Optimal. Leaf size=12 \[ -\operatorname {EllipticF}\left (\cos ^{-1}\left (\frac {x}{\sqrt {2}}\right ),2\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {420} \[ -F\left (\left .\cos ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |2\right ) \]
Antiderivative was successfully verified.
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Rule 420
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {2-x^2} \sqrt {-1+x^2}} \, dx &=-F\left (\left .\cos ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |2\right )\\ \end {align*}
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Mathematica [B] time = 0.02, size = 47, normalized size = 3.92 \[ \frac {\sqrt {1-x^2} \sqrt {1-\frac {x^2}{2}} \operatorname {EllipticF}\left (\sin ^{-1}(x),\frac {1}{2}\right )}{\sqrt {-x^4+3 x^2-2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {x^{2} - 1} \sqrt {-x^{2} + 2}}{x^{4} - 3 \, x^{2} + 2}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {x^{2} - 1} \sqrt {-x^{2} + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 28, normalized size = 2.33 \[ \frac {\sqrt {-x^{2}+1}\, \EllipticF \left (\frac {\sqrt {2}\, x}{2}, \sqrt {2}\right )}{\sqrt {x^{2}-1}} \]
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 \[ \int \frac {1}{\sqrt {x^{2} - 1} \sqrt {-x^{2} + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.08 \[ \int \frac {1}{\sqrt {x^2-1}\,\sqrt {2-x^2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {\left (x - 1\right ) \left (x + 1\right )} \sqrt {2 - x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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