On Saturday, December 11, 2021 at 12:00:02 PM UTC-5, Бальтазар Лин wrote:
This can be finally solved by Mathematica 13, not only by various forks of Axiom.
https://www.wolframcloud.com/obj/d9af14f6-3b98-43c4-b996-11dedc9d9f10
As they say 20 years only were needed. Thanks to https://resources.wolframcloud.com/FunctionRepository/resources/IntegrateAlgebraic/
Maple 2021.0, from March, 2021,
kernelopts(version);
Maple 2021.0, X86 64 LINUX, Mar 5 2021, Build ID 1523359
int(x/sqrt(x^4 + 10*x^2 - 96*x - 71), x, method=trager);
-1/8*ln(-x^8+(x^4+10*x^2-96*x-71)^(1/2)*x^6-20*x^6+15*(x^4+10*x^2-96*x-71)^(1/2)*x^4
+128*x^5-80*(x^4+10*x^2-96*x-71)^(1/2)*x^3-54*x^4+27*(x^4+10*x^2-96*x-71)^(1/2)*x^2+
1408*x^3-528*(x^4+10*x^2-96*x-71)^(1/2)*x-3124*x^2+781*(x^4+10*x^2-96*x-71)^(1/2)-\
10001)
simplify(%);
-1/8*ln((x^6+15*x^4-80*x^3+27*x^2-528*x+781)*(x^4+10*x^2-96*x-71)^(1/2)-x^8-20*x^6+
128*x^5-54*x^4+1408*x^3-3124*x^2-10001)
--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)