Optimal. Leaf size=23 \[ \frac {1}{2} x \sqrt {1-x^2}+\frac {1}{2} \sin ^{-1}(x) \]
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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 = {201, 222}
\begin {gather*} \frac {\text {ArcSin}(x)}{2}+\frac {1}{2} \sqrt {1-x^2} x \end {gather*}
Antiderivative was successfully verified.
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Rule 201
Rule 222
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 = 0.03, size = 37, normalized size = 1.61 \begin {gather*} \frac {1}{2} x \sqrt {1-x^2}-\tan ^{-1}\left (\frac {\sqrt {1-x^2}}{1+x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 18, normalized size = 0.78
method | result | size |
default | \(\frac {\arcsin \left (x \right )}{2}+\frac {x \sqrt {-x^{2}+1}}{2}\) | \(18\) |
risch | \(-\frac {x \left (x^{2}-1\right )}{2 \sqrt {-x^{2}+1}}+\frac {\arcsin \left (x \right )}{2}\) | \(23\) |
meijerg | \(\frac {i \left (-2 i \sqrt {\pi }\, x \sqrt {-x^{2}+1}-2 i \sqrt {\pi }\, \arcsin \left (x \right )\right )}{4 \sqrt {\pi }}\) | \(32\) |
trager | \(\frac {x \sqrt {-x^{2}+1}}{2}+\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {-x^{2}+1}+x \right )}{2}\) | \(41\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.51, size = 17, normalized size = 0.74 \begin {gather*} \frac {1}{2} \, \sqrt {-x^{2} + 1} x + \frac {1}{2} \, \arcsin \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.84, size = 31, normalized size = 1.35 \begin {gather*} \frac {1}{2} \, \sqrt {-x^{2} + 1} x - \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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Sympy [A]
time = 0.06, size = 15, normalized size = 0.65 \begin {gather*} \frac {x \sqrt {1 - x^{2}}}{2} + \frac {\operatorname {asin}{\left (x \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.47, size = 17, normalized size = 0.74 \begin {gather*} \frac {1}{2} \, \sqrt {-x^{2} + 1} x + \frac {1}{2} \, \arcsin \left (x\right ) \end {gather*}
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 \begin {gather*} \frac {\mathrm {asin}\left (x\right )}{2}+\frac {x\,\sqrt {1-x^2}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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