expr1[x_] := PDF[JohnsonDistribution["SU", γ, δ, ξ, λ],
  x]

Integrate[expr1[x]*Exp[I*t*x], {x, -Infinity, Infinity}]

expr1 = PDF[JohnsonDistribution["SU", γ, δ, ξ, λ], x]

expr1[x_] := PDF[JohnsonDistribution["SU", γ, δ, ξ, λ], x]

CharacteristicFunction[JohnsonDistribution["SU", γ, δ, ξ, λ], t]

emCF = With[{dist = #},
    RandomVariate[dist, 100] // Exp[I t #] & /@ # & // Mean //
     Set[approx, #] &;
    ReIm[approx]] &;

Plot[Evaluate[emCF[NormalDistribution[0, 1]]], {t, -1, 1}]

Plot[ReIm@CharacteristicFunction[NormalDistribution[0, 1], t], {t, -1,
   1}]

Manipulate[
 Plot[Evaluate[emCF[JohnsonDistribution["SU", a, b, c, d]]], {t, -1,
   1}], {a, -5, 5}, {b, 0.01, 5}, {c, -5, 5}, {d, 0.01, 5}]