# Let's do some calculus! (49)

Calculus Level 5

\begin{align} & \large I_1 = \int_0^{\infty} \dfrac{\ln^2 (x)}{e^x} x^{{1}/{4}} \mathrm{d}x \\ & \large I_2 = \int_0^{\infty} \dfrac{x^{-{1}/{4}}}{e^x} \mathrm{d}x \end{align}

Let $$I_1$$ and $$I_2$$ be as defined above. If the closed form of $$I_1 I_2$$ can be represented as

$\dfrac{a}{b \sqrt c} \pi^3 - \sqrt{c} \pi^2 + \dfrac{d}{c \sqrt{c}} \pi^2 \ln(c) + \dfrac{\pi^2 \gamma}{c \sqrt c} + \dfrac{\pi \gamma^2}{c \sqrt c} + \gamma \pi \left( \dfrac{d \ln(c)}{\sqrt c} - c \sqrt c \right) - e \sqrt{c} \pi \ln(c) + \dfrac{f \pi \ln^2 (c)}{c \sqrt{c}} + c \sqrt{c} \pi G$

where alphabets in lowercase from $$a$$ to $$f$$ are distinct positive integers with coprime-pairs $$(a,b) , (c,d)$$ and $$(c,f)$$ and $$c$$ is not a perfect power, evaluate $$a+b+c+d+e+f$$.

Notations:

Inspiration.