sHello to everyone, i would like a suggestion on how to evaluate these integrals using the hypergeometric functions:
$$\int_0^x \frac{\cos x}{\sqrt{1+a \ sinx+b \ cosx}} dx$$ where $a,b \in \mathbb{R}$.
Unfortunately, even after a lot of work, I did not get a compact formulation using the hypergeometric functions.
Thanks for your time and for your interest.
Tip: With some passage you get the following condition $$\int_0^x \frac{\cos x}{\sqrt{1+a \ sinx+b \ cosx}} dx=\sum_{n=0}^\infty \frac{(-1)^{n} (\frac{1}{2})_n}{n!} ({a^{2}+b^{2}})^{\frac{n}{2}}\int_0^x [sin(x+\alpha)]^{n} \ \ dsinx$$ Where $$\alpha=atan(\frac{b}{a})$$ Unfortunately, however, the latter does not seem easy to solve. It would be interesting to be able to solve it with the use of hypergeometric features.
It is easy to express the integral in terms of Elliptic Integrals of first and second kind (below).
The Elliptic Integrals are related to some hypergeometric functions, but the relationships are complicated. See : http://functions.wolfram.com/EllipticIntegrals/EllipticF/26/01/ and http://functions.wolfram.com/EllipticIntegrals/EllipticE2/26/01/