  We consider the additive decomposition problem in primitive towers and
present an algorithm to decompose a function in an S-primitive tower as a sum
of a derivative in the tower and a remainder which is minimal in some sense.
Special instances of S-primitive towers include differential fields generated
by finitely many logarithmic functions and logarithmic integrals. A function in
an S-primitive tower is integrable in the tower if and only if the remainder is
equal to zero. The additive decomposition is achieved by viewing our towers not
as a traditional chain of extension fields, but rather as a direct sum of
certain subrings. Furthermore, we can determine whether or not a function in an
S-primitive tower has an elementary integral without solving any differential
equations. We also show that a kind of S-primitive towers, known as logarithmic
towers, can be embedded into a particular extension where we can obtain a finer
remainder.
