Integrand size = 16, antiderivative size = 44 \[ \int \frac {1}{\sqrt {2+2 x^2-3 x^4}} \, dx=\frac {\operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{1+\sqrt {7}}} x\right ),\frac {1}{3} \left (-4-\sqrt {7}\right )\right )}{\sqrt {-1+\sqrt {7}}} \]
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Time = 0.05 (sec) , antiderivative size = 44, 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} \[ \int \frac {1}{\sqrt {2+2 x^2-3 x^4}} \, dx=\frac {\operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{1+\sqrt {7}}} x\right ),\frac {1}{3} \left (-4-\sqrt {7}\right )\right )}{\sqrt {\sqrt {7}-1}} \]
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Rule 430
Rule 1109
Rubi steps \begin{align*} \text {integral}& = \left (2 \sqrt {3}\right ) \int \frac {1}{\sqrt {2+2 \sqrt {7}-6 x^2} \sqrt {-2+2 \sqrt {7}+6 x^2}} \, dx \\ & = \frac {F\left (\sin ^{-1}\left (\sqrt {\frac {3}{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 = 49, normalized size of antiderivative = 1.11 \[ \int \frac {1}{\sqrt {2+2 x^2-3 x^4}} \, dx=-\frac {i \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {3}{-1+\sqrt {7}}} x\right ),\frac {1}{3} \left (-4+\sqrt {7}\right )\right )}{\sqrt {1+\sqrt {7}}} \]
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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 = 0.37 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.91
method | result | size |
default | \(\frac {2 \sqrt {1-\left (-\frac {1}{2}+\frac {\sqrt {7}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}-\frac {\sqrt {7}}{2}\right ) x^{2}}\, F\left (\frac {x \sqrt {-2+2 \sqrt {7}}}{2}, \frac {i \sqrt {6}}{6}+\frac {i \sqrt {42}}{6}\right )}{\sqrt {-2+2 \sqrt {7}}\, \sqrt {-3 x^{4}+2 x^{2}+2}}\) | \(84\) |
elliptic | \(\frac {2 \sqrt {1-\left (-\frac {1}{2}+\frac {\sqrt {7}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}-\frac {\sqrt {7}}{2}\right ) x^{2}}\, F\left (\frac {x \sqrt {-2+2 \sqrt {7}}}{2}, \frac {i \sqrt {6}}{6}+\frac {i \sqrt {42}}{6}\right )}{\sqrt {-2+2 \sqrt {7}}\, \sqrt {-3 x^{4}+2 x^{2}+2}}\) | \(84\) |
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none
Time = 0.07 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.82 \[ \int \frac {1}{\sqrt {2+2 x^2-3 x^4}} \, dx=\frac {1}{6} \, {\left (\sqrt {7} + 1\right )} \sqrt {\sqrt {7} - 1} F(\arcsin \left (\frac {1}{2} \, \sqrt {2} x \sqrt {\sqrt {7} - 1}\right )\,|\,-\frac {1}{3} \, \sqrt {7} - \frac {4}{3}) \]
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\[ \int \frac {1}{\sqrt {2+2 x^2-3 x^4}} \, dx=\int \frac {1}{\sqrt {- 3 x^{4} + 2 x^{2} + 2}}\, dx \]
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\[ \int \frac {1}{\sqrt {2+2 x^2-3 x^4}} \, dx=\int { \frac {1}{\sqrt {-3 \, x^{4} + 2 \, x^{2} + 2}} \,d x } \]
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\[ \int \frac {1}{\sqrt {2+2 x^2-3 x^4}} \, dx=\int { \frac {1}{\sqrt {-3 \, x^{4} + 2 \, x^{2} + 2}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt {2+2 x^2-3 x^4}} \, dx=\int \frac {1}{\sqrt {-3\,x^4+2\,x^2+2}} \,d x \]
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