3.1379 \(\int \frac{\sqrt [3]{x}}{1-x^6} \, dx\)

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 - (ArcTan[(x^(2/3) + Cos[Pi/9])*Csc[Pi/9]]*S
in[(2*Pi)/9])/3

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Rubi [A]  time = 0.304326, 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, Rules used = {329, 275, 294, 634, 618, 204, 628, 31} \[ -\frac{1}{6} \log \left (1-x^{2/3}\right )+\frac{1}{12} \log \left (x^{4/3}+x^{2/3}+1\right )-\frac{\tan ^{-1}\left (\frac{2 x^{2/3}+1}{\sqrt{3}}\right )}{2 \sqrt{3}}-\frac{1}{6} \cos \left (\frac{2 \pi }{9}\right ) \log \left (x^{4/3}+2 x^{2/3} \cos \left (\frac{\pi }{9}\right )+1\right )+\frac{1}{6} \cos \left (\frac{\pi }{9}\right ) \log \left (x^{4/3}-2 x^{2/3} \sin \left (\frac{\pi }{18}\right )+1\right )-\frac{1}{6} \sin \left (\frac{\pi }{18}\right ) \log \left (x^{4/3}-2 x^{2/3} \cos \left (\frac{2 \pi }{9}\right )+1\right )+\frac{1}{3} \cos \left (\frac{\pi }{18}\right ) \tan ^{-1}\left (\csc \left (\frac{2 \pi }{9}\right ) \left (x^{2/3}-\cos \left (\frac{2 \pi }{9}\right )\right )\right )+\frac{1}{3} \sin \left (\frac{\pi }{9}\right ) \tan ^{-1}\left (\sec \left (\frac{\pi }{18}\right ) \left (x^{2/3}-\sin \left (\frac{\pi }{18}\right )\right )\right )-\frac{1}{3} \sin \left (\frac{2 \pi }{9}\right ) \tan ^{-1}\left (\csc \left (\frac{\pi }{9}\right ) \left (x^{2/3}+\cos \left (\frac{\pi }{9}\right )\right )\right ) \]

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 - (ArcTan[(x^(2/3) + Cos[Pi/9])*Csc[Pi/9]]*S
in[(2*Pi)/9])/3

Rule 329

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/c^n)^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 275

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 294

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Module[{r = Numerator[Rt[-(a/b), n]], s = Denominator[Rt
[-(a/b), n]], k, u}, Simp[u = Int[(r*Cos[((2*k - 1)*m*Pi)/n] + s*Cos[((2*k - 1)*(m + 1)*Pi)/n]*x)/(r^2 + 2*r*s
*Cos[((2*k - 1)*Pi)/n]*x + s^2*x^2), x]; (r^(m + 1)*Int[1/(r - s*x), x])/(a*n*s^m) - Dist[(2*(-r)^(m + 1))/(a*
n*s^m), Sum[u, {k, 1, (n - 1)/2}], x], x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 1)/2, 0] && IGtQ[m, 0] && LtQ[m, n
- 1] && NegQ[a/b]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{\sqrt [3]{x}}{1-x^6} \, dx &=3 \operatorname{Subst}\left (\int \frac{x^3}{1-x^{18}} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{3}{2} \operatorname{Subst}\left (\int \frac{x}{1-x^9} \, dx,x,x^{2/3}\right )\\ &=\frac{1}{6} \operatorname{Subst}\left (\int \frac{1}{1-x} \, dx,x,x^{2/3}\right )-\frac{1}{3} \operatorname{Subst}\left (\int \frac{\frac{1}{2}-\frac{x}{2}}{1+x+x^2} \, dx,x,x^{2/3}\right )-\frac{1}{3} \operatorname{Subst}\left (\int \frac{\cos \left (\frac{\pi }{9}\right )+x \cos \left (\frac{2 \pi }{9}\right )}{1+x^2+2 x \cos \left (\frac{\pi }{9}\right )} \, dx,x,x^{2/3}\right )-\frac{1}{3} \operatorname{Subst}\left (\int \frac{-x \cos \left (\frac{\pi }{9}\right )-\sin \left (\frac{\pi }{18}\right )}{1+x^2-2 x \sin \left (\frac{\pi }{18}\right )} \, dx,x,x^{2/3}\right )-\frac{1}{3} \operatorname{Subst}\left (\int \frac{-\cos \left (\frac{2 \pi }{9}\right )+x \sin \left (\frac{\pi }{18}\right )}{1+x^2-2 x \cos \left (\frac{2 \pi }{9}\right )} \, dx,x,x^{2/3}\right )\\ &=-\frac{1}{6} \log \left (1-x^{2/3}\right )+\frac{1}{12} \operatorname{Subst}\left (\int \frac{1+2 x}{1+x+x^2} \, dx,x,x^{2/3}\right )-\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{1+x+x^2} \, dx,x,x^{2/3}\right )+\frac{1}{6} \cos \left (\frac{\pi }{9}\right ) \operatorname{Subst}\left (\int \frac{2 x-2 \sin \left (\frac{\pi }{18}\right )}{1+x^2-2 x \sin \left (\frac{\pi }{18}\right )} \, dx,x,x^{2/3}\right )-\frac{1}{6} \cos \left (\frac{2 \pi }{9}\right ) \operatorname{Subst}\left (\int \frac{2 x+2 \cos \left (\frac{\pi }{9}\right )}{1+x^2+2 x \cos \left (\frac{\pi }{9}\right )} \, dx,x,x^{2/3}\right )+\frac{1}{3} \left (\cos \left (\frac{2 \pi }{9}\right ) \left (1-\sin \left (\frac{\pi }{18}\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1+x^2-2 x \cos \left (\frac{2 \pi }{9}\right )} \, dx,x,x^{2/3}\right )-\frac{1}{6} \sin \left (\frac{\pi }{18}\right ) \operatorname{Subst}\left (\int \frac{2 x-2 \cos \left (\frac{2 \pi }{9}\right )}{1+x^2-2 x \cos \left (\frac{2 \pi }{9}\right )} \, dx,x,x^{2/3}\right )+\frac{1}{3} \left (\left (1+\cos \left (\frac{\pi }{9}\right )\right ) \sin \left (\frac{\pi }{18}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1+x^2-2 x \sin \left (\frac{\pi }{18}\right )} \, dx,x,x^{2/3}\right )-\frac{1}{3} \left (\sin \left (\frac{\pi }{9}\right ) \sin \left (\frac{2 \pi }{9}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1+x^2+2 x \cos \left (\frac{\pi }{9}\right )} \, dx,x,x^{2/3}\right )\\ &=-\frac{1}{6} \log \left (1-x^{2/3}\right )+\frac{1}{12} \log \left (1+x^{2/3}+x^{4/3}\right )-\frac{1}{6} \cos \left (\frac{2 \pi }{9}\right ) \log \left (1+x^{4/3}+2 x^{2/3} \cos \left (\frac{\pi }{9}\right )\right )+\frac{1}{6} \cos \left (\frac{\pi }{9}\right ) \log \left (1+x^{4/3}-2 x^{2/3} \sin \left (\frac{\pi }{18}\right )\right )-\frac{1}{6} \log \left (1+x^{4/3}-2 x^{2/3} \cos \left (\frac{2 \pi }{9}\right )\right ) \sin \left (\frac{\pi }{18}\right )+\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+2 x^{2/3}\right )-\frac{1}{3} \left (2 \cos \left (\frac{2 \pi }{9}\right ) \left (1-\sin \left (\frac{\pi }{18}\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-x^2-4 \sin ^2\left (\frac{2 \pi }{9}\right )} \, dx,x,2 x^{2/3}-2 \cos \left (\frac{2 \pi }{9}\right )\right )-\frac{1}{3} \left (2 \left (1+\cos \left (\frac{\pi }{9}\right )\right ) \sin \left (\frac{\pi }{18}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-x^2-4 \cos ^2\left (\frac{\pi }{18}\right )} \, dx,x,2 x^{2/3}-2 \sin \left (\frac{\pi }{18}\right )\right )+\frac{1}{3} \left (2 \sin \left (\frac{\pi }{9}\right ) \sin \left (\frac{2 \pi }{9}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-x^2-4 \sin ^2\left (\frac{\pi }{9}\right )} \, dx,x,2 \left (x^{2/3}+\cos \left (\frac{\pi }{9}\right )\right )\right )\\ &=-\frac{\tan ^{-1}\left (\frac{1+2 x^{2/3}}{\sqrt{3}}\right )}{2 \sqrt{3}}-\frac{1}{6} \log \left (1-x^{2/3}\right )+\frac{1}{12} \log \left (1+x^{2/3}+x^{4/3}\right )-\frac{1}{6} \cos \left (\frac{2 \pi }{9}\right ) \log \left (1+x^{4/3}+2 x^{2/3} \cos \left (\frac{\pi }{9}\right )\right )+\frac{1}{6} \cos \left (\frac{\pi }{9}\right ) \log \left (1+x^{4/3}-2 x^{2/3} \sin \left (\frac{\pi }{18}\right )\right )+\frac{1}{3} \tan ^{-1}\left (\left (x^{2/3}-\cos \left (\frac{2 \pi }{9}\right )\right ) \csc \left (\frac{2 \pi }{9}\right )\right ) \cot \left (\frac{2 \pi }{9}\right ) \left (1-\sin \left (\frac{\pi }{18}\right )\right )-\frac{1}{6} \log \left (1+x^{4/3}-2 x^{2/3} \cos \left (\frac{2 \pi }{9}\right )\right ) \sin \left (\frac{\pi }{18}\right )+\frac{1}{3} \tan ^{-1}\left (\sec \left (\frac{\pi }{18}\right ) \left (x^{2/3}-\sin \left (\frac{\pi }{18}\right )\right )\right ) \sin \left (\frac{\pi }{9}\right )-\frac{1}{3} \tan ^{-1}\left (\left (x^{2/3}+\cos \left (\frac{\pi }{9}\right )\right ) \csc \left (\frac{\pi }{9}\right )\right ) \sin \left (\frac{2 \pi }{9}\right )\\ \end{align*}

Mathematica [C]  time = 0.0060654, size = 20, normalized size = 0.08 \[ \frac{3}{4} x^{4/3} \, _2F_1\left (\frac{2}{9},1;\frac{11}{9};x^6\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^(1/3)/(1 - x^6),x]

[Out]

(3*x^(4/3)*Hypergeometric2F1[2/9, 1, 11/9, x^6])/4

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Maple [C]  time = 0.023, size = 162, normalized size = 0.7 \begin{align*} -{\frac{1}{6}\ln \left ( -1+\sqrt [3]{x} \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}\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}{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 ( \sqrt [3]{x}+1 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(1/3)/(-x^6+1),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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 ) \end{align*}

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) - integrate(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 [B]  time = 1.99812, size = 1872, normalized size = 7.67 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(sqrt(3)*cos(1/9*pi) - sin(1/9*pi))*arctan(-(8*(2*cos(1/9*pi)^3 - cos(1/9*pi))*sin(1/9*pi) + 2*(2*sqrt(3)*
cos(1/9*pi)^2 + 2*cos(1/9*pi)*sin(1/9*pi) - sqrt(3))*x^(2/3) - 2*(2*sqrt(3)*cos(1/9*pi)^2 + 2*cos(1/9*pi)*sin(
1/9*pi) - sqrt(3))*sqrt((2*sqrt(3)*cos(1/9*pi)*sin(1/9*pi) + 2*cos(1/9*pi)^2 - 1)*x^(2/3) + x^(4/3) + 1) + sqr
t(3))/(16*cos(1/9*pi)^4 - 16*cos(1/9*pi)^2 + 3)) - 1/3*(sqrt(3)*cos(1/9*pi) + sin(1/9*pi))*arctan(1/16*(16*cos
(1/9*pi)^2 - 8*x^(2/3) + sqrt(-128*(2*cos(1/9*pi)^2 - 1)*x^(2/3) + 64*x^(4/3) + 64) - 8)/(cos(1/9*pi)*sin(1/9*
pi))) - 1/12*(sqrt(3)*sin(1/9*pi) + cos(1/9*pi))*log(64*(2*sqrt(3)*cos(1/9*pi)*sin(1/9*pi) + 2*cos(1/9*pi)^2 -
1)*x^(2/3) + 64*x^(4/3) + 64) + 1/6*cos(1/9*pi)*log(-16*(2*sqrt(3)*cos(1/9*pi)*sin(1/9*pi) - 2*cos(1/9*pi)^2
+ 1)*x^(2/3) + 16*x^(4/3) + 16) + 1/12*(sqrt(3)*sin(1/9*pi) - cos(1/9*pi))*log(-128*(2*cos(1/9*pi)^2 - 1)*x^(2
/3) + 64*x^(4/3) + 64) + 2/3*arctan(-1/2*(16*(2*cos(1/9*pi)^3 - cos(1/9*pi))*sin(1/9*pi) - 4*(2*sqrt(3)*cos(1/
9*pi)^2 - 2*cos(1/9*pi)*sin(1/9*pi) - sqrt(3))*x^(2/3) + (2*sqrt(3)*cos(1/9*pi)^2 - 2*cos(1/9*pi)*sin(1/9*pi)
- sqrt(3))*sqrt(-16*(2*sqrt(3)*cos(1/9*pi)*sin(1/9*pi) - 2*cos(1/9*pi)^2 + 1)*x^(2/3) + 16*x^(4/3) + 16) - 2*s
qrt(3))/(16*cos(1/9*pi)^4 - 16*cos(1/9*pi)^2 + 3))*sin(1/9*pi) - 1/6*sqrt(3)*arctan(2/3*sqrt(3)*x^(2/3) + 1/3*
sqrt(3)) + 1/12*log(x^(4/3) + x^(2/3) + 1) - 1/6*log(x^(2/3) - 1)

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

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 [A]  time = 4.55058, size = 293, normalized size = 1.2 \begin{align*} \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 )} \log \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 )} \log \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 )} \log \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} \, \log \left (x^{\frac{4}{3}} + x^{\frac{2}{3}} + 1\right ) - \frac{1}{6} \, \log \left ({\left | x^{\frac{2}{3}} - 1 \right |}\right ) \end{align*}

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*arctan((x^(2/3) - cos(2/9*pi))/s
in(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)*log(-2*x^(2/3)*cos(4/9*pi) + x^(4/3) + 1) - 1/6*(cos(2/9*pi)^2 - sin(2/9
*pi)^2)*log(-2*x^(2/3)*cos(2/9*pi) + x^(4/3) + 1) - 1/6*(cos(1/9*pi)^2 - sin(1/9*pi)^2)*log(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*log(x^(4/3) + x^(2/3) + 1) - 1/6*l
og(abs(x^(2/3) - 1))