Integrand size = 16, antiderivative size = 141 \[ \int \frac {1}{\sqrt {-2+2 x^2+3 x^4}} \, dx=\frac {\sqrt {\frac {2-\left (1-\sqrt {7}\right ) x^2}{2-\left (1+\sqrt {7}\right ) x^2}} \sqrt {-2+\left (1+\sqrt {7}\right ) x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{7} x}{\sqrt {-2+\left (1+\sqrt {7}\right ) x^2}}\right ),\frac {1}{14} \left (7+\sqrt {7}\right )\right )}{2 \sqrt [4]{7} \sqrt {\frac {1}{2-\left (1+\sqrt {7}\right ) x^2}} \sqrt {-2+2 x^2+3 x^4}} \]
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Time = 0.02 (sec) , antiderivative size = 141, 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} \[ \int \frac {1}{\sqrt {-2+2 x^2+3 x^4}} \, dx=\frac {\sqrt {\frac {2-\left (1-\sqrt {7}\right ) x^2}{2-\left (1+\sqrt {7}\right ) x^2}} \sqrt {\left (1+\sqrt {7}\right ) x^2-2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{7} x}{\sqrt {\left (1+\sqrt {7}\right ) x^2-2}}\right ),\frac {1}{14} \left (7+\sqrt {7}\right )\right )}{2 \sqrt [4]{7} \sqrt {\frac {1}{2-\left (1+\sqrt {7}\right ) x^2}} \sqrt {3 x^4+2 x^2-2}} \]
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Rule 1112
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {\frac {2-\left (1-\sqrt {7}\right ) x^2}{2-\left (1+\sqrt {7}\right ) x^2}} \sqrt {-2+\left (1+\sqrt {7}\right ) x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{7} x}{\sqrt {-2+\left (1+\sqrt {7}\right ) x^2}}\right )|\frac {1}{14} \left (7+\sqrt {7}\right )\right )}{2 \sqrt [4]{7} \sqrt {\frac {1}{2-\left (1+\sqrt {7}\right ) x^2}} \sqrt {-2+2 x^2+3 x^4}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 10.05 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.59 \[ \int \frac {1}{\sqrt {-2+2 x^2+3 x^4}} \, dx=-\frac {i \sqrt {2-2 x^2-3 x^4} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {3}{1+\sqrt {7}}} x\right ),-\frac {4}{3}-\frac {\sqrt {7}}{3}\right )}{\sqrt {-1+\sqrt {7}} \sqrt {-2+2 x^2+3 x^4}} \]
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Result contains complex when optimal does not.
Time = 0.58 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.60
method | result | size |
default | \(\frac {2 \sqrt {1-\left (\frac {1}{2}-\frac {\sqrt {7}}{2}\right ) x^{2}}\, \sqrt {1-\left (\frac {1}{2}+\frac {\sqrt {7}}{2}\right ) x^{2}}\, F\left (\frac {\sqrt {2-2 \sqrt {7}}\, x}{2}, \frac {i \sqrt {6}}{6}+\frac {i \sqrt {42}}{6}\right )}{\sqrt {2-2 \sqrt {7}}\, \sqrt {3 x^{4}+2 x^{2}-2}}\) | \(84\) |
elliptic | \(\frac {2 \sqrt {1-\left (\frac {1}{2}-\frac {\sqrt {7}}{2}\right ) x^{2}}\, \sqrt {1-\left (\frac {1}{2}+\frac {\sqrt {7}}{2}\right ) x^{2}}\, F\left (\frac {\sqrt {2-2 \sqrt {7}}\, x}{2}, \frac {i \sqrt {6}}{6}+\frac {i \sqrt {42}}{6}\right )}{\sqrt {2-2 \sqrt {7}}\, \sqrt {3 x^{4}+2 x^{2}-2}}\) | \(84\) |
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none
Time = 0.08 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.35 \[ \int \frac {1}{\sqrt {-2+2 x^2+3 x^4}} \, dx=-\frac {1}{12} \, {\left (\sqrt {7} \sqrt {2} \sqrt {-2} - \sqrt {2} \sqrt {-2}\right )} \sqrt {\sqrt {7} + 1} F(\arcsin \left (\frac {1}{2} \, \sqrt {2} x \sqrt {\sqrt {7} + 1}\right )\,|\,\frac {1}{3} \, \sqrt {7} - \frac {4}{3}) \]
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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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