I once filed Luc Devroye, Non-Uniform Random Variates (1986), having "recipes". May be you can find it as well or find a backup for his site http://cgm.cs.mcgill.ca/~luc/rnbookindex.html through wayback machine

Likewise you may try x = 1+xi, y=x + eta, z=y+zeta (see attached file, it is Maple 2017, I do not have Maple 2021)

MP_232070.mw

I guess you mean "how to prove it", here a suggestion

Task:=Int(diff(u(x, y), x)*diff(u(x, y), x, y) +
  diff(u(x, y), x, x)*diff(u(x, y), y), x);
expand(%);
A,B:=op(1,%), op(2, %);

X:=IntegrationTools:-Parts(A, diff(u(x,y),x));
Y:=IntegrationTools:-Parts(B, diff(u(x,y),y));

Task=X+Y; combine(%);
lhs(%) - rhs(%);
combine(%);

From that it follows obviously.

For example you can save a sheet as html and pick images from there (it is easy to find the according subfolder).

Can one not handle this by providing an (initialized) array and handle it in the code? May be that is oldish.

I think it is 2.695678478*hypergeom([1., 1., 1.645396648],[2., 2.],-1.549434760*exp(-677.0138340*t))*exp(-677.0138340*t)+.95671881+1887.991395*t

find MP_231348.mws attached

Added 03. Jan 2021:

Changing coordinates by 1+57154/36887*exp(-342569/506*t)=s instead of 1+57154/36887*exp(-342569/506*t)=1+s gives the solution involving LerchPhi (instead of the more complicated 3F2 hypergeom) or likewise a 2F1, which is valid as well:

SolNum := -.4073509876e-1+62.97977396*t+4.176777934/(exp(677.0138340*t)+1.549434760)^.6453966478*exp(436.9424590*t)*(1.-1.*Re(hypergeom([-.6453966478, 1.],[.3546033522],1.+1.549434760*exp(-677.0138340*t))))

NB: Using Re = real part avoids spurious numerical imaginaries for LerchPhi.

Correction 03. Jan 2021

I made an error through copy + paste and took 62.97... instead of 862.97... for the constant.

So 800*t has to be added to both the posted anti-derivatives

@mmcdara 

If you say "expand" then it is written as polynomial and it should work

Plot it, "see" there is no solution and ask yourself whether you ask the correct question

That does not converge I think:

If you look at your E2 (in your sheet, but without feeding z) it is ~ e0 + rational in exp(z), exp(-z), then squared. For z ---> +- infinity that fraction tends to some constant*exp(-+z), but e0 survives. Thus the integral over the real line can not be finite. Now your z is just a linear function of your actual integration variable t. But a linear transform only changes the magnitude.

Likewise you may use MultiSeries:-asympt for your integrand to find similar reasoning.

Just use convert(left, Int), dito for the other term

Not sure if that helps for your situation - but I (always) use a pdf-printer, it works for any application (except for some odd web pages). For the example given by tomleslie for Maple 2020 the result is also complete and of size ~ 350 KB only for somewhat dated "pdfcreator"

Your AAA is linear in y, hence the integral is also linear in y.

The following works in Maple 18 (and should work in Maple 2015 as well)

Download MP_230226.mw

If you integrate rationalize(expr) it works fine, seems not restricted to Reals

I guess that this may be caused by a = +-1 and Maple does not handle that *removable* singularity (limit is quite simple).

You may try

MultiSeries:-series(1/z^4/(1-z), z=0, 2);

                           -4    -3      -2
                          z   + z   + O(z  )

For example:

L:=[1,2,3,4,5]:
convert(L, `+`);
                                  15

But you should try to understand / learn Maple's syntax and notations, studying your lectures (or the book).