Default Sivir core item is Duskblade of Draktharr, which is giving you invisibility after kill or assist. You can protect only yourself with it mythic passive, but if you choose Shurelya's Battlesong you literally getting second ULT - On The Hunt. With it you would be able to save your teammates life's or chase down enemies more effective.

SHURELYA'S BATTLESONG

VS

DUSKBLADE OF DRAKTHARR

You get: 20 Ability Haste; 40 Ability Power; 200 Health; 100% Base Mana Regeneration; (Mythic Passive: Empowers each of your other Legendary items with 5 ability haste) It means that you don't lose abitily haste, and also gain some hp and mana regeneration, because of bonus hp it would be harder to kill you.Also item's active is very useful for your team in long teamfights or chases.

You get: 20 Ability Haste; 60 Attack Damage; 18 Lethality; (Mythic Passive: Empowers each of your other Legendary items with 5 ability haste) With Duskblade of Draktharr you can get 18 lethality, and also item's passive which is very useful for you in long teamfights.

SORCERY

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Pros

+ ctetur adipiscing el + high mobility for entire team + additional mana regeneration and 200 hp + constant poke in teamfights + you will have your Shurelya's Battlesong or On The Hunt every teamfight (or both) + Safe splitpushing

Safe splitpushing means that you can run away from the danger with Shurelya's Battlesong or On The Hunt.If you use both of them, you and your team will have 2x movement speed multiplier (100%), but you can also save one of them for long teamfights

Cons

- lower damage - no invisibility from Duskblade of Draktharr - a bit lower armour penetration - lower sustain because of runes

If you are buying Duskblade of Draktharr you will get 60 damage points on your Boomerang Blade, and if you choose Shurelya's Battlesong it gives you 40 ap, so you'll get 20 damage points (because of 50% ap scales) on your Boomerang Blade