Playing with WA I have a trouble to understand something why we have (based on a numerical result from WA): $$\int_0^e \operatorname{W}(x)^i\Big(-\operatorname{W}(x)\Big)^{-i} \, dx = \int_0^e \exp(x)^i(-\exp(x))^{-i}\,dx=\int_0^e \sin(x)^i(-\sin(x))^{-i} \, dx = e^{1+\pi}$$
Always with Wolfram alpha we have :
$$\int_0^e\tan(x)^i(-\tan(x))^{-i} \, dx = \int_0^e \cos(x)^i(-\cos(x))^{-i} \, dx = e^{1-\pi} + \pi\sinh(\pi)$$
My question :
Is Wolfram alpha wrong or there exists a deeper formula ?
If it's good how to prove it ?
Maybe it's trivial I don't know I have not the skill to decide this .
Thanks a lot for share your knowledge .
Observe that after simplification,
$$\int_0^e(-1)^{-i}dx=e\cdot(e^{i\pi})^{-i}=e^{1+\pi}.$$
For the other integrals, decompose the interval to handle the phase jumps.