求一不定积分

wayne

咱们论坛之前出现过各种大量的，定积分的case，其技巧性往往也特别的多。这次来一个不定积分
\[\int\frac{\sqrt{x^2+1}}{x^3+1}\dif x\]

.·.·.

这是FullSimplify的结果
\(\frac{1}{6} \left(-2 \sqrt{2} \log \left(\sqrt{2} \sqrt{x^2+1}-x+1\right)+i \left(\log \left(((x-1) x+1) \left(2 \sqrt{x^2+1}-2 i \sqrt{3} \sqrt{x^2+1}+x \left(\sqrt{x^2+1}-i \sqrt{3} \sqrt{x^2+1}+\left(\sqrt{3}-3 i\right) x-2 i\right)-3 i+\sqrt{3}\right)\right)-\log \left(((x-1) x+1) \left(2 \sqrt{x^2+1}+2 i \sqrt{3} \sqrt{x^2+1}+x \left(\sqrt{x^2+1}+i \sqrt{3} \sqrt{x^2+1}+\left(\sqrt{3}+3 i\right) x+2 i\right)+3 i+\sqrt{3}\right)\right)\right)-2 i \tanh ^{-1}\left(\frac{x \left(4 \sqrt{3} \sqrt{x^2+1}+x \left(-4 \sqrt{3} \sqrt{x^2+1}+2 x \left(2 \sqrt{3} \sqrt{x^2+1}+3 x+i \sqrt{3}-3\right)+i \sqrt{3}+9\right)+2 i \sqrt{3}-6\right)}{8+x \left(x \left(2 x \left(x-2 i \sqrt{3}-2\right)+i \sqrt{3}+9\right)-4 i \sqrt{3}-4\right)}\right)+2 i \tanh ^{-1}\left(\frac{x \left(6 \sqrt{x^2+1}+2 i \sqrt{3} \sqrt{x^2+1}+x \left(-6 \sqrt{x^2+1}-2 i \sqrt{3} \sqrt{x^2+1}+x \left(6 \sqrt{x^2+1}+2 i \sqrt{3} \sqrt{x^2+1}+3 \left(\sqrt{3}+i\right) x-6 i-2 \sqrt{3}\right)+3 i+5 \sqrt{3}\right)-6 i-2 \sqrt{3}\right)}{x \left(x \left(x \left(\left(\sqrt{3}+i\right) x+4 i-4 \sqrt{3}\right)+3 i+5 \sqrt{3}\right)+4 i-4 \sqrt{3}\right)+4 \left(\sqrt{3}+i\right)}\right)+2 \sqrt{2} \log (x+1)\right)\)
看了一下用时

1. In[1]:= Timing[FullSimplify[Integrate[Sqrt[x^2+1]/(x^3+1),x]]]
   
2. Out[1]= {110.766,1/6 (2 I ArcTanh[(x (-6 I-2 Sqrt[3]+6 Sqrt[1+x^2]+2 I Sqrt[3] Sqrt[1+x^2]+x (3 I+5 Sqrt[3]-6 Sqrt[1+x^2]-2 I Sqrt[3] Sqrt[1+x^2]+x (-6 I-2 Sqrt[3]+3 (I+Sqrt[3]) x+6 Sqrt[1+x^2]+2 I Sqrt[3] Sqrt[1+x^2]))))/(4 (I+Sqrt[3])+x (4 I-4 Sqrt[3]+x (3 I+5 Sqrt[3]+x (4 I-4 Sqrt[3]+(I+Sqrt[3]) x))))]-2 I ArcTanh[(x (-6+2 I Sqrt[3]+4 Sqrt[3] Sqrt[1+x^2]+x (9+I Sqrt[3]-4 Sqrt[3] Sqrt[1+x^2]+2 x (-3+I Sqrt[3]+3 x+2 Sqrt[3] Sqrt[1+x^2]))))/(8+x (-4-4 I Sqrt[3]+x (9+I Sqrt[3]+2 x (-2-2 I Sqrt[3]+x))))]+2 Sqrt[2] Log[1+x]-2 Sqrt[2] Log[1-x+Sqrt[2] Sqrt[1+x^2]]+I (Log[(1+(-1+x) x) (-3 I+Sqrt[3]+2 Sqrt[1+x^2]-2 I Sqrt[3] Sqrt[1+x^2]+x (-2 I+(-3 I+Sqrt[3]) x+Sqrt[1+x^2]-I Sqrt[3] Sqrt[1+x^2]))]-Log[(1+(-1+x) x) (3 I+Sqrt[3]+2 Sqrt[1+x^2]+2 I Sqrt[3] Sqrt[1+x^2]+x (2 I+(3 I+Sqrt[3]) x+Sqrt[1+x^2]+I Sqrt[3] Sqrt[1+x^2]))]))}

复制代码

如果不化简

1. In[1]:= Timing[Integrate[Sqrt[x^2+1]/(x^3+1),x]]
   
2. Out[1]= {2.32813,(1/(18 Sqrt[2]))((1/Sqrt[1/3-I/Sqrt[3]])2 (3-I Sqrt[3]) ArcTan[(x (6+2 I Sqrt[3]+(-3-3 I Sqrt[3]) x^3-2 I Sqrt[6-6 I Sqrt[3]] Sqrt[1+x^2]+x^2 (6+2 I Sqrt[3]-2 I Sqrt[6-6 I Sqrt[3]] Sqrt[1+x^2])+I x (3 I-5 Sqrt[3]+2 Sqrt[6-6 I Sqrt[3]] Sqrt[1+x^2])))/(4 (-I+Sqrt[3])-4 (I+Sqrt[3]) x+(-3 I+5 Sqrt[3]) x^2-4 (I+Sqrt[3]) x^3+(-I+Sqrt[3]) x^4)]-(1/Sqrt[1/3+I/Sqrt[3]])2 I (-3 I+Sqrt[3]) ArcTan[(x (-6+2 I Sqrt[3]+(3-3 I Sqrt[3]) x^3-2 I Sqrt[6+6 I Sqrt[3]] Sqrt[1+x^2]+x^2 (-6+2 I Sqrt[3]-2 I Sqrt[6+6 I Sqrt[3]] Sqrt[1+x^2])+x (3-5 I Sqrt[3]+2 I Sqrt[6+6 I Sqrt[3]] Sqrt[1+x^2])))/(4 (I+Sqrt[3])-4 (-I+Sqrt[3]) x+(3 I+5 Sqrt[3]) x^2-4 (-I+Sqrt[3]) x^3+(I+Sqrt[3]) x^4)]+12 Log[1+x]+((-3 I+Sqrt[3]) Log[16 (1-x+x^2)^2])/Sqrt[1/3+I/Sqrt[3]]+((3 I+Sqrt[3]) Log[16 (1-x+x^2)^2])/Sqrt[1/3-I/Sqrt[3]]-12 Log[1-x+Sqrt[2] Sqrt[1+x^2]]-(1/Sqrt[1/3-I/Sqrt[3]])(3 I+Sqrt[3]) Log[(1-x+x^2) (3 I+Sqrt[3]+(3 I+Sqrt[3]) x^2+2 I Sqrt[2-2 I Sqrt[3]] Sqrt[1+x^2]+I x (2+Sqrt[2-2 I Sqrt[3]] Sqrt[1+x^2]))]-(1/Sqrt[1/3+I/Sqrt[3]])(-3 I+Sqrt[3]) Log[(1-x+x^2) (-3 I+Sqrt[3]+(-3 I+Sqrt[3]) x^2-2 I Sqrt[2+2 I Sqrt[3]] Sqrt[1+x^2]-I x (2+Sqrt[2+2 I Sqrt[3]] Sqrt[1+x^2]))])}

复制代码

普通化简

1. In[1]:= Timing[Simplify[Integrate[Sqrt[x^2+1]/(x^3+1),x]]]
   
2. Out[1]= {3.57813,(1/(18 Sqrt[2]))((1/Sqrt[1/3-I/Sqrt[3]])2 (3-I Sqrt[3]) ArcTan[(x (6+2 I Sqrt[3]+(-3-3 I Sqrt[3]) x^3-2 I Sqrt[6-6 I Sqrt[3]] Sqrt[1+x^2]+x^2 (6+2 I Sqrt[3]-2 I Sqrt[6-6 I Sqrt[3]] Sqrt[1+x^2])+I x (3 I-5 Sqrt[3]+2 Sqrt[6-6 I Sqrt[3]] Sqrt[1+x^2])))/(4 (-I+Sqrt[3])-4 (I+Sqrt[3]) x+(-3 I+5 Sqrt[3]) x^2-4 (I+Sqrt[3]) x^3+(-I+Sqrt[3]) x^4)]-(1/Sqrt[1/3+I/Sqrt[3]])2 I (-3 I+Sqrt[3]) ArcTan[(x (-6+2 I Sqrt[3]+(3-3 I Sqrt[3]) x^3-2 I Sqrt[6+6 I Sqrt[3]] Sqrt[1+x^2]+x^2 (-6+2 I Sqrt[3]-2 I Sqrt[6+6 I Sqrt[3]] Sqrt[1+x^2])+x (3-5 I Sqrt[3]+2 I Sqrt[6+6 I Sqrt[3]] Sqrt[1+x^2])))/(4 (I+Sqrt[3])-4 (-I+Sqrt[3]) x+(3 I+5 Sqrt[3]) x^2-4 (-I+Sqrt[3]) x^3+(I+Sqrt[3]) x^4)]+12 Log[1+x]+((-3 I+Sqrt[3]) Log[16 (1-x+x^2)^2])/Sqrt[1/3+I/Sqrt[3]]+((3 I+Sqrt[3]) Log[16 (1-x+x^2)^2])/Sqrt[1/3-I/Sqrt[3]]-12 Log[1-x+Sqrt[2] Sqrt[1+x^2]]-(1/Sqrt[1/3-I/Sqrt[3]])(3 I+Sqrt[3]) Log[(1-x+x^2) (3 I+Sqrt[3]+(3 I+Sqrt[3]) x^2+2 I Sqrt[2-2 I Sqrt[3]] Sqrt[1+x^2]+I x (2+Sqrt[2-2 I Sqrt[3]] Sqrt[1+x^2]))]-(1/Sqrt[1/3+I/Sqrt[3]])(-3 I+Sqrt[3]) Log[(1-x+x^2) (-3 I+Sqrt[3]+(-3 I+Sqrt[3]) x^2-2 I Sqrt[2+2 I Sqrt[3]] Sqrt[1+x^2]-I x (2+Sqrt[2+2 I Sqrt[3]] Sqrt[1+x^2]))])}

复制代码

Expand
\(-\frac{1}{3} \sqrt{2} \log \left(\sqrt{2} \sqrt{x^2+1}-x+1\right)-\frac{1}{6} i \log \left(((x-1) x+1) \left(2 \sqrt{x^2+1}+2 i \sqrt{3} \sqrt{x^2+1}+x \left(\sqrt{x^2+1}+i \sqrt{3} \sqrt{x^2+1}+\left(\sqrt{3}+3 i\right) x+2 i\right)+3 i+\sqrt{3}\right)\right)+\frac{1}{6} i \log \left(((x-1) x+1) \left(2 \sqrt{x^2+1}-2 i \sqrt{3} \sqrt{x^2+1}+x \left(\sqrt{x^2+1}-i \sqrt{3} \sqrt{x^2+1}+\left(\sqrt{3}-3 i\right) x-2 i\right)-3 i+\sqrt{3}\right)\right)-\frac{1}{3} i \tanh ^{-1}\left(\frac{x \left(4 \sqrt{3} \sqrt{x^2+1}+x \left(-4 \sqrt{3} \sqrt{x^2+1}+2 x \left(2 \sqrt{3} \sqrt{x^2+1}+3 x+i \sqrt{3}-3\right)+i \sqrt{3}+9\right)+2 i \sqrt{3}-6\right)}{8+x \left(x \left(2 x \left(x-2 i \sqrt{3}-2\right)+i \sqrt{3}+9\right)-4 i \sqrt{3}-4\right)}\right)+\frac{1}{3} i \tanh ^{-1}\left(\frac{x \left(6 \sqrt{x^2+1}+2 i \sqrt{3} \sqrt{x^2+1}+x \left(-6 \sqrt{x^2+1}-2 i \sqrt{3} \sqrt{x^2+1}+x \left(6 \sqrt{x^2+1}+2 i \sqrt{3} \sqrt{x^2+1}+3 \left(\sqrt{3}+i\right) x-6 i-2 \sqrt{3}\right)+3 i+5 \sqrt{3}\right)-6 i-2 \sqrt{3}\right)}{x \left(x \left(x \left(\left(\sqrt{3}+i\right) x+4 i-4 \sqrt{3}\right)+3 i+5 \sqrt{3}\right)+4 i-4 \sqrt{3}\right)+4 \left(\sqrt{3}+i\right)}\right)+\frac{1}{3} \sqrt{2} \log (x+1)\)
ExpandAll
\(\frac{1}{3} \tan ^{-1}\left(\frac{3 i \sqrt{3} x^4}{i x^4+\sqrt{3} x^4+4 i x^3-4 \sqrt{3} x^3+3 i x^2+5 \sqrt{3} x^2+4 i x-4 \sqrt{3} x+4 i+4 \sqrt{3}}-\frac{3 x^4}{i x^4+\sqrt{3} x^4+4 i x^3-4 \sqrt{3} x^3+3 i x^2+5 \sqrt{3} x^2+4 i x-4 \sqrt{3} x+4 i+4 \sqrt{3}}+\frac{6 i \sqrt{x^2+1} x^3}{i x^4+\sqrt{3} x^4+4 i x^3-4 \sqrt{3} x^3+3 i x^2+5 \sqrt{3} x^2+4 i x-4 \sqrt{3} x+4 i+4 \sqrt{3}}-\frac{2 i \sqrt{3} x^3}{i x^4+\sqrt{3} x^4+4 i x^3-4 \sqrt{3} x^3+3 i x^2+5 \sqrt{3} x^2+4 i x-4 \sqrt{3} x+4 i+4 \sqrt{3}}-\frac{2 \sqrt{3} \sqrt{x^2+1} x^3}{i x^4+\sqrt{3} x^4+4 i x^3-4 \sqrt{3} x^3+3 i x^2+5 \sqrt{3} x^2+4 i x-4 \sqrt{3} x+4 i+4 \sqrt{3}}+\frac{6 x^3}{i x^4+\sqrt{3} x^4+4 i x^3-4 \sqrt{3} x^3+3 i x^2+5 \sqrt{3} x^2+4 i x-4 \sqrt{3} x+4 i+4 \sqrt{3}}+\frac{5 i \sqrt{3} x^2}{i x^4+\sqrt{3} x^4+4 i x^3-4 \sqrt{3} x^3+3 i x^2+5 \sqrt{3} x^2+4 i x-4 \sqrt{3} x+4 i+4 \sqrt{3}}+\frac{2 \sqrt{3} \sqrt{x^2+1} x^2}{i x^4+\sqrt{3} x^4+4 i x^3-4 \sqrt{3} x^3+3 i x^2+5 \sqrt{3} x^2+4 i x-4 \sqrt{3} x+4 i+4 \sqrt{3}}-\frac{6 i \sqrt{x^2+1} x^2}{i x^4+\sqrt{3} x^4+4 i x^3-4 \sqrt{3} x^3+3 i x^2+5 \sqrt{3} x^2+4 i x-4 \sqrt{3} x+4 i+4 \sqrt{3}}-\frac{3 x^2}{i x^4+\sqrt{3} x^4+4 i x^3-4 \sqrt{3} x^3+3 i x^2+5 \sqrt{3} x^2+4 i x-4 \sqrt{3} x+4 i+4 \sqrt{3}}+\frac{6 i \sqrt{x^2+1} x}{i x^4+\sqrt{3} x^4+4 i x^3-4 \sqrt{3} x^3+3 i x^2+5 \sqrt{3} x^2+4 i x-4 \sqrt{3} x+4 i+4 \sqrt{3}}-\frac{2 i \sqrt{3} x}{i x^4+\sqrt{3} x^4+4 i x^3-4 \sqrt{3} x^3+3 i x^2+5 \sqrt{3} x^2+4 i x-4 \sqrt{3} x+4 i+4 \sqrt{3}}-\frac{2 \sqrt{3} \sqrt{x^2+1} x}{i x^4+\sqrt{3} x^4+4 i x^3-4 \sqrt{3} x^3+3 i x^2+5 \sqrt{3} x^2+4 i x-4 \sqrt{3} x+4 i+4 \sqrt{3}}+\frac{6 x}{i x^4+\sqrt{3} x^4+4 i x^3-4 \sqrt{3} x^3+3 i x^2+5 \sqrt{3} x^2+4 i x-4 \sqrt{3} x+4 i+4 \sqrt{3}}\right)+\frac{1}{3} i \tanh ^{-1}\left(-\frac{6 x^4}{2 x^4-4 i \sqrt{3} x^3-4 x^3+i \sqrt{3} x^2+9 x^2-4 i \sqrt{3} x-4 x+8}-\frac{2 i \sqrt{3} x^3}{2 x^4-4 i \sqrt{3} x^3-4 x^3+i \sqrt{3} x^2+9 x^2-4 i \sqrt{3} x-4 x+8}-\frac{4 \sqrt{3} \sqrt{x^2+1} x^3}{2 x^4-4 i \sqrt{3} x^3-4 x^3+i \sqrt{3} x^2+9 x^2-4 i \sqrt{3} x-4 x+8}+\frac{6 x^3}{2 x^4-4 i \sqrt{3} x^3-4 x^3+i \sqrt{3} x^2+9 x^2-4 i \sqrt{3} x-4 x+8}+\frac{4 \sqrt{3} \sqrt{x^2+1} x^2}{2 x^4-4 i \sqrt{3} x^3-4 x^3+i \sqrt{3} x^2+9 x^2-4 i \sqrt{3} x-4 x+8}-\frac{i \sqrt{3} x^2}{2 x^4-4 i \sqrt{3} x^3-4 x^3+i \sqrt{3} x^2+9 x^2-4 i \sqrt{3} x-4 x+8}-\frac{9 x^2}{2 x^4-4 i \sqrt{3} x^3-4 x^3+i \sqrt{3} x^2+9 x^2-4 i \sqrt{3} x-4 x+8}-\frac{2 i \sqrt{3} x}{2 x^4-4 i \sqrt{3} x^3-4 x^3+i \sqrt{3} x^2+9 x^2-4 i \sqrt{3} x-4 x+8}-\frac{4 \sqrt{3} \sqrt{x^2+1} x}{2 x^4-4 i \sqrt{3} x^3-4 x^3+i \sqrt{3} x^2+9 x^2-4 i \sqrt{3} x-4 x+8}+\frac{6 x}{2 x^4-4 i \sqrt{3} x^3-4 x^3+i \sqrt{3} x^2+9 x^2-4 i \sqrt{3} x-4 x+8}\right)-\frac{1}{6} i \log \left(3 i x^4+\sqrt{3} x^4-i x^3-\sqrt{3} x^3+i \sqrt{3} \sqrt{x^2+1} x^3+\sqrt{x^2+1} x^3+4 i x^2+2 \sqrt{3} x^2+i \sqrt{3} \sqrt{x^2+1} x^2+\sqrt{x^2+1} x^2-i x-\sqrt{3} x-i \sqrt{3} \sqrt{x^2+1} x-\sqrt{x^2+1} x+3 i+\sqrt{3}+2 i \sqrt{3} \sqrt{x^2+1}+2 \sqrt{x^2+1}\right)-\frac{1}{3} \sqrt{2} \log \left(-x+\sqrt{2} \sqrt{x^2+1}+1\right)+\frac{1}{6} i \log \left(-3 i x^4+\sqrt{3} x^4+i x^3-\sqrt{3} x^3-i \sqrt{3} \sqrt{x^2+1} x^3+\sqrt{x^2+1} x^3-4 i x^2+2 \sqrt{3} x^2-i \sqrt{3} \sqrt{x^2+1} x^2+\sqrt{x^2+1} x^2+i x-\sqrt{3} x+i \sqrt{3} \sqrt{x^2+1} x-\sqrt{x^2+1} x-3 i+\sqrt{3}-2 i \sqrt{3} \sqrt{x^2+1}+2 \sqrt{x^2+1}\right)+\frac{1}{3} \log (x+1) \sqrt{2}\)
然后用了一款神奇的软件

1. (3) -> integrate(sqrt(x^2+1)/(x^3+1),x)
   
2. 
   
3.    (3)
   
4.                               +------+
   
5.                  +-+          | 2                  +-+    2
   
6.         +-+    (\|2  + x + 1)\|x  + 1  + (- x - 1)\|2  - x  - x - 2
   
7.        \|2 log(----------------------------------------------------)
   
8.                                      +------+
   
9.                                      | 2         2
   
10.                              (x + 1)\|x  + 1  - x  - x
    
11.      +
    
12.                           +------+
    
13.                           | 2           2
    
14.                 (2 x - 1)\|x  + 1  - 2 x  + x - 1
    
15.        - 2 atan(---------------------------------)
    
16.                          +------+
    
17.                          | 2
    
18.                         \|x  + 1  - x - 1
    
19.   /
    
20.      3
    
21.                                          Type: Union(Expression(Integer),...)

复制代码

wayne

某软件 秒杀的结果是：

\[\frac{\sqrt{2} \log (\frac{(\sqrt{2} + x + 1) \sqrt{x^2 + 1} + (- x - 1)
\sqrt{2} - x^2 - x - 2}{(x + 1) \sqrt{x^2 + 1} - x^2 - x}) - 2 \text{atan}
(\frac{(2 x - 1) \sqrt{x^2 + 1} - 2 x^2 + x - 1}{\sqrt{x^2 + 1} - x - 1})}{3}\]

.·.·.

wayne 发表于 2018-12-2 09:29
某软件 秒杀的结果是：

\[\frac{\sqrt{2} \log (\frac{(\sqrt{2} + x + 1) \sqrt{x^2 + 1} + (- x - 1)
...

为什么这款软件到了你这里就这么好看了

还有
试过Mathematica做FullSimplify
发现这两个不定积分的确只是差一个常数
然鹅Mathematica根本没有意识到这一点
我限制了$x\in\text{Integers}$进行化简（里面还有一句Refine）都没有用

wayne

事实上，经过测试。这些积分都是有较为简洁但不简单的初等原函数的

\[\int\frac{\sqrt{px+q}}{ax+b}\dif x\]
\[\int\frac{\sqrt{px+q}}{cx^2+bx+c}\dif x\]
\[\int\frac{\sqrt{px^2+qx+r}}{ax+b}\dif x\]
\[\int\frac{\sqrt{px^2+qx+r}}{ax^2+bx+c}\dif x\]

wayne

还是认认真真的计算一下吧，$\int\frac{\sqrt{x^2+1}}{x^3+1}dx$
变量代换$x=tant$,得到 $\int\frac{dt}{cos^3t+sin^3t}$, 继续变量代换$s=tan(t/2)$, 得到$\int-\frac{2 \left(s^2+1\right)^2}{\left(s^2-2 s-1\right) \left(s^4+2 s^3+2 s^2-2 s+1\right)}ds = \frac{1}{3} \left(2 \tan ^{-1}\left(\frac{s^2+s}{1-s}\right)-\sqrt{2} \left(\log \left(-s+\sqrt{2}+1\right)-\log \left(s+\sqrt{2}-1\right)\right)\right)$

kastin

令 `x=\sinh t`，有 `\dif x=\cosh t \dif t`，那么
\begin{align*}\int \frac{\sqrt{x^2+1}}{x^3+1}\dif x&=\int\frac{\cosh^2 t}{\sinh^3t+1}\dif t\\
&=\int\frac{1+\sinh^2 t}{1+\sinh^3t}\dif t\\
&=\frac 23\int\frac{\dif t}{1+\sinh t}+\frac 13\int\frac{1+\sinh t}{1-\sinh t+\sinh^2t}\dif t\end{align*}剩下就好办了。

笨笨

楼上的学哥都是高手啊，这个怎么求啊，思考了很长时间未果\[\int^{\oo}_\sqrt{33}\frac1{\sqrt{x^3-11x^2+11x+121}}\dif x=\frac1{6\sqrt2\pi^2}\Gamma\left(\frac1{11}\right)\Gamma\left(\frac3{11}\right)\Gamma\left(\frac4{11}\right)\Gamma\left(\frac5{11}\right)\Gamma\left(\frac9{11}\right)\]

chyanog

1. int=Integrate[Sqrt[x^2+1]/(x^3+1),x];
   
2. Assuming[0<x<1,
   
3. (ComplexExpand[Re[int],TargetFunctions->{Re, Im}]/.(f:Cos|Sin)[x_]:>FullSimplify[Sign[f[x]]]Sqrt[TrigReduce[f[x]^2]]//Simplify)//.{ ArcTan[x_]+ ArcTan[y_]:>ArcTan[(x+y)/(1-x y)],ArcTan[x_]-ArcTan[y_]:>ArcTan[(x-y)/(1+x y)]}//FullSimplify]

复制代码

这个化简起来有点费劲
$\frac{1}{3} \left(\sqrt{2} \log \left(\frac{\sqrt{2} \sqrt{x^2+1}+x-1}{x+1}\right)+\cot ^{-1}\left(\frac{((x-3) x+1) \left(x^2+x+1\right)}{(x-1) x \left(2 \left(\sqrt{3}-2 \sqrt{x^2+1}\right)+\sqrt{3} (x-1) x\right)+\sqrt{3}}\right)\right)$

用Mathematica的一个package Rubi(Rule-based Integration)可以直接得到很简洁的结果，顺便算一下bbs.emath.ac.cn/thread-2613-1-1.html (integrate(x/sqrt(x^4+10*x^2-96*x-71),x))

kastin

续上7楼：
\begin{align*}\frac 23\int\frac{\dif t}{1+\sinh t}+\frac 13\int\frac{1+\sinh t}{1-\sinh t+\sinh^2t}&=\frac{2\sqrt{2}}{3}\mathrm{arctanh}\frac{\tanh(t/2)-1}{\sqrt{2}}-\frac 23\arctan(\mathrm{sech}t-\tanh t)+c\\
&=\frac{\sqrt{2}}{3}\ln\frac{\sqrt{x^2+1}+x-\sqrt{2}+1}{\sqrt{x^2+1}+x+\sqrt{2}+1}+\frac 23\arctan\frac{x-1}{\sqrt{x^2+1}}+c\end{align*}