Volume of a spherical tetrahedron

Question

In the paper Jun Murakami, The volume formulas for a spherical tetrahedron a formula for the volume of a spherical tetrahedron is given. I am trying to work through the details for the specific example where the dihedral angles are $\theta_1=\pi/4$, $\theta_2=\pi/4$, $\theta_3=\pi/4$, $\theta_4=\pi/6$, $\theta_5=\pi/6$ and $\theta_6=\pi/6$. I get lost in a sea of root threes. Could someone help ?