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]
[Out]
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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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