Can't integrate a highly oscillatory integrand

Question

I have a long formula

-0.0579484 kx ((Sin[-3.2 kx] Cos[Sqrt[0.0232723 + kx^2 + ky^2 + kz^2] t])/Sqrt[0.0232723 + kx^2 + ky^2 + kz^2] (Sqrt[0.0232723 + kx^2 + ky^2 + kz^2]/(0.0232723 + kx^2 + ky^2 + 0.00008649 kz^2) + 1/Sqrt[0.0232723 + kx^2 + ky^2 + kz^2] (kx^2 + ky^2 + kz^2)/(-kx^2 - ky^2 - 0.00008649 kz^2)) + (Cos[-3.2 kx] Sin[Sqrt[0.0232723 + kx^2 + ky^2 + kz^2] t])/Sqrt[0.0232723 + kx^2 + ky^2 + kz^2] ((-0.999957 kz)/(0.0232723 + kx^2 + ky^2 + 0.00008649 kz^2) + (-0.999957 kz)/(-kx^2 - ky^2 - 0.00008649 kz^2)) + (Sin[-3.2 kx] Cos[-0.999957 kz t] + Cos[-3.2 kx] Sin[-0.999957  kz t])/(-0.0232723 - kx^2 - ky^2 - 0.00008649 kz^2) + Sin[-3.2 kx] 0.0232723/(0.0232723 + kx^2 + ky^2 + kz^2)1/(-kx^2 - ky^2 - 0.00008649 kz^2))

which I want to integrate over $\int kx \, ky \, kz$ and plot over $t$.

I can integrate the elements of the sum separately and add them at the end. The elements are

I:

-0.0579484 kx ((Sin[3.2 kx] Cos[Sqrt[0.0232723 + kx^2 + ky^2 + kz^2] t])/Sqrt[0.0232723 + kx^2 + ky^2 + kz^2] Sqrt[0.0232723 + kx^2 + ky^2 + kz^2]/(0.0232723 + kx^2 + ky^2 + 0.00008649 kz^2))

II:

-0.0579484 kx ((Sin[3.2 kx] Cos[Sqrt[0.0232723 + kx^2 + ky^2 + kz^2] t])/Sqrt[0.0232723 + kx^2 + ky^2 + kz^2] 1/Sqrt[0.0232723 + kx^2 + ky^2 + kz^2] (kx^2 + ky^2 + kz^2)/(-kx^2 - ky^2 - 0.00008649 kz^2))

III:

-0.0579484 kx ((Cos[3.2 kx] Sin[Sqrt[0.0232723 + kx^2 + ky^2 + kz^2] t])/Sqrt[0.0232723 + kx^2 + ky^2 + kz^2] (0.999957 kz)/(0.0232723 + kx^2 + ky^2 + 0.00008649 kz^2))

IV:

-0.0579484 kx ((Cos[3.2 kx] Sin[Sqrt[0.0232723 + kx^2 + ky^2 + kz^2] t])/Sqrt[0.0232723 + kx^2 + ky^2 + kz^2] (0.999957 kz)/(-kx^2 - ky^2 - 0.00008649 kz^2))

V:

-0.0579484 kx ((Sin[3.2 kx] Cos[0.999957 kz t] + Cos[3.2 kx] Sin[0.999957  kz t])/(0.0232723 - kx^2 - ky^2 - 0.00008649 kz^2))

VI:

-0.0579484 kx (Sin[3.2 kx] 0.0232723/(0.0232723 + kx^2 + ky^2 + kz^2)1/(-kx^2 - ky^2 - 0.00008649 kz^2))

The V and VI is the only part which I know how to compute with correct results. The method is as follows: First, I Integrate the formula V analytically over $ky$ and $kz$

f = Assuming[{kz \[Element] Reals && kz != 0 && kx \[Element] Reals && ky \[Element] Reals && t \[Element] Reals}, Integrate[-0.0579484 kx ((Sin[3.2 kx] Cos[0.999957 kz t] + Cos[3.2 kx] Sin[0.999957  kz t])/(0.0232723 - kx^2 - ky^2 - 0.00008649 kz^2)), {kz, -\[Infinity], \[Infinity]}, {ky, -\[Infinity], \[Infinity]}]]

Then, I numerically integrate the result over $kx$

Nf[t_] := NIntegrate[fV2, {kx, -\[Infinity], \[Infinity]}, Method -> "LevinRule"];

(Integrating first over $kx$ and $ky$ and then numerically over $kz$ works too)

Finally I plot the result as

Plot[Nf[t], {t, -2, 2}, PlotRange -> {{-0.2, 0.2}, All}]

The above method was provided here https://mathematica.stackexchange.com/a/276353/88922 and works only for the V and VI case.

For I-IV I tried integrating over $kx$, $ky$, $kz$ separately and for I-II Mathematica doesn't provide any result, for III-IV integrating over $kx$ or $kz$ gives 0, over $ky$ - nothing.

Method of V works on VI as wellAny ideas how to approach the topic? How to integrate the I-IV formulas over $kx$, $ky$, $kz$?

P.S. I couldn't integrate analytically over all of the three k variables at once - no results, and integrating fully numerically is too slow.

Answer 1

Integrate manages one instance eg $$\int dx (\sum \frac{\sin \dots}{\sqrt{\dots}}\dots)$$ but the result is an $\text{Ei}$-function of the other variables. So there is no chance to perform further interation steps $$\int( \dots )\ dy \ dz$$ .

NIntegrate needs a threefold integral for any time t, because the algorithm does not accept variables.

Its impossible to put it inside a Plot statement for unspecified t-points, because on has no control of execution times and memory used. A ListPlot for an NIntegrate list over a list of t-values may work.

So, a way is to NIntegrate for a list of t-values over a 3d lattice with relatively few points per sin-period and interpolate the 4d lattice separately to get an interpolating function. For a visual Plot management, relatively small numbers of lattice point numbers are satisfying the objectives.

Answer 2

It diverges if I am not mistaken. Here are my arguments. Let us expand the integrand

Expand[-0.0579484 kx ((Sin[-3.2 kx] Cos[
     Sqrt[0.0232723 + kx^2 + ky^2 + kz^2] t])/
  Sqrt[0.0232723 + kx^2 + ky^2 + 
    kz^2] (Sqrt[
     0.0232723 + kx^2 + ky^2 + kz^2]/(0.0232723 + kx^2 + ky^2 + 
      0.00008649 kz^2) + 
   1/Sqrt[0.0232723 + kx^2 + ky^2 + kz^2] (kx^2 + ky^2 + 
       kz^2)/(-kx^2 - ky^2 - 
       0.00008649 kz^2)) + (Cos[-3.2 kx] Sin[
     Sqrt[0.0232723 + kx^2 + ky^2 + kz^2] t])/
  Sqrt[0.0232723 + kx^2 + ky^2 + 
    kz^2] ((-0.999957 kz)/(0.0232723 + kx^2 + ky^2 + 
      0.00008649 kz^2) + (-0.999957 kz)/(-kx^2 - ky^2 - 
      0.00008649 kz^2)) + (Sin[-3.2 kx] Cos[-0.999957 kz t] + 
   Cos[-3.2 kx] Sin[-0.999957 kz t])/(-0.0232723 - kx^2 - ky^2 - 
   0.00008649 kz^2) + 
Sin[-3.2 kx] .0232723/(0.0232723 + kx^2 + ky^2 + kz^2) 1/(-kx^2 - 
    ky^2 - 0.00008649 kz^2))]

(0.00134859 kx Sin[ 3.2 kx])/((-kx^2 - ky^2 - 0.00008649 kz^2) (0.0232723 + kx^2 + ky^2 + kz^2)) + ( 0.0579484 kx Cos[0.999957 kz t] Sin[3.2 kx])/(-0.0232723 - kx^2 - ky^2 - 0.00008649 kz^2) + ( 0.0579484 kx Cos[Sqrt[0.0232723 + kx^2 + ky^2 + kz^2] t] Sin[ 3.2 kx])/(0.0232723 + kx^2 + ky^2 + 0.00008649 kz^2) + ( 0.0579484 kx^3 Cos[Sqrt[0.0232723 + kx^2 + ky^2 + kz^2] t] Sin[ 3.2 kx])/((-kx^2 - ky^2 - 0.00008649 kz^2) (0.0232723 + kx^2 + ky^2 + kz^2)) + ( 0.0579484 kx ky^2 Cos[Sqrt[0.0232723 + kx^2 + ky^2 + kz^2] t] Sin[ 3.2 kx])/((-kx^2 - ky^2 - 0.00008649 kz^2) (0.0232723 + kx^2 + ky^2 + kz^2)) + ( 0.0579484 kx kz^2 Cos[Sqrt[0.0232723 + kx^2 + ky^2 + kz^2] t] Sin[ 3.2 kx])/((-kx^2 - ky^2 - 0.00008649 kz^2) (0.0232723 + kx^2 + ky^2 + kz^2)) + ( 0.0579484 kx Cos[3.2 kx] Sin[0.999957 kz t])/(-0.0232723 - kx^2 - ky^2 - 0.00008649 kz^2) + ( 0.0579459 kx kz Cos[3.2 kx] Sin[ Sqrt[0.0232723 + kx^2 + ky^2 + kz^2] t])/((-kx^2 - ky^2 - 0.00008649 kz^2) Sqrt[0.0232723 + kx^2 + ky^2 + kz^2]) + ( 0.0579459 kx kz Cos[3.2 kx] Sin[ Sqrt[0.0232723 + kx^2 + ky^2 + kz^2] t])/((0.0232723 + kx^2 + ky^2 + 0.00008649 kz^2) Sqrt[0.0232723 + kx^2 + ky^2 + kz^2])

and consider a term of the above sum

0.0579484*kx*Cos[0.999957 kz t]*Sin[3.2 kx]/(-0.0232723 - kx^2 - ky^2 - 0.00008649*kz^2)

Then

Integrate[Rationalize[(0.0579484` kx Cos[0.999957` kz t] Sin[3.2` kx])/
(-0.0232723` -   kx^2 - ky^2 - 0.00008649` kz^2), 0], {ky, -Infinity, Infinity}, 
 Assumptions -> {kx, kz} > -Infinity]

(144871 kx \[Pi] Cos[(999957 kz t)/1000000] Sin[(16 kx)/5])/(250 Sqrt[ 2327230 + 100000000 kx^2 + 8649 kz^2])

The above expression behaves as $c_1\sin \left(\frac {16} 5 kx\right)$ when $kx\to \infty$ and $c_2\sin \left(\frac {16} 5 kx\right)$ when $kx\to -\infty$ , where $c_1,c_2$ are real-valued constants. This implies the divergence of its integral over $kx$ from $-\infty$ to $\infty$ if $kz$ and $t$ are fixed.