Integrand size = 19, antiderivative size = 155 \[ \int \frac {x}{x+\sqrt {x+\sqrt {1+x}}} \, dx=x-2 \sqrt {x+\sqrt {1+x}}-\log \left (1+2 \sqrt {1+x}-2 \sqrt {x+\sqrt {1+x}}\right )+4 \text {RootSum}\left [1+3 \text {$\#$1}-5 \text {$\#$1}^2+2 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {\log \left (\sqrt {1+x}-\sqrt {x+\sqrt {1+x}}+\text {$\#$1}\right ) \text {$\#$1}+\log \left (\sqrt {1+x}-\sqrt {x+\sqrt {1+x}}+\text {$\#$1}\right ) \text {$\#$1}^3}{3-10 \text {$\#$1}+6 \text {$\#$1}^2+4 \text {$\#$1}^3}\&\right ] \]
Time = 0.00 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.03 \[ \int \frac {x}{x+\sqrt {x+\sqrt {1+x}}} \, dx=x-2 \sqrt {x+\sqrt {1+x}}-\log \left (-1-2 \sqrt {1+x}+2 \sqrt {x+\sqrt {1+x}}\right )+4 \text {RootSum}\left [1+3 \text {$\#$1}-5 \text {$\#$1}^2+2 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {\log \left (-\sqrt {1+x}+\sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}+\log \left (-\sqrt {1+x}+\sqrt {x+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^3}{3-10 \text {$\#$1}+6 \text {$\#$1}^2+4 \text {$\#$1}^3}\&\right ] \]
x - 2*Sqrt[x + Sqrt[1 + x]] - Log[-1 - 2*Sqrt[1 + x] + 2*Sqrt[x + Sqrt[1 + x]]] + 4*RootSum[1 + 3*#1 - 5*#1^2 + 2*#1^3 + #1^4 & , (Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x]] - #1]*#1 + Log[-Sqrt[1 + x] + Sqrt[x + Sqrt[1 + x] ] - #1]*#1^3)/(3 - 10*#1 + 6*#1^2 + 4*#1^3) & ]
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.
| \(\displaystyle \int \frac {x}{x+\sqrt {x+\sqrt {x+1}}} \, dx\) |
| \(\Big \downarrow \) 7267 |
| \(\displaystyle 2 \int -\frac {x \sqrt {x+1}}{-x-\sqrt {x+\sqrt {x+1}}}d\sqrt {x+1}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (\frac {(x+1)^{3/2}}{x+\sqrt {x+\sqrt {x+1}}}-\frac {\sqrt {x+1}}{x+\sqrt {x+\sqrt {x+1}}}\right )d\sqrt {x+1}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 \left (\frac {1}{4} \int \frac {1}{(x+1)^2-3 (x+1)-\sqrt {x+1}+2}d\sqrt {x+1}+\frac {1}{2} \int \frac {\sqrt {x+1}}{(x+1)^2-3 (x+1)-\sqrt {x+1}+2}d\sqrt {x+1}+\int \frac {x+1}{(x+1)^2-3 (x+1)-\sqrt {x+1}+2}d\sqrt {x+1}+\int \frac {\sqrt {x+1} \sqrt {x+\sqrt {x+1}}}{(x+1)^2-3 (x+1)-\sqrt {x+1}+2}d\sqrt {x+1}-\int \frac {(x+1)^{3/2} \sqrt {x+\sqrt {x+1}}}{(x+1)^2-3 (x+1)-\sqrt {x+1}+2}d\sqrt {x+1}+\frac {x+1}{2}+\frac {1}{4} \log \left ((x+1)^2-3 (x+1)-\sqrt {x+1}+2\right )\right )\) |
3.22.36.3.1 Defintions of rubi rules used
Int[u_, x_Symbol] :> With[{lst = SubstForFractionalPowerOfLinear[u, x]}, Si mp[lst[[2]]*lst[[4]] Subst[Int[lst[[1]], x], x, lst[[3]]^(1/lst[[2]])], x ] /; !FalseQ[lst] && SubstForFractionalPowerQ[u, lst[[3]], x]]
Time = 0.03 (sec) , antiderivative size = 502, normalized size of antiderivative = 3.24
| method | result | size |
| derivativedivides | \(-\sqrt {x +\sqrt {1+x}}+\sqrt {1+x}+2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}+2 \textit {\_Z}^{3}-5 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )}{\sum }\frac {\left (2 \textit {\_R}^{3}+\textit {\_R}^{2}-\textit {\_R} \right ) \ln \left (\sqrt {x +\sqrt {1+x}}-\sqrt {1+x}-\textit {\_R} \right )}{4 \textit {\_R}^{3}+6 \textit {\_R}^{2}-10 \textit {\_R} +3}\right )-2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-2 \textit {\_Z}^{3}+\textit {\_Z}^{2}+5 \textit {\_Z} -1\right )}{\sum }\frac {\left (2 \textit {\_R}^{3}-3 \textit {\_R}^{2}-\textit {\_R} +6\right ) \ln \left (\sqrt {x +\sqrt {1+x}}-\sqrt {1+x}-\textit {\_R} \right )}{4 \textit {\_R}^{3}-6 \textit {\_R}^{2}+2 \textit {\_R} +5}\right )+\frac {5}{2 \left (-1-2 \sqrt {1+x}+2 \sqrt {x +\sqrt {1+x}}\right )}+\ln \left (-1-2 \sqrt {1+x}+2 \sqrt {x +\sqrt {1+x}}\right )+2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-3 \textit {\_Z}^{2}-\textit {\_Z} +2\right )}{\sum }\frac {\textit {\_R} \ln \left (\sqrt {1+x}-\textit {\_R} \right )}{4 \textit {\_R}^{3}-6 \textit {\_R} -1}\right )-4 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-3 \textit {\_Z}^{2}-\textit {\_Z} +2\right )}{\sum }\frac {\textit {\_R}^{3} \ln \left (\sqrt {1+x}-\textit {\_R} \right )}{4 \textit {\_R}^{3}-6 \textit {\_R} -1}\right )+1+x +2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-3 \textit {\_Z}^{2}-\textit {\_Z} +2\right )}{\sum }\frac {\left (3 \textit {\_R}^{3}+\textit {\_R}^{2}-2 \textit {\_R} \right ) \ln \left (\sqrt {1+x}-\textit {\_R} \right )}{4 \textit {\_R}^{3}-6 \textit {\_R} -1}\right )-2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}+2 \textit {\_Z}^{3}-5 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )}{\sum }\frac {\left (\textit {\_R}^{3}+\textit {\_R}^{2}-2 \textit {\_R} \right ) \ln \left (\sqrt {x +\sqrt {1+x}}-\sqrt {1+x}-\textit {\_R} \right )}{4 \textit {\_R}^{3}+6 \textit {\_R}^{2}-10 \textit {\_R} +3}\right )+2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-2 \textit {\_Z}^{3}+\textit {\_Z}^{2}+5 \textit {\_Z} -1\right )}{\sum }\frac {\left (\textit {\_R}^{3}-\textit {\_R}^{2}+2\right ) \ln \left (\sqrt {x +\sqrt {1+x}}-\sqrt {1+x}-\textit {\_R} \right )}{4 \textit {\_R}^{3}-6 \textit {\_R}^{2}+2 \textit {\_R} +5}\right )\) | \(502\) |
| default | \(-\sqrt {x +\sqrt {1+x}}+\sqrt {1+x}+2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}+2 \textit {\_Z}^{3}-5 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )}{\sum }\frac {\left (2 \textit {\_R}^{3}+\textit {\_R}^{2}-\textit {\_R} \right ) \ln \left (\sqrt {x +\sqrt {1+x}}-\sqrt {1+x}-\textit {\_R} \right )}{4 \textit {\_R}^{3}+6 \textit {\_R}^{2}-10 \textit {\_R} +3}\right )-2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-2 \textit {\_Z}^{3}+\textit {\_Z}^{2}+5 \textit {\_Z} -1\right )}{\sum }\frac {\left (2 \textit {\_R}^{3}-3 \textit {\_R}^{2}-\textit {\_R} +6\right ) \ln \left (\sqrt {x +\sqrt {1+x}}-\sqrt {1+x}-\textit {\_R} \right )}{4 \textit {\_R}^{3}-6 \textit {\_R}^{2}+2 \textit {\_R} +5}\right )+\frac {5}{2 \left (-1-2 \sqrt {1+x}+2 \sqrt {x +\sqrt {1+x}}\right )}+\ln \left (-1-2 \sqrt {1+x}+2 \sqrt {x +\sqrt {1+x}}\right )+2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-3 \textit {\_Z}^{2}-\textit {\_Z} +2\right )}{\sum }\frac {\textit {\_R} \ln \left (\sqrt {1+x}-\textit {\_R} \right )}{4 \textit {\_R}^{3}-6 \textit {\_R} -1}\right )-4 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-3 \textit {\_Z}^{2}-\textit {\_Z} +2\right )}{\sum }\frac {\textit {\_R}^{3} \ln \left (\sqrt {1+x}-\textit {\_R} \right )}{4 \textit {\_R}^{3}-6 \textit {\_R} -1}\right )+1+x +2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-3 \textit {\_Z}^{2}-\textit {\_Z} +2\right )}{\sum }\frac {\left (3 \textit {\_R}^{3}+\textit {\_R}^{2}-2 \textit {\_R} \right ) \ln \left (\sqrt {1+x}-\textit {\_R} \right )}{4 \textit {\_R}^{3}-6 \textit {\_R} -1}\right )-2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}+2 \textit {\_Z}^{3}-5 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )}{\sum }\frac {\left (\textit {\_R}^{3}+\textit {\_R}^{2}-2 \textit {\_R} \right ) \ln \left (\sqrt {x +\sqrt {1+x}}-\sqrt {1+x}-\textit {\_R} \right )}{4 \textit {\_R}^{3}+6 \textit {\_R}^{2}-10 \textit {\_R} +3}\right )+2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-2 \textit {\_Z}^{3}+\textit {\_Z}^{2}+5 \textit {\_Z} -1\right )}{\sum }\frac {\left (\textit {\_R}^{3}-\textit {\_R}^{2}+2\right ) \ln \left (\sqrt {x +\sqrt {1+x}}-\sqrt {1+x}-\textit {\_R} \right )}{4 \textit {\_R}^{3}-6 \textit {\_R}^{2}+2 \textit {\_R} +5}\right )\) | \(502\) |
-(x+(1+x)^(1/2))^(1/2)+(1+x)^(1/2)+2*sum((2*_R^3+_R^2-_R)/(4*_R^3+6*_R^2-1 0*_R+3)*ln((x+(1+x)^(1/2))^(1/2)-(1+x)^(1/2)-_R),_R=RootOf(_Z^4+2*_Z^3-5*_ Z^2+3*_Z+1))-2*sum((2*_R^3-3*_R^2-_R+6)/(4*_R^3-6*_R^2+2*_R+5)*ln((x+(1+x) ^(1/2))^(1/2)-(1+x)^(1/2)-_R),_R=RootOf(_Z^4-2*_Z^3+_Z^2+5*_Z-1))+5/2/(-1- 2*(1+x)^(1/2)+2*(x+(1+x)^(1/2))^(1/2))+ln(-1-2*(1+x)^(1/2)+2*(x+(1+x)^(1/2 ))^(1/2))+2*sum(_R/(4*_R^3-6*_R-1)*ln((1+x)^(1/2)-_R),_R=RootOf(_Z^4-3*_Z^ 2-_Z+2))-4*sum(_R^3/(4*_R^3-6*_R-1)*ln((1+x)^(1/2)-_R),_R=RootOf(_Z^4-3*_Z ^2-_Z+2))+1+x+2*sum((3*_R^3+_R^2-2*_R)/(4*_R^3-6*_R-1)*ln((1+x)^(1/2)-_R), _R=RootOf(_Z^4-3*_Z^2-_Z+2))-2*sum((_R^3+_R^2-2*_R)/(4*_R^3+6*_R^2-10*_R+3 )*ln((x+(1+x)^(1/2))^(1/2)-(1+x)^(1/2)-_R),_R=RootOf(_Z^4+2*_Z^3-5*_Z^2+3* _Z+1))+2*sum((_R^3-_R^2+2)/(4*_R^3-6*_R^2+2*_R+5)*ln((x+(1+x)^(1/2))^(1/2) -(1+x)^(1/2)-_R),_R=RootOf(_Z^4-2*_Z^3+_Z^2+5*_Z-1))
Timed out. \[ \int \frac {x}{x+\sqrt {x+\sqrt {1+x}}} \, dx=\text {Timed out} \]
Not integrable
Time = 2.09 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.10 \[ \int \frac {x}{x+\sqrt {x+\sqrt {1+x}}} \, dx=\int \frac {x}{x + \sqrt {x + \sqrt {x + 1}}}\, dx \]
Not integrable
Time = 0.23 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.11 \[ \int \frac {x}{x+\sqrt {x+\sqrt {1+x}}} \, dx=\int { \frac {x}{x + \sqrt {x + \sqrt {x + 1}}} \,d x } \]
Not integrable
Time = 1.05 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.11 \[ \int \frac {x}{x+\sqrt {x+\sqrt {1+x}}} \, dx=\int { \frac {x}{x + \sqrt {x + \sqrt {x + 1}}} \,d x } \]
Not integrable
Time = 0.00 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.11 \[ \int \frac {x}{x+\sqrt {x+\sqrt {1+x}}} \, dx=\int \frac {x}{x+\sqrt {x+\sqrt {x+1}}} \,d x \]