Integrand size = 28, antiderivative size = 132 \[ \int \frac {\log ^2\left (f x^p\right ) \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )}{x} \, dx=\frac {\log ^3\left (f x^p\right ) \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )}{3 p}-\frac {b n \log ^3\left (f x^p\right ) \log \left (1+\frac {e x^m}{d}\right )}{3 p}-\frac {b n \log ^2\left (f x^p\right ) \operatorname {PolyLog}\left (2,-\frac {e x^m}{d}\right )}{m}+\frac {2 b n p \log \left (f x^p\right ) \operatorname {PolyLog}\left (3,-\frac {e x^m}{d}\right )}{m^2}-\frac {2 b n p^2 \operatorname {PolyLog}\left (4,-\frac {e x^m}{d}\right )}{m^3} \]
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Time = 0.12 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {2531, 2375, 2421, 2430, 6724} \[ \int \frac {\log ^2\left (f x^p\right ) \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )}{x} \, dx=\frac {\log ^3\left (f x^p\right ) \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )}{3 p}+\frac {2 b n p \log \left (f x^p\right ) \operatorname {PolyLog}\left (3,-\frac {e x^m}{d}\right )}{m^2}-\frac {b n \log ^2\left (f x^p\right ) \operatorname {PolyLog}\left (2,-\frac {e x^m}{d}\right )}{m}-\frac {b n \log ^3\left (f x^p\right ) \log \left (\frac {e x^m}{d}+1\right )}{3 p}-\frac {2 b n p^2 \operatorname {PolyLog}\left (4,-\frac {e x^m}{d}\right )}{m^3} \]
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Rule 2375
Rule 2421
Rule 2430
Rule 2531
Rule 6724
Rubi steps \begin{align*} \text {integral}& = \frac {\log ^3\left (f x^p\right ) \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )}{3 p}-\frac {(b e m n) \int \frac {x^{-1+m} \log ^3\left (f x^p\right )}{d+e x^m} \, dx}{3 p} \\ & = \frac {\log ^3\left (f x^p\right ) \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )}{3 p}-\frac {b n \log ^3\left (f x^p\right ) \log \left (1+\frac {e x^m}{d}\right )}{3 p}+(b n) \int \frac {\log ^2\left (f x^p\right ) \log \left (1+\frac {e x^m}{d}\right )}{x} \, dx \\ & = \frac {\log ^3\left (f x^p\right ) \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )}{3 p}-\frac {b n \log ^3\left (f x^p\right ) \log \left (1+\frac {e x^m}{d}\right )}{3 p}-\frac {b n \log ^2\left (f x^p\right ) \text {Li}_2\left (-\frac {e x^m}{d}\right )}{m}+\frac {(2 b n p) \int \frac {\log \left (f x^p\right ) \text {Li}_2\left (-\frac {e x^m}{d}\right )}{x} \, dx}{m} \\ & = \frac {\log ^3\left (f x^p\right ) \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )}{3 p}-\frac {b n \log ^3\left (f x^p\right ) \log \left (1+\frac {e x^m}{d}\right )}{3 p}-\frac {b n \log ^2\left (f x^p\right ) \text {Li}_2\left (-\frac {e x^m}{d}\right )}{m}+\frac {2 b n p \log \left (f x^p\right ) \text {Li}_3\left (-\frac {e x^m}{d}\right )}{m^2}-\frac {\left (2 b n p^2\right ) \int \frac {\text {Li}_3\left (-\frac {e x^m}{d}\right )}{x} \, dx}{m^2} \\ & = \frac {\log ^3\left (f x^p\right ) \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )}{3 p}-\frac {b n \log ^3\left (f x^p\right ) \log \left (1+\frac {e x^m}{d}\right )}{3 p}-\frac {b n \log ^2\left (f x^p\right ) \text {Li}_2\left (-\frac {e x^m}{d}\right )}{m}+\frac {2 b n p \log \left (f x^p\right ) \text {Li}_3\left (-\frac {e x^m}{d}\right )}{m^2}-\frac {2 b n p^2 \text {Li}_4\left (-\frac {e x^m}{d}\right )}{m^3} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(456\) vs. \(2(132)=264\).
Time = 0.16 (sec) , antiderivative size = 456, normalized size of antiderivative = 3.45 \[ \int \frac {\log ^2\left (f x^p\right ) \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )}{x} \, dx=\frac {1}{4} b m n p^2 \log ^4(x)-\frac {1}{3} b m n p \log ^3(x) \log \left (f x^p\right )+\frac {a \log ^3\left (f x^p\right )}{3 p}+\frac {2}{3} b n p^2 \log ^3(x) \log \left (1+\frac {d x^{-m}}{e}\right )-b n p \log ^2(x) \log \left (f x^p\right ) \log \left (1+\frac {d x^{-m}}{e}\right )-b n p^2 \log ^3(x) \log \left (d+e x^m\right )+\frac {b n p^2 \log ^2(x) \log \left (-\frac {e x^m}{d}\right ) \log \left (d+e x^m\right )}{m}+2 b n p \log ^2(x) \log \left (f x^p\right ) \log \left (d+e x^m\right )-\frac {2 b n p \log (x) \log \left (-\frac {e x^m}{d}\right ) \log \left (f x^p\right ) \log \left (d+e x^m\right )}{m}-b n \log (x) \log ^2\left (f x^p\right ) \log \left (d+e x^m\right )+\frac {b n \log \left (-\frac {e x^m}{d}\right ) \log ^2\left (f x^p\right ) \log \left (d+e x^m\right )}{m}+\frac {1}{3} b p^2 \log ^3(x) \log \left (c \left (d+e x^m\right )^n\right )-b p \log ^2(x) \log \left (f x^p\right ) \log \left (c \left (d+e x^m\right )^n\right )+b \log (x) \log ^2\left (f x^p\right ) \log \left (c \left (d+e x^m\right )^n\right )-\frac {b n p \log (x) \left (p \log (x)-2 \log \left (f x^p\right )\right ) \operatorname {PolyLog}\left (2,-\frac {d x^{-m}}{e}\right )}{m}+\frac {b n \left (-p \log (x)+\log \left (f x^p\right )\right )^2 \operatorname {PolyLog}\left (2,1+\frac {e x^m}{d}\right )}{m}+\frac {2 b n p \log \left (f x^p\right ) \operatorname {PolyLog}\left (3,-\frac {d x^{-m}}{e}\right )}{m^2}+\frac {2 b n p^2 \operatorname {PolyLog}\left (4,-\frac {d x^{-m}}{e}\right )}{m^3} \]
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\[\int \frac {\ln \left (f \,x^{p}\right )^{2} \left (a +b \ln \left (c \left (d +e \,x^{m}\right )^{n}\right )\right )}{x}d x\]
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Leaf count of result is larger than twice the leaf count of optimal. 281 vs. \(2 (127) = 254\).
Time = 0.32 (sec) , antiderivative size = 281, normalized size of antiderivative = 2.13 \[ \int \frac {\log ^2\left (f x^p\right ) \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )}{x} \, dx=-\frac {6 \, b n p^{2} {\rm polylog}\left (4, -\frac {e x^{m}}{d}\right ) - 3 \, {\left (b m^{3} \log \left (c\right ) + a m^{3}\right )} \log \left (f\right )^{2} \log \left (x\right ) - 3 \, {\left (b m^{3} p \log \left (c\right ) + a m^{3} p\right )} \log \left (f\right ) \log \left (x\right )^{2} - {\left (b m^{3} p^{2} \log \left (c\right ) + a m^{3} p^{2}\right )} \log \left (x\right )^{3} + 3 \, {\left (b m^{2} n p^{2} \log \left (x\right )^{2} + 2 \, b m^{2} n p \log \left (f\right ) \log \left (x\right ) + b m^{2} n \log \left (f\right )^{2}\right )} {\rm Li}_2\left (-\frac {e x^{m} + d}{d} + 1\right ) - {\left (b m^{3} n p^{2} \log \left (x\right )^{3} + 3 \, b m^{3} n p \log \left (f\right ) \log \left (x\right )^{2} + 3 \, b m^{3} n \log \left (f\right )^{2} \log \left (x\right )\right )} \log \left (e x^{m} + d\right ) + {\left (b m^{3} n p^{2} \log \left (x\right )^{3} + 3 \, b m^{3} n p \log \left (f\right ) \log \left (x\right )^{2} + 3 \, b m^{3} n \log \left (f\right )^{2} \log \left (x\right )\right )} \log \left (\frac {e x^{m} + d}{d}\right ) - 6 \, {\left (b m n p^{2} \log \left (x\right ) + b m n p \log \left (f\right )\right )} {\rm polylog}\left (3, -\frac {e x^{m}}{d}\right )}{3 \, m^{3}} \]
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\[ \int \frac {\log ^2\left (f x^p\right ) \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )}{x} \, dx=\int \frac {\left (a + b \log {\left (c \left (d + e x^{m}\right )^{n} \right )}\right ) \log {\left (f x^{p} \right )}^{2}}{x}\, dx \]
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\[ \int \frac {\log ^2\left (f x^p\right ) \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )}{x} \, dx=\int { \frac {{\left (b \log \left ({\left (e x^{m} + d\right )}^{n} c\right ) + a\right )} \log \left (f x^{p}\right )^{2}}{x} \,d x } \]
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\[ \int \frac {\log ^2\left (f x^p\right ) \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )}{x} \, dx=\int { \frac {{\left (b \log \left ({\left (e x^{m} + d\right )}^{n} c\right ) + a\right )} \log \left (f x^{p}\right )^{2}}{x} \,d x } \]
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Timed out. \[ \int \frac {\log ^2\left (f x^p\right ) \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )}{x} \, dx=\int \frac {{\ln \left (f\,x^p\right )}^2\,\left (a+b\,\ln \left (c\,{\left (d+e\,x^m\right )}^n\right )\right )}{x} \,d x \]
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