\(\int \frac {\sqrt {x-\sqrt {1+x^2}}}{x^2+\sqrt {1+x^2}} \, dx\) [1524]

Optimal result

Integrand size = 33, antiderivative size = 105 \[ \int \frac {\sqrt {x-\sqrt {1+x^2}}}{x^2+\sqrt {1+x^2}} \, dx=\text {RootSum}\left [1-2 \text {$\#$1}^2-2 \text {$\#$1}^4-2 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {x-\sqrt {1+x^2}}-\text {$\#$1}\right ) \text {$\#$1}+\log \left (\sqrt {x-\sqrt {1+x^2}}-\text {$\#$1}\right ) \text {$\#$1}^5}{-1-2 \text {$\#$1}^2-3 \text {$\#$1}^4+2 \text {$\#$1}^6}\&\right ] \]

[Out]

Unintegrable

Rubi [F]

\[ \int \frac {\sqrt {x-\sqrt {1+x^2}}}{x^2+\sqrt {1+x^2}} \, dx=\int \frac {\sqrt {x-\sqrt {1+x^2}}}{x^2+\sqrt {1+x^2}} \, dx \]

[In]

Int[Sqrt[x - Sqrt[1 + x^2]]/(x^2 + Sqrt[1 + x^2]),x]

[Out]

-1/2*((1 - Sqrt[5])*(I*(3 - Sqrt[5]) + 2*Sqrt[-2 + Sqrt[5]])*ArcTan[(2^(1/4)*Sqrt[x - Sqrt[1 + x^2]])/Sqrt[Sqr
t[3 - Sqrt[5]] - I*Sqrt[-1 + Sqrt[5]]]])/(2^(3/4)*Sqrt[-5*(2 - Sqrt[5])*(Sqrt[3 - Sqrt[5]] - I*Sqrt[-1 + Sqrt[
5]])]) + ((1 - Sqrt[5])*(I*(3 - Sqrt[5]) - 2*Sqrt[-2 + Sqrt[5]])*ArcTan[(2^(1/4)*Sqrt[x - Sqrt[1 + x^2]])/Sqrt
[Sqrt[3 - Sqrt[5]] + I*Sqrt[-1 + Sqrt[5]]]])/(2*2^(3/4)*Sqrt[-5*(2 - Sqrt[5])*(Sqrt[3 - Sqrt[5]] + I*Sqrt[-1 +
Sqrt[5]])]) - ((5 + Sqrt[5])*(3 + Sqrt[5] - 2*Sqrt[2 + Sqrt[5]])*ArcTan[(2^(1/4)*Sqrt[x - Sqrt[1 + x^2]])/Sqr
t[-Sqrt[1 + Sqrt[5]] + Sqrt[3 + Sqrt[5]]]])/(10*2^(3/4)*Sqrt[(2 + Sqrt[5])*(-Sqrt[1 + Sqrt[5]] + Sqrt[3 + Sqrt
[5]])]) + ((5 + Sqrt[5])*(3 + Sqrt[5] + 2*Sqrt[2 + Sqrt[5]])*ArcTan[(2^(1/4)*Sqrt[x - Sqrt[1 + x^2]])/Sqrt[Sqr
t[1 + Sqrt[5]] + Sqrt[3 + Sqrt[5]]]])/(10*2^(3/4)*Sqrt[(2 + Sqrt[5])*(Sqrt[1 + Sqrt[5]] + Sqrt[3 + Sqrt[5]])])
+ ((1 - Sqrt[5])*(I*(3 - Sqrt[5]) + 2*Sqrt[-2 + Sqrt[5]])*ArcTanh[(2^(1/4)*Sqrt[x - Sqrt[1 + x^2]])/Sqrt[Sqrt
[3 - Sqrt[5]] - I*Sqrt[-1 + Sqrt[5]]]])/(2*2^(3/4)*Sqrt[-5*(2 - Sqrt[5])*(Sqrt[3 - Sqrt[5]] - I*Sqrt[-1 + Sqrt
[5]])]) - ((1 - Sqrt[5])*(I*(3 - Sqrt[5]) - 2*Sqrt[-2 + Sqrt[5]])*ArcTanh[(2^(1/4)*Sqrt[x - Sqrt[1 + x^2]])/Sq
rt[Sqrt[3 - Sqrt[5]] + I*Sqrt[-1 + Sqrt[5]]]])/(2*2^(3/4)*Sqrt[-5*(2 - Sqrt[5])*(Sqrt[3 - Sqrt[5]] + I*Sqrt[-1
+ Sqrt[5]])]) + ((5 + Sqrt[5])*(3 + Sqrt[5] - 2*Sqrt[2 + Sqrt[5]])*ArcTanh[(2^(1/4)*Sqrt[x - Sqrt[1 + x^2]])/
Sqrt[-Sqrt[1 + Sqrt[5]] + Sqrt[3 + Sqrt[5]]]])/(10*2^(3/4)*Sqrt[(2 + Sqrt[5])*(-Sqrt[1 + Sqrt[5]] + Sqrt[3 + S
qrt[5]])]) - ((5 + Sqrt[5])*(3 + Sqrt[5] + 2*Sqrt[2 + Sqrt[5]])*ArcTanh[(2^(1/4)*Sqrt[x - Sqrt[1 + x^2]])/Sqrt
[Sqrt[1 + Sqrt[5]] + Sqrt[3 + Sqrt[5]]]])/(10*2^(3/4)*Sqrt[(2 + Sqrt[5])*(Sqrt[1 + Sqrt[5]] + Sqrt[3 + Sqrt[5]
])]) + (I*Defer[Int][(Sqrt[1 + x^2]*Sqrt[x - Sqrt[1 + x^2]])/(I*Sqrt[-1 + Sqrt[5]] - Sqrt[2]*x), x])/Sqrt[5*(-
1 + Sqrt[5])] + Defer[Int][(Sqrt[1 + x^2]*Sqrt[x - Sqrt[1 + x^2]])/(Sqrt[1 + Sqrt[5]] - Sqrt[2]*x), x]/Sqrt[5*
(1 + Sqrt[5])] + (I*Defer[Int][(Sqrt[1 + x^2]*Sqrt[x - Sqrt[1 + x^2]])/(I*Sqrt[-1 + Sqrt[5]] + Sqrt[2]*x), x])
/Sqrt[5*(-1 + Sqrt[5])] + Defer[Int][(Sqrt[1 + x^2]*Sqrt[x - Sqrt[1 + x^2]])/(Sqrt[1 + Sqrt[5]] + Sqrt[2]*x),
x]/Sqrt[5*(1 + Sqrt[5])]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {x^2 \sqrt {x-\sqrt {1+x^2}}}{-1-x^2+x^4}-\frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{-1-x^2+x^4}\right ) \, dx \\ & = \int \frac {x^2 \sqrt {x-\sqrt {1+x^2}}}{-1-x^2+x^4} \, dx-\int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{-1-x^2+x^4} \, dx \\ & = \int \left (\frac {\left (1+\frac {1}{\sqrt {5}}\right ) \sqrt {x-\sqrt {1+x^2}}}{-1-\sqrt {5}+2 x^2}+\frac {\left (1-\frac {1}{\sqrt {5}}\right ) \sqrt {x-\sqrt {1+x^2}}}{-1+\sqrt {5}+2 x^2}\right ) \, dx-\int \left (-\frac {2 \sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{\sqrt {5} \left (1+\sqrt {5}-2 x^2\right )}-\frac {2 \sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{\sqrt {5} \left (-1+\sqrt {5}+2 x^2\right )}\right ) \, dx \\ & = \frac {2 \int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{1+\sqrt {5}-2 x^2} \, dx}{\sqrt {5}}+\frac {2 \int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{-1+\sqrt {5}+2 x^2} \, dx}{\sqrt {5}}+\frac {1}{5} \left (5-\sqrt {5}\right ) \int \frac {\sqrt {x-\sqrt {1+x^2}}}{-1+\sqrt {5}+2 x^2} \, dx+\frac {1}{5} \left (5+\sqrt {5}\right ) \int \frac {\sqrt {x-\sqrt {1+x^2}}}{-1-\sqrt {5}+2 x^2} \, dx \\ & = \frac {2 \int \left (\frac {i \sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{2 \sqrt {-1+\sqrt {5}} \left (i \sqrt {-1+\sqrt {5}}-\sqrt {2} x\right )}+\frac {i \sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{2 \sqrt {-1+\sqrt {5}} \left (i \sqrt {-1+\sqrt {5}}+\sqrt {2} x\right )}\right ) \, dx}{\sqrt {5}}+\frac {2 \int \left (\frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{2 \sqrt {1+\sqrt {5}} \left (\sqrt {1+\sqrt {5}}-\sqrt {2} x\right )}+\frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{2 \sqrt {1+\sqrt {5}} \left (\sqrt {1+\sqrt {5}}+\sqrt {2} x\right )}\right ) \, dx}{\sqrt {5}}+\frac {1}{5} \left (5-\sqrt {5}\right ) \int \left (\frac {i \sqrt {x-\sqrt {1+x^2}}}{2 \sqrt {-1+\sqrt {5}} \left (i \sqrt {-1+\sqrt {5}}-\sqrt {2} x\right )}+\frac {i \sqrt {x-\sqrt {1+x^2}}}{2 \sqrt {-1+\sqrt {5}} \left (i \sqrt {-1+\sqrt {5}}+\sqrt {2} x\right )}\right ) \, dx+\frac {1}{5} \left (5+\sqrt {5}\right ) \int \left (\frac {\sqrt {1+\sqrt {5}} \sqrt {x-\sqrt {1+x^2}}}{2 \left (-1-\sqrt {5}\right ) \left (\sqrt {1+\sqrt {5}}-\sqrt {2} x\right )}+\frac {\sqrt {1+\sqrt {5}} \sqrt {x-\sqrt {1+x^2}}}{2 \left (-1-\sqrt {5}\right ) \left (\sqrt {1+\sqrt {5}}+\sqrt {2} x\right )}\right ) \, dx \\ & = \frac {1}{2} \left (i \sqrt {\frac {1}{5} \left (-1+\sqrt {5}\right )}\right ) \int \frac {\sqrt {x-\sqrt {1+x^2}}}{i \sqrt {-1+\sqrt {5}}-\sqrt {2} x} \, dx+\frac {1}{2} \left (i \sqrt {\frac {1}{5} \left (-1+\sqrt {5}\right )}\right ) \int \frac {\sqrt {x-\sqrt {1+x^2}}}{i \sqrt {-1+\sqrt {5}}+\sqrt {2} x} \, dx+\frac {i \int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{i \sqrt {-1+\sqrt {5}}-\sqrt {2} x} \, dx}{\sqrt {5 \left (-1+\sqrt {5}\right )}}+\frac {i \int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{i \sqrt {-1+\sqrt {5}}+\sqrt {2} x} \, dx}{\sqrt {5 \left (-1+\sqrt {5}\right )}}-\frac {1}{2} \sqrt {\frac {1}{5} \left (1+\sqrt {5}\right )} \int \frac {\sqrt {x-\sqrt {1+x^2}}}{\sqrt {1+\sqrt {5}}-\sqrt {2} x} \, dx-\frac {1}{2} \sqrt {\frac {1}{5} \left (1+\sqrt {5}\right )} \int \frac {\sqrt {x-\sqrt {1+x^2}}}{\sqrt {1+\sqrt {5}}+\sqrt {2} x} \, dx+\frac {\int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{\sqrt {1+\sqrt {5}}-\sqrt {2} x} \, dx}{\sqrt {5 \left (1+\sqrt {5}\right )}}+\frac {\int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{\sqrt {1+\sqrt {5}}+\sqrt {2} x} \, dx}{\sqrt {5 \left (1+\sqrt {5}\right )}} \\ & = \frac {1}{2} \left (i \sqrt {\frac {1}{5} \left (-1+\sqrt {5}\right )}\right ) \text {Subst}\left (\int \frac {1+x^2}{\sqrt {x} \left (\sqrt {2}+2 i \sqrt {-1+\sqrt {5}} x-\sqrt {2} x^2\right )} \, dx,x,x-\sqrt {1+x^2}\right )+\frac {1}{2} \left (i \sqrt {\frac {1}{5} \left (-1+\sqrt {5}\right )}\right ) \text {Subst}\left (\int \frac {1+x^2}{\sqrt {x} \left (-\sqrt {2}+2 i \sqrt {-1+\sqrt {5}} x+\sqrt {2} x^2\right )} \, dx,x,x-\sqrt {1+x^2}\right )+\frac {i \int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{i \sqrt {-1+\sqrt {5}}-\sqrt {2} x} \, dx}{\sqrt {5 \left (-1+\sqrt {5}\right )}}+\frac {i \int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{i \sqrt {-1+\sqrt {5}}+\sqrt {2} x} \, dx}{\sqrt {5 \left (-1+\sqrt {5}\right )}}-\frac {1}{2} \sqrt {\frac {1}{5} \left (1+\sqrt {5}\right )} \text {Subst}\left (\int \frac {1+x^2}{\sqrt {x} \left (\sqrt {2}+2 \sqrt {1+\sqrt {5}} x-\sqrt {2} x^2\right )} \, dx,x,x-\sqrt {1+x^2}\right )-\frac {1}{2} \sqrt {\frac {1}{5} \left (1+\sqrt {5}\right )} \text {Subst}\left (\int \frac {1+x^2}{\sqrt {x} \left (-\sqrt {2}+2 \sqrt {1+\sqrt {5}} x+\sqrt {2} x^2\right )} \, dx,x,x-\sqrt {1+x^2}\right )+\frac {\int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{\sqrt {1+\sqrt {5}}-\sqrt {2} x} \, dx}{\sqrt {5 \left (1+\sqrt {5}\right )}}+\frac {\int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{\sqrt {1+\sqrt {5}}+\sqrt {2} x} \, dx}{\sqrt {5 \left (1+\sqrt {5}\right )}} \\ & = \frac {1}{2} \left (i \sqrt {\frac {1}{5} \left (-1+\sqrt {5}\right )}\right ) \text {Subst}\left (\int \left (-\frac {1}{\sqrt {2} \sqrt {x}}+\frac {2+i \sqrt {2 \left (-1+\sqrt {5}\right )} x}{\sqrt {x} \left (\sqrt {2}+2 i \sqrt {-1+\sqrt {5}} x-\sqrt {2} x^2\right )}\right ) \, dx,x,x-\sqrt {1+x^2}\right )+\frac {1}{2} \left (i \sqrt {\frac {1}{5} \left (-1+\sqrt {5}\right )}\right ) \text {Subst}\left (\int \left (\frac {1}{\sqrt {2} \sqrt {x}}+\frac {2-i \sqrt {2 \left (-1+\sqrt {5}\right )} x}{\sqrt {x} \left (-\sqrt {2}+2 i \sqrt {-1+\sqrt {5}} x+\sqrt {2} x^2\right )}\right ) \, dx,x,x-\sqrt {1+x^2}\right )+\frac {i \int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{i \sqrt {-1+\sqrt {5}}-\sqrt {2} x} \, dx}{\sqrt {5 \left (-1+\sqrt {5}\right )}}+\frac {i \int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{i \sqrt {-1+\sqrt {5}}+\sqrt {2} x} \, dx}{\sqrt {5 \left (-1+\sqrt {5}\right )}}-\frac {1}{2} \sqrt {\frac {1}{5} \left (1+\sqrt {5}\right )} \text {Subst}\left (\int \left (-\frac {1}{\sqrt {2} \sqrt {x}}+\frac {2+\sqrt {2 \left (1+\sqrt {5}\right )} x}{\sqrt {x} \left (\sqrt {2}+2 \sqrt {1+\sqrt {5}} x-\sqrt {2} x^2\right )}\right ) \, dx,x,x-\sqrt {1+x^2}\right )-\frac {1}{2} \sqrt {\frac {1}{5} \left (1+\sqrt {5}\right )} \text {Subst}\left (\int \left (\frac {1}{\sqrt {2} \sqrt {x}}+\frac {2-\sqrt {2 \left (1+\sqrt {5}\right )} x}{\sqrt {x} \left (-\sqrt {2}+2 \sqrt {1+\sqrt {5}} x+\sqrt {2} x^2\right )}\right ) \, dx,x,x-\sqrt {1+x^2}\right )+\frac {\int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{\sqrt {1+\sqrt {5}}-\sqrt {2} x} \, dx}{\sqrt {5 \left (1+\sqrt {5}\right )}}+\frac {\int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{\sqrt {1+\sqrt {5}}+\sqrt {2} x} \, dx}{\sqrt {5 \left (1+\sqrt {5}\right )}} \\ & = \frac {1}{2} \left (i \sqrt {\frac {1}{5} \left (-1+\sqrt {5}\right )}\right ) \text {Subst}\left (\int \frac {2+i \sqrt {2 \left (-1+\sqrt {5}\right )} x}{\sqrt {x} \left (\sqrt {2}+2 i \sqrt {-1+\sqrt {5}} x-\sqrt {2} x^2\right )} \, dx,x,x-\sqrt {1+x^2}\right )+\frac {1}{2} \left (i \sqrt {\frac {1}{5} \left (-1+\sqrt {5}\right )}\right ) \text {Subst}\left (\int \frac {2-i \sqrt {2 \left (-1+\sqrt {5}\right )} x}{\sqrt {x} \left (-\sqrt {2}+2 i \sqrt {-1+\sqrt {5}} x+\sqrt {2} x^2\right )} \, dx,x,x-\sqrt {1+x^2}\right )+\frac {i \int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{i \sqrt {-1+\sqrt {5}}-\sqrt {2} x} \, dx}{\sqrt {5 \left (-1+\sqrt {5}\right )}}+\frac {i \int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{i \sqrt {-1+\sqrt {5}}+\sqrt {2} x} \, dx}{\sqrt {5 \left (-1+\sqrt {5}\right )}}-\frac {1}{2} \sqrt {\frac {1}{5} \left (1+\sqrt {5}\right )} \text {Subst}\left (\int \frac {2+\sqrt {2 \left (1+\sqrt {5}\right )} x}{\sqrt {x} \left (\sqrt {2}+2 \sqrt {1+\sqrt {5}} x-\sqrt {2} x^2\right )} \, dx,x,x-\sqrt {1+x^2}\right )-\frac {1}{2} \sqrt {\frac {1}{5} \left (1+\sqrt {5}\right )} \text {Subst}\left (\int \frac {2-\sqrt {2 \left (1+\sqrt {5}\right )} x}{\sqrt {x} \left (-\sqrt {2}+2 \sqrt {1+\sqrt {5}} x+\sqrt {2} x^2\right )} \, dx,x,x-\sqrt {1+x^2}\right )+\frac {\int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{\sqrt {1+\sqrt {5}}-\sqrt {2} x} \, dx}{\sqrt {5 \left (1+\sqrt {5}\right )}}+\frac {\int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{\sqrt {1+\sqrt {5}}+\sqrt {2} x} \, dx}{\sqrt {5 \left (1+\sqrt {5}\right )}} \\ & = \left (i \sqrt {\frac {1}{5} \left (-1+\sqrt {5}\right )}\right ) \text {Subst}\left (\int \frac {2+i \sqrt {2 \left (-1+\sqrt {5}\right )} x^2}{\sqrt {2}+2 i \sqrt {-1+\sqrt {5}} x^2-\sqrt {2} x^4} \, dx,x,\sqrt {x-\sqrt {1+x^2}}\right )+\left (i \sqrt {\frac {1}{5} \left (-1+\sqrt {5}\right )}\right ) \text {Subst}\left (\int \frac {2-i \sqrt {2 \left (-1+\sqrt {5}\right )} x^2}{-\sqrt {2}+2 i \sqrt {-1+\sqrt {5}} x^2+\sqrt {2} x^4} \, dx,x,\sqrt {x-\sqrt {1+x^2}}\right )+\frac {i \int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{i \sqrt {-1+\sqrt {5}}-\sqrt {2} x} \, dx}{\sqrt {5 \left (-1+\sqrt {5}\right )}}+\frac {i \int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{i \sqrt {-1+\sqrt {5}}+\sqrt {2} x} \, dx}{\sqrt {5 \left (-1+\sqrt {5}\right )}}-\sqrt {\frac {1}{5} \left (1+\sqrt {5}\right )} \text {Subst}\left (\int \frac {2+\sqrt {2 \left (1+\sqrt {5}\right )} x^2}{\sqrt {2}+2 \sqrt {1+\sqrt {5}} x^2-\sqrt {2} x^4} \, dx,x,\sqrt {x-\sqrt {1+x^2}}\right )-\sqrt {\frac {1}{5} \left (1+\sqrt {5}\right )} \text {Subst}\left (\int \frac {2-\sqrt {2 \left (1+\sqrt {5}\right )} x^2}{-\sqrt {2}+2 \sqrt {1+\sqrt {5}} x^2+\sqrt {2} x^4} \, dx,x,\sqrt {x-\sqrt {1+x^2}}\right )+\frac {\int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{\sqrt {1+\sqrt {5}}-\sqrt {2} x} \, dx}{\sqrt {5 \left (1+\sqrt {5}\right )}}+\frac {\int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{\sqrt {1+\sqrt {5}}+\sqrt {2} x} \, dx}{\sqrt {5 \left (1+\sqrt {5}\right )}} \\ & = \frac {i \int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{i \sqrt {-1+\sqrt {5}}-\sqrt {2} x} \, dx}{\sqrt {5 \left (-1+\sqrt {5}\right )}}+\frac {i \int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{i \sqrt {-1+\sqrt {5}}+\sqrt {2} x} \, dx}{\sqrt {5 \left (-1+\sqrt {5}\right )}}+\frac {\int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{\sqrt {1+\sqrt {5}}-\sqrt {2} x} \, dx}{\sqrt {5 \left (1+\sqrt {5}\right )}}+\frac {\int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{\sqrt {1+\sqrt {5}}+\sqrt {2} x} \, dx}{\sqrt {5 \left (1+\sqrt {5}\right )}}+\frac {1}{2} \left (\sqrt {\frac {1}{5} \left (1+\sqrt {5}\right )} \left (i \left (3-\sqrt {5}\right )-2 \sqrt {-2+\sqrt {5}}\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {3-\sqrt {5}}+i \sqrt {-1+\sqrt {5}}-\sqrt {2} x^2} \, dx,x,\sqrt {x-\sqrt {1+x^2}}\right )-\frac {1}{2} \left (\sqrt {\frac {1}{5} \left (1+\sqrt {5}\right )} \left (i \left (3-\sqrt {5}\right )-2 \sqrt {-2+\sqrt {5}}\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {3-\sqrt {5}}+i \sqrt {-1+\sqrt {5}}+\sqrt {2} x^2} \, dx,x,\sqrt {x-\sqrt {1+x^2}}\right )-\frac {1}{2} \left (\sqrt {\frac {1}{5} \left (1+\sqrt {5}\right )} \left (i \left (3-\sqrt {5}\right )+2 \sqrt {-2+\sqrt {5}}\right )\right ) \text {Subst}\left (\int \frac {1}{-\sqrt {3-\sqrt {5}}+i \sqrt {-1+\sqrt {5}}-\sqrt {2} x^2} \, dx,x,\sqrt {x-\sqrt {1+x^2}}\right )+\frac {1}{2} \left (\sqrt {\frac {1}{5} \left (1+\sqrt {5}\right )} \left (i \left (3-\sqrt {5}\right )+2 \sqrt {-2+\sqrt {5}}\right )\right ) \text {Subst}\left (\int \frac {1}{-\sqrt {3-\sqrt {5}}+i \sqrt {-1+\sqrt {5}}+\sqrt {2} x^2} \, dx,x,\sqrt {x-\sqrt {1+x^2}}\right )+\frac {1}{2} \left (\sqrt {\frac {1}{5} \left (-1+\sqrt {5}\right )} \left (3+\sqrt {5}-2 \sqrt {2+\sqrt {5}}\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+\sqrt {5}}-\sqrt {3+\sqrt {5}}-\sqrt {2} x^2} \, dx,x,\sqrt {x-\sqrt {1+x^2}}\right )-\frac {1}{2} \left (\sqrt {\frac {1}{5} \left (-1+\sqrt {5}\right )} \left (3+\sqrt {5}-2 \sqrt {2+\sqrt {5}}\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+\sqrt {5}}-\sqrt {3+\sqrt {5}}+\sqrt {2} x^2} \, dx,x,\sqrt {x-\sqrt {1+x^2}}\right )-\frac {1}{2} \left (\sqrt {\frac {1}{5} \left (-1+\sqrt {5}\right )} \left (3+\sqrt {5}+2 \sqrt {2+\sqrt {5}}\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+\sqrt {5}}+\sqrt {3+\sqrt {5}}-\sqrt {2} x^2} \, dx,x,\sqrt {x-\sqrt {1+x^2}}\right )+\frac {1}{2} \left (\sqrt {\frac {1}{5} \left (-1+\sqrt {5}\right )} \left (3+\sqrt {5}+2 \sqrt {2+\sqrt {5}}\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+\sqrt {5}}+\sqrt {3+\sqrt {5}}+\sqrt {2} x^2} \, dx,x,\sqrt {x-\sqrt {1+x^2}}\right ) \\ & = \frac {\left (i \left (3-\sqrt {5}\right )+2 \sqrt {-2+\sqrt {5}}\right ) \arctan \left (\frac {\sqrt [4]{2} \sqrt {x-\sqrt {1+x^2}}}{\sqrt {\sqrt {3-\sqrt {5}}-i \sqrt {-1+\sqrt {5}}}}\right )}{2 \sqrt [4]{2} \sqrt {\frac {5 \left (\sqrt {3-\sqrt {5}}-i \sqrt {-1+\sqrt {5}}\right )}{1+\sqrt {5}}}}-\frac {\left (i \left (3-\sqrt {5}\right )-2 \sqrt {-2+\sqrt {5}}\right ) \arctan \left (\frac {\sqrt [4]{2} \sqrt {x-\sqrt {1+x^2}}}{\sqrt {\sqrt {3-\sqrt {5}}+i \sqrt {-1+\sqrt {5}}}}\right )}{2 \sqrt [4]{2} \sqrt {\frac {5 \left (\sqrt {3-\sqrt {5}}+i \sqrt {-1+\sqrt {5}}\right )}{1+\sqrt {5}}}}-\frac {\left (3+\sqrt {5}-2 \sqrt {2+\sqrt {5}}\right ) \sqrt {\frac {-1+\sqrt {5}}{5 \left (-\sqrt {1+\sqrt {5}}+\sqrt {3+\sqrt {5}}\right )}} \arctan \left (\frac {\sqrt [4]{2} \sqrt {x-\sqrt {1+x^2}}}{\sqrt {-\sqrt {1+\sqrt {5}}+\sqrt {3+\sqrt {5}}}}\right )}{2 \sqrt [4]{2}}+\frac {\left (3+\sqrt {5}+2 \sqrt {2+\sqrt {5}}\right ) \sqrt {\frac {-1+\sqrt {5}}{5 \left (\sqrt {1+\sqrt {5}}+\sqrt {3+\sqrt {5}}\right )}} \arctan \left (\frac {\sqrt [4]{2} \sqrt {x-\sqrt {1+x^2}}}{\sqrt {\sqrt {1+\sqrt {5}}+\sqrt {3+\sqrt {5}}}}\right )}{2 \sqrt [4]{2}}-\frac {\left (i \left (3-\sqrt {5}\right )+2 \sqrt {-2+\sqrt {5}}\right ) \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt {x-\sqrt {1+x^2}}}{\sqrt {\sqrt {3-\sqrt {5}}-i \sqrt {-1+\sqrt {5}}}}\right )}{2 \sqrt [4]{2} \sqrt {\frac {5 \left (\sqrt {3-\sqrt {5}}-i \sqrt {-1+\sqrt {5}}\right )}{1+\sqrt {5}}}}+\frac {\left (i \left (3-\sqrt {5}\right )-2 \sqrt {-2+\sqrt {5}}\right ) \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt {x-\sqrt {1+x^2}}}{\sqrt {\sqrt {3-\sqrt {5}}+i \sqrt {-1+\sqrt {5}}}}\right )}{2 \sqrt [4]{2} \sqrt {\frac {5 \left (\sqrt {3-\sqrt {5}}+i \sqrt {-1+\sqrt {5}}\right )}{1+\sqrt {5}}}}+\frac {\left (3+\sqrt {5}-2 \sqrt {2+\sqrt {5}}\right ) \sqrt {\frac {-1+\sqrt {5}}{5 \left (-\sqrt {1+\sqrt {5}}+\sqrt {3+\sqrt {5}}\right )}} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt {x-\sqrt {1+x^2}}}{\sqrt {-\sqrt {1+\sqrt {5}}+\sqrt {3+\sqrt {5}}}}\right )}{2 \sqrt [4]{2}}-\frac {\left (3+\sqrt {5}+2 \sqrt {2+\sqrt {5}}\right ) \sqrt {\frac {-1+\sqrt {5}}{5 \left (\sqrt {1+\sqrt {5}}+\sqrt {3+\sqrt {5}}\right )}} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt {x-\sqrt {1+x^2}}}{\sqrt {\sqrt {1+\sqrt {5}}+\sqrt {3+\sqrt {5}}}}\right )}{2 \sqrt [4]{2}}+\frac {i \int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{i \sqrt {-1+\sqrt {5}}-\sqrt {2} x} \, dx}{\sqrt {5 \left (-1+\sqrt {5}\right )}}+\frac {i \int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{i \sqrt {-1+\sqrt {5}}+\sqrt {2} x} \, dx}{\sqrt {5 \left (-1+\sqrt {5}\right )}}+\frac {\int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{\sqrt {1+\sqrt {5}}-\sqrt {2} x} \, dx}{\sqrt {5 \left (1+\sqrt {5}\right )}}+\frac {\int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{\sqrt {1+\sqrt {5}}+\sqrt {2} x} \, dx}{\sqrt {5 \left (1+\sqrt {5}\right )}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {x-\sqrt {1+x^2}}}{x^2+\sqrt {1+x^2}} \, dx=\text {RootSum}\left [1-2 \text {$\#$1}^2-2 \text {$\#$1}^4-2 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {x-\sqrt {1+x^2}}-\text {$\#$1}\right ) \text {$\#$1}+\log \left (\sqrt {x-\sqrt {1+x^2}}-\text {$\#$1}\right ) \text {$\#$1}^5}{-1-2 \text {$\#$1}^2-3 \text {$\#$1}^4+2 \text {$\#$1}^6}\&\right ] \]

[In]

Integrate[Sqrt[x - Sqrt[1 + x^2]]/(x^2 + Sqrt[1 + x^2]),x]

[Out]

RootSum[1 - 2*#1^2 - 2*#1^4 - 2*#1^6 + #1^8 & , (Log[Sqrt[x - Sqrt[1 + x^2]] - #1]*#1 + Log[Sqrt[x - Sqrt[1 +
x^2]] - #1]*#1^5)/(-1 - 2*#1^2 - 3*#1^4 + 2*#1^6) & ]

Maple [N/A] (verified)

Not integrable

Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.26

\[\int \frac {\sqrt {x -\sqrt {x^{2}+1}}}{x^{2}+\sqrt {x^{2}+1}}d x\]

[In]

int((x-(x^2+1)^(1/2))^(1/2)/(x^2+(x^2+1)^(1/2)),x)

[Out]

int((x-(x^2+1)^(1/2))^(1/2)/(x^2+(x^2+1)^(1/2)),x)

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 0.96 (sec) , antiderivative size = 2627, normalized size of antiderivative = 25.02 \[ \int \frac {\sqrt {x-\sqrt {1+x^2}}}{x^2+\sqrt {1+x^2}} \, dx=\text {Too large to display} \]

[In]

integrate((x-(x^2+1)^(1/2))^(1/2)/(x^2+(x^2+1)^(1/2)),x, algorithm="fricas")

[Out]

-1/10*sqrt(5)*sqrt(-2*sqrt(5) + 2*sqrt(-3/4*(sqrt(5) + 2*sqrt(sqrt(5) + 2) + 3)^2 - 3/4*(sqrt(5) - 2*sqrt(sqrt
(5) + 2) + 3)^2 - 1/2*(sqrt(5) + 2*sqrt(sqrt(5) + 2) + 3)*(sqrt(5) - 2*sqrt(sqrt(5) + 2) - 9) + 6*sqrt(5) - 12
*sqrt(sqrt(5) + 2) + 26) + 6)*log(1/8*((sqrt(5)*(sqrt(5) - 2*sqrt(sqrt(5) + 2) + 3) - 4*sqrt(5))*(sqrt(5) + 2*
sqrt(sqrt(5) + 2) + 3)^2 - 4*sqrt(5)*(sqrt(5) - 2*sqrt(sqrt(5) + 2) + 3)^2 + (sqrt(5)*(sqrt(5) - 2*sqrt(sqrt(5
) + 2) + 3)^2 - 12*sqrt(5)*(sqrt(5) - 2*sqrt(sqrt(5) + 2) + 3) + 40*sqrt(5))*(sqrt(5) + 2*sqrt(sqrt(5) + 2) +
3) + 2*((sqrt(5)*(sqrt(5) - 2*sqrt(sqrt(5) + 2) + 3) - 4*sqrt(5))*(sqrt(5) + 2*sqrt(sqrt(5) + 2) + 3) - 4*sqrt
(5)*(sqrt(5) - 2*sqrt(sqrt(5) + 2) + 3) + 8*sqrt(5))*sqrt(-3/4*(sqrt(5) + 2*sqrt(sqrt(5) + 2) + 3)^2 - 3/4*(sq
rt(5) - 2*sqrt(sqrt(5) + 2) + 3)^2 - 1/2*(sqrt(5) + 2*sqrt(sqrt(5) + 2) + 3)*(sqrt(5) - 2*sqrt(sqrt(5) + 2) -
9) + 6*sqrt(5) - 12*sqrt(sqrt(5) + 2) + 26) + 40*sqrt(5)*(sqrt(5) - 2*sqrt(sqrt(5) + 2) + 3) - 48*sqrt(5))*sqr
t(-2*sqrt(5) + 2*sqrt(-3/4*(sqrt(5) + 2*sqrt(sqrt(5) + 2) + 3)^2 - 3/4*(sqrt(5) - 2*sqrt(sqrt(5) + 2) + 3)^2 -
1/2*(sqrt(5) + 2*sqrt(sqrt(5) + 2) + 3)*(sqrt(5) - 2*sqrt(sqrt(5) + 2) - 9) + 6*sqrt(5) - 12*sqrt(sqrt(5) + 2
) + 26) + 6) + 20*sqrt(x - sqrt(x^2 + 1))) + 1/10*sqrt(5)*sqrt(-2*sqrt(5) + 2*sqrt(-3/4*(sqrt(5) + 2*sqrt(sqrt
(5) + 2) + 3)^2 - 3/4*(sqrt(5) - 2*sqrt(sqrt(5) + 2) + 3)^2 - 1/2*(sqrt(5) + 2*sqrt(sqrt(5) + 2) + 3)*(sqrt(5)
- 2*sqrt(sqrt(5) + 2) - 9) + 6*sqrt(5) - 12*sqrt(sqrt(5) + 2) + 26) + 6)*log(-1/8*((sqrt(5)*(sqrt(5) - 2*sqrt
(sqrt(5) + 2) + 3) - 4*sqrt(5))*(sqrt(5) + 2*sqrt(sqrt(5) + 2) + 3)^2 - 4*sqrt(5)*(sqrt(5) - 2*sqrt(sqrt(5) +
2) + 3)^2 + (sqrt(5)*(sqrt(5) - 2*sqrt(sqrt(5) + 2) + 3)^2 - 12*sqrt(5)*(sqrt(5) - 2*sqrt(sqrt(5) + 2) + 3) +
40*sqrt(5))*(sqrt(5) + 2*sqrt(sqrt(5) + 2) + 3) + 2*((sqrt(5)*(sqrt(5) - 2*sqrt(sqrt(5) + 2) + 3) - 4*sqrt(5))
*(sqrt(5) + 2*sqrt(sqrt(5) + 2) + 3) - 4*sqrt(5)*(sqrt(5) - 2*sqrt(sqrt(5) + 2) + 3) + 8*sqrt(5))*sqrt(-3/4*(s
qrt(5) + 2*sqrt(sqrt(5) + 2) + 3)^2 - 3/4*(sqrt(5) - 2*sqrt(sqrt(5) + 2) + 3)^2 - 1/2*(sqrt(5) + 2*sqrt(sqrt(5
) + 2) + 3)*(sqrt(5) - 2*sqrt(sqrt(5) + 2) - 9) + 6*sqrt(5) - 12*sqrt(sqrt(5) + 2) + 26) + 40*sqrt(5)*(sqrt(5)
- 2*sqrt(sqrt(5) + 2) + 3) - 48*sqrt(5))*sqrt(-2*sqrt(5) + 2*sqrt(-3/4*(sqrt(5) + 2*sqrt(sqrt(5) + 2) + 3)^2
- 3/4*(sqrt(5) - 2*sqrt(sqrt(5) + 2) + 3)^2 - 1/2*(sqrt(5) + 2*sqrt(sqrt(5) + 2) + 3)*(sqrt(5) - 2*sqrt(sqrt(5
) + 2) - 9) + 6*sqrt(5) - 12*sqrt(sqrt(5) + 2) + 26) + 6) + 20*sqrt(x - sqrt(x^2 + 1))) - 1/10*sqrt(5)*sqrt(-2
*sqrt(5) - 2*sqrt(-3/4*(sqrt(5) + 2*sqrt(sqrt(5) + 2) + 3)^2 - 3/4*(sqrt(5) - 2*sqrt(sqrt(5) + 2) + 3)^2 - 1/2
*(sqrt(5) + 2*sqrt(sqrt(5) + 2) + 3)*(sqrt(5) - 2*sqrt(sqrt(5) + 2) - 9) + 6*sqrt(5) - 12*sqrt(sqrt(5) + 2) +
26) + 6)*log(1/8*((sqrt(5)*(sqrt(5) - 2*sqrt(sqrt(5) + 2) + 3) - 4*sqrt(5))*(sqrt(5) + 2*sqrt(sqrt(5) + 2) + 3
)^2 - 4*sqrt(5)*(sqrt(5) - 2*sqrt(sqrt(5) + 2) + 3)^2 + (sqrt(5)*(sqrt(5) - 2*sqrt(sqrt(5) + 2) + 3)^2 - 12*sq
rt(5)*(sqrt(5) - 2*sqrt(sqrt(5) + 2) + 3) + 40*sqrt(5))*(sqrt(5) + 2*sqrt(sqrt(5) + 2) + 3) - 2*((sqrt(5)*(sqr
t(5) - 2*sqrt(sqrt(5) + 2) + 3) - 4*sqrt(5))*(sqrt(5) + 2*sqrt(sqrt(5) + 2) + 3) - 4*sqrt(5)*(sqrt(5) - 2*sqrt
(sqrt(5) + 2) + 3) + 8*sqrt(5))*sqrt(-3/4*(sqrt(5) + 2*sqrt(sqrt(5) + 2) + 3)^2 - 3/4*(sqrt(5) - 2*sqrt(sqrt(5
) + 2) + 3)^2 - 1/2*(sqrt(5) + 2*sqrt(sqrt(5) + 2) + 3)*(sqrt(5) - 2*sqrt(sqrt(5) + 2) - 9) + 6*sqrt(5) - 12*s
qrt(sqrt(5) + 2) + 26) + 40*sqrt(5)*(sqrt(5) - 2*sqrt(sqrt(5) + 2) + 3) - 48*sqrt(5))*sqrt(-2*sqrt(5) - 2*sqrt
(-3/4*(sqrt(5) + 2*sqrt(sqrt(5) + 2) + 3)^2 - 3/4*(sqrt(5) - 2*sqrt(sqrt(5) + 2) + 3)^2 - 1/2*(sqrt(5) + 2*sqr
t(sqrt(5) + 2) + 3)*(sqrt(5) - 2*sqrt(sqrt(5) + 2) - 9) + 6*sqrt(5) - 12*sqrt(sqrt(5) + 2) + 26) + 6) + 20*sqr
t(x - sqrt(x^2 + 1))) + 1/10*sqrt(5)*sqrt(-2*sqrt(5) - 2*sqrt(-3/4*(sqrt(5) + 2*sqrt(sqrt(5) + 2) + 3)^2 - 3/4
*(sqrt(5) - 2*sqrt(sqrt(5) + 2) + 3)^2 - 1/2*(sqrt(5) + 2*sqrt(sqrt(5) + 2) + 3)*(sqrt(5) - 2*sqrt(sqrt(5) + 2
) - 9) + 6*sqrt(5) - 12*sqrt(sqrt(5) + 2) + 26) + 6)*log(-1/8*((sqrt(5)*(sqrt(5) - 2*sqrt(sqrt(5) + 2) + 3) -
4*sqrt(5))*(sqrt(5) + 2*sqrt(sqrt(5) + 2) + 3)^2 - 4*sqrt(5)*(sqrt(5) - 2*sqrt(sqrt(5) + 2) + 3)^2 + (sqrt(5)*
(sqrt(5) - 2*sqrt(sqrt(5) + 2) + 3)^2 - 12*sqrt(5)*(sqrt(5) - 2*sqrt(sqrt(5) + 2) + 3) + 40*sqrt(5))*(sqrt(5)
+ 2*sqrt(sqrt(5) + 2) + 3) - 2*((sqrt(5)*(sqrt(5) - 2*sqrt(sqrt(5) + 2) + 3) - 4*sqrt(5))*(sqrt(5) + 2*sqrt(sq
rt(5) + 2) + 3) - 4*sqrt(5)*(sqrt(5) - 2*sqrt(sqrt(5) + 2) + 3) + 8*sqrt(5))*sqrt(-3/4*(sqrt(5) + 2*sqrt(sqrt(
5) + 2) + 3)^2 - 3/4*(sqrt(5) - 2*sqrt(sqrt(5) + 2) + 3)^2 - 1/2*(sqrt(5) + 2*sqrt(sqrt(5) + 2) + 3)*(sqrt(5)
- 2*sqrt(sqrt(5) + 2) - 9) + 6*sqrt(5) - 12*sqrt(sqrt(5) + 2) + 26) + 40*sqrt(5)*(sqrt(5) - 2*sqrt(sqrt(5) + 2
) + 3) - 48*sqrt(5))*sqrt(-2*sqrt(5) - 2*sqrt(-3/4*(sqrt(5) + 2*sqrt(sqrt(5) + 2) + 3)^2 - 3/4*(sqrt(5) - 2*sq
rt(sqrt(5) + 2) + 3)^2 - 1/2*(sqrt(5) + 2*sqrt(sqrt(5) + 2) + 3)*(sqrt(5) - 2*sqrt(sqrt(5) + 2) - 9) + 6*sqrt(
5) - 12*sqrt(sqrt(5) + 2) + 26) + 6) + 20*sqrt(x - sqrt(x^2 + 1))) + sqrt(1/10*sqrt(5) + 1/5*sqrt(sqrt(5) + 2)
+ 3/10)*log(1/4*((sqrt(5) - 2*sqrt(sqrt(5) + 2) + 3)^3 + (sqrt(5) + 2*sqrt(sqrt(5) + 2) + 3)^2*(sqrt(5) - 2*s
qrt(sqrt(5) + 2) - 1) + ((sqrt(5) - 2*sqrt(sqrt(5) + 2) + 3)^2 - 12*sqrt(5) + 24*sqrt(sqrt(5) + 2) + 4)*(sqrt(
5) + 2*sqrt(sqrt(5) + 2) + 3) - 12*(sqrt(5) - 2*sqrt(sqrt(5) + 2) + 3)^2 + 28*sqrt(5) - 56*sqrt(sqrt(5) + 2) +
44)*sqrt(1/10*sqrt(5) + 1/5*sqrt(sqrt(5) + 2) + 3/10) + 2*sqrt(x - sqrt(x^2 + 1))) - sqrt(1/10*sqrt(5) + 1/5*
sqrt(sqrt(5) + 2) + 3/10)*log(-1/4*((sqrt(5) - 2*sqrt(sqrt(5) + 2) + 3)^3 + (sqrt(5) + 2*sqrt(sqrt(5) + 2) + 3
)^2*(sqrt(5) - 2*sqrt(sqrt(5) + 2) - 1) + ((sqrt(5) - 2*sqrt(sqrt(5) + 2) + 3)^2 - 12*sqrt(5) + 24*sqrt(sqrt(5
) + 2) + 4)*(sqrt(5) + 2*sqrt(sqrt(5) + 2) + 3) - 12*(sqrt(5) - 2*sqrt(sqrt(5) + 2) + 3)^2 + 28*sqrt(5) - 56*s
qrt(sqrt(5) + 2) + 44)*sqrt(1/10*sqrt(5) + 1/5*sqrt(sqrt(5) + 2) + 3/10) + 2*sqrt(x - sqrt(x^2 + 1))) - sqrt(1
/10*sqrt(5) - 1/5*sqrt(sqrt(5) + 2) + 3/10)*log(1/4*((sqrt(5) - 2*sqrt(sqrt(5) + 2) + 3)^3 - 8*(sqrt(5) - 2*sq
rt(sqrt(5) + 2) + 3)^2 - 12*sqrt(5) + 24*sqrt(sqrt(5) + 2) - 28)*sqrt(1/10*sqrt(5) - 1/5*sqrt(sqrt(5) + 2) + 3
/10) + 2*sqrt(x - sqrt(x^2 + 1))) + sqrt(1/10*sqrt(5) - 1/5*sqrt(sqrt(5) + 2) + 3/10)*log(-1/4*((sqrt(5) - 2*s
qrt(sqrt(5) + 2) + 3)^3 - 8*(sqrt(5) - 2*sqrt(sqrt(5) + 2) + 3)^2 - 12*sqrt(5) + 24*sqrt(sqrt(5) + 2) - 28)*sq
rt(1/10*sqrt(5) - 1/5*sqrt(sqrt(5) + 2) + 3/10) + 2*sqrt(x - sqrt(x^2 + 1)))

Sympy [N/A]

Not integrable

Time = 0.87 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.25 \[ \int \frac {\sqrt {x-\sqrt {1+x^2}}}{x^2+\sqrt {1+x^2}} \, dx=\int \frac {\sqrt {x - \sqrt {x^{2} + 1}}}{x^{2} + \sqrt {x^{2} + 1}}\, dx \]

[In]

integrate((x-(x**2+1)**(1/2))**(1/2)/(x**2+(x**2+1)**(1/2)),x)

[Out]

Integral(sqrt(x - sqrt(x**2 + 1))/(x**2 + sqrt(x**2 + 1)), x)

Maxima [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.28 \[ \int \frac {\sqrt {x-\sqrt {1+x^2}}}{x^2+\sqrt {1+x^2}} \, dx=\int { \frac {\sqrt {x - \sqrt {x^{2} + 1}}}{x^{2} + \sqrt {x^{2} + 1}} \,d x } \]

[In]

integrate((x-(x^2+1)^(1/2))^(1/2)/(x^2+(x^2+1)^(1/2)),x, algorithm="maxima")

[Out]

integrate(sqrt(x - sqrt(x^2 + 1))/(x^2 + sqrt(x^2 + 1)), x)

Giac [N/A]

Not integrable

Time = 0.39 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.28 \[ \int \frac {\sqrt {x-\sqrt {1+x^2}}}{x^2+\sqrt {1+x^2}} \, dx=\int { \frac {\sqrt {x - \sqrt {x^{2} + 1}}}{x^{2} + \sqrt {x^{2} + 1}} \,d x } \]

[In]

integrate((x-(x^2+1)^(1/2))^(1/2)/(x^2+(x^2+1)^(1/2)),x, algorithm="giac")

[Out]

integrate(sqrt(x - sqrt(x^2 + 1))/(x^2 + sqrt(x^2 + 1)), x)

Mupad [N/A]

Not integrable

Time = 6.03 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.28 \[ \int \frac {\sqrt {x-\sqrt {1+x^2}}}{x^2+\sqrt {1+x^2}} \, dx=\int \frac {\sqrt {x-\sqrt {x^2+1}}}{\sqrt {x^2+1}+x^2} \,d x \]

[In]

int((x - (x^2 + 1)^(1/2))^(1/2)/((x^2 + 1)^(1/2) + x^2),x)

[Out]

int((x - (x^2 + 1)^(1/2))^(1/2)/((x^2 + 1)^(1/2) + x^2), x)