There's a video on YouTube by CoS Gaming (this is his updated AoT version linked above). He posted a video of doing a run around on plaguelands doing patrols. He shows which patrols to do, the order, and the best and worst case scenario for the patrols you'll randomly get. The nightfall buff is necessary first and foremost as mentioned in the other comment. But by doing his method of running patrols, I was able to get Dead Orbit up to rank 18 as of 4/29/17 and I just joined about 2 weeks ago and I have only played for about 4 hours this week and less than 10 last week.
Most of it is RNG. But you will still be depending on strange coins for the weekends to get that added boost in your ranks. I'll guess between doing that method, regularly playing the game as you do, and bankrupting Xur by taking every piece of heavy ammo in existence, you can level up a faction in less than 2 weeks...
[spoiler]FYI : Being a completionist sucks, I know because I'm trying to finish the book by August as well... [url=http://www.example.com][/url]but you only need about 70% of the nodes unlocked for the book to register than you did 100% / Rank 7 and get you the shirt...[/spoiler]
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Thank you so much for all this info but I actually ended up getting all my factions to rank 25. This wasn't really necessary since, like you said, all the nodes don't need to be completed for the book to get to 100%. The entire process wasn't as daunting as I thought it would be and took a pretty short amount of time. What I really wish was clarified prior to the launch of AoT was that not everything needed to be completed. I drove myself nuts thinking that I had to complete everything on all 3 characters, -- which was a big problem for me since I basically only ever use my Titan -- complete all that Trials stuff, and run around doing a bunch of other activities that I don't really enjoy. In the end, I'm grateful that Bungie allowed players to get to 100% by doing what they enjoy and/or are capable of, but I still wish that they had made that point a little more clear during the livestream.