# Using Assumptions in Expressions that Evaluate to be Elliptic Integrals

I have the following integral that I am trying to evaluate in Mathematica:

$\int \sqrt{\alpha + m g l \cos(q)} dq$. If $\alpha > m g l$, then the result is a complete elliptic integral of the second kind and Mathematica outputs EllipticE[m]. If $\alpha < m g l$, then Mathematica outputs and incomplete elliptic integral of the second kind, i.e. EllipticE[$\phi$,m].

I am trying to obtain a general expression for the integral when $\alpha > m g l$, but Mathematica keeps giving me a conditional expression, even after using the Assumptions command with $\alpha > m g l$ and $\alpha \in$ Reals. Does anyone know what's going on here?

• FullSimplify[Integrate[Sqrt[a + b Cos@q], q], Assumptions -> (a > b > 0)] returns 2 Sqrt[a + b] EllipticE[q/2, (2 b)/(a + b)] May 8, 2015 at 17:28

In version 8 the integral is very simply solved without any explicit assuptions:

\$Version

(*
Out= "8.0 for Microsoft Windows (64-bit) (October 7, 2011)"
*)

Integrate[Sqrt[a + b Cos[q]], q]

(*
Out= (2 Sqrt[a + b Cos[q]] EllipticE[q/2, (2 b)/(a + b)])/Sqrt[(a +
b Cos[q])/(a + b)]
*)