Any special way to integrate $\int_{0}^{\infty}t^2\exp(-t)\sin(t)$ quickly?

Since $\int_0^\infty t^2 e^{-zt}dt=\frac{2}{z^3}$ if $\Re z>0$, the desired integral is $$\Im\int_0^\infty t^2\exp\Big[-(1-i)t\Big]dt=\Im\frac{2}{(1-i)^3}=\Im\frac{1}{\sqrt{2}}e^{3\pi i/4}=\frac{1}{\sqrt{2}}\sin\frac{3\pi}{4}=\frac{1}{2}.$$