Gil-Pelaez (1951) proves the Fourier inversion formula \begin{align*} F(x) &= \frac{1}{2} + \frac{1}{2\pi} \int_0^\infty \frac{e^{itx}\phi(-t)-e^{-itx}\phi(t)}{it}dt \\ &= \frac{1}{2} - \frac{1}{\pi} \int_0^\infty \Im\left(\frac{e^{-itx}\phi(t)}{t}\right)dt, \end{align*} where $F$ is the cdf of a random variable and $\phi$ the characteristic function, $\phi(t)=\int_\mathbb{R}e^{itx}dF(x)$.

Is it possible to relate the integral (or its principal value) $$\int_0^\infty \Im\left(\frac{e^{-itx}\phi(t)}{t^n}\right)dt$$ for $n\in\mathbb{N}$ to the cdf $F$?

Gil-Pelaez' original proof wouldn't work because it would then involve integrals like $\int_0^\infty \frac{\sin(x)}{x^n}dx$ and $\int_0^\infty \frac{\cos(x)}{x^n}dx$ which don't necessarily exist but perhaps somebody has seen a different proof that is adaptable for powers in the denominator?