Inspired by problem 3, IMO 1973

Calculus Level 5

\[ \large x^4+ a x^3 + x^2 + b x+1=0\]Let \(A\) be the set of points \((a,b)\) for which the above equation has no real root. Area of \(A\) can be expressed as\[\frac{p\sqrt{q}}{r}+ s \ \tanh ^{-1} \left( \sqrt{\frac{t}{u}} \right)\]Find the value of \(p+q+r+s+t+u\).

Details and Assumptions:

  • \(a\), \(b\) are real numbers.
  • \(p\), \(q\), \(r\), \(s\), \(t\) and \(u\) are positive integers, \(q\) is square free and \(\gcd(p, r)=\gcd(t, u)=1\).

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