(1)

\begin{align*} \int\frac{1}{\left(1+x^{2}\right)^{2}}dx & =\int\left(\frac{1+x^{2}}{\left(1+x^{2}\right)^{2}}-\frac{x^{2}}{\left(1+x^{2}\right)^{2}}\right)dx\\ & =\int\left(\frac{1}{1+x^{2}}+\frac{x}{2}\frac{-2x}{\left(1+x^{2}\right)^{2}}\right)dx\\ & =\tan^{\bullet}x+C+\frac{x}{2}\frac{1}{1+x^{2}}-\frac{1}{2}\int\left(\frac{1}{1+x^{2}}\right)dx\\ & =\tan^{\bullet}x+C+\frac{x}{2\left(1+x^{2}\right)}-\frac{1}{2}\tan^{\bullet}x\\ & =\frac{1}{2}\tan^{\bullet}x+\frac{x}{2\left(1+x^{2}\right)}+C \end{align*}

(2)

\begin{align*} \int\frac{x}{\left(1+x^{2}\right)^{2}}dx & =\int\frac{1}{2\left(1+x^{2}\right)^{2}}d\left(1+x^{2}\right)\\ & =-\frac{1}{2\left(1+x^{2}\right)}+C \end{align*}

(3)

\begin{align*} \int\frac{x^{2}}{\left(1+x^{2}\right)^{2}}dx & =\int\left(\frac{1+x^{2}}{\left(1+x^{2}\right)^{2}}-\frac{1}{\left(1+x^{2}\right)^{2}}\right)dx\\ & =\int\left(\frac{1}{1+x^{2}}-\frac{1}{\left(1+x^{2}\right)^{2}}\right)dx\\ & =\tan^{\bullet}x-\left(\frac{1}{2}\tan^{\bullet}x+\frac{x}{2\left(1+x^{2}\right)}\right)+C\\ & =\frac{1}{2}\tan^{\bullet}x-\frac{x}{2\left(1+x^{2}\right)}+C \end{align*}

(4)

\begin{align*} \int\frac{x^{3}}{\left(1+x^{2}\right)^{2}}dx & =\int\left(\frac{x\left(x^{2}+1\right)}{\left(1+x^{2}\right)^{2}}-\frac{x}{\left(1+x^{2}\right)^{2}}\right)dx\\ & =\int\left(\frac{x}{1+x^{2}}-\frac{x}{\left(1+x^{2}\right)^{2}}\right)dx\\ & =\int\frac{1}{2\left(1+x^{2}\right)}d\left(1+x^{2}\right)-\int\frac{x}{\left(1+x^{2}\right)^{2}}dx\\ & =\frac{1}{2}\log\left(1+x^{2}\right)-\left(-\frac{1}{2\left(1+x^{2}\right)}\right)+C\\ & =\frac{1}{2}\log\left(1+x^{2}\right)+\frac{1}{2\left(1+x^{2}\right)}+C \end{align*}

(5)

\begin{align*} \int\frac{x^{4}}{\left(1+x^{2}\right)^{2}}dx & =\int\left(\frac{\left(1+x^{2}\right)^{2}}{\left(1+x^{2}\right)^{2}}-\frac{2x^{2}+1}{\left(1+x^{2}\right)^{2}}\right)dx\\ & =\int\left(1-2\frac{x^{2}+1}{\left(1+x^{2}\right)^{2}}+\frac{1}{\left(1+x^{2}\right)^{2}}\right)dx\\ & =\int\left(1-2\frac{1}{\left(1+x^{2}\right)}+\frac{1}{\left(1+x^{2}\right)^{2}}\right)dx\\ & =x-2\tan^{\bullet}x+\frac{1}{2}\tan^{\bullet}x+\frac{x}{2\left(1+x^{2}\right)}+C\\ & =x-\frac{3}{2}\tan^{\bullet}x+\frac{x}{2\left(1+x^{2}\right)}+C \end{align*}