Integrand size = 23, antiderivative size = 54 \[ \int \frac {a+b \log \left (c x^n\right )}{x \left (d+e x^r\right )} \, dx=-\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {d x^{-r}}{e}\right )}{d r}+\frac {b n \operatorname {PolyLog}\left (2,-\frac {d x^{-r}}{e}\right )}{d r^2} \]
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Time = 0.05 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2379, 2438} \[ \int \frac {a+b \log \left (c x^n\right )}{x \left (d+e x^r\right )} \, dx=\frac {b n \operatorname {PolyLog}\left (2,-\frac {d x^{-r}}{e}\right )}{d r^2}-\frac {\log \left (\frac {d x^{-r}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d r} \]
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Rule 2379
Rule 2438
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {d x^{-r}}{e}\right )}{d r}+\frac {(b n) \int \frac {\log \left (1+\frac {d x^{-r}}{e}\right )}{x} \, dx}{d r} \\ & = -\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {d x^{-r}}{e}\right )}{d r}+\frac {b n \text {Li}_2\left (-\frac {d x^{-r}}{e}\right )}{d r^2} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 108, normalized size of antiderivative = 2.00 \[ \int \frac {a+b \log \left (c x^n\right )}{x \left (d+e x^r\right )} \, dx=\frac {b n r^2 \log ^2(x)-2 r \left (a+b \log \left (c x^n\right )\right ) \log \left (d-d x^r\right )+2 b n r \log (x) \left (\log \left (d-d x^r\right )-\log \left (d+e x^r\right )\right )+2 b n \log \left (-\frac {e x^r}{d}\right ) \log \left (d+e x^r\right )+2 b n \operatorname {PolyLog}\left (2,1+\frac {e x^r}{d}\right )}{2 d r^2} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.00 (sec) , antiderivative size = 246, normalized size of antiderivative = 4.56
method | result | size |
risch | \(\frac {b \ln \left (d +e \,x^{r}\right ) n \ln \left (x \right )}{r d}-\frac {b \ln \left (d +e \,x^{r}\right ) \ln \left (x^{n}\right )}{r d}-\frac {b \ln \left (x^{r}\right ) n \ln \left (x \right )}{r d}+\frac {b \ln \left (x^{r}\right ) \ln \left (x^{n}\right )}{r d}+\frac {b n \ln \left (x \right )^{2}}{2 d}-\frac {b n \ln \left (x \right ) \ln \left (1+\frac {e \,x^{r}}{d}\right )}{r d}-\frac {b n \,\operatorname {Li}_{2}\left (-\frac {e \,x^{r}}{d}\right )}{r^{2} d}+\left (-\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}+b \ln \left (c \right )+a \right ) \left (-\frac {\ln \left (d +e \,x^{r}\right )}{r d}+\frac {\ln \left (x^{r}\right )}{r d}\right )\) | \(246\) |
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Time = 0.28 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.72 \[ \int \frac {a+b \log \left (c x^n\right )}{x \left (d+e x^r\right )} \, dx=\frac {b n r^{2} \log \left (x\right )^{2} - 2 \, b n r \log \left (x\right ) \log \left (\frac {e x^{r} + d}{d}\right ) - 2 \, b n {\rm Li}_2\left (-\frac {e x^{r} + d}{d} + 1\right ) - 2 \, {\left (b r \log \left (c\right ) + a r\right )} \log \left (e x^{r} + d\right ) + 2 \, {\left (b r^{2} \log \left (c\right ) + a r^{2}\right )} \log \left (x\right )}{2 \, d r^{2}} \]
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Time = 148.70 (sec) , antiderivative size = 452, normalized size of antiderivative = 8.37 \[ \int \frac {a+b \log \left (c x^n\right )}{x \left (d+e x^r\right )} \, dx=\text {Too large to display} \]
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\[ \int \frac {a+b \log \left (c x^n\right )}{x \left (d+e x^r\right )} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x^{r} + d\right )} x} \,d x } \]
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\[ \int \frac {a+b \log \left (c x^n\right )}{x \left (d+e x^r\right )} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x^{r} + d\right )} x} \,d x } \]
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Timed out. \[ \int \frac {a+b \log \left (c x^n\right )}{x \left (d+e x^r\right )} \, dx=\int \frac {a+b\,\ln \left (c\,x^n\right )}{x\,\left (d+e\,x^r\right )} \,d x \]
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