I don't understand why...

[split grotto]

In this problem for simplifications sake

1. I multiplied $\sec^2{x}$ on top and bottom.
   Ok, up until this point everything is clear.
   But after that I had to assume
   $\tan{x}=z$ then using that I have to differentiate it and substitute the $\sec^{2}{x}dx$.
   My question is:
2. Why and how can someone know what to assume in this state:
   $$\int \frac{\sec^{2}{x}dx}{5+2\tan^{2}x}$$

And,
What happened at the last line?
I know the formula for:
$$\int \frac{dx}{a^{2}+x^{2}}=\frac{1}{a} \tan^{-1}{\frac{x}{a}}$$

But I'm not understanding how to implement it in this scenario.

[loud silo]

whoami