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?


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