Integrand size = 24, antiderivative size = 96 \[ \int \frac {3-x^2}{\sqrt {3+3 x^2-x^4}} \, dx=-\sqrt {\frac {1}{2} \left (-3+\sqrt {21}\right )} E\left (\arcsin \left (\sqrt {\frac {2}{3+\sqrt {21}}} x\right )|\frac {1}{2} \left (-5-\sqrt {21}\right )\right )+\sqrt {9+2 \sqrt {21}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {2}{3+\sqrt {21}}} x\right ),\frac {1}{2} \left (-5-\sqrt {21}\right )\right ) \]
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Time = 0.12 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1194, 538, 435, 430} \[ \int \frac {3-x^2}{\sqrt {3+3 x^2-x^4}} \, dx=\sqrt {9+2 \sqrt {21}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {2}{3+\sqrt {21}}} x\right ),\frac {1}{2} \left (-5-\sqrt {21}\right )\right )-\sqrt {\frac {1}{2} \left (\sqrt {21}-3\right )} E\left (\arcsin \left (\sqrt {\frac {2}{3+\sqrt {21}}} x\right )|\frac {1}{2} \left (-5-\sqrt {21}\right )\right ) \]
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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 {3+\sqrt {21}-2 x^2} \sqrt {-3+\sqrt {21}+2 x^2}} \, dx \\ & = \left (3+\sqrt {21}\right ) \int \frac {1}{\sqrt {3+\sqrt {21}-2 x^2} \sqrt {-3+\sqrt {21}+2 x^2}} \, dx-\int \frac {\sqrt {-3+\sqrt {21}+2 x^2}}{\sqrt {3+\sqrt {21}-2 x^2}} \, dx \\ & = -\sqrt {\frac {1}{2} \left (-3+\sqrt {21}\right )} E\left (\sin ^{-1}\left (\sqrt {\frac {2}{3+\sqrt {21}}} x\right )|\frac {1}{2} \left (-5-\sqrt {21}\right )\right )+\frac {1}{2} \sqrt {36+8 \sqrt {21}} F\left (\sin ^{-1}\left (\sqrt {\frac {2}{3+\sqrt {21}}} x\right )|\frac {1}{2} \left (-5-\sqrt {21}\right )\right ) \\ \end{align*}
Result contains complex when optimal does not.
Time = 10.14 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.07 \[ \int \frac {3-x^2}{\sqrt {3+3 x^2-x^4}} \, dx=-\frac {i \left (\left (3+\sqrt {21}\right ) E\left (i \text {arcsinh}\left (\sqrt {\frac {2}{-3+\sqrt {21}}} x\right )|\frac {1}{2} \left (-5+\sqrt {21}\right )\right )-\left (-3+\sqrt {21}\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {2}{-3+\sqrt {21}}} x\right ),\frac {1}{2} \left (-5+\sqrt {21}\right )\right )\right )}{\sqrt {2 \left (3+\sqrt {21}\right )}} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 203 vs. \(2 (74 ) = 148\).
Time = 2.09 (sec) , antiderivative size = 204, normalized size of antiderivative = 2.12
method | result | size |
default | \(\frac {18 \sqrt {1-\left (-\frac {1}{2}+\frac {\sqrt {21}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}-\frac {\sqrt {21}}{6}\right ) x^{2}}\, F\left (\frac {x \sqrt {-18+6 \sqrt {21}}}{6}, \frac {i \sqrt {3}}{2}+\frac {i \sqrt {7}}{2}\right )}{\sqrt {-18+6 \sqrt {21}}\, \sqrt {-x^{4}+3 x^{2}+3}}+\frac {36 \sqrt {1-\left (-\frac {1}{2}+\frac {\sqrt {21}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}-\frac {\sqrt {21}}{6}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-18+6 \sqrt {21}}}{6}, \frac {i \sqrt {3}}{2}+\frac {i \sqrt {7}}{2}\right )-E\left (\frac {x \sqrt {-18+6 \sqrt {21}}}{6}, \frac {i \sqrt {3}}{2}+\frac {i \sqrt {7}}{2}\right )\right )}{\sqrt {-18+6 \sqrt {21}}\, \sqrt {-x^{4}+3 x^{2}+3}\, \left (3+\sqrt {21}\right )}\) | \(204\) |
elliptic | \(\frac {18 \sqrt {1-\left (-\frac {1}{2}+\frac {\sqrt {21}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}-\frac {\sqrt {21}}{6}\right ) x^{2}}\, F\left (\frac {x \sqrt {-18+6 \sqrt {21}}}{6}, \frac {i \sqrt {3}}{2}+\frac {i \sqrt {7}}{2}\right )}{\sqrt {-18+6 \sqrt {21}}\, \sqrt {-x^{4}+3 x^{2}+3}}+\frac {36 \sqrt {1-\left (-\frac {1}{2}+\frac {\sqrt {21}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}-\frac {\sqrt {21}}{6}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-18+6 \sqrt {21}}}{6}, \frac {i \sqrt {3}}{2}+\frac {i \sqrt {7}}{2}\right )-E\left (\frac {x \sqrt {-18+6 \sqrt {21}}}{6}, \frac {i \sqrt {3}}{2}+\frac {i \sqrt {7}}{2}\right )\right )}{\sqrt {-18+6 \sqrt {21}}\, \sqrt {-x^{4}+3 x^{2}+3}\, \left (3+\sqrt {21}\right )}\) | \(204\) |
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Time = 0.10 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.11 \[ \int \frac {3-x^2}{\sqrt {3+3 x^2-x^4}} \, dx=\frac {-6 i \, \sqrt {2} x \sqrt {\sqrt {21} + 3} F(\arcsin \left (\frac {\sqrt {2} \sqrt {\sqrt {21} + 3}}{2 \, x}\right )\,|\,\frac {1}{2} \, \sqrt {21} - \frac {5}{2}) + {\left (i \, \sqrt {21} \sqrt {2} x + 3 i \, \sqrt {2} x\right )} \sqrt {\sqrt {21} + 3} E(\arcsin \left (\frac {\sqrt {2} \sqrt {\sqrt {21} + 3}}{2 \, x}\right )\,|\,\frac {1}{2} \, \sqrt {21} - \frac {5}{2}) + 4 \, \sqrt {-x^{4} + 3 \, x^{2} + 3}}{4 \, x} \]
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\[ \int \frac {3-x^2}{\sqrt {3+3 x^2-x^4}} \, dx=- \int \frac {x^{2}}{\sqrt {- x^{4} + 3 x^{2} + 3}}\, dx - \int \left (- \frac {3}{\sqrt {- x^{4} + 3 x^{2} + 3}}\right )\, dx \]
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\[ \int \frac {3-x^2}{\sqrt {3+3 x^2-x^4}} \, dx=\int { -\frac {x^{2} - 3}{\sqrt {-x^{4} + 3 \, x^{2} + 3}} \,d x } \]
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\[ \int \frac {3-x^2}{\sqrt {3+3 x^2-x^4}} \, dx=\int { -\frac {x^{2} - 3}{\sqrt {-x^{4} + 3 \, x^{2} + 3}} \,d x } \]
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Timed out. \[ \int \frac {3-x^2}{\sqrt {3+3 x^2-x^4}} \, dx=\int -\frac {x^2-3}{\sqrt {-x^4+3\,x^2+3}} \,d x \]
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