I want to solve an optimization that minimize log_det( I + inv(X) ) under the semidefinite constraint and some linear constraints to X. For the background of this problem, see this reference.

As mentioned in this reference, though log_det( I + inv(X) ) is a convex function of X, numerically solving the problem is not easy.

I tried to express the function in CVX as log_det( I + inv(X) ) and log_det( X + I ) - log_det( X ). However, CVX cannot accept these expressions.

I want to know whether CVX is able to handle the function. If not, is there any other softwares that can handle the function?

The trick in the scalar case linked by @Mark_L_Stone is to rewrite the argument to the log-function as a harmonic mean, so the inversion can be used to flip the sign of the log-function, i.e.:
t >= log(1 + 1/x) = log(1 / (x / (x+1)) ) = log(1 / (1-1/(x+1)) )
so that it can be rewritten:
-t <= log( 1 - 1/(x+1) )
and then you just substitute out 1/(x+1) for a dummy variable using a quadratic cone.

This generalize to the matrix case step-by-step as follows:
t >= log(det( I + inv(X) )) = log(det( inv(X*inv(X+I)) ) = log(det( inv(I - inv(X+I)) ))
so that it can be rewritten:
-t <= log(det( I - inv(X+I) ))
but I am missing a matrix-equivalent for substituting out inv(X+I).

is equivalent to -t \leq \log(\det( I - {\rm inv}(X+I) ))

is equivalent to -t \leq \log(\det( I - U )) U \succeq {\rm inv}(X + I)

is equivalent to -t \leq \log(\det( I - U )) \left[\begin{array}{cc}X+I & I\\I & U\end{array}\right] \succeq 0

The first step follows from the scalar to matrix generalization above which you can prove more rigidly. The second step follows by applying \log(\det( I - U )) \leq \log(\det( I - V )) on I \succeq U \succeq V to our case with V = {\rm inv}(X + I). The last step is the Schur complement lemma.

Thanks @hfriberg. I implemented your idea in CVX and it worked.
The equivalent optimization problem is

\begin{align}
\mathrm{max} \ & \ \log \det ( I - U ) \\
\mathrm{s.t.} \ & \ X \succeq 0 \ \text{and some linear constrains to} \ X \\
& \ I - U \succeq 0 , \\
& \ \left[
\begin{array}{cc}
X + I & I \\ I & U
\end{array}
\right] \succeq 0
\end{align}

Here, I add I - U \succeq 0 to ensure continuity of the function.
At first I also considered to apply Schur complement and introduce a variable V \succeq X ^ {-1}, but the optimization became minimizing a log_det function, will is concave. The first equivalence in the trick converts the target to be maximing a log_det function. That’s really ingenious

The I - U \succeq 0 part is redundant in cvx and may lower performance. Arguments to log_det are implicitly constrained this way already, as you’ll find in the cvx reference:

When used inside a CVX specification, log_det constrains its argument to be symmetric (if real) or Hermitian (if complex) and positive definite.