Integrand size = 16, antiderivative size = 9 \[ \int \frac {1-\log (x)}{x (x+\log (x))} \, dx=\log \left (1+\frac {\log (x)}{x}\right ) \]
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Time = 0.05 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6844, 31} \[ \int \frac {1-\log (x)}{x (x+\log (x))} \, dx=\log \left (\frac {\log (x)}{x}+1\right ) \]
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Rule 31
Rule 6844
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{1+x} \, dx,x,\frac {\log (x)}{x}\right ) \\ & = \log \left (1+\frac {\log (x)}{x}\right ) \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.11 \[ \int \frac {1-\log (x)}{x (x+\log (x))} \, dx=-\log (x)+\log (x+\log (x)) \]
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Time = 1.04 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.22
method | result | size |
default | \(-\ln \left (x \right )+\ln \left (x +\ln \left (x \right )\right )\) | \(11\) |
norman | \(-\ln \left (x \right )+\ln \left (x +\ln \left (x \right )\right )\) | \(11\) |
risch | \(-\ln \left (x \right )+\ln \left (x +\ln \left (x \right )\right )\) | \(11\) |
parallelrisch | \(-\ln \left (x \right )+\ln \left (x +\ln \left (x \right )\right )\) | \(11\) |
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none
Time = 0.30 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.11 \[ \int \frac {1-\log (x)}{x (x+\log (x))} \, dx=\log \left (x + \log \left (x\right )\right ) - \log \left (x\right ) \]
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Time = 0.05 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.89 \[ \int \frac {1-\log (x)}{x (x+\log (x))} \, dx=- \log {\left (x \right )} + \log {\left (x + \log {\left (x \right )} \right )} \]
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none
Time = 0.31 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.11 \[ \int \frac {1-\log (x)}{x (x+\log (x))} \, dx=\log \left (x + \log \left (x\right )\right ) - \log \left (x\right ) \]
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none
Time = 0.31 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.56 \[ \int \frac {1-\log (x)}{x (x+\log (x))} \, dx=-\log \left (x\right ) + \log \left (-x - \log \left (x\right )\right ) \]
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Time = 1.56 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.11 \[ \int \frac {1-\log (x)}{x (x+\log (x))} \, dx=\ln \left (x+\ln \left (x\right )\right )-\ln \left (x\right ) \]
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