Optimal. Leaf size=146 \[ \frac{\sqrt{\frac{4-\left (3-\sqrt{33}\right ) x^2}{4-\left (3+\sqrt{33}\right ) x^2}} \sqrt{\left (3+\sqrt{33}\right ) x^2-4} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{33} x}{\sqrt{\left (3+\sqrt{33}\right ) x^2-4}}\right ),\frac{1}{22} \left (11+\sqrt{33}\right )\right )}{2 \sqrt{2} \sqrt [4]{33} \sqrt{\frac{1}{4-\left (3+\sqrt{33}\right ) x^2}} \sqrt{3 x^4+3 x^2-2}} \]
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Rubi [A] time = 0.0385048, antiderivative size = 146, 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 = {1098} \[ \frac{\sqrt{\frac{4-\left (3-\sqrt{33}\right ) x^2}{4-\left (3+\sqrt{33}\right ) x^2}} \sqrt{\left (3+\sqrt{33}\right ) x^2-4} F\left (\sin ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{33} x}{\sqrt{\left (3+\sqrt{33}\right ) x^2-4}}\right )|\frac{1}{22} \left (11+\sqrt{33}\right )\right )}{2 \sqrt{2} \sqrt [4]{33} \sqrt{\frac{1}{4-\left (3+\sqrt{33}\right ) x^2}} \sqrt{3 x^4+3 x^2-2}} \]
Antiderivative was successfully verified.
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Rule 1098
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{-2+3 x^2+3 x^4}} \, dx &=\frac{\sqrt{\frac{4-\left (3-\sqrt{33}\right ) x^2}{4-\left (3+\sqrt{33}\right ) x^2}} \sqrt{-4+\left (3+\sqrt{33}\right ) x^2} F\left (\sin ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{33} x}{\sqrt{-4+\left (3+\sqrt{33}\right ) x^2}}\right )|\frac{1}{22} \left (11+\sqrt{33}\right )\right )}{2 \sqrt{2} \sqrt [4]{33} \sqrt{\frac{1}{4-\left (3+\sqrt{33}\right ) x^2}} \sqrt{-2+3 x^2+3 x^4}}\\ \end{align*}
Mathematica [C] time = 0.0648364, size = 83, normalized size = 0.57 \[ -\frac{i \sqrt{-6 x^4-6 x^2+4} \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{\frac{6}{3+\sqrt{33}}} x\right ),-\frac{7}{4}-\frac{\sqrt{33}}{4}\right )}{\sqrt{\sqrt{33}-3} \sqrt{3 x^4+3 x^2-2}} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.176, size = 84, normalized size = 0.6 \begin{align*} 2\,{\frac{\sqrt{1- \left ( -1/4\,\sqrt{33}+3/4 \right ){x}^{2}}\sqrt{1- \left ( 1/4\,\sqrt{33}+3/4 \right ){x}^{2}}{\it EllipticF} \left ( 1/2\,\sqrt{3-\sqrt{33}}x,i/4\sqrt{6}+i/4\sqrt{22} \right ) }{\sqrt{3-\sqrt{33}}\sqrt{3\,{x}^{4}+3\,{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} + 3 \, 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{1}{\sqrt{3 \, x^{4} + 3 \, 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} + 3 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} + 3 \, x^{2} - 2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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