Evaluate: `int1/((sqrt(e^(5x))((e^(2x)+e^(-2x))^3)^(1/4))`

Explanation

`I=int(1)/(sqrt(e^(5x))root(4)((e^(2x)+e^(-2x))^(3)))dx`
`=int(1)/(sqrt(e^(5x))e^(3x//2)(1+e^(-4x))^(3//4))dx`
`=int(1)/(e^(4x)(1+e^(-4x))^(3//4))dx`
Let `1+e^(-4x)=t`
` :. -4e^(-4x)dx=dt`
` :. I= -(1)/(4)int(dt)/(t^(3//4))`
`= - (1)/(4)(t^(1//4))/((1//4))+c`
`= -(1+e^(-4x))^(1//4)+c`