Optimal. Leaf size=24 \[ \frac {x}{\sqrt {3-x^2}}-\sin ^{-1}\left (\frac {x}{\sqrt {3}}\right ) \]
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Rubi [A]
time = 0.00, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {294, 222}
\begin {gather*} \frac {x}{\sqrt {3-x^2}}-\text {ArcSin}\left (\frac {x}{\sqrt {3}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 222
Rule 294
Rubi steps
\begin {align*} \int \frac {x^2}{\left (3-x^2\right )^{3/2}} \, dx &=\frac {x}{\sqrt {3-x^2}}-\int \frac {1}{\sqrt {3-x^2}} \, dx\\ &=\frac {x}{\sqrt {3-x^2}}-\sin ^{-1}\left (\frac {x}{\sqrt {3}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 30, normalized size = 1.25 \begin {gather*} \frac {x}{\sqrt {3-x^2}}-\tan ^{-1}\left (\frac {x}{\sqrt {3-x^2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 22, normalized size = 0.92
method | result | size |
default | \(-\arcsin \left (\frac {x \sqrt {3}}{3}\right )+\frac {x}{\sqrt {-x^{2}+3}}\) | \(22\) |
risch | \(-\arcsin \left (\frac {x \sqrt {3}}{3}\right )+\frac {x}{\sqrt {-x^{2}+3}}\) | \(22\) |
meijerg | \(\frac {i \left (-\frac {i \sqrt {\pi }\, x \sqrt {3}}{3 \sqrt {-\frac {x^{2}}{3}+1}}+i \sqrt {\pi }\, \arcsin \left (\frac {x \sqrt {3}}{3}\right )\right )}{\sqrt {\pi }}\) | \(40\) |
trager | \(-\frac {x \sqrt {-x^{2}+3}}{x^{2}-3}-\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {-x^{2}+3}+x \right )\) | \(48\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.24, size = 21, normalized size = 0.88 \begin {gather*} \frac {x}{\sqrt {-x^{2} + 3}} - \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 = 0.62, size = 41, normalized size = 1.71 \begin {gather*} \frac {{\left (x^{2} - 3\right )} \arctan \left (\frac {\sqrt {-x^{2} + 3}}{x}\right ) - \sqrt {-x^{2} + 3} x}{x^{2} - 3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 49 vs.
\(2 (19) = 38\).
time = 0.18, size = 49, normalized size = 2.04 \begin {gather*} - \frac {x^{2} \operatorname {asin}{\left (\frac {\sqrt {3} x}{3} \right )}}{x^{2} - 3} - \frac {x \sqrt {3 - x^{2}}}{x^{2} - 3} + \frac {3 \operatorname {asin}{\left (\frac {\sqrt {3} x}{3} \right )}}{x^{2} - 3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.64, size = 29, normalized size = 1.21 \begin {gather*} -\frac {\sqrt {-x^{2} + 3} x}{x^{2} - 3} - \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 = 0.30, size = 54, normalized size = 2.25 \begin {gather*} -\mathrm {asin}\left (\frac {\sqrt {3}\,x}{3}\right )-\frac {\sqrt {3-x^2}}{2\,\left (x-\sqrt {3}\right )}-\frac {\sqrt {3-x^2}}{2\,\left (x+\sqrt {3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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