Optimal. Leaf size=18 \[ \frac {x}{2 \sqrt {\frac {1}{2-x^2}}} \]
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Rubi [A]
time = 0.01, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {1972, 391}
\begin {gather*} \frac {x}{2 \sqrt {\frac {1}{2-x^2}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 391
Rule 1972
Rubi steps
\begin {align*} \int \left (1-x^2\right ) \sqrt {\frac {1}{2-x^2}} \, dx &=\left (\sqrt {\frac {1}{2-x^2}} \sqrt {2-x^2}\right ) \int \frac {1-x^2}{\sqrt {2-x^2}} \, dx\\ &=\frac {x}{2 \sqrt {\frac {1}{2-x^2}}}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 18, normalized size = 1.00 \begin {gather*} \frac {x}{2 \sqrt {\frac {1}{2-x^2}}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.23, size = 20, normalized size = 1.11
method | result | size |
gosper | \(-\frac {\left (x^{2}-2\right ) x \sqrt {-\frac {1}{x^{2}-2}}}{2}\) | \(20\) |
default | \(-\frac {\left (x^{2}-2\right ) x \sqrt {-\frac {1}{x^{2}-2}}}{2}\) | \(20\) |
trager | \(-\frac {\left (x^{2}-2\right ) x \sqrt {-\frac {1}{x^{2}-2}}}{2}\) | \(20\) |
risch | \(-\frac {\left (x^{2}-2\right ) x \sqrt {-\frac {1}{x^{2}-2}}}{2}\) | \(20\) |
meijerg | \(\sqrt {\frac {1}{-x^{2}+2}}\, \sqrt {-x^{2}+2}\, \arcsin \left (\frac {\sqrt {2}\, x}{2}\right )-\frac {i \sqrt {\frac {1}{-x^{2}+2}}\, \sqrt {-x^{2}+2}\, \left (\frac {i \sqrt {\pi }\, x \sqrt {2}\, \sqrt {1-\frac {x^{2}}{2}}}{2}-i \sqrt {\pi }\, \arcsin \left (\frac {\sqrt {2}\, x}{2}\right )\right )}{\sqrt {\pi }}\) | \(89\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 20, normalized size = 1.11 \begin {gather*} -\frac {1}{2} \, {\left (x^{3} - 2 \, x\right )} \sqrt {-\frac {1}{x^{2} - 2}} \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. 29 vs.
\(2 (12) = 24\).
time = 0.41, size = 29, normalized size = 1.61 \begin {gather*} - \frac {x^{3} \sqrt {- \frac {1}{x^{2} - 2}}}{2} + x \sqrt {- \frac {1}{x^{2} - 2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.88, size = 18, normalized size = 1.00 \begin {gather*} -\frac {1}{2} \, \sqrt {-x^{2} + 2} x \mathrm {sgn}\left (x^{2} - 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.50, size = 19, normalized size = 1.06 \begin {gather*} -\frac {x\,\left (x^2-2\right )\,\sqrt {-\frac {1}{x^2-2}}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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