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

Optimal. Leaf size=196 \[ -\frac{\log \left (x^2-\sqrt{1+2 \sqrt{2}} x+\sqrt{2}\right )}{4 \sqrt{2 \left (1+2 \sqrt{2}\right )}}+\frac{\log \left (x^2+\sqrt{1+2 \sqrt{2}} x+\sqrt{2}\right )}{4 \sqrt{2 \left (1+2 \sqrt{2}\right )}}-\frac{1}{2} \sqrt{\frac{1}{14} \left (1+2 \sqrt{2}\right )} \tan ^{-1}\left (\frac{\sqrt{1+2 \sqrt{2}}-2 x}{\sqrt{2 \sqrt{2}-1}}\right )+\frac{1}{2} \sqrt{\frac{1}{14} \left (1+2 \sqrt{2}\right )} \tan ^{-1}\left (\frac{2 x+\sqrt{1+2 \sqrt{2}}}{\sqrt{2 \sqrt{2}-1}}\right ) \]

[Out]

-(Sqrt[(1 + 2*Sqrt[2])/14]*ArcTan[(Sqrt[1 + 2*Sqrt[2]] - 2*x)/Sqrt[-1 + 2*Sqrt[2
]]])/2 + (Sqrt[(1 + 2*Sqrt[2])/14]*ArcTan[(Sqrt[1 + 2*Sqrt[2]] + 2*x)/Sqrt[-1 +
2*Sqrt[2]]])/2 - Log[Sqrt[2] - Sqrt[1 + 2*Sqrt[2]]*x + x^2]/(4*Sqrt[2*(1 + 2*Sqr
t[2])]) + Log[Sqrt[2] + Sqrt[1 + 2*Sqrt[2]]*x + x^2]/(4*Sqrt[2*(1 + 2*Sqrt[2])])

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Rubi [A]  time = 0.286099, antiderivative size = 196, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417 \[ -\frac{\log \left (x^2-\sqrt{1+2 \sqrt{2}} x+\sqrt{2}\right )}{4 \sqrt{2 \left (1+2 \sqrt{2}\right )}}+\frac{\log \left (x^2+\sqrt{1+2 \sqrt{2}} x+\sqrt{2}\right )}{4 \sqrt{2 \left (1+2 \sqrt{2}\right )}}-\frac{1}{2} \sqrt{\frac{1}{14} \left (1+2 \sqrt{2}\right )} \tan ^{-1}\left (\frac{\sqrt{1+2 \sqrt{2}}-2 x}{\sqrt{2 \sqrt{2}-1}}\right )+\frac{1}{2} \sqrt{\frac{1}{14} \left (1+2 \sqrt{2}\right )} \tan ^{-1}\left (\frac{2 x+\sqrt{1+2 \sqrt{2}}}{\sqrt{2 \sqrt{2}-1}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(2 - x^2 + x^4)^(-1),x]

[Out]

-(Sqrt[(1 + 2*Sqrt[2])/14]*ArcTan[(Sqrt[1 + 2*Sqrt[2]] - 2*x)/Sqrt[-1 + 2*Sqrt[2
]]])/2 + (Sqrt[(1 + 2*Sqrt[2])/14]*ArcTan[(Sqrt[1 + 2*Sqrt[2]] + 2*x)/Sqrt[-1 +
2*Sqrt[2]]])/2 - Log[Sqrt[2] - Sqrt[1 + 2*Sqrt[2]]*x + x^2]/(4*Sqrt[2*(1 + 2*Sqr
t[2])]) + Log[Sqrt[2] + Sqrt[1 + 2*Sqrt[2]]*x + x^2]/(4*Sqrt[2*(1 + 2*Sqrt[2])])

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Rubi in Sympy [A]  time = 8.09686, size = 178, normalized size = 0.91 \[ - \frac{\sqrt{2} \log{\left (x^{2} - x \sqrt{1 + 2 \sqrt{2}} + \sqrt{2} \right )}}{8 \sqrt{1 + 2 \sqrt{2}}} + \frac{\sqrt{2} \log{\left (x^{2} + x \sqrt{1 + 2 \sqrt{2}} + \sqrt{2} \right )}}{8 \sqrt{1 + 2 \sqrt{2}}} + \frac{\sqrt{2} \operatorname{atan}{\left (\frac{2 x - \sqrt{1 + 2 \sqrt{2}}}{\sqrt{-1 + 2 \sqrt{2}}} \right )}}{4 \sqrt{-1 + 2 \sqrt{2}}} + \frac{\sqrt{2} \operatorname{atan}{\left (\frac{2 x + \sqrt{1 + 2 \sqrt{2}}}{\sqrt{-1 + 2 \sqrt{2}}} \right )}}{4 \sqrt{-1 + 2 \sqrt{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(x**4-x**2+2),x)

[Out]

-sqrt(2)*log(x**2 - x*sqrt(1 + 2*sqrt(2)) + sqrt(2))/(8*sqrt(1 + 2*sqrt(2))) + s
qrt(2)*log(x**2 + x*sqrt(1 + 2*sqrt(2)) + sqrt(2))/(8*sqrt(1 + 2*sqrt(2))) + sqr
t(2)*atan((2*x - sqrt(1 + 2*sqrt(2)))/sqrt(-1 + 2*sqrt(2)))/(4*sqrt(-1 + 2*sqrt(
2))) + sqrt(2)*atan((2*x + sqrt(1 + 2*sqrt(2)))/sqrt(-1 + 2*sqrt(2)))/(4*sqrt(-1
+ 2*sqrt(2)))

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Mathematica [C]  time = 0.119972, size = 91, normalized size = 0.46 \[ \frac{i \tan ^{-1}\left (\frac{x}{\sqrt{\frac{1}{2} \left (-1+i \sqrt{7}\right )}}\right )}{\sqrt{\frac{7}{2} \left (-1+i \sqrt{7}\right )}}-\frac{i \tan ^{-1}\left (\frac{x}{\sqrt{\frac{1}{2} \left (-1-i \sqrt{7}\right )}}\right )}{\sqrt{\frac{7}{2} \left (-1-i \sqrt{7}\right )}} \]

Antiderivative was successfully verified.

[In]  Integrate[(2 - x^2 + x^4)^(-1),x]

[Out]

((-I)*ArcTan[x/Sqrt[(-1 - I*Sqrt[7])/2]])/Sqrt[(7*(-1 - I*Sqrt[7]))/2] + (I*ArcT
an[x/Sqrt[(-1 + I*Sqrt[7])/2]])/Sqrt[(7*(-1 + I*Sqrt[7]))/2]

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Maple [B]  time = 0.039, size = 386, normalized size = 2. \[{\frac{\ln \left ({x}^{2}+\sqrt{2}-x\sqrt{1+2\,\sqrt{2}} \right ) \sqrt{1+2\,\sqrt{2}}\sqrt{2}}{56}}-{\frac{\ln \left ({x}^{2}+\sqrt{2}-x\sqrt{1+2\,\sqrt{2}} \right ) \sqrt{1+2\,\sqrt{2}}}{14}}+{\frac{ \left ( 1+2\,\sqrt{2} \right ) \sqrt{2}}{28\,\sqrt{-1+2\,\sqrt{2}}}\arctan \left ({\frac{2\,x-\sqrt{1+2\,\sqrt{2}}}{\sqrt{-1+2\,\sqrt{2}}}} \right ) }-{\frac{1+2\,\sqrt{2}}{7\,\sqrt{-1+2\,\sqrt{2}}}\arctan \left ({\frac{2\,x-\sqrt{1+2\,\sqrt{2}}}{\sqrt{-1+2\,\sqrt{2}}}} \right ) }+{\frac{\sqrt{2}}{2\,\sqrt{-1+2\,\sqrt{2}}}\arctan \left ({\frac{2\,x-\sqrt{1+2\,\sqrt{2}}}{\sqrt{-1+2\,\sqrt{2}}}} \right ) }-{\frac{\ln \left ({x}^{2}+\sqrt{2}+x\sqrt{1+2\,\sqrt{2}} \right ) \sqrt{1+2\,\sqrt{2}}\sqrt{2}}{56}}+{\frac{\ln \left ({x}^{2}+\sqrt{2}+x\sqrt{1+2\,\sqrt{2}} \right ) \sqrt{1+2\,\sqrt{2}}}{14}}+{\frac{ \left ( 1+2\,\sqrt{2} \right ) \sqrt{2}}{28\,\sqrt{-1+2\,\sqrt{2}}}\arctan \left ({\frac{2\,x+\sqrt{1+2\,\sqrt{2}}}{\sqrt{-1+2\,\sqrt{2}}}} \right ) }-{\frac{1+2\,\sqrt{2}}{7\,\sqrt{-1+2\,\sqrt{2}}}\arctan \left ({\frac{2\,x+\sqrt{1+2\,\sqrt{2}}}{\sqrt{-1+2\,\sqrt{2}}}} \right ) }+{\frac{\sqrt{2}}{2\,\sqrt{-1+2\,\sqrt{2}}}\arctan \left ({\frac{2\,x+\sqrt{1+2\,\sqrt{2}}}{\sqrt{-1+2\,\sqrt{2}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x^4-x^2+2),x)

[Out]

1/56*ln(x^2+2^(1/2)-x*(1+2*2^(1/2))^(1/2))*(1+2*2^(1/2))^(1/2)*2^(1/2)-1/14*ln(x
^2+2^(1/2)-x*(1+2*2^(1/2))^(1/2))*(1+2*2^(1/2))^(1/2)+1/28/(-1+2*2^(1/2))^(1/2)*
arctan((2*x-(1+2*2^(1/2))^(1/2))/(-1+2*2^(1/2))^(1/2))*(1+2*2^(1/2))*2^(1/2)-1/7
/(-1+2*2^(1/2))^(1/2)*arctan((2*x-(1+2*2^(1/2))^(1/2))/(-1+2*2^(1/2))^(1/2))*(1+
2*2^(1/2))+1/2/(-1+2*2^(1/2))^(1/2)*arctan((2*x-(1+2*2^(1/2))^(1/2))/(-1+2*2^(1/
2))^(1/2))*2^(1/2)-1/56*ln(x^2+2^(1/2)+x*(1+2*2^(1/2))^(1/2))*(1+2*2^(1/2))^(1/2
)*2^(1/2)+1/14*ln(x^2+2^(1/2)+x*(1+2*2^(1/2))^(1/2))*(1+2*2^(1/2))^(1/2)+1/28/(-
1+2*2^(1/2))^(1/2)*arctan((2*x+(1+2*2^(1/2))^(1/2))/(-1+2*2^(1/2))^(1/2))*(1+2*2
^(1/2))*2^(1/2)-1/7/(-1+2*2^(1/2))^(1/2)*arctan((2*x+(1+2*2^(1/2))^(1/2))/(-1+2*
2^(1/2))^(1/2))*(1+2*2^(1/2))+1/2/(-1+2*2^(1/2))^(1/2)*arctan((2*x+(1+2*2^(1/2))
^(1/2))/(-1+2*2^(1/2))^(1/2))*2^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{4} - x^{2} + 2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/(x^4 - x^2 + 2), x)

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Fricas [A]  time = 0.21233, size = 501, normalized size = 2.56 \[ \frac{98^{\frac{3}{4}} \sqrt{7}{\left (\sqrt{7}{\left (\sqrt{2} - 4\right )} \log \left (14 \, \sqrt{2} x^{2} + 14 \cdot 98^{\frac{1}{4}} x \sqrt{\frac{\sqrt{2} - 4}{4 \, \sqrt{2} - 9}} + 28\right ) - \sqrt{7}{\left (\sqrt{2} - 4\right )} \log \left (14 \, \sqrt{2} x^{2} - 14 \cdot 98^{\frac{1}{4}} x \sqrt{\frac{\sqrt{2} - 4}{4 \, \sqrt{2} - 9}} + 28\right ) - 28 \, \sqrt{2} \arctan \left (\frac{7 \,{\left (2 \, \sqrt{2} - 1\right )}}{98^{\frac{1}{4}} \sqrt{7} \sqrt{\frac{1}{2}} \sqrt{\sqrt{2}{\left (\sqrt{2} x^{2} + 98^{\frac{1}{4}} x \sqrt{\frac{\sqrt{2} - 4}{4 \, \sqrt{2} - 9}} + 2\right )}}{\left (\sqrt{2} - 4\right )} \sqrt{\frac{\sqrt{2} - 4}{4 \, \sqrt{2} - 9}} + 98^{\frac{1}{4}} \sqrt{7}{\left (\sqrt{2} x - 4 \, x\right )} \sqrt{\frac{\sqrt{2} - 4}{4 \, \sqrt{2} - 9}} - 7 \, \sqrt{7}}\right ) - 28 \, \sqrt{2} \arctan \left (\frac{7 \,{\left (2 \, \sqrt{2} - 1\right )}}{98^{\frac{1}{4}} \sqrt{7} \sqrt{\frac{1}{2}} \sqrt{\sqrt{2}{\left (\sqrt{2} x^{2} - 98^{\frac{1}{4}} x \sqrt{\frac{\sqrt{2} - 4}{4 \, \sqrt{2} - 9}} + 2\right )}}{\left (\sqrt{2} - 4\right )} \sqrt{\frac{\sqrt{2} - 4}{4 \, \sqrt{2} - 9}} + 98^{\frac{1}{4}} \sqrt{7}{\left (\sqrt{2} x - 4 \, x\right )} \sqrt{\frac{\sqrt{2} - 4}{4 \, \sqrt{2} - 9}} + 7 \, \sqrt{7}}\right )\right )}}{2744 \,{\left (\sqrt{2} - 4\right )} \sqrt{\frac{\sqrt{2} - 4}{4 \, \sqrt{2} - 9}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2744*98^(3/4)*sqrt(7)*(sqrt(7)*(sqrt(2) - 4)*log(14*sqrt(2)*x^2 + 14*98^(1/4)*
x*sqrt((sqrt(2) - 4)/(4*sqrt(2) - 9)) + 28) - sqrt(7)*(sqrt(2) - 4)*log(14*sqrt(
2)*x^2 - 14*98^(1/4)*x*sqrt((sqrt(2) - 4)/(4*sqrt(2) - 9)) + 28) - 28*sqrt(2)*ar
ctan(7*(2*sqrt(2) - 1)/(98^(1/4)*sqrt(7)*sqrt(1/2)*sqrt(sqrt(2)*(sqrt(2)*x^2 + 9
8^(1/4)*x*sqrt((sqrt(2) - 4)/(4*sqrt(2) - 9)) + 2))*(sqrt(2) - 4)*sqrt((sqrt(2)
- 4)/(4*sqrt(2) - 9)) + 98^(1/4)*sqrt(7)*(sqrt(2)*x - 4*x)*sqrt((sqrt(2) - 4)/(4
*sqrt(2) - 9)) - 7*sqrt(7))) - 28*sqrt(2)*arctan(7*(2*sqrt(2) - 1)/(98^(1/4)*sqr
t(7)*sqrt(1/2)*sqrt(sqrt(2)*(sqrt(2)*x^2 - 98^(1/4)*x*sqrt((sqrt(2) - 4)/(4*sqrt
(2) - 9)) + 2))*(sqrt(2) - 4)*sqrt((sqrt(2) - 4)/(4*sqrt(2) - 9)) + 98^(1/4)*sqr
t(7)*(sqrt(2)*x - 4*x)*sqrt((sqrt(2) - 4)/(4*sqrt(2) - 9)) + 7*sqrt(7))))/((sqrt
(2) - 4)*sqrt((sqrt(2) - 4)/(4*sqrt(2) - 9)))

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Sympy [A]  time = 0.843832, size = 24, normalized size = 0.12 \[ \operatorname{RootSum}{\left (1568 t^{4} + 28 t^{2} + 1, \left ( t \mapsto t \log{\left (- 112 t^{3} + 6 t + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x**4-x**2+2),x)

[Out]

RootSum(1568*_t**4 + 28*_t**2 + 1, Lambda(_t, _t*log(-112*_t**3 + 6*_t + x)))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{4} - x^{2} + 2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/(x^4 - x^2 + 2), x)