Integrand size = 14, antiderivative size = 61 \[ \int \frac {x^m}{\sqrt {\log \left (a x^n\right )}} \, dx=\frac {\sqrt {\pi } x^{1+m} \left (a x^n\right )^{-\frac {1+m}{n}} \text {erfi}\left (\frac {\sqrt {1+m} \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{\sqrt {1+m} \sqrt {n}} \]
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Time = 0.03 (sec) , antiderivative size = 61, 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} \[ \int \frac {x^m}{\sqrt {\log \left (a x^n\right )}} \, dx=\frac {\sqrt {\pi } x^{m+1} \left (a x^n\right )^{-\frac {m+1}{n}} \text {erfi}\left (\frac {\sqrt {m+1} \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{\sqrt {m+1} \sqrt {n}} \]
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Rule 2211
Rule 2235
Rule 2347
Rubi steps \begin{align*} \text {integral}& = \frac {\left (x^{1+m} \left (a x^n\right )^{-\frac {1+m}{n}}\right ) \text {Subst}\left (\int \frac {e^{\frac {(1+m) x}{n}}}{\sqrt {x}} \, dx,x,\log \left (a x^n\right )\right )}{n} \\ & = \frac {\left (2 x^{1+m} \left (a x^n\right )^{-\frac {1+m}{n}}\right ) \text {Subst}\left (\int e^{\frac {(1+m) x^2}{n}} \, dx,x,\sqrt {\log \left (a x^n\right )}\right )}{n} \\ & = \frac {\sqrt {\pi } x^{1+m} \left (a x^n\right )^{-\frac {1+m}{n}} \text {erfi}\left (\frac {\sqrt {1+m} \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{\sqrt {1+m} \sqrt {n}} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00 \[ \int \frac {x^m}{\sqrt {\log \left (a x^n\right )}} \, dx=\frac {\sqrt {\pi } x^{1+m} \left (a x^n\right )^{-\frac {1+m}{n}} \text {erfi}\left (\frac {\sqrt {1+m} \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{\sqrt {1+m} \sqrt {n}} \]
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\[\int \frac {x^{m}}{\sqrt {\ln \left (a \,x^{n}\right )}}d x\]
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\[ \int \frac {x^m}{\sqrt {\log \left (a x^n\right )}} \, dx=\int { \frac {x^{m}}{\sqrt {\log \left (a x^{n}\right )}} \,d x } \]
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\[ \int \frac {x^m}{\sqrt {\log \left (a x^n\right )}} \, dx=\int \frac {x^{m}}{\sqrt {\log {\left (a x^{n} \right )}}}\, dx \]
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\[ \int \frac {x^m}{\sqrt {\log \left (a x^n\right )}} \, dx=\int { \frac {x^{m}}{\sqrt {\log \left (a x^{n}\right )}} \,d x } \]
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\[ \int \frac {x^m}{\sqrt {\log \left (a x^n\right )}} \, dx=\int { \frac {x^{m}}{\sqrt {\log \left (a x^{n}\right )}} \,d x } \]
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Timed out. \[ \int \frac {x^m}{\sqrt {\log \left (a x^n\right )}} \, dx=\int \frac {x^m}{\sqrt {\ln \left (a\,x^n\right )}} \,d x \]
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