Why does this pattern of "nasty" integrals stop?

We have (typo corrected),

$$\begin{aligned} \pi &=\int_{-\infty}^{\infty}\frac{(x-1)^2}{\color{blue}{(2x - 1)}^2 + (x^2 - x)^2}\,dx,\quad\text{(by Mark S.)}\\[1.8mm] \pi &=\int_{-\infty}^{\infty}\frac{(x+1)^2}{\color{blue}{(x + 1)}^2 + (x^2 + x)^2}\,dx\\[1.8mm] \pi &=\int_{-\infty}^{\infty}\frac{(x+1)^2}{\color{blue}{(x^2 - x - 1) }^2 + (x^2 + x)^2}\,dx\\[1.8mm] \color{red}\pi &=\int_{-\infty}^{\infty}\frac{(x+1)^2}{\color{blue}{(x^3 + 2x^2 - x - 1)}^2 + (x^2 + x)^2}\,dx\\[1.8mm] \pi &=\int_{-\infty}^{\infty}\frac{(x-1)^2}{\color{blue}{(x^3 - 3x^2 + 1)}^2 + (x^2 - x)^2}\,dx\\[1.8mm] ?? &=\int_{-\infty}^{\infty}\frac{(x\pm1)^2}{\color{blue}{(x^5 + 3x^4 - 3x^3 - 4x^2 + x + 1)}^2 + (x^2 \pm x)^2}\,dx \end{aligned}$$

where those in blue are the minimal polynomials of $x=\frac{1}{2\cos(2\pi/p)}$ for $p=1,3,5,7,9,11$.

Note: The red pi is the notorious one in the post, A nasty integral of a rational function, $$\int_0^{\infty} \frac{x^8 - 4x^6 + 9x^4 - 5x^2 + 1}{x^{12} - 10 x^{10} + 37x^8 - 42x^6 + 26x^4 - 8x^2 + 1} \, dx = \frac{\pi}{2}$$ as well as in this post after some manipulation.

Q1: Why did the "pattern" of using minimal polynomials work then stop at $p=11$, and how can we make it continue by adjusting the other parameters?

$\color{green}{Update:}$ Based on an insight from an old post, using the "negative" case of $p=7$, its denominator is still a sextic with a solvable Galois group and we find,

$$\int_{-\infty}^{\infty}\frac{(x\color{red}-1)^2}{\color{blue}{(x^3 + 2x^2 - x - 1)}^2 + (x^2 \color{red}- x)^2}\,dx=\pi\sqrt{\frac{u}{\color{green}{12833}}}$$

where $u$ is a root of a monic nonic also with a solvable Galois group,

$$\small -\color{green}{12833}^3*1782434241^2 - 41120374319577904376201744753 u - 354521093943488815427187669 u^2 - 550802363395052799639795 u^3 - 176617825075778391189 u^4 + 116970252692553921 u^5 - 20201478347596 u^6 + 1625465206 u^7 - 63997 u^8 + u^9=0$$

The denominator of $p=9$ also is solvable. However, for $p=11$, it no longer is.

Q2: Was the pattern interrupted because the denominator of $p=11$ no longer has a solvable Galois group?

Solution 1:

Partial answer. Mathematica 11.0.1.0 (64-bit Windows version) seems to symbolically evaluate $$ \int_{-\infty}^{\infty} \frac{(x - 1)^2}{(1 + x - 4 x^2 - 3 x^3 + 3 x^4 + x^5)^2 + (x^2 - x)^2} \, \mathrm{d}x =2\pi y $$ where $ y $ is a root (the 5th root in Mathematica's ordering), of \begin{align*} \small F(y)=55936138949897200689844509841956235222126377325 - 2082209926471466695895506312399091645554188710590 y + 49399208260228586110040380712822122163293326842296 y^2 - 904097593617672391563622547821611243428330356636656 y^3 + 14127632726315977701496334077804393041066245226028208 y^4 - 192534883415138070802102412843131348551007666040509024 y^5 + 2248032708977729589700543648210682328879792825038892288 y^6 - 22010013756272539692699272127690186099540607721493676800 y^7 + 177728824048935169179013735666882776433001119535910888192 y^8 - 1170270214760621202108304618484485542592211842152325435904 y^9 + 6226689208769791815298929222960276164825821302955689534464 y^{10} - 26437408929821178291367173439675032999610116594417230776320 y^{11} + 87275205150008062398776420803782617539547332212906935361536 y^{12} - 209632027731557385765045313738415590487122817707525011718144 y^{13} + 284829590179494874220555955086122649413365826411704845058048 y^{14} + 245738741392479529396402731465119601079938307163739661565952 y^{15} - 2744252632383133719563152613313766366008892259189754592296960 y^{16} + 9042239242455966498125021473251288480014205602523431668940800 y^{17} - 19642481348541153825949628077511851598796849639028033440972800 y^{18} + 31384454408136427453055038714389257858518560896664228069376000 y^{19} - 37847103175390150688294536889184184478935891337063789625344000 y^{20} + 34290036775233047407263179281808801381553538237009356390400000 y^{21} - 22732262960008031643227099738915285612779131750417374904320000 y^{22} + 10440433388762840105269721355193655567662001399784538112000000 y^{23} - 2974656530310569079556114222635017838466182586996359168000000 y^{24} + \color{blue}{831141777440}^5 y^{25}=0 \end{align*}

The discriminant $ d $ of the integrand's denominator

$$G(x)=(1 + x - 4 x^2 - 3 x^3 + 3 x^4 + x^5)^2 + (x^2 - x)^2$$

is $d=-2^5\times\color{blue}{831141777440}$. The discriminant of the $25$-deg $F(y)$ is divisible by $d^{65}$. However, its constant term is not integrally divisible by $ d $.

Moreover, $ y' = 831141777440 y $ is an algebraic integer. The discriminant of the minimal polynomial of $ y' $ is divisible by $d^{246}$. The constant term is divisible by $ d^{10} $, but the quotient is not a perfect power of an integer.

The positive case ++ passes the first test, but also fails on the second. The constant term is divisible by $ d^{990} $, but the quotient is not a perfect power.