The following series I'm interested in $$\sum_{n=2}^\infty(-1)^{n+1}\frac{\psi(n)}n\log(n)$$

where $\psi(n)$ is digamma function

arose in the evaluation of an integral I posted on MSE, http://math.stackexchange.com/questions/857301/evaluation-of-int-01-frac-log1x1x-log-left-log-left-frac1x-rig and as it can be easily seen, after a while all
gets reduced to computing the foregoing series. The investigations that were done so far led nowhere, no closed form could be found. My intuition tells me there is a closed form, and that's the main reason
for that I also posted the question here. To summarize, I have the following questions:

$$a). \text{Is this series known in literature? If yes, could you name some sources?} $$

$$b). \text{How would you recommend me to tackle this series? }$$

$$c). \text{I would appreciate if anyone would do some research on it. }$$

Please consider this question comes from a person with no background in mathematics.

The following series I'm interested in $$\sum_{n=2}^\infty(-1)^{n+1}\frac{\psi(n)}n\log(n)$$

where $\psi(n)$ is digamma function

arose in the evaluation of an integral I posted on MSE, https://math.stackexchange.com/questions/857301/evaluation-of-int-01-frac-log1x1x-log-left-log-left-frac1x-rig and as it can be easily seen, after a while all
gets reduced to computing the foregoing series. The investigations that were done so far led nowhere, no closed form could be found. My intuition tells me there is a closed form, and that's the main reason
for that I also posted the question here. To summarize, I have the following questions:

$$a). \text{Is this series known in literature? If yes, could you name some sources?} $$

$$b). \text{How would you recommend me to tackle this series? }$$

$$c). \text{I would appreciate if anyone would do some research on it. }$$

Please consider this question comes from a person with no background in mathematics.

The following series I'm interested in $$\sum_{n=2}^\infty(-1)^{n+1}\frac{\psi(n)}n\log(n)$$

arose in the evaluation of an integral I posted on MSE, http://math.stackexchange.com/questions/857301/evaluation-of-int-01-frac-log1x1x-log-left-log-left-frac1x-rig and as it can be easily seen, after a while all
gets reduced to computing the foregoing series. The investigations that were done so far led nowhere, no closed form could be found. My intuition tells me there is a closed form, and that's the main reason
for that I also posted the question here. To summarize, I have the following questions:

$$a). \text{Is this series known in literature? If yes, could you name some sources?} $$

$$b). \text{How would you recommend me to tackle this series? }$$

$$c). \text{I would appreciate if anyone would do some research on it. }$$

Please consider this question comes from a person with no background in mathematics.

The following series I'm interested in $$\sum_{n=2}^\infty(-1)^{n+1}\frac{\psi(n)}n\log(n)$$

where $\psi(n)$ is digamma function

arose in the evaluation of an integral I posted on MSE, http://math.stackexchange.com/questions/857301/evaluation-of-int-01-frac-log1x1x-log-left-log-left-frac1x-rig and as it can be easily seen, after a while all
gets reduced to computing the foregoing series. The investigations that were done so far led nowhere, no closed form could be found. My intuition tells me there is a closed form, and that's the main reason
for that I also posted the question here. To summarize, I have the following questions:

$$a). \text{Is this series known in literature? If yes, could you name some sources?} $$

$$b). \text{How would you recommend me to tackle this series? }$$

$$c). \text{I would appreciate if anyone would do some research on it. }$$

Please consider this question comes from a person with no background in mathematics.