Optimal. Leaf size=6 \[ -\text{EllipticF}\left (\cos ^{-1}(x),3\right ) \]
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Rubi [A] time = 0.0104999, antiderivative size = 6, normalized size of antiderivative = 1., 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 1095
Rule 420
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*}
Mathematica [B] time = 0.0264863, size = 53, normalized size = 8.83 \[ \frac{\sqrt{2-3 x^2} \sqrt{1-x^2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right ),\frac{2}{3}\right )}{\sqrt{-9 x^4+15 x^2-6}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 42, normalized size = 7. \begin{align*}{\frac{1}{2}\sqrt{-{x}^{2}+1}\sqrt{-6\,{x}^{2}+4}{\it EllipticF} \left ( x,{\frac{\sqrt{6}}{2}} \right ){\frac{1}{\sqrt{-3\,{x}^{4}+5\,{x}^{2}-2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-3 \, x^{4} + 5 \, x^{2} - 2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-3 \, x^{4} + 5 \, x^{2} - 2}}{3 \, x^{4} - 5 \, x^{2} + 2}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{- 3 x^{4} + 5 x^{2} - 2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-3 \, x^{4} + 5 \, x^{2} - 2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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