Devil May Cry 4 - Secret Mission 09 -- Unbreakable
When: Mission 15: Fortuna Castle
Where: Library
Objective: Eliminate all enemies without taking damage and without being captured by a Fault.
You're up against two waves of Assaults and Chimera Seeds here, and you have to wipe them all out without taking any damage or being captured by a Fault. The biggest threat are the Chimera-infected Assaults, as the Chimera swing their tentacles every which way, making them difficult to avoid. Even a small nick spells failure, so you can't be too careful.
With that said, this mission can be cleared quite easily with some patience and a bit of luck. With the shotgun out, hug the edge of the area and double-jump in one direction while firing at the enemies below. Do not use lock-on at all during this, just stick to the edge of the map, jump and Air Hike in one direction and continuously tap the Long-range attack button. As soon as you hit the ground, jump up into the air again and continue shooting. This way there is no need to worry about the Faults at all, because they'll only appear as soon as you hit the ground, and you'll be up in the air again before they can emerge completely.
There might be a few close calls, and this could take a couple of tries, but it is likely the safest and easiest method to get this done. Once you've killed off the two waves of enemies, collect the Blue Orb Fragment that spawns to clear the mission.