How do I approach $\int e^{\sin x}\cdot(1+\tan x\sec x)dx$

Question

$$\int e^{\sin x}\cdot(1+\tan x\sec x)dx$$ I tried the trig identity $$1+\tan x\sec x=1+\frac{\sin x}{\cos^2x}$$ Which gives $$\frac{\cos^2x+\sin x}{\cos^2x}$$ This is where I stopped

Answer 1

Note

$$ (\sec x)'=\tan x \sec x $$

Thus

\begin{align} I&=\int e^{\sin x}\,dx+\int e^{\sin x}\,d(\sec x) \\ &=\int e^{\sin x}\,dx+e^{\sin x}\sec x-\int e^{\sin x}\,dx \\ &=e^{\sin x}\sec x+C \\ \end{align}