up to this point I've determined c this way:
The issue here is that I cannot figure out how to proceed the same way with the other two variables a and b. Is there something I'm missing from the start? Sorry if it's hard to understand I can elaborate if necessary.
On
Just observe that (almost everywhere) $$\frac{ax+b}{x^2+1}+\frac cx=\frac{ax^2+bx}{x^3+x}+\frac{c(x^2+1)}{x^3+x} =\frac{(a+c)x^2+bx+c}{x^3+x}$$ and compare coefficients of the numerator.
On
If you clear denominators by multiplying through by $x(x^2+1)$ you get:
$$x+1=(ax+b)x+c(x^2+1)$$
The so-called cover-up rule then invites you to set $x=0$, which eliminates $ax+b$ and identifies $c$. You can then easily compare coefficients, or alternatively set $x=i=\sqrt {-1}$ to obtain $i+1=-a+ib$ and equate real and imaginary parts.
The $x=0$ and $x=i$ choices respectively make $x$ and $x^2+1$ equal to $0$. The trick can be particularly useful in more complex examples (see material on partial fractions).
Hint:
$$\frac{ax+b}{x^2+1}+\frac{c}{x} = \frac{ax^2+bx+cx^2+c}{x^3+x}$$ and you want this to be $$\frac{x+1}{x^3+x}$$