Background
Recently I came across a matrix differential equation of this form:
$\mathbf{X}'(t) = \exp(\mathbf{D}t)[\mathbf{S} + \exp(\mathbf{D}t)]^{-1}$,
where $t \in \mathbb{R}$ and $\mathbf{X}, \mathbf{S}, \mathbf{D} \in \mathbb{R}^{n \times n}$. Further, $\mathbf{S}$ is symmetric while $\mathbf{D}$ is diagonal (both constant with respect to $t$). If $n = 1$, the resulting scalar equation would be easy to solve:
$X(t) = \frac{\ln[S + \exp(Dt)]}{D} + C$,
where now all variables are scalar constants/functions. However, for greater integer values of $n$, my understanding is $\mathbf{X}(t) = \mathbf{D}^{-1}\ln[\mathbf{S} + \exp(\mathbf{D}t)]$ is no longer correct because the chain rule cannot be applied to matrix-by-scalar derivatives. Can I conclude then that the matrix ODE has no analytical solution because, if it did, it would be inconsistent with the solution to the scalar equation?
Additional Information
I have also checked numerically that neither
$\mathbf{X}(t) = \mathbf{D}^{-1}\ln[\mathbf{S} + \exp(\mathbf{D}t)]$
nor
$\mathbf{X}(t) = \ln[\mathbf{S} + \exp(\mathbf{D}t)]\mathbf{D}^{-1}$
are solutions to the ODE for specific matrices, but I just used a simple forward Euler method so I'm not totally confident in my results.