Prove that the function `f(x)=(log)_e(x^2+1)-e^(-x)+1` is strictly increasing `AAx in Rdot`

Explanation

`f(x) = log_(e)(x^(2)+1)-e^(-x)+1`
`therefore f(x) =(2x)/(1+x^(2))+e^(-x)=e^(-x)+(2)/(x+(1)/(x))`
For `xlt0, x+(1)/(x)lt-2`
`therefore-(1)/(2)lt(1)/(x+(1)/(x))lt0`
`rarr -1lt(2)/(x+(1)/(x))lt0`
Also, `e^(-x)gt1`
`therefore e^(-x)+(2x)/(1+x^(2))gt0`
`therefore` f(x) is strictly increasing function `forall` x in R