Integrand size = 10, antiderivative size = 10 \[ \int \frac {1}{x \log ^3(c x)} \, dx=-\frac {1}{2 \log ^2(c x)} \]
[Out]
Time = 0.01 (sec) , antiderivative size = 10, 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, 30} \[ \int \frac {1}{x \log ^3(c x)} \, dx=-\frac {1}{2 \log ^2(c x)} \]
[In]
[Out]
Rule 30
Rule 2339
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{x^3} \, dx,x,\log (c x)\right ) \\ & = -\frac {1}{2 \log ^2(c x)} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \log ^3(c x)} \, dx=-\frac {1}{2 \log ^2(c x)} \]
[In]
[Out]
Time = 0.02 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.90
method | result | size |
derivativedivides | \(-\frac {1}{2 \ln \left (x c \right )^{2}}\) | \(9\) |
default | \(-\frac {1}{2 \ln \left (x c \right )^{2}}\) | \(9\) |
norman | \(-\frac {1}{2 \ln \left (x c \right )^{2}}\) | \(9\) |
risch | \(-\frac {1}{2 \ln \left (x c \right )^{2}}\) | \(9\) |
parallelrisch | \(-\frac {1}{2 \ln \left (x c \right )^{2}}\) | \(9\) |
[In]
[Out]
none
Time = 0.32 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.80 \[ \int \frac {1}{x \log ^3(c x)} \, dx=-\frac {1}{2 \, \log \left (c x\right )^{2}} \]
[In]
[Out]
Time = 0.04 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \log ^3(c x)} \, dx=- \frac {1}{2 \log {\left (c x \right )}^{2}} \]
[In]
[Out]
none
Time = 0.19 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.80 \[ \int \frac {1}{x \log ^3(c x)} \, dx=-\frac {1}{2 \, \log \left (c x\right )^{2}} \]
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.80 \[ \int \frac {1}{x \log ^3(c x)} \, dx=-\frac {1}{2 \, \log \left (c x\right )^{2}} \]
[In]
[Out]
Time = 0.23 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.80 \[ \int \frac {1}{x \log ^3(c x)} \, dx=-\frac {1}{2\,{\ln \left (c\,x\right )}^2} \]
[In]
[Out]