For the integration of SMS gateway in your website I would recommend you Amazon SNS
which will not only provide a highly reliable service but also provide the service at a very affordable rate as compared to the rest of others. At the time of writing this post amazon charges only $0.00645(Rs 0.46) per sms to USA and $0.00223(Rs 0.16) per sms to India. For a full list of rates depending on the country you can click here
As a 2nd alternative I would recommend you Twilio which is also a well known company to offer such services. Current rate to USA is $ 0.0075(Rs0.53) per sms and to India is $ 0.0100(Re 0.71) per sms. For a full list of rates depending on the country you can click here .