Speed up Nintegrate for certain range of function's input

Questions
,
1

This is a part of my code

SE[x_, g_, c_] := 1/x NIntegrate[Rationalize[(
6 (10^-10 Sqrt[xP])^2 10^-11 (10^-7 Sqrt[(
   10^-38 c^2 xP + (10^-10 Sqrt[xP])/x)/(
   0.1` + 10^-38  c^2)])^3)/((10^-38 c^2 xP + (10^-10 Sqrt[xP])/
  x)^4/(0.1` + 10^-38  c^2)^4)], {xP, 1.`/g, 10^12}, 
PrecisionGoal -> 10.`, MinRecursion -> 3.`, MaxRecursion -> 15.`, 
Method -> {Automatic, "SymbolicProcessing" -> 0.`}]

For which this part

Sqrt[NIntegrate[(SE[x, 10^-23, 10^14])^2, {x, 10^-5, 
 1}]] // AbsoluteTiming

Taking too much time for my range of values for c->(10^9-10^15) but when I decrease the value of c the integral becomes faster. I have to make the integral atleast 10-100 times more faster

2

Have you tried pre-calculating the first integral symbolically?

SE[x_, g_, c_] = 
 1/x Integrate[
   Simplify[(6 (10^-10 Sqrt[
           xP])^2 10^-11 (10^-7 Sqrt[(10^-38 c^2 xP + (10^-10 Sqrt[
                  xP])/x)/(1/10 + 
              10^-38 c^2)])^3)/((10^-38 c^2 xP + (10^-10 Sqrt[xP])/
           x)^4/(1/10 + 10^-38 c^2)^4)], {xP, 1/g, 10^12}, 
   Assumptions -> And[c > 0, x > 0, xP > 0, g > 0]]
3

Symbolical evolution seems more faster but it gives a solution for g>10^-12, but my g can be low as 10^-24

4

Also, one important reason I have not gone for symbolic integration is that my integrand in its complete form

SE[x_, g_, c_] = 
 1/x Integrate[
   Simplify[(6 (10^-10 Sqrt[
            xP])^2 10^-11 (10^-7 Sqrt[(10^-38 c^2 xP + (10^-10 Sqrt[
                   xP])/x)/(1/10 + 
               10^-38 c^2)])^3)/((10^-38 c^2 xP + (10^-10 Sqrt[xP])/
            x)^4/(1/10 + 10^-38 c^2)^4)*
     HeavisideTheta[10^-38 c^2 xP + (10^-10 Sqrt[xP])/x - 1/g]], {xP, 
    1/g, 10^12}, Assumptions -> And[c > 0, x > 0, xP > 0, g > 0]]

contains a HeavisideTheta function

5

When integrating, I would recommend UnitStep instead of HeavisideTheta. This gives me a symbolic output:

With[{g = 10^-23, c = 10^14}, 
 Integrate[
  Simplify[(6 (10^-10 Sqrt[
           xP])^2 10^-11 (10^-7 Sqrt[(10^-38 c^2 xP + (10^-10 Sqrt[
                  xP])/x)/(1/10 + 
              10^-38 c^2)])^3)/((10^-38 c^2 xP + (10^-10 Sqrt[xP])/
           x)^4/(1/10 + 10^-38 c^2)^4)*
    UnitStep[10^-38 c^2 xP + (10^-10 Sqrt[xP])/x - 1/g]],
  {xP, 1/g, 10^12},
  Assumptions -> And[c > 0, x > 0, xP > 0, g > 0]]]
6

But I can’t assume {g = 10^-23, c = 10^14} as a general solution since it will give the results for a point but in principle g,c have to vary for a range of values g=>(10^-2,10^-24) and c=>(10^10-10^15)

7

If Mathematica doesn’t do it automatically, you can easily work out the point where the HeavisideTheta discontinuity lies, and the conditions under which that point is to the left, within, or to the right of the integration interval [1/g, 10^12].