2.4 \(\int \log \left (\sqrt {x^2+1} x+1\right ) \,dx\)

2.4.1 Mathematica

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

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

2.4.2 Rubi

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

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

2.4.3 Maple

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

\[ \ln \left ( 1+x\sqrt {{x}^{2}+1} \right ) x+{\frac {1}{\sqrt {2\,\sqrt {5}+2}}\arctan \left ( 2\,{\frac {x}{\sqrt {2\,\sqrt {5}+2}}} \right ) } +{\frac {\sqrt {5}}{\sqrt {2\,\sqrt {5}+2}}\arctan \left ( 2\,{\frac {x }{\sqrt {2\,\sqrt {5}+2}}} \right ) }-{\frac {1}{\sqrt {-2+2\,\sqrt {5} }}{\rm arctanh} \left (2\,{\frac {x}{\sqrt {-2+2\,\sqrt {5}}}}\right )}+ {\frac {\sqrt {5}}{\sqrt {-2+2\,\sqrt {5}}}{\rm arctanh} \left (2\,{ \frac {x}{\sqrt {-2+2\,\sqrt {5}}}}\right )}-2\,x-{\frac {3\,\sqrt {5} }{10\,\sqrt {2+\sqrt {5}}}\arctan \left ( {\frac {1}{\sqrt {2+\sqrt {5} }} \left ( \sqrt {{x}^{2}+1}-x \right ) } \right ) }-{\frac {1}{2\,\sqrt {2+\sqrt {5}}}\arctan \left ( {\frac {1}{\sqrt {2+\sqrt {5}}} \left ( \sqrt {{x}^{2}+1}-x \right ) } \right ) }-{\frac {3\,\sqrt {5}}{10\, \sqrt {\sqrt {5}-2}}{\rm arctanh} \left ({\frac {1}{\sqrt {\sqrt {5}-2} } \left ( \sqrt {{x}^{2}+1}-x \right ) }\right )}+{\frac {1}{2\,\sqrt { \sqrt {5}-2}}{\rm arctanh} \left ({\frac {1}{\sqrt {\sqrt {5}-2}} \left ( \sqrt {{x}^{2}+1}-x \right ) }\right )}-{\frac {1}{2\,\sqrt {2+ \sqrt {5}}}{\rm arctanh} \left ({\frac {1}{\sqrt {2+\sqrt {5}}} \left ( \sqrt {{x}^{2}+1}-x \right ) }\right )}-{\frac {\sqrt {5}}{2\,\sqrt {2+ \sqrt {5}}}{\rm arctanh} \left ({\frac {1}{\sqrt {2+\sqrt {5}}} \left ( \sqrt {{x}^{2}+1}-x \right ) }\right )}+{\frac {1}{2\,\sqrt {\sqrt {5}-2 }}\arctan \left ( {\frac {1}{\sqrt {\sqrt {5}-2}} \left ( \sqrt {{x}^{2} +1}-x \right ) } \right ) }-{\frac {\sqrt {5}}{2\,\sqrt {\sqrt {5}-2}} \arctan \left ( {\frac {1}{\sqrt {\sqrt {5}-2}} \left ( \sqrt {{x}^{2}+1 }-x \right ) } \right ) }+{\frac {2\,\sqrt {2+\sqrt {5}}\sqrt {5}}{5} \arctan \left ( {\frac {1}{\sqrt {2+\sqrt {5}}} \left ( \sqrt {{x}^{2}+1 }-x \right ) } \right ) }-{\frac {2\,\sqrt {\sqrt {5}-2}\sqrt {5}}{5} {\rm arctanh} \left ({\frac {1}{\sqrt {\sqrt {5}-2}} \left ( \sqrt {{x}^ {2}+1}-x \right ) }\right )} \]

2.4.4 Fricas

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

\[ {-{4 \ {\sqrt {2}} \ {\sqrt {{{\sqrt {5}}+1}}} \ {\arctan \left ( {{{{{\left ( {{\sqrt {2}} \ {\sqrt {{{{x} \sp {2}}+1}}}}+{x \ {\sqrt {2}}} \right )} \ {\sqrt {{{\sqrt {5}}+1}}} \ {\sqrt {{{{\left ( -{{16} \ x \ {\sqrt {5}}} -{{32} \ {{x} \sp {3}}} -{{16} \ x} \right )} \ {\sqrt {{{{x} \sp {2}}+1}} }}+{{\left ( {{16} \ {{x} \sp {2}}}+8 \right )} \ {\sqrt {5}}}+{{32} \ {{x} \sp {4}}}+{{32} \ {{x} \sp {2}}}+8}}}} -{4 \ {\sqrt {2}} \ {\sqrt {{{{x} \ sp {2}}+1}}} \ {\sqrt {{{\sqrt {5}}+1}}}}} \over 8}} \right )}} -{{\sqrt {2}} \ {\sqrt {{{\sqrt {5}} -1}}} \ {\log \left ( {{{{\left ( {{\left ( {{\sqrt {2 }} \ {\sqrt {5}}}+{\sqrt {2}} \right )} \ {\sqrt {{{{x} \sp {2}}+1}}}} -{x \ {\sqrt {2}} \ {\sqrt {5}}} -{x \ {\sqrt {2}}} \right )} \ {\sqrt {{{\sqrt {5}} -1}}}} -{4 \ x \ {\sqrt {{{{x} \sp {2}}+1}}}}+{4 \ {{x} \sp {2}}}+4} } \right )}}+{{\sqrt {2}} \ {\sqrt {{{\sqrt {5}} -1}}} \ {\log \left ( {{{{\sqrt {2}} \ {\sqrt {{{\sqrt {5}} -1}}}}+{2 \ x}}} \right )}}+{4 \ x \ {\log \left ( {{{x \ {\sqrt {{{{x} \sp {2}}+1}}}}+1}} \right )}} -{{\sqrt {2}} \ { \sqrt {{{\sqrt {5}} -1}}} \ {\log \left ( {{-{{\sqrt {2}} \ {\sqrt {{{\sqrt {5}} -1}}}}+{2 \ x}}} \right )}}+{{\sqrt {2}} \ {\sqrt {{{\sqrt {5}} -1}}} \ {\log \left ( {{{{\left ( {{\left ( -{{\sqrt {2}} \ {\sqrt {5}}} -{\sqrt {2}} \right )} \ {\sqrt {{{{x} \sp {2}}+1}}}}+{x \ {\sqrt {2}} \ {\sqrt {5}}}+{ x \ {\sqrt {2}}} \right )} \ {\sqrt {{{\sqrt {5}} -1}}}} -{4 \ x \ {\sqrt {{{{x} \sp {2}}+1}}}}+{4 \ {{x} \sp {2}}}+4}} \right )}} -{4 \ {\sqrt {2}} \ {\sqrt {{{\sqrt {5}}+1}}} \ {\arctan \left ( {{{{{\left ( {{\sqrt {2}} \ {\ sqrt {5}}} -{\sqrt {2}} \right )} \ {\sqrt {{{\sqrt {5}}+1}}} \ {\sqrt {{{2 \ {\sqrt {5}}}+{4 \ {{x} \sp {2}}}+2}}}}+{{\left ( -{2 \ x \ {\sqrt {2}} \ {\sqrt {5}}}+{2 \ x \ {\sqrt {2}}} \right )} \ {\sqrt {{{\sqrt {5}}+1}}}} } \over 8}} \right )}} -{8 \ x}} \over 4 \]

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)
 

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

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 \left ( 1+x\sqrt {{x}^{2}+1} \right ) x-2\,x+{\frac {{\frac {\sqrt {5}}{2}}-{\frac {5}{2}}}{2\,\sqrt {1/2\,\sqrt {5}-1/2}+4\, \left ( 1/2\, \sqrt {5}-1/2 \right ) ^{3/2}}\ln \left ( x-{\frac {\sqrt {2}\sqrt { \sqrt {5}-1}}{2}} \right ) }-{\frac {{\frac {\sqrt {5}}{2}}-{\frac {5}{2 }}}{2\,\sqrt {1/2\,\sqrt {5}-1/2}+4\, \left ( 1/2\,\sqrt {5}-1/2 \right ) ^{3/2}}\ln \left ( x+{\frac {\sqrt {2}\sqrt {\sqrt {5}-1}}{2} } \right ) }-{\frac {{\frac {\sqrt {5}}{2}}+{\frac {5}{2}}}{2\,\sqrt {-1 /2\,\sqrt {5}-1/2}+4\, \left ( -1/2\,\sqrt {5}-1/2 \right ) ^{3/2}}\ln \left ( x-{\frac {\sqrt {2}\sqrt {-\sqrt {5}-1}}{2}} \right ) }+{\frac {{\frac {\sqrt {5}}{2}}+{\frac {5}{2}}}{2\,\sqrt {-1/2\,\sqrt {5}-1/2}+ 4\, \left ( -1/2\,\sqrt {5}-1/2 \right ) ^{3/2}}\ln \left ( x+{\frac { \sqrt {2}\sqrt {-\sqrt {5}-1}}{2}} \right ) }+{\frac {\sqrt {{\frac { \sqrt {5}}{2}}-{\frac {1}{2}}}+2\, \left ( 1/2\,\sqrt {5}-1/2 \right ) ^{ 3/2}}{ \left ( 2\,\sqrt {1/2\,\sqrt {5}-1/2}+4\, \left ( 1/2\,\sqrt {5}- 1/2 \right ) ^{3/2} \right ) \sqrt {{\frac {\sqrt {5}}{2}}+{\frac {1}{2}} }} \left ( \ln \left ( x-{\frac {\sqrt {2}\sqrt {\sqrt {5}-1}}{2}} \right ) -\ln \left ( {\frac {\sqrt {2}x\sqrt {\sqrt {5}-1}}{2}}+{ \frac {\sqrt {2}\sqrt {\sqrt {5}+1}}{2}\sqrt {{x}^{2}+1}}+1 \right ) \right ) }+{\frac {\sqrt {{\frac {\sqrt {5}}{2}}-{\frac {1}{2}}}+2\, \left ( 1/2\,\sqrt {5}-1/2 \right ) ^{3/2}}{ \left ( 2\,\sqrt {1/2\, \sqrt {5}-1/2}+4\, \left ( 1/2\,\sqrt {5}-1/2 \right ) ^{3/2} \right ) \sqrt {{\frac {\sqrt {5}}{2}}+{\frac {1}{2}}}} \left ( \ln \left ( x+{ \frac {\sqrt {2}\sqrt {\sqrt {5}-1}}{2}} \right ) -\ln \left ( {\frac { \sqrt {2}\sqrt {\sqrt {5}+1}}{2}\sqrt {{x}^{2}+1}}-{\frac {\sqrt {2}x \sqrt {\sqrt {5}-1}}{2}}+1 \right ) \right ) }-{\frac {\sqrt {-{\frac { \sqrt {5}}{2}}-{\frac {1}{2}}}+2\, \left ( -1/2\,\sqrt {5}-1/2 \right ) ^ {3/2}}{ \left ( 2\,\sqrt {-1/2\,\sqrt {5}-1/2}+4\, \left ( -1/2\,\sqrt { 5}-1/2 \right ) ^{3/2} \right ) \sqrt {{\frac {1}{2}}-{\frac {\sqrt {5}}{ 2}}}} \left ( \ln \left ( {\frac {\sqrt {2}\sqrt {-\sqrt {5}+1}}{2} \sqrt {{x}^{2}+1}}-{\frac {\sqrt {2}x\sqrt {-\sqrt {5}-1}}{2}}+1 \right ) -\ln \left ( x+{\frac {\sqrt {2}\sqrt {-\sqrt {5}-1}}{2}} \right ) \right ) }-{\frac {\sqrt {-{\frac {\sqrt {5}}{2}}-{\frac {1}{2 }}}+2\, \left ( -1/2\,\sqrt {5}-1/2 \right ) ^{3/2}}{ \left ( 2\,\sqrt {- 1/2\,\sqrt {5}-1/2}+4\, \left ( -1/2\,\sqrt {5}-1/2 \right ) ^{3/2} \right ) \sqrt {{\frac {1}{2}}-{\frac {\sqrt {5}}{2}}}} \left ( \ln \left ( {\frac {\sqrt {2}x\sqrt {-\sqrt {5}-1}}{2}}+{\frac {\sqrt {2} \sqrt {-\sqrt {5}+1}}{2}\sqrt {{x}^{2}+1}}+1 \right ) -\ln \left ( x-{ \frac {\sqrt {2}\sqrt {-\sqrt {5}-1}}{2}} \right ) \right ) } \]