Find the indefinite integral.
(e^(9 x))/(2 + e^(9 x)) dx
did you notice that the derivative of the denominator is 9 e^(9x) which just happens to be sitting in the numerator as a multiple thereof.
Which is the ln derivative pattern.
So without any great fanfare.....
∫(e^(9x))/(2 + e^(9x)) dx
= (1/9) ln (2 + e^(9x) + c