We have a variety of options when it comes to compression and encryption in SQL Server. When using both compression and encryption you have to understand how each of these work and when they wil lwork together and when they won’t be able to work together to make using both technologies useful.
The trick to making compression and encryption work together is to ensure that the compression is done first and the data encryption is done second. This is most easily done by using TDE for compression and page level compression for data compression. This is becaues when using these two technologies, no matter in which order you have enabled them, SQL Server will compress the data first and encrypt the data second. This even happens if you have a database which is encrypted with TDE and you then enable data compression on the tables. This is because when the data is compressed it is rewritten as compressed data and then encrypted post compression.
Using application level encryption you can still compress data using native data compression feature of SQL Server, however the amount of data compression that you will typically get in this situation will be much less than by using TDE and data compression. The same applies if you use TDE and then backup the database using native (or third party) backup compression. This is because when backups of a TDE encrypted database are taken the database pages are not decrypted when backed up. They are backed up in the same encrypted state that they are normally in, then compressed. By it’s nature encrypted data is very unique so data compression doesn’t do much good against encrypted data.