Wolfram'sThe Integratorreturns the following:

Code:(x + Log[Cos[x] - Sin[x]] + (4 Sqrt[Cos[x]] ((1 - I) x EllipticPi[-I, -ArcSin[Sqrt[Tan[-]]], -1] + 2 x (1 + I) EllipticPi[I, -ArcSin[Sqrt[Tan[-]]], 2 1 -1] - EllipticPi[------------, -1 + Sqrt[2] x -ArcSin[Sqrt[Tan[-]]], -1] - 2 1 EllipticPi[-(-----------), 1 + Sqrt[2] x x -ArcSin[Sqrt[Tan[-]]], -1]) Sqrt[Tan[-]]) / 2 2 x 2 (Sqrt[Cos[x] Sec[-] ] Sqrt[Sin[x]])) / 2 2