3.31.25 \(\int \frac {1-x^3+x^6}{\sqrt [3]{x^2+x^4} (-1+x^6)} \, dx\) [3025]

Optimal. Leaf size=429 \[ -\frac {\text {ArcTan}\left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{x^2+x^4}}\right )}{2 \sqrt {3}}-\frac {1}{2} \sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{x^2+x^4}}\right )-\frac {\sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x}{-x+2^{2/3} \sqrt [3]{x^2+x^4}}\right )}{4 \sqrt [3]{2}}-\frac {\text {ArcTan}\left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{x^2+x^4}}\right )}{4 \sqrt [3]{2} \sqrt {3}}+\frac {1}{2} \log \left (-x+\sqrt [3]{x^2+x^4}\right )-\frac {1}{6} \log \left (x+\sqrt [3]{x^2+x^4}\right )+\frac {\log \left (-2 x+2^{2/3} \sqrt [3]{x^2+x^4}\right )}{12 \sqrt [3]{2}}-\frac {\log \left (2 x+2^{2/3} \sqrt [3]{x^2+x^4}\right )}{4 \sqrt [3]{2}}+\frac {1}{12} \log \left (x^2-x \sqrt [3]{x^2+x^4}+\left (x^2+x^4\right )^{2/3}\right )-\frac {1}{4} \log \left (x^2+x \sqrt [3]{x^2+x^4}+\left (x^2+x^4\right )^{2/3}\right )+\frac {\log \left (-2 x^2+2^{2/3} x \sqrt [3]{x^2+x^4}-\sqrt [3]{2} \left (x^2+x^4\right )^{2/3}\right )}{8 \sqrt [3]{2}}-\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{x^2+x^4}+\sqrt [3]{2} \left (x^2+x^4\right )^{2/3}\right )}{24 \sqrt [3]{2}} \]

[Out]

-1/6*3^(1/2)*arctan(3^(1/2)*x/(-x+2*(x^4+x^2)^(1/3)))-1/2*arctan(3^(1/2)*x/(x+2*(x^4+x^2)^(1/3)))*3^(1/2)-1/8*
3^(1/2)*arctan(3^(1/2)*x/(-x+2^(2/3)*(x^4+x^2)^(1/3)))*2^(2/3)-1/24*3^(1/2)*arctan(3^(1/2)*x/(x+2^(2/3)*(x^4+x
^2)^(1/3)))*2^(2/3)+1/2*ln(-x+(x^4+x^2)^(1/3))-1/6*ln(x+(x^4+x^2)^(1/3))+1/24*ln(-2*x+2^(2/3)*(x^4+x^2)^(1/3))
*2^(2/3)-1/8*ln(2*x+2^(2/3)*(x^4+x^2)^(1/3))*2^(2/3)+1/12*ln(x^2-x*(x^4+x^2)^(1/3)+(x^4+x^2)^(2/3))-1/4*ln(x^2
+x*(x^4+x^2)^(1/3)+(x^4+x^2)^(2/3))+1/16*ln(-2*x^2+2^(2/3)*x*(x^4+x^2)^(1/3)-2^(1/3)*(x^4+x^2)^(2/3))*2^(2/3)-
1/48*ln(2*x^2+2^(2/3)*x*(x^4+x^2)^(1/3)+2^(1/3)*(x^4+x^2)^(2/3))*2^(2/3)

________________________________________________________________________________________

Rubi [F]
time = 1.82, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {1-x^3+x^6}{\sqrt [3]{x^2+x^4} \left (-1+x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(-2*x*(1 + x^2)^(1/3)*AppellF1[1/6, 1, 1/3, 7/6, (-2*x^2)/(1 - I*Sqrt[3]), -x^2])/(x^2 + x^4)^(1/3) - (2*x*(1
+ x^2)^(1/3)*AppellF1[1/6, 1, 1/3, 7/6, (-2*x^2)/(1 + I*Sqrt[3]), -x^2])/(x^2 + x^4)^(1/3) + ((I - Sqrt[3])*x^
2*(1 + x^2)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, -x^2, (-2*x^2)/(1 - I*Sqrt[3])])/(4*(I + Sqrt[3])*(x^2 + x^4)^(1/
3)) + ((I + Sqrt[3])*x^2*(1 + x^2)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, -x^2, (-2*x^2)/(1 + I*Sqrt[3])])/(4*(I - S
qrt[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) +
(x^(2/3)*(1 + x^2)^(1/3)*Defer[Subst][Defer[Int][1/((-1 + x)*(1 + x^6)^(1/3)), 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/((1 + x)*(1 + x^6)^(1/3)), x], x, x^(1/3)])/(2*(x
^2 + x^4)^(1/3)) + ((1 + I*Sqrt[3])*x^(2/3)*(1 + x^2)^(1/3)*Defer[Subst][Defer[Int][1/((-1 - I*Sqrt[3] + 2*x)*
(1 + x^6)^(1/3)), x], x, x^(1/3)])/(2*(x^2 + x^4)^(1/3)) - ((1 - I*Sqrt[3])*x^(2/3)*(1 + x^2)^(1/3)*Defer[Subs
t][Defer[Int][1/((1 - I*Sqrt[3] + 2*x)*(1 + x^6)^(1/3)), 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/((-1 + I*Sqrt[3] + 2*x)*(1 + x^6)^(1/3)), x], x, x^(1/3
)])/(2*(x^2 + x^4)^(1/3)) - ((1 + I*Sqrt[3])*x^(2/3)*(1 + x^2)^(1/3)*Defer[Subst][Defer[Int][1/((1 + I*Sqrt[3]
+ 2*x)*(1 + x^6)^(1/3)), x], x, x^(1/3)])/(6*(x^2 + x^4)^(1/3))

Rubi steps

\begin {align*} \int \frac {1-x^3+x^6}{\sqrt [3]{x^2+x^4} \left (-1+x^6\right )} \, dx &=\frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \int \frac {1-x^3+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^9+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-x^9}{\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 (3 x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {2-x^9}{\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} \, _2F_1\left (\frac {1}{6},\frac {1}{3};\frac {7}{6};-x^2\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}{2 \sqrt [3]{1+x^6} \left (1-x^9\right )}-\frac {3}{2 \sqrt [3]{1+x^6} \left (1+x^9\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^4}}\\ &=\frac {3 x \sqrt [3]{1+x^2} \, _2F_1\left (\frac {1}{6},\frac {1}{3};\frac {7}{6};-x^2\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} \left (1-x^9\right )} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x^2+x^4}}-\frac {\left (9 x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{1+x^6} \left (1+x^9\right )} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x^2+x^4}}\\ &=\frac {3 x \sqrt [3]{1+x^2} \, _2F_1\left (\frac {1}{6},\frac {1}{3};\frac {7}{6};-x^2\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}{9 (-1+x) \sqrt [3]{1+x^6}}+\frac {2+x}{9 \left (1+x+x^2\right ) \sqrt [3]{1+x^6}}+\frac {2+x^3}{3 \sqrt [3]{1+x^6} \left (1+x^3+x^6\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x^2+x^4}}-\frac {\left (9 x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \left (\frac {1}{9 (1+x) \sqrt [3]{1+x^6}}+\frac {2-x}{9 \left (1-x+x^2\right ) \sqrt [3]{1+x^6}}+\frac {2-x^3}{3 \sqrt [3]{1+x^6} \left (1-x^3+x^6\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x^2+x^4}}\\ &=\frac {3 x \sqrt [3]{1+x^2} \, _2F_1\left (\frac {1}{6},\frac {1}{3};\frac {7}{6};-x^2\right )}{\sqrt [3]{x^2+x^4}}+\frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{6 \sqrt [3]{x^2+x^4}}-\frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {2+x}{\left (1+x+x^2\right ) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{6 \sqrt [3]{x^2+x^4}}-\frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{(1+x) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x^2+x^4}}-\frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {2-x}{\left (1-x+x^2\right ) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x^2+x^4}}-\frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {2+x^3}{\sqrt [3]{1+x^6} \left (1+x^3+x^6\right )} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x^2+x^4}}-\frac {\left (3 x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {2-x^3}{\sqrt [3]{1+x^6} \left (1-x^3+x^6\right )} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x^2+x^4}}\\ &=\frac {3 x \sqrt [3]{1+x^2} \, _2F_1\left (\frac {1}{6},\frac {1}{3};\frac {7}{6};-x^2\right )}{\sqrt [3]{x^2+x^4}}+\frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{6 \sqrt [3]{x^2+x^4}}-\frac {\left (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\right ) \sqrt [3]{1+x^6}}+\frac {1+i \sqrt {3}}{\left (1+i \sqrt {3}+2 x\right ) \sqrt [3]{1+x^6}}\right ) \, dx,x,\sqrt [3]{x}\right )}{6 \sqrt [3]{x^2+x^4}}-\frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{(1+x) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x^2+x^4}}-\frac {\left (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\right ) \sqrt [3]{1+x^6}}+\frac {-1+i \sqrt {3}}{\left (-1+i \sqrt {3}+2 x\right ) \sqrt [3]{1+x^6}}\right ) \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x^2+x^4}}-\frac {\left (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^3\right ) \sqrt [3]{1+x^6}}+\frac {1+i \sqrt {3}}{\left (1+i \sqrt {3}+2 x^3\right ) \sqrt [3]{1+x^6}}\right ) \, dx,x,\sqrt [3]{x}\right )}{2 \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-i \sqrt {3}}{\left (-1-i \sqrt {3}+2 x^3\right ) \sqrt [3]{1+x^6}}+\frac {-1+i \sqrt {3}}{\left (-1+i \sqrt {3}+2 x^3\right ) \sqrt [3]{1+x^6}}\right ) \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x^2+x^4}}\\ &=\frac {3 x \sqrt [3]{1+x^2} \, _2F_1\left (\frac {1}{6},\frac {1}{3};\frac {7}{6};-x^2\right )}{\sqrt [3]{x^2+x^4}}+\frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{6 \sqrt [3]{x^2+x^4}}-\frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{(1+x) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{2 \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 (-1-i \sqrt {3}+2 x\right ) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x^2+x^4}}-\frac {\left (3 \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^3\right ) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{2 \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 (1-i \sqrt {3}+2 x\right ) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{6 \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 (1-i \sqrt {3}+2 x^3\right ) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{2 \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 (-1+i \sqrt {3}+2 x\right ) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x^2+x^4}}-\frac {\left (3 \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^3\right ) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{2 \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 (1+i \sqrt {3}+2 x\right ) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{6 \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 (1+i \sqrt {3}+2 x^3\right ) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x^2+x^4}}\\ &=\frac {3 x \sqrt [3]{1+x^2} \, _2F_1\left (\frac {1}{6},\frac {1}{3};\frac {7}{6};-x^2\right )}{\sqrt [3]{x^2+x^4}}+\frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{6 \sqrt [3]{x^2+x^4}}-\frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{(1+x) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{2 \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 (-1-i \sqrt {3}+2 x\right ) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x^2+x^4}}-\frac {\left (3 \left (-1-i \sqrt {3}\right ) x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \left (\frac {i-\sqrt {3}}{2 \left (i+\sqrt {3}+2 i x^6\right ) \sqrt [3]{1+x^6}}+\frac {x^3}{\sqrt [3]{1+x^6} \left (1-i \sqrt {3}+2 x^6\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{2 \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 (1-i \sqrt {3}+2 x\right ) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{6 \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 \left (\frac {i+\sqrt {3}}{2 \left (-i+\sqrt {3}-2 i x^6\right ) \sqrt [3]{1+x^6}}+\frac {x^3}{\sqrt [3]{1+x^6} \left (1+i \sqrt {3}+2 x^6\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{2 \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 (-1+i \sqrt {3}+2 x\right ) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x^2+x^4}}-\frac {\left (3 \left (-1+i \sqrt {3}\right ) x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \left (\frac {-i-\sqrt {3}}{2 \left (-i+\sqrt {3}-2 i x^6\right ) \sqrt [3]{1+x^6}}+\frac {x^3}{\sqrt [3]{1+x^6} \left (1+i \sqrt {3}+2 x^6\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{2 \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 (1+i \sqrt {3}+2 x\right ) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{6 \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 \left (\frac {-i+\sqrt {3}}{2 \left (i+\sqrt {3}+2 i x^6\right ) \sqrt [3]{1+x^6}}+\frac {x^3}{\sqrt [3]{1+x^6} \left (1-i \sqrt {3}+2 x^6\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x^2+x^4}}\\ &=\frac {3 x \sqrt [3]{1+x^2} \, _2F_1\left (\frac {1}{6},\frac {1}{3};\frac {7}{6};-x^2\right )}{\sqrt [3]{x^2+x^4}}+\frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{6 \sqrt [3]{x^2+x^4}}-\frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{(1+x) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{2 \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 (-1-i \sqrt {3}+2 x\right ) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x^2+x^4}}-\frac {\left (3 \left (-1-i \sqrt {3}\right ) x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {x^3}{\sqrt [3]{1+x^6} \left (1-i \sqrt {3}+2 x^6\right )} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x^2+x^4}}-\frac {\left (3 \left (i-\sqrt {3}\right ) \left (-1-i \sqrt {3}\right ) x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (i+\sqrt {3}+2 i x^6\right ) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{4 \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 (1-i \sqrt {3}+2 x\right ) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{6 \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 {x^3}{\sqrt [3]{1+x^6} \left (1+i \sqrt {3}+2 x^6\right )} \, dx,x,\sqrt [3]{x}\right )}{2 \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 (-1+i \sqrt {3}+2 x\right ) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x^2+x^4}}-\frac {\left (3 \left (-1+i \sqrt {3}\right ) x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {x^3}{\sqrt [3]{1+x^6} \left (1+i \sqrt {3}+2 x^6\right )} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x^2+x^4}}-\frac {\left (3 \left (-i-\sqrt {3}\right ) \left (-1+i \sqrt {3}\right ) x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-i+\sqrt {3}-2 i x^6\right ) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{4 \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 (1+i \sqrt {3}+2 x\right ) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{6 \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 {x^3}{\sqrt [3]{1+x^6} \left (1-i \sqrt {3}+2 x^6\right )} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x^2+x^4}}-\frac {\left (\left (1+i \sqrt {3}\right ) \left (-i+\sqrt {3}\right ) x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (i+\sqrt {3}+2 i x^6\right ) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{4 \sqrt [3]{x^2+x^4}}-\frac {\left (\left (1-i \sqrt {3}\right ) \left (i+\sqrt {3}\right ) x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-i+\sqrt {3}-2 i x^6\right ) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{4 \sqrt [3]{x^2+x^4}}\\ &=-\frac {2 x \sqrt [3]{1+x^2} F_1\left (\frac {1}{6};1,\frac {1}{3};\frac {7}{6};-\frac {2 x^2}{1-i \sqrt {3}},-x^2\right )}{\sqrt [3]{x^2+x^4}}-\frac {2 x \sqrt [3]{1+x^2} F_1\left (\frac {1}{6};1,\frac {1}{3};\frac {7}{6};-\frac {2 x^2}{1+i \sqrt {3}},-x^2\right )}{\sqrt [3]{x^2+x^4}}+\frac {3 x \sqrt [3]{1+x^2} \, _2F_1\left (\frac {1}{6},\frac {1}{3};\frac {7}{6};-x^2\right )}{\sqrt [3]{x^2+x^4}}+\frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{6 \sqrt [3]{x^2+x^4}}-\frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{(1+x) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{2 \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 (-1-i \sqrt {3}+2 x\right ) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x^2+x^4}}-\frac {\left (3 \left (-1-i \sqrt {3}\right ) x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {x}{\sqrt [3]{1+x^3} \left (1-i \sqrt {3}+2 x^3\right )} \, dx,x,x^{2/3}\right )}{4 \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 (1-i \sqrt {3}+2 x\right ) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{6 \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 {x}{\sqrt [3]{1+x^3} \left (1+i \sqrt {3}+2 x^3\right )} \, dx,x,x^{2/3}\right )}{4 \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 (-1+i \sqrt {3}+2 x\right ) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x^2+x^4}}-\frac {\left (3 \left (-1+i \sqrt {3}\right ) x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {x}{\sqrt [3]{1+x^3} \left (1+i \sqrt {3}+2 x^3\right )} \, dx,x,x^{2/3}\right )}{4 \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 (1+i \sqrt {3}+2 x\right ) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{6 \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 {x}{\sqrt [3]{1+x^3} \left (1-i \sqrt {3}+2 x^3\right )} \, dx,x,x^{2/3}\right )}{4 \sqrt [3]{x^2+x^4}}\\ &=-\frac {2 x \sqrt [3]{1+x^2} F_1\left (\frac {1}{6};1,\frac {1}{3};\frac {7}{6};-\frac {2 x^2}{1-i \sqrt {3}},-x^2\right )}{\sqrt [3]{x^2+x^4}}-\frac {2 x \sqrt [3]{1+x^2} F_1\left (\frac {1}{6};1,\frac {1}{3};\frac {7}{6};-\frac {2 x^2}{1+i \sqrt {3}},-x^2\right )}{\sqrt [3]{x^2+x^4}}+\frac {\left (i-\sqrt {3}\right ) x^2 \sqrt [3]{1+x^2} F_1\left (\frac {2}{3};\frac {1}{3},1;\frac {5}{3};-x^2,-\frac {2 x^2}{1-i \sqrt {3}}\right )}{4 \left (i+\sqrt {3}\right ) \sqrt [3]{x^2+x^4}}+\frac {\left (i+\sqrt {3}\right ) x^2 \sqrt [3]{1+x^2} F_1\left (\frac {2}{3};\frac {1}{3},1;\frac {5}{3};-x^2,-\frac {2 x^2}{1+i \sqrt {3}}\right )}{4 \left (i-\sqrt {3}\right ) \sqrt [3]{x^2+x^4}}+\frac {3 x \sqrt [3]{1+x^2} \, _2F_1\left (\frac {1}{6},\frac {1}{3};\frac {7}{6};-x^2\right )}{\sqrt [3]{x^2+x^4}}+\frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{6 \sqrt [3]{x^2+x^4}}-\frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{(1+x) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{2 \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 (-1-i \sqrt {3}+2 x\right ) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{2 \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 (1-i \sqrt {3}+2 x\right ) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{6 \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 (-1+i \sqrt {3}+2 x\right ) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{2 \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 (1+i \sqrt {3}+2 x\right ) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{6 \sqrt [3]{x^2+x^4}}\\ \end {align*}

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Mathematica [A]
time = 3.48, size = 471, normalized size = 1.10 \begin {gather*} \frac {x^{2/3} \sqrt [3]{1+x^2} \left (8 \sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [3]{x}}{\sqrt [3]{x}-2 \sqrt [3]{1+x^2}}\right )-24 \sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [3]{x}}{\sqrt [3]{x}+2 \sqrt [3]{1+x^2}}\right )+6\ 2^{2/3} \sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [3]{x}}{\sqrt [3]{x}-2^{2/3} \sqrt [3]{1+x^2}}\right )-2\ 2^{2/3} \sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [3]{x}}{\sqrt [3]{x}+2^{2/3} \sqrt [3]{1+x^2}}\right )+24 \log \left (-\sqrt [3]{x}+\sqrt [3]{1+x^2}\right )-8 \log \left (\sqrt [3]{x}+\sqrt [3]{1+x^2}\right )+2\ 2^{2/3} \log \left (-2 \sqrt [3]{x}+2^{2/3} \sqrt [3]{1+x^2}\right )-6\ 2^{2/3} \log \left (2 \sqrt [3]{x}+2^{2/3} \sqrt [3]{1+x^2}\right )+4 \log \left (x^{2/3}-\sqrt [3]{x} \sqrt [3]{1+x^2}+\left (1+x^2\right )^{2/3}\right )-12 \log \left (x^{2/3}+\sqrt [3]{x} \sqrt [3]{1+x^2}+\left (1+x^2\right )^{2/3}\right )+3\ 2^{2/3} \log \left (-2 x^{2/3}+2^{2/3} \sqrt [3]{x} \sqrt [3]{1+x^2}-\sqrt [3]{2} \left (1+x^2\right )^{2/3}\right )-2^{2/3} \log \left (2 x^{2/3}+2^{2/3} \sqrt [3]{x} \sqrt [3]{1+x^2}+\sqrt [3]{2} \left (1+x^2\right )^{2/3}\right )\right )}{48 \sqrt [3]{x^2+x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x^(2/3)*(1 + x^2)^(1/3)*(8*Sqrt[3]*ArcTan[(Sqrt[3]*x^(1/3))/(x^(1/3) - 2*(1 + x^2)^(1/3))] - 24*Sqrt[3]*ArcTa
n[(Sqrt[3]*x^(1/3))/(x^(1/3) + 2*(1 + x^2)^(1/3))] + 6*2^(2/3)*Sqrt[3]*ArcTan[(Sqrt[3]*x^(1/3))/(x^(1/3) - 2^(
2/3)*(1 + x^2)^(1/3))] - 2*2^(2/3)*Sqrt[3]*ArcTan[(Sqrt[3]*x^(1/3))/(x^(1/3) + 2^(2/3)*(1 + x^2)^(1/3))] + 24*
Log[-x^(1/3) + (1 + x^2)^(1/3)] - 8*Log[x^(1/3) + (1 + x^2)^(1/3)] + 2*2^(2/3)*Log[-2*x^(1/3) + 2^(2/3)*(1 + x
^2)^(1/3)] - 6*2^(2/3)*Log[2*x^(1/3) + 2^(2/3)*(1 + x^2)^(1/3)] + 4*Log[x^(2/3) - x^(1/3)*(1 + x^2)^(1/3) + (1
+ x^2)^(2/3)] - 12*Log[x^(2/3) + x^(1/3)*(1 + x^2)^(1/3) + (1 + x^2)^(2/3)] + 3*2^(2/3)*Log[-2*x^(2/3) + 2^(2
/3)*x^(1/3)*(1 + x^2)^(1/3) - 2^(1/3)*(1 + x^2)^(2/3)] - 2^(2/3)*Log[2*x^(2/3) + 2^(2/3)*x^(1/3)*(1 + x^2)^(1/
3) + 2^(1/3)*(1 + x^2)^(2/3)]))/(48*(x^2 + x^4)^(1/3))

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 235.32, size = 15564, normalized size = 36.28

method	result	size
trager	\(\text {Expression too large to display}\)	\(15564\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (re
sidue poly has multiple non-linear factors)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{6} - x^{3} + 1}{\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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^6-x^3+1}{{\left (x^4+x^2\right )}^{1/3}\,\left (x^6-1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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