3.593 \(\int \frac{1}{x^5 \left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx\)

Optimal. Leaf size=140 \[ -\frac{\log \left (\frac{\sqrt [3]{2} x}{\sqrt [3]{1-x^3}}+1\right )}{3\ 2^{2/3}}-\frac{\tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{2^{2/3} \sqrt{3}}-\frac{\left (1-x^3\right )^{4/3}}{4 x^4}+\frac{\log \left (-\frac{\sqrt [3]{2} x}{\sqrt [3]{1-x^3}}+\frac{2^{2/3} x^2}{\left (1-x^3\right )^{2/3}}+1\right )}{6\ 2^{2/3}} \]

[Out]

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

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Rubi [A]  time = 0.218828, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364 \[ -\frac{\log \left (\frac{\sqrt [3]{2} x}{\sqrt [3]{1-x^3}}+1\right )}{3\ 2^{2/3}}-\frac{\tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{2^{2/3} \sqrt{3}}-\frac{\left (1-x^3\right )^{4/3}}{4 x^4}+\frac{\log \left (-\frac{\sqrt [3]{2} x}{\sqrt [3]{1-x^3}}+\frac{2^{2/3} x^2}{\left (1-x^3\right )^{2/3}}+1\right )}{6\ 2^{2/3}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^5*(1 - x^3)^(2/3)*(1 + x^3)),x]

[Out]

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

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Rubi in Sympy [A]  time = 22.5287, size = 121, normalized size = 0.86 \[ - \frac{\sqrt [3]{2} \log{\left (\frac{\sqrt [3]{2} x}{\sqrt [3]{- x^{3} + 1}} + 1 \right )}}{6} + \frac{\sqrt [3]{2} \log{\left (\frac{2^{\frac{2}{3}} x^{2}}{\left (- x^{3} + 1\right )^{\frac{2}{3}}} - \frac{\sqrt [3]{2} x}{\sqrt [3]{- x^{3} + 1}} + 1 \right )}}{12} - \frac{\sqrt [3]{2} \sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (- \frac{2 \sqrt [3]{2} x}{3 \sqrt [3]{- x^{3} + 1}} + \frac{1}{3}\right ) \right )}}{6} - \frac{\left (- x^{3} + 1\right )^{\frac{4}{3}}}{4 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**5/(-x**3+1)**(2/3)/(x**3+1),x)

[Out]

-2**(1/3)*log(2**(1/3)*x/(-x**3 + 1)**(1/3) + 1)/6 + 2**(1/3)*log(2**(2/3)*x**2/
(-x**3 + 1)**(2/3) - 2**(1/3)*x/(-x**3 + 1)**(1/3) + 1)/12 - 2**(1/3)*sqrt(3)*at
an(sqrt(3)*(-2*2**(1/3)*x/(3*(-x**3 + 1)**(1/3)) + 1/3))/6 - (-x**3 + 1)**(4/3)/
(4*x**4)

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Mathematica [C]  time = 44.1434, size = 1024, normalized size = 7.31 \[ \frac{81 \left (x^3+1\right )^2 \, _4F_3\left (\frac{2}{3},2,2,2;1,1,\frac{8}{3};\frac{2 x^3}{x^3-1}\right ) x^3+5 \left (9 x^9+x^6-9 x^3+\left (9 x^9-20 x^6-13 x^3+4\right ) \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};\frac{2 x^3}{x^3-1}\right )-1\right )+216 \left (x^9+x^6\right ) \, _3F_2\left (\frac{2}{3},2,2;1,\frac{8}{3};\frac{2 x^3}{x^3-1}\right )}{60 x^9 \left (1-x^3\right )^{4/3} \left (x^3-1\right ) \left (x^3+1\right ) \left (-\frac{81 \left (x^3+1\right )^2 \, _4F_3\left (\frac{2}{3},2,2,2;1,1,\frac{8}{3};\frac{2 x^3}{x^3-1}\right ) x^3+5 \left (9 x^9+x^6-9 x^3+\left (9 x^9-20 x^6-13 x^3+4\right ) \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};\frac{2 x^3}{x^3-1}\right )-1\right )+216 \left (x^9+x^6\right ) \, _3F_2\left (\frac{2}{3},2,2;1,\frac{8}{3};\frac{2 x^3}{x^3-1}\right )}{15 x^5 \left (1-x^3\right )^{2/3} \left (x^3-1\right )}+\frac{81 \left (x^3+1\right )^2 \, _4F_3\left (\frac{2}{3},2,2,2;1,1,\frac{8}{3};\frac{2 x^3}{x^3-1}\right ) x^3+5 \left (9 x^9+x^6-9 x^3+\left (9 x^9-20 x^6-13 x^3+4\right ) \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};\frac{2 x^3}{x^3-1}\right )-1\right )+216 \left (x^9+x^6\right ) \, _3F_2\left (\frac{2}{3},2,2;1,\frac{8}{3};\frac{2 x^3}{x^3-1}\right )}{30 x^2 \left (1-x^3\right )^{5/3} \left (x^3-1\right )}-\frac{81 \left (x^3+1\right )^2 \, _4F_3\left (\frac{2}{3},2,2,2;1,1,\frac{8}{3};\frac{2 x^3}{x^3-1}\right ) x^3+5 \left (9 x^9+x^6-9 x^3+\left (9 x^9-20 x^6-13 x^3+4\right ) \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};\frac{2 x^3}{x^3-1}\right )-1\right )+216 \left (x^9+x^6\right ) \, _3F_2\left (\frac{2}{3},2,2;1,\frac{8}{3};\frac{2 x^3}{x^3-1}\right )}{20 x^2 \left (1-x^3\right )^{2/3} \left (x^3-1\right )^2}+\frac{486 \left (x^3+1\right ) \, _4F_3\left (\frac{2}{3},2,2,2;1,1,\frac{8}{3};\frac{2 x^3}{x^3-1}\right ) x^5+162 \left (x^3+1\right )^2 \left (\frac{6 x^2}{x^3-1}-\frac{6 x^5}{\left (x^3-1\right )^2}\right ) \, _4F_3\left (\frac{5}{3},3,3,3;2,2,\frac{11}{3};\frac{2 x^3}{x^3-1}\right ) x^3+243 \left (x^3+1\right )^2 \, _4F_3\left (\frac{2}{3},2,2,2;1,1,\frac{8}{3};\frac{2 x^3}{x^3-1}\right ) x^2+5 \left (81 x^8+6 x^5-27 x^2+\left (81 x^8-120 x^5-39 x^2\right ) \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};\frac{2 x^3}{x^3-1}\right )+\frac{\left (x^3-1\right ) \left (9 x^9-20 x^6-13 x^3+4\right ) \left (\frac{6 x^2}{x^3-1}-\frac{6 x^5}{\left (x^3-1\right )^2}\right ) \left (\frac{1}{1-\frac{2 x^3}{x^3-1}}-\, _2F_1\left (\frac{2}{3},1;\frac{5}{3};\frac{2 x^3}{x^3-1}\right )\right )}{3 x^3}\right )+216 \left (9 x^8+6 x^5\right ) \, _3F_2\left (\frac{2}{3},2,2;1,\frac{8}{3};\frac{2 x^3}{x^3-1}\right )+216 \left (x^9+x^6\right ) \left (\frac{6 x^2}{x^3-1}-\frac{6 x^5}{\left (x^3-1\right )^2}\right ) \, _3F_2\left (\frac{5}{3},3,3;2,\frac{11}{3};\frac{2 x^3}{x^3-1}\right )}{60 x^4 \left (1-x^3\right )^{2/3} \left (x^3-1\right )}\right )} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/(x^5*(1 - x^3)^(2/3)*(1 + x^3)),x]

[Out]

(5*(-1 - 9*x^3 + x^6 + 9*x^9 + (4 - 13*x^3 - 20*x^6 + 9*x^9)*Hypergeometric2F1[2
/3, 1, 5/3, (2*x^3)/(-1 + x^3)]) + 216*(x^6 + x^9)*HypergeometricPFQ[{2/3, 2, 2}
, {1, 8/3}, (2*x^3)/(-1 + x^3)] + 81*x^3*(1 + x^3)^2*HypergeometricPFQ[{2/3, 2,
2, 2}, {1, 1, 8/3}, (2*x^3)/(-1 + x^3)])/(60*x^9*(1 - x^3)^(4/3)*(-1 + x^3)*(1 +
x^3)*(-(5*(-1 - 9*x^3 + x^6 + 9*x^9 + (4 - 13*x^3 - 20*x^6 + 9*x^9)*Hypergeomet
ric2F1[2/3, 1, 5/3, (2*x^3)/(-1 + x^3)]) + 216*(x^6 + x^9)*HypergeometricPFQ[{2/
3, 2, 2}, {1, 8/3}, (2*x^3)/(-1 + x^3)] + 81*x^3*(1 + x^3)^2*HypergeometricPFQ[{
2/3, 2, 2, 2}, {1, 1, 8/3}, (2*x^3)/(-1 + x^3)])/(20*x^2*(1 - x^3)^(2/3)*(-1 + x
^3)^2) + (5*(-1 - 9*x^3 + x^6 + 9*x^9 + (4 - 13*x^3 - 20*x^6 + 9*x^9)*Hypergeome
tric2F1[2/3, 1, 5/3, (2*x^3)/(-1 + x^3)]) + 216*(x^6 + x^9)*HypergeometricPFQ[{2
/3, 2, 2}, {1, 8/3}, (2*x^3)/(-1 + x^3)] + 81*x^3*(1 + x^3)^2*HypergeometricPFQ[
{2/3, 2, 2, 2}, {1, 1, 8/3}, (2*x^3)/(-1 + x^3)])/(30*x^2*(1 - x^3)^(5/3)*(-1 +
x^3)) - (5*(-1 - 9*x^3 + x^6 + 9*x^9 + (4 - 13*x^3 - 20*x^6 + 9*x^9)*Hypergeomet
ric2F1[2/3, 1, 5/3, (2*x^3)/(-1 + x^3)]) + 216*(x^6 + x^9)*HypergeometricPFQ[{2/
3, 2, 2}, {1, 8/3}, (2*x^3)/(-1 + x^3)] + 81*x^3*(1 + x^3)^2*HypergeometricPFQ[{
2/3, 2, 2, 2}, {1, 1, 8/3}, (2*x^3)/(-1 + x^3)])/(15*x^5*(1 - x^3)^(2/3)*(-1 + x
^3)) + (5*(-27*x^2 + 6*x^5 + 81*x^8 + ((-1 + x^3)*(4 - 13*x^3 - 20*x^6 + 9*x^9)*
((-6*x^5)/(-1 + x^3)^2 + (6*x^2)/(-1 + x^3))*((1 - (2*x^3)/(-1 + x^3))^(-1) - Hy
pergeometric2F1[2/3, 1, 5/3, (2*x^3)/(-1 + x^3)]))/(3*x^3) + (-39*x^2 - 120*x^5
+ 81*x^8)*Hypergeometric2F1[2/3, 1, 5/3, (2*x^3)/(-1 + x^3)]) + 216*(6*x^5 + 9*x
^8)*HypergeometricPFQ[{2/3, 2, 2}, {1, 8/3}, (2*x^3)/(-1 + x^3)] + 216*(x^6 + x^
9)*((-6*x^5)/(-1 + x^3)^2 + (6*x^2)/(-1 + x^3))*HypergeometricPFQ[{5/3, 3, 3}, {
2, 11/3}, (2*x^3)/(-1 + x^3)] + 486*x^5*(1 + x^3)*HypergeometricPFQ[{2/3, 2, 2,
2}, {1, 1, 8/3}, (2*x^3)/(-1 + x^3)] + 243*x^2*(1 + x^3)^2*HypergeometricPFQ[{2/
3, 2, 2, 2}, {1, 1, 8/3}, (2*x^3)/(-1 + x^3)] + 162*x^3*(1 + x^3)^2*((-6*x^5)/(-
1 + x^3)^2 + (6*x^2)/(-1 + x^3))*HypergeometricPFQ[{5/3, 3, 3, 3}, {2, 2, 11/3},
(2*x^3)/(-1 + x^3)])/(60*x^4*(1 - x^3)^(2/3)*(-1 + x^3))))

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Maple [F]  time = 0.082, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{5} \left ({x}^{3}+1 \right ) } \left ( -{x}^{3}+1 \right ) ^{-{\frac{2}{3}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^5/(-x^3+1)^(2/3)/(x^3+1),x)

[Out]

int(1/x^5/(-x^3+1)^(2/3)/(x^3+1),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (x^{3} + 1\right )}{\left (-x^{3} + 1\right )}^{\frac{2}{3}} x^{5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^3 + 1)*(-x^3 + 1)^(2/3)*x^5),x, algorithm="maxima")

[Out]

integrate(1/((x^3 + 1)*(-x^3 + 1)^(2/3)*x^5), x)

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Fricas [A]  time = 1.44013, size = 396, normalized size = 2.83 \[ -\frac{4^{\frac{2}{3}} \sqrt{3}{\left (2 \, \sqrt{3} \left (-1\right )^{\frac{1}{3}} x^{4} \log \left (\frac{3 \cdot 4^{\frac{2}{3}} \left (-1\right )^{\frac{2}{3}}{\left (5 \, x^{4} - x\right )}{\left (-x^{3} + 1\right )}^{\frac{2}{3}} - 12 \cdot 4^{\frac{1}{3}}{\left (2 \, x^{5} - x^{2}\right )}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} - 2 \, \left (-1\right )^{\frac{1}{3}}{\left (19 \, x^{6} - 16 \, x^{3} + 1\right )}}{x^{6} + 2 \, x^{3} + 1}\right ) - 4 \, \sqrt{3} \left (-1\right )^{\frac{1}{3}} x^{4} \log \left (-\frac{6 \cdot 4^{\frac{1}{3}} \left (-1\right )^{\frac{1}{3}}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} x^{2} - 3 \cdot 4^{\frac{2}{3}}{\left (-x^{3} + 1\right )}^{\frac{2}{3}} x - 2 \, \left (-1\right )^{\frac{2}{3}}{\left (x^{3} + 1\right )}}{x^{3} + 1}\right ) + 12 \, \left (-1\right )^{\frac{1}{3}} x^{4} \arctan \left (\frac{3 \cdot 4^{\frac{1}{3}} \sqrt{3} \left (-1\right )^{\frac{1}{3}}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} x^{2} + 3 \cdot 4^{\frac{2}{3}} \sqrt{3}{\left (-x^{3} + 1\right )}^{\frac{2}{3}} x - \sqrt{3} \left (-1\right )^{\frac{2}{3}}{\left (x^{3} + 1\right )}}{3 \,{\left (3 \cdot 4^{\frac{1}{3}} \left (-1\right )^{\frac{1}{3}}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} x^{2} + \left (-1\right )^{\frac{2}{3}}{\left (x^{3} + 1\right )}\right )}}\right ) - 9 \cdot 4^{\frac{1}{3}} \sqrt{3}{\left (x^{3} - 1\right )}{\left (-x^{3} + 1\right )}^{\frac{1}{3}}\right )}}{432 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^3 + 1)*(-x^3 + 1)^(2/3)*x^5),x, algorithm="fricas")

[Out]

-1/432*4^(2/3)*sqrt(3)*(2*sqrt(3)*(-1)^(1/3)*x^4*log((3*4^(2/3)*(-1)^(2/3)*(5*x^
4 - x)*(-x^3 + 1)^(2/3) - 12*4^(1/3)*(2*x^5 - x^2)*(-x^3 + 1)^(1/3) - 2*(-1)^(1/
3)*(19*x^6 - 16*x^3 + 1))/(x^6 + 2*x^3 + 1)) - 4*sqrt(3)*(-1)^(1/3)*x^4*log(-(6*
4^(1/3)*(-1)^(1/3)*(-x^3 + 1)^(1/3)*x^2 - 3*4^(2/3)*(-x^3 + 1)^(2/3)*x - 2*(-1)^
(2/3)*(x^3 + 1))/(x^3 + 1)) + 12*(-1)^(1/3)*x^4*arctan(1/3*(3*4^(1/3)*sqrt(3)*(-
1)^(1/3)*(-x^3 + 1)^(1/3)*x^2 + 3*4^(2/3)*sqrt(3)*(-x^3 + 1)^(2/3)*x - sqrt(3)*(
-1)^(2/3)*(x^3 + 1))/(3*4^(1/3)*(-1)^(1/3)*(-x^3 + 1)^(1/3)*x^2 + (-1)^(2/3)*(x^
3 + 1))) - 9*4^(1/3)*sqrt(3)*(x^3 - 1)*(-x^3 + 1)^(1/3))/x^4

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{5} \left (- \left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac{2}{3}} \left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**5/(-x**3+1)**(2/3)/(x**3+1),x)

[Out]

Integral(1/(x**5*(-(x - 1)*(x**2 + x + 1))**(2/3)*(x + 1)*(x**2 - x + 1)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (x^{3} + 1\right )}{\left (-x^{3} + 1\right )}^{\frac{2}{3}} x^{5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^3 + 1)*(-x^3 + 1)^(2/3)*x^5),x, algorithm="giac")

[Out]

integrate(1/((x^3 + 1)*(-x^3 + 1)^(2/3)*x^5), x)