Optimal. Leaf size=148 \[ \frac{\sqrt{-\left (2-\sqrt{10}\right ) x^2-2} \sqrt{\frac{\left (2+\sqrt{10}\right ) x^2+2}{\left (2-\sqrt{10}\right ) x^2+2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{2^{3/4} \sqrt [4]{5} x}{\sqrt{-\left (2-\sqrt{10}\right ) x^2-2}}\right ),\frac{1}{10} \left (5-\sqrt{10}\right )\right )}{2 \sqrt [4]{10} \sqrt{\frac{1}{\left (2-\sqrt{10}\right ) x^2+2}} \sqrt{3 x^4-4 x^2-2}} \]
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Rubi [A] time = 0.0288026, antiderivative size = 148, 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 = {1098} \[ \frac{\sqrt{-\left (2-\sqrt{10}\right ) x^2-2} \sqrt{\frac{\left (2+\sqrt{10}\right ) x^2+2}{\left (2-\sqrt{10}\right ) x^2+2}} F\left (\sin ^{-1}\left (\frac{2^{3/4} \sqrt [4]{5} x}{\sqrt{-\left (2-\sqrt{10}\right ) x^2-2}}\right )|\frac{1}{10} \left (5-\sqrt{10}\right )\right )}{2 \sqrt [4]{10} \sqrt{\frac{1}{\left (2-\sqrt{10}\right ) x^2+2}} \sqrt{3 x^4-4 x^2-2}} \]
Antiderivative was successfully verified.
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Rule 1098
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{-2-4 x^2+3 x^4}} \, dx &=\frac{\sqrt{-2-\left (2-\sqrt{10}\right ) x^2} \sqrt{\frac{2+\left (2+\sqrt{10}\right ) x^2}{2+\left (2-\sqrt{10}\right ) x^2}} F\left (\sin ^{-1}\left (\frac{2^{3/4} \sqrt [4]{5} x}{\sqrt{-2-\left (2-\sqrt{10}\right ) x^2}}\right )|\frac{1}{10} \left (5-\sqrt{10}\right )\right )}{2 \sqrt [4]{10} \sqrt{\frac{1}{2+\left (2-\sqrt{10}\right ) x^2}} \sqrt{-2-4 x^2+3 x^4}}\\ \end{align*}
Mathematica [C] time = 0.0638357, size = 81, normalized size = 0.55 \[ -\frac{i \sqrt{-3 x^4+4 x^2+2} \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{1+\sqrt{\frac{5}{2}}} x\right ),\frac{1}{3} \left (2 \sqrt{10}-7\right )\right )}{\sqrt{2+\sqrt{10}} \sqrt{3 x^4-4 x^2-2}} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.182, size = 84, normalized size = 0.6 \begin{align*} 2\,{\frac{\sqrt{1- \left ( -1-1/2\,\sqrt{10} \right ){x}^{2}}\sqrt{1- \left ( -1+1/2\,\sqrt{10} \right ){x}^{2}}{\it EllipticF} \left ( 1/2\,\sqrt{-4-2\,\sqrt{10}}x,i/3\sqrt{15}-i/3\sqrt{6} \right ) }{\sqrt{-4-2\,\sqrt{10}}\sqrt{3\,{x}^{4}-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} - 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} - 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} - 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} - 4 \, x^{2} - 2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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