Integral of x*arcsin(2*x) dx

1. Use integration by parts:

   $$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$

   Let $u{\left(x \right)} = \operatorname{asin}{\left(2 x \right)}$ and let $\operatorname{dv}{\left(x \right)} = x$.

   Then $\operatorname{du}{\left(x \right)} = \frac{2}{\sqrt{1 - 4 x^{2}}}$.

   To find $v{\left(x \right)}$:

   1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$:

      $$\int x\, dx = \frac{x^{2}}{2}$$

   Now evaluate the sub-integral.

   TrigSubstitutionRule(theta=_theta, func=sin(_theta)/2, rewritten=sin(_theta)**2/8, substep=ConstantTimesRule(constant=1/8, other=sin(_theta)**2, substep=RewriteRule(rewritten=1/2 - cos(2*_theta)/2, substep=AddRule(substeps=[ConstantRule(constant=1/2, context=1/2, symbol=_theta), ConstantTimesRule(constant=-1/2, other=cos(2*_theta), substep=URule(u_var=_u, u_func=2*_theta, constant=1/2, substep=ConstantTimesRule(constant=1/2, other=cos(_u), substep=TrigRule(func='cos', arg=_u, context=cos(_u), symbol=_u), context=cos(_u), symbol=_u), context=cos(2*_theta), symbol=_theta), context=-cos(2*_theta)/2, symbol=_theta)], context=1/2 - cos(2*_theta)/2, symbol=_theta), context=sin(_theta)**2, symbol=_theta), context=sin(_theta)**2/8, symbol=_theta), restriction=(x > -1/2) & (x < 1/2), context=x**2/sqrt(1 - 4*x**2), symbol=x)

2. Now simplify:

   $$\begin{cases} \frac{x^{2} \operatorname{asin}{\left(2 x \right)}}{2} + \frac{x \sqrt{1 - 4 x^{2}}}{8} - \frac{\operatorname{asin}{\left(2 x \right)}}{16} & \text{for}\: x > - \frac{1}{2} \wedge x < \frac{1}{2} \end{cases}$$

3. Add the constant of integration:

   $$\begin{cases} \frac{x^{2} \operatorname{asin}{\left(2 x \right)}}{2} + \frac{x \sqrt{1 - 4 x^{2}}}{8} - \frac{\operatorname{asin}{\left(2 x \right)}}{16} & \text{for}\: x > - \frac{1}{2} \wedge x < \frac{1}{2} \end{cases}+ \mathrm{constant}$$

The answer is:

$$\begin{cases} \frac{x^{2} \operatorname{asin}{\left(2 x \right)}}{2} + \frac{x \sqrt{1 - 4 x^{2}}}{8} - \frac{\operatorname{asin}{\left(2 x \right)}}{16} & \text{for}\: x > - \frac{1}{2} \wedge x < \frac{1}{2} \end{cases}+ \mathrm{constant}$$