Probably pdsolve is designed to find only one solution not more.We can find more solution using packages PDEtools:

Download By_PDE-Tools.mw

int(convert(cos(2*x)/(1+2*sin(3*x)^2), exp), x = 0 .. Pi);

# 0

Using Rubi on Mathematica I have got a answer for case 3D.

See attached file for details:

Integral_3D.mw

 To reproduce the computation below in Maple 2018.1 you need to install theMaplesoft Physics Updates (version 134 or higher)

Download PDE.mw

UPDATED:

Physics package was fixed and now with version 195 works fine.

UPDATED 15.11.2018:

Solution with Physics:-Version 200:

PDE_with_Physics-version_200.mw

Thumb if You like.

Yes it's a Bug in KroneckerDelta function for m=4 it should be 0 not 1.

with(Physics, KroneckerDelta);

[seq(KroneckerDelta[m, 0], m = 0 .. 10)];

#[1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0]

but It should be: [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]

See workaround in attached file:

Download bug.mw

Thumb if You like.

infolevel[IntegrationTools] := 3;

`assuming`([int(GAMMA(a, s)*exp(-b*s), s = 0 .. infinity)], [a > 0, b > 0]);

#Definite Integration:   Integrating expression on s=0..infinity
Definite Integration:   Using the integrators [distribution, piecewise, series, o, polynomial, ln, lookup, cook, ratpoly, elliptic, elliptictrig, meijergspecial, improper, asymptotic, ftoc, ftocms, meijerg, contour]
IntegralTransform LookUp Integrator:   Integral might be a laplace transform with expr=GAMMA(a,s) and s=b.
LookUp Integrator:   unable to find the specified integral in the table
Definite Integration:   Returning integral unevaluated.
 

Then we see Maple can't find in Table a Laplace Transform(?) ,but:

inttrans:-laplace(GAMMA(a, s), s, b);#Works fine

#GAMMA(a)*(1-(1+b)^(-a))/b

Thumb if You like.

If is indefine integral then is not possible.

More info in attached file.

integral.mw

INT := int(exp(-a^2-b^2-c^2), a = b^2/(4*c) .. infinity);

int(INT, b = -infinity .. infinity, c = 0 .. infinity, numeric);

#0.9786008283

Add option to roots: sqrt(2).

roots(x^3-x^2-8*x+8, sqrt(2));

#[[-2*sqrt(2), 1], [1, 1], [2*sqrt(2), 1]]

Thumb if You like.

Use:

Student[Calculus1][DiffTutor]();

Enter a function: a(x)^b(x) a click AllSteps and Close then you see all steps.

Download test_ver2.mw

At first read  this paper and everything will be clearer.

We can use (A.7 and A.9) and then:

Dirac(t) = Limit(exp(-t^2/epsilon^2)/(epsilon*sqrt(Pi)), epsilon = 0, right);

f := `assuming`([int((diff(exp(-t^2/epsilon^2)/(epsilon*sqrt(Pi)), t))*cos(t)*exp(-s*t), t = 0 .. infinity)], [s > 0, epsilon >= 0]);

limit(f, epsilon = 0, right);

# - infinity

Conclusion: Laplace((diff(Dirac(t), t))*cos(t), t, s) not exist in “generalized function” sense.

See attached file:

Laplace_Transfrom-Dirac-Heaviside-Derivative.mw

We can speed-up computation using dsolve, solving differential equation from definition: HeunC function.

Download test.mw

It's not a bug it's so designed.

More info in Help,see:

?sum,details

Use add than sum:

for i from 0 to 4 do add(1, j = i .. 1) end do;

Download way.mw