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 $$\int_0^{e^{-2\pi/\sqrt{10}}} q\prod_{j=1}^\infty \frac{(1-q^j)^3(1-q^{10j})^8}{(1-q^{5j})^7} \text{d} q = \frac14\left(\sqrt{10-4\sqrt{5}}-1\right).$$ 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.