Caboodle of tricks

Calculus Level 5

\[\large \displaystyle \int_{0}^{\infty}{\frac{{x}^{4} \ln(x)}{{(1+{x}^{4})}^{2}}} \, dx\]

The above integral can be written as \(\displaystyle \frac{{\pi}^{a}}{{b}^{c/d}} - \frac{{\pi}^{e}}{{f}^{g/h}}\) for positive integers \(a,b,c,d,e,f,g,h\) where \(b\) and \(f\) are prime numbers, with \( \gcd(c,d) =\gcd(g,h) = 1 \).

Give your answer as \(\displaystyle a + b + c + d + e + f + g + h\).

Details and Assumptions:

  • All of the letters need not represent distinct numbers.

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