Solving Exponential integral with Numerical Integration

Question

Integral( exp(-x) - exp(-x - b*Sqrt[-A + x^2)), {x, Sqrt(A), Infinity})

$$\int_{\sqrt{A}}^\infty e^{-x}-e^{-x-b\sqrt{-A+x^2}}\,dx$$ which numerical integration technique is applicable on this form?

Answer 1

Hint:

$\int_\sqrt{A}^\infty\left(e^{-x}-e^{-x-b\sqrt{-A+x^2}}\right)~dx$

$=\int_\sqrt{A}^\infty e^{-x}~dx-\int_\sqrt{A}^\infty e^{-x-b\sqrt{x^2-A}}~dx$

$=\left[-e^{-x}\right]_\sqrt{A}^\infty-\int_0^\infty e^{-\sqrt{A}\cosh x-b\sqrt{(\sqrt{A}\cosh x)^2-A}}~d(\sqrt{A}\cosh x)$

$=e^{-\sqrt{A}}-\sqrt{A}\int_0^\infty e^{-\sqrt{A}\cosh x-b\sqrt{A}\sinh x}\sinh x~dx$

$=e^{-\sqrt{A}}+\dfrac{\sqrt{A}}{2}\int_0^\infty e^{-\sqrt{A}\cosh x-b\sqrt{A}\sinh x-x}~dx-\dfrac{\sqrt{A}}{2}\int_0^\infty e^{-\sqrt{A}\cosh x-b\sqrt{A}\sinh x+x}~dx$

Which relates to some special functions in e.g. http://dlmf.nist.gov/10.9