Mma refuses to try on my "new" laptop, but Derive, Macsyma, and Maple all fail. After a good deal of hand polishing (or maybe hand wringing), / 2 | 2 log(x + 1) | atan (x) dx = atan(x) (- ----------- + x atan(x) + log(2)) | 2 / %i x 1 - imagpart(li (---- + -)). 2 2 2 Note that when x is tan(pi*rational), the dilog dehisces, and d ==== %pi n \ j j 2 %pi j n tan(-----) > (- 1) Psi (-) sin(---------) d / 1 d d / ==== | 2 j = 1 | atan (x) dx = ----------------------------------- | 2 / d 0 %pi n %pi n sec(-----) %pi n tan(-----) d d %pi n (2 log(----------) - ----------------) 2 d - --------------------------------------------, d -d/2 < n < d/2, where Psi_1 is the trigamma function. In particular, 1 / 2 | 2 %pi log(2) %pi | atan (x) dx = ---------- + ---- - %catalan ~ 0.2452812034667; | 4 16 / 0 1 ------- sqrt(3) 1 / sqrt(3) Psi (-) 2 | 2 %pi log(3) 1 3 sqrt(3) %pi | atan (x) dx = ---------- - --------------- + ------------ | 6 9 12 / 0 ~ .0568881659798, --rwg