Optimal. Leaf size=154 \[ 2^{2/3} \log \left (2^{2/3} \sqrt [3]{x^4-x^3+1}+2 x\right )+2^{2/3} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2^{2/3} \sqrt [3]{x^4-x^3+1}-x}\right )+\frac {3 \left (x^4-x^3+1\right )^{2/3}}{2 x^2}-\frac {\log \left (-2 x^2+2^{2/3} \sqrt [3]{x^4-x^3+1} x-\sqrt [3]{2} \left (x^4-x^3+1\right )^{2/3}\right )}{\sqrt [3]{2}} \]
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Rubi [F] time = 0.82, 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 {\left (-3+x^4\right ) \left (1-x^3+x^4\right )^{2/3}}{x^3 \left (1+x^3+x^4\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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\begin {align*} \int \frac {\left (-3+x^4\right ) \left (1-x^3+x^4\right )^{2/3}}{x^3 \left (1+x^3+x^4\right )} \, dx &=\int \left (-\frac {3 \left (1-x^3+x^4\right )^{2/3}}{x^3}+\frac {(3+4 x) \left (1-x^3+x^4\right )^{2/3}}{1+x^3+x^4}\right ) \, dx\\ &=-\left (3 \int \frac {\left (1-x^3+x^4\right )^{2/3}}{x^3} \, dx\right )+\int \frac {(3+4 x) \left (1-x^3+x^4\right )^{2/3}}{1+x^3+x^4} \, dx\\ &=-\left (3 \int \frac {\left (1-x^3+x^4\right )^{2/3}}{x^3} \, dx\right )+\int \left (\frac {3 \left (1-x^3+x^4\right )^{2/3}}{1+x^3+x^4}+\frac {4 x \left (1-x^3+x^4\right )^{2/3}}{1+x^3+x^4}\right ) \, dx\\ &=-\left (3 \int \frac {\left (1-x^3+x^4\right )^{2/3}}{x^3} \, dx\right )+3 \int \frac {\left (1-x^3+x^4\right )^{2/3}}{1+x^3+x^4} \, dx+4 \int \frac {x \left (1-x^3+x^4\right )^{2/3}}{1+x^3+x^4} \, dx\\ \end {align*}
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Mathematica [F] time = 0.33, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-3+x^4\right ) \left (1-x^3+x^4\right )^{2/3}}{x^3 \left (1+x^3+x^4\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.80, size = 154, normalized size = 1.00 \begin {gather*} \frac {3 \left (1-x^3+x^4\right )^{2/3}}{2 x^2}+2^{2/3} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{-x+2^{2/3} \sqrt [3]{1-x^3+x^4}}\right )+2^{2/3} \log \left (2 x+2^{2/3} \sqrt [3]{1-x^3+x^4}\right )-\frac {\log \left (-2 x^2+2^{2/3} x \sqrt [3]{1-x^3+x^4}-\sqrt [3]{2} \left (1-x^3+x^4\right )^{2/3}\right )}{\sqrt [3]{2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 74.09, size = 413, normalized size = 2.68 \begin {gather*} \frac {2 \cdot 4^{\frac {1}{3}} \sqrt {3} x^{2} \arctan \left (\frac {3 \cdot 4^{\frac {2}{3}} \sqrt {3} {\left (x^{9} - 4 \, x^{8} - 5 \, x^{7} + 2 \, x^{5} - 4 \, x^{4} + x\right )} {\left (x^{4} - x^{3} + 1\right )}^{\frac {2}{3}} - 6 \cdot 4^{\frac {1}{3}} \sqrt {3} {\left (x^{10} - 16 \, x^{9} + 19 \, x^{8} + 2 \, x^{6} - 16 \, x^{5} + x^{2}\right )} {\left (x^{4} - x^{3} + 1\right )}^{\frac {1}{3}} - \sqrt {3} {\left (x^{12} - 33 \, x^{11} + 111 \, x^{10} - 71 \, x^{9} + 3 \, x^{8} - 66 \, x^{7} + 111 \, x^{6} + 3 \, x^{4} - 33 \, x^{3} + 1\right )}}{3 \, {\left (x^{12} + 3 \, x^{11} - 105 \, x^{10} + 109 \, x^{9} + 3 \, x^{8} + 6 \, x^{7} - 105 \, x^{6} + 3 \, x^{4} + 3 \, x^{3} + 1\right )}}\right ) + 2 \cdot 4^{\frac {1}{3}} x^{2} \log \left (-\frac {3 \cdot 4^{\frac {2}{3}} {\left (x^{4} - x^{3} + 1\right )}^{\frac {1}{3}} x^{2} + 6 \, {\left (x^{4} - x^{3} + 1\right )}^{\frac {2}{3}} x + 4^{\frac {1}{3}} {\left (x^{4} + x^{3} + 1\right )}}{x^{4} + x^{3} + 1}\right ) - 4^{\frac {1}{3}} x^{2} \log \left (-\frac {6 \cdot 4^{\frac {1}{3}} {\left (x^{5} - 5 \, x^{4} + x\right )} {\left (x^{4} - x^{3} + 1\right )}^{\frac {2}{3}} - 4^{\frac {2}{3}} {\left (x^{8} - 16 \, x^{7} + 19 \, x^{6} + 2 \, x^{4} - 16 \, x^{3} + 1\right )} - 24 \, {\left (x^{6} - 2 \, x^{5} + x^{2}\right )} {\left (x^{4} - x^{3} + 1\right )}^{\frac {1}{3}}}{x^{8} + 2 \, x^{7} + x^{6} + 2 \, x^{4} + 2 \, x^{3} + 1}\right ) + 9 \, {\left (x^{4} - x^{3} + 1\right )}^{\frac {2}{3}}}{6 \, 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^{4} - x^{3} + 1\right )}^{\frac {2}{3}} {\left (x^{4} - 3\right )}}{{\left (x^{4} + x^{3} + 1\right )} x^{3}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 68.68, size = 1089, normalized size = 7.07
method | result | size |
risch | \(\frac {3 \left (x^{4}-x^{3}+1\right )^{\frac {2}{3}}}{2 x^{2}}-\ln \left (\frac {2 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )^{3} x^{3}-2 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-4\right )^{2} x^{3}+4 \left (x^{4}-x^{3}+1\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )^{2} x +\RootOf \left (\textit {\_Z}^{3}-4\right )^{2} \left (x^{4}-x^{3}+1\right )^{\frac {1}{3}} x^{2}-8 \left (x^{4}-x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right ) x^{2}+2 \RootOf \left (\textit {\_Z}^{3}-4\right ) x^{4}-2 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right ) x^{4}-2 \RootOf \left (\textit {\_Z}^{3}-4\right ) x^{3}+2 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right ) x^{3}-2 \left (x^{4}-x^{3}+1\right )^{\frac {2}{3}} x +2 \RootOf \left (\textit {\_Z}^{3}-4\right )-2 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right )}{x^{4}+x^{3}+1}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )+2 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right ) \ln \left (-\frac {3 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )^{3} x^{3}+2 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-4\right )^{2} x^{3}+4 \left (x^{4}-x^{3}+1\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )^{2} x -5 \RootOf \left (\textit {\_Z}^{3}-4\right )^{2} \left (x^{4}-x^{3}+1\right )^{\frac {1}{3}} x^{2}-8 \left (x^{4}-x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right ) x^{2}-3 \RootOf \left (\textit {\_Z}^{3}-4\right ) x^{4}-2 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right ) x^{4}+9 \RootOf \left (\textit {\_Z}^{3}-4\right ) x^{3}+6 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right ) x^{3}+10 \left (x^{4}-x^{3}+1\right )^{\frac {2}{3}} x -3 \RootOf \left (\textit {\_Z}^{3}-4\right )-2 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right )}{x^{4}+x^{3}+1}\right )-2 \ln \left (\frac {2 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )^{3} x^{3}-2 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-4\right )^{2} x^{3}+4 \left (x^{4}-x^{3}+1\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )^{2} x +\RootOf \left (\textit {\_Z}^{3}-4\right )^{2} \left (x^{4}-x^{3}+1\right )^{\frac {1}{3}} x^{2}-8 \left (x^{4}-x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right ) x^{2}+2 \RootOf \left (\textit {\_Z}^{3}-4\right ) x^{4}-2 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right ) x^{4}-2 \RootOf \left (\textit {\_Z}^{3}-4\right ) x^{3}+2 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right ) x^{3}-2 \left (x^{4}-x^{3}+1\right )^{\frac {2}{3}} x +2 \RootOf \left (\textit {\_Z}^{3}-4\right )-2 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right )}{x^{4}+x^{3}+1}\right ) \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right )\) | \(1089\) |
trager | \(\text {Expression too large to display}\) | \(1509\) |
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^{4} - x^{3} + 1\right )}^{\frac {2}{3}} {\left (x^{4} - 3\right )}}{{\left (x^{4} + x^{3} + 1\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 (x^4-3\right )\,{\left (x^4-x^3+1\right )}^{2/3}}{x^3\,\left (x^4+x^3+1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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