Is this integral reducible to an elliptic integral? I believe if $k=0$ the following integral is reducible to an elliptic integral. If $k > 0$ is it possible to reduce it to an elliptic integral or some other special function?
$$\int_\rho^x \sqrt{1 + (\alpha \cos(t) - k)^2}\, dt$$
 A: According to Maple, it can be done... $\int \sqrt{1+(a \cos t - k)^2}\,dt$ is reported as  

-((-cos(t)^2*a^2+2*cos(t)*a*k-k^2-1)*(cos(t)^2-1))^(1/2)*(a^2*((cos(t)-1)*(cos(t)-(k+I)/a)*(cos(t)-(k-I)/a)+(-(k-I)/a+1)*((-k+I-a)*(cos(t)-1)/((-k+I+a)*(1+cos(t))))^(1/2)*(1+cos(t))^2*((-2*a*cos(t)+2*I+2*k)/((k+I-a)*(1+cos(t))))^(1/2)*((2*a*cos(t)+2*I-2*k)/((-k+I+a)*(1+cos(t))))^(1/2)*((k-I)*EllipticF(((-k+I-a)*(cos(t)-1)/((-k+I+a)*(1+cos(t))))^(1/2), ((a+k+I)*(-k+I+a)/((k+I-a)*(-k+I-a)))^(1/2))/(a*((k-I)/a+1))-(1/2)*(-(k+I)/a+1)*EllipticE(((-k+I-a)*(cos(t)-1)/((-k+I+a)*(1+cos(t))))^(1/2), ((a+k+I)*(-k+I+a)/((k+I-a)*(-k+I-a)))^(1/2))-2*k*EllipticPi(((-k+I-a)*(cos(t)-1)/((-k+I+a)*(1+cos(t))))^(1/2), (-(k-I)/a+1)/(-1-(k-I)/a), ((a+k+I)*(-k+I+a)/((k+I-a)*(-k+I-a)))^(1/2))/(a*((k-I)/a+1))))/((cos(t)-1)*(1+cos(t))*(-a*cos(t)+I+k)*(a*cos(t)+I-k))^(1/2)+2*a*k*(-(k-I)/a+1)*((-k+I-a)*(cos(t)-1)/((-k+I+a)*(1+cos(t))))^(1/2)*(1+cos(t))^2*((-2*a*cos(t)+2*I+2*k)/((k+I-a)*(1+cos(t))))^(1/2)*((2*a*cos(t)+2*I-2*k)/((-k+I+a)*(1+cos(t))))^(1/2)*(-EllipticF(((-k+I-a)*(cos(t)-1)/((-k+I+a)*(1+cos(t))))^(1/2), ((a+k+I)*(-k+I+a)/((k+I-a)*(-k+I-a)))^(1/2))+2*EllipticPi(((-k+I-a)*(cos(t)-1)/((-k+I+a)*(1+cos(t))))^(1/2), ((k-I)/a-1)/((k-I)/a+1), ((a+k+I)*(-k+I+a)/((k+I-a)*(-k+I-a)))^(1/2)))/(((k-I)/a+1)*((cos(t)-1)*(1+cos(t))*(-a*cos(t)+I+k)*(a*cos(t)+I-k))^(1/2))-k^2*(-(k-I)/a+1)*((-k+I-a)*(cos(t)-1)/((-k+I+a)*(1+cos(t))))^(1/2)*(1+cos(t))^2*((-2*a*cos(t)+2*I+2*k)/((k+I-a)*(1+cos(t))))^(1/2)*((2*a*cos(t)+2*I-2*k)/((-k+I+a)*(1+cos(t))))^(1/2)*EllipticF(((-k+I-a)*(cos(t)-1)/((-k+I+a)*(1+cos(t))))^(1/2), ((a+k+I)*(-k+I+a)/((k+I-a)*(-k+I-a)))^(1/2))/(((k-I)/a+1)*((cos(t)-1)*(1+cos(t))*(-a*cos(t)+I+k)*(a*cos(t)+I-k))^(1/2))-(-(k-I)/a+1)*((-k+I-a)*(cos(t)-1)/((-k+I+a)*(1+cos(t))))^(1/2)*(1+cos(t))^2*((-2*a*cos(t)+2*I+2*k)/((k+I-a)*(1+cos(t))))^(1/2)*((2*a*cos(t)+2*I-2*k)/((-k+I+a)*(1+cos(t))))^(1/2)*EllipticF(((-k+I-a)*(cos(t)-1)/((-k+I+a)*(1+cos(t))))^(1/2), ((a+k+I)*(-k+I+a)/((k+I-a)*(-k+I-a)))^(1/2))/(((k-I)/a+1)*((cos(t)-1)*(1+cos(t))*(-a*cos(t)+I+k)*(a*cos(t)+I-k))^(1/2)))/(sin(t)*(cos(t)^2*a^2-2*cos(t)*a*k+k^2+1)^(1/2))

