Indefinite integral of $x^3+x+1/x^4+x^2+1$

Question

I had come across a question as follows:

Compute the integral $$\int_{}{(x^3+x+1)/(x^4+x^2+1)}dx$$

I tried a lot manipulating the numerator but I couldn't convert the terms into standard forms. Factorisation of the denominator didn't prove to be helpful either. I have run out of ideas. Would someone please help?

Thanks in advance!

Answer 1

Extended hint: $$ \frac{x^3+x+1}{x^4+x^2+1}=\frac12\left[\frac1{x^2-x+1}+\frac{2x+1}{x^2+x+1}\right]. $$

Answer 2

Hint $$x^4+x^2+1=(x^4+2x^2+1)-x^2$$

Use the formula for difference of squares.