Integrand size = 16, antiderivative size = 45 \[ \int \frac {1}{\sqrt {2+5 x^2-2 x^4}} \, dx=\sqrt {\frac {2}{-5+\sqrt {41}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {2 x}{\sqrt {5+\sqrt {41}}}\right ),\frac {1}{8} \left (-33-5 \sqrt {41}\right )\right ) \]
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Time = 0.04 (sec) , antiderivative size = 45, 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} \[ \int \frac {1}{\sqrt {2+5 x^2-2 x^4}} \, dx=\sqrt {\frac {2}{\sqrt {41}-5}} \operatorname {EllipticF}\left (\arcsin \left (\frac {2 x}{\sqrt {5+\sqrt {41}}}\right ),\frac {1}{8} \left (-33-5 \sqrt {41}\right )\right ) \]
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Rule 430
Rule 1109
Rubi steps \begin{align*} \text {integral}& = \left (2 \sqrt {2}\right ) \int \frac {1}{\sqrt {5+\sqrt {41}-4 x^2} \sqrt {-5+\sqrt {41}+4 x^2}} \, dx \\ & = \sqrt {\frac {2}{-5+\sqrt {41}}} F\left (\sin ^{-1}\left (\frac {2 x}{\sqrt {5+\sqrt {41}}}\right )|\frac {1}{8} \left (-33-5 \sqrt {41}\right )\right ) \\ \end{align*}
Result contains complex when optimal does not.
Time = 10.04 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.16 \[ \int \frac {1}{\sqrt {2+5 x^2-2 x^4}} \, dx=-i \sqrt {\frac {2}{5+\sqrt {41}}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {2 x}{\sqrt {-5+\sqrt {41}}}\right ),-\frac {33}{8}+\frac {5 \sqrt {41}}{8}\right ) \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (31 ) = 62\).
Time = 0.61 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.69
method | result | size |
default | \(\frac {2 \sqrt {1-\left (-\frac {5}{4}+\frac {\sqrt {41}}{4}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{4}-\frac {\sqrt {41}}{4}\right ) x^{2}}\, F\left (\frac {x \sqrt {-5+\sqrt {41}}}{2}, \frac {5 i}{4}+\frac {i \sqrt {41}}{4}\right )}{\sqrt {-5+\sqrt {41}}\, \sqrt {-2 x^{4}+5 x^{2}+2}}\) | \(76\) |
elliptic | \(\frac {2 \sqrt {1-\left (-\frac {5}{4}+\frac {\sqrt {41}}{4}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{4}-\frac {\sqrt {41}}{4}\right ) x^{2}}\, F\left (\frac {x \sqrt {-5+\sqrt {41}}}{2}, \frac {5 i}{4}+\frac {i \sqrt {41}}{4}\right )}{\sqrt {-5+\sqrt {41}}\, \sqrt {-2 x^{4}+5 x^{2}+2}}\) | \(76\) |
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none
Time = 0.08 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.91 \[ \int \frac {1}{\sqrt {2+5 x^2-2 x^4}} \, dx=\frac {1}{16} \, {\left (\sqrt {41} \sqrt {2} + 5 \, \sqrt {2}\right )} \sqrt {\sqrt {41} - 5} F(\arcsin \left (\frac {1}{2} \, x \sqrt {\sqrt {41} - 5}\right )\,|\,-\frac {5}{8} \, \sqrt {41} - \frac {33}{8}) \]
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\[ \int \frac {1}{\sqrt {2+5 x^2-2 x^4}} \, dx=\int \frac {1}{\sqrt {- 2 x^{4} + 5 x^{2} + 2}}\, dx \]
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\[ \int \frac {1}{\sqrt {2+5 x^2-2 x^4}} \, dx=\int { \frac {1}{\sqrt {-2 \, x^{4} + 5 \, x^{2} + 2}} \,d x } \]
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\[ \int \frac {1}{\sqrt {2+5 x^2-2 x^4}} \, dx=\int { \frac {1}{\sqrt {-2 \, x^{4} + 5 \, x^{2} + 2}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt {2+5 x^2-2 x^4}} \, dx=\int \frac {1}{\sqrt {-2\,x^4+5\,x^2+2}} \,d x \]
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