Optimal. Leaf size=42 \[ \frac{\text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{\sqrt{7}-1}} x\right ),\frac{1}{3} \left (\sqrt{7}-4\right )\right )}{\sqrt{1+\sqrt{7}}} \]
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Rubi [A] time = 0.0473294, antiderivative size = 42, 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, 419} \[ \frac{F\left (\sin ^{-1}\left (\sqrt{\frac{2}{-1+\sqrt{7}}} x\right )|\frac{1}{3} \left (-4+\sqrt{7}\right )\right )}{\sqrt{1+\sqrt{7}}} \]
Antiderivative was successfully verified.
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Rule 1095
Rule 419
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{3-2 x^2-2 x^4}} \, dx &=\left (2 \sqrt{2}\right ) \int \frac{1}{\sqrt{-2+2 \sqrt{7}-4 x^2} \sqrt{2+2 \sqrt{7}+4 x^2}} \, dx\\ &=\frac{F\left (\sin ^{-1}\left (\sqrt{\frac{2}{-1+\sqrt{7}}} x\right )|\frac{1}{3} \left (-4+\sqrt{7}\right )\right )}{\sqrt{1+\sqrt{7}}}\\ \end{align*}
Mathematica [C] time = 0.0440687, size = 51, normalized size = 1.21 \[ -\frac{i \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{7}}} x\right ),-\frac{4}{3}-\frac{\sqrt{7}}{3}\right )}{\sqrt{\sqrt{7}-1}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.208, size = 84, normalized size = 2. \begin{align*} 3\,{\frac{\sqrt{1- \left ( 1/3\,\sqrt{7}+1/3 \right ){x}^{2}}\sqrt{1- \left ( -1/3\,\sqrt{7}+1/3 \right ){x}^{2}}{\it EllipticF} \left ( 1/3\,x\sqrt{3+3\,\sqrt{7}},i/6\sqrt{42}-i/6\sqrt{6} \right ) }{\sqrt{3+3\,\sqrt{7}}\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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