Integrand size = 17, antiderivative size = 48 \[ \int \sqrt {\left (1-x^2\right ) \left (3+x^2\right )} \, dx=\frac {1}{3} x \sqrt {3-2 x^2-x^4}-\frac {2 E\left (\arcsin (x)\left |-\frac {1}{3}\right .\right )}{\sqrt {3}}+\frac {4 \operatorname {EllipticF}\left (\arcsin (x),-\frac {1}{3}\right )}{\sqrt {3}} \]
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Time = 0.03 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {1976, 1105, 1194, 538, 435, 430} \[ \int \sqrt {\left (1-x^2\right ) \left (3+x^2\right )} \, dx=\frac {4 \operatorname {EllipticF}\left (\arcsin (x),-\frac {1}{3}\right )}{\sqrt {3}}-\frac {2 E\left (\arcsin (x)\left |-\frac {1}{3}\right .\right )}{\sqrt {3}}+\frac {1}{3} \sqrt {-x^4-2 x^2+3} x \]
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Rule 430
Rule 435
Rule 538
Rule 1105
Rule 1194
Rule 1976
Rubi steps \begin{align*} \text {integral}& = \int \sqrt {3-2 x^2-x^4} \, dx \\ & = \frac {1}{3} x \sqrt {3-2 x^2-x^4}+\frac {1}{3} \int \frac {6-2 x^2}{\sqrt {3-2 x^2-x^4}} \, dx \\ & = \frac {1}{3} x \sqrt {3-2 x^2-x^4}+\frac {2}{3} \int \frac {6-2 x^2}{\sqrt {2-2 x^2} \sqrt {6+2 x^2}} \, dx \\ & = \frac {1}{3} x \sqrt {3-2 x^2-x^4}-\frac {2}{3} \int \frac {\sqrt {6+2 x^2}}{\sqrt {2-2 x^2}} \, dx+8 \int \frac {1}{\sqrt {2-2 x^2} \sqrt {6+2 x^2}} \, dx \\ & = \frac {1}{3} x \sqrt {3-2 x^2-x^4}-\frac {2 E\left (\sin ^{-1}(x)|-\frac {1}{3}\right )}{\sqrt {3}}+\frac {4 F\left (\sin ^{-1}(x)|-\frac {1}{3}\right )}{\sqrt {3}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.05 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.23 \[ \int \sqrt {\left (1-x^2\right ) \left (3+x^2\right )} \, dx=\frac {1}{3} \left (x \sqrt {3-2 x^2-x^4}-2 i E\left (\left .i \text {arcsinh}\left (\frac {x}{\sqrt {3}}\right )\right |-3\right )-4 i \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {x}{\sqrt {3}}\right ),-3\right )\right ) \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 113 vs. \(2 (44 ) = 88\).
Time = 1.53 (sec) , antiderivative size = 114, normalized size of antiderivative = 2.38
method | result | size |
default | \(\frac {x \sqrt {-x^{4}-2 x^{2}+3}}{3}+\frac {2 \sqrt {-x^{2}+1}\, \sqrt {3 x^{2}+9}\, F\left (x , \frac {i \sqrt {3}}{3}\right )}{3 \sqrt {-x^{4}-2 x^{2}+3}}+\frac {2 \sqrt {-x^{2}+1}\, \sqrt {3 x^{2}+9}\, \left (F\left (x , \frac {i \sqrt {3}}{3}\right )-E\left (x , \frac {i \sqrt {3}}{3}\right )\right )}{3 \sqrt {-x^{4}-2 x^{2}+3}}\) | \(114\) |
elliptic | \(\frac {x \sqrt {-x^{4}-2 x^{2}+3}}{3}+\frac {2 \sqrt {-x^{2}+1}\, \sqrt {3 x^{2}+9}\, F\left (x , \frac {i \sqrt {3}}{3}\right )}{3 \sqrt {-x^{4}-2 x^{2}+3}}+\frac {2 \sqrt {-x^{2}+1}\, \sqrt {3 x^{2}+9}\, \left (F\left (x , \frac {i \sqrt {3}}{3}\right )-E\left (x , \frac {i \sqrt {3}}{3}\right )\right )}{3 \sqrt {-x^{4}-2 x^{2}+3}}\) | \(114\) |
risch | \(-\frac {x \left (x^{2}-1\right ) \left (x^{2}+3\right )}{3 \sqrt {-\left (x^{2}-1\right ) \left (x^{2}+3\right )}}+\frac {2 \sqrt {-x^{2}+1}\, \sqrt {3 x^{2}+9}\, F\left (x , \frac {i \sqrt {3}}{3}\right )}{3 \sqrt {-x^{4}-2 x^{2}+3}}+\frac {2 \sqrt {-x^{2}+1}\, \sqrt {3 x^{2}+9}\, \left (F\left (x , \frac {i \sqrt {3}}{3}\right )-E\left (x , \frac {i \sqrt {3}}{3}\right )\right )}{3 \sqrt {-x^{4}-2 x^{2}+3}}\) | \(124\) |
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none
Time = 0.11 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.92 \[ \int \sqrt {\left (1-x^2\right ) \left (3+x^2\right )} \, dx=\frac {2 i \, x E(\arcsin \left (\frac {1}{x}\right )\,|\,-3) + 4 i \, x F(\arcsin \left (\frac {1}{x}\right )\,|\,-3) + \sqrt {-x^{4} - 2 \, x^{2} + 3} {\left (x^{2} + 2\right )}}{3 \, x} \]
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\[ \int \sqrt {\left (1-x^2\right ) \left (3+x^2\right )} \, dx=\int \sqrt {\left (1 - x^{2}\right ) \left (x^{2} + 3\right )}\, dx \]
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\[ \int \sqrt {\left (1-x^2\right ) \left (3+x^2\right )} \, dx=\int { \sqrt {-{\left (x^{2} + 3\right )} {\left (x^{2} - 1\right )}} \,d x } \]
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\[ \int \sqrt {\left (1-x^2\right ) \left (3+x^2\right )} \, dx=\int { \sqrt {-{\left (x^{2} + 3\right )} {\left (x^{2} - 1\right )}} \,d x } \]
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Timed out. \[ \int \sqrt {\left (1-x^2\right ) \left (3+x^2\right )} \, dx=\int \sqrt {-\left (x^2-1\right )\,\left (x^2+3\right )} \,d x \]
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