Optimal. Leaf size=10 \[ \frac{\text{EllipticF}\left (\sin ^{-1}(x),-2\right )}{\sqrt{2}} \]
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Rubi [A] time = 0.007433, 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{1-x^2} \sqrt{2+4 x^2}} \, dx &=\frac{F\left (\left .\sin ^{-1}(x)\right |-2\right )}{\sqrt{2}}\\ \end{align*}
Mathematica [C] time = 0.030298, size = 58, normalized size = 5.8 \[ -\frac{i \sqrt{1-x^2} \sqrt{2 x^2+1} \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{2} x\right ),-\frac{1}{2}\right )}{2 \sqrt{-2 x^4+x^2+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.023, size = 14, normalized size = 1.4 \begin{align*}{\frac{{\it EllipticF} \left ( x,i\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{4 \, x^{2} + 2} \sqrt{-x^{2} + 1}}\,{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{4 \, x^{2} + 2} \sqrt{-x^{2} + 1}}{2 \,{\left (2 \, x^{4} - x^{2} - 1\right )}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\sqrt{2} \int \frac{1}{\sqrt{1 - x^{2}} \sqrt{2 x^{2} + 1}}\, dx}{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{4 \, x^{2} + 2} \sqrt{-x^{2} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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