please help me with integration of e^tanx.

I don't think there's an analytical solution.

If you try it with Wolfram's online integrator, it gives the result in terms of the Exponential Integral Ei function, which I believe is transcendental.

If you had limits of integration, we may be able to do something 'non-elementary' to evaluate. But, as jcaron said, it has no nice closed indefinite form.

May I ask... you do not happen to be studying the (second) fundamental theorem of calculus, are you? Sometimes a student is given something like:

.

Then, instead of exhibiting an understanding of the theorem, by brute force they try to evaluate the integral. Which, a lot of times, is not integrable by elementary means learned in a typical calc class.

Just a thought.