Having trouble figuring out the total time it takes for something following the path of a brachistochrone. Starting at some point along and falling due to gravity to the lowest point. Equation I have is (a^.5/g^.5)*(1-cos(t))^.5/(cos(t_o)-cos(t))^.5 a and g are constants and t_o is the initial angle at which the particle is at in relation to the brachistochrone. Can not get this function to integrate on Mathematica or Calculator; no consistent answers. Mathematica returns part of the integrate as an imaginary number and I don't know why.