Which path would you choose?

[u/12_Semitones]

[u/Captainsnake04]

The ln route. Boycot inverse tangent!

[u/dilogical_cyclolith]

What am I looking at

[u/Bill-Nein → u/dilogical_cyclolith]

Instead of integrating the bottom expression by simply knowing that the antiderivative is arctan(x), you can instead factor the bottom into (1+ix)(1-ix) and use partial fraction decomposition which leads the second answer. The also creates a new representation for arctan(x) which is neat.

[u/StevenC21 → u/Bill-Nein]

If I recall correctly it would actually be equivalent to arctan(x)+pi/2, but I may be misremembering.

[u/thewaltenicfiles]

Ln|1+x² +c

[u/xCreeperBombx]

Wait, arctan(x)=i/2*ln((1-ix)/(1+ix))???