Integrand size = 24, antiderivative size = 124 \[ \int (f x)^{-1-n} \log ^2\left (c \left (d+e x^n\right )^p\right ) \, dx=\frac {2 e p x^{1+n} (f x)^{-1-n} \log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{d n}-\frac {x (f x)^{-1-n} \left (d+e x^n\right ) \log ^2\left (c \left (d+e x^n\right )^p\right )}{d n}+\frac {2 e p^2 x^{1+n} (f x)^{-1-n} \operatorname {PolyLog}\left (2,1+\frac {e x^n}{d}\right )}{d n} \]
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Time = 0.08 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {2506, 2504, 2444, 2441, 2352} \[ \int (f x)^{-1-n} \log ^2\left (c \left (d+e x^n\right )^p\right ) \, dx=-\frac {x (f x)^{-n-1} \left (d+e x^n\right ) \log ^2\left (c \left (d+e x^n\right )^p\right )}{d n}+\frac {2 e p x^{n+1} (f x)^{-n-1} \log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{d n}+\frac {2 e p^2 x^{n+1} (f x)^{-n-1} \operatorname {PolyLog}\left (2,\frac {e x^n}{d}+1\right )}{d n} \]
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Rule 2352
Rule 2441
Rule 2444
Rule 2504
Rule 2506
Rubi steps \begin{align*} \text {integral}& = \left (x^{1+n} (f x)^{-1-n}\right ) \int x^{-1-n} \log ^2\left (c \left (d+e x^n\right )^p\right ) \, dx \\ & = \frac {\left (x^{1+n} (f x)^{-1-n}\right ) \text {Subst}\left (\int \frac {\log ^2\left (c (d+e x)^p\right )}{x^2} \, dx,x,x^n\right )}{n} \\ & = -\frac {x (f x)^{-1-n} \left (d+e x^n\right ) \log ^2\left (c \left (d+e x^n\right )^p\right )}{d n}+\frac {\left (2 e p x^{1+n} (f x)^{-1-n}\right ) \text {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{x} \, dx,x,x^n\right )}{d n} \\ & = \frac {2 e p x^{1+n} (f x)^{-1-n} \log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{d n}-\frac {x (f x)^{-1-n} \left (d+e x^n\right ) \log ^2\left (c \left (d+e x^n\right )^p\right )}{d n}-\frac {\left (2 e^2 p^2 x^{1+n} (f x)^{-1-n}\right ) \text {Subst}\left (\int \frac {\log \left (-\frac {e x}{d}\right )}{d+e x} \, dx,x,x^n\right )}{d n} \\ & = \frac {2 e p x^{1+n} (f x)^{-1-n} \log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{d n}-\frac {x (f x)^{-1-n} \left (d+e x^n\right ) \log ^2\left (c \left (d+e x^n\right )^p\right )}{d n}+\frac {2 e p^2 x^{1+n} (f x)^{-1-n} \text {Li}_2\left (1+\frac {e x^n}{d}\right )}{d n} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.21 \[ \int (f x)^{-1-n} \log ^2\left (c \left (d+e x^n\right )^p\right ) \, dx=-\frac {x^{1+n} (f x)^{-1-n} \left (x^{-n} \log ^2\left (c \left (d+e x^n\right )^p\right )-2 e p \left (-\frac {p \log \left (-\frac {d x^{-n}}{e}\right ) \log \left (-e-d x^{-n}\right )}{d}+\frac {p \log ^2\left (-e-d x^{-n}\right )}{2 d}-\frac {\log \left (-e-d x^{-n}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{d}-\frac {p \operatorname {PolyLog}\left (2,\frac {e+d x^{-n}}{e}\right )}{d}\right )\right )}{n} \]
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\[\int \left (f x \right )^{-1-n} {\ln \left (c \left (d +e \,x^{n}\right )^{p}\right )}^{2}d x\]
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Time = 0.37 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.59 \[ \int (f x)^{-1-n} \log ^2\left (c \left (d+e x^n\right )^p\right ) \, dx=-\frac {2 \, e f^{-n - 1} n p^{2} x^{n} \log \left (x\right ) \log \left (\frac {e x^{n} + d}{d}\right ) - 2 \, e f^{-n - 1} n p x^{n} \log \left (c\right ) \log \left (x\right ) + 2 \, e f^{-n - 1} p^{2} x^{n} {\rm Li}_2\left (-\frac {e x^{n} + d}{d} + 1\right ) + d f^{-n - 1} \log \left (c\right )^{2} + {\left (e f^{-n - 1} p^{2} x^{n} + d f^{-n - 1} p^{2}\right )} \log \left (e x^{n} + d\right )^{2} + 2 \, {\left (d f^{-n - 1} p \log \left (c\right ) - {\left (e n p^{2} \log \left (x\right ) - e p \log \left (c\right )\right )} f^{-n - 1} x^{n}\right )} \log \left (e x^{n} + d\right )}{d n x^{n}} \]
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\[ \int (f x)^{-1-n} \log ^2\left (c \left (d+e x^n\right )^p\right ) \, dx=\int \left (f x\right )^{- n - 1} \log {\left (c \left (d + e x^{n}\right )^{p} \right )}^{2}\, dx \]
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\[ \int (f x)^{-1-n} \log ^2\left (c \left (d+e x^n\right )^p\right ) \, dx=\int { \left (f x\right )^{-n - 1} \log \left ({\left (e x^{n} + d\right )}^{p} c\right )^{2} \,d x } \]
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\[ \int (f x)^{-1-n} \log ^2\left (c \left (d+e x^n\right )^p\right ) \, dx=\int { \left (f x\right )^{-n - 1} \log \left ({\left (e x^{n} + d\right )}^{p} c\right )^{2} \,d x } \]
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Timed out. \[ \int (f x)^{-1-n} \log ^2\left (c \left (d+e x^n\right )^p\right ) \, dx=\int \frac {{\ln \left (c\,{\left (d+e\,x^n\right )}^p\right )}^2}{{\left (f\,x\right )}^{n+1}} \,d x \]
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