Integrand size = 38, antiderivative size = 105 \[ \int \frac {\left (-1+x^6\right )^{2/3} \left (1+x^6\right ) \left (-1-x^3+x^6\right )}{x^6 \left (-1+x^3+x^6\right )} \, dx=\frac {\left (-1+x^6\right )^{2/3} \left (-1-5 x^3+x^6\right )}{5 x^5}-\frac {2 \arctan \left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{-1+x^6}}\right )}{\sqrt {3}}-\frac {2}{3} \log \left (x+\sqrt [3]{-1+x^6}\right )+\frac {1}{3} \log \left (x^2-x \sqrt [3]{-1+x^6}+\left (-1+x^6\right )^{2/3}\right ) \]
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Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 0.69 (sec) , antiderivative size = 380, normalized size of antiderivative = 3.62, number of steps used = 25, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {6860, 252, 251, 372, 371, 281, 1452, 441, 440, 476, 525, 524} \[ \int \frac {\left (-1+x^6\right )^{2/3} \left (1+x^6\right ) \left (-1-x^3+x^6\right )}{x^6 \left (-1+x^3+x^6\right )} \, dx=-\frac {2 \left (1-\sqrt {5}\right ) \left (x^6-1\right )^{2/3} x \operatorname {AppellF1}\left (\frac {1}{6},-\frac {2}{3},1,\frac {7}{6},x^6,\frac {2 x^6}{3-\sqrt {5}}\right )}{\left (3-\sqrt {5}\right ) \left (1-x^6\right )^{2/3}}-\frac {2 \left (1+\sqrt {5}\right ) \left (x^6-1\right )^{2/3} x \operatorname {AppellF1}\left (\frac {1}{6},1,-\frac {2}{3},\frac {7}{6},\frac {2 x^6}{3+\sqrt {5}},x^6\right )}{\left (3+\sqrt {5}\right ) \left (1-x^6\right )^{2/3}}+\frac {\left (x^6-1\right )^{2/3} x^4 \operatorname {AppellF1}\left (\frac {2}{3},-\frac {2}{3},1,\frac {5}{3},x^6,\frac {2 x^6}{3-\sqrt {5}}\right )}{\left (3-\sqrt {5}\right ) \left (1-x^6\right )^{2/3}}+\frac {\left (x^6-1\right )^{2/3} x^4 \operatorname {AppellF1}\left (\frac {2}{3},-\frac {2}{3},1,\frac {5}{3},x^6,\frac {2 x^6}{3+\sqrt {5}}\right )}{\left (3+\sqrt {5}\right ) \left (1-x^6\right )^{2/3}}+\frac {\left (x^6-1\right )^{2/3} x \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},\frac {1}{6},\frac {7}{6},x^6\right )}{\left (1-x^6\right )^{2/3}}-\frac {\left (x^6-1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {5}{6},-\frac {2}{3},\frac {1}{6},x^6\right )}{5 \left (1-x^6\right )^{2/3} x^5}-\frac {\left (x^6-1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{3},\frac {2}{3},x^6\right )}{\left (1-x^6\right )^{2/3} x^2} \]
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Rule 251
Rule 252
Rule 281
Rule 371
Rule 372
Rule 440
Rule 441
Rule 476
Rule 524
Rule 525
Rule 1452
Rule 6860
Rubi steps \begin{align*} \text {integral}& = \int \left (\left (-1+x^6\right )^{2/3}+\frac {\left (-1+x^6\right )^{2/3}}{x^6}+\frac {2 \left (-1+x^6\right )^{2/3}}{x^3}-\frac {2 \left (1+2 x^3\right ) \left (-1+x^6\right )^{2/3}}{-1+x^3+x^6}\right ) \, dx \\ & = 2 \int \frac {\left (-1+x^6\right )^{2/3}}{x^3} \, dx-2 \int \frac {\left (1+2 x^3\right ) \left (-1+x^6\right )^{2/3}}{-1+x^3+x^6} \, dx+\int \left (-1+x^6\right )^{2/3} \, dx+\int \frac {\left (-1+x^6\right )^{2/3}}{x^6} \, dx \\ & = -\left (2 \int \left (\frac {2 \left (-1+x^6\right )^{2/3}}{1-\sqrt {5}+2 x^3}+\frac {2 \left (-1+x^6\right )^{2/3}}{1+\sqrt {5}+2 x^3}\right ) \, dx\right )+\frac {\left (-1+x^6\right )^{2/3} \int \left (1-x^6\right )^{2/3} \, dx}{\left (1-x^6\right )^{2/3}}+\frac {\left (-1+x^6\right )^{2/3} \int \frac {\left (1-x^6\right )^{2/3}}{x^6} \, dx}{\left (1-x^6\right )^{2/3}}+\text {Subst}\left (\int \frac {\left (-1+x^3\right )^{2/3}}{x^2} \, dx,x,x^2\right ) \\ & = -\frac {\left (-1+x^6\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {5}{6},-\frac {2}{3},\frac {1}{6},x^6\right )}{5 x^5 \left (1-x^6\right )^{2/3}}+\frac {x \left (-1+x^6\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},\frac {1}{6},\frac {7}{6},x^6\right )}{\left (1-x^6\right )^{2/3}}-4 \int \frac {\left (-1+x^6\right )^{2/3}}{1-\sqrt {5}+2 x^3} \, dx-4 \int \frac {\left (-1+x^6\right )^{2/3}}{1+\sqrt {5}+2 x^3} \, dx+\frac {\left (-1+x^6\right )^{2/3} \text {Subst}\left (\int \frac {\left (1-x^3\right )^{2/3}}{x^2} \, dx,x,x^2\right )}{\left (1-x^6\right )^{2/3}} \\ & = -\frac {\left (-1+x^6\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {5}{6},-\frac {2}{3},\frac {1}{6},x^6\right )}{5 x^5 \left (1-x^6\right )^{2/3}}-\frac {\left (-1+x^6\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{3},\frac {2}{3},x^6\right )}{x^2 \left (1-x^6\right )^{2/3}}+\frac {x \left (-1+x^6\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},\frac {1}{6},\frac {7}{6},x^6\right )}{\left (1-x^6\right )^{2/3}}-4 \int \left (\frac {\left (1+\sqrt {5}\right ) \left (-1+x^6\right )^{2/3}}{2 \left (3+\sqrt {5}-2 x^6\right )}+\frac {x^3 \left (-1+x^6\right )^{2/3}}{-3-\sqrt {5}+2 x^6}\right ) \, dx-4 \int \left (\frac {\left (-1+\sqrt {5}\right ) \left (-1+x^6\right )^{2/3}}{2 \left (-3+\sqrt {5}+2 x^6\right )}+\frac {x^3 \left (-1+x^6\right )^{2/3}}{-3+\sqrt {5}+2 x^6}\right ) \, dx \\ & = -\frac {\left (-1+x^6\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {5}{6},-\frac {2}{3},\frac {1}{6},x^6\right )}{5 x^5 \left (1-x^6\right )^{2/3}}-\frac {\left (-1+x^6\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{3},\frac {2}{3},x^6\right )}{x^2 \left (1-x^6\right )^{2/3}}+\frac {x \left (-1+x^6\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},\frac {1}{6},\frac {7}{6},x^6\right )}{\left (1-x^6\right )^{2/3}}-4 \int \frac {x^3 \left (-1+x^6\right )^{2/3}}{-3-\sqrt {5}+2 x^6} \, dx-4 \int \frac {x^3 \left (-1+x^6\right )^{2/3}}{-3+\sqrt {5}+2 x^6} \, dx+\left (2 \left (1-\sqrt {5}\right )\right ) \int \frac {\left (-1+x^6\right )^{2/3}}{-3+\sqrt {5}+2 x^6} \, dx-\left (2 \left (1+\sqrt {5}\right )\right ) \int \frac {\left (-1+x^6\right )^{2/3}}{3+\sqrt {5}-2 x^6} \, dx \\ & = -\frac {\left (-1+x^6\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {5}{6},-\frac {2}{3},\frac {1}{6},x^6\right )}{5 x^5 \left (1-x^6\right )^{2/3}}-\frac {\left (-1+x^6\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{3},\frac {2}{3},x^6\right )}{x^2 \left (1-x^6\right )^{2/3}}+\frac {x \left (-1+x^6\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},\frac {1}{6},\frac {7}{6},x^6\right )}{\left (1-x^6\right )^{2/3}}-2 \text {Subst}\left (\int \frac {x \left (-1+x^3\right )^{2/3}}{-3-\sqrt {5}+2 x^3} \, dx,x,x^2\right )-2 \text {Subst}\left (\int \frac {x \left (-1+x^3\right )^{2/3}}{-3+\sqrt {5}+2 x^3} \, dx,x,x^2\right )+\frac {\left (2 \left (1-\sqrt {5}\right ) \left (-1+x^6\right )^{2/3}\right ) \int \frac {\left (1-x^6\right )^{2/3}}{-3+\sqrt {5}+2 x^6} \, dx}{\left (1-x^6\right )^{2/3}}-\frac {\left (2 \left (1+\sqrt {5}\right ) \left (-1+x^6\right )^{2/3}\right ) \int \frac {\left (1-x^6\right )^{2/3}}{3+\sqrt {5}-2 x^6} \, dx}{\left (1-x^6\right )^{2/3}} \\ & = -\frac {2 \left (1-\sqrt {5}\right ) x \left (-1+x^6\right )^{2/3} \operatorname {AppellF1}\left (\frac {1}{6},-\frac {2}{3},1,\frac {7}{6},x^6,\frac {2 x^6}{3-\sqrt {5}}\right )}{\left (3-\sqrt {5}\right ) \left (1-x^6\right )^{2/3}}-\frac {2 \left (1+\sqrt {5}\right ) x \left (-1+x^6\right )^{2/3} \operatorname {AppellF1}\left (\frac {1}{6},1,-\frac {2}{3},\frac {7}{6},\frac {2 x^6}{3+\sqrt {5}},x^6\right )}{\left (3+\sqrt {5}\right ) \left (1-x^6\right )^{2/3}}-\frac {\left (-1+x^6\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {5}{6},-\frac {2}{3},\frac {1}{6},x^6\right )}{5 x^5 \left (1-x^6\right )^{2/3}}-\frac {\left (-1+x^6\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{3},\frac {2}{3},x^6\right )}{x^2 \left (1-x^6\right )^{2/3}}+\frac {x \left (-1+x^6\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},\frac {1}{6},\frac {7}{6},x^6\right )}{\left (1-x^6\right )^{2/3}}-\frac {\left (2 \left (-1+x^6\right )^{2/3}\right ) \text {Subst}\left (\int \frac {x \left (1-x^3\right )^{2/3}}{-3-\sqrt {5}+2 x^3} \, dx,x,x^2\right )}{\left (1-x^6\right )^{2/3}}-\frac {\left (2 \left (-1+x^6\right )^{2/3}\right ) \text {Subst}\left (\int \frac {x \left (1-x^3\right )^{2/3}}{-3+\sqrt {5}+2 x^3} \, dx,x,x^2\right )}{\left (1-x^6\right )^{2/3}} \\ & = -\frac {2 \left (1-\sqrt {5}\right ) x \left (-1+x^6\right )^{2/3} \operatorname {AppellF1}\left (\frac {1}{6},-\frac {2}{3},1,\frac {7}{6},x^6,\frac {2 x^6}{3-\sqrt {5}}\right )}{\left (3-\sqrt {5}\right ) \left (1-x^6\right )^{2/3}}-\frac {2 \left (1+\sqrt {5}\right ) x \left (-1+x^6\right )^{2/3} \operatorname {AppellF1}\left (\frac {1}{6},1,-\frac {2}{3},\frac {7}{6},\frac {2 x^6}{3+\sqrt {5}},x^6\right )}{\left (3+\sqrt {5}\right ) \left (1-x^6\right )^{2/3}}+\frac {x^4 \left (-1+x^6\right )^{2/3} \operatorname {AppellF1}\left (\frac {2}{3},-\frac {2}{3},1,\frac {5}{3},x^6,\frac {2 x^6}{3-\sqrt {5}}\right )}{\left (3-\sqrt {5}\right ) \left (1-x^6\right )^{2/3}}+\frac {x^4 \left (-1+x^6\right )^{2/3} \operatorname {AppellF1}\left (\frac {2}{3},-\frac {2}{3},1,\frac {5}{3},x^6,\frac {2 x^6}{3+\sqrt {5}}\right )}{\left (3+\sqrt {5}\right ) \left (1-x^6\right )^{2/3}}-\frac {\left (-1+x^6\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {5}{6},-\frac {2}{3},\frac {1}{6},x^6\right )}{5 x^5 \left (1-x^6\right )^{2/3}}-\frac {\left (-1+x^6\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{3},\frac {2}{3},x^6\right )}{x^2 \left (1-x^6\right )^{2/3}}+\frac {x \left (-1+x^6\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},\frac {1}{6},\frac {7}{6},x^6\right )}{\left (1-x^6\right )^{2/3}} \\ \end{align*}
Time = 1.80 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-1+x^6\right )^{2/3} \left (1+x^6\right ) \left (-1-x^3+x^6\right )}{x^6 \left (-1+x^3+x^6\right )} \, dx=\frac {\left (-1+x^6\right )^{2/3} \left (-1-5 x^3+x^6\right )}{5 x^5}-\frac {2 \arctan \left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{-1+x^6}}\right )}{\sqrt {3}}-\frac {2}{3} \log \left (x+\sqrt [3]{-1+x^6}\right )+\frac {1}{3} \log \left (x^2-x \sqrt [3]{-1+x^6}+\left (-1+x^6\right )^{2/3}\right ) \]
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Time = 16.60 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.99
method | result | size |
pseudoelliptic | \(\frac {-10 \ln \left (\frac {x +\left (x^{6}-1\right )^{\frac {1}{3}}}{x}\right ) x^{5}+3 \left (x^{6}-5 x^{3}-1\right ) \left (x^{6}-1\right )^{\frac {2}{3}}-5 x^{5} \left (2 \sqrt {3}\, \arctan \left (\frac {\left (x -2 \left (x^{6}-1\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 x}\right )-\ln \left (\frac {x^{2}-x \left (x^{6}-1\right )^{\frac {1}{3}}+\left (x^{6}-1\right )^{\frac {2}{3}}}{x^{2}}\right )\right )}{15 x^{5}}\) | \(104\) |
risch | \(\frac {x^{12}-5 x^{9}-2 x^{6}+5 x^{3}+1}{5 x^{5} \left (x^{6}-1\right )^{\frac {1}{3}}}-\frac {2 \ln \left (\frac {3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{6}-2 x^{6}-9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{3}+9 \left (x^{6}-1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x +9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}} x^{2}+6 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{3}-3 x \left (x^{6}-1\right )^{\frac {2}{3}}-3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )+2}{x^{6}+x^{3}-1}\right )}{3}+\frac {2 \ln \left (-\frac {3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{6}-2 x^{6}+9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{3}+9 \left (x^{6}-1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x -9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{3}+3 x^{2} \left (x^{6}-1\right )^{\frac {1}{3}}+2 x^{3}-3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )+2}{x^{6}+x^{3}-1}\right )}{3}-2 \ln \left (-\frac {3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{6}-2 x^{6}+9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{3}+9 \left (x^{6}-1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x -9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{3}+3 x^{2} \left (x^{6}-1\right )^{\frac {1}{3}}+2 x^{3}-3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )+2}{x^{6}+x^{3}-1}\right ) \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )\) | \(429\) |
trager | \(\text {Expression too large to display}\) | \(612\) |
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Time = 10.98 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.35 \[ \int \frac {\left (-1+x^6\right )^{2/3} \left (1+x^6\right ) \left (-1-x^3+x^6\right )}{x^6 \left (-1+x^3+x^6\right )} \, dx=-\frac {10 \, \sqrt {3} x^{5} \arctan \left (\frac {17707979315346691547103487078601066282657059082726673278841963389300888497059669011634 \, \sqrt {3} {\left (x^{6} - 1\right )}^{\frac {1}{3}} x^{2} + 18779074824464902023518972945875034013564452605964125877184864112405550428883609929964 \, \sqrt {3} {\left (x^{6} - 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (8791266734992875261237504664599259772605087326251698970792557525513888268399719816592 \, x^{6} + 9326814489551980499445247598236243638058784087870425269964007887066219234311690275757 \, x^{3} - 8791266734992875261237504664599259772605087326251698970792557525513888268399719816592\right )}}{3 \, {\left (9923243904393545413458713816471868889492119646716071835561526356358143878603884871272 \, x^{6} - 8320283165512251371852516195766181258618636197629327742451887394495075584367754599527 \, x^{3} - 9923243904393545413458713816471868889492119646716071835561526356358143878603884871272\right )}}\right ) + 5 \, x^{5} \log \left (\frac {x^{6} + x^{3} + 3 \, {\left (x^{6} - 1\right )}^{\frac {1}{3}} x^{2} + 3 \, {\left (x^{6} - 1\right )}^{\frac {2}{3}} x - 1}{x^{6} + x^{3} - 1}\right ) - 3 \, {\left (x^{6} - 5 \, x^{3} - 1\right )} {\left (x^{6} - 1\right )}^{\frac {2}{3}}}{15 \, x^{5}} \]
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Timed out. \[ \int \frac {\left (-1+x^6\right )^{2/3} \left (1+x^6\right ) \left (-1-x^3+x^6\right )}{x^6 \left (-1+x^3+x^6\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (-1+x^6\right )^{2/3} \left (1+x^6\right ) \left (-1-x^3+x^6\right )}{x^6 \left (-1+x^3+x^6\right )} \, dx=\int { \frac {{\left (x^{6} - x^{3} - 1\right )} {\left (x^{6} + 1\right )} {\left (x^{6} - 1\right )}^{\frac {2}{3}}}{{\left (x^{6} + x^{3} - 1\right )} x^{6}} \,d x } \]
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\[ \int \frac {\left (-1+x^6\right )^{2/3} \left (1+x^6\right ) \left (-1-x^3+x^6\right )}{x^6 \left (-1+x^3+x^6\right )} \, dx=\int { \frac {{\left (x^{6} - x^{3} - 1\right )} {\left (x^{6} + 1\right )} {\left (x^{6} - 1\right )}^{\frac {2}{3}}}{{\left (x^{6} + x^{3} - 1\right )} x^{6}} \,d x } \]
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Timed out. \[ \int \frac {\left (-1+x^6\right )^{2/3} \left (1+x^6\right ) \left (-1-x^3+x^6\right )}{x^6 \left (-1+x^3+x^6\right )} \, dx=\int -\frac {{\left (x^6-1\right )}^{2/3}\,\left (x^6+1\right )\,\left (-x^6+x^3+1\right )}{x^6\,\left (x^6+x^3-1\right )} \,d x \]
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