##tan^3x/3 + tan^5x/5 + C## where C = constant of integration
Using the Pythagorean identity, ##sec^2x = 1 + tan^2x## we have
##int tan^2 x sec^4x dx = int tan^2x sec^2 x sec^2x dx = inttan^2x(1+tan^2x)sec^2xdx##
Now let ##tanx = u, ##so ##(du)/dx = sec^2x or du = sec^2x dx##
Now integrating with respect to u, we have
##intu^2(1+u^2)du == int (u^2 + u^4) du = u^3/3 + u^5/5 + C ##
where C = constant of integration.
Resubstituting u = tanx, we have the final integration answer as
##tan^3x/3 +tan^5x/5 + C##
