Optimal. Leaf size=92 \[ \frac{\left (\sqrt{6} x^2+3\right ) \sqrt{\frac{2 x^4+2 x^2+3}{\left (\sqrt{6} x^2+3\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\sqrt [4]{\frac{2}{3}} x\right ),\frac{1}{12} \left (6-\sqrt{6}\right )\right )}{2 \sqrt [4]{6} \sqrt{2 x^4+2 x^2+3}} \]
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Rubi [A] time = 0.0095349, antiderivative size = 92, 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} \[ \frac{\left (\sqrt{6} x^2+3\right ) \sqrt{\frac{2 x^4+2 x^2+3}{\left (\sqrt{6} x^2+3\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{\frac{2}{3}} x\right )|\frac{1}{12} \left (6-\sqrt{6}\right )\right )}{2 \sqrt [4]{6} \sqrt{2 x^4+2 x^2+3}} \]
Antiderivative was successfully verified.
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Rule 1103
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{3+2 x^2+2 x^4}} \, dx &=\frac{\left (3+\sqrt{6} x^2\right ) \sqrt{\frac{3+2 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}{12} \left (6-\sqrt{6}\right )\right )}{2 \sqrt [4]{6} \sqrt{3+2 x^2+2 x^4}}\\ \end{align*}
Mathematica [C] time = 0.0848206, size = 144, normalized size = 1.57 \[ -\frac{i \sqrt{1-\frac{2 x^2}{-1-i \sqrt{5}}} \sqrt{1-\frac{2 x^2}{-1+i \sqrt{5}}} \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{-\frac{2}{-1-i \sqrt{5}}} x\right ),\frac{-1-i \sqrt{5}}{-1+i \sqrt{5}}\right )}{\sqrt{2} \sqrt{-\frac{1}{-1-i \sqrt{5}}} \sqrt{2 x^4+2 x^2+3}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.778, size = 87, normalized size = 1. \begin{align*} 3\,{\frac{\sqrt{1- \left ( -1/3+i/3\sqrt{5} \right ){x}^{2}}\sqrt{1- \left ( -1/3-i/3\sqrt{5} \right ){x}^{2}}{\it EllipticF} \left ( 1/3\,x\sqrt{-3+3\,i\sqrt{5}},1/3\,\sqrt{-6+3\,i\sqrt{5}} \right ) }{\sqrt{-3+3\,i\sqrt{5}}\sqrt{2\,{x}^{4}+2\,{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} + 2 \, 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} + 2 \, 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} + 2 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} + 2 \, x^{2} + 3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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