While studying analytic number theory, the following indefinite integrals fascinated me the most:
$$\int_{0}^{\infty}\dfrac{\cos(nx)}{(x^2 + 1)^{z + 1}}\log x\,dx + \dfrac{\pi}{2}\int_{0}^{\infty}\dfrac{\sin(nx)}{(x^2 + 1)^{z + 1}}\,dx \tag{1}\label{1}$$
$$\int_{0}^{\infty}\dfrac{\sin(nx)}{(x^2 + 1)^{z + 1}}\log x\,dx +\dfrac{1}{\pi}\int_{0}^{\infty}\dfrac{\cos(nx)}{(x^2 + 1)^{z + 1}}\log^2 x\,dx \tag{2}\label{2}$$
where $z \in \mathbb{N}$. I'm looking for a nice closed form of $\eqref{1}$ and $\eqref{2}$ (however I'm not totally sure if there even exists one, but according to my intuition and references there should exist one).
Is it possible to find a closed form (obviously which is not in terms of integrals)?
I have tried integrating, using contour integration and residue theorem but it didn't get me to any nice closed form. I'm looking for a neat detailed answer with proper references for the tools used.
Thanks in advance