x tan(x)とx tanh(x)の積分

x tan(x)とx tanh(x)の積分

(1)

\[ \int x\tan^{\pm1}\left(x\right)=\left(i\right)^{\pm1}\left(\frac{x^{2}}{2}-ixLi_{1}\left(\mp e^{2ix}\right)+\frac{1}{2}Li_{2}\left(\mp e^{2ix}\right)\right) \]

(2)

\[ \int x\tanh^{\pm1}\left(x\right)=\frac{x^{2}}{2}-xLi_{1}\left(\mp e^{-2x}\right)-\frac{1}{2}Li_{2}\left(\mp e^{-2x}\right) \]

(1)

\begin{align*} \int x\tan^{\pm1}\left(x\right) & =\int x\left(i\right)^{\mp1}\frac{e^{ix}\mp e^{-ix}}{e^{ix}\pm e^{-ix}}dx\\ & =\left(i\right)^{\mp1}\int x\frac{e^{2ix}\mp1}{e^{2ix}\pm1}dx\\ & =\left(i\right)^{\mp1}\int x\left(1\mp\frac{2}{e^{2ix}\pm1}\right)dx\\ & =\left(i\right)^{\mp1}\int x\left(1-\frac{2}{1\pm e^{2ix}}\right)dx\\ & =\left(i\right)^{\mp1}\left(\frac{x^{2}}{2}-2\int\frac{x}{1\pm e^{2ix}}dx\right)\\ & =\left(i\right)^{\mp1}\left(\frac{x^{2}}{2}-2\int\frac{1}{1\pm e^{2ix}}\frac{1}{2i}\log e^{2ix}\frac{e^{-2ix}}{2i}de^{2ix}\right)\\ & =\left(i\right)^{\mp1}\left(\frac{x^{2}}{2}+\frac{1}{2}\int\log e^{2ix}\frac{1}{e^{2ix}\left(1\pm e^{2ix}\right)}de^{2ix}\right)\\ & =\left(i\right)^{\mp1}\left(\frac{x^{2}}{2}+\frac{1}{2}\int\log e^{2ix}\left(\frac{1}{e^{2ix}}\mp\frac{1}{1\pm e^{2ix}}\right)de^{2ix}\right)\\ & =\left(i\right)^{\mp1}\left(\frac{x^{2}}{2}+\frac{1}{2}\int\log e^{2ix}d\left(\log e^{2ix}\right)\mp\frac{1}{2}\int\log e^{2ix}\frac{1}{1\pm e^{2ix}}de^{2ix}\right)\\ & =\left(i\right)^{\mp1}\left(\frac{x^{2}}{2}+\frac{1}{4}\left(\log e^{2ix}\right)^{2}\mp\frac{1}{2}\left(\pm\log e^{2ix}\log\left(1\pm e^{2ix}\right)\mp\int\frac{\log\left(1\pm e^{2ix}\right)}{e^{2ix}}de^{2ix}\right)\right)\\ & =\left(i\right)^{\mp1}\left(\frac{x^{2}}{2}-x^{2}\mp\frac{1}{2}\left(\mp2ixLi_{1}\left(\mp e^{2ix}\right)\pm\int\frac{Li_{1}\left(\mp e^{2ix}\right)}{e^{2ix}}de^{2ix}\right)\right)\\ & =\left(i\right)^{\mp1}\left(\frac{x^{2}}{2}-x^{2}\mp\frac{1}{2}\left(\mp2ixLi_{1}\left(\mp e^{2ix}\right)\pm Li_{2}\left(\mp e^{2ix}\right)\right)\right)\\ & =\left(i\right)^{\pm1}\left(\frac{x^{2}}{2}-ixLi_{1}\left(\mp e^{2ix}\right)+\frac{1}{2}Li_{2}\left(\mp e^{2ix}\right)\right) \end{align*}

(2)

\begin{align*} \int x\tanh^{\pm1}\left(x\right) & =\int x\left(-i\tan\left(ix\right)\right)^{\pm1}dx\\ & =\left(-i\right)^{\pm1}\int x\tan^{\pm1}\left(ix\right)dx\\ & =-\left(-i\right)^{\pm1}\int ix\tan^{\pm1}\left(ix\right)d\left(ix\right)\\ & =-\left(-i\right)^{\pm1}\left(i\right)^{\pm1}\left(-\frac{x^{2}}{2}+xLi_{1}\left(\mp e^{-2x}\right)+\frac{1}{2}Li_{2}\left(\mp e^{-2x}\right)\right)\\ & =\frac{x^{2}}{2}-xLi_{1}\left(\mp e^{-2x}\right)-\frac{1}{2}Li_{2}\left(\mp e^{-2x}\right) \end{align*}

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x tan(x)とx tanh(x)の積分

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