Optimal. Leaf size=47 \[ -\frac {1}{6} \sqrt {3+\sqrt {3}} \operatorname {EllipticF}\left (\cos ^{-1}\left (\sqrt {\frac {1}{3} \left (3-\sqrt {3}\right )} x\right ),\frac {1}{2} \left (1+\sqrt {3}\right )\right ) \]
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Rubi [A] time = 0.06, antiderivative size = 47, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.15, size = 81, normalized size = 1.72 \[ \frac {\sqrt {-2 \sqrt {3} x^2+3 \sqrt {3}-3} \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {\sqrt {1+\sqrt {3}} x}{\sqrt [4]{3}}\right ),2-\sqrt {3}\right )}{3^{3/4} \sqrt {4 \sqrt {3} x^2-6 \sqrt {3}+6}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {\sqrt {3} x^{2} - 3 \, x^{2} + 3} \sqrt {\sqrt {3} {\left (2 \, x^{2} - 3\right )} + 3} {\left (\sqrt {3} + 1\right )}}{6 \, {\left (2 \, x^{4} - 6 \, x^{2} + 3\right )}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {x^{2} {\left (\sqrt {3} - 3\right )} + 3} \sqrt {2 \, \sqrt {3} x^{2} - 3 \, \sqrt {3} + 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.22, size = 207, normalized size = 4.40 \[ \frac {\sqrt {\sqrt {3}\, x^{2}-3 x^{2}+3}\, \sqrt {2 \sqrt {3}\, x^{2}+3-3 \sqrt {3}}\, \sqrt {2}\, \sqrt {-\left (4 \sqrt {3}\, x^{2}-6 x^{2}-3 \sqrt {3}+3\right ) \left (\sqrt {3}-1\right )}\, \sqrt {-\left (2 \sqrt {3}\, x^{2}+3-3 \sqrt {3}\right ) \left (\sqrt {3}-1\right )}\, \left (-3+\sqrt {3}\right ) \EllipticF \left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\left (2 \sqrt {3}-3\right ) \left (\sqrt {3}-1\right )}\, x}{3 \sqrt {3}-3}, \frac {\sqrt {\left (\sqrt {3}-1\right ) \left (1+\sqrt {3}\right )}}{\sqrt {3}-1}\right )}{18 \left (\sqrt {3}-1\right )^{2} \left (2 \sqrt {3}\, x^{4}-2 x^{4}-6 \sqrt {3}\, x^{2}+6 x^{2}+3 \sqrt {3}-3\right ) \sqrt {\left (2 \sqrt {3}-3\right ) \left (\sqrt {3}-1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {x^{2} {\left (\sqrt {3} - 3\right )} + 3} \sqrt {2 \, \sqrt {3} x^{2} - 3 \, \sqrt {3} + 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{\sqrt {\left (\sqrt {3}-3\right )\,x^2+3}\,\sqrt {2\,\sqrt {3}\,x^2-3\,\sqrt {3}+3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {- 3 x^{2} + \sqrt {3} x^{2} + 3} \sqrt {2 \sqrt {3} x^{2} - 3 \sqrt {3} + 3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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