第三类完全椭圆积分的变换

青青子衿

\begin{align*}
&\qquad\int_{0}^{1}\frac{\mathrm{d}t}{\left(1-\beta^2u^8
t^{2}\right)\sqrt{\left(1-t^{2}\right)\left(1-u^8t^{2}\right)}}-\int_{0}^{1}\frac{\mathrm{d}t}{\sqrt{\left(1-t^{2}\right)\left(1-u^8t^{2}\right)}}\\
&=\frac{\beta}{\sqrt{(1-\beta^2)(1-u^8\beta^2)}}\left(\int_{0}^{1}\frac{\mathrm{d}w}{\sqrt{\left(1-w^{2}\right)\left(1-u^8w^{2}\right)}}\int_{0}^{\beta}\sqrt{\frac{1-u^8\omega^{2}}{1-\omega^{2}}}\mathrm{d}\omega\right.\\
&\qquad\qquad\qquad\qquad\qquad\qquad\left.-\int_{0}^{1}\sqrt{\frac{1-u^8w^{2}}{1-w^{2}}}\mathrm{d}w\int_{0}^{\beta}\frac{\mathrm{d}\omega}{\sqrt{\left(1-\omega^{2}\right)\left(1-u^8\omega^{2}\right)}}\right)\\
\end{align*}

\begin{align*}
&\qquad\int_{0}^{1}\frac{\mathrm{d}t}{\left(1-\gamma^2v^8
t^{2}\right)\sqrt{\left(1-t^{2}\right)\left(1-v^8t^{2}\right)}}-\int_{0}^{1}\frac{\mathrm{d}t}{\sqrt{\left(1-t^{2}\right)\left(1-v^8t^{2}\right)}}\\
&=\frac{\gamma}{\sqrt{(1-\gamma^2)(1-v^8\gamma^2)}}\left(\int_{0}^{1}\frac{\mathrm{d}w}{\sqrt{\left(1-w^{2}\right)\left(1-v^8w^{2}\right)}}\int_{0}^{\gamma}\sqrt{\frac{1-v^8\omega^{2}}{1-\omega^{2}}}\mathrm{d}\omega\right.\\
&\qquad\qquad\qquad\qquad\qquad\qquad\left.-\int_{0}^{1}\sqrt{\frac{1-v^8w^{2}}{1-w^{2}}}\mathrm{d}w\int_{0}^{\gamma}\frac{\mathrm{d}\omega}{\sqrt{\left(1-\omega^{2}\right)\left(1-v^8\omega^{2}\right)}}\right)\\
\end{align*}

\begin{align*}
\cdot\mathcal{E}(\gamma,V)
&=\frac{\frac{v^8(1-v^8)\frac{\partial\>\!U}{\partial\>\!u}}{u^8(1-u^8)\frac{\partial\>\!V}{\partial\>\!v}}\cdot\frac{\mathrm{d}u}{\mathrm{d}v}}{M}\cdot\mathcal{E}(\beta,U)
\\
&\qquad\quad+\dfrac{2V(1-V)}{\frac{\partial\>\!V}{\partial\>\!s}}\cdot\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(\beta,U)\\
&\qquad\qquad\quad-\dfrac{2V(1-V)\frac{\partial\>\!\Phi(\beta,U)}{\partial\>\!\beta}}{M\frac{\partial\>\!V}{\partial\>\!s}}\cdot\left(\frac{\beta(1-\beta^2)\frac{\partial\>\!U}{\partial\>\!s}}{2(1-U)}+\dfrac{\frac{\partial\>\!\gamma}{\partial\>\!s}-\frac{\gamma\cdot(1-\gamma^2)}{2(1-V)}\cdot\frac{\partial\>\!V}{\partial\>\!s}}{\frac{\partial\>\!\gamma}{\partial\>\!\beta}}\right)\\

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

\end{align*}


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青青子衿

\begin{gather*}
\int_{0}^{1}\frac{\mathrm{d}t}{(1-V\gamma^{2}t^{2})\sqrt{(1-t^{2})(1-Vt^{2})}}\\
=\int_{0}^{1}\frac{\frac{\mathfrak{n}}{\beta\>\!M}\!\cdot\!\frac{\gamma}{\gamma'_{\beta}} }{(1-U\beta^{2}t^{2})\sqrt{(1-t^{2})(1-Ut^{2})}}\mathrm{d}t
+\int_{0}^{1}\frac{\frac{1}{M}\left(1-\frac{\mathfrak{n}}{\beta}\!\cdot\!\frac{\gamma}{\gamma'_{\beta}}+\frac{\rho(\beta)}{M}\!\cdot\!\frac{\gamma}{\gamma'_{\beta}}\right)}{\sqrt{(1-t^{2})(1-Ut^{2})}}\mathrm{d}t\\
\end{gather*}

1. s=1
2. U=\frac{3-\sqrt{6}}{6}
3. V=\frac{1}{54}\left(11\sqrt{6}+27\right)
4. \beta=\sqrt{\frac{2}{3}}
5. \gamma=\frac{(1+2s)\beta\Upsilon_{51}\Upsilon_{52}}{\Gamma_{51}\Gamma_{52}}
6. \Upsilon_{51}=1-\frac{1+2s+s^{2}\sqrt{1+2s}-(1+s)\sqrt{(1+2s)(1+s^{2})}}{2(1+2s)}\beta^{2}
7. \Upsilon_{52}=1-\frac{1+2s-s^{2}\sqrt{1+2s}-(1+s)\sqrt{(1+2s)(1+s^{2})}}{2(1+2s)}\beta^{2}
8. \Gamma_{51}=1-\frac{1-s-s^{2}+s\sqrt{1+2s}+s\sqrt{1+s^{2}}-\sqrt{(1+2s)(1+s^{2})}}{2}\beta^{2}
9. \Gamma_{52}=1-\frac{1-s-s^{2}-s\sqrt{1+2s}-s\sqrt{1+s^{2}}-\sqrt{(1+2s)(1+s^{2})}}{2}\beta^{2}
10. P_{51}=1-\frac{1+2s-(1+s)\sqrt{(1+2s)(1+s^{2})}}{2(1+2s)}\beta^{2}
11. Q_{51}=1-\frac{1+2s+s(2-s)\sqrt{1+2s}-(1-s)\sqrt{(1+2s)(1+s^{2})}}{2(1+2s)}\beta^{2}
12. Q_{52}=1-\frac{1+2s-s(2-s)\sqrt{1+2s}-(1-s)\sqrt{(1+2s)(1+s^{2})}}{2(1+2s)}\beta^{2}
13. Q_{53}=1-\frac{1+s+s^{2}+s\sqrt{1+2s}-s\sqrt{1+s^{2}}-\sqrt{(1+2s)(1+s^{2})}}{2}\beta^{2}
14. Q_{54}=1-\frac{1+s+s^{2}-s\sqrt{1+2s}+s\sqrt{1+s^{2}}-\sqrt{(1+2s)(1+s^{2})}}{2}\beta^{2}
15. \int_{0}^{1}\frac{\frac{Q_{51}Q_{52}Q_{53}Q_{54}}{(1+2s)\Gamma_{51}\Gamma_{52}}}{(1-V\gamma^{2}t^{2})\sqrt{(1-t^{2})(1-Vt^{2})}}dt
16. \int_{0}^{1}\frac{5\Upsilon_{51}\Upsilon_{52}}{(1-U\beta^{2}t^{2})\sqrt{(1-t^{2})(1-Ut^{2})}}dt-\int_{0}^{1}\frac{4P_{51}}{\sqrt{(1-t^{2})(1-Ut^{2})}}dt
17. \int_{0}^{1}\frac{\frac{11\left(28\sqrt{6}-55\right)}{5913}}{(1-\frac{213123+86999\sqrt{6}}{431649}t^{2})\sqrt{(1-t^{2})(1-\frac{27+11\sqrt{6}}{54}t^{2})}}dt
18. \int_{0}^{1}\frac{\frac{5\left(19+8\sqrt{6}\right)}{27}}{(1-\frac{3-\sqrt{6}}{9}t^{2})\sqrt{(1-t^{2})(1-\frac{3-\sqrt{6}}{6}t^{2})}}dt-\int_{0}^{1}\frac{\frac{8\left(3+\sqrt{6}\right)}{9}}{\sqrt{(1-t^{2})(1-\frac{3-\sqrt{6}}{6}t^{2})}}dt
19. \int_{0}^{1}\frac{\frac{11\left(28\sqrt{6}-55\right)}{1971}}{(1-\frac{213123+86999\sqrt{6}}{431649}\left(\frac{t(6+(4\sqrt{6}-6)t^{2}+(5-2\sqrt{6})t^{4})}{2+2(1+\sqrt{6})t^{2}+t^{4}}\right)^{2})\sqrt{(1-t^{2})(1-\frac{3-\sqrt{6}}{6}t^{2})}}dt

复制代码

\begin{gather*}
\int_{0}^{1}\frac{\frac{Q_{5,1}Q_{5,2}Q_{5,3}Q_{5,4} }{(1+2s)\varGamma_{5,1}\varGamma_{5,2}}}{(1-V_5{\raise.3ex\hbox{$\gamma$}}_5^{2}t^{2})\sqrt{(1-t^{2})(1-V_5t^{2})}}\mathrm{d}t\\
=\int_{0}^{1}\frac{5\varUpsilon_{5,1}\varUpsilon_{5,2}}{(1-U_5\beta^{2}t^{2})\sqrt{(1-t^{2})(1-U_5t^{2})}}\mathrm{d}t
-\int_{0}^{1}\frac{4P_{5,1}}{\sqrt{(1-t^{2})(1-U_5t^{2})}}\mathrm{d}t\\
\\
\quad
\begin{split}
U_5&=\tfrac{1+2s-(1+s-s^{2})\delta_{5,3}}{2(1+2s)}\\
V_5&=\tfrac{(1+2s)^{3}-(1-11s-s^{2})\delta_{5,3}}{2(1+2s)^{3}}\\
\gamma_5&=\tfrac{(1+2s)\beta\,\varUpsilon_{5,1}\varUpsilon_{5,2}}{\varGamma_{5,1}\varGamma_{5,2}}\\
\end{split}
\qquad\begin{split}
\delta_{5,1}&={\scriptsize\sqrt{1+2s}}\\
\delta_{5,2}&={\scriptsize\sqrt{1+s^{2}}}\\
\delta_{5,3}&={\scriptsize\sqrt{\Tiny(1+2s)(1+s^{2})}}\\
\end{split}
\\
\\
\quad\begin{split}
\varUpsilon_{5,1}&=
1-\tfrac{1+2 s+s^2\delta_{5,1}-(1+s)\delta_{5,3}}{2(1+2 s)}\beta^2\\
\varUpsilon_{5,2}&=
1-\tfrac{1+2 s-s^2\delta_{5,1}-(1+s)\delta_{5,3}}{2(1+2 s)}\beta ^2\\

\varGamma_{5,1}&=
1-\tfrac{1-s-s^{2}+s\delta_{5,1}+s\delta_{5,2}-\delta_{5,3}}{2}\beta^{2}\\
\varGamma_{5,2}&=
1-\tfrac{1-s-s^{2}-s\delta_{5,1}-s\delta_{5,2}-\delta_{5,3}}{2}\beta^{2}

\end{split}\\
\\
\begin{split}
P_{5,1}&=1-\tfrac{1+2s-(1+s)\delta_{5,3}}{2(1+2s)}\beta^{2}\\
\end{split}\\
\\
\quad\quad\begin{split}
Q_{5,1}&=1-\tfrac{1+2s+s(2-s)\delta_{5,1}-(1-s)\delta_{5,3}}{2(1+2s)}\beta^{2}\\
Q_{5,2}&=1-\tfrac{1+2s-s(2-s)\delta_{5,1}-(1-s)\delta_{5,3}}{2(1+2s)}\beta^{2}\\
Q_{5,3}&=
1-\tfrac{1+s+s^{2}+s\delta_{5,1}-s\delta_{5,2}-\delta_{5,3}}{2}\beta^{2}
\\
Q_{5,4}&=1-\tfrac{1+s+s^{2}-s\delta_{5,1}+s\delta_{5,2}-\delta_{5,3}}{2}\beta^{2}\\
\end{split}\\
\\
\end{gather*}



\begin{align*}
P(x)&=\tfrac{1 - \frac{650911-48238 \sqrt{6}}{184174} x + \frac{943225-353856 \sqrt{6}}{276261} x^2 - \frac{2897484-971387 \sqrt{6}}{1105044} x^3 + \frac{2706115-1015473 \sqrt{6}}{3315132} x^4 + \frac{7 (34245 \sqrt{6}-76123)}{6630264} x^5}
{1-\frac{72493971-18338113\sqrt{6}}{40373016}x+\frac{65100391+620933\sqrt{6}}{20186508}x^{2}+\frac{21016181\sqrt{6}-42017958}{60559524}x^{3}+\frac{25346393-5977069\sqrt{6}}{121119048}x^{4}-\frac{49(404283-139021\sqrt{6})}{1453428576}x^{5}}\\
Q(x)&=\tfrac{1}{\sqrt{x(1-x)(1-\frac{3-\sqrt{6}}{6}x)}}\\
\\
I&=\int P(x)Q(x)\mathrm{d}x
\end{align*}




\begin{gather*}
\qquad\begin{split}
\Omega_4(x)&=-\int_{0}^{x}\frac{

p_{4}

}{\sqrt{(1-t^{2})(1-U_4t^{2})}}\mathrm{d}t\\
&\qquad\>+\int_{0}^{x}\frac{

4\varUpsilon_{4,1}

}{(1-U_4\beta^{2}t^{2})\sqrt{(1-t^{2})(1-U_4t^{2})}}\mathrm{d}t\\
&\qquad\>\>\>\>-\int_{0}^{y_4(x)}\frac{M_4q_4}{(1-V_4{\raise.3ex\hbox{$\gamma$}}_4^{2}t^{2})\sqrt{(1-t^{2})(1-V_4t^{2})}}\mathrm{d}t\\
&=\frac{\varUpsilon_{4,1}\beta}{\sqrt{(1-\beta^{2})(1-U_4\beta^{2})}}\operatorname{artanh}\left(\varrho_4(x)R_4(x)\right)\\
\end{split}\\
\\
\qquad\begin{split}
y_4(x)&=\tfrac{\frac{1}{M_4}x(1+\tau_{4,1}x^{2})}
{1+\omega_{4,1}x^{2}+\omega_{4,2}x^{2}}\\
\varrho_4(x)&=\tfrac{w_4x(1+\varsigma_{4,1}x^2+\varsigma_{4,2}x^4)}{1+\varpi_{4,1}x^2+\varpi_{4,2}x^4+\varpi_{4,3}x^6+\varpi_{4,4}x^8}\\
R_4(x)&={\scriptsize\beta\sqrt{(1-\beta^{2})(1-U_4\beta^{2})(1-x^{2})(1-U_4x^{2})}}\\
\end{split}\\
\\
\\
\qquad\begin{split}
U_4&=c^4\\
V_4&=\tfrac{8c(1+c^2)}{(1+c)^{4}}\\
M_4&=\tfrac{1}{(1+c)^{2}}\\

\gamma_4&=\tfrac{\frac{1}{M_4}\beta\,\varUpsilon_{4,1}}
{\varGamma_{4,1}\varGamma_{4,2}}\\
\end{split}\qquad
\begin{split}
\tau_{4,1}&={\scriptsize\,c^2}\\
\omega_{4,1}&={\scriptsize2c(1+c+c^2)}\\
\omega_{4,2}&={\scriptsize\,c^{4}}\\

\varUpsilon_{4,1}&={\scriptsize1+c^2\beta^{2}}\\
\varGamma_{4,1}&={\scriptsize1+\tfrac{c(1+c+\sqrt{1+c^{2}})^{2}}{2}\beta^{2}}\\
\varGamma_{4,2}&={\scriptsize1+\tfrac{c(1+c-\sqrt{1+c^{2}})^{2}}{2}\beta^{2}}\\
\end{split}\\
\\

\qquad\begin{split}
p_4&=3P_{4}\\

q_4&=\tfrac{Q_{4,1}Q_{4,2}Q_{4,3}}{\varGamma_{4,1}\varGamma_{4,2}}\\

\end{split}\qquad
\begin{split}
P_{4}&={\scriptsize1+\tfrac{c^2}{3}\beta^{2}}\\
Q_{4,1}&={\scriptsize1-c^2\beta^{2}}\\
Q_{4,2}&={\scriptsize1-\tfrac{c(1-c-\sqrt{1+c^2})^{2}}{2}\beta^{2}}\\
Q_{4,3}&={\scriptsize1-\tfrac{c(1-c+\sqrt{1+c^2})^{2}}{2}\beta^{2}}
\end{split}\\
\\
\qquad\>\>\begin{split}
w_4&=\tfrac{4\varUpsilon_{41}-p_{4}-q_{4}}{\varUpsilon_{41}\beta^{2}}=
\tfrac{4c\left(1+c+c^{2}+c^{3}\beta^{2}\right)}{\varGamma_{4,1}\varGamma_{4,2}}\\

\varsigma_{4,1}&=\tfrac{c^3(1-2 \beta ^2-2c\beta ^2-2c^2\beta ^2-2 c^3\beta ^2-2c^4\beta ^2+ c^4\beta ^4)}{1+c+c^2+c^3\beta^2}\\

\varsigma_{4,2}&=\tfrac{c^7\beta ^2(1+c\beta ^2+ c^2\beta ^2+c^3\beta ^2)}{1+c+c^2+c^3\beta ^2}\\

\varpi_{4,1}&=\tfrac{2c}{\varGamma_{4,1}\varGamma_{4,2}}\big({\scriptsize1 + c + c^2}\\
&\qquad\>{\scriptsize-(2 + 2c + 2c^2 - c^3 + 2c^4 + 2c^5 + 2c^6)\beta^2}\\
&\qquad\quad{\scriptsize-c^3(4 + c + c^2 + c^3 + 4c^4)\beta^4 + 3c^7\beta^6}\big)\\
\varpi_{4,2}&=\tfrac{c^4}{\varGamma_{4,1}\varGamma_{4,2}}\big({\scriptsize1 - 2 (4 + c + c^2 + c^3 + 4 c^4)\beta^2}\\
&\qquad\>{\scriptsize+2(4+4c+4c^{2}+4c^{3}+13c^{4})\beta^{4}}\\
&\qquad\quad{\scriptsize+2(4c^{5}+4c^{6}+4c^{7}+4c^{8})\beta^{4}}\\
&\qquad\quad\>\>{\scriptsize- 2 c^4 (4 + c + c^2 + c^3 + 4 c^4)\beta^6 + c^8\beta^8}\big)\\
\varpi_{4,3}&=\tfrac{2c^8}{\varGamma_{4,1}\varGamma_{4,2}}{\scriptsize\beta^2}\big({\scriptsize3 - (4 + c + c^2 + c^3 + 4 c^4)\beta^2}\\
&\qquad\>{\scriptsize-
   c (2 + 2 c + 2 c^2 - c^3 + 2 c^4 + 2 c^5 + 2 c^6)\beta^4}\\
&\qquad\quad\>\>{\scriptsize+
   c^5 (1 + c + c^2)\beta^6}\big)\\
\varpi_{4,4}&={\scriptsize\,c^{12}\beta^4}\\
\end{split}
\end{gather*}

1. c=0.547
2. U_{4}=c^{4}
3. V_{4}=\frac{8c(1+c^{2})}{(1+c)^{4}}
4. \beta=0.643
5. \gamma_{4}=\frac{(1+c)^{2}\beta\Upsilon_{41}}{\Gamma_{41}\Gamma_{42}}
6. \Upsilon_{41}=1+c^{2}\beta^{2}
7. \Gamma_{41}=1+\frac{c(1+c-\sqrt{1+c^{2}})^{2}}{2}\beta^{2}
8. \Gamma_{42}=1+\frac{c(1+c+\sqrt{1+c^{2}})^{2}}{2}\beta^{2}
9. P_{4}=1+\frac{c^{2}}{3}\beta^{2}
10. Q_{41}=1-c^{2}\beta^{2}
11. Q_{42}=1-\frac{c(1-c-\sqrt{1+c^{2}})^{2}}{2}\beta^{2}
12. Q_{43}=1-\frac{c(1-c+\sqrt{1+c^{2}})^{2}}{2}\beta^{2}
13. p_{4}=3P_{4}
14. q_{4}=\frac{Q_{41}Q_{42}Q_{43}}{\Gamma_{41}\Gamma_{42}}
15. w_{4}=\frac{4c\left(1+c+c^{2}+c^{3}\beta^{2}\right)}{1+2c(1+c+c^{2})\beta^{2}+c^{4}\beta^{4}}
16. \frac{4c\left(1+c+c^{2}+c^{3}\beta^{2}\right)}{\Gamma_{41}\Gamma_{42}}-\frac{4\Upsilon_{41}-p_{4}-q_{4}}{\Upsilon_{41}\beta^{2}}
17. \varsigma_{41}=\frac{c^{3}(1-2\beta^{2}-2c\beta^{2}-2c^{2}\beta^{2}-2c^{3}\beta^{2}-2c^{4}\beta^{2}+c^{4}\beta^{4})}{1+c+c^{2}+c^{3}\beta^{2}}
18. \varsigma_{42}=\frac{c^{7}\beta^{2}(1+c\beta^{2}+c^{2}\beta^{2}+c^{3}\beta^{2})}{1+c+c^{2}+c^{3}\beta^{2}}
19. \varpi_{41}=\frac{2c\left(1+c+c^{2}-(2+2c+2c^{2}-c^{3}+2c^{4}+2c^{5}+2c^{6})\beta^{2}-c^{3}(4+c+c^{2}+c^{3}+4c^{4})\beta^{4}+3c^{7}\beta^{6}\right)}{1+2c\beta^{2}+2c^{2}\beta^{2}+2c^{3}\beta^{2}+c^{4}\beta^{4}}
20. \varpi_{42}=\frac{c^{4}\left(1-2(4+c+c^{2}+c^{3}+4c^{4})\beta^{2}+2(4+4c+4c^{2}+4c^{3}+13c^{4}+4c^{5}+4c^{6}+4c^{7}+4c^{8})\beta^{4}-2c^{4}(4+c+c^{2}+c^{3}+4c^{4})\beta^{6}+c^{8}\beta^{8}\right)}{1+2c\beta^{2}+2c^{2}\beta^{2}+2c^{3}\beta^{2}+c^{4}\beta^{4}}
21. \varpi_{43}=\frac{2c^{8}\beta^{2}\left(3-(4+c+c^{2}+c^{3}+4c^{4})\beta^{2}-c(2+2c+2c^{2}-c^{3}+2c^{4}+2c^{5}+2c^{6})\beta^{4}+c^{5}(1+c+c^{2})\beta^{6}\right)}{1+2c\beta^{2}+2c^{2}\beta^{2}+2c^{3}\beta^{2}+c^{4}\beta^{4}}
22. \varpi_{44}=c^{12}\beta^{4}
23. V_{4}\gamma_{4}^{2}-\frac{8c(1+c^{2})\beta^{2}(1+c^{2}\beta^{2})^{2}}{\left(1+2c(1+c+c^{2})\beta^{2}+c^{4}\beta^{4}\right)^{2}}
24. M_{4}=\frac{1}{(1+c)^{2}}
25. \tau_{41}=c^{2}
26. \omega_{41}=2c(1+c+c^{2})
27. \omega_{42}=c^{4}
28. -\int_{0}^{x}\frac{p_{4}}{\sqrt{(1-t^{2})(1-U_{4}t^{2})}}dt+\int_{0}^{x}\frac{4\Upsilon_{41}}{(1-U_{4}\beta^{2}t^{2})\sqrt{(1-t^{2})(1-U_{4}t^{2})}}dt-\int_{0}^{\frac{\frac{1}{M_{4}}x(1+\tau_{41}x^{2})}{1+\omega_{41}x^{2}+\omega_{42}x^{4}}}\frac{M_{4}q_{4}}{\left(1-V_{4}\gamma_{4}^{2}\ t^{2}\right)\sqrt{(1-t^{2})(1-V_{4}t^{2})}}dt
29. \frac{\Upsilon_{41}\beta}{\sqrt{\left(1-\beta^{2}\right)\left(1-U_{4}\beta^{2}\right)}}\operatorname{artanh}\left(\frac{w_{4}\beta\sqrt{(1-\beta^{2})(1-c^{4}\beta^{2})}\left(1+\varsigma_{41}x^{2}+\varsigma_{42}x^{4}\right)x}{1+\varpi_{41}x^{2}+\varpi_{42}x^{4}+\varpi_{43}x^{6}+\varpi_{44}x^{8}}\sqrt{(1-x^{2})(1-U_{4}x^{2})}\right)
30. \Upsilon_{41}w_{4}\beta^{2}x
31. \frac{\Upsilon_{41}w_{4}\beta^{2}\left(1+\varsigma_{41}+\varsigma_{42}\right)\sqrt{2(U_{4}-1)(x-1)}}{1+\varpi_{41}+\varpi_{42}+\varpi_{43}+\varpi_{44}}
32. \frac{\Upsilon_{41}w_{4}\beta^{2}U_{4}\left(U_{4}^{2}+U_{4}\varsigma_{41}+\varsigma_{42}\right)\sqrt{2(1-U_{4})\left(\sqrt{U_{4}}x-1\right)}}{U_{4}^{4}+U_{4}^{3}\varpi_{41}+U_{4}^{2}\varpi_{42}+U_{4}\varpi_{43}+\varpi_{44}}
33. \frac{\Upsilon_{41}w_{4}\beta^{2}\sqrt{U_{4}}\varsigma_{42}}{\varpi_{44}x}
34. \frac{w_{4}\beta^{2}U_{4}(1-\beta^{2})(1-U_{4}\beta^{2})\left(\beta^{4}U_{4}^{2}+\beta^{2}U_{4}\varsigma_{41}+\varsigma_{42}\right)}{\beta^{8}U_{4}^{4}+\beta^{6}U_{4}^{3}\varpi_{41}+\beta^{4}U_{4}^{2}\varpi_{42}+\beta^{2}U_{4}\varpi_{43}+\varpi_{44}}
35. \int_{0}^{x}\left(\frac{4\Upsilon_{41}}{(1-U_{4}\beta^{2}t^{2})\sqrt{(1-t^{2})(1-U_{4}t^{2})}}-\frac{p_{4}}{\sqrt{(1-t^{2})(1-U_{4}t^{2})}}-\frac{q_{4}}{\left(1-V_{4}\gamma_{4}^{2}\ \left(\frac{\frac{1}{M_{4}}t(1+\tau_{41}t^{2})}{1+\omega_{41}t^{2}+\omega_{42}t^{4}}\right)^{2}\right)\sqrt{(1-t^{2})(1-U_{4}t^{2})}}\right)dt
36. \frac{\sqrt{2(1-U_{4})(1-x)}}{\sqrt{(1-x^{2})(1-U_{4}x^{2})}}
37. \frac{4\Upsilon_{41}}{1-U_{4}\beta^{2}x^{2}}-p_{4}-\frac{q_{4}}{1-V_{4}\gamma_{4}^{2}\ \left(\frac{\frac{1}{M_{4}}x(1+\tau_{41}x^{2})}{1+\omega_{41}x^{2}+\omega_{42}x^{4}}\right)^{2}}
38. y=\frac{\Upsilon_{41}w_{4}\beta^{2}\left(1+\varsigma_{41}+\varsigma_{42}\right)\left(U_{4}-1\right)}{1+\varpi_{41}+\varpi_{42}+\varpi_{43}+\varpi_{44}}
39. y=\frac{4\Upsilon_{41}}{1-U_{4}\beta^{2}}-p_{4}-\frac{q_{4}}{1-V_{4}\gamma_{4}^{2}}
40. \frac{\sqrt{\frac{2(1-U_{4})\left(\sqrt{U_{4}}x-1\right)}{U_{4}}}}{\sqrt{(1-x^{2})(1-U_{4}x^{2})}}
41. y=\frac{\Upsilon_{41}w_{4}U_{4}\left(U_{4}^{2}+U_{4}\varsigma_{41}+\varsigma_{42}\right)(1-U_{4})}{U_{4}^{4}+U_{4}^{3}\varpi_{41}+U_{4}^{2}\varpi_{42}+U_{4}\varpi_{43}+\varpi_{44}}\beta^{2}
42. y=\frac{4\Upsilon_{41}}{1-\beta^{2}}-p_{4}-\frac{q_{4}}{1-V_{4}\gamma_{4}^{2}}
43. x^{2}=\frac{1}{U_{4}}
44. \frac{\sqrt{U_{4}}x^{2}}{\sqrt{(1-x^{2})(1-U_{4}x^{2})}}
45. y=-\frac{\Upsilon_{41}w_{4}\beta^{2}U_{4}\varsigma_{42}}{\varpi_{44}}
46. y=-p_{4}-q_{4}
47. \frac{f\left(x\right)-\Upsilon_{41}w_{4}\beta^{2}}{x^{2}}
48. y=4\Upsilon_{41}U_{4}\beta^{2}-\frac{(1+U_{4})(p_{4}+q_{4}-4\Upsilon_{41})}{2}-\frac{q_{4}V_{4}\gamma_{4}^{2}}{M_{4}^{2}}
49. \frac{1}{x^{2}}\left(\frac{f\left(x\right)-\Upsilon_{41}w_{4}\beta^{2}}{x^{2}}-\frac{\Upsilon_{41}w_{4}\beta^{2}\left(2a_{4}^{2}-3-3U_{4}+6\varsigma_{41}-6\varpi_{41}\right)}{2}\right)
50. f\left(x\right)=\frac{1}{\sqrt{(1-x^{2})(1-U_{4}x^{2})}}\left(\frac{4\Upsilon_{41}}{1-U_{4}\beta^{2}x^{2}}-p_{4}-\frac{q_{4}}{1-V_{4}\gamma_{4}^{2}\ \left(\frac{\frac{1}{M_{4}}x(1+\tau_{41}x^{2})}{1+\omega_{41}x^{2}+\omega_{42}x^{4}}\right)^{2}}\right)
51. a_{4}=w_{4}\beta\sqrt{(1-\beta^{2})(1-c^{4}\beta^{2})}
52. y=-\frac{q_{4}V_{4}\gamma_{4}^{2}\left(1+U_{4}+4\tau_{41}-4\omega_{41}\right)}{2M_{4}^{2}}-\frac{q_{4}V_{4}^{2}\gamma_{4}^{4}}{M_{4}^{4}}-\frac{(3+2U_{4}+3U_{4}^{2})(p_{4}+q_{4}-4\Upsilon_{41})}{8}+2\Upsilon_{41}U_{4}\beta^{2}\left(1+U_{4}+2U_{4}\beta^{2}\right)
53. y=-\frac{\Upsilon_{41}w_{4}\beta^{2}\left(5+20a_{4}^{2}-8a_{4}^{4}-10U_{4}+20a_{4}^{2}U_{4}+5U_{4}^{2}\right)}{8}+\frac{5\Upsilon_{41}w_{4}\beta^{2}\left(2\varpi_{41}^{2}-2\varpi_{41}\varsigma_{41}-2a_{4}^{2}\varpi_{41}+U_{4}\varpi_{41}+\varpi_{41}-2\varpi_{42}+2a_{4}^{2}\varsigma_{41}-U_{4}\varsigma_{41}-\varsigma_{41}+2\varsigma_{42}\right)}{2}

复制代码

.
【模变换构造的伪椭圆积分】
\begin{gather*}
\int_0^{x}\tfrac{1-\frac{1814418748+230951805\sqrt{2}}{546257732}t^{2}+\frac{1636531702-928528863\sqrt{2}}{1092515464}t^{4}+\frac{281941167\sqrt{2}-337091756}{546257732}t^{6}+\frac{1029008528-225365571\sqrt{2}}{4370061856}t^{8}+\frac{969\sqrt{2}-560}{70208}t^{10}}{\left(1-\frac{t^{2}}{36}\right)\left(1+\frac{16(19612471197\sqrt{2}-6863885278)}{247960177936}t^{2}+\frac{48(18375386402+6537490399\sqrt{2})}{247960177936}t^{4}+\frac{4(19612471197\sqrt{2}-6863885278)}{247960177936}t^{6}+\frac{15497511121t^{8}}{247960177936}\right)\sqrt{(1-t^{2})(1-\frac{1}{4}t^{2})}}\mathrm{d}t\\
\\
=\tfrac{3\sqrt{70}(8721\sqrt{2}-3943)}{307160}\operatorname{artanh}\left(\tfrac{\frac{4\sqrt{70}(3943+8721\sqrt{2})}{373467}x\left(1+\frac{6399\sqrt{2}-6697}{39492}x^{2}+\frac{969\sqrt{2}-560}{157968}x^{4}\right)\sqrt{(1-x^{2})(1-\frac{1}{4}x^{2})}}{1+\frac{295891+678879\sqrt{2}}{746934}x^{2}-\frac{2945268\sqrt{2}-1638757}{161337744}x^{4}+\frac{10141454-1944267\sqrt{2}}{322675488}x^{6}+\frac{x^{8}}{5184}}\right)
\end{gather*}

\begin{align*}
&\qquad\qquad\quad\int_{0}^{x}\left(\tfrac{76}{(1-\frac{1}{36}t^{2})\sqrt{(1-t^{2})(1-\frac{1}{4}t^{2})}}-\tfrac{55}{\sqrt{(1-t^{2})(1-\frac{1}{4}t^{2})}}\right)\mathrm{d}t\\
&\qquad-\int_{0}^{\scriptsize\frac{\frac{3+2\sqrt{2}}{2}x(1+\frac{1}{2}x^{2})}{1+\frac{2+3\sqrt{2}}{2}x^{2}+\frac{1}{4}x^{4}}}
\tfrac{\frac{34(564411-389270\sqrt{2})}{124489}}{\left(1-\frac{77976(136153\sqrt{2}-77976)}{15497511121}\ t^{2}\right)\sqrt{(1-t^{2})(1-24(17\sqrt{2}-24)t^{2})}}\mathrm{d}t\\
\\
&=\tfrac{57\sqrt{70}}{70}\operatorname{artanh}\left(\tfrac{\frac{4\sqrt{70}(3943+8721\sqrt{2})}{373467}x\left(1+\frac{6399\sqrt{2}-6697}{39492}x^{2}+\frac{969\sqrt{2}-560}{157968}x^{4}\right)\sqrt{(1-x^{2})(1-\frac{1}{4}x^{2})}}{1+\frac{295891+678879\sqrt{2}}{746934}x^{2}-\frac{2945268\sqrt{2}-1638757}{161337744}x^{4}+\frac{10141454-1944267\sqrt{2}}{322675488}x^{6}+\frac{x^{8}}{5184}}\right)
\end{align*}

1. -\int_{0}^{x}\frac{\frac{55}{18}}{\sqrt{(1-t^{2})(1-\frac{1}{4}t^{2})}}dt+\int_{0}^{x}\frac{\frac{38}{9}}{(1-\frac{1}{36}t^{2})\sqrt{(1-t^{2})(1-\frac{1}{4}t^{2})}}dt-\int_{0}^{\frac{\frac{3+2\sqrt{2}}{2}x(1+\frac{1}{2}x^{2})}{1+\frac{2+3\sqrt{2}}{2}x^{2}+\frac{1}{4}x^{4}}}\frac{\frac{17(564411-389270\sqrt{2})}{1120401}}{\left(1-\frac{77976(136153\sqrt{2}-77976)}{15497511121}\ t^{2}\right)\sqrt{(1-t^{2})(1-24(17\sqrt{2}-24)t^{2})}}dt
2. \frac{19\sqrt{70}}{420}\operatorname{artanh}\left(\frac{\frac{4\sqrt{70}(3943+8721\sqrt{2})}{373467}x\left(1+\frac{6399\sqrt{2}-6697}{39492}x^{2}+\frac{969\sqrt{2}-560}{157968}x^{4}\right)\sqrt{(1-x^{2})(1-\frac{1}{4}x^{2})}}{1+\frac{295891+678879\sqrt{2}}{746934}x^{2}-\frac{2945268\sqrt{2}-1638757}{161337744}x^{4}+\frac{10141454-1944267\sqrt{2}}{322675488}x^{6}+\frac{x^{8}}{5184}}\right)
3. \frac{19\sqrt{3}\left(5273+1970\sqrt{2}\right)\sqrt{x-2}}{2\left(252507+155342\sqrt{2}\right)}
4. -\int_{0}^{x}\frac{p_{4}}{\sqrt{(1-t^{2})(1-U_{4}t^{2})}}dt+\int_{0}^{x}\frac{4\Upsilon_{41}}{(1-U_{4}\beta^{2}t^{2})\sqrt{(1-t^{2})(1-U_{4}t^{2})}}dt-\int_{0}^{\frac{\frac{1}{M_{4}}x(1+\tau_{41}x^{2})}{1+\omega_{41}x^{2}+\omega_{42}x^{4}}}\frac{M_{4}q_{4}}{\left(1-V_{4}\gamma_{4}^{2}\ t^{2}\right)\sqrt{(1-t^{2})(1-V_{4}t^{2})}}dt
5. U_{4}=\frac{1}{4}
6. V_{4}=24(17\sqrt{2}-24)
7. \beta=\frac{1}{3}
8. p_{4}=\frac{55}{18}
9. \Upsilon_{41}=\frac{19}{18}
10. M_{4}=6-4\sqrt{2}
11. \tau_{41}=\frac{1}{2}
12. \omega_{41}=\frac{2+3\sqrt{2}}{2}
13. \omega_{42}=\frac{1}{4}
14. \gamma_{4}=\frac{57\left(867+560\sqrt{2}\right)}{124489}
15. q_{4}=\frac{17\left(136153-38988\sqrt{2}\right)}{2240802}
16. A\operatorname{artanh}\left(\frac{w_{4}r_{40}x\left(1+\varsigma_{41}x^{2}+\varsigma_{42}x^{4}\right)\sqrt{(1-x^{2})(1-U_{4}x^{2})}}{1+\varpi_{41}x^{2}+\varpi_{42}x^{4}+\varpi_{43}x^{6}+\varpi_{44}x^{8}}\right)
17. A=\frac{19\sqrt{70}}{420}
18. w_{4}=\frac{36\left(3943+8721\sqrt{2}\right)}{124489}
19. r_{40}=\frac{\sqrt{70}}{27}
20. \varsigma_{41}=\frac{6399\sqrt{2}-6697}{39492}
21. \varsigma_{42}=\frac{969\sqrt{2}-560}{157968}
22. \varpi_{41}=\frac{295891+678879\sqrt{2}}{746934}
23. \varpi_{42}=-\frac{2945268\sqrt{2}-1638757}{161337744}
24. \varpi_{43}=\frac{10141454-1944267\sqrt{2}}{322675488}
25. \varpi_{44}=\frac{1}{5184}

复制代码

青青子衿

1. s=0.687
2. \beta=0.531
3. \varpi_{31}=\frac{s(2+s)\left(1+2s-2(1+2s+s^{2}+2s^{3}+s^{4})\beta^{2}+s^{3}(2+s)\beta^{4}\right)}{(1+2s)\left(1+s(2+s)\beta^{2}\right)}
4. \varpi_{32}=\frac{\beta^{2}s^{4}(2+s)^{2}(1+2s+s^{2}\beta^{2})}{(1+2s)^{2}\left(1+s(2+s)\beta^{2}\right)}
5. y_{3}=\frac{(1+2s)x\left(1+\frac{s^{2}}{1+2s}x^{2}\right)}{1+s(2+s)x^{2}}
6. \gamma_{3}=\frac{(1+2s)\beta\left(1+\frac{s^{2}}{1+2s}\beta^{2}\right)}{1+s(2+s)\beta^{2}}
7. \int_{0}^{x}\left(\frac{3\left(1+\frac{s^{2}}{1+2s}\beta^{2}\right)}{\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}{\sqrt{\left(1-t^{2}\right)\left(1-\frac{s^{3}(2+s)}{1+2s}t^{2}\right)}}\right)dt-\int_{0}^{y_{3}}\frac{\frac{(1-s^{2}\beta^{2})\left(1-\frac{s(2+s)}{1+2s}\beta^{2}\right)}{(1+2s)(1+s(2+s)\beta^{2})}}{\left(1-s\frac{(2+s)^{3}}{(1+2s)^{3}}\gamma_{3}^{2}t^{2}\right)\sqrt{\left(1-t^{2}\right)\left(1-s\frac{(2+s)^{3}}{(1+2s)^{3}}t^{2}\right)}}dt
8. \frac{\left(1+\frac{s^{2}}{1+2s}\beta^{2}\right)\beta}{\sqrt{\left(1-\beta^{2}\right)\left(1-\frac{s^{3}(2+s)}{1+2s}\beta^{2}\right)}}\operatorname{artanh}\left(\frac{\frac{2s(2+s)}{1+s(2+s)\beta^{2}}\ x\ }{1+\varpi_{31}x^{2}+\varpi_{32}x^{4}}\beta\sqrt{\left(1-\beta^{2}\right)\left(1-\frac{s^{3}(2+s)}{1+2s}\beta^{2}\right)\left(1-x^{2}\right)\left(1-\frac{s^{3}(2+s)}{1+2s}x^{2}\right)}\right)

复制代码

\begin{align*}
\Omega_4(x)&=\tfrac{\varUpsilon_{4,1}\beta}{\sqrt{(1-\beta^{2})(1-U_4\beta^{2})}}\operatorname{artanh}\left(\varrho_4(x)R_4(x)\right)\\
\varrho_4(x)&=\tfrac{w_4x(1+\varsigma_{4,1}x^2+\varsigma_{4,2}x^4)}{1+\varpi_{4,1}x^2+\varpi_{4,2}x^4+\varpi_{4,3}x^6+\varpi_{4,4}x^8}\\
R_4(x)&={\scriptsize\beta\sqrt{(1-\beta^{2})(1-U_4\beta^{2})(1-x^{2})(1-U_4x^{2})}}\\
a_4&=w_4R_4(0)=w_4{\scriptsize\beta\sqrt{(1-\beta^{2})(1-U_4\beta^{2})}}\\
d_4&={\small\tfrac{1}{\sqrt{U_4}}}\\
\\

L_{4,1}&=\lim_{\>x\to1^{-}}\sqrt{\scriptsize2(1-U_4)(1-x)}\,\Omega_4'(x)\\&={\small-\tfrac{\varUpsilon_{4,1}w_{4}\left(1+\varsigma_{4,1}+\varsigma_{4,2}\right)\left(1-U_4\right)}{1+\varpi_{4,1}+\varpi_{4,2}+\varpi_{4,3}+\varpi_{4,4}}\,\beta^{2}}\\
&={\small\tfrac{4\varUpsilon_{4,1}}{1-U_4\beta^{2}}-p_{4}-\tfrac{q_{4}}{1-V_4\gamma_{4}^{2}}}\\

L_{4,2}&={\kern-2ex\lim_{\quad\,x\to(d_{4})^{+}}\kern-.3ex}\sqrt{\tfrac{2(1-U_4)(\sqrt{\scriptsize\,\!U_4}\>\!x-1)}{U_4}}\,\Omega_4'(x)\\
&={\small\tfrac{\varUpsilon_{4,1}w_{4}U_4\left(U_4^{2}+U_4\varsigma_{4,1}+\varsigma_{4,2}\right)(1-U_4)}{U_4^{4}+U_4^{3}\varpi_{4,1}+U_4^{2}\varpi_{4,2}+U_4\varpi_{4,3}+\varpi_{4,4}}\beta^{2}}\\
&={\small\tfrac{4\varUpsilon_{4,1}}{1-\beta^{2}}-p_{4}-\tfrac{q_{4}}{1-V_4\gamma_{4}^{2}}}\\

L_{4,3}&={\small\varrho_4(\tfrac{1}{  \sqrt{U_4}\beta\,})R_4(\tfrac{1}{  \sqrt{U_4}\beta\,})}\\
&={\small\tfrac{w_{4}U_4(1-\beta^{2})(1-U_4\beta^{2})\left(U_4^{2}\beta^{4}+U_4\beta^{2}\varsigma_{4,1}+\varsigma_{4,2}\right)}{U_4^{4}\beta^{8}+U_4^{3}\beta^{6}\varpi_{4,1}+U_4^{2}\beta^{4}\varpi_{4,2}+U_4\beta^{2}\varpi_{4,3}+\varpi_{4,4}}\beta^{2}}={\small1}\\

L_{4,4}&=\lim_{\>x\to+\infty}\sqrt{\small\,\!U_4}\>\!x^2\,\Omega_4'(x)=-{\small\tfrac{\varUpsilon_{4,1}w_{4}U_4\beta^{2}\varsigma_{4,2}}{\varpi_{4,4}}}\\
&={\small-\,p_{4}-q_{4}}\\

L_{4,5}&=\lim_{\>x\to0}\tfrac{\Omega_4'(x)-\varUpsilon_{4,1}w_{4}\beta^{2}}{x^{2}}\\

&={\small\tfrac{\varUpsilon_{4,1}w_{4}\beta^{2}(2a_4^2-3-3U_4+6\varsigma_{4,1}-6\varpi_{4,1})}{2}}\\
&={\scriptsize4\varUpsilon_{4,1}U_{4}\beta^{2}}-{\small\tfrac{(1+U_{4})(p_{4}+q_{4}-4\varUpsilon_{4,1})}{2}}-{\small\tfrac{q_{4}V_4\gamma_{4}^{2}}{M_4^{2}}}\\

L_{4,6}&=\lim_{\>x\to0}\tfrac{1}{x^{2}}\left(\tfrac{\Omega_4'(x)-\varUpsilon_{4,1}w_{4}\beta^{2}}{x^{2}}-{\scriptsize\,\!L_{4,5}}\right)\\
&=-\tfrac{\varUpsilon_{4,1}w_{4}\beta^{2}\left(5+20a_4^{2}-8a_4^{4}-10U_4+20a_4^{2}U_4+5U_4^{2}\right)}{8}\\
&\qquad+\tfrac{5\varUpsilon_{4,1}w_{4}\beta^{2}\left(2\varpi_{4,1}^{2}-2\varpi_{4,1}\varsigma_{4,1}-2a_4^{2}\varpi_{4,1}+U_4\varpi_{4,1}\right)}{2}\\
&\qquad\>\>\>\>+\tfrac{5\varUpsilon_{4,1}w_{4}\beta^{2}\left(\varpi_{4,1}-2\varpi_{4,2}+2a_4^{2}\varsigma_{4,1}-U_4\varsigma_{4,1}-\varsigma_{4,1}+2\varsigma_{4,2}\right)}{2}\\

&=-{\small\tfrac{q_4V_4\gamma_4^{2}\left(1+U_4+4\tau_{4,1}-4 \omega_{4,1}\right)}{2M_4^{2}}}\\
&\qquad\,-{\small\tfrac{q_4V_4^{2}\gamma_4^{4}}{M_4^{4}}-\tfrac{(3+2U_4+3U_4^{2})(p_4+q_4-4 \varUpsilon _{41})}{8}}\\
&\qquad\>\>\>\>+{\scriptsize2\varUpsilon _{41} U_4\beta ^2(1+U_4+2 U_4\beta ^2)}\\

\end{align*}

1. {-((Subscript[\[CurlyCapitalUpsilon], 41] Subscript[w,
2.       4] (1 + Subscript[\[Stigma], 41] + Subscript[\[Stigma],
3.         42]) (1 - Subscript[U, 4]))/(
4.      1 + Subscript[\[CurlyPi], 41] + Subscript[\[CurlyPi], 42] +
5.       Subscript[\[CurlyPi], 43] + Subscript[\[CurlyPi],
6.       44])) \[Beta]^2,
7.    (4 Subscript[\[CurlyCapitalUpsilon], 41])/(
8.     1 - Subscript[U, 4] \[Beta]^2) - Subscript[p, 4] - Subscript[q,
9.     4]/(1 - Subscript[V, 4] Subscript[\[Gamma], 4]^2),
10.    (Subscript[\[CurlyCapitalUpsilon], 41] Subscript[w, 4] Subscript[U,
11.       4] (Subscript[U, 4]^2 +
12.        Subscript[U, 4]*Subscript[\[Stigma], 41] + Subscript[\[Stigma],
13.         42]) (1 - Subscript[U, 4]))/(
14.     Subscript[U, 4]^4 + Subscript[U, 4]^3 Subscript[\[CurlyPi], 41] +
15.      Subscript[U, 4]^2 Subscript[\[CurlyPi], 42] +
16.      Subscript[U, 4]*Subscript[\[CurlyPi], 43] + Subscript[\[CurlyPi],
17.       44]) \[Beta]^2,
18.    (4 Subscript[\[CurlyCapitalUpsilon], 41])/(1 - \[Beta]^2) -
19.     Subscript[p, 4] - Subscript[q, 4]/(
20.     1 - Subscript[V, 4] Subscript[\[Gamma], 4]^2),
21.    (Subscript[w, 4] Subscript[U,
22.      4] (1 - \[Beta]^2) (1 -
23.        Subscript[U, 4] \[Beta]^2) (Subscript[U, 4]^2 \[Beta]^4 +
24.        Subscript[U, 4] \[Beta]^2*Subscript[\[Stigma], 41] +
25.        Subscript[\[Stigma], 42]))/(
26.     Subscript[U, 4]^4 \[Beta]^8 +
27.      Subscript[U, 4]^3 \[Beta]^6 Subscript[\[CurlyPi], 41] +
28.      Subscript[U, 4]^2 \[Beta]^4 Subscript[\[CurlyPi], 42] +
29.      Subscript[U, 4] \[Beta]^2*Subscript[\[CurlyPi], 43] +
30.      Subscript[\[CurlyPi], 44]) \[Beta]^2,
31.    -((Subscript[\[CurlyCapitalUpsilon], 41] Subscript[w, 4] Subscript[
32.      U, 4] \[Beta]^2 Subscript[\[Stigma], 42])/Subscript[\[CurlyPi],
33.     44]),
34.    -Subscript[p, 4] - Subscript[q, 4],
35.    (Subscript[\[CurlyCapitalUpsilon], 41] Subscript[w,
36.     4] \[Beta]^2 (2 Subscript[a, 4]^2 - 3 - 3 Subscript[U, 4] +
37.       6 Subscript[\[Stigma], 41] - 6 Subscript[\[CurlyPi], 41]))/2,
38.    2 Subscript[\[CurlyCapitalUpsilon],
39.      41] (1 + Subscript[U, 4] + 2 Subscript[U, 4] \[Beta]^2) - ((1 +
40.        Subscript[U, 4]) (Subscript[p, 4] + Subscript[q, 4]))/2 - (
41.     Subscript[q, 4] Subscript[V, 4] Subscript[\[Gamma], 4]^2)/
42.     Subscript[M, 4]^2,
43.    (-((Subscript[\[CurlyCapitalUpsilon], 41] Subscript[w,
44.        4] \[Beta]^2 (5 + 20 Subscript[a, 4]^2 - 8 Subscript[a, 4]^4 -
45.          10 Subscript[U, 4] + 20 Subscript[a, 4]^2 Subscript[U, 4] +
46.          5 Subscript[U, 4]^2))/8)
47.      + (5 Subscript[\[CurlyCapitalUpsilon], 41] Subscript[w,
48.       4] \[Beta]^2 (2 Subscript[\[CurlyPi], 41]^2 -
49.         2 Subscript[\[CurlyPi], 41] Subscript[\[Stigma], 41] -
50.         2 Subscript[a, 4]^2 Subscript[\[CurlyPi], 41] +
51.         Subscript[U, 4] Subscript[\[CurlyPi], 41]))/2
52.      + (5 Subscript[\[CurlyCapitalUpsilon], 41] Subscript[w,
53.       4] \[Beta]^2 (Subscript[\[CurlyPi], 41] -
54.         2 Subscript[\[CurlyPi], 42] +
55.         2 Subscript[a, 4]^2 Subscript[\[Stigma], 41] -
56.         Subscript[U, 4] Subscript[\[Stigma], 41] -
57.         Subscript[\[Stigma], 41] + 2 Subscript[\[Stigma], 42]))/2),
58.    (-((Subscript[q, 4] Subscript[V, 4] Subscript[\[Gamma],
59.        4]^2 (1 + Subscript[U, 4] + 4 Subscript[\[Tau], 41] -
60.          4 Subscript[\[Omega], 41]))/(2 Subscript[M, 4]^2))
61.      - (Subscript[q, 4] Subscript[V, 4]^2 Subscript[\[Gamma], 4]^4)/
62.      Subscript[M, 4]^4 - ((3 + 2 Subscript[U, 4] + 3
63. \!\(\*SubsuperscriptBox[\(U\), \(4\), \(2\)]\)) (Subscript[p, 4] +
64.         Subscript[q, 4] - 4 Subscript[\[CurlyCapitalUpsilon], 41]))/
65.      8
66.      + 2 Subscript[\[CurlyCapitalUpsilon], 41] Subscript[U,
67.       4] \[Beta]^2 (1 + Subscript[U, 4] +
68.         2 Subscript[U, 4] \[Beta]^2 ))} /. {
69.    Subscript[a, 4] -> (4 Sqrt[70] (3943 + 8721 Sqrt[2]))/373467,
70.    Subscript[\[CurlyCapitalUpsilon], 41] -> 19/18,
71.    Subscript[w, 4] -> (36 (3943 + 8721 Sqrt[2]))/124489,
72.    Subscript[\[Stigma], 41] -> (6399 Sqrt[2] - 6697)/39492,
73.    Subscript[\[Stigma], 42] -> (969 Sqrt[2] - 560)/157968,
74.    Subscript[\[CurlyPi], 41] -> (295891 + 678879 Sqrt[2])/746934,
75.    Subscript[\[CurlyPi],
76.     42] -> -((2945268 Sqrt[2] - 1638757)/161337744),
77.    Subscript[\[CurlyPi], 43] -> (10141454 - 1944267 Sqrt[2])/
78.     322675488, Subscript[\[CurlyPi], 44] -> 1/5184,
79.    Subscript[U, 4] -> 1/4, Subscript[V, 4] -> 24 (17 Sqrt[2] - 24),
80.    \[Beta] -> 1/3, Subscript[p, 4] -> 55/18,
81.    Subscript[q, 4] -> (17 (136153 - 38988 Sqrt[2]))/2240802,
82.    Subscript[\[Gamma], 4] -> (57 (867 + 560 Sqrt[2]))/124489,
83.    Subscript[M, 4] -> 6 - 4 Sqrt[2],
84.    Subscript[\[Tau], 41] -> 1/2,
85.    Subscript[\[Omega], 41] -> (2 + 3 Sqrt[2])/2} // N

复制代码

\begin{align*}
{T\kern-.4ex\hbox{R}\kern-.4ex}
\end{align*}

青青子衿

2. U_{5}=\frac{3-\sqrt{6}}{6}
3. V_{5}=\frac{27+11\sqrt{6}}{54}
4. \Upsilon_{5}=\frac{19+8\sqrt{6}}{27}
5. \beta=\frac{\sqrt{6}}{3}
6. \gamma_{5}=\frac{132+35\sqrt{6}}{219}
7. p_{5}=\frac{8\left(3+\sqrt{6}\right)}{9}
8. q_{5}=\frac{11\left(28\sqrt{6}-55\right)}{1971}
9. \tau_{51}=\frac{2\sqrt{6}-3}{3}
10. \tau_{52}=\frac{5-2\sqrt{6}}{6}
11. \omega_{51}=1+\sqrt{6}
12. \omega_{52}=\frac{1}{2}
13. M_{5}=\frac{1}{3}
14. \int_{0}^{x}\left(\frac{5\Upsilon_{5}}{(1-U_{5}\beta^{2}t^{2})\sqrt{(1-t^{2})(1-U_{5}t^{2})}}-\frac{p_{5}}{\sqrt{(1-t^{2})(1-U_{5}t^{2})}}-\frac{q_{5}}{(1-V_{5}\gamma_{5}^{2}\left(\frac{\frac{1}{M_{5}}t(1+\tau_{51}t^{2}+\tau_{52}t^{4})}{1+\omega_{51}t^{2}+\omega_{52}t^{4}}\right)^{2})\sqrt{(1-t^{2})(1-U_{5}t^{2})}}\right)dt
15. \int_{0}^{x}\left(\frac{\frac{5\left(19+8\sqrt{6}\right)}{27}}{(1-\frac{3-\sqrt{6}}{9}t^{2})\sqrt{(1-t^{2})(1-\frac{3-\sqrt{6}}{6}t^{2})}}-\frac{\frac{8\left(3+\sqrt{6}\right)}{9}}{\sqrt{(1-t^{2})(1-\frac{3-\sqrt{6}}{6}t^{2})}}-\frac{\frac{11\left(28\sqrt{6}-55\right)}{1971}}{(1-\frac{213123+86999\sqrt{6}}{431649}\left(\frac{t(6+(4\sqrt{6}-6)t^{2}+(5-2\sqrt{6})t^{4})}{2+2(1+\sqrt{6})t^{2}+t^{4}}\right)^{2})\sqrt{(1-t^{2})(1-\frac{3-\sqrt{6}}{6}t^{2})}}\right)dt
16. f\left(x\right)=\frac{1}{\sqrt{(1-x^{2})(1-U_{5}x^{2})}}\left(\frac{5\Upsilon_{5}}{1-U_{5}\beta^{2}x^{2}}-p_{5}-\frac{q_{5}}{1-V_{5}\gamma_{5}^{2}\ \left(\frac{\frac{1}{M_{5}}x(1+\tau_{51}x^{2}+\tau_{52}x^{4})}{1+\omega_{51}x^{2}+\omega_{52}x^{4}}\right)^{2}}\right)
17. \frac{f\left(x\right)-\left(5\Upsilon_{5}-p_{5}-q_{5}\right)}{x^{2}}
18. y=5\Upsilon_{5}U_{5}\beta^{2}-\frac{(1+U_{5})(p_{5}+q_{5}-5\Upsilon_{5})}{2}-\frac{q_{5}V_{5}\gamma_{5}^{2}}{M_{5}^{2}}
19. \frac{1}{x^{2}}\left(\frac{f\left(x\right)-\left(5\Upsilon_{5}-p_{5}-q_{5}\right)}{x^{2}}-\left(5\Upsilon_{5}U_{5}\beta^{2}-\frac{(1+U_{5})(p_{5}+q_{5}-5\Upsilon_{5})}{2}-\frac{q_{5}V_{5}\gamma_{5}^{2}}{M_{5}^{2}}\right)\right)
20. y=-\frac{q_{5}V_{5}\gamma_{5}^{2}(1+U_{5}+4\tau_{51}-4\omega_{51})}{2M_{5}^{2}}-\frac{q_{5}V_{5}^{2}\gamma_{5}^{4}}{M_{5}^{4}}-\frac{(3+2U_{5}+3U_{5}^{2})(p_{5}+q_{5}-5\Upsilon_{5})}{8}+\frac{5\Upsilon_{5}U_{5}\beta^{2}\left(1+U_{5}+2U_{5}\beta^{2}\right)}{2}
21. \sqrt{2\left(1-U_{5}\right)\left(1-x\right)}f\left(x\right)
22. y=\frac{5\Upsilon_{5}}{1-U_{5}\beta^{2}}-p_{5}-\frac{q_{5}}{1-V_{5}\gamma_{5}^{2}\ }
23. \frac{5\Upsilon_{5}}{1-U_{5}\beta^{2}x^{2}}-p_{5}-\frac{q_{5}}{1-V_{5}\gamma_{5}^{2}\ \left(\frac{\frac{1}{M_{5}}x(1+\tau_{51}x^{2}+\tau_{52}x^{4})}{1+\omega_{51}x^{2}+\omega_{52}x^{4}}\right)^{2}}
24. y=\frac{5\Upsilon_{5}}{1-\beta^{2}}-p_{5}-\frac{q_{5}}{1-V_{5}\gamma_{5}^{2}\ \left(\frac{\frac{1}{M_{5}}\left(\frac{1}{\sqrt{U_{5}}}\right)(1+\tau_{51}\left(\frac{1}{\sqrt{U_{5}}}\right)^{2}+\tau_{52}\left(\frac{1}{\sqrt{U_{5}}}\right)^{4})}{1+\omega_{51}\left(\frac{1}{\sqrt{U_{5}}}\right)^{2}+\omega_{52}\left(\frac{1}{\sqrt{U_{5}}}\right)^{4}}\right)^{2}}
25. x^{2}=\frac{1}{U_{5}}
26. \frac{\frac{1}{M_{5}}\left(\frac{1}{\sqrt{U_{5}}}\right)(1+\tau_{51}\left(\frac{1}{\sqrt{U_{5}}}\right)^{2}+\tau_{52}\left(\frac{1}{\sqrt{U_{5}}}\right)^{4})}{1+\omega_{51}\left(\frac{1}{\sqrt{U_{5}}}\right)^{2}+\omega_{52}\left(\frac{1}{\sqrt{U_{5}}}\right)^{4}}
27. 3\sqrt{54-22\sqrt{6}}
28. x^{2}=\frac{1}{U_{5}\beta^{2}}
29. y=-p_{5}
30. w_{5}=\frac{6\left(21\sqrt{6}-23\right)}{73}
31. \frac{5\Upsilon_{5}-p_{5}-q_{5}}{\Upsilon_{5}\beta^{2}}

复制代码

\begin{gather*}
{\large\int}_{0}^{x}\tfrac{\frac{2\sqrt{26508-5242\sqrt{6}}}{219}\left(1-\frac{2(9102-1001\sqrt{6})}{6351}t^{2}+\frac{36050-10117\sqrt{6}}{12702}t^{4}-\frac{55926-19073\sqrt{6}}{38106}t^{6}-\frac{8443\sqrt{6}-20065}{76212}t^{8}+\frac{221-5\sqrt{6}}{38106}t^{10}\right)}{\left(1-\frac{3-\sqrt{6}}{9}t^{2}\right)\left(1-\frac{2(16869-599\sqrt{6})}{15987}t^{2}+\frac{8(3242-75\sqrt{6})}{15987}t^{4}-\frac{17571-967\sqrt{6}}{47961}t^{6}+\frac{4129+1540\sqrt{6}}{191844}t^{8}\right)\sqrt{(1-t^{2})(1-\frac{3-\sqrt{6}}{6}t^{2})}}\mathrm{d}t\\
\\
=
\operatorname{arctanh}\left(\tfrac{\frac{2\sqrt{26508-5242\sqrt{6}}}{219}x\left(1-\frac{129-31\sqrt{6}}{261}x^{2}+\frac{2(3\sqrt{6}-5)}{261}x^{4}\right)\sqrt{(1-x^{2})(1-\frac{3-\sqrt{6}}{6}x^{2})}}{1-\frac{3(15-\sqrt{6})}{73}x^{2}+\frac{5(202\sqrt{6}-475)}{3942}x^{4}-\frac{4(296\sqrt{6}-717)}{5913}x^{6}+\frac{40\sqrt{6}-89}{11826}x^{8}}\right)\quad
\end{gather*}

1. D[ArcTanh[((2 Sqrt[26508 - 5242 Sqrt[6]])/
2.      219 t (1 - (129 - 31 Sqrt[6])/261 t^2 + (2 (3 Sqrt[6] - 5))/
3.         261 t^4) Sqrt[(1 - t^2) (1 - (3 - Sqrt[6])/6 t^2)])/(
4.     1 - (3 (15 - Sqrt[6]))/73 t^2 + (5 (202 Sqrt[6] - 475))/
5.       3942 t^4 - (4 (296 Sqrt[6] - 717))/5913 t^6 + (40 Sqrt[6] - 89)/
6.       11826 t^8)],
7.    t]^2 - (((2 Sqrt[26508 - 5242 Sqrt[6]])/
8.     219 (1 - (2 (9102 - 1001 Sqrt[6]))/6351 t^2 + (
9.        36050 - 10117 Sqrt[6])/12702 t^4 - (55926 - 19073 Sqrt[6])/
10.        38106 t^6 - (8443 Sqrt[6] - 20065)/76212 t^8 + (
11.        221 - 5 Sqrt[6] )/38106 t^10))/((1 - (3 - Sqrt[6])/
12.        9 t^2) (1 - (2 (16869 - 599 Sqrt[6]))/15987 t^2 + (
13.        8 (3242 - 75 Sqrt[6]) )/15987 t^4 - (17571 - 967 Sqrt[6])/
14.        47961 t^6 + (4129 + 1540 Sqrt[6])/191844 t^8) Sqrt[(1 -
15.        t^2) (1 - (3 - Sqrt[6])/6 t^2)] ))^2 // Factor

复制代码

\begin{gather*}
\scriptsize{
\begin{split}
P_{7,1}&=\tfrac{4(1-\xi+\xi^{2})\delta_{7}-2(4-10\xi+17\xi^{2}-18\xi^{3}+10\xi^{4}-4\xi^{5})}{3}\\
P_{7,2}&=\tfrac{(2-\xi+2\xi^{2})(4-14\xi+28\xi^{2}-35\xi^{3}+28\xi^{4}-14\xi^{5}+4\xi^{6})-2(1-\xi)(1-\xi+\xi^{2})(2-\xi+2\xi^{2})\delta_{7}}{3}\\
\end{split}
}\\
\\
\scriptsize{
\begin{split}
Q_{7,1}&=2 (1 - \xi) (3 - 2 \xi + 2 \xi^2) \delta_{7} - 2 (6 - 18 \xi )\\
&\qquad- 2(35 \xi^2 - 40 \xi^3 + 30 \xi^4 - 14 \xi^5 + 4 \xi^6)\\

Q_{7,2}&=40 - 104 \xi + 26 \xi^2 + 568 \xi^3 - 1861 \xi^4\\
&\qquad+ 3416 \xi^5- 4294 \xi^6 + 3912 \xi^7 - 2600 \xi^8\\
&\qquad\>\>+ 1232 \xi^9 - 384 \xi^{10}+ 64 \xi^{11} \\
&\qquad\quad-2 (1 - \xi) (10 - \xi - 31 \xi^2 + 95 \xi^3- 130 \xi^4)\delta_{7}\\
&\qquad\quad\>\>\>-2 (1 - \xi) (112 \xi^5 - 56 \xi^6 + 16 \xi^7) \delta_{7}\\

Q_{7,3}&=4 (1 - \xi) (4 + 32 \xi - 178 \xi^2 + 522 \xi^3 - 1025 \xi^4)\delta_{7}\\
&\qquad+4 (1 - \xi)(1516 \xi^5 - 1736 \xi^6 + 1552 \xi^7 - 1056 \xi^8)\delta_{7}\\
&\qquad\>\>\>+4 (1 - \xi)(528 \xi^9 - 176 \xi^{10} + 32 \xi^{11}) \delta_{7}\\
&\qquad\quad- 4 (8 + 44 \xi - 488 \xi^2 + 2136 \xi^3 - 6076 \xi^4)\\
&\qquad\quad\>\>\>-4(12838 \xi^5 - 21363 \xi^6 + 28832 \xi^7 - 31972 \xi^8)\\
&\qquad\qquad-4(29190 \xi^9 - 21792 \xi^{10} + 13096 \xi^{11})\\
&\qquad\quad+4(6160 \xi^{12}-2160 \xi^{13}+ 512 \xi^{14}-64 \xi^{15})\\

Q_{7,4}&=
\xi (320 - 1952 \xi + 5360 \xi^2 - 4816 \xi^3 - 19796 \xi^4)\\
&\qquad+\xi(102452 \xi^5 - 269276 \xi^6 + 503335 \xi^7 - 729820 \xi^8)\\
&\qquad\>\>\>+\xi (850228 \xi^9 - 807988 \xi^{10} + 628880 \xi^{11} )\\
&\qquad\quad-\xi(399024 \xi^{12}- 203552 \xi^{13}+81280 \xi^{14})\\
&\qquad\quad\>\>\>+\xi(24192 \xi^{15}- 4864 \xi^{16} + 512 \xi^{17}) \\
&\qquad\qquad- 4 \xi(1 - \xi)  (40 - 144 \xi+ 170 \xi^2 + 437 \xi^3)\delta_{7}\\
&\qquad\quad\>\>\>+4 \xi(1 - \xi)  (2403 \xi^4- 5845 \xi^5+9571 \xi^6 )\delta_{7}\\
&\qquad\quad-4 \xi(1 - \xi)(11623 \xi^7- 10806 \xi^8 + 7744 \xi^9)\delta_{7}\\
&\qquad\>\>+4 \xi(1 - \xi)(4216 \xi^{10} - 1680 \xi^{11} + 448 \xi^{12} - 64 \xi^{13}) \delta_{7}\\

Q_{7,5}&=
2 \xi^2 (1 - \xi) (192 - 1456 \xi + 5992 \xi^2 - 16832 \xi^3)\delta_{7}\\
&\qquad+2 \xi^2 (1 - \xi) (35392 \xi^4 - 58268 \xi^5 + 76975 \xi^6 )\delta_{7}\\
&\qquad\>\>\>-2 \xi^2 (1 - \xi) (82614 \xi^7-72338 \xi^8 + 51496 \xi^9 )\delta_{7}\\
&\qquad\quad+2 \xi^2 (1 - \xi) (29456 \xi^{10}- 13216 \xi^{11}+4448 \xi^{12}) \delta_{7}\\
&\qquad\quad\>\>\>-2 \xi^2 (1 - \xi) (1024 \xi^{13}-128 \xi^{14})\delta_{7}\\
&\qquad\qquad- 2 \xi^2 (384 - 3872 \xi + 20608 \xi^2 - 74872 \xi^3)\\
&\qquad\quad\>\>\>- 2 \xi^2(205184 \xi^4- 446460 \xi^5 + 794682 \xi^6)\\
&\qquad\quad+ 2 \xi^2(1178478 \xi^7- 1472331 \xi^8+1559180 \xi^9) \\
&\qquad\>\>\>- 2 \xi^2(1402450 \xi^{10} - 1069642 \xi^{11}+ 687764 \xi^{12})\\
&\qquad+ 2 \xi^2(368784 \xi^{13} -161952 \xi^{14}+56544 \xi^{15})\\
&\quad\>\>- 2 \xi^2(14912 \xi^{16} - 2688 \xi^{17} + 256 \xi^{18})\\

Q_{7,6}&=
\xi^3Q_{7,6,1}Q_{7,6,2}- 2 \xi^3 (1 - \xi)  (1
     - \xi + \xi^2)Q_{7,6,3}Q_{7,6,4} \delta_{7}
\end{split}
}\\
\\
\scriptsize{
\begin{split}
Q_{7,6,1}&=4 - 14 \xi + 28 \xi^2 - 35 \xi^3 + 28 \xi^4 - 14 \xi^5 + 4 \xi^6\\
Q_{7,6,2}&=64 - 448 \xi+ 1680 \xi^2 - 4256 \xi^3 + 7952 \xi^4\\
&\qquad- 11424 \xi^5 + 12865 \xi^6 - 11424 \xi^7 + 7952 \xi^8\\
&\qquad\quad- 4256 \xi^9 + 1680 \xi^{10} - 448 \xi^{11} + 64 \xi^{12}\\
Q_{7,6,3}&=8 - 28 \xi + 56 \xi^2 - 71 \xi^3 + 56 \xi^4 - 28 \xi^5 + 8 \xi^6\\
Q_{7,6,4}&=8 - 28 \xi + 56 \xi^2 - 69 \xi^3 + 56 \xi^4 - 28 \xi^5 + 8 \xi^6
\end{split}}\\
\\
\end{gather*}

青青子衿

\begin{gather*}
\int_{0}^{x}\left(\tfrac{\frac{1008+379\sqrt{7}}{216}}{(1-\frac{8-3\sqrt{7}}{24}t^{2})\sqrt{(1-t^{2})(1-\frac{8-3\sqrt{7}}{16}t^{2})}}-\tfrac{\frac{2(17+6\sqrt{7})}{9}}{\sqrt{(1-t^{2})(1-\frac{8-3\sqrt{7}}{16}t^{2})}}\right)\mathrm{d}t\\
-\int_{0}^{\Tiny\frac{\sqrt{7}x\Big(1+{\tiny\frac{3\sqrt{7}-6}{4}}x^{2}+{\tiny\frac{8-3\sqrt{7}}{4}}x^{4}+{\tiny\frac{127\sqrt{7}-336}{448}}x^{6}\Big)}{1+\sqrt{7}x^{2}+{\tiny\frac{21-6\sqrt{7}}{16}}x^{4}+{\tiny\frac{8\sqrt{7}-21}{64}}x^{6}}}
\tfrac{\frac{811055-63624\sqrt{7}}{13379688}}{(1-\frac{914151008+345479643\sqrt{7}}{1879315224}t^{2})\sqrt{(1-t^{2})(1-\frac{8+3\sqrt{7}}{16}t^{2})}}\mathrm{d}t\\
\\
\quad={\raise2px\tiny\tfrac{\sqrt{9437394288+3566854809\sqrt{7}}}{72954}\operatorname{artanh}\left(\tfrac{\frac{\sqrt{2393411160-787082247\sqrt{7}}}{17698}x\left(1-\frac{6932-2121\sqrt{7}}{16452}x^{2}+\frac{64320\sqrt{7}-170045}{263232}x^{4}-\frac{48499\sqrt{7}-128315}{263232}x^{6}+\frac{127139-48054\sqrt{7}}{3158784}x^{8}\right)\sqrt{(1-x^{2})(1-\frac{8-3\sqrt{7}}{16}x^{2})}}{1-\frac{10395\sqrt{7}-22739}{17698}x^{2}-\frac{10595321-3952008\sqrt{7}}{1699008}x^{4}+\frac{322797878-121974345\sqrt{7}}{61164288}x^{6}-\frac{24402131-9222568\sqrt{7}}{163104768}x^{8}+\frac{156525261\sqrt{7}-414126907}{489314304}x^{10}+\frac{15220468\sqrt{7}-40269573}{7829028864}x^{12}}\right)}
\end{gather*}

1. \left(\int_{0}^{x}\frac{\frac{1008+379\sqrt{7}}{216}}{(1-\frac{8-3\sqrt{7}}{24}t^{2})\sqrt{(1-t^{2})(1-\frac{8-3\sqrt{7}}{16}t^{2})}}dt-\int_{0}^{x}\frac{\frac{2\left(17+6\sqrt{7}\right)}{9}}{\sqrt{(1-t^{2})(1-\frac{8-3\sqrt{7}}{16}t^{2})}}dt\right)-\int_{0}^{\frac{\sqrt{7}x\left(1+\frac{3\sqrt{7}-6}{4}x^{2}+\frac{8-3\sqrt{7}}{4}x^{4}+\frac{127\sqrt{7}-336}{448}x^{6}\right)}{1+\sqrt{7}x^{2}+\frac{21-6\sqrt{7}}{16}x^{4}+\frac{8\sqrt{7}-21}{64}x^{6}}}\frac{\frac{811055-63624\sqrt{7}}{13379688}}{(1-\frac{914151008+345479643\sqrt{7}}{1879315224}t^{2})\sqrt{(1-t^{2})(1-\frac{8+3\sqrt{7}}{16}t^{2})}}dt
2. \frac{\sqrt{9437394288+3566854809\sqrt{7}}}{72954}\operatorname{artanh}\left(\frac{\frac{\sqrt{2393411160-787082247\sqrt{7}}}{17698}x\left(1-\frac{6932-2121\sqrt{7}}{16452}x^{2}+\frac{64320\sqrt{7}-170045}{263232}x^{4}-\frac{48499\sqrt{7}-128315}{263232}x^{6}+\frac{127139-48054\sqrt{7}}{3158784}x^{8}\right)\sqrt{(1-x^{2})(1-\frac{8-3\sqrt{7}}{16}x^{2})}}{1-\frac{10395\sqrt{7}-22739}{17698}x^{2}-\frac{10595321-3952008\sqrt{7}}{1699008}x^{4}+\frac{322797878-121974345\sqrt{7}}{61164288}x^{6}-\frac{24402131-9222568\sqrt{7}}{163104768}x^{8}+\frac{156525261\sqrt{7}-414126907}{489314304}x^{10}+\frac{15220468\sqrt{7}-40269573}{7829028864}x^{12}}\right)

复制代码

青青子衿

\begin{gather*}
\int_{0}^{x}\frac{1}{\left(1-\frac{1}{\beta^{2}}t^{2}\right)\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}dt+\int_{0}^{x}\frac{1}{\left(1-k^{2}\beta^{2}t^{2}\right)\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}dt\\

=\int_{0}^{x}\frac{1}{\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}dt+\frac{\beta}{\sqrt{\left(1-\beta^{2}\right)\left(1-k^{2}\beta^{2}\right)}}\operatorname{arctanh}\left(\frac{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{gather*}

\begin{gather*}
3\int_{0}^{1}\frac{\frac{1}{\beta}\sqrt{\left(1-\beta^{2}\right)\left(1-k^{2}\beta^{2}\right)}}{\left(1-k^{2}\beta^{2}t^{2}\right)\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}dt\\
=\int_{0}^{1}\frac{\frac{1}{\varkappa}\sqrt{\left(1-\varkappa^{2}\right)\left(1-k^{2}\varkappa^{2}\right)}}{\left(1-k^{2}\varkappa^{2}t^{2}\right)\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}dt
+\int_{0}^{1}\frac{L}{\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}dt\\
\\
\varkappa=\frac{\beta(3-4\beta^{2}-4k^{2}\beta^{2}+6k^{2}\beta^{4}-k^{4}\beta^{8})}{1-6k^{2}\beta^{4}+4k^{2}\beta^{6}+4k^{4}\beta^{6}-3k^{4}\beta^{8}}\\
\\
L=\tfrac{3\sqrt{\left(1-\beta^{2}\right)\left(1-k^{2}\beta^{2}\right)}}{\beta}
-\tfrac{\sqrt{\left(1-\varkappa^{2}\right)\left(1-k^{2}\varkappa^{2}\right)}}{\varkappa}
+\tfrac{2k^{2}\beta^{2}\left(\beta+\varkappa\right)\sqrt{\left(1-k^{2}\beta^{2}\right)\left(1-\beta^{2}\right)}}{1-k^{2}\beta^{4}}
\end{gather*}

\begin{gather*}
3\int_{\frac{1}{k}}^{+\infty}\frac{\frac{1}{\beta}\sqrt{\left(\beta^{2}-1\right)\left(k^{2}\beta^{2}-1\right)}}{\left(1-\frac{1}{\beta^{2}}t^{2}\right)\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}dt\\
=\int_{\frac{1}{k}}^{+\infty}\frac{\frac{1}{\varkappa}\sqrt{\left(\varkappa^{2}-1\right)\left(k^{2}\varkappa^{2}-1\right)}}{\left(1-\frac{1}{\varkappa^{2}}t^{2}\right)\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}dt+\int_{\frac{1}{k}}^{+\infty}\frac{J}{\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}dt\\
\\
\varkappa=\frac{\beta(3-4\beta^{2}-4k^{2}\beta^{2}+6k^{2}\beta^{4}-k^{4}\beta^{8})}{1-6k^{2}\beta^{4}+4k^{2}\beta^{6}+4k^{4}\beta^{6}-3k^{4}\beta^{8}}\\
\\
J=\tfrac{3\sqrt{\left(\beta^{2}-1\right)\left(k^{2}\beta^{2}-1\right)}}{\beta}-\tfrac{\sqrt{\left(\varkappa^{2}-1\right)\left(k^{2}\varkappa^{2}-1\right)}}{\varkappa}+\tfrac{2\sqrt{\left(\beta^{2}-1\right)\left(k^{2}\beta^{2}-1\right)}}{k^{2}\beta^{4}-1}\left(\tfrac{1}{\beta}+\tfrac{1}{\varkappa}\right)
\end{gather*}

1. \int_{0}^{1}\frac{\frac{3}{\beta}\sqrt{\left(1-\beta^{2}\right)\left(1-k^{2}\beta^{2}\right)}}{\left(1-k^{2}\beta^{2}t^{2}\right)\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}dt-\int_{0}^{1}\frac{\frac{3}{\beta}\sqrt{\left(1-\beta^{2}\right)\left(1-k^{2}\beta^{2}\right)}}{\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}dt
2. \int_{0}^{1}\frac{1}{\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}dt\left(3\int_{0}^{\beta}\sqrt{\frac{1-k^{2}t^{2}}{1-t^{2}}}dt\right)-3\int_{0}^{\beta}\frac{1}{\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}dt\int_{0}^{1}\sqrt{\frac{1-k^{2}t^{2}}{1-t^{2}}}dt
3. \int_{0}^{1}\frac{1}{\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}dt\left(\int_{0}^{\varkappa}\sqrt{\frac{1-k^{2}t^{2}}{1-t^{2}}}dt+\frac{2k^{2}\beta^{2}\left(\beta+\varkappa\right)\sqrt{\left(1-k^{2}\beta^{2}\right)\left(1-\beta^{2}\right)}}{1-k^{2}\beta^{4}}\right)-\int_{0}^{\varkappa}\frac{1}{\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}dt\int_{0}^{1}\sqrt{\frac{1-k^{2}t^{2}}{1-t^{2}}}dt
4. \int_{0}^{1}\frac{\frac{1}{\varkappa}\sqrt{\left(1-\varkappa^{2}\right)\left(1-k^{2}\varkappa^{2}\right)}}{\left(1-k^{2}\varkappa^{2}t^{2}\right)\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}dt-\int_{0}^{1}\frac{\frac{1}{\varkappa}\sqrt{\left(1-\varkappa^{2}\right)\left(1-k^{2}\varkappa^{2}\right)}}{\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}dt+\int_{0}^{1}\frac{\frac{2k^{2}\beta^{2}\left(\beta+\varkappa\right)\sqrt{\left(1-k^{2}\beta^{2}\right)\left(1-\beta^{2}\right)}}{1-k^{2}\beta^{4}}}{\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}dt
5. \varkappa=\frac{\beta(3-4\beta^{2}-4k^{2}\beta^{2}+6k^{2}\beta^{4}-k^{4}\beta^{8})}{1-6k^{2}\beta^{4}+4k^{2}\beta^{6}+4k^{4}\beta^{6}-3k^{4}\beta^{8}}
6. 3\int_{0}^{\beta}\frac{1}{\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}dt-\int_{0}^{\varkappa}\frac{1}{\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}dt
7. l=\frac{3\sqrt{\left(1-\beta^{2}\right)\left(1-k^{2}\beta^{2}\right)}}{\beta}-\frac{\sqrt{\left(1-\varkappa^{2}\right)\left(1-k^{2}\varkappa^{2}\right)}}{\varkappa}+\frac{2k^{2}\beta^{2}\left(\beta+\varkappa\right)\sqrt{\left(1-k^{2}\beta^{2}\right)\left(1-\beta^{2}\right)}}{1-k^{2}\beta^{4}}
8. \int_{0}^{1}\frac{\frac{3}{\beta}\sqrt{\left(1-\beta^{2}\right)\left(1-k^{2}\beta^{2}\right)}}{\left(1-k^{2}\beta^{2}t^{2}\right)\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}dt-\int_{0}^{1}\frac{\frac{1}{\varkappa}\sqrt{\left(1-\varkappa^{2}\right)\left(1-k^{2}\varkappa^{2}\right)}}{\left(1-k^{2}\varkappa^{2}t^{2}\right)\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}dt-\int_{0}^{1}\frac{l}{\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}dt
9. \beta=0.47
10. k=0.85

复制代码

\begin{gather*}
\int_{0}^{x}\tfrac{{\mathrm{d}}t}{\left(1-{\raise2px\scriptsize\frac{2+\sqrt{3}}{2}}t^{2}\right)\sqrt{\left(1-t^{2}\right)\left(1-{\raise2px\scriptsize\frac{2+\sqrt{3}}{4}}t^{2}\right)}}
+\int_{0}^{x}\tfrac{{\mathrm{d}}t}{\left(1-{\raise2px\scriptsize\frac{1}{2}}t^{2}\right)\sqrt{\left(1-t^{2}\right)\left(1-{\raise2px\scriptsize\frac{2+\sqrt{3}}{4}}t^{2}\right)}}\\
=\int_{0}^{x}\tfrac{{\mathrm{d}}t}{\sqrt{\left(1-t^{2}\right)\left(1-{\raise2px\scriptsize\frac{2+\sqrt{3}}{4}}t^{2}\right)}}
+\tfrac{2}{\,\,3^{1/4}}\operatorname{arctanh}\left(\tfrac{3^{1/4}x}{2\sqrt{\left(1-x^{2}\right)\left(1-{\raise2px\scriptsize\frac{2+\sqrt{3}}{4}}x^{2}\right)}}\right)
\end{gather*}


\begin{align*}
{\Large\int}_{0}^{x}\frac{\frac{\frac{485121}{17143558}}{1-\frac{17032129}{51767148} t^2}-\frac{\frac{3}{2}}{1-\frac{1}{12}t^2}+\frac{6336}{4127}}{\sqrt{\left(1-t^2\right) \left(1-\frac{1}{3}t^2\right)}}{\mathrm{d}}t
=\tfrac{1}{\sqrt{11}}\operatorname{artanh}\left(\tfrac{19008 \sqrt{11} x \sqrt{\left(1-x^2\right) \left(1-\frac{1}{3}x^2\right)}}{299088-79944 x^2-4127 x^4}\right)
\end{align*}



\begin{gather*}
\int_{0}^{1}\frac{\frac{3}{\beta}\sqrt{(1-\beta^{2})(1-k^{2}\beta^{2})}}{(1-k^{2}\beta^{2}t^{2})\sqrt{(1-t^{2})(1-k^{2}t^{2})}}{\mathrm{d}}t\\
=\int_{0}^{1}\frac{\frac{1}{\varkappa}\sqrt{(1-\varkappa^{2})(1-k^{2}\varkappa^{2})}}{(1-k^{2}\varkappa^{2}t^{2})\sqrt{(1-t^{2})(1-k^{2}t^{2})}}{\mathrm{d}}t
+\int_{0}^{1}\frac{\frac{1}{1-k^{2}\beta^{3}\varkappa}\left(\frac{3\sqrt{(1-\beta^{2})(1-k^{2}\beta^{2})}}{\beta}-\frac{\sqrt{(1-\varkappa^{2})(1-k^{2}\varkappa^{2})}}{\varkappa}\right)}{\sqrt{(1-t^{2})(1-k^{2}t^{2})}}{\mathrm{d}}t\\
\\
\varkappa=\frac{\beta\left(3-4\beta^{2}-4k^{2}\beta^{2}+6k^{2}\beta^{4}-k^{4}\beta^{8}\right)}{1-6k^{2}\beta^{4}+4k^{2}\beta^{6}+4k^{4}\beta^{6}-3k^{4}\beta^{8}}
\end{gather*}

\begin{gather*}
\int_{0}^{x}\frac{\frac{3}{\beta}\sqrt{(1-\beta^{2})(1-k^{2}\beta^{2})}}{(1-k^{2}\beta^{2}t^{2})\sqrt{(1-t^{2})(1-k^{2}t^{2})}}{\mathrm{d}}t\\
=\int_{0}^{x}\frac{\frac{1}{\varkappa}\sqrt{(1-\varkappa^{2})(1-k^{2}\varkappa^{2})}}{(1-k^{2}\varkappa^{2}t^{2})\sqrt{(1-t^{2})(1-k^{2}t^{2})}}{\mathrm{d}}t

+\int_{0}^{x}\frac{\frac{1}{1-k^{2}\beta^{3}\varkappa}\left(\frac{3\sqrt{(1-\beta^{2})(1-k^{2}\beta^{2})}}{\beta}-\frac{\sqrt{(1-\varkappa^{2})(1-k^{2}\varkappa^{2})}}{\varkappa}\right)}{\sqrt{(1-t^{2})(1-k^{2}t^{2})}}{\mathrm{d}}t\\
-\operatorname{artanh}\left(\frac{\frac{2k^{2}\beta^{2}(\beta+\varkappa)\sqrt{(1-\beta^{2})(1-k^{2}\beta^{2})}}{1-k^{2}\beta^{4}}x\sqrt{(1-x^{2})(1-k^{2}x^{2})}}{1-k^{2}(1-\beta^{3}\varkappa-(1-\beta^{2})\sqrt{(1-\beta^{2})(1-\varkappa^{2})})x^{2}-k^{4}\beta^{3}\varkappa x^{4}}\right)\\
\\
\varkappa=\frac{\beta\left(3-4\beta^{2}-4k^{2}\beta^{2}+6k^{2}\beta^{4}-k^{4}\beta^{8}\right)}{1-6k^{2}\beta^{4}+4k^{2}\beta^{6}+4k^{4}\beta^{6}-3k^{4}\beta^{8}}\\


\end{gather*}

1. D[ArcTanh[((
2.        2 k^2 \[Beta]^2 (\[Beta] + \[CurlyKappa]) Sqrt[(1 - \[Beta]^2) \
3. (1 - k^2 \[Beta]^2)])/(1 - k^2 \[Beta]^4)
4.         x Sqrt[(1 - x^2) (1 - k^2 x^2)])/(
5.       1 - k^2 (1 - \[Beta]^3 \[CurlyKappa] - (1 - \[Beta]^2) Sqrt[(1 \
6. - \[Beta]^2) (1 - \[CurlyKappa]^2)]) x^2 -
7.        k^4 \[Beta]^3 \[CurlyKappa]*
8.         x^4)] /. \[CurlyKappa] -> (\[Beta] (3 - 4 \[Beta]^2 -
9.          4 k^2 \[Beta]^2 + 6 k^2 \[Beta]^4 - k^4 \[Beta]^8))/(
10.       1 - 6 k^2 \[Beta]^4 + 4 k^2 \[Beta]^6 + 4 k^4 \[Beta]^6 -
11.        3 k^4 \[Beta]^8) /. {k -> 1/Sqrt[3], \[Beta] -> 1/2} // Factor,
12.    x] // Factor
13. Sqrt[(1 - \[CurlyKappa]^2) (1 -
14.        k^2 \[CurlyKappa]^2)]/\[CurlyKappa]/((1 -
15.        k^2 \[CurlyKappa]^2 t^2) Sqrt[(1 - t^2) (1 - k^2 t^2)]) - (
16.     3 Sqrt[(1 - \[Beta]^2) (1 - k^2 \[Beta]^2)])/\[Beta]/((1 -
17.        k^2 \[Beta]^2 t^2) Sqrt[(1 - t^2) (1 - k^2 t^2)]) + (
18.     1/(1 - k^2 \[Beta]^3 \[CurlyKappa]) ((
19.        3 Sqrt[(1 - \[Beta]^2) (1 - k^2 \[Beta]^2)])/\[Beta] -
20.        Sqrt[(1 - \[CurlyKappa]^2) (1 -
21.           k^2 \[CurlyKappa]^2)]/\[CurlyKappa]))/
22.     Sqrt[(1 - t^2) (1 - k^2 t^2)] /. \[CurlyKappa] -> (\[Beta] (3 -
23.        4 \[Beta]^2 - 4 k^2 \[Beta]^2 + 6 k^2 \[Beta]^4 -
24.        k^4 \[Beta]^8))/(
25.     1 - 6 k^2 \[Beta]^4 + 4 k^2 \[Beta]^6 + 4 k^4 \[Beta]^6 -
26.      3 k^4 \[Beta]^8) /. {k -> 1/Sqrt[3], \[Beta] -> 1/2} // Factor

复制代码

dlmf.nist.gov/19.7#E7

1. (4 x Sqrt[(1 - x^2) (1 - k^2 x^2)] Det[( {
2.       {x^2, 0, 0, 0, 0, 1},
3.       {-1, x^2, 0, 0, 0, -2 (1 + k^2)},
4.       {0, -1, x^4, 0, 0, 5 k^2},
5.       {0, 0, -1, x^2, 0, -5 k^4},
6.       {0, 0, 0, -1, x^2, 2 k^4 (1 + k^2)},
7.       {0, 0, 0, 0, -1, -k^6}
8.      } )])/Det[( {
9.      {x^4, 0, 0, 0, 0, 0, 1},
10.      {-1, x^2, 0, 0, 0, 0, -20 k^2},
11.      {0, -1, x^2, 0, 0, 0, 32 k^2 (1 + k^2)},
12.      {0, 0, -1, x^2, 0, 0, -2 k^2 (8 + 29 k^2 + 8 k^4)},
13.      {0, 0, 0, -1, x^2, 0, 32 k^4 (1 + k^2)},
14.      {0, 0, 0, 0, -1, x^4, -20 k^6},
15.      {0, 0, 0, 0, 0, -1, k^8}
16.     } )] - (
17.   4 x Sqrt[(1 - x^2) (1 - k^2 x^2)] (1 - 2 x^2 - 2 k^2 x^2 +
18.      5 k^2 x^4 - 5 k^4 x^8 + 2 k^4 x^10 + 2 k^6 x^10 - k^6 x^12))/(
19.   1 - 20 k^2 x^4 + 32 k^2 x^6 + 32 k^4 x^6 - 16 k^2 x^8 -
20.    58 k^4 x^8 - 16 k^6 x^8 + 32 k^4 x^10 + 32 k^6 x^10 - 20 k^6 x^12 +
21.     k^8 x^16) // Factor
22. (x*Det[( {
23.       {x^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5},
24.       {-1, x^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, -20 (1 + k^2)},
25.       {0, -1, x^2, 0, 0, 0, 0, 0, 0, 0, 0, 2 (8 + 47 k^2 + 8 k^4)},
26.       {0, 0, -1, x^2, 0, 0, 0, 0, 0, 0, 0, -80 k^2 (1 + k^2)},
27.       {0, 0, 0, -1, x^2, 0, 0, 0, 0, 0, 0, -105 k^4},
28.       {0, 0, 0, 0, -1, x^2, 0, 0, 0, 0, 0, 360 k^4 (1 + k^2)},
29.       {0, 0, 0, 0, 0, -1, x^2, 0, 0, 0,
30.        0, -60 k^4 (4 + 13 k^2 + 4 k^4)},
31.       {0, 0, 0, 0, 0, 0, -1, x^2, 0, 0, 0,
32.        16 k^4 (1 + k^2) (4 + 31 k^2 + 4 k^4)},
33.       {0, 0, 0, 0, 0, 0, 0, -1, x^2, 0,
34.        0, -5 k^6 (32 + 89 k^2 + 32 k^4)},
35.       {0, 0, 0, 0, 0, 0, 0, 0, -1, x^2, 0, 140 k^8 (1 + k^2)},
36.       {0, 0, 0, 0, 0, 0, 0, 0, 0, -1, x^4, -50 k^10},
37.       {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, k^12}
38.      } )])/Det[( {
39.      {x^4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1},
40.      {-1, x^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, -50 k^2},
41.      {0, -1, x^2, 0, 0, 0, 0, 0, 0, 0, 0, 140 k^2 (1 + k^2)},
42.      {0, 0, -1, x^2, 0, 0, 0, 0, 0, 0,
43.       0, -5 k^2 (32 + 89 k^2 + 32 k^4)},
44.      {0, 0, 0, -1, x^2, 0, 0, 0, 0, 0, 0,
45.       16 k^2 (1 + k^2) (4 + 31 k^2 + 4 k^4)},
46.      {0, 0, 0, 0, -1, x^2, 0, 0, 0, 0,
47.       0, -60 k^4 (4 + 13 k^2 + 4 k^4)},
48.      {0, 0, 0, 0, 0, -1, x^2, 0, 0, 0, 0, 360 k^6 (1 + k^2)},
49.      {0, 0, 0, 0, 0, 0, -1, x^2, 0, 0, 0, -105 k^8},
50.      {0, 0, 0, 0, 0, 0, 0, -1, x^2, 0, 0, -80 k^8 (1 + k^2)},
51.      {0, 0, 0, 0, 0, 0, 0, 0, -1, x^2, 0,
52.       2 k^8 (8 + 47 k^2 + 8 k^4)},
53.      {0, 0, 0, 0, 0, 0, 0, 0, 0, -1, x^2, -20 k^10 (1 + k^2)},
54.      {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 5 k^12}
55.     } )] - (x (5 - 20 x^2 - 20 k^2 x^2 + 16 x^4 + 94 k^2 x^4 +
56.        16 k^4 x^4 - 80 k^2 x^6 - 80 k^4 x^6 - 105 k^4 x^8 +
57.        360 k^4 x^10 + 360 k^6 x^10 - 240 k^4 x^12 - 780 k^6 x^12 -
58.        240 k^8 x^12 + 64 k^4 x^14 + 560 k^6 x^14 + 560 k^8 x^14 +
59.        64 k^10 x^14 - 160 k^6 x^16 - 445 k^8 x^16 - 160 k^10 x^16 +
60.        140 k^8 x^18 + 140 k^10 x^18 - 50 k^10 x^20 + k^12 x^24))/(1 -
61.      50 k^2 x^4 + 140 k^2 x^6 + 140 k^4 x^6 - 160 k^2 x^8 -
62.      445 k^4 x^8 - 160 k^6 x^8 + 64 k^2 x^10 + 560 k^4 x^10 +
63.      560 k^6 x^10 + 64 k^8 x^10 - 240 k^4 x^12 - 780 k^6 x^12 -
64.      240 k^8 x^12 + 360 k^6 x^14 + 360 k^8 x^14 - 105 k^8 x^16 -
65.      80 k^8 x^18 - 80 k^10 x^18 + 16 k^8 x^20 + 94 k^10 x^20 +
66.      16 k^12 x^20 - 20 k^10 x^22 - 20 k^12 x^22 +
67.      5 k^12 x^24) // Factor

复制代码

\begin{gather*}
\int_{0}^{1}\frac{{\mathrm{d}}t}{\left(1-U\beta^{2}t^{2}\right)\sqrt{\left(1-t^{2}\right)\left(1-Ut^{2}\right)}}=\int_{0}^{1}\frac{\frac{1}{1-U\beta^{3}}}{\sqrt{\left(1-t^{2}\right)\left(1-Ut^{2}\right)}}{\mathrm{d}}t\\
\\
U=\frac{1-2\beta}{\beta^{3}(\beta-2)}
\end{gather*}

青青子衿

青青子衿 发表于 2024-4-20 18:29
\begin{gather*}
\int_{0}^{1}\frac{{\mathrm{d}}t}{\left(1-U\beta^{2}t^{2}\right)\sqrt{\left(1-t^{2}\right)\left(1-Ut^{2}\right)}}=\int_{0}^{1}\frac{\frac{1}{1-U\beta^{3}}}{\sqrt{\left(1-t^{2}\right)\left(1-Ut^{2}\right)}}{\mathrm{d}}t\\
\\
U=\frac{1-2\beta}{\beta^{3}(\beta-2)}
\end{gather*}

\begin{gather*}
\int_{0}^{1}\frac{{\mathrm{d}t}}{\sqrt{(1-t^{2})(1-Ut^{2})}}=\frac{3}{2}\int_{0}^{\beta}\frac{{\mathrm{d}t}}{\sqrt{(1-t^{2})(1-Ut^{2})}}\\
\\
\int_{0}^{1}\sqrt{\frac{1-Ut^{2}}{1-t^{2}}}{\mathrm{d}}t

=

\frac{3}{2}\int_{0}^{\beta}\sqrt{\frac{1-Ut^{2}}{1-t^{2}}}{\mathrm{d}}t+\phi\\
\\
\int_{0}^{1}\frac{{\mathrm{d}t}}{(1-U\beta^{2}t^{2})\sqrt{(1-t^{2})(1-Ut^{2})}}=\int_{0}^{1}\frac{\chi}{\sqrt{(1-t^{2})(1-Ut^{2})}}{\mathrm{d}t}\\
\\

\left\{\begin{split}
\beta&=\tfrac{2s}{1+s^{2}}\\
U&=\tfrac{(1+s^{2})^{3}(3s^{2}-1)}{16s^{6}}\\
\phi&=\mathfrak{n}\cdot\left(\tfrac{2U\left(1-U\right)
\frac{\beta'_{s}}{U'_{s}}
}{\sqrt{\left(1-\beta^{2}\right)\left(1-U\beta^{2}\right)}}-\tfrac{U\beta\sqrt{1-\beta^{2}}}{\sqrt{1-U\beta^{2}}}\right)\\
&=\tfrac{1-3s^{2}}{4s^{3}}\\
\chi&=1-\tfrac{\beta\phi}{\mathfrak{n}\sqrt{(1-\beta^{2})(1-U\beta^{2})}}\\
&=1-\tfrac{2(1-3s^{2})}{3(1-s)^{2}(1+s)^{2}}
\end{split}\right.
\end{gather*}

1. \int_{0}^{1}\sqrt{\frac{1-\frac{(1+x^{2})^{3}(3x^{2}-1)}{16x^{6}}t^{2}}{1-t^{2}}}dt
2. \frac{3}{2}\int_{0}^{\frac{2x}{1+x^{2}}}\sqrt{\frac{1-\frac{(1+x^{2})^{3}(3x^{2}-1)}{16x^{6}}t^{2}}{1-t^{2}}}dt+\frac{1-3x^{2}}{4x^{3}}
3. \int_{0}^{1}\frac{1}{\left(1-\frac{(1+x^{2})^{3}(3x^{2}-1)}{16x^{6}}\left(\frac{2x}{1+x^{2}}\right)^{2}t^{2}\right)\sqrt{\left(1-t^{2}\right)\left(1-\frac{(1+x^{2})^{3}(3x^{2}-1)}{16x^{6}}t^{2}\right)}}dt
4. \int_{0}^{1}\frac{1-\frac{2(1-3x^{2})}{3(1-x)^{2}(1+x)^{2}}}{\sqrt{\left(1-t^{2}\right)\left(1-\frac{(1+x^{2})^{3}(3x^{2}-1)}{16x^{6}}t^{2}\right)}}dt

复制代码

1. \frac{2}{3}\left(\int_{0}^{1}\sqrt{\frac{1-\frac{(1+x^{2})^{3}(3x^{2}-1)}{16x^{6}}t^{2}}{1-t^{2}}}dt\right)
2. \int_{0}^{\frac{2x}{1+x^{2}}}\sqrt{\frac{1-\frac{(1+x^{2})^{3}(3x^{2}-1)}{16x^{6}}t^{2}}{1-t^{2}}}dt-\frac{\frac{(1+x^{2})^{3}(3x^{2}-1)}{16x^{6}}\frac{2x}{1+x^{2}}\sqrt{1-\left(\frac{2x}{1+x^{2}}\right)^{2}}}{\sqrt{1-\frac{(1+x^{2})^{3}(3x^{2}-1)}{16x^{6}}\left(\frac{2x}{1+x^{2}}\right)^{2}}}+\frac{2\frac{(1+x^{2})^{3}(3x^{2}-1)}{16x^{6}}\left(1-\frac{(1+x^{2})^{3}(3x^{2}-1)}{16x^{6}}\right)\frac{\left(\frac{d}{dx}\left(\frac{2x}{1+x^{2}}\right)\right)}{\left(\frac{d}{dx}\left(\frac{(1+x^{2})^{3}(3x^{2}-1)}{16x^{6}}\right)\right)}}{\sqrt{\left(1-\left(\frac{2x}{1+x^{2}}\right)^{2}\right)\left(1-\frac{(1+x^{2})^{3}(3x^{2}-1)}{16x^{6}}\left(\frac{2x}{1+x^{2}}\right)^{2}\right)}}

复制代码

\begin{gather*}
\frac{2}{3}\int_{0}^{1}\sqrt{\frac{1-Ut^{2}}{1-t^{2}}}{\mathrm{d}}t
=\int_{0}^{\beta}\sqrt{\frac{1-Ut^{2}}{1-t^{2}}}{\mathrm{d}}t+\frac{2U\left(1-U\right) \frac{\beta'_{s}}{U'_{s}}
}{\sqrt{\left(1-\beta^{2}\right)\left(1-U\beta^{2}\right)}}

-\frac{U\beta\sqrt{1-\beta^{2}}}{\sqrt{1-U\beta^{2}}}\\
\\
\chi=1-\frac{\beta}{\sqrt{(1-\beta^{2})(1-U\beta^{2})}}\left(\frac{2U\left(1-U\right)
\frac{\beta'_{s}}{U'_{s}}
}{\sqrt{\left(1-\beta^{2}\right)\left(1-U\beta^{2}\right)}}-\frac{U\beta\sqrt{1-\beta^{2}}}{\sqrt{1-U\beta^{2}}}\right)\\

\chi=\frac{1}{1-\beta^{2}}\left(1-\frac{1-U}{1-U\beta^{2}}\cdot\frac{\left(U\beta^{2}\right)'_{s}}{U'_{s}}\right)
\end{gather*}

\begin{align*}
2\mathfrak{n}U\sqrt{\frac{1-U\beta^{2}}{1-\beta^{2}}}\frac{\beta'_{s}}{U'_{s}}+2U\frac{\phi'_{s}}{U'_{s}}=\phi\\
\end{align*}

\begin{align*}
\chi'_s+\frac{\left(U\beta^{2}\right)'_s}{2U\beta^{2}(1-U\beta^{2})}=\frac{\chi}{2U(1-\beta^{2})}\left(\frac{(1-U\beta^{4})\left(\left(U\beta^{2}\right)'_s\right)}{\beta^{2}(1-U\beta^{2})}-\beta^{2}U'_{s}\right)
\end{align*}

青青子衿

\begin{align*}
&\quad\>\>\frac{\mathrm{d}}{\mathrm{d}m}\left(\int_{0}^{1}\frac{\mathrm{d}t}{\sqrt{t\left(1-t\right)\left(1-mt\right)}}\right)\\

&=\frac{1}{2m}\left(\frac{1}{1-m}\int_{0}^{1}\sqrt{\frac{1-mt}{t(1-t)}}\mathrm{d}t-\int_{0}^{1}\frac{\mathrm{d}t}{\sqrt{t(1-t)(1-mt)}}\right)\\
\\

&\quad\>\>\frac{\mathrm{d}}{\mathrm{d}m}\left(\int_{0}^{1}\sqrt{\frac{1-mt}{t(1-t)}}\mathrm{d}t\right)\\


&=\frac{1}{2m}\left(\int_{0}^{1}\sqrt{\frac{1-mt}{t(1-t)}}\mathrm{d}t-\int_{0}^{1}\frac{\mathrm{d}t}{\sqrt{t(1-t)(1-mt)}}\right)\\
\\

&\quad\>\>\frac{\mathrm{d}}{\mathrm{d}m}\left(\int_{0}^{1}\frac{\mathrm{d}t}{\left(1-nt\right)\sqrt{t(1-t)(1-mt)}}\right)\\

&=\frac{1}{2(m-n)}\left(\frac{1}{1-m}\int_{0}^{1}\sqrt{\frac{1-mt}{t(1-t)}}\mathrm{d}t-\int_{0}^{1}\frac{\mathrm{d}t}{\left(1-nt\right)\sqrt{t(1-t)(1-mt)}}\right)\\

\end{align*}

\begin{align*}
&\quad\>\>\frac{\mathrm{d}}{\mathrm{d}n}\left(\int_{0}^{1}\frac{\mathrm{d}t}{\left(1-nt\right)\sqrt{t(1-t)(1-mt)}}\right)\\

&=\frac{1}{2n(m-n)(1-n)}\left(\int_{0}^{1}\frac{m-n^{2}}{\left(1-nt\right)\sqrt{t(1-t)(1-mt)}}{\mathrm{d}t}\right.\\

&\qquad\left.
-\int_{0}^{1}\frac{m-n}{\sqrt{t(1-t)(1-mt)}}{\mathrm{d}t}
-n\int_{0}^{1}\sqrt{\frac{1-mt}{t(1-t)}}{\mathrm{d}t}\right)\\
\end{align*}

1. \frac{d}{dm}\left(\int_{0}^{1}\sqrt{\frac{1-mt}{t(1-t)}}dt\right)
2. \frac{1}{2m}\left(\int_{0}^{1}\sqrt{\frac{1-mt}{t(1-t)}}dt-\int_{0}^{1}\frac{1}{\sqrt{t(1-t)(1-mt)}}dt\right)
3. \frac{d}{dm}\left(\int_{0}^{1}\frac{1}{\left(1-nt\right)\sqrt{t(1-t)(1-mt)}}dt\right)
4. \frac{1}{2(m-n)}\left(\frac{1}{1-m}\int_{0}^{1}\sqrt{\frac{1-mt}{t(1-t)}}dt-\int_{0}^{1}\frac{1}{\left(1-nt\right)\sqrt{t(1-t)(1-mt)}}dt\right)
5. \frac{d}{dn}\left(\int_{0}^{1}\frac{1}{\left(1-nt\right)\sqrt{t(1-t)(1-mt)}}dt\right)
6. \frac{1}{2n(m-n)(1-n)}\left(\int_{0}^{1}\frac{m-n^{2}}{\left(1-nt\right)\sqrt{t(1-t)(1-mt)}}dt-\int_{0}^{1}\frac{m-n}{\sqrt{t(1-t)(1-mt)}}dt-n\int_{0}^{1}\sqrt{\frac{1-mt}{t(1-t)}}dt\right)
7. m=0.76
8. n=0.48
9. \frac{d}{dm}\left(\int_{0}^{x}\sqrt{\frac{1-mt}{t(1-t)}}dt\right)
10. \frac{1}{2m}\left(\int_{0}^{x}\sqrt{\frac{1-mt}{t(1-t)}}dt-\int_{0}^{x}\frac{1}{\sqrt{t(1-t)(1-mt)}}dt\right)
11. \frac{d}{dm}\left(\int_{0}^{x}\frac{1}{\left(1-nt\right)\sqrt{t(1-t)(1-mt)}}dt\right)
12. \frac{1}{2(m-n)}\left(\frac{1}{1-m}\int_{0}^{x}\sqrt{\frac{1-mt}{t(1-t)}}dt-\int_{0}^{x}\frac{1}{\left(1-nt\right)\sqrt{t(1-t)(1-mt)}}dt-\frac{2m\sqrt{x(1-x)}}{(1-m)\sqrt{1-mx}}\right)
13. \frac{d}{dn}\left(\int_{0}^{x}\frac{1}{\left(1-nt\right)\sqrt{t(1-t)(1-mt)}}dt\right)
14. \frac{1}{2n(m-n)(1-n)}\left(\int_{0}^{x}\frac{m-n^{2}}{\left(1-nt\right)\sqrt{t(1-t)(1-mt)}}dt-\int_{0}^{x}\frac{m-n}{\sqrt{t(1-t)(1-mt)}}dt-n\int_{0}^{x}\sqrt{\frac{1-mt}{t(1-t)}}dt+\frac{2n^{2}\sqrt{x(1-x)(1-mx)}}{1-nx}\right)

复制代码

\begin{align*}
&\quad\>\>\frac{\mathrm{d}}{\mathrm{d}n}\left(\int_{0}^{x}\frac{\mathrm{d}t}{\left(1-nt\right)\sqrt{t(1-t)(1-mt)}}\right)\\

&=\frac{1}{2n(m-n)(1-n)}\left(\int_{0}^{x}\frac{m-n^{2}}{\left(1-nt\right)\sqrt{t(1-t)(1-mt)}}{\mathrm{d}t}\right.\\

&\qquad\left.
-\int_{0}^{x}\frac{m-n}{\sqrt{t(1-t)(1-mt)}}{\mathrm{d}t}
-n\int_{0}^{x}\sqrt{\frac{1-mt}{t(1-t)}}{\mathrm{d}t}\right.\\

&\qquad\qquad\left.+\frac{2n^{2}\sqrt{x(1-x)(1-mx)}}{1-nx}\right)
\end{align*}