Optimal. Leaf size=61 \[ -\frac {4 x}{9}-\frac {2 x^3}{27}+\frac {4}{9} \sqrt {1-x^2} \sin ^{-1}(x)+\frac {2}{9} x^2 \sqrt {1-x^2} \sin ^{-1}(x)+\frac {1}{3} x^3 \sin ^{-1}(x)^2 \]
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Rubi [A]
time = 0.06, 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 = {4723, 4795,
4767, 8, 30} \begin {gather*} \frac {1}{3} x^3 \text {ArcSin}(x)^2+\frac {2}{9} \sqrt {1-x^2} x^2 \text {ArcSin}(x)+\frac {4}{9} \sqrt {1-x^2} \text {ArcSin}(x)-\frac {2 x^3}{27}-\frac {4 x}{9} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 4723
Rule 4767
Rule 4795
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 = 0.01, size = 42, normalized size = 0.69 \begin {gather*} \frac {1}{27} \left (-2 x \left (6+x^2\right )+6 \sqrt {1-x^2} \left (2+x^2\right ) \sin ^{-1}(x)+9 x^3 \sin ^{-1}(x)^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 37, normalized size = 0.61
method | result | size |
default | \(\frac {x^{3} \arcsin \left (x \right )^{2}}{3}+\frac {2 \arcsin \left (x \right ) \left (x^{2}+2\right ) \sqrt {-x^{2}+1}}{9}-\frac {2 x^{3}}{27}-\frac {4 x}{9}\) | \(37\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 2.21, size = 47, normalized size = 0.77 \begin {gather*} \frac {1}{3} \, x^{3} \arcsin \left (x\right )^{2} - \frac {2}{27} \, x^{3} + \frac {2}{9} \, {\left (\sqrt {-x^{2} + 1} x^{2} + 2 \, \sqrt {-x^{2} + 1}\right )} \arcsin \left (x\right ) - \frac {4}{9} \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.45, size = 36, normalized size = 0.59 \begin {gather*} \frac {1}{3} \, x^{3} \arcsin \left (x\right )^{2} - \frac {2}{27} \, x^{3} + \frac {2}{9} \, {\left (x^{2} + 2\right )} \sqrt {-x^{2} + 1} \arcsin \left (x\right ) - \frac {4}{9} \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.14, size = 54, normalized size = 0.89 \begin {gather*} \frac {x^{3} \operatorname {asin}^{2}{\left (x \right )}}{3} - \frac {2 x^{3}}{27} + \frac {2 x^{2} \sqrt {1 - x^{2}} \operatorname {asin}{\left (x \right )}}{9} - \frac {4 x}{9} + \frac {4 \sqrt {1 - x^{2}} \operatorname {asin}{\left (x \right )}}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.07, size = 57, normalized size = 0.93 \begin {gather*} \frac {1}{3} \, {\left (x^{2} - 1\right )} x \arcsin \left (x\right )^{2} + \frac {1}{3} \, x \arcsin \left (x\right )^{2} - \frac {2}{9} \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} \arcsin \left (x\right ) - \frac {2}{27} \, {\left (x^{2} - 1\right )} x + \frac {2}{3} \, \sqrt {-x^{2} + 1} \arcsin \left (x\right ) - \frac {14}{27} \, x \end {gather*}
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 \begin {gather*} \int x^2\,{\mathrm {asin}\left (x\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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