Vtyjkyuk

So I found this problem:

Calculate $$int fracx^4+1x^12-1 dx$$ where $xin(1, +infty)$

and I don't have any ideea how to solve it. I tried to write $$x^12-1=(x^6+1)(x+1)(x-1)(x^2+x+1)(x^2-x+1)$$ but with $x^4+1$ I had no idea what to do, an obvious thing would be to add some terms (an $x^5$ would be helpful) and then discard them, but again I couldn't do anything. What should I do?

You can use that $$x^4+1=(x^2+1)^2-2x^2=…$$
You will get $$fracx^4+1x^12-1=frac-2 x-112 left(x^2+x+1right)-frac13 left(x^2+1right)+frac2 x-112
left(x^2-x+1right)+frac-x^2-16 left(x^4-x^2+1right)+frac16
(x-1)-frac16 (x+1)$$

Use $$x^6+1=(x^2+1)(x^4-x^2+1)$$
$$I=displaystyleintdfracx^4+1left(x-1right)left(x+1right)left(x^2+1right)left(x^2-x+1right)left(x^2+x+1right)left(x^4-x^2+1right),mathrmdx$$

Apply partial Factor decomposition you get

$$I=displaystyleintleft(-dfracx^2+16left(x^4-x^2+1right)-dfrac2x+112left(x^2+x+1right)+dfrac2x-112left(x^2-x+1right)-dfrac13left(x^2+1right)-dfrac16left(x+1right)+dfrac16left(x-1right)right)mathrmdx$$
$$I=-classsteps-nodecssIdsteps-node-1dfrac16displaystyleintdfracx^2+1x^4-x^2+1,mathrmdx-classsteps-nodecssIdsteps-node-2dfrac112displaystyleintdfrac2x+1x^2+x+1,mathrmdx+classsteps-nodecssIdsteps-node-3dfrac112displaystyleintdfrac2x-1x^2-x+1,mathrmdx-classsteps-nodecssIdsteps-node-4dfrac13displaystyleintdfrac1x^2+1,mathrmdx-classsteps-nodecssIdsteps-node-5dfrac16displaystyleintdfrac1x+1,mathrmdx+classsteps-nodecssIdsteps-node-6dfrac16displaystyleintdfrac1x-1,mathrmdx$$