2/((x^4-1)x)=2/((x^2-1)(x^2+1)x)=2/((x-1)(x+1)(x^2+1)x)
2/((x-1)(x+1)(x^2+1)x)=A/x+B/(x-1)+C/(x+1)+(Dx+E)/(x^2+1)=
=(A(x-1)(x+1)(x^2+1))/(x(x-1)(x+1)(x^2+1))+(Bx(x+1)(x^2+1))/(x(x-1)(x+1)(x^2+1))+
+(Cx(x-1)(x^2+1))/(x(x-1)(x+1)(x^2+1))+((Dx+E)x(x-1)(x+1))/(x(x-1)(x+1)(x^2+1))=
=(A(x^4-1))/(x(x-1)(x+1)(x^2+1))+(Bx(x^3+x^2+x+1))/(x(x-1)(x+1)(x^2+1))+
+(Cx(x^3-x^2+x-1))/(x(x-1)(x+1)(x^2+1))+((Dx+E)(x^3-x))/(x(x-1)(x+1)(x^2+1))=
=(Ax^4-A)/(x(x-1)(x+1)(x^2+1))+(Bx^4+Bx^3+Bx^2+Bx)/(x(x-1)(x+1)(x^2+1))+
+(Cx^4-Cx^3+Cx^2-Cx)/(x(x-1)(x+1)(x^2+1))+(Dx^4+Ex^3-Dx^2-Ex)/(x(x-1)(x+1)(x^2+1))=
=(x^4(A+B+C+D)+x^3(B-C+E))/(x(x-1)(x+1)(x^2+1))+
+(x^2(B+C-D)+x(B-C-E)+(-A))/(x(x-1)(x+1)(x^2+1))
A+B+C+D=0
B-C+E=0
B+C-D=0
B-C-E=0
-A=2 => A=-2
[II-IV] => 2E=0 => E=0
B+C+D=2
B-C=0 => B=C
B+C-D=0
2C+D=2
2C-D=0
4C=2 => C=1/2 => B=1/2
D=2C => D=1
I=int 2/((x^4-1)x)dx= -2int dx/x +1/2int dx/(x-1) + 1/2int dx/(x+1) + int (xdx)/(x^2+1)
I=-2ln|x|+1/2ln|x-1|+1/2ln|x+1|+1/2ln|x^2+1|+C
Note:
int (xdx)/(x^2+1) = int (1/2(2xdx))/(x^2+1) = 1/2int (d(x^2+1))/(x^2+1) = 1/2ln|x^2+1|