Integrand size = 16, antiderivative size = 148 \[ \int \frac {1}{\sqrt {-3+4 x^2+2 x^4}} \, dx=\frac {\sqrt {\frac {3-\left (2-\sqrt {10}\right ) x^2}{3-\left (2+\sqrt {10}\right ) x^2}} \sqrt {-3+\left (2+\sqrt {10}\right ) x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {2^{3/4} \sqrt [4]{5} x}{\sqrt {-3+\left (2+\sqrt {10}\right ) x^2}}\right ),\frac {1}{10} \left (5+\sqrt {10}\right )\right )}{2^{3/4} \sqrt {3} \sqrt [4]{5} \sqrt {\frac {1}{3-\left (2+\sqrt {10}\right ) x^2}} \sqrt {-3+4 x^2+2 x^4}} \]
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Time = 0.02 (sec) , antiderivative size = 148, 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 = {1112} \[ \int \frac {1}{\sqrt {-3+4 x^2+2 x^4}} \, dx=\frac {\sqrt {\frac {3-\left (2-\sqrt {10}\right ) x^2}{3-\left (2+\sqrt {10}\right ) x^2}} \sqrt {\left (2+\sqrt {10}\right ) x^2-3} \operatorname {EllipticF}\left (\arcsin \left (\frac {2^{3/4} \sqrt [4]{5} x}{\sqrt {\left (2+\sqrt {10}\right ) x^2-3}}\right ),\frac {1}{10} \left (5+\sqrt {10}\right )\right )}{2^{3/4} \sqrt {3} \sqrt [4]{5} \sqrt {\frac {1}{3-\left (2+\sqrt {10}\right ) x^2}} \sqrt {2 x^4+4 x^2-3}} \]
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Rule 1112
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {\frac {3-\left (2-\sqrt {10}\right ) x^2}{3-\left (2+\sqrt {10}\right ) x^2}} \sqrt {-3+\left (2+\sqrt {10}\right ) x^2} F\left (\sin ^{-1}\left (\frac {2^{3/4} \sqrt [4]{5} x}{\sqrt {-3+\left (2+\sqrt {10}\right ) x^2}}\right )|\frac {1}{10} \left (5+\sqrt {10}\right )\right )}{2^{3/4} \sqrt {3} \sqrt [4]{5} \sqrt {\frac {1}{3-\left (2+\sqrt {10}\right ) x^2}} \sqrt {-3+4 x^2+2 x^4}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 10.05 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.56 \[ \int \frac {1}{\sqrt {-3+4 x^2+2 x^4}} \, dx=-\frac {i \sqrt {3-4 x^2-2 x^4} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {2}{2+\sqrt {10}}} x\right ),-\frac {7}{3}-\frac {2 \sqrt {10}}{3}\right )}{\sqrt {-2+\sqrt {10}} \sqrt {-3+4 x^2+2 x^4}} \]
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Result contains complex when optimal does not.
Time = 0.60 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.57
method | result | size |
default | \(\frac {3 \sqrt {1-\left (\frac {2}{3}-\frac {\sqrt {10}}{3}\right ) x^{2}}\, \sqrt {1-\left (\frac {2}{3}+\frac {\sqrt {10}}{3}\right ) x^{2}}\, F\left (\frac {\sqrt {6-3 \sqrt {10}}\, x}{3}, \frac {i \sqrt {6}}{3}+\frac {i \sqrt {15}}{3}\right )}{\sqrt {6-3 \sqrt {10}}\, \sqrt {2 x^{4}+4 x^{2}-3}}\) | \(84\) |
elliptic | \(\frac {3 \sqrt {1-\left (\frac {2}{3}-\frac {\sqrt {10}}{3}\right ) x^{2}}\, \sqrt {1-\left (\frac {2}{3}+\frac {\sqrt {10}}{3}\right ) x^{2}}\, F\left (\frac {\sqrt {6-3 \sqrt {10}}\, x}{3}, \frac {i \sqrt {6}}{3}+\frac {i \sqrt {15}}{3}\right )}{\sqrt {6-3 \sqrt {10}}\, \sqrt {2 x^{4}+4 x^{2}-3}}\) | \(84\) |
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none
Time = 0.08 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.34 \[ \int \frac {1}{\sqrt {-3+4 x^2+2 x^4}} \, dx=-\frac {1}{18} \, {\left (\sqrt {10} \sqrt {3} \sqrt {-3} - 2 \, \sqrt {3} \sqrt {-3}\right )} \sqrt {\sqrt {10} + 2} F(\arcsin \left (\frac {1}{3} \, \sqrt {3} x \sqrt {\sqrt {10} + 2}\right )\,|\,\frac {2}{3} \, \sqrt {10} - \frac {7}{3}) \]
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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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