Given the standard hyperbola $\displaystyle \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ I need to find the area bounded by the curve and the line $\displaystyle x = 2a$

Here is the visual representation

Solving for y gives me $\displaystyle y = b\sqrt{\frac{x^2}{a^2} - 1}$

I then set up the integral $\displaystyle A = 2b\int_{a}^{2a}\sqrt{\frac{x^2}{a^2} - 1} \,\,dx$

I set $\displaystyle x = a \sec{\theta} \Rightarrow \,\,dx = a\sec{\theta}\tan{\theta}\,\, d\theta$, substituted and changed limits of integration accordingly $\displaystyle A = 2b\int_{0}^{\frac{\pi}{3}} \sqrt{\sec^2{\theta} - 1}\, a\sec{\theta}\tan{\theta} \,\, d\theta$

Using a trig identity I can simplify down to $\displaystyle A = 2ab\int_{0}^{\frac{\pi}{3}} \tan^2\theta\sec\theta \,\,d\theta $

And that is where I am stuck, I tried another u-substitution to no avail.