Integrate $\int \dfrac{e^x}{1+\cos x}dx$

Question

Integrate $\displaystyle\int \dfrac{e^x}{1+\cos x}dx$

My Effort; I couldn't nothing.

Answer 1

Write $\cos(x)=\dfrac{1-\tan^2(x/2)}{1+\tan^2(x/2)}$ and then it's all easy.

Answer 2

$$\cos2A=2\cos^2A-1\iff1+\cos2A=?$$

Now $\dfrac{d\tan(ax)}{dx}=a\sec^2(ax)$

Answer 3

$$\frac1{1+\cos x}=\frac{1-\cos x}{\sin^2x}$$

and now split the integral in two almost immediate ones.