H[u_?NumericQ] := NIntegrate[u^(5/2)/2*(L[u]/(1 - L[u]) + Log[1 - L[u]]),
{t, 0, 10^3}, {s, -10^3, 10^3}, Exclusions -> {0, 0}] // Chop

L[u_] := Simplify[1/(64 [Pi]^2 t^5 Sqrt[u]) (-4 I t (s + I t^2 + s Log[4])
+ (s^2 + 4 I s t - 4 t^2 + t^4) Log[-2 + (I s)/t + t]
- (s^2 - 4 I s t - 4 t^2 + t^4) Log[2 + (I s)/t + t]),
Assumptions -> {u [Element] Reals, s [Element] Reals,
t [Element] Reals}]

sol[r_] :=
Simplify[r^2*
NIntegrate[Re[H[u]]/u^3, {u, 1, r}], Assumptions -> u [Element] Reals]

Plot[sol[r], {r, 1, 100}, AxesLabel -> {r, sol},PlotRange->All]