Integrand size = 16, antiderivative size = 90 \[ \int \frac {1}{\sqrt {3+4 x^2+2 x^4}} \, dx=\frac {\left (3+\sqrt {6} x^2\right ) \sqrt {\frac {3+4 x^2+2 x^4}{\left (3+\sqrt {6} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt [4]{\frac {2}{3}} x\right ),\frac {1}{2}-\frac {1}{\sqrt {6}}\right )}{2 \sqrt [4]{6} \sqrt {3+4 x^2+2 x^4}} \]
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Time = 0.01 (sec) , antiderivative size = 90, 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 = {1117} \[ \int \frac {1}{\sqrt {3+4 x^2+2 x^4}} \, dx=\frac {\left (\sqrt {6} x^2+3\right ) \sqrt {\frac {2 x^4+4 x^2+3}{\left (\sqrt {6} x^2+3\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt [4]{\frac {2}{3}} x\right ),\frac {1}{2}-\frac {1}{\sqrt {6}}\right )}{2 \sqrt [4]{6} \sqrt {2 x^4+4 x^2+3}} \]
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Rule 1117
Rubi steps \begin{align*} \text {integral}& = \frac {\left (3+\sqrt {6} x^2\right ) \sqrt {\frac {3+4 x^2+2 x^4}{\left (3+\sqrt {6} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{\frac {2}{3}} x\right )|\frac {1}{2}-\frac {1}{\sqrt {6}}\right )}{2 \sqrt [4]{6} \sqrt {3+4 x^2+2 x^4}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 10.07 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.60 \[ \int \frac {1}{\sqrt {3+4 x^2+2 x^4}} \, dx=-\frac {i \sqrt {1-\frac {2 x^2}{-2-i \sqrt {2}}} \sqrt {1-\frac {2 x^2}{-2+i \sqrt {2}}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {2}{-2-i \sqrt {2}}} x\right ),\frac {-2-i \sqrt {2}}{-2+i \sqrt {2}}\right )}{\sqrt {2} \sqrt {-\frac {1}{-2-i \sqrt {2}}} \sqrt {3+4 x^2+2 x^4}} \]
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Result contains complex when optimal does not.
Time = 0.62 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.97
method | result | size |
default | \(\frac {3 \sqrt {1-\left (-\frac {2}{3}+\frac {i \sqrt {2}}{3}\right ) x^{2}}\, \sqrt {1-\left (-\frac {2}{3}-\frac {i \sqrt {2}}{3}\right ) x^{2}}\, F\left (\frac {x \sqrt {-6+3 i \sqrt {2}}}{3}, \frac {\sqrt {3+6 i \sqrt {2}}}{3}\right )}{\sqrt {-6+3 i \sqrt {2}}\, \sqrt {2 x^{4}+4 x^{2}+3}}\) | \(87\) |
elliptic | \(\frac {3 \sqrt {1-\left (-\frac {2}{3}+\frac {i \sqrt {2}}{3}\right ) x^{2}}\, \sqrt {1-\left (-\frac {2}{3}-\frac {i \sqrt {2}}{3}\right ) x^{2}}\, F\left (\frac {x \sqrt {-6+3 i \sqrt {2}}}{3}, \frac {\sqrt {3+6 i \sqrt {2}}}{3}\right )}{\sqrt {-6+3 i \sqrt {2}}\, \sqrt {2 x^{4}+4 x^{2}+3}}\) | \(87\) |
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none
Time = 0.07 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.40 \[ \int \frac {1}{\sqrt {3+4 x^2+2 x^4}} \, dx=-\frac {1}{6} \, {\left (\sqrt {-2} + 2\right )} \sqrt {\sqrt {-2} - 2} F(\arcsin \left (\frac {1}{3} \, \sqrt {3} x \sqrt {\sqrt {-2} - 2}\right )\,|\,\frac {2}{3} \, \sqrt {-2} + \frac {1}{3}) \]
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\[ \int \frac {1}{\sqrt {3+4 x^2+2 x^4}} \, dx=\int \frac {1}{\sqrt {2 x^{4} + 4 x^{2} + 3}}\, dx \]
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\[ \int \frac {1}{\sqrt {3+4 x^2+2 x^4}} \, dx=\int { \frac {1}{\sqrt {2 \, x^{4} + 4 \, x^{2} + 3}} \,d x } \]
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\[ \int \frac {1}{\sqrt {3+4 x^2+2 x^4}} \, dx=\int { \frac {1}{\sqrt {2 \, x^{4} + 4 \, x^{2} + 3}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt {3+4 x^2+2 x^4}} \, dx=\int \frac {1}{\sqrt {2\,x^4+4\,x^2+3}} \,d x \]
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