The triple integral:
$$\int_{0}^{1} \int_{0}^{1} \int_{0}^{1} \frac{dx dy dz}{(1+x^2+y^2+z^2)^2}=\frac{\pi^2}{32}$$
can be confirmed using Mathematica.
But how can one show it by hand? I provide two answers below: 1,2. Are there any other methods?
[Context added later by Jack:]
See also this calculation from Wolfram Alpha:
[Added later by OP:] The triple integral was fabricated by the method of the integral representation of $1/p^2=\int_{0}^{\infty} t e^{pt} dt.$ See my first solution in the Answer below. The final result was confirmed numerically by Mathematica. More interestingly Mathematical gives this integral $I=\pi^2/12-A$ in analytic mode where A is Ahmed's Integral! which equals $5\pi^2/96$.