Optimal. Leaf size=47 \[ -\frac{1}{6} \sqrt{3+\sqrt{3}} \text{EllipticF}\left (\cos ^{-1}\left (\sqrt{\frac{1}{3} \left (3-\sqrt{3}\right )} x\right ),\frac{1}{2} \left (1+\sqrt{3}\right )\right ) \]
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Rubi [A] time = 0.0616624, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.024, Rules used = {420} \[ -\frac{1}{6} \sqrt{3+\sqrt{3}} F\left (\cos ^{-1}\left (\sqrt{\frac{1}{3} \left (3-\sqrt{3}\right )} x\right )|\frac{1}{2} \left (1+\sqrt{3}\right )\right ) \]
Antiderivative was successfully verified.
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Rule 420
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{3-3 \sqrt{3}+2 \sqrt{3} x^2} \sqrt{3+\left (-3+\sqrt{3}\right ) x^2}} \, dx &=-\frac{1}{6} \sqrt{3+\sqrt{3}} F\left (\cos ^{-1}\left (\sqrt{\frac{1}{3} \left (3-\sqrt{3}\right )} x\right )|\frac{1}{2} \left (1+\sqrt{3}\right )\right )\\ \end{align*}
Mathematica [A] time = 0.147288, size = 81, normalized size = 1.72 \[ \frac{\sqrt{-2 \sqrt{3} x^2+3 \sqrt{3}-3} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1+\sqrt{3}} x}{\sqrt [4]{3}}\right ),2-\sqrt{3}\right )}{3^{3/4} \sqrt{4 \sqrt{3} x^2-6 \sqrt{3}+6}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.181, size = 207, normalized size = 4.4 \begin{align*}{\frac{\sqrt{2} \left ( -3+\sqrt{3} \right ) }{18\, \left ( \sqrt{3}-1 \right ) ^{2} \left ( 2\,{x}^{4}\sqrt{3}-2\,{x}^{4}-6\,\sqrt{3}{x}^{2}+6\,{x}^{2}+3\,\sqrt{3}-3 \right ) \sqrt{ \left ( 2\,\sqrt{3}-3 \right ) \left ( \sqrt{3}-1 \right ) }}\sqrt{\sqrt{3}{x}^{2}-3\,{x}^{2}+3}\sqrt{3-3\,\sqrt{3}+2\,\sqrt{3}{x}^{2}}\sqrt{- \left ( 4\,\sqrt{3}{x}^{2}-6\,{x}^{2}-3\,\sqrt{3}+3 \right ) \left ( \sqrt{3}-1 \right ) }\sqrt{- \left ( 3-3\,\sqrt{3}+2\,\sqrt{3}{x}^{2} \right ) \left ( \sqrt{3}-1 \right ) }{\it EllipticF} \left ({\frac{x\sqrt{2}\sqrt{3}\sqrt{ \left ( 2\,\sqrt{3}-3 \right ) \left ( \sqrt{3}-1 \right ) }}{3\,\sqrt{3}-3}},{\frac{\sqrt{ \left ( \sqrt{3}-1 \right ) \left ( 1+\sqrt{3} \right ) }}{\sqrt{3}-1}} \right ) } \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{x^{2}{\left (\sqrt{3} - 3\right )} + 3} \sqrt{2 \, \sqrt{3} x^{2} - 3 \, \sqrt{3} + 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{\sqrt{\sqrt{3} x^{2} - 3 \, x^{2} + 3} \sqrt{\sqrt{3}{\left (2 \, x^{2} - 3\right )} + 3}{\left (\sqrt{3} + 1\right )}}{6 \,{\left (2 \, x^{4} - 6 \, x^{2} + 3\right )}}, 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^{2} + \sqrt{3} x^{2} + 3} \sqrt{2 \sqrt{3} x^{2} - 3 \sqrt{3} + 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{x^{2}{\left (\sqrt{3} - 3\right )} + 3} \sqrt{2 \, \sqrt{3} x^{2} - 3 \, \sqrt{3} + 3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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