Optimal. Leaf size=141 \[ \frac{\sqrt{\frac{2-\left (1-\sqrt{7}\right ) x^2}{2-\left (1+\sqrt{7}\right ) x^2}} \sqrt{\left (1+\sqrt{7}\right ) x^2-2} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{7} x}{\sqrt{\left (1+\sqrt{7}\right ) x^2-2}}\right ),\frac{1}{14} \left (7+\sqrt{7}\right )\right )}{2 \sqrt [4]{7} \sqrt{\frac{1}{2-\left (1+\sqrt{7}\right ) x^2}} \sqrt{3 x^4+2 x^2-2}} \]
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Rubi [A] time = 0.0302293, antiderivative size = 141, 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} \[ \frac{\sqrt{\frac{2-\left (1-\sqrt{7}\right ) x^2}{2-\left (1+\sqrt{7}\right ) x^2}} \sqrt{\left (1+\sqrt{7}\right ) x^2-2} F\left (\sin ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{7} x}{\sqrt{\left (1+\sqrt{7}\right ) x^2-2}}\right )|\frac{1}{14} \left (7+\sqrt{7}\right )\right )}{2 \sqrt [4]{7} \sqrt{\frac{1}{2-\left (1+\sqrt{7}\right ) x^2}} \sqrt{3 x^4+2 x^2-2}} \]
Antiderivative was successfully verified.
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Rule 1098
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{-2+2 x^2+3 x^4}} \, dx &=\frac{\sqrt{\frac{2-\left (1-\sqrt{7}\right ) x^2}{2-\left (1+\sqrt{7}\right ) x^2}} \sqrt{-2+\left (1+\sqrt{7}\right ) x^2} F\left (\sin ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{7} x}{\sqrt{-2+\left (1+\sqrt{7}\right ) x^2}}\right )|\frac{1}{14} \left (7+\sqrt{7}\right )\right )}{2 \sqrt [4]{7} \sqrt{\frac{1}{2-\left (1+\sqrt{7}\right ) x^2}} \sqrt{-2+2 x^2+3 x^4}}\\ \end{align*}
Mathematica [C] time = 0.0515857, size = 83, normalized size = 0.59 \[ -\frac{i \sqrt{-3 x^4-2 x^2+2} \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{\frac{3}{1+\sqrt{7}}} x\right ),-\frac{4}{3}-\frac{\sqrt{7}}{3}\right )}{\sqrt{\sqrt{7}-1} \sqrt{3 x^4+2 x^2-2}} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.178, size = 84, normalized size = 0.6 \begin{align*} 2\,{\frac{\sqrt{1- \left ( -1/2\,\sqrt{7}+1/2 \right ){x}^{2}}\sqrt{1- \left ( 1/2\,\sqrt{7}+1/2 \right ){x}^{2}}{\it EllipticF} \left ( 1/2\,\sqrt{2-2\,\sqrt{7}}x,i/6\sqrt{6}+i/6\sqrt{42} \right ) }{\sqrt{2-2\,\sqrt{7}}\sqrt{3\,{x}^{4}+2\,{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} + 2 \, 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} + 2 \, 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} + 2 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} + 2 \, x^{2} - 2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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