Another Integral!

Calculus Level 5

$\int_{0}^{1}\frac{1-x}{(1+x)(\ln x)}\mathrm dx=\ln\Gamma\left(a\right)-\ln\Gamma\left(\frac{b}{c}\right)-\frac12\ln\pi$

The equation above is true for constants $$a,b,c,$$ with coprime positive integers $$b,c$$ and $$a<\dfrac{b}{c}$$. Find $$b^c+a$$.

This came up when my Uncle and I were discussing Integrals, so I cannot vouch for its originality.
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