Can I apply for my citizenship or do I have to wait 5 years after the case was dismissed?

A charge of a criminal offense, dismissed without a conviction, may not be problematic. Depending upon all of the details of the court's disposition ending in dismissal, the arrest for the offense of rape may not stand in the way of eligibility to apply to become a naturalized citizen. There really is no substitute for engaging an immigration attorney and allowing him/her to review court-certified copies of all of the records relating to the criminal case - that would include the police report, the charging document (indictment, accusation or similar document, your plea in court, the court's decision and all papers regarding compliance with anything the court may have required prior to its dismissal of the case. Among other things, the attorney will be able to confirm that not only was there no conviction, but there also was not acknowledgement of the elements of the crime that the USCIS may treat the same as a conviction. Some immigration law firms, including mine, offer legal services on a "flat fee" basis so that a client will know the total expense from the very beginning, and a few immigration law firms, including mine, offer an initial consultation free of charge.