\documentclass[a4paper]{article}
\usepackage{amsmath}
\DeclareMathOperator{\arcsec}{arcsec}
\begin{document}
\begin{eqnarray*}
f (x)
 & = & \frac{1}{(x + a)^{3} \sqrt{x^{2} - a^{2}}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
x
 & = & a \sec t \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\sec t
 & = & \frac{x}{a} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
t
 & = & \arcsec \frac{x}{a} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
F (x)
 & = & \int \frac{1}{(x + a)^{3} \sqrt{x^{2} - a^{2}}} \, \mbox{d}x \\
& = & \int \frac{1}{(a \sec t + a)^{3} \sqrt{(a \sec t)^{2} - a^{2}}} \, \mbox{d}(a \sec t) \\
& = & \int \frac{1}{(a \sec t + a)^{3} \sqrt{a^{2} \sec^{2} t - a^{2}}} a \sec^{2} t \sin t \, \mbox{d}t \\
& = & \frac{1}{a^{3}} \int \frac{\sec^{2} t \sin t}{(\sec t + 1)^{3} \sqrt{\sec^{2} t - 1}} \, \mbox{d}t \\
& = & \frac{1}{a^{3}} \int \frac{\sec^{2} t \sin t}{(\sec t + 1)^{3} \sqrt{\frac{1}{\cos^{2} t} - \frac{\cos^{2} t}{\cos^{2} t}}} \, \mbox{d}t \\
& = & \frac{1}{a^{3}} \int \frac{\sec^{2} t \sin t}{(\sec t + 1)^{3} \sqrt{\frac{1 - \cos^{2} t}{\cos^{2} t}}} \, \mbox{d}t \\
& = & \frac{1}{a^{3}} \int \frac{\sec^{2} t \sin t}{(\sec t + 1)^{3} \sqrt{\frac{\sin^{2} t}{\cos^{2} t}}} \, \mbox{d}t \\
& = & \frac{1}{a^{3}} \int \frac{\sec^{2} t \sin t}{(\sec t + 1)^{3} \frac{\sin t}{\cos t}} \, \mbox{d}t \\
& = & \frac{1}{a^{3}} \int \frac{\sec t}{(\sec t + 1)^{3}} \, \mbox{d}t \\
& = & \frac{1}{a^{3}} \int \frac{\cos^{2} t}{(1 + \cos t)^{3}} \, \mbox{d}t \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
F (x)
 & = & \frac{1}{a^{3}} \int \frac{\cos^{2} t}{(1 + \cos t)^{3}} \, \mbox{d}t \\
& = & \frac{1}{a^{3}} \left ( - \frac{4}{5} \cot^{5} t + \frac{4}{5} \csc^{5} t - \frac{1}{3} \cot^{3} t - \frac{5}{3} \csc^{3} t + \csc t \right ) + c \\
& = & - \frac{4}{5 a^{3}} \cot^{5} t + \frac{4}{5 a^{3}} \csc^{5} t - \frac{1}{3 a^{3}} \cot^{3} t - \frac{5}{3 a^{3}} \csc^{3} t
 + \frac{1}{a^{3}} \csc t + c \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\sin \alpha
 & = & \sqrt{1 - \frac{1}{\sec^{2} \alpha}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\cos \alpha
 & = & \frac{1}{\sec \alpha} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
F (x)
 & = & - \frac{4}{5 a^{3}} \cot^{5} t + \frac{4}{5 a^{3}} \csc^{5} t - \frac{1}{3 a^{3}} \cot^{3} t - \frac{5}{3 a^{3}} \csc^{3} t
 + \frac{1}{a^{3}} \csc t + c \\
& = & - \frac{4}{5 a^{3}} \frac{\cos^{5} t}{\sin^{5} t} + \frac{4}{5 a^{3}} \frac{1}{\sin^{5} t} - \frac{1}{3 a^{3}} \frac{\cos^{3} t}{\sin^{3} t}
 - \frac{5}{3 a^{3}} \frac{1}{\sin^{3} t} + \frac{1}{a^{3}} \frac{1}{\sin t} + c \\
& = & - \frac{4}{5 a^{3}} \frac{\frac{1}{\sec^{5} t}}{\left ( \sqrt{1 - \frac{1}{\sec^{2} t}} \, \right )^{5}}
 + \frac{4}{5 a^{3}} \frac{1}{\left ( \sqrt{1 - \frac{1}{\sec^{2} t}} \, \right )^{5}}
 - \frac{1}{3 a^{3}} \frac{\frac{1}{\sec^{3} t}}{\left ( \sqrt{1 - \frac{1}{\sec^{2} t}} \, \right )^{3}} \\
 & & \qquad - \, \frac{5}{3 a^{3}} \frac{1}{\left ( \sqrt{1 - \frac{1}{\sec^{2} t}} \, \right )^{3}}
 + \frac{1}{a^{3}} \frac{1}{\sqrt{1 - \frac{1}{\sec^{2} t}}} + c \\
& = & - \frac{4}{5 a^{3}} \frac{1}{\left ( \sqrt{\sec^{2} t - 1} \, \right )^{5}}
 + \frac{4}{5 a^{3}} \frac{\sec^{5} t}{\left ( \sqrt{\sec^{2} t - 1} \, \right )^{5}}
 - \frac{1}{3 a^{3}} \frac{1}{\left ( \sqrt{\sec^{2} t - 1} \, \right )^{3}} \\
 & & \qquad - \, \frac{5}{3 a^{3}} \frac{\sec^{3} t}{\left ( \sqrt{\sec^{2} t - 1} \, \right )^{3}}
 + \frac{1}{a^{3}} \frac{\sec t}{\sqrt{\sec^{2} t - 1}} + c \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
F (x)
 & = & - \frac{4}{5 a^{3}} \frac{1}{\left ( \sqrt{\sec^{2} t - 1} \, \right )^{5}}
 + \frac{4}{5 a^{3}} \frac{\sec^{5} t}{\left ( \sqrt{\sec^{2} t - 1} \, \right )^{5}}
 - \frac{1}{3 a^{3}} \frac{1}{\left ( \sqrt{\sec^{2} t - 1} \, \right )^{3}} \\
 & & \qquad - \, \frac{5}{3 a^{3}} \frac{\sec^{3} t}{\left ( \sqrt{\sec^{2} t - 1} \, \right )^{3}}
 + \frac{1}{a^{3}} \frac{\sec t}{\sqrt{\sec^{2} t - 1}} + c \\
& = & - \frac{4}{5 a^{3}} \frac{1}{\left ( \sqrt{\sec^{2} \arcsec \frac{x}{a} - 1} \, \right )^{5}}
 + \frac{4}{5 a^{3}} \frac{\sec^{5} \arcsec \frac{x}{a}}{\left ( \sqrt{\sec^{2} \arcsec \frac{x}{a} - 1} \, \right )^{5}}
 - \frac{1}{3 a^{3}} \frac{1}{\left ( \sqrt{\sec^{2} \arcsec \frac{x}{a} - 1} \, \right )^{3}} \\
 & & \qquad - \, \frac{5}{3 a^{3}} \frac{\sec^{3} \arcsec \frac{x}{a}}{\left ( \sqrt{\sec^{2} \arcsec \frac{x}{a} - 1} \, \right )^{3}}
 + \frac{1}{a^{3}} \frac{\sec \arcsec \frac{x}{a}}{\sqrt{\sec^{2} \arcsec \frac{x}{a} - 1}} + c \\
& = & - \frac{4}{5 a^{3}} \frac{1}{\left ( \sqrt{\left ( \frac{x}{a} \right )^{2} - 1} \, \right )^{5}}
 + \frac{4}{5 a^{3}} \frac{\left ( \frac{x}{a} \right )^{5}}{\left ( \sqrt{\left ( \frac{x}{a} \right )^{2} - 1} \, \right )^{5}}
 - \frac{1}{3 a^{3}} \frac{1}{\left ( \sqrt{\left ( \frac{x}{a} \right )^{2} - 1} \, \right )^{3}} \\
 & & \qquad - \, \frac{5}{3 a^{3}} \frac{\left ( \frac{x}{a} \right )^{3}}{\left ( \sqrt{\left ( \frac{x}{a} \right )^{2} - 1} \, \right )^{3}}
 + \frac{1}{a^{3}} \frac{\frac{x}{a}}{\sqrt{\left ( \frac{x}{a} \right )^{2} - 1}} + c \\
& = & - \frac{4}{5 a^{3}} \frac{1}{\left ( \sqrt{\frac{x^{2}}{a^{2}} - 1} \, \right )^{5}}
 + \frac{4}{5 a^{3}} \frac{\frac{x^{5}}{a^{5}}}{\left ( \sqrt{\frac{x^{2}}{a^{2}} - 1} \, \right )^{5}}
 - \frac{1}{3 a^{3}} \frac{1}{\left ( \sqrt{\frac{x^{2}}{a^{2}} - 1} \, \right )^{3}} \\
 & & \qquad - \, \frac{5}{3 a^{3}} \frac{\frac{x^{3}}{a^{3}}}{\left ( \sqrt{\frac{x^{2}}{a^{2}} - 1} \, \right )^{3}}
 + \frac{1}{a^{3}} \frac{\frac{x}{a}}{\sqrt{\frac{x^{2}}{a^{2}} - 1}} + c \\
& = & - \frac{4}{5 a^{3}} \frac{a^{5}}{\left ( \sqrt{x^{2} - a^{2}} \, \right )^{5}}
 + \frac{4}{5 a^{3}} \frac{x^{5}}{\left ( \sqrt{x^{2} - a^{2}} \, \right )^{5}}
 - \frac{1}{3 a^{3}} \frac{a^{3}}{\left ( \sqrt{x^{2} - a^{2}} \, \right )^{3}} \\
 & & \qquad - \, \frac{5}{3 a^{3}} \frac{x^{3}}{\left ( \sqrt{x^{2} - a^{2}} \, \right )^{3}}
 + \frac{1}{a^{3}} \frac{x}{\sqrt{x^{2} - a^{2}}} + c \\
& = & - \frac{4}{5 a^{3}} \frac{a^{5}}{(x^{2} - a^{2})^{2} \sqrt{x^{2} - a^{2}}}
 + \frac{4}{5 a^{3}} \frac{x^{5}}{(x^{2} - a^{2})^{2} \sqrt{x^{2} - a^{2}}}
 - \frac{1}{3 a^{3}} \frac{a^{3} (x^{2} - a^{2})}{(x^{2} - a^{2})^{2} \sqrt{x^{2} - a^{2}}} \\
 & & \qquad - \, \frac{5}{3 a^{3}} \frac{x^{3} (x^{2} - a^{2})}{(x^{2} - a^{2})^{2} \sqrt{x^{2} - a^{2}}}
 + \frac{1}{a^{3}} \frac{x (x^{2} - a^{2})^{2}}{(x^{2} - a^{2})^{2} \sqrt{x^{2} - a^{2}}} + c \\
& = & \frac{- \frac{4}{5 a^{3}} a^{5}
 + \frac{4}{5 a^{3}} x^{5}
 - \frac{1}{3 a^{3}} a^{3} (x^{2} - a^{2})
 - \frac{5}{3 a^{3}} x^{3} (x^{2} - a^{2})
 + \frac{1}{a^{3}} x (x^{2} - a^{2})^{2}}{(x^{2} - a^{2})^{2} \sqrt{x^{2} - a^{2}}} + c \\
& = & \frac{- \frac{4}{5} a^{2}
 + \frac{4}{5 a^{3}} x^{5}
 - \frac{1}{3} x^{2}
 + \frac{1}{3} a^{2}
 - \frac{5}{3 a^{3}} x^{5}
 + \frac{5}{3 a} x^{3}
 + \frac{1}{a^{3}} x (x^{4} - 2 a^{2} x^{2} + a^{4})}{(x^{2} - a^{2}) (x^{2} - a^{2}) \sqrt{x^{2} - a^{2}}} + c \\
& = & \frac{- \frac{4}{5} a^{2}
 + \frac{4}{5 a^{3}} x^{5}
 - \frac{1}{3} x^{2}
 + \frac{1}{3} a^{2}
 - \frac{5}{3 a^{3}} x^{5}
 + \frac{5}{3 a} x^{3}
 + \frac{1}{a^{3}} x^{5}
 - \frac{2}{a} x^{3}
 + a x}{(x^{2} - a^{2})^{2} \sqrt{x^{2} - a^{2}}} + c \\
& = & \frac{\frac{2}{15 a^{3}} x^{5} - \frac{1}{3 a} x^{3} - \frac{1}{3} x^{2} + a x - \frac{7}{15} a^{2}}
{(x + a) (x - a) (x + a) (x - a) \sqrt{x^{2} - a^{2}}} + c \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
& = & \frac{\frac{2}{15 a^{3}} x^{5} - \frac{1}{3 a} x^{3} - \frac{1}{3} x^{2} + a x - \frac{7}{15} a^{2}}
{(x + a) (x - a) (x + a) (x - a) \sqrt{x^{2} - a^{2}}} + c \\
& = & \frac{1}{15 a^{3}} \frac{2 x^{5} - 5 a^{2} x^{3} - 5 a^{3} x^{2} + 15 a^{4} x - 7 a^{5}}{(x + a)^{2} (x - a)^{2} \sqrt{x^{2} - a^{2}}} + c \\
& = & \frac{1}{15 a^{3}} \frac{(2 x^{3} + 4 a x^{2} + a^{2} x - 7 a^{3}) (x - a)^{2}}{(x + a)^{2} (x - a)^{2} \sqrt{x^{2} - a^{2}}} + c \\
& = & \frac{1}{15 a^{3}} \frac{2 x^{3} + 4 a x^{2} + a^{2} x - 7 a^{3}}{(x + a)^{2} \sqrt{x^{2} - a^{2}}} + c \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
f (x)
 & = & \frac{\mbox{d}(F (x))}{\mbox{d}x} \\
& = & \frac{\mbox{d}}{\mbox{d}x} \left ( \frac{1}{15 a^{3}} \frac{2 x^{3} + 4 a x^{2} + a^{2} x - 7 a^{3}}{(x + a)^{2} \sqrt{x^{2} - a^{2}}} + c \right ) \\
& = & \frac{1}{15 a^{3}} \Bigl ( (6 x^{2} + 8 a x + a^{2}) (x + a)^{2} \sqrt{x^{2} - a^{2}} \\
 & & \qquad \qquad \qquad
 - \, (2 x^{3} + 4 a x^{2} + a^{2} x - 7 a^{3}) (2 (x + a) \sqrt{x^{2} - a^{2}} + (x + a)^{2} \frac{1}{2} \frac{1}{\sqrt{x^{2} - a^{2}}} (2x)) \Bigr ) \\
 & & \qquad \qquad \qquad \qquad \qquad \cdot ((x + a)^{4} (x^{2} - a^{2}))^{-1} + 0 \\
& = & \frac{1}{15 a^{3}} \Bigl ( (6 x^{2} + 8 a x + a^{2}) (x + a) (x^{2} - a^{2})
 - (2 x^{3} + 4 a x^{2} + a^{2} x - 7 a^{3}) (2 (x^{2} - a^{2}) + (x + a) x) \Bigr ) \\
 & & \qquad \qquad \qquad \qquad \qquad \cdot ((x + a)^{3} (x^{2} - a^{2}) \sqrt{x^{2} - a^{2}})^{-1} \\
& = & \frac{1}{15 a^{3}} \Bigl ( (6 x^{2} + 8 a x + a^{2}) (x + a) (x + a) (x - a) \\
 & & \qquad \qquad \qquad
 - \, (2 x^{3} + 4 a x^{2} + a^{2} x - 7 a^{3}) (2 (x + a) (x - a) + (x + a) x) \Bigr ) \\
 & & \qquad \qquad \qquad \qquad \qquad \cdot ((x + a)^{3} (x + a) (x - a) \sqrt{x^{2} - a^{2}})^{-1} \\
& = & \frac{1}{15 a^{3}} \Bigl ( (6 x^{2} + 8 a x + a^{2}) (x + a) (x - a)
 - (2 x^{3} + 4 a x^{2} + a^{2} x - 7 a^{3}) (2 (x - a) + x) \Bigr ) \\
 & & \qquad \qquad \qquad \qquad \qquad \cdot ((x + a)^{3} (x - a) \sqrt{x^{2} - a^{2}})^{-1} \\
& = & \frac{1}{15 a^{3}} \Bigl ( (6 x^{2} + 8 a x + a^{2}) (x^{2} - a^{2})
 - (2 x^{3} + 4 a x^{2} + a^{2} x - 7 a^{3}) (3 x - 2 a) \Bigr ) \\
 & & \qquad \qquad \qquad \qquad \qquad \cdot ((x + a)^{3} (x - a) \sqrt{x^{2} - a^{2}})^{-1} \\
& = & \frac{1}{15 a^{3}} \Bigl (
 6 x^{4} + 8 a x^{3} + a^{2} x^{2}
 - 6 a^{2} x^{2} - 8 a^{3} x - a^{4} \\
 & & \qquad \qquad \qquad
 - \, 6 x^{4} - 12 a x^{3} - 3 a^{2} x^{2} + 21 a^{3} x
 + 4 a x^{3} + 8 a^{2} x^{2} + 2 a^{3} x - 14 a^{4} \Bigr ) \\
 & & \qquad \qquad \qquad \qquad \qquad \cdot ((x + a)^{3} (x - a) \sqrt{x^{2} - a^{2}})^{-1} \\
& = & \frac{x - a}{(x + a)^{3} (x - a) \sqrt{x^{2} - a^{2}}} \\
& = & \frac{1}{(x + a)^{3} \sqrt{x^{2} - a^{2}}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
F (x)
 & = & \frac{1}{15 a^{3}} \frac{2 x^{3} + 4 a x^{2} + a^{2} x - 7 a^{3}}{(x + a)^{2} \sqrt{x^{2} - a^{2}}} + c \\
\end{eqnarray*}
\end{document}
