Integrand size = 10, antiderivative size = 80 \[ \int \frac {1}{\log ^{\frac {5}{2}}\left (a x^n\right )} \, dx=\frac {4 \sqrt {\pi } x \left (a x^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{3 n^{5/2}}-\frac {2 x}{3 n \log ^{\frac {3}{2}}\left (a x^n\right )}-\frac {4 x}{3 n^2 \sqrt {\log \left (a x^n\right )}} \]
[Out]
Time = 0.02 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2334, 2337, 2211, 2235} \[ \int \frac {1}{\log ^{\frac {5}{2}}\left (a x^n\right )} \, dx=\frac {4 \sqrt {\pi } x \left (a x^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{3 n^{5/2}}-\frac {4 x}{3 n^2 \sqrt {\log \left (a x^n\right )}}-\frac {2 x}{3 n \log ^{\frac {3}{2}}\left (a x^n\right )} \]
[In]
[Out]
Rule 2211
Rule 2235
Rule 2334
Rule 2337
Rubi steps \begin{align*} \text {integral}& = -\frac {2 x}{3 n \log ^{\frac {3}{2}}\left (a x^n\right )}+\frac {2 \int \frac {1}{\log ^{\frac {3}{2}}\left (a x^n\right )} \, dx}{3 n} \\ & = -\frac {2 x}{3 n \log ^{\frac {3}{2}}\left (a x^n\right )}-\frac {4 x}{3 n^2 \sqrt {\log \left (a x^n\right )}}+\frac {4 \int \frac {1}{\sqrt {\log \left (a x^n\right )}} \, dx}{3 n^2} \\ & = -\frac {2 x}{3 n \log ^{\frac {3}{2}}\left (a x^n\right )}-\frac {4 x}{3 n^2 \sqrt {\log \left (a x^n\right )}}+\frac {\left (4 x \left (a x^n\right )^{-1/n}\right ) \text {Subst}\left (\int \frac {e^{\frac {x}{n}}}{\sqrt {x}} \, dx,x,\log \left (a x^n\right )\right )}{3 n^3} \\ & = -\frac {2 x}{3 n \log ^{\frac {3}{2}}\left (a x^n\right )}-\frac {4 x}{3 n^2 \sqrt {\log \left (a x^n\right )}}+\frac {\left (8 x \left (a x^n\right )^{-1/n}\right ) \text {Subst}\left (\int e^{\frac {x^2}{n}} \, dx,x,\sqrt {\log \left (a x^n\right )}\right )}{3 n^3} \\ & = \frac {4 \sqrt {\pi } x \left (a x^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{3 n^{5/2}}-\frac {2 x}{3 n \log ^{\frac {3}{2}}\left (a x^n\right )}-\frac {4 x}{3 n^2 \sqrt {\log \left (a x^n\right )}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.04 \[ \int \frac {1}{\log ^{\frac {5}{2}}\left (a x^n\right )} \, dx=-\frac {2 x \left (a x^n\right )^{-1/n} \left (2 n \Gamma \left (\frac {1}{2},-\frac {\log \left (a x^n\right )}{n}\right ) \left (-\frac {\log \left (a x^n\right )}{n}\right )^{3/2}+\left (a x^n\right )^{\frac {1}{n}} \left (n+2 \log \left (a x^n\right )\right )\right )}{3 n^2 \log ^{\frac {3}{2}}\left (a x^n\right )} \]
[In]
[Out]
\[\int \frac {1}{\ln \left (a \,x^{n}\right )^{\frac {5}{2}}}d x\]
[In]
[Out]
Exception generated. \[ \int \frac {1}{\log ^{\frac {5}{2}}\left (a x^n\right )} \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
\[ \int \frac {1}{\log ^{\frac {5}{2}}\left (a x^n\right )} \, dx=\int \frac {1}{\log {\left (a x^{n} \right )}^{\frac {5}{2}}}\, dx \]
[In]
[Out]
\[ \int \frac {1}{\log ^{\frac {5}{2}}\left (a x^n\right )} \, dx=\int { \frac {1}{\log \left (a x^{n}\right )^{\frac {5}{2}}} \,d x } \]
[In]
[Out]
\[ \int \frac {1}{\log ^{\frac {5}{2}}\left (a x^n\right )} \, dx=\int { \frac {1}{\log \left (a x^{n}\right )^{\frac {5}{2}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {1}{\log ^{\frac {5}{2}}\left (a x^n\right )} \, dx=\int \frac {1}{{\ln \left (a\,x^n\right )}^{5/2}} \,d x \]
[In]
[Out]