Optimal. Leaf size=122 \[ \frac {5}{243} \log \left (\sqrt [3]{x^3-x}-x\right )+\frac {5 \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3-x}+x}\right )}{81 \sqrt {3}}-\frac {5}{486} \log \left (\sqrt [3]{x^3-x} x+\left (x^3-x\right )^{2/3}+x^2\right )+\frac {1}{648} \sqrt [3]{x^3-x} \left (81 x^7-9 x^5-12 x^3-20 x\right ) \]
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Rubi [A] time = 0.26, antiderivative size = 240, normalized size of antiderivative = 1.97, number of steps used = 14, number of rules used = 12, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {2021, 2024, 2032, 329, 275, 331, 292, 31, 634, 618, 204, 628} \begin {gather*} -\frac {1}{54} \sqrt [3]{x^3-x} x^3-\frac {5}{162} \sqrt [3]{x^3-x} x+\frac {1}{8} \sqrt [3]{x^3-x} x^7-\frac {1}{72} \sqrt [3]{x^3-x} x^5+\frac {5 \left (x^2-1\right )^{2/3} x^{2/3} \log \left (1-\frac {x^{2/3}}{\sqrt [3]{x^2-1}}\right )}{243 \left (x^3-x\right )^{2/3}}-\frac {5 \left (x^2-1\right )^{2/3} x^{2/3} \log \left (\frac {x^{4/3}}{\left (x^2-1\right )^{2/3}}+\frac {x^{2/3}}{\sqrt [3]{x^2-1}}+1\right )}{486 \left (x^3-x\right )^{2/3}}+\frac {5 \left (x^2-1\right )^{2/3} x^{2/3} \tan ^{-1}\left (\frac {\frac {2 x^{2/3}}{\sqrt [3]{x^2-1}}+1}{\sqrt {3}}\right )}{81 \sqrt {3} \left (x^3-x\right )^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 204
Rule 275
Rule 292
Rule 329
Rule 331
Rule 618
Rule 628
Rule 634
Rule 2021
Rule 2024
Rule 2032
Rubi steps
\begin {align*} \int x^6 \sqrt [3]{-x+x^3} \, dx &=\frac {1}{8} x^7 \sqrt [3]{-x+x^3}-\frac {1}{12} \int \frac {x^7}{\left (-x+x^3\right )^{2/3}} \, dx\\ &=-\frac {1}{72} x^5 \sqrt [3]{-x+x^3}+\frac {1}{8} x^7 \sqrt [3]{-x+x^3}-\frac {2}{27} \int \frac {x^5}{\left (-x+x^3\right )^{2/3}} \, dx\\ &=-\frac {1}{54} x^3 \sqrt [3]{-x+x^3}-\frac {1}{72} x^5 \sqrt [3]{-x+x^3}+\frac {1}{8} x^7 \sqrt [3]{-x+x^3}-\frac {5}{81} \int \frac {x^3}{\left (-x+x^3\right )^{2/3}} \, dx\\ &=-\frac {5}{162} x \sqrt [3]{-x+x^3}-\frac {1}{54} x^3 \sqrt [3]{-x+x^3}-\frac {1}{72} x^5 \sqrt [3]{-x+x^3}+\frac {1}{8} x^7 \sqrt [3]{-x+x^3}-\frac {10}{243} \int \frac {x}{\left (-x+x^3\right )^{2/3}} \, dx\\ &=-\frac {5}{162} x \sqrt [3]{-x+x^3}-\frac {1}{54} x^3 \sqrt [3]{-x+x^3}-\frac {1}{72} x^5 \sqrt [3]{-x+x^3}+\frac {1}{8} x^7 \sqrt [3]{-x+x^3}-\frac {\left (10 x^{2/3} \left (-1+x^2\right )^{2/3}\right ) \int \frac {\sqrt [3]{x}}{\left (-1+x^2\right )^{2/3}} \, dx}{243 \left (-x+x^3\right )^{2/3}}\\ &=-\frac {5}{162} x \sqrt [3]{-x+x^3}-\frac {1}{54} x^3 \sqrt [3]{-x+x^3}-\frac {1}{72} x^5 \sqrt [3]{-x+x^3}+\frac {1}{8} x^7 \sqrt [3]{-x+x^3}-\frac {\left (10 x^{2/3} \left (-1+x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x^3}{\left (-1+x^6\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{81 \left (-x+x^3\right )^{2/3}}\\ &=-\frac {5}{162} x \sqrt [3]{-x+x^3}-\frac {1}{54} x^3 \sqrt [3]{-x+x^3}-\frac {1}{72} x^5 \sqrt [3]{-x+x^3}+\frac {1}{8} x^7 \sqrt [3]{-x+x^3}-\frac {\left (5 x^{2/3} \left (-1+x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x}{\left (-1+x^3\right )^{2/3}} \, dx,x,x^{2/3}\right )}{81 \left (-x+x^3\right )^{2/3}}\\ &=-\frac {5}{162} x \sqrt [3]{-x+x^3}-\frac {1}{54} x^3 \sqrt [3]{-x+x^3}-\frac {1}{72} x^5 \sqrt [3]{-x+x^3}+\frac {1}{8} x^7 \sqrt [3]{-x+x^3}-\frac {\left (5 x^{2/3} \left (-1+x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x}{1-x^3} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{81 \left (-x+x^3\right )^{2/3}}\\ &=-\frac {5}{162} x \sqrt [3]{-x+x^3}-\frac {1}{54} x^3 \sqrt [3]{-x+x^3}-\frac {1}{72} x^5 \sqrt [3]{-x+x^3}+\frac {1}{8} x^7 \sqrt [3]{-x+x^3}-\frac {\left (5 x^{2/3} \left (-1+x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{243 \left (-x+x^3\right )^{2/3}}+\frac {\left (5 x^{2/3} \left (-1+x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1-x}{1+x+x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{243 \left (-x+x^3\right )^{2/3}}\\ &=-\frac {5}{162} x \sqrt [3]{-x+x^3}-\frac {1}{54} x^3 \sqrt [3]{-x+x^3}-\frac {1}{72} x^5 \sqrt [3]{-x+x^3}+\frac {1}{8} x^7 \sqrt [3]{-x+x^3}+\frac {5 x^{2/3} \left (-1+x^2\right )^{2/3} \log \left (1-\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{243 \left (-x+x^3\right )^{2/3}}-\frac {\left (5 x^{2/3} \left (-1+x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1+2 x}{1+x+x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{486 \left (-x+x^3\right )^{2/3}}+\frac {\left (5 x^{2/3} \left (-1+x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{162 \left (-x+x^3\right )^{2/3}}\\ &=-\frac {5}{162} x \sqrt [3]{-x+x^3}-\frac {1}{54} x^3 \sqrt [3]{-x+x^3}-\frac {1}{72} x^5 \sqrt [3]{-x+x^3}+\frac {1}{8} x^7 \sqrt [3]{-x+x^3}+\frac {5 x^{2/3} \left (-1+x^2\right )^{2/3} \log \left (1-\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{243 \left (-x+x^3\right )^{2/3}}-\frac {5 x^{2/3} \left (-1+x^2\right )^{2/3} \log \left (1+\frac {x^{4/3}}{\left (-1+x^2\right )^{2/3}}+\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{486 \left (-x+x^3\right )^{2/3}}-\frac {\left (5 x^{2/3} \left (-1+x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{81 \left (-x+x^3\right )^{2/3}}\\ &=-\frac {5}{162} x \sqrt [3]{-x+x^3}-\frac {1}{54} x^3 \sqrt [3]{-x+x^3}-\frac {1}{72} x^5 \sqrt [3]{-x+x^3}+\frac {1}{8} x^7 \sqrt [3]{-x+x^3}+\frac {5 x^{2/3} \left (-1+x^2\right )^{2/3} \tan ^{-1}\left (\frac {1+\frac {2 x^{2/3}}{\sqrt [3]{-1+x^2}}}{\sqrt {3}}\right )}{81 \sqrt {3} \left (-x+x^3\right )^{2/3}}+\frac {5 x^{2/3} \left (-1+x^2\right )^{2/3} \log \left (1-\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{243 \left (-x+x^3\right )^{2/3}}-\frac {5 x^{2/3} \left (-1+x^2\right )^{2/3} \log \left (1+\frac {x^{4/3}}{\left (-1+x^2\right )^{2/3}}+\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{486 \left (-x+x^3\right )^{2/3}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 72, normalized size = 0.59 \begin {gather*} \frac {x \sqrt [3]{x \left (x^2-1\right )} \left (20 \, _2F_1\left (-\frac {1}{3},\frac {2}{3};\frac {5}{3};x^2\right )+\sqrt [3]{1-x^2} \left (27 x^6-3 x^4-4 x^2-20\right )\right )}{216 \sqrt [3]{1-x^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.42, size = 122, normalized size = 1.00 \begin {gather*} \frac {1}{648} \sqrt [3]{-x+x^3} \left (-20 x-12 x^3-9 x^5+81 x^7\right )+\frac {5 \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-x+x^3}}\right )}{81 \sqrt {3}}+\frac {5}{243} \log \left (-x+\sqrt [3]{-x+x^3}\right )-\frac {5}{486} \log \left (x^2+x \sqrt [3]{-x+x^3}+\left (-x+x^3\right )^{2/3}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.86, size = 117, normalized size = 0.96 \begin {gather*} \frac {5}{243} \, \sqrt {3} \arctan \left (-\frac {44032959556 \, \sqrt {3} {\left (x^{3} - x\right )}^{\frac {1}{3}} x + \sqrt {3} {\left (16754327161 \, x^{2} - 2707204793\right )} - 10524305234 \, \sqrt {3} {\left (x^{3} - x\right )}^{\frac {2}{3}}}{81835897185 \, x^{2} - 1102302937}\right ) + \frac {1}{648} \, {\left (81 \, x^{7} - 9 \, x^{5} - 12 \, x^{3} - 20 \, x\right )} {\left (x^{3} - x\right )}^{\frac {1}{3}} + \frac {5}{486} \, \log \left (-3 \, {\left (x^{3} - x\right )}^{\frac {1}{3}} x + 3 \, {\left (x^{3} - x\right )}^{\frac {2}{3}} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 127, normalized size = 1.04 \begin {gather*} -\frac {1}{648} \, {\left (20 \, {\left (\frac {1}{x^{2}} - 1\right )}^{3} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + 72 \, {\left (\frac {1}{x^{2}} - 1\right )}^{2} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} - 93 \, {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {4}{3}} - 40 \, {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}}\right )} x^{8} - \frac {5}{243} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {5}{486} \, \log \left ({\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {2}{3}} + {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {5}{243} \, \log \left ({\left | {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 3.10, size = 33, normalized size = 0.27
method | result | size |
meijerg | \(\frac {3 \mathrm {signum}\left (x^{2}-1\right )^{\frac {1}{3}} x^{\frac {22}{3}} \hypergeom \left (\left [-\frac {1}{3}, \frac {11}{3}\right ], \left [\frac {14}{3}\right ], x^{2}\right )}{22 \left (-\mathrm {signum}\left (x^{2}-1\right )\right )^{\frac {1}{3}}}\) | \(33\) |
trager | \(\frac {x \left (81 x^{6}-9 x^{4}-12 x^{2}-20\right ) \left (x^{3}-x \right )^{\frac {1}{3}}}{648}+\frac {5 \ln \left (4959 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{2}+6768 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {2}{3}}-22833 \left (x^{3}-x \right )^{\frac {1}{3}} \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x +17718 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}-19836 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}+7611 \left (x^{3}-x \right )^{\frac {2}{3}}-5355 x \left (x^{3}-x \right )^{\frac {1}{3}}-1705 x^{2}-9711 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+1085\right )}{243}+\frac {5 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \ln \left (-6354 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{2}-6768 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {2}{3}}-16065 \left (x^{3}-x \right )^{\frac {1}{3}} \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x +20715 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}+25416 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}+5355 \left (x^{3}-x \right )^{\frac {2}{3}}-7611 x \left (x^{3}-x \right )^{\frac {1}{3}}+1550 x^{2}-4494 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-465\right )}{81}\) | \(322\) |
risch | \(\frac {x \left (81 x^{6}-9 x^{4}-12 x^{2}-20\right ) \left (x \left (x^{2}-1\right )\right )^{\frac {1}{3}}}{648}+\frac {\left (\frac {5 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) \ln \left (-\frac {47 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )^{2} x^{4}+3207 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) x^{4}-235 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )^{2} x^{2}+2925 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {1}{3}} x^{2}+6930 x^{4}+2925 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {2}{3}}-5601 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) x^{2}+5238 \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {1}{3}} x^{2}+188 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )^{2}-2925 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {1}{3}}+5238 \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {2}{3}}-11340 x^{2}+2394 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )-5238 \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {1}{3}}+4410}{\left (-1+x \right ) \left (1+x \right )}\right )}{1458}-\frac {5 \ln \left (-\frac {47 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )^{2} x^{4}-2643 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) x^{4}-235 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )^{2} x^{2}-2925 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {1}{3}} x^{2}-10620 x^{4}-2925 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {2}{3}}+2781 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) x^{2}-12312 \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {1}{3}} x^{2}+188 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )^{2}+2925 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {1}{3}}-12312 \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {2}{3}}+13806 x^{2}-138 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )+12312 \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {1}{3}}-3186}{\left (-1+x \right ) \left (1+x \right )}\right ) \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )}{1458}-\frac {5 \ln \left (-\frac {47 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )^{2} x^{4}-2643 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) x^{4}-235 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )^{2} x^{2}-2925 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {1}{3}} x^{2}-10620 x^{4}-2925 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {2}{3}}+2781 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) x^{2}-12312 \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {1}{3}} x^{2}+188 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )^{2}+2925 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {1}{3}}-12312 \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {2}{3}}+13806 x^{2}-138 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )+12312 \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {1}{3}}-3186}{\left (-1+x \right ) \left (1+x \right )}\right )}{243}\right ) \left (x \left (x^{2}-1\right )\right )^{\frac {1}{3}} \left (x^{2} \left (x^{2}-1\right )^{2}\right )^{\frac {1}{3}}}{x \left (x^{2}-1\right )}\) | \(803\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int {\left (x^{3} - x\right )}^{\frac {1}{3}} x^{6}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^6\,{\left (x^3-x\right )}^{1/3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{6} \sqrt [3]{x \left (x - 1\right ) \left (x + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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