\(\int \frac {1+x^6}{\sqrt [3]{x^2+x^4} (-1+x^6)} \, dx\) [2833]

Optimal result

Integrand size = 24, antiderivative size = 288 \[ \int \frac {1+x^6}{\sqrt [3]{x^2+x^4} \left (-1+x^6\right )} \, dx=-\frac {\arctan \left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{x^2+x^4}}\right )}{\sqrt {3}}-\frac {\arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{x^2+x^4}}\right )}{\sqrt {3}}-\frac {\arctan \left (\frac {\sqrt {3} x}{-x+2^{2/3} \sqrt [3]{x^2+x^4}}\right )}{2 \sqrt [3]{2} \sqrt {3}}-\frac {\arctan \left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{x^2+x^4}}\right )}{2 \sqrt [3]{2} \sqrt {3}}-\frac {2}{3} \text {arctanh}\left (\frac {x}{\sqrt [3]{x^2+x^4}}\right )-\frac {\text {arctanh}\left (\frac {\sqrt [3]{2} x}{\sqrt [3]{x^2+x^4}}\right )}{3 \sqrt [3]{2}}-\frac {1}{3} \text {arctanh}\left (\frac {x^2+\left (x^2+x^4\right )^{2/3}}{x \sqrt [3]{x^2+x^4}}\right )-\frac {\text {arctanh}\left (\frac {\sqrt [3]{2} x^2+\frac {\left (x^2+x^4\right )^{2/3}}{\sqrt [3]{2}}}{x \sqrt [3]{x^2+x^4}}\right )}{6 \sqrt [3]{2}} \]

[Out]

-1/3*3^(1/2)*arctan(3^(1/2)*x/(-x+2*(x^4+x^2)^(1/3)))-1/3*arctan(3^(1/2)*x/(x+2*(x^4+x^2)^(1/3)))*3^(1/2)-1/12
*3^(1/2)*arctan(3^(1/2)*x/(-x+2^(2/3)*(x^4+x^2)^(1/3)))*2^(2/3)-1/12*3^(1/2)*arctan(3^(1/2)*x/(x+2^(2/3)*(x^4+
x^2)^(1/3)))*2^(2/3)-2/3*arctanh(x/(x^4+x^2)^(1/3))-1/6*arctanh(2^(1/3)*x/(x^4+x^2)^(1/3))*2^(2/3)-1/3*arctanh
((x^2+(x^4+x^2)^(2/3))/x/(x^4+x^2)^(1/3))-1/12*arctanh((2^(1/3)*x^2+1/2*(x^4+x^2)^(2/3)*2^(2/3))/x/(x^4+x^2)^(
1/3))*2^(2/3)

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]

-1/6*(x^(2/3)*(1 + x^2)^(1/3)*Defer[Subst][Defer[Int][(1 + x^6)^(2/3)/(1 - x), x], x, x^(1/3)])/(x^2 + x^4)^(1
/3) - (x^(2/3)*(1 + x^2)^(1/3)*Defer[Subst][Defer[Int][(1 + x^6)^(2/3)/(1 + x), x], x, x^(1/3)])/(6*(x^2 + x^4
)^(1/3)) - ((1 + I*Sqrt[3])*x^(2/3)*(1 + x^2)^(1/3)*Defer[Subst][Defer[Int][(1 + x^6)^(2/3)/(1 - (-1)^(1/9)*x)
, x], x, x^(1/3)])/(6*(x^2 + x^4)^(1/3)) - ((1 + I*Sqrt[3])*x^(2/3)*(1 + x^2)^(1/3)*Defer[Subst][Defer[Int][(1
+ x^6)^(2/3)/(1 + (-1)^(1/9)*x), x], x, x^(1/3)])/(6*(x^2 + x^4)^(1/3)) - ((1 - I*Sqrt[3])*x^(2/3)*(1 + x^2)^
(1/3)*Defer[Subst][Defer[Int][(1 + x^6)^(2/3)/(1 - (-1)^(2/9)*x), x], x, x^(1/3)])/(6*(x^2 + x^4)^(1/3)) - ((1
- I*Sqrt[3])*x^(2/3)*(1 + x^2)^(1/3)*Defer[Subst][Defer[Int][(1 + x^6)^(2/3)/(1 + (-1)^(2/9)*x), x], x, x^(1/
3)])/(6*(x^2 + x^4)^(1/3)) - (x^(2/3)*(1 + x^2)^(1/3)*Defer[Subst][Defer[Int][(1 + x^6)^(2/3)/(1 - (-1)^(1/3)*
x), x], x, x^(1/3)])/(6*(x^2 + x^4)^(1/3)) - (x^(2/3)*(1 + x^2)^(1/3)*Defer[Subst][Defer[Int][(1 + x^6)^(2/3)/
(1 + (-1)^(1/3)*x), x], x, x^(1/3)])/(6*(x^2 + x^4)^(1/3)) - ((1 + I*Sqrt[3])*x^(2/3)*(1 + x^2)^(1/3)*Defer[Su
bst][Defer[Int][(1 + x^6)^(2/3)/(1 - (-1)^(4/9)*x), x], x, x^(1/3)])/(6*(x^2 + x^4)^(1/3)) - ((1 + I*Sqrt[3])*
x^(2/3)*(1 + x^2)^(1/3)*Defer[Subst][Defer[Int][(1 + x^6)^(2/3)/(1 + (-1)^(4/9)*x), x], x, x^(1/3)])/(6*(x^2 +
x^4)^(1/3)) - ((1 - I*Sqrt[3])*x^(2/3)*(1 + x^2)^(1/3)*Defer[Subst][Defer[Int][(1 + x^6)^(2/3)/(1 - (-1)^(5/9
)*x), x], x, x^(1/3)])/(6*(x^2 + x^4)^(1/3)) - ((1 - I*Sqrt[3])*x^(2/3)*(1 + x^2)^(1/3)*Defer[Subst][Defer[Int
][(1 + x^6)^(2/3)/(1 + (-1)^(5/9)*x), x], x, x^(1/3)])/(6*(x^2 + x^4)^(1/3)) - (x^(2/3)*(1 + x^2)^(1/3)*Defer[
Subst][Defer[Int][(1 + x^6)^(2/3)/(1 - (-1)^(2/3)*x), x], x, x^(1/3)])/(6*(x^2 + x^4)^(1/3)) - (x^(2/3)*(1 + x
^2)^(1/3)*Defer[Subst][Defer[Int][(1 + x^6)^(2/3)/(1 + (-1)^(2/3)*x), x], x, x^(1/3)])/(6*(x^2 + x^4)^(1/3)) -
((1 + I*Sqrt[3])*x^(2/3)*(1 + x^2)^(1/3)*Defer[Subst][Defer[Int][(1 + x^6)^(2/3)/(1 - (-1)^(7/9)*x), x], x, x
^(1/3)])/(6*(x^2 + x^4)^(1/3)) - ((1 + I*Sqrt[3])*x^(2/3)*(1 + x^2)^(1/3)*Defer[Subst][Defer[Int][(1 + x^6)^(2
/3)/(1 + (-1)^(7/9)*x), x], x, x^(1/3)])/(6*(x^2 + x^4)^(1/3)) - ((1 - I*Sqrt[3])*x^(2/3)*(1 + x^2)^(1/3)*Defe
r[Subst][Defer[Int][(1 + x^6)^(2/3)/(1 - (-1)^(8/9)*x), x], x, x^(1/3)])/(6*(x^2 + x^4)^(1/3)) - ((1 - I*Sqrt[
3])*x^(2/3)*(1 + x^2)^(1/3)*Defer[Subst][Defer[Int][(1 + x^6)^(2/3)/(1 + (-1)^(8/9)*x), x], x, x^(1/3)])/(6*(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 (x^{2/3} \sqrt [3]{1+x^2}\right ) \int \frac {\left (1+x^2\right )^{2/3} \left (1-x^2+x^4\right )}{x^{2/3} \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 {\left (1+x^6\right )^{2/3} \left (1-x^6+x^{12}\right )}{-1+x^{18}} \, 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 {\left (1+x^6\right )^{2/3}}{18 (1-x)}-\frac {\left (1+x^6\right )^{2/3}}{18 (1+x)}-\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \left (1+x^6\right )^{2/3}}{18 \left (1-\sqrt [9]{-1} x\right )}-\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \left (1+x^6\right )^{2/3}}{18 \left (1+\sqrt [9]{-1} x\right )}-\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \left (1+x^6\right )^{2/3}}{18 \left (1-(-1)^{2/9} x\right )}-\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \left (1+x^6\right )^{2/3}}{18 \left (1+(-1)^{2/9} x\right )}-\frac {\left (1+x^6\right )^{2/3}}{18 \left (1-\sqrt [3]{-1} x\right )}-\frac {\left (1+x^6\right )^{2/3}}{18 \left (1+\sqrt [3]{-1} x\right )}-\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \left (1+x^6\right )^{2/3}}{18 \left (1-(-1)^{4/9} x\right )}-\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \left (1+x^6\right )^{2/3}}{18 \left (1+(-1)^{4/9} x\right )}-\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \left (1+x^6\right )^{2/3}}{18 \left (1-(-1)^{5/9} x\right )}-\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \left (1+x^6\right )^{2/3}}{18 \left (1+(-1)^{5/9} x\right )}-\frac {\left (1+x^6\right )^{2/3}}{18 \left (1-(-1)^{2/3} x\right )}-\frac {\left (1+x^6\right )^{2/3}}{18 \left (1+(-1)^{2/3} x\right )}-\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \left (1+x^6\right )^{2/3}}{18 \left (1-(-1)^{7/9} x\right )}-\frac {\left (1+\sqrt [3]{-1}+(-1)^{2/3}\right ) \left (1+x^6\right )^{2/3}}{18 \left (1+(-1)^{7/9} x\right )}-\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \left (1+x^6\right )^{2/3}}{18 \left (1-(-1)^{8/9} x\right )}-\frac {\left (1-\sqrt [3]{-1}-(-1)^{2/3}\right ) \left (1+x^6\right )^{2/3}}{18 \left (1+(-1)^{8/9} x\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^4}} \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.95 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.16 \[ \int \frac {1+x^6}{\sqrt [3]{x^2+x^4} \left (-1+x^6\right )} \, dx=-\frac {x^{2/3} \sqrt [3]{1+x^2} \left (-4 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{x}}{\sqrt [3]{x}-2 \sqrt [3]{1+x^2}}\right )+4 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{x}}{\sqrt [3]{x}+2 \sqrt [3]{1+x^2}}\right )+2^{2/3} \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{x}}{-\sqrt [3]{x}+2^{2/3} \sqrt [3]{1+x^2}}\right )+2^{2/3} \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{x}}{\sqrt [3]{x}+2^{2/3} \sqrt [3]{1+x^2}}\right )+8 \text {arctanh}\left (\frac {\sqrt [3]{x}}{\sqrt [3]{1+x^2}}\right )+2\ 2^{2/3} \text {arctanh}\left (\frac {\sqrt [3]{2} \sqrt [3]{x}}{\sqrt [3]{1+x^2}}\right )+4 \text {arctanh}\left (\frac {\sqrt [3]{x} \sqrt [3]{1+x^2}}{x^{2/3}+\left (1+x^2\right )^{2/3}}\right )+2^{2/3} \text {arctanh}\left (\frac {2^{2/3} \sqrt [3]{x} \sqrt [3]{1+x^2}}{2 x^{2/3}+\sqrt [3]{2} \left (1+x^2\right )^{2/3}}\right )\right )}{12 \sqrt [3]{x^2+x^4}} \]

[In]

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

[Out]

-1/12*(x^(2/3)*(1 + x^2)^(1/3)*(-4*Sqrt[3]*ArcTan[(Sqrt[3]*x^(1/3))/(x^(1/3) - 2*(1 + x^2)^(1/3))] + 4*Sqrt[3]
*ArcTan[(Sqrt[3]*x^(1/3))/(x^(1/3) + 2*(1 + x^2)^(1/3))] + 2^(2/3)*Sqrt[3]*ArcTan[(Sqrt[3]*x^(1/3))/(-x^(1/3)
+ 2^(2/3)*(1 + x^2)^(1/3))] + 2^(2/3)*Sqrt[3]*ArcTan[(Sqrt[3]*x^(1/3))/(x^(1/3) + 2^(2/3)*(1 + x^2)^(1/3))] +
8*ArcTanh[x^(1/3)/(1 + x^2)^(1/3)] + 2*2^(2/3)*ArcTanh[(2^(1/3)*x^(1/3))/(1 + x^2)^(1/3)] + 4*ArcTanh[(x^(1/3)
*(1 + x^2)^(1/3))/(x^(2/3) + (1 + x^2)^(2/3))] + 2^(2/3)*ArcTanh[(2^(2/3)*x^(1/3)*(1 + x^2)^(1/3))/(2*x^(2/3)
+ 2^(1/3)*(1 + x^2)^(2/3))]))/(x^2 + x^4)^(1/3)

Maple [A] (verified)

Time = 35.54 (sec) , antiderivative size = 388, normalized size of antiderivative = 1.35

method	result	size
pseudoelliptic	\(\frac {\ln \left (\frac {\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {2}{3}}-\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}} x +x^{2}}{x^{2}}\right )}{6}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (-2 \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}}+x \right ) \sqrt {3}}{3 x}\right )}{3}-\frac {\ln \left (\frac {\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {2}{3}}+\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}} x +x^{2}}{x^{2}}\right )}{6}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}}+x \right ) \sqrt {3}}{3 x}\right )}{3}+\frac {2^{\frac {2}{3}} \ln \left (\frac {-2^{\frac {1}{3}} x +\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}}}{x}\right )}{12}-\frac {2^{\frac {2}{3}} \ln \left (\frac {2^{\frac {2}{3}} x^{2}+2^{\frac {1}{3}} \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}} x +\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {2}{3}}}{x^{2}}\right )}{24}+\frac {\sqrt {3}\, 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3}\, \left (\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}} 2^{\frac {2}{3}}+x \right )}{3 x}\right )}{12}-\frac {\ln \left (\frac {\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}}+x}{x}\right )}{3}+\frac {\ln \left (\frac {\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}}-x}{x}\right )}{3}-\frac {2^{\frac {2}{3}} \ln \left (\frac {2^{\frac {1}{3}} x +\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}}}{x}\right )}{12}+\frac {2^{\frac {2}{3}} \ln \left (\frac {2^{\frac {2}{3}} x^{2}-2^{\frac {1}{3}} \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}} x +\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {2}{3}}}{x^{2}}\right )}{24}-\frac {\sqrt {3}\, 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3}\, \left (-\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}} 2^{\frac {2}{3}}+x \right )}{3 x}\right )}{12}\)	\(388\)
trager	\(\text {Expression too large to display}\)	\(5994\)

[In]

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

[Out]

1/6*ln(((x^2*(x^2+1))^(2/3)-(x^2*(x^2+1))^(1/3)*x+x^2)/x^2)-1/3*3^(1/2)*arctan(1/3*(-2*(x^2*(x^2+1))^(1/3)+x)*
3^(1/2)/x)-1/6*ln(((x^2*(x^2+1))^(2/3)+(x^2*(x^2+1))^(1/3)*x+x^2)/x^2)+1/3*3^(1/2)*arctan(1/3*(2*(x^2*(x^2+1))
^(1/3)+x)*3^(1/2)/x)+1/12*2^(2/3)*ln((-2^(1/3)*x+(x^2*(x^2+1))^(1/3))/x)-1/24*2^(2/3)*ln((2^(2/3)*x^2+2^(1/3)*
(x^2*(x^2+1))^(1/3)*x+(x^2*(x^2+1))^(2/3))/x^2)+1/12*3^(1/2)*2^(2/3)*arctan(1/3*3^(1/2)*((x^2*(x^2+1))^(1/3)*2
^(2/3)+x)/x)-1/3*ln(((x^2*(x^2+1))^(1/3)+x)/x)+1/3*ln(((x^2*(x^2+1))^(1/3)-x)/x)-1/12*2^(2/3)*ln((2^(1/3)*x+(x
^2*(x^2+1))^(1/3))/x)+1/24*2^(2/3)*ln((2^(2/3)*x^2-2^(1/3)*(x^2*(x^2+1))^(1/3)*x+(x^2*(x^2+1))^(2/3))/x^2)-1/1
2*3^(1/2)*2^(2/3)*arctan(1/3*3^(1/2)*(-(x^2*(x^2+1))^(1/3)*2^(2/3)+x)/x)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 456 vs. \(2 (223) = 446\).

Time = 1.75 (sec) , antiderivative size = 456, normalized size of antiderivative = 1.58 \[ \int \frac {1+x^6}{\sqrt [3]{x^2+x^4} \left (-1+x^6\right )} \, dx=\frac {1}{12} \, \sqrt {6} 2^{\frac {1}{6}} \left (-1\right )^{\frac {1}{3}} \arctan \left (\frac {2^{\frac {1}{6}} {\left (4 \, \sqrt {6} 2^{\frac {2}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{4} + x^{2}\right )}^{\frac {2}{3}} {\left (x^{2} + 2 \, x + 1\right )} + 8 \, \sqrt {6} \left (-1\right )^{\frac {1}{3}} {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}} {\left (x^{3} - 2 \, x^{2} + x\right )} - \sqrt {6} 2^{\frac {1}{3}} {\left (x^{5} - 8 \, x^{4} - 2 \, x^{3} - 8 \, x^{2} + x\right )}\right )}}{6 \, {\left (x^{5} + 8 \, x^{4} - 2 \, x^{3} + 8 \, x^{2} + x\right )}}\right ) + \frac {1}{12} \cdot 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \log \left (-\frac {4 \cdot 2^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}} x - 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{3} + 2 \, x^{2} + x\right )} + 4 \, {\left (x^{4} + x^{2}\right )}^{\frac {2}{3}}}{x^{3} - 2 \, x^{2} + x}\right ) - \frac {1}{24} \cdot 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \log \left (\frac {2^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{3} - 2 \, x^{2} + x\right )} + 2 \cdot 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{4} + x^{2}\right )}^{\frac {2}{3}} + 4 \, {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}} x}{x^{3} - 2 \, x^{2} + x}\right ) - \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {4 \, \sqrt {3} {\left (x^{4} + x^{2}\right )}^{\frac {2}{3}} {\left (x^{2} + x + 1\right )} - 4 \, \sqrt {3} {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}} {\left (x^{3} - x^{2} + x\right )} - \sqrt {3} {\left (x^{5} - 4 \, x^{4} + x^{3} - 4 \, x^{2} + x\right )}}{3 \, {\left (x^{5} + 4 \, x^{4} + x^{3} + 4 \, x^{2} + x\right )}}\right ) + \frac {1}{3} \, \log \left (\frac {x^{3} - x^{2} + 2 \, {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}} x + x - 2 \, {\left (x^{4} + x^{2}\right )}^{\frac {2}{3}}}{x^{3} + x^{2} + x}\right ) + \frac {1}{6} \, \log \left (\frac {x^{3} - x^{2} + 2 \, {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}} x + x - 2 \, {\left (x^{4} + x^{2}\right )}^{\frac {2}{3}}}{x^{3} - x^{2} + x}\right ) \]

[In]

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

[Out]

1/12*sqrt(6)*2^(1/6)*(-1)^(1/3)*arctan(1/6*2^(1/6)*(4*sqrt(6)*2^(2/3)*(-1)^(2/3)*(x^4 + x^2)^(2/3)*(x^2 + 2*x
+ 1) + 8*sqrt(6)*(-1)^(1/3)*(x^4 + x^2)^(1/3)*(x^3 - 2*x^2 + x) - sqrt(6)*2^(1/3)*(x^5 - 8*x^4 - 2*x^3 - 8*x^2
+ x))/(x^5 + 8*x^4 - 2*x^3 + 8*x^2 + x)) + 1/12*2^(2/3)*(-1)^(1/3)*log(-(4*2^(1/3)*(-1)^(2/3)*(x^4 + x^2)^(1/
3)*x - 2^(2/3)*(-1)^(1/3)*(x^3 + 2*x^2 + x) + 4*(x^4 + x^2)^(2/3))/(x^3 - 2*x^2 + x)) - 1/24*2^(2/3)*(-1)^(1/3
)*log((2^(1/3)*(-1)^(2/3)*(x^3 - 2*x^2 + x) + 2*2^(2/3)*(-1)^(1/3)*(x^4 + x^2)^(2/3) + 4*(x^4 + x^2)^(1/3)*x)/
(x^3 - 2*x^2 + x)) - 1/3*sqrt(3)*arctan(1/3*(4*sqrt(3)*(x^4 + x^2)^(2/3)*(x^2 + x + 1) - 4*sqrt(3)*(x^4 + x^2)
^(1/3)*(x^3 - x^2 + x) - sqrt(3)*(x^5 - 4*x^4 + x^3 - 4*x^2 + x))/(x^5 + 4*x^4 + x^3 + 4*x^2 + x)) + 1/3*log((
x^3 - x^2 + 2*(x^4 + x^2)^(1/3)*x + x - 2*(x^4 + x^2)^(2/3))/(x^3 + x^2 + x)) + 1/6*log((x^3 - x^2 + 2*(x^4 +
x^2)^(1/3)*x + x - 2*(x^4 + x^2)^(2/3))/(x^3 - x^2 + x))

Sympy [F]

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

[In]

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

[Out]

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