Integrand size = 16, antiderivative size = 63 \[ \int \frac {1}{\sqrt {-3-5 x^2+2 x^4}} \, dx=\frac {\sqrt {-3+x^2} \sqrt {1+2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {7} x}{\sqrt {-3+x^2}}\right ),\frac {1}{7}\right )}{\sqrt {7} \sqrt {-3-5 x^2+2 x^4}} \]
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Time = 0.01 (sec) , antiderivative size = 63, 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} \[ \int \frac {1}{\sqrt {-3-5 x^2+2 x^4}} \, dx=\frac {\sqrt {x^2-3} \sqrt {2 x^2+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {7} x}{\sqrt {x^2-3}}\right ),\frac {1}{7}\right )}{\sqrt {7} \sqrt {2 x^4-5 x^2-3}} \]
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Rule 1111
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {-3+x^2} \sqrt {1+2 x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {7} x}{\sqrt {-3+x^2}}\right )|\frac {1}{7}\right )}{\sqrt {7} \sqrt {-3-5 x^2+2 x^4}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 10.03 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.03 \[ \int \frac {1}{\sqrt {-3-5 x^2+2 x^4}} \, dx=-\frac {i \sqrt {1-\frac {x^2}{3}} \sqrt {1+2 x^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {2} x\right ),-\frac {1}{6}\right )}{\sqrt {2} \sqrt {-3-5 x^2+2 x^4}} \]
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Result contains complex when optimal does not.
Time = 0.56 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.84
method | result | size |
default | \(-\frac {i \sqrt {2}\, \sqrt {2 x^{2}+1}\, \sqrt {-3 x^{2}+9}\, F\left (i \sqrt {2}\, x , \frac {i \sqrt {6}}{6}\right )}{6 \sqrt {2 x^{4}-5 x^{2}-3}}\) | \(53\) |
elliptic | \(-\frac {i \sqrt {2}\, \sqrt {2 x^{2}+1}\, \sqrt {-3 x^{2}+9}\, F\left (i \sqrt {2}\, x , \frac {i \sqrt {6}}{6}\right )}{6 \sqrt {2 x^{4}-5 x^{2}-3}}\) | \(53\) |
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none
Time = 0.07 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.27 \[ \int \frac {1}{\sqrt {-3-5 x^2+2 x^4}} \, dx=-\frac {1}{3} \, \sqrt {3} \sqrt {-3} F(\arcsin \left (\frac {1}{3} \, \sqrt {3} x\right )\,|\,-6) \]
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\[ \int \frac {1}{\sqrt {-3-5 x^2+2 x^4}} \, dx=\int \frac {1}{\sqrt {2 x^{4} - 5 x^{2} - 3}}\, dx \]
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\[ \int \frac {1}{\sqrt {-3-5 x^2+2 x^4}} \, dx=\int { \frac {1}{\sqrt {2 \, x^{4} - 5 \, x^{2} - 3}} \,d x } \]
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\[ \int \frac {1}{\sqrt {-3-5 x^2+2 x^4}} \, dx=\int { \frac {1}{\sqrt {2 \, x^{4} - 5 \, x^{2} - 3}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt {-3-5 x^2+2 x^4}} \, dx=\int \frac {1}{\sqrt {2\,x^4-5\,x^2-3}} \,d x \]
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