Optimal. Leaf size=6 \[ -\text{EllipticF}\left (\cos ^{-1}(x),2\right ) \]
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Rubi [A] time = 0.0060251, antiderivative size = 6, 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 = {420} \[ -F\left (\left .\cos ^{-1}(x)\right |2\right ) \]
Antiderivative was successfully verified.
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Rule 420
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{1-x^2} \sqrt{-1+2 x^2}} \, dx &=-F\left (\left .\cos ^{-1}(x)\right |2\right )\\ \end{align*}
Mathematica [B] time = 0.0245561, size = 27, normalized size = 4.5 \[ \frac{\sqrt{1-2 x^2} \text{EllipticF}\left (\sin ^{-1}(x),2\right )}{\sqrt{2 x^2-1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 25, normalized size = 4.2 \begin{align*}{{\it EllipticF} \left ( x,\sqrt{2} \right ) \sqrt{-2\,{x}^{2}+1}{\frac{1}{\sqrt{2\,{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{2 \, x^{2} - 1} \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{2 \, x^{2} - 1} \sqrt{-x^{2} + 1}}{2 \, x^{4} - 3 \, x^{2} + 1}, 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} - 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{2 \, x^{2} - 1} \sqrt{-x^{2} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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