An improper integral : $\int_{0}^\infty {\ln(a^2+x^2)\over{b^2+x^2}}dx$

How to evaluate the following improper integral:$$\int_{0}^\infty {\ln(a^2+x^2)\over{b^2+x^2}}dx,$$ where $a,b>0$.

I tried to suppose $$f(a)=\int_0^\infty {\ln(a^2+x^2)\over{b^2+x^2}}dx,$$ based on the convergence theorem, and then I tried $${df(a)\over da}=\int_0^\infty {2a\over {(a^2+x^2)(b^2+x^2)}}dx = {\pi\over b(b+a)},$$and then $$f(a)={\pi\over b}\ln(b+a)+C,$$where $C$ is a constant, but I don't know how to find the constant $C$. Could anyone tell me that, and explain why? Or could anyone find other methods to evaluate the integral? If you could, please explain. Thanks.