The integral I am dealing with is:
$$\frac{3}{2 \Gamma \left(\frac{1}{3}\right)}\int_{z-1}^z \log \left(\frac{1}{z-y}\right) \exp \left(-\left| y\right| ^{3}\right) \, dy$$
where $z\in \mathbb{R}$
Due to its complexity, I tried Mathematica to solve the integral. It gave back the same form. Does that mean it cant be solve? Or are there any other methods to solve such complex integrals? A handbook?
Assuming $z\geq 1$, we have: $$ \int_{z-1}^{z}\log\left(\frac{1}{z-y}\right) e^{-y^3}\,dy = -\frac{d}{d\alpha}\left.\int_{z-1}^{z}(z-y)^{\alpha}e^{-y^3}\,dy\,\right|_{\alpha=0} $$ hence the original integral depends on the derivatives of incomplete gamma functions.