Hello! Let $I$ be the given integral.Set $t=-x$ to get $I=\int_{\frac{\pi}{2}}^{-\frac{\pi}{2}} -\frac{\cos ^7(-t)}{e^{-t}+1}dt=\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{e^t\cos ^7t}{e^t+1}dt$. Thus $2I=I+I=\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{\cos ^7x}{e^x+1}+\frac{e^x\cos^7x}{e^x+1}dx=\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \cos ^7xdx$. We have $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \cos^7xdx=\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \cos x\cdot (1-\sin ^2x)^3\overset{u=\sin x}=\int_{-1}^{1} (1-u^2)^3du=$ $=\int_{-1}^{1} 1-3u^2+3u^4-u^6du=\left[u-u^3+\frac{3}{5}u^5-\frac{1}{7}u^7\right]_{-1}^{1}=2\left(1-1+\frac{3}{5}-\frac{1}{7}\right)=\frac{32}{35}$ thus $2I=\frac{32}{35}\Rightarrow \boxed{I=\frac{16}{35}}$.