Indefinite Integral for rational trigonometric functions without hypergeometric functions

Question

How to find indefinite integrals for the rational trigonometric functions of the form

$\qquad\Lambda^{pq}_m = \int \frac{\cos^p x\sin^q x}{(a+b\sin x +c\cos x)^m} dx$

where $(p,q,m)>0$ using Mathematica?

Any suggestions for preventing the results from being given in terms of AppellF1, but rather to obtain them as simpler summation functions for the ranges

$\qquad 0 \leq p \leq 5$, $0 \leq q \leq 1$ and $3 \leq m \leq 9$.

For example: ClearAll[x, a, b, c, p, q, m];Integrate[1/(a + b Cos[x] + c Sin[x])^m, x]

yields an output in terms of AppellF1

Answer 1

If I do it this way

Lambda[p_, q_, m_] := 
 Integrate[(Cos[x]^p Sin[x]^q)/(a + b Cos[x] + c Sin[x])^m, x]

Table[Lambda[p, q, m], {p, 0, 5}, {q, 0, 1}, {m, 3, 9}]

That will take awhile and will produce very large output. I did not wait for it to finish, but spot checking a few, such as

Lambda[0, 0, 3]
Lambda[2, 1, 5]
Lambda[5, 1, 9]

all produced standard Trig functions in output. If you want to use values for p,q,m other than integers, you are probably out of luck.