Integrand size = 10, antiderivative size = 5 \[ \int \frac {1}{x \log (c x)} \, dx=\log (\log (c x)) \]
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Time = 0.01 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2339, 29} \[ \int \frac {1}{x \log (c x)} \, dx=\log (\log (c x)) \]
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Rule 29
Rule 2339
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{x} \, dx,x,\log (c x)\right ) \\ & = \log (\log (c x)) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \log (c x)} \, dx=\log (\log (c x)) \]
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Time = 0.02 (sec) , antiderivative size = 6, normalized size of antiderivative = 1.20
method | result | size |
derivativedivides | \(\ln \left (\ln \left (x c \right )\right )\) | \(6\) |
default | \(\ln \left (\ln \left (x c \right )\right )\) | \(6\) |
norman | \(\ln \left (\ln \left (x c \right )\right )\) | \(6\) |
risch | \(\ln \left (\ln \left (x c \right )\right )\) | \(6\) |
parallelrisch | \(\ln \left (\ln \left (x c \right )\right )\) | \(6\) |
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none
Time = 0.30 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \log (c x)} \, dx=\log \left (\log \left (c x\right )\right ) \]
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Time = 0.04 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \log (c x)} \, dx=\log {\left (\log {\left (c x \right )} \right )} \]
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none
Time = 0.19 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \log (c x)} \, dx=\log \left (\log \left (c x\right )\right ) \]
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none
Time = 0.28 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \log (c x)} \, dx=\log \left (\log \left (c x\right )\right ) \]
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Time = 0.20 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \log (c x)} \, dx=\ln \left (\ln \left (c\,x\right )\right ) \]
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