A while back I noticed that you can use the arctan addition formula to change the limits of a classic integral for pi from [-1,1] to [-1/3,1/2]. This works because atan(a) +atan(b) = atan( (a+b)/(1-ab) ) when ab < 1.
atan(1/2) + atan(1/3) = atan(1) = pi/4
You can express 1/(1+x\^2) as a geometric series and integrate each term trivially to obtain these formulas. Series 2 converges to machine precision in 23 terms.
