# 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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