Optimal. Leaf size=9 \[ \log \left (\log \left (\frac{x}{a}\right )+1\right ) \]
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Rubi [A] time = 0.0180361, antiderivative size = 9, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2302, 29} \[ \log \left (\log \left (\frac{x}{a}\right )+1\right ) \]
Antiderivative was successfully verified.
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Rule 2302
Rule 29
Rubi steps
\begin{align*} \int \frac{1}{x \left (1+\log \left (\frac{x}{a}\right )\right )} \, dx &=\operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,1+\log \left (\frac{x}{a}\right )\right )\\ &=\log \left (1+\log \left (\frac{x}{a}\right )\right )\\ \end{align*}
Mathematica [A] time = 0.0161651, size = 9, normalized size = 1. \[ \log \left (\log \left (\frac{x}{a}\right )+1\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 10, normalized size = 1.1 \begin{align*} \ln \left ( 1+\ln \left ({\frac{x}{a}} \right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.931764, size = 12, normalized size = 1.33 \begin{align*} \log \left (\log \left (\frac{x}{a}\right ) + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.27744, size = 26, normalized size = 2.89 \begin{align*} \log \left (\log \left (\frac{x}{a}\right ) + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.095479, size = 7, normalized size = 0.78 \begin{align*} \log{\left (\log{\left (\frac{x}{a} \right )} + 1 \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.05264, size = 77, normalized size = 8.56 \begin{align*} \frac{1}{2} \, \log \left (\frac{1}{4} \,{\left (\pi{\left (\mathrm{sgn}\left (a\right ) - 1\right )} + \pi{\left (\mathrm{sgn}\left (x\right ) - 1\right )} + 4 \, \pi \left \lfloor -\frac{\pi{\left (\mathrm{sgn}\left (a\right ) - 1\right )} + \pi{\left (\mathrm{sgn}\left (x\right ) - 1\right )}}{4 \, \pi } + \frac{1}{2} \right \rfloor \right )}^{2} +{\left (\log \left (\frac{{\left | x \right |}}{{\left | a \right |}}\right ) + 1\right )}^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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