Let \(\displaystyle f(x)=\int_{0}^{x} g(t)\,\log_{e}\!\left(\dfrac{1-t}{1+t}\right)\,dt \), where \( g \) is a continuous odd function.
If \(\displaystyle \int_{-\pi/2}^{\pi/2} \left( f(x) + \dfrac{x^{2}\cos x}{1+e^{x}} \right) dx = \left(\dfrac{\pi}{\alpha}\right)^{2}-\alpha \), then \( \alpha \) is equal to __________ .
