Optimal. Leaf size=73 \[ -\frac {3 x^2}{8}+\frac {3}{4} x \sqrt {1-x^2} \sin ^{-1}(x)-\frac {3}{8} \sin ^{-1}(x)^2+\frac {3}{4} x^2 \sin ^{-1}(x)^2-\frac {1}{2} x \sqrt {1-x^2} \sin ^{-1}(x)^3+\frac {1}{8} \sin ^{-1}(x)^4 \]
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Rubi [A]
time = 0.10, 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 = {4795, 4737,
4723, 30} \begin {gather*} -\frac {1}{2} x \sqrt {1-x^2} \text {ArcSin}(x)^3+\frac {3}{4} x^2 \text {ArcSin}(x)^2+\frac {3}{4} x \sqrt {1-x^2} \text {ArcSin}(x)+\frac {\text {ArcSin}(x)^4}{8}-\frac {3 \text {ArcSin}(x)^2}{8}-\frac {3 x^2}{8} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 4723
Rule 4737
Rule 4795
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 \begin {gather*} \frac {1}{8} \left (-3 x^2+6 x \sqrt {1-x^2} \sin ^{-1}(x)+\left (-3+6 x^2\right ) \sin ^{-1}(x)^2-4 x \sqrt {1-x^2} \sin ^{-1}(x)^3+\sin ^{-1}(x)^4\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 69, normalized size = 0.95
method | result | size |
default | \(\frac {\arcsin \left (x \right )^{3} \left (-x \sqrt {-x^{2}+1}+\arcsin \left (x \right )\right )}{2}+\frac {3 \arcsin \left (x \right )^{2} \left (x^{2}-1\right )}{4}+\frac {3 \arcsin \left (x \right ) \left (x \sqrt {-x^{2}+1}+\arcsin \left (x \right )\right )}{4}-\frac {3 \arcsin \left (x \right )^{2}}{8}-\frac {3 x^{2}}{8}-\frac {3 \arcsin \left (x \right )^{4}}{8}\) | \(69\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.58, size = 49, normalized size = 0.67 \begin {gather*} \frac {1}{8} \, \arcsin \left (x\right )^{4} + \frac {3}{8} \, {\left (2 \, x^{2} - 1\right )} \arcsin \left (x\right )^{2} - \frac {3}{8} \, x^{2} - \frac {1}{4} \, {\left (2 \, x \arcsin \left (x\right )^{3} - 3 \, x \arcsin \left (x\right )\right )} \sqrt {-x^{2} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.23, size = 66, normalized size = 0.90 \begin {gather*} \frac {3 x^{2} \operatorname {asin}^{2}{\left (x \right )}}{4} - \frac {3 x^{2}}{8} - \frac {x \sqrt {1 - x^{2}} \operatorname {asin}^{3}{\left (x \right )}}{2} + \frac {3 x \sqrt {1 - x^{2}} \operatorname {asin}{\left (x \right )}}{4} + \frac {\operatorname {asin}^{4}{\left (x \right )}}{8} - \frac {3 \operatorname {asin}^{2}{\left (x \right )}}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.74, size = 60, normalized size = 0.82 \begin {gather*} -\frac {1}{2} \, \sqrt {-x^{2} + 1} x \arcsin \left (x\right )^{3} + \frac {1}{8} \, \arcsin \left (x\right )^{4} + \frac {3}{4} \, {\left (x^{2} - 1\right )} \arcsin \left (x\right )^{2} + \frac {3}{4} \, \sqrt {-x^{2} + 1} x \arcsin \left (x\right ) - \frac {3}{8} \, x^{2} + \frac {3}{8} \, \arcsin \left (x\right )^{2} + \frac {3}{16} \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 = -0.01 \begin {gather*} \int \frac {x^2\,{\mathrm {asin}\left (x\right )}^3}{\sqrt {1-x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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