Integrate $\int{\frac{x^2-1}{x^4+3x^3+5x^2+3x+1}}dx$ The answer of this integral is 
$$\int{\frac{x^2-1}{x^4+3x^3+5x^2+3x+1}}dx$$
$$=\frac{2}{\sqrt{3}}arctan(\frac{2}{\sqrt{3}}(x+\frac{1}{x})+\sqrt{3})+C$$
But I can't figure out how could I solve it. I tried to use partial fraction, but the denominator 
$$x^4+3x^3+5x^2+3x+1$$
can't be easily factored.
I also tried to use WolframAlpha to know how to solve it, but it can't give a useful answer for this integral: 

\begin{align}
&\int{\frac{x^2-1}{x^4+3x^3+5x^2+3x+1}}dx\\
=&\int \frac{ 1 - \frac{1}{x^2}}{x^2+3x+5+\frac{3}{x}+\frac{1}{x^2}}dx 
=\int \frac{ d\left( x+\frac{1}{x}\right) }{\left( x+\frac{1}{x}+\frac{3}{2}\right)^2 + \frac 34 }  \\
=&\ \frac{2}{\sqrt{3}}\arctan\left[\frac{2}{\sqrt{3}}\left(x+\frac{1}{x}\right)+\sqrt{3}\right]+C
\end{align}
