Optimal. Leaf size=88 \[ \frac {\left (\sqrt {6} x^2+2\right ) \sqrt {\frac {3 x^4-4 x^2+2}{\left (\sqrt {6} x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{\frac {3}{2}} x\right )|\frac {1}{2}+\frac {1}{\sqrt {6}}\right )}{2 \sqrt [4]{6} \sqrt {3 x^4-4 x^2+2}} \]
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Rubi [A] time = 0.01, antiderivative size = 88, 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 = {1103} \[ \frac {\left (\sqrt {6} x^2+2\right ) \sqrt {\frac {3 x^4-4 x^2+2}{\left (\sqrt {6} x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{\frac {3}{2}} x\right )|\frac {1}{2}+\frac {1}{\sqrt {6}}\right )}{2 \sqrt [4]{6} \sqrt {3 x^4-4 x^2+2}} \]
Antiderivative was successfully verified.
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Rule 1103
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {2-4 x^2+3 x^4}} \, dx &=\frac {\left (2+\sqrt {6} x^2\right ) \sqrt {\frac {2-4 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}{2}+\frac {1}{\sqrt {6}}\right )}{2 \sqrt [4]{6} \sqrt {2-4 x^2+3 x^4}}\\ \end {align*}
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Mathematica [C] time = 0.08, size = 144, normalized size = 1.64 \[ -\frac {i \sqrt {1-\frac {3 x^2}{2-i \sqrt {2}}} \sqrt {1-\frac {3 x^2}{2+i \sqrt {2}}} F\left (i \sinh ^{-1}\left (\sqrt {-\frac {3}{2-i \sqrt {2}}} x\right )|\frac {2-i \sqrt {2}}{2+i \sqrt {2}}\right )}{\sqrt {3} \sqrt {-\frac {1}{2-i \sqrt {2}}} \sqrt {3 x^4-4 x^2+2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.81, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{\sqrt {3 \, x^{4} - 4 \, x^{2} + 2}}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {3 \, x^{4} - 4 \, x^{2} + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.12, size = 87, normalized size = 0.99 \[ \frac {2 \sqrt {-\left (1+\frac {i \sqrt {2}}{2}\right ) x^{2}+1}\, \sqrt {-\left (1-\frac {i \sqrt {2}}{2}\right ) x^{2}+1}\, \EllipticF \left (\frac {\sqrt {4+2 i \sqrt {2}}\, x}{2}, \frac {\sqrt {3-6 i \sqrt {2}}}{3}\right )}{\sqrt {4+2 i \sqrt {2}}\, \sqrt {3 x^{4}-4 x^{2}+2}} \]
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 \[ \int \frac {1}{\sqrt {3 \, x^{4} - 4 \, x^{2} + 2}}\,{d x} \]
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 \[ \int \frac {1}{\sqrt {3\,x^4-4\,x^2+2}} \,d x \]
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 \[ \int \frac {1}{\sqrt {3 x^{4} - 4 x^{2} + 2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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