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 is part of Ordered Disorder.
This came up when my Uncle and I were discussing Integrals, so I cannot vouch for its originality.

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