Some area here

$\int \left(\sqrt{\cos (x)}-\left(\sqrt{\sec (x)}-1\right)\right) \, dx$ ==> $x+2 E\left(\left.\frac{x}{2}\right|2\right)-2 \sqrt{\cos (x)} \sqrt{\sec (x)} F\left(\left.\frac{x}{2}\right|2\right)$ $\int_{-\cos ^{-1}\left(\frac{1}{2} \left(3-\sqrt{5}\right)\right)}^{\cos ^{-1}\left(\frac{1}{2} \left(3-\sqrt{5}\right)\right)} \left(\sqrt{\cos (x)}-\left(\sqrt{\sec (x)}-1\right)\right) \, dx$ ==> $2 \left(-2 F\left(\left.\frac{1}{4} \left(2 \tan ^{-1}\left(-\sqrt{\frac{3}{2 \sqrt{5}}-\frac{1}{2}}\right)+\pi \right)\right|2\right)+2 E\left(\left.\frac{1}{4} \left(2 \tan ^{-1}\left(-\sqrt{\frac{3}{2 \sqrt{5}}-\frac{1}{2}}\right)+\pi \right)\right|2\right)+\cos ^{-1}\left(\frac{1}{2} \left(3-\sqrt{5}\right)\right)\right)$

Approximately, 1.69516286386.

F and E are the elliptic integrals of the first and second kinds respectively.