What is the distribution of $\int_0^t e^{(W_s-s/2)}ds$

Question

I knew that the $\int_0^t e^{(W_s-s/2)}dW_s$ simply follows the distribution of $e^{(Wt-t/2)} - 1$. But how about changing the $dW_s$ to $ds$? how can I solve $\int_0^t e^{(W_s-s/2)}ds$? I tried to build a $F(s,W_s)$ based on Ito's lemma, but does not make it because it always gives me $F_s + \frac{1}{2}F_{xx}=0$. So how can I solve it? Thanks!