Integrate $\frac{\log(x^2+4)}{(x^2+1)^2}$. 
Using residue calculus show that
$$\int_0^{\infty}\frac{\log(x^2+4)}{(x^2+1)^2}dx=\frac{\pi}2\log 3-\frac{\pi}6.$$

I was thinking of using some keyhole or semi-circular contour here. But the problem is apart from poles at $x=-i$ and $x=i$, the logarithm has singularities when $x=\pm 4i$.
I consider $C_R$, a semicircle contour oriented clockwise with radius $R$ centered at origin. The semicircle resides in the lower half plane, so that it encloses $x=-i$ as a pole.  I set
$$\int_{C_R} \frac{\log(2-ix)}{(x^2+1)^2}dx$$
But it seems like it leads to wrong answer.

One thing you could do to avoid a keyhole is to denote
$$I[a] = \int_0^\infty \frac{\log\left(a(x^2+1)+3\right)}{(x^2+1)^2}\:dx \implies I'[a] = \frac{1}{2}\int_{-\infty}^\infty \frac{1}{x^2+1}\cdot \frac{1}{ax^2+a+3}\:dx$$
then apply a semicircular contour and calculate the residues as normal.
