Optimal. Leaf size=14 \[ -F\left (\left .\cos ^{-1}\left (\sqrt {\frac {2}{3}} x\right )\right |3\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 14, 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} \[ -F\left (\left .\cos ^{-1}\left (\sqrt {\frac {2}{3}} x\right )\right |3\right ) \]
Antiderivative was successfully verified.
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Rule 420
Rule 1095
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-3+5 x^2-2 x^4}} \, dx &=\left (2 \sqrt {2}\right ) \int \frac {1}{\sqrt {6-4 x^2} \sqrt {-4+4 x^2}} \, dx\\ &=-F\left (\left .\cos ^{-1}\left (\sqrt {\frac {2}{3}} x\right )\right |3\right )\\ \end {align*}
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Mathematica [B] time = 0.02, size = 53, normalized size = 3.79 \[ \frac {\sqrt {3-2 x^2} \sqrt {1-x^2} F\left (\sin ^{-1}\left (\sqrt {\frac {2}{3}} x\right )|\frac {3}{2}\right )}{\sqrt {-4 x^4+10 x^2-6}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.04, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-2 \, x^{4} + 5 \, x^{2} - 3}}{2 \, x^{4} - 5 \, x^{2} + 3}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {-2 \, x^{4} + 5 \, x^{2} - 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 50, normalized size = 3.57 \[ \frac {\sqrt {6}\, \sqrt {-6 x^{2}+9}\, \sqrt {-x^{2}+1}\, \EllipticF \left (\frac {\sqrt {6}\, x}{3}, \frac {\sqrt {6}}{2}\right )}{6 \sqrt {-2 x^{4}+5 x^{2}-3}} \]
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 \[ \int \frac {1}{\sqrt {-2 \, x^{4} + 5 \, x^{2} - 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.07 \[ \int \frac {1}{\sqrt {-2\,x^4+5\,x^2-3}} \,d x \]
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 \[ \int \frac {1}{\sqrt {- 2 x^{4} + 5 x^{2} - 3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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