Optimal. Leaf size=12 \[ \frac{\text{EllipticF}\left (\sin ^{-1}(x),-\frac{3}{2}\right )}{\sqrt{2}} \]
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Rubi [A] time = 0.0119981, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {1095, 419} \[ \frac{F\left (\sin ^{-1}(x)|-\frac{3}{2}\right )}{\sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 1095
Rule 419
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{2+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\\ &=\frac{F\left (\sin ^{-1}(x)|-\frac{3}{2}\right )}{\sqrt{2}}\\ \end{align*}
Mathematica [C] time = 0.0227184, size = 63, normalized size = 5.25 \[ -\frac{i \sqrt{1-x^2} \sqrt{3 x^2+2} \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right ),-\frac{2}{3}\right )}{\sqrt{3} \sqrt{-3 x^4+x^2+2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.049, size = 41, normalized size = 3.4 \begin{align*}{\frac{{\it EllipticF} \left ( x,{\frac{i}{2}}\sqrt{6} \right ) }{2}\sqrt{-{x}^{2}+1}\sqrt{6\,{x}^{2}+4}{\frac{1}{\sqrt{-3\,{x}^{4}+{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} + 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} + x^{2} + 2}}{3 \, x^{4} - 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} + 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} + x^{2} + 2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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