expectation of log(x+a) when X follows a beta distribution Is there a closed form expression for the expectation of $\log(x+a)$ (with $a>0$, the case $a=0$ is obvious) when X follows a beta distribution?
 A: closed form --- yes, simple --- no:
$$I(\alpha,\beta;a)=\int_0^1 dx\; \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}x^{\alpha-1}(1-x)^{\beta-1}\ln(x+a)=$$
$$\frac{\alpha}{a(\alpha+\beta)}  \; _3{F}_2\left(1,1,\alpha+1;2,\alpha+\beta+1;-1/a\right)+\ln a$$
a few special cases:
$$I(1/2,1/2;a)=2\ln\left(1+\sqrt{1+1/a}\right)+\ln(a/4)$$
$$I(1/2,1;a)=\ln(1+a)-2+2\sqrt{a}\,{\rm arccot}\sqrt{a}$$
$$I(1,1,a)=\ln(1+a)+a\ln(1+1/a)-1$$
A: Here is the answer done with Maple. The Maple code $$ with(Statistics):
X := RandomVariable(Beta(alpha, beta)):$$
$$Z := log(X+a)\,\,assuming \,\,a > 0:
simplify(Mean(Z));$$ outputs $$\left(\Gamma  \left( -1+\alpha \right) \Gamma  \left( \beta \right) {\alpha}
^{2}\ln  \left( {a}^{-1} \right) \sin \left( \alpha\,\pi  \right) -
\Gamma  \left( -1+\alpha \right) \Gamma  \left( \beta \right) {\alpha}
^{2}\Psi \left( \alpha+\beta \right) \sin \left( \alpha\,\pi  \right) 
+\Gamma  \left( -1+\alpha \right) \Gamma  \left( \beta \right) {\alpha
}^{2}\Psi \left( \alpha \right) \sin \left( \alpha\,\pi  \right) +
\Gamma  \left( -1+\alpha \right) \Gamma  \left( \beta \right) {\alpha}
^{2}\ln  \left( a \right) \sin \left( \alpha\,\pi  \right) +\Gamma 
 \left( -1+\alpha \right) \Gamma  \left( \beta \right) {\alpha}^{2}
{\mbox{$_3$F$_2$}(1,1,-\alpha-\beta+2;\,2,-\alpha+2;\,-a)}a\sin
 \left( \alpha\,\pi  \right) +\Gamma  \left( -1+\alpha \right) \Gamma 
 \left( \beta \right) \alpha\,
{\mbox{$_3$F$_2$}(1,1,-\alpha-\beta+2;\,2,-\alpha+2;\,-a)}a\beta\,\sin
 \left( \alpha\,\pi  \right) -\Gamma  \left( -1+\alpha \right) \Gamma 
 \left( \beta \right) \alpha\,
{\mbox{$_3$F$_2$}(1,1,-\alpha-\beta+2;\,2,-\alpha+2;\,-a)}a\sin
 \left( \alpha\,\pi  \right) +\Gamma  \left( -1+\alpha \right) \Gamma 
 \left( \beta \right) \alpha\,\Psi \left( \alpha+\beta \right) \sin
 \left( \alpha\,\pi  \right) -\Gamma  \left( -1+\alpha \right) \Gamma 
 \left( \beta \right) \alpha\,\ln  \left( {a}^{-1} \right) \sin
 \left( \alpha\,\pi  \right) -\Gamma  \left( -1+\alpha \right) \Gamma 
 \left( \beta \right) \alpha\,\ln  \left( a \right) \sin \left( \alpha
\,\pi  \right) -\Gamma  \left( -1+\alpha \right) \Gamma  \left( \beta
 \right) \alpha\,\Psi \left( \alpha \right) \sin \left( \alpha\,\pi 
 \right) +\pi \, \left( {a}^{-1} \right) ^{-\alpha}
{\mbox{$_2$F$_1$}(\alpha,1-\beta;\,\alpha+1;\,-a)}\Gamma  \left( 
\alpha+\beta \right) \right)$$ $$(\sin \left( \alpha\,\pi  \right) \rm{B} \left( \alpha,\beta \right) 
\Gamma  \left( \alpha+\beta \right) \alpha
)^{-1}
,
 $$ where the function  $\Psi(x)$ is described here http://www.maplesoft.com/support/help/Maple/view.aspx?path=Psi .
