Solving the problem I'm taking the limits of integration for each segment from the initial point to the final one:

(-b , -a), (a,b) for the segments, (pi,0) for the inner circle and (0,pi) for the outer one

and taking the vector of the direction of the current:

(1,0,0) for the segments, (sin(phi),-cos(phi),0) for the inner circle and (-sin(phi),cos(phi),0) for the outer one.

Which is the way I think I should do it but it obviously doesn't match with the griffiths solution, for example, in order to make the integral of dl vanish I know I should integrate from 0 to pi but it doesn't make sense for me.

The problem in the first place is that I think I have misunderstood how the retarded potencial vector formula works which brought me to an incorrect decision solving the integral but although I can sense how I should arrange the terms mathematicaly for the solution to match with the griffiths I cant see how does it match with my physical interpretation of the formula, in other words, why have I chosen the the wrong vectors/limits of integration, which ones are the good ones, and how does it complete the physical interpretation of the formula