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Tutorial:
A 10 to 15 minute loosening-up period is always going to be appropriate. Take it from someone who is into advanced anal play (e.g., fisting); even those who are capable of taking exceptionally large objects in their rectum need a bit of a warm-up period. You should use that opportunity to accomplish two things with your dog: 1.) You will warm him up for anal play, and 2.) You can clean out any remaining fecal residue.
That second point is important, because penetrating him with something larger than a finger while there are still 'solids' present in his rectum may produce discomfort for your dog, and will increase the likelihood that he will clench or resist. I think this is one error that some people make when it comes to anal sex with canines. Don't assume that 'messiness' just comes with the territory. It doesn't have to, and -- along with going too fast, too hard, or too soon -- the presence of fecal matter during anal penetration may be the primary culprit contributing to the discomfort that some dogs experience with anal play. Trying putting on some latex gloves, use a generous amount of lube, and slowly penetrate his bum with your fingers. During this process, you may be able to manually 'extract' some substances if they are present. When you remove your fingers and you find residue, it may be a good idea to rinse off the glove, dry it off with a paper towel, and then re-apply a small amount of lube before re-entering.
Of course, make sure he's had a full bowel movement before hand. Some dogs end up being residue-free with a regular poop schedule; some require a little extra touch. A lot of this really depends on understanding all the nuances of your dog's body and rhythm (and, to some extent, diet, exercise, and overall health). Anal play NEVER has to hurt when it is done correctly, and the fact that this is an erogenous zone with a lot of nerve endings means that it always has the potential to be pleasurable for your dog (this may be especially true for males, given the prostate and all).
It will, of course, be helpful to engage in daily anal play, even if you aren't penetrating him with your penis. This will increase the speed with which he can open up quite a bit, and you may gradually move up in size over time. As mentioned previously, the sphincter is a muscle like any other. You can strengthen the muscle and increase its elasticity by giving it a good work out, but it will always return to its 'resting' state. However, by working the muscle frequently, it will be able to expand faster; long intervals between workouts will result in a longer time to reach the desired level of 'openness.'
It goes without saying that you should be doing all of this at a comfortable pace. Your dog will give clear signals of discomfort if you are being overzealous. Aside from some enthusiastic tail wagging, my pup is always still while I penetrate him before anal play, and during anal sex. He is also very relaxed (i.e., not clenching his hole). Make sure you give a lot of positive reinforcement before, during, and after any kind of play with your dog -- he will be more relaxed, and it will give him an opportunity to experience anal play as a pleasurable event. Remember that your dog's interpretation of, and prior experiences with, any activity that you engage him in will predict future experiences -- and will influence the quality of that activity for the both of you. Establish a healthy precedent. It all takes time, trust, patience, and love.