∫cos⁡ec22x2xdx is equal to

Let I=∫cos⁡ec22x2xdx Put 2x=y⇒122x2xlog⁡2dx=dy⇒2xdx=2ylog⁡2dyI=2log⁡2∫cos⁡ec2yydy=2log⁡2y∫cosec2⁡y−∫dydy∫cos⁡ec2ydydy=2log⁡2−ycot⁡y+∫cot⁡ydy=2log⁡2[−ycot⁡y+log⁡sin⁡y]+C=2log⁡2log⁡sin⁡2x−2xcot⁡2x+C