DIRK solvers are rather thoroughly investigated in the recent paper

Without any special needs, I found the suggested default solver â€śESDIRK4(3)6L[2]SAâ€ť pretty good.
Speed greatly depends on how fast the underlying implicit equation can be solved (by Newton?) and how good the adaptive time step control can be tuned.

I do atmospheric chemical kinetics involving 10,000s ODEs, which are very stiff. For this problem, one of the best methods is CVODE BDF with the banded matrix solver. CVODE is written in C but has an excellent Fortran interface.