Optimal. Leaf size=44 \[ -\frac{\text{EllipticF}\left (\cos ^{-1}\left (\sqrt{\frac{1}{3} \left (3-\sqrt{3}\right )} x\right ),\frac{1}{2} \left (1+\sqrt{3}\right )\right )}{\sqrt{2} \sqrt [4]{3}} \]
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Rubi [A] time = 0.0607096, antiderivative size = 44, 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, 420} \[ -\frac{F\left (\cos ^{-1}\left (\sqrt{\frac{1}{3} \left (3-\sqrt{3}\right )} x\right )|\frac{1}{2} \left (1+\sqrt{3}\right )\right )}{\sqrt{2} \sqrt [4]{3}} \]
Antiderivative was successfully verified.
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Rule 1095
Rule 420
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{-3+6 x^2-2 x^4}} \, dx &=\left (2 \sqrt{2}\right ) \int \frac{1}{\sqrt{6+2 \sqrt{3}-4 x^2} \sqrt{-6+2 \sqrt{3}+4 x^2}} \, dx\\ &=-\frac{F\left (\cos ^{-1}\left (\sqrt{\frac{1}{3} \left (3-\sqrt{3}\right )} x\right )|\frac{1}{2} \left (1+\sqrt{3}\right )\right )}{\sqrt{2} \sqrt [4]{3}}\\ \end{align*}
Mathematica [A] time = 0.0272034, size = 81, normalized size = 1.84 \[ \frac{\sqrt{-2 x^2-\sqrt{3}+3} \sqrt{\left (\sqrt{3}-3\right ) x^2+3} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{1+\frac{1}{\sqrt{3}}} x\right ),2-\sqrt{3}\right )}{\sqrt{6} \sqrt{-2 x^4+6 x^2-3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.173, size = 82, normalized size = 1.9 \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{\sqrt{-2 \, x^{4} + 6 \, x^{2} - 3}}{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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