Integrand size = 30, antiderivative size = 151 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (-2+2 x^3+x^6\right )}{x^6 \left (1+x^3\right )^2} \, dx=\frac {\left (-1+x^3\right )^{2/3} \left (2-15 x^3-22 x^6\right )}{5 x^5 \left (1+x^3\right )}+\frac {7\ 2^{2/3} \arctan \left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{-1+x^3}}\right )}{\sqrt {3}}-\frac {7}{3} 2^{2/3} \log \left (-2 x+2^{2/3} \sqrt [3]{-1+x^3}\right )+\frac {7 \log \left (2 x^2+2^{2/3} x \sqrt [3]{-1+x^3}+\sqrt [3]{2} \left (-1+x^3\right )^{2/3}\right )}{3 \sqrt [3]{2}} \]
[Out]
Leaf count is larger than twice the leaf count of optimal. \(468\) vs. \(2(151)=302\).
Time = 1.33 (sec) , antiderivative size = 468, normalized size of antiderivative = 3.10, number of steps used = 97, number of rules used = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.033, Rules used = {6874, 270, 283, 245, 2181, 386, 384, 480, 372, 371, 455, 43, 58, 631, 210, 31, 478, 544, 598, 502, 2174, 206, 648, 642, 2178, 2177, 2183, 1600, 399, 495, 52} \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (-2+2 x^3+x^6\right )}{x^6 \left (1+x^3\right )^2} \, dx=\frac {8\ 2^{2/3} \arctan \left (\frac {1-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{x^3-1}}}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {8\ 2^{2/3} \arctan \left (\frac {\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {13\ 2^{2/3} \arctan \left (\frac {\frac {2 \sqrt [3]{2} x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {8\ 2^{2/3} \arctan \left (\frac {1-2^{2/3} \sqrt [3]{x^3-1}}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {x \left (x^3-1\right )^{2/3}}{x^3+1}-\frac {5}{9} 2^{2/3} \log \left (x^3+1\right )+\frac {5 \log \left (x^3+1\right )}{3 \sqrt [3]{2}}+\frac {8}{9} 2^{2/3} \log \left (1-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{x^3-1}}\right )-\frac {4}{9} 2^{2/3} \log \left (\frac {2^{2/3} (1-x)^2}{\left (x^3-1\right )^{2/3}}+\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{x^3-1}}+1\right )+\frac {2}{9} 2^{2/3} \log \left (\sqrt [3]{2} x-\sqrt [3]{x^3-1}\right )-\frac {43 \log \left (\sqrt [3]{2} x-\sqrt [3]{x^3-1}\right )}{9 \sqrt [3]{2}}+\frac {4}{3} 2^{2/3} \log \left (\sqrt [3]{x^3-1}+\sqrt [3]{2}\right )-\frac {4}{3} 2^{2/3} \log \left (2^{2/3} \sqrt [3]{x^3-1}-x+1\right )-\frac {2 \left (x^3-1\right )^{5/3}}{5 x^5}-\frac {3 \left (x^3-1\right )^{2/3}}{x^2}+\frac {4}{9} 2^{2/3} \log \left ((1-x) (x+1)^2\right ) \]
[In]
[Out]
Rule 31
Rule 43
Rule 52
Rule 58
Rule 206
Rule 210
Rule 245
Rule 270
Rule 283
Rule 371
Rule 372
Rule 384
Rule 386
Rule 399
Rule 455
Rule 478
Rule 480
Rule 495
Rule 502
Rule 544
Rule 598
Rule 631
Rule 642
Rule 648
Rule 1600
Rule 2174
Rule 2177
Rule 2178
Rule 2181
Rule 2183
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {2 \left (-1+x^3\right )^{2/3}}{x^6}+\frac {6 \left (-1+x^3\right )^{2/3}}{x^3}-\frac {\left (-1+x^3\right )^{2/3}}{3 (1+x)^2}-\frac {8 \left (-1+x^3\right )^{2/3}}{3 (1+x)}+\frac {(-1+x) \left (-1+x^3\right )^{2/3}}{\left (1-x+x^2\right )^2}+\frac {(-15+8 x) \left (-1+x^3\right )^{2/3}}{3 \left (1-x+x^2\right )}\right ) \, dx \\ & = -\left (\frac {1}{3} \int \frac {\left (-1+x^3\right )^{2/3}}{(1+x)^2} \, dx\right )+\frac {1}{3} \int \frac {(-15+8 x) \left (-1+x^3\right )^{2/3}}{1-x+x^2} \, dx-2 \int \frac {\left (-1+x^3\right )^{2/3}}{x^6} \, dx-\frac {8}{3} \int \frac {\left (-1+x^3\right )^{2/3}}{1+x} \, dx+6 \int \frac {\left (-1+x^3\right )^{2/3}}{x^3} \, dx+\int \frac {(-1+x) \left (-1+x^3\right )^{2/3}}{\left (1-x+x^2\right )^2} \, dx \\ & = -\frac {4}{3} \left (-1+x^3\right )^{2/3}-\frac {3 \left (-1+x^3\right )^{2/3}}{x^2}-\frac {2 \left (-1+x^3\right )^{5/3}}{5 x^5}-\frac {1}{3} \int \left (\frac {\left (-1+x^3\right )^{2/3}}{\left (1+x^3\right )^2}-\frac {2 x \left (-1+x^3\right )^{2/3}}{\left (1+x^3\right )^2}+\frac {3 x^2 \left (-1+x^3\right )^{2/3}}{\left (1+x^3\right )^2}-\frac {2 x^3 \left (-1+x^3\right )^{2/3}}{\left (1+x^3\right )^2}+\frac {x^4 \left (-1+x^3\right )^{2/3}}{\left (1+x^3\right )^2}\right ) \, dx+\frac {1}{3} \int \left (-\frac {15 \left (-1+x^3\right )^{2/3}}{1+x^3}-\frac {7 x \left (-1+x^3\right )^{2/3}}{1+x^3}+\frac {8 x^2 \left (-1+x^3\right )^{2/3}}{1+x^3}\right ) \, dx+\frac {8}{3} \int \frac {x}{\sqrt [3]{-1+x^3}} \, dx-\frac {8}{3} \int \frac {-1+x}{(1+x) \sqrt [3]{-1+x^3}} \, dx+6 \int \frac {1}{\sqrt [3]{-1+x^3}} \, dx+\int \left (-\frac {2 (1+x)^2 \left (-1+x^3\right )^{2/3}}{\left (1+x^3\right )^2}+\frac {(1+x)^3 \left (-1+x^3\right )^{2/3}}{\left (1+x^3\right )^2}\right ) \, dx \\ & = -\frac {4}{3} \left (-1+x^3\right )^{2/3}-\frac {3 \left (-1+x^3\right )^{2/3}}{x^2}-\frac {2 \left (-1+x^3\right )^{5/3}}{5 x^5}+2 \sqrt {3} \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )-3 \log \left (-x+\sqrt [3]{-1+x^3}\right )-\frac {1}{3} \int \frac {\left (-1+x^3\right )^{2/3}}{\left (1+x^3\right )^2} \, dx-\frac {1}{3} \int \frac {x^4 \left (-1+x^3\right )^{2/3}}{\left (1+x^3\right )^2} \, dx+\frac {2}{3} \int \frac {x \left (-1+x^3\right )^{2/3}}{\left (1+x^3\right )^2} \, dx+\frac {2}{3} \int \frac {x^3 \left (-1+x^3\right )^{2/3}}{\left (1+x^3\right )^2} \, dx-2 \int \frac {(1+x)^2 \left (-1+x^3\right )^{2/3}}{\left (1+x^3\right )^2} \, dx-\frac {7}{3} \int \frac {x \left (-1+x^3\right )^{2/3}}{1+x^3} \, dx-\frac {8}{3} \int \frac {1}{\sqrt [3]{-1+x^3}} \, dx+\frac {8}{3} \int \frac {x^2 \left (-1+x^3\right )^{2/3}}{1+x^3} \, dx-5 \int \frac {\left (-1+x^3\right )^{2/3}}{1+x^3} \, dx+\frac {16}{3} \int \frac {1}{(1+x) \sqrt [3]{-1+x^3}} \, dx+\frac {\left (8 \sqrt [3]{1-x^3}\right ) \int \frac {x}{\sqrt [3]{1-x^3}} \, dx}{3 \sqrt [3]{-1+x^3}}-\int \frac {x^2 \left (-1+x^3\right )^{2/3}}{\left (1+x^3\right )^2} \, dx+\int \frac {(1+x)^3 \left (-1+x^3\right )^{2/3}}{\left (1+x^3\right )^2} \, dx \\ & = -\frac {4}{3} \left (-1+x^3\right )^{2/3}-\frac {3 \left (-1+x^3\right )^{2/3}}{x^2}-\frac {2 \left (-1+x^3\right )^{5/3}}{5 x^5}-\frac {x \left (-1+x^3\right )^{2/3}}{3 \left (1+x^3\right )}+\frac {x^2 \left (-1+x^3\right )^{2/3}}{3 \left (1+x^3\right )}+\frac {4\ 2^{2/3} \arctan \left (\frac {1-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {8 \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3}}+2 \sqrt {3} \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )+\frac {4 x^2 \sqrt [3]{1-x^3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},x^3\right )}{3 \sqrt [3]{-1+x^3}}+\frac {2}{3} 2^{2/3} \log \left ((1-x) (1+x)^2\right )-\frac {5}{3} \log \left (-x+\sqrt [3]{-1+x^3}\right )-2\ 2^{2/3} \log \left (1-x+2^{2/3} \sqrt [3]{-1+x^3}\right )-\frac {1}{9} \int \frac {x \left (-2+4 x^3\right )}{\sqrt [3]{-1+x^3} \left (1+x^3\right )} \, dx-\frac {2}{9} \int \frac {x}{\sqrt [3]{-1+x^3}} \, dx+\frac {2}{9} \int \frac {1}{\sqrt [3]{-1+x^3} \left (1+x^3\right )} \, dx+\frac {2}{9} \int \frac {-1+3 x^3}{\sqrt [3]{-1+x^3} \left (1+x^3\right )} \, dx-\frac {1}{3} \text {Subst}\left (\int \frac {(-1+x)^{2/3}}{(1+x)^2} \, dx,x,x^3\right )+\frac {8}{9} \text {Subst}\left (\int \frac {(-1+x)^{2/3}}{1+x} \, dx,x,x^3\right )-2 \int \frac {\left (-1+x^3\right )^{2/3}}{\left (1-x+x^2\right )^2} \, dx-\frac {7}{3} \int \frac {x}{\sqrt [3]{-1+x^3}} \, dx+\frac {14}{3} \int \frac {x}{\sqrt [3]{-1+x^3} \left (1+x^3\right )} \, dx-5 \int \frac {1}{\sqrt [3]{-1+x^3}} \, dx+10 \int \frac {1}{\sqrt [3]{-1+x^3} \left (1+x^3\right )} \, dx+\int \frac {(1+x) \left (-1+x^3\right )^{2/3}}{\left (1-x+x^2\right )^2} \, dx \\ & = -\frac {3 \left (-1+x^3\right )^{2/3}}{x^2}-\frac {2 \left (-1+x^3\right )^{5/3}}{5 x^5}+\frac {\left (-1+x^3\right )^{2/3}}{3 \left (1+x^3\right )}-\frac {x \left (-1+x^3\right )^{2/3}}{3 \left (1+x^3\right )}+\frac {x^2 \left (-1+x^3\right )^{2/3}}{3 \left (1+x^3\right )}+\frac {4\ 2^{2/3} \arctan \left (\frac {1-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {23 \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3}}+2 \sqrt {3} \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )+\frac {46\ 2^{2/3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{2} x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{9 \sqrt {3}}+\frac {4 x^2 \sqrt [3]{1-x^3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},x^3\right )}{3 \sqrt [3]{-1+x^3}}+\frac {2}{3} 2^{2/3} \log \left ((1-x) (1+x)^2\right )+\frac {23}{27} 2^{2/3} \log \left (1+x^3\right )-\frac {23}{9} 2^{2/3} \log \left (\sqrt [3]{2} x-\sqrt [3]{-1+x^3}\right )+\frac {5}{6} \log \left (-x+\sqrt [3]{-1+x^3}\right )-2\ 2^{2/3} \log \left (1-x+2^{2/3} \sqrt [3]{-1+x^3}\right )-\frac {1}{9} \int \left (\frac {4 x}{\sqrt [3]{-1+x^3}}-\frac {6 x}{\sqrt [3]{-1+x^3} \left (1+x^3\right )}\right ) \, dx-\frac {2}{9} \text {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x} (1+x)} \, dx,x,x^3\right )+\frac {2}{3} \int \frac {1}{\sqrt [3]{-1+x^3}} \, dx-\frac {8}{9} \int \frac {1}{\sqrt [3]{-1+x^3} \left (1+x^3\right )} \, dx-\frac {14}{9} \int \frac {1}{(1+x) \sqrt [3]{-1+x^3}} \, dx-\frac {16}{9} \text {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x} (1+x)} \, dx,x,x^3\right )-2 \int \left (\frac {\left (-1+x^3\right )^{2/3}}{\left (1+x^3\right )^2}+\frac {2 x \left (-1+x^3\right )^{2/3}}{\left (1+x^3\right )^2}+\frac {x^2 \left (-1+x^3\right )^{2/3}}{\left (1+x^3\right )^2}\right ) \, dx-\frac {14}{3} \text {Subst}\left (\int \frac {1}{1-2 x^3} \, dx,x,\frac {1-x}{\sqrt [3]{-1+x^3}}\right )-\frac {\left (2 \sqrt [3]{1-x^3}\right ) \int \frac {x}{\sqrt [3]{1-x^3}} \, dx}{9 \sqrt [3]{-1+x^3}}-\frac {\left (7 \sqrt [3]{1-x^3}\right ) \int \frac {x}{\sqrt [3]{1-x^3}} \, dx}{3 \sqrt [3]{-1+x^3}}+\int \left (\frac {\left (-1+x^3\right )^{2/3}}{\left (1+x^3\right )^2}+\frac {3 x \left (-1+x^3\right )^{2/3}}{\left (1+x^3\right )^2}+\frac {3 x^2 \left (-1+x^3\right )^{2/3}}{\left (1+x^3\right )^2}+\frac {x^3 \left (-1+x^3\right )^{2/3}}{\left (1+x^3\right )^2}\right ) \, dx \\ & = -\frac {3 \left (-1+x^3\right )^{2/3}}{x^2}-\frac {2 \left (-1+x^3\right )^{5/3}}{5 x^5}+\frac {\left (-1+x^3\right )^{2/3}}{3 \left (1+x^3\right )}-\frac {x \left (-1+x^3\right )^{2/3}}{3 \left (1+x^3\right )}+\frac {x^2 \left (-1+x^3\right )^{2/3}}{3 \left (1+x^3\right )}-\frac {7 \arctan \left (\frac {1-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{3 \sqrt [3]{2} \sqrt {3}}+\frac {4\ 2^{2/3} \arctan \left (\frac {1-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {7 \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+2 \sqrt {3} \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )+\frac {14\ 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 {x^2 \sqrt [3]{1-x^3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},x^3\right )}{18 \sqrt [3]{-1+x^3}}-\frac {7 \log \left ((1-x) (1+x)^2\right )}{18 \sqrt [3]{2}}+\frac {2}{3} 2^{2/3} \log \left ((1-x) (1+x)^2\right )-\frac {\log \left (1+x^3\right )}{9 \sqrt [3]{2}}+\frac {1}{3} 2^{2/3} \log \left (1+x^3\right )-\frac {7}{3} 2^{2/3} \log \left (\sqrt [3]{2} x-\sqrt [3]{-1+x^3}\right )+\frac {1}{2} \log \left (-x+\sqrt [3]{-1+x^3}\right )+\frac {7 \log \left (1-x+2^{2/3} \sqrt [3]{-1+x^3}\right )}{6 \sqrt [3]{2}}-2\ 2^{2/3} \log \left (1-x+2^{2/3} \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 {4}{9} \int \frac {x}{\sqrt [3]{-1+x^3}} \, dx+\frac {2}{3} \int \frac {x}{\sqrt [3]{-1+x^3} \left (1+x^3\right )} \, dx-\frac {14}{9} \text {Subst}\left (\int \frac {1}{1-\sqrt [3]{2} x} \, dx,x,\frac {1-x}{\sqrt [3]{-1+x^3}}\right )-\frac {14}{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 )-2 \int \frac {\left (-1+x^3\right )^{2/3}}{\left (1+x^3\right )^2} \, dx-2 \int \frac {x^2 \left (-1+x^3\right )^{2/3}}{\left (1+x^3\right )^2} \, dx-\frac {8}{3} \text {Subst}\left (\int \frac {1}{2^{2/3}-\sqrt [3]{2} x+x^2} \, dx,x,\sqrt [3]{-1+x^3}\right )+3 \int \frac {x \left (-1+x^3\right )^{2/3}}{\left (1+x^3\right )^2} \, dx+3 \int \frac {x^2 \left (-1+x^3\right )^{2/3}}{\left (1+x^3\right )^2} \, dx-4 \int \frac {x \left (-1+x^3\right )^{2/3}}{\left (1+x^3\right )^2} \, dx+\frac {\text {Subst}\left (\int \frac {1}{\sqrt [3]{2}+x} \, dx,x,\sqrt [3]{-1+x^3}\right )}{3 \sqrt [3]{2}}+\frac {1}{3} \left (4\ 2^{2/3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{2}+x} \, dx,x,\sqrt [3]{-1+x^3}\right )+\int \frac {\left (-1+x^3\right )^{2/3}}{\left (1+x^3\right )^2} \, dx+\int \frac {x^3 \left (-1+x^3\right )^{2/3}}{\left (1+x^3\right )^2} \, dx \\ & = -\frac {3 \left (-1+x^3\right )^{2/3}}{x^2}-\frac {2 \left (-1+x^3\right )^{5/3}}{5 x^5}+\frac {\left (-1+x^3\right )^{2/3}}{3 \left (1+x^3\right )}-\frac {x \left (-1+x^3\right )^{2/3}}{1+x^3}-\frac {7 \arctan \left (\frac {1-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{3 \sqrt [3]{2} \sqrt {3}}+\frac {4\ 2^{2/3} \arctan \left (\frac {1-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {7 \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+2 \sqrt {3} \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )+\frac {14\ 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 {x^2 \sqrt [3]{1-x^3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},x^3\right )}{18 \sqrt [3]{-1+x^3}}-\frac {7 \log \left ((1-x) (1+x)^2\right )}{18 \sqrt [3]{2}}+\frac {2}{3} 2^{2/3} \log \left ((1-x) (1+x)^2\right )-\frac {\log \left (1+x^3\right )}{9 \sqrt [3]{2}}+\frac {1}{3} 2^{2/3} \log \left (1+x^3\right )+\frac {7}{9} 2^{2/3} \log \left (1-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{-1+x^3}}\right )-\frac {7}{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}}+\frac {4}{3} 2^{2/3} \log \left (\sqrt [3]{2}+\sqrt [3]{-1+x^3}\right )+\frac {1}{2} \log \left (-x+\sqrt [3]{-1+x^3}\right )+\frac {7 \log \left (1-x+2^{2/3} \sqrt [3]{-1+x^3}\right )}{6 \sqrt [3]{2}}-2\ 2^{2/3} \log \left (1-x+2^{2/3} \sqrt [3]{-1+x^3}\right )-\frac {2}{9} \int \frac {1}{(1+x) \sqrt [3]{-1+x^3}} \, dx+\frac {1}{3} \int \frac {-1+3 x^3}{\sqrt [3]{-1+x^3} \left (1+x^3\right )} \, dx-\frac {2}{3} \int \frac {1}{\sqrt [3]{-1+x^3} \left (1+x^3\right )} \, dx-\frac {2}{3} \text {Subst}\left (\int \frac {(-1+x)^{2/3}}{(1+x)^2} \, 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 {4}{3} \int \frac {x}{\sqrt [3]{-1+x^3}} \, dx+\frac {4}{3} \int \frac {1}{\sqrt [3]{-1+x^3} \left (1+x^3\right )} \, dx-\frac {7}{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 {7 \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 {1}{3} \left (8\ 2^{2/3}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-2^{2/3} \sqrt [3]{-1+x^3}\right )-\frac {\left (4 \sqrt [3]{1-x^3}\right ) \int \frac {x}{\sqrt [3]{1-x^3}} \, dx}{9 \sqrt [3]{-1+x^3}}-\int \frac {x}{\sqrt [3]{-1+x^3}} \, dx+\text {Subst}\left (\int \frac {(-1+x)^{2/3}}{(1+x)^2} \, dx,x,x^3\right ) \\ & = -\frac {3 \left (-1+x^3\right )^{2/3}}{x^2}-\frac {2 \left (-1+x^3\right )^{5/3}}{5 x^5}-\frac {x \left (-1+x^3\right )^{2/3}}{1+x^3}+\frac {8\ 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 {7 \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+2 \sqrt {3} \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )+\frac {5\ 2^{2/3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{2} x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+2^{2/3} \sqrt {3} \arctan \left (\frac {1-2^{2/3} \sqrt [3]{-1+x^3}}{\sqrt {3}}\right )-\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 {4}{9} 2^{2/3} \log \left ((1-x) (1+x)^2\right )+\frac {1}{3} 2^{2/3} \log \left (1+x^3\right )+\frac {7}{9} 2^{2/3} \log \left (1-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{-1+x^3}}\right )-\frac {7 \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 (\sqrt [3]{2} x-\sqrt [3]{-1+x^3}\right )}{3 \sqrt [3]{2}}-\frac {8}{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}}+\frac {4}{3} 2^{2/3} \log \left (\sqrt [3]{2}+\sqrt [3]{-1+x^3}\right )+\frac {1}{2} \log \left (-x+\sqrt [3]{-1+x^3}\right )-\frac {4}{3} 2^{2/3} \log \left (1-x+2^{2/3} \sqrt [3]{-1+x^3}\right )-\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 {4}{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}{\sqrt [3]{-1+x} (1+x)} \, dx,x,x^3\right )-\frac {4}{3} \int \frac {1}{\sqrt [3]{-1+x^3} \left (1+x^3\right )} \, dx+\frac {1}{3} \left (7\ 2^{2/3}\right ) \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 {\sqrt [3]{1-x^3} \int \frac {x}{\sqrt [3]{1-x^3}} \, dx}{\sqrt [3]{-1+x^3}}+\frac {\left (4 \sqrt [3]{1-x^3}\right ) \int \frac {x}{\sqrt [3]{1-x^3}} \, dx}{3 \sqrt [3]{-1+x^3}}+\int \frac {1}{\sqrt [3]{-1+x^3}} \, dx \\ & = -\frac {3 \left (-1+x^3\right )^{2/3}}{x^2}-\frac {2 \left (-1+x^3\right )^{5/3}}{5 x^5}-\frac {x \left (-1+x^3\right )^{2/3}}{1+x^3}+\frac {8\ 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 {7\ 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 {13\ 2^{2/3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{2} x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3}}+2^{2/3} \sqrt {3} \arctan \left (\frac {1-2^{2/3} \sqrt [3]{-1+x^3}}{\sqrt {3}}\right )+\frac {4}{9} 2^{2/3} \log \left ((1-x) (1+x)^2\right )+\frac {\log \left (1+x^3\right )}{3 \sqrt [3]{2}}+\frac {1}{9} 2^{2/3} \log \left (1+x^3\right )+\frac {8}{9} 2^{2/3} \log \left (1-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{-1+x^3}}\right )-\frac {7 \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 (\sqrt [3]{2} x-\sqrt [3]{-1+x^3}\right )}{3 \sqrt [3]{2}}-\frac {7}{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}}+\frac {4}{3} 2^{2/3} \log \left (\sqrt [3]{2}+\sqrt [3]{-1+x^3}\right )-\frac {4}{3} 2^{2/3} \log \left (1-x+2^{2/3} \sqrt [3]{-1+x^3}\right )-\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 {2}{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 {\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 {\text {Subst}\left (\int \frac {1}{\sqrt [3]{2}+x} \, dx,x,\sqrt [3]{-1+x^3}\right )}{\sqrt [3]{2}}+\frac {1}{3} 2^{2/3} \text {Subst}\left (\int \frac {1}{\sqrt [3]{2}+x} \, dx,x,\sqrt [3]{-1+x^3}\right )+\text {Subst}\left (\int \frac {1}{2^{2/3}-\sqrt [3]{2} x+x^2} \, dx,x,\sqrt [3]{-1+x^3}\right ) \\ & = -\frac {3 \left (-1+x^3\right )^{2/3}}{x^2}-\frac {2 \left (-1+x^3\right )^{5/3}}{5 x^5}-\frac {x \left (-1+x^3\right )^{2/3}}{1+x^3}+\frac {8\ 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 {7\ 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 {13\ 2^{2/3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{2} x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3}}+2^{2/3} \sqrt {3} \arctan \left (\frac {1-2^{2/3} \sqrt [3]{-1+x^3}}{\sqrt {3}}\right )+\frac {4}{9} 2^{2/3} \log \left ((1-x) (1+x)^2\right )+\frac {\log \left (1+x^3\right )}{3 \sqrt [3]{2}}+\frac {1}{9} 2^{2/3} \log \left (1+x^3\right )+\frac {8}{9} 2^{2/3} \log \left (1-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{-1+x^3}}\right )-\frac {4}{9} 2^{2/3} \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 )+\frac {\log \left (\sqrt [3]{2} x-\sqrt [3]{-1+x^3}\right )}{3 \sqrt [3]{2}}-\frac {7}{3} 2^{2/3} \log \left (\sqrt [3]{2} x-\sqrt [3]{-1+x^3}\right )+\frac {4}{3} 2^{2/3} \log \left (\sqrt [3]{2}+\sqrt [3]{-1+x^3}\right )-\frac {4}{3} 2^{2/3} \log \left (1-x+2^{2/3} \sqrt [3]{-1+x^3}\right )+\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 {1}{3} \left (2\ 2^{2/3}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-2^{2/3} \sqrt [3]{-1+x^3}\right )+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 {3 \left (-1+x^3\right )^{2/3}}{x^2}-\frac {2 \left (-1+x^3\right )^{5/3}}{5 x^5}-\frac {x \left (-1+x^3\right )^{2/3}}{1+x^3}+\frac {8\ 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 {8\ 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 {13\ 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 {2^{2/3} \arctan \left (\frac {1-2^{2/3} \sqrt [3]{-1+x^3}}{\sqrt {3}}\right )}{3 \sqrt {3}}+2^{2/3} \sqrt {3} \arctan \left (\frac {1-2^{2/3} \sqrt [3]{-1+x^3}}{\sqrt {3}}\right )+\frac {4}{9} 2^{2/3} \log \left ((1-x) (1+x)^2\right )+\frac {\log \left (1+x^3\right )}{3 \sqrt [3]{2}}+\frac {1}{9} 2^{2/3} \log \left (1+x^3\right )+\frac {8}{9} 2^{2/3} \log \left (1-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{-1+x^3}}\right )-\frac {4}{9} 2^{2/3} \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 )+\frac {\log \left (\sqrt [3]{2} x-\sqrt [3]{-1+x^3}\right )}{3 \sqrt [3]{2}}-\frac {7}{3} 2^{2/3} \log \left (\sqrt [3]{2} x-\sqrt [3]{-1+x^3}\right )+\frac {4}{3} 2^{2/3} \log \left (\sqrt [3]{2}+\sqrt [3]{-1+x^3}\right )-\frac {4}{3} 2^{2/3} \log \left (1-x+2^{2/3} \sqrt [3]{-1+x^3}\right ) \\ \end{align*}
Time = 0.45 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (-2+2 x^3+x^6\right )}{x^6 \left (1+x^3\right )^2} \, dx=\frac {\left (-1+x^3\right )^{2/3} \left (2-15 x^3-22 x^6\right )}{5 x^5 \left (1+x^3\right )}+\frac {7\ 2^{2/3} \arctan \left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{-1+x^3}}\right )}{\sqrt {3}}-\frac {7}{3} 2^{2/3} \log \left (-2 x+2^{2/3} \sqrt [3]{-1+x^3}\right )+\frac {7 \log \left (2 x^2+2^{2/3} x \sqrt [3]{-1+x^3}+\sqrt [3]{2} \left (-1+x^3\right )^{2/3}\right )}{3 \sqrt [3]{2}} \]
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Time = 13.60 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.87
method | result | size |
pseudoelliptic | \(\frac {35 x^{5} \left (x^{3}+1\right ) \left (-2 \arctan \left (\frac {\sqrt {3}\, \left (x +2^{\frac {2}{3}} \left (x^{3}-1\right )^{\frac {1}{3}}\right )}{3 x}\right ) \sqrt {3}+\ln \left (\frac {2^{\frac {2}{3}} x^{2}+2^{\frac {1}{3}} x \left (x^{3}-1\right )^{\frac {1}{3}}+\left (x^{3}-1\right )^{\frac {2}{3}}}{x^{2}}\right )-2 \ln \left (\frac {-2^{\frac {1}{3}} x +\left (x^{3}-1\right )^{\frac {1}{3}}}{x}\right )\right ) 2^{\frac {2}{3}}-6 \left (x^{3}-1\right )^{\frac {2}{3}} \left (22 x^{6}+15 x^{3}-2\right )}{30 x^{8}+30 x^{5}}\) | \(131\) |
risch | \(\text {Expression too large to display}\) | \(836\) |
trager | \(\text {Expression too large to display}\) | \(1151\) |
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Leaf count of result is larger than twice the leaf count of optimal. 294 vs. \(2 (117) = 234\).
Time = 1.94 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.95 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (-2+2 x^3+x^6\right )}{x^6 \left (1+x^3\right )^2} \, dx=-\frac {70 \, \sqrt {3} \left (-4\right )^{\frac {1}{3}} {\left (x^{8} + x^{5}\right )} \arctan \left (\frac {3 \, \sqrt {3} \left (-4\right )^{\frac {2}{3}} {\left (5 \, x^{7} + 4 \, x^{4} - x\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} + 6 \, \sqrt {3} \left (-4\right )^{\frac {1}{3}} {\left (19 \, x^{8} - 16 \, x^{5} + x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}} - \sqrt {3} {\left (71 \, x^{9} - 111 \, x^{6} + 33 \, x^{3} - 1\right )}}{3 \, {\left (109 \, x^{9} - 105 \, x^{6} + 3 \, x^{3} + 1\right )}}\right ) - 70 \, \left (-4\right )^{\frac {1}{3}} {\left (x^{8} + x^{5}\right )} \log \left (-\frac {3 \, \left (-4\right )^{\frac {2}{3}} {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{2} - 6 \, {\left (x^{3} - 1\right )}^{\frac {2}{3}} x + \left (-4\right )^{\frac {1}{3}} {\left (x^{3} + 1\right )}}{x^{3} + 1}\right ) + 35 \, \left (-4\right )^{\frac {1}{3}} {\left (x^{8} + x^{5}\right )} \log \left (-\frac {6 \, \left (-4\right )^{\frac {1}{3}} {\left (5 \, x^{4} - x\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} - \left (-4\right )^{\frac {2}{3}} {\left (19 \, x^{6} - 16 \, x^{3} + 1\right )} - 24 \, {\left (2 \, x^{5} - x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x^{6} + 2 \, x^{3} + 1}\right ) + 18 \, {\left (22 \, x^{6} + 15 \, x^{3} - 2\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{90 \, {\left (x^{8} + x^{5}\right )}} \]
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\[ \int \frac {\left (-1+x^3\right )^{2/3} \left (-2+2 x^3+x^6\right )}{x^6 \left (1+x^3\right )^2} \, dx=\int \frac {\left (\left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {2}{3}} \left (x^{6} + 2 x^{3} - 2\right )}{x^{6} \left (x + 1\right )^{2} \left (x^{2} - x + 1\right )^{2}}\, dx \]
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\[ \int \frac {\left (-1+x^3\right )^{2/3} \left (-2+2 x^3+x^6\right )}{x^6 \left (1+x^3\right )^2} \, dx=\int { \frac {{\left (x^{6} + 2 \, x^{3} - 2\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{3} + 1\right )}^{2} x^{6}} \,d x } \]
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\[ \int \frac {\left (-1+x^3\right )^{2/3} \left (-2+2 x^3+x^6\right )}{x^6 \left (1+x^3\right )^2} \, dx=\int { \frac {{\left (x^{6} + 2 \, x^{3} - 2\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{3} + 1\right )}^{2} x^{6}} \,d x } \]
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Timed out. \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (-2+2 x^3+x^6\right )}{x^6 \left (1+x^3\right )^2} \, dx=\int \frac {{\left (x^3-1\right )}^{2/3}\,\left (x^6+2\,x^3-2\right )}{x^6\,{\left (x^3+1\right )}^2} \,d x \]
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