3.168 \(\int \frac{1}{x^5 \left (3+4 x^3+x^6\right )} \, dx\)

Optimal. Leaf size=126 \[ -\frac{1}{12 x^4}+\frac{1}{12} \log \left (x^2-x+1\right )-\frac{\log \left (x^2-\sqrt [3]{3} x+3^{2/3}\right )}{108 \sqrt [3]{3}}+\frac{4}{9 x}-\frac{1}{6} \log (x+1)+\frac{\log \left (x+\sqrt [3]{3}\right )}{54 \sqrt [3]{3}}-\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{2 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{3}-2 x}{3^{5/6}}\right )}{18\ 3^{5/6}} \]

[Out]

-1/(12*x^4) + 4/(9*x) - ArcTan[(1 - 2*x)/Sqrt[3]]/(2*Sqrt[3]) + ArcTan[(3^(1/3)
- 2*x)/3^(5/6)]/(18*3^(5/6)) - Log[1 + x]/6 + Log[3^(1/3) + x]/(54*3^(1/3)) + Lo
g[1 - x + x^2]/12 - Log[3^(2/3) - 3^(1/3)*x + x^2]/(108*3^(1/3))

_______________________________________________________________________________________

Rubi [A]  time = 0.210512, 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 \[ -\frac{1}{12 x^4}+\frac{1}{12} \log \left (x^2-x+1\right )-\frac{\log \left (x^2-\sqrt [3]{3} x+3^{2/3}\right )}{108 \sqrt [3]{3}}+\frac{4}{9 x}-\frac{1}{6} \log (x+1)+\frac{\log \left (x+\sqrt [3]{3}\right )}{54 \sqrt [3]{3}}-\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{2 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{3}-2 x}{3^{5/6}}\right )}{18\ 3^{5/6}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^5*(3 + 4*x^3 + x^6)),x]

[Out]

-1/(12*x^4) + 4/(9*x) - ArcTan[(1 - 2*x)/Sqrt[3]]/(2*Sqrt[3]) + ArcTan[(3^(1/3)
- 2*x)/3^(5/6)]/(18*3^(5/6)) - Log[1 + x]/6 + Log[3^(1/3) + x]/(54*3^(1/3)) + Lo
g[1 - x + x^2]/12 - Log[3^(2/3) - 3^(1/3)*x + x^2]/(108*3^(1/3))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 34.4844, size = 116, normalized size = 0.92 \[ - \frac{\log{\left (x + 1 \right )}}{6} + \frac{3^{\frac{2}{3}} \log{\left (x + \sqrt [3]{3} \right )}}{162} + \frac{\log{\left (x^{2} - x + 1 \right )}}{12} - \frac{3^{\frac{2}{3}} \log{\left (x^{2} - \sqrt [3]{3} x + 3^{\frac{2}{3}} \right )}}{324} + \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} - \frac{1}{3}\right ) \right )}}{6} + \frac{\sqrt [6]{3} \operatorname{atan}{\left (\sqrt{3} \left (- \frac{2 \cdot 3^{\frac{2}{3}} x}{9} + \frac{1}{3}\right ) \right )}}{54} + \frac{4}{9 x} - \frac{1}{12 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**5/(x**6+4*x**3+3),x)

[Out]

-log(x + 1)/6 + 3**(2/3)*log(x + 3**(1/3))/162 + log(x**2 - x + 1)/12 - 3**(2/3)
*log(x**2 - 3**(1/3)*x + 3**(2/3))/324 + 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))/54 + 4/(9*x) - 1/(12*x**4)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0996501, size = 118, normalized size = 0.94 \[ \frac{1}{324} \left (-\frac{27}{x^4}+27 \log \left (x^2-x+1\right )-3^{2/3} \log \left (\sqrt [3]{3} x^2-3^{2/3} x+3\right )+\frac{144}{x}-54 \log (x+1)+2\ 3^{2/3} \log \left (3^{2/3} x+3\right )+6 \sqrt [6]{3} \tan ^{-1}\left (\frac{\sqrt [3]{3}-2 x}{3^{5/6}}\right )+54 \sqrt{3} \tan ^{-1}\left (\frac{2 x-1}{\sqrt{3}}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^5*(3 + 4*x^3 + x^6)),x]

[Out]

(-27/x^4 + 144/x + 6*3^(1/6)*ArcTan[(3^(1/3) - 2*x)/3^(5/6)] + 54*Sqrt[3]*ArcTan
[(-1 + 2*x)/Sqrt[3]] - 54*Log[1 + x] + 2*3^(2/3)*Log[3 + 3^(2/3)*x] + 27*Log[1 -
x + x^2] - 3^(2/3)*Log[3 - 3^(2/3)*x + 3^(1/3)*x^2])/324

_______________________________________________________________________________________

Maple [A]  time = 0.014, size = 94, normalized size = 0.8 \[{\frac{{3}^{{\frac{2}{3}}}\ln \left ( \sqrt [3]{3}+x \right ) }{162}}-{\frac{{3}^{{\frac{2}{3}}}\ln \left ({3}^{{\frac{2}{3}}}-\sqrt [3]{3}x+{x}^{2} \right ) }{324}}-{\frac{\sqrt [6]{3}}{54}\arctan \left ({\frac{\sqrt{3}}{3} \left ({\frac{2\,{3}^{2/3}x}{3}}-1 \right ) } \right ) }-{\frac{\ln \left ( 1+x \right ) }{6}}-{\frac{1}{12\,{x}^{4}}}+{\frac{4}{9\,x}}+{\frac{\ln \left ({x}^{2}-x+1 \right ) }{12}}+{\frac{\sqrt{3}}{6}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^5/(x^6+4*x^3+3),x)

[Out]

1/162*3^(2/3)*ln(3^(1/3)+x)-1/324*3^(2/3)*ln(3^(2/3)-3^(1/3)*x+x^2)-1/54*3^(1/6)
*arctan(1/3*3^(1/2)*(2/3*3^(2/3)*x-1))-1/6*ln(1+x)-1/12/x^4+4/9/x+1/12*ln(x^2-x+
1)+1/6*3^(1/2)*arctan(1/3*(2*x-1)*3^(1/2))

_______________________________________________________________________________________

Maxima [A]  time = 0.866539, size = 130, normalized size = 1.03 \[ -\frac{1}{324} \cdot 3^{\frac{2}{3}} \log \left (x^{2} - 3^{\frac{1}{3}} x + 3^{\frac{2}{3}}\right ) + \frac{1}{162} \cdot 3^{\frac{2}{3}} \log \left (x + 3^{\frac{1}{3}}\right ) + \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) - \frac{1}{54} \cdot 3^{\frac{1}{6}} \arctan \left (\frac{1}{3} \cdot 3^{\frac{1}{6}}{\left (2 \, x - 3^{\frac{1}{3}}\right )}\right ) + \frac{16 \, x^{3} - 3}{36 \, x^{4}} + \frac{1}{12} \, \log \left (x^{2} - x + 1\right ) - \frac{1}{6} \, \log \left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^6 + 4*x^3 + 3)*x^5),x, algorithm="maxima")

[Out]

-1/324*3^(2/3)*log(x^2 - 3^(1/3)*x + 3^(2/3)) + 1/162*3^(2/3)*log(x + 3^(1/3)) +
1/6*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - 1)) - 1/54*3^(1/6)*arctan(1/3*3^(1/6)*(2*
x - 3^(1/3))) + 1/36*(16*x^3 - 3)/x^4 + 1/12*log(x^2 - x + 1) - 1/6*log(x + 1)

_______________________________________________________________________________________

Fricas [A]  time = 0.263886, size = 171, normalized size = 1.36 \[ \frac{3^{\frac{1}{6}}{\left (9 \cdot 3^{\frac{5}{6}} x^{4} \log \left (x^{2} - x + 1\right ) - 18 \cdot 3^{\frac{5}{6}} x^{4} \log \left (x + 1\right ) - \sqrt{3} x^{4} \log \left (3^{\frac{1}{3}} x^{2} - 3^{\frac{2}{3}} x + 3\right ) + 2 \, \sqrt{3} x^{4} \log \left (3^{\frac{2}{3}} x + 3\right ) + 54 \cdot 3^{\frac{1}{3}} x^{4} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) - 6 \, x^{4} \arctan \left (\frac{2}{3} \cdot 3^{\frac{1}{6}} x - \frac{1}{3} \, \sqrt{3}\right ) + 3 \cdot 3^{\frac{5}{6}}{\left (16 \, x^{3} - 3\right )}\right )}}{324 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^6 + 4*x^3 + 3)*x^5),x, algorithm="fricas")

[Out]

1/324*3^(1/6)*(9*3^(5/6)*x^4*log(x^2 - x + 1) - 18*3^(5/6)*x^4*log(x + 1) - sqrt
(3)*x^4*log(3^(1/3)*x^2 - 3^(2/3)*x + 3) + 2*sqrt(3)*x^4*log(3^(2/3)*x + 3) + 54
*3^(1/3)*x^4*arctan(1/3*sqrt(3)*(2*x - 1)) - 6*x^4*arctan(2/3*3^(1/6)*x - 1/3*sq
rt(3)) + 3*3^(5/6)*(16*x^3 - 3))/x^4

_______________________________________________________________________________________

Sympy [A]  time = 4.61768, size = 141, normalized size = 1.12 \[ - \frac{\log{\left (x + 1 \right )}}{6} + \left (\frac{1}{12} - \frac{\sqrt{3} i}{12}\right ) \log{\left (x + \frac{4782978 \left (\frac{1}{12} - \frac{\sqrt{3} i}{12}\right )^{2}}{547} + \frac{1028869776 \left (\frac{1}{12} - \frac{\sqrt{3} i}{12}\right )^{5}}{547} \right )} + \left (\frac{1}{12} + \frac{\sqrt{3} i}{12}\right ) \log{\left (x + \frac{1028869776 \left (\frac{1}{12} + \frac{\sqrt{3} i}{12}\right )^{5}}{547} + \frac{4782978 \left (\frac{1}{12} + \frac{\sqrt{3} i}{12}\right )^{2}}{547} \right )} + \operatorname{RootSum}{\left (472392 t^{3} - 1, \left ( t \mapsto t \log{\left (\frac{1028869776 t^{5}}{547} + \frac{4782978 t^{2}}{547} + x \right )} \right )\right )} + \frac{16 x^{3} - 3}{36 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**5/(x**6+4*x**3+3),x)

[Out]

-log(x + 1)/6 + (1/12 - sqrt(3)*I/12)*log(x + 4782978*(1/12 - sqrt(3)*I/12)**2/5
47 + 1028869776*(1/12 - sqrt(3)*I/12)**5/547) + (1/12 + sqrt(3)*I/12)*log(x + 10
28869776*(1/12 + sqrt(3)*I/12)**5/547 + 4782978*(1/12 + sqrt(3)*I/12)**2/547) +
RootSum(472392*_t**3 - 1, Lambda(_t, _t*log(1028869776*_t**5/547 + 4782978*_t**2
/547 + x))) + (16*x**3 - 3)/(36*x**4)

_______________________________________________________________________________________

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^5),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError