Evaluate: ∫ √(secx/2 - 1) dx

$I = \int \sqrt{\sec\frac{x}{2} - 1}\,dx = \int \sqrt{\frac{1 - \cos\frac{x}{2}}{\cos\frac{x}{2}}}\,dx = \int \sqrt{\frac{(1 - \cos\frac{x}{2})(1 + \cos\frac{x}{2})}{\cos\frac{x}{2}(1 + \cos\frac{x}{2})}}\,dx = \int \frac{\sin\frac{x}{2}}{\sqrt{\cos^2\frac{x}{2} + \cos\frac{x}{2}}}\,dx$ Let $\cos\frac{x}{2} = t \Rightarrow -\sin\frac{x}{2} \cdot \frac{1}{2}\,dx = dt \Rightarrow \sin\frac{x}{2}\,dx = -2\,dt$ $\therefore I = -2\int \frac{dt}{\sqrt{t^2 + t}} = -2\int \frac{dt}{\sqrt{\left(t + \frac{1}{2}\right)^2 - \left(\frac{1}{2}\right)^2}} = -2\log_e\left|\left(t + \frac{1}{2}\right) + \sqrt{t^2 + t}\right| + c$ $= -2\log_e\left|\left(\cos\frac{x}{2} + \frac{1}{2}\right) + \sqrt{\cos^2\frac{x}{2} + \cos\frac{x}{2}}\right| + c$