<s>[INST] How can I integrate this function? $\frac{\sqrt{e^{2x+2y+z}}}{(1+e^x+e^y+e^{x+y+z})^2}$ I want to evaluate the following integration
$$
\int_{-\infty}^{\infty} dx \int_{-\infty}^{\infty} dy \int_{-\infty}^{\infty} dz \frac{\sqrt{e^{2x+2y+z}}}{(1+e^x+e^y+e^{x+y+z})^2}.
$$
According to Mathematica12, the answer is $\pi^2$.
How can I get the answer?

 [/INST] Let $x= \ln X$, $y=\ln Y$ and $z=2\ln Z$. Then
$$ I = \int_0^\infty\int_0^\infty\int_0^\infty\frac{XYZ}{(1+X+Y+XYZ^2)^2} \frac{dX}{X}\frac{dY}{Y}\frac{2\cdot dZ}{Z}.$$
This is a rational function, and thus can be easily integrated. For instance integrate first against X:
$$ I = 2\int_0^\infty\int_0^\infty \frac{1}{1+YZ^2}\frac{1}{1+Y}dY dZ. $$
Then integrate against $Z$:
$$ I = 2\int_0^\infty \frac{\pi}{2\sqrt{Y}}\frac{1}{1+Y}dY . $$
Then, let $u=\sqrt{Y}$ to obtain
$$I= \pi\int_0^\infty \frac{2u }{u}\frac{1}{1+u^2}du=\pi^2.$$
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