Is it possible to find the indefinite integral of:
$$ y = \frac{x^2}{x^2 + a} e^{-(x+b)^2} $$
I have tried integration by parts but haven't been successful. Maybe its possible considering a special function?
2026-05-14 12:36:17.1778762177
Integral of exponential times $\frac{x^2}{x^2+a}$
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As $\dfrac{x^2}{x^2+a} = 1 - \dfrac{a}{x^2+a},$ then we can rewrite the integral in question as $$ \int \dfrac{x^2}{x^2+a} e^{-(x+b)^2}dx=\int e^{-(x+b)^2}dx - a\int \dfrac{e^{-(x+b)^2}}{x^2+a}dx.$$
Since $\int e^{-(x+b)^2}dx = \frac{1}{2}\sqrt{\pi} \operatorname{erf}(x+b),$ then it's clear you'll need special functions. I don't believe there's a well known formulation in terms of special functions.