Integrand size = 25, antiderivative size = 151 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (2+x^6\right )}{x^6 \left (-1+x^3\right )^2} \, dx=\frac {\left (1+x^3\right )^{2/3} \left (2+10 x^3-17 x^6\right )}{5 x^5 \left (-1+x^3\right )}+\frac {5\ 2^{2/3} \arctan \left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{1+x^3}}\right )}{\sqrt {3}}-\frac {5}{3} 2^{2/3} \log \left (-2 x+2^{2/3} \sqrt [3]{1+x^3}\right )+\frac {5 \log \left (2 x^2+2^{2/3} x \sqrt [3]{1+x^3}+\sqrt [3]{2} \left (1+x^3\right )^{2/3}\right )}{3 \sqrt [3]{2}} \]
[Out]
Leaf count is larger than twice the leaf count of optimal. \(545\) vs. \(2(151)=302\).
Time = 1.30 (sec) , antiderivative size = 545, normalized size of antiderivative = 3.61, number of steps used = 93, number of rules used = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.240, Rules used = {6874, 2181, 386, 384, 480, 371, 455, 43, 57, 631, 210, 31, 478, 544, 245, 598, 502, 2174, 206, 648, 642, 2178, 2177, 270, 283, 2183, 1600, 21, 399, 495, 52} \[ \int \frac {\left (1+x^3\right )^{2/3} \left (2+x^6\right )}{x^6 \left (-1+x^3\right )^2} \, dx=2^{2/3} \sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [3]{2} x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )-\frac {2\ 2^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} (x+1)}{\sqrt [3]{x^3+1}}}{\sqrt {3}}\right )}{\sqrt {3}}+2^{2/3} \sqrt {3} \arctan \left (\frac {\frac {\sqrt [3]{2} (x+1)}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )-\frac {2^{2/3} \arctan \left (\frac {\frac {\sqrt [3]{2} (x+1)}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {2\ 2^{2/3} \arctan \left (\frac {2^{2/3} \sqrt [3]{x^3+1}+1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {x \left (x^3+1\right )^{2/3}}{1-x^3}+\frac {2}{9} 2^{2/3} \log \left (1-x^3\right )-\frac {2}{27} 2^{2/3} \log \left (x^3-1\right )+\frac {\log \left (x^3-1\right )}{27 \sqrt [3]{2}}-\frac {1}{3} 2^{2/3} \log \left (\frac {2^{2/3} (x+1)^2}{\left (x^3+1\right )^{2/3}}-\frac {\sqrt [3]{2} (x+1)}{\sqrt [3]{x^3+1}}+1\right )+\frac {2}{3} 2^{2/3} \log \left (\frac {\sqrt [3]{2} (x+1)}{\sqrt [3]{x^3+1}}+1\right )+2^{2/3} \log \left (\sqrt [3]{2}-\sqrt [3]{x^3+1}\right )+\frac {2}{9} 2^{2/3} \log \left (\sqrt [3]{2} x-\sqrt [3]{x^3+1}\right )-\frac {31 \log \left (\sqrt [3]{2} x-\sqrt [3]{x^3+1}\right )}{9 \sqrt [3]{2}}+\frac {\log \left (-2^{2/3} \sqrt [3]{x^3+1}+x+1\right )}{\sqrt [3]{2}}-\frac {3 \log \left (2^{2/3} \sqrt [3]{x^3+1}-x-1\right )}{\sqrt [3]{2}}-\frac {2 \left (x^3+1\right )^{5/3}}{5 x^5}-\frac {2 \left (x^3+1\right )^{2/3}}{x^2}+\frac {\log \left (-(1-x)^2 (x+1)\right )}{\sqrt [3]{2}}-\frac {\log \left ((1-x)^2 (x+1)\right )}{3 \sqrt [3]{2}} \]
[In]
[Out]
Rule 21
Rule 31
Rule 43
Rule 52
Rule 57
Rule 206
Rule 210
Rule 245
Rule 270
Rule 283
Rule 371
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 {\left (1+x^3\right )^{2/3}}{3 (-1+x)^2}-\frac {2 \left (1+x^3\right )^{2/3}}{-1+x}+\frac {2 \left (1+x^3\right )^{2/3}}{x^6}+\frac {4 \left (1+x^3\right )^{2/3}}{x^3}+\frac {(1+x) \left (1+x^3\right )^{2/3}}{\left (1+x+x^2\right )^2}+\frac {(11+6 x) \left (1+x^3\right )^{2/3}}{3 \left (1+x+x^2\right )}\right ) \, dx \\ & = \frac {1}{3} \int \frac {\left (1+x^3\right )^{2/3}}{(-1+x)^2} \, dx+\frac {1}{3} \int \frac {(11+6 x) \left (1+x^3\right )^{2/3}}{1+x+x^2} \, dx-2 \int \frac {\left (1+x^3\right )^{2/3}}{-1+x} \, dx+2 \int \frac {\left (1+x^3\right )^{2/3}}{x^6} \, dx+4 \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 \\ & = -\left (1+x^3\right )^{2/3}-\frac {2 \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 {11 \left (1+x^3\right )^{2/3}}{1-x^3}-\frac {5 x \left (1+x^3\right )^{2/3}}{1-x^3}-\frac {6 x^2 \left (1+x^3\right )^{2/3}}{1-x^3}\right ) \, dx+\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-2 \int \frac {x}{\sqrt [3]{1+x^3}} \, dx-2 \int \frac {1+x}{(-1+x) \sqrt [3]{1+x^3}} \, dx+4 \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 \\ & = -\left (1+x^3\right )^{2/3}-\frac {2 \left (1+x^3\right )^{2/3}}{x^2}-\frac {2 \left (1+x^3\right )^{5/3}}{5 x^5}+\frac {4 \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-x^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},-x^3\right )-2 \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-\frac {5}{3} \int \frac {x \left (1+x^3\right )^{2/3}}{1-x^3} \, dx-2 \int \frac {1}{\sqrt [3]{1+x^3}} \, dx+2 \int \frac {(1-x)^2 \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}}{1-x^3} \, dx+\frac {11}{3} \int \frac {\left (1+x^3\right )^{2/3}}{1-x^3} \, dx-4 \int \frac {1}{(-1+x) \sqrt [3]{1+x^3}} \, dx-\int \frac {(1-x)^3 \left (1+x^3\right )^{2/3}}{\left (1-x^3\right )^2} \, dx+\int \frac {x^2 \left (1+x^3\right )^{2/3}}{\left (-1+x^3\right )^2} \, dx \\ & = -\left (1+x^3\right )^{2/3}-\frac {2 \left (1+x^3\right )^{2/3}}{x^2}+\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 {2 \left (1+x^3\right )^{5/3}}{5 x^5}+\frac {2 \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+2^{2/3} \sqrt {3} \arctan \left (\frac {1+\frac {\sqrt [3]{2} (1+x)}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )-x^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},-x^3\right )+\frac {\log \left (-(1-x)^2 (1+x)\right )}{\sqrt [3]{2}}-\log \left (-x+\sqrt [3]{1+x^3}\right )-\frac {3 \log \left (-1-x+2^{2/3} \sqrt [3]{1+x^3}\right )}{\sqrt [3]{2}}+\frac {1}{9} \int \frac {x \left (2+4 x^3\right )}{\left (-1+x^3\right ) \sqrt [3]{1+x^3}} \, dx+\frac {2}{9} \int \frac {x}{\sqrt [3]{1+x^3}} \, dx-\frac {2}{9} \int \frac {1}{\left (-1+x^3\right ) \sqrt [3]{1+x^3}} \, dx+\frac {2}{9} \int \frac {1+3 x^3}{\left (-1+x^3\right ) \sqrt [3]{1+x^3}} \, dx+\frac {1}{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+x)^{2/3}}{1-x} \, dx,x,x^3\right )+\frac {5}{3} \int \frac {x}{\sqrt [3]{1+x^3}} \, dx+2 \int \frac {\left (1+x^3\right )^{2/3}}{\left (1+x+x^2\right )^2} \, dx-\frac {10}{3} \int \frac {x}{\left (1-x^3\right ) \sqrt [3]{1+x^3}} \, dx-\frac {11}{3} \int \frac {1}{\sqrt [3]{1+x^3}} \, dx+\frac {22}{3} \int \frac {1}{\left (1-x^3\right ) \sqrt [3]{1+x^3}} \, dx-\int \frac {(1-x) \left (1+x^3\right )^{2/3}}{\left (1+x+x^2\right )^2} \, dx \\ & = -\frac {2 \left (1+x^3\right )^{2/3}}{x^2}+\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 {2 \left (1+x^3\right )^{5/3}}{5 x^5}-\frac {5 \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {34\ 2^{2/3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{9 \sqrt {3}}+2^{2/3} \sqrt {3} \arctan \left (\frac {1+\frac {\sqrt [3]{2} (1+x)}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )-\frac {1}{18} x^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},-x^3\right )+\frac {\log \left (-(1-x)^2 (1+x)\right )}{\sqrt [3]{2}}+\frac {11 \log \left (1-x^3\right )}{9 \sqrt [3]{2}}+\frac {\log \left (-1+x^3\right )}{27 \sqrt [3]{2}}-\frac {17}{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 )-\frac {3 \log \left (-1-x+2^{2/3} \sqrt [3]{1+x^3}\right )}{\sqrt [3]{2}}+\frac {1}{9} \int \left (\frac {4 x}{\sqrt [3]{1+x^3}}+\frac {6 x}{\left (-1+x^3\right ) \sqrt [3]{1+x^3}}\right ) \, dx+\frac {2}{9} \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt [3]{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}{\left (-1+x^3\right ) \sqrt [3]{1+x^3}} \, dx-\frac {10}{9} \int \frac {1}{(1-x) \sqrt [3]{1+x^3}} \, dx-\frac {4}{3} \text {Subst}\left (\int \frac {1}{(1-x) \sqrt [3]{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 {10}{3} \text {Subst}\left (\int \frac {1}{1+2 x^3} \, dx,x,\frac {1+x}{\sqrt [3]{1+x^3}}\right )-\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 {2 \left (1+x^3\right )^{2/3}}{x^2}+\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 {2 \left (1+x^3\right )^{5/3}}{5 x^5}-\frac {\arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {10\ 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 {5 \arctan \left (\frac {1+\frac {\sqrt [3]{2} (1+x)}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{3 \sqrt [3]{2} \sqrt {3}}+2^{2/3} \sqrt {3} \arctan \left (\frac {1+\frac {\sqrt [3]{2} (1+x)}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )-\frac {1}{18} x^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},-x^3\right )+\frac {\log \left (-(1-x)^2 (1+x)\right )}{\sqrt [3]{2}}-\frac {5 \log \left ((1-x)^2 (1+x)\right )}{18 \sqrt [3]{2}}+\frac {2}{9} 2^{2/3} \log \left (1-x^3\right )+\frac {\log \left (-1+x^3\right )}{27 \sqrt [3]{2}}-\frac {2}{27} 2^{2/3} \log \left (-1+x^3\right )-\frac {5}{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 {5 \log \left (1+x-2^{2/3} \sqrt [3]{1+x^3}\right )}{6 \sqrt [3]{2}}-\frac {3 \log \left (-1-x+2^{2/3} \sqrt [3]{1+x^3}\right )}{\sqrt [3]{2}}+\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}{\left (-1+x^3\right ) \sqrt [3]{1+x^3}} \, dx+\frac {10}{9} \text {Subst}\left (\int \frac {1}{1+\sqrt [3]{2} x} \, dx,x,\frac {1+x}{\sqrt [3]{1+x^3}}\right )+\frac {10}{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+2 \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}}-2^{2/3} \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 {2 \left (1+x^3\right )^{2/3}}{x^2}+\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 {2 \left (1+x^3\right )^{5/3}}{5 x^5}-\frac {\arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {10\ 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 {5 \arctan \left (\frac {1+\frac {\sqrt [3]{2} (1+x)}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{3 \sqrt [3]{2} \sqrt {3}}+2^{2/3} \sqrt {3} \arctan \left (\frac {1+\frac {\sqrt [3]{2} (1+x)}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )+\frac {1}{6} x^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},-x^3\right )+\frac {\log \left (-(1-x)^2 (1+x)\right )}{\sqrt [3]{2}}-\frac {5 \log \left ((1-x)^2 (1+x)\right )}{18 \sqrt [3]{2}}+\frac {2}{9} 2^{2/3} \log \left (1-x^3\right )+\frac {\log \left (-1+x^3\right )}{27 \sqrt [3]{2}}-\frac {2}{27} 2^{2/3} \log \left (-1+x^3\right )+\frac {5}{9} 2^{2/3} \log \left (1+\frac {\sqrt [3]{2} (1+x)}{\sqrt [3]{1+x^3}}\right )+\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{1+x^3}\right )}{3 \sqrt [3]{2}}+2^{2/3} \log \left (\sqrt [3]{2}-\sqrt [3]{1+x^3}\right )-\frac {5}{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 {5 \log \left (1+x-2^{2/3} \sqrt [3]{1+x^3}\right )}{6 \sqrt [3]{2}}-\frac {3 \log \left (-1-x+2^{2/3} \sqrt [3]{1+x^3}\right )}{\sqrt [3]{2}}-\frac {2}{9} \int \frac {1}{(1-x) \sqrt [3]{1+x^3}} \, dx-\frac {1}{3} \int \frac {1+3 x^3}{\left (1-x^3\right ) \sqrt [3]{1+x^3}} \, dx-\frac {2}{3} \int \frac {1}{\left (1-x^3\right ) \sqrt [3]{1+x^3}} \, 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 {1}{\left (1-x^3\right ) \sqrt [3]{1+x^3}} \, dx+\frac {4}{3} \int \frac {x \left (-1+x^3\right )}{\left (1-x^3\right ) \sqrt [3]{1+x^3}} \, dx+\frac {5}{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 {5 \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 )-\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 )-\int \frac {x \left (-1+x^3\right )}{\left (1-x^3\right ) \sqrt [3]{1+x^3}} \, dx-\text {Subst}\left (\int \frac {(1+x)^{2/3}}{(1-x)^2} \, dx,x,x^3\right ) \\ & = -\frac {2 \left (1+x^3\right )^{2/3}}{x^2}+\frac {x \left (1+x^3\right )^{2/3}}{1-x^3}-\frac {2 \left (1+x^3\right )^{5/3}}{5 x^5}-\frac {\arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {11\ 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+\frac {\sqrt [3]{2} (1+x)}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+2^{2/3} \sqrt {3} \arctan \left (\frac {1+\frac {\sqrt [3]{2} (1+x)}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )+\frac {7\ 2^{2/3} \arctan \left (\frac {1+2^{2/3} \sqrt [3]{1+x^3}}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {1}{6} x^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},-x^3\right )+\frac {\log \left (-(1-x)^2 (1+x)\right )}{\sqrt [3]{2}}-\frac {\log \left ((1-x)^2 (1+x)\right )}{3 \sqrt [3]{2}}-\frac {\log \left (1-x^3\right )}{9 \sqrt [3]{2}}+\frac {1}{3} 2^{2/3} \log \left (1-x^3\right )+\frac {\log \left (-1+x^3\right )}{27 \sqrt [3]{2}}-\frac {2}{27} 2^{2/3} \log \left (-1+x^3\right )-\frac {5 \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 {5}{9} 2^{2/3} \log \left (1+\frac {\sqrt [3]{2} (1+x)}{\sqrt [3]{1+x^3}}\right )+\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{1+x^3}\right )}{3 \sqrt [3]{2}}+2^{2/3} \log \left (\sqrt [3]{2}-\sqrt [3]{1+x^3}\right )+\frac {\log \left (\sqrt [3]{2} x-\sqrt [3]{1+x^3}\right )}{3 \sqrt [3]{2}}-2\ 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 {\log \left (1+x-2^{2/3} \sqrt [3]{1+x^3}\right )}{\sqrt [3]{2}}-\frac {3 \log \left (-1-x+2^{2/3} \sqrt [3]{1+x^3}\right )}{\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 {4}{9} \text {Subst}\left (\int \frac {1}{(1-x) \sqrt [3]{1+x}} \, dx,x,x^3\right )+\frac {2}{3} \text {Subst}\left (\int \frac {1}{(1-x) \sqrt [3]{1+x}} \, dx,x,x^3\right )-\frac {4}{3} \int \frac {x}{\sqrt [3]{1+x^3}} \, dx-\frac {4}{3} \int \frac {1}{\left (1-x^3\right ) \sqrt [3]{1+x^3}} \, dx+\frac {1}{3} \left (5\ 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 )+\int \frac {1}{\sqrt [3]{1+x^3}} \, dx+\int \frac {x}{\sqrt [3]{1+x^3}} \, dx \\ & = -\frac {2 \left (1+x^3\right )^{2/3}}{x^2}+\frac {x \left (1+x^3\right )^{2/3}}{1-x^3}-\frac {2 \left (1+x^3\right )^{5/3}}{5 x^5}+2^{2/3} \sqrt {3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )-\frac {5\ 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 {2^{2/3} \arctan \left (\frac {1+\frac {\sqrt [3]{2} (1+x)}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+2^{2/3} \sqrt {3} \arctan \left (\frac {1+\frac {\sqrt [3]{2} (1+x)}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )+\frac {7\ 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)^2 (1+x)\right )}{\sqrt [3]{2}}-\frac {\log \left ((1-x)^2 (1+x)\right )}{3 \sqrt [3]{2}}+\frac {2}{9} 2^{2/3} \log \left (1-x^3\right )+\frac {\log \left (-1+x^3\right )}{27 \sqrt [3]{2}}-\frac {2}{27} 2^{2/3} \log \left (-1+x^3\right )-\frac {5 \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 {2}{3} 2^{2/3} \log \left (1+\frac {\sqrt [3]{2} (1+x)}{\sqrt [3]{1+x^3}}\right )+\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{1+x^3}\right )}{3 \sqrt [3]{2}}+2^{2/3} \log \left (\sqrt [3]{2}-\sqrt [3]{1+x^3}\right )+\frac {\log \left (\sqrt [3]{2} x-\sqrt [3]{1+x^3}\right )}{3 \sqrt [3]{2}}-\frac {5}{3} 2^{2/3} \log \left (\sqrt [3]{2} x-\sqrt [3]{1+x^3}\right )+\frac {\log \left (1+x-2^{2/3} \sqrt [3]{1+x^3}\right )}{\sqrt [3]{2}}-\frac {3 \log \left (-1-x+2^{2/3} \sqrt [3]{1+x^3}\right )}{\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 {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 {2 \left (1+x^3\right )^{2/3}}{x^2}+\frac {x \left (1+x^3\right )^{2/3}}{1-x^3}-\frac {2 \left (1+x^3\right )^{5/3}}{5 x^5}+2^{2/3} \sqrt {3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )-\frac {5\ 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 {2^{2/3} \arctan \left (\frac {1+\frac {\sqrt [3]{2} (1+x)}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+2^{2/3} \sqrt {3} \arctan \left (\frac {1+\frac {\sqrt [3]{2} (1+x)}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )+\frac {7\ 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)^2 (1+x)\right )}{\sqrt [3]{2}}-\frac {\log \left ((1-x)^2 (1+x)\right )}{3 \sqrt [3]{2}}+\frac {2}{9} 2^{2/3} \log \left (1-x^3\right )+\frac {\log \left (-1+x^3\right )}{27 \sqrt [3]{2}}-\frac {2}{27} 2^{2/3} \log \left (-1+x^3\right )-\frac {1}{3} 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 {2}{3} 2^{2/3} \log \left (1+\frac {\sqrt [3]{2} (1+x)}{\sqrt [3]{1+x^3}}\right )+2^{2/3} \log \left (\sqrt [3]{2}-\sqrt [3]{1+x^3}\right )+\frac {\log \left (\sqrt [3]{2} x-\sqrt [3]{1+x^3}\right )}{3 \sqrt [3]{2}}-\frac {5}{3} 2^{2/3} \log \left (\sqrt [3]{2} x-\sqrt [3]{1+x^3}\right )+\frac {\log \left (1+x-2^{2/3} \sqrt [3]{1+x^3}\right )}{\sqrt [3]{2}}-\frac {3 \log \left (-1-x+2^{2/3} \sqrt [3]{1+x^3}\right )}{\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 {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 {2 \left (1+x^3\right )^{2/3}}{x^2}+\frac {x \left (1+x^3\right )^{2/3}}{1-x^3}-\frac {2 \left (1+x^3\right )^{5/3}}{5 x^5}+2^{2/3} \sqrt {3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )-\frac {2\ 2^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} (1+x)}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {2^{2/3} \arctan \left (\frac {1+\frac {\sqrt [3]{2} (1+x)}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+2^{2/3} \sqrt {3} \arctan \left (\frac {1+\frac {\sqrt [3]{2} (1+x)}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )+\frac {2\ 2^{2/3} \arctan \left (\frac {1+2^{2/3} \sqrt [3]{1+x^3}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\log \left (-(1-x)^2 (1+x)\right )}{\sqrt [3]{2}}-\frac {\log \left ((1-x)^2 (1+x)\right )}{3 \sqrt [3]{2}}+\frac {2}{9} 2^{2/3} \log \left (1-x^3\right )+\frac {\log \left (-1+x^3\right )}{27 \sqrt [3]{2}}-\frac {2}{27} 2^{2/3} \log \left (-1+x^3\right )-\frac {1}{3} 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 {2}{3} 2^{2/3} \log \left (1+\frac {\sqrt [3]{2} (1+x)}{\sqrt [3]{1+x^3}}\right )+2^{2/3} \log \left (\sqrt [3]{2}-\sqrt [3]{1+x^3}\right )+\frac {\log \left (\sqrt [3]{2} x-\sqrt [3]{1+x^3}\right )}{3 \sqrt [3]{2}}-\frac {5}{3} 2^{2/3} \log \left (\sqrt [3]{2} x-\sqrt [3]{1+x^3}\right )+\frac {\log \left (1+x-2^{2/3} \sqrt [3]{1+x^3}\right )}{\sqrt [3]{2}}-\frac {3 \log \left (-1-x+2^{2/3} \sqrt [3]{1+x^3}\right )}{\sqrt [3]{2}} \\ \end{align*}
Time = 0.44 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (2+x^6\right )}{x^6 \left (-1+x^3\right )^2} \, dx=\frac {\left (1+x^3\right )^{2/3} \left (2+10 x^3-17 x^6\right )}{5 x^5 \left (-1+x^3\right )}+\frac {5\ 2^{2/3} \arctan \left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{1+x^3}}\right )}{\sqrt {3}}-\frac {5}{3} 2^{2/3} \log \left (-2 x+2^{2/3} \sqrt [3]{1+x^3}\right )+\frac {5 \log \left (2 x^2+2^{2/3} x \sqrt [3]{1+x^3}+\sqrt [3]{2} \left (1+x^3\right )^{2/3}\right )}{3 \sqrt [3]{2}} \]
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Time = 14.03 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.87
method | result | size |
pseudoelliptic | \(\frac {25 x^{5} \left (x^{3}-1\right ) \left (-2 \arctan \left (\frac {\sqrt {3}\, \left (x +2^{\frac {2}{3}} \left (x^{3}+1\right )^{\frac {1}{3}}\right )}{3 x}\right ) \sqrt {3}+\ln \left (\frac {2^{\frac {2}{3}} x^{2}+2^{\frac {1}{3}} x \left (x^{3}+1\right )^{\frac {1}{3}}+\left (x^{3}+1\right )^{\frac {2}{3}}}{x^{2}}\right )-2 \ln \left (\frac {-2^{\frac {1}{3}} x +\left (x^{3}+1\right )^{\frac {1}{3}}}{x}\right )\right ) 2^{\frac {2}{3}}-6 \left (x^{3}+1\right )^{\frac {2}{3}} \left (17 x^{6}-10 x^{3}-2\right )}{30 x^{8}-30 x^{5}}\) | \(131\) |
risch | \(\text {Expression too large to display}\) | \(938\) |
trager | \(\text {Expression too large to display}\) | \(1127\) |
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Leaf count of result is larger than twice the leaf count of optimal. 297 vs. \(2 (117) = 234\).
Time = 1.89 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.97 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (2+x^6\right )}{x^6 \left (-1+x^3\right )^2} \, dx=-\frac {50 \, \sqrt {3} \left (-4\right )^{\frac {1}{3}} {\left (x^{8} - x^{5}\right )} \arctan \left (\frac {3 \, \sqrt {3} \left (-4\right )^{\frac {2}{3}} {\left (5 \, x^{7} - 4 \, x^{4} - x\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}} + 6 \, \sqrt {3} \left (-4\right )^{\frac {1}{3}} {\left (19 \, x^{8} + 16 \, x^{5} + x^{2}\right )} {\left (x^{3} + 1\right )}^{\frac {1}{3}} - \sqrt {3} {\left (71 \, x^{9} + 111 \, x^{6} + 33 \, x^{3} + 1\right )}}{3 \, {\left (109 \, x^{9} + 105 \, x^{6} + 3 \, x^{3} - 1\right )}}\right ) - 50 \, \left (-4\right )^{\frac {1}{3}} {\left (x^{8} - x^{5}\right )} \log \left (\frac {3 \, \left (-4\right )^{\frac {2}{3}} {\left (x^{3} + 1\right )}^{\frac {1}{3}} x^{2} - 6 \, {\left (x^{3} + 1\right )}^{\frac {2}{3}} x + \left (-4\right )^{\frac {1}{3}} {\left (x^{3} - 1\right )}}{x^{3} - 1}\right ) + 25 \, \left (-4\right )^{\frac {1}{3}} {\left (x^{8} - x^{5}\right )} \log \left (-\frac {6 \, \left (-4\right )^{\frac {1}{3}} {\left (5 \, x^{4} + x\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}} - \left (-4\right )^{\frac {2}{3}} {\left (19 \, x^{6} + 16 \, x^{3} + 1\right )} - 24 \, {\left (2 \, x^{5} + x^{2}\right )} {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{x^{6} - 2 \, x^{3} + 1}\right ) + 18 \, {\left (17 \, x^{6} - 10 \, x^{3} - 2\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{90 \, {\left (x^{8} - x^{5}\right )}} \]
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\[ \int \frac {\left (1+x^3\right )^{2/3} \left (2+x^6\right )}{x^6 \left (-1+x^3\right )^2} \, dx=\int \frac {\left (\left (x + 1\right ) \left (x^{2} - x + 1\right )\right )^{\frac {2}{3}} \left (x^{6} + 2\right )}{x^{6} \left (x - 1\right )^{2} \left (x^{2} + x + 1\right )^{2}}\, dx \]
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\[ \int \frac {\left (1+x^3\right )^{2/3} \left (2+x^6\right )}{x^6 \left (-1+x^3\right )^2} \, dx=\int { \frac {{\left (x^{6} + 2\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (x^{3} - 1\right )}^{2} x^{6}} \,d x } \]
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\[ \int \frac {\left (1+x^3\right )^{2/3} \left (2+x^6\right )}{x^6 \left (-1+x^3\right )^2} \, dx=\int { \frac {{\left (x^{6} + 2\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (x^{3} - 1\right )}^{2} x^{6}} \,d x } \]
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Timed out. \[ \int \frac {\left (1+x^3\right )^{2/3} \left (2+x^6\right )}{x^6 \left (-1+x^3\right )^2} \, dx=\int \frac {{\left (x^3+1\right )}^{2/3}\,\left (x^6+2\right )}{x^6\,{\left (x^3-1\right )}^2} \,d x \]
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