Unless there's some snaky trick, this seems to be a specially nasty exercise in decomposition of rational functions:
, so :
, and now multiply
by the common denominator, choose nice values of x in both sides and find the coefficients
A,B,C,D. It looks horrible, indeed...but who knows? Perhaps there's another way...
Tonio
After Tonio's factorization, we can write the in terms of the sum of the two quadratic factors, i.e:
We just observe that then subtract .
For , we write as in terms of the two quadratic factors, and we do it as follows -- let:
, then putting gives you the value:
Therefore, we have:
Therefore, we have:
Isn't that just an absolute beauty? Beats all the sunsets one can see or imagine, haha! Now: