Integrand size = 8, antiderivative size = 19 \[ \int \frac {\log (c x)}{x^3} \, dx=-\frac {1}{4 x^2}-\frac {\log (c x)}{2 x^2} \]
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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, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2341} \[ \int \frac {\log (c x)}{x^3} \, dx=-\frac {\log (c x)}{2 x^2}-\frac {1}{4 x^2} \]
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Rule 2341
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{4 x^2}-\frac {\log (c x)}{2 x^2} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {\log (c x)}{x^3} \, dx=-\frac {1}{4 x^2}-\frac {\log (c x)}{2 x^2} \]
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Time = 0.02 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.68
method | result | size |
norman | \(\frac {-\frac {1}{4}-\frac {\ln \left (x c \right )}{2}}{x^{2}}\) | \(13\) |
parallelrisch | \(\frac {-1-2 \ln \left (x c \right )}{4 x^{2}}\) | \(14\) |
risch | \(-\frac {1}{4 x^{2}}-\frac {\ln \left (x c \right )}{2 x^{2}}\) | \(16\) |
parts | \(-\frac {1}{4 x^{2}}-\frac {\ln \left (x c \right )}{2 x^{2}}\) | \(16\) |
derivativedivides | \(c^{2} \left (-\frac {\ln \left (x c \right )}{2 x^{2} c^{2}}-\frac {1}{4 x^{2} c^{2}}\right )\) | \(26\) |
default | \(c^{2} \left (-\frac {\ln \left (x c \right )}{2 x^{2} c^{2}}-\frac {1}{4 x^{2} c^{2}}\right )\) | \(26\) |
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none
Time = 0.29 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.68 \[ \int \frac {\log (c x)}{x^3} \, dx=-\frac {2 \, \log \left (c x\right ) + 1}{4 \, x^{2}} \]
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Time = 0.04 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89 \[ \int \frac {\log (c x)}{x^3} \, dx=- \frac {\log {\left (c x \right )}}{2 x^{2}} - \frac {1}{4 x^{2}} \]
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none
Time = 0.18 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \frac {\log (c x)}{x^3} \, dx=-\frac {\log \left (c x\right )}{2 \, x^{2}} - \frac {1}{4 \, x^{2}} \]
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none
Time = 0.34 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \frac {\log (c x)}{x^3} \, dx=-\frac {\log \left (c x\right )}{2 \, x^{2}} - \frac {1}{4 \, x^{2}} \]
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Time = 0.03 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.58 \[ \int \frac {\log (c x)}{x^3} \, dx=-\frac {\ln \left (c\,x\right )+\frac {1}{2}}{2\,x^2} \]
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