Let $\alpha,\beta,\gamma\in\mathbb{C}$. I'd like to calculate the integral $$\int_{-\infty}^{0}t^\alpha e^{\beta t}(1-e^{-2t})^\gamma dt.$$ For some special cases like when $\gamma=0$ or if $\gamma\in\mathbb{R}$, I can find the value of the integral explicitly. However, I can't find how to compute this integral for the general case.
How can I find it?