Optimal. Leaf size=65 \[ \frac{\sqrt{x^2-1} \sqrt{3 x^2+2} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{2}} x}{\sqrt{x^2-1}}\right ),\frac{2}{5}\right )}{\sqrt{5} \sqrt{3 x^4-x^2-2}} \]
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Rubi [A] time = 0.006997, antiderivative size = 65, 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 = {1097} \[ \frac{\sqrt{x^2-1} \sqrt{3 x^2+2} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{2}} x}{\sqrt{x^2-1}}\right )|\frac{2}{5}\right )}{\sqrt{5} \sqrt{3 x^4-x^2-2}} \]
Antiderivative was successfully verified.
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Rule 1097
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{-2-x^2+3 x^4}} \, dx &=\frac{\sqrt{-1+x^2} \sqrt{2+3 x^2} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{2}} x}{\sqrt{-1+x^2}}\right )|\frac{2}{5}\right )}{\sqrt{5} \sqrt{-2-x^2+3 x^4}}\\ \end{align*}
Mathematica [C] time = 0.0232928, size = 60, normalized size = 0.92 \[ -\frac{i \sqrt{1-x^2} \sqrt{3 x^2+2} \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right ),-\frac{2}{3}\right )}{\sqrt{9 x^4-3 x^2-6}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.05, size = 53, normalized size = 0.8 \begin{align*}{-{\frac{i}{6}}\sqrt{6}{\it EllipticF} \left ({\frac{i}{2}}x\sqrt{6},{\frac{i}{3}}\sqrt{6} \right ) \sqrt{6\,{x}^{2}+4}\sqrt{-{x}^{2}+1}{\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{1}{\sqrt{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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