How to speed up this integration?

Question

I try to evaluate this integral:

eq[k_, q_, b_, f_] = (1/Sqrt[h^2 - k^2])* Sinh[Sqrt[h^2 - k^2]*(q - b)]*(1/(h*b))*(m^2*y[b]^2 - (1/(h*b))* D[y[b], b]^2) + (1/Sqrt[h^2 - k^2])*Sinh[Sqrt[h^2 - k^2]*(q - f)]*(1/(h*b))*(m^2*y[f]^2 - (1/(h*f))* D[y[f], f]^2); 

Where $m= h = 10^{14}$ GeV, and the initial integration limit $a= -10^{14}$ GeV.

  y[b_]:= b^(1/2 + Sqrt[1 + m^2/h^2]) + b^(1/2 - Sqrt[1 + m^2/h^2]); 
    
    y[f_]:= f^(1/2 + Sqrt[1 + m^2/h^2]) +  f^(1/2 - Sqrt[1 + m^2/h^2]); 

To double integrate eq once with respect for b and once with respect for f, I use a symbolic integration

i1[k_, f_,q_]=Map[Integrate[#, {b, a, q}] &, eq[k, q, b, f] // FullSimplify // Expand]

i2[k_,q_]=Map[Integrate[#, {f, a, q}] &, i1[k,f,q] // FullSimplify // Expand]

Then I want to plot i2[k,q] in the limit where $q\to 0$, so I use:

i3[k_]:= Limit[i2[k,q],q \to 0]

The problem is that this integration takes so much time, especially i2 it has not even been evaluated. So is there a way to speed up this integration and make its evaluation easier?

Hint

I'm using MA12.0. And in a previous thread

How to make NIntegrate with large integration limit?

I tried NIntegrate but it doesn’t run with large initial integration limit $a$ and I could not take the limit of q.

Edit

Here are my trials to do i1 and i2 in one step based on the answer Of Ulrich Neumann and to avoid the non covergence error message:

Map[Integrate[#, {b, a, q}, {f, a, q}, Assumptions -> {a < 0}] &, expr]

This gives a Condtitional expression: Re[q] $\leq$ a || a Re $\frac{1}{-a+q}) \leq -1 $ || q is not [Element] R

So I think this is the problem a should be ( > 0 ), so that q is an integer. So I try:

Limit[Map[Integrate[#, {b, a, q}, {f, a, q}, Assumptions -> { q \[Element] Integers && a > 0}] &, expr],q->0]

Since I initially want to determine the integral at the q->0 Limit, but this gives Indeterminate

Answer 1

Assuming m==h and a<0<q Mathematica 12.2 states non-convergent integral:

m=h
y[b_] := b^(1/2 + Sqrt[1 + m^2/h^2]) + b^(1/2 - Sqrt[1 + m^2/h^2]);
expr = (1/Sqrt[h^2 - k^2])*Sinh[Sqrt[h^2 - k^2]*(q - b)]*(1/(h*b))*(m^2*y[b]^2 - (1/(h*b))*D[y[b], b]^2) 
+ (1/Sqrt[h^2 - k^2])*Sinh[Sqrt[h^2 - k^2]*(q - f)]*(1/(h*b))*(m^2*y[f]^2 
- (1/(h*f))*D[y[f], f]^2);  // Expand
expr[[0]] = List (* list the parts of `expr`

Integration

Map[Integrate[#, {b, a, q}, {f, a, q}, Assumptions -> {a < 0 < q}] &, expr] 

doesn't converge, gives error message

Integrate::idiv: Integral of (1/(4 m^2 Sqrt[-k^2+m^2]))b^(-3-2 Sqrt[2]) (9-4 Sqrt[2]-14 b^(2 Sqrt[2])+(9+4 Sqrt[2]) b^(4 Sqrt[2])-4 b^3 (1+b^(2<<1>><<5>><<1>><<1>>]))^2 m^3) (a-q) Sinh[Sqrt[-k^2+m^2] (-b+q)] does not converge on {a,q}.