Optimal. Leaf size=175 \[ -\frac {1}{2} \left (\frac {3}{2}\right )^{2/3} \log \left (\sqrt [3]{2} 3^{2/3} \sqrt [3]{x^3-x+1}-3 x\right )+\frac {3 \sqrt [6]{3} \tan ^{-1}\left (\frac {3^{5/6} x}{2 \sqrt [3]{2} \sqrt [3]{x^3-x+1}+\sqrt [3]{3} x}\right )}{2\ 2^{2/3}}-\frac {3 \left (x^3-x+1\right )^{2/3}}{4 x^2}+\frac {1}{4} \left (\frac {3}{2}\right )^{2/3} \log \left (\sqrt [3]{2} 3^{2/3} \sqrt [3]{x^3-x+1} x+2^{2/3} \sqrt [3]{3} \left (x^3-x+1\right )^{2/3}+3 x^2\right ) \]
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Rubi [F] time = 2.14, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {(-3+2 x) \left (1-x+x^3\right )^{2/3}}{x^3 \left (-2+2 x+x^3\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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\begin {align*} \int \frac {(-3+2 x) \left (1-x+x^3\right )^{2/3}}{x^3 \left (-2+2 x+x^3\right )} \, dx &=\int \left (\frac {3 \left (1-x+x^3\right )^{2/3}}{2 x^3}+\frac {\left (1-x+x^3\right )^{2/3}}{2 x^2}+\frac {\left (1-x+x^3\right )^{2/3}}{2 x}+\frac {\left (-5-x-x^2\right ) \left (1-x+x^3\right )^{2/3}}{2 \left (-2+2 x+x^3\right )}\right ) \, dx\\ &=\frac {1}{2} \int \frac {\left (1-x+x^3\right )^{2/3}}{x^2} \, dx+\frac {1}{2} \int \frac {\left (1-x+x^3\right )^{2/3}}{x} \, dx+\frac {1}{2} \int \frac {\left (-5-x-x^2\right ) \left (1-x+x^3\right )^{2/3}}{-2+2 x+x^3} \, dx+\frac {3}{2} \int \frac {\left (1-x+x^3\right )^{2/3}}{x^3} \, dx\\ &=\frac {1}{2} \int \left (-\frac {5 \left (1-x+x^3\right )^{2/3}}{-2+2 x+x^3}-\frac {x \left (1-x+x^3\right )^{2/3}}{-2+2 x+x^3}-\frac {x^2 \left (1-x+x^3\right )^{2/3}}{-2+2 x+x^3}\right ) \, dx+\frac {\left (1-x+x^3\right )^{2/3} \int \frac {\left (\frac {\sqrt [3]{18-2 \sqrt {69}}+\sqrt [3]{2 \left (9+\sqrt {69}\right )}}{6^{2/3}}+x\right )^{2/3} \left (\frac {1}{18} \left (-6+\sqrt [3]{2} \left (27-3 \sqrt {69}\right )^{2/3}+6 \sqrt [3]{3} \left (\frac {2}{9-\sqrt {69}}\right )^{2/3}\right )-\frac {\left (\sqrt [3]{9-\sqrt {69}}+\sqrt [3]{9+\sqrt {69}}\right ) x}{\sqrt [3]{2} 3^{2/3}}+x^2\right )^{2/3}}{x^2} \, dx}{2 \left (\frac {\sqrt [3]{18-2 \sqrt {69}}+\sqrt [3]{2 \left (9+\sqrt {69}\right )}}{6^{2/3}}+x\right )^{2/3} \left (\frac {1}{18} \left (-6+\sqrt [3]{2} \left (27-3 \sqrt {69}\right )^{2/3}+6 \sqrt [3]{3} \left (\frac {2}{9-\sqrt {69}}\right )^{2/3}\right )-\frac {\left (\sqrt [3]{9-\sqrt {69}}+\sqrt [3]{9+\sqrt {69}}\right ) x}{\sqrt [3]{2} 3^{2/3}}+x^2\right )^{2/3}}+\frac {\left (1-x+x^3\right )^{2/3} \int \frac {\left (\frac {\sqrt [3]{18-2 \sqrt {69}}+\sqrt [3]{2 \left (9+\sqrt {69}\right )}}{6^{2/3}}+x\right )^{2/3} \left (\frac {1}{18} \left (-6+\sqrt [3]{2} \left (27-3 \sqrt {69}\right )^{2/3}+6 \sqrt [3]{3} \left (\frac {2}{9-\sqrt {69}}\right )^{2/3}\right )-\frac {\left (\sqrt [3]{9-\sqrt {69}}+\sqrt [3]{9+\sqrt {69}}\right ) x}{\sqrt [3]{2} 3^{2/3}}+x^2\right )^{2/3}}{x} \, dx}{2 \left (\frac {\sqrt [3]{18-2 \sqrt {69}}+\sqrt [3]{2 \left (9+\sqrt {69}\right )}}{6^{2/3}}+x\right )^{2/3} \left (\frac {1}{18} \left (-6+\sqrt [3]{2} \left (27-3 \sqrt {69}\right )^{2/3}+6 \sqrt [3]{3} \left (\frac {2}{9-\sqrt {69}}\right )^{2/3}\right )-\frac {\left (\sqrt [3]{9-\sqrt {69}}+\sqrt [3]{9+\sqrt {69}}\right ) x}{\sqrt [3]{2} 3^{2/3}}+x^2\right )^{2/3}}+\frac {\left (3 \left (1-x+x^3\right )^{2/3}\right ) \int \frac {\left (\frac {\sqrt [3]{18-2 \sqrt {69}}+\sqrt [3]{2 \left (9+\sqrt {69}\right )}}{6^{2/3}}+x\right )^{2/3} \left (\frac {1}{18} \left (-6+\sqrt [3]{2} \left (27-3 \sqrt {69}\right )^{2/3}+6 \sqrt [3]{3} \left (\frac {2}{9-\sqrt {69}}\right )^{2/3}\right )-\frac {\left (\sqrt [3]{9-\sqrt {69}}+\sqrt [3]{9+\sqrt {69}}\right ) x}{\sqrt [3]{2} 3^{2/3}}+x^2\right )^{2/3}}{x^3} \, dx}{2 \left (\frac {\sqrt [3]{18-2 \sqrt {69}}+\sqrt [3]{2 \left (9+\sqrt {69}\right )}}{6^{2/3}}+x\right )^{2/3} \left (\frac {1}{18} \left (-6+\sqrt [3]{2} \left (27-3 \sqrt {69}\right )^{2/3}+6 \sqrt [3]{3} \left (\frac {2}{9-\sqrt {69}}\right )^{2/3}\right )-\frac {\left (\sqrt [3]{9-\sqrt {69}}+\sqrt [3]{9+\sqrt {69}}\right ) x}{\sqrt [3]{2} 3^{2/3}}+x^2\right )^{2/3}}\\ &=-\left (\frac {1}{2} \int \frac {x \left (1-x+x^3\right )^{2/3}}{-2+2 x+x^3} \, dx\right )-\frac {1}{2} \int \frac {x^2 \left (1-x+x^3\right )^{2/3}}{-2+2 x+x^3} \, dx-\frac {5}{2} \int \frac {\left (1-x+x^3\right )^{2/3}}{-2+2 x+x^3} \, dx+\frac {\left (1-x+x^3\right )^{2/3} \int \frac {\left (\frac {\sqrt [3]{18-2 \sqrt {69}}+\sqrt [3]{2 \left (9+\sqrt {69}\right )}}{6^{2/3}}+x\right )^{2/3} \left (\frac {1}{18} \left (-6+\sqrt [3]{2} \left (27-3 \sqrt {69}\right )^{2/3}+6 \sqrt [3]{3} \left (\frac {2}{9-\sqrt {69}}\right )^{2/3}\right )-\frac {\left (\sqrt [3]{9-\sqrt {69}}+\sqrt [3]{9+\sqrt {69}}\right ) x}{\sqrt [3]{2} 3^{2/3}}+x^2\right )^{2/3}}{x^2} \, dx}{2 \left (\frac {\sqrt [3]{18-2 \sqrt {69}}+\sqrt [3]{2 \left (9+\sqrt {69}\right )}}{6^{2/3}}+x\right )^{2/3} \left (\frac {1}{18} \left (-6+\sqrt [3]{2} \left (27-3 \sqrt {69}\right )^{2/3}+6 \sqrt [3]{3} \left (\frac {2}{9-\sqrt {69}}\right )^{2/3}\right )-\frac {\left (\sqrt [3]{9-\sqrt {69}}+\sqrt [3]{9+\sqrt {69}}\right ) x}{\sqrt [3]{2} 3^{2/3}}+x^2\right )^{2/3}}+\frac {\left (1-x+x^3\right )^{2/3} \int \frac {\left (\frac {\sqrt [3]{18-2 \sqrt {69}}+\sqrt [3]{2 \left (9+\sqrt {69}\right )}}{6^{2/3}}+x\right )^{2/3} \left (\frac {1}{18} \left (-6+\sqrt [3]{2} \left (27-3 \sqrt {69}\right )^{2/3}+6 \sqrt [3]{3} \left (\frac {2}{9-\sqrt {69}}\right )^{2/3}\right )-\frac {\left (\sqrt [3]{9-\sqrt {69}}+\sqrt [3]{9+\sqrt {69}}\right ) x}{\sqrt [3]{2} 3^{2/3}}+x^2\right )^{2/3}}{x} \, dx}{2 \left (\frac {\sqrt [3]{18-2 \sqrt {69}}+\sqrt [3]{2 \left (9+\sqrt {69}\right )}}{6^{2/3}}+x\right )^{2/3} \left (\frac {1}{18} \left (-6+\sqrt [3]{2} \left (27-3 \sqrt {69}\right )^{2/3}+6 \sqrt [3]{3} \left (\frac {2}{9-\sqrt {69}}\right )^{2/3}\right )-\frac {\left (\sqrt [3]{9-\sqrt {69}}+\sqrt [3]{9+\sqrt {69}}\right ) x}{\sqrt [3]{2} 3^{2/3}}+x^2\right )^{2/3}}+\frac {\left (3 \left (1-x+x^3\right )^{2/3}\right ) \int \frac {\left (\frac {\sqrt [3]{18-2 \sqrt {69}}+\sqrt [3]{2 \left (9+\sqrt {69}\right )}}{6^{2/3}}+x\right )^{2/3} \left (\frac {1}{18} \left (-6+\sqrt [3]{2} \left (27-3 \sqrt {69}\right )^{2/3}+6 \sqrt [3]{3} \left (\frac {2}{9-\sqrt {69}}\right )^{2/3}\right )-\frac {\left (\sqrt [3]{9-\sqrt {69}}+\sqrt [3]{9+\sqrt {69}}\right ) x}{\sqrt [3]{2} 3^{2/3}}+x^2\right )^{2/3}}{x^3} \, dx}{2 \left (\frac {\sqrt [3]{18-2 \sqrt {69}}+\sqrt [3]{2 \left (9+\sqrt {69}\right )}}{6^{2/3}}+x\right )^{2/3} \left (\frac {1}{18} \left (-6+\sqrt [3]{2} \left (27-3 \sqrt {69}\right )^{2/3}+6 \sqrt [3]{3} \left (\frac {2}{9-\sqrt {69}}\right )^{2/3}\right )-\frac {\left (\sqrt [3]{9-\sqrt {69}}+\sqrt [3]{9+\sqrt {69}}\right ) x}{\sqrt [3]{2} 3^{2/3}}+x^2\right )^{2/3}}\\ \end {align*}
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Mathematica [F] time = 0.34, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(-3+2 x) \left (1-x+x^3\right )^{2/3}}{x^3 \left (-2+2 x+x^3\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.58, size = 175, normalized size = 1.00 \begin {gather*} -\frac {3 \left (1-x+x^3\right )^{2/3}}{4 x^2}+\frac {3 \sqrt [6]{3} \tan ^{-1}\left (\frac {3^{5/6} x}{\sqrt [3]{3} x+2 \sqrt [3]{2} \sqrt [3]{1-x+x^3}}\right )}{2\ 2^{2/3}}-\frac {1}{2} \left (\frac {3}{2}\right )^{2/3} \log \left (-3 x+\sqrt [3]{2} 3^{2/3} \sqrt [3]{1-x+x^3}\right )+\frac {1}{4} \left (\frac {3}{2}\right )^{2/3} \log \left (3 x^2+\sqrt [3]{2} 3^{2/3} x \sqrt [3]{1-x+x^3}+2^{2/3} \sqrt [3]{3} \left (1-x+x^3\right )^{2/3}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 15.74, size = 426, normalized size = 2.43 \begin {gather*} -\frac {4 \cdot 4^{\frac {1}{6}} \sqrt {3} \left (-9\right )^{\frac {1}{3}} x^{2} \arctan \left (\frac {4^{\frac {1}{6}} \sqrt {3} {\left (4 \cdot 4^{\frac {2}{3}} \left (-9\right )^{\frac {2}{3}} {\left (4 \, x^{7} + 7 \, x^{5} - 7 \, x^{4} - 2 \, x^{3} + 4 \, x^{2} - 2 \, x\right )} {\left (x^{3} - x + 1\right )}^{\frac {2}{3}} + 12 \, \left (-9\right )^{\frac {1}{3}} {\left (55 \, x^{8} - 50 \, x^{6} + 50 \, x^{5} + 4 \, x^{4} - 8 \, x^{3} + 4 \, x^{2}\right )} {\left (x^{3} - x + 1\right )}^{\frac {1}{3}} - 4^{\frac {1}{3}} {\left (377 \, x^{9} - 600 \, x^{7} + 600 \, x^{6} + 204 \, x^{5} - 408 \, x^{4} + 196 \, x^{3} + 24 \, x^{2} - 24 \, x + 8\right )}\right )}}{6 \, {\left (487 \, x^{9} - 480 \, x^{7} + 480 \, x^{6} + 12 \, x^{5} - 24 \, x^{4} + 20 \, x^{3} - 24 \, x^{2} + 24 \, x - 8\right )}}\right ) - 2 \cdot 4^{\frac {2}{3}} \left (-9\right )^{\frac {1}{3}} x^{2} \log \left (-\frac {6 \cdot 4^{\frac {1}{3}} \left (-9\right )^{\frac {2}{3}} {\left (x^{3} - x + 1\right )}^{\frac {1}{3}} x^{2} + 4^{\frac {2}{3}} \left (-9\right )^{\frac {1}{3}} {\left (x^{3} + 2 \, x - 2\right )} - 36 \, {\left (x^{3} - x + 1\right )}^{\frac {2}{3}} x}{x^{3} + 2 \, x - 2}\right ) + 4^{\frac {2}{3}} \left (-9\right )^{\frac {1}{3}} x^{2} \log \left (-\frac {18 \cdot 4^{\frac {2}{3}} \left (-9\right )^{\frac {1}{3}} {\left (4 \, x^{4} - x^{2} + x\right )} {\left (x^{3} - x + 1\right )}^{\frac {2}{3}} - 4^{\frac {1}{3}} \left (-9\right )^{\frac {2}{3}} {\left (55 \, x^{6} - 50 \, x^{4} + 50 \, x^{3} + 4 \, x^{2} - 8 \, x + 4\right )} - 54 \, {\left (7 \, x^{5} - 4 \, x^{3} + 4 \, x^{2}\right )} {\left (x^{3} - x + 1\right )}^{\frac {1}{3}}}{x^{6} + 4 \, x^{4} - 4 \, x^{3} + 4 \, x^{2} - 8 \, x + 4}\right ) + 36 \, {\left (x^{3} - x + 1\right )}^{\frac {2}{3}}}{48 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{3} - x + 1\right )}^{\frac {2}{3}} {\left (2 \, x - 3\right )}}{{\left (x^{3} + 2 \, x - 2\right )} x^{3}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 18.88, size = 635, normalized size = 3.63
method | result | size |
risch | \(-\frac {3 \left (x^{3}-x +1\right )^{\frac {2}{3}}}{4 x^{2}}+\frac {3 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+36 \textit {\_Z}^{2}\right ) \ln \left (-\frac {5 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+18\right )^{3} x^{3}+18 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+36 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}+18\right )^{2} x^{3}+14 \left (x^{3}-x +1\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+18\right )^{2} x +7 \left (x^{3}-x +1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}+18\right )^{2} x^{2}+48 \left (x^{3}-x +1\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+18\right ) x^{2}+10 \RootOf \left (\textit {\_Z}^{3}+18\right ) x^{3}+36 x^{3} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+36 \textit {\_Z}^{2}\right )+6 \left (x^{3}-x +1\right )^{\frac {2}{3}} x -10 \RootOf \left (\textit {\_Z}^{3}+18\right ) x -36 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+36 \textit {\_Z}^{2}\right ) x +10 \RootOf \left (\textit {\_Z}^{3}+18\right )+36 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+36 \textit {\_Z}^{2}\right )}{x^{3}+2 x -2}\right )}{2}+\frac {\RootOf \left (\textit {\_Z}^{3}+18\right ) \ln \left (-\frac {-\RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+18\right )^{3} x^{3}-3 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+36 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}+18\right )^{2} x^{3}+9 \left (x^{3}-x +1\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+18\right ) x^{2}+5 \RootOf \left (\textit {\_Z}^{3}+18\right ) x^{3}+15 x^{3} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+36 \textit {\_Z}^{2}\right )+9 \left (x^{3}-x +1\right )^{\frac {2}{3}} x -2 \RootOf \left (\textit {\_Z}^{3}+18\right ) x -6 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+36 \textit {\_Z}^{2}\right ) x +2 \RootOf \left (\textit {\_Z}^{3}+18\right )+6 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+36 \textit {\_Z}^{2}\right )}{x^{3}+2 x -2}\right )}{4}\) | \(635\) |
trager | \(\text {Expression too large to display}\) | \(1234\) |
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 \frac {{\left (x^{3} - x + 1\right )}^{\frac {2}{3}} {\left (2 \, x - 3\right )}}{{\left (x^{3} + 2 \, x - 2\right )} x^{3}}\,{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 \frac {\left (2\,x-3\right )\,{\left (x^3-x+1\right )}^{2/3}}{x^3\,\left (x^3+2\,x-2\right )} \,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 \frac {\left (2 x - 3\right ) \left (x^{3} - x + 1\right )^{\frac {2}{3}}}{x^{3} \left (x^{3} + 2 x - 2\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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