Optimal. Leaf size=29 \[ 2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^3-1}}\right )-2 \tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^3-1}}\right ) \]
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Rubi [F] time = 1.04, 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 {x^2 \left (-4+x^3\right )}{\left (-1+x^3\right )^{3/4} \left (1-x^3+x^4\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {x^2 \left (-4+x^3\right )}{\left (-1+x^3\right )^{3/4} \left (1-x^3+x^4\right )} \, dx &=\int \left (\frac {1}{\left (-1+x^3\right )^{3/4}}+\frac {x}{\left (-1+x^3\right )^{3/4}}-\frac {1+x+4 x^2-x^3}{\left (-1+x^3\right )^{3/4} \left (1-x^3+x^4\right )}\right ) \, dx\\ &=\int \frac {1}{\left (-1+x^3\right )^{3/4}} \, dx+\int \frac {x}{\left (-1+x^3\right )^{3/4}} \, dx-\int \frac {1+x+4 x^2-x^3}{\left (-1+x^3\right )^{3/4} \left (1-x^3+x^4\right )} \, dx\\ &=\frac {\left (1-x^3\right )^{3/4} \int \frac {1}{\left (1-x^3\right )^{3/4}} \, dx}{\left (-1+x^3\right )^{3/4}}+\frac {\left (1-x^3\right )^{3/4} \int \frac {x}{\left (1-x^3\right )^{3/4}} \, dx}{\left (-1+x^3\right )^{3/4}}-\int \left (\frac {1}{\left (-1+x^3\right )^{3/4} \left (1-x^3+x^4\right )}+\frac {x}{\left (-1+x^3\right )^{3/4} \left (1-x^3+x^4\right )}+\frac {4 x^2}{\left (-1+x^3\right )^{3/4} \left (1-x^3+x^4\right )}-\frac {x^3}{\left (-1+x^3\right )^{3/4} \left (1-x^3+x^4\right )}\right ) \, dx\\ &=\frac {x \left (1-x^3\right )^{3/4} \, _2F_1\left (\frac {1}{3},\frac {3}{4};\frac {4}{3};x^3\right )}{\left (-1+x^3\right )^{3/4}}+\frac {x^2 \left (1-x^3\right )^{3/4} \, _2F_1\left (\frac {2}{3},\frac {3}{4};\frac {5}{3};x^3\right )}{2 \left (-1+x^3\right )^{3/4}}-4 \int \frac {x^2}{\left (-1+x^3\right )^{3/4} \left (1-x^3+x^4\right )} \, dx-\int \frac {1}{\left (-1+x^3\right )^{3/4} \left (1-x^3+x^4\right )} \, dx-\int \frac {x}{\left (-1+x^3\right )^{3/4} \left (1-x^3+x^4\right )} \, dx+\int \frac {x^3}{\left (-1+x^3\right )^{3/4} \left (1-x^3+x^4\right )} \, dx\\ \end {align*}
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Mathematica [F] time = 0.20, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^2 \left (-4+x^3\right )}{\left (-1+x^3\right )^{3/4} \left (1-x^3+x^4\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 2.70, size = 29, normalized size = 1.00 \begin {gather*} 2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^3-1}}\right )-2 \tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^3-1}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 48, normalized size = 1.66 \begin {gather*} -2 \, \arctan \left (\frac {{\left (x^{3} - 1\right )}^{\frac {1}{4}}}{x}\right ) - \log \left (\frac {x + {\left (x^{3} - 1\right )}^{\frac {1}{4}}}{x}\right ) + \log \left (-\frac {x - {\left (x^{3} - 1\right )}^{\frac {1}{4}}}{x}\right ) \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} - 4\right )} x^{2}}{{\left (x^{4} - x^{3} + 1\right )} {\left (x^{3} - 1\right )}^{\frac {3}{4}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.95, size = 150, normalized size = 5.17 \begin {gather*} -\ln \left (\frac {2 \left (x^{3}-1\right )^{\frac {3}{4}} x +2 x^{2} \sqrt {x^{3}-1}+2 \left (x^{3}-1\right )^{\frac {1}{4}} x^{3}+x^{4}+x^{3}-1}{x^{4}-x^{3}+1}\right )+\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{3}-1}\, x^{2}+\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{4}+2 \left (x^{3}-1\right )^{\frac {3}{4}} x -2 \left (x^{3}-1\right )^{\frac {1}{4}} x^{3}+\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}-\RootOf \left (\textit {\_Z}^{2}+1\right )}{x^{4}-x^{3}+1}\right ) \end {gather*}
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} - 4\right )} x^{2}}{{\left (x^{4} - x^{3} + 1\right )} {\left (x^{3} - 1\right )}^{\frac {3}{4}}}\,{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 = -0.03 \begin {gather*} \int \frac {x^2\,\left (x^3-4\right )}{{\left (x^3-1\right )}^{3/4}\,\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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