This is mechanized in Mathematica:

```
CDF[TransformedDistribution[x1/(x1 + x2 + x3),{x1\[Distributed]UniformDistribution[{0, 1}],
x2\[Distributed]UniformDistribution[{0, 1}],x3\[Distributed] UniformDistribution[{0, 1}]}],t]
```

$$\begin{array}{cc}
\{ &
\begin{array}{cc}
1 & t>1 \\
-\frac{t}{t-1} & 0<t\leq \frac{1}{3} \\
\frac{5 t^2+2 t-1}{6 t^2} & \frac{1}{2}<t\leq 1 \\
\frac{21 t^3-27 t^2+9 t-1}{6 (t-1) t^2} & \frac{1}{3}<t\leq \frac{1}{2} \\
\end{array}
\\
\end{array} $$

In the case $n=4$ Mathematica produces

$$\begin{array}{cc}
\{ &
\begin{array}{cc}
1 & t\geq 1 \\
-\frac{3 t}{2 (t-1)} & 0<t\leq \frac{1}{4} \\
\frac{25 t^3-3 t^2+3 t-1}{24 t^3} & \frac{1}{2}\leq t<1 \\
\frac{-23 t^4+68 t^3-66 t^2+20 t-2}{24 (t-1) t^3} & \frac{1}{3}\leq t<\frac{1}{2} \\
\frac{220 t^4-256 t^3+96 t^2-16 t+1}{24 (t-1) t^3} & \frac{1}{4}<t<\frac{1}{3} \\
\end{array}
\\
\end{array} $$