On Saturday, March 8, 2014 4:22:30 AM UTC-10, clicl...@freenet.de wrote:
> My interest in Example 44 (like the similar but simpler 62, 64, 66, and > 118) was mostly in the asymmetric LN evaluation. Please fuse the ATAN's > in numbers 62, 64, 66 as you see fit; I find no continuity problems: > > ATAN(1 + SQRT(2)*SQRT(TAN(x))) - ATAN(1 - SQRT(2)*SQRT(TAN(x))) > = ATAN((TAN(x) - 1)/(SQRT(2)*SQRT(TAN(x)))) + pi/2 > > [...]
I revised the optimal antiderivatives for examples 62, 64, 66, 90, 91, 112 and 118 of Chapter 5 of the Timofeev integration test suite unifying arctangents as you suggested, and posted the results on the Rubi website at
Note that the derivative of each side of the above equation are equal, so the unified arctangents can be used for antiderivatives. However, the two sides are not always equal assuming standard CCW (counter-clockwise) continuity is used along the arctangent's branch cut on the imaginary line. Interestingly the equation IS everywhere valid in Derive, since Derive uses CW continuity on the arctangent's branch cut...