In[]:=
g[x_,i_]:=D[LogisticSigmoid[x/s],{x,i}]
In[]:=
integrand[i_,j_]:=g[t,i]g[t,j]g[t,1]
In[]:=
ntgl[i_,j_]:=Integrate[Simplify[FunctionExpand[integrand[j,i,s]]],t]
In[]:=
ntgl[1,2]
Out[]=
3t
s

3
6
1+
t
s


3
s
In[]:=
defNtgl[i_,j_]:=Integrate[Simplify[FunctionExpand[integrand[j,i]]],{t,-Infinity,Infinity}]
In[]:=
Timing[ntgl[5,3]]
Out[]=
0.131821,-
1
2310
11
1+
t
s


8
s
-17-187
t
s

-935
2t
s

-3575
3t
s

+10175
4t
s

-64757
5t
s

+56903
6t
s

-15785
7t
s

+770
8t
s


In[]:=
Timing[defNtgl[5,3]]
In[]:=
Timing[x=Limit[ntgl[5,3],{t-Infinity}]]
Out[]=
0.976187,
17
2310
8
s
if s>0

In[]:=
Timing[x=Limit[ntgl[5,3],{tInfinity}]]
Out[]=
0.64695,
0 if s>0
