Integrand size = 16, antiderivative size = 42 \[ \int \frac {1}{\sqrt {3-2 x^2-2 x^4}} \, dx=\frac {\operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {2}{-1+\sqrt {7}}} x\right ),\frac {1}{3} \left (-4+\sqrt {7}\right )\right )}{\sqrt {1+\sqrt {7}}} \]
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Time = 0.04 (sec) , antiderivative size = 42, 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} \[ \int \frac {1}{\sqrt {3-2 x^2-2 x^4}} \, dx=\frac {\operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {2}{-1+\sqrt {7}}} x\right ),\frac {1}{3} \left (-4+\sqrt {7}\right )\right )}{\sqrt {1+\sqrt {7}}} \]
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Rule 430
Rule 1109
Rubi steps \begin{align*} \text {integral}& = \left (2 \sqrt {2}\right ) \int \frac {1}{\sqrt {-2+2 \sqrt {7}-4 x^2} \sqrt {2+2 \sqrt {7}+4 x^2}} \, dx \\ & = \frac {F\left (\sin ^{-1}\left (\sqrt {\frac {2}{-1+\sqrt {7}}} x\right )|\frac {1}{3} \left (-4+\sqrt {7}\right )\right )}{\sqrt {1+\sqrt {7}}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 10.04 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.21 \[ \int \frac {1}{\sqrt {3-2 x^2-2 x^4}} \, dx=-\frac {i \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {2}{1+\sqrt {7}}} x\right ),-\frac {4}{3}-\frac {\sqrt {7}}{3}\right )}{\sqrt {-1+\sqrt {7}}} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (34 ) = 68\).
Time = 0.32 (sec) , antiderivative size = 84, normalized size of antiderivative = 2.00
method | result | size |
default | \(\frac {3 \sqrt {1-\left (\frac {1}{3}+\frac {\sqrt {7}}{3}\right ) x^{2}}\, \sqrt {1-\left (\frac {1}{3}-\frac {\sqrt {7}}{3}\right ) x^{2}}\, F\left (\frac {x \sqrt {3+3 \sqrt {7}}}{3}, \frac {i \sqrt {42}}{6}-\frac {i \sqrt {6}}{6}\right )}{\sqrt {3+3 \sqrt {7}}\, \sqrt {-2 x^{4}-2 x^{2}+3}}\) | \(84\) |
elliptic | \(\frac {3 \sqrt {1-\left (\frac {1}{3}+\frac {\sqrt {7}}{3}\right ) x^{2}}\, \sqrt {1-\left (\frac {1}{3}-\frac {\sqrt {7}}{3}\right ) x^{2}}\, F\left (\frac {x \sqrt {3+3 \sqrt {7}}}{3}, \frac {i \sqrt {42}}{6}-\frac {i \sqrt {6}}{6}\right )}{\sqrt {3+3 \sqrt {7}}\, \sqrt {-2 x^{4}-2 x^{2}+3}}\) | \(84\) |
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none
Time = 0.08 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.86 \[ \int \frac {1}{\sqrt {3-2 x^2-2 x^4}} \, dx=\frac {1}{6} \, \sqrt {\sqrt {7} + 1} {\left (\sqrt {7} - 1\right )} F(\arcsin \left (\frac {1}{3} \, \sqrt {3} x \sqrt {\sqrt {7} + 1}\right )\,|\,\frac {1}{3} \, \sqrt {7} - \frac {4}{3}) \]
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\[ \int \frac {1}{\sqrt {3-2 x^2-2 x^4}} \, dx=\int \frac {1}{\sqrt {- 2 x^{4} - 2 x^{2} + 3}}\, dx \]
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\[ \int \frac {1}{\sqrt {3-2 x^2-2 x^4}} \, dx=\int { \frac {1}{\sqrt {-2 \, x^{4} - 2 \, x^{2} + 3}} \,d x } \]
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\[ \int \frac {1}{\sqrt {3-2 x^2-2 x^4}} \, dx=\int { \frac {1}{\sqrt {-2 \, x^{4} - 2 \, x^{2} + 3}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt {3-2 x^2-2 x^4}} \, dx=\int \frac {1}{\sqrt {-2\,x^4-2\,x^2+3}} \,d x \]
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