Why is the solution to a non-homogenous linear ODE written in terms of a general fundamental solution and not a matrix exponential?

Why is the solution to a non-homogenous linear ODE written in terms of a general fundamental solution and not a matrix exponential? Generally, I see the solution to a non-homogenous linear ODE

$$
dot{x} = Ax + b(t)\
x(0) = x_0
$$

written as

$$
x(t) = Phi(t)Phi^{-1}(0)x_0 + int_0^t Phi(t)Phi^{-1}(tau)b(tau)dtau
$$

where

$$
Phi^prime(t)=APhi(t).
$$

At the same time, I thought $Phi(t)=mathbb{e}^{At}$, which means that $Phi(0)=I$ and the above could at least be simplified to

$$
x(t) = Phi(t)x_0 + int_0^t Phi(t)Phi^{-1}(tau)b(tau)dtau
$$

if not further to

$$
x(t) = mathbb{e}^{At}x_0 + mathbb{e}^{At}int_0^t (mathbb{e}^{Atau})^{-1}b(tau)dtau.
$$

Why is it, then, that the non-homgenous solution is written as originally stated in terms of $Phi$? Is there a change of basis or generalization that I’m not aware of, or is there a mistake in this simplication?