Optimal. Leaf size=90 \[ \frac{\left (\sqrt{6} x^2+2\right ) \sqrt{\frac{3 x^4+4 x^2+2}{\left (\sqrt{6} x^2+2\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\sqrt [4]{\frac{3}{2}} x\right ),\frac{1}{2}-\frac{1}{\sqrt{6}}\right )}{2 \sqrt [4]{6} \sqrt{-3 x^4-4 x^2-2}} \]
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Rubi [A] time = 0.0094641, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {1103} \[ \frac{\left (\sqrt{6} x^2+2\right ) \sqrt{\frac{3 x^4+4 x^2+2}{\left (\sqrt{6} x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{\frac{3}{2}} x\right )|\frac{1}{2}-\frac{1}{\sqrt{6}}\right )}{2 \sqrt [4]{6} \sqrt{-3 x^4-4 x^2-2}} \]
Antiderivative was successfully verified.
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Rule 1103
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{-2-4 x^2-3 x^4}} \, dx &=\frac{\left (2+\sqrt{6} x^2\right ) \sqrt{\frac{2+4 x^2+3 x^4}{\left (2+\sqrt{6} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{\frac{3}{2}} x\right )|\frac{1}{2}-\frac{1}{\sqrt{6}}\right )}{2 \sqrt [4]{6} \sqrt{-2-4 x^2-3 x^4}}\\ \end{align*}
Mathematica [C] time = 0.0870506, size = 144, normalized size = 1.6 \[ -\frac{i \sqrt{1-\frac{3 x^2}{-2-i \sqrt{2}}} \sqrt{1-\frac{3 x^2}{-2+i \sqrt{2}}} \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{-\frac{3}{-2-i \sqrt{2}}} x\right ),\frac{-2-i \sqrt{2}}{-2+i \sqrt{2}}\right )}{\sqrt{3} \sqrt{-\frac{1}{-2-i \sqrt{2}}} \sqrt{-3 x^4-4 x^2-2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.445, size = 87, normalized size = 1. \begin{align*} 2\,{\frac{\sqrt{1- \left ( -1-i/2\sqrt{2} \right ){x}^{2}}\sqrt{1- \left ( -1+i/2\sqrt{2} \right ){x}^{2}}{\it EllipticF} \left ( 1/2\,\sqrt{-4-2\,i\sqrt{2}}x,1/3\,\sqrt{3-6\,i\sqrt{2}} \right ) }{\sqrt{-4-2\,i\sqrt{2}}\sqrt{-3\,{x}^{4}-4\,{x}^{2}-2}}} \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{-3 \, x^{4} - 4 \, 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{-3 \, x^{4} - 4 \, x^{2} - 2}}{3 \, x^{4} + 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{- 3 x^{4} - 4 x^{2} - 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{-3 \, x^{4} - 4 \, x^{2} - 2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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