Optimal. Leaf size=146 \[ \frac {\sqrt {\frac {4-\left (3-\sqrt {33}\right ) x^2}{4-\left (3+\sqrt {33}\right ) x^2}} \sqrt {\left (3+\sqrt {33}\right ) x^2-4} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{33} x}{\sqrt {\left (3+\sqrt {33}\right ) x^2-4}}\right )|\frac {1}{22} \left (11+\sqrt {33}\right )\right )}{2 \sqrt {2} \sqrt [4]{33} \sqrt {\frac {1}{4-\left (3+\sqrt {33}\right ) x^2}} \sqrt {3 x^4+3 x^2-2}} \]
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Rubi [A] time = 0.04, antiderivative size = 146, 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 = {1098} \[ \frac {\sqrt {\frac {4-\left (3-\sqrt {33}\right ) x^2}{4-\left (3+\sqrt {33}\right ) x^2}} \sqrt {\left (3+\sqrt {33}\right ) x^2-4} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{33} x}{\sqrt {\left (3+\sqrt {33}\right ) x^2-4}}\right )|\frac {1}{22} \left (11+\sqrt {33}\right )\right )}{2 \sqrt {2} \sqrt [4]{33} \sqrt {\frac {1}{4-\left (3+\sqrt {33}\right ) x^2}} \sqrt {3 x^4+3 x^2-2}} \]
Antiderivative was successfully verified.
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Rule 1098
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-2+3 x^2+3 x^4}} \, dx &=\frac {\sqrt {\frac {4-\left (3-\sqrt {33}\right ) x^2}{4-\left (3+\sqrt {33}\right ) x^2}} \sqrt {-4+\left (3+\sqrt {33}\right ) x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{33} x}{\sqrt {-4+\left (3+\sqrt {33}\right ) x^2}}\right )|\frac {1}{22} \left (11+\sqrt {33}\right )\right )}{2 \sqrt {2} \sqrt [4]{33} \sqrt {\frac {1}{4-\left (3+\sqrt {33}\right ) x^2}} \sqrt {-2+3 x^2+3 x^4}}\\ \end {align*}
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Mathematica [C] time = 0.06, size = 83, normalized size = 0.57 \[ -\frac {i \sqrt {-6 x^4-6 x^2+4} F\left (i \sinh ^{-1}\left (\sqrt {\frac {6}{3+\sqrt {33}}} x\right )|-\frac {7}{4}-\frac {\sqrt {33}}{4}\right )}{\sqrt {\sqrt {33}-3} \sqrt {3 x^4+3 x^2-2}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.82, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{\sqrt {3 \, x^{4} + 3 \, x^{2} - 2}}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {3 \, x^{4} + 3 \, x^{2} - 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.04, size = 84, normalized size = 0.58 \[ \frac {2 \sqrt {-\left (-\frac {\sqrt {33}}{4}+\frac {3}{4}\right ) x^{2}+1}\, \sqrt {-\left (\frac {\sqrt {33}}{4}+\frac {3}{4}\right ) x^{2}+1}\, \EllipticF \left (\frac {\sqrt {3-\sqrt {33}}\, x}{2}, \frac {i \sqrt {6}}{4}+\frac {i \sqrt {22}}{4}\right )}{\sqrt {3-\sqrt {33}}\, \sqrt {3 x^{4}+3 x^{2}-2}} \]
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 \[ \int \frac {1}{\sqrt {3 \, x^{4} + 3 \, x^{2} - 2}}\,{d x} \]
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 \[ \int \frac {1}{\sqrt {3\,x^4+3\,x^2-2}} \,d x \]
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 \[ \int \frac {1}{\sqrt {3 x^{4} + 3 x^{2} - 2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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