Integrand size = 8, antiderivative size = 10 \[ \int \frac {\log (e x)}{x} \, dx=\frac {1}{2} \log ^2(e x) \]
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Time = 0.00 (sec) , antiderivative size = 10, 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 = {2338} \[ \int \frac {\log (e x)}{x} \, dx=\frac {1}{2} \log ^2(e x) \]
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Rule 2338
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \log ^2(e x) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {\log (e x)}{x} \, dx=\frac {1}{2} \log ^2(e x) \]
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Time = 0.13 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.90
method | result | size |
derivativedivides | \(\frac {\ln \left (e x \right )^{2}}{2}\) | \(9\) |
default | \(\frac {\ln \left (e x \right )^{2}}{2}\) | \(9\) |
norman | \(\frac {\ln \left (e x \right )^{2}}{2}\) | \(9\) |
risch | \(\frac {\ln \left (e x \right )^{2}}{2}\) | \(9\) |
parts | \(\ln \left (e x \right ) \ln \left (x \right )-\frac {\ln \left (x \right )^{2}}{2}\) | \(15\) |
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none
Time = 0.27 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.80 \[ \int \frac {\log (e x)}{x} \, dx=\frac {1}{2} \, \log \left (e x\right )^{2} \]
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Time = 0.04 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.70 \[ \int \frac {\log (e x)}{x} \, dx=\frac {\log {\left (e x \right )}^{2}}{2} \]
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none
Time = 0.19 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.80 \[ \int \frac {\log (e x)}{x} \, dx=\frac {1}{2} \, \log \left (e x\right )^{2} \]
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none
Time = 0.30 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.80 \[ \int \frac {\log (e x)}{x} \, dx=\frac {1}{2} \, \log \left (e x\right )^{2} \]
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Time = 1.31 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.80 \[ \int \frac {\log (e x)}{x} \, dx=\frac {{\ln \left (e\,x\right )}^2}{2} \]
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