Definite Integral to Infinity of Power of x by Logarithm of x over One plus x

Theorem

 * $\ds \int_0^\infty \frac {x^{p - 1} \ln x} {1 + x} \rd x = -\pi^2 \csc p \pi \cot p \pi$

where:
 * $p$ is a real number with $0 < p < 1$
 * $\csc$ denotes the cosecant function
 * $\cot$ denotes the cotangent function.