Let $\mathbf{\Phi}(t)$ and $\mathbf A(t)$ be matrices satisfying the differential equation $$ \mathbf{\Phi}'(t)=\mathbf A(t)\mathbf{\Phi}(t)\ . $$ If I am not mistaken, if $\mathbf A$ and its integral commutes, i.e.
$$ \mathbf A(t) \cdot \int_0^t \mathbf A(\tau)d\tau = \int_0^t \mathbf A(\tau)d\tau \ \cdot \mathbf A(t)\ , $$
then $\mathbf \Phi(t)$ can be written in the form $$\mathbf \Phi(t)=\exp\left( \int_0^t \mathbf A(\tau)d\tau \right)\ .$$
Evidently, such condition is satisfied when $\mathbf A$ is a symmetric (or Hermitian). I want to know if there's a similar theory when the highlighted condition is not satisfied.
In particular, I am interested in the case where $\mathbf A(t)$ is anti Hermitian and $\int_0^t \mathbf A(\tau)d\tau $ is Hermitian. Any help or references would be highly appreciated.
Edit: As some users had mentioned, some of my claims might be incorrect. I must apologize for that. Let me summarize my question to avoid misinterpreting and confusion
Is there a general method to find $\mathbf \Phi(t)$ when $A(t)$ and $B(t)=\int_0^t A(\tau) d\tau$ do not commute?
The exponential works under the very strong hypothesis that all of the $A(t), t \in \mathbb{R}$ commute. I am not aware of a weaker general hypothesis than this.
Otherwise, the solution can be expressed as a Dyson series or a Magnus series. The former is common in physics, and more specifically in perturbation theory. The latter is arguably better, though (e.g. if the $A(t)$ lie in some Lie algebra then so do the terms of the Magnus series).