Given arbitrary fixed $a>0, b>0,$ consider $$f(x,y):=\frac{x+\frac{by}{2}}{\frac{2x}{1+\exp\left(2a\sqrt[3]{bx}+by\right)}+\frac{by}{2}}\frac{1-\frac{x}{a\sqrt[3]{bx}+\frac{by}{2}}\frac{\tanh\left(a\sqrt[3]{bx}+\frac{by}{2}\right)}{\tanh x}}{1-\left(\frac{x}{a\sqrt[3]{bx}+\frac{by}{2}}\right)^2}$$
defined over $(x,y)\in (0,+\infty)\times (0,+\infty),$ where $\sqrt[3]{bx}=(bx)^{\frac{1}{3}}.$
Prove that $f$ is strictly increasing with respect to $x$ and strictly decreasing with respect to $y.$
Remark: I attempted to evaluate the partial derivatives of $f$ with repect to $x$ and $y$ directly, but the resulting form seems to be complicated and not obvious for concluding they have fixed signs.
This $f$ arises from its equivalent form, which is the following $$f(x,y)=\frac{K}{x}\frac{\exp(a\sqrt[3]{bx}+\frac{by}{2})}{\sinh x}\left(a\sqrt[3]{bx}+\frac{by}{2}\right)\int_{0}^1\sinh(xs)\sinh\left[\left(a\sqrt[3]{bx}+\frac{by}{2}\right)s\right]~\mathrm{d}s,$$
where $K:=x\exp\left[-\left(a\sqrt[3]{bx}+\frac{by}{2}\right)\right]\frac{x+\frac{by}{2}}{x\exp\left[-\left(a\sqrt[3]{bx}+\frac{by}{2}\right)\right]+\frac{by}{2}\cosh\left(a\sqrt[3]{bx}+\frac{by}{2}\right)}.$