Here is how to formulate x\log|I+ Y(Y+xI)^{-1}|, x \geq 0,Y \in S^{+}, Y+xI \in S^{++}
or equivalently, x\log|I+ (Y+xI)^{-1}Y|, x \geq 0,Y \in S^{+}, Y+xI \in S^{++}, which are jointly concave in x and Y.
Equivalence of the two problems holds due to Y and (Y+xI)^{-1} commuting due to being diagnoliizable and having the same eigenvectors.
x*log(det(eye(n) + Y*inv(x*eye(n)+Y))) and
x*log(det(eye(n) + inv(x*eye(n)+Y)*Y))
can both be formulated as
-2*quantum_rel_entr(x*eye(n)+Y,x*eye(n)+2*Y) - quantum_rel_entr(x*eye(n)+2*Y,x*eye(n)+Y)
which turns out to be the quantum (matrix) analog of the scalar formulation
x*log(1+y/(x+y)) = -2*rel_entr(x+y,x+2*y) - rel_entr(x+2*y,x+y)
from Xlog( 1+ Y/(X+Y) ): DCP rules, and build-in functions in cvx