Integrand size = 14, antiderivative size = 27 \[ \int \frac {\log ^2\left (a x^n\right )^p}{x} \, dx=\frac {\log \left (a x^n\right ) \log ^2\left (a x^n\right )^p}{n (1+2 p)} \]
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Time = 0.02 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {15, 30} \[ \int \frac {\log ^2\left (a x^n\right )^p}{x} \, dx=\frac {\log \left (a x^n\right ) \log ^2\left (a x^n\right )^p}{n (2 p+1)} \]
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Rule 15
Rule 30
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \left (x^2\right )^p \, dx,x,\log \left (a x^n\right )\right )}{n} \\ & = \frac {\left (\log ^{-2 p}\left (a x^n\right ) \log ^2\left (a x^n\right )^p\right ) \text {Subst}\left (\int x^{2 p} \, dx,x,\log \left (a x^n\right )\right )}{n} \\ & = \frac {\log \left (a x^n\right ) \log ^2\left (a x^n\right )^p}{n (1+2 p)} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {\log ^2\left (a x^n\right )^p}{x} \, dx=\frac {\log \left (a x^n\right ) \log ^2\left (a x^n\right )^p}{n (1+2 p)} \]
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Time = 5.66 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.11
method | result | size |
derivativedivides | \(\frac {\ln \left (a \,x^{n}\right ) {\mathrm e}^{p \ln \left (\ln \left (a \,x^{n}\right )^{2}\right )}}{n \left (1+2 p \right )}\) | \(30\) |
default | \(\frac {\ln \left (a \,x^{n}\right ) {\mathrm e}^{p \ln \left (\ln \left (a \,x^{n}\right )^{2}\right )}}{n \left (1+2 p \right )}\) | \(30\) |
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Time = 0.31 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.41 \[ \int \frac {\log ^2\left (a x^n\right )^p}{x} \, dx=\frac {{\left (n \log \left (x\right ) + \log \left (a\right )\right )} {\left (n^{2} \log \left (x\right )^{2} + 2 \, n \log \left (a\right ) \log \left (x\right ) + \log \left (a\right )^{2}\right )}^{p}}{2 \, n p + n} \]
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\[ \int \frac {\log ^2\left (a x^n\right )^p}{x} \, dx=\int \frac {\left (\log {\left (a x^{n} \right )}^{2}\right )^{p}}{x}\, dx \]
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Exception generated. \[ \int \frac {\log ^2\left (a x^n\right )^p}{x} \, dx=\text {Exception raised: RuntimeError} \]
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Time = 0.31 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.78 \[ \int \frac {\log ^2\left (a x^n\right )^p}{x} \, dx=\frac {{\left (n \log \left (x\right ) \mathrm {sgn}\left (\log \left (a x^{n}\right )\right ) + \log \left (a\right ) \mathrm {sgn}\left (\log \left (a x^{n}\right )\right )\right )}^{2 \, p + 1}}{n {\left (2 \, p + 1\right )} \mathrm {sgn}\left (\log \left (a x^{n}\right )\right )} \]
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Time = 1.48 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {\log ^2\left (a x^n\right )^p}{x} \, dx=\frac {\ln \left (a\,x^n\right )\,{\left ({\ln \left (a\,x^n\right )}^2\right )}^p}{n\,\left (2\,p+1\right )} \]
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