5

The integral $$ F(x) = \int \frac{x}{\sqrt{x^4+10x^2-96x-71}} \, dx ,$$ has a closed-form expression, namely, $$F(x) = -\frac{1}{8} \ln\left((x^6 + 15x^4-80x^3 + 27x^2-528x + 781)\sqrt{x^4 + 10x^2-96x - 71} - (x^8 + 20x^6 -128 x^5 + 54x^4-1408x^3 + 3124x^2 + 10001)\right) \\ +C. $$

It seems that most, if not all, computer integrators such as Mathematica fail to find its antiderivative. What makes this integral so hard to solve, and how do you begin solving it?

Michael Hardy
  • 1
pacosta
  • 251
  • $\sqrt{...}$ of fourth order polynomial = nasty (however it is known that integrals of your type, with an additional x in the numerator can in principle always reduced to elementary functions) – tired Mar 22 '18 at 00:26
  • 2
    For sure, realizing that $$x^4+10x^2-96x-71=(x^2-11)^2 - 12(x+4)^2$$ is pretty useful in tackling such integral by integration by parts and substitutions. – Jack D'Aurizio Mar 22 '18 at 00:26
  • Mathematic does get this in closed form, as a truly vicious expression involving elliptic functions. The problem is, Mathematica is not all that good at clever simplifications when it comes to square roots and tric functions, much less elliptic functions. – Mark Fischler Mar 22 '18 at 00:38
  • If you factor the expression under the radical as$ (x-\sqrt{3}+2\sqrt{2(\sqrt{3}-1)}) (x-\sqrt{3}-2\sqrt{2(\sqrt{3}-1)})(x+\sqrt{3}+2i\sqrt{2(\sqrt{3}+1)})(x+\sqrt{3}-2i\sqrt{2(\sqrt{3}+1)})$ the problem becomes easier. But I can't even get this expression to show up in a comment... – Mark Fischler Mar 22 '18 at 00:42
  • This integral is awsome (awful?). Where did you get it from? – Mark Fischler Mar 22 '18 at 01:01

0 Answers0