integral problem $\int \frac{2 \lambda a}{\mathbf{ (e^{at}-1)\lambda \sigma^2+2ae^{-at}}}dt $

Question

Does anybody know how to tackle the below integral? I am analyzing a formula derivation where this appears as the final calculation, but I don't know how to get it solved

$$\int \frac{2 \lambda a}{\mathbf{ (e^{at}-1)\lambda \sigma^2+2ae^{-at}}}dt $$

this should result in $$\frac{2ab \ \big{[} \ at+log(2a)+log \big{(} \mathbf{(e^{at}-1)\lambda \sigma^2 +2ae^{-at}}\big{)} \ \big{]}}{\sigma^2}$$

Answer 1

HINT:

Note that the integral can be written

$$\int \frac{2 \lambda a}{ (e^{at}-1)\lambda \sigma^2+2ae^{-at}}\,dt=\int \frac{2 \lambda a e^{at}}{e^{at} (e^{at}-1)\lambda \sigma^2+2a}\,dt$$

Now, enforce the substitution $x=e^{at}$ with $dx=ae^{at}\,dt$ and evaluate the integral

$$\int \frac{2\lambda}{\lambda \sigma^2x(x-1)+2a}\,dx$$