\[ \int \cos ^{-1}\left (\sqrt {\frac {x}{1+x}}\right ) \, dx \]
Optimal antiderivative \[ \left (1+x \right ) \left (\arccos \left (\sqrt {\frac {x}{1+x}}\right )+\sqrt {\frac {1}{1+x}}\, \sqrt {\frac {x}{1+x}}\right ) \]
command
integrate(acos((x/(1+x))**(1/2)),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ x \operatorname {acos}{\left (\sqrt {\frac {x}{x + 1}} \right )} - 2 \left (\begin {cases} - \frac {\sqrt {\frac {x}{x + 1}}}{2 \sqrt {- \frac {x}{x + 1} + 1}} + \frac {\operatorname {asin}{\left (\sqrt {\frac {x}{x + 1}} \right )}}{2} & \text {for}\: \sqrt {\frac {x}{x + 1}} > -1 \wedge \sqrt {\frac {x}{x + 1}} < 1 \end {cases}\right ) \]
Sympy 1.8 under Python 3.8.8 output
\[ \int \operatorname {acos}{\left (\sqrt {\frac {x}{x + 1}} \right )}\, dx \]________________________________________________________________________________________