Definite double integral.

I have a problem and final answer that I got from a journal. I was trying to solve it but never reach the final answer. Here is the problem . I have tried up to this so far The last integral I tried to solve with mathematica and it gives very different result that is very far from the actual one $$-\frac{\csc (\beta ) \csc (\theta ) \sqrt{\cos (2 \beta )+\cos (2 \theta )} \tanh ^{-1}\left(\frac{\sec (\beta ) \sqrt{\cos (2 \beta )+\cos (2 \theta )}}{\sqrt{2}}\right)}{\sqrt{2-2 \cot ^2(\beta ) \cot ^2(\theta )}}$$

I am not sure where, I am doing mistake. Shall I go for any other method?

• Looks fine to me: now make the change of variables $t=\sin\theta$ to solve the last integral. – Aretino May 8 at 13:02
• @Aretino I have solved that. It was quite complex. the main trick was in U-substitution. I have a new problem now. Can you please check this out? Nobody can answer as usual. math.stackexchange.com/questions/3245142/… – T. an May 31 at 0:51
• Isn't this the same question as math.stackexchange.com/questions/3207674/… ? – David K Jul 25 at 11:55