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

Optimal. Leaf size=39 \[ -\frac{\tan ^{-1}\left (\frac{1-2 x^4}{\sqrt{3}}\right )}{4 \sqrt{3}}-\frac{1}{8} \log \left (x^8-x^4+1\right ) \]

[Out]

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

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Rubi [A]  time = 0.0923199, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217 \[ -\frac{\tan ^{-1}\left (\frac{1-2 x^4}{\sqrt{3}}\right )}{4 \sqrt{3}}-\frac{1}{8} \log \left (x^8-x^4+1\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 14.0539, size = 34, normalized size = 0.87 \[ - \frac{\log{\left (x^{8} - x^{4} + 1 \right )}}{8} + \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x^{4}}{3} - \frac{1}{3}\right ) \right )}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-log(x**8 - x**4 + 1)/8 + sqrt(3)*atan(sqrt(3)*(2*x**4/3 - 1/3))/12

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Mathematica [A]  time = 0.0199397, size = 39, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{2 x^4-1}{\sqrt{3}}\right )}{4 \sqrt{3}}-\frac{1}{8} \log \left (x^8-x^4+1\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.005, size = 33, normalized size = 0.9 \[ -{\frac{\ln \left ({x}^{8}-{x}^{4}+1 \right ) }{8}}+{\frac{\sqrt{3}}{12}\arctan \left ({\frac{ \left ( 2\,{x}^{4}-1 \right ) \sqrt{3}}{3}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(-x^4+1)/(x^8-x^4+1),x)

[Out]

-1/8*ln(x^8-x^4+1)+1/12*3^(1/2)*arctan(1/3*(2*x^4-1)*3^(1/2))

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Maxima [A]  time = 0.825658, size = 43, normalized size = 1.1 \[ \frac{1}{12} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{4} - 1\right )}\right ) - \frac{1}{8} \, \log \left (x^{8} - x^{4} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.253155, size = 49, normalized size = 1.26 \[ -\frac{1}{24} \, \sqrt{3}{\left (\sqrt{3} \log \left (x^{8} - x^{4} + 1\right ) - 2 \, \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{4} - 1\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/24*sqrt(3)*(sqrt(3)*log(x^8 - x^4 + 1) - 2*arctan(1/3*sqrt(3)*(2*x^4 - 1)))

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Sympy [A]  time = 0.329236, size = 37, normalized size = 0.95 \[ - \frac{\log{\left (x^{8} - x^{4} + 1 \right )}}{8} + \frac{\sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x^{4}}{3} - \frac{\sqrt{3}}{3} \right )}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-log(x**8 - x**4 + 1)/8 + sqrt(3)*atan(2*sqrt(3)*x**4/3 - sqrt(3)/3)/12

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GIAC/XCAS [A]  time = 0.273112, size = 43, normalized size = 1.1 \[ \frac{1}{12} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{4} - 1\right )}\right ) - \frac{1}{8} \,{\rm ln}\left (x^{8} - x^{4} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^4 - 1)) - 1/8*ln(x^8 - x^4 + 1)