Optimal. Leaf size=110 \[ \frac{\sqrt{\frac{\left (9-\sqrt{57}\right ) x^2+6}{\left (9+\sqrt{57}\right ) x^2+6}} \left (\left (9+\sqrt{57}\right ) x^2+6\right ) \text{EllipticF}\left (\tan ^{-1}\left (\sqrt{\frac{1}{6} \left (9+\sqrt{57}\right )} x\right ),\frac{1}{4} \left (3 \sqrt{57}-19\right )\right )}{\sqrt{6 \left (9+\sqrt{57}\right )} \sqrt{2 x^4+9 x^2+3}} \]
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Rubi [A] time = 0.091197, antiderivative size = 110, 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} \[ \frac{\sqrt{\frac{\left (9-\sqrt{57}\right ) x^2+6}{\left (9+\sqrt{57}\right ) x^2+6}} \left (\left (9+\sqrt{57}\right ) x^2+6\right ) F\left (\tan ^{-1}\left (\sqrt{\frac{1}{6} \left (9+\sqrt{57}\right )} x\right )|\frac{1}{4} \left (-19+3 \sqrt{57}\right )\right )}{\sqrt{6 \left (9+\sqrt{57}\right )} \sqrt{2 x^4+9 x^2+3}} \]
Antiderivative was successfully verified.
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Rule 1099
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{3+9 x^2+2 x^4}} \, dx &=\frac{\sqrt{\frac{6+\left (9-\sqrt{57}\right ) x^2}{6+\left (9+\sqrt{57}\right ) x^2}} \left (6+\left (9+\sqrt{57}\right ) x^2\right ) F\left (\tan ^{-1}\left (\sqrt{\frac{1}{6} \left (9+\sqrt{57}\right )} x\right )|\frac{1}{4} \left (-19+3 \sqrt{57}\right )\right )}{\sqrt{6 \left (9+\sqrt{57}\right )} \sqrt{3+9 x^2+2 x^4}}\\ \end{align*}
Mathematica [C] time = 0.0721278, size = 97, normalized size = 0.88 \[ -\frac{i \sqrt{\frac{-4 x^2+\sqrt{57}-9}{\sqrt{57}-9}} \sqrt{4 x^2+\sqrt{57}+9} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{2 x}{\sqrt{9+\sqrt{57}}}\right ),\frac{23}{4}+\frac{3 \sqrt{57}}{4}\right )}{2 \sqrt{2 x^4+9 x^2+3}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.236, size = 82, normalized size = 0.8 \begin{align*} 6\,{\frac{\sqrt{1- \left ( -3/2+1/6\,\sqrt{57} \right ){x}^{2}}\sqrt{1- \left ( -3/2-1/6\,\sqrt{57} \right ){x}^{2}}{\it EllipticF} \left ( 1/6\,x\sqrt{-54+6\,\sqrt{57}},3/4\,\sqrt{6}+1/4\,\sqrt{38} \right ) }{\sqrt{-54+6\,\sqrt{57}}\sqrt{2\,{x}^{4}+9\,{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} + 9 \, 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} + 9 \, 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} + 9 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} + 9 \, x^{2} + 3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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