Longyan - Divine Fox

If you don't have Divine Fox yet but want to try her out, she is available as a Trial Operative in story chapter SP9-8 "Winds of Warmth" (Main Story -> Journey -> Ulster)

Longyan offers both strong single-target damage and strong AoE-damage, but this comes at the cost of her benefitting a lot from certain support effects (Aptitudes) which severely limits her options for teambuilding, and Longyan also has mobility so bad it can only be described as tragic, which can give her extremely low survivability as well, and her awful mobility and interruptible Standard Skill can make her performance highly inconsistent.
Longyan has extremely good synergy with the 4* support operative Fritia - Little Sunshine, which can make Longyan easy for new players to make a good team around her, although for endgame content Longyan should ideally always be paired with Tess - Phantasia, which means she has relatively poor teambuilding options for endgame content, and being the only team that deals two different damage types means Longyan needs both resistances to be good in order to perform optimally, limiting which bosses she's good against.

The Basics

Role: Main DPS
Deals Thermal Damage
Weapon Type: Dual SMG
Deals Ballistic damage
Mobility is dia-f**king-bolical

How to build Divine Fox

Weapon:
Porcelain Grace / Petal Dust
Weapon Parts: Max ATK
Logistics Squad: Moon
Basic: +10.0% ATK
+10.6% Crit DMG Amplifier
+106 Alignment Index is also very good
Special: +8.5% Thermal Damage
Weapon Parts: ATK
Good Support Combos to support Longyan with:
Tess - Phantasia + Marian - Aquila
Tess - Phantasia + Yao - Nightglow
Vidya - Celestial Swan + Fenny - Starshine

How to use Divine Fox

Against Single-targets / bosses:

If her Ultimate Skill is available, use it.
Spam her Standard Skill off-cooldown
Otherwise, just hold down the trigger.

With her Standard Skills Neuronic buff, she regains ammo naturally, so you don't normally need to manually reload - just hold down the trigger.

Against groups:

1. If her Ultimate Skill is available, use it.
2. Spam her Standard Skill off-cooldown
3. Otherwise, hold down her ADS button to deal AoE damage.

Her ADS attack doesn't use any bullets, so she doesn't need to reload when using this rotation.

Standard Shot / Passive Effect

Whenever Longyan fires or uses ADS, she enters the "Brocade Pelt" status.

Whilst in Brocade Pelt status:

Left clicking causes Longyan to shoot at the target.
Right clicking causes Longyan to deal AoE damage, hitting up to 5 targets.
This is a shot, but doesn't use any bullets.
Longyan gains 1 "Fox Favor" per LMB shot fired, and 10 per every RMB shot fired, up to a max of 60.

For both regular shots and her ADS attacks, the damage of Elemental Aptitudes is increased to 300% of their normal amount.

Longyan literally cannot go ADS - attempting to go ADS will instead trigger the corresponding ADS AoE damage attack.

After shooting 3 times in a row (i.e. 3 bursts of 20 shots each), the next usage will perform a Chain Finisher, which:

Counts as a "skill shot", meaning that it does proc Aptitude effects like a normal shot
Deals more damage (of the same type as the attack that triggered it, so LMB deals extra single-target, and RMB deals extra AoE)
Always crits
Doesn't use any bullets
Increases the damage of all Elemental Aptitudes it procs to 1,000% of their normal damage
With its first Neuronic, it fully replenishes her magazine (this is not a reload, and will not proc effects that trigger upon reloading)
Consumes all Fox Favour, decreasing her Standard Skill Cooldown by 0.05s per Fox Favor, which will typically be the max of 3s cooldown reduction.
With her second Neuronic, it also gains 1% Final DMG per Fox Favor, which is potentially a very strong buff.

These 60 shots can be against anything - Longyan doesn't actually need to hit anything to proc her Chain Finisher.

Longyan's Deiwos will give enemies a "Cinder Mark" after Longyan damages them whilst in Brocade Pelt status.
When enemies have 30 Cinder Marks (i.e. after 30 shots hit them), they are removed to deal some Thermal DMG and give Longyan 1 Tail Flame (which is used for her ult).
Unlike the Chain Finisher above, Cinder Marks are only inflicted on enemies, so Longyan does need to actually hit an enemy with her shots to proc it.

Note that, at least for me, this is currently bugged, and does not proc after 30 shots, instead usually proc'ing after slightly more - usually 32-33, but it can and does vary.

Standard Skill

Longyan dashses forward and deals Thermal Ballistic DMG to up to 5 enemies, which:

Counts as skill damage, but also counts as a shot, and hence does proc Aptitude effects
Always crits
Doesn't use any bullets
Increases the damage of Elemental Aptitudes to 1,000% of their normal damage

After the final hit of her Standard Skill, Longyan also gains 2 "Tail Flames" which are used for her ultimate.

Longyan's Standard Skill can be interrupted
Since Longyan only gains the 2 Tail Flames after the skill finishes, if this skill is interrupted at all then you won't get any Tail Flames.

Ultimate Skill

Longyan's Ultimate uses 0 U-Energy, but instead she needs to have 9 Tail Flames (gained from her Standard Skill and/or Deiwos) to be able to use it.

Longyan Ult uses all her Tail Flames to deal damage 8 times, followed by 1 stronger shot, which can damage up to 10 enemies, and also:

Counts as skill damage, but also counts as a shot, and hence does proc Aptitude effects
Always crits
Doesn't use any bullets
Increases the damage of Elemental Aptitudes to 1,000% of their normal damage.

Much like a lot of the recent operatives who have Ultimates that deal AoE damage, with its second Neuronic effect, Longyan's ult deals more damage the fewer targets there are, dealing +15% Final DMG per target less than 10, which is a huge damage boost against bosses and other single-targets.

Support Skill

Longyan deals AoE damage in front of her.
If it hits a target, it will proc another, large AoE, which can itself proc another even larger AoE if it hits.
With its Neuronics effects, the first and second stike will decrease movement speed for 2s, and the third strike will also Knock Back the targets.

Like all Main DPS operatives, Longyan's Support Skill isn't very good, and you really don't want to be using her as a support operative.

Weapon

Porcelain Grace
Her signature 5* weapon, and her best weapon.
This gives ATK and Thermal DMG, but as usual for a 5* weapon, the most useful part of the weapon is the "Thermal DMG ignores X% of target's Thermal Resistance".
Petal Dust
Increases her ATK and Thermal DMG.

There is no other operative who uses Thermal-damage Dual SMGs, so if you don't have either or Longyan's signature weapons, then you can use one of the signature 4* weapons of another Dual SMG operative if you wish, but since none of them will proc her Deiwos effect or several of her Manifestation effects, they're all much worse than either of Longyan's actual signature weapons.
The default 3* Dual SMG that Longyan comes with (Optimal Solution) doesn't perform that well for her, as it isn't a Thermal-damage weapon, and hence doesn't proc her Deiwos effect or several of her Manifestation effects.
Bubu - Specular Soul's 4* weapon (Serrated Pines) does marginally out-perform the default 3* weapon Longyan comes with, as Longyan can benefit from its Ballistic damage buff.
Bubu's 5* doesn't work well with Longyan.
Longyan doesn't work well with either of Lyfe - Infinite Sight's signature SMGs.

For weapon Parts Longyan wants maximum ATK.

Logistics

Squad

Use Moon squad - it was designed specifically for her and is her best choice.

Moon squad gives her ballistic damage 20% DEFpen when she has an Elemental Aptitude.

When she's in the Brocade Pelt state, she gets +40% Thermal DMG.
When not in the Brocade Pelt state, she gets +40% Standard Skill and Ultimate DMG.
Practically speaking, Longyan will always get +40% damage from Moon squad, her state merely depends what type of buff it is.
Since Longyan enters Brocade Pelt status automatically when shooting, Longyan always has the appropriate buffs, meaning this is functionally a permanent +40% damage buff.

Stats

Longyan wants:

Basic Stats +10.0% ATK
+106 Alignment Index or +10.6% Crit DMG Amplifier
Special Stats +8.5% Thermal DMG

ATK to maximize her damage, and Thermal DMG for more damage.

Longyan can't heal, can't create shields, basically all of her damage always crits anyway, she doesn't use U-Energy, and doesn't normally have any S-Energy issues, so most logistics stats aren't useful to her at all.
Note that Longyan's default shots (single-target or AoE via her ADS attack) don't normally crit, so Crit Rate does help their damage when not shooting at a weakspot (since hits against weakspots always crit), but since the rest of her damage always crits anyway, Crit Rate doesn't increase the damage of most of her rotation, and hence is a very bad stat for Longyan.
Crit Rate does help her damage, it's just such a tiny buff it's essentially negligible.

Skill Haste

TL:DR:
If Longyan is supported by Tess - Phantasia, then Skill Haste is a little bit worse than Alignment Index or Crit DMG Amplifier.
If Longyan isn't supported by Tess - Phantasia, then Skill Haste can potentially out-perform Alignment Index/Crit DMG Amplifier, but in practice performs about the same.
Since Phantasia is Longyans best support, and you should always try to support Longyan with Tess if possible, you generally want Alignment Index or Crit DMG Amplifier over Skill Haste, as that's what's best for Longyan's ideal team, and in practice tends to work out either the same or slightly better anyway.

There's a longer explanation of this in its own section at the bottom of this page.

3x Max Skill Haste rolls can work well, but I don't recommend it - get Alignment Index or Crit DMG Amplifier instead.

Alignment Index and Crit DMG Amplifier

TL:DR:
If Longyan is supported by Tess - Phantasia:
With max investment, Crit DMG Amplifier is a bit better than Alignment Index.
Without high investment (5* weapon, M5, very good ATK rolls on her Logistics) then Alignment Index performs about the same as Crit DMG Amplifier.
With low investment, Alignment Index will generally out-perform Crit DMG Amplifier.
If Longyan isn't supported by Tess - Phantasia, then without high investment, Alignment Index will out-perform Crit DMG Amplifier.

With Longyan at max investment, and when supported by her ideal supports of Tess - Phantasia and Marian - Aquila, swapping Longyan's logistics
from +10.0% ATK and +106 Alignment Index
to +10.0% ATK and +10.6% Crit DMG Amplifier
would result in her total [Final] damage increasing by around 2.5%.

There's a longer explanation of this in its own section at the bottom of this page, but the difference between them is small.

Manifestation

wth M1, Longyan's ballistic DMG ignores 15% of the enemies Thermal Resistance
This only works if Longyan has a Thermal-damage weapon equipped, but that should hopefully always be the case.
This is a strong damage buff.
With M2, Longyan damaging an enemy with Thermal DMG reduces their DEF by 5% for 5s, up to 20%.
This is a good damage buff, and since it debuffs the enemy rather than just buffing Longyan it buffs the damage that Tess - Phantasia does, which makes it very good.
With M3, Longyan's Ult increases its Final DMG by 8% per hit, which resets after the ult finishes.
This isn't quite linear, as her 9th/final shot deals more damage than the first 8.
Overall, M3 works out to a 43.8% Final DMG increase for her ult, which is strong, but it is only for her ult.
M4 gives the usual +10% Damage to her Ultimate
This is a decent damage increase, but M4 is pretty expensive, and it's just a damage increase.
M5 increases Longyan's ATK by 10%, and increases the ATK of your entire squad by another 10% if Longyan has an Aptitude effect.
This makes Tess - Phantasia and Fritia - Little Sunshine comparatively stronger as supports, since they are the only support that can give Aptitude effects and hence proc the conditional buff, but the unconditional +10% ATK is decent in itself.

If you don't want to get her all the way to M5, then I'd say M3 is a good stopping point.

Good Supports for Longyan - Divine Fox

Tess - Phantasia works perfectly with Longyan.
Tess provides some great ATK and Ballistic damage buffs, which affect all of Longyan's damage.
Longyan deals damage very frequently, which gives Tess' Aptitude effect a lot of chances to add extra damage.
Tess' Aptitude effect can also proc Longyan's Logistics buff for more damage.
Longyan provides a significant increase to the damage Tess deals as well.
Tess and Longyan work perfectly, and Phantasia is Longyan's best support by a large margin.
Vidya - Celestial Swan
Vidya massive reduces the enemies DEF and Resistance, and buffs the teams damage a little too.
Vidya only debuffs the damage resistance of the damage type the enemy has most recently received.
As such, if using Celestial Swan, then make sure the other non-Vidya support isn't off-element to Longyan.
(i.e. They either deal Thermal damage or do not deal damage at all).

Whilst Vidya and Tess are Longyan's best supports, since Longyan deals Thermal damage and Tess deals Frost damage, you can't support Longyan with both Tess and Vidya at the same time.
Whatever supports you use for Longyan, one of them should be either Phantasia or Celestial Swan.
On their own, both Tess and Vidya are amazing supports for Longyan, but in a three-person team the lack of a strong thermal ballistic support to pair alongside Vidya means that Longyan's best support is Phantasia by a notable margin.

Other great supports for Longyan include:

Marian - Aquila
Aquila provides very strong generic damage buffs, which makes her an excellent support.
Aquila is off-element to Longyan (Aquila constantly deals Kinetic damage).
Yao - Nightglow
Yao buffs ballistic damage, ATK, and gives extra Crit DMG Amplifier, which are three stats Longyan really wants, so Yao adds a lot of damage to Longyan.
Longyan has quite a low base Crit Damage value of just 30%, but even with that Yao is still very strong.
Yao is off-element to Longyan (Yao constantly deals Electric damage).

Good supports for Longyan include:

Fenny - Starshine
Starshine provides a lot of very strong ballistic damage buffs, however Fenny's buffs are only active when the deployed operative has one of her Special Clips equipped.
Having a special Clip equipped though prevents Aptitude effects from working, so Starshine's buffs block Tess - Phantasia's Aptiutude damage, which is a huge loss.
Starshine is thus mutually exclusive with Phantasia, however Phantasia is so strong that this means that you'll only want to use Starshine when not using Phantasia.
Vidya - Celestial Swan + Fenny - Starshine is a great combo, but it is out-performed by some of the Tess combos.
Fenny - Coronet
Coronet's Support Skill is just a 30% RoF buff, however +30% RoF allows Longyan to proc Phantasia's Aptitude buff a lot more, and Longyan buffs Phantasia's damage too.
This means that Coronet does actually work surprisingly well alongside Longyan + Phantasia.
Coronet performs very poorly when Longyan isn't supported by Phantasia though.
The low uptime of Coronet's Support Skill buff means that she works best when Longyan is fighting enemies that she can't just continually shoot or can easily defeat.
Having some downtime to allow Coronet's Support Skill to recharge is extremely useful.
Phantasia + Coronet can perform very well when Coronet has some downtime to recharge, but isn't as strong otherwise.
Eatchel - The Cub
A Blitzing Fangs Eatchel provides healing as well as a decent ATK buff.
Not the strongest support, but still good, and Longyan's horrendous mobility means she really appreciates Eatchel's constant healing.
Since Eatchel doesn't deal damage, she works alongside Celestial Swan.
Fritia - Little Sunshine
Whilst she's only a 4* operative, Fritia can do a lot of hits for Longyan, due to Longyan's extremely high average Rate-of-Fire.
Longyan also inherently boosts the damage of all Aptitude effects, including Fritia's, and Fritia's Aptitude effect can proc Longyan's Logistics buff for more damage.
Fritia is also on-element to Longyan, and can thus be used alongside Celestial Swan.
Fritia is only a 4* operative, and does generally perform worse than the other 5* supports listed, but she is still strong for Longyan.
Acacia - Kaguya
Kaguya is a good resistance debuffer - not nearly as strong as Vidya - Celestial Swan, but if you aren't using Vidya then Kaguya can be used as a much weaker substitute.
Kaguya deals Frost-damage.

Longyan benefits from all of the buffs that Tess - Phantasia can provide, whilst also boosting Tess' Aptitude damage as well, and Phantasia is thus Longyan's best support, and since Tess and Vidya are mutually exclusive, Longyan's best support pairs are thus:

Tess - Phantasia + Marian - Aquila
Tess - Phantasia + Yao - Nightglow
Vidya - Celestial Swan + Fenny - Starshine

Little Sunshine does work well, but is out-classed by the 5* supports.

Tess - Phantasia + Fritia - Little Sunshine
Vidya - Celestial Swan + Fritia - Little Sunshine

Cope Supports like Mauxir - Shadow Ka, Vidya -Agave, Acacia - [Redacted], Yao - Quiet Quitter, etc. are still fine if they're all you have, but Longyan really wants an Aptitude effect, so you should always try to pair her with either Tess - Phantasia or Fritia - Little Sunshine, although again; Vidya - Celestial Swan is so strong that she works well too.

Logistics: Skill Haste

Since her Chain Finisher gives a huge amount of cooldown to her Standard Skill, if a Skill Haste mod doesn't let Longyan use her Standard Skill an extra time before the Chain Finisher would fully recharge it anyway, then that mod is not useful, as it hasn't allowed her to use her skill an additional time - unless it's worth it to wait a little bit to use her Standard Skill an extra time so you can then instantly recharge it with her Chain Finisher - It usually isn't worth it to do that though.

Skill Haste is theoretically a good stat for Longyan since it could increase the rate at which she can use her Standard Skill, and thus how quickly she generates Tail Flames, and thus how often she can use her ult.

If Longyan can't get enough Skill Haste to get a second Standard Skill in the middle of her chain, then her chain will be
Standard Skill -> shoot 60 shots -> Chain Finisher -> Standard Skill -> shoot 60 shots -> Chain Finisher -> Standard Skill -> ult [This is her default way of playing]

With enough Skill Haste, her ideal chain would be:
Standard Skill -> shoot 30+ shots -> Standard Skill -> shoot <30 shots -> Chain Finisher -> Standard Skill -> shoot ~30 shots -> ult

Since her ult will completely reset both rotations back to the beginning, Skill Haste doesn't let you get more Standard Skills off per rotation - it's 3 for both.
The 1st rotation has 3 Standard Skill casts, two Chain Finishers, and 120 shots.
The 2nd rotation has 3 Standard Skill casts, one Chain Finisher, and 90 shots.
The first rotation gets 10 Tail Flames, wasting the second Tail Flame from the third Standard Skill cast.

The first chain is obviously slower as it has more casts and shots, but it will also deal more damage.
The second chain can have ~90 shots back-to-back with no Chain Finisher, so you'll likely want to manually reload in there somewhere.
Since the first chain doesn't need to manually reload as it does Chain Finishers instead, having to manually reload is a disadvantage to the second chain.
With very good accuracy (and maybe luck) you don't need to manually reload, but in practice you will have to do so every now and then.

Even with three max roll (+19.3) Skill Haste rolls, Longyan can't cleanly do the second chain.
Even after she finishes shooting her shots, her Standard Skill still hasn't quite fully recharged - it still has around 0.5s of cooldown left before it can be used.
Longyan has to stop shooting and just wait for her Standard Skill to finish recharging before she can use it again, probably filling the time with a dodge or two, or possibly a reload.
In practice it'll probably be a reload, as Longyan can have 90 shots back-to-back with no Chain Finisher to reload, so she'll likely want to reload there.
This extra time spent manually reloading is time that the first rotation uses to deal damage instead.
This wait for the Standard Skill to recharge is extremely annoying, and having to stop and wait for her skill to recharge will make the default rotation better simply because it's less infuriating to play, but the wait does make the chain longer, reducing the benefit of doing it.

In addition, this delay is small with three max-roll Skill Haste mods, so with non-max Skill Haste mods, the delay will be even longer, which makes it difficult to swap from the default rotation to the faster Skill Haste-dependent rotation, as you need a lot of Skill Haste in order for the delay to not be prohibitively long.
With a good amount of Skill Haste, it can work, but it's really not any better.

If Longyan is being supported by Tess - Phantasia, then the extra shots in the first rotation do make up for the rotation being slightly longer due to just how much damage Tess adds to Longyan.
If Longyan is not supported by Tess - Phantasia, then the faster second rotation [with the Skill Haste] is slightly better.
Since Phantasia is Longyan's best support, you'll ideally support Longyan with Phantasia, and hence Skill Haste is not what I recommend, but if you have enough of it to be able to use Longyan's Standard Skill more often, then it's not a bad choice.
I don't recommend re-rolling for Skill Haste, but if you get a lot of it just from RNG, then you can potentially use it to good effect.

I still recommend you try and get your logistics with +10.0% ATK and either Alignment Index or Crit DMG Amplifier though.

Logistics: Alignment Index and Crit DMG Amplifier

Alignment Index gives Longyan more ATK; a max-roll of +106 Alignment Index translates to give Longyan +5.3% ATK, so three max-roll stats gives her +15.9% ATK.

Longyan does have quite a lot of %ATK from various sources (her 5* weapon, her logistics, her Manifestations, her natural 300 Alignment Index from her Neuronics...), but Longyan is likely to get a decent amount of %ATK from her supports, for example Tess - Phantasia provides 35% ATK via her Logistics, and Marian - Aquila can provide an extra 33.2% ATK as well, further saturating Longyan's ATK stat.

Crit DMG Amplifier might seem like a good stat since most of Longyan's rotation always crits, but her Dual SMGs have a base Crit DMG value of just 30%, so with three maxed Crit DMG Amplifier mods, Longyan's crits would go from 130% to 139.54%, which is functionally a (139.54%/130%)= +7.33% Final Damage boost.

The only other source of Crit DMG Amplifier that Longyan is likely to get in endgame content (i.e. Neural Sim or Supreme Battleground) is from the Support/Ultimate skill of Yao - Nightglow, however whilst Nightglow is a fantastic support for Longyan, and the amount Yao provides is very significant, Marian - Aquila is generally the better support for Longyan, and hence Yao's Crit buffs are something you should be aware of, but probably aren't something you should build around, as Yao isn't Longyan's ideal support.
Adding the Crit Weakness from Yao's Support Skill to a Longyan with 3x max-roll Crit DMG Amplifier mods is effectively around a 6.4% Final Damage boost.
Adding the Crit Weakness from Yao's Ultimate Skill to a Longyan with 3x max-roll Crit DMG Amplifier mods is effectively around a 10.7% Final Damage boost.

ATK and Crit DMG essentially scale multiplicatively with all other stats, so when comparing them you can basically just ignore all other stats, and which of the two stats is better for Longyan is essentially determined by how much of each Longyan actually has.

In order for a +7.33% Final damage boost to equal a +15.9% ATK buff, Longyan would need to have around 216.9% ATK (or +116.9% ATK).
I.e. 216.9% +15.9% = 216.9% * 1.0733
With good investment (and/or some supports that give her some extra ATK), Longyan can get this much, but if you aren't able to get Longyan;

Her 5* weapon
Some, or ideally all of her Manifestations (notably M5, which can give +20% ATK)
Some decent ATK rolls on her Logistics

then even with Tess - Phantasia's +35% ATK from her Arcane Logistics squad, Alignment Index will probably perform about the same as Crit DMG Amplifier.
If you aren't able to get any of the above, then both Alignment Index and Crit DMG Amplifier both perform fine, and are similar enough that I recommend you use whichever you roll for first - once you have a good roll for either Alignment Index or Crit DMG Amplifier, don't waste re-rolls trying to get the other.

With high investment, Crit DMG Amplifier does beat Alignment Index, and hence you should aim for Crit DMG Amplifier, but if you aren't going to fully invest in Longyan, Alignment Index will probably perform about the same, so it's generally fine to settle with whichever you get first.

Whilst Crit DMG Amplifier is a little better at max investment, since the difference between them is small, unless you are a whale who is willing to buy a ton of supports for Longyan just to get a perfect Crit DMG Amplifier roll, it's generally fine to use whichever of the two stats you happen to roll first.