\(\int \frac {-1+x^6}{\sqrt [3]{x^2+x^4} (1+x^6)} \, dx\) [2124]

Optimal result

Integrand size = 24, antiderivative size = 154 \[ \int \frac {-1+x^6}{\sqrt [3]{x^2+x^4} \left (1+x^6\right )} \, dx=-\frac {\left (x^2+x^4\right )^{2/3}}{x \left (1+x^2\right )}+\frac {\arctan \left (\frac {3^{2/3} x \sqrt [3]{x^2+x^4}}{\sqrt [3]{3} x^2-\left (x^2+x^4\right )^{2/3}}\right )}{3^{2/3}}-\frac {2 \text {arctanh}\left (\frac {\sqrt [6]{3} x}{\sqrt [3]{x^2+x^4}}\right )}{3 \sqrt [6]{3}}-\frac {\text {arctanh}\left (\frac {\sqrt [6]{3} x^2+\frac {\left (x^2+x^4\right )^{2/3}}{\sqrt [6]{3}}}{x \sqrt [3]{x^2+x^4}}\right )}{3 \sqrt [6]{3}} \]

[Out]

-(x^4+x^2)^(2/3)/x/(x^2+1)+1/3*arctan(3^(2/3)*x*(x^4+x^2)^(1/3)/(3^(1/3)*x^2-(x^4+x^2)^(2/3)))*3^(1/3)-2/9*arc
tanh(3^(1/6)*x/(x^4+x^2)^(1/3))*3^(5/6)-1/9*arctanh((3^(1/6)*x^2+1/3*(x^4+x^2)^(2/3)*3^(5/6))/x/(x^4+x^2)^(1/3
))*3^(5/6)

Rubi [F]

\[ \int \frac {-1+x^6}{\sqrt [3]{x^2+x^4} \left (1+x^6\right )} \, dx=\int \frac {-1+x^6}{\sqrt [3]{x^2+x^4} \left (1+x^6\right )} \, dx \]

[In]

Int[(-1 + x^6)/((x^2 + x^4)^(1/3)*(1 + x^6)),x]

[Out]

(-2*x*(1 + x^2)^(1/3)*AppellF1[1/6, 1/3, 1, 7/6, -x^2, (2*x^2)/(1 - I*Sqrt[3])])/(x^2 + x^4)^(1/3) - (2*x*(1 +
x^2)^(1/3)*AppellF1[1/6, 1/3, 1, 7/6, -x^2, (2*x^2)/(1 + I*Sqrt[3])])/(x^2 + x^4)^(1/3) + (3*x*(1 + x^2)^(1/3
)*Hypergeometric2F1[1/6, 1/3, 7/6, -x^2])/(x^2 + x^4)^(1/3) - ((I/3)*x^(2/3)*(1 + x^2)^(1/3)*Defer[Subst][Defe
r[Int][1/((I - x)*(1 + x^6)^(1/3)), x], x, x^(1/3)])/(x^2 + x^4)^(1/3) - ((I/3)*x^(2/3)*(1 + x^2)^(1/3)*Defer[
Subst][Defer[Int][1/((I + x)*(1 + x^6)^(1/3)), x], x, x^(1/3)])/(x^2 + x^4)^(1/3) - (Sqrt[1 - I*Sqrt[3]]*x^(2/
3)*(1 + x^2)^(1/3)*Defer[Subst][Defer[Int][1/((Sqrt[1 - I*Sqrt[3]] - Sqrt[2]*x)*(1 + x^6)^(1/3)), x], x, x^(1/
3)])/(3*(x^2 + x^4)^(1/3)) - (Sqrt[1 + I*Sqrt[3]]*x^(2/3)*(1 + x^2)^(1/3)*Defer[Subst][Defer[Int][1/((Sqrt[1 +
I*Sqrt[3]] - Sqrt[2]*x)*(1 + x^6)^(1/3)), x], x, x^(1/3)])/(3*(x^2 + x^4)^(1/3)) - (Sqrt[1 - I*Sqrt[3]]*x^(2/
3)*(1 + x^2)^(1/3)*Defer[Subst][Defer[Int][1/((Sqrt[1 - I*Sqrt[3]] + Sqrt[2]*x)*(1 + x^6)^(1/3)), x], x, x^(1/
3)])/(3*(x^2 + x^4)^(1/3)) - (Sqrt[1 + I*Sqrt[3]]*x^(2/3)*(1 + x^2)^(1/3)*Defer[Subst][Defer[Int][1/((Sqrt[1 +
I*Sqrt[3]] + Sqrt[2]*x)*(1 + x^6)^(1/3)), x], x, x^(1/3)])/(3*(x^2 + x^4)^(1/3))

Rubi steps \begin{align*} \text {integral}& = \frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \int \frac {-1+x^6}{x^{2/3} \sqrt [3]{1+x^2} \left (1+x^6\right )} \, dx}{\sqrt [3]{x^2+x^4}} \\ & = \frac {\left (3 x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {-1+x^{18}}{\sqrt [3]{1+x^6} \left (1+x^{18}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^4}} \\ & = \frac {\left (3 x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \left (\frac {1}{\sqrt [3]{1+x^6}}-\frac {2}{\sqrt [3]{1+x^6} \left (1+x^{18}\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^4}} \\ & = \frac {\left (3 x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^4}}-\frac {\left (6 x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{1+x^6} \left (1+x^{18}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^4}} \\ & = \frac {3 x \sqrt [3]{1+x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{3},\frac {7}{6},-x^2\right )}{\sqrt [3]{x^2+x^4}}-\frac {\left (6 x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \left (\frac {1}{9 \left (1+x^2\right ) \sqrt [3]{1+x^6}}+\frac {2-x^2}{9 \left (1-x^2+x^4\right ) \sqrt [3]{1+x^6}}+\frac {2-x^6}{3 \sqrt [3]{1+x^6} \left (1-x^6+x^{12}\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^4}} \\ & = \frac {3 x \sqrt [3]{1+x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{3},\frac {7}{6},-x^2\right )}{\sqrt [3]{x^2+x^4}}-\frac {\left (2 x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{x^2+x^4}}-\frac {\left (2 x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {2-x^2}{\left (1-x^2+x^4\right ) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{x^2+x^4}}-\frac {\left (2 x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {2-x^6}{\sqrt [3]{1+x^6} \left (1-x^6+x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^4}} \\ & = \frac {3 x \sqrt [3]{1+x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{3},\frac {7}{6},-x^2\right )}{\sqrt [3]{x^2+x^4}}-\frac {\left (2 x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \left (\frac {i}{2 (i-x) \sqrt [3]{1+x^6}}+\frac {i}{2 (i+x) \sqrt [3]{1+x^6}}\right ) \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{x^2+x^4}}-\frac {\left (2 x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \left (\frac {-1-i \sqrt {3}}{\left (-1-i \sqrt {3}+2 x^2\right ) \sqrt [3]{1+x^6}}+\frac {-1+i \sqrt {3}}{\left (-1+i \sqrt {3}+2 x^2\right ) \sqrt [3]{1+x^6}}\right ) \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{x^2+x^4}}-\frac {\left (2 x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \left (\frac {-1-i \sqrt {3}}{\sqrt [3]{1+x^6} \left (-1-i \sqrt {3}+2 x^6\right )}+\frac {-1+i \sqrt {3}}{\sqrt [3]{1+x^6} \left (-1+i \sqrt {3}+2 x^6\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^4}} \\ & = \frac {3 x \sqrt [3]{1+x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{3},\frac {7}{6},-x^2\right )}{\sqrt [3]{x^2+x^4}}-\frac {\left (i x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{(i-x) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{x^2+x^4}}-\frac {\left (i x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{(i+x) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{x^2+x^4}}-\frac {\left (2 \left (-1-i \sqrt {3}\right ) x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-1-i \sqrt {3}+2 x^2\right ) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{x^2+x^4}}-\frac {\left (2 \left (-1-i \sqrt {3}\right ) x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{1+x^6} \left (-1-i \sqrt {3}+2 x^6\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^4}}-\frac {\left (2 \left (-1+i \sqrt {3}\right ) x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-1+i \sqrt {3}+2 x^2\right ) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{x^2+x^4}}-\frac {\left (2 \left (-1+i \sqrt {3}\right ) x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{1+x^6} \left (-1+i \sqrt {3}+2 x^6\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^4}} \\ & = -\frac {2 x \sqrt [3]{1+x^2} \operatorname {AppellF1}\left (\frac {1}{6},\frac {1}{3},1,\frac {7}{6},-x^2,\frac {2 x^2}{1-i \sqrt {3}}\right )}{\sqrt [3]{x^2+x^4}}-\frac {2 x \sqrt [3]{1+x^2} \operatorname {AppellF1}\left (\frac {1}{6},\frac {1}{3},1,\frac {7}{6},-x^2,\frac {2 x^2}{1+i \sqrt {3}}\right )}{\sqrt [3]{x^2+x^4}}+\frac {3 x \sqrt [3]{1+x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{3},\frac {7}{6},-x^2\right )}{\sqrt [3]{x^2+x^4}}-\frac {\left (i x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{(i-x) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{x^2+x^4}}-\frac {\left (i x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{(i+x) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{x^2+x^4}}-\frac {\left (2 \left (-1-i \sqrt {3}\right ) x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \left (\frac {\sqrt {1+i \sqrt {3}}}{2 \left (-1-i \sqrt {3}\right ) \left (\sqrt {1+i \sqrt {3}}-\sqrt {2} x\right ) \sqrt [3]{1+x^6}}+\frac {\sqrt {1+i \sqrt {3}}}{2 \left (-1-i \sqrt {3}\right ) \left (\sqrt {1+i \sqrt {3}}+\sqrt {2} x\right ) \sqrt [3]{1+x^6}}\right ) \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{x^2+x^4}}-\frac {\left (2 \left (-1+i \sqrt {3}\right ) x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \left (\frac {\sqrt {1-i \sqrt {3}}}{2 \left (-1+i \sqrt {3}\right ) \left (\sqrt {1-i \sqrt {3}}-\sqrt {2} x\right ) \sqrt [3]{1+x^6}}+\frac {\sqrt {1-i \sqrt {3}}}{2 \left (-1+i \sqrt {3}\right ) \left (\sqrt {1-i \sqrt {3}}+\sqrt {2} x\right ) \sqrt [3]{1+x^6}}\right ) \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{x^2+x^4}} \\ & = -\frac {2 x \sqrt [3]{1+x^2} \operatorname {AppellF1}\left (\frac {1}{6},\frac {1}{3},1,\frac {7}{6},-x^2,\frac {2 x^2}{1-i \sqrt {3}}\right )}{\sqrt [3]{x^2+x^4}}-\frac {2 x \sqrt [3]{1+x^2} \operatorname {AppellF1}\left (\frac {1}{6},\frac {1}{3},1,\frac {7}{6},-x^2,\frac {2 x^2}{1+i \sqrt {3}}\right )}{\sqrt [3]{x^2+x^4}}+\frac {3 x \sqrt [3]{1+x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{3},\frac {7}{6},-x^2\right )}{\sqrt [3]{x^2+x^4}}-\frac {\left (i x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{(i-x) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{x^2+x^4}}-\frac {\left (i x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{(i+x) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt [3]{x^2+x^4}}+\frac {\left (\left (-1+i \sqrt {3}\right ) x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {1-i \sqrt {3}}-\sqrt {2} x\right ) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt {1-i \sqrt {3}} \sqrt [3]{x^2+x^4}}+\frac {\left (\left (-1+i \sqrt {3}\right ) x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {1-i \sqrt {3}}+\sqrt {2} x\right ) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt {1-i \sqrt {3}} \sqrt [3]{x^2+x^4}}+\frac {\left (\left (-1-i \sqrt {3}\right ) x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {1+i \sqrt {3}}-\sqrt {2} x\right ) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt {1+i \sqrt {3}} \sqrt [3]{x^2+x^4}}+\frac {\left (\left (-1-i \sqrt {3}\right ) x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {1+i \sqrt {3}}+\sqrt {2} x\right ) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{3 \sqrt {1+i \sqrt {3}} \sqrt [3]{x^2+x^4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.81 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.22 \[ \int \frac {-1+x^6}{\sqrt [3]{x^2+x^4} \left (1+x^6\right )} \, dx=-\frac {x^{2/3} \left (9 \sqrt [3]{x}-3 \sqrt [3]{3} \sqrt [3]{1+x^2} \arctan \left (\frac {3^{2/3} \sqrt [3]{x} \sqrt [3]{1+x^2}}{\sqrt [3]{3} x^{2/3}-\left (1+x^2\right )^{2/3}}\right )+2\ 3^{5/6} \sqrt [3]{1+x^2} \text {arctanh}\left (\frac {\sqrt [6]{3} \sqrt [3]{x}}{\sqrt [3]{1+x^2}}\right )+3^{5/6} \sqrt [3]{1+x^2} \text {arctanh}\left (\frac {3^{5/6} \sqrt [3]{x} \sqrt [3]{1+x^2}}{3 x^{2/3}+3^{2/3} \left (1+x^2\right )^{2/3}}\right )\right )}{9 \sqrt [3]{x^2+x^4}} \]

[In]

Integrate[(-1 + x^6)/((x^2 + x^4)^(1/3)*(1 + x^6)),x]

[Out]

-1/9*(x^(2/3)*(9*x^(1/3) - 3*3^(1/3)*(1 + x^2)^(1/3)*ArcTan[(3^(2/3)*x^(1/3)*(1 + x^2)^(1/3))/(3^(1/3)*x^(2/3)
- (1 + x^2)^(2/3))] + 2*3^(5/6)*(1 + x^2)^(1/3)*ArcTanh[(3^(1/6)*x^(1/3))/(1 + x^2)^(1/3)] + 3^(5/6)*(1 + x^2
)^(1/3)*ArcTanh[(3^(5/6)*x^(1/3)*(1 + x^2)^(1/3))/(3*x^(2/3) + 3^(2/3)*(1 + x^2)^(2/3))]))/(x^2 + x^4)^(1/3)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(294\) vs. \(2(124)=248\).

Time = 46.41 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.92

method	result	size
pseudoelliptic	\(\frac {3^{\frac {5}{6}} \ln \left (\frac {-3^{\frac {1}{6}} \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}} x +3^{\frac {1}{3}} x^{2}+\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {2}{3}}}{x^{2}}\right ) \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}}-2 \,3^{\frac {5}{6}} \ln \left (\frac {3^{\frac {1}{6}} x +\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}}}{x}\right ) \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}}+2 \,3^{\frac {5}{6}} \ln \left (\frac {-3^{\frac {1}{6}} x +\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}}}{x}\right ) \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}}-3^{\frac {5}{6}} \ln \left (\frac {3^{\frac {1}{6}} \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}} x +3^{\frac {1}{3}} x^{2}+\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {2}{3}}}{x^{2}}\right ) \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}}-6 \,3^{\frac {1}{3}} \arctan \left (\frac {x \sqrt {3}-2 \,3^{\frac {1}{3}} \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}}}{3 x}\right ) \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}}+6 \,3^{\frac {1}{3}} \arctan \left (\frac {x \sqrt {3}+2 \,3^{\frac {1}{3}} \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}}}{3 x}\right ) \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}}-18 x}{18 \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}}}\)	\(295\)
trager	\(\text {Expression too large to display}\)	\(6673\)
risch	\(\text {Expression too large to display}\)	\(7488\)

[In]

int((x^6-1)/(x^4+x^2)^(1/3)/(x^6+1),x,method=_RETURNVERBOSE)

[Out]

1/18*(3^(5/6)*ln((-3^(1/6)*(x^2*(x^2+1))^(1/3)*x+3^(1/3)*x^2+(x^2*(x^2+1))^(2/3))/x^2)*(x^2*(x^2+1))^(1/3)-2*3
^(5/6)*ln((3^(1/6)*x+(x^2*(x^2+1))^(1/3))/x)*(x^2*(x^2+1))^(1/3)+2*3^(5/6)*ln((-3^(1/6)*x+(x^2*(x^2+1))^(1/3))
/x)*(x^2*(x^2+1))^(1/3)-3^(5/6)*ln((3^(1/6)*(x^2*(x^2+1))^(1/3)*x+3^(1/3)*x^2+(x^2*(x^2+1))^(2/3))/x^2)*(x^2*(
x^2+1))^(1/3)-6*3^(1/3)*arctan(1/3*(x*3^(1/2)-2*3^(1/3)*(x^2*(x^2+1))^(1/3))/x)*(x^2*(x^2+1))^(1/3)+6*3^(1/3)*
arctan(1/3*(x*3^(1/2)+2*3^(1/3)*(x^2*(x^2+1))^(1/3))/x)*(x^2*(x^2+1))^(1/3)-18*x)/(x^2*(x^2+1))^(1/3)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1438 vs. \(2 (125) = 250\).

Time = 2.03 (sec) , antiderivative size = 1438, normalized size of antiderivative = 9.34 \[ \int \frac {-1+x^6}{\sqrt [3]{x^2+x^4} \left (1+x^6\right )} \, dx=\text {Too large to display} \]

[In]

integrate((x^6-1)/(x^4+x^2)^(1/3)/(x^6+1),x, algorithm="fricas")

[Out]

-1/36*(3^(5/6)*(x^3 + sqrt(-3)*(x^3 + x) + x)*log(-(2*3^(5/6)*(13*x^5 - 45*x^4 + 65*x^3 - 45*x^2 + sqrt(-3)*(1
3*x^5 - 45*x^4 + 65*x^3 - 45*x^2 + 13*x) + 13*x) + 12*(x^4 + x^2)^(2/3)*(26*x^2 + sqrt(3)*(15*x^2 - 26*x + 15)
- 45*x + 26) + 3*3^(1/3)*(15*x^5 - 52*x^4 + 75*x^3 - 52*x^2 + sqrt(-3)*(15*x^5 - 52*x^4 + 75*x^3 - 52*x^2 + 1
5*x) + 15*x) + 6*(x^4 + x^2)^(1/3)*(3^(2/3)*(15*x^3 - 26*x^2 - sqrt(-3)*(15*x^3 - 26*x^2 + 15*x) + 15*x) + 3^(
1/6)*(26*x^3 - 45*x^2 - sqrt(-3)*(26*x^3 - 45*x^2 + 26*x) + 26*x)))/(x^5 - x^3 + x)) - 3^(5/6)*(x^3 + sqrt(-3)
*(x^3 + x) + x)*log((2*3^(5/6)*(13*x^5 - 45*x^4 + 65*x^3 - 45*x^2 + sqrt(-3)*(13*x^5 - 45*x^4 + 65*x^3 - 45*x^
2 + 13*x) + 13*x) - 12*(x^4 + x^2)^(2/3)*(26*x^2 - sqrt(3)*(15*x^2 - 26*x + 15) - 45*x + 26) - 3*3^(1/3)*(15*x
^5 - 52*x^4 + 75*x^3 - 52*x^2 + sqrt(-3)*(15*x^5 - 52*x^4 + 75*x^3 - 52*x^2 + 15*x) + 15*x) - 6*(x^4 + x^2)^(1
/3)*(3^(2/3)*(15*x^3 - 26*x^2 - sqrt(-3)*(15*x^3 - 26*x^2 + 15*x) + 15*x) - 3^(1/6)*(26*x^3 - 45*x^2 - sqrt(-3
)*(26*x^3 - 45*x^2 + 26*x) + 26*x)))/(x^5 - x^3 + x)) + 3^(5/6)*(x^3 - sqrt(-3)*(x^3 + x) + x)*log(-(2*3^(5/6)
*(13*x^5 - 45*x^4 + 65*x^3 - 45*x^2 - sqrt(-3)*(13*x^5 - 45*x^4 + 65*x^3 - 45*x^2 + 13*x) + 13*x) + 12*(x^4 +
x^2)^(2/3)*(26*x^2 + sqrt(3)*(15*x^2 - 26*x + 15) - 45*x + 26) + 3*3^(1/3)*(15*x^5 - 52*x^4 + 75*x^3 - 52*x^2
- sqrt(-3)*(15*x^5 - 52*x^4 + 75*x^3 - 52*x^2 + 15*x) + 15*x) + 6*(x^4 + x^2)^(1/3)*(3^(2/3)*(15*x^3 - 26*x^2
+ sqrt(-3)*(15*x^3 - 26*x^2 + 15*x) + 15*x) + 3^(1/6)*(26*x^3 - 45*x^2 + sqrt(-3)*(26*x^3 - 45*x^2 + 26*x) + 2
6*x)))/(x^5 - x^3 + x)) - 3^(5/6)*(x^3 - sqrt(-3)*(x^3 + x) + x)*log((2*3^(5/6)*(13*x^5 - 45*x^4 + 65*x^3 - 45
*x^2 - sqrt(-3)*(13*x^5 - 45*x^4 + 65*x^3 - 45*x^2 + 13*x) + 13*x) - 12*(x^4 + x^2)^(2/3)*(26*x^2 - sqrt(3)*(1
5*x^2 - 26*x + 15) - 45*x + 26) - 3*3^(1/3)*(15*x^5 - 52*x^4 + 75*x^3 - 52*x^2 - sqrt(-3)*(15*x^5 - 52*x^4 + 7
5*x^3 - 52*x^2 + 15*x) + 15*x) - 6*(x^4 + x^2)^(1/3)*(3^(2/3)*(15*x^3 - 26*x^2 + sqrt(-3)*(15*x^3 - 26*x^2 + 1
5*x) + 15*x) - 3^(1/6)*(26*x^3 - 45*x^2 + sqrt(-3)*(26*x^3 - 45*x^2 + 26*x) + 26*x)))/(x^5 - x^3 + x)) - 2*3^(
5/6)*(x^3 + x)*log((2*3^(5/6)*(13*x^5 - 45*x^4 + 65*x^3 - 45*x^2 + 13*x) - 6*(x^4 + x^2)^(2/3)*(26*x^2 + sqrt(
3)*(15*x^2 - 26*x + 15) - 45*x + 26) + 3*3^(1/3)*(15*x^5 - 52*x^4 + 75*x^3 - 52*x^2 + 15*x) + 6*(x^4 + x^2)^(1
/3)*(3^(2/3)*(15*x^3 - 26*x^2 + 15*x) + 3^(1/6)*(26*x^3 - 45*x^2 + 26*x)))/(x^5 - x^3 + x)) + 2*3^(5/6)*(x^3 +
x)*log(-(2*3^(5/6)*(13*x^5 - 45*x^4 + 65*x^3 - 45*x^2 + 13*x) + 6*(x^4 + x^2)^(2/3)*(26*x^2 - sqrt(3)*(15*x^2
- 26*x + 15) - 45*x + 26) - 3*3^(1/3)*(15*x^5 - 52*x^4 + 75*x^3 - 52*x^2 + 15*x) - 6*(x^4 + x^2)^(1/3)*(3^(2/
3)*(15*x^3 - 26*x^2 + 15*x) - 3^(1/6)*(26*x^3 - 45*x^2 + 26*x)))/(x^5 - x^3 + x)) + 36*(x^4 + x^2)^(2/3))/(x^3
+ x)

Sympy [F]

\[ \int \frac {-1+x^6}{\sqrt [3]{x^2+x^4} \left (1+x^6\right )} \, dx=\int \frac {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )}{\sqrt [3]{x^{2} \left (x^{2} + 1\right )} \left (x^{2} + 1\right ) \left (x^{4} - x^{2} + 1\right )}\, dx \]

[In]

integrate((x**6-1)/(x**4+x**2)**(1/3)/(x**6+1),x)

[Out]

Integral((x - 1)*(x + 1)*(x**2 - x + 1)*(x**2 + x + 1)/((x**2*(x**2 + 1))**(1/3)*(x**2 + 1)*(x**4 - x**2 + 1))
, x)

Maxima [F]

\[ \int \frac {-1+x^6}{\sqrt [3]{x^2+x^4} \left (1+x^6\right )} \, dx=\int { \frac {x^{6} - 1}{{\left (x^{6} + 1\right )} {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}}} \,d x } \]

[In]

integrate((x^6-1)/(x^4+x^2)^(1/3)/(x^6+1),x, algorithm="maxima")

[Out]

integrate((x^6 - 1)/((x^6 + 1)*(x^4 + x^2)^(1/3)), x)

Giac [F]

\[ \int \frac {-1+x^6}{\sqrt [3]{x^2+x^4} \left (1+x^6\right )} \, dx=\int { \frac {x^{6} - 1}{{\left (x^{6} + 1\right )} {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}}} \,d x } \]

[In]

integrate((x^6-1)/(x^4+x^2)^(1/3)/(x^6+1),x, algorithm="giac")

[Out]

integrate((x^6 - 1)/((x^6 + 1)*(x^4 + x^2)^(1/3)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {-1+x^6}{\sqrt [3]{x^2+x^4} \left (1+x^6\right )} \, dx=\int \frac {x^6-1}{{\left (x^4+x^2\right )}^{1/3}\,\left (x^6+1\right )} \,d x \]

[In]

int((x^6 - 1)/((x^2 + x^4)^(1/3)*(x^6 + 1)),x)

[Out]

int((x^6 - 1)/((x^2 + x^4)^(1/3)*(x^6 + 1)), x)