integrales de funciones trigonometricas hiperbolicas

[Wes Loewer]

I was asked privately about the integral ∫tanh(x) dx

The Prime gives
∫(tanh(x),x) --> ln((e^x)^2+1)-x
which is correct.

By hand I would have done the following:

∫tanh(x) dx = ∫sinh(x)/cosh(x) dx = ln(cosh(x))+C

but this can be rewritten as

= ln((e^x+e^-x)/2) + C
= ln((e^(2x)+1)/(2e^x)) + C
= ln(e^2x)+1) - ln(2e^x) + C
= ln(e^2x)+1) - ln(2) - ln(e^x) + C
= ln(e^2x)+1) - x + (C-ln(2))
= ln(e^2x)+1) - x + D

Once again, the correct answers can be rewritten such that they differ by only a constant.

I've learned over the years that whenever a CAS's antiderivative looks different than mine, it's usually that they differ by a constant, or my antiderivative is wrong. :-)