Approximation of the error function

Calculus Level 5

As it turns out, a good approximation for the error function $$\textrm{erf}(x)$$ is the function $$\tanh\left(\dfrac{2x}{\sqrt{\pi}}\right)$$.

How good of an approximation? Tell me yourself by finding the total area bounded by these two functions. That is, find the value of:

$\int_{-\infty}^{\infty} \left|\textrm{erf}(x) -\tanh\left(\frac{2x}{\sqrt{\pi}}\right)\right| dx$

Please round to 3 decimal places.

Note: $$\displaystyle \textrm{erf}(x)= \dfrac{2}{\sqrt{\pi}}\int_{0}^{x} e^{-t^2} dt$$ and $$\tanh(x)= \dfrac{\sinh(x)}{\cosh(x)}$$.

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