Optimal. Leaf size=104 \[ \frac{\sqrt{\frac{\left (3-\sqrt{3}\right ) x^2+3}{\left (3+\sqrt{3}\right ) x^2+3}} \left (\left (3+\sqrt{3}\right ) x^2+3\right ) \text{EllipticF}\left (\tan ^{-1}\left (\sqrt{\frac{1}{3} \left (3+\sqrt{3}\right )} x\right ),\sqrt{3}-1\right )}{\sqrt{3 \left (3+\sqrt{3}\right )} \sqrt{2 x^4+6 x^2+3}} \]
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Rubi [A] time = 0.0642674, antiderivative size = 104, 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 = {1099} \[ \frac{\sqrt{\frac{\left (3-\sqrt{3}\right ) x^2+3}{\left (3+\sqrt{3}\right ) x^2+3}} \left (\left (3+\sqrt{3}\right ) x^2+3\right ) F\left (\tan ^{-1}\left (\sqrt{\frac{1}{3} \left (3+\sqrt{3}\right )} x\right )|-1+\sqrt{3}\right )}{\sqrt{3 \left (3+\sqrt{3}\right )} \sqrt{2 x^4+6 x^2+3}} \]
Antiderivative was successfully verified.
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Rule 1099
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{3+6 x^2+2 x^4}} \, dx &=\frac{\sqrt{\frac{3+\left (3-\sqrt{3}\right ) x^2}{3+\left (3+\sqrt{3}\right ) x^2}} \left (3+\left (3+\sqrt{3}\right ) x^2\right ) F\left (\tan ^{-1}\left (\sqrt{\frac{1}{3} \left (3+\sqrt{3}\right )} x\right )|-1+\sqrt{3}\right )}{\sqrt{3 \left (3+\sqrt{3}\right )} \sqrt{3+6 x^2+2 x^4}}\\ \end{align*}
Mathematica [C] time = 0.0747967, size = 90, normalized size = 0.87 \[ -\frac{i \sqrt{\frac{-2 x^2+\sqrt{3}-3}{\sqrt{3}-3}} \sqrt{2 x^2+\sqrt{3}+3} \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{1-\frac{1}{\sqrt{3}}} x\right ),2+\sqrt{3}\right )}{\sqrt{4 x^4+12 x^2+6}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.229, size = 82, normalized size = 0.8 \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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