Optimal. Leaf size=12 \[ \frac{\text{EllipticF}\left (\sin ^{-1}(x),-\frac{1}{3}\right )}{\sqrt{3}} \]
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Rubi [A] time = 0.0103334, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {1095, 419} \[ \frac{F\left (\sin ^{-1}(x)|-\frac{1}{3}\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 1095
Rule 419
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{3-2 x^2-x^4}} \, dx &=2 \int \frac{1}{\sqrt{2-2 x^2} \sqrt{6+2 x^2}} \, dx\\ &=\frac{F\left (\sin ^{-1}(x)|-\frac{1}{3}\right )}{\sqrt{3}}\\ \end{align*}
Mathematica [C] time = 0.0174583, size = 18, normalized size = 1.5 \[ -i \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{x}{\sqrt{3}}\right ),-3\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.044, size = 43, normalized size = 3.6 \begin{align*}{\frac{{\it EllipticF} \left ( x,{\frac{i}{3}}\sqrt{3} \right ) }{3}\sqrt{-{x}^{2}+1}\sqrt{3\,{x}^{2}+9}{\frac{1}{\sqrt{-{x}^{4}-2\,{x}^{2}+3}}}} \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^{4} - 2 \, x^{2} + 3}}\,{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^{4} - 2 \, x^{2} + 3}}{x^{4} + 2 \, x^{2} - 3}, 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{- x^{4} - 2 x^{2} + 3}}\, 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^{4} - 2 \, x^{2} + 3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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