Optimal. Leaf size=6 \[ -F\left (\left .\cos ^{-1}(x)\right |3\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 6, 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} \[ -F\left (\left .\cos ^{-1}(x)\right |3\right ) \]
Antiderivative was successfully verified.
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Rule 420
Rule 1095
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-2+5 x^2-3 x^4}} \, dx &=\left (2 \sqrt {3}\right ) \int \frac {1}{\sqrt {6-6 x^2} \sqrt {-4+6 x^2}} \, dx\\ &=-F\left (\left .\cos ^{-1}(x)\right |3\right )\\ \end {align*}
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Mathematica [B] time = 0.03, size = 53, normalized size = 8.83 \[ \frac {\sqrt {2-3 x^2} \sqrt {1-x^2} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{2}} x\right )|\frac {2}{3}\right )}{\sqrt {-9 x^4+15 x^2-6}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.91, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-3 \, x^{4} + 5 \, x^{2} - 2}}{3 \, x^{4} - 5 \, x^{2} + 2}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {-3 \, x^{4} + 5 \, x^{2} - 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 42, normalized size = 7.00 \[ \frac {\sqrt {-x^{2}+1}\, \sqrt {-6 x^{2}+4}\, \EllipticF \left (x , \frac {\sqrt {6}}{2}\right )}{2 \sqrt {-3 x^{4}+5 x^{2}-2}} \]
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 \[ \int \frac {1}{\sqrt {-3 \, x^{4} + 5 \, x^{2} - 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.17 \[ \int \frac {1}{\sqrt {-3\,x^4+5\,x^2-2}} \,d x \]
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 \[ \int \frac {1}{\sqrt {- 3 x^{4} + 5 x^{2} - 2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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