Optimal. Leaf size=70 \[ -\frac {1}{3} \sqrt {3+2 \sqrt {3}} \tan ^{-1}\left (\frac {\left (2 x+\sqrt {3}+1\right )^2}{2 \sqrt {3 \left (3+2 \sqrt {3}\right )} \sqrt {4 x^4-4 \sqrt {3} x^2-1}}\right ) \]
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Rubi [A] time = 0.13, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {1740, 203} \[ -\frac {1}{3} \sqrt {3+2 \sqrt {3}} \tan ^{-1}\left (\frac {\left (2 x+\sqrt {3}+1\right )^2}{2 \sqrt {3 \left (3+2 \sqrt {3}\right )} \sqrt {4 x^4-4 \sqrt {3} x^2-1}}\right ) \]
Antiderivative was successfully verified.
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Rule 203
Rule 1740
Rubi steps
\begin {align*} \int \frac {1+\sqrt {3}+2 x}{\left (1-\sqrt {3}+2 x\right ) \sqrt {-1-4 \sqrt {3} x^2+4 x^4}} \, dx &=-\left (\left (4 \left (2+\sqrt {3}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{12 \left (1-\sqrt {3}\right ) \left (1+\sqrt {3}\right )^3+6 \left (1+\sqrt {3}\right )^4+2 x^2} \, dx,x,\frac {\left (1+\sqrt {3}+2 x\right )^2}{\sqrt {-1-4 \sqrt {3} x^2+4 x^4}}\right )\right )\\ &=-\frac {1}{3} \sqrt {3+2 \sqrt {3}} \tan ^{-1}\left (\frac {\left (1+\sqrt {3}+2 x\right )^2}{2 \sqrt {3 \left (3+2 \sqrt {3}\right )} \sqrt {-1-4 \sqrt {3} x^2+4 x^4}}\right )\\ \end {align*}
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Mathematica [C] time = 6.39, size = 881, normalized size = 12.59 \[ -\frac {\sqrt {\frac {\sqrt {3}-1-\frac {4}{-2 x+\sqrt {3}+1}}{-3+\sqrt {3}-i \sqrt {4-2 \sqrt {3}}}} \left (-2 x+\sqrt {3}+1\right )^2 \left (\left (-\frac {2 i \left (2 \sqrt {3} \sqrt {i \left (\sqrt {3}+1-\frac {8}{-2 x+\sqrt {3}+1}\right )+\sqrt {4-2 \sqrt {3}}}-i \sqrt {6} \sqrt {-\frac {2 i \left (\left (-1+\sqrt {3}\right ) x-4 \sqrt {3}+7\right )}{-2 x+\sqrt {3}+1}+2 \sqrt {4-2 \sqrt {3}}-\sqrt {12-6 \sqrt {3}}}-i \sqrt {-\frac {4 i \left (\left (-1+\sqrt {3}\right ) x-4 \sqrt {3}+7\right )}{-2 x+\sqrt {3}+1}+4 \sqrt {4-2 \sqrt {3}}-2 \sqrt {12-6 \sqrt {3}}}\right )}{-2 x+\sqrt {3}+1}+i \sqrt {3} \sqrt {i \left (\sqrt {3}+1-\frac {8}{-2 x+\sqrt {3}+1}\right )+\sqrt {4-2 \sqrt {3}}}+i \sqrt {i \left (\sqrt {3}+1-\frac {8}{-2 x+\sqrt {3}+1}\right )+\sqrt {4-2 \sqrt {3}}}+\sqrt {-\frac {4 i \left (\left (-1+\sqrt {3}\right ) x-4 \sqrt {3}+7\right )}{-2 x+\sqrt {3}+1}+4 \sqrt {4-2 \sqrt {3}}-2 \sqrt {12-6 \sqrt {3}}}\right ) F\left (\sin ^{-1}\left (\frac {\sqrt {\sqrt {4-2 \sqrt {3}}-i \left (\sqrt {3}+1-\frac {8}{-2 x+\sqrt {3}+1}\right )}}{2^{3/4} \sqrt [4]{2-\sqrt {3}}}\right )|\frac {2 \sqrt {4-2 \sqrt {3}}}{\sqrt {4-2 \sqrt {3}}+i \left (-3+\sqrt {3}\right )}\right )+4 \sqrt {\sqrt {4-2 \sqrt {3}}-i \left (\sqrt {3}+1-\frac {8}{-2 x+\sqrt {3}+1}\right )} \sqrt {\frac {6 x^2-3 \sqrt {3}+6}{\left (-2 x+\sqrt {3}+1\right )^2}} \Pi \left (\frac {2 \sqrt {4-2 \sqrt {3}}}{\sqrt {4-2 \sqrt {3}}-i \left (-3+\sqrt {3}\right )};\sin ^{-1}\left (\frac {\sqrt {\sqrt {4-2 \sqrt {3}}-i \left (\sqrt {3}+1-\frac {8}{-2 x+\sqrt {3}+1}\right )}}{2^{3/4} \sqrt [4]{2-\sqrt {3}}}\right )|\frac {2 \sqrt {4-2 \sqrt {3}}}{\sqrt {4-2 \sqrt {3}}+i \left (-3+\sqrt {3}\right )}\right )\right )}{\sqrt {2} \left (\sqrt {4-2 \sqrt {3}}-i \left (-3+\sqrt {3}\right )\right ) \sqrt {\sqrt {4-2 \sqrt {3}}-i \left (\sqrt {3}+1-\frac {8}{-2 x+\sqrt {3}+1}\right )} \sqrt {4 x^4-4 \sqrt {3} x^2-1}} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 1.41, size = 114, normalized size = 1.63 \[ \frac {1}{6} \, \sqrt {2 \, \sqrt {3} + 3} \arctan \left (-\frac {{\left (36 \, x^{4} - 60 \, x^{3} + 18 \, x^{2} - \sqrt {3} {\left (16 \, x^{4} - 40 \, x^{3} + 6 \, x^{2} - 10 \, x + 1\right )} + 6\right )} \sqrt {4 \, x^{4} - 4 \, \sqrt {3} x^{2} - 1} \sqrt {2 \, \sqrt {3} + 3}}{88 \, x^{6} - 168 \, x^{5} + 132 \, x^{4} - 176 \, x^{3} - 66 \, x^{2} - 42 \, x - 11}\right ) \]
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 \[ \int \frac {2 \, x + \sqrt {3} + 1}{\sqrt {4 \, x^{4} - 4 \, \sqrt {3} x^{2} - 1} {\left (2 \, x - \sqrt {3} + 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.15, size = 337, normalized size = 4.81 \[ \frac {\sqrt {-\left (-4-2 \sqrt {3}\right ) x^{2}+1}\, \sqrt {-\left (-2 \sqrt {3}+4\right ) x^{2}+1}\, \EllipticF \left (\left (i+i \sqrt {3}\right ) x , i \sqrt {1-\sqrt {3}\, \left (-2 \sqrt {3}+4\right )}\right )}{\left (i+i \sqrt {3}\right ) \sqrt {4 x^{4}-4 \sqrt {3}\, x^{2}-1}}+2 \sqrt {3}\, \left (-\frac {\sqrt {-\left (-4-2 \sqrt {3}\right ) x^{2}+1}\, \sqrt {-\left (-2 \sqrt {3}+4\right ) x^{2}+1}\, \EllipticPi \left (\sqrt {-4-2 \sqrt {3}}\, x , \frac {1}{\left (-4-2 \sqrt {3}\right ) \left (\frac {\sqrt {3}}{2}-\frac {1}{2}\right )^{2}}, \frac {\sqrt {-2 \sqrt {3}+4}}{\sqrt {-4-2 \sqrt {3}}}\right )}{2 \sqrt {-4-2 \sqrt {3}}\, \left (\frac {\sqrt {3}}{2}-\frac {1}{2}\right ) \sqrt {4 x^{4}-4 \sqrt {3}\, x^{2}-1}}-\frac {\arctanh \left (\frac {-4 \sqrt {3}\, x^{2}+8 \left (\frac {\sqrt {3}}{2}-\frac {1}{2}\right )^{2} x^{2}-4 \sqrt {3}\, \left (\frac {\sqrt {3}}{2}-\frac {1}{2}\right )^{2}-2}{2 \sqrt {4 \left (\frac {\sqrt {3}}{2}-\frac {1}{2}\right )^{4}-4 \sqrt {3}\, \left (\frac {\sqrt {3}}{2}-\frac {1}{2}\right )^{2}-1}\, \sqrt {4 x^{4}-4 \sqrt {3}\, x^{2}-1}}\right )}{4 \sqrt {4 \left (\frac {\sqrt {3}}{2}-\frac {1}{2}\right )^{4}-4 \sqrt {3}\, \left (\frac {\sqrt {3}}{2}-\frac {1}{2}\right )^{2}-1}}\right ) \]
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 \[ \int \frac {2 \, x + \sqrt {3} + 1}{\sqrt {4 \, x^{4} - 4 \, \sqrt {3} x^{2} - 1} {\left (2 \, x - \sqrt {3} + 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {2\,x+\sqrt {3}+1}{\sqrt {4\,x^4-4\,\sqrt {3}\,x^2-1}\,\left (2\,x-\sqrt {3}+1\right )} \,d x \]
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 \[ \int \frac {2 x + 1 + \sqrt {3}}{\left (2 x - \sqrt {3} + 1\right ) \sqrt {4 x^{4} - 4 \sqrt {3} x^{2} - 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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