I'm stuck solving the following integral
$$\int_{-\infty}^t \sin(\omega_o(t-t'))(1+\tanh(\frac{t'}{\tau}))dt'$$
it seems to me it should converge at$-\infty$ because of the $\tanh(t'/\tau)$ part.
I've tried with some contours (it seemed like the right way to go) but that didn't work, I can't think of a good substitution, and all online integral calculators can't seem to solve it.
The problem is originally generated by trying to solve $\ddot{x}+\omega_0x=f_0/2(1+\tanh(t/\tau))$ using Green's functions
any suggestion?