Optimal. Leaf size=30 \[ \sqrt {-4 x^2-9}-3 \tan ^{-1}\left (\frac {1}{3} \sqrt {-4 x^2-9}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {266, 50, 63, 204} \[ \sqrt {-4 x^2-9}-3 \tan ^{-1}\left (\frac {1}{3} \sqrt {-4 x^2-9}\right ) \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 204
Rule 266
Rubi steps
\begin {align*} \int \frac {\sqrt {-9-4 x^2}}{x} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {\sqrt {-9-4 x}}{x} \, dx,x,x^2\right )\\ &=\sqrt {-9-4 x^2}-\frac {9}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-9-4 x} x} \, dx,x,x^2\right )\\ &=\sqrt {-9-4 x^2}+\frac {9}{4} \operatorname {Subst}\left (\int \frac {1}{-\frac {9}{4}-\frac {x^2}{4}} \, dx,x,\sqrt {-9-4 x^2}\right )\\ &=\sqrt {-9-4 x^2}-3 \tan ^{-1}\left (\frac {1}{3} \sqrt {-9-4 x^2}\right )\\ \end {align*}
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Mathematica [A] time = 0.00, size = 30, normalized size = 1.00 \[ \sqrt {-4 x^2-9}-3 \tan ^{-1}\left (\frac {1}{3} \sqrt {-4 x^2-9}\right ) \]
Antiderivative was successfully verified.
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fricas [C] time = 1.10, size = 52, normalized size = 1.73 \[ \sqrt {-4 \, x^{2} - 9} - \frac {3}{2} i \, \log \left (-\frac {6 \, {\left (i \, \sqrt {-4 \, x^{2} - 9} - 3\right )}}{x}\right ) + \frac {3}{2} i \, \log \left (-\frac {6 \, {\left (-i \, \sqrt {-4 \, x^{2} - 9} - 3\right )}}{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.17, size = 24, normalized size = 0.80 \[ \sqrt {-4 \, x^{2} - 9} - 3 \, \arctan \left (\frac {1}{3} \, \sqrt {-4 \, x^{2} - 9}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 25, normalized size = 0.83 \[ 3 \arctan \left (\frac {3}{\sqrt {-4 x^{2}-9}}\right )+\sqrt {-4 x^{2}-9} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 2.93, size = 35, normalized size = 1.17 \[ \sqrt {-4 \, x^{2} - 9} + 3 i \, \log \left (\frac {6 \, \sqrt {4 \, x^{2} + 9}}{{\left | x \right |}} + \frac {18}{{\left | x \right |}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.73, size = 24, normalized size = 0.80 \[ \sqrt {-4\,x^2-9}-3\,\mathrm {atan}\left (\frac {\sqrt {-4\,x^2-9}}{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.26, size = 44, normalized size = 1.47 \[ \frac {2 i x}{\sqrt {1 + \frac {9}{4 x^{2}}}} - 3 i \operatorname {asinh}{\left (\frac {3}{2 x} \right )} + \frac {9 i}{2 x \sqrt {1 + \frac {9}{4 x^{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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