$$ F(x)+x=x\exp{\left(\sum _{k>0}\frac{H(x^k)}{k}\right)}\qquad\qquad (1)\\ F'(x)+1=\exp{\left(\sum _{k>0}\frac{H(x^k)}{k}\right)}+x\left(\sum_{k>0}x^{k-1}H'(x^k)\right)\exp{\left(\sum _{k>0}\frac{H(x^k)}{k}\right)}\\ F'(x)+1=\left(1+\sum_{k>0}x^{k}H'(x^k)\right)\exp{\left(\sum _{k>0}\frac{H(x^k)}{k}\right)}\qquad (2)\\ $$
$\frac{(2)}{(1)}$： $$ \frac{F'(x)+1}{F(x)+x}=\frac{1+\sum_{k>0}x^{k}H'(x^k)}{x}\\ xF'(x)+x=(x+F(x))\left(1+\sum_{k>0}x^{k}H'(x^k)\right)\\ xF'(x)=F(x)+(F(x)+x)\sum_{k>0}x^{k}H'(x^k) $$