stuck in a definite integral problem

stuck in a definite integral problem

$$\int \limits
_{0}^{\infty}\ln\left({x+\frac{1}{x}}\right)\cdot\frac{dx}{1+x^2}$$ we are
asked to solve this definite integral so here's what i did $$\int
\limits_{0}^{\infty}\ln \left({\frac{x^2
+1}{x}}\right)\cdot\frac{dx}{1+x^2} = \int\limits_{0}^{\infty}\ln
\left(x^2+1\right)\cdot\frac{dx}{1+x^2} - \int \limits_{0}^{\infty}\ln
(x).\frac{dx}{1+x^2}$$ now how to proceed after that ? should i intregrate
both seperate functions by substituting $(x^2+1)$ and what should i
susbtitute in other integral , by-parts integration is making troubles
when substituting $\infty$ , now what to do ?