# Integral of product of two error functions

Is there a simple formula for the integral $$\int_a^{\infty}\!\text{erf}(\alpha x)\,\text{erf}(\beta x)\:\frac{dx}{x^2}$$ where $a, \alpha, \beta > 0$?

• i think only when $a=0$ – tired Feb 6 '16 at 1:32
• Thanks. i've been able to find a simplified formulae when a=0. while integration from $0$ to $a$ can be done by a simple quadratre. for $a=0$, this integral is equal to : $$\frac{2}{\pi} \Gamma(\frac{1}{2}) \left( \beta \, _2F_1(1/2;1/2;3/2;\alpha^2/\beta^2)+ \alpha \, _2F_1(1/2;1/2;3/2;\beta^2/\alpha^2)\right)$$ where _2F_1 is the hypergeommetric function. – Yassine Feb 6 '16 at 15:44