The problem is delicate. In Maple there is no "complex diff". The function abs is nowhere complex-differentiable, not even at real points. The abs(1,z) is a compromise between real and complex differentiation. When using diff with abs in a nonreal context, there will be always problems.

@mzivari It's just a simple change of variables, taking into account the constraints.

@mzivari  Yes, it is nonnegative; not easy to prove though (some work is needed for this).
f >= 0 (in the domain) iff  fuv >=0  for u,v >=0,  where:

f := (x, y, z) -> 1 - x^4 + ln(x^4) + z^4 - y^4 - y^4*(ln(z^4) - ln(y^4)):
fuv:=f(sqrt(1+u), sqrt(1+u+v), sqrt(1+2*u+v)); # >=0 for u,v>=0

plot3d(fuv, u=0..10, v=0..10); # graphically should be clear

@chrisc The integral could exist in some distributional sense. If you know how this distribution is defined, maybe Maple could be used to retrive the desired result.

This seems to be a "try to guess what I mean" challenge.

@Carl Love  Warning: modp(a,b) = irem(a,b)  only for a>=0. Also, `mod` could be mods.
 

@mclaine The assumptions are important but unfortunately Maple is far from perfect here.
Note first that
simplify(f__1 - f __2, symbolic);
returns 0, but then you will have to decide in what conditions the equality is actually true.

The equality holds when lambda__g * a > 0, even when lambda__g and a  are complex, but Maple is not able to simplify to 0, not even:

combine(simplify(f__1 - f__2)) assuming lambda__g * a>0;
 

@one man  The method skips zeros too. Using the zer function I have provided, 
zer(3614)=106.546511887030462;
zer(3682)=107.544351296277895;
so, 68 zeros are missing!

I have a procedure that can find all of them, the first stage being to isolate the zeros. After that, fsolve can be invoked.
 

@mmcdara It is an interesting workaround. Unfortunately it has limited efficiency. Yesterday when I tried to find all the zeros using NextZero with adjusted options, I managed to find the first 117 (I think, probably not a coincidence)  zeros, but not beyond (up to 154 as in OP's question). I needed higher precision for this. Your approach has the same problem (just try 120 instead of 100). Also, for Digits:=15, it does not run.
My conclusion is that NextZero must be revised to cover such examples.

@mmcdara I the last example, setting Digits:=15 is enough to work.
For OP's example I could not find any setting!

@mmcdara Note first that the function takes obviously negative values but on short intervals.
The f__eps does not help. Actually for Digits=10 many roots are skipped!
Please compare with the exact roots:

restart: 
Digits := 10:
zer:= k -> evalf(sqrt((-1)^(k+1)*arcsin(10000/10001)+k*Pi)):
f      := proc (x) options operator, arrow; x*(1+1.0001*sin(x^2)) end proc:
eps    := 1:
f__eps := proc (x) options operator, arrow; x*(1+1.0001*sin(x^2))-eps end proc:

sol := NULL:
s := 1.0:
s := RootFinding:-NextZero(f, s): 
sol := sol, s:
printf("%3d  %2.8f\n", 1, s):
for j from 2 to 100 do 
  s__eps := RootFinding:-NextZero(f__eps, s): 
  s := RootFinding:-NextZero(f, s__eps): 
  sol := sol, s:
  printf("%3d  %2.8f  exact=%2.8f\n", j, s, zer(j)):
end do:

  2  3.31382449  exact=2.17405854
  3  4.15507136  exact=3.31382449
  4  4.85260789  exact=3.31808918
  5  5.46177523  exact=4.15507136
  6  6.00950696  exact=4.15847342
  7  6.51132546  exact=4.85260789
  8  6.97714444  exact=4.85552123
  9  7.41375275  exact=5.46177523
 10  7.82604084  exact=5.46436380
 11  8.21767002  exact=6.00950697
 12  8.59146587  exact=6.01185970
 13  8.94966318  exact=6.51132546
 14  9.29406566  exact=6.51349694
 15  9.62615405  exact=6.97714444
 16  9.94716176  exact=6.97917099
 17  10.25812908  exact=7.41375275
 18  10.55994307  exact=7.41565998
 19  10.85336735  exact=7.82604084
 20  11.13906496  exact=7.82784762
 21  11.41761593  exact=8.21767002
 22  11.68953116  exact=8.21939071
 23  11.95526345  exact=8.59146587
 24  12.21521630  exact=8.59311171
 25  12.46975119  exact=8.94966318
 26  12.71919337  exact=8.95124317
 27  12.96383683  exact=9.29406566
 28  13.20394830  exact=9.29558710
 29  13.43977068  exact=9.62615405
 30  13.67152593  exact=9.62762301
 31  13.89941749  exact=9.94716176
 32  14.12363239  exact=9.94858332
 33  14.34434304  exact=10.25812909
 34  14.56170877  exact=10.25950756
 35  14.77587722  exact=10.55994307
 36  14.98698546  exact=10.56128215
 37  15.19516102  exact=10.85336736
 38  15.40052284  exact=10.85467024
 39  15.60318201  exact=11.13906497
 40  15.80324252  exact=11.14033444
 41  16.00080184  exact=11.41761593
 42  16.19595150  exact=11.41885444
 43  16.38877757  exact=11.68953117
 44  16.57936113  exact=11.69074087
 45  16.76777865  exact=11.95526345
 46  16.95410234  exact=11.95644627
 47  17.13840049  exact=12.21521631
 48  17.32073776  exact=12.21637395
 49  17.50117545  exact=12.46975119
 50  17.67977170  exact=12.47088521
 51  17.85658177  exact=12.71919338
 52  18.03165821  exact=12.72030515
 53  18.20505104  exact=12.96383684
 54  18.37680790  exact=12.96492763
 55  18.54697425  exact=13.20394830
 56  18.71559348  exact=13.20501927
 57  18.88270702  exact=13.43977069
 58  19.04835452  exact=13.44082286
 59  19.21257388  exact=13.67152594
 60  19.37540142  exact=13.67256027
 61  19.53687195  exact=13.89941749
 62  19.69701883  exact=13.90043487
 63  19.85587410  exact=14.12363239
 64  20.01346850  exact=14.12463363
 65  20.16983160  exact=14.34434304
 66  20.32499181  exact=14.34532887
 67  20.47897648  exact=14.56170878
 68  20.63181191  exact=14.56267989
 69  20.78352348  exact=14.77587723
 70  20.93413560  exact=14.77683427
 71  21.08367185  exact=14.98698546
 72  21.23215495  exact=14.98792902
 73  21.37960686  exact=15.19516102
 74  21.52604875  exact=15.19609166
 75  21.67150110  exact=15.40052284
 76  21.81598371  exact=15.40144107
 77  21.95951572  exact=15.60318202
 78  22.10211564  exact=15.60408832
 79  22.24380141  exact=15.80324253
 80  22.38459038  exact=15.80413735
 81  22.52449937  exact=16.00080185
 82  22.66354467  exact=16.00168562
 83  22.80174209  exact=16.19595150
 84  22.93910695  exact=16.19682463
 85  23.07565412  exact=16.38877757
 86  23.21139803  exact=16.38964043
 87  23.34635268  exact=16.57936114
 88  23.48053170  exact=16.58021408
 89  23.61394830  exact=16.76777865
 90  23.74661533  exact=16.76862201
 91  23.87854528  exact=16.95410234
 92  24.00975031  exact=16.95493643
 93  24.14024224  exact=17.13840050
 94  24.27003257  exact=17.13922562
 95  24.39913249  exact=17.32073777
 96  24.52755290  exact=17.32155420
 97  24.65530443  exact=17.50117545
 98  24.78239742  exact=17.50198347
 99  24.90884195  exact=17.67977170
100  25.03464784  exact=17.68057156
 

@Carl Love I am not so sure that an implementation with objects would make a big difference without a massive redesign.
If X1,X2 are RVs, a PDF of f(X1,X2) is always problematic, the resulting integral could be very hard do compute, even numerically. After all, what we have here is just a bug, probably very easy to fix, but hard to find without the source.

Nice example, vote up!
NextZero has the parameters guardDigits, maxdistance, initialDigits
but is is very hard to guess a successfull combination. In other cases it works.
It is sad that such simple examples exist!
 

So, for the first F, you have the equations

F(a, b) + a = F(a, b + 1),  F(a, b) = F(a*(a + 1)/2, 2),  a,b in N.

The general solution for the first equation is:

F (a, b)  =  f(a) + a*b

where f is a function of a single variable. Plugging in the second one ==>

f(a*(a + 1)/2) + a*(a + 1) - f(a) - a*b = 0

But this is obviously impossible for arbitrary a,b in N, so, F does not exist.

The same for the second F.

@Mikhail Drugov The package is not bad. This function seems to be not very often used with custom RVs.

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