Here is the fragment from the help  ?plot,structure :

"GRID(a..b, c..d, A)
The GRID structure represents surfaces in 3-D space defined by a uniform sampling over a rectangular (aligned) region of the plane.  The GRID structure takes the form GRID(a..b, c..d, A) where a..b is the x-range, c..d is the y-range, and A is a two-dimensional Array.  If you have an m-by-n grid, then element A[i, j] is the function value at grid point (i, j), for i in 1..m and j in 1..n.  The Array A may be replaced by a list of the form [[z11,...z1n], [z21,...z2n],...[zm1...zmn]] where zij is the function value at grid point (i, j)."

(i, j) are not values of x and y, but indexes for function z values. In your example i=1..4, j=1..3

Here are two simple steps to get the desired simplification. We take the second step only if we are not satisfied with the first one:

simplify(evalc(w)); # The first step
numer(%)/expand(denom(%)); # The second step

You've spent a lot of time looking for a one-line code. You could use it to do more useful things.

The standard way is to use the  piecewise command:

restart;
f:=piecewise(x<0,1/x, x>=0,sqrt(x));
plot(f, x=-4..4, -4..2, scaling=constrained, discont);

If you don't like the notation Maple uses, you can enter the notation you want. For example, you could write e^x instead of exp(x) (but first you must to assign  e:=exp(1) ). Some conversions are also useful,  for example  convert(..., elementary)  or   convert(..., factorial)  or   convert(..., ln)  ans so on.

You wrote that you don't like the black border when we use the  legend  option. We can show the legends without using this option and without any borders. Here are two solutions to the problem. In the left picture below, the equations of the graphs are written next to the corresponding lines. In the right picture, the equations are written next to small segments of the corresponding color (as done when using the  legend option):

restart;
p:=eval([x,y],solve({y=1/x,y=1/x^2},{x,y})):
A:=plot([1/x,1/x^2], x=0..4, 0..4, color=[red,blue]):
T:=plots:-textplot([[1,1,typeset("(", p[1],",",p[2], ")")],[0.5,3.4,y=1/x^2],[2.8,0.3,y=1/x]],align={above,right}, font=[times,bold,14]):
P:=plots:-pointplot([1,1], symbol=solidcircle, symbolsize=15):
T1:=plots:-textplot([[1.2,1.1,typeset("(", p[1],",",p[2], ")")],[3.5,2,y=1/x^2],[3.5,2.6,y=1/x]], font=[times,bold,14]):
A1:=plot([[[2.5,2],[3,2]],[[2.5,2.6],[3,2.6]]], color=[blue,red]):
P1:=plots:-display(A,T,P):
P2:=plots:-display(A,T1,P,A1):
plots:-display(<P1 | P2>);

The  plot  command has no options for constructing asymptotes, but you can first automatically calculate the asymptotes using the  Student:-Calculus1:-Asymptotes  command, and then plot them using the formulas already found. In your example below, for plotting I have replaced the asymptote  x=0  with  x=0.05  so that it does not coincide with the vertical axis:
 

Download asymp.mw

The cause of the problem is easy to see if your code is written in 1Dmath:

Download Prj_1Dmath.mw

Should be Pi  not  pi:

evalc(exp(2*I*k*Pi)) assuming k::integer;
evalc(exp((2*k+1)*I*Pi)) assuming k::integer;
evalc(exp(I*Pi/2));
evalc(exp(3*I*Pi/2));
evalc(exp(I*Pi/3));

Another easy way is to replace the curve with a polygon with more sides (I used 500 ones):

Download PPP.mw

You have presented the original expression  (*)  as a picture. But if you want to write it in Maple language, then you can use the  conjugate command. With further conversions, the following problem arises related to the use of square brackets. [M,A]  in Maple means list of objects  M  and  A . If you just write  [M,A] - 3*[M,B] = 7*[M,C] , Maple will automatically do these linear operations on lists and you get  [-2*M, -3*B+A] = [7*M, 7*C] . Therefore, in the conversion below, instead of square brackets, we use a function  f  of two variables. You yourself can easily replace it with any function you need.

restart;
expr:=conjugate(MA)-3*conjugate(MB) = 7*conjugate(MC);
expr1:=subsindets(expr,function,t->op(1,t));
expr2:=subsindets(expr1,symbol,t->f(seq(substring(t,i),i=1..2)));

I don't know what the square brackets in your question mean.

You missed two multiplication signs in the line  eq14_2 .

The direct way to get these undefined constants is to use the  dsolve  command instead of the  int  one, since the operation of indefinite integration is inverse to differentiation.

Examples:

restart;
dsolve(diff(F(x),x)=x);
dsolve(diff(F(x),x)=x*sin(x));

Here you can draw an analogy with the exponentiation operation, which in the general case does not return all exponent values, but only the principal value, for example  (-8)^(1/3). To get all values we can use the  solve  command instead:

solve(z^3=-8);

Another possible approach is to use the simplest custom procedure, for example

restart;
my_int:=proc(expr, var)
int(expr, var)+C;
end proc:

# Examples
my_int(x, x);
my_int(t*sin(t), t);

Example:

A:=Array(1..2,1..2,1..2,(i,j,k)->i*k+j):
B:=Array(1..2,1..2,1..2,(i,j,k)->A[j,i,k]):

restart;  
plot3d([x*exp(-x^2-y^2),piecewise(x>=-1 and x<=1 and y>=-1 and y<=2,exp(-x^2-y^2),
undefined)],x=-3..3, y=-2..2, color=[red,blue], style=[surface,wireframe]);

The system  Sys  has infinitely many integer solutions, since instead of  x , you can take any integer:

Sys:={x*y*z + y*z + y = -1, x*y*z + x*z + z = 0, x*y*z + x*y + x = 0}:
solve(Sys);

                                              {x = x, y = -1, z = 0}