1/(x^3+x)=(A/(x^2+1))+(B/x) in partial fractions.

So multiplying through by x(x^2+1) we get Ax+B(x^2+1)=1.

Put x=0 and we get B=1. Put x=1 and we get A+2B=1, so A=-1. This implies that:

(1/x)-(1/(x^2+1)) is equivalent to 1/(x^3+x), which is false.

The fallacy arises because x=0 can't be substituted in B/x, and because no matter what values for x are substituted, the equation containing A and B always equals 1.

For example, x=-1 gives -A+2B=1 while x=1 gives A+2B=1. This implies that A+2B=-A+2B, so A=0.

We must conclude then that the original expression is flawed or partial fractions cannot be applied.