Integrand size = 24, antiderivative size = 88 \[ \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{x} \, dx=\frac {\log ^2\left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 m}-\frac {b n \log ^2\left (f x^m\right ) \log \left (1+\frac {e x}{d}\right )}{2 m}-b n \log \left (f x^m\right ) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )+b m n \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right ) \]
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Time = 0.05 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2472, 2354, 2421, 6724} \[ \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{x} \, dx=\frac {\log ^2\left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 m}-b n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right ) \log \left (f x^m\right )-\frac {b n \log \left (\frac {e x}{d}+1\right ) \log ^2\left (f x^m\right )}{2 m}+b m n \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right ) \]
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Rule 2354
Rule 2421
Rule 2472
Rule 6724
Rubi steps \begin{align*} \text {integral}& = \frac {\log ^2\left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 m}-\frac {(b e n) \int \frac {\log ^2\left (f x^m\right )}{d+e x} \, dx}{2 m} \\ & = \frac {\log ^2\left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 m}-\frac {b n \log ^2\left (f x^m\right ) \log \left (1+\frac {e x}{d}\right )}{2 m}+(b n) \int \frac {\log \left (f x^m\right ) \log \left (1+\frac {e x}{d}\right )}{x} \, dx \\ & = \frac {\log ^2\left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 m}-\frac {b n \log ^2\left (f x^m\right ) \log \left (1+\frac {e x}{d}\right )}{2 m}-b n \log \left (f x^m\right ) \text {Li}_2\left (-\frac {e x}{d}\right )+(b m n) \int \frac {\text {Li}_2\left (-\frac {e x}{d}\right )}{x} \, dx \\ & = \frac {\log ^2\left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 m}-\frac {b n \log ^2\left (f x^m\right ) \log \left (1+\frac {e x}{d}\right )}{2 m}-b n \log \left (f x^m\right ) \text {Li}_2\left (-\frac {e x}{d}\right )+b m n \text {Li}_3\left (-\frac {e x}{d}\right ) \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.45 \[ \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{x} \, dx=\frac {1}{2} \left (\frac {a \log ^2\left (f x^m\right )}{m}-b m \log ^2(x) \log \left (c (d+e x)^n\right )+2 b \log (x) \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )+b m n \log ^2(x) \log \left (1+\frac {e x}{d}\right )-2 b n \log (x) \log \left (f x^m\right ) \log \left (1+\frac {e x}{d}\right )-2 b n \log \left (f x^m\right ) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )+2 b m n \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )\right ) \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 4.60 (sec) , antiderivative size = 756, normalized size of antiderivative = 8.59
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\[ \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{x} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )} \log \left (f x^{m}\right )}{x} \,d x } \]
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Timed out. \[ \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{x} \, dx=\text {Timed out} \]
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\[ \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{x} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )} \log \left (f x^{m}\right )}{x} \,d x } \]
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\[ \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{x} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )} \log \left (f x^{m}\right )}{x} \,d x } \]
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Timed out. \[ \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{x} \, dx=\int \frac {\ln \left (f\,x^m\right )\,\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}{x} \,d x \]
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