# The question is: how to get S knowing E?  All the attemps below after S assigned (suggested by Claude) fail factoring E.  

restart;

interface(showassumed = 0): 
assume(a>=0, b>=0, c>=0);
assume(x::real, y::real,z::real);

S := sqrt(x^2+(a+sqrt(y^2+(b+sqrt(z^2+c^2))^2))^2):
E := expand(S):
simplify(E,sqrt):
combine(E,sqrt):
simplify(combine(E, sqrt), sqrt):
evalindets(E, 'radical', f -> simplify(f, sqrt)):
simplify(radnormal(E), sqrt):

The Abel integral equation of the form

g(x) =  Int( f(y) / (x-y)^alpha, y = 0 .. x ) assuming alpha>0, alpha<1;

is linear and has a well known explicit solution for f(x).  

It is a particular form of Volterra equation of the first kind. 

It occurs in many problems of physics with alpha = 1/2.

However intsolve gives an error message:

> eq := g(x) =  Int(f(y)/(x-y)^alpha, y = 0 .. x) assuming alpha>0, alpha<1

> intsolve(eq, f(x));
Error, (in intsolve) numeric exception: division by zero

Replacing (x-y)^alpha by sqrt(x-y) gives the same error message.

Suppose a sum of sines, each depending on a distinct variable x[i]:

>  S := sum(sin(x[i]),i=1..N);

Differentiating wrt to x[i]

> diff(S,x[i]);

should yield cos([x[i]) if i belongs to the interval 1..N, and zero otherwise, but Maple gives

> diff(S,x[i]);
                          N            
                        -----          
                         \             
                          )            
                         /    cos(x[i])
                        -----          
                        i = 1          

which is wrong of course.
 

In a sum I would expect 0 when the minimum index is larger than the maximum index.

Yet Maple 2017.3 does this only for special cases:

> for i from 0 to 4 do
     sum(1,j=i..1);
  end do;
                               2
                               1
                               0
                               -1
                               -2

Hi experts!

Suppose we want to plot a function depending on integral evaluation, like

> S := Int(sin(sin(x+u)+sin(y-u)),u=0..Pi);
               /Pi                                 
             ...