<s>[INST] How can one integrate $\int\frac{1}{(x+1)^4(x^2+1)} dx$? 
How can one integrate $\displaystyle\int\frac{1}{(x+1)^4(x^2+1)}\  dx$?

Attempt:
I tried partial fraction decomposition (PFD) and got lost. The method of u-substitution didn't work for me either. 
What else can I do? Can one calculate the integral without PFD?
 [/INST] We use a variant of the Heaviside method. Shift by one and consider
$$\frac{1}{z^4(z^2-2z+2)}\text{.}$$
Develop $1/(z^2-2z+2)$ in series about $z=0$, keeping the remainder exactly as you go:
$$\frac{1}{z^2-2z+2}=\frac{1}{2}+\frac{z}{2}+\frac{z^2}{4}-\frac{z^4}{4(z^2-2z+2)}\text{.}$$
Then
$$\frac{1}{z^4(z^2-2z+2)}=\frac{2 + 2z + z^2}{4z^4}-\frac{1}{4(z^2-2z+2)}\text{.}$$
Can you take it from here?
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