Optimal. Leaf size=88 \[ \frac {\left (\sqrt {6} x^2+2\right ) \sqrt {\frac {3 x^4-x^2+2}{\left (\sqrt {6} x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{\frac {3}{2}} x\right )|\frac {1}{24} \left (12+\sqrt {6}\right )\right )}{2 \sqrt [4]{6} \sqrt {-3 x^4+x^2-2}} \]
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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 = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {1103} \[ \frac {\left (\sqrt {6} x^2+2\right ) \sqrt {\frac {3 x^4-x^2+2}{\left (\sqrt {6} x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{\frac {3}{2}} x\right )|\frac {1}{24} \left (12+\sqrt {6}\right )\right )}{2 \sqrt [4]{6} \sqrt {-3 x^4+x^2-2}} \]
Antiderivative was successfully verified.
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Rule 1103
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-2+x^2-3 x^4}} \, dx &=\frac {\left (2+\sqrt {6} x^2\right ) \sqrt {\frac {2-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}{24} \left (12+\sqrt {6}\right )\right )}{2 \sqrt [4]{6} \sqrt {-2+x^2-3 x^4}}\\ \end {align*}
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Mathematica [C] time = 0.08, size = 142, normalized size = 1.61 \[ -\frac {i \sqrt {1-\frac {6 x^2}{1-i \sqrt {23}}} \sqrt {1-\frac {6 x^2}{1+i \sqrt {23}}} F\left (i \sinh ^{-1}\left (\sqrt {-\frac {6}{1-i \sqrt {23}}} x\right )|\frac {1-i \sqrt {23}}{1+i \sqrt {23}}\right )}{\sqrt {6} \sqrt {-\frac {1}{1-i \sqrt {23}}} \sqrt {-3 x^4+x^2-2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.74, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-3 \, x^{4} + x^{2} - 2}}{3 \, x^{4} - x^{2} + 2}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {-3 \, x^{4} + x^{2} - 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.05, size = 85, normalized size = 0.97 \[ \frac {2 \sqrt {-\left (\frac {1}{4}-\frac {i \sqrt {23}}{4}\right ) x^{2}+1}\, \sqrt {-\left (\frac {1}{4}+\frac {i \sqrt {23}}{4}\right ) x^{2}+1}\, \EllipticF \left (\frac {\sqrt {1-i \sqrt {23}}\, x}{2}, \frac {\sqrt {-33+3 i \sqrt {23}}}{6}\right )}{\sqrt {1-i \sqrt {23}}\, \sqrt {-3 x^{4}+x^{2}-2}} \]
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 \[ \int \frac {1}{\sqrt {-3 \, x^{4} + x^{2} - 2}}\,{d x} \]
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 \[ \int \frac {1}{\sqrt {-3\,x^4+x^2-2}} \,d x \]
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 \[ \int \frac {1}{\sqrt {- 3 x^{4} + x^{2} - 2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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