My first thought was that it of course all depended on whether I used many smites or not. If I throw only five smites for 30 seconds, then maybe the total mana gain would be less than what I'd get from the Archangel buff. Where was the breaking point? How many Smites would I have to use to gain more mana from the reduced mana cost than from the archangel buff?
There were several factors to consider;
Evangelism increases healing done to Smite by 20%, while Archangel increases all healing by 15%.
What about Holy Fire who we also use, is affected by Evangelism as well and further increases the healing done to Smite for a couple of seconds by 20%.
What fights allow us to spam smites for 30 seconds straight anyway?
Do we want to use Archangel on cooldown? Maybe while questing, but most people tend to save it for bigger incoming damage during raids.
This is the difference between theoretical and practical calculations, and I was mostly interested in the theoretical results, because knowing them meant mastering its practical values - or so I think. Therefor I decided to remove Holy Fire from the calculation since the mana gain through Evangelism is fairly low (at best 816 mana during a 30 second time span as we will soon see). I also decided that the 20% Evangelism buff and 15% overall healing buff were close enough to cancel out (to ease up on the calculations). The Archangel buff has a duration, but on the other hand it takes a while to gather a 5 stack with Evangelism. My guesses are that to calculate which is more mana effective in the long run, this wouldn't make big enough of an impact to warrant the headache the calculations would give me. And since this is theoretical, we can ignore the fact that most fights probably won't allow us to use every gcd on Smite 30 seconds straight (at least not in progress raiding).
Evangelism lowers the cost of Smite by a fixed number, meaning it is unaffected by buffs (only slightly affected by race). Each buff lowers the cost of the next Smite by an additional 6%. To me that was 185 mana. Beyond the first smite, it would look like this;
Holy Fire by comparison;
How many Smites and Holy Fires we'll manage to cram into 30 seconds depends on your haste. Unbuffed I have 1483 (12,7%) haste, and using a regular rotation I can throw 9 Smites and 3 Holy Fire, or 12 Smites within 30 seconds.
9 Smites and 3 Holy Fire equals 6366 mana "gained" through reduced mana cost.
12 Smites equal 8325 "gained" through reduced mana cost.Remember that this does not take into consideration the increased hps of Smite through Evangelism and HF buff.
And now for the final conclusion;
Completely unbuffed, I gain 5954 mana from Archangel. This is completely dependant on your max mana, and will differ greatly once you're fully raid buffed, but let's look at it like this first. It would mean that I, unbuffed, need to throw at least 10 Smites (6475 mana "gained"), or at least 9 Smites and 2 HF (5958 mana "gained") within 30 seconds to gain more mana than using the Archangel on cooldown. That is just barely what I have time with.
So what does this all mean?
There are two practical areas where this has an impact, questing and raiding (instancing).
Questing: While questing the healing aspect of AEA matters less, and you can fully focus on being as mana efficient as possible. As long as you can expect to keep your Smites rolling, you'll gain more mana from doing so than using up your Archangel buff. Most of the time there will be some sort of downtime between mobs however, which means you will want to use up your Archangel on cooldown.
Raiding: As mentioned, there are many factors that need to be calculated to get an exact value of whether it is best to use Archangel or keep on Smiting. Regarding haste; Eventhough we can only expect the numbers to differ perhaps 1, max 2 spells between characters, if one has stacked haste (like me) and the other hasn't, it still means that raid buffs like Bloodlust, if you've got PI up or otherwise get a huge buff in haste, will greatly favor spamming Smite, since the more Smites we can cram into 30 seconds, the more mana we will "gain" over using Archangel. In reality however, dps is very rarely the reason we use Atonement healing in raid - most of the time we want to consider not overhealing by spamming Smites, keeping Archangel for the right moment and a thousand other things that probably will interfer with your perfect Smite + HF rotation. Regarding max mana: Although I might need 10 Smites unbuffed, the number is a lot higher during a raid where my max mana is higher and Archangel might give me 7 or 8k mana instead of 6k. Overall we can safely conclude that it is nearly always better to use up the Archangel buff whenever you need it than to keep the Evangelism buff as long as possible.