What is the integral of the matrix exponential
$$\int_0^{2 \pi} dt \exp(A \cos(t) + B \sin(t))$$
with matrix $A$ and $B$.
The commutator $$\left[A,B\right]\neq0$$ is non-vanishing.
I approached the problem so far with Zassenhaus formula (and alike formulas). Also, the power series expansion can be derived, which does not yield a familiar matrix function but resembles features of the zeroth order modified Bessel function.
Solving the integral $$\int_0^{2 \pi} dt \exp(A \cos(t))\exp(B \sin(t))$$ would be also of interest.
A simple second order accurate solution can be obtained by exploiting second order Lie-Trotter splitting of the matrix exponential: $$\int_0^{2 \pi} dt \exp\left[\epsilon \left(A \cos(t) + B \sin(t)\right)\right] \\ \approx \int_0^{2 \pi} dt \left[\exp(\epsilon A \cos(t))\exp(\epsilon B \sin(t))+\exp(\epsilon B \cos(t))\exp(\epsilon A \sin(t))\right]/2 + \mathcal{O}(\epsilon^3) \\ = 2 \pi I_0(\epsilon \sqrt{A^2 + B^2}) + \mathcal{O}(\epsilon^3) $$ I used power expansion for the proof and to the best of my knowledge this result is not published anywhere. Note that $\int_0^{2 \pi} dt \exp(A \cos(t) + B \sin(t)) = 2 \pi I_0(\epsilon \sqrt{A^2 + B^2}) $ for a vanishing commutator $\left[A,B\right]=0$.