`int(e^x+e^-x)^(1/4)(e^x-e^-x)dx`

Put `u=e^x+e^-x`, so `du=(e^x-e^-x)dx`

Then

`int(e^x+e^-x)^(1/4)(e^x-e^-x)dx=intu^(1/4)du`

`=(4u^(5//4))/5+K`

`=(4(e^x+e^-x)^(5//4))/5+K`