Hi All, I was trying to derive a formula by evaluating an integral. I defined all the restrictions for that setting and I wanted to have the solution in terms of a simplified formula. But Mathematica did not give the solution like last time (with some modification). I was wondering if there is anyone who can help me solve this problem.

Thank you in advance for your kind attention and support.

Here is the code for the equation A with Cp and infectable functions defined in it.

Cp[te_, Di_, Dv_, D0_] :=
 Which[
  Di + D0 <= Dv + te || Dv + te <= 0, 0,
  D0 < Dv + te && D0 + te < 0 && Di > Dv + te, Inf/(Inf + Ip),
  D0 < Dv + te && D0 + te >= 0 && Di > Dv + te, (D0 Inf)/(
  D0 Inf - Ip te),
  Di <= Dv + 
     te && ((te == 0 && D0 >= Dv) || (D0 >= Dv + te && te >= 0)), Di/
  Dv,
  D0 >= Dv + te && D0 + te < 0 && Di <= Dv + te, (Di Inf)/(
  D0 Ip + Inf (Dv + te)),
  D0 >= Dv + te && te < 0 && D0 + te >= 0 && Di <= Dv + te, (Di Inf)/(
  Dv Inf + Inf te - Ip te),
  D0 < Dv + te && D0 + Di >= Dv + te && D0 + te < 0 && 
   Di <= Dv + te, (Inf (D0 + Di - Dv - te))/(D0 (Inf + Ip)),
  D0 < Dv + te && D0 + Di >= Dv + te && te < 0 && D0 + te >= 0 && 
   Di <= Dv + te, (Inf (D0 + Di - Dv - te))/(D0 Inf - Ip te),
  te == 0 && D0 < Dv && D0 + Di >= Dv, (D0 + Di - Dv)/D0,
  D0 < Dv + te && D0 + Di >= Dv + te && te >= 0, 
  1 + (Di - Dv)/(D0 - te),
  D0 >= Dv + te && D0 + te < 0 && Di > Dv + te && Dv + te > 0, (
  Inf (Dv + te))/(D0 Ip + Inf (Dv + te)),
  D0 >= Dv + te && D0 + te >= 0 && Di > Dv + te && Dv + te > 0, (
  Inf (Dv + te))/(Dv Inf + (Inf - Ip) te),
  True, 0]
infectable[t_] := UnitStep[t]*UnitStep[D0 - t]
{$Assumptions = 
   Di < Dv && Dv < D0 && Di > 0 && Di \[Element] Reals && 
    D0 \[Element] Reals && Inf > 0 && Inf \[Element] Reals && 
    Ip > 0  && Ip \[Element] Reals  };
A = FullSimplify[(Inf/N/D0)*phi*tau*
   Integrate[
    Integrate[
     Integrate[
      infectable[ti] Cp[te, Di, Dv, D0], {tx, te + Dv, ti + Di}], {ti,
       te + Dv - Di, te + Dv}], {te, -Dv, D0 - Dv + Di}]]

It used to work when the Cp was defined as (Cor) for the integral (A).

Cor[te_, Di_, Dv_, D0_] :=
 Which[Di > Dv + te && Dv + te > 0, 1,
  te >= 0 && Dv + te <= D0, Di/Dv,
  te < 0 && Di <= Dv + te && Dv + te <= D0, Di/(Dv + te),
  te < 0 && Dv + te > D0, (Di - Dv - te + D0)/D0,
  Di + D0 < Dv + te || Dv + te <= 0, 0,
  True, 1 + (-Di + Dv)/(te - D0)]

infectable[t_] := UnitStep[t]*UnitStep[D0 - t]

{$Assumptions = 
   Di < Dv && Dv < D0 && Di > 0 && Di \[Element] Reals && 
    D0 \[Element] Reals && Inf > 0 && Inf \[Element] Reals && 
    Ip > 0  && Ip \[Element] Reals  };
A = FullSimplify[(Inf/N/D0)*phi*tau*
   Integrate[
    Integrate[
     Integrate[
      infectable[ti] Cor[te, Di, Dv, D0], {tx, te + Dv, 
       ti + Di}], {ti, te + Dv - Di, te + Dv}], {te, -Dv, 
     D0 - Dv + Di}]]

Here is the result from that equation

1/(4 D0 Dv N)
  Inf phi tau (Di (2 D0 Di^2 + (3 Di - 2 Dv) (Di - Dv) Dv) + 
    2 Dv ((-Di + Dv)^3 Log[1 - Di/Dv] + Di^3 Log[Dv/Di]))

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