Maple 9.5 gets int( arctan(x)^2,x ); 2 2 2 I arctan(x) 2 (1 + x I) -------------- - 2 I arctan(x) + 2 arctan(x) ln(1 + ----------) 2 2 (1 + x I) 1 + x 1 + ---------- 2 1 + x 2 (1 + x I) - polylog(2, - ----------) I 2 1 + x which ( without hand polish ) at least gives the same numerical value for the definite integral...but I can't make the square root of -1 disappear like the imaginary quantity it is... evalc( um ); 2 1/2 1/2 Pi 3 -polylog(2, - 1/2 - 1/2 I 3 ) I + -------- + 1/6 Pi ln(3) 108 2 - 1/18 I Pi
evalf( um );
-9 0.0568881662 - 0.1 10 I -r On Fri, 8 Oct 2004, R. William Gosper wrote:
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