(1)

\[ I_{n}=\int\sin^{n}xdx \] とおくと積分漸化式は
\begin{align*} I_{n+2} & =-\frac{1}{n+2}\cos x\sin^{n+1}x+\frac{n+1}{n+2}I_{n}\\ \frac{\Gamma\left(\frac{n+2}{2}+1\right)}{\Gamma\left(\frac{n+2}{2}+\frac{1}{2}\right)}I_{n+2}-\frac{\Gamma\left(\frac{n}{2}+1\right)}{\Gamma\left(\frac{n}{2}+\frac{1}{2}\right)}I_{n} & =-\frac{1}{2}\frac{\Gamma\left(\frac{n+2}{2}\right)}{\Gamma\left(\frac{n+2}{2}+\frac{1}{2}\right)}\cos x\sin^{n+1}x \end{align*} となる。
\(m_{\pm}=\frac{1\pm1}{2}\)とおく。
\begin{align*} I_{2n+m_{\pm}} & =\frac{\Gamma\left(\frac{2n+m_{\pm}}{2}+\frac{1}{2}\right)}{\Gamma\left(\frac{2n+m_{\pm}}{2}+1\right)}\left\{ \sum_{k=0}^{n-1}\left(\frac{\Gamma\left(\frac{2\left(k+1\right)+m_{\pm}}{2}+1\right)}{\Gamma\left(\frac{2\left(k+1\right)+m_{\pm}}{2}+\frac{1}{2}\right)}I_{2\left(k+1\right)+m_{\pm}}-\frac{\Gamma\left(\frac{2k+m_{\pm}}{2}+1\right)}{\Gamma\left(\frac{2k+m_{\pm}}{2}+\frac{1}{2}\right)}I_{2k+m_{\pm}}\right)+\frac{\Gamma\left(\frac{m_{\pm}}{2}+1\right)}{\Gamma\left(\frac{m_{\pm}}{2}+\frac{1}{2}\right)}I_{m_{\pm}}\right\} \\ & =\frac{\Gamma\left(n+\frac{m_{\pm}}{2}+\frac{1}{2}\right)}{\Gamma\left(n+\frac{m_{\pm}}{2}+1\right)}\left\{ -\frac{1}{2}\sum_{k=0}^{n-1}\left(\frac{\Gamma\left(\frac{2(k+1)+m_{\pm}}{2}\right)}{\Gamma\left(\frac{2(k+1)+m_{\pm}}{2}+\frac{1}{2}\right)}\cos x\sin^{2k+m_{\pm}+1}x\right)+\frac{\Gamma\left(\frac{m_{\pm}}{2}+1\right)}{\Gamma\left(\frac{m_{\pm}}{2}+\frac{1}{2}\right)}I_{m_{\pm}}\right\} \\ & =\frac{\Gamma\left(n+\frac{1}{2}+\frac{m_{\pm}}{2}\right)}{\Gamma\left(n+1+\frac{m_{\pm}}{2}\right)}\left\{ -\frac{1}{2}\sum_{k=0}^{n-1}\left(\frac{\Gamma\left(k+1+\frac{m_{\pm}}{2}\right)}{\Gamma\left(k+\frac{3}{2}+\frac{m_{\pm}}{2}\right)}\cos x\sin^{2k+1+m_{\pm}}x\right)+\frac{\Gamma\left(1+\frac{m_{\pm}}{2}\right)}{\Gamma\left(\frac{1}{2}+\frac{m_{\pm}}{2}\right)}I_{m_{\pm}}\right\} \end{align*}

(2)

\begin{align*} I_{-2n-m_{\pm}} & =\frac{\Gamma\left(-n+\frac{1}{2}-\frac{m_{\pm}}{2}\right)}{\Gamma\left(-n+1-\frac{m_{\pm}}{2}\right)}\left\{ -\frac{1}{2}\sum_{k=0}^{-n-1}\left(\frac{\Gamma\left(k+1-\frac{m_{\pm}}{2}\right)}{\Gamma\left(k+\frac{3}{2}-\frac{m_{\pm}}{2}\right)}\cos x\sin^{2k+1-m_{\pm}}x\right)+\frac{\Gamma\left(1-\frac{m_{\pm}}{2}\right)}{\Gamma\left(\frac{1}{2}-\frac{m_{\pm}}{2}\right)}I_{-m_{\pm}}\right\} \\ & =\frac{\Gamma\left(-n+\frac{1}{2}-\frac{m_{\pm}}{2}\right)}{\Gamma\left(-n+1-\frac{m_{\pm}}{2}\right)}\left\{ \frac{1}{2}\sum_{k=-n}^{-1}\left(\frac{\Gamma\left(k+1-\frac{m_{\pm}}{2}\right)}{\Gamma\left(k+\frac{3}{2}-\frac{m_{\pm}}{2}\right)}\cos x\sin^{2k+1-m_{\pm}}x\right)+\frac{\Gamma\left(1-\frac{m_{\pm}}{2}\right)}{\Gamma\left(\frac{1}{2}-\frac{m_{\pm}}{2}\right)}I_{-m_{\pm}}\right\} \\ & =\frac{\Gamma\left(-n+\frac{1}{2}-\frac{m_{\pm}}{2}\right)}{\Gamma\left(-n+1-\frac{m_{\pm}}{2}\right)}\left\{ \frac{1}{2}\sum_{k=0}^{n-1}\left(\frac{\Gamma\left(-n+k+1-\frac{m_{\pm}}{2}\right)}{\Gamma\left(-n+k+\frac{3}{2}-\frac{m_{\pm}}{2}\right)}\cos x\sin^{2\left(k-n\right)+1-m_{\pm}}x\right)+\frac{\Gamma\left(1-\frac{m_{\pm}}{2}\right)}{\Gamma\left(\frac{1}{2}-\frac{m_{\pm}}{2}\right)}I_{-m_{\pm}}\right\} \\ & =\frac{\Gamma\left(1+n-1+\frac{m_{\pm}}{2}\right)\sin\left(\left(-n+1-\frac{m_{\pm}}{2}\right)\pi\right)}{\Gamma\left(1+n-\frac{1}{2}+\frac{m_{\pm}}{2}\right)\sin\left(\left(-n+\frac{1}{2}-\frac{m_{\pm}}{2}\right)\pi\right)}\left\{ \frac{1}{2}\sum_{k=0}^{n-1}\left(\frac{\Gamma\left(1+n-k-\frac{3}{2}+\frac{m_{\pm}}{2}\right)\sin\left(\left(-n+k+\frac{3}{2}-\frac{m_{\pm}}{2}\right)\pi\right)}{\Gamma\left(1+n-k-1+\frac{m_{\pm}}{2}\right)\sin\left(\left(-n+k+1-\frac{m_{\pm}}{2}\right)\pi\right)}\cos x\sin^{2\left(k-n\right)+1-m_{\pm}}x\right)+\frac{\Gamma\left(1-\frac{m_{\pm}}{2}\right)}{\Gamma\left(\frac{1}{2}-\frac{m_{\pm}}{2}\right)}I_{-m_{\pm}}\right\} \\ & =\frac{\Gamma\left(n+\frac{m_{\pm}}{2}\right)\sin\left(\frac{m_{\pm}}{2}\pi\right)}{\Gamma\left(n+\frac{1}{2}+\frac{m_{\pm}}{2}\right)\cos\left(\frac{m_{\pm}}{2}\pi\right)}\left\{ -\frac{1}{2}\sum_{k=0}^{n-1}\left(\frac{\Gamma\left(n-k-\frac{1}{2}+\frac{m_{\pm}}{2}\right)\cos\left(\frac{m_{\pm}}{2}\pi\right)}{\Gamma\left(n-k+\frac{m_{\pm}}{2}\right)\sin\left(\frac{m_{\pm}}{2}\pi\right)}\cos x\sin^{2\left(k-n\right)+1-m_{\pm}}x\right)+\frac{\Gamma\left(1-\frac{m_{\pm}}{2}\right)}{\Gamma\left(\frac{1}{2}-\frac{m_{\pm}}{2}\right)}I_{-m_{\pm}}\right\} \\ & =-\frac{\Gamma\left(n+\frac{m_{\pm}}{2}\right)}{\Gamma\left(n+\frac{1}{2}+\frac{m_{\pm}}{2}\right)}\left\{ \frac{1}{2}\sum_{k=0}^{n-1}\left(\frac{\Gamma\left(n-k-\frac{1}{2}+\frac{m_{\pm}}{2}\right)}{\Gamma\left(n-k+\frac{m_{\pm}}{2}\right)}\cos x\sin^{2\left(k-n\right)+1-m_{\pm}}x\right)-\tan\left(\frac{m_{\pm}}{2}\pi\right)\frac{\Gamma\left(1-\frac{m_{\pm}}{2}\right)}{\Gamma\left(\frac{1}{2}-\frac{m_{\pm}}{2}\right)}I_{-m_{\pm}}\right\} \\ & =-\frac{\Gamma\left(n+\frac{m_{\pm}}{2}\right)}{\Gamma\left(n+\frac{1}{2}+\frac{m_{\pm}}{2}\right)}\left\{ \frac{1}{2}\sum_{k=0}^{n-1}\left(\frac{\Gamma\left(n-k-\frac{1}{2}+\frac{m_{\pm}}{2}\right)}{\Gamma\left(n-k+\frac{m_{\pm}}{2}\right)}\cos x\sin^{2\left(k-n\right)+1-m_{\pm}}x\right)-\frac{1}{\sqrt{\pi}}I_{-m_{\pm}}\delta_{1,m_{\pm}}\right\} \end{align*}

(3)

\[ I_{n}=\int\cos^{n}xdx \] とおくと積分漸化式は
\begin{align*} I_{n+2} & =\frac{1}{n+2}\sin x\cos^{n+1}x+\frac{n+1}{n+2}I_{n}\\ \frac{\Gamma\left(\frac{n+2}{2}+1\right)}{\Gamma\left(\frac{n+2}{2}+\frac{1}{2}\right)}I_{n+2}-\frac{\Gamma\left(\frac{n}{2}+1\right)}{\Gamma\left(\frac{n}{2}+\frac{1}{2}\right)}I_{n} & =\frac{1}{2}\frac{\Gamma\left(\frac{n+2}{2}\right)}{\Gamma\left(\frac{n+2}{2}+\frac{1}{2}\right)}\sin x\cos^{n+1}x \end{align*} となる。
\(m_{\pm}=\frac{1\pm1}{2}\)とおく。
\begin{align*} I_{2n+m_{\pm}} & =\frac{\Gamma\left(\frac{2n+m_{\pm}}{2}+\frac{1}{2}\right)}{\Gamma\left(\frac{2n+m_{\pm}}{2}+1\right)}\left\{ \sum_{k=0}^{n-1}\left(\frac{\Gamma\left(\frac{2\left(k+1\right)+m_{\pm}}{2}+1\right)}{\Gamma\left(\frac{2\left(k+1\right)+m_{\pm}}{2}+\frac{1}{2}\right)}I_{2\left(k+1\right)+m_{\pm}}-\frac{\Gamma\left(\frac{2k+m_{\pm}}{2}+1\right)}{\Gamma\left(\frac{2k+m_{\pm}}{2}+\frac{1}{2}\right)}I_{2k+m_{\pm}}\right)+\frac{\Gamma\left(\frac{m_{\pm}}{2}+1\right)}{\Gamma\left(\frac{m_{\pm}}{2}+\frac{1}{2}\right)}I_{m_{\pm}}\right\} \\ & =\frac{\Gamma\left(n+\frac{m_{\pm}}{2}+\frac{1}{2}\right)}{\Gamma\left(n+\frac{m_{\pm}}{2}+1\right)}\left\{ \frac{1}{2}\sum_{k=0}^{n-1}\left(\frac{\Gamma\left(\frac{2(k+1)+m_{\pm}}{2}\right)}{\Gamma\left(\frac{2(k+1)+m_{\pm}}{2}+\frac{1}{2}\right)}\sin x\cos^{2k+m_{\pm}+1}x\right)+\frac{\Gamma\left(\frac{m_{\pm}}{2}+1\right)}{\Gamma\left(\frac{m_{\pm}}{2}+\frac{1}{2}\right)}I_{m_{\pm}}\right\} \\ & =\frac{\Gamma\left(n+\frac{1}{2}+\frac{m_{\pm}}{2}\right)}{\Gamma\left(n+1+\frac{m_{\pm}}{2}\right)}\left\{ \frac{1}{2}\sum_{k=0}^{n-1}\left(\frac{\Gamma\left(k+1+\frac{m_{\pm}}{2}\right)}{\Gamma\left(k+\frac{3}{2}+\frac{m_{\pm}}{2}\right)}\sin x\cos^{2k+m_{\pm}+1}x\right)+\frac{\Gamma\left(1+\frac{m_{\pm}}{2}\right)}{\Gamma\left(\frac{1}{2}+\frac{m_{\pm}}{2}\right)}I_{m_{\pm}}\right\} \end{align*}

(4)

\begin{align*} I_{-2n-m_{\pm}} & =\frac{\Gamma\left(-n+\frac{1}{2}-\frac{m_{\pm}}{2}\right)}{\Gamma\left(-n+1-\frac{m_{\pm}}{2}\right)}\left\{ \frac{1}{2}\sum_{k=0}^{-n-1}\left(\frac{\Gamma\left(k+1-\frac{m_{\pm}}{2}\right)}{\Gamma\left(k+\frac{3}{2}-\frac{m_{\pm}}{2}\right)}\sin x\cos^{2k-m_{\pm}+1}x\right)+\frac{\Gamma\left(1-\frac{m_{\pm}}{2}\right)}{\Gamma\left(\frac{1}{2}-\frac{m_{\pm}}{2}\right)}I_{-m_{\pm}}\right\} \\ & =\frac{\Gamma\left(-n+\frac{1}{2}-\frac{m_{\pm}}{2}\right)}{\Gamma\left(-n+1-\frac{m_{\pm}}{2}\right)}\left\{ -\frac{1}{2}\sum_{k=-n}^{-1}\left(\frac{\Gamma\left(k+1-\frac{m_{\pm}}{2}\right)}{\Gamma\left(k+\frac{3}{2}-\frac{m_{\pm}}{2}\right)}\sin x\cos^{2k-m_{\pm}+1}x\right)+\frac{\Gamma\left(1-\frac{m_{\pm}}{2}\right)}{\Gamma\left(\frac{1}{2}-\frac{m_{\pm}}{2}\right)}I_{-m_{\pm}}\right\} \\ & =\frac{\Gamma\left(-n+\frac{1}{2}-\frac{m_{\pm}}{2}\right)}{\Gamma\left(-n+1-\frac{m_{\pm}}{2}\right)}\left\{ -\frac{1}{2}\sum_{k=0}^{n-1}\left(\frac{\Gamma\left(-n+k+1-\frac{m_{\pm}}{2}\right)}{\Gamma\left(-n+k+\frac{3}{2}-\frac{m_{\pm}}{2}\right)}\sin x\cos^{2\left(k-n\right)-m_{\pm}+1}x\right)+\frac{\Gamma\left(1-\frac{m_{\pm}}{2}\right)}{\Gamma\left(\frac{1}{2}-\frac{m_{\pm}}{2}\right)}I_{-m_{\pm}}\right\} \\ & =\frac{\Gamma\left(1+n-1+\frac{m_{\pm}}{2}\right)\sin\left(-n+1-\frac{m_{\pm}}{2}\right)}{\Gamma\left(1+n-\frac{1}{2}+\frac{m_{\pm}}{2}\right)\sin\left(-n+\frac{1}{2}-\frac{m_{\pm}}{2}\right)}\left\{ -\frac{1}{2}\sum_{k=0}^{n-1}\left(\frac{\Gamma\left(1+n-k-\frac{3}{2}+\frac{m_{\pm}}{2}\right)\sin\left(-n+k+\frac{3}{2}-\frac{m_{\pm}}{2}\right)}{\Gamma\left(1+n-k-1+\frac{m_{\pm}}{2}\right)\sin\left(-n+k+1-\frac{m_{\pm}}{2}\right)}\sin x\cos^{2\left(k-n\right)-m_{\pm}+1}x\right)+\frac{\Gamma\left(1-\frac{m_{\pm}}{2}\right)}{\Gamma\left(\frac{1}{2}-\frac{m_{\pm}}{2}\right)}I_{-m_{\pm}}\right\} \\ & =\frac{\Gamma\left(n+\frac{m_{\pm}}{2}\right)\sin\left(\frac{m_{\pm}}{2}\pi\right)}{\Gamma\left(n+\frac{1}{2}+\frac{m_{\pm}}{2}\right)\cos\left(\frac{m_{\pm}}{2}\pi\right)}\left\{ \frac{1}{2}\sum_{k=0}^{n-1}\left(\frac{\Gamma\left(n-k-\frac{1}{2}+\frac{m_{\pm}}{2}\right)\cos\left(\frac{m_{\pm}}{2}\pi\right)}{\Gamma\left(n-k+\frac{m_{\pm}}{2}\right)\sin\left(\frac{m_{\pm}}{2}\pi\right)}\sin x\cos^{2\left(k-n\right)-m_{\pm}+1}x\right)+\frac{\Gamma\left(1-\frac{m_{\pm}}{2}\right)}{\Gamma\left(\frac{1}{2}-\frac{m_{\pm}}{2}\right)}I_{-m_{\pm}}\right\} \\ & =\frac{\Gamma\left(n+\frac{m_{\pm}}{2}\right)}{\Gamma\left(n+\frac{1}{2}+\frac{m_{\pm}}{2}\right)}\left\{ \frac{1}{2}\sum_{k=0}^{n-1}\left(\frac{\Gamma\left(n-k-\frac{1}{2}+\frac{m_{\pm}}{2}\right)}{\Gamma\left(n-k+\frac{m_{\pm}}{2}\right)}\sin x\cos^{2\left(k-n\right)-m_{\pm}+1}x\right)+\tan\left(\frac{m_{\pm}}{2}\pi\right)\frac{\Gamma\left(1-\frac{m_{\pm}}{2}\right)}{\Gamma\left(\frac{1}{2}-\frac{m_{\pm}}{2}\right)}I_{-m_{\pm}}\right\} \\ & =\frac{\Gamma\left(n+\frac{m_{\pm}}{2}\right)}{\Gamma\left(n+\frac{1}{2}+\frac{m_{\pm}}{2}\right)}\left\{ \frac{1}{2}\sum_{k=0}^{n-1}\left(\frac{\Gamma\left(n-k-\frac{1}{2}+\frac{m_{\pm}}{2}\right)}{\Gamma\left(n-k+\frac{m_{\pm}}{2}\right)}\sin x\cos^{2\left(k-n\right)-m_{\pm}+1}x\right)+\frac{1}{\sqrt{\pi}}I_{-m_{\pm}}\delta_{1,m_{\pm}}\right\} \end{align*}

(5)

\[ I_{n}=\int\tan^{n}xdx \] とおくと積分漸化式は
\begin{align*} I_{n+2} & =\frac{1}{n+1}\tan^{n+1}x-I_{n}\\ i^{n+2}I_{n+2}-i^{n}I_{n} & =\frac{i^{n+2}}{n+1}\tan^{n+1}x \end{align*} となる。
\(m_{\pm}=\frac{1\pm1}{2}\)とおく。
\begin{align*} I_{2n+m_{\pm}} & =\frac{1}{i^{2n+m_{\pm}}}\left\{ \sum_{k=0}^{n-1}\left(i^{2\left(k+1\right)+m_{\pm}}I_{2\left(k+1\right)+m_{\pm}}-i^{2k+m_{\pm}}I_{2k+m_{\pm}}\right)+i^{m_{\pm}}I_{m_{\pm}}\right\} \\ & =\frac{1}{i^{2n+m_{\pm}}}\left\{ \sum_{k=0}^{n-1}\left(\frac{i^{2\left(k+1\right)+m_{\pm}}}{2k+m_{\pm}+1}\tan^{2k+m_{\pm}+1}x\right)+i^{m_{\pm}}I_{m_{\pm}}\right\} \\ & =\frac{1}{\left(-1\right)^{n}}\left\{ \sum_{k=0}^{n-1}\left(\frac{\left(-1\right)^{k+1}}{2k+m_{\pm}+1}\tan^{2k+m_{\pm}+1}x\right)+I_{m_{\pm}}\right\} \end{align*}

(6)

\begin{align*} I_{-2n-m_{\pm}} & =\frac{1}{\left(-1\right)^{-n}}\left\{ \sum_{k=0}^{-n-1}\left(\frac{\left(-1\right)^{k+1}}{2k-m_{\pm}+1}\tan^{2k-m_{\pm}+1}x\right)+I_{-m_{\pm}}\right\} \\ & =\frac{1}{\left(-1\right)^{-n}}\left\{ -\sum_{k=-n}^{-1}\left(\frac{\left(-1\right)^{k+1}}{2k-m_{\pm}+1}\tan^{2k-m_{\pm}+1}x\right)+I_{-m_{\pm}}\right\} \\ & =\frac{1}{\left(-1\right)^{-n}}\left\{ -\sum_{k=0}^{n-1}\left(\frac{\left(-1\right)^{k-n+1}}{2\left(k-n\right)-m_{\pm}+1}\tan^{2\left(k-n\right)-m_{\pm}+1}x\right)+I_{-m_{\pm}}\right\} \end{align*}

(1)

\begin{align*} \int\sinh^{2n+m_{\pm}}xdx & =\frac{1}{i^{2n+m_{\pm}+1}}\int\sin^{2n+m_{\pm}}\left(ix\right)d\left(ix\right)\\ & =\frac{1}{i^{2n+m_{\pm}+1}}\frac{\Gamma\left(n+\frac{1}{2}+\frac{m_{\pm}}{2}\right)}{\Gamma\left(n+1+\frac{m_{\pm}}{2}\right)}\left\{ -\frac{1}{2}\sum_{k=0}^{n-1}\left(\frac{\Gamma\left(k+1+\frac{m_{\pm}}{2}\right)}{\Gamma\left(k+\frac{3}{2}+\frac{m_{\pm}}{2}\right)}\cos\left(ix\right)\sin^{2k+1+m_{\pm}}\left(ix\right)\right)+\frac{\Gamma\left(1+\frac{m_{\pm}}{2}\right)}{\Gamma\left(\frac{1}{2}+\frac{m_{\pm}}{2}\right)}\int\sin^{m_{\pm}}\left(ix\right)d\left(ix\right)\right\} \\ & =\frac{1}{i^{2n+m_{\pm}+1}}\frac{\Gamma\left(n+\frac{1}{2}+\frac{m_{\pm}}{2}\right)}{\Gamma\left(n+1+\frac{m_{\pm}}{2}\right)}\left\{ -\frac{1}{2}\sum_{k=0}^{n-1}\left(i^{2k+1+m_{\pm}}\frac{\Gamma\left(k+1+\frac{m_{\pm}}{2}\right)}{\Gamma\left(k+\frac{3}{2}+\frac{m_{\pm}}{2}\right)}\cosh x\sinh^{2k+1+m_{\pm}}x\right)+i^{m_{\pm}+1}\frac{\Gamma\left(1+\frac{m_{\pm}}{2}\right)}{\Gamma\left(\frac{1}{2}+\frac{m_{\pm}}{2}\right)}\int\sinh^{m_{\pm}}xdx\right\} \\ & =\frac{\Gamma\left(n+\frac{1}{2}+\frac{m_{\pm}}{2}\right)}{\Gamma\left(n+1+\frac{m_{\pm}}{2}\right)}\left\{ -\frac{1}{2}\sum_{k=0}^{n-1}\left(\left(-1\right)^{k-n}\frac{\Gamma\left(k+1+\frac{m_{\pm}}{2}\right)}{\Gamma\left(k+\frac{3}{2}+\frac{m_{\pm}}{2}\right)}\cosh x\sinh^{2k+1+m_{\pm}}x\right)+\left(-1\right)^{n}\frac{\Gamma\left(1+\frac{m_{\pm}}{2}\right)}{\Gamma\left(\frac{1}{2}+\frac{m_{\pm}}{2}\right)}\int\sinh^{m_{\pm}}xdx\right\} \end{align*}

(2)

\begin{align*} \int\sinh^{-2n-m_{\pm}}xdx & =\frac{1}{i^{-2n-m_{\pm}+1}}\int\sin^{-2n-m_{\pm}}\left(ix\right)d\left(ix\right)\\ & =-\frac{1}{i^{-2n-m_{\pm}+1}}\frac{\Gamma\left(n+\frac{m_{\pm}}{2}\right)}{\Gamma\left(n+\frac{1}{2}+\frac{m_{\pm}}{2}\right)}\left\{ \frac{1}{2}\sum_{k=0}^{n-1}\left(\frac{\Gamma\left(n-k-\frac{1}{2}+\frac{m_{\pm}}{2}\right)}{\Gamma\left(n-k+\frac{m_{\pm}}{2}\right)}\cos\left(ix\right)\sin^{2\left(k-n\right)+1-m_{\pm}}\left(ix\right)\right)-\frac{1}{\sqrt{\pi}}\delta_{1,m_{\pm}}\int\sin^{-m_{\pm}}\left(ix\right)d\left(ix\right)\right\} \\ & =-\frac{1}{i^{-2n-m_{\pm}+1}}\frac{\Gamma\left(n+\frac{m_{\pm}}{2}\right)}{\Gamma\left(n+\frac{1}{2}+\frac{m_{\pm}}{2}\right)}\left\{ \frac{1}{2}\sum_{k=0}^{n-1}\left(i^{2\left(k-n\right)+1-m_{\pm}}\frac{\Gamma\left(n-k-\frac{1}{2}+\frac{m_{\pm}}{2}\right)}{\Gamma\left(n-k+\frac{m_{\pm}}{2}\right)}\cosh x\sinh^{2\left(k-n\right)+1-m_{\pm}}x\right)-\frac{1}{i^{-m_{\pm}+1}}\frac{1}{\sqrt{\pi}}\delta_{1,m_{\pm}}\int\sinh^{-m_{\pm}}\left(x\right)dx\right\} \\ & =-\frac{\Gamma\left(n+\frac{m_{\pm}}{2}\right)}{\Gamma\left(n+\frac{1}{2}+\frac{m_{\pm}}{2}\right)}\left\{ \frac{1}{2}\sum_{k=0}^{n-1}\left(\left(-1\right)^{k}\frac{\Gamma\left(n-k-\frac{1}{2}+\frac{m_{\pm}}{2}\right)}{\Gamma\left(n-k+\frac{m_{\pm}}{2}\right)}\cosh x\sinh^{2\left(k-n\right)+1-m_{\pm}}x\right)-\left(-1\right)^{n}\frac{1}{\sqrt{\pi}}\delta_{1,m_{\pm}}\int\sinh^{-m_{\pm}}\left(x\right)dx\right\} \end{align*}

(3)

\begin{align*} \int\cosh^{2n+m_{\pm}}xdx & =\frac{1}{i}\int\cos^{2n+m_{\pm}}\left(ix\right)d\left(ix\right)\\ & =\frac{1}{i}\frac{\Gamma\left(n+\frac{1}{2}+\frac{m_{\pm}}{2}\right)}{\Gamma\left(n+1+\frac{m_{\pm}}{2}\right)}\left\{ \frac{1}{2}\sum_{k=0}^{n-1}\left(\frac{\Gamma\left(k+1+\frac{m_{\pm}}{2}\right)}{\Gamma\left(k+\frac{3}{2}+\frac{m_{\pm}}{2}\right)}\sin\left(ix\right)\cos^{2k+m_{\pm}+1}\left(ix\right)\right)+\frac{\Gamma\left(1+\frac{m_{\pm}}{2}\right)}{\Gamma\left(\frac{1}{2}+\frac{m_{\pm}}{2}\right)}\int\cos^{m_{\pm}}\left(ix\right)d\left(ix\right)\right\} \\ & =\frac{1}{i}\frac{\Gamma\left(n+\frac{1}{2}+\frac{m_{\pm}}{2}\right)}{\Gamma\left(n+1+\frac{m_{\pm}}{2}\right)}\left\{ \frac{1}{2}\sum_{k=0}^{n-1}\left(i\frac{\Gamma\left(k+1+\frac{m_{\pm}}{2}\right)}{\Gamma\left(k+\frac{3}{2}+\frac{m_{\pm}}{2}\right)}\sinh x\cosh^{2k+m_{\pm}+1}x\right)+i\frac{\Gamma\left(1+\frac{m_{\pm}}{2}\right)}{\Gamma\left(\frac{1}{2}+\frac{m_{\pm}}{2}\right)}\int\cosh^{m_{\pm}}xdx\right\} \\ & =\frac{\Gamma\left(n+\frac{1}{2}+\frac{m_{\pm}}{2}\right)}{\Gamma\left(n+1+\frac{m_{\pm}}{2}\right)}\left\{ \frac{1}{2}\sum_{k=0}^{n-1}\left(\frac{\Gamma\left(k+1+\frac{m_{\pm}}{2}\right)}{\Gamma\left(k+\frac{3}{2}+\frac{m_{\pm}}{2}\right)}\sinh x\cosh^{2k+m_{\pm}+1}x\right)+\frac{\Gamma\left(1+\frac{m_{\pm}}{2}\right)}{\Gamma\left(\frac{1}{2}+\frac{m_{\pm}}{2}\right)}\int\cosh^{m_{\pm}}xdx\right\} \end{align*}

(4)

\begin{align*} \int\cosh^{-2n-m_{\pm}}xdx & =\frac{1}{i}\int\cos^{-2n-m_{\pm}}\left(ix\right)d\left(ix\right)\\ & =\frac{1}{i}\frac{\Gamma\left(n+\frac{m_{\pm}}{2}\right)}{\Gamma\left(n+\frac{1}{2}+\frac{m_{\pm}}{2}\right)}\left\{ \frac{1}{2}\sum_{k=0}^{n-1}\left(\frac{\Gamma\left(n-k-\frac{1}{2}+\frac{m_{\pm}}{2}\right)}{\Gamma\left(n-k+\frac{m_{\pm}}{2}\right)}\sin\left(ix\right)\cos^{2\left(k-n\right)-m_{\pm}+1}\left(ix\right)\right)+\frac{1}{\sqrt{\pi}}\delta_{1,m_{\pm}}\int\cos^{-m_{\pm}}\left(ix\right)d\left(ix\right)\right\} \\ & =\frac{1}{i}\frac{\Gamma\left(n+\frac{m_{\pm}}{2}\right)}{\Gamma\left(n+\frac{1}{2}+\frac{m_{\pm}}{2}\right)}\left\{ \frac{1}{2}\sum_{k=0}^{n-1}\left(i\frac{\Gamma\left(n-k-\frac{1}{2}+\frac{m_{\pm}}{2}\right)}{\Gamma\left(n-k+\frac{m_{\pm}}{2}\right)}\sinh x\cosh^{2\left(k-n\right)-m_{\pm}+1}x\right)+i\frac{1}{\sqrt{\pi}}\delta_{1,m_{\pm}}\int\cosh^{-m_{\pm}}xdx\right\} \\ & =\frac{\Gamma\left(n+\frac{m_{\pm}}{2}\right)}{\Gamma\left(n+\frac{1}{2}+\frac{m_{\pm}}{2}\right)}\left\{ \frac{1}{2}\sum_{k=0}^{n-1}\left(\frac{\Gamma\left(n-k-\frac{1}{2}+\frac{m_{\pm}}{2}\right)}{\Gamma\left(n-k+\frac{m_{\pm}}{2}\right)}\sinh x\cosh^{2\left(k-n\right)-m_{\pm}+1}x\right)+\frac{1}{\sqrt{\pi}}\delta_{1,m_{\pm}}\int\cosh^{-m_{\pm}}xdx\right\} \end{align*}

(5)

\begin{align*} \int\tanh^{2n+m_{\pm}}xdx & =\frac{1}{i^{2n+m_{\pm}+1}}\int\tan^{2n+m_{\pm}}\left(ix\right)d\left(ix\right)\\ & =\frac{1}{i^{2n+m_{\pm}+1}}\frac{1}{\left(-1\right)^{n}}\left\{ \sum_{k=0}^{n-1}\left(\frac{\left(-1\right)^{k+1}}{2k+m_{\pm}+1}\tan^{2k+m_{\pm}+1}\left(ix\right)\right)+\int\tan^{m_{\pm}}\left(ix\right)d\left(ix\right)\right\} \\ & =\frac{1}{i^{2n+m_{\pm}+1}}\frac{1}{\left(-1\right)^{n}}\left\{ \sum_{k=0}^{n-1}\left(i^{2k+m_{\pm}+1}\frac{\left(-1\right)^{k+1}}{2k+m_{\pm}+1}\tanh^{2k+m_{\pm}+1}x\right)+i^{m_{\pm}+1}\int\tanh^{m_{\pm}}xdx\right\} \\ & =\frac{1}{\left(-1\right)^{n}}\left\{ \sum_{k=0}^{n-1}\left(\left(-1\right)^{k-n}\frac{\left(-1\right)^{k+1}}{2k+m_{\pm}+1}\tanh^{2k+m_{\pm}+1}x\right)+\left(-1\right)^{n}\int\tanh^{m_{\pm}}xdx\right\} \\ & =-\sum_{k=0}^{n-1}\left(\frac{1}{2k+m_{\pm}+1}\tanh^{2k+m_{\pm}+1}x\right)+\int\tanh^{m_{\pm}}xdx \end{align*}

(6)

\begin{align*} \int\tanh^{-2n-m_{\pm}}xdx & =\frac{1}{i^{-2n-m_{\pm}+1}}\int\tan^{-2n-m_{\pm}}\left(ix\right)d\left(ix\right)\\ & =\frac{1}{i^{-2n-m_{\pm}+1}}\frac{1}{\left(-1\right)^{-n}}\left\{ -\sum_{k=0}^{n-1}\left(\frac{\left(-1\right)^{k-n+1}}{2\left(k-n\right)-m_{\pm}+1}\tan^{2\left(k-n\right)-m_{\pm}+1}\left(ix\right)\right)+\int\tan^{-m_{\pm}}\left(ix\right)d\left(ix\right)\right\} \\ & =\frac{1}{i^{-2n-m_{\pm}+1}}\frac{1}{\left(-1\right)^{-n}}\left\{ -\sum_{k=0}^{n-1}\left(i^{2\left(k-n\right)-m_{\pm}+1}\frac{\left(-1\right)^{k-n+1}}{2\left(k-n\right)-m_{\pm}+1}\tanh^{2\left(k-n\right)-m_{\pm}+1}x\right)+i^{-m_{\pm}+1}\int\tanh^{-m_{\pm}}xdx\right\} \\ & =\frac{1}{\left(-1\right)^{-n}}\left\{ -\sum_{k=0}^{n-1}\left(\left(-1\right)^{k}\frac{\left(-1\right)^{k-n+1}}{2\left(k-n\right)-m_{\pm}+1}\tanh^{2\left(k-n\right)-m_{\pm}+1}x\right)+\left(-1\right)^{n}\int\tanh^{-m_{\pm}}xdx\right\} \\ & =\left\{ \sum_{k=0}^{n-1}\left(\frac{1}{2\left(k-n\right)-m_{\pm}+1}\tanh^{2\left(k-n\right)-m_{\pm}+1}x\right)+\int\tanh^{-m_{\pm}}xdx\right\} \end{align*}