Integral of $$$\operatorname{asin}{\left(\frac{2 x}{x^{2} + 1} \right)}$$$

Find $$$\int \operatorname{asin}{\left(\frac{2 x}{x^{2} + 1} \right)}\, dx$$$.

For the integral $$$\int{\operatorname{asin}{\left(\frac{2 x}{x^{2} + 1} \right)} d x}$$$, use integration by parts $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$.

Let $$$\operatorname{u}=\operatorname{asin}{\left(\frac{2 x}{x^{2} + 1} \right)}$$$ and $$$\operatorname{dv}=dx$$$.

Then $$$\operatorname{du}=\left(\operatorname{asin}{\left(\frac{2 x}{x^{2} + 1} \right)}\right)^{\prime }dx=\frac{2 \left(1 - x^{2}\right)}{\left(x^{2} + 1\right) \sqrt{x^{4} - 2 x^{2} + 1}} dx$$$ (steps can be seen ») and $$$\operatorname{v}=\int{1 d x}=x$$$ (steps can be seen »).

So,

$${\color{red}{\int{\operatorname{asin}{\left(\frac{2 x}{x^{2} + 1} \right)} d x}}}={\color{red}{\left(\operatorname{asin}{\left(\frac{2 x}{x^{2} + 1} \right)} \cdot x-\int{x \cdot \frac{2 \left(1 - x^{2}\right)}{\left(x^{2} + 1\right) \sqrt{x^{4} - 2 x^{2} + 1}} d x}\right)}}={\color{red}{\left(x \operatorname{asin}{\left(\frac{2 x}{x^{2} + 1} \right)} - \int{\left(- \frac{2 x \left(x - 1\right) \left(x + 1\right)}{\left(x^{2} + 1\right) \left|{x - 1}\right| \left|{x + 1}\right|}\right)d x}\right)}}$$

Let $$$u=x^{2} + 1$$$.

Then $$$du=\left(x^{2} + 1\right)^{\prime }dx = 2 x dx$$$ (steps can be seen »), and we have that $$$x dx = \frac{du}{2}$$$.

The integral can be rewritten as

$$x \operatorname{asin}{\left(\frac{2 x}{x^{2} + 1} \right)} - {\color{red}{\int{\left(- \frac{2 x \left(x - 1\right) \left(x + 1\right)}{\left(x^{2} + 1\right) \left|{x - 1}\right| \left|{x + 1}\right|}\right)d x}}} = x \operatorname{asin}{\left(\frac{2 x}{x^{2} + 1} \right)} - {\color{red}{\int{\frac{2 - u}{u \left(u - 2\right)} d u}}}$$

Simplify the integrand:

$$x \operatorname{asin}{\left(\frac{2 x}{x^{2} + 1} \right)} - {\color{red}{\int{\frac{2 - u}{u \left(u - 2\right)} d u}}} = x \operatorname{asin}{\left(\frac{2 x}{x^{2} + 1} \right)} - {\color{red}{\int{\left(- \frac{1}{u}\right)d u}}}$$

Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=-1$$$ and $$$f{\left(u \right)} = \frac{1}{u}$$$:

$$x \operatorname{asin}{\left(\frac{2 x}{x^{2} + 1} \right)} - {\color{red}{\int{\left(- \frac{1}{u}\right)d u}}} = x \operatorname{asin}{\left(\frac{2 x}{x^{2} + 1} \right)} - {\color{red}{\left(- \int{\frac{1}{u} d u}\right)}}$$

The integral of $$$\frac{1}{u}$$$ is $$$\int{\frac{1}{u} d u} = \ln{\left(\left|{u}\right| \right)}$$$:

$$x \operatorname{asin}{\left(\frac{2 x}{x^{2} + 1} \right)} + {\color{red}{\int{\frac{1}{u} d u}}} = x \operatorname{asin}{\left(\frac{2 x}{x^{2} + 1} \right)} + {\color{red}{\ln{\left(\left|{u}\right| \right)}}}$$

Recall that $$$u=x^{2} + 1$$$:

$$x \operatorname{asin}{\left(\frac{2 x}{x^{2} + 1} \right)} + \ln{\left(\left|{{\color{red}{u}}}\right| \right)} = x \operatorname{asin}{\left(\frac{2 x}{x^{2} + 1} \right)} + \ln{\left(\left|{{\color{red}{\left(x^{2} + 1\right)}}}\right| \right)}$$

Therefore,

$$\int{\operatorname{asin}{\left(\frac{2 x}{x^{2} + 1} \right)} d x} = x \operatorname{asin}{\left(\frac{2 x}{x^{2} + 1} \right)} + \ln{\left(x^{2} + 1 \right)}$$

Add the constant of integration:

$$\int{\operatorname{asin}{\left(\frac{2 x}{x^{2} + 1} \right)} d x} = x \operatorname{asin}{\left(\frac{2 x}{x^{2} + 1} \right)} + \ln{\left(x^{2} + 1 \right)}+C$$

$$$\int \operatorname{asin}{\left(\frac{2 x}{x^{2} + 1} \right)}\, dx = \left(x \operatorname{asin}{\left(\frac{2 x}{x^{2} + 1} \right)} + \ln\left(x^{2} + 1\right)\right) + C$$$A