TITLE: Value of a certain integral
QUESTION [7 upvotes]: How to find the value of the integral
$$
\int_o^{\infty} \! \frac{x^8}{1+x^4+x^6+x^{10}} \, \mathrm{d}x
$$
given to be $\frac{1}{12}(3\sqrt{2}-1) \pi$ by WolframAlpha and, in general, is there a procedure to find the value of the definite integral of a rational function of the form $\dfrac{x^l}{p(x)}$ where $deg(p) > l > 1$ from $0$ to $\infty$ ?

REPLY [2 votes]: For the given integral, the denominator can be factored so you get 
$1+x^4+x^6+x^{10}=1+x^4+x^6(1+x^4)=(1+x^4)(1+x^6)$. 
Then, using partial fractions you can split the denominator 
$\dfrac{x^8}{(1+x^4)(1+x^6)}=\dfrac{x^2+1}{2(1+x^4)}+\dfrac{x^4-x^2-1}{2(1+x^6)}$
Which gives you two solvable (although challenging) integrals. In general, this would be my approach to solving any integral of the given form, but if it failed I would probably try using a series to approximate the functions and integrate term by term.