3.646 \(\int \frac{x^6}{(1-x^3)^{4/3} (1+x^3)} \, dx\)

Optimal. Leaf size=153 \[ \frac{x}{2 \sqrt [3]{1-x^3}}-\frac{\log \left (x^3+1\right )}{12 \sqrt [3]{2}}+\frac{\log \left (-\sqrt [3]{1-x^3}-\sqrt [3]{2} x\right )}{4 \sqrt [3]{2}}-\frac{1}{2} \log \left (\sqrt [3]{1-x^3}+x\right )+\frac{\tan ^{-1}\left (\frac{1-\frac{2 x}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{\sqrt{3}}-\frac{\tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{2 \sqrt [3]{2} \sqrt{3}} \]

[Out]

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

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Rubi [C]  time = 0.0180263, antiderivative size = 26, normalized size of antiderivative = 0.17, 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} \[ \frac{1}{7} x^7 F_1\left (\frac{7}{3};\frac{4}{3},1;\frac{10}{3};x^3,-x^3\right ) \]

Warning: Unable to verify antiderivative.

[In]

Int[x^6/((1 - x^3)^(4/3)*(1 + x^3)),x]

[Out]

(x^7*AppellF1[7/3, 4/3, 1, 10/3, x^3, -x^3])/7

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^6}{\left (1-x^3\right )^{4/3} \left (1+x^3\right )} \, dx &=\frac{1}{7} x^7 F_1\left (\frac{7}{3};\frac{4}{3},1;\frac{10}{3};x^3,-x^3\right )\\ \end{align*}

Mathematica [C]  time = 0.138084, size = 142, normalized size = 0.93 \[ \frac{1}{24} \left (-6 x^4 F_1\left (\frac{4}{3};\frac{1}{3},1;\frac{7}{3};x^3,-x^3\right )+\frac{12 x}{\sqrt [3]{1-x^3}}+2^{2/3} \left (\log \left (\frac{2^{2/3} x^2}{\left (x^3-1\right )^{2/3}}-\frac{\sqrt [3]{2} x}{\sqrt [3]{x^3-1}}+1\right )-2 \log \left (\frac{\sqrt [3]{2} x}{\sqrt [3]{x^3-1}}+1\right )-2 \sqrt{3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{2} x}{\sqrt [3]{x^3-1}}-1}{\sqrt{3}}\right )\right )\right ) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^6/((1 - x^3)^(4/3)*(1 + x^3)),x]

[Out]

((12*x)/(1 - x^3)^(1/3) - 6*x^4*AppellF1[4/3, 1/3, 1, 7/3, x^3, -x^3] + 2^(2/3)*(-2*Sqrt[3]*ArcTan[(-1 + (2*2^
(1/3)*x)/(-1 + x^3)^(1/3))/Sqrt[3]] + Log[1 + (2^(2/3)*x^2)/(-1 + x^3)^(2/3) - (2^(1/3)*x)/(-1 + x^3)^(1/3)] -
2*Log[1 + (2^(1/3)*x)/(-1 + x^3)^(1/3)]))/24

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Maple [F]  time = 0.047, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{6}}{{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^6/(-x^3+1)^(4/3)/(x^3+1),x)

[Out]

int(x^6/(-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^{6}}{{\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^6/(-x^3+1)^(4/3)/(x^3+1),x, algorithm="maxima")

[Out]

integrate(x^6/((x^3 + 1)*(-x^3 + 1)^(4/3)), x)

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Fricas [B]  time = 1.85947, size = 655, normalized size = 4.28 \begin{align*} -\frac{2 \, \sqrt{6} 2^{\frac{1}{6}}{\left (x^{3} - 1\right )} \arctan \left (-\frac{2^{\frac{1}{6}}{\left (\sqrt{6} 2^{\frac{1}{3}} x - 2 \, \sqrt{6}{\left (-x^{3} + 1\right )}^{\frac{1}{3}}\right )}}{6 \, x}\right ) - 2 \cdot 2^{\frac{2}{3}}{\left (x^{3} - 1\right )} \log \left (\frac{2^{\frac{1}{3}} x +{\left (-x^{3} + 1\right )}^{\frac{1}{3}}}{x}\right ) + 2^{\frac{2}{3}}{\left (x^{3} - 1\right )} \log \left (\frac{2^{\frac{2}{3}} x^{2} - 2^{\frac{1}{3}}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} x +{\left (-x^{3} + 1\right )}^{\frac{2}{3}}}{x^{2}}\right ) - 8 \, \sqrt{3}{\left (x^{3} - 1\right )} \arctan \left (-\frac{\sqrt{3} x - 2 \, \sqrt{3}{\left (-x^{3} + 1\right )}^{\frac{1}{3}}}{3 \, x}\right ) + 8 \,{\left (x^{3} - 1\right )} \log \left (\frac{x +{\left (-x^{3} + 1\right )}^{\frac{1}{3}}}{x}\right ) - 4 \,{\left (x^{3} - 1\right )} \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 ) + 12 \,{\left (-x^{3} + 1\right )}^{\frac{2}{3}} x}{24 \,{\left (x^{3} - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(-x^3+1)^(4/3)/(x^3+1),x, algorithm="fricas")

[Out]

-1/24*(2*sqrt(6)*2^(1/6)*(x^3 - 1)*arctan(-1/6*2^(1/6)*(sqrt(6)*2^(1/3)*x - 2*sqrt(6)*(-x^3 + 1)^(1/3))/x) - 2
*2^(2/3)*(x^3 - 1)*log((2^(1/3)*x + (-x^3 + 1)^(1/3))/x) + 2^(2/3)*(x^3 - 1)*log((2^(2/3)*x^2 - 2^(1/3)*(-x^3
+ 1)^(1/3)*x + (-x^3 + 1)^(2/3))/x^2) - 8*sqrt(3)*(x^3 - 1)*arctan(-1/3*(sqrt(3)*x - 2*sqrt(3)*(-x^3 + 1)^(1/3
))/x) + 8*(x^3 - 1)*log((x + (-x^3 + 1)^(1/3))/x) - 4*(x^3 - 1)*log((x^2 - (-x^3 + 1)^(1/3)*x + (-x^3 + 1)^(2/
3))/x^2) + 12*(-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^{6}}{\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**6/(-x**3+1)**(4/3)/(x**3+1),x)

[Out]

Integral(x**6/((-(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^{6}}{{\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^6/(-x^3+1)^(4/3)/(x^3+1),x, algorithm="giac")

[Out]

integrate(x^6/((x^3 + 1)*(-x^3 + 1)^(4/3)), x)