How to simplify the result of this indefinite integral

Question

There is an indefinite integral:

$$\int \frac{x}{\sqrt{x^4+10 x^2-96 x-71}} \, dx$$

Mathematica 13 can handle it and evaluate this integral analytically if we enter:

Integrate[x/Sqrt[x^4 + 10*x^2 - 96*x - 71], x] // ToRadicals

But the result is too large and I don't know how to simplify it (I tried Simplify and FullSimplify, but didn't help).

In Maple 2021 it's much easier. You just need to write:

simplify(int(x/sqrt(x^4 + 10*x^2 - 96*x - 71), x, method = Trager))

and you will get:

$\frac{1}{8}\ln \! \left(\left(x^{6}+15 x^{4}-80 x^{3}+27 x^{2}-528 x +781\right) \sqrt{x^{4}+10 x^{2}-96 x -71}+x^{8}+20 x^{6}-128 x^{5}+54 x^{4}-1408 x^{3}+3124 x^{2}+10001\right)$

Answer 1

There is code for handling indefinte algebraic integrals that is new to version 13. It is based on code IntegrateAlgebraic in the Wolfram Function Repository and was written by the same person (Sam Blake).

We use a time constraint to keep it from grinding its gears and in some cases this might be too small to get a result. In particular there will be examploes, such as from this post, where some processors will complete the computation in this code but others will not.

As a workaround when encountering this issue, this time constraint is configurable as a system option.

In[669]:= "IntegrateOptions" /. SystemOptions[]

Out[669]= {"IntegrateAlgebraicTimeConstraint" -> 10, 
 "RubiTimeConstraint" -> 5, "UseIntegrateAlgebraic" -> True, 
 "UseRubiIntegrate" -> True}

If you want to increase the value above 10 you might do for example:

SetSystemOptions[
  "IntegrateOptions" -> {"IntegrateAlgebraicTimeConstraint" -> 100}];

Answer 2

$Version

(* "13.0.1 for Mac OS X x86 (64-bit) (January 28, 2022)" *)

Clear["Global`*"]

int = Integrate[x/Sqrt[x^4 + 10*x^2 - 96*x - 71], x]

(* -(1/8) Log[-10001 - 3124 x^2 + 1408 x^3 - 54 x^4 + 128 x^5 - 20 x^6 - x^8 + 
   Sqrt[-71 - 96 x + 10 x^2 + 
     x^4] (781 - 528 x + 27 x^2 - 80 x^3 + 15 x^4 + x^6)] *)

intR = Integrate[x/Sqrt[x^4 + 10*x^2 - 96*x - 71], x, 
   Assumptions -> x ∈ Reals];

The results are identical with or without the Assumptions

int === intR

(* True *)

The result that you report from Maple

intMaple = 1/8*
      Log[(x^6 + 15 x^4 - 80 x^3 + 27 x^2 - 528 x + 781)*
      Sqrt[x^4 + 10 x^2 - 96 x - 71] +
     x^8 + 20 x^6 - 128 x^5 + 54 x^4 - 1408 x^3 + 3124 x^2 + 10001];

There is little difference in their complexity

LeafCount /@ {int, intMaple}

(* {78, 76} *)

They are equivalent anti-derivatives

D[int, x] == D[intMaple, x] // Simplify

(* True *)

Since you are taking the indefinite integral (anti-derivative), results are equivalent if they differ by only a constant (which can be complex).

The two results differ by a piecewise complex constant

ReImPlot[int - intMaple, {x, -10, 10},
 PlotLegends -> Placed["ReIm", {.8, .4}]]

int - intMaple /. x -> 5/2 // FullSimplify

(* (I π)/8 + Log[1/(4 3^(7/8))] *)

% // N

(* -2.34758 + 0.392699 I *)

int - intMaple /. x -> 5 // FullSimplify

(* -((I π)/8) + Log[1/(4 3^(7/8))] *)

% // N

(* -2.34758 - 0.392699 I *)

Answer 3

Your result is only correct for a real variable.

In MMA variables are assumed to be complex. If you want to restrict the variable to be real, you need to specify this, like:

Integrate[x/Sqrt[x^4 + 10*x^2 - 96*x - 71], x, 
 Assumptions -> x \[Element] Reals]

Answer 4

Below is the solution I obtained.