Check yours syntax,because is wrong!

Execute in command line for more information about itegration:  ?int

Probably you want as a example:

int(x*y, [x = 4 .. 16, y = 0 .. 2]);

#240

Wrong address.Your question is pure math. Ask that in math forums.

Laplace transform of yours examples not exist. This is only my opinion.

Not sure what is expected from that input.

A correct syntax is:

L1 := inttrans:-laplace(psi1(t)*(diff(z1(t), t)), t, s);
L2 := inttrans:-laplace((diff(psi1(t), t))^2, t, s);

#Return unevaluated

You made a  simple mistake in the equation (18).You  wrote xi(1),but  it should be: 1/xi(1).

See attached file:

feijisuan.mw

You forgot to deliver a inital conditions to DEplot function.

with(DEtools):

DE3 := {diff(y(x), x) = y(x)-z(x), diff(z(x), x) = z(x)-2*y(x)};

DEplot(DE3, [y(x), z(x)], x = 0 .. 3, y = 0 .. 2, z = -4 .. 4, [[y(0) = 1, z(0) = 1]], arrows = medium, numpoints = 200);

Download DEplot.mw

For more information execute: ?DEplot in  Maple command line.

On Maple 2018.1  definition doublefactorial is the same as 2016 nothing has changed.

From Maple help: ?factorial;

"Note: In Maple, !! is used for repeated factorials and so it does not indicate the double factorial."

then use: doublefactorial(n);

Maybe you can use alias function:

alias(`bi-factorial` = doublefactorial);

`bi-factorial`(15);

#2027025

doublefactorial(15);

#2027025

What version of Maple do you use?.

If you are using an older version, the latest version of Maple int correctly solves.

Maybe you can try and see what hapens ?

infolevel[IntegrationTools] := 3;
lambda[1] := 3/10; lambda[2] := 2*(1/10); alpha := 4:
int(2*alpha^2*Z*exp(lambda[1]*Z)/((exp(lambda[1]*Z)-1+alpha)^2*(exp(lambda[2]*Z)-1+alpha)), Z = 0 .. infinity, method = ftocms);

Download aquestion_ver_3.mw

Substitution x = t^2:

IntegrationTools:-Change(Int(1/(1+sqrt(x)), x = 0 .. 1), x = t^2);

value(%);

#2-2*ln(2)

Or using convert(expr,elementary):

Int(1/(1+sqrt(x)), x = 0 .. 1) = int(1/(1+sqrt(x)), x = 0 .. 1);

#Int(1/(1+sqrt(x)), x = 0 .. 1) = MeijerG([[0, 0, 1/2], []], [[1/2, 0], [-1]], 1)/Pi

Substitution: 1=x

convert(MeijerG([[0, 0, 1/2], []], [[1/2, 0], [-1]], x)/Pi, elementary);

#-ln(1-x)/x-2*arctanh(sqrt(x))/x+2/sqrt(x)

limit(%, x = 1);

#2-2*ln(2)

It's a bug in int !

Download function_evaluation_goes_wrong_v2.mw

Only for: z>0,j>0,Re(z),Re(j)

Download Ei.mw

interface(typesetting = extended);
F := n-> ifactors(ithprime(n)-2)[2, 1, 1] ;
F(11);
print(F);

#                               29
#n -> ifactors(ithprime(n) - 2)[2, 1, 1]

On Maple 2018.1

I a little speed up computation.The plot is now gererated by Maple about 1 minute.

If N=0 then integral is probably singular ,I changed q = 0.1e-2 to q = 0.1e-1.

Download PLOT_fun.mw

A simple example than yours and  we convert to piecewise to understand more:

convert(signum(0, x), piecewise);

#piecewise(x < 0, -1, x = 0, 0, 0 < x, 1)

convert(signum(0, x,1), piecewise);

#piecewise(x < 0, -1, 0 <= x, 1)

Using derivatives:

diff(signum(0, x),x);

convert(%, piecewise);

#signum(1, x)

#piecewise(x = 0, undefined, 0)

diff(signum(0, x, 1),x);

convert(%, piecewise);

#signum(1, x,1)

#piecewise(x = 0, undefined, 0)

_Envsignum0 := 0; convert(signum(0, x), piecewise);

#piecewise(x < 0, -1, x = 0, 0, 0 < x, 1)

_Envsignum0 := 1; convert(signum(0, x), piecewise);

#piecewise(x < 0, -1, 0 <= x, 1)

_Envsignum0 := 2; convert(signum(0, x), piecewise);

#piecewise(x < 0, -1, x = 0, 2, 0 < x, 1)

_Envsignum0 := -2; convert(signum(0, x), piecewise);

#piecewise(x < 0, -1, x = 0, -2, 0 < x, 1)

1.

convert(Ei(1, -ln(a-2)), Sum, dummy = n);

#-gamma-ln(-ln(a-2))+Sum((-1)^n*(-ln(a-2))^(n+1)/(factorial(n+1)*(n+1)), n = 0 .. infinity)

2.

convert(Ei(a, b), Sum, dummy = n);

eval(%, b = 1);

value(%);

limit(%, a = 1);

evalf(%);

#0.2193839344

Probably your equation have no analytical solution.

Download rootof_ver2.mw

Download no_change_ver2.mw