Optimal. Leaf size=20 \[ -\frac{1}{3} \tanh ^{-1}\left (\frac{1}{3} \sqrt{9-4 x^2}\right ) \]
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Rubi [A] time = 0.0101745, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {266, 63, 206} \[ -\frac{1}{3} \tanh ^{-1}\left (\frac{1}{3} \sqrt{9-4 x^2}\right ) \]
Antiderivative was successfully verified.
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Rule 266
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{x \sqrt{9-4 x^2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{9-4 x} x} \, dx,x,x^2\right )\\ &=-\left (\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{\frac{9}{4}-\frac{x^2}{4}} \, dx,x,\sqrt{9-4 x^2}\right )\right )\\ &=-\frac{1}{3} \tanh ^{-1}\left (\frac{1}{3} \sqrt{9-4 x^2}\right )\\ \end{align*}
Mathematica [A] time = 0.0027887, size = 20, normalized size = 1. \[ -\frac{1}{3} \tanh ^{-1}\left (\frac{1}{3} \sqrt{9-4 x^2}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 15, normalized size = 0.8 \begin{align*} -{\frac{1}{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 = 2.48907, size = 34, normalized size = 1.7 \begin{align*} -\frac{1}{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 = 1.27376, size = 47, normalized size = 2.35 \begin{align*} \frac{1}{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 = 1.05755, size = 26, normalized size = 1.3 \begin{align*} \begin{cases} - \frac{\operatorname{acosh}{\left (\frac{3}{2 x} \right )}}{3} & \text{for}\: \frac{9}{4 \left |{x^{2}}\right |} > 1 \\\frac{i \operatorname{asin}{\left (\frac{3}{2 x} \right )}}{3} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 2.64758, size = 42, normalized size = 2.1 \begin{align*} -\frac{1}{6} \, \log \left (\sqrt{-4 \, x^{2} + 9} + 3\right ) + \frac{1}{6} \, \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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