Optimal. Leaf size=18 \[ \frac {x}{2 \sqrt {\frac {1}{2-x^2}}} \]
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Rubi [A] time = 0.05, 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 = {6720, 383} \[ \frac {x}{2 \sqrt {\frac {1}{2-x^2}}} \]
Antiderivative was successfully verified.
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Rule 383
Rule 6720
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 = 0.01, size = 18, normalized size = 1.00 \[ \frac {x}{2 \sqrt {\frac {1}{2-x^2}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.79, size = 20, normalized size = 1.11 \[ -\frac {1}{2} \, {\left (x^{3} - 2 \, x\right )} \sqrt {-\frac {1}{x^{2} - 2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.48, size = 18, normalized size = 1.00 \[ -\frac {1}{2} \, \sqrt {-x^{2} + 2} x \mathrm {sgn}\left (x^{2} - 2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 20, normalized size = 1.11 \[ -\frac {\left (x^{2}-2\right ) \sqrt {-\frac {1}{x^{2}-2}}\, x}{2} \]
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 \[ -\int {\left (x^{2} - 1\right )} \sqrt {-\frac {1}{x^{2} - 2}}\,{d x} \]
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 \[ -\frac {x\,\left (x^2-2\right )\,\sqrt {-\frac {1}{x^2-2}}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.50, size = 26, normalized size = 1.44 \[ - \frac {x^{3} \sqrt {\frac {1}{2 - x^{2}}}}{2} + x \sqrt {\frac {1}{2 - x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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