@spalinowy 

I gave you only a comment,my point of view. Wait for the answer.

And so at the end you'll end up with numerical calculations.

You have a Transcendental equations that why Maple can't solve(Any CAS can't solve).Transcendental equations have no analytical solution.

Try numeric solver like fsolve, DirectSearch package,or something else.

See:https://en.wikipedia.org/wiki/Transcendental_equation

        https://www.mapleprimes.com/questions/88377-Direct-Search-Optimization-Package

@digerdiga 

I updated my answer.Its the same answer like done by  the user Carl Love.Probably a better simplification can not be done.

Ei_ver2.mw

@digerdiga 

restart;

assume(R::real, z::real); is(-R+sqrt(R^2+z^2) < 0);

#false

@digerdiga 

`assuming`([convert(signum(0, R-sqrt(R^2+z^2), 1), piecewise)], [R-sqrt(R^2+z^2) < 0]);

# -1

or:

`assuming`([signum(0, R-sqrt(R^2+z^2), 1)], [R-sqrt(R^2+z^2) < 0])

#-1

Maple have some limitation about regarding the resolution of the case and falsehood.

restart:

assume(R > 0, z > 0, R in real, z in real);

coulditbe(R-sqrt(R^2+z^2) < 0);#FAIL

is(R-sqrt(R^2+z^2) < 0);#FAIL

Maybe this helps: http://www.cim.mcgill.ca/~migueltt/ltp/ltp.html

@one pound 

From Wikipedia, using property: Ei(1,z) = - Ei(-z) and faster converging series by Ramanujan:

For example, for z = 10,and for  40 terms  an answer correct to 16 significant figures.

series.mw

@kuwait1 

A numeric check gives a big numbers in domain x=(-5,5) and z=(-5,5).Integral is divergent.

r := 2.5;
Q := proc (x, z) options operator, arrow; evalf(Int(1/(x*r^2*cos(x-y)+z*r^2*sin(z-y))^5, y = 0 .. Pi, method = _Gquad, epsilon = 1/100000)) end proc;
seq(seq(lprint(Q(x, z)), x = -5 .. 5, .25), z = -5 .. 5, .25);

-0.3091653839e55
0.7846201518e57
0.2353821795e58
-0.7841865263e57
-0.7756899663e57
0.1123370081e58
...

@ssllys You are welcome :)

@mmcdara . Thanks for checking :)

@pr0t3ctabl3 

All  my code in answer works on Maple 2017.3 and 2018. I don't have earlier version of Maple, so I will not check it.

@pr0t3ctabl3 

I edited the answer.It should work now.

All code was exectuted in Maple 2018.

@mmcdara 

31165-118782-3-PB.pdf

 look here: https://www.mapleprimes.com/questions/147800-Code-For-Homotopy-Analysis-Method and choose what you need.

If you discovered a strange result produced by Maple,
it would be useful to exhibit the simplest version where
this abnormal result is still present.Can you add actual code so people can copy-paste easily?.