Integral of Gamma Functions

Question

Is the following integral able to be expressed by closed form or special functions?

$$\int_{0}^{1}\log\left(\frac{\Gamma(at+b)}{\Gamma(ct+b)}\right)\frac{dt}{t}$$, for any non-negative numbers $a,b,c$.

Appreciate!

Answer 1

Maple expresses it as $$ \ln \left( a \right) \ln \left( b \right) -1/2\, \left( \ln \left( a \right) \right) ^{2}-{\it polylog} \left( 2,-{\frac {b}{a}} \right) -\ln \left( c \right) \ln \left( b \right) +1/2\, \left( \ln \left( c \right) \right) ^{2}+{\it polylog} \left( 2,-{\frac {b }{c}} \right) -a\gamma+c\gamma-\sum _{k=1}^{\infty }{\frac {1}{2k } \left( 2\,\ln \left( \left( k+b \right) ^{-1} \right) \ln \left( a \right) k-2\,\ln \left( \left( k+b \right) ^{-1} \right) \ln \left( c \right) k+ \left( \ln \left( a \right) \right) ^{2}k- \left( \ln \left( c \right) \right) ^{2}k+2\,{\it polylog} \left( 2 ,-{\frac {k+b}{a}} \right) k-2\,{\it polylog} \left( 2,-{\frac {k+b}{c }} \right) k-2\,a+2\,c \right) } $$

EDIT: Here is the appropriate Maple command:

int(log(GAMMA(a*t+b)/GAMMA(c*t+b))/t,t=0..1) assuming a>0, b>0, c>0;

I would be interested to know what command user64494 tried.