Integrand size = 8, antiderivative size = 21 \[ \int \log (c (d+e x)) \, dx=-x+\frac {(d+e x) \log (c (d+e x))}{e} \]
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Time = 0.01 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2436, 2332} \[ \int \log (c (d+e x)) \, dx=\frac {(d+e x) \log (c (d+e x))}{e}-x \]
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Rule 2332
Rule 2436
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}(\int \log (c x) \, dx,x,d+e x)}{e} \\ & = -x+\frac {(d+e x) \log (c (d+e x))}{e} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \log (c (d+e x)) \, dx=-x+\frac {(d+e x) \log (c (d+e x))}{e} \]
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Time = 0.08 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.24
method | result | size |
risch | \(x \ln \left (c \left (e x +d \right )\right )-x +\frac {d \ln \left (e x +d \right )}{e}\) | \(26\) |
norman | \(x \ln \left (c \left (e x +d \right )\right )+\frac {d \ln \left (c \left (e x +d \right )\right )}{e}-x\) | \(28\) |
parallelrisch | \(\frac {\ln \left (c \left (e x +d \right )\right ) x e -e x +d \ln \left (c \left (e x +d \right )\right )+d}{e}\) | \(32\) |
parts | \(x \ln \left (c \left (e x +d \right )\right )-e \left (\frac {x}{e}-\frac {d \ln \left (e x +d \right )}{e^{2}}\right )\) | \(33\) |
derivativedivides | \(\frac {\left (c e x +c d \right ) \ln \left (c e x +c d \right )-c e x -c d}{c e}\) | \(36\) |
default | \(\frac {\left (c e x +c d \right ) \ln \left (c e x +c d \right )-c e x -c d}{c e}\) | \(36\) |
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none
Time = 0.27 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.19 \[ \int \log (c (d+e x)) \, dx=-\frac {e x - {\left (e x + d\right )} \log \left (c e x + c d\right )}{e} \]
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Time = 0.07 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.24 \[ \int \log (c (d+e x)) \, dx=- e \left (- \frac {d \log {\left (d + e x \right )}}{e^{2}} + \frac {x}{e}\right ) + x \log {\left (c \left (d + e x\right ) \right )} \]
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Time = 0.19 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.48 \[ \int \log (c (d+e x)) \, dx=\frac {{\left (e x + d\right )} c \log \left ({\left (e x + d\right )} c\right ) - {\left (e x + d\right )} c}{c e} \]
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Time = 0.30 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.48 \[ \int \log (c (d+e x)) \, dx=\frac {{\left (e x + d\right )} c \log \left ({\left (e x + d\right )} c\right ) - {\left (e x + d\right )} c}{c e} \]
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Time = 0.07 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.19 \[ \int \log (c (d+e x)) \, dx=x\,\ln \left (c\,\left (d+e\,x\right )\right )-x+\frac {d\,\ln \left (d+e\,x\right )}{e} \]
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