The LinearOperators[IntegrateSols] function performs "accurate integration". That is, it solves the following problem. Let y satisfy L(y)=0 and g satisfy delta(g)=y, where delta means the usual derivative in the differential case and the first difference in the shift case. The routine builds an annihilator S for g of the same degree as that of L, and an operator K such that g=K(y) if both exist. Otherwise, it returns NULL.

An Ore operator is a structure that consists of the keyword OrePoly with a sequence of coefficients starting with the one of degree zero. The coefficients must be rational functions in x. For example, in the differential case with the differential operator D, OrePoly(2/x, x, x+1, 1) represents the operator $\frac{2}{x}+x\mathrm{D}+\left(x+1\right){\mathrm{D}}^2+{\mathrm{D}}^3$.

There are routines in the package that convert between Ore operators and the corresponding Maple expressions. See LinearOperators[converters].

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

$\mathrm{expr}≔\mathrm{sqrt}\left(x\right){\log \left(x\right)}^2$

$\mathrm{expr}≔\sqrt{x}{\ln \left(x\right)}^2$

$L≔\mathrm{FactoredAnnihilator}\left(\mathrm{expr}\,x\,'\mathrm{differential}'\right)$

$L≔\mathrm{FactoredOrePoly}\left(\left[\frac{3}{2x}\,1\right]\,\left[\frac{1}{2x}\,1\right]\,\left[-\frac{1}{2x}\,1\right]\right)$

$L≔\mathrm{FactoredOrePolyToOrePoly}\left(L\,x\,'\mathrm{differential}'\right)$

$L≔\mathrm{OrePoly}\left(-\frac{1}{8x^3}\,\frac{1}{4x^2}\,\frac{3}{2x}\,1\right)$

$\mathrm{IntegrateSols}\left(L\,x\,'\mathrm{differential}'\right)$

$\mathrm{OrePoly}\left(-\frac{27}{8x^3}\,\frac{13}{4x^2}\,-\frac{3}{2x}\,1\right),\mathrm{OrePoly}\left(\frac{26x}{27}\,-\frac{4x^2}{9}\,\frac{8x^3}{27}\right)$

Abramov, S. A., and van Hoeij, M. "Integration of Solutions of Linear Functional Equations." Integral Transforms and Special Functions. (1999): 3-12.