You just need to add the Tesla.Middleware.FormUrlencoded plug to your pipeline, then specify a map for the body of the request:

request_body = %{
  "user[email]" => "link@hotmail.com",
  "user[cellphone]" => 12345,
  "user[country_code]" => 49
}

So, something like this:

defmodule MyClient do

  use Tesla

  plug Tesla.Middleware.BaseUrl, "https://api.mysite.com" 
  plug Tesla.Middleware.FormUrlencoded

  def go do

    request_body = %{
      "user[email]" => "link@hotmail.com",
      "user[cellphone]" => 12345,
      "user[country_code]" => 49
    }
    path = "/protected/json/new"

    {:ok, response} = post(path, request_body)


  end

end

On the server side, I see the post body:

user%5Bcellphone%5D=12345&user%5Bcountry_code%5D=49&user%5Bemail%5D=link%40hotmail.com

That is different than the post body that your curl command produces:

user[email]=link@hotmail.com&user[cellphone]=12345&user[country_code]=49

However, the post body produced by curl does not conform to the spec for the Content-Type application/x-www-form-urlencoded, which is what the -d flag automatically sets as the Content-Type header in the curl post request:

 -d, --data <data>
        (HTTP) Sends the specified data in a POST request  to  the
         HTTP  server,  in  the same way that a browser does when a
         user has filled in an HTML form  and  presses  the  submit
         button.  This  will  cause  curl  to  pass the data to the
         server  using  the content-type application/x-www-form-urlencoded. 

The spec (see 17.13.4 Form content types) requires that all non letter/number characters, i.e. all non alphanumeric characters, be escaped with % character codes:

application/x-www-form-urlencoded
…Forms submitted with this content type must be encoded as follows:

1. …names and values are escaped. Space characters are replaced by +, and then reserved characters are escaped as described in RFC1738, section 2.2: Non-alphanumeric characters are replaced by %HH, a percent sign and two hexadecimal digits representing the ASCII code of the character. Line breaks are represented as “CR LF” pairs (i.e., %0D%0A).
2. The control names/values are listed in the order they appear in the document. The name is separated from the value by = and name/value pairs are separated from each other by &.

curl does allow you to specify that you want the post data to be percent encoded using the --data-urlencode flag instead of -d , but that still doesn’t produce the same post body as tesla. This is what I get:

user[email]=link%40hotmail.com%26user%5Bcellphone%5D%3D12345%26user%5Bcountry_code%5D%3D49

For some reason, curl does not escape the brackets everywhere as can be seen in user[email]. Confounding. Edit: Ah. To get curl to percent encode the leading name in the data, you have to prepend the data with an equals sign:

--data-urlencode '=user[email]=link@hotmail.com&user[cellphone]=12345&user[country_code]=49'

and the server receives the post body:

user%5Bemail%5D%3Dlink%40hotmail.com%26user%5Bcellphone%5D%3D12345%26user%5Bcountry_code%5D%3D49

Unfortunately, that still does not conform to the spec: curl replaces the equal signs and the ampersands with percent escapes, which confuses the server and the server can’t extract the key/value pairs. Compare to tesla:

user%5Bcellphone%5D=12345&user%5Bcountry_code%5D=49&user%5Bemail%5D=link%40hotmail.com

Tesla get’s it right and curl doesn’t.