Fransén-Robinson Constant and an intriguing series

Question

Whilst confirming a well known result for the Fransén-Robinson constant $\int_0^\infty \frac{1}{\Gamma(x)} dx$ I reduced the calculation to: $$\int_0^\infty\frac{1}{\pi}\sin(\pi x) \sum_{n \ge 1} \frac{(-1)^n}{n!(n+1-x)} dx$$ As I know the goal of my calculations this narrowed down to the following equality: $$\sum_{n\ge 1} \frac{1}{n!} \int_{n+1}^\infty \frac{\sin(\pi x)}{\pi x} dx=\int_0^1 \frac{e^{-t}}{\pi^2+\log^2(t)} dt$$ Where I am unfortunately stuck. Any help, especially tips, will be well appreciated :)