3.110 \(\int \frac{(1-2 x) (1-x^3)^{2/3}}{(1-x+x^2)^2} \, dx\)

Optimal. Leaf size=199 \[ \frac{\left (1-x^3\right )^{2/3}}{x^2-x+1}+\frac{\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{\sqrt [3]{2}}-\frac{\log \left (-\sqrt [3]{1-x^3}-\sqrt [3]{2} x\right )}{\sqrt [3]{2}}+\log \left (\sqrt [3]{1-x^3}+x\right )-\frac{2 \tan ^{-1}\left (\frac{1-\frac{2 x}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{\sqrt{3}}+\frac{2^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{\sqrt{3}}+\frac{2^{2/3} \tan ^{-1}\left (\frac{2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt{3}}\right )}{\sqrt{3}} \]

[Out]

(1 - x^3)^(2/3)/(1 - x + x^2) - (2*ArcTan[(1 - (2*x)/(1 - x^3)^(1/3))/Sqrt[3]])/Sqrt[3] + (2^(2/3)*ArcTan[(1 -
(2*2^(1/3)*x)/(1 - x^3)^(1/3))/Sqrt[3]])/Sqrt[3] + (2^(2/3)*ArcTan[(1 + 2^(2/3)*(1 - x^3)^(1/3))/Sqrt[3]])/Sq
rt[3] + Log[2^(1/3) - (1 - x^3)^(1/3)]/2^(1/3) - Log[-(2^(1/3)*x) - (1 - x^3)^(1/3)]/2^(1/3) + Log[x + (1 - x^
3)^(1/3)]

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Rubi [F]  time = 0.945257, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{(1-2 x) \left (1-x^3\right )^{2/3}}{\left (1-x+x^2\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

(-4*Defer[Int][(1 - x^3)^(2/3)/(1 + I*Sqrt[3] - 2*x)^2, x])/3 + (4*(1 + I*Sqrt[3])*Defer[Int][(1 - x^3)^(2/3)/
(1 + I*Sqrt[3] - 2*x)^2, x])/3 - (4*Defer[Int][(1 - x^3)^(2/3)/(-1 + I*Sqrt[3] + 2*x)^2, x])/3 + (4*(1 - I*Sqr
t[3])*Defer[Int][(1 - x^3)^(2/3)/(-1 + I*Sqrt[3] + 2*x)^2, x])/3

Rubi steps

\begin{align*} \int \frac{(1-2 x) \left (1-x^3\right )^{2/3}}{\left (1-x+x^2\right )^2} \, dx &=\int \left (\frac{\left (1-x^3\right )^{2/3}}{\left (1-x+x^2\right )^2}-\frac{2 x \left (1-x^3\right )^{2/3}}{\left (1-x+x^2\right )^2}\right ) \, dx\\ &=-\left (2 \int \frac{x \left (1-x^3\right )^{2/3}}{\left (1-x+x^2\right )^2} \, dx\right )+\int \frac{\left (1-x^3\right )^{2/3}}{\left (1-x+x^2\right )^2} \, dx\\ &=-\left (2 \int \left (-\frac{2 \left (1+i \sqrt{3}\right ) \left (1-x^3\right )^{2/3}}{3 \left (1+i \sqrt{3}-2 x\right )^2}+\frac{2 i \left (1-x^3\right )^{2/3}}{3 \sqrt{3} \left (1+i \sqrt{3}-2 x\right )}-\frac{2 \left (1-i \sqrt{3}\right ) \left (1-x^3\right )^{2/3}}{3 \left (-1+i \sqrt{3}+2 x\right )^2}+\frac{2 i \left (1-x^3\right )^{2/3}}{3 \sqrt{3} \left (-1+i \sqrt{3}+2 x\right )}\right ) \, dx\right )+\int \left (-\frac{4 \left (1-x^3\right )^{2/3}}{3 \left (1+i \sqrt{3}-2 x\right )^2}+\frac{4 i \left (1-x^3\right )^{2/3}}{3 \sqrt{3} \left (1+i \sqrt{3}-2 x\right )}-\frac{4 \left (1-x^3\right )^{2/3}}{3 \left (-1+i \sqrt{3}+2 x\right )^2}+\frac{4 i \left (1-x^3\right )^{2/3}}{3 \sqrt{3} \left (-1+i \sqrt{3}+2 x\right )}\right ) \, dx\\ &=-\left (\frac{4}{3} \int \frac{\left (1-x^3\right )^{2/3}}{\left (1+i \sqrt{3}-2 x\right )^2} \, dx\right )-\frac{4}{3} \int \frac{\left (1-x^3\right )^{2/3}}{\left (-1+i \sqrt{3}+2 x\right )^2} \, dx+\frac{1}{3} \left (4 \left (1-i \sqrt{3}\right )\right ) \int \frac{\left (1-x^3\right )^{2/3}}{\left (-1+i \sqrt{3}+2 x\right )^2} \, dx+\frac{1}{3} \left (4 \left (1+i \sqrt{3}\right )\right ) \int \frac{\left (1-x^3\right )^{2/3}}{\left (1+i \sqrt{3}-2 x\right )^2} \, dx\\ \end{align*}

Mathematica [F]  time = 0.34793, size = 0, normalized size = 0. \[ \int \frac{(1-2 x) \left (1-x^3\right )^{2/3}}{\left (1-x+x^2\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [F]  time = 0.088, size = 0, normalized size = 0. \begin{align*} \int{\frac{1-2\,x}{ \left ({x}^{2}-x+1 \right ) ^{2}} \left ( -{x}^{3}+1 \right ) ^{{\frac{2}{3}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 14.5052, size = 5191, normalized size = 26.09 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/72*(8*4^(1/3)*sqrt(3)*(x^2 - x + 1)*arctan(-1/6*(3822*4^(2/3)*sqrt(3)*(50*x^4 - 74*x^3 - 207*x^2 + 143*x +
19)*(-x^3 + 1)^(2/3) + 7644*4^(1/3)*sqrt(3)*(19*x^5 - 150*x^4 + 43*x^3 + 112*x^2 + 57*x - 50)*(-x^3 + 1)^(1/3)
- 7*sqrt(39)*(6*4^(1/3)*sqrt(3)*(1150*x^4 - 3974*x^3 - 1911*x^2 + 1522*x + 3898)*(-x^3 + 1)^(2/3) - 4^(2/3)*s
qrt(3)*(1778*x^6 - 6366*x^5 - 8412*x^4 + 17254*x^3 + 15117*x^2 - 4227*x - 16105) + 12*sqrt(3)*(437*x^5 - 1539*
x^4 - 333*x^3 - 2074*x^2 + 372*x + 3261)*(-x^3 + 1)^(1/3))*sqrt((6*4^(1/3)*(5*x^4 + 4*x^3 - 3*x^2 - 4*x + 1)*(
-x^3 + 1)^(2/3) + 4^(2/3)*(19*x^6 + 15*x^5 - 12*x^4 - 25*x^3 - 12*x^2 + 15*x + 1) - 12*(4*x^5 + 3*x^4 - 2*x^3
- 5*x^2 + 1)*(-x^3 + 1)^(1/3))/(x^6 - 3*x^5 + 6*x^4 - 7*x^3 + 6*x^2 - 3*x + 1)) + 6*sqrt(3)*(29494*x^6 - 17582
*x^5 + 153824*x^4 - 266248*x^3 - 129950*x^2 + 238106*x - 29747))/(138718*x^6 - 463746*x^5 - 296508*x^4 - 11507
2*x^3 + 1093704*x^2 - 70446*x - 256859)) + 8*4^(1/3)*sqrt(3)*(x^2 - x + 1)*arctan(1/6*(3822*4^(2/3)*sqrt(3)*(1
9*x^4 - 181*x^3 + 36*x^2 + 169*x - 31)*(-x^3 + 1)^(2/3) - 7644*4^(1/3)*sqrt(3)*(31*x^5 + 57*x^4 - 131*x^3 - 11
9*x^2 + 93*x + 19)*(-x^3 + 1)^(1/3) + 7*sqrt(39)*(6*4^(1/3)*sqrt(3)*(3385*x^4 + 3574*x^3 - 1911*x^2 - 2948*x +
124)*(-x^3 + 1)^(2/3) + 4^(2/3)*sqrt(3)*(13027*x^6 + 16539*x^5 - 8961*x^4 - 32644*x^3 - 2361*x^2 + 17139*x -
239) - 12*sqrt(3)*(2748*x^5 + 3450*x^4 - 4126*x^3 - 2385*x^2 + 1539*x - 76)*(-x^3 + 1)^(1/3))*sqrt((6*4^(1/3)*
(x^4 - 4*x^3 - 3*x^2 + 4*x + 5)*(-x^3 + 1)^(2/3) + 4^(2/3)*(x^6 + 15*x^5 - 12*x^4 - 25*x^3 - 12*x^2 + 15*x + 1
9) + 12*(x^5 - 5*x^3 - 2*x^2 + 3*x + 4)*(-x^3 + 1)^(1/3))/(x^6 - 3*x^5 + 6*x^4 - 7*x^3 + 6*x^2 - 3*x + 1)) + 6
*sqrt(3)*(53953*x^6 - 12994*x^5 - 396521*x^4 + 169424*x^3 + 300029*x^2 - 62294*x - 41597))/(52723*x^6 + 682854
*x^5 - 325173*x^4 - 1353400*x^3 + 193623*x^2 + 640446*x - 16073)) + 16*4^(1/3)*sqrt(3)*(x^2 - x + 1)*arctan(1/
6*(7644*4^(2/3)*sqrt(3)*(5*x^4 - 107*x^3 - 243*x^2 + 26*x + 157)*(-x^3 + 1)^(2/3) - 7644*4^(1/3)*sqrt(3)*(307*
x^5 + 300*x^4 - 140*x^3 - 221*x^2 - 186*x - 98)*(-x^3 + 1)^(1/3) + 7*sqrt(39)*4^(1/3)*(6*4^(1/3)*sqrt(3)*(3109
*x^4 + 400*x^3 - 3822*x^2 + 1426*x + 3622)*(-x^3 + 1)^(2/3) + 4^(2/3)*sqrt(3)*(15505*x^6 + 11493*x^5 - 22383*x
^4 - 22720*x^3 - 5454*x^2 + 13032*x + 10888) - 12*sqrt(3)*(2111*x^5 + 3450*x^4 - 941*x^3 - 1111*x^2 - 372*x -
2624)*(-x^3 + 1)^(1/3)) + 6*sqrt(3)*(307479*x^6 + 239258*x^5 - 543668*x^4 - 607716*x^3 + 19112*x^2 + 232000*x
+ 343788))/(933353*x^6 + 1472754*x^5 + 285042*x^4 - 1008596*x^3 - 1598208*x^2 - 560184*x + 468980)) + 48*sqrt(
3)*(x^2 - x + 1)*arctan((4*sqrt(3)*(-x^3 + 1)^(1/3)*x^2 + 2*sqrt(3)*(-x^3 + 1)^(2/3)*x - sqrt(3)*(x^3 - 1))/(9
*x^3 - 1)) - 3*4^(1/3)*(x^2 - x + 1)*log(39626496*(6*4^(1/3)*(5*x^4 + 4*x^3 - 3*x^2 - 4*x + 1)*(-x^3 + 1)^(2/3
) + 4^(2/3)*(19*x^6 + 15*x^5 - 12*x^4 - 25*x^3 - 12*x^2 + 15*x + 1) - 12*(4*x^5 + 3*x^4 - 2*x^3 - 5*x^2 + 1)*(
-x^3 + 1)^(1/3))/(x^6 - 3*x^5 + 6*x^4 - 7*x^3 + 6*x^2 - 3*x + 1)) - 3*4^(1/3)*(x^2 - x + 1)*log(9906624*(6*4^(
1/3)*(5*x^4 + 4*x^3 - 3*x^2 - 4*x + 1)*(-x^3 + 1)^(2/3) + 4^(2/3)*(19*x^6 + 15*x^5 - 12*x^4 - 25*x^3 - 12*x^2
+ 15*x + 1) - 12*(4*x^5 + 3*x^4 - 2*x^3 - 5*x^2 + 1)*(-x^3 + 1)^(1/3))/(x^6 - 3*x^5 + 6*x^4 - 7*x^3 + 6*x^2 -
3*x + 1)) + 3*4^(1/3)*(x^2 - x + 1)*log(39626496*(6*4^(1/3)*(x^4 - 4*x^3 - 3*x^2 + 4*x + 5)*(-x^3 + 1)^(2/3) +
4^(2/3)*(x^6 + 15*x^5 - 12*x^4 - 25*x^3 - 12*x^2 + 15*x + 19) + 12*(x^5 - 5*x^3 - 2*x^2 + 3*x + 4)*(-x^3 + 1)
^(1/3))/(x^6 - 3*x^5 + 6*x^4 - 7*x^3 + 6*x^2 - 3*x + 1)) + 3*4^(1/3)*(x^2 - x + 1)*log(9906624*(6*4^(1/3)*(x^4
- 4*x^3 - 3*x^2 + 4*x + 5)*(-x^3 + 1)^(2/3) + 4^(2/3)*(x^6 + 15*x^5 - 12*x^4 - 25*x^3 - 12*x^2 + 15*x + 19) +
12*(x^5 - 5*x^3 - 2*x^2 + 3*x + 4)*(-x^3 + 1)^(1/3))/(x^6 - 3*x^5 + 6*x^4 - 7*x^3 + 6*x^2 - 3*x + 1)) - 24*(x
^2 - x + 1)*log(3*(-x^3 + 1)^(1/3)*x^2 + 3*(-x^3 + 1)^(2/3)*x + 1) - 72*(-x^3 + 1)^(2/3))/(x^2 - x + 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int - \frac{\left (1 - x^{3}\right )^{\frac{2}{3}}}{x^{4} - 2 x^{3} + 3 x^{2} - 2 x + 1}\, dx - \int \frac{2 x \left (1 - x^{3}\right )^{\frac{2}{3}}}{x^{4} - 2 x^{3} + 3 x^{2} - 2 x + 1}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(-(1 - x**3)**(2/3)/(x**4 - 2*x**3 + 3*x**2 - 2*x + 1), x) - Integral(2*x*(1 - x**3)**(2/3)/(x**4 - 2
*x**3 + 3*x**2 - 2*x + 1), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{{\left (-x^{3} + 1\right )}^{\frac{2}{3}}{\left (2 \, x - 1\right )}}{{\left (x^{2} - x + 1\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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