Evaluate: ∫1/(sin4x + sin2xcos2x + cos4x)dx

Let I = ∫1/(sin4x + sin2xcos2x + cos4x)dx = ∫1/(sin4x + 2sin2xcos2x - sin2xcos2x + cos4x)dx = ∫1/((sin2x + cos2x)² - sin2xcos2x)dx
Multiplying top and bottom by sec²(2x) we get
= ∫sec²(2x)/sec²(2x)((sin2x + cos2x)² - sin2xcos2x)dx
Let u = tan(2x) ⇒ du = 2sec²(2x)dx
= ∫(1/2)du/((1 + u²) - (u²/4)) = ∫(1/2)du/(1 + (3/4)u²) = (1/2) × (1/√3)tan⁻¹(√3/2 u) = (1/√3)tan⁻¹(√3/2 tan(2x)) + C