Optimal. Leaf size=318 \[ -\log \left (\sqrt [3]{x^6-x^4-x^3-x}+x\right )+\sqrt [3]{2} \log \left (\sqrt [3]{x^6-x^4-x^3-x}+\sqrt [3]{2} x\right )+\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{x^6-x^4-x^3-x}}{\sqrt [3]{x^6-x^4-x^3-x}-2 x}\right )-\sqrt [3]{2} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{x^6-x^4-x^3-x}}{\sqrt [3]{x^6-x^4-x^3-x}-2 \sqrt [3]{2} x}\right )+\frac {1}{2} \log \left (x^2-\sqrt [3]{x^6-x^4-x^3-x} x+\left (x^6-x^4-x^3-x\right )^{2/3}\right )-\frac {\log \left (2^{2/3} x^2-\sqrt [3]{2} \sqrt [3]{x^6-x^4-x^3-x} x+\left (x^6-x^4-x^3-x\right )^{2/3}\right )}{2^{2/3}} \]
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Rubi [F] time = 15.70, 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 {\left (2-2 x+2 x^2-3 x^3+3 x^4\right ) \sqrt [3]{-x-x^3-x^4+x^6}}{(1+x) \left (-1+2 x-2 x^2+x^3\right ) \left (-1-x^3+x^5\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\left (2-2 x+2 x^2-3 x^3+3 x^4\right ) \sqrt [3]{-x-x^3-x^4+x^6}}{(1+x) \left (-1+2 x-2 x^2+x^3\right ) \left (-1-x^3+x^5\right )} \, dx &=\frac {\sqrt [3]{-x-x^3-x^4+x^6} \int \frac {\sqrt [3]{x} \left (2-2 x+2 x^2-3 x^3+3 x^4\right ) \sqrt [3]{-1-x^2-x^3+x^5}}{(1+x) \left (-1+2 x-2 x^2+x^3\right ) \left (-1-x^3+x^5\right )} \, dx}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}\\ &=\frac {\left (3 \sqrt [3]{-x-x^3-x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {x^3 \left (2-2 x^3+2 x^6-3 x^9+3 x^{12}\right ) \sqrt [3]{-1-x^6-x^9+x^{15}}}{\left (1+x^3\right ) \left (-1+2 x^3-2 x^6+x^9\right ) \left (-1-x^9+x^{15}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}\\ &=\frac {\left (3 \sqrt [3]{-x-x^3-x^4+x^6}\right ) \operatorname {Subst}\left (\int \left (-\frac {\sqrt [3]{-1-x^6-x^9+x^{15}}}{3 (-1+x)}-\frac {2 \sqrt [3]{-1-x^6-x^9+x^{15}}}{3 (1+x)}+\frac {2 (-2+x) \sqrt [3]{-1-x^6-x^9+x^{15}}}{3 \left (1-x+x^2\right )}+\frac {(2+x) \sqrt [3]{-1-x^6-x^9+x^{15}}}{3 \left (1+x+x^2\right )}+\frac {\left (1-2 x^3\right ) \sqrt [3]{-1-x^6-x^9+x^{15}}}{1-x^3+x^6}+\frac {x^6 \left (-3+5 x^6\right ) \sqrt [3]{-1-x^6-x^9+x^{15}}}{-1-x^9+x^{15}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}\\ &=-\frac {\sqrt [3]{-x-x^3-x^4+x^6} \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1-x^6-x^9+x^{15}}}{-1+x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}+\frac {\sqrt [3]{-x-x^3-x^4+x^6} \operatorname {Subst}\left (\int \frac {(2+x) \sqrt [3]{-1-x^6-x^9+x^{15}}}{1+x+x^2} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}-\frac {\left (2 \sqrt [3]{-x-x^3-x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1-x^6-x^9+x^{15}}}{1+x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}+\frac {\left (2 \sqrt [3]{-x-x^3-x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {(-2+x) \sqrt [3]{-1-x^6-x^9+x^{15}}}{1-x+x^2} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}+\frac {\left (3 \sqrt [3]{-x-x^3-x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {\left (1-2 x^3\right ) \sqrt [3]{-1-x^6-x^9+x^{15}}}{1-x^3+x^6} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}+\frac {\left (3 \sqrt [3]{-x-x^3-x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {x^6 \left (-3+5 x^6\right ) \sqrt [3]{-1-x^6-x^9+x^{15}}}{-1-x^9+x^{15}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}\\ &=-\frac {\sqrt [3]{-x-x^3-x^4+x^6} \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1-x^6-x^9+x^{15}}}{-1+x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}+\frac {\sqrt [3]{-x-x^3-x^4+x^6} \operatorname {Subst}\left (\int \left (\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{-1-x^6-x^9+x^{15}}}{1-i \sqrt {3}+2 x}+\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{-1-x^6-x^9+x^{15}}}{1+i \sqrt {3}+2 x}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}-\frac {\left (2 \sqrt [3]{-x-x^3-x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1-x^6-x^9+x^{15}}}{1+x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}+\frac {\left (2 \sqrt [3]{-x-x^3-x^4+x^6}\right ) \operatorname {Subst}\left (\int \left (\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{-1-x^6-x^9+x^{15}}}{-1-i \sqrt {3}+2 x}+\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{-1-x^6-x^9+x^{15}}}{-1+i \sqrt {3}+2 x}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}+\frac {\left (3 \sqrt [3]{-x-x^3-x^4+x^6}\right ) \operatorname {Subst}\left (\int \left (-\frac {2 \sqrt [3]{-1-x^6-x^9+x^{15}}}{-1-i \sqrt {3}+2 x^3}-\frac {2 \sqrt [3]{-1-x^6-x^9+x^{15}}}{-1+i \sqrt {3}+2 x^3}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}+\frac {\left (3 \sqrt [3]{-x-x^3-x^4+x^6}\right ) \operatorname {Subst}\left (\int \left (-\frac {3 x^6 \sqrt [3]{-1-x^6-x^9+x^{15}}}{-1-x^9+x^{15}}+\frac {5 x^{12} \sqrt [3]{-1-x^6-x^9+x^{15}}}{-1-x^9+x^{15}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}\\ &=-\frac {\sqrt [3]{-x-x^3-x^4+x^6} \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1-x^6-x^9+x^{15}}}{-1+x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}-\frac {\left (2 \sqrt [3]{-x-x^3-x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1-x^6-x^9+x^{15}}}{1+x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}-\frac {\left (6 \sqrt [3]{-x-x^3-x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1-x^6-x^9+x^{15}}}{-1-i \sqrt {3}+2 x^3} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}-\frac {\left (6 \sqrt [3]{-x-x^3-x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1-x^6-x^9+x^{15}}}{-1+i \sqrt {3}+2 x^3} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}-\frac {\left (9 \sqrt [3]{-x-x^3-x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {x^6 \sqrt [3]{-1-x^6-x^9+x^{15}}}{-1-x^9+x^{15}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}+\frac {\left (15 \sqrt [3]{-x-x^3-x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {x^{12} \sqrt [3]{-1-x^6-x^9+x^{15}}}{-1-x^9+x^{15}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}+\frac {\left (\left (1-i \sqrt {3}\right ) \sqrt [3]{-x-x^3-x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1-x^6-x^9+x^{15}}}{1-i \sqrt {3}+2 x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}+\frac {\left (2 \left (1-i \sqrt {3}\right ) \sqrt [3]{-x-x^3-x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1-x^6-x^9+x^{15}}}{-1+i \sqrt {3}+2 x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}+\frac {\left (\left (1+i \sqrt {3}\right ) \sqrt [3]{-x-x^3-x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1-x^6-x^9+x^{15}}}{1+i \sqrt {3}+2 x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}+\frac {\left (2 \left (1+i \sqrt {3}\right ) \sqrt [3]{-x-x^3-x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1-x^6-x^9+x^{15}}}{-1-i \sqrt {3}+2 x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}\\ &=-\frac {\sqrt [3]{-x-x^3-x^4+x^6} \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1-x^6-x^9+x^{15}}}{-1+x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}-\frac {\left (2 \sqrt [3]{-x-x^3-x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1-x^6-x^9+x^{15}}}{1+x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}-\frac {\left (6 \sqrt [3]{-x-x^3-x^4+x^6}\right ) \operatorname {Subst}\left (\int \left (\frac {\sqrt [3]{1-i \sqrt {3}} \sqrt [3]{-1-x^6-x^9+x^{15}}}{3 \left (-1+i \sqrt {3}\right ) \left (\sqrt [3]{1-i \sqrt {3}}+\sqrt [3]{-2} x\right )}+\frac {\sqrt [3]{1-i \sqrt {3}} \sqrt [3]{-1-x^6-x^9+x^{15}}}{3 \left (-1+i \sqrt {3}\right ) \left (\sqrt [3]{1-i \sqrt {3}}-\sqrt [3]{2} x\right )}+\frac {\sqrt [3]{1-i \sqrt {3}} \sqrt [3]{-1-x^6-x^9+x^{15}}}{3 \left (-1+i \sqrt {3}\right ) \left (\sqrt [3]{1-i \sqrt {3}}-(-1)^{2/3} \sqrt [3]{2} x\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}-\frac {\left (6 \sqrt [3]{-x-x^3-x^4+x^6}\right ) \operatorname {Subst}\left (\int \left (\frac {\sqrt [3]{1+i \sqrt {3}} \sqrt [3]{-1-x^6-x^9+x^{15}}}{3 \left (-1-i \sqrt {3}\right ) \left (\sqrt [3]{1+i \sqrt {3}}+\sqrt [3]{-2} x\right )}+\frac {\sqrt [3]{1+i \sqrt {3}} \sqrt [3]{-1-x^6-x^9+x^{15}}}{3 \left (-1-i \sqrt {3}\right ) \left (\sqrt [3]{1+i \sqrt {3}}-\sqrt [3]{2} x\right )}+\frac {\sqrt [3]{1+i \sqrt {3}} \sqrt [3]{-1-x^6-x^9+x^{15}}}{3 \left (-1-i \sqrt {3}\right ) \left (\sqrt [3]{1+i \sqrt {3}}-(-1)^{2/3} \sqrt [3]{2} x\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}-\frac {\left (9 \sqrt [3]{-x-x^3-x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {x^6 \sqrt [3]{-1-x^6-x^9+x^{15}}}{-1-x^9+x^{15}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}+\frac {\left (15 \sqrt [3]{-x-x^3-x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {x^{12} \sqrt [3]{-1-x^6-x^9+x^{15}}}{-1-x^9+x^{15}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}+\frac {\left (\left (1-i \sqrt {3}\right ) \sqrt [3]{-x-x^3-x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1-x^6-x^9+x^{15}}}{1-i \sqrt {3}+2 x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}+\frac {\left (2 \left (1-i \sqrt {3}\right ) \sqrt [3]{-x-x^3-x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1-x^6-x^9+x^{15}}}{-1+i \sqrt {3}+2 x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}+\frac {\left (\left (1+i \sqrt {3}\right ) \sqrt [3]{-x-x^3-x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1-x^6-x^9+x^{15}}}{1+i \sqrt {3}+2 x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}+\frac {\left (2 \left (1+i \sqrt {3}\right ) \sqrt [3]{-x-x^3-x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1-x^6-x^9+x^{15}}}{-1-i \sqrt {3}+2 x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}\\ &=-\frac {\sqrt [3]{-x-x^3-x^4+x^6} \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1-x^6-x^9+x^{15}}}{-1+x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}-\frac {\left (2 \sqrt [3]{-x-x^3-x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1-x^6-x^9+x^{15}}}{1+x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}-\frac {\left (9 \sqrt [3]{-x-x^3-x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {x^6 \sqrt [3]{-1-x^6-x^9+x^{15}}}{-1-x^9+x^{15}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}+\frac {\left (15 \sqrt [3]{-x-x^3-x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {x^{12} \sqrt [3]{-1-x^6-x^9+x^{15}}}{-1-x^9+x^{15}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}+\frac {\left (2 \sqrt [3]{-x-x^3-x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1-x^6-x^9+x^{15}}}{\sqrt [3]{1-i \sqrt {3}}+\sqrt [3]{-2} x} \, dx,x,\sqrt [3]{x}\right )}{\left (1-i \sqrt {3}\right )^{2/3} \sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}+\frac {\left (2 \sqrt [3]{-x-x^3-x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1-x^6-x^9+x^{15}}}{\sqrt [3]{1-i \sqrt {3}}-\sqrt [3]{2} x} \, dx,x,\sqrt [3]{x}\right )}{\left (1-i \sqrt {3}\right )^{2/3} \sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}+\frac {\left (2 \sqrt [3]{-x-x^3-x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1-x^6-x^9+x^{15}}}{\sqrt [3]{1-i \sqrt {3}}-(-1)^{2/3} \sqrt [3]{2} x} \, dx,x,\sqrt [3]{x}\right )}{\left (1-i \sqrt {3}\right )^{2/3} \sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}+\frac {\left (\left (1-i \sqrt {3}\right ) \sqrt [3]{-x-x^3-x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1-x^6-x^9+x^{15}}}{1-i \sqrt {3}+2 x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}+\frac {\left (2 \left (1-i \sqrt {3}\right ) \sqrt [3]{-x-x^3-x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1-x^6-x^9+x^{15}}}{-1+i \sqrt {3}+2 x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}+\frac {\left (2 \sqrt [3]{-x-x^3-x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1-x^6-x^9+x^{15}}}{\sqrt [3]{1+i \sqrt {3}}+\sqrt [3]{-2} x} \, dx,x,\sqrt [3]{x}\right )}{\left (1+i \sqrt {3}\right )^{2/3} \sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}+\frac {\left (2 \sqrt [3]{-x-x^3-x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1-x^6-x^9+x^{15}}}{\sqrt [3]{1+i \sqrt {3}}-\sqrt [3]{2} x} \, dx,x,\sqrt [3]{x}\right )}{\left (1+i \sqrt {3}\right )^{2/3} \sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}+\frac {\left (2 \sqrt [3]{-x-x^3-x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1-x^6-x^9+x^{15}}}{\sqrt [3]{1+i \sqrt {3}}-(-1)^{2/3} \sqrt [3]{2} x} \, dx,x,\sqrt [3]{x}\right )}{\left (1+i \sqrt {3}\right )^{2/3} \sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}+\frac {\left (\left (1+i \sqrt {3}\right ) \sqrt [3]{-x-x^3-x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1-x^6-x^9+x^{15}}}{1+i \sqrt {3}+2 x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}+\frac {\left (2 \left (1+i \sqrt {3}\right ) \sqrt [3]{-x-x^3-x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1-x^6-x^9+x^{15}}}{-1-i \sqrt {3}+2 x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1-x^2-x^3+x^5}}\\ \end {align*}
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Mathematica [F] time = 1.93, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (2-2 x+2 x^2-3 x^3+3 x^4\right ) \sqrt [3]{-x-x^3-x^4+x^6}}{(1+x) \left (-1+2 x-2 x^2+x^3\right ) \left (-1-x^3+x^5\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 5.05, size = 318, normalized size = 1.00 \begin {gather*} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{-x-x^3-x^4+x^6}}{-2 x+\sqrt [3]{-x-x^3-x^4+x^6}}\right )-\sqrt [3]{2} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{-x-x^3-x^4+x^6}}{-2 \sqrt [3]{2} x+\sqrt [3]{-x-x^3-x^4+x^6}}\right )-\log \left (x+\sqrt [3]{-x-x^3-x^4+x^6}\right )+\sqrt [3]{2} \log \left (\sqrt [3]{2} x+\sqrt [3]{-x-x^3-x^4+x^6}\right )+\frac {1}{2} \log \left (x^2-x \sqrt [3]{-x-x^3-x^4+x^6}+\left (-x-x^3-x^4+x^6\right )^{2/3}\right )-\frac {\log \left (2^{2/3} x^2-\sqrt [3]{2} x \sqrt [3]{-x-x^3-x^4+x^6}+\left (-x-x^3-x^4+x^6\right )^{2/3}\right )}{2^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 {{\left (x^{6} - x^{4} - x^{3} - x\right )}^{\frac {1}{3}} {\left (3 \, x^{4} - 3 \, x^{3} + 2 \, x^{2} - 2 \, x + 2\right )}}{{\left (x^{5} - x^{3} - 1\right )} {\left (x^{3} - 2 \, x^{2} + 2 \, x - 1\right )} {\left (x + 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 121.57, size = 3515, normalized size = 11.05
method | result | size |
trager | \(\text {Expression too large to display}\) | \(3515\) |
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 {{\left (x^{6} - x^{4} - x^{3} - x\right )}^{\frac {1}{3}} {\left (3 \, x^{4} - 3 \, x^{3} + 2 \, x^{2} - 2 \, x + 2\right )}}{{\left (x^{5} - x^{3} - 1\right )} {\left (x^{3} - 2 \, x^{2} + 2 \, x - 1\right )} {\left (x + 1\right )}}\,{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 = -0.00 \begin {gather*} \int -\frac {{\left (x^6-x^4-x^3-x\right )}^{1/3}\,\left (3\,x^4-3\,x^3+2\,x^2-2\,x+2\right )}{\left (x+1\right )\,\left (-x^5+x^3+1\right )\,\left (x^3-2\,x^2+2\,x-1\right )} \,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 {\sqrt [3]{x \left (x^{5} - x^{3} - x^{2} - 1\right )} \left (3 x^{4} - 3 x^{3} + 2 x^{2} - 2 x + 2\right )}{\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} - x + 1\right ) \left (x^{5} - x^{3} - 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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