Optimal. Leaf size=23 \[ \frac {1}{2} \sqrt {1-x^2} x+\frac {1}{2} \sin ^{-1}(x) \]
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Rubi [A] time = 0.00, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {195, 216} \[ \frac {1}{2} \sqrt {1-x^2} x+\frac {1}{2} \sin ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 195
Rule 216
Rubi steps
\begin {align*} \int \sqrt {1-x^2} \, dx &=\frac {1}{2} x \sqrt {1-x^2}+\frac {1}{2} \int \frac {1}{\sqrt {1-x^2}} \, dx\\ &=\frac {1}{2} x \sqrt {1-x^2}+\frac {1}{2} \sin ^{-1}(x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 20, normalized size = 0.87 \[ \frac {1}{2} \left (\sqrt {1-x^2} x+\sin ^{-1}(x)\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 31, normalized size = 1.35 \[ \frac {1}{2} \, \sqrt {-x^{2} + 1} x - \arctan \left (\frac {\sqrt {-x^{2} + 1} - 1}{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.01, size = 17, normalized size = 0.74 \[ \frac {1}{2} \, \sqrt {-x^{2} + 1} x + \frac {1}{2} \, \arcsin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 18, normalized size = 0.78 \[ \frac {\sqrt {-x^{2}+1}\, x}{2}+\frac {\arcsin \relax (x )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.34, size = 17, normalized size = 0.74 \[ \frac {1}{2} \, \sqrt {-x^{2} + 1} x + \frac {1}{2} \, \arcsin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 17, normalized size = 0.74 \[ \frac {\mathrm {asin}\relax (x)}{2}+\frac {x\,\sqrt {1-x^2}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.21, size = 15, normalized size = 0.65 \[ \frac {x \sqrt {1 - x^{2}}}{2} + \frac {\operatorname {asin}{\relax (x )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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