Hi
I have been doing same on my Linux laptop. There I have used cURL program which can load a web form, fill in forms and POST/GET it back to authentication web server.
So on your phone you should create a Tasker job to test first any web page (like Google). And if it is not page you (actually a Tasker) were looking for then it could be a redirected to login page.
Then you should load that page with your laptop by using Perl script named "formfind". That would tell what fields you should be filling and if that is POSTed or GETed to web page.
And depending this authentication page then Tasker should fill this information by using POST or GET.
It requires you to understand how HTTP protocol works and how forms are filled. And I do not know if Tasker can dig information from web page you do fill manually at first time and later use it on automated login.
If you visit regularly on same WiFi then you could dig all this information manually (like with laptop) and create own Tasker POST or GET script to be used. I can help you if you get output from formfind into your login page. On Linux run command:
That would output all fields to be filled on your HTTP POST or GET.
Regards, Henri