Optimal. Leaf size=153 \[ \frac{\sqrt{-\left (3-\sqrt{33}\right ) x^2-6} \sqrt{\frac{\left (3+\sqrt{33}\right ) x^2+6}{\left (3-\sqrt{33}\right ) x^2+6}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{33} x}{\sqrt{-\left (3-\sqrt{33}\right ) x^2-6}}\right ),\frac{1}{22} \left (11-\sqrt{33}\right )\right )}{2\ 3^{3/4} \sqrt [4]{11} \sqrt{\frac{1}{\left (3-\sqrt{33}\right ) x^2+6}} \sqrt{2 x^4-3 x^2-3}} \]
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Rubi [A] time = 0.0235416, antiderivative size = 153, 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} \[ \frac{\sqrt{-\left (3-\sqrt{33}\right ) x^2-6} \sqrt{\frac{\left (3+\sqrt{33}\right ) x^2+6}{\left (3-\sqrt{33}\right ) x^2+6}} F\left (\sin ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{33} x}{\sqrt{-\left (3-\sqrt{33}\right ) x^2-6}}\right )|\frac{1}{22} \left (11-\sqrt{33}\right )\right )}{2\ 3^{3/4} \sqrt [4]{11} \sqrt{\frac{1}{\left (3-\sqrt{33}\right ) x^2+6}} \sqrt{2 x^4-3 x^2-3}} \]
Antiderivative was successfully verified.
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Rule 1098
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{-3-3 x^2+2 x^4}} \, dx &=\frac{\sqrt{-6-\left (3-\sqrt{33}\right ) x^2} \sqrt{\frac{6+\left (3+\sqrt{33}\right ) x^2}{6+\left (3-\sqrt{33}\right ) x^2}} F\left (\sin ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{33} x}{\sqrt{-6-\left (3-\sqrt{33}\right ) x^2}}\right )|\frac{1}{22} \left (11-\sqrt{33}\right )\right )}{2\ 3^{3/4} \sqrt [4]{11} \sqrt{\frac{1}{6+\left (3-\sqrt{33}\right ) x^2}} \sqrt{-3-3 x^2+2 x^4}}\\ \end{align*}
Mathematica [C] time = 0.0565178, size = 78, normalized size = 0.51 \[ -\frac{i \sqrt{-4 x^4+6 x^2+6} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{2 x}{\sqrt{\sqrt{33}-3}}\right ),\frac{1}{4} \left (\sqrt{33}-7\right )\right )}{\sqrt{3+\sqrt{33}} \sqrt{2 x^4-3 x^2-3}} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.185, size = 84, normalized size = 0.6 \begin{align*} 6\,{\frac{\sqrt{1- \left ( -1/2-1/6\,\sqrt{33} \right ){x}^{2}}\sqrt{1- \left ( -1/2+1/6\,\sqrt{33} \right ){x}^{2}}{\it EllipticF} \left ( 1/6\,\sqrt{-18-6\,\sqrt{33}}x,i/4\sqrt{22}-i/4\sqrt{6} \right ) }{\sqrt{-18-6\,\sqrt{33}}\sqrt{2\,{x}^{4}-3\,{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} - 3 \, 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} - 3 \, 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} - 3 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} - 3 \, x^{2} - 3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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