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

Optimal. Leaf size=293 \[ \frac{1}{2} x \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};x^3\right )+\frac{\log \left (2^{2/3}-\frac{1-x}{\sqrt [3]{1-x^3}}\right )}{6\ 2^{2/3}}-\frac{\log \left (\frac{2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac{\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{6\ 2^{2/3}}+\frac{\log \left (\frac{\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{3\ 2^{2/3}}-\frac{\log \left (\frac{(1-x)^2}{\left (1-x^3\right )^{2/3}}+\frac{2^{2/3} (1-x)}{\sqrt [3]{1-x^3}}+2 \sqrt [3]{2}\right )}{12\ 2^{2/3}}+\frac{\tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{2^{2/3} \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{\frac{\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}{\sqrt{3}}\right )}{2\ 2^{2/3} \sqrt{3}} \]

[Out]

ArcTan[(1 - (2*2^(1/3)*(1 - x))/(1 - x^3)^(1/3))/Sqrt[3]]/(2^(2/3)*Sqrt[3]) + ArcTan[(1 + (2^(1/3)*(1 - x))/(1
- x^3)^(1/3))/Sqrt[3]]/(2*2^(2/3)*Sqrt[3]) + (x*Hypergeometric2F1[1/3, 2/3, 4/3, x^3])/2 + Log[2^(2/3) - (1 -
x)/(1 - x^3)^(1/3)]/(6*2^(2/3)) - Log[1 + (2^(2/3)*(1 - x)^2)/(1 - x^3)^(2/3) - (2^(1/3)*(1 - x))/(1 - x^3)^(
1/3)]/(6*2^(2/3)) + Log[1 + (2^(1/3)*(1 - x))/(1 - x^3)^(1/3)]/(3*2^(2/3)) - Log[2*2^(1/3) + (1 - x)^2/(1 - x^
3)^(2/3) + (2^(2/3)*(1 - x))/(1 - x^3)^(1/3)]/(12*2^(2/3))

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Rubi [C]  time = 0.0088603, antiderivative size = 21, normalized size of antiderivative = 0.07, 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/((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{1}{\left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, 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 = 0.118103, size = 111, normalized size = 0.38 \[ -\frac{4 x F_1\left (\frac{1}{3};\frac{2}{3},1;\frac{4}{3};x^3,-x^3\right )}{\left (1-x^3\right )^{2/3} \left (x^3+1\right ) \left (x^3 \left (3 F_1\left (\frac{4}{3};\frac{2}{3},2;\frac{7}{3};x^3,-x^3\right )-2 F_1\left (\frac{4}{3};\frac{5}{3},1;\frac{7}{3};x^3,-x^3\right )\right )-4 F_1\left (\frac{1}{3};\frac{2}{3},1;\frac{4}{3};x^3,-x^3\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [F]  time = 0.032, 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(1/(-x^3+1)^(2/3)/(x^3+1),x)

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{{\left (-x^{3} + 1\right )}^{\frac{1}{3}}}{x^{6} - 1}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-(-x^3 + 1)^(1/3)/(x^6 - 1), x)

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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