Cumulative distribution function for $ n $ degrees of freedom
(see mathworld.wolfram.com):
$$
F(x) = \int_{-\infty}^x,f(y),dy
= \frac{1}{2},
+,\frac{1}{2},sgn(x),
\left[ I\left(1,\frac{n}{2},\frac{1}{2}\right)
I\left(\frac{n}{n+y^2}, \frac{n}{2},\frac{1}{2}\right)\right]
$$
where $ I(z; a, b) $ is the regularized incomplete beta function.
Cumulative Student t-distribution
Cumulative distribution function for $ n $ degrees of freedom (see mathworld.wolfram.com): $$ F(x) = \int_{-\infty}^x,f(y),dy = \frac{1}{2}, +,\frac{1}{2},sgn(x), \left[ I\left(1,\frac{n}{2},\frac{1}{2}\right)