Integrand size = 10, antiderivative size = 10 \[ \int \frac {\log ^3(c x)}{x} \, dx=\frac {1}{4} \log ^4(c x) \]
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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} \[ \int \frac {\log ^3(c x)}{x} \, dx=\frac {1}{4} \log ^4(c x) \]
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Rule 30
Rule 2339
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int x^3 \, dx,x,\log (c x)\right ) \\ & = \frac {1}{4} \log ^4(c x) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {\log ^3(c x)}{x} \, dx=\frac {1}{4} \log ^4(c x) \]
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Time = 0.22 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.90
method | result | size |
derivativedivides | \(\frac {\ln \left (x c \right )^{4}}{4}\) | \(9\) |
default | \(\frac {\ln \left (x c \right )^{4}}{4}\) | \(9\) |
norman | \(\frac {\ln \left (x c \right )^{4}}{4}\) | \(9\) |
risch | \(\frac {\ln \left (x c \right )^{4}}{4}\) | \(9\) |
parts | \(\ln \left (x c \right )^{3} \ln \left (x \right )-\frac {3 \ln \left (x c \right )^{2} \ln \left (x \right )^{2}}{2}+\ln \left (x c \right ) \ln \left (x \right )^{3}-\frac {\ln \left (x \right )^{4}}{4}\) | \(38\) |
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Time = 0.30 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.80 \[ \int \frac {\log ^3(c x)}{x} \, dx=\frac {1}{4} \, \log \left (c x\right )^{4} \]
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Time = 0.04 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.70 \[ \int \frac {\log ^3(c x)}{x} \, dx=\frac {\log {\left (c x \right )}^{4}}{4} \]
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none
Time = 0.22 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.80 \[ \int \frac {\log ^3(c x)}{x} \, dx=\frac {1}{4} \, \log \left (c x\right )^{4} \]
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Time = 0.31 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.80 \[ \int \frac {\log ^3(c x)}{x} \, dx=\frac {1}{4} \, \log \left (c x\right )^{4} \]
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Time = 0.24 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.80 \[ \int \frac {\log ^3(c x)}{x} \, dx=\frac {{\ln \left (c\,x\right )}^4}{4} \]
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