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

Optimal. Leaf size=132 \[ -\frac{\log \left (x^3+1\right )}{3 \sqrt [3]{2}}+\frac{\log \left (-\sqrt [3]{1-x^3}-\sqrt [3]{2} x\right )}{\sqrt [3]{2}}-\frac{1}{2} \log \left (\sqrt [3]{1-x^3}+x\right )+\frac{\tan ^{-1}\left (\frac{1-\frac{2 x}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{\sqrt{3}}-\frac{2^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{\sqrt{3}} \]

[Out]

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

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Rubi [C]  time = 0.0082518, antiderivative size = 21, normalized size of antiderivative = 0.16, number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {429} \[ x F_1\left (\frac{1}{3};-\frac{2}{3},1;\frac{4}{3};x^3,-x^3\right ) \]

Warning: Unable to verify antiderivative.

[In]

Int[(1 - x^3)^(2/3)/(1 + x^3),x]

[Out]

x*AppellF1[1/3, -2/3, 1, 4/3, x^3, -x^3]

Rule 429

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*x*AppellF1[1/n, -p,
-q, 1 + 1/n, -((b*x^n)/a), -((d*x^n)/c)], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n
, -1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rubi steps

\begin{align*} \int \frac{\left (1-x^3\right )^{2/3}}{1+x^3} \, dx &=x F_1\left (\frac{1}{3};-\frac{2}{3},1;\frac{4}{3};x^3,-x^3\right )\\ \end{align*}

Mathematica [C]  time = 0.0937368, size = 111, normalized size = 0.84 \[ -\frac{4 x \left (1-x^3\right )^{2/3} F_1\left (\frac{1}{3};-\frac{2}{3},1;\frac{4}{3};x^3,-x^3\right )}{\left (x^3+1\right ) \left (x^3 \left (3 F_1\left (\frac{4}{3};-\frac{2}{3},2;\frac{7}{3};x^3,-x^3\right )+2 F_1\left (\frac{4}{3};\frac{1}{3},1;\frac{7}{3};x^3,-x^3\right )\right )-4 F_1\left (\frac{1}{3};-\frac{2}{3},1;\frac{4}{3};x^3,-x^3\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-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*AppellF1[4/3, 1/3, 1, 7/3, x^3, -x^3])))

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Maple [F]  time = 0.038, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3}+1} \left ( -{x}^{3}+1 \right ) ^{{\frac{2}{3}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((-x^3+1)^(2/3)/(x^3+1),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}}}{x^{3} + 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.6769, size = 532, normalized size = 4.03 \begin{align*} -\frac{1}{3} \cdot 4^{\frac{1}{3}} \sqrt{3} \arctan \left (-\frac{\sqrt{3} x - 4^{\frac{1}{3}} \sqrt{3}{\left (-x^{3} + 1\right )}^{\frac{1}{3}}}{3 \, x}\right ) + \frac{1}{3} \, \sqrt{3} \arctan \left (-\frac{\sqrt{3} x - 2 \, \sqrt{3}{\left (-x^{3} + 1\right )}^{\frac{1}{3}}}{3 \, x}\right ) + \frac{1}{3} \cdot 4^{\frac{1}{3}} \log \left (\frac{4^{\frac{2}{3}} x + 2 \,{\left (-x^{3} + 1\right )}^{\frac{1}{3}}}{x}\right ) - \frac{1}{6} \cdot 4^{\frac{1}{3}} \log \left (\frac{2 \cdot 4^{\frac{1}{3}} x^{2} - 4^{\frac{2}{3}}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} x + 2 \,{\left (-x^{3} + 1\right )}^{\frac{2}{3}}}{x^{2}}\right ) - \frac{1}{3} \, \log \left (\frac{x +{\left (-x^{3} + 1\right )}^{\frac{1}{3}}}{x}\right ) + \frac{1}{6} \, \log \left (\frac{x^{2} -{\left (-x^{3} + 1\right )}^{\frac{1}{3}} x +{\left (-x^{3} + 1\right )}^{\frac{2}{3}}}{x^{2}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*4^(1/3)*sqrt(3)*arctan(-1/3*(sqrt(3)*x - 4^(1/3)*sqrt(3)*(-x^3 + 1)^(1/3))/x) + 1/3*sqrt(3)*arctan(-1/3*(
sqrt(3)*x - 2*sqrt(3)*(-x^3 + 1)^(1/3))/x) + 1/3*4^(1/3)*log((4^(2/3)*x + 2*(-x^3 + 1)^(1/3))/x) - 1/6*4^(1/3)
*log((2*4^(1/3)*x^2 - 4^(2/3)*(-x^3 + 1)^(1/3)*x + 2*(-x^3 + 1)^(2/3))/x^2) - 1/3*log((x + (-x^3 + 1)^(1/3))/x
) + 1/6*log((x^2 - (-x^3 + 1)^(1/3)*x + (-x^3 + 1)^(2/3))/x^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((-(x - 1)*(x**2 + x + 1))**(2/3)/((x + 1)*(x**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}}}{x^{3} + 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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