Optimal. Leaf size=28 \[ \frac{1}{8 (3 x+2)}+\frac{\log (x)}{16}-\frac{1}{16} \log (3 x+2) \]
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Rubi [A] time = 0.0089128, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {44} \[ \frac{1}{8 (3 x+2)}+\frac{\log (x)}{16}-\frac{1}{16} \log (3 x+2) \]
Antiderivative was successfully verified.
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Rule 44
Rubi steps
\begin{align*} \int \frac{1}{x (4+6 x)^2} \, dx &=\int \left (\frac{1}{16 x}-\frac{3}{8 (2+3 x)^2}-\frac{3}{16 (2+3 x)}\right ) \, dx\\ &=\frac{1}{8 (2+3 x)}+\frac{\log (x)}{16}-\frac{1}{16} \log (2+3 x)\\ \end{align*}
Mathematica [A] time = 0.0226824, size = 26, normalized size = 0.93 \[ \frac{1}{16} \left (\frac{2}{3 x+2}+\log (-6 x)-\log (6 x+4)\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 23, normalized size = 0.8 \begin{align*}{\frac{1}{16+24\,x}}+{\frac{\ln \left ( x \right ) }{16}}-{\frac{\ln \left ( 2+3\,x \right ) }{16}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08183, size = 30, normalized size = 1.07 \begin{align*} \frac{1}{8 \,{\left (3 \, x + 2\right )}} - \frac{1}{16} \, \log \left (3 \, x + 2\right ) + \frac{1}{16} \, \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.68688, size = 88, normalized size = 3.14 \begin{align*} -\frac{{\left (3 \, x + 2\right )} \log \left (3 \, x + 2\right ) -{\left (3 \, x + 2\right )} \log \left (x\right ) - 2}{16 \,{\left (3 \, x + 2\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.113678, size = 19, normalized size = 0.68 \begin{align*} \frac{\log{\left (x \right )}}{16} - \frac{\log{\left (x + \frac{2}{3} \right )}}{16} + \frac{1}{24 x + 16} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19607, size = 34, normalized size = 1.21 \begin{align*} \frac{1}{8 \,{\left (3 \, x + 2\right )}} + \frac{1}{16} \, \log \left ({\left | -\frac{2}{3 \, x + 2} + 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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