Integrand size = 14, antiderivative size = 97 \[ \int \log \left (1+x \sqrt {1+x^2}\right ) \, dx=-2 x+\sqrt {2 \left (1+\sqrt {5}\right )} \arctan \left (\sqrt {-2+\sqrt {5}} \left (x+\sqrt {1+x^2}\right )\right )-\sqrt {2 \left (-1+\sqrt {5}\right )} \text {arctanh}\left (\sqrt {2+\sqrt {5}} \left (x+\sqrt {1+x^2}\right )\right )+x \log \left (1+x \sqrt {1+x^2}\right ) \]
-2*x+x*ln(1+x*(x^2+1)^(1/2))-arctanh((x+(x^2+1)^(1/2))*(2+5^(1/2))^(1/2))* (-2+2*5^(1/2))^(1/2)+arctan((x+(x^2+1)^(1/2))*(-2+5^(1/2))^(1/2))*(2+2*5^( 1/2))^(1/2)
Time = 0.27 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.94 \[ \int \log \left (1+x \sqrt {1+x^2}\right ) \, dx=-2 x+\frac {\left (5+\sqrt {5}\right ) \arctan \left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )}{\sqrt {10 \left (1+\sqrt {5}\right )}}+\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} \arctan \left (\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}} \sqrt {1+x^2}\right )-\frac {\left (-5+\sqrt {5}\right ) \text {arctanh}\left (\sqrt {\frac {2}{-1+\sqrt {5}}} x\right )}{\sqrt {10 \left (-1+\sqrt {5}\right )}}-\sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )} \text {arctanh}\left (\sqrt {-\frac {1}{2}+\frac {\sqrt {5}}{2}} \sqrt {1+x^2}\right )+x \log \left (1+x \sqrt {1+x^2}\right ) \]
-2*x + ((5 + Sqrt[5])*ArcTan[Sqrt[2/(1 + Sqrt[5])]*x])/Sqrt[10*(1 + Sqrt[5 ])] + Sqrt[(1 + Sqrt[5])/2]*ArcTan[Sqrt[1/2 + Sqrt[5]/2]*Sqrt[1 + x^2]] - ((-5 + Sqrt[5])*ArcTanh[Sqrt[2/(-1 + Sqrt[5])]*x])/Sqrt[10*(-1 + Sqrt[5])] - Sqrt[(-1 + Sqrt[5])/2]*ArcTanh[Sqrt[-1/2 + Sqrt[5]/2]*Sqrt[1 + x^2]] + x*Log[1 + x*Sqrt[1 + x^2]]
Leaf count is larger than twice the leaf count of optimal. \(332\) vs. \(2(97)=194\).
Time = 0.85 (sec) , antiderivative size = 332, normalized size of antiderivative = 3.42, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3028, 7293, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \log \left (\sqrt {x^2+1} x+1\right ) \, dx\) |
| \(\Big \downarrow \) 3028 |
| \(\displaystyle x \log \left (\sqrt {x^2+1} x+1\right )-\int \frac {x \left (2 x^2+1\right )}{x^3+x+\sqrt {x^2+1}}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle x \log \left (\sqrt {x^2+1} x+1\right )-\int \left (\frac {2 x^3}{x^3+x+\sqrt {x^2+1}}+\frac {x}{x^3+x+\sqrt {x^2+1}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \sqrt {\frac {2}{5} \left (\sqrt {5}-1\right )} \arctan \left (\sqrt {\frac {2}{\sqrt {5}-1}} \sqrt {x^2+1}\right )+\sqrt {\frac {2}{5 \left (\sqrt {5}-1\right )}} \arctan \left (\sqrt {\frac {2}{\sqrt {5}-1}} \sqrt {x^2+1}\right )+2 \sqrt {\frac {1}{5} \left (2+\sqrt {5}\right )} \arctan \left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )-\sqrt {\frac {1}{10} \left (1+\sqrt {5}\right )} \arctan \left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )-\sqrt {\frac {2}{5} \left (1+\sqrt {5}\right )} \text {arctanh}\left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x^2+1}\right )+\sqrt {\frac {2}{5 \left (1+\sqrt {5}\right )}} \text {arctanh}\left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x^2+1}\right )+\sqrt {\frac {1}{10} \left (\sqrt {5}-1\right )} \text {arctanh}\left (\sqrt {\frac {2}{\sqrt {5}-1}} x\right )+2 \sqrt {\frac {1}{5} \left (\sqrt {5}-2\right )} \text {arctanh}\left (\sqrt {\frac {2}{\sqrt {5}-1}} x\right )+x \log \left (\sqrt {x^2+1} x+1\right )-2 x\) |
-2*x - Sqrt[(1 + Sqrt[5])/10]*ArcTan[Sqrt[2/(1 + Sqrt[5])]*x] + 2*Sqrt[(2 + Sqrt[5])/5]*ArcTan[Sqrt[2/(1 + Sqrt[5])]*x] + Sqrt[2/(5*(-1 + Sqrt[5]))] *ArcTan[Sqrt[2/(-1 + Sqrt[5])]*Sqrt[1 + x^2]] + Sqrt[(2*(-1 + Sqrt[5]))/5] *ArcTan[Sqrt[2/(-1 + Sqrt[5])]*Sqrt[1 + x^2]] + 2*Sqrt[(-2 + Sqrt[5])/5]*A rcTanh[Sqrt[2/(-1 + Sqrt[5])]*x] + Sqrt[(-1 + Sqrt[5])/10]*ArcTanh[Sqrt[2/ (-1 + Sqrt[5])]*x] + Sqrt[2/(5*(1 + Sqrt[5]))]*ArcTanh[Sqrt[2/(1 + Sqrt[5] )]*Sqrt[1 + x^2]] - Sqrt[(2*(1 + Sqrt[5]))/5]*ArcTanh[Sqrt[2/(1 + Sqrt[5]) ]*Sqrt[1 + x^2]] + x*Log[1 + x*Sqrt[1 + x^2]]
3.1.4.3.1 Defintions of rubi rules used
Int[Log[u_], x_Symbol] :> Simp[x*Log[u], x] - Int[SimplifyIntegrand[x*(D[u, x]/u), x], x] /; InverseFunctionFreeQ[u, x]
Leaf count of result is larger than twice the leaf count of optimal. \(412\) vs. \(2(75)=150\).
Time = 0.18 (sec) , antiderivative size = 413, normalized size of antiderivative = 4.26
| method | result | size |
| parts | \(x \ln \left (1+x \sqrt {x^{2}+1}\right )-2 x +\frac {\left (3+\sqrt {5}\right ) \sqrt {5}\, \arctan \left (\frac {2 x}{\sqrt {2+2 \sqrt {5}}}\right )}{5 \sqrt {2+2 \sqrt {5}}}-\frac {\left (\sqrt {5}-3\right ) \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 x}{\sqrt {-2+2 \sqrt {5}}}\right )}{5 \sqrt {-2+2 \sqrt {5}}}-\frac {\sqrt {5}\, \operatorname {arctanh}\left (\frac {\sqrt {x^{2}+1}-x}{\sqrt {2+\sqrt {5}}}\right )}{5 \sqrt {2+\sqrt {5}}}-\frac {\sqrt {5}\, \arctan \left (\frac {\sqrt {x^{2}+1}-x}{\sqrt {-2+\sqrt {5}}}\right )}{5 \sqrt {-2+\sqrt {5}}}-\frac {\sqrt {-2+\sqrt {5}}\, \sqrt {5}\, \operatorname {arctanh}\left (\frac {\sqrt {x^{2}+1}-x}{\sqrt {-2+\sqrt {5}}}\right )}{5}+\frac {\sqrt {2+\sqrt {5}}\, \sqrt {5}\, \arctan \left (\frac {\sqrt {x^{2}+1}-x}{\sqrt {2+\sqrt {5}}}\right )}{5}+\frac {2 \sqrt {5}\, \arctan \left (\frac {2 x}{\sqrt {2+2 \sqrt {5}}}\right )}{5 \sqrt {2+2 \sqrt {5}}}+\frac {2 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 x}{\sqrt {-2+2 \sqrt {5}}}\right )}{5 \sqrt {-2+2 \sqrt {5}}}-\frac {\left (3+\sqrt {5}\right ) \sqrt {5}\, \operatorname {arctanh}\left (\frac {\sqrt {x^{2}+1}-x}{\sqrt {2+\sqrt {5}}}\right )}{10 \sqrt {2+\sqrt {5}}}+\frac {\left (\sqrt {5}-3\right ) \sqrt {5}\, \arctan \left (\frac {\sqrt {x^{2}+1}-x}{\sqrt {-2+\sqrt {5}}}\right )}{10 \sqrt {-2+\sqrt {5}}}-\frac {\left (\sqrt {5}-1\right ) \sqrt {5}\, \operatorname {arctanh}\left (\frac {\sqrt {x^{2}+1}-x}{\sqrt {-2+\sqrt {5}}}\right )}{10 \sqrt {-2+\sqrt {5}}}+\frac {\left (\sqrt {5}+1\right ) \sqrt {5}\, \arctan \left (\frac {\sqrt {x^{2}+1}-x}{\sqrt {2+\sqrt {5}}}\right )}{10 \sqrt {2+\sqrt {5}}}\) | \(413\) |
| default | \(x \ln \left (1+x \sqrt {x^{2}+1}\right )-\frac {\left (\sqrt {5}+1\right ) \sqrt {5}\, \arctan \left (\frac {2 x}{\sqrt {2+2 \sqrt {5}}}\right )}{5 \sqrt {2+2 \sqrt {5}}}+\frac {\left (\sqrt {5}-1\right ) \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 x}{\sqrt {-2+2 \sqrt {5}}}\right )}{5 \sqrt {-2+2 \sqrt {5}}}-2 x +\frac {2 \left (3+\sqrt {5}\right ) \sqrt {5}\, \arctan \left (\frac {2 x}{\sqrt {2+2 \sqrt {5}}}\right )}{5 \sqrt {2+2 \sqrt {5}}}-\frac {2 \left (\sqrt {5}-3\right ) \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 x}{\sqrt {-2+2 \sqrt {5}}}\right )}{5 \sqrt {-2+2 \sqrt {5}}}-\frac {\left (\sqrt {5}+1\right ) \sqrt {5}\, \operatorname {arctanh}\left (\frac {\sqrt {x^{2}+1}-x}{\sqrt {2+\sqrt {5}}}\right )}{10 \sqrt {2+\sqrt {5}}}+\frac {\left (\sqrt {5}-1\right ) \sqrt {5}\, \arctan \left (\frac {\sqrt {x^{2}+1}-x}{\sqrt {-2+\sqrt {5}}}\right )}{10 \sqrt {-2+\sqrt {5}}}+\frac {\left (\sqrt {5}-3\right ) \sqrt {5}\, \operatorname {arctanh}\left (\frac {\sqrt {x^{2}+1}-x}{\sqrt {-2+\sqrt {5}}}\right )}{10 \sqrt {-2+\sqrt {5}}}-\frac {\left (3+\sqrt {5}\right ) \sqrt {5}\, \arctan \left (\frac {\sqrt {x^{2}+1}-x}{\sqrt {2+\sqrt {5}}}\right )}{10 \sqrt {2+\sqrt {5}}}-\frac {2 \sqrt {5}\, \operatorname {arctanh}\left (\frac {\sqrt {x^{2}+1}-x}{\sqrt {2+\sqrt {5}}}\right )}{5 \sqrt {2+\sqrt {5}}}-\frac {2 \sqrt {5}\, \arctan \left (\frac {\sqrt {x^{2}+1}-x}{\sqrt {-2+\sqrt {5}}}\right )}{5 \sqrt {-2+\sqrt {5}}}-\frac {2 \sqrt {-2+\sqrt {5}}\, \sqrt {5}\, \operatorname {arctanh}\left (\frac {\sqrt {x^{2}+1}-x}{\sqrt {-2+\sqrt {5}}}\right )}{5}+\frac {2 \sqrt {2+\sqrt {5}}\, \sqrt {5}\, \arctan \left (\frac {\sqrt {x^{2}+1}-x}{\sqrt {2+\sqrt {5}}}\right )}{5}\) | \(423\) |
x*ln(1+x*(x^2+1)^(1/2))-2*x+1/5*(3+5^(1/2))*5^(1/2)/(2+2*5^(1/2))^(1/2)*ar ctan(2*x/(2+2*5^(1/2))^(1/2))-1/5*(5^(1/2)-3)*5^(1/2)/(-2+2*5^(1/2))^(1/2) *arctanh(2*x/(-2+2*5^(1/2))^(1/2))-1/5*5^(1/2)/(2+5^(1/2))^(1/2)*arctanh(( (x^2+1)^(1/2)-x)/(2+5^(1/2))^(1/2))-1/5*5^(1/2)/(-2+5^(1/2))^(1/2)*arctan( ((x^2+1)^(1/2)-x)/(-2+5^(1/2))^(1/2))-1/5*(-2+5^(1/2))^(1/2)*5^(1/2)*arcta nh(((x^2+1)^(1/2)-x)/(-2+5^(1/2))^(1/2))+1/5*(2+5^(1/2))^(1/2)*5^(1/2)*arc tan(((x^2+1)^(1/2)-x)/(2+5^(1/2))^(1/2))+2/5*5^(1/2)/(2+2*5^(1/2))^(1/2)*a rctan(2*x/(2+2*5^(1/2))^(1/2))+2/5*5^(1/2)/(-2+2*5^(1/2))^(1/2)*arctanh(2* x/(-2+2*5^(1/2))^(1/2))-1/10*(3+5^(1/2))*5^(1/2)/(2+5^(1/2))^(1/2)*arctanh (((x^2+1)^(1/2)-x)/(2+5^(1/2))^(1/2))+1/10*(5^(1/2)-3)*5^(1/2)/(-2+5^(1/2) )^(1/2)*arctan(((x^2+1)^(1/2)-x)/(-2+5^(1/2))^(1/2))-1/10*(5^(1/2)-1)*5^(1 /2)/(-2+5^(1/2))^(1/2)*arctanh(((x^2+1)^(1/2)-x)/(-2+5^(1/2))^(1/2))+1/10* (5^(1/2)+1)*5^(1/2)/(2+5^(1/2))^(1/2)*arctan(((x^2+1)^(1/2)-x)/(2+5^(1/2)) ^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 451 vs. \(2 (75) = 150\).
Time = 0.26 (sec) , antiderivative size = 451, normalized size of antiderivative = 4.65 \[ \int \log \left (1+x \sqrt {1+x^2}\right ) \, dx=-\frac {1}{4} \, \sqrt {2} \sqrt {-\sqrt {5} - 1} \log \left (4 \, x^{2} - \sqrt {x^{2} + 1} {\left ({\left (\sqrt {5} \sqrt {2} - \sqrt {2}\right )} \sqrt {-\sqrt {5} - 1} + 4 \, x\right )} + {\left (\sqrt {5} \sqrt {2} x - \sqrt {2} x\right )} \sqrt {-\sqrt {5} - 1} + 4\right ) + \frac {1}{4} \, \sqrt {2} \sqrt {-\sqrt {5} - 1} \log \left (4 \, x^{2} + \sqrt {x^{2} + 1} {\left ({\left (\sqrt {5} \sqrt {2} - \sqrt {2}\right )} \sqrt {-\sqrt {5} - 1} - 4 \, x\right )} - {\left (\sqrt {5} \sqrt {2} x - \sqrt {2} x\right )} \sqrt {-\sqrt {5} - 1} + 4\right ) + \frac {1}{4} \, \sqrt {2} \sqrt {\sqrt {5} - 1} \log \left (4 \, x^{2} - 4 \, \sqrt {x^{2} + 1} x + {\left (\sqrt {5} \sqrt {2} x - \sqrt {x^{2} + 1} {\left (\sqrt {5} \sqrt {2} + \sqrt {2}\right )} + \sqrt {2} x\right )} \sqrt {\sqrt {5} - 1} + 4\right ) - \frac {1}{4} \, \sqrt {2} \sqrt {\sqrt {5} - 1} \log \left (4 \, x^{2} - 4 \, \sqrt {x^{2} + 1} x - {\left (\sqrt {5} \sqrt {2} x - \sqrt {x^{2} + 1} {\left (\sqrt {5} \sqrt {2} + \sqrt {2}\right )} + \sqrt {2} x\right )} \sqrt {\sqrt {5} - 1} + 4\right ) + x \log \left (\sqrt {x^{2} + 1} x + 1\right ) + \frac {1}{4} \, \sqrt {2} \sqrt {\sqrt {5} - 1} \log \left (2 \, x + \sqrt {2} \sqrt {\sqrt {5} - 1}\right ) - \frac {1}{4} \, \sqrt {2} \sqrt {\sqrt {5} - 1} \log \left (2 \, x - \sqrt {2} \sqrt {\sqrt {5} - 1}\right ) + \frac {1}{4} \, \sqrt {2} \sqrt {-\sqrt {5} - 1} \log \left (2 \, x + \sqrt {2} \sqrt {-\sqrt {5} - 1}\right ) - \frac {1}{4} \, \sqrt {2} \sqrt {-\sqrt {5} - 1} \log \left (2 \, x - \sqrt {2} \sqrt {-\sqrt {5} - 1}\right ) - 2 \, x \]
-1/4*sqrt(2)*sqrt(-sqrt(5) - 1)*log(4*x^2 - sqrt(x^2 + 1)*((sqrt(5)*sqrt(2 ) - sqrt(2))*sqrt(-sqrt(5) - 1) + 4*x) + (sqrt(5)*sqrt(2)*x - sqrt(2)*x)*s qrt(-sqrt(5) - 1) + 4) + 1/4*sqrt(2)*sqrt(-sqrt(5) - 1)*log(4*x^2 + sqrt(x ^2 + 1)*((sqrt(5)*sqrt(2) - sqrt(2))*sqrt(-sqrt(5) - 1) - 4*x) - (sqrt(5)* sqrt(2)*x - sqrt(2)*x)*sqrt(-sqrt(5) - 1) + 4) + 1/4*sqrt(2)*sqrt(sqrt(5) - 1)*log(4*x^2 - 4*sqrt(x^2 + 1)*x + (sqrt(5)*sqrt(2)*x - sqrt(x^2 + 1)*(s qrt(5)*sqrt(2) + sqrt(2)) + sqrt(2)*x)*sqrt(sqrt(5) - 1) + 4) - 1/4*sqrt(2 )*sqrt(sqrt(5) - 1)*log(4*x^2 - 4*sqrt(x^2 + 1)*x - (sqrt(5)*sqrt(2)*x - s qrt(x^2 + 1)*(sqrt(5)*sqrt(2) + sqrt(2)) + sqrt(2)*x)*sqrt(sqrt(5) - 1) + 4) + x*log(sqrt(x^2 + 1)*x + 1) + 1/4*sqrt(2)*sqrt(sqrt(5) - 1)*log(2*x + sqrt(2)*sqrt(sqrt(5) - 1)) - 1/4*sqrt(2)*sqrt(sqrt(5) - 1)*log(2*x - sqrt( 2)*sqrt(sqrt(5) - 1)) + 1/4*sqrt(2)*sqrt(-sqrt(5) - 1)*log(2*x + sqrt(2)*s qrt(-sqrt(5) - 1)) - 1/4*sqrt(2)*sqrt(-sqrt(5) - 1)*log(2*x - sqrt(2)*sqrt (-sqrt(5) - 1)) - 2*x
Timed out. \[ \int \log \left (1+x \sqrt {1+x^2}\right ) \, dx=\text {Timed out} \]
\[ \int \log \left (1+x \sqrt {1+x^2}\right ) \, dx=\int { \log \left (\sqrt {x^{2} + 1} x + 1\right ) \,d x } \]
x*log(sqrt(x^2 + 1)*x + 1) - 2*x + arctan(x) + integrate((2*x^2 + 1)/(x^2 + (x^3 + x)*sqrt(x^2 + 1) + 1), x)
Leaf count of result is larger than twice the leaf count of optimal. 235 vs. \(2 (75) = 150\).
Time = 0.40 (sec) , antiderivative size = 235, normalized size of antiderivative = 2.42 \[ \int \log \left (1+x \sqrt {1+x^2}\right ) \, dx=x \log \left (\sqrt {x^{2} + 1} x + 1\right ) + \frac {1}{2} \, \sqrt {2 \, \sqrt {5} + 2} \arctan \left (-\frac {x - \sqrt {x^{2} + 1} + \frac {1}{x - \sqrt {x^{2} + 1}}}{\sqrt {2 \, \sqrt {5} - 2}}\right ) + \frac {1}{2} \, \sqrt {2 \, \sqrt {5} + 2} \arctan \left (\frac {x}{\sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}}}\right ) - \frac {1}{4} \, \sqrt {2 \, \sqrt {5} - 2} \log \left (-x + \sqrt {x^{2} + 1} + \sqrt {2 \, \sqrt {5} + 2} - \frac {1}{x - \sqrt {x^{2} + 1}}\right ) + \frac {1}{4} \, \sqrt {2 \, \sqrt {5} - 2} \log \left ({\left | x + \sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} \right |}\right ) - \frac {1}{4} \, \sqrt {2 \, \sqrt {5} - 2} \log \left ({\left | x - \sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} \right |}\right ) + \frac {1}{4} \, \sqrt {2 \, \sqrt {5} - 2} \log \left ({\left | -x + \sqrt {x^{2} + 1} - \sqrt {2 \, \sqrt {5} + 2} - \frac {1}{x - \sqrt {x^{2} + 1}} \right |}\right ) - 2 \, x \]
x*log(sqrt(x^2 + 1)*x + 1) + 1/2*sqrt(2*sqrt(5) + 2)*arctan(-(x - sqrt(x^2 + 1) + 1/(x - sqrt(x^2 + 1)))/sqrt(2*sqrt(5) - 2)) + 1/2*sqrt(2*sqrt(5) + 2)*arctan(x/sqrt(1/2*sqrt(5) + 1/2)) - 1/4*sqrt(2*sqrt(5) - 2)*log(-x + s qrt(x^2 + 1) + sqrt(2*sqrt(5) + 2) - 1/(x - sqrt(x^2 + 1))) + 1/4*sqrt(2*s qrt(5) - 2)*log(abs(x + sqrt(1/2*sqrt(5) - 1/2))) - 1/4*sqrt(2*sqrt(5) - 2 )*log(abs(x - sqrt(1/2*sqrt(5) - 1/2))) + 1/4*sqrt(2*sqrt(5) - 2)*log(abs( -x + sqrt(x^2 + 1) - sqrt(2*sqrt(5) + 2) - 1/(x - sqrt(x^2 + 1)))) - 2*x
Time = 1.50 (sec) , antiderivative size = 666, normalized size of antiderivative = 6.87 \[ \int \log \left (1+x \sqrt {1+x^2}\right ) \, dx=x\,\ln \left (x\,\sqrt {x^2+1}+1\right )-2\,x+\frac {\ln \left (x-\frac {\sqrt {2}\,\sqrt {\sqrt {5}-1}}{2}\right )\,\left (\frac {\sqrt {5}}{2}-\frac {5}{2}\right )}{2\,\sqrt {\frac {\sqrt {5}}{2}-\frac {1}{2}}+4\,{\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}^{3/2}}-\frac {\ln \left (x+\frac {\sqrt {2}\,\sqrt {\sqrt {5}-1}}{2}\right )\,\left (\frac {\sqrt {5}}{2}-\frac {5}{2}\right )}{2\,\sqrt {\frac {\sqrt {5}}{2}-\frac {1}{2}}+4\,{\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}^{3/2}}-\frac {\ln \left (x-\frac {\sqrt {2}\,\sqrt {-\sqrt {5}-1}}{2}\right )\,\left (\frac {\sqrt {5}}{2}+\frac {5}{2}\right )}{2\,\sqrt {-\frac {\sqrt {5}}{2}-\frac {1}{2}}+4\,{\left (-\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}^{3/2}}+\frac {\ln \left (x+\frac {\sqrt {2}\,\sqrt {-\sqrt {5}-1}}{2}\right )\,\left (\frac {\sqrt {5}}{2}+\frac {5}{2}\right )}{2\,\sqrt {-\frac {\sqrt {5}}{2}-\frac {1}{2}}+4\,{\left (-\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}^{3/2}}+\frac {\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 {x^2+1}\,\sqrt {\sqrt {5}+1}}{2}+1\right )\right )\,\left (\sqrt {\frac {\sqrt {5}}{2}-\frac {1}{2}}+2\,{\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}^{3/2}\right )}{\left (2\,\sqrt {\frac {\sqrt {5}}{2}-\frac {1}{2}}+4\,{\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}^{3/2}\right )\,\sqrt {\frac {\sqrt {5}}{2}+\frac {1}{2}}}+\frac {\left (\ln \left (x+\frac {\sqrt {2}\,\sqrt {\sqrt {5}-1}}{2}\right )-\ln \left (\frac {\sqrt {2}\,\sqrt {x^2+1}\,\sqrt {\sqrt {5}+1}}{2}-\frac {\sqrt {2}\,x\,\sqrt {\sqrt {5}-1}}{2}+1\right )\right )\,\left (\sqrt {\frac {\sqrt {5}}{2}-\frac {1}{2}}+2\,{\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}^{3/2}\right )}{\left (2\,\sqrt {\frac {\sqrt {5}}{2}-\frac {1}{2}}+4\,{\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}^{3/2}\right )\,\sqrt {\frac {\sqrt {5}}{2}+\frac {1}{2}}}-\frac {\left (\ln \left (\frac {\sqrt {2}\,\sqrt {x^2+1}\,\sqrt {1-\sqrt {5}}}{2}-\frac {\sqrt {2}\,x\,\sqrt {-\sqrt {5}-1}}{2}+1\right )-\ln \left (x+\frac {\sqrt {2}\,\sqrt {-\sqrt {5}-1}}{2}\right )\right )\,\left (\sqrt {-\frac {\sqrt {5}}{2}-\frac {1}{2}}+2\,{\left (-\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}^{3/2}\right )}{\left (2\,\sqrt {-\frac {\sqrt {5}}{2}-\frac {1}{2}}+4\,{\left (-\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}^{3/2}\right )\,\sqrt {\frac {1}{2}-\frac {\sqrt {5}}{2}}}-\frac {\left (\ln \left (\frac {\sqrt {2}\,x\,\sqrt {-\sqrt {5}-1}}{2}+\frac {\sqrt {2}\,\sqrt {x^2+1}\,\sqrt {1-\sqrt {5}}}{2}+1\right )-\ln \left (x-\frac {\sqrt {2}\,\sqrt {-\sqrt {5}-1}}{2}\right )\right )\,\left (\sqrt {-\frac {\sqrt {5}}{2}-\frac {1}{2}}+2\,{\left (-\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}^{3/2}\right )}{\left (2\,\sqrt {-\frac {\sqrt {5}}{2}-\frac {1}{2}}+4\,{\left (-\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}^{3/2}\right )\,\sqrt {\frac {1}{2}-\frac {\sqrt {5}}{2}}} \]
x*log(x*(x^2 + 1)^(1/2) + 1) - 2*x + (log(x - (2^(1/2)*(5^(1/2) - 1)^(1/2) )/2)*(5^(1/2)/2 - 5/2))/(2*(5^(1/2)/2 - 1/2)^(1/2) + 4*(5^(1/2)/2 - 1/2)^( 3/2)) - (log(x + (2^(1/2)*(5^(1/2) - 1)^(1/2))/2)*(5^(1/2)/2 - 5/2))/(2*(5 ^(1/2)/2 - 1/2)^(1/2) + 4*(5^(1/2)/2 - 1/2)^(3/2)) - (log(x - (2^(1/2)*(- 5^(1/2) - 1)^(1/2))/2)*(5^(1/2)/2 + 5/2))/(2*(- 5^(1/2)/2 - 1/2)^(1/2) + 4 *(- 5^(1/2)/2 - 1/2)^(3/2)) + (log(x + (2^(1/2)*(- 5^(1/2) - 1)^(1/2))/2)* (5^(1/2)/2 + 5/2))/(2*(- 5^(1/2)/2 - 1/2)^(1/2) + 4*(- 5^(1/2)/2 - 1/2)^(3 /2)) + ((log(x - (2^(1/2)*(5^(1/2) - 1)^(1/2))/2) - log((2^(1/2)*x*(5^(1/2 ) - 1)^(1/2))/2 + (2^(1/2)*(x^2 + 1)^(1/2)*(5^(1/2) + 1)^(1/2))/2 + 1))*(( 5^(1/2)/2 - 1/2)^(1/2) + 2*(5^(1/2)/2 - 1/2)^(3/2)))/((2*(5^(1/2)/2 - 1/2) ^(1/2) + 4*(5^(1/2)/2 - 1/2)^(3/2))*(5^(1/2)/2 + 1/2)^(1/2)) + ((log(x + ( 2^(1/2)*(5^(1/2) - 1)^(1/2))/2) - log((2^(1/2)*(x^2 + 1)^(1/2)*(5^(1/2) + 1)^(1/2))/2 - (2^(1/2)*x*(5^(1/2) - 1)^(1/2))/2 + 1))*((5^(1/2)/2 - 1/2)^( 1/2) + 2*(5^(1/2)/2 - 1/2)^(3/2)))/((2*(5^(1/2)/2 - 1/2)^(1/2) + 4*(5^(1/2 )/2 - 1/2)^(3/2))*(5^(1/2)/2 + 1/2)^(1/2)) - ((log((2^(1/2)*(x^2 + 1)^(1/2 )*(1 - 5^(1/2))^(1/2))/2 - (2^(1/2)*x*(- 5^(1/2) - 1)^(1/2))/2 + 1) - log( x + (2^(1/2)*(- 5^(1/2) - 1)^(1/2))/2))*((- 5^(1/2)/2 - 1/2)^(1/2) + 2*(- 5^(1/2)/2 - 1/2)^(3/2)))/((2*(- 5^(1/2)/2 - 1/2)^(1/2) + 4*(- 5^(1/2)/2 - 1/2)^(3/2))*(1/2 - 5^(1/2)/2)^(1/2)) - ((log((2^(1/2)*x*(- 5^(1/2) - 1)^(1 /2))/2 + (2^(1/2)*(x^2 + 1)^(1/2)*(1 - 5^(1/2))^(1/2))/2 + 1) - log(x -...