Simplification of a combination of 6 values of the gamma function

Question

I'm trying to simplify this combination of gamma functions: $$\frac{\Gamma\left(\frac{2}{25}\right)\Gamma\left(\frac{7}{25}\right)\Gamma\left(\frac{12}{25}\right)}{\Gamma\left(\frac{2}{5}\right)\Gamma\left(\frac{3}{25}\right)\Gamma\left(\frac{8}{25}\right)}$$ I tried to apply Gauss's multiplication formula, as it was done in this answer, but without any success. It is possible to simplify this expression at all?

Answer 1

Using the Gauss Multiplication Formula, which is proven in this answer, $$ \Gamma\left(\frac2{25}\right)\Gamma\left(\frac{7}{25}\right)\Gamma\left(\frac{12}{25}\right)\Gamma\left(\frac{17}{25}\right)\Gamma\left(\frac{22}{25}\right)=4\pi^25^{1/10}\Gamma\left(\frac25\right) $$ Using the Euler Reflection Formula, proven at the end of this answer, $$ \Gamma\left(\frac3{25}\right)\Gamma\left(\frac{22}{25}\right)=\frac\pi{\sin\left(\frac{3\pi}{25}\right)} $$ and $$ \Gamma\left(\frac8{25}\right)\Gamma\left(\frac{17}{25}\right)=\frac\pi{\sin\left(\frac{8\pi}{25}\right)} $$ Putting these together, we get $$ \frac{\Gamma\left(\frac2{25}\right)\Gamma\left(\frac{7}{25}\right)\Gamma\left(\frac{12}{25}\right)}{\Gamma\left(\frac25\right)\Gamma\left(\frac3{25}\right)\Gamma\left(\frac8{25}\right)}=4\cdot5^{1/10}\sin\left(\frac{3\pi}{25}\right)\sin\left(\frac{8\pi}{25}\right) $$

Answer 2

Using Maple, I get

$$ {\frac {{5}^{1/10} \left( 2\,\cos \left( {\frac {8\,\pi }{25}} \right) +2\,\cos \left( {\frac {4\,\pi }{25}} \right) +1 \right) }{ 2\;\cos \left( {\frac {6\,\pi }{25}} \right) +2\;\cos \left( \frac{\pi}{5} \right) }} $$