\documentclass[a4paper]{article}
\usepackage{amsmath}
\DeclareMathOperator{\artanh}{artanh}
\begin{document}
\begin{eqnarray*}
f (x)
 & = & \left ( \frac{x^{2}}{a^{2} - x^{2}} \right )^{4} \\
\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 )^{4} \, \mbox{d}x \\
& = & \int \left ( \frac{(a \sin t)^{2}}{a^{2} - (a \sin t)^{2}} \right )^{4} \, \mbox{d}(a \sin t) \\
& = & a \int \left ( \frac{a^{2} \sin^{2} t}{a^{2} - a^{2} \sin^{2} t} \right )^{4} \cos t \, \mbox{d}t \\
& = & a \int \left ( \frac{\sin^{2} t}{1 - \sin^{2} t} \right )^{4} \cos t \, \mbox{d}t \\
& = & a \int \left ( \frac{1 - \cos^{2} t}{\cos^{2} t} \right )^{4} \cos t \, \mbox{d}t \\
& = & a \int \left ( \frac{1}{\cos^{2} t} - \frac{\cos^{2} t}{\cos^{2} t} \right )^{4} \cos t \, \mbox{d}t \\
& = & a \int (\sec^{2} t - 1)^{4} \cos t \, \mbox{d}t \\
& = & a \int (\sec^{8} t - 4 \sec^{6} t + 6 \sec^{4} t - 4 \sec^{2} t + 1) \cos t \, \mbox{d}t \\
& = & a \int (\sec^{7} t - 4 \sec^{5} t + 6 \sec^{3} t - 4 \sec t + \cos t) \, \mbox{d}t \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
F (x)
 & = & a \int (\sec^{7} t - 4 \sec^{5} t + 6 \sec^{3} t - 4 \sec t + \cos t) \, \mbox{d}t \\
& = & a \biggl ( \frac{1}{6} \frac{\sin t}{\cos^{6} t}
 + \frac{5}{24} \frac{\sin t}{\cos^{4} t}
 + \frac{5}{16} \frac{\sin t}{\cos^{2} t}
 + \frac{5}{16} \ln \left | \, \frac{1 + \sin t}{\cos t} \, \right |
 - \frac{\sin t}{\cos^{4} t}
 - \frac{3}{2} \frac{\sin t}{\cos^{2} t}
 - \frac{3}{2} \ln \left | \, \frac{1 + \sin t}{\cos t} \, \right | \\
 & & \qquad \qquad + \, 3 \ln \left | \, \frac{1 + \sin t}{\cos t} \, \right |
 + 3 \frac{\sin t}{\cos^{2} t}
 - 4 \ln \left | \, \frac{1 + \sin t}{\cos t} \, \right |
 + \sin t \biggr ) + c \\
& = & a \left (
 - \frac{35}{16} \ln \left | \, \frac{1 + \sin t}{\sqrt{1 - \sin^{2} t}} \, \right |
 + \frac{1}{6} \frac{\sin t}{\cos^{6} t}
 - \frac{19}{24} \frac{\sin t}{\cos^{4} t}
 + \frac{29}{16} \frac{\sin t}{\cos^{2} t}
 + \sin t \right ) + c \\
& = & a \left (
 - \frac{35}{16} \ln \left | \, \frac{1 + \sin t}{\sqrt{1 - \sin^{2} t}} \, \right |
 + \frac{1}{6} \frac{\sin t}{(1 - \sin^{2} t)^{3}}
 - \frac{19}{24} \frac{\sin t}{(1 - \sin^{2} t)^{2}}
 + \frac{29}{16} \frac{\sin t}{1 - \sin^{2} t}
 + \sin t \right ) + c \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
F (x)
 & = & a \left (
 - \frac{35}{16} \ln \left | \, \frac{1 + \sin t}{\sqrt{1 - \sin^{2} t}} \, \right |
 + \frac{1}{6} \frac{\sin t}{(1 - \sin^{2} t)^{3}}
 - \frac{19}{24} \frac{\sin t}{(1 - \sin^{2} t)^{2}}
 + \frac{29}{16} \frac{\sin t}{1 - \sin^{2} t}
 + \sin t \right ) + c \\
& = & a \left (
 - \frac{35}{16} \ln \left | \, \frac{1 + \sin \arcsin \frac{x}{a}}{\sqrt{1 - \left ( \sin \arcsin \frac{x}{a} \right )^{2}}} \, \right |
 + \frac{1}{6} \frac{\sin \arcsin \frac{x}{a}}{\left ( 1 - \left ( \sin \arcsin \frac{x}{a} \right )^{2} \right )^{3}}
 - \frac{19}{24} \frac{\sin \arcsin \frac{x}{a}}{\left ( 1 - \left ( \sin \arcsin \frac{x}{a} \right )^{2} \right )^{2}} \right . \\
 & & \qquad \qquad \left . + \, \frac{29}{16} \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{35}{16} \ln \left | \, \frac{1 + \frac{x}{a}}{\sqrt{1 - \left ( \frac{x}{a} \right )^{2}}} \, \right |
 + \frac{1}{6} \frac{\frac{x}{a}}{\left ( 1 - \left ( \frac{x}{a} \right )^{2} \right )^{3}}
 - \frac{19}{24} \frac{\frac{x}{a}}{\left ( 1 - \left ( \frac{x}{a} \right )^{2} \right )^{2}}
 + \frac{29}{16} \frac{\frac{x}{a}}{1 - \left ( \frac{x}{a} \right )^{2}}
 + \frac{x}{a} \right ) + c \\
& = & - \frac{35}{16} a \ln \left | \, \frac{a + x}{\sqrt{a^{2} - x^{2}}} \, \right |
 + \frac{1}{6} a^{6} \frac{x}{(a^{2} - x^{2})^{3}}
 - \frac{19}{24} a^{4} \frac{x}{(a^{2} - x^{2})^{2}}
 + \frac{29}{16} a^{2} \frac{x}{a^{2} - x^{2}}
 + x + c \\
& = & - \frac{35}{16} a \ln \left | \, \sqrt{\frac{(a + x) (a + x)}{(a - x) (a + x)}} \, \right |
 + \frac{1}{6} a^{6} \frac{x}{(a^{2} - x^{2})^{3}}
 - \frac{19}{24} a^{4} \frac{x}{(a^{2} - x^{2})^{2}}
 + \frac{29}{16} a^{2} \frac{x}{a^{2} - x^{2}}
 + x + c \\
& = & - \frac{35}{32} a \ln \left | \, \frac{a + x}{a - x} \, \right |
 + \frac{1}{6} a^{6} \frac{x}{(a^{2} - x^{2})^{3}}
 - \frac{19}{24} a^{4} \frac{x}{(a^{2} - x^{2})^{2}}
 + \frac{29}{16} a^{2} \frac{x}{a^{2} - x^{2}}
 + x + c \\
& = & - \frac{35}{32} a \ln \left | \, \frac{1 + \frac{x}{a}}{1 - \frac{x}{a}} \, \right |
 + \frac{1}{6} a^{6} \frac{x}{(a^{2} - x^{2})^{3}}
 - \frac{19}{24} a^{4} \frac{x}{(a^{2} - x^{2})^{2}}
 + \frac{29}{16} 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{35}{32} a \ln \left | \, \frac{1 + \frac{x}{a}}{1 - \frac{x}{a}} \, \right |
 + \frac{1}{6} a^{6} \frac{x}{(a^{2} - x^{2})^{3}}
 - \frac{19}{24} a^{4} \frac{x}{(a^{2} - x^{2})^{2}}
 + \frac{29}{16} a^{2} \frac{x}{a^{2} - x^{2}}
 + x + c \\
& = & - \frac{35}{16} a \artanh \frac{x}{a}
 + \frac{1}{6} a^{6} \frac{x}{(a^{2} - x^{2})^{3}}
 - \frac{19}{24} a^{4} \frac{x}{(a^{2} - x^{2})^{2}}
 + \frac{29}{16} 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{35}{16} a \artanh \frac{x}{a}
 + \frac{1}{6} a^{6} \frac{x}{(a^{2} - x^{2})^{3}}
 - \frac{19}{24} a^{4} \frac{x}{(a^{2} - x^{2})^{2}}
 + \frac{29}{16} a^{2} \frac{x}{a^{2} - x^{2}}
 + x + c \right ) \\
& = & - \frac{35}{16} a \frac{1}{1 - \left ( \frac{x}{a} \right )^{2}} \frac{1}{a}
 + \frac{1}{6} a^{6} \frac{1 (a^{2} - x^{2})^{3} - 3x (a^{2} - x^{2})^{2} (- 2x)}{(a^{2} - x^{2})^{6}} \\
 & & \qquad - \, \frac{19}{24} a^{4} \frac{1 (a^{2} - x^{2})^{2} - 2x (a^{2} - x^{2}) (- 2x)}{(a^{2} - x^{2})^{4}}
 + \frac{29}{16} a^{2} \frac{1 (a^{2} - x^{2}) - x (- 2x)}{(a^{2} - x^{2})^{2}}
 + 1 + 0 \\
& = & - \frac{35}{16} \frac{a^{2}}{a^{2} - x^{2}} \frac{(a^{2} - x^{2})^{3}}{(a^{2} - x^{2})^{3}}
 + \frac{1}{6} a^{6} \frac{a^{2} - x^{2} + 6x^{2}}{(a^{2} - x^{2})^{4}}
 - \frac{19}{24} a^{4} \frac{(a^{2} - x^{2})^{2} + 4x^{2} (a^{2} - x^{2})}{(a^{2} - x^{2})^{4}} \\
 & & \qquad + \, \frac{29}{16} 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{(a^{2} - x^{2})^{4}}{(a^{2} - x^{2})^{4}} \\
& = & \Biggl ( - \frac{35}{16} a^{2} (a^{6} - 3 a^{4} x^{2} + 3 a^{2} x^{4} - x^{6})
 + \frac{1}{6} a^{6} (a^{2} + 5x^{2})
 - \frac{19}{24} a^{4} (a^{4} - 2 a^{2} x^{2} + x^{4} + 4 a^{2} x^{2} - 4 x^{4}) \Biggr . \\
 & & \qquad \Biggl . + \, \frac{29}{16} a^{2} (a^{2} + x^{2}) (a^{4} - 2 a^{2} x^{2} + x^{4})
 + a^{8} - 4 a^{6} x^{2} + 6 a^{4} x^{4} - 4 a^{2} x^{6} + x^{8} \Biggr ) (a^{2} - x^{2})^{-4} \\
& = & \Biggl ( - \frac{35}{16} a^{8} + \frac{105}{16} a^{6} x^{2} - \frac{105}{16} a^{4} x^{4} + \frac{35}{16} a^{2} x^{6}
 + \frac{1}{6} a^{8} + \frac{5}{6} a^{6} x^{2}
 - \frac{19}{24} a^{4} (a^{4} + 2 a^{2} x^{2} - 3 x^{4}) \Biggr . \\
 & & \qquad \Biggl . + \, \frac{29}{16} a^{2} (a^{6} - 2 a^{4} x^{2} + a^{2} x^{4} + a^{4} x^{2} - 2 a^{2} x^{4} + x^{6})
 + a^{8} - 4 a^{6} x^{2} + 6 a^{4} x^{4} - 4 a^{2} x^{6} + x^{8} \Biggr ) \\
 & & \qquad \qquad \cdot (a^{2} - x^{2})^{-4} \\
& = & \Biggl ( - \frac{35}{16} a^{8} + \frac{105}{16} a^{6} x^{2} - \frac{105}{16} a^{4} x^{4} + \frac{35}{16} a^{2} x^{6}
 + \frac{1}{6} a^{8} + \frac{5}{6} a^{6} x^{2}
 - \frac{19}{24} a^{8} - \frac{19}{12} a^{6} x^{2} + \frac{19}{8} a^{4} x^{4} \Biggr . \\
 & & \qquad \Biggl . + \, \frac{29}{16} a^{8} - \frac{29}{16} a^{6} x^{2} - \frac{29}{16} a^{4} x^{4} + \frac{29}{16} a^{2} x^{6}
 + a^{8} - 4 a^{6} x^{2} + 6 a^{4} x^{4} - 4 a^{2} x^{6} + x^{8} \Biggr ) (a^{2} - x^{2})^{-4} \\
& = & \frac{x^{8}}{(a^{2} - x^{2})^{4}} \\
& = & \left ( \frac{x^{2}}{a^{2} - x^{2}} \right )^{4} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
F (x)
 & = & - \frac{35}{16} a \artanh \frac{x}{a}
 + x
 + \frac{29}{16} a^{2} \frac{x}{a^{2} - x^{2}}
 - \frac{19}{24} a^{4} \frac{x}{(a^{2} - x^{2})^{2}} \\
 & & \qquad + \, \frac{1}{6} a^{6} \frac{x}{(a^{2} - x^{2})^{3}} + c \\
& = & - \frac{35}{32} a \ln \left | \, \frac{a + x}{a - x} \, \right |
 + x
 + \frac{29}{16} a^{2} \frac{x}{a^{2} - x^{2}}
 - \frac{19}{24} a^{4} \frac{x}{(a^{2} - x^{2})^{2}} \\
 & & \qquad + \, \frac{1}{6} a^{6} \frac{x}{(a^{2} - x^{2})^{3}} + c \\
\end{eqnarray*}
\end{document}
