超椭圆积分的化简

青青子衿

高本庆编著. 椭圆函数及其应用[M]. 北京：国防工业出版社, 1991.11.
P142
\begin{gather*}
\int_{0}^{x}\frac{1}{\sqrt{t\left(1-t^2\right)\left(1-k^2t^2\right)}}{\mathrm{d}}t\\
=\frac{1}{\sqrt{2\left(1+k\right)}}\int_{0}^{\sqrt{\frac{2\left(1+k\right)x}{\left(1+x\right)\left(1+kx\right)}}}\left\{\frac{1}{\sqrt{\left(1-t^{2}\right)\left[1-\frac{\left(1+\sqrt{k}\right)^{2}}{2\left(1+k\right)}t^{2}\right]}}
+\frac{1}{\sqrt{\left(1-t^{2}\right)\left[1-\frac{\left(1-\sqrt{k}\right)^{2}}{2\left(1+k\right)}t^{2}\right]}}\right\}{\mathrm{d}}t
\end{gather*}

[Paul F. Byrd, Morris D. Friedman (auth.) -
Handbook of Elliptic Integrals for Engineers and Physicists (1954)
lntegrands involving √(τ(1-τ²)(1-n²τ²)) ,(0<Y≤1<1/n), T595.00]

Weierstrass Elliptic Functions
\begin{align*}
\wp(u+v)=\dfrac{2\Big[\wp(u)\wp(v)+\frac{1}{4}g_2\Big]^2+2g_3\Big[\wp(u)+\wp(v)\Big]}{2\Big[\wp(u)\wp(v)-\frac{1}{4}g_2\Big]\Big[\wp(u)+\wp(v)\Big]-g_3+\wp'(u)\wp'(v)}
\end{align*}

青青子衿

\begin{align*}
\int_{0}^{x}\frac{1}{\sqrt{\left(1-a^{2}t^{2}\right)\left(1-b^{2}t^{2}\right)\left(1-c^{2}t^{2}\right)}}\mathrm{d}t\quad\qquad\\
\\
=
\frac{1}{\sqrt{c^{2}-a^{2}}}\int_{0}^{x\cdot\sqrt{\frac{c^2-a^2}{1-a^2x^2}}}\frac{1}{\sqrt{\left(1-t^{2}\right)\left(1-\frac{b^{2}-a^{2}}{c^{2}-a^{2}}t^{2}\right)}}\mathrm{d}t
\end{align*}

\begin{align*}
g'(x)^2 &=4g^3(x)-4(a^2+b^2+c^2)g^2(x)+4(a^2b^2+a^2c^2+b^2c^2)g(x)-4a^2b^2c^2\\
g'(x)^2 &= 4\Big[g(x)-\alpha\,\Big]^3 - g_2 \Big[g(x)-\alpha\,\Big] - g_3\\
g'(x)^2 &= 4g^3(x)-12\alpha\,g^2(x) + (12\alpha^2-g_2)g(x) - (g_3-g_2\alpha+4\alpha^3)\\
\alpha&=\dfrac{1}{3}\left(a^2+b^2+c^2\right)\\
g_2&=\dfrac{4}{3}(a^4+b^4+c^4-a^2b^2-b^2c^2-c^2a^2)\\
g_3&=\dfrac{4}{27} \left(a^2+b^2-2 c^2\right) \left(2 a^2-b^2-c^2\right) \left(a^2-2 b^2+c^2\right)


\end{align*}

1. f\left(x\right)=\int_{0}^{x}\frac{1}{\sqrt{\left(u-a\right)\left(u-b\right)\left(u-c\right)\left(u-d\right)}}du
2. g\left(x\right)=\int_{-\frac{\beta}{\alpha}}^{-\frac{\delta x-\beta}{\gamma x-\alpha}}\frac{\alpha\delta-\beta\gamma}{\sqrt{\left(\left(\alpha-a\gamma\right)t+\beta-a\delta\right)\left(\left(\alpha-b\gamma\right)t+\beta-b\delta\right)\left(\left(\alpha-c\gamma\right)t+\beta-c\delta\right)\left(\left(\alpha-d\gamma\right)t+\beta-d\delta\right)}}dt

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\begin{align*}
\frac{2\left(\left(\sqrt{3}-1\right)x^{2}-\left(\sqrt{3}-2\right)\right)}{\left(1-\sqrt{3}-2x^{2}\right)^{2}}\equiv\frac{1}{1+\left(1+\sqrt{3}\right)x^{2}}
\end{align*}

\begin{gather*}
\int_{0}^{x}\frac{\mathrm{d}t}{\sqrt{1+t^{6}}}=\frac{1}{2\sqrt[4]{3}}\int_{0}^{\frac{2\sqrt[4]{3}x\sqrt{1+x^{2}}}{1+\left(1+\sqrt{3}\right)x^{2}}}\frac{\mathrm{d}t}{\sqrt{\left(1-t^{2}\right)\left[1-\left(\frac{1+\sqrt{3}}{2\sqrt{2}}\right)^{2}t^{2}\right]}}\\
-\sqrt{\frac{1+\sqrt{3}}{2}}\le x\le\sqrt{\frac{1+\sqrt{3}}{2}}
\end{gather*}

\begin{gather*}
\int_{0}^{x}\frac{1}{\sqrt{1+t^{6}}}dt\\
=\frac{1}{2\sqrt[4]{3}}\int_{0}^{\frac{\left(\sqrt{3}-1\right)x^{2}-1}{\sqrt{x^{4}-x^{2}+1}}}\frac{\mathrm{d}t}{\sqrt{\left(1-t^{2}\right)\left[1-\left(\frac{1+\sqrt{3}}{2\sqrt{2}}\right)^{2}t^{2}\right]}}+\frac{1}{2\sqrt[4]{3}}\int_{0}^{1}\frac{\mathrm{d}t}{\sqrt{\left(1-t^{2}\right)\left[1-\left(\frac{1+\sqrt{3}}{2\sqrt{2}}\right)^{2}t^{2}\right]}}
\end{gather*}


\begin{align*}
\\
&\qquad\int\frac{25-10x-x^2}{\sqrt{x (x+5) (x+20) \left(x^4+10 x^3-245 x^2-450 x+10125\right)}}\mathrm{d}x\\
\\
&=\frac{1}{\sqrt{15}}\int_{C_0}^{\frac{x^{6}+30x^{5}+675x^{4}+14750x^{3}+110625x^{2}-675000x-15187500}{60(x+5)\left(x^{2}+10x-225\right)^{2}}}\frac{\mathrm{d}t}{\sqrt{4t^{3}-\frac{24}{5}t-\frac{4}{5}}}
\end{align*}

青青子衿

青青子衿 发表于 2021-1-29 19:41
\begin{align*}
\int_{0}^{x}\frac{1}{\sqrt{\left(1-a^{2}t^{2}\right)\left(1-b^{2}t^{2}\right)\left(1-c^{2}t^{2}\right)}}\mathrm{d}t\quad\qquad\\
\\
=
\frac{1}{\sqrt{c^{2}-a^{2}}}\int_{0}^{x\cdot\sqrt{\frac{c^2-a^2}{1-a^2x^2}}}\frac{1}{\sqrt{\left(1-t^{2}\right)\left(1-\frac{b^{2}-a^{2}}{c^{2}-a^{2}}t^{2}\right)}}\mathrm{d}t
\end{align*}

\begin{align*}
I&=\int_{0}^{x}\frac{\mathrm{d}t}{\sqrt{1+t^{8}}}\\
&=\frac{1}{2\sqrt{2+\sqrt{2}}}\int_{0}^{\frac{\sqrt{2+\sqrt{2}}x}{1+x^{2}}}\frac{\mathrm{d}t}{\sqrt{\left(1-t^{2}\right)\big(1-(\sqrt{2}-1)^{2}t^{2}\big)}}\\
&\qquad+\frac{1}{2\sqrt{2+\sqrt{2}}}\int_{0}^{\frac{x\sqrt{(10-7\sqrt{2})x^{4}+2(5\sqrt{2}-7)x^{2}+10-7\sqrt{2}}}{(3-2\sqrt{2})x^{4}+(3\sqrt{2}-4)x^{2}+3-2\sqrt{2}}}
\frac{\mathrm{d}t}{\sqrt{\left(1-t^{2}\right)\big(1-(\sqrt{2\sqrt{2}-2}\,)^{2}t^{2}\big)}}

\end{align*}

1. D[1/(2Sqrt[2+Sqrt[2]]) EllipticF[ArcSin[(Sqrt[2+Sqrt[2]]x)/(1+x^2)],(Sqrt[2]-1)^2]+1/(2I Sqrt[2+Sqrt[2]])*EllipticF[ArcSin[(I*(x^2-1))/(Sqrt[2-Sqrt[2]]x)],(Sqrt[2]-1)^2],x]/.x->1.1
2. D[1/(2Sqrt[2+Sqrt[2]]) EllipticF[ArcSin[(Sqrt[2+Sqrt[2]]x)/(1+x^2)],(Sqrt[2]-1)^2]+1/(2Sqrt[2+Sqrt[2]]) EllipticF[ArcTan[(x^2-1)/(Sqrt[2-Sqrt[2]]x)],2(Sqrt[2]-1)],x]/.x->1.1
3. 1/Sqrt[1+x^8] /.x->1.1

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\begin{align*}
&\qquad\int_{0}^{x}\frac{1}{\sqrt{1+t^{12}}}dt\\
&=\frac{1+i\sqrt{3}}{2\sqrt{3\sqrt{3}}}\int_{c_{1}}^{\frac{i}{\sqrt[4]{3}}\sqrt{\frac{x^{4}+1}{2x^{2}}-\frac{i\sqrt{3}\left(x^{4}-1\right)}{2x^{2}}}}\frac{\mathrm{d}t}{\sqrt{1-t^{4}}}\\
&\qquad\quad-\frac{1-i\sqrt{3}}{2\sqrt{3\sqrt{3}}}\int_{c_{2}}^{\frac{i}{\sqrt[4]{3}}\sqrt{\frac{x^{4}+1}{2x^{2}}+\frac{i\sqrt{3}\left(x^{4}-1\right)}{2x^{2}}}}\frac{\mathrm{d}t}{\sqrt{1-t^{4}}}dt
\end{align*}


\begin{align*}
\int_{0}^{x}\frac{\mathrm{d}t}{\sqrt{\left(1-t^{2}\right)\left(1-k^2t^2\right)}}=
\frac{2}{1+k}\int_{0}^{\frac{\sqrt{(1+x)(1+kx)}-\sqrt{(1-x)(1-kx)}}{2}}\frac{\mathrm{d}t}{\sqrt{\left(1-t^{2}\right)(1-\left.4kt^2\middle/(1+k)\right.^{2})}}
\end{align*}

2. \int_{0}^{x}\frac{1}{\sqrt{\left(1-t^{2}\right)\left(1-\left(kt\right)^{2}\right)}}dt
3. \frac{2}{1+k}\int_{0}^{\frac{\sqrt{(1+x)(1+kx)}-\sqrt{(1-x)(1-kx)}}{2}}\frac{1}{\sqrt{\left(1-t^{2}\right)\left(1-\left(\frac{2\sqrt{k}t}{1+k}\right)^{2}\right)}}dt

复制代码

\begin{align*}
\int_{0}^{x}\frac{1+t^{4}}{\sqrt{1+t^{12}}}\>\!\mathrm{d}t&=\frac{1}{2}\int_{x^{2}+\frac{1}{x^{2}}}^{+\infty}\sqrt{\frac{t}{t^{4}-7t^{2}+12}}\>\!\mathrm{d}t\\
\int_{0}^{x}\frac{t^{2}}{\sqrt{1+t^{12}}}\>\!\mathrm{d}t&=\frac{1}{2}\int_{x^{2}+\frac{1}{x^{2}}}^{+\infty}\frac{1}{\sqrt{t(t^{2}-3)(t^{2}-4)}}\>\!\mathrm{d}t\\
\end{align*}

\begin{align*}
\int_{0}^{x}\frac{{\mathrm{d}t}}{\sqrt{1+t^{12}}}
&=\frac{1}{2}\int_{x^{2}+\frac{1}{x^{2}}}^{+\infty}\frac{{\mathrm{d}t}}{\sqrt{t^{3}-3t}}\\
&\qquad+\frac{1}{2}\int_{\frac{x^{2}+\frac{1}{x^{2}}}{\sqrt[4]{12}}+\frac{\sqrt[4]{12}}{x^{2}+\frac{1}{x^{2}}}}^{+\infty}\frac{\frac{1}{2\sqrt[8]{12}}}{\sqrt{(t+2)\left(t^{2}-2-\frac{7}{2\sqrt{3}}\right)}}{\mathrm{d}t}\\
&\qquad\quad+\frac{1}{2}\int_{\frac{x^{2}+\frac{1}{x^{2}}}{\sqrt[4]{12}}+\frac{\sqrt[4]{12}}{x^{2}+\frac{1}{x^{2}}}}^{+\infty}\frac{\frac{1}{2\sqrt[8]{12}}}{\sqrt{(t-2)\left(t^{2}-2-\frac{7}{2\sqrt{3}}\right)}}{\mathrm{d}t}
\end{align*}

青青子衿

青青子衿 发表于 2023-8-1 18:11
\begin{align*}
\int_{0}^{x}\frac{\mathrm{d}t}{\sqrt{\left(1-t^{2}\right)\left(1-k^2t^2\right)}}=
\frac{2}{1+k}\int_{0}^{\frac{\sqrt{(1+x)(1+kx)}-\sqrt{(1-x)(1-kx)}}{2}}\frac{\mathrm{d}t}{\sqrt{\left(1-t^{2}\right)(1-\left.4kt^2\middle/(1+k)\right.^{2})}}
\end{align*}

\begin{align*}
E(\varphi;k)&=\int_0^{F(\varphi;k)}\Big(1-k^2\operatorname{sn}^2(\xi,k)\Big)\mathrm{d}\xi=\int_0^{\varphi}\sqrt{1-k^2\sin^2u}\,\mathrm{d}u\\

\Pi(\varphi;\beta,k)&=\int_0^{F(\varphi;k)}\frac{\mathrm{d}\xi}{1-\beta^2\operatorname{sn}^2(\xi,k)}=\int_0^{\varphi}\frac{\mathrm{d}u}{(1-\beta^2\sin^2u)\sqrt{1-k^2\sin^2u}}\\

\end{align*}



\begin{align*}
2\frac{\mathrm{d}B}{\mathrm{d}\beta}&=2\frac{\mathrm{d}\Big(\sqrt{\chi}\,\Pi(\beta;\varphi,k)\Big)}{\mathrm{d}\beta}\\
&=2\frac{d}{d\beta}\left(\sqrt{\left(1+\beta\right)\left(1+\frac{k^{2}}{\beta}\right)}\int_{0}^{\varphi}\frac{\mathrm{d}u}{\left(1+\beta\sin^{2}u\right)\sqrt{1-k^{2}\sin^{2}u}}\right)\\
&=\frac{\sin\left(\varphi\right)\cos\left(\varphi\right)\sqrt{1-k^{2}\sin^{2}u}}{\left(1+\beta\sin^{2}u\right)\sqrt{\left(1+\beta\right)\left(1+\frac{k^{2}}{\beta}\right)}}-\frac{\displaystyle\,k^{2}\int_{0}^{\varphi}\frac{1}{\sqrt{1-k^{2}\sin^{2}u}}\mathrm{d}u}{\beta^{2}\sqrt{\left(1+\beta\right)\left(1+\frac{k^{2}}{\beta}\right)}}-\frac{\displaystyle\,k^{2}\int_{0}^{\varphi}\frac{\sin^{2}u}{\sqrt{1-k^{2}\sin^{2}u}}\mathrm{d}u}{\beta\sqrt{\left(1+\beta\right)\left(1+\frac{k^{2}}{\beta}\right)}}
\end{align*}


\begin{align*}
&\qquad\qquad\qquad\qquad(k^2+\ell^2=1)\\
\\
\Xi&=\tfrac{\sqrt{1-{\ell}^{2}\sin^{2}\psi}}{\sin\left(\psi\right)\cos\left(\psi\right)}\int_{0}^{\varphi}\frac{\mathrm{d}u}{\left(1+\frac{\cos^{2}\psi}{\sin^{2}\psi}\sin^{2}u\right)\sqrt{1-k^{2}\sin^{2}u}}\\

&\qquad+\tfrac{\sqrt{1-k^{2}\sin^{2}\varphi}}{\sin\left(\varphi\right)\cos\left(\varphi\right)}\int_{0}^{\psi}\frac{\mathrm{d}u}{\left(1+\frac{\cos^{2}\varphi}{\sin^{2}\varphi}\sin^{2}u\right)\sqrt{1-{\ell}^{2}\sin^{2}u}}\\
\\
&=\frac{\pi}{2}+\tfrac{\sin\left(\psi\right)\sqrt{1-{\ell}^{2}\sin^{2}\psi}}{\cos\psi}\int_{0}^{\varphi}\frac{\mathrm{d}u}{\sqrt{1-k^{2}\sin^{2}u}}\\
&\quad\>\>\>\>\>\>+\tfrac{\sin\left(\varphi\right)\sqrt{1-k^{2}\sin^{2}\varphi}}{\cos\varphi}\int_{0}^{\psi}\frac{\mathrm{d}u}{\sqrt{1-{\ell}^{2}\sin^{2}u}}\\
&\qquad+\int_{0}^{\varphi}\frac{\mathrm{d}u}{\sqrt{1-k^{2}\sin^{2}u}}\int_{0}^{\psi}\frac{\mathrm{d}u}{\sqrt{1-{\ell}^{2}\sin^{2}u}}\\
&\qquad\quad-\int_{0}^{\varphi}\frac{\mathrm{d}u}{\sqrt{1-k^{2}\sin^{2}u}}\int_{0}^{\psi}\sqrt{1-{\ell}^{2}\sin^{2}u}\>\>\!\mathrm{d}u\\
&\qquad\qquad-\int_{0}^{\varphi}\sqrt{1-k^{2}\sin^{2}u}\>\>\!\mathrm{d}u\int_{0}^{\psi}\frac{\mathrm{d}u}{\sqrt{1-{\ell}^{2}\sin^{2}u}}\\

\end{align*}


\begin{align*}
\mathcal{V}(\psi,\varphi)&=\frac{\cos(\varphi)\sqrt{1-k^{2}\sin^{2}\varphi}}{\sin\varphi}\,\Big(\Pi(\psi;k\sin\varphi,k)-F(\psi;k)\Big)\\
\\
&=\int_{0}^{\psi}\frac{k^{2}\sin(\varphi)\cos(\varphi)\sqrt{1-k^{2}\sin^{2}\varphi}\sin^{2}u}{\left(1-k^{2}\sin^{2}(\varphi)\sin^{2}(u)\right)\sqrt{1-k^{2}\sin^{2}u}}du\\

&=\left(\int_{0}^{\psi}\frac{\mathrm{d}u}{\sqrt{1-k^{2}\sin^{2}u}}\right)\left(\int_{0}^{\varphi}\sqrt{1-k^{2}\sin^{2}u}\>\!\>\!\mathrm{d}u\right)\\
&\qquad\qquad-\frac{1}{2}\int_{\varphi\ominus\psi}^{\varphi\oplus\psi}\int_{0}^{v}\sqrt{\frac{1-k^{2}\sin^{2}u}{1-k^{2}\sin^{2}v}}\>\!\mathrm{d}u\mathrm{d}v\\
\\
\varphi\oplus\psi&=\arcsin\left(\tfrac{\sin(\varphi)\cos(\psi)\sqrt{1-k^{2}\sin^{2}\psi}+\sin(\psi)\cos(\varphi)\sqrt{1-k^{2}\sin^{2}\varphi}}{1-k^{2}\sin^{2}(\varphi)\sin^{2}(\psi)}\right)\\
\varphi\ominus\psi&=\arcsin\left(\tfrac{\sin(\varphi)\cos(\psi)\sqrt{1-k^{2}\sin^{2}\psi}-\sin(\psi)\cos(\varphi)\sqrt{1-k^{2}\sin^{2}\varphi}}{1-k^{2}\sin^{2}(\varphi)\sin^{2}(\psi)}\right)\\
\\
\mathcal{I}(\psi,\varphi)&=\int_{\varphi\ominus\psi}^{\varphi\oplus\psi}\int_{0}^{v}\sqrt{\frac{1-k^{2}\sin^{2}u}{1-k^{2}\sin^{2}v}}\>\!\mathrm{d}u\mathrm{d}v\\
&=\ln\left(\frac{\vartheta _4\left(\frac{\pi  (F(\varphi ;\,k)+F(\psi;\,k))}{2 K(k)},\exp \left(-\frac{\pi  K(\sqrt{1-k^2}\>\!)}{K(k)}\right)\right)}{\vartheta _4\left(\frac{\pi  (F(\varphi;\,k)-F(\psi;\,k))}{2 K(k)},\exp \left(-\frac{\pi  K(\sqrt{1-k^2}\>\!)}{K(k)}\right)\right)}\right)\\
&\qquad+\frac{2 E(k)F(\varphi;k) F(\psi ;k)}{K(k)}
\end{align*}


\begin{align*}
\mathcal{W}(\varphi,\psi)&=\mathcal{V}(\varphi,\psi)-\mathcal{V}(\psi,\varphi)\\
&=\int_{0}^{\varphi}\frac{k^{2}\sin(\psi)\cos(\psi)\sqrt{1-k^{2}\sin^{2}\psi}\sin^{2}u}{\left(1-k^{2}\sin^{2}(\psi)\sin^{2}(u)\right)\sqrt{1-k^{2}\sin^{2}u}}\mathrm{d}u\\
&\qquad-
\int_{0}^{\psi}\frac{k^{2}\sin(\varphi)\cos(\varphi)\sqrt{1-k^{2}\sin^{2}\varphi}\sin^{2}u}{\left(1-k^{2}\sin^{2}(\varphi)\sin^{2}(u)\right)\sqrt{1-k^{2}\sin^{2}u}}\mathrm{d}u\\

&=\left(\int_{0}^{\psi}\sqrt{1-k^{2}\sin^{2}u}\>\!\>\!\mathrm{d}u\right)\left(\int_{0}^{\varphi}\frac{\mathrm{d}u}{\sqrt{1-k^{2}\sin^{2}u}}\right)\\

&\qquad-\left(\int_{0}^{\varphi}\sqrt{1-k^{2}\sin^{2}u}\>\!\>\!\mathrm{d}u\right)\left(\int_{0}^{\psi}\frac{\mathrm{d}u}{\sqrt{1-k^{2}\sin^{2}u}}\right)\\
\end{align*}

青青子衿

青青子衿 发表于 2023-12-6 12:09
\begin{align*}
E(\varphi;k)&=\int_0^{F(\varphi;k)}\Big(1-k^2\operatorname{sn}^2(\xi,k)\Big)\mathrm{d}\xi=\int_0^{\varphi}\sqrt{1-k^2\sin^2u}\,\mathrm{d}u\\

\Pi(\varphi;\beta,k)&=\int_0^{F(\varphi;k)}\frac{\mathrm{d}\xi}{1-\beta^2\operatorname{sn}^2(\xi,k)}=\int_0^{\varphi}\frac{\mathrm{d}u}{(1-\beta^2\sin^2u)\sqrt{1-k^2\sin^2u}}\\

\end{align*}

\begin{align*}
\frac{\partial }{\partial u}\left(\frac{\operatorname{sn}(u)\operatorname{cn}(u)\operatorname{dn}(u)}{\operatorname{sn}^2(u)-\operatorname{sn}^2(v)}\right)
-\frac{\partial }{\partial v}\left(\frac{\operatorname{sn}(v)\operatorname{cn}(v)\operatorname{dn}(v)}{\operatorname{sn}^2(v)-\operatorname{sn}^2(u)}\right)
&=k^2 \left(\operatorname{sn}^2(u)-\operatorname{sn}^2(v)\right)\\
\\
\sqrt{(1-x^2)(1-k^2x^2)}\frac{\partial }{\partial x}\left(\frac{x\sqrt{(1-x^2)(1-k^2x^2)}}{x^2-y^2}\right)\quad
\\
-\sqrt{(1-y^2)(1-k^2y^2)}\frac{\partial }{\partial y}\left(\frac{y\sqrt{(1-y^2)(1-k^2y^2)}}{y^2-x^2}\right)
&=k^2 \left(x^2-y^2\right)

\end{align*}

青青子衿

\begin{align*}
\frac{\partial\,\Phi(y,V)}{\partial\,s}&=\frac{\partial\,\Phi(y,V)}{\partial\,V}\cdot\frac{\partial\,V}{\partial\,s}+\frac{\partial\,\Phi(y,V)}{\partial\,y}\cdot\frac{\partial\,y}{\partial\,s}\\
&=\left(\dfrac{\mathcal{E}(y,V)}{{2V(1-V)}}-\dfrac{\Phi(y,V)}{2V}-\tfrac{y\cdot(1-y^2)}{2(1-V)}\cdot\frac{\partial\,\Phi(y,V)}{\partial\,y}\right)\cdot\tfrac{\partial\,V}{\partial\,s}\\
&\qquad\quad+\frac{\partial\,\Phi(y,V)}{\partial\,y}\cdot\frac{\partial\,y}{\partial\,s}\\
&=\dfrac{\frac{\partial\,V}{\partial\,s}}{2V(1-V)}\cdot\mathcal{E}(y,V)-\dfrac{\frac{\partial\,V}{\partial\,s}}{2VM}\cdot\Phi(x,U)\\
&\qquad\qquad+\frac{\partial\,\Phi(y,V)}{\partial\,y}\cdot\left(\frac{\partial\,y}{\partial\,s}-\dfrac{y\!\cdot\!(1-y^2)}{2(1-V)}\cdot\frac{\partial\,V}{\partial\,s}\right)\\
&=\dfrac{\frac{\partial\,V}{\partial\,s}}{2V(1-V)}\cdot\mathcal{E}(y,V)-\dfrac{\frac{\partial\,V}{\partial\,s}}{2VM}\cdot\Phi(x,U)\\
&\qquad\qquad+\frac{\frac{\partial\,\Phi(x,U)}{\partial\,x}}{M}\cdot\dfrac{\frac{\partial\,y}{\partial\,s}-\frac{y\cdot(1-y^2)}{2(1-V)}\cdot\frac{\partial\,V}{\partial\,s}}{\frac{\partial\,y}{\partial\,x}}\\

\end{align*}
\begin{align*}
\\
\frac{\partial}{\partial\>\!s}\left(\frac{\Phi(x,U)}{M}\right)&=\frac{\partial\>\!(\frac{1}{M})}{\partial\>\!s}\cdot\Phi(x,U)+\frac{\frac{\partial\,U}{\partial\>\!s}}{M}\cdot\frac{\partial\,\Phi(x,U)}{\partial\,U}\\
&=\left(-\frac{\frac{\partial\>\!M}{\partial\>\!s}}{M^2}-\frac{\frac{\partial\>\!U}{\partial\>\!s}}{2UM}\right)\cdot\Phi(x,U)\\
&\qquad\quad+\frac{\frac{\partial\>\!U}{\partial\>\!s}}{2U\left(1-U\right)M}\cdot\mathcal{E}(x,U)\\
&\qquad\qquad\quad-\frac{\frac{\partial\,\Phi(x,U)}{\partial\,x}}{M}\cdot\frac{x(1-x^2)\frac{\partial\>\!U}{\partial\>\!s}}{2(1-U)}\\

\end{align*}

\begin{align*}
\frac{\partial\>\!\Delta}{\partial\>\!s}&=\frac{\partial\,\Phi(y,V)}{\partial\>\!s}
-\frac{\partial}{\partial\>\!s}\left(\frac{\Phi(x,U)}{M}\right)\\
&=\left(\frac{\frac{\partial\>\!M}{\partial\>\!s}}{M^2}+\frac{\frac{\partial\>\!U}{\partial\>\!s}}{2UM}-\dfrac{\frac{\partial\,V}{\partial\,s}}{2VM}\right)\cdot\Phi(x,U)\\


&\qquad\quad+\dfrac{\frac{\partial\>\!V}{\partial\>\!s}}{2V(1-V)}\cdot\mathcal{E}(y,V)-\frac{\frac{\partial\>\!U}{\partial\>\!s}}{2U\left(1-U\right)M}\cdot\mathcal{E}(x,U)\\
&\qquad\qquad\quad+\frac{\frac{\partial\,\Phi(x,U)}{\partial\>\!x}}{M}\cdot\left(\frac{x(1-x^2)\frac{\partial\>\!U}{\partial\>\!s}}{2(1-U)}+\dfrac{\frac{\partial\,y}{\partial\,s}-\frac{y\cdot(1-y^2)}{2(1-V)}\cdot\frac{\partial\,V}{\partial\,s}}{\frac{\partial\,y}{\partial\,x}}\right)\\

\end{align*}

\begin{align*}
\dfrac{\frac{\partial\>\!V}{\partial\>\!s}}{2V(1-V)}\cdot\mathcal{E}(y,V)
&=\frac{\frac{\partial\>\!U}{\partial\>\!s}}{2U\left(1-U\right)M}\cdot\mathcal{E}(x,U)
\\
&\qquad\quad+\left(\dfrac{\frac{\partial\>\!V}{\partial\>\!s}}{2VM}-\frac{\frac{\partial\>\!U}{\partial\>\!s}}{2UM}-\frac{\frac{\partial\>\!M}{\partial\>\!s}}{M^2}\right)\cdot\Phi(x,U)\\
&\qquad\qquad\quad-\frac{\frac{\partial\>\!\Phi(x,U)}{\partial\>\!x}}{M}\cdot\left(\frac{x(1-x^2)\frac{\partial\>\!U}{\partial\>\!s}}{2(1-U)}+\dfrac{\frac{\partial\>\!y}{\partial\>\!s}-\frac{y\cdot(1-y^2)}{2(1-V)}\cdot\frac{\partial\>\!V}{\partial\>\!s}}{\frac{\partial\>\!y}{\partial\>\!x}}\right)\\

\end{align*}

1. \Phi\left(x,\varphi\right)=\int_{0}^{x}\frac{1}{\sqrt{\left(1-t^{2}\right)\left(1-\varphi t^{2}\right)}}dt
2. \Phi_{V}\left(\varphi,x\right)=\frac{d}{dx}\left(\Phi\left(\varphi,x\right)\right)
3. \Phi_{x}\left(x,\varphi\right)=\frac{d}{dx}\left(\Phi\left(x,\varphi\right)\right)
4. V\left(x,\nu\right)=x\left(\frac{x+2}{2x+1}\right)^{3}
5. V_{s}\left(x,\nu\right)=\frac{d}{dx}\left(V\left(x,\nu\right)\right)
6. Y\left(x,\varphi\right)=\frac{x\left(1+2\varphi+\varphi^{2}x^{2}\right)}{1+2\varphi x^{2}+\varphi^{2}x^{2}}
7. Y_{s}\left(\varphi,x\right)=\frac{d}{dx}\left(Y\left(\varphi,x\right)\right)
8. \Phi_{sY}\left(\varphi,x,\nu\right)=\frac{d}{dx}\Phi\left(Y\left(\varphi,x\right),V\left(x,\nu\right)\right)
9. s=0.68
10. \Phi_{V}\left(x,s\right)
11. \frac{\int_{0}^{x}\sqrt{\frac{1-st^{2}}{1-t^{2}}}dt}{2s(1-s)}-\frac{\int_{0}^{x}\frac{1}{\sqrt{\left(1-t^{2}\right)\left(1-st^{2}\right)}}dt}{2s}-\frac{x\sqrt{1-x^{2}}}{2(1-s)\sqrt{1-sx^{2}}}
12. \Phi_{sY}\left(x,s,\nu\right)
13. \Phi_{V}\left(Y\left(x,s\right),V\left(s,\nu\right)\right)\cdot V_{s}\left(s,\nu\right)+\Phi_{x}\left(Y\left(x,s\right),V\left(s,\nu\right)\right)\cdot Y_{s}\left(x,s\right)
14. M\left(x,\nu\right)=\frac{1}{2x+1}
15. U\left(x,\nu\right)=\frac{x^{3}\left(x+2\right)}{2x+1}
16. \varepsilon\left(x,\varphi\right)=\int_{0}^{x}\sqrt{\frac{1-\varphi t^{2}}{1-t^{2}}}dt
17. Y_{x}\left(x,\varphi\right)=\frac{d}{dx}\left(Y\left(x,\varphi\right)\right)
18. \left(\frac{\varepsilon\left(Y\left(x,s\right),V\left(s,\nu\right)\right)}{2V\left(s,\nu\right)(1-V\left(s,\nu\right))}-\frac{\Phi\left(x,U\left(s,\nu\right)\right)}{2V\left(s,\nu\right)M\left(s,\nu\right)}\right)\cdot V_{s}\left(s,\nu\right)+\frac{\Phi_{x}\left(x,U\left(s,\nu\right)\right)}{M\left(s,\nu\right)}\cdot\left(\frac{Y_{s}\left(x,s\right)-\frac{Y\left(x,s\right)\left(1-Y\left(x,s\right)^{2}\right)}{2(1-V\left(s,\nu\right))}V_{s}\left(s,\nu\right)}{Y_{x}\left(x,s\right)}\right)
19. \frac{(1+2s)^{2}\varepsilon\left(Y\left(x,s\right),V\left(s,\nu\right)\right)}{s\left(1-s^{2}\right)(2+s)}-\frac{(s-1)^{2}\Phi\left(x,U\left(s,\nu\right)\right)}{s(2+s)}-\frac{x(1-x^{2})\left(2+4s+3s^{2}+2s^{3}x^{2}+s^{4}x^{2}\right)}{(1-s^{2})\left(1+2sx^{2}+s^{2}x^{2}\right)}\Phi_{x}\left(x,U\left(s,\nu\right)\right)
20. \Phi_{sM}\left(\varphi,x,\nu\right)=\frac{d}{dx}\left(\frac{\Phi\left(\varphi,U\left(x,\nu\right)\right)}{M\left(x,\nu\right)}\right)
21. M_{s}\left(x,\nu\right)=\frac{d}{dx}\left(M\left(x,\nu\right)\right)
22. U_{s}\left(x,\nu\right)=\frac{d}{dx}\left(U\left(x,\nu\right)\right)
23. \Phi_{sM}\left(x,s,\nu\right)
24. \Phi\left(x,U\left(s,\nu\right)\right)\cdot\left(-\frac{M_{s}\left(s,\nu\right)}{M\left(s,\nu\right)^{2}}-\frac{U_{s}\left(s,\nu\right)}{2U\left(s,\nu\right)M\left(s,\nu\right)}\right)+\frac{U_{s}\left(s,\nu\right)}{2U\left(s,\nu\right)\left(1-U\left(s,\nu\right)\right)M\left(s,\nu\right)}\cdot\varepsilon\left(x,U\left(s,\nu\right)\right)-\frac{\Phi_{x}\left(x,U\left(s,\nu\right)\right)}{M\left(s,\nu\right)}\cdot\frac{x\left(1-x^{2}\right)U_{s}\left(s,\nu\right)}{2\left(1-U\left(s,\nu\right)\right)}
25. \frac{3(1+2s)\varepsilon\left(x,U\left(s,\nu\right)\right)}{s(s+2)(1-s^{2})}-\frac{\left(3+2s+s^{2}\right)\Phi\left(x,U\left(s,\nu\right)\right)}{s(2+s)}-\frac{3s^{2}x\left(1-x^{2}\right)\Phi_{x}\left(x,U\left(s,\nu\right)\right)}{(1-s^{2})}

复制代码

\begin{align*}
\frac{\partial\,\Phi(y,V)}{\partial\>\!s}&=\frac{(1+2s)^{2}}{s(1-s^{2})(2+s)}\int_{0}^{\frac{x(1+2s+s^{2}x^{2})}{1+2sx^{2}+s^{2}x^{2}}}\sqrt{\small{\frac{1-s({\scriptsize\,\!\frac{2+s}{1+2s}})^{3}t^{2}}{1-t^{2}}}}\mathrm{d}t\\

&\qquad\>\>-\frac{(1-s)^{2}}{s(2+s)}\int_{0}^{x}\frac{\mathrm{d}t}{\sqrt{(1-t^{2})(1-{\scriptsize{\frac{s^{3}(2+s)}{1+2s}}}t^{2})}}\\
&\qquad\qquad-\frac{x\left(2+4s+3s^{2}+2s^{3}x^{2}+s^{4}x^{2}\right)\sqrt{1-x^{2}}}{(1-s^{2})\left(1+2sx^{2}+s^{2}x^{2}\right)\sqrt{1-{\scriptsize{\frac{s^{3}(2+s)}{1+2s}}}x^{2}}}
\end{align*}

青青子衿

\begin{gather*}

\rho_3\left(x,s\right)=\frac{x(1+2s+s^{2}x^{2})}{1+2sx^{2}+s^{2}x^{2}}\>\>\>\\
\\
\int_{0}^{x}\frac{\mathrm{d}t}{\sqrt{(1-t^{2})(1-{\kern-.1ex\raise.3ex\hbox{$\tiny\frac{s^{3}(2+s)}{1+2s}$}\kern-.1ex}\,t^{2})}}=\frac{1}{1+2s}\int_{0}^{\rho_3\left(x,s\right)}
\frac{\mathrm{d}t}{\sqrt{(1-t^{2})(1-{\kern-.1ex\raise.3ex\hbox{$\tiny\,s\!\cdot\!\left({\frac{2+s}{1+2s}}\right)\>\!\!^{3}$}\kern-.1ex}\,t^{2})}}
\\
\\
\begin{split}
&\>\int_{0}^{x}\frac{\scriptsize\frac{2(1-s^{2})}{1+2s}}{\sqrt{(1-t^{2})(1-{\kern-.1ex\raise.3ex\hbox{$\tiny\frac{s^{3}(2+s)}{1+2s}$}\kern-.1ex}\,t^{2})}}\mathrm{d}t\\
&\quad\>\>\>-\frac{3}{1+2s}\int_{0}^{x}\sqrt{\frac{1-{\kern-.1ex\raise.1ex\hbox{$\scriptsize\frac{s^{3}(2+s)}{1+2s}$}\kern-.1ex}\,t^{2}}{1-t^{2}}}\mathrm{d}t\\
&\qquad\quad+\int_{0}^{\rho_3\left(x,s\right)}\sqrt{\frac{1-{\kern-.1ex\raise.1ex\hbox{$\scriptsize\,s\!\cdot\!\left({\frac{2+s}{1+2s}}\right)\>\!\!^{3}$}\kern-.1ex}\,t^{2}}{1-t^{2}}}\mathrm{d}t\\
&=\frac{{\kern-.1ex\raise.1ex\hbox{$\tiny\frac{2s\>\!(2+s)}{1+2s}$}\kern-.1ex}\,x\sqrt{\left(1-x^{2}\right)\left(1-{\kern-.1ex\raise.3ex\hbox{$\tiny\frac{s^{3}(2+s)}{1+2s}$}\kern-.1ex}\,x^{2}\right)}}
{1+
{\kern-.1ex\raise.2ex\hbox{$\scriptsize\>\!s(2+s)$}\kern-.1ex}
\,x^{2}
}
\end{split}
\end{gather*}


\begin{align*}
\int_{0}^{\frac{x(1+2s+s^{2}x^{2})}{1+2sx^{2}+s^{2}x^{2}}}\sqrt{\frac{1-s\left({\small\frac{2+s}{1+2s}}\right)\>\!\!^{3}\,t^{2}}{1-t^{2}}}\mathrm{d}t
&=\frac{3}{1+2s}\int_{0}^{x}\sqrt{\frac{1-{\small\frac{s^{3}(2+s)}{1+2s}}\,t^{2}}{1-t^{2}}}\mathrm{d}t\\
&\qquad-\frac{2(1-s^{2})}{1+2s}\int_{0}^{x}\frac{\mathrm{d}t}{\sqrt{\left(1-t^{2}\right)\left(1-{\scriptsize\frac{s^{3}(2+s)}{1+2s}}t^{2}\right)}}\\
&\qquad\qquad+\frac{2s(2+s)x\sqrt{\left(1-x^{2}\right)\left(1-{\scriptsize\frac{s^{3}(2+s)}{1+2s}}x^{2}\right)}}{(1+2s)\left(1+s(2+s)x^{2}\right)}
\end{align*}

1. \varepsilon\left(Y\left(x,s\right),V\left(s,\nu\right)\right)
2. \frac{3}{1+2s}\varepsilon\left(x,U\left(s,\nu\right)\right)-\frac{2\left(1-s^{2}\right)}{1+2s}\Phi\left(x,U\left(s,\nu\right)\right)+\frac{2s(2+s)x}{(1+2s)\left(1+s(2+s)x^{2}\right)\Phi_{x}\left(x,U\left(s,\nu\right)\right)}
3. \int_{0}^{\frac{x(1+2s+s^{2}x^{2})}{1+2sx^{2}+s^{2}x^{2}}}\sqrt{\frac{1-s\left(\frac{2+s}{1+2s}\right)^{3}t^{2}}{1-t^{2}}}dt
4. \frac{3}{1+2s}\int_{0}^{x}\sqrt{\frac{1-\frac{s^{3}(2+s)}{1+2s}t^{2}}{1-t^{2}}}dt-\frac{2(1-s^{2})}{1+2s}\int_{0}^{x}\frac{1}{\sqrt{\left(1-t^{2}\right)\left(1-\frac{s^{3}(2+s)}{1+2s}t^{2}\right)}}dt+\frac{2s(2+s)x\sqrt{\left(1-x^{2}\right)\left(1-\frac{s^{3}(2+s)}{1+2s}x^{2}\right)}}{(1+2s)\left(1+s(2+s)x^{2}\right)}

复制代码

\begin{align*}
&\qquad\qquad\int_{0}^{1}\frac{\mathrm{d}t}{\left(1-s\!\left({\small\frac{2+s}{1+2s}}\right)\>\!\!^{3}
\!\left({\scriptsize{\frac{\beta(1+2s+s^{2}\beta^{2})}{1+2s\beta^{2}+s^{2}\beta^{2}}\,}\!}\right)\>\!\!^{2}t^{2}\right)\sqrt{\left(1-t^{2}\right)\left(1-s\!\left({\scriptsize\frac{2+s}{1+2s}}\right)\>\!\!^{3}t^{2}\right)}}\\

&=\frac{3(1+2s+s^{2}\beta^{2})(1+s(2+s)\beta^{2})}{\left(1-s^{2}\beta^{2}\right)\left(1-{\small\frac{s(2+s)}{1+2s}}\beta^{2}\right)}
\int_{0}^{1}\frac{\mathrm{d}t}{\left(1-{\small\frac{s^{3}(2+s)}{1+2s}}\beta^{2}t^{2}\right)\sqrt{\left(1-t^{2}\right)\left(1-{\scriptsize\frac{s^{3}(2+s)}{1+2s}}t^{2}\right)}}\\
&\qquad\qquad-\frac{2(1+2s)\left(1+s(2+s)\beta^{2}\right)}{\left(1-s^{2}\beta^{2}\right)\left(1-{\small\frac{s(2+s)}{1+2s}}\beta^{2}\right)}\int_{0}^{1}\frac{\mathrm{d}t}{\sqrt{\left(1-t^{2}\right)\left(1-{\scriptsize\frac{s^{3}(2+s)}{1+2s}}t^{2}\right)}}
\end{align*}

1. NIntegrate[(
2.     3 (1 + 2 s + s^2 \[Beta]^2) (1 + s (2 + s) \[Beta]^2))/((1 -
3.        s^2 \[Beta]^2) (1 - (s (2 + s) \[Beta]^2)/(
4.        1 + 2 s)))*1/((1 - (s^3 (2 + s))/(
5.         1 + 2 s) \[Beta]^2 t^2) Sqrt[(1 -
6.         t^2) (1 - (s^3 (2 + s))/(1 + 2 s) t^2)] ) - (
7.     2 (1 + 2 s) (1 + s (2 + s) \[Beta]^2))/((1 - s^2 \[Beta]^2) (1 - (
8.        s (2 + s) \[Beta]^2)/(1 + 2 s)))*1/
9.     Sqrt[(1 - t^2) (1 - (s^3 (2 + s))/(1 + 2 s) t^2)]  /. { \[Beta] ->
10.      0.3156214514134524613561537152462096,
11.    s -> 0.623451461461441692682625245249246}, {t, 0, 1},
12. WorkingPrecision -> 30]
13. NIntegrate[
14. 1/((1 - s ((s + 2)/(
15.        2 s + 1))^3 ((\[Beta] (1 + 2 s + s^2 \[Beta]^2))/(
16.        1 + 2 s \[Beta]^2 + s^2 \[Beta]^2))^2 t^2) Sqrt[(1 - t^2) (1 -
17.       s ((s + 2)/(2 s + 1))^3 t^2)] ) /. { \[Beta] ->
18.     0.3156214514134524613561537152462096,
19.    s -> 0.623451461461441692682625245249246}, {t, 0, 1},
20. WorkingPrecision -> 30]
21. NIntegrate[(
22.     3 (1 + 2 s) (s (2 + s) + \[Beta]^2) (s^2 + (1 +
23.           2 s) \[Beta]^2))/((s^2 - \[Beta]^2) (s (2 + s) - (1 +
24.           2 s) \[Beta]^2))*1/((1 - \[Beta]^2 t^2) Sqrt[(1 -
25.         t^2) (1 - (s^3 (2 + s))/(1 + 2 s) t^2)] ) - (
26.     2 s (2 + s) (1 +
27.        2 s) (s^2 + (1 + 2 s) \[Beta]^2))/((s^2 - \[Beta]^2) (s (2 +
28.           s) - (1 + 2 s) \[Beta]^2))*1/
29.     Sqrt[(1 - t^2) (1 - (s^3 (2 + s))/(1 + 2 s) t^2)]  /. { \[Beta] ->
30.      0.3156214514134524613561537152462096,
31.    s -> 0.623451461461441692682625245249246}, {t, 0, 1},
32. WorkingPrecision -> 30]
33. NIntegrate[
34. 1/((1 - ((\[Beta] (s^2 + 2 s + \[Beta]^2))/(
35.        s^2 + 2 s*\[Beta]^2 + \[Beta]^2)*t)^2) Sqrt[(1 - t^2) (1 -
36.       s ((s + 2)/(2 s + 1))^3 t^2)] ) /. { \[Beta] ->
37.     0.3156214514134524613561537152462096,
38.    s -> 0.623451461461441692682625245249246}, {t, 0, 1},
39. WorkingPrecision -> 30]

复制代码

\begin{align*}

\\
\Pi(x,k\beta,k)&=\int_{0}^{x}\frac{\mathrm{d}t}{\left(1-k^2\beta^{2}t^{2}\right)\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}\\
\\
&=\int_{0}^{x}\frac{\mathrm{d}t}{\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}\\
&\qquad-\int_{0}^{x}\frac{\mathrm{d}t}{\left(1-\frac{1}{\beta^{2}}t^{2}\right)\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}\\
&\qquad\quad\>+\tfrac{\beta}{\sqrt{\left(1-\beta^{2}\right)\left(1-k^2\beta^{2}\right)}}\operatorname{artanh}\left(\tfrac{x\sqrt{\left(1-\beta^{2}\right)\left(1-k^2\beta^{2}\right)}}{\beta\sqrt{\left(1-x^{2}\right)\left(1-k^{2}x^{2}\right)}}\right)\\
\end{align*}

青青子衿

青青子衿 发表于 2024-1-27 13:10
\begin{align*}
&\qquad\qquad\int_{0}^{1}\frac{\mathrm{d}t}{\left(1-s\!\left({\small\frac{2+s}{1+2s}}\right)\>\!\!^{3}
\!\left({\scriptsize{\frac{\beta(1+2s+s^{2}\beta^{2})}{1+2s\beta^{2}+s^{2}\beta^{2}}\,}\!}\right)\>\!\!^{2}t^{2}\right)\sqrt{\left(1-t^{2}\right)\left(1-s\!\left({\scriptsize\frac{2+s}{1+2s}}\right)\>\!\!^{3}t^{2}\right)}}\\

&=\frac{3(1+2s+s^{2}\beta^{2})(1+s(2+s)\beta^{2})}{\left(1-s^{2}\beta^{2}\right)\left(1-{\small\frac{s(2+s)}{1+2s}}\beta^{2}\right)}
\int_{0}^{1}\frac{\mathrm{d}t}{\left(1-{\small\frac{s^{3}(2+s)}{1+2s}}\beta^{2}t^{2}\right)\sqrt{\left(1-t^{2}\right)\left(1-{\scriptsize\frac{s^{3}(2+s)}{1+2s}}t^{2}\right)}}\\
&\qquad\qquad-\frac{2(1+2s)\left(1+s(2+s)\beta^{2}\right)}{\left(1-s^{2}\beta^{2}\right)\left(1-{\small\frac{s(2+s)}{1+2s}}\beta^{2}\right)}\int_{0}^{1}\frac{\mathrm{d}t}{\sqrt{\left(1-t^{2}\right)\left(1-{\scriptsize\frac{s^{3}(2+s)}{1+2s}}t^{2}\right)}}
\end{align*}

\begin{align*}
&\qquad\int_{0}^{\frac{x(1+2s+s^{2}x^{2})}{1+2sx^{2}+s^{2}x^{2}}}\frac{\frac{(1-\beta^{2}s^{2})(1+2s-2\beta^{2}s-\beta^{2}s^{2})}{(1+2s)(1+2\beta^{2}s+\beta^{2}s^{2})}}{\left(1-s\!\left({\small\frac{2+s}{1+2s}}\right)\>\!\!^{3}
\!\left({\scriptsize{\frac{\beta(1+2s+\beta^{2}s^{2})}{1+2\beta^{2}s+\beta^{2}s^{2}}\,}\!}\right)\>\!\!^{2}t^{2}\right)\sqrt{\left(1-t^{2}\right)\left(1-s\!\left({\scriptsize\frac{2+s}{1+2s}}\right)\>\!\!^{3}t^{2}\right)}}\mathrm{d}t\\

&=
\int_{0}^{x}\left(\frac{3(1+2s+s^{2}\beta^{2})}{\left(1-{\small\frac{s^{3}(2+s)}{1+2s}}\beta^{2}t^{2}\right)\sqrt{\left(1-t^{2}\right)\left(1-{\scriptsize\frac{s^{3}(2+s)}{1+2s}}t^{2}\right)}}-\frac{2(1+2s)}{\sqrt{\left(1-t^{2}\right)\left(1-{\scriptsize\frac{s^{3}(2+s)}{1+2s}}t^{2}\right)}}\right)\mathrm{d}t\\

&\qquad\quad-\frac{P_1}{Q_1}\operatorname{artanh}\left(\frac{R_1}{S_1}\right)\\
\\
\\
P_1&=\beta(1+2s+\beta^{2}s^{2})\sqrt{1+2s}\\
Q_1&=\sqrt{\small\left(1-\beta^{2}\right)\left(1+2s-s^{3}(2+s)\beta^{2}\right)}\\

R_1&=2\beta s(s+2)(1+2s)x\sqrt{\small(1-\beta^{2})(1+2s-2\beta^{2}s^{3}-\beta^{2}s^{4})(1-x^{2})(1+2s-2s^{3}x^{2}-s^{4}x^{2})}\\
S_1&=(1+2s)^{2}(1+2\beta^{2}s+\beta^{2}s^{2})+T_1x^{2}+\beta^{2}s^{4}(2+s)^{2}(1+2s+\beta^{2}s^{2})x^{4}\\
T_1&=s(2+s)(1+2s)(1-2\beta^{2}+2s-4\beta^{2}s-2\beta^{2}s^{2}-4\beta^{2}s^{3}+2\beta^{4}s^{3}-2\beta^{2}s^{4}+\beta^{4}s^{4})\\
\\
\\

&\qquad\int_{0}^{\frac{x(1+2s+s^{2}x^{2})}{1+2sx^{2}+s^{2}x^{2}}}\frac{\frac{(1-\beta^{2}s^{2})(1+2s-2\beta^{2}s-\beta^{2}s^{2})}{(1+2s)(1+2s+\beta^{2}s^{2})}}{\left(1-
\!\left({\scriptsize{\frac{1+2\beta^{2}s+\beta^{2}s^{2}}{\beta(1+2s+\beta^{2}s^{2})}\,}\!}\right)\>\!\!^{2}t^{2}\right)\sqrt{\left(1-t^{2}\right)\left(1-s\!\left({\scriptsize\frac{2+s}{1+2s}}\right)\>\!\!^{3}t^{2}\right)}}\mathrm{d}t\\
&=
\int_{0}^{x}\left(\frac{3(1+2\beta^{2}s+\beta^{2}s^{2})}{\left(1-{\small\frac{1}{\beta^{2}}}t^{2}\right)\sqrt{\left(1-t^{2}\right)\left(1-{\scriptsize\frac{s^{3}(2+s)}{1+2s}}t^{2}\right)}}-\frac{2\beta^{2}s(2+s)}{\sqrt{\left(1-t^{2}\right)\left(1-{\scriptsize\frac{s^{3}(2+s)}{1+2s}}t^{2}\right)}}\right)\mathrm{d}t\\
&\qquad\quad-\frac{P_2}{Q_2}\operatorname{artanh}\left(\frac{R_2}{S_2}\right)\\
\\
\\
P_2&=
\beta(1+2\beta^{2}s+\beta^{2}s^{2})\sqrt{1+2s}
\\
Q_2&=\sqrt{\small\left(1-\beta^{2}\right)\left(1+2s-s^{3}(2+s)\beta^{2}\right)}\\


R_2&=2\beta\,\!x\sqrt{\small(1-\beta^{2})(1+2s-2\beta^{2}s^{3}-\beta^{2}s^{4})(1-x^{2})(1+2s-2s^{3}x^{2}-s^{4}x^{2})}\\
S_2
&=\beta^{2}(1+2s+\beta^{2}s^{2})+T_2x^{2}+s^{2}(1+2\beta^{2}s+\beta^{2}s^{2})x^{4}\\


T_2&=(1-2\beta^{2}+2s-4\beta^{2}s-2\beta^{2}s^{2}-4\beta^{2}s^{3}+2\beta^{4}s^{3}-2\beta^{2}s^{4}+\beta^{4}s^{4})\\

\end{align*}

2. \int_{0}^{\frac{x(1+2s+s^{2}x^{2})}{1+2sx^{2}+s^{2}x^{2}}}\frac{\frac{(1-\beta^{2}s^{2})(1+2s-2\beta^{2}s-\beta^{2}s^{2})}{(1+2s)(1+2\beta^{2}s+\beta^{2}s^{2})}}{\left(1-s\left(\frac{2+s}{1+2s}\right)^{3}\left(\frac{\beta(1+2s+s^{2}\beta^{2})}{1+2s\beta^{2}+s^{2}\beta^{2}}\right)^{2}t^{2}\right)\sqrt{\left(1-t^{2}\right)\left(1-s\left(\frac{2+s}{1+2s}\right)^{3}t^{2}\right)}}dt
3. \int_{0}^{x}\left(\frac{3(1+2s+s^{2}\beta^{2})}{\left(1-\frac{s^{3}(2+s)}{1+2s}\beta^{2}t^{2}\right)\sqrt{\left(1-t^{2}\right)\left(1-\frac{s^{3}(2+s)}{1+2s}t^{2}\right)}}-\frac{2(1+2s)}{\sqrt{\left(1-t^{2}\right)\left(1-\frac{s^{3}(2+s)}{1+2s}t^{2}\right)}}\right)dt-\frac{\beta(1+2s+\beta^{2}s^{2})\sqrt{1+2s}}{\sqrt{\left(1-\beta^{2}\right)\left(1+2s-s^{3}(2+s)\beta^{2}\right)}}\operatorname{artanh}\left(\frac{R_{1}\left(x,1\right)}{S_{1}\left(x,1\right)}\right)
4. R_{1}\left(x,\varphi\right)=2\beta s(s+2)(1+2s)x\sqrt{\left(1-\beta^{2}\right)\left(1+2s-2\beta^{2}s^{3}-\beta^{2}s^{4}\right)\left(1-x^{2}\right)\left(1+2s-2s^{3}x^{2}-s^{4}x^{2}\right)}
5. S_{1}\left(x,\varphi\right)=(1+2s)^{2}\left(1+2\beta^{2}s+\beta^{2}s^{2}\right)+T_{1}x^{2}+\beta^{2}s^{4}(2+s)^{2}\left(1+2s+\beta^{2}s^{2}\right)x^{4}
6. T_{1}=s(2+s)(1+2s)\left(1-2\beta^{2}+2s-4\beta^{2}s-2\beta^{2}s^{2}-4\beta^{2}s^{3}+2\beta^{4}s^{3}-2\beta^{2}s^{4}+\beta^{4}s^{4}\right)
7. s=0.77
8. \beta=0.54
9. \int_{0}^{\frac{x(1+2s+s^{2}x^{2})}{1+2sx^{2}+s^{2}x^{2}}}\frac{\frac{(1-\beta^{2}s^{2})(1+2s-2\beta^{2}s-\beta^{2}s^{2})}{(1+2s)(1+2s+\beta^{2}s^{2})}}{\left(1-\left(\frac{1+2s\beta^{2}+s^{2}\beta^{2}}{\beta(1+2s+s^{2}\beta^{2})}t\right)^{2}\right)\sqrt{\left(1-t^{2}\right)\left(1-s\left(\frac{2+s}{1+2s}\right)^{3}t^{2}\right)}}dt
10. \int_{0}^{x}\left(\frac{3(1+2\beta^{2}s+\beta^{2}s^{2})}{\left(1-\frac{1}{\beta^{2}}t^{2}\right)\sqrt{\left(1-t^{2}\right)\left(1-\frac{s^{3}(2+s)}{1+2s}t^{2}\right)}}-\frac{2\beta^{2}s(2+s)}{\sqrt{\left(1-t^{2}\right)\left(1-\frac{s^{3}(2+s)}{1+2s}t^{2}\right)}}\right)dt-\frac{\beta\left(1+2\beta^{2}s+\beta^{2}s^{2}\right)\sqrt{1+2s}}{\sqrt{\left(1-\beta^{2}\right)\left(1+2s-s^{3}(2+s)\beta^{2}\right)}}\tanh^{-1}\left(\frac{R_{2}\left(x,1\right)}{S_{2}\left(x,1\right)}\right)
11. R_{2}\left(x,\varphi\right)=2\beta x\sqrt{\left(1-\beta^{2}\right)\left(1+2s-2\beta^{2}s^{3}-\beta^{2}s^{4}\right)\left(1-x^{2}\right)\left(1+2s-2s^{3}x^{2}-s^{4}x^{2}\right)}
12. S_{2}\left(x,\varphi\right)=\beta^{2}\left(1+2s+\beta^{2}s^{2}\right)+T_{2}x^{2}+s^{2}\left(1+2\beta^{2}s+\beta^{2}s^{2}\right)x^{4}
13. T_{2}=1-2\beta^{2}+2s-4\beta^{2}s-2\beta^{2}s^{2}-4\beta^{2}s^{3}+2\beta^{4}s^{3}+\beta^{4}s^{4}-2\beta^{2}s^{4}

复制代码

\begin{align*}
{\large\int}\frac{5875-15000 x^2+2428 x^4+376 x^6}{(15-2 x) (15+2 x)(55225+26540 x^2+8836 x^4) \sqrt{25-29 x^2+4 x^4}} \mathrm{d}x
\end{align*}

青青子衿

\begin{align*}
\qquad\int_{x}^{+\infty}\frac{{\mathrm{d}}t}{\sqrt{A^{4}+2B^{2}t^{2}+t^{4}}}=
\frac{1}{2A}\int_{0}^{\frac{2Ax}{A^{2}+x^{2}}}\frac{{\mathrm{d}}t}{\sqrt{\left(1-t^{2}\right)\left(1-{\raise2px\scriptsize\frac{A^{2}-B^{2}}{2A^{2}}}t^{2}\right)}}\qquad(x\geqslant\,\!A)
\end{align*}

\begin{gather*}
\int_{x}^{+\infty}\frac{{\mathrm{d}}t}{\sqrt{A^{4}+2B^{2}t^{2}+t^{4}}}
\\
=\frac{1}{2A}\int_{0}^{\frac{A^{2}-x^{2}}{\sqrt{A^{4}+2B^{2}x^{2}+x^{4}}}}\frac{{\mathrm{d}}t}{\sqrt{\left(1-t^{2}\right)\left(1-{\raise2px\scriptsize\frac{A^{2}-B^{2}}{2A^{2}}}t^{2}\right)}}+\frac{1}{2A}\int_{0}^{1}\frac{{\mathrm{d}}t}{\sqrt{\left(1-t^{2}\right)\left(1-{\raise2px\scriptsize\frac{A^{2}-B^{2}}{2A^{2}}}t^{2}\right)}}\\
\\
(x\geqslant0)
\end{gather*}


\begin{align*}
{\boldsymbol{K}(k_5)}
&={\boldsymbol{K}(\tfrac{\sqrt{2}-\sqrt{10}+2 \sqrt{\sqrt{5}-1}}{4})}\\
&=\tfrac{\sqrt{\sqrt{2+\sqrt{5}}\,\Gamma \left(\frac{1}{20}\right) \Gamma \left(\frac{3}{20}\right) \Gamma \left(\frac{7}{20}\right) \Gamma \left(\frac{9}{20}\right)}}{4 \sqrt{10 \pi }}\\
&=\tfrac{(1+\sqrt{5}\,) \sqrt{\sqrt{2+\sqrt{5}}}-\sqrt{2 \sqrt{10+10 \sqrt{5}}}}{2^{37/10} \cdot5^{7/8}}\cdot\tfrac{\,\Gamma ^2\!\left(\frac{1}{20}\right)}{\Gamma \left(\frac{1}{5}\right) \Gamma \left(\frac{2}{5}\right)}\sqrt{\pi}
\end{align*}


\begin{align*}
I&=\int_{x}^{+\infty}\frac{1}{\sqrt{112+21t^{2}+t^{4}}}\mathrm{d}t\\

&=\frac{1}{4\sqrt[4]{7}}\int_{0}^{\frac{4\sqrt{7}-x^{2}}{\sqrt{112+21x^{2}+x^{4}}}}\frac{1}{\sqrt{(1-t^{2})(1-\left({\raise2px\Tiny\frac{3\sqrt{2}-\sqrt{14}}{8}}\right)^{2}t^{2})}}\mathrm{d}t\\
&\qquad
+\frac{1}{4\sqrt[4]{7}}\int_{0}^{1}\frac{1}{\sqrt{(1-t^{2})(1-\left({\raise2px\Tiny\frac{3\sqrt{2}-\sqrt{14}}{8}}\right)^{2}t^{2})}}\mathrm{d}t\\
&=\frac{1}{2}\int_{s}^{+\infty}\frac{(1+t^{7})^{3/7}+t^{3}(1+t^{7})^{1/7}-t}{(1+t^{7})^{6/7}}\mathrm{d}t
\end{align*}

\begin{align*}
x=\frac{2\left(s^{2}(1+s^{7})^{3/7}-(1+s^{7})^{2/7}-s^{3}\right)}{s(1+s^{7})^{1/7}\sqrt{(1+s^{7})^{3/7}+s^{3}(1+s^{7})^{1/7}-s}}
\end{align*}

1. \int_{x}^{+\infty}\frac{1}{\sqrt{112+21t^{2}+t^{4}}}dt
2. \frac{1}{4\sqrt[4]{7}}\int_{0}^{\frac{4\sqrt{7}-x^{2}}{\sqrt{112+21x^{2}+x^{4}}}}\frac{1}{\sqrt{\left(1-t^{2}\right)\left(1-\left(\frac{3\sqrt{2}-\sqrt{14}}{8}\right)^{2}t^{2}\right)}}dt+\frac{1}{4\sqrt[4]{7}}\int_{0}^{1}\frac{1}{\sqrt{\left(1-t^{2}\right)\left(1-\left(\frac{3\sqrt{2}-\sqrt{14}}{8}\right)^{2}t^{2}\right)}}dt
3. \int_{\frac{2\left(s^{2}(1+s^{7})^{3/7}-(1+s^{7})^{2/7}-s^{3}\right)}{s(1+s^{7})^{1/7}\sqrt{s^{3}(1+s^{7})^{1/7}-s+(1+s^{7})^{3/7}}}}^{+\infty}\frac{1}{\sqrt{112+21t^{2}+t^{4}}}dt
4. \int_{s}^{+\infty}\frac{(1+t^{7})^{3/7}+t^{3}(1+t^{7})^{1/7}-t}{2(1+t^{7})^{6/7}}dt

复制代码

青青子衿

青青子衿 发表于 2024-4-10 20:13
\begin{align*}
I&=\int_{x}^{+\infty}\frac{1}{\sqrt{112+21t^{2}+t^{4}}}\mathrm{d}t\\

&=\frac{1}{4\sqrt[4]{7}}\int_{0}^{\frac{4\sqrt{7}-x^{2}}{\sqrt{112+21x^{2}+x^{4}}}}\frac{1}{\sqrt{(1-t^{2})(1-\left({\raise2px\Tiny\frac{3\sqrt{2}-\sqrt{14}}{8}}\right)^{2}t^{2})}}\mathrm{d}t\\
&\qquad
+\frac{1}{4\sqrt[4]{7}}\int_{0}^{1}\frac{1}{\sqrt{(1-t^{2})(1-\left({\raise2px\Tiny\frac{3\sqrt{2}-\sqrt{14}}{8}}\right)^{2}t^{2})}}\mathrm{d}t\\
&=\frac{1}{2}\int_{s}^{+\infty}\frac{(1+t^{7})^{3/7}+t^{3}(1+t^{7})^{1/7}-t}{(1+t^{7})^{6/7}}\mathrm{d}t
\end{align*}

\begin{align*}
&\qquad\qquad\qquad\qquad\qquad\qquad\int_{0}^{x}\frac{{\mathrm{d}}t}{\sqrt{1+t^{5}}}\\
&=\frac{1}{4}\int_{1-\frac{2\sqrt{x}}{1+x}}^{1}\frac{1-t}{\sqrt{t(1-\cos^{2}(\frac{\pi}{10})(1-t)^{2})(1-\cos^{2}(\frac{3\pi}{10})(1-t)^{2})}}{\mathrm{d}}t\\
&\qquad\quad-\frac{1}{4}\int_{1}^{1+\frac{2\sqrt{x}}{1+x}}
\frac{1-t}{\sqrt{t(1-\cos^{2}(\frac{\pi}{10})(1-t)^{2})(1-\cos^{2}(\frac{3\pi}{10})(1-t)^{2})}}{\mathrm{d}}t
\end{align*}

\begin{gather*}
\int_{0}^{X}\frac{\mathrm{d}t}{\sqrt{t\left(1-t\right)\left(1+at\right)\left(1+bt\right)\left(1-abt\right)}}\\
=\int_{0}^{x}\frac{\frac{1}{\sqrt{\left(1+a\right)\left(1+b\right)}}}{\sqrt{\left(1-t^{2}\right)\left(1-\frac{(\sqrt{a}+\sqrt{b})^{2}}{(1+a)(1+b)}t^{2}\right)}}{\mathrm{d}}t
+\int_{0}^{x}\frac{\frac{1}{\sqrt{\left(1+a\right)\left(1+b\right)}}}{\sqrt{\left(1-t^{2}\right)\left(1-\frac{(\sqrt{a}-\sqrt{b})^{2}}{(1+a)(1+b)}t^{2}\right)}}{\mathrm{d}}t\\
\\
X=\left(\frac{\frac{2x}{\sqrt{(1+a)(1+b)}}}{\sqrt{1-\frac{(\sqrt{a}+\sqrt{b})^{2}}{(1+a)(1+b)}x^{2}}+\sqrt{1-\frac{(\sqrt{a}-\sqrt{b})^{2}}{(1+a)(1+b)}x^{2}}}\right)^{2}
\end{gather*}


\begin{gather*}
\left\{
\begin{split}
p_{1}&=\sqrt{1-p^{2}}\\
q_{1}&=\sqrt{1-q^{2}}\\
r_{1}&=\sqrt{1-r^{2}}\\
r_{p}&=\sqrt{p^{2}-r^{2}}\\
r_{q}&=\sqrt{q^{2}-r^{2}}\\
\end{split}
\right.\\
\\
\left\{
\begin{split}
b&=\frac{2(1-r_{1}^{2})(r_{1}^{2}-p_{1}^{2}q_{1}^{2})-\frac{4r_{1}^{2}(1-(1-p_{1}^{2}q_{1}^{2})x_{3})}{x_{3}(1-x_{3})}}{(1-r_{1})^{2}(r_{1}-p_{1}q_{1})^{2}}\\
c&=\left(\frac{(1+r_{1})(r_{1}+p_{1}q_{1})}{(1-r_{1})(r_{1}-p_{1}q_{1})}\right)^{2}\\
x_{1}&=\frac{-b-\sqrt{b^{2}-4c}}{2}\\
x_{2}&=\frac{-b+\sqrt{b^{2}-4c}}{2}\\
\end{split}
\right.\\
\\
\left\{
\begin{split}
\varkappa&=\frac{r_{1}-p_{1}q_{1}}{r_{1}+p_{1}q_{1}}\\
\lambda&=\sqrt{\frac{(1-r_{1})(r_{1}-p_{1}q_{1})(pqr_{1}+r_{p}r_{q})}{(1+r_{1})(r_{1}+p_{1}q_{1})(pqr_{1}-r_{p}r_{q})}}\\
\mu&=\sqrt{\frac{(1-r_{1})(r_{1}-p_{1}q_{1})(pqr_{1}-r_{p}r_{q})}{(1+r_{1})(r_{1}+p_{1}q_{1})(pqr_{1}+r_{p}r_{q})}}\\
\end{split}
\right.\\
\end{gather*}


\begin{align*}
\begin{split}
&\qquad\>\int_{0}^{x_{3}}\frac{{\mathrm{d}}t}{\sqrt{t(1-t)(1-p^{2}t)(1-q^{2}t)(1-r^{2}t)}}\\
&=\int_{0}^{x_{1}}\frac{\frac{(\varkappa+\lambda\mu)(1+\varkappa)}{2(\varkappa-\lambda\mu)}(1+\lambda\mu t)}{\sqrt{t(1-t)(1-\varkappa^{2}t)(1-\lambda^{2}t)(1-\mu^{2}t)}}{\mathrm{d}}t\\
&\qquad-\int_{x_{2}}^{+\infty}\frac{\frac{(\varkappa+\lambda\mu)(1+\varkappa)}{2(\varkappa-\lambda\mu)}(1+\lambda\mu t)}{\sqrt{t(1-t)(1-\varkappa^{2}t)(1-\lambda^{2}t)(1-\mu^{2}t)}}{\mathrm{d}}t
\end{split}
\end{align*}


\begin{gather*}
\color{black}{\int_{0}^{\frac{\color{red}1}{\color{red}6}}\frac{{\mathrm{d}t}}{\sqrt{t(1-t)(1-(\frac{2}{3})^{2}t)(1-(\frac{1}{2})^{2}t)(1-(\frac{1}{3})^{2}t)}}}\\

\color{black}{=\int_{0}^{\frac{(17+12\sqrt{2})(47+8\sqrt{30})(3727-24\sqrt{24115})}{289}}\frac{\frac{3\sqrt{2}(8-\sqrt{30})}{17}+\frac{3(17\sqrt{2}-24)(616-111\sqrt{30})}{289}t}{\sqrt{t(1-t)(1-(\frac{47-8\sqrt{30}}{17})^{2}t)(1-(3-2\sqrt{2})^{2}t)(1-(\frac{(3-2\sqrt{2})(47-8\sqrt{30})}{17})^{2}t)}}{\mathrm{d}t}}\\

\color{black}{-\int_{\frac{(17+12\sqrt{2})(47+8\sqrt{30})(3727+24\sqrt{24115})}{289}}^{+\infty}\frac{\frac{3\sqrt{2}(8-\sqrt{30})}{17}+\frac{3(17\sqrt{2}-24)(616-111\sqrt{30})}{289}t}{\sqrt{t(1-t)(1-(\frac{47-8\sqrt{30}}{17})^{2}t)(1-(3-2\sqrt{2})^{2}t)(1-(\frac{(3-2\sqrt{2})(47-8\sqrt{30})}{17})^{2}t)}}{\mathrm{d}t}}

\end{gather*}

1. \int_{0}^{\frac{(17+12\sqrt{2})(47+8\sqrt{30})(3727-24\sqrt{24115})}{289}}\frac{\frac{3\sqrt{2}(8-\sqrt{30})}{17}+\frac{3(17\sqrt{2}-24)(616-111\sqrt{30})}{289}t}{\sqrt{t(1-t)(1-(\frac{47-8\sqrt{30}}{17})^{2}t)(1-(3-2\sqrt{2})^{2}t)(1-(\frac{(3-2\sqrt{2})(47-8\sqrt{30})}{17})^{2}t)}}dt
2. \int_{0}^{\frac{1}{6}}\frac{1}{\sqrt{t(1-t)(1-(\frac{2}{3})^{2}t)(1-(\frac{1}{2})^{2}t)(1-(\frac{1}{3})^{2}t)}}dt+\int_{\frac{(17+12\sqrt{2})(47+8\sqrt{30})(3727+24\sqrt{24115})}{289}}^{+\infty}\frac{\frac{3\sqrt{2}(8-\sqrt{30})}{17}+\frac{3(17\sqrt{2}-24)(616-111\sqrt{30})}{289}t}{\sqrt{t(1-t)(1-(\frac{47-8\sqrt{30}}{17})^{2}t)(1-(3-2\sqrt{2})^{2}t)(1-(\frac{(3-2\sqrt{2})(47-8\sqrt{30})}{17})^{2}t)}}dt
3. \int_{0}^{1-\frac{2\sqrt{x}}{1+x}}\frac{1-t}{\sqrt{t\left(1-\cos\left(\frac{\pi}{10}\right)^{2}\left(1-t\right)^{2}\right)\left(1-\cos\left(\frac{3\pi}{10}\right)^{2}\left(1-t\right)^{2}\right)}}dt
4. \frac{\sqrt{\frac{1+\cos\left(\frac{\pi}{10}\right)}{\cos\left(\frac{\pi}{10}\right)}}}{\sin\left(\frac{\pi}{10}\right)\sin\left(\frac{3\pi}{10}\right)}\int_{0}^{\frac{\cos\left(\frac{\pi}{10}\right)}{1+\cos\left(\frac{\pi}{10}\right)}\left(1-\frac{2\sqrt{x}}{1+x}\right)}\frac{1-\frac{1+\cos\left(\frac{\pi}{10}\right)}{\cos\left(\frac{\pi}{10}\right)}t}{\sqrt{t\left(1-t\right)\left(1+\frac{1+\cos\left(\frac{\pi}{10}\right)}{1-\cos\left(\frac{\pi}{10}\right)}t\right)\left(1+\frac{\cos\left(\frac{3\pi}{10}\right)\left(1+\cos\left(\frac{\pi}{10}\right)\right)}{\cos\left(\frac{\pi}{10}\right)\left(1-\cos\left(\frac{3\pi}{10}\right)\right)}t\right)\left(1-\frac{\cos\left(\frac{3\pi}{10}\right)\left(1+\cos\left(\frac{\pi}{10}\right)\right)}{\cos\left(\frac{\pi}{10}\right)\left(1+\cos\left(\frac{3\pi}{10}\right)\right)}t\right)}}dt

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青青子衿

\begin{gather*}
\int_{0}^{x}\frac{{\mathrm{d}}t}{\sqrt{1+t^{12}}}\\

=\frac{1}{\sqrt{6\sqrt{3}}}\int_{0}^{X_{1}}\frac{{\mathrm{d}}t}{\sqrt{\left(1-t^{2}\right)\left(1-\frac{1}{2}t^{2}\right)}}+\frac{1}{\sqrt{2\sqrt{3}}}\int_{0}^{X_{2}}\frac{{\mathrm{d}}t}{\sqrt{\left(1-t^{2}\right)\left(1-\frac{1}{2}t^{2}\right)}}\\
\\
\left\{
\begin{split}
X_{1}=\sqrt{\tfrac{2\sqrt{3}x^{2}+\sqrt{(1+x^{4})(1+\sqrt{3}x^{2}+x^{4})}-\sqrt{(1+x^{4})(1-\sqrt{3}x^{2}+x^{4})}}{\sqrt{3}x^{2}+\sqrt{1-x^{4}+x^{8}}+\sqrt{(1+x^{4})(1+\sqrt{3}x^{2}+x^{4})}}}\\
X_{2}=\sqrt{\tfrac{2\sqrt{3}x^{2}-\sqrt{(1+x^{4})(1+\sqrt{3}x^{2}+x^{4})}+\sqrt{(1+x^{4})(1-\sqrt{3}x^{2}+x^{4})}}{\sqrt{3}x^{2}+\sqrt{1-x^{4}+x^{8}}+\sqrt{(1+x^{4})(1-\sqrt{3}x^{2}+x^{4})}}}
\end{split}
\right.

\end{gather*}

1. \int_{0}^{x}\frac{1}{\sqrt{1+t^{12}}}dt
2. \frac{1}{\sqrt{6\sqrt{3}}}\int_{0}^{X_{1}}\frac{1}{\sqrt{\left(1-t^{2}\right)\left(1-\frac{1}{2}t^{2}\right)}}dt+\frac{1}{\sqrt{2\sqrt{3}}}\int_{0}^{X_{2}}\frac{1}{\sqrt{\left(1-t^{2}\right)\left(1-\frac{1}{2}t^{2}\right)}}dt
3. X_{1}=\sqrt{\frac{2\sqrt{3}x^{2}+\sqrt{(1+x^{4})(1+\sqrt{3}x^{2}+x^{4})}-\sqrt{(1+x^{4})(1-\sqrt{3}x^{2}+x^{4})}}{\sqrt{3}x^{2}+\sqrt{1-x^{4}+x^{8}}+\sqrt{(1+x^{4})(1+\sqrt{3}x^{2}+x^{4})}}}
4. X_{2}=\sqrt{\frac{2\sqrt{3}x^{2}-\sqrt{(1+x^{4})(1+\sqrt{3}x^{2}+x^{4})}+\sqrt{(1+x^{4})(1-\sqrt{3}x^{2}+x^{4})}}{\sqrt{3}x^{2}+\sqrt{1-x^{4}+x^{8}}+\sqrt{(1+x^{4})(1-\sqrt{3}x^{2}+x^{4})}}}

复制代码

\begin{gather*}
\int_{0}^{x}\frac{1}{\sqrt{1+t^{8}}}dt\\
=\frac{1}{2\sqrt{2+\sqrt{2}}}\int_{0}^{\frac{\sqrt{2+\sqrt{2}}x}{1+x^{2}}}\frac{{\mathrm{d}}t}{\sqrt{(1-t^{2})(1-(\sqrt{2}-1)^{2}t^{2})}}\\
+\frac{1}{\sqrt{4+2\sqrt{2}}}\int_{0}^{\frac{1+\sqrt{2}x^{2}+x^{4}-\sqrt{1+x^{8}}}{x\sqrt{2(2-\sqrt{2})(1+\sqrt{2}x^{2}+x^{4})}}}\frac{{\mathrm{d}}t}{\sqrt{(1-t^{2})(1-(\sqrt{2}-1)^{2}t^{2})}}
\end{gather*}

1. \int_{0}^{x}\frac{1}{\sqrt{1+t^{8}}}dt
2. \frac{1}{2\sqrt{2+\sqrt{2}}}\int_{0}^{\frac{\sqrt{2+\sqrt{2}}x}{1+x^{2}}}\frac{1}{\sqrt{\left(1-t^{2}\right)\left(1-(\sqrt{2}-1)^{2}t^{2}\right)}}dt+\frac{1}{\sqrt{4+2\sqrt{2}}}\int_{0}^{\frac{1+\sqrt{2}x^{2}+x^{4}-\sqrt{1+x^{8}}}{x\sqrt{2(2-\sqrt{2})(1+\sqrt{2}x^{2}+x^{4})}}}\frac{1}{\sqrt{\left(1-t^{2}\right)\left(1-(\sqrt{2}-1)^{2}t^{2}\right)}}dt
3. \frac{1}{2\sqrt{2+\sqrt{2}}}\int_{0}^{\frac{\sqrt{2+\sqrt{2}}x}{1+x^{2}}}\frac{1}{\sqrt{\left(1-t^{2}\right)\left(1-(\sqrt{2}-1)^{2}t^{2}\right)}}dt+\frac{1}{2\sqrt{2+\sqrt{2}}}\int_{0}^{\frac{\sqrt{2+\sqrt{2}}x}{\sqrt{1+\sqrt{2}x^{2}+x^{4}}}}\frac{1}{\sqrt{\left(1-t^{2}\right)\left(1-(\sqrt{2\sqrt{2}-2})^{2}t^{2}\right)}}dt

复制代码

\begin{gather*}
\int_{0}^{X_{1}}\frac{1}{\sqrt{t(1-t)(1-k_{1}^{2}t)}}dt+\int_{0}^{X_{1}}\frac{1}{\sqrt{t(1-t)(1-k_{2}^{2}t)}}dt+\int_{0}^{X_{1}}\frac{1}{\sqrt{t(1-t)(1-k_{3}^{2}t)}}dt\\
?\\
=\int_{0}^{X_{1}}\frac{1}{\sqrt{t(1-t)(1-l_{1}^{2}t)(1-l_{2}^{2}t)(1-l_{3}^{2}t)(1-l_{4}^{2}t)(1-l_{5}^{2}t)}}dt
\end{gather*}


\begin{align*}
&\>\>\int\frac{t+1}{\sqrt{t(2644t^{4}-359800t^{3}-663555t^{2}-359800t+2644)}}{\mathrm{d}}t\\
\\
&\>\>\int\frac{t+2}{\sqrt{(t^2+8)(t^3+6t-7)}}{\mathrm{d}}t\\
\\
&\>\>\int\frac{t}{\sqrt{(t^2-3)(t^3-6t+6)}}{\mathrm{d}}t\\
\end{align*}


\begin{align*}
&\>\>\int_{0}^{x}\frac{1+t}{\sqrt{t(2644t^{4}-359800t^{3}-663555t^{2}-359800t+2644)}}{\mathrm{d}}t\\
&=\int_{\frac{2x^{2}+3x+2}{6x^{2}+11x+6}}^{\frac{1}{3}}\frac{1}{\sqrt{(1-3t)(7-23t)(61597-185716t+1453t^{2})}}{\mathrm{d}}t
\end{align*}

青青子衿

青青子衿 发表于 2024-6-2 05:23
\begin{align*}

&\>\>\int\frac{t+2}{\sqrt{(t^2+8)(t^3+6t-7)}}{\mathrm{d}}t\\

\end{align*}

\begin{gather*}
\int_{x}^{+\infty}\frac{t}{\sqrt{t^{5}-5t^{3}+5t}}{\mathrm{d}t}\\
\\
=\int_{0}^{\frac{\sqrt{2x\sqrt{5+2\sqrt{5}}}}{\sqrt{x^{2}+\sqrt{5+2\sqrt{5}}x+\sqrt{5}}}}\frac{\frac{\sqrt{10\sqrt{25-10\sqrt{5}}}}{10}}{\sqrt{(1-t^{2})(1-(\raise{1pt}{\scriptsize{\frac{\sqrt{10}-\sqrt{2}+2\sqrt{\sqrt{5}-1}}{4}}})^{2}t^{2})}}{\mathrm{d}t}\\
+\int_{0}^{\frac{\sqrt{2x\sqrt{5+2\sqrt{5}}}}{\sqrt{x^{2}+\sqrt{5+2\sqrt{5}}x+\sqrt{5}}}}\frac{\frac{\sqrt{10\sqrt{25-10\sqrt{5}}}}{10}}{\sqrt{(1-t^{2})(1-(\raise{1pt}{\scriptsize{\frac{\sqrt{2}-\sqrt{10}+2\sqrt{\sqrt{5}-1}}{4}}})^{2}t^{2})}}{\mathrm{d}t}
\end{gather*}

\begin{align*}
&\qquad\qquad\qquad\int_{x}^{+\infty}\frac{{\mathrm{d}}t}{\sqrt{t^{6}+at^{3}+b}}\\

&=
\int_{0}^{\frac{b^{1/6}x}{(x-b^{1/6})^{2}}}\frac{\frac{1}{2b^{1/12}}}{\sqrt{(a+2\sqrt{b})t^{3}+\sqrt{b}(3t+1)^{2}}}{\mathrm{d}}t\\

&\qquad
-\int_{0}^{\frac{b^{1/6}x}{(x+b^{1/6})^{2}}}\frac{\frac{1}{2b^{1/12}}}{\sqrt{(a-2\sqrt{b})t^{3}+\sqrt{b}(3t-1)^{2}}}{\mathrm{d}}t
\end{align*}


\begin{align*}

\int_{x}^{+\infty}\frac{{\mathrm{d}}t}{1+t^{3}}=
\int_{0}^{\frac{x}{(x-1)^{2}}}\frac{{\mathrm{d}}t}{2(1+t)\sqrt{1+4t}}-\int_{0}^{\frac{x}{(x+1)^{2}}}\frac{{\mathrm{d}}t}{2(1-3t)}
\end{align*}

青青子衿

青青子衿 发表于 2024-6-20 22:29
\begin{gather*}
\int_{x}^{+\infty}\frac{t}{\sqrt{t^{5}-5t^{3}+5t}}{\mathrm{d}t}\\
\end{gather*}

\begin{align*}
\int_{3}^{4}\frac{{\mathrm{d}}t}{\sqrt{1+t^{20}}}

&=\int_{\alpha_{1}}^{\beta_{1}}\frac{\raise{0.5pt}{\scriptsize{\frac{\sqrt{10(5+\sqrt{5})}}{40}\cdot\frac{\sqrt{10\sqrt{25-10\sqrt{5}}}}{10}}}}{\sqrt{(1-t^{2})(1-(\raise{0.5pt}{\scriptsize{\frac{\sqrt{2}-\sqrt{10}+2\sqrt{\sqrt{5}-1}}{4}}})^{2}t^{2})}}{\mathrm{d}}t\\

&\qquad-\int_{\alpha_{2}}^{\beta_{2}}\frac{\raise{0.5pt}{\scriptsize{\frac{\sqrt{5}-1}{8}\cdot\frac{\sqrt{10\sqrt{25-10\sqrt{5}}}}{10}}}}{\sqrt{(1-t^{2})(1-(\raise{0.5pt}{\scriptsize{\frac{\sqrt{10}-\sqrt{2}+2\sqrt{\sqrt{5}-1}}{4}}})^{2}t^{2})}}{\mathrm{d}}t\\

&\qquad\quad+\int_{\alpha_{3}}^{\beta_{3}}\frac{\raise{0.5pt}{\scriptsize{\frac{\sqrt{10(5+\sqrt{5})}}{40}\cdot\frac{\sqrt{10\sqrt{25-10\sqrt{5}}}}{10}}}}{\sqrt{(1-t^{2})(1-(\raise{0.5pt}{\scriptsize{\frac{\sqrt{10}-\sqrt{2}+2\sqrt{\sqrt{5}-1}}{4}}})^{2}t^{2})}}{\mathrm{d}}t\\

&\qquad\qquad-\int_{\alpha_{4}}^{\beta_{4}}\frac{\raise{0.5pt}{\scriptsize{\frac{\sqrt{5}-1}{8}\cdot\frac{\sqrt{10\sqrt{25-10\sqrt{5}}}}{10}}}}{\sqrt{(1-t^{2})(1-(\raise{0.5pt}{\scriptsize{\frac{\sqrt{2}-\sqrt{10}+2\sqrt{\sqrt{5}-1}}{4}}})^{2}t^{2})}}{\mathrm{d}}t
\end{align*}


\begin{align*}
\beta_{10}&=\tfrac{2279414528913914627817589015107017803710978235000564629444000\sqrt{\scriptsize2(1+\sqrt{5})}}{11859579919098989643080084429778414413008199844171387925441441}\\
&\qquad-\tfrac{5195513461956811140853727887119422093445241184804762400000000}{11859579919098989643080084429778414413008199844171387925441441}\\
\beta_{11}&=-\left(\tfrac{2351772659466457732075531923419613323854370776011145728000000}{11859579919098989643080084429778414413008199844171387925441441}\right.\\
&\qquad-\tfrac{589866642330551541791694091673277720588055285772817844134000\sqrt{\scriptsize2(1+\sqrt{5})}}{11859579919098989643080084429778414413008199844171387925441441}\big)\\
\beta_{12}
&=-\left(\tfrac{1521861836504716764064394773399783244308434019443009907200000}{11859579919098989643080084429778414413008199844171387925441441}\right.\\
&\qquad-\tfrac{225518691780476199225028756148476721089776079946230639044000\sqrt{\scriptsize2(1+\sqrt{5})}}{11859579919098989643080084429778414413008199844171387925441441}\big)\\
\beta_{13}&=\tfrac{1627137457011642796054930161772953029988642878354922250249200\sqrt{\scriptsize2(1+\sqrt{5})}}{11859579919098989643080084429778414413008199844171387925441441}\\
&\qquad-\tfrac{613507087330241619589187027322321566828354970016535577600000}{11859579919098989643080084429778414413008199844171387925441441}\\
\beta_{1}&=\sqrt{\beta_{10}+5^{1/4}\beta_{11}+5^{1/2}\beta_{12}+5^{3/4}\beta_{13}}\\
\alpha_{10}&=\tfrac{16726307363358463800490984604959387365663679974501407768404177257940693035053918784000\sqrt{\scriptsize2(1+\sqrt{5})}}{377084446625457307041668933338985315563266836741439934499136087902684540706815585729281}\\
&\qquad-\tfrac{40271810567826226278553963162328172145007378959894673201342321455403119219128900000000}{377084446625457307041668933338985315563266836741439934499136087902684540706815585729281}\\
\alpha_{11}&=-\left(\tfrac{10171211040697214887732011202913860444253802838454724629639435565088477309669632000000}{377084446625457307041668933338985315563266836741439934499136087902684540706815585729281}\right.\\
&\qquad-\tfrac{2019317162745492464433222820629387517028430638961787458026957318250842805307973376000\sqrt{\scriptsize2(1+\sqrt{5})}}{377084446625457307041668933338985315563266836741439934499136087902684540706815585729281}\big)\\
\alpha_{12}&=-\left(\tfrac{9167292255340104789702638612889982053361393515388301925121484563836811134661543200000}{377084446625457307041668933338985315563266836741439934499136087902684540706815585729281}\right.\\
&\qquad-\tfrac{453058786858655593163731028191533081761239867295798754519296854058819438197776384000\sqrt{\scriptsize2(1+\sqrt{5})}}{377084446625457307041668933338985315563266836741439934499136087902684540706815585729281}\big)\\
\alpha_{13}&=\tfrac{24851717989176467600488271399936188741764045666722821419676585888145247119190309238800\sqrt{\scriptsize2(1+\sqrt{5})}}{377084446625457307041668933338985315563266836741439934499136087902684540706815585729281}\\
&\qquad-\tfrac{2204559143393324810307145862921249072632794113505753188550411532817534570431334400000}{377084446625457307041668933338985315563266836741439934499136087902684540706815585729281}\\
\alpha_{1}&=\sqrt{\alpha_{10}+5^{1/4}\alpha_{11}+5^{1/2}\alpha_{12}+5^{3/4}\alpha_{13}}\\

\beta_{20}&=-\tfrac{86229978741286629981443967130736940090}{29226313301389985169939299906350357921}\\
&\qquad-\tfrac{28196604852246818567886332186902535355\sqrt{\scriptsize2(1+\sqrt{5})}}{29226313301389985169939299906350357921}\\
\beta_{21}&=\tfrac{46654077730815231019000851895387170120}{29226313301389985169939299906350357921}\\
&\qquad+\tfrac{25573944703916057721076020609842251869\sqrt{\scriptsize2(1+\sqrt{5})}}{29226313301389985169939299906350357921}\\
\beta_{22}&=-\tfrac{38543181576087283220049530230441163196}{29226313301389985169939299906350357921}\\
&\qquad-\tfrac{12443002302015215306658527450635743471\sqrt{\scriptsize2(1+\sqrt{5})}}{29226313301389985169939299906350357921}\\
\beta_{23}&=\tfrac{20854015850812361167160087805577671048}{29226313301389985169939299906350357921}\\
&\qquad+\tfrac{11764493805467831604577455145690648863\sqrt{\scriptsize2(1+\sqrt{5})}}{29226313301389985169939299906350357921}\\
\beta_{2}&=\sqrt{\beta_{20}+5^{1/4}\beta_{21}+5^{1/2}\beta_{22}+5^{3/4}\beta_{23}}\\
\alpha_{20}&=-\tfrac{333228095857186815587582353638477775358538563840}{1009191758958765945322486883888562945986478385921}\\
&\qquad-\tfrac{69385658675983999571422025145608482316130388480\sqrt{\scriptsize2(1+\sqrt{5})}}{1009191758958765945322486883888562945986478385921}\\
\alpha_{21}&=\tfrac{90340246516990943002107136464364349521575034880}{1009191758958765945322486883888562945986478385921}\\
&\qquad+\tfrac{166181874057432127650930161457599356247304628256\sqrt{\scriptsize2(1+\sqrt{5})}}{1009191758958765945322486883888562945986478385921}\\
\alpha_{22}&=-\tfrac{148769738384174042342985598069246384198857661696}{1009191758958765945322486883888562945986478385921}\\
&\qquad-\tfrac{29180927452671004958932177754051551140814980096\sqrt{\scriptsize2(1+\sqrt{5})}}{1009191758958765945322486883888562945986478385921}\\
\alpha_{23}&=\tfrac{40334938835760476674086058603974646761118973952}{1009191758958765945322486883888562945986478385921}\\
&\qquad+\tfrac{80882320820007796766741810531898212578054506512\sqrt{\scriptsize2(1+\sqrt{5})}}{1009191758958765945322486883888562945986478385921}\\
\alpha_{2}&=\sqrt{\alpha_{20}+5^{1/4}\alpha_{21}+5^{1/2}\alpha_{22}+5^{3/4}\alpha_{23}}\\

\beta_{30}&=-\tfrac{5195513461956811140853727887119422093445241184804762400000000}{11859579919098989643080084429778414413008199844171387925441441}\\
&\qquad-\tfrac{2279414528913914627817589015107017803710978235000564629444000\sqrt{\scriptsize2(1+\sqrt{5})}}{11859579919098989643080084429778414413008199844171387925441441}\\
\beta_{31}&=\tfrac{2351772659466457732075531923419613323854370776011145728000000}{11859579919098989643080084429778414413008199844171387925441441}\\
&\qquad+\tfrac{589866642330551541791694091673277720588055285772817844134000\sqrt{\scriptsize2(1+\sqrt{5})}}{11859579919098989643080084429778414413008199844171387925441441}\\
\beta_{32}&=-\tfrac{1521861836504716764064394773399783244308434019443009907200000}{11859579919098989643080084429778414413008199844171387925441441}\\
&\qquad-\tfrac{225518691780476199225028756148476721089776079946230639044000\sqrt{\scriptsize2(1+\sqrt{5})}}{11859579919098989643080084429778414413008199844171387925441441}\\
\beta_{33}&=\tfrac{613507087330241619589187027322321566828354970016535577600000}{11859579919098989643080084429778414413008199844171387925441441}\\
&\qquad+\tfrac{1627137457011642796054930161772953029988642878354922250249200\sqrt{\scriptsize2(1+\sqrt{5})}}{11859579919098989643080084429778414413008199844171387925441441}\\
\beta_{3}&=\sqrt{\beta_{30}+5^{1/4}\beta_{31}+5^{1/2}\beta_{32}+5^{3/4}\beta_{33}}\\
\alpha_{30}&=-\tfrac{40271810567826226278553963162328172145007378959894673201342321455403119219128900000000}{377084446625457307041668933338985315563266836741439934499136087902684540706815585729281}\\
&\qquad-\tfrac{16726307363358463800490984604959387365663679974501407768404177257940693035053918784000\sqrt{\scriptsize2(1+\sqrt{5})}}{377084446625457307041668933338985315563266836741439934499136087902684540706815585729281}\\
\alpha_{31}&=\tfrac{10171211040697214887732011202913860444253802838454724629639435565088477309669632000000}{377084446625457307041668933338985315563266836741439934499136087902684540706815585729281}\\
&\qquad+\tfrac{2019317162745492464433222820629387517028430638961787458026957318250842805307973376000\sqrt{\scriptsize2(1+\sqrt{5})}}{377084446625457307041668933338985315563266836741439934499136087902684540706815585729281}\\
\alpha_{32}&=-\tfrac{9167292255340104789702638612889982053361393515388301925121484563836811134661543200000}{377084446625457307041668933338985315563266836741439934499136087902684540706815585729281}\\
&\qquad-\tfrac{453058786858655593163731028191533081761239867295798754519296854058819438197776384000\sqrt{\scriptsize2(1+\sqrt{5})}}{377084446625457307041668933338985315563266836741439934499136087902684540706815585729281}\\
\alpha_{33}&=\tfrac{2204559143393324810307145862921249072632794113505753188550411532817534570431334400000}{377084446625457307041668933338985315563266836741439934499136087902684540706815585729281}\\
&\qquad+\tfrac{24851717989176467600488271399936188741764045666722821419676585888145247119190309238800\sqrt{\scriptsize2(1+\sqrt{5})}}{377084446625457307041668933338985315563266836741439934499136087902684540706815585729281}\\
\alpha_{3}&=\sqrt{\alpha_{30}+5^{1/4}\alpha_{31}+5^{1/2}\alpha_{32}+5^{3/4}\alpha_{33}}\\

\beta_{40}&=-\left(\tfrac{86229978741286629981443967130736940090}{29226313301389985169939299906350357921}\right.\\
&\qquad-\tfrac{28196604852246818567886332186902535355\sqrt{\scriptsize2(1+\sqrt{5})}}{29226313301389985169939299906350357921}\big)\\
\beta_{41}&=\tfrac{25573944703916057721076020609842251869\sqrt{\scriptsize2(1+\sqrt{5})}}{29226313301389985169939299906350357921}\\
&\qquad-\tfrac{46654077730815231019000851895387170120}{29226313301389985169939299906350357921}\\
\beta_{42}&=-\left(\tfrac{38543181576087283220049530230441163196}{29226313301389985169939299906350357921}\right.\\
&\qquad-\tfrac{12443002302015215306658527450635743471\sqrt{\scriptsize2(1+\sqrt{5})}}{29226313301389985169939299906350357921}\big)\\
\beta_{43}&=\tfrac{11764493805467831604577455145690648863\sqrt{\scriptsize2(1+\sqrt{5})}}{29226313301389985169939299906350357921}\\
&\qquad-\tfrac{20854015850812361167160087805577671048}{29226313301389985169939299906350357921}\\
\beta_{4}&=\sqrt{\beta_{40}+5^{1/4}\beta_{41}+5^{1/2}\beta_{42}+5^{3/4}\beta_{43}}\\
\alpha_{40}&=-\left(\tfrac{333228095857186815587582353638477775358538563840}{1009191758958765945322486883888562945986478385921}\right.\\
&\qquad-\tfrac{69385658675983999571422025145608482316130388480\sqrt{\scriptsize2(1+\sqrt{5})}}{1009191758958765945322486883888562945986478385921}\big)\\
\alpha_{41}&=\tfrac{166181874057432127650930161457599356247304628256\sqrt{\scriptsize2(1+\sqrt{5})}}{1009191758958765945322486883888562945986478385921}\\
&\qquad-\tfrac{90340246516990943002107136464364349521575034880}{1009191758958765945322486883888562945986478385921}\\
\alpha_{42}&=-\left(\tfrac{148769738384174042342985598069246384198857661696}{1009191758958765945322486883888562945986478385921}\right.\\
&\qquad-\tfrac{29180927452671004958932177754051551140814980096\sqrt{\scriptsize2(1+\sqrt{5})}}{1009191758958765945322486883888562945986478385921}\big)\\
\alpha_{43}&=\tfrac{80882320820007796766741810531898212578054506512\sqrt{\scriptsize2(1+\sqrt{5})}}{1009191758958765945322486883888562945986478385921}\\
&\qquad-\tfrac{40334938835760476674086058603974646761118973952}{1009191758958765945322486883888562945986478385921}\\
\alpha_{4}&=\sqrt{\alpha_{40}+5^{1/4}\alpha_{41}+5^{1/2}\alpha_{42}+5^{3/4}\alpha_{43}}\\
\end{align*}