My task is to evaluate the following:
$$\tag{ab > 0}\int_a^b\dfrac{\tanh(x)}{2\cdot\sqrt{\cosh(x)-1}}dx$$
I tried to do the Weierstraß-Substitution but it did not work out for me.
Wolfram alpha only gave me a complex solution however the solution I got from the university is: $$\arctan\left(\sqrt{\cosh(x)-1}\right)\Bigg\vert_a^b$$
I would be very happy if someone could show me how to get to that solution and why$$a \cdot b > 0$$ is given as an information.
Substitute $u = \sqrt{\cosh x -1}$, then $\cosh x = u^2 + 1$, and by the chain rule
$$du = \dfrac{\sinh x}{2\sqrt{\cosh x-1}}dx$$
Therefore
$$ \int \frac{\tanh x}{\sqrt{\cosh x - 1}} dx = \int \frac{\sinh x}{2\cosh x\sqrt{\cosh x-1}} dx = \int\frac{du}{u^2+1} = \arctan u + C $$