Optimal. Leaf size=10 \[ \frac {\operatorname {EllipticF}\left (\sin ^{-1}(x),2\right )}{\sqrt {2}} \]
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Rubi [A] time = 0.01, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {419} \[ \frac {F\left (\left .\sin ^{-1}(x)\right |2\right )}{\sqrt {2}} \]
Antiderivative was successfully verified.
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Rule 419
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {2-4 x^2} \sqrt {1-x^2}} \, dx &=\frac {F\left (\left .\sin ^{-1}(x)\right |2\right )}{\sqrt {2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 10, normalized size = 1.00 \[ \frac {\operatorname {EllipticF}\left (\sin ^{-1}(x),2\right )}{\sqrt {2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.63, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-x^{2} + 1} \sqrt {-4 \, x^{2} + 2}}{2 \, {\left (2 \, x^{4} - 3 \, x^{2} + 1\right )}}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {-x^{2} + 1} \sqrt {-4 \, x^{2} + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 11, normalized size = 1.10 \[ \frac {\sqrt {2}\, \EllipticF \left (x , \sqrt {2}\right )}{2} \]
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 \[ \int \frac {1}{\sqrt {-x^{2} + 1} \sqrt {-4 \, x^{2} + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.10 \[ \int \frac {1}{\sqrt {1-x^2}\,\sqrt {2-4\,x^2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.85, size = 39, normalized size = 3.90 \[ \frac {\sqrt {2} \left (\begin {cases} \frac {\sqrt {2} F\left (\operatorname {asin}{\left (\sqrt {2} x \right )}\middle | \frac {1}{2}\right )}{2} & \text {for}\: x > - \frac {\sqrt {2}}{2} \wedge x < \frac {\sqrt {2}}{2} \end {cases}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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