Integrand size = 22, antiderivative size = 96 \[ \int \frac {3-x^2}{\sqrt {3+x^2-x^4}} \, dx=-\sqrt {\frac {1}{2} \left (-1+\sqrt {13}\right )} E\left (\arcsin \left (\sqrt {\frac {2}{1+\sqrt {13}}} x\right )|\frac {1}{6} \left (-7-\sqrt {13}\right )\right )+\sqrt {7+2 \sqrt {13}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {2}{1+\sqrt {13}}} x\right ),\frac {1}{6} \left (-7-\sqrt {13}\right )\right ) \]
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Time = 0.08 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {1194, 538, 435, 430} \[ \int \frac {3-x^2}{\sqrt {3+x^2-x^4}} \, dx=\sqrt {7+2 \sqrt {13}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {2}{1+\sqrt {13}}} x\right ),\frac {1}{6} \left (-7-\sqrt {13}\right )\right )-\sqrt {\frac {1}{2} \left (\sqrt {13}-1\right )} E\left (\arcsin \left (\sqrt {\frac {2}{1+\sqrt {13}}} x\right )|\frac {1}{6} \left (-7-\sqrt {13}\right )\right ) \]
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Rule 430
Rule 435
Rule 538
Rule 1194
Rubi steps \begin{align*} \text {integral}& = 2 \int \frac {3-x^2}{\sqrt {1+\sqrt {13}-2 x^2} \sqrt {-1+\sqrt {13}+2 x^2}} \, dx \\ & = \left (5+\sqrt {13}\right ) \int \frac {1}{\sqrt {1+\sqrt {13}-2 x^2} \sqrt {-1+\sqrt {13}+2 x^2}} \, dx-\int \frac {\sqrt {-1+\sqrt {13}+2 x^2}}{\sqrt {1+\sqrt {13}-2 x^2}} \, dx \\ & = -\sqrt {\frac {1}{2} \left (-1+\sqrt {13}\right )} E\left (\sin ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {13}}} x\right )|\frac {1}{6} \left (-7-\sqrt {13}\right )\right )+\sqrt {7+2 \sqrt {13}} F\left (\sin ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {13}}} x\right )|\frac {1}{6} \left (-7-\sqrt {13}\right )\right ) \\ \end{align*}
Result contains complex when optimal does not.
Time = 10.13 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.07 \[ \int \frac {3-x^2}{\sqrt {3+x^2-x^4}} \, dx=-\frac {i \left (\left (1+\sqrt {13}\right ) E\left (i \text {arcsinh}\left (\sqrt {\frac {2}{-1+\sqrt {13}}} x\right )|\frac {1}{6} \left (-7+\sqrt {13}\right )\right )-\left (-5+\sqrt {13}\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {2}{-1+\sqrt {13}}} x\right ),\frac {1}{6} \left (-7+\sqrt {13}\right )\right )\right )}{\sqrt {2 \left (1+\sqrt {13}\right )}} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 199 vs. \(2 (74 ) = 148\).
Time = 2.32 (sec) , antiderivative size = 200, normalized size of antiderivative = 2.08
method | result | size |
default | \(\frac {18 \sqrt {1-\left (-\frac {1}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, F\left (\frac {x \sqrt {-6+6 \sqrt {13}}}{6}, \frac {i \sqrt {3}}{6}+\frac {i \sqrt {39}}{6}\right )}{\sqrt {-6+6 \sqrt {13}}\, \sqrt {-x^{4}+x^{2}+3}}+\frac {36 \sqrt {1-\left (-\frac {1}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-6+6 \sqrt {13}}}{6}, \frac {i \sqrt {3}}{6}+\frac {i \sqrt {39}}{6}\right )-E\left (\frac {x \sqrt {-6+6 \sqrt {13}}}{6}, \frac {i \sqrt {3}}{6}+\frac {i \sqrt {39}}{6}\right )\right )}{\sqrt {-6+6 \sqrt {13}}\, \sqrt {-x^{4}+x^{2}+3}\, \left (1+\sqrt {13}\right )}\) | \(200\) |
elliptic | \(\frac {18 \sqrt {1-\left (-\frac {1}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, F\left (\frac {x \sqrt {-6+6 \sqrt {13}}}{6}, \frac {i \sqrt {3}}{6}+\frac {i \sqrt {39}}{6}\right )}{\sqrt {-6+6 \sqrt {13}}\, \sqrt {-x^{4}+x^{2}+3}}+\frac {36 \sqrt {1-\left (-\frac {1}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-6+6 \sqrt {13}}}{6}, \frac {i \sqrt {3}}{6}+\frac {i \sqrt {39}}{6}\right )-E\left (\frac {x \sqrt {-6+6 \sqrt {13}}}{6}, \frac {i \sqrt {3}}{6}+\frac {i \sqrt {39}}{6}\right )\right )}{\sqrt {-6+6 \sqrt {13}}\, \sqrt {-x^{4}+x^{2}+3}\, \left (1+\sqrt {13}\right )}\) | \(200\) |
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none
Time = 0.10 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.09 \[ \int \frac {3-x^2}{\sqrt {3+x^2-x^4}} \, dx=\frac {-2 i \, \sqrt {2} x \sqrt {\sqrt {13} + 1} F(\arcsin \left (\frac {\sqrt {2} \sqrt {\sqrt {13} + 1}}{2 \, x}\right )\,|\,\frac {1}{6} \, \sqrt {13} - \frac {7}{6}) + {\left (i \, \sqrt {13} \sqrt {2} x + i \, \sqrt {2} x\right )} \sqrt {\sqrt {13} + 1} E(\arcsin \left (\frac {\sqrt {2} \sqrt {\sqrt {13} + 1}}{2 \, x}\right )\,|\,\frac {1}{6} \, \sqrt {13} - \frac {7}{6}) + 4 \, \sqrt {-x^{4} + x^{2} + 3}}{4 \, x} \]
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\[ \int \frac {3-x^2}{\sqrt {3+x^2-x^4}} \, dx=- \int \frac {x^{2}}{\sqrt {- x^{4} + x^{2} + 3}}\, dx - \int \left (- \frac {3}{\sqrt {- x^{4} + x^{2} + 3}}\right )\, dx \]
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\[ \int \frac {3-x^2}{\sqrt {3+x^2-x^4}} \, dx=\int { -\frac {x^{2} - 3}{\sqrt {-x^{4} + x^{2} + 3}} \,d x } \]
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\[ \int \frac {3-x^2}{\sqrt {3+x^2-x^4}} \, dx=\int { -\frac {x^{2} - 3}{\sqrt {-x^{4} + x^{2} + 3}} \,d x } \]
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Timed out. \[ \int \frac {3-x^2}{\sqrt {3+x^2-x^4}} \, dx=-\int \frac {x^2-3}{\sqrt {-x^4+x^2+3}} \,d x \]
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