<s>[INST] Integrate $\int \frac{x^2-2}{(x^2+2)^3}dx$ $$\int \frac{x^2-2}{(x^2+2)^3}dx$$
What I did :
Method $(1) $ Re writing $x^2-2 = (x^2+2)-4 $ and partial fractions.
Method $(2) $ Substituting $x^2 = 2\tan^2 \theta $ 
Is there any other easy methods ? 
Some substitution  ? 
 [/INST] If
$$I_n=\int\frac{1}{(ax^2+b)^n}dx\quad,\quad n\ge 2$$
$$I_1=\frac{\sqrt{\frac{b}{a}}}{b}\tan^{-1}\left(\sqrt{\frac{a}{b}}x\right)$$
then
$$I_n=\frac{2n-3}{2b(n-1)}I_{n-1}+\frac{x}{2b(n-1)(ax^2+b)^{n-1}}$$
set $a=1$ and $b=2$ apply
$$\frac{x^2-2}{(x^2+2)^3}=\frac{1}{(x^2+2)^2}-\frac{4}{(x^2+2)^3}$$
we have 
$$\int \frac{x^2-2}{(x^2+2)^3}dx=-\left(\frac{\sqrt{2}}{16}\tan^{-1}\left(\frac{\sqrt{2}x}{2}\right)+\frac{x}{8x^2+16}+\frac{x}{2(x^2+2)^2}\right)+c$$
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