When writing an essay or article, it is essential to use different words and phrases to avoid repetition and enhance clarity. Some synonyms for "according to claim" include as stated by, as alleged by, as asserted by, as purported by, as contended by, and as maintained by. These phrases can be used interchangeably to convey the idea that a particular claim is being made by someone or something. Additionally, phrases such as in accordance with, consistent with, and in line with can also convey similar meanings and add variety to your writing. Varying your language use will make your writing more engaging and professional.