  In a recent work, we developed three new compact numerical quadrature
formulas for finite-range periodic supersingular integrals $I[f]=\intBar^b_a
f(x)\,dx$, where $f(x)=g(x)/(x-t)^3,$ assuming that $g\in C^\infty[a,b]$ and
$f(x)$ is $T$-periodic, $T=b-a$. With $h=T/n$, these numerical quadrature
formulas read
  \begin{align*}
  \widehat{T}{}^{(0)}_n[f]&=h\sum^{n-1}_{j=1}f(t+jh)
  -\frac{\pi^2}{3}\,g'(t)\,h^{-1}+\frac{1}{6}\,g'''(t)\,h,
  \widehat{T}{}^{(1)}_n[f]&=h\sum^n_{j=1}f(t+jh-h/2)
  -\pi^2\,g'(t)\,h^{-1},
  \widehat{T}{}^{(2)}_n[f]&=2h\sum^n_{j=1}f(t+jh-h/2)-
  \frac{h}{2}\sum^{2n}_{j=1}f(t+jh/2-h/4).
  \end{align*}
  We also showed that these formulas have spectral accuracy; that is,
  $$\widehat{T}{}^{(s)}_n[f]-I[f]=O(n^{-\mu})\quad\text{as $n\to\infty$}\quad
\forall \mu>0.$$
  In the present work, we continue our study of these formulas for the special
case in which
  $f(x)=\frac{\cos\frac{\pi(x-t)}{T}}{\sin^3\frac{\pi(x-t)}{T}}\,u(x)$, where
$u(x)$ is in $C^\infty(\mathbb{R})$ and is $T$-periodic. Actually, we prove
that $\widehat{T}{}^{(s)}_n[f]$,
  $s=0,1,2,$ are exact for a class of singular integrals involving $T$-periodic
trigonometric polynomials of degree at most $n-1$; that is, $$
\widehat{T}{}^{(s)}_n[f]=I[f]\quad\text{when\ \
$f(x)=\frac{\cos\frac{\pi(x-t)}{T}}{\sin^3\frac{\pi(x-t)}{T}}\,\sum^{n-1}_{m=-(n-1)}
c_m\exp(\mrm{i}2m\pi x/T)$.}$$ We also prove that, when $u(z)$ is analytic in a
strip $\big|\text{Im}\,z\big|<\sigma$ of the complex $z$-plane, the errors in
all three $\widehat{T}{}^{(s)}_n[f]$ are $O(e^{-2n\pi\sigma/T})$ as
$n\to\infty$, for all practical purposes.
