Optimal. Leaf size=44 \[ -\frac {F\left (\cos ^{-1}\left (\sqrt {\frac {1}{3} \left (3-\sqrt {3}\right )} x\right )|\frac {1}{2} \left (1+\sqrt {3}\right )\right )}{\sqrt {2} \sqrt [4]{3}} \]
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Rubi [A] time = 0.06, antiderivative size = 44, normalized size of antiderivative = 1.00, 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 420
Rule 1095
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*}
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Mathematica [A] time = 0.03, size = 81, normalized size = 1.84 \[ \frac {\sqrt {-2 x^2-\sqrt {3}+3} \sqrt {\left (\sqrt {3}-3\right ) x^2+3} F\left (\sin ^{-1}\left (\sqrt {1+\frac {1}{\sqrt {3}}} x\right )|2-\sqrt {3}\right )}{\sqrt {6} \sqrt {-2 x^4+6 x^2-3}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.82, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-2 \, x^{4} + 6 \, x^{2} - 3}}{2 \, x^{4} - 6 \, x^{2} + 3}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {-2 \, x^{4} + 6 \, x^{2} - 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 82, normalized size = 1.86 \[ \frac {3 \sqrt {-\left (1-\frac {\sqrt {3}}{3}\right ) x^{2}+1}\, \sqrt {-\left (1+\frac {\sqrt {3}}{3}\right ) x^{2}+1}\, \EllipticF \left (\frac {\sqrt {9-3 \sqrt {3}}\, x}{3}, \frac {\sqrt {6}}{2}+\frac {\sqrt {2}}{2}\right )}{\sqrt {9-3 \sqrt {3}}\, \sqrt {-2 x^{4}+6 x^{2}-3}} \]
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 \[ \int \frac {1}{\sqrt {-2 \, x^{4} + 6 \, x^{2} - 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{\sqrt {-2\,x^4+6\,x^2-3}} \,d x \]
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 \[ \int \frac {1}{\sqrt {- 2 x^{4} + 6 x^{2} - 3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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