Evaluating $\int_{0}^\infty \frac{\log x \, dx}{\sqrt x(x^2+a^2)^2}$ using contour integration I need your help with this integral:
$$\int_{0}^\infty \frac{\log x \, dx}{\sqrt x(x^2+a^2)^2}$$
where $a>0$. I have tried some complex integration methods, but none seems adequate for this particular one.
Is there a specific method for this kind of integrals? What contour should I use?

Hint:
$$\forall a>0,\; s\in(-1,1),\qquad J(a,s) = \int_{0}^{+\infty}\frac{x^{s}}{x^2+a^2}\,dx = \frac{\pi}{2\sin\frac{\pi s}{2}}\,a^{s-1}.$$
Consider the partial derivatives with respect to $a$ and $s$, then evaluate at $s=-\frac{1}{2}$.
