What is anti-derivative of this function? 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.
 A: $$\begin{cases}
I=\int \left(\frac{d}{dx}g(x)\right)\cdot\tanh\left(n\cdot f(x)\cdot\frac{d}{dx}f(x)\right)dx \\
g(x) = \frac{1}{1+n\cdot f(x)^2}
\end{cases} 
$$ 
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.
