Finite truncated tanh–sinh, exp–sinh, and sinh–sinh trapezoid rules

Input a real function $f$ that is continuous on the interior of the selected target domain, a step $h>0$, nonnegative integers $N_{-},N_{+}$, and exactly one domain mode. The finite rule evaluates $f$ only at interior points:

1. Finite interval $(a,b)$, $a<b$. Put $m=(a+b)\mathrm{/}2$, $d=(b-a)\mathrm{/}2$, $s(t)=\sinh t=(e^t-e^{-t})\mathrm{/}2$, $c(t)=\cosh t=(e^t+e^{-t})\mathrm{/}2$, and $\psi (t)=\tanh ((\pi \mathrm{/}2)s(t))$, where $\tanh u=\sinh u\mathrm{/}\cosh u$. Set $x(t)=m+d\psi (t)$, $x^{\mathrm{\prime }}(t)=d(\pi \mathrm{/}2)c(t)\mathrm{/}{\cosh }^2((\pi \mathrm{/}2)s(t))$, and $\mathrm{\Phi }(t)=f(x(t))x^{\mathrm{\prime }}(t)$.
2. Positive half-line $(0,\mathrm{\infty })$. Here $f$ is continuous on $(0,\mathrm{\infty })$. Set $x(t)=\exp ((\pi \mathrm{/}2)s(t))$, $x^{\mathrm{\prime }}(t)=x(t)(\pi \mathrm{/}2)c(t)$, and $\mathrm{\Phi }(t)=f(x(t))x^{\mathrm{\prime }}(t)$.
3. Whole line $\mathbb{R}$. Here $f$ is continuous on $\mathbb{R}$. Set $x(t)=\sinh ((\pi \mathrm{/}2)s(t))$, $x^{\mathrm{\prime }}(t)=\cosh ((\pi \mathrm{/}2)s(t))(\pi \mathrm{/}2)c(t)$, and $\mathrm{\Phi }(t)=f(x(t))x^{\mathrm{\prime }}(t)$.

Return the finite sum $T_{h,N_{-},N_{+}}(f)=h{\sum }_{k=-N_{-}}^{N_{+}}\mathrm{\Phi }(kh)\mathrm{.}$ The algorithm makes no accuracy claim from integrability alone. Analytic-strip and tail hypotheses required for an error certificate are stated in A finite double-exponential trapezoid has separate strip and tail errors ↗.