\(\int \frac {-1+x^6}{\sqrt [3]{-x^2+x^4} (1+x^6)} \, dx\) [2805]

Optimal result

Integrand size = 26, antiderivative size = 274 \[ \int \frac {-1+x^6}{\sqrt [3]{-x^2+x^4} \left (1+x^6\right )} \, dx=-\frac {2}{3} \arctan \left (\frac {x}{\sqrt [3]{-x^2+x^4}}\right )-\frac {\arctan \left (\frac {\sqrt [3]{2} x}{\sqrt [3]{-x^2+x^4}}\right )}{3 \sqrt [3]{2}}-\frac {1}{3} \arctan \left (\frac {x \sqrt [3]{-x^2+x^4}}{-x^2+\left (-x^2+x^4\right )^{2/3}}\right )-\frac {\arctan \left (\frac {2^{2/3} x \sqrt [3]{-x^2+x^4}}{-2 x^2+\sqrt [3]{2} \left (-x^2+x^4\right )^{2/3}}\right )}{6 \sqrt [3]{2}}-\frac {\text {arctanh}\left (\frac {\frac {x^2}{\sqrt {3}}+\frac {\left (-x^2+x^4\right )^{2/3}}{\sqrt {3}}}{x \sqrt [3]{-x^2+x^4}}\right )}{\sqrt {3}}-\frac {\text {arctanh}\left (\frac {\frac {\sqrt [3]{2} x^2}{\sqrt {3}}+\frac {\left (-x^2+x^4\right )^{2/3}}{\sqrt [3]{2} \sqrt {3}}}{x \sqrt [3]{-x^2+x^4}}\right )}{2 \sqrt [3]{2} \sqrt {3}} \]

[Out]

-2/3*arctan(x/(x^4-x^2)^(1/3))-1/6*arctan(2^(1/3)*x/(x^4-x^2)^(1/3))*2^(2/3)-1/3*arctan(x*(x^4-x^2)^(1/3)/(-x^
2+(x^4-x^2)^(2/3)))-1/12*arctan(2^(2/3)*x*(x^4-x^2)^(1/3)/(-2*x^2+2^(1/3)*(x^4-x^2)^(2/3)))*2^(2/3)-1/3*arctan
h((1/3*3^(1/2)*x^2+1/3*(x^4-x^2)^(2/3)*3^(1/2))/x/(x^4-x^2)^(1/3))*3^(1/2)-1/12*3^(1/2)*arctanh((1/3*2^(1/3)*x
^2*3^(1/2)+1/6*(x^4-x^2)^(2/3)*2^(2/3)*3^(1/2))/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)^(1/18)*(1 + I*Sqrt[3])*x^(2/3)*(-1 + x^2)^(1/3)*Defer[Subst][Defer[Int][(-1 + x^6)^(2/3)/((-1)^(1/18) -
x), x], x, x^(1/3)])/(6*(-x^2 + x^4)^(1/3)) + ((-1)^(1/18)*(1 + I*Sqrt[3])*x^(2/3)*(-1 + x^2)^(1/3)*Defer[Subs
t][Defer[Int][(-1 + x^6)^(2/3)/((-1)^(1/18) + x), x], x, x^(1/3)])/(6*(-x^2 + x^4)^(1/3)) + ((-1)^(1/18)*(1 -
I*Sqrt[3])*x^(2/3)*(-1 + x^2)^(1/3)*Defer[Subst][Defer[Int][(-1 + x^6)^(2/3)/((-1)^(1/18) - (-1)^(1/9)*x), x],
x, x^(1/3)])/(6*(-x^2 + x^4)^(1/3)) + ((-1)^(1/18)*(1 - I*Sqrt[3])*x^(2/3)*(-1 + x^2)^(1/3)*Defer[Subst][Defe
r[Int][(-1 + x^6)^(2/3)/((-1)^(1/18) + (-1)^(1/9)*x), x], x, x^(1/3)])/(6*(-x^2 + x^4)^(1/3)) + ((-1)^(1/18)*x
^(2/3)*(-1 + x^2)^(1/3)*Defer[Subst][Defer[Int][(-1 + x^6)^(2/3)/((-1)^(1/18) - (-1)^(2/9)*x), x], x, x^(1/3)]
)/(6*(-x^2 + x^4)^(1/3)) + ((-1)^(1/18)*x^(2/3)*(-1 + x^2)^(1/3)*Defer[Subst][Defer[Int][(-1 + x^6)^(2/3)/((-1
)^(1/18) + (-1)^(2/9)*x), x], x, x^(1/3)])/(6*(-x^2 + x^4)^(1/3)) + ((-1)^(1/18)*(1 + I*Sqrt[3])*x^(2/3)*(-1 +
x^2)^(1/3)*Defer[Subst][Defer[Int][(-1 + x^6)^(2/3)/((-1)^(1/18) - (-1)^(1/3)*x), x], x, x^(1/3)])/(6*(-x^2 +
x^4)^(1/3)) + ((-1)^(1/18)*(1 + I*Sqrt[3])*x^(2/3)*(-1 + x^2)^(1/3)*Defer[Subst][Defer[Int][(-1 + x^6)^(2/3)/
((-1)^(1/18) + (-1)^(1/3)*x), x], x, x^(1/3)])/(6*(-x^2 + x^4)^(1/3)) + ((-1)^(1/18)*(1 - I*Sqrt[3])*x^(2/3)*(
-1 + x^2)^(1/3)*Defer[Subst][Defer[Int][(-1 + x^6)^(2/3)/((-1)^(1/18) - (-1)^(4/9)*x), x], x, x^(1/3)])/(6*(-x
^2 + x^4)^(1/3)) + ((-1)^(1/18)*(1 - I*Sqrt[3])*x^(2/3)*(-1 + x^2)^(1/3)*Defer[Subst][Defer[Int][(-1 + x^6)^(2
/3)/((-1)^(1/18) + (-1)^(4/9)*x), x], x, x^(1/3)])/(6*(-x^2 + x^4)^(1/3)) + ((-1)^(1/18)*x^(2/3)*(-1 + x^2)^(1
/3)*Defer[Subst][Defer[Int][(-1 + x^6)^(2/3)/((-1)^(1/18) - (-1)^(5/9)*x), x], x, x^(1/3)])/(6*(-x^2 + x^4)^(1
/3)) + ((-1)^(1/18)*x^(2/3)*(-1 + x^2)^(1/3)*Defer[Subst][Defer[Int][(-1 + x^6)^(2/3)/((-1)^(1/18) + (-1)^(5/9
)*x), x], x, x^(1/3)])/(6*(-x^2 + x^4)^(1/3)) + ((-1)^(1/18)*(1 + I*Sqrt[3])*x^(2/3)*(-1 + x^2)^(1/3)*Defer[Su
bst][Defer[Int][(-1 + x^6)^(2/3)/((-1)^(1/18) - (-1)^(2/3)*x), x], x, x^(1/3)])/(6*(-x^2 + x^4)^(1/3)) + ((-1)
^(1/18)*(1 + I*Sqrt[3])*x^(2/3)*(-1 + x^2)^(1/3)*Defer[Subst][Defer[Int][(-1 + x^6)^(2/3)/((-1)^(1/18) + (-1)^
(2/3)*x), x], x, x^(1/3)])/(6*(-x^2 + x^4)^(1/3)) + ((-1)^(1/18)*(1 - I*Sqrt[3])*x^(2/3)*(-1 + x^2)^(1/3)*Defe
r[Subst][Defer[Int][(-1 + x^6)^(2/3)/((-1)^(1/18) - (-1)^(7/9)*x), x], x, x^(1/3)])/(6*(-x^2 + x^4)^(1/3)) + (
(-1)^(1/18)*(1 - I*Sqrt[3])*x^(2/3)*(-1 + x^2)^(1/3)*Defer[Subst][Defer[Int][(-1 + x^6)^(2/3)/((-1)^(1/18) + (
-1)^(7/9)*x), x], x, x^(1/3)])/(6*(-x^2 + x^4)^(1/3)) + ((-1)^(1/18)*x^(2/3)*(-1 + x^2)^(1/3)*Defer[Subst][Def
er[Int][(-1 + x^6)^(2/3)/((-1)^(1/18) - (-1)^(8/9)*x), x], x, x^(1/3)])/(6*(-x^2 + x^4)^(1/3)) + ((-1)^(1/18)*
x^(2/3)*(-1 + x^2)^(1/3)*Defer[Subst][Defer[Int][(-1 + x^6)^(2/3)/((-1)^(1/18) + (-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 (\sqrt [18]{-1}+(-1)^{7/18}+(-1)^{13/18}\right ) \left (-1+x^6\right )^{2/3}}{18 \left (\sqrt [18]{-1}-x\right )}+\frac {\left (\sqrt [18]{-1}+(-1)^{7/18}+(-1)^{13/18}\right ) \left (-1+x^6\right )^{2/3}}{18 \left (\sqrt [18]{-1}+x\right )}+\frac {\left (\sqrt [18]{-1}-(-1)^{7/18}-(-1)^{13/18}\right ) \left (-1+x^6\right )^{2/3}}{18 \left (\sqrt [18]{-1}-\sqrt [9]{-1} x\right )}+\frac {\left (\sqrt [18]{-1}-(-1)^{7/18}-(-1)^{13/18}\right ) \left (-1+x^6\right )^{2/3}}{18 \left (\sqrt [18]{-1}+\sqrt [9]{-1} x\right )}+\frac {\sqrt [18]{-1} \left (-1+x^6\right )^{2/3}}{18 \left (\sqrt [18]{-1}-(-1)^{2/9} x\right )}+\frac {\sqrt [18]{-1} \left (-1+x^6\right )^{2/3}}{18 \left (\sqrt [18]{-1}+(-1)^{2/9} x\right )}+\frac {\left (\sqrt [18]{-1}+(-1)^{7/18}+(-1)^{13/18}\right ) \left (-1+x^6\right )^{2/3}}{18 \left (\sqrt [18]{-1}-\sqrt [3]{-1} x\right )}+\frac {\left (\sqrt [18]{-1}+(-1)^{7/18}+(-1)^{13/18}\right ) \left (-1+x^6\right )^{2/3}}{18 \left (\sqrt [18]{-1}+\sqrt [3]{-1} x\right )}+\frac {\left (\sqrt [18]{-1}-(-1)^{7/18}-(-1)^{13/18}\right ) \left (-1+x^6\right )^{2/3}}{18 \left (\sqrt [18]{-1}-(-1)^{4/9} x\right )}+\frac {\left (\sqrt [18]{-1}-(-1)^{7/18}-(-1)^{13/18}\right ) \left (-1+x^6\right )^{2/3}}{18 \left (\sqrt [18]{-1}+(-1)^{4/9} x\right )}+\frac {\sqrt [18]{-1} \left (-1+x^6\right )^{2/3}}{18 \left (\sqrt [18]{-1}-(-1)^{5/9} x\right )}+\frac {\sqrt [18]{-1} \left (-1+x^6\right )^{2/3}}{18 \left (\sqrt [18]{-1}+(-1)^{5/9} x\right )}+\frac {\left (\sqrt [18]{-1}+(-1)^{7/18}+(-1)^{13/18}\right ) \left (-1+x^6\right )^{2/3}}{18 \left (\sqrt [18]{-1}-(-1)^{2/3} x\right )}+\frac {\left (\sqrt [18]{-1}+(-1)^{7/18}+(-1)^{13/18}\right ) \left (-1+x^6\right )^{2/3}}{18 \left (\sqrt [18]{-1}+(-1)^{2/3} x\right )}+\frac {\left (\sqrt [18]{-1}-(-1)^{7/18}-(-1)^{13/18}\right ) \left (-1+x^6\right )^{2/3}}{18 \left (\sqrt [18]{-1}-(-1)^{7/9} x\right )}+\frac {\left (\sqrt [18]{-1}-(-1)^{7/18}-(-1)^{13/18}\right ) \left (-1+x^6\right )^{2/3}}{18 \left (\sqrt [18]{-1}+(-1)^{7/9} x\right )}+\frac {\sqrt [18]{-1} \left (-1+x^6\right )^{2/3}}{18 \left (\sqrt [18]{-1}-(-1)^{8/9} x\right )}+\frac {\sqrt [18]{-1} \left (-1+x^6\right )^{2/3}}{18 \left (\sqrt [18]{-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 [C] (verified)

Result contains complex when optimal does not.

Time = 0.95 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.22 \[ \int \frac {-1+x^6}{\sqrt [3]{-x^2+x^4} \left (1+x^6\right )} \, dx=-\frac {x^{2/3} \sqrt [3]{-1+x^2} \left (8 \arctan \left (\frac {\sqrt [3]{x}}{\sqrt [3]{-1+x^2}}\right )+2\ 2^{2/3} \arctan \left (\frac {\sqrt [3]{2} \sqrt [3]{x}}{\sqrt [3]{-1+x^2}}\right )+4 \arctan \left (\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{x}}{2 \sqrt [3]{-1+x^2}}\right )+4 i \sqrt {3} \arctan \left (\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{x}}{2 \sqrt [3]{-1+x^2}}\right )+4 \arctan \left (\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{x}}{2 \sqrt [3]{-1+x^2}}\right )-4 i \sqrt {3} \arctan \left (\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{x}}{2 \sqrt [3]{-1+x^2}}\right )+2^{2/3} \arctan \left (\frac {2^{2/3} \sqrt [3]{x} \sqrt [3]{-1+x^2}}{-2 x^{2/3}+\sqrt [3]{2} \left (-1+x^2\right )^{2/3}}\right )+2^{2/3} \sqrt {3} \text {arctanh}\left (\frac {2^{2/3} \sqrt {3} \sqrt [3]{x} \sqrt [3]{-1+x^2}}{2 x^{2/3}+\sqrt [3]{2} \left (-1+x^2\right )^{2/3}}\right )\right )}{12 \sqrt [3]{x^2 \left (-1+x^2\right )}} \]

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

Maple [A] (verified)

Time = 82.27 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.22

method	result	size
pseudoelliptic	\(\frac {\left (\left (\ln \left (\frac {-\sqrt {3}\, 2^{\frac {1}{3}} \left (x^{4}-x^{2}\right )^{\frac {1}{3}} x +2^{\frac {2}{3}} x^{2}+\left (x^{4}-x^{2}\right )^{\frac {2}{3}}}{x^{2}}\right )-\ln \left (\frac {\sqrt {3}\, 2^{\frac {1}{3}} \left (x^{4}-x^{2}\right )^{\frac {1}{3}} x +2^{\frac {2}{3}} x^{2}+\left (x^{4}-x^{2}\right )^{\frac {2}{3}}}{x^{2}}\right )\right ) 2^{\frac {2}{3}}+4 \ln \left (\frac {-\left (x^{4}-x^{2}\right )^{\frac {1}{3}} \sqrt {3}\, x +\left (x^{4}-x^{2}\right )^{\frac {2}{3}}+x^{2}}{x^{2}}\right )-4 \ln \left (\frac {\left (x^{4}-x^{2}\right )^{\frac {1}{3}} \sqrt {3}\, x +\left (x^{4}-x^{2}\right )^{\frac {2}{3}}+x^{2}}{x^{2}}\right )\right ) \sqrt {3}}{24}+\frac {\left (4 \arctan \left (\frac {\left (x^{4}-x^{2}\right )^{\frac {1}{3}} 2^{\frac {2}{3}}}{2 x}\right )-2 \arctan \left (\frac {-\left (x^{4}-x^{2}\right )^{\frac {1}{3}} 2^{\frac {2}{3}}+x \sqrt {3}}{x}\right )+2 \arctan \left (\frac {\left (x^{4}-x^{2}\right )^{\frac {1}{3}} 2^{\frac {2}{3}}+x \sqrt {3}}{x}\right )\right ) 2^{\frac {2}{3}}}{24}+\frac {2 \arctan \left (\frac {\left (x^{4}-x^{2}\right )^{\frac {1}{3}}}{x}\right )}{3}-\frac {\arctan \left (\frac {x \sqrt {3}-2 \left (x^{4}-x^{2}\right )^{\frac {1}{3}}}{x}\right )}{3}+\frac {\arctan \left (\frac {x \sqrt {3}+2 \left (x^{4}-x^{2}\right )^{\frac {1}{3}}}{x}\right )}{3}\)	\(334\)
trager	\(\text {Expression too large to display}\)	\(12283\)

[In]

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

[Out]

1/24*((ln((-3^(1/2)*2^(1/3)*(x^4-x^2)^(1/3)*x+2^(2/3)*x^2+(x^4-x^2)^(2/3))/x^2)-ln((3^(1/2)*2^(1/3)*(x^4-x^2)^
(1/3)*x+2^(2/3)*x^2+(x^4-x^2)^(2/3))/x^2))*2^(2/3)+4*ln((-(x^4-x^2)^(1/3)*3^(1/2)*x+(x^4-x^2)^(2/3)+x^2)/x^2)-
4*ln(((x^4-x^2)^(1/3)*3^(1/2)*x+(x^4-x^2)^(2/3)+x^2)/x^2))*3^(1/2)+1/24*(4*arctan(1/2*(x^4-x^2)^(1/3)/x*2^(2/3
))-2*arctan((-(x^4-x^2)^(1/3)*2^(2/3)+x*3^(1/2))/x)+2*arctan(((x^4-x^2)^(1/3)*2^(2/3)+x*3^(1/2))/x))*2^(2/3)+2
/3*arctan((x^4-x^2)^(1/3)/x)-1/3*arctan((x*3^(1/2)-2*(x^4-x^2)^(1/3))/x)+1/3*arctan((x*3^(1/2)+2*(x^4-x^2)^(1/
3))/x)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 9.20 (sec) , antiderivative size = 2527, normalized size of antiderivative = 9.22 \[ \int \frac {-1+x^6}{\sqrt [3]{-x^2+x^4} \left (1+x^6\right )} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/48*2^(2/3)*(-1)^(1/6)*(sqrt(-3) - 1)*log(-2*(8*(x^4 - x^2)^(2/3)*((512*I - 59)*x^2 - (118*I + 1024)*x - 512*
I + 59) + 4*2^(1/3)*(x^4 - x^2)^(1/3)*((-1)^(5/6)*(59*x^3 + 1024*x^2 + sqrt(-3)*(59*x^3 + 1024*x^2 - 59*x) - 5
9*x) + 2*(-1)^(1/3)*(256*x^3 - 59*x^2 + sqrt(-3)*(256*x^3 - 59*x^2 - 256*x) - 256*x)) + 2^(2/3)*(4*(-1)^(2/3)*
(128*x^5 - 59*x^4 - 768*x^3 + 59*x^2 - sqrt(-3)*(128*x^5 - 59*x^4 - 768*x^3 + 59*x^2 + 128*x) + 128*x) - (-1)^
(1/6)*(59*x^5 + 2048*x^4 - 354*x^3 - 2048*x^2 - sqrt(-3)*(59*x^5 + 2048*x^4 - 354*x^3 - 2048*x^2 + 59*x) + 59*
x)))/(x^5 + 2*x^3 + x)) - 1/48*2^(2/3)*(-1)^(1/6)*(sqrt(-3) + 1)*log(-2*(8*(x^4 - x^2)^(2/3)*((512*I - 59)*x^2
- (118*I + 1024)*x - 512*I + 59) + 4*2^(1/3)*(x^4 - x^2)^(1/3)*((-1)^(5/6)*(59*x^3 + 1024*x^2 - sqrt(-3)*(59*
x^3 + 1024*x^2 - 59*x) - 59*x) + 2*(-1)^(1/3)*(256*x^3 - 59*x^2 - sqrt(-3)*(256*x^3 - 59*x^2 - 256*x) - 256*x)
) + 2^(2/3)*(4*(-1)^(2/3)*(128*x^5 - 59*x^4 - 768*x^3 + 59*x^2 + sqrt(-3)*(128*x^5 - 59*x^4 - 768*x^3 + 59*x^2
+ 128*x) + 128*x) - (-1)^(1/6)*(59*x^5 + 2048*x^4 - 354*x^3 - 2048*x^2 + sqrt(-3)*(59*x^5 + 2048*x^4 - 354*x^
3 - 2048*x^2 + 59*x) + 59*x)))/(x^5 + 2*x^3 + x)) - 1/48*2^(2/3)*(-1)^(1/6)*(sqrt(-3) - 1)*log(-2*(8*(x^4 - x^
2)^(2/3)*(-(512*I + 59)*x^2 + (118*I - 1024)*x + 512*I + 59) - 4*2^(1/3)*(x^4 - x^2)^(1/3)*((-1)^(5/6)*(59*x^3
+ 1024*x^2 + sqrt(-3)*(59*x^3 + 1024*x^2 - 59*x) - 59*x) - 2*(-1)^(1/3)*(256*x^3 - 59*x^2 + sqrt(-3)*(256*x^3
- 59*x^2 - 256*x) - 256*x)) + 2^(2/3)*(4*(-1)^(2/3)*(128*x^5 - 59*x^4 - 768*x^3 + 59*x^2 - sqrt(-3)*(128*x^5
- 59*x^4 - 768*x^3 + 59*x^2 + 128*x) + 128*x) + (-1)^(1/6)*(59*x^5 + 2048*x^4 - 354*x^3 - 2048*x^2 - sqrt(-3)*
(59*x^5 + 2048*x^4 - 354*x^3 - 2048*x^2 + 59*x) + 59*x)))/(x^5 + 2*x^3 + x)) + 1/48*2^(2/3)*(-1)^(1/6)*(sqrt(-
3) + 1)*log(-2*(8*(x^4 - x^2)^(2/3)*(-(512*I + 59)*x^2 + (118*I - 1024)*x + 512*I + 59) - 4*2^(1/3)*(x^4 - x^2
)^(1/3)*((-1)^(5/6)*(59*x^3 + 1024*x^2 - sqrt(-3)*(59*x^3 + 1024*x^2 - 59*x) - 59*x) - 2*(-1)^(1/3)*(256*x^3 -
59*x^2 - sqrt(-3)*(256*x^3 - 59*x^2 - 256*x) - 256*x)) + 2^(2/3)*(4*(-1)^(2/3)*(128*x^5 - 59*x^4 - 768*x^3 +
59*x^2 + sqrt(-3)*(128*x^5 - 59*x^4 - 768*x^3 + 59*x^2 + 128*x) + 128*x) + (-1)^(1/6)*(59*x^5 + 2048*x^4 - 354
*x^3 - 2048*x^2 + sqrt(-3)*(59*x^5 + 2048*x^4 - 354*x^3 - 2048*x^2 + 59*x) + 59*x)))/(x^5 + 2*x^3 + x)) + 1/24
*2^(2/3)*(-1)^(1/6)*log(-2*(4*(x^4 - x^2)^(2/3)*((512*I - 59)*x^2 - (118*I + 1024)*x - 512*I + 59) - 4*2^(1/3)
*(x^4 - x^2)^(1/3)*((-1)^(5/6)*(59*x^3 + 1024*x^2 - 59*x) + 2*(-1)^(1/3)*(256*x^3 - 59*x^2 - 256*x)) - 2^(2/3)
*(4*(-1)^(2/3)*(128*x^5 - 59*x^4 - 768*x^3 + 59*x^2 + 128*x) - (-1)^(1/6)*(59*x^5 + 2048*x^4 - 354*x^3 - 2048*
x^2 + 59*x)))/(x^5 + 2*x^3 + x)) - 1/24*2^(2/3)*(-1)^(1/6)*log(-2*(4*(x^4 - x^2)^(2/3)*(-(512*I + 59)*x^2 + (1
18*I - 1024)*x + 512*I + 59) + 4*2^(1/3)*(x^4 - x^2)^(1/3)*((-1)^(5/6)*(59*x^3 + 1024*x^2 - 59*x) - 2*(-1)^(1/
3)*(256*x^3 - 59*x^2 - 256*x)) - 2^(2/3)*(4*(-1)^(2/3)*(128*x^5 - 59*x^4 - 768*x^3 + 59*x^2 + 128*x) + (-1)^(1
/6)*(59*x^5 + 2048*x^4 - 354*x^3 - 2048*x^2 + 59*x)))/(x^5 + 2*x^3 + x)) - 1/12*sqrt(-2*I*sqrt(3) + 2)*log((60
8*x^5 + 5014*x^4 - 1824*x^3 - 5014*x^2 + (x^4 - x^2)^(2/3)*(10028*x^2 - (608*x^2 - sqrt(3)*(608*I*x^2 + 2507*I
*x - 608*I) + 2507*x - 608)*sqrt(-2*I*sqrt(3) + 2) - 2432*x - 10028) - 2*sqrt(3)*(-304*I*x^5 - 2507*I*x^4 + 91
2*I*x^3 + 2507*I*x^2 - 304*I*x) + (2507*x^5 - 1216*x^4 - 7521*x^3 + 1216*x^2 + 2507*x)*sqrt(-2*I*sqrt(3) + 2)
- (x^4 - x^2)^(1/3)*(1216*x^3 + 5014*x^2 + 2*sqrt(3)*(-608*I*x^3 - 2507*I*x^2 + 608*I*x) - (2507*x^3 - 608*x^2
+ sqrt(3)*(2507*I*x^3 - 608*I*x^2 - 2507*I*x) - 2507*x)*sqrt(-2*I*sqrt(3) + 2) - 1216*x) + 608*x)/(x^5 - x^3
+ x)) + 1/12*sqrt(-2*I*sqrt(3) + 2)*log((608*x^5 + 5014*x^4 - 1824*x^3 - 5014*x^2 + (x^4 - x^2)^(2/3)*(10028*x
^2 + (608*x^2 + sqrt(3)*(-608*I*x^2 - 2507*I*x + 608*I) + 2507*x - 608)*sqrt(-2*I*sqrt(3) + 2) - 2432*x - 1002
8) - 2*sqrt(3)*(-304*I*x^5 - 2507*I*x^4 + 912*I*x^3 + 2507*I*x^2 - 304*I*x) - (2507*x^5 - 1216*x^4 - 7521*x^3
+ 1216*x^2 + 2507*x)*sqrt(-2*I*sqrt(3) + 2) - (x^4 - x^2)^(1/3)*(1216*x^3 + 5014*x^2 + 2*sqrt(3)*(-608*I*x^3 -
2507*I*x^2 + 608*I*x) + (2507*x^3 - 608*x^2 - sqrt(3)*(-2507*I*x^3 + 608*I*x^2 + 2507*I*x) - 2507*x)*sqrt(-2*
I*sqrt(3) + 2) - 1216*x) + 608*x)/(x^5 - x^3 + x)) + 1/12*sqrt(2*I*sqrt(3) + 2)*log((608*x^5 + 5014*x^4 - 1824
*x^3 - 5014*x^2 + 4*(x^4 - x^2)^(2/3)*(2507*x^2 - 608*x - 2507) - 2*sqrt(3)*(304*I*x^5 + 2507*I*x^4 - 912*I*x^
3 - 2507*I*x^2 + 304*I*x) - (2507*x^5 - 1216*x^4 - 7521*x^3 + 1216*x^2 - (x^4 - x^2)^(2/3)*(608*x^2 + sqrt(3)*
(608*I*x^2 + 2507*I*x - 608*I) + 2507*x - 608) + (x^4 - x^2)^(1/3)*(2507*x^3 - 608*x^2 - sqrt(3)*(2507*I*x^3 -
608*I*x^2 - 2507*I*x) - 2507*x) + 2507*x)*sqrt(2*I*sqrt(3) + 2) - 2*(x^4 - x^2)^(1/3)*(608*x^3 + 2507*x^2 + s
qrt(3)*(608*I*x^3 + 2507*I*x^2 - 608*I*x) - 608*x) + 608*x)/(x^5 - x^3 + x)) - 1/12*sqrt(2*I*sqrt(3) + 2)*log(
(608*x^5 + 5014*x^4 - 1824*x^3 - 5014*x^2 + 4*(x^4 - x^2)^(2/3)*(2507*x^2 - 608*x - 2507) - 2*sqrt(3)*(304*I*x
^5 + 2507*I*x^4 - 912*I*x^3 - 2507*I*x^2 + 304*I*x) + (2507*x^5 - 1216*x^4 - 7521*x^3 + 1216*x^2 - (x^4 - x^2)
^(2/3)*(608*x^2 - sqrt(3)*(-608*I*x^2 - 2507*I*x + 608*I) + 2507*x - 608) + (x^4 - x^2)^(1/3)*(2507*x^3 - 608*
x^2 + sqrt(3)*(-2507*I*x^3 + 608*I*x^2 + 2507*I*x) - 2507*x) + 2507*x)*sqrt(2*I*sqrt(3) + 2) - 2*(x^4 - x^2)^(
1/3)*(608*x^3 + 2507*x^2 + sqrt(3)*(608*I*x^3 + 2507*I*x^2 - 608*I*x) - 608*x) + 608*x)/(x^5 - x^3 + x)) - 1/3
*arctan(-2*(1910654896*x^5 - 17610113139*x^4 - 5731964688*x^3 + 17610113139*x^2 - 6654713*(x^4 - x^2)^(2/3)*(2
507*x^2 + 1216*x - 2507) + 6654713*(x^4 - x^2)^(1/3)*(1216*x^3 - 2507*x^2 - 1216*x) + 1910654896*x)/(157566178
43*x^5 + 24725904448*x^4 - 47269853529*x^3 - 24725904448*x^2 + 15756617843*x))

Sympy [F]

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

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

[In]

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

[Out]

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