How to simplify the integration? If any integration is in form $$\int \frac{1}{1+x^2}dx$$ it easily follows the $\tan^{-1}x$ but How to simplify if we have
$$\int \frac{1}{(1+x^2)^2}dx$$

Since 
$$\left(\frac{x}{1+x^2}\right)'=\frac{2-(1+x^2)}{(1+x^2)^2}=\frac{2}{(1+x^2)^2}-\frac{1}{1+x^2}$$
one has
$$2\int \frac{1}{(1+x^2)^2}dx=\frac{x}{1+x^2}+\int \frac{1}{1+x^2}dx$$
$$\Rightarrow \int\frac{1}{(1+x^2)^2}dx=\frac{x}{2(1+x^2)}+\frac{\tan^{-1} x}{2}+C$$
