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I am working with some dataset which looks very similar to the negative of $\tan^{-1}(x)$ to me:

enter image description here

The only thing is that this curve need not be "anti-symmetric" around zero. I have tried fitting functions of the form $-a\tan^{-1}(bx) + c$ but it doesn't always produce good fit because of the very fact that data can be asymmetric. Does anybody know any good function which would qualitatively look like negative $\tan^{-1}$ but is asymmetric about zero? Most probably, it would involve parameters which one can change to change the amount of asymmetry. Any ideas?

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    What do you mean by anti-symmetric? Later you say asymmetric? You should plot from -20 to 20 so we can see what you mean. – 2017-01-20
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    By that I mean that in general $f(-x) \neq =-f(x)$ – 2017-01-20
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    @SnehalShekatkar, is this your entire data set (except possible some high values)? If so, I don't think an arctangent is necessarily a great fit. – 2017-01-20

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Following your line of reasoining, you would have to define it as $f(x) = a - b \sigma (x)$. That means introducing 2 parameters and, in addition, choosing a sigmoid function $\sigma$ which doesn't have to be an arctan.

There is a good choice of sigmoids on https://en.wikipedia.org/wiki/Sigmoid_function#/media/File:Gjl-t(x).svg

However, a sigmoid is a natural choice only if we see a flat asymptote on both half-axes in the data. Your data show the flat asymptote only for positive values of $x$, from what I can see.

If these are all your data, I'm not sure I would use a sigmoid. I would try something like $f(x) = a + \frac{1}{1 + \exp\left( b x\right)}$, for positive $a$ and $b$.