※exceptionは処理を行わないで上に投げてるだけです。

本体

[TestHTTP.cs]

using System;
using System.IO;
using System.Collections.Generic;
using System.Threading;
using System.Threading.Tasks;
using System.Net;
using System.Net.Http;
using System.Net.Http.Headers;

namespace Test {
  public partial class TestHTTP {
    // "http://proxy.foo.com:8080"
    public String proxyurl = null;

    public TestHTTP () {;}

    private Dictionary<String, String> defaultRequestHeaders =
      new Dictionary<string, string> {
        {@"User-Agent", @"Mozilla/4.0 (compatible; MSIE 6.0; Windows XP)"},
        {@"Accept-Language", @"ja-JP"}
      };
    protected CookieContainer cookieContainer = null;

    protected class SSHttpClientHandler : HttpClientHandler {
      public SSHttpClientHandler (String proxyurl) {
        if (proxyurl != null) {
          Proxy = new WebProxy (proxyurl);
          UseProxy = true;
        } else {
          Proxy = null;
          UseProxy = false;
        }
        // CookieContainer =
      }
    }

    protected class SendClientHandler : DelegatingHandler {
      public SendClientHandler () : base (new HttpClientHandler ()) {;}
      public SendClientHandler (HttpMessageHandler innerHandler) : base (innerHandler) {;}

      protected override async Task<HttpResponseMessage> SendAsync (HttpRequestMessage request, CancellationToken cancellationToken) {
        HttpResponseMessage response = await base.SendAsync (request, cancellationToken).ConfigureAwait (false);
        return response;
      }
    }

    private TimeSpan waittime = Timeout.InfiniteTimeSpan;
    //TimeSpan.FromSeconds (800.0);

    protected HttpResponseHeaders GetHead (Uri uri, Dictionary<string, string> send, CancellationTokenSource cancellationTokenSource) {
      using (HttpClientHandler handler = new SSHttpClientHandler (proxyurl)) {
        using (HttpClient client = new HttpClient (new SendClientHandler (handler))) {
          foreach (KeyValuePair<String, String> h in defaultRequestHeaders)
            client.DefaultRequestHeaders.Add (h.Key, h.Value);
          client.Timeout = waittime;
          client.MaxResponseContentBufferSize = 256000;

          HttpRequestMessage request = new HttpRequestMessage ();
          request.Method = HttpMethod.Head;
          request.RequestUri = uri;

          if (send != null)
            request.Content = new FormUrlEncodedContent (send);

          try {
            Task<HttpResponseMessage> hTask = client.SendAsync (request, cancellationTokenSource.Token);
            hTask.Wait ();
            cookieContainer = handler.CookieContainer;
            return hTask.Result.Headers;
          } catch (HttpRequestException e) {
            Exception ex = e;
            while (ex != null) {
              ex = ex.InnerException;
            }
            throw;
          } catch (TaskCanceledException e) {
            throw;
          } catch (AggregateException e) {
            Exception ex = e;
            while (ex != null) {
              ex = ex.InnerException;
            }
            throw;
          }
        }
      }
    }

    protected async Task<String> WebAsync (Uri uri, HttpMethod method, Dictionary<string, string> send, CancellationTokenSource cancellationTokenSource) {
      using (HttpClientHandler handler = new SSHttpClientHandler (proxyurl)) {
        using (HttpClient client = new HttpClient (new SendClientHandler (handler))) {
          foreach (KeyValuePair<String, String> h in defaultRequestHeaders)
            client.DefaultRequestHeaders.Add (h.Key, h.Value);
          client.Timeout = waittime;
          client.MaxResponseContentBufferSize = 256000;

          HttpRequestMessage request = new HttpRequestMessage ();
          request.Method = method;
          request.RequestUri = uri;

          if (send != null)
            request.Content = new FormUrlEncodedContent (send);

          try {
            Task<HttpResponseMessage> hTask = client.SendAsync (request, cancellationTokenSource.Token);
            hTask.Wait ();
            HttpResponseMessage response = hTask.Result;
            cookieContainer = handler.CookieContainer;
            if (response.IsSuccessStatusCode)
              using (StreamReader reader = new StreamReader (await response.Content.ReadAsStreamAsync ()))
                return reader.ReadToEndAsync ().Result;
          } catch (HttpRequestException e) {
            Exception ex = e;
            while (ex != null) {
              ex = ex.InnerException;
            }
            throw;
          } catch (TaskCanceledException e) {
            throw;
          } catch (AggregateException e) {
            Exception ex = e;
            while (ex != null) {
              ex = ex.InnerException;
            }
            throw;
          }
        }
      }
      return null;
    }

    public HttpResponseHeaders GetWebHead (Uri uri, Dictionary<string, string> send, CancellationTokenSource cancellationTokenSource) {
      return GetHead (uri, send, cancellationTokenSource);
    }

    public async Task<String> GetWebAsync (Uri uri, Dictionary<string, string> send, CancellationTokenSource cancellationTokenSource) {
      return await WebAsync (uri, HttpMethod.Get, send, cancellationTokenSource);
    }

    public async Task<String> PostWebAsync (Uri uri, Dictionary<string, string> send, CancellationTokenSource cancellationTokenSource) {
      return await WebAsync (uri, HttpMethod.Post, send, cancellationTokenSource);
    }
  }
}

呼び出し側

private System.Windows.Forms.TextBox log;
private System.Windows.Forms.Button getweb;
private System.Windows.Forms.Button head;

[Test.cs]

    private void getweb_Click (Object sender, EventArgs e) {
      TestHTTP http = new TestHTTP ();
      // Uri uri = new Uri (@"http://www.google.co.jp/");
      Uri uri = new Uri (@"http://www.yahoo.co.jp/");
      String r = http.GetWebAsync (uri, null, new CancellationTokenSource ()).Result;
      LogOut (r);
    }

    private void head_Click (Object sender, EventArgs e) {
      TestHTTP http = new TestHTTP ();
      Uri uri = new Uri (@"http://www.yahoo.co.jp/");
      HttpResponseHeaders r = http.GetWebHead (uri, null, new CancellationTokenSource ());
      LogOut (r.ToString ());
    }