Integrand size = 16, antiderivative size = 90 \[ \int \frac {1}{\sqrt {2+5 x^2+4 x^4}} \, dx=\frac {\left (1+\sqrt {2} x^2\right ) \sqrt {\frac {2+5 x^2+4 x^4}{\left (1+\sqrt {2} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt [4]{2} x\right ),\frac {1}{16} \left (8-5 \sqrt {2}\right )\right )}{2\ 2^{3/4} \sqrt {2+5 x^2+4 x^4}} \]
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Time = 0.01 (sec) , antiderivative size = 90, 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 = {1117} \[ \int \frac {1}{\sqrt {2+5 x^2+4 x^4}} \, dx=\frac {\left (\sqrt {2} x^2+1\right ) \sqrt {\frac {4 x^4+5 x^2+2}{\left (\sqrt {2} x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt [4]{2} x\right ),\frac {1}{16} \left (8-5 \sqrt {2}\right )\right )}{2\ 2^{3/4} \sqrt {4 x^4+5 x^2+2}} \]
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Rule 1117
Rubi steps \begin{align*} \text {integral}& = \frac {\left (1+\sqrt {2} x^2\right ) \sqrt {\frac {2+5 x^2+4 x^4}{\left (1+\sqrt {2} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac {1}{16} \left (8-5 \sqrt {2}\right )\right )}{2\ 2^{3/4} \sqrt {2+5 x^2+4 x^4}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 10.07 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.63 \[ \int \frac {1}{\sqrt {2+5 x^2+4 x^4}} \, dx=-\frac {i \sqrt {1-\frac {8 x^2}{-5-i \sqrt {7}}} \sqrt {1-\frac {8 x^2}{-5+i \sqrt {7}}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (2 \sqrt {-\frac {2}{-5-i \sqrt {7}}} x\right ),\frac {-5-i \sqrt {7}}{-5+i \sqrt {7}}\right )}{2 \sqrt {2} \sqrt {-\frac {1}{-5-i \sqrt {7}}} \sqrt {2+5 x^2+4 x^4}} \]
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Result contains complex when optimal does not.
Time = 0.62 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.97
method | result | size |
default | \(\frac {2 \sqrt {1-\left (-\frac {5}{4}+\frac {i \sqrt {7}}{4}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{4}-\frac {i \sqrt {7}}{4}\right ) x^{2}}\, F\left (\frac {x \sqrt {-5+i \sqrt {7}}}{2}, \frac {\sqrt {9+5 i \sqrt {7}}}{4}\right )}{\sqrt {-5+i \sqrt {7}}\, \sqrt {4 x^{4}+5 x^{2}+2}}\) | \(87\) |
elliptic | \(\frac {2 \sqrt {1-\left (-\frac {5}{4}+\frac {i \sqrt {7}}{4}\right ) x^{2}}\, \sqrt {1-\left (-\frac {5}{4}-\frac {i \sqrt {7}}{4}\right ) x^{2}}\, F\left (\frac {x \sqrt {-5+i \sqrt {7}}}{2}, \frac {\sqrt {9+5 i \sqrt {7}}}{4}\right )}{\sqrt {-5+i \sqrt {7}}\, \sqrt {4 x^{4}+5 x^{2}+2}}\) | \(87\) |
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none
Time = 0.08 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.40 \[ \int \frac {1}{\sqrt {2+5 x^2+4 x^4}} \, dx=-\frac {1}{32} \, \sqrt {2} {\left (\sqrt {-7} + 5\right )} \sqrt {\sqrt {-7} - 5} F(\arcsin \left (\frac {1}{2} \, x \sqrt {\sqrt {-7} - 5}\right )\,|\,\frac {5}{16} \, \sqrt {-7} + \frac {9}{16}) \]
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\[ \int \frac {1}{\sqrt {2+5 x^2+4 x^4}} \, dx=\int \frac {1}{\sqrt {4 x^{4} + 5 x^{2} + 2}}\, dx \]
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\[ \int \frac {1}{\sqrt {2+5 x^2+4 x^4}} \, dx=\int { \frac {1}{\sqrt {4 \, x^{4} + 5 \, x^{2} + 2}} \,d x } \]
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\[ \int \frac {1}{\sqrt {2+5 x^2+4 x^4}} \, dx=\int { \frac {1}{\sqrt {4 \, x^{4} + 5 \, x^{2} + 2}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt {2+5 x^2+4 x^4}} \, dx=\int \frac {1}{\sqrt {4\,x^4+5\,x^2+2}} \,d x \]
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