As Henning Makholm pointed out, the ANF is unique. I will illustrate how you determine the ANF given the truth table for the function
for the case of three variables.
The ANF of $f(x_1,x_2,x_3)$ is
$$\begin{align*}
f(x_1,x_2,x_3) &= a_0\\
&\quad \oplus a_1x_1 \oplus a_2x_2 \oplus a_3x_3\\
&\quad \oplus a_{12}x_1x_2 \oplus a_{13}x_1x_3 \oplus a_{23}x_2x_3\\
&\quad
\oplus a_{123}x_1x_2x_3
\end{align*}$$
and we need to determine the $a$'s. We proceed as follows (working exactly
in reverse order of the method outlined by Henning) by determining
$a_{123}$, the coefficient of highest degree term, first and working our
way downwards.
$a_{123}$ equals the parity of the Hamming weight of the function.
That is, $a_{123}=1$ if the column labeled $f$ in the truth table has an
odd number of $1$s in it, and $a_{123}=0$ if the column labeled $f$ in
the truth table has an even number of $1$s in it.
We obtain $a_{12}$ by considering the four rows corresponding to $x_3=0$
as the truth table of a function
$$\begin{align*}f_{12}(x_1,x_2) &= f(x_1,x_2,0)\\
&= a_0\\
&\quad \oplus a_1x_1 \oplus a_2x_2\\
&\quad \oplus a_{12}x_1x_2
\end{align*}$$
and applying the same technique as in the previous step.
$a_{12}$ is the parity of the Hamming weight of the
column labeled $f_{12}$ in this shorter truth table.
Similarly, we obtain $a_{13}$ and $a_{23}$ by considering
truth tables for $f_{13}(x_1,x_3) = f(x_1,0,x_3)$ and
$f_{23}(x_2,x_3) = f(0,x_2,x_3)$ respectively
and finding the parity of the Hamming weights of the reduced functions.
Similarly, we obtain $a_{1}$, $a_{2}$, and $a_3$ by considering
truth tables for $f_{1}(x_1) = f(x_1,0,0)$,
$f_{2}(x_2) = f(0,x_2,0)$, and $f_{3}(x_3) = f(0,0,x_3)$ respectively
and finding the parity of the Hamming weights of the reduced functions
Finally, $a_0 = f(0,0,0)$ can be just read off the truth table.
It is, of course possible to proceed in opposite order,
but the method outlined above is related to Reed-Muller decoding
for which you might find software available. Look also for
canonical ring sum expansion.