Ramanujan--Fine integrals for level 10

We investigate the question of when an eta quotient is a derivative of a formal power series with integer coefficients and present an analysis in the case of level 10. As a consequence, we establish and classify an infinite number of integral evaluations such as $${\int }_0^{e^{-2\pi \mathrm{/}\sqrt{10}}}q\prod _{j=1}^{\mathrm{\infty }}\frac{(1-q^j{)}^3(1-q^{10j}{)}^8}{(1-q^{5j}{)}^7}\text{d}q=\frac{1}{4}(\sqrt{10-4\sqrt{5}}-1)\mathrm{.}$$ We describe how the results were found and give reasons for why it is reasonable to conjecture that the list is complete for level 10.