If that was in Mathematica 8 (M8) ... it's still here in Mathematica 10.0.1

In[1]:= Integrate[ChebyshevT[1, z] (1 - z^2)^(-1/2), {z, a, b}, Assumptions -> -1 < a && a < 0 && b > a && b < +1]
Out[1]= Sqrt[1 - a^2] - Sqrt[1 - b^2]

In[2]:= Integrate[ChebyshevT[1, z] (1 - z^2)^(-1/2), {z, a, b},  Assumptions -> -1 < a && a > 0 && b > a && b < +1]
Out[2]= Sqrt[1 - a^2] - Sqrt[1 - b^2]

In[3]:= Integrate[ChebyshevT[1, z] (1 - z^2)^(-1/2), {z, a, b},  Assumptions -> -1 < a && b > a && b < +1]
Out[3]= ConditionalExpression[Sqrt[1 - a^2] - Sqrt[1 - b^2], a > 0]

In[4]:= $Version
Out[4]= "10.0 for Microsoft Windows (64-bit) (September 9, 2014)"

In[5]:= Integrate[ChebyshevT[1, z] (1 - z^2)^(-1/2), {z, a, b},  Assumptions -> -1 < a < b < 1]
Out[5]= ConditionalExpression[Sqrt[1 - a^2] - Sqrt[1 - b^2], a > 0]

In[6]:= Integrate[ChebyshevT[1, z] (1 - z^2)^(-1/2), {z, a, b},  Assumptions -> -1 < a && ((a < 0) || (a > 0)) && a < b && b < 1]
Out[6]= ConditionalExpression[Sqrt[1 - a^2] - Sqrt[1 - b^2], a > 0]

In[8]:= Integrate[ChebyshevT[1, z] (1 - z^2)^(-1/2), {z, a, b}, Assumptions -> -1 < a && a != 0 && a < b && b < 1]
Out[8]= ConditionalExpression[Sqrt[1 - a^2] - Sqrt[1 - b^2], a > 0]