I need to show the following inequality: Let $A,B$ be $d\times d$ symmetric matrices and define $|A|:=Tr(A^\top A)$. Then, the following inequality holds: $$ |(Tr B-\log\det (Id+B))-(Tr A-\log\det (Id+A))| \leq c|B-A|(|A|+|B-A|) $$ whenever $|A|,|B|\leq c^\prime$ and $c,c^\prime$ some positive constants.
There is also the hint to use the formula $\int\exp(-2^{-1}z^\top(Id+\epsilon x)z)dz=(2\pi)^{d/2}\det(Id+\epsilon x)^{-1/2}.$
I have tried writing $\log\det (Id+B)$ using this formula, but have a problem with $log\int$ since we can not interchange this (as far as I know). What I do know, ist that
\begin{align*} |(Tr B-\log\det (Id+B))-(Tr A-\log\det (Id+A))| &= |Tr(B-A) - \log\det (Id+B)(Id+a)^{-1} |\\&\leq c_1* |B-A| + | \log\det (Id+B)(Id+a)^{-1}| \end{align*}
for some positive constant $c_1$ since the trace is a linear function and we can consider $|\cdot|$ as a norm.
I would be very greatful for any hints or reading material.