Logarithmic expansion involving reciprocal powers

Shew that if $a{x}^2$ and $\frac{a}{x^2}$ are each less than unity, then $a\left(x^2+\frac{1}{x^2}\right)-\frac{a^2}{2}\left(x^4+\frac{1}{x^4}\right)+\frac{a^3}{3}\left(x^6+\frac{1}{x^6}\right)-\cdots ={\log }_e\left(1+a{x}^2+a^2+\frac{a}{x^2}\right)$.

H. S. Hall & S. R. Knight, Higher Algebra (1891) · Ch. XVII. Exponential and Logarithmic Series, Examples XVII, No. 11