Let $\mathrm{H1}\,\mathrm{H2}$ be hyperexponential functions of x over a field K of characteristic 0. The AreSimilar(H1,H2,x) command returns true if $\mathrm{H1}\left(x\right)$ and $\mathrm{H2}\left(x\right)$ are similar. Otherwise, it returns false.

H1 and H2 are similar if their ratio can be written as the product of a rational function and a constant in some extension of K.

$\mathrm{with}\left(\mathrm{DEtools}\right)\:$

$H≔\frac{{\ⅇ}^{\∫\frac{2x-7}{{\left(x\+4\right)}^2}\ⅆx}\left(x^6\+16x^5\+103x^4\+327x^3\+647x^2\+737x\+194\right)}{{\left(x-1\right)}^2{\left(x\+2\right)}^4{\left(x\+4\right)}^2}$

$H≔\frac{{\ⅇ}^{\∫\frac{2x-7}{{\left(x\+4\right)}^2}\ⅆx}\left(x^6\+16x^5\+103x^4\+327x^3\+647x^2\+737x\+194\right)}{{\left(x-1\right)}^2{\left(x\+2\right)}^4{\left(x\+4\right)}^2}$

$\mathrm{H1}\,\mathrm{H2}≔\mathrm{ReduceHyperexp}\left(H\,x\right)\:$

$\mathrm{H1}$

$-\frac{\left(24x^3\+143x^2\+292x\+216\right){\ⅇ}^{\∫\left(-\frac{15}{{\left(x\+4\right)}^2}\right)\ⅆx}}{{\left(x\+2\right)}^3\left(x-1\right)}$

$\mathrm{H2}$

$\frac{\left(x^3\+17x^2\+88x-231\right){\ⅇ}^{\∫\frac{-23-2x}{{\left(x\+4\right)}^2}\ⅆx}}{x-1}$

$\mathrm{AreSimilar}\left(H\,\mathrm{H2}\,x\right)$

$\mathrm{true}$

$\mathrm{AreSimilar}\left(\mathrm{H1}\,\mathrm{H2}\,x\right)$

$\mathrm{true}$

Geddes, Keith; Le, Ha; and Li, Ziming. "Differential rational normal forms and a reduction algorithm for hyperexponential functions." Proceedings of ISSAC 2004. ACM Press. (2004): 183-190.