Problem: pooling confidence intervals when calculating SIRs?

Question

I try to pool my confidence intervals for SIRs after using imputed datasets. A reprex with a simple example is below. The fraction of missing in my dataset was 10% and I made 10 multiple imputed datasets, leading to 10 SIRS with 10 confidence intervals.

library(popEpi)
data(popmort)
data(sire)
c <- lexpand( sire, status = status, birth = bi_date, exit = ex_date, entry = dg_date,
              breaks = list(per = 1950:2013, age = 1:100, fot = c(0,10,20,Inf)),
              aggre = list(fot, agegroup = age, year = per, sex) )
## SMR due other causes: status = 2
sr <- sir( coh.data = c, coh.obs = 'from0to2', coh.pyrs = 'pyrs',
           ref.data = popmort, ref.rate = 'haz',
           adjust = c('agegroup', 'year', 'sex'),
           EAR = T)

sr

#E.g., second imputation
sire2 <- sire
sire2$status[1:1000] <- 1

c <- lexpand( sire2, status = status, birth = bi_date, exit = ex_date, entry = dg_date,
              breaks = list(per = 1950:2013, age = 1:100, fot = c(0,10,20,Inf)),
              aggre = list(fot, agegroup = age, year = per, sex) )
## SMR due other causes: status = 2
sr2 <- sir( coh.data = c, coh.obs = 'from0to2', coh.pyrs = 'pyrs',
           ref.data = popmort, ref.rate = 'haz',
           adjust = c('agegroup', 'year', 'sex'),
           EAR = T)

sr2

sr_bind <- rbind(sr,sr2)

I tried to find a function with Rubin's rules to pool confidence intervals, but I could not find anything that would help in this specific example.