This paper presents explicit closed-form solutions and computational methods for the indefinite integral with positive integers . By extending partial fraction decomposition to higherorder poles through complex factorization, we derive a reduction formula that recursively resolves cases to the fundamental solution expressed in logarithmic-arctangent form. Singularity analysis reveals connections to elliptic functions for fractional exponents, validated through Padé approximants and nonlinear ODEs. The methodology provides efficient computation of symmetric rational integrals with applications in nonlinear systems, demonstrated through comprehensive examples.
Tat Leung Yee (Sat,) studied this question.