Integral of x²/(x²+1)² dx

1. There are multiple ways to do this integral.

   Method #1

   1. Rewrite the integrand:

      $$\frac{x^{2}}{\left(x^{2} + 1\right)^{2}} = \frac{1}{x^{2} + 1} - \frac{1}{\left(x^{2} + 1\right)^{2}}$$

   2. Integrate term-by-term:

      1. The integral of $\frac{1}{x^{2} + 1}$ is $\operatorname{atan}{\left(x \right)}$.

      1. The integral of a constant times a function is the constant times the integral of the function:

         $$\int \left(- \frac{1}{\left(x^{2} + 1\right)^{2}}\right)\, dx = - \int \frac{1}{\left(x^{2} + 1\right)^{2}}\, dx$$

         TrigSubstitutionRule(theta=_theta, func=tan(_theta), rewritten=cos(_theta)**2, substep=RewriteRule(rewritten=cos(2*_theta)/2 + 1/2, substep=AddRule(substeps=[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)/2, symbol=_u), context=cos(2*_theta), symbol=_theta), context=cos(2*_theta)/2, symbol=_theta), ConstantRule(constant=1/2, context=1/2, symbol=_theta)], context=cos(2*_theta)/2 + 1/2, symbol=_theta), context=cos(_theta)**2, symbol=_theta), restriction=True, context=(x**2 + 1)**(-2), symbol=x)

         So, the result is: $- \frac{x}{2 \left(x^{2} + 1\right)} - \frac{\operatorname{atan}{\left(x \right)}}{2}$

      The result is: $- \frac{x}{2 \left(x^{2} + 1\right)} + \frac{\operatorname{atan}{\left(x \right)}}{2}$

   Method #2

   1. Rewrite the integrand:

      $$\frac{x^{2}}{\left(x^{2} + 1\right)^{2}} = \frac{x^{2}}{x^{4} + 2 x^{2} + 1}$$

   2. Rewrite the integrand:

      $$\frac{x^{2}}{x^{4} + 2 x^{2} + 1} = \frac{1}{x^{2} + 1} - \frac{1}{\left(x^{2} + 1\right)^{2}}$$

   3. Integrate term-by-term:

      1. The integral of $\frac{1}{x^{2} + 1}$ is $\operatorname{atan}{\left(x \right)}$.

      1. The integral of a constant times a function is the constant times the integral of the function:

         $$\int \left(- \frac{1}{\left(x^{2} + 1\right)^{2}}\right)\, dx = - \int \frac{1}{\left(x^{2} + 1\right)^{2}}\, dx$$

         TrigSubstitutionRule(theta=_theta, func=tan(_theta), rewritten=cos(_theta)**2, substep=RewriteRule(rewritten=cos(2*_theta)/2 + 1/2, substep=AddRule(substeps=[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)/2, symbol=_u), context=cos(2*_theta), symbol=_theta), context=cos(2*_theta)/2, symbol=_theta), ConstantRule(constant=1/2, context=1/2, symbol=_theta)], context=cos(2*_theta)/2 + 1/2, symbol=_theta), context=cos(_theta)**2, symbol=_theta), restriction=True, context=(x**2 + 1)**(-2), symbol=x)

         So, the result is: $- \frac{x}{2 \left(x^{2} + 1\right)} - \frac{\operatorname{atan}{\left(x \right)}}{2}$

      The result is: $- \frac{x}{2 \left(x^{2} + 1\right)} + \frac{\operatorname{atan}{\left(x \right)}}{2}$

   Method #3

   1. Rewrite the integrand:

      $$\frac{x^{2}}{\left(x^{2} + 1\right)^{2}} = \frac{x^{2}}{x^{4} + 2 x^{2} + 1}$$

   2. Rewrite the integrand:

      $$\frac{x^{2}}{x^{4} + 2 x^{2} + 1} = \frac{1}{x^{2} + 1} - \frac{1}{\left(x^{2} + 1\right)^{2}}$$

   3. Integrate term-by-term:

      1. The integral of $\frac{1}{x^{2} + 1}$ is $\operatorname{atan}{\left(x \right)}$.

      1. The integral of a constant times a function is the constant times the integral of the function:

         $$\int \left(- \frac{1}{\left(x^{2} + 1\right)^{2}}\right)\, dx = - \int \frac{1}{\left(x^{2} + 1\right)^{2}}\, dx$$

         TrigSubstitutionRule(theta=_theta, func=tan(_theta), rewritten=cos(_theta)**2, substep=RewriteRule(rewritten=cos(2*_theta)/2 + 1/2, substep=AddRule(substeps=[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)/2, symbol=_u), context=cos(2*_theta), symbol=_theta), context=cos(2*_theta)/2, symbol=_theta), ConstantRule(constant=1/2, context=1/2, symbol=_theta)], context=cos(2*_theta)/2 + 1/2, symbol=_theta), context=cos(_theta)**2, symbol=_theta), restriction=True, context=(x**2 + 1)**(-2), symbol=x)

         So, the result is: $- \frac{x}{2 \left(x^{2} + 1\right)} - \frac{\operatorname{atan}{\left(x \right)}}{2}$

      The result is: $- \frac{x}{2 \left(x^{2} + 1\right)} + \frac{\operatorname{atan}{\left(x \right)}}{2}$

2. Now simplify:

   $$\frac{- x + \left(x^{2} + 1\right) \operatorname{atan}{\left(x \right)}}{2 \left(x^{2} + 1\right)}$$

3. Add the constant of integration:

   $$\frac{- x + \left(x^{2} + 1\right) \operatorname{atan}{\left(x \right)}}{2 \left(x^{2} + 1\right)}+ \mathrm{constant}$$

The answer is:

$$\frac{- x + \left(x^{2} + 1\right) \operatorname{atan}{\left(x \right)}}{2 \left(x^{2} + 1\right)}+ \mathrm{constant}$$