Integrand size = 16, antiderivative size = 20 \[ \int \frac {1}{\sqrt {2-x^2-3 x^4}} \, dx=\frac {\operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{2}} x\right ),-\frac {2}{3}\right )}{\sqrt {3}} \]
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Time = 0.01 (sec) , antiderivative size = 20, 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} \[ \int \frac {1}{\sqrt {2-x^2-3 x^4}} \, dx=\frac {\operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{2}} x\right ),-\frac {2}{3}\right )}{\sqrt {3}} \]
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Rule 430
Rule 1109
Rubi steps \begin{align*} \text {integral}& = \left (2 \sqrt {3}\right ) \int \frac {1}{\sqrt {4-6 x^2} \sqrt {6+6 x^2}} \, dx \\ & = \frac {F\left (\sin ^{-1}\left (\sqrt {\frac {3}{2}} x\right )|-\frac {2}{3}\right )}{\sqrt {3}} \\ \end{align*}
Time = 10.03 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {2-x^2-3 x^4}} \, dx=\frac {\operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{2}} x\right ),-\frac {2}{3}\right )}{\sqrt {3}} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (18 ) = 36\).
Time = 0.33 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.45
method | result | size |
default | \(\frac {\sqrt {6}\, \sqrt {-6 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {x \sqrt {6}}{2}, \frac {i \sqrt {6}}{3}\right )}{6 \sqrt {-3 x^{4}-x^{2}+2}}\) | \(49\) |
elliptic | \(\frac {\sqrt {6}\, \sqrt {-6 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {x \sqrt {6}}{2}, \frac {i \sqrt {6}}{3}\right )}{6 \sqrt {-3 x^{4}-x^{2}+2}}\) | \(49\) |
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none
Time = 0.08 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85 \[ \int \frac {1}{\sqrt {2-x^2-3 x^4}} \, dx=\frac {1}{3} \, \sqrt {3} F(\arcsin \left (\frac {1}{2} \, \sqrt {3} \sqrt {2} x\right )\,|\,-\frac {2}{3}) \]
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\[ \int \frac {1}{\sqrt {2-x^2-3 x^4}} \, dx=\int \frac {1}{\sqrt {- 3 x^{4} - x^{2} + 2}}\, dx \]
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\[ \int \frac {1}{\sqrt {2-x^2-3 x^4}} \, dx=\int { \frac {1}{\sqrt {-3 \, x^{4} - x^{2} + 2}} \,d x } \]
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\[ \int \frac {1}{\sqrt {2-x^2-3 x^4}} \, dx=\int { \frac {1}{\sqrt {-3 \, x^{4} - x^{2} + 2}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt {2-x^2-3 x^4}} \, dx=\int \frac {1}{\sqrt {-3\,x^4-x^2+2}} \,d x \]
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