Integrand size = 10, antiderivative size = 37 \[ \int \frac {\log ^3(c x)}{x^2} \, dx=-\frac {6}{x}-\frac {6 \log (c x)}{x}-\frac {3 \log ^2(c x)}{x}-\frac {\log ^3(c x)}{x} \]
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Time = 0.02 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2342, 2341} \[ \int \frac {\log ^3(c x)}{x^2} \, dx=-\frac {\log ^3(c x)}{x}-\frac {3 \log ^2(c x)}{x}-\frac {6 \log (c x)}{x}-\frac {6}{x} \]
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Rule 2341
Rule 2342
Rubi steps \begin{align*} \text {integral}& = -\frac {\log ^3(c x)}{x}+3 \int \frac {\log ^2(c x)}{x^2} \, dx \\ & = -\frac {3 \log ^2(c x)}{x}-\frac {\log ^3(c x)}{x}+6 \int \frac {\log (c x)}{x^2} \, dx \\ & = -\frac {6}{x}-\frac {6 \log (c x)}{x}-\frac {3 \log ^2(c x)}{x}-\frac {\log ^3(c x)}{x} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00 \[ \int \frac {\log ^3(c x)}{x^2} \, dx=-\frac {6}{x}-\frac {6 \log (c x)}{x}-\frac {3 \log ^2(c x)}{x}-\frac {\log ^3(c x)}{x} \]
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Time = 0.03 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.78
method | result | size |
norman | \(\frac {-6-3 \ln \left (x c \right )^{2}-\ln \left (x c \right )^{3}-6 \ln \left (x c \right )}{x}\) | \(29\) |
parallelrisch | \(\frac {-6-3 \ln \left (x c \right )^{2}-\ln \left (x c \right )^{3}-6 \ln \left (x c \right )}{x}\) | \(29\) |
risch | \(-\frac {6}{x}-\frac {6 \ln \left (x c \right )}{x}-\frac {3 \ln \left (x c \right )^{2}}{x}-\frac {\ln \left (x c \right )^{3}}{x}\) | \(38\) |
parts | \(-\frac {\ln \left (x c \right )^{3}}{x}+3 c \left (-\frac {\ln \left (x c \right )^{2}}{x c}-\frac {2 \ln \left (x c \right )}{x c}-\frac {2}{x c}\right )\) | \(51\) |
derivativedivides | \(c \left (-\frac {\ln \left (x c \right )^{3}}{x c}-\frac {3 \ln \left (x c \right )^{2}}{x c}-\frac {6 \ln \left (x c \right )}{x c}-\frac {6}{x c}\right )\) | \(52\) |
default | \(c \left (-\frac {\ln \left (x c \right )^{3}}{x c}-\frac {3 \ln \left (x c \right )^{2}}{x c}-\frac {6 \ln \left (x c \right )}{x c}-\frac {6}{x c}\right )\) | \(52\) |
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Time = 0.32 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.73 \[ \int \frac {\log ^3(c x)}{x^2} \, dx=-\frac {\log \left (c x\right )^{3} + 3 \, \log \left (c x\right )^{2} + 6 \, \log \left (c x\right ) + 6}{x} \]
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Time = 0.06 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.84 \[ \int \frac {\log ^3(c x)}{x^2} \, dx=- \frac {\log {\left (c x \right )}^{3}}{x} - \frac {3 \log {\left (c x \right )}^{2}}{x} - \frac {6 \log {\left (c x \right )}}{x} - \frac {6}{x} \]
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Time = 0.19 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.73 \[ \int \frac {\log ^3(c x)}{x^2} \, dx=-\frac {\log \left (c x\right )^{3} + 3 \, \log \left (c x\right )^{2} + 6 \, \log \left (c x\right ) + 6}{x} \]
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Time = 0.30 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00 \[ \int \frac {\log ^3(c x)}{x^2} \, dx=-\frac {\log \left (c x\right )^{3}}{x} - \frac {3 \, \log \left (c x\right )^{2}}{x} - \frac {6 \, \log \left (c x\right )}{x} - \frac {6}{x} \]
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Time = 0.23 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.73 \[ \int \frac {\log ^3(c x)}{x^2} \, dx=-\frac {{\ln \left (c\,x\right )}^3+3\,{\ln \left (c\,x\right )}^2+6\,\ln \left (c\,x\right )+6}{x} \]
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