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

Optimal. Leaf size=126 \[ -\frac{1}{15 x^5}+\frac{2}{9 x^2}-\frac{1}{12} \log \left (x^2-x+1\right )+\frac{\log \left (x^2-\sqrt [3]{3} x+3^{2/3}\right )}{108\ 3^{2/3}}+\frac{1}{6} \log (x+1)-\frac{\log \left (x+\sqrt [3]{3}\right )}{54\ 3^{2/3}}-\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{2 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{3}-2 x}{3^{5/6}}\right )}{54 \sqrt [6]{3}} \]

[Out]

-1/(15*x^5) + 2/(9*x^2) - ArcTan[(1 - 2*x)/Sqrt[3]]/(2*Sqrt[3]) + ArcTan[(3^(1/3
) - 2*x)/3^(5/6)]/(54*3^(1/6)) + Log[1 + x]/6 - Log[3^(1/3) + x]/(54*3^(2/3)) -
Log[1 - x + x^2]/12 + Log[3^(2/3) - 3^(1/3)*x + x^2]/(108*3^(2/3))

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Rubi [A]  time = 0.202867, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 10, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.625 \[ -\frac{1}{15 x^5}+\frac{2}{9 x^2}-\frac{1}{12} \log \left (x^2-x+1\right )+\frac{\log \left (x^2-\sqrt [3]{3} x+3^{2/3}\right )}{108\ 3^{2/3}}+\frac{1}{6} \log (x+1)-\frac{\log \left (x+\sqrt [3]{3}\right )}{54\ 3^{2/3}}-\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{2 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{3}-2 x}{3^{5/6}}\right )}{54 \sqrt [6]{3}} \]

Antiderivative was successfully verified.

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

[Out]

-1/(15*x^5) + 2/(9*x^2) - ArcTan[(1 - 2*x)/Sqrt[3]]/(2*Sqrt[3]) + ArcTan[(3^(1/3
) - 2*x)/3^(5/6)]/(54*3^(1/6)) + Log[1 + x]/6 - Log[3^(1/3) + x]/(54*3^(2/3)) -
Log[1 - x + x^2]/12 + Log[3^(2/3) - 3^(1/3)*x + x^2]/(108*3^(2/3))

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Rubi in Sympy [A]  time = 34.2344, size = 117, normalized size = 0.93 \[ \frac{\log{\left (x + 1 \right )}}{6} - \frac{\sqrt [3]{3} \log{\left (x + \sqrt [3]{3} \right )}}{162} - \frac{\log{\left (x^{2} - x + 1 \right )}}{12} + \frac{\sqrt [3]{3} \log{\left (x^{2} - \sqrt [3]{3} x + 3^{\frac{2}{3}} \right )}}{324} + \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} - \frac{1}{3}\right ) \right )}}{6} + \frac{3^{\frac{5}{6}} \operatorname{atan}{\left (\sqrt{3} \left (- \frac{2 \cdot 3^{\frac{2}{3}} x}{9} + \frac{1}{3}\right ) \right )}}{162} + \frac{2}{9 x^{2}} - \frac{1}{15 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(x + 1)/6 - 3**(1/3)*log(x + 3**(1/3))/162 - log(x**2 - x + 1)/12 + 3**(1/3)*
log(x**2 - 3**(1/3)*x + 3**(2/3))/324 + 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))/162 + 2/(9*x**2) - 1/(15*x**5)

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Mathematica [A]  time = 0.110343, size = 118, normalized size = 0.94 \[ \frac{-\frac{108}{x^5}+\frac{360}{x^2}-135 \log \left (x^2-x+1\right )+5 \sqrt [3]{3} \log \left (\sqrt [3]{3} x^2-3^{2/3} x+3\right )+270 \log (x+1)-10 \sqrt [3]{3} \log \left (3^{2/3} x+3\right )+10\ 3^{5/6} \tan ^{-1}\left (\frac{\sqrt [3]{3}-2 x}{3^{5/6}}\right )+270 \sqrt{3} \tan ^{-1}\left (\frac{2 x-1}{\sqrt{3}}\right )}{1620} \]

Antiderivative was successfully verified.

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

[Out]

(-108/x^5 + 360/x^2 + 10*3^(5/6)*ArcTan[(3^(1/3) - 2*x)/3^(5/6)] + 270*Sqrt[3]*A
rcTan[(-1 + 2*x)/Sqrt[3]] + 270*Log[1 + x] - 10*3^(1/3)*Log[3 + 3^(2/3)*x] - 135
*Log[1 - x + x^2] + 5*3^(1/3)*Log[3 - 3^(2/3)*x + 3^(1/3)*x^2])/1620

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Maple [A]  time = 0.014, size = 94, normalized size = 0.8 \[ -{\frac{\sqrt [3]{3}\ln \left ( \sqrt [3]{3}+x \right ) }{162}}+{\frac{\sqrt [3]{3}\ln \left ({3}^{{\frac{2}{3}}}-\sqrt [3]{3}x+{x}^{2} \right ) }{324}}-{\frac{{3}^{{\frac{5}{6}}}}{162}\arctan \left ({\frac{\sqrt{3}}{3} \left ({\frac{2\,{3}^{2/3}x}{3}}-1 \right ) } \right ) }+{\frac{\ln \left ( 1+x \right ) }{6}}-{\frac{1}{15\,{x}^{5}}}+{\frac{2}{9\,{x}^{2}}}-{\frac{\ln \left ({x}^{2}-x+1 \right ) }{12}}+{\frac{\sqrt{3}}{6}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^6/(x^6+4*x^3+3),x)

[Out]

-1/162*3^(1/3)*ln(3^(1/3)+x)+1/324*3^(1/3)*ln(3^(2/3)-3^(1/3)*x+x^2)-1/162*3^(5/
6)*arctan(1/3*3^(1/2)*(2/3*3^(2/3)*x-1))+1/6*ln(1+x)-1/15/x^5+2/9/x^2-1/12*ln(x^
2-x+1)+1/6*3^(1/2)*arctan(1/3*(2*x-1)*3^(1/2))

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Maxima [A]  time = 0.883354, size = 130, normalized size = 1.03 \[ -\frac{1}{162} \cdot 3^{\frac{5}{6}} \arctan \left (\frac{1}{3} \cdot 3^{\frac{1}{6}}{\left (2 \, x - 3^{\frac{1}{3}}\right )}\right ) + \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + \frac{1}{324} \cdot 3^{\frac{1}{3}} \log \left (x^{2} - 3^{\frac{1}{3}} x + 3^{\frac{2}{3}}\right ) - \frac{1}{162} \cdot 3^{\frac{1}{3}} \log \left (x + 3^{\frac{1}{3}}\right ) + \frac{10 \, x^{3} - 3}{45 \, x^{5}} - \frac{1}{12} \, \log \left (x^{2} - x + 1\right ) + \frac{1}{6} \, \log \left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/162*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)) + 1/324*3^(1/3)*log(x^2 - 3^(1/3)*x + 3^(2/3)) - 1/162*3^(1/3)*lo
g(x + 3^(1/3)) + 1/45*(10*x^3 - 3)/x^5 - 1/12*log(x^2 - x + 1) + 1/6*log(x + 1)

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Fricas [A]  time = 0.270312, size = 230, normalized size = 1.83 \[ -\frac{9^{\frac{2}{3}} \sqrt{3}{\left (5 \, \sqrt{3} \left (-1\right )^{\frac{1}{3}} x^{5} \log \left (9^{\frac{2}{3}} x^{2} + 3 \cdot 9^{\frac{1}{3}} \left (-1\right )^{\frac{1}{3}} x + 9 \, \left (-1\right )^{\frac{2}{3}}\right ) + 45 \cdot 9^{\frac{1}{3}} \sqrt{3} x^{5} \log \left (x^{2} - x + 1\right ) - 10 \, \sqrt{3} \left (-1\right )^{\frac{1}{3}} x^{5} \log \left (9^{\frac{1}{3}} x - 3 \, \left (-1\right )^{\frac{1}{3}}\right ) - 90 \cdot 9^{\frac{1}{3}} \sqrt{3} x^{5} \log \left (x + 1\right ) + 30 \, \left (-1\right )^{\frac{1}{3}} x^{5} \arctan \left (-\frac{1}{9} \, \left (-1\right )^{\frac{2}{3}}{\left (2 \cdot 9^{\frac{1}{3}} \sqrt{3} x + 3 \, \sqrt{3} \left (-1\right )^{\frac{1}{3}}\right )}\right ) - 270 \cdot 9^{\frac{1}{3}} x^{5} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) - 12 \cdot 9^{\frac{1}{3}} \sqrt{3}{\left (10 \, x^{3} - 3\right )}\right )}}{14580 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/14580*9^(2/3)*sqrt(3)*(5*sqrt(3)*(-1)^(1/3)*x^5*log(9^(2/3)*x^2 + 3*9^(1/3)*(
-1)^(1/3)*x + 9*(-1)^(2/3)) + 45*9^(1/3)*sqrt(3)*x^5*log(x^2 - x + 1) - 10*sqrt(
3)*(-1)^(1/3)*x^5*log(9^(1/3)*x - 3*(-1)^(1/3)) - 90*9^(1/3)*sqrt(3)*x^5*log(x +
1) + 30*(-1)^(1/3)*x^5*arctan(-1/9*(-1)^(2/3)*(2*9^(1/3)*sqrt(3)*x + 3*sqrt(3)*
(-1)^(1/3))) - 270*9^(1/3)*x^5*arctan(1/3*sqrt(3)*(2*x - 1)) - 12*9^(1/3)*sqrt(3
)*(10*x^3 - 3))/x^5

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Sympy [A]  time = 4.36458, size = 136, normalized size = 1.08 \[ \frac{\log{\left (x + 1 \right )}}{6} + \left (- \frac{1}{12} + \frac{\sqrt{3} i}{12}\right ) \log{\left (x + \frac{88573}{6562} - \frac{88573 \sqrt{3} i}{6562} + \frac{119042784 \left (- \frac{1}{12} + \frac{\sqrt{3} i}{12}\right )^{4}}{3281} \right )} + \left (- \frac{1}{12} - \frac{\sqrt{3} i}{12}\right ) \log{\left (x + \frac{88573}{6562} + \frac{119042784 \left (- \frac{1}{12} - \frac{\sqrt{3} i}{12}\right )^{4}}{3281} + \frac{88573 \sqrt{3} i}{6562} \right )} + \operatorname{RootSum}{\left (1417176 t^{3} + 1, \left ( t \mapsto t \log{\left (\frac{119042784 t^{4}}{3281} - \frac{531438 t}{3281} + x \right )} \right )\right )} + \frac{10 x^{3} - 3}{45 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(x + 1)/6 + (-1/12 + sqrt(3)*I/12)*log(x + 88573/6562 - 88573*sqrt(3)*I/6562
+ 119042784*(-1/12 + sqrt(3)*I/12)**4/3281) + (-1/12 - sqrt(3)*I/12)*log(x + 885
73/6562 + 119042784*(-1/12 - sqrt(3)*I/12)**4/3281 + 88573*sqrt(3)*I/6562) + Roo
tSum(1417176*_t**3 + 1, Lambda(_t, _t*log(119042784*_t**4/3281 - 531438*_t/3281
+ x))) + (10*x**3 - 3)/(45*x**5)

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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^6),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError