Integrand size = 16, antiderivative size = 52 \[ \int \frac {1}{\sqrt {3+5 x^2+2 x^4}} \, dx=\frac {\left (1+x^2\right ) \sqrt {\frac {3+2 x^2}{1+x^2}} \operatorname {EllipticF}\left (\arctan (x),\frac {1}{3}\right )}{\sqrt {3} \sqrt {3+5 x^2+2 x^4}} \]
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Time = 0.01 (sec) , antiderivative size = 52, 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} \[ \int \frac {1}{\sqrt {3+5 x^2+2 x^4}} \, dx=\frac {\left (x^2+1\right ) \sqrt {\frac {2 x^2+3}{x^2+1}} \operatorname {EllipticF}\left (\arctan (x),\frac {1}{3}\right )}{\sqrt {3} \sqrt {2 x^4+5 x^2+3}} \]
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Rule 1113
Rubi steps \begin{align*} \text {integral}& = \frac {\left (1+x^2\right ) \sqrt {\frac {3+2 x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac {1}{3}\right )}{\sqrt {3} \sqrt {3+5 x^2+2 x^4}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 10.03 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.12 \[ \int \frac {1}{\sqrt {3+5 x^2+2 x^4}} \, dx=-\frac {i \sqrt {1+x^2} \sqrt {3+2 x^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {2}{3}} x\right ),\frac {3}{2}\right )}{\sqrt {6+10 x^2+4 x^4}} \]
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Result contains complex when optimal does not.
Time = 0.81 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.96
method | result | size |
default | \(-\frac {i \sqrt {6}\, \sqrt {6 x^{2}+9}\, \sqrt {x^{2}+1}\, F\left (\frac {i x \sqrt {6}}{3}, \frac {\sqrt {6}}{2}\right )}{6 \sqrt {2 x^{4}+5 x^{2}+3}}\) | \(50\) |
elliptic | \(-\frac {i \sqrt {6}\, \sqrt {6 x^{2}+9}\, \sqrt {x^{2}+1}\, F\left (\frac {i x \sqrt {6}}{3}, \frac {\sqrt {6}}{2}\right )}{6 \sqrt {2 x^{4}+5 x^{2}+3}}\) | \(50\) |
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none
Time = 0.07 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.33 \[ \int \frac {1}{\sqrt {3+5 x^2+2 x^4}} \, dx=-\frac {1}{2} \, \sqrt {-2} F(\arcsin \left (\frac {1}{3} \, \sqrt {3} \sqrt {-2} x\right )\,|\,\frac {3}{2}) \]
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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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