$\displaystyle\int \dfrac{dx}{x^5 + 1}$.

Afim de decompor $\dfrac{1}{x^5 + 1}$ em frações parciais, calculemos as raízes quintas de $-1$, que estão graficamente representadas abaixo:

Logo $\dfrac{1}{x^5 + 1} = \dfrac{Ax + B}{x^2 - \left(2\cos \dfrac{\pi}{5}\right)x + 1} + \dfrac{Cx + D}{x^2 - \left(2\cos \dfrac{3\pi}{5}\right)x + 1} + \dfrac{E}{x + 1}$. ${\large (I)}$

Donde, resolvendo o sistema:

${\tiny \begin{cases}B + D + E = 1\\ \left[4 - 4\cos \left(\dfrac{3\pi}{5}\right)\right]A + \left[4 - 4\cos \left(\dfrac{3\pi}{5}\right)\right]B + \left[4 - 4\cos \left(\dfrac{\pi}{5}\right)\right]C + \left[4 - 4\cos \left(\dfrac{\pi}{5}\right)\right]D + \left[4 - 4\cos \left(\dfrac{3\pi}{5}\right)\right]\left[4 - 4\cos \left(\dfrac{\pi}{5}\right)\right]E = 1\\ \left[30 - 24\cos \left(\dfrac{3\pi}{5}\right)\right]A + \left[15 - 12\cos \left(\dfrac{3\pi}{5}\right)\right]B + \left[30 - 24\cos \left(\dfrac{\pi}{5}\right)\right]C + \left[15 - 12\cos \left(\dfrac{\pi}{5}\right)\right]D + \left[5 - 4\cos \left(\dfrac{3\pi}{5}\right)\right]\left[5 - 4\cos \left(\dfrac{\pi}{5}\right)\right]E = 1\\ \left[120 - 72\cos \left(\dfrac{3\pi}{5}\right)\right]A + \left[40 - 24\cos \left(\dfrac{3\pi}{5}\right)\right]B + \left[120 - 72\cos \left(\dfrac{\pi}{5}\right)\right]C + \left[40 - 24\cos \left(\dfrac{\pi}{5}\right)\right]D + \left[10 - 6\cos \left(\dfrac{3\pi}{5}\right)\right]\left[10 - 6\cos \left(\dfrac{\pi}{5}\right)\right]E = 1\\ \left[10 + 8\cos \left(\dfrac{3\pi}{5}\right)\right]A - \left[5 + 4\cos \left(\dfrac{3\pi}{5}\right)\right]B + \left[10 + 8\cos \left(\dfrac{\pi}{5}\right)\right]C - \left[5 + 4\cos \left(\dfrac{\pi}{5}\right)\right]D + \left[5 + 4\cos \left(\dfrac{3\pi}{5}\right)\right]\left[5 + 4\cos \left(\dfrac{\pi}{5}\right)\right]E = 1\end{cases}}$,

obtemos:

${\tiny \begin{cases}A = \dfrac{\left( 24 {\cos^2 { \dfrac{\pi }{5} }}+92 \cos{ \dfrac{\pi }{5} }+200\right) \cos{ \dfrac{3 \pi }{5} }+32 {\cos^2 { \dfrac{\pi }{5} }}+170 \cos{ \dfrac{\pi }{5} }+285}{\left( 288 \cos{ \dfrac{\pi }{5} }+480\right) {\cos^2 { \dfrac{3 \pi }{5} }}+\left( 720-288 {\cos^2 { \dfrac{\pi }{5} }}\right) \cos{ \dfrac{3 \pi }{5} }-480 {\cos^2 { \dfrac{\pi }{5} }}-720 \cos{ \dfrac{\pi }{5} }}\\ B = -\dfrac{\left( 288 {\cos^2 { \dfrac{\pi }{5} }}+420 \cos{ \dfrac{\pi }{5} }-200\right) \cos{ \dfrac{3 \pi }{5} }+480 {\cos^2 { \dfrac{\pi }{5} }}+592 \cos{ \dfrac{\pi }{5} }-285}{\left( 288 \cos{ \dfrac{\pi }{5} }+480\right) {\cos^2 { \dfrac{3 \pi }{5} }}+\left( 720-288 {\cos^2 { \dfrac{\pi }{5} }}\right) \cos{ \dfrac{3 \pi }{5} }-480 {\cos^2 { \dfrac{\pi }{5} }}-720 \cos{ \dfrac{\pi }{5} }}\\ C = -\dfrac{\left( 24 \cos{ \dfrac{\pi }{5} }+32\right) {\cos^2 { \dfrac{3 \pi }{5} }}+\left( 92 \cos{ \dfrac{\pi }{5} }+170\right) \cos{ \dfrac{3 \pi }{5} }+200 \cos{ \dfrac{\pi }{5} }+285}{\left( 288 \cos{ \dfrac{\pi }{5} }+480\right) {\cos^2 { \dfrac{3 \pi }{5} }}+\left( 720-288 {\cos^2 { \dfrac{\pi }{5} }}\right) \cos{ \dfrac{3 \pi }{5} }-480 {\cos^2 { \dfrac{\pi }{5} }}-720 \cos{ \dfrac{\pi }{5} }}\\ D = \dfrac{\left( 288 \cos{ \dfrac{\pi }{5} }+480\right) {\cos^2 { \dfrac{3 \pi }{5} }}+\left( 420 \cos{ \dfrac{\pi }{5} }+592\right) \cos{ \dfrac{3 \pi }{5} }-200 \cos{ \dfrac{\pi }{5} }-285}{\left( 288 \cos{ \dfrac{\pi }{5} }+480\right) {\cos^2 { \dfrac{3 \pi }{5} }}+\left( 720-288 {\cos^2 { \dfrac{\pi }{5} }}\right) \cos{ \dfrac{3 \pi }{5} }-480 {\cos^2 { \dfrac{\pi }{5} }}-720 \cos{ \dfrac{\pi }{5} }}\\ E = -\dfrac{3}{\left( 12 \cos{ \dfrac{\pi }{5} }+20\right) \cos{ \dfrac{3 \pi }{5} }+20 \cos{ \dfrac{\pi }{5} }+30}\end{cases}}$. ${\large (II)}$

De ${\large (I)}$ obtemos:

${\tiny \dfrac{1}{x^5 + 1} = \dfrac{A}{2} \cdot \dfrac{2x - 2\cos \dfrac{\pi}{5}}{x^2 - \left(2\cos \dfrac{\pi}{5}\right)x + 1} + \dfrac{B + A\cos \dfrac{\pi}{5}}{\sin^2 \dfrac{\pi}{5}} \cdot \dfrac{1}{\left(\dfrac{x - \cos \dfrac{\pi}{5}}{\sin \dfrac{\pi}{5}}\right)^2 + 1} + \dfrac{C}{2} \cdot \dfrac{2x - 2\cos \dfrac{3\pi}{5}}{x^2 - \left(2\cos \dfrac{3\pi}{5}\right)x + 1} + \dfrac{D + C\cos \dfrac{3\pi}{5}}{\sin^2 \dfrac{3\pi}{5}} \cdot \dfrac{1}{\left(\dfrac{x - \cos \dfrac{3\pi}{5}}{\sin \dfrac{3\pi}{5}}\right)^2 + 1} + \dfrac{E}{x + 1}}$. ${\large (III)}$
 
Substituindo ${\large (II)}$ em ${\large (III)}$:

$\fbox{$\begin{array}{l}{\tiny \displaystyle\int \dfrac{dx}{x^5 + 1} \overset{x\ \neq\ \cos \dfrac{\pi}{5}}{=} \dfrac{\left( 24 {\cos^2 { \dfrac{\pi }{5} }}+92 \cos{ \dfrac{\pi }{5} }+200\right) \cos{ \dfrac{3 \pi }{5} }+32 {\cos^2 { \dfrac{\pi }{5} }}+170 \cos{ \dfrac{\pi }{5} }+285}{2 \left[ \left( 288 \cos{ \dfrac{\pi }{5} }+480\right) {\cos^2 { \dfrac{3 \pi }{5} }}+\left( 720-288 {\cos^2 { \dfrac{\pi }{5} }}\right) \cos{ \dfrac{3 \pi }{5} }-480 {\cos^2 { \dfrac{\pi }{5} }}-720 \cos{ \dfrac{\pi }{5} }\right] } \log \left|x^2 - \left(2\cos \dfrac{\pi}{5}\right)x + 1\right| +}\\ \\ {\tiny \dfrac{\dfrac{\cos{ \dfrac{\pi }{5} } \left[ \left( 24 {\cos^2 { \dfrac{\pi }{5} }}+92 \cos{ \dfrac{\pi }{5} }+200\right) \cos{ \dfrac{3 \pi }{5} }+32 {\cos^2 { \dfrac{\pi }{5} }}+170 \cos{ \dfrac{\pi }{5} }+285\right] }{\left( 288 \cos{ \dfrac{\pi }{5} }+480\right) {\cos^2 { \dfrac{3 \pi }{5} }}+\left( 720-288 {\cos^2 { \dfrac{\pi }{5} }}\right) \cos{ \dfrac{3 \pi }{5} }-480 {\cos^2 { \dfrac{\pi }{5} }}-720 \cos{ \dfrac{\pi }{5} }} - \dfrac{\left( 288 {\cos^2 { \dfrac{\pi }{5} }}+420 \cos{ \dfrac{\pi }{5} }-200\right) \cos{ \dfrac{3 \pi }{5} }+480 {\cos^2 { \dfrac{\pi }{5} }}+592 \cos{ \dfrac{\pi }{5} }-285}{\left( 288 \cos{ \dfrac{\pi }{5} }+480\right) {\cos^2 { \dfrac{3 \pi }{5} }}+\left( 720-288 {\cos^2 { \dfrac{\pi }{5} }}\right) \cos{ \dfrac{3 \pi }{5} }-480 {\cos^2 { \dfrac{\pi }{5} }}-720 \cos{ \dfrac{\pi }{5} }}}{\left(\sin \dfrac{\pi}{5}\right) ⋅ \arctan^{-1} \dfrac{x - \cos \dfrac{\pi}{5}}{\sin \dfrac{\pi}{5}}} -} \\ \\ {\tiny -\dfrac{\left( 24 \cos{ \dfrac{\pi }{5} }+32\right) {\cos^2 { \dfrac{3 \pi }{5} }}+\left( 92 \cos{ \dfrac{\pi }{5} }+170\right) \cos{ \dfrac{3 \pi }{5} }+200 \cos{ \dfrac{\pi }{5} }+285}{2 \left[ \left( 288 \cos{ \dfrac{\pi }{5} }+480\right) {\cos^2 { \dfrac{3 \pi }{5} }}+\left( 720-288 {\cos^2 { \dfrac{\pi }{5} }}\right] \cos{ \dfrac{3 \pi }{5} }-480 {\cos^2 { \dfrac{\pi }{5} }}-720 \cos{ \dfrac{\pi }{5} }\right) } \log \left|x^2 - \left(2\cos \dfrac{3\pi}{2}\right)x + 1\right| +}\\ \\ {\tiny +\dfrac{\dfrac{\left( 288 \cos{ \dfrac{\pi }{5} }+480\right) {\cos^2 { \dfrac{3 \pi }{5} }}+\left( 420 \cos{ \dfrac{\pi }{5} }+592\right) \cos{ \dfrac{3 \pi }{5} }-200 \cos{ \dfrac{\pi }{5} }-285}{\left( 288 \cos{ \dfrac{\pi }{5} }+480\right) {\cos^2 { \dfrac{3 \pi }{5} }}+\left( 720-288 {\cos^2 { \dfrac{\pi }{5} }}\right) \cos{ \dfrac{3 \pi }{5} }-480 {\cos^2 { \dfrac{\pi }{5} }}-720 \cos{ \dfrac{\pi }{5} - \dfrac{\cos{ \dfrac{3 \pi }{5} } \left( \left( 24 \cos{ \dfrac{\pi }{5} }+32\right) {\cos^2 { \dfrac{3 \pi }{5} }}+\left( 92 \cos{ \dfrac{\pi }{5} }+170\right) \cos{ \dfrac{3 \pi }{5} }+200 \cos{ \dfrac{\pi }{5} }+285\right) }{\left( 288 \cos{ \dfrac{\pi }{5} }+480\right) {\cos^2 { \dfrac{3 \pi }{5} }}+\left( 720-288 {\cos^2 { \dfrac{\pi }{5} }}\right) \cos{ \dfrac{3 \pi }{5} }-480 {\cos^2 { \dfrac{\pi }{5} }}-720 \cos{ \dfrac{\pi }{5} }}}}}{\left(\sin \dfrac{3\pi}{5}\right) ⋅ \arctan^{-1} \dfrac{x - \cos \dfrac{3\pi}{5}}{\sin \dfrac{3\pi}{5}}} -}\\ \\ {\tiny -\dfrac{3 \log \left|x + 1\right|}{\left( 12 \cos{ \dfrac{\pi }{5} }+20\right) \cos{ \dfrac{3 \pi }{5} }+20 \cos{ \dfrac{\pi }{5} }+30} + c}\\ \\ {\tiny c\ \in\ \mathbb{R}}\end{array}$}$.