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

Optimal. Leaf size=186 \[ -\frac{(-1)^{5/18} \left (3 \log \left (\sqrt [9]{-1}-x\right )+\log (2)\right )}{9 \sqrt{3}}+\frac{(-1)^{13/18} \log \left (-\sqrt [3]{2} \left (x+(-1)^{8/9}\right )\right )}{3 \sqrt{3}}-\frac{(-1)^{13/18} \log \left (-2^{2/3} \left (\left ((-1)^{8/9}-x\right ) x+(-1)^{7/9}\right )\right )}{6 \sqrt{3}}+\frac{(-1)^{5/18} \log \left (2^{2/3} \left (x \left (x+\sqrt [9]{-1}\right )+(-1)^{2/9}\right )\right )}{6 \sqrt{3}}-\frac{1}{3} (-1)^{13/18} \tan ^{-1}\left (\frac{2 \sqrt [9]{-1} x+1}{\sqrt{3}}\right )+\frac{1}{3} (-1)^{5/18} \tan ^{-1}\left (\frac{1-2 (-1)^{8/9} x}{\sqrt{3}}\right ) \]

[Out]

-((-1)^(13/18)*ArcTan[(1 + 2*(-1)^(1/9)*x)/Sqrt[3]])/3 + ((-1)^(5/18)*ArcTan[(1
- 2*(-1)^(8/9)*x)/Sqrt[3]])/3 - ((-1)^(5/18)*(Log[2] + 3*Log[(-1)^(1/9) - x]))/(
9*Sqrt[3]) + ((-1)^(13/18)*Log[-(2^(1/3)*((-1)^(8/9) + x))])/(3*Sqrt[3]) - ((-1)
^(13/18)*Log[-(2^(2/3)*((-1)^(7/9) + ((-1)^(8/9) - x)*x))])/(6*Sqrt[3]) + ((-1)^
(5/18)*Log[2^(2/3)*((-1)^(2/9) + x*((-1)^(1/9) + x))])/(6*Sqrt[3])

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Rubi [C]  time = 0.572612, antiderivative size = 375, normalized size of antiderivative = 2.02, number of steps used = 13, number of rules used = 7, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.583 \[ -\frac{i \log \left (2^{2/3} x^2+\sqrt [3]{2 \left (1-i \sqrt{3}\right )} x+\left (1-i \sqrt{3}\right )^{2/3}\right )}{3 \sqrt [3]{2} \sqrt{3} \left (1-i \sqrt{3}\right )^{2/3}}+\frac{i \log \left (2^{2/3} x^2+\sqrt [3]{2 \left (1+i \sqrt{3}\right )} x+\left (1+i \sqrt{3}\right )^{2/3}\right )}{3 \sqrt [3]{2} \sqrt{3} \left (1+i \sqrt{3}\right )^{2/3}}+\frac{i \log \left (-\sqrt [3]{2} x+\sqrt [3]{1-i \sqrt{3}}\right )}{3 \sqrt{3} \left (\frac{1}{2} \left (1-i \sqrt{3}\right )\right )^{2/3}}-\frac{i \log \left (-\sqrt [3]{2} x+\sqrt [3]{1+i \sqrt{3}}\right )}{3 \sqrt{3} \left (\frac{1}{2} \left (1+i \sqrt{3}\right )\right )^{2/3}}-\frac{i \tan ^{-1}\left (\frac{1+\frac{2 x}{\sqrt [3]{\frac{1}{2} \left (1-i \sqrt{3}\right )}}}{\sqrt{3}}\right )}{3 \left (\frac{1}{2} \left (1-i \sqrt{3}\right )\right )^{2/3}}+\frac{i \tan ^{-1}\left (\frac{1+\frac{2 x}{\sqrt [3]{\frac{1}{2} \left (1+i \sqrt{3}\right )}}}{\sqrt{3}}\right )}{3 \left (\frac{1}{2} \left (1+i \sqrt{3}\right )\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]  Int[(1 - x^3 + x^6)^(-1),x]

[Out]

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

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Rubi in Sympy [A]  time = 88.5601, size = 335, normalized size = 1.8 \[ \frac{2^{\frac{2}{3}} \sqrt{3} i \log{\left (\sqrt [3]{2} x - \sqrt [3]{1 - \sqrt{3} i} \right )}}{9 \left (1 - \sqrt{3} i\right )^{\frac{2}{3}}} - \frac{2^{\frac{2}{3}} \sqrt{3} i \log{\left (\sqrt [3]{2} x - \sqrt [3]{1 + \sqrt{3} i} \right )}}{9 \left (1 + \sqrt{3} i\right )^{\frac{2}{3}}} - \frac{2^{\frac{2}{3}} \sqrt{3} i \log{\left (x^{2} + \frac{2^{\frac{2}{3}} x \sqrt [3]{1 - \sqrt{3} i}}{2} + \frac{\sqrt [3]{2} \left (1 - \sqrt{3} i\right )^{\frac{2}{3}}}{2} \right )}}{18 \left (1 - \sqrt{3} i\right )^{\frac{2}{3}}} + \frac{2^{\frac{2}{3}} \sqrt{3} i \log{\left (x^{2} + \frac{2^{\frac{2}{3}} x \sqrt [3]{1 + \sqrt{3} i}}{2} + \frac{\sqrt [3]{2} \left (1 + \sqrt{3} i\right )^{\frac{2}{3}}}{2} \right )}}{18 \left (1 + \sqrt{3} i\right )^{\frac{2}{3}}} - \frac{2^{\frac{2}{3}} i \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 \sqrt [3]{2} x}{3 \sqrt [3]{1 - \sqrt{3} i}} + \frac{1}{3}\right ) \right )}}{3 \left (1 - \sqrt{3} i\right )^{\frac{2}{3}}} + \frac{2^{\frac{2}{3}} i \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 \sqrt [3]{2} x}{3 \sqrt [3]{1 + \sqrt{3} i}} + \frac{1}{3}\right ) \right )}}{3 \left (1 + \sqrt{3} i\right )^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2**(2/3)*sqrt(3)*I*log(2**(1/3)*x - (1 - sqrt(3)*I)**(1/3))/(9*(1 - sqrt(3)*I)**
(2/3)) - 2**(2/3)*sqrt(3)*I*log(2**(1/3)*x - (1 + sqrt(3)*I)**(1/3))/(9*(1 + sqr
t(3)*I)**(2/3)) - 2**(2/3)*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)/(18*(1 - sqrt(3)*I)**(2/3)) + 2**(2/3)*
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)/(18*(1 + sqrt(3)*I)**(2/3)) - 2**(2/3)*I*atan(sqrt(3)*(2*2**(1/3)*
x/(3*(1 - sqrt(3)*I)**(1/3)) + 1/3))/(3*(1 - sqrt(3)*I)**(2/3)) + 2**(2/3)*I*ata
n(sqrt(3)*(2*2**(1/3)*x/(3*(1 + sqrt(3)*I)**(1/3)) + 1/3))/(3*(1 + sqrt(3)*I)**(
2/3))

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Mathematica [C]  time = 0.0141045, size = 42, normalized size = 0.23 \[ \frac{1}{3} \text{RootSum}\left [\text{$\#$1}^6-\text{$\#$1}^3+1\&,\frac{\log (x-\text{$\#$1})}{2 \text{$\#$1}^5-\text{$\#$1}^2}\&\right ] \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.006, size = 37, normalized size = 0.2 \[{\frac{1}{3}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{6}-{{\it \_Z}}^{3}+1 \right ) }{\frac{\ln \left ( x-{\it \_R} \right ) }{2\,{{\it \_R}}^{5}-{{\it \_R}}^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x^6-x^3+1),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/(x^6 - x^3 + 1), x)

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Fricas [A]  time = 0.28018, size = 876, normalized size = 4.71 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/18*sqrt(3)*(4*(sqrt(3)*cos(2/3*arctan(1/(sqrt(3) + 2))) - sin(2/3*arctan(1/(sq
rt(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(-sq
rt(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)*cos(2/3*arctan(1/(sqrt(3) + 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)))
)) + 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/(sqrt(3) + 2)))^2) - (sqrt(3)*sin(2/3*arctan(1/(sqrt(3
) + 2))) + cos(2/3*arctan(1/(sqrt(3) + 2))))*log(-sqrt(3)*x*cos(2/3*arctan(1/(sq
rt(3) + 2))) + x^2 + cos(2/3*arctan(1/(sqrt(3) + 2)))^2 - x*sin(2/3*arctan(1/(sq
rt(3) + 2))) + sin(2/3*arctan(1/(sqrt(3) + 2)))^2) + (sqrt(3)*sin(2/3*arctan(1/(
sqrt(3) + 2))) - cos(2/3*arctan(1/(sqrt(3) + 2))))*log(x^2 + cos(2/3*arctan(1/(s
qrt(3) + 2)))^2 + 2*x*sin(2/3*arctan(1/(sqrt(3) + 2))) + sin(2/3*arctan(1/(sqrt(
3) + 2)))^2) - 8*arctan((sqrt(3)*sin(2/3*arctan(1/(sqrt(3) + 2))) + cos(2/3*arct
an(1/(sqrt(3) + 2))))/(sqrt(3)*cos(2/3*arctan(1/(sqrt(3) + 2))) + 2*x + 2*sqrt(s
qrt(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))))

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Sympy [A]  time = 0.467334, size = 20, normalized size = 0.11 \[ \operatorname{RootSum}{\left (19683 t^{6} - 243 t^{3} + 1, \left ( t \mapsto t \log{\left (729 t^{4} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x**6-x**3+1),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.291177, size = 849, normalized size = 4.56 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x^6 - x^3 + 1),x, 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)*si
n(4/9*pi)^4 + 4*cos(4/9*pi)^3*sin(4/9*pi) - 4*cos(4/9*pi)*sin(4/9*pi)^3 - sqrt(3
)*cos(4/9*pi) - sin(4/9*pi))*arctan(-((sqrt(3)*i + 1)*cos(4/9*pi) - 2*x)/((sqrt(
3)*i + 1)*sin(4/9*pi))) - 1/9*(sqrt(3)*cos(2/9*pi)^4 - 6*sqrt(3)*cos(2/9*pi)^2*s
in(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*p
i)*sin(2/9*pi)^3 - sqrt(3)*cos(2/9*pi) - sin(2/9*pi))*arctan(-((sqrt(3)*i + 1)*c
os(2/9*pi) - 2*x)/((sqrt(3)*i + 1)*sin(2/9*pi))) - 1/9*(sqrt(3)*cos(1/9*pi)^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 + sqrt(3)*cos(1/9*pi) - sin(1/9*pi))*a
rctan(((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 - 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*(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 - 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*(4*s
qrt(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 + sqrt(3)*sin(1/9*pi) + c
os(1/9*pi))*ln((sqrt(3)*i*cos(1/9*pi) + cos(1/9*pi))*x + x^2 + 1)