@lcz 

Yes, you are correct I overlooked the indefinite v. definite distinction.  Still the integral I was trying to evaluate is quite simple, and the substitution so obvious, that it is beyond my comprehension given what I have seen Maple do with much more complicated definite integrals and sums, why it is not able to immediately do this integral.

@Thomas Richard 

I thought the purpose of the Risch algorithm (https://en.wikipedia.org/wiki/Risch_algorithm) was to be able to do all integrals that can in fact be done?

@Thomas Richard 

But why doesn't Maple see this substitution?  It is fairly simple.

@acer 

Thank you

Mark

@vv What changed between Maple 2019 and Maple 2020 to where the integral is evaluated directly?

@Carl Love Very well done, thank you.

@Mariusz Iwaniuk The lowest eigenvalues are as in my original post:0.184358, 0.70747, 1.2065, 1.5625.  You skipped a few eigenvalues.

@Carl Love What exactly does this statement mean in the F procedure?

Sol(parameters= [:-e= e]);

Is that an interval range? I get the same results you get using just Sol(parameters=[e].

So this is basically a shooting procedure.  But it is interesting that in my worksheet, the dsolve without the parameters automatically does the same thing except that it seemingly can only find the lowest eigenvalue.  

@Mariusz Iwaniuk I have solved each of my problems and there are an infinite numer of eigenvalues.  There are the boundary conditions, plus there is the adjustable parameter e which is the scaled energy eigenvalue for the problem.  The worksheet shows a solution for each problem, but there are higher eigenvalues, my problem is trying to get them automatically.

@Carl Love 

Thanks, I thought of that but couldn't remember the print command.  

@acer 

Most of the suggestions don't work.  But when I highlight the gamma symbol or pi the help menu opens to those pages, when I highlight Psi (digamma function) or Phi whether a greek symbol or LerchPhi, nothing happens.

I suspect many of the key combinations I need for Maple have been allocated to other programs ot the operating system.  Anyway thanks for the help. 

@acer 

No, I am using a Mac Pro OS X El Capitan, Maple 2015.  I tried just about every F key and also simultaneously holding down command, control buttons, and option buttons.  But so far no luck.

Mark

@acer 

I just set my interface to typesetting=standard on my preferences.

@Preben Alsholm 

I confirm

@vv 

I thought the problem was that Maple was treating some things as zero that really weren't - thus leading to the divide by zero exception.