<s>[INST] Evaluating the rational integral $\int \frac{x^2+3}{x^6(x^2+1)}dx $ 
Evaluate $$\int \frac{x^2+3}{x^6(x^2+1)}dx .$$

I am unable to break into partial fractions so I don't think it is the way to go. Neither is $x=\tan \theta$ substitution. Please give some hints. Thanks.
 [/INST] That rational expression, like all rational expressions, can be broken into partial fractions. You may be making the common mistake of trying to change the expression to
$$\frac{x^2+3}{x^6(x^2+1)}=\frac{A}{x^6}+\frac{Bx+C}{x^2+1}$$
However, you need to include all the lesser powers of the irreducible factors. In other words, you want
$$\frac{x^2+3}{x^6(x^2+1)}=\frac{A}{x}+\frac{B}{x^2}+\frac{C}{x^3}+\frac{D}{x^4}+\frac{E}{x^5}+\frac{F}{x^6}+\frac{Gx+H}{x^2+1}$$
Try that, then come back if you cannot finish.
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