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$\ds{%
I \equiv \int_{-\pi/2}^{\pi/2}{x\sin\pars{x} \over 1 + \cos^{4}\pars{x}}\,\dd x:\
{\large ?}}$
$\large\tt\mbox{Hint:}$
\begin{align}
I &= 2\int_{0}^{\pi/2}x\sin\pars{x}\,{1 \over 2\expo{\ic\pi/2}}\bracks{%
{1 \over \cos^{2}\pars{x} - \expo{\ic\pi/2}} -
{1 \over \cos^{2}\pars{x} + \expo{\ic\pi/2}}}\,\dd x
\\[3mm]&=
2\Im\int_{0}^{\pi/2}{x\sin\pars{x} \over \cos^{2}\pars{x} - \expo{\ic\pi/2}}\,\dd x
=
2\Im\int_{0}^{\pi/2}{x\sin\pars{x} \over 2\expo{\ic\pi/4}}\bracks{%
{1 \over \cos\pars{x} - \expo{\ic\pi/4}} -
{1 \over \cos\pars{x} + \expo{\ic\pi/4}}}\,\dd x
\\[3mm]&=
\Im\int_{0}^{\pi/2}x\sin\pars{x}\,{\root{2} \over 2}\pars{1 - \ic}
\braces{2\ic\,\Im\bracks{1 \over \cos\pars{x} - \expo{\ic\pi/4}}}\,\dd x
\\[3mm]&=
\root{2}\Im\int_{0}^{\pi/2}{x\sin\pars{x} \over \cos\pars{x} - \expo{\ic\pi/4}}
\,\dd x
\end{align}
G&R-$7^{\ul{\rm a}}$ ed. has an identity $\pars{~{\bf 2.647}.2,\ \mbox{pag.}\ 224~}$ which seems close to this integral but unfortunately it's only valid for $\color{#0000ff}{\large m \not= 1}$:
$$
\int{x^{n}\sin\pars{x}\,\dd x \over \bracks{a + b\cos\pars{x}}^{m}}
=
{x^{n} \over \pars{m - 1}\bracks{a + b\cos\pars{x}}^{m - 1}}
-
{n \over \pars{m - 1}b}\int{x^{n - 1}\,\dd x \over \bracks{a + b\cos\pars{x}}^{m - 1}}
$$