Oloa (2010, pers. comm., Jan. 20, 2010) has considered the following integrals containing nested radicals of 1/2 plus terms in and :
(1)
 
(2)

which he terms Ramanujan logtrigonometric integrals because they involve terms like Ramanujan's nested radicals of 1/2.
The special case
(3)

was known to Euler.
Amazingly, the general integrals have closed forms
(4)
 
(5)

for .