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Is there a better way to get a closed form solution in terms of sigma? I've already attempted a significant amount of simplification up to this point and am unsure how to go on. The below "full simplify" doesn't run on my machine:

FullSimplify[
 Log[1/(1 +
     Integrate[(-((-3*K^2 + 6*K + 1^2)/(2*K^(3/2)*(K + 1)^3)))*
       ((1/2)*K*
          Erfc[(-(σ^2/2) + Log[1/K])/(Sqrt[2]*σ)] -
         (1/2)*
          Erfc[(σ^2/2 + Log[1/K])/(Sqrt[2]*σ)]), {K, 0, 1}] +
     Integrate[(-((-3*K^2 + 6*K + 1^2)/(2*K^(3/2)*(K + 1)^3)))*
       ((-(1/2))*K*
          Erfc[-((-(σ^2/2) + Log[1/K])/(Sqrt[2]*σ))] +
         (1/2)*
          Erfc[-((σ^2/2 + Log[1/K])/(Sqrt[2]*σ))]),
      {K, 1, Infinity}])], Assumptions -> {σ > 0}]
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1 Answer 1

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When using Rubi, you can see in detail which parts of your expression seem to not have an analytic solution. For your first integral, one of these parts is

Mathematica graphics

Even when stripping the analytic σ and reducing the expression, it appears that this similar form has no antiderivative which can be calculated by either Mathematica or with Rubi

Mathematica graphics

Therefore, I tend to say you'll hit a dead end trying to find an analytic solution to your much more complex expression and you are left with numerics.

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