They are all the same deformation, which has nothing to do with integral calculation. . .

Red line: ((1+x 2)-x 2)/(x 4 (1+x 2))

=( 1+x^2)/(x^4( 1+x^2))-x^2/(x^4( 1+x^2))

= 1/x^4- 1/(x^2( 1+x^2))

= 1/x^4-(( 1+x^2)-x^2)/(x^2( 1+x^2))

Blue line: (continued) =1/x4-(1+x2)/(x2 (1+x2))+x2/(x2 (1+x2)

= 1/x^4- 1/x^2+ 1/( 1+x^2)

Green line: x √ (x-1) = (x-1+) √ (x-1) = (x-1) √ (x-1)+