Integrand size = 14, antiderivative size = 51 \[ \int \frac {x^2}{\sqrt {\log \left (a x^n\right )}} \, dx=\frac {\sqrt {\frac {\pi }{3}} x^3 \left (a x^n\right )^{-3/n} \text {erfi}\left (\frac {\sqrt {3} \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{\sqrt {n}} \]
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Time = 0.03 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {2347, 2211, 2235} \[ \int \frac {x^2}{\sqrt {\log \left (a x^n\right )}} \, dx=\frac {\sqrt {\frac {\pi }{3}} x^3 \left (a x^n\right )^{-3/n} \text {erfi}\left (\frac {\sqrt {3} \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{\sqrt {n}} \]
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Rule 2211
Rule 2235
Rule 2347
Rubi steps \begin{align*} \text {integral}& = \frac {\left (x^3 \left (a x^n\right )^{-3/n}\right ) \text {Subst}\left (\int \frac {e^{\frac {3 x}{n}}}{\sqrt {x}} \, dx,x,\log \left (a x^n\right )\right )}{n} \\ & = \frac {\left (2 x^3 \left (a x^n\right )^{-3/n}\right ) \text {Subst}\left (\int e^{\frac {3 x^2}{n}} \, dx,x,\sqrt {\log \left (a x^n\right )}\right )}{n} \\ & = \frac {\sqrt {\frac {\pi }{3}} x^3 \left (a x^n\right )^{-3/n} \text {erfi}\left (\frac {\sqrt {3} \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{\sqrt {n}} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00 \[ \int \frac {x^2}{\sqrt {\log \left (a x^n\right )}} \, dx=\frac {\sqrt {\frac {\pi }{3}} x^3 \left (a x^n\right )^{-3/n} \text {erfi}\left (\frac {\sqrt {3} \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{\sqrt {n}} \]
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\[\int \frac {x^{2}}{\sqrt {\ln \left (a \,x^{n}\right )}}d x\]
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Exception generated. \[ \int \frac {x^2}{\sqrt {\log \left (a x^n\right )}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {x^2}{\sqrt {\log \left (a x^n\right )}} \, dx=\int \frac {x^{2}}{\sqrt {\log {\left (a x^{n} \right )}}}\, dx \]
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\[ \int \frac {x^2}{\sqrt {\log \left (a x^n\right )}} \, dx=\int { \frac {x^{2}}{\sqrt {\log \left (a x^{n}\right )}} \,d x } \]
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\[ \int \frac {x^2}{\sqrt {\log \left (a x^n\right )}} \, dx=\int { \frac {x^{2}}{\sqrt {\log \left (a x^{n}\right )}} \,d x } \]
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Timed out. \[ \int \frac {x^2}{\sqrt {\log \left (a x^n\right )}} \, dx=\int \frac {x^2}{\sqrt {\ln \left (a\,x^n\right )}} \,d x \]
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