This integral 

for n integer and n>=0. 

Maple finds it if I tell it this in the assumptions.  But if I remove the assumption that n>=0 and just keep the assumption that n is integer, it hangs. (at least I waited 5 minutes and gave up).

Is this something to be expected?  If I try the same thing in Mathematica, i.e. telling it n is an integer, but not that it is n>=0, it returns result  immediately, with condition that the result is valid for n>=0.

restart;
int(x^n*exp(-x),x=0..infinity) assuming n::integer

seems to hang.

In Mathematica:

My question is  if this behaviour of int hanging is to be expected, since it was not told than n>=0? Should it have returned result of n!  like Mathematica, with the assumption given in the result as well?

Since for n<0 the integral does not converge, and maple knows this

restart;
int(x^(-2)*exp(-x),x=0..infinity)

            infinity

May be this is just a design issue in int and it was stuck trying to evaluate the integral for negative integers, and could not determine if it converges or not?

Any thoughts?

Maple 2020.2

Is there a workaround for this?

restart;
int(sqrt(x)*sin(sqrt(3)*ln(x)/2),x)

The answer according to Mathematica is

Maple 2020.2 on windows 10

Why this fails in solve in Maple 2020.2?

restart;

A:=-ln(u)/2 + ln(3*u - 2)/6;
B:=_C1 + ln(x);
sol := solve(A-B= 0,u) assuming real

No error if I try the above code in Maple 2019.2.

Also, the error goes away if I replace assuming real  with assuming x::real

restart;

A:=-ln(u)/2 + ln(3*u - 2)/6;
B:=_C1 + ln(x);
sol := solve(A-B= 0,u) assuming x::real

Is this a bug in solve?

Maple 2020.2 on windows 10.

Why this error shows up when adding assuming?

restart;
expr:= ln(c^2*y/sqrt(c^2)+sqrt(c^2*y^2+1));
simplify(expr,size=true,evaluate_known_functions=false);
simplify(expr,size=true,evaluate_known_functions=false) assuming real;  #error

Is this to be expected?

Maple 2020.2

I think Maple's simplify could need much more improvement. 

Here is another example, which I can't get Maple to simplify to zero for positive x, when it is clearly zero there

restart;
ode:=x = (diff(y(x),x)^2+1)^(1/2)*diff(y(x),x);
mysol:=y(x)=_C1+int(  sqrt(-2+2*sqrt(4*a^2+1))/2,a=0..x);
check:=odetest(mysol,ode);

The above is zero for x>0

But I tried every assumption or option on it, and it will not give zero.

simplify(check) assuming x>0;
simplify(check,symbolic);
simplify(check,symbolic,sqrt) assuming x>0;
simplify(check,sqrt) assuming x>0;
simplify(check,radical) assuming x>0;
simplify(check,power) assuming x>0;
etc...

Here it is in Mathematica

ClearAll[x];
check = x - (Sqrt[2 + 2 Sqrt[4 x^2 + 1]]) (Sqrt[-2 + 2 Sqrt[4 x^2 + 1]])/4;
Simplify[check, Assumptions -> x > 0]

Why Maple can't simplify this to zero?  Is there some other specific trick one must use each time?

Maple 2020.2 on windows 10