3.647 \(\int \frac{x^3}{(1-x^3)^{4/3} (1+x^3)} \, dx\)

Optimal. Leaf size=106 \[ \frac{x}{2 \sqrt [3]{1-x^3}}+\frac{\log \left (x^3+1\right )}{12 \sqrt [3]{2}}-\frac{\log \left (-\sqrt [3]{1-x^3}-\sqrt [3]{2} x\right )}{4 \sqrt [3]{2}}+\frac{\tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{2 \sqrt [3]{2} \sqrt{3}} \]

[Out]

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

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Rubi [C]  time = 0.018378, antiderivative size = 38, normalized size of antiderivative = 0.36, 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} \[ \frac{x^4 \, _2F_1\left (\frac{4}{3},\frac{4}{3};\frac{7}{3};\frac{2 x^3}{x^3+1}\right )}{4 \left (x^3+1\right )^{4/3}} \]

Warning: Unable to verify antiderivative.

[In]

Int[x^3/((1 - x^3)^(4/3)*(1 + x^3)),x]

[Out]

(x^4*Hypergeometric2F1[4/3, 4/3, 7/3, (2*x^3)/(1 + x^3)])/(4*(1 + x^3)^(4/3))

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 )^{4/3} \left (1+x^3\right )} \, dx &=\frac{x^4 \, _2F_1\left (\frac{4}{3},\frac{4}{3};\frac{7}{3};\frac{2 x^3}{1+x^3}\right )}{4 \left (1+x^3\right )^{4/3}}\\ \end{align*}

Mathematica [C]  time = 0.0092013, size = 38, normalized size = 0.36 \[ \frac{x^4 \, _2F_1\left (\frac{4}{3},\frac{4}{3};\frac{7}{3};\frac{2 x^3}{x^3+1}\right )}{4 \left (x^3+1\right )^{4/3}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^3/((1 - x^3)^(4/3)*(1 + x^3)),x]

[Out]

(x^4*Hypergeometric2F1[4/3, 4/3, 7/3, (2*x^3)/(1 + x^3)])/(4*(1 + x^3)^(4/3))

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Maple [F]  time = 0.052, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{3}}{{x}^{3}+1} \left ( -{x}^{3}+1 \right ) ^{-{\frac{4}{3}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(-x^3+1)^(4/3)/(x^3+1),x)

[Out]

int(x^3/(-x^3+1)^(4/3)/(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{4}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(-x^3+1)^(4/3)/(x^3+1),x, algorithm="maxima")

[Out]

integrate(x^3/((x^3 + 1)*(-x^3 + 1)^(4/3)), x)

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Fricas [B]  time = 21.7687, size = 876, normalized size = 8.26 \begin{align*} -\frac{2 \, \sqrt{6} 2^{\frac{1}{6}} \left (-1\right )^{\frac{1}{3}}{\left (x^{3} - 1\right )} \arctan \left (\frac{2^{\frac{1}{6}}{\left (6 \, \sqrt{6} 2^{\frac{2}{3}} \left (-1\right )^{\frac{2}{3}}{\left (5 \, x^{7} + 4 \, x^{4} - x\right )}{\left (-x^{3} + 1\right )}^{\frac{2}{3}} - 12 \, \sqrt{6} \left (-1\right )^{\frac{1}{3}}{\left (19 \, x^{8} - 16 \, x^{5} + x^{2}\right )}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} - \sqrt{6} 2^{\frac{1}{3}}{\left (71 \, x^{9} - 111 \, x^{6} + 33 \, x^{3} - 1\right )}\right )}}{6 \,{\left (109 \, x^{9} - 105 \, x^{6} + 3 \, x^{3} + 1\right )}}\right ) - 2 \cdot 2^{\frac{2}{3}} \left (-1\right )^{\frac{1}{3}}{\left (x^{3} - 1\right )} \log \left (\frac{6 \cdot 2^{\frac{1}{3}} \left (-1\right )^{\frac{2}{3}}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} x^{2} - 2^{\frac{2}{3}} \left (-1\right )^{\frac{1}{3}}{\left (x^{3} + 1\right )} + 6 \,{\left (-x^{3} + 1\right )}^{\frac{2}{3}} x}{x^{3} + 1}\right ) + 2^{\frac{2}{3}} \left (-1\right )^{\frac{1}{3}}{\left (x^{3} - 1\right )} \log \left (-\frac{3 \cdot 2^{\frac{2}{3}} \left (-1\right )^{\frac{1}{3}}{\left (5 \, x^{4} - x\right )}{\left (-x^{3} + 1\right )}^{\frac{2}{3}} - 2^{\frac{1}{3}} \left (-1\right )^{\frac{2}{3}}{\left (19 \, x^{6} - 16 \, x^{3} + 1\right )} + 12 \,{\left (2 \, x^{5} - x^{2}\right )}{\left (-x^{3} + 1\right )}^{\frac{1}{3}}}{x^{6} + 2 \, x^{3} + 1}\right ) + 36 \,{\left (-x^{3} + 1\right )}^{\frac{2}{3}} x}{72 \,{\left (x^{3} - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(-x^3+1)^(4/3)/(x^3+1),x, algorithm="fricas")

[Out]

-1/72*(2*sqrt(6)*2^(1/6)*(-1)^(1/3)*(x^3 - 1)*arctan(1/6*2^(1/6)*(6*sqrt(6)*2^(2/3)*(-1)^(2/3)*(5*x^7 + 4*x^4
- x)*(-x^3 + 1)^(2/3) - 12*sqrt(6)*(-1)^(1/3)*(19*x^8 - 16*x^5 + x^2)*(-x^3 + 1)^(1/3) - sqrt(6)*2^(1/3)*(71*x
^9 - 111*x^6 + 33*x^3 - 1))/(109*x^9 - 105*x^6 + 3*x^3 + 1)) - 2*2^(2/3)*(-1)^(1/3)*(x^3 - 1)*log((6*2^(1/3)*(
-1)^(2/3)*(-x^3 + 1)^(1/3)*x^2 - 2^(2/3)*(-1)^(1/3)*(x^3 + 1) + 6*(-x^3 + 1)^(2/3)*x)/(x^3 + 1)) + 2^(2/3)*(-1
)^(1/3)*(x^3 - 1)*log(-(3*2^(2/3)*(-1)^(1/3)*(5*x^4 - x)*(-x^3 + 1)^(2/3) - 2^(1/3)*(-1)^(2/3)*(19*x^6 - 16*x^
3 + 1) + 12*(2*x^5 - x^2)*(-x^3 + 1)^(1/3))/(x^6 + 2*x^3 + 1)) + 36*(-x^3 + 1)^(2/3)*x)/(x^3 - 1)

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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{4}{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)**(4/3)/(x**3+1),x)

[Out]

Integral(x**3/((-(x - 1)*(x**2 + x + 1))**(4/3)*(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{4}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(-x^3+1)^(4/3)/(x^3+1),x, algorithm="giac")

[Out]

integrate(x^3/((x^3 + 1)*(-x^3 + 1)^(4/3)), x)