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 = 0.0151617, antiderivative size = 30, normalized size of antiderivative = 1., 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 266
Rule 50
Rule 63
Rule 204
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*}
Mathematica [A] time = 0.0043035, size = 30, normalized size = 1. \[ \sqrt{-4 x^2-9}-3 \tan ^{-1}\left (\frac{1}{3} \sqrt{-4 x^2-9}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 25, normalized size = 0.8 \begin{align*} \sqrt{-4\,{x}^{2}-9}+3\,\arctan \left ( 3\,{\frac{1}{\sqrt{-4\,{x}^{2}-9}}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 3.30422, size = 47, normalized size = 1.57 \begin{align*} \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 1.289, size = 142, normalized size = 4.73 \begin{align*} \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 1.24054, size = 44, normalized size = 1.47 \begin{align*} \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}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 2.42821, size = 35, normalized size = 1.17 \begin{align*} i \, \sqrt{4 \, x^{2} + 9} - 3 \, \arctan \left (\frac{1}{3} i \, \sqrt{4 \, x^{2} + 9}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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