$\tan^6x$

$\begin{aligned} I & = \int_{-\frac \pi 2}^\frac \pi 2 \frac {dx}{\tan^6 x + 1} & \small \color{#3D99F6} \text{The integrand is even.} \\ & = 2 \int_0 ^\frac \pi 2 \frac {dx}{\tan^6 x + 1} & \small \color{#3D99F6} \text{Using }\int_a^b f(x) \ dx = \int_a^b f(a+b-x) \ dx \\ & = \int_0 ^\frac \pi 2 \left( \frac 1{\tan^6 x + 1}+\frac 1{\tan^6 \left(\frac \pi 2 - x \right) + 1} \right) dx \\ & = \int_0 ^\frac \pi 2 \left( \frac 1{\tan^6 x + 1}+\frac 1{\cot^6 x + 1} \right) dx \\ & = \int_0 ^\frac \pi 2 \left( \frac 1{\tan^6 x + 1}+\frac {\tan^6 x}{1+\tan^6 x} \right) dx \\ & = \int_0 ^\frac \pi 2 dx = \boxed{\dfrac \pi 2} \end{aligned}$