Integrand size = 16, antiderivative size = 27 \[ \int \frac {(d x)^{-1+n}}{\log \left (c x^n\right )} \, dx=\frac {x^{1-n} (d x)^{-1+n} \operatorname {LogIntegral}\left (c x^n\right )}{c n} \]
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Time = 0.03 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2345, 2344, 2335} \[ \int \frac {(d x)^{-1+n}}{\log \left (c x^n\right )} \, dx=\frac {x^{1-n} (d x)^{n-1} \operatorname {LogIntegral}\left (c x^n\right )}{c n} \]
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Rule 2335
Rule 2344
Rule 2345
Rubi steps \begin{align*} \text {integral}& = \left (x^{1-n} (d x)^{-1+n}\right ) \int \frac {x^{-1+n}}{\log \left (c x^n\right )} \, dx \\ & = \frac {\left (x^{1-n} (d x)^{-1+n}\right ) \text {Subst}\left (\int \frac {1}{\log (c x)} \, dx,x,x^n\right )}{n} \\ & = \frac {x^{1-n} (d x)^{-1+n} \text {li}\left (c x^n\right )}{c n} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {(d x)^{-1+n}}{\log \left (c x^n\right )} \, dx=\frac {x^{1-n} (d x)^{-1+n} \operatorname {LogIntegral}\left (c x^n\right )}{c n} \]
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\[\int \frac {\left (d x \right )^{n -1}}{\ln \left (c \,x^{n}\right )}d x\]
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none
Time = 0.30 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.74 \[ \int \frac {(d x)^{-1+n}}{\log \left (c x^n\right )} \, dx=\frac {d^{n - 1} {\rm Ei}\left (n \log \left (x\right ) + \log \left (c\right )\right )}{c n} \]
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\[ \int \frac {(d x)^{-1+n}}{\log \left (c x^n\right )} \, dx=\int \frac {\left (d x\right )^{n - 1}}{\log {\left (c x^{n} \right )}}\, dx \]
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\[ \int \frac {(d x)^{-1+n}}{\log \left (c x^n\right )} \, dx=\int { \frac {\left (d x\right )^{n - 1}}{\log \left (c x^{n}\right )} \,d x } \]
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\[ \int \frac {(d x)^{-1+n}}{\log \left (c x^n\right )} \, dx=\int { \frac {\left (d x\right )^{n - 1}}{\log \left (c x^{n}\right )} \,d x } \]
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Timed out. \[ \int \frac {(d x)^{-1+n}}{\log \left (c x^n\right )} \, dx=\int \frac {{\left (d\,x\right )}^{n-1}}{\ln \left (c\,x^n\right )} \,d x \]
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