Evaluate `int(dx)/((1+sinx)^(1//2))`

Answer

`sqrt(2) log|(tan ((x)/(4)+(pi)/(x)))|+C`

Explanation

`int(dx)/((1+sinx)^(1//2))=int(dx)/("cos"(x)/(2)+"sin"(x)/(2))`
`=(1)/(sqrt(2))int(dx)/("sin"((x)/(2)+(pi)/(4)))`
`=(1)/(sqrt(2))int"cosec"((x)/(2)+(pi)/(4))dx`
`=(1)/(sqrt(2))(log|tan((x)/(4)+(pi)/(8))|)/((1)/(2))+C`
`=sqrt(2) log|(tan ((x)/(4)+(pi)/(x)))|+C`