Azir and Attackspeed; Where are you trying to go with this?

Azir gets attack speed for free from Cooldown reduction (50% per 40% gained). And Azir's soldiers inflict a flat damage that scales with AP - that's perfect for attack speed right?

The problem is, there are very few attack speed items that Azir can actually build into, because the bulk of items inflict either AD or on-hit effects, and Azir's soldiers don't benefit from either.

I took a per-gold analysis of the top items for Azir, and they mostly have drawbacks...

Here are 5 interesting items on Azir. (3 attackspeed items and 2 comparable pure-AP items) These calculations assume a Lv.18 Azir with only the selected item bought, calculating from the damage of the Soldier autoattacks. Q, E and R are set aside, because these skills scale solely off AP (and health for E.)

The basic Azir has **118 DPS on his soldiers ** at level 18.

#1, Nashor's Tooth, provides a total of 236 DPS per soldier, up 118, for 2920 gold. That's 24g per DPS, and 60 AP for other character benefits.

But Nashor's Tooth does not proc the On-Hit passive from soldier attacks.

#2, Guinsoo's Rageblade, provides a total of 217 DPS per soldier, up 99, for 2600 gold. That's 26g per DPS, and 72 AP for other character benefits.

But the AD from Guinsoo's Rageblade is essentially worthless. You must also maintain your rage, though that is rather easy with soldier-summoning.

#3, Morellonomicon, provides a total of 188 DPS per soldier, up 70, for 2200 gold. That's 31g per DPS, and 80 AP for other character benefits.

#4, Rabadon's Deathcap, provides a total of 194 DPS per soldier, up 99, for 3300 gold. That's 43g per DPS, and 156 AP for other character benefits.

#5, Sword of the Divine, provides a total of 161 DPS per soldier, up 43, for 2150 gold. That's 50g per DPS, and Sword of the Divine also provides a unique Active - Soldiers will attack at +100% speed(+.56 attacks a second) for a full 3 seconds, because their attacks do not count as basic attacks and therefore do not consume the active effect.

But Soldiers cannot Critically hit!

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From this collection of information, we can glean that Nashor's Tooth and Guinsoo's Rageblade, despite having inefficiencies, are probably ideal build items for Azir... but Azir doesn't have any real "clean" items that fit his kit perfectly.

What have you all figured out so far?