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

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

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Rubi [F]  time = 2.18, 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)])/(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)) + ((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)) - ((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)])/(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)) - ((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))

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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {Subst}\left (\int \left (-\frac {3}{2 \sqrt [3]{1+x^6} \left (1-x^9\right )}-\frac {1}{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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 (i-\sqrt {3}\right ) \left (-1-i \sqrt {3}\right ) x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 (-i-\sqrt {3}\right ) \left (-1+i \sqrt {3}\right ) x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 (1+i \sqrt {3}\right ) \left (-i+\sqrt {3}\right ) x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {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 (3 \left (1-i \sqrt {3}\right ) \left (i+\sqrt {3}\right ) x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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}}\\ \end {align*}

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Mathematica [F]  time = 0.38, size = 0, normalized size = 0.00 \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]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 1.38, size = 429, normalized size = 1.00 \begin {gather*} -\frac {1}{2} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{x^2+x^4}}\right )-\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{x^2+x^4}}\right )}{2 \sqrt {3}}-\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{-x+2^{2/3} \sqrt [3]{x^2+x^4}}\right )}{4 \sqrt [3]{2} \sqrt {3}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{x^2+x^4}}\right )}{4 \sqrt [3]{2}}+\frac {1}{6} \log \left (-x+\sqrt [3]{x^2+x^4}\right )-\frac {1}{2} \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 )}{4 \sqrt [3]{2}}-\frac {\log \left (2 x+2^{2/3} \sqrt [3]{x^2+x^4}\right )}{12 \sqrt [3]{2}}+\frac {1}{4} \log \left (x^2-x \sqrt [3]{x^2+x^4}+\left (x^2+x^4\right )^{2/3}\right )-\frac {1}{12} \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 )}{24 \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 )}{8 \sqrt [3]{2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{6} + x^{3} + 1}{{\left (x^{6} - 1\right )} {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}}}\,{d x} \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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maple [F]  time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {x^{6}+x^{3}+1}{\left (x^{4}+x^{2}\right )^{\frac {1}{3}} \left (x^{6}-1\right )}\, dx\]

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)

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{6} + x^{3} + 1}{{\left (x^{6} - 1\right )} {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}}}\,{d x} \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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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^3 + x^6 + 1)/((x^2 + x^4)^(1/3)*(x^6 - 1)),x)

[Out]

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

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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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