$\int_1^{\infty}\sqrt{\frac{\log x}{x^4+1}}dx=$??

Consider the terrible-horrible-not-good-very-bad integral $ $ I=\int_1^{\infty}\sqrt{\frac{\log x}{x^4+1}}dx$ $ Where of course $ \log x$ denotes the natural logarithm.

I don’t know where to even begin, because I can’t think of any series that would give the integral. I’m sure the integrand doesn’t have any elementary antiderivative, and I have no idea what an appropriate substitution for Feynman integration would be. I thought that it might be beneficial to try simplify it with a substitution of $ \log x=t$ , but I can’t see that getting anywhere. Please help.