Optimal. Leaf size=10 \[ \text{EllipticF}\left (\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right ),-2\right ) \]
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Rubi [A] time = 0.0061151, antiderivative size = 10, 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} \[ F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right ) \]
Antiderivative was successfully verified.
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Rule 419
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{2-x^2} \sqrt{1+x^2}} \, dx &=F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )\\ \end{align*}
Mathematica [C] time = 0.0183213, size = 19, normalized size = 1.9 \[ -\frac{i \text{EllipticF}\left (i \sinh ^{-1}(x),-\frac{1}{2}\right )}{\sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.022, size = 14, normalized size = 1.4 \begin{align*}{\it EllipticF} \left ({\frac{x\sqrt{2}}{2}},i\sqrt{2} \right ) \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{-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{-x^{2} + 2}}{x^{4} - x^{2} - 2}, 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*} \int \frac{1}{\sqrt{2 - x^{2}} \sqrt{x^{2} + 1}}\, dx \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{-x^{2} + 2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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