Optimal. Leaf size=48 \[ \sqrt{\frac{1}{6} \left (2+\sqrt{10}\right )} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{1}{2} \left (\sqrt{10}-2\right )} x\right ),\frac{1}{3} \left (-7-2 \sqrt{10}\right )\right ) \]
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Rubi [A] time = 0.133588, antiderivative size = 48, 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} \[ \sqrt{\frac{1}{6} \left (2+\sqrt{10}\right )} F\left (\sin ^{-1}\left (\sqrt{\frac{1}{2} \left (-2+\sqrt{10}\right )} x\right )|\frac{1}{3} \left (-7-2 \sqrt{10}\right )\right ) \]
Antiderivative was successfully verified.
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Rule 1095
Rule 419
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{2+4 x^2-3 x^4}} \, dx &=\left (2 \sqrt{3}\right ) \int \frac{1}{\sqrt{4+2 \sqrt{10}-6 x^2} \sqrt{-4+2 \sqrt{10}+6 x^2}} \, dx\\ &=\sqrt{\frac{1}{6} \left (2+\sqrt{10}\right )} F\left (\sin ^{-1}\left (\sqrt{\frac{1}{2} \left (-2+\sqrt{10}\right )} x\right )|\frac{1}{3} \left (-7-2 \sqrt{10}\right )\right )\\ \end{align*}
Mathematica [C] time = 0.0562925, size = 49, normalized size = 1.02 \[ -\frac{i \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{1+\sqrt{\frac{5}{2}}} x\right ),\frac{1}{3} \left (2 \sqrt{10}-7\right )\right )}{\sqrt{2+\sqrt{10}}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.24, size = 84, normalized size = 1.8 \begin{align*} 2\,{\frac{\sqrt{1- \left ( -1+1/2\,\sqrt{10} \right ){x}^{2}}\sqrt{1- \left ( -1-1/2\,\sqrt{10} \right ){x}^{2}}{\it EllipticF} \left ( 1/2\,x\sqrt{-4+2\,\sqrt{10}},i/3\sqrt{6}+i/3\sqrt{15} \right ) }{\sqrt{-4+2\,\sqrt{10}}\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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