These are three functions that I try to integrate by hand, however, I try many methods and they do not work. Can you help me with that?
The first function is:
$$\int_{0}^{2\pi} \frac{1}{\sqrt{1- C\cos(t)}^3}dt\qquad \text{with } 0 < C <1$$
The second function is:
$$\int_{0}^{2\pi } \frac{\cos(t)}{\sqrt{1- C \cos(t)}}dt \qquad \text{with } 0 < C <1$$
The third function is:
$$\int_{0}^{2\pi } \frac{\cos(t)}{\sqrt{1- C \cos(t)}^3}dt \qquad \text{with } 0 < C <1$$
I have tried finding the answer through Maple, however, I could not understand the answer. Here is the answer for the three functions. The first one is: $$\,{\frac {-4}{\sqrt {C+1} \left( -1+C \right) }{\it EllipticE} \left( {\frac {\sqrt {2}\sqrt {C}}{\sqrt {C+1}}} \right) }$$
The second one is: $$\,{\frac {-4}{\sqrt {C+1}C} \left( C{\it EllipticE} \left( {\frac { \sqrt {2}\sqrt {C}}{\sqrt {C+1}}} \right) -{\it EllipticK} \left( { \frac {\sqrt {2}\sqrt {C}}{\sqrt {C+1}}} \right) +{\it EllipticE} \left( {\frac {\sqrt {2}\sqrt {C}}{\sqrt {C+1}}} \right) \right) }$$
And the third one is: $$-{\frac {4}{ \left( -3+3\,C \right) {C}^{3}} \left( {\it EllipticK} \left( {\sqrt {2}\sqrt {C}{\frac {1}{\sqrt {C+1}}}} \right) {C}^{3}-{ \it EllipticK} \left( {\sqrt {2}\sqrt {C}{\frac {1}{\sqrt {C+1}}}} \right) {C}^{2}-5\,{\it EllipticE} \left( {\frac {\sqrt {2}\sqrt {C} }{\sqrt {C+1}}} \right) {C}^{2}+8\,{\it EllipticK} \left( {\frac { \sqrt {2}\sqrt {C}}{\sqrt {C+1}}} \right) C-8\,{\it EllipticK} \left( {\frac {\sqrt {2}\sqrt {C}}{\sqrt {C+1}}} \right) +8\,{\it EllipticE} \left( {\frac {\sqrt {2}\sqrt {C}}{\sqrt {C+1}}} \right) \right) { \frac {1}{\sqrt {C+1}}}}$$
I am trying to understand how Maple can generate these answers, so any help is wonderful for me. Thank you so much!
Techniques for evaluation of elliptic integrals were worked out in the 19th century. The book I know about is
Hancock, H., Elliptic integrals., New York: J. Wiley and Sons, 104 S. (1917). ZBL46.1469.01.
Many years ago I got the softcover reprint of this from Dover.
Hancock spends a chapter or two on evaluating integrals in terms of the standard elliptic integrals $E, F, K, \Pi$. I can only assume that Maple uses something like these methods.