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

Optimal. Leaf size=382 \[ -\frac{x^2}{2}-\frac{i \log \left (2^{2/3} x^2+\sqrt [3]{2 \left (1-i \sqrt{3}\right )} x+\left (1-i \sqrt{3}\right )^{2/3}\right )}{3\ 2^{2/3} \sqrt{3} \sqrt [3]{1-i \sqrt{3}}}+\frac{i \log \left (2^{2/3} x^2+\sqrt [3]{2 \left (1+i \sqrt{3}\right )} x+\left (1+i \sqrt{3}\right )^{2/3}\right )}{3\ 2^{2/3} \sqrt{3} \sqrt [3]{1+i \sqrt{3}}}+\frac{i \log \left (-\sqrt [3]{2} x+\sqrt [3]{1-i \sqrt{3}}\right )}{3 \sqrt{3} \sqrt [3]{\frac{1}{2} \left (1-i \sqrt{3}\right )}}-\frac{i \log \left (-\sqrt [3]{2} x+\sqrt [3]{1+i \sqrt{3}}\right )}{3 \sqrt{3} \sqrt [3]{\frac{1}{2} \left (1+i \sqrt{3}\right )}}+\frac{i \tan ^{-1}\left (\frac{1+\frac{2 x}{\sqrt [3]{\frac{1}{2} \left (1-i \sqrt{3}\right )}}}{\sqrt{3}}\right )}{3 \sqrt [3]{\frac{1}{2} \left (1-i \sqrt{3}\right )}}-\frac{i \tan ^{-1}\left (\frac{1+\frac{2 x}{\sqrt [3]{\frac{1}{2} \left (1+i \sqrt{3}\right )}}}{\sqrt{3}}\right )}{3 \sqrt [3]{\frac{1}{2} \left (1+i \sqrt{3}\right )}} \]

[Out]

-x^2/2 + ((I/3)*ArcTan[(1 + (2*x)/((1 - I*Sqrt[3])/2)^(1/3))/Sqrt[3]])/((1 - I*S
qrt[3])/2)^(1/3) - ((I/3)*ArcTan[(1 + (2*x)/((1 + I*Sqrt[3])/2)^(1/3))/Sqrt[3]])
/((1 + I*Sqrt[3])/2)^(1/3) + ((I/3)*Log[(1 - I*Sqrt[3])^(1/3) - 2^(1/3)*x])/(Sqr
t[3]*((1 - I*Sqrt[3])/2)^(1/3)) - ((I/3)*Log[(1 + I*Sqrt[3])^(1/3) - 2^(1/3)*x])
/(Sqrt[3]*((1 + I*Sqrt[3])/2)^(1/3)) - ((I/3)*Log[(1 - I*Sqrt[3])^(2/3) + (2*(1
- I*Sqrt[3]))^(1/3)*x + 2^(2/3)*x^2])/(2^(2/3)*Sqrt[3]*(1 - I*Sqrt[3])^(1/3)) +
((I/3)*Log[(1 + I*Sqrt[3])^(2/3) + (2*(1 + I*Sqrt[3]))^(1/3)*x + 2^(2/3)*x^2])/(
2^(2/3)*Sqrt[3]*(1 + I*Sqrt[3])^(1/3))

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Rubi [A]  time = 0.777977, antiderivative size = 382, 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 \[ -\frac{x^2}{2}-\frac{i \log \left (2^{2/3} x^2+\sqrt [3]{2 \left (1-i \sqrt{3}\right )} x+\left (1-i \sqrt{3}\right )^{2/3}\right )}{3\ 2^{2/3} \sqrt{3} \sqrt [3]{1-i \sqrt{3}}}+\frac{i \log \left (2^{2/3} x^2+\sqrt [3]{2 \left (1+i \sqrt{3}\right )} x+\left (1+i \sqrt{3}\right )^{2/3}\right )}{3\ 2^{2/3} \sqrt{3} \sqrt [3]{1+i \sqrt{3}}}+\frac{i \log \left (-\sqrt [3]{2} x+\sqrt [3]{1-i \sqrt{3}}\right )}{3 \sqrt{3} \sqrt [3]{\frac{1}{2} \left (1-i \sqrt{3}\right )}}-\frac{i \log \left (-\sqrt [3]{2} x+\sqrt [3]{1+i \sqrt{3}}\right )}{3 \sqrt{3} \sqrt [3]{\frac{1}{2} \left (1+i \sqrt{3}\right )}}+\frac{i \tan ^{-1}\left (\frac{1+\frac{2 x}{\sqrt [3]{\frac{1}{2} \left (1-i \sqrt{3}\right )}}}{\sqrt{3}}\right )}{3 \sqrt [3]{\frac{1}{2} \left (1-i \sqrt{3}\right )}}-\frac{i \tan ^{-1}\left (\frac{1+\frac{2 x}{\sqrt [3]{\frac{1}{2} \left (1+i \sqrt{3}\right )}}}{\sqrt{3}}\right )}{3 \sqrt [3]{\frac{1}{2} \left (1+i \sqrt{3}\right )}} \]

Antiderivative was successfully verified.

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

[Out]

-x^2/2 + ((I/3)*ArcTan[(1 + (2*x)/((1 - I*Sqrt[3])/2)^(1/3))/Sqrt[3]])/((1 - I*S
qrt[3])/2)^(1/3) - ((I/3)*ArcTan[(1 + (2*x)/((1 + I*Sqrt[3])/2)^(1/3))/Sqrt[3]])
/((1 + I*Sqrt[3])/2)^(1/3) + ((I/3)*Log[(1 - I*Sqrt[3])^(1/3) - 2^(1/3)*x])/(Sqr
t[3]*((1 - I*Sqrt[3])/2)^(1/3)) - ((I/3)*Log[(1 + I*Sqrt[3])^(1/3) - 2^(1/3)*x])
/(Sqrt[3]*((1 + I*Sqrt[3])/2)^(1/3)) - ((I/3)*Log[(1 - I*Sqrt[3])^(2/3) + (2*(1
- I*Sqrt[3]))^(1/3)*x + 2^(2/3)*x^2])/(2^(2/3)*Sqrt[3]*(1 - I*Sqrt[3])^(1/3)) +
((I/3)*Log[(1 + I*Sqrt[3])^(2/3) + (2*(1 + I*Sqrt[3]))^(1/3)*x + 2^(2/3)*x^2])/(
2^(2/3)*Sqrt[3]*(1 + I*Sqrt[3])^(1/3))

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Rubi in Sympy [A]  time = 125.338, size = 340, normalized size = 0.89 \[ - \frac{x^{2}}{2} + \frac{\sqrt [3]{2} \sqrt{3} i \log{\left (\sqrt [3]{2} x - \sqrt [3]{1 - \sqrt{3} i} \right )}}{9 \sqrt [3]{1 - \sqrt{3} i}} - \frac{\sqrt [3]{2} \sqrt{3} i \log{\left (\sqrt [3]{2} x - \sqrt [3]{1 + \sqrt{3} i} \right )}}{9 \sqrt [3]{1 + \sqrt{3} i}} - \frac{\sqrt [3]{2} \sqrt{3} i \log{\left (x^{2} + \frac{2^{\frac{2}{3}} x \sqrt [3]{1 - \sqrt{3} i}}{2} + \frac{\sqrt [3]{2} \left (1 - \sqrt{3} i\right )^{\frac{2}{3}}}{2} \right )}}{18 \sqrt [3]{1 - \sqrt{3} i}} + \frac{\sqrt [3]{2} \sqrt{3} i \log{\left (x^{2} + \frac{2^{\frac{2}{3}} x \sqrt [3]{1 + \sqrt{3} i}}{2} + \frac{\sqrt [3]{2} \left (1 + \sqrt{3} i\right )^{\frac{2}{3}}}{2} \right )}}{18 \sqrt [3]{1 + \sqrt{3} i}} + \frac{\sqrt [3]{2} i \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 \sqrt [3]{2} x}{3 \sqrt [3]{1 - \sqrt{3} i}} + \frac{1}{3}\right ) \right )}}{3 \sqrt [3]{1 - \sqrt{3} i}} - \frac{\sqrt [3]{2} i \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 \sqrt [3]{2} x}{3 \sqrt [3]{1 + \sqrt{3} i}} + \frac{1}{3}\right ) \right )}}{3 \sqrt [3]{1 + \sqrt{3} i}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x**2/2 + 2**(1/3)*sqrt(3)*I*log(2**(1/3)*x - (1 - sqrt(3)*I)**(1/3))/(9*(1 - sq
rt(3)*I)**(1/3)) - 2**(1/3)*sqrt(3)*I*log(2**(1/3)*x - (1 + sqrt(3)*I)**(1/3))/(
9*(1 + sqrt(3)*I)**(1/3)) - 2**(1/3)*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)**(1/3)) +
2**(1/3)*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)**(1/3)) + 2**(1/3)*I*atan(sqrt(3)*(2
*2**(1/3)*x/(3*(1 - sqrt(3)*I)**(1/3)) + 1/3))/(3*(1 - sqrt(3)*I)**(1/3)) - 2**(
1/3)*I*atan(sqrt(3)*(2*2**(1/3)*x/(3*(1 + sqrt(3)*I)**(1/3)) + 1/3))/(3*(1 + sqr
t(3)*I)**(1/3))

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Mathematica [C]  time = 0.0202578, size = 48, normalized size = 0.13 \[ \frac{1}{3} \text{RootSum}\left [\text{$\#$1}^6-\text{$\#$1}^3+1\&,\frac{\log (x-\text{$\#$1})}{2 \text{$\#$1}^4-\text{$\#$1}}\&\right ]-\frac{x^2}{2} \]

Antiderivative was successfully verified.

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

[Out]

-x^2/2 + RootSum[1 - #1^3 + #1^6 & , Log[x - #1]/(-#1 + 2*#1^4) & ]/3

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Maple [C]  time = 0.008, size = 44, normalized size = 0.1 \[ -{\frac{{x}^{2}}{2}}+{\frac{1}{3}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{6}-{{\it \_Z}}^{3}+1 \right ) }{\frac{{\it \_R}\,\ln \left ( x-{\it \_R} \right ) }{2\,{{\it \_R}}^{5}-{{\it \_R}}^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4*(-x^3+1)/(x^6-x^3+1),x)

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.282208, size = 1447, normalized size = 3.79 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.492278, size = 32, normalized size = 0.08 \[ - \frac{x^{2}}{2} - \operatorname{RootSum}{\left (19683 t^{6} + 243 t^{3} + 1, \left ( t \mapsto t \log{\left (- 6561 t^{5} - 27 t^{2} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.286148, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done