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  ? 

A standard method for dealing with a numerator that is a power of positive definite quadratic polynomial is the following.
Observe that for any positive integer $n$ we have
$$
D\frac{x}{(x^2+2)^n}=-\frac{(2n-1)x^2-2}{(x^2+2)^{n+1}}.\qquad(*)
$$
You can write the right hand side as a linear combination of $1/(x^2+2)^n$ and
$1/(x^2+2)^{n+1}$.
Interpreting $(*)$ as an integration formula then gives you a linear equation involving $\int (x^2+2)^{-(n+1)}\,dx$, $\int (x^2+2)^{-n}\,dx$ and $x/(x^2+2)^n$.
So:


*

*Setting $n=2$ in the resulting equation allows you to integrate $1/(x^2+2)^3$ if you know how to integrate $1/(x^2+2)^2$.

*Setting $n=1$ in the resulting equation allows you to integrate $1/(x^2+2)^2$ if you know how to integrate $1/(x^2+2)$.

*But you know how to integrate $1/(x^2+2)$, don't you?

