\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]

↓

\[\tan^{-1} \left(\sqrt{\sqrt[3]{\frac{1 - x}{x + 1} \cdot \left(\frac{1 - x}{x + 1} \cdot \frac{1 - x}{x + 1}\right)}}\right) \cdot 2\]

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Derivation

Initial program 0.0
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
Initial simplification0.0
\[\leadsto \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right) \cdot 2\]
Using strategy rm
Applied add-cbrt-cube0.0
\[\leadsto \tan^{-1} \left(\sqrt{\color{blue}{\sqrt[3]{\left(\frac{1 - x}{1 + x} \cdot \frac{1 - x}{1 + x}\right) \cdot \frac{1 - x}{1 + x}}}}\right) \cdot 2\]
Final simplification0.0
\[\leadsto \tan^{-1} \left(\sqrt{\sqrt[3]{\frac{1 - x}{x + 1} \cdot \left(\frac{1 - x}{x + 1} \cdot \frac{1 - x}{x + 1}\right)}}\right) \cdot 2\]

herbie shell --seed 2018254 
(FPCore (x)
  :name "arccos"
  (* 2 (atan (sqrt (/ (- 1 x) (+ 1 x))))))