∫ e^x sec x(1+tan x) d x equals :

$\int e^x \sec x(1+\tan x) d x$

$= \int e^x \sec x+e^x \sec x \tan x d x$

So $\int e^x \sec x+e^x \sec x \tan x d x$

$\frac{d}{d x}(\sec x)=\sec x \tan x$

so this is of the form

$\int e^x f(x)+e^x f'(x) d x=e^x f(x)$

$f(x)=\sec x$

$=e^x \sec x+c$