Optimal. Leaf size=90 \[ \frac{\left (\sqrt{6} x^2+3\right ) \sqrt{\frac{2 x^4-2 x^2+3}{\left (\sqrt{6} x^2+3\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\sqrt [4]{\frac{2}{3}} x\right ),\frac{1}{12} \left (6+\sqrt{6}\right )\right )}{2 \sqrt [4]{6} \sqrt{-2 x^4+2 x^2-3}} \]
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Rubi [A] time = 0.0096004, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {1103} \[ \frac{\left (\sqrt{6} x^2+3\right ) \sqrt{\frac{2 x^4-2 x^2+3}{\left (\sqrt{6} x^2+3\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{\frac{2}{3}} x\right )|\frac{1}{12} \left (6+\sqrt{6}\right )\right )}{2 \sqrt [4]{6} \sqrt{-2 x^4+2 x^2-3}} \]
Antiderivative was successfully verified.
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Rule 1103
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{-3+2 x^2-2 x^4}} \, dx &=\frac{\left (3+\sqrt{6} x^2\right ) \sqrt{\frac{3-2 x^2+2 x^4}{\left (3+\sqrt{6} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{\frac{2}{3}} x\right )|\frac{1}{12} \left (6+\sqrt{6}\right )\right )}{2 \sqrt [4]{6} \sqrt{-3+2 x^2-2 x^4}}\\ \end{align*}
Mathematica [C] time = 0.072151, size = 144, normalized size = 1.6 \[ -\frac{i \sqrt{1-\frac{2 x^2}{1-i \sqrt{5}}} \sqrt{1-\frac{2 x^2}{1+i \sqrt{5}}} \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{-\frac{2}{1-i \sqrt{5}}} x\right ),\frac{1-i \sqrt{5}}{1+i \sqrt{5}}\right )}{\sqrt{2} \sqrt{-\frac{1}{1-i \sqrt{5}}} \sqrt{-2 x^4+2 x^2-3}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.667, size = 87, normalized size = 1. \begin{align*} 3\,{\frac{\sqrt{1- \left ( 1/3-i/3\sqrt{5} \right ){x}^{2}}\sqrt{1- \left ( 1/3+i/3\sqrt{5} \right ){x}^{2}}{\it EllipticF} \left ( 1/3\,\sqrt{3-3\,i\sqrt{5}}x,1/3\,\sqrt{-6+3\,i\sqrt{5}} \right ) }{\sqrt{3-3\,i\sqrt{5}}\sqrt{-2\,{x}^{4}+2\,{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} + 2 \, 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} + 2 \, x^{2} - 3}}{2 \, x^{4} - 2 \, 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} + 2 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} + 2 \, x^{2} - 3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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