三角関数・双曲線関数の一次結合の逆数の積分

三角関数の一次結合の逆数の積分

\begin{align*} \int\frac{1}{\alpha\sin z+\beta\cos z+\gamma}dz & =\begin{cases} -\frac{2}{\sqrt{\alpha^{2}+\beta^{2}-\gamma^{2}}}\tanh^{\circ}\frac{\left(\gamma-\beta\right)\tan\frac{z}{2}+\alpha}{\sqrt{\alpha^{2}+\beta^{2}-\gamma^{2}}}+C & \beta\ne\gamma\;\land\;\alpha^{2}+\beta^{2}\ne\gamma^{2}\\ \frac{1}{\alpha}\Log\left(\alpha\tan\frac{z}{2}+\beta\right)+C & \beta=\gamma\;\land\;\alpha\ne0\\ \frac{1}{\beta}\tan\frac{z}{2}+C & \beta=\gamma\;\land\;\alpha=0\\ -\frac{2}{\left(\gamma-\beta\right)\tan\frac{z}{2}+\alpha}+C & \alpha^{2}+\beta^{2}=\gamma^{2} \end{cases} \end{align*}

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\(\beta\ne\gamma\;\land\;\alpha^{2}+\beta^{2}\ne\gamma^{2}\)のとき

\begin{align*} \int\frac{1}{\alpha\sin z+\beta\cos z+\gamma}dz & =\int\frac{1}{\alpha\frac{2t}{1+t^{2}}+\beta\frac{1-t^{2}}{1+t^{2}}+\gamma}\frac{2}{1+t^{2}}dt\cnd{t=\tan\frac{z}{2}}\\ & =\int\frac{2}{2\alpha t+\beta\left(1-t^{2}\right)+\gamma\left(1+t^{2}\right)}dt\\ & =\int\frac{2}{\left(\gamma-\beta\right)t^{2}+2\alpha t+\gamma+\beta}dt\\ & =\int\frac{2}{\left(\gamma-\beta\right)t^{2}+2\alpha t+\gamma+\beta}dt\\ & =\int\frac{2}{\left(\gamma-\beta\right)\left(t+\frac{\alpha}{c-\beta}\right)^{2}-\frac{\alpha^{2}}{\left(\gamma-\beta\right)}+\gamma+\beta}dt\\ & =\frac{2}{\gamma-\beta}\int\frac{1}{\left(t+\frac{\alpha}{\gamma-\beta}\right)^{2}-\frac{\alpha^{2}+\beta^{2}-\gamma^{2}}{\left(\gamma-\beta\right)^{2}}}dt\\ & =2\int\frac{1}{\left(\left(\gamma-\beta\right)t+\alpha\right)^{2}-\left(\alpha^{2}+\beta^{2}-\gamma^{2}\right)}\left(\gamma-\beta\right)dt\\ & =-\frac{2}{\sqrt{\alpha^{2}+\beta^{2}-\gamma^{2}}}\tanh^{\circ}\frac{\left(\gamma-\beta\right)t+\alpha}{\sqrt{\alpha^{2}+\beta^{2}-\gamma^{2}}}+C\\ & =-\frac{2}{\sqrt{\alpha^{2}+\beta^{2}-\gamma^{2}}}\tanh^{\circ}\frac{\left(\gamma-\beta\right)\tan\frac{z}{2}+\alpha}{\sqrt{\alpha^{2}+\beta^{2}-\gamma^{2}}}+C \end{align*}

\(\beta=\gamma\;\land\;\alpha\ne0\)のとき

\begin{align*} \int\frac{1}{\alpha\sin z+\beta\cos z+\gamma}dz & =\lim_{\gamma\rightarrow\beta}\int\frac{1}{\alpha\sin z+\beta\cos z+\gamma}dz\\ & =2\lim_{\gamma\rightarrow\beta}\int\frac{1}{\left(\left(\gamma-\beta\right)t+\alpha\right)^{2}-\left(\alpha^{2}+\beta^{2}-\gamma^{2}\right)}\left(\gamma-\beta\right)dt\cnd{t=\tan\frac{z}{2}}\\ & =2\lim_{\gamma\rightarrow\beta}\int\frac{\left(\gamma-\beta\right)}{\left(\gamma-\beta\right)^{2}t^{2}+2\alpha\left(\gamma-\beta\right)t+\left(\gamma^{2}-\beta^{2}\right)}dt\\ & =2\lim_{\gamma\rightarrow\beta}\int\frac{1}{\left(\gamma-\beta\right)t^{2}+2\alpha t+\left(\gamma+\beta\right)}dt\\ & =\int\frac{1}{\alpha t+\beta}dt\\ & =\frac{1}{\alpha}\Log\left(\alpha t+\beta\right)+C\\ & =\frac{1}{\alpha}\Log\left(\alpha\tan\frac{z}{2}+\beta\right)+C \end{align*}

\(\beta=\gamma\;\land\;\alpha=0\)のとき

\begin{align*} \int\frac{1}{\alpha\sin z+\beta\cos z+\gamma}dz & =\lim_{\alpha\rightarrow0}\frac{1}{\alpha}\Log\left(\alpha\tan\frac{z}{2}+\beta\right)+C\\ & =\lim_{\alpha\rightarrow0}\frac{\Log\left(\alpha\tan\frac{z}{2}+\beta\right)-\Log\beta}{\alpha}+C\\ & =\lim_{\alpha\rightarrow0}\frac{\left(\alpha\tan\frac{z}{2}+\beta\right)^{-1}\tan\frac{z}{2}}{1}+C\\ & =\frac{1}{\beta}\tan\frac{z}{2}+C \end{align*}

\begin{align*} \int\frac{1}{\alpha\sin z+\beta\cos z+\gamma}dz & =\lim_{\alpha\rightarrow0}\lim_{\gamma\rightarrow\beta}\int\frac{1}{\alpha\sin z+\beta\cos z+\gamma}dz\\ & =\lim_{\alpha\rightarrow0}\int\frac{1}{\alpha t+\beta}dt\cnd{t=\tan\frac{z}{2}}\\ & =\frac{1}{\beta}\int dt\\ & =\frac{t}{\beta}+C\\ & =\frac{1}{\beta}\tan\frac{z}{2}+C \end{align*}

\(\alpha^{2}+\beta^{2}=\gamma^{2}\)のとき

\begin{align*} \int\frac{1}{\alpha\sin z+\beta\cos z+\gamma}dz & =2\int\frac{1}{\left(\left(\gamma-\beta\right)t+\alpha\right)^{2}-\left(\alpha^{2}+\beta^{2}-\gamma^{2}\right)}\left(\gamma-\beta\right)dt\cnd{t=\tan\frac{z}{2}}\\ & =2\int\frac{1}{\left(\left(\gamma-\beta\right)t+\alpha\right)^{2}}\left(\gamma-\beta\right)dt\\ & =\frac{2}{\gamma-\beta}\int\frac{1}{\left(t+\frac{\alpha}{\gamma-\beta}\right)^{2}}dt\\ & =-\frac{2}{\gamma-\beta}\frac{1}{\left(t+\frac{\alpha}{\gamma-\beta}\right)}+C\\ & =-\frac{2}{\left(\gamma-\beta\right)t+\alpha}+C\\ & =-\frac{2}{\left(\gamma-\beta\right)\tan\frac{z}{2}+\alpha}+C \end{align*}

双曲線関数の一次結合の逆数の積分

\begin{align*} \int\frac{1}{\alpha\sinh z+\beta\cosh z+\gamma}dz & =\begin{cases} \frac{2}{\sqrt{-\alpha^{2}+\beta^{2}-\gamma^{2}}}\tan^{\circ}\frac{\left(\beta-\gamma\right)\tanh\frac{z}{2}+\alpha}{\sqrt{-\alpha^{2}+\beta^{2}-\gamma^{2}}}+C & \beta\ne\gamma\;\land\;\beta^{2}\ne\alpha^{2}+\gamma^{2}\\ \frac{1}{\alpha}\Log\left(\tanh\frac{z}{2}+\frac{\beta}{\alpha}\right)+C & \beta=\gamma\;\land\;\alpha\ne0\\ \frac{1}{\beta}\tanh\frac{z}{2}+C & \beta=\gamma\;\land\;\alpha=0\\ \frac{2}{\left(\gamma-\beta\right)\tanh\frac{z}{2}-\alpha}+C & \beta^{2}=\alpha^{2}+\gamma^{2} \end{cases} \end{align*}

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\(\beta\ne\gamma\;\land\;\beta^{2}\ne\alpha^{2}+\gamma^{2}\)のとき

\begin{align*} \int\frac{1}{\alpha\sinh z+\beta\cosh z+\gamma}dz & =-i\int\frac{1}{-i\alpha\sin\left(iz\right)+\beta\cos\left(iz\right)+\gamma}idz\\ & =-i\cdot-\frac{2}{\sqrt{\left(-i\alpha\right)^{2}+\beta^{2}-\gamma^{2}}}\tanh^{\circ}\frac{\left(\gamma-\beta\right)\tan\frac{iz}{2}-i\alpha}{\sqrt{\left(-i\alpha\right)^{2}+\beta^{2}-\gamma^{2}}}+C\\ & =i\frac{2}{\sqrt{-\alpha^{2}+\beta^{2}-\gamma^{2}}}\tanh^{\circ}i\frac{\left(\gamma-\beta\right)\tanh\frac{z}{2}-\alpha}{\sqrt{-\alpha^{2}+\beta^{2}-\gamma^{2}}}+C\\ & =\frac{2}{\sqrt{-\alpha^{2}+\beta^{2}-\gamma^{2}}}\tan^{\circ}\frac{\left(\beta-\gamma\right)\tanh\frac{z}{2}+\alpha}{\sqrt{-\alpha^{2}+\beta^{2}-\gamma^{2}}}+C \end{align*}

\(\beta=\gamma\;\land\;\alpha\ne0\)のとき

\begin{align*} \int\frac{1}{\alpha\sinh z+\beta\cosh z+\gamma}dz & =\lim_{\gamma\rightarrow\beta}\int\frac{1}{\alpha\sinh z+\beta\cosh z+\gamma}dz\\ & -i\int\frac{1}{-i\alpha\sin\left(iz\right)+\beta\cos\left(iz\right)+\beta}idz\\ & =-i\cdot\frac{1}{-i\alpha}\Log\left(\tan\frac{zi}{2}+\frac{\beta}{-i\alpha}\right)+C\\ & =\frac{1}{\alpha}\Log\left(i\tanh\frac{z}{2}+\frac{\beta}{\alpha}i\right)+C\\ & =\frac{1}{\alpha}\Log\left(\tanh\frac{z}{2}+\frac{\beta}{\alpha}\right)+C \end{align*}

\(\beta=\gamma\;\land\;\alpha=0\)のとき

\begin{align*} \int\frac{1}{\alpha\sinh z+\beta\cosh z+\gamma}dz & =\lim_{\alpha\rightarrow0}\lim_{\gamma\rightarrow\beta}\int\frac{1}{\alpha\sinh z+\beta\cosh z+\gamma}dz\\ & =-i\lim_{\alpha\rightarrow0}\lim_{\gamma\rightarrow\beta}\int\frac{1}{-i\alpha\sin\left(iz\right)+\beta\cos\left(iz\right)+\gamma}idz\\ & =-i\frac{1}{\beta}\tan\frac{iz}{2}+C\\ & =\frac{1}{\beta}\tanh\frac{z}{2}+C \end{align*}

\(\beta^{2}=\alpha^{2}+\gamma^{2}\)のとき

\begin{align*} \int\frac{1}{\alpha\sinh z+\beta\cosh z+\gamma}dz & =-i\int\frac{1}{-i\alpha\sin\left(iz\right)+\beta\cos\left(iz\right)+\gamma}idz\\ & =-i\cdot-\frac{2}{\left(\gamma-\beta\right)\tan\frac{zi}{2}-i\alpha}+C\\ & =\frac{2}{\left(\gamma-\beta\right)\tanh\frac{z}{2}-\alpha}+C \end{align*}

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三角関数・双曲線関数の一次結合の逆数の積分

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