Integrand size = 14, antiderivative size = 59 \[ \int \frac {\sqrt {\log \left (a x^n\right )}}{x^2} \, dx=\frac {\sqrt {n} \sqrt {\pi } \left (a x^n\right )^{\frac {1}{n}} \text {erf}\left (\frac {\sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{2 x}-\frac {\sqrt {\log \left (a x^n\right )}}{x} \]
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Time = 0.04 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2342, 2347, 2211, 2236} \[ \int \frac {\sqrt {\log \left (a x^n\right )}}{x^2} \, dx=\frac {\sqrt {\pi } \sqrt {n} \left (a x^n\right )^{\frac {1}{n}} \text {erf}\left (\frac {\sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{2 x}-\frac {\sqrt {\log \left (a x^n\right )}}{x} \]
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Rule 2211
Rule 2236
Rule 2342
Rule 2347
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {\log \left (a x^n\right )}}{x}+\frac {1}{2} n \int \frac {1}{x^2 \sqrt {\log \left (a x^n\right )}} \, dx \\ & = -\frac {\sqrt {\log \left (a x^n\right )}}{x}+\frac {\left (a x^n\right )^{\frac {1}{n}} \text {Subst}\left (\int \frac {e^{-\frac {x}{n}}}{\sqrt {x}} \, dx,x,\log \left (a x^n\right )\right )}{2 x} \\ & = -\frac {\sqrt {\log \left (a x^n\right )}}{x}+\frac {\left (a x^n\right )^{\frac {1}{n}} \text {Subst}\left (\int e^{-\frac {x^2}{n}} \, dx,x,\sqrt {\log \left (a x^n\right )}\right )}{x} \\ & = \frac {\sqrt {n} \sqrt {\pi } \left (a x^n\right )^{\frac {1}{n}} \text {erf}\left (\frac {\sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{2 x}-\frac {\sqrt {\log \left (a x^n\right )}}{x} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.10 \[ \int \frac {\sqrt {\log \left (a x^n\right )}}{x^2} \, dx=-\frac {2 \log \left (a x^n\right )+n \left (a x^n\right )^{\frac {1}{n}} \Gamma \left (\frac {1}{2},\frac {\log \left (a x^n\right )}{n}\right ) \sqrt {\frac {\log \left (a x^n\right )}{n}}}{2 x \sqrt {\log \left (a x^n\right )}} \]
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\[\int \frac {\sqrt {\ln \left (a \,x^{n}\right )}}{x^{2}}d x\]
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Exception generated. \[ \int \frac {\sqrt {\log \left (a x^n\right )}}{x^2} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {\sqrt {\log \left (a x^n\right )}}{x^2} \, dx=\int \frac {\sqrt {\log {\left (a x^{n} \right )}}}{x^{2}}\, dx \]
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\[ \int \frac {\sqrt {\log \left (a x^n\right )}}{x^2} \, dx=\int { \frac {\sqrt {\log \left (a x^{n}\right )}}{x^{2}} \,d x } \]
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\[ \int \frac {\sqrt {\log \left (a x^n\right )}}{x^2} \, dx=\int { \frac {\sqrt {\log \left (a x^{n}\right )}}{x^{2}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {\log \left (a x^n\right )}}{x^2} \, dx=\int \frac {\sqrt {\ln \left (a\,x^n\right )}}{x^2} \,d x \]
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