How to integrate the following integral?

Question

I need to solve the following integral $$\int\frac{\sec^2(x)}{\sec^2(x)+a}dx$$ where $a$ is some positive constant. Thanks in advance.

Answer 1

$\textbf{Hint:}$ $$ \dfrac{\sec^2(x)}{\sec^2(x) + a} = \dfrac{\sec^2(x) + a - a}{\sec^2(x) + a} = 1 - \dfrac{a}{\sec^2(x) + a} $$ and $\sec^2(x) = 1 + \tan^2(x)$.

Answer 2

Hint: \begin{align} \frac{\sec^2 x}{\sec^2 x + a} = \frac{\sec^2 x}{1+a+\tan^2 x} \end{align} then $u = \tan x$.