I am trying to Integrate the following Integral (all of the variables are reals) $$\int Exp(-(\cos^{-1}\left[\frac{\text{n}_0 (\text{v}_0-\text{x}_0)+\text{n}_1 (\text{v}_1-\text{x}_1)+\text{n}_2 (u \text{v}+\text{v}_2-\text{x}_2)}{\sqrt{\text{n}_0^2+\text{n}_1^2+\text{n}_2^2} \sqrt{(\text{v}_0-\text{x}_0)^2+(\text{v}_1-\text{x}_1)^2+(u \text{v}+\text{v}_2-\text{x}_2)^2}}\right]/m)^2) \, du$$

The formula, simplified, in Mathematica input:

Integrate[Exp[-(ArcCos[(a + b*u)/(c*Sqrt[d + (e + b*u)^2])]/m)^2], u]


If I simply enter that into Mathematica, it instantly returns the same expression. If I enter it as an Integral with set limits (0 and t), it calculates on and on and on....

1. Find out if there is a closed Form for this (obviously including Erf terms)
• @swish I am absolutely sure. This is an integral over specular lighting (Gauss Specular lighting) along a part of the z-axis. The formula is $e^{-\angle(H,L)^2}$ - I just realized I forgot the ^2 >_< – CBenni Jan 22 '13 at 15:22
• Should be slightly easier as there are fewer new variables than old. Also your a+b*u and e+b*u appear to be not quite right. I think it should be n2*(a+b*u) in the numerator and a+b*u in the denominator (n2 being the constant in the original formulation). – Daniel Lichtblau Jan 22 '13 at 15:41