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