Integrand size = 16, antiderivative size = 43 \[ \int \frac {1}{\sqrt {3-3 x^2-2 x^4}} \, dx=\sqrt {\frac {2}{3+\sqrt {33}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {2 x}{\sqrt {-3+\sqrt {33}}}\right ),\frac {1}{4} \left (-7+\sqrt {33}\right )\right ) \]
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Time = 0.03 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1109, 430} \[ \int \frac {1}{\sqrt {3-3 x^2-2 x^4}} \, dx=\sqrt {\frac {2}{3+\sqrt {33}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {2 x}{\sqrt {-3+\sqrt {33}}}\right ),\frac {1}{4} \left (-7+\sqrt {33}\right )\right ) \]
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Rule 430
Rule 1109
Rubi steps \begin{align*} \text {integral}& = \left (2 \sqrt {2}\right ) \int \frac {1}{\sqrt {-3+\sqrt {33}-4 x^2} \sqrt {3+\sqrt {33}+4 x^2}} \, dx \\ & = \sqrt {\frac {2}{3+\sqrt {33}}} F\left (\sin ^{-1}\left (\frac {2 x}{\sqrt {-3+\sqrt {33}}}\right )|\frac {1}{4} \left (-7+\sqrt {33}\right )\right ) \\ \end{align*}
Result contains complex when optimal does not.
Time = 10.04 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.21 \[ \int \frac {1}{\sqrt {3-3 x^2-2 x^4}} \, dx=-i \sqrt {\frac {2}{-3+\sqrt {33}}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {2 x}{\sqrt {3+\sqrt {33}}}\right ),-\frac {7}{4}-\frac {\sqrt {33}}{4}\right ) \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (35 ) = 70\).
Time = 0.33 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.95
method | result | size |
default | \(\frac {6 \sqrt {1-\left (\frac {1}{2}+\frac {\sqrt {33}}{6}\right ) x^{2}}\, \sqrt {1-\left (\frac {1}{2}-\frac {\sqrt {33}}{6}\right ) x^{2}}\, F\left (\frac {x \sqrt {18+6 \sqrt {33}}}{6}, \frac {i \sqrt {22}}{4}-\frac {i \sqrt {6}}{4}\right )}{\sqrt {18+6 \sqrt {33}}\, \sqrt {-2 x^{4}-3 x^{2}+3}}\) | \(84\) |
elliptic | \(\frac {6 \sqrt {1-\left (\frac {1}{2}+\frac {\sqrt {33}}{6}\right ) x^{2}}\, \sqrt {1-\left (\frac {1}{2}-\frac {\sqrt {33}}{6}\right ) x^{2}}\, F\left (\frac {x \sqrt {18+6 \sqrt {33}}}{6}, \frac {i \sqrt {22}}{4}-\frac {i \sqrt {6}}{4}\right )}{\sqrt {18+6 \sqrt {33}}\, \sqrt {-2 x^{4}-3 x^{2}+3}}\) | \(84\) |
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none
Time = 0.07 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.35 \[ \int \frac {1}{\sqrt {3-3 x^2-2 x^4}} \, dx=\frac {1}{24} \, {\left (\sqrt {11} \sqrt {6} - \sqrt {6} \sqrt {3}\right )} \sqrt {\sqrt {11} \sqrt {3} + 3} F(\arcsin \left (\frac {1}{6} \, \sqrt {6} \sqrt {\sqrt {11} \sqrt {3} + 3} x\right )\,|\,\frac {1}{4} \, \sqrt {11} \sqrt {3} - \frac {7}{4}) \]
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\[ \int \frac {1}{\sqrt {3-3 x^2-2 x^4}} \, dx=\int \frac {1}{\sqrt {- 2 x^{4} - 3 x^{2} + 3}}\, dx \]
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\[ \int \frac {1}{\sqrt {3-3 x^2-2 x^4}} \, dx=\int { \frac {1}{\sqrt {-2 \, x^{4} - 3 \, x^{2} + 3}} \,d x } \]
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\[ \int \frac {1}{\sqrt {3-3 x^2-2 x^4}} \, dx=\int { \frac {1}{\sqrt {-2 \, x^{4} - 3 \, x^{2} + 3}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt {3-3 x^2-2 x^4}} \, dx=\int \frac {1}{\sqrt {-2\,x^4-3\,x^2+3}} \,d x \]
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