\int{\frac{x^3}{a^2-x^2}}dx=-\int{\frac{x(a^2-x^2)}{a^2-x^2}}dx+\int{\frac{a^2x}{a^2-x^2}}dx=-\int{x}dx+a^2\int{\frac{xdx}{a^2-x^2}}=-\frac{x^2}{2}+a^2\int{\frac{xdx}{a^2-x^2}}