Evaluate the following $\int \frac {(\sinh^3 x+1)\cosh x}{\sinh^2 x+1}dx$
Keep in mind I only learnt the following identity for hyperbolic functions so far $\sinh^2x+1=\cosh^2 x$ and I only know how to differentiate cosh and sinh.
Here is my attempt at a solution
$\int \frac {(\sinh^3 x+1)\cosh x}{\sinh^2 x+1}dx=\int \frac {(\sinh^3 x+1)\cosh x}{\cosh^2 x}dx=\int \frac {(\sinh^3 x+1)}{\cosh x}dx=\int \frac {(\sinh x+1)(\sinh^2 x - \sinh x +1)}{\cosh x}dx=\int \frac {(\sinh x+1)(\cosh^2 x - \sinh x)}{\cosh x}dx=\int \frac {\cosh^2 x \sinh x-\sinh^2 x+\cosh^2 x - \sinh x}{\cosh x}dx=\int\cosh x \sinh x-\frac{\sinh^2 x}{\cosh x}+\cosh x-\frac{\sinh x}{\cosh x}=0.5\sinh^2 x+\sinh x-\ln|\cosh x|-\int\frac{\sinh^2 x}{\cosh x}dx$
I find myself unable to integrate $\int\frac{\sinh^2 x}{\cosh x}dx$, Hints only thanks in advance!