@mary120 To plot phi vs x just use

odeplot(ans1,[x(phi),phi],1e-2..0.5);

This goes through the origin because of the initial condition x(0)=0. But your plot doesn't go through the origin so I don't understand what you really want. Note the slope is wrong, so you probably need diff(x(phi),phi)=-integrand

Integral-New.mw

Here is something, but it is unclear how the x axis is constructed - what does x=0 mean on your plot?

something.mw

@Carl Love My original example was a graph with some randomly chosen edges. Later @Anthrazit wanted it to be a complete graph, and so I gave a version that does the complete graph for the same points, which does give the same path and distance as you find, see the worksheet TravelingSalesmanCompleteGraph.mw above.

I was also assuming that the start and end vertices were prescribed, and were to be given as the first and last in the list (since @Anthrazit mentioned something about nearest and outermost bolts), rather than just the shortest of all Hamiltonian paths. 

@Saha   You have gamma(Theta - Theta[a]) instead of gamma*(Theta - Theta[a])

@sand15 Yes! That syntax error can be avoided by diff(..., [eta$(m-2)]) , which returns the undifferentiated function. But probably not what the OP intended.

If I comment out with # the line setting values to various variables, then look at the output of the main loop I notice:

rps1.mw

1. Theta' is being interpreted as a derivative with respect to x. Probably you want to use declare(prime=eta)

2. You have multiplication before square brackets [] in two places, which Maple does not know how to interpret - if these are just for grouping you should use ().

3. The derivative with respect to eta m-2 times has not worked - the "d" should appear as upright, not italics. Use the calculus palette to form this, or use diff( ... , eta $ (m-2))

4. You have both Theta and Theta[a] and Theta with other subscripts. If it has subscripts you should not use Theta without subscripts as well.

5. If you are intending Theta without subscripts to be a function of eta you need to write Theta(eta)  or use declare so that Maple does not think it is a constant when it is differentiating.

6. use add instead of sum for just adding a finite number of terms.

Perhaps after fixing these, it will be clearer what you want to do, but at the moment I do not understand.

In Windows, if you click on the plot, the controls appear.

@Anthrazit Here's the CompleteGraph version (usually in traveling salesman problems there are not roads between every combination of cities). If you want shortest paths between two vertices that don't have to visit all vertices, DijkstrasAlgorithm or BellmanFordAlgorithm can be used - both can find shortest paths from vertex 1 to all other vertices, and then you can use an undirected graph - a path will not revisit an edge). If that is what you want I can set it up.

TravelingSalesmanCompleteGraph.mw

@C_R Yes, I forgot the abs; it works OK with it. I only used it without abs for a single "period" where I knew the sign, when I was playing around with it. I agree that the integral to infinity without abs is not relevant to @Mariusz Iwaniuk's integral, but I'm not sure whether or not the answer is wrong.

@Rouben Rostamian  PDEtools:-dchange gives the simpler result that you found, but comparing this with the IntegrationTools:-Change result shows that they are equivalent, so this is not a bug.

I1:=Int(sin(x^4)/(sqrt(x)+x^2),x=0..infinity);
PDEtools:-dchange(x=u^(1/4),I1);

dchange.mw

warnlevel = 0 works for me in Maple 2023.2.1

warnlevel.mw

@jalal To draw an arc representing the angle A in the triangle ABS, you can first find the cross product of the vector AB and AS to use as a rotation axis. Then find the rotation matrix tor rotation about this axis by arbitrary angle t. Apply this to a shortened version of vector AB, and its components (offset by A) are a parametric curve for the arc with parameter t, where t runs from zero to the angle A.

triangle(ABS, [A, B, S]);
angleSAB := FindAngle(ABS, A);
vAB := Vector(coordinates(B) - coordinates(A));
vAS := Vector(coordinates(S) - coordinates(A));
normalA := LinearAlgebra:-CrossProduct(vAB, vAS);
arcBAS := convert(0.2*((Student:-LinearAlgebra:-RotationMatrix(t, normalA)) . vAB), list);
plotarcBAS := spacecurve(coordinates(A) + arcBAS, t = 0 .. angleSAB, color = red, thickness = 2);
display(draw(ABS), plotarcBAS);

The code is at the end of the document.

rotation.mw

@MaPal93 Not sure why that wasn't working. I added simplify and it worked.

 MaPal93approx(3).mw

[But now I can delete that line and it still works.]

@MaPal93 You did pretty well with the quadratic.

It's not easy to explain exactly how I came up with it, because I just played around with different things. I tried some things where P(x) was linear playing with the interpolation point, and I tried some polynomials on a transformed x-axis -> w = 1 - exp(-x), now from 0 to 1, which preserves derivatives at the origin - it looks close to linear but it was hard to improve. Most of these were below f(x). Then I tried some simple rational functions that preserve the slope at the origin. At some point I had something higher than f(x) and then added back the exp(), then simplified some more.

Once I had the general form (f0-finf)*exp(-a*x)/(1+b*x), I solved for b that gave the correct slope at the origin. That left only a as a parameter. I just used Explore on the plot and adjusted the slider for a. That looked good, so I played with the slider on the relative error plot - there is a range of a that works well, and 0.126 seemed roughly the best. That was close enough to 1/8, for which b was then 0.837, and 5/6 = 0.833 was close enough and (perhaps) nicer.

Simple and resonably accurate

Download MaPal93approx.mw

@MaPal93 I remain confused about what you actually want:

1. a simple function (a few parameters) that gives good agreement for the function at 0 and infinity and for the derivative at 0, but is not necessarily that accurate.

2. an accurate approximation over a limited range.

3. an accurate approximation over the full range.

My objective was (1), because you asked for a 1-line function and you seemed to want to interpret it easily. That was also what I did in that paper. But now you are comparing as though you want (2), which you did ask for at some point and @acer gave a nice answer to. For (3) you can just use the function itself.

Since the comparison you provided is for (2) then a fair comparison requires the same number of parameters, which is 8 here in each case (though maybe 9 for the rational function; I'm not sure). (That's more that I was originally thinking of for (1).) Now plot them out to 200 and you will see that mine is OK at infinity (by design), but has the oscillation. The second one is a polynomial and will become large at infinity. The rational function can go to a constant value at infinity if the degree of the numerator and denominator polynomials are the same, and @acer chose [4,4], presumably for this reason. As @acer pointed out, the rational functions are usually better than the polynomials anyway, and you see that here.

The numapprox routines are focussed on accuracy over the whole range and they do well for that. They should also do well for derivatives at zero since they are based on some series expansions around that point. I'm assuming they are based on evaluating the function at evenly spaced points across the chosen interval or some criterion that treats all the interval evenly. If you get to choose the interpolation points then you can do better than if they are evenly spaced. In Gaussian quadrature, you optimize this, so (from memory) an (n/2)-degree polynomial with optimally chosen points can do as well as an n-degree polynomial at evenly spaced points. This is not quadrature, so I don't think the Laguerre points are optimal, but the basic idea is that they should spread out with more near the origin (where the function is changing) and fewer at large values (where it isn't changing much). But that again is assuming you want to approximate out to infinity. And then why not (3)?