Integrand size = 6, antiderivative size = 19 \[ \int x \log (c x) \, dx=-\frac {x^2}{4}+\frac {1}{2} x^2 \log (c x) \]
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Time = 0.00 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2341} \[ \int x \log (c x) \, dx=\frac {1}{2} x^2 \log (c x)-\frac {x^2}{4} \]
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Rule 2341
Rubi steps \begin{align*} \text {integral}& = -\frac {x^2}{4}+\frac {1}{2} x^2 \log (c x) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int x \log (c x) \, dx=-\frac {x^2}{4}+\frac {1}{2} x^2 \log (c x) \]
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Time = 0.01 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.84
method | result | size |
norman | \(-\frac {x^{2}}{4}+\frac {x^{2} \ln \left (x c \right )}{2}\) | \(16\) |
risch | \(-\frac {x^{2}}{4}+\frac {x^{2} \ln \left (x c \right )}{2}\) | \(16\) |
parallelrisch | \(-\frac {x^{2}}{4}+\frac {x^{2} \ln \left (x c \right )}{2}\) | \(16\) |
parts | \(-\frac {x^{2}}{4}+\frac {x^{2} \ln \left (x c \right )}{2}\) | \(16\) |
derivativedivides | \(\frac {\frac {x^{2} c^{2} \ln \left (x c \right )}{2}-\frac {x^{2} c^{2}}{4}}{c^{2}}\) | \(26\) |
default | \(\frac {\frac {x^{2} c^{2} \ln \left (x c \right )}{2}-\frac {x^{2} c^{2}}{4}}{c^{2}}\) | \(26\) |
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none
Time = 0.27 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int x \log (c x) \, dx=\frac {1}{2} \, x^{2} \log \left (c x\right ) - \frac {1}{4} \, x^{2} \]
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Time = 0.04 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.74 \[ \int x \log (c x) \, dx=\frac {x^{2} \log {\left (c x \right )}}{2} - \frac {x^{2}}{4} \]
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none
Time = 0.19 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int x \log (c x) \, dx=\frac {1}{2} \, x^{2} \log \left (c x\right ) - \frac {1}{4} \, x^{2} \]
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none
Time = 0.29 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int x \log (c x) \, dx=\frac {1}{2} \, x^{2} \log \left (c x\right ) - \frac {1}{4} \, x^{2} \]
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Time = 0.03 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.58 \[ \int x \log (c x) \, dx=\frac {x^2\,\left (\ln \left (c\,x\right )-\frac {1}{2}\right )}{2} \]
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