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This is another question about the problem of collecting of the logariths.

Suppose the function

F[s2_, Ss_, t1_, m_, m2_] := Evaluate[-6 ((s2^6 + (Ss - 7 t1) s2^5 + (-4 Ss^2 + 13 t1 Ss + 16 t1^2) s2^4 + 2 (Ss^3 - 13 t1 Ss^2 - 10 t1^2 Ss - 8 t1^3) s2^3 + t1 (18 Ss^3 + 6 t1 Ss^2 - 4 t1^2 Ss + 7 t1^3) s2^2 + t1^2 (18 Ss^3 + 22 t1 Ss^2 + 11 t1^2 Ss - t1^3) s2 + Ss t1^3 (2 Ss^2 + 2 t1 Ss - t1^2)) m^6 - Ss (s2 - t1) ((11 t1 - 2 Ss) s2^4 + 2 (Ss^2 - 16 t1 Ss - 7 t1^2) s2^3 + 6 (4 Ss^2 t1 - t1^3) s2^2 + 2 t1^2 (15 Ss^2 + 16 t1 Ss + 5 t1^2) s2 + t1^3 (4 Ss^2 + 2 t1 Ss - t1^2)) m^4 + Ss^2 (s2 - t1)^2 (s2^4 - (Ss + 9 t1) s2^3 + 3 (3 Ss - t1) t1 s2^2 + t1^2 (15 Ss + 11 t1) s2 + Ss t1^3) m^2 + Ss^3 (s2 - t1)^3 t1 (s2^2 - 4 t1 s2 + t1^2)) Log[((-m^2 + m2^2 + s2 - Sqrt[m^4 - 2 (m2^2 + s2) m^2 + (m2^2 - s2)^2]) (s2 - t1))/(2 s2)] s2^3 + 6 ((s2^6 + (Ss - 7 t1) s2^5 + (-4 Ss^2 + 13 t1 Ss + 16 t1^2) s2^4 + 2 (Ss^3 - 13 t1 Ss^2 - 10 t1^2 Ss - 8 t1^3) s2^3 + t1 (18 Ss^3 + 6 t1 Ss^2 - 4 t1^2 Ss + 7 t1^3) s2^2 + t1^2 (18 Ss^3 + 22 t1 Ss^2 + 11 t1^2 Ss - t1^3) s2 + Ss t1^3 (2 Ss^2 + 2 t1 Ss - t1^2)) m^6 - Ss (s2 - t1) ((11 t1 - 2 Ss) s2^4 + 2 (Ss^2 - 16 t1 Ss - 7 t1^2) s2^3 + 6 (4 Ss^2 t1 - t1^3) s2^2 + 2 t1^2 (15 Ss^2 + 16 t1 Ss + 5 t1^2) s2 + t1^3 (4 Ss^2 + 2 t1 Ss - t1^2)) m^4 + Ss^2 (s2 - t1)^2 (s2^4 - (Ss + 9 t1) s2^3 + 3 (3 Ss - t1) t1 s2^2 + t1^2 (15 Ss + 11 t1) s2 + Ss t1^3) m^2 + Ss^3 (s2 - t1)^3 t1 (s2^2 - 4 t1 s2 + t1^2)) Log[((-m^2 + m2^2 + s2 + Sqrt[m^4 - 2 (m2^2 + s2) m^2 + (m2^2 - s2)^2]) (s2 - t1))/(2 s2)] s2^3]

I need to collect the logarithms with the same prefactors to the form Log[.../...]. If I call already statically defined function F inside the construction

Simplify[F[s2_, Ss_, t1_, m_, m2_]]/. -Log[x_] + Log[y_] -> Log[y/x]

the logarithms become to be collected. But if the function F is defined dynamically (in my case - because of an integration, see the precise example below), the construction doesn't work. I.e., for

F[] := Evaluate[Simplify[Integrate[]]/. -Log[x_] + Log[y_] -> Log[y/x]] F[]

it doesn't work.

Only the construction FullSimplify works, but it requires so many time that I doesn't have the wish to wait. The Simplify construction, on the contrary, works very fast.

So, does anyone know some alternative ways to force Mathematica to collect the logarithms, especially working for dynamically defined functions?

I'm using Mathematica 11.

Precise example when Simplify doesn't work, but FullSimplify does

Just to illustrate the problem completely, I add the integral:

 f[Ss_,s2_,t1_,t2_,m_,m2_] =1/(32 \[Pi]^3 Ss^3 (s2-t1) (m^2-s2+t1-t2)) (-(((s2^2 t2-s2 t2 (Ss+t1)+Ss t1 (t1-s2)-Ss t1 t2) (s2^2-6 s2 (Ss-t1+t2)+Ss^2+2 Ss (t2-t1)-(t1-t2)^2))/(s2-t1)^2)+(s2-Ss) (s2^2+2 s2 (Ss-2 t1+2 t2)+Ss^2+(t1-t2)^2)-(2 (2 s2-2 Ss+t1-t2) (-2 Ss t1 t2 (s2-t1) (2 s2^2-2 s2 (Ss+t1)-Ss t1)+t2^2 (s2^4-2 s2^3 (Ss+t1)+s2^2 (Ss-t1)^2+4 s2 Ss t1 (Ss+t1)+Ss^2 t1^2)+Ss^2 t1^2 (s2-t1)^2))/(s2-t1)^4-(2 (s2^2 t2-s2 t2 (Ss+t1)+Ss t1 (t1-s2)-Ss t1 t2) (2 Ss t1 t2 (s2-t1) (-4 s2^2+4 s2 (Ss+t1)+Ss t1)+t2^2 (s2^4-2 s2^3 (Ss+t1)+s2^2 (Ss^2-6 Ss t1+t1^2)+8 s2 Ss t1 (Ss+t1)+Ss^2 t1^2)+Ss^2 t1^2 (s2-t1)^2))/(s2-t1)^6) F[Ss_,s2_,t1_,m_,m2_]=Assuming[s2> (m+m2)^2 && s2-t1 > 0 && m > 0 && m2 > 0,Simplify[Integrate[f[Ss,s2,t1,t2,m,m2],{t2,t2lower,t2upper}] ]]/.-Log[x_]+Log[y_]:> Log[y/x]

Here, the code defining the limits of integration is

lambda[a_, b_, c_] := a^2 + b^2 + c^2 - 2*(a*b + a*c + b*c) t2lower = Simplify[m^2 - 1/(2*s2)*(s2 - t1)*(s2 + m^2 - m2^2) - 1/(2*s2)*(s2 - t1)*Sqrt[lambda[s2, m^2, m2^2]]] t2upper = Simplify[m^2 - 1/(2*s2)*(s2 - t1)*(s2 + m^2 - m2^2) + 1/(2*s2)*(s2 - t1)*Sqrt[lambda[s2, m^2, m2^2]]]

For this construction, the logs collecting including Simplify doesn't work. Changing it to FullSimplify works, but it takes a plenty of time. I want to avoid this.

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  • $\begingroup$ I'm not sure I understand. Can you please give an example where FullSimplify works but Simplify does not? The origin of the expression should not matter, only its content. $\endgroup$ Commented Aug 1, 2017 at 21:19
  • $\begingroup$ @Mr.Wizard : in my specific case it doesn't. $\endgroup$ Commented Aug 1, 2017 at 21:21
  • $\begingroup$ Two general notes: (1) you can leave out Evaluate and define F with Set, i.e. F[ . . . ] = def and (2) you should be using RuleDelayed when you have named patterns on the left-hand-side, unless you specifically know otherwise, i.e. -Log[x_] + Log[y_] :> Log[y/x] $\endgroup$ Commented Aug 1, 2017 at 21:22
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    $\begingroup$ @Mr.Wizard : I've added the precise code to the question, for which Simplify doesn't collect the logarithms (with /. -Log[x_] + Log[y_] :> Log[y/x]), while FullSimplify (with a plenty of time) does. $\endgroup$ Commented Aug 1, 2017 at 21:51
  • $\begingroup$ @Mr.Wizard : I've corrected the example (there weren't definitions for the limits of the integration). $\endgroup$ Commented Aug 1, 2017 at 23:27

1 Answer 1

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In version 10.1

Assuming[s2 > (m + m2)^2 && s2 - t1 > 0 && m > 0 && m2 > 0, Integrate[f[Ss, s2, t1, t2, m, m2], {t2, t2lower, t2upper}]]

Yields for me this expression, which I replicate for reference:

expr = ConditionalExpression[-((-6 m^4 s2^6 t2lower - 6 m^2 s2^7 t2lower - 6 s2^8 t2lower - 6 m^4 s2^5 Ss t2lower - 6 m^2 s2^6 Ss t2lower - 6 s2^7 Ss t2lower + 24 m^4 s2^4 Ss^2 t2lower + 12 m^2 s2^5 Ss^2 t2lower + 6 s2^6 Ss^2 t2lower - 12 m^4 s2^3 Ss^3 t2lower + 6 s2^5 Ss^3 t2lower + 42 m^4 s2^5 t1 t2lower + 48 m^2 s2^6 t1 t2lower + 54 s2^7 t1 t2lower - 78 m^4 s2^4 Ss t1 t2lower - 6 m^2 s2^5 Ss t1 t2lower + 156 m^4 s2^3 Ss^2 t1 t2lower - 48 m^2 s2^4 Ss^2 t1 t2lower + 6 s2^5 Ss^2 t1 t2lower - 108 m^4 s2^2 Ss^3 t1 t2lower + 36 m^2 s2^3 Ss^3 t1 t2lower - 30 s2^4 Ss^3 t1 t2lower - 96 m^4 s2^4 t1^2 t2lower - 138 m^2 s2^5 t1^2 t2lower - 186 s2^6 t1^2 t2lower + 120 m^4 s2^3 Ss t1^2 t2lower + 48 m^2 s2^4 Ss t1^2 t2lower + 54 s2^5 Ss t1^2 t2lower - 36 m^4 s2^2 Ss^2 t1^2 t2lower - 48 s2^4 Ss^2 t1^2 t2lower - 108 m^4 s2 Ss^3 t1^2 t2lower + 36 m^2 s2^2 Ss^3 t1^2 t2lower + 24 s2^3 Ss^3 t1^2 t2lower + 96 m^4 s2^3 t1^3 t2lower + 192 m^2 s2^4 t1^3 t2lower + 330 s2^5 t1^3 t2lower + 24 m^4 s2^2 Ss t1^3 t2lower - 48 m^2 s2^3 Ss t1^3 t2lower - 96 s2^4 Ss t1^3 t2lower - 132 m^4 s2 Ss^2 t1^3 t2lower + 96 m^2 s2^2 Ss^2 t1^3 t2lower + 48 s2^3 Ss^2 t1^3 t2lower - 12 m^4 Ss^3 t1^3 t2lower - 60 m^2 s2 Ss^3 t1^3 t2lower + 24 s2^2 Ss^3 t1^3 t2lower - 42 m^4 s2^2 t1^4 t2lower - 138 m^2 s2^3 t1^4 t2lower - 330 s2^4 t1^4 t2lower - 66 m^4 s2 Ss t1^4 t2lower + 6 m^2 s2^2 Ss t1^4 t2lower + 54 s2^3 Ss t1^4 t2lower - 12 m^4 Ss^2 t1^4 t2lower - 60 m^2 s2 Ss^2 t1^4 t2lower - 6 s2^2 Ss^2 t1^4 t2lower - 12 m^2 Ss^3 t1^4 t2lower - 30 s2 Ss^3 t1^4 t2lower + 6 m^4 s2 t1^5 t2lower + 48 m^2 s2^2 t1^5 t2lower + 186 s2^3 t1^5 t2lower + 6 m^4 Ss t1^5 t2lower + 6 m^2 s2 Ss t1^5 t2lower - 6 s2 Ss^2 t1^5 t2lower + 6 Ss^3 t1^5 t2lower - 6 m^2 s2 t1^6 t2lower - 54 s2^2 t1^6 t2lower - 6 s2 Ss t1^6 t2lower + 6 s2 t1^7 t2lower - 3 m^2 s2^6 t2lower^2 - 6 s2^7 t2lower^2 - 3 m^2 s2^5 Ss t2lower^2 - 6 s2^6 Ss t2lower^2 + 12 m^2 s2^4 Ss^2 t2lower^2 + 18 s2^5 Ss^2 t2lower^2 - 6 m^2 s2^3 Ss^3 t2lower^2 - 6 s2^4 Ss^3 t2lower^2 + 21 m^2 s2^5 t1 t2lower^2 + 48 s2^6 t1 t2lower^2 - 39 m^2 s2^4 Ss t1 t2lower^2 - 39 s2^5 Ss t1 t2lower^2 + 78 m^2 s2^3 Ss^2 t1 t2lower^2 + 42 s2^4 Ss^2 t1 t2lower^2 - 54 m^2 s2^2 Ss^3 t1 t2lower^2 - 30 s2^3 Ss^3 t1 t2lower^2 - 48 m^2 s2^4 t1^2 t2lower^2 - 138 s2^5 t1^2 t2lower^2 + 60 m^2 s2^3 Ss t1^2 t2lower^2 + 123 s2^4 Ss t1^2 t2lower^2 - 18 m^2 s2^2 Ss^2 t1^2 t2lower^2 - 96 s2^3 Ss^2 t1^2 t2lower^2 - 54 m^2 s2 Ss^3 t1^2 t2lower^2 + 18 s2^2 Ss^3 t1^2 t2lower^2 + 48 m^2 s2^3 t1^3 t2lower^2 + 192 s2^4 t1^3 t2lower^2 + 12 m^2 s2^2 Ss t1^3 t2lower^2 - 72 s2^3 Ss t1^3 t2lower^2 - 66 m^2 s2 Ss^2 t1^3 t2lower^2 - 6 m^2 Ss^3 t1^3 t2lower^2 + 18 s2 Ss^3 t1^3 t2lower^2 - 21 m^2 s2^2 t1^4 t2lower^2 - 138 s2^3 t1^4 t2lower^2 - 33 m^2 s2 Ss t1^4 t2lower^2 - 42 s2^2 Ss t1^4 t2lower^2 - 6 m^2 Ss^2 t1^4 t2lower^2 + 30 s2 Ss^2 t1^4 t2lower^2 + 3 m^2 s2 t1^5 t2lower^2 + 48 s2^2 t1^5 t2lower^2 + 3 m^2 Ss t1^5 t2lower^2 + 39 s2 Ss t1^5 t2lower^2 + 6 Ss^2 t1^5 t2lower^2 - 6 s2 t1^6 t2lower^2 - 3 Ss t1^6 t2lower^2 - 2 s2^6 t2lower^3 - 2 s2^5 Ss t2lower^3 + 8 s2^4 Ss^2 t2lower^3 - 4 s2^3 Ss^3 t2lower^3 + 14 s2^5 t1 t2lower^3 - 26 s2^4 Ss t1 t2lower^3 + 52 s2^3 Ss^2 t1 t2lower^3 - 36 s2^2 Ss^3 t1 t2lower^3 - 32 s2^4 t1^2 t2lower^3 + 40 s2^3 Ss t1^2 t2lower^3 - 12 s2^2 Ss^2 t1^2 t2lower^3 - 36 s2 Ss^3 t1^2 t2lower^3 + 32 s2^3 t1^3 t2lower^3 + 8 s2^2 Ss t1^3 t2lower^3 - 44 s2 Ss^2 t1^3 t2lower^3 - 4 Ss^3 t1^3 t2lower^3 - 14 s2^2 t1^4 t2lower^3 - 22 s2 Ss t1^4 t2lower^3 - 4 Ss^2 t1^4 t2lower^3 + 2 s2 t1^5 t2lower^3 + 2 Ss t1^5 t2lower^3 + 6 m^4 s2^6 t2upper + 6 m^2 s2^7 t2upper + 6 s2^8 t2upper + 6 m^4 s2^5 Ss t2upper + 6 m^2 s2^6 Ss t2upper + 6 s2^7 Ss t2upper - 24 m^4 s2^4 Ss^2 t2upper - 12 m^2 s2^5 Ss^2 t2upper - 6 s2^6 Ss^2 t2upper + 12 m^4 s2^3 Ss^3 t2upper - 6 s2^5 Ss^3 t2upper - 42 m^4 s2^5 t1 t2upper - 48 m^2 s2^6 t1 t2upper - 54 s2^7 t1 t2upper + 78 m^4 s2^4 Ss t1 t2upper + 6 m^2 s2^5 Ss t1 t2upper - 156 m^4 s2^3 Ss^2 t1 t2upper + 48 m^2 s2^4 Ss^2 t1 t2upper - 6 s2^5 Ss^2 t1 t2upper + 108 m^4 s2^2 Ss^3 t1 t2upper - 36 m^2 s2^3 Ss^3 t1 t2upper + 30 s2^4 Ss^3 t1 t2upper + 96 m^4 s2^4 t1^2 t2upper + 138 m^2 s2^5 t1^2 t2upper + 186 s2^6 t1^2 t2upper - 120 m^4 s2^3 Ss t1^2 t2upper - 48 m^2 s2^4 Ss t1^2 t2upper - 54 s2^5 Ss t1^2 t2upper + 36 m^4 s2^2 Ss^2 t1^2 t2upper + 48 s2^4 Ss^2 t1^2 t2upper + 108 m^4 s2 Ss^3 t1^2 t2upper - 36 m^2 s2^2 Ss^3 t1^2 t2upper - 24 s2^3 Ss^3 t1^2 t2upper - 96 m^4 s2^3 t1^3 t2upper - 192 m^2 s2^4 t1^3 t2upper - 330 s2^5 t1^3 t2upper - 24 m^4 s2^2 Ss t1^3 t2upper + 48 m^2 s2^3 Ss t1^3 t2upper + 96 s2^4 Ss t1^3 t2upper + 132 m^4 s2 Ss^2 t1^3 t2upper - 96 m^2 s2^2 Ss^2 t1^3 t2upper - 48 s2^3 Ss^2 t1^3 t2upper + 12 m^4 Ss^3 t1^3 t2upper + 60 m^2 s2 Ss^3 t1^3 t2upper - 24 s2^2 Ss^3 t1^3 t2upper + 42 m^4 s2^2 t1^4 t2upper + 138 m^2 s2^3 t1^4 t2upper + 330 s2^4 t1^4 t2upper + 66 m^4 s2 Ss t1^4 t2upper - 6 m^2 s2^2 Ss t1^4 t2upper - 54 s2^3 Ss t1^4 t2upper + 12 m^4 Ss^2 t1^4 t2upper + 60 m^2 s2 Ss^2 t1^4 t2upper + 6 s2^2 Ss^2 t1^4 t2upper + 12 m^2 Ss^3 t1^4 t2upper + 30 s2 Ss^3 t1^4 t2upper - 6 m^4 s2 t1^5 t2upper - 48 m^2 s2^2 t1^5 t2upper - 186 s2^3 t1^5 t2upper - 6 m^4 Ss t1^5 t2upper - 6 m^2 s2 Ss t1^5 t2upper + 6 s2 Ss^2 t1^5 t2upper - 6 Ss^3 t1^5 t2upper + 6 m^2 s2 t1^6 t2upper + 54 s2^2 t1^6 t2upper + 6 s2 Ss t1^6 t2upper - 6 s2 t1^7 t2upper + 3 m^2 s2^6 t2upper^2 + 6 s2^7 t2upper^2 + 3 m^2 s2^5 Ss t2upper^2 + 6 s2^6 Ss t2upper^2 - 12 m^2 s2^4 Ss^2 t2upper^2 - 18 s2^5 Ss^2 t2upper^2 + 6 m^2 s2^3 Ss^3 t2upper^2 + 6 s2^4 Ss^3 t2upper^2 - 21 m^2 s2^5 t1 t2upper^2 - 48 s2^6 t1 t2upper^2 + 39 m^2 s2^4 Ss t1 t2upper^2 + 39 s2^5 Ss t1 t2upper^2 - 78 m^2 s2^3 Ss^2 t1 t2upper^2 - 42 s2^4 Ss^2 t1 t2upper^2 + 54 m^2 s2^2 Ss^3 t1 t2upper^2 + 30 s2^3 Ss^3 t1 t2upper^2 + 48 m^2 s2^4 t1^2 t2upper^2 + 138 s2^5 t1^2 t2upper^2 - 60 m^2 s2^3 Ss t1^2 t2upper^2 - 123 s2^4 Ss t1^2 t2upper^2 + 18 m^2 s2^2 Ss^2 t1^2 t2upper^2 + 96 s2^3 Ss^2 t1^2 t2upper^2 + 54 m^2 s2 Ss^3 t1^2 t2upper^2 - 18 s2^2 Ss^3 t1^2 t2upper^2 - 48 m^2 s2^3 t1^3 t2upper^2 - 192 s2^4 t1^3 t2upper^2 - 12 m^2 s2^2 Ss t1^3 t2upper^2 + 72 s2^3 Ss t1^3 t2upper^2 + 66 m^2 s2 Ss^2 t1^3 t2upper^2 + 6 m^2 Ss^3 t1^3 t2upper^2 - 18 s2 Ss^3 t1^3 t2upper^2 + 21 m^2 s2^2 t1^4 t2upper^2 + 138 s2^3 t1^4 t2upper^2 + 33 m^2 s2 Ss t1^4 t2upper^2 + 42 s2^2 Ss t1^4 t2upper^2 + 6 m^2 Ss^2 t1^4 t2upper^2 - 30 s2 Ss^2 t1^4 t2upper^2 - 3 m^2 s2 t1^5 t2upper^2 - 48 s2^2 t1^5 t2upper^2 - 3 m^2 Ss t1^5 t2upper^2 - 39 s2 Ss t1^5 t2upper^2 - 6 Ss^2 t1^5 t2upper^2 + 6 s2 t1^6 t2upper^2 + 3 Ss t1^6 t2upper^2 + 2 s2^6 t2upper^3 + 2 s2^5 Ss t2upper^3 - 8 s2^4 Ss^2 t2upper^3 + 4 s2^3 Ss^3 t2upper^3 - 14 s2^5 t1 t2upper^3 + 26 s2^4 Ss t1 t2upper^3 - 52 s2^3 Ss^2 t1 t2upper^3 + 36 s2^2 Ss^3 t1 t2upper^3 + 32 s2^4 t1^2 t2upper^3 - 40 s2^3 Ss t1^2 t2upper^3 + 12 s2^2 Ss^2 t1^2 t2upper^3 + 36 s2 Ss^3 t1^2 t2upper^3 - 32 s2^3 t1^3 t2upper^3 - 8 s2^2 Ss t1^3 t2upper^3 + 44 s2 Ss^2 t1^3 t2upper^3 + 4 Ss^3 t1^3 t2upper^3 + 14 s2^2 t1^4 t2upper^3 + 22 s2 Ss t1^4 t2upper^3 + 4 Ss^2 t1^4 t2upper^3 - 2 s2 t1^5 t2upper^3 - 2 Ss t1^5 t2upper^3 - 6 (Ss^3 (s2 - t1)^3 t1 (s2^2 - 4 s2 t1 + t1^2) + m^2 Ss^2 (s2 - t1)^2 (s2^4 + 3 s2^2 (3 Ss - t1) t1 + Ss t1^3 - s2^3 (Ss + 9 t1) + s2 t1^2 (15 Ss + 11 t1)) - m^4 Ss (s2 - t1) (s2^4 (-2 Ss + 11 t1) + 2 s2^3 (Ss^2 - 16 Ss t1 - 7 t1^2) + t1^3 (4 Ss^2 + 2 Ss t1 - t1^2) + 2 s2 t1^2 (15 Ss^2 + 16 Ss t1 + 5 t1^2) + 6 s2^2 (4 Ss^2 t1 - t1^3)) + m^6 (s2^6 + s2^5 (Ss - 7 t1) + Ss t1^3 (2 Ss^2 + 2 Ss t1 - t1^2) + s2^4 (-4 Ss^2 + 13 Ss t1 + 16 t1^2) + 2 s2^3 (Ss^3 - 13 Ss^2 t1 - 10 Ss t1^2 - 8 t1^3) + s2 t1^2 (18 Ss^3 + 22 Ss^2 t1 + 11 Ss t1^2 - t1^3) + s2^2 t1 (18 Ss^3 + 6 Ss^2 t1 - 4 Ss t1^2 + 7 t1^3))) Log[-m^2 + s2 - t1 + t2lower] + 6 (Ss^3 (s2 - t1)^3 t1 (s2^2 - 4 s2 t1 + t1^2) + m^2 Ss^2 (s2 - t1)^2 (s2^4 + 3 s2^2 (3 Ss - t1) t1 + Ss t1^3 - s2^3 (Ss + 9 t1) + s2 t1^2 (15 Ss + 11 t1)) - m^4 Ss (s2 - t1) (s2^4 (-2 Ss + 11 t1) + 2 s2^3 (Ss^2 - 16 Ss t1 - 7 t1^2) + t1^3 (4 Ss^2 + 2 Ss t1 - t1^2) + 2 s2 t1^2 (15 Ss^2 + 16 Ss t1 + 5 t1^2) + 6 s2^2 (4 Ss^2 t1 - t1^3)) + m^6 (s2^6 + s2^5 (Ss - 7 t1) + Ss t1^3 (2 Ss^2 + 2 Ss t1 - t1^2) + s2^4 (-4 Ss^2 + 13 Ss t1 + 16 t1^2) + 2 s2^3 (Ss^3 - 13 Ss^2 t1 - 10 Ss t1^2 - 8 t1^3) + s2 t1^2 (18 Ss^3 + 22 Ss^2 t1 + 11 Ss t1^2 - t1^3) + s2^2 t1 (18 Ss^3 + 6 Ss^2 t1 - 4 Ss t1^2 + 7 t1^3))) Log[-m^2 + s2 - t1 + t2upper])/(192 \[Pi]^3 Ss^3 (s2 - t1)^7)), (Re[( m^2 - s2 + t1 - t2lower)/( t2lower - t2upper)] < -1 && (Re[(-m^2 + s2 - t1 + t2lower)/( t2lower - t2upper)] == 0 || Re[(-m^2 + s2 - t1 + t2lower)/(t2lower - t2upper)] >= 1)) || ( m^2 - s2 + t1 - t2lower)/(t2lower - t2upper) \[NotElement] Reals || (Re[(m^2 - s2 + t1 - t2lower)/(t2lower - t2upper)] >= 0 && Re[(-m^2 + s2 - t1 + t2lower)/(t2lower - t2upper)] < 0)]

I think this might do what you want. There are only two Log expressions and they are contiguous.

Position[expr[[1]], _Log] 
{{5, 265, 3}, {5, 266, 3}}

If we extract them, simplify, and replace them in the original:

old = expr[[1, 5, 265 ;; 266]]; new = Simplify[old] /. -Log[x_] + Log[y_] :> Log[y/x]; expr /. old -> new

In which appears Log[(-m^2 + s2 - t1 + t2upper)/(-m^2 + s2 - t1 + t2lower)]

You might be able to use Collect[expr[[1]], _Log] first if the expression is not of this form. Or all this might be a terrible idea, as I think I'm falling asleep. Good luck. :^)

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