You can see what to do in the scalar case at Can CVX solve this kind of function {x-log(1+0.01*x/(x+1))} - #15 by Mark_L_Stone
if this actually is concave for the matrix case, maybe (maybe!!) there is some way to do it using one or more of determinantal identities, Schur complement, or quantum functions in
or
https://qetlab.com/ .