Math Help - Integration: arctan(x) rule derivation

1. Integration: arctan(x) rule derivation

Hello,
the task is to derive integration rule (2) from rule (1).

(1) $\int \frac{1}{x^2+1} dx = \arctan (x)$

(2) $\int \frac{1}{x^2+c^2} dx = \frac{1}{c} \arctan (\frac{x}{c})$

$c$ is constant and $>0$

I tried to massage rule (1) with basic operations, but I always ended up with too many x's and c's. Also, I couldn't find any help using forum's search function. So, any help is welcome. Mucho thanks in advance!

2. let $cu=x \implies cdu = dx$

$\displaystyle \int \frac{dx}{x^2+c}=\int \frac{c du }{c^2u^2+c^2}=\frac{1}{c}\int \frac{du}{u^2+1}$