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

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 = 0.854118, antiderivative size = 418, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 9, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.391 \[ -\frac{1}{2 x^2}+\frac{\left (3+i \sqrt{3}\right ) \log \left (2^{2/3} x^2+\sqrt [3]{2 \left (1-i \sqrt{3}\right )} x+\left (1-i \sqrt{3}\right )^{2/3}\right )}{18 \sqrt [3]{2} \left (1-i \sqrt{3}\right )^{2/3}}+\frac{\left (3-i \sqrt{3}\right ) \log \left (2^{2/3} x^2+\sqrt [3]{2 \left (1+i \sqrt{3}\right )} x+\left (1+i \sqrt{3}\right )^{2/3}\right )}{18 \sqrt [3]{2} \left (1+i \sqrt{3}\right )^{2/3}}-\frac{\left (3+i \sqrt{3}\right ) \log \left (-\sqrt [3]{2} x+\sqrt [3]{1-i \sqrt{3}}\right )}{9 \sqrt [3]{2} \left (1-i \sqrt{3}\right )^{2/3}}-\frac{\left (3-i \sqrt{3}\right ) \log \left (-\sqrt [3]{2} x+\sqrt [3]{1+i \sqrt{3}}\right )}{9 \sqrt [3]{2} \left (1+i \sqrt{3}\right )^{2/3}}+\frac{\left (\sqrt{3}+i\right ) \tan ^{-1}\left (\frac{1+\frac{2 x}{\sqrt [3]{\frac{1}{2} \left (1-i \sqrt{3}\right )}}}{\sqrt{3}}\right )}{3 \sqrt [3]{2} \left (1-i \sqrt{3}\right )^{2/3}}-\frac{\left (-\sqrt{3}+i\right ) \tan ^{-1}\left (\frac{1+\frac{2 x}{\sqrt [3]{\frac{1}{2} \left (1+i \sqrt{3}\right )}}}{\sqrt{3}}\right )}{3 \sqrt [3]{2} \left (1+i \sqrt{3}\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]  Int[(1 - x^3)/(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 = 129.123, size = 342, normalized size = 0.82 \[ - \frac{2^{\frac{2}{3}} \sqrt{3} i \sqrt [3]{1 - \sqrt{3} i} \log{\left (\sqrt [3]{2} x - \sqrt [3]{1 - \sqrt{3} i} \right )}}{18} + \frac{2^{\frac{2}{3}} \sqrt{3} i \sqrt [3]{1 + \sqrt{3} i} \log{\left (\sqrt [3]{2} x - \sqrt [3]{1 + \sqrt{3} i} \right )}}{18} + \frac{2^{\frac{2}{3}} \sqrt{3} i \sqrt [3]{1 - \sqrt{3} i} \log{\left (x^{2} + \frac{2^{\frac{2}{3}} x \sqrt [3]{1 - \sqrt{3} i}}{2} + \frac{\sqrt [3]{2} \left (1 - \sqrt{3} i\right )^{\frac{2}{3}}}{2} \right )}}{36} - \frac{2^{\frac{2}{3}} \sqrt{3} i \sqrt [3]{1 + \sqrt{3} i} \log{\left (x^{2} + \frac{2^{\frac{2}{3}} x \sqrt [3]{1 + \sqrt{3} i}}{2} + \frac{\sqrt [3]{2} \left (1 + \sqrt{3} i\right )^{\frac{2}{3}}}{2} \right )}}{36} + \frac{2^{\frac{2}{3}} i \sqrt [3]{1 - \sqrt{3} i} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 \sqrt [3]{2} x}{3 \sqrt [3]{1 - \sqrt{3} i}} + \frac{1}{3}\right ) \right )}}{6} - \frac{2^{\frac{2}{3}} i \sqrt [3]{1 + \sqrt{3} i} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 \sqrt [3]{2} x}{3 \sqrt [3]{1 + \sqrt{3} i}} + \frac{1}{3}\right ) \right )}}{6} - \frac{1}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2**(2/3)*sqrt(3)*I*(1 - sqrt(3)*I)**(1/3)*log(2**(1/3)*x - (1 - sqrt(3)*I)**(1/
3))/18 + 2**(2/3)*sqrt(3)*I*(1 + sqrt(3)*I)**(1/3)*log(2**(1/3)*x - (1 + sqrt(3)
*I)**(1/3))/18 + 2**(2/3)*sqrt(3)*I*(1 - sqrt(3)*I)**(1/3)*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 - 2**(2/3)*sqr
t(3)*I*(1 + sqrt(3)*I)**(1/3)*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 + 2**(2/3)*I*(1 - sqrt(3)*I)**(1/3)*atan(sq
rt(3)*(2*2**(1/3)*x/(3*(1 - sqrt(3)*I)**(1/3)) + 1/3))/6 - 2**(2/3)*I*(1 + sqrt(
3)*I)**(1/3)*atan(sqrt(3)*(2*2**(1/3)*x/(3*(1 + sqrt(3)*I)**(1/3)) + 1/3))/6 - 1
/(2*x**2)

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Mathematica [C]  time = 0.0204511, size = 47, normalized size = 0.11 \[ -\frac{1}{3} \text{RootSum}\left [\text{$\#$1}^6-\text{$\#$1}^3+1\&,\frac{\text{$\#$1} \log (x-\text{$\#$1})}{2 \text{$\#$1}^3-1}\&\right ]-\frac{1}{2 x^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.009, size = 46, normalized size = 0.1 \[ -{\frac{1}{3}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{6}-{{\it \_Z}}^{3}+1 \right ) }{\frac{{{\it \_R}}^{3}\ln \left ( x-{\it \_R} \right ) }{2\,{{\it \_R}}^{5}-{{\it \_R}}^{2}}}}-{\frac{1}{2\,{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-x^3+1)/x^3/(x^6-x^3+1),x)

[Out]

-1/3*sum(_R^3/(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. \[ -\frac{1}{2 \, x^{2}} - \int \frac{x^{3}}{x^{6} - x^{3} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.279619, size = 930, normalized size = 2.22 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x^3 - 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(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/(sqr
t(3) + 2)))^2) + 8*x^2*arctan(cos(2/3*arctan(1/(sqrt(3) + 2)))/(x + sqrt(x^2 + c
os(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)))))*sin(2/3*arct
an(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))) + sin(2/3*arctan(1/(sqrt
(3) + 2)))^2) - sin(2/3*arctan(1/(sqrt(3) + 2))))) - 4*(sqrt(3)*x^2*cos(2/3*arct
an(1/(sqrt(3) + 2))) + x^2*sin(2/3*arctan(1/(sqrt(3) + 2))))*arctan(-(sqrt(3)*si
n(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)))
)) - (sqrt(3)*x^2*sin(2/3*arctan(1/(sqrt(3) + 2))) + x^2*cos(2/3*arctan(1/(sqrt(
3) + 2))))*log(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/(sq
rt(3) + 2)))^2) + (sqrt(3)*x^2*sin(2/3*arctan(1/(sqrt(3) + 2))) - x^2*cos(2/3*ar
ctan(1/(sqrt(3) + 2))))*log(-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) - 3*sqrt(3))/x^2

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Sympy [A]  time = 0.562082, size = 32, normalized size = 0.08 \[ - \operatorname{RootSum}{\left (19683 t^{6} + 243 t^{3} + 1, \left ( t \mapsto t \log{\left (- 1458 t^{4} - 9 t + x \right )} \right )\right )} - \frac{1}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-RootSum(19683*_t**6 + 243*_t**3 + 1, Lambda(_t, _t*log(-1458*_t**4 - 9*_t + x))
) - 1/(2*x**2)

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GIAC/XCAS [A]  time = 0.297666, size = 867, normalized size = 2.07 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/9*(2*sqrt(3)*cos(4/9*pi)^4 - 12*sqrt(3)*cos(4/9*pi)^2*sin(4/9*pi)^2 + 2*sqrt(3
)*sin(4/9*pi)^4 + 8*cos(4/9*pi)^3*sin(4/9*pi) - 8*cos(4/9*pi)*sin(4/9*pi)^3 + sq
rt(3)*cos(4/9*pi) + sin(4/9*pi))*arctan(-((sqrt(3)*i + 1)*cos(4/9*pi) - 2*x)/((s
qrt(3)*i + 1)*sin(4/9*pi))) + 1/9*(2*sqrt(3)*cos(2/9*pi)^4 - 12*sqrt(3)*cos(2/9*
pi)^2*sin(2/9*pi)^2 + 2*sqrt(3)*sin(2/9*pi)^4 + 8*cos(2/9*pi)^3*sin(2/9*pi) - 8*
cos(2/9*pi)*sin(2/9*pi)^3 + sqrt(3)*cos(2/9*pi) + 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*(2*sqrt(3)*cos(1
/9*pi)^4 - 12*sqrt(3)*cos(1/9*pi)^2*sin(1/9*pi)^2 + 2*sqrt(3)*sin(1/9*pi)^4 - 8*
cos(1/9*pi)^3*sin(1/9*pi) + 8*cos(1/9*pi)*sin(1/9*pi)^3 - sqrt(3)*cos(1/9*pi) +
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*(8*sqrt(3)*cos(4/9*pi)^3*sin(4/9*pi) - 8*sqrt(3)*cos(4/9*pi)*sin(4
/9*pi)^3 - 2*cos(4/9*pi)^4 + 12*cos(4/9*pi)^2*sin(4/9*pi)^2 - 2*sin(4/9*pi)^4 +
sqrt(3)*sin(4/9*pi) - cos(4/9*pi))*ln(-(sqrt(3)*i*cos(4/9*pi) + cos(4/9*pi))*x +
x^2 + 1) + 1/18*(8*sqrt(3)*cos(2/9*pi)^3*sin(2/9*pi) - 8*sqrt(3)*cos(2/9*pi)*si
n(2/9*pi)^3 - 2*cos(2/9*pi)^4 + 12*cos(2/9*pi)^2*sin(2/9*pi)^2 - 2*sin(2/9*pi)^4
+ sqrt(3)*sin(2/9*pi) - cos(2/9*pi))*ln(-(sqrt(3)*i*cos(2/9*pi) + cos(2/9*pi))*
x + x^2 + 1) - 1/18*(8*sqrt(3)*cos(1/9*pi)^3*sin(1/9*pi) - 8*sqrt(3)*cos(1/9*pi)
*sin(1/9*pi)^3 + 2*cos(1/9*pi)^4 - 12*cos(1/9*pi)^2*sin(1/9*pi)^2 + 2*sin(1/9*pi
)^4 - sqrt(3)*sin(1/9*pi) - cos(1/9*pi))*ln((sqrt(3)*i*cos(1/9*pi) + cos(1/9*pi)
)*x + x^2 + 1) - 1/2/x^2