {1-x^2} \left (x^2-7\right )-9 \left (1-x^2\right )^{3/2} \cos ^{-1}(x)^2+6 x \left (x^2-3\right ) \cos ^{-1}(x)\right ) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \sqrt {1-x^2} \cos ^{-1}(x)^2 \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 1.15, size = 41, normalized size = 0.62 \[ \frac {2}{9} \, {\left (x^{3} - 3 \, x\right )} \arccos \relax (x) + \frac {1}{27} \, {\left (9 \, {\left (x^{2} - 1\right )} \arccos \relax (x)^{2} - 2 \, x^{2} + 14\right )} \sqrt {-x^{2} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.05, size = 53, normalized size = 0.80 \[ \frac {2}{9} \, x^{3} \arccos \relax (x) - \frac {1}{3} \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} \arccos \relax (x)^{2} - \frac {2}{27} \, \sqrt {-x^{2} + 1} x^{2} - \frac {2}{3} \, x \arccos \relax (x) + \frac {14}{27} \, \sqrt {-x^{2} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.32, size = 158, normalized size = 2.39
method | result | size |
default | \(-\frac {\left (6 i \arccos \relax (x )+9 \arccos \relax (x )^{2}-2\right ) \left (4 i x^{3}-4 \sqrt {-x^{2}+1}\, x^{2}-3 i x +\sqrt {-x^{2}+1}\right )}{216}+\frac {\left (\arccos \relax (x )^{2}-2+2 i \arccos \relax (x )\right ) \left (i x -\sqrt {-x^{2}+1}\right )}{8}-\frac {\left (\arccos \relax (x )^{2}-2-2 i \arccos \relax (x )\right ) \left (i x +\sqrt {-x^{2}+1}\right )}{8}+\frac {\left (-6 i \arccos \relax (x )+9 \arccos \relax (x )^{2}-2\right ) \left (4 i x^{3}+4 \sqrt {-x^{2}+1}\, x^{2}-3 i x -\sqrt {-x^{2}+1}\right )}{216}\) | \(158\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.96, size = 52, normalized size = 0.79 \[ -\frac {1}{3} \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} \arccos \relax (x)^{2} - \frac {2}{27} \, \sqrt {-x^{2} + 1} x^{2} + \frac {2}{9} \, {\left (x^{3} - 3 \, x\right )} \arccos \relax (x) + \frac {14}{27} \, \sqrt {-x^{2} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int x\,{\mathrm {acos}\relax (x)}^2\,\sqrt {1-x^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.04, size = 78, normalized size = 1.18 \[ \frac {2 x^{3} \operatorname {acos}{\relax (x )}}{9} + \frac {x^{2} \sqrt {1 - x^{2}} \operatorname {acos}^{2}{\relax (x )}}{3} - \frac {2 x^{2} \sqrt {1 - x^{2}}}{27} - \frac {2 x \operatorname {acos}{\relax (x )}}{3} - \frac {\sqrt {1 - x^{2}} \operatorname {acos}^{2}{\relax (x )}}{3} + \frac {14 \sqrt {1 - x^{2}}}{27} \]
Verification of antiderivative is not currently implemented for this CAS.
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