\documentclass[a4paper]{article}
\usepackage{amsmath}
\DeclareMathOperator{\artanh}{artanh}
\begin{document}
\begin{eqnarray*}
f (x)
 & = & \left ( \frac{x^{2}}{a^{2} - x^{2}} \right )^{2} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
x
 & = & a \sin t \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
t
 & = & \arcsin \frac{x}{a} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
F (x)
 & = & \int \left ( \frac{x^{2}}{a^{2} - x^{2}} \right )^{2} \, \mbox{d}x \\
& = & \int \left ( \frac{(a \sin t)^{2}}{a^{2} - (a \sin t)^{2}} \right )^{2} \, \mbox{d}(a \sin t) \\
& = & a \int \left ( \frac{a^{2} \sin^{2} t}{a^{2} - a^{2} \sin^{2} t} \right )^{2} \cos t \, \mbox{d}t \\
& = & a \int \left ( \frac{\sin^{2} t}{1 - \sin^{2} t} \right )^{2} \cos t \, \mbox{d}t \\
& = & a \int \left ( \frac{1 - \cos^{2} t}{\cos^{2} t} \right )^{2} \cos t \, \mbox{d}t \\
& = & a \int \left ( \frac{1}{\cos^{2} t} - \frac{\cos^{2} t}{\cos^{2} t} \right )^{2} \cos t \, \mbox{d}t \\
& = & a \int (\sec^{2} t - 1)^{2} \cos t \, \mbox{d}t \\
& = & a \int (\sec^{4} t - 2 \sec^{2} t + 1) \cos t \, \mbox{d}t \\
& = & a \int (\sec^{3} t - 2 \sec t + \cos t) \, \mbox{d}t \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
F (x)
 & = & a \int (\sec^{3} t - 2 \sec t + \cos t) \, \mbox{d}t \\
& = & a \left ( \frac{1}{2} \ln \left | \, \frac{1 + \sin t}{\cos t} \, \right |
 + \frac{1}{2} \frac{\sin t}{\cos^{2} t}
 - 2 \ln \left | \, \frac{1 + \sin t}{\cos t} \, \right |
 + \sin t \right ) + c \\
& = & a \left ( - \frac{3}{2} \ln \left | \, \frac{1 + \sin t}{\cos t} \, \right |
 + \frac{1}{2} \frac{\sin t}{\cos^{2} t}
 + \sin t \right ) + c \\
& = & a \left ( - \frac{3}{2} \ln \left | \, \frac{1 + \sin t}{\sqrt{1 - \sin^{2} t}} \, \right |
 + \frac{1}{2} \frac{\sin t}{1 - \sin^{2} t}
 + \sin t \right ) + c \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
F (x)
 & = & a \left ( - \frac{3}{2} \ln \left | \, \frac{1 + \sin t}{\sqrt{1 - \sin^{2} t}} \, \right |
 + \frac{1}{2} \frac{\sin t}{1 - \sin^{2} t}
 + \sin t \right ) + c \\
& = & a \left ( - \frac{3}{2} \ln \left | \, \frac{1 + \sin \arcsin \frac{x}{a}}{\sqrt{1 - \left ( \sin \arcsin \frac{x}{a} \right )^{2}}} \, \right |
 + \frac{1}{2} \frac{\sin \arcsin \frac{x}{a}}{1 - \left ( \sin \arcsin \frac{x}{a} \right )^{2}}
 + \sin \arcsin \frac{x}{a} \right ) + c \\
& = & a \left ( - \frac{3}{2} \ln \left | \, \frac{1 + \frac{x}{a}}{\sqrt{1 - \left ( \frac{x}{a} \right )^{2}}} \, \right |
 + \frac{1}{2} \frac{\frac{x}{a}}{1 - \left ( \frac{x}{a} \right )^{2}}
 + \frac{x}{a} \right ) + c \\
& = & - \frac{3}{2} a \ln \left | \, \frac{a + x}{\sqrt{a^{2} - x^{2}}} \, \right |
 + \frac{1}{2} a^{2} \frac{x}{a^{2} - x^{2}}
 + x + c \\
& = & - \frac{3}{2} a \ln \left | \, \sqrt{\frac{(a + x) (a + x)}{(a - x) (a + x)}} \, \right |
 + \frac{1}{2} a^{2} \frac{x}{a^{2} - x^{2}}
 + x + c \\
& = & - \frac{3}{4} a \ln \left | \, \frac{a + x}{a - x} \, \right |
 + \frac{1}{2} a^{2} \frac{x}{a^{2} - x^{2}}
 + x + c \\
& = & - \frac{3}{4} a \ln \left | \, \frac{1 + \frac{x}{a}}{1 - \frac{x}{a}} \, \right |
 + \frac{1}{2} a^{2} \frac{x}{a^{2} - x^{2}}
 + x + c \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\artanh x
 & = & \frac{1}{2} \ln \frac{1 + x}{1 - x} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
F (x)
 & = & - \frac{3}{4} a \ln \left | \, \frac{1 + \frac{x}{a}}{1 - \frac{x}{a}} \, \right |
 + \frac{1}{2} a^{2} \frac{x}{a^{2} - x^{2}}
 + x + c \\
& = & - \frac{3}{2} a \artanh \frac{x}{a}
 + \frac{1}{2} a^{2} \frac{x}{a^{2} - x^{2}}
 + x + c \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
f (x)
 & = & \frac{\mbox{d}(F (x))}{\mbox{d}x} \\
& = & \frac{\mbox{d}}{\mbox{d}x} \left ( - \frac{3}{2} a \artanh \frac{x}{a} + \frac{1}{2} a^{2} \frac{x}{a^{2} - x^{2}}
 + x + c \right ) \\
& = & - \frac{3}{2} a \frac{1}{1 - \left ( \frac{x}{a} \right )^{2}} \frac{1}{a}
 + \frac{1}{2} a^{2} \frac{1 (a^{2} - x^{2}) - x (- 2x)}{(a^{2} - x^{2})^{2}}
 + 1 + 0 \\
& = & - \frac{3}{2} \frac{a^{2}}{a^{2} - x^{2}} \frac{a^{2} - x^{2}}{a^{2} - x^{2}}
 + \frac{1}{2} a^{2} \frac{a^{2} - x^{2} + 2x^{2}}{(a^{2} - x^{2})^{2}}
 + \frac{(a^{2} - x^{2})^{2}}{(a^{2} - x^{2})^{2}} \\
& = & \frac{- \frac{3}{2} a^{2} (a^{2} - x^{2}) + \frac{1}{2} a^{2} (a^{2} + x^{2}) + (a^{2} - x^{2})^{2}}{(a^{2} - x^{2})^{2}} \\
& = & \frac{- \frac{3}{2} a^{4} + \frac{3}{2} a^{2} x^{2} + \frac{1}{2} a^{4} + \frac{1}{2} a^{2} x^{2} + a^{4} - 2 a^{2} x^{2} + x^{4}}{(a^{2} - x^{2})^{2}} \\
& = & \frac{x^{4}}{(a^{2} - x^{2})^{2}} \\
& = & \left ( \frac{x^{2}}{a^{2} - x^{2}} \right )^{2} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
F (x)
 & = & - \frac{3}{2} a \artanh \frac{x}{a}
 + x + \frac{1}{2} a^{2} \frac{x}{a^{2} - x^{2}} + c \\
& = & - \frac{3}{4} a \ln \left | \, \frac{a + x}{a - x} \, \right |
 + x + \frac{1}{2} a^{2} \frac{x}{a^{2} - x^{2}} + c \\
\end{eqnarray*}
\end{document}
