Optimal. Leaf size=36 \[ \frac{\sqrt{x^2+1} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{2} x\right ),-\frac{1}{2}\right )}{2 \sqrt{-x^2-1}} \]
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Rubi [A] time = 0.0152354, antiderivative size = 36, normalized size of antiderivative = 1., 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 = {421, 419} \[ \frac{\sqrt{x^2+1} F\left (\sin ^{-1}\left (\sqrt{2} x\right )|-\frac{1}{2}\right )}{2 \sqrt{-x^2-1}} \]
Antiderivative was successfully verified.
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Rule 421
Rule 419
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{2-4 x^2} \sqrt{-1-x^2}} \, dx &=\frac{\sqrt{1+x^2} \int \frac{1}{\sqrt{2-4 x^2} \sqrt{1+x^2}} \, dx}{\sqrt{-1-x^2}}\\ &=\frac{\sqrt{1+x^2} F\left (\sin ^{-1}\left (\sqrt{2} x\right )|-\frac{1}{2}\right )}{2 \sqrt{-1-x^2}}\\ \end{align*}
Mathematica [A] time = 0.0298108, size = 36, normalized size = 1. \[ \frac{\sqrt{x^2+1} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{2} x\right ),-\frac{1}{2}\right )}{2 \sqrt{-x^2-1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.019, size = 34, normalized size = 0.9 \begin{align*}{{\frac{i}{2}}{\it EllipticF} \left ( ix,i\sqrt{2} \right ) \sqrt{2}\sqrt{-{x}^{2}-1}{\frac{1}{\sqrt{{x}^{2}+1}}}} \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 = 8.71455, size = 44, normalized size = 1.22 \begin{align*} \frac{\sqrt{2} \left (\begin{cases} - \frac{\sqrt{2} i 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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