calculus - Integral $int_0^infty frac{log^2 x cos ax}{x^n-1}dx$

Hi I am trying to calculate
$$I:=\int\limits_0^\infty \frac{\log^2 x \cos (ax)}{x^n-1}\mathrm dx,\quad \Re(n)>1, \, a\in \mathbb{R}.$$
Note if we set $a=0$ we get a similar integral given by
$$\int\limits_0^\infty \frac{\log^2 x }{x^n-1}\mathrm dx=-\frac{2\pi^3\cot(\pi/n)\csc^2(\pi/n)}{n^3},\quad \Re(n)>1.$$
I was trying to write I as
$$I=\Re \bigg[\int\limits_0^\infty \frac{ e^{i ax}\log^2 x}{x^n-1}\mathrm dx\bigg]=\Re\bigg[\int\limits_\infty^0\frac{e^{iax}\log^2 x}{1-x^n}\mathrm dx\bigg]=\Re\bigg[\int\limits_\infty^0e^{iax}\log^2 x\sum_{m=0}^\infty x^{nm} \mathrm dx\bigg].$$
But was unsure of where to go from here. How can we calculate $I$? It is clear that this method is not going to work.