Integrand size = 14, antiderivative size = 69 \[ \int \frac {1}{x^3 \log ^{\frac {3}{2}}\left (a x^n\right )} \, dx=-\frac {2 \sqrt {2 \pi } \left (a x^n\right )^{2/n} \text {erf}\left (\frac {\sqrt {2} \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{n^{3/2} x^2}-\frac {2}{n x^2 \sqrt {\log \left (a x^n\right )}} \]
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Time = 0.04 (sec) , antiderivative size = 69, 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 = {2343, 2347, 2211, 2236} \[ \int \frac {1}{x^3 \log ^{\frac {3}{2}}\left (a x^n\right )} \, dx=-\frac {2 \sqrt {2 \pi } \left (a x^n\right )^{2/n} \text {erf}\left (\frac {\sqrt {2} \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{n^{3/2} x^2}-\frac {2}{n x^2 \sqrt {\log \left (a x^n\right )}} \]
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Rule 2211
Rule 2236
Rule 2343
Rule 2347
Rubi steps \begin{align*} \text {integral}& = -\frac {2}{n x^2 \sqrt {\log \left (a x^n\right )}}-\frac {4 \int \frac {1}{x^3 \sqrt {\log \left (a x^n\right )}} \, dx}{n} \\ & = -\frac {2}{n x^2 \sqrt {\log \left (a x^n\right )}}-\frac {\left (4 \left (a x^n\right )^{2/n}\right ) \text {Subst}\left (\int \frac {e^{-\frac {2 x}{n}}}{\sqrt {x}} \, dx,x,\log \left (a x^n\right )\right )}{n^2 x^2} \\ & = -\frac {2}{n x^2 \sqrt {\log \left (a x^n\right )}}-\frac {\left (8 \left (a x^n\right )^{2/n}\right ) \text {Subst}\left (\int e^{-\frac {2 x^2}{n}} \, dx,x,\sqrt {\log \left (a x^n\right )}\right )}{n^2 x^2} \\ & = -\frac {2 \sqrt {2 \pi } \left (a x^n\right )^{2/n} \text {erf}\left (\frac {\sqrt {2} \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{n^{3/2} x^2}-\frac {2}{n x^2 \sqrt {\log \left (a x^n\right )}} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.96 \[ \int \frac {1}{x^3 \log ^{\frac {3}{2}}\left (a x^n\right )} \, dx=\frac {2 \left (-1+\sqrt {2} \left (a x^n\right )^{2/n} \Gamma \left (\frac {1}{2},\frac {2 \log \left (a x^n\right )}{n}\right ) \sqrt {\frac {\log \left (a x^n\right )}{n}}\right )}{n x^2 \sqrt {\log \left (a x^n\right )}} \]
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\[\int \frac {1}{x^{3} \ln \left (a \,x^{n}\right )^{\frac {3}{2}}}d x\]
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Exception generated. \[ \int \frac {1}{x^3 \log ^{\frac {3}{2}}\left (a x^n\right )} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {1}{x^3 \log ^{\frac {3}{2}}\left (a x^n\right )} \, dx=\int \frac {1}{x^{3} \log {\left (a x^{n} \right )}^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {1}{x^3 \log ^{\frac {3}{2}}\left (a x^n\right )} \, dx=\int { \frac {1}{x^{3} \log \left (a x^{n}\right )^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {1}{x^3 \log ^{\frac {3}{2}}\left (a x^n\right )} \, dx=\int { \frac {1}{x^{3} \log \left (a x^{n}\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{x^3 \log ^{\frac {3}{2}}\left (a x^n\right )} \, dx=\int \frac {1}{x^3\,{\ln \left (a\,x^n\right )}^{3/2}} \,d x \]
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