What prescription med were you using? While some dogs can eventually develop liver problems with some anti-inflammatories, it's pretty rare. I've used deramaxx long-term on two dogs with severe arthritis and they never had any liver function issues.
Dogs can metabolize aspirin, but many vets don't recommend it because it can cause severe stomach problems. My vet had one dog develop a bleeding ulcer after only a few days of aspirin therapy. After that, he never recommended it again.
Ibuprofin (motrin, advil) is not at all safe for dogs. Tylenol is OK, but it's really not effective for the severe inflammation of arthritis.
I would go ahead and use the prescription meds, and have liver function tests run annually to make sure everything is OK in that department.
Please do try a Glucosamine/Chondroitin supplement daily. It might have very very good results. I know for a fact it has with my dog. She had a broken leg years ago (in her past life before I rescued her) Since then she has limped, and 'hitched' that leg from time to time, especially when the weather turned damp and cold. I started her with the supplement a few months ago in the Spring. I have not looked back!
For testimony, she is leaping over hurdles 3-4 feet tall! She did that same thing today, chasing a flying bird! She took my breath away! She could not have done that before without suffering for it.
I use ordinary supplements in capsule form, on sale for Humans. They are cheaper. But I get the exact dosage for her weight. She weighs 28lbs and gets 500mg Glucosamine, and 200mg Chondrotin daily. That's the max. dose for her weight.