3.1377 \(\int \frac{\sqrt [3]{x}}{1-x^6} \, dx\)

Optimal. Leaf size=244 \[ -\frac{1}{6} \log \left (1-x^{2/3}\right )+\frac{1}{12} \log \left (x^{4/3}+x^{2/3}+1\right )-\frac{\tan ^{-1}\left (\frac{2 x^{2/3}+1}{\sqrt{3}}\right )}{2 \sqrt{3}}-\frac{1}{6} \cos \left (\frac{2 \pi }{9}\right ) \log \left (x^{4/3}+2 x^{2/3} \cos \left (\frac{\pi }{9}\right )+1\right )+\frac{1}{6} \cos \left (\frac{\pi }{9}\right ) \log \left (x^{4/3}-2 x^{2/3} \sin \left (\frac{\pi }{18}\right )+1\right )-\frac{1}{6} \sin \left (\frac{\pi }{18}\right ) \log \left (x^{4/3}-2 x^{2/3} \cos \left (\frac{2 \pi }{9}\right )+1\right )+\frac{1}{3} \cos \left (\frac{\pi }{18}\right ) \tan ^{-1}\left (\csc \left (\frac{2 \pi }{9}\right ) \left (x^{2/3}-\cos \left (\frac{2 \pi }{9}\right )\right )\right )+\frac{1}{3} \sin \left (\frac{\pi }{9}\right ) \tan ^{-1}\left (\sec \left (\frac{\pi }{18}\right ) \left (x^{2/3}-\sin \left (\frac{\pi }{18}\right )\right )\right )-\frac{1}{3} \sin \left (\frac{2 \pi }{9}\right ) \tan ^{-1}\left (\csc \left (\frac{\pi }{9}\right ) \left (x^{2/3}+\cos \left (\frac{\pi }{9}\right )\right )\right ) \]

[Out]

-ArcTan[(1 + 2*x^(2/3))/Sqrt[3]]/(2*Sqrt[3]) + (ArcTan[(x^(2/3) - Cos[(2*Pi)/9])
*Csc[(2*Pi)/9]]*Cos[Pi/18])/3 - Log[1 - x^(2/3)]/6 + Log[1 + x^(2/3) + x^(4/3)]/
12 - (Cos[(2*Pi)/9]*Log[1 + x^(4/3) + 2*x^(2/3)*Cos[Pi/9]])/6 + (Cos[Pi/9]*Log[1
+ x^(4/3) - 2*x^(2/3)*Sin[Pi/18]])/6 - (Log[1 + x^(4/3) - 2*x^(2/3)*Cos[(2*Pi)/
9]]*Sin[Pi/18])/6 + (ArcTan[Sec[Pi/18]*(x^(2/3) - Sin[Pi/18])]*Sin[Pi/9])/3 - (A
rcTan[(x^(2/3) + Cos[Pi/9])*Csc[Pi/9]]*Sin[(2*Pi)/9])/3

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Rubi [A]  time = 0.551223, antiderivative size = 244, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.533 \[ -\frac{1}{6} \log \left (1-x^{2/3}\right )+\frac{1}{12} \log \left (x^{4/3}+x^{2/3}+1\right )-\frac{\tan ^{-1}\left (\frac{2 x^{2/3}+1}{\sqrt{3}}\right )}{2 \sqrt{3}}-\frac{1}{6} \cos \left (\frac{2 \pi }{9}\right ) \log \left (x^{4/3}+2 x^{2/3} \cos \left (\frac{\pi }{9}\right )+1\right )+\frac{1}{6} \cos \left (\frac{\pi }{9}\right ) \log \left (x^{4/3}-2 x^{2/3} \sin \left (\frac{\pi }{18}\right )+1\right )-\frac{1}{6} \sin \left (\frac{\pi }{18}\right ) \log \left (x^{4/3}-2 x^{2/3} \cos \left (\frac{2 \pi }{9}\right )+1\right )+\frac{1}{3} \cos \left (\frac{\pi }{18}\right ) \tan ^{-1}\left (\csc \left (\frac{2 \pi }{9}\right ) \left (x^{2/3}-\cos \left (\frac{2 \pi }{9}\right )\right )\right )+\frac{1}{3} \sin \left (\frac{\pi }{9}\right ) \tan ^{-1}\left (\sec \left (\frac{\pi }{18}\right ) \left (x^{2/3}-\sin \left (\frac{\pi }{18}\right )\right )\right )-\frac{1}{3} \sin \left (\frac{2 \pi }{9}\right ) \tan ^{-1}\left (\csc \left (\frac{\pi }{9}\right ) \left (x^{2/3}+\cos \left (\frac{\pi }{9}\right )\right )\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^(1/3)/(1 - x^6),x]

[Out]

-ArcTan[(1 + 2*x^(2/3))/Sqrt[3]]/(2*Sqrt[3]) + (ArcTan[(x^(2/3) - Cos[(2*Pi)/9])
*Csc[(2*Pi)/9]]*Cos[Pi/18])/3 - Log[1 - x^(2/3)]/6 + Log[1 + x^(2/3) + x^(4/3)]/
12 - (Cos[(2*Pi)/9]*Log[1 + x^(4/3) + 2*x^(2/3)*Cos[Pi/9]])/6 + (Cos[Pi/9]*Log[1
+ x^(4/3) - 2*x^(2/3)*Sin[Pi/18]])/6 - (Log[1 + x^(4/3) - 2*x^(2/3)*Cos[(2*Pi)/
9]]*Sin[Pi/18])/6 + (ArcTan[Sec[Pi/18]*(x^(2/3) - Sin[Pi/18])]*Sin[Pi/9])/3 - (A
rcTan[(x^(2/3) + Cos[Pi/9])*Csc[Pi/9]]*Sin[(2*Pi)/9])/3

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Rubi in Sympy [A]  time = 113.974, size = 298, normalized size = 1.22 \[ - \frac{\log{\left (- x^{\frac{2}{3}} + 1 \right )}}{6} + \frac{\log{\left (x^{\frac{4}{3}} + x^{\frac{2}{3}} + 1 \right )}}{12} - \frac{\log{\left (x^{\frac{4}{3}} + 2 x^{\frac{2}{3}} \cos{\left (\frac{\pi }{9} \right )} + 1 \right )} \cos{\left (\frac{2 \pi }{9} \right )}}{6} - \frac{\log{\left (x^{\frac{4}{3}} - 2 x^{\frac{2}{3}} \cos{\left (\frac{2 \pi }{9} \right )} + 1 \right )} \cos{\left (\frac{4 \pi }{9} \right )}}{6} + \frac{\log{\left (x^{\frac{4}{3}} - 2 x^{\frac{2}{3}} \cos{\left (\frac{4 \pi }{9} \right )} + 1 \right )} \cos{\left (\frac{\pi }{9} \right )}}{6} - \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x^{\frac{2}{3}}}{3} + \frac{1}{3}\right ) \right )}}{6} + \frac{\sqrt{2} \sqrt{\sin{\left (\frac{7 \pi }{18} \right )} + 1} \sin{\left (\frac{\pi }{18} \right )} \operatorname{atan}{\left (\frac{\sqrt{2} \left (x^{\frac{2}{3}} - \sin{\left (\frac{\pi }{18} \right )}\right )}{\sqrt{\sin{\left (\frac{7 \pi }{18} \right )} + 1}} \right )}}{3} - \frac{\sqrt{2} \sqrt{- \sin{\left (\frac{5 \pi }{18} \right )} + 1} \sin{\left (\frac{7 \pi }{18} \right )} \operatorname{atan}{\left (\frac{\sqrt{2} \left (x^{\frac{2}{3}} + \cos{\left (\frac{\pi }{9} \right )}\right )}{\sqrt{- \sin{\left (\frac{5 \pi }{18} \right )} + 1}} \right )}}{3} + \frac{\sqrt{2} \sqrt{- \sin{\left (\frac{\pi }{18} \right )} + 1} \sin{\left (\frac{5 \pi }{18} \right )} \operatorname{atan}{\left (\frac{\sqrt{2} \left (x^{\frac{2}{3}} - \cos{\left (\frac{2 \pi }{9} \right )}\right )}{\sqrt{- \sin{\left (\frac{\pi }{18} \right )} + 1}} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(1/3)/(-x**6+1),x)

[Out]

-log(-x**(2/3) + 1)/6 + log(x**(4/3) + x**(2/3) + 1)/12 - log(x**(4/3) + 2*x**(2
/3)*cos(pi/9) + 1)*cos(2*pi/9)/6 - log(x**(4/3) - 2*x**(2/3)*cos(2*pi/9) + 1)*co
s(4*pi/9)/6 + log(x**(4/3) - 2*x**(2/3)*cos(4*pi/9) + 1)*cos(pi/9)/6 - sqrt(3)*a
tan(sqrt(3)*(2*x**(2/3)/3 + 1/3))/6 + sqrt(2)*sqrt(sin(7*pi/18) + 1)*sin(pi/18)*
atan(sqrt(2)*(x**(2/3) - sin(pi/18))/sqrt(sin(7*pi/18) + 1))/3 - sqrt(2)*sqrt(-s
in(5*pi/18) + 1)*sin(7*pi/18)*atan(sqrt(2)*(x**(2/3) + cos(pi/9))/sqrt(-sin(5*pi
/18) + 1))/3 + sqrt(2)*sqrt(-sin(pi/18) + 1)*sin(5*pi/18)*atan(sqrt(2)*(x**(2/3)
- cos(2*pi/9))/sqrt(-sin(pi/18) + 1))/3

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Mathematica [A]  time = 0.67361, size = 448, normalized size = 1.84 \[ \frac{1}{12} \left (\log \left (x^{2/3}-\sqrt [3]{x}+1\right )+\log \left (x^{2/3}+\sqrt [3]{x}+1\right )+2 \cos \left (\frac{\pi }{9}\right ) \log \left (x^{2/3}-2 \sqrt [3]{x} \cos \left (\frac{2 \pi }{9}\right )+1\right )+2 \cos \left (\frac{\pi }{9}\right ) \log \left (x^{2/3}+2 \sqrt [3]{x} \cos \left (\frac{2 \pi }{9}\right )+1\right )-2 \cos \left (\frac{2 \pi }{9}\right ) \log \left (x^{2/3}-2 \sqrt [3]{x} \sin \left (\frac{\pi }{18}\right )+1\right )-2 \cos \left (\frac{2 \pi }{9}\right ) \log \left (x^{2/3}+2 \sqrt [3]{x} \sin \left (\frac{\pi }{18}\right )+1\right )-2 \sin \left (\frac{\pi }{18}\right ) \log \left (x^{2/3}-2 \sqrt [3]{x} \cos \left (\frac{\pi }{9}\right )+1\right )-2 \sin \left (\frac{\pi }{18}\right ) \log \left (x^{2/3}+2 \sqrt [3]{x} \cos \left (\frac{\pi }{9}\right )+1\right )-2 \log \left (1-\sqrt [3]{x}\right )-2 \log \left (\sqrt [3]{x}+1\right )-2 \sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{x}-1}{\sqrt{3}}\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{x}+1}{\sqrt{3}}\right )-4 \cos \left (\frac{\pi }{18}\right ) \tan ^{-1}\left (\csc \left (\frac{\pi }{9}\right ) \left (\sqrt [3]{x}+\cos \left (\frac{\pi }{9}\right )\right )\right )+4 \sin \left (\frac{2 \pi }{9}\right ) \tan ^{-1}\left (\sec \left (\frac{\pi }{18}\right ) \left (\sqrt [3]{x}+\sin \left (\frac{\pi }{18}\right )\right )\right )-4 \sin \left (\frac{2 \pi }{9}\right ) \tan ^{-1}\left (\sqrt [3]{x} \sec \left (\frac{\pi }{18}\right )-\tan \left (\frac{\pi }{18}\right )\right )-4 \cos \left (\frac{\pi }{18}\right ) \tan ^{-1}\left (\cot \left (\frac{\pi }{9}\right )-\sqrt [3]{x} \csc \left (\frac{\pi }{9}\right )\right )+4 \sin \left (\frac{\pi }{9}\right ) \tan ^{-1}\left (\csc \left (\frac{2 \pi }{9}\right ) \left (\sqrt [3]{x}-\cos \left (\frac{2 \pi }{9}\right )\right )\right )-4 \sin \left (\frac{\pi }{9}\right ) \tan ^{-1}\left (\csc \left (\frac{2 \pi }{9}\right ) \left (\sqrt [3]{x}+\cos \left (\frac{2 \pi }{9}\right )\right )\right )\right ) \]

Warning: Unable to verify antiderivative.

[In]  Integrate[x^(1/3)/(1 - x^6),x]

[Out]

(-2*Sqrt[3]*ArcTan[(-1 + 2*x^(1/3))/Sqrt[3]] + 2*Sqrt[3]*ArcTan[(1 + 2*x^(1/3))/
Sqrt[3]] - 4*ArcTan[(x^(1/3) + Cos[Pi/9])*Csc[Pi/9]]*Cos[Pi/18] - 4*ArcTan[Cot[P
i/9] - x^(1/3)*Csc[Pi/9]]*Cos[Pi/18] - 2*Log[1 - x^(1/3)] - 2*Log[1 + x^(1/3)] +
Log[1 - x^(1/3) + x^(2/3)] + Log[1 + x^(1/3) + x^(2/3)] + 2*Cos[Pi/9]*Log[1 + x
^(2/3) - 2*x^(1/3)*Cos[(2*Pi)/9]] + 2*Cos[Pi/9]*Log[1 + x^(2/3) + 2*x^(1/3)*Cos[
(2*Pi)/9]] - 2*Cos[(2*Pi)/9]*Log[1 + x^(2/3) - 2*x^(1/3)*Sin[Pi/18]] - 2*Cos[(2*
Pi)/9]*Log[1 + x^(2/3) + 2*x^(1/3)*Sin[Pi/18]] - 2*Log[1 + x^(2/3) - 2*x^(1/3)*C
os[Pi/9]]*Sin[Pi/18] - 2*Log[1 + x^(2/3) + 2*x^(1/3)*Cos[Pi/9]]*Sin[Pi/18] + 4*A
rcTan[(x^(1/3) - Cos[(2*Pi)/9])*Csc[(2*Pi)/9]]*Sin[Pi/9] - 4*ArcTan[(x^(1/3) + C
os[(2*Pi)/9])*Csc[(2*Pi)/9]]*Sin[Pi/9] + 4*ArcTan[Sec[Pi/18]*(x^(1/3) + Sin[Pi/1
8])]*Sin[(2*Pi)/9] - 4*ArcTan[x^(1/3)*Sec[Pi/18] - Tan[Pi/18]]*Sin[(2*Pi)/9])/12

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Maple [C]  time = 0.046, size = 162, normalized size = 0.7 \[{\frac{1}{12}\ln \left ({x}^{{\frac{2}{3}}}+\sqrt [3]{x}+1 \right ) }+{\frac{\sqrt{3}}{6}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,\sqrt [3]{x}+1 \right ) } \right ) }-{\frac{1}{6}\ln \left ( \sqrt [3]{x}-1 \right ) }-{\frac{1}{6}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{6}+{{\it \_Z}}^{3}+1 \right ) }{\frac{-{{\it \_R}}^{3}+1}{2\,{{\it \_R}}^{5}+{{\it \_R}}^{2}}\ln \left ( \sqrt [3]{x}-{\it \_R} \right ) }}+{\frac{1}{12}\ln \left ({x}^{{\frac{2}{3}}}-\sqrt [3]{x}+1 \right ) }-{\frac{\sqrt{3}}{6}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,\sqrt [3]{x}-1 \right ) } \right ) }-{\frac{1}{6}\ln \left ( 1+\sqrt [3]{x} \right ) }+{\frac{1}{6}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{6}-{{\it \_Z}}^{3}+1 \right ) }{\frac{{{\it \_R}}^{3}+1}{2\,{{\it \_R}}^{5}-{{\it \_R}}^{2}}\ln \left ( \sqrt [3]{x}-{\it \_R} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(1/3)/(-x^6+1),x)

[Out]

1/12*ln(x^(2/3)+x^(1/3)+1)+1/6*3^(1/2)*arctan(1/3*(2*x^(1/3)+1)*3^(1/2))-1/6*ln(
x^(1/3)-1)-1/6*sum((-_R^3+1)/(2*_R^5+_R^2)*ln(x^(1/3)-_R),_R=RootOf(_Z^6+_Z^3+1)
)+1/12*ln(x^(2/3)-x^(1/3)+1)-1/6*3^(1/2)*arctan(1/3*(2*x^(1/3)-1)*3^(1/2))-1/6*l
n(1+x^(1/3))+1/6*sum((_R^3+1)/(2*_R^5-_R^2)*ln(x^(1/3)-_R),_R=RootOf(_Z^6-_Z^3+1
))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{\frac{1}{3}} + 1\right )}\right ) - \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{\frac{1}{3}} - 1\right )}\right ) + \int \frac{x^{\frac{4}{3}} + 2 \, x^{\frac{1}{3}}}{6 \,{\left (x^{2} + x + 1\right )}}\,{d x} - \int \frac{x^{\frac{4}{3}} - 2 \, x^{\frac{1}{3}}}{6 \,{\left (x^{2} - x + 1\right )}}\,{d x} + \frac{1}{12} \, \log \left (x^{\frac{2}{3}} + x^{\frac{1}{3}} + 1\right ) + \frac{1}{12} \, \log \left (x^{\frac{2}{3}} - x^{\frac{1}{3}} + 1\right ) - \frac{1}{6} \, \log \left (x^{\frac{1}{3}} + 1\right ) - \frac{1}{6} \, \log \left (x^{\frac{1}{3}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x^(1/3)/(x^6 - 1),x, algorithm="maxima")

[Out]

1/6*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^(1/3) + 1)) - 1/6*sqrt(3)*arctan(1/3*sqrt(3)
*(2*x^(1/3) - 1)) + integrate(1/6*(x^(4/3) + 2*x^(1/3))/(x^2 + x + 1), x) - inte
grate(1/6*(x^(4/3) - 2*x^(1/3))/(x^2 - x + 1), x) + 1/12*log(x^(2/3) + x^(1/3) +
1) + 1/12*log(x^(2/3) - x^(1/3) + 1) - 1/6*log(x^(1/3) + 1) - 1/6*log(x^(1/3) -
1)

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Fricas [A]  time = 0.264349, size = 671, normalized size = 2.75 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x^(1/3)/(x^6 - 1),x, algorithm="fricas")

[Out]

1/36*sqrt(3)*(2*sqrt(3)*cos(1/9*pi)*log(-3*(2*sqrt(3)*cos(1/9*pi)*sin(1/9*pi) -
2*cos(1/9*pi)^2 + 1)*x^(2/3) + 3*x^(4/3) + 3) - 8*sqrt(3)*arctan((2*sqrt(3)*cos(
1/9*pi)*sin(1/9*pi) + 6*cos(1/9*pi)^2 - 3)/(2*sqrt(3)*cos(1/9*pi)^2 - 6*cos(1/9*
pi)*sin(1/9*pi) + 2*sqrt(3)*x^(2/3) - sqrt(3) + 2*sqrt(-3*(2*sqrt(3)*cos(1/9*pi)
*sin(1/9*pi) - 2*cos(1/9*pi)^2 + 1)*x^(2/3) + 3*x^(4/3) + 3)))*sin(1/9*pi) - 4*(
sqrt(3)*sin(1/9*pi) + 3*cos(1/9*pi))*arctan(-2*sqrt(3)*cos(1/9*pi)*sin(1/9*pi)/(
2*sqrt(3)*cos(1/9*pi)^2 - sqrt(3)*x^(2/3) - sqrt(3) - sqrt(-6*(2*cos(1/9*pi)^2 -
1)*x^(2/3) + 3*x^(4/3) + 3))) + 4*(sqrt(3)*sin(1/9*pi) - 3*cos(1/9*pi))*arctan(
(2*sqrt(3)*cos(1/9*pi)*sin(1/9*pi) - 6*cos(1/9*pi)^2 + 3)/(2*sqrt(3)*cos(1/9*pi)
^2 + 6*cos(1/9*pi)*sin(1/9*pi) + 2*sqrt(3)*x^(2/3) - sqrt(3) + 2*sqrt(3*(2*sqrt(
3)*cos(1/9*pi)*sin(1/9*pi) + 2*cos(1/9*pi)^2 - 1)*x^(2/3) + 3*x^(4/3) + 3))) - (
sqrt(3)*cos(1/9*pi) + 3*sin(1/9*pi))*log(3*(2*sqrt(3)*cos(1/9*pi)*sin(1/9*pi) +
2*cos(1/9*pi)^2 - 1)*x^(2/3) + 3*x^(4/3) + 3) - (sqrt(3)*cos(1/9*pi) - 3*sin(1/9
*pi))*log(-6*(2*cos(1/9*pi)^2 - 1)*x^(2/3) + 3*x^(4/3) + 3) + sqrt(3)*log(x^(4/3
) + x^(2/3) + 1) - 2*sqrt(3)*log(x^(2/3) - 1) - 6*arctan(2/3*sqrt(3)*x^(2/3) + 1
/3*sqrt(3)))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(1/3)/(-x**6+1),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 1.10577, size = 293, normalized size = 1.2 \[ \frac{2}{3} \, \arctan \left (\frac{x^{\frac{2}{3}} - \cos \left (\frac{4}{9} \, \pi \right )}{\sin \left (\frac{4}{9} \, \pi \right )}\right ) \cos \left (\frac{4}{9} \, \pi \right ) \sin \left (\frac{4}{9} \, \pi \right ) + \frac{2}{3} \, \arctan \left (\frac{x^{\frac{2}{3}} - \cos \left (\frac{2}{9} \, \pi \right )}{\sin \left (\frac{2}{9} \, \pi \right )}\right ) \cos \left (\frac{2}{9} \, \pi \right ) \sin \left (\frac{2}{9} \, \pi \right ) - \frac{2}{3} \, \arctan \left (\frac{x^{\frac{2}{3}} + \cos \left (\frac{1}{9} \, \pi \right )}{\sin \left (\frac{1}{9} \, \pi \right )}\right ) \cos \left (\frac{1}{9} \, \pi \right ) \sin \left (\frac{1}{9} \, \pi \right ) - \frac{1}{6} \,{\left (\cos \left (\frac{4}{9} \, \pi \right )^{2} - \sin \left (\frac{4}{9} \, \pi \right )^{2}\right )}{\rm ln}\left (-2 \, x^{\frac{2}{3}} \cos \left (\frac{4}{9} \, \pi \right ) + x^{\frac{4}{3}} + 1\right ) - \frac{1}{6} \,{\left (\cos \left (\frac{2}{9} \, \pi \right )^{2} - \sin \left (\frac{2}{9} \, \pi \right )^{2}\right )}{\rm ln}\left (-2 \, x^{\frac{2}{3}} \cos \left (\frac{2}{9} \, \pi \right ) + x^{\frac{4}{3}} + 1\right ) - \frac{1}{6} \,{\left (\cos \left (\frac{1}{9} \, \pi \right )^{2} - \sin \left (\frac{1}{9} \, \pi \right )^{2}\right )}{\rm ln}\left (2 \, x^{\frac{2}{3}} \cos \left (\frac{1}{9} \, \pi \right ) + x^{\frac{4}{3}} + 1\right ) - \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{\frac{2}{3}} + 1\right )}\right ) + \frac{1}{12} \,{\rm ln}\left (x^{\frac{4}{3}} + x^{\frac{2}{3}} + 1\right ) - \frac{1}{6} \,{\rm ln}\left ({\left | x^{\frac{2}{3}} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x^(1/3)/(x^6 - 1),x, algorithm="giac")

[Out]

2/3*arctan((x^(2/3) - cos(4/9*pi))/sin(4/9*pi))*cos(4/9*pi)*sin(4/9*pi) + 2/3*ar
ctan((x^(2/3) - cos(2/9*pi))/sin(2/9*pi))*cos(2/9*pi)*sin(2/9*pi) - 2/3*arctan((
x^(2/3) + cos(1/9*pi))/sin(1/9*pi))*cos(1/9*pi)*sin(1/9*pi) - 1/6*(cos(4/9*pi)^2
- sin(4/9*pi)^2)*ln(-2*x^(2/3)*cos(4/9*pi) + x^(4/3) + 1) - 1/6*(cos(2/9*pi)^2
- sin(2/9*pi)^2)*ln(-2*x^(2/3)*cos(2/9*pi) + x^(4/3) + 1) - 1/6*(cos(1/9*pi)^2 -
sin(1/9*pi)^2)*ln(2*x^(2/3)*cos(1/9*pi) + x^(4/3) + 1) - 1/6*sqrt(3)*arctan(1/3
*sqrt(3)*(2*x^(2/3) + 1)) + 1/12*ln(x^(4/3) + x^(2/3) + 1) - 1/6*ln(abs(x^(2/3)
- 1))