calculus and analysis – Evaluate a certain three-dimensional constrained integral


The result of the three-dimensional integration

Integrate(9081072000 (Subscript((Lambda), 1) - Subscript((Lambda), 
2))^2 (Subscript((Lambda), 1) - Subscript((Lambda), 
3))^2 (Subscript((Lambda), 2) - Subscript((Lambda), 3))^2 (-1 + 
2 Subscript((Lambda), 1) + Subscript((Lambda), 2) + 
Subscript((Lambda), 3))^2 (-1 + Subscript((Lambda), 1) + 
2 Subscript((Lambda), 2) + Subscript((Lambda), 3))^2 (-1 + 
Subscript((Lambda), 1) + Subscript((Lambda), 2) + 
2 Subscript((Lambda), 3))^2 Boole(Subscript((Lambda), 1) > Subscript((Lambda), 2) && 
Subscript((Lambda), 2) > Subscript((Lambda), 3) && 
Subscript((Lambda), 3) > 
 1 - Subscript((Lambda), 1) - Subscript((Lambda), 2) - 
  Subscript((Lambda), 3) && 
Subscript((Lambda), 1) - Subscript((Lambda), 3) < 
 2 Sqrt(Subscript((Lambda), 
   2) (1 - Subscript((Lambda), 1) - Subscript((Lambda), 2) - 
     Subscript((Lambda), 3)))), {Subscript((Lambda), 3), 0, 1}, {Subscript((Lambda), 2), 0, 1}, {Subscript((Lambda), 1), 0, 1}),

that is,

3Dintegral

for the two-qubit Hilbert-Schmidt absolute separability probability
apparently can be expressed as

begin{equation} label{HSabs}
frac{29902415923}{497664}+frac{-3217542976+5120883075 pi -16386825840 tan
^{-1}left(sqrt{2}right)}{32768 sqrt{2}} =
end{equation}

begin{equation}
frac{32(29902415923 – 24433216974 sqrt{2})+248874917445 sqrt{2}(5 pi – 16 tan ^{-1}left(sqrt{2}right))}{2^{16} cdot 3^5} approx 0.00365826
end{equation}

QuantumComputingStackExchangeQuestion

Can this be explicitly confirmed using Mathematica?