Let $X$ be smooth algebraic variety over $\mathbb C$ and $D_1, D_2$ are snc divisors such that $D_1\cup D_2$ is also snc.

Is it true that there is an exact sequence

$$H^*(X, D_1\cup D_2)\to H^*(X, D_1)\to H^*(D_2, D_1\cap D_2)\to H^{*+1}(X, D_1\cup D_2)\to\dots$$

The motivation is that it is standard long exact sequence for the pair $(Y, D_2)$ where $Y=(X, D_1)$.

If it is not true, then what else exact sequence there is to compute $H^*(X, D_1\cup D_2)$?

The cohomology is singular cohomology with rational coefficients.