Deadly integral $\int_0^1\frac{x^{2}+x+1}{x^{4}+x^{3}+x^{2}+x+1}dx$. How to solve this question

$$\int\limits_0^1\frac{x^{2}+x+1}{x^{4}+x^{3}+x^{2}+x+1}dx$$

. Please help me in solving this short way 
my approach is in the answer
Is it correct and can it be solved in a shorter way ?

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\begin{align}
&\color{#f00}{\int_{0}^{1}{x^{2} + x + 1 \over x^{4} + x^{3} + x^{2} + x + 1}
\,\dd x} =
\int_{0}^{1}{\pars{x^{2} + x + 1}\pars{x - 1} \over
             \pars{x^{4} + x^{3} + x^{2} + x + 1}\pars{x - 1}}\,\dd x =
\int_{0}^{1}{1 - x^{3} \over 1 - x^{5}}\,\dd x
\\[3mm] \stackrel{x^{5}\ \to x}{=} &
\int_{0}^{1}{1 - x^{3/5} \over 1 - x}\,{1 \over 5}\,x^{-4/5}\,\dd x =
{1 \over 5}\bracks{%
\int_{0}^{1}{1 - x^{-1/5} \over 1 - x}\,\dd x -
\int_{0}^{1}{1 - x^{-4/5} \over 1 - x}\,\dd x}
\end{align}
However,
$\ds{\Psi\pars{z} + \gamma = \int_{0}^{1}{1 - t^{z - 1} \over 1 - t}\,\dd t\,,\ \Re\pars{z} > 0}$. $\Psi$ and $\gamma$ are the digamma function and the
Euler-Mascheroni constant, respectively. See, for example, Abramowitz $\&$ Stegun Table. Then,
\begin{align}
&\color{#f00}{\int_{0}^{1}{x^{2} + x + 1 \over x^{4} + x^{3} + x^{2} + x + 1}
\,\dd x} =
{1 \over 5}\braces{\bracks{\Psi\pars{-\,{1 \over 5} + 1} + \gamma} -
\bracks{\Psi\pars{-\,{4 \over 5} + 1} + \gamma}}
\\[3mm] = &
{1 \over 5}\bracks{\Psi\pars{4 \over 5} - \Psi\pars{1 \over 5}} =
{1 \over 5}\,\pi\cot\pars{\pi\,{1 \over 5}}
\end{align}
In the last step we used the PolyGamma's Euler identity.
\begin{align}
&\color{#f00}{\int_{0}^{1}{x^{2} + x + 1 \over x^{4} + x^{3} + x^{2} + x + 1}
\,\dd x} =
{1 \over 5}\,\pi\cot\pars{\pi \over 5} =
{1 \over 5}\,\pi\root{1 + {2 \over \root{5}}}
\\[3mm] = &
\color{#f00}{{1 \over 25}\,\pi\root{10\root{5} + 25}} \approx 0.8648
\end{align}
