If you're using the simpler definitions it seems to me the $\frac{1}{2}$ terms in your title and related formulas should really be $\frac{1}{2}\sqrt{\frac{\pi}{2}}=\sqrt{\frac{\pi}{8}}$.

I don't understand how you derived the final result $-\frac{\sqrt\pi}{2\sqrt2}-2\int_0^\infty\frac{C(x)}{e^{2\pi x}-1}dx+2\int_0^\infty \frac{S(x)}{e^{2\pi x}-1}dx$ in the derivation based on the Abel-Plana formula in the question above. Also I believe there's one spot where $\frac{1}{2}$ should be $\frac{\sqrt\pi}{2\sqrt2}$ in the derivation.

It seems to me your final result is missing the two integrals $\int_0^\infty\left(C(x)-\frac{\sqrt{\pi }}{2 \sqrt{2}}\right)\,dx+\int_0^\infty\left(S(x)-\frac{\sqrt{\pi }}{2 \sqrt{2}}\right)\,dx$. Are you saying these two integrals sum to zero?

@StevenClark We can actually evaluate the integrals as seen here. The link has the evaluated integral for:$$\int_0^\infty C(x)+S(x) -\sqrt{\frac \pi 2}dx$$, the sum of the two integrals. The value at $\infty$ of the integral is 0, so the link just has the value at x=0. However, I will edit for the right value. Let me work on this.

The formula still has some holes in it, so do you have any other ideas?

I obtained $2\int_0^\infty\frac{S(x)}{e^{2 \pi x}-1}\,dx-2\int_0^\infty\frac{C(x)}{e^{2 \pi x}-1}\,dx-\frac{1}{2}-\frac{\sqrt{\pi }}{2 \sqrt{2}}$. I'm not sure it's equivalent but I noticed you deleted the $2$ preceding the two integrals, changed the sign of one of the two integrals, and added $\frac{1}{2}$ instead of subtracting $\frac{1}{2}$.

@StevenClark Maybe we can head to the chat room? The concern here is that your integral evaluates to-1.206… while the sum over 400 terms is -1.50…. Maybe from the formula conditions?

I see you are interested in the problem and helping to possibly evaluate it. Thanks again.

One last thing, the Wolfram Alpja version uses the Normalized Fresnel Integrals found at the bottom of the “definition” section of the link.

I just subtracted the $\frac{1}{2}$ term from your earlier formula where the $\frac{1}{2}$ term was based on your earlier comment and observational convergence of $\int_0^y\left(C(x)+S(x) -\sqrt{\frac \pi 2}\right) dx\to -\frac{1}{2}$ as $y\to\infty$. I see you updated the formula in your question in a similar manner. I thought I was close to a result for $2\int_0^\infty\frac{S(x)-C(x)}{e^{2 \pi x}-1}\,dx$ but I found an error in my derivation so back to the drawing board.

With respect to the discrepancy between the evaluation of the sum and the numerical evaluation of the integral, I tried increasing the precision of the evaluations and also evaluating the sum out to $100,000$ terms but it didn't seem to resolve the discrepancy.

@StevenClark The formula seems to have conditions for which it can be used. If the formula does work, then what techniques did you use to try to evaluate? Also see related problems like at the start of the question

With respect to derivation of a closed form result for the integral, I was investigating integration by parts and I thought I had a result but I discovered I accidentally wrote the denominator as E^(2 Pi x - 1) instead of E^(2 Pi x) - 1.