How does wolfram alpha compute this integral?

Question

Wolfram alpha can find that

$\int_1^\infty\frac{x^8-8x^7+24x^6-32x^5+19x^4-12x^3+17x^2-10x+2}{x^{12}-12x^{11}+56x^{10}-120x^9+82x^8+112x^7-182x^6-92x^5+381x^4-356x^3+170x^2-44x+5}dx=\frac{\pi}{2}$

http://www.wolframalpha.com/input/?i=int_1%5Einf%28%28x%5E8-8x%5E7%2B24x%5E6-32x%5E5%2B19x%5E4-12x%5E3%2B17x%5E2-10x%2B2%29%2F%28x%5E12-12x%5E11%2B56x%5E10-120x%5E9%2B82x%5E8%2B112x%5E7-182x%5E6-92x%5E5%2B381x%5E4-356x%5E3%2B170x%5E2-44x%2B5%29%29

Is it possible to compute it by manual?

Thanks in advance.

Answer 1

This is equivalent to American Mathematical Monthly Problem 11148 published in April 2005. Let $u=x-1$. Then rewrite the integral as $$\int_{0}^{\infty}{\frac{u^8-4u^6+9u^4-5u^2+1}{u^{12}-10u^{10}+37u^8-42u^6+26u^4-8u^2+1}}du.$$

The value of this integral is equal to the value of the original integral. The method to solve this equivalent integral can be found in the link below.

http://www.mat.uniroma2.it/~tauraso/AMM/AMM11148.pdf