Question: Do you know how to evaluate these integrals ?

The integrals are

v1:=int((1/(2*Pi) + Q* cos(k*phi))^2/ (1+ beta^2*((1/(2*Pi)) + Q* cos(k*phi) )^2),phi=-Pi..Pi);

v2:=int((1/(2*Pi) + Q* cos(k*phi))^2*cos(k*phi)/ (1+ beta^2*((1/(2*Pi)) + Q* cos(k*phi) )^2),phi=-Pi..Pi);

v3:=int((1/(2*Pi) + Q* cos(k*phi))^3*cos(k*phi)/ ((1+ beta^2*((1/(2*Pi)) + Q* cos(k*phi) )^2))^2,phi=-Pi..Pi);

v4 := int(exp((-1/2)*((psi-theta-k*sin(theta-phi))/sigma)^2),phi = - Pi ..Pi);

 v5:=int((exp((-1/2)*((psi-theta-k*sin(theta-phi))/sigma)^2))  * (1/(2*Pi) + Q*cos(k*phi)),phi=-Pi..Pi);

I can't get any output at all from maple for any of these integrands.  In the case of v1, v2 atleast when I do a convert to exponential form of the integrand I get an output but the output doesn't make much sense. In particular I get many RootOf expressions and I can't simplify the output to an intelligible form. I'd particularly like to be able to evaluate either v4 or v5. Any ideas or suggestions you have are welcomed as I am completely stuck.