Integrand size = 16, antiderivative size = 77 \[ \int \frac {(d x)^{-1+n}}{\log ^3\left (c x^n\right )} \, dx=-\frac {(d x)^n}{2 d n \log ^2\left (c x^n\right )}-\frac {(d x)^n}{2 d n \log \left (c x^n\right )}+\frac {x^{1-n} (d x)^{-1+n} \operatorname {LogIntegral}\left (c x^n\right )}{2 c n} \]
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Time = 0.05 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2343, 2345, 2344, 2335} \[ \int \frac {(d x)^{-1+n}}{\log ^3\left (c x^n\right )} \, dx=\frac {x^{1-n} (d x)^{n-1} \operatorname {LogIntegral}\left (c x^n\right )}{2 c n}-\frac {(d x)^n}{2 d n \log ^2\left (c x^n\right )}-\frac {(d x)^n}{2 d n \log \left (c x^n\right )} \]
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Rule 2335
Rule 2343
Rule 2344
Rule 2345
Rubi steps \begin{align*} \text {integral}& = -\frac {(d x)^n}{2 d n \log ^2\left (c x^n\right )}+\frac {1}{2} \int \frac {(d x)^{-1+n}}{\log ^2\left (c x^n\right )} \, dx \\ & = -\frac {(d x)^n}{2 d n \log ^2\left (c x^n\right )}-\frac {(d x)^n}{2 d n \log \left (c x^n\right )}+\frac {1}{2} \int \frac {(d x)^{-1+n}}{\log \left (c x^n\right )} \, dx \\ & = -\frac {(d x)^n}{2 d n \log ^2\left (c x^n\right )}-\frac {(d x)^n}{2 d n \log \left (c x^n\right )}+\frac {1}{2} \left (x^{1-n} (d x)^{-1+n}\right ) \int \frac {x^{-1+n}}{\log \left (c x^n\right )} \, dx \\ & = -\frac {(d x)^n}{2 d n \log ^2\left (c x^n\right )}-\frac {(d x)^n}{2 d n \log \left (c x^n\right )}+\frac {\left (x^{1-n} (d x)^{-1+n}\right ) \text {Subst}\left (\int \frac {1}{\log (c x)} \, dx,x,x^n\right )}{2 n} \\ & = -\frac {(d x)^n}{2 d n \log ^2\left (c x^n\right )}-\frac {(d x)^n}{2 d n \log \left (c x^n\right )}+\frac {x^{1-n} (d x)^{-1+n} \text {li}\left (c x^n\right )}{2 c n} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.79 \[ \int \frac {(d x)^{-1+n}}{\log ^3\left (c x^n\right )} \, dx=\frac {x^{-n} (d x)^n \left (-c x^n \left (1+\log \left (c x^n\right )\right )+\log ^2\left (c x^n\right ) \operatorname {LogIntegral}\left (c x^n\right )\right )}{2 c d n \log ^2\left (c x^n\right )} \]
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\[\int \frac {\left (d x \right )^{n -1}}{\ln \left (c \,x^{n}\right )^{3}}d x\]
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none
Time = 0.30 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.09 \[ \int \frac {(d x)^{-1+n}}{\log ^3\left (c x^n\right )} \, dx=-\frac {{\left (n \log \left (x\right ) + \log \left (c\right ) + 1\right )} d^{n - 1} x^{n} - \frac {{\left (n^{2} \log \left (x\right )^{2} + 2 \, n \log \left (c\right ) \log \left (x\right ) + \log \left (c\right )^{2}\right )} d^{n - 1} {\rm Ei}\left (n \log \left (x\right ) + \log \left (c\right )\right )}{c}}{2 \, {\left (n^{3} \log \left (x\right )^{2} + 2 \, n^{2} \log \left (c\right ) \log \left (x\right ) + n \log \left (c\right )^{2}\right )}} \]
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\[ \int \frac {(d x)^{-1+n}}{\log ^3\left (c x^n\right )} \, dx=\int \frac {\left (d x\right )^{n - 1}}{\log {\left (c x^{n} \right )}^{3}}\, dx \]
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\[ \int \frac {(d x)^{-1+n}}{\log ^3\left (c x^n\right )} \, dx=\int { \frac {\left (d x\right )^{n - 1}}{\log \left (c x^{n}\right )^{3}} \,d x } \]
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\[ \int \frac {(d x)^{-1+n}}{\log ^3\left (c x^n\right )} \, dx=\int { \frac {\left (d x\right )^{n - 1}}{\log \left (c x^{n}\right )^{3}} \,d x } \]
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Timed out. \[ \int \frac {(d x)^{-1+n}}{\log ^3\left (c x^n\right )} \, dx=\int \frac {{\left (d\,x\right )}^{n-1}}{{\ln \left (c\,x^n\right )}^3} \,d x \]
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