\int_{-1}^{-3} \frac{1}{x^4+4x+5} \,dx=\int_{-1}^{-3} \frac{1}{(x+2)^2+1} \,dx=\arctan(x+2)\,|_{-1}^{-3}=-\frac{\pi}{4}-\frac{\pi}{4}=\boxed{-\frac{\pi}{2}}