Optimal. Leaf size=36 \[ \frac {1}{2} x \sqrt {-4 x^2-9}-\frac {9}{4} \tan ^{-1}\left (\frac {2 x}{\sqrt {-4 x^2-9}}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {195, 217, 203} \[ \frac {1}{2} x \sqrt {-4 x^2-9}-\frac {9}{4} \tan ^{-1}\left (\frac {2 x}{\sqrt {-4 x^2-9}}\right ) \]
Antiderivative was successfully verified.
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Rule 195
Rule 203
Rule 217
Rubi steps
\begin {align*} \int \sqrt {-9-4 x^2} \, dx &=\frac {1}{2} x \sqrt {-9-4 x^2}-\frac {9}{2} \int \frac {1}{\sqrt {-9-4 x^2}} \, dx\\ &=\frac {1}{2} x \sqrt {-9-4 x^2}-\frac {9}{2} \operatorname {Subst}\left (\int \frac {1}{1+4 x^2} \, dx,x,\frac {x}{\sqrt {-9-4 x^2}}\right )\\ &=\frac {1}{2} x \sqrt {-9-4 x^2}-\frac {9}{4} \tan ^{-1}\left (\frac {2 x}{\sqrt {-9-4 x^2}}\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 36, normalized size = 1.00 \[ \frac {1}{4} \left (2 x \sqrt {-4 x^2-9}-9 \tan ^{-1}\left (\frac {2 x}{\sqrt {-4 x^2-9}}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [C] time = 0.88, size = 59, normalized size = 1.64 \[ \frac {1}{2} \, \sqrt {-4 \, x^{2} - 9} x - \frac {9}{8} i \, \log \left (-\frac {8 \, x + 4 i \, \sqrt {-4 \, x^{2} - 9}}{x}\right ) + \frac {9}{8} i \, \log \left (-\frac {8 \, x - 4 i \, \sqrt {-4 \, x^{2} - 9}}{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {-4 \, x^{2} - 9}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 29, normalized size = 0.81 \[ \frac {\sqrt {-4 x^{2}-9}\, x}{2}-\frac {9 \arctan \left (\frac {2 x}{\sqrt {-4 x^{2}-9}}\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 2.99, size = 19, normalized size = 0.53 \[ \frac {1}{2} \, \sqrt {-4 \, x^{2} - 9} x + \frac {9}{4} i \, \operatorname {arsinh}\left (\frac {2}{3} \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.03, size = 28, normalized size = 0.78 \[ \frac {x\,\sqrt {-4\,x^2-9}}{2}-\frac {9\,\mathrm {atan}\left (\frac {2\,x}{\sqrt {-4\,x^2-9}}\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.39, size = 34, normalized size = 0.94 \[ \frac {x \sqrt {- 4 x^{2} - 9}}{2} - \frac {9 \operatorname {atan}{\left (\frac {2 x}{\sqrt {- 4 x^{2} - 9}} \right )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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