What is anti-derivative of this function?

Question

Let $f(x)$ be an arbitrary continuous function, $n\in \mathbb{N}$ and
$$g(x) = \frac{1}{1+n\cdot f(x)^2}$$ then what is anti-derivative of this: $$ \int \left(\frac{d}{dx}g(x)\right)\cdot\tanh\left(n\cdot f(x)\cdot\frac{d}{dx}f(x)\right)dx = ? $$ or $$ \int \left(\frac{d}{dx}g(x)\right)\cdot\tanh\left(n\cdot f(x)\right)\cdot\tanh\left(n\cdot\frac{d}{dx}f(x)\right)dx = ? $$ Both integral equals(in the limit when n goes to infinity). Also we know that $\int \left(\frac{d}{dx}g(x)\right)dx = g(x)$ and $\int \left(n.f(x).\frac{d}{dx}f(x)\right)dx = n\cdot\frac{f(x)^2}{2}$
I used Mathcad and Maple for simplifying this anti-derivative, but they can't solve this problem.

Answer 1

$$\begin{cases} I=\int \left(\frac{d}{dx}g(x)\right)\cdot\tanh\left(n\cdot f(x)\cdot\frac{d}{dx}f(x)\right)dx \\ g(x) = \frac{1}{1+n\cdot f(x)^2} \end{cases} $$ FIRST CASE :

$g(x)$ is the given (known) function and one want to express the next integral in terms of $g(x)$ : $$ I=\int \left(\frac{dg}{dx}\tanh\left(\frac{n}{2}\frac{d(f^2)}{dx}\right)\right)dx $$ $g(x) = \frac{1}{1+n\cdot f(x)^2}\quad\implies\quad f(x)^2=\frac{1}{n(g-1)}$

$\frac{d(f^2)}{dx}= -\frac{1}{n(g-1)^2}\frac{dg}{dx}$ $$ I=\int \left(\frac{dg}{dx}\tanh\left(-\frac{n}{2}\frac{1}{n(g-1)^2}\frac{dg}{dx} \right)\right)dx $$ $$ I=-\frac12\int \left(\frac{dg}{dx}\tanh\left(\frac{1}{(g-1)^2}\frac{dg}{dx} \right)\right)dx $$ So, as expected, the integral is expressed with $g(x)$ only. Further calculus requiers the explicit form of $g(x)$ which is not specified in the wording of the question.

SECOND CASE :

$f(x)$ is the given (known) function and one want to express the next integral in terms of $f(x)$ : $$ I=\int \left(\frac{dg}{dx}\tanh\left(\frac{n}{2}\frac{d(f^2)}{dx}\right)\right)dx $$ $\frac{dg}{dx}= -\frac{2n}{(1+nf^2)^2}\frac{d(f^2)}{dx}$ $$ I=\int \left(-\frac{2n}{(1+nf^2)^2}\frac{d(f^2)}{dx}\tanh\left(\frac{n}{2}\frac{d(f^2)}{dx}\right)\right)dx $$ $$ I=-2n\int \left(-\frac{1}{(1+nf^2)^2}\frac{d(f^2)}{dx}\tanh\left(\frac{n}{2}\frac{d(f^2)}{dx}\right)\right)dx $$ So, as expected, the integral is expressed with $f(x)$ only. Further calculus requiers the explicit form of $f(x)$ which is not specified in the wording of the question.