<s>[INST] Is there any simpler form of this function Assume that $n$ is a positive integer. Is there any simple form of this hypergeometric value $$_2\mathrm{F}_1\left[\frac{1}{2},1,\frac{3+n}{2},-1\right]?$$
 [/INST] write $F(n)$ for your formula.  Then
$$
F \left( 0 \right) =\frac{1}{4}\,\pi \\
F \left( 2 \right) =-\frac{3}{2}+\frac{3}{4}\,\pi \\
F \left( 4 \right) =-5+{\frac {15}{8}}\,\pi \\
F \left( 6 \right) =-{\frac {77}{6}}+{\frac {35}{8}}\,\pi \\
F \left( 8 \right) =-30+{\frac {315}{32}}\,\pi \\
F \left( 10 \right) =-{\frac {671}{10}}+{\frac {693}{32}}\,\pi 
$$
and
$$
F \left( 1 \right) =-2+2\,\sqrt {2}\\
F \left( 3 \right) =-{\frac {20}{3}}+\frac{16}{3}\,\sqrt {2}\\
F \left( 5 \right) =-{\frac {86}{5}}+{\frac {64}{5}}\,\sqrt {2}\\
F \left( 7 \right) =-{\frac {1416}{35}}+{\frac {1024}{35}}\,\sqrt {2}\\
F \left( 9 \right) =-{\frac {5734}{63}}+{\frac {4096}{63}}\,\sqrt {2}\\
F \left( 11 \right) =-{\frac {46124}{231}}+{\frac {32768}{231}}\,
\sqrt {2}
$$
Now, can you guess the patterns?
</s>