calculus and analysis - How to make Mathematica show intermediate steps in integral

I want to know how Mathematica evaluates this integral:

$PV \int _ {0} ^ {i \infty} d \tau \, \frac{e^{-\frac{\tau^2}{2 M^2}}(b^2 - 3 τ^2)^2 (|b^2+\tau^2|-(b^2-\tau^2))}{\tau (\tau^2 - b^2)}$, where the Cauchy Principal Value is taken and $b>0 \, \& \, M>0$.

Mathematica gives me as result

$b^2(-i \pi b^2-\gamma b^2-18 M^2+4b^2e^{-\frac{b^{2}}{2M^2}}(i \pi + Ei(\frac{b^2}{2M^2}))+2 b^2 \log (M)+b^2 \log (2))$

where $\gamma$ is the Euler constant

I am far from sure that the above result is correct, so I would like to know the intermediate steps that Mathematica uses to get this result. Any help (either on the intermediate steps or the validity of the result)?

Answer

First of all correcting the sign in the exponential we get

f2 = Integrate[

  Exp[r^2/(2 M^2)] (b^2 - 3 r^2)^2 (Abs[b^2 + r^2] - (b^2 - r^2))/(
   r (r^2 - b^2)), {r, 0, I \[Infinity]}, 
  Assumptions -> {M > 0, b > 0]

(* Out[278]= 
18 b^2 M^2 + 4 b^4 E^(b^2/(2 M^2)) ExpIntegralEi[-(b^2/(2 M^2))]
*)

Taking the pricipal value has no influence as there is no singularity on the imaginary r-axis.

Maybe this is already the result expected by user127054.

As for a step-by-step approach I would recommend taking the indefinite integral which would give the antiderivative. But in this case MMA does not provide the antiderivative.

More details:

Transforming the integral to the real axis by letting r -> I s we have for the integrand (to be taken between s = 0 and s = +inf)

I Exp[r^2/(2 M^2)] (b^2 - 3 r^2)^2 (Abs[b^2 + r^2] - (b^2 - r^2))/(
   r (r^2 - b^2)) /. r -> I s // Simplify

(*
Out[294]= (E^(-(s^2/(
  2 M^2))) (b^2 + 3 s^2)^2 (b^2 + s^2 - Abs[b^2 - s^2]))/(s (b^2 + s^2))
*)

It seems that there is a singularity at s = 0.

But let us look at the series expansion and observe that b^2 > s^2 since s->0 and b>0:

Simplify[Series[%, {s, 0, 2}] // Normal, b^2 > s^2]

(*
Out[295]= 5 s^3 + b^2 (2 s - s^3/(2 M^2))
*)

Hence there is no singularity at s = 0 and therefore none on the positive s axis. Hence taking the principal has no effect on the result.

Series expansion in b of the result are (for M->1)

at b = 0

Series[f2 /. M -> 1, {b, 0, 4}, Assumptions -> b > 0] // Normal

(*
Out[304]= 18 b^2 + b^4 (4 EulerGamma - 4 Log[2] + 8 Log[b])
*)

at b = inf

Series[f2 /. M -> 1, {b, \[Infinity], 4}, Assumptions -> b > 0] // Normal


(*
Out[306]= 10 b^2 + 16 - 64/b^2 + 384/b^4
*)