From: clicliclic@freenet.de   
      
   Have some fun with GPT-5 mini! Its answers to my questions are on the   
   verge of becoming hallucinatory in phrases like "Reducible polynomial   
   [...] e.g., P(x) = (ax^2+bx+c)(dx^2+ex+f)", "*\C389\mile* (instead of   
   \C389\duard) Goursat", "what *later authors* call   
   \E2809C\pseudo\E28091\elliptic\E2809D\", "the classical Goursat   
   substitution", and "the explicit elementary primitive *Goursat   
   obtained*".   
      
   I have no idea how severely GPT-5 mini has been dumbed down from GPT-5   
   and I didn't check any of the mathematical details. UTF-8 entities in   
   the answers are represented by hex bytes between backslashes: \E2889A\ =   
   square root sign, \E288AB\ = integral sign, etc.   
      
   = = =   
      
   What are pseudo-elliptic integrals?   
      
   - GPT-5 mini:   
      
   Pseudo-elliptic integrals are integrals that look like elliptic   
   integrals (involving square roots of polynomials of degree 3 or 4) but   
   are not genuine elliptic integrals because their integrands can be   
   transformed into elementary functions or simpler standard integrals by   
   algebraic substitutions, factorization, or decomposition. In practice   
   the term is used in a few related senses:   
      
   Informal / historical usage: an integral whose integrand contains a   
   square root of a cubic or quartic (the hallmark of elliptic integrals)   
   but which, after an elementary algebraic manipulation, reduces to a   
   combination of elementary functions (log, arctan, rational functions) or   
   to elliptic integrals of lower complexity. Such integrals   
   \E2809C\mimic\E2809D\ elliptic form but are not transcendental in the   
   same way elliptic integrals are.   
      
   More precise algebraic viewpoint: integrals of the form \E288AB\R(x,   
   \E2889A\P(x)) dx where P(x) is cubic or quartic and R is a rational   
   function, but where the algebraic curve y^2 = P(x) has genus 0 (not   
   genus 1). Genus 0 curves are birationally equivalent to the Riemann   
   sphere; the square root can be parameterized rationally and the integral   
   becomes elementary. Genuine elliptic integrals arise when the curve has   
   genus 1 (an elliptic curve).   
      
   Examples and tests   
      
   Reducible polynomial: if P(x) factors so that the square root simplifies   
   (e.g., P(x) = (ax^2+bx+c)(dx^2+ex+f) and substitution reduces the   
   integrand), the integral may be pseudo-elliptic.   
      
   Double root / rational parameterization: if P(x) has a multiple root or   
   admits a rational parameterization (equivalently the curve y^2 = P(x)   
   has genus 0), then \E288AB\R(x, \E2889A\P(x)) dx is elementary. For   
   instance, \E288AB\ dx / \E2889A\(x^3 + ax^2 + bx + c) can be   
   pseudo-elliptic when the cubic has a rational parametrization (e.g., a   
   double root), otherwise it is genuinely elliptic.   
      
   Liouville/Risch perspective: symbolic-integration algorithms detect when   
   algebraic integrals reduce to elementary antiderivatives; those that   
   pass are pseudo-elliptic.   
      
   Concrete small examples   
      
   If P(x) = (x \E28892\ r)^2(x + s), then \E2889A\P(x) = (x \E28892\   
   r)\E2889A\(x + s) and substitution u = \E2889A\(x + s) rationalizes the   
   root and yields elementary integrals.   
      
   \E288AB\ dx / \E2889A\(x^4 + 2x^3 + x^2) = \E288AB\ dx /   
   (x\E2889A\(x^2+2x+1)) simplifies because x^4+2x^3+x^2 = x^2(x+1)^2,   
   giving elementary antiderivative.   
      
   Summary   
      
   \E2809C\Pseudo-elliptic\E2809D\ is a descriptive label for integrals   
   that superficially have elliptic form but reduce to elementary (or   
   otherwise simpler) forms because the underlying algebraic curve has   
   genus 0 or admits simplifying factorization/parameterization. Genuine   
   elliptic integrals correspond to genus 1 curves and cannot be reduced   
   this way.   
      
   - - -   
      
   How did \C389\mile Goursat contribute to our understanding of   
   pseudo-elliptic integrals?   
      
   - GPT-5 mini:   
      
   \C389\mile Goursat (1858\E28093\1936) contributed to the theory of   
   integrals involving algebraic functions, including what later authors   
   call \E2809C\pseudo\E28091\elliptic\E2809D\ integrals, by clarifying   
   when integrals of the form \E288AB\R(x, \E2889A\P(x)) dx (R rational, P   
   polynomial of degree 3 or 4) reduce to elementary functions. His main   
   contributions relevant here are:   
      
   Systematic use of algebraic function theory and birational   
   transformations to determine when the curve y^2 = P(x) has genus 0 so   
   the square root can be rationally parametrized. Goursat\E28099\s work   
   helped make precise that integrals over genus\E28091\0 algebraic curves   
   are reducible to elementary integrals, while genus\E28091\1 curves   
   produce genuine elliptic integrals.   
      
   Development and exposition of methods for reducing integrals of   
   algebraic functions by rational substitutions and factorization. In his   
   textbooks and papers he collected transformation tricks and criteria   
   that identify \E2809C\pseudo\E28091\elliptic\E2809D\ cases (e.g.,   
   multiple roots, reducible quartics, or quartics that are biquadratic)   
   and show how to rationalize the integrand.   
      
   Influence on later rigorous formulations: Goursat\E28099\s treatments of   
   algebraic integrals and his clear presentation of transformations and   
   canonical forms fed into the modern algebraic\E28091\geometric viewpoint   
   (genus, birational equivalence) used to classify when an   
   elliptic\E28091\looking integral is actually elementary.   
      
   In short: Goursat helped move the subject from ad\E28091\hoc   
   manipulations toward a structured algebraic classification that explains   
   why some integrals that look elliptic are only   
   \E2809C\pseudo\E28091\elliptic\E2809D\ \E28094\ reducible to elementary   
   functions because the underlying curve has genus 0 or admits rational   
   parameterizations. If you want references, his exposition appears in his   
   writings on algebraic functions and in his analysis texts from the late   
   19th/early 20th century.   
      
   - - -   
      
   Is the integrand x/((x^3+8)*sqrt(x^3-1)) pseudo-elliptic?   
      
   - GPT-5 mini:   
      
   We need to check whether the algebraic curve y^2 = x^3 - 1 (cubic) gives   
   a genus-0 or genus-1 curve and whether the integrand can be reduced by   
   an algebraic substitution to elementary functions.   
      
   The curve y^2 = x^3 - 1 is a nonsingular cubic \E28692\ genus 1 (an   
   elliptic curve). So generic integrals involving \E2889A\(x^3\E28892\1)   
   are elliptic, not elementary.   
      
   Try to see if the integrand simplifies to an elementary form. Consider   
   substitution y = \E2889A\(x^3\E28892\1). Differentiate: dy = (3x^2 /   
   (2\E2889A\(x^3\E28892\1))) dx = (3x^2/(2y)) dx, so dx = 2y dy/(3x^2).   
   That mixes x and y \E28094\ not helpful directly.   
      
   Look for algebraic substitution that rationalizes   
   \E2889A\(x^3\E28892\1). A rational parametrization exists only if the   
   cubic has a rational double root (singular) \E28094\ it does not. So no   
   global rational parameterization.   
      
   Try elementary algebraic manipulation: integrand = x   
   /((x^3+8)\E2889A\(x^3\E28892\1)). Let t = x^3. Then dt = 3x^2 dx, not   
   matching numerator x dx. So that fails.   
      
   Check for substitution x = u^k that might help: letting x = 1/u yields   
      
   [continued in next message]   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)