I use Wipe Out Products called "Accelerator" and "Patch Out". Usually takes 3 applications to get clean. Once in awhile, I use the sprayfoam wipeout on a rifle, but I usually only use it for handguns.
Apply 2 patches of accelerator, then 2 patches of patch out. Sit 15 minutes then dry patch out.
After the first application with patches most of the big mess will be gone, so swap your jags for 2 wool mops. Wool mops are way nicer than cotton - they retain loft, generate suds better, dry quicker and last longer. I like to set up 2 rods - one with a mop for accelerator and the other with a mop for patch out.
Mop in the accelerator then mop in the patchout. Run the patch out mop up and down a bit, to work up suds. Sit 30 minutes to an hour, then dry patch out. Repeat mops again and dry patch to complete. If the gun will sit awhile before the next shoot, finish with a lightly oiled patch.
If you're not shooting a hand lapped barrel, you may need to scrub a bit with a brush after mopping, but don't spend much effort. Let the chemical do the work.
I spend the waiting times going through my targets, prepping brass and other reloading chores....
If that's not good enough, try boretech eliminator, witches brew or slip2000 copper cutter. These are all very good copper removers. Just don't leave them in the barrel long, or bad things can happen.