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

Optimal. Leaf size=383 \[ \frac {1}{2} x^2 \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {5}{3};x^3\right )-\frac {\log \left (x^3+1\right )}{3 \sqrt [3]{2}}-\frac {\log \left (\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{3 \sqrt [3]{2}}+\frac {1}{3} 2^{2/3} \log \left (\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )+\frac {\log \left (-\sqrt [3]{1-x^3}-\sqrt [3]{2} x\right )}{\sqrt [3]{2}}-\frac {1}{2} \log \left (\sqrt [3]{1-x^3}+x\right )+\frac {\log \left (2^{2/3} \sqrt [3]{1-x^3}+x-1\right )}{2 \sqrt [3]{2}}-\frac {2^{2/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\tan ^{-1}\left (\frac {\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}+\frac {\tan ^{-1}\left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {2^{2/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\log \left ((1-x) (x+1)^2\right )}{6 \sqrt [3]{2}} \]

[Out]

1/2*x^2*hypergeom([1/3, 2/3],[5/3],x^3)-1/12*ln((1-x)*(1+x)^2)*2^(2/3)-1/6*ln(x^3+1)*2^(2/3)-1/6*ln(1+2^(2/3)*
(1-x)^2/(-x^3+1)^(2/3)-2^(1/3)*(1-x)/(-x^3+1)^(1/3))*2^(2/3)+1/3*ln(1+2^(1/3)*(1-x)/(-x^3+1)^(1/3))*2^(2/3)+1/
2*ln(-2^(1/3)*x-(-x^3+1)^(1/3))*2^(2/3)-1/2*ln(x+(-x^3+1)^(1/3))+1/4*ln(-1+x+2^(2/3)*(-x^3+1)^(1/3))*2^(2/3)-1
/3*arctan(1/3*(1-2*2^(1/3)*(1-x)/(-x^3+1)^(1/3))*3^(1/2))*2^(2/3)*3^(1/2)-1/6*arctan(1/3*(1+2^(1/3)*(1-x)/(-x^
3+1)^(1/3))*3^(1/2))*3^(1/2)*2^(2/3)+1/3*arctan(1/3*(1-2*x/(-x^3+1)^(1/3))*3^(1/2))*3^(1/2)-1/3*arctan(1/3*(1-
2*2^(1/3)*x/(-x^3+1)^(1/3))*3^(1/2))*2^(2/3)*3^(1/2)

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Rubi [F]  time = 0.40, 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 = {} \[ \int \frac {(1-x) \left (1-x^3\right )^{2/3}}{1+x^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [C]  time = 0.16, size = 138, normalized size = 0.36 \[ -\frac {4 \left (1-x^3\right )^{2/3} x F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};x^3,-x^3\right )}{\left (x^3+1\right ) \left (x^3 \left (3 F_1\left (\frac {4}{3};-\frac {2}{3},2;\frac {7}{3};x^3,-x^3\right )+2 F_1\left (\frac {4}{3};\frac {1}{3},1;\frac {7}{3};x^3,-x^3\right )\right )-4 F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};x^3,-x^3\right )\right )}-\frac {1}{2} x^2 F_1\left (\frac {2}{3};-\frac {2}{3},1;\frac {5}{3};x^3,-x^3\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/2*(x^2*AppellF1[2/3, -2/3, 1, 5/3, x^3, -x^3]) - (4*x*(1 - x^3)^(2/3)*AppellF1[1/3, -2/3, 1, 4/3, x^3, -x^3
])/((1 + x^3)*(-4*AppellF1[1/3, -2/3, 1, 4/3, x^3, -x^3] + x^3*(3*AppellF1[4/3, -2/3, 2, 7/3, x^3, -x^3] + 2*A
ppellF1[4/3, 1/3, 1, 7/3, x^3, -x^3])))

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fricas [F]  time = 11.79, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (-x^{3} + 1\right )}^{\frac {2}{3}} {\left (x - 1\right )}}{x^{3} + 1}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.11, size = 0, normalized size = 0.00 \[ \int \frac {\left (-x +1\right ) \left (-x^{3}+1\right )^{\frac {2}{3}}}{x^{3}+1}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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