Optimal. Leaf size=10 \[ \frac{\text{EllipticF}\left (\sin ^{-1}(x),2\right )}{\sqrt{2}} \]
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Rubi [A] time = 0.0073752, antiderivative size = 10, normalized size of antiderivative = 1., 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*}
Mathematica [A] time = 0.0089306, size = 10, normalized size = 1. \[ \frac{\text{EllipticF}\left (\sin ^{-1}(x),2\right )}{\sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.019, size = 11, normalized size = 1.1 \begin{align*}{\frac{{\it EllipticF} \left ( x,\sqrt{2} \right ) \sqrt{2}}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-x^{2} + 1} \sqrt{-4 \, x^{2} + 2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 8.18606, size = 39, normalized size = 3.9 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-x^{2} + 1} \sqrt{-4 \, x^{2} + 2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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