Is it worthwhile to use the formulas of a stratified two stage sample when computing a confidence interval for a proportion? Or can one be satisfied by stating that "stratification compensates multistage sampling, since the former yields a smaller standard error and the latter yields a bigger standard error than in the case of a simple random sample" and therefore just use the formulas of a simple random sample?

If it is worthwhile to use the extended formulas, can anyone give me some references of books or papers with these formulas (for proportions)?