Hi friends,
Few days back I was facing difficulty in integrating recaptcha with rails so once after integrated it and I though it would be helpful for you guyz, if I mention it step-by-step as I did.

1. Open your Gemfile and add this code:
 

gem "recaptcha", require: "recaptcha/rails"

2. Now run bundle

bundle install

3. Then I inherited devise registration controller and overridden the create method of it.

class User::RegistrationsController < Devise::RegistrationsController
  #
  # Other Codes

   def create
    ## To build the resource
    build_resource(sign_up_params)   
    ## Verifying Captcha
    if verify_recaptcha(model: resource) 
      super
    else
      render 'new'
    end
  end

  #
  # Other Codes
end

4. Now you need to get the public and secret key by registering into Google recaptcha account.

The link is https://www.google.com/recaptcha/admin

5. Now add the configurations for recaptcha at /config/initializer/recaptcha.rb

Recaptcha.configure do |config|
   config.public_key  = 'Replace with your public key'
   config.private_key = 'Replace with private key'
end

6. Add the recaptcha tags to app/views/devise/registrations/new

<div><%= recaptcha_tags %></div>

7. Now we need to tell the route to go to our registration controller instead of devise registration controller. So routes needs to be modified to:

devise_for :users, :controllers => {:registrations => "user/registrations"}

Thus you are now ready to use recaptcha with devise in your rails application.