Optimal. Leaf size=29 \[ \frac {1}{2} x \sqrt {3-x^2}+\frac {3}{2} \sin ^{-1}\left (\frac {x}{\sqrt {3}}\right ) \]
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Rubi [A]
time = 0.00, antiderivative size = 29, 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 {3}{2} \text {ArcSin}\left (\frac {x}{\sqrt {3}}\right )+\frac {1}{2} \sqrt {3-x^2} x \end {gather*}
Antiderivative was successfully verified.
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Rule 201
Rule 222
Rubi steps
\begin {align*} \int \sqrt {3-x^2} \, dx &=\frac {1}{2} x \sqrt {3-x^2}+\frac {3}{2} \int \frac {1}{\sqrt {3-x^2}} \, dx\\ &=\frac {1}{2} x \sqrt {3-x^2}+\frac {3}{2} \sin ^{-1}\left (\frac {x}{\sqrt {3}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 35, normalized size = 1.21 \begin {gather*} \frac {1}{2} x \sqrt {3-x^2}+\frac {3}{2} \tan ^{-1}\left (\frac {x}{\sqrt {3-x^2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 23, normalized size = 0.79
method | result | size |
default | \(\frac {3 \arcsin \left (\frac {x \sqrt {3}}{3}\right )}{2}+\frac {x \sqrt {-x^{2}+3}}{2}\) | \(23\) |
risch | \(-\frac {x \left (x^{2}-3\right )}{2 \sqrt {-x^{2}+3}}+\frac {3 \arcsin \left (\frac {x \sqrt {3}}{3}\right )}{2}\) | \(28\) |
meijerg | \(\frac {3 i \left (-\frac {2 i \sqrt {\pi }\, x \sqrt {3}\, \sqrt {-\frac {x^{2}}{3}+1}}{3}-2 i \sqrt {\pi }\, \arcsin \left (\frac {x \sqrt {3}}{3}\right )\right )}{4 \sqrt {\pi }}\) | \(40\) |
trager | \(\frac {x \sqrt {-x^{2}+3}}{2}+\frac {3 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {-x^{2}+3}+x \right )}{2}\) | \(41\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.78, size = 22, normalized size = 0.76 \begin {gather*} \frac {1}{2} \, \sqrt {-x^{2} + 3} x + \frac {3}{2} \, \arcsin \left (\frac {1}{3} \, \sqrt {3} x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.80, size = 29, normalized size = 1.00 \begin {gather*} \frac {1}{2} \, \sqrt {-x^{2} + 3} x - \frac {3}{2} \, \arctan \left (\frac {\sqrt {-x^{2} + 3}}{x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.07, size = 24, normalized size = 0.83 \begin {gather*} \frac {x \sqrt {3 - x^{2}}}{2} + \frac {3 \operatorname {asin}{\left (\frac {\sqrt {3} x}{3} \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 22, normalized size = 0.76 \begin {gather*} \frac {1}{2} \, \sqrt {-x^{2} + 3} x + \frac {3}{2} \, \arcsin \left (\frac {1}{3} \, \sqrt {3} x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 22, normalized size = 0.76 \begin {gather*} \frac {3\,\mathrm {asin}\left (\frac {\sqrt {3}\,x}{3}\right )}{2}+\frac {x\,\sqrt {3-x^2}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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