Hi!
I tried to simply your problem, using the linearity of the integral.
First,
IntegListe =
Apply[List,
ExpandAll[
K*(a*K*Exp[-(a + ((g*(1 - c*g))*(L/(1 + (L - 1)*Exp[-(t - T)])))/
K)*t] + (g*(1 -
c*g))*(L/(1 + (L - 1)*Exp[-(t - T)])))/(a*
K + (g*(1 - c*g))*(L/(1 + (L - 1)*Exp[-(t - T)])))]]
gives 3 functions, which can be integrated separately, and added afterwards. As for the third one,
we have:
In[21]:= Integrate[IntegListe[[3]], {t, T, Q},
Assumptions -> Q > T > 0]
Out[21]= (c g^2 K L (T + Log[(g (-1 + c g) - a K) L] -
Log[-a K (E^Q + E^T (-1 + L)) + E^Q g (-1 + c g) L]))/(
a K + g (1 - c g) L)
so, it has a closed form. The second member has a closed form, too, contrary to the first one:
In[24]:= Integrate[FullSimplify[IntegListe[[1]]], {t, T, Q},
Assumptions -> Q > T > 0]
Out[24]= Integrate[(
a E^(-a t + (E^t g (-1 + c g) L t)/(K (E^t + E^T (-1 + L))))
K^2 (E^t + E^T (-1 + L)))/(
a E^T K (-1 + L) + E^t (a K + g (1 - c g) L)), {t, T, Q},
Assumptions -> Q > T > 0]
Consequently, I think you should focus on this first function (simplify it, approximate by a convergent series, etc).
Hope this helps!
Claude