3.162 \(\int \frac{x^4}{3+4 x^3+x^6} \, dx\)

Optimal. Leaf size=112 \[ -\frac{1}{12} \log \left (x^2-x+1\right )+\frac{\log \left (x^2-\sqrt [3]{3} x+3^{2/3}\right )}{4 \sqrt [3]{3}}+\frac{1}{6} \log (x+1)-\frac{\log \left (x+\sqrt [3]{3}\right )}{2 \sqrt [3]{3}}+\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{2 \sqrt{3}}-\frac{1}{2} \sqrt [6]{3} \tan ^{-1}\left (\frac{\sqrt [3]{3}-2 x}{3^{5/6}}\right ) \]

[Out]

ArcTan[(1 - 2*x)/Sqrt[3]]/(2*Sqrt[3]) - (3^(1/6)*ArcTan[(3^(1/3) - 2*x)/3^(5/6)]
)/2 + Log[1 + x]/6 - Log[3^(1/3) + x]/(2*3^(1/3)) - Log[1 - x + x^2]/12 + Log[3^
(2/3) - 3^(1/3)*x + x^2]/(4*3^(1/3))

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Rubi [A]  time = 0.135999, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 8, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5 \[ -\frac{1}{12} \log \left (x^2-x+1\right )+\frac{\log \left (x^2-\sqrt [3]{3} x+3^{2/3}\right )}{4 \sqrt [3]{3}}+\frac{1}{6} \log (x+1)-\frac{\log \left (x+\sqrt [3]{3}\right )}{2 \sqrt [3]{3}}+\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{2 \sqrt{3}}-\frac{1}{2} \sqrt [6]{3} \tan ^{-1}\left (\frac{\sqrt [3]{3}-2 x}{3^{5/6}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^4/(3 + 4*x^3 + x^6),x]

[Out]

ArcTan[(1 - 2*x)/Sqrt[3]]/(2*Sqrt[3]) - (3^(1/6)*ArcTan[(3^(1/3) - 2*x)/3^(5/6)]
)/2 + Log[1 + x]/6 - Log[3^(1/3) + x]/(2*3^(1/3)) - Log[1 - x + x^2]/12 + Log[3^
(2/3) - 3^(1/3)*x + x^2]/(4*3^(1/3))

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Rubi in Sympy [A]  time = 21.6207, size = 104, normalized size = 0.93 \[ \frac{\log{\left (x + 1 \right )}}{6} - \frac{3^{\frac{2}{3}} \log{\left (x + \sqrt [3]{3} \right )}}{6} - \frac{\log{\left (x^{2} - x + 1 \right )}}{12} + \frac{3^{\frac{2}{3}} \log{\left (x^{2} - \sqrt [3]{3} x + 3^{\frac{2}{3}} \right )}}{12} - \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} - \frac{1}{3}\right ) \right )}}{6} - \frac{\sqrt [6]{3} \operatorname{atan}{\left (\sqrt{3} \left (- \frac{2 \cdot 3^{\frac{2}{3}} x}{9} + \frac{1}{3}\right ) \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4/(x**6+4*x**3+3),x)

[Out]

log(x + 1)/6 - 3**(2/3)*log(x + 3**(1/3))/6 - log(x**2 - x + 1)/12 + 3**(2/3)*lo
g(x**2 - 3**(1/3)*x + 3**(2/3))/12 - sqrt(3)*atan(sqrt(3)*(2*x/3 - 1/3))/6 - 3**
(1/6)*atan(sqrt(3)*(-2*3**(2/3)*x/9 + 1/3))/2

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Mathematica [A]  time = 0.0461831, size = 107, normalized size = 0.96 \[ \frac{1}{12} \left (-\log \left (x^2-x+1\right )+3^{2/3} \log \left (\sqrt [3]{3} x^2-3^{2/3} x+3\right )+2 \log (x+1)-2\ 3^{2/3} \log \left (3^{2/3} x+3\right )-6 \sqrt [6]{3} \tan ^{-1}\left (\frac{\sqrt [3]{3}-2 x}{3^{5/6}}\right )-2 \sqrt{3} \tan ^{-1}\left (\frac{2 x-1}{\sqrt{3}}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^4/(3 + 4*x^3 + x^6),x]

[Out]

(-6*3^(1/6)*ArcTan[(3^(1/3) - 2*x)/3^(5/6)] - 2*Sqrt[3]*ArcTan[(-1 + 2*x)/Sqrt[3
]] + 2*Log[1 + x] - 2*3^(2/3)*Log[3 + 3^(2/3)*x] - Log[1 - x + x^2] + 3^(2/3)*Lo
g[3 - 3^(2/3)*x + 3^(1/3)*x^2])/12

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Maple [A]  time = 0.009, size = 84, normalized size = 0.8 \[ -{\frac{{3}^{{\frac{2}{3}}}\ln \left ( \sqrt [3]{3}+x \right ) }{6}}+{\frac{{3}^{{\frac{2}{3}}}\ln \left ({3}^{{\frac{2}{3}}}-\sqrt [3]{3}x+{x}^{2} \right ) }{12}}+{\frac{\sqrt [6]{3}}{2}\arctan \left ({\frac{\sqrt{3}}{3} \left ({\frac{2\,{3}^{2/3}x}{3}}-1 \right ) } \right ) }+{\frac{\ln \left ( 1+x \right ) }{6}}-{\frac{\ln \left ({x}^{2}-x+1 \right ) }{12}}-{\frac{\sqrt{3}}{6}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(x^6+4*x^3+3),x)

[Out]

-1/6*3^(2/3)*ln(3^(1/3)+x)+1/12*3^(2/3)*ln(3^(2/3)-3^(1/3)*x+x^2)+1/2*3^(1/6)*ar
ctan(1/3*3^(1/2)*(2/3*3^(2/3)*x-1))+1/6*ln(1+x)-1/12*ln(x^2-x+1)-1/6*3^(1/2)*arc
tan(1/3*(2*x-1)*3^(1/2))

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Maxima [A]  time = 0.845666, size = 113, normalized size = 1.01 \[ \frac{1}{12} \cdot 3^{\frac{2}{3}} \log \left (x^{2} - 3^{\frac{1}{3}} x + 3^{\frac{2}{3}}\right ) - \frac{1}{6} \cdot 3^{\frac{2}{3}} \log \left (x + 3^{\frac{1}{3}}\right ) - \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + \frac{1}{2} \cdot 3^{\frac{1}{6}} \arctan \left (\frac{1}{3} \cdot 3^{\frac{1}{6}}{\left (2 \, x - 3^{\frac{1}{3}}\right )}\right ) - \frac{1}{12} \, \log \left (x^{2} - x + 1\right ) + \frac{1}{6} \, \log \left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(x^6 + 4*x^3 + 3),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.267202, size = 163, normalized size = 1.46 \[ -\frac{1}{36} \cdot 3^{\frac{1}{6}}{\left (3 \, \sqrt{3} \left (-1\right )^{\frac{1}{3}} \log \left (-3^{\frac{2}{3}} \left (-1\right )^{\frac{2}{3}} x + 3^{\frac{1}{3}} x^{2} - 3 \, \left (-1\right )^{\frac{1}{3}}\right ) + 3^{\frac{5}{6}} \log \left (x^{2} - x + 1\right ) - 6 \, \sqrt{3} \left (-1\right )^{\frac{1}{3}} \log \left (3^{\frac{2}{3}} x + 3 \, \left (-1\right )^{\frac{2}{3}}\right ) - 2 \cdot 3^{\frac{5}{6}} \log \left (x + 1\right ) + 18 \, \left (-1\right )^{\frac{1}{3}} \arctan \left (-\frac{1}{9} \, \sqrt{3} \left (-1\right )^{\frac{1}{3}}{\left (2 \cdot 3^{\frac{2}{3}} x - 3 \, \left (-1\right )^{\frac{2}{3}}\right )}\right ) + 6 \cdot 3^{\frac{1}{3}} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(x^6 + 4*x^3 + 3),x, algorithm="fricas")

[Out]

-1/36*3^(1/6)*(3*sqrt(3)*(-1)^(1/3)*log(-3^(2/3)*(-1)^(2/3)*x + 3^(1/3)*x^2 - 3*
(-1)^(1/3)) + 3^(5/6)*log(x^2 - x + 1) - 6*sqrt(3)*(-1)^(1/3)*log(3^(2/3)*x + 3*
(-1)^(2/3)) - 2*3^(5/6)*log(x + 1) + 18*(-1)^(1/3)*arctan(-1/9*sqrt(3)*(-1)^(1/3
)*(2*3^(2/3)*x - 3*(-1)^(2/3))) + 6*3^(1/3)*arctan(1/3*sqrt(3)*(2*x - 1)))

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Sympy [A]  time = 2.18963, size = 134, normalized size = 1.2 \[ \frac{\log{\left (x + 1 \right )}}{6} + \left (- \frac{1}{12} - \frac{\sqrt{3} i}{12}\right ) \log{\left (x + \frac{2592 \left (- \frac{1}{12} - \frac{\sqrt{3} i}{12}\right )^{5}}{5} + \frac{168 \left (- \frac{1}{12} - \frac{\sqrt{3} i}{12}\right )^{2}}{5} \right )} + \left (- \frac{1}{12} + \frac{\sqrt{3} i}{12}\right ) \log{\left (x + \frac{168 \left (- \frac{1}{12} + \frac{\sqrt{3} i}{12}\right )^{2}}{5} + \frac{2592 \left (- \frac{1}{12} + \frac{\sqrt{3} i}{12}\right )^{5}}{5} \right )} + \operatorname{RootSum}{\left (24 t^{3} + 1, \left ( t \mapsto t \log{\left (\frac{2592 t^{5}}{5} + \frac{168 t^{2}}{5} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4/(x**6+4*x**3+3),x)

[Out]

log(x + 1)/6 + (-1/12 - sqrt(3)*I/12)*log(x + 2592*(-1/12 - sqrt(3)*I/12)**5/5 +
168*(-1/12 - sqrt(3)*I/12)**2/5) + (-1/12 + sqrt(3)*I/12)*log(x + 168*(-1/12 +
sqrt(3)*I/12)**2/5 + 2592*(-1/12 + sqrt(3)*I/12)**5/5) + RootSum(24*_t**3 + 1, L
ambda(_t, _t*log(2592*_t**5/5 + 168*_t**2/5 + x)))

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GIAC/XCAS [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/(x^6 + 4*x^3 + 3),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError