3.579 \(\int \frac{1}{x^9 \sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx\)

Optimal. Leaf size=175 \[ -\frac{\log \left (\frac{\sqrt [3]{2} x}{\sqrt [3]{1-x^3}}+1\right )}{3 \sqrt [3]{2}}+\frac{\tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{\sqrt [3]{2} \sqrt{3}}-\frac{\left (1-x^3\right )^{8/3}}{8 x^8}-\frac{\left (1-x^3\right )^{5/3}}{5 x^5}-\frac{\left (1-x^3\right )^{2/3}}{2 x^2}+\frac{\log \left (-\frac{\sqrt [3]{2} x}{\sqrt [3]{1-x^3}}+\frac{2^{2/3} x^2}{\left (1-x^3\right )^{2/3}}+1\right )}{6 \sqrt [3]{2}} \]

[Out]

-(1 - x^3)^(2/3)/(2*x^2) - (1 - x^3)^(5/3)/(5*x^5) - (1 - x^3)^(8/3)/(8*x^8) + A
rcTan[(1 - (2*2^(1/3)*x)/(1 - x^3)^(1/3))/Sqrt[3]]/(2^(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)]/(6*2^(1/3)) - Log[1
+ (2^(1/3)*x)/(1 - x^3)^(1/3)]/(3*2^(1/3))

_______________________________________________________________________________________

Rubi [A]  time = 0.23688, antiderivative size = 175, 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 \[ -\frac{\log \left (\frac{\sqrt [3]{2} x}{\sqrt [3]{1-x^3}}+1\right )}{3 \sqrt [3]{2}}+\frac{\tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{\sqrt [3]{2} \sqrt{3}}-\frac{\left (1-x^3\right )^{8/3}}{8 x^8}-\frac{\left (1-x^3\right )^{5/3}}{5 x^5}-\frac{\left (1-x^3\right )^{2/3}}{2 x^2}+\frac{\log \left (-\frac{\sqrt [3]{2} x}{\sqrt [3]{1-x^3}}+\frac{2^{2/3} x^2}{\left (1-x^3\right )^{2/3}}+1\right )}{6 \sqrt [3]{2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^9*(1 - x^3)^(1/3)*(1 + x^3)),x]

[Out]

-(1 - x^3)^(2/3)/(2*x^2) - (1 - x^3)^(5/3)/(5*x^5) - (1 - x^3)^(8/3)/(8*x^8) + A
rcTan[(1 - (2*2^(1/3)*x)/(1 - x^3)^(1/3))/Sqrt[3]]/(2^(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)]/(6*2^(1/3)) - Log[1
+ (2^(1/3)*x)/(1 - x^3)^(1/3)]/(3*2^(1/3))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 24.3662, size = 148, normalized size = 0.85 \[ - \frac{2^{\frac{2}{3}} \log{\left (\frac{\sqrt [3]{2} x}{\sqrt [3]{- x^{3} + 1}} + 1 \right )}}{6} + \frac{2^{\frac{2}{3}} \log{\left (\frac{2^{\frac{2}{3}} x^{2}}{\left (- x^{3} + 1\right )^{\frac{2}{3}}} - \frac{\sqrt [3]{2} x}{\sqrt [3]{- x^{3} + 1}} + 1 \right )}}{12} + \frac{2^{\frac{2}{3}} \sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (- \frac{2 \sqrt [3]{2} x}{3 \sqrt [3]{- x^{3} + 1}} + \frac{1}{3}\right ) \right )}}{6} - \frac{\left (- x^{3} + 1\right )^{\frac{2}{3}}}{2 x^{2}} - \frac{\left (- x^{3} + 1\right )^{\frac{5}{3}}}{5 x^{5}} - \frac{\left (- x^{3} + 1\right )^{\frac{8}{3}}}{8 x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**9/(-x**3+1)**(1/3)/(x**3+1),x)

[Out]

-2**(2/3)*log(2**(1/3)*x/(-x**3 + 1)**(1/3) + 1)/6 + 2**(2/3)*log(2**(2/3)*x**2/
(-x**3 + 1)**(2/3) - 2**(1/3)*x/(-x**3 + 1)**(1/3) + 1)/12 + 2**(2/3)*sqrt(3)*at
an(sqrt(3)*(-2*2**(1/3)*x/(3*(-x**3 + 1)**(1/3)) + 1/3))/6 - (-x**3 + 1)**(2/3)/
(2*x**2) - (-x**3 + 1)**(5/3)/(5*x**5) - (-x**3 + 1)**(8/3)/(8*x**8)

_______________________________________________________________________________________

Mathematica [A]  time = 0.174408, size = 133, normalized size = 0.76 \[ \frac{-2 \log \left (\frac{\sqrt [3]{2} x}{\sqrt [3]{x^3-1}}+1\right )-2 \sqrt{3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{2} x}{\sqrt [3]{x^3-1}}-1}{\sqrt{3}}\right )+\log \left (-\frac{\sqrt [3]{2} x}{\sqrt [3]{x^3-1}}+\frac{2^{2/3} x^2}{\left (x^3-1\right )^{2/3}}+1\right )}{6 \sqrt [3]{2}}-\frac{\left (1-x^3\right )^{2/3} \left (17 x^6-2 x^3+5\right )}{40 x^8} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/(x^9*(1 - x^3)^(1/3)*(1 + x^3)),x]

[Out]

-((1 - x^3)^(2/3)*(5 - 2*x^3 + 17*x^6))/(40*x^8) + (-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)])/(6*2^(1
/3))

_______________________________________________________________________________________

Maple [F]  time = 0.043, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{9} \left ({x}^{3}+1 \right ) }{\frac{1}{\sqrt [3]{-{x}^{3}+1}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^9/(-x^3+1)^(1/3)/(x^3+1),x)

[Out]

int(1/x^9/(-x^3+1)^(1/3)/(x^3+1),x)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (x^{3} + 1\right )}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} x^{9}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^3 + 1)*(-x^3 + 1)^(1/3)*x^9),x, algorithm="maxima")

[Out]

integrate(1/((x^3 + 1)*(-x^3 + 1)^(1/3)*x^9), x)

_______________________________________________________________________________________

Fricas [A]  time = 1.84806, size = 402, normalized size = 2.3 \[ \frac{\sqrt{3} 2^{\frac{2}{3}}{\left (40 \, \sqrt{3} \left (-1\right )^{\frac{1}{3}} x^{8} \log \left (\frac{6 \, \left (-1\right )^{\frac{2}{3}}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} x^{2} + 3 \cdot 2^{\frac{2}{3}}{\left (-x^{3} + 1\right )}^{\frac{2}{3}} x - 2^{\frac{1}{3}} \left (-1\right )^{\frac{1}{3}}{\left (x^{3} + 1\right )}}{x^{3} + 1}\right ) - 20 \, \sqrt{3} \left (-1\right )^{\frac{1}{3}} x^{8} \log \left (\frac{2^{\frac{2}{3}} \left (-1\right )^{\frac{2}{3}}{\left (19 \, x^{6} - 16 \, x^{3} + 1\right )} - 6 \, \left (-1\right )^{\frac{1}{3}}{\left (5 \, x^{4} - x\right )}{\left (-x^{3} + 1\right )}^{\frac{2}{3}} - 12 \cdot 2^{\frac{1}{3}}{\left (2 \, x^{5} - x^{2}\right )}{\left (-x^{3} + 1\right )}^{\frac{1}{3}}}{x^{6} + 2 \, x^{3} + 1}\right ) - 120 \, \left (-1\right )^{\frac{1}{3}} x^{8} \arctan \left (-\frac{6 \, \sqrt{3} \left (-1\right )^{\frac{2}{3}}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} x^{2} - 6 \, \sqrt{3} 2^{\frac{2}{3}}{\left (-x^{3} + 1\right )}^{\frac{2}{3}} x - \sqrt{3} 2^{\frac{1}{3}} \left (-1\right )^{\frac{1}{3}}{\left (x^{3} + 1\right )}}{3 \,{\left (6 \, \left (-1\right )^{\frac{2}{3}}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} x^{2} + 2^{\frac{1}{3}} \left (-1\right )^{\frac{1}{3}}{\left (x^{3} + 1\right )}\right )}}\right ) - 9 \, \sqrt{3} 2^{\frac{1}{3}}{\left (17 \, x^{6} - 2 \, x^{3} + 5\right )}{\left (-x^{3} + 1\right )}^{\frac{2}{3}}\right )}}{2160 \, x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^3 + 1)*(-x^3 + 1)^(1/3)*x^9),x, algorithm="fricas")

[Out]

1/2160*sqrt(3)*2^(2/3)*(40*sqrt(3)*(-1)^(1/3)*x^8*log((6*(-1)^(2/3)*(-x^3 + 1)^(
1/3)*x^2 + 3*2^(2/3)*(-x^3 + 1)^(2/3)*x - 2^(1/3)*(-1)^(1/3)*(x^3 + 1))/(x^3 + 1
)) - 20*sqrt(3)*(-1)^(1/3)*x^8*log((2^(2/3)*(-1)^(2/3)*(19*x^6 - 16*x^3 + 1) - 6
*(-1)^(1/3)*(5*x^4 - x)*(-x^3 + 1)^(2/3) - 12*2^(1/3)*(2*x^5 - x^2)*(-x^3 + 1)^(
1/3))/(x^6 + 2*x^3 + 1)) - 120*(-1)^(1/3)*x^8*arctan(-1/3*(6*sqrt(3)*(-1)^(2/3)*
(-x^3 + 1)^(1/3)*x^2 - 6*sqrt(3)*2^(2/3)*(-x^3 + 1)^(2/3)*x - sqrt(3)*2^(1/3)*(-
1)^(1/3)*(x^3 + 1))/(6*(-1)^(2/3)*(-x^3 + 1)^(1/3)*x^2 + 2^(1/3)*(-1)^(1/3)*(x^3
+ 1))) - 9*sqrt(3)*2^(1/3)*(17*x^6 - 2*x^3 + 5)*(-x^3 + 1)^(2/3))/x^8

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{9} \sqrt [3]{- \left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**9/(-x**3+1)**(1/3)/(x**3+1),x)

[Out]

Integral(1/(x**9*(-(x - 1)*(x**2 + x + 1))**(1/3)*(x + 1)*(x**2 - x + 1)), x)

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (x^{3} + 1\right )}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} x^{9}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^3 + 1)*(-x^3 + 1)^(1/3)*x^9),x, algorithm="giac")

[Out]

integrate(1/((x^3 + 1)*(-x^3 + 1)^(1/3)*x^9), x)