3.30 \(\int \frac{1-x^3}{x^2 \left (1-x^3+x^6\right )} \, dx\)

Optimal. Leaf size=416 \[ \frac{\left (3+i \sqrt{3}\right ) \log \left (2^{2/3} x^2+\sqrt [3]{2 \left (1-i \sqrt{3}\right )} x+\left (1-i \sqrt{3}\right )^{2/3}\right )}{18\ 2^{2/3} \sqrt [3]{1-i \sqrt{3}}}+\frac{\left (3-i \sqrt{3}\right ) \log \left (2^{2/3} x^2+\sqrt [3]{2 \left (1+i \sqrt{3}\right )} x+\left (1+i \sqrt{3}\right )^{2/3}\right )}{18\ 2^{2/3} \sqrt [3]{1+i \sqrt{3}}}-\frac{1}{x}-\frac{\left (3+i \sqrt{3}\right ) \log \left (-\sqrt [3]{2} x+\sqrt [3]{1-i \sqrt{3}}\right )}{9\ 2^{2/3} \sqrt [3]{1-i \sqrt{3}}}-\frac{\left (3-i \sqrt{3}\right ) \log \left (-\sqrt [3]{2} x+\sqrt [3]{1+i \sqrt{3}}\right )}{9\ 2^{2/3} \sqrt [3]{1+i \sqrt{3}}}-\frac{\left (\sqrt{3}+i\right ) \tan ^{-1}\left (\frac{1+\frac{2 x}{\sqrt [3]{\frac{1}{2} \left (1-i \sqrt{3}\right )}}}{\sqrt{3}}\right )}{3\ 2^{2/3} \sqrt [3]{1-i \sqrt{3}}}+\frac{\left (-\sqrt{3}+i\right ) \tan ^{-1}\left (\frac{1+\frac{2 x}{\sqrt [3]{\frac{1}{2} \left (1+i \sqrt{3}\right )}}}{\sqrt{3}}\right )}{3\ 2^{2/3} \sqrt [3]{1+i \sqrt{3}}} \]

[Out]

-x^(-1) - ((I + Sqrt[3])*ArcTan[(1 + (2*x)/((1 - I*Sqrt[3])/2)^(1/3))/Sqrt[3]])/
(3*2^(2/3)*(1 - I*Sqrt[3])^(1/3)) + ((I - Sqrt[3])*ArcTan[(1 + (2*x)/((1 + I*Sqr
t[3])/2)^(1/3))/Sqrt[3]])/(3*2^(2/3)*(1 + I*Sqrt[3])^(1/3)) - ((3 + I*Sqrt[3])*L
og[(1 - I*Sqrt[3])^(1/3) - 2^(1/3)*x])/(9*2^(2/3)*(1 - I*Sqrt[3])^(1/3)) - ((3 -
I*Sqrt[3])*Log[(1 + I*Sqrt[3])^(1/3) - 2^(1/3)*x])/(9*2^(2/3)*(1 + I*Sqrt[3])^(
1/3)) + ((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^(2/3)*(1 - I*Sqrt[3])^(1/3)) + ((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^(2/3)*(1 +
I*Sqrt[3])^(1/3))

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Rubi [A]  time = 0.697243, antiderivative size = 416, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 8, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.348 \[ \frac{\left (3+i \sqrt{3}\right ) \log \left (2^{2/3} x^2+\sqrt [3]{2 \left (1-i \sqrt{3}\right )} x+\left (1-i \sqrt{3}\right )^{2/3}\right )}{18\ 2^{2/3} \sqrt [3]{1-i \sqrt{3}}}+\frac{\left (3-i \sqrt{3}\right ) \log \left (2^{2/3} x^2+\sqrt [3]{2 \left (1+i \sqrt{3}\right )} x+\left (1+i \sqrt{3}\right )^{2/3}\right )}{18\ 2^{2/3} \sqrt [3]{1+i \sqrt{3}}}-\frac{1}{x}-\frac{\left (3+i \sqrt{3}\right ) \log \left (-\sqrt [3]{2} x+\sqrt [3]{1-i \sqrt{3}}\right )}{9\ 2^{2/3} \sqrt [3]{1-i \sqrt{3}}}-\frac{\left (3-i \sqrt{3}\right ) \log \left (-\sqrt [3]{2} x+\sqrt [3]{1+i \sqrt{3}}\right )}{9\ 2^{2/3} \sqrt [3]{1+i \sqrt{3}}}-\frac{\left (\sqrt{3}+i\right ) \tan ^{-1}\left (\frac{1+\frac{2 x}{\sqrt [3]{\frac{1}{2} \left (1-i \sqrt{3}\right )}}}{\sqrt{3}}\right )}{3\ 2^{2/3} \sqrt [3]{1-i \sqrt{3}}}+\frac{\left (-\sqrt{3}+i\right ) \tan ^{-1}\left (\frac{1+\frac{2 x}{\sqrt [3]{\frac{1}{2} \left (1+i \sqrt{3}\right )}}}{\sqrt{3}}\right )}{3\ 2^{2/3} \sqrt [3]{1+i \sqrt{3}}} \]

Antiderivative was successfully verified.

[In]  Int[(1 - x^3)/(x^2*(1 - x^3 + x^6)),x]

[Out]

-x^(-1) - ((I + Sqrt[3])*ArcTan[(1 + (2*x)/((1 - I*Sqrt[3])/2)^(1/3))/Sqrt[3]])/
(3*2^(2/3)*(1 - I*Sqrt[3])^(1/3)) + ((I - Sqrt[3])*ArcTan[(1 + (2*x)/((1 + I*Sqr
t[3])/2)^(1/3))/Sqrt[3]])/(3*2^(2/3)*(1 + I*Sqrt[3])^(1/3)) - ((3 + I*Sqrt[3])*L
og[(1 - I*Sqrt[3])^(1/3) - 2^(1/3)*x])/(9*2^(2/3)*(1 - I*Sqrt[3])^(1/3)) - ((3 -
I*Sqrt[3])*Log[(1 + I*Sqrt[3])^(1/3) - 2^(1/3)*x])/(9*2^(2/3)*(1 + I*Sqrt[3])^(
1/3)) + ((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^(2/3)*(1 - I*Sqrt[3])^(1/3)) + ((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^(2/3)*(1 +
I*Sqrt[3])^(1/3))

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Rubi in Sympy [A]  time = 108.539, size = 338, normalized size = 0.81 \[ - \frac{\sqrt [3]{2} \sqrt{3} i \left (1 - \sqrt{3} i\right )^{\frac{2}{3}} \log{\left (\sqrt [3]{2} x - \sqrt [3]{1 - \sqrt{3} i} \right )}}{18} + \frac{\sqrt [3]{2} \sqrt{3} i \left (1 + \sqrt{3} i\right )^{\frac{2}{3}} \log{\left (\sqrt [3]{2} x - \sqrt [3]{1 + \sqrt{3} i} \right )}}{18} + \frac{\sqrt [3]{2} \sqrt{3} i \left (1 - \sqrt{3} i\right )^{\frac{2}{3}} \log{\left (x^{2} + \frac{2^{\frac{2}{3}} x \sqrt [3]{1 - \sqrt{3} i}}{2} + \frac{\sqrt [3]{2} \left (1 - \sqrt{3} i\right )^{\frac{2}{3}}}{2} \right )}}{36} - \frac{\sqrt [3]{2} \sqrt{3} i \left (1 + \sqrt{3} i\right )^{\frac{2}{3}} \log{\left (x^{2} + \frac{2^{\frac{2}{3}} x \sqrt [3]{1 + \sqrt{3} i}}{2} + \frac{\sqrt [3]{2} \left (1 + \sqrt{3} i\right )^{\frac{2}{3}}}{2} \right )}}{36} - \frac{\sqrt [3]{2} i \left (1 - \sqrt{3} i\right )^{\frac{2}{3}} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 \sqrt [3]{2} x}{3 \sqrt [3]{1 - \sqrt{3} i}} + \frac{1}{3}\right ) \right )}}{6} + \frac{\sqrt [3]{2} i \left (1 + \sqrt{3} i\right )^{\frac{2}{3}} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 \sqrt [3]{2} x}{3 \sqrt [3]{1 + \sqrt{3} i}} + \frac{1}{3}\right ) \right )}}{6} - \frac{1}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-x**3+1)/x**2/(x**6-x**3+1),x)

[Out]

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

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Mathematica [C]  time = 0.0203759, size = 47, normalized size = 0.11 \[ -\frac{1}{3} \text{RootSum}\left [\text{$\#$1}^6-\text{$\#$1}^3+1\&,\frac{\text{$\#$1}^2 \log (x-\text{$\#$1})}{2 \text{$\#$1}^3-1}\&\right ]-\frac{1}{x} \]

Antiderivative was successfully verified.

[In]  Integrate[(1 - x^3)/(x^2*(1 - x^3 + x^6)),x]

[Out]

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

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Maple [C]  time = 0.01, size = 46, normalized size = 0.1 \[ -{\frac{1}{3}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{6}-{{\it \_Z}}^{3}+1 \right ) }{\frac{{{\it \_R}}^{4}\ln \left ( x-{\it \_R} \right ) }{2\,{{\it \_R}}^{5}-{{\it \_R}}^{2}}}}-{x}^{-1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-x^3+1)/x^2/(x^6-x^3+1),x)

[Out]

-1/3*sum(_R^4/(2*_R^5-_R^2)*ln(x-_R),_R=RootOf(_Z^6-_Z^3+1))-1/x

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\frac{1}{x} - \int \frac{x^{4}}{x^{6} - x^{3} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x^3 - 1)/((x^6 - x^3 + 1)*x^2),x, algorithm="maxima")

[Out]

-1/x - integrate(x^4/(x^6 - x^3 + 1), x)

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Fricas [A]  time = 0.282379, size = 1463, normalized size = 3.52 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x^3 - 1)/((x^6 - x^3 + 1)*x^2),x, algorithm="fricas")

[Out]

1/18*sqrt(3)*(2*x*cos(2/3*arctan(1/(sqrt(3) + 2)))*log(cos(2/3*arctan(1/(sqrt(3)
+ 2)))^4 + sin(2/3*arctan(1/(sqrt(3) + 2)))^4 + 2*sqrt(3)*x*cos(2/3*arctan(1/(s
qrt(3) + 2)))*sin(2/3*arctan(1/(sqrt(3) + 2))) + x*cos(2/3*arctan(1/(sqrt(3) + 2
)))^2 + (2*cos(2/3*arctan(1/(sqrt(3) + 2)))^2 - x)*sin(2/3*arctan(1/(sqrt(3) + 2
)))^2 + x^2) - 8*x*arctan((2*sqrt(3)*cos(2/3*arctan(1/(sqrt(3) + 2)))*sin(2/3*ar
ctan(1/(sqrt(3) + 2))) - 3*cos(2/3*arctan(1/(sqrt(3) + 2)))^2 + 3*sin(2/3*arctan
(1/(sqrt(3) + 2)))^2)/(sqrt(3)*cos(2/3*arctan(1/(sqrt(3) + 2)))^2 - sqrt(3)*sin(
2/3*arctan(1/(sqrt(3) + 2)))^2 + 6*cos(2/3*arctan(1/(sqrt(3) + 2)))*sin(2/3*arct
an(1/(sqrt(3) + 2))) + 2*sqrt(3)*x + 2*sqrt(3)*sqrt(cos(2/3*arctan(1/(sqrt(3) +
2)))^4 + sin(2/3*arctan(1/(sqrt(3) + 2)))^4 + 2*sqrt(3)*x*cos(2/3*arctan(1/(sqrt
(3) + 2)))*sin(2/3*arctan(1/(sqrt(3) + 2))) + x*cos(2/3*arctan(1/(sqrt(3) + 2)))
^2 + (2*cos(2/3*arctan(1/(sqrt(3) + 2)))^2 - x)*sin(2/3*arctan(1/(sqrt(3) + 2)))
^2 + x^2)))*sin(2/3*arctan(1/(sqrt(3) + 2))) + 4*(sqrt(3)*x*cos(2/3*arctan(1/(sq
rt(3) + 2))) - x*sin(2/3*arctan(1/(sqrt(3) + 2))))*arctan(-2*cos(2/3*arctan(1/(s
qrt(3) + 2)))*sin(2/3*arctan(1/(sqrt(3) + 2)))/(cos(2/3*arctan(1/(sqrt(3) + 2)))
^2 - sin(2/3*arctan(1/(sqrt(3) + 2)))^2 - x - sqrt(cos(2/3*arctan(1/(sqrt(3) + 2
)))^4 + sin(2/3*arctan(1/(sqrt(3) + 2)))^4 - 2*x*cos(2/3*arctan(1/(sqrt(3) + 2))
)^2 + 2*(cos(2/3*arctan(1/(sqrt(3) + 2)))^2 + x)*sin(2/3*arctan(1/(sqrt(3) + 2))
)^2 + x^2))) - 4*(sqrt(3)*x*cos(2/3*arctan(1/(sqrt(3) + 2))) + x*sin(2/3*arctan(
1/(sqrt(3) + 2))))*arctan(-(2*sqrt(3)*cos(2/3*arctan(1/(sqrt(3) + 2)))*sin(2/3*a
rctan(1/(sqrt(3) + 2))) + 3*cos(2/3*arctan(1/(sqrt(3) + 2)))^2 - 3*sin(2/3*arcta
n(1/(sqrt(3) + 2)))^2)/(sqrt(3)*cos(2/3*arctan(1/(sqrt(3) + 2)))^2 - sqrt(3)*sin
(2/3*arctan(1/(sqrt(3) + 2)))^2 - 6*cos(2/3*arctan(1/(sqrt(3) + 2)))*sin(2/3*arc
tan(1/(sqrt(3) + 2))) + 2*sqrt(3)*x + 2*sqrt(3)*sqrt(cos(2/3*arctan(1/(sqrt(3) +
2)))^4 + sin(2/3*arctan(1/(sqrt(3) + 2)))^4 - 2*sqrt(3)*x*cos(2/3*arctan(1/(sqr
t(3) + 2)))*sin(2/3*arctan(1/(sqrt(3) + 2))) + x*cos(2/3*arctan(1/(sqrt(3) + 2))
)^2 + (2*cos(2/3*arctan(1/(sqrt(3) + 2)))^2 - x)*sin(2/3*arctan(1/(sqrt(3) + 2))
)^2 + x^2))) + (sqrt(3)*x*sin(2/3*arctan(1/(sqrt(3) + 2))) - x*cos(2/3*arctan(1/
(sqrt(3) + 2))))*log(cos(2/3*arctan(1/(sqrt(3) + 2)))^4 + sin(2/3*arctan(1/(sqrt
(3) + 2)))^4 - 2*sqrt(3)*x*cos(2/3*arctan(1/(sqrt(3) + 2)))*sin(2/3*arctan(1/(sq
rt(3) + 2))) + x*cos(2/3*arctan(1/(sqrt(3) + 2)))^2 + (2*cos(2/3*arctan(1/(sqrt(
3) + 2)))^2 - x)*sin(2/3*arctan(1/(sqrt(3) + 2)))^2 + x^2) - (sqrt(3)*x*sin(2/3*
arctan(1/(sqrt(3) + 2))) + x*cos(2/3*arctan(1/(sqrt(3) + 2))))*log(cos(2/3*arcta
n(1/(sqrt(3) + 2)))^4 + sin(2/3*arctan(1/(sqrt(3) + 2)))^4 - 2*x*cos(2/3*arctan(
1/(sqrt(3) + 2)))^2 + 2*(cos(2/3*arctan(1/(sqrt(3) + 2)))^2 + x)*sin(2/3*arctan(
1/(sqrt(3) + 2)))^2 + x^2) - 6*sqrt(3))/x

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Sympy [A]  time = 0.548836, size = 31, normalized size = 0.07 \[ - \operatorname{RootSum}{\left (19683 t^{6} + 243 t^{3} + 1, \left ( t \mapsto t \log{\left (6561 t^{5} + 54 t^{2} + x \right )} \right )\right )} - \frac{1}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-x**3+1)/x**2/(x**6-x**3+1),x)

[Out]

-RootSum(19683*_t**6 + 243*_t**3 + 1, Lambda(_t, _t*log(6561*_t**5 + 54*_t**2 +
x))) - 1/x

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GIAC/XCAS [A]  time = 0.294574, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x^3 - 1)/((x^6 - x^3 + 1)*x^2),x, algorithm="giac")

[Out]

Done