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

Optimal. Leaf size=50 \[ \frac{\log \left (x^4+\sqrt{3} x^2+1\right )}{4 \sqrt{3}}-\frac{\log \left (x^4-\sqrt{3} x^2+1\right )}{4 \sqrt{3}} \]

[Out]

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

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Rubi [A]  time = 0.0802674, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{\log \left (x^4+\sqrt{3} x^2+1\right )}{4 \sqrt{3}}-\frac{\log \left (x^4-\sqrt{3} x^2+1\right )}{4 \sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 20.2581, size = 42, normalized size = 0.84 \[ - \frac{\sqrt{3} \log{\left (x^{4} - \sqrt{3} x^{2} + 1 \right )}}{12} + \frac{\sqrt{3} \log{\left (x^{4} + \sqrt{3} x^{2} + 1 \right )}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(3)*log(x**4 - sqrt(3)*x**2 + 1)/12 + sqrt(3)*log(x**4 + sqrt(3)*x**2 + 1)/
12

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Mathematica [A]  time = 0.0259039, size = 44, normalized size = 0.88 \[ \frac{\log \left (x^4+\sqrt{3} x^2+1\right )-\log \left (-x^4+\sqrt{3} x^2-1\right )}{4 \sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.015, size = 39, normalized size = 0.8 \[ -{\frac{\ln \left ( 1+{x}^{4}-{x}^{2}\sqrt{3} \right ) \sqrt{3}}{12}}+{\frac{\ln \left ( 1+{x}^{4}+{x}^{2}\sqrt{3} \right ) \sqrt{3}}{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(-x^4+1)/(x^8-x^4+1),x)

[Out]

-1/12*ln(1+x^4-x^2*3^(1/2))*3^(1/2)+1/12*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. \[ -\int \frac{{\left (x^{4} - 1\right )} x}{x^{8} - x^{4} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.24828, size = 59, normalized size = 1.18 \[ \frac{1}{12} \, \sqrt{3} \log \left (\frac{6 \, x^{6} + 6 \, x^{2} + \sqrt{3}{\left (x^{8} + 5 \, x^{4} + 1\right )}}{x^{8} - x^{4} + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*sqrt(3)*log((6*x^6 + 6*x^2 + sqrt(3)*(x^8 + 5*x^4 + 1))/(x^8 - x^4 + 1))

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Sympy [A]  time = 0.288936, size = 42, normalized size = 0.84 \[ - \frac{\sqrt{3} \log{\left (x^{4} - \sqrt{3} x^{2} + 1 \right )}}{12} + \frac{\sqrt{3} \log{\left (x^{4} + \sqrt{3} x^{2} + 1 \right )}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(3)*log(x**4 - sqrt(3)*x**2 + 1)/12 + sqrt(3)*log(x**4 + sqrt(3)*x**2 + 1)/
12

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GIAC/XCAS [A]  time = 0.286236, size = 42, normalized size = 0.84 \[ -\frac{1}{12} \, \sqrt{3}{\rm ln}\left (\frac{x^{2} - \sqrt{3} + \frac{1}{x^{2}}}{x^{2} + \sqrt{3} + \frac{1}{x^{2}}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/12*sqrt(3)*ln((x^2 - sqrt(3) + 1/x^2)/(x^2 + sqrt(3) + 1/x^2))