Solving an integral I find:
\begin{equation} I = \int \exp \left\{-N\left ( \epsilon r+\log(1+r) +\frac{rx^2(1-r\tau + \tau r^2 (\tau +1))}{1+r}+\frac{ry^2(-1+r\tau + \tau r^2 (\tau -1))}{1+r}\right) \right \} \end{equation}
where $-1<\tau<1$, $r>0$ and $x,y\in \mathbb{R}$.
However the correct result is:
\begin{equation} A = \int \exp \left\{-N\left ( \epsilon r+\log(1+r) +\frac{rx^2}{1+r(1+\tau)}+\frac{ry^2}{1+r(1-\tau)}\right)\right\} \end{equation}
When I compare the two expressions by plugging in numerical values I do not obtain the same. However when $\tau=0$ I do recover the correct result.
$$\boxed{\frac{rx^2}{1+r(1+\tau)}+\frac{ry^2}{1+r(1-\tau)}} \text{ and } \boxed{\frac{rx^2(1-r\tau + \tau r^2 (\tau +1))}{1+r}+\frac{ry^2(-1+r\tau + \tau r^2 (\tau -1))}{1+r}}$$
I do not see any mistakes in my computation and do not even know how to compare the two answers in a clever way. Is there a smart way to put everything on the same denominator and quickly see which terms are missing?
Any advice or remark is always appreciated. Thank you.
You don't need to transform the fractions to see that they are different. You just need to consider them as functions of $\tau$, for example.
Then you will see that forms: $$ \frac{A}{B+C\tau}\qquad\text{and}\qquad E+F\tau+G\tau^2 $$
cannot be the same.