<s>[INST] 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.
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\begin{align}
&\bbox[10px,#ffd]{\int_{0}^{1}{x^{2} + x + 1 \over
x^{4} + x^{3} + x^{2} + x + 1}\,\dd x} =
\int_{0}^{1}{\pars{x^{3} - 1}/\pars{x - 1} \over
\pars{x^{5} - 1}/\pars{x - 1}}\,\dd x
\\[5mm] = &\
\int_{0}^{1}{1 - x^{3} \over 1 - x^{5}}\,\dd x
\,\,\,\stackrel{x^{5}\ \mapsto\ x}{=}\,\,\,
{1 \over 5}\int_{0}^{1}{x^{-4/5} - x^{-1/5} \over 1 - x}\,\dd x
\\[5mm] = &\
{1 \over 5}\pars{\int_{0}^{1}{1 - x^{-1/5} \over 1 - x}\,\dd x -
\int_{0}^{1}{1 - x^{-4/5} \over 1 - x}\,\dd x} =
{1 \over 5}\bracks{\Psi\pars{4 \over 5} - \Psi\pars{1 \over 5}}
\\[5mm] = &\
{1 \over 5}\bracks{\pi\cot\pars{\pi{1 \over 5}}} =
\bbx{{1 \over 5}\root{1 + {2 \over \root{5}}}\,\pi} \approx 0.8648
\end{align}
</s>