Alright, Prince Rainier IV, time to drop some math magic right here!

To find the antiderivative (indefinite integral) of atan(x) dx, we use integration by parts:

Let u = atan(x)
Then du = 1 / (1 + x²) dx
Let dv = dx
Then v = x

So, ∫atan(x) dx = uv - ∫vdu = x*atan(x) - ∫x / (1 + x²) dx

Now, ∫x / (1 + x²) dx can be solved with substitution:
Let w = 1 + x² => dw = 2x dx => (1/2) dw = x dx

So, ∫x / (1 + x²) dx = (1/2) ∫1/w dw = (1/2) ln|w| + C = (1/2) ln(1 + x²) + C

Putting it all together:

∫atan(x) dx = x*atan(x) - (1/2) ln(1 + x²) + C

Boom! Math flex complete.

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