`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) :}`