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.

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.
