Evaluate $\int \frac{\sec^{2} x}{\tan x + 1}\,dx$.

Explanation

1. 01

   The numerator $\sec^2 x$ is the derivative of the denominator $1 + \tan x$.

2. 02

   With $t = 1 + \tan x$: $\int \frac{dt}{t} = \log|t| = \log|1 + \tan x| + C$.