Here is the full list of all 52 Summon Night 6: Lost Borders trophies - there are 40 bronze, 9 silver, 2 gold and 1 platinum.
Completed [Obstinate Sword]
Completed [The Pirate, the Angel and the Lost]
Completed [Heroes Falling From the Sky]
Achieved a Brave Order for the first time
Won the first battle
Acquired all trophies
Completed [Demon Lord Attack!]
Completed [Prisoners of Delusion]
Completed [Troublesome Girl]
Completed [So Long, Tranquility]
Completed [Raving Madness]
Completed [Thou Shalt Not Deceive]
Completed [Brutal Sword and Magical Sword]
Completed [An Encounter of Past and Future]
Completed [World Cobbled Together]
Completed [An Administrator's Choice]
Completed [By the Side of One Yet Unknown]
Completed [Silhouette of Thanatos]
Completed [Clash of the Kings]
Completed [Lamp that Illuminates the Darkness]
Completed [The Price Called Bonds]
Completed [Get Those Days Back!]
Completed [From the Cage of Despair]
Completed [Truth to Be Mocked]
Completed [The Line Between Reason and Savagery]
Completed [End of Infancy]
Completed [The Chosen Future]
Completed [End of the Illusion, and the Beginning]
Completed [Demon of the Mountain Pass]
Completed [The Violent Man and the Runaway Daughter]
Welcomed Karma End
Cleared all events and battles at S rank
Achieved all Brave Orders
Used Summon Burst for the first time
Generated all types of Summon Bursts
Generated Chain Attacks 30 times
Generated a Summon Assist using 8 people
Defeated 100 enemies
Defeated 400 enemies
Cleared the lowest level of the Endless Domain
Used Raj, Amu or Ist's special skills a total of 30 times
Acquired all of a character's skills
Upgraded a weapon to the maximum
Upgraded a Summon Beast to the maximum
Acquired all Summonite leaves
Acquired all Summon Beasts
Created all possible dishes
Completed all of a character's Night Conversations
Completed all Night Conversations
Fished the Cove King
Completed all Fishing Orders
Completed all Town Quests at the Eucross