More indefinite integrals from Riccati equations

Two new methods for obtaining indefinite integrals of a special function using Riccati equations are presented. One method uses quadratic fragments of the Riccati equation, the solutions of which are given by simple quadratic equations. This method is applied to cylinder functions, parabolic cylinder functions and the general solution of the Mathieu equation. No such indefinite integrals for general Mathieu functions seem to have been presented previously. The second method obtains indefinite integrals by assuming simple algebraic forms involving constants for the Riccati variable and then choosing the values of these constants to give simple and interesting integrals. This method is illustrated here for cylinder functions and Associated Legendre functions. All integrals obtained have been checked using Mathematica.