Integrand size = 14, antiderivative size = 9 \[ \int \frac {1}{x \left (1+\log \left (\frac {x}{a}\right )\right )} \, dx=\log \left (1+\log \left (\frac {x}{a}\right )\right ) \]
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Time = 0.02 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2339, 29} \[ \int \frac {1}{x \left (1+\log \left (\frac {x}{a}\right )\right )} \, dx=\log \left (\log \left (\frac {x}{a}\right )+1\right ) \]
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Rule 29
Rule 2339
Rubi steps \begin{align*} \text {integral}& = \text {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*}
Time = 0.04 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \left (1+\log \left (\frac {x}{a}\right )\right )} \, dx=\log \left (1+\log \left (\frac {x}{a}\right )\right ) \]
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Time = 0.03 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.11
method | result | size |
derivativedivides | \(\ln \left (1+\ln \left (\frac {x}{a}\right )\right )\) | \(10\) |
default | \(\ln \left (1+\ln \left (\frac {x}{a}\right )\right )\) | \(10\) |
norman | \(\ln \left (1+\ln \left (\frac {x}{a}\right )\right )\) | \(10\) |
risch | \(\ln \left (1+\ln \left (\frac {x}{a}\right )\right )\) | \(10\) |
parallelrisch | \(\ln \left (1+\ln \left (\frac {x}{a}\right )\right )\) | \(10\) |
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none
Time = 0.24 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \left (1+\log \left (\frac {x}{a}\right )\right )} \, dx=\log \left (\log \left (\frac {x}{a}\right ) + 1\right ) \]
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Time = 0.04 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.78 \[ \int \frac {1}{x \left (1+\log \left (\frac {x}{a}\right )\right )} \, dx=\log {\left (\log {\left (\frac {x}{a} \right )} + 1 \right )} \]
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none
Time = 0.20 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \left (1+\log \left (\frac {x}{a}\right )\right )} \, dx=\log \left (\log \left (\frac {x}{a}\right ) + 1\right ) \]
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none
Time = 0.29 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \left (1+\log \left (\frac {x}{a}\right )\right )} \, dx=\log \left (\log \left (\frac {x}{a}\right ) + 1\right ) \]
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Time = 0.28 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \left (1+\log \left (\frac {x}{a}\right )\right )} \, dx=\ln \left (\ln \left (\frac {x}{a}\right )+1\right ) \]
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