Option 1 -> "taken out" means to remove or extract something, which doesn't fit the context of attempting a task.
Option 2 -> "taken off" means to depart or remove, not suitable for beginning or undertaking a task.
Option 3 -> "taken up" means to begin or undertake something (a task, hobby, or challenge), which perfectly fits the context.
Option 4 -> "taken into" is not a complete phrasal verb in this context and doesn't make grammatical sense with "task."
Hence, Option 3: taken up -> "Take up" is the correct phrasal verb meaning to begin or undertake a task or responsibility. The sentence indicates that several people have attempted or begun the challenging task of writing the country's history, though few completed it successfully. -> correct