Integrand size = 16, antiderivative size = 65 \[ \int \frac {1}{\sqrt {-2-x^2+3 x^4}} \, dx=\frac {\sqrt {-1+x^2} \sqrt {2+3 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {5}{2}} x}{\sqrt {-1+x^2}}\right ),\frac {2}{5}\right )}{\sqrt {5} \sqrt {-2-x^2+3 x^4}} \]
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Time = 0.00 (sec) , antiderivative size = 65, 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 = {1111} \[ \int \frac {1}{\sqrt {-2-x^2+3 x^4}} \, dx=\frac {\sqrt {x^2-1} \sqrt {3 x^2+2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {5}{2}} x}{\sqrt {x^2-1}}\right ),\frac {2}{5}\right )}{\sqrt {5} \sqrt {3 x^4-x^2-2}} \]
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Rule 1111
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {-1+x^2} \sqrt {2+3 x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {5}{2}} x}{\sqrt {-1+x^2}}\right )|\frac {2}{5}\right )}{\sqrt {5} \sqrt {-2-x^2+3 x^4}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 10.03 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.92 \[ \int \frac {1}{\sqrt {-2-x^2+3 x^4}} \, dx=-\frac {i \sqrt {1-x^2} \sqrt {2+3 x^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right ),-\frac {2}{3}\right )}{\sqrt {-6-3 x^2+9 x^4}} \]
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Result contains complex when optimal does not.
Time = 0.58 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.82
method | result | size |
default | \(-\frac {i \sqrt {6}\, \sqrt {6 x^{2}+4}\, \sqrt {-x^{2}+1}\, F\left (\frac {i x \sqrt {6}}{2}, \frac {i \sqrt {6}}{3}\right )}{6 \sqrt {3 x^{4}-x^{2}-2}}\) | \(53\) |
elliptic | \(-\frac {i \sqrt {6}\, \sqrt {6 x^{2}+4}\, \sqrt {-x^{2}+1}\, F\left (\frac {i x \sqrt {6}}{2}, \frac {i \sqrt {6}}{3}\right )}{6 \sqrt {3 x^{4}-x^{2}-2}}\) | \(53\) |
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none
Time = 0.07 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.14 \[ \int \frac {1}{\sqrt {-2-x^2+3 x^4}} \, dx=-\frac {1}{2} \, \sqrt {-2} F(\arcsin \left (x\right )\,|\,-\frac {3}{2}) \]
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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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