Elegant way to show $\frac{\cosh(z)}{4z^3-z}$ is holomorphic?

Question

We are calculating integrals. To use Cauchy's Theorem for punctured domains I need to show that $$f(z)=\frac{\cosh(z)}{4z^3-z}$$ is holomorphic. (The integral to be calculated is over the disk around $0$ with $r=\frac{1}{4}$)

I've tried to reshape $f$ to $f(x,iy)=u(x,y)+iv(x,y)$ to use the CR-Equations. However both using polar coordinates and just straight algebra I get a huge mess, massive sums with 6+ terms.

Is there an algebraic trick or some other way to show that f is holomorphic?

Answer 1

Since it is the quotient of two holomorphic functions, it is holomorphic.