Math Is Fun Forum

Please help with the proof of the following definite integral equation

∫x(Ф(x))^3 φ(x)dx=-3(1+2a/π) /(8√π)   from minus infinite to plus infinite   

where Ф(x)=∫1/√2π exp(- t^2/2) dt from minus infinite to x
φ(x)=1/√2π exp(- x^2/2)   
a=arcsin(1/3)

To be exact, φ(x) is the probability density function of standard normal distribution, Ф(x) is the cumulative probability function of standard normal distribution.

Please see the attached image for clear equation about the question.