3.1017 \(\int \frac {1-x^2}{(1-x+x^2) (1-x^3)^{2/3}} \, dx\)

Optimal. Leaf size=103 \[ -\frac {\log \left (-x^3+2 (1-x)^3+1\right )}{2\ 2^{2/3}}+\frac {3 \log \left (\sqrt [3]{1-x^3}+\sqrt [3]{2} (1-x)\right )}{2\ 2^{2/3}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{2^{2/3}} \]

[Out]

-1/4*ln(1+2*(1-x)^3-x^3)*2^(1/3)+3/4*ln(2^(1/3)*(1-x)+(-x^3+1)^(1/3))*2^(1/3)+1/2*arctan(1/3*(1-2*2^(1/3)*(1-x
)/(-x^3+1)^(1/3))*3^(1/2))*3^(1/2)*2^(1/3)

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Rubi [F]  time = 0.53, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {1-x^2}{\left (1-x+x^2\right ) \left (1-x^3\right )^{2/3}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

-(x*Hypergeometric2F1[1/3, 2/3, 4/3, x^3]) - (1 + I*Sqrt[3])*Defer[Int][1/((-1 - I*Sqrt[3] + 2*x)*(1 - x^3)^(2
/3)), x] - (1 - I*Sqrt[3])*Defer[Int][1/((-1 + I*Sqrt[3] + 2*x)*(1 - x^3)^(2/3)), x]

Rubi steps

\begin {align*} \int \frac {1-x^2}{\left (1-x+x^2\right ) \left (1-x^3\right )^{2/3}} \, dx &=\int \left (-\frac {1}{\left (1-x^3\right )^{2/3}}+\frac {2-x}{\left (1-x+x^2\right ) \left (1-x^3\right )^{2/3}}\right ) \, dx\\ &=-\int \frac {1}{\left (1-x^3\right )^{2/3}} \, dx+\int \frac {2-x}{\left (1-x+x^2\right ) \left (1-x^3\right )^{2/3}} \, dx\\ &=-x \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};x^3\right )+\int \left (\frac {-1-i \sqrt {3}}{\left (-1-i \sqrt {3}+2 x\right ) \left (1-x^3\right )^{2/3}}+\frac {-1+i \sqrt {3}}{\left (-1+i \sqrt {3}+2 x\right ) \left (1-x^3\right )^{2/3}}\right ) \, dx\\ &=-x \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};x^3\right )+\left (-1-i \sqrt {3}\right ) \int \frac {1}{\left (-1-i \sqrt {3}+2 x\right ) \left (1-x^3\right )^{2/3}} \, dx+\left (-1+i \sqrt {3}\right ) \int \frac {1}{\left (-1+i \sqrt {3}+2 x\right ) \left (1-x^3\right )^{2/3}} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.19, size = 0, normalized size = 0.00 \[ \int \frac {1-x^2}{\left (1-x+x^2\right ) \left (1-x^3\right )^{2/3}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 8.81, size = 289, normalized size = 2.81 \[ -\frac {1}{6} \cdot 4^{\frac {1}{6}} \sqrt {3} \arctan \left (\frac {4^{\frac {1}{6}} \sqrt {3} {\left (2 \cdot 4^{\frac {2}{3}} {\left (x^{5} - x^{4} - 3 \, x^{3} + 3 \, x^{2} + x - 1\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + 4 \, {\left (x^{4} - 4 \, x^{3} + 5 \, x^{2} - 4 \, x + 1\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (x^{6} - 7 \, x^{5} + 10 \, x^{4} - 7 \, x^{3} + 10 \, x^{2} - 7 \, x + 1\right )}\right )}}{6 \, {\left (3 \, x^{6} - 9 \, x^{5} + 6 \, x^{4} - x^{3} + 6 \, x^{2} - 9 \, x + 3\right )}}\right ) - \frac {1}{24} \cdot 4^{\frac {2}{3}} \log \left (\frac {2 \cdot 4^{\frac {1}{3}} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} {\left (x^{2} - 3 \, x + 1\right )} - 4^{\frac {2}{3}} {\left (x^{4} - 3 \, x^{2} + 1\right )} - 8 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}} {\left (x^{2} - x\right )}}{x^{4} - 2 \, x^{3} + 3 \, x^{2} - 2 \, x + 1}\right ) + \frac {1}{12} \cdot 4^{\frac {2}{3}} \log \left (-\frac {4^{\frac {2}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} {\left (x - 1\right )} - 4^{\frac {1}{3}} {\left (x^{2} - x + 1\right )} - 2 \, {\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{x^{2} - x + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*4^(1/6)*sqrt(3)*arctan(1/6*4^(1/6)*sqrt(3)*(2*4^(2/3)*(x^5 - x^4 - 3*x^3 + 3*x^2 + x - 1)*(-x^3 + 1)^(1/3
) + 4*(x^4 - 4*x^3 + 5*x^2 - 4*x + 1)*(-x^3 + 1)^(2/3) + 4^(1/3)*(x^6 - 7*x^5 + 10*x^4 - 7*x^3 + 10*x^2 - 7*x
+ 1))/(3*x^6 - 9*x^5 + 6*x^4 - x^3 + 6*x^2 - 9*x + 3)) - 1/24*4^(2/3)*log((2*4^(1/3)*(-x^3 + 1)^(2/3)*(x^2 - 3
*x + 1) - 4^(2/3)*(x^4 - 3*x^2 + 1) - 8*(-x^3 + 1)^(1/3)*(x^2 - x))/(x^4 - 2*x^3 + 3*x^2 - 2*x + 1)) + 1/12*4^
(2/3)*log(-(4^(2/3)*(-x^3 + 1)^(1/3)*(x - 1) - 4^(1/3)*(x^2 - x + 1) - 2*(-x^3 + 1)^(2/3))/(x^2 - x + 1))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {x^{2} - 1}{{\left (-x^{3} + 1\right )}^{\frac {2}{3}} {\left (x^{2} - x + 1\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 8.03, size = 1026, normalized size = 9.96 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*RootOf(_Z^3-2)*ln(-(2*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)^4*x+4*RootOf(Root
Of(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)^2*RootOf(_Z^3-2)^3*x+2*(-x^3+1)^(2/3)*RootOf(RootOf(_Z^3-2)^2+2*_Z*Ro
otOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)^2+RootOf(_Z^3-2)^2*x^2+2*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^
2)*RootOf(_Z^3-2)*x^2-RootOf(_Z^3-2)^2*x-2*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)*
x-2*(-x^3+1)^(1/3)*RootOf(_Z^3-2)*x-4*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*(-x^3+1)^(1/3)*x+Roo
tOf(_Z^3-2)^2+2*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)+2*(-x^3+1)^(1/3)*RootOf(_Z^
3-2)+4*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*(-x^3+1)^(1/3))/(x^2-x+1))-1/2*ln((2*RootOf(RootOf(
_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)^4*x+4*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)
^2*RootOf(_Z^3-2)^3*x+2*(-x^3+1)^(2/3)*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)^2-Ro
otOf(_Z^3-2)^2*x^2-2*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)*x^2+3*RootOf(_Z^3-2)^2
*x+6*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)*x-2*(-x^3+1)^(1/3)*RootOf(_Z^3-2)*x-Ro
otOf(_Z^3-2)^2-2*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)+2*(-x^3+1)^(1/3)*RootOf(_Z
^3-2)+2*(-x^3+1)^(2/3))/(x^2-x+1))*RootOf(_Z^3-2)-ln((2*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*Ro
otOf(_Z^3-2)^4*x+4*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)^2*RootOf(_Z^3-2)^3*x+2*(-x^3+1)^(2/3)*R
ootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)^2-RootOf(_Z^3-2)^2*x^2-2*RootOf(RootOf(_Z^3-
2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)*x^2+3*RootOf(_Z^3-2)^2*x+6*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf
(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)*x-2*(-x^3+1)^(1/3)*RootOf(_Z^3-2)*x-RootOf(_Z^3-2)^2-2*RootOf(RootOf(_Z^3-2)^2
+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)+2*(-x^3+1)^(1/3)*RootOf(_Z^3-2)+2*(-x^3+1)^(2/3))/(x^2-x+1))*RootO
f(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {x^{2} - 1}{{\left (-x^{3} + 1\right )}^{\frac {2}{3}} {\left (x^{2} - x + 1\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ -\int \frac {x^2-1}{{\left (1-x^3\right )}^{2/3}\,\left (x^2-x+1\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {x^{2}}{x^{2} \left (1 - x^{3}\right )^{\frac {2}{3}} - x \left (1 - x^{3}\right )^{\frac {2}{3}} + \left (1 - x^{3}\right )^{\frac {2}{3}}}\, dx - \int \left (- \frac {1}{x^{2} \left (1 - x^{3}\right )^{\frac {2}{3}} - x \left (1 - x^{3}\right )^{\frac {2}{3}} + \left (1 - x^{3}\right )^{\frac {2}{3}}}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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