`int e^x/(1-tan(e^x)) dx` Use integration tables to find the indefinite integral.

`inte^x/(1-tan(e^x))dx`

First, let's apply integral substitution:`u=e^x`

`=>du=e^xdx`

`=int1/(1-tan(u))du`

Now from the integration tables:

`int1/(1+-tan(u))du=1/2(u+-ln|cos(u)+-sin(u)|)+C`

Using the above,

`=1/2(u-ln|cos(u)-sin(u)|)+C`

Substitute back `u=e^x`

`=1/2(e^x-ln|cos(e^x)-sin(e^x)|)+C`

`inte^x/(1-tan(e^x))dx`

First, let's apply integral substitution:`u=e^x`

`=>du=e^xdx`

`=int1/(1-tan(u))du`

Now from the integration tables:

`int1/(1+-tan(u))du=1/2(u+-ln|cos(u)+-sin(u)|)+C`

Using the above,

`=1/2(u-ln|cos(u)-sin(u)|)+C`

Substitute back `u=e^x`

`=1/2(e^x-ln|cos(e^x)-sin(e^x)|)+C`