1a) Prove that $$e^x\operatorname{sech} x\equiv\frac{2e^{2x}}{e^{2x}+1}$$
b) Find $$\frac{d}{dx}[\arcsin(\tanh x)]$$ Simplify your answer as far as possible.
c) Hence, or otherwise, solve $$\int e^x\arcsin(\tanh x)dx$$
I did parts a and b but then got stuck on part c. Can someone please explain how I go about solving this? Thanks.
BTW, for part b I got an answer of $\operatorname{sech}x$.
Hints:
First use integration by parts (along with your knowledge found b).
Second use part a to rewrite the resulting integral.
Third set $u=e^{2x}$ and evaluate.
And of course, $+C$ must be remembered!