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

Optimal. Leaf size=177 \[ \frac {1}{2} x^2 \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {5}{3};x^3\right )+\frac {1}{2} \left (1-x^3\right )^{2/3}-\frac {1}{2} \log \left (\sqrt [3]{1-x^3}+x\right )+\frac {3 \log \left (2^{2/3} \sqrt [3]{1-x^3}+x-1\right )}{2 \sqrt [3]{2}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}{\sqrt {3}}\right )}{\sqrt [3]{2}}+\frac {\tan ^{-1}\left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\log \left ((1-x) (x+1)^2\right )}{2 \sqrt [3]{2}} \]

[Out]

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

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Rubi [F]  time = 0.07, 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 = {} \[ \int \frac {\left (1-x+x^2\right ) \left (1-x^3\right )^{2/3}}{1+x^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [F]  time = 0.15, size = 0, normalized size = 0.00 \[ \int \frac {\left (1-x+x^2\right ) \left (1-x^3\right )^{2/3}}{1+x^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [F]  time = 4.60, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{x + 1}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((-x^3 + 1)^(2/3)/(x + 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[ \int \frac {\left (x^{2}-x +1\right ) \left (-x^{3}+1\right )^{\frac {2}{3}}}{x^{3}+1}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((x^2-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 \[ \int \frac {{\left (-x^{3} + 1\right )}^{\frac {2}{3}} {\left (x^{2} - x + 1\right )}}{x^{3} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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