2.4 $\int \log (\sqrt{x^2+1}x+1)dx$

2.4.1 Mathematica

ClearAll[x] 
integrand = Log[1 + x Sqrt[1 + x^2]]; 
res = Integrate[integrand, x]; 
TeXForm[res]
 

$$x\log (\sqrt{x^2+1}x+1)-\frac{\sqrt{2(\sqrt{5}-1)}{\tan }^{-1}\left(\sqrt{\frac{2}{\sqrt{5}-1}}\sqrt{x^2+1}\right)}{1-\sqrt{5}}-\sqrt{\frac{2}{1+\sqrt{5}}}{\tanh }^{-1}\left(\sqrt{\frac{2}{1+\sqrt{5}}}\sqrt{x^2+1}\right)-2x+\frac{(5+\sqrt{5}){\tan }^{-1}\left(\sqrt{\frac{2}{1+\sqrt{5}}}x\right)}{\sqrt{10(1+\sqrt{5})}}-\frac{(\sqrt{5}-5){\tanh }^{-1}\left(\sqrt{\frac{2}{\sqrt{5}-1}}x\right)}{\sqrt{10(\sqrt{5}-1)}}$$

2.4.2 Rubi

<< Rubi` 
ClearAll[x] 
integrand = Log[1 + x Sqrt[1 + x^2]] 
res = Int[integrand, x]; 
TeXForm[res]
 

$$x\log (\sqrt{x^2+1}x+1)+\sqrt{\frac{2}{5}(\sqrt{5}-1)}{\tan }^{-1}\left(\sqrt{\frac{2}{\sqrt{5}-1}}\sqrt{x^2+1}\right)+\sqrt{\frac{2}{5(\sqrt{5}-1)}}{\tan }^{-1}\left(\sqrt{\frac{2}{\sqrt{5}-1}}\sqrt{x^2+1}\right)-\sqrt{\frac{2}{5}(1+\sqrt{5})}{\tanh }^{-1}\left(\sqrt{\frac{2}{1+\sqrt{5}}}\sqrt{x^2+1}\right)+\sqrt{\frac{2}{5(1+\sqrt{5})}}{\tanh }^{-1}\left(\sqrt{\frac{2}{1+\sqrt{5}}}\sqrt{x^2+1}\right)-2x+2\sqrt{\frac{1}{5}(2+\sqrt{5})}{\tan }^{-1}\left(\sqrt{\frac{2}{1+\sqrt{5}}}x\right)-\sqrt{\frac{1}{10}(1+\sqrt{5})}{\tan }^{-1}\left(\sqrt{\frac{2}{1+\sqrt{5}}}x\right)+\sqrt{\frac{1}{10}(\sqrt{5}-1)}{\tanh }^{-1}\left(\sqrt{\frac{2}{\sqrt{5}-1}}x\right)+2\sqrt{\frac{1}{5}(\sqrt{5}-2)}{\tanh }^{-1}\left(\sqrt{\frac{2}{\sqrt{5}-1}}x\right)$$

2.4.3 Maple

restart; 
integrand:=ln(1 + x*sqrt(1 + x^2)); 
res:=int(integrand,x); 
latex(res)
 

$$\ln (1+x\sqrt{x^2+1})x+\frac{1}{\sqrt{2\sqrt{5}+2}}\arctan \left(2\frac{x}{\sqrt{2\sqrt{5}+2}}\right)+\frac{\sqrt{5}}{\sqrt{2\sqrt{5}+2}}\arctan \left(2\frac{x}{\sqrt{2\sqrt{5}+2}}\right)-\frac{1}{\sqrt{-2+2\sqrt{5}}}\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left(2\frac{x}{\sqrt{-2+2\sqrt{5}}}\right)+\frac{\sqrt{5}}{\sqrt{-2+2\sqrt{5}}}\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left(2\frac{x}{\sqrt{-2+2\sqrt{5}}}\right)-2x-\frac{3\sqrt{5}}{10\sqrt{2+\sqrt{5}}}\arctan \left(\frac{1}{\sqrt{2+\sqrt{5}}}(\sqrt{x^2+1}-x)\right)-\frac{1}{2\sqrt{2+\sqrt{5}}}\arctan \left(\frac{1}{\sqrt{2+\sqrt{5}}}(\sqrt{x^2+1}-x)\right)-\frac{3\sqrt{5}}{10\sqrt{\sqrt{5}-2}}\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left(\frac{1}{\sqrt{\sqrt{5}-2}}(\sqrt{x^2+1}-x)\right)+\frac{1}{2\sqrt{\sqrt{5}-2}}\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left(\frac{1}{\sqrt{\sqrt{5}-2}}(\sqrt{x^2+1}-x)\right)-\frac{1}{2\sqrt{2+\sqrt{5}}}\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left(\frac{1}{\sqrt{2+\sqrt{5}}}(\sqrt{x^2+1}-x)\right)-\frac{\sqrt{5}}{2\sqrt{2+\sqrt{5}}}\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left(\frac{1}{\sqrt{2+\sqrt{5}}}(\sqrt{x^2+1}-x)\right)+\frac{1}{2\sqrt{\sqrt{5}-2}}\arctan \left(\frac{1}{\sqrt{\sqrt{5}-2}}(\sqrt{x^2+1}-x)\right)-\frac{\sqrt{5}}{2\sqrt{\sqrt{5}-2}}\arctan \left(\frac{1}{\sqrt{\sqrt{5}-2}}(\sqrt{x^2+1}-x)\right)+\frac{2\sqrt{2+\sqrt{5}}\sqrt{5}}{5}\arctan \left(\frac{1}{\sqrt{2+\sqrt{5}}}(\sqrt{x^2+1}-x)\right)-\frac{2\sqrt{\sqrt{5}-2}\sqrt{5}}{5}\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left(\frac{1}{\sqrt{\sqrt{5}-2}}(\sqrt{x^2+1}-x)\right)$$

2.4.4 Fricas

set output tex off 
setSimplifyDenomsFlag(true) 
integrand := log(1 + x*sqrt(1 + x^2)); 
ii:=integrate(integrand,x); 
latex(ii)
 

$$\frac{-4\sqrt{2}\sqrt{\sqrt{5}+1}\arctan \left(\frac{(\sqrt{2}\sqrt{x^2+1}+x\sqrt{2})\sqrt{\sqrt{5}+1}\sqrt{(-16x\sqrt{5}-32x^3-16x)\sqrt{x^2+1}+(16x^2+8)\sqrt{5}+32x^4+32x^2+8}-4\sqrt{2}\sqrt{xsp2+1}\sqrt{\sqrt{5}+1}}{8}\right)-\sqrt{2}\sqrt{\sqrt{5}-1}\log \left(((\sqrt{2}\sqrt{5}+\sqrt{2})\sqrt{x^2+1}-x\sqrt{2}\sqrt{5}-x\sqrt{2})\sqrt{\sqrt{5}-1}-4x\sqrt{x^2+1}+4x^2+4\right)+\sqrt{2}\sqrt{\sqrt{5}-1}\log \left(\sqrt{2}\sqrt{\sqrt{5}-1}+2x\right)+4x\log \left(x\sqrt{x^2+1}+1\right)-\sqrt{2}\sqrt{\sqrt{5}-1}\log \left(-\sqrt{2}\sqrt{\sqrt{5}-1}+2x\right)+\sqrt{2}\sqrt{\sqrt{5}-1}\log \left(((-\sqrt{2}\sqrt{5}-\sqrt{2})\sqrt{x^2+1}+x\sqrt{2}\sqrt{5}+x\sqrt{2})\sqrt{\sqrt{5}-1}-4x\sqrt{x^2+1}+4x^2+4\right)-4\sqrt{2}\sqrt{\sqrt{5}+1}\arctan \left(\frac{(\sqrt{2}sqrt5-\sqrt{2})\sqrt{\sqrt{5}+1}\sqrt{2\sqrt{5}+4x^2+2}+(-2x\sqrt{2}\sqrt{5}+2x\sqrt{2})\sqrt{\sqrt{5}+1}}{8}\right)-8x}{4}$$

2.4.5 Maxima

integrand : log(1 + x*sqrt(1 + x^2)); 
ii : integrate(integrand,x); 
latex(ii)
 

$$\text{did not solve}$$

2.4.6 XCAS

integrand := log(1 + x*sqrt(1 + x^2)); 
ii := integrate(integrand,x); 
latex(ii)
 

$$-\frac{1}{4}\sqrt{2(\sqrt{5}-1)}\ln |x-\sqrt{\frac{-1+\sqrt{5}}{2}}|+\frac{1}{4}\sqrt{2(\sqrt{5}-1)}\ln |x+\sqrt{\frac{-1+\sqrt{5}}{2}}|+\frac{1}{2}\sqrt{2(\sqrt{5}+1)}\arctan \left(\frac{x}{\sqrt{-\frac{-1-\sqrt{5}}{2}}}\right)-2x-2(-\frac{1}{8}\sqrt{2(\sqrt{5}-1)}\ln |\sqrt{x^2+1}-x+\frac{1}{\sqrt{x^2+1}-x}-\sqrt{\frac{4+4\sqrt{5}}{2}}|+\frac{1}{8}\sqrt{2(\sqrt{5}-1)}\ln (\sqrt{x^2+1}-x+\frac{1}{\sqrt{x^2+1}-x}+\sqrt{\frac{4+4\sqrt{5}}{2}})-\frac{1}{4}\sqrt{2(\sqrt{5}+1)}\arctan \left(\frac{\sqrt{x^2+1}-x+\frac{1}{\sqrt{x^2+1}-x}}{\sqrt{-\frac{4-4\sqrt{5}}{2}}}\right))+x\ln (1+x\sqrt{1+x^2})$$

2.4.7 Sympy

>python 
Python 3.7.3 (default, Mar 27 2019, 22:11:17) 
[GCC 7.3.0] :: Anaconda, Inc. on linux 
 
from sympy import * 
x = symbols('x') 
integrand = log(1 + x*sqrt(1 + x**2)); 
ii = integrate(integrand,x); 
latex(ii)
 

$$\text{did not solve}$$

2.4.8 MuPad

evalin(symengine,'int(log(1 + x*sqrt(1 + x^2)),x)')
 

$$\ln (1+x\sqrt{x^2+1})x-2x+\frac{\frac{\sqrt{5}}{2}-\frac{5}{2}}{2\sqrt{1/2\sqrt{5}-1/2}+4{(1/2\sqrt{5}-1/2)}^{3/2}}\ln (x-\frac{\sqrt{2}\sqrt{\sqrt{5}-1}}{2})-\frac{\frac{\sqrt{5}}{2}-\frac{5}{2}}{2\sqrt{1/2\sqrt{5}-1/2}+4{(1/2\sqrt{5}-1/2)}^{3/2}}\ln (x+\frac{\sqrt{2}\sqrt{\sqrt{5}-1}}{2})-\frac{\frac{\sqrt{5}}{2}+\frac{5}{2}}{2\sqrt{-1/2\sqrt{5}-1/2}+4{(-1/2\sqrt{5}-1/2)}^{3/2}}\ln (x-\frac{\sqrt{2}\sqrt{-\sqrt{5}-1}}{2})+\frac{\frac{\sqrt{5}}{2}+\frac{5}{2}}{2\sqrt{-1/2\sqrt{5}-1/2}+4{(-1/2\sqrt{5}-1/2)}^{3/2}}\ln (x+\frac{\sqrt{2}\sqrt{-\sqrt{5}-1}}{2})+\frac{\sqrt{\frac{\sqrt{5}}{2}-\frac{1}{2}}+2{(1/2\sqrt{5}-1/2)}^{3/2}}{(2\sqrt{1/2\sqrt{5}-1/2}+4{(1/2\sqrt{5}-1/2)}^{3/2})\sqrt{\frac{\sqrt{5}}{2}+\frac{1}{2}}}(\ln (x-\frac{\sqrt{2}\sqrt{\sqrt{5}-1}}{2})-\ln (\frac{\sqrt{2}x\sqrt{\sqrt{5}-1}}{2}+\frac{\sqrt{2}\sqrt{\sqrt{5}+1}}{2}\sqrt{x^2+1}+1))+\frac{\sqrt{\frac{\sqrt{5}}{2}-\frac{1}{2}}+2{(1/2\sqrt{5}-1/2)}^{3/2}}{(2\sqrt{1/2\sqrt{5}-1/2}+4{(1/2\sqrt{5}-1/2)}^{3/2})\sqrt{\frac{\sqrt{5}}{2}+\frac{1}{2}}}(\ln (x+\frac{\sqrt{2}\sqrt{\sqrt{5}-1}}{2})-\ln (\frac{\sqrt{2}\sqrt{\sqrt{5}+1}}{2}\sqrt{x^2+1}-\frac{\sqrt{2}x\sqrt{\sqrt{5}-1}}{2}+1))-\frac{\sqrt{-\frac{\sqrt{5}}{2}-\frac{1}{2}}+2{(-1/2\sqrt{5}-1/2)}^{3/2}}{(2\sqrt{-1/2\sqrt{5}-1/2}+4{(-1/2\sqrt{5}-1/2)}^{3/2})\sqrt{\frac{1}{2}-\frac{\sqrt{5}}{2}}}(\ln (\frac{\sqrt{2}\sqrt{-\sqrt{5}+1}}{2}\sqrt{x^2+1}-\frac{\sqrt{2}x\sqrt{-\sqrt{5}-1}}{2}+1)-\ln (x+\frac{\sqrt{2}\sqrt{-\sqrt{5}-1}}{2}))-\frac{\sqrt{-\frac{\sqrt{5}}{2}-\frac{1}{2}}+2{(-1/2\sqrt{5}-1/2)}^{3/2}}{(2\sqrt{-1/2\sqrt{5}-1/2}+4{(-1/2\sqrt{5}-1/2)}^{3/2})\sqrt{\frac{1}{2}-\frac{\sqrt{5}}{2}}}(\ln (\frac{\sqrt{2}x\sqrt{-\sqrt{5}-1}}{2}+\frac{\sqrt{2}\sqrt{-\sqrt{5}+1}}{2}\sqrt{x^2+1}+1)-\ln (x-\frac{\sqrt{2}\sqrt{-\sqrt{5}-1}}{2}))$$