Dear Gianluca,
I tried 'Collect' and 'Simplify', but the fractions in the results are not the simplest.
In[66]:= eq3 = Collect[eq1, E^_, Simplify]
Out[66]= (-A2 + pinf (s + \[Theta]c))/(s (s + \[Theta]c)) - (
A1 Dc E^((lh - x) Sqrt[(s + \[Theta]c)/
Dc]) (\[Alpha]1 Sqrt[Dp (s + \[Theta]c)] + Sqrt[
s (s + \[Theta]c)]))/((Sqrt[s] +
Sqrt[Dp] \[Alpha]1) (s + \[Theta]c) (-Dc s +
Dp (s + \[Theta]c))) + (E^((s (lh - x))/Sqrt[
Dp s]) (Sqrt[
Dp] \[Alpha]1 (Dc s - Dp (s + \[Theta]c)) (-A2 +
pinf (s + \[Theta]c)) +
A1 s (Sqrt[Dc Dp] s + Sqrt[Dc Dp] \[Theta]c +
Dc \[Alpha]1 Sqrt[Dp (s + \[Theta]c)])))/(s (Sqrt[s] +
Sqrt[Dp] \[Alpha]1) (s + \[Theta]c) (-Dc s +
Dp (s + \[Theta]c)))