3.353 \(\int \frac{x}{1-x^4+x^8} \, dx\)

Optimal. Leaf size=82 \[ -\frac{1}{4} \tan ^{-1}\left (\sqrt{3}-2 x^2\right )+\frac{1}{4} \tan ^{-1}\left (2 x^2+\sqrt{3}\right )-\frac{\log \left (x^4-\sqrt{3} x^2+1\right )}{8 \sqrt{3}}+\frac{\log \left (x^4+\sqrt{3} x^2+1\right )}{8 \sqrt{3}} \]

[Out]

-ArcTan[Sqrt[3] - 2*x^2]/4 + ArcTan[Sqrt[3] + 2*x^2]/4 - Log[1 - Sqrt[3]*x^2 + x
^4]/(8*Sqrt[3]) + Log[1 + Sqrt[3]*x^2 + x^4]/(8*Sqrt[3])

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Rubi [A]  time = 0.108086, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429 \[ -\frac{1}{4} \tan ^{-1}\left (\sqrt{3}-2 x^2\right )+\frac{1}{4} \tan ^{-1}\left (2 x^2+\sqrt{3}\right )-\frac{\log \left (x^4-\sqrt{3} x^2+1\right )}{8 \sqrt{3}}+\frac{\log \left (x^4+\sqrt{3} x^2+1\right )}{8 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[x/(1 - x^4 + x^8),x]

[Out]

-ArcTan[Sqrt[3] - 2*x^2]/4 + ArcTan[Sqrt[3] + 2*x^2]/4 - Log[1 - Sqrt[3]*x^2 + x
^4]/(8*Sqrt[3]) + Log[1 + Sqrt[3]*x^2 + x^4]/(8*Sqrt[3])

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Rubi in Sympy [A]  time = 18.366, size = 70, normalized size = 0.85 \[ - \frac{\sqrt{3} \log{\left (x^{4} - \sqrt{3} x^{2} + 1 \right )}}{24} + \frac{\sqrt{3} \log{\left (x^{4} + \sqrt{3} x^{2} + 1 \right )}}{24} + \frac{\operatorname{atan}{\left (2 x^{2} - \sqrt{3} \right )}}{4} + \frac{\operatorname{atan}{\left (2 x^{2} + \sqrt{3} \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(3)*log(x**4 - sqrt(3)*x**2 + 1)/24 + sqrt(3)*log(x**4 + sqrt(3)*x**2 + 1)/
24 + atan(2*x**2 - sqrt(3))/4 + atan(2*x**2 + sqrt(3))/4

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Mathematica [C]  time = 0.104553, size = 83, normalized size = 1.01 \[ \frac{i \left (\sqrt{-1-i \sqrt{3}} \tan ^{-1}\left (\frac{1}{2} \left (1-i \sqrt{3}\right ) x^2\right )-\sqrt{-1+i \sqrt{3}} \tan ^{-1}\left (\frac{1}{2} \left (1+i \sqrt{3}\right ) x^2\right )\right )}{2 \sqrt{6}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[x/(1 - x^4 + x^8),x]

[Out]

((I/2)*(Sqrt[-1 - I*Sqrt[3]]*ArcTan[((1 - I*Sqrt[3])*x^2)/2] - Sqrt[-1 + I*Sqrt[
3]]*ArcTan[((1 + I*Sqrt[3])*x^2)/2]))/Sqrt[6]

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Maple [A]  time = 0.009, size = 65, normalized size = 0.8 \[{\frac{\arctan \left ( 2\,{x}^{2}-\sqrt{3} \right ) }{4}}+{\frac{\arctan \left ( 2\,{x}^{2}+\sqrt{3} \right ) }{4}}-{\frac{\ln \left ( 1+{x}^{4}-{x}^{2}\sqrt{3} \right ) \sqrt{3}}{24}}+{\frac{\ln \left ( 1+{x}^{4}+{x}^{2}\sqrt{3} \right ) \sqrt{3}}{24}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(x^8-x^4+1),x)

[Out]

1/4*arctan(2*x^2-3^(1/2))+1/4*arctan(2*x^2+3^(1/2))-1/24*ln(1+x^4-x^2*3^(1/2))*3
^(1/2)+1/24*ln(1+x^4+x^2*3^(1/2))*3^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x/(x^8 - x^4 + 1), x)

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Fricas [A]  time = 0.265392, size = 159, normalized size = 1.94 \[ -\frac{1}{24} \, \sqrt{3}{\left (4 \, \sqrt{3} \arctan \left (\frac{\sqrt{3}}{2 \, \sqrt{3} x^{2} + 2 \, \sqrt{3} \sqrt{x^{4} + \sqrt{3} x^{2} + 1} + 3}\right ) + 4 \, \sqrt{3} \arctan \left (\frac{\sqrt{3}}{2 \, \sqrt{3} x^{2} + 2 \, \sqrt{3} \sqrt{x^{4} - \sqrt{3} x^{2} + 1} - 3}\right ) - \log \left (x^{4} + \sqrt{3} x^{2} + 1\right ) + \log \left (x^{4} - \sqrt{3} x^{2} + 1\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/24*sqrt(3)*(4*sqrt(3)*arctan(sqrt(3)/(2*sqrt(3)*x^2 + 2*sqrt(3)*sqrt(x^4 + sq
rt(3)*x^2 + 1) + 3)) + 4*sqrt(3)*arctan(sqrt(3)/(2*sqrt(3)*x^2 + 2*sqrt(3)*sqrt(
x^4 - sqrt(3)*x^2 + 1) - 3)) - log(x^4 + sqrt(3)*x^2 + 1) + log(x^4 - sqrt(3)*x^
2 + 1))

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Sympy [A]  time = 0.617644, size = 70, normalized size = 0.85 \[ - \frac{\sqrt{3} \log{\left (x^{4} - \sqrt{3} x^{2} + 1 \right )}}{24} + \frac{\sqrt{3} \log{\left (x^{4} + \sqrt{3} x^{2} + 1 \right )}}{24} + \frac{\operatorname{atan}{\left (2 x^{2} - \sqrt{3} \right )}}{4} + \frac{\operatorname{atan}{\left (2 x^{2} + \sqrt{3} \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(x**8-x**4+1),x)

[Out]

-sqrt(3)*log(x**4 - sqrt(3)*x**2 + 1)/24 + sqrt(3)*log(x**4 + sqrt(3)*x**2 + 1)/
24 + atan(2*x**2 - sqrt(3))/4 + atan(2*x**2 + sqrt(3))/4

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{x^{8} - x^{4} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x/(x^8 - x^4 + 1), x)