Optimal. Leaf size=40 \[ \frac {\sqrt {1-2 x^2} E\left (\sin ^{-1}\left (\sqrt {2} x\right )|\frac {1}{2}\right )}{\sqrt {2} \sqrt {2 x^2-1}} \]
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Rubi [A] time = 0.01, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {427, 424} \[ \frac {\sqrt {1-2 x^2} E\left (\sin ^{-1}\left (\sqrt {2} x\right )|\frac {1}{2}\right )}{\sqrt {2} \sqrt {2 x^2-1}} \]
Antiderivative was successfully verified.
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Rule 424
Rule 427
Rubi steps
\begin {align*} \int \frac {\sqrt {1-x^2}}{\sqrt {-1+2 x^2}} \, dx &=\frac {\sqrt {1-2 x^2} \int \frac {\sqrt {1-x^2}}{\sqrt {1-2 x^2}} \, dx}{\sqrt {-1+2 x^2}}\\ &=\frac {\sqrt {1-2 x^2} E\left (\sin ^{-1}\left (\sqrt {2} x\right )|\frac {1}{2}\right )}{\sqrt {2} \sqrt {-1+2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 35, normalized size = 0.88 \[ \frac {\sqrt {1-2 x^2} E\left (\sin ^{-1}\left (\sqrt {2} x\right )|\frac {1}{2}\right )}{\sqrt {4 x^2-2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.76, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-x^{2} + 1}}{\sqrt {2 \, x^{2} - 1}}, x\right ) \]
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 \[ \int \frac {\sqrt {-x^{2} + 1}}{\sqrt {2 \, x^{2} - 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 32, normalized size = 0.80 \[ \frac {\left (\EllipticE \left (x , \sqrt {2}\right )+\EllipticF \left (x , \sqrt {2}\right )\right ) \sqrt {-2 x^{2}+1}}{2 \sqrt {2 x^{2}-1}} \]
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 \[ \int \frac {\sqrt {-x^{2} + 1}}{\sqrt {2 \, x^{2} - 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\sqrt {1-x^2}}{\sqrt {2\,x^2-1}} \,d x \]
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 \[ \int \frac {\sqrt {- \left (x - 1\right ) \left (x + 1\right )}}{\sqrt {2 x^{2} - 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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