Indefinite integrals of some special functions from a new method

A substantial number of indefinite integrals of special functions are presented, which have been obtained using a new method presented in a companion paper [Conway JT. A Lagrangian method for deriving new indefinite integrals of special functions. Integral Transforms Spec Funct. 2015; submitted to]. The method was originally derived from the Euler–Lagrange equations but an elementary proof is also presented in [Conway JT. A Lagrangian method for deriving new indefinite integrals of special functions. Integral Transforms Spec Funct. 2015; submitted to]. Sample results are presented here for Bessel functions, Airy functions and hypergeometric functions. More extensive results are given for the complete elliptic integrals of the first and second kinds. Results are presented which link integrals of some products of associated Legendre functions and the complete elliptic integral of the second kind with the Golden Ratio. The method is applicable to any elementary or special function which satisfies a linear ordinary differential equation of the second order.