Optimal. Leaf size=20 \[ \frac{\text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{3}} x\right ),-\frac{3}{2}\right )}{\sqrt{2}} \]
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Rubi [A] time = 0.0110002, antiderivative size = 20, 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} \[ \frac{F\left (\sin ^{-1}\left (\sqrt{\frac{2}{3}} x\right )|-\frac{3}{2}\right )}{\sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 1095
Rule 419
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{3+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\\ &=\frac{F\left (\sin ^{-1}\left (\sqrt{\frac{2}{3}} x\right )|-\frac{3}{2}\right )}{\sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.0253478, size = 20, normalized size = 1. \[ \frac{\text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{3}} x\right ),-\frac{3}{2}\right )}{\sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.064, size = 47, normalized size = 2.4 \begin{align*}{\frac{\sqrt{6}}{6}\sqrt{-6\,{x}^{2}+9}\sqrt{{x}^{2}+1}{\it EllipticF} \left ({\frac{x\sqrt{6}}{3}},{\frac{i}{2}}\sqrt{6} \right ){\frac{1}{\sqrt{-2\,{x}^{4}+{x}^{2}+3}}}} \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{-2 \, x^{4} + x^{2} + 3}}\,{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{-2 \, x^{4} + x^{2} + 3}}{2 \, x^{4} - x^{2} - 3}, 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{- 2 x^{4} + x^{2} + 3}}\, 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{-2 \, x^{4} + x^{2} + 3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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