I am trying to find a good way to simplify (or even solve) the following integral:
$$ \int_0^{-\log \varepsilon} \cosh^3r \frac{\partial^2}{\partial r^2} \log (\det E) dr $$
where $r > 0$ is a real variable, and $0 < \varepsilon << 1$ is a small parameter, and $E = E(r)$ denotes a family of positive definite (Hermitian) matrices.
There are several thoughts that I have in mind, but I am not sure whether these would turn out to be fruitful. For example, one thing that might be helpful is to use the identity $$ \frac{\partial}{\partial r} \log \det E = \text{tr}(E^{-1}\frac{\partial}{\partial r} E)\,, $$ Alternatively one could try to (formally) do the integration by expanding $\cosh^3 r$ using the series expansion $$ \cosh r = \sum^\infty_{n = 0} \frac{r^{2n}}{(2n)!} $$ Yet another approach might be to use $2\cosh r = e^r + e^{-r}$ is symmetric, so maybe one can divide the integral into two parts?