# Collect terms with same-kind coefficients and factor exponential based on that

In the following expression, how can I group apart the terms in the exponential with a and b coefficients so that I can express the expression as the product of two exponentials with same-kind coefficients?

expr = C * E^(a1*w[i] + b1*x[i] + (a2 + b2) p[i])


I would like to be able to simplify the above expression to get

C * E^(a1*w[i] + a2*p[i]) * E^(b1*x[i] + b2*p[i])


Edit:

Thanks to J. M. needs help. I realize that in attempting to simplify my original problem I was asking for something non-sensical. So here is a bigger part the expression I'm trying to simplify

toSimplify = (E^((a2 + b2) p[i] + a1 w[i] + b x[i]) w[i])/(1 + E^(a2 p[i] + a1 w[i]))^2


using

fA[i] = E^(a2*p[i] + a1*w[i])/(1 + E^(a2*p[i] + a1*w[i]))
fB[i] = E^(b2*p[i] + b1*w[i])


So that I can get something like this

desired = fA[i]*((1 - fA[i])*fB[id]*w[i]

• Did you try evaluating E^(a1*w[i] + a2*p[i])*E^(b1*x[i] + b2*p[i]) yourself to see what happens? – J. M. will be back soon Apr 10 '18 at 1:56
• @J.M. Thanks for pointing that out. I have edited my question to something more meaningful. – Maturin Apr 10 '18 at 3:38

One can use the ComplexityFunction option in FullSimplify. In your example the following works:

toSimplify = (E^((a2 + b2) p[i] + a1 w[i] + b x[i]) w[i])/(1 +E^(a2 p[i] + a1 w[i]))^2
varsa = a1 | a2
varsb = b | b2
f[e_] := LeafCount[e] +
1000 Total@ Boole@(! Or[FreeQ[#, varsa], FreeQ[#, varsb]] & /@ List @@ e)
FullSimplify[toSimplify, ComplexityFunction -> f]


$$\frac{w(i) e^{b x(i)+\text{b2} p(i)}}{2 \cosh (\text{a1} w(i)+\text{a2} p(i))+2}$$

You might make the following steps:

expr1 = MapAt[Expand,
toSimplify, {1, 2}] /. {a2 p[i] + a1 w[i] -> X} /.
b2 p[i] + b x[i] -> Y

(*   (E^(X + Y) w[i])/(1 + E^X)^2   *)


Then

expr2 = expr1 /. (Exp[X + Y]*a_)/(1 + Exp[X])^2 ->
fA*HoldForm[Exp[Y]/(1 + Exp[X])]*a and

expr3 = ReleaseHold[expr2] /. E^Y -> fB

(* (fA fB w[i])/(1 + E^X)  *)


I guess that it is not exactly what you had in mind, but that's how you have defined the transformations, and I followed them strictly.

Have fun