I should know this as I did compression algorithms for my dissertation.. but I've forgotten most of it now as it seems so long ago and I've had no use for the subject since.
The only things I remember these days are tearing my hair out trying to code my own algorithm.. how I began to hate Huffman coding & trees .. and about the only use I've had since is how knowledge in that area aided understanding of the error protection & correction algorithms used by DSL.
TBH I didn't enjoy it we had limited choice for dissertation subjects and to this day I'm still not sure what they expected from us other than probably attempt to come up with some algorithm which was better than the tried and trusted algorithms which were already out there.
Anyhows back on topic, rar is more efficient than zip for file data. I have the comparison data somewhere up in the loft in a pile of folders. Quite often though with various algorithms there is a trade-off with archive time-v-storage but I cba to get deep into the topic again. (Win)rar is also proprietary and not sure if it will work on all O/S. My copy of it must date back to the days I did my degree so not sure how its moved on since other than these days then you must pay for it and they nolonger offer trial or cutdown versions.
There's one more alternative - 7z. 7zip combines numerous archiving algorithms and reports to be more efficient than RAR but tbh I don't know that much about it and have never used it as it is newer than the other 2 more common formats.
At the end of the day zip tends to remain most popular as it does a good job, is supported across all platforms and is free.