Evaluate: `int(e^x)/((1+e^x)(2+e^x))\ dx`

Explanation

Let `I=int(e^(x))/((1+e^(x)).(2+e^(x))).dx`
Suppose `e^(x)=t`
`:.e^(x)dx=dt`
`:.I=int(dt)/((1+t)(2+t))`
Let `(1)/((1+t)(2+t))=(A)/(1+t)+(B)/(2+t)`
`:.1=A=(2+t)+B(1+t)`
Put `t=-1implies1=A(2-1)`
`:.A=1`
Put `t=-2implies1=B(-2+1)`
`:.B=-1`
`:.(1)/((1+t)(2+t))+(1)/(1+t)-(1)/(2+t)`
`:.I=int((1)/(1+t)-(1)/(2+t)).dt`
`=int(1)/(1+t)dt-int(1)/(2+t).dt`
`=log|1+t|-log|2+t|+C`
`=log|(1)/(2)+(t)/(t)|+C`
`:.I=log|(1)/(2)+e^(x)/(e^(x))|+C`