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Es bastante claro que el OP ha evaluado la integral
$\ds{\color{#f00}{\int_{0}^{\pi/2}x^{2}\root{\cot\pars{x}}\,\dd x}}$. Sin embargo, el OP es$\texttt{still looking for a simple evaluation}$, el cual comenzará a partir de $\ds{\,\mrm{I}\pars{a,b}\ \pars{~\mbox{see below}~}}$. De aquí en adelante nos presente un "relativamente simple" evaluación de la misma.
$\ds{\mrm{I}\pars{a,b} =
{\pi\,2^{b}\,\cos\pars{\pi/2}\Gamma(1 - b) \más
\Gamma\pars{a/2 - b/2 + 1}\Gamma\pars{-a/2 - b/2 + 1}}\,,\qquad\Gamma\pars{\mitad} = \raíz{\pi}}$
\begin{align}
\mrm{I}\pars{a,\half} & =
\root{2}\pi^{3/2}\,\,
{\cos\pars{\pi a/2} \over \Gamma\pars{3/4 + a/2}\Gamma\pars{3/4 - a/2}}
\end{align}
Tenga en cuenta que
$$
\color{#f00}{\int_{0}^{\pi/2}x^{2}\raíz{\cuna\pars{x}}\,\dd x} =
-\,{\raíz{2} \más de 8}\llaves{2\bracks{\epsilon^{2}}\mrm{I}\pars{\mitad + \epsilon,\mitad}}
$$
donde
\begin{align}
\mrm{I}\pars{\half + \epsilon,\half} & =
\pi^{3/2}\,\,{\cos\pars{\pi\epsilon/2} - \sin\pars{\pi\epsilon/2} \over
\Gamma\pars{1 + \epsilon/2}\Gamma\pars{1/2 - \epsilon/2}} =
\pi^{3/2}\,\,{\cos\pars{\pi\epsilon/2} - \sin\pars{\pi\epsilon/2} \over
\pars{\epsilon/2}!\pars{-1/2 - \epsilon/2}!}
\\[5mm] & =
\pi^{3/2}\,{1 \over \pars{-1/2}!}{-1/2 \choose \epsilon/2}
\bracks{\cos\pars{\pi\epsilon/2} - \sin\pars{\pi\epsilon/2}}
\\[5mm] & =
\pi\,{-1/2 \choose \epsilon/2}
\bracks{\cos\pars{\pi\epsilon \over 2} - \sin\pars{\pi\epsilon \over 2}}
\end{align}
Sólo necesitamos la binomial y la 'dentro de corchetes término" expansión de la a a $\ds{\epsilon^{2}}$. El binomio de expansión, hasta el fin de $\ds{\epsilon^{2}}$, es sencillo pero laborioso: se simplifica mediante el uso de la Digamma valor $\ds{\Psi\pars{1/2} = -\gamma - 2\ln\pars{2}}$. $\ds{\gamma}$: De Euler-Mascheroni Constante.
$$
\left\{\begin{array}{rcl}
\ds{-1/2 \choose \epsilon/2} & \ds{=} &
\ds{1 - \ln\pars{2}\,\epsilon +
\bracks{\half\,\ln^{2}\pars{2} - {\pi^{2} \over 12}}\epsilon^{2} + \,\mrm{O}\pars{\epsilon^{3}}}
\\[3mm]
\ds{\cos\pars{\pi\epsilon \over 2} - \sin\pars{\pi\epsilon \over 2}} & \ds{=} & \ds{1 - {\pi \over 2}\,\epsilon - {\pi^{2} \over 8}\,\epsilon^{2} + \,\mrm{O}\pars{\epsilon^{3}}}
\end{array}\right.
$$
\begin{align}
&\color{#f00}{\int_{0}^{\pi/2}x^{2}\root{\cot\pars{x}}\,\dd x}
\\[5mm] = &\
-\,{\root{2} \over 8}\,\times 2\times \pi\braces{%
1\times\pars{-\,{\pi^{2} \over 8}} +
\bracks{-\ln\pars{2}}\pars{-\,{\pi \over 2}} +
\bracks{\half\,\ln^{2}\pars{2} - {\pi^{2} \over 12}}\times 1}
\\[5mm] & =
\color{#f00}{{\root{2} \over 8}\bracks{{5\pi^{3} \over 12} -
\pi^{2}\ln\pars{2} - \pi\ln^{2}\pars{2}}} \approx 0.8077
\end{align}
