<s>[INST] Find $\int\frac{x}{(1+x)(x^4+4)}dx$ Find $$\int\frac{x}{(1+x)(x^4+4)}dx$$

I tried to do partial fractions.
$$\int\frac{x}{(1+x)(x^4+4)}dx=\int\frac{x+1-1}{(1+x)(x^4+4)}dx=\int\frac{dx}{x^4+4}-\int\frac{dx}{(1+x)(x^4+4)}$$
I am stuck now.I do not know how to integrate now.
 [/INST] HINT:
$$x^4+4=(x^2+2)^2-(2x)^2=\cdots$$
Now use Partial Fraction Decomposition,
$$\dfrac x{(x+1)(x^4+4)}=\dfrac A{x+1}+\dfrac{Bx+C}{x^2+2x+2}+\dfrac{Dx+E}{x^ 2-2x+2}$$
Write $$\dfrac{Dx+E}{x^ 2-2x+2}=\dfrac{D(x-1)}{(x-1)^2+1^2}+\dfrac{D+E}{(x-1)^2+1^2}$$
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