Is it possible to represent this integral in terms of elementary functions?
$$\int \textrm{ln}(1+e^{x^{2}})dx$$
I saw a challenge on this site (integral challenge) and still can not figure out how to solve it (if it's solvable).
Addition
The full integral (the one I want to calculate) is $$\int \left [\textrm{ln}(1+e^{x^{2}})+2\frac{[e^{x^{2}}(2x^{2}-1)-1]}{(e^{x^{2}}+1)^{2}}\right ]dx$$
Perhaps combining the two terms gives rise to an elementary primitive.
One method may be to expand the logarithm as seen by: \begin{align} \int \ln(1 + e^{x^2}) \, dx &= \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n} \, \int e^{n \, x^2} \, dx \\ &= \frac{\sqrt{\pi}}{2} \, \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n \, \sqrt{n}} \, erfi(\sqrt{n} \, x), \end{align} where $erfi(x)$ is the imaginary error function.