Optimal. Leaf size=12 \[ \frac{\text{EllipticF}\left (\sin ^{-1}(x),-\frac{1}{2}\right )}{\sqrt{2}} \]
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Rubi [A] time = 0.0065566, antiderivative size = 12, 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} \[ \frac{F\left (\sin ^{-1}(x)|-\frac{1}{2}\right )}{\sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 419
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{1-x^2} \sqrt{2+x^2}} \, dx &=\frac{F\left (\sin ^{-1}(x)|-\frac{1}{2}\right )}{\sqrt{2}}\\ \end{align*}
Mathematica [C] time = 0.0194594, size = 18, normalized size = 1.5 \[ -i \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right ),-2\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.022, size = 14, normalized size = 1.2 \begin{align*}{\frac{{\it EllipticF} \left ( x,{\frac{i}{2}}\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} + 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{x^{2} + 2} \sqrt{-x^{2} + 1}}{x^{4} + x^{2} - 2}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.17779, size = 19, normalized size = 1.58 \begin{align*} \begin{cases} \frac{\sqrt{2} F\left (\operatorname{asin}{\left (x \right )}\middle | - \frac{1}{2}\right )}{2} & \text{for}\: x > -1 \wedge x < 1 \end{cases} \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} + 2} \sqrt{-x^{2} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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