gistfile1.txt

ClearAll[parts, ii]
parts[expr_Plus] := Distribute[Unevaluated[parts[expr]]]
parts[a_ x_] /; NumericQ[a] := a parts[x]
parts[ii[Sin[t] rest_]] := 
 Cos[x] (rest /. t -> x) + ii[Cos[t] D[rest, t] // Simplify]
parts[ii[Cos[t] rest_]] := -Sin[x] (rest /. t -> x) - 
  ii[Sin[t] D[rest, t] // Simplify]
parts[expr_] := expr

Nest[parts, ii[Log[t] Sin[t]/t], 10];
CoefficientList[% /. ii[_] :> 0, {Cos[x], Sin[x], Log[x]}]
Nest[parts, ii[Log[t] Cos[t]/t], 10];
CoefficientList[% /. ii[_] :> 0, {Cos[x], Sin[x], Log[x]}]
Nest[parts, ii[Sin[t]/t], 10];
CoefficientList[% /. ii[_] :> 0, {Cos[x], Sin[x], Log[x]}]
Nest[parts, ii[Cos[t]/t], 10];
CoefficientList[% /. ii[_] :> 0, {Cos[x], Sin[x], Log[x]}]