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

Optimal. Leaf size=167 \[ -\frac {\left (x^4-x^2\right )^{2/3}}{x \left (x^2-1\right )}-\frac {2 \tan ^{-1}\left (\frac {\sqrt [6]{3} x}{\sqrt [3]{x^4-x^2}}\right )}{3 \sqrt [6]{3}}-\frac {\tan ^{-1}\left (\frac {3^{5/6} x \sqrt [3]{x^4-x^2}}{3^{2/3} \left (x^4-x^2\right )^{2/3}-3 x^2}\right )}{3 \sqrt [6]{3}}-\frac {\tanh ^{-1}\left (\frac {\frac {x^2}{\sqrt [3]{3}}+\frac {\left (x^4-x^2\right )^{2/3}}{3^{2/3}}}{x \sqrt [3]{x^4-x^2}}\right )}{3^{2/3}} \]

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Rubi [F]  time = 2.64, 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^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^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) + (x^(2/3)*(-1 + x^2)^(1/3)*Defer[Subst][Defer[I
nt][1/((-1 + x)*(-1 + x^6)^(1/3)), x], x, x^(1/3)])/(3*(-x^2 + x^4)^(1/3)) - (x^(2/3)*(-1 + x^2)^(1/3)*Defer[S
ubst][Defer[Int][1/((1 + x)*(-1 + x^6)^(1/3)), x], x, x^(1/3)])/(3*(-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)])/(3*
(-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)])/(3*(-x^2 + x^4)^(1/3)) + ((1 - I*Sqrt[3])*x^(2/3)*(-1 + x^2)^(1/3)*Defe
r[Subst][Defer[Int][1/((-1 + I*Sqrt[3] + 2*x)*(-1 + x^6)^(1/3)), x], x, x^(1/3)])/(3*(-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)])/(3*(-x^2 + x^4)^(1/3))

Rubi steps

\begin {align*} \int \frac {1+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^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^{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}{\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 (6 x^{2/3} \sqrt [3]{-1+x^2}\right ) \operatorname {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 {\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 (6 x^{2/3} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{9 \left (-1+x^2\right ) \sqrt [3]{-1+x^6}}+\frac {-2+x}{18 \left (1-x+x^2\right ) \sqrt [3]{-1+x^6}}+\frac {-2-x}{18 \left (1+x+x^2\right ) \sqrt [3]{-1+x^6}}+\frac {-2+x^3}{6 \sqrt [3]{-1+x^6} \left (1-x^3+x^6\right )}+\frac {-2-x^3}{6 \sqrt [3]{-1+x^6} \left (1+x^3+x^6\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 (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 )}{3 \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 )}{3 \sqrt [3]{-x^2+x^4}}+\frac {\left (2 x^{2/3} \sqrt [3]{-1+x^2}\right ) \operatorname {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 (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 )}{\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 )}{\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 \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 )}{3 \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 )}{3 \sqrt [3]{-x^2+x^4}}+\frac {\left (2 x^{2/3} \sqrt [3]{-1+x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{2 (-1+x) \sqrt [3]{-1+x^6}}-\frac {1}{2 (1+x) \sqrt [3]{-1+x^6}}\right ) \, dx,x,\sqrt [3]{x}\right )}{3 \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 )}{\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 )}{\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 )}{3 \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 )}{3 \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 )}{3 \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 )}{\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 )}{3 \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 )}{\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 )}{3 \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 )}{\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 )}{3 \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 )}{\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 )}{3 \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 )}{3 \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 )}{3 \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 )}{\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 )}{3 \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 )}{\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 )}{3 \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 )}{\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 )}{3 \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 )}{\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 )}{3 \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 )}{3 \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 )}{3 \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 )}{\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 )}{3 \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 )}{\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 )}{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 )}{3 \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 )}{\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 )}{3 \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 )}{\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 )}{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 ) \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 )}{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 ) \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 )}{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 (\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}{\sqrt [3]{1-x^6} \left (-i+\sqrt {3}-2 i 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}{\sqrt [3]{1-x^6} \left (i+\sqrt {3}+2 i 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 ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{1-x^6} \left (i+\sqrt {3}+2 i 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 ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{1-x^6} \left (-i+\sqrt {3}-2 i 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 \frac {1}{(-1+x) \sqrt [3]{-1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{3 \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 )}{3 \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 )}{3 \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 )}{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 )}{3 \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 )}{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 )}{3 \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 )}{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 )}{3 \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 )}{2 \sqrt [3]{-x^2+x^4}}\\ &=-\frac {2 x \sqrt [3]{1-x^2} F_1\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} F_1\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} \, _2F_1\left (\frac {1}{6},\frac {1}{3};\frac {7}{6};x^2\right )}{\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 )}{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 {x}{\sqrt [3]{1-x^3} \left (1+i \sqrt {3}+2 x^3\right )} \, dx,x,x^{2/3}\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 {x}{\sqrt [3]{1-x^3} \left (1+i \sqrt {3}+2 x^3\right )} \, dx,x,x^{2/3}\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 {x}{\sqrt [3]{1-x^3} \left (1-i \sqrt {3}+2 x^3\right )} \, dx,x,x^{2/3}\right )}{2 \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 )}{3 \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 )}{3 \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 )}{3 \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 )}{3 \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 )}{3 \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 )}{3 \sqrt [3]{-x^2+x^4}}\\ &=-\frac {2 x \sqrt [3]{1-x^2} F_1\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} F_1\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} \, _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 )}{3 \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 )}{3 \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 )}{3 \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 )}{3 \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 )}{3 \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 )}{3 \sqrt [3]{-x^2+x^4}}\\ \end {align*}

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Mathematica [F]  time = 0.78, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1+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^6)/((-x^2 + x^4)^(1/3)*(-1 + x^6)),x]

[Out]

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

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IntegrateAlgebraic [A]  time = 0.54, size = 167, normalized size = 1.00 \begin {gather*} -\frac {\left (-x^2+x^4\right )^{2/3}}{x \left (-1+x^2\right )}-\frac {2 \tan ^{-1}\left (\frac {\sqrt [6]{3} x}{\sqrt [3]{-x^2+x^4}}\right )}{3 \sqrt [6]{3}}-\frac {\tan ^{-1}\left (\frac {3^{5/6} x \sqrt [3]{-x^2+x^4}}{-3 x^2+3^{2/3} \left (-x^2+x^4\right )^{2/3}}\right )}{3 \sqrt [6]{3}}-\frac {\tanh ^{-1}\left (\frac {\frac {x^2}{\sqrt [3]{3}}+\frac {\left (-x^2+x^4\right )^{2/3}}{3^{2/3}}}{x \sqrt [3]{-x^2+x^4}}\right )}{3^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 7.67, size = 2054, normalized size = 12.30

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/72*(16*3^(5/6)*(x^3 - x)*arctan(-1/3*(5200566*3^(5/6)*(x^4 - x^2)^(1/3)*(960*x^3 + 419*x^2 - 960*x) - 931*3^
(5/6)*(2*3^(5/6)*(x^4 - x^2)^(2/3)*(201120*x^2 + 1557961*x - 201120) + 2*sqrt(3)*(x^4 - x^2)^(1/3)*(1557961*x^
3 - 603360*x^2 - 1557961*x) + 175561*3^(1/6)*(x^5 + x^3 + x)) + 5200566*3^(1/6)*(x^4 - x^2)^(2/3)*(419*x^2 - 2
880*x - 419) + 2514*sqrt(3)*(100560*x^5 + 1557961*x^4 - 502800*x^3 - 1557961*x^2 + 100560*x))/(73560059*x^5 -
9479471040*x^4 - 367800295*x^3 + 9479471040*x^2 + 73560059*x)) - 8*3^(5/6)*(x^3 - x)*arctan(1/3*(1862*sqrt(3)*
(8*(x^4 - x^2)^(2/3)*(3^(5/6)*(1024558870866960*x^6 - 2840455578302701*x^5 + 221502398520960*x^4 + 48794976323
39105*x^3 - 221502398520960*x^2 - 2840455578302701*x - 1024558870866960) - 18*3^(5/6)*(30538688294400*x^6 + 26
1995401277240*x^5 - 513898276545299*x^4 - 714476479787080*x^3 + 513898276545299*x^2 + 261995401277240*x - 3053
8688294400)) + 2*(x^4 - x^2)^(1/3)*(sqrt(3)*(9732856787649299*x^7 + 33905739302897760*x^6 - 95756305292621940*
x^5 - 77616433764537120*x^4 + 95756305292621940*x^3 + 33905739302897760*x^2 - 9732856787649299*x) - 2600283*sq
rt(3)*(5229447840*x^7 - 9925022695*x^6 - 16872865920*x^5 + 22235880413*x^4 + 16872865920*x^3 - 9925022695*x^2
- 5229447840*x)) + 3^(1/6)*(1509469307073299*x^9 + 65809436013483840*x^8 - 74031236443268180*x^7 - 23367646126
3875840*x^6 + 131458310508730071*x^5 + 233676461263875840*x^4 - 74031236443268180*x^3 - 65809436013483840*x^2
+ 1509469307073299*x) - 6*3^(1/6)*(864612490925040*x^9 + 498731334053101*x^8 + 11166292173984960*x^7 - 2787581
6538793188*x^6 - 31843321748145360*x^5 + 27875816538793188*x^4 + 11166292173984960*x^3 - 498731334053101*x^2 +
864612490925040*x))*sqrt((3^(2/3)*(x^5 + x^3 + x) + 12*3^(2/3)*(x^4 - x^2) + 6*(x^4 - x^2)^(2/3)*(3^(1/3)*(x^
2 - 1) + 3*3^(1/3)*x) + 18*(x^4 - x^2)^(1/3)*(x^3 + x^2 - x))/(x^5 + x^3 + x)) + 10401132*(x^4 - x^2)^(2/3)*(3
^(1/6)*(4268995295279*x^6 - 21514334541120*x^5 - 29545286126340*x^4 + 68726754959040*x^3 + 29545286126340*x^2
- 21514334541120*x - 4268995295279) - 3*3^(1/6)*(763467207360*x^6 + 2042237245205*x^5 - 4977855152640*x^4 + 46
76326938953*x^3 + 4977855152640*x^2 + 2042237245205*x - 763467207360)) + 48*sqrt(3)*(181136316848795880*x^9 +
83939366796831719*x^8 - 4891577093539842480*x^7 - 503636200780990314*x^6 + 11051108405021256120*x^5 + 50363620
0780990314*x^4 - 4891577093539842480*x^3 - 83939366796831719*x^2 + 181136316848795880*x) - 2600283*sqrt(3)*(36
02552045521*x^9 - 9955710305280*x^8 + 17741462866034*x^7 + 59734261831680*x^6 - 10265061413421*x^5 - 597342618
31680*x^4 + 17741462866034*x^3 + 9955710305280*x^2 + 3602552045521*x) - 10401132*(x^4 - x^2)^(1/3)*(3^(5/6)*(2
920267143121*x^7 + 2687453530560*x^6 - 2634089693748*x^5 - 12246111927360*x^4 + 2634089693748*x^3 + 2687453530
560*x^2 - 2920267143121*x) - 3^(5/6)*(2855342875200*x^7 + 5579433413501*x^6 - 30080363166720*x^5 - 23965852712
839*x^4 + 30080363166720*x^3 + 5579433413501*x^2 - 2855342875200*x)))/(6837784281928633319*x^9 - 9417576913539
8261760*x^8 - 96817417641248917346*x^7 + 565054614812389570560*x^6 + 241499325255998267925*x^5 - 5650546148123
89570560*x^4 - 96817417641248917346*x^3 + 94175769135398261760*x^2 + 6837784281928633319*x)) - 8*3^(5/6)*(x^3
- x)*arctan(1/3*(1862*sqrt(3)*(8*(x^4 - x^2)^(2/3)*(3^(5/6)*(1024558870866960*x^6 - 2840455578302701*x^5 + 221
502398520960*x^4 + 4879497632339105*x^3 - 221502398520960*x^2 - 2840455578302701*x - 1024558870866960) + 18*3^
(5/6)*(30538688294400*x^6 + 261995401277240*x^5 - 513898276545299*x^4 - 714476479787080*x^3 + 513898276545299*
x^2 + 261995401277240*x - 30538688294400)) + 2*(x^4 - x^2)^(1/3)*(sqrt(3)*(9732856787649299*x^7 + 339057393028
97760*x^6 - 95756305292621940*x^5 - 77616433764537120*x^4 + 95756305292621940*x^3 + 33905739302897760*x^2 - 97
32856787649299*x) + 2600283*sqrt(3)*(5229447840*x^7 - 9925022695*x^6 - 16872865920*x^5 + 22235880413*x^4 + 168
72865920*x^3 - 9925022695*x^2 - 5229447840*x)) + 3^(1/6)*(1509469307073299*x^9 + 65809436013483840*x^8 - 74031
236443268180*x^7 - 233676461263875840*x^6 + 131458310508730071*x^5 + 233676461263875840*x^4 - 7403123644326818
0*x^3 - 65809436013483840*x^2 + 1509469307073299*x) + 6*3^(1/6)*(864612490925040*x^9 + 498731334053101*x^8 + 1
1166292173984960*x^7 - 27875816538793188*x^6 - 31843321748145360*x^5 + 27875816538793188*x^4 + 111662921739849
60*x^3 - 498731334053101*x^2 + 864612490925040*x))*sqrt((3^(2/3)*(x^5 + x^3 + x) - 12*3^(2/3)*(x^4 - x^2) - 6*
(x^4 - x^2)^(2/3)*(3^(1/3)*(x^2 - 1) - 3*3^(1/3)*x) + 18*(x^4 - x^2)^(1/3)*(x^3 - x^2 - x))/(x^5 + x^3 + x)) +
10401132*(x^4 - x^2)^(2/3)*(3^(1/6)*(4268995295279*x^6 - 21514334541120*x^5 - 29545286126340*x^4 + 6872675495
9040*x^3 + 29545286126340*x^2 - 21514334541120*x - 4268995295279) + 3*3^(1/6)*(763467207360*x^6 + 204223724520
5*x^5 - 4977855152640*x^4 + 4676326938953*x^3 + 4977855152640*x^2 + 2042237245205*x - 763467207360)) + 48*sqrt
(3)*(181136316848795880*x^9 + 83939366796831719*x^8 - 4891577093539842480*x^7 - 503636200780990314*x^6 + 11051
108405021256120*x^5 + 503636200780990314*x^4 - 4891577093539842480*x^3 - 83939366796831719*x^2 + 1811363168487
95880*x) + 2600283*sqrt(3)*(3602552045521*x^9 - 9955710305280*x^8 + 17741462866034*x^7 + 59734261831680*x^6 -
10265061413421*x^5 - 59734261831680*x^4 + 17741462866034*x^3 + 9955710305280*x^2 + 3602552045521*x) + 10401132
*(x^4 - x^2)^(1/3)*(3^(5/6)*(2920267143121*x^7 + 2687453530560*x^6 - 2634089693748*x^5 - 12246111927360*x^4 +
2634089693748*x^3 + 2687453530560*x^2 - 2920267143121*x) + 3^(5/6)*(2855342875200*x^7 + 5579433413501*x^6 - 30
080363166720*x^5 - 23965852712839*x^4 + 30080363166720*x^3 + 5579433413501*x^2 - 2855342875200*x)))/(683778428
1928633319*x^9 - 94175769135398261760*x^8 - 96817417641248917346*x^7 + 565054614812389570560*x^6 + 24149932525
5998267925*x^5 - 565054614812389570560*x^4 - 96817417641248917346*x^3 + 94175769135398261760*x^2 + 68377842819
28633319*x)) - 3*3^(1/3)*(x^3 - x)*log(10401132*(3^(2/3)*(x^5 + x^3 + x) + 12*3^(2/3)*(x^4 - x^2) + 6*(x^4 - x
^2)^(2/3)*(3^(1/3)*(x^2 - 1) + 3*3^(1/3)*x) + 18*(x^4 - x^2)^(1/3)*(x^3 + x^2 - x))/(x^5 + x^3 + x)) - 3*3^(1/
3)*(x^3 - x)*log(2600283*(3^(2/3)*(x^5 + x^3 + x) + 12*3^(2/3)*(x^4 - x^2) + 6*(x^4 - x^2)^(2/3)*(3^(1/3)*(x^2
- 1) + 3*3^(1/3)*x) + 18*(x^4 - x^2)^(1/3)*(x^3 + x^2 - x))/(x^5 + x^3 + x)) + 3*3^(1/3)*(x^3 - x)*log(104011
32*(3^(2/3)*(x^5 + x^3 + x) - 12*3^(2/3)*(x^4 - x^2) - 6*(x^4 - x^2)^(2/3)*(3^(1/3)*(x^2 - 1) - 3*3^(1/3)*x) +
18*(x^4 - x^2)^(1/3)*(x^3 - x^2 - x))/(x^5 + x^3 + x)) + 3*3^(1/3)*(x^3 - x)*log(2600283*(3^(2/3)*(x^5 + x^3
+ x) - 12*3^(2/3)*(x^4 - x^2) - 6*(x^4 - x^2)^(2/3)*(3^(1/3)*(x^2 - 1) - 3*3^(1/3)*x) + 18*(x^4 - x^2)^(1/3)*(
x^3 - x^2 - x))/(x^5 + x^3 + x)) - 72*(x^4 - x^2)^(2/3))/(x^3 - x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{6} + 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+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)

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maple [C]  time = 11.18, size = 758, normalized size = 4.54

method	result	size
risch	\(-\frac {x}{\left (x^{2} \left (x^{2}-1\right )\right )^{\frac {1}{3}}}-\frac {\ln \left (-\frac {\RootOf \left (\textit {\_Z}^{6}+243\right )^{4} x^{5}+12 \RootOf \left (\textit {\_Z}^{6}+243\right )^{4} x^{4}+\RootOf \left (\textit {\_Z}^{6}+243\right )^{4} x^{3}-54 \left (x^{4}-x^{2}\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{6}+243\right )^{2} x^{3}-12 \RootOf \left (\textit {\_Z}^{6}+243\right )^{4} x^{2}-54 \left (x^{4}-x^{2}\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{6}+243\right )^{2} x^{2}+\RootOf \left (\textit {\_Z}^{6}+243\right )^{4} x +162 x^{2} \left (x^{4}-x^{2}\right )^{\frac {2}{3}}+54 \RootOf \left (\textit {\_Z}^{6}+243\right )^{2} \left (x^{4}-x^{2}\right )^{\frac {1}{3}} x +486 \left (x^{4}-x^{2}\right )^{\frac {2}{3}} x -162 \left (x^{4}-x^{2}\right )^{\frac {2}{3}}}{\left (x^{2}-x +1\right ) x \left (x^{2}+x +1\right )}\right ) \RootOf \left (\textit {\_Z}^{6}+243\right )^{4}}{162}+\frac {\ln \left (-\frac {\RootOf \left (\textit {\_Z}^{6}+243\right )^{4} x^{5}+12 \RootOf \left (\textit {\_Z}^{6}+243\right )^{4} x^{4}+\RootOf \left (\textit {\_Z}^{6}+243\right )^{4} x^{3}-54 \left (x^{4}-x^{2}\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{6}+243\right )^{2} x^{3}-12 \RootOf \left (\textit {\_Z}^{6}+243\right )^{4} x^{2}-54 \left (x^{4}-x^{2}\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{6}+243\right )^{2} x^{2}+\RootOf \left (\textit {\_Z}^{6}+243\right )^{4} x +162 x^{2} \left (x^{4}-x^{2}\right )^{\frac {2}{3}}+54 \RootOf \left (\textit {\_Z}^{6}+243\right )^{2} \left (x^{4}-x^{2}\right )^{\frac {1}{3}} x +486 \left (x^{4}-x^{2}\right )^{\frac {2}{3}} x -162 \left (x^{4}-x^{2}\right )^{\frac {2}{3}}}{\left (x^{2}-x +1\right ) x \left (x^{2}+x +1\right )}\right ) \RootOf \left (\textit {\_Z}^{6}+243\right )}{18}-\frac {\RootOf \left (\textit {\_Z}^{6}+243\right ) \ln \left (\frac {-6 \left (x^{4}-x^{2}\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{6}+243\right )^{5} x^{3}+\RootOf \left (\textit {\_Z}^{6}+243\right )^{4} x^{5}+6 \left (x^{4}-x^{2}\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{6}+243\right )^{5} x^{2}-12 \RootOf \left (\textit {\_Z}^{6}+243\right )^{4} x^{4}+6 \RootOf \left (\textit {\_Z}^{6}+243\right )^{5} \left (x^{4}-x^{2}\right )^{\frac {1}{3}} x +\RootOf \left (\textit {\_Z}^{6}+243\right )^{4} x^{3}-54 \left (x^{4}-x^{2}\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{6}+243\right )^{2} x^{3}+12 \RootOf \left (\textit {\_Z}^{6}+243\right )^{4} x^{2}+27 \RootOf \left (\textit {\_Z}^{6}+243\right ) x^{5}+54 \left (x^{4}-x^{2}\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{6}+243\right )^{2} x^{2}+\RootOf \left (\textit {\_Z}^{6}+243\right )^{4} x -324 \RootOf \left (\textit {\_Z}^{6}+243\right ) x^{4}+324 x^{2} \left (x^{4}-x^{2}\right )^{\frac {2}{3}}+54 \RootOf \left (\textit {\_Z}^{6}+243\right )^{2} \left (x^{4}-x^{2}\right )^{\frac {1}{3}} x +27 \RootOf \left (\textit {\_Z}^{6}+243\right ) x^{3}-972 \left (x^{4}-x^{2}\right )^{\frac {2}{3}} x +324 \RootOf \left (\textit {\_Z}^{6}+243\right ) x^{2}-324 \left (x^{4}-x^{2}\right )^{\frac {2}{3}}+27 \RootOf \left (\textit {\_Z}^{6}+243\right ) x}{\left (x^{2}-x +1\right ) x \left (x^{2}+x +1\right )}\right )}{9}\)	\(758\)
trager	\(-\frac {\left (x^{4}-x^{2}\right )^{\frac {2}{3}}}{x \left (x^{2}-1\right )}+\frac {\ln \left (\frac {-\RootOf \left (\textit {\_Z}^{6}+243\right )^{4} x^{5}+12 \RootOf \left (\textit {\_Z}^{6}+243\right )^{4} x^{4}-\RootOf \left (\textit {\_Z}^{6}+243\right )^{4} x^{3}+54 \left (x^{4}-x^{2}\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{6}+243\right )^{2} x^{3}-12 \RootOf \left (\textit {\_Z}^{6}+243\right )^{4} x^{2}-54 \left (x^{4}-x^{2}\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{6}+243\right )^{2} x^{2}-\RootOf \left (\textit {\_Z}^{6}+243\right )^{4} x +162 x^{2} \left (x^{4}-x^{2}\right )^{\frac {2}{3}}-54 \RootOf \left (\textit {\_Z}^{6}+243\right )^{2} \left (x^{4}-x^{2}\right )^{\frac {1}{3}} x -486 \left (x^{4}-x^{2}\right )^{\frac {2}{3}} x -162 \left (x^{4}-x^{2}\right )^{\frac {2}{3}}}{\left (x^{2}-x +1\right ) x \left (x^{2}+x +1\right )}\right ) \RootOf \left (\textit {\_Z}^{6}+243\right )^{4}}{162}+\frac {\ln \left (\frac {-\RootOf \left (\textit {\_Z}^{6}+243\right )^{4} x^{5}+12 \RootOf \left (\textit {\_Z}^{6}+243\right )^{4} x^{4}-\RootOf \left (\textit {\_Z}^{6}+243\right )^{4} x^{3}+54 \left (x^{4}-x^{2}\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{6}+243\right )^{2} x^{3}-12 \RootOf \left (\textit {\_Z}^{6}+243\right )^{4} x^{2}-54 \left (x^{4}-x^{2}\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{6}+243\right )^{2} x^{2}-\RootOf \left (\textit {\_Z}^{6}+243\right )^{4} x +162 x^{2} \left (x^{4}-x^{2}\right )^{\frac {2}{3}}-54 \RootOf \left (\textit {\_Z}^{6}+243\right )^{2} \left (x^{4}-x^{2}\right )^{\frac {1}{3}} x -486 \left (x^{4}-x^{2}\right )^{\frac {2}{3}} x -162 \left (x^{4}-x^{2}\right )^{\frac {2}{3}}}{\left (x^{2}-x +1\right ) x \left (x^{2}+x +1\right )}\right ) \RootOf \left (\textit {\_Z}^{6}+243\right )}{18}+\frac {\RootOf \left (\textit {\_Z}^{6}+243\right ) \ln \left (\frac {6 \left (x^{4}-x^{2}\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{6}+243\right )^{5} x^{3}+\RootOf \left (\textit {\_Z}^{6}+243\right )^{4} x^{5}-6 \left (x^{4}-x^{2}\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{6}+243\right )^{5} x^{2}-12 \RootOf \left (\textit {\_Z}^{6}+243\right )^{4} x^{4}-6 \RootOf \left (\textit {\_Z}^{6}+243\right )^{5} \left (x^{4}-x^{2}\right )^{\frac {1}{3}} x +\RootOf \left (\textit {\_Z}^{6}+243\right )^{4} x^{3}-54 \left (x^{4}-x^{2}\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{6}+243\right )^{2} x^{3}+12 \RootOf \left (\textit {\_Z}^{6}+243\right )^{4} x^{2}-27 \RootOf \left (\textit {\_Z}^{6}+243\right ) x^{5}+54 \left (x^{4}-x^{2}\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{6}+243\right )^{2} x^{2}+\RootOf \left (\textit {\_Z}^{6}+243\right )^{4} x +324 \RootOf \left (\textit {\_Z}^{6}+243\right ) x^{4}+324 x^{2} \left (x^{4}-x^{2}\right )^{\frac {2}{3}}+54 \RootOf \left (\textit {\_Z}^{6}+243\right )^{2} \left (x^{4}-x^{2}\right )^{\frac {1}{3}} x -27 \RootOf \left (\textit {\_Z}^{6}+243\right ) x^{3}-972 \left (x^{4}-x^{2}\right )^{\frac {2}{3}} x -324 \RootOf \left (\textit {\_Z}^{6}+243\right ) x^{2}-324 \left (x^{4}-x^{2}\right )^{\frac {2}{3}}-27 \RootOf \left (\textit {\_Z}^{6}+243\right ) x}{\left (x^{2}-x +1\right ) x \left (x^{2}+x +1\right )}\right )}{9}\)	\(771\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x/(x^2*(x^2-1))^(1/3)-1/162*ln(-(RootOf(_Z^6+243)^4*x^5+12*RootOf(_Z^6+243)^4*x^4+RootOf(_Z^6+243)^4*x^3-54*(
x^4-x^2)^(1/3)*RootOf(_Z^6+243)^2*x^3-12*RootOf(_Z^6+243)^4*x^2-54*(x^4-x^2)^(1/3)*RootOf(_Z^6+243)^2*x^2+Root
Of(_Z^6+243)^4*x+162*x^2*(x^4-x^2)^(2/3)+54*RootOf(_Z^6+243)^2*(x^4-x^2)^(1/3)*x+486*(x^4-x^2)^(2/3)*x-162*(x^
4-x^2)^(2/3))/(x^2-x+1)/x/(x^2+x+1))*RootOf(_Z^6+243)^4+1/18*ln(-(RootOf(_Z^6+243)^4*x^5+12*RootOf(_Z^6+243)^4
*x^4+RootOf(_Z^6+243)^4*x^3-54*(x^4-x^2)^(1/3)*RootOf(_Z^6+243)^2*x^3-12*RootOf(_Z^6+243)^4*x^2-54*(x^4-x^2)^(
1/3)*RootOf(_Z^6+243)^2*x^2+RootOf(_Z^6+243)^4*x+162*x^2*(x^4-x^2)^(2/3)+54*RootOf(_Z^6+243)^2*(x^4-x^2)^(1/3)
*x+486*(x^4-x^2)^(2/3)*x-162*(x^4-x^2)^(2/3))/(x^2-x+1)/x/(x^2+x+1))*RootOf(_Z^6+243)-1/9*RootOf(_Z^6+243)*ln(
(-6*(x^4-x^2)^(1/3)*RootOf(_Z^6+243)^5*x^3+RootOf(_Z^6+243)^4*x^5+6*(x^4-x^2)^(1/3)*RootOf(_Z^6+243)^5*x^2-12*
RootOf(_Z^6+243)^4*x^4+6*RootOf(_Z^6+243)^5*(x^4-x^2)^(1/3)*x+RootOf(_Z^6+243)^4*x^3-54*(x^4-x^2)^(1/3)*RootOf
(_Z^6+243)^2*x^3+12*RootOf(_Z^6+243)^4*x^2+27*RootOf(_Z^6+243)*x^5+54*(x^4-x^2)^(1/3)*RootOf(_Z^6+243)^2*x^2+R
ootOf(_Z^6+243)^4*x-324*RootOf(_Z^6+243)*x^4+324*x^2*(x^4-x^2)^(2/3)+54*RootOf(_Z^6+243)^2*(x^4-x^2)^(1/3)*x+2
7*RootOf(_Z^6+243)*x^3-972*(x^4-x^2)^(2/3)*x+324*RootOf(_Z^6+243)*x^2-324*(x^4-x^2)^(2/3)+27*RootOf(_Z^6+243)*
x)/(x^2-x+1)/x/(x^2+x+1))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{6} + 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+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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^6+1}{\left (x^6-1\right )\,{\left (x^4-x^2\right )}^{1/3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

[Out]

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

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