Resolving an integral equal to the exponential generating function involving the Riemann zeta function

Question

It is well-known that

$$-\gamma-\psi\left(1-x\right)=\sum_{n=1}^{\infty}\zeta\left(n+1\right)x^{n}$$

Using the OGF to EGF integral transformation, then

$$\frac{1}{2\pi}\int_{-\pi}^\pi (-\gamma-\psi(1-xe^{-it}))e^{\large e^{it}}dt=\sum_{n=1}^{\infty}\frac{\zeta\left(n+1\right)}{n!}x^{n}$$

Can we make any headway at resolving this integral into a new form?