z = tg(x), dz = dx/cos^2(x), dx = cos^2(x)*dz = dz/(1 + z^2) ===>

int tg^4(x) = int (z^4)/(1 + z^2) = int{(z^4 - 1 + 1)/z^2 + 1)} =
int{z^2 - 1} + int{1/(z^2 + 1) = z^3/3 - z + arctg(z) + C ===>
int tg^4(x) = (tg(x))^3/ 3 - tg(x) + arctg(tg(x)) + C


както так вроде...