[(sinx)^2-(cosx)^2]/[sinx^4+cosx^4]
=-cos2x/[(sinx^2+cosx^2)^2-2sinx^2cosx^2]
=-cos2x/1-(sin2x)^2/2
∫[(sinx)^2-(cosx)^2]dx/[sinx^4+cosx^4]
=∫-cos2xdx/[1-(sin2x)^2/2]
=-∫dsin2x/(2-(sin2x)^2)
=[-1/(2√2)]∫(√2+sin2x+√2-sin2x)dsin2x/[2-(sin2x)^2]
=(-1/2√2)ln|(√2+sin2x)/(√2-sin2x)|+C