Calculus of a Single Variable, Chapter 5, 5.2, Section 5.2, Problem 39

int (sec(x)tan(x))/(sec(x)-1)dx=
We will use the following formula: int (f'(x))/(f(x))dx=ln|f(x)|+C
The formula tells us that if we have integral of rational function where the numerator is equal to the derivative of the denominator, then the integral is equal to natural logarithm of the denominator plus some constant. The proof of the formula can be obtained by simply integrating the right-hand side.
Since (sec(x)-1)'=sec(x)tan(x) we can apply the formula to obtain the final result.
ln|sec(x)-1|+C