Using icdf() function with custom distribution

According to MathWorks here :

"User-defined custom distributions, created using files and function handles, are supported by the Statistics and Machine Learning Toolbox™ functions pdf, cdf, icdf, and mle, and the Statistics and Machine Learning Toolbox Distribution Fitter app."

Details of problem:

I have a rather complicated probability distribution that I am working with. I have provided a picture of the pdf and cdf below. I need to find the quantile function by inverting the cdf. I have already done this by writing a function handle for the cdf and then inverting it using fzero. This works but its slow and is not an ideal method for numerically inverting a cdf. Therefore, I believe this is not the best approach to take. On the other hand, the icdf function probably uses more appropriate techniques for inverting cdf's and I would like to use that instead.

My question:

If I understand the above statement from mathworks, I should be able to use the icdf function given a function handle for my distribution. I have tried to make this work but with no success. How can one create/define a custom distribution such as mine and use it with the statistics toolbox functions above? Any examples would be helpful. I can certainly provide my code if requested as well. Also, I do not need the ability to fit data using the distribution fitting tool in case that matters.

Thanks.