Assume the partial fraction form
#1/{(P^2-1)^2(P+1)}=1/{(P-1)^2(P+1)^3} #
#=A/{P-1}+B/{(P-1)^2}+C/{P+1}+D/(P+1)^2+E/(P+1)^3#
multiplying both sides by #(P-1)^2(P+1)^3# leads to
#1 = (P-1)(P+1)^3A+(P+1)^3B+(P-1)^2(P+1)^2C+(P-1)^2(P+1)D+(P-1)^2E #
#= (P^4+2P^3-2P-1)A+(P^3+3P^2+3P+1)B+(P^4-2P^2+1)C+(P^3-P^2-P+1)D+(P^2-2P+1)E#
# = (A+C)P^4+(2A+B+D)P^3+(3B-2C-D+E)P^2+(-2A+3B-D-2E)P+(-A+B+C+D+E)#
Comparing both sides we get the set of simultaneous equations
# ((1,0,1,0,0),(2,1,0,1,0),(0,3,-2,-1,1),(-2,3,0,-1,-2),(-1,1,1,1,1))((A),(B),(C),(D),(E)) = ((0),(0),(0),(0),(1))#
It is easiest to solve this by Gauss elimination (any other method of solving a set of simultaneous equations will do, but Gauss elimination is definitely one of the most efficient). Start with the augmented matrix
# ((1,0,1,0,0,|,0),(2,1,0,1,0,|,0),(0,3,-2,-1,1,|,0),(-2,3,0,-1,-2,|,0),(-1,1,1,1,1,|,1))#
We carry out row reductions to take this to a row-echelon form
#R_2-2R_1, R_4+2R_1,R_5+R_1#:
# ((1,0,1,0,0,|,0),(0,1,-2,1,0,|,0),(0,3,-2,-1,1,|,0),(0,3,2,-1,-2,|,0),(0,1,2,1,1,|,1))#
#R_3-3R_2,R_4-3R_2, R_5-R_2 #:
# ((1,0,1,0,0,|,0),(0,1,-2,1,0,|,0),(0,0,4,-4,1,|,0),(0,0,8,-4,-2,|,0),(0,0,4,0,1,|,1))#
#R_4-2R_3,R_5-R_3# :
# ((1,0,1,0,0,|,0),(0,1,-2,1,0,|,0),(0,0,4,-4,1,|,0),(0,0,0,4,-4,|,0),(0,0,0,4,0,|,1))#
#R_5-R_4# :
# ((1,0,1,0,0,|,0),(0,1,-2,1,0,|,0),(0,0,4,-4,1,|,0),(0,0,0,4,-4,|,0),(0,0,0,0,4,|,1))#
We can now easily determine the parameters by back-substitution
#4E= 1 implies E = 1/4#
#4D-4E=0 implies D = E = 1/4#
#4C-4D+E = 0 implies 4C = 4times (1/4)-1/4 = 3/4 implies C=3/16#
#B-2C+D = 0 implies B = 2times 3/16-1/4=1/8#
#A+C = 0 implies A=-3/16#
So, finally
#1/{(P^2-1)^2(P+1)}= -3/16 1/{P-1}+1/8 1/{(P-1)^2}+3/16 1/{P+1}+1/4 1/(P+1)^2+1/4 1/(P+1)^3#
