The integral $\; \int_{0}^{\pi} \sqrt{1+4sin^{2} \large\frac{x}{2}-4 sin \large\frac{x}{2}} dx\;$ equals :

Question

$(a)\;4 \sqrt{3} -4 \qquad(b)\;4 \sqrt{3} -4-\large\frac{\pi}{3}\qquad(c)\;\pi -4\qquad(d)\;\large\frac{2\pi}{3}-4 -4 \sqrt{3}$