I would like to find a simple way to compute the inverse of a matrix $M\in\mathbb{R}^{(N+1)\times(N+1)}$ given by
$$M=\begin{bmatrix}3D_1 & 0 & 0 & \dots & 0 & 1 \\ 0 & 3D_2 & 0 & \dots & 0 & 1 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & 0 & \dots & 3D_N & 1 \\ 1 & 1 &1 & \dots & 1 & 0 \end{bmatrix},$$
where $D_1, \dots, D_N$ are strictly positive constant terms.
On Wikipedia, I found something like the following. However, I am looking for something simpler.
Apply the Woodbury matrix identity $$ \left(A + UCV \right)^{-1} = A^{-1} - A^{-1}U \left(C^{-1} + VA^{-1}U \right)^{-1} VA^{-1} $$ with $$ A = \pmatrix{3D_1 \\ &\ddots \\ && 3D_n\\ &&& -1}, \quad U = \pmatrix{1&0\\\vdots&\vdots\\1&0\\1&1}, \quad V = \pmatrix{0&\cdots&0&1\\1&\cdots&1&0}, $$ and $C = I$ (where $I$ denotes the $2\times 2$ identity matrix).