Integrand size = 19, antiderivative size = 50 \[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{f x} \, dx=\frac {\log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{f n}+\frac {p \operatorname {PolyLog}\left (2,1+\frac {e x^n}{d}\right )}{f n} \]
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Time = 0.03 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {12, 2504, 2441, 2352} \[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{f x} \, dx=\frac {\log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{f n}+\frac {p \operatorname {PolyLog}\left (2,\frac {e x^n}{d}+1\right )}{f n} \]
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Rule 12
Rule 2352
Rule 2441
Rule 2504
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx}{f} \\ & = \frac {\text {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{x} \, dx,x,x^n\right )}{f n} \\ & = \frac {\log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{f n}-\frac {(e p) \text {Subst}\left (\int \frac {\log \left (-\frac {e x}{d}\right )}{d+e x} \, dx,x,x^n\right )}{f n} \\ & = \frac {\log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{f n}+\frac {p \text {Li}_2\left (1+\frac {e x^n}{d}\right )}{f n} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.92 \[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{f x} \, dx=\frac {\log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )+p \operatorname {PolyLog}\left (2,\frac {d+e x^n}{d}\right )}{f n} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 1.28 (sec) , antiderivative size = 182, normalized size of antiderivative = 3.64
method | result | size |
risch | \(\frac {\ln \left (x \right ) \ln \left (\left (d +e \,x^{n}\right )^{p}\right )}{f}+\frac {\left (\frac {i \pi \,\operatorname {csgn}\left (i \left (d +e \,x^{n}\right )^{p}\right ) {\operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )}^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i \left (d +e \,x^{n}\right )^{p}\right ) \operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i \pi {\operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )}^{3}}{2}+\frac {i \pi {\operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )}{2}+\ln \left (c \right )\right ) \ln \left (x \right )}{f}-\frac {p \operatorname {dilog}\left (\frac {d +e \,x^{n}}{d}\right )}{f n}-\frac {p \ln \left (x \right ) \ln \left (\frac {d +e \,x^{n}}{d}\right )}{f}\) | \(182\) |
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none
Time = 0.31 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.26 \[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{f x} \, dx=\frac {n p \log \left (e x^{n} + d\right ) \log \left (x\right ) - n p \log \left (x\right ) \log \left (\frac {e x^{n} + d}{d}\right ) + n \log \left (c\right ) \log \left (x\right ) - p {\rm Li}_2\left (-\frac {e x^{n} + d}{d} + 1\right )}{f n} \]
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\[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{f x} \, dx=\frac {\int \frac {\log {\left (c \left (d + e x^{n}\right )^{p} \right )}}{x}\, dx}{f} \]
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\[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{f x} \, dx=\int { \frac {\log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{f x} \,d x } \]
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\[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{f x} \, dx=\int { \frac {\log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{f x} \,d x } \]
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Timed out. \[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{f x} \, dx=\int \frac {\ln \left (c\,{\left (d+e\,x^n\right )}^p\right )}{f\,x} \,d x \]
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