Solving an "impossible" integral, $\int \frac{dx}{a^{x^2}+b^{x^2}} $

First a saw this problem$$\int \frac{dx}{a^{x^2}+b^{x^2}} $$where a,b are natural numbers

This problem would be easier if $a=b$then error function appears here

I have tried with Wolfram alpha but he show me that"(no result found in terms of standard mathematical functions)"

But I was wondering is there a solution when $a≠b$ , at least when the interval from $[-∞,∞]$Could anybody help or give a hints to solve?