Verify the integration formula:

$$\int \operatorname{sech} x \, dx =\sin^{-1}(\tanh x)+C$$

I know that I have to manipulate the equation, but I am not sure how. Here is what I have done so far:

\begin{align}
& \int \operatorname{sech} x \, dx \\[6pt]
= {} & \int{\operatorname{sech}^3x\over \operatorname{sech}^2 x} dx \\[6pt]
= {} & \int{\operatorname{sech}^3x\over 1-\operatorname{tanh}^2x} \,dx
\end{align}

$$u= \operatorname{tanh} x, \quad du= \operatorname{sech}^2x \, dx$$

I am not even sure if I am even doing anything right up to this point. If someone could steer me in the right direction that would be great.