<s>[INST] How should I evaluate $\int_0^{\infty}\int_0^{\infty}\frac{\sqrt[3]{x}}{1+\sqrt[3]{x}}e^{-y\pi\left(x^2+1/x^2+1\right)}\ \mathrm{d}y\ \mathrm{d}x$? $$\int_0^{\infty}\int_0^{\infty}\frac{\sqrt[3]{x}}{1+\sqrt[3]{x}}e^{-y\pi\left(x^2+1/x^2+1\right)}\ \mathrm{d}y\ \mathrm{d}x$$
$$\int_0^{\infty}\frac{\sqrt[3]{x}}{1+\sqrt[3]{x}}\frac{1}{\pi\left(x^2+1/x^2+1\right)}\ \mathrm{d}x$$
$$\frac1{\pi}\int_0^{\infty}\frac{\sqrt[3]{x}}{1+\sqrt[3]{x}}\frac{x^2}{x^4+x^2+1}\ \mathrm{d}x$$
then substituting $t^3=x$
$$\frac3{\pi}\int_0^{\infty}\frac{t^9}{(1+t)(t^{12}+t^6+1)}\ \mathrm{d}t$$
what should i do after this should i write $\frac{1}{1+t}$ as $\sum(-1)^kt^k$?
 [/INST] After some truly horrific partial fraction decomposition (I'm trusting WolframAlpha (check near the bottom of the page) for now, until I can verify it), we get
\begin{align}
\frac{ t^3 }{3 (t^6 - t^3 + 1)} + \frac{t}{6 (t^6 - t^3 + 1)} - \frac{t}{2 (t^6 + t^3 + 1)} - \frac{1}{6 (t^6 - t^3 + 1)} + \frac{1}{2 (t^6 + t^3 + 1)} + \frac{t^5}{3 (t^6 - t^3 + 1)} - \frac{t^4}{3 (t^6 - t^3 + 1)} - \frac{t^2}{6 (t^6 - t^3 + 1)} + \frac{t^2}{2 (t^6 + t^3 + 1)} - \frac{1}{3 (t + 1)}.
\end{align}
If I plug this guy into FriCAS, it simplifies to
\begin{align}
\frac{t^5 - t^4 + t^3 + t^2 - t}{3}.
\end{align}
This still looks like it'll diverge, so something else funky is going on here.
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