Integrand size = 14, antiderivative size = 41 \[ \int \frac {\log \left (c \left (d+e x^n\right )\right )}{x} \, dx=\frac {\log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )\right )}{n}+\frac {\operatorname {PolyLog}\left (2,1+\frac {e x^n}{d}\right )}{n} \]
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Time = 0.03 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {2504, 2441, 2352} \[ \int \frac {\log \left (c \left (d+e x^n\right )\right )}{x} \, dx=\frac {\log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )\right )}{n}+\frac {\operatorname {PolyLog}\left (2,\frac {e x^n}{d}+1\right )}{n} \]
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Rule 2352
Rule 2441
Rule 2504
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\log (c (d+e x))}{x} \, dx,x,x^n\right )}{n} \\ & = \frac {\log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )\right )}{n}-\frac {e \text {Subst}\left (\int \frac {\log \left (-\frac {e x}{d}\right )}{d+e x} \, dx,x,x^n\right )}{n} \\ & = \frac {\log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )\right )}{n}+\frac {\text {Li}_2\left (1+\frac {e x^n}{d}\right )}{n} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.95 \[ \int \frac {\log \left (c \left (d+e x^n\right )\right )}{x} \, dx=\frac {\log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )\right )+\operatorname {PolyLog}\left (2,\frac {d+e x^n}{d}\right )}{n} \]
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Time = 1.25 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.93
method | result | size |
derivativedivides | \(\frac {\operatorname {dilog}\left (-\frac {e \,x^{n}}{d}\right )+\ln \left (c e \,x^{n}+c d \right ) \ln \left (-\frac {e \,x^{n}}{d}\right )}{n}\) | \(38\) |
default | \(\frac {\operatorname {dilog}\left (-\frac {e \,x^{n}}{d}\right )+\ln \left (c e \,x^{n}+c d \right ) \ln \left (-\frac {e \,x^{n}}{d}\right )}{n}\) | \(38\) |
risch | \(\ln \left (x \right ) \ln \left (d +e \,x^{n}\right )+\left (\frac {i \pi \,\operatorname {csgn}\left (i \left (d +e \,x^{n}\right )\right ) {\operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )\right )}^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i \left (d +e \,x^{n}\right )\right ) \operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i \pi {\operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )\right )}^{3}}{2}+\frac {i \pi {\operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )\right )}^{2} \operatorname {csgn}\left (i c \right )}{2}+\ln \left (c \right )\right ) \ln \left (x \right )-\frac {\operatorname {dilog}\left (\frac {d +e \,x^{n}}{d}\right )}{n}-\ln \left (x \right ) \ln \left (\frac {d +e \,x^{n}}{d}\right )\) | \(154\) |
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Time = 0.32 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.32 \[ \int \frac {\log \left (c \left (d+e x^n\right )\right )}{x} \, dx=\frac {n \log \left (c e x^{n} + c d\right ) \log \left (x\right ) - n \log \left (x\right ) \log \left (\frac {e x^{n} + d}{d}\right ) - {\rm Li}_2\left (-\frac {e x^{n} + d}{d} + 1\right )}{n} \]
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\[ \int \frac {\log \left (c \left (d+e x^n\right )\right )}{x} \, dx=\int \frac {\log {\left (c d + c e x^{n} \right )}}{x}\, dx \]
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\[ \int \frac {\log \left (c \left (d+e x^n\right )\right )}{x} \, dx=\int { \frac {\log \left ({\left (e x^{n} + d\right )} c\right )}{x} \,d x } \]
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\[ \int \frac {\log \left (c \left (d+e x^n\right )\right )}{x} \, dx=\int { \frac {\log \left ({\left (e x^{n} + d\right )} c\right )}{x} \,d x } \]
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Timed out. \[ \int \frac {\log \left (c \left (d+e x^n\right )\right )}{x} \, dx=\int \frac {\ln \left (c\,\left (d+e\,x^n\right )\right )}{x} \,d x \]
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