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

Optimal. Leaf size=255 \[ -\frac {1}{2} \text {RootSum}\left [\text {$\#$1}^8-8 \text {$\#$1}^7+32 \text {$\#$1}^6-80 \text {$\#$1}^5+128 \text {$\#$1}^4-128 \text {$\#$1}^3+80 \text {$\#$1}^2-32 \text {$\#$1}+8\& ,\frac {\text {$\#$1}^3 \log \left (\text {$\#$1}+\sqrt {x-\sqrt {x^2+1}}-1\right )-2 \text {$\#$1}^2 \log \left (\text {$\#$1}+\sqrt {x-\sqrt {x^2+1}}-1\right )+\text {$\#$1} \log \left (\text {$\#$1}+\sqrt {x-\sqrt {x^2+1}}-1\right )}{\text {$\#$1}^6-6 \text {$\#$1}^5+18 \text {$\#$1}^4-32 \text {$\#$1}^3+32 \text {$\#$1}^2-16 \text {$\#$1}+4}\& \right ]-\frac {1}{\left (\sqrt {x^2+1}-x-1\right ) \left (\sqrt {x-\sqrt {x^2+1}}+1\right )}+\frac {1}{2} \tan ^{-1}\left (\sqrt {x-\sqrt {x^2+1}}\right )+\frac {1}{2} \tanh ^{-1}\left (\sqrt {x-\sqrt {x^2+1}}\right ) \]

________________________________________________________________________________________

Rubi [F]  time = 13.04, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {1-\sqrt {x-\sqrt {1+x^2}}}{x^4-2 x^2 \sqrt {1+x^2}} \, dx \end {gather*}

Verification is not applicable to the result.

[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*} \int \frac {1-\sqrt {x-\sqrt {1+x^2}}}{x^4-2 x^2 \sqrt {1+x^2}} \, dx &=\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} \operatorname {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 {\tan ^{-1}\left (\frac {x}{\sqrt {2 \left (-1+\sqrt {2}\right )}}\right )}{8 \sqrt {-1+\sqrt {2}}}-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {2 \left (1+\sqrt {2}\right )}}\right )}{8 \sqrt {1+\sqrt {2}}}-\frac {1}{4} \operatorname {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\\ &=\frac {\sqrt {1+x^2}}{2 x}+\frac {1}{2 x \sqrt {x-\sqrt {1+x^2}}}-\frac {\tan ^{-1}\left (\frac {x}{\sqrt {2 \left (-1+\sqrt {2}\right )}}\right )}{8 \sqrt {-1+\sqrt {2}}}-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {2 \left (1+\sqrt {2}\right )}}\right )}{8 \sqrt {1+\sqrt {2}}}-\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^{3/2}}{-1+x^2} \, dx,x,x-\sqrt {1+x^2}\right )+\frac {\int \left (\frac {i \sqrt {\frac {1}{2} \left (-1+\sqrt {2}\right )} \sqrt {x-\sqrt {1+x^2}}}{\left (-4+4 \sqrt {2}\right ) \left (i \sqrt {2 \left (-1+\sqrt {2}\right )}-x\right )}+\frac {i \sqrt {\frac {1}{2} \left (-1+\sqrt {2}\right )} \sqrt {x-\sqrt {1+x^2}}}{\left (-4+4 \sqrt {2}\right ) \left (i \sqrt {2 \left (-1+\sqrt {2}\right )}+x\right )}\right ) \, dx}{2 \sqrt {2}}+\frac {\int \left (\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \sqrt {x-\sqrt {1+x^2}}}{\left (4+4 \sqrt {2}\right ) \left (\sqrt {2 \left (1+\sqrt {2}\right )}-x\right )}+\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \sqrt {x-\sqrt {1+x^2}}}{\left (4+4 \sqrt {2}\right ) \left (\sqrt {2 \left (1+\sqrt {2}\right )}+x\right )}\right ) \, dx}{2 \sqrt {2}}-\frac {\int \left (\frac {i \sqrt {\frac {1}{2} \left (-1+\sqrt {2}\right )} \sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{\left (-4+4 \sqrt {2}\right ) \left (i \sqrt {2 \left (-1+\sqrt {2}\right )}-x\right )}+\frac {i \sqrt {\frac {1}{2} \left (-1+\sqrt {2}\right )} \sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{\left (-4+4 \sqrt {2}\right ) \left (i \sqrt {2 \left (-1+\sqrt {2}\right )}+x\right )}\right ) \, dx}{\sqrt {2}}-\frac {\int \left (\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{\left (4+4 \sqrt {2}\right ) \left (\sqrt {2 \left (1+\sqrt {2}\right )}-x\right )}+\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{\left (4+4 \sqrt {2}\right ) \left (\sqrt {2 \left (1+\sqrt {2}\right )}+x\right )}\right ) \, dx}{\sqrt {2}}-\frac {1}{2} \left (4-3 \sqrt {2}\right ) \operatorname {Subst}\left (\int \frac {1}{-4+4 \sqrt {2}-\left (-6+4 \sqrt {2}\right ) x^2} \, dx,x,\frac {x}{\sqrt {1+x^2}}\right )-\frac {1}{4} \left (2-\sqrt {2}\right ) \int \left (\frac {i \sqrt {\frac {1}{2} \left (-1+\sqrt {2}\right )} \sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{\left (-4+4 \sqrt {2}\right ) \left (i \sqrt {2 \left (-1+\sqrt {2}\right )}-x\right )}+\frac {i \sqrt {\frac {1}{2} \left (-1+\sqrt {2}\right )} \sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{\left (-4+4 \sqrt {2}\right ) \left (i \sqrt {2 \left (-1+\sqrt {2}\right )}+x\right )}\right ) \, dx-\frac {1}{4} \left (2+\sqrt {2}\right ) \int \left (\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{\left (-4-4 \sqrt {2}\right ) \left (\sqrt {2 \left (1+\sqrt {2}\right )}-x\right )}+\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{\left (-4-4 \sqrt {2}\right ) \left (\sqrt {2 \left (1+\sqrt {2}\right )}+x\right )}\right ) \, dx-\frac {1}{2} \left (4+3 \sqrt {2}\right ) \operatorname {Subst}\left (\int \frac {1}{-4-4 \sqrt {2}-\left (-6-4 \sqrt {2}\right ) x^2} \, dx,x,\frac {x}{\sqrt {1+x^2}}\right )-\frac {1}{4} \left (-10+7 \sqrt {2}\right ) \int \frac {1}{\sqrt {1+x^2} \left (-4+4 \sqrt {2}+2 x^2\right )} \, dx+\frac {1}{4} \left (10+7 \sqrt {2}\right ) \int \frac {1}{\sqrt {1+x^2} \left (-4-4 \sqrt {2}+2 x^2\right )} \, dx\\ &=\frac {\sqrt {1+x^2}}{2 x}+\frac {1}{2 x \sqrt {x-\sqrt {1+x^2}}}-\sqrt {x-\sqrt {1+x^2}}-\frac {\tan ^{-1}\left (\frac {x}{\sqrt {2 \left (-1+\sqrt {2}\right )}}\right )}{8 \sqrt {-1+\sqrt {2}}}+\frac {1}{4} \sqrt {-1+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {\frac {1}{2} \left (-1+\sqrt {2}\right )} x}{\sqrt {1+x^2}}\right )-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {2 \left (1+\sqrt {2}\right )}}\right )}{8 \sqrt {1+\sqrt {2}}}+\frac {1}{4} \sqrt {1+\sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} x}{\sqrt {1+x^2}}\right )-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {x} \left (-1+x^2\right )} \, dx,x,x-\sqrt {1+x^2}\right )+\frac {i \int \frac {\sqrt {x-\sqrt {1+x^2}}}{i \sqrt {2 \left (-1+\sqrt {2}\right )}-x} \, dx}{16 \sqrt {-1+\sqrt {2}}}+\frac {i \int \frac {\sqrt {x-\sqrt {1+x^2}}}{i \sqrt {2 \left (-1+\sqrt {2}\right )}+x} \, dx}{16 \sqrt {-1+\sqrt {2}}}-\frac {i \int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{i \sqrt {2 \left (-1+\sqrt {2}\right )}-x} \, dx}{8 \sqrt {-1+\sqrt {2}}}-\frac {i \int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{i \sqrt {2 \left (-1+\sqrt {2}\right )}+x} \, dx}{8 \sqrt {-1+\sqrt {2}}}-\frac {1}{16} \left (i \sqrt {-1+\sqrt {2}}\right ) \int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{i \sqrt {2 \left (-1+\sqrt {2}\right )}-x} \, dx-\frac {1}{16} \left (i \sqrt {-1+\sqrt {2}}\right ) \int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{i \sqrt {2 \left (-1+\sqrt {2}\right )}+x} \, dx+\frac {\int \frac {\sqrt {x-\sqrt {1+x^2}}}{\sqrt {2 \left (1+\sqrt {2}\right )}-x} \, dx}{16 \sqrt {1+\sqrt {2}}}+\frac {\int \frac {\sqrt {x-\sqrt {1+x^2}}}{\sqrt {2 \left (1+\sqrt {2}\right )}+x} \, dx}{16 \sqrt {1+\sqrt {2}}}-\frac {\int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{\sqrt {2 \left (1+\sqrt {2}\right )}-x} \, dx}{8 \sqrt {1+\sqrt {2}}}-\frac {\int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{\sqrt {2 \left (1+\sqrt {2}\right )}+x} \, dx}{8 \sqrt {1+\sqrt {2}}}+\frac {1}{16} \sqrt {1+\sqrt {2}} \int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{\sqrt {2 \left (1+\sqrt {2}\right )}-x} \, dx+\frac {1}{16} \sqrt {1+\sqrt {2}} \int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{\sqrt {2 \left (1+\sqrt {2}\right )}+x} \, dx-\frac {1}{4} \left (-10+7 \sqrt {2}\right ) \operatorname {Subst}\left (\int \frac {1}{-4+4 \sqrt {2}-\left (-6+4 \sqrt {2}\right ) x^2} \, dx,x,\frac {x}{\sqrt {1+x^2}}\right )+\frac {1}{4} \left (10+7 \sqrt {2}\right ) \operatorname {Subst}\left (\int \frac {1}{-4-4 \sqrt {2}-\left (-6-4 \sqrt {2}\right ) x^2} \, dx,x,\frac {x}{\sqrt {1+x^2}}\right )\\ &=\frac {\sqrt {1+x^2}}{2 x}+\frac {1}{2 x \sqrt {x-\sqrt {1+x^2}}}-\sqrt {x-\sqrt {1+x^2}}-\frac {\tan ^{-1}\left (\frac {x}{\sqrt {2 \left (-1+\sqrt {2}\right )}}\right )}{8 \sqrt {-1+\sqrt {2}}}+\frac {1}{4} \sqrt {-1+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {\frac {1}{2} \left (-1+\sqrt {2}\right )} x}{\sqrt {1+x^2}}\right )+\frac {1}{8} \sqrt {-7+5 \sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {\frac {1}{2} \left (-1+\sqrt {2}\right )} x}{\sqrt {1+x^2}}\right )-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {2 \left (1+\sqrt {2}\right )}}\right )}{8 \sqrt {1+\sqrt {2}}}+\frac {1}{4} \sqrt {1+\sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} x}{\sqrt {1+x^2}}\right )-\frac {1}{8} \sqrt {7+5 \sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} x}{\sqrt {1+x^2}}\right )+\frac {i \operatorname {Subst}\left (\int \frac {1+x^2}{\sqrt {x} \left (1+2 i \sqrt {2 \left (-1+\sqrt {2}\right )} x-x^2\right )} \, dx,x,x-\sqrt {1+x^2}\right )}{16 \sqrt {-1+\sqrt {2}}}+\frac {i \operatorname {Subst}\left (\int \frac {1+x^2}{\sqrt {x} \left (-1+2 i \sqrt {2 \left (-1+\sqrt {2}\right )} x+x^2\right )} \, dx,x,x-\sqrt {1+x^2}\right )}{16 \sqrt {-1+\sqrt {2}}}-\frac {i \int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{i \sqrt {2 \left (-1+\sqrt {2}\right )}-x} \, dx}{8 \sqrt {-1+\sqrt {2}}}-\frac {i \int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{i \sqrt {2 \left (-1+\sqrt {2}\right )}+x} \, dx}{8 \sqrt {-1+\sqrt {2}}}-\frac {1}{16} \left (i \sqrt {-1+\sqrt {2}}\right ) \int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{i \sqrt {2 \left (-1+\sqrt {2}\right )}-x} \, dx-\frac {1}{16} \left (i \sqrt {-1+\sqrt {2}}\right ) \int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{i \sqrt {2 \left (-1+\sqrt {2}\right )}+x} \, dx+\frac {\operatorname {Subst}\left (\int \frac {1+x^2}{\sqrt {x} \left (1+2 \sqrt {2 \left (1+\sqrt {2}\right )} x-x^2\right )} \, dx,x,x-\sqrt {1+x^2}\right )}{16 \sqrt {1+\sqrt {2}}}+\frac {\operatorname {Subst}\left (\int \frac {1+x^2}{\sqrt {x} \left (-1+2 \sqrt {2 \left (1+\sqrt {2}\right )} x+x^2\right )} \, dx,x,x-\sqrt {1+x^2}\right )}{16 \sqrt {1+\sqrt {2}}}-\frac {\int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{\sqrt {2 \left (1+\sqrt {2}\right )}-x} \, dx}{8 \sqrt {1+\sqrt {2}}}-\frac {\int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{\sqrt {2 \left (1+\sqrt {2}\right )}+x} \, dx}{8 \sqrt {1+\sqrt {2}}}+\frac {1}{16} \sqrt {1+\sqrt {2}} \int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{\sqrt {2 \left (1+\sqrt {2}\right )}-x} \, dx+\frac {1}{16} \sqrt {1+\sqrt {2}} \int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{\sqrt {2 \left (1+\sqrt {2}\right )}+x} \, dx-\operatorname {Subst}\left (\int \frac {1}{-1+x^4} \, dx,x,\sqrt {x-\sqrt {1+x^2}}\right )\\ &=\frac {\sqrt {1+x^2}}{2 x}+\frac {1}{2 x \sqrt {x-\sqrt {1+x^2}}}-\sqrt {x-\sqrt {1+x^2}}-\frac {\tan ^{-1}\left (\frac {x}{\sqrt {2 \left (-1+\sqrt {2}\right )}}\right )}{8 \sqrt {-1+\sqrt {2}}}+\frac {1}{4} \sqrt {-1+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {\frac {1}{2} \left (-1+\sqrt {2}\right )} x}{\sqrt {1+x^2}}\right )+\frac {1}{8} \sqrt {-7+5 \sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {\frac {1}{2} \left (-1+\sqrt {2}\right )} x}{\sqrt {1+x^2}}\right )-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {2 \left (1+\sqrt {2}\right )}}\right )}{8 \sqrt {1+\sqrt {2}}}+\frac {1}{4} \sqrt {1+\sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} x}{\sqrt {1+x^2}}\right )-\frac {1}{8} \sqrt {7+5 \sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} x}{\sqrt {1+x^2}}\right )+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {x-\sqrt {1+x^2}}\right )+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {x-\sqrt {1+x^2}}\right )+\frac {i \operatorname {Subst}\left (\int \left (-\frac {1}{\sqrt {x}}+\frac {2 \left (1+i \sqrt {2 \left (-1+\sqrt {2}\right )} x\right )}{\sqrt {x} \left (1+2 i \sqrt {2 \left (-1+\sqrt {2}\right )} x-x^2\right )}\right ) \, dx,x,x-\sqrt {1+x^2}\right )}{16 \sqrt {-1+\sqrt {2}}}+\frac {i \operatorname {Subst}\left (\int \left (\frac {1}{\sqrt {x}}+\frac {2 \left (1-i \sqrt {2 \left (-1+\sqrt {2}\right )} x\right )}{\sqrt {x} \left (-1+2 i \sqrt {2 \left (-1+\sqrt {2}\right )} x+x^2\right )}\right ) \, dx,x,x-\sqrt {1+x^2}\right )}{16 \sqrt {-1+\sqrt {2}}}-\frac {i \int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{i \sqrt {2 \left (-1+\sqrt {2}\right )}-x} \, dx}{8 \sqrt {-1+\sqrt {2}}}-\frac {i \int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{i \sqrt {2 \left (-1+\sqrt {2}\right )}+x} \, dx}{8 \sqrt {-1+\sqrt {2}}}-\frac {1}{16} \left (i \sqrt {-1+\sqrt {2}}\right ) \int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{i \sqrt {2 \left (-1+\sqrt {2}\right )}-x} \, dx-\frac {1}{16} \left (i \sqrt {-1+\sqrt {2}}\right ) \int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{i \sqrt {2 \left (-1+\sqrt {2}\right )}+x} \, dx+\frac {\operatorname {Subst}\left (\int \left (-\frac {1}{\sqrt {x}}+\frac {2 \left (1+\sqrt {2 \left (1+\sqrt {2}\right )} x\right )}{\sqrt {x} \left (1+2 \sqrt {2 \left (1+\sqrt {2}\right )} x-x^2\right )}\right ) \, dx,x,x-\sqrt {1+x^2}\right )}{16 \sqrt {1+\sqrt {2}}}+\frac {\operatorname {Subst}\left (\int \left (\frac {1}{\sqrt {x}}+\frac {2 \left (1-\sqrt {2 \left (1+\sqrt {2}\right )} x\right )}{\sqrt {x} \left (-1+2 \sqrt {2 \left (1+\sqrt {2}\right )} x+x^2\right )}\right ) \, dx,x,x-\sqrt {1+x^2}\right )}{16 \sqrt {1+\sqrt {2}}}-\frac {\int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{\sqrt {2 \left (1+\sqrt {2}\right )}-x} \, dx}{8 \sqrt {1+\sqrt {2}}}-\frac {\int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{\sqrt {2 \left (1+\sqrt {2}\right )}+x} \, dx}{8 \sqrt {1+\sqrt {2}}}+\frac {1}{16} \sqrt {1+\sqrt {2}} \int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{\sqrt {2 \left (1+\sqrt {2}\right )}-x} \, dx+\frac {1}{16} \sqrt {1+\sqrt {2}} \int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{\sqrt {2 \left (1+\sqrt {2}\right )}+x} \, dx\\ &=\frac {\sqrt {1+x^2}}{2 x}+\frac {1}{2 x \sqrt {x-\sqrt {1+x^2}}}-\sqrt {x-\sqrt {1+x^2}}-\frac {\tan ^{-1}\left (\frac {x}{\sqrt {2 \left (-1+\sqrt {2}\right )}}\right )}{8 \sqrt {-1+\sqrt {2}}}+\frac {1}{4} \sqrt {-1+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {\frac {1}{2} \left (-1+\sqrt {2}\right )} x}{\sqrt {1+x^2}}\right )+\frac {1}{8} \sqrt {-7+5 \sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {\frac {1}{2} \left (-1+\sqrt {2}\right )} x}{\sqrt {1+x^2}}\right )+\frac {1}{2} \tan ^{-1}\left (\sqrt {x-\sqrt {1+x^2}}\right )-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {2 \left (1+\sqrt {2}\right )}}\right )}{8 \sqrt {1+\sqrt {2}}}+\frac {1}{4} \sqrt {1+\sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} x}{\sqrt {1+x^2}}\right )-\frac {1}{8} \sqrt {7+5 \sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} x}{\sqrt {1+x^2}}\right )+\frac {1}{2} \tanh ^{-1}\left (\sqrt {x-\sqrt {1+x^2}}\right )-\frac {i \int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{i \sqrt {2 \left (-1+\sqrt {2}\right )}-x} \, dx}{8 \sqrt {-1+\sqrt {2}}}-\frac {i \int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{i \sqrt {2 \left (-1+\sqrt {2}\right )}+x} \, dx}{8 \sqrt {-1+\sqrt {2}}}+\frac {i \operatorname {Subst}\left (\int \frac {1+i \sqrt {2 \left (-1+\sqrt {2}\right )} x}{\sqrt {x} \left (1+2 i \sqrt {2 \left (-1+\sqrt {2}\right )} x-x^2\right )} \, dx,x,x-\sqrt {1+x^2}\right )}{8 \sqrt {-1+\sqrt {2}}}+\frac {i \operatorname {Subst}\left (\int \frac {1-i \sqrt {2 \left (-1+\sqrt {2}\right )} x}{\sqrt {x} \left (-1+2 i \sqrt {2 \left (-1+\sqrt {2}\right )} x+x^2\right )} \, dx,x,x-\sqrt {1+x^2}\right )}{8 \sqrt {-1+\sqrt {2}}}-\frac {1}{16} \left (i \sqrt {-1+\sqrt {2}}\right ) \int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{i \sqrt {2 \left (-1+\sqrt {2}\right )}-x} \, dx-\frac {1}{16} \left (i \sqrt {-1+\sqrt {2}}\right ) \int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{i \sqrt {2 \left (-1+\sqrt {2}\right )}+x} \, dx-\frac {\int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{\sqrt {2 \left (1+\sqrt {2}\right )}-x} \, dx}{8 \sqrt {1+\sqrt {2}}}-\frac {\int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{\sqrt {2 \left (1+\sqrt {2}\right )}+x} \, dx}{8 \sqrt {1+\sqrt {2}}}+\frac {\operatorname {Subst}\left (\int \frac {1+\sqrt {2 \left (1+\sqrt {2}\right )} x}{\sqrt {x} \left (1+2 \sqrt {2 \left (1+\sqrt {2}\right )} x-x^2\right )} \, dx,x,x-\sqrt {1+x^2}\right )}{8 \sqrt {1+\sqrt {2}}}+\frac {\operatorname {Subst}\left (\int \frac {1-\sqrt {2 \left (1+\sqrt {2}\right )} x}{\sqrt {x} \left (-1+2 \sqrt {2 \left (1+\sqrt {2}\right )} x+x^2\right )} \, dx,x,x-\sqrt {1+x^2}\right )}{8 \sqrt {1+\sqrt {2}}}+\frac {1}{16} \sqrt {1+\sqrt {2}} \int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{\sqrt {2 \left (1+\sqrt {2}\right )}-x} \, dx+\frac {1}{16} \sqrt {1+\sqrt {2}} \int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{\sqrt {2 \left (1+\sqrt {2}\right )}+x} \, dx\\ &=\frac {\sqrt {1+x^2}}{2 x}+\frac {1}{2 x \sqrt {x-\sqrt {1+x^2}}}-\sqrt {x-\sqrt {1+x^2}}-\frac {\tan ^{-1}\left (\frac {x}{\sqrt {2 \left (-1+\sqrt {2}\right )}}\right )}{8 \sqrt {-1+\sqrt {2}}}+\frac {1}{4} \sqrt {-1+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {\frac {1}{2} \left (-1+\sqrt {2}\right )} x}{\sqrt {1+x^2}}\right )+\frac {1}{8} \sqrt {-7+5 \sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {\frac {1}{2} \left (-1+\sqrt {2}\right )} x}{\sqrt {1+x^2}}\right )+\frac {1}{2} \tan ^{-1}\left (\sqrt {x-\sqrt {1+x^2}}\right )-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {2 \left (1+\sqrt {2}\right )}}\right )}{8 \sqrt {1+\sqrt {2}}}+\frac {1}{4} \sqrt {1+\sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} x}{\sqrt {1+x^2}}\right )-\frac {1}{8} \sqrt {7+5 \sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} x}{\sqrt {1+x^2}}\right )+\frac {1}{2} \tanh ^{-1}\left (\sqrt {x-\sqrt {1+x^2}}\right )-\frac {i \int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{i \sqrt {2 \left (-1+\sqrt {2}\right )}-x} \, dx}{8 \sqrt {-1+\sqrt {2}}}-\frac {i \int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{i \sqrt {2 \left (-1+\sqrt {2}\right )}+x} \, dx}{8 \sqrt {-1+\sqrt {2}}}+\frac {i \operatorname {Subst}\left (\int \frac {1+i \sqrt {2 \left (-1+\sqrt {2}\right )} x^2}{1+2 i \sqrt {2 \left (-1+\sqrt {2}\right )} x^2-x^4} \, dx,x,\sqrt {x-\sqrt {1+x^2}}\right )}{4 \sqrt {-1+\sqrt {2}}}+\frac {i \operatorname {Subst}\left (\int \frac {1-i \sqrt {2 \left (-1+\sqrt {2}\right )} x^2}{-1+2 i \sqrt {2 \left (-1+\sqrt {2}\right )} x^2+x^4} \, dx,x,\sqrt {x-\sqrt {1+x^2}}\right )}{4 \sqrt {-1+\sqrt {2}}}-\frac {1}{16} \left (i \sqrt {-1+\sqrt {2}}\right ) \int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{i \sqrt {2 \left (-1+\sqrt {2}\right )}-x} \, dx-\frac {1}{16} \left (i \sqrt {-1+\sqrt {2}}\right ) \int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{i \sqrt {2 \left (-1+\sqrt {2}\right )}+x} \, dx-\frac {\int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{\sqrt {2 \left (1+\sqrt {2}\right )}-x} \, dx}{8 \sqrt {1+\sqrt {2}}}-\frac {\int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{\sqrt {2 \left (1+\sqrt {2}\right )}+x} \, dx}{8 \sqrt {1+\sqrt {2}}}+\frac {\operatorname {Subst}\left (\int \frac {1+\sqrt {2 \left (1+\sqrt {2}\right )} x^2}{1+2 \sqrt {2 \left (1+\sqrt {2}\right )} x^2-x^4} \, dx,x,\sqrt {x-\sqrt {1+x^2}}\right )}{4 \sqrt {1+\sqrt {2}}}+\frac {\operatorname {Subst}\left (\int \frac {1-\sqrt {2 \left (1+\sqrt {2}\right )} x^2}{-1+2 \sqrt {2 \left (1+\sqrt {2}\right )} x^2+x^4} \, dx,x,\sqrt {x-\sqrt {1+x^2}}\right )}{4 \sqrt {1+\sqrt {2}}}+\frac {1}{16} \sqrt {1+\sqrt {2}} \int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{\sqrt {2 \left (1+\sqrt {2}\right )}-x} \, dx+\frac {1}{16} \sqrt {1+\sqrt {2}} \int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{\sqrt {2 \left (1+\sqrt {2}\right )}+x} \, dx\\ &=\frac {\sqrt {1+x^2}}{2 x}+\frac {1}{2 x \sqrt {x-\sqrt {1+x^2}}}-\sqrt {x-\sqrt {1+x^2}}-\frac {\tan ^{-1}\left (\frac {x}{\sqrt {2 \left (-1+\sqrt {2}\right )}}\right )}{8 \sqrt {-1+\sqrt {2}}}+\frac {1}{4} \sqrt {-1+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {\frac {1}{2} \left (-1+\sqrt {2}\right )} x}{\sqrt {1+x^2}}\right )+\frac {1}{8} \sqrt {-7+5 \sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {\frac {1}{2} \left (-1+\sqrt {2}\right )} x}{\sqrt {1+x^2}}\right )+\frac {1}{2} \tan ^{-1}\left (\sqrt {x-\sqrt {1+x^2}}\right )-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {2 \left (1+\sqrt {2}\right )}}\right )}{8 \sqrt {1+\sqrt {2}}}+\frac {1}{4} \sqrt {1+\sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} x}{\sqrt {1+x^2}}\right )-\frac {1}{8} \sqrt {7+5 \sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} x}{\sqrt {1+x^2}}\right )+\frac {1}{2} \tanh ^{-1}\left (\sqrt {x-\sqrt {1+x^2}}\right )-\frac {i \int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{i \sqrt {2 \left (-1+\sqrt {2}\right )}-x} \, dx}{8 \sqrt {-1+\sqrt {2}}}-\frac {i \int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{i \sqrt {2 \left (-1+\sqrt {2}\right )}+x} \, dx}{8 \sqrt {-1+\sqrt {2}}}-\frac {1}{16} \left (i \sqrt {-1+\sqrt {2}}\right ) \int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{i \sqrt {2 \left (-1+\sqrt {2}\right )}-x} \, dx-\frac {1}{16} \left (i \sqrt {-1+\sqrt {2}}\right ) \int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{i \sqrt {2 \left (-1+\sqrt {2}\right )}+x} \, dx-\frac {\int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{\sqrt {2 \left (1+\sqrt {2}\right )}-x} \, dx}{8 \sqrt {1+\sqrt {2}}}-\frac {\int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{\sqrt {2 \left (1+\sqrt {2}\right )}+x} \, dx}{8 \sqrt {1+\sqrt {2}}}+\frac {1}{16} \sqrt {1+\sqrt {2}} \int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{\sqrt {2 \left (1+\sqrt {2}\right )}-x} \, dx+\frac {1}{16} \sqrt {1+\sqrt {2}} \int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{\sqrt {2 \left (1+\sqrt {2}\right )}+x} \, dx+\frac {\left (3 i-2 i \sqrt {2}-\sqrt {2 \left (-7+5 \sqrt {2}\right )}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {3-2 \sqrt {2}}+i \sqrt {2 \left (-1+\sqrt {2}\right )}-x^2} \, dx,x,\sqrt {x-\sqrt {1+x^2}}\right )}{8 \sqrt {-7+5 \sqrt {2}}}-\frac {\left (3 i-2 i \sqrt {2}-\sqrt {2 \left (-7+5 \sqrt {2}\right )}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {3-2 \sqrt {2}}+i \sqrt {2 \left (-1+\sqrt {2}\right )}+x^2} \, dx,x,\sqrt {x-\sqrt {1+x^2}}\right )}{8 \sqrt {-7+5 \sqrt {2}}}-\frac {\left (3 i-2 i \sqrt {2}+\sqrt {2 \left (-7+5 \sqrt {2}\right )}\right ) \operatorname {Subst}\left (\int \frac {1}{-\sqrt {3-2 \sqrt {2}}+i \sqrt {2 \left (-1+\sqrt {2}\right )}-x^2} \, dx,x,\sqrt {x-\sqrt {1+x^2}}\right )}{8 \sqrt {-7+5 \sqrt {2}}}+\frac {\left (3 i-2 i \sqrt {2}+\sqrt {2 \left (-7+5 \sqrt {2}\right )}\right ) \operatorname {Subst}\left (\int \frac {1}{-\sqrt {3-2 \sqrt {2}}+i \sqrt {2 \left (-1+\sqrt {2}\right )}+x^2} \, dx,x,\sqrt {x-\sqrt {1+x^2}}\right )}{8 \sqrt {-7+5 \sqrt {2}}}-\frac {\left (3+2 \sqrt {2}-\sqrt {2 \left (7+5 \sqrt {2}\right )}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {2 \left (1+\sqrt {2}\right )}-\sqrt {3+2 \sqrt {2}}-x^2} \, dx,x,\sqrt {x-\sqrt {1+x^2}}\right )}{8 \sqrt {7+5 \sqrt {2}}}+\frac {\left (3+2 \sqrt {2}-\sqrt {2 \left (7+5 \sqrt {2}\right )}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {2 \left (1+\sqrt {2}\right )}-\sqrt {3+2 \sqrt {2}}+x^2} \, dx,x,\sqrt {x-\sqrt {1+x^2}}\right )}{8 \sqrt {7+5 \sqrt {2}}}+\frac {\left (3+2 \sqrt {2}+\sqrt {2 \left (7+5 \sqrt {2}\right )}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {2 \left (1+\sqrt {2}\right )}+\sqrt {3+2 \sqrt {2}}-x^2} \, dx,x,\sqrt {x-\sqrt {1+x^2}}\right )}{8 \sqrt {7+5 \sqrt {2}}}-\frac {\left (3+2 \sqrt {2}+\sqrt {2 \left (7+5 \sqrt {2}\right )}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {2 \left (1+\sqrt {2}\right )}+\sqrt {3+2 \sqrt {2}}+x^2} \, dx,x,\sqrt {x-\sqrt {1+x^2}}\right )}{8 \sqrt {7+5 \sqrt {2}}}\\ &=\frac {\sqrt {1+x^2}}{2 x}+\frac {1}{2 x \sqrt {x-\sqrt {1+x^2}}}-\sqrt {x-\sqrt {1+x^2}}-\frac {\tan ^{-1}\left (\frac {x}{\sqrt {2 \left (-1+\sqrt {2}\right )}}\right )}{8 \sqrt {-1+\sqrt {2}}}+\frac {1}{4} \sqrt {-1+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {\frac {1}{2} \left (-1+\sqrt {2}\right )} x}{\sqrt {1+x^2}}\right )+\frac {1}{8} \sqrt {-7+5 \sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {\frac {1}{2} \left (-1+\sqrt {2}\right )} x}{\sqrt {1+x^2}}\right )+\frac {1}{2} \tan ^{-1}\left (\sqrt {x-\sqrt {1+x^2}}\right )+\frac {\left (3 i-2 i \sqrt {2}+\sqrt {2 \left (-7+5 \sqrt {2}\right )}\right ) \tan ^{-1}\left (\frac {\sqrt {x-\sqrt {1+x^2}}}{\sqrt {\sqrt {3-2 \sqrt {2}}-i \sqrt {2 \left (-1+\sqrt {2}\right )}}}\right )}{8 \sqrt {-\left (\left (7-5 \sqrt {2}\right ) \left (\sqrt {3-2 \sqrt {2}}-i \sqrt {2 \left (-1+\sqrt {2}\right )}\right )\right )}}-\frac {\left (3 i-2 i \sqrt {2}-\sqrt {2 \left (-7+5 \sqrt {2}\right )}\right ) \tan ^{-1}\left (\frac {\sqrt {x-\sqrt {1+x^2}}}{\sqrt {\sqrt {3-2 \sqrt {2}}+i \sqrt {2 \left (-1+\sqrt {2}\right )}}}\right )}{8 \sqrt {-\left (\left (7-5 \sqrt {2}\right ) \left (\sqrt {3-2 \sqrt {2}}+i \sqrt {2 \left (-1+\sqrt {2}\right )}\right )\right )}}+\frac {\left (3+2 \sqrt {2}-\sqrt {2 \left (7+5 \sqrt {2}\right )}\right ) \tan ^{-1}\left (\frac {\sqrt {x-\sqrt {1+x^2}}}{\sqrt {-\sqrt {2 \left (1+\sqrt {2}\right )}+\sqrt {3+2 \sqrt {2}}}}\right )}{8 \sqrt {\left (7+5 \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 ) \tan ^{-1}\left (\frac {\sqrt {x-\sqrt {1+x^2}}}{\sqrt {\sqrt {2 \left (1+\sqrt {2}\right )}+\sqrt {3+2 \sqrt {2}}}}\right )}{8 \sqrt {\left (7+5 \sqrt {2}\right ) \left (\sqrt {2 \left (1+\sqrt {2}\right )}+\sqrt {3+2 \sqrt {2}}\right )}}-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {2 \left (1+\sqrt {2}\right )}}\right )}{8 \sqrt {1+\sqrt {2}}}+\frac {1}{4} \sqrt {1+\sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} x}{\sqrt {1+x^2}}\right )-\frac {1}{8} \sqrt {7+5 \sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} x}{\sqrt {1+x^2}}\right )+\frac {1}{2} \tanh ^{-1}\left (\sqrt {x-\sqrt {1+x^2}}\right )-\frac {\left (3 i-2 i \sqrt {2}+\sqrt {2 \left (-7+5 \sqrt {2}\right )}\right ) \tanh ^{-1}\left (\frac {\sqrt {x-\sqrt {1+x^2}}}{\sqrt {\sqrt {3-2 \sqrt {2}}-i \sqrt {2 \left (-1+\sqrt {2}\right )}}}\right )}{8 \sqrt {-\left (\left (7-5 \sqrt {2}\right ) \left (\sqrt {3-2 \sqrt {2}}-i \sqrt {2 \left (-1+\sqrt {2}\right )}\right )\right )}}+\frac {\left (3 i-2 i \sqrt {2}-\sqrt {2 \left (-7+5 \sqrt {2}\right )}\right ) \tanh ^{-1}\left (\frac {\sqrt {x-\sqrt {1+x^2}}}{\sqrt {\sqrt {3-2 \sqrt {2}}+i \sqrt {2 \left (-1+\sqrt {2}\right )}}}\right )}{8 \sqrt {-\left (\left (7-5 \sqrt {2}\right ) \left (\sqrt {3-2 \sqrt {2}}+i \sqrt {2 \left (-1+\sqrt {2}\right )}\right )\right )}}-\frac {\left (3+2 \sqrt {2}-\sqrt {2 \left (7+5 \sqrt {2}\right )}\right ) \tanh ^{-1}\left (\frac {\sqrt {x-\sqrt {1+x^2}}}{\sqrt {-\sqrt {2 \left (1+\sqrt {2}\right )}+\sqrt {3+2 \sqrt {2}}}}\right )}{8 \sqrt {\left (7+5 \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 ) \tanh ^{-1}\left (\frac {\sqrt {x-\sqrt {1+x^2}}}{\sqrt {\sqrt {2 \left (1+\sqrt {2}\right )}+\sqrt {3+2 \sqrt {2}}}}\right )}{8 \sqrt {\left (7+5 \sqrt {2}\right ) \left (\sqrt {2 \left (1+\sqrt {2}\right )}+\sqrt {3+2 \sqrt {2}}\right )}}-\frac {i \int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{i \sqrt {2 \left (-1+\sqrt {2}\right )}-x} \, dx}{8 \sqrt {-1+\sqrt {2}}}-\frac {i \int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{i \sqrt {2 \left (-1+\sqrt {2}\right )}+x} \, dx}{8 \sqrt {-1+\sqrt {2}}}-\frac {1}{16} \left (i \sqrt {-1+\sqrt {2}}\right ) \int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{i \sqrt {2 \left (-1+\sqrt {2}\right )}-x} \, dx-\frac {1}{16} \left (i \sqrt {-1+\sqrt {2}}\right ) \int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{i \sqrt {2 \left (-1+\sqrt {2}\right )}+x} \, dx-\frac {\int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{\sqrt {2 \left (1+\sqrt {2}\right )}-x} \, dx}{8 \sqrt {1+\sqrt {2}}}-\frac {\int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{\sqrt {2 \left (1+\sqrt {2}\right )}+x} \, dx}{8 \sqrt {1+\sqrt {2}}}+\frac {1}{16} \sqrt {1+\sqrt {2}} \int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{\sqrt {2 \left (1+\sqrt {2}\right )}-x} \, dx+\frac {1}{16} \sqrt {1+\sqrt {2}} \int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{\sqrt {2 \left (1+\sqrt {2}\right )}+x} \, dx\\ \end {align*}

________________________________________________________________________________________

Mathematica [B]  time = 4.16, size = 777, normalized size = 3.05 \begin {gather*} \frac {\left (x^2-2 \sqrt {x^2+1}\right ) x^2 \left (-2 \text {RootSum}\left [\text {$\#$1}^8-4 \text {$\#$1}^6-2 \text {$\#$1}^4-4 \text {$\#$1}^2+1\&,\frac {\text {$\#$1}^3 \log \left (\sqrt {x-\sqrt {x^2+1}}-\text {$\#$1}\right )}{\text {$\#$1}^6-3 \text {$\#$1}^4-\text {$\#$1}^2-1}\&\right ]+2 \text {RootSum}\left [\text {$\#$1}^8+4 \text {$\#$1}^6-2 \text {$\#$1}^4+4 \text {$\#$1}^2+1\&,\frac {\text {$\#$1}^3 \log \left (\sqrt {x-\sqrt {x^2+1}}-\text {$\#$1}\right )}{\text {$\#$1}^6+3 \text {$\#$1}^4-\text {$\#$1}^2+1}\&\right ]+\frac {2 \sqrt {x^2+1} \left (\sqrt {x^2+1}-x\right ) \left (\left (\sqrt {x^2+1}-x\right ) x \text {RootSum}\left [\text {$\#$1}^8-4 \text {$\#$1}^6-2 \text {$\#$1}^4-4 \text {$\#$1}^2+1\&,\frac {\text {$\#$1}^3 \log \left (\sqrt {x-\sqrt {x^2+1}}-\text {$\#$1}\right )}{\text {$\#$1}^6-3 \text {$\#$1}^4-\text {$\#$1}^2-1}\&\right ]+\left (\sqrt {x^2+1}-x\right ) x \text {RootSum}\left [\text {$\#$1}^8+4 \text {$\#$1}^6-2 \text {$\#$1}^4+4 \text {$\#$1}^2+1\&,\frac {\text {$\#$1}^3 \log \left (\sqrt {x-\sqrt {x^2+1}}-\text {$\#$1}\right )}{\text {$\#$1}^6+3 \text {$\#$1}^4-\text {$\#$1}^2+1}\&\right ]-2 \sqrt {x-\sqrt {x^2+1}}+x^2 \log \left (1-\sqrt {x-\sqrt {x^2+1}}\right )-x^2 \log \left (\sqrt {x-\sqrt {x^2+1}}+1\right )-\sqrt {x^2+1} x \log \left (1-\sqrt {x-\sqrt {x^2+1}}\right )+\sqrt {x^2+1} x \log \left (\sqrt {x-\sqrt {x^2+1}}+1\right )-2 x^2 \tan ^{-1}\left (\sqrt {x-\sqrt {x^2+1}}\right )+2 \sqrt {x^2+1} x \tan ^{-1}\left (\sqrt {x-\sqrt {x^2+1}}\right )\right )}{x \left (-2 x^3+2 \sqrt {x^2+1} x^2+\sqrt {x^2+1}-2 x\right )}+\sqrt {1+\sqrt {2}} \tan ^{-1}\left (\frac {x}{\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {x^2+1}}\right )-\sqrt {\sqrt {2}-1} \tanh ^{-1}\left (\frac {x}{\sqrt {2 \left (\sqrt {2}-1\right )} \sqrt {x^2+1}}\right )-\frac {\tan ^{-1}\left (\frac {x}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )}{\sqrt {\sqrt {2}-1}}-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {2 \left (1+\sqrt {2}\right )}}\right )}{\sqrt {1+\sqrt {2}}}\right )}{8 \left (x^4-2 x^2 \sqrt {x^2+1}\right )}+\frac {\sqrt {x^2+1}}{2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.34, size = 255, normalized size = 1.00 \begin {gather*} -\frac {1}{\left (-1-x+\sqrt {1+x^2}\right ) \left (1+\sqrt {x-\sqrt {1+x^2}}\right )}+\frac {1}{2} \tan ^{-1}\left (\sqrt {x-\sqrt {1+x^2}}\right )+\frac {1}{2} \tanh ^{-1}\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 ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {\sqrt {x - \sqrt {x^{2} + 1}} - 1}{x^{4} - 2 \, \sqrt {x^{2} + 1} x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

maple [F]  time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {1-\sqrt {x -\sqrt {x^{2}+1}}}{x^{4}-2 x^{2} \sqrt {x^{2}+1}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\frac {1}{6 \, x^{3}} - \int \frac {{\left (x^{2} - 2 \, \sqrt {x^{2} + 1}\right )} \sqrt {x - \sqrt {x^{2} + 1}}}{x^{6} - 4 \, \sqrt {x^{2} + 1} x^{4} + 4 \, x^{4} + 4 \, x^{2}}\,{d x} - \int -\frac {x^{4} - 4 \, x^{2} - 4}{2 \, {\left (x^{8} - 4 \, \sqrt {x^{2} + 1} x^{6} + 4 \, x^{6} + 4 \, x^{4}\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\sqrt {x-\sqrt {x^2+1}}-1}{2\,x^2\,\sqrt {x^2+1}-x^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________