In[]:=
$Version
Out[]=
14.0.0 for Microsoft Windows (64-bit) (December 13, 2023)
In[]:=
ClearAll["`*"];Remove["`*"];
Remove
:There are no symbols matching "`*.
In[]:=
func=-

-
2
(-m+r)
2
2
s

π
2
2
r
1+
2
1/4
(-1)
v
A
BesselJ1,
3/4
(-1)
r
v
A

rBesselJ0,
3/4
(-1)
r
v
A

Asρ
//FullSimplify
Out[]=
-
2
(m-r)
2
2
s

π
2
4
r
Hypergeometric0F1Regularized3,
A
2
r
4v

4svρHypergeometric0F1Regularized1,
A
2
r
4v

In[]:=
IntegrateChangeVariables[Inactive[Integrate][func,{r,0,Infinity}],t,t10^5*r]//ExpandAll(*Rescaling*)
Out[]=
∞
∫
0
-
2
m
2
2
s
+
mt
100000
2
s
-
2
t
20000000000
2
s

π
2
4
t
Hypergeometric0F1Regularized3,
A
2
t
40000000000v

40000000000000000000000000svρHypergeometric0F1Regularized1,
A
2
t
40000000000v

t
In[]:=
ρ=1000;​​v=10^(-6);​​m=10^(-4);​​s=2*10^(-5);​​newfunc=
-
2
m
2
2
s
+
mt
100000
2
s
-
2
t
20000000000
2
s

π
2
4
t
Hypergeometric0F1Regularized3,
A
2
t
40000000000v

40000000000000000000000000svρHypergeometric0F1Regularized1,
A
2
t
40000000000v

;
In[]:=
LogLogPlot[Abs[NIntegrate[newfunc,{t,0,Infinity}]],{A,10^-10,10^10},PlotRangeAll]
Out[]=