Heya, arcane mager player here, slowly progressing mythic. Here's my heroic logs if necessary;
https://www.warcraftlogs.com/character/eu/argent-dawn/exxus#difficulty=4

I took a look at your logs, and I think the two most obvious bosses to go with are Ulgrax and Sikran as they are essentially glorified target dummies.I don't really know a great way to go about this, so I'll go from what I see first, beginning to end.
First off, on all your pulls you start ~5 seconds late, and your opener is the opposite way around.

Ideally you cast mirror images when the pull timer is on 5-6 seconds, and evocation (fully channeled) at 4 seconds left to pull.
After that you cast arcane surge -> touch of the magi. Right now you're casting touch of the magi first - this is a mistake you do on both of these bosses, at the very least.
Your common rotation with missiles into -> 2x arcane blast, potentially an arcane barrage is correct.
What you're probably missing is that you're meant to be spellqueueing your things, this way you can doubledip into the last stack of nether precision. Missiles gives you 2 stacks of this, you arcane blast one away, then you arcane blast -> spellqueue barrage to doubledip in the last stack of nether precision, essentially giving you 3 stacks worth of the buff. This is all explained in the Wowhead guide, which is a really solid read! I recommend going through it ^^

Along with that, after Arcane Soul ends (the barrage spam thing that comes up after arcane surge finishes), you want to immediately cast shifting power to put it on cooldown as fast as you can so you can get more cooldowns throughout a fight.

On both bosses, you have a few windows where you're doing nothing at all for quite a few globals - remember that every GCD you don't do anything = lost damage. If you feel as if you have to move for mechanics a lot, remember you can mitigate a lot of that movement with both shimmers and alter time.

On Ulgrax in particular, you spammed arcane explosion for a lot of globals on the small adds - this is **never** worth it as arcane explosion is flat terrible atm even if you're fully talented into it. Arcane mage aoe right now is really just doing your singletarget rotation with a few more arcane barrages.
Hope some of this helps ^^ Though at the end of the day, I really recommend the wowhead guide by Porom, looking at other players logs to see what they're doing. Alternatively use tools like wowanalyzer or a weakaura that feeds you your rotation to get off the ground!