Integrand size = 16, antiderivative size = 104 \[ \int \frac {1}{\sqrt {3+6 x^2+2 x^4}} \, dx=\frac {\sqrt {\frac {3+\left (3-\sqrt {3}\right ) x^2}{3+\left (3+\sqrt {3}\right ) x^2}} \left (3+\left (3+\sqrt {3}\right ) x^2\right ) \operatorname {EllipticF}\left (\arctan \left (\sqrt {\frac {1}{3} \left (3+\sqrt {3}\right )} x\right ),-1+\sqrt {3}\right )}{\sqrt {3 \left (3+\sqrt {3}\right )} \sqrt {3+6 x^2+2 x^4}} \]
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Time = 0.05 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {1113} \[ \int \frac {1}{\sqrt {3+6 x^2+2 x^4}} \, dx=\frac {\sqrt {\frac {\left (3-\sqrt {3}\right ) x^2+3}{\left (3+\sqrt {3}\right ) x^2+3}} \left (\left (3+\sqrt {3}\right ) x^2+3\right ) \operatorname {EllipticF}\left (\arctan \left (\sqrt {\frac {1}{3} \left (3+\sqrt {3}\right )} x\right ),-1+\sqrt {3}\right )}{\sqrt {3 \left (3+\sqrt {3}\right )} \sqrt {2 x^4+6 x^2+3}} \]
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Rule 1113
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {\frac {3+\left (3-\sqrt {3}\right ) x^2}{3+\left (3+\sqrt {3}\right ) x^2}} \left (3+\left (3+\sqrt {3}\right ) x^2\right ) F\left (\tan ^{-1}\left (\sqrt {\frac {1}{3} \left (3+\sqrt {3}\right )} x\right )|-1+\sqrt {3}\right )}{\sqrt {3 \left (3+\sqrt {3}\right )} \sqrt {3+6 x^2+2 x^4}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 10.07 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.87 \[ \int \frac {1}{\sqrt {3+6 x^2+2 x^4}} \, dx=-\frac {i \sqrt {\frac {-3+\sqrt {3}-2 x^2}{-3+\sqrt {3}}} \sqrt {3+\sqrt {3}+2 x^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {1-\frac {1}{\sqrt {3}}} x\right ),2+\sqrt {3}\right )}{\sqrt {6+12 x^2+4 x^4}} \]
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Time = 0.63 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.79
method | result | size |
default | \(\frac {3 \sqrt {1-\left (-1+\frac {\sqrt {3}}{3}\right ) x^{2}}\, \sqrt {1-\left (-1-\frac {\sqrt {3}}{3}\right ) x^{2}}\, F\left (\frac {x \sqrt {-9+3 \sqrt {3}}}{3}, \frac {\sqrt {6}}{2}+\frac {\sqrt {2}}{2}\right )}{\sqrt {-9+3 \sqrt {3}}\, \sqrt {2 x^{4}+6 x^{2}+3}}\) | \(82\) |
elliptic | \(\frac {3 \sqrt {1-\left (-1+\frac {\sqrt {3}}{3}\right ) x^{2}}\, \sqrt {1-\left (-1-\frac {\sqrt {3}}{3}\right ) x^{2}}\, F\left (\frac {x \sqrt {-9+3 \sqrt {3}}}{3}, \frac {\sqrt {6}}{2}+\frac {\sqrt {2}}{2}\right )}{\sqrt {-9+3 \sqrt {3}}\, \sqrt {2 x^{4}+6 x^{2}+3}}\) | \(82\) |
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none
Time = 0.08 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.33 \[ \int \frac {1}{\sqrt {3+6 x^2+2 x^4}} \, dx=-\frac {1}{6} \, {\left (\sqrt {3} + 3\right )} \sqrt {\sqrt {3} - 3} F(\arcsin \left (\frac {1}{3} \, \sqrt {3} x \sqrt {\sqrt {3} - 3}\right )\,|\,\sqrt {3} + 2) \]
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\[ \int \frac {1}{\sqrt {3+6 x^2+2 x^4}} \, dx=\int \frac {1}{\sqrt {2 x^{4} + 6 x^{2} + 3}}\, dx \]
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\[ \int \frac {1}{\sqrt {3+6 x^2+2 x^4}} \, dx=\int { \frac {1}{\sqrt {2 \, x^{4} + 6 \, x^{2} + 3}} \,d x } \]
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\[ \int \frac {1}{\sqrt {3+6 x^2+2 x^4}} \, dx=\int { \frac {1}{\sqrt {2 \, x^{4} + 6 \, x^{2} + 3}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt {3+6 x^2+2 x^4}} \, dx=\int \frac {1}{\sqrt {2\,x^4+6\,x^2+3}} \,d x \]
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