Oloa (2010, pers. comm., Jan. 20, 2010) has considered the following integrals containing nested radicals of 1/2 plus terms in θ^2 and ln^2 cos θ: R_n^- | = | 2/π integral_0^(π/2) (θ^2 + ln^2 cos θ)^((-2)^(-n - 1)) sqrt(1/2 + 1/2 sqrt(1/2 + ... + 1/2 sqrt((ln^2 cos θ)/(θ^2 + ln^2 cos θ))))d θ R_n^+ | = | 2/π integral_0^(π/2) (θ^2 + ln^2 cos θ)^(2^(-n - 1)) sqrt(1/2 + 1/2 sqrt(1/2 + ... + 1/2 sqrt((ln^2 cos θ)/(θ^2 + ln^2 cos θ))))d θ, which he terms Ramanujan log-trigonometric integrals because they involve terms like Ramanujan's nested radicals of 1/2.
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