Infinite Cosine Product Integral

At the age of 17, Bernard Mares proposed the definite integral (Borwein and Bailey 2003, p. 26; Bailey et al. 2006)

(1) (2)

(OEIS A091473). Although this is within of ,

(3)

(OEIS A091494), it is not equal to it. Apparently, no closed-form solution is known for .

Interestingly, the integral

(4) (5)

(Borwein et al. 2004, pp. 101-102) has a value fairly close to , but no other similar relationships seem to hold for other multipliers of the form or .

The identity

(6)

can be expanded to yield

(7)

In fact,

(8)

where is a Borwein integral.