A tricky integral I have been working on, and probably doesn't have a solution in terms of known functions, is:
$\int_{0}^{\infty} e^{- \lambda \left(\sqrt{(z+a)^2+b^2}+\sqrt{(z+c)^2+d^2}~\right)}\mathrm{d}z$ Taken separately the integrals have solutions:
$\int_{0}^{\infty} e^{- \lambda \left(\sqrt{(z+a)^2+b^2}~\right)}\mathrm{d}z = b~ \mathrm{K}_1 ( \lambda b , \textrm{sinh}^{-1}(a/b) ) $ where the $K_1$ is an incomplete modified Bessel function of the second kind.
Can anyone think of a way to extend this in the case where there are two square roots in the exponential?
Cheers