Integrand size = 4, antiderivative size = 10 \[ \int \log (c x) \, dx=-x+x \log (c x) \]
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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, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2332} \[ \int \log (c x) \, dx=x \log (c x)-x \]
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Rule 2332
Rubi steps \begin{align*} \text {integral}& = -x+x \log (c x) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \log (c x) \, dx=-x+x \log (c x) \]
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Time = 0.01 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.10
method | result | size |
norman | \(-x +x \ln \left (x c \right )\) | \(11\) |
risch | \(-x +x \ln \left (x c \right )\) | \(11\) |
parallelrisch | \(-x +x \ln \left (x c \right )\) | \(11\) |
parts | \(-x +x \ln \left (x c \right )\) | \(11\) |
derivativedivides | \(\frac {x c \ln \left (x c \right )-x c}{c}\) | \(17\) |
default | \(\frac {x c \ln \left (x c \right )-x c}{c}\) | \(17\) |
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none
Time = 0.30 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \log (c x) \, dx=x \log \left (c x\right ) - x \]
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Time = 0.04 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.70 \[ \int \log (c x) \, dx=x \log {\left (c x \right )} - x \]
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none
Time = 0.19 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.60 \[ \int \log (c x) \, dx=\frac {c x \log \left (c x\right ) - c x}{c} \]
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none
Time = 0.29 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.60 \[ \int \log (c x) \, dx=\frac {c x \log \left (c x\right ) - c x}{c} \]
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Time = 0.02 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.80 \[ \int \log (c x) \, dx=x\,\left (\ln \left (c\,x\right )-1\right ) \]
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