Optimal. Leaf size=143 \[ \frac{\sqrt{\frac{3-\left (1-\sqrt{7}\right ) x^2}{3-\left (1+\sqrt{7}\right ) x^2}} \sqrt{\left (1+\sqrt{7}\right ) x^2-3} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{7} x}{\sqrt{\left (1+\sqrt{7}\right ) x^2-3}}\right ),\frac{1}{14} \left (7+\sqrt{7}\right )\right )}{\sqrt{6} \sqrt [4]{7} \sqrt{\frac{1}{3-\left (1+\sqrt{7}\right ) x^2}} \sqrt{2 x^4+2 x^2-3}} \]
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Rubi [A] time = 0.0233424, antiderivative size = 143, 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} \[ \frac{\sqrt{\frac{3-\left (1-\sqrt{7}\right ) x^2}{3-\left (1+\sqrt{7}\right ) x^2}} \sqrt{\left (1+\sqrt{7}\right ) x^2-3} F\left (\sin ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{7} x}{\sqrt{\left (1+\sqrt{7}\right ) x^2-3}}\right )|\frac{1}{14} \left (7+\sqrt{7}\right )\right )}{\sqrt{6} \sqrt [4]{7} \sqrt{\frac{1}{3-\left (1+\sqrt{7}\right ) x^2}} \sqrt{2 x^4+2 x^2-3}} \]
Antiderivative was successfully verified.
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Rule 1098
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{-3+2 x^2+2 x^4}} \, dx &=\frac{\sqrt{\frac{3-\left (1-\sqrt{7}\right ) x^2}{3-\left (1+\sqrt{7}\right ) x^2}} \sqrt{-3+\left (1+\sqrt{7}\right ) x^2} F\left (\sin ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{7} x}{\sqrt{-3+\left (1+\sqrt{7}\right ) x^2}}\right )|\frac{1}{14} \left (7+\sqrt{7}\right )\right )}{\sqrt{6} \sqrt [4]{7} \sqrt{\frac{1}{3-\left (1+\sqrt{7}\right ) x^2}} \sqrt{-3+2 x^2+2 x^4}}\\ \end{align*}
Mathematica [C] time = 0.0515208, size = 83, normalized size = 0.58 \[ -\frac{i \sqrt{-2 x^4-2 x^2+3} \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{7}}} x\right ),-\frac{4}{3}-\frac{\sqrt{7}}{3}\right )}{\sqrt{\sqrt{7}-1} \sqrt{2 x^4+2 x^2-3}} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.18, size = 84, normalized size = 0.6 \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\,\sqrt{3-3\,\sqrt{7}}x,i/6\sqrt{6}+i/6\sqrt{42} \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{1}{\sqrt{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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