∫ 1/ (x^4+2x^2+1)
Notice the PFD?
$\displaystyle \int\frac{1}{x^{4}+2x^{2}+1}dx=\int\frac{1}{(x^{2} +1)^{2}}dx$
Let $\displaystyle x=tan(u), \;\ u=tan^{-1}(x), \;\ du=\frac{1}{1+x^{2}}dx$
Make the subs and get:
$\displaystyle \int\frac{1}{1+tan^{2}(u)}du$
$\displaystyle \int\frac{1}{sec^{2}(u)}du$
$\displaystyle \int cos^{2}(u)du$
Continue?.