3.46 \(\int \frac{a+b x+c x^2}{\left (1-x+x^2\right ) \sqrt [3]{1-x^3}} \, dx\)

Optimal. Leaf size=326 \[ \frac{(a-b-2 c) \left (-3 \log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )-2 \sqrt{3} \tan ^{-1}\left (\frac{2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt{3}}\right )\right )}{12 \sqrt [3]{2}}-\frac{(a-b-2 c) \left (2 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )-3 \log \left (\sqrt [3]{1-x^3}+\sqrt [3]{2} x\right )\right )}{12 \sqrt [3]{2}}+\frac{(a+b) \left (-3 \log \left (\sqrt [3]{1-x^3}-\sqrt [3]{2} (x-1)\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{2} (x-1)}{\sqrt [3]{1-x^3}}+1}{\sqrt{3}}\right )+\log \left (-3 x^3+6 x^2-6 x+3\right )\right )}{4 \sqrt [3]{2}}-\frac{1}{6} c \left (-2 \log \left (\frac{x}{\sqrt [3]{1-x^3}}+1\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 x}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )+\log \left (-\frac{x}{\sqrt [3]{1-x^3}}+\frac{x^2}{\left (1-x^3\right )^{2/3}}+1\right )\right ) \]

[Out]

-(c*(2*Sqrt[3]*ArcTan[(1 - (2*x)/(1 - x^3)^(1/3))/Sqrt[3]] + Log[1 + x^2/(1 - x^
3)^(2/3) - x/(1 - x^3)^(1/3)] - 2*Log[1 + x/(1 - x^3)^(1/3)]))/6 + ((a - b - 2*c
)*(-2*Sqrt[3]*ArcTan[(1 + 2^(2/3)*(1 - x^3)^(1/3))/Sqrt[3]] - 3*Log[2^(1/3) - (1
- x^3)^(1/3)]))/(12*2^(1/3)) + ((a + b)*(2*Sqrt[3]*ArcTan[(1 + (2*2^(1/3)*(-1 +
x))/(1 - x^3)^(1/3))/Sqrt[3]] + Log[3 - 6*x + 6*x^2 - 3*x^3] - 3*Log[-(2^(1/3)*
(-1 + x)) + (1 - x^3)^(1/3)]))/(4*2^(1/3)) - ((a - b - 2*c)*(2*Sqrt[3]*ArcTan[(1
- (2*2^(1/3)*x)/(1 - x^3)^(1/3))/Sqrt[3]] - 3*Log[2^(1/3)*x + (1 - x^3)^(1/3)])
)/(12*2^(1/3))

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Rubi [C]  time = 1.52529, antiderivative size = 576, normalized size of antiderivative = 1.77, number of steps used = 7, number of rules used = 3, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.094 \[ -\frac{\log \left (2\ 2^{2/3} \sqrt [3]{1-x^3}+2 x-i \sqrt{3}+1\right ) \left (3 i b-\sqrt{3} \left (2 a+b-i \sqrt{3} c-c\right )\right )}{4 \sqrt [3]{2} \left (\sqrt{3}+i\right )}-\frac{\log \left (2\ 2^{2/3} \sqrt [3]{1-x^3}+2 x+i \sqrt{3}+1\right ) \left (\sqrt{3} \left (2 a+b+i \sqrt{3} c-c\right )+3 i b\right )}{4 \sqrt [3]{2} \left (-\sqrt{3}+i\right )}-\frac{\tan ^{-1}\left (\frac{2-\frac{\sqrt [3]{2} \left (2 x-i \sqrt{3}+1\right )}{\sqrt [3]{1-x^3}}}{2 \sqrt{3}}\right ) \left (2 a-i \sqrt{3} b+b-\left (1+i \sqrt{3}\right ) c\right )}{2 \sqrt [3]{2} \left (\sqrt{3}+i\right )}+\frac{\tan ^{-1}\left (\frac{2-\frac{\sqrt [3]{2} \left (2 x+i \sqrt{3}+1\right )}{\sqrt [3]{1-x^3}}}{2 \sqrt{3}}\right ) \left (2 a+i \sqrt{3} b+b+i \sqrt{3} c-c\right )}{2 \sqrt [3]{2} \left (-\sqrt{3}+i\right )}+\frac{\log \left (-\left (-2 x-i \sqrt{3}+1\right )^2 \left (2 x-i \sqrt{3}+1\right )\right ) \left (3 i b-\sqrt{3} \left (2 a+b-i \sqrt{3} c-c\right )\right )}{12 \sqrt [3]{2} \left (\sqrt{3}+i\right )}+\frac{\log \left (-\left (-2 x+i \sqrt{3}+1\right )^2 \left (2 x+i \sqrt{3}+1\right )\right ) \left (\sqrt{3} \left (2 a+b+i \sqrt{3} c-c\right )+3 i b\right )}{12 \sqrt [3]{2} \left (-\sqrt{3}+i\right )}+\frac{1}{2} c \log \left (\sqrt [3]{1-x^3}+x\right )-\frac{c \tan ^{-1}\left (\frac{1-\frac{2 x}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

-((c*ArcTan[(1 - (2*x)/(1 - x^3)^(1/3))/Sqrt[3]])/Sqrt[3]) - ((2*a + b - I*Sqrt[
3]*b - (1 + I*Sqrt[3])*c)*ArcTan[(2 - (2^(1/3)*(1 - I*Sqrt[3] + 2*x))/(1 - x^3)^
(1/3))/(2*Sqrt[3])])/(2*2^(1/3)*(I + Sqrt[3])) + ((2*a + b + I*Sqrt[3]*b - c + I
*Sqrt[3]*c)*ArcTan[(2 - (2^(1/3)*(1 + I*Sqrt[3] + 2*x))/(1 - x^3)^(1/3))/(2*Sqrt
[3])])/(2*2^(1/3)*(I - Sqrt[3])) + (((3*I)*b - Sqrt[3]*(2*a + b - c - I*Sqrt[3]*
c))*Log[-((1 - I*Sqrt[3] - 2*x)^2*(1 - I*Sqrt[3] + 2*x))])/(12*2^(1/3)*(I + Sqrt
[3])) + (((3*I)*b + Sqrt[3]*(2*a + b - c + I*Sqrt[3]*c))*Log[-((1 + I*Sqrt[3] -
2*x)^2*(1 + I*Sqrt[3] + 2*x))])/(12*2^(1/3)*(I - Sqrt[3])) + (c*Log[x + (1 - x^3
)^(1/3)])/2 - (((3*I)*b - Sqrt[3]*(2*a + b - c - I*Sqrt[3]*c))*Log[1 - I*Sqrt[3]
+ 2*x + 2*2^(2/3)*(1 - x^3)^(1/3)])/(4*2^(1/3)*(I + Sqrt[3])) - (((3*I)*b + Sqr
t[3]*(2*a + b - c + I*Sqrt[3]*c))*Log[1 + I*Sqrt[3] + 2*x + 2*2^(2/3)*(1 - x^3)^
(1/3)])/(4*2^(1/3)*(I - Sqrt[3]))

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x+a)/(x**2-x+1)/(-x**3+1)**(1/3),x)

[Out]

Timed out

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Mathematica [A]  time = 0.165259, size = 0, normalized size = 0. \[ \int \frac{a+b x+c x^2}{\left (1-x+x^2\right ) \sqrt [3]{1-x^3}} \, dx \]

Verification is Not applicable to the result.

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

[Out]

Integrate[(a + b*x + c*x^2)/((1 - x + x^2)*(1 - x^3)^(1/3)), x]

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Maple [F]  time = 0.183, size = 0, normalized size = 0. \[ \int{\frac{c{x}^{2}+bx+a}{{x}^{2}-x+1}{\frac{1}{\sqrt [3]{-{x}^{3}+1}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x+a)/(x^2-x+1)/(-x^3+1)^(1/3),x)

[Out]

int((c*x^2+b*x+a)/(x^2-x+1)/(-x^3+1)^(1/3),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{c x^{2} + b x + a}{{\left (-x^{3} + 1\right )}^{\frac{1}{3}}{\left (x^{2} - x + 1\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((c*x^2 + b*x + a)/((-x^3 + 1)^(1/3)*(x^2 - x + 1)), x)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{a + b x + c x^{2}}{\sqrt [3]{- \left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (x^{2} - x + 1\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+b*x+a)/(x**2-x+1)/(-x**3+1)**(1/3),x)

[Out]

Integral((a + b*x + c*x**2)/((-(x - 1)*(x**2 + x + 1))**(1/3)*(x**2 - x + 1)), x
)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{c x^{2} + b x + a}{{\left (-x^{3} + 1\right )}^{\frac{1}{3}}{\left (x^{2} - x + 1\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((c*x^2 + b*x + a)/((-x^3 + 1)^(1/3)*(x^2 - x + 1)), x)