Evaluate $\int_0^{\infty } \frac{\tan ^{-1}\left(\sqrt{a^2+x^2}\right)}{\left(x^2+1\right)\sqrt{a^2+x^2}} \, dx$

While reading Borwein's review on experimental mathematics, I found a curious identity which is similar to the well-known Ahmed integral. $$\int_0^{\infty } \frac{\tan ^{-1}\left(\sqrt{a^2+x^2}\right)}{\left(x^2+1\right)\sqrt{a^2+x^2}} \, dx=\frac{\pi \left(2 \tan ^{-1}\left(\sqrt{a^2-1}\right)-\tan^{-1}\left(\sqrt{a^4-1}\right)\right)}{2 \sqrt{a^2-1}}, \ a>1$$ However, the original solution of Ahmed integral (depending on symmetry) does not seem to work here. Therefore I'd like to post it here to seek for suggestions. Any help will be appreciated!

from Hot Weekly Questions - Mathematics Stack Exchange