Optimal. Leaf size=61 \[ -\frac {2 x^3}{27}+\frac {1}{3} x^3 \sin ^{-1}(x)^2+\frac {2}{9} \sqrt {1-x^2} x^2 \sin ^{-1}(x)+\frac {4}{9} \sqrt {1-x^2} \sin ^{-1}(x)-\frac {4 x}{9} \]
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Rubi [A] time = 0.09, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {4627, 4707, 4677, 8, 30} \[ -\frac {2 x^3}{27}+\frac {1}{3} x^3 \sin ^{-1}(x)^2+\frac {2}{9} \sqrt {1-x^2} x^2 \sin ^{-1}(x)+\frac {4}{9} \sqrt {1-x^2} \sin ^{-1}(x)-\frac {4 x}{9} \]
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 4627
Rule 4677
Rule 4707
Rubi steps
\begin {align*} \int x^2 \sin ^{-1}(x)^2 \, dx &=\frac {1}{3} x^3 \sin ^{-1}(x)^2-\frac {2}{3} \int \frac {x^3 \sin ^{-1}(x)}{\sqrt {1-x^2}} \, dx\\ &=\frac {2}{9} x^2 \sqrt {1-x^2} \sin ^{-1}(x)+\frac {1}{3} x^3 \sin ^{-1}(x)^2-\frac {2 \int x^2 \, dx}{9}-\frac {4}{9} \int \frac {x \sin ^{-1}(x)}{\sqrt {1-x^2}} \, dx\\ &=-\frac {2 x^3}{27}+\frac {4}{9} \sqrt {1-x^2} \sin ^{-1}(x)+\frac {2}{9} x^2 \sqrt {1-x^2} \sin ^{-1}(x)+\frac {1}{3} x^3 \sin ^{-1}(x)^2-\frac {4 \int 1 \, dx}{9}\\ &=-\frac {4 x}{9}-\frac {2 x^3}{27}+\frac {4}{9} \sqrt {1-x^2} \sin ^{-1}(x)+\frac {2}{9} x^2 \sqrt {1-x^2} \sin ^{-1}(x)+\frac {1}{3} x^3 \sin ^{-1}(x)^2\\ \end {align*}
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Mathematica [A] time = 0.03, size = 42, normalized size = 0.69 \[ \frac {1}{27} \left (9 x^3 \sin ^{-1}(x)^2-2 \left (x^2+6\right ) x+6 \sqrt {1-x^2} \left (x^2+2\right ) \sin ^{-1}(x)\right ) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^2 \sin ^{-1}(x)^2 \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 1.05, size = 36, normalized size = 0.59 \[ \frac {1}{3} \, x^{3} \arcsin \relax (x)^{2} - \frac {2}{27} \, x^{3} + \frac {2}{9} \, {\left (x^{2} + 2\right )} \sqrt {-x^{2} + 1} \arcsin \relax (x) - \frac {4}{9} \, x \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.90, size = 57, normalized size = 0.93 \[ \frac {1}{3} \, {\left (x^{2} - 1\right )} x \arcsin \relax (x)^{2} + \frac {1}{3} \, x \arcsin \relax (x)^{2} - \frac {2}{9} \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} \arcsin \relax (x) - \frac {2}{27} \, {\left (x^{2} - 1\right )} x + \frac {2}{3} \, \sqrt {-x^{2} + 1} \arcsin \relax (x) - \frac {14}{27} \, x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.36, size = 37, normalized size = 0.61
method | result | size |
default | \(\frac {x^{3} \arcsin \relax (x )^{2}}{3}+\frac {2 \arcsin \relax (x ) \left (x^{2}+2\right ) \sqrt {-x^{2}+1}}{9}-\frac {2 x^{3}}{27}-\frac {4 x}{9}\) | \(37\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.97, size = 47, normalized size = 0.77 \[ \frac {1}{3} \, x^{3} \arcsin \relax (x)^{2} - \frac {2}{27} \, x^{3} + \frac {2}{9} \, {\left (\sqrt {-x^{2} + 1} x^{2} + 2 \, \sqrt {-x^{2} + 1}\right )} \arcsin \relax (x) - \frac {4}{9} \, x \]
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 \[ \int x^2\,{\mathrm {asin}\relax (x)}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.66, size = 54, normalized size = 0.89 \[ \frac {x^{3} \operatorname {asin}^{2}{\relax (x )}}{3} - \frac {2 x^{3}}{27} + \frac {2 x^{2} \sqrt {1 - x^{2}} \operatorname {asin}{\relax (x )}}{9} - \frac {4 x}{9} + \frac {4 \sqrt {1 - x^{2}} \operatorname {asin}{\relax (x )}}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
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