This question is a sequel of this. I tried very hard but i am not able to simplify or beautify this result-$$\sum_{n=0}^\infty (AB)^{n^2}(A/B)^n=\int_{-\infty}^\infty \frac{e^{-t^2}}{\sqrt{\pi}}\frac{1}{1- (A/B)e^{-2t\sqrt{\log AB}}}dt$$ to this result- $$\sum_{n=0}^\infty a^{\frac{n(n+1)}{2}}b^{\frac{n(n-1)}{2}}=1+\int_0^\infty \frac{2ae^{-t^{2}/2}}{\sqrt{2\pi}}\left[\frac{1-a\sqrt{ab}\operatorname{cosh}(\sqrt{\operatorname{log}(ab)}t)}{a^3b-2a\sqrt{ab}\operatorname{cosh}(\sqrt{\operatorname{log}(ab)}t)+1}\right]dt+\int_0^\infty \frac{2be^{-t^{2}/2}}{\sqrt{2\pi}}\left[\frac{1-b\sqrt{ab}\operatorname{cosh}(\sqrt{\operatorname{log}(ab)}t)}{ab^3-2b\sqrt{ab}\operatorname{cosh}(\sqrt{\operatorname{log}(ab)}t)+1}\right]dt$$ I would appreciate a full length solution since I am already very confused

I saw these kind of integrals in Ramanujan's lost notebook and Jacobi theta functions, if anybody can shed light on how and from where these big and beautiful integrals come, i would appreciate that too :)

This integral representations were taken from this (it might help)


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