\documentclass[a4paper]{article}
\usepackage{amsmath}
\DeclareMathOperator{\artanh}{artanh}
\begin{document}
\begin{eqnarray*}
f (x)
 & = & \left ( \frac{x^{2}}{a^{2} - x^{2}} \right )^{5} \\
\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 )^{5} \, \mbox{d}x \\
& = & \int \left ( \frac{(a \sin t)^{2}}{a^{2} - (a \sin t)^{2}} \right )^{5} \, \mbox{d}(a \sin t) \\
& = & a \int \left ( \frac{a^{2} \sin^{2} t}{a^{2} - a^{2} \sin^{2} t} \right )^{5} \cos t \, \mbox{d}t \\
& = & a \int \left ( \frac{\sin^{2} t}{1 - \sin^{2} t} \right )^{5} \cos t \, \mbox{d}t \\
& = & a \int \left ( \frac{1 - \cos^{2} t}{\cos^{2} t} \right )^{5} \cos t \, \mbox{d}t \\
& = & a \int \left ( \frac{1}{\cos^{2} t} - \frac{\cos^{2} t}{\cos^{2} t} \right )^{5} \cos t \, \mbox{d}t \\
& = & a \int (\sec^{2} t - 1)^{5} \cos t \, \mbox{d}t \\
& = & a \int (\sec^{10} t - 5 \sec^{8} t + 10 \sec^{6} t - 10 \sec^{4} t + 5 \sec^{2} t - 1) \cos t \, \mbox{d}t \\
& = & a \int (\sec^{9} t - 5 \sec^{7} t + 10 \sec^{5} t - 10 \sec^{3} t + 5 \sec t - \cos t) \, \mbox{d}t \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
F (x)
 & = & a \int (\sec^{9} t - 5 \sec^{7} t + 10 \sec^{5} t - 10 \sec^{3} t + 5 \sec t - \cos t) \, \mbox{d}t \\
& = & a \biggl ( \frac{1}{8} \frac{\sin t}{\cos^{8} t}
 + \frac{7}{48} \frac{\sin t}{\cos^{6} t}
 + \frac{35}{192} \frac{\sin t}{\cos^{4} t}
 + \frac{35}{128} \frac{\sin t}{\cos^{2} t}
 + \frac{35}{128} \ln \left | \, \frac{1 + \sin t}{\cos t} \, \right |
 - \frac{5}{6} \frac{\sin t}{\cos^{6} t}
 - \frac{25}{24} \frac{\sin t}{\cos^{4} t} \\
 & & \qquad \qquad - \, \frac{25}{16} \frac{\sin t}{\cos^{2} t}
 - \frac{25}{16} \ln \left | \, \frac{1 + \sin t}{\cos t} \, \right |
 + \frac{5}{2} \frac{\sin t}{\cos^{4} t}
 + \frac{15}{4} \frac{\sin t}{\cos^{2} t}
 + \frac{15}{4} \ln \left | \, \frac{1 + \sin t}{\cos t} \, \right | \\
 & & \qquad \qquad \qquad - \, 5 \ln \left | \, \frac{1 + \sin t}{\cos t} \, \right |
 - 5 \frac{\sin t}{\cos^{2} t}
 + 5 \ln \left | \, \frac{1 + \sin t}{\cos t} \, \right |
 - \sin t \biggr ) + c \\
& = & a \left ( \frac{315}{128} \ln \left | \, \frac{1 + \sin t}{\cos t} \, \right |
 + \frac{1}{8} \frac{\sin t}{\cos^{8} t}
 - \frac{11}{16} \frac{\sin t}{\cos^{6} t}
 + \frac{105}{64} \frac{\sin t}{\cos^{4} t}
 - \frac{325}{128} \frac{\sin t}{\cos^{2} t}
 - \sin t \right ) + c \\
& = & a \biggl ( \frac{315}{128} \ln \left | \, \frac{1 + \sin t}{\sqrt{1 - \sin^{2} t}} \, \right |
 + \frac{1}{8} \frac{\sin t}{(1 - \sin^{2} t)^{4}}
 - \frac{11}{16} \frac{\sin t}{(1 - \sin^{2} t)^{3}}
 + \frac{105}{64} \frac{\sin t}{(1 - \sin^{2} t)^{2}} \\
 & & \qquad \qquad - \, \frac{325}{128} \frac{\sin t}{1 - \sin^{2} t}
 - \sin t \biggr ) + c \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
F (x)
 & = & a \biggl ( \frac{315}{128} \ln \left | \, \frac{1 + \sin t}{\sqrt{1 - \sin^{2} t}} \, \right |
 + \frac{1}{8} \frac{\sin t}{(1 - \sin^{2} t)^{4}}
 - \frac{11}{16} \frac{\sin t}{(1 - \sin^{2} t)^{3}}
 + \frac{105}{64} \frac{\sin t}{(1 - \sin^{2} t)^{2}} \\
 & & \qquad \qquad - \, \frac{325}{128} \frac{\sin t}{1 - \sin^{2} t}
 - \sin t \biggr ) + c \\
& = & a \biggl ( \frac{315}{128} \ln \left | \, \frac{1 + \sin \arcsin \frac{x}{a}}{\sqrt{1 - \left ( \sin \arcsin \frac{x}{a} \right )^{2}}} \, \right |
 + \frac{1}{8} \frac{\sin \arcsin \frac{x}{a}}{\left ( 1 - \left ( \sin \arcsin \frac{x}{a} \right )^{2} \right )^{4}}
 - \frac{11}{16} \frac{\sin \arcsin \frac{x}{a}}{\left ( 1 - \left ( \sin \arcsin \frac{x}{a} \right )^{2} \right )^{3}} \\
 & & \qquad \qquad + \, \frac{105}{64} \frac{\sin \arcsin \frac{x}{a}}{\left ( 1 - \left ( \sin \arcsin \frac{x}{a} \right )^{2} \right )^{2}}
 - \frac{325}{128} \frac{\sin \arcsin \frac{x}{a}}{1 - \left ( \sin \arcsin \frac{x}{a} \right )^{2}}
 - \sin \arcsin \frac{x}{a} \biggr ) + c \\
& = & a \biggl ( \frac{315}{128} \ln \left | \, \frac{1 + \frac{x}{a}}{\sqrt{1 - \left ( \frac{x}{a} \right )^{2}}} \, \right |
 + \frac{1}{8} a^{8} \frac{\frac{x}{a}}{\left ( 1 - \left ( \frac{x}{a} \right )^{2} \right )^{4}}
 - \frac{11}{16} a^{6} \frac{\frac{x}{a}}{\left ( 1 - \left ( \frac{x}{a} \right )^{2} \right )^{3}}
 + \frac{105}{64} a^{4} \frac{\frac{x}{a}}{\left ( 1 - \left ( \frac{x}{a} \right )^{2} \right )^{2}} \\
 & & \qquad \qquad - \, \frac{325}{128} a^{2} \frac{\frac{x}{a}}{1 - \left ( \frac{x}{a} \right )^{2}}
 - \frac{x}{a} \biggr ) + c \\
& = & \frac{315}{128} a \ln \left | \, \frac{a + x}{\sqrt{a^{2} - x^{2}}} \, \right |
 + \frac{1}{8} a^{8} \frac{x}{(a^{2} - x^{2})^{4}}
 - \frac{11}{16} a^{6} \frac{x}{(a^{2} - x^{2})^{3}}
 + \frac{105}{64} a^{4} \frac{x}{(a^{2} - x^{2})^{2}} \\
 & & \qquad \qquad - \, \frac{325}{128} a^{2} \frac{x}{a^{2} - x^{2}}
 - x + c \\
& = & \frac{315}{128} a \ln \left | \, \sqrt{\frac{(a + x) (a + x)}{(a - x) (a + x)}} \, \right |
 + \frac{1}{8} a^{8} \frac{x}{(a^{2} - x^{2})^{4}}
 - \frac{11}{16} a^{6} \frac{x}{(a^{2} - x^{2})^{3}}
 + \frac{105}{64} a^{4} \frac{x}{(a^{2} - x^{2})^{2}} \\
 & & \qquad \qquad - \, \frac{325}{128} a^{2} \frac{x}{a^{2} - x^{2}}
 - x + c \\
& = & \frac{315}{256} a \ln \left | \, \frac{a + x}{a - x} \, \right |
 + \frac{1}{8} a^{8} \frac{x}{(a^{2} - x^{2})^{4}}
 - \frac{11}{16} a^{6} \frac{x}{(a^{2} - x^{2})^{3}}
 + \frac{105}{64} a^{4} \frac{x}{(a^{2} - x^{2})^{2}} \\
 & & \qquad \qquad - \, \frac{325}{128} a^{2} \frac{x}{a^{2} - x^{2}}
 - x + c \\
& = & \frac{315}{256} a \ln \left | \, \frac{1 + \frac{x}{a}}{1 - \frac{x}{a}} \, \right |
 + \frac{1}{8} a^{8} \frac{x}{(a^{2} - x^{2})^{4}}
 - \frac{11}{16} a^{6} \frac{x}{(a^{2} - x^{2})^{3}}
 + \frac{105}{64} a^{4} \frac{x}{(a^{2} - x^{2})^{2}} \\
 & & \qquad \qquad - \, \frac{325}{128} 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{315}{256} a \ln \left | \, \frac{1 + \frac{x}{a}}{1 - \frac{x}{a}} \, \right |
 + \frac{1}{8} a^{8} \frac{x}{(a^{2} - x^{2})^{4}}
 - \frac{11}{16} a^{6} \frac{x}{(a^{2} - x^{2})^{3}}
 + \frac{105}{64} a^{4} \frac{x}{(a^{2} - x^{2})^{2}} \\
 & & \qquad \qquad - \, \frac{325}{128} a^{2} \frac{x}{a^{2} - x^{2}}
 - x + c \\
& = & \frac{315}{128} a \artanh \frac{x}{a}
 + \frac{1}{8} a^{8} \frac{x}{(a^{2} - x^{2})^{4}}
 - \frac{11}{16} a^{6} \frac{x}{(a^{2} - x^{2})^{3}}
 + \frac{105}{64} a^{4} \frac{x}{(a^{2} - x^{2})^{2}}
 - \frac{325}{128} 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} \biggl ( \frac{315}{128} a \artanh \frac{x}{a}
 + \frac{1}{8} a^{8} \frac{x}{(a^{2} - x^{2})^{4}}
 - \frac{11}{16} a^{6} \frac{x}{(a^{2} - x^{2})^{3}}
 + \frac{105}{64} a^{4} \frac{x}{(a^{2} - x^{2})^{2}} \\
 & & \qquad \qquad - \, \frac{325}{128} a^{2} \frac{x}{a^{2} - x^{2}}
 - x + c \biggr ) \\
& = & \frac{315}{128} a \frac{1}{1 - \left ( \frac{x}{a} \right )^{2}} \frac{1}{a}
 + \frac{1}{8} a^{8} \frac{1 (a^{2} - x^{2})^{4} - 4x (a^{2} - x^{2})^{3} (- 2x)}{(a^{2} - x^{2})^{8}} \\
 & & \qquad - \, \frac{11}{16} a^{6} \frac{1 (a^{2} - x^{2})^{3} - 3x (a^{2} - x^{2})^{2} (- 2x)}{(a^{2} - x^{2})^{6}}
 + \frac{105}{64} a^{4} \frac{1 (a^{2} - x^{2})^{2} - 2x (a^{2} - x^{2}) (- 2x)}{(a^{2} - x^{2})^{4}} \\
 & & \qquad \qquad - \, \frac{325}{128} a^{2} \frac{1 (a^{2} - x^{2}) - x (- 2x)}{(a^{2} - x^{2})^{2}}
 - 1 + 0 \\
& = & \frac{315}{128} \frac{a^{2}}{a^{2} - x^{2}} \frac{(a^{2} - x^{2})^{4}}{(a^{2} - x^{2})^{4}}
 + \frac{1}{8} a^{8} \frac{a^{2} - x^{2} + 8x^{2}}{(a^{2} - x^{2})^{5}}
 - \frac{11}{16} a^{6} \frac{a^{2} - x^{2} + 6x^{2}}{(a^{2} - x^{2})^{4}} \frac{a^{2} - x^{2}}{a^{2} - x^{2}} \\
 & & \qquad + \, \frac{105}{64} a^{4} \frac{a^{2} - x^{2} + 4x^{2}}{(a^{2} - x^{2})^{3}} \frac{(a^{2} - x^{2})^{2}}{(a^{2} - x^{2})^{2}}
 - \frac{325}{128} a^{2} \frac{a^{2} - x^{2} + 2x^{2}}{(a^{2} - x^{2})^{2}} \frac{(a^{2} - x^{2})^{3}}{(a^{2} - x^{2})^{3}}
 - \frac{(a^{2} - x^{2})^{5}}{(a^{2} - x^{2})^{5}} \\
& = & \Biggl ( \frac{315}{128} a^{2} (a^{8} - 4 a^{6} x^{2} + 6 a^{4} x^{4} - 4 a^{2} x^{6} + x^{8})
 + \frac{1}{8} a^{8} (a^{2} + 7x^{2})
 - \frac{11}{16} a^{6} (a^{2} + 5x^{2}) (a^{2} - x^{2}) \Biggr . \\
 & & \qquad + \, \frac{105}{64} a^{4} (a^{2} + 3x^{2}) (a^{4} - 2 a^{2} x^{2} + x^{4})
 - \frac{325}{128} a^{2} (a^{2} + x^{2}) (a^{6} - 3 a^{4} x^{2} + 3 a^{2} x^{4} - x^{6}) \\
 & & \qquad \qquad \Biggl . - \, (a^{10} - 5 a^{8} x^{2} + 10 a^{6} x^{4} - 10 a^{4} x^{6} + 5 a^{2} x^{8} - x^{10}) \Biggr ) (a^{2} - x^{2})^{-5} \\
& = & \Biggl ( \frac{315}{128} a^{10} - \frac{315}{32} a^{8} x^{2} + \frac{945}{64} a^{6} x^{4} - \frac{315}{32} a^{4} x^{6} + \frac{315}{128} a^{2} x^{8}
 + \frac{1}{8} a^{10} + \frac{7}{8} a^{8} x^{2} \Biggr . \\
 & & \qquad - \, \frac{11}{16} a^{6} (a^{4} - a^{2} x^{2} + 5 a^{2} x^{2} - 5 x^{4})
 + \frac{105}{64} a^{4} (a^{6} - 2 a^{4} x^{2} + a^{2} x^{4} + 3 a^{4} x^{2} - 6 a^{2} x^{4} + 3 x^{6}) \\
 & & \qquad \qquad - \, \frac{325}{128} a^{2} (a^{8} - 3 a^{6} x^{2} + 3 a^{4} x^{4} - a^{2} x^{6} + a^{6} x^{2} - 3 a^{4} x^{4} + 3 a^{2} x^{6} - x^{8}) \\
 & & \qquad \qquad \qquad \Biggl . - \, a^{10} + 5 a^{8} x^{2} - 10 a^{6} x^{4} + 10 a^{4} x^{6} - 5 a^{2} x^{8} + x^{10} \Biggr ) (a^{2} - x^{2})^{-5} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
f (x)
 & = & \frac{\mbox{d}(F (x))}{\mbox{d}x} \\
& = & \Biggl ( \frac{315}{128} a^{10} - \frac{315}{32} a^{8} x^{2} + \frac{945}{64} a^{6} x^{4} - \frac{315}{32} a^{4} x^{6} + \frac{315}{128} a^{2} x^{8}
 + \frac{1}{8} a^{10} + \frac{7}{8} a^{8} x^{2} \Biggr . \\
 & & \qquad - \, \frac{11}{16} a^{6} (a^{4} - a^{2} x^{2} + 5 a^{2} x^{2} - 5 x^{4})
 + \frac{105}{64} a^{4} (a^{6} - 2 a^{4} x^{2} + a^{2} x^{4} + 3 a^{4} x^{2} - 6 a^{2} x^{4} + 3 x^{6}) \\
 & & \qquad \qquad - \, \frac{325}{128} a^{2} (a^{8} - 3 a^{6} x^{2} + 3 a^{4} x^{4} - a^{2} x^{6} + a^{6} x^{2} - 3 a^{4} x^{4} + 3 a^{2} x^{6} - x^{8}) \\
 & & \qquad \qquad \qquad \Biggl . - \, a^{10} + 5 a^{8} x^{2} - 10 a^{6} x^{4} + 10 a^{4} x^{6} - 5 a^{2} x^{8} + x^{10} \Biggr ) (a^{2} - x^{2})^{-5} \\
& = & \Biggl ( \frac{315}{128} a^{10} - \frac{315}{32} a^{8} x^{2} + \frac{945}{64} a^{6} x^{4} - \frac{315}{32} a^{4} x^{6} + \frac{315}{128} a^{2} x^{8}
 + \frac{1}{8} a^{10} + \frac{7}{8} a^{8} x^{2} \Biggr . \\
 & & \qquad - \, \frac{11}{16} a^{6} (a^{4} + 4 a^{2} x^{2} - 5 x^{4})
 + \frac{105}{64} a^{4} (a^{6} + a^{4} x^{2} - 5 a^{2} x^{4} + 3 x^{6}) \\
 & & \qquad \qquad - \, \frac{325}{128} a^{2} (a^{8} - 2 a^{6} x^{2} + 2 a^{2} x^{6} - x^{8}) \\
 & & \qquad \qquad \qquad \Biggl . - \, a^{10} + 5 a^{8} x^{2} - 10 a^{6} x^{4} + 10 a^{4} x^{6} - 5 a^{2} x^{8} + x^{10} \Biggr ) (a^{2} - x^{2})^{-5} \\
& = & \Biggl ( \frac{315}{128} a^{10} - \frac{315}{32} a^{8} x^{2} + \frac{945}{64} a^{6} x^{4} - \frac{315}{32} a^{4} x^{6} + \frac{315}{128} a^{2} x^{8}
 + \frac{1}{8} a^{10} + \frac{7}{8} a^{8} x^{2} \Biggr . \\
 & & \qquad - \, \frac{11}{16} a^{10} - \frac{11}{4} a^{8} x^{2} + \frac{55}{16} a^{6} x^{4}
 + \frac{105}{64} a^{10} + \frac{105}{64} a^{8} x^{2} - \frac{525}{64} a^{6} x^{4} + \frac{315}{64} a^{4} x^{6} \\
 & & \qquad \qquad - \, \frac{325}{128} a^{10} + \frac{325}{64} a^{8} x^{2} - \frac{325}{64} a^{4} x^{6} + \frac{325}{128} a^{2} x^{8} \\
 & & \qquad \qquad \qquad \Biggl . - \, a^{10} + 5 a^{8} x^{2} - 10 a^{6} x^{4} + 10 a^{4} x^{6} - 5 a^{2} x^{8} + x^{10} \Biggr ) (a^{2} - x^{2})^{-5} \\
& = & \frac{x^{10}}{(a^{2} - x^{2})^{5}} \\
& = & \left ( \frac{x^{2}}{a^{2} - x^{2}} \right )^{5} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
F (x)
 & = & \frac{315}{128} a \artanh \frac{x}{a}
 - x
 - \frac{325}{128} a^{2} \frac{x}{a^{2} - x^{2}}
 + \frac{105}{64} a^{4} \frac{x}{(a^{2} - x^{2})^{2}} \\
 & & \qquad - \, \frac{11}{16} a^{6} \frac{x}{(a^{2} - x^{2})^{3}}
 + \frac{1}{8} a^{8} \frac{x}{(a^{2} - x^{2})^{4}}
 + c \\
& = & \frac{315}{256} a \ln \left | \, \frac{a + x}{a - x} \, \right |
 - x
 - \frac{325}{128} a^{2} \frac{x}{a^{2} - x^{2}}
 + \frac{105}{64} a^{4} \frac{x}{(a^{2} - x^{2})^{2}} \\
 & & \qquad - \, \frac{11}{16} a^{6} \frac{x}{(a^{2} - x^{2})^{3}}
 + \frac{1}{8} a^{8} \frac{x}{(a^{2} - x^{2})^{4}}
 + c \\
\end{eqnarray*}
\end{document}
