Integrand size = 40, antiderivative size = 63 \[ \int \frac {1+\sqrt {3}+x}{\left (1-\sqrt {3}+x\right ) \sqrt {-4-4 \sqrt {3} x^2+x^4}} \, dx=-\frac {1}{3} \sqrt {3+2 \sqrt {3}} \arctan \left (\frac {\left (1+\sqrt {3}+x\right )^2}{\sqrt {3 \left (3+2 \sqrt {3}\right )} \sqrt {-4-4 \sqrt {3} x^2+x^4}}\right ) \]
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Time = 0.10 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {1754, 209} \[ \int \frac {1+\sqrt {3}+x}{\left (1-\sqrt {3}+x\right ) \sqrt {-4-4 \sqrt {3} x^2+x^4}} \, dx=-\frac {1}{3} \sqrt {3+2 \sqrt {3}} \arctan \left (\frac {\left (x+\sqrt {3}+1\right )^2}{\sqrt {3 \left (3+2 \sqrt {3}\right )} \sqrt {x^4-4 \sqrt {3} x^2-4}}\right ) \]
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Rule 209
Rule 1754
Rubi steps \begin{align*} \text {integral}& = -\left (\left (4 \left (2+\sqrt {3}\right )\right ) \text {Subst}\left (\int \frac {1}{6 \left (1-\sqrt {3}\right ) \left (1+\sqrt {3}\right )^3+3 \left (1+\sqrt {3}\right )^4+4 x^2} \, dx,x,\frac {\left (1+\sqrt {3}+x\right )^2}{\sqrt {-4-4 \sqrt {3} x^2+x^4}}\right )\right ) \\ & = -\frac {1}{3} \sqrt {3+2 \sqrt {3}} \arctan \left (\frac {\left (1+\sqrt {3}+x\right )^2}{\sqrt {3 \left (3+2 \sqrt {3}\right )} \sqrt {-4-4 \sqrt {3} x^2+x^4}}\right ) \\ \end{align*}
Time = 8.20 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.22 \[ \int \frac {1+\sqrt {3}+x}{\left (1-\sqrt {3}+x\right ) \sqrt {-4-4 \sqrt {3} x^2+x^4}} \, dx=-\frac {1}{3} \sqrt {3+2 \sqrt {3}} \arctan \left (\frac {\sqrt {-9+6 \sqrt {3}} \sqrt {-4-4 \sqrt {3} x^2+x^4}}{-2+\left (2-2 \sqrt {3}\right ) x+\left (-2+\sqrt {3}\right ) x^2}\right ) \]
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Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 2.31 (sec) , antiderivative size = 311, normalized size of antiderivative = 4.94
method | result | size |
elliptic | \(\frac {\sqrt {1-\left (-\frac {\sqrt {3}}{2}-1\right ) x^{2}}\, \sqrt {1-\left (1-\frac {\sqrt {3}}{2}\right ) x^{2}}\, F\left (x \left (\frac {i}{2}+\frac {i \sqrt {3}}{2}\right ), i \sqrt {1-4 \sqrt {3}\, \left (1-\frac {\sqrt {3}}{2}\right )}\right )}{\left (\frac {i}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {-4+x^{4}-4 \sqrt {3}\, x^{2}}}+2 \sqrt {3}\, \left (-\frac {\operatorname {arctanh}\left (\frac {-4 \left (\sqrt {3}-1\right )^{2} \sqrt {3}-8-4 \sqrt {3}\, x^{2}+2 x^{2} \left (\sqrt {3}-1\right )^{2}}{2 \sqrt {\left (\sqrt {3}-1\right )^{4}-4 \left (\sqrt {3}-1\right )^{2} \sqrt {3}-4}\, \sqrt {-4+x^{4}-4 \sqrt {3}\, x^{2}}}\right )}{2 \sqrt {\left (\sqrt {3}-1\right )^{4}-4 \left (\sqrt {3}-1\right )^{2} \sqrt {3}-4}}-\frac {\sqrt {1-\left (-\frac {\sqrt {3}}{2}-1\right ) x^{2}}\, \sqrt {1-\left (1-\frac {\sqrt {3}}{2}\right ) x^{2}}\, \Pi \left (\sqrt {-\frac {\sqrt {3}}{2}-1}\, x , \frac {1}{\left (-\frac {\sqrt {3}}{2}-1\right ) \left (\sqrt {3}-1\right )^{2}}, \frac {\sqrt {1-\frac {\sqrt {3}}{2}}}{\sqrt {-\frac {\sqrt {3}}{2}-1}}\right )}{\sqrt {-\frac {\sqrt {3}}{2}-1}\, \left (\sqrt {3}-1\right ) \sqrt {-4+x^{4}-4 \sqrt {3}\, x^{2}}}\right )\) | \(311\) |
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Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (45) = 90\).
Time = 0.38 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.78 \[ \int \frac {1+\sqrt {3}+x}{\left (1-\sqrt {3}+x\right ) \sqrt {-4-4 \sqrt {3} x^2+x^4}} \, dx=\frac {1}{6} \, \sqrt {2 \, \sqrt {3} + 3} \arctan \left (-\frac {{\left (9 \, x^{4} - 30 \, x^{3} + 18 \, x^{2} - 2 \, \sqrt {3} {\left (2 \, x^{4} - 10 \, x^{3} + 3 \, x^{2} - 10 \, x + 2\right )} + 24\right )} \sqrt {x^{4} - 4 \, \sqrt {3} x^{2} - 4} \sqrt {2 \, \sqrt {3} + 3}}{11 \, x^{6} - 42 \, x^{5} + 66 \, x^{4} - 176 \, x^{3} - 132 \, x^{2} - 168 \, x - 88}\right ) \]
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\[ \int \frac {1+\sqrt {3}+x}{\left (1-\sqrt {3}+x\right ) \sqrt {-4-4 \sqrt {3} x^2+x^4}} \, dx=\int \frac {x + 1 + \sqrt {3}}{\left (x - \sqrt {3} + 1\right ) \sqrt {x^{4} - 4 \sqrt {3} x^{2} - 4}}\, dx \]
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\[ \int \frac {1+\sqrt {3}+x}{\left (1-\sqrt {3}+x\right ) \sqrt {-4-4 \sqrt {3} x^2+x^4}} \, dx=\int { \frac {x + \sqrt {3} + 1}{\sqrt {x^{4} - 4 \, \sqrt {3} x^{2} - 4} {\left (x - \sqrt {3} + 1\right )}} \,d x } \]
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\[ \int \frac {1+\sqrt {3}+x}{\left (1-\sqrt {3}+x\right ) \sqrt {-4-4 \sqrt {3} x^2+x^4}} \, dx=\int { \frac {x + \sqrt {3} + 1}{\sqrt {x^{4} - 4 \, \sqrt {3} x^{2} - 4} {\left (x - \sqrt {3} + 1\right )}} \,d x } \]
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Timed out. \[ \int \frac {1+\sqrt {3}+x}{\left (1-\sqrt {3}+x\right ) \sqrt {-4-4 \sqrt {3} x^2+x^4}} \, dx=\int \frac {x+\sqrt {3}+1}{\sqrt {x^4-4\,\sqrt {3}\,x^2-4}\,\left (x-\sqrt {3}+1\right )} \,d x \]
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