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Infinite cosine product integral

At the age of 17, Bernard Mares proposed the definite integral (Borwein and Bailey2003, 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.

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