Hello everyone and thanks for your time. I would like a suggestion on how to evaluate these integrals using the hypergeometric series (rewrite them in the form of hypergeometric series)
$$\int_0^x \frac{\sin (x)}{\sqrt{a-\sin(x)}}\, dx$$ where $$0<a<1.$$ (Sorry if I put the extreme of integration coincides with the variable of integration, but I put it to show that it too is variable).
$$\int_0^x \frac{\cos (x)}{\sqrt{\cos(x)-b}}\, dx$$ where $$0<b<1$$ and $$\int_0^x \frac{\sin (x)}{c+d\sqrt{a-\sin(x)}+e\sqrt{\cos(x)-b}}\, dx$$
$$\int_0^x \frac{\cos (x)}{c+d\sqrt{a-\sin(x)}+e\sqrt{\cos(x)-b}}\, dx$$ where $c,d,e \in \mathbb{R}$ and $c,d,e>0.$
Of course I know that derive the expression in hypergeometric series of each integral would lose much time,so if someone can suggest me how to do, or show me the process I would be very grateful. Unfortunately, my whole knowledge on i ipergeometrici are still insufficient and too new to be able to be applied easily,and that's why I can not pass by the integral Readily to the generalized hypergeometric series. Thank you again for your interest.