Optimal. Leaf size=61 \[ \frac {3 x^2}{16}+\frac {x^4}{16}-\frac {3}{8} x \sqrt {1-x^2} \sin ^{-1}(x)-\frac {1}{4} x^3 \sqrt {1-x^2} \sin ^{-1}(x)+\frac {3}{16} \sin ^{-1}(x)^2 \]
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Rubi [A]
time = 0.07, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {4795, 4737, 30}
\begin {gather*} \frac {x^4}{16}+\frac {3 x^2}{16}-\frac {3}{8} \sqrt {1-x^2} x \sin ^{-1}(x)-\frac {1}{4} \sqrt {1-x^2} x^3 \sin ^{-1}(x)+\frac {3}{16} \sin ^{-1}(x)^2 \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 4737
Rule 4795
Rubi steps
\begin {align*} \int \frac {x^4 \sin ^{-1}(x)}{\sqrt {1-x^2}} \, dx &=-\frac {1}{4} x^3 \sqrt {1-x^2} \sin ^{-1}(x)+\frac {\int x^3 \, dx}{4}+\frac {3}{4} \int \frac {x^2 \sin ^{-1}(x)}{\sqrt {1-x^2}} \, dx\\ &=\frac {x^4}{16}-\frac {3}{8} x \sqrt {1-x^2} \sin ^{-1}(x)-\frac {1}{4} x^3 \sqrt {1-x^2} \sin ^{-1}(x)+\frac {3 \int x \, dx}{8}+\frac {3}{8} \int \frac {\sin ^{-1}(x)}{\sqrt {1-x^2}} \, dx\\ &=\frac {3 x^2}{16}+\frac {x^4}{16}-\frac {3}{8} x \sqrt {1-x^2} \sin ^{-1}(x)-\frac {1}{4} x^3 \sqrt {1-x^2} \sin ^{-1}(x)+\frac {3}{16} \sin ^{-1}(x)^2\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 43, normalized size = 0.70 \begin {gather*} \frac {1}{16} \left (x^2 \left (3+x^2\right )-2 x \sqrt {1-x^2} \left (3+2 x^2\right ) \sin ^{-1}(x)+3 \sin ^{-1}(x)^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 2.05, size = 47, normalized size = 0.77 \begin {gather*} \frac {-3 x \text {ArcSin}\left [x\right ] \sqrt {1-x^2}}{8}+\frac {3 x^2}{16}-\frac {x^3 \text {ArcSin}\left [x\right ] \sqrt {1-x^2}}{4}+\frac {x^4}{16}+\frac {3 \text {ArcSin}\left [x\right ]^2}{16} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 54, normalized size = 0.89
method | result | size |
default | \(\frac {\arcsin \left (x \right ) \left (-2 \sqrt {-x^{2}+1}\, x^{3}-3 x \sqrt {-x^{2}+1}+3 \arcsin \left (x \right )\right )}{8}-\frac {3 \arcsin \left (x \right )^{2}}{16}+\frac {\left (2 x^{2}+3\right )^{2}}{64}\) | \(54\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.33, size = 52, normalized size = 0.85 \begin {gather*} \frac {1}{16} \, x^{4} + \frac {3}{16} \, x^{2} - \frac {1}{8} \, {\left (2 \, \sqrt {-x^{2} + 1} x^{3} + 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 = 0.32, size = 39, normalized size = 0.64 \begin {gather*} \frac {1}{16} \, x^{4} - \frac {1}{8} \, {\left (2 \, x^{3} + 3 \, x\right )} \sqrt {-x^{2} + 1} \arcsin \left (x\right ) + \frac {3}{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 = 0.27, size = 53, normalized size = 0.87 \begin {gather*} \frac {x^{4}}{16} - \frac {x^{3} \sqrt {1 - x^{2}} \operatorname {asin}{\left (x \right )}}{4} + \frac {3 x^{2}}{16} - \frac {3 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 = 0.01, size = 77, normalized size = 1.26 \begin {gather*} \frac {\left (-x^{2}+1\right )^{2}}{16}+\frac {3}{16} \arcsin ^{2}x-\frac {5}{8} x \sqrt {-x^{2}+1} \arcsin x+\frac {1}{4} x \sqrt {-x^{2}+1} \left (-x^{2}+1\right ) \arcsin x-\frac {5}{16} \left (-x^{2}+1\right )+\frac 1{8}+\frac {1}{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 \frac {x^4\,\mathrm {asin}\left (x\right )}{\sqrt {1-x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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