You should try your problem in the integrator

This site contains a list of integrals provided by Gosper involving Glaisher's constant.

Some more jewels "stolen" from Bill Gosper: First and Second some Stieltjes integrals.

This is a connection to Khintchine constant homepage.

This site contains a list of integrals involving Euler's constant. Here is a link to Euler's constant homepage.

This site contains a list of integrals involving Catalan's constant.

This site contains a list of integrals involving Apery's constant. Here is a link to Apery's constant homepage.

Nice collection by A. Dieckmann.