Optimal. Leaf size=38 \[ \frac {1}{3} \left (1-x^2\right )^{3/2}+\frac {1}{2} x \sqrt {1-x^2}+\frac {1}{2} \sin ^{-1}(x) \]
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Rubi [A] time = 0.01, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {641, 195, 216} \begin {gather*} \frac {1}{3} \left (1-x^2\right )^{3/2}+\frac {1}{2} x \sqrt {1-x^2}+\frac {1}{2} \sin ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
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Rule 195
Rule 216
Rule 641
Rubi steps
\begin {align*} \int (1-x) \sqrt {1-x^2} \, dx &=\frac {1}{3} \left (1-x^2\right )^{3/2}+\int \sqrt {1-x^2} \, dx\\ &=\frac {1}{2} x \sqrt {1-x^2}+\frac {1}{3} \left (1-x^2\right )^{3/2}+\frac {1}{2} \int \frac {1}{\sqrt {1-x^2}} \, dx\\ &=\frac {1}{2} x \sqrt {1-x^2}+\frac {1}{3} \left (1-x^2\right )^{3/2}+\frac {1}{2} \sin ^{-1}(x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 32, normalized size = 0.84 \begin {gather*} \frac {1}{6} \left (3 \sin ^{-1}(x)-\sqrt {1-x^2} \left (2 x^2-3 x-2\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.14, size = 46, normalized size = 1.21 \begin {gather*} \frac {1}{6} \left (-2 x^2+3 x+2\right ) \sqrt {1-x^2}-\tan ^{-1}\left (\frac {\sqrt {1-x^2}}{x+1}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 40, normalized size = 1.05 \begin {gather*} -\frac {1}{6} \, {\left (2 \, x^{2} - 3 \, x - 2\right )} \sqrt {-x^{2} + 1} - \arctan \left (\frac {\sqrt {-x^{2} + 1} - 1}{x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 25, normalized size = 0.66 \begin {gather*} -\frac {1}{6} \, {\left ({\left (2 \, x - 3\right )} x - 2\right )} \sqrt {-x^{2} + 1} + \frac {1}{2} \, \arcsin \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 29, normalized size = 0.76 \begin {gather*} \frac {\sqrt {-x^{2}+1}\, x}{2}+\frac {\arcsin \relax (x )}{2}+\frac {\left (-x^{2}+1\right )^{\frac {3}{2}}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.00, size = 28, normalized size = 0.74 \begin {gather*} \frac {1}{3} \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} + \frac {1}{2} \, \sqrt {-x^{2} + 1} x + \frac {1}{2} \, \arcsin \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 25, normalized size = 0.66 \begin {gather*} \frac {\mathrm {asin}\relax (x)}{2}+\sqrt {1-x^2}\,\left (-\frac {x^2}{3}+\frac {x}{2}+\frac {1}{3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.22, size = 39, normalized size = 1.03 \begin {gather*} - \frac {x^{2} \sqrt {1 - x^{2}}}{3} + \frac {x \sqrt {1 - x^{2}}}{2} + \frac {\sqrt {1 - x^{2}}}{3} + \frac {\operatorname {asin}{\relax (x )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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