BOSS ANALYSIS : Dark Anima throws a little bit extra into the mix, mostly via side-effects of moves, and quite a few at that! Her Attack consists of damage, Poison, Petrification (then Shatter), Curse, and Zombie to a single person. Her usual Pain move causes death, which is unblockable via Ribbon or Deathproof. There's Mega-Graviton, a new one for her; it's really just a loss of 43.75% (nearly half) of your maximum HP as well as Slow, Silence, Darkness, Sleep, and a 5-turn Doom. Ugh. And, finally, we have her Overdrive, Oblivion - that's 16 hits to everyone, plus an MP-draining effect and a Dispel effect. Basically, expect death from that.
MAIN STRATEGY : Ribbon, Ribbon, RIBBON! You will need that ability a LOT for this fight. With it, you'll nullify most of her annoying ailments, including Attack's Petrify-Shatter combo. Auto-Haste will ward off the Slow effects, and Auto-Protect can help to lessen the damage you get. Auto-Phoenix will be essential as well, since it's basically the only way to circumvent Pain's Death effect without casting Auto-Life every few turns.
So, basically, come in, all guns blazing - the faster you get started, the less time you have to worry about Pain. While, sure, Auto-Phoenix > Pain, Pain > 0 Phoenix Downs, you dig? If anything, that should keep you close to alive for Mega-Graviton; if not (basically, under 5,000 HP, or under 4,376/9,999 technically), Curaga needs to be used, and quickly. Basically try to prepare for anything that can come on the next turn.
In the end, though, you will have to Quick Hit, and Quick Hit a LOT. 8,000,000 HP is a ton to take down, and you only have 7 Aeons with which to deflect Oblivion with; 1,145,000 damage between Oblivions is your main target. A likely-unreachable target, but it's something to aim for; with a ton of Quick Hits, high Strength, high base Agility, and Auto-Haste, you might actually have a shot at that. If not, when Overdrives come, you'll have to send out Auto-Life or Hyper Mighty G quickly in lieu of your Aeons. If you have to get hit by the eighth-and-on Oblivions, be sure to recover your MP via Megalixirs.
It's ultimately a very simple fight to deal with once you prevent the main ailments and throw up a few good abilities on your armor ... albeit a LENGTHY fight.
ALTERNATE STRATEGY - OVERDRIVE ABUSE : This simple enough. With Break Damage Limit and high enough Strength (or Magic and maybe Trio of 9999 with Lulu), you are able to do massive damage to this boss in a small number of turns. In doing so, you must first enter the battle with as many Overdrive gauges full as possible. Have your main Overdrive user use an Overdrive, then have others Entrust their gauges to that person and keep attacking until done.
Question is, who to use? Tidus has the most reliable one in Blitz Ace, hitting 9 times when done right. Wakka has the luck-based 2x-2x-2x Attack Reels, which hits 12 times. Lulu hits the most if she uses one of her low-level (Fire, Water, etc.) spells with Fury, hitting up to 16 times, but it's more likely 8 if you have to use Flare or Ultima. In any case, there's pretty much no hope of doing it with under nine or ten Overdrives, so it will take a while, but it's probably worth it if you're desperate.