<s>[INST] High dimensional generalizations of $\int_{\sqrt{2} }^{\sqrt{3} } \frac{\arctan(y)}{(y^2-1)\sqrt{y^2-2} } dy =\frac{5\pi^2}{96}$ Let $Q=n_1+n_2+n_3+1$,$\mathbf{s}=(n_1,n_2,n_3)$.
Define
$$
A(\mathbf{s})
=\int_{D}\prod_{n=1}^{Q-1}\frac{1}{1+x_n^2}\int_{1}^{\infty}\left(Q+\sum_{n=1}^{Q-1}x_n^2 +y^2\right)^{-1}\mathrm{d}y
\text{d}x_i.
$$
Where $D=[0,\infty]^{n_1}\times[0,1]^{n_2}
\times[1,\infty]^{n_3}\subset\mathbb{R}^{Q-1}$,
$\mathrm{d}x_i
=\prod_{n=1}^{Q-1} \text{d}x_n$.

*

*Special case.
For  $\mathbf{s}=(0,0,n_3)$, we have
$$
A(\mathbf{s})
=\frac{1}{Q}\left ( \frac{\pi}{4}  \right )^Q.
$$

*Question 1
Prove
$$
\pi^{-Q}A(\mathbf{s})\in\mathbb{Q}.
$$

My ultimate goal is to evaluate $A(\mathbf{s})$. Numerical calculations suggest that
$$A(1,0,0)=\frac{\pi^2}{12} \quad A(0,1,0)=\frac{5\pi^2}{96}\quad A(0,0,1)=\frac{\pi^2}{32}$$
$$A(2,0,0)=\frac{\pi^3}{32} \quad A(1,1,0)=\frac{\pi^3}{80}$$
$$A(0,3,0)=\frac{93\pi^4}{35840} \qquad A(0,4,0)=\frac{193\pi^5}{322560}$$
Actually, if we explicit calculate the multiple integrals, it yields
$$
\begin{aligned}
&\int_{\sqrt{2} }^{\sqrt{3} } 
\frac{\arctan(y)}{(y^2-1)\sqrt{y^2-2} } \text{d}y
=\frac{5\pi^2}{96},\\
&\frac{\pi}{6} \int_{\sqrt{3} }^{\sqrt{5} } 
\frac{\arctan(y)}{(y^2-1)\sqrt{y^2-2} } \text{d}y
-\int_{2}^{\sqrt{5} } 
\frac{\displaystyle{\arctan (y)\arctan \sqrt{\frac{y^2-4}{y^2-2} } } }{(y^2-1)\sqrt{y^2-2} }
\text{d}y=\frac{11\pi^3}{5760},\\
&\frac{\pi}{6} \int_{\sqrt{3} }^{\sqrt{5} } 
\frac{\arctan\left(y\sqrt{2+y^2} \right)}{(y^2-1)\sqrt{y^2-2} } \text{d}y
-\int_{2}^{\sqrt{5} } 
\frac{\displaystyle{\arctan\left(y\sqrt{2+y^2}\right)\arctan \sqrt{\frac{y^2-4}{y^2-2} } } }{(y^2-1)\sqrt{y^2-2} }
\text{d}y=\frac{\pi^3}{420}.
\end{aligned}
$$
Are there any other simple results? For $Q=4$, we may meet some 'troubles', such as this one:
$$
\int_{0}^{1} \int_{1}^{\sqrt{2} } 
\frac{u\left(\pi-2\arctan\sqrt{u^4-1}-2\arctan \sqrt{\frac{u^2-1}{u^2+1} }
\right) \arctan \sqrt{4+u^2+v^2}  }
{(1+v^2)\sqrt{1+u^2}(2+u^2) \sqrt{4+u^2+v^2}  } \text{d}u\text{d}v.
$$
I can hardly convert into a 'simple' form.

*

*Question 2.
Can we evaluate a more general family of this kind of integrals?
$$A(\alpha,\mathbf{s})
=\int_{D}\prod_{n=1}^{Q-1}\frac{1}{\alpha^2+x_n^2}\int_{1}^{\infty}\left(\alpha^2 Q+\sum_{n=1}^{Q-1}x_n^2 +y^2\right)^{-1}\mathrm{d}y
\text{d}x_i.$$
 [/INST] I've found a direct way.
Let me give an example,
$$
I=\int_{0}^{1} \int_{0}^{1} \int_{0}^{1} 
\frac{1}{(1+x^2)(1+y^2)(1+z^2)\sqrt{4+x^2+y^2+z^2} }\text{d}x\text{d}y\text{d}z.
$$
Use the fact that
$$
\frac{1}{\sqrt{x} } =\frac{2}{\sqrt{\pi}}\int_{0}^{\infty}e^{-xt^2}\text{d}t
$$,
and rewrite the integral into
$$I = \frac{2}{\sqrt{\pi}} 
\int_{0}^{\infty}e^{-4t^2}
\left ( \int_{0}^{1} \frac{e^{-x^2t^2}}{1+x^2} \text{d}x \right )^3
 \text{d}t.$$
The inner integral is simply equal to
$$\int_{0}^{1} \frac{e^{-x^2t^2}}{1+x^2} \text{d}x 
=\frac{\pi}{4}e^{t^2}\left(1-\operatorname{erf}(t)^2\right).$$
Then,$$I=\frac{\pi^3}{32}\cdot\frac{1}{\sqrt{\pi}} 
\int_{0}^{\infty}e^{-t^2}\left(1-\operatorname{erf}(t)^2\right)^3\text{d}t.$$
Note that the integrand has a polynomial of $\operatorname{erf}(t)$. That is to say, if we set
$$P(x)=a_n x^n+a_{n-1} x^{n-1}+...+a_{1} x+a_0$$,
we get
$$\begin{aligned}
I & = \frac{\pi^3}{32}\cdot\frac{1}{\sqrt{\pi} } 
\int_{0}^{\infty}e^{-t^2}P(\operatorname{erf}(t))\text{d}t \\
& = \frac{\pi^3}{32}\cdot\frac{1}{\sqrt{\pi} }
\int_{0}^{\infty}e^{-t^2}\sum_{k=0}^{n}a_k\operatorname{erf}(t)^k\text{d}t\\
&= \frac{\pi^3}{32}\cdot\frac{1}{\sqrt{\pi} }
\sum_{k=0}^{n}a_k \int_{0}^{\infty}e^{-t^2}\operatorname{erf}(t)^k\text{d}t\\
&=\frac{\pi^3}{32}\cdot\frac{1}{\sqrt{\pi} }
\sum_{k=0}^{n}a_k\cdot\frac{\sqrt{\pi} }{2(k+1)} \\
&=\frac{\pi^3}{64} \sum_{k=0}^{n} \frac{a_k}{k+1} 
\end{aligned}$$
The finite sum is easy to evaluate, and we conclude that
$$I=\frac{\pi^3}{140} .$$

These four integrals have the similar expression:
$$\begin{aligned}
&\int_{0}^{1} \int_{0}^{1} \int_{0}^{1} 
\frac{1}{(1+x^2+y^2+z^2)^2}\text{d}x\text{d}y\text{d}z\\
&\int_{0}^{1} \int_{0}^{1} \int_{0}^{1} 
\frac{1}{(1+x^2)(2+x^2+y^2+z^2)^{3/2}}\text{d}x\text{d}y\text{d}z\\
&\int_{0}^{1} \int_{0}^{1} \int_{0}^{1} 
\frac{1}{(1+x^2)(1+y^2)(3+x^2+y^2+z^2)}\text{d}x\text{d}y\text{d}z\\
&\int_{0}^{1} \int_{0}^{1} \int_{0}^{1} 
\frac{1}{(1+x^2)(1+y^2)(1+z^2)\sqrt{4+x^2+y^2+z^2}}\text{d}x\text{d}y\text{d}z.
\end{aligned}$$
Their limits can rewrite as $[1,\infty],[0,\infty]$ and still have similar results, of course.
</s>