Integrand size = 22, antiderivative size = 80 \[ \int (f x)^{-1-n} \log \left (c \left (d+e x^n\right )^p\right ) \, dx=\frac {e p x^n (f x)^{-n} \log (x)}{d f}-\frac {e p x^n (f x)^{-n} \log \left (d+e x^n\right )}{d f n}-\frac {(f x)^{-n} \log \left (c \left (d+e x^n\right )^p\right )}{f n} \]
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Time = 0.02 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {2505, 19, 272, 36, 29, 31} \[ \int (f x)^{-1-n} \log \left (c \left (d+e x^n\right )^p\right ) \, dx=-\frac {(f x)^{-n} \log \left (c \left (d+e x^n\right )^p\right )}{f n}+\frac {e p x^n \log (x) (f x)^{-n}}{d f}-\frac {e p x^n (f x)^{-n} \log \left (d+e x^n\right )}{d f n} \]
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Rule 19
Rule 29
Rule 31
Rule 36
Rule 272
Rule 2505
Rubi steps \begin{align*} \text {integral}& = -\frac {(f x)^{-n} \log \left (c \left (d+e x^n\right )^p\right )}{f n}+\frac {(e p) \int \frac {x^{-1+n} (f x)^{-n}}{d+e x^n} \, dx}{f} \\ & = -\frac {(f x)^{-n} \log \left (c \left (d+e x^n\right )^p\right )}{f n}+\frac {\left (e p x^n (f x)^{-n}\right ) \int \frac {1}{x \left (d+e x^n\right )} \, dx}{f} \\ & = -\frac {(f x)^{-n} \log \left (c \left (d+e x^n\right )^p\right )}{f n}+\frac {\left (e p x^n (f x)^{-n}\right ) \text {Subst}\left (\int \frac {1}{x (d+e x)} \, dx,x,x^n\right )}{f n} \\ & = -\frac {(f x)^{-n} \log \left (c \left (d+e x^n\right )^p\right )}{f n}+\frac {\left (e p x^n (f x)^{-n}\right ) \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^n\right )}{d f n}-\frac {\left (e^2 p x^n (f x)^{-n}\right ) \text {Subst}\left (\int \frac {1}{d+e x} \, dx,x,x^n\right )}{d f n} \\ & = \frac {e p x^n (f x)^{-n} \log (x)}{d f}-\frac {e p x^n (f x)^{-n} \log \left (d+e x^n\right )}{d f n}-\frac {(f x)^{-n} \log \left (c \left (d+e x^n\right )^p\right )}{f n} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.71 \[ \int (f x)^{-1-n} \log \left (c \left (d+e x^n\right )^p\right ) \, dx=-\frac {(f x)^{-n} \left (-e n p x^n \log (x)+e p x^n \log \left (d+e x^n\right )+d \log \left (c \left (d+e x^n\right )^p\right )\right )}{d f n} \]
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\[\int \left (f x \right )^{-1-n} \ln \left (c \left (d +e \,x^{n}\right )^{p}\right )d x\]
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none
Time = 0.31 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.94 \[ \int (f x)^{-1-n} \log \left (c \left (d+e x^n\right )^p\right ) \, dx=\frac {e f^{-n - 1} n p x^{n} \log \left (x\right ) - d f^{-n - 1} \log \left (c\right ) - {\left (e f^{-n - 1} p x^{n} + d f^{-n - 1} p\right )} \log \left (e x^{n} + d\right )}{d n x^{n}} \]
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Exception generated. \[ \int (f x)^{-1-n} \log \left (c \left (d+e x^n\right )^p\right ) \, dx=\text {Exception raised: TypeError} \]
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none
Time = 0.21 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.89 \[ \int (f x)^{-1-n} \log \left (c \left (d+e x^n\right )^p\right ) \, dx=\frac {e p {\left (\frac {\log \left (x\right )}{d f^{n}} - \frac {\log \left (\frac {e x^{n} + d}{e}\right )}{d f^{n} n}\right )}}{f} - \frac {\log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{\left (f x\right )^{n} f n} \]
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\[ \int (f x)^{-1-n} \log \left (c \left (d+e x^n\right )^p\right ) \, dx=\int { \left (f x\right )^{-n - 1} \log \left ({\left (e x^{n} + d\right )}^{p} c\right ) \,d x } \]
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Timed out. \[ \int (f x)^{-1-n} \log \left (c \left (d+e x^n\right )^p\right ) \, dx=\int \frac {\ln \left (c\,{\left (d+e\,x^n\right )}^p\right )}{{\left (f\,x\right )}^{n+1}} \,d x \]
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