I am considering the vector space of derivates of the LogisticSigmoid orthogonalized using the inner product induced by the first derivative. I ran into trouble with integrand(6,4) below. If you look below at out64, you will see that the indefinite integral is computed; however, applying FunctionExpand, yields a result, directOut, which contains one term
Log-1+
1
1+
-
t
s


which is imaginary. This must be wrong since everything is real.​
In[]:=
g[x_,i_]:=D[LogisticSigmoid[x/s],{x,i}]
In[]:=
integrand[i_,j_]:=g[t,i]g[t,j]g[t,1]
In[]:=
in64=integrand[6,4]
In[]:=
out64=Integrate[in64,t]
In[]:=
step2=Simplify[FunctionExpand[in64]]​​
In[]:=
step3=Integrate[step2,t]
In[]:=
directOut=Simplify[FunctionExpand[out64]]