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

Optimal. Leaf size=180 \[ \frac{1}{4} \sqrt{3 \left (2-\sqrt{3}\right )} \log \left (x^2-\sqrt{2-\sqrt{3}} x+1\right )-\frac{1}{4} \sqrt{3 \left (2-\sqrt{3}\right )} \log \left (x^2+\sqrt{2-\sqrt{3}} x+1\right )+\frac{1}{2} \sqrt{3 \left (2-\sqrt{3}\right )} \tan ^{-1}\left (\frac{\sqrt{2+\sqrt{3}}-2 x}{\sqrt{2-\sqrt{3}}}\right )-\frac{1}{2} \sqrt{3 \left (2-\sqrt{3}\right )} \tan ^{-1}\left (\frac{2 x+\sqrt{2+\sqrt{3}}}{\sqrt{2-\sqrt{3}}}\right ) \]

[Out]

(Sqrt[3*(2 - Sqrt[3])]*ArcTan[(Sqrt[2 + Sqrt[3]] - 2*x)/Sqrt[2 - Sqrt[3]]])/2 -
(Sqrt[3*(2 - Sqrt[3])]*ArcTan[(Sqrt[2 + Sqrt[3]] + 2*x)/Sqrt[2 - Sqrt[3]]])/2 +
(Sqrt[3*(2 - Sqrt[3])]*Log[1 - Sqrt[2 - Sqrt[3]]*x + x^2])/4 - (Sqrt[3*(2 - Sqrt
[3])]*Log[1 + Sqrt[2 - Sqrt[3]]*x + x^2])/4

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Rubi [A]  time = 0.27579, antiderivative size = 180, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{1}{4} \sqrt{3 \left (2-\sqrt{3}\right )} \log \left (x^2-\sqrt{2-\sqrt{3}} x+1\right )-\frac{1}{4} \sqrt{3 \left (2-\sqrt{3}\right )} \log \left (x^2+\sqrt{2-\sqrt{3}} x+1\right )+\frac{1}{2} \sqrt{3 \left (2-\sqrt{3}\right )} \tan ^{-1}\left (\frac{\sqrt{2+\sqrt{3}}-2 x}{\sqrt{2-\sqrt{3}}}\right )-\frac{1}{2} \sqrt{3 \left (2-\sqrt{3}\right )} \tan ^{-1}\left (\frac{2 x+\sqrt{2+\sqrt{3}}}{\sqrt{2-\sqrt{3}}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[3*(2 - Sqrt[3])]*ArcTan[(Sqrt[2 + Sqrt[3]] - 2*x)/Sqrt[2 - Sqrt[3]]])/2 -
(Sqrt[3*(2 - Sqrt[3])]*ArcTan[(Sqrt[2 + Sqrt[3]] + 2*x)/Sqrt[2 - Sqrt[3]]])/2 +
(Sqrt[3*(2 - Sqrt[3])]*Log[1 - Sqrt[2 - Sqrt[3]]*x + x^2])/4 - (Sqrt[3*(2 - Sqrt
[3])]*Log[1 + Sqrt[2 - Sqrt[3]]*x + x^2])/4

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Rubi in Sympy [A]  time = 54.4499, size = 216, normalized size = 1.2 \[ - \frac{\left (- 2 \sqrt{3} + 3\right ) \log{\left (x^{2} - x \sqrt{- \sqrt{3} + 2} + 1 \right )}}{4 \sqrt{- \sqrt{3} + 2}} + \frac{\left (- 2 \sqrt{3} + 3\right ) \log{\left (x^{2} + x \sqrt{- \sqrt{3} + 2} + 1 \right )}}{4 \sqrt{- \sqrt{3} + 2}} + \frac{\sqrt{3} \left (- 3 \sqrt{3} + 6\right )^{2} \operatorname{atan}{\left (\frac{x \left (-4 + 2 \sqrt{3}\right ) - \left (-2 + \sqrt{3}\right ) \sqrt{\sqrt{3} + 2}}{\sqrt{- 15 \sqrt{3} + 26}} \right )}}{18 \sqrt{- 15 \sqrt{3} + 26}} + \frac{\sqrt{3} \left (- 3 \sqrt{3} + 6\right )^{2} \operatorname{atan}{\left (\frac{x \left (-4 + 2 \sqrt{3}\right ) + \left (-2 + \sqrt{3}\right ) \sqrt{\sqrt{3} + 2}}{\sqrt{- 15 \sqrt{3} + 26}} \right )}}{18 \sqrt{- 15 \sqrt{3} + 26}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-2*sqrt(3) + 3)*log(x**2 - x*sqrt(-sqrt(3) + 2) + 1)/(4*sqrt(-sqrt(3) + 2)) +
(-2*sqrt(3) + 3)*log(x**2 + x*sqrt(-sqrt(3) + 2) + 1)/(4*sqrt(-sqrt(3) + 2)) + s
qrt(3)*(-3*sqrt(3) + 6)**2*atan((x*(-4 + 2*sqrt(3)) - (-2 + sqrt(3))*sqrt(sqrt(3
) + 2))/sqrt(-15*sqrt(3) + 26))/(18*sqrt(-15*sqrt(3) + 26)) + sqrt(3)*(-3*sqrt(3
) + 6)**2*atan((x*(-4 + 2*sqrt(3)) + (-2 + sqrt(3))*sqrt(sqrt(3) + 2))/sqrt(-15*
sqrt(3) + 26))/(18*sqrt(-15*sqrt(3) + 26))

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Mathematica [C]  time = 0.0687416, size = 89, normalized size = 0.49 \[ \frac{1}{4} \text{RootSum}\left [\text{$\#$1}^8-\text{$\#$1}^4+1\&,\frac{\sqrt{3} \text{$\#$1}^4 \log (x-\text{$\#$1})-3 \text{$\#$1}^4 \log (x-\text{$\#$1})-2 \sqrt{3} \log (x-\text{$\#$1})+3 \log (x-\text{$\#$1})}{2 \text{$\#$1}^7-\text{$\#$1}^3}\&\right ] \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.013, size = 62, normalized size = 0.3 \[{\frac{1}{8}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}-{{\it \_Z}}^{4}+1 \right ) }{\frac{ \left ( -6\,{{\it \_R}}^{4}+2\,\sqrt{3}{{\it \_R}}^{4}+ \left ( -3+\sqrt{3} \right ) \left ( \sqrt{3}-1 \right ) \right ) \ln \left ( x-{\it \_R} \right ) }{2\,{{\it \_R}}^{7}-{{\it \_R}}^{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*sum(1/(2*_R^7-_R^3)*(-6*_R^4+2*3^(1/2)*_R^4+(-3+3^(1/2))*(3^(1/2)-1))*ln(x-_
R),_R=RootOf(_Z^8-_Z^4+1))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: NotImplementedError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: PolynomialError

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GIAC/XCAS [A]  time = 0.290372, size = 177, normalized size = 0.98 \[ \frac{1}{4} \,{\left (\sqrt{6} - 3 \, \sqrt{2}\right )} \arctan \left (\frac{4 \, x + \sqrt{6} + \sqrt{2}}{\sqrt{6} - \sqrt{2}}\right ) + \frac{1}{4} \,{\left (\sqrt{6} - 3 \, \sqrt{2}\right )} \arctan \left (\frac{4 \, x - \sqrt{6} - \sqrt{2}}{\sqrt{6} - \sqrt{2}}\right ) + \frac{1}{8} \,{\left (\sqrt{6} - 3 \, \sqrt{2}\right )}{\rm ln}\left (x^{2} + \frac{1}{2} \, x{\left (\sqrt{6} - \sqrt{2}\right )} + 1\right ) - \frac{1}{8} \,{\left (\sqrt{6} - 3 \, \sqrt{2}\right )}{\rm ln}\left (x^{2} - \frac{1}{2} \, x{\left (\sqrt{6} - \sqrt{2}\right )} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(sqrt(6) - 3*sqrt(2))*arctan((4*x + sqrt(6) + sqrt(2))/(sqrt(6) - sqrt(2)))
+ 1/4*(sqrt(6) - 3*sqrt(2))*arctan((4*x - sqrt(6) - sqrt(2))/(sqrt(6) - sqrt(2))
) + 1/8*(sqrt(6) - 3*sqrt(2))*ln(x^2 + 1/2*x*(sqrt(6) - sqrt(2)) + 1) - 1/8*(sqr
t(6) - 3*sqrt(2))*ln(x^2 - 1/2*x*(sqrt(6) - sqrt(2)) + 1)