3.1363 \(\int \frac{x^4}{1+x^6} \, dx\)

Optimal. Leaf size=80 \[ \frac{\log \left (x^2-\sqrt{3} x+1\right )}{4 \sqrt{3}}-\frac{\log \left (x^2+\sqrt{3} x+1\right )}{4 \sqrt{3}}-\frac{1}{6} \tan ^{-1}\left (\sqrt{3}-2 x\right )+\frac{1}{3} \tan ^{-1}(x)+\frac{1}{6} \tan ^{-1}\left (2 x+\sqrt{3}\right ) \]

[Out]

-ArcTan[Sqrt[3] - 2*x]/6 + ArcTan[x]/3 + ArcTan[Sqrt[3] + 2*x]/6 + Log[1 - Sqrt[
3]*x + x^2]/(4*Sqrt[3]) - Log[1 + Sqrt[3]*x + x^2]/(4*Sqrt[3])

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Rubi [A]  time = 0.683001, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.546 \[ \frac{\log \left (x^2-\sqrt{3} x+1\right )}{4 \sqrt{3}}-\frac{\log \left (x^2+\sqrt{3} x+1\right )}{4 \sqrt{3}}-\frac{1}{6} \tan ^{-1}\left (\sqrt{3}-2 x\right )+\frac{1}{3} \tan ^{-1}(x)+\frac{1}{6} \tan ^{-1}\left (2 x+\sqrt{3}\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^4/(1 + x^6),x]

[Out]

-ArcTan[Sqrt[3] - 2*x]/6 + ArcTan[x]/3 + ArcTan[Sqrt[3] + 2*x]/6 + Log[1 - Sqrt[
3]*x + x^2]/(4*Sqrt[3]) - Log[1 + Sqrt[3]*x + x^2]/(4*Sqrt[3])

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Rubi in Sympy [A]  time = 90.3598, size = 68, normalized size = 0.85 \[ \frac{\sqrt{3} \log{\left (x^{2} - \sqrt{3} x + 1 \right )}}{12} - \frac{\sqrt{3} \log{\left (x^{2} + \sqrt{3} x + 1 \right )}}{12} + \frac{\operatorname{atan}{\left (x \right )}}{3} + \frac{\operatorname{atan}{\left (2 x - \sqrt{3} \right )}}{6} + \frac{\operatorname{atan}{\left (2 x + \sqrt{3} \right )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4/(x**6+1),x)

[Out]

sqrt(3)*log(x**2 - sqrt(3)*x + 1)/12 - sqrt(3)*log(x**2 + sqrt(3)*x + 1)/12 + at
an(x)/3 + atan(2*x - sqrt(3))/6 + atan(2*x + sqrt(3))/6

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Mathematica [A]  time = 0.0202885, size = 73, normalized size = 0.91 \[ \frac{1}{12} \left (\sqrt{3} \log \left (x^2-\sqrt{3} x+1\right )-\sqrt{3} \log \left (x^2+\sqrt{3} x+1\right )-2 \tan ^{-1}\left (\sqrt{3}-2 x\right )+4 \tan ^{-1}(x)+2 \tan ^{-1}\left (2 x+\sqrt{3}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^4/(1 + x^6),x]

[Out]

(-2*ArcTan[Sqrt[3] - 2*x] + 4*ArcTan[x] + 2*ArcTan[Sqrt[3] + 2*x] + Sqrt[3]*Log[
1 - Sqrt[3]*x + x^2] - Sqrt[3]*Log[1 + Sqrt[3]*x + x^2])/12

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Maple [A]  time = 0.002, size = 61, normalized size = 0.8 \[{\frac{\arctan \left ( x \right ) }{3}}+{\frac{\arctan \left ( 2\,x-\sqrt{3} \right ) }{6}}+{\frac{\arctan \left ( 2\,x+\sqrt{3} \right ) }{6}}+{\frac{\ln \left ( 1+{x}^{2}-x\sqrt{3} \right ) \sqrt{3}}{12}}-{\frac{\ln \left ( 1+{x}^{2}+x\sqrt{3} \right ) \sqrt{3}}{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(x^6+1),x)

[Out]

1/3*arctan(x)+1/6*arctan(2*x-3^(1/2))+1/6*arctan(2*x+3^(1/2))+1/12*ln(1+x^2-x*3^
(1/2))*3^(1/2)-1/12*ln(1+x^2+x*3^(1/2))*3^(1/2)

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Maxima [A]  time = 1.59595, size = 81, normalized size = 1.01 \[ -\frac{1}{12} \, \sqrt{3} \log \left (x^{2} + \sqrt{3} x + 1\right ) + \frac{1}{12} \, \sqrt{3} \log \left (x^{2} - \sqrt{3} x + 1\right ) + \frac{1}{6} \, \arctan \left (2 \, x + \sqrt{3}\right ) + \frac{1}{6} \, \arctan \left (2 \, x - \sqrt{3}\right ) + \frac{1}{3} \, \arctan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(x^6 + 1),x, algorithm="maxima")

[Out]

-1/12*sqrt(3)*log(x^2 + sqrt(3)*x + 1) + 1/12*sqrt(3)*log(x^2 - sqrt(3)*x + 1) +
1/6*arctan(2*x + sqrt(3)) + 1/6*arctan(2*x - sqrt(3)) + 1/3*arctan(x)

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Fricas [A]  time = 0.237147, size = 126, normalized size = 1.58 \[ -\frac{1}{12} \, \sqrt{3} \log \left (x^{2} + \sqrt{3} x + 1\right ) + \frac{1}{12} \, \sqrt{3} \log \left (x^{2} - \sqrt{3} x + 1\right ) + \frac{1}{3} \, \arctan \left (x\right ) - \frac{1}{3} \, \arctan \left (\frac{1}{2 \, x + \sqrt{3} + 2 \, \sqrt{x^{2} + \sqrt{3} x + 1}}\right ) - \frac{1}{3} \, \arctan \left (\frac{1}{2 \, x - \sqrt{3} + 2 \, \sqrt{x^{2} - \sqrt{3} x + 1}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(x^6 + 1),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.690425, size = 68, normalized size = 0.85 \[ \frac{\sqrt{3} \log{\left (x^{2} - \sqrt{3} x + 1 \right )}}{12} - \frac{\sqrt{3} \log{\left (x^{2} + \sqrt{3} x + 1 \right )}}{12} + \frac{\operatorname{atan}{\left (x \right )}}{3} + \frac{\operatorname{atan}{\left (2 x - \sqrt{3} \right )}}{6} + \frac{\operatorname{atan}{\left (2 x + \sqrt{3} \right )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4/(x**6+1),x)

[Out]

sqrt(3)*log(x**2 - sqrt(3)*x + 1)/12 - sqrt(3)*log(x**2 + sqrt(3)*x + 1)/12 + at
an(x)/3 + atan(2*x - sqrt(3))/6 + atan(2*x + sqrt(3))/6

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GIAC/XCAS [A]  time = 0.226319, size = 81, normalized size = 1.01 \[ -\frac{1}{12} \, \sqrt{3}{\rm ln}\left (x^{2} + \sqrt{3} x + 1\right ) + \frac{1}{12} \, \sqrt{3}{\rm ln}\left (x^{2} - \sqrt{3} x + 1\right ) + \frac{1}{6} \, \arctan \left (2 \, x + \sqrt{3}\right ) + \frac{1}{6} \, \arctan \left (2 \, x - \sqrt{3}\right ) + \frac{1}{3} \, \arctan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(x^6 + 1),x, algorithm="giac")

[Out]

-1/12*sqrt(3)*ln(x^2 + sqrt(3)*x + 1) + 1/12*sqrt(3)*ln(x^2 - sqrt(3)*x + 1) + 1
/6*arctan(2*x + sqrt(3)) + 1/6*arctan(2*x - sqrt(3)) + 1/3*arctan(x)