A Remark on the Convolution of the Generalised Logistic Random Variables, Matthew Oladejo Ojo (2003).
Probability density function $f(x)$ for each of $n$ independent random variables gives the probability distribution $f_n(y)$ for the sum:
http://ilorentz.org/beenakker/MO/logistic.png
Some explicit expressions of $f_n(y)$ for this case $p=q=1$:
$$f_{1}(y)=\frac{e^y}{\left(e^y+1\right)^2}$$
$$f_{2}(y)=\frac{e^y \left(e^y (y-2)+y+2\right)}{\left(e^y-1\right)^3}$$
$$f_{3}(y)=\frac{e^{2 y} \left[-2 y^2+\left(y^2+6\right) \cosh y-6 y \sinh y+6\right]}{\left(e^y+1\right)^4}$$
$$f_{4}(y)=\tfrac{1}{3}e^{5y/2}(e^y-1)^{-5}[y (11 y^2-36) \cosh(y/2) + y (36 + y^2) \cos(3 y/2) - 24 (2 y^2-2 + (2 + y^2) \cosh y) \sinh(y/2)]$$