Integrand size = 12, antiderivative size = 9 \[ \int \frac {1}{x (1-\log (x))} \, dx=-\log (1-\log (x)) \]
[Out]
Time = 0.02 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2339, 29} \[ \int \frac {1}{x (1-\log (x))} \, dx=-\log (1-\log (x)) \]
[In]
[Out]
Rule 29
Rule 2339
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {1}{x} \, dx,x,1-\log (x)\right ) \\ & = -\log (1-\log (x)) \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.78 \[ \int \frac {1}{x (1-\log (x))} \, dx=-\log (-1+\log (x)) \]
[In]
[Out]
Time = 0.04 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.89
method | result | size |
norman | \(-\ln \left (-1+\ln \left (x \right )\right )\) | \(8\) |
risch | \(-\ln \left (-1+\ln \left (x \right )\right )\) | \(8\) |
parallelrisch | \(-\ln \left (-1+\ln \left (x \right )\right )\) | \(8\) |
derivativedivides | \(-\ln \left (1-\ln \left (x \right )\right )\) | \(10\) |
default | \(-\ln \left (1-\ln \left (x \right )\right )\) | \(10\) |
[In]
[Out]
none
Time = 0.23 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.78 \[ \int \frac {1}{x (1-\log (x))} \, dx=-\log \left (\log \left (x\right ) - 1\right ) \]
[In]
[Out]
Time = 0.04 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.78 \[ \int \frac {1}{x (1-\log (x))} \, dx=- \log {\left (\log {\left (x \right )} - 1 \right )} \]
[In]
[Out]
none
Time = 0.21 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.78 \[ \int \frac {1}{x (1-\log (x))} \, dx=-\log \left (\log \left (x\right ) - 1\right ) \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 22 vs. \(2 (9) = 18\).
Time = 0.33 (sec) , antiderivative size = 22, normalized size of antiderivative = 2.44 \[ \int \frac {1}{x (1-\log (x))} \, dx=-\frac {1}{2} \, \log \left (\frac {1}{4} \, \pi ^{2} {\left (\mathrm {sgn}\left (x\right ) - 1\right )}^{2} + {\left (\log \left ({\left | x \right |}\right ) - 1\right )}^{2}\right ) \]
[In]
[Out]
Time = 0.26 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.78 \[ \int \frac {1}{x (1-\log (x))} \, dx=-\ln \left (\ln \left (x\right )-1\right ) \]
[In]
[Out]