3.165 \(\int \frac{1}{3+4 x^3+x^6} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.117778, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 8, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667 \[ -\frac{1}{12} \log \left (x^2-x+1\right )+\frac{\log \left (x^2-\sqrt [3]{3} x+3^{2/3}\right )}{12\ 3^{2/3}}+\frac{1}{6} \log (x+1)-\frac{\log \left (x+\sqrt [3]{3}\right )}{6\ 3^{2/3}}-\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{2 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{3}-2 x}{3^{5/6}}\right )}{6 \sqrt [6]{3}} \]

Antiderivative was successfully verified.

[In]  Int[(3 + 4*x^3 + x^6)^(-1),x]

[Out]

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

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Rubi in Sympy [A]  time = 17.9273, size = 104, normalized size = 0.93 \[ \frac{\log{\left (x + 1 \right )}}{6} - \frac{\sqrt [3]{3} \log{\left (x + \sqrt [3]{3} \right )}}{18} - \frac{\log{\left (x^{2} - x + 1 \right )}}{12} + \frac{\sqrt [3]{3} \log{\left (x^{2} - \sqrt [3]{3} x + 3^{\frac{2}{3}} \right )}}{36} + \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} - \frac{1}{3}\right ) \right )}}{6} + \frac{3^{\frac{5}{6}} \operatorname{atan}{\left (\sqrt{3} \left (- \frac{2 \cdot 3^{\frac{2}{3}} x}{9} + \frac{1}{3}\right ) \right )}}{18} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(x**6+4*x**3+3),x)

[Out]

log(x + 1)/6 - 3**(1/3)*log(x + 3**(1/3))/18 - log(x**2 - x + 1)/12 + 3**(1/3)*l
og(x**2 - 3**(1/3)*x + 3**(2/3))/36 + sqrt(3)*atan(sqrt(3)*(2*x/3 - 1/3))/6 + 3*
*(5/6)*atan(sqrt(3)*(-2*3**(2/3)*x/9 + 1/3))/18

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Mathematica [A]  time = 0.0499714, size = 107, normalized size = 0.96 \[ \frac{1}{36} \left (-3 \log \left (x^2-x+1\right )+\sqrt [3]{3} \log \left (\sqrt [3]{3} x^2-3^{2/3} x+3\right )+6 \log (x+1)-2 \sqrt [3]{3} \log \left (3^{2/3} x+3\right )+2\ 3^{5/6} \tan ^{-1}\left (\frac{\sqrt [3]{3}-2 x}{3^{5/6}}\right )+6 \sqrt{3} \tan ^{-1}\left (\frac{2 x-1}{\sqrt{3}}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(3 + 4*x^3 + x^6)^(-1),x]

[Out]

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

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Maple [A]  time = 0.009, size = 84, normalized size = 0.8 \[ -{\frac{\sqrt [3]{3}\ln \left ( \sqrt [3]{3}+x \right ) }{18}}+{\frac{\sqrt [3]{3}\ln \left ({3}^{{\frac{2}{3}}}-\sqrt [3]{3}x+{x}^{2} \right ) }{36}}-{\frac{{3}^{{\frac{5}{6}}}}{18}\arctan \left ({\frac{\sqrt{3}}{3} \left ({\frac{2\,{3}^{2/3}x}{3}}-1 \right ) } \right ) }+{\frac{\ln \left ( 1+x \right ) }{6}}-{\frac{\ln \left ({x}^{2}-x+1 \right ) }{12}}+{\frac{\sqrt{3}}{6}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x^6+4*x^3+3),x)

[Out]

-1/18*3^(1/3)*ln(3^(1/3)+x)+1/36*3^(1/3)*ln(3^(2/3)-3^(1/3)*x+x^2)-1/18*3^(5/6)*
arctan(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)*a
rctan(1/3*(2*x-1)*3^(1/2))

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Maxima [A]  time = 0.867097, size = 113, normalized size = 1.01 \[ -\frac{1}{18} \cdot 3^{\frac{5}{6}} \arctan \left (\frac{1}{3} \cdot 3^{\frac{1}{6}}{\left (2 \, x - 3^{\frac{1}{3}}\right )}\right ) + \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + \frac{1}{36} \cdot 3^{\frac{1}{3}} \log \left (x^{2} - 3^{\frac{1}{3}} x + 3^{\frac{2}{3}}\right ) - \frac{1}{18} \cdot 3^{\frac{1}{3}} \log \left (x + 3^{\frac{1}{3}}\right ) - \frac{1}{12} \, \log \left (x^{2} - x + 1\right ) + \frac{1}{6} \, \log \left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x^6 + 4*x^3 + 3),x, algorithm="maxima")

[Out]

-1/18*3^(5/6)*arctan(1/3*3^(1/6)*(2*x - 3^(1/3))) + 1/6*sqrt(3)*arctan(1/3*sqrt(
3)*(2*x - 1)) + 1/36*3^(1/3)*log(x^2 - 3^(1/3)*x + 3^(2/3)) - 1/18*3^(1/3)*log(x
+ 3^(1/3)) - 1/12*log(x^2 - x + 1) + 1/6*log(x + 1)

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Fricas [A]  time = 0.263153, size = 178, normalized size = 1.59 \[ -\frac{1}{324} \cdot 9^{\frac{2}{3}} \sqrt{3}{\left (\sqrt{3} \left (-1\right )^{\frac{1}{3}} \log \left (9^{\frac{2}{3}} x^{2} + 3 \cdot 9^{\frac{1}{3}} \left (-1\right )^{\frac{1}{3}} x + 9 \, \left (-1\right )^{\frac{2}{3}}\right ) + 9^{\frac{1}{3}} \sqrt{3} \log \left (x^{2} - x + 1\right ) - 2 \, \sqrt{3} \left (-1\right )^{\frac{1}{3}} \log \left (9^{\frac{1}{3}} x - 3 \, \left (-1\right )^{\frac{1}{3}}\right ) - 2 \cdot 9^{\frac{1}{3}} \sqrt{3} \log \left (x + 1\right ) + 6 \, \left (-1\right )^{\frac{1}{3}} \arctan \left (-\frac{1}{9} \, \left (-1\right )^{\frac{2}{3}}{\left (2 \cdot 9^{\frac{1}{3}} \sqrt{3} x + 3 \, \sqrt{3} \left (-1\right )^{\frac{1}{3}}\right )}\right ) - 6 \cdot 9^{\frac{1}{3}} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x^6 + 4*x^3 + 3),x, algorithm="fricas")

[Out]

-1/324*9^(2/3)*sqrt(3)*(sqrt(3)*(-1)^(1/3)*log(9^(2/3)*x^2 + 3*9^(1/3)*(-1)^(1/3
)*x + 9*(-1)^(2/3)) + 9^(1/3)*sqrt(3)*log(x^2 - x + 1) - 2*sqrt(3)*(-1)^(1/3)*lo
g(9^(1/3)*x - 3*(-1)^(1/3)) - 2*9^(1/3)*sqrt(3)*log(x + 1) + 6*(-1)^(1/3)*arctan
(-1/9*(-1)^(2/3)*(2*9^(1/3)*sqrt(3)*x + 3*sqrt(3)*(-1)^(1/3))) - 6*9^(1/3)*arcta
n(1/3*sqrt(3)*(2*x - 1)))

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Sympy [A]  time = 4.6758, size = 124, normalized size = 1.11 \[ \frac{\log{\left (x + 1 \right )}}{6} + \left (- \frac{1}{12} + \frac{\sqrt{3} i}{12}\right ) \log{\left (x + \frac{13}{10} - \frac{13 \sqrt{3} i}{10} + \frac{23328 \left (- \frac{1}{12} + \frac{\sqrt{3} i}{12}\right )^{4}}{5} \right )} + \left (- \frac{1}{12} - \frac{\sqrt{3} i}{12}\right ) \log{\left (x + \frac{13}{10} + \frac{23328 \left (- \frac{1}{12} - \frac{\sqrt{3} i}{12}\right )^{4}}{5} + \frac{13 \sqrt{3} i}{10} \right )} + \operatorname{RootSum}{\left (1944 t^{3} + 1, \left ( t \mapsto t \log{\left (\frac{23328 t^{4}}{5} - \frac{78 t}{5} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x**6+4*x**3+3),x)

[Out]

log(x + 1)/6 + (-1/12 + sqrt(3)*I/12)*log(x + 13/10 - 13*sqrt(3)*I/10 + 23328*(-
1/12 + sqrt(3)*I/12)**4/5) + (-1/12 - sqrt(3)*I/12)*log(x + 13/10 + 23328*(-1/12
- sqrt(3)*I/12)**4/5 + 13*sqrt(3)*I/10) + RootSum(1944*_t**3 + 1, Lambda(_t, _t
*log(23328*_t**4/5 - 78*_t/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(1/(x^6 + 4*x^3 + 3),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError