Integrand size = 16, antiderivative size = 46 \[ \int \frac {1}{\sqrt {2-3 x^2-3 x^4}} \, dx=\sqrt {\frac {2}{3+\sqrt {33}}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {6}{-3+\sqrt {33}}} x\right ),\frac {1}{4} \left (-7+\sqrt {33}\right )\right ) \]
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Time = 0.06 (sec) , antiderivative size = 46, 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} \[ \int \frac {1}{\sqrt {2-3 x^2-3 x^4}} \, dx=\sqrt {\frac {2}{3+\sqrt {33}}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {6}{-3+\sqrt {33}}} x\right ),\frac {1}{4} \left (-7+\sqrt {33}\right )\right ) \]
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Rule 430
Rule 1109
Rubi steps \begin{align*} \text {integral}& = \left (2 \sqrt {3}\right ) \int \frac {1}{\sqrt {-3+\sqrt {33}-6 x^2} \sqrt {3+\sqrt {33}+6 x^2}} \, dx \\ & = \sqrt {\frac {2}{3+\sqrt {33}}} F\left (\sin ^{-1}\left (\sqrt {\frac {6}{-3+\sqrt {33}}} x\right )|\frac {1}{4} \left (-7+\sqrt {33}\right )\right ) \\ \end{align*}
Result contains complex when optimal does not.
Time = 10.06 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.20 \[ \int \frac {1}{\sqrt {2-3 x^2-3 x^4}} \, dx=-i \sqrt {\frac {2}{-3+\sqrt {33}}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {6}{3+\sqrt {33}}} x\right ),-\frac {7}{4}-\frac {\sqrt {33}}{4}\right ) \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 79 vs. \(2 (37 ) = 74\).
Time = 0.36 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.74
method | result | size |
default | \(\frac {2 \sqrt {1-\left (\frac {3}{4}+\frac {\sqrt {33}}{4}\right ) x^{2}}\, \sqrt {1-\left (\frac {3}{4}-\frac {\sqrt {33}}{4}\right ) x^{2}}\, F\left (\frac {x \sqrt {3+\sqrt {33}}}{2}, \frac {i \sqrt {22}}{4}-\frac {i \sqrt {6}}{4}\right )}{\sqrt {3+\sqrt {33}}\, \sqrt {-3 x^{4}-3 x^{2}+2}}\) | \(80\) |
elliptic | \(\frac {2 \sqrt {1-\left (\frac {3}{4}+\frac {\sqrt {33}}{4}\right ) x^{2}}\, \sqrt {1-\left (\frac {3}{4}-\frac {\sqrt {33}}{4}\right ) x^{2}}\, F\left (\frac {x \sqrt {3+\sqrt {33}}}{2}, \frac {i \sqrt {22}}{4}-\frac {i \sqrt {6}}{4}\right )}{\sqrt {3+\sqrt {33}}\, \sqrt {-3 x^{4}-3 x^{2}+2}}\) | \(80\) |
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none
Time = 0.08 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.89 \[ \int \frac {1}{\sqrt {2-3 x^2-3 x^4}} \, dx=\frac {1}{24} \, {\left (\sqrt {33} \sqrt {2} - 3 \, \sqrt {2}\right )} \sqrt {\sqrt {33} + 3} F(\arcsin \left (\frac {1}{2} \, x \sqrt {\sqrt {33} + 3}\right )\,|\,\frac {1}{4} \, \sqrt {33} - \frac {7}{4}) \]
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\[ \int \frac {1}{\sqrt {2-3 x^2-3 x^4}} \, dx=\int \frac {1}{\sqrt {- 3 x^{4} - 3 x^{2} + 2}}\, dx \]
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\[ \int \frac {1}{\sqrt {2-3 x^2-3 x^4}} \, dx=\int { \frac {1}{\sqrt {-3 \, x^{4} - 3 \, x^{2} + 2}} \,d x } \]
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\[ \int \frac {1}{\sqrt {2-3 x^2-3 x^4}} \, dx=\int { \frac {1}{\sqrt {-3 \, x^{4} - 3 \, x^{2} + 2}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt {2-3 x^2-3 x^4}} \, dx=\int \frac {1}{\sqrt {-3\,x^4-3\,x^2+2}} \,d x \]
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