cosの逆数(sec)の三乗の積分

$$ \begin{eqnarray} \int \frac{1}{\cos^3{\left(\theta\right)}} \mathrm{d}\theta &=&\int \sec^3{\left(\theta\right)} \mathrm{d}\theta \;\cdots\;\sec{\left(\theta\right)}=\frac{1}{\cos{\left(\theta\right)}} \\&=&\int \frac{1}{\cos{\left(\theta\right)}}\;\frac{1}{\cos^2{\left(\theta\right)}} \mathrm{d}\theta \\&=&\int \frac{1}{\cos{\left(\theta\right)}}\;\left\{\tan{\left(\theta\right)}\right\}^\prime \mathrm{d}\theta \;\cdots\;\left\{\tan{\left(\theta\right)}\right\}^\prime =\left\{\frac{\sin{\left(\theta\right)}}{\cos{\left(\theta\right)}}\right\}^\prime =\href{https://shikitenkai.blogspot.com/2020/02/blog-post.html}{\left\{\sin{\left(\theta\right)}\right\}^\prime\frac{1}{\cos{\left(\theta\right)}}+\sin{\left(\theta\right)}\left\{\frac{1}{\cos{\left(\theta\right)}}\right\}^\prime} =\cos{\left(\theta\right)}\frac{1}{\cos{\left(\theta\right)}}+\sin{\left(\theta\right)}\frac{\sin{\left(\theta\right)}}{\cos^2{\left(\theta\right)}} =\frac{\cos{\left(\theta\right)}}{\cos{\left(\theta\right)}}+\frac{\sin^2{\left(\theta\right)}}{\cos^2{\left(\theta\right)}} =\frac{\cos^{2}{\left(\theta\right)}+\sin^{2}{\left(\theta\right)}}{\cos^{2}{\left(\theta\right)}} =\frac{1}{\cos^{2}{\left(\theta\right)}} \\&=&\frac{\tan{\left(\theta\right)}}{\cos{\left(\theta\right)}} -\int \frac{\sin{\left(\theta\right)}}{\cos^2{\left(\theta\right)}}\;\tan{\left(\theta\right)} \mathrm{d}\theta \;\cdots\;\href{https://shikitenkai.blogspot.com/2020/02/blog-post_7.html}{\int f^\prime g\;\mathrm{d}x=\left[fg\right]- \int f g^\prime \mathrm{d}x} \\&=&\frac{\tan{\left(\theta\right)}}{\cos{\left(\theta\right)}} -\int \frac{1}{\cos{\left(\theta\right)}}\tan^2{\left(\theta\right)} \mathrm{d}\theta \\&=&\frac{\tan{\left(\theta\right)}}{\cos{\left(\theta\right)}} -\int \frac{1}{\cos{\left(\theta\right)}}\;\frac{\sin^2{\left(\theta\right)}}{\cos^2{\left(\theta\right)}} \mathrm{d}\theta \\&=&\frac{\tan{\left(\theta\right)}}{\cos{\left(\theta\right)}} -\int \frac{\sin^2{\left(\theta\right)}}{\cos^3{\left(\theta\right)}} \mathrm{d}\theta \\&=&\frac{\tan{\left(\theta\right)}}{\cos{\left(\theta\right)}} -\int \frac{1-\cos^2{\left(\theta\right)}}{\cos^3{\left(\theta\right)}} \mathrm{d}\theta \\&=&\frac{\tan{\left(\theta\right)}}{\cos{\left(\theta\right)}} -\int \left\{ \frac{1}{\cos^3{\left(\theta\right)}}-\frac{\cos^2{\left(\theta\right)}}{\cos^3{\left(\theta\right)}} \right\} \mathrm{d}\theta \\&=&\frac{\tan{\left(\theta\right)}}{\cos{\left(\theta\right)}} -\int \left\{ \frac{1}{\cos^3{\left(\theta\right)}}-\frac{1}{\cos{\left(\theta\right)}} \right\} \mathrm{d}\theta \\&=&\frac{\tan{\left(\theta\right)}}{\cos{\left(\theta\right)}} -\int \frac{1}{\cos^3{\left(\theta\right)}} \mathrm{d}\theta +\int \frac{1}{\cos{\left(\theta\right)}} \mathrm{d}\theta \\ 2\int \frac{1}{\cos^3{\left(\theta\right)}} \mathrm{d}\theta &=&\frac{\tan{\left(\theta\right)}}{\cos{\left(\theta\right)}} +\int \frac{1}{\cos{\left(\theta\right)}}\mathrm{d}\theta \\&=&\frac{\tan{\left(\theta\right)}}{\cos{\left(\theta\right)}} +\frac{1}{2}\ln{\left| \frac{1+\sin{\left(\theta\right)}}{1-\sin{\left(\theta\right)}} \right|} \;\cdots\;\href{https://shikitenkai.blogspot.com/2020/07/cossec.html}{\int \frac{1}{\cos{\left(\theta\right)}}\mathrm{d}\theta =\frac{1}{2}\ln{\left| \frac{1+\sin{\left(\theta\right)}}{1-\sin{\left(\theta\right)}} \right|}+C\;(C:積分定数) } \\ \int \frac{1}{\cos^3{\left(\theta\right)}}\mathrm{d}\theta &=&\frac{1}{2}\left\{ \frac{\tan{\left(\theta\right)}}{\cos{\left(\theta\right)}} +\frac{1}{2}\ln{\left| \frac{1+\sin{\left(\theta\right)}}{1-\sin{\left(\theta\right)}} \right|} \right\}+C\;\cdots\;C:積分定数 \\&=&\frac{1}{2}\left\{ \frac{\tan{\left(\theta\right)}}{\cos{\left(\theta\right)}} +\ln{\left| \frac{1}{\cos{\left(\theta\right)}}+\tan{\left(\theta\right)} \right|} \right\}+C \;\cdots\;\href{https://shikitenkai.blogspot.com/2020/07/cossec.html}{\frac{ 1+\sin{\left(\theta\right)} }{ 1-\sin{\left(\theta\right)} }=\left\{ \frac{1}{\cos{\left(\theta\right)}} +\tan{\left(\theta\right)} \right\}^2 } \\&=&\frac{1}{2}\left\{ \sec{\left(\theta\right)}\tan{\left(\theta\right)} +\ln{\left| \sec{\left(\theta\right)}+\tan{\left(\theta\right)} \right|} \right\}+C \;\cdots\;\sec{\left(\theta\right)}=\frac{1}{\cos{\left(\theta\right)}} \end{eqnarray} $$