Optimal. Leaf size=30 \[ \sqrt{9-4 x^2}-3 \tanh ^{-1}\left (\frac{1}{3} \sqrt{9-4 x^2}\right ) \]
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Rubi [A] time = 0.0152687, 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, 206} \[ \sqrt{9-4 x^2}-3 \tanh ^{-1}\left (\frac{1}{3} \sqrt{9-4 x^2}\right ) \]
Antiderivative was successfully verified.
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Rule 266
Rule 50
Rule 63
Rule 206
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 \tanh ^{-1}\left (\frac{1}{3} \sqrt{9-4 x^2}\right )\\ \end{align*}
Mathematica [A] time = 0.0042257, size = 30, normalized size = 1. \[ \sqrt{9-4 x^2}-3 \tanh ^{-1}\left (\frac{1}{3} \sqrt{9-4 x^2}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 25, normalized size = 0.8 \begin{align*} \sqrt{-4\,{x}^{2}+9}-3\,{\it Artanh} \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 [A] time = 2.27514, size = 47, normalized size = 1.57 \begin{align*} \sqrt{-4 \, x^{2} + 9} - 3 \, \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 [A] time = 1.55815, size = 70, normalized size = 2.33 \begin{align*} \sqrt{-4 \, x^{2} + 9} + 3 \, \log \left (\frac{\sqrt{-4 \, x^{2} + 9} - 3}{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.30778, size = 76, normalized size = 2.53 \begin{align*} \begin{cases} i \sqrt{4 x^{2} - 9} - 3 \log{\left (x \right )} + \frac{3 \log{\left (x^{2} \right )}}{2} + 3 i \operatorname{asin}{\left (\frac{3}{2 x} \right )} & \text{for}\: \frac{4 \left |{x^{2}}\right |}{9} > 1 \\\sqrt{9 - 4 x^{2}} + \frac{3 \log{\left (x^{2} \right )}}{2} - 3 \log{\left (\sqrt{1 - \frac{4 x^{2}}{9}} + 1 \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.41552, size = 54, normalized size = 1.8 \begin{align*} \sqrt{-4 \, x^{2} + 9} - \frac{3}{2} \, \log \left (\sqrt{-4 \, x^{2} + 9} + 3\right ) + \frac{3}{2} \, \log \left (-\sqrt{-4 \, x^{2} + 9} + 3\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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