If $I=\int \frac{dx}{\sec x+\cos ecx},$ then $I$ equals

$\frac{1}{2}[\sin x-\cos x]-\frac{1}{\sqrt{2}}\log |\cos ec(x+\frac{\pi }{4})-\cot (x+\frac{\pi }{4})|+C$

$\frac{1}{2}(\cos x+\sin x-\frac{1}{\sqrt{2}}\log (\cos ecx-\cos x))+C$

$\frac{1}{2}(\sin x+\cos x+\frac{1}{2}\log |\cos ecx-\cos x|)+C$

$\frac{1}{2}(\sin x-\cos x-\frac{1}{\sqrt{2}}\log |\cos ecx+\cot x|)+C$