Integrand size = 16, antiderivative size = 110 \[ \int \frac {1}{\sqrt {3+8 x^2+2 x^4}} \, dx=\frac {\sqrt {\frac {3+\left (4-\sqrt {10}\right ) x^2}{3+\left (4+\sqrt {10}\right ) x^2}} \left (3+\left (4+\sqrt {10}\right ) x^2\right ) \operatorname {EllipticF}\left (\arctan \left (\sqrt {\frac {1}{3} \left (4+\sqrt {10}\right )} x\right ),-\frac {2}{3} \left (5-2 \sqrt {10}\right )\right )}{\sqrt {3 \left (4+\sqrt {10}\right )} \sqrt {3+8 x^2+2 x^4}} \]
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Time = 0.05 (sec) , antiderivative size = 110, 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 = {1113} \[ \int \frac {1}{\sqrt {3+8 x^2+2 x^4}} \, dx=\frac {\sqrt {\frac {\left (4-\sqrt {10}\right ) x^2+3}{\left (4+\sqrt {10}\right ) x^2+3}} \left (\left (4+\sqrt {10}\right ) x^2+3\right ) \operatorname {EllipticF}\left (\arctan \left (\sqrt {\frac {1}{3} \left (4+\sqrt {10}\right )} x\right ),-\frac {2}{3} \left (5-2 \sqrt {10}\right )\right )}{\sqrt {3 \left (4+\sqrt {10}\right )} \sqrt {2 x^4+8 x^2+3}} \]
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Rule 1113
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {\frac {3+\left (4-\sqrt {10}\right ) x^2}{3+\left (4+\sqrt {10}\right ) x^2}} \left (3+\left (4+\sqrt {10}\right ) x^2\right ) F\left (\tan ^{-1}\left (\sqrt {\frac {1}{3} \left (4+\sqrt {10}\right )} x\right )|-\frac {2}{3} \left (5-2 \sqrt {10}\right )\right )}{\sqrt {3 \left (4+\sqrt {10}\right )} \sqrt {3+8 x^2+2 x^4}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 10.06 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.89 \[ \int \frac {1}{\sqrt {3+8 x^2+2 x^4}} \, dx=-\frac {i \sqrt {\frac {-4+\sqrt {10}-2 x^2}{-4+\sqrt {10}}} \sqrt {4+\sqrt {10}+2 x^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {2}{4+\sqrt {10}}} x\right ),\frac {13}{3}+\frac {4 \sqrt {10}}{3}\right )}{\sqrt {6+16 x^2+4 x^4}} \]
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Time = 0.66 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.75
method | result | size |
default | \(\frac {3 \sqrt {1-\left (-\frac {4}{3}+\frac {\sqrt {10}}{3}\right ) x^{2}}\, \sqrt {1-\left (-\frac {4}{3}-\frac {\sqrt {10}}{3}\right ) x^{2}}\, F\left (\frac {x \sqrt {-12+3 \sqrt {10}}}{3}, \frac {2 \sqrt {6}}{3}+\frac {\sqrt {15}}{3}\right )}{\sqrt {-12+3 \sqrt {10}}\, \sqrt {2 x^{4}+8 x^{2}+3}}\) | \(82\) |
elliptic | \(\frac {3 \sqrt {1-\left (-\frac {4}{3}+\frac {\sqrt {10}}{3}\right ) x^{2}}\, \sqrt {1-\left (-\frac {4}{3}-\frac {\sqrt {10}}{3}\right ) x^{2}}\, F\left (\frac {x \sqrt {-12+3 \sqrt {10}}}{3}, \frac {2 \sqrt {6}}{3}+\frac {\sqrt {15}}{3}\right )}{\sqrt {-12+3 \sqrt {10}}\, \sqrt {2 x^{4}+8 x^{2}+3}}\) | \(82\) |
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none
Time = 0.07 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.33 \[ \int \frac {1}{\sqrt {3+8 x^2+2 x^4}} \, dx=-\frac {1}{6} \, {\left (\sqrt {10} + 4\right )} \sqrt {\sqrt {10} - 4} F(\arcsin \left (\frac {1}{3} \, \sqrt {3} x \sqrt {\sqrt {10} - 4}\right )\,|\,\frac {4}{3} \, \sqrt {10} + \frac {13}{3}) \]
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\[ \int \frac {1}{\sqrt {3+8 x^2+2 x^4}} \, dx=\int \frac {1}{\sqrt {2 x^{4} + 8 x^{2} + 3}}\, dx \]
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\[ \int \frac {1}{\sqrt {3+8 x^2+2 x^4}} \, dx=\int { \frac {1}{\sqrt {2 \, x^{4} + 8 \, x^{2} + 3}} \,d x } \]
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\[ \int \frac {1}{\sqrt {3+8 x^2+2 x^4}} \, dx=\int { \frac {1}{\sqrt {2 \, x^{4} + 8 \, x^{2} + 3}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt {3+8 x^2+2 x^4}} \, dx=\int \frac {1}{\sqrt {2\,x^4+8\,x^2+3}} \,d x \]
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