This note (PDF version) discusses a generalization of some formulas in "A Class of Logarithmic Integrals" by Victor Adamchik.  Adamchik's paper proves 6 different general propositions allowing Adamchik to solve integrals such as

Generally, I like Adamchik's paper.  The most interesting proposition is Proposition 1 which Adamchik attributes to G. Almkvist and A. Meurman.  The proofs for Adamchik's Proposition 3, Proposition 5, and Proposition 6 are more drawn out and tortured than necessary.  My theorem below is proved in less space and generalizes half the propositions in Adamchik's paper.

Definition of Two Functions

Definition. For convenience, define

Function has properties

The following theorem generalizes Adamchik's Proposition 3, Proposition 5, and Proposition 6.

Generalization of Adamchik's Propositions 3, 5, and 6

Theorem.

where coefficients are determined by

Proof. Since

We get

Therefore,

Comment. The first few are

Adamchik's Proposition 4

The proof for Adamchik's Proposition 4 can also be simpler:

Theorem.

Proof.

References

The first reference is Victor Adamchik's paper. The other
papers also discuss expression of definite integrals
in terms of derivatives of products of and functions.

Adamchik, Victor (1997) "A Class of Logarithmic Integrals", Proceedings of ISSAC '97, 1-8. ACM, New York.

Geddes, K. O. and Scott, T. C. (1989) "Recipes for Classes of Definite Integrals Involving Exponentials and Logarithms", Computers and Mathematics, Kaltofen, E. and Watt, S. M. (eds.), 192-201, Springer-Verlag.

Williams, Kenneth S. and Zhang, Nan Yue (1993) "Special Values of the Lerch Zeta Function and the Evaluation of Certain Integrals", Proc. Amer. Math. Soc., 119(1), 35-49.

Zhang, Nan Yue and Williams, Kenneth S. (1993) "Applications of the Hurwitz Zeta Function to the Evaluation of Certain Integrals", Canad. Math. Bull., 36(3), 373-384.