3.28.46 \(\int \frac {1-\sqrt {x-\sqrt {1+x^2}}}{x^4-2 x^2 \sqrt {1+x^2}} \, dx\) [2746]

3.28.46.1 Optimal result

Integrand size = 42, antiderivative size = 255 \[ \int \frac {1-\sqrt {x-\sqrt {1+x^2}}}{x^4-2 x^2 \sqrt {1+x^2}} \, dx=-\frac {1}{\left (-1-x+\sqrt {1+x^2}\right ) \left (1+\sqrt {x-\sqrt {1+x^2}}\right )}+\frac {1}{2} \arctan \left (\sqrt {x-\sqrt {1+x^2}}\right )+\frac {1}{2} \text {arctanh}\left (\sqrt {x-\sqrt {1+x^2}}\right )-\frac {1}{2} \text {RootSum}\left [8-32 \text {$\#$1}+80 \text {$\#$1}^2-128 \text {$\#$1}^3+128 \text {$\#$1}^4-80 \text {$\#$1}^5+32 \text {$\#$1}^6-8 \text {$\#$1}^7+\text {$\#$1}^8\&,\frac {\log \left (-1+\sqrt {x-\sqrt {1+x^2}}+\text {$\#$1}\right ) \text {$\#$1}-2 \log \left (-1+\sqrt {x-\sqrt {1+x^2}}+\text {$\#$1}\right ) \text {$\#$1}^2+\log \left (-1+\sqrt {x-\sqrt {1+x^2}}+\text {$\#$1}\right ) \text {$\#$1}^3}{4-16 \text {$\#$1}+32 \text {$\#$1}^2-32 \text {$\#$1}^3+18 \text {$\#$1}^4-6 \text {$\#$1}^5+\text {$\#$1}^6}\&\right ] \]

output

Unintegrable
 

3.28.46.2 Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.13 \[ \int \frac {1-\sqrt {x-\sqrt {1+x^2}}}{x^4-2 x^2 \sqrt {1+x^2}} \, dx=\frac {x^2 \left (x^2-2 \sqrt {1+x^2}\right ) \left (-\frac {2}{\left (-1-x+\sqrt {1+x^2}\right ) \left (1+\sqrt {x-\sqrt {1+x^2}}\right )}+\arctan \left (\sqrt {x-\sqrt {1+x^2}}\right )+\text {arctanh}\left (\sqrt {x-\sqrt {1+x^2}}\right )-\text {RootSum}\left [8-32 \text {$\#$1}+80 \text {$\#$1}^2-128 \text {$\#$1}^3+128 \text {$\#$1}^4-80 \text {$\#$1}^5+32 \text {$\#$1}^6-8 \text {$\#$1}^7+\text {$\#$1}^8\&,\frac {\log \left (-1+\sqrt {x-\sqrt {1+x^2}}+\text {$\#$1}\right ) \text {$\#$1}-2 \log \left (-1+\sqrt {x-\sqrt {1+x^2}}+\text {$\#$1}\right ) \text {$\#$1}^2+\log \left (-1+\sqrt {x-\sqrt {1+x^2}}+\text {$\#$1}\right ) \text {$\#$1}^3}{4-16 \text {$\#$1}+32 \text {$\#$1}^2-32 \text {$\#$1}^3+18 \text {$\#$1}^4-6 \text {$\#$1}^5+\text {$\#$1}^6}\&\right ]\right )}{2 \left (x^4-2 x^2 \sqrt {1+x^2}\right )} \]

input

Integrate[(1 - Sqrt[x - Sqrt[1 + x^2]])/(x^4 - 2*x^2*Sqrt[1 + x^2]),x]
 

output

(x^2*(x^2 - 2*Sqrt[1 + x^2])*(-2/((-1 - x + Sqrt[1 + x^2])*(1 + Sqrt[x - S 
qrt[1 + x^2]])) + ArcTan[Sqrt[x - Sqrt[1 + x^2]]] + ArcTanh[Sqrt[x - Sqrt[ 
1 + x^2]]] - RootSum[8 - 32*#1 + 80*#1^2 - 128*#1^3 + 128*#1^4 - 80*#1^5 + 
 32*#1^6 - 8*#1^7 + #1^8 & , (Log[-1 + Sqrt[x - Sqrt[1 + x^2]] + #1]*#1 - 
2*Log[-1 + Sqrt[x - Sqrt[1 + x^2]] + #1]*#1^2 + Log[-1 + Sqrt[x - Sqrt[1 + 
 x^2]] + #1]*#1^3)/(4 - 16*#1 + 32*#1^2 - 32*#1^3 + 18*#1^4 - 6*#1^5 + #1^ 
6) & ]))/(2*(x^4 - 2*x^2*Sqrt[1 + x^2]))
 

3.28.46.3 Rubi [F]

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.

\(\displaystyle \int \frac {1-\sqrt {x-\sqrt {x^2+1}}}{x^4-2 x^2 \sqrt {x^2+1}} \, dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {\sqrt {x-\sqrt {x^2+1}}}{x^2 \left (2 \sqrt {x^2+1}-x^2\right )}-\frac {1}{x^2 \left (2 \sqrt {x^2+1}-x^2\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\arctan \left (\frac {x}{\sqrt {2 \left (-1+\sqrt {2}\right )}}\right )}{8 \sqrt {-1+\sqrt {2}}}+\frac {1}{8} \sqrt {-7+5 \sqrt {2}} \arctan \left (\frac {\sqrt {\frac {1}{2} \left (-1+\sqrt {2}\right )} x}{\sqrt {x^2+1}}\right )+\frac {1}{4} \sqrt {-1+\sqrt {2}} \arctan \left (\frac {\sqrt {\frac {1}{2} \left (-1+\sqrt {2}\right )} x}{\sqrt {x^2+1}}\right )+\frac {1}{2} \arctan \left (\sqrt {x-\sqrt {x^2+1}}\right )+\frac {\left (3 i-2 i \sqrt {2}+\sqrt {2 \left (-7+5 \sqrt {2}\right )}\right ) \arctan \left (\frac {\sqrt {x-\sqrt {x^2+1}}}{\sqrt {\sqrt {3-2 \sqrt {2}}-i \sqrt {2 \left (-1+\sqrt {2}\right )}}}\right )}{8 \sqrt {5 \sqrt {6-4 \sqrt {2}}-7 \sqrt {3-2 \sqrt {2}}-i \left (10-7 \sqrt {2}\right ) \sqrt {-1+\sqrt {2}}}}-\frac {\left (3 i-2 i \sqrt {2}-\sqrt {2 \left (-7+5 \sqrt {2}\right )}\right ) \arctan \left (\frac {\sqrt {x-\sqrt {x^2+1}}}{\sqrt {\sqrt {3-2 \sqrt {2}}+i \sqrt {2 \left (-1+\sqrt {2}\right )}}}\right )}{8 \sqrt {5 \sqrt {6-4 \sqrt {2}}-7 \sqrt {3-2 \sqrt {2}}+i \left (10-7 \sqrt {2}\right ) \sqrt {-1+\sqrt {2}}}}+\frac {\left (3+2 \sqrt {2}-\sqrt {2 \left (7+5 \sqrt {2}\right )}\right ) \arctan \left (\sqrt {-\frac {x-\sqrt {x^2+1}}{\sqrt {2 \left (1+\sqrt {2}\right )}-\sqrt {3+2 \sqrt {2}}}}\right )}{8 \sqrt {\left (1+\sqrt {2}\right ) \left (3+2 \sqrt {2}\right ) \left (-\sqrt {2 \left (1+\sqrt {2}\right )}+\sqrt {3+2 \sqrt {2}}\right )}}-\frac {\left (3+2 \sqrt {2}+\sqrt {2 \left (7+5 \sqrt {2}\right )}\right ) \arctan \left (\sqrt {\frac {x-\sqrt {x^2+1}}{\sqrt {2 \left (1+\sqrt {2}\right )}+\sqrt {3+2 \sqrt {2}}}}\right )}{8 \sqrt {\left (1+\sqrt {2}\right ) \left (3+2 \sqrt {2}\right ) \left (\sqrt {2 \left (1+\sqrt {2}\right )}+\sqrt {3+2 \sqrt {2}}\right )}}-\frac {\text {arctanh}\left (\frac {x}{\sqrt {2 \left (1+\sqrt {2}\right )}}\right )}{8 \sqrt {1+\sqrt {2}}}-\frac {1}{8} \sqrt {7+5 \sqrt {2}} \text {arctanh}\left (\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} x}{\sqrt {x^2+1}}\right )+\frac {1}{4} \sqrt {1+\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} x}{\sqrt {x^2+1}}\right )+\frac {1}{2} \text {arctanh}\left (\sqrt {x-\sqrt {x^2+1}}\right )-\frac {\left (3 i-2 i \sqrt {2}+\sqrt {2 \left (-7+5 \sqrt {2}\right )}\right ) \text {arctanh}\left (\frac {\sqrt {x-\sqrt {x^2+1}}}{\sqrt {\sqrt {3-2 \sqrt {2}}-i \sqrt {2 \left (-1+\sqrt {2}\right )}}}\right )}{8 \sqrt {5 \sqrt {6-4 \sqrt {2}}-7 \sqrt {3-2 \sqrt {2}}-i \left (10-7 \sqrt {2}\right ) \sqrt {-1+\sqrt {2}}}}+\frac {\left (3 i-2 i \sqrt {2}-\sqrt {2 \left (-7+5 \sqrt {2}\right )}\right ) \text {arctanh}\left (\frac {\sqrt {x-\sqrt {x^2+1}}}{\sqrt {\sqrt {3-2 \sqrt {2}}+i \sqrt {2 \left (-1+\sqrt {2}\right )}}}\right )}{8 \sqrt {5 \sqrt {6-4 \sqrt {2}}-7 \sqrt {3-2 \sqrt {2}}+i \left (10-7 \sqrt {2}\right ) \sqrt {-1+\sqrt {2}}}}-\frac {\left (3+2 \sqrt {2}-\sqrt {2 \left (7+5 \sqrt {2}\right )}\right ) \text {arctanh}\left (\sqrt {-\frac {x-\sqrt {x^2+1}}{\sqrt {2 \left (1+\sqrt {2}\right )}-\sqrt {3+2 \sqrt {2}}}}\right )}{8 \sqrt {\left (1+\sqrt {2}\right ) \left (3+2 \sqrt {2}\right ) \left (-\sqrt {2 \left (1+\sqrt {2}\right )}+\sqrt {3+2 \sqrt {2}}\right )}}+\frac {\left (3+2 \sqrt {2}+\sqrt {2 \left (7+5 \sqrt {2}\right )}\right ) \text {arctanh}\left (\sqrt {\frac {x-\sqrt {x^2+1}}{\sqrt {2 \left (1+\sqrt {2}\right )}+\sqrt {3+2 \sqrt {2}}}}\right )}{8 \sqrt {\left (1+\sqrt {2}\right ) \left (3+2 \sqrt {2}\right ) \left (\sqrt {2 \left (1+\sqrt {2}\right )}+\sqrt {3+2 \sqrt {2}}\right )}}-\frac {1}{16} i \sqrt {-1+\sqrt {2}} \int \frac {\sqrt {x^2+1} \sqrt {x-\sqrt {x^2+1}}}{i \sqrt {2 \left (-1+\sqrt {2}\right )}-x}dx-\frac {i \int \frac {\sqrt {x^2+1} \sqrt {x-\sqrt {x^2+1}}}{i \sqrt {2 \left (-1+\sqrt {2}\right )}-x}dx}{8 \sqrt {-1+\sqrt {2}}}+\frac {1}{16} \sqrt {1+\sqrt {2}} \int \frac {\sqrt {x^2+1} \sqrt {x-\sqrt {x^2+1}}}{\sqrt {2 \left (1+\sqrt {2}\right )}-x}dx-\frac {\int \frac {\sqrt {x^2+1} \sqrt {x-\sqrt {x^2+1}}}{\sqrt {2 \left (1+\sqrt {2}\right )}-x}dx}{8 \sqrt {1+\sqrt {2}}}-\frac {1}{16} i \sqrt {-1+\sqrt {2}} \int \frac {\sqrt {x^2+1} \sqrt {x-\sqrt {x^2+1}}}{x+i \sqrt {2 \left (-1+\sqrt {2}\right )}}dx-\frac {i \int \frac {\sqrt {x^2+1} \sqrt {x-\sqrt {x^2+1}}}{x+i \sqrt {2 \left (-1+\sqrt {2}\right )}}dx}{8 \sqrt {-1+\sqrt {2}}}+\frac {1}{16} \sqrt {1+\sqrt {2}} \int \frac {\sqrt {x^2+1} \sqrt {x-\sqrt {x^2+1}}}{x+\sqrt {2 \left (1+\sqrt {2}\right )}}dx-\frac {\int \frac {\sqrt {x^2+1} \sqrt {x-\sqrt {x^2+1}}}{x+\sqrt {2 \left (1+\sqrt {2}\right )}}dx}{8 \sqrt {1+\sqrt {2}}}+\frac {\sqrt {x^2+1}}{2 x}-\sqrt {x-\sqrt {x^2+1}}+\frac {1}{2 x \sqrt {x-\sqrt {x^2+1}}}\)

input

Int[(1 - Sqrt[x - Sqrt[1 + x^2]])/(x^4 - 2*x^2*Sqrt[1 + x^2]),x]
 

output

$Aborted
 

3.28.46.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

3.28.46.4 Maple [N/A] (verified)

Not integrable

Time = 0.02 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.14

\[\int \frac {1-\sqrt {x -\sqrt {x^{2}+1}}}{x^{4}-2 x^{2} \sqrt {x^{2}+1}}d x\]

input

int((1-(x-(x^2+1)^(1/2))^(1/2))/(x^4-2*x^2*(x^2+1)^(1/2)),x)
 

output

int((1-(x-(x^2+1)^(1/2))^(1/2))/(x^4-2*x^2*(x^2+1)^(1/2)),x)
 

3.28.46.5 Fricas [F(-1)]

Timed out. \[ \int \frac {1-\sqrt {x-\sqrt {1+x^2}}}{x^4-2 x^2 \sqrt {1+x^2}} \, dx=\text {Timed out} \]

input

integrate((1-(x-(x^2+1)^(1/2))^(1/2))/(x^4-2*x^2*(x^2+1)^(1/2)),x, algorit 
hm="fricas")
 

output

Timed out
 

3.28.46.6 Sympy [N/A]

Not integrable

Time = 8.06 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.21 \[ \int \frac {1-\sqrt {x-\sqrt {1+x^2}}}{x^4-2 x^2 \sqrt {1+x^2}} \, dx=- \int \frac {\sqrt {x - \sqrt {x^{2} + 1}}}{x^{4} - 2 x^{2} \sqrt {x^{2} + 1}}\, dx - \int \left (- \frac {1}{x^{4} - 2 x^{2} \sqrt {x^{2} + 1}}\right )\, dx \]

input

integrate((1-(x-(x**2+1)**(1/2))**(1/2))/(x**4-2*x**2*(x**2+1)**(1/2)),x)
 

output

-Integral(sqrt(x - sqrt(x**2 + 1))/(x**4 - 2*x**2*sqrt(x**2 + 1)), x) - In 
tegral(-1/(x**4 - 2*x**2*sqrt(x**2 + 1)), x)
 

3.28.46.7 Maxima [N/A]

Not integrable

Time = 0.33 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.43 \[ \int \frac {1-\sqrt {x-\sqrt {1+x^2}}}{x^4-2 x^2 \sqrt {1+x^2}} \, dx=\int { -\frac {\sqrt {x - \sqrt {x^{2} + 1}} - 1}{x^{4} - 2 \, \sqrt {x^{2} + 1} x^{2}} \,d x } \]

input

integrate((1-(x-(x^2+1)^(1/2))^(1/2))/(x^4-2*x^2*(x^2+1)^(1/2)),x, algorit 
hm="maxima")
 

output

-1/6/x^3 - integrate((x^2 - 2*sqrt(x^2 + 1))*sqrt(x - sqrt(x^2 + 1))/(x^6 
- 4*sqrt(x^2 + 1)*x^4 + 4*x^4 + 4*x^2), x) - integrate(-1/2*(x^4 - 4*x^2 - 
 4)/(x^8 - 4*sqrt(x^2 + 1)*x^6 + 4*x^6 + 4*x^4), x)
 

3.28.46.8 Giac [N/A]

Not integrable

Time = 0.45 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.15 \[ \int \frac {1-\sqrt {x-\sqrt {1+x^2}}}{x^4-2 x^2 \sqrt {1+x^2}} \, dx=\int { -\frac {\sqrt {x - \sqrt {x^{2} + 1}} - 1}{x^{4} - 2 \, \sqrt {x^{2} + 1} x^{2}} \,d x } \]

input

integrate((1-(x-(x^2+1)^(1/2))^(1/2))/(x^4-2*x^2*(x^2+1)^(1/2)),x, algorit 
hm="giac")
 

output

integrate(-(sqrt(x - sqrt(x^2 + 1)) - 1)/(x^4 - 2*sqrt(x^2 + 1)*x^2), x)
 

3.28.46.9 Mupad [N/A]

Not integrable

Time = 8.67 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.15 \[ \int \frac {1-\sqrt {x-\sqrt {1+x^2}}}{x^4-2 x^2 \sqrt {1+x^2}} \, dx=\int \frac {\sqrt {x-\sqrt {x^2+1}}-1}{2\,x^2\,\sqrt {x^2+1}-x^4} \,d x \]

input

int(((x - (x^2 + 1)^(1/2))^(1/2) - 1)/(2*x^2*(x^2 + 1)^(1/2) - x^4),x)
 

output

int(((x - (x^2 + 1)^(1/2))^(1/2) - 1)/(2*x^2*(x^2 + 1)^(1/2) - x^4), x)