https://en.wikipedia.org/wiki/Borwein_integral Mma 10.4 gets In[562]:= Integrate[Product[Sinc[x/(2 k - 1)], {k, 8}], {x, 0,∞}] Out[562]= (467807924713440738696537864469 π/ 935615849440640907310521750000 surprisingly (suspiciously?) quickly, but has run all night on Integrate[2 Cos[x] Product[Sinc[x/(2 k - 1)], {k, 18}], {x, 0,∞}] // tim which is only a quarter of the way to where it gets interesting. In[421]:= ContinuedFraction[467807924713440738696537864469/ 935615849440640907310521750000] Out[421]= {0, 2, 33999077543, 5, 16, 1, 2, 1, 2, 1, 1, 1, 5, 1, 2, 1, 15, 1, 13, 2, 16, 1, 2, 3, 1, 7, 1, 10, 3, 1, 1, 2, 1, 1, 3, 3, 1, 11, 10, 1, 1, 3, 1, 19} AFAI can tell, http://schmid-werren.ch/hanspeter/publications/ 2014elemath.pdf mentioned by Wikipedia tells *where* (the Cos[x] case) gets interesting, but not *how* interesting. Has anybody actually computed that huge rational? We wonder how far from π/2 the integral manages to stray as we tip over the 8, It's a bit surprising, though well known, that Product[Cos[x/(2*k)], {k, Infinity}] == Product[Sinc[x/(2*k-1)], {k, Infinity}] Might the left infinite product be any easier to integrate? Not likely! All the integrals of the finite cosine products diverge! Mathematica is so far unhelpful finding the integral. I may need my homebrew Euler-Maclaurin function. --rwg In fact, I've so far failed to qualitatively accelerate the infinite product for a given x. This gets it from zeta 2 to zeta 8 type convergence, In[323]:= sncprd[x_?NumericQ] := Block[{p = 1/(\[Pi]^6 x^6) 30 Sqrt[ 42] (Cos[(\[Pi] x)/30^(1/4)] - Cosh[(\[Pi] x)/30^(1/4)]) (Cos[(3^(1/3) \[Pi] x)/( Sqrt[2] 35^(1/6))] - Cosh[(\[Pi] x)/(Sqrt[2] 105^(1/6))]) Sin[(\[Pi] x)/Sqrt[ 6]] Sinh[(\[Pi] x)/(Sqrt[2] 105^(1/6))], n = 1}, While[p != (p *= Sinc[x/n]/((1 - x^2/(6 n^2)) (1 + x^4/(120 n^4)) (1 + x^6/( 840 n^6)))), ++n]; Print@n; p] In[740]:= sncprd[\[Pi]/2`22] During evaluation of In[740]:= 87 Out[740]= 0.48690721697316073847 In[741]:= sncprd[\[Pi]/2`33] // tim During evaluation of In[741]:= 1454 During evaluation of In[741]:= 0.095577,0 Out[741]= 0.48690721697316075167428625082 In[742]:= sncprd[\[Pi]/2`39] // tim During evaluation of In[742]:= 6751 During evaluation of In[742]:= 0.302979,0 Out[742]= 0.48690721697316075167428628848113831 In[743]:= sncprd[\[Pi]/2`44] // tim During evaluation of In[743]:= 24261 During evaluation of In[743]:= 1.299224,0 Out[743]= 0.486907216973160751674286288481949287233 but that's still zero bits/term. The integer printouts are number of terms until the cumulative product stops changing. Note that this is several digits short of the truth. The plot strongly resembles some band-limited, interpolated row of Pascal's triangle. It sucks that I can't attach a plot.