How do I evaluate the integral $\int_0^1\frac{x^2+x+1}{x^4+x^3+x^2+x+1}dx$. stuck on this integral
$$\int_0^1\dfrac{(x^2+x+1)}{(x^4+x^3+x^2+x+1)}\ dx$$
I was attempting to evaluate  the infinity sum S = $ 1- \frac{1}{4}  + \frac {1}{6} - \frac {1}{9} + \frac {1}{11} -\frac {1}{14}+  ........ $ 
what I then  did was define the S to be equal to $$ \int_0^1 (1-x^3 +x^5-x^8+x^{10}-x^{13}+........) dx $$
I simplified this  and got the above integral I tried to do partial fraction but did not succeed.

If your purpose is to evaluate
$$ S=\sum_{k\geq 0}\left[\frac{1}{10k+1}-\frac{1}{10k+4}+\frac{1}{10k+6}-\frac{1}{10k+9} \right]$$
you do not need an indefinite integral, just an integral over $(0,1)$: that's a huge difference, in some cases. 
Actually, since $1+9=4+6=10$, you may just invoke the identities
$$ \sum_{k\geq 0}\left[\frac{1}{10k+1}-\frac{1}{10k+9}\right]=\frac{\pi}{10}\cot\frac{\pi}{10} $$
$$ \sum_{k\geq 0}\left[\frac{1}{10k+4}-\frac{1}{10k+6}\right]=\frac{\pi}{10}\cot\frac{4\pi}{10} $$
which follow from Herglotz' trick / the reflection formula for the digamma function.
In particular
$$ S = \frac{\pi}{5}\sqrt{1+\frac{2}{\sqrt{5}}}.$$
