If $f\left(x\right)=\frac{\log x}{x^3}$, then its antiderivative $F\left(x\right)$

given by

$F\left(x\right)=\int \frac{\log x}{x^3}dx=\log x\int \frac{1}{x^3}dx-$$\int \frac{1}{x}\left(\int \frac{1}{x^3}dx\right)dx$ 

$=-\frac{1}{2x^2}\log x+\frac{1}{2}\int \frac{1}{x^3}dx+\text{ Constant }$

$\begin{array}{l}=-\frac{1}{2x^2}\log x-\frac{1}{4x^2}+C\\ =-\frac{1}{2x^2}\left[\log x+\frac{1}{2}\log e\right]+C\\ =-\frac{1}{2x^2}\log x\sqrt{e}+C\end{array}$