Generalized sine integral $ \int_0^\infty \sin^m(x^n)/x^p\,\mathrm dx$

I have seen that both the integrals $$ \color{black}{ \int_0^\infty \operatorname{sinc}(x^n)\,\mathrm{d}x = \frac{1}{n-1} \cos\left( \frac{\pi}{2n}\right)\Gamma \left(\frac{1}{n} \right) } $$ and $$ \color{black}{ \int_0^\infty \operatorname{sinc}^n(x)\,\mathrm{d}x = \frac{\pi}{2^n (n-1)!} \sum_{k=0}^{\lfloor n/2\rfloor} (-1)^k {n \choose k} (n-2k)^{n-1} } $$ beeing computed here. Is there a similar expression for the more general $$ \int_0^\infty \frac{\sin^m(x^n)}{x^p}\,\mathrm{d}x $$

sine integral? I do not know precielly the restriction on the constant. I assume that they have to be positive. From my research $m$ and $p$ does not neccecary have to be equal, for example $$ \int_0^\infty \frac{\sin^3(x)}{x^2}\,\mathrm{d}x = \frac{4}{3} \log 3 \qquad \text{and} \qquad \int_0^\infty \frac{\sin^3(x^2)}{x^2}\,\mathrm{d}x = \frac{1}{4} \sqrt{\frac{\pi}{2}}\left( \sqrt{3} - 3\right) $$ but any closer restrictions on $n,m$ and $p$ I have not been able to gather.

Solution 1:

Through the substitution $x=z^{1/m}$ the problem boils down to finding $$ I(m,\alpha)=\int_{0}^{+\infty}\frac{\sin(x)^m}{x^\alpha}\,dx $$ and assuming $m\in\mathbb{N}$ we have that $\sin(x)^m$ can be expressed as a finite Fourier sine/cosine series.
So the problem boils down to finding, through the Laplace (inverse) transform, $$ \int_{0}^{+\infty}\frac{\sin(x)}{x^\alpha}\,dx = \Gamma(1-\alpha)\cos\left(\frac{\pi\alpha}{2}\right),\qquad \int_{0}^{+\infty}\frac{\cos(x)}{x^\alpha}\,dx=\Gamma(1-\alpha)\sin\left(\frac{\pi\alpha}{2}\right)$$ then extend the range of validity of such identities through integration by parts.