第三类椭圆积分的次参合并公式

青青子衿

\begin{align*}
\theta_{1}&=\arcsin\left(\frac{\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)\\
P&=\sqrt{\big(1-k^{2}\sin\!^{2}\>\!\theta_{1}\big)\big(1-k^{2}\sin\!^{2}\>\!\xi\,\big)}-k^{2}\sin(\theta_{1})\cos(\theta_{1})\sin(\xi)\cos(\xi)\\
Q&=\sqrt{\big(1-k^{2}\sin\!^{2}\>\!\theta_{1}\big)\big(1-k^{2}\sin\!^{2}\>\!\xi\,\big)}+k^{2}\sin(\theta_{1})\cos(\theta_{1})\sin(\xi)\cos(\xi)\\
R&=\cos(\theta_{1})\sin(\xi)\sqrt{1-k^{2}\sin\!^{2}\>\!\xi\,}-\cos(\xi)\sin(\theta_{1})\sqrt{1-k^{2}\sin\!^{2}\>\,\!\theta_{1}\,}\\
S&=\cos(\theta_{1})\sin(\xi)\sqrt{1-k^{2}\sin\!^{2}\>\!\xi\,}+\cos(\xi)\sin(\theta_{1})\sqrt{1-k^{2}\sin\!^{2}\>\,\!\theta_{1}\,}\\
T&=\frac{\cos(\varphi)\sqrt{1-k^{2}\sin\!^{2}\>\!\varphi\,}}{\sin(\varphi)}
+\frac{\cos(\psi)\sqrt{1-k^{2}\sin\!^{2}\>\!\psi\,}}{\sin(\psi)}\\
&\qquad\quad
-\frac{\cos(\theta_{1})\sqrt{1-k^{2}\sin\!^{2}\>\!\theta_{1}\,}}{\sin(\theta_{1})}
+k^{2}\sin(\varphi)\sin(\psi)\sin(\theta_{1})\\
\\
\Pi_1+\Pi_2-\Pi_3&=\int_{0}^{\xi}\frac{\frac{\cos(\varphi)}{\sin(\varphi)}\sqrt{1-k^{2}\sin\!^{2}\varphi}}{(1-k^{2}\sin\!^{2}(\varphi)\sin\!^{2}(t))\sqrt{1-k^{2}\sin\!^2t}}\mathrm{d}t\\
&\qquad\quad+\int_{0}^{\xi}\frac{\frac{\cos(\psi)}{\sin(\psi)}\sqrt{1-k^{2}\sin\!^{2}\psi}}{(1-k^{2}\sin\!^{2}(\psi)\sin\!^{2}(t))\sqrt{1-k^{2}\sin\!^2t}}\mathrm{d}t\\
&\qquad\qquad\qquad-\int_{0}^{\xi}\frac{\frac{\cos(\theta_{1})}{\sin(\theta_{1})}\sqrt{1-k^{2}\sin\!^{2}\theta_{1}}}{(1-k^{2}\sin\!^{2}(\theta_{1})\sin\!^{2}(t))\sqrt{1-k^{2}\sin\!^2t}}\mathrm{d}t\\
&=T\int_{0}^{\xi}\frac{\mathrm{d}t}{\sqrt{1-k^{2}\sin\!^{2}t}}+\frac{1}{2}\ln\left(\frac{PQ+k^{2}\sin(\varphi)\sin(\psi)\sin(\xi)QR}{PQ+k^{2}\sin(\varphi)\sin(\psi)\sin(\xi)PS}\right)
\end{align*}

1. \theta_{1}=\arcsin\left(\frac{\sin\left(\varphi\right)\cos\left(\psi\right)\sqrt{1-k^{2}\sin\left(\psi\right)^{2}}+\sin\left(\psi\right)\cos\left(\varphi\right)\sqrt{1-k^{2}\sin\left(\varphi\right)^{2}}}{1-k^{2}\sin\left(\varphi\right)^{2}\sin\left(\psi\right)^{2}}\right)
2. P=\sqrt{\left(1-k^{2}\sin\left(\theta_{1}\right)^{2}\right)\left(1-k^{2}\sin\left(\xi\right)^{2}\right)}-k^{2}\sin\left(\theta_{1}\right)\cos\left(\theta_{1}\right)\sin\left(\xi\right)\cos\left(\xi\right)
3. Q=\sqrt{\left(1-k^{2}\sin\left(\theta_{1}\right)^{2}\right)\left(1-k^{2}\sin\left(\xi\right)^{2}\right)}+k^{2}\sin\left(\theta_{1}\right)\cos\left(\theta_{1}\right)\sin\left(\xi\right)\cos\left(\xi\right)
4. R=\cos\left(\theta_{1}\right)\sin\left(\xi\right)\sqrt{1-k^{2}\sin\left(\xi\right)^{2}}-\cos\left(\xi\right)\sin\left(\theta_{1}\right)\sqrt{1-k^{2}\sin\left(\theta_{1}\right)^{2}}
5. S=\cos\left(\theta_{1}\right)\sin\left(\xi\right)\sqrt{1-k^{2}\sin\left(\xi\right)^{2}}+\cos\left(\xi\right)\sin\left(\theta_{1}\right)\sqrt{1-k^{2}\sin\left(\theta_{1}\right)^{2}}
6. T=\frac{\cos\left(\varphi\right)\sqrt{1-k^{2}\sin\left(\varphi\right)^{2}}}{\sin\left(\varphi\right)}+\frac{\cos\left(\psi\right)\sqrt{1-k^{2}\sin\left(\psi\right)^{2}}}{\sin\left(\psi\right)}-\frac{\cos\left(\theta_{1}\right)\sqrt{1-k^{2}\sin\left(\theta_{1}\right)^{2}}}{\sin\left(\theta_{1}\right)}+k^{2}\sin\left(\varphi\right)\sin\left(\psi\right)\sin\left(\theta_{1}\right)
7. \int_{0}^{\xi}\frac{\frac{\cos\left(\varphi\right)}{\sin\left(\varphi\right)}\sqrt{1-k^{2}\sin\left(\varphi\right)^{2}}}{\left(1-k^{2}\sin\left(\varphi\right)^{2}\sin\left(t\right)^{2}\right)\sqrt{1-k^{2}\sin\left(t\right)^{2}}}dt+\int_{0}^{\xi}\frac{\frac{\cos\left(\psi\right)}{\sin\left(\psi\right)}\sqrt{1-k^{2}\sin\left(\psi\right)^{2}}}{\left(1-k^{2}\sin\left(\psi\right)^{2}\sin\left(t\right)^{2}\right)\sqrt{1-k^{2}\sin\left(t\right)^{2}}}dt-\int_{0}^{\xi}\frac{\frac{\cos\left(\theta_{1}\right)}{\sin\left(\theta_{1}\right)}\sqrt{1-k^{2}\sin\left(\theta_{1}\right)^{2}}}{\left(1-k^{2}\sin\left(\theta_{1}\right)^{2}\sin\left(t\right)^{2}\right)\sqrt{1-k^{2}\sin\left(t\right)^{2}}}dt
8. T\int_{0}^{\xi}\frac{1}{\sqrt{1-k^{2}\sin\left(t\right)^{2}}}dt+\frac{1}{2}\ln\left(\frac{PQ+k^{2}\sin\left(\varphi\right)\sin\left(\psi\right)\sin\left(\xi\right)QR}{PQ+k^{2}\sin\left(\varphi\right)\sin\left(\psi\right)\sin\left(\xi\right)PS}\right)

复制代码

青青子衿

\begin{align*}
r_{1}&=\frac{p\sqrt{\left(1-q^{2}\right)\left(1-k^{2}q^{2}\right)}+q\sqrt{\left(1-p^{2}\right)\left(1-k^{2}p^{2}\right)}}{1-k^{2}p^{2}q^{2}}\\

P_1&=\sqrt{\left(1-k^{2}r_{1}^{2}\right)\left(1-k^{2}x^{2}\right)}-k^{2}r_{1}x\sqrt{\left(1-r_{1}^{2}\right)\left(1-x^{2}\right)}\\

Q_1&=\sqrt{\left(1-k^{2}r_{1}^{2}\right)\left(1-k^{2}x^{2}\right)}+k^{2}r_{1}x\sqrt{\left(1-r_{1}^{2}\right)\left(1-x^{2}\right)}\\

R_1&=x\sqrt{\left(1-r_{1}^{2}\right)\left(1-k^{2}x^{2}\right)}-r_{1}\sqrt{\left(1-x^{2}\right)\left(1-k^{2}r_{1}^{2}\right)}\\

S_1&=x\sqrt{\left(1-r_{1}^{2}\right)\left(1-k^{2}x^{2}\right)}+r_{1}\sqrt{\left(1-x^{2}\right)\left(1-k^{2}r_{1}^{2}\right)}\\

T_1&=\frac{\sqrt{\left(1-p^{2}\right)\left(1-k^{2}p^{2}\right)}}{p}+\frac{\sqrt{\left(1-q^{2}\right)\left(1-k^{2}q^{2}\right)}}{q}\\
&\qquad\qquad-\frac{\sqrt{\left(1-r_{1}^{2}\right)\left(1-k^{2}r_{1}^{2}\right)}}{r_{1}}+k^{2}pqr_{1}\\
\\
\Omega(kp)+\Omega(kq)-\Omega(r_1)&=\int_{0}^{x}\frac{\frac{1}{p}\sqrt{\left(1-p^{2}\right)\left(1-k^{2}p^{2}\right)}}{\left(1-k^{2}p^{2}t^{2}\right)\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}\mathrm{d}t\\
&\qquad\quad+\int_{0}^{x}\frac{\frac{1}{q}\sqrt{\left(1-q^{2}\right)\left(1-k^{2}q^{2}\right)}}{\left(1-k^{2}q^{2}t^{2}\right)\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}\mathrm{d}t\\

&\qquad\qquad\>-\int_{0}^{x}\frac{\frac{1}{r_{1}}\sqrt{\left(1-r_{1}^{2}\right)\left(1-k^{2}r_{1}^{2}\right)}}{\left(1-k^{2}r_{1}^{2}t^{2}\right)\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}\mathrm{d}t\\

&=
T_1\int_{0}^{x}\frac{\mathrm{d}t}{\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}+\frac{1}{2}\ln\left(\frac{P_1Q_1+k^{2}pqxQ_1R_1}{P_1Q_1+k^{2}pqxP_1S_1}\right)\\
\\
\mathcal{F}(p)+\mathcal{F}(q)-\mathcal{F}(r_1)&=
\int_{0}^{p}\frac{\mathrm{d}t}{\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}dt+\int_{0}^{q}\frac{\mathrm{d}t}{\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}\\
&\qquad\qquad\qquad-\int_{0}^{r_{1}}\frac{\mathrm{d}t}{\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}

\end{align*}

1. r_{1}=\frac{p\sqrt{\left(1-q^{2}\right)\left(1-k^{2}q^{2}\right)}+q\sqrt{\left(1-p^{2}\right)\left(1-k^{2}p^{2}\right)}}{1-k^{2}p^{2}q^{2}}
2. P\left(x,\alpha\right)=\sqrt{\left(1-k^{2}r_{1}^{2}\right)\left(1-k^{2}x^{2}\right)}-k^{2}r_{1}x\sqrt{\left(1-r_{1}^{2}\right)\left(1-x^{2}\right)}
3. Q\left(x,\alpha\right)=\sqrt{\left(1-k^{2}r_{1}^{2}\right)\left(1-k^{2}x^{2}\right)}+k^{2}r_{1}x\sqrt{\left(1-r_{1}^{2}\right)\left(1-x^{2}\right)}
4. R\left(x,\alpha\right)=x\sqrt{\left(1-r_{1}^{2}\right)\left(1-k^{2}x^{2}\right)}-r_{1}\sqrt{\left(1-x^{2}\right)\left(1-k^{2}r_{1}^{2}\right)}
5. S\left(x,\alpha\right)=x\sqrt{\left(1-r_{1}^{2}\right)\left(1-k^{2}x^{2}\right)}+r_{1}\sqrt{\left(1-x^{2}\right)\left(1-k^{2}r_{1}^{2}\right)}
6. T=\frac{\sqrt{\left(1-p^{2}\right)\left(1-k^{2}p^{2}\right)}}{p}+\frac{\sqrt{\left(1-q^{2}\right)\left(1-k^{2}q^{2}\right)}}{q}-\frac{\sqrt{\left(1-r_{1}^{2}\right)\left(1-k^{2}r_{1}^{2}\right)}}{r_{1}}+k^{2}pqr_{1}
7. \int_{0}^{x}\frac{\frac{1}{p}\sqrt{\left(1-p^{2}\right)\left(1-k^{2}p^{2}\right)}}{\left(1-k^{2}p^{2}t^{2}\right)\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}dt+\int_{0}^{x}\frac{\frac{1}{q}\sqrt{\left(1-q^{2}\right)\left(1-k^{2}q^{2}\right)}}{\left(1-k^{2}q^{2}t^{2}\right)\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}dt-\int_{0}^{x}\frac{\frac{1}{r_{1}}\sqrt{\left(1-r_{1}^{2}\right)\left(1-k^{2}r_{1}^{2}\right)}}{\left(1-k^{2}r_{1}^{2}t^{2}\right)\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}dt
8. T\int_{0}^{x}\frac{1}{\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}dt+\frac{1}{2}\ln\left(\frac{P\left(x,1\right)Q\left(x,1\right)+k^{2}pqxQ\left(x,1\right)R\left(x,1\right)}{P\left(x,1\right)Q\left(x,1\right)+k^{2}pqxP\left(x,1\right)S\left(x,1\right)}\right)
9. \int_{0}^{p}\frac{1}{\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}dt+\int_{0}^{q}\frac{1}{\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}dt-\int_{0}^{r_{1}}\frac{1}{\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}dt

复制代码

\begin{align*}
r_{2}&=\frac{p\sqrt{\left(1-q^{2}\right)\left(1-k^{2}q^{2}\right)}+q\sqrt{\left(1-p^{2}\right)\left(1-k^{2}p^{2}\right)}}{k^{2}p^{2}q^{2}-1}\\

P_2&=\sqrt{\left(1-k^{2}r_{2}^{2}\right)\left(1-k^{2}x^{2}\right)}-k^{2}r_{2}x\sqrt{\left(1-r_{2}^{2}\right)\left(1-x^{2}\right)}\\
Q_2&=\sqrt{\left(1-k^{2}r_{2}^{2}\right)\left(1-k^{2}x^{2}\right)}+k^{2}r_{2}x\sqrt{\left(1-r_{2}^{2}\right)\left(1-x^{2}\right)}\\

R_2&=x\sqrt{\left(1-r_{2}^{2}\right)\left(1-k^{2}x^{2}\right)}-r_{2}\sqrt{\left(1-x^{2}\right)\left(1-k^{2}r_{2}^{2}\right)}\\

S_2&=x\sqrt{\left(1-r_{2}^{2}\right)\left(1-k^{2}x^{2}\right)}+r_{2}\sqrt{\left(1-x^{2}\right)\left(1-k^{2}r_{2}^{2}\right)}\\

T_2&=\frac{\sqrt{\left(1-p^{2}\right)\left(1-k^{2}p^{2}\right)}}{p}+\frac{\sqrt{\left(1-q^{2}\right)\left(1-k^{2}q^{2}\right)}}{q}\\
&\qquad\qquad-\frac{\sqrt{\left(1-r_{2}^{2}\right)\left(1-k^{2}r_{2}^{2}\right)}}{r_{2}}+\frac{r_{2}}{pq}\\

\Omega(\tfrac{1}{p})+\Omega(\tfrac{1}{q})-\Omega(r_2)&=\int_{0}^{x}\frac{\frac{1}{p}\sqrt{\left(1-p^{2}\right)\left(1-k^{2}p^{2}\right)}}{\left(1-\frac{1}{p^{2}}t^{2}\right)\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}\mathrm{d}t\\
&\qquad\quad+\int_{0}^{x}\frac{\frac{1}{q}\sqrt{\left(1-q^{2}\right)\left(1-k^{2}q^{2}\right)}}{\left(1-\frac{1}{q^{2}}t^{2}\right)\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}\mathrm{d}t\\

&\qquad\qquad\>-\int_{0}^{x}\frac{\frac{1}{r_{2}}\sqrt{\left(1-r_{2}^{2}\right)\left(1-k^{2}r_{2}^{2}\right)}}{\left(1-k^{2}r_{2}^{2}t^{2}\right)\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}\mathrm{d}t\\

&=T_2\int_{0}^{x}\frac{\mathrm{d}t}{\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}
+\frac{1}{2}\ln\left(\frac{pqP_2Q_2+xQ_2R_2}{pqP_2Q_2+xP_2S_2}\right)\\
\\
\mathcal{F}(\tfrac{1}{kp})+\mathcal{F}(\tfrac{1}{kq})-\mathcal{F}(r_2)&=\int_{0}^{\frac{1}{kp}}\frac{\mathrm{d}t}{\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}+\int_{0}^{\frac{1}{kq}}\frac{\mathrm{d}t}{\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}\\
&\qquad\qquad\qquad\quad-\int_{0}^{r_{2}}\frac{\mathrm{d}t}{\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}

\end{align*}

1. r_{1}=\frac{k^{2}p^{2}q^{2}-1}{p\sqrt{\left(k^{2}q^{2}-1\right)\left(k^{2}q^{2}-k^{2}\right)}+q\sqrt{\left(k^{2}p^{2}-1\right)\left(k^{2}p^{2}-k^{2}\right)}}
2. P\left(x,\alpha\right)=\frac{1}{r_{1}}\sqrt{\left(r_{1}^{2}-1\right)\left(1-k^{2}x^{2}\right)}-\frac{1}{r_{1}^{2}}x\sqrt{\left(k^{2}r_{1}^{2}-1\right)\left(1-x^{2}\right)}
3. Q\left(x,\alpha\right)=\frac{1}{r_{1}}\sqrt{\left(r_{1}^{2}-1\right)\left(1-k^{2}x^{2}\right)}+\frac{1}{r_{1}^{2}}x\sqrt{\left(k^{2}r_{1}^{2}-1\right)\left(1-x^{2}\right)}
4. R\left(x,\alpha\right)=\frac{x}{kr_{1}}\sqrt{\left(k^{2}r_{1}^{2}-1\right)\left(1-k^{2}x^{2}\right)}-\frac{1}{kr_{1}^{2}}\sqrt{\left(1-x^{2}\right)\left(r_{1}^{2}-1\right)}
5. S\left(x,\alpha\right)=\frac{x}{kr_{1}}\sqrt{\left(k^{2}r_{1}^{2}-1\right)\left(1-k^{2}x^{2}\right)}+\frac{1}{kr_{1}^{2}}\sqrt{\left(1-x^{2}\right)\left(r_{1}^{2}-1\right)}
6. T=\frac{1}{p}\sqrt{\left(p^{2}-1\right)\left(k^{2}p^{2}-1\right)}+\frac{1}{q}\sqrt{\left(q^{2}-1\right)\left(k^{2}q^{2}-1\right)}-\frac{1}{r_{1}}\sqrt{\left(r_{1}^{2}-1\right)\left(k^{2}r_{1}^{2}-1\right)}+\frac{1}{kpqr_{1}}
7. \int_{0}^{x}\frac{\frac{1}{p}\sqrt{\left(p^{2}-1\right)\left(k^{2}p^{2}-1\right)}}{\left(1-\frac{1}{p^{2}}t^{2}\right)\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}dt+\int_{0}^{x}\frac{\frac{1}{q}\sqrt{\left(q^{2}-1\right)\left(k^{2}q^{2}-1\right)}}{\left(1-\frac{1}{q^{2}}t^{2}\right)\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}dt-\int_{0}^{x}\frac{\frac{1}{r_{1}}\sqrt{\left(r_{1}^{2}-1\right)\left(k^{2}r_{1}^{2}-1\right)}}{\left(1-\frac{1}{r_{1}^{2}}t^{2}\right)\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}dt
8. T\int_{0}^{x}\frac{1}{\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}dt+\frac{1}{2}\ln\left(\frac{pqP\left(x,1\right)Q\left(x,1\right)+xQ\left(x,1\right)R\left(x,1\right)}{pqP\left(x,1\right)Q\left(x,1\right)+xP\left(x,1\right)S\left(x,1\right)}\right)
9. \int_{0}^{\frac{1}{kp}}\frac{1}{\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}dt+\int_{0}^{\frac{1}{kq}}\frac{1}{\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}dt-\int_{0}^{\frac{1}{kr_{1}}}\frac{1}{\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}dt
10. p=3.53
11. q=4.73
12. k=0.48

复制代码

青青子衿

\begin{gather*}
\int_{0}^{x}\tfrac{\frac{((1-k^{2}\beta^{4})^{2}-4k^{2}\beta^{2}(1-\beta^{2})^{2})((1-k^{2}\beta^{4})^{2}-4\beta^{2}(1-k^{2}\beta^{2})^{2})}{\varkappa(1-6k^{2}\beta^{4}+4k^{2}\beta^{6}+4k^{4}\beta^{6}-3k^{4}\beta^{8})^{2}(1-k^{2}\varkappa^{2}t^{2})}-\frac{3}{\beta(1-k^{2}\beta^{2}t^{2})}+\frac{8(1-\beta^{2})(1-k^{2}\beta^{2})}{\beta(3-4\beta^{2}-4k^{2}\beta^{2}+6k^{2}\beta^{4}-k^{4}\beta^{8})}}{\sqrt{(1-t^{2})(1-k^{2}t^{2})}}{\mathrm{d}}t\\
\\
=\tfrac{1}{\sqrt{(1-\beta^{2})(1-k^{2}\beta^{2})}}\operatorname{artanh}\left(\tfrac{\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-\frac{2k^{2}\beta^{2}(3-6\beta^{2}-6k^{2}\beta^{2}+4\beta^{4}+10k^{2}\beta^{4}+4k^{4}\beta^{4}-6k^{2}\beta^{6}-6k^{4}\beta^{6}+3k^{4}\beta^{8})}{1-6k^{2}\beta^{4}+4k^{2}\beta^{6}+4k^{4}\beta^{6}-3k^{4}\beta^{8}}x^{2}-\frac{k^{4}\beta^{4}(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}}x^{4}}\right)\\
\\
\varkappa=\tfrac{\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}}\\
\\
\overset{\beta\>\!=\>\!\sqrt{3}-1}{}\>\>
{\Large{\Bigg\downarrow}}
\>\>\overset{k\>\!=\>\!\frac{\sqrt{2}+\sqrt{6}}{4}}{}\\
\begin{split}
\int_{0}^{x}\tfrac{{\small3+2\sqrt{3}}-\frac{3(1+\sqrt{3})}{2(1-\frac{t^{2}}{2})}}{\sqrt{(1-t^{2})(1-\frac{2+\sqrt{3}}{4}t^{2})}}{\mathrm{d}}t
&
=\tfrac{\raise0.8px\small1+\sqrt{3}}{3^{1/4}}\operatorname{artanh}\left(\tfrac{
{\raise0.8px\small\frac{3^{3/4}}{2}}x\sqrt{(1-x^{2})(1-{\raise0.8px\small\frac{2+\sqrt{3}}{4}}x^{2})}}
{1-{\raise0.8px\small\frac{4-\sqrt{3}}{4}}x^{2}-{\raise0.8px\small\frac{1+\sqrt{3}}{8}}x^{4}}\right)\\
\\
\int_{0}^{x}\tfrac{1-\frac{1+\sqrt{3}}{2}t^{2}}{\left(1-\frac{t^{2}}{2}\right)\sqrt{(1-t^{2})(1-\frac{2+\sqrt{3}}{4}t^{2})}}{\mathrm{d}}t&
=\tfrac{2}{3^{3/4}}\operatorname{artanh}\left(\tfrac{
{\raise0.8px\small\frac{3^{3/4}}{2}}x\sqrt{(1-x^{2})(1-{\raise0.8px\small\frac{2+\sqrt{3}}{4}}x^{2})}}
{1-{\raise0.8px\small\frac{4-\sqrt{3}}{4}}x^{2}-{\raise0.8px\small\frac{1+\sqrt{3}}{8}}x^{4}}\right)
\end{split}
\end{gather*}

1. \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}}
2. \int_{0}^{x}\frac{\frac{((1-k^{2}\beta^{4})^{2}-4k^{2}\beta^{2}(1-\beta^{2})^{2})((1-k^{2}\beta^{4})^{2}-4\beta^{2}(1-k^{2}\beta^{2})^{2})}{\varkappa(1-6k^{2}\beta^{4}+4k^{2}\beta^{6}+4k^{4}\beta^{6}-3k^{4}\beta^{8})^{2}(1-k^{2}\varkappa^{2}t^{2})}-\frac{3}{\beta(1-k^{2}\beta^{2}t^{2})}+\frac{8(1-\beta^{2})(1-k^{2}\beta^{2})}{\beta(3-4\beta^{2}-4k^{2}\beta^{2}+6k^{2}\beta^{4}-k^{4}\beta^{8})}}{\sqrt{(1-t^{2})(1-k^{2}t^{2})}}dt
3. \frac{1}{\sqrt{(1-\beta^{2})(1-k^{2}\beta^{2})}}\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-\frac{2k^{2}\beta^{2}(3-6\beta^{2}-6k^{2}\beta^{2}+4\beta^{4}+10k^{2}\beta^{4}+4k^{4}\beta^{4}-6k^{2}\beta^{6}-6k^{4}\beta^{6}+3k^{4}\beta^{8})}{1-6k^{2}\beta^{4}+4k^{2}\beta^{6}+4k^{4}\beta^{6}-3k^{4}\beta^{8}}x^{2}-\frac{k^{4}\beta^{4}(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}}x^{4}}\right)

复制代码

青青子衿

\begin{align*}
&\>\>\>\>\>\>\Omega_{3}(x,\beta,s)\\
&=\int_{0}^{x}\left(\tfrac{3\varUpsilon_{3}}{\left(1-U_{3}\beta^{2}t^{2}\right)\sqrt{\left(1-t^{2}\right)\left(1-U_{3}t^{2}\right)}}
-\tfrac{2}{\sqrt{\left(1-t^{2}\right)\left(1-U_{3}t^{2}\right)}}\right){\mathrm{d}}t\\

&\qquad-\int_{0}^{y_{3}}
\tfrac{M_{3}q_{3}}{\left(1-V_{3}{\raise1.5px\gamma}_{3}^{2}t^{2}\right)\sqrt{\left(1-t^{2}\right)\left(1-V_{3}t^{2}\right)}}{\mathrm{d}}t\\

&=\tfrac{\varUpsilon_{3}\beta}{\sqrt{\left(1-\beta^{2}\right)\left(1-U_{3}\beta^{2}\right)}}\operatorname{artanh}\left(
\tfrac{\frac{2s(2+s)}{1+s(2+s)\beta^{2}}\ x\ \beta\sqrt{\left(1-\beta^{2}\right)\left(1-U_{3}\beta^{2}\right)\left(1-x^{2}\right)\left(1-U_{3}x^{2}\right)}}{1+\frac{s(2+s)(1+2s-2(1+2s+s^{2}+2s^{3}+s^{4})\beta^{2}+s^{3}(2+s)\beta^{4})}{(1+2s)(1+s(2+s)\beta^{2})}x^{2}
+\frac{s^{4}\beta^{2}(2+s)^{2}(1+2s+s^{2}\beta^{2})}{(1+2s)^{2}(1+s(2+s)\beta^{2})}x^{4}}\right)\\
\\
\\
&\>\>\>\>\>\>
\Omega_{3}\left(x,\beta,{\small-}\tfrac{2 + s}{1 + 2 s}\right)
=\Omega^{*}_{3}\left(x,\beta,s\right)\\
&=\int_{0}^{x}\left(\tfrac{3\varUpsilon_{3}^{*}}{\left(1-V_{3}\beta^{2}t^{2}\right)\sqrt{\left(1-t^{2}\right)\left(1-V_{3}t^{2}\right)}}
-\tfrac{2}{\sqrt{\left(1-t^{2}\right)\left(1-V_{3}t^{2}\right)}}\right){\mathrm{d}}t\\

&\qquad-\int_{0}^{y_{3}^{*}}
\tfrac{M_{3}^{*}q_{3}^{*}}{\left(1-U_{3}{{\raise1px\gamma}_{3}^{*}}^{2}t^{2}\right)\sqrt{\left(1-t^{2}\right)\left(1-U_{3}t^{2}\right)}}{\mathrm{d}}t\\

&=\tfrac{\varUpsilon_{3}^{*}\beta}{\sqrt{\left(1-\beta^{2}\right)\left(1-V_{3}\beta^{2}\right)}}\operatorname{artanh}\left(\tfrac{-\frac{\frac{6s(2+s)}{(1+2s)^{2}}}{1-\frac{3s(2+s)}{(1+2s)^{2}}\beta^{2}}\ x\ \beta\sqrt{\left(1-\beta^{2}\right)\left(1-V_{3}\beta^{2}\right)\left(1-x^{2}\right)\left(1-V_{3}x^{2}\right)}}{1-\frac{s(2+s)(3(1+2s)^{3}+2(1-22s-39s^{2}-22s^{3}+s^{4})\beta^{2}+3s(2+s)^{3}\beta^{4})}{(1+2s)^{3}((1+2s)^{2}-3s(2+s)\beta^{2})}x^{2}-\frac{s^{2}\beta^{2}(2+s)^{4}(3(1+2s)-(2+s)^{2}\beta^{2})}{(1+2s)^{4}((1+2s)^{2}-3s(2+s)\beta^{2})}x^{4}}\right)\\
\\
&\>\>\>\>\>\>
\Omega^{*}_{3}\left(y_{3},
\gamma_{3},s\right)=\Omega^{*}_{3}\left(\tfrac{(1+2s)x(1+\frac{s^{2}}{1+2s}x^{2})}{1+s(2+s)x^{2}},
\tfrac{(1+2s)\beta(1+\frac{s^{2}}{1+2s}\beta^{2})}{1+s(2+s)\beta^{2}},
s\right)\\
&=\int_{0}^{y_{3}}\tfrac{3(\varUpsilon_{3}^{*}\circ\gamma_{3})}{\left(1-V_{3}{\raise1.5px\gamma}_{3}^{2}t^{2}\right)\sqrt{\left(1-t^{2}\right)\left(1-V_{3}t^{2}\right)}}{\mathrm{d}}t
\\

&\qquad
-\int_{0}^{x}
\tfrac{\frac{2}{M_3}}{\sqrt{\left(1-t^{2}\right)\left(1-U_{3}t^{2}\right)}}
{\mathrm{d}}t-\int_{0}^{y_{3}^{*}\circ\>\!y_{3}}
\tfrac{M_{3}^{*}(q_{3}^{*}\circ\gamma_{3})}{\left(1-U_{3}\left({\raise1.5px\gamma}_{3}^{*}\circ\gamma_{3}\right)^{2}t^{2}\right)\sqrt{\left(1-t^{2}\right)\left(1-U_{3}t^{2}\right)}}{\mathrm{d}}t\\

&=\tfrac{(\varUpsilon_{3}^{*}\circ\gamma_{3})\gamma_{3}}{\sqrt{\left(1-\gamma_{3}^{2}\right)\left(1-V_{3}\gamma_{3}^{2}\right)}}\operatorname{artanh}\left(\tfrac{-\frac{\frac{6s(2+s)}{(1+2s)^{2}}}{1-\frac{3s(2+s)}{(1+2s)^{2}}\gamma_{3}^{2}}\ y_{3}\gamma_{3}\sqrt{\left(1-\gamma_{3}^{2}\right)\left(1-V_{3}\gamma_{3}^{2}\right)\left(1-y_{3}^{2}\right)\left(1-V_{3}y_{3}^{2}\right)}}{1-\frac{s(2+s)(3(1+2s)^{3}+2(1-22s-39s^{2}-22s^{3}+s^{4})\gamma_{3}^{2}+3s(2+s)^{3}\gamma_{3}^{4})}{(1+2s)^{3}((1+2s)^{2}-3s(2+s)\gamma_{3}^{2})}y_{3}^{2}-\frac{s^{2}\gamma_{3}^{2}(2+s)^{4}(3(1+2s)-(2+s)^{2}\gamma_{3}^{2})}{(1+2s)^{4}((1+2s)^{2}-3s(2+s)\gamma_{3}^{2})}y_{3}^{4}}\right)\\

&=\tfrac{(\varUpsilon_{3}^{*}\circ\gamma_{3})\frac{\varUpsilon_{3}\beta}{M_{3}q_{3}}}{\sqrt{\left(1-\beta^{2}\right)\left(1-U_{3}\beta^{2}\right)}}\operatorname{artanh}\left(\tfrac{-\frac{\frac{6s(2+s)}{(1+2s)^{2}}}{1-\frac{3s(2+s)}{(1+2s)^{2}}\gamma_{3}^{2}}\ y_{3}\gamma_{3}\sqrt{\left(1-\gamma_{3}^{2}\right)\left(1-V_{3}\gamma_{3}^{2}\right)\left(1-y_{3}^{2}\right)\left(1-V_{3}y_{3}^{2}\right)}}{1-\frac{s(2+s)(3(1+2s)^{3}+2(1-22s-39s^{2}-22s^{3}+s^{4})\gamma_{3}^{2}+3s(2+s)^{3}\gamma_{3}^{4})}{(1+2s)^{3}((1+2s)^{2}-3s(2+s)\gamma_{3}^{2})}y_{3}^{2}-\frac{s^{2}\gamma_{3}^{2}(2+s)^{4}(3(1+2s)-(2+s)^{2}\gamma_{3}^{2})}{(1+2s)^{4}((1+2s)^{2}-3s(2+s)\gamma_{3}^{2})}y_{3}^{4}}\right)\\


\end{align*}

1. \int_{0}^{x}\left(\frac{\frac{3\left(1+\frac{s^{2}}{1+2s}\beta^{2}\right)}{1-\frac{s^{3}(2+s)}{1+2s}\beta^{2}t^{2}}-2}{\sqrt{\left(1-t^{2}\right)\left(1-\frac{s^{3}(2+s)}{1+2s}t^{2}\right)}}\right)dt-\int_{0}^{\frac{(1+2s)x(1+\frac{s^{2}}{1+2s}x^{2})}{(1+s(2+s)x^{2})}}\left(\frac{\frac{\frac{1-\frac{2s(1+s+s^{2})}{1+2s}\beta^{2}+\frac{s^{3}(2+s)}{1+2s}\beta^{4}}{(1+2s)(1+s(2+s)\beta^{2})}}{1-\frac{s(2+s)^{3}}{(1+2s)^{3}}\left(\frac{(1+2s)\beta(1+\frac{s^{2}}{1+2s}\beta^{2})}{(1+s(2+s)\beta^{2})}\right)^{2}t^{2}}}{\sqrt{\left(1-t^{2}\right)\left(1-\frac{s(2+s)^{3}}{(1+2s)^{3}}t^{2}\right)}}\right)dt
2. \int_{0}^{x}\left(\frac{\frac{3\left(1-\frac{(2+S)^{2}}{3(1+2S)}\beta^{2}\right)}{1-\frac{S(2+S)^{3}}{(1+2S)^{3}}\beta^{2}t^{2}}-2}{\sqrt{\left(1-t^{2}\right)\left(1-\frac{S(2+S)^{3}}{(1+2S)^{3}}t^{2}\right)}}\right)dt-\int_{0}^{\frac{-\frac{3}{1+2S}x\left(1-\frac{(2+S)^{2}}{3(1+2S)}x^{2}\right)}{1-\frac{3S(2+S)}{(1+2S)^{2}}x^{2}}}\left(\frac{\frac{\frac{1-\frac{2(2+S)(1+S+S^{2})}{(1+2S)^{2}}\beta^{2}+\frac{S(2+S)^{3}}{(1+2S)^{3}}\beta^{4}}{-\frac{3}{1+2S}\left(1-\frac{3S(2+S)}{(1+2S)^{2}}\beta^{2}\right)}}{1-\frac{S^{3}(2+S)}{1+2S}\left(\frac{\frac{3}{1+2S}\beta\left(1-\frac{(2+S)^{2}}{3(1+2S)}\beta^{2}\right)}{1-\frac{3S(2+S)}{(1+2S)^{2}}\beta^{2}}\right)^{2}t^{2}}}{\sqrt{\left(1-t^{2}\right)\left(1-\frac{S^{3}(2+S)}{1+2S}t^{2}\right)}}\right)dt
3. \frac{\beta\left(1+\frac{s^{2}}{1+2s}\beta^{2}\right)}{\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)\ x\ \beta}{1+s(2+s)\beta^{2}}\ \ \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)}}{1+\frac{s(2+s)(1+2s-2(1+2s+s^{2}+2s^{3}+s^{4})\beta^{2}+s^{3}(2+s)\beta^{4})}{(1+2s)(1+s(2+s)\beta^{2})}x^{2}+\frac{s^{4}(2+s)^{2}\beta^{2}(1+2s+s^{2}\beta^{2})}{(1+2s)^{2}(1+s(2+s)\beta^{2})}x^{4}}\right)
4. \frac{\beta\left(1-\frac{(2+S)^{2}}{3(1+2S)}\beta^{2}\right)}{\sqrt{\left(1-\beta^{2}\right)\left(1-\frac{S(2+S)^{3}}{(1+2S)^{3}}\beta^{2}\right)}}\operatorname{artanh}\left(\frac{-\frac{\frac{6S(2+S)}{(1+2S)^{2}}\ x\ \beta}{1-\frac{3S(2+S)}{(1+2S)^{2}}\beta^{2}}\sqrt{\left(1-\beta^{2}\right)\left(1-\frac{S(2+S)^{3}}{(1+2S)^{3}}\beta^{2}\right)\left(1-x^{2}\right)\left(1-\frac{S(2+S)^{3}}{(1+2S)^{3}}x^{2}\right)}}{1-\frac{S(2+S)(3(1+2S)^{3}+2(1-22S-39S^{2}-22S^{3}+S^{4})\beta^{2}+3S(2+S)^{3}\beta^{4})}{(1+2S)^{3}((1+2S)^{2}-3S(2+S)\beta^{2})}x^{2}-\frac{S^{2}\beta^{2}(2+S)^{4}(3(1+2S)-(2+S)^{2}\beta^{2})}{(1+2S)^{4}((1+2S)^{2}-3S(2+S)\beta^{2})}x^{4}}\right)
5. s=0.88
6. \beta=0.54
7. S=-\frac{2+s}{1+2s}

复制代码

\begin{align*}
&\>\>\>\>\>\>\tfrac{1}{M_{3}q_{3}}\Omega_{3}(x,\beta,s)+\tfrac{1}{3(\varUpsilon_{3}^{*}\circ\gamma_{3})}\Omega^{*}_{3}\left(y_{3},
\gamma_{3},s\right)\\
&=\int_{0}^{x}\left(\tfrac{\frac{3\varUpsilon_{3}}{M_{3}q_{3}}}{\left(1-U_{3}\beta^{2}t^{2}\right)\sqrt{\left(1-t^{2}\right)\left(1-U_{3}t^{2}\right)}}
-\tfrac{\frac{2}{M_{3}q_{3}}+\frac{2}{3M_3(\varUpsilon_{3}^{*}\circ\gamma_{3})}}{\sqrt{\left(1-t^{2}\right)\left(1-U_{3}t^{2}\right)}}\right){\mathrm{d}}t\\

&\qquad-\int_{0}^{y_{3}^{*}\circ\>\!y_{3}}
\tfrac{\frac{M_{3}^{*}(q_{3}^{*}\circ\gamma_{3})}{3(\varUpsilon_{3}^{*}\circ\gamma_{3})}}{\left(1-U_{3}\left({\raise1.5px\gamma}_{3}^{*}\circ\gamma_{3}\right)^{2}t^{2}\right)\sqrt{\left(1-t^{2}\right)\left(1-U_{3}t^{2}\right)}}{\mathrm{d}}t\\

&=\tfrac{\frac{\varUpsilon_{3}\beta}{M_{3}q_{3}}}{\sqrt{\left(1-\beta^{2}\right)\left(1-U_{3}\beta^{2}\right)}}\operatorname{artanh}\left(
\tfrac{\frac{2s(2+s)}{1+s(2+s)\beta^{2}}\ x\ \beta\sqrt{\left(1-\beta^{2}\right)\left(1-U_{3}\beta^{2}\right)\left(1-x^{2}\right)\left(1-U_{3}x^{2}\right)}}{1+\frac{s(2+s)(1+2s-2(1+2s+s^{2}+2s^{3}+s^{4})\beta^{2}+s^{3}(2+s)\beta^{4})}{(1+2s)(1+s(2+s)\beta^{2})}x^{2}
+\frac{s^{4}\beta^{2}(2+s)^{2}(1+2s+s^{2}\beta^{2})}{(1+2s)^{2}(1+s(2+s)\beta^{2})}x^{4}}\right)\\

&\qquad+\tfrac{\frac{\varUpsilon_{3}\beta}{3M_{3}q_{3}}}{\sqrt{\left(1-\beta^{2}\right)\left(1-U_{3}\beta^{2}\right)}}
\operatorname{artanh}\left(\tfrac{-\frac{\frac{6s(2+s)}{(1+2s)^{2}}}{1-\frac{3s(2+s)}{(1+2s)^{2}}\gamma_{3}^{2}}\ y_{3}\gamma_{3}\sqrt{\left(1-\gamma_{3}^{2}\right)\left(1-V_{3}\gamma_{3}^{2}\right)\left(1-y_{3}^{2}\right)\left(1-V_{3}y_{3}^{2}\right)}}{1-\frac{s(2+s)(3(1+2s)^{3}+2(1-22s-39s^{2}-22s^{3}+s^{4})\gamma_{3}^{2}+3s(2+s)^{3}\gamma_{3}^{4})}{(1+2s)^{3}((1+2s)^{2}-3s(2+s)\gamma_{3}^{2})}y_{3}^{2}-\frac{s^{2}\gamma_{3}^{2}(2+s)^{4}(3(1+2s)-(2+s)^{2}\gamma_{3}^{2})}{(1+2s)^{4}((1+2s)^{2}-3s(2+s)\gamma_{3}^{2})}y_{3}^{4}}\right)\\

\end{align*}



\begin{gather*}
\int_{0}^{\frac{2x\sqrt{(1-x^{2})(1-Ux^{2})}}{1-Ux^{4}}}\frac{{\mathrm{d}}t}{(1-UBt^{2})\sqrt{(1-t^{2})(1-Ut^{2})}}\\

-2\int_{0}^{x}\frac{{\mathrm{d}}t}{(1-UBt^{2})\sqrt{\left(1-t^{2}\right)\left(1-Ut^{2}\right)}}\\

=\sqrt{\tfrac{B}{\left(1-B\right)\left(1-UB\right)}}\operatorname{artanh}\left(\tfrac{2Ux^{3}\ \sqrt{B(1-B)(1-UB)(1-x^{2})(1-Ux^{2})}}{1-3UBx^{2}-Ux^{4}+2UBx^{4}+2U^{2}Bx^{4}-U^{2}Bx^{6}}\right)
\end{gather*}



\begin{gather*}
\int_{0}^{\frac{x\ (3-4x^{2}-4Ux^{2}+6Ux^{4}-U^{2}x^{8})}{1-6Ux^{4}+4Ux^{6}+4U^{2}x^{6}-3U^{2}x^{8}}}\frac{{\mathrm{d}}t}{(1-UBt^{2})\sqrt{\left(1-t^{2}\right)\left(1-Ut^{2}\right)}}\\
-3\int_{0}^{x}\frac{{\mathrm{d}}t}{(1-UBt^{2})\sqrt{\left(1-t^{2}\right)\left(1-Ut^{2}\right)}}\\
\\
=\sqrt{\tfrac{B}{(1-UB)(1-B)}}
\operatorname{artanh}\left(\tfrac{8Ux^{3}(1-x^{2})(1-Ux^{2})\sqrt{B(1-B)(1-UB)(1-x^{2})(1-Ux^{2})}}
{\Tiny{1-6UBx^{2}-3U(2-4B-4UB+UB^{2})x^{4}+4U(1-2B+U-5UB+UB^{2}-2U^{2}B+U^{2}B^{2})x^{6}-3U^{2}(1-4B-4UB+2UB^{2})x^{8}-6U^{3}Bx^{10}+U^{4}B^{2}x^{12}}}\right)
\end{gather*}

1. \int_{0}^{\frac{2x\sqrt{\left(1-x^{2}\right)\left(1-Ux^{2}\right)}}{1-Ux^{4}}}\frac{1}{(1-UBt^{2})\sqrt{\left(1-t^{2}\right)\left(1-Ut^{2}\right)}}dt-2\int_{0}^{x}\frac{1}{(1-UBt^{2})\sqrt{\left(1-t^{2}\right)\left(1-Ut^{2}\right)}}dt
2. \sqrt{\frac{B}{\left(1-B\right)\left(1-UB\right)}}\operatorname{artanh}\left(\frac{2U\ x^{3}\ \sqrt{B(1-B)(1-UB)(1-x^{2})(1-Ux^{2})}}{1-3UBx^{2}-Ux^{4}+2UBx^{4}+2U^{2}Bx^{4}-U^{2}Bx^{6}}\right)
3. \int_{0}^{\frac{x\ (3-4x^{2}-4Ux^{2}+6Ux^{4}-U^{2}x^{8})}{1-6Ux^{4}+4Ux^{6}+4U^{2}x^{6}-3U^{2}x^{8}}}\frac{1}{(1-UBt^{2})\sqrt{\left(1-t^{2}\right)\left(1-Ut^{2}\right)}}dt-3\int_{0}^{x}\frac{1}{(1-UBt^{2})\sqrt{\left(1-t^{2}\right)\left(1-Ut^{2}\right)}}dt
4. \sqrt{\frac{B}{\left(1-UB\right)\left(1-B\right)}}\operatorname{arctanh}\left(\frac{8Ux^{3}(1-x^{2})(1-Ux^{2})\sqrt{B(1-B)(1-UB)(1-x^{2})(1-Ux^{2})}}{1-6UBx^{2}-3U(2-4B-4UB+UB^{2})x^{4}+4U(1-2B+U-5UB+UB^{2}-2U^{2}B+U^{2}B^{2})x^{6}-3U^{2}(1-4B-4UB+2UB^{2})x^{8}-6U^{3}Bx^{10}+U^{4}B^{2}x^{12}}\right)

复制代码

\begin{align*}
&\>\>\>\>\>\>\tfrac{1}{\varUpsilon_{3}}\Omega_{3}(x,\beta,s)+\tfrac{M_{3}q_{3}}{3\varUpsilon_{3}(\varUpsilon_{3}^{*}\circ\gamma_{3})}\Omega^{*}_{3}\left(y_{3},
\gamma_{3},s\right)\\
&=\int_{0}^{x}\left(\tfrac{3}{\left(1-U_{3}\beta^{2}t^{2}\right)\sqrt{\left(1-t^{2}\right)\left(1-U_{3}t^{2}\right)}}
-\tfrac{\frac{2}{\varUpsilon_{3}}+\frac{2q_{3}}{3\varUpsilon_{3}(\varUpsilon_{3}^{*}\circ\gamma_{3})}}{\sqrt{\left(1-t^{2}\right)\left(1-U_{3}t^{2}\right)}}\right){\mathrm{d}}t\\

&\qquad+\int_{0}^{y_{3}^{*}\circ\>\!y_{3}}
\tfrac{\frac{q_3(q_{3}^{*}\circ\gamma_{3})}{9\varUpsilon_{3}(\varUpsilon_{3}^{*}\circ\gamma_{3})}}{\left(1-U_{3}\left({\raise1.5px\gamma}_{3}^{*}\circ\gamma_{3}\right)^{2}t^{2}\right)\sqrt{\left(1-t^{2}\right)\left(1-U_{3}t^{2}\right)}}{\mathrm{d}}t\\

&=\int_{0}^{x}\frac{\frac{3}{1-U_{3}\beta^{2}t^{2}}-\frac{2(q_{3}+3(\varUpsilon_{3}^{*}\circ\gamma_{3}))}{3\varUpsilon_{3}(\varUpsilon_{3}^{*}\circ\gamma_{3})}-\frac{\frac{q_{3}(q_{3}^{*}\circ\gamma_{3})}{3\varUpsilon_{3}(\varUpsilon_{3}^{*}\circ\gamma_{3})}}{1-U_{3}\varkappa_{3}^{2}t^{2}}}{\sqrt{\left(1-t^{2}\right)\left(1-U_{3}t^{2}\right)}}{\mathrm{d}}t\\

&=\tfrac{\beta}{\sqrt{(1-\beta^{2})(1-U_{3}\beta^{2})}}\operatorname{artanh}\left(\tfrac{\frac{2U_{3}\beta^{2}(\beta+\varkappa_{3})\sqrt{(1-\beta^{2})(1-U_{3}\beta^{2})}}{1-U_{3}\beta^{4}}x\sqrt{(1-x^{2})(1-U_{3}x^{2})}}{1-\frac{2U_{3}\beta^{2}(3-6\beta^{2}-6U_{3}\beta^{2}+4\beta^{4}+10U_{3}\beta^{4}+4U_{3}^{2}\beta^{4}-6U_{3}\beta^{6}-6U_{3}^{2}\beta^{6}+3U_{3}^{2}\beta^{8})}{1-6U_{3}\beta^{4}+4U_{3}\beta^{6}+4U_{3}^{2}\beta^{6}-3U_{3}^{2}\beta^{8}}x^{2}-\frac{U_{3}^{2}\beta^{4}(3-4\beta^{2}-4U_{3}\beta^{2}+6U_{3}\beta^{4}-U_{3}^{2}\beta^{8})}{1-6U_{3}\beta^{4}+4U_{3}\beta^{6}+4U_{3}^{2}\beta^{6}-3U_{3}^{2}\beta^{8}}x^{4}}\right)\\
\\
\end{align*}

1. s=0.88
2. \beta=0.54
3. S=-\frac{2+s}{1+2s}
4. \gamma_{3}=\frac{(1+2s)\beta(1+\frac{s^{2}}{1+2s}\beta^{2})}{(1+s(2+s)\beta^{2})}
5. Y_{3}=\frac{(1+2s)x(1+\frac{s^{2}}{1+2s}x^{2})}{(1+s(2+s)x^{2})}
6. U_{3}=\frac{s^{3}(2+s)}{1+2s}
7. V_{3}=\frac{s(2+s)^{3}}{(1+2s)^{3}}
8. \Upsilon_{3}=1+\frac{s^{2}}{1+2s}\beta^{2}
9. M_{3}=\frac{1}{1+2s}
10. q_{3}=\frac{1-\frac{2s(1+s+s^{2})}{1+2s}\beta^{2}+\frac{s^{3}(2+s)}{1+2s}\beta^{4}}{1+s(2+s)\beta^{2}}
11. \int_{0}^{x}\frac{\frac{3\Upsilon_{3}}{1-U_{3}\beta^{2}t^{2}}-2}{\sqrt{\left(1-t^{2}\right)\left(1-U_{3}t^{2}\right)}}dt-\int_{0}^{Y_{3}}\frac{\frac{M_{3}q_{3}}{1-V_{3}\gamma_{3}^{2}t^{2}}}{\sqrt{\left(1-t^{2}\right)\left(1-V_{3}t^{2}\right)}}dt
12. \frac{\Upsilon_{3}\beta}{\sqrt{\left(1-\beta^{2}\right)\left(1-U_{3}\beta^{2}\right)}}\operatorname{artanh}\left(\frac{\frac{2s(2+s)\ x\ \beta}{1+s(2+s)\beta^{2}}\ \ \sqrt{\left(1-\beta^{2}\right)\left(1-U_{3}\beta^{2}\right)\left(1-x^{2}\right)\left(1-U_{3}x^{2}\right)}}{1+\frac{s(2+s)(1+2s-2(1+2s+s^{2}+2s^{3}+s^{4})\beta^{2}+s^{3}(2+s)\beta^{4})}{(1+2s)(1+s(2+s)\beta^{2})}x^{2}+\frac{s^{4}(2+s)^{2}\beta^{2}(1+2s+s^{2}\beta^{2})}{(1+2s)^{2}(1+s(2+s)\beta^{2})}x^{4}}\right)
13. \Upsilon_{s3}=1-\frac{(2+s)^{2}}{3(1+2s)}\beta^{2}
14. M_{s3}=-\frac{1+2s}{3}
15. q_{s3}=\frac{1-\frac{2(2+s)(1+s+s^{2})}{(1+2s)^{2}}\beta^{2}+\frac{s(2+s)^{3}}{(1+2s)^{3}}\beta^{4}}{1-\frac{3s(2+s)}{(1+2s)^{2}}\beta^{2}}
16. \gamma_{s3}=\frac{-\frac{3}{1+2s}\beta\left(1-\frac{(2+s)^{2}}{3(1+2s)}\beta^{2}\right)}{1-\frac{3s(2+s)}{(1+2s)^{2}}\beta^{2}}
17. Y_{s3}=\frac{-\frac{3}{1+2s}x\left(1-\frac{(2+s)^{2}}{3(1+2s)}x^{2}\right)}{1-\frac{3s(2+s)}{(1+2s)^{2}}x^{2}}
18. \int_{0}^{x}\frac{\frac{3\Upsilon_{s3}}{1-V_{3}\beta^{2}t^{2}}-2}{\sqrt{\left(1-t^{2}\right)\left(1-V_{3}t^{2}\right)}}dt-\int_{0}^{Y_{s3}}\frac{\frac{M_{s3}q_{s3}}{1-U_{3}\gamma_{s3}^{2}t^{2}}}{\sqrt{\left(1-t^{2}\right)\left(1-U_{3}t^{2}\right)}}dt
19. \frac{\Upsilon_{s3}\beta}{\sqrt{\left(1-\beta^{2}\right)\left(1-V_{3}\beta^{2}\right)}}\operatorname{artanh}\left(\frac{-\frac{\frac{6s(2+s)}{(1+2s)^{2}}\ x\ \beta}{1-\frac{3s(2+s)}{(1+2s)^{2}}\beta^{2}}\sqrt{\left(1-\beta^{2}\right)\left(1-V_{3}\beta^{2}\right)\left(1-x^{2}\right)\left(1-V_{3}x^{2}\right)}}{1-\frac{s(2+s)(3(1+2s)^{3}+2(1-22s-39s^{2}-22s^{3}+s^{4})\beta^{2}+3s(2+s)^{3}\beta^{4})}{(1+2s)^{3}((1+2s)^{2}-3s(2+s)\beta^{2})}x^{2}-\frac{s^{2}\beta^{2}(2+s)^{4}(3(1+2s)-(2+s)^{2}\beta^{2})}{(1+2s)^{4}((1+2s)^{2}-3s(2+s)\beta^{2})}x^{4}}\right)
20. \Upsilon_{sg3}=1-\frac{(2+s)^{2}}{3(1+2s)}\gamma_{3}^{2}
21. q_{sg3}=\frac{1-\frac{2(2+s)(1+s+s^{2})}{(1+2s)^{2}}\gamma_{3}^{2}+\frac{s(2+s)^{3}}{(1+2s)^{3}}\gamma_{3}^{4}}{1-\frac{3s(2+s)}{(1+2s)^{2}}\gamma_{3}^{2}}
22. \gamma_{sg3}=\frac{-\frac{3}{1+2s}\gamma_{3}\left(1-\frac{(2+s)^{2}}{3(1+2s)}\gamma_{3}^{2}\right)}{1-\frac{3s(2+s)}{(1+2s)^{2}}\gamma_{3}^{2}}
23. Y_{sY3}=\frac{-\frac{3}{1+2s}Y_{3}\left(1-\frac{(2+s)^{2}}{3(1+2s)}Y_{3}^{2}\right)}{1-\frac{3s(2+s)}{(1+2s)^{2}}Y_{3}^{2}}
24. \int_{0}^{Y_{3}}\frac{\frac{3\Upsilon_{sg3}}{1-V_{3}\gamma_{3}^{2}t^{2}}-2}{\sqrt{\left(1-t^{2}\right)\left(1-V_{3}t^{2}\right)}}dt-\int_{0}^{Y_{sY3}}\frac{\frac{M_{s3}q_{sg3}}{1-U_{3}\gamma_{sg3}^{2}t^{2}}}{\sqrt{\left(1-t^{2}\right)\left(1-U_{3}t^{2}\right)}}dt
25. \frac{\Upsilon_{sg3}\gamma_{3}}{\sqrt{\left(1-\gamma_{3}^{2}\right)\left(1-V_{3}\gamma_{3}^{2}\right)}}\operatorname{artanh}\left(\frac{-\frac{\frac{6s(2+s)}{(1+2s)^{2}}\ Y_{3}\ \gamma_{3}}{1-\frac{3s(2+s)}{(1+2s)^{2}}\gamma_{3}^{2}}\sqrt{\left(1-\gamma_{3}^{2}\right)\left(1-V_{3}\gamma_{3}^{2}\right)\left(1-Y_{3}^{2}\right)\left(1-V_{3}Y_{3}^{2}\right)}}{1-\frac{s(2+s)(3(1+2s)^{3}+2(1-22s-39s^{2}-22s^{3}+s^{4})\gamma_{3}^{2}+3s(2+s)^{3}\gamma_{3}^{4})}{(1+2s)^{3}((1+2s)^{2}-3s(2+s)\gamma_{3}^{2})}Y_{3}^{2}-\frac{s^{2}\gamma_{3}^{2}(2+s)^{4}(3(1+2s)-(2+s)^{2}\gamma_{3}^{2})}{(1+2s)^{4}((1+2s)^{2}-3s(2+s)\gamma_{3}^{2})}Y_{3}^{4}}\right)
26. \int_{0}^{x}\frac{\frac{\frac{3\Upsilon_{3}}{M_{3}q_{3}}}{1-U_{3}\beta^{2}t^{2}}-\frac{2}{M_{3}q_{3}}-\frac{2}{3M_{3}\Upsilon_{sg3}}}{\sqrt{\left(1-t^{2}\right)\left(1-U_{3}t^{2}\right)}}dt-\int_{0}^{Y_{sY3}}\frac{\frac{\frac{M_{s3}q_{sg3}}{3\Upsilon_{sg3}}}{1-U_{3}\gamma_{sg3}^{2}t^{2}}}{\sqrt{\left(1-t^{2}\right)\left(1-U_{3}t^{2}\right)}}dt
27. G=\frac{\frac{\Upsilon_{3}\beta}{M_{3}q_{3}}}{\sqrt{\left(1-\beta^{2}\right)\left(1-U_{3}\beta^{2}\right)}}\operatorname{artanh}\left(\frac{\frac{2s(2+s)\ x\ \beta}{1+s(2+s)\beta^{2}}\ \ \sqrt{\left(1-\beta^{2}\right)\left(1-U_{3}\beta^{2}\right)\left(1-x^{2}\right)\left(1-U_{3}x^{2}\right)}}{1+\frac{s(2+s)(1+2s-2(1+2s+s^{2}+2s^{3}+s^{4})\beta^{2}+s^{3}(2+s)\beta^{4})}{(1+2s)(1+s(2+s)\beta^{2})}x^{2}+\frac{s^{4}(2+s)^{2}\beta^{2}(1+2s+s^{2}\beta^{2})}{(1+2s)^{2}(1+s(2+s)\beta^{2})}x^{4}}\right)+\frac{\frac{\Upsilon_{3}\beta}{3M_{3}q_{3}}}{\sqrt{\left(1-\beta^{2}\right)\left(1-U_{3}\beta^{2}\right)}}\operatorname{artanh}\left(\frac{-\frac{\frac{6s(2+s)}{(1+2s)^{2}}\ Y_{3}\ \gamma_{3}}{1-\frac{3s(2+s)}{(1+2s)^{2}}\gamma_{3}^{2}}\sqrt{\left(1-\gamma_{3}^{2}\right)\left(1-V_{3}\gamma_{3}^{2}\right)\left(1-Y_{3}^{2}\right)\left(1-V_{3}Y_{3}^{2}\right)}}{1-\frac{s(2+s)(3(1+2s)^{3}+2(1-22s-39s^{2}-22s^{3}+s^{4})\gamma_{3}^{2}+3s(2+s)^{3}\gamma_{3}^{4})}{(1+2s)^{3}((1+2s)^{2}-3s(2+s)\gamma_{3}^{2})}Y_{3}^{2}-\frac{s^{2}\gamma_{3}^{2}(2+s)^{4}(3(1+2s)-(2+s)^{2}\gamma_{3}^{2})}{(1+2s)^{4}((1+2s)^{2}-3s(2+s)\gamma_{3}^{2})}Y_{3}^{4}}\right)
28. \frac{-\frac{3}{1+2s}Y_{3}\left(1-\frac{(2+s)^{2}}{3(1+2s)}Y_{3}^{2}\right)}{1-\frac{3s(2+s)}{(1+2s)^{2}}Y_{3}^{2}}-\frac{-x\ (3-4x^{2}-4U_{3}x^{2}+6U_{3}x^{4}-U_{3}^{2}x^{8})}{1-6U_{3}x^{4}+4U_{3}x^{6}+4U_{3}^{2}x^{6}-3U_{3}^{2}x^{8}}
29. -3\int_{0}^{x}\frac{\frac{1}{1-U_{3}\gamma_{sg3}^{2}t^{2}}}{\sqrt{\left(1-t^{2}\right)\left(1-U_{3}t^{2}\right)}}dt-\int_{0}^{Y_{sY3}}\frac{\frac{1}{1-U_{3}\gamma_{sg3}^{2}t^{2}}}{\sqrt{\left(1-t^{2}\right)\left(1-U_{3}t^{2}\right)}}dt
30. H=\sqrt{\frac{B}{\left(1-UB\right)\left(1-B\right)}}\operatorname{arctanh}\left(\frac{8Ux^{3}(1-x^{2})(1-Ux^{2})\sqrt{B(1-B)(1-UB)(1-x^{2})(1-Ux^{2})}}{1-6UBx^{2}-3U(2-4B-4UB+UB^{2})x^{4}+4U(1-2B+U-5UB+UB^{2}-2U^{2}B+U^{2}B^{2})x^{6}-3U^{2}(1-4B-4UB+2UB^{2})x^{8}-6U^{3}Bx^{10}+U^{4}B^{2}x^{12}}\right)
31. B=\gamma_{sg3}^{2}
32. U=U_{3}
33. \frac{G-\frac{M_{s3}q_{sg3}}{3\Upsilon_{sg3}}H}{\frac{\Upsilon_{3}\beta}{M_{3}q_{3}}}\beta
34. \frac{\frac{M_{s3}q_{sg3}}{\Upsilon_{sg3}}\sqrt{\frac{B}{\left(1-UB\right)\left(1-B\right)}}}{\frac{\Upsilon_{3}\beta}{M_{3}q_{3}}}
35. \frac{1}{\sqrt{\left(1-\beta^{2}\right)\left(1-U_{3}\beta^{2}\right)}}
36. \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}}
37. -\int_{0}^{x}\frac{\frac{\beta((1-k^{2}\beta^{4})^{2}-4k^{2}\beta^{2}(1-\beta^{2})^{2})((1-k^{2}\beta^{4})^{2}-4\beta^{2}(1-k^{2}\beta^{2})^{2})}{\varkappa(1-6k^{2}\beta^{4}+4k^{2}\beta^{6}+4k^{4}\beta^{6}-3k^{4}\beta^{8})^{2}(1-k^{2}\varkappa^{2}t^{2})}-\frac{3}{1-k^{2}\beta^{2}t^{2}}+\frac{8(1-\beta^{2})(1-k^{2}\beta^{2})}{3-4\beta^{2}-4k^{2}\beta^{2}+6k^{2}\beta^{4}-k^{4}\beta^{8}}}{\sqrt{(1-t^{2})(1-k^{2}t^{2})}}dt
38. \frac{-\beta}{\sqrt{(1-\beta^{2})(1-k^{2}\beta^{2})}}\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-\frac{2k^{2}\beta^{2}(3-6\beta^{2}-6k^{2}\beta^{2}+4\beta^{4}+10k^{2}\beta^{4}+4k^{4}\beta^{4}-6k^{2}\beta^{6}-6k^{4}\beta^{6}+3k^{4}\beta^{8})}{1-6k^{2}\beta^{4}+4k^{2}\beta^{6}+4k^{4}\beta^{6}-3k^{4}\beta^{8}}x^{2}-\frac{k^{4}\beta^{4}(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}}x^{4}}\right)
39. k=\sqrt{U_{3}}
40. \int_{0}^{x}\frac{\frac{3}{1-U_{3}\beta^{2}t^{2}}-\frac{2(q_{3}+3\Upsilon_{sg3})}{3\Upsilon_{3}\Upsilon_{sg3}}-\frac{\frac{q_{3}q_{sg3}}{3\Upsilon_{3}\Upsilon_{sg3}}}{1-U_{3}\gamma_{sg3}^{2}t^{2}}}{\sqrt{\left(1-t^{2}\right)\left(1-U_{3}t^{2}\right)}}dt
41. \int_{0}^{x}\frac{3}{\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{\frac{\beta}{\varkappa}\frac{\sqrt{\left(1-\varkappa^{2}\right)\left(1-k^{2}\varkappa^{2}\right)}}{\sqrt{\left(1-\beta^{2}\right)\left(1-k^{2}\beta^{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}^{x}\frac{\frac{L\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
42. 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}}
43. M_{3}M_{s3}
44. \frac{2k^{2}\beta^{3}\left(\beta+\varkappa\right)}{1-k^{2}\beta^{4}}
45. \frac{2q_{3}+q_{3}q_{sg3}+6\Upsilon_{sg3}-9\Upsilon_{3}\Upsilon_{sg3}}{3\Upsilon_{3}\Upsilon_{sg3}}

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青青子衿

青青子衿 发表于 2024-4-24 13:33
\begin{align*}
&\>\>\>\>\>\>\tfrac{1}{\varUpsilon_{3}}\Omega_{3}(x,\beta,s)+\tfrac{M_{3}q_{3}}{3\varUpsilon_{3}(\varUpsilon_{3}^{*}\circ\gamma_{3})}\Omega^{*}_{3}\left(y_{3},
\gamma_{3},s\right)\\
&=\int_{0}^{x}\left(\tfrac{3}{\left(1-U_{3}\beta^{2}t^{2}\right)\sqrt{\left(1-t^{2}\right)\left(1-U_{3}t^{2}\right)}}
-\tfrac{\frac{2}{\varUpsilon_{3}}+\frac{2q_{3}}{3\varUpsilon_{3}(\varUpsilon_{3}^{*}\circ\gamma_{3})}}{\sqrt{\left(1-t^{2}\right)\left(1-U_{3}t^{2}\right)}}\right){\mathrm{d}}t\\

&\qquad+\int_{0}^{y_{3}^{*}\circ\>\!y_{3}}
\tfrac{\frac{q_3(q_{3}^{*}\circ\gamma_{3})}{9\varUpsilon_{3}(\varUpsilon_{3}^{*}\circ\gamma_{3})}}{\left(1-U_{3}\left({\raise1.5px\gamma}_{3}^{*}\circ\gamma_{3}\right)^{2}t^{2}\right)\sqrt{\left(1-t^{2}\right)\left(1-U_{3}t^{2}\right)}}{\mathrm{d}}t\\

&=\int_{0}^{x}\frac{\frac{3}{1-U_{3}\beta^{2}t^{2}}-\frac{2(q_{3}+3(\varUpsilon_{3}^{*}\circ\gamma_{3}))}{3\varUpsilon_{3}(\varUpsilon_{3}^{*}\circ\gamma_{3})}-\frac{\frac{q_{3}(q_{3}^{*}\circ\gamma_{3})}{3\varUpsilon_{3}(\varUpsilon_{3}^{*}\circ\gamma_{3})}}{1-U_{3}\varkappa_{3}^{2}t^{2}}}{\sqrt{\left(1-t^{2}\right)\left(1-U_{3}t^{2}\right)}}{\mathrm{d}}t\\

&=\tfrac{\beta}{\sqrt{(1-\beta^{2})(1-U_{3}\beta^{2})}}\operatorname{artanh}\left(\tfrac{\frac{2U_{3}\beta^{2}(\beta+\varkappa_{3})\sqrt{(1-\beta^{2})(1-U_{3}\beta^{2})}}{1-U_{3}\beta^{4}}x\sqrt{(1-x^{2})(1-U_{3}x^{2})}}{1-\frac{2U_{3}\beta^{2}(3-6\beta^{2}-6U_{3}\beta^{2}+4\beta^{4}+10U_{3}\beta^{4}+4U_{3}^{2}\beta^{4}-6U_{3}\beta^{6}-6U_{3}^{2}\beta^{6}+3U_{3}^{2}\beta^{8})}{1-6U_{3}\beta^{4}+4U_{3}\beta^{6}+4U_{3}^{2}\beta^{6}-3U_{3}^{2}\beta^{8}}x^{2}-\frac{U_{3}^{2}\beta^{4}(3-4\beta^{2}-4U_{3}\beta^{2}+6U_{3}\beta^{4}-U_{3}^{2}\beta^{8})}{1-6U_{3}\beta^{4}+4U_{3}\beta^{6}+4U_{3}^{2}\beta^{6}-3U_{3}^{2}\beta^{8}}x^{4}}\right)\\
\\
\end{align*}

\begin{align*}
&\phantom{\>=\>\>\>}\frac{\sqrt{(1-\beta^2)(1-k^{2}\beta^2)}}{\beta}\,\Big(\Pi(x;k\beta,k)-F(x;k)\Big)\\
\\
&=\int_{0}^{x}\frac{k^{2}\beta\sqrt{(1-\beta^2)(1-k^2\beta^2)}\>t^2}{\left(1-k^{2}\beta^2t^2\right)\sqrt{(1-t^2)(1-k^2t^2)}}{\mathrm{d}}t\\

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

1. \frac{\sqrt{(1-\beta^{2})(1-k^{2}\beta^{2})}}{\beta}\left(\int_{0}^{x}\frac{1}{(1-k^{2}\beta^{2}t^{2})\sqrt{(1-t^{2})(1-k^{2}t^{2})}}dt-\int_{0}^{x}\frac{1}{\sqrt{(1-t^{2})(1-k^{2}t^{2})}}dt\right)
2. \int_{0}^{x}\frac{k^{2}\beta\sqrt{(1-\beta^{2})(1-k^{2}\beta^{2})}t^{2}}{(1-k^{2}\beta^{2}t^{2})\sqrt{(1-t^{2})(1-k^{2}t^{2})}}dt
3. k=0.9
4. \beta=0.4
5. \left(\int_{0}^{x}\frac{1}{\sqrt{(1-t^{2})(1-k^{2}t^{2})}}dt\right)\left(\int_{0}^{\beta}\sqrt{\frac{1-k^{2}t^{2}}{1-t^{2}}}dt\right)-\frac{1}{2}\int_{Q}^{P}\int_{0}^{v}\sqrt{\frac{1-k^{2}u^{2}}{(1-u^{2})(1-v^{2})(1-k^{2}v^{2})}}dudv
6. \left(\int_{0}^{x}\frac{1}{\sqrt{(1-t^{2})(1-k^{2}t^{2})}}dt\right)\left(\int_{0}^{\beta}\sqrt{\frac{1-k^{2}t^{2}}{1-t^{2}}}dt\right)-\frac{1}{2}\int_{Q}^{P}\frac{\int_{0}^{v}\sqrt{\frac{1-k^{2}u^{2}}{1-u^{2}}}du}{\sqrt{(1-v^{2})(1-k^{2}v^{2})}}dv
7. P=\frac{x\sqrt{(1-\beta^{2})(1-k^{2}\beta^{2})}+\beta\sqrt{(1-x^{2})(1-k^{2}x^{2})}}{1-k^{2}\beta^{2}x^{2}}
8. Q=\frac{x\sqrt{(1-\beta^{2})(1-k^{2}\beta^{2})}-\beta\sqrt{(1-x^{2})(1-k^{2}x^{2})}}{1-k^{2}\beta^{2}x^{2}}

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青青子衿

青青子衿 发表于 2024-1-30 17:40
\begin{align*}
r_{1}&=\frac{p\sqrt{\left(1-q^{2}\right)\left(1-k^{2}q^{2}\right)}+q\sqrt{\left(1-p^{2}\right)\left(1-k^{2}p^{2}\right)}}{1-k^{2}p^{2}q^{2}}\\

P_1&=\sqrt{\left(1-k^{2}r_{1}^{2}\right)\left(1-k^{2}x^{2}\right)}-k^{2}r_{1}x\sqrt{\left(1-r_{1}^{2}\right)\left(1-x^{2}\right)}\\

Q_1&=\sqrt{\left(1-k^{2}r_{1}^{2}\right)\left(1-k^{2}x^{2}\right)}+k^{2}r_{1}x\sqrt{\left(1-r_{1}^{2}\right)\left(1-x^{2}\right)}\\

R_1&=x\sqrt{\left(1-r_{1}^{2}\right)\left(1-k^{2}x^{2}\right)}-r_{1}\sqrt{\left(1-x^{2}\right)\left(1-k^{2}r_{1}^{2}\right)}\\

S_1&=x\sqrt{\left(1-r_{1}^{2}\right)\left(1-k^{2}x^{2}\right)}+r_{1}\sqrt{\left(1-x^{2}\right)\left(1-k^{2}r_{1}^{2}\right)}\\

T_1&=\frac{\sqrt{\left(1-p^{2}\right)\left(1-k^{2}p^{2}\right)}}{p}+\frac{\sqrt{\left(1-q^{2}\right)\left(1-k^{2}q^{2}\right)}}{q}\\
&\qquad\qquad-\frac{\sqrt{\left(1-r_{1}^{2}\right)\left(1-k^{2}r_{1}^{2}\right)}}{r_{1}}+k^{2}pqr_{1}\\
\\
\Omega(kp)+\Omega(kq)-\Omega(r_1)&=\int_{0}^{x}\frac{\frac{1}{p}\sqrt{\left(1-p^{2}\right)\left(1-k^{2}p^{2}\right)}}{\left(1-k^{2}p^{2}t^{2}\right)\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}\mathrm{d}t\\
&\qquad\quad+\int_{0}^{x}\frac{\frac{1}{q}\sqrt{\left(1-q^{2}\right)\left(1-k^{2}q^{2}\right)}}{\left(1-k^{2}q^{2}t^{2}\right)\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}\mathrm{d}t\\

&\qquad\qquad\>-\int_{0}^{x}\frac{\frac{1}{r_{1}}\sqrt{\left(1-r_{1}^{2}\right)\left(1-k^{2}r_{1}^{2}\right)}}{\left(1-k^{2}r_{1}^{2}t^{2}\right)\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}\mathrm{d}t\\

&=
T_1\int_{0}^{x}\frac{\mathrm{d}t}{\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}+\frac{1}{2}\ln\left(\frac{P_1Q_1+k^{2}pqxQ_1R_1}{P_1Q_1+k^{2}pqxP_1S_1}\right)\\
\\
\mathcal{F}(p)+\mathcal{F}(q)-\mathcal{F}(r_1)&=
\int_{0}^{p}\frac{\mathrm{d}t}{\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}dt+\int_{0}^{q}\frac{\mathrm{d}t}{\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}\\
&\qquad\qquad\qquad-\int_{0}^{r_{1}}\frac{\mathrm{d}t}{\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}

\end{align*}

\begin{align*}
r_{1}&=\frac{p\sqrt{\left(1-q^{2}\right)\left(1-k^{2}q^{2}\right)}+q\sqrt{\left(1-p^{2}\right)\left(1-k^{2}p^{2}\right)}}{1-k^{2}p^{2}q^{2}}\\
T_1&=\frac{\sqrt{\left(1-p^{2}\right)\left(1-k^{2}p^{2}\right)}}{p}+\frac{\sqrt{\left(1-q^{2}\right)\left(1-k^{2}q^{2}\right)}}{q}\\
&\qquad\qquad-\frac{\sqrt{\left(1-r_{1}^{2}\right)\left(1-k^{2}r_{1}^{2}\right)}}{r_{1}}+k^{2}pqr_{1}\\
\\
\Omega(kp)+\Omega(kq)-\Omega(r_1)&=\int_{0}^{x}\frac{\frac{1}{p}\sqrt{\left(1-p^{2}\right)\left(1-k^{2}p^{2}\right)}}{\left(1-k^{2}p^{2}t^{2}\right)\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}\mathrm{d}t\\
&\qquad\quad+\int_{0}^{x}\frac{\frac{1}{q}\sqrt{\left(1-q^{2}\right)\left(1-k^{2}q^{2}\right)}}{\left(1-k^{2}q^{2}t^{2}\right)\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}\mathrm{d}t\\
&\qquad\qquad\>-\int_{0}^{x}\frac{\frac{1}{r_{1}}\sqrt{\left(1-r_{1}^{2}\right)\left(1-k^{2}r_{1}^{2}\right)}}{\left(1-k^{2}r_{1}^{2}t^{2}\right)\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}\mathrm{d}t\\
&=
T_1\int_{0}^{x}\frac{\mathrm{d}t}{\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}\\
&\qquad-\operatorname{arctanh}\left(\frac{k^{2}pqr_{1}x\sqrt{\left(1-x^{2}\right)\left(1-k^{2}x^{2}\right)}}{1-k^{2}r_{1}^{2}x^{2}+k^{2}pqx^{2}\sqrt{\left(1-r_{1}^{2}\right)\left(1-k^{2}r_{1}^{2}\right)}}\right)\\
\\
\mathcal{F}(p)+\mathcal{F}(q)-\mathcal{F}(r_1)&=
\int_{0}^{p}\frac{\mathrm{d}t}{\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}dt+\int_{0}^{q}\frac{\mathrm{d}t}{\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}\\
&\qquad\qquad\qquad-\int_{0}^{r_{1}}\frac{\mathrm{d}t}{\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}

\end{align*}

1. r_{1}=\frac{p\sqrt{\left(1-q^{2}\right)\left(1-k^{2}q^{2}\right)}+q\sqrt{\left(1-p^{2}\right)\left(1-k^{2}p^{2}\right)}}{1-k^{2}p^{2}q^{2}}
2. T=\frac{\sqrt{\left(1-p^{2}\right)\left(1-k^{2}p^{2}\right)}}{p}+\frac{\sqrt{\left(1-q^{2}\right)\left(1-k^{2}q^{2}\right)}}{q}-\frac{\sqrt{\left(1-r_{1}^{2}\right)\left(1-k^{2}r_{1}^{2}\right)}}{r_{1}}+k^{2}pqr_{1}
3. \int_{0}^{x}\frac{\frac{1}{p}\sqrt{\left(1-p^{2}\right)\left(1-k^{2}p^{2}\right)}}{\left(1-k^{2}p^{2}t^{2}\right)\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}dt+\int_{0}^{x}\frac{\frac{1}{q}\sqrt{\left(1-q^{2}\right)\left(1-k^{2}q^{2}\right)}}{\left(1-k^{2}q^{2}t^{2}\right)\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}dt-\int_{0}^{x}\frac{\frac{1}{r_{1}}\sqrt{\left(1-r_{1}^{2}\right)\left(1-k^{2}r_{1}^{2}\right)}}{\left(1-k^{2}r_{1}^{2}t^{2}\right)\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}dt
4. T\int_{0}^{x}\frac{1}{\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}dt-\operatorname{arctanh}\left(\frac{k^{2}pqr_{1}x\sqrt{\left(1-x^{2}\right)\left(1-k^{2}x^{2}\right)}}{(1-k^{2}r_{1}^{2}x^{2})+k^{2}pqx^{2}\sqrt{\left(1-r_{1}^{2}\right)\left(1-k^{2}r_{1}^{2}\right)}}\right)
5. \int_{0}^{p}\frac{1}{\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}dt+\int_{0}^{q}\frac{1}{\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}dt-\int_{0}^{r_{1}}\frac{1}{\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}dt

复制代码

青青子衿

\begin{align*}
&=\int_{0}^{x}\frac{p\sqrt{\left(1-p^{2}\right)\left(1-k^{2}p^{2}\right)}}{\left(p^{2}-t^{2}\right)\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}{\mathrm{d}}t\\
&\qquad+\int_{0}^{x}\frac{q\sqrt{\left(1-q^{2}\right)\left(1-k^{2}q^{2}\right)}}{\left(q^{2}-t^{2}\right)\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}{\mathrm{d}}t\\

&\qquad\quad-\int_{0}^{x}\frac{r_{1}\sqrt{\scriptsize(1-r_{1}^{2})(1-k^{2}r_{1}^{2})}}{\left(r_{1}^{2}-t^{2}\right)\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}{\mathrm{d}}t\\

&\qquad\qquad\quad+\int_{0}^{x}\frac{k^{2}pqr_{1}}{\sqrt{\left(1-t^{2}\right)\left(1-k^{2}t^{2}\right)}}{\mathrm{d}}t\\



&=\operatorname{arctanh}\left(\frac{k^{2}pqr_{1}x\sqrt{(1-x^{2})(1-k^{2}x^{2})}}{1-k^{2}x^{2}(r_{1}^{2}-pq\sqrt{\scriptsize(1-r_{1}^{2})(1-k^{2}r_{1}^{2})} )}\right)\\

&\qquad+\operatorname{arctanh}\left(\frac{x\sqrt{(1-p^{2})(1-k^{2}p^{2})}}{p\sqrt{(1-x^{2})(1-k^{2}x^{2})}}\right)\\
&\qquad\qquad+\operatorname{arctanh}\left(\frac{x\sqrt{(1-q^{2})(1-k^{2}q^{2})}}{q\sqrt{(1-x^{2})(1-k^{2}x^{2})}}\right)\\
&\qquad\qquad\qquad-\operatorname{arctanh}\left(\frac{x\sqrt{(1-r_{1}^{2})(1-k^{2}r_{1}^{2})}}{r_{1}\sqrt{(1-x^{2})(1-k^{2}x^{2})}}\right)

\end{align*}




\begin{gather*}
\int_x^{+\infty}\left(\frac{P}{p-x}+\frac{Q}{q-x}-\frac{R}{r-x}-\frac{P-Q}{p-q}\right)\frac{{\mathrm{d}}x}{\sqrt{B+A x+x^3}}\\
=-2\operatorname{artanh}\left(\frac{\frac{P-Q}{p-q}x+\frac{pQ-qP}{p-q}}{\sqrt{B+Ax+x^3}}\right)\\
\\
\left\{\begin{split}
P^2&=p^3+Ap+B\\
Q^2&=q^3+Aq+B\\
r&=\tfrac{(p+q) (A+p q)+2 (B-P Q)}{(p-q)^2}\\
&=\left(\tfrac{P-Q}{p-q}\right)^2-p-q\\
R&=\frac{\begin{subarray}{l}P\left(4B+A(p+3q)+q^{2}(3p+q)\right)\\
\qquad\quad-Q\left(4B+A(3p+q)+p^{2}(p+3q)\right)\end{subarray}}{(p-q)^{3}}\\
&=\left(\tfrac{P-Q}{p-q}\right)^{3}-\tfrac{pP-qQ}{p-q}+\tfrac{2(pQ-qP)}{p-q}

\end{split}\right.
\end{gather*}



\begin{align*}
\int\frac{x-3}{\left(x^3-9 x^2-81 x-351\right) \sqrt{x^3+27 x-27}}{\mathrm{d}}x\\
\int\frac{x^2-9}{\left(x^3-9 x^2-81 x-351\right) \sqrt{x^3+27 x-27}}{\mathrm{d}}x\\
\int\frac{x^3+297}{\left(x^3-9 x^2-81 x-351\right) \sqrt{x^3+27 x-27}}{\mathrm{d}}x\\
\end{align*}

\begin{align*}
\int\frac{x-8}{(x^3+6 x^2+72 x+148)\sqrt{x^3-12 x-11}}{\mathrm{d}}x\\
\int\frac{x^2+26}{(x^3+6 x^2+72 x+148)\sqrt{x^3-12 x-11}}{\mathrm{d}}x\\
\int\frac{x^3-242}{(x^3+6 x^2+72 x+148)\sqrt{x^3-12 x-11}}{\mathrm{d}}x
\end{align*}