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I was looking at the integral described in this paper, which claims it is simple using the Risch algorithm and shows the result

$$\displaystyle \int \frac{x (x+1) \left(e^{2 x^2} x^2+2 e^{3 x^2} x \left(x-\left(2 x^3+2 x^2+x+1\right) \log (x+1)\right)-\log ^2(x+1)\right)}{\left((x+1) \log ^2(x+1)-\left(x^3+x^2\right) e^{2 x^2}\right)^2}~dx$$

I tried it with Mathematica (I thought that parts of the Risch algorithm were implemented), but to no avail with Windows10, x86, V13.

  Integrate[ x (x + 1) ((x^2 E^(2 x^2) - Log[x + 1]^2 + 
  2 x E^(3 x^2) (x - 
     1 (2 x^3 + 2 x^2 + x + 1) Log[x + 1])))/((x + 1) Log[
     x + 1]^2 - (x^3 + x^2) E^(2 x^2))^2, x]

I also tried to download and use the IntegrateAlgebraic, but didn't have any luck with it. Lastly, I was going to try in on FriCAS Online since it has most of the algorithm implemented, but couldn't get that to work.

Is there some way to coax MMA to solve this?

Moo
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1 Answers1

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First, you do not need to download IntegrateAlgebraic anymore. It is integrated now in V 13 Integrate. By default, it is on

You can turn it off if you want, using the command

SetSystemOptions["IntegrateOptions" -> "UseIntegrateAlgebraic" -> False];

Mathematica has large parts of Risch implemented as far as I know, and it can indeed do this integral. After fixing typos you had from the paper.

ClearAll[x]
numer = x*(x + 1) *((x^2 *Exp[2*x^2] - Log[x + 1]^2)^2 + 
    2 *x* Exp[3*x^2]*(x - (2 *x^3 + 2 *x^2 + x + 1)* Log[x + 1]))
denom = ((x + 1)*Log[x + 1]^2 - (x^3 + x^2) Exp[2 x^2])^2 
Integrate[numer/denom, x]

Mathematica graphics

Compare to the paper

enter image description here

Nasser
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    Thanks @Nasser for the comments regarding IntegrateAlgebraic. I should note that IntegrateAlgebraic will return instantly on such integrands, as it only attempts to integrate algebraic expressions. For integrals of this form it would use an implementation of the transcendental Risch algorithm and follow very similar computations to those given for this example in Geddes et al. – Sam Blake Mar 09 '23 at 06:19