Tips for Complicated Indefinite Integrals

Question

I would like to have an analytical expression for some complicated indefinite integrals (example below). I am able to solve these integrals numerically, and I am able to plot them in Mathematica, however I don't know how to obtain an analytical expression for them.

1) Is it possible to evaluate the indefinite integral in the example below?

2) Is it possible to tell whether Mathematica is timing out, or if the integral simply cannot be done analytically? What does it mean when Mathematica spits back the unevaluated integral still with an integral sign?

3) More generally, are there any tips or tricks to successfully obtaining a result, or to speed up the calculation/stop it from timing out? I am thinking along the lines of: specifying assumptions, time constraints, rewriting special functions, Pade approximations, Taylor expansions, polynomial fits, etc.

Any help would be greatly appreciated :-)

Eii1[R_, ri_, 
ae_] = -(1/(32 ae^6 (1 + E^(ri/ae))^4 \[Pi] R ri^3))
 a0^2 (-E^((ri/ae))
    ri (-4 ae (-1 + E^(ri/ae) + 2 E^((2 ri)/ae)) + (1 - 
       8 E^(ri/ae) + 3 E^((2 ri)/ae)) ri) + 
 32 ae^3 DiracDelta[ri]) ((4 ae E^(R/ae) (R - ri))/(
 E^(R/ae) + E^(ri/ae)) - 4 R ri - 4 ae Sqrt[R^2 + ri^2] + (
 4 ae (1 + E^((R + ri)/ae)) R^2 - 
  4 ae E^((R + ri)/ae) R Sqrt[R^2 + ri^2] + 
  4 ae ri (ri + E^((R + ri)/ae) ri - 
     E^((R + ri)/ae) Sqrt[R^2 + ri^2]))/((1 + E^((R + ri)/
    ae)) Sqrt[R^2 + ri^2]) - 
 2 ae (R - ri) Log[1 + E^((R - ri)/ae)] + 
 2 ae (R + ri) Log[1 + E^((R + ri)/ae)] - 
 2 ae^2 PolyLog[2, -E^(((R - ri)/ae))] + 
 2 ae^2 PolyLog[2, -E^(((R + ri)/ae))]);

indefiniteEii1[R_, ri_, ae_] = 
Integrate[Eii1[R, ri, ae] 2 \[Pi] ri^2, ri];

definiteEii1[R_, ae_] = 
Integrate[Eii1[R, ri, ae] 2 \[Pi] ri^2, {ri, 0, \[Infinity]}];

Plot[(Eii1[1, ri, 1]/a0^2) 2 \[Pi] ri^2, {ri, 0, 50}, 
PlotRange -> Full]

Answer 1

What does it mean when Mathematica spits back the unevaluated integral still with an integral sign?

That is Mathematica's way of telling you that she cannot find an analytical expression for the indefinite integral. Notice also the error message Integrate::idiv which states that the definite integral you request does not exist.

Is it possible to evaluate the indefinite integral in the example below?

Sometimes, giving additional Assumptions helps...

And Albert Rich's Mathematica packages Rubi often finds indefinite integrals where Mathematica alobe has to resign... No, I've just checked it. It cannot. But Rubi may still serve as an answer for your follow-up question:

More generally, are there any tips or tricks to successfully obtaining a result, or to speed up the calculation/stop it from timing out?

Btw.: You know that there are indefinite integrals that cannot be written down in closed-form, don't you?