Integrand size = 24, antiderivative size = 200 \[ \int (f x)^{-1-2 n} \log ^2\left (c \left (d+e x^n\right )^p\right ) \, dx=\frac {e^2 p^2 x^{1+2 n} (f x)^{-1-2 n} \log (x)}{d^2}-\frac {e p x^{1+n} (f x)^{-1-2 n} \left (d+e x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )}{d^2 n}-\frac {x (f x)^{-1-2 n} \log ^2\left (c \left (d+e x^n\right )^p\right )}{2 n}-\frac {e^2 p x^{1+2 n} (f x)^{-1-2 n} \log \left (c \left (d+e x^n\right )^p\right ) \log \left (1-\frac {d}{d+e x^n}\right )}{d^2 n}+\frac {e^2 p^2 x^{1+2 n} (f x)^{-1-2 n} \operatorname {PolyLog}\left (2,\frac {d}{d+e x^n}\right )}{d^2 n} \]
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Time = 0.18 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {2506, 2504, 2445, 2458, 2389, 2379, 2438, 2351, 31} \[ \int (f x)^{-1-2 n} \log ^2\left (c \left (d+e x^n\right )^p\right ) \, dx=-\frac {e^2 p x^{2 n+1} (f x)^{-2 n-1} \log \left (1-\frac {d}{d+e x^n}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{d^2 n}-\frac {e p x^{n+1} (f x)^{-2 n-1} \left (d+e x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )}{d^2 n}-\frac {x (f x)^{-2 n-1} \log ^2\left (c \left (d+e x^n\right )^p\right )}{2 n}+\frac {e^2 p^2 x^{2 n+1} (f x)^{-2 n-1} \operatorname {PolyLog}\left (2,\frac {d}{e x^n+d}\right )}{d^2 n}+\frac {e^2 p^2 x^{2 n+1} \log (x) (f x)^{-2 n-1}}{d^2} \]
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Rule 31
Rule 2351
Rule 2379
Rule 2389
Rule 2438
Rule 2445
Rule 2458
Rule 2504
Rule 2506
Rubi steps \begin{align*} \text {integral}& = \left (x^{1+2 n} (f x)^{-1-2 n}\right ) \int x^{-1-2 n} \log ^2\left (c \left (d+e x^n\right )^p\right ) \, dx \\ & = \frac {\left (x^{1+2 n} (f x)^{-1-2 n}\right ) \text {Subst}\left (\int \frac {\log ^2\left (c (d+e x)^p\right )}{x^3} \, dx,x,x^n\right )}{n} \\ & = -\frac {x (f x)^{-1-2 n} \log ^2\left (c \left (d+e x^n\right )^p\right )}{2 n}+\frac {\left (e p x^{1+2 n} (f x)^{-1-2 n}\right ) \text {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{x^2 (d+e x)} \, dx,x,x^n\right )}{n} \\ & = -\frac {x (f x)^{-1-2 n} \log ^2\left (c \left (d+e x^n\right )^p\right )}{2 n}+\frac {\left (p x^{1+2 n} (f x)^{-1-2 n}\right ) \text {Subst}\left (\int \frac {\log \left (c x^p\right )}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^2} \, dx,x,d+e x^n\right )}{n} \\ & = -\frac {x (f x)^{-1-2 n} \log ^2\left (c \left (d+e x^n\right )^p\right )}{2 n}+\frac {\left (p x^{1+2 n} (f x)^{-1-2 n}\right ) \text {Subst}\left (\int \frac {\log \left (c x^p\right )}{\left (-\frac {d}{e}+\frac {x}{e}\right )^2} \, dx,x,d+e x^n\right )}{d n}-\frac {\left (e p x^{1+2 n} (f x)^{-1-2 n}\right ) \text {Subst}\left (\int \frac {\log \left (c x^p\right )}{x \left (-\frac {d}{e}+\frac {x}{e}\right )} \, dx,x,d+e x^n\right )}{d n} \\ & = -\frac {e p x^{1+n} (f x)^{-1-2 n} \left (d+e x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )}{d^2 n}-\frac {x (f x)^{-1-2 n} \log ^2\left (c \left (d+e x^n\right )^p\right )}{2 n}-\frac {e^2 p x^{1+2 n} (f x)^{-1-2 n} \log \left (c \left (d+e x^n\right )^p\right ) \log \left (1-\frac {d}{d+e x^n}\right )}{d^2 n}+\frac {\left (e p^2 x^{1+2 n} (f x)^{-1-2 n}\right ) \text {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x}{e}} \, dx,x,d+e x^n\right )}{d^2 n}+\frac {\left (e^2 p^2 x^{1+2 n} (f x)^{-1-2 n}\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {d}{x}\right )}{x} \, dx,x,d+e x^n\right )}{d^2 n} \\ & = \frac {e^2 p^2 x^{1+2 n} (f x)^{-1-2 n} \log (x)}{d^2}-\frac {e p x^{1+n} (f x)^{-1-2 n} \left (d+e x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )}{d^2 n}-\frac {x (f x)^{-1-2 n} \log ^2\left (c \left (d+e x^n\right )^p\right )}{2 n}-\frac {e^2 p x^{1+2 n} (f x)^{-1-2 n} \log \left (c \left (d+e x^n\right )^p\right ) \log \left (1-\frac {d}{d+e x^n}\right )}{d^2 n}+\frac {e^2 p^2 x^{1+2 n} (f x)^{-1-2 n} \text {Li}_2\left (\frac {d}{d+e x^n}\right )}{d^2 n} \\ \end{align*}
Time = 0.25 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.44 \[ \int (f x)^{-1-2 n} \log ^2\left (c \left (d+e x^n\right )^p\right ) \, dx=\frac {(f x)^{-2 n} \left (e^2 n^2 p^2 x^{2 n} \log ^2(x)+e^2 p^2 x^{2 n} \log ^2\left (e+d x^{-n}\right )-2 e^2 p^2 x^{2 n} \log \left (e-e x^{-n}\right )-2 e^2 p^2 x^{2 n} \log \left (e+d x^{-n}\right ) \log \left (e-e x^{-n}\right )-2 d e p x^n \log \left (c \left (d+e x^n\right )^p\right )+2 e^2 p x^{2 n} \log \left (e-e x^{-n}\right ) \log \left (c \left (d+e x^n\right )^p\right )-d^2 \log ^2\left (c \left (d+e x^n\right )^p\right )+2 e^2 n p x^{2 n} \log (x) \left (p+p \log \left (e+d x^{-n}\right )-p \log \left (e-e x^{-n}\right )-\log \left (c \left (d+e x^n\right )^p\right )+p \log \left (1+\frac {e x^n}{d}\right )\right )+2 e^2 p^2 x^{2 n} \operatorname {PolyLog}\left (2,-\frac {e x^n}{d}\right )\right )}{2 d^2 f n} \]
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\[\int \left (f x \right )^{-1-2 n} {\ln \left (c \left (d +e \,x^{n}\right )^{p}\right )}^{2}d x\]
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Time = 0.33 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.40 \[ \int (f x)^{-1-2 n} \log ^2\left (c \left (d+e x^n\right )^p\right ) \, dx=\frac {2 \, e^{2} f^{-2 \, n - 1} n p^{2} x^{2 \, n} \log \left (x\right ) \log \left (\frac {e x^{n} + d}{d}\right ) + 2 \, e^{2} f^{-2 \, n - 1} p^{2} x^{2 \, n} {\rm Li}_2\left (-\frac {e x^{n} + d}{d} + 1\right ) - 2 \, d e f^{-2 \, n - 1} p x^{n} \log \left (c\right ) - d^{2} f^{-2 \, n - 1} \log \left (c\right )^{2} + 2 \, {\left (e^{2} n p^{2} - e^{2} n p \log \left (c\right )\right )} f^{-2 \, n - 1} x^{2 \, n} \log \left (x\right ) + {\left (e^{2} f^{-2 \, n - 1} p^{2} x^{2 \, n} - d^{2} f^{-2 \, n - 1} p^{2}\right )} \log \left (e x^{n} + d\right )^{2} - 2 \, {\left (d e f^{-2 \, n - 1} p^{2} x^{n} + d^{2} f^{-2 \, n - 1} p \log \left (c\right ) + {\left (e^{2} n p^{2} \log \left (x\right ) + e^{2} p^{2} - e^{2} p \log \left (c\right )\right )} f^{-2 \, n - 1} x^{2 \, n}\right )} \log \left (e x^{n} + d\right )}{2 \, d^{2} n x^{2 \, n}} \]
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\[ \int (f x)^{-1-2 n} \log ^2\left (c \left (d+e x^n\right )^p\right ) \, dx=\int \left (f x\right )^{- 2 n - 1} \log {\left (c \left (d + e x^{n}\right )^{p} \right )}^{2}\, dx \]
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\[ \int (f x)^{-1-2 n} \log ^2\left (c \left (d+e x^n\right )^p\right ) \, dx=\int { \left (f x\right )^{-2 \, n - 1} \log \left ({\left (e x^{n} + d\right )}^{p} c\right )^{2} \,d x } \]
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\[ \int (f x)^{-1-2 n} \log ^2\left (c \left (d+e x^n\right )^p\right ) \, dx=\int { \left (f x\right )^{-2 \, n - 1} \log \left ({\left (e x^{n} + d\right )}^{p} c\right )^{2} \,d x } \]
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Timed out. \[ \int (f x)^{-1-2 n} \log ^2\left (c \left (d+e x^n\right )^p\right ) \, dx=\int \frac {{\ln \left (c\,{\left (d+e\,x^n\right )}^p\right )}^2}{{\left (f\,x\right )}^{2\,n+1}} \,d x \]
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