  This paper proposes ideas to speed up the process of creative telescoping,
particularly when the telescoper is reducible. One can interpret telescoping as
computing an annihilator $L \in D$ for an element $m$ in a $D$-module $M$. The
main idea is to look for submodules of $M$. If $N$ is a non-trivial submodule
of $M$, constructing the minimal operator $R$ of the image of $m$ in $M/N$
gives a right-factor of $L$ in $D$. Then $L = L' R$ where the left-factor $L'$
is the telescoper of $R(m) \in N$. To expedite computing $L'$, compute the
action of $D$ on a natural basis of $N$, then obtain $L'$ with a cyclic vector
computation.
  The next main idea is that when $N$ has automorphisms, use them to construct
submodules. An automorphism with distinct eigenvalues can be used to decompose
$N$ as a direct sum $N_1 \oplus \cdots \oplus N_k$. Then $L'$ is the LCLM
(Least Common Left Multiple) of $L_1, \ldots, L_k$ where $L_i$ is the
telescoper of the projection of $R(m)$ on $N_i$. An LCLM can greatly increase
the degrees of coefficients, so $L'$ and $L$ can be much larger expressions
than the factors $L_1,\ldots,L_k$ and $R$. Examples show that computing each
factor $L_i$ and $R$ seperately can save a lot of CPU time compared to
computing $L$ in expanded form with standard creative telescoping.
