Integrate[(2 z - 4 z (9 - x^2 - y^2 - z^2))/(
2 Sqrt[z^2 + (9 - x^2 - y^2 - z^2)^2])*(1/
e^((x + y + z - \[Sqrt](z^2 + (9 - x^2 - y^2 - z^2)^2) - 10)/
20) - 1/e^((x + y + z + \[Sqrt](z^2 + (9 - x^2 - y^2 - z^2)^2) -
10)/20))*(z^2 + (9 - x^2 - y^2 - z^2)^2)^2 DiracDelta[
2 Sqrt[z^2 + (9 - x^2 - y^2 - z^2)^2] - 4], {x, -2*\[Pi],
2*\[Pi]}, {y, -2*\[Pi], 2*\[Pi]}, {z, -2*\[Pi], 2*\[Pi]}]
2 Sqrt[z^2 + (9 - x^2 - y^2 - z^2)^2])*(1/
e^((x + y + z - \[Sqrt](z^2 + (9 - x^2 - y^2 - z^2)^2) - 10)/
20) - 1/e^((x + y + z + \[Sqrt](z^2 + (9 - x^2 - y^2 - z^2)^2) -
10)/20))*(z^2 + (9 - x^2 - y^2 - z^2)^2)^2 DiracDelta[
2 Sqrt[z^2 + (9 - x^2 - y^2 - z^2)^2] - 4], {x, -2*\[Pi],
2*\[Pi]}, {y, -2*\[Pi], 2*\[Pi]}, {z, -2*\[Pi], 2*\[Pi]}]
