$$\int{\sqrt {(-\sin^2 t + \cos^2 t - \tan^2 t)}}~\textrm{d}t$$
I'm aware of a few trig identities, such as ${\cos^2 t - \sin^2 t} = \cos (2t)$ and $\tan^2 t = \frac{\sin^2 t}{\cos^2 t}$ but these don't seem to help simplify the problem.
No simple $u$-substitution seems to prevent itself, and my attempt to integrate by parts has resulted in an even more difficult integrand.
WolframAlpha and a few different integral calculators cannot seem to solve this.
By enforcing the substitution $t=\arcsin u$ we are left with $$ \int\sqrt{1-4u^2+2u^4}\frac{du}{1-u^2}=\int\sqrt{2-\frac{1}{(1-u^2)^2}}\,du $$ or, by setting $\frac{1}{1-u^2}=v$, $$\int \frac{\sqrt{2-v^2}}{2v^{3/2}\sqrt{v-1}}\,dv$$ which boils down to an elliptic integral. So, no simple answer to this question.