Integrand size = 16, antiderivative size = 72 \[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x^3} \, dx=-\frac {e n p x^{-2+n} \operatorname {Hypergeometric2F1}\left (1,-\frac {2-n}{n},2 \left (1-\frac {1}{n}\right ),-\frac {e x^n}{d}\right )}{2 d (2-n)}-\frac {\log \left (c \left (d+e x^n\right )^p\right )}{2 x^2} \]
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Time = 0.02 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2505, 371} \[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x^3} \, dx=-\frac {\log \left (c \left (d+e x^n\right )^p\right )}{2 x^2}-\frac {e n p x^{n-2} \operatorname {Hypergeometric2F1}\left (1,-\frac {2-n}{n},2 \left (1-\frac {1}{n}\right ),-\frac {e x^n}{d}\right )}{2 d (2-n)} \]
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Rule 371
Rule 2505
Rubi steps \begin{align*} \text {integral}& = -\frac {\log \left (c \left (d+e x^n\right )^p\right )}{2 x^2}+\frac {1}{2} (e n p) \int \frac {x^{-3+n}}{d+e x^n} \, dx \\ & = -\frac {e n p x^{-2+n} \, _2F_1\left (1,-\frac {2-n}{n};2 \left (1-\frac {1}{n}\right );-\frac {e x^n}{d}\right )}{2 d (2-n)}-\frac {\log \left (c \left (d+e x^n\right )^p\right )}{2 x^2} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.86 \[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x^3} \, dx=\frac {\frac {e n p x^n \operatorname {Hypergeometric2F1}\left (1,\frac {-2+n}{n},2-\frac {2}{n},-\frac {e x^n}{d}\right )}{d (-2+n)}-\log \left (c \left (d+e x^n\right )^p\right )}{2 x^2} \]
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\[\int \frac {\ln \left (c \left (d +e \,x^{n}\right )^{p}\right )}{x^{3}}d x\]
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\[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x^3} \, dx=\int { \frac {\log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{x^{3}} \,d x } \]
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Result contains complex when optimal does not.
Time = 6.77 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.08 \[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x^3} \, dx=- \frac {\log {\left (c \left (d + e x^{n}\right )^{p} \right )}}{2 x^{2}} + \frac {d^{\frac {2}{n}} d^{1 - \frac {2}{n}} e e^{- \frac {2}{n}} e^{-1 + \frac {2}{n}} p \Phi \left (\frac {d x^{- n} e^{i \pi }}{e}, 1, \frac {2}{n}\right ) \Gamma \left (- \frac {2}{n}\right )}{d n x^{2} \Gamma \left (1 - \frac {2}{n}\right )} \]
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\[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x^3} \, dx=\int { \frac {\log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{x^{3}} \,d x } \]
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\[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x^3} \, dx=\int { \frac {\log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{x^{3}} \,d x } \]
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Timed out. \[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x^3} \, dx=\int \frac {\ln \left (c\,{\left (d+e\,x^n\right )}^p\right )}{x^3} \,d x \]
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