Integrating hyperbolic tanh x - tanh^3 x

I have just shown by writing in terms of sinh x and cosh x that

tanh^2 x = 1 - sech^2 x and

d/dx (tanh x) = sech^2 x

The question is to integrate (tanh x - tanh^3 (x)) which I could use a little help with.

I have got this into the form
sinh x / cosh^3 x or tanhx sech^2 x and tried integrating by parts but I got minus the exact same integral. If someone could just point out where I'm going wrong.

Thanks

tanhx differentiates to sech²x
tanh²x differentiates to 2 sech²xtanhx

does this help you?

Yeah thanks, just as you posted that I did product rule on tanh^2 x and got it.
Thanks!