3.31 \(\int \frac{-1+\sqrt{3}+2 x^4}{1-x^4+x^8} \, dx\)

Optimal. Leaf size=135 \[ -\frac{\log \left (x^2-\sqrt{2-\sqrt{3}} x+1\right )}{2 \sqrt{2}}+\frac{\log \left (x^2+\sqrt{2-\sqrt{3}} x+1\right )}{2 \sqrt{2}}-\frac{\tan ^{-1}\left (\frac{\sqrt{2+\sqrt{3}}-2 x}{\sqrt{2-\sqrt{3}}}\right )}{\sqrt{2}}+\frac{\tan ^{-1}\left (\frac{2 x+\sqrt{2+\sqrt{3}}}{\sqrt{2-\sqrt{3}}}\right )}{\sqrt{2}} \]

[Out]

-(ArcTan[(Sqrt[2 + Sqrt[3]] - 2*x)/Sqrt[2 - Sqrt[3]]]/Sqrt[2]) + ArcTan[(Sqrt[2
+ Sqrt[3]] + 2*x)/Sqrt[2 - Sqrt[3]]]/Sqrt[2] - Log[1 - Sqrt[2 - Sqrt[3]]*x + x^2
]/(2*Sqrt[2]) + Log[1 + Sqrt[2 - Sqrt[3]]*x + x^2]/(2*Sqrt[2])

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Rubi [A]  time = 0.250058, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24 \[ -\frac{\log \left (x^2-\sqrt{2-\sqrt{3}} x+1\right )}{2 \sqrt{2}}+\frac{\log \left (x^2+\sqrt{2-\sqrt{3}} x+1\right )}{2 \sqrt{2}}-\frac{\tan ^{-1}\left (\frac{\sqrt{2+\sqrt{3}}-2 x}{\sqrt{2-\sqrt{3}}}\right )}{\sqrt{2}}+\frac{\tan ^{-1}\left (\frac{2 x+\sqrt{2+\sqrt{3}}}{\sqrt{2-\sqrt{3}}}\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Int[(-1 + Sqrt[3] + 2*x^4)/(1 - x^4 + x^8),x]

[Out]

-(ArcTan[(Sqrt[2 + Sqrt[3]] - 2*x)/Sqrt[2 - Sqrt[3]]]/Sqrt[2]) + ArcTan[(Sqrt[2
+ Sqrt[3]] + 2*x)/Sqrt[2 - Sqrt[3]]]/Sqrt[2] - Log[1 - Sqrt[2 - Sqrt[3]]*x + x^2
]/(2*Sqrt[2]) + Log[1 + Sqrt[2 - Sqrt[3]]*x + x^2]/(2*Sqrt[2])

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Rubi in Sympy [A]  time = 51.7622, size = 202, normalized size = 1.5 \[ \frac{\left (- \sqrt{3} + 1\right ) \log{\left (x^{2} - x \sqrt{- \sqrt{3} + 2} + 1 \right )}}{4 \sqrt{- \sqrt{3} + 2}} - \frac{\left (- \sqrt{3} + 1\right ) \log{\left (x^{2} + x \sqrt{- \sqrt{3} + 2} + 1 \right )}}{4 \sqrt{- \sqrt{3} + 2}} + \frac{\sqrt{2} \left (- \sqrt{3} + 3\right )^{2} \operatorname{atan}{\left (\frac{\sqrt{6} \left (x \left (- \frac{\sqrt{3}}{3} + 1\right ) - \frac{\left (-3 + \sqrt{3}\right ) \sqrt{\sqrt{3} + 2}}{6}\right )}{- \sqrt{3} + 2} \right )}}{12 \left (- \sqrt{3} + 2\right )} + \frac{\sqrt{2} \left (- \sqrt{3} + 3\right )^{2} \operatorname{atan}{\left (\frac{\sqrt{6} \left (x \left (- \frac{\sqrt{3}}{3} + 1\right ) + \frac{\left (-3 + \sqrt{3}\right ) \sqrt{\sqrt{3} + 2}}{6}\right )}{- \sqrt{3} + 2} \right )}}{12 \left (- \sqrt{3} + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-1+2*x**4+3**(1/2))/(x**8-x**4+1),x)

[Out]

(-sqrt(3) + 1)*log(x**2 - x*sqrt(-sqrt(3) + 2) + 1)/(4*sqrt(-sqrt(3) + 2)) - (-s
qrt(3) + 1)*log(x**2 + x*sqrt(-sqrt(3) + 2) + 1)/(4*sqrt(-sqrt(3) + 2)) + sqrt(2
)*(-sqrt(3) + 3)**2*atan(sqrt(6)*(x*(-sqrt(3)/3 + 1) - (-3 + sqrt(3))*sqrt(sqrt(
3) + 2)/6)/(-sqrt(3) + 2))/(12*(-sqrt(3) + 2)) + sqrt(2)*(-sqrt(3) + 3)**2*atan(
sqrt(6)*(x*(-sqrt(3)/3 + 1) + (-3 + sqrt(3))*sqrt(sqrt(3) + 2)/6)/(-sqrt(3) + 2)
)/(12*(-sqrt(3) + 2))

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Mathematica [C]  time = 0.0522052, size = 71, normalized size = 0.53 \[ \frac{1}{4} \text{RootSum}\left [\text{$\#$1}^8-\text{$\#$1}^4+1\&,\frac{2 \text{$\#$1}^4 \log (x-\text{$\#$1})+\sqrt{3} \log (x-\text{$\#$1})-\log (x-\text{$\#$1})}{2 \text{$\#$1}^7-\text{$\#$1}^3}\&\right ] \]

Antiderivative was successfully verified.

[In]  Integrate[(-1 + Sqrt[3] + 2*x^4)/(1 - x^4 + x^8),x]

[Out]

RootSum[1 - #1^4 + #1^8 & , (-Log[x - #1] + Sqrt[3]*Log[x - #1] + 2*Log[x - #1]*
#1^4)/(-#1^3 + 2*#1^7) & ]/4

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Maple [C]  time = 0.076, size = 47, normalized size = 0.4 \[{\frac{1}{4}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}-{{\it \_Z}}^{4}+1 \right ) }{\frac{ \left ( -1+2\,{{\it \_R}}^{4}+\sqrt{3} \right ) \ln \left ( x-{\it \_R} \right ) }{2\,{{\it \_R}}^{7}-{{\it \_R}}^{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-1+2*x^4+3^(1/2))/(x^8-x^4+1),x)

[Out]

1/4*sum(1/(2*_R^7-_R^3)*(-1+2*_R^4+3^(1/2))*ln(x-_R),_R=RootOf(_Z^8-_Z^4+1))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{2 \, x^{4} + \sqrt{3} - 1}{x^{8} - x^{4} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x^4 + sqrt(3) - 1)/(x^8 - x^4 + 1),x, algorithm="maxima")

[Out]

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

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x^4 + sqrt(3) - 1)/(x^8 - x^4 + 1),x, algorithm="fricas")

[Out]

Exception raised: NotImplementedError

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Sympy [A]  time = 3.84787, size = 136, normalized size = 1.01 \[ \frac{\sqrt{2} \left (2 \operatorname{atan}{\left (\frac{x \left (\sqrt{6} + 2 \sqrt{2}\right )}{1 + \sqrt{3}} \right )} + 2 \operatorname{atan}{\left (\frac{x^{3} \left (\sqrt{6} + 2 \sqrt{2}\right )}{1 + \sqrt{3}} - \sqrt{2} x \right )}\right )}{4} - \frac{\sqrt{2} \log{\left (x^{2} - \frac{\sqrt{2} x \left (2 + 2 \sqrt{3}\right )}{4 \left (\sqrt{3} + 2\right )} + 1 \right )}}{4} + \frac{\sqrt{2} \log{\left (x^{2} + \frac{\sqrt{2} x \left (2 + 2 \sqrt{3}\right )}{4 \left (\sqrt{3} + 2\right )} + 1 \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-1+2*x**4+3**(1/2))/(x**8-x**4+1),x)

[Out]

sqrt(2)*(2*atan(x*(sqrt(6) + 2*sqrt(2))/(1 + sqrt(3))) + 2*atan(x**3*(sqrt(6) +
2*sqrt(2))/(1 + sqrt(3)) - sqrt(2)*x))/4 - sqrt(2)*log(x**2 - sqrt(2)*x*(2 + 2*s
qrt(3))/(4*(sqrt(3) + 2)) + 1)/4 + sqrt(2)*log(x**2 + sqrt(2)*x*(2 + 2*sqrt(3))/
(4*(sqrt(3) + 2)) + 1)/4

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GIAC/XCAS [A]  time = 0.291177, size = 144, normalized size = 1.07 \[ \frac{1}{2} \, \sqrt{2} \arctan \left (\frac{4 \, x + \sqrt{6} + \sqrt{2}}{\sqrt{6} - \sqrt{2}}\right ) + \frac{1}{2} \, \sqrt{2} \arctan \left (\frac{4 \, x - \sqrt{6} - \sqrt{2}}{\sqrt{6} - \sqrt{2}}\right ) + \frac{1}{4} \, \sqrt{2}{\rm ln}\left (x^{2} + \frac{1}{2} \, x{\left (\sqrt{6} - \sqrt{2}\right )} + 1\right ) - \frac{1}{4} \, \sqrt{2}{\rm ln}\left (x^{2} - \frac{1}{2} \, x{\left (\sqrt{6} - \sqrt{2}\right )} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x^4 + sqrt(3) - 1)/(x^8 - x^4 + 1),x, algorithm="giac")

[Out]

1/2*sqrt(2)*arctan((4*x + sqrt(6) + sqrt(2))/(sqrt(6) - sqrt(2))) + 1/2*sqrt(2)*
arctan((4*x - sqrt(6) - sqrt(2))/(sqrt(6) - sqrt(2))) + 1/4*sqrt(2)*ln(x^2 + 1/2
*x*(sqrt(6) - sqrt(2)) + 1) - 1/4*sqrt(2)*ln(x^2 - 1/2*x*(sqrt(6) - sqrt(2)) + 1
)