Optimal. Leaf size=12 \[ \frac{\text{EllipticF}\left (\sin ^{-1}(x),\frac{1}{2}\right )}{\sqrt{2}} \]
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Rubi [A] time = 0.0074622, antiderivative size = 12, 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} \[ \frac{F\left (\sin ^{-1}(x)|\frac{1}{2}\right )}{\sqrt{2}} \]
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 [A] time = 0.005636, size = 12, normalized size = 1. \[ \frac{\text{EllipticF}\left (\sin ^{-1}(x),\frac{1}{2}\right )}{\sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.022, size = 13, normalized size = 1.1 \begin{align*}{\frac{\sqrt{2}}{2}{\it EllipticF} \left ( x,{\frac{\sqrt{2}}{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} + 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} - 3 \, 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{- \left (x - 1\right ) \left (x + 1\right )} \sqrt{2 - x^{2}}}\, 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} + 2} \sqrt{-x^{2} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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