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

Optimal result

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

[Out]

Unintegrable

Rubi [F]

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

[In]

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

[Out]

Sqrt[1 + x^2]/(2*x) + 1/(2*x*Sqrt[x - Sqrt[1 + x^2]]) - Sqrt[x - Sqrt[1 + x^2]] - ArcTan[x/Sqrt[2*(-1 + Sqrt[2
])]]/(8*Sqrt[-1 + Sqrt[2]]) + (Sqrt[-1 + Sqrt[2]]*ArcTan[(Sqrt[(-1 + Sqrt[2])/2]*x)/Sqrt[1 + x^2]])/4 + (Sqrt[
-7 + 5*Sqrt[2]]*ArcTan[(Sqrt[(-1 + Sqrt[2])/2]*x)/Sqrt[1 + x^2]])/8 + ArcTan[Sqrt[x - Sqrt[1 + x^2]]]/2 + ((3*
I - (2*I)*Sqrt[2] + Sqrt[2*(-7 + 5*Sqrt[2])])*ArcTan[Sqrt[x - Sqrt[1 + x^2]]/Sqrt[Sqrt[3 - 2*Sqrt[2]] - I*Sqrt
[2*(-1 + Sqrt[2])]]])/(8*Sqrt[5*Sqrt[6 - 4*Sqrt[2]] - 7*Sqrt[3 - 2*Sqrt[2]] - I*(10 - 7*Sqrt[2])*Sqrt[-1 + Sqr
t[2]]]) - ((3*I - (2*I)*Sqrt[2] - Sqrt[2*(-7 + 5*Sqrt[2])])*ArcTan[Sqrt[x - Sqrt[1 + x^2]]/Sqrt[Sqrt[3 - 2*Sqr
t[2]] + I*Sqrt[2*(-1 + Sqrt[2])]]])/(8*Sqrt[5*Sqrt[6 - 4*Sqrt[2]] - 7*Sqrt[3 - 2*Sqrt[2]] + I*(10 - 7*Sqrt[2])
*Sqrt[-1 + Sqrt[2]]]) + ((3 + 2*Sqrt[2] - Sqrt[2*(7 + 5*Sqrt[2])])*ArcTan[Sqrt[-((x - Sqrt[1 + x^2])/(Sqrt[2*(
1 + Sqrt[2])] - Sqrt[3 + 2*Sqrt[2]]))]])/(8*Sqrt[(1 + Sqrt[2])*(3 + 2*Sqrt[2])*(-Sqrt[2*(1 + Sqrt[2])] + Sqrt[
3 + 2*Sqrt[2]])]) - ((3 + 2*Sqrt[2] + Sqrt[2*(7 + 5*Sqrt[2])])*ArcTan[Sqrt[(x - Sqrt[1 + x^2])/(Sqrt[2*(1 + Sq
rt[2])] + Sqrt[3 + 2*Sqrt[2]])]])/(8*Sqrt[(1 + Sqrt[2])*(3 + 2*Sqrt[2])*(Sqrt[2*(1 + Sqrt[2])] + Sqrt[3 + 2*Sq
rt[2]])]) - ArcTanh[x/Sqrt[2*(1 + Sqrt[2])]]/(8*Sqrt[1 + Sqrt[2]]) + (Sqrt[1 + Sqrt[2]]*ArcTanh[(Sqrt[(1 + Sqr
t[2])/2]*x)/Sqrt[1 + x^2]])/4 - (Sqrt[7 + 5*Sqrt[2]]*ArcTanh[(Sqrt[(1 + Sqrt[2])/2]*x)/Sqrt[1 + x^2]])/8 + Arc
Tanh[Sqrt[x - Sqrt[1 + x^2]]]/2 - ((3*I - (2*I)*Sqrt[2] + Sqrt[2*(-7 + 5*Sqrt[2])])*ArcTanh[Sqrt[x - Sqrt[1 +
x^2]]/Sqrt[Sqrt[3 - 2*Sqrt[2]] - I*Sqrt[2*(-1 + Sqrt[2])]]])/(8*Sqrt[5*Sqrt[6 - 4*Sqrt[2]] - 7*Sqrt[3 - 2*Sqrt
[2]] - I*(10 - 7*Sqrt[2])*Sqrt[-1 + Sqrt[2]]]) + ((3*I - (2*I)*Sqrt[2] - Sqrt[2*(-7 + 5*Sqrt[2])])*ArcTanh[Sqr
t[x - Sqrt[1 + x^2]]/Sqrt[Sqrt[3 - 2*Sqrt[2]] + I*Sqrt[2*(-1 + Sqrt[2])]]])/(8*Sqrt[5*Sqrt[6 - 4*Sqrt[2]] - 7*
Sqrt[3 - 2*Sqrt[2]] + I*(10 - 7*Sqrt[2])*Sqrt[-1 + Sqrt[2]]]) - ((3 + 2*Sqrt[2] - Sqrt[2*(7 + 5*Sqrt[2])])*Arc
Tanh[Sqrt[-((x - Sqrt[1 + x^2])/(Sqrt[2*(1 + Sqrt[2])] - Sqrt[3 + 2*Sqrt[2]]))]])/(8*Sqrt[(1 + Sqrt[2])*(3 + 2
*Sqrt[2])*(-Sqrt[2*(1 + Sqrt[2])] + Sqrt[3 + 2*Sqrt[2]])]) + ((3 + 2*Sqrt[2] + Sqrt[2*(7 + 5*Sqrt[2])])*ArcTan
h[Sqrt[(x - Sqrt[1 + x^2])/(Sqrt[2*(1 + Sqrt[2])] + Sqrt[3 + 2*Sqrt[2]])]])/(8*Sqrt[(1 + Sqrt[2])*(3 + 2*Sqrt[
2])*(Sqrt[2*(1 + Sqrt[2])] + Sqrt[3 + 2*Sqrt[2]])]) - ((I/8)*Defer[Int][(Sqrt[1 + x^2]*Sqrt[x - Sqrt[1 + x^2]]
)/(I*Sqrt[2*(-1 + Sqrt[2])] - x), x])/Sqrt[-1 + Sqrt[2]] - (I/16)*Sqrt[-1 + Sqrt[2]]*Defer[Int][(Sqrt[1 + x^2]
*Sqrt[x - Sqrt[1 + x^2]])/(I*Sqrt[2*(-1 + Sqrt[2])] - x), x] - Defer[Int][(Sqrt[1 + x^2]*Sqrt[x - Sqrt[1 + x^2
]])/(Sqrt[2*(1 + Sqrt[2])] - x), x]/(8*Sqrt[1 + Sqrt[2]]) + (Sqrt[1 + Sqrt[2]]*Defer[Int][(Sqrt[1 + x^2]*Sqrt[
x - Sqrt[1 + x^2]])/(Sqrt[2*(1 + Sqrt[2])] - x), x])/16 - ((I/8)*Defer[Int][(Sqrt[1 + x^2]*Sqrt[x - Sqrt[1 + x
^2]])/(I*Sqrt[2*(-1 + Sqrt[2])] + x), x])/Sqrt[-1 + Sqrt[2]] - (I/16)*Sqrt[-1 + Sqrt[2]]*Defer[Int][(Sqrt[1 +
x^2]*Sqrt[x - Sqrt[1 + x^2]])/(I*Sqrt[2*(-1 + Sqrt[2])] + x), x] - Defer[Int][(Sqrt[1 + x^2]*Sqrt[x - Sqrt[1 +
x^2]])/(Sqrt[2*(1 + Sqrt[2])] + x), x]/(8*Sqrt[1 + Sqrt[2]]) + (Sqrt[1 + Sqrt[2]]*Defer[Int][(Sqrt[1 + x^2]*S
qrt[x - Sqrt[1 + x^2]])/(Sqrt[2*(1 + Sqrt[2])] + x), x])/16

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {1}{x^2 \left (-x^2+2 \sqrt {1+x^2}\right )}+\frac {\sqrt {x-\sqrt {1+x^2}}}{x^2 \left (-x^2+2 \sqrt {1+x^2}\right )}\right ) \, dx \\ & = -\int \frac {1}{x^2 \left (-x^2+2 \sqrt {1+x^2}\right )} \, dx+\int \frac {\sqrt {x-\sqrt {1+x^2}}}{x^2 \left (-x^2+2 \sqrt {1+x^2}\right )} \, dx \\ & = -\int \left (\frac {\sqrt {1+x^2}}{2 x^2}+\frac {1}{4+4 x^2-x^4}+\frac {2 \sqrt {1+x^2}}{-4-4 x^2+x^4}-\frac {x^2 \sqrt {1+x^2}}{2 \left (-4-4 x^2+x^4\right )}\right ) \, dx+\int \left (\frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{2 x^2}+\frac {\sqrt {x-\sqrt {1+x^2}}}{4+4 x^2-x^4}+\frac {2 \sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{-4-4 x^2+x^4}-\frac {x^2 \sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{2 \left (-4-4 x^2+x^4\right )}\right ) \, dx \\ & = -\left (\frac {1}{2} \int \frac {\sqrt {1+x^2}}{x^2} \, dx\right )+\frac {1}{2} \int \frac {x^2 \sqrt {1+x^2}}{-4-4 x^2+x^4} \, dx+\frac {1}{2} \int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{x^2} \, dx-\frac {1}{2} \int \frac {x^2 \sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{-4-4 x^2+x^4} \, dx-2 \int \frac {\sqrt {1+x^2}}{-4-4 x^2+x^4} \, dx+2 \int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{-4-4 x^2+x^4} \, dx-\int \frac {1}{4+4 x^2-x^4} \, dx+\int \frac {\sqrt {x-\sqrt {1+x^2}}}{4+4 x^2-x^4} \, dx \\ & = \frac {\sqrt {1+x^2}}{2 x}-\frac {1}{2} \int \frac {-4-5 x^2}{\sqrt {1+x^2} \left (-4-4 x^2+x^4\right )} \, dx-\frac {1}{2} \int \left (\frac {\left (1+\frac {1}{\sqrt {2}}\right ) \sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{-4-4 \sqrt {2}+2 x^2}+\frac {\left (1-\frac {1}{\sqrt {2}}\right ) \sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{-4+4 \sqrt {2}+2 x^2}\right ) \, dx-\frac {1}{2} \text {Subst}\left (\int \frac {\left (1+x^2\right )^2}{\sqrt {x} \left (-1+x^2\right )^2} \, dx,x,x-\sqrt {1+x^2}\right )+2 \int \left (-\frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{2 \sqrt {2} \left (4+4 \sqrt {2}-2 x^2\right )}-\frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{2 \sqrt {2} \left (-4+4 \sqrt {2}+2 x^2\right )}\right ) \, dx+\frac {\int \frac {1}{2-2 \sqrt {2}-x^2} \, dx}{4 \sqrt {2}}-\frac {\int \frac {1}{2+2 \sqrt {2}-x^2} \, dx}{4 \sqrt {2}}-\frac {\int \frac {\sqrt {1+x^2}}{-4-4 \sqrt {2}+2 x^2} \, dx}{\sqrt {2}}+\frac {\int \frac {\sqrt {1+x^2}}{-4+4 \sqrt {2}+2 x^2} \, dx}{\sqrt {2}}+\int \left (\frac {\sqrt {x-\sqrt {1+x^2}}}{2 \sqrt {2} \left (4+4 \sqrt {2}-2 x^2\right )}+\frac {\sqrt {x-\sqrt {1+x^2}}}{2 \sqrt {2} \left (-4+4 \sqrt {2}+2 x^2\right )}\right ) \, dx \\ & = \frac {\sqrt {1+x^2}}{2 x}+\frac {1}{2 x \sqrt {x-\sqrt {1+x^2}}}-\frac {\arctan \left (\frac {x}{\sqrt {2 \left (-1+\sqrt {2}\right )}}\right )}{8 \sqrt {-1+\sqrt {2}}}-\frac {\text {arctanh}\left (\frac {x}{\sqrt {2 \left (1+\sqrt {2}\right )}}\right )}{8 \sqrt {1+\sqrt {2}}}-\frac {1}{4} \text {Subst}\left (\int \frac {2 x^{3/2}}{-1+x^2} \, dx,x,x-\sqrt {1+x^2}\right )-\frac {1}{2} \int \left (\frac {-5-\frac {7}{\sqrt {2}}}{\sqrt {1+x^2} \left (-4-4 \sqrt {2}+2 x^2\right )}+\frac {-5+\frac {7}{\sqrt {2}}}{\sqrt {1+x^2} \left (-4+4 \sqrt {2}+2 x^2\right )}\right ) \, dx+\frac {\int \frac {\sqrt {x-\sqrt {1+x^2}}}{4+4 \sqrt {2}-2 x^2} \, dx}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {x-\sqrt {1+x^2}}}{-4+4 \sqrt {2}+2 x^2} \, dx}{2 \sqrt {2}}-\frac {\int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{4+4 \sqrt {2}-2 x^2} \, dx}{\sqrt {2}}-\frac {\int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{-4+4 \sqrt {2}+2 x^2} \, dx}{\sqrt {2}}-\frac {1}{2} \left (4-3 \sqrt {2}\right ) \int \frac {1}{\sqrt {1+x^2} \left (-4+4 \sqrt {2}+2 x^2\right )} \, dx-\frac {1}{4} \left (2-\sqrt {2}\right ) \int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{-4+4 \sqrt {2}+2 x^2} \, dx-\frac {1}{4} \left (2+\sqrt {2}\right ) \int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{-4-4 \sqrt {2}+2 x^2} \, dx-\frac {1}{2} \left (4+3 \sqrt {2}\right ) \int \frac {1}{\sqrt {1+x^2} \left (-4-4 \sqrt {2}+2 x^2\right )} \, dx \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (verified)

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

[In]

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

[Out]

(x^2*(x^2 - 2*Sqrt[1 + x^2])*(-2/((-1 - x + Sqrt[1 + x^2])*(1 + Sqrt[x - Sqrt[1 + x^2]])) + ArcTan[Sqrt[x - Sq
rt[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]))

Maple [N/A] (verified)

Not integrable

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

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

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [N/A]

Not integrable

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

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

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

[In]

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

[Out]

-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)

Giac [N/A]

Not integrable

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

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

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

[In]

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

[Out]

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