Finding integral of functions involving e raised to another function.

Finding integral of functions involving e raised to another function.

$$\int_0^{\pi/4}(1+e^{\tan\theta})\sec^2\theta\, d\theta$$
I have tried to let $u=\sec^2\theta$ so that $du=\tan\theta \, d\theta$.
After doing that I was unable to figure a way to substitute $u$ and $du$
back into my integrand. The integrand I ended up with was
$(1+e^{du/d\theta}) u\, d\theta$. This does not seem right to me because I
have never learned to deal with the exponent of $e$ if it's even possible.
What other approach can I use that involves $u$-substitution?