Integral relating gamma function's linear subspaces

Question

I'm trying to solve the equation: $$\int_0^\infty \frac{\cos(s\ln(t)+\frac{π(4n+1)}{4})}{\sqrt{t}e^t}\mathrm{d}t=0$$ For the variable $s\in\mathbb{R}$, where $n\in\mathbb{Z}$ is any nonzero integer. This is related to a more complicated integral equation: $$\int_0^\infty \left(\frac{\cos(s\ln(t))-\sin(s\ln(t))}{\sqrt{t}e^t}\right)\mathrm{d}t=0$$ This equation involves certain linear subspaces of the complete gamma function. More specifically, the subspace of all the vomplef numbers of the form: $a(1+i)$ where $a$ is any real number. Thanks for the help.

Answer 1

As @MariuszIwaniuk notes, these integrals are computable. In fact, not only by Mathematica [sic], but by human beings: replace the cosine by its expression in terms of exponentials. Grouping things gives a linear combination of two integrals of the form $\int_0^\infty t^{s-1/2}\,e^{-t}\;dt/t$, with the coefficients of the linear combination depending on $n$.