If $\displaystyle \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\left(\dfrac{x^2\cos x}{1+x^2}+\dfrac{1+\sin^2 x}{1+e^{\sin 2x}}\right)dx = \dfrac{\pi}{4}(\pi+a)-2$, then the value of $a$ is: