Use the trig substitution x =tan(t)

you will end up with the integral of cos^4(t) which can use a reduction formula on or

write cos^4(t) =cos^2(t) - cos^2(t)sin^2(t) = cos^2(t) - 1/4 sin^2(2t)

cos^2(t) - 1/4 sin^2(2t) = (1+cos(2t)/2 -1/8(1- cos(4t))