Optimal. Leaf size=90 \[ \frac{\left (\sqrt{6} x^2+3\right ) \sqrt{\frac{2 x^4-6 x^2+3}{\left (\sqrt{6} x^2+3\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\sqrt [4]{\frac{2}{3}} x\right ),\frac{1}{4} \left (2+\sqrt{6}\right )\right )}{2 \sqrt [4]{6} \sqrt{2 x^4-6 x^2+3}} \]
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Rubi [A] time = 0.0095625, 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 = {1096} \[ \frac{\left (\sqrt{6} x^2+3\right ) \sqrt{\frac{2 x^4-6 x^2+3}{\left (\sqrt{6} x^2+3\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{\frac{2}{3}} x\right )|\frac{1}{4} \left (2+\sqrt{6}\right )\right )}{2 \sqrt [4]{6} \sqrt{2 x^4-6 x^2+3}} \]
Antiderivative was successfully verified.
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Rule 1096
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{3-6 x^2+2 x^4}} \, dx &=\frac{\left (3+\sqrt{6} x^2\right ) \sqrt{\frac{3-6 x^2+2 x^4}{\left (3+\sqrt{6} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{\frac{2}{3}} x\right )|\frac{1}{4} \left (2+\sqrt{6}\right )\right )}{2 \sqrt [4]{6} \sqrt{3-6 x^2+2 x^4}}\\ \end{align*}
Mathematica [A] time = 0.0748645, size = 81, normalized size = 0.9 \[ \frac{\sqrt{-2 x^2-\sqrt{3}+3} \sqrt{\left (\sqrt{3}-3\right ) x^2+3} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{1+\frac{1}{\sqrt{3}}} x\right ),2-\sqrt{3}\right )}{\sqrt{6} \sqrt{2 x^4-6 x^2+3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.202, size = 82, normalized size = 0.9 \begin{align*} 3\,{\frac{\sqrt{1- \left ( 1+1/3\,\sqrt{3} \right ){x}^{2}}\sqrt{1- \left ( 1-1/3\,\sqrt{3} \right ){x}^{2}}{\it EllipticF} \left ( 1/3\,x\sqrt{9+3\,\sqrt{3}},1/2\,\sqrt{6}-1/2\,\sqrt{2} \right ) }{\sqrt{9+3\,\sqrt{3}}\sqrt{2\,{x}^{4}-6\,{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} - 6 \, 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{1}{\sqrt{2 \, x^{4} - 6 \, 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} - 6 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} - 6 \, x^{2} + 3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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