Integrand size = 8, antiderivative size = 15 \[ \int \frac {\log (c x)}{x^2} \, dx=-\frac {1}{x}-\frac {\log (c x)}{x} \]
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Time = 0.00 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2341} \[ \int \frac {\log (c x)}{x^2} \, dx=-\frac {\log (c x)}{x}-\frac {1}{x} \]
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Rule 2341
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{x}-\frac {\log (c x)}{x} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {\log (c x)}{x^2} \, dx=-\frac {1}{x}-\frac {\log (c x)}{x} \]
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Time = 0.02 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87
method | result | size |
norman | \(\frac {-1-\ln \left (x c \right )}{x}\) | \(13\) |
parallelrisch | \(\frac {-1-\ln \left (x c \right )}{x}\) | \(13\) |
risch | \(-\frac {1}{x}-\frac {\ln \left (x c \right )}{x}\) | \(16\) |
parts | \(-\frac {1}{x}-\frac {\ln \left (x c \right )}{x}\) | \(16\) |
derivativedivides | \(c \left (-\frac {\ln \left (x c \right )}{x c}-\frac {1}{x c}\right )\) | \(24\) |
default | \(c \left (-\frac {\ln \left (x c \right )}{x c}-\frac {1}{x c}\right )\) | \(24\) |
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none
Time = 0.30 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.73 \[ \int \frac {\log (c x)}{x^2} \, dx=-\frac {\log \left (c x\right ) + 1}{x} \]
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Time = 0.04 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.67 \[ \int \frac {\log (c x)}{x^2} \, dx=- \frac {\log {\left (c x \right )}}{x} - \frac {1}{x} \]
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none
Time = 0.20 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {\log (c x)}{x^2} \, dx=-\frac {\log \left (c x\right )}{x} - \frac {1}{x} \]
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none
Time = 0.33 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {\log (c x)}{x^2} \, dx=-\frac {\log \left (c x\right )}{x} - \frac {1}{x} \]
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Time = 0.04 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.73 \[ \int \frac {\log (c x)}{x^2} \, dx=-\frac {\ln \left (c\,x\right )+1}{x} \]
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