Optimal. Leaf size=12 \[ \frac{\text{EllipticF}\left (\sin ^{-1}(x),-\frac{2}{3}\right )}{\sqrt{3}} \]
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Rubi [A] time = 0.010715, 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} \[ \frac{F\left (\sin ^{-1}(x)|-\frac{2}{3}\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 1095
Rule 419
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{3-x^2-2 x^4}} \, dx &=\left (2 \sqrt{2}\right ) \int \frac{1}{\sqrt{4-4 x^2} \sqrt{6+4 x^2}} \, dx\\ &=\frac{F\left (\sin ^{-1}(x)|-\frac{2}{3}\right )}{\sqrt{3}}\\ \end{align*}
Mathematica [C] time = 0.0224095, size = 65, normalized size = 5.42 \[ -\frac{i \sqrt{1-x^2} \sqrt{2 x^2+3} \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{\frac{2}{3}} x\right ),-\frac{3}{2}\right )}{\sqrt{2} \sqrt{-2 x^4-x^2+3}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.049, size = 43, normalized size = 3.6 \begin{align*}{\frac{{\it EllipticF} \left ( x,{\frac{i}{3}}\sqrt{6} \right ) }{3}\sqrt{-{x}^{2}+1}\sqrt{6\,{x}^{2}+9}{\frac{1}{\sqrt{-2\,{x}^{4}-{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{-2 \, x^{4} - 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{-2 \, x^{4} - x^{2} + 3}}{2 \, x^{4} + 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{- 2 x^{4} - 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{-2 \, x^{4} - x^{2} + 3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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