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

Optimal. Leaf size=294 \[ \frac{1}{2} x \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};x^3\right )-\frac{\log \left (2^{2/3}-\frac{1-x}{\sqrt [3]{1-x^3}}\right )}{6\ 2^{2/3}}+\frac{\log \left (\frac{2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac{\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{6\ 2^{2/3}}-\frac{\log \left (\frac{\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{3\ 2^{2/3}}+\frac{\log \left (\frac{(1-x)^2}{\left (1-x^3\right )^{2/3}}+\frac{2^{2/3} (1-x)}{\sqrt [3]{1-x^3}}+2 \sqrt [3]{2}\right )}{12\ 2^{2/3}}-\frac{\tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{2^{2/3} \sqrt{3}}-\frac{\tan ^{-1}\left (\frac{\frac{\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}{\sqrt{3}}\right )}{2\ 2^{2/3} \sqrt{3}} \]

[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 = 0.0185345, antiderivative size = 26, normalized size of antiderivative = 0.09, number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {510} \[ \frac{1}{4} x^4 F_1\left (\frac{4}{3};\frac{2}{3},1;\frac{7}{3};x^3,-x^3\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

Rule 510

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

Rubi steps

\begin{align*} \int \frac{x^3}{\left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx &=\frac{1}{4} x^4 F_1\left (\frac{4}{3};\frac{2}{3},1;\frac{7}{3};x^3,-x^3\right )\\ \end{align*}

Mathematica [C]  time = 0.0327678, size = 26, normalized size = 0.09 \[ \frac{1}{4} x^4 F_1\left (\frac{4}{3};\frac{2}{3},1;\frac{7}{3};x^3,-x^3\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [F]  time = 0.036, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{3}}{{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/(-x^3+1)^(2/3)/(x^3+1),x)

[Out]

int(x^3/(-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{x^{3}}{{\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(x^3/(-x^3+1)^(2/3)/(x^3+1),x, algorithm="maxima")

[Out]

integrate(x^3/((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^{3}}{x^{6} - 1}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Integral(x**3/((-(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{x^{3}}{{\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(x^3/(-x^3+1)^(2/3)/(x^3+1),x, algorithm="giac")

[Out]

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