Optimal. Leaf size=19 \[ -\frac {F\left (\cos ^{-1}\left (\frac {x}{\sqrt {3}}\right )|\frac {6}{5}\right )}{\sqrt {5}} \]
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Rubi [A] time = 0.01, antiderivative size = 19, 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} \[ -\frac {F\left (\cos ^{-1}\left (\frac {x}{\sqrt {3}}\right )|\frac {6}{5}\right )}{\sqrt {5}} \]
Antiderivative was successfully verified.
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Rule 420
Rule 1095
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-3+7 x^2-2 x^4}} \, dx &=\left (2 \sqrt {2}\right ) \int \frac {1}{\sqrt {12-4 x^2} \sqrt {-2+4 x^2}} \, dx\\ &=-\frac {F\left (\cos ^{-1}\left (\frac {x}{\sqrt {3}}\right )|\frac {6}{5}\right )}{\sqrt {5}}\\ \end {align*}
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Mathematica [B] time = 0.02, size = 58, normalized size = 3.05 \[ \frac {\sqrt {1-2 x^2} \sqrt {1-\frac {x^2}{3}} F\left (\sin ^{-1}\left (\sqrt {2} x\right )|\frac {1}{6}\right )}{\sqrt {2} \sqrt {-2 x^4+7 x^2-3}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.78, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-2 \, x^{4} + 7 \, x^{2} - 3}}{2 \, x^{4} - 7 \, x^{2} + 3}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {-2 \, x^{4} + 7 \, x^{2} - 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 48, normalized size = 2.53 \[ \frac {\sqrt {3}\, \sqrt {-3 x^{2}+9}\, \sqrt {-2 x^{2}+1}\, \EllipticF \left (\frac {\sqrt {3}\, x}{3}, \sqrt {6}\right )}{3 \sqrt {-2 x^{4}+7 x^{2}-3}} \]
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 \[ \int \frac {1}{\sqrt {-2 \, x^{4} + 7 \, x^{2} - 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.05 \[ \int \frac {1}{\sqrt {-2\,x^4+7\,x^2-3}} \,d x \]
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 \[ \int \frac {1}{\sqrt {- 2 x^{4} + 7 x^{2} - 3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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