Using comparision test from this video, we have:

0 <= (x + 2)/((2*x^3 + 1)*ln(x)) and (x + 2)/((2*x^3 + 1)*ln(x)) <= 1/ln(x)

int(1/ln(x), x = 1 .. infinity);

#infinity

It can be done only numerically.

a := 0;
b := 1;
int(exp(-1/sqrt(x^2 + 1)), x = a .. b, numeric);

#0.4160094117

If I execute your code Maple give me: "Error, mismatched or missing bracket/operator" .Please check your input for bad syntax?

@Viva 

I don't have any  suggestions ,because Maple can't solve solvelable integrals.

Maple is weak for solving  indefinite and define integrals.

2_ver2.mw

Mathematica 13.0 also can't integrate.

Better post this question to this webpage.

int(A,theta=0..eta)

@C_R 

The problem in Maple is: '"automatic simplification' and 'evaluation'".

See here , here and here.

Cross post here. and here

1.What you need PDEtools for that?

2.You want to solve a Nonlinear Volterra Integro Differential Equation ?, if yes post the equation.

I'm voting to close this question as off-topic because there is no well-posed question in this post; the OP is simply asking for somebody to act as a free coding service.

I'm voting to close this question as off-topic because there is no well-posed question in this post; the OP is simply asking for somebody to act as a free coding service.

This line of code dosen't work:

sum(sum((k[1]+3)!*((-1)^(k-k[1]-1)/2)*Y[k[1]+3]/(k[1]!*(k-k[1])!,k[1]=0..k)

?

@Axel Vogt 

I don't have reference.

I'm played with this sum a few minutes and I found  generalization.

restart;

n := 2;

Sum(Beta(k, c)/(a*k + b)^n, k = 1 .. infinity) = GAMMA(c)*hypergeom([1, 1, seq(1 + b/a, j = 1 .. n)], [seq(2 + b/a, j = 1 .. n), 1 + c], 1)/(GAMMA(1 + c)*(a + b)^n);

@escorpsy 

(sum((-1)^k*z^(1 + 2*k)*GAMMA(1/2 + k)/((1 + 2*k)*sqrt(Pi)*GAMMA(1 + k)), k = 0 .. infinity, formal) assuming (0 < z));

#arcsinh(z)

(sum((-1)^k*sqrt(Pi)*z^(2 + 2*k)*GAMMA(1 + k)/(2*(1 + k)*GAMMA(3/2 + k)), k = 0 .. infinity, formal) assuming (0 < z));

#arcsinh(z)^2

And what is the question ?