求一不定积分

wayne

zeroieme 发表于 2018-12-3 21:57
试过MMA自带函数能算，就是觉得那个结果复杂。期待Rubi有个更漂亮表达式。

Rubi的安装其实并不麻烦，自己可以试一试的，如图：

1. PacletInstall["https://github.com/RuleBasedIntegration/Rubi/releases/download/4.16.0.4/Rubi-4.16.0.4.paclet"]

复制代码

[如果github在线的地址下载不了，可以离线下载了再load]

wayne

.·.·. 发表于 2018-12-2 11:25
为什么这款软件到了你这里就这么好看了

还有

Sagemath 也可以调用fricas，而且notebook的排版也很不错。参考链接：http://doc.sagemath.org/html/en/ ... ation/integral.html
比如：

zeroieme

wayne 发表于 2018-12-26 18:27
Sagemath 也可以调用fricas，而且notebook的排版也很不错。参考链接：http://doc.sagemath.org/html/en ...

终于在这论坛里看见Sagemath了。

葡萄糖

请教一下这个题
\begin{align*}
\int_0^x\arctan\left(1+\sqrt{1+t^2}\,\right)\mathrm{d}t
\end{align*}

wayne

用sageMath调用fricas计算的

1. integrate(arctan(1+sqrt(x^2+1)),x,algorithm="fricas")

复制代码

\[\newcommand{\Bold}[1]{\mathbf{#1}}\frac{4 \, {\left(\sqrt{5} x + x\right)} \sqrt{\frac{\sqrt{5} + 5}{\sqrt{5} + 3}} \arctan\left(\sqrt{x^{2} + 1} + 1\right) + 4 \, {\left(\sqrt{5} + 1\right)} \sqrt{\frac{\sqrt{5} + 5}{\sqrt{5} + 3}} \log\left(-x + \sqrt{x^{2} + 1}\right) - 5^{\frac{1}{4}} {\left(\sqrt{5} + 1\right)} \log\left(\frac{15 \, \sqrt{5} x^{2} + 35 \, x^{2} + 2 \cdot 5^{\frac{1}{4}} {\left(7 \, \sqrt{5} x + 15 \, x\right)} \sqrt{\frac{\sqrt{5} + 5}{\sqrt{5} + 3}} + 5 \, \sqrt{5} {\left(3 \, \sqrt{5} + 7\right)}}{5 \, {\left(3 \, \sqrt{5} + 7\right)}}\right) + 5^{\frac{1}{4}} {\left(\sqrt{5} + 1\right)} \log\left(\frac{15 \, \sqrt{5} x^{2} + 35 \, x^{2} - 2 \cdot 5^{\frac{1}{4}} {\left(7 \, \sqrt{5} x + 15 \, x\right)} \sqrt{\frac{\sqrt{5} + 5}{\sqrt{5} + 3}} + 5 \, \sqrt{5} {\left(3 \, \sqrt{5} + 7\right)}}{5 \, {\left(3 \, \sqrt{5} + 7\right)}}\right) + 5^{\frac{1}{4}} {\left(\sqrt{5} + 1\right)} \log\left(\frac{10 \, x^{4} + 15 \, x^{2} + 5^{\frac{1}{4}} {\left(3 \, x^{2} + \sqrt{5} {\left(x^{2} + 11\right)} + 25\right)} \sqrt{\frac{\sqrt{5} + 5}{\sqrt{5} + 3}} + 2 \, \sqrt{5} {\left(2 \, x^{4} + 3 \, x^{2} + 5\right)} - {\left(10 \, x^{3} + 5^{\frac{1}{4}} {\left(\sqrt{5} x + 3 \, x\right)} \sqrt{\frac{\sqrt{5} + 5}{\sqrt{5} + 3}} + 4 \, \sqrt{5} {\left(x^{3} + x\right)} + 10 \, x\right)} \sqrt{x^{2} + 1} + 2 \, \sqrt{5} {\left(2 \, \sqrt{5} + 5\right)} + 25}{2 \, \sqrt{5} + 5}\right) - 5^{\frac{1}{4}} {\left(\sqrt{5} + 1\right)} \log\left(\frac{10 \, x^{4} + 15 \, x^{2} - 5^{\frac{1}{4}} {\left(3 \, x^{2} + \sqrt{5} {\left(x^{2} + 11\right)} + 25\right)} \sqrt{\frac{\sqrt{5} + 5}{\sqrt{5} + 3}} + 2 \, \sqrt{5} {\left(2 \, x^{4} + 3 \, x^{2} + 5\right)} - {\left(10 \, x^{3} - 5^{\frac{1}{4}} {\left(\sqrt{5} x + 3 \, x\right)} \sqrt{\frac{\sqrt{5} + 5}{\sqrt{5} + 3}} + 4 \, \sqrt{5} {\left(x^{3} + x\right)} + 10 \, x\right)} \sqrt{x^{2} + 1} + 2 \, \sqrt{5} {\left(2 \, \sqrt{5} + 5\right)} + 25}{2 \, \sqrt{5} + 5}\right) - 8 \cdot 5^{\frac{1}{4}} \arctan\left(\frac{5^{\frac{1}{4}} {\left(\sqrt{5} + 1\right)}}{\sqrt{\frac{1}{5}} {\left(\sqrt{5} + 1\right)} \sqrt{\frac{15 \, \sqrt{5} x^{2} + 35 \, x^{2} + 2 \cdot 5^{\frac{1}{4}} {\left(7 \, \sqrt{5} x + 15 \, x\right)} \sqrt{\frac{\sqrt{5} + 5}{\sqrt{5} + 3}} + 5 \, \sqrt{5} {\left(3 \, \sqrt{5} + 7\right)}}{3 \, \sqrt{5} + 7}} \sqrt{\frac{\sqrt{5} + 5}{\sqrt{5} + 3}} + {\left(\sqrt{5} x + x\right)} \sqrt{\frac{\sqrt{5} + 5}{\sqrt{5} + 3}} + 2 \cdot 5^{\frac{1}{4}}}\right) - 8 \cdot 5^{\frac{1}{4}} \arctan\left(\frac{5^{\frac{1}{4}} {\left(\sqrt{5} + 1\right)}}{\sqrt{\frac{1}{5}} {\left(\sqrt{5} + 1\right)} \sqrt{\frac{15 \, \sqrt{5} x^{2} + 35 \, x^{2} - 2 \cdot 5^{\frac{1}{4}} {\left(7 \, \sqrt{5} x + 15 \, x\right)} \sqrt{\frac{\sqrt{5} + 5}{\sqrt{5} + 3}} + 5 \, \sqrt{5} {\left(3 \, \sqrt{5} + 7\right)}}{3 \, \sqrt{5} + 7}} \sqrt{\frac{\sqrt{5} + 5}{\sqrt{5} + 3}} + {\left(\sqrt{5} x + x\right)} \sqrt{\frac{\sqrt{5} + 5}{\sqrt{5} + 3}} - 2 \cdot 5^{\frac{1}{4}}}\right) - 8 \cdot 5^{\frac{1}{4}} \arctan\left(-\frac{2 \, {\left(\sqrt{5} + 1\right)} \sqrt{\frac{\sqrt{5} + 5}{\sqrt{5} + 3}} + 5^{\frac{1}{4}} {\left(\sqrt{5} + 3\right)}}{\sqrt{x^{2} + 1} {\left(\sqrt{5} x + x\right)} \sqrt{\frac{\sqrt{5} + 5}{\sqrt{5} + 3}} - {\left(\sqrt{5} + 1\right)} \sqrt{\frac{10 \, x^{4} + 15 \, x^{2} + 5^{\frac{1}{4}} {\left(3 \, x^{2} + \sqrt{5} {\left(x^{2} + 11\right)} + 25\right)} \sqrt{\frac{\sqrt{5} + 5}{\sqrt{5} + 3}} + 2 \, \sqrt{5} {\left(2 \, x^{4} + 3 \, x^{2} + 5\right)} - {\left(10 \, x^{3} + 5^{\frac{1}{4}} {\left(\sqrt{5} x + 3 \, x\right)} \sqrt{\frac{\sqrt{5} + 5}{\sqrt{5} + 3}} + 4 \, \sqrt{5} {\left(x^{3} + x\right)} + 10 \, x\right)} \sqrt{x^{2} + 1} + 2 \, \sqrt{5} {\left(2 \, \sqrt{5} + 5\right)} + 25}{2 \, \sqrt{5} + 5}} \sqrt{\frac{\sqrt{5} + 5}{\sqrt{5} + 3}} - {\left(x^{2} + \sqrt{5} {\left(x^{2} + 1\right)} + 1\right)} \sqrt{\frac{\sqrt{5} + 5}{\sqrt{5} + 3}} - 5^{\frac{1}{4}} {\left(\sqrt{5} - 1\right)}}\right) + 8 \cdot 5^{\frac{1}{4}} \arctan\left(-\frac{2 \, {\left(\sqrt{5} + 1\right)} \sqrt{\frac{\sqrt{5} + 5}{\sqrt{5} + 3}} - 5^{\frac{1}{4}} {\left(\sqrt{5} + 3\right)}}{\sqrt{x^{2} + 1} {\left(\sqrt{5} x + x\right)} \sqrt{\frac{\sqrt{5} + 5}{\sqrt{5} + 3}} - {\left(\sqrt{5} + 1\right)} \sqrt{\frac{10 \, x^{4} + 15 \, x^{2} - 5^{\frac{1}{4}} {\left(3 \, x^{2} + \sqrt{5} {\left(x^{2} + 11\right)} + 25\right)} \sqrt{\frac{\sqrt{5} + 5}{\sqrt{5} + 3}} + 2 \, \sqrt{5} {\left(2 \, x^{4} + 3 \, x^{2} + 5\right)} - {\left(10 \, x^{3} - 5^{\frac{1}{4}} {\left(\sqrt{5} x + 3 \, x\right)} \sqrt{\frac{\sqrt{5} + 5}{\sqrt{5} + 3}} + 4 \, \sqrt{5} {\left(x^{3} + x\right)} + 10 \, x\right)} \sqrt{x^{2} + 1} + 2 \, \sqrt{5} {\left(2 \, \sqrt{5} + 5\right)} + 25}{2 \, \sqrt{5} + 5}} \sqrt{\frac{\sqrt{5} + 5}{\sqrt{5} + 3}} - {\left(x^{2} + \sqrt{5} {\left(x^{2} + 1\right)} + 1\right)} \sqrt{\frac{\sqrt{5} + 5}{\sqrt{5} + 3}} + 5^{\frac{1}{4}} {\left(\sqrt{5} - 1\right)}}\right)}{4 \, {\left(\sqrt{5} + 1\right)} \sqrt{\frac{\sqrt{5} + 5}{\sqrt{5} + 3}}}\]

wayne

用Rubi计算的结果要简洁一些,不过带了虚数在里面：

\[\frac{1}{8} \left(-\sqrt{2 \left(\sqrt{5}+1\right)} \log \left(x^2+\sqrt{2 \left(\sqrt{5}-1\right)} x+\sqrt{5}\right)+\sqrt{2 \left(\sqrt{5}+1\right)} \log \left(x^2-\sqrt{2 \left(\sqrt{5}-1\right)} x+\sqrt{5}\right)+8 x \tan ^{-1}\left(\sqrt{x^2+1}+1\right)+(2-2 i) \sqrt{4+2 i} \tanh ^{-1}\left(\frac{x}{\sqrt{1+\frac{i}{2}} \sqrt{x^2+1}}\right)+(2+2 i) \sqrt{4-2 i} \tanh ^{-1}\left(\frac{x}{\sqrt{1-\frac{i}{2}} \sqrt{x^2+1}}\right)-2 \sqrt{2 \left(\sqrt{5}-1\right)} \tan ^{-1}\left(\frac{\sqrt{2 \left(\sqrt{5}-1\right)}-2 x}{\sqrt{2 \left(\sqrt{5}+1\right)}}\right)+2 \sqrt{2 \left(\sqrt{5}-1\right)} \tan ^{-1}\left(\frac{2 x+\sqrt{2 \left(\sqrt{5}-1\right)}}{\sqrt{2 \left(\sqrt{5}+1\right)}}\right)-8 \sinh ^{-1}(x)\right)\]