Optimal. Leaf size=104 \[ \frac {\sqrt {\frac {\left (3-\sqrt {3}\right ) x^2+3}{\left (3+\sqrt {3}\right ) x^2+3}} \left (\left (3+\sqrt {3}\right ) x^2+3\right ) F\left (\tan ^{-1}\left (\sqrt {\frac {1}{3} \left (3+\sqrt {3}\right )} x\right )|-1+\sqrt {3}\right )}{\sqrt {3 \left (3+\sqrt {3}\right )} \sqrt {2 x^4+6 x^2+3}} \]
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Rubi [A] time = 0.06, antiderivative size = 104, 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 = {1099} \[ \frac {\sqrt {\frac {\left (3-\sqrt {3}\right ) x^2+3}{\left (3+\sqrt {3}\right ) x^2+3}} \left (\left (3+\sqrt {3}\right ) x^2+3\right ) F\left (\tan ^{-1}\left (\sqrt {\frac {1}{3} \left (3+\sqrt {3}\right )} x\right )|-1+\sqrt {3}\right )}{\sqrt {3 \left (3+\sqrt {3}\right )} \sqrt {2 x^4+6 x^2+3}} \]
Antiderivative was successfully verified.
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Rule 1099
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {3+6 x^2+2 x^4}} \, dx &=\frac {\sqrt {\frac {3+\left (3-\sqrt {3}\right ) x^2}{3+\left (3+\sqrt {3}\right ) x^2}} \left (3+\left (3+\sqrt {3}\right ) x^2\right ) F\left (\tan ^{-1}\left (\sqrt {\frac {1}{3} \left (3+\sqrt {3}\right )} x\right )|-1+\sqrt {3}\right )}{\sqrt {3 \left (3+\sqrt {3}\right )} \sqrt {3+6 x^2+2 x^4}}\\ \end {align*}
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Mathematica [C] time = 0.06, size = 90, normalized size = 0.87 \[ -\frac {i \sqrt {\frac {-2 x^2+\sqrt {3}-3}{\sqrt {3}-3}} \sqrt {2 x^2+\sqrt {3}+3} F\left (i \sinh ^{-1}\left (\sqrt {1-\frac {1}{\sqrt {3}}} x\right )|2+\sqrt {3}\right )}{\sqrt {4 x^4+12 x^2+6}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.91, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{\sqrt {2 \, x^{4} + 6 \, x^{2} + 3}}, x\right ) \]
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 = 0.08, size = 82, normalized size = 0.79 \[ \frac {3 \sqrt {-\left (-1+\frac {\sqrt {3}}{3}\right ) x^{2}+1}\, \sqrt {-\left (-1-\frac {\sqrt {3}}{3}\right ) x^{2}+1}\, \EllipticF \left (\frac {\sqrt {-9+3 \sqrt {3}}\, x}{3}, \frac {\sqrt {6}}{2}+\frac {\sqrt {2}}{2}\right )}{\sqrt {-9+3 \sqrt {3}}\, \sqrt {2 x^{4}+6 x^{2}+3}} \]
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 = -0.01 \[ \int \frac {1}{\sqrt {2\,x^4+6\,x^2+3}} \,d x \]
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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