The solution to $\operatorname{sinc}(x)$$=a,0<a<\frac 2\pi$ involves inverting $ax-\sin(x)$ near $x=\frac\pi2 $ by transforming into $f_a(x)=a\left(x+\frac\pi2\right)-\sin \left(x+\frac\pi2\right)-\frac{\pi a}2+1=1+ax-\cos(x)$ with interval $L=f_a\left(\frac\pi2\right)=\frac{\pi a}2+1$. Define $h_a(f_a(x))=x$ on this interval with a complex Fourier series
$$h_a(x)=\frac{\pi^2 a+8}{16L}+\frac1{2L}\sum_{0\ne n\in\Bbb Z}e^\frac{\pi i n x}L\int_0^\frac\pi2te^{-\frac{\pi i n}Lf_a(t)}df_a(t)$$
integrating by parts:
$$h_a(x)=\frac{4\pi x+a\pi^2+8}{8(a\pi+2)}-\sum_{0\ne n\in\Bbb Z}\frac{ie^\frac{\pi i n (x-1)}L}{2\pi n}\int_0^\frac\pi2e^{-\frac{i\pi n}L(at-\cos(t)}dt=\frac{4\pi x+a\pi^2+8}{8(a\pi+2)}+\sum_{n=1}^\infty\frac1{\pi n}\int_0^\frac\pi2\sin\left(\frac{\pi n (x-1)}L-\frac{\pi n}L(a t-\cos(t)\right)dt$$
Here is a plot the complex Fourier series result that $f_a(h_a(x))=x$ and a numerical test for the second series inverting $\operatorname{sinc}(x)$ at a point. We find $\operatorname{sinc}^{-1}(x)=\frac\pi2+h_x\left(1-\frac{\pi x}2\right)$. However, no matter what is tried,
$$\int_0^\frac\pi2e^{i(ut+v\cos(t))}dt\text{ or }\int_0^\frac\pi2\sin(w+ut+v\cos(t))dt\tag 1$$
seemingly cannot be evaluated using Bessel, Struve, Anger Weber, and Lommel functions or otherwise as a closed form. For example,
$$\begin{align}\int_0^\frac\pi2 e^{i (u t-v\cos(t))}=\int_0^\frac\pi 2 \cos(ut)\cos(v\cos(t))-i\cos(ut)\sin(v\cos(t))+i\sin(ut)\cos(v\cos(t))+\sin(ut)\sin(v\cos(t))dt= \cos\left(\frac{\pi u}2\right) s_{0,u}(v)-u\sin\left(\frac{\pi u}2\right)s_{-1,u}(v)+\color{red}{\int_0^\frac\pi 2 \sin(u t)\sin(v\cos(t))dt}+i\color{red}{\int_0^\frac\pi 2\sin(ut)\cos(v\cos(t))dt}\end{align}$$
with Lommel $s_{a,b}(z)$ in Gradshteyn and Ryzhik 3.715,3.716 and can simplify into Anger Weber or Bessel functions. Additionally, a Jacobi Anger expansion uses the Bessel J function:
$$\int_0^\frac\pi2 e^{iv\cos(t)}e^{i ut}dt=-\sum_{m\in\Bbb Z}\frac{i^{m+1}}{m+u} J_m(v)\left(e^{\pi i \frac{m+u}2}-1\right)$$
but this is not a closed form for the Fourier series coefficients.
How can we evaluate $(1)$, the integrals in red, or the above series?
