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

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

[Out]

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

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Rubi [A]  time = 0.128514, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 8, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.571 \[ \frac{1}{12} \log \left (x^2-x+1\right )-\frac{\log \left (x^2-\sqrt [3]{3} x+3^{2/3}\right )}{12 \sqrt [3]{3}}-\frac{1}{6} \log (x+1)+\frac{\log \left (x+\sqrt [3]{3}\right )}{6 \sqrt [3]{3}}-\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{2 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{3}-2 x}{3^{5/6}}\right )}{2\ 3^{5/6}} \]

Antiderivative was successfully verified.

[In]  Int[x/(3 + 4*x^3 + x^6),x]

[Out]

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

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Rubi in Sympy [A]  time = 20.7523, size = 104, normalized size = 0.93 \[ - \frac{\log{\left (x + 1 \right )}}{6} + \frac{3^{\frac{2}{3}} \log{\left (x + \sqrt [3]{3} \right )}}{18} + \frac{\log{\left (x^{2} - x + 1 \right )}}{12} - \frac{3^{\frac{2}{3}} \log{\left (x^{2} - \sqrt [3]{3} x + 3^{\frac{2}{3}} \right )}}{36} + \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} - \frac{1}{3}\right ) \right )}}{6} + \frac{\sqrt [6]{3} \operatorname{atan}{\left (\sqrt{3} \left (- \frac{2 \cdot 3^{\frac{2}{3}} x}{9} + \frac{1}{3}\right ) \right )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-log(x + 1)/6 + 3**(2/3)*log(x + 3**(1/3))/18 + log(x**2 - x + 1)/12 - 3**(2/3)*
log(x**2 - 3**(1/3)*x + 3**(2/3))/36 + 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))/6

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Mathematica [A]  time = 0.0446223, size = 108, normalized size = 0.96 \[ \frac{1}{36} \left (3 \log \left (x^2-x+1\right )-3^{2/3} \log \left (\sqrt [3]{3} x^2-3^{2/3} x+3\right )-6 \log (x+1)+2\ 3^{2/3} \log \left (3^{2/3} x+3\right )+6 \sqrt [6]{3} \tan ^{-1}\left (\frac{\sqrt [3]{3}-2 x}{3^{5/6}}\right )+6 \sqrt{3} \tan ^{-1}\left (\frac{2 x-1}{\sqrt{3}}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(6*3^(1/6)*ArcTan[(3^(1/3) - 2*x)/3^(5/6)] + 6*Sqrt[3]*ArcTan[(-1 + 2*x)/Sqrt[3]
] - 6*Log[1 + x] + 2*3^(2/3)*Log[3 + 3^(2/3)*x] + 3*Log[1 - x + x^2] - 3^(2/3)*L
og[3 - 3^(2/3)*x + 3^(1/3)*x^2])/36

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Maple [A]  time = 0.008, size = 84, normalized size = 0.8 \[{\frac{{3}^{{\frac{2}{3}}}\ln \left ( \sqrt [3]{3}+x \right ) }{18}}-{\frac{{3}^{{\frac{2}{3}}}\ln \left ({3}^{{\frac{2}{3}}}-\sqrt [3]{3}x+{x}^{2} \right ) }{36}}-{\frac{\sqrt [6]{3}}{6}\arctan \left ({\frac{\sqrt{3}}{3} \left ({\frac{2\,{3}^{2/3}x}{3}}-1 \right ) } \right ) }-{\frac{\ln \left ( 1+x \right ) }{6}}+{\frac{\ln \left ({x}^{2}-x+1 \right ) }{12}}+{\frac{\sqrt{3}}{6}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(x^6+4*x^3+3),x)

[Out]

1/18*3^(2/3)*ln(3^(1/3)+x)-1/36*3^(2/3)*ln(3^(2/3)-3^(1/3)*x+x^2)-1/6*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 = 0.875692, size = 113, normalized size = 1.01 \[ -\frac{1}{36} \cdot 3^{\frac{2}{3}} \log \left (x^{2} - 3^{\frac{1}{3}} x + 3^{\frac{2}{3}}\right ) + \frac{1}{18} \cdot 3^{\frac{2}{3}} \log \left (x + 3^{\frac{1}{3}}\right ) + \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) - \frac{1}{6} \cdot 3^{\frac{1}{6}} \arctan \left (\frac{1}{3} \cdot 3^{\frac{1}{6}}{\left (2 \, x - 3^{\frac{1}{3}}\right )}\right ) + \frac{1}{12} \, \log \left (x^{2} - x + 1\right ) - \frac{1}{6} \, \log \left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/36*3^(2/3)*log(x^2 - 3^(1/3)*x + 3^(2/3)) + 1/18*3^(2/3)*log(x + 3^(1/3)) + 1
/6*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - 1)) - 1/6*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 = 0.273926, size = 126, normalized size = 1.12 \[ \frac{1}{36} \cdot 3^{\frac{1}{6}}{\left (3^{\frac{5}{6}} \log \left (x^{2} - x + 1\right ) - 2 \cdot 3^{\frac{5}{6}} \log \left (x + 1\right ) - \sqrt{3} \log \left (3^{\frac{1}{3}} x^{2} - 3^{\frac{2}{3}} x + 3\right ) + 2 \, \sqrt{3} \log \left (3^{\frac{2}{3}} x + 3\right ) + 6 \cdot 3^{\frac{1}{3}} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) - 6 \, \arctan \left (\frac{2}{3} \cdot 3^{\frac{1}{6}} x - \frac{1}{3} \, \sqrt{3}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 4.49702, size = 119, normalized size = 1.06 \[ - \frac{\log{\left (x + 1 \right )}}{6} + \left (\frac{1}{12} - \frac{\sqrt{3} i}{12}\right ) \log{\left (x + 90 \left (\frac{1}{12} - \frac{\sqrt{3} i}{12}\right )^{2} + 11664 \left (\frac{1}{12} - \frac{\sqrt{3} i}{12}\right )^{5} \right )} + \left (\frac{1}{12} + \frac{\sqrt{3} i}{12}\right ) \log{\left (x + 11664 \left (\frac{1}{12} + \frac{\sqrt{3} i}{12}\right )^{5} + 90 \left (\frac{1}{12} + \frac{\sqrt{3} i}{12}\right )^{2} \right )} + \operatorname{RootSum}{\left (648 t^{3} - 1, \left ( t \mapsto t \log{\left (11664 t^{5} + 90 t^{2} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(x**6+4*x**3+3),x)

[Out]

-log(x + 1)/6 + (1/12 - sqrt(3)*I/12)*log(x + 90*(1/12 - sqrt(3)*I/12)**2 + 1166
4*(1/12 - sqrt(3)*I/12)**5) + (1/12 + sqrt(3)*I/12)*log(x + 11664*(1/12 + sqrt(3
)*I/12)**5 + 90*(1/12 + sqrt(3)*I/12)**2) + RootSum(648*_t**3 - 1, Lambda(_t, _t
*log(11664*_t**5 + 90*_t**2 + 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/(x^6 + 4*x^3 + 3),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError