Optimal. Leaf size=73 \[ -\frac {3 x^2}{8}-\frac {1}{2} x \sqrt {1-x^2} \sin ^{-1}(x)^3+\frac {3}{4} x^2 \sin ^{-1}(x)^2+\frac {3}{4} x \sqrt {1-x^2} \sin ^{-1}(x)+\frac {1}{8} \sin ^{-1}(x)^4-\frac {3}{8} \sin ^{-1}(x)^2 \]
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Rubi [A] time = 0.15, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {4707, 4641, 4627, 30} \[ -\frac {3 x^2}{8}-\frac {1}{2} x \sqrt {1-x^2} \sin ^{-1}(x)^3+\frac {3}{4} x^2 \sin ^{-1}(x)^2+\frac {3}{4} x \sqrt {1-x^2} \sin ^{-1}(x)+\frac {1}{8} \sin ^{-1}(x)^4-\frac {3}{8} \sin ^{-1}(x)^2 \]
Antiderivative was successfully verified.
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Rule 30
Rule 4627
Rule 4641
Rule 4707
Rubi steps
\begin {align*} \int \frac {x^2 \sin ^{-1}(x)^3}{\sqrt {1-x^2}} \, dx &=-\frac {1}{2} x \sqrt {1-x^2} \sin ^{-1}(x)^3+\frac {1}{2} \int \frac {\sin ^{-1}(x)^3}{\sqrt {1-x^2}} \, dx+\frac {3}{2} \int x \sin ^{-1}(x)^2 \, dx\\ &=\frac {3}{4} x^2 \sin ^{-1}(x)^2-\frac {1}{2} x \sqrt {1-x^2} \sin ^{-1}(x)^3+\frac {1}{8} \sin ^{-1}(x)^4-\frac {3}{2} \int \frac {x^2 \sin ^{-1}(x)}{\sqrt {1-x^2}} \, dx\\ &=\frac {3}{4} x \sqrt {1-x^2} \sin ^{-1}(x)+\frac {3}{4} x^2 \sin ^{-1}(x)^2-\frac {1}{2} x \sqrt {1-x^2} \sin ^{-1}(x)^3+\frac {1}{8} \sin ^{-1}(x)^4-\frac {3 \int x \, dx}{4}-\frac {3}{4} \int \frac {\sin ^{-1}(x)}{\sqrt {1-x^2}} \, dx\\ &=-\frac {3 x^2}{8}+\frac {3}{4} x \sqrt {1-x^2} \sin ^{-1}(x)-\frac {3}{8} \sin ^{-1}(x)^2+\frac {3}{4} x^2 \sin ^{-1}(x)^2-\frac {1}{2} x \sqrt {1-x^2} \sin ^{-1}(x)^3+\frac {1}{8} \sin ^{-1}(x)^4\\ \end {align*}
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Mathematica [A] time = 0.02, size = 60, normalized size = 0.82 \[ \frac {1}{8} \left (-3 x^2-4 x \sqrt {1-x^2} \sin ^{-1}(x)^3+\left (6 x^2-3\right ) \sin ^{-1}(x)^2+6 x \sqrt {1-x^2} \sin ^{-1}(x)+\sin ^{-1}(x)^4\right ) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^2 \sin ^{-1}(x)^3}{\sqrt {1-x^2}} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.95, size = 49, normalized size = 0.67 \[ \frac {1}{8} \, \arcsin \relax (x)^{4} + \frac {3}{8} \, {\left (2 \, x^{2} - 1\right )} \arcsin \relax (x)^{2} - \frac {3}{8} \, x^{2} - \frac {1}{4} \, {\left (2 \, x \arcsin \relax (x)^{3} - 3 \, x \arcsin \relax (x)\right )} \sqrt {-x^{2} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.97, size = 60, normalized size = 0.82 \[ -\frac {1}{2} \, \sqrt {-x^{2} + 1} x \arcsin \relax (x)^{3} + \frac {1}{8} \, \arcsin \relax (x)^{4} + \frac {3}{4} \, {\left (x^{2} - 1\right )} \arcsin \relax (x)^{2} + \frac {3}{4} \, \sqrt {-x^{2} + 1} x \arcsin \relax (x) - \frac {3}{8} \, x^{2} + \frac {3}{8} \, \arcsin \relax (x)^{2} + \frac {3}{16} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.32, size = 69, normalized size = 0.95
method | result | size |
default | \(\frac {\arcsin \relax (x )^{3} \left (-\sqrt {-x^{2}+1}\, x +\arcsin \relax (x )\right )}{2}+\frac {3 \arcsin \relax (x )^{2} \left (x^{2}-1\right )}{4}+\frac {3 \arcsin \relax (x ) \left (\sqrt {-x^{2}+1}\, x +\arcsin \relax (x )\right )}{4}-\frac {3 \arcsin \relax (x )^{2}}{8}-\frac {3 x^{2}}{8}-\frac {3 \arcsin \relax (x )^{4}}{8}\) | \(69\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2} \arcsin \relax (x)^{3}}{\sqrt {-x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^2\,{\mathrm {asin}\relax (x)}^3}{\sqrt {1-x^2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.32, size = 66, normalized size = 0.90 \[ \frac {3 x^{2} \operatorname {asin}^{2}{\relax (x )}}{4} - \frac {3 x^{2}}{8} - \frac {x \sqrt {1 - x^{2}} \operatorname {asin}^{3}{\relax (x )}}{2} + \frac {3 x \sqrt {1 - x^{2}} \operatorname {asin}{\relax (x )}}{4} + \frac {\operatorname {asin}^{4}{\relax (x )}}{8} - \frac {3 \operatorname {asin}^{2}{\relax (x )}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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