Oloa (2010, pers. comm., Jan. 20, 2010) has considered the following integrals containing nested radicals of 1/2 plus terms in 
and 
:
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(1)
|
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(2)
|
which he terms Ramanujan log-trigonometric 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
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(4)
|
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(5)
|
for 
.
