Optimal. Leaf size=20 \[ \frac{1}{3} \tan ^{-1}\left (\frac{1}{3} \sqrt{-4 x^2-9}\right ) \]
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Rubi [A] time = 0.0101194, 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, 204} \[ \frac{1}{3} \tan ^{-1}\left (\frac{1}{3} \sqrt{-4 x^2-9}\right ) \]
Antiderivative was successfully verified.
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Rule 266
Rule 63
Rule 204
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} \tan ^{-1}\left (\frac{1}{3} \sqrt{-9-4 x^2}\right )\\ \end{align*}
Mathematica [A] time = 0.0023916, size = 20, normalized size = 1. \[ \frac{1}{3} \tan ^{-1}\left (\frac{1}{3} \sqrt{-4 x^2-9}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 15, normalized size = 0.8 \begin{align*} -{\frac{1}{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 = 1.89725, size = 34, normalized size = 1.7 \begin{align*} -\frac{1}{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 = 1.27353, size = 123, normalized size = 6.15 \begin{align*} -\frac{1}{6} i \, \log \left (-\frac{2 \,{\left (i \, \sqrt{-4 \, x^{2} - 9} + 3\right )}}{3 \, x}\right ) + \frac{1}{6} i \, \log \left (-\frac{2 \,{\left (-i \, \sqrt{-4 \, x^{2} - 9} + 3\right )}}{3 \, x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 1.02936, size = 8, normalized size = 0.4 \begin{align*} \frac{i \operatorname{asinh}{\left (\frac{3}{2 x} \right )}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 2.67627, size = 19, normalized size = 0.95 \begin{align*} \frac{1}{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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