The integration of (sinx+cosx+tanx+cosecx+secx+cotx)^-1
Integration is very lengthy and requires a lot of Time and space. It is impossible to upload a solution here so I would suggest you to go to the following link where You will be able to find Indefinite Integral of any function with all the steps and even they provide you options to exp[ort It in the form of Latex so Link is given Below.
https://www.integral-calculator.com/