# 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,

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?