Optimal. Leaf size=110 \[ \frac {\sqrt {\frac {\left (9-\sqrt {57}\right ) x^2+6}{\left (9+\sqrt {57}\right ) x^2+6}} \left (\left (9+\sqrt {57}\right ) x^2+6\right ) F\left (\tan ^{-1}\left (\sqrt {\frac {1}{6} \left (9+\sqrt {57}\right )} x\right )|\frac {1}{4} \left (-19+3 \sqrt {57}\right )\right )}{\sqrt {6 \left (9+\sqrt {57}\right )} \sqrt {2 x^4+9 x^2+3}} \]
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Rubi [A] time = 0.09, antiderivative size = 110, 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} \[ \frac {\sqrt {\frac {\left (9-\sqrt {57}\right ) x^2+6}{\left (9+\sqrt {57}\right ) x^2+6}} \left (\left (9+\sqrt {57}\right ) x^2+6\right ) F\left (\tan ^{-1}\left (\sqrt {\frac {1}{6} \left (9+\sqrt {57}\right )} x\right )|\frac {1}{4} \left (-19+3 \sqrt {57}\right )\right )}{\sqrt {6 \left (9+\sqrt {57}\right )} \sqrt {2 x^4+9 x^2+3}} \]
Antiderivative was successfully verified.
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Rule 1099
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {3+9 x^2+2 x^4}} \, dx &=\frac {\sqrt {\frac {6+\left (9-\sqrt {57}\right ) x^2}{6+\left (9+\sqrt {57}\right ) x^2}} \left (6+\left (9+\sqrt {57}\right ) x^2\right ) F\left (\tan ^{-1}\left (\sqrt {\frac {1}{6} \left (9+\sqrt {57}\right )} x\right )|\frac {1}{4} \left (-19+3 \sqrt {57}\right )\right )}{\sqrt {6 \left (9+\sqrt {57}\right )} \sqrt {3+9 x^2+2 x^4}}\\ \end {align*}
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Mathematica [C] time = 0.07, size = 97, normalized size = 0.88 \[ -\frac {i \sqrt {\frac {-4 x^2+\sqrt {57}-9}{\sqrt {57}-9}} \sqrt {4 x^2+\sqrt {57}+9} F\left (i \sinh ^{-1}\left (\frac {2 x}{\sqrt {9+\sqrt {57}}}\right )|\frac {23}{4}+\frac {3 \sqrt {57}}{4}\right )}{2 \sqrt {2 x^4+9 x^2+3}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.77, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{\sqrt {2 \, x^{4} + 9 \, x^{2} + 3}}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {2 \, x^{4} + 9 \, x^{2} + 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 82, normalized size = 0.75 \[ \frac {6 \sqrt {-\left (-\frac {3}{2}+\frac {\sqrt {57}}{6}\right ) x^{2}+1}\, \sqrt {-\left (-\frac {3}{2}-\frac {\sqrt {57}}{6}\right ) x^{2}+1}\, \EllipticF \left (\frac {\sqrt {-54+6 \sqrt {57}}\, x}{6}, \frac {3 \sqrt {6}}{4}+\frac {\sqrt {38}}{4}\right )}{\sqrt {-54+6 \sqrt {57}}\, \sqrt {2 x^{4}+9 x^{2}+3}} \]
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 \[ \int \frac {1}{\sqrt {2 \, x^{4} + 9 \, x^{2} + 3}}\,{d x} \]
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 \[ \int \frac {1}{\sqrt {2\,x^4+9\,x^2+3}} \,d x \]
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 \[ \int \frac {1}{\sqrt {2 x^{4} + 9 x^{2} + 3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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