Optimal. Leaf size=16 \[ \frac{1}{2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{2} x\right ),-\frac{1}{2}\right ) \]
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Rubi [A] time = 0.0066735, antiderivative size = 16, normalized size of antiderivative = 1., 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 = {419} \[ \frac{1}{2} F\left (\sin ^{-1}\left (\sqrt{2} x\right )|-\frac{1}{2}\right ) \]
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{1}{2} F\left (\sin ^{-1}\left (\sqrt{2} x\right )|-\frac{1}{2}\right )\\ \end{align*}
Mathematica [A] time = 0.0062921, size = 16, normalized size = 1. \[ \frac{1}{2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{2} x\right ),-\frac{1}{2}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.019, size = 15, normalized size = 0.9 \begin{align*}{\frac{{\it EllipticF} \left ( x\sqrt{2},{\frac{i}{2}}\sqrt{2} \right ) }{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} + 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 = 8.5156, size = 41, normalized size = 2.56 \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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