I have been trying for a while now to solve the following integral:
$$ \int_0^t e^{-ct} \left(\int_0^t(1-M_r t)^a e^{ct} dt\right) (1-M_r t)^{-(a+1)} dt $$
I know the answer is: $$ \frac {1}{M_r^2} [(1-M_r t) \ln( (1-M_r t) -1)] $$
the only approach I can think of is to use integration by parts and setting the integral within the integral to be the function which is differentiated. However, this does not seem to lead to the answer. Does anybody have ANY idea? A reference, term I could search. Literally anything that could point me to the right trail would be helpful.