Optimal. Leaf size=24 \[ \frac{1}{4} \log (2 \log (x)+3)+\frac{1}{4 (2 \log (x)+3)} \]
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Rubi [A] time = 0.0405776, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {2365, 43} \[ \frac{1}{4} \log (2 \log (x)+3)+\frac{1}{4 (2 \log (x)+3)} \]
Antiderivative was successfully verified.
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Rule 2365
Rule 43
Rubi steps
\begin{align*} \int \frac{1+\log (x)}{x (3+2 \log (x))^2} \, dx &=\operatorname{Subst}\left (\int \frac{1+x}{(3+2 x)^2} \, dx,x,\log (x)\right )\\ &=\operatorname{Subst}\left (\int \left (-\frac{1}{2 (3+2 x)^2}+\frac{1}{2 (3+2 x)}\right ) \, dx,x,\log (x)\right )\\ &=\frac{1}{4 (3+2 \log (x))}+\frac{1}{4} \log (3+2 \log (x))\\ \end{align*}
Mathematica [A] time = 0.0266525, size = 20, normalized size = 0.83 \[ \frac{1}{4} \left (\log (2 \log (x)+3)+\frac{1}{2 \log (x)+3}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 21, normalized size = 0.9 \begin{align*}{\frac{1}{12+8\,\ln \left ( x \right ) }}+{\frac{\ln \left ( 3+2\,\ln \left ( x \right ) \right ) }{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.10067, size = 27, normalized size = 1.12 \begin{align*} \frac{1}{4 \,{\left (2 \, \log \left (x\right ) + 3\right )}} + \frac{1}{4} \, \log \left (2 \, \log \left (x\right ) + 3\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.95505, size = 80, normalized size = 3.33 \begin{align*} \frac{{\left (2 \, \log \left (x\right ) + 3\right )} \log \left (2 \, \log \left (x\right ) + 3\right ) + 1}{4 \,{\left (2 \, \log \left (x\right ) + 3\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.120399, size = 17, normalized size = 0.71 \begin{align*} \frac{\log{\left (\log{\left (x \right )} + \frac{3}{2} \right )}}{4} + \frac{1}{8 \log{\left (x \right )} + 12} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21718, size = 46, normalized size = 1.92 \begin{align*} \frac{1}{4 \,{\left (2 \, \log \left (x\right ) + 3\right )}} + \frac{1}{8} \, \log \left (\pi ^{2}{\left (\mathrm{sgn}\left (x\right ) - 1\right )}^{2} +{\left (2 \, \log \left ({\left | x \right |}\right ) + 3\right )}^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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