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

Optimal. Leaf size=418 \[ -\frac{1}{2 x^2}+\frac{\left (3-i \sqrt{3}\right ) \log \left (2^{2/3} x^2+\sqrt [3]{2 \left (1-i \sqrt{3}\right )} x+\left (1-i \sqrt{3}\right )^{2/3}\right )}{18 \sqrt [3]{2} \left (1-i \sqrt{3}\right )^{2/3}}+\frac{\left (3+i \sqrt{3}\right ) \log \left (2^{2/3} x^2+\sqrt [3]{2 \left (1+i \sqrt{3}\right )} x+\left (1+i \sqrt{3}\right )^{2/3}\right )}{18 \sqrt [3]{2} \left (1+i \sqrt{3}\right )^{2/3}}-\frac{\left (3-i \sqrt{3}\right ) \log \left (-\sqrt [3]{2} x+\sqrt [3]{1-i \sqrt{3}}\right )}{9 \sqrt [3]{2} \left (1-i \sqrt{3}\right )^{2/3}}-\frac{\left (3+i \sqrt{3}\right ) \log \left (-\sqrt [3]{2} x+\sqrt [3]{1+i \sqrt{3}}\right )}{9 \sqrt [3]{2} \left (1+i \sqrt{3}\right )^{2/3}}-\frac{\left (-\sqrt{3}+i\right ) \tan ^{-1}\left (\frac{1+\frac{2 x}{\sqrt [3]{\frac{1}{2} \left (1-i \sqrt{3}\right )}}}{\sqrt{3}}\right )}{3 \sqrt [3]{2} \left (1-i \sqrt{3}\right )^{2/3}}+\frac{\left (\sqrt{3}+i\right ) \tan ^{-1}\left (\frac{1+\frac{2 x}{\sqrt [3]{\frac{1}{2} \left (1+i \sqrt{3}\right )}}}{\sqrt{3}}\right )}{3 \sqrt [3]{2} \left (1+i \sqrt{3}\right )^{2/3}} \]

[Out]

-1/(2*x^2) - ((I - Sqrt[3])*ArcTan[(1 + (2*x)/((1 - I*Sqrt[3])/2)^(1/3))/Sqrt[3]
])/(3*2^(1/3)*(1 - I*Sqrt[3])^(2/3)) + ((I + Sqrt[3])*ArcTan[(1 + (2*x)/((1 + I*
Sqrt[3])/2)^(1/3))/Sqrt[3]])/(3*2^(1/3)*(1 + I*Sqrt[3])^(2/3)) - ((3 - I*Sqrt[3]
)*Log[(1 - I*Sqrt[3])^(1/3) - 2^(1/3)*x])/(9*2^(1/3)*(1 - I*Sqrt[3])^(2/3)) - ((
3 + I*Sqrt[3])*Log[(1 + I*Sqrt[3])^(1/3) - 2^(1/3)*x])/(9*2^(1/3)*(1 + I*Sqrt[3]
)^(2/3)) + ((3 - I*Sqrt[3])*Log[(1 - I*Sqrt[3])^(2/3) + (2*(1 - I*Sqrt[3]))^(1/3
)*x + 2^(2/3)*x^2])/(18*2^(1/3)*(1 - I*Sqrt[3])^(2/3)) + ((3 + I*Sqrt[3])*Log[(1
+ I*Sqrt[3])^(2/3) + (2*(1 + I*Sqrt[3]))^(1/3)*x + 2^(2/3)*x^2])/(18*2^(1/3)*(1
+ I*Sqrt[3])^(2/3))

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Rubi [A]  time = 0.754704, antiderivative size = 418, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 8, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5 \[ -\frac{1}{2 x^2}+\frac{\left (3-i \sqrt{3}\right ) \log \left (2^{2/3} x^2+\sqrt [3]{2 \left (1-i \sqrt{3}\right )} x+\left (1-i \sqrt{3}\right )^{2/3}\right )}{18 \sqrt [3]{2} \left (1-i \sqrt{3}\right )^{2/3}}+\frac{\left (3+i \sqrt{3}\right ) \log \left (2^{2/3} x^2+\sqrt [3]{2 \left (1+i \sqrt{3}\right )} x+\left (1+i \sqrt{3}\right )^{2/3}\right )}{18 \sqrt [3]{2} \left (1+i \sqrt{3}\right )^{2/3}}-\frac{\left (3-i \sqrt{3}\right ) \log \left (-\sqrt [3]{2} x+\sqrt [3]{1-i \sqrt{3}}\right )}{9 \sqrt [3]{2} \left (1-i \sqrt{3}\right )^{2/3}}-\frac{\left (3+i \sqrt{3}\right ) \log \left (-\sqrt [3]{2} x+\sqrt [3]{1+i \sqrt{3}}\right )}{9 \sqrt [3]{2} \left (1+i \sqrt{3}\right )^{2/3}}-\frac{\left (-\sqrt{3}+i\right ) \tan ^{-1}\left (\frac{1+\frac{2 x}{\sqrt [3]{\frac{1}{2} \left (1-i \sqrt{3}\right )}}}{\sqrt{3}}\right )}{3 \sqrt [3]{2} \left (1-i \sqrt{3}\right )^{2/3}}+\frac{\left (\sqrt{3}+i\right ) \tan ^{-1}\left (\frac{1+\frac{2 x}{\sqrt [3]{\frac{1}{2} \left (1+i \sqrt{3}\right )}}}{\sqrt{3}}\right )}{3 \sqrt [3]{2} \left (1+i \sqrt{3}\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^3*(1 - x^3 + x^6)),x]

[Out]

-1/(2*x^2) - ((I - Sqrt[3])*ArcTan[(1 + (2*x)/((1 - I*Sqrt[3])/2)^(1/3))/Sqrt[3]
])/(3*2^(1/3)*(1 - I*Sqrt[3])^(2/3)) + ((I + Sqrt[3])*ArcTan[(1 + (2*x)/((1 + I*
Sqrt[3])/2)^(1/3))/Sqrt[3]])/(3*2^(1/3)*(1 + I*Sqrt[3])^(2/3)) - ((3 - I*Sqrt[3]
)*Log[(1 - I*Sqrt[3])^(1/3) - 2^(1/3)*x])/(9*2^(1/3)*(1 - I*Sqrt[3])^(2/3)) - ((
3 + I*Sqrt[3])*Log[(1 + I*Sqrt[3])^(1/3) - 2^(1/3)*x])/(9*2^(1/3)*(1 + I*Sqrt[3]
)^(2/3)) + ((3 - I*Sqrt[3])*Log[(1 - I*Sqrt[3])^(2/3) + (2*(1 - I*Sqrt[3]))^(1/3
)*x + 2^(2/3)*x^2])/(18*2^(1/3)*(1 - I*Sqrt[3])^(2/3)) + ((3 + I*Sqrt[3])*Log[(1
+ I*Sqrt[3])^(2/3) + (2*(1 + I*Sqrt[3]))^(1/3)*x + 2^(2/3)*x^2])/(18*2^(1/3)*(1
+ I*Sqrt[3])^(2/3))

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Rubi in Sympy [A]  time = 115.871, size = 359, normalized size = 0.86 \[ - \frac{2^{\frac{2}{3}} \left (3 - \sqrt{3} i\right ) \log{\left (\sqrt [3]{2} x - \sqrt [3]{1 - \sqrt{3} i} \right )}}{18 \left (1 - \sqrt{3} i\right )^{\frac{2}{3}}} - \frac{2^{\frac{2}{3}} \left (3 + \sqrt{3} i\right ) \log{\left (\sqrt [3]{2} x - \sqrt [3]{1 + \sqrt{3} i} \right )}}{18 \left (1 + \sqrt{3} i\right )^{\frac{2}{3}}} + \frac{2^{\frac{2}{3}} \left (3 - \sqrt{3} i\right ) \log{\left (x^{2} + \frac{2^{\frac{2}{3}} x \sqrt [3]{1 - \sqrt{3} i}}{2} + \frac{\sqrt [3]{2} \left (1 - \sqrt{3} i\right )^{\frac{2}{3}}}{2} \right )}}{36 \left (1 - \sqrt{3} i\right )^{\frac{2}{3}}} + \frac{2^{\frac{2}{3}} \left (3 + \sqrt{3} i\right ) \log{\left (x^{2} + \frac{2^{\frac{2}{3}} x \sqrt [3]{1 + \sqrt{3} i}}{2} + \frac{\sqrt [3]{2} \left (1 + \sqrt{3} i\right )^{\frac{2}{3}}}{2} \right )}}{36 \left (1 + \sqrt{3} i\right )^{\frac{2}{3}}} + \frac{2^{\frac{2}{3}} \left (\sqrt{3} - i\right ) \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 \sqrt [3]{2} x}{3 \sqrt [3]{1 - \sqrt{3} i}} + \frac{1}{3}\right ) \right )}}{6 \left (1 - \sqrt{3} i\right )^{\frac{2}{3}}} + \frac{2^{\frac{2}{3}} \left (\sqrt{3} + i\right ) \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 \sqrt [3]{2} x}{3 \sqrt [3]{1 + \sqrt{3} i}} + \frac{1}{3}\right ) \right )}}{6 \left (1 + \sqrt{3} i\right )^{\frac{2}{3}}} - \frac{1}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2**(2/3)*(3 - sqrt(3)*I)*log(2**(1/3)*x - (1 - sqrt(3)*I)**(1/3))/(18*(1 - sqrt
(3)*I)**(2/3)) - 2**(2/3)*(3 + sqrt(3)*I)*log(2**(1/3)*x - (1 + sqrt(3)*I)**(1/3
))/(18*(1 + sqrt(3)*I)**(2/3)) + 2**(2/3)*(3 - sqrt(3)*I)*log(x**2 + 2**(2/3)*x*
(1 - sqrt(3)*I)**(1/3)/2 + 2**(1/3)*(1 - sqrt(3)*I)**(2/3)/2)/(36*(1 - sqrt(3)*I
)**(2/3)) + 2**(2/3)*(3 + sqrt(3)*I)*log(x**2 + 2**(2/3)*x*(1 + sqrt(3)*I)**(1/3
)/2 + 2**(1/3)*(1 + sqrt(3)*I)**(2/3)/2)/(36*(1 + sqrt(3)*I)**(2/3)) + 2**(2/3)*
(sqrt(3) - I)*atan(sqrt(3)*(2*2**(1/3)*x/(3*(1 - sqrt(3)*I)**(1/3)) + 1/3))/(6*(
1 - sqrt(3)*I)**(2/3)) + 2**(2/3)*(sqrt(3) + I)*atan(sqrt(3)*(2*2**(1/3)*x/(3*(1
+ sqrt(3)*I)**(1/3)) + 1/3))/(6*(1 + sqrt(3)*I)**(2/3)) - 1/(2*x**2)

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Mathematica [C]  time = 0.0205042, size = 65, normalized size = 0.16 \[ -\frac{1}{3} \text{RootSum}\left [\text{$\#$1}^6-\text{$\#$1}^3+1\&,\frac{\text{$\#$1}^3 \log (x-\text{$\#$1})-\log (x-\text{$\#$1})}{2 \text{$\#$1}^5-\text{$\#$1}^2}\&\right ]-\frac{1}{2 x^2} \]

Antiderivative was successfully verified.

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

[Out]

-1/(2*x^2) - RootSum[1 - #1^3 + #1^6 & , (-Log[x - #1] + Log[x - #1]*#1^3)/(-#1^
2 + 2*#1^5) & ]/3

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Maple [C]  time = 0.012, size = 50, normalized size = 0.1 \[{\frac{1}{3}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{6}-{{\it \_Z}}^{3}+1 \right ) }{\frac{ \left ( -{{\it \_R}}^{3}+1 \right ) \ln \left ( x-{\it \_R} \right ) }{2\,{{\it \_R}}^{5}-{{\it \_R}}^{2}}}}-{\frac{1}{2\,{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(x^6-x^3+1),x)

[Out]

1/3*sum((-_R^3+1)/(2*_R^5-_R^2)*ln(x-_R),_R=RootOf(_Z^6-_Z^3+1))-1/2/x^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\frac{1}{2 \, x^{2}} - \int \frac{x^{3} - 1}{x^{6} - x^{3} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2/x^2 - integrate((x^3 - 1)/(x^6 - x^3 + 1), x)

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Fricas [A]  time = 0.277702, size = 927, normalized size = 2.22 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/18*sqrt(3)*(2*x^2*cos(2/3*arctan(1/(sqrt(3) - 2)))*log(sqrt(3)*x*cos(2/3*arcta
n(1/(sqrt(3) - 2))) + x^2 + cos(2/3*arctan(1/(sqrt(3) - 2)))^2 + x*sin(2/3*arcta
n(1/(sqrt(3) - 2))) + sin(2/3*arctan(1/(sqrt(3) - 2)))^2) - 8*x^2*arctan((sqrt(3
)*sin(2/3*arctan(1/(sqrt(3) - 2))) - cos(2/3*arctan(1/(sqrt(3) - 2))))/(sqrt(3)*
cos(2/3*arctan(1/(sqrt(3) - 2))) + 2*x + 2*sqrt(sqrt(3)*x*cos(2/3*arctan(1/(sqrt
(3) - 2))) + x^2 + cos(2/3*arctan(1/(sqrt(3) - 2)))^2 + x*sin(2/3*arctan(1/(sqrt
(3) - 2))) + sin(2/3*arctan(1/(sqrt(3) - 2)))^2) + sin(2/3*arctan(1/(sqrt(3) - 2
)))))*sin(2/3*arctan(1/(sqrt(3) - 2))) - 4*(sqrt(3)*x^2*cos(2/3*arctan(1/(sqrt(3
) - 2))) + x^2*sin(2/3*arctan(1/(sqrt(3) - 2))))*arctan(-(sqrt(3)*sin(2/3*arctan
(1/(sqrt(3) - 2))) + cos(2/3*arctan(1/(sqrt(3) - 2))))/(sqrt(3)*cos(2/3*arctan(1
/(sqrt(3) - 2))) - 2*x - 2*sqrt(-sqrt(3)*x*cos(2/3*arctan(1/(sqrt(3) - 2))) + x^
2 + cos(2/3*arctan(1/(sqrt(3) - 2)))^2 + x*sin(2/3*arctan(1/(sqrt(3) - 2))) + si
n(2/3*arctan(1/(sqrt(3) - 2)))^2) - sin(2/3*arctan(1/(sqrt(3) - 2))))) - 4*(sqrt
(3)*x^2*cos(2/3*arctan(1/(sqrt(3) - 2))) - x^2*sin(2/3*arctan(1/(sqrt(3) - 2))))
*arctan(cos(2/3*arctan(1/(sqrt(3) - 2)))/(x + sqrt(x^2 + cos(2/3*arctan(1/(sqrt(
3) - 2)))^2 - 2*x*sin(2/3*arctan(1/(sqrt(3) - 2))) + sin(2/3*arctan(1/(sqrt(3) -
2)))^2) - sin(2/3*arctan(1/(sqrt(3) - 2))))) + (sqrt(3)*x^2*sin(2/3*arctan(1/(s
qrt(3) - 2))) - x^2*cos(2/3*arctan(1/(sqrt(3) - 2))))*log(-sqrt(3)*x*cos(2/3*arc
tan(1/(sqrt(3) - 2))) + x^2 + cos(2/3*arctan(1/(sqrt(3) - 2)))^2 + x*sin(2/3*arc
tan(1/(sqrt(3) - 2))) + sin(2/3*arctan(1/(sqrt(3) - 2)))^2) - (sqrt(3)*x^2*sin(2
/3*arctan(1/(sqrt(3) - 2))) + x^2*cos(2/3*arctan(1/(sqrt(3) - 2))))*log(x^2 + co
s(2/3*arctan(1/(sqrt(3) - 2)))^2 - 2*x*sin(2/3*arctan(1/(sqrt(3) - 2))) + sin(2/
3*arctan(1/(sqrt(3) - 2)))^2) - 3*sqrt(3))/x^2

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Sympy [A]  time = 0.567722, size = 31, normalized size = 0.07 \[ \operatorname{RootSum}{\left (19683 t^{6} + 243 t^{3} + 1, \left ( t \mapsto t \log{\left (729 t^{4} + 9 t + x \right )} \right )\right )} - \frac{1}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

RootSum(19683*_t**6 + 243*_t**3 + 1, Lambda(_t, _t*log(729*_t**4 + 9*_t + x))) -
1/(2*x**2)

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GIAC/XCAS [A]  time = 0.323989, size = 867, normalized size = 2.07 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/9*(sqrt(3)*cos(4/9*pi)^4 - 6*sqrt(3)*cos(4/9*pi)^2*sin(4/9*pi)^2 + sqrt(3)*sin
(4/9*pi)^4 + 4*cos(4/9*pi)^3*sin(4/9*pi) - 4*cos(4/9*pi)*sin(4/9*pi)^3 + 2*sqrt(
3)*cos(4/9*pi) + 2*sin(4/9*pi))*arctan(-((sqrt(3)*i + 1)*cos(4/9*pi) - 2*x)/((sq
rt(3)*i + 1)*sin(4/9*pi))) + 1/9*(sqrt(3)*cos(2/9*pi)^4 - 6*sqrt(3)*cos(2/9*pi)^
2*sin(2/9*pi)^2 + sqrt(3)*sin(2/9*pi)^4 + 4*cos(2/9*pi)^3*sin(2/9*pi) - 4*cos(2/
9*pi)*sin(2/9*pi)^3 + 2*sqrt(3)*cos(2/9*pi) + 2*sin(2/9*pi))*arctan(-((sqrt(3)*i
+ 1)*cos(2/9*pi) - 2*x)/((sqrt(3)*i + 1)*sin(2/9*pi))) + 1/9*(sqrt(3)*cos(1/9*p
i)^4 - 6*sqrt(3)*cos(1/9*pi)^2*sin(1/9*pi)^2 + sqrt(3)*sin(1/9*pi)^4 - 4*cos(1/9
*pi)^3*sin(1/9*pi) + 4*cos(1/9*pi)*sin(1/9*pi)^3 - 2*sqrt(3)*cos(1/9*pi) + 2*sin
(1/9*pi))*arctan(((sqrt(3)*i + 1)*cos(1/9*pi) + 2*x)/((sqrt(3)*i + 1)*sin(1/9*pi
))) + 1/18*(4*sqrt(3)*cos(4/9*pi)^3*sin(4/9*pi) - 4*sqrt(3)*cos(4/9*pi)*sin(4/9*
pi)^3 - cos(4/9*pi)^4 + 6*cos(4/9*pi)^2*sin(4/9*pi)^2 - sin(4/9*pi)^4 + 2*sqrt(3
)*sin(4/9*pi) - 2*cos(4/9*pi))*ln(-(sqrt(3)*i*cos(4/9*pi) + cos(4/9*pi))*x + x^2
+ 1) + 1/18*(4*sqrt(3)*cos(2/9*pi)^3*sin(2/9*pi) - 4*sqrt(3)*cos(2/9*pi)*sin(2/
9*pi)^3 - cos(2/9*pi)^4 + 6*cos(2/9*pi)^2*sin(2/9*pi)^2 - sin(2/9*pi)^4 + 2*sqrt
(3)*sin(2/9*pi) - 2*cos(2/9*pi))*ln(-(sqrt(3)*i*cos(2/9*pi) + cos(2/9*pi))*x + x
^2 + 1) - 1/18*(4*sqrt(3)*cos(1/9*pi)^3*sin(1/9*pi) - 4*sqrt(3)*cos(1/9*pi)*sin(
1/9*pi)^3 + cos(1/9*pi)^4 - 6*cos(1/9*pi)^2*sin(1/9*pi)^2 + sin(1/9*pi)^4 - 2*sq
rt(3)*sin(1/9*pi) - 2*cos(1/9*pi))*ln((sqrt(3)*i*cos(1/9*pi) + cos(1/9*pi))*x +
x^2 + 1) - 1/2/x^2