Reduction formula for Integral I am trying to find a reduction formula for $$\int\frac{\sin ^n(x)}{x} dx  .$$
Some numerical calculations seem to indicate a reduction formula should exist, but the usual tricks (break up the $\sin ^n(x)$ etc)  don't seem to help.
Any pointers/solutions would be great. 

This is not an answer, but can lead you to the answer
$\text{Si}=\int\frac{\sin (x)}{x} \,dx;\;\text{Ci}=\int\frac{\cos (x)}{x} \,dx$
Some solutions of 
$$\int\frac{\sin ^n(x)}{x} \,dx$$
for $n=1..10$
$$
\begin{array}{l|l}
n & integral\\
\hline
 1 & \text{Si}(x) \\
 2 & \frac{\log (x)}{2}-\frac{\text{Ci}(2 x)}{2} \\
 3 & \frac{1}{4} (3 \text{Si}(x)-\text{Si}(3 x)) \\
 4 & \frac{1}{8} (-4 \text{Ci}(2 x)+\text{Ci}(4 x)+3 \log (x)) \\
 5 & \frac{1}{16} (10 \text{Si}(x)-5 \text{Si}(3 x)+\text{Si}(5 x)) \\
 6 & \frac{1}{32} (-15 \text{Ci}(2 x)+6 \text{Ci}(4 x)-\text{Ci}(6 x)+10 \log (x)) \\
 7 & \frac{1}{64} (35 \text{Si}(x)-21 \text{Si}(3 x)+7 \text{Si}(5 x)-\text{Si}(7 x)) \\
 8 & \frac{1}{128} (-56 \text{Ci}(2 x)+28 \text{Ci}(4 x)-8 \text{Ci}(6 x)+\text{Ci}(8 x)+35 \log (x)) \\
 9 & \frac{1}{256} (126 \text{Si}(x)-84 \text{Si}(3 x)+36 \text{Si}(5 x)-9 \text{Si}(7 x)+\text{Si}(9 x)) \\
 10 & \frac{1}{512} (-210 \text{Ci}(2 x)+120 \text{Ci}(4 x)-45 \text{Ci}(6 x)+10 \text{Ci}(8 x)-\text{Ci}(10 x)+126 \log (x)) \\
\end{array}
$$
