<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] The claim is non-trivial. Let $$g(n_1,n_2) = \int_1^\infty\int_{{{[0,\pi /4]}^{{n_1}}}{{\times [\pi /4,\pi /2]}^{n_2}}} {\frac{d{x_i}}{{{1+a^2 + {{\sec }^2}{x_1} + ... + {{\sec }^2}{x_n}}}}} da$$
We need to prove $$\tag{*}g(n_1,n_2)\in \pi^{1+n_1+n_2} \mathbb{Q}$$

The solution is largely in parallel to the one I wrote here. One should read that answer first. For any (measurable) set $A\subset \mathbb{R}^{2n}$, let
$$\nu(A) = \int_1^\infty \int_{A} \frac{dx_i}{(1+a^2+x_1^2+\cdots+x_{2n}^2)^{n+1}} da$$
With $T,R$ as in the linked answer, mutatis mutandis, we have (with $n=n_1+n_2$), $$\tag{1}\nu({T^{{n_1}}} \times {R^{{n_2}}}) = \sum\limits_{k = 0}^{{n_2}} {\binom{n_2}{k}\frac{{{{( - 1)}^k}}}{{{2^{{n_1} + k}}}}\nu({{[0,1]}^{2{n_1} + {n_2} + k}} \times {{[0,\infty ]}^{{n_2} - k}})} $$
The following lemma proved there

(Lemma) Let $n_1,n_2$ be nonnegative integers, $n=n_1+n_2$, $m,r>0$. If $mr=n+1$, then $$\int_{{{[0,1]}^{{n_1}}}{{\times[0,\infty ]}^{{n_2}}}} {\frac{1}{{{{(1 + {x_1}^r + ... + {x_n}^r)}^m}}}d{x_i}} = \frac{r}{{\Gamma (m)}}\frac{{\Gamma {{(1 + \frac{1}{r})}^{n + 1}}}}{{{n_1} + 1}} $$

implies $$\tag{2}\nu({{[0,1]}^{2{n_1} + {n_2} + k}} \times {{[0,\infty ]}^{{n_2} - k}}) = \frac{\pi^{n+1}}{2^{2n+1}n!} \frac{1}{(2n_1+n_2+k+2)(2n_1+n_2+k+1)}$$
On the other hand, polar coordinates on each pairs $(x_1,x_2), (x_3,x_4),\cdots$ gives,
$$\begin{aligned}\nu({T^{{n_1}}} \times {R^{{n_2}}}) &= \int_1^\infty\int_{{{[0,\sec {\theta _i}]}^n} \times {{[0,\pi /4]}^{{n_1}}} \times {{[\pi/4,\pi /2]}^{{n_2}}}} {\frac{{{r_1}...{r_n}d{r_i}d{\theta _i}}}{{{{(1 + {r_1}^2 + ... + {r_n}^2)}^{n+1}}}}} da\\
&=\frac{1}{2^nn!} {\sum\limits_{i,j \ge 0} {{{(\frac{\pi }{4})}^{{n_1} + {n_2} - i - j}}{{( - 1)}^{i + j}}\binom{n_1}{i}\binom{n_2}{j}g(i,j)} }  \end{aligned}$$
Compare this with $(1),(2)$ will give a recurrence of $g(i,j)$, which allows one to calculate all $g(i,j)$ starting from inital value $g(0,0) = \frac{\pi}{4}$, proving $(*)$.
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