restart;
expr:=a^2*(2*a^2*p^3-a^2*((p^2+1)^2*(a^2-1))^(1/2)-((p^2+1)^2*(a^2-1))^(1/2)*p^2+2*p*a^2-2*p^3+((p^2+1)^2*(a^2-1))^(1/2)-2*p)/((p^2+1)^2*(a^2-1))^(1/2)/(p^3-((p^2+1)^2*(a^2-1))^(1/2)+p)/(a^2-p^2-1);
int(expr,p)

Gives

why does Maple give division by zero?

Here is the result from integration package on Mathematica

ClearAll[a, p];
expr = a^2*(2*a^2*p^3 - 
       a^2*((p^2 + 1)^2*(a^2 - 1))^(1/2) - ((p^2 + 1)^2*(a^2 - 1))^(1/
           2)*p^2 + 2*p*a^2 - 2*p^3 + ((p^2 + 1)^2*(a^2 - 1))^(1/2) - 
       2*p)/((p^2 + 1)^2*(a^2 - 1))^(1/
        2)/(p^3 - ((p^2 + 1)^2*(a^2 - 1))^(1/2) + p)/(a^2 - p^2 - 1)
<< Rubi`
Int[expr, p]

Which it can integrate. Result is a little long. (removed since looks too long)

But my question really is not why Maple could not integrate it, but why the division by zero? 

Maple 2020.1

This is a programming question.

The goal is to solve an equation such as eq:=x^2+2*x-1=0; and obtain the solution as list with x= on each solution, like this

                       sol := [x = sqrt(2) - 1, x = -1 - sqrt(2)]

The command  to start with is sol:=solve(eq,x) which gives 

                       sol := sqrt(2) - 1, -1 - sqrt(2)

But to have x= show up, the command is modifed to be sol:=solve(eq,{x}) by adding {} around the variable to solve for, and now Maple gives 

                       sol := {x = sqrt(2) - 1}, {x = -1 - sqrt(2)}

Which is not yet the goal.. Changing the command to sol:=[solve(eq,{x})]  gives 

                       sol := [{x = sqrt(2) - 1}, {x = -1 - sqrt(2)}]

Which is still not the goal. Changing the command to sol:=solve(eq,[x])  gets closer to the goal.  it gives

                      sol := [[x = sqrt(2) - 1], [x = -1 - sqrt(2)]]

To remove the extra pair [ ] the list is Flattened like this

eq:=x^2+2*x-1=0;
sol:=solve(eq,[x]);
sol:=ListTools:-Flatten(sol)

Which gives me what I want, which is one list (not list of lists), and with x= in there

                             sol := [x = sqrt(2) - 1, x = -1 - sqrt(2)]

Is there a better way to obtain the above form of result than what I have above?

I need to check if expression is "special" kind of polynomial. Where powers are allowed to be non-integers and can be fractions. This is not polynomial in the mathematical sense ofcourse so can not use type(expr,polynom) on it, and did not see a way to tell type(expr,polynom) to accept non-integer exponents.

For an example, given p(x):=x^2+x^(1/3)+3+sqrt(x)+x^Pi+1/x*sin(x): it will return false, due to sin(x) term there. Without this term, it will return true, since all other terms have the form x^anything.

Currently, this is what I do

expr:=x^2+x^(1/3)+3+sqrt(x)+x^Pi+1/x*sin(x):

if type(expr,polynom(anything,x)) then
   print("Yes, normal polynomial");
else
   what_is_left:=remove(Z->type(Z,{`^`('identical'(x),algebraic),'identical'(x)}),expr);
   if has(what_is_left,x) then
      print("Not special polynomial");
   else
      print("Yes, special polynomial");
   fi;
fi;

While with

expr:=x^2+x^(1/3)+3+sqrt(x)+x^Pi+1/x:

It will print "Yes, special polynomial"

Is the above a good way to do this, or do you suggest a better way? It seems to work on the few tests  I did on it so far.

It is always "polynomial" in one symbol, such as x. So if it contains any function of x, other than  x^exponent, where exponent can only be numeric, or other symbol, it will fail the test. So this below will pass the a above test as well

expr:=x^2+x^(1/3)+3+sqrt(x)+x^a+1/x:

I was just praising Maple for not rewriting/simplifying  expressions automatically without the explicit user asking for it, when I found the following strange result

expr:=arccos(a-p) does not cause any change in the input. Good.

But when I change the letter ordering to expr:=arccos(p-a) now Maple changed it to Pi - arccos(a - p)

I have no idea why. Is there an option to tell Maple not to do that, even if it is mathematically correct? 

restart;
expr:=arccos(a-p);

restart;
expr:=arccos(p-a);

It seems to use lexicographical ordering to re-write things. Is it possible to turn that off?  Notice that I did not ask for simplification or anything. 

Maple 2020.1, Physics 724

I was checking my solution against Maple. Maple gives solution with integral and RootOf. I am not able to simplify it to better compare.

restart;
ode:=y(x)=ln(cos(diff(y(x),x)))+diff(y(x),x)*tan(diff(y(x),x));
sol:=dsolve(ode);

It is the second solution above I want to simplify/evaluate. So I tried

restart;
ode:=y(x)=ln(cos(diff(y(x),x)))+diff(y(x),x)*tan(diff(y(x),x));
sol:=dsolve(ode);
sol:=[sol][2];
DEtools:-remove_RootOf(sol)

But this did nothing. Then I tried adding useint in the dsolve command

restart;
ode:=y(x)=ln(cos(diff(y(x),x)))+diff(y(x),x)*tan(diff(y(x),x));
sol:=dsolve(ode,useint);

But this also kept the integral there. Then I tried eval

restart;
ode:=y(x)=ln(cos(diff(y(x),x)))+diff(y(x),x)*tan(diff(y(x),x));
sol:=[dsolve(ode,'explicit')];
eval(sol[2])

The solution I obtained is 

y(x) = -ln(_C1^2 - 2*_C1*x + x^2 + 1)/2 + ((-2*x + 2*_C1)*arctan(-x + _C1))/2

Which does verify to zero OK using odetest.

Any suggestions/tricks to use to get an more explicit solution from Maple I am overlooking?

Maple 2020.1 , Physics 724

I can post/attach my full solution if needed.