calculus and analysis - Integrate and NIntegrate disagreeing over branch cut of Sqrt?

Bug introduced after 8.0.4 and before 9.0.1, and fixed in 11.1.0.

In Mathematica 10.2 I'm trying to integrate this piecewise continuous function, but Integrate and NIntegrate seem to disagree on the branch cut of Sqrt[]:

 Integrate[Sqrt[Exp[I*t]^2 - 1], {t, 0, Pi}]
 (* 2 Sqrt[2] - I Pi *)


 NIntegrate[Sqrt[Exp[I*t]^2 - 1], {t, 0, Pi}]
 (* 1.06568 - 4.51028*10^-16 I *)

It seems that NIntegrate's answer is correct if Sqrt[] has its branch cut on the negative real axis. Is there a way to force Integrate to use the same branch cut?

Comments

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