Let H be a hyperexponential function of x over a field K of characteristic 0. The MultiplicativeDecomposition[i](H,x) calling sequence constructs the ith multiplicative decomposition for H, $i=\left\{1\,2\right\}$.

If the MultiplicativeDecomposition command is called without an index, the first multiplicative decomposition is constructed.

A multiplicative decomposition of H is a pair of rational functions $F,V$ such that $H\left(x\right)=V\left(x\right){\ⅇ}^{\int F\left(x\right)\ⅆx}$. Let R be the rational certificate of H, i.e., $R=\frac{\frac{\ⅆ}{\ⅆx}H\left(x\right)}{H\left(x\right)}$. Let $F,V$ be a differential rational normal form of R. Then $F,V$ is a multiplicative decomposition of H. Hence, each differential rational normal form $F,V$ of the certificate R of H is also a multiplicative decomposition of H.

The construction of MultiplicativeDecomposition[i](H,x) is based on ${\mathrm{RationalCanonicalForm}}_i\left(\frac{\frac{\ⅆH}{\ⅆx}}{H}\,x\right)$, for $i=1,2$.

The output is of the form $V\left(x\right){\ⅇ}^{\int F\left(x\right)\ⅆx}$ where V and F are rational function of x over K.

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

$R≔\frac{4}{x-2}+\frac{4}{x+1}-\frac{3}{{\left(x+1\right)}^2}-\frac{9}{{\left(x-1\right)}^2}-\frac{9x^2+12}{x^3+4x-2}+\frac{1}{{\left(x^3+4x-2\right)}^2}$

$R≔\frac{4}{x-2}+\frac{4}{x+1}-\frac{3}{{\left(x+1\right)}^2}-\frac{9}{{\left(x-1\right)}^2}-\frac{9x^2+12}{x^3+4x-2}+\frac{1}{{\left(x^3+4x-2\right)}^2}$

$H≔\exp \left(\mathrm{Int}\left(R\,x\right)\right)$

$H≔{\ⅇ}^{\int \left(\frac{4}{x-2}+\frac{4}{x+1}-\frac{3}{{\left(x+1\right)}^2}-\frac{9}{{\left(x-1\right)}^2}-\frac{9x^2+12}{x^3+4x-2}+\frac{1}{{\left(x^3+4x-2\right)}^2}\right)\ⅆx}$

$\mathrm{MultiplicativeDecomposition}\left[1\right]\left(H\,x\right)$

$\frac{{\left(x+1\right)}^4{\left(x-2\right)}^4{\ⅇ}^{\int \frac{-12x^8-12x^7-108x^6-48x^5-239x^4+48x^3-50x^2+144x-47}{{\left(x+1\right)}^2{\left(x-1\right)}^2{\left(x^3+4x-2\right)}^2}\ⅆx}}{{\left(x^3+4x-2\right)}^3}$

$\mathrm{MultiplicativeDecomposition}\left[2\right]\left(H\,x\right)$

${\left(x-2\right)}^4{\ⅇ}^{\int \frac{-5x^9-16x^8-14x^7-134x^6+39x^5-331x^4+96x^3+32x^2+16x-7}{{\left(x+1\right)}^2{\left(x-1\right)}^2{\left(x^3+4x-2\right)}^2}\ⅆx}$

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