Integrand size = 30, antiderivative size = 254 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (4+x^3\right )}{x^6 \left (-2-x^3+x^6\right )} \, dx=\frac {\left (8-13 x^3\right ) \left (-1+x^3\right )^{2/3}}{20 x^5}-\frac {\arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{2} \sqrt [3]{-1+x^3}}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {2^{2/3} \arctan \left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{-1+x^3}}\right )}{\sqrt {3}}+\frac {\log \left (-x+\sqrt [3]{2} \sqrt [3]{-1+x^3}\right )}{6\ 2^{2/3}}-\frac {1}{3} 2^{2/3} \log \left (-2 x+2^{2/3} \sqrt [3]{-1+x^3}\right )+\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{-1+x^3}+\sqrt [3]{2} \left (-1+x^3\right )^{2/3}\right )}{3 \sqrt [3]{2}}-\frac {\log \left (x^2+\sqrt [3]{2} x \sqrt [3]{-1+x^3}+2^{2/3} \left (-1+x^3\right )^{2/3}\right )}{12\ 2^{2/3}} \]
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Time = 1.03 (sec) , antiderivative size = 505, normalized size of antiderivative = 1.99, number of steps used = 36, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.767, Rules used = {6860, 270, 283, 245, 2178, 2177, 2174, 372, 371, 2183, 399, 384, 495, 502, 206, 31, 648, 631, 210, 642, 455, 52, 58} \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (4+x^3\right )}{x^6 \left (-2-x^3+x^6\right )} \, dx=\frac {2^{2/3} \arctan \left (\frac {1-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{x^3-1}}}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {2^{2/3} \arctan \left (\frac {\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {2\ 2^{2/3} \arctan \left (\frac {\frac {2 \sqrt [3]{2} x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {\arctan \left (\frac {\frac {2^{2/3} x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {2^{2/3} \arctan \left (\frac {1-2^{2/3} \sqrt [3]{x^3-1}}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {\log \left (x^3-2\right )}{12\ 2^{2/3}}+\frac {1}{9} 2^{2/3} \log \left (x^3+1\right )-\frac {\log \left (x^3+1\right )}{9 \sqrt [3]{2}}+\frac {1}{9} 2^{2/3} \log \left (1-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{x^3-1}}\right )-\frac {\log \left (\frac {2^{2/3} (1-x)^2}{\left (x^3-1\right )^{2/3}}+\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{x^3-1}}+1\right )}{9 \sqrt [3]{2}}+\frac {\log \left (\frac {x}{\sqrt [3]{2}}-\sqrt [3]{x^3-1}\right )}{4\ 2^{2/3}}-\frac {1}{3} 2^{2/3} \log \left (\sqrt [3]{2} x-\sqrt [3]{x^3-1}\right )+\frac {\log \left (\sqrt [3]{x^3-1}+\sqrt [3]{2}\right )}{3 \sqrt [3]{2}}-\frac {\log \left (2^{2/3} \sqrt [3]{x^3-1}-x+1\right )}{3 \sqrt [3]{2}}-\frac {2 \left (x^3-1\right )^{5/3}}{5 x^5}-\frac {\left (x^3-1\right )^{2/3}}{4 x^2}+\frac {\log \left ((1-x) (x+1)^2\right )}{9 \sqrt [3]{2}} \]
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Rule 31
Rule 52
Rule 58
Rule 206
Rule 210
Rule 245
Rule 270
Rule 283
Rule 371
Rule 372
Rule 384
Rule 399
Rule 455
Rule 495
Rule 502
Rule 631
Rule 642
Rule 648
Rule 2174
Rule 2177
Rule 2178
Rule 2183
Rule 6860
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {2 \left (-1+x^3\right )^{2/3}}{x^6}+\frac {\left (-1+x^3\right )^{2/3}}{2 x^3}-\frac {\left (-1+x^3\right )^{2/3}}{3 (1+x)}+\frac {(-2+x) \left (-1+x^3\right )^{2/3}}{3 \left (1-x+x^2\right )}+\frac {\left (-1+x^3\right )^{2/3}}{2 \left (-2+x^3\right )}\right ) \, dx \\ & = -\left (\frac {1}{3} \int \frac {\left (-1+x^3\right )^{2/3}}{1+x} \, dx\right )+\frac {1}{3} \int \frac {(-2+x) \left (-1+x^3\right )^{2/3}}{1-x+x^2} \, dx+\frac {1}{2} \int \frac {\left (-1+x^3\right )^{2/3}}{x^3} \, dx+\frac {1}{2} \int \frac {\left (-1+x^3\right )^{2/3}}{-2+x^3} \, dx-2 \int \frac {\left (-1+x^3\right )^{2/3}}{x^6} \, dx \\ & = -\frac {1}{6} \left (-1+x^3\right )^{2/3}-\frac {\left (-1+x^3\right )^{2/3}}{4 x^2}-\frac {2 \left (-1+x^3\right )^{5/3}}{5 x^5}+\frac {1}{3} \int \frac {x}{\sqrt [3]{-1+x^3}} \, dx-\frac {1}{3} \int \frac {-1+x}{(1+x) \sqrt [3]{-1+x^3}} \, dx+\frac {1}{3} \int \left (-\frac {2 \left (-1+x^3\right )^{2/3}}{1+x^3}-\frac {x \left (-1+x^3\right )^{2/3}}{1+x^3}+\frac {x^2 \left (-1+x^3\right )^{2/3}}{1+x^3}\right ) \, dx+2 \left (\frac {1}{2} \int \frac {1}{\sqrt [3]{-1+x^3}} \, dx\right )+\frac {1}{2} \int \frac {1}{\left (-2+x^3\right ) \sqrt [3]{-1+x^3}} \, dx \\ & = -\frac {1}{6} \left (-1+x^3\right )^{2/3}-\frac {\left (-1+x^3\right )^{2/3}}{4 x^2}-\frac {2 \left (-1+x^3\right )^{5/3}}{5 x^5}-\frac {\arctan \left (\frac {1+\frac {2^{2/3} x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{2\ 2^{2/3} \sqrt {3}}-\frac {\log \left (-2+x^3\right )}{12\ 2^{2/3}}+\frac {\log \left (\frac {x}{\sqrt [3]{2}}-\sqrt [3]{-1+x^3}\right )}{4\ 2^{2/3}}+2 \left (\frac {\arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {1}{4} \log \left (-x+\sqrt [3]{-1+x^3}\right )\right )-\frac {1}{3} \int \frac {1}{\sqrt [3]{-1+x^3}} \, dx-\frac {1}{3} \int \frac {x \left (-1+x^3\right )^{2/3}}{1+x^3} \, dx+\frac {1}{3} \int \frac {x^2 \left (-1+x^3\right )^{2/3}}{1+x^3} \, dx+\frac {2}{3} \int \frac {1}{(1+x) \sqrt [3]{-1+x^3}} \, dx-\frac {2}{3} \int \frac {\left (-1+x^3\right )^{2/3}}{1+x^3} \, dx+\frac {\sqrt [3]{1-x^3} \int \frac {x}{\sqrt [3]{1-x^3}} \, dx}{3 \sqrt [3]{-1+x^3}} \\ & = -\frac {1}{6} \left (-1+x^3\right )^{2/3}-\frac {\left (-1+x^3\right )^{2/3}}{4 x^2}-\frac {2 \left (-1+x^3\right )^{5/3}}{5 x^5}+\frac {\arctan \left (\frac {1-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}-\frac {\arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {\arctan \left (\frac {1+\frac {2^{2/3} x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {x^2 \sqrt [3]{1-x^3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},x^3\right )}{6 \sqrt [3]{-1+x^3}}+\frac {\log \left ((1-x) (1+x)^2\right )}{6 \sqrt [3]{2}}-\frac {\log \left (-2+x^3\right )}{12\ 2^{2/3}}+\frac {\log \left (\frac {x}{\sqrt [3]{2}}-\sqrt [3]{-1+x^3}\right )}{4\ 2^{2/3}}+2 \left (\frac {\arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {1}{4} \log \left (-x+\sqrt [3]{-1+x^3}\right )\right )+\frac {1}{6} \log \left (-x+\sqrt [3]{-1+x^3}\right )-\frac {\log \left (1-x+2^{2/3} \sqrt [3]{-1+x^3}\right )}{2 \sqrt [3]{2}}+\frac {1}{9} \text {Subst}\left (\int \frac {(-1+x)^{2/3}}{1+x} \, dx,x,x^3\right )-\frac {1}{3} \int \frac {x}{\sqrt [3]{-1+x^3}} \, dx-\frac {2}{3} \int \frac {1}{\sqrt [3]{-1+x^3}} \, dx+\frac {2}{3} \int \frac {x}{\sqrt [3]{-1+x^3} \left (1+x^3\right )} \, dx+\frac {4}{3} \int \frac {1}{\sqrt [3]{-1+x^3} \left (1+x^3\right )} \, dx \\ & = -\frac {\left (-1+x^3\right )^{2/3}}{4 x^2}-\frac {2 \left (-1+x^3\right )^{5/3}}{5 x^5}+\frac {\arctan \left (\frac {1-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}-\frac {\arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {2\ 2^{2/3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{2} x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {\arctan \left (\frac {1+\frac {2^{2/3} x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {x^2 \sqrt [3]{1-x^3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},x^3\right )}{6 \sqrt [3]{-1+x^3}}+\frac {\log \left ((1-x) (1+x)^2\right )}{6 \sqrt [3]{2}}-\frac {\log \left (-2+x^3\right )}{12\ 2^{2/3}}+\frac {1}{9} 2^{2/3} \log \left (1+x^3\right )+\frac {\log \left (\frac {x}{\sqrt [3]{2}}-\sqrt [3]{-1+x^3}\right )}{4\ 2^{2/3}}-\frac {1}{3} 2^{2/3} \log \left (\sqrt [3]{2} x-\sqrt [3]{-1+x^3}\right )+2 \left (\frac {\arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {1}{4} \log \left (-x+\sqrt [3]{-1+x^3}\right )\right )+\frac {1}{2} \log \left (-x+\sqrt [3]{-1+x^3}\right )-\frac {\log \left (1-x+2^{2/3} \sqrt [3]{-1+x^3}\right )}{2 \sqrt [3]{2}}-\frac {2}{9} \int \frac {1}{(1+x) \sqrt [3]{-1+x^3}} \, dx-\frac {2}{9} \text {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x} (1+x)} \, dx,x,x^3\right )-\frac {2}{3} \text {Subst}\left (\int \frac {1}{1-2 x^3} \, dx,x,\frac {1-x}{\sqrt [3]{-1+x^3}}\right )-\frac {\sqrt [3]{1-x^3} \int \frac {x}{\sqrt [3]{1-x^3}} \, dx}{3 \sqrt [3]{-1+x^3}} \\ & = -\frac {\left (-1+x^3\right )^{2/3}}{4 x^2}-\frac {2 \left (-1+x^3\right )^{5/3}}{5 x^5}+\frac {2^{2/3} \arctan \left (\frac {1-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {\arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {2\ 2^{2/3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{2} x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {\arctan \left (\frac {1+\frac {2^{2/3} x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {\log \left ((1-x) (1+x)^2\right )}{9 \sqrt [3]{2}}-\frac {\log \left (-2+x^3\right )}{12\ 2^{2/3}}-\frac {\log \left (1+x^3\right )}{9 \sqrt [3]{2}}+\frac {1}{9} 2^{2/3} \log \left (1+x^3\right )+\frac {\log \left (\frac {x}{\sqrt [3]{2}}-\sqrt [3]{-1+x^3}\right )}{4\ 2^{2/3}}-\frac {1}{3} 2^{2/3} \log \left (\sqrt [3]{2} x-\sqrt [3]{-1+x^3}\right )+2 \left (\frac {\arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {1}{4} \log \left (-x+\sqrt [3]{-1+x^3}\right )\right )+\frac {1}{2} \log \left (-x+\sqrt [3]{-1+x^3}\right )-\frac {\log \left (1-x+2^{2/3} \sqrt [3]{-1+x^3}\right )}{3 \sqrt [3]{2}}-\frac {2}{9} \text {Subst}\left (\int \frac {1}{1-\sqrt [3]{2} x} \, dx,x,\frac {1-x}{\sqrt [3]{-1+x^3}}\right )-\frac {2}{9} \text {Subst}\left (\int \frac {2+\sqrt [3]{2} x}{1+\sqrt [3]{2} x+2^{2/3} x^2} \, dx,x,\frac {1-x}{\sqrt [3]{-1+x^3}}\right )-\frac {1}{3} \text {Subst}\left (\int \frac {1}{2^{2/3}-\sqrt [3]{2} x+x^2} \, dx,x,\sqrt [3]{-1+x^3}\right )+\frac {\text {Subst}\left (\int \frac {1}{\sqrt [3]{2}+x} \, dx,x,\sqrt [3]{-1+x^3}\right )}{3 \sqrt [3]{2}} \\ & = -\frac {\left (-1+x^3\right )^{2/3}}{4 x^2}-\frac {2 \left (-1+x^3\right )^{5/3}}{5 x^5}+\frac {2^{2/3} \arctan \left (\frac {1-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {\arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {2\ 2^{2/3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{2} x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {\arctan \left (\frac {1+\frac {2^{2/3} x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {\log \left ((1-x) (1+x)^2\right )}{9 \sqrt [3]{2}}-\frac {\log \left (-2+x^3\right )}{12\ 2^{2/3}}-\frac {\log \left (1+x^3\right )}{9 \sqrt [3]{2}}+\frac {1}{9} 2^{2/3} \log \left (1+x^3\right )+\frac {1}{9} 2^{2/3} \log \left (1-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{-1+x^3}}\right )+\frac {\log \left (\frac {x}{\sqrt [3]{2}}-\sqrt [3]{-1+x^3}\right )}{4\ 2^{2/3}}-\frac {1}{3} 2^{2/3} \log \left (\sqrt [3]{2} x-\sqrt [3]{-1+x^3}\right )+\frac {\log \left (\sqrt [3]{2}+\sqrt [3]{-1+x^3}\right )}{3 \sqrt [3]{2}}+2 \left (\frac {\arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {1}{4} \log \left (-x+\sqrt [3]{-1+x^3}\right )\right )+\frac {1}{2} \log \left (-x+\sqrt [3]{-1+x^3}\right )-\frac {\log \left (1-x+2^{2/3} \sqrt [3]{-1+x^3}\right )}{3 \sqrt [3]{2}}-\frac {1}{3} \text {Subst}\left (\int \frac {1}{1+\sqrt [3]{2} x+2^{2/3} x^2} \, dx,x,\frac {1-x}{\sqrt [3]{-1+x^3}}\right )-\frac {\text {Subst}\left (\int \frac {\sqrt [3]{2}+2\ 2^{2/3} x}{1+\sqrt [3]{2} x+2^{2/3} x^2} \, dx,x,\frac {1-x}{\sqrt [3]{-1+x^3}}\right )}{9 \sqrt [3]{2}}-\frac {1}{3} 2^{2/3} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-2^{2/3} \sqrt [3]{-1+x^3}\right ) \\ & = -\frac {\left (-1+x^3\right )^{2/3}}{4 x^2}-\frac {2 \left (-1+x^3\right )^{5/3}}{5 x^5}+\frac {2^{2/3} \arctan \left (\frac {1-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {\arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {2\ 2^{2/3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{2} x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {\arctan \left (\frac {1+\frac {2^{2/3} x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {2^{2/3} \arctan \left (\frac {1-2^{2/3} \sqrt [3]{-1+x^3}}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {\log \left ((1-x) (1+x)^2\right )}{9 \sqrt [3]{2}}-\frac {\log \left (-2+x^3\right )}{12\ 2^{2/3}}-\frac {\log \left (1+x^3\right )}{9 \sqrt [3]{2}}+\frac {1}{9} 2^{2/3} \log \left (1+x^3\right )+\frac {1}{9} 2^{2/3} \log \left (1-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{-1+x^3}}\right )-\frac {\log \left (1+\frac {2^{2/3} (1-x)^2}{\left (-1+x^3\right )^{2/3}}+\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{-1+x^3}}\right )}{9 \sqrt [3]{2}}+\frac {\log \left (\frac {x}{\sqrt [3]{2}}-\sqrt [3]{-1+x^3}\right )}{4\ 2^{2/3}}-\frac {1}{3} 2^{2/3} \log \left (\sqrt [3]{2} x-\sqrt [3]{-1+x^3}\right )+\frac {\log \left (\sqrt [3]{2}+\sqrt [3]{-1+x^3}\right )}{3 \sqrt [3]{2}}+2 \left (\frac {\arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {1}{4} \log \left (-x+\sqrt [3]{-1+x^3}\right )\right )+\frac {1}{2} \log \left (-x+\sqrt [3]{-1+x^3}\right )-\frac {\log \left (1-x+2^{2/3} \sqrt [3]{-1+x^3}\right )}{3 \sqrt [3]{2}}+\frac {1}{3} 2^{2/3} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{-1+x^3}}\right ) \\ & = -\frac {\left (-1+x^3\right )^{2/3}}{4 x^2}-\frac {2 \left (-1+x^3\right )^{5/3}}{5 x^5}+\frac {2^{2/3} \arctan \left (\frac {1-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {2^{2/3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {\arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {2\ 2^{2/3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{2} x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {\arctan \left (\frac {1+\frac {2^{2/3} x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {2^{2/3} \arctan \left (\frac {1-2^{2/3} \sqrt [3]{-1+x^3}}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {\log \left ((1-x) (1+x)^2\right )}{9 \sqrt [3]{2}}-\frac {\log \left (-2+x^3\right )}{12\ 2^{2/3}}-\frac {\log \left (1+x^3\right )}{9 \sqrt [3]{2}}+\frac {1}{9} 2^{2/3} \log \left (1+x^3\right )+\frac {1}{9} 2^{2/3} \log \left (1-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{-1+x^3}}\right )-\frac {\log \left (1+\frac {2^{2/3} (1-x)^2}{\left (-1+x^3\right )^{2/3}}+\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{-1+x^3}}\right )}{9 \sqrt [3]{2}}+\frac {\log \left (\frac {x}{\sqrt [3]{2}}-\sqrt [3]{-1+x^3}\right )}{4\ 2^{2/3}}-\frac {1}{3} 2^{2/3} \log \left (\sqrt [3]{2} x-\sqrt [3]{-1+x^3}\right )+\frac {\log \left (\sqrt [3]{2}+\sqrt [3]{-1+x^3}\right )}{3 \sqrt [3]{2}}+2 \left (\frac {\arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {1}{4} \log \left (-x+\sqrt [3]{-1+x^3}\right )\right )+\frac {1}{2} \log \left (-x+\sqrt [3]{-1+x^3}\right )-\frac {\log \left (1-x+2^{2/3} \sqrt [3]{-1+x^3}\right )}{3 \sqrt [3]{2}} \\ \end{align*}
Time = 0.53 (sec) , antiderivative size = 247, normalized size of antiderivative = 0.97 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (4+x^3\right )}{x^6 \left (-2-x^3+x^6\right )} \, dx=\frac {1}{120} \left (\frac {6 \left (8-13 x^3\right ) \left (-1+x^3\right )^{2/3}}{x^5}-10 \sqrt [3]{2} \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{2} \sqrt [3]{-1+x^3}}\right )+40\ 2^{2/3} \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{-1+x^3}}\right )+10 \sqrt [3]{2} \log \left (-x+\sqrt [3]{2} \sqrt [3]{-1+x^3}\right )-40\ 2^{2/3} \log \left (-2 x+2^{2/3} \sqrt [3]{-1+x^3}\right )+20\ 2^{2/3} \log \left (2 x^2+2^{2/3} x \sqrt [3]{-1+x^3}+\sqrt [3]{2} \left (-1+x^3\right )^{2/3}\right )-5 \sqrt [3]{2} \log \left (x^2+\sqrt [3]{2} x \sqrt [3]{-1+x^3}+2^{2/3} \left (-1+x^3\right )^{2/3}\right )\right ) \]
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Time = 23.47 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.96
method | result | size |
pseudoelliptic | \(\frac {-40 \sqrt {3}\, 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3}\, \left (x +2^{\frac {2}{3}} \left (x^{3}-1\right )^{\frac {1}{3}}\right )}{3 x}\right ) x^{5}+10 \sqrt {3}\, 2^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3}\, \left (x +2 \,2^{\frac {1}{3}} \left (x^{3}-1\right )^{\frac {1}{3}}\right )}{3 x}\right ) x^{5}-40 \,2^{\frac {2}{3}} \ln \left (\frac {-2^{\frac {1}{3}} x +\left (x^{3}-1\right )^{\frac {1}{3}}}{x}\right ) x^{5}+20 \,2^{\frac {2}{3}} \ln \left (\frac {2^{\frac {2}{3}} x^{2}+2^{\frac {1}{3}} x \left (x^{3}-1\right )^{\frac {1}{3}}+\left (x^{3}-1\right )^{\frac {2}{3}}}{x^{2}}\right ) x^{5}-5 \,2^{\frac {1}{3}} x^{5} \ln \left (2\right )+10 \,2^{\frac {1}{3}} x^{5} \ln \left (\frac {-2^{\frac {2}{3}} x +2 \left (x^{3}-1\right )^{\frac {1}{3}}}{x}\right )-5 \,2^{\frac {1}{3}} x^{5} \ln \left (\frac {2^{\frac {2}{3}} x \left (x^{3}-1\right )^{\frac {1}{3}}+2^{\frac {1}{3}} x^{2}+2 \left (x^{3}-1\right )^{\frac {2}{3}}}{x^{2}}\right )-78 x^{3} \left (x^{3}-1\right )^{\frac {2}{3}}+48 \left (x^{3}-1\right )^{\frac {2}{3}}}{120 x^{5}}\) | \(243\) |
trager | \(\text {Expression too large to display}\) | \(1891\) |
risch | \(\text {Expression too large to display}\) | \(1913\) |
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Leaf count of result is larger than twice the leaf count of optimal. 521 vs. \(2 (189) = 378\).
Time = 3.77 (sec) , antiderivative size = 521, normalized size of antiderivative = 2.05 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (4+x^3\right )}{x^6 \left (-2-x^3+x^6\right )} \, dx=-\frac {80 \, \sqrt {3} \left (-4\right )^{\frac {1}{3}} x^{5} \arctan \left (\frac {3 \, \sqrt {3} \left (-4\right )^{\frac {2}{3}} {\left (5 \, x^{7} + 4 \, x^{4} - x\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} + 6 \, \sqrt {3} \left (-4\right )^{\frac {1}{3}} {\left (19 \, x^{8} - 16 \, x^{5} + x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}} - \sqrt {3} {\left (71 \, x^{9} - 111 \, x^{6} + 33 \, x^{3} - 1\right )}}{3 \, {\left (109 \, x^{9} - 105 \, x^{6} + 3 \, x^{3} + 1\right )}}\right ) - 20 \cdot 4^{\frac {1}{6}} \sqrt {3} x^{5} \arctan \left (\frac {4^{\frac {1}{6}} {\left (12 \cdot 4^{\frac {2}{3}} \sqrt {3} {\left (2 \, x^{7} - 5 \, x^{4} + 2 \, x\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} + 4^{\frac {1}{3}} \sqrt {3} {\left (91 \, x^{9} - 168 \, x^{6} + 84 \, x^{3} - 8\right )} + 12 \, \sqrt {3} {\left (19 \, x^{8} - 22 \, x^{5} + 4 \, x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}\right )}}{6 \, {\left (53 \, x^{9} - 48 \, x^{6} - 12 \, x^{3} + 8\right )}}\right ) - 10 \cdot 4^{\frac {2}{3}} x^{5} \log \left (\frac {6 \cdot 4^{\frac {1}{3}} {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{2} + 4^{\frac {2}{3}} {\left (x^{3} - 2\right )} - 12 \, {\left (x^{3} - 1\right )}^{\frac {2}{3}} x}{x^{3} - 2}\right ) + 5 \cdot 4^{\frac {2}{3}} x^{5} \log \left (\frac {6 \cdot 4^{\frac {2}{3}} {\left (2 \, x^{4} - x\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (19 \, x^{6} - 22 \, x^{3} + 4\right )} + 6 \, {\left (5 \, x^{5} - 4 \, x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x^{6} - 4 \, x^{3} + 4}\right ) - 80 \, \left (-4\right )^{\frac {1}{3}} x^{5} \log \left (-\frac {3 \, \left (-4\right )^{\frac {2}{3}} {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{2} - 6 \, {\left (x^{3} - 1\right )}^{\frac {2}{3}} x + \left (-4\right )^{\frac {1}{3}} {\left (x^{3} + 1\right )}}{x^{3} + 1}\right ) + 40 \, \left (-4\right )^{\frac {1}{3}} x^{5} \log \left (-\frac {6 \, \left (-4\right )^{\frac {1}{3}} {\left (5 \, x^{4} - x\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} - \left (-4\right )^{\frac {2}{3}} {\left (19 \, x^{6} - 16 \, x^{3} + 1\right )} - 24 \, {\left (2 \, x^{5} - x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x^{6} + 2 \, x^{3} + 1}\right ) + 36 \, {\left (13 \, x^{3} - 8\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{720 \, x^{5}} \]
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\[ \int \frac {\left (-1+x^3\right )^{2/3} \left (4+x^3\right )}{x^6 \left (-2-x^3+x^6\right )} \, dx=\int \frac {\left (\left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {2}{3}} \left (x^{3} + 4\right )}{x^{6} \left (x + 1\right ) \left (x^{3} - 2\right ) \left (x^{2} - x + 1\right )}\, dx \]
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\[ \int \frac {\left (-1+x^3\right )^{2/3} \left (4+x^3\right )}{x^6 \left (-2-x^3+x^6\right )} \, dx=\int { \frac {{\left (x^{3} + 4\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{6} - x^{3} - 2\right )} x^{6}} \,d x } \]
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\[ \int \frac {\left (-1+x^3\right )^{2/3} \left (4+x^3\right )}{x^6 \left (-2-x^3+x^6\right )} \, dx=\int { \frac {{\left (x^{3} + 4\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{6} - x^{3} - 2\right )} x^{6}} \,d x } \]
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Timed out. \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (4+x^3\right )}{x^6 \left (-2-x^3+x^6\right )} \, dx=\int -\frac {{\left (x^3-1\right )}^{2/3}\,\left (x^3+4\right )}{x^6\,\left (-x^6+x^3+2\right )} \,d x \]
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