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= (Ax²+1+Bx²+Bx+Cx+C)/{(x+1)(x²+1)}

on comparing the coefficient of both side of same power

A+B = 0

B+C = 1

A+C = 2

A =1/2 ; B =-1/2 ; C= 3/2

put the value of A,B,C

= 1/2{1/(x+1) + (3-x)/(x²+1)}