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

Optimal. Leaf size=113 \[ -\frac{1}{12} \log \left (x^2-x+1\right )+\frac{1}{4} \sqrt [3]{3} \log \left (x^2-\sqrt [3]{3} x+3^{2/3}\right )+x+\frac{1}{6} \log (x+1)-\frac{1}{2} \sqrt [3]{3} \log \left (x+\sqrt [3]{3}\right )-\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{2 \sqrt{3}}+\frac{1}{2} 3^{5/6} \tan ^{-1}\left (\frac{\sqrt [3]{3}-2 x}{3^{5/6}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.146049, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 9, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.562 \[ -\frac{1}{12} \log \left (x^2-x+1\right )+\frac{1}{4} \sqrt [3]{3} \log \left (x^2-\sqrt [3]{3} x+3^{2/3}\right )+x+\frac{1}{6} \log (x+1)-\frac{1}{2} \sqrt [3]{3} \log \left (x+\sqrt [3]{3}\right )-\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{2 \sqrt{3}}+\frac{1}{2} 3^{5/6} \tan ^{-1}\left (\frac{\sqrt [3]{3}-2 x}{3^{5/6}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^6/(3 + 4*x^3 + x^6),x]

[Out]

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

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Rubi in Sympy [A]  time = 28.8504, size = 105, normalized size = 0.93 \[ x + \frac{\log{\left (x + 1 \right )}}{6} - \frac{\sqrt [3]{3} \log{\left (x + \sqrt [3]{3} \right )}}{2} - \frac{\log{\left (x^{2} - x + 1 \right )}}{12} + \frac{\sqrt [3]{3} \log{\left (x^{2} - \sqrt [3]{3} x + 3^{\frac{2}{3}} \right )}}{4} + \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} - \frac{1}{3}\right ) \right )}}{6} + \frac{3^{\frac{5}{6}} \operatorname{atan}{\left (\sqrt{3} \left (- \frac{2 \cdot 3^{\frac{2}{3}} x}{9} + \frac{1}{3}\right ) \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x + log(x + 1)/6 - 3**(1/3)*log(x + 3**(1/3))/2 - log(x**2 - x + 1)/12 + 3**(1/3
)*log(x**2 - 3**(1/3)*x + 3**(2/3))/4 + 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))/2

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Mathematica [A]  time = 0.0505266, size = 111, normalized size = 0.98 \[ \frac{1}{12} \left (-\log \left (x^2-x+1\right )+3 \sqrt [3]{3} \log \left (\sqrt [3]{3} x^2-3^{2/3} x+3\right )+12 x+2 \log (x+1)-6 \sqrt [3]{3} \log \left (3^{2/3} x+3\right )+6\ 3^{5/6} \tan ^{-1}\left (\frac{\sqrt [3]{3}-2 x}{3^{5/6}}\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{2 x-1}{\sqrt{3}}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.009, size = 85, normalized size = 0.8 \[ x-{\frac{\sqrt [3]{3}\ln \left ( \sqrt [3]{3}+x \right ) }{2}}+{\frac{\sqrt [3]{3}\ln \left ({3}^{{\frac{2}{3}}}-\sqrt [3]{3}x+{x}^{2} \right ) }{4}}-{\frac{{3}^{{\frac{5}{6}}}}{2}\arctan \left ({\frac{\sqrt{3}}{3} \left ({\frac{2\,{3}^{2/3}x}{3}}-1 \right ) } \right ) }+{\frac{\ln \left ( 1+x \right ) }{6}}-{\frac{\ln \left ({x}^{2}-x+1 \right ) }{12}}+{\frac{\sqrt{3}}{6}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^6/(x^6+4*x^3+3),x)

[Out]

x-1/2*3^(1/3)*ln(3^(1/3)+x)+1/4*3^(1/3)*ln(3^(2/3)-3^(1/3)*x+x^2)-1/2*3^(5/6)*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 = 0.852332, size = 115, normalized size = 1.02 \[ -\frac{1}{2} \cdot 3^{\frac{5}{6}} \arctan \left (\frac{1}{3} \cdot 3^{\frac{1}{6}}{\left (2 \, x - 3^{\frac{1}{3}}\right )}\right ) + \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + \frac{1}{4} \cdot 3^{\frac{1}{3}} \log \left (x^{2} - 3^{\frac{1}{3}} x + 3^{\frac{2}{3}}\right ) - \frac{1}{2} \cdot 3^{\frac{1}{3}} \log \left (x + 3^{\frac{1}{3}}\right ) + x - \frac{1}{12} \, \log \left (x^{2} - x + 1\right ) + \frac{1}{6} \, \log \left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*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)) + 1/4*3^(1/3)*log(x^2 - 3^(1/3)*x + 3^(2/3)) - 1/2*3^(1/3)*log(x +
3^(1/3)) + x - 1/12*log(x^2 - x + 1) + 1/6*log(x + 1)

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Fricas [A]  time = 0.265852, size = 142, normalized size = 1.26 \[ -\frac{1}{36} \, \sqrt{3}{\left (3 \, \sqrt{3} \left (-3\right )^{\frac{1}{3}} \log \left (x^{2} + \left (-3\right )^{\frac{1}{3}} x + \left (-3\right )^{\frac{2}{3}}\right ) - 6 \, \sqrt{3} \left (-3\right )^{\frac{1}{3}} \log \left (x - \left (-3\right )^{\frac{1}{3}}\right ) - 12 \, \sqrt{3} x + \sqrt{3} \log \left (x^{2} - x + 1\right ) - 2 \, \sqrt{3} \log \left (x + 1\right ) + 18 \, \left (-3\right )^{\frac{1}{3}} \arctan \left (-\frac{1}{9} \, \sqrt{3} \left (-3\right )^{\frac{2}{3}}{\left (2 \, x + \left (-3\right )^{\frac{1}{3}}\right )}\right ) - 6 \, \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/36*sqrt(3)*(3*sqrt(3)*(-3)^(1/3)*log(x^2 + (-3)^(1/3)*x + (-3)^(2/3)) - 6*sqr
t(3)*(-3)^(1/3)*log(x - (-3)^(1/3)) - 12*sqrt(3)*x + sqrt(3)*log(x^2 - x + 1) -
2*sqrt(3)*log(x + 1) + 18*(-3)^(1/3)*arctan(-1/9*sqrt(3)*(-3)^(2/3)*(2*x + (-3)^
(1/3))) - 6*arctan(1/3*sqrt(3)*(2*x - 1)))

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Sympy [A]  time = 2.21808, size = 126, normalized size = 1.12 \[ x + \frac{\log{\left (x + 1 \right )}}{6} + \left (- \frac{1}{12} - \frac{\sqrt{3} i}{12}\right ) \log{\left (x - \frac{121}{246} - \frac{121 \sqrt{3} i}{246} + \frac{864 \left (- \frac{1}{12} - \frac{\sqrt{3} i}{12}\right )^{4}}{41} \right )} + \left (- \frac{1}{12} + \frac{\sqrt{3} i}{12}\right ) \log{\left (x - \frac{121}{246} + \frac{864 \left (- \frac{1}{12} + \frac{\sqrt{3} i}{12}\right )^{4}}{41} + \frac{121 \sqrt{3} i}{246} \right )} + \operatorname{RootSum}{\left (8 t^{3} + 3, \left ( t \mapsto t \log{\left (\frac{864 t^{4}}{41} + \frac{242 t}{41} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x + log(x + 1)/6 + (-1/12 - sqrt(3)*I/12)*log(x - 121/246 - 121*sqrt(3)*I/246 +
864*(-1/12 - sqrt(3)*I/12)**4/41) + (-1/12 + sqrt(3)*I/12)*log(x - 121/246 + 864
*(-1/12 + sqrt(3)*I/12)**4/41 + 121*sqrt(3)*I/246) + RootSum(8*_t**3 + 3, Lambda
(_t, _t*log(864*_t**4/41 + 242*_t/41 + 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^6/(x^6 + 4*x^3 + 3),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError