Optimal. Leaf size=90 \[ \frac {\left (\sqrt {6} x^2+2\right ) \sqrt {\frac {3 x^4-6 x^2+2}{\left (\sqrt {6} x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{\frac {3}{2}} x\right )|\frac {1}{4} \left (2+\sqrt {6}\right )\right )}{2 \sqrt [4]{6} \sqrt {3 x^4-6 x^2+2}} \]
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Rubi [A] time = 0.02, antiderivative size = 90, normalized size of antiderivative = 1.00, 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 = {1096} \[ \frac {\left (\sqrt {6} x^2+2\right ) \sqrt {\frac {3 x^4-6 x^2+2}{\left (\sqrt {6} x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{\frac {3}{2}} x\right )|\frac {1}{4} \left (2+\sqrt {6}\right )\right )}{2 \sqrt [4]{6} \sqrt {3 x^4-6 x^2+2}} \]
Antiderivative was successfully verified.
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Rule 1096
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {2-6 x^2+3 x^4}} \, dx &=\frac {\left (2+\sqrt {6} x^2\right ) \sqrt {\frac {2-6 x^2+3 x^4}{\left (2+\sqrt {6} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{\frac {3}{2}} x\right )|\frac {1}{4} \left (2+\sqrt {6}\right )\right )}{2 \sqrt [4]{6} \sqrt {2-6 x^2+3 x^4}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 85, normalized size = 0.94 \[ \frac {\sqrt {-3 x^2-\sqrt {3}+3} \sqrt {\left (\sqrt {3}-3\right ) x^2+2} F\left (\sin ^{-1}\left (\sqrt {\frac {1}{2} \left (3+\sqrt {3}\right )} x\right )|2-\sqrt {3}\right )}{\sqrt {6} \sqrt {3 x^4-6 x^2+2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.74, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{\sqrt {3 \, x^{4} - 6 \, x^{2} + 2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {3 \, x^{4} - 6 \, x^{2} + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 82, normalized size = 0.91 \[ \frac {2 \sqrt {-\left (\frac {\sqrt {3}}{2}+\frac {3}{2}\right ) x^{2}+1}\, \sqrt {-\left (-\frac {\sqrt {3}}{2}+\frac {3}{2}\right ) x^{2}+1}\, \EllipticF \left (\frac {\sqrt {6+2 \sqrt {3}}\, x}{2}, \frac {\sqrt {6}}{2}-\frac {\sqrt {2}}{2}\right )}{\sqrt {6+2 \sqrt {3}}\, \sqrt {3 x^{4}-6 x^{2}+2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {3 \, x^{4} - 6 \, x^{2} + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{\sqrt {3\,x^4-6\,x^2+2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {3 x^{4} - 6 x^{2} + 2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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