If I attempt to simplify the sum
ArcTan[1/x] + ArcTan[x] with
ArcTan[ Simplify[ TrigExpand[ Tan[ ArcTan[1/x] + ArcTan[x]]]]]
During evaluation of In:= Power::infy: Infinite expression 1/0 encountered. >> During evaluation of In:= Power::infy: Infinite expression 1/0 encountered. >> During evaluation of In:= Infinity::indet: Indeterminate expression ComplexInfinity + ComplexInfinity encountered. >> Indeterminate
even though the result is $\pm\pi/2$. The error messages occur during the call to
TrigExpand, and I can't think of a way to avoid them. Also, I tried in vain the assumption
x > 0, so that the result is not ambiguous.
The full expression to simplify is
(1/(16 a))Q^2 (-16 ArcTan[r1/a] + 1/(2 r1^2)(16 \[Pi] r1^2 + 32 r1 a + \[Pi] a^2 + 8 \[Pi] r1^2 Log[r1] + 4 ArcTan[a/r1] (7 a^2 - 8 r1^2 Log[r1]) - 2 ArcTan[r1/(2 a) - a/(2 r1)] (a^2 + 8 r1^2 Log[r1])))
I know that the
Logs vanish after simplification and I was trying to prove this with Mathematica. It is possible if I split the expression into several parts, then simplify with the transformation rule for the addition of
ArcTan, because none of the other methods work in this case, but the transformation rule aborts when it meets an infinite expression.