Optimal. Leaf size=59 \[ -\frac {5 x^2}{16}+\frac {x^4}{16}+\frac {3}{8} x \sqrt {1-x^2} \sin ^{-1}(x)+\frac {1}{4} x \left (1-x^2\right )^{3/2} \sin ^{-1}(x)+\frac {3}{16} \sin ^{-1}(x)^2 \]
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Rubi [A]
time = 0.03, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {4743, 4741,
4737, 30, 14} \begin {gather*} \frac {1}{4} \left (1-x^2\right )^{3/2} x \text {ArcSin}(x)+\frac {3}{8} \sqrt {1-x^2} x \text {ArcSin}(x)+\frac {3 \text {ArcSin}(x)^2}{16}+\frac {x^4}{16}-\frac {5 x^2}{16} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 30
Rule 4737
Rule 4741
Rule 4743
Rubi steps
\begin {align*} \int \left (1-x^2\right )^{3/2} \sin ^{-1}(x) \, dx &=\frac {1}{4} x \left (1-x^2\right )^{3/2} \sin ^{-1}(x)-\frac {1}{4} \int x \left (1-x^2\right ) \, dx+\frac {3}{4} \int \sqrt {1-x^2} \sin ^{-1}(x) \, dx\\ &=\frac {3}{8} x \sqrt {1-x^2} \sin ^{-1}(x)+\frac {1}{4} x \left (1-x^2\right )^{3/2} \sin ^{-1}(x)-\frac {1}{4} \int \left (x-x^3\right ) \, dx-\frac {3 \int x \, dx}{8}+\frac {3}{8} \int \frac {\sin ^{-1}(x)}{\sqrt {1-x^2}} \, dx\\ &=-\frac {5 x^2}{16}+\frac {x^4}{16}+\frac {3}{8} x \sqrt {1-x^2} \sin ^{-1}(x)+\frac {1}{4} x \left (1-x^2\right )^{3/2} \sin ^{-1}(x)+\frac {3}{16} \sin ^{-1}(x)^2\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 42, normalized size = 0.71 \begin {gather*} \frac {1}{16} \left (-5 x^2+x^4-2 x \sqrt {1-x^2} \left (-5+2 x^2\right ) \sin ^{-1}(x)+3 \sin ^{-1}(x)^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 54, normalized size = 0.92
method | result | size |
default | \(\frac {\arcsin \left (x \right ) \left (-2 \sqrt {-x^{2}+1}\, x^{3}+5 x \sqrt {-x^{2}+1}+3 \arcsin \left (x \right )\right )}{8}-\frac {3 \arcsin \left (x \right )^{2}}{16}+\frac {\left (2 x^{2}-5\right )^{2}}{64}\) | \(54\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.51, size = 50, normalized size = 0.85 \begin {gather*} \frac {1}{16} \, x^{4} - \frac {5}{16} \, x^{2} + \frac {1}{8} \, {\left (2 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} x + 3 \, \sqrt {-x^{2} + 1} x + 3 \, \arcsin \left (x\right )\right )} \arcsin \left (x\right ) - \frac {3}{16} \, \arcsin \left (x\right )^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.48, size = 39, normalized size = 0.66 \begin {gather*} \frac {1}{16} \, x^{4} - \frac {1}{8} \, {\left (2 \, x^{3} - 5 \, x\right )} \sqrt {-x^{2} + 1} \arcsin \left (x\right ) - \frac {5}{16} \, x^{2} + \frac {3}{16} \, \arcsin \left (x\right )^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.33, size = 53, normalized size = 0.90 \begin {gather*} \frac {x^{4}}{16} - \frac {x^{3} \sqrt {1 - x^{2}} \operatorname {asin}{\left (x \right )}}{4} - \frac {5 x^{2}}{16} + \frac {5 x \sqrt {1 - x^{2}} \operatorname {asin}{\left (x \right )}}{8} + \frac {3 \operatorname {asin}^{2}{\left (x \right )}}{16} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.23, size = 50, normalized size = 0.85 \begin {gather*} \frac {1}{4} \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} x \arcsin \left (x\right ) + \frac {3}{8} \, \sqrt {-x^{2} + 1} x \arcsin \left (x\right ) + \frac {1}{16} \, {\left (x^{2} - 1\right )}^{2} - \frac {3}{16} \, x^{2} + \frac {3}{16} \, \arcsin \left (x\right )^{2} + \frac {9}{128} \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 \mathrm {asin}\left (x\right )\,{\left (1-x^2\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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