Optimal. Leaf size=27 \[ -\frac{(3 x+2) \tanh ^{-1}(3 x+1)}{\sqrt{9 x^2+12 x+4}} \]
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Rubi [B] time = 0.0123664, antiderivative size = 55, normalized size of antiderivative = 2.04, number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {646, 36, 29, 31} \[ \frac{(3 x+2) \log (x)}{2 \sqrt{9 x^2+12 x+4}}-\frac{(3 x+2) \log (3 x+2)}{2 \sqrt{9 x^2+12 x+4}} \]
Antiderivative was successfully verified.
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Rule 646
Rule 36
Rule 29
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{x \sqrt{4+12 x+9 x^2}} \, dx &=\frac{(6+9 x) \int \frac{1}{x (6+9 x)} \, dx}{\sqrt{4+12 x+9 x^2}}\\ &=\frac{(6+9 x) \int \frac{1}{x} \, dx}{6 \sqrt{4+12 x+9 x^2}}-\frac{(3 (6+9 x)) \int \frac{1}{6+9 x} \, dx}{2 \sqrt{4+12 x+9 x^2}}\\ &=\frac{(2+3 x) \log (x)}{2 \sqrt{4+12 x+9 x^2}}-\frac{(2+3 x) \log (2+3 x)}{2 \sqrt{4+12 x+9 x^2}}\\ \end{align*}
Mathematica [A] time = 0.011526, size = 31, normalized size = 1.15 \[ \frac{(3 x+2) (\log (x)-\log (3 x+2))}{2 \sqrt{(3 x+2)^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.106, size = 28, normalized size = 1. \begin{align*}{\frac{ \left ( 2+3\,x \right ) \left ( \ln \left ( x \right ) -\ln \left ( 2+3\,x \right ) \right ) }{2}{\frac{1}{\sqrt{ \left ( 2+3\,x \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.46949, size = 32, normalized size = 1.19 \begin{align*} -\frac{1}{2} \, \left (-1\right )^{12 \, x + 8} \log \left (\frac{12 \, x}{{\left | x \right |}} + \frac{8}{{\left | x \right |}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.03214, size = 43, normalized size = 1.59 \begin{align*} -\frac{1}{2} \, \log \left (3 \, x + 2\right ) + \frac{1}{2} \, \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.127484, size = 12, normalized size = 0.44 \begin{align*} \frac{\log{\left (x \right )}}{2} - \frac{\log{\left (x + \frac{2}{3} \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10481, size = 28, normalized size = 1.04 \begin{align*} -\frac{1}{2} \,{\left (\log \left ({\left | 3 \, x + 2 \right |}\right ) - \log \left ({\left | x \right |}\right )\right )} \mathrm{sgn}\left (3 \, x + 2\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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