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Clarified relationship between ver 9 and ver 10 solutions
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bbgodfrey
bbgodfrey
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Between them, the two solutions appear to cover all of p - k space. Nonetheless, it seems unlikely thatis unclear whether patching the two together and then integrating would be successful, due to branchpoints at z = 1 and z = -1, where z is the argument of ArcTanh or ArcCoth in the expressions above.

VersionAnswer to question in a comment above: Although version 10.1 produces symbolic solutions that lookappear different from the version 9.0.1 solutions.

Theirtheir 3D plots appearare identical respectively to the fourth and second plots above, although their asserted ranges of validity differ. Note that Even though a straightforward application of FullSimplify does not succeed in showing themthe respective version 9.0.1 and version 10.1 solutions to be identical, it appears that they are from identities in Wolfram MathWorld.

Between them, the two solutions appear to cover all of p - k space. Nonetheless, it seems unlikely that patching the two together and then integrating would be successful.

Version 10.1 produces symbolic solutions that look different.

Their 3D plots appear identical respectively the fourth and second plots above, although their asserted ranges of validity differ. Note that a straightforward application of FullSimplify does not succeed in showing them to be identical.

Between them, the two solutions cover all of p - k space. Nonetheless, it is unclear whether patching the two together and then integrating would be successful, due to branchpoints at z = 1 and z = -1, where z is the argument of ArcTanh or ArcCoth in the expressions above.

Answer to question in a comment above: Although version 10.1 produces symbolic solutions that appear different from the version 9.0.1 solutions.

their 3D plots are identical respectively to the fourth and second plots above, although their asserted ranges of validity differ. Even though a straightforward application of FullSimplify does not succeed in showing the respective version 9.0.1 and version 10.1 solutions to be identical, it appears that they are from identities in Wolfram MathWorld.

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bbgodfrey
bbgodfrey
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With version 9.0.1,

f[x_] := (p^2 + k^2 - 2 p k x)/(x - (p^2 + k^2 + 1 - ((p^2 - k^2)^2)/4)/(2 p k)); 
ans9 = Integrate[f[x], {x, -1, 1}, PrincipalValue -> True]
(* ConditionalExpression[1/2 k p (-8 + ((-4 + (k^2 - p^2)^2) 
   ArcCoth[(8 k p)/(-4 + k^4 - 4 p^2 + p^4 - 2 k^2 (2 + p^2))])/(k p)), 
   k^4 + p^4 < 4 + 4 p^2 + 2 k^2 (2 + p^2) && (k - p)^2 (-2 + k + p) (2 + k + p) > 4] *)

Some insight can be gained by plotting the solution ad its region of validity as specified by ConditionalExpression,

RegionPlot[Evaluate[ans9[[2]]], {k, -5, 5}, {p, -5, 5}, FrameLabel -> {k, p}]

enter image description here

Plot3D[Evaluate[ans9[[1]]], {k, -5, 5}, {p, -5, 5}, PlotPoints -> 100,
  PlotRange -> All, Mesh -> None, AxesLabel -> {k, p, z}]

enter image description here

Thus, the solution is real and continuous over a wider range than that given by the ConditionalExpression. Nonetheless, the actual range is bounded by singularities, so it is not surprising that the second Integrate in the question does not yield an answer. Of course, even without the singularities, Integrate might fail, if no known symbolic solution exists.

Because, I presume, k and p are meant to be Reals, it is reasonable to inform Integrate of this.

ans9r = Integrate[f[x], {x, -1, 1}, PrincipalValue -> True, 
  Assumptions -> k ∈ Reals && p ∈ Reals]
(* ConditionalExpression[1/2 k p (-8 + ((-4 + (k^2 - p^2)^2) 
   ArcTanh[(8 k p)/(-4 + k^4 - 4 p^2 + p^4 - 2 k^2 (2 + p^2))])/(k p)),
   (-4 + (k - p)^2) (k + p)^2 <= 4 && (k - p)^2 (-2 + k + p) (2 + k + p) <= 4] *)

which produces a different but related solution (perhaps, a different branch).

enter image description here

enter image description here

Between them, the two solutions appear to cover all of p - k space. Nonetheless, it seems unlikely that patching the two together and then integrating would be successful.

Version 10.1 produces symbolic solutions that look different.

(* ConditionalExpression[1/4 (-(-4 + (k^2 - p^2)^2) Log[-k p] + (k^2 - p^2)^2 (Log[k p]
   - Log[4 - (-4 + (k - p)^2) (k + p)^2] + Log[4 - (k - p)^2 (-2 + k + p) (2 + k + p)])
   - 4 (4 k p + Log[k p] - Log[4 - (-4 + (k - p)^2) (k + p)^2] + 
   Log[4 - (k - p)^2 (-2 + k + p) (2 + k + p)])),
   (-4 + k^4 - 4 p^2 + p^4 - 2 k^2 (2 + p^2))/(k p) ∈ Reals] *)

and

(* ConditionalExpression[1/4 (-16 k p - (-4 + (k^2 - p^2)^2) 
   Log[4 - (-4 + (k - p)^2) (k + p)^2] + (-4 + (k^2 - p^2)^2) 
   Log[4 - (k - p)^2 (-2 + k + p) (2 + k + p)]), 
   k^4 + p^4 <= 4 + 4 p^2 + 2 k^2 (2 + p^2) && (-4 + (k - p)^2) (k + p)^2 <= 4] *)

Their 3D plots appear identical respectively the fourth and second plots above, although their asserted ranges of validity differ. Note that a straightforward application of FullSimplify does not succeed in showing them to be identical.