Prove that $\sum_{n=1}^{\infty}(-1)^{n+1}\frac{\ln(2n+1)}{2n+1} = \frac{\pi}{4}(\gamma - \ln \pi) + \pi \ln \Gamma(\frac{3}{4})$

Question

$$\sum_{n=1}^\infty(-1)^{n+1}\frac{\ln(2n+1)}{2n+1} =\frac\pi4(\gamma-\ln\pi)+\pi\ln\Gamma(3/4)$$ This sum was obtained by Malmsten in 1842. I researched this series and found no proof. I also saw that this series is the derivative of Dirichlet's beta function at point 1. I tried to turn the series into an integral from 0 to 1, but I did not succeed.

Thanks for any help.