How do I solve $ \int \frac{e^x}{ 1+e^{2x}}\, dx$?

Question

So... I have been struggling with this expression $$\frac{e^x}{1+e^{2x}}$$ for a while now... I allways seem to get into a loop and can't get out!

Example: I substitute $u=e^x$ and then get $du= e^x du$. So the new expression looks like this --> $$\frac{1}{1+u^2} \cdot du$$ . After this I substitute $m=u^2$ and get $$dm = 2u \cdot du.$$ So the new expression looks like this -> $$\frac{1}{2\sqrt{m}(1+m)}$$.. After this Im stuck and can't get out.. I have tried to substitute other expressions first in u but still I end up somewhat simmilar to this situation!

Please can someone explain me what am I doing wrong! Can someone evaluate this expression for me showing me the steps?! I would be greatful!

Answer 1

Hint: $$\int \frac{1}{1+u^2}du=\arctan(u)$$

Answer 2

Hint:

What's the derivative of $\arctan(u)$?