Optimal. Leaf size=12 \[ \frac {F\left (\sin ^{-1}(x)|-\frac {2}{3}\right )}{\sqrt {3}} \]
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Rubi [A] time = 0.01, antiderivative size = 12, 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, 419} \[ \frac {F\left (\sin ^{-1}(x)|-\frac {2}{3}\right )}{\sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 419
Rule 1095
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {3-x^2-2 x^4}} \, dx &=\left (2 \sqrt {2}\right ) \int \frac {1}{\sqrt {4-4 x^2} \sqrt {6+4 x^2}} \, dx\\ &=\frac {F\left (\sin ^{-1}(x)|-\frac {2}{3}\right )}{\sqrt {3}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 65, normalized size = 5.42 \[ -\frac {i \sqrt {1-x^2} \sqrt {2 x^2+3} F\left (i \sinh ^{-1}\left (\sqrt {\frac {2}{3}} x\right )|-\frac {3}{2}\right )}{\sqrt {2} \sqrt {-2 x^4-x^2+3}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.67, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-2 \, x^{4} - x^{2} + 3}}{2 \, x^{4} + x^{2} - 3}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {-2 \, x^{4} - x^{2} + 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 43, normalized size = 3.58 \[ \frac {\sqrt {-x^{2}+1}\, \sqrt {6 x^{2}+9}\, \EllipticF \left (x , \frac {i \sqrt {6}}{3}\right )}{3 \sqrt {-2 x^{4}-x^{2}+3}} \]
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 \[ \int \frac {1}{\sqrt {-2 \, x^{4} - x^{2} + 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.08 \[ \int \frac {1}{\sqrt {-2\,x^4-x^2+3}} \,d x \]
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 \[ \int \frac {1}{\sqrt {- 2 x^{4} - x^{2} + 3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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